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Commun. Math. Phys. 293, 1–36 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0885-2 Communications in Mathe...

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Commun. Math. Phys. 293, 1–36 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0885-2

Communications in

Mathematical Physics

Stability of Isentropic Navier–Stokes Shocks in the High-Mach Number Limit∗ Jeffrey Humpherys1 , Olivier Lafitte2 , Kevin Zumbrun3 1 Department of Mathematics, Brigham Young University, Provo, UT 84602, USA.

E-mail: [email protected]

2 LAGA, Institut Galilee, Universite Paris 13, 93 430 Villetaneuse, France, and CEA Saclay, DM2S/DIR,

91 191 Gif sur Yvette Cedex, France. E-mail: [email protected] 3 Department of Mathematics, Indiana University, Bloomington, IN 47402, USA. E-mail: [email protected] Received: 6 March 2008 / Accepted: 30 April 2009 Published online: 2 September 2009 – © Springer-Verlag 2009

Abstract: By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier–Stokes equations with γ -law pressure (i) in the limit as Mach number M goes to infinity, for any γ ≥ 1 (proved analytically), and (ii) for M ≥ 2, 500, γ ∈ [1, 2.5] or M ≥ 13, 000, γ ∈ [2.5, 3] (demonstrated numerically). This builds on and completes earlier studies by Matsumura–Nishihara and Barker–Humpherys– Rudd–Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, independent of shock amplitude, of viscous shock waves for γ -law gas dynamics in the range γ ∈ [1, 3]. Other γ -values may be treated similarly, but have not been checked numerically. The main idea is to establish convergence of the Evans function in the high-Mach number limit to that of a pressureless, or “infinitely compressible”, gas with additional upstream boundary condition determined by a boundary-layer analysis. Recall that low-Mach number behavior is formally incompressible. Contents 1. 2.

3.

Introduction . . . . . . . . . . Preliminaries . . . . . . . . . . 2.1 Profile equation . . . . . . 2.2 Eigenvalue equations . . . 2.3 Preliminary estimates . . . 2.4 Evans function formulation Description of the Main Results 3.1 Limiting equations . . . .

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∗ This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.

2

J. Humpherys, O. Lafitte, K. Zumbrun

3.2 Limiting Evans function . . . . . . . . . 3.3 Physical interpretation . . . . . . . . . 3.4 Analytical results . . . . . . . . . . . . 3.5 Numerical computations . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . 4. Boundary-Layer Analysis . . . . . . . . . . 4.1 Preliminary transformation . . . . . . . 4.2 Dynamic triangularization . . . . . . . 4.3 Fast/Slow dynamics . . . . . . . . . . . 5. Proof of the Main Theorem . . . . . . . . . 5.1 Boundary estimate . . . . . . . . . . . 5.2 Convergence to D 0 . . . . . . . . . . . 6. Numerical Convergence . . . . . . . . . . . 7. Discussion and Open Problems . . . . . . . Appendix A. Proofs of Preliminary Estimates . . Appendix B. Nonvanishing of D 0 . . . . . . . . Appendix C. Quantitative Conjugation Estimates Appendix D. An Illuminating Example . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The isentropic compressible Navier-Stokes equations in one spatial dimension expressed in Lagrangian coordinates take the form vt − u x = 0, u x u t + p(v)x = , v x

(1.1)

where v is specific volume, u is velocity, and p pressure. We assume an adiabatic pressure law p(v) = a0 v −γ

(1.2)

corresponding to a γ -law gas, for some constants a0 > 0 and γ ≥ 1. In the thermodynamical rarified gas approximation, γ > 1 is the average over constituent particles of γ = (n + 2)/n, where n is the number of internal degrees of freedom of an individual particle [4]: n = 3 (γ = 5/3) for monatomic, n = 5 (γ = 7/5) for diatomic gas. For dense fluids, γ is typically determined phenomenologically [19]. In general, γ is usually taken within 1 ≤ γ ≤ 3 in models of gas-dynamical flow, whether phenomenological or derived by statistical mechanics [42,43,45]. It is well known that these equations support viscous shock waves, or asymptoticallyconstant traveling-wave solutions (v, u)(x, t) = (v, ˆ u)(x ˆ − st),

lim (v, ˆ u)(z) ˆ = (v, u)± ,

z→±∞

(1.3)

in agreement with physically-observed phenomena. In nature, such waves are seen to be quite stable, even for large variations in pressure between v± . However, it is a long-standing mathematical question to what extent this is reflected in the continuum-mechanical model (1.1), that is, for which choice of parameters (v± , u ± , γ ) are solutions of (1.3)

Stability of Isentropic Navier–Stokes Shocks

3

time-evolutionarily stable in the sense of PDE; see, for example, the discussions in [3,25]. The first result on this problem was obtained by Matsumura and Nishihara in 1985 [36] using clever energy estimates on the integrals of perturbations in v and u, by which they established stability with respect to the restricted class of perturbations with “zero mass”, i.e., perturbations whose integral is zero, for shocks with sufficiently small amplitude | p(v+ ) − p(v− )| ≤ C(v− , γ ), with C → ∞ as γ → 1, but C 1, with optimal rates. For concreteness, we state below the specialization of this result to system (1.1), along with an extension of Raoofi [40] asserting phase-asymptotic stability, or convergence to a specific translate of the unperturbed wave, under the additional assumption that the initial perturbation has small L 1 -first moment. These results hold for general hyperbolic-parabolic systems and waves under the additional assumptions of transverality/hyperbolic stability. Under our hypotheses, transversality/hyperbolic stability hold always for traveling waves of (1.1), as discussed in detail in the introduction of [34]. Proposition 1.1 ([34]). For any p such that p < 0 and p (0) > 0, in particular for p as in (1.2) with γ ≥ 1, let (v, ˆ u)(x ˆ − st) be a spectrally stable traveling-wave (1.3) of (1.1). Then, for any solution (v, ˜ u)(x, ˜ t) of (1.1) with L 1 ∩ H 3 initial difference 1 The result of [23] is obtained by energy estimates combining the techniques of [36] with those of [14,15]; a similar approach has been used in [32] to obtain small-amplitude zero-mass stability of Boltzmann shocks. See [12,39] for an alternative approach based on asymptotic ODE methods. 2 “Refined” because the linearized operator L does not possess a spectral gap, hence e Lt decays time-algebraically and not exponentially; see [49,50] for further discussion.

4


E 0 := (v, ˜ u)(·, ˜ 0) − (v, ˆ u)

ˆ L 1 ∩H 3 sufficiently small and some uniform C > 0, (v, ˜ u) ˜ exists for all t ≥ 0, with

(v, ˜ u)(·, ˜ t) − (v, ˆ u)(· ˆ − st) L 1 ∩H 3 ≤ C E 0 (stability).

(1.4)

˙ ≤ C E 0 (1 + t)−1/2 such that Moreover, there exists |α(t)| ≤ C E 0 , |α(t)|

(v, ˜ u)(·, ˜ t) − (v, ˆ u)(· ˆ − st − α(t)) L p ≤ C E 0 (1 + t)−(1/2)(1−1/ p)

(1.5)

for all 1 ≤ p ≤ ∞ (asymptotic orbital stability). Proposition 1.2 ([40]). For (v, ˜ u), ˜ (v, ˆ u) ˆ as in Proposition 1.1, if the initial difference has in addition a sufficiently small L 1 -first moment ˜ u)(·, ˜ 0) − (v, ˆ u)|

ˆ L1 , E 1 := |x| |(v,

(1.6)

then also α converges to a limit α∞ as t → +∞, with ˙ ≤ C() max{E 0 , E 1 }(1 + t)−1/2+ , |α − α∞ |(t), (1 + t)1/2 |α(t)|

(1.7)

> 0 arbitrary (phase-asymptotic orbital stability). Remark 1.3. Under the additional assumption that the initial perturbation and its first two derivatives decay as E 0 (1 + |x|)−3/2 , E 0 sufficiently small, one may obtain also sharp pointwise bounds on the solution, along with the sharp rate (1.7), = 0 for the phase; see [20,21]. Such a localization condition, or the weaker (1.6) is necessary in order to obtain a rate of convergence for α; smallness in L 1 ∪ H 3 is not enough, as may be seen by the example of a small initial perturbation localized arbitrarily far from x = 0, which takes arbitrarily long to reach and substantially affect the location of the shock profile. Propositions 1.1–1.2 reduce the problem of large-amplitude stability to the study of the associated eigenvalue equation (L − λ)u = 0, a standard analytically and numerically well-posed (boundary value) problem in ODE, which can be attacked by the large body of techniques developed for asymptotic, exact, and numerical study of ODE. In particular, there exist well-developed and efficient numerical algorithms to determine the number of unstable roots for any specific linearized operator L, independent of its origins in the PDE setting; see, e.g., [8–11,24] and references therein. In this sense, the problem of determining stability of any single wave is satisfactorily resolved, or, for that matter, of any compact family of waves. To determine stability of a family of waves across an unbounded parameter regime, however, is another matter. It is this issue that we confront in attempting to decide the stability of general isentropic Navier–Stokes shocks. As pointed out in [23,50], zero-mass stability implies (and in a practical sense is roughly equivalent to) spectral stability. Thus, the original results of Matsumura and Nishida [36] imply small-amplitude shock stability for general γ and large-amplitude stability as γ → 1. Recently, Barker, Humpherys, Rudd, and Zumbrun [2] have carried out a numerical Evans function study indicating stability on the large, but still bounded, parameter range γ ∈ [1, 3], 1 ≤ M ≤ 3,000, where M is the Mach number associated with the shock. For discussion of the Evans function, see Sect. 2.4. Recall that Mach number is an alternative measure of shock strength, with 1 corresponding to | p(v+ ) − p(v− )| = 0 and M → ∞ corresponding to | p(v+ ) − p(v− )| → ∞; see Appendix A, [2], or Sect. 2.1 below. Mach 3, 000 is far beyond the hypersonic regime


5

M ∼ 101 encountered in current aerodynamics. However, the mathematical question of stability across arbitrary γ , M has remained open up to now. In this paper, we resolve this question, using a combination of asymptotic ODE estimates and numerical Evans function calculations to conclude, first, stability of isentropic Navier–Stokes shocks in the large Mach number limit M → ∞ for any γ ≥ 1, and, second, stability for all M ≥ 2, 500 for γ ∈ [1, 2.5] (for γ ∈ [1, 2], we obtain in fact stability for M ≥ 500) and for all M ≥ 13, 000 for γ ∈ [2.5, 3]. The first result is obtained analytically, the second by a systematic numerical study. Together with the numerical results of [2] (supplemented with additional computations for γ ∈ [2.5, 3]), this gives convincing numerical evidence of unconditional stability for γ ∈ [1, 3], independent of shock amplitude. As in [2], our numerical study is not a numerical proof, but contains the necessary ingredients for one; see discussion, Sect. 6. The restriction to γ ∈ [1, 3] is an arbitrary one coming from the choice of parameters on which the numerical study [2] was carried out; stability for other γ can be easily checked as well. (Note that all analytical results are for any γ ≥ 1.) In particular, we establish without aid of numerics the following theorem, an immediate consequence of Corollary 3.3 and Proposition 3.5 below. The rest of our results, both analytical and numerical, are stated after some preliminary preparations and discussion in Sect. 3. Theorem 1.4. For any γ ≥ 1, isentropic Navier–Stokes shocks are spectrally stable for Mach number M sufficiently large (equivalently, v+ sufficiently small, taking without loss of generality v− > v+ > 0 in (1.3)), hence nonlinearly stable in L 1 ∩ H 3 , with bounds (1.4)–(1.5), (1.7). Our method of analysis is straightforward, though somewhat delicate to carry out. Working with the rescaled and conveniently rearranged versions of the equations introduced in [2], we observe that the associated eigenvalue equations converge uniformly as Mach number goes to infinity on a “regular region” x ≤ L, for any fixed L > 0, to a limiting system that is well-behaved (hence treatable by the standard methods of [2,33,39]) in the sense that its coefficient matrix converges uniformly exponentially in x to limits at x = ±∞, but is underdetermined at x = +∞. On the complementary “singular region” x ≥ L, the convergence is only pointwise due to a fast “inner structure” featuring rapid variation of the converging coefficient matrices near x = +∞, but the behavior at x = +∞ is of course determinate. Performing a boundary-layer analysis on the singular region and matching across x = L, we are able to show convergence of the Evans function of the original system as the Mach number goes to infinity to an Evans function of the limiting system with an appropriately imposed additional condition at x = +∞, upstream of the shock. This reduces the question of stability in the high-Mach number limit to existence or nonexistence of zeroes of the limiting Evans function on eλ ≥ 0, a question that can be resolved by routine numerical computation as in [2], or by energy estimates as in Appendix B. The limiting system can be recognized as the eigenvalue equation associated with a pressureless (γ = 0) gas, that is, the “infinitely-compressible” limit one might expect as the Mach number goes to infinity. Recall that behavior in the low Mach number limit is incompressible [18,27,29]. However, the upstream boundary condition has to our knowledge no such simple interpretation. Indeed, to carry out the boundary-layer analysis by which we derive this condition is the main technical difficulty of the paper. Our results on the large-amplitude limit allow us to complete a global stability analysis for traveling-waves of a nontrivial, physically relevant, and much-studied system of equations, allowing us to conclude for this canonical model that isentropic

6


Navier–Stokes shocks are stable for a γ -law gas with γ ∈ [1, 3]. More, the large amplitude limit appears to serve as an organizing center governing behavior also of shocks of large but relatively modest size. See, for example, Figs. 2–4, from which universal stability may be deduced essentially by inspection. Indeed, together with the smallamplitude limit, this appears to essentially govern by interpolation behavior of shocks of all amplitudes. The study of this limit thus has importance apart from the specific physical interest of the high-Mach number regime. Besides their independent interest, the results of this paper seem significant as prototypes for future analyses. Our calculations use some properties specific to the structure of (1.1). In particular, we make use of surprisingly strong energy estimates carried out in [2] in confining unstable eigenvalues to a bounded set independent of shock strength (or Mach number). Also, we use the extremely simple structure of the eigenvalue equation to carry out the key analysis of the eigenvalue flow in the singular region near x = +∞ essentially by hand. However, these appear to be conveniences rather than essential aspects of the analysis. It is our hope that the basic argument structure of this paper together with [2] can serve as a blueprint for the global study of large-amplitude stability in more general situations. In particular, we expect that the analysis will carry over to the full (nonisentropic) equations of gas dynamics with ideal gas equation of state.

2. Preliminaries We begin by recalling a number of preliminary steps carried out in [2]. Making the standard change of coordinates x → x − st, we consider instead stationary solutions (v, u)(x, t) ≡ (v, ˆ u)(x) ˆ of vt − svx − u x = 0, u x u t − su x + (a0 v −γ )x = . v x

(2.1)

Under the rescaling (x, t, v, u) → (−εsx, εs 2 t, v/ε, −u/(εs)),

(2.2)

where ε is chosen so that 0 < v+ < v− = 1, our system takes the convenient form vt + vx − u x = 0, u x u t + u x + (av −γ )x = , v x where a = a0 ε−γ −1 s −2 .

2.1. Profile equation. Steady shock profiles of (2.3) satisfy v − u = 0, u u + (av −γ ) = , v

(2.3)


7

subject to boundary conditions (v, u)(±∞) = (v± , u ± ), or, simplifying, v −γ v + (av ) = . v Integrating from −∞ to x, we get the profile equation v = H (v, v+ ) := v(v − 1 + a(v −γ − 1)),

(2.4)

where a is found by setting x = +∞, thus yielding the Rankine-Hugoniot condition a=−

v+ − 1

−γ v+

−1

γ

= v+

1 − v+ γ. 1 − v+

(2.5) γ

Evidently, a → γ −1 in the weak shock limit v+ → 1, while a ∼ v+ in the strong shock limit v+ → 0. The associated Mach number M may be computed as in [2], Appendix A, as M = (γ a)−1/2

(2.6)

−γ /2

→ +∞ as v+ → 0 and M → 1 as v+ → 1; that is, the so that M ∼ γ −1/2 v+ high-Mach number limit corresponds to the limit v+ → 0. 2.2. Eigenvalue equations. Linearizing (2.3) about the profile (v, ˆ u), ˆ we obtain the eigenvalue problem λv + v − u = 0, h(v) ˆ u λu + u − v = , vˆ γ +1 vˆ

(2.7)

where h(v) ˆ = −vˆ γ +1 + a(γ − 1) + (a + 1)vˆ γ .

(2.8)

We seek nonstable eigenvalues λ ∈ { e(λ) ≥ 0} \ {0}, i.e., λ for which (2.7) possess a nontrivial solution (v, u) decaying at plus and minus spatial infinity. As pointed in [23,50], by divergence form of the equations, such solutions necessarily satisfy out +∞ +∞ v(x)d x = −∞ −∞ u(x)d x = 0, from which we may deduce that x x u(z)dz, v(x) ˜ = v(z)dz u(x) ˜ = −∞

−∞

and their derivatives decay exponentially as x → ∞. Substituting and then integrating, we find that (u, ˜ v) ˜ satisfies the integrated eigenvalue equations (suppressing the tilde) λv + v − u = 0, (2.9a) h(v) ˆ u λu + u − γ +1 v = . (2.9b) vˆ vˆ This new eigenvalue problem differs spectrally from (2.7) only at λ = 0, hence spectral stability of (2.7) is equivalent to spectral stability of (2.9). Moreover, since (2.9) has no eigenvalue at λ = 0, one can expect more uniform stability estimates for the integrated equations in the vicinity of λ = 0 [14,36,50].

8


2.3. Preliminary estimates. The following estimates established in [2] indicate the suitability of the rescaling chosen in Sect. 2.1. For completeness, we recall their proofs in Appendix A. Proposition 2.1 ([2]). For each γ ≥ 1, 0 < v+ ≤ 1, (2.4) has a unique (up to translation) monotone decreasing solution vˆ decaying to its endstates with a uniform exponential 1 1 rate, independent of v+ , γ . In particular, for 0 < v+ ≤ 12 and v(0) ˆ := v+ + 12 , 3x 1 |v(x) ˆ − v+ | ≤ e− 4 x ≥ 0, (2.10a) 12 x+12 1 e 2 x ≤ 0. (2.10b) |v(x) ˆ − v− | ≤ 4 Proposition 2.2 ([2]). Nonstable eigenvalues λ of (2.9), i.e., eigenvalues with nonnegative real part, are confined for any 0 < v+ ≤ 1 to the region := {λ : e(λ) + |m(λ)| ≤

1 √ γ+ 2

2 }.

(2.11)

2.4. Evans function formulation. Following [2], we may express (2.9) concisely as a first-order system ⎛

0 A(x, λ) = ⎝ 0 λvˆ

W = A(x, λ)W, ⎞ ⎛ ⎞ λ 1 u ⎠ , W = ⎝v ⎠ , = d , 0 1 dx λvˆ f (v) ˆ −λ v

(2.12) (2.13)

where f (v) ˆ = vˆ − vˆ −γ h(v) ˆ = 2vˆ − a(γ − 1)vˆ −γ − (a + 1), with h as in (2.8) and a as in (2.5), or, equivalently, 1 − v+ v+ γ 1 − v+ γ f (v) ˆ = 2vˆ − (γ − 1) − γ γ v+ − 1. v ˆ 1 − v+ 1 − v+

(2.14)

(2.15)

Eigenvalues of (2.9) correspond to nontrivial solutions W for which the boundary conditions W (±∞) = 0 are satisfied. Because A(x, λ) as a function of vˆ is asymptotically constant in x, the behavior near x = ±∞ of solutions of (2.13) is governed by the limiting constant-coefficient systems W = A± (λ)W,

A± (λ) := A(±∞, λ),

(2.16)

from which we readily find on the (nonstable) domain λ ≥ 0, λ = 0 of interest that there is a one-dimensional unstable manifold W1− (x) of solutions decaying at x = −∞ and a two-dimensional stable manifold W2+ (x) ∧ W3+ (x) of solutions decaying at x = +∞, analytic in λ, with asymptotic behavior µ± (λ)x ± W± V j (λ) j (x, λ) ∼ e

(2.17)


9

as x → ±∞, where µ± (λ) and V j± (λ) are eigenvalues and associated analytically chosen eigenvectors of the limiting coefficient matrices A± (λ). A standard choice of eigenvectors V j± [8,11,13,22], uniquely specifying W ± j (up to constant factor) is obtained by Kato’s ODE [26], a linear, analytic ODE whose solution can be alternatively characterized by the property that there exist corresponding left eigenvectors V˜ j± such that (V˜ j · V j )± ≡ constant, (V˜ j · V˙ j )± ≡ 0,

(2.18)

where “ ˙ ” denotes d/dλ; for further discussion, see [13,22,26]. Defining the Evans function D associated with operator L as the analytic function D(λ) := det(W1− W2+ W3+ )|x=0 ,

(2.19)

we find that eigenvalues of L correspond to zeroes of D both in location and multiplicity; moreover, the Evans function extends analytically to λ = 0, i.e., to all of λ ≥ 0. See [1,13,33,49] for further details. Equivalently, following [2,38], we may express the Evans function as +

1 · W − D(λ) = W (2.20) 1 |x=0 , + (x) spans the one-dimensional unstable manifold of solutions decaying at where W 1 x = +∞ (necessarily orthogonal to the span of W2+ (x) and W3+ (x)) of the adjoint eigenvalue ODE, . = −A(x, λ)∗ W W

(2.21)

The simpler representation (2.20) is the one that we shall use here. 3. Description of the Main Results We can now state precisely our main results.

3.1. Limiting equations. Under the strategic rescaling (2.2), both profile and eigenvalues equations converge pointwise as v+ → 0 to limiting equations at v+ = 0. The limiting profile equation (the limit of (2.4)) is evidently v = v(v − 1),

(3.1)

with explicit solution vˆ0 (x) =

1 − tanh(x/2) , 2

(3.2)

while the limiting eigenvalue system (the limit of (2.13)) is W = A0 (x, λ)W, ⎛ 0 λ 0 A0 (x, λ) = ⎝ 0 λvˆ0 λvˆ0

⎞

1 ⎠, 1 f 0 (vˆ0 ) − λ

(3.3) (3.4)

10


where f 0 (vˆ0 ) = 2vˆ0 − 1 = − tanh(x/2).

(3.5)

Indeed, this convergence is uniform on any interval vˆ0 ≥ > 0, or, equivalently, x ≤ L, for L any positive constant, where the sequence is therefore a regular perturbation of its limit. We will call x ∈ (−∞, L] the “regular region” or “regular side”. For vˆ0 → 0 on the other hand, or x → ∞, the limit is less well-behaved, as may be

seen by the fact that ∂ f /∂ vˆ ∼ vˆ −1 as vˆ → v+ , a consequence of the appearance of vvˆ+ in the expression (2.15) for f . Similarly, in contrast to v, ˆ A(x, λ) does not converge to A+ (λ) as x → +∞ with uniform exponential rate independent of v+ , γ , but rather as C vˆ −1 e−x/2 . We call x ∈ [L , +∞) therefore the “singular region” or “singular side”. (This is not a singular perturbation in the usual sense but a weaker type of singularity, at least as we have framed the problem here.) 3.2. Limiting Evans function. We should now like to define a limiting Evans function following the asymptotic Evans function framework introduced in [39] and establish convergence to this function in the v+ → 0 limit, thus reducing the stability problem as v+ → 0 to the study of the (fixed) limiting Evans function. However, we face certain difficulties due to the (mild) singularity of the limit, as can be seen even at the first step of defining an Evans function for the limiting system. For, the limiting coefficient matrix ⎛ ⎞ 0 λ 1 1 ⎠ A0+ (λ) := A0 (+∞, λ) = ⎝0 0 (3.6) 0 0 −1 − λ is nonhyperbolic (in ODE sense) for all λ, having eigenvalues 0, 0, −1 − λ; in particular, the stable manifold drops to dimension one in the limit v+ → 0. Thus, the subspace in which W2+ and W3+ should be initialized at x = +∞ is not self-determined by (3.6), but must be deduced by a careful study of the double limit v+ → 0, x → +∞. But, the computation ⎛ ⎞ 0 λ 1 1 ⎠ = A0+ (λ) = lim lim A(x, λ) lim A(+∞, λ) = ⎝0 0 x→∞ v+ →0 v+ →0 0 0 −γ − λ shows that these limits do not commute, except in the special case γ = 1 already treated in [36] by other methods. The rigorous treatment of this issue is the main work of the paper. However, the end result can be easily motivated on heuristic grounds. A study of limv+ →0 A(+∞, λ) on the set eλ ≥ 0 of interest reveals that eigenmodes decouple into a single “fast” (stable subspace) decaying mode (∗, ∗, 1)T associated with eigenvalue −γ − λ of strictly negative real part and a two-dimensional (center) subspace of neutral modes (r, 0)T associated 0λ with Jordan block , of which there is only a single genuine eigenvector (1, 0, 0)T . 00 For v+ small, therefore, A+ (λ) has also a single fast, decaying, eigenmode with eigenvalue near −γ − 1 and two slow eigenmodes with eigenvalues near zero, one decaying and the other growing (recall, Sect. 2.4, that the stable subspace of A+ has dimension two for eλ ≥ 0, λ = 0 and the unstable subspace dimension one).


11

Focusing on the single slow decaying eigenvector of A+ , and considering its limiting behavior as v+ → 0, we see immediately that it must converge in direction to ±(1, 0, 0)T . For, the sequence of direction vectors, since continuously varying and restricted to a compact set, has a nonempty, connected set of accumulation points, and these must be eigenvectors of limv+ →0 A+ with eigenvalues near zero. Since ±(1, 0, 0)T are the unique candidates, we obtain the result. Indeed, both growing and decaying slow eigenmodes must converge to this common direction, making the limiting analysis trivial. The same argument shows that ±(1, 0, 0)T is the limiting direction of the slow stable eigenmode of A0 (x, λ) as x → +∞, that is, in the alternate limit lim lim A(x, λ).

x→∞ v+ →0

Since V2+ := (1, 0, 0)T is the common limit of the slow decaying eigenmode in either of the two alternative limits limv+ →0 A+ and limv+ →0 A+ , it thus seems a reasonable choice to use this limiting slow direction to define an Evans function for the limiting system (3.4). On the other hand, the stable eigenmode V3 := (b−1 (λ/b + 1), b−1 , 1)T , b = −1 − λ, of A0+ is forced on us by the system itself, independent of the limiting process. Combining these two observations, we require that solutions W20+ and W30+ of the limiting eigenvalue system (3.4) lie asympotically in directions V2 and V3 , respectively, thus determining a limiting, or “reduced” Evans function D 0 (λ) := det(W10− W20+ W30+ )|x=0 ,

(3.7)

or alternatively 10+ · W 0− D 0 (λ) = W 1

|x=0

,

(3.8)

0+ defined analogously as a solution of the adjoint limiting system lying asympwith W 1 totically at x = +∞ in direction 1 := (0, 1, b¯ −1 )T = (0, 1, (1 + λ¯ )−1 )T V

(3.9)

orthogonal to the span of V2 and V3 , where “¯” denotes complex conjugate. (The prescription of W10− in the regular region is straightforward: it must lie on the one-dimensional unstable manifold of A0− as in the v+ > 0 case.) 3.3. Physical interpretation. Alternatively, the limiting equations may be derived by γ taking a formal limit as v+ → 0 of the rescaled equations (2.3), recalling that a ∼ v+ , to obtain a limiting evolution equation vt + vx − u x = 0, u x ut + u x = v x

(3.10)

12


corresponding to a pressureless gas, or γ = 0, then deriving profile and eigenvalue equations from (3.10) in the usual way. This gives some additional insight on behavior, of which we make important mathematical use in Appendix B. Physically, it has the interpretation that, in the high-Mach number limit v+ → 0, effects of pressure are concentrated near x = +∞ on the infinite-density side, as encoded in the special upstream boundary condition (u, u , v, v ) → c(1, 0, 0, 0) as x → +∞ for the integrated eigenvalue equation, which may be seen to be equivalent to the conditions imposed on W +j in the previous subsection.

3.4. Analytical results. Defining D 0 as in (3.7)–(3.8), we have the following main theorems, all valid for arbitrary γ ≥ 1. Theorem 3.1. For each γ ≥ 1 and λ in any compact subset of eλ ≥ 0, D(λ) converges uniformly to D 0 (λ) as v+ → 0. Corollary 3.2. For each γ ≥ 1 and any compact subset of eλ ≥ 0, D is nonvanishing on for v+ sufficiently small if D 0 is nonvanishing on , and is nonvanishing on the interior of only if D 0 is nonvanishing there. Proof. Standard properties of uniform limit of analytic functions. Corollary 3.3. For each γ ≥ 1, isentropic Navier–Stokes shocks are spectrally stable in the high-Mach number limit v+ → 0 if D 0 is nonvanishing on the wedge : e(λ) + |m(λ)| ≤

1 √ γ+ 2

2 , eλ ≥ 0

(3.11)

and only if D 0 is nonvanishing on the interior of . Proof. Corollary 3.2 together with Proposition 2.2. Remark 3.4. Likewise, on any compact subset of eλ ≥ 0, |D| is uniformly bounded from zero for v+ sufficiently small (M sufficiently large) if and only if |D 0 | is uniformly bounded from zero. Thus, isentropic Navier–Stokes shocks are “uniformly stable” for sufficiently small v+ , in the sense that |D| is bounded from below independent of v+ , if and only if D 0 is nonvanishing on as defined in (3.11). The following result completes our abstract stability analysis. The proof, given in Appendix B, is by an energy estimate analogous to that of [36].3 Proposition 3.5. The limiting Evans function D 0 (note: independent of γ ) is nonzero on

eλ ≥ 0. From Proposition 3.5 and Corollary 3.3 we obtain Main Theorem 1.4 stated in the Introduction. 3 Stability for γ = 1, proved in [36], already implies nonvanishing of D 0 outside the imaginary interval √ √ [−i 3/2, +i 3/2], by Corollary 3.3.


13

3.5. Numerical computations. Unfortunately, the energy estimate used to establish Proposition 3.5, though mathematically elegant, yields only the stated, abstract result and not quantitative estimates. A simpler and more general approach, that does yield quantitative information, is to compute the reduced Evans function numerically. We carry out this by-now routine numerical computation using the methods of [2]. Specifically, we map a semicircle ∂({ eλ ≥ 0} ∩ {|λ| ≤ 10}) enclosing for γ ∈ [1, 3] by D 0 and compute the winding number of its image about the origin to determine the number of zeroes of D 0 within the semicircle, and thus within . For details of the numerical algorithm, see [2,11,23]. The result is displayed in Fig. 1, clearly indicating stability. More precisely, the minimum of |D| on the semicircle is found to be ≈ 0.2433. Together with Rouchés Theorem, this gives explicit bounds on the size of the Mach number for which shocks are stable, as displayed in Table 1, Sect. 6. Specifically, computing numerically the first value of M at which |D − D 0 |/|D 0 | < 1/2 on the entire semicircle, we obtain the lower bounds M ≥ 500 for γ ∈ [1, 2], M ≥ 2, 400 for γ ∈ [2, 2.5], and M ≥ 13,000 for γ ∈ [2.5, 3], all corresponding approximately to v+ = 10−3 . Recall by Rouchés Theorem that winding number is unaffected by relative error |D − D 0 |/|D 0 | < 1. This is essentially a convergence study, since we rely on the assumed convergence of the relative error to zero in concluding that relative error remains < 1 also for Mach numbers larger than this minimum value. In Fig. 2, we superimpose on the image of the semicircle by D 0 its (again numerically computed) image by the full Evans function D, for a monatomic gas γ = 5/3 at successively higher Mach numbers v+ =1e-1,1e-2,1e-3,1e-4,1e-5,1e-6, graphically demonstrating the convergence of D to D 0 as v+ approaches zero and (numerically) verifying the conclusions of Table 1. In fact, we can see a great deal more from Fig. 2. For, note that the displayed contours are, to the scale visible by eye, “monotone” in v+ , or nested, one within the other (they do not appear so at smaller scales). Thus, lower-Mach number contours are essentially “trapped” within higher-Mach number contours, with the worst-case, outmost contour corresponding to the limiting Evans function D 0 . From this observation, we may conclude with confidence stability down to the smallest value M ≈ 5.5 displayed in the figure, corresponding (see Table 2) to v+ = 10−1 . Note, further, that the low-Mach number contours appear to be shrinking to a point as v+ → 1. This is indeed the case, as can be confirmed by the small-amplitude analysis of [39]; see Remark 3.6. That is, a great deal of topological information is encoded in the analytic family of Evans functions indexed by v+ , from which stability may be deduced almost by inspection. Behavior for other γ ∈ [1, 3] is similar. See, for example, the case γ = 3 displayed in Fig. 4, which is virtually identical to Fig. 2.4 Such topological information does not seem to be available from other methods of investigating stability such as direct discretation of the linearized operator about the wave [28] or studies based on linearized time-evolution or power methods [5,6]. This represents in our view a significant advantage of the Evans function formulation. Remark 3.6. Recall that the Evans function is not determined uniquely, but only up to a nonvanishing analytic factor [1,13]. The simple contour structure in Fig. 2 is thus partly due to a favorable choice of D induced by the initialization at ±∞ by Kato’s ODE [26], as described in [2,11,24]. A canonical algorithm for tracking bases of evolving subspaces, this in some sense minimizes “action”; see [22] for further discussion. 4 In particular, Fig. 4 indicates stability down to v = 10−1 , or Mach number ∼ 20, from which we may + conclude unconditional stability on the whole range γ ∈ [1, 3] of [2].

14


4 3 2

Im

1 0 −1 −2 −3 −4 −2

−1

0

1

2

3

4

5

Re Fig. 1. The image of the semi-circular contour via the Evans function for the reduced system. Note that the winding number of this graph is zero. Hence, there are no unstable eigenvalues in the semi-circle 4 3 2

Im

1 0 −1 −2 −3 −4 −2

−1

0

1

2

3

4

5

Re

Fig. 2. Convergence to the limiting Evans function as v+ → 0 for a monatomic gas, γ = 5/3. The contours depicted, going from inner to outer, are images of the semicircle under D for v+ =1e-1,1e-2, 1e-3,1e-4,1e-5,1e-6. The outermost contour is the image under D 0 , which is nearly indistinguishable from the image for v+ =1e-6

In particular, this leads to the surprisingly simple constant small-amplitude limit, as we show by explicit computation in Appendix D for the case of a Burgers shock. The small-amplitude analysis of [39] shows that the Evans function in the general case converges to a multiple of the Burgers Evans function, yielding the result in the general case. Remark 3.7. Note that the limiting equations, and the limiting Evans function D 0 are both independent of γ . To study high-Mach number stability for γ > 3, therefore,


15

requires only to examine D 0 on successively larger semicircles. Thus, our methods in combination with the those of [2], allow us to determine stability in principle over any bounded interval in γ , for γ ≥ 1 and for all Mach numbers M ≥ 1. Remark 3.8. As Fig. 2 suggests, an alternative method for determining stability, without reference to D 0 , is to compute the full Evans function for sufficiently high Mach number. That is, nonvanishing of D 0 , and thus stability of sufficiently high Mach number shocks for γ ∈ [1, 3], can already be concluded by large-but-finite Mach number study of [2] together with the fact that a limit D → D 0 exists (see also Remark 3.4). 3.6. Conclusions. The analytical result of Theorem 1.4 guarantees stability for γ ≥ 1, M sufficiently large. For γ ∈ [1, 2.5], our numerical results indicate stability for M ≥ 2,500 by a crude Rouche bound, and indeed much lower if further structure is taken into account. Together with the small and intermediate Mach number studies of [2,36] for M ≤ 3, 000, this yields unconditional stability of isentropic Navier–Stokes shocks for γ ∈ [1, 2.5] and M ≥ 1. Additional intermediate-strength computations supplementing [2] extend this result to γ ∈ [1, 3]. There is no inherent restriction to γ ∈ [1, 3]; as discussed in Remark 3.7, numerical computations can be carried out for any value of γ to determine stability (or instability) for all M ≥ 1. Indeed, our method of analysis indicates that the large-γ limit is quite analogous to the high-Mach number limit v+ → 0, suggesting the possibility to establish still more general results encompassing all γ ≥ 1, M ≥ 1. Our numerical results reveal also an unexpected “universality” of behavior in the high-Mach number regime, beyond just convergence to the limiting system. Namely, we see (cf. Figs. 2 and 4) that behavior for a given v+ is virtually independent of the value of γ . This also indicates that v+ and not M is the more useful measure of shock strength in this regime. 4. Boundary-Layer Analysis We now carry out the main work of the paper, analyzing the flow of (2.13) in the singular region. Our starting point is the observation that ⎛ ⎞ 0 λ 1 ⎠ 0 1 A(x, λ) = ⎝ 0 (4.1) λvˆ λvˆ f (v) ˆ −λ is approximately block upper-triangular for vˆ sufficiently small, with diagonal blocks

0λ ˆ − λ that are uniformly spectrally separated on eλ ≥ 0, as follows and f (v) 00 by f (v) ˆ ≤ 2vˆ − 1 ≤ −1/2.

(4.2)

We exploit this structure by a judicious coordinate change converting (2.13) to a system in exact upper triangular form, for which the decoupled “slow” upper lefthand 2 × 2 block undergoes a regular perturbation that can be analyzed by standard tools introduced in [39]. Meanwhile, the fast, lower righthand 1 × 1 block, since scalar, may be solved exactly.

16

J. Humpherys, O. Lafitte, K. Zumbrun 4 3 2

Im

1 0 −1 −2 −3 −4 −2

−1

0

1

2

3

4

5

Re Fig. 3. Convergence to the limiting Evans function as v+ → 0 for γ = 1. The contours depicted, going from inner to outer, are images of the semicircle under D for v+ =1e-1,1e-2,1e-3,1e-4,1e-5,1e-6. The outermost contour is the image under D 0 4 3 2

Im

1 0 −1 −2 −3 −4 −2

−1

0

1

2

3

4

5

Re Fig. 4. Convergence to the limiting Evans function as v+ → 0 for γ = 3. The contours depicted, going from inner to outer, are images of the semicircle under D for v+ =1e-1,1e-2,1e-3,1e-4,1e-5,1e-6. The outermost contour is the image under D 0

The global structure of this argument loosely follows the general strategy introduced in [39] of first decoupling fast and slow modes, then treating slow modes by regular perturbation methods. However, there are interesting departures that may be of use in other degenerate situations. First, we only partially decouple the system, to block-triangular rather than block-diagonal form as in more standard cases, and second, we introduce a more stable method of block-reduction taking account of usually negligible derivative terms in the definition of block-triangularizing transformations, which, if ignored, would in this case lead to unacceptably large errors.


17

4.1. Preliminary transformation. We first block upper-triangularize by a static (constant) coordinate transformation the limiting matrix ⎛ ⎞ 0 λ 1 ⎠ 0 1 A+ = A(+∞, λ) = ⎝ 0 (4.3) λv+ λv+ f (v+ ) − λ at x = +∞ using special block lower-triangular transformations I 0 I −1 R+ := , L + := R+ = λv+ θ+ 1 −λv+ θ+

0 , 1

(4.4)

where I denotes the 2 × 2 identity matrix and θ+ ∈ C1×2 is a 1 × 2 row vector. Lemma 4.1. For each γ ≥ 1, on any compact subset of eλ ≥ 0, for each v+ > 0 sufficiently small, there exists a unique θ+ = θ+ (v+ , λ) such that Aˆ + := L + A+ R+ is upper block-triangular, 11 λ(J + v+ 11θ+ ) , (4.5) Aˆ + = 0 f (v+ ) − λ − λv+ θ+ 11 01 1 where J = and 11 = , satisfying a uniform bound 00 1 |θ+ | ≤ C.

(4.6)

Proof. Setting the (2, 1) block of Aˆ + to zero, we obtain the matrix equation θ+ (bI − λJ ) = −11T + λv+ θ+ 11θ+ , where b = f (v+ ) − λ, or, equivalently, the fixed-point equation θ+ = −11T + λv+ θ+ 11θ+ (bI − λJ )−1 .

(4.7)

By det(bI − λJ ) = b2 = 0, (bI − λJ )−1 is uniformly bounded on compact subsets of

eλ ≥ 0 (indeed, it is uniformly bounded on all of eλ ≥ 0), whence, for |λ| bounded and v+ sufficiently small, there exists a unique solution by the Contraction Mapping Theorem, which, moreover, satisfies (4.6). 4.2. Dynamic triangularization. Defining now Y := L + W and ˆ A(x, λ) = L + A(x, λ)R+ λ(J + v+ 11θ+ ) = λ(vˆ − v+ )11T + λv+ ( f (v) ˆ − f (v+ ))θ+

11 , f (v) ˆ − λ − λv+ θ+ 11

we have converted (2.13) to an asymptotically block upper-triangular system ˆ λ)Y, Y = A(x,

(4.8)

18


ˆ with Aˆ + = A(+∞, λ) as in (4.5). Our next step is to choose a dynamic transformation of the same form I 0 I 0 −1 ˜ ˜ ˜ R := ˜ , L := R = (4.9) ˜ 1 , 1 − ˜ uniformly expoconverting (4.8) to an exactly block upper-triangular system, with nentially decaying at x = +∞: that is, a regular perturbation of the identity. Lemma 4.2. For each γ ≥ 1, on any compact subset of eλ ≥ 0, for L sufficiently ˜ = (x, ˜ large and each v+ > 0 sufficiently small, there exists a unique λ, v+ ) such ˆ that A˜ := L˜ A(x, λ) R˜ + L˜ R˜ is upper block-triangular, ˜ 11 λ(J + v+ 11θ+ ) + 11 ˜ (4.10) A= ˜ 1 , 0 f (v) ˆ − λ − λv+ θ+ 11 − 1 ˜ and (L) = 0, satisfying a uniform bound ˜ | (x, λ, v+ )| ≤ Ce−ηx ,

η > 0, x ≥ L ,

(4.11)

independent of the choice of L, v+ . Proof. Setting the (2, 1) block of A˜ to zero and computing I 0 0 0 0 L˜ R˜ = = ˜ ˜ ˜ − 0 I −

0 , 0

we obtain the matrix equation ˜ − ˜ (bI − λ(J + v+ 11θ+ )) = ζ + 1 ˜ 1 , ˜

(4.12)

where b(x, λ, v+ ) := f (v) ˆ − λ − λv+ θ+ 11 and the forcing term ˆ − f (v+ ))θ+ ζ := λ(vˆ − v+ )11T + λv+ ( f (v) by derivative estimate d f /d vˆ ≤ C vˆ −1 together with the Mean Value Theorem is uniformly exponentially decaying: |ζ | ≤ C|vˆ − v+ | ≤ C2 e−ηx ,

η > 0.

(4.13)

˜ Initializing (L) = 0, we obtain by Duhamel’s Principle/Variation of Constants the representation (supressing the argument λ) x ˜ ˜ 1 )(y) ˜ (x) = S y→x (ζ + 1 dy, (4.14) L

where

S y→x

is the solution operator for the homogeneous equation ˜ − ˜ (bI − λ(J + v+ 11θ+ )) = 0,

or, explicitly, S y→x = e

x y

b(y)dy −λ(J +v+ 11θ+ )(x−y)

e

.


19

For |λ| bounded and v+ sufficiently small, we have by matrix perturbation theory that the eigenvalues of −λ(J + v+ 11θ+ ) are small and the entries are bounded, hence |e−λ(J +v+ 11θ+ )z | ≤ Cez for z ≥ 0. Recalling the uniform spectral gap eb = f (v) ˆ − eλ ≤ −1/2 for eλ ≥ 0, we thus have |S y→x | ≤ Ce−η(x−y)

(4.15)

for some C, η > 0. Combining (4.13) and (4.15), we obtain x x y→x S ζ (y) dy ≤ C2 e−η(x−y) e−(η/2)y dy L

L

= C3 e−(η/2)x .

(4.16)

˜ Defining (x) =: θ˜ (x)e−(η/2)x and recalling (4.14) we thus have x ˜θ (x) = f + e(η/2)x ˜ 1θ˜ (y) dy, S y→x e−ηy θ1 where f := e(η/2)x

x L

(4.17)

L

S y→x ζ (y) dy is uniformly bounded, | f | ≤ C3 , and x (η/2)x S y→x e−ηy θ˜ 11θ˜ (y) dy e L

is contractive with arbitrarily small contraction constant > 0 in L ∞ [L , +∞) for |θ˜ | ≤ 2C3 for L sufficiently large, by the calculation x (η/2)x x y→x −ηy y→x −ηy ˜ e ˜1 11θ˜1 (y) − e(η/2)x ˜ θ θ S e S e 1 1 θ (y) 2 2 L L x ≤ e(η/2)x Ce−η(x−y) e−ηy dy θ˜1 − θ˜2 ∞ max θ˜ j ∞ j L x ≤ e−(η/2)L Ce−(η/2)(x−y) dy θ˜1 − θ˜2 ∞ max θ˜ j ∞ j

L

= C3 e

−(η/2)L

θ˜1 − θ˜2 ∞ max θ˜ j ∞ . j

It follows by the Contraction Mapping Principle that there exists a unique solution θ˜ ˜ of fixed point equation (4.17) with |θ(x)| ≤ 2C3 for x ≥ L, or, equivalently (redefining the unspecified constant η), (4.11). Remark 4.3. The above calculation is the most delicate part of the analysis, and the main technical point of the paper. The interested reader may verify that a “quasi-static” ˜ in (4.12) as an error, as is typically used in situations transformation treating term of slowly-varying coefficients (see for example [33,39]), would lead to unacceptable errors of magnitude O(|( f (v)) ˆ ||vˆ − v+ |) = O(|d f /d v|| ˆ vˆ − v+ |) = O(|v| ˆ −1 |vˆ − v+ |). One may think of the exact ODE solution (4.9) as “averaging” the effects of rapidlyvarying coefficients by integration of (4.12). Note that success of this approach depends

20


in principle only on spectral separation at infinity and not everywhere along the profile; thus, it may be considered as interpolating between the tracking and conjugation lemmas of [50,33,48,49,39], used respectively to treat slowly-varying- and asymptotically constant- coefficient systems. This seems likely to be of use also in other delicate situations. Remark 4.4. One may combine the static and the dynamic triangularization used in this section. We may define ˜ + λv+ θ+ , =

(4.18)

˜ ˜ and obtain the two different terms and λv+ θ+ as follows. Consider A being defined as I 0 L R + L Aˆ R, where L = and R := L −1 . One obtains − 1 A˜ =

λJ + 11 − + ( f (v) ˆ − λ − 11) − λ J + λv1 ˆ 1T

11 . f (v) ˆ − λ − 11

A necessary and sufficient condition for A˜ to be upper block-triangular is that solve ˆ − λ − .11) − λ J + λv1 ˆ 1T . = ( f (v)

(4.19)

If a solution is bounded at +∞ and → 0, then the limit of , called + , solves ( f (v+ ) − λ − + 11) + − λ + J + λv+ 11T = 0. The unique solution of this equation that is uniformly bounded by C|λ|v+ for 0 < v+ < 1/C and λ in a compact subset of ˜ = − λv+ θ+ and replacing in Eq. (4.19),

eλ ≥ 0 is + = λv+ θ+ . Considering now ˜ one obtains the equation (4.12) for . The preliminary, static, transformation is thus a way of subtracting the behavior at infinity (and choosing this behavior) in the dynamic triangularization and can be used in general. For each possible behavior at infinity, however, we have a different solution ˜ and our construction is the only one that ensures the uniform bounds in v+ at infinity. 4.3. Fast/Slow dynamics. Making now the further change of coordinates ˜ Z = LY and computing ˜ ) = LY ˜ + L˜ Y = ( L˜ A+ + L˜ )Y = ( L˜ A+ R˜ + L˜ R)Z ˜ , ( LY we find that we have converted (4.8) to a block-triangular system ˜ 11 ˜ = λ(J + v+ 11θ+ ) + 11 Z = AZ ˜ 1 Z, 0 f (v) ˆ − λ − λv+ θ+ 11 − 1 related to the original eigenvalue system (2.13) by I I 0 W = L Z , R := R+ R˜ = , L := R −1 = − 1

0 . 1

(4.20)

(4.21)


21

Since it is triangular, (4.20) may be solved completely if we can solve the component systems associated with its diagonal blocks. The fast system z =

˜ 1 z f (v) ˆ − λ − λv+ θ+ 11 − 1

associated to the lower righthand block features rapidly-varying coefficients. However, because it is scalar, it can be solved explicitly by exponentiation. The slow system ˜ z z = λ(J + v+ 11θ+ ) + 11

(4.22)

associated to the upper lefthand block, on the other hand, by (4.11), is an exponentially decaying perturbation of a constant-coefficient system z = λ(J + v+ 11θ+ )z

(4.23)

that can be explicitly solved by exponentiation, and thus can be well-estimated by comparison with (4.23). A rigorous version of this statement is given by the conjugation lemma of [37]: Proposition 4.5 ([37]). Let M(x, λ) = M+ (λ) + (x, λ), with M+ continuous in λ and | (x, λ)| ≤ Ce−ηx , for λ in some compact set . Then, there exists a globally invertible matrix P(x, λ) = I + Q(x, λ) such that the coordinate change z = Pv converts the variable-coefficient ODE z = M(x, λ)z to a constant-coefficient equation v = M+ (λ)v, satisfying for any L, 0 < ηˆ < η a uniform bound ˆ for x ≥ L. |Q(x, λ)| ≤ C(L , η, ˆ η, max |(M+ )i j |, dim M+ )e−ηx

Proof. See [37,49], or Appendix C.

(4.24)

By Proposition 4.5, the solution operator for (4.22) is given by P(x, λ)eλ(J +v+ 11θ+ (λ,v+ ))(x−y) P(y, λ)−1 ,

(4.25)

where P is a uniformly small perturbation of the identity for x ≥ L and L > 0 sufficiently large.

5. Proof of the Main Theorem With these preparations, we turn now to the proof of the main theorem.

22


5.1. Boundary estimate. We begin by establishing the following key estimates on + (L), that is, the value of the dual mode W + appearing in (2.20) at the boundary W 1 1 x = L between regular and singular regions. Lemma 5.1. For each γ ≥ 1, for λ on any compact subset of eλ ≥ 0, and L > 0 + normalized as in [13,39,2], sufficiently large, with W 1 1 | ≤ Ce−ηL 1+ (L) − V |W

(5.1)

as v+ → 0, uniformly in λ, where C, η > 0 are independent of L and 1 := (0, 1, (1 + λ¯ )−1 )T V is the limiting direction vector (3.9) appearing in the definition of D 0 . Corollary 5.2. For each γ ≥ 1, under the hypotheses of Lemma 5.1, 1 | ≤ Ce−ηL |W˜ 10+ (L) − V

(5.2)

10+ (L)| ≤ Ce−ηL 1+ (L) − W |W

(5.3)

and

0+ is the solution as v+ → 0, uniformly in λ, where C, η > 0 are independent of L and W 1 of the limiting adjoint eigenvalue system appearing in definition (3.8) of D 0 . Proof of Lemma 5.1. Making the coordinate-change Z˜ := R ∗ W˜ ,

(5.4)

R as in (4.21), reduces the adjoint equation W˜ = −A∗ W˜ to block lower-triangular form, Z˜ = − A˜ ∗ Z˜ ∗ ˜ − λJ + λv+ 11θ+ ) + 11 0 Z , (5.5) = ˜ 1)∗ − f (v) ˆ + λ¯ + λ¯ v+ (θ+ 11)∗ + ( 1 −11T with “¯” denoting complex conjugate. Denoting by V˜1+ a suitably normalized element of the one-dimensional (slow) stable subspace of − A˜ ∗ , we find, similarly as in the discussion of Sect. 3.2 that, without loss of generality, V˜1+ → (0, 1, (γ + λ¯ )−1 )T

(5.6)

0, while the associated eigenvalue µ˜ +1 → 0, uniformly for λ on an compact

eλ ≥ 0. The dual mode Z˜ 1+ = R ∗ W˜ 1+ is uniquely determined by the prop-

as v+ → subset of erty that it is asymptotic as x → +∞ to the corresponding constant-coefficient solution + eµ˜ 1 V˜1+ (the standard normalization of [2,13,39]). By lower block-triangular form (5.5), the equations for the slow variable z˜ T := ( Z˜ 1 , Z˜ 2 ) decouples as a slow system ∗ ˜ z˜ (5.7) z˜ = − λ(J + v+ 11θ+ ) + 11


23

dual to (4.22), with solution operator ¯

P ∗ (x, λ)−1 e−λ(J +v+ 11θ+ )

∗ )(x−y)

P(y, λ)∗

(5.8)

dual to—that is, the adjoint inverse of—(4.25), i.e. (fixing y = L, say), having solutions of general form ¯

z˜ (λ, x) = P ∗ (x, λ)−1 e−λ(J +v+ 11θ+ )

∗ )x

v, ˜

(5.9)

v˜ ∈ C2 arbitrary. Denoting by Z˜ 1+ (L) := R ∗ W˜ 1+ (L), therefore, the unique (up to constant factor) decaying solution at +∞, and v˜1+ := ((V˜ + )1 , (V˜ + )2 )T , we thus have evidently 1

1

¯

z˜ 1+ (x, λ) = P ∗ (x, λ)−1 e−λ(J +v+ 11θ+ )

∗ )x

v˜1+ ,

which, as v+ → 0, is uniformly bounded by |˜z 1+ (x, λ)| ≤ Cex

(5.10)

for arbitrarily small > 0 and, by (5.6), converges for x less than or equal to any fixed X simply to lim z˜ 1+ (x, λ) = P ∗ (x, λ)−1 (0, 1)T .

v+ →0

(5.11)

Defining by q˜ := ( Z˜ 1+ )3 the fast coordinate of Z˜ 1+ , we have, by (5.5), ˜ 1)∗ q˜ = 11T z˜ 1+ , ˆ − λ¯ − (λv+ θ+ 11 + 1 q˜ + f (v) whence, by Duhamel’s principle, any decaying solution is given by +∞ x ˜ q(x, ˜ λ) = e y b(z,λ,v+ )dz 11T z 1+ (y) dy, x

where

¯ λ, v+ ) = − f (v) ˜ λ, v+ ) := −b(y, ˜ 1)∗ , ˆ − λ¯ − (λv+ θ+ 11 + 1 b(y,

b as in (4.12). Recalling, for eλ ≥ 0, that eb˜ = − b ≥ 1/2, combining (5.10) and (5.11), and noting that b˜ converges uniformly on y ≤ Y as v+ → 0 for any Y > 0 to ˜ 0 11)∗ ˆ + λ¯ + ( b˜0 (y, λ) := − f 0 (v) = (1 + λ¯ ) + O(e−ηy ), we obtain by the Lebesgue Dominated Convergence Theorem that +∞ L ˜ q(L ˜ , λ) → e y b0 (z,λ)dz 11T (0, 1)T dy L+∞ L −ηz ¯ = e(1+λ)(L−y)+ y O(e )dz dy L

= (1 + λ¯ )−1 (1 + O(e−ηL )).

24


Recalling, finally, (5.11), and the fact that |P − I d|(L , λ), |R − I d|(L , λ) ≤ Ce−ηL for v+ sufficiently small, we obtain (5.1) as claimed.

Proof of Corollary 5.2. Applying Proposition 4.5 to the limiting adjoint system, ⎛ ⎞ 0 0 0 0 0 ⎠ W˜ + O(e−ηx )W˜ , W˜ = −(A0 )∗ W˜ = ⎝−λ¯ −1 −1 1 + λ¯ we find that, up to an I d + O(e−ηx ) coordinate change, W˜ 10+ (x) is given by the exact solution W˜ ≡ V˜1 of the limiting, constant-coefficient system ⎛ ⎞ 0 0 0 0 0 ⎠ W˜ . W˜ = −(A0 )∗ W˜ = ⎝−λ¯ −1 −1 1 + λ¯ This yields immediately (5.2), which, together with (5.1), yields (5.3).

Remark 5.3. Noting that (5.2) is sharp, we see from (5.3) that the error between W˜ 1+ (L) and W˜ 10+ (L) is already within the error tolerance of the numerical scheme used to approx 0+ is initialized at x = L with approximate value V˜1 [2,11,39]. imate D 0 , in which W 1 Thus, so long as the flow on the regular region x ≤ L well-approximates the exact limiting flow as v+ → 0, we can expect convergence of D to D 0 based on the known convergence of the numerical approximation scheme. 5.2. Convergence to D 0 . As hinted by Remark 5.3, the rest of our analysis is standard if not entirely routine. Lemma 5.4. For each γ ≥ 1, on x ≤ L for any fixed L > 0, there exists a coordinatechange W = T Z conjugating (2.13) to the limiting equations (3.4), T = T (x, λ, v+ ), satisfying a uniform bound |T − I d| ≤ C(L)v+

(5.12)

for all v+ > 0 sufficiently small. Proof. For x ∈ (−∞, 0], this is a consequence of the Convergence Lemma of [39], a variation on Proposition 4.5, together with uniform convergence of the profile and eigenvalue equations. For x ∈ [0, L], it is essentially continuous dependence; more precisely, observing that |A − A0 | ≤ C1 (L)v+ for x ∈ [0, L], setting S := T − I d, and writing the homological equation expressing conjugacy of (2.13) and (3.4), we obtain S − (AS − S A0 ) = (A − A0 ), which, considered as an inhomogeneous linear matrix-valued equation, yields an exponential growth bound S(x) ≤ eC x (S(0) + C −1 C1 (L)v+ ) for some C > 0, giving the result.


25

Proof of Theorem 3.1. Lemma 5.4, together with convergence as v+ → 0 of the unstable subspace of A− to the unstable subspace of A0− at the same rate O(v+ ) (as follows by spectral separation of the unstable eigenvalue of A0 and standard matrix perturbation theory) yields |W1− (0, λ) − W10− (0, λ)| ≤ C(L)v+ .

(5.13)

Likewise, Lemma 5.4 gives |W˜ 1+ (0, λ) − W˜ 10+ (0, λ)| ≤ C(L)v+ |W˜ 1+ (0, λ)| +|S0L→0 ||W˜ 1+ (L , λ) − W˜ 10+ (L , λ)|,

(5.14)

y→x

where S0 denotes the solution operator of the limiting adjoint eigenvalue equation W˜ = −(A0 )∗ W˜ . Applying Proposition 4.5 to the limiting system, we obtain |S0L→0 | ≤ C2 e−A+ L ≤ C2 L|λ| 0

by direct computation of e−A+ L , where C2 is independent of L > 0. Together with (5.3) and (5.14), this gives 0

|W˜ 1+ (0, λ) − W˜ 10+ (0, λ)| ≤ C(L)v+ |W˜ 1+ (0, λ)| + L|λ|C2 Ce−ηL , hence, for |λ| bounded, |W˜ 1+ (0, λ) − W˜ 10+ (0, λ)| ≤ C3 (L)v+ |W˜ 10+ (0, λ)| + LC4 e−ηL ≤ C5 (L)v+ + LC4 e−ηL .

(5.15)

Taking first L → ∞ and then v+ → 0, we obtain therefore convergence of W1+ (0, λ) and W˜ 1+ (0, λ) to W10+ (0, λ) and W˜ 10+ (0, λ), yielding the result by definitions (2.20) and (3.8). 6. Numerical Convergence Having established analytically the convergence of D to D 0 as v+ → 0 (M → ∞), we turn finally to numerics to obtain quantitative information yielding a concrete stability threshold. Specifically, for fixed γ , we numerically compute the “Rouché bound” consisting of the value of v+ at which the maximum relative error |D − D 0 |/|D 0 | over the semicircular contour ∂{ eλ ≥ 0, |λ| ≤ 10} around which we perform our winding number calculations becomes 1/2. Recall that Rouché’s Theorem guarantees for relative error < 1 that the winding number of D is equal to the winding number of D 0 , which we have shown to be zero, hence we may conservatively conclude stability for v+ less than or equal to this bound, or M greater than or equal to the corresponding Mach number. Computations are performed using the algorithm of [2]; results are displayed in Table 1. More detailed results are displayed for the monatomic gas case γ = 5/3 in Table 2. Results are similar for other γ ∈ [1, 3], as may be seen by comparing Figs. 2–4. From the quantitative gap and conjugation estimates given in Appendix C, which in turn yield a quantitative version of the Convergence Lemma of [39], one could in principle establish quantitative convergence rates for |D − D 0 |, by tracking constants carefully through the estimates of the previous sections. Indeed, one could do much better than the rather crude bounds stated for the general case by taking into account the

26

J. Humpherys, O. Lafitte, K. Zumbrun Table 1. Rouché bounds for various γ

γ

v+

Relative Error

Mach Number

3.0 2.5 2.0 1.5 1.0

1.27e-3 1.36e-3 1.49e-3 1.75e-3 2.8e-3

.5009 .5006 .5001 .4999 .4995

12765 2423 474 95.5 18.9

Table 2. Maximum relative and absolute differences between D and D 0 , for γ = 5/3 and λ on the semicircle of radius 10 v+

Mach Number

Relative Difference

Absolute Difference

1.0 (−6) 1.0 (−5) 1.0 (−4) 1.0 (−3) 1.0 (−2) 1.0 (−1)

7.71 (4) 1.13 (4) 1.64 (3) 2.44 (2) 36.1 5.50

0.1221 0.1236 0.1487 0.4098 0.9046 1.2386

0.0601 0.1445 0.4714 1.3464 2.8253 3.8688

eigenstructure of the actual matrices A± , A0± appearing in our analysis, and (crucial for good estimates, since bounds grow exponentially with dimension n) by observing that it suffices to use the gap and not the full conjugation lemma for the estimation of the single dual mode at plus infinity. That is, there are contained in our analysis, as in the study of [2], all of the ingredients needed for a numerical proof. Given the fundamental nature of the problem studied, this would be a very interesting program to carry out.

7. Discussion and Open Problems Besides long-time stability, our results have application also to existence of shock layers in the small-viscosity limit, which likewise reduces to the question of stability of the Evans function [17,41]. Indeed, spectral stability has been a key missing piece in several directions [48,49]. Our methods should have application also to spectral stability of large-amplitude noncharacteristic boundary layers, completing the investigations of [16,37,44]. It may be hoped that they will extend also to full gas dynamics and multi-dimensions, two important directions for further investigation. As discussed in the text, the problems of numerical proof and of stability in the large-γ limit are two other interesting directions for further study. More speculatively, our results suggest the possibility of a large-variation version of the results obtained by quite different methods in [7] on general viscous solutions (including not only noninteracting shocks, but shocks, rarefactions, and their interactions), and, through the physical insight provided into the high-Mach number limit, perhaps even a hint toward possible methods of analysis. Note that the results of [7] include not only convergence in the small-viscosity limit but also bounded L 1 stability for constant viscosity of general small-variation solutions. This would be an extremely interesting direction for further investigation.


27

Appendix A. Proofs of Preliminary Estimates Proof of Proposition 2.1. Existence and monotonicity follow trivially by the fact that (2.4) is a scalar first-order ODE with convex righthand side. Exponential convergence as x → +∞ follows, for example, by the computation (v+ − 1)(v −γ − 1) H (v, v+ ) = v (v − 1) − −γ v+ − 1 1 − v+ v+ γ = v (v − v+ ) + − 1 γ v 1 − v+ γ 1 − vv+ 1 − v+ = (v − v+ ) v − , γ 1 − vv+ 1 − v+ yielding v−γ ≤

H (v, v+ ) ≤ v − (1 − v+ ) v − v+ γ

by the elementary estimate 1 ≤ 1−x 1−x ≤ γ for 0 ≤ x ≤ 1. Convergence as x → −∞ follows by a similar, but simpler computation; see [2]. Lemma A.1. For each γ ≥ 1, the following identity holds for eλ ≥ 0: 1 ( e(λ) + |m(λ)|) v|u| ˆ 2− vˆ x |u|2 + |u |2 2 R R R √ h(v) ˆ ≤ 2 |v ||u| + v|u ˆ ||u|. γ R vˆ R

(A.1)

Proof. We multiply (2.9b) by vˆ u¯ and integrate along x. This yields h(v) ˆ ˆ 2+ vu ˆ u¯ + |u |2 = v u. ¯ λ v|u| γ R R R R vˆ We get (A.1) by taking the√ real and imaginary parts and adding them together, and noting that | e(z)| + |m(z)| ≤ 2|z|. Lemma A.2. For each γ ≥ 1, the following identity holds for eλ ≥ 0: h(v) ˆ |v |2 1 aγ |v |2 . + |u |2 = 2 e(λ)2 |v|2 + e(λ) + 2 R vˆ γ +1 vˆ γ +1 R R R vˆ

(A.2)

Proof. We multiply (2.9b) by v¯ and integrate along x. This yields h(v) ˆ 2 1 1 u (λv + v )v¯ . u v¯ − |v | = v ¯ = λ u v¯ + γ +1 R R R vˆ R vˆ R vˆ Using (2.9a) on the right-hand side, integrating by parts, and taking the real part gives h(v) ˆ |v |2 vˆ x 2 |v .

e λ u v¯ + u v¯ = + | +

e(λ) γ +1 2vˆ 2 R R R vˆ R vˆ

28


The right hand side can be rewritten as 1 h(v) ˆ |v |2 aγ 2 |v u v¯ = + | +

e(λ)

e λ u v¯ + . 2 R vˆ γ +1 vˆ γ +1 R R R vˆ

(A.3)

Now we manipulate the left-hand side. Note that ¯ λ u v¯ + u v¯ = (λ + λ) u v¯ − u(λ¯ v¯ + v¯ ) R R R R = −2 e(λ) u v¯ − u u¯ R R = −2 e(λ) (λv + v )v¯ + |u |2 . R

R

Hence, by taking the real part we get u v¯ = |u |2 − 2 e(λ)2 |v|2 .

e λ u v¯ + R

R

R

R

This combines with (A.3) to give (A.2). Lemma A.3. For each γ ≥ 1, for h(v) ˆ as in (2.8), we have h(v) ˆ 1 − v+ sup γ = γ γ ≤ γ. vˆ 1 − v+ vˆ

(A.4)

Proof. Defining g(v) ˆ := h(v) ˆ vˆ −γ = −vˆ + a(γ − 1)vˆ −γ + (a + 1),

(A.5)

ˆ = −1 − aγ (γ − 1)vˆ −γ −1 < 0 for 0 < v+ ≤ vˆ ≤ v− = 1, hence the we have g (v) maximum of g on vˆ ∈ [v+ , v− ] is achieved at vˆ = v+ . Substituting (2.5) into (A.5) and simplifying yields (A.4). Proof of Proposition 2.2. Using Young’s inequality twice on right-hand side of (A.1) together with (A.4), we get 1 ( e(λ) + |m(λ)|) v|u| ˆ 2− vˆ x |u|2 + |u |2 2 R R R √ h(v) ˆ ≤ 2 |v ||u| + v|u ˆ ||u| γ R vˆ R √ h(v) ˆ 2 ( 2)2 h(v) ˆ 1 2 2 ≤θ |v | + v|u| ˆ + v|u ˆ | + v|u| ˆ 2 γ +1 γ 4θ 4 R R vˆ R vˆ R 1 h(v) ˆ 2 γ 2 + 0, and in particular e(u u) ¯ (1+)x x /e ≤ Cex → 0 1 − vˆ ≤ Ce as x → −∞. It follows that the boundary contribution at x = a vanishes also as a → −∞, hence, in the limit as a → −∞, b → +∞, +∞ +∞ 2 vˆ |u |

e(λ) |u|2 + |v|2 d x + d x = 0. (B.5) 1 − vˆ −∞ −∞ 1 − vˆ


31

But, for eλ ≥ 0, this implies u ≡ 0, or u ≡ constant, which, by u(−∞) = 0, implies u ≡ 0. This reduces (B.1a) to v = λv, yielding the explicit solution v = Ceλx . By v(±∞) = 0, therefore, v ≡ 0 for eλ ≥ 0. It follows that there are no nontrivial solutions of (B.1), (B.2) for eλ ≥ 0. Remark B.1. The above is equivalent to multiplying the system by the energy estimate 1 0 special symmetrizer , then taking the L 2 inner product with (v, u)T . The 0 v/(1 ˆ − v) ˆ analog of the high-frequency estimates of Appendix A would be obtained using the alter 1 − vˆ 0 native symmetrizer optimized for its effect on second-order derivative term 0 vˆ u /v. ˆ This may clarify somewhat the strategy of the energy estimates used in [2,36].

Appendix C. Quantitative Conjugation Estimates Consider a general first-order system W = A(x, λ)W.

(C.1)

Proposition C.1 (Quantitative Gap Lemma [13,50]). Let V + and µ+ be an eigenvector and associated eigenvalue of A+ (λ) and suppose that there exist complementary generalized eigenprojections (i.e., A-invariant projections) P and Q such that |Pe(A+ −µ

+ )x

ˆ | ≤ C1 e−ηx x ≤ 0,

−µ+ )x

ˆ |Qe(A+ | ≤ C1 e−ηx x ≥ 0, −ηx |(A − A+ )(x)| ≤ C2 e x ≥ 0,

(C.2)

with 0 ≤ ηˆ < η. Then, there exists a solution W = eµ x V (x, λ) of (C.1) with +

C1 C2 e−ηx |V (x, λ) − V + (λ)| ≤ for x ≥ L |V + (λ)| (η − η)(1 ˆ − )

(C.3)

provided (η − η) ˆ −1 C1 C2 e−ηL ≤ < 1. Proof. Writing V = (A+ − µ+ )V + (A − A+ )V and imposing the limiting behavior V (+∞, λ) = V + , we seek a solution in the form V = T V ,

+∞

Pe(A+ −µ )(x−y) (A − A+ )V (y)dy T V (x) := V − x x + Qe(A+ −µ )(x−y) (A − A+ )V (y)dy, + +

L

+

32


from which the result follows by a straightforward Contraction Mapping argument, using (C.2) to compute that +∞ + Pe(A+ −µ )(x−y) (A − A+ )(V1 − V2 )(y)dy |T V1 − T V2 |(x) = − x x + Qe(A+ −µ )(x−y) (A − A+ )(V1 − V2 )(y)dy + L +∞ ˆ ≤ C1 C2 e−η(x−y) e−ηy dy V1 − V2 L ∞ [L ,+∞) L

ˆ e−(η−η)L ˆ C1 C2 e−ηx

V1 − V2 L ∞ [L ,+∞) , = η − ηˆ

and thus T V1 − T V2 L ∞ [L ,+∞) ≤

C1 C2 e−ηL

V1 η−ηˆ

− V2 L ∞ [L ,+∞) .

Corollary C.2. Let V + and µ+ be an eigenvector and associated eigenvalue of A+ (λ), where A+ is n × n with at most k eigenvalues of real part < µ+ and max |(A+ − µ)i j | ≤ C0 ;

|(A − A+ )(x)| ≤ C2 e−ηx x ≥ 0,

(C.4)

0 < ηˆ < η. Then, there exists a solution W = eµ x V (x, λ) of (C.1) with +

ˆ 16nn!(C0 )n C2 e−ηx |V (x, λ) − V + (λ)| ≤ for x ≥ L, |V + (λ)| δ n (η − η)(1 ˆ − )

δ :=

η−ηˆ 2k+2 ,

provided

ˆ 16nn!(C0 )n C2 e−ηL δ n (η−η) ˆ

(C.5)

≤ < 1.

Proof. Without loss of generality, take µ ≡ 0. Dividing [−η, −η] ˆ into k + 1 equal subintervals, we find by the pigeonhole principle that at least one subinterval contains the real part of no eigenvalue of A+ . Denoting the midpoint of this interval by −η˜ > η, ˆ we have min | eσ (A+ ) − η| ˜ ≥ δ :=

η − ηˆ . 2k + 2

(C.6)

Defining P to be the total eigenprojection of A+ associated with eigenvalues of real part greater than ηˆ and Q the total eigenprojection associated with eigenvalues of real part less than η, ˆ and estimating Pe A+ x , Qe A+ x using the inverse Laplace transform representation 1 e A+ x = e zx (z − A+ )−1 dz, 2πi with chosen to be a rectangle of side 4nC0 centered about the real axis, with one vertical side passing through eλ ≡ −η˜ and the other respectively lying respectively to the right and to the left, and estimating |(λ − A+ )−1 | ≤ n!C0n−1 δ −n crudely by Kramer’s rule, we obtain (C.2) with C1 = 16nn!C0n δ −n , whence the result follows by Proposition C.1.


33

Corollary C.3 (Quantitative Conjugation Lemma). Proposition 4.5 holds for 0 < < 1 with C(L , η, ˆ η, max |(M+ )i j |, dim M+ ) = n := (dim M+ )2 , k :=

(dim M+ )2 −dim M , 2

when

ˆ 16nn!(C0 )n C2 e−ηx , δ n (η − η)(1 ˆ − )

ˆ 16nn!(C0 )n C2 e−ηL δ n (η−η) ˆ

≤ .

Proof. Writing the homological equation expressing conjugacy of variable- and constant-coefficient systems following [37], we have P = M+ P − P M+ + M. Considering this as an asymptotically constant-coefficient system on the n 2 -dimensional vector space of matrices P, noting that the linear operator M+ P := M+ P − P M+ , as a Sylvester matrix, has at least n zero eigenvalues and equal numbers of stable and unstable eigenvalues, we see that the number of its stable eigenvalues is not more than 2 k := n 2−n , whence the result follows by Corollary C.2.

Appendix D. An Illuminating Example Consider Burgers’ equation, u t + (u 2 )x = u x x , and the family of stationary viscous shock solutions uˆ (x) := − tanh(x/2),

lim uˆ (z) = ∓

z→±∞

(D.1)

of amplitude |u + − u − | = 2 going to zero as → 0. The linearized eigenvalue equation u = (uˆ u) + λu about u¯ , expressed in the x integrated variable w(x) := −∞ u(y)dy, appears as w = uˆ w + λw.

(D.2)

This reduces by the linearized Hopf–Cole transformation w = sech(x/2)z to the constant-coefficient linear oscillator equation z = (λ + 2 /4)z, yielding exact solutions √ 2 w ± (x, λ) = sech(x/2)e∓ /4+λx (D.3) decaying, respectively, as x → ±∞, with asymptotic behavior (D.4) W ± (x, λ) ∼ eµ± (λ)x V± (λ), where µ± (λ) := ∓(/2 + 2 /4 + λ and V± := (1, µ± (λ))T are the eigenvalues and eigenvectors of the limiting constant-coefficient equations at plus and minus spatial T. infinity written as a first-order system, W± := (w, w )± − + Defining an Evans function D(λ) := det(W , W )|x=0 , we may compute explicitly (D.5) D(λ) = −2 2 /4 + λ.

34


However, this is not “the” Evans function D(λ) := det(W − , W + )|x=0 specified in Sect. 2.4, which is constructed, rather, from a special basis W± = c± (λ)W ± ∼ eµ± x V ± , where V± = c± (λ)V ± are “Kato” eigenvectors determined uniquely (up to a constant factor independent of λ) by the property that there exist corresponding left eigenvectors V˜ ± such that (V˜ · V )± ≡ constant, (V˜ · V˙ )± ≡ 0,

(D.6)

where “ ˙ ” denotes d/dλ; see [26,13,22] for further discussion. Computing dual eigenvectors V˜ ± = (λ + µ2 )−1 (λ, µ± ) satisfying (V˜ · V)± ≡ 1, and setting V ± = c± V ± , V˜ ± = V ± /c± , we find after a brief calculation that (D.6) is equivalent to the complex ODE, ± V˜ · V˙ µ˙ c˙± = − c± = − c± , (D.7) 2µ − ± V˜ · V which may be solved by exponentiation, yielding the general solution c± (λ) = C( 2 /4 + λ)−1/4 .

(D.8)

Initializing at a fixed nonzero point, without loss of generality c± (1) = 1,5 and noting that D (λ) = c− c+ D (λ), we thus obtain the remarkable formula (D.9) D (λ) ≡ −2 2 /4 + 1. That is, with the “Kato” normalization, the Evans function associated with a Burgers shock is not only stable (nonvanishing), but identically constant. Moreover, in the weak shock limit, → 0, D (λ) converges uniformly to the constant function D0 (λ) ≡ −2.6 References 1. Alexander, J., Gardner, R., Jones, C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990) 2. Barker, B., Humpherys, J., Rudd, K., Zumbrun, K.: Stability of viscous shocks in isentropic gas dynamics. Commun. Math. Phys. 281(1), 231–249 (2008) 3. Barmin, A.A., Egorushkin, S.A.: Stability of shock waves. Adv. Mech. 15(1–2), 3–37 (1992) 4. Batchelor, G.K.: An introduction to fluid dynamics. Cambridge Mathematical Library. Cambridge: Cambridge University Press, 1999 5. Bertozzi, A.L., Brenner, M.P.: Linear stability and transient growth in driven contact lines. Phys. Fluids 9(3), 530–539 (1997) 6. Bertozzi, A.L., Münch, A., Fanton, X., Cazabat, A.M.: Contact line stability and undercompressive shocks in driven thin film flow. Phys. Rev. Lett. 81(23), 5169–5172 (1998) 7. Bianchini, S., Bressan, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2) 161(1), 223–342 (2005) 5 In the numerics of Sect. 6, we initialize always at λ = 10. 6 Not to be confused with the limiting Evans function D 0 in the strong shock limit v → 0. +


35

8. Bridges, T.J., Derks, G., Gottwald, G.: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D 172(1-4), 190–216 (2002) 9. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, Bloomington, 1998 10. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. Math. Comp. 70(235), 1071–1088 (2001) 11. Brin, L.Q., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. In: Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). Mat. Contemp. 22, 19–32, (2002). 12. Freistühler, H., Szmolyan, P.: Spectral stability of small shock waves. Arch. Ration. Mech. Anal. 164(4), 287–309 (2002) 13. Gardner, R.A., Zumbrun, K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998) 14. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344 (1986) 15. Goodman, J.: Remarks on the stability of viscous shock waves. In: Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990), Philadelphia, PA: SIAM, 1991, pp. 66–72 16. Grenier, E., Rousset, F.: Stability of one-dimensional boundary layers by using Green’s functions. Comm. Pure Appl. Math. 54(11), 1343–1385 (2001) 17. Guès, C.M.I.O., Métivier, G., Williams, M., Zumbrun, K.: Navier-Stokes regularization of multidimensional Euler shocks. Ann. Sci. École Norm. Sup. (4) 39(1), 75–175 (2006) 18. Hoff, D.: The zero-Mach limit of compressible flows. Commun. Math. Phys. 192(3), 543–554 (1998) 19. Hoover, W.G.: Structure of a shock-wave front in a liquid. Phys. Rev. Lett. 42(23), 1531–1534 (1979) 20. Howard, P., Raoofi, M.: Pointwise asymptotic behavior of perturbed viscous shock profiles. Adv. Differ. Eqs. 11(9), 1031–1080 (2006) 21. Howard, P., Raoofi, M., Zumbrun, K.: Sharp pointwise bounds for perturbed viscous shock waves. J. Hyperbolic Differ. Eq. 3(2), 297–373 (2006) 22. Humpherys, J., Sandstede, B., Zumbrun, K.: Efficient computation of analytic bases in Evans function analysis of large systems. Numer. Math. 103(4), 631–642 (2006) 23. Humpherys, J., Zumbrun, K.: Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems. Z. Angew. Math. Phys. 53(1), 20–34 (2002) 24. Humpherys, J., Zumbrun, K.: An efficient shooting algorithm for Evans function calculations in large systems. Phys. D 220(2), 116–126 (2006) 25. Il’in, A.M., Ole˘ınik, O.A.: Behavior of solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time. Dokl. Akad. Nauk SSSR 120, 25–28 (1958); (see also AMS Translations 42(2), 19–23 (1964). 26. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995, reprint of the 1980 edition 27. Klainerman, S., Majda, A.: Compressible and incompressible fluids. Comm. Pure Appl. Math. 35(5), 629– 651 (1982) 28. Kreiss, G., Liefvendahl, M.: Numerical investigation of examples of unstable viscous shock waves. In: Hyperbolic problems: theory, numerics, applications, Vol. I, II (Magdeburg, 2000), Volume 141 of Internat. Ser. Numer. Math., Basel: Birkhäuser, 2001 pp. 613–621 29. Kreiss, H.-O., Lorenz, J., Naughton, M.J.: Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations. Adv. in Appl. Math. 12(2), 187–214 (1991) 30. Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(328), v–108 (1985) 31. Liu, T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50(11), 1113–1182 (1997) 32. Liu, T.-P., Yu, S.-H.: Boltzmann equation: micro-macro decompositions and positivity of shock profiles. Commun. Math. Phys. 246(1), 133–179 (2004) 33. Mascia, C., Zumbrun, K.: Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Ration. Mech. Anal. 169(3), 177–263 (2003) 34. Mascia, C., Zumbrun, K.: Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Ration. Mech. Anal. 172(1), 93–131 (2004) 35. Mascia, C., Zumbrun, K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Comm. Pure Appl. Math. 57(7), 841–876 (2004) 36. Matsumura, A., Nishihara, K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985) 37. Métivier, G., Zumbrun, K.: Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. 175(826), vi+107 (2005)

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38. Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340(1656), 47–94 (1992) 39. Plaza, R., Zumbrun, K.: An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete Contin. Dyn. Syst. 10(4), 885–924 (2004) 40. Raoofi, M.: L p asymptotic behavior of perturbed viscous shock profiles. J. Hyperbolic Differ. Equ. 2(3), 595–644 (2005) 41. Rousset, F.: Viscous approximation of strong shocks of systems of conservation laws. SIAM J. Math. Anal. 35(2), 492–519 (electronic), (2003) 42. Serre, D.: Systems of conservation laws. 1. Cambridge: Cambridge University Press, 1999, translated from the 1996 French original by I. N. Sneddon 43. Serre, D.: Systems of conservation laws. 2. Cambridge: Cambridge University Press, 2000. translated from the 1996 French original by I. N. Sneddon 44. Serre, D., Zumbrun, K.: Boundary layer stability in real vanishing viscosity limit. Commun. Math. Phys. 221(2), 267–292 (2001) 45. Smoller, J.: Shock waves and reaction-diffusion equations. 2nd ed. New York: Springer-Verlag, 1994 46. Szepessy, A., Xin, Z.P.: Nonlinear stability of viscous shock waves. Arch. Ration. Mech. Anal. 122(1), 53–103 (1993) 47. Zumbrun, K.: Refined wave-tracking and nonlinear stability of viscous Lax shocks. Meth. Appl. Anal. 7(4), 747–768 (2000) 48. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In: Advances in the theory of shock waves, Volume 47 of Progr. Nonlinear Differential Equations Appl. Boston, MA: Birkhäuser Boston, 2001, pp. 307–516 49. Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 311–533 with an appendix by Helge Kristian Jenssen and Gregory Lyng 50. Zumbrun, K., Howard, P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871 (1998) Communicated by P. Constantin

Commun. Math. Phys. 293, 37–83 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0940-z

Communications in


Strichartz Estimates on Schwarzschild Black Hole Backgrounds Jeremy Marzuola1 , Jason Metcalfe2 , Daniel Tataru3 , Mihai Tohaneanu4 1 Department of Applied Physics and Applied Mathematics, Columbia University,

New York, NY 10027, USA

2 Department of Mathematics, University of North Carolina, Chapel Hill,

NC 27599-3250, USA

3 Department of Mathematics, University of California, Berkeley,

CA 94720-3840, USA

4 Department of Mathematics, Purdue University, West Lafayette,

IN 47907-2067, USA Received: 9 April 2008 / Accepted: 13 August 2009 Published online: 4 November 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of [29], to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere. 1. Introduction The aim of this article is to contribute to the understanding of the global-in-time dispersive properties of solutions to wave equations on Schwarzschild black hole backgrounds. Precisely, we consider two robust ways to measure dispersion, namely the local energy estimates and the Strichartz estimates. Let us begin with the local energy estimates. For solutions to the constant coefficient wave equation in 3 + 1 dimensions, u = 0,

u(0) = u 0 , u t (0) = u 1 ,

we have the original estimates of Morawetz [33], 1 t 1 | ∇ u|2 (t, x) dt d x ∇u 0 2L 2 + u 1 2L 2 , 3 |x| 0 R

(1.1)

where ∇ denotes the angular derivative. To prove this one multiplies the wave equation by the multiplier (∂r + r1 )u and integrates by parts. Within dyadic spatial regions one can also control u, ∂t u and ∂r u. Precisely, we have the local energy estimates The authors were supported in part by the NSF grants DMS0354539 and DMS0301122. 1 There is another estimate commonly referred to as a Morawetz estimate. This corresponds to using the multiplier (t 2 + r 2 )∂t + 2tr ∂r . We will reserve the term Morawetz estimate for (1.1) and shall call the latter

estimate the Morawetz conformal estimate.

38

J. Marzuola, J. Metcalfe, D. Tataru, M. Tohaneanu 1

3

R − 2 ∇u L 2 (R×B(0,R)) + R − 2 u L 2 (R×B(0,R)) ∇u 0 L 2 + u 1 L 2 .

(1.2)

See for instance [20,22,40–42]. One can also consider the inhomogeneous problem, u = f,

u(0) = u 0 , u t (0) = u 1 .

In view of (1.2) we define the local energy space L E M for the solution u by j 3j u L E M = sup 2− 2 ∇u L 2 (A j ) + 2− 2 u L 2 (A j ) , j∈Z

(1.3)

(1.4)

where A j = R × {2 j ≤ |x| ≤ 2 j+1 }. For the inhomogeneous term f we introduce a dual type norm j 2 2 f L 2 (A j ) . f L E ∗M = j∈Z

Then we have: Theorem 1.1. The solution u to (1.3) satisfies the following estimate: u L E M ∇u 0 L 2 + u 1 L 2 + f L E ∗M .

(1.5)

One may ask whether similar bounds also hold for perturbations of the Minkowski space-time. Indeed, in the case of small long range perturbations the same bounds as above were established very recently by two of the authors, see [30, Prop. 2.2] or [28, (2.23)] (with no obstacle, = ∅). See also [1,27] for related local energy estimates for small perturbations of the d’Alembertian. For large perturbations one faces additional difficulties, due on one hand to trapping for large frequencies and on the other hand to eigenvalues and resonances for low frequencies. The Schwarzschild space-time, considered in the present paper, is a very interesting example of a large perturbation of the Minkowski space-time, where trapping causes significant difficulties. The Schwarzschild space-time M is a spherically symmetric solution to Einstein’s equations with an additional Killing vector field K , which models the exterior of a massive spherically symmetric body. Factoring out the S2 component it can be represented via the Penrose diagram in Fig. 1. The radius r of the S2 spheres is intrinsically determined and is a smooth function on M which has a single critical point at the center. The regions I and I represent the exterior of the black hole, respectively its symmetric twin, and are characterized by the relation r > 2M. We can represent I as I = R × (2M, ∞) × S2 with a metric whose line element is 2M 2M −1 2 2 2 ds = − 1 − dt + 1 − dr + r 2 dω2 , r r

(1.6)

where dω2 is the measure on the sphere S2 . The Killing vector field K is given by K = ∂t , which is time-like within I . The differential dt is intrinsic, but the function t is only defined up to translations on I .

Strichartz Estimates on Schwarzschild Black Hole Backgrounds

39

Fig. 1. The Penrose diagram for the Kruskal extension of the Schwarzschild solution

The regions I I and I I represent the black hole, respectively its symmetric twin, the white hole, and are characterized by the relation r < 2M. The same metric as in (1.6) can be used. The Killing vector field K is still given by K = ∂t , which is now space-like. Light rays can enter the black hole but not leave it. By symmetry light rays can leave the white hole but not enter it. The surface r = 2M is called the event horizon. While the singularity at r = 0 is a true metric singularity, we note that the apparent singularity at r = 2M is merely a coordinate singularity. Indeed, denote r ∗ = r + 2M log(r − 2M) − 3M − 2M log M, so that

2M −1 dr ∗ = 1 − dr, r

r ∗ (3M) = 0

and set v = t + r ∗ . Then in the (r, v, ω) coordinates the metric in region I is expressed in the form 2M 2 dv 2 + 2dvdr + r 2 dω2 , ds = − 1 − r which extends analytically into the black hole region I + I I . In particular, given a choice of the function t in region I , this uniquely determines the function t in the region I I via the same change of coordinates. In a symmetric fashion we set w = t − r ∗ . Then in the (r, w, ω) coordinates the metric is expressed in the form 2M ds 2 = − 1 − dw 2 − 2dwdr + r 2 dω2 , r which extends analytically into the white hole region I + I I . One can also introduce global nonsingular coordinates by rewriting the metric in the Kruskal-Szekeres coordinate system, v

v = e 4M ,

w

w = −e− 4M .

40

J. Marzuola, J. Metcalfe, D. Tataru, M. Tohaneanu

However, this is of less interest for our purposes here. Further information on the Schwarzschild space can be found in a number of excellent texts. We refer the interested reader to, e.g., [18,31 and 51]. As far as the results in this paper are concerned, for large r the Schwarzschild spacetime can be viewed as a small perturbation of the Minkowski space-time. The difficulties in our analysis are caused by the dynamics for small r , where trapping occurs. The presence of trapped rays, i.e. rays which do not escape either to infinity or to the singularity r = 0, are known to be a significant obstacle to proving local energy, dispersive, and Strichartz estimates and, in some cases, are known to necessitate a loss of regularity. See, e.g., [10 and 37]. There are two places where trapping occurs on the Schwarzschild manifold. The first is the surface r = 3M which is called the photon sphere. Null geodesics which are initially tangent to the photon sphere will remain on the surface for all times. Microlocally the energy is preserved near such periodic orbits. However what allows for local energy estimates near the photon sphere is the fact that these periodic orbits are hyperbolic. The second is at the event horizon r = 2M, where the trapped geodesics are the vertical ones in the (r, v, ω) coordinates. However, this second family of trapped rays turns out to cause no difficulty in the decay estimates since in the high frequency limit the energy decays exponentially along it as v → ∞. This is due to the fact that the frequency decays exponentially along the Hamilton flow, and in the physics literature it is well-known as the red shift effect. To describe the decay properties of solutions to the wave equation in the Schwarzschild space, it is convenient to use coordinates which make good use of the Killing vector field and are nonsingular along the event horizon. The (r, v, ω) coordinates would satisfy these requirements. However the level sets of v are null surfaces, which would cause some minor difficulties. This is why in I + I I we introduce the function v˜ defined by v˜ = v − µ(r ), where µ is a smooth function of r . In the (v, ˜ r, ω) coordinates the metric has the form 2M 2M 2 2 ds = − 1 − d v˜ + 2 1 − 1 − µ (r ) d vdr ˜ r r 2M (µ (r ))2 dr 2 + r 2 dω2 . + 2µ (r ) − 1 − r On the function µ we impose the following two conditions: (i) µ(r ) ≥ r ∗ for r > 2M, with equality for r > 5M/2. (ii) The surfaces v˜ = const are space-like, i.e. 2M µ (r ) > 0, µ (r ) > 0. 2− 1− r The first condition (i) insures that the (r, v, ˜ ω) coordinates coincide with the (r, t, ω) coordinates in r > 5M/2. This is convenient but not required for any of our results. What is important is that in these coordinates the metric is asymptotically flat as r → ∞. −1 near In the proof of the Strichartz estimates, it is also required that µ (r ) = 1 − 2M r r = 3M, which in other words says that we can work in the (r, t) coordinates near the photon sphere. However, this may be merely an artifact of our method.


41

Fig. 2. The Schwarzschild space partition represented on the Penrose diagram

We introduce a symmetric function v˜1 in I + I I , as well as the functions w˜ and w˜ 1 in I + I I , respectively I + I I . Given a parameter 0 < r0 < 2M we partition the Schwarzschild space into seven regions M = M R ∪ M L ∪ MR ∪ ML ∪ MT ∪ MC ∪ M B as in Fig. 2. The right/left top/bottom regions are M L = {v˜1 ≥ 0, r ≥ r0 } ⊂ I + I I, M R = {v˜ ≥ 0, r ≥ r0 } ⊂ I + I I, MR = {w˜ ≤ 0, r ≥ r0 } ⊂ I + I I , ML = {w˜ 1 ≤ 0, r ≥ r0 } ⊂ I + I I , the top and bottom regions are MT = {r < r0 } ∩ I I,

M B = {r < r0 } ∩ I I ,

and the central region MC is the remainder of M. Moreover, define − R = M R ∩ {v˜ = 0}, +R = M R ∩ {r = r0 }. and similarly for the other regions. In what follows we consider the Cauchy problem g φ = f,

φ|0 = φ0 ,

K˜ φ|0 = φ1 ,

(1.7)

where for convenience we choose the initial surface 0 to be the horizontal surface of symmetry 0 = {t = 0} ∩ (I + I ) and K˜ is smooth, everywhere timelike and equals K on 0 outside MC . Observe that we cannot use K on all of 0 since it is degenerate at the center (i.e. on the bifurcate sphere). Equation (1.7) can be solved as follows: (i) Solve the equation in MC with Cauchy data on 0 . Since MC is compact and has forward and backward space-like boundaries, this is a purely local problem.

42


(ii) Solve the equation in M R with Cauchy data on − R . The forward boundary of M R is +R , which is space-like. This is the most interesting part, where we are interested in the decay properties as v˜ → ∞. In a similar manner solve the equation in M L , MR and ML . (iii) Solve the equation in MT with initial data on the space-like surface T = {r = r0 } ∩ I I . Here one can track the solution up to the singularity and encounter a mix of local and global features. This part of the analysis is not pursued in the present article. A significant role in our analysis is played by the Killing vector field K , which in the (r, v) ˜ coordinates equals ∂v˜ . This is time-like outside the black hole but space-like inside it. Furthermore, it is degenerate at the center. Using the Killing vector field outside the black hole we obtain a conserved energy E 0 [φ] for solutions φ to the homogeneous equation g φ = 0. On surfaces t = const in the (r, t) coordinates the energy E 0 [φ](t) has the form

∞ 2M −1 2M 2 2 2 1− (∂r φ) +| ∇ φ| r 2 dr dω. (1.8) (∂t φ) + 1 − E 0 [φ] = r r S 2 2M Since the vector field K is degenerate at the center, so is the corresponding energy E 0 at r = 2M. Hence it would be natural to replace it with a nondegenerate energy, which on the initial surface 0 can be expressed as E[φ](0 ) =

S2

∞ 2M

3 1 2M − 2 2M 2 2 2 2 1− (∂t φ) + 1 − (∂r φ) +| ∇ φ| r 2 dr dω. r r (1.9)

Unfortunately this is no longer conserved, and this is one of the difficulties which we face in our analysis. We remark that a related form of a nondegenerate energy expression was introduced in [14] and proved to be bounded in the exterior region on surfaces t = const. Part of the novelty of our approach is to prove bounds not only in the exterior region, but also inside the event horizon. This is natural if one considers the fact that the singularity at r = 2M is merely a removable coordinate singularity. In order to do this, it is no longer suitable to measure the evolution of the energy on the surfaces t = const (see below). Thus the above energy E[φ](0 ) is relegated to a secondary role here and is used only to measure the size of the initial data. A priori the energy E[φ](t) of φ only determines its Cauchy data at time t modulo constants. However, in what follows we implicitly assume that φ decays at ∞, in which case φ can also be estimated via a Hardy-type inequality,

2M 1− r

− 1 2

r

1 2M 2 1− φ r dr dω (∂r φ)2 r 2 dr dω. r

−2 2

2

(1.10)

This is proved in a standard manner; the details are left to the reader. We shall now further describe our main estimates in the region M R : the local energy decay, the WKB analysis which yields a local energy decay with only a logarithmic loss, and finally the Strichartz estimates.


43

For the initial energy on − R we use 2 2 2 |∂ r 2 dr dω. ) = φ| + |∂ φ| + | ∇ φ| E[φ]( − r v ˜ R − R

For the final energy on +R we set |∂r φ|2 + |∂v˜ φ|2 + | ∇ φ|2 r02 d vdω. ˜ E[φ]( +R ) = +R

We also track the energy on the space-like slices v˜ = const, |∂r φ|2 + |∂v˜ φ|2 + | ∇ φ|2 r 2 dr dω. E[φ](v˜0 ) = M R ∩{v= ˜ v˜0 }

Thus E[φ]( − R ) = E[φ](0). For the local energy estimates one may first consider a direct analogue of the Minkowski bound (1.5). Unfortunately such a bound is hopeless due to the trapping which occurs at r = 3M. Instead, for our first result we define a weaker preliminary local energy space L E 0 with norm

1 2 2 3M 2 1 1 2 2 2 1 2 φ L E 0 = |∂ φ| + 1 − |∂v˜ φ| + | ∇ φ| + 4 φ r dr d vdω. ˜ 2 r r r2 r r MR r (1.11) Compared to the L E M norm we note the power loss in the angular and v˜ derivatives at r = 3M. The L E 0 norm is also weaker than L E M as r → ∞, but this is merely for convenience. At the same time we would like to also consider the inhomogeneous problem g φ = f . To measure the inhomogeneous term f , we introduce the norm L E 0∗ , which is stronger than L E ∗M : f 2L E ∗ = 0

3M −2 2 2 2 1− r f r dr d vdω. ˜ r MR

(1.12)

Again the important difference is at r = 3M. Our first local energy estimate is the following: Theorem 1.2. Let φ solve the inhomogeneous wave equation g φ = f on the Schwarzschild manifold. Then we have 2 E[φ]( +R ) + sup E[φ](v) ˜ + φ2L E 0 E[φ]( − R ) + f L E ∗ . v≥0 ˜

0

(1.13)

Here we made no effort to optimize the weights at r = 3M and r = ∞. This is done later in the paper. On the other hand the above estimate follows from a relatively simple application of the classical positive commutator method. The advantage of having even such a weaker estimate is that it is sufficient in order to allow localization near the interesting regions r = 3M and r = ∞, which can then be studied in greater detail using specific tools.

44


The first related results regarding the solution of the wave equation on Schwarzschild backgrounds were obtained in [50 and 24] which proved uniform boundedness in region I (including the event horizon). The first pointwise decay result (without, however, a rate of decay) was obtained in [49]. Heuristics from [36] suggest that solutions to the wave equation in the Schwarzschild case should locally decay like v −3 . For spherically symmetric data a v −3+ decay rate was obtained in [16], and under the additional assumption of the initial data vanishing near the event horizon, the v −3 decay rate was proved in [23]. In general the best known decay rate, proved in [14], is v −1 (see also [7]). We also refer the reader to [38], where optimal pointwise decay rates for each spherical harmonic are established for a closely related problem. Estimates related to (1.13) were first proved in [25] for radially symmetric Schrödinger equations on Schwarzschild backgrounds. In [2–4], those estimates are extended to allow for general data for the wave equation. The same authors, in [5,6], have provided studies that give improved estimates near the photon sphere r = 3M. Moreover, we note that variants of these bounds have played an important role in the works [7 and 14] which prove analogues of the Morawetz conformal estimates on Schwarzschild backgrounds. This allows one to deduce a uniform decay rate for the local energy away from the event horizon, though there is necessarily a loss of regularity due to the trapping that occurs at the photon sphere. Instead in this paper we restrict ourselves to time translation invariant estimates, and we aim to clarify/streamline these as much as possible. All of the above articles use the conserved (degenerate) energy E 0 [φ] on time slices, obtained using the Killing vector field ∂t . As such, their estimates are degenerate near the event horizon. Further progress was made in [14], where an additional vector field was introduced near the event horizon, in connection to the red shift effect. This led to bounds in the exterior region involving a nondegenerate form of the energy related to (1.9). The approach of [2,7,14 and 25] is to write the equation using the Regge-Wheeler tortoise coordinate and to expand in spherical harmonics. For the equation corresponding to each spherical harmonic, one uses a multiplier which changes sign at the critical point of the effective potential. Here we work in the coordinates (r, v, ˜ ω), though this is not of particular significance, and we do not expand into spherical harmonics. We prove (1.13) using a positive commutator argument which requires a single differential multiplier. We hope that this makes the methods more robust for other potential applications. During final preparations of this article, localized energy estimates proved without using the spherical harmonic decomposition also appeared in [15]. The methods contained therein are somewhat different from ours. Compared to the stronger norms L E M , L E ∗M the weights in (1.13) have a polynomial singularity at r = 3M, which corresponds to the family of trapped geodesics on the photon sphere. As a consequence of the results we prove later, see Theorem 3.2, the latter fact can be remedied to produce a stronger estimate. Theorem 1.3. Let φ solve the inhomogeneous wave equation g φ = f on the Schwarzschild manifold. Then (1.13) still holds if the coefficient (1 − 3M/r )2 in the L E 0 and the L E 0∗ norms is replaced by 3M −2 1 − ln 1 − . r


45

Now we have only a logarithmic singularity at r = 3M. The result above is only stated in this form for the reader’s convenience. The full result in Theorem 3.2 is stronger but also more complicated to state since it provides a more precise microlocal local energy estimate. The logarithmic loss is not surprising, since it is characteristic of geometries with trapped hyperbolic orbits (see for instance [9,12,34]). Indeed, a similar estimate in the semiclassical setting is obtained in [13] using entirely different techniques. Note, however, that the aforementioned estimate only involves logarithmic loss of the frequency; our result is stronger since it also implies bounds for (ln |r ∗ |)−1 u L 2 , which are necessary in order to prove Strichartz estimates. There are two regions on which the analysis is distinct. The metric is asymptotically flat, and thus, near infinity, one can retrieve the classical Morawetz type estimate. On the other hand, around the photon sphere r = 3M we take an expansion into spherical harmonics as well as a time Fourier transform. Then it remains to study an ordinary differential equation which is essentially similar to (∂x2 − λ2 (x 2 + ))u = f,

|| 1, |x| 1.

For this we use a rough WKB approximation in the hyperbolic region combined with energy estimates in the elliptic region. Airy type dynamics occur near the zeroes of the potential. Even though it is weaker, the initial bound in Theorem 1.2 plays a key role in the analysis. Precisely, it allows us to glue together the estimates in the two regions described above. We next consider the Strichartz estimates. For solutions to the constant coefficient wave equation on R × R3 , the well-known Strichartz estimates state that |Dx |−ρ1 ∇u L p1 L qx1 ∇u(0) L 2 + |Dx |ρ2 f t

p

q

L t 2 L x2

.

(1.14)

Here the exponents (ρi , pi , qi ) are subject to the scaling relation 3 1 3 + = −ρ p q 2

(1.15)

and the dispersion relation 1 1 1 + ≤ , p q 2

2 < p ≤ ∞.

(1.16)

All pairs (ρ, p, q) satisfying (1.15) and (1.16) are called Strichartz pairs. Those for which the equality holds in (1.16) are called sharp Strichartz pairs. Such estimates first appeared in the seminal works [8,43,44] and as stated include contributions from, e.g., [17,19,26,35 and 21]. If one allows variable coefficients, such estimates are well-understood locallyin-time. For smooth coefficients, this was first shown in [32] and later for C 2 coefficients in [39 and 45–47]. Globally-in-time, the problem is more delicate. Even a small, smooth, compactly supported perturbation of the flat metric may refocus a group of rays and produce caustics. Thus, constructing a parametrix for incoming rays proves to be quite difficult. At the same time, one needs to contend with the possibility of trapped rays at high frequencies and with eigenfunctions/resonances at low frequencies.

46


Global-in-time estimates were shown for small, long range perturbations of the metric in [29] using an outgoing parametrix. In order to keep the parametrix outgoing one must allow evolution both forward and backward in time. This construction is based on an earlier argument in [48] for the Schrödinger equation. The smallness assumption, however, precludes trapping and does not permit a direct application to the current setup. On the other hand, a second result of [29] asserts that even for large, long range perturbations of the metric one can still establish global-in-time Strichartz estimates provided that a strong form of the local energy estimates holds. This switches the burden to the question of proving local energy estimates. The result in [29] cannot be applied directly to the present problem due to the logarithmic losses in the local energy estimates near the trapped rays. However, it can be applied for the near infinity part of the solution. In a bounded spatial region, on the other hand, we take advantage of the local energy estimates to localize the problem to bounded sets, in which estimates are shown using the local-in-time Strichartz estimates of [39,45]. Thus we obtain Theorem 1.4. If φ solves g φ = f in M R then for all nonsharp Strichartz pairs (ρ1 , p1 , q1 ) and (ρ2 , p2 , q2 ) we have E[φ]( +R ) + sup E[φ](v) ˜ + ∇φ2 p1

−ρ ,q L v˜ H˙ x 1 1

v≥0 ˜

2 E[φ]( − R ) + f p

ρ ,q2

2 L v˜ 2 H˙ x

. (1.17)

Here the Sobolev-type spaces H˙ s, p coincide with the usual H˙ s, p homogeneous spaces in R3 expressed in polar coordinates (r, ω). As a corollary of this result one can consider the global solvability question for the energy critical semilinear wave equation in the Schwarzschild space, g φ = ±φ 5 in M (1.18) φ = φ0 , K˜ φ = φ1 in 0 . Theorem 1.5. Let r0 > 0. Then there exists > 0 so that for each initial data (φ0 , φ1 ) which satisfies E[φ](0 ) ≤ , Eq. (1.18) admits an unique solution φ in the region {r > r0 } which satisfies the bound E[φ](r0 ) + φ H˙ s, p ({r >r0 }) E[φ](0 ) for all indices s, p satisfying 1 4 =s+ , p 2

0≤s
r0 }) appearing above, in view of the ambiguity due to the choice of coordinates. In a compact neighbourhood of the center region MC this is nothing but the classical H s, p norm. By compactness, different choices of coordinates lead to equivalent norms. Consider now the upper exterior region M R (as well as its three other mirror images). Using the coordinates (v, ˜ x) with x = ωr , we define H˙ s, p (M R ) as the restrictions to R+ × {|x| > r0 } of functions in the homogeneous Sobolev space H˙ s, p (R × R3 ).


47

2. The Morawetz-Type Estimate In this section, we shall prove Theorem 1.2. We note that the estimate (1.13) is trivial over a finite v˜ interval by energy estimates for the wave equation; the difficulty consists in proving a global bound in v. ˜ By the same token, once we prove (1.13) for some choice of r0 < 2M, we can trivially make the transition to any r0 < 2M due to the local theory. Thus in the arguments which follow we reserve the right to take r0 sufficiently close to 2M. We consider solutions to the inhomogeneous wave equation on the Schwarzschild manifold in M R , which is given by g φ = ∇ α ∂α φ = f. Here ∇ represents the metric connection. Associated to this equation is an energymomentum tensor given by 1 Q αβ [φ] = ∂α φ∂β φ − gαβ ∂ γ φ∂γ φ. 2 A simple calculation yields the most important property of Q αβ , namely that if φ solves the homogeneous wave equation then Q αβ [φ] is divergence-free: ∇ α Q αβ [φ] = 0, if ∇ α ∂α φ = 0. More generally, we have ∇ α Q αβ [φ] = ∂β φ g φ. In order to prove Theorem 1.2, we shall contract Q αβ with a vector field X to form the momentum density Pα [φ, X ] = Q αβ [φ]X β . Computing the divergence of this vector field, we have ∇ α Pα [φ, X ] = g φ X φ + Q αβ [φ]π αβ , where παβ =

1 (∇α X β + ∇β X α ) 2

is the deformation tensor of X . If X is the Killing vector field K then the above divergence vanishes, ∇ α Pα [φ, K ] = 0 if g φ = 0.

(2.1)

This gives rise to the E 0 [φ] conservation law outside the black hole. Naively, one may seek vector fields X so that the quadratic form Q αβ [φ]π αβ is positive definite. However, this may not always be possible to achieve. Instead we note that it may be just as good to have the symbol of this quadratic form positive on the characteristic set of g . Then it would be possible to make the above quadratic form positive after adding a Lagrangian correction term of the form q∂ γ φ∂γ φ. Such a term

48


can be conveniently expressed in divergence form modulo lower order terms. Precisely, for a vector field X , a scalar function q and a 1-form m we define 1 1 Pα [φ, X, q, m] = Pα [φ, X ] + qφ∂α φ − ∂α qφ 2 + m α φ 2 , 2 2 where m allows us to modify the lower order terms in the divergence formula. Then we obtain the modified divergence relation ∇ α Pα [φ, X, q, m] = g φ (X φ + qφ) + Q[φ, X, q, m],

(2.2) 1 Q[φ, X, q, m] = Q αβ [φ]π αβ + q∂ α φ ∂α φ + m α φ ∂ α φ + (∇ α m α − ∇ α ∂α q) φ 2 . 2

Theorem 1.2 is proved by making appropriate choices for X , q and m so that the quadratic form Q[φ, X, q, m] defined by the divergence relation is positive definite. In what follows we assume that X , q and m are all spherically symmetric and invariant with respect to the Killing vector field K . Lemma 2.1. There exist smooth, spherically symmetric, K -invariant X , q, and m in r ≥ 2M satisfying the following properties: (i) X is bounded2 , |q(r )| r −1 , |q (r )| r −2 and m has compact support in r . (ii) The quadratic form Q[φ, X, q, m] is positive definite, 3M 2 −2 Q[φ, X, q, m] r −2 |∂r φ|2 + 1 − (r |∂v˜ φ|2 + r −1 | ∇ φ|2 ) + r −4 φ 2 . r (iii) X (2M) points toward the black hole, X (dr )(2M) < 0, and m, dr (2M) > 0. We postpone the proof of the lemma and use it to conclude the proof of Theorem 1.2. Let X , q and m be as in the lemma. We extend them smoothly beyond the event horizon preserving the spherical symmetry and the K -invariance. By (2.1) we can modify the vector field X without changing the quadratic form Q in (2.3), ∇ α Pα [φ, X + C K , q, m] = g φ ((X + C K )φ + qφ) + Q[φ, X, q, m]. Here C is a large constant. We integrate this relation in the region D = {0 < v˜ < v˜0 , r > r0 } using the (r, v, ˜ ω) coordinates. This yields ˜ g φ ((X + C K )φ + qφ) + Q[φ, X, q, m] r 2 dr d vdω D

v= ˜ v˜0 d v, ˜ P[φ, X + C K , q, m]r 2 dr dω v=0 ˜ 2 − dr, P[φ, X + C K , q, m]r0 d vdω. ˜

=

r =r0

2 In the (r, v) ˜ coordinates


49

We claim that if C is large enough and r0 sufficiently close to 2M then the integrals on the right have the correct sign, E[φ](v˜1 )− d v, ˜ P[φ, X +C K , q, m]r 2 dr dω C E[φ](v˜1 ), v˜1 ≥ 0, (2.3) v= ˜ v˜1

dr, P[φ, X + C K , q, m] |∂r φ|2 + |∂v˜ φ|2 + |∂ω φ|2 + φ 2 ,

r = r0 .

(2.4)

If these bounds hold then the conclusion of the theorem follows by (ii) and CauchySchwarz. Indeed, a direct computation yields 1 2M 2M 2 2 d v, ˜ P[φ, ∂v˜ ] = − 2µ − 1 − µ |∂v˜ φ| + 1 − |∂r φ|2 2 r r + r −2 |∂ω φ|2 , respectively 2M dr, P[φ, ∂v˜ ] = |∂v˜ φ| + 1 − (∂r − µ ∂v˜ )φ∂v˜ φ. r 2

On the other hand

2M 2M 2 d v, ˜ P[φ, ∂r ] = 1 − 1 − µ |∂r φ| − 2µ − 1 − µ2 ∂v˜ φ∂r φ, r r

while

1 2M 2M − 2µ − 1 − µ2 |∂v˜ φ|2 − 1 − |∂r φ|2 2 r r + r −2 |∂ω φ|2 .

dr, P[φ, ∂r ] = −

We compute d v, ˜ P[φ, X + C K ] = (X (d v) ˜ + C)d v, ˜ P[φ, ∂v˜ ] + X (dr )d v, ˜ P[φ, ∂r ]. For large enough C we have X (d v) ˜ + C C. Therefore the first term on the right is negative definite for r > 2M. More precisely, it is only the coefficient of the |∂r φ|2 term which degenerates at r = 2M. However, due to condition (iii) in the lemma we have X (dr )(2M) < 0; therefore we pick up a negative |∂r φ|2 coefficient at r = 2M. Thus we obtain 2M 2 2 −2 2 −d v, ˜ P[φ, X + C K ] ≈ C |∂v˜ φ| + 1 − |∂r φ| + r |∂ω φ| + |∂r φ|2 , r r > 2M. Since all the coefficients in the quadratic form on the left are continuous, it follows that the above relation extends to r > r0 for some r0 < 2M depending on C, namely 0 < 2M − r0 C −1 .

(2.5)

50


In order to prove (2.3) it remains to estimate the lower order terms P[φ, 0, q, m] in terms of the positive contribution above. Since |q| r −1 and m has compact support in r , we can bound 1 2 |d v, ˜ P[φ, 0, q, m]| r −1 |φ| |∂v˜ φ|2 + |∂r φ|2 + r −2 |φ|2 . Then by Cauchy-Schwarz it suffices to estimate ∞ ∞ 2M −2 2 2 − 21 C 1− + 1 |∂r φ|2 r 2 dr r |φ| r dr C r r0 r0 which is a routine Hardy-type inequality. We next turn our attention to (2.4) and begin with the principal part dr, P[φ, X + C K ] = (X (d v) ˜ + C)dr, P[φ, ∂v˜ ] + X (dr )dr, P[φ, ∂r ]. Examining the expressions for the two terms above, we see that for r0 subject to (2.5) we have 2M |∂r φ|2 , dr, P[φ, X + C K ] C|∂v˜ φ|2 + |∂ω φ|2 − 1 − r = r0 . r0 Next we consider the lower order terms. The contribution of m is 1 m, dr φ 2 φ 2 2 due to condition (iii) in the lemma. The contribution of q is 1 qφdr, dφ − φ 2 dr, dq. 2 The coefficient of the second term is 1 − 2M r q , which is negligible for r0 close to 2M. In the first term we have 2M 2M ∂r φ + 1 − 1 − µ ∂v˜ φ. dr, dφ = 1 − r r All terms involving 1 − 2M are negligible, and since q is bounded we get r qφ∂v˜ φ C|∂v˜ φ|2 + φ 2 for large enough C. Proof of Lemma 2.1. It is convenient to look for X in the (r, t) coordinates, where we choose the vector field X of the form X = X 1 + δ X 2, with 2M ∂r , X 1 = a(r ) 1 − r

δ 1,

2M 2M −1 X 2 = b(r ) 1 − ∂t ∂r − 1 − r r


51

will be chosen to be smooth. Note that X is a smooth vector field and a and b 1 − 2M r in the nonsingular coordinates (r, v), since in these coordinates we have 2M 2M X 2 = b(r ) 1 − ∂r + ∂v , ∂r . X 1 = a(r ) 1− r r We remark that the vector field X 2 is closely related to the vector field Y introduced earlier in [14] in order to take advantage of the red shift effect. However, in their construction Y is in a form which is nonsmooth near the event horizon and which is restricted to the exterior region. The primary role played by X 2 here is to ensure that X + C K is time-like near the event horizon. The red-shift effect largely takes care of the rest. For convenience, we set 2M 1 2 t1 (r ) = 1 − r ∂ a(r ) . r r r2 A direct computation yields 2M 2 r − 3M a (r )(∂r φ)2 + a(r ) | ∇ φ|2 ∇ α Pα [φ, X 1 ] = 1 − r r2 1 − t1 (r )∂ γ φ ∂γ φ + X 1 φg φ, 2

(2.6)

respectively 2 1 2M ∂r φ − ∂t φ 1− ∇ Pα [φ, X 2 ] = b (r ) 2 r 2M r − 3M 1 1− b (r ) | ∇ φ|2 + b(r ) − r2 2 r 2M 1 1− b(r )∂ γ φ ∂γ φ + X 2 φg φ, − r r α

where γ

∂ φ ∂γ φ = −

2M 1− r

−1

(2.7)

2M 2 2 (∂r φ) − | ∇ φ| . (∂t φ) − 1 − r 2

We choose a so that the first line of the right side of (2.6) is positive. This requires that a (r ) r −2 ,

a(3M) = 0.

(2.8)

We choose b so that the first line of the right-hand side of (2.7) is positive. Precisely, we take b supported in r ≤ 3M with b=−

b0 (r ) 1−

2M r

,

r ∈ [2M, 3M],

with b0 smooth, decreasing in [2M, 3M) and supported in {r ≤ 3M}. In particular this guarantees that b0 (2M) > 0, which is later used to verify the condition (iii) in the lemma.

52


The exact choice of b0 is not important, and in effect b only plays a role very close to the event horizon r = 2M. Even though b is singular at 2M, the second term of the coefficient of | ∇ φ|2 in the second line of (2.7) is nonsingular. Hence if δ is sufficiently small this term is controlled by the first line in (2.6). Taking the above choices into account, we have 2 2M 2 1 2M Q[φ, X, 0, 0] = 1 − a (r )(∂r φ)2 + δ b (r ) 1− ∂r φ − ∂t φ r 2 r (2.9) 2 (r − 3M) 2 γ +O − q ∂ φ ∂ φ | ∇ φ| 0 γ r3 where

q0 (r ) =

1 1 2M . t1 (r ) + δ b(r ) 1 − 2 r r

The last term in (2.9) is a Lagrangian expression and is accounted for via the q term. The first three terms give a nonnegative quadratic form in ∇φ. This form is in effect positive definite for r < 3M, where b > 0. However for larger r it controls ∂r φ and ∇ φ but not ∂t φ. This can be easily remedied with the Lagrangian term. Precisely, we choose q of the form q = q 0 + δ1 q 1 ,

q1 (r ) = χ{r >5M/2}

(r − 3M)2 , r4

where χ{r >5M/2} is a smooth nonnegative cutoff which is supported in {r > 5M/2} and equals 1 for r > 3M. The positive parameter δ1 is chosen so that δ1 δ. Then the only nonnegligible contribution of δ1 q1 is the one involving ∂t φ. We obtain 2 2M 2 1 2M −2 2 ∂r φ − ∂t φ Q[φ, X, q, 0] = 1 − O(r )(∂r φ) + δ b (r ) 1− r 2 r 2M −1 (r − 3M)2 1 2 | ∇ φ| 1 − +O + δ q |∂t φ|2 − ∇ α ∂α q φ 2 . 1 1 3 r r 2 (2.10) The contribution of q1 can be made arbitrarily small by taking δ1 small. Hence it will be neglected in the sequel. At this stage it would be convenient to be able to choose a so that ∇ α ∂α t1 (r ) < 0. A direct computation yields ∇ α ∂α t1 (r ) = −La with 1 La(r ) = − 2 ∂r r

2M 2 2M 1 2 1− 1− r ∂r ∂r r a(r ) . r r 2r 2

Unfortunately it turns out that the condition La > 0 and (2.8) are incompatible, in the sense that there is no smooth a which satisfies both. However, one can find a with a logarithmic blow-up at 2M which satisfies both requirements. Such an example is r − 2M . a(r ) = r −2 (r − 3M)(r + 2M) + 6M 2 log M


53

This is in no way unique, it is merely the simplest we were able to produce. One verifies directly that a (r ) r −2 ,

La(r ) r −4 .

To eliminate the singularity of a above we replace it by a (r ) =

1 f (R), r2

where is a small parameter,

R = (r − 3M)(r + 2M) + 6M 2 log

r − 2M , M

and f (R) = −1 f ( R), where f is a smooth nondecreasing function such that f (R) = R on [−1, ∞] and f = −2 on (−∞, −3]. The condition (2.8) is satisfied uniformly with respect to small ; therefore the choice of δ is independent of the choice of . With this modification of a we recompute

2M −1 2 La = f ( R)La + O() f ( R) + O 1 − f ( R). r This is still positive except for the region { R < −1}. To control it we introduce an m term in the divergence relation as follows: Let γ (r ) be a function to be chosen later. We set 2M 2 2M m t = δb (r ) 1 − γ, m ω = 0. γ, m r = δb (r ) 1 − r r Then

2M 2M 1− ∂r φ − ∂t φ , m α ∂ α φ = δb (r )γ (r ) 1 − r r

while α

∇ m α = δr

−2

∂r

2M 1− r

2

2

r b (r )γ (r ) .

Hence, completing the square we obtain 2M 2 (r − 3M)2 | ∇ φ|2 Q[φ, X, q, m] = 1 − O(r −2 )(∂r φ)2 + O r r3 2M −1 + δ1 q 1 1 − |∂t φ|2 + nφ 2 r 2 1 2M 2M ∂r φ − ∂t φ + 1 − γφ , + δ b (r ) 1− 2 r r

54


where the coefficient n is given by 2M b (r ) 2M 2 1 −2 2 2M −1 ∂r r b(r ) 1 − −δ 1− γ (r )2 n=La − δr ∂r r 1 − 2 r r 2 r

2 2M 2M 2 1 −2 1 2 + δr γ ∂r r 1 − b (r ) + δγ 1 − b (r ). 2 r 2 r We assume that γ is supported in {r < 3M} and satisfies γ > −1.

0 ≤ γ ≤ 1, Then for r > 3M we have

n = La r −4 , while for r ≤ 3M we can write

2M 2 n = La + δγ (r ) 1 − b (r ) + O(δ). r

If R > −1 then, using the bound from below on γ , we further have n ≥ La + O(δ), which is positive provided that δ is sufficiently small. On the other hand in the region { R ≤ −1}, we have

1 2M −1 2M 2 2 n≥ δ 1 − b (r )γ (r ) + O(δ)+ O() f ( R) + O 1 − f ( R). 2 r r The γ term can be taken positive, while all the other terms may be negative so they must be controlled by it. The restriction we face in the choice of γ comes from the fact that 0 ≤ γ ≤ 1. Hence we need to verify that 2M −1 2 I = δ + | f ( R)| + 1 − | f ( R)| δ. r R≤−1 −1

Indeed, the interval of integration has size ≤ e−c ; therefore the above integral can be bounded by I e−c

−1

+ ,

which suffices provided that is small enough. Finally, note that 2M 2M X (dr )(2M) = a(r ) 1 − + δb(r ) 1 − (2M) < 0, r r

2M 2 m, dr (2M) = δb (r ) 1 − γ (2M) > 0. r So (iii) is also satisfied.


55

3. Log-Loss Local Energy Estimates The aim of this section is to prove a local energy estimate for solutions to the wave equation on the Schwarzschild space which is stronger than the one in Theorem 1.2. Consequently, we strengthen the norm L E 0 to a norm L E and we relax the norm L E 0∗ to a norm L E ∗ which satisfy the following natural bounds: φ2L E 0 φ2L E φ2L E M ,

(3.11)

f 2L E ∗ f 2L E ∗ f 2L E ∗ .

(3.12)

respectively

0

M

We note that these bounds uniquely determine the topology of the L E and L E ∗ spaces away from the photon sphere and from infinity. This is due to the fact that the local energy estimates in Theorem 1.2 have no loss in any bounded region away from the photon sphere. To define the L E, respectively L E ∗ , norms we consider a smooth partition of unity 1 = χeh (r ) + χ ps (r ) + χ∞ (r ), where χeh is supported in {r < 11M/4}, χ ps is supported in {5M/2 < r < 5M} and χ∞ is supported in {r > 4M}. Then we set φ2L E = χeh φ2L E M + χ ps φ2L E ps + χ∞ φ2L E M ,

(3.13)

φ2L E ∗ = χeh φ2L E ∗ + χ ps φ2L E ∗ps + χ∞ φ2L E ∗ .

(3.14)

respectively

M

M

The norms L E ps and L E ∗ps near the photon sphere are defined in Sect. 3.1 below, see (3.20), respectively (3.21); their topologies coincide with L E M , respectively L E ∗M , away from the photon sphere. With these notations, the main result of this section can be phrased in a manner similar to Theorem 1.2: Theorem 3.2. For all functions φ which solve g φ = f in M R we have 2 ˜ + E[φ]( +R ) + φ2L E E[φ]( − sup E[φ](v) R ) + f L E ∗ .

(3.15)

v>0 ˜

We continue with the setup and estimates near the photon sphere in Sect. 3.1, the setup and estimates near infinity in Sect. 3.2 and finally the proof of the theorem in Sect. 3.3.

56


3.1. The analysis near the photon sphere. Here it is convenient to work in the ReggeWheeler coordinates given by r ∗ = r + 2M log(r − 2M) − 3M − 2M log M. Then r = 3M corresponds to r ∗ = 0, and a neighbourhood of r = 3M away from infinity and the event horizon corresponds to a compact set in r ∗ . In these coordinates the operator g has the form 2M r − 2M r 1− g r −1 = L RW = ∂t2 −∂r2∗ − ∂ω + V (r ), V (r ) =r −1 ∂r2∗ r. r r3 (3.16) For r ∗ in a compact set the energy has the form E[φ] ≈ (∂t φ)2 + (∂r ∗ φ)2 + (∂ω φ)2 dr dω, and the initial local smoothing norms are expressed as φ2L E 0 ≈ (∂r ∗ φ)2 + r ∗ 2 ((∂ω φ)2 + (∂t φ)2 ) + φ 2 dr dωdt, respectively

f 2L E ∗ 0

On the other hand

r ∗ −2 f 2 dr dωdt.

≈

φ2L E M

≈

f 2L E ∗ ≈ M

(∂r ∗ φ)2 + (∂ω φ)2 + (∂t φ)2 + φ 2 dr dωdt, f 2 dr dωdt.

In the sequel we work with spatial spherically symmetric pseudodifferential operators in the (r ∗ , ω) coordinates where ω ∈ S2 . We denote by ξ the dual variable to r ∗ , 1 and by λ the spectral parameter for (−S2 ) 2 . Thus the role of the Fourier variable is played by the pair (ξ, λ), and all our symbols are of the form a(r ∗ , ξ, λ). To such a symbol we associate the corresponding Weyl operator Aw . Since there is no symbol dependence on ω, one can view this operator as a combination of a one dimen1 sional Weyl operator and the spectral projectors λ associated to the operator (−S2 ) 2 , namely Aw = a w (λ)λ . λ

All of our L 2 estimates admit orthogonal decompositions with respect to spherical harmonics, therefore in order to prove them it suffices to work with the fixed λ operators a w (λ), and treat λ as a parameter. However, in the proof of the Strichartz estimates later


57

on we need kernel bounds for operators of the form Aw , which is why we think of λ as a second Fourier variable and track the symbol regularity with respect to λ as well. Of course, this is meaningless for λ in a compact set; only the asymptotic behavior as λ → ∞ is relevant. Let γ0 : R → R+ be a smooth increasing function so that 1 y < 1, γ0 (y) = y y ≥ 2. Let γ1 : R+ → R+ be a smooth increasing function so that 1 2 γ1 (y) = y y < 1/2, 1 y ≥ 1. Let γ : R2 → R+ be a smooth function with the following properties: ⎧ 1 ⎨ √z < C, (y) y < z/2, z ≥ C, γ 0 γ (y, z) = √ ⎩ 1 2 z 2 γ1 (y /z) y ≥ z/2, z ≥ C, where C is a large constant. In the sequel z is a discrete parameter, so the lack of smoothness at z = C is of no consequence. Consider the symbol a ps (r ∗ , ξ, λ) = γ (− ln(r ∗ 2 + λ−2 ξ 2 ), ln λ), and its inverse ∗ a −1 ps (r , ξ, λ) =

γ (− ln(r ∗ 2

1 . + λ−2 ξ 2 ), ln λ)

We note that if λ is small then they both equal 1, while if λ is large then they satisfy the bounds 1

1 ≤ a ps (r ∗ , ξ, λ) ≤ a ps (r ∗ , 0, λ) ≤ (ln λ) 2 ,

(3.17)

1

∗ −1 ∗ (ln λ)− 2 ≤ a −1 ps (r , 0, λ) ≤ a ps (r , ξ, λ) ≤ 1.

√

We also observe that the region where y 2 > z corresponds to r ∗ 2 + λ−2 ξ 2 < e− Thus differentiating the two symbols we obtain the following bounds √

β

|∂rα∗ ∂ξ ∂λν a ps (r ∗ , ξ, λ)| ≤ cα,β,ν λ−β−ν (r ∗ 2 + λ−2 ξ 2 + e−

ln λ − α+β 2

)

,

ln λ .

(3.18)

respectively √

β

∗ −2 ∗ −β−ν ∗ 2 |∂rα∗ ∂ξ ∂λν a −1 (r +λ−2 ξ 2 +e− ps (r , ξ, λ)| ≤ cα,β,ν a ps (r , ξ, λ)λ

ln λ − α+β 2

)

, (3.19)

where α + β + ν > 0. These show that we have a good operator calculus for the corresponding pseudodifferential operators. In particular in terms of the classical symbol classes we have δ a ps , a −1 ps ∈ S1,0 ,

δ > 0.

58


Then we introduce the Weyl operators A ps =

aw ps (λ)λ ,

λ

respectively A−1 ps =

w (a −1 ps ) (λ)λ .

λ

By (3.18) and (3.19) one easily sees that these operators are approximate inverses. More precisely for small λ, ln λ < C, they are both the identity, while for large λ, −1 w −1 a w ps (λ)(a ps ) (λ) − I L 2 →L 2 λ e

√

ln λ

,

ln λ ≥ C.

Choosing C large enough we insure that the bound above is always much smaller than 1. We use these two operators in order to define the improved local smoothing norms −1 φ L E ps = A−1 ps φ H 1 ≈ A ps ∇t,x φ L 2 ,

(3.20)

f L E ∗ps = A ps f L 2 .

(3.21)

t,x

Due to the inequalities (3.17) we have a bound from above for a w ps (λ), ∗ ∗ a w ps (λ) f L 2 a ps (r , 0, λ) f L 2 | ln |r || f L 2 , w respectively a bound from below for (a −1 ps ) (λ), w −1 ∗ ∗ −1 (a −1 f L 2 ps ) (λ) f L 2 a ps (r , 0, λ) f L 2 | ln |r ||

for f supported near r ∗ = 0. In particular this shows that for f supported near the photon sphere we have φ L E ps | ln |r ∗ ||−1 ∇φ L 2 ,

f L E ∗ps | ln |r ∗ || f L 2 ,

(3.22)

which makes Theorem 1.3 a direct consequence of Theorem 3.2. Our main estimate near the photon sphere is Proposition 3.3. a) Let φ be a function supported in {5M/2 < r < 5M} which solves g φ = f . Then φ2L E ps f 2L E ∗ps .

(3.23)

b) Let f ∈ L E ∗ps be supported in {11M/4 < r < 4M}. Then there is a function φ supported in {5M/2 < r < 5M} so that sup E[φ] + φ2L E ps + g φ − f 2L E ∗ f 2L E ∗ps . t

0

(3.24)


59

Proof. Due to (3.16) recast the problem in Regge-Wheeler coordinates. Denot we can ing u = r φ, g = 1 − 2M r r f , we have L RW u = g. Also it is easy to verify that for φ and f supported in a fixed compact set in r ∗ we have φ L E ps ≈ u L E ps ,

f L E ∗ps ≈ g L E ∗ps .

Hence in the proposition we can replace φ and f by u and g, and g by L RW . To prove part (a) we expand in spherical harmonics with respect to the angular variable and take a time Fourier transform. We are left with the ordinary differential equation (∂r2∗ + Vλ,τ (r ∗ ))u = g,

(3.25)

where Vλ,τ (r ∗ ) = τ 2 −

r − 2M 2 λ + V. r3

Depending on the relative sizes of λ and τ we consider several cases. In the easier cases it suffices to replace the bound (3.23) with a simpler bound ∂r ∗ u L 2 + (|τ | + |λ|)u L 2 g L 2 .

(3.26)

Case 1. λ, τ 1. Then we solve (3.25) as a Cauchy problem with data on one side and obtain a pointwise bound, |u| + |u r ∗ | g L 2 , which easily implies (3.26). Case 2. λ τ . Then Vλ,τ (r ∗ ) ≈ τ 2 for r ∗ in a compact set; therefore (3.25) is hyperbolic in nature. Hence we can solve (3.25) as a Cauchy problem with data on one side and obtain τ |u| + |u r ∗ | g L 2 , which implies (3.26). Case 3. λ τ . Then Vλ,τ (r ∗ ) ≈ −λ2 for r ∗ in a compact set; therefore (3.25) is elliptic. Then we solve (3.25) as an elliptic problem with Dirichlet boundary conditions on a compact interval and obtain 3

1

λ 2 |u| + λ 2 |u r ∗ | g L 2 , which again gives (3.26). Case 4. λ ≈ τ 1. In this case (3.26) is no longer true, and we need to prove (3.23), which in this case can be written in the form w w ∂r ∗ u L 2 + λ(a −1 ps ) (λ)u L 2 a ps (λ)g L 2 ,

(3.27)

where u, g are subject to (3.25). The ∂r ∗ u term above is present in order to estimate the high frequencies |ξ | λ. For lower frequencies it is controlled by the second term on the left of (3.27).

60


The potential V in (3.25) can be treated perturbatively in (3.23) and is negligible. The remaining part of Vλ,τ (r ∗ ) has a nondegenerate minimum at r = 3M which corresponds to r ∗ = 0. Hence we express it in the form Vλ,τ (r ∗ ) = λ2 (W (r ∗ ) + ), where W is smooth and has a nondegenerate zero minimum at r ∗ = 0 and || 1. We now prove the following: Proposition 3.4. Let W be a smooth function satisfying W (0) = W (0) = 0, W (0) > 0, and || 1. Let w be a solution of the ordinary differential equation (∂r2∗ + λ2 (W (r ∗ ) + ))w(r ∗ ) = g, supported near r ∗ = 0. Then (3.27) holds. It would be convenient to replace the norm on the right in (3.27) by a ps (r ∗ , 0, λ)g L 2 . This is not entirely possible since this is a stronger norm. However, we can split g into a component g1 with a ps (r ∗ , 0, λ)g1 ∈ L 2 plus a high frequency part: Lemma 3.5. Each function g ∈ L 2 supported near the photon sphere can be expressed in the form g = g1 + λ−2 ∂r2∗ g2 with g1 and g2 supported near the photon sphere so that √

a ps (r ∗ , 0, λ)g1 L 2 + |r ∗ 2 + e−

ln λ

1

| 8 g2 L 2 + λ−2 ∂r ∗ g2 L 2 a w ps (λ)g L 2 . (3.28)

Proof. The symbols a ps (r ∗ , 0, λ) and a ps (r ∗ , ξ, λ) are comparable provided that √

ln(r ∗ 2 + e−

ln λ

√

) ≈ ln(r ∗ 2 + e−

ln λ

+ λ−2 ξ 2 ).

This includes a region of the form √ 1 D = ln(λ−2 ξ 2 ) < ln(r ∗ 2 + e− ln λ ) . 8 We note that the factor 18 , arising also in the exponent of the second term in (3.28), is somewhat arbitrary. A small choice leads to a better bound in (3.28). If χ is a smooth function which is 1 in (−∞, −1] and 0 in [0, ∞) then we define a smooth characteristic function χ D of the domain D by χ D (r ∗ , ξ, λ) = χ (ln(λ−2 ξ 2 ) −

√ 1 ln(r ∗ 2 + e− ln λ )). 8

One can directly compute the regularity of χ D , 0 χ D ∈ S1,δ ,

δ > 0.

To obtain the decomposition of g we set g2 = q w g,


61

where the symbol of q is q(r ∗ , ξ, λ) = λ2 ξ −2 (1 − χ D ). w w Since (a −1 ps ) (λ) is an approximate inverse for a ps (λ), the estimate for g2 in the lemma can be written in the form √

(r ∗ 2 + e−

ln λ

1

w −2 w −1 w ) 8 q w (a −1 ps ) (λ) f L 2 + λ ∂r ∗ q (a ps ) (λ) f L 2 f L 2 . (3.29)

In the first term it suffices to look at the principal symbol of the operator product since −1+δ the remainder belongs to O P S1,δ for all δ > 0. To verify that the product of the −1 symbols is bounded we note that a ps is bounded. For the other two factors we consider two cases. If |ξ | λ then both factors are bounded. On the other hand if |ξ | λ then in the support of q we have √

λ−2 ξ 2 (r ∗ 2 + e−

ln λ

1

)8 ,

which gives √

q (r ∗ 2 + e−

ln λ − 18

)

.

The estimate for the second term in (3.29) is similar but simpler. It remains to consider the bound for g1 , which is given by g1 = (1 + λ−2 Dr2∗ q w )g,

Dr ∗ =

1 ∂r ∗ . i

As above, the bound for g1 can be written in the form w a ps (r ∗ , 0, λ)(1 + λ−2 Dr2∗ q w )(a −1 ps ) (λ) f L 2 f L 2 . δ , S 0 , and S δ for all δ > 0. Hence The three operators above belong respectively to S1,δ 1,δ 1,δ δ , and it suffices to show that its principal symbol is bounded. the product belongs to S1,δ But the principal symbol of the product is given by ∗ a ps (r ∗ , 0, λ)χ D a −1 ps (r , ξ, λ),

which is bounded due to the choice of D. Finally we remark that as constructed the functions g1 and g2 are not necessarily supported near the photon sphere. This is easily rectified by replacing them with truncated versions, g1 := χ1 (r ∗ )g1 ,

g2 := χ1 (r ∗ )g2 ,

where χ1 is a smooth compactly supported cutoff which equals 1 in the support of g. It is clear that the bound (3.28) is still valid after truncation.

62


Using the above decomposition of g we write u in the form u = λ−2 g2 + u. ˜ For the first term we use the above lemma to estimate λλ−2 g2 L 2 + λ−2 ∂r ∗ g2 L 2 a w ps (λ)g L 2 , which is stronger than what we need. For u˜ we write the equation (∂r2∗ + λ2 (W + ))u˜ = g, ˜

g˜ = g1 − (W + )g2 .

(3.30)

For g˜ we only use a weighted L 1 bound, 1

˜ L 1 a w (λ−1 + |W + |)− 4 g ps (λ)g L 2 , which is obtained from the weighted L 2 bounds on g1 and g2 by Cauchy-Schwarz. For u˜ on the other hand, it suffices to obtain a pointwise bound: Lemma 3.6. For each λ−1 < σ < 1 and each function u˜ with compact support, we have 1

1

w λ(a −1 ˜ L 2 (σ + |W + |)− 4 ∂r ∗ u ˜ L ∞ + λ(σ + |W + |) 4 u ˜ L∞ . ps ) (λ)u

Proof. Since W has a nondegenerate zero minimum at 0, if > −σ then σ + |W + | ≈ σ + || + W . Hence without any restriction in generality we can replace (, σ ) by (0, σ + ||). Thus in the sequel we can assume that either = 0 or < −σ . We consider three cases: √

Case 1. ||, σ < e− ln λ . We consider an almost orthogonal partition of u˜ in dyadic regions with respect to r ∗ : √ 1 u˜ s , s0 = e− 2 ln λ . u˜ = u˜ <s0 + s0 ≤s 0,

where the second term accounts for the region where |ξ | > λs δ . This yields w −1 −δ λ(a −1 ps ) (λ)u˜ s L 2 | ln s| λu˜ s L 2 + s ∂r ∗ u˜ s L 2 .

In the support of u˜ s we have σ + |W + | ≈ s 2 ; therefore by Cauchy-Schwarz we obtain 1

1

w −1 4 ˜ L ∞ + s 1−δ (σ + |W +|)− 4 ∂r ∗ u ˜ L∞ . λ(a −1 ps ) (λ)u˜ s L 2 | ln s| λ(σ + |W + |) u


63

The summation with respect to s follows due to the L 2 almost orthogonality of the w functions (a −1 ps ) (λ)u˜ s , −1 w u˜ s λ (a ps ) (λ) s0 ≤s 4M}. Then χ∞ φ2L E M φ2L E 0 + E[φ](0).

70


b) Let f ∈ L E ∗M be supported in {r > 4M}. Then there is a function φ supported in {r > 3M} which solves g φ = f in {r M} so that sup E[φ](t) + φ2L E M + g φ − f 2L 2 f 2L E ∗ . M

t

(3.45)

Proof. a) For R > 0 we denote by χ>R a smooth cutoff function which is supported in {|x| > R} and equals 1 in {|x| ≥ 2R}. If R > 4M then (χ∞ − χ>R )φ2L E M φ2L E 0 . It remains to show that for a fixed sufficiently large R we have χ>R φ2L E M φ2L E 0 + E[φ](0). For this we notice that χ>R φ solves the equation g (χ>R φ) = f 1 (x)∇φ + f 2 (x)φ,

(3.46)

where f 1 and f 2 are supported in {R < |x| < 2R}. If R is sufficiently large then outside the ball {|x| ≤ R} the operator g is a small long range perturbation of the d’Alembertian. Then the estimate (1.5) applies, see e.g. [30, Prop. 2.2] or [28, (2.23)] (with no obstacle, = ∅) and we have χ>R φ2L E M E[χ>R φ](0) + g (χ>R φ)2L E ∗

M

E[φ](0) + [g , χ>R ]φ2L 2 E[φ](0) + φ2L E 0 ,

where in the last two steps we have used the compact support of g (χ>R φ) = [g , χ>R ]φ. b) Let R be large enough, as in part (a). For |x| > R the Schwarzchild metric g is a small long range perturbation of the Minkowski metric, according to the definition in [29]. We consider a second metric g˜ in R3+1 which coincides with g in {|x| > R} but which is globally a small long range perturbation of the Minkowski metric. Let ψ be the forward solution to g˜ ψ = f . Then we set φ = χ>R ψ. The estimate (1.5) holds for the metric g, ˜ therefore we obtain sup E[ψ](t) + ψ L E M f 2L E ∗ . t

M

Then the same bound holds as well for φ. Furthermore, we can compute the error g φ − f = (χ>R − 1) f + [g , χ>R ]ψ. This has compact spatial support, and can be easily estimated in L 2 as in part (a).


71

3.3. Proof of Theorem 3.2. Given f ∈ L E ∗ we split it into f = χeh f + χ ps f + χ∞ f. For the last two terms we use part (b) of Propositions 3.3, 3.10 to produce approximate solutions φ ps and φ∞ near the photon sphere, respectively near infinity. Adding them up we obtain an approximate solution φ0 = φ ps + φ∞ for the equation g φ = f . Due to (3.24) and (3.45) we obtain for φ0 the bound sup E[φ0 ](v) ˜ + φ0 2L E f 2L E ∗ ,

(3.47)

v˜

while the error f 1 = g (φ ps + φ∞ ) − f is supported away from r = 3M and r = ∞ and satisfies f 1 L E 0∗ ≈ f 1 L 2 f L E ∗ . Then we find φ = φ0 + φ1 by solving g φ1 = f 1 ∈ L E 0∗ ,

φ1 [0] = φ[0] − φ0 [0].

By Theorem 1.2 we obtain the L E 0 bound for φ1 . It remains to improve this to an L E bound for φ1 . By part (a) of Proposition 3.10 we can estimate χ∞ φ L E M . Near the photon sphere we would like to apply part (a) of Proposition 3.3 to χ ps φ. However we cannot proceed in an identical manner because part (a) of Proposition 3.3 does not involve the Cauchy data of φ at t = 0, and instead applies to functions φ defined on the full real axis in t. To address this issue we extend φ1 backward in t to the set MR , by solving the homogeneous problem g φ1 = 0 in MR , with matching Cauchy data on the common boundary of M R and MR . The extended function φ1 belongs to both L E(M R ) and L E(MR ), and now we can estimate χ ps φ1 via part (a) of Proposition 3.3. 4. Strichartz Estimates In this section we prove Theorem 1.4. The theorem follows from the following two propositions. The first gives the result for the right-hand side, f , in the dual local energy space: Proposition 4.11. Let (ρ, p, q) be a nonsharp Strichartz pair. Then for each φ ∈ L E with g φ ∈ L E ∗ + L 1v˜ L 2 we have ∇φ2L p H˙ −ρ,q E[φ](0) + φ2L E + g φ2L E ∗ +L 1 L 2 . v˜

v˜

(4.48)

The second one allows us to use L p2 L q2 in the right-hand side of the wave equation.

72


Proposition 4.12. There is a parametrix K for g so that for all nonsharp Strichartz pairs (ρ1 , p1 , q1 ) and (ρ2 , p2 , q2 ) we have ˜ + E[K f ]( +R ) + K f 2L E + ∇ K f 2 p1 sup E[K f ](v)

L v˜ H˙ −ρ1 ,q1

v˜

f 2 p

L v˜ 2 H˙ ρ2 ,q2

,

(4.49) and the error estimate g K f − f L E ∗ +L 1 L 2 f v˜

p

L v˜ 2 H˙ ρ2 ,q2

.

(4.50)

We first show how to use the propositions in order to prove the theorem. Proof of Theorem 1.4. Suppose that g φ = f with f ∈ L p2 H˙ ρ2 ,q2 . We write φ as

φ = φ1 + K f with K as in Proposition 4.12. By (4.49) the K f term satisfies all the required estimates; therefore it remains to consider φ1 . Using also (4.50) we obtain g φ1 2L E ∗ +L 1 L 2 + E[φ1 ](0) E[φ](0) + f 2 p L

v˜

˙ ρ2 ,q2 2H

.

Then Theorem 3.2 combined with Duhamel’s formula yields ˜ E[φ](0) + f 2 p φ1 2L E + g φ1 2L E ∗ +L 1 L 2 + sup E[φ1 ](v) v˜

v˜

Finally the L p1 H˙ −ρ1 ,q1 bound for ∇φ1 follows by Proposition 4.11.

L

˙ ρ2 ,q2 2H

.

We continue with the proofs of the two propositions. Proof of Proposition 4.11. By Duhamel’s formula and Theorem 3.2 we can neglect the L 1 L 2 part of g φ. Hence in the sequel we assume that g φ ∈ L E ∗ . We use cutoffs to split the space into three regions, namely near the event horizon, near the photon sphere and near infinity, φ = χeh φ + χ ps φ + χ∞ φ. Due to the definition of the L E and L E ∗ norms we have E[φ](0) + φ2L E + g φ2L E ∗ E[χeh φ](0) + χeh φ2H 1 + g (χeh φ)2L 2 + E[χ ps φ](0) + χ ps φ2L E ps + g (χ ps φ)2L E ∗ps +E[χ∞ φ](0) + χ∞ φ2L E M + g (χ∞ φ)2L E ∗ . M

Proving this requires commuting g with the cutoffs. However this is straightforward since the L E and L E ∗ norms are equivalent to the H 1 , respectively L 2 , norm in the support of ∇χeh , ∇χ ps and ∇χ∞ . p It remains to prove the L v˜ H˙ −ρ,q bound for each of the three terms in ∇φ. We consider the three cases separately:


73

I. The estimate near the event horizon. This is the easiest case. Given φ supported in {r < 11M/4}, we partition it on the unit scale with respect to v, ˜ χ (v˜ − j)φ, φ= j∈Z

where χ is a suitable smooth compactly supported bump function. Commuting the cutoffs with g one easily obtains the square summability relation χ (v˜ − j)φ2H 1 + g (χ (v˜ − j)φ)2L 2 + E[χ (v˜ − j)φ](0) j∈N

φ2H 1 + g φ2L 2 + E[φ](0), where the energy term on the left is nonzero only for finitely many j. Since each of the functions χ (v˜ − j)φ have compact support, they satisfy the Strichartz estimates due to the local theory; see [32,39,47]. The above square summability with respect to j guarantees that the local estimates can be added up. II. The estimate near the photon sphere. For φ supported in {5M/2 < r < 5M} we need to show that ∇φ2L p H −ρ,q E[φ](0) + φ2L E ps + g φ2L E ∗ps . v˜

We use again the Regge-Wheeler coordinates. Then the operator g is replaced by L RW . The potential V can be neglected due to the straightforward bound V φ L E ∗ps φ L E ps . Indeed, for φ at spherical frequency λ we have 1

1

V φ L E ∗ps | ln(2 + λ)| 2 φ L 2 λ| ln(2 + λ)|− 2 φ L 2 φ L E ps . We introduce the auxiliary function ψ = A−1 ps φ. By the definition of the L E ps norm we have ψ H 1 φ L E ps .

(4.51)

L RW ψ L 2 φ L E ps + L RW φ L 2 .

(4.52)

We also claim that

2 Since A−1 ps is L bounded, this is a consequence of the commutator bound 2 [A−1 ps , L RW ] : L E ps → L ,

or equivalently 1 2 [A−1 ps , L RW ]A ps : H → L .

(4.53)

74


It suffices to consider the first term in the symbol calculus, as the remainder belongs to δ , mapping H δ to L 2 for all δ > 0. The symbol of the first term is O P S1,δ 2 −3 2 q(ξ, r ∗ , λ) = {a −1 ps (λ), ξ + r (r − 2M)λ }a ps (λ), 1+δ . For a better estimate we compute the Poisson bracket and a-priori we have q ∈ S1,δ

q(ξ, r ∗ , λ) = a −1 ps (λ)γ y (y, ln λ)

4ξr ∗ − 2ξ ∂r ∗ (r −3 (r − 2M)) , r ∗ 2 + λ−2 ξ 2

where y = r ∗ 2 + λ−2 ξ 2 . The first two factors on the right are bounded. The third is bounded by λ since ∂r ∗ (r −3 (r − 2M)) vanishes at r ∗ = 0. In addition, q is supported in 0 |ξ | λ. Hence we obtain q ∈ λS1−δ,δ . Then the commutator bound (4.53) follows. Given (4.51) and (4.52), we argue as in the first case, namely we localize ψ to time intervals of unit length and then apply the local Strichartz estimates. By summing over these strips we obtain ∇ψ L p H −ρ,q φ L E ps + L RW φ L 2 for all sharp Strichartz pairs (ρ, p, q). To return to φ we invert A−1 ps , φ = A ps ψ + (1 − A ps A−1 ps )φ. The second term is much more regular, ∇(1 − A ps A−1 ps )φ L 2 H 1−δ φ L E ps ,

δ > 0;

therefore it satisfies all the Strichartz estimates simply by Sobolev embeddings. For the main term A ps ψ we take advantage of the fact that we only seek to prove the nonsharp Strichartz estimates for φ. The nonsharp Strichartz estimates for ψ are obtained from the sharp ones via Sobolev embeddings, 3 3 + ρ2 = + ρ1 , ρ1 < ρ2 . q2 q1

∇ψ H −ρ2 ,q2 ∇ψ H −ρ1 ,q1 ,

To obtain the nonsharp estimates for φ instead, we need a slightly stronger form of the above bound, namely Lemma 4.13. Assume that 1 < q1 < q2 < ∞. Then 3 3 + ρ2 = + ρ1 . q2 q1

A ps u H −ρ2 ,q2 u H −ρ1 ,q1 , Proof. We need to prove that the operator B = O p w (ξ 2 + λ2 + 1)−

ρ2 2

A ps O p w (ξ 2 + λ2 + 1)

maps L q1 into L q2 . The principal symbol of B is b0 (r ∗ , ξ, λ) = (ξ 2 + λ2 + 1)

ρ1 −ρ2 2

a ps (r ∗ , ξ, λ),

ρ1 2

(4.54)


75

and by the pdo calculus the remainder is easy to estimate, ρ −ρ2 −1+δ

1 B − b0w ∈ O P S1,0

,

δ > 0.

The conclusion of the lemma will follow from the Hardy-Littlewood-Sobolev inequality if we prove a suitable pointwise bound on the kernel K of b0w , namely |K (r1∗ , ω1 , r2∗ , ω2 )| (|r1∗ − r2∗ ||ω1 − ω2 |2 )

−1+ q1 − q1 1

2

.

(4.55)

For fixed r ∗ we consider a smooth dyadic partition of unity in frequency as follows:

µ χ{λ≈µ} χ{|ξ |ν0 } + χ{|ξ |≈ν} , 1 = χ{|ξ |>λ} + ν=ν0

µ dyadic

where ν0 = ν0 (λ, r ∗ ) is given by √ ln ν0 (λ, r ∗ ) = ln λ + max{ln r ∗ , − ln λ}. This leads to a similar decomposition for b0 , namely

µ bµν . b0 = b00 + bµ,R φ2L E M + g (χ>R φ)2L E ∗ M

M

+ E[χR which is supported in r > R and equals 1 in the support of χ>R . Then we define >R K∞ f = χ˜ >R ψ∞ . >R satisfies the appropriate bounds, It remains to show that K ∞ >R >R >R sup E[K ∞ f ](t) + K ∞ f 2L E M + ∇ K ∞ f 2L p1 H˙ −ρ1 ,q1 χ>R f 2 p L

t

respectively the error estimate >R g K ∞ f − χ>R f L E ∗M χ>R f

L p2 H˙ ρ2 ,q2

.

These are easily obtained by applying the following lemma to ψ∞ :

˙ ρ2 ,q2 2H

,

80


Lemma 4.16. Let f ∈ L p2 H˙ ρ2 ,q2 . Then the forward solution ψ to g˜ ψ = f satisfies the bound sup E[ψ](t) + ψ2L E M + ∇ψ2L p1 H˙ −ρ1 ,q1 f 2 p L

t

˙ ρ2 ,q2 2H

.

(4.65)

It remains to prove the lemma. This largely follows from [29, Theorem 6], but there is an interesting technical issue that needs clarification. Precisely, [29, Theorem 6] shows that we have the bound sup E[ψ](t) + ∇ψ2X 0 + ∇ψ2L p1 H˙ −ρ1 ,q1 f 2 p L

t

˙ ρ2 ,q2 2H

.

(4.66)

By Lemma 4.14, we are left with proving that 3j

sup 2− 2 ψ L 2 (A j ) f j∈Z

L p2 H˙ ρ2 ,q2

.

(4.67)

We note that this does not follow from Lemma 4.14; this is a forbidden endpoint of the Hardy inequality in [29, Lemma 1(b)]. However, the bound (4.67) can still be obtained, although in a roundabout way. Precisely, from (4.66) we have sup E[ψ](t) f 2 p L

t

(4.68)

˙ ρ2 ,q2 2H

for the forward in time evolution, and similarly for the backward in time problem. On the other hand, a straightforward modification of the classical Morawetz estimates (see e.g. [27]) for the wave equation shows that the solutions to the homogeneous wave equation g˜ ψ = 0 satisfy sup 2−3 j ψ2L 2 (A j∈Z

j)

E[ψ](0).

(4.69)

Denote by 1t>s H (t, s) the forward fundamental solution for g˜ and by H (t, s) its backward extension to a solution to the homogeneous equation, g˜ H (t, s) = 0. Combining the bounds (4.68) and (4.69) shows that sup 2

−3 j

j

2 H (t, s) f (s)ds 2 R

L (A j )

f 2 p L

˙ ρ2 ,q2 2H

.

Since p2 < 2, by the Christ-Kiselev lemma [11], it follows that sup 2−3 j j

which is exactly (4.67).

∞ t

2 H (t, s) f (s)ds

L 2 (A

j)

f 2 p L

˙ ρ2 ,q2 2H

,


81

5. The Critical NLW In this section we prove Theorem 1.5. We first consider (1.18) in the compact region MC . We denote by ψ the solution to the homogeneous equation ψ|0 = φ0 , K˜ ψ|0 = φ1 ,

g ψ = 0,

and by T f the solution to the inhomogeneous problem T F|0 = 0, K˜ T f |0 = 0.

g (T f ) = f,

Then we can rewrite the nonlinear equation (1.18) in the form φ = ψ ± T (φ 5 ).

(5.70)

We define Sobolev spaces in MC by restricting to MC functions in the same Sobolev space which are compactly supported in a larger open set. By the local Strichartz estimates we have ψ

1

H 2 ,4 (MC )

E[φ](0 )

and T f

1

H 2 ,4 (MC )

f

1 4

H 2 , 3 (MC )

.

At the same time we have the multiplicative estimate φ 5

1 4

H 2 , 3 (MC )

φ5

1

H 2 ,4 (MC )

.

Then for small initial data we can use the contraction principle to solve (5.70) and obtain 1 a solution φ ∈ H 2 ,4 (MC ). In addition, still by local Strichartz estimates, the solution φ will have finite energy on any space-like surface, in particular on the forward and backward space-like boundary of MC . Thus we obtain E[φ]( − R ) E[φ](0 ). It remains to solve (1.18) in M R (and its other three symmetrical copies). Using the (v, ˜ r, ω) coordinates in M R we define ψ and T as above, but with Cauchy data on − R. By the global Strichartz estimates in Theorem 1.4, for (s, p) as in the theorem we have ψ L p H˙ s, p (M R ) E[φ]( − R) and T f L p H˙ s, p (M R ) f L 1 L 2 . In particular we can take p = 5 which corresponds to s = dings we have φ L 5 L 10 φ

3

H˙ 10 ,5

;

3 10 .

By Sobolev embed-

82


therefore φ 5 L 1 L 2 φ5

3

H˙ 10 ,5

.

Hence we can solve (5.70) using the contraction principle and obtain a solution φ ∈ 3 H˙ 10 ,5 . This implies that φ 5 ∈ L 1 L 2 , which yields all of the other Strichartz estimates, as well as the energy bound on the forward boundary +R of M R . This concludes the proof of the theorem. Acknowledgement. The authors are grateful to M. Dafermos and I. Rodnianski for pointing out their novel way of taking advantage of the red shift effect in [14], and to N. Burq and M. Zworski for useful conversations concerning the analysis near trapped null geodesics. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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21. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) 22. Kenig, C.E., Ponce, G., Vega, L.: On the Zakharov and Zakharov-Schulman systems. J. Funct. Anal. 127, 204–234 (1995) 23. Kronthaler, J.: Decay Rates for Spherical Scalar Waves in the Schwarzschild Geometry. http://arXiv.org/ abs/0709.3703v1[gr-qc], 2007 24. Kay, B., Wald, R.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quant. Grav. 4, 893–898 (1987) 25. Laba, I., Soffer, A.: Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds. Helv. Phys. Acta 72, 274–294 (1999) 26. Lindblad, H., Sogge, C.D.: On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal. 130, 357–426 (1995) 27. Metcalfe, J., Sogge, C.D.: Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods. SIAM J. Math. Anal. 38, 391–420 (2006) 28. Metcalfe, J., Sogge, C.D.: Global existence of null-form wave equations in exterior domains. Math. Z. 256, 521–549 (2007) 29. Metcalfe, J., Tataru, D.: Global parametrices and dispersive estimates for variable coefficient wave equations. http://arXiv.org/abs/0707.1191v2[math.Ap], 2007 30. Metcalfe, J., Tataru, D.: Global parametrices and dispersive estimates for variable coefficient wave equations in exterior domains. http://arXiv.org/abs/0806.3409v2[math.Ap], 2009 31. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. San Francisco, CA: W. H. Freeman and Co., 1973 32. Mockenhaupt, G., Seeger, A., Sogge, C.D.: Local smoothing of Fourier integral operators and CarlesonSjölin estimates. J. Amer. Math. Soc. 6, 65–130 (1993) 33. Morawetz, C.: Time decay for the nonlinear Klein-Gordon equations. Proc. Roy. Soc. Ser. A. 306, 291–296 (1968) 34. Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering. http://arXiv.org/abs/0706. 3242v2[math-ph], 2007 35. Pecher, H.: Nonlinear small data scattering for the wave and Klein-Gordon equations. Math. Z. 185, 261–270 (1984) 36. Price, R.: Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations. Phys. Rev. D (3) 5, 2419–2438 (1972) 37. Ralston, J.V.: Solutions of the wave equation with localized energy. Comm. Pure Appl. Math. 22, 807–823 (1969) 38. Schlag, W., Soffer, A., Staubach, W.: Decay for the wave and Schroedinger evolutions on manifolds with conical ends, Part II. http://arXiv.org/abs/0801.2001v1[math.Ap], 2008 39. Smith, H.F.: A parametrix construction for wave equations with C 1,1 coefficients. Ann. Inst. Fourier (Grenoble) 48, 797–835 (1998) 40. Smith, H.F., Sogge, C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Part. Diff. Eqs. 25, 2171–2183 (2000) 41. Sterbenz, J.: Angular regularity and Strichartz estimates for the wave equation. With an appendix by I. Rodnianski. Int. Math. Res. Not. 2005, 187–231 (2005) 42. Strauss, W.A.: Dispersal of waves vanishing on the boundary of an exterior domain. Comm. Pure Appl. Math. 28, 265–278 (1975) 43. Strichartz, R.S.: A priori estimates for the wave equation and some applications. J. Funct. Anal. 5, 218–235 (1970) 44. Strichartz, R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977) 45. Tataru, D.: Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation. Amer. J. Math. 122, 349–376 (2000) 46. Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients II. Amer. J. Math. 123, 385–423 (2001) 47. Tataru, D.: Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III. J. Amer. Math. Soc. 15, 419–442 (2002) 48. Tataru, D.: Parametrices and dispersive estimates for Schroedinger operators with variable coefficients. Amer. J. Math. 130, 571–634 (2008) 49. Twainy, F.: The Time Decay of Solutions to the Scalar Wave Equation in Schwarzschild Background. Thesis, University of California San Diego, 1989 50. Wald, R.M.: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20, 1056–1058 (1979) 51. Wald, R.M.: General Relativity. Chicago, IL: University of Chicago Press, 1984 Communicated by S. Zelditch


Communications in


Axiomatic Quantum Field Theory in Curved Spacetime Stefan Hollands1 , Robert M. Wald2 1 School of Mathematics, Cardiff University, Cardiff, Wales CF24 4AG, UK.


2 Enrico Fermi Institute and Department of Physics, University of Chicago,

Chicago, IL 60637, USA. E-mail: [email protected] Received: 22 May 2008 / Accepted: 6 May 2009 Published online: 1 September 2009 – © Springer-Verlag 2009

Abstract: The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed. 1. Introduction The Wightman axioms [27] of quantum field theory in Minkowski spacetime are generally believed to express the fundamental properties that quantum fields possess. In essence, these axioms require that the following key properties hold: (1) The states of the theory are unit rays in a Hilbert space, H, that carries a unitary representation of the Poincaré group. (2) The 4-momentum (defined by the action of the Poincaré group on the Hilbert space) is positive, i.e., its spectrum is contained within the closed future light cone (“spectrum condition”). (3) There exists a unique, Poincaré invariant state (“the vacuum”). (4) The quantum fields are operator-valued distributions defined on a

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dense domain D ⊂ H that is both Poincaré invariant and invariant under the action of the fields and their adjoints. (5) The fields transform in a covariant manner under the action of Poincaré transformations. (6) At spacelike separations, quantum fields either commute or anticommute. During the past 40 years, considerable progress has been made in understanding both the physical and mathematical properties of quantum fields in curved spacetime. Although gravity itself is treated classically, this theory incorporates some key aspects of general relativity and thereby should provide a more fundamental base for quantum field theory. Much of the progress has occurred in the analysis of free (i.e., nonself-interacting) fields, but in the past decade, major progress also has been made in the perturbative analysis of interacting quantum fields. Significant insights have thereby been obtained into the nature of quantum field phenomena in strong gravitational fields. In addition, some important insights have been obtained into the nature of quantum field theory itself. One of the key insights is that—apart from stationary spacetimes or spacetimes with other very special properties—there is no unique, natural notion of a “vacuum state” or of “particles”. Indeed, unless the spacetime is asymptotically stationary at early or late times, there will not, in general, even be an asymptotic notion of particle states. Consequently, it is essential that quantum field theory in curved spacetime be formulated in terms of the local field observables as opposed, e.g., to S-matrices. Since quantum field theory in curved spacetime should be much closer to a true theory of nature than quantum field theory in Minkowski spacetime, it is of interest to attempt to abstract the fundamental features of quantum field theory in curved spacetime in a manner similar to the way the Wightman axioms abstract what are generally believed to be the fundamental features of quantum field theory in Minkowski spacetime. The Wightman axioms are entirely compatible with the focus on local field observables, as needed for a formulation of quantum field theory in curved spacetime. However, most of the properties of quantum fields stated in the Wightman axioms are very special to Minkowski spacetime and cannot be generalized straightforwardly to curved spacetime. Specifically, a curved spacetime cannot possess Poincaré symmetry—indeed a generic curved spacetime will not possess any symmetries at all—so one certainly cannot require “Poincaré invariance/covariance” or invariance under any other type of spacetime symmetry. Thus, no direct analog of properties (3) and (5) can be imposed in curved spacetime, and the key aspects of properties (1) and (2) (as well as an important aspect of (4)) also do not make sense. In fact, the situation with regard to importing properties (1), (2), and (4) to curved spacetime is even worse than would be suggested by merely the absence of symmetries: There exist unitarily inequivalent Hilbert space constructions of free quantum fields in spacetimes with a noncompact Cauchy surface and (in the absence of symmetries of the spacetime) none appears “preferred”. Thus, it is not appropriate even to assume, as in (1), that states are unit rays in a single Hilbert space, nor is it appropriate to assume, as in (4), that the (smeared) quantum fields are operators on this unique Hilbert space. With regard to (2), although energy and momentum in curved spacetime cannot be defined via the action of a symmetry group, the stress-energy tensor of a quantum field in curved spacetime should be well defined as a distributional observable on spacetime, so one might hope that it might be possible to, say, integrate the (smeared) energy density of a quantum field over a Cauchy surface and replace the Minkowski spacetime spectrum condition by the condition that the total energy of the quantum field in any state always is non-negative. However, this is not a natural thing to do, since the “total energy” defined in this way is highly slice/smearing dependent, and it is well known in classical general

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relativity that in asymptotically flat spacetimes, the integrated energy density of matter may bear little relationship to the true total mass-energy. Furthermore, it is well known that the energy density of a quantum field (in flat or curved spacetime) can be negative, and, in some simple examples involving free fields in curved spacetime, the integrated energy density is found to be negative. Consequently, there is no analog of property (2) in curved spacetime that can be formulated in terms of the “total energy-momentum” of the quantum field. Thus, of all of the properties of quantum fields in Minkowski spacetime stated in the Wightman axioms, only property (6) has a straightforward generalization to curved spacetimes! Nevertheless, it has been understood for quite some time that the difficulties in the formulation of quantum field theory in curved spacetime that arise from the existence of unitarily inequivalent Hilbert space constructions of the theory can be overcome by simply formulating the theory via the algebraic framework [14]. Instead of starting from the postulate that the states of the theory comprise a Hilbert space and that the (smeared) quantum fields are operators on this Hilbert space, one starts with the assumption that the (smeared) quantum fields (together with the identity element 1) generate a *-algebra, A. States are then simply expectation functionals . ω : A → C on the algebra, i.e., linear functionals that are positive in the sense that A∗ Aω ≥ 0 for all A ∈ A. The GNS construction then assures us that given a state, ω, one can find a Hilbert space H that carries a representation, π , of the *-algebra A, such that there exists a vector | ∈ H for which Aω = |π(A)| for all A ∈ A. All of the operators, π(A), are automatically defined on a common dense invariant domain, D ⊂ H, and each vector | ∈ D defines a state via A = |π(A)|. Thus, by simply adopting the algebraic viewpoint, we effectively incorporate into quantum field theory in curved spacetime the portions of the content of properties (1) and (4) above that do not refer to Poincaré symmetry. It is often said that in special relativity one has invariance under “special coordinate transformations” (i.e., Poincaré transformations), whereas in general relativity, one has invariance under “general coordinate transformations” (i.e., all diffeomorphisms). Thus, one might be tempted to think that the Minkowski spacetime requirements of invariance/covariance under Poincaré transformations could be generalized to curved spacetime by requiring a similar “invariance/covariance under arbitrary diffeomorphisms”. However, such thoughts are based upon a misunderstanding of the true meaning of “special covariance” and “general covariance”. By explicitly incorporating the flat spacetime metric, ηab , into the formulation of special relativity, it can easily be seen that special relativity can be formulated in as “generally covariant” a manner as general relativity. However, the act of formulating special relativity in a generally covariant manner does not provide one with any additional symmetries or other useful conditions on physical theories in flat spacetime. The point is that in special relativity, Poincaré transformations are symmetries of the spacetime structure, and we impose a nontrivial requirement on a physical theory when we demand that its formulation respect these symmetries. However, a generic curved spacetime will not possess any symmetries at all, so no corresponding conditions on a physical theory can be imposed. The demand that a theory be “generally covariant” (i.e., that its formulation is invariant under arbitrary diffeomorphisms) can always be achieved by explicitly incorporating any “background structure” into the formulation of the theory. If one considers a fixed, curved spacetime without symmetries, no useful conditions can be imposed upon a quantum field theory by attempting to require some sort of “invariance” of the theory under diffeomorphisms. However, there is a meaningful notion of “general covariance” that can be very usefully and powerfully applied to quantum field theory in curved spacetime. The

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basic idea behind this notion is that the only “background structure” that should occur in the theory is the spacetime manifold and metric modulo diffeomorphisms, together with the time and space orientations and (if spinors are present in the theory) spin structure. The quantum fields should be “covariant” in that their construction should only make use of this background structure. Indeed, since the smeared quantum fields are associated with local regions of spacetime (namely, the support of the test function used for the smearing), it seems natural to demand that the quantum fields be locally constructed from the background structure in the sense that the quantum fields in any neighborhood O be covariantly constructed from the background structure within O. This idea may be formulated in a precise manner as follows [7,19,20]. First, in order to assure a well defined dynamics and in order to avoid causal pathologies, we restrict consideration to globally hyperbolic spacetimes (M, gab ). (We consider theories in arbitrary spacetime dimension D ≡ dim M ≥ 2.) If spinors are present in the theory, we also demand that M admit a spin structure. It is essential that the quantum field theory be defined on all D-dimensional globally hyperbolic spacetimes admitting a spin structure, since in essence, we can only tell whether the quantum field is “locally and covariantly constructed out of the metric” if we can see how the theory changes when we change the metric in an arbitrary way. The “background structure”, M, of the theory is taken to consist of the manifold M, the metric gab , the spacetime orientation— which may be represented by a nowhere vanishing D-form, ea1 ...a D on M—and a time orientation—which may be represented e.g. by the equivalence class of a time function T : M → R—i.e., we have M = (M, g, T, e).

(1)

(If spinors are present in the theory, and M admits more than one spin structure, then the choice of spin-structure over M also should be understood to be included in M.) For each choice of M, we assume that there is specified a *-algebra A(M) that is generated by a countable list of quantum fields φ (i) and their “adjoints” φ (i)∗ . These fields may be of arbitrary tensorial or spinorial type, and they are smeared with arbitrarily chosen smooth, compact support fields of dual tensorial or spinorial type. In order to determine if the quantum field theory and quantum fields φ (i) are “locally and covariantly constructed out of the background structure M”, we consider the following situation: Let (M, g) and (M , g ) be two globally hyberbolic spacetimes that have the property that there exists a one-to-one (but not necessarily onto) map ρ : M → M that preserves all of the background structure. In other words, ρ is an isometric imbedding that is orientation and time orientation preserving (and, if spinors are present, the choices of spin structure on M and M correspond under ρ). We further assume that ρ is causality preserving in the sense that if x1 , x2 ∈ M cannot be connected by a causal curve in M, then ρ(x1 ) and ρ(x2 ) cannot be connected by a causal curve in M . We say that the quantum field theory is locally and covariantly constructed from M (or, for short, that the theory is local and covariant) if (i) for every such M, M , and ρ we have a corresponding *-isomorphism χρ between A(M) and the subalgebra of A(M ) generated by the quantum fields φ (i) and φ (i)∗ smeared with test fields with support in ρ[M] and (ii) if ρ is a similar background structure and causality preserving map taking M to M , then χρ ◦ρ = χρ ◦ χρ . We further say that the quantum field φ (i) is locally and covariantly constructed from M (or, for short, that φ (i) is local and covariant) if for every such M, M , and ρ, we have χρ φ (i) ( f ) = φ (i) (ρ∗ ( f )),

(2)


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where ρ∗ ( f ) denotes the natural push-forward action of ρ on the tensor/spinor field f on M. Note that in contrast to the notion of Poincaré covariance—which applies to quantum field theory on a single spacetime (namely, Minkowski spacetime)—the notion that a quantum field theory or quantum field is local and covariant is a condition that applies to the formulation of quantum field theory on different spacetimes. Nevertheless, the close relationship between these notions can be seen as follows: Suppose that we have a local and covariant quantum field theory, with local and covariant quantum fields φ (i) . Let M and M both be the background structure of Minkowski spacetime, and let ρ be a proper Poincaré transformation. Then ρ preserves all of the background structure, so for each proper Poincaré transformation, we obtain a *-isomorphism χρ : A → A, where A here denotes the quantum field algebra for Minkowski spacetime. Furthermore, if ρ and ρ are proper Poincaré transformations, we have χρ◦ρ = χρ ◦ χρ . Thus, every local and covariant quantum field theory in curved spacetime gives rise to a Poincaré invariant theory in this sense when restricted to Minkowski spacetime. Furthermore, if φ (i) is a local and covariant quantum field, then in Minkowski spacetime it transforms covariantly via Eq. (2) under proper Poincaré transformations. Note also that, more generally, in any curved spacetime with symmetries, a local and covariant quantum field theory will be similarly invariant under these symmetries, and a local and covariant quantum field will transform covariantly under these symmetries. But even for spacetimes without any symmetries, the requirement that the quantum field theory and quantum fields be local and covariant imposes a very powerful restriction akin to requiring Poincaré invariance/covariance in Minkowski spacetime. From these considerations, it can be seen that if we adopt the above algebraic framework for quantum field theory in curved spacetime and if we additionally demand that the quantum field theory and the quantum fields φ (i) be local and covariant, then we obtain satisfactory generalizations to curved spacetime of properties (1), (4), and (5) of the Wightman axioms in Minkowski spacetime. Since we already noted that (6) has a trivial generalization to curved spacetime, only properties (2) and (3) remain to be generalized to curved spacetime. We have already noted above that there is no analog of property (2) in curved spacetime that can be formulated in terms of the “total energy-momentum” of the quantum field. However, it is possible to reformulate the spectrum condition in Minkowski spacetime in terms of purely local properties of the quantum fields. Specifically, the “positive frequency” (and, thereby, positive energy) properties of states are characterized by the short-distance singularity structure of the n-point functions of the quantum fields, as described by their wavefront set. One thereby obtains a microlocal spectrum condition [5,6,24] that is formulated purely in terms of the local in spacetime properties of the quantum fields. This microlocal spectrum condition has a natural generalization to curved spacetime (see Sect. 2 below), thus providing the desired generalization of property (2) to curved spacetime. Consequently, only property (3) remains to be generalized. In Minkowski spacetime, the existence of a unique, Poincaré invariant state has very powerful consequences, so it is clear that a key portion of the content of quantum field theory in Minkowski spacetime would be missing if we failed to impose an analogous condition in curved spacetime. However, as already mentioned above, one of the clear lessons of the study of free quantum fields in curved spacetime is that, in a general curved spacetime, there does not exist a unique, “preferred” vacuum or other state. Furthermore, even if a prescription for finding a unique “preferred state” on each spacetime could be found, since

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generic curved spacetimes do not have any symmetries and states on different spacetimes cannot be meaningfully compared, there would appear to be no sensible “invariance” properties that such a preferred state could have. We do not believe that property (3) can be generalized to curved spacetime by a condition that postulates the existence of a preferred state with special properties. In addition, we question the fundamental status of demanding the existence of a state that is invariant under the symmetries of the spacetime. For example, it is well known that the free massless Klein-Gordon field in two-dimensional Minkowski spacetime does not admit a Poincaré invariant state. However, there is absolutely nothing wrong with the quantum field algebra of this field; the quantum field theory is “Poincaré covariant” and the quantum field transforms in a “Poincaré covariant manner” in the sense described above. Furthermore, there is no shortage of physically acceptable (“Hadamard”) states. Thus, the only thing unusual about this quantum field theory is that it happens not to admit a Poincaré invariant state. We do not feel that this is an appropriate reason to exclude the free massless Klein-Gordon field in two-dimensional Minkowski from being considered to be a legitimate quantum field theory. Similar remarks apply to the free Klein-Gordon field of negative m 2 in Minkowski spacetime of all dimensions. The classical and quantum dynamics of this field are entirely well posed and causal, although they are unstable in the sense of admitting solutions/states where the field grows exponentially with time. This instability provides legitimate grounds for arguing that the free Klein-Gordon field of negative m 2 does not occur in nature, but we do not feel that the absence of a Poincaré covariant state constitutes legitimate grounds for rejecting this theory as a quantum field theory; see Sect. 6 below for further discussion. For the above reasons, we seek a replacement of property (3) that does not require the existence of states of a special type. The main purpose of this paper is to propose that the appropriate replacement of property (3) for quantum field theory in curved spacetime is to postulate the existence of a suitable operator product expansion [26,29,31] of the quantum fields. The type of operator product expansion that we shall postulate is known to hold in free field theory and to hold order by order in perturbation theory on any Lorentzian curved spacetime [15]. We thus propose to elevate this operator product expansion to the status of a fundamental property of quantum fields1 . Although the assumption of the existence of an operator product expansion in quantum field theory in curved spacetime is remarkably different in nature from the assumption of the existence of a Poincaré covariant state in quantum field theory in Minkowski spacetime, we will show in Sects. 4 and 5 below that it can do “much of the same work” as the latter assumption. It is shown in [21] how to exploit consistency conditions on the OPE in a framework closely related to that presented here. In particular, it is shown how perturbations of a quantum field theory can be characterized and calculated via consistency conditions arising from the OPE. We have an additional motivation for proposing to elevate the operator product expansion to the status of a fundamental property of quantum fields. For free quantum fields in curved spacetime, an entirely satisfactory *-algebra, A0 , of observables has been constructed [5,6,19], which includes all Wick powers and time-ordered products. However, the elements of A0 correspond to unbounded operators, and there does not seem to be any 1 Various axiomatic approaches to conformal field theories based on the operator product expansion have been proposed previously, see e.g. the one based upon the notion of “vertex operator algebras” given in [2, 11,23]. However, in contrast to our approach, these approaches incorporate in an essential way the conformal symmetry of the underlying space. In some approaches to quantum field theory on Minkowski space, the OPE is not postulated, but instead derived [3,4,10].


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natural algebra of bounded elements (with, e.g., a C ∗ -structure) corresponding to A0 . Furthermore, A0 does not appear to have any natural topology (apart from a topology that can be defined a posteriori by using the allowed states as semi-norms). Fortunately, a topology is not actually needed to define A0 because the relations that hold in A0 can be expressed in terms of finite sums of finite products of fields. However, it is inconceivable that the relations that define an interacting field algebra will all be expressible in terms of finite sums of finite products of the fields. Thus, without a natural topology and without finitely expressible relations, it is far from clear as to how an interacting field algebra might be defined. We claim that an operator-product expansion effectively provides the needed “relations” between the quantum fields, and we will propose in this paper that these relations are sufficient to define a quantum field theory. In other words, we believe that an interacting quantum field theory is, in essence, defined via its operator-product expansion. From this perspective, it seems natural to view the operator product-expansion as a fundamental aspect of the quantum field theory. In the next section, we will describe our framework for quantum field theory in curved spacetime. In particular, we will provide a precise statement of what we mean by an operator-product expansion and the properties that we will assume that it possesses. In Sect. 3, we will state our axioms for quantum field theory in curved spacetime and explain how it is constructed from the operator-product expansion. Finally, in Sects. 4 and 5 we will show that our axioms have much of the same power as the Wightman axioms by establishing “normal” (anti-)commutation relations and proving curved spacetime versions of the spin-statistics theorem and PCT theorem. Some further implications of our new perspective on quantum field theory are discussed in Sect. 6. 2. General Framework for the Formulation of QFT We will now explain in much more detail our proposed framework for defining quantum field theory. We fix a dimension D ≥ 2 of the spacetime and consider all D-dimensional globally hyperbolic spacetimes (M, gab ). As explained in the previous section, we will assume that each spacetime is equipped with an orientation, specified by a nowhere vanishing D-form ea1 ...a D on M, and a time orientation, specified by (the equivalence class of) a globally defined time function T : M → R. The set of background data specified this way will be denoted M = (M, g, T, e).

(3)

In certain cases, more background structure may be prescribed, such as a choice of bundles in which the quantum fields live. For example, if spinors are present and if M admits more than one spin structure, then a choice of spin-structure over M is assumed to be given as part of the background structure2 , and is understood to also be part of M. It should be emphasized that two spacetimes with the same manifold and metric, 2 It is convenient to think of the background structure as a category [7], whose objects are the tuples M. Morphisms in the category of tuples M are isometric, causality, orientation, and other background structure preserving embeddings,

ρ : M → M .

(4)

Thus, ρ is a diffeomorphism M → M such that g = ρ ∗ g , such that ρ ∗ T represents the same timeorientation as T , such that ρ ∗ e represents the same orientation as e, and such that the causal relations in (M, g) inherited from (M , g ) coincide with the original ones. Furthermore, if M also includes the choice of a spin structure, then ρ must also preserve the spin structures.

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but, e.g., with different time-orientations define distinct background structures. Below, we will consider quantum field theories associated with background structures, and we stress that, at this stage, the quantum field theories associated with different background structures (e.g., ones that merely differ in the choice of, say, time-orientation) need not have any relation whatsoever. The quantum fields present in a given theory will be assumed to correspond to sections of vector bundles over M. We will denote the various quantum fields by φ (i) , with i ∈ I , where I is a suitable indexing set, and we will write V (i) for the vector bundle over M, of which φ (i) corresponds to a section. It should be emphasized that i ∈ I labels all of the quantum fields present in the theory, not just the “fundamental” ones. Thus, even if we were considering the theory of a single scalar field ϕ, there will be infinitely many composite fields of various tensorial types corresponding to all monomials in ϕ and its derivatives, each of which would be labeled by a different index i. It will be convenient to also include a field denoted φ (1) in the list of quantum fields, which will play the role of the identity element, 1, in the quantum field algebra. We assume that each field φ (i) has been assigned a Bose/Fermi parity F(i) = 0, 1 modulo two. We further assume that there is an operation : I → I i → i ,

(5)

having the property that V (i ) = V (i), where for any vector space E, the vector space E consists of all anti-linear maps E v → C, with E v denoting the dual space of E. In particular, if i is associated with, say the vector bundle V (i) of spinors with P primed and U unprimed indices, then V (i ) is the bundle of spinors with U primed and P unprimed spinor indices. We demand that the star operation squares to the identity3 , i = i. We ∗ also require that φ (1 ) = φ (1) , i.e. that 1∗ = 1. As in many other approaches to quantum field theory, we will use the smeared fields φ (i) ( f )—with i ∈ I , and f a compactly supported test section in the dual vector bundle V (i)v to V (i)—to generate a *-algebra of observables, A(M). However, in most other algebraic approaches to quantum field theory, A(M) is assumed, a priori, to possess a particular topological and/or other structure (e.g., C*-algebra structure) and the algebraic relations within A(M)—together, perhaps, with specified actions of symmetry groups on A(M)—are assumed to encode all of the information about the quantum field theory under consideration. In particular, since the state space, S(M), is normally taken to consist of all positive linear maps on A(M), it is clear that S(M) cannot contain any information about the quantum field theory that is not already contained in A(M). We shall not proceed in this manner because it is far from clear to us what topological and other structure A(M) should be assumed to possess a priori in order to describe the quantum field theory. Instead, we shall view the theory as being specified by providing both an algebra of observables, A(M), and a space of allowed states, S(M). Essentially the only information about the theory contained in the algebra of obervables, A(M), will be the list of fields appearing in the theory and the relations that can be written as polynomial expressions in the fields and their derivatives. In our approach, the information normally encoded in the topology of A(M) will now be encoded in S(M). Of course, the semi-norms provided by S(M) could be used, a posteriori, to define a topology on A(M), but it is not clear that this topology would encode all of the information in S(M); in any case, we find it simpler and more natural to consider the quantum field theory to be defined by the pair {A(M), S(M)}. 3 This operation gives I the structure of an involutive category.


93

The key idea of this paper is that we will obtain the pair {A(M), S(M)} in a natural (i.e., functorial) manner from the space of field labels I and another datum, namely, the collection of “operator product expansion (OPE) coefficients”. The OPE coefficients are a family (i )···(i ) C(M) ≡ C( j)1 n (x1 , . . . , xn ; y) : i 1 , . . . , i n , j ∈ I, n ∈ N , (6) (i )···(i n )

where each C( j)1

is a distribution on M n+1 , valued in the vector bundle π

E = V (i 1 ) × · · · × V (i n ) × V ( j)v −→ M n+1

(7)

that is defined in some open neighborhood of the diagonal in M n+1 . Thus, given C(M), we will construct both the algebra A(M) and the state space S(M) S(M) | C(M) dual pairing

| A(M)

Thus, in our framework, a quantum field theory is uniquely specified by providing a list of quantum fields I and a corresponding list of OPE coefficients C(M). The OPE coefficients will be required to satisfy certain general properties, which, in effect, become the “axioms” of quantum field theory in curved spacetime. Most of the remainder of this section will be devoted to formulating these axioms. However, before providing the axioms for C(M), we briefly outline how the algebra A(M) and the state space S(M) are constructed from C(M) for any background structure M = (M, g, T, e). The algebra A(M) is constructed by starting with the free algebra Free(M) generated by all expressions of the form φ (i) ( f ) with i ∈ I , and f a compactly supported test section in the dual vector bundle to V (i) . We define an antilinear *-operation on Free(M) by requiring that its action on the generators be given by [φ (i) ( f )]∗ = φ (i ) ( f¯),

(8)

where f¯ ∈ Sect 0 [V¯ (i)] is the conjugate test section to f ∈ Sect 0 (V (i)). The *-algebra A(M) is taken to be the resulting free *-algebra factored by a 2-sided ideal generated by a set of polynomial relations in the fields and their derivatives. These relations consist of certain “universal” relations that do not depend on the particular theory under consideration (such as linearity of φ (i) ( f ) in f and (anti-)commutation relations) together with certain relations that may arise from the OPE coefficients C(M). A precise enumeration of the relations that define A(M) will be given at the beginning of Sect. 3. The state space S(M) is a subspace of the space of all linear, functionals ω : A(M) → C that are positive in the sense that ω(A∗ A) ≡ A∗ Aω ≥ 0 for all A ∈ A(M). This subspace is specified as follows: First, we require that for any state ω ∈ S(M), the OPE coefficients in the collection C(M) in Eq. (6) appear in the expansion of the expectation value of the product of fields φ (i1 ) (x1 ) · · · φ (in ) (xn )ω in terms of the fields φ ( j) (y)ω , (i )...(i ) C( j)1 n (x1 , . . . , xn ; y) φ ( j) (y) . (9) φ (i1 ) (x1 ) · · · φ (in ) (xn ) ≈ ω

j

ω

94


Here “≈” means that this equation holds in a suitably strong sense as an asymptotic relation in the limit that x1 , . . . , xn → y. A precise definition of what is meant by this asymptotic relation will be given in Eq. (39) below. Secondly, we require that ω satisfy a microlocal spectrum condition that, in essence, states that the singularities of φ (i1 ) (x1 ) . . . φ (in ) (xn )ω are of “positive frequency type” in the cotangent space T(x∗ 1 ,...,xn ) M n . The precise form of this condition will be formulated in terms of the wave front set [22] (see Eqs. (23) and (24) below). We turn now to the formulation of the conditions that we shall impose on the OPE coefficients C(M). As indicated above, in our framework, these conditions play the role of axioms for quantum field theory. Each operator product coefficient C((ij)1 )···(in ) in C(M) is a distribution4 on M n+1 , valued in the vector bundle V (i 1 ) × · · · × V (i n ) × V ( j)v that is defined in some open neighborhood of the diagonal in M n+1 . We will impose the following requirements on these coefficients: C1) C2) C3) C4) C5) C6) C7) C8) C9)

Locality and Covariance, Identity element, Compatibility with the -operation, Commutativity/Anti-Commutativity, Scaling Degree, Asymptotic positivity, Spectrum condition, Associativity, Analytic dependence upon the metric.

Before formulating these conditions in detail, for each i ∈ I we define the dimension, dim(i) ∈ R, of the field φ (i) by5 1 (i)(i ) , (10) sup dim(i) := sd C(1) 2 backgrounds M where “sd” denotes the scaling degree of a distribution (see Appendix A) and it is understood that the scaling degree is taken about a point on the diagonal. In other words, dim(i) measures the rate at which the coefficient of the identity 1 in the operator product expansion of φ (i) (x1 )φ (i ) (x2 ) blows up as x1 → x2 . It will follow immediately from Condition (C3) below that dim(i ) = dim(i). Note also that dim(1) = 0. For distributions u 1 and u 2 on M n+1 , we introduce the equivalence relation u1 ∼ u2

(11)

to mean that the scaling degree of the distribution u = u 1 − u 2 about any point on the total diagonal is −∞. However, it should be noted that in the formulation of Condition 4 More precisely, each OPE coefficient is an equivalence class of distributions, where two distributions are considered equivalent if their difference satisfies Eq. (29) below for all δ > 0 and all T. Indeed, the OPE coefficients are more properly thought of as a sequence of (equivalence classes of) distributions, such that the difference between the n th and m th terms in the sequence satisfies Eq. (29) for δ = min(m, n). However, to avoid such an extremely cumbersome formulation of our axioms and results, we will treat each OPE coefficient as a distribution. 5 Note that, when V (i) is not equal to M × C, i.e., when φ (i) is not a scalar field, then quantities like (i)(i )

are, by definition, distributions taking values in a vector bundle. What we mean by the scaling degree C(1) here and in similar equations in the following such as Eq. (20) is the maximum of the scaling degrees of all “components” of such a bundle-valued distribution.


95

(C8) below and in the precise definition of the operator product expansion, Eq. (39), we will need to consider limits where different points approach the total diagonal at different rates. We will then introduce a stronger notion of equivalence that, in effect, requires a scaling degree of −∞ under these possibly different rates of approach. It is this stronger notion that was meant in Eq. (9) above. (C1) Covariance. Let ρ : M → M be a causality preserving isometric embedding preserving the orientations, spin-structures, and all other background structure, i.e. ρ is a morphism in the category of background structures. We postulate that for each member of the above collection (6), we have (i )...(i n )

(ρ ∗ × · · · ρ ∗ × ρ∗−1 ) C( j)1

(i )...(i n )

[M ] ∼ C( j)1

[M].

(12)

(C2) Identity element. We require that (i )...(1)...(i n )

C( j)1

(i )...(i k )(i k+1 )...(i n )

(x1 , . . . , xn ; y) = C( j)1

(x1 , . . . xk−1 , xk+1 , . . . xn ; y), (13)

where the identity 1 is in the k th place. When n = 1, the condition involves the OPE coefficient C (i)(1) ( j)(x1 , x2 ; y) = C (i) ( j)(x1 ; y). This coefficient should be thought of as describing the operator product expansion around the reference point y of product of the field labelled by i and localized at the point x1 with the identity operator. These coefficients should hence merely implement a Taylor expansion around y, and to express this idea, we impose the following further conditions on those coefficients. As in a Taylor series, we demand that these coefficients depend only polynomially on the Riemannian normal coordinates of x1 relative to y (and are thus in particular smooth), and that (i)

(i)

C( j) (x; x) = δ( j) id(i) (x),

(14)

where id(i) (x) is the identity map in the fiber over x of the vector bundle V (i). Since a Taylor expansion of an operator at x1 around another point y involves the derivatives of the operators considered at y, and because derivatives tend to increase the dimension, (i) we further demand that C( j) = 0 if dim( j) < dim(i). Finally, if we Taylor expand a quantity at x1 successively around a second point x2 , and then a third point x3 , this should be equivalent to expanding it in one stroke around the third point. Thus, we require that we have (i) (i) (k) C(k) (x1 ; x2 )C( j) (x2 ; x3 ), (15) C( j) (x1 ; x3 ) = (k)

where we note that the sum is only over the (finitely many) field labels k such that dim(k) ≤ dim( j). (C3) Compatibility with . This relation encodes the fact that the underlying theory will have an operation analogous to the hermitian adjoint of a linear operator. The requirement is (i )...(i 1 )

C((ij)1 )...(in ) (x1 , . . . , xn , y) ∼ C( jn ) where π is the permutation

◦ π(x1 , . . . , xn , y),

1 2 ... n n + 1 . π= n n − 1 ... 1 n + 1

(16)

(17)

96


(C4) Commutativity/Anti-Commutativity. Let σ be the permutation 1 ... k k + 1 ... n + 1 σ = , 1 ... k + 1 k ... n + 1

(18)

and let F(i) be the Bose/Fermi parity of φ (i) . Then we have C((σj)i1 )...(σ in ) ◦ σ (x1 , . . . , xn , y) = −(−1) F(ik )F(ik+1 ) C((ij)1 )...(in ) (x1 , . . . , xn , y) (19) whenever xk and xk+1 are spacelike separated (and in the neighborhood of the diagonal in M n+1 , where the OPE coefficients are actually defined). (C5) Scaling Degree. We require that (i 1 )...(i n ) sd C(k) ≤ dim(i 1 ) + · · · + dim(i n ) − dim(k).

(20)

(C6) Asymptotic Positivity. Let i ∈ I be any given index. Then, for D ≥ 3 we postulate that dim(i) ≥ 0, and that dim(i) = 0 if and only if i = 1. Note that, because we are taking the supremum over all spacetimes in Eq. (10), our requirement that dim(i) > 0 (i)(i ) for i = 1 does not imply that the scaling degree of C(1) for i = 1 is positive for all spacetimes, since the coefficient may e.g. “accidentally” happen to have a lower scaling degree for certain spacetimes of high symmetry, as happens for certain supersymmetric theories on Minkowski spacetime. (i)(i ) On a spacetime M where the scaling degree of C(1) is dim(i), we know that if we scale the arguments of this distribution together by a factor of λ, and multiply by a power of λ less than 2dim(i), then the resulting family of distributions cannot be bounded as λ → 0. For our applications below, it is convenient to have a slightly stronger property, which we now explain. Let X a be a vector field on M locally defined near y such that ∇a X b = −δab at y. Let t be the flow of this field, which scales points by a factor of e−t relative to y along the flow lines of X a . If f is a compactly supported test section in V (i), we set f λ = λ−D ∗log λ f . This family of test sections becomes more and more sharply peaked at y as λ → 0. We postulate that, for any δ > 0 and any X a as above, there exists an f such that

2dim(i)−δ

(i)(i )

¯ (21) lim λ C(1) (x1 , x2 , y) f λ (x1 ) f λ (x2 ) dµ1 dµ2

= ∞, λ→0

M×M

uniformly in y in some neighborhood. This statement is slightly stronger than the statement that the scaling degree of our distribution is dim(i) on M, since the latter would only imply that the rescaled distributions under the limit sign in Eq. (21) contain a subsequence that is unbounded in λ for some test section F(x1 , x2 ) in V (i) × V (i ), not necessarily of the form f (x1 ) f¯(x2 ). The reason for the terminology “asymptotic positivity axiom” arises from Lemma 2 below. An alternative essentially equivalent formulation of this condition, which is related to “quantum inequalities”, is given in Appendix B. In D = 2 spacetime dimensions, the above form of the asymptotic positivity condition is in general too restrictive. The reason is that in D = 2, there are usually many fields φ (i) of dimension dim(i) = 0 different from the identity operator. For example,


97

for a free Klein-Gordon field, the basic field ϕ and all its Wick-powers have vanishing dimension—in fact, their OPE-coefficients have a logarithmic scaling behavior. A possible way to deal with this example would be to consider only composite fields containing derivatives, as this subspace of fields is closed under the OPE. For this subspace of fields, the asymptotic positivity condition would then hold as stated. Another possibility is to introduce a suitably refined measure of the degree of divergence of the OPE coefficients also taking into account logarithms. Such a concept would clearly be sensible for free or conformal field theories in D = 2, and it would also be adequate in perturbation theory (to arbitrary but finite orders). A suitable refinement of the above asymptotic positivity condition could then be defined, and all proofs given in the remainder of this paper would presumably still hold true, with minor modifications. For simplicity, however, we will not discuss this issue further in this paper, and we will stick with the asymptotic positivity condition in the above form. (C7) Spectrum condition. The spectrum condition roughly says that the singularities of a field product ought to be of “positive frequency type,” and is completely analogous the condition imposed on states that we will impose below: We demand that, near the diagonal, the wave front set (see Appendix A) of the OPE coefficient satisfies (i )...(i n )

WF(C( j)1

) ⊂ n (M) × Z ∗ M,

(22)

where the last factor Z ∗ M is the zero section of T ∗ M and corresponds to the reference point y in the OPE, and where the set n (M) ⊂ T ∗ M n \{0} is defined as follows. Consider embedded graphs G( x , y, p) ∈ Gm,n in the spacetime manifold M which have the following properties. Each graph G has n so-called “external vertices”, x1 , . . . , xn ∈ M, and m so-called “internal” or “interaction vertices” y1 , . . . , ym ∈ M. These vertices are of arbitrary valence, and are joined by edges, e, which are null-geodesic curves γe : (0, 1) → M. It is assumed that an ordering of the vertices is defined, and that the ordering among the external vertices is x1 < · · · < xn , while the ordering of the remaining interaction vertices is unconstrained. If e is an edge joining two vertices, then s(e) (the source) and t (e) (the target) are the two vertices γe (0) and γe (1), where the curve is oriented in such a way that it starts at the smaller vertex relative to the fixed vertex ordering. Each edge carries a future directed, tangent parallel covector field, pe , meaning that ∇γ˙e pe = 0, and pe ∈ ∂ V + . With this notation set up, we define x , y, p) ∈ Gm,n n,m (M, g) = (x1 , k1 ; . . . ; xn , kn ) ∈ T ∗ M n \{0}|∃ decorated graph G( such that yi ∈ J + ({x1 , . . . , xn }) ∩ J − ({x1 , . . . , xn }) for all 1 ≤ i ≤ m, pe − pe for all xi and such that ki = e:t (e)=xi

e:s(e)=xi

such that 0 =

e:s(e)=yi

pe −

pe for all yi ,

(23)

e:t (e)=yi

where J ± (U ) is the causal future resp. past of a set U ⊂ M, defined as the set of points that can be reached from U via a future resp. past directed causal curve. We set m,n . (24) n = m≥0

98


S0

S1

S3

S2

S4

S6

S5

Fig. 1. T = {S0 , S1 , . . . , S6 }, S0 = {1, 2, 3, 4}, S1 = {1, 2}, S2 = {3, 4}, S3 = {1}, S4 = {2}, S5 = {3}, S6 = {4}

Note in particular that the microlocal spectrum condition implies that the dependence of our OPE coefficients (6) on the reference point y is smooth. When the spacetime is real analytic, we require a similar condition for the “analytic” wave front set [22] WF A of the OPE coefficient. Our formulation of the microlocal spectrum condition is a weaker condition than that previously proposed in [5], based on earlier work of [24]. The microlocal spectrum condition of [5] is satisfied by the correlation functions of suitable Hadamard states in linear field theory, but need not hold even perturbatively for interacting fields. In essence, our formulation allows for the presence of interaction vertices, thus weakening the condition relative to the free field case. Our condition can be shown to hold for perturbative interacting fields [15]. (C8) Associativity. Following [15], a notion of associativity is formulated by considering configurations (x1 , . . . , xn ) of points in M n (where n > 2) approaching a point y ∈ M at different rates. For example, if we have 3 points (x1 , x2 , x3 ), we may consider all points coming close to each other at the same rate, i.e., assume that their mutual distances are of order ε, where ε → 0. Alternatively, we may consider a situation in which, say, x1 , x2 approach each other very fast, say, at rate ε2 , while x3 approaches x1 , x2 at a slower rate, say at rate ε. The first situation corresponds, intuitively, to the process of performing the OPE of a triple product of operators “at once”, while the second situation corresponds to first performing an OPE in the factors corresponding to x1 , x2 , and then successively performing a second OPE between the resulting fields and the third field situated at x3 . Obviously, for an arbitrary number n of points, there are many different possibilities in which configurations can come close. We classify the different possibilities in terms of “merger trees,” T. Each merger tree will give rise to a separate associativity condition. For this, one constructs curves in M n parametrized by ε, which are in M0n (the space M n minus all its diagonals) for ε > 0, and which tend to a point on the diagonal as ε → 0. These curves are labeled by trees T that characterize the subsequent mergers of the points in the configuration as ε → 0. A convenient way to formally describe a tree T (or more generally, the disjoint union of trees, a “forest”) is by a nested set T = {S1 , . . . , Sk } of subsets Si ⊂ {1, . . . , n}. “Nested” means that two sets are either disjoint, or one is a proper subset of the other. We agree that the sets {1}, . . . , {n} are always contained in the tree (or forest). Each set Si in T represents a node of a tree, i.e.,


root

y

99

S0

εv1

εv2

S1 ε2 v3

x1 ε

S2 ε2 v4 ε2 v6

x2 ε

ε2 v5

x4 ε

x3 ε

Fig. 2. T = {S0 , S1 , . . . , S6 }, x1 (ε) = εv1 + ε 2 v3 , x2 (ε) = εv1 + ε 2 v4 , x3 (ε) = εv2 + ε 2 v5 , x4 (ε) = εv2 + ε 2 v6

the set of vertices Vert(T) is given by the sets in T, and Si ⊂ S j means that the node corresponding to Si can be reached by moving downward from the node represented by S j . The root(s) of the tree(s) correspond to the maximal elements, i.e., the sets that are not subsets of any other set. If the set {1, . . . , n} ∈ T, then there is in fact only one tree, while if there are several maximal elements, then there are several trees in the forest, each maximal element corresponding to the root of the respective tree. The leaves correspond to the sets {1}, . . . , {n}, i.e., the minimal elements. For example for a configuration of n = 4 points, a tree might look like in the following figure, and the corresponding nested set of subsets is also given. In the following, we will consider only T with a single root. The desired curves x(ε) tending to the diagonal are associated with T and are constructed as follows. First, we construct Riemannian normal coordinates around the reference point y, so that each point in a convex normal neighborhood of y may be identified with a tangent vector v ∈ Ty M. We then choose a tetrad and further identify Ty M ∼ = R D , so that v is in fact D viewed as an element in R . With each set S ∈ T, we now associate a vector v S ∈ R D , which we collect in a tuple v = (v S1 , . . . , v Sr ) ∈ (R D )|T| , T = {S1 , . . . , Sr },

(25)

and we agree that v{1,...,n} = 0, and where |T| is the number of nodes of the tree, i.e. the number of elements of the set T. For ε > 0, we define a mapping ψT (ε) : (R D )|T| → (R D )|T| , (v S1 , . . . , v Sr ) → (x S1 (ε), . . . , x Sr (ε))

(26)

by the formula x S (ε) =

εdepth(S ) v S ,

(27)

S :S S

where depth(S ) is defined as the number of nodes that connect S with the root of the tree T. For ε sufficiently small, and v in a ball B1 (0)|T| , the vectors x S (ε) ∈ R D may be identified with points in M via the exponential map. If the vectors v S satisfy the condition that, v S = v S for any S , S that are connected to a common S by an edge, then the vector (x{1} (ε), . . . , x{n} (ε)) ∈ M n does not lie on any of the diagonals, i.e., any pair of entries are distinct from each other. Its value as ε → 0 approaches the diagonal of M n . The i th point in the configuration x{i} (ε) is obtained starting from y

100


by following the branches of the tree towards the i th leaf, moving along the first edge by an amount ε in the direction of the corresponding v S , then the second by an amount ε2 in the direction of the corresponding v S , and so forth, until the i th leaf is reached. The curve (x{1} (ε), . . . , x{n} (ε)) ∈ M n thus represents a configuration of points which merge hierarchically according to the structure of the tree T, as ε → 0. That is, the outermost branches of the tree merge at the highest order in ε, i.e., at rate εdepth of branch , then the next level at a lower order, and so forth, while the branches closest to the root merge at the slowest rate, ε. The following figure illustrates our definition. Thus, the points are scaled towards the diagonal of M n , even though possibly at different speeds, and the limiting element as ε → 0 is the element (y, . . . , y) on the diagonal6 . Using the maps ψT (ε) we can define an asymptotic equivalence relation ∼T,δ for distributions u defined on M |T| . For points within a convex normal neighborhood, and sufficiently small ε > 0, we can define the pull-back u ◦ ψT (ε). This may be viewed as a distribution in the variables v S ∈ R D , S ∈ T. We now define u ∼T,δ 0

:⇐⇒

lim ε−δ u ◦ ψT (ε) = 0 for δ > 0,

ε→0+

(29)

in the sense of distributions defined on a neighborhood of the origin in (R D )|T| . We write u ≈ 0 if u ∼T,δ 0 for all T and all δ. The condition that u ≈ 0 is stronger than the previously defined condition u ∼ 0 [see Eq. (11)], which corresponds to the requirement that u ∼T,δ 0 for all δ only for the trivial tree T = {{1}, . . . {n}, {1, . . . , n}}. We can now state the requirement of associativity. Recall that if T is a tree with n leaves, then ψT (ε) gives a curve in the configuration space of n points in M which represents the process of a subsequent hierarchical merger of the points according to the structure of the tree. If a subset of points in (x{1} (ε), . . . , x{n} (ε)) merges first, then one intuitively expects that one should be able to perform the OPE in those points first, and then subsequently perform OPE’s of the other points in the hierarchical order represented by the tree. We will impose this as the associativity requirement. For example, if we have 4 points, and the tree corresponds to the nested set of subsets T = {{1, 2}, {3, 4}, {1, 2, 3, 4}} as in the above figure, the pairs of points x{1} (ε), x{2} (ε) respectively x{3} (ε), x{4} (ε) approach each other at order ε2 , while the two groups then approach each other at a slower rate ε. We postulate that7 (i )(i )(i 3 )(i 4 )

C(i51) 2 i 6 ,i 7

(x1 , x2 , x3 , x4 ; y) ∼T,δ

(i )(i 2 )

C(i61)

(i )(i 4 )

(x1 , x2 ; x6 ) C(i73)

(i )(i 7 )

(x3 , x4 ; x7 ) C(i56)

(x6 , x7 ; y),

(30)

where the sums are finite but carried out to sufficiently large order (depending on δ > 0). For the same product of operators, consider alternatively the tree T = {{1, 2, 3}, 6 However, if the vector (x (ε), . . . , x (ε)) ∈ M n is alternatively viewed as an element of the “Ful{1} {n} ton-MacPherson compactification” Mcn of the configuration space M n , then its limiting value may be viewed alternatively as lying in the boundary ∂ Mcn of the compactification, and the vectors v may be viewed as defining a coordinate system of that boundary, which thereby has the structure of a stratifold ∂ Mcn ∼ M[T], (28) = T

with each face T corresponding to a lower dimensional subspace associated with a given merger tree [1]. 7 Here, the distribution on the left side is viewed as a distribution in x , . . . , x , y with a trivial dependence 1 7 on x5 , x6 , x7 .


101

{1, 2, 3, 4}}. The corresponding associativity relation for the OPE coefficient is now (i )(i 2 )(i 3 )(i 4 )

C(i51)

(x1 , x2 , x3 , x4 ; y) ∼T ,δ

(i )(i )(i ) (i )(i ) C(i61) 2 3 (x1 , x2 , x3 ; x6 ) C(i56) 4 (x6 , x4 ; y) i6

(31) It is important to note, however, that there is in general no simple relation between the right hand sides of Eqs. (30), (31) for different trees T and T . The corresponding relation for arbitrary numbers of points, and arbitrary types of trees is a straightforward generalization of this case, the only challenge being to introduce an appropriate notation to express the subsequent OPE’s. For this, we consider maps i : T → I which associate with every node S ∈ T of the tree an element i S ∈ I , the index set labelling the fields. If S ∈ T, we let S(1), S(2), . . . S(r ) be the branches of this tree, i.e. the nodes connected to S by a single upward edge. With these notations in place, the generalization of Eqs. (30) and (31) for an arbitrary number n of points, and an arbitrary tree T is as follows. Let T be an arbitrary tree on n elements, and let δ > 0 be an arbitrary real number. Then we have8 C((ij)1 )...(in ) (x1 , . . . , xn ;

y) ∼T,δ

i∈Map(T,I )

(i )...(i S(r ) ) x S(1) , . . . , x S(r ) ; x S C(i SS(1) )

,

S∈T

(32) where the sums are over i with the properties that i {1} = i 1 , . . . , i {n} = i n , i {1,...,n} = j.

(33)

The sum over i is finite, with dim(i S ) ≤ , where is a number depending on the tree T and the real number δ. Furthermore, it is understood that x{1} = x1 , . . . , x{n} = xn , and that x{1,...,n} = y. (C9) Analytic and smooth dependence. Due to requirement (C1), the OPE coefficients may be regarded as functionals of the spacetime metric. We require that the distributions C((ij)1 )...(in ) have an analytic dependence upon the spacetime metric. For this, let g (s) be a 1-parameter family of analytic metrics, depending analytically on s ∈ R. Then the corre(i )...(i ) sponding OPE-coefficients C( j)1 n are distributions in x1 , . . . , xn , y that also depend on the parameter s. We demand that the dependence on s is “analytic”. It is technically (i )...(i ) somewhat involved to define what one precisely means by this, because C( j)1 n itself is not analytic, but instead a distribution in the spacetime points. The appropriate way to make this definition rigorous was provided in [16,20]. Similarly, if the spacetime is only smooth, we require a corresponding smooth variation of the OPE coefficients under smooth variations of the metric. 8 Here, the distribution on the left is regarded as a distribution in x ; S ∈ T, with a trivial dependence on S the x S with S not equal to {1}, . . . , {n}, {1, . . . , n}.

102


3. Construction of the QFT from the OPE Coefficients Now that we have stated in detail all the desired properties of the OPE coefficients, we are ready to give the precise definition of quantum field theory. A quantum field theory in curved spacetime associated with a collection of OPE coefficients satisfying the above properties is the pair {A(M), S(M)} consisting of a *-algebra A(M) and a space of states S(M) on A(M) that is canonically defined by the operator product coefficients, C(M), for any choice of the background spacetime structure M. The algebra A(M) is defined as follows. To begin, let Free(M) be the free *-algebra generated by all expressions of the form φ (i) ( f ) with i ∈ I , and f a compactly supported test section in the vector bundle V (i) associated with the tensor or spinor character of φ (i) . The algebra A(M) is obtained by factoring Free(M) by a set of relations, which are as follows. A1) Identity. We have φ (1) ( f ) =

f dµ · 1.

A2) Linearity. For any complex numbers a1 , a2 , any test sections f 1 , f 2 , and any field φ (i) , we have φ (i) (a1 f 1 + a2 f 2 ) = a1 φ (i) ( f 1 ) + a2 φ (i) ( f 2 ). The linearity condition might be viewed as saying that, informally, φ (i) (x) f (x)dµ (34) φ (i) ( f ) = M

is a pointlike field that is averaged against a smooth weighting function. We shall often use the informal pointlike fields as a notational device, with the understanding that all identities are supposed to be valid after formally smearing with a test function. A3) Star operation. For any field φ (i) , and any test section f ∈ Sect 0 (V (i)), let f¯ ∈ Sect 0 (V¯ (i)), be the conjugate test section. Then we require that [φ (i) ( f )]∗ = φ (i ) ( f¯).

(35)

A4) Relations arising from the OPE. Let K ⊂ I be a subset of the index set, and let c(i) , i ∈ K be scalar valued differential operators [i.e., differential operators taking a section V (i) to a scalar function on M], such that (i )( j) sd < 0. (36) (c¯(i ) ⊗ c( j) )C(1) i, j∈K

Then we impose the relation 0=

c(i) φ (i) ( f )

(37)

i∈K

for all f ∈ C0∞ (M). Here the differential operators act in the sense of distributions, i.e., c(i) φ (i) ( f ) by definition means φ (i) (c(i) t f ), where t means the dual of a differential operator defined with respect to the volume element e associated with the metric. This relation can be intuitively understood as follows. Let O( f ) be the smeared quantum field defined by the right side of Eq. (37). If we consider the OPE of the quantity O(x1 )O(x2 )∗ , then the term involving the identity operator will have a negative scaling degree due to (36). Above, in the asymptotic positivity requirement, we demanded that


103

any one of our fields φ (i) except the identity field must have a non-negative scaling degree. It is natural to extend this postulate also to linear combinations of such fields such as O. In other words, O should vanish if its scaling degree is negative. This is what our requirement states. The above requirement serves to eliminate any redundancies in the field content arising e.g. from initially viewing, say, a field ϕ and ϕ as independent fields, or from initially specifying a set of linearly dependent fields. More nontrivially, this requirement should also serve to impose field equations in A(M). For example, in λϕ 4 -theory, we expect that a field equation of the form ϕ − m 2 ϕ − λϕ 3 = 0 should hold, where ϕ 3 is a composite field in the theory that should appear in the operator product expansion of three factors of ϕ. If such a field equation holds, then clearly ϕ − m 2 ϕ − λϕ 3 should have a trivial OPE with itself. In particular, the OPE coefficient multiplying the identity operator in this expansion should have arbitrarily low scaling degree. Thus, in this example, if we take c(1) = − m 2 , c(2) = −λ, and φ (1) = ϕ, φ (2) = ϕ 3 , then Eq. (36) should hold. Our requirement effectively demands that field equations hold if and only if they are implied by the OPE condition Eq. (36). A5) (Anti-)commutation relations. Let φ (i1 ) and φ (i2 ) be fields, and let f 1 and f 2 be test sections corresponding to their respective spinor or tensor character, whose supports are assumed to be spacelike separated. Then the relation φ (i1 ) ( f 1 )φ (i2 ) ( f 2 ) + (−1) F(i1 )F(i2 ) φ (i2 ) ( f 2 )φ (i1 ) ( f 1 ) = 0

(38)

holds in A(M). Having defined the algebra A(M), we next define the state space S(M) to consist of all those linear functionals . ω : A(M) → C with the following properties: S1) Positivity. The functional should be of positive type, meaning that A∗ Aω ≥ 0 for each A ∈ A(M). Physically, Aω is interpreted as the expectation value of the observable A in ω. S2) OPE. The operator product expansion holds as an asymptotic relation. By this we mean more precisely the following. Let φ (i1 ) , . . . , φ (in ) be any collection of fields, let δ > 0 be arbitrary but fixed, and let T be any merger tree as described in the associativity condition. Let ∼δ,T be the associated asymptotic equality relation between distributions of n spacetime points that are defined in a neighborhood of the diagonal which was defined in the associativity condition [see Eq. (29]. Then we require that (i )...(i ) φ (i1 ) (x1 ) · · · φ (in ) (xn ) ∼δ,T C( j)1 n (x1 , . . . , xn ; y) φ ( j) (y) , (39) ω

j

ω

where the sum is carried out over all j such that dim( j) ≤ , where is a number depending upon the tree T, and the specified accuracy, δ. S3) Spectrum condition. We have WF φ (i1 ) (x1 ) . . . φ (in ) (xn )ω ⊂ n (M),

(40)

where the set n (M) ⊂ T ∗ M n \{0} was defined above in Eq. (23). By definition, 1 (M) = ∅, so the microlocal spectrum condition says in particular that φ (i) (x)ω is smooth in x.

104


As part of our definition of a quantum field theory, we make the final requirement that there is at least one state, i.e., S(M) = ∅ for all M.

(41)

If the state space were empty, then this is a sign that the OPE is inconsistent, and does not define a physically acceptable quantum field theory. Remarks. (1) The OPE coefficients enter the construction of the algebra A(M) only via condition (A4). However, they provide a strong restriction on the state space S(M) via condition (S2). (2) If ω ∈ S(M) and A ∈ A(M), then ω(A∗ · A) is a positive linear functional on A(M) [i.e., satisfying (S1)] which can also be shown to satisfy (S3). This functional can be identified with a vector state in the Hilbert space representation of A(M) obtained by applying the GNS construction to ω, and is therefore in the domain of all smeared field operators. It is natural to expect that, in a reasonable quantum field theory, the OPE [i.e. (S2)] should hold in such states as well, but this does not appear to follow straightforwardly within our axiomatic setting. (3) There are some apparent redundancies in our assumptions in that commutativity/anti-commutativity conditions have been imposed separately on the OPE coefficients and the algebra (see Conditions (C4) and (A5)), and microlocal spectrum conditions have been imposed separately on the OPE coefficients and the states (see Conditions (C7) and (S3)). It is possible that our assumptions could be reformulated in such a way as to eliminate these redundancies, e.g., it is possible that Condition (A5) might follow from Condition (C4) with perhaps somewhat stronger assumptions about states. However, we shall not pursue these possibilities here. The construction of the pair {A(M), S(M)} obviously depends only upon the data entering that construction, namely the set of all operator product coefficients C(M), as well as the assignments i → V (i) and i → F(i) of the index set enumerating the fields with tensor/spinor character, and with Bose/Fermi character. Thus, any transformation on field space preserving the OPE and the Bose/Fermi character will evidently give rise to a corresponding isomorphism between the algebras, and a corresponding map between the state spaces. We now give a more formal statement of this obvious fact, and then point out some applications. Let ψ : M → M , not necessarily an isometry at this stage. Furthermore, for each ( j) pair of indices (i, j) ∈ I × I , assume that we are given a C-linear bundle map z (i) from V (i), viewed as a bundle over M, to V ( j), viewed as a bundle over M . We make the following requirements about the pair (ψ, z): First, we require that the following diagram commutes: ( j)

z (i)

V (i) −−−−→ V ( j) ⏐ ⏐ ⏐π , ⏐ πM M

(42)

ψ

M −−−−→ M where π M respectively π M are the bundle projections associated with the vector bundles V (i) and V ( j) over M respectively M that characterize the spinor/tensor character of the field labelled by i and j.


105

(i)

We furthermore require that z ( j) = 0 unless dim( j) ≤ dim(i), and unless F(i) = F( j). Recalling that V (i ) is required to be equal to the hermitian conjugate bundle V (i), and denoting by con j the operation of conjugation mapping between these bundles, we also require as a consistency condition that ( j )

( j)

conjM ◦ z (i) = z (i ) ◦ conjM ,

(43)

for all indices i, j ∈ I . We say that a collection of OPE coefficients C(M) on M = (M, g, T, e) and a collection C (M ) on M = (M , g , T , e ) are equivalent under (z, ψ) if ( j) ( j )...( jn ) (i ) (i ) (i )...(i ) (z ( j11 ) × · · · × z ( jnn ) × (z v −1 )(k) ) C(k)1 [M] ∼ C( j)1 n [M ] ◦ ψ, (44) j1 ,..., jn ,k

see Eq. (11). Note that the sums are finite, since we are assuming that the number of ( j) fields less than a fixed dimension is finite, and note that (z v −1 )(k) denotes the inverse matrix of dual maps between the dual bundles. As before, let A(M) be the algebra and S(M) the state space defined from C(M), and similarly let A (M ) and S (M ) be defined from C (M ). By simply going through the definitions it is then clear that the following (almost trivial) lemma holds. Lemma 1. Under the consistency conditions (43) and (44), and assuming that ψ preserves all background structure (i.e., ψ is an isometric embedding preserving the causality relations, orientations, and spin structures) the map α(ψ,z) : A(M) → A (M ) ( j) (i) (i) α(ψ,z) : φM ( f ) → φM [(z v )( j) f ◦ ψ −1 ], (45) j v defines a linear *-homomorphism. The dual map α(ψ,z) between the corresponding state spaces defines a map S (M ) → S(M).

Another way of stating this result is to view the pairs (z, ψ) as described above as morphisms in the category whose objects are the OPE-coefficient systems C(M). The above lemma then says that the constructions of S(M) and of A(M) from C(M) are functorial in nature. We now discuss some applications of the lemma: Application 1. Consider the case where C(M) = C (M) for all M, z ((i)j) = δ((i)j) ψ∗ , ψ : M → M is an isometric embedding preserving orientations and any other background structure, and ψ∗ is the natural bundle map (push-forward) associated with ψ. Then (43) obviously holds, while (44) holds because of the locality and covariance property of the OPE coefficients. The map α(ψ,z) : A(M) → A(M ), whose existence is guaranteed by the lemma, then corresponds to the map χρ discussed in the Introduction. In particular, the assignment Background Structures → Algebras, M → A(M)

(46)

is functorial, in the sense that if ρ is an arrow in the category of background structures— i.e., an orientation and causality preserving isometric embedding from one spacetime

106


into another—then χρ is the corresponding arrow in the category of *-algebras—i.e., an injective *-homomorphism. Functoriality means that the assignment Isometric Embeddings → Algebra Homomorphisms, ρ → χρ

(47)

respects composition of arrows in the respective categories. Thus, in the terminology of the introduction, the assumptions of our framework define a local, covariant quantum field theory, and all fields φ (i) are local, covariant quantum fields. Furthermore, A : M → A(M) is a functor in the sense of [7]. Application 2. As the second application, consider an internal symmetry, i.e., consider the case that M = M , C(M) = C (M), ψ = id. Then α(id,z) acts upon A(M) as a *-automorphism, i.e., an internal symmetry. More generally, there could be an entire group G. In this case, we get an action of G on A(M) by ∗-automorphisms α(id,z) satisfying the composition law α(id,z) ◦ α(id,z ) = α(id,z◦z ) . Here the composition is defined as (i) ( j) z ( j) ◦ z (k) . (48) (z ◦ z )(i) (k) = j

The sum over j is finite because we are assuming that there are only a finite number of indices up to a given dimension. Application 3. Field redefinitions are covered by the case M = M , ψ = id. The lemma then states that there is a *-homomorphism, A(M) → A (M ). In other words, if we (i) change the definition of the field by a “mixing matrix” z ( j) and make a corresponding change of the OPE coefficients as in Eq. (44), then we obtain an equivalent theory. In renormalized perturbation theory, such changes arise naturally from changes in the renormalization conditions. Another simple consequence of our axioms is the following lemma. As above in (C6), let X a be a vector field on M locally defined near y such that ∇a X b = −δab at y. Let t be the flow of this field9 , which scales points relative to y by a factor of e−t . If f is a compactly supported test section in V (i), we set f λ = λ−D ∗log λ f for λ > 0. Lemma 2. For i = 1, there exists an M and test section f such that (i)(i ) lim λ2dim(i)−δ C(1) (x1 , x2 , y) f λ (x1 ) f¯λ (x2 ) dµ1 dµ2 = +∞, λ→0

(49)

M×M

for all sufficiently small δ > 0. Proof. Let ω be an arbitrary state. Then we have φ (i) ( f λ )φ (i ) ( f¯λ )ω ≥ 0, from the star axiom (C3), and the positivity of any state. Let M be such that the scaling degree (i)(i ) of C(1) equals 2dim(i). By the scaling degree and asymptotic positivity axioms (C5) and (C6), for sufficiently small δ > 0, the quantity 2dim(i) − δ is bigger than the scaling (i)(i ) degree of C( j) for any j = 1. Hence using Eq. (39) and 1ω = 1, we have (i)(i ) 2dim(i)−δ (i) (i ) ¯ φ ( f λ )φ ( f λ ) − C ( f λ , f¯λ ) = 0. (50) lim λ

λ→0

ω

(1)

9 It follows that we can write −1 log λ (x) = y + λ (x − y) in a suitable coordinate system covering y.


107

By Axiom (C6), we can choose f so that the second term tends to ∞ in absolute value (i)(i ) as λ → 0. However, the first term is always non-negative. Therefore, C(1) ( f λ , f¯λ ) → +∞. 4. Normal (Anti-)Commutation Relations In our axioms, we assumed that every field φ (i) was either a Bose or Fermi field, i.e., that there was a consistent assignment i → F(i) ∈ Z2 such that (19) holds. A priori, there is no relation between F(i) and the Bose/Fermi character of the corresponding hermitian conjugate field, i.e., F(i ). We will now prove that, in fact, F(i) = F(i ) as a consequence of our axioms. Theorem 1. We have “normal (anti-)commutation relations,” in the sense that F(i) = F(i )

(51)

for all i ∈ I . Proof. Let f and h be compactly supported test sections with support in a convex normal neighborhood of a point y ∈ M. Let ω be a quantum state, i.e., a positive normalized linear functional A(M) → C. Using (35), we see that positivity immediately implies that ¯ (i ) ( f¯) ≥ 0. (52) φ (i) ( f )φ (i) (h)φ (i ) (h)φ ω

Assume now that the supports of f, h are spacelike separated. Then, using the (anti-) commutation relations Eq. (38), it follows that ¯ (i) (h) ≥ 0, p φ (i) ( f )φ (i ) ( f¯)φ (i ) (h)φ (53) ω

F(i)F(i )+F(i )2

. Clearly, if we could show that the expectation value where p = (−1) in this expression were positive for some test sections, f, h, in some spacetime, then it would follow that p = +1, i.e. F(i)F(i ) + F(i )2 = 0 mod 2,

(54)

and by reversing the roles of i and i , it would also follow that F(i)F(i ) + F(i)2 = 0 mod 2,

(55)

from which the statement F(i) = F(i ) modulo 2 would follow. Clearly, it suffices to show that the expectation value is asymptotically positive for a 1-parameter family of test sections f λ , h λ whose supports are scaled towards y ∈ M as λ → 0. To show this, we consider the particular merger tree T of Fig. 1, and the corresponding associativity condition. This tree corresponds to the scaling map T (ε) : x(1) → x(ε), with x1 (ε) = Exp y εv1 + ε2 v3 , x2 (ε) = Exp y εv1 + ε2 v4 ,

108


x3 (ε) = Exp y εv2 + ε2 v5 , x4 (ε) = Exp y εv2 + ε2 v6 , x5 (ε) = Exp y εv1 , x6 (ε) = Exp y εv2 ,

(56)

x7 (ε) = y. The corresponding associativity condition together with (S2) yields −δ (i) (i ) (i ) (i) φ (x1 (ε))φ (x2 (ε))φ (x3 (ε))φ (x4 (ε)) lim ε ε→0

−

j1 , j2 , j3

ω (i)(i ) (i )(i) C( j1 ) (x1 (ε), x2 (ε); x5 (ε)) C( j2 ) (x3 (ε), x4 (ε); x6 (ε))

( j )( j ) ×C( j31) 2 (x5 (ε), x6 (ε);

y) φ

( j3 )

= 0.

(y)

(57)

ω

This is to be understood in the sense of distributions in v1 , . . . , v6 . The sums go over all indices with dim( jk ) ≤ , where depends on δ. We now use Axioms (C5) and (C6) to analyze the scaling of the individual terms under the sum. It follows that

) (i )(i) lim εα C((i)(i j1 ) (x 1 (ε), x 2 (ε); x 5 (ε)) C ( j2 ) (x 3 (ε), x 4 (ε);

ε→0

( j )( j2 )

x6 (ε)) C( j31)

(58)

(x5 (ε), x6 (ε); y) = 0

if α > 8dim(i) − dim( j1 ) − dim( j2 ) − dim( j3 ) in the sense of distributions. Thus, the term under the sum with the potentially most singular behavior as ε → 0 is the one where dim( j1 ) = dim( j2 ) = dim( j3 ) is minimal, i.e. equal to 0, by Axiom (C6). If these dimensions vanish, then by Axiom (C6), we have jk = 1. Because 1ω = 1, and because the OPE coefficients involving only identity operators are equal to 1 by the identity axiom, we have 8dim(i)+δ (i) (i ) (i ) (i) φ (x1 (ε))φ (x2 (ε))φ (x3 (ε))φ (x4 (ε)) lim ε ε→0

(i)(i ) (i )(i) −C(1) (x1 (ε), x2 (ε), x5 (ε)) C(1) (x3 (ε), x4 (ε), x6 (ε))

ω

= 0,

(59)

for some δ > 0. We now integrate this expression against the test section f (v3 ) f¯(v4 ) ¯ 5 )h(v6 ), where f, h are of compact support, and change integration variables10 . Then h(v we get for the terms under the limit sign (i)(i ) (i )(i) (i) (i ) ¯ (i ) ¯ (i) φ ( f ε )φ ( f ε )φ (h ε )φ (h ε ) − C(1) ( f ε , f¯ε )C(1) (h¯ ε , h ε ). (60) ω

10 We should also integrate against a test function in v , v . But the result (60) is already smooth in these 1 2 variables, so we can omit this smearing.


109

Here, we have defined f ε (x) = ε−2D f ◦ α1 (ε, x) and h ε (x) = ε−2D h ◦ α2 (ε, x) with αi (ε, . ) : M → R D are the maps that are defined in a sufficiently small neighborhood of y by α1 (ε, x) = ε−1 v1 + ε−2 Exp−1 y (x), α2 (ε, x) = ε

−1

v2 + ε

−2

Exp−1 y (x).

(61) (62)

(i)(i ) Finally, we use the Lemma 2 to conclude that there exist f, h such that both C(1) (i)(i ) ( f ε , f¯ε ) → +∞ and C(1) (h ε , h¯ ε ) → +∞ for some spacetime and some subsequence of ε → 0. In view of Eq. (60), it follows that

lim φ (i) ( f ε )φ (i ) ( f¯ε )φ (i ) (h¯ ε )φ (i) (h ε ) = +∞,

ε→0

ω

(63)

so the expectation value (53) is positive for the choice of test sections f, h given by f ε , h ε , see Eq. (61), for sufficiently small ε. These test sections will have spacelike separated support as long as Exp y v1 and Exp y v2 are spacelike, which we may assume to be the case. 5. The Spin-statistics Theorem and the PCT-theorem In this section, we prove that appropriate versions of the spin-statistics theorem and the PCT theorem hold in curved spacetime under our axiom scheme. We explicitly discuss the case when the spacetime dimension is even, D = 2m and discuss the case of odd dimensions briefly in Remark 2 below the PCT theorem. The key ingredient in both proofs is a relation, proven in [16], between the OPE coefficients C(M) on the background structure M = (M, g, T, e), and the OPE coefficients C(M) on the background structure M = (M, g, −T, e)

(64)

consisting of the same manifold M, the same metric g, the same orientation e, but the opposite time orientation T . For even spacetime dimensions D = 2m, this relation is11 : C((ij)1 )...(in ) [M]

∼

(i )...(i ) C( j1 ) n [M] ·

i−F( j) (−1)−U ( j) nk=1 i F(ik ) (−1)U (ik ) m even, n −F( j)+U ( j)−P( j) F(i )−U (i )+P(i ) k k k i m odd, k=1 i (65)

√ where i = −1. Here, we recall that with each quantum field φ (i) there is associated a bundle V (i) over M corresponding to the tensor or spinor character of the quantum field. In even spacetime dimensions D = 2m, such a bundle V (i) is a tensor product ⊗U (i)

V (i) = S−

⊗P(i)

⊗ S+

,

(66)

11 Note that the “bar” symbol is referring to the P T -reversed background structure in the term on the left side, while it means hermitian conjugation on the right side.

110


where the first factor corresponds to the U (i) “unprimed-” and the second to the P(i) “primed” spinor indices. More precisely, the bundles S± are defined as the ±1 eigenspaces of the chirality operator12 =

1 (−1)(m−1)(2m−1)/2 ea1 a2 ...a D γa1 γa2 · · · γa D , D!

2 = id S ,

(67)

acting on a 2m -dimensional complex vector bundle S over M of “Dirac spinors”. This bundle S corresponds to a fundamental representation of the Clifford algebra (in the tangent bundle) generated by the curved space gamma-matrices γa . There exists a linear isomorphism c : S → S, where S is the conjugate bundle of anti-linear maps S v → C. Owing to the relation = (−1)m−1 c c−1 , it follows that for even m, the bundles S± and S ∓ are isomorphic via c, while for odd m the bundles S± and S ± are isomorphic, and we will hence always identify these bundles. Thus, for even m the roles of primed and unprimed spinor indices [i.e., respective tensor factors in Eq. (66)] are exchanged when passing to the hermitian adjoint φ (i ) of a quantum field φ (i) , while for odd m the roles are not exchanged. We also note that the coefficients on the right side of Eq. (65) are sections in the (tensor product of the) spin bundles VM (i) referring to the time function T associated with M, while the coefficients on the left side are sections in the spinor bundles VM (i ) defined via the opposite time orientation −T associated with M. As explained in [16], there is a natural identification map between these bundles, and this identification map is understood in (65). The proof of (65) makes use of the microlocal, analytical, and causal properties of the OPE coefficients and proceeds via analytic continuation [16]. Since it is the main input in the proofs of both the spin-statistics theorem and the PCT theorem, we now outline, following [16], how (65) is proven within our axiomatic setting. We first consider the case where g is analytic. Let y ∈ M and introduce Riemannian normal coordinates x = (x 0 , . . . , x D−1 ) ∈ R D about y. In this neighborhood of y, consider the 1-parameter family of metrics g (s) for all |s| ≤ 1 defined by g (s) = gµν (sx) d x µ d x ν .

(68)

Note that this family, in effect, interpolates between the given metric g = g (1) and the flat Minkowski metric η = g (0) . We can expand g (s) in a power series in s about s = 0, which takes the form (s) gµν = ηµν +

∞

s n pµνβ1 ...βn−2 (y) x β1 . . . x βn−2 ,

(69)

n=2

where each p is a curvature polynomial p(y) = p[Rµνσρ (y), . . . , ∇(α1 · · · ∇α(n−2) ) Rµνσρ (y)]. It can then be shown, using Axiom (C1), that each OPE coefficient has an asymptotic expansion of the form (i )...(i n )

C( j)1

(x1 , . . . , xn , y) =

∞

(i )...(i n )

qk (y) · (Wk )( j)1

(x1 , . . . , xn ),

(70)

k=0

where qk = (qk )µ1 ...µk is a curvature polynomial of the same general form as the p, and where Wk = (Wk )µ1 ...µk are distributions defined on a neighborhood of 0 in (R D )n , 12 Here, the orientation D-form is normalized so that g a1 b1 . . . g a D b D e a1 ...a D eb1 ...b D = −D!.


111

valued in the tensor product of (R D )⊗k with the spinor representation corresponding to the index structure of the quantum fields in the operator product considered. They transform covariantly under the connected component SpinR (D − 1, 1)0 of the spin group of D-dimensional Minkowski space. Consider now the map ρ defined in a suitable convex normal neighborhood, O, of y by (x 0 , . . . , x D−1 ) → (−x 0 , . . . , −x D−1 ). In Minkowski spacetime, this map would define an isometry which preserves spacetime orientation but reverses time orientation. In a general curved spacetime, this map does not define an isometry. Nevertheless, we may view ρ as a map ρ : (O, g (s) , e, T ) → (O, g (−s) , e, −T ).

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Viewed in this manner, it is easily seen that ρ preserves all background structure, i.e., it is a causality preserving isometry that preserves orientations. Consequently, by the covariance axiom (C1), the relation (65) is equivalent to a corresponding relation between the OPE-coefficients on the spacetimes (M, g (s) , e, T ) and (M, g (−s) , e, T ), i.e., spacetimes with different metrics but the same orientation and time orientation. If one now differentiates this relation m-times with respect to s and puts s = 0 afterwards, then one can prove that (65) is equivalent to the relation Wk (x1 , . . . , xn ) =

∗

(−1) π Wk (−xn , . . . , −x1 ) · k

i−F( j) (−1)−U ( j) nk=1 i F(ik ) (−1)U (ik ) m even, n −F( j)−U ( j)+P( j) F(i )−U (i )+P(i ) k k k m odd, i k=1 i (72)

for all k = 0, 1, 2, . . .. Here, π is the permutation (17), which acts by permuting the implicit spinor/indices associated with the spacetime points xi . We have thus reduced the proof of (65) to the proof of a statement about Minkowski distributions Wk that transform covariantly under SpinR (D −1, 1)0 . To prove it, one next shows that Wk can be analytically continued, and that the analytic continuation transforms covariantly under the connected component of the identity in the complexified spin group SpinC (D −1, 1)0 . For this, one first proves, using the microlocal condition on the OPE-coefficients, that, near 0, the analytic wave front set [22], WF A , of Wk satisfies WF A (Wk ) ⊂ K .

(73)

Here, K is a conic set defined in terms of the Minkowskian metric η and orientation e, T , by

K = (y1 , k1 ; . . . ; yn , kn ) ∈ T ∗ (×n Br )\{0} ∃ pi j ∈ V¯+ , n ≥ j > i ≥ 1 : ki =

j: j>i

pi j −

p ji for all i ,

(74)

j: j

where V¯ + is the closure of the forward light cone V + in Minkowski space (defined with respect to the time orientation T ), V ± = {k ∈ R D | ηµν k µ k ν > 0, k µ ∇µ T > 0}.

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The relation (73) is important because a theorem of [22] now guarantees that Wk is the distributional boundary value Wk (x1 , . . . , xn ) =

B. V.

(y1 ,...,yn )∈K v →0

Wk (x1 + iy1 , . . . , xn + iyn )

(76)

of a holomorphic function Wk (z 1 , . . . , z n ) that is defined in the “half-space” Wk : Br (0)n + iK v → C, some r > 0,

(77)

v where Br (0) is a ball of radius r in R D , where K is the “dual cone” of all covectors D n (y1 , . . . , yn ) ∈ (R ) with the property that ki · yi > 0 for all (k1 , . . . , kn ) ∈ K . Using the “edge of the wedge-theorem” [27], one proves that the holomorphic function Wk (z 1 , . . . , z n ) transforms covariantly under the spin group SpinR (D − 1, 1)0 . As explained in more detail in [16], one can use this in turn to prove the desired relation (72): For D = 2m and m even, we consider the chirality element in Eq. (67) in flat space, which is an element of the connected component of the identity of the complexified spin group SpinC (D − 1, 1)0 . It corresponds to the reflection element ρ : R D → R D , x → −x of the complexified Lorentz group S O(D − 1, 1; C) under the standard covering homomorphism between these groups. This is an immediate consequence of the relation γa −1 = −γa . Using the method of analytic continuation in overlapping patches, it can be shown that Wk may be continued to a single valued analytic function on an extension of the domain in Eq. (77), and it can be shown that this continuation transforms covariantly under . As explained above, acts as +id S+ on each tensor factor corresponding to a primed spinor index, and as −id S− on each tensor factor associated with an unprimed spinor index. Therefore, if we apply the transformation law of Wk under the element , then we obtain relation (72) for complex spacetime arguments, except that the order of the complex spacetime arguments z 1 , . . . , z n is reversed, and except for the factors relating to the Bose-Fermi character of the fields involved. In order to be able to take the limit Im z i → 0 from within K v , we must pass to so-called “Jost points” (z 1 , . . . , z n ) in the extended domain of holomorphicity. For such points, the (anti-)commutativity may be used, effectively allowing to permute the spacetime arguments in Wk in such a way that one can take the limit to real points from within K v as required in Eq. (76) afterwards. When permuting the arguments, we pick up the factors related to the Bose/Fermi character of the fields. For D = 2m and m odd, we consider instead the element i of the connected component of the identity of the complexified spin group SpinC (D − 1, 1)0 . This element again covers the reflection ρ(x) = −x on D-dimensional Minkowski space. It acts as +i id S+ on each tensor factor corresponding to a primed spinor index, and as −i id S− on each tensor factor associated with an unprimed spinor index. Again it can be shown that Wk may be continued analytically to a domain extending that in Eq. (77), and that it transforms covariantly under i on the extended domain. The additional factor of i gives rise to the different factors in Eq. (72) compared to the case when m is even. The rest of the argument is identical to that case. This proves the PCT-relation (65) for analytic spacetimes, and even dimensions D = 2m. The validity of the corresponding relation for smooth spacetimes then follows from the smoothness of the OPE-coefficients under smooth variations of the metric, since any smooth metric can be viewed as the limiting member of a smooth 1-parameter family of metrics g (λ) that are analytic for λ > 0 and smooth for λ = 0. The differences in the statement and proof of the PCT-relation (65) for odd spacetime dimension D are described in Remark 2 below, following the proof of the PCT-Theorem.


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We now are ready to state and prove the spin-statistics theorem within our framework. The statement and proof of this theorem closely parallel the Minkowski spacetime version: Theorem 2 (Spin-Statistics Theorem). If our axioms hold, then the spin statistics relation F(i) = U (i) + P(i) mod 2,

(78)

also holds, i.e. fields with integer spin (= one half the number of primed + unprimed spinor indices) have Bose statistics, while fields of half integer spin have Fermi statistics. Proof. Let i ∈ I , and, as above, we restrict consideration to the even dimensional case (i)(i ) D = 2m. Consider the PCT-relation (65) for the OPE-coefficient C(1) . This condition can be written as i F(i)+F(i ) (−1)U (i )+U (i) m even, (i)(i ) (i )(i) C(1) (x1 , x2 , y)M ∼ C(1) (x2 , x1 ; y)M · F(i)+U (i)−P(i)+F(i )+U (i )−P(i ) i m odd, (79) where we have used the hermitian conjugation axiom, and where we have used that F(1) = 0 since the identity is always a Bose field, by the identity axiom. When m is even, then U (i ) = P(i) because conjugation of a spinor exchanges the number of primed and unprimed indices. Furthermore, F(i) = F(i ) by Theorem 1, so we obtain (i)(i )

C(1)

(i )(i)

(x1 , x2 , y)M ∼ (−1) F(i)+U (i)+P(i) C(1)

(x2 , x1 ; y)M .

(80)

When m is odd, U (i ) = U (i) and P(i ) = P(i), because conjugation of a spinor does not change the number of primed and unprimed spinor indices in that case. Using this, we again obtain the expression Eq. (80) when m is odd. We now smear this expression with the test section f λ (x1 ) f¯λ (x2 ), where f λ (x) = λ−D f [y + λ−1 (x − y)].

(81)

We are taking here an f of compact support in a sufficiently small neighborhood of y covered by some coordinate system, and y + λ−1 (x − y) is computed in this arbitrary coordinate system. Furthermore, by Lemma 2, we may assume that Eq. (49) is satisfied for this choice for all sufficiently small δ > 0. Denote by dim(i) the dimension of the field labeled by i. It then follows from Eq. (80) that (i)(i ) 2dim(i)−δ λ C(1) (x1 , x2 ; y)M f λ (x1 ) f¯λ (x2 ) dµ1 dµ2 (i )(i) − p λ2dim(i)−δ C(1) (x2 , x1 ; y)M f λ (x1 ) f¯λ (x2 ) dµ1 dµ2 → 0 (82) as λ → 0, for all sufficiently small δ > 0 where p = (−1) F(i)+U (i)+P(i) . The first term goes to +∞ by Lemma 2, and hence the second term must be unbounded, too. In fact, arguing as in the proof of Lemma 2 (now for M), the second term goes to − p·∞. The two terms can only cancel for small λ if we have p = +1, meaning that F(i)+ P(i)+U (i) = 0 modulo 2. Hence the spin-statistics relation must hold.

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Next, we state and prove the PCT theorem, the formulation of which is quite different from the Minkowski spacetime version (see the discussion below). Again, we restrict consideration here to D = 2m, and describe the differences occurring in odd dimensions in Remark 2 below: Theorem 3 (PCT-Theorem). Given a background structure M = (M, g, e, T ), we write M = (M, g, e, −T ). In spacetime dimension D = 2m, define the anti-linear map PCT : A(M) → A(M) by θM i F(i) (−1)U (i) when m is even, (i) (i) PCT ∗ θM : φM ( f ) → φ ( f ) · F(i)+U (i)−P(i) (83) M when m is odd. i PCT is an anti-linear *-isomorphism such that the diagram Then θM PCT θM

A(M) −−−−→ A(M) ⏐ ⏐ ⏐χρ χρ ⏐ PCT θM

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A(M ) −−−−→ A(M ) as well as the diagram

PCT v θM

S(M ) −−−−→ S(M ) ⏐ ⏐ ⏐χ v ⏐ χρv ρ

(85)

PCT v θM

S(M) −−−−→ S(M) commute for every isometric, causality and orientation preserving embedding ρ : M → M . Here χρv denotes the dual of the linear map χρ , and θ PCT v denotes the dual of θ PCT . Proof. The proof of this theorem is, in essence, an application of lemma 1. In the notation of lemma 1, we choose M = (M, g, e, T ), we choose M = M, we take ψ = id (i) (i ) and we choose z ( j) = δ( j) ψ( j) where ψ( j) is the natural anti-linear bundle map from VM ( j) to VM ( j ) that is implicit in the formula (83), composed with the multiplication map by i F(i) (−1)U (i) when m is even and by i F(i)+U (i)−P(i) when m is odd. From this definition we then have (−1) F(i)/2+F(i )/2+U (i)−U (i ) · ψ(i ) ◦ conjM m even, conjM ◦ ψ(i) = F(i)+F(i )−U (i)−U (i )+P(i)+P(i ) (86) · ψ(i ) ◦ conjM m odd, i where con jM is the anti-linear map that sends a spinor to the hermitian conjugate spinor on M. Now for even m the number of unprimed spinor indices associated with V (i) is precisely equal to the number of primed indices P(i ) associated with V (i ), because V (i ) is assumed to be equal to V (i). Thus, P(i) = U (i ). Furthermore, we have F(i) = F(i ) by Theorem 1, and F(i) = U (i) + P(i) mod 2 by the spin-statistics theorem. Consequently, when m is even we have shown conjM ◦ ψ(i) = ψ(i ) ◦ conjM .

(87)


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This immediately implies the required compatibility condition (43) required for lemma 1. When m is odd, then U (i) = U (i ) and P(i) = P(i ), and the compatibility condition again holds because of F(i) = F(i ) and the spin-statistics theorem. Thus, we have shown that the first input in lemma 1 holds. The second input is the compatibility of the OPE coefficients on M and M , Eq. (44). That condition is essentially equivalent to the relation (65), except that the latter relation also involves an additional complex conjugation of the OPE-coefficient. However it is immediately seen that this will result only in the following difference in the conclusion of lemma 1: Instead of the linear *-homomorphism as provided by this lemma, we now find that the PCT-map θ PCT defined by Eq. (83) yields an anti-linear *-homomophism. Remarks. (1) The above formulation of the PCT-theorem was suggested in [16]. As noted in [9] the theorem can be stated in the language of functors by saying that the functors M → A(M) and M → A(M) = A(M) are equivalent. (2) In odd spacetime dimensions D = 2m + 1, the chirality operator of Eq. (67) is proportional to the identity in S. Thus, in this case there is no decomposition S = S+ ⊕ S− as in the even dimensional case, and there is consequently no difference between “primed” and “unprimed” spinors. If we denote the number of spinor indices of a quantum field by N (i) (i.e., the bundle associated with the label i of the is V (i) = S ⊗N (i) ), then the factors in formula (65) are now field −F( j)−N ( j) F(i )+N (i k ) . In the proof of this formula, one must now consider the k i i 0 1 map ρ : (x , x , . . . , x D−1 ) → (−x 0 , −x 1 , . . . , +x D−1 ). This corresponds again to a change of time orientation in Minkowski spacetime which leaves the spacetime orientation invariant. The rest of the proof is similar. We now explain the relation of the above formulation of the PCT theorem to the usual PCT theorem in Minkowski spacetime (see e.g. [27]). Changing T → −T while keeping e unchanged is equivalent to changing parity (i.e., the spatial orientation s of a Cauchy surface induced by e and T via dT ∧ s = e) and time (i.e., the time function). Furthermore, the field appearing on the right side of Eq. (83) is usually referred to as the (i) “charge conjugate field” to φM ( f ). Thus our formulation of the PCT theorem asserts that the theory is indeed invariant under simultaneous PCT-reversal in the sense that the theory on M is “the same” as the theory on M with the fields replaced by their charge conjugates. However, note that our PCT theorem relates theories on the two different background structures, M and M. By contrast, the usual PCT theorem in Minkowski spacetime provides a symmetry of the theory defined on a single background structure, namely Minkowski spacetime with a fixed choice of orientation and time orientation. Indeed, the usual formulation of the PCT theorem in Minkowski spacetime asserts the existence of an anti-unitary operator : H → H on the Hilbert space, H, of physical states such that, if ρ denotes the isometry on the, say, even-dimensional Minkowski spacetime defined by ρ : (x 0 , x 1 , . . . , x D−1 ) → (−x 0 , −x 1 , . . . , −x D−1 ),

(88)

and if φ C is the charge conjugate field associated with φ defined by Eq. (83), then Ad φ(ρ(x)) ≡ φ(ρ(x))† = φ C (x). The relationship between these formulations can be seen as follows. Start with our formulation of the PCT theorem. The isometry ρ maps Minkowski spacetime M = (R D , η, e, T ) with a given choice of orientation e and time orientation T , to M = (R D , η, e, −T ). Thus, by “Application 1” of Lemma 1, we know that there is a

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*-homomorphism χρ : A(M) → A(M) mapping the quantum fields φM ( f ) on M to ¯ Thus, if we define the “same” quantum fields φM (ρ∗ ( f )) on M. PCT Ad := χρ−1 ◦ θM ,

(89)

we obtain a result that is essentially equivalent to the usual Minkowski version (as suitably reformulated in an algebraic setting). Conversely, if we start with the usual formulation of the PCT theorem and if we define quantum field theory on A(M) in terms of quantum field theory on M by means of the map χρ , then we obtain a version essentially equivalent to our formulation by setting PCT θM = χρ ◦ Ad .

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Although the above formulations are essentially equivalent in Minkowski spacetime, in a general spacetime, there does not exist any discrete isometry analogous to ρ. Thus, in general we only have a PCT theorem describing the relation between the theory defined on different backgrounds M and M. Of course in the case of a spacetime that admits an isometry ρ mapping (e, T ) to (e, −T ) (as occurs, e.g. in Schwarzschild and deSitter spacetimes), then a “same background structure” version of the PCT theorem can be given via Eq. (89). The example of a Robertson-Walker spacetime g = −dt 2 + a(t)2 dx · dx

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with a(t) a strictly increasing function of t, may be useful for clarifying the physical meaning of our formulation of the PCT theorem in a general curved spacetime. If we choose the time orientation T = t, then the above metric describes an expanding universe, while if we take T = −t, it describes a corresponding contracting universe (with opposite choice of spatial orientation since we keep e fixed). In essence, our formulation of the PCT theorem relates phenomena/processes occuring in the expanding universe Eq. (91) to corresponding processes (involving the charge conjugate fields and also a reversal of parity) in the corresponding contracting universe. Since the metric Eq. (91) has no time reflection isometry ρ, there are no relations implied by the PCT theorem between phenomena/processes occurring in the expanding universe, Eq. (91) with T = t. As a concrete illustration of this, suppose that it were possible to give a definition of “particle masses” in curved spacetime—although it is far from obvious that any such useful notion exists. The PCT theorem would then imply that the mass of a particle in an expanding universe must be equal to the mass of the corresponding antiparticle in a contracting universe. However, it would make no statement about the masses of particles and antiparticles in the same universe13 . 13 It is worth noting that the “third Sakharov necessary condition” for baryogenesis in the early universe (namely, “interactions out of thermal equilibrium”) is based upon the (now seen to be unjustified) assumption that particle and antiparticle masses are equal in an expanding universe. However, to the extent that particle and antiparticle masses might differ in an expanding universe (even assuming that a useful notion of “particle mass” can be defined) as a result of the lack of a time reflection symmetry, it would probably not even be possible to define a notion of “thermal equilibrium” as a result of the lack of a time translation symmetry.


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6. Conclusions and Outlook In this paper we have proposed a new axiomatic framework for quantum field theory on curved spacetime. We demonstrated that our new framework captures much of the same content as the Wightman axioms by proving curved spacetime analogs of the spinstatistics theorem and PCT theorem. In this section, we discuss some of the implications and potential ramifications of the viewpoint suggested by this new framework. First, we address the issue of why we even seek an axiomatic framework for quantum field theory in curved spacetime at all. Since gravity is being treated classically, quantum field theory in curved spacetime cannot be a fundamental theory of nature, i.e., it must have a limited domain of validity. In particular, we have focused considerable attention in this paper on the OPE of quantum fields in curved spacetime, but the OPE is a statement about the arbitrarily-short-distance singularity structure of products of fields. One would not expect that quantum field theory in curved spacetime would give an accurate description of nature at separations smaller than, say, the Planck scale. Consequently, why should one seek a set of mathematically consistent rules governing quantum field theory that are rigorously applicable only in a regime where the theory is not expected to be valid? Our response to this question is that an exactly similar situation arises for classical field theory. Classical field theory also is not a fundamental theory of nature, and its description of nature makes essential use of differentiability/smoothness properties of the classical fields at short distance scales; one could not even write down the partial differential equations governing the evolution of classical fields without such short-distance-scale assumptions. However, if quantum field theory is any guide, the description of physical fields as smooth tensor fields is drastically wrong at short distance scales. Nevertheless, classical field theory has been found to give a very accurate description of nature within its domain of validity, and we have obtained a great deal of insight into nature by obtaining a mathematically precise formulation of classical field theory. It is our belief that there exists a mathematically consistent framework for quantum field theory in curved spacetime, and that by obtaining and studying this framework, we will not only get an accurate description of nature within the domain of validity of this theory, but we will also get important insights and clues concerning the nature of quantum gravity. We began our quest for the mathematical framework of quantum field theory in curved spacetime by seeking to generalize the Wightman axioms to curved spacetime in as conservative a manner as possible. As described much more fully in the Introduction, there are three key ingredients of the Wightman axioms that do not generalize straightforwardly to curved spacetime: (1) Poincaré invariance; (2) the spectrum condition; (3) existence of a Poincaré invariant state. We have seen in this paper that quantum field theory can be generalized to curved spacetime by replacing these ingredients by the following: (1’) quantum fields are locally and covariantly defined; (2’) the microlocal spectrum condition; (3’) existence of an OPE. Although the formulation of conditions (1’) and (2’) differs significantly from the formulation of conditions (1) and (2), the basic content of these conditions is essentially the same. Indeed, there would be no essential difference in the formulation of axiomatic quantum field theory in Minkowski spacetime if one replaced (1) and (2) with (1’) and (2’). By contrast, as we shall elucidate further below, the replacement of (3) by (3’) leads to a radically different viewpoint on quantum field theory. The most important aspect of this difference is that the existence of a “preferred state” no longer plays any role in the formulation of the theory. States are inherently

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non-local in character, and the replacement of (3) by (3’)—along with the replacements of (1) with (1’) and (2) with (2’)—yields a formulation of quantum field theory that is entirely local in nature. In this way, the formulation of quantum field theory becomes much more analogous to the formulation of classical field theory. Indeed, one can view a classical field theory as being specified by providing the list of fields φ (i) occuring in the theory and the list of local, partial differential relations satisfied by these fields. Solutions to the classical field theory are then sections of the appropriate vector bundles that satisfy the partial differential relations as well as regularity conditions (e.g., smoothness). Similarly, in our framework, a quantum field theory is specified by providing the list of fields φ (i) occurring in the theory and the list of local, OPE relations satisfied by these fields. Thus, the OPE relations play a role completely analogous to the role of field equations in classical field theory. States—which are the analogs of solutions in classical field theory—are positive linear maps on the algebra A defined in Sect. 3 that satisfy the OPE relations as well as regularity conditions (in this case, the microlocal spectrum condition). We note that in classical field theory, the field equations always manifest all of the symmetries of the theory, even in cases where there are no solutions that manifest these symmetries. Similarly, in our formulation of quantum field theory, the OPE relations that define the theory should always respect the symmetries of the theory [30], even if no states happen to respect these symmetries. Our viewpoint on quantum field theory is more restrictive than standard viewpoints in that we require the existence of an OPE. On the other hand, it is less restrictive in that we do not require the existence of a ground state. This latter point is best illustrated by considering a free Klein-Gordon field ϕ in Minkowski spacetime (2 − m 2 )ϕ = 0,

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where the mass term, m 2 , is allowed to be positive, zero, or negative. In the standard viewpoint, a quantum field theory of the free Klein-Gordon field does not exist in any dimension when m 2 < 0 and does not exist in D = 2 when m 2 = 0 on account of the non-existence of a Poincaré invariant state. However, there is no difficulty in specifying OPE relations that satisfy our axioms for all values of m 2 and all D ≥ 2. In particular, for D = 4 we can choose the OPE-coefficient C of the identity in the OPE of ϕ(x1 )ϕ(x2 ) to be given by C(x1 , x2 ; y) = 1 1 2 2 2 2 2 2 2 2 + m j[m x ] log[µ (x + i0t)] + m h[m x ] , (93) 4π 2 x 2 + i0t where x 2 = (x1 − x2 )2 and t = x10 − x20 . Here µ is an arbitrarily chosen mass scale and √ j (z) ≡ 2i 1√z J1 (i z) is an analytic function of z, where J1 denotes the Bessel function of order 1. Furthermore, h(z) is the analytic function defined by h(z) = −π

∞ k=0

[ψ(k + 1) + ψ(k + 2)]

(z/4)k , k!(k + 1)!

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with ψ the psi-function. This formula for the OPE coefficient—as well as the corresponding formulas for all of the other OPE coefficients—is as well defined for negative m 2 as for positive m 2 . Existence of states satisfying all of the OPE relations for negative m 2 can be proven by the deformation argument of [12], using the fact that such states exist for positive m 2 .


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Although, in our framework, the Klein-Gordon field with negative m 2 now joins the ranks of legitimate quantum field theories, this theory is not physically viable because, in all states, field quantities will grow exponentially in time14 . The potential importance of the above example is that it explicitly demonstrates that the local OPE coefficients can have a much more regular behavior under variations of the parameters of the theory as compared with state-dependent quantities, such as vacuum expectation values. The OPE coefficients in the above example are analytic in m 2 . On the other hand, the 2-point function of the global vacuum state is, of course, defined only for m 2 ≥ 0 and is given by 0|ϕ(x1 )ϕ(x2 )|0 = 1 1 2 2 2 2 2 2 2 2 + m j[m x ] log[m (x + i0t)] + m h[m x ] . (95) 4π 2 x 2 + i0t This behaves non-analytically in m 2 at m 2 = 0 on account of the log m 2 term. In other words, in free Klein-Gordon theory, vacuum expectation values cannot be constructed perturbatively by expanding about m 2 = 0—as should be expected, since no vacuum state exists for m 2 < 0—but there is no difficulty in perturbatively constructing the OPE coefficients by expanding about m 2 = 0. The above considerations raise the possibility that the well known failure of convergence of perturbation series in interacting quantum field theory may be due to the non-analytic dependence of states on the parameters of the theory, rather than any nonanalytic dependence of the fields themselves, i.e., that the OPE coefficients may vary analytically with the parameters of the theory. In other words, we are suggesting the possibility that the perturbation series for OPE coefficients may converge, and, thus, that, within our framework, it may be possible to perturbatively construct15 interacting quantum field theories. In order to do so, it will be necessary to define the basis fields φ (i) appropriately (see below and [18]) and also to parametrize the theory appropriately (since a theory with an analytic dependence on a parameter could always be made to appear non-analytic by a non-analytic reparametrization). Aside from the free Klein-Gordon example above, the only evidence we have in favor of convergence of perturbative expansions for OPE coefficients is the example of super-renormalizable theories, such as λϕ 4 -theory in two spacetime dimensions [17]. Here, only finitely many terms in a perturbative expansion can contribute to any OPE coefficient up to any given scaling degree, so convergence (up to any given scaling degree) is trivial. By contrast, for λϕ 4 -theory in two spacetime dimensions, the rigorously constructed, nonperturbative ground state n-point functions can be proven to be non-analytic at λ = 0 [25]. In cases—such as free Klein-Gordon theory above—where the OPE coefficients can be chosen to be analytic in the parameters of the theory, it seems natural to require that the theory be defined so that this analytic dependence holds. This requirement has some potentially major ramifications. Since vacuum expectation values of a product of fields (i.e., a correlation function) would be expected to have a non-analytic dependence on 14 In this respect, the quantum field of the Klein-Gordon field with negative m 2 behaves very similarly to the corresponding classical theory. The classical Klein-Gordon field with negative m 2 has a well posed initial value formulation and causal propagation (despite frequently expressed claims to the contrary). However, the classical Klein-Gordon field with negative m 2 is not physically viable since it is unstable, i.e., it admits solutions that grow exponentially with time. 15 However, we are not suggesting that it should be possible to perturbatively construct states of the theory. Even if one had the complete list of OPE coefficients, it would not be obvious how to construct states.

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the parameters of the theory, it follows that if the OPE coefficients have an analytic dependence on these parameters, then, even in Minkowski spacetime, some of the fields appearing on the right side of the OPE of a product of fields must acquire a nonvanishing vacuum expectation value, at least for some values of the parameters. This point is well illustrated by the above Klein-Gordon example. It is natural to identify the next term in the OPE of ϕ(x1 )ϕ(x2 ) [i.e., the term beyond the identity term, whose coefficient is given by Eq. (93)] as being ϕ 2 (with unit coefficient), i.e., ϕ(x1 )ϕ(x2 ) ∼ C(x1 , x2 ; y)1 + ϕ 2 (y) + ... .

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This corresponds to the usual “point-splitting” definition of ϕ 2 , except that C(x1 , x2 ; y) now replaces16 0|ϕ(x1 )ϕ(x2 )|0. If we take the vacuum expectation value of this formula (for m 2 ≥ 0, when a vacuum state exists) and compare it with Eq. (95), we obtain 0|ϕ 2 (y)|0 = −

m2 log(m 2 /µ2 ). 16π 2

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Thus, we cannot set 0|ϕ 2 |0 = 0 for all values of m 2 . A similar calculation for the stress-energy tensor of ϕ yields 0|Tab (y)|0 =

m4 log(m 2 /µ2 )ηab . 64π 2

(98)

As in other approaches, the freedom to choose the arbitrary mass scale µ in Eq. (93) gives rise to a freedom to choose the value of the “cosmological constant term” in Eq. (98). However, unlike other approaches, there is no freedom to adjust the value of the cosmological constant when m 2 = 0 (i.e., we unambiguously obtain 0|Tab |0 = 0 in Minkowski spacetime in that case), and the m 2 -dependence of the cosmological constant is fixed (since µ is not allowed to depend upon m 2 ). A much more interesting possibility arises for interacting field theories, such as nonabelian gauge theories. In such theories, it is expected that there are “non-perturbative effects” that vary with the coupling parameter g as exp(−1/g 2 ). Such non-perturbative effects can potentially be very small compared with the natural scales appearing in the theory. If such non-perturbative terms occur the vacuum expectation values of products of fields and if—as we have speculated above—the OPE coefficients have an analytic dependence on the coupling parameter, then composite fields—such as the stress-energy tensor—must acquire nonvanishing vacuum expectation values that vary as exp(−1/g 2 ). This possibility appears worthy of further investigation. Acknowledgements. This research was supported in part by NSF Grant PHY04-56619 to the University of Chicago. 16 The point-split expression using 0|ϕ(x )ϕ(x )|0 yields the “normal ordered” quantity : ϕ 2 :. From the 1 2 point of view of quantum field theory in curved spacetime it is much more natural to define ϕ 2 via Eq. (96) than by normal ordering, since there is no generalization of normal ordering to curved spacetime that is compatible with a local and covariant definition of ϕ 2 [19]. Indeed, it follows from the results of [19] that Eq. (96) is the unique way to define ϕ 2 compatible with desired properties, with the only ambiguities in the definition of ϕ 2 arising from different allowed choices of C(x1 , x2 ; y). These correspond to a field redefinition of ϕ 2 by addition of a multiple of the identity.


121

A. Definition of the Scaling Degree and Wave Front Set of a Distribution In this Appendix, we recall the notion of scaling degree and of the wave front set of a distribution, which play an important role in the body of the paper. Quite generally, let u be a distribution on Rn . We say that u has scaling degree d at x = 0, if d is the smallest real number such that λδ u( f λ ) → 0 as λ → 0+, for all δ > d. Here, f λ (x) = λ−n f (λ−1 x) denotes the function of compact support that is obtained by rescaling a smooth test function f around x = 0, making it more and more sharply peaked at that point. The scaling degree at an arbitrary point is obtained by simply translating the distribution u or the test function f by the desired amount. We write sd x (u) = d for the scaling degree at a point x. We next recall the definition of the wave front set of a distribution u on Rn . Let χ be any smooth function of compact support. Then χ u is evidently a distribution of compact support, and its Fourier transform χ u(k) defines an entire function of k ∈ Rn . For any distribution v of compact support, we define its corresponding “singular set”, (v) as the collection of all k ∈ Rn such that | v (λk)| ≥ Cλ N ,

(99)

for some C > 0, and some N , and all λ > λ0 for some λ0 . We define the wave front set WF x (u) at a point x ∈ Rn as the intersection WF x (u) = (χ u), (100) χ :x∈supp χ

and we define WF(u) as the union WF(u) =

WF x (u).

(101)

x∈Rn

Each set WF x (u) is a conic set, in the sense that if k ∈ WF x (u), then so is tk for any t > 0, and k = 0 is never in WF x (u). It immediately follows from the definition that WF x (u) = ∅ if and only if u can be represented by a smooth function in an open neighborhood of x. In this sense, the wave-front set tells one at which points a distribution is singular. It also contains information about the most singular directions in local momentum space, which are represented by k ∈ WF x (u). It turns out that both the scaling degree of a distribution at a point x, as well as the wave front set at x are invariantly defined. By this one means the following. Let ρ : V → U be a smooth diffeomorphism between open sets U, V ⊂ Rn . Let u be a distribution supported in U , and let ρ ∗ u be the pulled back distribution in V , where the pull-back is defined by analogy with the pull back of a smooth function. Then it is easy to show that sd x (ρ ∗ u) = sdρ(x) (u). Furthermore, if x = ρ(x), and if we define ρ ∗ (x , k ) = (x, k), where k = [dρ(x)]v k , then one can show WF x (ρ ∗ u) = ρ ∗ WFρ(x) (u).

(102)

These relations imply that the scaling degree and the wave front set can be invariantly defined on an arbitrary manifold X , and that the wave-front set should be viewed as a subset of T ∗ X . In the body of the paper, we frequently consider the case X = M n+1 , and the scaling degree at the point (y, y, . . . , y), i.e., points on the total diagonal. To save writing, this is simply denoted sd u.

122


B. Equivalent Formulation of Condition (C6) In this Appendix, we relate the scaling degree (C5) and asymptotic positivity (C6) assumptions to other properties of the quantum field theory. The first is essentially just a repetition of a result obtained in [8]: Theorem 4. For any x ∈ M, and any (scalar) field T not equal to a multiple of the identity operator, we define the convex set Sx ⊂ R by Sx = {T (x) | a normalized state}

(103)

Then, Sx = R for at least one spacetime M. Remarks. The statement means that pointlike hermitian fields T (x) are unbounded from above and below, even though their classical counterpart (if the theory has a classical limit) might be manifestly non-negative, such as the Wick square T = ϕ 2 , or the energy density T = Tab u a u b of a free Klein-Gordon field ϕ. Proof. Choose any state . We may assume that T ( f ) = 0, because if not, we just need to replace T (x) by T (x) − T (x) 1. Also, we may assume without loss of generality that T ∗ = T , for otherwise we can just pass to the hermitian and anti-hermitian parts of T . Define A = cos α 1 + sin α

T( f ) 1/2

T ( f )T ( f )

,

(104)

for a real testfunction f , and define a new normalized state by .α = A . A∗ . Then T ( f )α = a sin 2α + b(1 − cos 2α),

(105)

where 1/2

a = T ( f )T ( f ) , b =

1 T ( f )T ( f )T ( f ) . 2 T ( f )T ( f )

Minimizing over α gives inf T ( f ) ≤ b −

a 2 + b2 .

Replacing f by − f also gives sup T ( f ) ≥ −b +

a 2 + b2 .

(106)

(107)

(108)

Now, consider a test function f with f dµ = 1, f (x) = 0, and let f λ (y) = λ−D f (x + λ−1 (y − x)), λ > 0. By Lemma 2, there exists a spacetime, a choice of f , and a state such that |T ( f λ )T ( f λ ) | ≥ c−1 λ−2d+2δ for some c > 0, sufficiently small δ > 0, and all λ < λ0 . d > 0 is the dimension of the field T . Also, combining the operator product expansion and the scaling degree axiom (C5), we have |T ( f λ )T ( f λ )T ( f λ ) | ≤ cλ−3d−δ , for all λ < λ0 . Combining these inequalities gives |T ( f λ )T ( f λ ) |3/2 ≥ c−2 λ4δ |T ( f λ )T ( f λ )T ( f λ ) | for all λ < λ0 .

(109)


123

We now consider two cases: (a) there exists a sequence λn → 0 such that |an /bn | ≥ ε > 0 for some ε and all n, where an , bn are defined using the rescaled testfunctions f n ≡ f λn . Then, as an → +∞, it follows immediately from Eq. (107) that inf T ( f n ) becomes arbitrarily small as n → ∞. It similarly follows from Eq. (108) that sup T ( f n ) becomes arbitrarily large as n → ∞. (b) |an /bn | → 0 for each sequence λn → 0. Then |bn | → ∞, and we may in fact assume that bn → +∞, for otherwise the statement follows immediately from Eqs. (107), (108). From the above inequality have an /bn ≥ c−2 λ4δ n for sufficiently large n. It follows that √ (109), we then 1 (using 1 + x ≥ 1 + 2+ε x for sufficiently small x ≥ 0) bn −

an2 + bn2 ≤ −

1 an2 1 1 ≤− λ4δ λ6δ−2d , (110) n an ≤ − 2 2 + ε bn (2 + ε)c (2 + ε)c3 n

for sufficiently large n. We can choose δ < 3d/2. Then the right side tends to −∞, and we conclude that inf T ( f n ) becomes arbitrarily small as n → ∞. It similarly follows that sup T ( f n ) becomes arbitrarily large as n → ∞. Our next result is in some sense a converse to the above result: Theorem 5. Let the set Sx be equal to R for a given spacetime M, and all hermitian operators T not equal to a multiple of the identity. Then for any i, k ∈ I with i = 1, and any sections v(i) of V (i) we have that

(k)(k ) (k)(k ) (v(k) ⊗ v¯(k ) )] < sd[C(1) (v(k) ⊗ v¯(k ) )]. sd[C(i)

(111)

Remarks. In generic spacetimes, we expect that the scaling degree of the right side is equal to 2dim(k), where dim(k) is the dimension of the field φ (k) ; see the scaling degree axiom (C5). We also expect the quantity on the left side to be equal to 2dim(k) − dim(i); see again (C5). Thus, the result tells us that, in this situation, dim(i) > 0 unless φ (i) is the identity field. Thus, in this sense, the assumption of the theorem implies the asymptotic positivity axiom (C6), or—stated differently—the asymptotic positivity axiom is inconsistent with not having Sx = R. Proof. By assumption, we can find a state such that T (x) > A for each A ∈ R. Consider an arbitrary, but fixed, finite collection φ (1) , . . . , φ (n) of fields. Each field is valued in some vector bundle V (i). The set of expectation values of this collection of fields forms a subset which we denote

! n

(1) (n)

K x = (φ (x) , . . . , φ (x) ) states ⊂ V (i)x =: Vx . (112) i=1

Because the set of states is convex (i.e., any convex linear combination of normalized states is again a normalized state), the set K x is a convex subset of Vx . We claim that, in fact, K x = Vx . Assume that this were not the case. Then, since any proper convex subset of a finite dimensional vector space can be enclosed by a collection of planes, there exists a collection of dual vectors c(i) ∈ V (i)v and an A ∈ R such that c(i) v (i) < A, for all (v (1) , . . . , v (n) ) ∈ K x . (113) Re (i)

However, this would mean by definition that if T = Re for all states , a contradiction.

c(i) φ (i) , then T (x) < A

124


Assume that the statement of the theorem is not true. Let i = 1, . . . , n ∈ I be the field labels, with i = 1 for which the inequality (111) does not hold. From what we have just shown, if we define K x as above, then K x is equal to Vx . In particular, we may find states , , and nonzero v (i) with the property that φ (i) (x) = v (i) , φ (i) (x) = −v (i) .

(114)

Let f λ (x) = λ−D f (y + λ−1 (x − y)) be a test section in the dual of the space V (k), and let δ be a real number which is bigger than 2dim(k), but smaller than the left side of Eq. (111). Using the fact that φ (k) ( f λ )φ (k ) ( f¯λ ) ≥ 0 for all λ, we find from the operator product expansion that (k )(k) λδ C(i) ( f¯λ , f λ ; x)v (i) → +∞, (115) (i)

for at least one f and a subsequence of λ tending to 0. Applying a similar argument to the state gives that (k )(k) − λδ C(i) ( f¯λ , f λ ; x)v (i) → +∞, (116) (i)

for this subsequence of λ tending to 0. This is a contradiction, so the inequality (111) must hold.

References 1. Axelrod, S., Singer, I.M.: Chern-Simons perturbation theory. 2. J. Diff. Geom. 39, 173 (1994) 2. Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the monster. Proc. Nat. Acad. Sci. 83, 3068 (1986) 3. Bostelmann, H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005) 4. Bostelmann, H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301 (2005) 5. Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials on curved spacetimes. Commun. Math. Phys. 180, 633–652 (1996) 6. Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623–661 (2000) 7. Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: A new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003); see also K. Fredenhagen,: Locally covariant quantum field theory. In: Proc. Int. Conf., Math. Phys. (Lisbon, Portugal, 2003), Singapore, World Scientific, 2005 8. Fewster, C.J.: Energy Inequalities in Quantum Field Theory. Proceedings of XIVth International Congress on Mathematical Physics, ed. J.-C. Zambrini, Singapore: World Scientific, 2005, p. 559 9. Fredenhagen, K.: Locally Covariant Quantum Field Theory. Proceedings of XIVth International Congress on Mathematical Physics, ed. J.-C. Zambrini, Singapore: Worl Scientific, 2003, p. 29 10. Fredenhagen, K., Hertel, J.: Local Algebras Of Observables And Point - Like Localized Fields. Commun. Math. Phys. 80, 555 (1981); Fredenhagen, K., Jorss, M.: Conformal Haag-Kastler nets, point - like localized fields and the existence of operator product expansions. Commun. Math. Phys. 176, 541 (1996) 11. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Boston: Academic Press, 1988 12. Fulling, S.A., Narcowich, F.J., Wald, R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime, II. Ann. Phys. 136, 243 (1981) 13. Fulton, W., MacPherson, R.: A compactification of configuration spaces. Ann. Math. 139, 183 (1994) 14. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848 (1964)


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15. Hollands, S.: The operator product expansion for perturbative quantum field theory in curved spacetime. Commun. Math. Phys. 273, 1 (2007) 16. Hollands, S.: A general PCT theorem for the operator product expansion in curved spacetime. Commun. Math. Phys. 244, 209 (2004) 17. Hollands, S., Kopper, C.: In progress 18. Hollands, S., Wald, R.M.: In progress 19. Hollands, S., Wald, R.M.: Local wick polynomials and time ordered products of quantum fields in curved space. Commun. Math. Phys. 223, 289–326 (2001) 20. Hollands, S., Wald, R.M.: Existence of local covariant time-ordered-products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309–345 (2002) 21. Hollands, S.: Quantum field theory in terms of consistency conditions I: General framework, and perturbation theory via Hochschild cohomology. http://arxiv.org/abs/0802.2198v2[hep-th], 2008 22. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Berlin: Springer-Verlag, 1983 23. Kac, V.: Vertex Algebras for Beginners. Univ. Lect. Series 10, Providence, RI: Amer. Math. Soc., 1996 24. Radzikowski, M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529 (1996) 25. Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton Series in Physics, Princeton, NJ: Princeton Univ. Pr., 1991 26. Schroer, B., Swieca, J.A., Volkel, A.H.: Global operator expansions in conformally invariant relativistic quantum field theory. Phys. Rev. D 11, 1509 (1975) 27. Streater, R.F., Wightman, A.A.: PCT, Spin and Statistics and All That. New York: Benjamin, 1964 28. Wald, R.M.: Quantum Field Theory on Curved Spacetimes and Black Hole Thermodynamics. Chicago, IL: The University of Chicago Press, 1990 29. Wilson, K.G.: Nonlagrangian models of current algebra. Phys. Rev. 179, 1499 (1969) 30. Bernard, C., Duncan, A., LoSecco, J., Weinberg, S.: Exact spectral-function sum rules. Phys. Rev. D 12, 792–804 (1975), See the Appendix 31. Zimmermann, W.: Normal Products And The Short Distance Expansion In The Perturbation Theory Of Renormalizable Interactions. Annals Phys. 77, 570 (1973); seealso, Lect. Notes Phys. 558, Berlin Heidelberg-NewYork: Springer, 2008, p. 278 Communicated by G. W. Gibbons


Communications in


Focusing Components in Typical Chaotic Billiards Should be Absolutely Focusing Leonid A. Bunimovich1,2 , Alexander Grigo2 1 ABC Math Program, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A 2 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

E-mail: [email protected]; [email protected] Received: 9 June 2008 / Accepted: 17 November 2008 Published online: 27 September 2009 – © Springer-Verlag 2009

Abstract: We demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing. More precisely, we construct billiard tables with arbitrary long free path away from a non-absolutely focusing component such that a nonlinearly stable periodic orbit exists. Therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing. 1. Introduction and Statement of the Main Results The foundation of the theory of hyperbolic systems with singularities was laid in Sinai’s seminal paper [20] where hyperbolicity and ergodicity for billiards with smooth dispersing boundary was proven. The situation changes drastically if a billiard table has at least one focusing component. Billiards with focusing boundaries demonstrate behaviors from completely regular to strongly chaotic. In fact, Lazutkin proved in [16,17] that for any strictly convex billiard table with smooth enough boundary there exist caustics near the boundary, which prevent global ergodicity and hyperbolicity. However, certain classes of hyperbolic and ergodic billiards with focusing boundary components were found in [3,4]. The mechanism behind this is called the mechanism of defocusing. It relies on the fact that a focusing beam will eventually go through a conjugate point, and will be dispersing afterwards. Therefore, if the free path to the next reflection is sufficiently large an essentially analogous situation to the case of dispersing billiard tables arises. This method of placing focusing boundary components sufficiently far away from other boundary components is the only known general procedure of constructing hyperbolic billiard tables, [21]. Therefore, after the discovery of the mechanism of defocusing the question of which focusing components could be components of the boundary of a chaotic (hyperbolic) billiard was raised. Two dual classes of such focusing components were introduced in

128

L. A. Bunimovich, A. Grigo

[21 and 18]. Then a much more general class of focusing components, admissible for chaotic billiard tables, was introduced [5–7 and 14]. (Formally the class of focusing components introduced in [14] seems to be more restrictive than the one in [6]. However, these two classes coincide [7].) These focusing components are called absolutely focusing, and were shown in [8,12] to allow for hyperbolicity and ergodicity. It was conjectured in [6] that in hyperbolic billiards each focusing component of the boundary must be absolutely focusing, and it was outlined there how one can construct a stable periodic orbit if at least one focusing component is not absolutely focusing. In this paper we show along the lines described in [6,8] that as soon as the absolutely focusing property of focusing boundary components fails to hold the general procedure of designing chaotic billiard tables generally fails. Even if one makes the free path after a reflection from a non-absolutely focusing component arbitrarily large, then such billiards can still have elliptic periodic points. Thus these billiards have islands of stability and are not completely hyperbolic. These results, once again, indicate that the mechanism of defocusing plays a key role in generic Hamiltonian systems which demonstrate the coexistence of islands of stability (KAM-islands) and chaotic hyperbolic components. The main result of this paper is the following: Theorem 1. Let Γ be a C 5 non-absolutely focusing curve of minimal length which encloses an angle of no more than π , or a small enough extension of such a curve. Then for every L > 0 there exist an open set (in the sense of C 5 ) of billiard tables Q(L , Γ ), Γ ⊂ ∂ Q(L , Γ ), which all have a nonlinearly stable periodic orbit with free path of length at least L before and after a sequence of consecutive reflections off of Γ . The C 5 (C k ) closeness of two billiard tables simply means that there are parametrizations of their respective boundaries, which are piecewise C 5 (C k ) and are piecewise close in the C 5 (C k ) sense as functions from [0, 1] → R2 . Remark 1. We would like to mention at this point that the corresponding result for a linearly stable, non-resonant periodic orbit can be proved when assuming only C 3 –smoothness, cf. Theorem 2 below. This is the usual assumption in the theory of hyperbolic billiards, which deals with the first derivative of the billiard map (or flow) only. To deduce nonlinear stability, however, we need, for technical reasons, to assume C 5 smoothness to be able to apply KAM theory (Moser’s twist theorem [19]), because it takes higher order derivatives of the billiard map into account. Dynamics on KAM-islands is characterized by a balance between focusing (convergence of nearby orbits) and defocusing (their divergence) while on chaotic components defocusing dominates focusing. Recall that dispersing is just a special case of defocusing (when the focusing time is negative) and neutral components of the boundary cannot generate by themselves a chaotic behavior [2]. Therefore, our results show that apparently there are no other mechanisms of hyperbolicity in billiards besides dispersing and defocusing. If both dispersing and focusing components are present, then they should be arranged in such a way that any initially parallel (infinitesimal) beam of rays in the course of its dynamics arrives at any curved (dispersing or focusing) component of the boundary being dispersing. Then dispersing either takes over focusing, which occurs in the part of phase space where hyperbolicity emerges, or these two are balanced, [9,10]. This happens on the part with regular dynamics (KAM-island). Thus defocusing is a fundamental mechanism of hyperbolicity (at least in billiards) and its violation leads to the creation of KAM-islands.

Focusing Components Should be Absolutely Focusing

129

The structure of the paper is the following. In Sect. 2 we will review some basic properties of billiards. The construction of a linearly stable periodic orbit with a series of consecutive reflections off non-absolutely focusing components is described in Sect. 3. The nonlinear stability is first established for a very general setting in Sect. 4 and then applied in Sect. 5 to the linearly stable orbit constructed before, which will prove our main result Theorem 1. 2. Basic Facts about Billiards In this section the basic properties of billiards we will need are described. The notation used is chosen close to the one used in [11], which contains most of the results listed below. Let Q ⊂ R2 denote an open bounded domain with piecewise C 3 boundary ∂ Q. The dynamics generated by a point-like particle moving along straight lines inside Q and having specular reflections off the boundary ∂ Q is called billiard flow Φ t on the billiard table Q. The induced first return map F to the boundary ∂ Q, where only the state right after the reflection is considered, is called the associated billiard map. These constructions yield Φ t : Q × S 1 → Q × S 1 and F : ∂ Q × [−π/2, π/2] → ∂ Q × [−π/2, π/2], which are defined almost everywhere with respect to the Lebesgue measure. The natural coordinates for the billiard map are the arc length parameter s along the boundary, which we will assume to be oriented in the counterclockwise direction, and the angle of reflection ϕ relative to the normal direction. For the billiard flow the natural coordinates are the angle ω giving the direction of the velocity vector relative to the horizontal direction, and (x, y) denoting the position inside of Q. It is well known that the billiard map preserves the measure dµ = cos ϕ dϕ ds and that dν = dω d x d y is preserved by the billiard flow. (Indeed, even the volume forms are preserved.) In particular, the billiard map is symplectic in the coordinates (s, sin ϕ). When working with the billiard flow it is often more convenient to use the so called Jacobi coordinates (η, ξ, ω), which in infinitesimal form read dη = cos ω d x + sin ω dy and dξ = − sin ω d x + cos ω dy . The derivative of Φ t then becomes ∂(ξ t , ωt ) 1 0 t with Ut = and det Ut = 1, DΦ = 0 Ut ∂(ξ, ω) since η measures the distance in direction of the flow. We will refer to Ut as the reduced Jacobian of Φ t . Throughout the paper we denote by K = K(s) the (signed) curvature of the boundary at the point corresponding to the arc length parameter s. The sign of K is chosen such that it is negative for convex shaped boundary components, as in the case of a billiard inside a circle. Correspondingly, we shall call boundary components for which K < 0, K = 0, K > 0 focusing, neutral, dispersing, respectively. The derivative of the billiard map is given by 1 τ K + cos ϕ τ DF(s, ϕ) = − , cos ϕ1 τ K K1 + K cos ϕ1 + K1 cos ϕ τ K1 + cos ϕ1

130


where we set (s1 , ϕ1 ) = F(s, ϕ) and τ the distance (free path) between the two points along the straight line connecting them. In particular, the billiard map is of class C k−1 if the boundary components are of class C k . Another well known fact is that the (local) generating function of the billiard dynamics is (locally) given by the Euclidean distance along the straight line segments connecting the points of reflection. More precisely, consider a point (s0 , ϕ0 ) such that (sk , ϕk ):=F k (s0 , ϕ0 ) is well defined for 0 ≤ k ≤ n. Denote by Γi the boundary component on which the i th reflection occurs (i.e. on which (si , ϕi ) lies), so that Γi (si ) is the point of the i th reflection in the plane. Then L(s0 , . . . , sn ):=

n

Γi (si ) − Γi−1 (si−1 )

i=1

is the Euclidean length of the corresponding trajectory of the billiard flow, and ∂s0 L = − sin ϕ0 ,

∂si L = 0 for 1 ≤ i ≤ n − 1 ,

∂sn L = sin ϕn

hold, which is why L is the generating function (cf. Sects. 47 and 48 in [1] for details on generating functions). An important quantity to describe the billiard flow is the wavefront curvature B:=dω/ dξ , which obeys 1 1 = + t for a free flight of length t, Bt B0 2K at a point of reflection. B + = B − + R with R:= cos ϕ In geometric optics the second relation is called the mirror formula. In particular we have ct + dt B0 at bt Bt = for Ut = ct dt at + bt B0 which relates the wavefront curvature and the derivative of the flow. For a segment of a billiard trajectory γ we denote the wavefront curvature of an initially parallel beam of rays sent along γ by c a b , (1) with U (γ ) = Bout (γ ):= c d a where U (γ ) is the reduced Jacobian of the billiard flow along γ . 3. Construction of Linearly Stable Periodic Orbits In this section let Γ be a (focusing) C 3 curve of length l > 0, parametrized by its arc length. Definition 1 (Absolutely Focusing; see [5,6]). A closed, focusing component Γ is called absolutely focusing if every incoming infinitesimal beam of parallel rays leaves Γ , after a complete sequence of consecutive reflections, as a focusing beam.


131

Fig. 1. Illustration of the family of trajectories constructed in Lemma 1

It was shown in [7,8,12,14] that the defocusing mechanism applies to focusing boundary components which are absolutely focusing. Furthermore, it was also shown in [14] that every short enough piece of a focusing curve is absolutely focusing. This, in particular, motivates the following notion to characterize the transition between absolutely and non-absolutely focusing curves. Definition 2 (Non-Absolutely Focusing of Minimal Length). A focusing curve Γ is called non-absolutely focusing of minimal length, if every of its closed sub-arcs is absolutely focusing, but the curve itself is not absolutely focusing. Our approach is based on the one outlined in [6]. The first step is to show that for a non-absolutely focusing curve there exists an infinitesimal beam of parallel rays falling onto Γ which leaves Γ after its last reflection as arbitrary weakly focusing. In the second step such a beam is used to construct a stable periodic orbit. It is well known that the billiard map satisfies the so-called twist property ds1 /dϕ < 0, no matter which type of boundary components are considered. It is also well known that compositions of twist maps, and iterates of a twist map, are in general no longer twist maps. However, it was observed in [14] (Prop. 3.6) that when restricting the billiard map to absolutely focusing boundary components, then its iterates will still be twist maps. This property is the main technical step in the following key lemma, whose result is illustrated in Fig. 1. Lemma 1. Let Γ be a C k (k ≥ 3) non-absolutely focusing curve of minimal length enclosing an angle of no more than π , and let γ be a part of a billiard trajectory with N consecutive reflections off of Γ (possibly off its endpoints). Then there exists a family of trajectories (γ ) ≥0 with a C k−1 dependence on , such that γ0 = γ , and for all > 0 the trajectory γ has a sequence of N consecutive reflections off of Γ with no reflections off its endpoints. Proof. If all reflections of γ off of Γ are already in the interior of Γ , just set γ ≡ γ , so that we only consider such a γ with at least one reflection off of an endpoint of Γ in the following. If N = 1, then denote by γ the trajectory obtained by moving the point of reflection of γ by into the interior of Γ , while, say, keeping the angle of reflection constant. For N = 2, denote by γ the trajectory obtained by moving both points of reflection of γ by into the interior. The corresponding angles of reflection are then determined by the location of the new points of reflection. It remains to consider the case N ≥ 3. Since Γ does not enclose an angle of more than π , there are no back reflections possible. Hence the second point of reflection must be in the interior of Γ . Furthermore, by assumption on Γ , the sub-arc on which the 2nd up to the N th reflection take place is absolutely focusing.

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To construct γ proceed as shown in Fig. 1. Increase the (absolute value of the) angle of reflection ϕ2 by . Then Proposition 3.6 in [14] shows that the i th reflection points, 3 ≤ i ≤ N , all move closer to the second one. And since the billiard map is of class C k−1 , the implicit function theorem asserts in this case that the dependence of new points of reflection on is C k−1 . Since the location of the second reflection does not change, the general twist property applied to the first reflection point shows that it also moves towards the second reflection point. Therefore the family of rays γ depends C k−1 –smooth on , and γ for > 0 has reflections only off the interior of Γ , as desired. Remark 2. Although the result of Lemma 1 seems entirely obvious, especially when looking at Fig. 1, this is not so. In fact, this is the only point in this paper where the restriction to curves enclosing an angle of no more than π comes in. The reason is that, in general, changing the angle of incidence (at some point of reflection) in either direction may not move all points of reflection into the interior of Γ , as stated in Lemma 1, because of possibly existing back-reflections. This makes the argument in the construction of the family of orbits of Lemma 1 fail in general when applying it to curves enclosing an angle of more than π . However, there are certainly situations where the result of Lemma 1 is true for such curves as well, e.g. for extensions to a curve enclosing an angle of more than π which do not destroy the constructed family of orbits. It was conjectured in [6] that for any non-absolutely focusing curve Γ there exists an infinitesimal beam of parallel rays falling onto Γ and leaving it as parallel beam. The next statement proves this claim in a slightly more restricted setting. Proposition 1. Let the C k (k ≥ 3) curve Γ be a small enough extension of a nonabsolutely focusing curve of minimal length, which encloses an angle of no more than π . Then there exists a family of rays (γ ) ≥0 , depending C k−1 –smoothly on , such that they all have the same number of reflections off of Γ , and a parallel beam sent along γ leaves Γ as a focusing one (parallel one) if > 0 ( = 0). Proof. Consider first the case where Γ is non-absolutely focusing of minimal length, enclosing an angle of no more than π . Then, by definition, there exists an incoming infinitesimal beam of parallel rays, say γ0 , which leaves Γ after a sequence of consecutive reflections as either a parallel or dispersing beam. By Lemma 1 there is a C k−1 –smooth family (γ ) ≥0 such that for any > 0, the ray γ has reflections off the interior of Γ only. Since any sub-arc of Γ is absolutely focusing, a parallel incoming beam sent along γ must leave as a focusing one. Hence the continuous dependence on shows that the parallel incoming beam along γ0 must leave as a parallel one, and the family (γ ) is as desired. Clearly, the constructed family (γ ) ≥0 is not destroyed if we allow for small enough extension of Γ , hence the above construction carries over to this more general case. In order to construct a linearly stable periodic orbit it is more convenient to work with the billiard flow, rather than the billiard map. Later on we will see that this changes when we want to establish nonlinear stability. The next statement shows that if a curve Γ is non-absolutely focusing, then, regardless of how large the free paths before and after a series of consecutive reflections off


133

of Γ are, a linearly stable periodic orbit can exist. To establish later nonlinear stability we need this (linearly stable) periodic orbit to be non-resonant, that means that the eigenvalues λ = e±i α of the corresponding monodromy matrix satisfy λ2 , λ3 , λ4 = 1. Theorem 2. Let Γ be a C 3 non-absolutely focusing curve of minimal length which encloses an angle of no more than π , or a small enough extension of such a curve. Then for every L > 0 there exists a billiard table Q(L , Γ ) with Γ as a focusing boundary component which has a linearly stable, non-resonant periodic orbit γ with free path of length at least L before and after a sequence of consecutive reflections off of Γ . Proof. Let (γ ) ≥0 be as in Proposition 1. From (1) we conclude that the reduced Jacobian U along γ must read U11 + a U = c

U12 + b 1 U11 + d

with a , b , c , d = o(1) as 0 .

The values of U11 and U12 satisfy U12 c + b c = U11 d +

a + a d U11

because det U = 1. Since Bout (γ ) < 0 for all > 0 holds, we must have c < 0 hence c sgn U11 < 0 U11 + a for all small enough. Let L > 0 be arbitrary, and consider τ with τ > 6 (L + |Γ |) whose value will be chosen in the following. For every > 0 we can close up γ to a periodic orbit γ˜ using three plane mirrors such that the length of the free path before and after the sequence of reflections off of Γ is at least L, and the total length of the orbit away from Γ is τ . This is illustrated in Fig. 2. The monodromy matrix M along γ˜ then reads M = −U

1 0

U11 + a τ =− c 1

U12 + b + τ U11 + τ a , 1 U11 + d + τ c

where the minus sign is due to the three reflections at the plane mirrors. In particular, the trace of M is − tr M = U11 + a +

1 + d + τ c U11

134


Fig. 2. Construction of the periodic orbit using plane mirrors such that the resulting orbit has free paths before and after hitting Γ of length at least L

so that − sgn U11 tr M = |U11 | +

1 |U11 |

− τ |c | + (a + d ) sgn U11

holds for all > 0, where we used c sgn U11 < 0. Now let ∗ be so small that the relations 2 3

|U11 | +

1 |U11 |

1 3 ≤ |U11 | + + (a ∗ + d ∗ ) sgn U11 ≤ |U11 | 2

|U11 | +

1 |U11 |

and |c ∗ | 6 (L + |Γ |)
0, consider a family of C 5 functions L : U × U → R2 for | | < 0 which satisfy s 4 + s14 + O5 (s, s1 ) and ∂s ∂s1 L 0 (0, 0) = 0 ∂ =0 L (s, s1 ) = C 24

(2)

136


for some C = 0. Denote the family of area-preserving maps generated by L by T (see Sects. 47 and 48 of [1] for details of this construction) which we will write in coordinate form as T (s, y) ≡ (S (s, y), Y (s, y)) . Assume further that the map T0 has an elliptic fixed point (s∗ , y∗ ), T (s∗ , y∗ ) = (s∗ , y∗ ) with y∗ :=∂s1 L 0 (s∗ , s∗ ) = −∂s L 0 (s∗ , s∗ ), and denote the eigenvalues of DT0 (s∗ , y∗ ) by λ = e±iα . We will assume that the nonresonance conditions λ2 , λ3 , λ4 = 1 hold for the elliptic fixed point of T0 . Let us set j

L i j :=∂si ∂s1 L 0 (s∗ , s∗ ) to simplify the notations. Proposition 2. The derivative of T0 at the fixed point (s∗ , y∗ ) reads 1 1 L 20 DT0 (s∗ , y∗ ) = − , L 11 L 20 L 02 − L 211 L 02 and the family of maps T satisfies ⎞ ⎛ 1 3 s 1 L 11 ∂ =0 T (s + s∗ , y + y∗ ) = −C ⎝ L 02 3 L 20 s+y 3 ⎠ + O4 (s, y) 6 L s + L 11 11 for s and y in a neighborhood of zero. Proof. The definition of the maps T (s, y) ≡ (S (s, y), Y (s, y)) in terms of the generating function L , ∂s L (s, S (s, y)) = −y and ∂s1 L (s, S (s, y)) = Y (s, y),

(3)

immediately shows 1 DT0 (s∗ , y∗ ) = − L 11

L 20 L 20 L 02 − L 211

1 L 02

for the derivative of T0 at the fixed point (s∗ , y∗ ). Differentiating Eqs. (3) with respect to , and evaluating at = 0 yields [s − s∗ ]3 + ∂s ∂s1 L 0 (s, S0 (s, y)) ∂ =0 S (s, y) 6 +O4 (s − s∗ , y − y∗ ), [S0 (s, y) − s∗ ]3 + ∂s21 L 0 (s, S0 (s, y)) ∂ =0 S (s, y) ∂ =0 Y (s, y) = C 6 +O4 (s − s∗ , y − y∗ ), 0=C

(4)


137

where we used the special property of the generating functions stated in Eq. (2). Solving now for the -derivatives of S and Y we obtain ∂ =0 S (s + s∗ , y + y∗ ) = −C

s3 + O4 (s, y), 6 L 11 1 L 20 s + y 3 L 02 s 3 ∂ =0 Y (s + s∗ , y + y∗ ) = −C −C + O4 (s, y) 6 L 11 6 L 11 by Taylor-expansion and (4).

The point of the specific form of the generating functions (2) is now clear. By Proposition 2 the change of the map to first order in is of third order in (s, y). Comparing this to the general form of the Birkhoff coefficients A , as given in Lemma 2, we see that ∂ | =0 A only involves the third order term Im c21 . This we can further exploit to obtain the main result of this section, which is the following Theorem 3. Let (s∗ , y∗ ) be a non-resonant elliptic fixed point of a family of planar area-preserving maps T which are generated in a neighborhood of (s∗ , y∗ ) by L satisfying (2), i.e. s 4 + s14 + O5 (s, s1 ) for some C = 0 ∂ =0 L (s + s∗ , s1 + s∗ ) = C 24 and ∂s ∂s1 L 0 (s∗ , s∗ ) = 0. Then there exists an ∗ > 0 such that for every ∈ (− ∗ , ∗ ) \ {0} the point (s∗ , y∗ ) is a nonlinearly stable fixed point of T with a nonzero first Birkhoff coefficient. Proof. Without loss of generality assume that (s∗ , y∗ ) = (0, 0), so that we are exactly in the setting discussed so far in this section. Also, by rescaling by C we may also assume that C = 1. Combining the general expression of the Birkhoff coefficient of Lemma 2 with the specific structure of the considered family of maps as given in Proposition 2 we obtain ∂ =0 A = ∂ =0 Im c21 1 b a 10 01 = a10 −∂ a21 + 3 ∂ a03 − 3 ∂ | =0 b30 + ∂ | =0 b12 8 a01 =0 b10 =0 a01 a01 1 ∂ a30 − ∂ | =0 b21 + 3 ∂ | =0 b03 , − b10 ∂ a12 − 3 8 b10 =0 b10 =0 because only third order terms appear in ∂ | =0 T . Using the Taylor expansion we can express the various coefficients ai j and bi j in terms of L i j as a10 = −

L 20 1 a10 b01 − 1 L 02 , a01 = − , b10 = , b01 = − L 11 L 11 a01 L 11

138


Fig. 3. General billiard table Q with an elliptic, non-resonant periodic orbit γ

and ∂ =0 a30 ∂ =0 a21 ∂ =0 a12 ∂ =0 a03

= − 16

1 L 11

=

a01 6

,

=0, =0, =0,

∂ =0 b30 ∂ =0 b21 ∂ =0 b12 ∂ =0 b03

= − 16 = − 21 = =

− 21 − 16

L 02 L 11 L 220 L 311 L 20 L 311 1 L 311

− = = =

3 1 L 20 1 6 L3 = 6 11 1 2 2 a10 a01 , 1 2 2 a10 a01 , 1 3 6 a01 ,

3 ], [b01 + a10

again by using the result of Proposition 2. Therefore we obtain 4 ) a 2 (1 + a10 2 2 2 2 − a01 a10 b01 + a01 + 2 a10 a01 . 16 ∂ =0 A = − 01 a10 b01 − 1

Since we assume that the fixed point is elliptic, we must have | tr DT0 | = |a10 +b01 | < 2 and a01 = 0. With t:=

a10 + b01 ∈ (−1, 1) 2

the above becomes a2 ∂ =0 A = 01 8

2 1 + a10

2 a01 1 + 2 (t − a10 )2 >0. ≥ 1 − t 2 + (t − a10 )2 8

Hence there exists an ∗ > 0 such that A = 0 holds true for all ∈ (− ∗ , ∗ ) \ {0}. Moser’s twist theorem then implies (see Theorem 2.13 in [19]) that the fixed point (s∗ , y∗ ) is nonlinearly stable with a nonzero first Birkhoff coefficient for all maps T with ∈ (− ∗ , ∗ ) \ {0}. 5. Construction of Nonlinearly Stable Periodic Orbits In this section we consider the nonlinear stability problem for a linearly stable periodic orbit γ on a general billiard table Q. A typical situation is shown in Fig. 3. Let N denote the number of reflections of γ , and let si and ϕi , i = 1, . . . , N , denote the arc length parameter and angle of reflection at the i th reflection point of γ , respectively. As before, let Φ t and F denote the billiard flow and billiard map, respectively.


139

Fig. 4. Construction of the curves Γ

Since we assume that the periodic orbit γ is linearly stable the eigenvalues of the linearized billiard map are complex conjugate of modulus one, hence λ1 = λ¯ 2 = ei α ≡ λ are the eigenvalues of DF N (s1 , sin ϕ1 ) for some angle α. Furthermore we assume that γ is non-resonant, i.e. λ2 , λ3 , λ4 = 1 holds. In the non-resonant case a result due to Moser (see Theorems 2.12 and 2.13 in [19]) guarantees that there exists a real-analytic canonical change of coordinates (s, sin ϕ) → z ∈ C in a neighborhood of any of the (si , sin ϕi ) such that it conjugates F N to its Birkhoff normal form z → λ z ei A |z| + O(|z|4 ) 2

with A ∈ R the first Birkhoff coefficient. A sufficient condition for nonlinear stability of γ then is a nonzero value for A, cf. Theorem 2.13 in [19]. Applying this strategy to our general setup seems intractable because we would need a way to decide whether or not A vanishes. Therefore we will not consider the nonlinear stability problem of γ on the given table Q. Instead we will introduce a family of tables Q , which are almost identical to Q and have γ as a periodic orbit. We then want to know whether γ is nonlinearly stable for at least some tables in that family. A construction of such a family, which allows us to analyze A as a function of the table was given in [13,15] in the context of two-periodic orbits. It consists of a local perturbation of the boundary curve in normal direction, maintaining a third order contact, as shown in Fig. 4. Lemma 3 (Local perturbation in normal direction). Let Γ be a C 5 curve, parametrized by its arc length s, for s in some interval I containing a neighborhood of s = 0. Let φ : I → R be a C 5 function with φ(0) = φ (0) = φ (0) = φ (0) = 0. Then the parametrization of the curve Γφ (ξ ):=Γ (ξ ) + φ(ξ ) n(ξ ) by its arc length reads Γφ (s) = Γ (s) + φ (0) n 0

s4 + O(s 5 ) 24

in a neighborhood of s = 0. Proof. Denote by t (s) = Γ (s) and n(s) = t (s)⊥ the tangent and normal vector of Γ (s), respectively. The definition of the curvature K(s) of Γ (s) yields n (s) = K(s) t (s)

140


Fig. 5. An arbitrarily small local perturbation of the boundary (in the sense of the C 4 topology) of the billiard table that preserves the elliptic periodic orbit and renders it nonlinearly stable

hence Γφ (ξ ) = [1 + K(ξ ) φ(ξ )] t (ξ ) + φ (ξ ) n(ξ ) . In particular we obtain Γφ (ξ ) = [1 + K(ξ ) φ(ξ )]2 + φ (ξ )2 = 1 + O(ξ 4 ) for the length of the tangent vector of Γφ . Therefore S (ξ ) =

0

ξ

Γφ (ζ ) dζ = ξ + O(ξ 5 )

for the arc length of Γφ . The claim that Γφ and Γ have a third order contact (in terms of their arc length parametrizations) at the reference point Γ (0) follows now immediately from the definition of Γφ and Taylor expansion of φ and n. This perturbation of the boundary was used in [13,15] to study the stability of two-periodic orbits on strictly convex billiard tables. More precisely, it was shown that the first Birkhoff coefficient can be made nonzero using an arbitrary small local boundary perturbation of the above type. The following theorem generalizes this construction to arbitrary linearly stable, non-resonant periodic orbits on arbitrary billiard tables. Theorem 4. Consider a billiard table Q with a linearly stable, non-resonant N -periodic orbit γ , which has reflections off of C 5 boundary components. Let s1 , . . . , s N denote the arc length parameters corresponding to the N points of reflection. Pick any of the si , say si∗ . Then for all small enough δ > 0 there exists a billiard table Q δ,i∗ , which is C 5 close to Q and coincides with Q outside the δ-neighborhood Bδ (Γ (si∗ )) of Γ (si∗ ), and such that γ is a nonlinearly stable periodic orbit on Q δ,i∗ with nonzero Birkhoff coefficient. Proof. After possibly relabeling the points, we may assume that i ∗ = 1, and without loss of generality we may choose the arc length parametrization of ∂ Q such that s1 = 0. Denote the boundary components of the i th point of reflection by Γi . Let δ > 0 be arbitrary, but so small that Bδ (Γ (s1 )) contains no other points of reflection, and has intersection only with the boundary component Γ1 (see Fig. 5 for an illustration).


141

Choose a C ∞ function φ : R → R with support in [−δ/2, δ/2] such that φ(0) = = φ (0) = φ (0) = 0, φ (0) = 1, |φ(s)| ≤ 1. Let n(s) denote the unit normal vector at the point Γ1 (s). Define

φ (0)

Γ1 (t):=Γ1 (t) + φ(t) n(t) for all ∈ [−δ/2, δ/2]. Clearly the graphs of Γ1 and Γ1 coincide outside Bδ (Γ (0)). Moreover, for small enough Γ1 has no self-intersections and thus defines a new billiard table Q by replacing Γ1 by Γ1 . By Lemma 3 Γ1 (s) = Γ1 (s) +

s4 + O(s 5 ) 24

in a neighborhood of s = 0. In particular, the tangent line at s = 0 is preserved, and hence the orbit γ remains N -periodic for all Q . In fact, since the generating function of the N th iterate of the billiard map in a small enough neighborhood of the periodic orbit γ reads L (s1 , . . . , s N , s N +1 ) = Γ2 (s2 ) − Γ1 (s1 ) + · · · + Γ N (s N ) − Γ N −1 (s N −1 ) + Γ (s N +1 ) − Γ N (s N ) , 1

the above expansion of

Γ1

immediately implies

Γ2 (s2 ) − Γ1 (s1 ) ∂ =0 L (s1 , . . . , s N , s N +1 ) = − · ∂ =0 Γ1 (s1 ) Γ2 (s2 ) − Γ1 (s1 ) Γ1 (s N +1 ) − Γ N (s N ) + · ∂ =0 Γ1 (s N +1 ) Γ1 (s N +1 ) − Γ N (s N ) s 4 + s N4 +1 + O5 (s1 , s N +1 ), = − cos ϕ1 1 24 because only two terms in L really depend on . The boundary curves are of class C 5 in a neighborhood of the reflection points, hence the associated (N -fold iteration of the) billiard map is of class C 4 . Since cos ϕ1 > 0 we can apply Theorem 3 to conclude that for all small enough nonzero values of , the N -periodic orbit γ is nonlinearly stable on Q . Let Q ∗ (Γ, L) denote one such table on which the periodic orbit γ is nonlinearly stable with nonzero first Birkhoff coefficient. Now we are in the position to prove our main result as stated in Theorem 1. Proof of Theorem 1. Let L > 0 be arbitrary, and Γ as in the statement of Theorem 1. By Theorem 2 there exists a billiard table Q(L , Γ ) with Γ ⊂ ∂ Q which has a linearly stable, non-resonant periodic orbit γ with free path of length at least L before and after a sequence of consecutive reflections off of Γ . Applying now the result of Theorem 4 we may assume that this orbit is actually non-linearly stable with nonzero Birkhoff coefficient, otherwise we slightly modify the original table Q(L , Γ ) as stated in Theorem 4, which does not change the length of the free path γ before and after Γ to a value less than L.

142


Notice that by the implicit function theorem the periodic orbit γ persists under any (not just the ones of the type as in Lemma 3, cf. the proof of Theorem 4) sufficiently small enough C 5 perturbation of the boundary of the constructed billiard table. The perturbed periodic orbits will have the same period and are elliptic with a free path before and after Γ of length at least L. By Lemma 2 the first Birkhoff coefficient depends C 3 continuously on the map, hence C 4 -continuously on the boundary. Thus all the perturbed periodic orbits will have a nonzero Birkhoff coefficient, and hence are non-linearly stable, provided that the perturbation of the boundary is small enough. 6. Conclusion It is well known that if at least one focusing component is present, then in order to ensure hyperbolicity the boundary should be arranged in such a way that all beams of rays focus after any sequence of consecutive reflection off the focusing part of the boundary, and defocus (i.e. pass through a conjugate point) before the next reflection from the curved (non-flat) part of the boundary of the billiard table. So far, there are only two examples [9,10] of hyperbolic billiards where this condition of defocusing between any reflection from a focusing component and the next reflection off the curved part of the boundary is violated. However, both of these classes of billiards are very special. In fact, the absence of defocusing between some reflections from a focusing part of the boundary and the next reflection (off the curved part of the boundary) is allowed in [9] only in those parts of the phase space where the billiard dynamics is integrable. Therefore focusing does not dominate dispersing in this part of the phase space while on the complementary part of the phase space the defocusing mechanism in the usual way ensures hyperbolicity. Likewise, in [10] the billiard table has a very specially designed part where one can control that focusing is dominated by dispersing, although the beams do not defocus after reflections off the focusing component, while on the complement of this part of phase space, yet again, the defocusing mechanism generates hyperbolicity. Therefore, one may conclude that in all hyperbolic billiards with focusing boundary components constructed so far, it is the mechanism of defocusing which is responsible for hyperbolicity. Hence, the standard strategy to construct ergodic chaotic billiards is to choose all focusing components to be absolutely focusing, and to move them sufficiently far away from all other regular (smooth) components of the boundary, e.g. [12]. In this paper we have shown that for this strategy it is very essential that all focusing components are absolutely focusing, and conjecture that in two-dimensional billiards with at least one focusing, but non-absolutely focusing boundary component typically there are stable periodic orbits. References 1. Arnold, V.I.: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics. Vol. 60, second ed. New York: Springer-Verlag, 1989 2. Boldrighini, C., Keane, M., Marchetti, F.: Billiards in polygons. Ann. Probab. 6(4), 532–540 (1978) 3. Bunimovich, L.A.: On ergodic properties of certain billiards. Funk. Anal. i Priložen. 8(3), 73–74 (1974) 4. Bunimovich, L.A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65(3), 295–312 (1979) 5. Bunimovich, L.A.: Many-dimensional nowhere dispersing billiards with chaotic behavior. Phys. D 33(1–3), 58–64 (1988) 6. Bunimovich, L.A.: Conditions of stochasticity of two-dimensional billiards. Chaos 1(2), 183–187 (1991)


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7. Bunimovich, L.A.: On absolutely focusing mirrors. In: Ergodic Theory and Related Topics, III (Güstrow, 1990), Lecture Notes in Math., Vol. 1514, Berlin: Springer, 1992, pp. 62–82 8. Bunimovich, L.A.: Absolute focusing and ergodicity of billiards. Regul. Chaotic Dyn. 8(1), 15–28 (2003) 9. Bunimovich, L.A., Del Magno, G.: Track billiards. Commun. Math. Phys. 288, 699–713 (2009) 10. Bussolari, L., Lenci, M.: Hyperbolic billiards with nearly flat focusing boundaries, I. Physica D 237(18), 2272–2281 (2008) 11. Chernov, N., Markarian, R.: Chaotic billiards. In: Mathematical Surveys and Monographs. Vol. 127. Providence, RI: Amer. Math. Soc., 2006 12. Del Magno, G., Markarian, R.: On the Bernoulli property of planar hyperbolic billiards, 2006, available at http://www.ma.utexas.edu/mp_arc/c/06/06-164.pdf 13. Dias Carneiro, M.J., Oliffson Kamphorst, S., Pintode Carvalho, S.: Elliptic islands in strictly convex billiards. Erg. Th. Dynam. Syst. 23(3), 799–812 (2003) 14. Donnay, V.J.: Using integrability to produce chaos: billiards with positive entropy. Commun. Math. Phys. 141(2), 225–257 (1991) 15. Kamphorst, S.O., Pinto-de Carvalho, S.: The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards. Exp. Math. 14(3), 299–306 (2005) 16. Lazutkin, V.F.: Existence of a continuum of closed invariant curves for a convex billiard. Usp. Mat. Nauk 2(3(165)), 201–202 (1972) 17. Lazutkin, V.F.: Existence of caustics for the billiard problem in a convex domain. Izv. Akad. Nauk SSSR Ser. Mat. 37, 186–216 (1973) 18. Markarian, R.: Billiards with Pesin region of measure one. Commun. Math. Phys. 118(1), 87–97 (1988) 19. Moser, J.: Stable and Random Motions in Dynamical Systems. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 2001, with special emphasis on celestial mechanics, Reprint of the 1973 original, with a foreword by Philip J. Holmes 20. Sina˘ı, Y.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Usp. Mat. Nauk 25(2 (152)), 141–192 (1970) 21. Wojtkowski, M.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105(3), 391–414 (1986) Communicated by G. Gallavotti


Communications in


Gibbs Ensembles of Nonintersecting Paths Alexei Borodin1,2 , Senya Shlosman2,3 1 Department of Mathematics, Caltech, Pasadena, USA. E-mail: [email protected] 2 IITP, RAS, Moscow, Russia. E-mail: [email protected] 3 Centre de Physique Theorique, CNRS, Luminy, Marseille, France. E-mail: [email protected]

Received: 29 June 2008 / Accepted: 14 July 2009 Published online: 30 August 2009 – © Springer-Verlag 2009

Abstract: We consider a family of determinantal random point processes on the twodimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are non-translation-invariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices. 1. Introduction It is well known that the Gibbs random fields are defined via prescribing their conditional distributions. In the case of the nearest-neighbor interactions these distributions are given by some relatively simple relations. The computation of their correlation functions is on the other hand usually a very difficult problem, because it requires the passing to the thermodynamic limit. In comparison, the determinantal random fields (or random point processes, both terms are used) are defined in such a way that the correlation functions are given by relatively simple direct formulas, while the computation of the conditional distributions may again require taking the thermodynamic limit, since the dependence is usually long-range. The purpose of the present paper is the study of some 2D random fields n = n t = 0, 1, t ∈ Z2 , which are both determinantal and have in addition some kind of Gibbs property. We wanted to understand which properties of the kernel produce the Gibbsianity of the random field. It turns out that the property sought is some linear relation on the matrix elements of the kernel. Below we explain this statement for a certain class of 2D determinantal random fields. Every random assignment of the probabilities Pr field is specified by the (consistent) to the events n ti = 1, ti ∈ I ; n t j = 0, t j ∈ J , for any two non-intersecting finite sets

146

A. Borodin, S. Shlosman

I, J ⊂ Z2 . We say that a random field n is a determinantal random field with the kernel K (·, ·) , if for every finite set I = ti ∈ Z2 , Pr K n ti = 1, ti ∈ I = det K si , s j s ,s ∈I . (1) i

j

The inclusion-exclusion principle then implies that the probability of a more general event Pr K n ti = 1, ti ∈ I ; n t j = 0, t j ∈ J = (−1)h det K˜ si , s j , si ,s j ∈I ∪J

where

⎧ ⎨K K˜ si , s j = K ⎩ K

si , s j if si = s j , if si = s j ∈ I, (s i , si ) s j , s j − 1 if si = s j ∈ J,

and h is the number of “holes”, i.e. indices t j ∈ J. We will refer to the sites with values 1 as “particles”. The above formula is sometimes referred to as the complementation principle, cf. A.3 of [3]. In this paper we study random fields n, corresponding to the kernels K constructed as follows. Suppose that for every k ∈ Z1 the function ψk (u) is given, which can be one of the following four functions:

−1

−1 , 1 − αk− u −1 , 1 + βk+ u , 1 + βk− u −1 , 1 − αk+ u with positive constants α ± , β ± . Let us also fix a complex number z with z > 0 and denote C± any contour that joins z¯ and z and crosses the real axis at a point of R± . For t = (σ, x) , t = (τ, y) we define ⎧ −1 τ ⎪ ⎪ 1 du ⎪ ⎪ ψk (u) if σ < τ, ⎪ x−y+1 ⎪ u ⎪ 2πi C+ k=σ +1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1 du K (σ, x; τ, y) ≡ K σ,τ (x − y) = ifσ = τ, (2) x−y+1 ⎪ 2πi C+ u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ ⎪ ⎪ 1 du ⎪ ⎪ ψk (u) x−y+1 if σ > τ. ⎪ ⎩ 2πi C u − k=τ +1

Our main results concerning such determinantal random fields are threefold: 1. Due to the (particles or holes) interlacing property of our fields n, they can be interpreted as ensembles of non-intersecting infinite random lattice paths. (These ensembles are different for different kernels, and will be described in detail below.) 2. The collections ω = {ωi } of random lattice paths thus obtained are “Gibbs random paths ensembles”. They are defined by the action functional S K (ω) , which is local and is determined by the parameters of the kernel K . 3. The validity of the above two statements follows from simple linear relations that the matrix elements of the kernel K satisfy, and do not depend on K otherwise.

Gibbs Ensembles of Nonintersecting Paths

147

1.1. Gibbs path ensembles. We will define the Gibbs random path ensemble, corresponding to the additive functional S. Here S is a function defined on the set of all finite selfavoiding lattice paths ω, which has the additivity property: if ω = ω1 ∪ ω2 , ω1 ∩ ω2 = {t} ∈ ω, then S (ω) = S (ω1 ) + S (ω2 ). Let µ be a probability distribution on the set of families of non-intersecting double-infinite polygonal lattice paths ω = {ωi , −∞ < i < ∞}. Let ⊂ Z2 be a finite box, i.e. a finite connected subset of Z2 with connected complement. Let the paths ωi be fixed outside . We denote the restriction of ω to the complement of by ω¯ . Some of the paths from ω are entering and exiting . Let P be the set p1 , . . . , pk ∈ ∂ of all the entrance points to , while Q is the set of the exit points q1 , . . . , qk ∈ ∂ from . Let = 1 , . . . , k be a collection of non-intersecting lattice paths contained in , joining the points p1 , . . . , pk and q1 , . . . , qk . We denote the set of such k-tuples of paths by (P, Q). We say that the measure µ is a Gibbs measure with the action functional S, if for every triple (, P, Q) the conditional distributions of µ satisfy

exp {S ( ) + · · · + S ( )} 1 k µ ω¯ = , (3) Z (, P, Q) where Z (, P, Q) = ∈ (P,Q) exp {S ( 1 ) + · · · + S ( k )} is the partition function.

1.2. The interlacing property. This property of the random field n holds almost surely with respect to the measure Pr K , as we will show below. Its exact formulation is different at different locations and depends on the structure of the kernel K at this location. The picture on Fig. 1 illustrates the various cases which are described below. −1 If for some k we have ψk (u) = 1 − αk+ u , then in the strip Rk = (σ, x) ∈ Z2 : σ = k, k + 1 the following property holds Pr K -a.s.: For any two particles n (k,x1 ) = n (k,x2 ) = 1, x1 < x2 , of the configuration n, separated by a string of holes, n (k,x) = 0 for all x1 < x < x2 , we find on the neighboring line σ = k + 1 exactly one particle n (k+1,x) = 1 sitting in the set {(k + 1, x) : x1 ≤ x < x2 } , and the rest of points of this set host holes. This is the upward interlacing of particles. Consider now the correspondence π + , which n (k,x) = 1 the assigns to a particle particle n (k+1,π + (x)) = 1, where π + (x) = min y ≥ x : n (k+1,y) = 1 ≥ x. The correspondence π + is one-to-one with probability one. Let us connect each particle n (k,x) = 1 with the corresponding particle n (k+1,π + (x)) = 1 by the three-link path ωx,π + (x) = (k, x) , k + 21 , x ∪ k + 21 , x , k + 21 , π + (x) ∪ k + 21 , π + (x) , k + 1, π + (x) . (4) Then for different particles n (k,x ) = 1, n (k,x ) = 1 the connectors ωx ,π + (x ) , ωx ,π + (x ) do not intersect. We define the action S on each of these connectors by S ωx,π + (x) = π + (x) − x ln αk+ . (5) + Clearly, exp S ωx,π + (x) = (αk+ )π (x)−x , which can be seen as the product of factors αk+ over the unit pieces of the connector. −1 the situation is very similar, except the In the case when ψk (u) = 1 − αk− u −1 upward interlacing is replaced by the downward interlacing: The correspondence π + is

148


Fig. 1. The first two columns, together with the 5th and the 9th display ascending β-paths, the 3rd and the 4th – ascending α-paths, the 6th and the 10th – the descending β-paths, the 7th and the 8th – the descending α-paths

replaced by π − , which assigns to a particle n (k,x) = 1 the particle n (k+1,π − (x)) = 1, where π − (x) = max y ≤ x : n (k+1,y) = 1 ≤ x. Again, π − is one-to-one a.s., and the (downward) connectors ωx,π − (x) = (k, x) , k + 21 , x ∪ k + 21 , x , k + 21 , π − (x) ∪ k + 21 , π − (x) , k + 1, π − (x) (6) do not intersect. The action S is given by S ωx,π − (x) = x − π − (x) ln αk− . (7) For ψk (u) = 1 + βk+ u we have the upward interlacing of holes: If in the configuration n we have two holes n (k,x1 ) = n (k,x 2 ) = 0, x1 < x2 , separated by the string of particles n (k,x) = 1 for all x1 < x < x2 , then on the neighboring line σ = k + 1 we have Pr K -a.s. exactly one hole n (k+1,x) = 0 in the set {(k + 1, x) : x1 < x ≤ x2 } , while the rest of the points in this set is filled by the particles. We then define a correspondence χ + , assigning to every particle on the σ = k line a particle on the σ = k + 1 line, as follows. Take any string of consecutive particles n (k,x) = 1 for all x1 < x < x2 which is maximal, i.e. n (k,x1 ) = n (k,x2 ) = 0. We put χ + (x1 + 1) = min x ≥ x1 + 1 : n (k+1,x) = 1 , and then proceed inductively by putting χ + (x + 1) = min y > χ + (x) : n (k+1,y) = 1 .


149

The hole interlacing implies that χ + is a.s. well-defined, is one-to-one, and that for every particle n (k,x) = 1 we have either χ + (x) = x or χ + (x) = x +1. The particle connectors, which in this case are segments ωx,χ + (x) = (k, x) , k + 1, χ + (x) ,

(8)

clearly do not intersect each other. We put S ωx,χ + (x) =

0 ln βk+

if χ + (x) = x, if χ + (x) = x + 1.

(9)

This means that exp S ωx,χ + (x) is βk+ if the connector has slope 1, and 1 if the connector has slope 0. For ψk (u) = 1 + βk− u −1 we have likewise the downward interlacing of holes. The correspondence χ + is replaced by χ − , with the property that either χ − (x) = x or χ − (x) = x − 1. The connectors are non-intersecting segments ωx,χ − (x) = (k, x) , k + 1, χ − (x) ,

(10)

and we define S ωx,χ − (x) =

0 ln βk−

if χ − (x) = x, if χ − (x) = x − 1.

(11)

1.3. Main result. Now we can formulate our claims more precisely. Let us fix a sequence of functions ψk (u) , k ∈ Z1 , such that for every k the function ψk (u) is one of the four −1 −1 functions 1 − αk+ u , 1 − αk− u −1 , 1 + βk+ u , 1 + βk− u −1 , with α∗± , β∗± > 0. Let us also fix a complex number z, z > 0. Theorem 1. i) The kernel (2) defines a determinantal random field n on Z2 , which is invariant with respect to the shifts of the second coordinate. ii) The random field n possesses the interlacing property as defined in Sect. 1.2. In particular, there is a map ω, assigning to Pr K -a.e. realization of n a countable collection of non-intersecting lattice paths ωn = {ωi } , passing through all the particles of the configuration n. The construction of the collection ωn is given by (4), (6), (8) and (10). iii) The random paths ω thus constructed form the Gibbs Path Ensemble, as defined in Sect. 1.1. It corresponds to the action functional S, given by the formulas (5), (7), (9) and (11). iv) item For every k ∈ Z the matrix elements of the kernel K satisfy the following relations: −1 for the case ψk (u) = 1 − αk+ u , K k−1,τ (x − y) − δ

x=y τ =k−1

= K k,τ (x − y) − αk+ K k,τ (x − y − 1) ,

K σ,k (x − y) − δ x=y = K σ,k−1 (x − y) − αk+ K σ,k−1 (x − y − 1) ; σ =k

150


for the case ψk (u) = 1 + βk+ u , K k,τ (x − y) = K k−1,τ (x − y) − δ x=y τ =k−1 + + βk K k−1,τ (x − y − 1) − δ x=y+1 , τ =k−1 + K σ,k−1 (x − y) = βk K σ,k (x − y − 1) − δ σ =k + K σ,k (x − y) − δσ =k ; x=y+1

x=y

and similar relations for the α − , β − cases. The determinant identities, expressing the properties ii) and iii) above, are corollaries of these relations only, and thus hold true for any other kernel K , satisfying them. 2. Examples 1. Our first example will be Gibbs ensembles of the β-paths, introduced in (8) , (10). These are collections of non-intersecting infinite paths {ωi } on Z2 , such that if a path visits the point (σ, x) , then its next link is either the segment [(σ, x) , (σ + 1, x)] or the segment [(σ, x) , (σ + 1, x + 1)]. Now let = ( p, q) be a finite piece of such β-path, where p, q are the end-points of . We define the energy U ( ) of this path in the following way. Let − ( p, q) be the β-path, which is the lowest among all the β-paths connecting p and q. Then exp {−U ( )} is by definition the area surrounded by the loop ( p, q) ∪ − ( p, q). For a collection = { i } of finite paths we define H ( ) = i U ( i ). We call the measure µ on the ensemble ω of non-intersecting infinite β-paths the Gibbs measure corresponding to the energy H and the inverse temperature τ, if it has the following property. Let ⊂ Z2 be a finite volume, and the sets P = { p1 , . . . , pk ∈ ∂} , Q = {q1 , . . . , qk ∈ ∂} of the entrance points and exit points are fixed. Then the conditional distribution of µ on (P, Q) = { } under the condition that the path configuration ω¯ is fixed outside is given by the formula

exp {−τ H ( )} . (12) µ ω¯ = Z¯ (, P, Q) This definition is just a convenient rewriting of the relation (3) above. The advantage is that our function H here is manifestly translation-invariant. Our main result implies in particular that the determinantal random fields µκ,z defined by the kernel K = K (κ, z) with the functions ψk (u) = 1 + κekτ u , interpreted as path measures, are Gibbs measures with the energy H and the inverse temperature τ. Here κ > 0 is any real number. When the temperature τ −1 goes to zero, the Gibbs measures (12) are concentrated on ground-state configurations, which are local minima of the energy H. For low temperatures they are concentrated on configurations which are small perturbations of the ground state configurations, see Fig. 2. Note that the ground state configurations have their corner points confined to at most two nearest neighbor vertical lattice lines. One can say that for large τ our two-dimensional random field is essentially one-dimensional, and outside the strip of width ∼ τ −1 it is basically frozen. Along this vertical direction 2 it has the following correlation decay property: for every two finite subsets A, B ⊂ Z we have n A+x n B − n A n B → 0 as |x| → ∞. Here n A = (σ,x)∈A n σ,x .


151

Fig. 2. A ground state configuration of the β-paths, and a low-temperature configuration

Without loss of generality we can assume that |z| = 1. The parameter z then defines the “slope” of the “height function”, or, what is the same, the (constant) density of the paths in the path ensemble. Note now, that for every z the determinantal processes µκ,z are different for different values of κ. This follows just from the computation of the second correlation function for these processes. That means that there are continuum non-translation-invariant Gibbs measures (12), corresponding to the same slope and the same temperature. Of course, the field µκ,z is just a translate of the field µτ κ,z by the unit lattice vector. But the fields µκ,z with κ between 1 and τ are all different and are not related by the lattice shift transformation. One can understand better the role played by the parameter κ looking at the boundary conditions and the limiting behavior of the processes µκ,z . The restriction of the process to any vertical line σ = const gives the (translation-invariant) sine process with density arg z/π . Let us consider two such lines, say σ = ±k. Then the paths of the process µκ,z define in a natural way the coupling C (k, κ, τ ) between the two sine processes. Consider the limiting coupling C (κ, τ ) = limk→∞ C (k, κ, τ ). It turns out that the couplings C (κ, τ ) are still non-trivial (i.e. these couplings are not product-couplings), and are different for different κ. The two couplings C (κ, τ ) and C (τ κ, τ ) are related by the unit shift of one of the sine processes. The couplings C (κ, τ ) become trivial only in the limit when τ → 0. In this limit the fields µκ,z become fully translation invariant.

152


Fig. 3. The β-paths and the corresponding (deformed-)lozenge tiling

Fig. 4. The limit shape

A straightforward geometric interpretation of our ensemble of the β-paths is to relate them to the lozenge tilings of the plane. Our paths are then composed by the middle lines of all “vertical” plaquettes, see Fig. 3, cf. [9]. One may wonder about the existence of the asymptotic shape of the height function corresponding to the field µ1,z , i.e. to the choice ψk (u) = 1 + τ k u . However, this height function is almost frozen outside the strip of width ∼ τ −1 around the line σ = 0. If we scale this surface by the factor τ in all three dimensions, then the conjectural limit when τ → 0 is a non-random cylindrical surface. This surface has a gutter shape, see Fig. 4, and is given by the following geometric construction. To describe it we first recall the geometric construction (see [16] or [17]), used to obtain the limit shape CC K of the plane partitions by Cerf and Kenyon in [5]. For every positive unit vector n ∈ 2 = S 2 ∩ R3+ let ent (n) be the residual entropy of the lozenge tilings having the slope plane orthogonal to n (see Theorem 1.1 in [5]). Now define the


153

halfspaces K n = x ∈ R3 : (x, n) ≥ ent (n) . Let K = ∩n∈2 K n ; the boundary of the region K is precisely the surface CC K , describing the typical shape of a large plane partition. To describe the gutter shape we first define its slope, γ . This is determined by the frequency of the lines in our family (indeed, they are just the level lines of the height function). If z = eiϕ , then the density in question is the first correlation function of our ϕ determinantal process, and it is equal to πϕ . Therefore γ satisfies tan γ = 1−ϕ . Let us

define the vector m (γ ) = m x , m y , m z = − √1 , − √1 , tan γ . This is the direction of 2

2

our gutter. Consider the arc A (γ ) of vectors in the “triangle” 2 , which are orthogonal to m(γ ): A (γ ) = n ∈ 2 : (n, m (γ )) = 0 . Our gutter surface, G (γ ) , is defined to be the boundary of the convex region K (γ ) ≡ ∩n∈A(γ ) K n . Note that the surface G (γ ) consists of straight lines parallel to the vector m (γ ). The surfaces G (γ ) and CC K are tangent to each other along the common curve g (γ ) = G (γ )∩ CC K . Each of the curves g (γ ) is a smooth curve without straight pieces. Asymptotically

each of them approaches the Vershik curve CV : exp − √π x + exp − √π y = 1, 6 6 z = 0 , which belongs to the boundary of the curved part of the surface CC K . −1 2. Our second example isthe ensemble of αβ-paths, with ψ2k (u) = 1 − αk− u −1 and ψ2k+1 (u) = 1 + βk+ u . Again we will choose α and β to be geometric progressions, by putting αk− = (κekτ )−1 , βk+ = λekτ , with κ, λ, τ > 0. In the same way that the β-paths are related to the lozenge tiling, the αβ-paths are related to the domino tilings. The relation however is not so easy to explain; the corresponding construction, establishing the bijection between the two entities, is presented in [10], see also [13]. Figure 5 shows one collection of αβ-paths with κ = λ = 1 and τ = ∞ (which therefore should be called a ground state configuration). For k > 0 all the α-steps are zero height steps, while all the β-steps are ascending. For k < 0 all the β-steps are zero height steps, while all the α-steps are descending in a maximal possible way. Figure 6 shows the corresponding domino tiling. In domino tilings, the elementary moves correspond to finding a 2 × 2 square tiled by two dominoes, say horizontal, and replacing this pair of dominos by two vertical ones. If one assigns four weights a, b, c, d to the four possible positions of a single domino, then every move replacing a horizontal pair by a vertical one changes the overall weight by a constant factor. (If a, b are two horizontal weights, then the overall change will be cd by a factor ab .) In our case the overall weight after an elementary move depends on the κ τ parity of the 2 × 2 square, and is κ λ in one case, and λ e in the other. (The parity of the square is the parity of the sum of the coordinates of its lower-left corner.) This means

154


Fig. 5. A ground state configuration of αβ-paths

that the measures on the domino tilings that we have constructed, can not be obtained by assigning weights to single dominoes. Again, for τ = 0 our measures are non-translation-invariant, and by varying the ratio κ we obtain a whole continuum of different measures. λ In the case τ = 0, the two-parametric measure on lozenge tilings and the three-parametric measure on domino tilings are translation invariant with respect to all shifts of Z2 . These measures are well known; for lozenge tilings they were obtained in [11,14], and for domino tilings they were obtained in [4,6], see also [10]. As proved in [15], they are the only fully translation invariant ergodic measures. 3. Proof of the Main Result The proof of i of Theorem 1 is given in Sect. 4 below. We will start the proof of ii-iv by dealing with the special case when for each k ∈ Z1 the function ψk (u) is either (1 − αk u)−1 or (1 + βk u) , with αk , βk some positive constants. We will consider the general case at the end of the proof.


155

Fig. 6. The domino tiling, corresponding to the paths of Fig. 5

3.1. Linear relations. Let the sequence of functions ψk (u) be given, where for every k ∈ Z1 the function ψk (u) is either (1 − αk u)−1 or (1 + βk u) , with αk , βk some positive constants. In this subsection we will show that the kernel K σ,τ (x − y) satisfies the linear relations mentioned in Theorem 1. Indeed, if ψk (u) = (1 − αk u)−1 , then for σ = k − 1 and τ ≥ k,

K k−1,τ

1 (x − y) = 2πi

(1 − αk u) C+

τ

−1 ψ j (u)

k+1

du u x−y+1

= K k,τ (x − y) − αk K k,τ (x − y − 1) , the same for τ < k − 1: K k,τ (x − y) − αk K k,τ (x − y − 1) k−1 du 1 ψ j (u) (1 − αk u)−1 x−y+1 = 2πi C− u τ +1

−αk =

1 2πi

1 2πi

k−1

C− τ +1 k−1

ψ j (u) (1 − αk u)−1

ψ j (u)

C− τ +1

du u x−y+1

du u x−y

= K k−1,τ (x − y) ,

156


while for τ = k − 1 we have du 1 K k−1,k−1 (x − y) = , x−y+1 2πi C+ u K k,k−1 (x − y) − αk K k,k−1 (x − y − 1) du 1 du 1 = (1 − αk u)−1 x−y+1 − αk (1 − αk u)−1 x−y 2πi C− u 2πi C− u du 1 = , 2πi C− u x−y+1 which means that K k−1,k−1 (x − y) − K k,k−1 (x − y) + αk K k,k−1 (x − y − 1) = δx=y .

(13)

Altogether, these relations read = K k,τ (x − y) − αk K k,τ (x − y − 1) .

K k−1,τ (x − y) − δ

x=y τ =k−1

(14)

Also, if τ = k and σ ≤ k − 1, then 1 K σ,k (x − y) = 2πi

C+

τ

−1 ψ j (u)

(1 − αk u)

σ +1

du u x−y+1

= K σ,k−1 (x − y) − αk K σ,k−1 (x − y − 1) , and the same for σ > k. Since the diagonal elements K r,r (x − y) do not depend on r, for σ = k we have immediately from (13): K k,k (x − y) − K k,k−1 (x − y) + αk K k,k−1 (x − y − 1) = δx=y . Altogether, K σ,k (x − y) − δ x=y = K σ,k−1 (x − y) − αk K σ,k−1 (x − y − 1) .

(15)

σ =k

Likewise, for ψk (u) = (1 + βk u), σ = k and k − 1 > τ we have k−1 du 1 ψ j (u) (1 + βk u) x−y+1 K k,τ (x − y) = 2πi C− u τ +1

= K k−1,τ (x − y) + βk K k−1,τ (x − y − 1) , and similarly for k − 1 < τ. For τ = k − 1 we have du 1 K k,k−1 (x − y) = (1 + βk u) x−y+1 , 2πi C− u while K k−1,k−1 (x − y) + βk K k−1,k−1 (x − y − 1) =

1 2πi

C+

du

u

+ βk x−y+1

(16)

1 2πi

C+

du , u x−y (17)


157

so K k−1,k−1 (x − y) + βk K k−1,k−1 (x − y − 1) − K k,k−1 (x − y) du 1 = (1 + βk u) x−y+1 = δx=y + βk δx=y+1 . 2πi u Summarizing, we have K k,τ (x − y) = K k−1,τ (x − y) − δ

(18)

+ βk K k−1,τ (x − y − 1) − δ x=y+1 .

x=y τ =k−1

τ =k−1

(19) The last relation we obtain by considering for ψk (u) = (1 + βk u) the case when σ > k while τ = k − 1. Then we have

σ du 1 K σ,k−1 (x − y) = ψ j (u) (1 + βk u) 2πi C− u x−y+1 k+1

= K σ,k (x − y) + βk K σ,k (x − y − 1) . The same relation holds in the region σ < k, while for σ = k we use (18), which immediately implies that K k,k (x − y) + βk K k,k (x − y − 1) − K k,k−1 (x − y) = δx=y + βk δx=y+1 , thus getting us to

K σ,k−1 (x − y) = βk K σ,k (x − y − 1) − δ

σ =k x=y+1

+ K σ,k (x − y) − δσ =k .

(20)

x=y

3.2. Interlacing property. Simplest case. Let us start by checking the interlacing property in the simplest situations. In the case ψk (u) = (1 − αk u)−1 we will show that 1 ∗ 0 1 Pr K = 0, Pr K = 0, (21) 1 0 k−1,k ∗ 1 k−1,k 1∗ denotes the corresponding event in some two by two where the symbol 1 0 k−1,k square in the vertical strip {(k − 1, ∗) , (k, ∗)} . For the case σ = k with ψk (u) = (1 + βk u) we will show that 0 1 ∗ 0 Pr K = 0, Pr K = 0. 0 ∗ k−1,k 1 0 k−1,k

To save on notation, we will put k = 1, and we will write α, β, ψ instead of α1 , β1 , ψ1 . 1. The case ψ (u) = (1 − αu)−1 : Pr K

1 ∗ 1 0

= 0.

0,1

158


This relation is equivalent to showing that K 0,0 (0) K 0,0 (−1) K 0,1 (0) K 0,1 (1) = 0. det K 0,0 (1) K 0,0 (0) K 1,0 (0) K 1,0 (−1) K 1,1 (0) − 1 But this relation does hold, since the relation (15) implies that the last column is a linear combination of the remaining two. 2. The case ψ (u) = (1 − αu)−1 : Pr K

0 1 ∗ 1

= 0. 0,1

We have to check that K 0,0 (0) − 1 K 0,1 (1) K 0,1 (0) det K 1,0 (−1) K 1,1 (0) K 1,1 (−1) = 0. K 1,0 (0) K 1,1 (1) K 1,1 (0) But from (14) it follows that the first row is a combination of the remaining two. 3. The case ψ (u) = (1 + βu): Pr K

0 1 0 ∗

= 0.

(22)

0,1

We have thus to show the vanishing of K 0,0 (0) − 1 K 0,0 (−1) K 0,1 (−1) K 0,0 (0) − 1 K 0,1 (0) . det K 0,0 (1) K 1,0 (1) K 1,0 (0) K 1,1 (0) But the relation (19) tells us that the third row of the last determinant is a linear combination of the first two. 4. The case ψ (u) = (1 + βu): Pr K

∗ 0 1 0

= 0. 0,1

We thus need the vanishing of the determinant K 0,0 (0) K 0,1 (0) K 0,1 (−1) det K 1,0 (0) K 1,1 (0) − 1 K 1,1 (−1) . K 1,0 (1) K 1,1 (1) K 1,1 (0) − 1 But the first column is a combination of the second and the third, due to (20).


159

3.3. Elementary moves. We will prove the interlacing property in the general case by reducing the corresponding events to simpler ones. The reduction is done by applying to the configurations in question the transformations, which are called elementary moves. (For the case of lozenge tilings – see Case 1 below – an elementary move corresponds to the replacement of the tiling of the regular hexagon by three lozenges by a different one.) In this and the next section we will prove the set of identities, corresponding to the elementary moves of the paths. We will use them in Sect. 3.5. Since every move involves two adjacent columns of the lattice, we have four different types of moves, according to the four types – αα, αβ, βα, or ββ – of the column pairs. 1. We start with the case ψ1 (u) = (1 + β1 u) , ψ2 (u) = (1 + β2 u) . We will prove that β1 Pr K

∗ 0 1 1 1 ∗

= β2 Pr K

0,1,2

∗ 1 1 1 0 ∗

. 0,1,2

The corresponding determinant relation reads: K 0,0 (0) K 0,1 (0) K 0,1 (−1) K 0,2 (−1) K (0) K 1,1 (0) K 1,1 (−1) K 1,2 (−1) β1 det 1,0 K 1,0 (1) K 1,1 (1) K 1,1 (0) − 1 K 1,2 (0) K (1) K (1) K 2,1 (0) K 2,2 (0) 2,0 2,1 K 0,0 (0) K 0,1 (0) K 0,1 (−1) K 0,2 (−1) K (0) K 1,1 (0) − 1 K 1,1 (−1) K 1,2 (−1) . = β2 det 1,0 K 1,1 (1) K 1,1 (0) K 1,2 (0) K 1,0 (1) K (1) K (1) K (0) K (0) 2,0

2,1

2,1

(23)

2,2

Due to the relation (20), applied to the first determinant, K 0,0 (0) K (0) det 1,0 K 1,0 (1) K (1) 2,0

K 0,1 (0) β1 K 0,1 (−1) K 0,2 (−1) K 1,1 (0) β1 K 1,1 (−1) K 1,2 (−1) , K 1,1 (1) β1 K 1,1 (0) − 1 K 1,2 (0) K 2,1 (1) β1 K 2,1 (0) K 2,2 (0)

subtracting from the third column the first one and adding the second one, results in K 0,0 (0) K (0) det 1,0 K 1,0 (1) K (1) 2,0

K 0,1 (0) K 1,1 (0) K 1,1 (1) K 2,1 (1)

0 K 0,2 (−1) K 0,0 (0) K 0,1 (0) K 0,2 (−1) 1 K 1,2 (−1) = − det K 1,0 (1) K 1,1 (1) K 1,2 (0) . 0 K 1,2 (0) K 2,0 (1) K 2,1 (1) K 2,2 (0) 0 K 2,2 (0)

Due to the relation (19), applied to the second determinant, K 0,2 (−1) K 0,0 (0) K 0,1 (0) K 0,1 (−1) β K (0) β2 K 1,1 (0) − 1 β2 K 1,1 (−1) β2 K 1,2 (−1) det 2 1,0 , K 1,1 (1) K 1,1 (0) K 1,2 (0) K 1,0 (1) K (1) K 2,1 (1) K 2,1 (0) K 2,2 (0) 2,0

160


subtracting from the second row the last one and adding the third one, results in: K 0,0 (0) K 0,1 (0) K 0,1 (−1) K 0,2 (−1) 0 0 1 0 det K 1,2 (0) K 1,0 (1) K 1,1 (1) K 1,1 (0) K (1) K (1) K (0) K 2,2 (0) 2,0 2,1 2,1 K 0,0 (0) K 0,1 (0) K 0,2 (−1) = − det K 1,0 (1) K 1,1 (1) K 1,2 (0) . K 2,0 (1) K 2,1 (1) K 2,2 (0) But this is the same matrix as above. 2. Now we consider the case ψ1 (u) = (1 − α1 u)−1 , ψ2 (u) = (1 − α2 u)−1 . We have to prove that 0 ∗ ∗ 0 1 ∗ α1 Pr K = α2 Pr K , (24) ∗ 1 0 0,1,2 ∗ ∗ 0 0,1,2 or

K 0,0 (0) − 1 K 0,1 (1) K 0,2 (1) K 1,2 (0) α1 det K 1,0 (−1) K 1,1 (0) K 2,0 (−1) K 2,1 (0) K 2,2 (0) − 1 K 0,0 (0) − 1 K 0,1 (0) K 0,2 (1) K 1,1 (0) K 1,2 (1) . = α2 det K 1,0 (0) K 2,0 (−1) K 2,1 (−1) K 2,2 (0) − 1 Applying the relation (14) to the first two rows of the first determinant, we see that K 0,0 (0) − 1 K 0,1 (1) K 0,2 (1) det α1 K 1,0 (−1) α1 K 1,1 (0) α1 K 1,2 (0) K 2,0 (−1) K 2,1 (0) K 2,2 (0) − 1 K 0,0 (0) − 1 K 0,1 (1) K 0,2 (1) K 1,1 (1) K 1,2 (1) . = det K 1,0 (0) K 2,0 (−1) K 2,1 (0) K 2,2 (0) − 1

Applying now the relation (15) to the second and third columns of the second determinant, we see the same result: K 0,0 (0) − 1 α2 K 0,1 (0) K 0,2 (1) α2 K 1,1 (0) K 1,2 (1) det K 1,0 (0) K 2,0 (−1) α2 K 2,1 (−1) K 2,2 (0) − 1 K 0,0 (0) − 1 K 0,1 (1) K 0,2 (1) K 1,1 (1) K 1,2 (1) . = det K 1,0 (0) K 2,0 (−1) K 2,1 (0) K 2,2 (0) − 1 3. We go to the case ψ1 (u) = (1 + β1 u) , ψ2 (u) = (1 − α2 u)−1 . Here we need to see that ∗ ∗ ∗ ∗ 1 ∗ β1 Pr K = α2 Pr K , (25) 1 1 0 0,1,2 1 ∗ 0 0,1,2


which is the same as

β1 Pr K

∗ 0 ∗ 1 ∗ 0

161

= α2 Pr K

0,1,2

∗ 1 ∗ 1 ∗ 0

. 0,1,2

(The equivalence of the two identities follows from the simplest case of the interlacing property proved in the previous section.) Expressed via determinants, this is the relation K 0,0 (0) K 0,1 (−1) K 0,2 (0) K 1,2 (1) β1 det K 1,0 (1) K 1,1 (0) − 1 K 2,0 (0) K 2,1 (−1) K 2,2 (0) − 1 K 0,0 (0) K 0,1 (−1) K 0,2 (0) K 1,2 (1) . = −α2 det K 1,0 (1) K 1,1 (0) K 2,0 (0) K 2,1 (−1) K 2,2 (0) − 1 By (20), subtracting in the first determinant, K 0,0 (0) β1 K 0,1 (−1) K 0,2 (0) det K 1,0 (1) β1 K 1,1 (0) − 1 K 1,2 (1) K 2,0 (0) β1 K 2,1 (−1) K 2,2 (0) − 1 the first column from the second one, makes it into K 0,0 (0) −K 0,1 (0) K 0,2 (0) K 1,2 (1) . det K 1,0 (1) −K 1,1 (1) K 2,0 (0) −K 2,1 (0) K 2,2 (0) − 1 From (15), adding in the second determinant, K 0,0 (0) α2 K 0,1 (−1) K 0,2 (0) K 1,2 (1) , − det K 1,0 (1) α2 K 1,1 (0) K 2,0 (0) α2 K 2,1 (−1) K 2,2 (0) − 1 the third column to the second one results in K 0,0 (0) K 0,1 (0) K 0,2 (0) K 1,2 (1) , − det K 1,0 (1) K 1,1 (1) K 2,0 (0) K 2,1 (0) K 2,2 (0) − 1 which is what we need. 4. The remaining case is ψ1 (u) = (1 − α1 u)−1 , ψ2 (u) = (1 + β2 u) . Here we need to see that 0 ∗ 1 0 1 1 α1 Pr K = β2 Pr K , (26) ∗ 1 ∗ 0,1,2 ∗ ∗ ∗ 0,1,2 which is the same as α1 Pr K

0 ∗ 1 ∗ 1 ∗

= β2 Pr K

0,1,2

0 ∗ 1 ∗ 0 ∗

. 0,1,2

162


The determinant relation to be checked is K 0,0 (0) − 1 K 0,1 (1) K 0,2 (0) −α1 det K 1,0 (−1) K 1,1 (0) K 1,2 (−1) K 2,0 (0) K 2,1 (1) K 2,2 (0) K 0,0 (0) − 1 K 0,1 (1) K 0,2 (0) = β2 det K 1,0 (−1) K 1,1 (0) − 1 K 1,2 (−1) . K 2,0 (0) K 2,1 (1) K 2,2 (0) By (14), applied to the first determinant, K 0,0 (0) − 1 K 0,1 (1) K 0,2 (0) − det α1 K 1,0 (−1) α1 K 1,1 (0) α1 K 1,2 (−1) , K 2,0 (0) K 2,1 (1) K 2,2 (0) the addition of the first row to the second one makes it into K 0,0 (0) − 1 K 0,1 (1) K 0,2 (0) K 1,1 (1) K 1,2 (0) . − det K 1,0 (0) K 2,0 (0) K 2,1 (1) K 2,2 (0) Applying (19) to the determinant K 0,0 (0) − 1 K 0,1 (1) K 0,2 (0) det β2 K 1,0 (−1) β2 K 1,1 (0) − 1 β2 K 1,2 (−1) , K 2,0 (0) K 2,1 (1) K 2,2 (0) we turn it, after subtracting the third row from the second one, into K 0,0 (0) − 1 K 0,1 (1) K 0,2 (0) det −K 1,0 (0) −K 1,1 (1) −K 1,2 (0) . K 2,0 (0) K 2,1 (1) K 2,2 (0) That finishes our proof. 3.4. Moves in a general environment. Now we have to check that the identities of the previous subsection hold in a more general situation. For the ββ case it means for example that ∗ 0 1 ∗ 1 1 β1 Pr K ∪ n V = β2 Pr K ∪ nV , 1 1 ∗ 0,1,2 1 0 ∗ 0,1,2

∗∗ ∗∗

is an arbitrary finite set, disjoint from the set , and where V ⊂ 0,1,2 ∗01 the symbol ∪ n V denotes the event that we have the configuration 1 1 ∗ 0,1,2 ∗01 in our window, and all the sites in V are occupied by the particles. Con1 1 ∗ 0,1,2 Z2

sider the case when V is just a single site (ζ, z) ∈ Z2 . Let us see that the same relations which were used in Subsect. 3.3, work here as well.


163

We need to show that K 0,0 (0) K 0,1 (0) K 0,1 (−1) K 0,2 (−1) K 0,ζ (−z) K 1,2 (−1) K 1,ζ (−z) K 1,0 (0) K 1,1 (0) K 1,1 (−1) K 1,2 (0) K 1,ζ (1 − z) β1 det K 1,0 (1) K 1,1 (1) K 1,1 (0) − 1 K (1) K (1) K 2,1 (0) K 2,2 (0) K 2,ζ (1 − z) 2,0 2,1 K (z) K (z) K (z − 1) K (z − 1) K ζ,ζ (0) ζ,0 ζ,1 ζ,1 ζ,2 K 0,0 (0) K 0,1 (0) K 0,1 (−1) K 0,2 (−1) K 0,ζ (−z) K 1,2 (−1) K 1,ζ (−z) K 1,0 (0) K 1,1 (0) − 1 K 1,1 (−1) K 1,1 (1) K 1,1 (0) K 1,2 (0) K 1,ζ (1 − z) . = β2 det K 1,0 (1) K (1) K 2,1 (1) K 2,1 (0) K 2,2 (0) K 2,ζ (1 − z) 2,0 K (z) K (z) K (z − 1) K (z − 1) K (0) ζ,0

ζ,1

ζ,1

ζ,2

ζ,ζ

(27) It is immediate to see that the same strategy which was used in the simplest case 4 × 4 works: application of (20) turns the determinant K 0,0 (0) K 1,0 (0) det K 1,0 (1) K (1) 2,0 K (z) ζ,0

K 0,1 (0) β1 K 0,1 (−1) K 0,2 (−1) K 0,ζ (−z) K 1,1 (0) β1 K 1,1 (−1) K 1,2 (−1) K 1,ζ (−z) K 1,1 (1) β1 K 1,1 (0) − 1 K 1,2 (0) K 1,ζ (1 − z) K 2,1 (1) β1 K 2,1 (0) K 2,2 (0) K 2,ζ (1 − z) K ζ,1 (z) β1 K ζ,1 (z − 1) K ζ,2 (z − 1) K ζ,ζ (0)

into K 0,0 (0) K 1,0 (0) det K 1,0 (1) K (1) 2,0 K (z) ζ,0

K 0,1 (0) K 1,1 (0) K 1,1 (1) K 2,1 (1) K ζ,1 (z)

0 K 0,2 (−1) K 0,ζ (−z) 1 K 1,2 (−1) K 1,ζ (−z) 0 K 1,2 (0) K 1,ζ (1 − z) , 0 K 2,2 (0) K 2,ζ (1 − z) 0 K ζ,2 (z − 1) K ζ,ζ (0)

while the rhs of (27) is treated as the rhs of (23). So one sees in this way that the identities of Subsect. 3.3 work for all sets V in all the cases. 3.5. Interlacing property. General case. 1. The case ψ1 (u) = (1 − α1 u)−1 . We will show now that ⎧⎛ ⎧⎛ ⎞ ⎫ ⎞ ⎫ ⎪ ⎪ ⎪ ⎪ 1 ∗ 1 ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ 0 ∗ ⎟ ⎪ ⎬ ⎨⎜ 0 0 ⎟ ⎪ ⎬ ⎟ ⎟ ⎜ ⎜ = Pr K ⎜ ... ... ⎟ Pr K ⎜ ... ∗ ⎟ ⎪ ⎪ ⎠ ⎪ ⎠ ⎪ ⎝ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ∗ ⎪ ⎪ 0 0 ⎪ ⎩ ⎭ ⎩ 1 ∗ 0,1 1 1 0,1 ⎭ ⎧⎛ ⎧⎛ ⎞ ⎫ ⎞ ⎫ ⎪ ⎪ ⎪ ⎪ 1 ∗ 1 ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ 0 0 ⎟ ⎪ ⎬ ⎨⎜ 0 1 ⎟ ⎪ ⎬ ⎟ ⎟ ⎜ ⎜ +Pr K ⎜ ... ... ⎟ + · · · + Pr K ⎜ ... ... ⎟ , ⎪ ⎪ ⎝0 1⎠ ⎪ ⎝0 0⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ 1 0 ⎭ 1 0 0,1 ⎭ 0,1 (28)

164


that is, if we have a configuration with two particles, separated by a vertical string of n − 2 holes, then with probability one the next column to the right has in the lower n − 1 cells exactly one particle and n − 2 holes. (So our sum above has n − 1 terms.) This is the particle interlacing property. We will prove it simultaneously with the complementary (reflected) statement: ⎧⎛ ⎧⎛ ⎞ ⎫ ⎞ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ 1 ⎪ ⎪ 0 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ∗ 0 ⎟ ⎪ ⎬ ⎨⎜ 0 0 ⎟ ⎪ ⎬ ⎟ ⎟ ⎜ ⎜ = Pr K ⎜ ... ... ⎟ Pr K ⎜ ... ... ⎟ ⎪ ⎪ ⎝∗ 0⎠ ⎪ ⎝1 0⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∗ 1 ⎭ ⎩ ∗ 1 ⎭ 0,1 0,1 ⎧⎛ ⎧⎛ ⎞ ⎫ ⎞ ⎫ ⎪ ⎪ ⎪ ⎪ 0 1 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ 1 0 ⎟ ⎪ ⎬ ⎨⎜ 0 0 ⎟ ⎪ ⎬ ⎟ ⎟ ⎜ ⎜ + · · · + Pr K ⎜ ... ... ⎟ + Pr K ⎜ ... ... ⎟ . ⎪ ⎪ ⎝0 0⎠ ⎪ ⎝0 0⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∗ 1 ⎭ ⎩ ∗ 1 ⎭ 0,1 0,1 (29) The proof goes by induction on the length of the aforementioned string of the holes. The case of the empty string – i.e. n = 2 – was dealt with in Sect. 3.2. So suppose that we know already the relations (28) and (29) for all n < k. Let us prove them for n = k. First we can exclude the case in (28) when we have at least ⎛ two⎞particles in the second 0 1 ⎜0 0⎟ ⎟ ⎜ column. Indeed, that means that we have there a pattern ⎜ ... ... ⎟ with the string of ⎝0 0⎠ ∗ 1 0,1 holes in the second column of length less than k − 2, which is ruled out by induction hypothesis for (29). The same argument applies⎫to (29). ⎧⎛ ⎞ ⎪ ⎪ 1 ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ 0 0 ⎟ ⎪ ⎬ ⎟ ⎜ = 0, where we have k − 1 holes in It remains to show that Pr K ⎜ ... ... ⎟ ⎪ ⎝0 0⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 0 ⎭ 0,1

the right column. Here we note that the above probability depends on K only through one parameter, α1 . So without loss of generality we can assume that we are in the “αα” situation, i.e. that ψ0 (u) = (1 − α0 u)−1 . Let us write our event as a sum of four events: ⎛

1 ⎜0 ⎜ ⎜ ... ⎝0 1

⎞ ⎞ ⎞ ⎛ ⎛ ∗ 1 1 ∗ 1 1 ∗ 0⎟ ⎜∗ 0 0 ⎟ ⎜∗ 0 0 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ... ⎟ = ⎜ ∗ ... ... ⎟ + ⎜ ∗ ... ... ⎟ ⎠ ⎠ ⎝ ⎝ 0 1 0 0 0 0 0⎠ 0 0,1 ∗ 1 0 −1,0,1 ∗ 1 0 −1,0,1 ⎞ ⎞ ⎛ ⎛ 0 1 ∗ 0 1 ∗ ⎜∗ 0 0 ⎟ ⎜∗ 0 0 ⎟ ⎟ ⎟ ⎜ ⎜ + ⎜ ∗ ... ... ⎟ + ⎜ ∗ ... ... ⎟ . ⎝1 0 0 ⎠ ⎝0 0 0 ⎠ ∗ 1 0 −1,0,1 ∗ 1 0 −1,0,1

(30)


165

The first one has zero probability; this is our induction assumption. And to every one of the remaining events we can apply the move transformation (24), which makes the two particles in the middle column become one unit closer. After that we get a configuration, which has zero probability by induction hypothesis. This ends the proof of our statement. 2. The case ψ1 (u) = (1 + β1 u)−1 . We will show that ⎧⎛ ⎧⎛ ⎞ ⎫ ⎞ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ∗ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ 1 ∗ ⎟ ⎪ ⎬ ⎨⎜ 1 1 ⎟ ⎪ ⎬ ⎟ ⎟ ⎜ ⎜ = Pr K ⎜ ... ... ⎟ Pr K ⎜ ... ∗ ⎟ ⎪ ⎪ ⎝ 1 ∗⎠ ⎪ ⎝1 1⎠ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ∗ ⎭ ⎩ 0 ∗ ⎭ 0,1 0,1 ⎧⎛ ⎞ ⎫ ⎪ ⎪ 0 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ 1 0 ⎟ ⎪ ⎬ ⎟ ⎜ + Pr K ⎜ ... ... ⎟ + · · · + Pr K ⎪ ⎠ ⎪ ⎝ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎩ 0 ∗ ⎭ 0,1

⎧⎛ ⎪ 0 ⎪ ⎪ ⎪ ⎨⎜ 1 ⎜ ⎜ ... ⎪ ⎝1 ⎪ ⎪ ⎪ ⎩ 0

⎞ ⎫ ⎪ 1 ⎪ ⎪ 1⎟ ⎪ ⎬ ⎟ ... ⎟ . ⎪ 0⎠ ⎪ ⎪ ⎪ ∗ 0,1 ⎭ (31)

Here in the lhs we have the probability of the event that two holes are separated by the string of n −2 particles, while in the rhs we have a sum of probabilities of the n −1 events that the right column has exactly one hole in the upper n − 1 positions. This is the hole interlacing. Again, we will prove it by induction on n; the case n = 2 was established above, see (22). We will treat simultaneously the reflected event as well (compare with (28) and (29).) The presence of more than one hole in the right⎛column ⎞ in (31) is again ruled out 0 1 ⎜1 1⎟ ⎟ ⎜ by induction. To study the probability of the event ⎜ ... ... ⎟ we can without loss of ⎝1 1⎠ 0 ∗ 0,1 generality consider the case ψ0 (u) = (1 + β0 u)−1 , and we write ⎛

0 ⎜1 ⎜ ⎜ ... ⎝1 0

⎞ ⎞ ⎞ ⎛ ⎛ ∗ 0 1 ∗ 0 1 1 1⎟ ⎜0 1 1 ⎟ ⎜1 1 1 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ... ⎟ = ⎜ ∗ ... ... ⎟ + ⎜ ∗ ... ... ⎟ ⎝∗ 1 1 ⎠ ⎝∗ 1 1 ⎠ 1⎠ 0 0 ∗ −1,0,1 0 0 ∗ −1,0,1 ∗ 0,1 ⎞ ⎞ ⎛ ⎛ ∗ 0 1 ∗ 0 1 ⎜0 1 1 ⎟ ⎜1 1 1 ⎟ ⎟ ⎟ ⎜ ⎜ + ⎜ ∗ ... ... ⎟ + ⎜ ∗ ... ... ⎟ . ⎝∗ 1 1 ⎠ ⎝∗ 1 1 ⎠ 1 0 ∗ −1,0,1 1 0 ∗ −1,0,1

The first event is ruled out by induction, while the remaining three are movable, and the application of the corresponding move (the first one described in Sect. 3.3) reduces the length of the string of particles in the middle column (0th one) by one, so the remaining three events also have vanishing probability.

166


3.6. The downward paths. We will show now that the case of the functions ψ-s of the −1 and 1 + βk− u −1 can be reduced to the one when all ψk (u) are types 1 − αk− u −1 −1 of the form 1 − αk+ u or 1 + βk+ u . Indeed, we have the identities

−1

−1 = −αk− u 1 − (αk− )−1 u , 1 − αk− u −1

1 + βk− u −1 = βk− u −1 1 + (βk− )−1 u . Observe that multiplication of ψk (u) by a constant c leads to the conjugation of the kernel: ⎧ ⎨ cK (σ, x; τ, y) if σ ≥ k > τ, K (σ, x; τ, y) → c−1 K (σ, x; τ, y) if τ ≥ k > σ, ⎩ K (σ, x; τ, y) otherwise, which does not affect the determinants for the correlation functions. On the other hand, the multiplication of ψk (u) by u in the formula for the kernel is equivalent to the following transformation of the state space Z2 : (σ, x) if σ < k, (σ, x) → (σ, x + 1) if σ ≥ k. Under this transformation every configuration which was satisfying the downward interlacing (for particles or for holes) in the column {(σ, x) : σ = k, k + 1} will satisfy the upward interlacing, so the new configuration can still be associated with a collection of paths. It is straightforward to see that the change of the weight of a path affected by an elementary move is the same in both path configurations, compare the definitions (5), (7), (9), (11). That proves our statement. 3.7. The Gibbs property. Now we are in the position to check the Gibbs property of the field n viewed as the probability distribution over the lattice paths built from the patterns (4), (6), (8) and (10). After the preliminary work we did it is almost immediate. Indeed, we have already checked in Subsects. 3.3, 3.4 that the ratio of the probabilities of two configurations n V and n V which differ by an allowed move of one particle depends only on the type of the move and equals the exponent of the action difference for the corresponding paths. Let us take, in particular, any (finite simply-connected) box ⊂ Z2 , and fix the sets P and Q of the entrance and exit points of the paths on the boundary ∂. Note that in that case any allowed configuration of paths in can be obtained from any other by a sequence of elementary moves. This claim is essentially obvious; it follows from the fact that there is a minimal path joining any two points (if the set of paths joining the two points is nonempty), and induction on the number of paths. That finishes our proof. 4. Positivity ∞ Denote by the set of elements = (α, β, γ ) ∈ R∞ + × R+ × R+ such that ∞ ( i=1

αi < ∞,

∞ ( i=1

βi < ∞.


167

For ∈ , we denote by ψ the following meromorphic functions on C: ψ (u) = eγ u

∞ 1 + βju . 1 − αju

(32)

j=1

For + , − ∈ , we also set ψ + , − (u) = ψ + (u)ψ − (u −1 ). Coordinates of ± will be denoted as αi± , βi± , γ ± . Our goal is to prove the following statement, which is a slight generalization of Theorem 4.4 in [2]. Theorem 2. Fix a complex number z with z > 0 and denote by C± any contour that joins z¯ and z and crosses the real axis at a point of R± . Then for any doubly infinite sequences { + [k], − [k]}k∈Z of elements in , there exists a (unique) determinantal point process on Z × Z with the correlation kernel ⎧ −1 τ ⎪ 1 du ⎪ ⎪ ψ + [k], − [k] (u) , σ ≤ τ, ⎪ ⎨ 2πi x−y+1 u C+ k=σ +1 K (σ, x; τ, y) = (33) σ ⎪ 1 du ⎪ ⎪ ⎪ ψ + [k], − [k] (u) x−y+1 , σ > τ. ⎩ 2πi u C− k=τ +1

Comments. 1. The kernels considered in the previous sections are the ones with each of ψ + [k], − [k] (u) having the form either (1 − αk u)−1 or (1 + βk u). 2. The classical fact that lies at the foundation of this theorem is that functions ψ (u) are generating functions of the totally positive sequences. This statement was independently proved by Aissen-Edrei-Schoenberg-Whitney in 1951 [1,7], and by Thoma in 1964 [18]. An excellent exposition of deep relations of this result to representation theory of the infinite symmetric group can be found in Kerov’s book [12]. 3. The equal time restriction of the kernel above is equivalent to the discrete sine kernel on Z; for any τ ∈ Z, 1 dζ |z| y sin((arg z)(x − y)) K (τ, x; τ, y) = . = x x−y+1 2πi C+ ζ |z| π(x − y) In particular, the density of particles is equal to arg z/π everywhere. The kernels K (σ, x; τ, y) may be viewed as extensions of the discrete sine kernel. 4. The class of the random point processes afforded by this theorem is closed under • projections of Z × Z to A × Z, where A = {an }+∞ n=−∞ is any doubly infinite sequence of integers; • shifts and reflection of either of the two coordinates of Z × Z; • particle-hole inversion on any subset of the form B × Z, where B ⊂ Z. 5. The projection of the process to the set {1, . . . , T } × Z depends only on ± [k] with k = 1, . . . , T . We will give two proofs of the theorem; one is essentially a reduction to Theorem 4.4 from [2], while the second one is “more constructive” — it explains how to build our process from a deformation of the uniform measure on large plane partitions.

168


Proof 1. Observe that the change of the integration variable u = r v, r > 0, replaces the formula for the kernel by a similar one with z → z/r , all coordinates of k+ ’s multiplied by r , all coordinates of k− ’s divided by r , and the integral itself multiplied by r y−x . The prefactor r y−x cancels out in the determinants of the form det[K (σi , xi ; σ j , x j )], thus it can be removed. Hence, it suffices to prove the claim for z with |z| = 1. Observe further, that multiplication of ψ + [m], − [m] (u) by u n in the formula for the kernel above is equivalent to the following transformation of the state space Z × Z: (σ, x), σ < m, (σ, x) → (σ, x + n), σ ≥ m. On the other hand, multiplication of ψ + [m], − [m] (u) by a constant c leads to the conjugation of the kernel ⎧ ⎪ σ + 1 ≤ m ≤ τ, ⎨cK (σ, x; τ, y), K (σ, x; τ, y) → c−1 K (σ, x; τ, y), τ + 1 ≤ m ≤ σ, ⎪ ⎩ K (σ, x; τ, y), otherwise, which does not affect the determinants for the correlation functions. The identities 1 − αu = −αu · (1 − α −1 u −1 ),

1 + βu = βu · (1 + β −1 u −1 )

then show that we can freely replace parameters αi+ [k] = α and βi+ [k] = β by αi− [k] = α −1 and βi− [k] = β −1 and the other way around, and such changes do not affect the statement that the kernel defines a random point process. Using such replacements we can then choose the parameters in such a way that all αi± [k], βi± [k] are in the segment [0, 1]. Since the statement of the theorem is stable under limit transitions, we can assume that all the parameters are strictly smaller than 1 without loss of generality. But if the parameters satisfy the conditions |z| = 1,

αi± [k], βi± [k] < 1, for all i, k

then our claim is exactly Theorem 4.4 in [2].

Proof 2. The argument is based on the Schur process of [14] (see also [8]) and can be constructed as follows. We use the definitions and notation of [14]. Let us construct a deformation of the Schur process. More precisely, the Schur process is parameterized by two sequences {φ + [m], φ − [m]}m∈Z+ 1 of functions holomorphic 2 and nonvanishing in some neighborhood of the interior (resp., exterior) of the unit disc. In order for the process to assign positive weights, the functions φ ± have to be such that all minors of the triangular Toeplitz matrices with symbols φ + (u) and φ − (u −1 ) are nonnegative; this is exactly the content of Comment 2 above. The concrete example of the Schur process studied asymptotically in [14] and [8] corresponds to the choice (1 − q −m u)−1 , m < 0, + φ [m](u) = 1, m > 0, 1, m < 0, φ − [m](u) = (1 − q m u −1 )−1 , m > 0.


169

Let us choose N consecutive values of m, say M, M + 1, . . . , M + N − 1, and replace the corresponding functions φ ± as follows: )+ [M + k](u) = ψ + [k] (u), φ

)− [M + k](u) = ψ − [k] (u −1 ), φ

for k = 0, . . . , N − 1. Taking the point of the limit shape with τ = 0 (near the corner), one readily sees that such a modification produces no impact on the asymptotic analysis of [14] until the * (2) very last stage — the computation of the residue denoted as in Sect. 3.1.6. The residue is an integral of the expression in formula (26) without the factor (z − w) in the denominator and with z = w, where (t, z) is defined by the formula (20). The computation gives 1 2πi

zc

ti

z¯ c m=t j

dw 1 φ + [m](w −1 )φ − [m](w −1 ) w h i −h j +(ti −t j )/2+1 +1

for ti ≥ t j and 1 − 2πi

z¯ c

tj

z c m=t +1 i

φ + [m](w −1 )φ − [m](w −1 )

dw w h i −h j +(ti −t j )/2+1

for ti < t j . If we now choose M in such a way that ti and t j lie in the set M, M + )± [M + k](u) we arrive at 1, . . . , M + N − 1, then substituting the deformed functions φ the kernel (33) with the change of variables tj ti −1 −1 → (y, x). w → u , z c → z , (ti , t j ) → (τ, σ ), hi + , h j + 2 2 Thus, we showed that determinants made from the kernel (33) are limits of the correlation functions of certain point processes.

References 1. Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.: On the Generating Functions of Totally Positive Sequences. Proc. Natl. Acad. Sci. USA 37(5), 303–307 (1951) 2. Borodin, A.: Periodic Schur process and cylindric partitions. Duke Math. J. 140(3), 391–468 (2006) 3. Borodin, A., Okounkov, A., Olshanski, G.: Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13, 491–515 (2000) 4. Burton, R., Pemantle, R.: Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21(3), 1329–1371 (1993) 5. Cerf, R., Kenyon, R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Commun. Math. Phys. 222, 147–179 (2001) 6. Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Amer. Math. Soc. 14(2), 297– 346 (2001) 7. Edrei, A.: On the generating function of a doubly-infinite, totally positive sequence. Trans. Amer. Math. Soc. 74(3), 367–383 (1953) 8. Ferrari, P.L., Spohn, H.: Step fluctations for a faceted crystal. J. Stat. Phys. 113, 1–46 (2003) 9. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Rel. Fields 123(2), 225–280 (2002) 10. Johansson, K.: The arctic circle boundary and the Airy process. Ann. Probab. 33, 1–30 (2005) 11. Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33(5), 591–618 (1997)

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12. Kerov, S.V.: Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis. Translations of Mathematical Monographs, 219, Providence, RI: Amer. Math. Soc., 2003 13. Luby, M., Randall, D., Sinclair, A.J.: Markov chain algorithms for planar lattice structures. SIAM J. Comp. 31, 167–192 (2001) 14. Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16(3), 581–603 (2003) 15. Sheffield, S.: Random Surfaces. Astérisque 304 (2005) 16. Shlosman, S.: Geometric variational problems of statistical mechanics and of combinatorics. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41(3), 1364–1370 (2000) 17. Shlosman, S.: The Wulff construction in statistical mechanics and in combinatorics. Russ. Math. Surv. 56(4), 709–738 (2001) 18. Thoma, E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Zeitschr. 85, 40–61 (1964) Communicated by H. Spohn

Commun. Math. Phys. 293, 171–183 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0909-y

Communications in


Universal Approximation of Multi-copy States and Universal Quantum Lossless Data Compression Masahito Hayashi Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan. E-mail: [email protected] Received: 18 July 2008 / Accepted: 19 May 2009 Published online: 1 September 2009 – © Springer-Verlag 2009

Abstract: We have proven that there exists a quantum state approximating any multi-copy state universally when we measure the error by means of the normalized relative entropy. While the qubit case was proven by Krattenthaler and Slater (IEEE Trans. IT 46, 810–819 (2009)), the general case has been open for more than ten years. For a deeper analysis, we have solved the mini-max problem concerning ‘approximation error’ up to the second order. Furthermore, we have applied this result to quantum lossless data compression, and have constructed a universal quantum lossless data compression. 1. Introduction Does there exist a quantum state approximating the n-copy state universally? If we measure the difference by a measure satisfying the axioms for distance, such a state does not exist. However, if we measure the difference between two states in terms of the normalized relative entropy, such a state does exist. For the classical (i.e., diagonal or commutative) case, Clarke and Barron [1] showed that the mixture state of n-fold tensor product states approximates all of the n-fold tensor product states. More precisely, they proved the following. Consider the set of diagonal density matrices {ρθ |θ ∈ ⊂ Rm } on H := Cd , and a prior distribution w(θ ) on . When their diagonal elements are twice continuously differentiable for the parameter θ and several regularity conditions for ρθ hold, they showed that the mixture state σw,n := ρθ w(θ )dθ satisfies the relation n m + log det I (θ ) − log w(θ ), D(ρθ⊗n σw,n ) ∼ (1) = log 2 2π e where I (θ ) is the Fisher information matrix (see [2]). In this paper, logarithms are taken to base 2. In particular, Clarke and Barron [3] proved that the mini-max of √ the constant det I (θ) √ term is given when the prior w is chosen as the so-called Jeffreys’ prior . det I (θ )dθ If we replace the relative entropy by another measure satisfying the axioms for distance and additivity D(ρ ⊗n σ ⊗n ) = n D(ρσ ), no state satisfies the relation (1)

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and the measure D(ρ ⊗n σn ) behaves as O(n). That is, since the relative entropy does not satisfy the axioms for distance, such a state can exist. Concerning the quantum case, in 1996, Krattenthaler and Slater[4] derived its quantum extension for the qubit case. However, the general case has remained an open problem for more than ten years. Their paper did not provide a complete solution to the mini-max problem for the constant term. In this paper, we prove the existence of states σn on H⊗n satisfying D(ρ ⊗n σn ) ∼ =

d2 − 1 log n + O(1) 2

(2)

for all faithful states ρ on H, i.e., states ρ in the set S := {ρ| rank ρ = d}. Since the dimension of the state family S is d 2 − 1, the relation (2) can be regarded as a natural quantum extension of (1). More precisely, we calculate the following mini-max value: d2 − 1 log n , (3) min sup lim D(ρ ⊗n σn ) − {σn } ρ∈S n→∞ 2 which is the main result in this paper. Krattenthaler and Slater[4] treated the same problem for a restricted class of states {σn } in the qubit case. The term (3) expresses an approximation error in the sense of information processing. Recently, Hayashi [5] provided a universal coding for a classical-quantum channel. In his derivation, an approximating state σn was used instead of the true state ρ ⊗n . The reason that the code given in [5] works universally seems to be that this type of information-theoretic approximation (2) is valid. In the classical case, this relation is viewed as the asymptotic redundancy of prefixed variable-length lossless data compression[1,3]. Any distribution corresponds to the prefixed variable-length data compression via the Kraft inequality, and the minimum average length is the Shannon entropy[6]. When the code applied corresponds to the distribution σµ,n , the relation (1) expresses the asymptotic redundancy when the true state is ρ ⊗n . This fact means that a good information-theoretic approximation implies only a small amount of redundancy in the prefixed variable-length lossless data compression. That is, the relation (1) guarantees the existence of a universal variable-length lossless data compression in the classical case[1,3,7,8]. In the quantum case, it is known that there exists universal fixed-length data compression[9,10]. However, the quantum analogue of variable-length lossless data compression is not simple[11]. In order to determine the size of the memory storing the quantum information, we have to measure the length. This measurement demolishes the quantum state while the degree of the state demolition can be made infinitesimal[12]. Hence, it is impossible to interpret the length of the quantum case in the same way. Therefore, we would have to prepare a storage area of size equal to the maximum length of the given code in the framework of variable-length lossless data compression[13]. In order to avoid this problem, we regard the length of the code as the energy and treat the average energy because it is not necessary to measure the energy. In this paper, we formulate quantum variable-length lossless data compression in this way, and explain how the relative entropy D(ρ ⊗n σn ) expresses the asymptotic redundancy. This paper is organized as follows: In Sect. 2, we present some basic results of representation theory, and show (2). In Sect. 3, using the relations given in Sect. 2, we calculate (3). In Sect. 4, we numerically calculate (3) in the qubit case, and compare the result of Krattenthaler and Slater [4]. In Sect. 5, we treat quantum variable-length lossless data compression, and apply our result to this topic.

Universal Approximation of Multi-copy States

173

2. Representation Theory and Approximation of Multi-copy States In order to treat approximation in an information-theoretic sense, we consider the dual representation of the n-fold tensor product space by the special unitary group SU(d) and the n th symmetric group Sn . (Christandl[14] contains a good survey of representation theory for quantum information.) For this purpose, we focus on the Young diagram. When a vector of integers n = (n 1 , n 2 , . . . , n d ) satisfies the condition n 1 ≥ n 2 ≥ · · · ≥ n d ≥ 0 d and i=1 n i = n, the vector n is called a Young diagram (frame) of size n and depth d; the set of such vectors is denoted by Ynd . The size of this set is constrained by the inequality |Ynd | ≤ (n + 1)d−1 .

(4)

Since the sets {(n s(i) )|n ∈ Ynd } and {(n s (i) )|n ∈ Ynd } are distinct for any s = s ∈ Sd , n d−1 the relation |{n| i n i = n}| ∼ implies the following asymptotic behavior of the = (d−1)! d cardinality |Yn |: |Ynd | ∼ =

n d−1 . d!(d − 1)!

(5)

Using the Young diagram, we can characterize the irreducible decomposition of the natural action of SU (d) and Sn on the tensor space H⊗n : H⊗n = Un ⊗ Vn , n∈Ynd

where Un is the irreducible representation space of SU(d) characterized by n, and Vn is the irreducible representation space of the n th symmetric group Sn characterized by n. Here, we denote the representation of the n th symmetric group Sn by V : Sn s → Vs . According to Weyl’s dimension formula, the dimension of Un can be expressed as dim Un =

ni − n j + j − i d(d−1) p2 > · · · > pd , when ( p1 , . . . , pd ) = ( nn1 , . . . , nnd ), d(d−1) i< j ( pi − p j ) ∼ dim Un = n 2 . (7) d−1 d−2 1 2 3 · · · (d − 1) Then, we denote the projection to the subspace Un ⊗ Vn by In , and define the following. ρn :=

1 dim Un ⊗ Vn

In , σU,n :=

n∈Ynd

1 ρn . |Ynd |

For any state ρ and any Young diagram n ∈ Ynd , the following relation holds: dim Un ρn ≥ In ρ ⊗n In .

(8)

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Thus, (4), (6), and (8) yield the inequality (n + 1)

(d+2)(d−1) 2

σU,n ≥ ρ ⊗n .

(9)

Since σU,n is commutative with ρ ⊗n , we have (d + 2)(d − 1) log(n + 1) + log σU,n ≥ log ρ ⊗n . 2 Thus, we obtain (d + 2)(d − 1) log(n + 1). 2

D(ρ ⊗n σU,n ) = Tr ρ ⊗n (log ρ ⊗n − log σU,n ) ≤

Therefore, the state σU,n universally approximates the state ρ ⊗n in the sense of the normalized quantum relative entropy: 1 D(ρ ⊗n σU,n ) → 0. n Now, we calculate the value D(ρ ⊗n σU,n ) more precisely. In the following calculation, we focus on the set Y d := { p = ( p1 , p2 , . . . , pd−1 , 1 − p1 − · · · − pd−1 )| p1 > p2 > · · · > pd−1 > 1 − p1 − · · · − pd−1 > 0} of the probability distributions on {1, . . . , d}, and the density ρ( p) =

d

pi |ii|,

i=1 d is the standard orthonormal basis of H. Thus, for any state ρ, there where {|i}i=1 exist ∈ SU(d)/U (1)d−1 and p ∈ Y d such that ρ = ρ p, := U ρ( p)U† , where U is a representative of . In this calculation, it is essential to calculate the average of the random variable log |Ynd | + log dim Un + log dim Vn under the distribution Q p (n) := Tr ρ( p)⊗n In . In order to treat n∈Ynd Q p (n) log dim Vn asymptotically,

Matsumoto and Hayashi [15] introduced the quantity nn!! := Appendix D,1 they showed that

Q p (n)(log dim Vn − log

n∈Ynd

n! n 1 !n 2 !···n d ! .

In their

n! ) n!

sgn(s) i p δs(i) i< j ( pi − p j ) i ∼ log = δs(i) i< j ( pi − p j ) p s∈Sd

= log

i< j

i

i

sgn(s) i p δs(i) δs(i) i ( pi − p j ) − pi , log i< j ( pi − p j ) s∈Sd

(10)

i

1 Appendix D of [15] contains three systematic typos. In order to recover the correct meaning, replace S , n i > j, and n by Sd , i < j, and nπ , respectively.


175

δs(i) where δi := d − i and we have applied the formula = s∈Sd sgn(s) i pi i< j ( pi − p j ). In their Appendix C, they calculated n∈Ynd Q p (n) as

Q p (n) log

n∈Ynd

d −1 1 n! ∼ d −1 log n − log 2π e − log pi . = H ( p)n − n! 2 2 2

(11)

i

The combination of (5), (7), (10), and (11) yields

P(n)(log |Ynd | + log dim Un + log dim Vn )

n∈Ynd

∼ = H ( p)n + log

i< j

δs(i) sgn(s) i p δs(i) i log ( pi − p j ) − pi i< j ( pi − p j ) s∈Sd

d −1 1 d −1 log n − log 2π e − − log pi 2 2 2 i n d(d−1) i< j ( pi − p j ) n d−1 + log d−1 d−2 + log d!(d − 1)! 2 3 · · · (d − 1)1 2 d −1 d −1 ∼ log n − log 2π e = H ( p)n + 2 2 − log 2d−1 3d−2 · · · (d − 1)1 − log d!(d − 1)! + 2 log ( pi − p j )

−

i

i< j

δs(i) i pi

δs(i) 1 sgn(s) log pi − log pi 2 i< j ( pi − p j )

s∈Sd

∼ = H ( p)n +

i

d2

i

−1 log n + Cd − log d!(d − 1)! + C( p), 2

where d −1 log 2π e − log 2d−1 3d−2 · · · (d − 1)1 , 2 δs(i) sgn(s) i p δs(i) 1 i log pi + 2 log ( pi − p j ) − log pi . C( p) := − 2 i< j ( pi − p j ) Cd := −

s∈Sd

i

i< j

Hence, D(ρ ⊗n σU,n ) = − Tr ρ ⊗n log σU,n − n H (ρ) d2 − 1 ∼ log n + Cd − log d!(d − 1)! + C( p). = 2

i

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3. Mini-max Problem From the discussion of the above section, it is possible to reduce the asymptotic 2 approximation error D(ρ ⊗n σU,n ) to d 2−1 log n universally. In this section, we treat the mini-max problem concerning the constant term, which is the second-order term. The following is our main result. Theorem 1. We obtain the following mini-max value:

d2 − 1 ⊗n log n = Cd + log eC( p) d p, min sup lim D(ρ σn ) − {σn } ρ∈S n→∞ 2 Yd

(12)

where d p := dp1 dp2 . . . pd−1 . The above mini-max value is realized when we choose C( n n) . the mixture state σ J,n := n∈Ynd Jn (n)ρn with the distribution Jn (n) := e n

n ∈Ynd

eC(

n )

This mini-max value is also attained by the mixture state σ˜ J,n := Y d ρ( p)⊗n J ( p)d p, ⊗n mixture with where ρ( p) is† the respect to the invariant measure µ on SU(d), i.e., ⊗n µ(dU ) = ρ ⊗n d. In addition, when the state σ is σ (Uρ( p)U ) ˜ J,n , n J,n or σ p, SU(d) the limit of (12) exists. Since the Jeffreys’ prior gives the mini-max solution in the classical case, the distribution J ( p)d pd can be regarded as a quantum extension of Jeffreys’ prior. Proof. Since the state ρ ⊗n is invariant under permutation, D(ρ ⊗n σn ) = − Tr[ρ ⊗n log σn ] − n H (ρ) is equal to − Tr[ρ ⊗n log Vs σn Vs† ] − n H (ρ) for any s ∈ Sn . Thus, the operator convexity of the function x → − log x implies that − Tr[ρ ⊗n log σn ] − n H (ρ) =

1 Tr[ρ ⊗n (− log Vs σn Vs† )] − n H (ρ) |Sn |

s∈Sn

≥ − Tr[ρ ⊗n log

1 Vs σn Vs† ] − n H (ρ). |Sn |

s∈Sn

Since the state s∈Sn |S1n | Vs σn Vs† is invariant under the action of Sn , we can restrict our states σn in (12) to permutation-invariant states. Due to the unitary covariance and operator convexity of the function x → − log x, the invariant measure µ on SU(d) satisfies d2 − 1 ⊗n sup lim D(ρ σn ) − log n 2 ρ n→∞ d2 − 1 = sup sup lim − Tr ρ( p)⊗n log U ⊗n σn (U ⊗n )† − n H ( p) − log n 2 p U ∈SU(d) n→∞

d2 − 1 ≥ sup lim − log n Tr ρ( p)⊗n log U ⊗n σn (U ⊗n )† µ(dU ) − n H ( p) − 2 p n→∞ SU(d)

d2 − 1 ⊗n ⊗n ⊗n † = sup lim − Tr ρ( p) log n log U σn (U ) µ(dU ) − n H ( p) − 2 p n→∞ SU(d)

d2 − 1 U ⊗n σn (U ⊗n )† µ(dU ) − n H ( p) − ≥ sup lim − Tr ρ( p)⊗n log log n. 2 p n→∞ SU(d)


177

Since the state SU(d) U ⊗n σn (U ⊗n )† µ(dU ) is invariant under the action of SU(d), we can restrict our states σn in (12) to states invariant under the SU(d) action. That is, we obtain d2 − 1 inf sup lim D(ρ ⊗n σn ) − log n {σn } ρ n→∞ 2 d2 − 1 ⊗n = inf sup lim D(ρ σ Pn ,n ) − log n , {Pn } ρ n→∞ 2 where Pn is a probability measure on Ynd and σ Pn ,n := n∈Ynd Pn (n)ρn . Since

C( nn ) n ∈Ynd e n d−1

→

eC( p) d p, Yd

we have D(ρ( p)⊗n σ Pn ,n ) Pn (n) d2 − 1 ∼ log n + Cd − log d!(d − 1)! + C( p) − log 1 = 2 |Ynd |

2 d −1 ∼ log n + Cd − log d!(d − 1)! + log( eC( p ) d p ) = d 2 Y n d−1 + log J ( p) − Q p (n) log Pn (n) d!(d − 1)! n

2 d −1 ∼ log n + Cd + log eC( p ) d p = 2 Yd + Q p (n)(log J ( p) − log Pn (n) − (d − 1) log n).

(13)

n

Only the second group depends on p in (13). In order to evaluate the second group, we consider the joint distribution Q J,n ( p, n) := J ( p)Q p (n). The marginal distribution Q J,n (n)(= Y d Q J,n ( p, n)d p) approaches Jn (n) because the variable nn approaches p in probability Q p [16,12] and J ( p) is continuous. The variable p approaches nn in probability under the conditional distribution Q J,n ( p|n). Then,

Q p (n)Jn ( p) dp log J ( p) Yd Y d Jn ( p)P p (n)d p C( nn ) n n ∈Ynd e ∼ = log J ( ) = log Jn (n) + log C( p) d p n Yd e ∼ (14) = log Jn (n) + (d − 1) log n. On the other hand, since nn approaches p in the probability distribution Q p (n), the random variable log Jn (n) − (d − 1) log n approaches log J ( p) in the probability distribution Q p (n) [16,12]. That is, Q p (n)(log J ( p) − log Jn (n) − (d − 1) log n) → 0 (15) n

for any p.

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By using (14), the supremum of the second group of (13) concerning p is evaluated as follows. Q p (n)(log J ( p) − log Pn (n) − (d − 1) log n) sup p

≥ ∼ =

n

Yd

Q p (n)(log J ( p) − log Pn (n) − (d − 1) log n)J ( p)d p

n

Jn (n)(log Jn (n) − log Pn (n)) = D(Jn Pn ) ≥ 0.

n

Hence, lim

n→∞

n d2 − 1 log n) ≥ Cd + log Pn (n)(D(ρ( )⊗n σ Pn ,n ) − n 2 n

eC( p) d p. Yd

Due to (15), equality holds when Pn (n) = Jn (n). Therefore,

d2 − 1 eC( p) d p. min sup lim D(ρ ⊗n σn ) − log n = Cd + log {σn } ρ n→∞ 2 Yd Note that existence of the minimum value has been proven here. Since the state ρ( p)⊗n has the form n∈Ynd Q p (n)ρn , the state σ˜ J,n has the form n∈Ynd Q J,n (n)ρn . Because Q J,n (n) approaches Jn (n), the state σ˜ J,n also attains the mini-max value in (12). 4. Qubit Case In the qubit case, C2 and C( p) are calculated as follows: 1 C2 = − log 2π e − log 2, 2 C( p) = 2 log( p1 − p2 ) −

p1 p2 1 log p1 + log p2 − (log p1 + log p2 ). p1 − p2 p1 − p2 2

Therefore, our optimal prior has the form CeC( p) d pd = C

1 θ 4 cos θ tan cos θ dθ d tan θ 2

with the normalizing constant C under the parametrization p1 = cos2 θ2 and p2 = sin2 θ2 (0 ≤ θ ≤ π2 ). That is, the optimal prior distribution derived here coincides with the quasi-Bures distribution (21) in Slater [20] in the qubit case. Then, the term log Y 2 eC( p) d p can be numerically calculated as the value −0.50737. We obtain 3 min sup lim (D(ρ ⊗n σn ) − log n) = −3.5545. {σn } ρ∈S n→∞ 2 On the other hand, Krattenthaler and Slater [4] focus on the one-parameter family {ζn (u)| − ∞ < u < 1}, whose elements are given as mixtures of n-copy states under invariant measures qu , as stated in (1.7) of [4]. Their numerical calculation implies that min sup lim (D(ρ ⊗n ζn (u)) − u

ρ∈S n→∞

3 log n) = −2.3956 2

(16)


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in the logarithmic base 2. Note that they calculated the l.h.s. of (16) to be the value −1.66050 working with the natural logarithmic base. Their result does not contradict our result because we take the minimum with respect to all sequences of states {σn } whereas they take it with respect to specific sequences of states {ζn (u)}. The difference −2.3956 − (−3.5545) = 1.1589 expresses the improvement resulting from the choice of the optimal approximating state. 5. Minimization of the Average Energy In this section, we apply the result obtained to quantum variable-length lossless coding. First, we present a formulation of variable-length lossless coding. Remember that the fixed-length code is given as the pair of an encoder and a decoder, in which, the encoder is a trace preserving completely positive (TP-CP) map from the set of states on H⊗n to the set of states on (C2 )⊗k for fixed n and k, and the decoder is a trace preserving completely positive (TP-CP) map with the opposite direction[17,18]. In the variable-length code, the coding length is not determined. So, the encoded space is given as the direct sum product of the tensor product spaces of the qubit system: H⊕ :=

∞ (C2 )⊗k , k=0

which is often called the Fock space. Then, the encoder is given as a trace preserving completely positive (TP-CP) map from the set of states on H⊗n to the set of states on the Fock space H⊕ and the decoder is given as a TP-CP map with the opposite direction. When we use this space for storing quantum information, we cannot determine the length of the stored state without state demolition. Further, the encoder given by a TP-CP map is an isometry map and the decoder is given as its adjoint map; it can recover the original state perfectly. So, such a code is called a lossless code. Note that our definition is more general than that of Boström and Felbinger [11] because they assume that the compressed state of the basis state belongs to the space (C2 )⊗n , i.e., it is not a superposition of states of different length. Furthermore, when we store a quantum state in terms of a Fock state, we do not have to know the length of the stored state. From another viewpoint, the average energy is an important quantity for physical realization because a higher energy damages the communication channel and storage. Note that it is not necessary to measure the energy. In this case, it is natural to treat the Hamiltonian H := ∞ k=0 k Pk , where Pk is the projection to the space (C2 )⊗k . In this case, when the initial state is given by ρ, the average energy is given as Tr HUρU † . Hence, it is appropriate to consider the minimization of the average energy for a given ensemble {(ρi , pi )} on H⊗n . The following lemma gives a lower bound for the average energy. Theorem 2. For any lossless code Un on H⊗n , we can choose the density σ (Un ) on H⊗n such that − Tr ρ log σ (Un ) − logn log d ≤ Tr HUn ρUn†

(17)

for any state ρ of H⊗n . Its proof is given in Appendix A, Since the relation − Tr ρ p log σ (Un ) − H (ρ p ) = D(ρ p σ (Un )) holds for the mixture of the ensemble ρ p := i pi ρi , the average energy Tr HUn ρ p Un† is greater than H (ρ p ) − log2 (n log2 d).

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Next, we consider the small class of quantum lossless codes. When we concatenate two general quantum lossless codes, the concatenated lossless code is not necessarily determined. In order to avoid this problem, we consider the prefix quantum lossless d n on H⊗n code. A code Un on H⊗n is called a prefix when there exists a basis {|ei }i=1 k and a classical prefix code φ : {1, . . . , d n } → ∪∞ k=1 {0, 1} such that Un |ei = |φ1 (i) · · · φk (i) ∈ (C2 )⊗k ⊂ H⊕ . Note that the basis {|ei } does not necessarily have the tensor product form. Since the classical concatenated code of two classical prefix codes can be defined, the concatenated quantum lossless code of two prefix quantum lossless codes can be defined. Hence, it is natural to restrict our quantum lossless codes to prefix lossless codes. Due to the construction of the prefix code, the maximum length maxi |φ(i)| exists. Then, it is sufficient to store the subspace up to this size of Fock space. As is known in information theory [6], any classical prefix code φ satisfies the Kraft inequality 2−|φ(i)| ≤ 1, i

where |φ(i)| is the coding length. For any prefix quantum lossless code Un , we choose a classical prefix code φ such that Un |ei = |φ(i). Then, a quantum version of the Kraft inequality † c := Tr 2−i|Un HUn |i |ei ei | = 2−|φ(i)| ≤ 1 (18) i

holds. That is, the state σ (Un ) :=

i 1 c

i

†

2−i|Un HUn |i |ei ei | satisfies an inequality

− Tr ρ log σ (Un ) ≤ Tr HUn ρUn† , which is tighter than (17) Conversely, as is known in information theory [6], for any probability distribution pi , there exists a prefix classical code φ such that − log pi ≤ |φ(i)| ≤ − log pi + 1.

(19)

Hence, for a given state σ on H⊗n , based on its diagonalization σ = i pi |ii|, we choose the prefix classical code φ satisfying (19). Then, we can define the prefix quantum lossless code Un by Un |ei = |φ(i). The prefix quantum lossless code Un satisfies − Tr ρ log σ ≤ Tr HUn ρUn† ≤ − Tr ρ log σ + 1 for any state ρ on H⊗n . That is, a prefix quantum lossless code almost corresponds to a quantum state. Therefore we can identify a prefix quantum lossless code with its corresponding quantum state. Since the relation − Tr ρ p log σ (Un ) − H (ρ p ) = D(ρ p σ (Un )) holds for the mixture of the ensemble ρ p , the minimum of the average energy Tr HUn ρ p Un† among prefix quantum lossless codes is almost equal to the von


181

Neumann entropy H (ρ p ). Furthermore, the inequality (17) guarantees that even though we remove the prefix condition, it is impossible to increase the average energy very much. When we use a prefix quantum lossless code corresponding to a state σ and the true mixture state is ρ p , the average energy is − Tr ρ p log σ = H (ρ p ) + D(ρ p σ ). Therefore, the relative entropy D(ρ p σ ) can be regarded as the redundancy of the quantum lossless code σ with respect to ρ p . Then, a sequence of prefix quantum lossless codes corresponding to a state {σn } is called universal if the redundancy n1 D(ρ ⊗n σn ) goes to zero for all ρ. ⊗n and the prefix quantum lossless code σ Pn ,n := When the true mixture is ρ n∈Ynd Pn (n)ρn is applied, the asymptotic redundancy is given by (13). Hence, the 2 mini-max asymptotic redundancy is d 2−1 log n +Cd +log Y d eC( p) d p, which is attained when we choose the mini-max code σ J,n or σ˜ J,n . Therefore, the prefix quantum lossless code σ J,n and σ˜ J,n is a universal quantum lossless data compression. 6. Discussion We have found a sequence of states {σn } such that the relative entropy D(ρ ⊗n σn ) 2 behaves universally as d 2−1 log n. While this result was known for the case of a qubit, the general case has remained open for more than ten years. In this derivation, the calculation of Matsumoto and Hayashi [15] plays an essential role. Furthermore, we have solved the asymptotic mini-max problem: min sup lim

{σn } ρ∈S n→∞

D(ρ ⊗n σn ) −

d2 − 1 log n . 2

(20)

It has been checked that our optimal value is better than the result of Krattenthaler and Slater [4] for the qubit case. Our discussion is different from the original discussion of Clarke and Barron[1,3] in the following two points. In the first point, the state σn is restricted to be n i.i.d. distributions in their optimization problem, while the state σn runs over all density matrices in our optimization. In the second point, they treat any subfamily of distributions with several regularity conditions while we have restricted the set S to the set of all faithful density matrices. So, it is a future problem to extend the obtained result to the case when S is a continuous subfamily. Our method can be translated to the classical case when the family is taken as the full multinomial distribution family. In such a case, the derivation of the solution to the mini-max problem can be expected to be shorter than the original derivation[1,3]. The optimal prior distribution derived here coincides with the quasi-Bures distribution (21) in Slater [20] in the qubit case. Its geometrical properties in the qubit case will be discussed in the forthcoming paper[22]. Hence, it is also left as a future problem to clarify the geometrical properties of the optimal prior in the arbitrary-dimensional case. In addition, we have revisited quantum variable-length lossless data compression, and characterized the average energy (length) by (17) for the general case. This result (17) has a similar form to Theorem 5 in Chou and Cheng [19], but is a little different from the result of Chou and Cheng. In addition, our definition of quantum lossless code

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is more general, hence our result is stronger. Also they used the method of majorization, whereas we have not done so. Concerning the prefix case, we have derived the correspondence between the prefix quantum lossless code and a density matrix through a quantum version of the Kraft inequality (18). Using this relation, we have applied our result (2) and (12) to quantum variable-length lossless data compression. It has been established that it is possible to compress multi-copy states universally by means of the lossless code σ J,n or σ˜ J,n . We note that relations between the approximation obtained and other universal protocols have not been discussed in this paper. This relation will be treated in a future paper[21]. The universal approximation developed here may be applicable to other topics in quantum information theory[18]. Such applications may well prove to be of interest. Acknowledgements. The author expresses his appreciation to Professor Hiroshi Nagaoka for explaining Clarke and Barron’s result[1,3]. He is also grateful to Professor Keiji Matsumoto for helpful discussion concerning representation theory. He thanks Professor Paul Slater for informing him about the quasi-Bures distribution. He also appreciates the referees’ helpful comments concerning this manuscript. This research was partially supported by a Grant-in-Aid for Scientific Research on Priority Area ‘Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI)’, No. 18079014 and a MEXT Grant-in-Aid for Young Scientists (A) No. 20686026.

A. Proof of Theorem 2 Let A be the projection to the image of the code Un . We diagonalize AH A as AH A = d n d n −λ † ˜ (Un ) := i=1 2 i Un |ei ei |Un . Then, i=1 λi |ei ei |. Define the Hermitian matrix σ x the convexity of x → 2 implies that n

Tr σ˜ (Un ) =

d

n

2

−λi

d

=

i=1

n

2

ei |−H |ei

d ≤ ei |2−H |ei

i=1

= Tr 2−H

dn

i=1

|ei ei | ≤

i=1

max

rank P=d n

Tr 2−H P.

Define the function f (M) := maxrank P=M Tr 2−H P. When M = 21 + 22 + 23 + · · · + 2m (≥ 2m ), we have f (2m ) ≤ f (M) = m. Thus, f (d m ) ≤ n log d, which implies that Tr σ˜ (Un ) ≤ n log d. Define the state σ (Un ) :=

1 ˜ (Un ). σ˜ (Un ) σ

Then, any input state ρ satisfies n

Tr Un ρUn† H

=

Tr Un ρUn† AH A

=

ρUn†

d

λi |ei ei |Un = − Tr ρ log σ˜ (Un )

i=1

= −(Tr ρ log σ (Un )) − log Tr σ˜ (Un ) ≥ −(Tr ρ log σ (Un )) − logn log d.


183

References 1. Clarke, B.S., Barron, A.R.: Information-theoretic asymptotics of Bayes methods. IEEE Trans. Inform. Theory 36, 453–471 (1990) 2. Amari S., Nagaoka H.: Methods of Information Geometry. Providence, RI: Amer. Math. Soc. & Oxford University Press, 2000 3. Clarke, B.S., Barron, A.R.: Jeffreys’ prior is asymptotically least favorable under entropy risk. J. Stat. Plan. Inference 41(1), 37–61 (1994) 4. Krattenthaler, C., Slater, P.: Asymptotic Redundancies for Universal Quantum Coding. IEEE Trans. Inform. Theory 46, 801–819 (2000) 5. Hayashi, M.: Universal coding for classical-quantum channel. Commun. Math. Phys. 289, 1087–1098 (2009) 6. Han T.S., Kobayashi K.: Mathematics of Information and Encoding. Providence, RI: Amer Math. Soc. 2002 (Originally written in Japanese in 1999) 7. Lynch, T.J.: Sequence time coding for data compression. Proc. IEEE 54, 1490–1491 (1966) 8. Davisson, L.D.: Comments on ‘Sequence time coding for data compression’. Proc. IEEE 54, 2010 (1966) 9. Jozsa, R., Horodecki, M., Horodecki, P., Horodecki, R.: Universal Quantum Information Compression. Phys. Rev. Lett. 81, 1714 (1998) 10. Hayashi, M.: Exponents of quantum fixed-length pure state source coding. Phys. Rev. A 66, 032321 (2002) 11. Boström, K., Felbinger, T.: Lossless quantum data compression and variable-length coding. Phys. Rev. A 65, 032313 (2002) 12. Hayashi, M., Matsumoto, K.: Quantum universal variable-length source coding. Phys. Rev. A 66, 022311 (2002) 13. Koashi, M., Imoto, N.: Quantum Information is Incompressible Without Errors. Phys. Rev. Lett. 89, 097904 (2002) 14. Christandl M.: The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography. PhD thesis, February 2006, University of Cambridge. available at http://arXiv.org/abs/quant-ph/ 0604183v1, 2006 15. Matsumoto, K., Hayashi, M.: Universal distortion-free entanglement concentration. Phys. Rev. A 75, 062338 (2007) 16. Keyl, M., Werner, R.F.: Estimating the spectrum of a density operator. Phys. Rev. A 64, 052311 (2001) 17. Schumacher, B.: Quantum coding. Phys. Rev. A 51, 2738–2747 (1995) 18. Hayashi M.: Quantum Information: An Introduction. Berlin: Springer, 2006 (Originally written in Japanese in 2004) 19. Chou H.-H., Cheng J.: New lower bounds on the average base length of lossless quantum data compression. In: Proceedings of the 8th Quantum Communication, Computing, and Measurement, edited by O. Hirota, J.H. Shapiro, M. Sasaki, Tokyo: NICT Press, 2008, pp. 279–282 20. Slater, P.: Hall normalization constants for the Bures volumes of the n-state quantum systems. J. Phys. A: Math. Gen. 32, 8231–8246 (1999) 21. Hayashi M.: In preparation 22. Grosse H., Krattenthaler C., Slater P.B.: Asymptotic Redundancies for Universal Quantum Coding II. In preparation Communicated by M. B. Ruskai


Communications in


W -Symmetry of the Adèlic Grassmannian David Ben-Zvi1 , Thomas Nevins2 1 Department of Mathematics, University of Texas at Austin, Austin,

TX 78712, USA. E-mail: [email protected]

2 Department of Mathematics, University of Illinois at Urbana-Champaign,

Urbana, IL 61801, USA. E-mail: [email protected] Received: 7 August 2008 / Accepted: 14 July 2009 Published online: 29 September 2009 – © Springer-Verlag 2009

Abstract: We give a geometric construction of the W1+∞ vertex algebra as the infinitesimal form of a factorization structure on an adèlic Grassmannian. This gives a concise interpretation of the higher symmetries and Bäcklund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of “W1+∞ -geometry”: the geometry of D-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves. 1. Introduction There is a beautiful and well-known relationship between the conformal field theory of free fermions, the KP hierarchy, and the geometry of moduli spaces of curves and bundles. In particular, the deformation theory of curves and of line (or vector) bundles on them is realized algebraically by the Virasoro and Heisenberg (or Kac-Moody) algebra symmetries of the free fermion system. (For a sampling of perspectives on this story see [KNTY,W,DJM,SW,Mu,MP,T] and references therein.) The algebra W1+∞ (a central extension of differential operators with Laurent coefficients) and its more complicated nonlinear reductions, the Wn algebras, themselves play a well-established role as algebras of “higher symmetries” in integrable systems and string theory. For example, W-algebras arise as the additional symmetries of the KP and Toda hierarchies, and tau-functions satisfying W-constraints arise as partition functions of matrix models and topological string theories (see [vM]). Moreover, the search for W-geometry and W-gravity generalizing the moduli of curves and topological gravity has also been the topic of an extensive literature (see for example [H,Po,LO]). In the present paper, we establish a “global” (or exponentiated) geometric realization of these higher W1+∞ -symmetries. More precisely, in Sects. 3 and 4 (aimed primarily at algebraists) we give an algebro-geometric construction of a factorization space and prove that it naturally integrates the W1+∞ -symmetry of the free fermion system (in a sense described below). Our work gives a precise analog of earlier work on the Virasoro

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and Kac-Moody algebras, in which the role of the moduli of curves and bundles is played by the moduli of D-bundles (projective modules over differential operators) on a smooth complex curve X , or equivalently by the moduli of torsion-free sheaves on cusp quotients of X . Then in Sect. 5 (aimed at a broader audience of geometers and physicists) we discuss the roles of this construction in relation to integrable systems and conformal field theory. In the late 1980s, the beautiful paper [W] of Witten developed an algebro-geometric formulation of the chiral theory of a free fermion on a Riemann surface from an adèlic perspective; see also [T] for current developments. More recently, Beilinson and Drinfeld [BD2] introduced the notion of factorization space as a nonlinear (or integrated) counterpart to the notion of a vertex algebra or chiral CFT (see [FB] for a review). Factorization spaces provide a general geometric bridge between conformal field theory and the algebraic structure of adèle groups and Hecke correspondences. The Beilinson-Drinfeld Grassmannian, a factorization space built out of the affine Grassmannian of a Lie group G, simultaneously encodes the infinitesimal (Kac-Moody) and global (Hecke) symmetries of bundles on curves and is arguably the central figure in the geometric Langlands correspondence [BD1]. The case of G = G L 1 (C) encodes geometrically the vertex algebra of a free fermion, or in the language of integrable systems the KP flows together with the vertex operator [DJKM]. In Sect. 3, we establish an algebro-geometric formulation in the spirit of [W] (but using the technology of [BD2]) for W1+∞ -symmetry in conformal field theory. More precisely, for any smooth complex curve X we define an algebro-geometric variant GrD of the adèlic Grassmannian introduced by Wilson [Wi1,Wi2] in the study of the bispectral problem and the rational solutions of KP (see Remark 3.1 for a discussion of the relationship of GrD to Wilson’s construction). We then show that GrD is naturally a factorization space in which the G L 1 Beilinson-Drinfeld Grassmannian embeds as a factorization subspace. This is a special case of a more general construction in Sect. 3 of an adèlic Grassmannian Gr P associated to any D-bundle P on X ; the case P = Dn leads to multicomponent KP (and the W1+∞ (gl n ) vertex algebra). We show in Sect. 4 that the adèlic Grassmannian integrates the W1+∞ vertex algebra, realizing the latter as the algebra of infinitesimal symmetries of D-bundles: c Theorem 1.1. (See Theorem 4.2). For any c ∈ C, the chiral algebra W1+∞ (gln ) on X at level c associated to the W1+∞ (gl n )-vertex algebra is isomorphic to the chiral algebra c of level c delta functions along the unit section of the adèlic Grassmannian Gr n . δD n D

In Sect. 5 we discuss relations between the adèlic Grassmannian, integrable systems and conformal field theory. In particular we explain that the factorization structure of our adèlic Grassmannian encodes both the infinitesimal symmetries and the Hecke modifications of D-bundles: that is, the W1+∞ vertex algebra and the BäcklundDarboux transformations [BHY1,BHY3] of the KP hierarchy (and, more generally, the W1+∞ (gln )-algebras and Bäcklund-Darboux transformations or vertex operators of the multicomponent KP hierarchies). In other words, the factorization structure on the adèlic Grassmannian unites the vertex operators, Bäcklund transformations and additional symmetries of the KP hierarchy [vM] in a single geometric structure. The embedding of the G L n Beilinson-Drinfeld Grassmannian into the rank n adèlic Grassmannian globalizes n into W1+∞ (gln ). the inclusion of the Kac-Moody algebra gl Section 5 also contains an extended explanation of the interaction between our picture on the one hand and the free fermion theory and the Krichever construction on the other. Namely, we realize W1+∞ orbits on the Sato Grassmannian, and thereby the

W-Symmetry of the Adèlic Grassmannian

187

Orlov-Schulman additional symmetries of the KP hierarchy, via moduli of D-bundles or cusp line bundles. This generalizes the realization of Heisenberg and Virasoro orbits on the Grassmannian via moduli of curves and line bundles. We close with a discussion of the significance of the factorization space Gr P for W-geometry. For related studies of (and background on) the geometry of D-bundles we refer the reader to [BN1,BN2,BN3], which originated as an attempt to understand geometrically and generalize the relations between bispectrality, the adèlic Grassmannian, ideals in the Weyl algebra and Calogero-Moser spaces first uncovered and explored in work of Wilson and Berest-Wilson [Wi1,Wi2,BW1,BW2]. In [BN1], the Cannings-Holland description of D-modules on curves and their cuspidal quotients is revisited from a more algebro-geometric viewpoint (and the Morita equivalence of Smith-Stafford generalized to arbitrary dimension). In [BN2], two constructions of KP solutions from D-bundles are explained—one (which is most relevant to this paper) related to line bundles on cusp curves and the other related to Calogero-Moser systems. (This description of moduli spaces of D-bundles or ideals in D as Calogero-Moser spaces is established for arbitrary curves in [BN3].) As explained in [BN2], in genus zero the two constructions are evidently exchanged by the Fourier transform and thus, by [BW1], by Wilson’s bispectral involution [Wi1]. However, in higher genus the constructions are completely independent. 2. D-Bundles In this section we review the definition and features of D-bundles on a smooth complex algebraic curve X , following ideas of Cannings-Holland [CH1,CH2] as they are explained in [BN1]. Let D = D X denote the sheaf of differential operators on X (which can be viewed as functions on the quantization of the cotangent bundle T ∗ X ). Definition 2.1. A D-bundle M on X is a locally projective coherent right D X -module. Equivalently, a D-bundle is a torsion-free coherent right D X -module (since the sheaf of algebras D X locally has homological dimension one, [SS, Sect. 1.4(e)]). Moreover, D X possess a skew field of fractions (see [SS, Sect. 2.3]), from which it follows that any D-bundle has a well-defined rank. Thus D-bundles can be viewed alternatively as vector bundles or as torsion-free sheaves on the quantized cotangent bundle of X . Some obvious examples of D-bundles are the locally free (or induced) rank n D-bundles, of the form M = V ⊗ D for a rank n vector bundle V on X . In general, however, D-bundles of rank 1 are not locally free (isomorphic to D), but only generically free, behaving more like the ideal sheaves of a collection of points on an algebraic surface. Note that every right ideal in D X is torsion-free, hence a D-bundle of rank 1 (conversely any rank one D-bundle may locally be embedded as a right ideal in D X ). For example, let X = A1 , so that D X has as its ring of global sections the Weyl algebra Cz, ∂/{∂z − z∂ = 1}. The right ideal M0 generated by z 2 and 1 − z∂ agrees with D outside of 0 but is not locally free near z = 0. Let W (M0 ) = {θ ( f ) | θ ∈ M0 , f ∈ C[z]} ⊂ C[z]. Then W (M0 ) = C[z 2 , z 3 ], the subring of C[z] generated by z 2 and z 3 ; this is isomorphic to the coordinate ring of the cuspidal cubic curve y 2 = x 3 . Furthermore, M0 = D(C[z], W (M0 )), the space of differential operators that take C[z] into W (M0 ).

188


2.1. Grassmannian parametrization. In general, to describe D-modules it is convenient to extract linear algebra data using the de Rham (or Riemann-Hilbert) functor, which (on right D-modules) takes M to the sheaf of C-vector spaces which is its quotient by all total derivatives, or formally M → h(M) := M ⊗D O X . Note that if M = V ⊗ D is induced, we recover the underlying vector bundle V = h(M) in this fashion (though only as a sheaf of C-vector spaces rather than as an O X -module). In the example of M0 given above, h(M0 ) = W (M0 ). We can parametrize D-bundles by choosing generic trivializations: given a D-bundle M we can embed M into n = rk M copies of rational differential operators, M → Dn (K X ), so that M ⊗ K X = Dn (K X ). The module M is then determined by the finite collection of points xi at which M differs from DnX , and choices of subspaces of n copies of Laurent series Knxi at xi . This is succinctly explained in [BD2, Sect. 2.1]: D-submodules of a D-module N that are cosupported at a point x ∈ X are in canonical bijection (via the de Rham functor h) with subspaces of the stalk of the de Rham cohomology h(N )x at x that are open in a natural topology. In our case, we obtain collections of linearly compact open subspaces (or c-lattices, in the terminology of [BD2]) of n copies of Laurent series Knxi at xi with respect to the usual (Tate) topology, i.e., subspaces commensurable with n copies of Taylor series. These subspaces correspond to points of the “thin Sato Grassmannian” Gr(Knxi ) (not to be confused with the complementary original or “thick” Sato Grassmannian, see Sect. 5).

2.2. Cusps. D-bundles can alternatively be described by torsion-free sheaves on cuspidal curves. Namely, given a D-module M with a generic trivialization (identification with Dn ), we can find a subring (or rather subsheaf of rings) OY ⊂ O X whose left action on Dn (K X ) (by right D-automorphisms) preserves M. Equivalently, the C-subsheaf h(M) ⊂ h(D(K X )) is preserved by a ring OY ⊂ O X , which differs from O X only at the finitely many singularities xi of the embedding. Thus h(M) defines a torsionfree sheaf VY on the cuspidal curve Y = Spec OY , which has X → Y as a bijective normalization. In fact, as was shown by Smith and Stafford [SS] and generalized to arbitrary dimension in [BN1], passing from X to such a cuspidal quotients Y doesn’t change the category of D-modules: the sheaves of rings D X and DY are Morita equivalent. Thus, given a torsion-free sheaf VY on Y we can define a torsion-free DY -module MY = VY ⊗ DY by induction, and transport it to obtain a D-bundle M on X . Moreover, this process of “cusp-induction” reverses the above procedure M → VY . As a result, we obtain a geometric reinterpretation of the linear algebra data classifying D-bundles. Every D-bundle arises by cusp-induction from a torsion-free sheaf on a cuspidal quotient X → Y , but Y is not unique. We can consider M as associated to smaller and smaller subsheaves OY ⊂ OY ⊂ O X , introducing deeper and deeper cusps in the curves X → Y → Y . The collection of D-bundles on X is obtained as a direct limit over these deepening cusp curves of the collections of torsion-free sheaves. Under this limit the geometry of X “evaporates” and we are left with a reinterpretation of the linear algebra data above, characterizing a class of C-sheaves on X which arise from torsion-free sheaves on some cusp quotient.


189

3. Factorization In this section we study the factorization space structure on the symmetries of a D-bundle. Factorization spaces were introduced in [BD2]; we refer the reader to [FB, Sect. 20] for a leisurely exposition. In Sect. 3.1, we establish a factorization space structure for D-bundles on a curve. As we explain in Sect. 3.2, standard constructions then produce a chiral algebra which acts infinitesimally simply transitively near the unit section of our factorization space Gr P . In Sect. 3.3, we describe how to twist this chiral algebra by the determinant line bundle to obtain a family of chiral algebras at different levels.

3.1. The factorization space Gr P . Let X denote a smooth complex curve. We usually fix a D-bundle P in what follows. Recall the definition of a factorization space or factorization monoid from [KV, Def. 2.2.1]. We will define a factorization space, the P-adèlic Grassmannian Gr P of X , as follows. For each finite set I we define Gr IP (S), for a scheme S, as a set over X I (S), φ

the set of maps S − → X I . Such a map φ defines a divisor D ⊂ S × X . Then Gr IP (S) is the set of pairs (M, ι) consisting of: (1) An S-flat, locally finitely presented right D X ×S/S -module M, such that the restriction M| X ×{s} to each fiber is torsion-free (equivalently, locally projective).

(2) An isomorphism ι : M| X ×S D − → P| X ×S D such that the composite M → ι M| X ×S D − → P| X ×S D is injective with S-flat cokernel. Remark 3.1. As we have mentioned in the Introduction, our adèlic Grassmannian is not the same as the one introduced by Wilson [Wi1,Wi2]. Indeed, our space differs from Wilson’s in two significant respects. One of these is that Wilson’s space is closer to the colimit lim Gr IP of spaces in our system. This colimit naturally lives (see [BD2] or −→ I

[FB]) over the colimit lim X I of finite products of the curve, which is known as the −→ I

Ran space Ran(X ) of X . The space Ran(X ) is a contractible topological space that is not naturally by itself an object of algebraic geometry: it is better to work with it via the system {X I } I , and hence with the factorization space Gr IP . The second difference is that Wilson identifies the spaces h(M1 ) and h(M2 ) associated to two D-bundles M1 and M2 on A1 if they are identified by an element of C(x)× , whereas we keep the two separate. An alternative to our approach, closer in spirit to Wilson’s, would be to take the factorization subspace of Gr P parametrizing those M which “have index zero with respect to P at every point of X .” Suppose V ⊗ D is an induced D-bundle. Standard arguments using finite generation show: Lemma 3.2. Suppose (M, ι) ∈ Gr VI ⊗D (S). Then, locally on S, there exists an integer k such that V (−k D) ⊗ D ⊂ M ⊂ V (k D) ⊗ D. For any D-bundle P, the space Gr P is a factorization space: Proposition 3.3. The functors Gr IP assemble into a factorization monoid over X in the sense of [KV, Def. 2.2.1].

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Proof. We need to show: for any set I , Gr IP is naturally an ind-scheme of ind-finite type, formally smooth over X I , and comes equipped with a formally integrable connection over X I . The “factorization” properties themselves (parts (a) and (b) in the definition from [KV]) are standard. Briefly, we need to see that the fiber of Gr IP over an I -tuple x I of (S-)points of X depends only on the support of the tuple—not the multiplicities—and factorizes as a product for every disjoint union decomposition of the tuple. The first property is automatic for any functor defined in terms of data outside of x I . The second follows from the fact that modifications at points x I are local, and so modifications away from disjoint points can be glued together (explicitly, this follows from the description of modifications by collections of points of Grassmannians, as in the previous section). Choose an induced D-bundle V ⊗D and containments V (−E)⊗D ⊆ P ⊆ V (E)⊗D (which is possible by Lemma 3.2). Then P is cusp-induced from a cusp curve Y with homeomorphism X → Y that fails to be an isomorphism only at the support of E. Suppose Y has a cusp located “under” a point p in X . If z is a uniformizer at p, then for n sufficiently large, z n lies in OY . It follows that we can make sense of D-module inclusions O(−np) ⊗ P ⊂ P ⊂ O(np) ⊗ P (say, as subsheaves of P| X E ). Write P(k D) = O(k D) ⊗ P when this makes sense. We then give Gr P the ind-structure coming from Lemma 3.2: we let Gr IP (n) denote the subset of Gr IP of pairs (M, ι) for which we have P(−n D) ⊂ M ⊂ P(n D). There are now many ways to see that Gr IP (n) is a scheme of finite type. For example, applying the de Rham functor to M/P(−n D) ⊂ P(n D)/P(−n D) and applying Theorem 5.7 of [BN1], we see that choosing M is equivalent to choosing a sheaf of vector subspaces of h(P(n D)/P(−n D)). The functor of such choices is a closed subscheme of the relative Grassmannian of subspaces of H 0 h(P(n D)/P(−n D)) (this is essentially the Cannings-Holland picture of the adèlic Grassmannian, as explained and used to great effect in [Wi2]). It remains to show that Gr P is formally smooth and to describe the formally integrable connection over X I . We first show that Gr IP → X I is formally smooth. Let (M, ι) be an object of Gr IP (S) for a scheme S. We may assume that P(−n D) ⊂ M ⊂ P(n D) and that M is, in the terminology of [BN1], cusp-induced from a family F of OY -modules, where Y is a family of cusp curves over S given by a sheaf of sub-algebras OY ⊂ O X ×S . We then have inclusions of OY -modules h(P(−n D)) ⊂ F ⊂ h(P(n D)). Now, given a nilpotent thickening S ⊂ S and a divisor D on S × X that is a nilpotent thickening of the divisor D to which (M, ι) corresponds, we need to extend (M, ι) to a pair (M , ι ) over X × S . To do this, we first extend Y to a family Y of cusp curves over S “with cusps determined by D ,” i.e. cusps whose depths are determined by D . It then suffices to deform the map h(P(n D)) → h(P(n D))/F: the kernel of the cusp-induction of the deformed map will give M , and the inclusion in h(P(n D )) ⊗ D will give ι . But, by construction, as an OY -module h(P(n D))/F is a direct sum of skyscrapers: more precisely, it is isomorphic to the direct image to Y of ⊕i∈I O pi , where pi is the i th section of X × S → S determined by the map S → X I . So, as our deformation of h(P(n D))/F we can take ⊕i∈I O pi as an OY -module, and then deformations of this sum of skyscrapers, as well as the map from h(P(n D )), certainly exist. This proves the existence of (M , ι ). Hence Gr IP → X I is formally smooth. (Note that an alternative proof of this assertion uses the formally transitive action of a Lie algebra of matrix differential operators, as in the next section.) Finally, we describe the formally integrable connection over X I —note, however, that the existence of such a connection follows formally (as in [FB, 20.3.8]) from the existence of a unit for the factorization space. This follows [Ga, Sect. 5.2]. Namely, given an


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Artinian scheme A, a pair (M, ι) parametrized by a scheme S, and a map S × A → X I determining a divisor D A , we pull M back along the projection X × S × A → X × S. Since X × S × A D A = (X × S D) × A, this pullback comes equipped with an isomorphism to P over X × S × A D A , as desired. This canonical lift of infinitesimal extensions gives our connection. Suppose P = V ⊗ D is an induced D-bundle. Let Gr VI denote the Beilinson-Drinfeld Grassmannian associated to V : over a divisor D ⊂ X , this parametrizes vector bundles W equipped with an isomorphism W | X D ∼ = V | X D . Such an isomorphism induces a D-bundle isomorphism W ⊗ D| X D ∼ = P| X D , thus giving an embedding of unital factorization spaces: Gr VI → Gr IP .

(3.1)

We note that the relative tangent space of Gr IP → X I at (M, ι) is given by T(M,ι) (Gr IP / X I ) = HomD (M, P(∞ · D)/M)

(3.2)

by the same analysis as one uses for a Quot-scheme. Given a divisor D ⊂ X , let supp(D) denote the support of the divisor D and X D the formal completion of X along D. For a quasi-coherent sheaf F, let Fη denote the direct sum of stalks at points of supp(D). Lemma 3.4. There is a natural surjection: I / X I ). End(P(∞ · D))η T(M,ι) (GrD

If, in addition, P = Dn , this yields a surjection: I I H 0 (D(∞ · D)| X D ) ⊗ gl n T(M,ι) (GrD / X ).

In particular, if D consists of a single point x, we get a surjection {∗}

D(Kx ) ⊗ gl n T(M,ι) (GrD / X ). Proof. We use the short exact sequence: 0 → M → P(∞ · D) → P(∞ · D)/M → 0. Since M and P(∞ · D) are locally projective, the sheaf Ext group Ext 1D P(∞ · D), M vanishes, and we get a surjective map of sheaves HomD P(∞ · D), P(∞ · D) → Hom P(∞ · D), P(∞ · D)/M . Since Hom P(∞ · D), P(∞ · D)/M is supported along supp(D), the conclusions follow.

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3.2. Chiral algebra. The D-bundle P, equipped with its canonical isomorphism id with P over X D for any D, defines a “unit” section unit I : X I → Gr IP , compatible with the factorization isomorphisms and preserved by the relative connection. Let δ P denote {1} the O-push-forward of the right D-module unit 1! ω X on Gr XP = Gr P , and δ PI the pushI I I forward of unit! ω X I from Gr P . The sheaves δ P are right D-modules on X I , and we let δ I denote the corresponding left D-modules. P Corollary 3.5. The left D-modules δ PI form a factorization algebra on X . In particular δ P is a chiral algebra. Proof. This exactly follows Sect. 5.3.1 of [Ga].

Remark 3.6. We will identify this chiral algebra with a W-algebra below in the case P = Dn . It would be interesting to investigate the structure of δ P further in the case that P is not locally free. Proposition 3.7. Suppose P = Dn . The Lie algebra D(Kx ) ⊗ gl n acts continuously and {∗} formally transitively on the Grassmannian fiber GrDn (x), inducing an isomorphism {∗} TDn GrDn (x) = D(Kx )/D(Ox ) ⊗ gl n . Proof. In the Grassmannian description of the fiber GrDn (x) = Gr(Knx ), the action of D(Kx ) ⊗ gl n comes from its defining action by continuous endomorphisms of Knx . The tangent space to the Grassmannian Gr(Knx ) at Oxn is identified with Homcont (Oxn , Knx /Oxn ) = D(Kx )/D(Ox ) ⊗ gl n , n since a continuous homomorphism factors through a map (Ox /mkx )n → (m−k x /Ox ) for k sufficiently large, and all such homomorphisms may be realized by differential operators (or alternatively since Homcont (Oxn , Knx /Oxn ) = HomD (Dn , Knx /Oxn ⊗ D), since a homomorphism on either side is automatically OY -linear for a deep enough cusp Y ). The formal transitivity at other points follows similarly.

It follows that D(Kx ) ⊗ gl n also acts on the space of delta functions δDn (x)—we denote this action by actx : D(Kx ) ⊗ gl n ⊗ δDn (x) → δDn (x). This action induces an isomorphism of D(Kx ) ⊗ gl n -modules, δDn |x ∼ = U D(Kx ) ⊗ gl n ⊗U D(O )⊗gl C. x

n

(3.3)

π

3.3. Levels. Let M be a coherent D X ×S/S -module on X ×S → S. We define a line bundle L

det(M) on S as follows. Consider the bounded complex Rπ· DR· (M) = Rπ· (M ⊗ O) of coherent O S -modules, and form its determinant line bundle L

det(M) = det(Rπ· M ⊗ O). DX

DX


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Note that if S is smooth and M is in fact a right (absolute) D X ×S -module, then Rπ∗ M = L

Rπ· M ⊗ O ∈ mod −D S is the right D-module push-forward of M. Thus the deterDX

minant of de Rham cohomology line bundle det(M) is the D-module analogue of the determinant of cohomology line bundles on moduli stacks of bundles. Suppose that M is a cusp-induced D-module, so that M ∼ = M ⊗ DY ←X for a cuspidal OY

quotient π : Y → S of X × S → S over S. Then we have DR· M = h(M) = M, and det(M) is the determinant of cohomology, det(M) = det(Rπ ∗ M) of the O-module push-forward of M to S. It follows that the determinant line bun{∗} dle, restricted to a fiber GrD (x), is naturally identified with the “Plücker” determinant line bundle on the Grassmannian Gr(Knx ). The following proposition follows from the → I description of det on GrD n | → as a tensor product of local factors at the points of x (for x an algebraic treatment of factorization in the closely related context of -factors, see [BBE]): Proposition 3.8. The line bundle det on Gr P has a natural factorization structure over the factorization structure of Gr P . We can now define a new chiral algebra δ cP for every c ∈ C as level c delta functions along the unit section on Gr P . More precisely, the factorization line bundle det has a lift unit det of the unit section of Gr P : in other words, det is canonically trivialized along unit. As a consequence, there is a natural direct image functor from D X I -modules to det ⊗c -twisted D-modules on Gr IP . It follows that we may take ω X I , push it forward to Gr IP as a det ⊗c -twisted D-module, and take its O-module direct image to X I to obtain a new chiral algebra, denoted by δ cP . (Alternatively we can describe δ cP as the O-module restriction of the sheaf of det c -twisted differential operators to the unit section.) 4. W1+∞ and W1+∞(gl n ) c or, more generally, δ c : In this section, we explain how to identify the chiral algebra δD Dn it is the chiral algebra associated to the vertex algebra W1+∞ (respectively, W1+∞ (gl n )) at level c. We refer to [K] for a discussion of W1+∞ and [vdL] for W1+∞ (gln ). We begin in Sect. 4.1 by reviewing the definition of the chiral algebras W1+∞ and, more generally, c and δ c . W1+∞ (gl n ). In Sect. 4.2 we then identify them with δD Dn

4.1. Introducing W1+∞ . We recall (from [BD2], 2.5 and [BS]) the construction of the Lie* algebra (or conformal algebra [K]) gl(D) = End ∗ D. As a right D-module, gl(D) is the induced D-module D ⊗ D. We consider the sheaf D as a quasi-coherent O-module equipped with a Lie algebra structure given by differential morphisms. (It forms the Lie algebra of right D-module endomorphisms of D.) It follows that the induced D-module has a natural Lie* algebra structure, as explained in [BD2] or [FB, 19,1,7]. We let W1+∞ = Uch gl(D) denote the universal enveloping chiral algebra of gl(D). It follows that the fibers of W1+∞ are identified with the vacuum representation of D(Kx ), W1+∞ (x) ∼ = U D(Kx ) ⊗U D(Ox ) C.

(4.1)

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The Lie* algebra gl(D) has a central extension gl(D) which may be described as follows. Consider the short exact sequence 0 → O ω X → j∗ j ∗ O ω X → D → 0, where D is considered as a O-bimodule. Restriction to the diagonal defines a morphism O ω X → ∗ ω X , and we take the push-out exact sequence (and push forward to X ), obtaining a central extension → D → 0. 0 → ωX → D This ω X -extension of D corresponds to an ω X -central extension of the Lie* algebra gl(D). It is also shown in [BD2,BS] that the action of D(Kx ) on Kx gives rise to a dense embedding of D(Kx ) in gl(Kx ), the Tate endomorphisms of Kx , and that the Tate cen x ). The following tral extension of gl(Kx ) restricts to the above central extension D(K lemma is an immediate consequence: Lemma 4.1. The action of D(Kx ) on Gr(Kx ) lifts to an action of the central extension x ) on det with level one. D(K c the corresponding chiral enveloping algebra at level c. Repeating We denote W1+∞ c (gl ). the construction with D ⊗ gl n in place of D leads to the chiral algebra W∞ n c 4.2. Comparison of δD and W1+∞ . In this section we identify the chiral algebra δD n c (gl ): associated to the factorization space GrDn with the W-algebra W∞ n

Theorem 4.2. There is a natural isomorphism of chiral algebras W∞ (gl n ) ∼ = δD n . Moreover for any c ∈ C, this isomorphism lifts to an isomorphism of the chiral algebra c (gl ) at level c with the chiral algebra δ c of level c delta functions on Gr n . W∞ n D Dn We will explain two proofs of this theorem below. First, in Sect. 4.2.1, we explain that the theorem is a special case of a very general construction of chiral algebras from the unit section of a unital factorization space. Because this general construction seems to be a folk theorem that does not appear in the literature, we also explain in Sect. 4.2.2 a more concrete proof that follows closely the approach to Kac-Moody algebras explained in [Ga]. 4.2.1. Chiral Hopf algebras and factorization spaces We begin with some generalities. Let {S I → X I } I be a unital factorization space. Recall that, in particular, each S I is r a formally smooth ind-scheme of ind-finite type over X I . Write S = S {∗} − → X . Pulling back the tangent sheaf unit ∗ TS along the unit section gives a D X -module (using the flat connection on S/ X ) which we denote by L. The factorization structure on S equips L with a structure of Lie*-algebra on X . We assume that this Lie*-algebra is an ind-D-vector bundle—that is, that it is a colimit of D-vector bundles L j and the Lie*algebra structure is compatible with this realization (in the standard sense that L j and L k multiply to L j+k ). It then follows from [BD2, Lemma 2.5.7] that the dual D-module L ∨ is a pro-Lie! -coalgebra, [BD2, Sect. 2.5.7]. Write S for the formal completion of S along the unit section; then S is again a unital factorization space over X . This space can be completely reconstructed from the Lie*-algebra L: this is essentially just the factorization analog of the fact that a smooth


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formal group scheme can be reconstructed from its Lie algebra. In the factorization setting, this works as follows. To the Lie*-algebra one can associate a chiral algebra, the chiral enveloping algebra U ch (L) of L [BD2, Sect. 3.7]. The chiral enveloping algebra is naturally a cocommutative Hopf chiral algebra (defined in [BD2, Sect. 3.4.16]. The dual D-module U ch (L)∨ is then a commutative pro-Hopf chiral algebra, and, by the discussion in [BD2, Sect. 3.4.17], its formal spectrum gives a factorization space S(L) = { S(L) I → X I }. Under the hypothesis that L is an ind-D-vector bundle, more over, S(L) is isomorphic to S, the formal completion of our original factorization space along the unit section. As an immediate consequence, one has: Theorem 4.3. Suppose that {S I → X I } I is a unital factorization space and that the associated Lie*-algebra L is an ind-D-vector bundle. Then the delta-function chiral algebra δ = r∗ unit ! ω X is isomorphic to U ch (L). There is also an extension of the theorem to the twisted chiral algebra δ c associated to a factorization line bundle L I on S I . We leave it to the reader to formulate the analog of Theorem 4.3 in this case. Theorem 4.2 is an immediate corollary once we identify L with gl(Dn ); this is a consequence of Proposition 4.4. 4.2.2. Direct proof of Theorem 4.2 In this section we give a somewhat different, more direct and concrete, identification of the chiral algebra δDn associated to the factorization space GrDn with the W-algebra W1+∞ (gl n ). For simplicity of exposition, we restrict to the case n = 1 and c = 0, in other words to W1+∞ . The proof is modeled directly on the identification of the Kac-Moody vertex algebra with the factorization algebra of delta functions on the affine Grassmannian due to Beilinson and Drinfeld explained in [Ga]. The result is a formal consequence of the construction of an extension of the action of D(Kx ) on GrD (x) to a “factorization action” of the Lie* algebra gl(D) on the factorization space GrD (Proposition 4.4 below). We assume that X is affine in the construction, though the final result will not depend on this assumption. Recall ([BD2] 3.7, [Ga]) that to any right D-module L on X we can assign a collection L of left D-modules on X , with certain factorization isomorphisms. Consider the open subset j (I ) : U ⊂ X I × X of pairs ({xi }, x), where x = xi for any i ∈ I . We let L I denote the relative de Rham cohomology sheaf (I ) I (I )∗ L = p X I · (id h)( j∗ j O X I L) on X I , whose fiber at {xi } is the de Rham cohomology of L on X with poles allowed at the xi . The sheaf inherits a left D-module structure from that on O X I . The sheaves L I satisfy several good factorization properties with respect to maps of sets J → I ([BD2]), in particular for J I we have canonical isomorphisms !I LI. LJ ∼ = I The sheaf L contains as a D-submodule the de Rham cohomology of L with no I poles allowed, L = p X I ∗ (O X I L). We define L I as the quotient sheaf, and note that this sheaf is local in X , in the sense that the fiber at {xi } only depends on L in the neighborhood of the xi . In particular L {1} = p X ∗ ( j∗ j ∗ (O X L)/O X L) = L is the left D-module version of L. When L is a Lie* algebra, then the sheaf L I is a Lie I algebra in the tensor category of left D-modules, and L is a Lie sub-algebra. The sheaf

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U L I ⊗U L I C of induced (vacuum) representations then forms a factorization algebra, corresponding to the chiral enveloping algebra U ch (L) of L. In our case, the Lie* algebra L = gl(D) is an induced D-module, with de Rham I may be identified (as Lie cohomology h(gl(D)) = D. It follows that the sheaves gl(D) algebras in left D-modules) with the usual (sheaf or O-module) push-forward → I = p X I · ( j∗(I ) j (I )∗ O X I D) = p X I ∗ (D X I ×X/ X I (∗ x )), gl(D)

i.e. with differential operators with arbitrary poles along the universal divisor on X I × X over X I . The subsheaf gl(D) I then consists of global differential operators on X . The quotient sheaf has fiber at x the space D(Kx )/D(Ox ) of delta-functions at x, and has fiber at {xi } the global sections of the D-module on X which consists of delta functions at each of the points xi , without multiplicities. We now describe the “factorization action” of gl(D) on GrD , which is a multipoint generalization of the formally transitive action of D(Kx ) on GrD (x) and resulting description TD GrD |x = D(Kx )/D(Ox ) of the tangent space. Proposition 4.4. I in left D-modules over X I acts on Gr I , compatibly with 1. The Lie algebra gl(D) D connections and factorization structures, and with stabilizer at the unit section the Lie sub-algebra gl(D) I . 2. For x ∈ X , the Lie algebra gl(D)(x) is dense in D(Kx ), and its action on GrD (x) is the restriction of the formally transitive action of D(Kx ). → I = p Proof. Consider the sheaf of Lie algebras gl(D) X I ∗ D(∗ x ) of differential opera→ tors with poles along the universal divisor x . It acts by infinitesimal automorphisms on → the sheaf D(∗ x ), and hence on its functor of submodules, preserving the sub-functor → I . The action is of submodules cosupported on x . This is our desired action on GrD compatible with decompositions of the divisor and forgetting multiplicities, hence with the factorization structure. The formal transitivity follows from Lemma 3.4.

We are now ready to prove Theorem 4.2 Proof of Theorem 4.2. We begin by constructing a natural map act : j∗ j ∗ (gl(D) δD ) → ! δD satisfying 1. gl(D) acts on the chiral algebra δD by derivations. 2. The resulting action on δD (x) of the completed de Rham cohomology h x (gl(D)) coincides with the action D(Kx ) ⊗ δD (x) → δD (x) induced by the action actx of D(Kx ) on GrD (x). To construct the map act, it is equivalent to define the map obtained by taking de Rham cohomology along the first factor, that is a map ( j∗ j ∗ (D O)) δD → ∗ δD . This map is supported set-theoretically on the diagonal, and so is equivalent to a continuous D-module action of the completion p1∗ ( j∗ j ∗ (D O))along the diagonal on δD . Note that X = p1∗ ( j∗ j ∗ (D O)) ⊂ p1∗ ( j∗ j ∗ (D O)) gl(D)


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is dense since X is affine, and so it suffices to define a continuous D-module action of X on δ . Such an action is provided by Proposition 4.4. The statement that this gl(D) D action is by derivations of the chiral algebra δD follows from the compatibility of the X on δ is the restriction action with factorization. More precisely, the action of gl(D) D I on δ I for I = {1, 2}, and the chiral bracket on to the diagonal of the action of gl(D) D I along the diagonal, so that we have a commutative δD is simply the gluing data for δD diagram {1,2} ⊗ δ {1,2} ) = gl(D) {1,2} ⊗ j j ∗ (δ δ ) −→ j j ∗ (δ δ ) j∗ j ∗ (gl(D) ∗ ∗ D D D D D ↓ ↓ ↓ {1,2} ⊗ δ X ⊗δ ) = ! (gl(D) −→ ! δD . gl(D) ! D D

The compatibility with the one-point action follows from the compatibility of the action X (x). of gl(D) Next we construct a map of gl(D) in δD which on fibers is the embedding of the tangent space i x! gl(D) = D(Kx )/D(Ox ) into delta-functions on GrD (x). This is simply the image of the action map D(Kx ) · 1 ⊂ δD (x) on the unit, so the families version is the map act(unit)

j∗ j ∗ (gl(D) ω X ) −−−−−→ ! δD given by the action act on the unit, which factors through a map gl(D) → δD . It follows from the derivation property of act that this is a map of chiral modules over gl(D). Moreover we claim that the restriction of the chiral bracket of δD to j∗ j ∗ (gl(D) δD ) is the same as the action map act. This follows from the derivation property of act and the unit axiom, by comparing the two ways of multiplying gl(D) ω X δD . It now follows that the map gl(D) → δD lifts to a homomorphism of chiral algebras and gl(D)-modules W1+∞ → δD , which on fibers induces the isomorphism W1+∞ (x) ∼ = δD (x) arising from the actions of D(Kx ), (3.3) and (4.1). The theorem at level zero follows. Finally, the theorem at arbitrary level follows from the statement concerning the lifting of D(Kx ) to the determinant line bundle on GrD (x) at level one. 5. W1+∞-Symmetry, Integrable Systems and Conformal Field Theory In this section, we explain how the factorization space GrDn unifies geometric features of the free fermion CFT, the KP hierarchy, and the geometry of moduli of curves and bundles. We begin by reviewing the geometry of the Krichever construction (Sect. 5.1). We then explain, from the point of view of the geometry of D-bundles, the additional W1+∞ -symmetry of these systems (Sect. 5.2). The infinitesimal W1+∞ -symmetry is only part of the full symmetry encoded in the factorization space GrDn , however. In Sect. 5.3, we explain the relationship between global symmetries of bundles—the Hecke modifications—and of solitons—the Bäcklund-Darboux transformations. Finally, we explore (Sect. 5.4) the significance of GrDn for the interesting and still-mysterious subject of W-geometry. 5.1. The Krichever construction. Let us begin by briefly reviewing the Krichever construction of solutions of KP, the formalism of the Sato Grassmannian and the correlation functions of free fermions, following [KNTY]—see also [DJM,SW,Mu], as well as [W,MP,T,FB] and references therein for a purely algebro-geometric approach to free fermions.

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Let K = C((z)) denote the field of Laurent series, and O = C[[z]] the ring of Taylor series. The Sato Grassmannian GR [S] parametrizes subspaces of K which are complementary to subspaces commensurable with O (see [AMP,BN2] for algebraic constructions of GR. Note that this Grassmannian, which is a scheme, is quite different from the ind-scheme Gr(K), the thin Grassmannian, parametrizing subspaces commensurable with O.) The Lie algebras K and Der K = C((z))∂z of Laurent series (with zero bracket) and Laurent vector fields act on GR through their action (by multiplication and derivation) on K itself. The KP hierarchy appears as the infinitely many commuting flows coming from the action of K (or more precisely of its sub-algebra C[z −1 ]) on the Grassmannian (or more precisely on its big cell, in natural coordinates). Given a smooth projective complex curve (X, x) and a line bundle L on it, together with a formal coordinate and a formal trivialization of L at x, Krichever’s construction produces a point of the Sato Grassmannian, and hence a solution of the KP hierarchy which may be expressed using theta functions. Namely, we consider the space L(X \ x) of sections on the punctured curve, embedded in K using the coordinate and trivialization. Note that for this construction we only require X to be smooth at x, and can consider on an equal footing rank one torsion-free sheaves on singular curves with a marked smooth point. From the viewpoint of conformal field theory, we are considering the chiral theory of a free fermion on the Riemann surface X , twisted by the line bundle L. This theory has a one-dimensional space of conformal blocks (potential partition functions, or solutions to the conformal Ward identities), defining a point in the projectivized fermionic Fock space. This point is the image of the point in the Sato Grassmannian described above under the Plücker embedding. (See [KNTY,U,FB].) The KP flows (action of K) vary the line bundle L via translation in the Picard group of X (the positive half O merely changing the trivialization). This action is in fact infinitesimally transitive on the moduli space Pic(X, x) of line bundles on X with a formal trivialization at x, thereby giving a formal uniformization of the Picard group Pic(X ). Likewise, the action of Der K (for fixed line bundle L = O X ) or the action of the semi-direct product K ⊕ Der K = D≤1 (K) deforms the pointed curve (X, x) (with the Taylor vector fields Der O = C[[z]]∂z merely changing the coordinate and moving the point along X ). Again, these actions are infinitesimally transitive on the moduli space g,1 of pointed curves with formal coordinate or of the Pic(X, x)-bundle Pic g,1 of all M Krichever data, and give formal uniformizations of the moduli spaces Mg,1 and Picg,1 : see [Ko,AKDP,BS,TUY], or [FB] for a detailed exposition. We can thus consider the g,1 and moduli spaces Pic(X, x), M Pic g,1 as global (integrated) versions of homogeneous spaces for the corresponding Lie algebras. The same construction applies with GR = GR(K) replaced by the (isomorphic) Grassmannian GRn = GR(Kn ), line bundles replaced by vector bundles and K = gl1 (K) replaced by the loop algebra gln (K). The KP flows on GR are now replaced by the multicomponent KP flows on GRn , corresponding to maximal tori in gln (K) [KvdL] (see [Pl and BN2] for more on the geometry of multicomponent KP). From the CFT perspective, we are considering insertions of the bosonic current or the stress tensor as deformations of the conformal block associated to (X, L), which can be realized as conformal blocks for deformations of the pair (X, L). The Sato Grassmannian carries a canonical line bundle, the determinant line bundle, which defines the Plücker embedding of GR into the projective space of the fermionic Fock space (i.e. where our conformal blocks live). This line bundle restricts to the


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corresponding determinant or theta line bundles on the moduli of curves and bundles. The actions of the Lie algebras K, Der K and gln (K) on the Grassmannian (and on the corresponding moduli spaces) extend canonically to the actions of the Heisenberg, Virasoro 1 , V ir and gl n respectively on the determinant line and Kac-Moody central extensions gl bundles. 5.2. W1+∞ -orbits of Krichever data. The free fermion theory, the Sato Grassmannian and the determinant line bundle all carry an important additional symmetry: the action of the Lie algebra D(K) of differential operators with Laurent coefficients and of its central extension W1+∞ (or equivalently of the W1+∞ vertex algebra). From the point of view of the KP hierarchy, the action of W1+∞ is given by the Orlov-Schulman additional symmetries, and is given explicitly on tau functions (i.e. on conformal blocks) by the Adler-Shiota-van Moerbeke formula (see [vM] for a review). Note that this symmetry algebra contains both the Heisenberg and Virasoro algebras as zeroth-order and first-order operators. It is natural, therefore, to ask for a moduli space interpretation of (integrated) orbits of the W1+∞ action extending the HeisenbergVirasoro uniformization of moduli of curves and bundles. Thus, given a pointed curve (X, x) we consider the moduli functor BunD (X, x) of D-line bundles on X trivialized in the formal neighborhood of x. By the CanningsHolland correspondence (as in Sect. 2) this functor is the direct limit of the moduli functors of rank one torsion-free sheaves on cuspidal quotients of X equipped with trivializations near x (in other words, we allow singularities at all points other than x). It is then easy to see that the Krichever embedding on cusp curves gives an embedding BunD (X, x) → GR. Explicitly, this assigns to a D-line bundle M trivialized near x its de Rham cohomology on the punctured curve h(M| X \x ) ⊂ h(M| D × ), viewed as a subspace of its de Rham cohomology on the punctured disc (which is itself identified with C((z))). The determinant bundle on the Sato Grassmannian restricts to the usual determinant line for torsion-free sheaves, which, as we have explained in Sect. 3.3, is the determinant line of de Rham cohomology for D-bundles. Furthermore, we have: Proposition 5.1. The W1+∞ flows on the Sato Grassmannian span the tangent space to the embedding BunD (X, x) → GR of the moduli of D-line bundles on X , or equivalently rank one sheaves on cusp quotients of X , trivialized at x. Proof. Let D and D × denote the disc and punctured disc at x. Suppose M is a D-bundle equipped with a trivialization on D. First-order deformations of M are given by Ext 1D (M, M); since M is locally projective over D, we find that Ext 1D (M, M) = H 1 (X, EndD (M)). Using the long exact sequence 0 → EndD (M) → EndD (M)| X x → EndD (M)| X x /EndD (M) → 0 and the isomorphisms EndD (M)| X x /EndD (M) ∼ = End(M| D × )/End(M| D ) ∼ = D(K)/D(O) (the second one coming from the given trivialization of M over D), we find that H1 (X, End(M)) = End(M| X \x )\ End(M| D × )/ End(M| D ).

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Similarly, the tangent space to BunD (X, x) at a pair consisting of a D-bundle M and trivialization on D is End(M| X \x )\D(K). In particular the canonical action of D(K) on BunD (X, x) given by changing the transition functions on the punctured disc is infinitesimally transitive. Moreover, the de Rham functor identifies the action of D(K) on itself by right D-module automorphisms with its action on K as differential operators. Thus the embedding BunD (X, x) → GR is equivariant for D(K). Remark 5.2 (Isomonodromy and moving the curve). We now have two interpretations of the action of vector fields Der(K) (or the Virasoro algebra) on Krichever data: on the one hand this action deforms the pointed curve (X, x), but on the other hand it is tangent to the moduli BunD (X, x) of D-line bundles on a fixed pointed curve (X, x). This apparent discrepancy is accounted for by the isomonodromy connection on the moduli spaces BunD (X, x) (see [BF] for a parallel discussion of isomonodromy and the Segal-Sugawara construction). More precisely, as we vary (X, x) infinitesimally, each D-module on X has a unique extension to a D-module on the family, i.e., a unique isomonodromic deformation. This gives a canonical horizontal distribution on the family of moduli of D-bundles (independent of the trivialization near x) over the moduli Mg,1 of pointed curves. In coordinates (i.e., in terms of the Virasoro and W1+∞ uniformizations) this distribution is given by the embedding Der(K) → D(K). Thus, the effect of moving the curve infinitesimally on the Krichever data in the Sato Grassmannian can be realized by torsion-free sheaves on cusp quotients of the fixed curve. 5.3. Bäcklund-Darboux transformations and the adèlic Grassmannians. So far we have discussed only the infinitesimal symmetries of the Sato Grassmannian and KP hierarchy. We next turn our attention to global symmetries, the Bäcklund-Darboux transformations or, equivalently, Hecke modifications. The global symmetry transformations of soliton hierarchies are the BäcklundDarboux transformations, which in their classical form for KdV involve replacing a Lax operator L by L when the two are conjugate by a differential operator P, i.e. L = P Q and L = Q P for some differential operator Q. More generally, Bäcklund-Darboux transformations (for equations of KdV type) are the residual symmetries coming from the loop group symmetries of the corresponding Grassmannians (see e.g. [Pa]). On the other hand, the global symmetry transformations of moduli spaces of vector bundles or principal bundles on curves are the Hecke correspondences. Two bundles on a curve are related by a (simple) Hecke modification when we are given an isomorphism between their restriction to the complement of a point (this is also the origin of Tyurin coordinates on moduli of bundles). Since the Hecke correspondences are the residue of the loop group symmetries underlying moduli of bundles, it is not surprising that Bäcklund-Darboux transformations are expressed as Hecke modifications on the level of geometric solutions of soliton equations, though this connection does not appear to be very explicit in the literature (see however [LOZ]). The Hecke modifications of line bundles are directly related to the vertex operator of [DJKM] (the bosonic realization of a free fermion, see [DJM,KNTY]), which is itself a “Darboux operator in disguise” [vM]. As is explained in detail in [FB, Ch.20],


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the fermion vertex algebra, and specifically the fermion vertex operator, is obtained as distributions on (i.e., as the group algebra of) K× /O× (whose C-points are the integers Z). This group acts on the moduli of line bundles on a pointed curve, with the generator taking a line bundle L to its Hecke transform L(x). The corresponding element of the group algebra (the delta function at the generator) gives the vertex operator. While Hecke modifications of line bundles form a group, Hecke modifications for vector bundles or principal G-bundles do not (they correspond to cosets, or double cosets, of a loop group). The algebraic structure of composition of Hecke modifications is precisely captured by the notion of factorization space, and specifically the factorization space structure on the Beilinson-Drinfeld Grassmannian. This is an adèlic version of the affine Grassmannian G(K)/G(O), which in the G L 1 case reduces to the geometry of the group K× /O× above. The infinitesimal version of this structure is the Kac-Moody vertex algebra, which in the G L 1 case reduces to the Heisenberg algebra (i.e. the free boson, or the KP flows), while the fermionic vertex operator itself generalizes to the chiral Hecke algebra of Beilinson-Drinfeld (see [FB] for a discussion). Above we have described a factorization structure on the adèlic Grassmannian of any rank, and shown that its infinitesimal version is precisely the W1+∞ vertex algebra and its higher rank analogues W1+∞ (gln ). On the other hand, the global structure of the adèlic Grassmannian captures the Bäcklund-Darboux transformations of the KP hierarchy. We have already seen that the adèlic Grassmannian contains as a factorization subspace the Beilinson-Drinfeld Grassmannian for G L n (see (3.1)), so in particular in the rank one case contains the information of the fermionic vertex operator or simple Bäcklund-Darboux-Hecke transformations. The Bäcklund-Darboux transformations for the full KP hierarchy were defined in [BHY1], where it is observed that the adèlic Grassmannian (as originally defined by Wilson, as a space of “conditions” [Wi1]) is precisely the space of all BäcklundDarboux transformations of the trivial solution. (See [HvdL] for a related study of Bäcklund-Darboux transformations in terms of actions on the Grassmannian.) A point V ⊂ C((z)) in the Sato Grassmannian GR = GR(C((z))) is defined in [BHY1] to be a Bäcklund-Darboux transformation of a point W ⊂ C((z)) if there are polynomials f, g in z −1 such that f · V ⊂ W ⊂ g −1 · V. In the case when W corresponds to an algebro-geometric solution (X, x, L) of some n-KdV hierarchy, i.e. z −n extends to a global function on X \ x (for example the trivial solution C[z −1 ]), this implies that V differs from W by finite-dimensional conditions at some divisor D on X . It then follows that V is defined by a torsion-free sheaf on a curve obtained by adding cusps to X along D, and that this sheaf is identified with L off D. Equivalently, V is defined by a D-bundle on X which is identified with L⊗D off D—i.e. a Hecke modification of the D-bundle L ⊗ D. Conversely, all torsion-free sheaves on cusp quotients of X (or D-bundles on X ) define points V which are Bäcklund-Darboux transforms of the given solution W . In other words, the adèlic Grassmannian of L ⊗ D precisely parametrizes Bäcklund-Darboux transforms of the corresponding KP solution. Thus, we obtain a geometric approach to some of the results of [BHY2,BHY3] relating W1+∞ , bispectrality and the adèlic Grassmannian. (For a discussion of bispectrality for solutions of KP and its multicomponent versions in terms of D-bundles see [BN2].) In particular, the fact that the action of W1+∞ preserves the spaces of Darboux transformations of various solutions, or that the corresponding tau functions form representations of W1+∞ , are explained by the fact that the W1+∞ vertex algebra is the “Lie algebra” of

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the “group” of Bäcklund-Darboux transformations, namely the adèlic Grassmannian of Hecke modifications of the corresponding D-bundle. One advantage of the geometric approach is that it immediately extends to the multicomponent KP hierarchies (see [vdL] for a study of the W1+∞ symmetries of these hierarchies). Namely the W1+∞ (gln )-action on algebro-geometric solutions exponentiates to the space of Bäcklund-Darboux-Hecke transforms of the corresponding D-bundles, and in the case of solutions coming from the affine line we obtain a bispectral involution. It would be interesting to find explicit descriptions of the corresponding bispectral solutions and the action of the W1+∞ symmetries on them. 5.4. W-geometry. Conformal field theory has uncovered a fascinating new W-geometry. This mathematically mysterious geometric structure is expected to bear the same relation to the nonlinear Wn vertex algebras (the quantized symmetries of the n th KdV hierarchy) as the moduli of bundles and curves bear to the Kac-Moody and Virasoro (or W2 ) algebras. This geometry is further expected to have deep connections to higher Teichmüller theory, isomonodromy, quantization of higher Hitchin hamiltonians and geometric Langlands (for a sample see [H,Po,LO]). The present paper yields a precise mathematical framework for W-geometry in the limiting, linear case of W1+∞ (studied for example in [Po] and references therein), as we sketch below. Namely, we propose that W1+∞ geometry is the geometry of a noncommutative variety, the quantized cotangent bundles of a Riemann surface X (for general X ).1 This is a natural consequence of the interpretation of the W1+∞ Lie algebra as the quantization of the Poisson algebra of functions on the cotangent bundle of the punctured disc (or Hamiltonian vector fields on the cylinder). Note that the quantized algebra generalizes simultaneously the ring structure on functions and the Lie bracket on Hamiltonian vector fields, hence the corresponding deformation problem generalizes simultaneously the deformations of line bundles and of the underlying variety. Namely we consider the moduli stack BunD (X ) of all D-bundles on X , equipped with the line bundle defined by the determinant of de Rham cohomology, as a substitute for the moduli of bundles or curves with W1+∞ symmetry. We have shown that the choice of trivialization at a point x defines a space BunD (X, x) (and line bundle) which is embedded in the Sato Grassmannian (and determinant bundle) and is uniformized by W1+∞ . (An important distinction from the moduli of bundles, parallel to the moduli of torsion-free sheaves on a singular curve, is that not all D-bundles are locally trivial at a given x, though they are all generically trivial.) The tangent space to BunD (X ) at the trivial D-bundle is given by H1 (X, D), and in fact functions on the formal neighborhood of the trivial bundle are easily seen to be calculated as the conformal blocks of the W1+∞ vertex algebra on X , as expected from W1+∞ geometry. Analogous statements hold at an arbitrary D-bundle M with tangent space given by H1 (X, End(M)) and formal functions calculated as conformal blocks of the chiral algebra of endomorphisms of M. (More generally, one can define D-modules on BunD (X ) from conformal blocks of any representation of W1+∞ .) It would be very interesting to obtain a better geometric understanding of the stack BunD (X ) and its relation to the many conjectured aspects of W-geometry. In particular, for a curve of genus larger than 1 this stack is the noncommutative counterpart of the 1 In the following discussion we will suppress the distinctions between the different variants of the W 1+∞ algebra, which differ by the central extension or the Heisenberg sub-algebra, whose geometric interpretations are evident.


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stack of torsion-free sheaves on a Stein surface with no global functions, and so may be expected to have good geometric properties. (Note that the stacks that were studied in [BN2,BN3]—the Calogero-Moser spaces—are the moduli of filtered D-bundles, corresponding to a compactification of the noncommutative surface under consideration.) We also hope that this geometry might provide a hint to the mystery of Wn -geometry. Acknowledgements. The authors are grateful to Edward Frenkel for many helpful discussions, in particular on the subject of W-geometry. The present paper grew out of discussions that began in 2002; during the paper’s gestational period, the authors have been supported by MSRI postdoctoral fellowships, NSF postdoctoral fellowships, and NSF grants. DBZ is currently supported by NSF CAREER award DMS-0449830, and TN by NSF award DMS-0500221.

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[BN1] [BN2] [BN3] [BW1] [BW2] [CH1] [CH2] [DJKM] [DJM] [FB]

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Communicated by L. Takhtajan


Communications in


Bergman Kernel from Path Integral Michael R. Douglas1,2,3 , Semyon Klevtsov1,2,4 1 Simons Center for Geometry and Physics, Stony Brook University,

Stony Brook, NY 11794–3840, USA

2 NHETC and Department of Physics and Astronomy, Rutgers University,

Piscataway, NJ 08855–0849, USA. E-mail: [email protected]; [email protected] 3 I.H.E.S., Le Bois-Marie, Bures-sur-Yvette 91440, France 4 ITEP, Moscow 117259, Russia

Received: 18 September 2008 / Accepted: 15 July 2009 Published online: 3 September 2009 – © Springer-Verlag 2009

Abstract: We rederive the expansion of the Bergman kernel on Kähler manifolds developed by Tian, Yau, Zelditch, Lu and Catlin, using path integral and perturbation theory, and generalize it to supersymmetric quantum mechanics. One physics interpretation of this result is as an expansion of the projector of wave functions on the lowest Landau level, in the special case that the magnetic field is proportional to the Kähler form. This is relevant for the quantum Hall effect in curved space, and for its higher dimensional generalizations. Other applications include the theory of coherent states, the study of balanced metrics, noncommutative field theory, and a conjecture on metrics in black hole backgrounds discussed in [24]. We give a short overview of these various topics. From a conceptual point of view, this expansion is noteworthy as it is a geometric expansion, somewhat similar to the DeWitt-Seeley-Gilkey et al short time expansion for the heat kernel, but in this case describing the long time limit, without depending on supersymmetry.

1. Introduction A prototypical topic at the interface of geometry and theoretical physics is the study of quantum mechanics in curved space, i.e. on a Riemannian manifold M [1–4]. Many results in this area are of great interest both to physicists and to mathematicians, with some examples being the DeWitt-Seeley-Gilkey short time expansion of the heat kernel, and the relation between supersymmetric quantum mechanics and the Atiyah-Singer index theorem [5–7]. A more recent result which, although not well known by physicists, we feel also belongs in this category, is the expansion for the Bergman kernel on a Kähler manifold developed by Tian, Yau, Zelditch, Lu and Catlin [8–11]. It applies to Kähler quantization and gives an asymptotic expansion around the semiclassical limit. This has many uses in mathematics [8,12–14]; see the recent book [15].

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M. R. Douglas, S. Klevtsov

Here we will provide a physics derivation of the asymptotic expansion of the Bergman kernel using path integrals, and explain various possible applications of this result. In physics terms, perhaps the simplest way to define the Bergman kernel is in the context of quantum mechanics of a particle in a magnetic field, in which it is the projector on the lowest Landau level. It is not hard to see that the limit of large magnetic field is semiclassical, so that one can get an expansion in the inverse magnetic field strength using standard perturbative methods. Our basic result is to rederive the Tian-Yau-Zelditch et al expansion as the large time limit of the perturbative expansion for the quantum mechanical path integral. We also generalize it to N = 1 and N = 2 supersymmetric quantum mechanics. Let us state the basic result for (nonsupersymmetric) quantum mechanics. We consider a compact Kähler manifold M, and a particle in magnetic field, with the field strength proportional to the Kähler form on the manifold Fi j ∼ ωi j .

(1)

One can show (see below) that, just as for a constant magnetic field in flat space, in this situation the spectrum is highly degenerate, splitting into “Landau levels.” Let the lowest Landau level (LLL or ground state) be N -fold degenerate with a basis of orthonormal wave functions ψ I (x), then we define the projector on the LLL as ρ(x, x ) =

N

ψ I∗ (x )ψ I (x).

(2)

I =1

We could also regard this as a density matrix describing a mixed state in which each ground state appears with equal weight, describing the zero temperature state of maximum entropy. We then consider scaling up the magnetic field by a parameter k, as F → k F. Note that on a compact manifold, F must satisfy a Dirac quantization condition; thus we take k = 1, 2, 3, . . .. In the large k limit, the diagonal term then satisfies 2 1 1 ρ(x, x) ∼ k n 1 + R+ 2 R + |Riem|2 2k k 3 24 1 1 − |Ric|2 + R 2 + O((/k)3 ) (3) 6 8 as an asymptotic expansion [11] (see Appendix A for the precise definitions of different terms here). In some ways this expansion is similar to the well known short time expansion of the heat kernel, but note that it is a long time expansion, because it projects on the ground states. Unlike other analogous results for ground states, it does not require supersymmetry, either for its definition or computation. Of course, similar results can be obtained for supersymmetric theories, our point is that that they do not depend on supersymmetry (whether they depend ultimately on holomorphy is an interesting question discussed below). There are various other physics interpretations of this result. One familiar variation is to regard M as a phase space, and try to quantize it, following Berezin [16]. As a phase space, M must have a structure which can be used to define Poisson brackets; it is familiar [17] that this is a symplectic structure, i.e. a nondegenerate closed two-form ω. The definitions we just gave are then the standard recipe of geometric quantization [18].

Bergman Kernel from Path Integral

207

They lead to a finite dimensional Hilbert space, whose dimension is roughly the phase space volume of M in units of (2π )n . In this interpretation, the parameter k plays the role of 1/, and thus the large k limit is semiclassical. From this point of view, it is intuitive that one should be able to localize a wave function in a region of volume (2π )n ∼ 1/k n , and thus in the large k limit the density matrix ρ(x, x) should be computable in terms of the local geometry and magnetic field near x. To do this, given a point z ∈ M, one might seek a wave function ψz (z ) which is peaked around z. Given an orthonormal basis for H, a natural candidate is ψz (z ) = ψ I∗ (¯z )ψ I (z ). I

This is a coherent state (in the sense of [18]). It can be used to define the symbol of an operator, the star product [19], and related constructions. In this context the Bergman kernel is the “reproducing kernel” studied in [20], see [17] for a review. For recent work on applications of the Bergman kernel to quantization of Kähler manifolds see [21,22]. Another recent paper discussing the topic is [23]. Our original interest in this type of result came from the study of balanced metrics in [14], and a conjecture about their relevance for black holes in string theory stated in [24]. However, after realizing that these results and techniques do not seem to have direct analogs in the physics literature and could have other applications, we decided to provide a more general introduction as well. 2. Background Let us give a few mathematical and physical origins and applications of this type of result. 2.1. Particle in a magnetic field. We consider a particle of mass m (which later we set to one) and charge k, moving on a 2n-dimensional manifold M which carries a general metric gi j , and a magnetic field Fi j . It is described by a wave function ψ(x; t) which satisfies the Schrödinger equation, Hψ ≡

2 √ √ Di gg i j D j ψ = Eψ, 2m g

(4)

where Di = i∂i + k Ai is the covariant derivative appropriate for a scalar wavefunction with charge k, and E = i∂/∂t. If M is topologically nontrivial, as usual we need to work in coordinate patches related by gauge and coordinate transformations, and this expression applies in each coordinate patch. We can of course also consider the time-independent Schrödinger equation with E fixed, and seek the energy eigenstates H ψi (x) = E i ψi . The operator H is also called the magnetic Schrödinger operator in mathematical physics literature, see [25] for a recent review. Let us now consider the limit of large magnetic field or equivalently large k. The case of two-dimensional Euclidean space gi j = δi j with a constant magnetic field Fi j = Bi j is very familiar. The energy eigenstates break up into Landau levels, such that all states in the l th level have energy El = k B(l + 21 )/m. Within a Landau level, one can roughly localize a state to a region of volume /k B.

208


These results can be easily generalized to d = 2n dimensions. Choose coordinates such that the magnetic field lies in the 12, 34 planes and so forth, and B12 > 0, B34 > 0, etc. Then, considering the lowest Landau level (LLL) we have E=

(B12 + . . . + B2n−1,2n ), 2m

(5)

with states localized as before within each two-plane. In a general metric and magnetic field, while one might not at first expect this high degree of degeneracy, it is still possible. When the magnetic field is much larger than the curvature of the metric, the intuition that wave functions localize should still be valid. Then, we might estimate the energy of a wave function in the lowest Landau level localized around a point x as Eq. (5), where the components B12 , B34 and so on are evaluated in a local orthonormal frame. If the energy E in Eq. (5) is constant, then all states in the LLL will be degenerate, at least in the limit of large k. The proper generalization of the splitting of the components of B, Eq. (5) for nonconstant magnetic fields seems to be that the field strength should be nonzero only for mixed components Fa a¯ , with Fab = Fa¯ b¯ = 0 in the complex coordinates z a , z¯ a¯ (a, a¯ = 1, . . . , n) on the manifold. Mathematically it means that the underlying line bundle is holomorphic. In this case, the argument can be sharpened by using the identity [Di , D j ] = Fi j to rewrite the Hamiltonian as H = g a a¯ Fa a¯ + g a a¯ Da D¯ a¯ . This makes it clear that if the following combination is constant: g a a¯ Fa a¯ = const,

(6)

0 = D¯ a¯ ψ

(7)

every wave function satisfying

will be degenerate and lie in the LLL. This argument can hold away from the strict k → ∞ limit. The condition (6) is known as the hermitian Yang-Mills equation, and is essentially equivalent to the Maxwell equation in the case F 0,2 = 0. Recalling that the metric coefficients on the Kähler manifold are related to the Kähler form ω as ga a¯ = −iωa a¯ , gaa ¯ = iωaa ¯ , one can see that our choice (1) of the magnetic field strength Fa a¯ = kga a¯

(8)

does satisfy the condition (6). In fact the previous argument relies only upon the Maxwell equations and the condition F 2,0 = 0. This suggests that there exist more general magnetic field configurations than (8), for which the LLL is still highly degenerate and the expansion in large magnetic fields, analogous to (3), is possible. For example this includes the case when b1,1 (M) > 1. We will elaborate this question in a future publication. The previous physical condition for the field strength is equivalent to the mathematical condition that M be a complex manifold with complex structure J = B. For a tensor


209 j]

J ji to be a complex structure, it must satisfy the conditions J 2 = −1 and 0 = ∇[i Jk . The first is manifest, and given the expression for J in terms of the vector potential J ji = g ik ∂[k A j] , so is the second. Now, a standard trick to simplify the equations Eq. (7), is to do a “gauge transformation” with a complex parameter θ (x). While at first this might seem to violate physical requirements such as the unitarity of the Hamiltonian, in fact it is perfectly sensible as long as we generalize another ingredient in the standard definitions, namely the inner product on wave functions. Explicitly, we define the wave function in terms of another function s(x), as ψ(x) = eikθ(x) s(x),

Da ψ(x) = eikθ(x) (i∂a + k Aa − k∂a θ )s(x).

(9)

This would be a standard U (1) gauge transformation if θ (x) were real. By allowing complex θ (x), and assuming 0 = [ D¯ a¯ , D¯ b¯ ] ≡ Fa¯ b¯ (i.e. F 0,2 = 0), we can find a transformation which trivializes all the antiholomorphic derivatives, D¯ a¯ → ∂¯a¯ .

(10)

In this “gauge,” wave functions in the LLL can be expressed locally in terms of holomorphic functions. The only price we pay is that the usual inner product,

ψ|ψ =

√

gψ ∗ (x)ψ (x),

M

turns into an inner product which depends on an auxiliary real function, h(x) ≡ e−2Imθ(x) ,

(11)

as (s, s ) =

√

g h k (x) s¯ (x) ¯ s (x).

M

Taking into account the gauge transformations between coordinate patches, the s(x) are holomorphic sections of a holomorphic line bundle L k . In mathematics, one would say that s(x) is a section of L k evaluated in a specific frame, while the quantity h k (x) defines a hermitian metric on the line bundle L k .

210


2.2. The lowest Landau level. Since we have made a one-to-one correspondence between LLL wave functions and holomorphic sections of the line bundle, we can now find the total number of LLL states, which we denote N . The number of holomorphic sections dim H 0 (L k ) can be determined for large k by the index formula [6,7], N = dim H 0 (L k ) = e F ∧ Td(M) = a0 k n + a1 k n−1 + . . . , (12) M

where the coefficients ai are given by certain integrals involving the curvature of the metric. Now, given that there is a large degeneracy of ground states and thus a nontrivial LLL, it becomes interesting to study the projector on the LLL, or in other words the LLL density matrix P≡ |i i|. i;E i =E 0

If we shift H to set the ground state energy E 0 = 0, it can also be defined as the large time limit of propagation in Euclidean time. Regarded as a function on two variables, the projector P becomes the Bergman kernel P(x, x ) = lim x|e−T H |x . T →∞

Thus it can be defined as a path integral by the standard Feynman-Kac formula. Its probabilistic version has been used in [26] to obtain some estimates on the ground state density for the magnetic Schrödinger operator (4) as well as for the Pauli operator, which is its two dimensional analog for a particle with spin. The standard example in which the projector on LLL appears in physics is the Quantum Hall Effect, see for a review [27]. In the simplest case, one studies the dynamics of electrons on a two-dimensional plane with a constant orthogonal magnetic field. At low temperatures and high values of the field only the lowest energy level is important. It is also interesting to consider a partly filled ground state, with the number of fermionic particles K < N . In this case one has to introduce a potential V , then the particles form an incompressible droplet, whose edge dynamics is of particular interest. In recent years this problem has been much generalized: to Riemann surfaces in [28] and references therein, while higher dimensional examples include the case of S 4 [29], R4 [30] and CPn [31], see also [32] for a review. The case of CPn is the first nontrivial case in which we can make contact with the results of this paper. The choice made in [31] for the U (1) background field is Fa a¯ ∼ Ra a¯ ,

(13)

proportional to the Ricci tensor. Since for CPn the Ricci tensor is equal to the Kähler metric, one immediately recognizes Eq. (13) as the physical condition on the magnetic field Eq. (8). Using the local complex coordinates z 1 , . . . , z n , the LLL wave functions can be constructed explicitly as ψα ∼

z 1α1 z 2α2 · · · z nαn , α = 1, . . . , N (1 + |z|2 )k/2

(14)

up to a normalization constant [32]. As in Eq. (9) it has the form of the holomorphic function, weighted by a metric of the line bundle Eq. (11), or, equivalently, the magnetic potential.


211

The dynamics of the droplet is characterized in the following way. One starts with the diagonal density matrix ρ0 with K states occupied, then the fluctuations, preserving the number of states are given by unitary transfomation ρ0 → ρ = Uρ0 U † , and the equation of motion is the quantum Liouville equation i

∂ρ = [V, ρ]. ∂t

The form of the droplet is determined by the form of the minima of the confining potential. In [31] the case of the spherically symmetric potential V = V (r = z z¯ ) was studied. In the limit of a large number of states N (i.e. large magnetic field) and a large number of fermions K < N the density matrix has the form ρ(r 2 ) = (r 2 − Rd2 ), where Rd is the radius of the droplet and is the step function. In other words the density matrix is equal to a constant in the region, occupied by the droplet. This is due to the fact that the LLL is only partly filled. For the completely filled ground state the condition of the constant density matrix (Bergman kernel) turns out to have interesting mathematical consequences, that we describe in the next section. Partially filled LLL has also been considered from the mathematical standpoint in [33] for higher-dimensional complex projective spaces. For inhomogeneous magnetic fields in two dimensions there is also the closely related normal random matrix description of Hall droplets [34]. The edge dynamics of the droplet is described by Chern-Simons type action in higher dimensions [31]. One can generalize the above construction to nonabelian background gauge fields. Since CPn = SU (n + 1)/U (n) and Lie algebra of U (n) = U (1) × SU (n), then in addition to U (1) gauge field one can also turn on SU (n) gauge field. In [31] the case of constant SU (n) gauge field was considered, so that the field strength is proportional to the SU (n) component of the Riemann curvature. The wave functions (14) as well as the density matrix now carry additional indices, corresponding to SU (n) representation. The similar generalization of the Bergman kernel was considered recently in [35]. In addition to the line bundle L one can consider the more general hermitian vector bundle E with corresponding connection with the curvature R E . Then the corresponding Bergman kernel is given by the large time limit of the exponential of Dirac operator D squared, for which the expansion analogous to (3) exists 1 2 ρ(x) = lim e−T D (x, x) = k n + k n−1 ( R · 1 E + i R E ) + . . . . T →∞ 2 The second term was computed in [36]. It would be interesting to make further contact between these results and the higher dimensional Quantum Hall Effect.

2.3. Applications in Kähler geometry. The original mathematical motivation for this development, usually attributed to Tian and to Yau, seems to have been to use Bergman metrics to study the problem of existence and approximation of Kähler-Einstein metrics, which by definition have Ricci tensor proportional to the metric itself, on complex manifolds, see [37] for a recent review. In [8] Tian considered an algebraic manifold M of complex dimension n, embedded in some projective space CP N , N > n. Turning on the magnetic field is equivalent to

212


considering a bundle L, or its k th power L k for magnetic flux proportional to k, whose choice corresponds to a choice of “polarization” on M. The Kähler metric, polarized with respect to L, has the associated Kähler form ωg = iga a¯ dz a ∧ d z¯ a¯ in the same class as the Chern class C1 (L) of L. A particularly useful choice of ωg is to take it to be equal to the curvature of the line bundle (magnetic field strength). If the hermitian metric of L is h(z, z¯ ) then for L k the metric is h k and its curvature is kga a¯ = Fa a¯ = −∂a ∂¯a¯ log h k ,

(15)

exactly as in Eq. (8). Consider next some, not necessarily orthonormal, basis s0 (z), . . . , s Nk (z) on the space H 0 (M, L k ) of all global holomorphic sections of L k . One can think of sα as projective coordinates on CP Nk . Therefore a particular choice of the basis of sections defines a particular embedding of the manifold M into CP Nk . Different choices of the basis are related by the P G L(Nk + 1) transformation. The standard metric on the projective space is the Fubini-Study metric g F S = ∂ ∂¯ log α |sα |2 . One immediately realizes that the 1 1 k -multiple of its pullback k g F S | M to M is in the same Kähler class as the original metric g (2.3), since ⎛ ⎞ Nk 1 1 g F S | M = g + ∂ ∂¯ log ⎝h k sα s¯α ⎠ k k α=0

and the expression inside the logarithm is a globally defined function. This metric is called the Bergman metric. Now consider the orthonormal basis of sections sα with respect to the standard inner product √ k (sα , sβ ) = gh sα s¯β = δαβ . (16) M

In this case the original metric g and the corresponding Bergman metric k1 g F S | M differ by ∂ ∂¯ of the logarithm of the “density of states” function ρk (z) = h k

Nk

sα (z)¯sα (¯z ),

α=0

where now all the terms in the sum are correctly normalized. It is interesting to look at its structure for large k. Zelditch [9] and also Catlin [10] proved that there is an asymptotic expansion of the density function in 1/k in terms of local invariants of the metric g, such as the Riemann tensor and its contractions. These invariants were computed by Lu [11] up to third order in 1/k with the following result up to the second order in 1/k, 1 1 n−1 1 1 2 n n−2 1 2 2 ρk (z) = k + k R + |Riem| − |Ric| + R + O(k n−3 ). R+k 2 3 24 6 8 (17) One of the most important consequences of this expansion is the Tian-Yau-Zelditch theorem, stating that an arbitrary metric g in a given Kähler class can be approximated by Bergman metrics as k → ∞.


213

The computation is based on the global peak section method, developed by Tian [8], which is a technique to approximate sections of line bundle for large values of k; as well as on the earlier results by Boutet de Monvel and Sjöstrand, and by Fefferman [38]. Other methods to derive these results are the heat kernel approach of [35] and the reproducing kernel approach of [39]. Let us also mention their importance for the proof of holomorphic Morse inequalities [35,40]. In this paper, we reproduce the expansion (17) by taking the large time limit of the quantum mechanical path integral for a particle in magnetic field. The function ρk is nothing but the diagonal of the density matrix on the lowest Landau level. Based on the results of [8–11] Donaldson suggested to study the metrics with constant density function ρk (z) = const =

dim H 0 (M, L k ) . Vol M

Solving the previous equation for h k and plugging back into the orthonormality condition Eq. (16) we get the equation dim H 0 (M, L k ) Vol M

√ M

sα s¯β g = δαβ γ sγ s¯γ

on the sections of the line bundle. This is the orthonormality condition for the basis in H 0 (M, L k ). The embedding M → CP Nk satisfying this condition is called “balanced” [41], and the corresponding Kähler metric g is the “balanced metric” (see [42] for the first appearance of this concept). Using the expansion from Eq. (17) Donaldson was able to show [12–14] that under assumption of existence of constant scalar curvature metric in the given Kähler class, the metric, satisfying the previous equation, approaches the metric of constant scalar curvature as k → ∞. In [14] an iterative procedure was proposed to construct these metrics numerically. One starts with an arbitrary choice of basis, parameterized by a hermitian matrix G αβ , and defines the following integral operator: T (G)αβ

dim H 0 (M, L k ) = Vol M

√ M

g

sα s¯β −1 (G )γ δ s

γ s¯δ

.

The fixed point of this operator T (G) = G corresponds to the balanced embedding. It was shown in [12,14] that for any initial choice of the matrix G, the iterative procedure for T converges to the balanced embedding. This construction was recently used for approximating the Ricci flat metrics on Calabi-Yau hypersurfaces in projective spaces [43] and finding numerical solutions to the hermitian Yang-Mills equation on holomorphic vector bundles [44]. 3. Non-Supersymmetric Bergman Kernel 3.1. Density matrix. The euclidean path integral for a particle on a 2n-dimensional Kähler manifold M with the magnetic field is ρ(xi , x f ) = N

x(t f )=x f x(ti )=xi

ti 2), whose iterates form a discrete dynamical system on the torus. Since the matrix A has determinant 1, Lebesgue measure on the torus is preserved by A. We also have periodic orbits, giving rise to invariant “atomic” measures of finite (or countable) support, and many other varieties of measures invariant under A. The quantization of this map, discussed further in Sect. 3, associates to every integer N = (2π )−1 a unitary propagator M N (A) acting on an N -dimensional Hilbert space H N of states. The normalized eigenvectors of M N (A) give rise to distributions (e.g., Husimi measures) on T2 , such that weak-* limit points of these distributions (as → 0) are A-invariant probability measures, called quantum limits. It is natural to ask which of the myriad A-invariant measures can arise as quantum limits. In particular, how restrictive a condition is this? Quantum Ergodicity (or Schnirelman’s Theorem), proved for cat maps in [BouDB], asserts that “almost all” quantum limits are equal to Lebesgue measure, in an appropriate

232

S. Brooks

sense (a stronger notion of “almost all” is proved in [KR]). This does not exclude the possibility of very wild quantum limits coming from sparse subsequences; it is such exceptional sequences that we wish to understand. It was shown in [DBFN] that there do exist exceptional sequences; in particular, an explicit construction is given of quantum limits that are half Lebesgue and half atomic. On the other hand, [FN] shows (improving the result of [BonDB2]) that any quantum limit with an atomic component must have a Lebesgue component of equal or greater weight. As the example of [DBFN] shows, this bound is sharp. However, we can get further restrictions on quantum limits by studying their entropies. Loosely speaking, the entropy of an invariant measure quantifies the complexitiy of a generic orbit under the transformation. Alternatively, the entropy gauges the extent to which an invariant measure cannot concentrate in small balls (precise definitions are given in Sect. 2). Work of Anantharaman and Nonnenmacher (see [AN1] in the context of the Walsh-quantized Baker’s map, and [AN2] for the geodesic flow on a compact manifold of negative curvature) can be adapted to cat maps to show that the entropy of a quantum limit must be at least half of the maximal (i.e., Lebesgue) entropy; again, this bound is sharp due to [DBFN]. To state our result, which “combines” or “interpolates between” the two mentioned above, we recall the ergodic decomposition of invariant measures; more details can be found in a book on ergodic theory, such as [Pet] or [Rud]. The set of probability measures on T2 is weak-* compact and convex. If µ is an A-invariant measure, then for any x ∈ T2 we let N −1 1 δ An x , µx = lim N →∞ N n=0

where δx is the delta measure at x, and the limit (whose existence µ-a.e. is implied by the Pointwise Ergodic Theorem) is taken in the weak-* topology; thus, µx is an A-invariant measure obtained by averaging over the orbit of x. Any invariant measure µ can be decomposed into µx dµ(x) µ= T2

i.e., every invariant measure is the weighted average of its ergodic components (the terminology comes from the fact that µx is ergodic for µ-a.e. x). Entropy respects the ergodic decomposition, so that h(µx )dµ(x), (1) h(µ) = T2

where h(ν) denotes the entropy (with respect to A) of an A-invariant measure ν. We can now state our main result. Throughout, we set λ to be the absolute value of the “large” eigenvalue of A, so that λ > 1. We also write h max = log λ for the maximal entropy of A-invariant probability measures; it is also the entropy of Lebesgue measure with respect to A. Theorem 1. Let µ be a quantum limit for A, and fix 0 < h 0 < 21 h max . Then µ{x : h(µx ) ≤ h 0 } ≤ µ{x : h(µx ) ≥ h max − h 0 },

On the Entropy of Quantum Limits

233

Equivalently, say we decompose µ = αµlow + βµhigh + (1 − α − β)µ as a convex combination of invariant probability measures, where αµlow ( f ) := µx ( f )dµ(x), {x:h(µx )≤h 0 } µx ( f )dµ(x), βµhigh ( f ) := {x:h(µx )≥h max −h 0 }

so that the probability measure µlow (resp. µhigh ) is the average of the low entropy (resp. high entropy) components of µ; i.e., those ergodic components µx of µ with h(µx ) ≤ h 0 (resp. h(µx ) ≥ h max − h 0 ). Then Theorem 1 states that α ≤ β, so that ergodic components of high entropy carry at least as much weight as those of low entropy. The statement of Theorem 1 includes the Faure-Nonnenmacher result of [FN] as a special case by taking h 0 → 0, since all atomic components have entropy 0, while Lebesgue measure is the unique invariant probability measure of maximal entropy. It also includes the Anantharaman-Nonnenmacher bound, by virtue of (1). Our strategy of proof is based on the following general philosophy, also used in [FN] and [AN2,AN1]: “Localized states become delocalized at the Ehrenfest time T := logλ N ”. The precise meanings of “localized” and “delocalized” are what distinguish the results. For us, these terms will refer to a time scale (denoted “t”) that is independent of the semi-classical time scale T → ∞; that is, a state in H N localized in a small (but fixed) box of volume λ−t in T2 will evolve, at time T , into a state evenly distributed among like boxes (see Proposition 1 for the precise statement). The independence of the scale of the partition (t) from the semi-classical parameters (T , N ) is crucial for us to obtain information on the entropies of ergodic components. We remark that Theorem 1 also holds in the setting of the Walsh-quantized Baker’s map. For that case, the analog of the main estimate (Proposition 1) was proved in [AN1]. 2. Partitions and Characteristic Parallelograms 2 We begin with some definitions. A finite partition P 2of T is a finite collection of pair2 wise-disjoint subsets of T , such that P∈P P = T . For the remainder of the paper, the term “partition” will always mean a finite partition. Given two partitions P and Q, we denote their refinement by

P ∨ Q = {P ∩ Q : P ∈ P and Q ∈ Q}. We will often be interested in successive refinements under the action of A, and so for k1 , k2 ∈ Z, we write [P]kk21 :=

k2

A j P.

j=k1

We also write P Q if P refines Q; i.e., if P = P ∨ Q (alternatively, every Q ∈ Q is a union of elements of P). We say that P generates a σ -algebra B if [P]∞ −∞ = B. For a partition P and a point x ∈ T2 , we write [x]P for the atom of P containing x; i.e. the unique element P ∈ P such that x ∈ P. When P is implicitly understood and no confusion will arise, we will write [x]kk21 in place of [x][P ]k2 . k1

The starting point for our analysis is the SMB Theorem; for more details, see eg. [Pet,Rud].

234

S. Brooks

Theorem 2 (Shannon-McMillan-Breiman). Let ν be an A-invariant and ergodic measure, and fix a partition P that generates the Borel σ -algebra of T2 . Then | log ν([x]t0 )| = h(ν) t→∞ t lim

pointwise for ν-a.e. x, and in L 1 (T2 , ν). We can (and will) take the right-hand side to be the definition of the entropy of ν, and extend it to all A-invariant measures by setting h(µ) := h(µx )dµ(x) to be the (µ-)average of the entropies of the ergodic components of µ. It follows from the Kolmogorov-Sinai Theorem [Pet, Sect.5.3] that this value is independent of the choice of P, as long as P generates the Borel σ -algebra. We remark that this definition can be shown to coincide with other standard definitions of entropy (see [Pet] or [Rud] for example). Note that by A-invariance of the measures we can take a more symmetric refinement, t/2 replacing [x]t0 with [x]−t/2 . This will be more convenient in our argument, so we adopt the convention that, if t is odd, we write t/2 to mean t/2 , and similarly for −t/2 (this will have no effect on our estimates in the end). We will need the following version of the SMB Theorem for non-ergodic measures (see [Par, Theorem 2.5]): Theorem 3. Let µ be an A-invariant probability measure. Then a.e. ergodic component of µ has entropy less than h if and only if t/2

lim

t→∞

| log µ([x]−t/2 )| t

0, there exists t such that for every t/2 t > t , there is a subset R ⊂ [P]−t/2 such that µ R > 1 − , R∈R

µ(R) > e−(h+)t

for every R ∈ R.

Similarly, a.e. ergodic component of µ has entropy greater than h if and only if t/2

lim

t→∞

| log µ([x]−t/2 )| t

>h t/2

a.e. and in L 1 , which implies that for sufficiently large t, there is a subset L ⊂ [P]−t/2 such that µ L > 1 − , L∈L

µ(L) < e−(h−)t

for every L ∈ L.


235

2 −1 Fig. 1. P(2) and Q(1), for the matrix A = −1 1 . In this case, each contains 5 elements

Though the entropy is independent of the choice of the generating partition P in the SMB Theorem, it will be convenient for us to work with a particular family of partitions of T2 , which we now describe. For simplicity, we will assume that the eigenvalues of A are positive, and that the λ−1 -eigenvector has a “steep” slope of absolute value > 1. There is no loss of generality in this assumption; if the slope is not steep, simply reverse 2 the roles of the horizontal

and vertical axes ofT in what follows. ab a b k k We write A = , and Ak = . Note that all four entries ak , bk , ck , dk cd ck dk k are of order λ , as one sees by diagonalizing A. Definition 1. The partition of k th characteristic parallelograms, denoted P(k), is a partition of T2 whose boundaries consist of the horizontal axis and the image of the vertical axis under A−k ; precisely, elements of P(k) are of the form

bk 1 j + r1 − , 1 + r2 , 0 : 0 ≤ r1 , r2 < 1 |ak | ak ak

for j = 1, 2, . . . , |ak |. We set Q(k) := P(k) ∨ P(−k). To justify the expression given for elements of P(k), look at the image of the vertical axis under A−k . In R2 , this will be a straight line connecting the origin with the point (−bk , ak ) ∈ Z2 . Since det(Ak ) = 1, we must have gcd(bk , ak ) = 1, and so the projection of this line to the torus will intersect the horizontal axis at the points ( |ajk | , 0),

for j = 1, 2, . . . , |ak |. The slope tells us that this line connects the point ( |ajk | , 0) with

( |ajk | − abkk , 1) to form a boundary segment. These segments, together with the horizontal axis, form the boundaries of the partition, giving the |ak | parallelograms described in Definition 1. Note that the translation Tk : (q, p) → (q − bk /ak , p + 1) maps P(k) to itself, and again due to the coprimality of ak and bk , the successive images j |ak |−1 {Tk P ∈ P(k)} j=0 of one parallelogram P ∈ P(k) tile the torus. One reason to construct these partitions is that they refine nicely under the dynamics of A.

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Fig. 2. P ∈ P(3) on the left side, and A P on the right. The two elements of P(2) containing A P are outlined

Lemma 1. There exists k0 depending only on A, such that, for any k > k0 , we have P(k + 1) P(k) ∨ A−1 P(k). t/2

Hence, by induction, we have that [P(k0 )]0−t P(k0 +t) and [Q(k0 )]−t/2 Q(k0 +t/2) for all t. Proof. Pick some (k + 1)th characteristic parallelogram P ∈ P(k + 1). We first claim that A P is contained in the union of (at most) 2 elements of P(k). To see this, lift P to the parallelogram P˜ ⊂ R2 given by

bk+1 1 ˜ P = p0 + r 1 − , 1 + r2 , 0 : 0 ≤ r1 , r2 < 1 . ak+1 ak+1 Since A−k (0, 1) = (−bk , ak ), it follows that A(−bk+1 , ak+1 ) = (−bk , ak ), and we get

r1 r2 A P˜ = Apo + (−bk , ak ) + (a, c) : 0 ≤ r1 , r2 < 1 . ak+1 ak+1 We now use the following simple observation: since the boundary of P consists of A−(k+1) (0, 1) and short horizontal edges, and the boundaries of P(k) consist of A−k (0, 1) and short horizontal edges, the projection of the parallelogram A P˜ to T2 should look like a piece of an element of P(k), at least away from the short “top and bottom edges” (see Fig. 2). Since these top and bottom edges are only of length O(λ−k ), we find that A P˜ is contained in a slightly lengthened parallelogram, that is parallel to elements of P(k). Precisely, we observe that

ak bk a b ak+1 bk+1 k k+1 =A A=A = c d ck dk ck+1 dk+1 so that ak a = ak+1 − bk c.


237

This then implies 1

c (a, c) = (−bk , ak ) + ak+1 ak ak+1

1 ,0 , ak

which we use to circumvent the discrepancies at the top and bottom edges, by estimating r1 + ack r 2 (−bk , ak ) + (1, 0) A P˜ = Apo + ak+1 ak

bk ak c r2 − , 1 + (1, 0) r1 + ⊂ Apo + ak+1 ak+1 ak ak

1 bk ,0 (2) ⊂ Apo ± (r1 + o(1)) − , 1 + r2 ak ak for large enough k, since ak+1 → ∞ and limk→∞ aak+1 = λ−1 < 1 (the sign in the last k k line is the sign of aak+1 ). The projection of (2) to T2 is contained in at most 2 elements of P(k), as claimed; call them P1 , P2 . Thus

P ⊂ A−1 P1 ∪ A−1 P2 Recall that we want to show that P is equal to a union of elements of the refinement P(k) ∨ A−1 P(k). What we’ve shown is that P is contained in a union of 2 elements of A−1 P(k), so if we can show that there exist elements of P(k) whose union contains P but doesn’t intersect (A−1 P1 ∪ A−1 P2 )\P, then P is a union of elements of the refinement. But this is an elementary observation. Elements of P(k) are wider than those of P(k + 1), and for large k, it takes at most 2 elements of P(k) to cover any given element of P(k + 1); their union is a fattened parallelogram of the form

bk 1 U = p + r1 − , 1 + 2r2 , 0 : 0 ≤ r1 , r2 < 1 , ak ak −1 which has width 2|ak |−1 . This means that AU is a fattening A P of width O(|ak−1 | ). of b But P1 and P2 are separated by a horizontal distance of akk mod 1, which is bounded

−1 below as k → ∞. Therefore if k is sufficiently large, then AU has width O(|ak−1 | ) < bk ak mod 1, and thus AU ∩ (P1 ∪ P2 \A P) = ∅ (see Fig. 3). This means that U ∩

A−1 (P1 ∪ P2 ) = P, as required.

Remark. The partition Q(k0 ) generates the Borel σ -algebra of T2 if k0 is large, so we may use it in the SMB Theorem (see Lemma 7). To see that Q(k0 ) generates, observe that, as A is expansive, there exists = (A) > 0 such that, given any x, y ∈ T2 , there exists j ∈ Z so that d(A j x, A j y) > (in fact, can be taken to be of the order of λ−1 ). Now Q(k0 ) consists of parallelograms of diameter O(λ−k0 ) (and subsets of such parallelograms, along the horizontal axis), so taking k0 large enough (depending on , which depends only on A), the points A j x and A j y will lie in distinct elements of Q(k0 ) j whenever d(A j x, A j y) > . Therefore x and y lie in distinct elements of [Q(k0 )]− j .

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Fig. 3. On the left, P ∈ P(3) (dark outline) is the intersection of A−1 P1 ∪ A−1 P2 (shaded regions) with an element of P(2) (dotted outline). The diagram on the right is the image under A of the diagram on the left, showing the spacing between P1 and P2

The partitions P(k) are also useful because they encode some dynamical properties of the stable direction, as evidenced by the next lemma: Lemma 2. Let s∞ = limk→∞ (− abkk ), and write T∞ : (q, p) → (q + s∞ , p + 1) for the corresponding translation. Then for any constant C, and any partition elements P, P ∈ P(k), the number of intersections j

#{0 ≤ j < C|ak | : T∞ P ∩ P = ∅} = O A (C 2 ) as k → ∞. j

j

In other words, though the T∞ P don’t quite tile the torus as the Tk P do, we still have a uniform bound on the number of return times. Proof. The main point here is that s∞ is very close to − abkk for large k, such that the j

j

translations Tk and T∞ are still close to each other up to j = O(λk ) = O(|ak |). The j lemma then follows from the fact that the Tk P tile the torus evenly, and in particular do not return close to P often. Decompose the vector (0, 1) = s + u, where s and u are the stable and unstable components, respectively. We know that (−bk , ak ) = A−k (0, 1) = λk s + λ−k u, and so the λ−1 eigenspace (span of s) passes through the point (−bk , ak + O A (λ−k )). Thus s∞ = − abkk + O A (λ−2k ), and in particular, for 0 ≤ j < C|ak |, we have js∞ = j − abkk + O A (Cλ−k ) (Fig. 4). Now, by definition, T∞ and Tk can be realized on T2 as horizontal translations by s∞ j

and −bk /ak , respectively. Let j0 be such that T∞ P intersects P ; then any other such j must satisfy |( j − j0 )s∞ (mod 1)| ≤ |ak |−1 = O(λ−k ), whereby |( j − j0 ) abkk (mod 1)| =

O A (Cλ−k ). But since Tk P tiles the torus evenly for 0 ≤ j < |ak |, and each (disjoint) translate has width |ak |−1 , there are only O A (C) possible values of j in this range. The same is true for |ak | ≤ j < 2|ak | etc., since Tk is |ak |-periodic. Thus over our range 0 ≤ j < C|ak |, the number of intersections is bounded by C · O A (C) = O A (C 2 ), as required. j


239

Fig. 4. Here P ∈ P(2) is lightly outlined, and we show some translates of P under T∞ outlined in bold. 2 P, while the right diagram shows T 3 P intersecting P, The left diagram shows P disjoint from T∞ P and T∞ ∞ when T23 P is close to P (here T23 P borders P on the left)

Lemma 2 is a manisfestation of the nice equidistribution properties of the stable direction. We will use this, in conjunction with a similar property of the unstable direction, in Lemma 4, once we have set up the connection between quantum and classical dynamics in the next section. 3. Fourier Coefficients and the Main Estimate We briefly review some facts about the Weyl and Anti-Wick quantizations of observables on T2 that we will need; for more details, see [BouDB]. We write throughout e(·) = e2πi(·) , and em,n (q, p) := e2πi(nq−mp) for the (symplectic) characters on T2 . In order to quantize kinematics with T2 as our phase space, we seek a Hilbert space of states that are periodic in both position and momentum. This requires that Planck’s constant 2π = N −1 for N ∈ Z, and for each N our Hilbert space H N must be N -dimensional, isomorphic to L 2 ( N1 Z/Z). For every N , Weyl quantization then associates to a character em,n the unitary operator m n , , O pW N (em,n ) := U N N N where we have the unitary “translation” operators U N (q, p) ∈ H om(L 2 (R)) (coming from an irreducible unitary representation of the real Heisenberg group U N (q, p, φ) on L 2 (R)) given by 1 [U N (q, p)ψ] (y) := e( N qp)e(N py)ψ(y − q) 2 whose restriction to q, p ∈ N1 Z acts on H N . In fact, H N can be realized as an irreducible subrepresentation of L 2 (R) for the action of the subgroup {U N (q, p, φ) : q, p ∈ 1 1 N Z, φ ∈ R}; in this realization, H N is spanned by delta-functions at N Z/Z. One should think of U N as (up to a phase) translating the position of ψ by q, and its momentum by p.

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∞ 2 By linearity, we extend O p W N : C (T ) → H om(H N ) from characters to all ∞ 2 “observables”— that is, smooth functions— on T2 . We will write O pW N : C (R ) → H om(L 2 (R)) for the corresponding quantization of observables on the plane. One can show [BouDB] that W W ≤ O p N ( f ) (3) O p N ( f ) H om(H N )

H om(L 2 (R))

for all f ∈ C ∞ (T2 ) ⊂ C ∞ (R2 ), in the respective operator norms on H N and L 2 (R). For “quantizable”1 A ∈ S L(2, Z), the Weil representation (also called the metaplectic representation) gives a family of operators M N (A), each acting on H N , satisfying the exact Egorov property W M N (A)−1 O p W N ( f )M N (A) = O p N ( f ◦ A)

(4)

(see [Fol] for an explicit construction of the M N (A)). We also have an Anti-Wick quantization of observables, which satisfies a useful positivity property (see below), though it only satisfies an approximate Egorov condition. To describe this quantization, we first observe that for each N , there is a periodization S N : S(R) → H N given by SN ψ = (−1) N mn U N (m, n)ψ (m,n)∈Z2

as a tempered distribution supported on N1 Z/Z. The inner product on H N can be taken from the usual pairing of S(R) and S (R) as S N ψ|S N φH N = ψ|S N φ L 2 (R) . In particular, we can take the normalized Gaussian function g N (y) = (2N )1/4 exp(−π N y 2 ) in S(R), and form the coherent state centered at 0 by |0 := S N (g N ) ∈ H N . Similarly, we define the coherent state centered at x ∈ T2 by |x := S N (U N (x)g N ) ∈ H N , where we have used x to denote both a point in T2 , and the corresponding point in a fundamental domain in R2 . Note that the translation operators satisfy U N (x1 )U N (x2 ) = eiπ N x1 ,x2 sp , U N (x1 + x2 ), where we write (q1 , p1 ), (q2 , p2 )sp := p1 q2 −q1 p2 for the symplectic form. In particular, the translation operators commute up to a phase. The commutator vanishes whenever 1 This amounts to a parity condition ab ≡ cd ≡ 0 mod 2 [HB]. For a discussion of Bloch angles and a quantization of general matrices in S L(2, Z), see [BouDB].


241

2 x1 ∈ N1 Z and x2 ∈ Z2 , and so the translation operators on H N also commute with S N ; in particular, m n m n m n , |x = (eiπ N ( N , N ),xsp ) x + , (5) U N N N N for m, n ∈ Z. We also have by [BonDB1], x1 |x2 H N = x1 |x2 L 2 (R) + (−1) N mn x1 |U (m, n)|x2 L 2 (R) (m,n)=(0,0)

=e

− 12 π N |x1 −x2 |2

e

− 12 π N d(x1 ,x2 )2

+

1

eiϕ(m,n) e− 2 π N |x1 −[x2 +(m,n)]|

2

(m,n)=(0,0)

(6)

with the usual metric d(, ) on T2 , and so for all x1 = x2 , the states |x1 and |x2 become essentially orthogonal (exponentially) for large N . We define the Anti-Wick quantization of observables by O p NAW ( f )ψ = N f (x)x|ψ|xd x T2

for any f ∈

L ∞ (T2 ),

and the corresponding Husimi measures µψ ( f ) := ψ|O p NAW ( f )ψ = N f (x) |ψ|x|2 d x T2

which are clearly positive measures. A bit more work (see [BouDB, Lemma 3.8.ii]) shows that, for f ∈ C ∞ (T2 ), the operator O p NAW ( f ) is the restriction to the subrep p AW ( f ) on L 2 (R), just like the resentation H of the standard Anti-Wick operator O N

Weyl-quantized observables. Furthermore [BouDB]

N

AW −1 W AW ||O p W N ( f ) − O p N ( f )|| ≤ || O p N ( f ) − O p N ( f )|| = O f (N )

(7)

in the operator norms on H N and L 2 (R), respectively. Thus the two quantizations are asymptotically equivalent, and we will exploit this by passing between them in the semi-classical limit. We also have a resolution of the identity [BouDB]: I dH N = N |xx|d x = O p NAW (1) (8) T2

for all N , which can again be derived from the corresponding property of O p NAW (1) on L 2 (R). Moreover, we also have the following composition rule for the quantizations of smooth functions f 1 , f 2 ∈ C ∞ (T2 ): ||O p NAW ( f 1 )O p NAW ( f 2 ) − O p NAW ( f 1 f 2 )|| = O f1 , f2 (N −1 ),

(9)

n derived from the symbolic calculus on R2 as above: the operators U N ( m N , N ) on H N are 2 restrictions of the corresponding operators on L (R), which obey the standard composition rule of the Weyl calculus (see eg., [Mar, Sect. 2.6])

W W O pW N ( f1 ) O p N ( f2 ) = O p N ( f3 )

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with f 3 = f 1 # f 2 = f 1 f 2 + O f1 , f2 (N −1 ). Appealing again to (3) and the CalderónVaillancourt Theorem we get W W W W ||O p W N ( f 1 )O p N ( f 2 ) − O p N ( f 1 f 2 )|| ≤ || O p N ( f 3 ) − O p N ( f 1 f 2 )|| = O f1 , f2 (N −1 ). The change of quantization estimate (7) then shows that (9) is valid for the Anti-Wick quantization as well. Suppose now that we take a sequence {ψ j ∈ H N j }∞ j=1 , with each ψ j an eigenvector of the corresponding M N j (A), normalized so that ||ψ j || = 1. The corresponding Husimi measures µψ j form a sequence of probability measures on T2 . Naturally, we will assume that N j → ∞ (i.e., → 0) to work in the semiclassical regime. We will often drop the subscript and simply write N = N j for the semiclassical parameter. Since the set of probability measures on T2 is weak-* compact, we can find a subsequence of the ψ j whose Husimi measures converge weak-* to a limit measure µ; such a limit measure is called a quantum limit. By the Egorov property, any quantum limit must be an A-invariant probability measure. In order to get useful semiclassical estimates for the measure of partition elements, we will have to work with smooth functions approximating the characteristic functions of our partition Q = Q(k0 ) from Lemma 1. To construct these, observe that for every t and any η > 0, there exists a neighborhood Ut,η ⊃ R∈[Q]t/2 ∂ R such that −t/2

⎞

⎛ ⎜ µ ⎝Ut,η \

⎟ ∂ R ⎠ < η.

t/2

R∈[Q]−t/2

(We remove the boundaries of the partition from Ut,η , since the former may carry some fixed positive µ-measure.) We take a “smooth partition” to be a family {χ R,η } R∈[Q]t/2 −t/2

of smooth functions such that:

• χ R,η approximates the characteristic function of R: the support of χ R,η is contained in R ∪ Ut,η , and intersects only R and adjacent partition elements (i.e., the union of R such that R ∩ R = ∅); and moreover χ R,η χ R ,η ≡ 0 if R ∩ R = ∅. • 0 ≤ χ R,η ≤ 1, with χ R,η identically 1 on R\(R ∩ Ut,η ). 2 • t/2 χ R,η (x) = 1 for all x ∈ T . R∈[Q] −t/2

We then define the symmetric operators2 P˜R,η := O p NAW (χ R,η ) which play the role of smoothed “projections” to R. The composition rule (9) tells us that lim || P˜R,η P˜R ,η − O p NAW (χ R,η χ R ,η )|| = 0

N →∞

(N ) (N ) 2 More precisely, each P˜ R,η is a family of operators { P˜ R,η }, with each P˜ R,η acting on H N , but we will suppress the N and emphasize the dependence on R, η in the semiclassical limit.


243

and so, in particular, lim || P˜R,η P˜R ,η || = 0

N →∞

if R ∩ R = ∅. t/2 Another important consequence is that, for any R, R ∈ [Q]−t/2 , any η > 0, and any unit vector φ ∈ H N , we have P˜R,η φ| P˜R ,η φ = µφ (χ R,η χ R ,η ) + o N →∞ (1) ≤ min{µφ (χ R,η ), µφ (χ R ,η )} + o N →∞ (1), since 0 ≤ χ R,η , χ R ,η ≤ 1. In particular, || P˜R,η φ||2 ≤ φ| P˜R,η φ + o N →∞ (1)

(10)

(in fact, (10) holds even without a remainder term, since the operators P˜R,η actually t/2 satisfy 0 ≤ P˜R,η ≤ 1). Also, note that if R ⊂ [Q]−t/2 is a subcollection of the partition, we have 2 2 || P˜R,η φ|| ≤ (11) P˜R,η φ + o N →∞ (1), R∈R

R∈R

since all off-diagonal contributions to the right-hand side are of the form P˜R,η φ| P˜R ,η φ = φ|O p NAW (χ R,η χ R ,η )φ + o N →∞ (1) = µφ (χ R,η χ R ,η ) + o N →∞ (1) and χ R,η χ R ,η is a non-negative function. (The error terms o N →∞ (1) will depend on the smooth partition, but not on the unit vector φ ∈ H N .) t/2 If R ⊂ [Q]−t/2 , then we write P˜R,η = P˜R,η . R∈R

We also write #(R) for the cardinality of the collection R. We will denote Dt := sup R∈[Q]t/2 diam(R), which is O(λ−t/2 ) by Lemma 1, since any element of Q(k0 +t/2) −t/2

is contained in a parallelogram whose dimensions are O(λ−(k0 +t/2) ).

Remark. Note that the smoothing is done at the (-independent) time scale t, rather than using an -dependent smoothing (as in [Ana], for example). Though the latter gives sharper estimates in some respects, it is important for us to use -independent partitions to analyze ergodic components of the limit measure, and it is natural to smooth at this partition scale. In particular, the role of the parameter η is minimal in our setting; we will only require that η be small at the end of the argument (see the proof of Theorem 1 at the end of Sect. 4), and it will not effect the present estimates. We can now state our main estimate, which implements the “general philosophy” mentioned in the introduction on an -independent time scale t. We recall our convention that T = logλ N is the Ehrenfest time, while t is independent of N . We will use the notation f g to mean f = O(g). Unless otherwise indicated by a subscript, all implied constants depend only on the matrix A.

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Proposition 1. Let {ψ j ∈ H N j }∞ j=1 be a sequence of normalized states, with N j → ∞. t/2

Set Q = Q(k0 ) from Lemma 1. Then for t sufficiently large, any L , R ∈ [Q]−t/2 , and η > 0, we have M N (A T ) P˜R,η ψ j | P˜L ,η M N (A T ) P˜R,η ψ j λ−t || P˜R,η ψ j ||2 + o j→∞ (1).

(12)

t/2

Moreover, for R ⊂ [Q]−t/2 , we have M N (A T ) P˜R,η ψ j | P˜L ,η M N (A T ) P˜R,η ψ j λ−t · #(R) ·

|| P˜R,η ψ j ||2 + o j→∞ (1).

R∈R

Here and throughout, we will always use N = N j and T = T j = logλ N j to correspond to the appropriate semiclassical parameter of ψ j , without explicit reference. Before proving Proposition 1, we draw some conclusions to motivate the argument. Though not necessary for the statement of the proposition, we have in mind the case where the Husimi measures µψ j converge weak-* to a limit measure µ. If we set ν j,R ( f ) := M N (A T ) P˜R,η ψ j |O p NAW ( f )M N (A T ) P˜R,η ψ j to be the Husimi measure associated to M N (A T ) P˜R,η ψ j , then the left-hand side of (12) is simply ν j,R (χ L ,η ), which is an approximation for ν j,R (L). Similarly, the right-hand side of (12) should be thought of as a technical replacement for λ−t µψ j (R). So Proposition 1 essentially says that in the semi-classical limit, a state ( P˜R,η ψ j ) localized at a scale t is evenly distributed, by M N (A T ), among other partition elements of the same scale. This lets us study the quantum limit measure in a fixed partition. Since we wish to study the entropies of ergodic components, we will use the SMB Theorem to define an appropriate collection R, so that P˜R,η ψ j represents the “lowentropy piece” of ψ j (see Lemma 7). The (perhaps strange looking) quantity λ−t · #(R) will turn out to be the correct “high-entropy” bound for the measure of partition elements (see (21) in Sect. 4). It is in this sense that M N (A T ) delocalizes low entropy components into high entropy components. To prove Proposition 1, we follow the argument of [FN] (and [BonDB2]), and examine the Fourier coefficients of the (positive) measures ν j,R and the analogous measures ν j,R defined by ν j,R ( f ) := M N (A T ) P˜R,η ψ j |O p NAW ( f )M N (A T ) P˜R,η ψ j . Proposition 2. Under the hypotheses of Proposition 1, we have ν j,R (m, n) || P˜R,η ψ j ||2 + o j→∞ (1), |m|,|n|≤λt/2

|m|,|n|≤λt/2

ν j,R (m, n) #(R) · || P˜R,η ψ j ||2 + o j→∞ (1). R∈R

Proof. Observe that em,n ◦ A T (x) = e((m, n), A T xsp ) = e A−T (m,n) (x),


245

since A ∈ S L(2, Z) preserves the symplectic form. Now, as N j → ∞, the vector A−T (m, n) becomes asymptotically parallel to the λ−1 eigenvector. In fact, if we decompose (m, n) = αs (m, n)s + αu (m, n)u into stable and unstable components, then A−T (m, n) = λT αs (m, n)s + λ−T αu (m, n)u = λT αs (m, n)s + O(N −1 ). For the moment, fix (m, n). Then as j → ∞, we have, by the change of quantization estimate (7) and the Egorov property (4), ν j,R (m, n) = M N (A T ) P˜R,η ψ j |O p NAW (em,n )M N (A T ) P˜R,η ψ j = M N (A T ) P˜R,η ψ j |O p W (em,n )M N (A T ) P˜R,η ψ j + o(1) N

˜ = P˜R,η ψ j |O p W N (e A−T (m,n) ) PR,η ψ j + o(1). Using the resolution of the identity (8), we get W ˜ ˜ ν j,R (m, n) = N PR,η ψ j |xx|O p N (e A−T (m,n) ) PR,η ψ j d x + o(1) 2 T 1 −T ˜ ˜ = N PR,η ψ j |xx|U N ( A (m, n)) PR,η ψ j d x + o(1) N T2 1 −T ˜ ˜ ≤ N PR,η ψ j |xx − A (m, n)| PR,η ψ j d x + o(1), (13) N T2 where we use the unitarity of U N and (5) in the last line. Now let

λT ˜ αs (m, n)s : x ∈ R R N = R + c N αs (m, n)s = x + N be the translation of R by c N αs (m, n)s, where we set c N := small for all large enough N satisfying

λT N

. We claim that (13) is

R˜ N ∩ B(R, 4Dt ) = ∅,

(14)

where we denote B(R, r ) := {x ∈ T2 : d(x, R) < r }, with the usual metric d(, ) on T2 . To see this, we use (6) to note that P˜R,η ψ j |x decays exponentially as N → ∞, uni formly for any x ∈ / B(supp(χ R,η ), Dt ) ⊂ B(R, 2Dt ), and similarly x − N1 A−T (m, n) / B( R˜ N , 2Dt ), since | P˜R,η ψ j decays exponentially, uniformly for any x ∈

1 d x − A−T (m, n), R = d (x − c N αs (m, n)s, R) + O(N −2 ) N = d(x, R˜ N ) + O(N −2 ). Whenever (14) holds, the set B(R, 2Dt ) ∩ B( R˜ N , 2Dt ) is empty, and so the integrand of (13) is small for all x ∈ T2 . More generally, the argument shows that M N (A T ) P˜R,η ψ j |O p NAW (em,n )M N (A T ) P˜R ,η ψ j = o(1)

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whenever

R + c N αs (m, n)s ∩ B(R, 4Dt ) = ∅.

(15)

We now fix N and let (m, n) vary over our range, and wish to examine how often (15) fails to hold. Lemma 3. Let R, R ∈ [Q]−t/2 . Then for large enough (fixed) N , there are at most O A (1) integer pairs (m, n) in the range |m|, |n| ≤ λt/2 for which (15) does not hold. t/2

Proof. First, we observe that the values c ∈ R for which (R + cs) ∩ B(R, 4Dt ) = ∅

(16)

are contained in discrete intervals whose lengths are bounded above by 10Dt λ−t/2 . For this, note that if t is large enough, then any y ∈ R satisfies d(y + 10Dt s, R) ≥ 9Dt . Hence if there exists y1 ∈ B(R + cs, 4Dt ) ∩ R, then we have d(y1 + 10Dt s, R) ≥ 9Dt , while d(y1 + 10Dt s, y2 + 10Dt s) = d(y1 , y2 ) < 5Dt R

for all y2 ∈ + cs. The triangle inequality then shows that y2 + 10Dt s cannot be in B(R, 4Dt ). Our task is now to show that both • The number of such intervals satisfying (16) with c λt/2 is bounded uniformly in t. • For any such interval I of length O(λ−t/2 ), the number of integer pairs (m, n) such that c N αs (m, n) ∈ I is bounded uniformly in t. These will boil down to estimates concerning the equidistribution of the stable and unstable eigenspaces, respectively, based on the diophantine properties of the eigenvectors. In fact, we now show that the former follows from Lemma 2. Since Dt λ−t/2 , we know that R and B(R, 4Dt ) are each contained in a union of O A (1) elements of P(k0 + t/2) (say, P1 , . . . , Pl and P1 , . . . , Pl respectively). Therefore the intersection condition (16) can hold only when some (Pi + cs) intersects some Pi . Since the slope of s is within λ−2k0 +t of the slope of the parallelograms Pi and Pi , and this difference is less than the width λ−k0 +t/2 of the parallelograms, we must have either Pi + cs or Pi + c s intersecting Pi or one of the 4 parallelograms adjacent to Pi . So increasing the number l of possible parallelograms Pi by a factor of 5, it is sufficient to count the number of intersections between Pi + cs (resp. Pi + c s) and the 5l possible Pi ’s. Now λt/2 = O A (|ak0 +t/2 |), so Lemma 2 implies that such an intersection can only c occur for O A (1) possible values of c (resp. c ), corresponding to overlaps of T∞ Pi c (resp. T∞ Pi ) and Pi . But for any such intersecting pair, a given point x ∈ Pi +cs meets B(R, 4Dt ) for c in an interval of length ≤ 9Dt . In particular, any x ∈ R + cs meets B(R, 4Dt ) for c in an interval of length ≤ 9Dt , which means that R +cs∩B(R, 4Dt ) = ∅ for c in an interval of length ≤ 10Dt . Thus each of the O A (1) intersecting pairs yields at most one interval of length ≤ 10Dt satisfying (16), giving us the desired uniform bound.


247

The unstable equidistribution statement follows from Lemma 4. Let c ∈ R, and C be a constant independent of t. There are at most C = C (A, C) integer pairs (m, n) in the range |m|, |n| ≤ λt/2 such that |c N αs (m, n) − c| ≤ Cλ−t/2 . Essentially, this lemma says that there cannot be too many integral points within λt/2 of the origin, that are all contained in a thin strip of width Cλ−t/2 parallel to the unstable eigenvector. Intuitively, this is because the area of the strip remains bounded, while its slope guarantees that the strip is well-distributed mod Z2 , and cannot contain more than its fair share of integral points. Proof of Lemma 4. We may assume, of course, that there exists some (m 0 , n 0 ) satisfying these conditions. For any other (m, n) to satisfy them, we would have to have |m − m 0 | ≤ 2λt/2 and |n − n 0 | ≤ 2λt/2 , as well as |c N αs (m − m 0 , n − n 0 )| = c N |αs (m, n) − αs (m 0 , n 0 )| ≤ 2Cλ−t/2 . Therefore, up to altering constants and doubling the range under consideration, we may assume that c = 0. Moreover, since λ−1 ≤ c N ≤ 1 by definition of T = logλ N , we may further adjust the constants and ignore the factor c N . Our problem is now reduced to counting integer points in a thin strip (of width Cλ−t/2 ) around u, whose length is O A (λt/2 ). Notice3 that A−t/2 maps this strip into a ball, whose radius is O A (1) independent of t. Since A−t/2 ∈ S L(2, Z), integer points in the strip are mapped one-to-one to integer points in the ball, and it suffices to count integer points inside the latter. But a ball of radius O A (1) in R2 clearly contains O A (1) integer points, and we are done. Continuation of the Proof of Proposition 2. Note that we have a trivial bound | ν j,R (m, n)| ≤ || P˜R,η ψ j ||2 , since ν j,R is a positive measure of total mass ν j,R (1) = || P˜R,η ψ j ||2 , and the characters have absolute value |em,n (x)| = 1. Therefore Lemma 3 shows, by taking R = R , that ν j,R (m, n) || P˜R,η ψ j ||2 + o j→∞ (1), |m|,|n|≤λt/2

since for any sufficiently large N j , all but O A (1) terms on the left-hand side are o j→∞ (1). For the second part of Proposition 2, observe that ν j,R (m, n) |m|,|n|≤λt/2

T AW T = P˜R,η ψ j |O p N (em,n )M N (A ) P˜R ,η ψ j M N (A ) R∈R R ∈R |m|,|n|≤λt/2 ˜ ≤ P˜R,η ψ j |O p W N (e A−T (m,n) ) PR ,η ψ j + o j→∞ (1)

R,R ∈R |m|,|n|≤λt/2

R,R ∈R

max

|m|,|n|≤λt/2

˜ ˜ ,η ψ j + o j→∞ (1) (e ) P −T PR,η ψ j |O p W R A (m,n) N

3 We thank the referee for pointing out this trick to simplify the argument.

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S. Brooks

by using Lemma 3 to omit all but O A (1) terms from the sum over (m, n). Recalling that O pW N (e A−T (m,n) ) is unitary, we apply Cauchy-Schwarz to get ν j,R (m, n) ≤ C A || P˜R,η ψ j || · || P˜R ,η ψ j || + o j→∞ (1) R,R ∈R

|m|,|n|≤λt/2

= CA ≤ CA

2

|| P˜R,η ψ j ||

R∈R

1

R∈R

= C A · #(R) ·

+ o j→∞ (1)

|| P˜R,η ψ j ||

R∈R

+ o j→∞ (1)

2

|| P˜R,η ψ j ||

2

+ o j→∞ (1)

R∈R

as required.

In order to use Proposition 2 to prove Proposition 1, we use Fejér’s Kernel, defined by K M (θ ) :=

M−1 i 1 e(lθ ). M i=0 l=−i

θ 2 Recall that K M (θ ) = M −1 [sin( Mθ 2 )/ sin( 2 )] is non-negative, and that the convolution

(σ M f )(θ0 ) := 0

1

M−1 i 1 ˆ K M (θ ) f (θ0 − θ )d x = f (l)e(lθ0 ). M i=0 l=−i

(2)

We will use the two-dimensional Fejér kernel on T2 , given by K λt/2 (q, p) := K λt/2 (q) K λt/2 ( p), as a non-negative smoothed δ kernel. From the properties of K we see that (2) K (2) is non-negative, is an average of characters, and that K λt/2 (q, p) λt for all (q, p) in the box {|q|, | p| ≤ λ−t/2 }. Therefore, Proposition 2 implies that max K λ(2) ∗ ν (q , p ) || P˜R,η ψ j ||2 + o j→∞ (1). (17) #(R) 0 0 j,R t/2 q 0 , p0

R∈R

On the other hand, the positivity and lower bound of K (2) imply that (2) K λt/2 (q, p)dν j,R (q0 − q, p0 − p) λt dν j,R , T2

(18)

B

where B is the box {|q0 − q|, | p0 − p| ≤ λ−t/2 }. Combining (17) and (18), we see that −t 2 ˜ || PR,η ψ j || + o j→∞ (1). ν j,R (B) λ #(R) R∈R t/2

Since supp(χ L ,η ) is covered by at most O A (1) such boxes for any L ∈ [Q]−t/2 , Proposition 1 follows.


249

4. Entropies of Ergodic Components We will now use the main delocalization estimate of Proposition 1 to study the entropies of the ergodic components of our quantum limit µ, and to deduce Theorem 1. Since we t/2 wish to estimate the µ-measure of partition elements in [Q]−t/2 by using the invariance of ψ j under the propagator M N (A), we will have to do a certain amount of translating between the eigenstates ψ j and the semiclassical measure µ. As noted earlier, the independence of the partition scale t from the semiclassical scale T = logλ N will play a crucial role in enabling us to take the semiclassical limit while working with elements of a fixed partition. One technical problem we will face is the possibility of mass concentrated on the t/2 boundaries of the partition [Q]−t/2 . Our first lemma examines when this can happen. Lemma 5. Suppose that µ is an A-invariant probability measure on T2 , and that µ is non-atomic; that is, µ(E) = 0 for any countable set E ⊂ T2 . Then µ(∂ R) = 0 for any t/2 R ∈ [Q]−t/2 . More generally, the restriction of µ to the boundaries of the partition is supported on a countable set. t/2

Proof. The boundaries of [Q]−t/2 are contained in the images of the set J = T × {0} ∪ {0} × T under A j , for j = −(k0 + t/2), . . . , (k0 + t/2); so by A-invariance, it suffices to show that the restriction of µ to J has countable support. Now, the images A j J for j ∈ Z will intersect pair-wise at countably many points (in fact, since A ∈ S L(2, Z), these will be the rational points of T2 ), so there exists a countable set E = J ∩ Q2 such that the images A j (J \E) are all disjoint. By A-invariance, we have µ(A j (J \E)) = µ(J \E) for all j. Since µ is a probability measure, we must then have µ(J \E) = 0, and so µ| J has countable support. We will also need the following simple observation: t/2

Lemma 6. Any partition element R ∈ [Q]−t/2 is a convex polygon with O A (t) edges and vertices. Moreover, the number of elements meeting at any given vertex is also O A (t), t/2 t/2 and thus for each R ∈ [Q]−t/2 , the number of R ∈ [Q]−t/2 such that R ∩ R = ∅ is O A (t 2 ). t/2

Proof. The boundaries of [Q]−t/2 are made up of 2k0 +t +1 images A j J , as in Lemma 5, and hence have O A (t) distinct slopes. Therefore at most O A (t) distinct boundary segments can meet at any given vertex, which proves the second statement. The first statement is proved by induction. All elements of Q = Q(k0 ) are parallelograms or triangles, which are convex. Suppose now that the statement is true for t − 2. Since A and A−1 take convex polygons to convex polygons, the induction hypothesis t/2 implies that any element of [Q]−t/2 is the intersection of three convex polygons, which is also convex. A convex polygon can have at most two edges of the same slope, and since the number of distinct slopes is O A (t), the number of edges is O A (t) as well. For the final statement, observe that R ∩ R = ∅ if and only if R and R share a vertex. There are O A (t) possible vertices, and O A (t) choices for R at each vertex.

250

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We now return to Theorem 1. Recall that we decomposed our quantum limit into µ = αµlow + βµhigh + (1 − α − β)µ , where almost every ergodic component of the probability measure µlow (resp. µhigh ) has entropy ≤ h 0 (resp. ≥ h max − h 0 ), and the probability measure µ consists of the remaining ergodic components. Our goal is to show that α ≤ β. We observe that it is sufficient to prove this in the case where almost every ergodic component of µlow has entropy strictly less than h 0 . For we can apply this weaker statement to a sequence of bounds h 0 + converging to h 0 from above, to show that α ≤ β +δ; here β + δ is the weight of ergodic components with entropy ≥ h max − (h 0 + ), which converges to β as → 0. The following lemma is our main application of the SMB Theorem: Lemma 7. Fix > 0. Then there exists a small δ = δ() > 0 such that for every t suft/2 ficiently large (say, t > t (, δ)), there is a collection Rt ⊂ [Q]−t/2 , whose cardinality is bounded by #(Rt ) < e(h 0 −δ)t , satisfying (19) α− 1 − /α, and µlow (R) > e−(h 0 −δ)t for every R ∈ R t . Since R these partition elements are disjoint and µlow is a probability measure, the cardinality t is less than e(h 0 −δ)t . Moreover, the left side of (19) follows from of R t > α (1 − /α) = α − . t ≥ αµlow R µ R For the right side of (19), note that if t is sufficiently large, we also have δ

(µ − αµlow )(R) < e−(h 0 − 2 )t


251

t/2

for all R ∈ [Q]−t/2 outside of a collection whose union has total (µ − αµlow )-measure less than /2, since a.e. ergodic component of (µ − αµlow ) has entropy ≥ h 0 . Therefore t ) · e−(h 0 − 2δ )t + /2 < e− 2δ t + /2, t ≤ #(R R (µ − αµlow ) and hence µ

t = (µ − αµlow ) t < t − αµlow R R R

if t is large enough. We now repeat the argument with µ − βµhigh in place of αµlow and /2, δ/2 in place t c of cardinality < e(h max −h 0 −δ/2)t , with of , δ to construct L t c + /2, t c < (µ − βµhigh ) L 1 − β − /2 < µ L less than (h max − h 0 ). The since a.e. ergodic component of (µ − βµhigh ) has entropy c t < /2, and so L right side of the inequality implies that βµhigh µ

t ≥ βµhigh t > β − /2. L L

On the other hand, 1 − β − /2 < (µ − βµhigh ) (µ − βµhigh )

t c + /2 means that L

t = (1 − β) − (µ − βµhigh ) t c < L L

as required. We now show how to modify the construction to avoid large measure on ∂Rt and ∂Lt . Let E = {e1 , e2 , . . . , } be the (countable) set of atoms, ordered by decreasing measure; i.e. µ(ei ) ≥ µ(ei+1 ) for i ∈ N. We define t/2

Et,i := {R ∈ [Q]−t/2 : ei ∈ R} t and Et = i=1 Et,i ; that is, Et is the collection of partition elements containing or bordering on one of the first t atoms. By Lemma 6, we have #(Et,i ) = O(t), and so #(Et ) = O(t 2 ). Now set t ∪ Et , Rt := R t ∩ Etc . Lt := L Since neither ∂Rt nor ∂Lt contain any of the first t atoms, we have by Lemma 5, µ(∂Rt ) + µ(∂Lt ) ≤ 2

∞

µ(ei ) → 0

i=t+1

as t → ∞, and it remains to show that Rt and Lt still satisfy the necessary bounds.

252

S. Brooks

Since the cardinality of Et increases only polynomially in t, it is clear that the cardinality bounds are satisfied for Rt and Lct , as long as t is taken sufficiently large. Moreover, the bounds t > α − , R µ Rt ≥ µ t < (µ − βµhigh ) Lt ≤ (µ − βµhigh ) L hold trivially. Let t c : (µ − αµlow )(S) > e−(h 0 −δ)t }, S1 := {S ∈ Et ∩ R t c : (µ − αµlow )(S) ≤ e−(h 0 −δ)t }. S2 := {S ∈ Et ∩ R Since a.e. ergodic component of µ−αµlow has entropy ≥ h 0 , the SMB Theorem implies that (µ − αµlow ) S1 = ot→∞ (1). On the other hand, since #(S2 ) ≤ #(Et ) = O(t 2 ), we have (µ − αµlow ) S2 t 2 e−(h 0 −δ)t = ot→∞ (1). We now estimate t + (µ − αµlow ) t ) (µ − αµlow ) Rt = (µ − αµlow ) R (Rt \R t c ) < + (µ − αµlow ) (Et ∩ R = + (µ − αµlow ) S1 ∪ S2 < 2 for t sufficiently large. Similarly, define t : βµhigh (S) > e−(h max −h 0 −δ/2)t }, S1 := {S ∈ Et ∩ L t : βµhigh (S) ≤ e−(h max −h 0 −δ/2)t }, S2 := {S ∈ Et ∩ L and use the same argument to show that βµhigh S1 ∪ S2 = ot→∞ (1), which implies that t − βµhigh t \Lt ) Lt ≥ βµhigh L (L µ Lt ≥ βµhigh S1 ∪ > β − − βµhigh S2 > β − 2 for t sufficiently large.


253

The point of Lemma 7 is that we wish to consider P˜Rt ,η ψ j and P˜Lt ,η ψ j as vectors representative of αµlow and βµhigh , respectively. The general idea is then to use Proposition 1 to compare the norms of M N (A T ) P˜Rt ,η ψ j and P˜Lt ,η ψ j . Proof of Theorem 1. Since µ is a quantum limit, we can select a sequence {ψ j } of eigenvectors such that the Husimi measures µψ j converge weak-* to µ. Now pick > 0. By Lemma 7, for t sufficiently large, we have α − < µ Rt . Since µ (∂Rt ) = ot→∞ (1), we have for large t and η < , ⎛ ⎞ α− 0, we have 2 lim sup P˜Lct ,η ψ j |M N (A T ) P˜Rt ,η ψ j = ot→∞ (1). j→∞

Proof. We write 2 T ˜ ˜ PL ,η M N (A ) PRt ,η ψ j lim sup ψ j | j→∞ L ∈Lct ≤ lim sup || P˜L ,η M N (A T ) P˜Rt ,η ψ j ||2 j→∞

≤ lim sup j→∞

L ∈Lct

| P˜L ,η M N (A T ) P˜Rt ,η ψ j | P˜L ,η M N (A T ) P˜Rt ,η ψ j |

L ,L ∈Lct

≤ lim sup #{L , L ∈ Lct : L ∩ L = ∅} · maxc || P˜L ,η M N (A T ) P˜Rt ,η ψ j ||2 j→∞

L ∈Lt

since lim j→∞ || P˜L ,η P˜L ,η || = 0 whenever L ∩ L = ∅, by the definition of the smooth partition. The number of intersections is bounded by Lemmas 6 and 7 δ

maxc #{L ∈ Lct : L ∩ L = ∅} · #(Lct ) t 2 · e(h max −h 0 + 2 )t

L ∈Lt

(20)

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S. Brooks

so it remains to use (10) and estimate by Proposition 1, || P˜L ,η M N (A T ) P˜Rt ,η ψ j ||2 ≤ M N (A T ) P˜Rt ,η ψ j | P˜L ,η M N (A T ) P˜Rt ,η ψ j + o j→∞ (1) ≤ C(A)λ−t #(Rt ) || P˜R,η ψ j ||2 + o j→∞ (1) R∈Rt

≤ C(A)e−h max t #(Rt )||

P˜R,η ψ j ||2 + o j→∞ (1)

R∈Rt

≤ C(A)e

−h max t (h 0 −δ)t

e

|| P˜Rt ,η ψ j ||2 + o j→∞ (1)

≤ C(A)e−(h max −h 0 +δ)t + o j→∞ (1)

(21)

for some constant C(A) independent of t, where we use (11), the cardinality estimate for Rt of Lemma 7, and the fact that || P˜Rt ,η ψ j ||2 ≤ 1. Combining with the cardinality estimate (20), we conclude that 2 δ T lim sup ψ j | P˜L ,η M N (A ) P˜Rt ,η ψ j (t 2 e(h max −h 0 + 2 )t )(e−(h max −h 0 +δ)t ) j→∞ L ∈Lc t

δ

= t 2 e− 2 t = ot→∞ (1) as required.

Completion of the Proof of Theorem 1. Given > 0, we therefore find that for j and t sufficiently large (and η < ), α − 3 < P˜Lt ,η ψ j |M N (A T ) P˜Rt ,η ψ j + ≤ || P˜Lt ,η ψ j || · || P˜Rt ,η ψ j || + ≤ ψ j | P˜Lt ,η ψ j 2 ψ j | P˜Rt ,η ψ j 2 + as in (10), since 0 ≤ ( L∈Lt χ L ,η ), ( R∈Rt χ R,η ) ≤ 1. Therefore, if t is large enough that µ(∂Lt ), µ(∂Rt ) < , we get ⎛ ⎞1 ⎛ ⎞1 2 2 ⎝ ⎠ ⎝ ⎠ α − 4 < µ supp(χ L ,η ) µ supp(χ R,η ) 1

L∈Lt

1

R∈Rt

1 1 2 2 µ Rt + + η , ≤ µ Lt + + η and recalling that η < , we can finish by using Lemma 7 to get (α − 4)2 < βµhigh Lt + 2 + η αµlow Rt + 2 + η ≤ (β + 3)(α + 3), and letting → 0. Acknowledgements. This work was done as part of the author’s Ph.D. research under the direction of Elon Lindenstrauss, whose guidance and dedication made this paper possible. We would also like to thank Nalini Anantharaman, for valuble discussions and for her help in correcting a previous version of this paper; and the referee, whose comments and suggestions greatly improved its quality.


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Anantharaman, N.: Entropy and the localization of eigenfunctions. Ann. Math. 168(2), 435–475 (2008) Anantharaman, N., Nonnenmacher, S.: Entropy of semiclassical measures of the walshquantized baker’s map. Ann. Henri Poincaré 8(1), 37–74 (2007) Anantharaman, N., Nonnenmacher, S.: Half-delocalization of eigenfunctions for the laplacian on an anosov manifold. Ann. de l’Institut Fourier 57(7), 2465–2523 (2007) Bonechi, F., De Bièvre, S.: Exponential mixing and | ln | time scales in quantized hyperbolic maps on the torus. Commun. Math. Phys. 211(3), 659–686 (2000) Bonechi, F., De Bièvre, S.: Controlling strong scarring for quantized ergodic toral automorphisms. Duke Math J. 117, 571–587 (2003) Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178(1), 83–105 (1996) De Bièvre, S., Faure, F., Nonnenmacher, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239(3), 449–492 (2003) Faure, F., Nonnenmacher, S.: On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys. 245(1), 201–214 (2004) Folland, G.: Harmonic Analysis in Phase Space, Ann. Math. Stud. no. 122, Princeton, NJ: Princeton University Press, 1989 Hannay, J.H., Berry, M.V.: Quantization of linear maps—fresnel diffraction by a periodic grating. Physica D. 1, 267–291 (1980) Kurlberg, P., Rudnick, Z.: On quantum ergodicity for linear maps of the torus. Commun. Math. Phys. 222(1), 201–227 (2001) Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163(1), 165–219 (2006) Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext, Berlin, Heidelberg, Newyork: Springer-Verlag, 2002 Parry, W.: Entropy and Generators in Ergodic Theory. New York: W.A. Benjamin Inc., 1969 Petersen, K.: Ergodic Theory. Cambridge Studies in Advanced Mathematics no. 2, Cambridge: Cambridge University Press, 1989 Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press, 1978 Rudolph, D.: Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Oxford: Oxford University Press, 1990

Communicated by S. Zelditch


Communications in


Properties of Hyperkähler Manifolds and Their Twistor Spaces Ulf Lindström1 , Martin Roˇcek2 1 Department of Physics and Astronomy, Uppsala University, Box 803,

SE-751 08 Uppsala, Sweden. E-mail: [email protected]

2 C.N. Yang Institute for Theoretical Physics, SUNY, Stony Brook,

NY 11794-3840, USA. E-mail: [email protected]; [email protected] Received: 30 October 2008 / Accepted: 23 July 2009 Published online: 10 October 2009 – © Springer-Verlag 2009

Abstract: We describe the relation between supersymmetric σ -models on hyperkähler manifolds, projective superspace, and twistor space. We review the essential aspects and present a coherent picture with a number of new results.

Contents 1. 2.

Introduction and a Succinct Mathematical Summary . Review of Projective Superspace and SUSY σ -Models 2.1 Spinor derivatives . . . . . . . . . . . . . . . . . 2.2 Superfields . . . . . . . . . . . . . . . . . . . . . 2.3 SUSY σ -model Lagrangians . . . . . . . . . . . 3. Superspace Equations of Motion . . . . . . . . . . . 4. The N = 1 Superspace Lagrangian . . . . . . . . . . . 5. The 2-Form and the Meaning of the Lagrangian . . 6. Generalized ϒ ↔ ϒ Duality Transformations . . . . 7. O(2n)-Multiplets and Killing Spinors . . . . . . . . . 7.1 Supersymmetric σ -models and O(2n)-multiplets 7.2 Four-dimensional hyperkähler manifolds . . . . . 7.3 Higher dimensional hyperkähler manifolds . . . . 8. Properties of Twistor Space . . . . . . . . . . . . . . 9. Rotating the Complex Structures . . . . . . . . . . . 9.1 Rotating P1 . . . . . . . . . . . . . . . . . . . . 9.2 Rotating the hyperkähler structure on M . . . . . 9.3 The Kähler potential is a Hamiltonian . . . . . . 10. Normal Gauge . . . . . . . . . . . . . . . . . . . . . 11. Example: The Eguchi-Hansen Geometry . . . . . . . 12. Outlook . . . . . . . . . . . . . . . . . . . . . . . . .

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A. The Hyperkähler Quotient in Projective Superspace A.1 Isometries . . . . . . . . . . . . . . . . . . . . A.2 Quotients and duality . . . . . . . . . . . . . . B. Dualities and Contour Ambiguities . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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274 274 276 276 277

1. Introduction and a Succinct Mathematical Summary This paper collects the insights that we have gained over twenty years of studying supersymmetric σ -models and hyperkähler geometry. Many of our results have appeared elsewhere, both in our work and in the work of others. Here we want to present a coherent view of how supersymmetry naturally reveals the geometric structure; in particular, we are led to the twistor spaces of hyperkähler manifolds. Supersymmetric σ -models are described by an action functional for maps from a spacetime into a target manifold; we focus on the case when the target space of the σ -models is hyperkähler [1]. Supersymmetry is most naturally studied by extending the spacetime to a superspace with fermionic as well as bosonic dimensions. N = 2 supersymmetric σ -models in four spacetime dimensions (as well as their dimensional reductions in three and two dimensions)1 are best described in projective superspace2 [2–11]. Projective superspace naturally leads to twistor space [8,12–14,16]. We begin with a brief mathematical summary of some of our main results that also serves as an introduction to hyperkähler geometry. A hyperkähler space M supports three globally defined integrable complex structures I, J, K obeying the quaternion algebra: I J = −J I = K , plus cyclic permutations. Any linear combination of these, a I + b J + cK is again a Kähler structure on M if a 2 + b2 + c2 = 1, i.e., if {a, b, c} lies on a two-sphere S 2 P1 . The Twistor space Z of a hyperkähler space M is the product of M with this two-sphere Z = M × P1 . The two-sphere thus parameterizes the complex structures and we choose projective (inhomogeneous) coordinates ζ to describe it (in a patch including the north pole). A choice of ζ corresponds to a choice of a preferred complex structure, e.g., J . The corresponding Kähler form ω(1,1) is a (1, 1) two-form with respect to J . For this choice, the two remaining independent complex structures I and K can be used to construct the holomorphic and antiholomorphic symplectic two-forms ω(2,0) and ω(0,2) . These three two-forms are conveniently combined into [14] (ζ ) ≡ ω(2,0) + ζ ω(1,1) − ζ 2 ω(0,2) ,

(1.1)

which is a section of a two-form valued O(2) bundle on P1 . For the four-dimensional case, the statement that the hyperkähler space obeys the Monge-Ampère equation, 2 ω(2,0) ω(0,2) = (ω(1,1) )2 ,

(1.2)

2 = 0.

(1.3)

simply becomes the identity3

1 The formalism can also be developed in six dimensions [4] as well as five dimensions [15]; however, the four dimensional formalism is the most familiar. 2 The terminology “projective superspace” is historic; we are not actually considering a projective supermanifold. 3 For the four dimensional case, these ideas were found previously in a different context [17].

Properties of Hyperkähler Manifolds and Their Twistor Spaces

259

For higher-dimensional manifolds the corresponding identity n+1 = 0

(1.4)

results in a system of equations constraining the geometry to be hyperkähler. For ζ = 0, we can choose local Darboux coordinates (holomorphic with respect to the complex structure at the north pole) for ω2,0 ; as we smoothly rotate the P1 of complex structures, we find Darboux coordinates ϒ p (ζ ) and ϒ˜ p (ζ ) that are regular at ζ = 0: (ζ ) = i dϒ p (ζ ) d ϒ˜ p (ζ ),

(1.5)

where p = 1, . . . , n and the (real) dimension of M is 4n, and the exterior derivative acts only along M and not along the P1 . We introduce the real-structure R on P1 defined by complex conjugation composed with the antipodal map. From (1.1) we see that the two-form obeys the reality condition

since

(ζ ) = −ζ 2 R((ζ ));

(1.6)

1 p ¯ R(ϒ (ζ )) = ϒ − ζ

(1.7)

1 1 d ϒ¯˜ p − . i dϒ p (ζ ) d ϒ˜ p (ζ ) = i ζ 2 d ϒ¯ p − ζ ζ

(1.8)

p

we have

The reality relations (1.6, 1.8) show that ϒ and ϒ˜ are related to ϒ¯ and ϒ¯˜ by a symplec¯ ζ ) for this tomorphism up to the ζ 2 -factor. We introduce a generating function f (ϒ, ϒ; twisted symplectomorphism: ϒ˜ p = ζ

∂f 1 ∂f , ϒ¯˜ p = − ; ∂ϒ p ζ ∂ ϒ¯ p

(1.9)

then i dϒ p d ϒ˜ p = i ζ

∂2 f dϒ p d ϒ¯ q ≡ i ζ ∂ ∂¯ f, ∂ϒ p ∂ ϒ¯ q

(1.10)

where ∂ and ∂¯ are respectively holomorphic and anti-holomorphic exterior derivatives with respect to the complex structure J at the north pole of the P1 , and again act only on ∂f M and not along the P1 . The conditions above imply that ζ ∂ϒ p is regular at the north pole, and hence, for a contour encircling ζ = 0, dζ i ∂ f ζ = 0, i 2, (1.11) 2πiζ ∂ϒ p as well as the complex conjugate relation. As we shall see in subsequent sections, this beautiful mathematics follows from the σ -model. In particular, (1.11) are the equations of motion, and f is the projective superspace Lagrangian. Thus the function f has the rôle both of the superspace σ -model Lagrangian and as a generating function for north-south symplectomorphisms.

260


One of our new observations generalizes a result proven in [14] for the special case when the rotation of the complex structures is generated by an isometry of the manifold. In general, rotations of the sphere of complex structures correspond to nonholomorphic diffeomorphisms on the hyperkähler manifold. In twistor space we can compose such a rotation with the corresponding diffeomorphism to construct a symplectomorphism preserving (up to the ζ factor). Going to Darboux-coordinates for ω(2,0) we can analyze the effect of these rotations on the Kähler potential K . It does not transform simply under rotations of the complex structures but the net result is always a new K˜ . We find that for any hyperkähler manifold, the moment map for transformations with respect to rotations about an axis is the Kähler potential with respect to any complex structure in the equatorial plane normal to the axis. 2. Review of Projective Superspace and SUSY σ -Models The projective superspace4 approach to N = 2 supersymmetry has been discussed many times [2,3,6,8,14]; a concise but extensive review can be found in the appendices of [20]. Here we review the aspects relevant to this paper. We want to emphasize that the requirements of supersymmetry in spacetime naturally lead to the constructions that we describe, and lead us to uncover the geometric structures of the target space.

2.1. Spinor derivatives. Superspace is a space with both bosonic and fermionic coordinates; its essential properties are captured in the algebra of the fermionic derivatives. The algebra of N = 2 superspace derivatives in four (spacetime) dimensions is {Daα , Dbβ } = { D¯ αa˙ , D¯ βb˙ } = 0, {Daα , D¯ βb˙ } = iδab ∂a β˙ ,

(2.1)

where a, b = 1 . . . 2 are isospin indices and α, β and α, ˙ β˙ are left and right handed spinor indices respectively. Mathematically, the D’s are Grassmann odd derivations that are sections of the self-dual spin-bundle tensored with an associated SU (2) bundle, ¯ are sections of S− ⊗ C2 . The superspace derivatives D1α , D¯ 1 S+ ⊗ C2 , and the D’s α˙ generate an N = 1 subalgebra; we will often decompose representations of the full N = 2 algebra in terms of N = 1 representations. We may parameterize a P1 of maximal graded abelian subalgebras as5 ∇α (ζ ) = D2α + ζ D1α , ∇¯ α˙ (ζ ) = D¯ α1˙ − ζ D¯ α2˙ ,

(2.2)

where ζ is the inhomogeneous coordinate on P1 in a patch around the north pole and ∇¯ α˙ (ζ ) is the conjugate of ∇α (ζ ) with respect to the real structure R (complex conjugation composed with the antipodal map on P1 ): ¯ ) ≡ −ζ R(∇(ζ )) = −ζ ∇ ∗ (− 1 ). ∇(ζ ζ

(2.3)

4 A related [18] formalism is harmonic superspace, as described in [19] and references therein. 5 In many papers, e.g., [3,5,8,10,11], the role of D and D are interchanged. However, this leads 1 2

to inconvenient identifications of the holomorphic coordinates, and we choose conventions compatible with [16].


261

2.2. Superfields. Superfields are the generalizations of functions and sections of bundles to superspace. Superfields in projective superspace are by definition annihilated by the projective derivatives (2.2); they differ by their analytic properties on the P1 parameterized by ζ . The most general superfield that describes a scalar multiplet is the arctic multiplet ϒ, which is analytic around the north pole, and its conjugate antarctic ¯ which is analytic around the south pole [6]. The conjugate is again defined multiplet ϒ, with respect to the real structure R. In some cases, we impose a reality condition on ϒ. Other useful superfields are tropical; they may have singularities at both poles, but are regular in a region where the two coordinate patches overlap. These are also usually taken to be real. ¯ ) all anticommute, we may impose the conditions Because the derivatives ∇(ζ ), ∇(ζ ∇α (ζ )ϒ(ζ ) = ∇¯ α˙ (ζ )ϒ(ζ ) = 0;

(2.4)

D1α ϒi−1 + D2α ϒi = D¯ α2˙ ϒi−1 − D¯ α1˙ ϒi = 0,

(2.5)

these imply

where ϒ=

ϒi ζ i .

(2.6)

i=0

The relations (2.5) imply the constraints D¯ α1˙ ϒ0 = D¯ 1α˙ D¯ α1˙ ϒ1 = 0.

(2.7)

If we decompose ϒ into its N = 1 content, we see that only the coefficients (ϒ0 , ϒ1 ) (and their complex conjugates) are constrained as N = 1 superfields–the constraints (2.5) do not imply any constraints in N = 1 superspace for the remaining coefficients. 2.3. SUSY σ -model Lagrangians. Field theories describing maps from a spacetime into a target manifold M are called σ -models, and are generally described by a Lagrangian. The fields map points of spacetime to points of the target M. The projective superspace Lagrange density F of a σ -model with a real 4D-dimensional target M is a contour integral on P1 of an unconstrained function f (ϒ a , ϒ¯ a ; ζ ) of the multiplets ϒ a , a = 1 . . . D as well as the coordinate ζ : dζ a ¯a F(ϒi , ϒi ) = (2.8) f (ϒ a , ϒ¯ a ; ζ ); C 2πiζ the function f is real with respect to the real structure modulo terms that do not contribute to the contour integral, and F is real. For general polar multiplets, since all we know about ϒ, ϒ¯ is that they are analytic near the north and south pole respectively, this is a purely formal expression and the contour C is not yet defined; we will see how to make this into a sensible contour integral below. For other multiplets, the contour depends on f and in known examples turns out to be essentially unique. The Lagrangian is, e.g., L = D1α D1α D¯ 1α˙ D¯ α1˙ F; (2.9) 4 because of the constraints (2.4), the action d x L is invariant under the full N = 2 supersymmetry.

262


3. Superspace Equations of Motion The equations that describe the extrema of the action can be described in superspace. Since the N = 2 Lagrangian is written with an N = 1 measure (2.9), the equations of motion that follow from varying with respect to ϒ can best be understood by thinking of the N = 1 superspace content of the ζ -expansion of ϒ. The constraints (2.5, 2.7) for a general polar multiplet imply that as N = 1 superfields, all the ϒi , i ≥ 2 are unconstrained. The equations that follow from varying them are (we suppress the index a that labels the various ϒ superfields): ∂F = ∂ϒi

C

dζ i ζ 2πiζ

∂ ¯ ζ ) = 0, i ≥ 2. f (ϒ, ϒ; ∂ϒ

(3.1)

Here the contour should really be interpreted as enclosing ζ = 0; the auxiliary N = 1 superfields ϒi>1 are eliminated in such a way as to make sense of this contour. The equations that follow from varying with respect to the constrained N = 1 superfields ϒ1 and ϒ0 can be found by applying D¯ α2˙ and D¯ 2α˙ D¯ α2˙ to the ϒ2 equation and then using ∂f = 0 to re-express the equations in terms of D¯ α1˙ and the N = 2 constraint (2.4) ∇¯ α˙ ∂ϒ 1 1 α ˙ ¯ ¯ D Dα˙ , respectively. It is important to distinguish N = 1 and N = 2 on-shell constraints. When the conditions (3.1) are interpreted in N = 1 superspace, they serve only to eliminate unconstrained (auxiliary) N = 1 superfields, and so they do not put the N = 1 theory on-shell. When we impose N = 2 supersymmetry as described in the previous paragraph, field equations for the physical N = 1 superfields follow from (3.1), and the theory is fully on-shell. ∂ ¯ ζ ) and hence ∂ f ≡ ∂ f (ϒ, ϒ; ¯ ζ) The equations (3.1) simply imply that ∂ϒ f (ϒ, ϒ; ∂ϒ dϒ have at most simple poles; here ∂ is a holomorphic derivative without a term dζ ∂ζ along P1 and dϒ ≡ ζ i dϒi . Thus when one imposes Eqs. (3.1), ϒ˜ ≡ ζ

∂ ¯ ζ) f (ϒ, ϒ; ∂ϒ

(3.2)

is again an arctic multiplet. The conjugate equation ∂F = ∂ ϒ¯ i

C

dζ (−ζ )−i 2πiζ

∂ ¯ ζ ) = 0, i ≥ 2, f (ϒ, ϒ; ∂ ϒ¯

(3.3)

¯ ζ ) has at most simple zeros. Formally, the equations (3.1, similarly implies that ∂¯ f (ϒ, ϒ; 3.3) can be used to eliminate the components ϒi , ϒ¯ i , i ≥ 2 in terms of ϒ0 , ϒ1 , ϒ¯ 0 , ϒ¯ 1 . Given such a solution, ϒ and ϒ¯ become maps on P1 ; substituting back into (2.8), for a contour that encloses the relevant singularities, the formal expression now becomes well defined. In N = 1 superspace, Eqs. (3.1, 3.3) serve to eliminate the N = 2 superfields that are unconstrained as N = 1 superfields; thus the Lagrangian (2.8) results in a well defined N = 1 superspace action for the N = 1 superfields {ϒ0 , ϒ1 , ϒ¯ 0 , ϒ¯ 1 }, or equivalently, for the N = 1 (anti)chiral superfields {ϒ0 , ϒ˜ 0 , ϒ¯ 0 , ϒ¯˜ 0 }.


263

4. The N = 1 Superspace Lagrangian In N = 1 superspace, the σ -model superspace Lagrangian is the Kähler potential expressed as a function of chiral superfields that geometrically are identified as holomorphic coordinates. Here we find the N = 1 superspace Lagrangian that arises after solving Eqs. (3.1, 3.3); the Kähler potential can be written in terms of the N = 1 (anti)chiral superfields {z ≡ ϒ0 , u ≡ ϒ˜ 0 , z¯ ≡ ϒ¯ 0 , u¯ ≡ ϒ¯˜ 0 }: dζ dζ 1 dζ ¯ f −u ϒ − u¯ (−ζ )ϒ, (4.1) K (z, z¯ , u, u) ¯ = 2πiζ 2πiζ ζ 2πiζ C ON OS where O N ,S are the contours around the north and south poles; we can write dζ ¯ dζ ˜ u¯ = ˜ ϒ, ϒ, u= 2πiζ 2πiζ dζ dζ ¯ z= ϒ, z¯ = ϒ. 2πiζ 2πiζ

(4.2)

5. The 2-Form and the Meaning of the Lagrangian In this section we construct a 2-form that leads us to a geometric interpretation of the N = 2 superspace Lagrangian. As we shall see in subsequent sections, this 2-form captures the essential aspects of hyperkähler geometry. An essential observation is that (3.1, 3.3) imply that ≡ iζ ∂ ∂¯ f = iζ

∂2 ∂ϒ a ∂ ϒ¯ b¯

¯ ζ ) dϒ a d ϒ¯ b¯ f (ϒ, ϒ;

(5.1)

is a section of an O(2) bundle. The two-form plays a central role in our understanding of the mathematical structure of the model. It can also be written as ˜¯ ¯ ϒ, = idϒd ϒ˜ = iζ 2 d ϒd

(5.2)

¯ ϒ¯˜ are antarctic, Eq. (5.2) where ϒ˜¯ = − ζ1 ∂∂ϒ¯ f . Note that because ϒ, ϒ˜ are arctic and ϒ, implies that is a section of an O(2) bundle. Equation (5.2) has the form of a twisted symplectomorphism, and therefore there should exist a generating function for this transformation. Indeed, (3.2) and its conju¯ ζ ) as this generating gate allow us to identify the N = 2 superspace Lagrangian f (ϒ, ϒ; function.6 6. Generalized ϒ ↔ ϒ Duality Transformations Dualities of various sorts have been considered extensively in superspace. A rather trivial kind results in a diffeomorphism on the target manifold. In projective superspace, 6 Superspace Lagrangians with the interpretation of a generating function of a symplectomorphism have also been discovered in the context of σ -models with bihermitian target spaces [21].

264


one may generate such a diffeomorphism by relaxing the regularity constraint on ϒ and ˜ re-imposing it with an arctic Lagrange multiplier ϒ: ˜ ϒY ¯ ζ ) → f (Y, Y¯ ; ζ ) − + ϒ¯˜ Y¯ ζ ; f (ϒ, ϒ; ζ

(6.1)

˜ ϒ¯˜ imposes the constraints that Y, Y¯ are arctic and antarctic respecintegrating out ϒ, ¯˜ ζ ) which is the Legendre ˜ ϒ; tively; integrating out Y, Y¯ gives a dual Lagrangian f (ϒ, transform of f . This corresponds to simply interchanging the roles of ϒ and ϒ˜ above. The interpretation of the N = 2 superspace Lagrange density f as the generating function of a twisted symplectomorphism from holomorphic coordinates adapted to the complex structure at the north pole to those at the south pole allows us to generalize this duality. We can construct holomorphic symplectomorphisms of ϒ, ϒ˜ → χ , χ˜ and compose them with f to find the transformed N = 2 superspace Lagrange densities. Explicitly, we consider a generating function g(ϒ, χ ; ζ ) such that ϒ˜ =

∂g ∂g , χ˜ = − , ∂ϒ ∂χ

(6.2)

˜ χ , χ˜ are all arctic. By polar where the explicit ζ dependence of g is such that ϒ, ϒ, −1 ¯ conjugation we have g( ¯ ϒ, χ¯ ; ζ ) such that ∂ g¯ ∂ g¯ , χ¯˜ = − . ϒ¯˜ = ∂ χ¯ ∂ ϒ¯

(6.3)

Then the transformed Lagrange density h(χ , χ; ¯ ζ ) is given by ¯ , χ¯ ; −1 ); ζ ) + h = f (ϒ(χ , χ; ¯ ζ ), ϒ(χ ζ ¯ , χ¯ ; −1 ), χ¯ ; −1 ), −ζ g( ¯ ϒ(χ ζ ζ

1 g(ϒ(χ , χ; ¯ ζ ), χ ; ζ ) ζ (6.4)

¯ , χ; where ϒ(χ , χ; ¯ ζ ), ϒ(χ ¯ −1 ζ ) are determined by ¯ χ¯ ; −1 ) ¯ ϒ, ¯ ζ ) ∂ g( ¯ ζ) ∂g(ϒ, χ ; ζ ) ∂ f (ϒ, ϒ; 1 ∂ f (ϒ, ϒ; ζ = −ζ , = . ∂ϒ ∂ϒ ζ ∂ ϒ¯ ∂ ϒ¯

(6.5)

To check this, we need to see that χ˜ = −ζ

∂h ; ∂χ

(6.6)

using (6.4), we have: ∂h = −ζ −ζ ∂χ

∂ f ∂ϒ ∂ f ∂ ϒ¯ + ∂ϒ ∂χ ∂ ϒ¯ ∂χ

−

∂g ∂ϒ ∂g ∂ g¯ ∂ ϒ¯ − + ζ2 ; ∂ϒ ∂χ ∂χ ∂ ϒ¯ ∂χ

∂g ∂h from (6.5), this gives −ζ ∂χ = − ∂χ , and hence, from (6.2), we find (6.6).

(6.7)


265

7. O(2n)-Multiplets and Killing Spinors In this section, we consider projective superfields that are sections of certain bundles on the P1 . In particular, ϒ ≡ η(2n) may be a section of a O(2n) bundle7 over P1 [6,22]: 1 ¯ ). ϒ(ζ ) ≡ η(2n) (ζ ) = (−)n ζ 2n ϒ(− ζ

(7.1)

Thus η(2n) (ζ ) is a polynomial of order 2n in ζ . We show that σ -models described in terms of these O(2n)-multiplets admit certain local Killing spinors. These multiplets as well as other special multiplets were considered in [6].

7.1. Supersymmetric σ -models and O(2n)-multiplets. We begin with a review of O(2n)-multiplets and the generalized Legendre transform construction [6]. The formal expression for the σ -model Lagrangian (2.8) can be made well-defined without imposing the conditions (3.1, 3.3) if we impose certain constraints on ϒ. Here we focus on the constraint that ϒ is a section of an O(2n)-bundle. We may then impose the reality condition (7.1): ¯ −1 ); ϒ(ζ ) ≡ η(2n) (ζ ) = (−)n ζ 2n ϒ( ζ η(2n) (ζ ) =

2n

i=0 ζ

iη i

(7.2)

is a polynomial of order 2n in ζ obeying the constraints: η¯ i = (−1)n−i η2n−i .

(7.3)

Now we can find a suitable contour (see, e.g., the discussion in [23]) and compute the Lagrange density F(ηi ) = C

dζ f (η; ζ ). 2πiζ

(7.4)

As for the polar case, the Kähler potential is found by eliminating the N = 1 auxiliary superfields ηi , 2 ≤ i ≤ 2(n − 1) and performing a complex Legendre transform with respect to η1 and η2n−1 = (−1)n η¯ 1 : K (z, z¯ , u, u) ¯ = F (ηi (z, z¯ , u, u)) ¯ − u η1 (z, z¯ , u, u) ¯ − u¯ η¯ 1 (z, z¯ , u, u), ¯

(7.5)

¯ are found by solving (preserving the reality conditions (7.3)): where ηi (z, z¯ , u, u) z = η0 , u =

∂ F(ηi ) ∂ F(ηi ) , = 0, 2 ≤ j ≤ 2(n − 1). ∂η1 ∂η j

(7.6)

7 The O(2) case is special because it arises for hyperkähler manifolds admitting a triholomorphic torus action, and has been discussed extensively [3,14].

266


7.2. Four-dimensional hyperkähler manifolds. We begin by considering 4(real)dimensional manifolds; the generalization to higher dimensions is given later. We prove that a σ -model description in terms of a O(2n)-multiplet is possible if and only if the manifold admits a 2n-index Killing spinor8 . The metric of a hyperkähler manifold satisfies the Monge-Ampère equation; we can always find holomorphic coordinates such that this has the form K u u¯ K z z¯ − K u z¯ K z u¯ = 1.

(7.7)

This implies that we can write the line element as ds 2 = |kdz|2 + |k −1 du + k K z u¯ dz|2 ,

(7.8)

where −1

k ≡ K u u¯2 .

(7.9)

˙

We choose frames eˆ A B (here A, B˙ are target space spinor indices) eˆ++˙ = kd z¯ , ˙

eˆ−− = kdz,

d u¯ ˙ eˆ+− = k ∂¯ K u = + k K u z¯ d z¯ , k du + k K z u¯ dz , eˆ−+˙ = −k∂ K u¯ = − k

(7.10) (7.11)

(so that ds 2 = eˆ++˙ eˆ−−˙ − eˆ+−˙ eˆ−+˙ ). We compute the connection; it is self-dual, with ω A B = 0; the nonvanishing terms are K u z¯ K z u¯ ˙ ˙ +˙ +˙ − − ¯ ¯ ω +˙ = −ω −˙ = (∂ − ∂) ln(k), ω −˙ = K u u¯ ∂ , ω +˙ = −K u u¯ ∂ . K u u¯ K u u¯ (7.12) The dual vector fields are ˙

˙

e− + = −k −1 ∂z¯ + k K u z¯ ∂u¯ , e+˙ + = k∂u¯ , e+˙ − = k −1 ∂z − k K z u¯ ∂u , e− − = k∂u . (7.13) We now construct a rank 2n Killing spinor for an O(2n) multiplet η. The components of η are related to the components of the spinor by: ηi =

2n η+ · · · + − · · · − , i

2n−i

η≡

i

2n

ηi ζ i .

(7.14)

0

The Killing spinor equation is ˙

e A (A η B1 ...B2n ) = 0

(7.15)

because we work in a frame where the connection 1-form ω A B vanishes, or ˙

˙

e A − ηi−1 + e A + ηi = 0. 8 This was shown using different techniques in [24].

(7.16)


267

We begin by checking i = 0, 1. In the generalized Legendre transform construction above, we identify9 η0 = z, η1 = −K u , η2n = (−1)n z¯ , η2n−1 = (−1)n K u¯ .

(7.17)

Then (7.16) is trivially satisfied for i = 0. For i = 1, we have: ˙

˙

e+˙ − z − e+˙ + K u = k −1 − k K u u¯ = 0, e− − z − e− + K u = 0 − k −1 K u z¯ + k K u z¯ K u u¯ = 0. (7.18) The i = 2n, 2n + 1 equations are just the complex conjugates of the above. For 1 < i < 2n, we find equations that do not have a simple expression in terms of the Kählerpotential; however, we can easily prove that they are satisfied by studying the superspace description of the O(2n) multiplet η. The superspace constraints (2.5) can be written as D1α ηi−1 + D2α ηi = 0,

(7.19)

Daα

are the superspace spinor derivatives with isospin indices a and spinor indiwhere ces α. Note the similarity to (7.16). For i = 0, 1, 2n, 2n + 1, (7.19) is a set of relations between Daα xµ , where xµ = {z, u, z¯ , u}. ¯ Note that these relations are exactly the same α ˙ µ as those obeyed by e± x as a consequence of (7.16). In superspace, however, (7.19) is imposed as a constraint that defines η. When we eliminate the N = 1 auxiliary superfields ηi , 1 < i < 2n − 1, and the Legendre transform variables η1 , η2n−1 , we must consider ηi (xµ ). Then Eqs. (7.19) become: ∂µ ηi−1 D1α xµ + ∂µ ηi D2α xµ = 0.

(7.20) ˙

However, since the linear relations between the Daα xµ and the e±A xµ are the same, this implies relations between the ∂µ ηi−1 and ∂µ ηi that guarantee that the Killing spinor equation (7.16) is satisfied. The leading component of the Killing spinors discussed here is proportional to a coordinate; there is a closely related Killing tensor that can be constructed out of the spinors which may be easier to define globally. This is defined by the components of the derivative of the Killing spinor that do not vanish: ˙

˙

X BA1 ...B2n−1 ≡ ∇ A A η AB1 ...B2n−1 .

(7.21)

Because the connection is self-dual, these obey the Killing tensor equations [25], ˙

˙

˙

(B X BA)2 ...B2n ) = 0. ∇ AB˙ 1 X BA1 ...B2n−1 = 0, ∇(B 1

(7.22)

For n = 1, this is the well-known triholomorphic Killing vector that characterizes the O(2) geometries [26]. 7.3. Higher dimensional hyperkähler manifolds. For four dimensional hyperkähler manifolds, we were able to explicitly relate projective superspace and geometry; bolstered by our success, we can conjecture geometric results from projective superspace for the higher dimensional case: In projective superspace, higher dimensional target spaces arise when one considers models with more independent superfields. Depending on the type of multiplets in the model, we will get corresponding Killing spinors and Killing tensors. 9 In [6] and many other references, the role of z, u is interchanged with z¯ , u; ¯ also, in some references, the η’s are defined with an extra overall factor ζ −n .

268


8. Properties of Twistor Space For the reader’s convenience, we review the properties of twistor spaces summarized in Sect. 1.1 and relate them to the geometric structure that projective superspace revealed. The description of hyperkähler geometry that follows from the projective superspace formulation of N = 2 supersymmetric σ -models leads to a coherent picture in twistor space, where the P1 of graded abelian subalgebra of the N = 2 superalgebra is identified with the P1 of complex structures on the hyperkähler manifold. The fundamental object is the 2-form (5.1). In terms of the hyperkähler structure, it can be written as: = ω(2,0) + ζ ω(1,1) − ζ 2 ω(0,2) ,

(8.1)

where ω(2,0) is a nondegenerate holomorphic 2-form and ω(1,1) is the Kähler form with respect to the complex structure at the north pole of the P1 . One may always choose Darboux coordinates z, u for the holomorphic symplectic structure ω(2,0) ; extending these to arbitrary complex structures parametrized by a point ζ on the P1 lifts z, u to ˜ ) and leads us to write ϒ(ζ ), ϒ(ζ ˜ (ζ ) = idϒd ϒ,

(8.2)

˜ ) such that (ζ ) is projectively real, and hence a section of O(2) ⊗ with ϒ(ζ ), ϒ(ζ 2 (M). The reality condition implies the existence of a twisted symplectomorphism from the north pole to the south pole, and consequently the existence of the generating ¯ ζ ). This in particular proves that the projective superspace formalism function f (ϒ, ϒ; with polar superfields ϒ, ϒ¯ is completely general (at least locally in each patch of the hyperkähler manifold, though we see no obstruction to patching this together over the whole manifold using the general symplectic transformations of Sect. 6). An interesting feature of this way of thinking about hyperkähler geometry is that ¯ ζ )? and (2) What is it naturally leads to two separate problems: (1) What is f (ϒ, ϒ; ϒ(ζ )? In N = 2 language, the first is an off-shell problem and the second is the on-shell problem. It may be possible to solve the off-shell problem for, e.g., K 3, without solving the on-shell problem. This would still be very interesting, though it would not yield an explicit metric. The 2-form also allows us to find the system of partial differential equations that characterize hyperkähler geometry. For a 4D-dimensional hyperkähler manifold M, the form (8.2) clearly obeys D+1 = 0.

(8.3)

For D = 1, this reduces to the usual Monge-Ampère equation. For higher D, this gives a nice system of equations that implies the Monge-Ampère equation. For example, for D = 2, expanding in ζ , we find ω(2,0) ((ω(1,1) )2 − ω(2,0) ω(0,2) ) = 0, ω(1,1) ((ω(1,1) )2 − 6ω(2,0) ω(0,2) ) = 0,

(8.4)

ω(0,2) ((ω(1,1) )2 − ω(2,0) ω(0,2) ) = 0. This implies the Monge-Ampère equation, which in our conventions for general dimension D is 2D (1,1) 2D (ω(2,0) ω(0,2) ) D = 0. (ω ) − (8.5) D


269

9. Rotating the Complex Structures A crucial role both for the twistor structure and for the supersymmetric σ -models is played by rotations of the P1 combined with corresponding rotations of the hyperkähler structure on M. We consider the 2-form with ω(2,0) in Darboux coordinates ω(2,0) = i i j 2 i j dz dz : (ζ ) = idzdu + i∂ ∂¯ K ζ + id z¯ d uζ ¯ 2,

(9.1)

¯ As described in previous where ∂ ∂¯ K = K z z¯ dzd z¯ + K z u¯ dzd u¯ + K u z¯ dud z¯ + K u u¯ dud u. sections of this article, the form is a real section of an O(2) bundle, where the real structure is defined by complex conjugation composed with the antipodal map ζ¯ → −1/ζ , i and acts on an O(2n) section η = 2n 0 ζ ηi as: η(ζ ) = (−)n ζ 2n η( ¯ −1 ζ ).

(9.2)

η0 = −η¯ 2 , η1 = η¯ 1 .

(9.3)

For the O(2) case, we have

9.1. Rotating P1 . An SU (2) R-symmetry transformation in superspace is generated by Möbius transformations of ζ , and rotates the complex structures on the hyperkähler manifold. If we write ζ =

aζ + b , cζ + d

(9.4)

where ad − bc = 1 and d¯ = a, c¯ = −b for an SU (2) transformation, then for a = 1 + iα, b = β, and α = α, ¯ the infinitesimal transformation of ζ is ¯ 2. δζ = β + 2iαζ + βζ

(9.5)

An SU (2)-transformation is generated by α·J ≡

3

αi Ji ≡ α J3 + 21 β J− + 21 β¯ J+ ,

(9.6)

1

where the SU (2)-algebra is J± ≡ J1 ± i J2 , [J3 , J± ] = ±J± , [J+ , J− ] = 2J3 .

(9.7)

δζ = [2iα · J , ζ ],

(9.8)

Writing

we may represent the SU (2) generators as J− = −i∂ζ ,

J3 = ζ ∂ζ , J+ = −iζ 2 ∂ζ .

(9.9)

More generally, we can add a spin piece, and write J− = −i∂ζ , J3 = ζ ∂ζ − 21 h, J+ = −iζ 2 ∂ζ + i hζ.

(9.10)

270


An O(2n) multiplet transforms with h = 2n, and transforms with h = 2 (see, e.g., [3] and [11]). Then, from δ = −2i(α J3 + 21 β J− + 21 β¯ J+ ), we find δ(idzdu) = 2iα(idzdu) − β(i∂ ∂¯ K ),

(9.11)

¯ δ(i∂ ∂¯ K ) = −2β(id z¯ d u) ¯ + 2β(idzdu),

(9.12)

¯ ∂¯ K ). δ(id z¯ d u) ¯ = −2iα(id z¯ d u) ¯ + β(i∂

(9.13)

9.2. Rotating the hyperkähler structure on M. It is easy to find diffeomorphisms on M that satisfy (9.11): δz = iαhz + β K u , δu = iα(2 − h)u − β K z

(9.14)

clearly give the correct transformation. Notice the close relation to the Legendre transform construction: −K u ≡ η1 , so δz = iαhz−βη1 . This is exactly what we would expect from projective superspace; by changing h, we get different η and or ϒ multiplets. As the α transformations are holomorphic, ∂ and ∂¯ are invariant under them. Naively, K transforms as δα K = iα[hz K z + (2 − h)u K u ] + c.c.; we can cancel this by simply subtracting this from the variation of K ; thus we define δα K = iα[hz K z +(2−h)u K u ]+c.c.+δα K = 0. This may look odd, but as we shall see, it is very necessary and much more nontrivial below. Thus we focus on the β transformations. We write them as δβ z i = β i j K j , δβ z¯ i = 0,

(9.15)

where {z i } ≡ {u, z}. Note that here the naive variation of K vanishes: δβ K = β i j K j K i = 0. Consequently, we have: ¯ β K )]. (9.16) δβ (i∂ ∂¯ K ) = i[d(δβ z i )∂i ∂¯ K + dz i (δβ ∂i )∂¯ K + ∂d z¯ i (δβ ∂¯i )K + ∂ ∂(δ Because δβ z¯ i = 0, we have δβ ∂i = −(∂i δβ z j )∂ j , etc., and we find ¯ β K )] δβ (i∂ ∂¯ K ) = i[d(δβ z i )∂i ∂¯ K − dz i (∂i δβ z j )∂ j ∂¯ K − ∂(d z¯ i (∂¯i δβ z j )K j ) + ∂ ∂(δ ¯ β K )] ¯ β z i )∂i ∂¯ K −(∂δβ z i )∂i ∂¯ K − ∂((∂δ ¯ β z i )K i )+∂ ∂(δ = i[(∂δβ z i )∂i ∂¯ K +(∂δ ¯ β z i )∂i ∂¯ K − ∂((∂δ ¯ β K )] ¯ β z i )K i ) + ∂ ∂(δ = i[(∂δ ¯ β z i )K i ) + ∂ ∂(δ ¯ β K )]. ¯ β z i )∂¯ K i − ∂((∂δ = i[(∂δ

(9.17)

Now we substitute (9.15): ¯ β z i )∂¯ K i = iβ i j (∂¯ K j )∂¯ K i = iβ i j d z¯ j d z¯ i = −2iβd ud i(∂δ ¯ z¯ , ω(1,1) [ω(2,0) ]−1 ω(1,1)

(9.18)

−ω(0,2) .

where we use the quaternionic relation = Finally, we need to show that all remaining terms can cancel. In contrast to (9.18), which is a (2, 0) ¯ β z i )K i ) + ∂ ∂(δ ¯ K )] are both (1, 1) forms. We need form, the remaining terms i[−∂((∂δ β ¯ i )K i ) is both ∂ and ∂¯ closed; this is manifest for ∂. For ∂, ¯ we use to show that ∂((∂δz (9.18): ¯ ∂δ ¯ β z i )K i ) = ∂((∂δ ¯ β z i )∂¯ K i ) = ∂(−2βd ud ∂∂(( ¯ z¯ ) = 0. Thus there exists a

δβ K

(9.19)

such that the total variation δβ (i∂ ∂¯ K ) is given by (9.12).


271

9.3. The Kähler potential is a Hamiltonian. A remarkable feature allows us to interpret the Kähler potential K as a Hamiltonian function. The transformation (9.5) has a fixed ¯ then (9.11, 9.12, 9.13) imply that δ0 ≡ δα=0,β=β¯ point at ζ = ±i for α = 0, β = β; preserves [ω(2,0) + ω(0,2) ] = 21 [(ζ = i) + (ζ = −i)].

(9.20)

Thus δ0 is a symplectomorphism that preserves Re(ω(2,0) ), and hence is generated by a moment map; this moment map is precisely the i times the Kähler potential: [ω(2,0) + ω(0,2) ](δ0 z i , .) = id K .

(9.21)

This generalizes the observation in [14] that for manifolds with an isometry that rotates the complex structure, the Kähler potential can be viewed as the moment map of the rotation with respect to a complex structure preserved by the rotation; here we do not need an isometry.

10. Normal Gauge On any Kähler manifold, one can define a normal gauge for the Kähler potential [27]. In this gauge, one eliminates any purely holomorphic or antiholomorphic pieces using Kähler transformations, and uses holomorphic coordinate transformations to make the potential as close as possible to flat: K = z i z¯ i + O(z 2 z¯ 2 ),

(10.1)

i.e., all terms except for the flat term are at least quadratic in z and quadratic in z¯ ; these terms are all expressible in terms of the curvature and its derivatives, and the explicit expression is easily found by direct computation. Clearly, normal gauge is unique up to the choice of base point, and up to constant U (2) tranformations. For a Ricci-flat Kähler manifold, det gi j¯ = f (z) f¯(¯z );

(10.2)

in normal gauge, f (z) is constant, as follows from (10.1), which implies (∂z )n gi j¯

(z=¯z =0)

= (∂z¯ )n gi j¯

(z=¯z =0)

= 0 ∀n.

(10.3)

For a hyperkähler manifold (at least for real D=4), we have (ω1 )2 = (ω2 )2 = (ω3 )2 ∝ det gi j¯ , and hence ω(2,0) ω(0,2) is constant. However, since ω(2,0) is holomorphic, and its magnitude is constant, we conclude that it is in Darboux coordinates (up to a constant phase which can be absorbed by a constant U (1) transformation that preserves the normal gauge); thus: ω(2,0) = i dz 1 dz 2 .

(10.4)

272


11. Example: The Eguchi-Hansen Geometry In this section we derive the Eguchi-Hansen metric using the methods developed above. This is related to the general program of constructing hyperkähler metrics on cotangent bundles of symmetric spaces using projective superspace methods [28–31], and indeed can be applied to all of them. Other recent examples in the projective/twistor formalism include the explicit elliptic examples of [32] and the explicit linear deformations of hyperkähler manifolds given in [33]. The Eguchi-Hansen metric lives on the cotangent space P1 ; hence we start with the Fubini-Study Kähler potential for P1 and lift it to N = 2 superspace: ¯ f = ln(1 + ϒ ϒ).

(11.1)

The Eguchi-Hansen metric has a triholomorphic SU (2) isometry which can be realized by P SU (2) transformations of ϒ. We can therefore choose a particular form for ϒ and reach general points by acting with the isometry [29]. In particular, we make the ansatz that when we set z ≡ ϒ(0) = 0 then ϒ|z=0 = yζ

(11.2)

is a valid point on the manifold. We now act by a P SU (2) transformation which we parameterize so as to recover (11.2) as well as z = ϒ(0): z + yζ ; 1 − y z¯ ζ

ϒ→

(11.3)

note that this is a triholomorphic P SU (2) transformation that acts on ϒ, not a rotation of the P1 of complex structures parameterized by ζ . The conjugate is ϒ¯ =

y¯ − z¯ ζ . z y¯ − ζ

(11.4)

˜ Following the methods described above, to find we need to calculate ϒ: ϒ˜ = ζ

∂f ζ ϒ¯ (y¯ − z¯ ζ )(1 − y z¯ ζ ) = . = ∂ϒ (1 + z z¯ )(1 − y y) ¯ 1 + ϒ ϒ¯

(11.5)

A quick calculation reveals that i dϒd ϒ˜ is indeed a section of O(2); the structure is clarified if we introduce the second holomorphic coordinate ˜ u ≡ ϒ(0) =

y¯ , (1 + z z¯ )(1 − y y) ¯

(11.6)

which implies y=

1+

2(1 + z z¯ )u¯ 1 + 4u u(1 ¯ + z z¯ )2

.

(11.7)

This gives the standard for the Eguchi-Hansen Kähler form: (1,1)

¯ E H = i dϒd ϒ˜ = i dzdu + ζ ω E H + i ζ 2 d z¯ d u,

(11.8)


273

where (1,1) ωE H

1 + z z¯

= −i 1 + 4u u(1 ¯ + z z¯ )2

dzd z¯ dud u¯ + + (zdu + 2udz)(¯z d u¯ + 2ud ¯ z¯ ) . (1 + z z¯ )3 (11.9)

This can be made more familiar by the holomorphic symplectomorphism u=

1 2 z u , z= 2 u

(11.10)

which gives 1 1 (1,1) r 2 (du d u¯ + dz d z¯ ) + 4 (z du − u dz )(¯z d u¯ − u¯ d z¯ ) , ω E H = −i √ r 1 + r4 √ (11.11) r ≡ u u¯ + z z¯ . This calculation reveals an important feature of our approach and the virtue of using : we found the Kähler-form without evaluating any contour integral; in particular, there are no ambiguities about the orientation of the contour that can arise in a direct evaluation of the superspace Lagrangian. An example of such issues is given in Appendix B. 12. Outlook We have discussed the intimate relation between twistor space and supersymmetry as manifested in projective superspace. Our primary tools are N = 2 sigma models with hyperkähler target spaces, but gauging them also introduces gauge connections. These were mainly used here to describe quotient constructions and dualities, but may be studied in their own right in projective superspace. This leaves one obvious gap in the description of models: N = 2 supergravity. To a certain extent this gap is presently being filled (see [34] and references therein). A more immediate extension of the framework presented here is to include quaternionic Kähler manifolds. Such an extension is presently under way. We further note that projective superspace has recently been used to study linear perturbations of a class of hyperkähler metric in [33], where an extension to quaternionic Kähler metrics is also advertised. As our description is fully non-linear, a comparison should be fruitful. Acknowledgement. We thank the 2003, 2004, 2005, 2006, and 2008 Simons Workshops in Physics and Mathematics at C.N. Yang Institute for Theoretical Physics and the Department of Mathematics at Stony Brook for partial support and for a stimulating atmosphere. We are happy to thank Claude LeBrun, Blaine Lawson and Dennis Sullivan for many useful discussions over the years as well as Nigel Hitchin, Lionel Mason, David Skinner, and Rikard von Unge for more recent comments. We are also happy to thank Sergei Kuzenko for discussions of the example, and Stefan Vandoren for making [33] available to us prior to posting it on the arXiv. MR thanks the Institute for Theoretical Physics at the University of Amsterdam for hospitality during the spring of 2006. The work of UL is supported in part by VR grant 621-2006-3365 and by EU grant (Superstring theory) MRTN-2004-512194. MR is supported in part by NSF grant no. PHY-06-53342.

274


A. The Hyperkähler Quotient in Projective Superspace For completeness we review constructions having to do with gauge fields, quotients, and dualities in projective superspace. The hyperkähler quotient construction was discovered in [35] and its geometric interpretation was given in [14]. The tools to describe it in projective superspace were developed in [8], and the description was given in [20], though it has been known to us for a long time. Here we review it. A.1. Isometries. The polar multiplet ϒ has an infinite number of N = 1 superfields; consequently, it is difficult to extract the Kähler potential except in special circumstances. On the other hand, the space of polar multiplets has an algebraic structure: holomorphic functions of arctic multiplets are themselves arctic. This allows for a very direct realization of triholomorphic isometries of the hyperkähler geometry in projective superspace: they are simply symmetries of the projective superspace action (2.9) that are holomorphic in the arctic multiplets. As we explain below, the whole process of gauging triholomorphic isometries and performing hyperkähler quotients, when described in terms of polar multiplets in projective superspace is essentially the same procedure as for Kähler quotients described in terms of chiral superfields in N = 1 superspace [14,36]. A triholomorphic isometry acts without rotating the complex structures; therefore it is generated by a holomorphic vector field X (ϒ) that has no explicit dependence on ζ : ¯ δϒ = X (ϒ), δ ϒ¯ = X¯ (ϒ).

(A.1)

When we gauge a symmetry generated by such a vector field, we introduce a local parameter λ(ζ ): ¯ ¯ δϒ = λ(ζ )X (ϒ), δ ϒ¯ = λ¯ ( −1 ζ ) X (ϒ);

(A.2)

to preserve the holomorphic properties of ϒ, the parameter λ(ζ ) must itself be an arctic ¯ −1 ) must be antarctic. We are thus led to introduce a real superfield, and consequently, λ( ζ tropical field V = R(V); it has coefficients for all powers of ζ that are unconstrained as N = 1 superfields. It transforms as δV = i(λ¯ − λ).

(A.3)

This may be generalized to a nonabelian action, where V, λ, λ¯ all become matrix valued; for a finite transformation by an element g = eLiλX , we have:

¯ (A.4) eV = ei λ eV e−iλ . Having introduced the field V, we now show how it describes N = 2 super YangMills theory [8]. We split the tropical gauge multiplet factors regular at the north and south poles: eV = eV− eV+ ,

V+ =

∞

V+n ζ n ,

¯ +. V− = V

(A.5)

n=0

Because V is an analytic superfield, ∇eV = 0, and we may define a gauge-covariant analytic derivative D,

D ≡ ∇ + e−V− (∇eV− ) = ∇ − (∇eV+ )e−V+ .

(A.6)


275

Comparing powers of ζ for both expressions, we conclude that D has only a constant and a linear term (just as ∇), and hence defines the N = 2 gauge-covariant derivative (for a more detailed explanation see [8]). This structure is precisely the same as Ward’s twistor construction of self-dual Yang-Mills fields [37]. Observe that (A.6) depends crucially on the reality of V. We find the covariantly chiral gauge field strength W by computing ¯ (ζ ), ∂ (D ¯ (ζ ))} = ε W. {D α˙ α˙ β˙ β˙ ∂ζ

(A.7)

Note that W is ζ independent. We focus on the case when we start with a vector space, and quotient by a linear (or possibly affine) action; this has the virtue that the formal expression (2.8) for the superspace Lagrangian can be explicitly evaluated. Thus we start with ¯ V) = ϒe ¯ Vϒ f (ϒ, ϒ,

(A.8)

for any compact group acting linearly on the vector space coordinatized by ϒ. We define covariantly analytic polar multiplets, ¯ V− . ϒˆ¯ = ϒe

ϒˆ ≡ eV+ ϒ ,

(A.9)

In terms of these, the gauge-invariant Lagrange density (A.8) is quadratic; hence, the ζ integral dζ dζ ˆ ¯ Vϒ = F= ϒe ϒ¯ · ϒˆ (A.10) C 2πiζ C 2πiζ can be trivially evaluated, and the auxiliary superfields can be integrated out to get the gauge-invariant N = 1 superspace Lagrangian, L N =1 = zˆ¯ · zˆ − sˆ¯ · sˆ ,

(A.11)

where zˆ ≡ ϒˆ 0 are N = 1 gauge-covariantly (vector representation) chiral superfields and sˆ ≡ ϒˆ 1 are modified N = 1 gauge-covariantly complex linear superfields D¯ α˙ zˆ = 0 ,

D¯ 2 sˆ = Wˆ zˆ .

(A.12)

Here Wˆ is the N = 1 covariantly chiral projection of the N = 2 field strength W (A.7) in the representation that acts on zˆ and D is the N = 1 gauge-covariant derivative. We can go to chiral representation and replace zˆ , sˆ , Wˆ with ordinary chiral and linear superfields z, s, W by introducing the N = 1 gauge potential V : e V ≡ e V− e V+ , zˆ = e V+ z , sˆ = e V+ s , Wˆ = e V+ W e−V+ ,

(A.13)

where V± ≡ V0± is the N = 1 projection of the ζ -independent coefficients of V± . These substitutions lead to L N =1 = z¯ e V z − s¯ e V s , D¯ 2 s = W z.

(A.14) (A.15)

276


It is convenient to rewrite the N = 1 Lagrangian (A.14) in terms of chiral superfields; to do this, we impose the constraints (A.15) by chiral Lagrange multipliers u in a superpotential term u( D¯ 2 s − W z),

(A.16)

and integrate out s to obtain the nonabelian generalization of the N = 1 gauged Lagrangian (after relabeling z → z + , u → z − ): L N =1 = z¯ + e V z + − z − e−V z¯ − .

(A.17)

In addition, we are left with a superpotential term Tr [ W µ+ ] = z − W z + ,

(A.18)

where µ+ is just the holomorphic moment map. Observe that interchanging z + ↔ z − and changing the representation of V to its conjugate does not modify the gauged Lagrangian (A.17); this implies that in the original N = 2 Lagrangian F (A.10), we can take ϒ transforming in the conjugate representation (e.g., opposite charge for U(1)) without changing the final result. This interchange can be implemented directly in projective superspace by the ϒ ↔ ϒ duality transformation of Sect. 6 (ϒ˜ naturally transforms in the conjugate representation to ϒ). In the next subsection we integrate out the N = 2 gauge fields to find the quotient Lagrangian; in N = 1 superspace, integrating out the chiral superfield W imposes the moment map constraint µ+ = 0. A.2. Quotients and duality. Just as N = 2 isometries and gauging in projective superspace bear a striking resemblance to their N = 1 superspace analogs, so do N = 2 quotients and duality; indeed, the tensor multiplet projective superspace Lagrangian is just the Legendre transform of the polar multiplet Lagrangian. The procedure we follow is the same as in N = 1 superspace: we gauge the relevant isometries as above; to perform a quotient, we simply integrate out the gauge prepotential eV . Since this does not break the isometry, we are left with an action defined on the quotient space. To find the dual, we add a Lagrange multiplier η that constrains the gauge prepotential to be trivial10 , and again integrate out V; the dual field is then the Lagrange multiplier η. As in the N = 1 case, we only consider duality for abelian isometries. In that case, the Lagrange multiplier term that constrains V is η V, ζ

(A.19)

where η is the O(2) superfield that describes the N = 2 tensor multiplet [3]. B. Dualities and Contour Ambiguities The Eguchi-Hansen metric can also be described in terms of the O(2)-multiplet [3] (these are particular instances of the multiplets described in Sect. 7). A particularly nice 10 As explained in [14,38], this is the correct geometric way of understanding duality; when one chooses coordinates such that the killing vectors generating the isometries are constant, this gives the usual Legendre transform.


277

way of finding this description involves the quotient and duality described in the previous appendix. Starting from (11.1), one can write f V = ln(1 + eV ) −

η(2) V, ζ

(B.20)

where η(2) is an O(2)-multiplet; eliminating η(2) imposes the condition that V ∝ ¯ whereas eliminating V gives: ln(ϒ ϒ), η(2) η(2) η(2) η(2) fη = − ln − 1− ln 1 − . (B.21) ζ ζ ζ ζ The metric can be found by evaluating the ζ integral along a contour first given in [3] with the caveat that the opposite orientation must be used for the two terms in (B.21) to obtain a metric with definite signature. On the other hand, we can rewrite (11.1) in terms of the symplectic conjugate vari˜ ables ϒ: ¯ ¯˜ ˜ ˜ ˜ f = ln 1 + 1 − 4ϒ ϒ − 1 − 4ϒ˜ ϒ. (B.22) Performing the duality transformation to the O(2) multiplet η(2) as above, we obtain: η(2) η(2) η(2) η(2) ˜ ln − 1+ ln 1 + . (B.23) fη = − ζ ζ ζ ζ The difference in relative sign between the terms in (B.21) and (B.23) mean that we need to use different orientations of the contours when evaluating the metric in the two cases. Clearly the issue of contours, in particular their orientation, is a subtle one. In the definition of no ambiguities exist, as illustrated in Sect. 11. We thus determine the integration contours by requiring agreement with an derivation. It would be interesting to compare this idea to the discussions of contours presented in [23,33]. References 1. Alvarez-Gaumé, L., Freedman, D.Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model. Commun. Math. Phys. 80, 443 (1981) 2. Gates, S.J., Hull, C.M., Roˇcek, M.: Twisted multiplets and new supersymmetric nonlinear sigma models. Nucl. Phys. B 248, 157 (1984) 3. Karlhede, A., Lindström, U., Roˇcek, M.: Selfinteracting tensor multiplets in N = 2 superspace. Phys. Lett. B 147, 297 (1984) 4. Grundberg, J., Lindström, U.: Actions for linear multiplets in six-Dimensions. Class. Quant. Grav. 2, L33 (1985) 5. Karlhede, A., Lindström, U., Roˇcek, M.: Hyperkahler manifolds and nonlinear supermultiplets. Commun. Math. Phys. 108, 529 (1987) 6. Lindström, U., Roˇcek, M.: New hyperkahler metrics and new supermultiplets. Commun. Math. Phys. 115, 21 (1988) 7. Buscher, T., Lindström, U., Roˇcek, M.: New supersymmetric sigma models with wess-zumino terms. Phys. Lett. B 202, 94 (1988) 8. Lindström, U., Roˇcek, M.: N = 2 super yang-mills theory in projective superspace. Commun. Math. Phys. 128, 191 (1990) 9. Lindström, U., Ivanov, I.T., Roˇcek, M.: New N = 4 superfields and sigma models. Phys. Lett. B 328, 49 (1994) 10. Lindström, U., Kim, B.B., Roˇcek, M.: The Nonlinear multiplet revisited. Phys. Lett. B 342, 99 (1995)

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11. Gonzalez-Rey, F., Roˇcek, M., Wiles, S., Lindström, U., von Unge, R.: Feynman rules in N = 2 projective superspace. I: Massless hypermultiplets. Nucl. Phys. B 516, 426 (1998) 12. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31 (1976) 13. Salamon, S.: Quaternionic Kähler manifolds. Invent. Math. 67, 143–171 (1982) 14. Hitchin, N.J., Karlhede, A., Lindström, U., Roˇcek, M.: Hyperkahler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987) 15. Kuzenko, S.M.: On compactified harmonic / projective superspace, 5D superconformal theories, and all that. Nucl. Phys. B 745, 176–207 (2006) 16. Ivanov, I.T., Roˇcek, M.: Supersymmetric sigma models, twistors, and the Atiyah-Hitchin metric. Commun. Math. Phys. 182, 291 (1996) 17. Chakravarty, S., Mason, L., Newman, E.T.: Canonical structures on antiselfdual four manifolds and the diffeomorphism group. J. Math. Phys. 32, 1458 (1991) 18. Kuzenko, S.M.: Projective superspace as a double-punctured harmonic superspace. Int. J. Mod. Phys. A 14, 1737 (1999) 19. Galperin, A.S., Ivanov, E.A., Ogievetsky, V.I., Sokatchev, E.S.: Harmonic Superspace. Cambridge, UK: Cambridge Univ. Pr., 2001 20. de Wit, B., Roˇcek, M., Vandoren, S.: Hypermultiplets, hyperkaehler cones and quaternion-Kaehler geometry. JHEP 0102, 039 (2001) 21. Lindström, U., Roˇcek, M., von Unge, R., Zabzine, M.: Generalized Kaehler manifolds and off-shell supersymmetry. Commun. Math. Phys. 269, 833–849 (2007) 22. Ketov, S.V., Lokhvitsky, B.B., Tyutin, I.V.: Hyperkähler sigma models in extended superspace. Theor. Math. Phys. 71, 496 (1987) [Teor. Mat. Fiz. 71, 226 (1987)] 23. Houghton, C.J.: On the generalized Legendre transform and monopole metrics. JHEP 0002, 042 (2000) 24. Bielawski, R.: Twistor quotients of hyperkaehler manifolds. http://arxiv.org/abs/math/0006142v1[math. DG], 2000 25. Carter, B.: Killing tensor quantum numbers and conserved currents in curved space. Phys. Rev. D 16, 3395 (1977) 26. Howe, P.S., Karlhede, A., Lindström, U., Roˇcek, M.: The Geometry of duality. Phys. Lett. B 168, 89 (1986) 27. Gates, S.J., Grisaru, M.T., Roˇcek, M., Siegel, W.: Superspace, or one thousand and one lessons in supersymmetry. Front. Phys. 58, 1 (1983) 28. Gates, S.J., Kuzenko, S.M.: The CNM-hypermultiplet nexus. Nucl. Phys. B 543, 122 (1999) 29. Arai, M., Nitta, M.: Hyper-Kaehler sigma models on (co)tangent bundles with SO(n) isometry. Nucl. Phys. B 745, 208–235 (2006) 30. Arai, M., Kuzenko, S.M., Lindström, U.: Hyperkaehler sigma models on cotangent bundles of Hermitian symmetric spaces using projective superspace. JHEP 0702, 100 (2007) 31. Arai, M., Kuzenko, S.M., Lindström, U.: Polar supermultiplets, Hermitian symmetric spaces and hyperkahler metrics. JHEP 0712, 008 (2007) 32. Iona¸s, R.A.: Elliptic constructions of hyperkaehler metrics I: The Atiyah-Hitchin manifold. http://arxiv. org/abs/0712.3598v1[math.DG], 2007; Iona¸s, R.A.: Elliptic constructions of hyperkaehler metrics II: The quantum mechanics of a Swann bundle. http://arxiv.org/abs/0712.3600v1[math.DG], 2007; Iona¸s, R.A.: Elliptic constructions of hyperkaehler metrics III: Gravitons and Poncelet polygons. http://arxiv. org/abs/0712.3601v1[math.DG], 2007 33. Alexandrov, S., Pioline, B., Saueressig, F., Vandoren, S.: Linear perturbations of quaternionic metrics I. The Hyperkahler case. Lett. Math. Phys. 87, 225–265 (2009) 34. Kuzenko, S.M., Lindström, U., Roˇcek, M., Tartaglino-Mazzucchelli, G.: 4D N = 2 supergravity and projective superspace. JHEP 0809, 051 (2008) 35. Lindström, U., Roˇcek, M.: Scalar tensor duality and N = 1, N = 2 nonlinear sigma models. Nucl. Phys. B 222, 285 (1983) 36. Hull, C.M., Karlhede, A., Lindström, U., Roˇcek, M.: Nonlinear sigma models and their Gauging in and out of superspace. Nucl. Phys. B 266, 1 (1986) 37. Ward, R.S.: On Selfdual gauge fields. Phys. Lett. A 61, 81 (1977) 38. Roˇcek, M., Verlinde, E.P.: Duality, quotients, and currents. Nucl. Phys. B 373, 630 (1992) Communicated by A. Kapustin


Communications in


Asymptotic Behavior of Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum Changjiang Zhu Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 430079, P.R. China. E-mail: [email protected] Received: 2 November 2008 / Accepted: 17 July 2009 Published online: 10 September 2009 – © Springer-Verlag 2009

Abstract: In this paper, we study the one-dimensional Navier-Stokes equations connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coefficient µ(ρ) is proportional to ρ θ with θ > 0, where ρ is the density, we give the asymptotic behavior and the decay rate of the density function ρ(x, t). Furthermore, the behavior of the density function ρ(x, t) near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied. The analysis is based on some new mathematical techniques and some new useful estimates. This fills a final gap on studying Navier-Stokes equations with the viscosity coefficient µ(ρ) dependent on the density ρ. Contents 1. 2. 3.

Introduction and the Main Theorems . . . . . . Reformulation of the Problem . . . . . . . . . . A priori Estimates and the Asymptotic Behavior of the Density Function . . . . . . . . . . . . . 3.1 Uniform a priori estimates . . . . . . . . . 3.2 Asymptotic behavior of the density function 4. Decay Rate of Density Function . . . . . . . . . 5. The Proof of the Main Theorems . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction and the Main Theorems In this paper, we consider the asymptotic behavior, the decay rates and the behavior near the interfaces separating the gas from vacuum of the density function ρ(x, t) and the expanding rate of the interfaces of the one-dimensional Navier-Stokes equations for compressible isentropic flow with a jump to the vacuum initially when the viscosity

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coefficient depends on the density. The important feature of this problem is that the interfaces separating the gas and the vacuum propagate with finite speed. This model exhibits many interesting phenomena, such as gaseous stars problems in astrophysics, see [24]. For other physical significance and mathematical treatment of this kind of phenomenon, we refer to the excellent survey paper of Nishida, cf. [28]. The one-dimensional compressible Navier-Stokes equations with the viscosity depending on the density and the free boundaries can be written, in Eulerian coordinates, as follows: ⎧ ⎨ ρt + (ρu)x = 0, (1.1) ⎩ (ρu)t + (ρu 2 + P(ρ))x = (µ(ρ)u x )x , a(t) < x < b(t), t > 0, with initial data (ρ(x, 0), u(x, 0)) = (ρ0 (x), u 0 (x)), a ≤ x ≤ b,

(1.2)

and the free boundary conditions (P(ρ) − µ(ρ)u x )(a(t), t) = (P(ρ) − µ(ρ)u x )(b(t), t) = 0, t ≥ 0,

(1.3)

where ρ = ρ(x, t), u = u(x, t) and P(ρ) denote respectively the density, velocity and the pressure; µ(ρ) ≥ 0 is the viscosity coefficient which is possibly degenerate. a(t) and b(t) are the free boundaries defined by da(t) = u(a(t), t), t > 0, (1.4) dt a(0) = a, and

db(t) = u(b(t), t), t > 0, dt b(0) = b,

(1.5)

which are the interfaces separating the gas from the vacuum. In this paper, we will assume that the gas is polytropic for simplicity of presentation, i.e., the pressure P(ρ) obeys a gamma type law: P(ρ) = Aρ γ ,

A > 0, γ > 1.

(1.6)

The viscosity coefficient µ(ρ) is often assumed to be a positive constant. However, it is well-known that the viscosity of the gas is not constant and depends on the temperature. For example, we can get the viscosity µ(ρ) is proportional to the square root of the temperature when we use the Chapman-Enskog expansion to derive Navier-Stokes equations from the Boltzmann equation, cf. [3,7]. Especially, for isentropic flow, this dependence of the viscosity is translated into the dependence on the density. Precisely, the temperature T is of the order of ρ γ −1 for the perfect gas, where the pressure is proγ −1 portional to the product of the density and the temperature, which implies µ(ρ) = cρ 2 with c > 0. More generally, µ(ρ) is expected to vanish as a power of the ρ on the vacuum. In this paper, we consider the degenerate viscosity coefficient that vanishes for ρ = 0 at most like ρ θ for some θ > 0, i.e., µ(ρ) = cρ θ , c > 0, θ > 0. For more physical background, please refer to [38,39,45] and references therein.

(1.7)

Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

281

When the viscosity coefficient µ(ρ) is a positive constant and the initial density is away from vacuum, there has been a lot of investigation on the Navier-Stokes equations, both for smooth initial data or discontinuous initial data, and one-dimensional or multidimensional problems. For these results, please refer to [4,10–12,18,19,25,29,34–37] and references therein. For results concerning vanishing initial density, please refer to [13,23,30,31,41] and references therein. Precisely, Hoff and Serre in [13] showed that the continuous dependence fails on the initial data of the solutions to Navier-Stokes equations with vacuum and constant viscosity coefficient. The main reason for the failure at the vacuum comes from the kinematic viscosity coefficient being independent of the density. Okada in [30] proved the existence of the global weak solutions to the free boundary problem of one-dimensional Navier-Stokes equations with one boundary fixed and the other connected to vacuum. Similar results were obtained by Okada and Makino in [31] for the equations of spherically symmetric motion of viscous gases. A further understanding of the regularity and the asymptotic behavior of solutions near the interfaces between the gas and vacuum was given by Luo, Xin and Yang in [23,41]. When the viscosity coefficient µ(ρ) depends on the density and the initial density was assumed to be connected to vacuum with discontinuities, the local existence of weak solutions to the corresponding free boundary value problem (1.1)–(1.3) was studied by Liu, Xin, Yang in [22] and Makino in [24]. The global existence and uniqueness of weak solutions was first obtained by Okada, Matuš˙u-Neˇcasová and Makino in [32] for 0 < θ < 13 by using the techniques similar to [14]. This result was later generalized to the case when 0 < θ < 21 by Yang, Yao and Zhu in [43] and 0 < θ < 1 by Jiang, Xin and Zhang in [15], respectively. Recently, Qin, Yao and Zhao in [33] generalized furthermore these results to the case when 0 < θ ≤ 1. When the viscosity coefficient µ(ρ) depends on the density and the initial density was assumed to be connected to vacuum continuously, i.e., when we consider the free boundary value problem (1.1) and (1.2) with the following free boundary conditions: ρ(a(t), t) = ρ(b(t), t) = 0,

(1.8)

although there is no strictly positive lower bound for the density and the viscosity coefficient vanishes at vacuum, there has been a lot of investigation. The local existence of weak solutions was obtained by Yang and Zhao in [42] when 21 < θ ≤ γ − 13 . The global existence of weak solutions was first proved by Yang and Zhu in [44] for 0 < θ < 29 , and was later generalized to the case when 0 < θ < 13 by Vong, Yang and Zhu in [40], and 0 < θ < 1 by Fang and Zhang in [6]. Recently, Guo and Zhu in [9] studied first the asymptotic behavior and the decay rate of the density function ρ(x, t) with respect to the time t for any θ > 0 based on the following new mathematical entropy inequality: b(t) 1

1 ργ ρu 2 + u(ρ θ )x + ρ 2θ−3 ρx2 + dx 2 2 γ −1 a(t) t b(t) 4θ γ + ρ γ +θ−3 ρx2 dxds ≤ C, 0 < t < ∞, 2 0 a(t) (γ + θ )

(1.9)

where C is a uniform constant independent of the time t. This kind of very crucial new mathematical entropy inequality was derived by Kanel in [16] for the one-dimensional case and Bresch, Desjardins, Lin and Mellet, Vasseur for the multi-dimensional case, cf. [1,2,26] due to the specificity of the boundary conditions (1.8).

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Finally, we point out that there has been a lot of investigation on Navier-Stokes equations related to the viscosity coefficient µ(ρ) depending on the density and the initial density allowing to appear vacuum or not for one-dimensional or multi-dimensional problems, cf. [5,8,20,21] and references therein. To sum up, it is still an open problem how to get the asymptotic behavior and the decay rate estimates on the density function ρ(x, t) for the free boundary value problem (1.1)–(1.3). More precisely, only a gap and an open problem remain so far to study Navier-Stokes equations with the viscosity coefficient µ(ρ) depending on the density. In this paper, we will give a positive answer to this problem and fill this final gap. As we have mentioned above, the new mathematical entropy inequality (1.9) can be obtained due to the specificity of the boundary conditions (1.8). This uniform estimate on the derivative of the density function ρ(x, t) plays a very important role in studying the asymptotic behavior and the decay rate estimates on the density function ρ(x, t). It is well-known that, under the boundary conditions (1.3), the estimates similar to (1.9) depending on the time t on the derivative of the density function ρ(x, t) have been obtained by many authors in studying the global existence of the weak solutions, cf. [15,30,43]. However, to study the asymptotic behavior and the decay rate estimates on the density function ρ(x, t), we have to derive a uniform estimate similar to (1.9) independent of the time t on the derivative of the density function ρ(x, t). Under the boundary conditions (1.3), the boundary layers will occur, cf. (3.24), (4.9), because the density function ρ(x, t) does not vanish at the vacuum boundaries. This kind of phenomena is quite different from that of the boundary conditions (1.8), which have no boundary layers because the density function ρ(x, t) vanishes at the vacuum boundaries. The appearance of the boundary layers gives rise to new analysis difficulties. To overcome these new difficulties, in this paper, we introduce some new mathematical techniques and give some new useful estimates, cf. Lemma 3.4, Lemma 3.5 and Corollary 3.6. Throughout this paper, the assumptions on the initial data, θ and γ can be stated as follows: (A1) ρ0 (x) ∈ C 1 ([a(t), b(t)]), ρ02θ−3 (x)[(ρ0 (x))x ]2 ∈ L 1 ([a(t), b(t)]) and there exists a positive constant ρ0 > 0 such that ρ0 (x) ≥ ρ0 . (A2) For any given positive integer n satisfying n≥

2γ + θ , 2θ

(1.10)

1 assume ρ0 (x)u 2n 0 (x) ∈ L ([a(t), b(t)]). (A3) θ > 0, γ ≥ 1 + θ .

Remark 1.1. As we have mentioned above, the compressible Navier-Stokes equations are obtained as the second approximation of the Chapman-Enskog expansion to Boltzmann equations for a rarefied simple gas, cf. [3,7]. Here we assume the cut-off hard potentials and consider two important special cases: the hard sphere and the cut-off inverse power forces. Then the viscosity coefficient is given explicitly, i.e. for the first case we have mentioned above, and for the second case, the viscosity is proportional to the power s+3 2(s−1) (s ≥ 5) of the temperature (see [17]). Therefore in the case of the cut-off inverse power forces, we have γ ≥ 1 + θ for s ≥ 5, provided that the equation of state is that of ideal and polytropic gas. This shows Assumption (A3) is reasonable. Under the above Assumptions (A1)-(A3), we will study the asymptotic behavior, the decay rate and the behavior near the interfaces of the density function ρ(x, t) and


283

the expanding rate of the interfaces provided that the global weak solution to the initial boundary value problem (1.1)-(1.3) exists. To do this, we introduce first the definition of the global weak solution as follows, which is the same as the one in [32]. Definition 1.2 (Weak solution). A pair of functions (ρ(x, t), u(x, t)) is called a global weak solution to the initial boundary value problem (1.1)-(1.3), if there exist the solutions a(t), b(t) ∈ C([0, ∞)) of (1.4) and (1.5), such that for any t > 0, C(t) ≤ ρ(x, t) ≤ C,

(1.11)

ρ, u ∈ L ∞ ([a(t), b(t)] × [0, ∞)) ∩ C 1 ([0, ∞); L 2 ([a(t), b(t)])),

(1.12)

1

ρ 1+θ u x ∈ L ∞ ([a(t), b(t)] × [0, ∞)) ∩ C 2 ([0, ∞); L 2 ([a(t), b(t)])),

(1.13)

and lim (P(ρ) − µ(ρ)u x )(x, t) =

x→a(t)+

lim (P(ρ) − µ(ρ)u x )(x, t) = 0,

x→b(t)−

(1.14)

where C(t) is a positive constant dependent on t. Furthermore, the following equations hold:

∞ b(t) 0

and ∞ 0

b(t)

(ρϕt + ρuϕx )dxdt +

a(t)

b

ρ0 (x)ϕ(x, 0)dx = 0,

(1.15)

a

(ρuψt + (ρu + P(ρ) − µ(ρ)u x )ψx )dxdt + 2

a(t)

b

ρ0 (x)u 0 (x)ψ(x, 0)dx= 0,

a

(1.16) for any test functions ϕ, ψ ∈ C0∞ () with = {(x, t) : a(t) ≤ x ≤ b(t), t ≥ 0}. In what follows, we always use C (and Cn ) to denote a generic positive constant depending only on the initial data (and the given positive integer n), but independent of t. We now state the main theorems in this paper as follows: Theorem 1.3 (The asymptotic behavior of the density function). Under the Assumptions (A1)-(A3), let (ρ(x, t), u(x, t)) be a global weak solution to the free boundary value problem (1.1)-(1.3). Then we have the following asymptotic behavior on the density function ρ(x, t) lim

sup

t→∞ x∈[a(t),b(t)]

ρ(x, t) = 0.

(1.17)

Furthermore, we can get the decay rate, the behavior near the interfaces of the density function ρ(x, t) and the expanding rate of the interfaces as follows:

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C. J. Zhu

Theorem 1.4 (The decay rate, behavior near the interfaces and the expanding rate of the interfaces). Under the Assumptions (A1)–(A3), let (ρ(x, t), u(x, t)) be a global weak solution to the free boundary value problem (1.1)–(1.3). Then the following estimate holds: ρ(x, t) ≤ β ,0 0, (1 + t)λ 1 n

(1.18)

and n is defined by (1.10).

2θ β < . γ − 1 + 2θ (2γ + θ )(γ − 1 + 2θ )

Furthermore, for any x ∈ [a(t), b(t)], t > 0, ⎧ 1 1 1 ⎪ ⎨ ρ θ− 2 (x, t) − ρ θ− 2 (a(t), t) ≤ Cn |x − a(t)| 2 , ⎪ ⎩ ρ θ− 12 (x, t) − ρ θ− 12 (b(t), t) ≤ Cn |x − b(t)| 21 ,

(1.19)

(1.20)

and b(t) − a(t) ≥ Cn (1 + t)λ .

(1.21)

The rest of this paper is organized as follows. In Sect. 2, we reformulate the free boundary value problem (1.1)-(1.3) into a fixed boundary value problem by introducing the Lagrangian coordinates and restate the main theorems in the Lagrangian coordinates. In Sect. 3, we derive some crucial uniform estimates for studying the asymptotic behavior and the decay rate estimate on the density function ρ(x, t). In Sect. 4, the decay rate estimate on the density function ρ(x, t) will be given. In Sect. 5, we give the proof of the main theorems. 2. Reformulation of the Problem To solve the free boundary problem (1.1)–(1.3), it is convenient to convert the free boundaries to the fixed boundaries by using Lagrangian mass coordinates. To do this, let x y= ρ(z, t)dz, τ = t. (2.1) a(t)

b(t) Then the free boundaries x = a(t) and x = b(t) become x = 0 and x = a(t) ρ(z, t)dz =

b

b of mass, where a ρ0 (z)dz is the total mass. Without a ρ0 (z)dz by the conservation

b loss of generality, we assume a ρ0 (z)dz = 1. Hence, in the Lagrangian coordinates, the free boundary problem (1.1)–(1.3) becomes ⎧ ⎨ ρτ + ρ 2 u y = 0, (2.2) ⎩ u τ + P(ρ)y = (µ(ρ)ρu y )y , 0 < y < 1, τ > 0,


285

with initial data (ρ, u)(y, 0) = (ρ0 (y), u 0 (y)), 0 ≤ y ≤ 1,

(2.3)

and the boundary conditions (P(ρ) − µ(ρ)ρu y )(0, τ ) = (P(ρ) − µ(ρ)ρu y )(1, τ ) = 0, τ ≥ 0,

(2.4)

where P(ρ) = Aρ γ , µ(ρ) = cρ θ . Without loss of generality, we assume A = 1 and c = 1 for simplicity of presentation. Furthermore, in the Lagrangian coordinates, the assumptions corresponding to (A1)–(A3) in the Introduction are transformed into: 2 (y) ∈ L 1 ([0, 1]) and there exists a positive (B1) ρ0 (y) ∈ C 1 ([0, 1]), ρ02θ−2 (y)ρ0y constant ρ0 > 0 such that ρ0 (y) ≥ ρ0 . (B2) For any given positive integer n satisfying (1.10), assume u 0 (y) ∈ L 2n ([0, 1]). (B3) θ > 0, γ ≥ 1 + θ .

Under the above Assumptions (B1)-(B3), we will study the asymptotic behavior and the decay rate of the density function ρ(y, τ ) provided that the global weak solution to the initial boundary value problem (2.2)–(2.4) exists. In the Lagrangian coordinates, the definition of the weak solution to (2.2)–(2.4) corresponding to Definition 1.2 can be stated as follows: Definition 2.1 Weak solution. A pair of functions (ρ(y, τ ), u(y, τ )) is called a global weak solution to the initial boundary value problem (2.2)–(2.4), if C(τ ) ≤ ρ(y, τ ) ≤ C, 0 ≤ y ≤ 1, τ > 0,

(2.5)

ρ, u ∈ L ∞ ([0, 1] × [0, ∞)) ∩ C 1 ([0, ∞); L 2 (0, 1)),

(2.6)

1

ρ 1+θ u y ∈ L ∞ ([0, 1] × [0, ∞)) ∩ C 2 ([0, ∞); L 2 (0, 1)),

(2.7)

and lim (P(ρ) − ρ 1+θ u y )(y, τ ) = lim (P(ρ) − ρ 1+θ u y )(y, τ ) = 0, τ ≥ 0, (2.8) y→1−

y→0+

where C(τ ) is a positive constant dependent on τ . Furthermore, the following equations hold: ∞ 1 (ρϕτ + ρuϕy )dydτ + 0

0

1

ρ0 (y)ϕ(y, 0)dy = 0,

(2.9)

0

and 0

∞ 1 0

(uψτ + (P(ρ) − µ(ρ)ρu y )ψy )dydτ +

1

u 0 (y)ψ(y, 0)dy = 0, (2.10)

0

for any test functions ϕ, ψ ∈ C0∞ ( ) with = {(y, τ ) : 0 ≤ y ≤ 1, τ ≥ 0}.

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Remark 2.2 (Existence of the global weak solution). There exists the global weak solution satisfying Definition 1.2 (Definition 2.1). For example, we refer to Theorem 1.1 in [33]. In what follows, we also use C (and Cn ) to denote a generic positive constant depending only on the initial data (and the given positive integer n), but independent of τ . We now state the main theorems in Lagrangian coordinates as follows: Theorem 2.3 (The asymptotic behavior of the density function). Under the Assumptions (B1)-(B3), let (ρ(y, τ ), u(y, τ )) be a global weak solution to the fixed boundary value problem (2.2)-(2.4). Then we have the following asymptotic behavior of the density function ρ(y, τ ): lim

sup ρ(y, τ ) = 0.

τ →∞ y∈[0,1]

(2.11)

Furthermore, we can get the decay rate of the density function ρ(y, τ ) as follows: Theorem 2.4 (The decay rate of the density function). Under the Assumptions (B1)(B3), let (ρ(y, τ ), u(y, τ )) be a global weak solution to the fixed boundary value problem (2.2)-(2.4). Then the following decay rate estimate on the density function ρ(y, τ ) holds: ρ(y, τ ) ≤ where λ =

β γ −1+2θ ,

0 0, ρ(y, τ ) ≤ C.

(3.2)

θ ρ τ = −θρ 1+θ u y .

(3.3)

Proof. From (2.2)1 , we have

Integrating (3.3) with respect to τ over [0, τ ] yields ρ θ (y, τ ) = ρ0θ (y) − θ

τ

ρ 1+θ u y (y, s)ds.

(3.4)

0

Integrating the second equation of (2.2) with respect to y over [0, y], we have

y 0

u τ (ξ, τ )dξ + P(ρ) − P(ρ(0, τ )) + ρ 1+θ u y (0, τ ) = ρ 1+θ u y .

(3.5)

Substituting (3.5) into (3.4) and using the boundary conditions (2.4), one gets ρ θ (y, τ ) + θ

τ 0

P(ρ)(y, s)ds = ρ0θ (y) + θ

y

y

u 0 (ξ )dξ −

0

u(ξ, τ )dξ . (3.6)

0

By the Hölder inequality and Lemma 3.1, we have

y

0

u(ξ, τ )dξ ≤ C.

(3.7)

Thus

θ

τ

ρ (y, τ ) + θ

P(ρ)(y, s)ds ≤ C,

(3.8)

0

which implies (3.2). The proof of Lemma 3.2 is complete. Now we give the boundary estimates for the density function ρ(y, τ ), which will frequently be used later. Lemma 3.3 (The boundary estimates for the density function). Under the conditions of Theorem 2.3, we have for any τ > 0, ρ(d, τ ) = ρ0 (d)

γ −θ γ −θ

γ − θ + ρ0

(d)τ

1 γ −θ

,

d = 0 or 1.

(3.9)

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C. J. Zhu

Proof. By taking y = 1 in (3.5) and using the boundary conditions (2.4), we have d dτ

1

u(y, τ )dy = 0,

0

i.e.,

1

1

u(y, τ )dy =

0

u 0 (y)dy.

(3.10)

0

Hence (3.6) and (3.10) give

θ

ρ (d, τ ) + θ 0

τ

ρ γ (d, s)ds = ρ0θ (d), d = 0 or 1.

(3.11)

Since θ < γ , the differential-integral equation (3.11) immediately gives (3.9). And this completes the proof of Lemma 3.3. Lemma 3.4. Under the Assumptions (B1)-(B3), let (ρ(y, τ ), u(y, τ )) be a global weak solution to the fixed boundary value problem (2.2)-(2.4). Then for any positive integer n satisfying the condition (1.10), we have for any τ > 0,

1

τ

u 2n dy + n(2n − 1)

0

0

1

u 2n−2 ρ 1+θ u 2y dyds ≤ Cn ,

0

(3.12)

where Cn is a positive constant depending on n, but independent of τ . Proof. Multiplying the second equation of (2.2) by u 2n−1 and integrating the resulting equation over [0, 1] × [0, τ ], we have by the boundary conditions (2.4),

1

0

τ

u 2n dy + 2n(2n − 1) 0

1

= 0

1

u 2n−2 ρ 1+θ u 2y dyds

0

u 2n 0 dy + 2n(2n − 1)

τ 0

1

u 2n−2 ρ γ u y dyds.

(3.13)

0

Applying the Cauchy-Schwarz inequality to the last term in (3.13) yields

1

τ

u 2n dy + n(2n − 1)

0

0

τ

≤ C + n(2n − 1) 0

By using the Young inequality ab ≤

0

1

1


u 2n−2 ρ 2γ −1−θ dyds.

(3.14)

0

1 p 1 q 1 1 a + b , where + = 1, p, q > 1, a, b ≥ 0, p q p q

we have u 2n−2 ρ 2γ −1−θ ≤

1 n(γ −1−θ)+γ n − 1 γ 2n ρ ρ u . + n n

(3.15)


Substituting (3.15) into (3.14), we get 1 τ 1 2n u dy + n(2n − 1) u 2n−2 ρ 1+θ u 2y dyds 0 0 0 τ 1 ≤ C + (2n − 1) ρ n(γ −1−θ)+γ dyds + (n − 1)(2n − 1) 0

0

τ 0

289

1

ρ γ u 2n dyds,

0

(3.16) which implies by (3.2), 1 u 2n dy + n(2n − 1) 0

≤ C + (2n − 1)C

τ 0

τ

1

0


γ

max ρ ds + (n − 1)(2n − 1)

0 y∈[0,1]

τ

max ρ

0 y∈[0,1]

γ

1

u 2n dyds.

0

(3.17) Then the Gronwall inequality and (3.8) yield 1 u 2n dy ≤ Cn ,

(3.18)

0

where Cn is a positive constant depending on n, but independent of τ . Equations (3.17), (3.18) and (3.8) together imply (3.12) and this completes the proof of Lemma 3.4. The uniform estimate on the derivative of the density function ρ(y, τ ) in the following lemma is quite similar to the new mathematical entropy inequality (1.9) as we have mentioned in the Introduction. It will play a very important role in studying the asymptotic behavior and the decay rate on the density function ρ(y, τ ). Lemma 3.5 (A crucial estimate). Under the conditions of Theorem 2.3, we have the following uniform estimate on the derivative of the density function ρ(y, τ ): 1 τ 1 ρ 2θ−2 ρy2 dy + ρ γ +θ−2 ρy2 dyds ≤ Cn , 0 < τ < ∞. (3.19) 0

0

0

Proof. From (3.3) and (2.2), we have

θ ρ yτ = −θ u τ + (ρ γ )y .

(3.20)

Multiplying (3.20) by (ρ θ )y and integrating it over [0, 1] × [0, τ ], we have τ 1 2

1 1 2θ−2 2 1 1 2θ−2 (ρ0 )y dy − θ ρ ρy dy = ρ0 u τ ρ θ y dyds 2 0 2 0 0 0 τ 1 2 γ +θ−2 2 −θ γ ρ ρy dyds 1 1 1 0 0 2

1 = ρ02θ−2 (ρ0 )y dy − θ u ρ θ y dy + θ u 0 ρ0θ y dy 2 0 0 0 τ 1 τ 1 θ 2 γ +θ−2 2 +θ u ρ yτ dyds − θ γ ρ ρy dyds. (3.21) 0

0

0

0

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C. J. Zhu

Substituting (3.20) into (3.21), we have τ 1 1 1 2θ−2 2 ρ ρy dy + θ 2 γ ρ γ +θ−2 ρy2 dyds 2 0 0 0 1 1 2 1 1 2θ−2 = (ρ0 )y dy − θ 2 ρ0 uρ θ−1 ρy dy + θ 2 u 0 ρ0θ−1 (ρ0 )y dy 2 0 0 0 τ 1 τ 1 −θ 2 uu τ dyds − θ 2 (ρ γ )y udyds 0 0 1 0 0 1 2 1 1 2θ−2 = (ρ0 )y dy − θ 2 ρ0 uρ θ−1 ρy dy + θ 2 u 0 ρ0θ−1 (ρ0 )y dy 2 0 0 0 τ 1 1 2 1 2 1 2 1 2 2 γ − θ u dy + θ u 0 dy + θ ρ u y dyds 2 0 2 0 0 0 τ −θ 2 ρ γ (1, s)u(1, s) − ρ γ (0, s)u(0, s) ds =

7

0

Ji .

(3.22)

i=1

Now we estimate J1 -J7 as follows: First, by the Assumptions (B1)–(B3), Lemma 3.1, Lemma 3.2, (3.8) and the Cauchy-Schwarz inequality, we have ⎧ J1 ≤ C, ⎪ 1 ⎪ ⎪ 1 1 2θ−2 2 1 1 2θ−2 2 ⎪ 2 ⎪ ⎪ J ≤ ρ ρ dy + C u dy ≤ C + ρ ρy dy, ⎪ y ⎪ 2 4 0 4 0 ⎪ 0 ⎪ 1 1 ⎪ ⎪ 2θ−2 ⎪ 2 ⎪ J ≤ C ρ [(ρ ) ] dy + C u 20 dy ≤ C, 3 0 y ⎪ 0 ⎨ 0 0 (3.23) J4 ≤ C, ⎪ ⎪ J5 ≤ C, ⎪ ⎪ τ 1 τ 1 ⎪ ⎪ ⎪ 1+θ 2 ⎪ ⎪ J ≤ C ρ u dyds + C ρ 2γ −1−θ dyds 6 ⎪ y ⎪ ⎪ 0 0 0 0 τ ⎪ ⎪ ⎪ ⎩ ≤ C + C max ρ γ −1−θ ρ γ (y, s)ds ≤ C. 0

Secondly, by the Young inequality and Lemma 3.3, we have τ γ −θ

2 J7 = −θ ρ (1, s) ρ θ (1, s)u(1, s) − ρ γ −θ (0, s) ρ θ (0, s)u(0, s) ds τ0 nθ ρ (1, s)u n (1, s) + ρ nθ (0, s)u n (0, s) ds ≤C 0 τ n n +C ρ (γ −θ) n−1 (1, s) + ρ (γ −θ) n−1 (0, s) ds 0 τ nθ ρ (·, s)u n (·, s) ∞ ≤ Cn + C ds. (3.24) L ([0,1]) 0

Substituting (3.23) and (3.24) into (3.22), we have τ 1 1 ρ 2θ−2 ρy2 dy + ρ γ +θ−2 ρy2 dyds ≤ Cn + C J8 , 0

0

0

(3.25)


291

where

τ

J8 =

nθ ρ (·, s)u n (·, s)

0

L ∞ ([0,1])

ds.

(3.26)

By the embedding theorem W 1,1 ([0, 1]) → L ∞ ([0, 1]), we have τ 1 nθ n (ρ nθ u n )(y, s) dyds + ρ u y (y, s) dyds 0 0 τ 01 0 τ 1 τ 1 ≤C ρ γ u 2n dyds + C ρ 2nθ−γ dyds + C ρ 2nθ−γ −θ u 2n dyds 0 0 0 0 0 0 τ 1 1 τ 1 γ +θ−2 2 1+θ 2n−2 2 + ρ ρy dyds + C ρ u u y dyds 2 0 0 0 0 τ 1 +C ρ 2nθ−1−θ dyds 0 0 1 τ 1 γ +θ−2 2 ≤ Cn + ρ ρy dyds. (3.27) 2 0 0

J8 ≤

τ

1

Here we have used Lemma 3.2, Lemma 3.4, (3.8) and n ≥ 2γ2θ+θ . Substituting (3.27) into (3.25), we get (3.19). This proves Lemma 3.5. By Lemma 3.5, (3.26) and (3.27), we can get the following result: Corollary 3.6. Under the conditions of Theorem 2.3, we have for any τ > 0, 0

τ

||ρ nθ (·, s)u n (·, s)|| L ∞ ([0,1]) ds ≤ Cn .

3.2. Asymptotic behavior of the density function. To apply the uniform estimates obtained above to study the asymptotic behavior of the density function ρ(y, τ ) with respect to the time τ , we introduce the following lemma. The proof is quite simple and the detail is omitted. Lemma 3.7. Suppose that g(τ ) ≥ 0 for τ ≥ 0, g(τ ) ∈ L 1 (0, ∞) and g (τ ) ∈ L 1 (0, ∞), then lim g(τ ) = 0. τ →∞

Now we prove Theorem 2.3. Let g(τ ) =

1

ρ γ (y, τ )dy.

(3.28)

0

Integrating (3.8) with respect to y over [0, 1], we have

τ 0

which implies g(τ ) ∈ L 1 (0, ∞).

0

1

ρ γ (y, τ )dydτ ≤ C,

(3.29)

292

C. J. Zhu

Now we prove g (τ ) ∈ L 1 (0, ∞). By the second equation of (2.2) and using Cauchy-Schwarz inequality, we obtain ∞ ∞ 1 |g (τ )|dτ = γ ρ γ −1 |ρτ |dydτ 0 0 ∞ 0 1 =γ ρ γ +1 |u y |dydτ 0 0 ∞ 1 ∞ 1 ≤C ρ 1+θ u 2y dydτ + C ρ 2γ +1−θ dydτ ≤ C.

0

0

0

0

(3.30)

Here we have used Lemma 3.1, Lemma 3.2, (3.29) and γ ≥ 1 + θ . Consequently, lim g(τ ) = 0.

τ →∞

By (3.31), Lemma 3.2 or the Hölder inequality, we have 1 ρ α (y, τ )dy = 0 lim τ →∞ 0

(3.31)

(3.32)

for any 0 < α < ∞. Choosing m > θ > 0 and applying the Hölder inequality, Lemma 3.3 and Lemma 3.5, we have y m m 0 ≤ ρ (y, τ ) = ρ (0, τ ) + (ρ m )ξ dξ 0 1 21 1 21 m − γm 2m−2θ 2θ−2 2 −θ + ρ dy ρ ρy dy ≤ C(1 + τ ) θ 0 0 1 1 2 − m ≤ C(1 + τ ) γ −θ + Cn ρ 2m−2θ dy , (3.33) 0

which implies by (3.9) and (3.32), ρ(y, τ ) → 0,

as τ → ∞.

This proves Theorem 2.3. 4. Decay Rate of Density Function In this section, we will study the decay rate of the density function ρ(y, τ ) with respect to the time τ . To do this, introduce a new function w(y, τ ) defined as follows (cf. [9,27]) y 1 y 1 1 1 1 w(y, τ ) = u(y, τ ) − dξ + dξ dy. (4.1) 1 + τ 0 ρ(ξ, τ ) 1 + τ 0 0 ρ(ξ, τ ) By direct calculation, we have wy = u y −

1 , (1 + τ )ρ

(4.2)


293

and wτ + Here we have used the fact that 1

w = uτ . 1+τ

1

u(y, τ )dy =

0

(4.3)

u 0 (y)dy,

0

1 (see (3.10)). Assume now 0 u 0 (y)dy = 0 for the simplicity of presentation. Thus Eqs. (2.2) can be rewritten in terms of variables (ρ, w) in the form ⎧ ρ ⎪ ρτ + ρ 2 wy + = 0, ⎪ ⎪ ⎨ 1+τ w ρθ ⎪ 1+θ γ ⎪ ⎪ . ⎩ wτ + 1 + τ = ρ wy + 1 + τ − ρ y

(4.4)

Then we have Lemma 4.1. Let (ρ(y, τ ), u(y, τ )) be a global weak solution to the fixed boundary value problem (2.2)–(2.4). Then for any θ > 0, γ ≥ 1 + θ and any positive integer n satisfying the condition (1.10), there exists a positive constant β satisfying 0 1. τ 1 1 1 (1 + τ )β 1 γ −1 β β 2 β−1 (1 + τ ) w dy + ρ dy + 1 − (1 + s) w 2 dyds 2 γ −1 0 2 0 0 0 1 τ 1 γ −1−β τ + (1 + s)β ρ 1+θ wy2 dyds + (1 + s)β−1 ρ γ −1 dyds γ −1 0 0 0 0 (4.8) ≤ Cn .

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C. J. Zhu

Consequently, ρ(y, τ ) ≤

Cn , (1 + τ )λ

λ=

2θ β < . γ − 1 + 2θ (2γ + θ )(γ − 1 + 2θ )

(4.9)

Proof. Multiplying (4.4)2 by w, integrating the resulting equation with respect to y over [0, 1], using integration by parts, we obtain by the boundary conditions (2.4) 1 1 1 d 1 w 2 dy + w 2 dy 2 dτ 0 1+τ 0 1 1 1 1 1+θ θ (ρ wy )y wdy + (ρ )y wdy − (ρ γ )y wdy = 1+τ 0 0 0 1 1 1 1 ρ 1+θ wy2 dy − ρ θ wy dy + ρ γ wy dy =− 1+τ 0 0 0 1 θ ρ (1, τ )w(1, τ ) − ρ θ (0, τ )w(0, τ ) , + 1+τ

(4.10)

i.e., 1 d 2 dτ +

1

w 2 dy +

0 1

1 1+τ

ρ γ wy dy +

0

1

1

w 2 dy +

0

0

ρ 1+θ wy2 dy = −

1 1+τ

1

ρ θ wy dy

0

1 θ ρ (1, τ )w(1, τ ) − ρ θ (0, τ )w(0, τ ) . 1+τ

(4.11)

Now we will prove (4.6)–(4.8). Case I. 0 < θ < 1 (The proof of (4.6)). By (2.2)1 and (4.2), we have 1 1 1 = . − wy = u y − (1 + τ )ρ ρ τ (1 + τ )ρ

(4.12)

Thus we can estimate the first and second terms on the right-hand side in (4.11) as follows: 1 1 1 1 1 1 θ θ − dy ρ wy dy = − ρ − 1+τ 0 1+τ 0 ρ τ (1 + τ )ρ 1 1 1 1 = (ρ θ−1 )τ dy + ρ θ−1 dy, (θ − 1)(1 + τ ) 0 (1 + τ )2 0 (4.13) and

1 0

1 1 dy ρ wy dy = ρ − ρ τ (1 + τ )ρ 0 1 1 1 1 =− (ρ γ −1 )τ dy − ρ γ −1 dy. γ −1 0 1+τ 0 γ

1

γ

(4.14)


295

Substituting (4.13) and (4.14) into (4.11), we get 1 1

1 1 1 1 ρ γ −1 1 dy + w 2 dy + ρ 1+θ wy2 dy + ρ γ −1 dy 2 γ − 1 1+τ 1 + τ 0 0 0 0 1 1 1 1 (ρ θ−1 )τ dy + ρ θ−1 dy = (θ − 1)(1 + τ ) 0 (1 + τ )2 0 1 θ ρ (1, τ )u(1, τ ) − ρ θ (0, τ )u(0, τ ) . (4.15) + 1+τ d dτ

w2 +

Here we have used the fact that w(1, τ ) = u(1, τ ) and w(0, τ ) = u(0, τ ) due to (4.1). Multiplying (4.15) by (1 + τ )β , where β is defined by (4.5), we deduce that

1 1 (1 + τ )β−1 1 θ−1 (1 + τ )β 1 γ −1 (1 + τ )β w 2 dy + ρ dy + ρ dy 2 1−θ γ −1 0 0 0 1 1 β (1 + τ )β−1 w 2 dy + (1 + τ )β ρ 1+θ wy2 dy + 1− 2 0 0 1 1 γ −1−β θ −β (1 + τ )β−2 (1 + τ )β−1 ρ θ−1 dy + ρ γ −1 dy + 1−θ γ − 1 0 0

= (1 + τ )β−1 ρ θ (1, τ )u(1, τ ) − ρ θ (0, τ )u(0, τ ) . (4.16)

d dτ

By the Young inequality, Corollary 3.6 and (4.5), we have

τ

(1 + s)β−1 ρ θ (1, s)u(1, s) − ρ θ (0, s)u(0, s) ds 0 τ τ nθ n (β−1) n−1 |ρ (1, s)u n (1, s)| + |ρ nθ (0, s)u n (0, s)| ds ≤C (1 + s) ds + C 0 0 τ ≤ Cn + C ||ρ nθ (·, s)u n (·, s)|| L ∞ ([0,1]) ds ≤ Cn .

0

(4.17)

By using (1.10), (4.5) and (B3), we have 1−

β > 0, θ − β > 0, γ − 1 − β ≥ 0. 2

(4.18)

Integrating (4.16) with respect to τ over [0, τ ] and using (4.17) and (4.18), we deduce (4.6). Case II. θ = 1 (The proof of (4.7)). Under this case, the first term on the right-hand side in (4.11) can be rewritten as 1 − 1+τ

1 0

1 1 1 1 dy ρ wy dy = − ρ − 1 + τ 0 ρ τ (1 + τ )ρ 1 1 1 = (ln ρ)τ dy + . (1 + τ ) 0 (1 + τ )2 θ

(4.19)

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C. J. Zhu

Thus when θ = 1, similar to (4.16), we have

1 1 (1 + τ )β 1 γ −1 β 2 (1 + τ ) w dy + ρ dy 2 γ −1 0 0 1 β (1 + τ )β−1 w 2 dy + (1 + τ )β + 1− 2 0 1 1 γ −1−β (1 + τ )β−1 × ρwy2 dy + ρ γ −1 dy γ − 1 0 0 1 1 d (1 + τ )β−1 ln ρdy + (1 − β)(1 + τ )β−2 ln ρdy + (1 + τ )β−2 = dτ 0 0

d dτ

+(1 + τ )β−1 (ρ(1, τ )u(1, τ ) − ρ(0, τ )u(0, τ )).

(4.20)

By using ln x < x − 1 for any x > 0 and Lemma 3.2, we have

1

1

ln ρdy ≤

0

ρdy ≤ C.

0

Integrating (4.20) with respect to τ over [0, τ ], we deduce (4.7). Case III. θ > 1 (The proof of (4.8)). Rewrite (4.16) as

1 1 (1 + τ )β 1 γ −1 β 2 (1 + τ ) w dy + ρ dy 2 γ −1 0 0 1 β (1 + τ )β−1 w 2 dy + (1 + τ )β + 1− 2 0 1 1 γ − 1−β (1 + τ )β−1 × ρ 1+θ wy2 dy + ρ γ −1 dy γ −1 0 0 1 θ −β d (1 + τ )β−1 1 θ−1 β−2 (1 + τ ) ρ dy + ρ θ−1 dy = dτ θ −1 θ − 1 0 0

+ (1 + τ )β−1 ρ θ (1, τ )u(1, τ ) − ρ θ (0, τ )u(0, τ ) .

d dτ

(4.21)

By Lemma 3.2 and θ > 1, we have

1

ρ θ−1 dy ≤ C.

0

Integrating (4.21) with respect to τ over [0, τ ], we deduce (4.8). Now we turn to prove (4.9). By (4.6)-(4.8), we have for any θ > 0 and τ > 0, (1 + τ )β

1 0

ρ γ −1 (y, τ )dy ≤ Cn .

(4.22)


297

Choosing k with 2k = γ − 1 + 2θ and using Lemma 3.3, Lemma 3.5 and (4.22), we have y (ρ k )ξ (ξ, τ )dξ ρ k (y, τ ) = ρ k (0, τ ) + 0 1 21 1 21 k − γ −θ 2θ−2 2 2k−2θ +C ρ ρy dy ρ dy ≤ C(1 + τ ) ≤ C(1 + τ )

k − γ −θ β

0

0

β

+ Cn (1 + τ )− 2

≤ Cn (1 + τ )− 2 ,

(4.23)

which implies β

ρ(y, τ ) ≤ Cn (1 + τ )− 2k .

(4.24)

Here we have used β
0; we x ,σ ± ± ± and ± shall call x0 the time variable. We shall write x,σ,1 = ax,σ x,σ,2 = bx+,σ , with = (0, δ1 ). We define

Snβ, (x1 , ε1 , σ1 , ρ1 ; . . . ; xn , εn , σn , ρn ) = T{xε11,σ1 ,ρ1 · · · xεnn ,σn ,ρn } β, , (2.4) where xi ∈ [0, β] × , σi =↑↓, εi = ±, ρi = 1, 2 and T is the operator of fermionic time ordering, acting on a product of fermionic fields as: ε

ε

π(1) π(n) T(xε11,σ1 ,ρ1 · · · xεnn ,σn ,ρn ) = (−1)π xπ(1) ,σπ(1) ,ρπ(1) · · · xπ(n) ,σπ(n) ,ρπ(n) ,

(2.5)

where π is a permutation of {1, . . . , n}, chosen in such a way that xπ(1)0 ≥ · · · ≥ xπ(n)0 , and (−1)π is its sign. [If some of the time coordinates are equal to each other, the arbitrariness of the definition is solved by ordering each set of operators with the same time coordinate so that creation operators precede the annihilation operators.] Taking the limit → ∞ in (2.4) we get the finite temperature n-point Schwinger β functions, denoted by Sn (x1 , ε1 , σ1 , ρ1 ; . . . ; xn , εn , σn , ρn ), which describe the properties of the infinite volume system at finite temperature. Taking the β → ∞ limit

304

A. Giuliani, V. Mastropietro

of the finite temperature Schwinger functions, we get the zero temperature Schwinger functions, simply denoted by Sn (x1 , ε1 , σ1 , ρ1 ; . . . ; xn , εn , σn , ρn ), which by definition characterize the properties of the thermal ground state of (2.1) in the thermodynamic limit. 2.2. The non interacting case. In the non–interacting case U = 0 the Schwinger functions of any order n can be exactly computed as linear combinations of products of two–point Schwinger functions, via the well–known Wick rule. The two–point Schwin is ger function itself, also called the free propagator, for x = y and x − y = (±β, 0), equal to (see Appendix A for details): β, β, S0 (x − y)ρ,ρ ≡ S2 (x, σ, −, ρ; y, σ, +, ρ ) U =0 e−ik·(x−y) ik −v ∗ (k) 1 0 = , (2.6) 2 −v(k) ik0 β|| k02 + |v(k)| ρ,ρ k∈D β,L

where and Dβ,L = Dβ × D L ; 1. k = (k0 , k) 1 {k = nL1 b1 + nL2 b2 : 0 ≤ n 1 , n 2 ≤ 2. Dβ = {k0 = 2π β (n 0 + 2 ) :√n 0 ∈ Z} and D L = √ 2π 3) are a basis of the dual lattice ∗ ; L − 1}, where b1 = 2π 3 (1, 3), b2 = 3 (1, − √ 3 δi −δ1 ) is the disper = i=1 ei k( = 1 + 2e−i3/2k1 cos 23 k2 ; its modulus |v(k)| 3. v(k) sion relation, given by

2 √ √ 1 + 2 cos(3k1 /2) cos( 3k2 /2) + 4 sin2 (3k1 /2) cos2 ( 3k2 /2). (2.7) |vk | = the free propagator has a jump discontinuity, At x = y or x − y = (±β, 0), β, β, see Appendix A. Note that S0 (x) is antiperiodic in x0 , i.e. S0 (x0 + β, x ) = β, β, −S0 (x0 , x ), and that its Fourier transform Sˆ0 (k) is well–defined for any k ∈ Dβ,L , even in the thermodynamic limit L → ∞, since |k0 | ≥ πβ . We shall refer to this last property by saying that the inverse temperature β acts as an infrared cutoff for our theory. If we take β, L → ∞, the limiting propagator Sˆ0 (k) becomes singular at {k0 = 0} × {k = p± F }, where 2π 2π (2.8) ,± √ p F± = 3 3 3 are the Fermi points (also called Dirac points, for an analogy with massive QED2+1 close to the that will become clearer below). Note that the asymptotic behavior of v(k) ± 3 Fermi points is given by v( p F + k ) 2 (ik1 ± k2 ). In particular, if ω = ±, the Fourier transform of the 2-point Schwinger function close to the Fermi point pωF can be rewritten in the form: −1 (0) + ωk ) + r (k ) 1 −v (−ik −ik 0 ω ω 1 2 F , Sˆ0 (k0 , p F + k ) = ∗ Z 0 −v (0) −ik0 F (ik1 + ωk2 ) + rω (k ) (2.9)

Two-Dimensional Hubbard Model on the Honeycomb Lattice

305 (0)

where Z 0 = 1 is the free wave function renormalization and v F = 3/2 is the free 2 k | , for small values of k and for some positive Fermi velocity. Moreover, |rω (k )| ≤ C constant C. 2.3. The interacting case. We are now interested in what happens by adding a local interaction. In the case U = 0, the Schwinger functions are not exactly computable anymore. It is well–known that they can be written as formal power series in U , constructed in terms of Feynmann diagrams, using as the free propagator the function S0 (x) in (2.6). Our main result consists in a proof of convergence of this formal expansion for U small enough, after the implementation of suitable resummations of the original power series. Our main result can be described as follows. Theorem 1. Let us consider the 2D Hubbard model on the honeycomb lattice at half filling, defined by (2.1). There exist a constant U0 > 0 such that, if |U | ≤ U0 , the specific free energy f β (U ) and the finite temperature Schwinger functions are analytic functions of U , uniformly in β as β → ∞, and so are the specific ground state energy e(U ) and the zero temperature Schwinger functions. The Fourier transform of the zero temperature de f ˆ two point Schwinger function S(x)ρ,ρ = S2 (x, σ, −, ρ; 0, σ, +, ρ ), denoted by S(k), ± ± is singular only at the Fermi points k = p F = (0, p F ), see (2.8), and, close to the singularities, if ω = ±, it can be written as −1

−ik0 −v F (−ik1 + ωk2 ) ˆS(k0 , p Fω + k ) = 1 1 + R(k ) , + ωk ) −v (ik −ik F 0 Z 1 2 (2.10) with k = (k0 , k ), and with Z and v F two real constants such that Z = 1 + O(U 2 ),

vF =

3 + O(U 2 ). 2

(2.11)

Moreover the matrix R(k ) satisfies ||R(k )|| ≤ C|k |ϑ for some constants C, ϑ > 0 and for |k | small enough. Remarks. 1) Theorem 1 says that the location of the singularity does not change in the presence of interaction; on the contrary, the wave function renormalization and Fermi velocity are modified by the interaction. Note also that, in the presence of the interaction, the Fermi velocity remains the same in the two coordinate direction even though the model does not display 90o discrete rotational symmetry, but rather a 120o rotational symmetry. 2) The resulting theory is not quasi-free: the Wick rule is not valid anymore in the presence of interactions. However, the long distance asymptotics of the higher order Schwinger functions can be estimated by the same methods used to prove Theorem 1, and it is the same suggested by the Wick rule. 3) The fact that the interacting correlations decay as in the non-interacting case implies in particular the absence of magnetic long range order in the thermal ground state of the system at weak coupling (we recall that the thermal ground state is by definition the weak limit as β, || → ∞ of the grandcanonical Gibbs state e−β H , see the end of Sec.2.1). In fact, as a corollary of our construction, we find: 1 lim Sx · Sy ≤ C , (2.12) β,||→∞ β, | x − y|4

306


where, if x ∈ , the spin operator Sx is defined as: Sx = ax+ ,· σ ax− , with σi , + ,· − i = 1, 2, 3, the Pauli matrices; similarly, if x ∈ + δ1 , Sx = σ bx ,· σ bx ,· . Note that it is known, at least in the microcanonical ensemble [15], that the ground state is unique and it has zero total spin; however, so far, existence of Néel order in the ground state was neither proven nor ruled out. 4) Similarly to what was remarked in the previous item, one can exclude the existence of superconducting long range order in the thermal ground state: the Cooper pairs correlations decay to zero at infinity at least as fast as the spin-spin correlations in (2.12). 5) Our analysis can be extended in a straightforward way to the case of exponentially decaying interactions (instead of local interactions). However, if the decay is slower, the result may change. In particular, in the presence of 3D Coulomb interactions, the electron-electron interaction becomes marginal (instead of irrelevant), in a renormalization group sense [13]. 6) Previous analyses of the Hubbard model on the honeycomb lattice were performed only at a perturbative level, without any control on the convergence of the weak coupling expansion, and directly in the Quantum Field Theory approximation, consisting in the replacement of Sˆ0 (k) by its linear approximation around the Fermi points, see for instance [12] and references therein. The proof of the theorem is based on constructive fermionic Renormalization Group (RG) methods, see [2,17,21] for extensive reviews. It is worth remarking that the result summarized in Theorem 1 is one of the few rigorous constructions of the ground state properties (including correlations) of a weak coupling 2D Hubbard model. The only other example we are aware of is the Fermi liquid construction in [8], applicable to cases of weakly interacting 2D Fermi systems with a highly asymmetric interacting Fermi surface. Related results include the construction of the state at temperatures larger than a BCS-like critical temperature [4,7], or the computation of the first contribution to the ground state energy in a weak coupling limit [11,16,22]. The rest of the paper will be devoted to the proof of Theorem 1. In Sec. 3.1 we review the Grassmann integral representation for the free energy and the Schwinger functions. In Sec.3.2 we start to describe the integration procedure leading to the computation of the free energy, and in particular we describe how to integrate out the ultraviolet degrees of freedom. In Sec.3.3 we complete the proof of convergence of the series for the free energy and the ground state energy. In Sec.3.4 we describe the proof of convergence for the series for the Schwinger functions, with particular emphasis on the case of the two-point Schwinger function. In the Appendices we provide further details concerning the non-interacting theory, the ultraviolet integration, the thermodynamic and zero temperature limits. 3. Renormalization Group Analysis 3.1. Grassmann integration. In this subsection, for any β and L finite, we rewrite the partition function and the Schwinger functions of model (2.1) in terms of Grassmann functional integrals, defined as follows. Let M ∈ N and χ0 (t) be a smooth compact support function that is 1 for t ≤ a0 and 0 for t ≥ a0 γ , with γ > 1 and a0 a constant to be chosen below, see the lines ∗ preceding (3.29). Let Dβ,L = Dβ,L ∩ {k0 : χ0 (γ −M |k0 |) > 0}, with Dβ,L defined after (2.6). We consider the finite Grassmann algebra generated by the Grassmannian variables

Two-Dimensional Hubbard Model on the Honeycomb Lattice σ =↑↓, ρ=1,2

ˆ± } { k,σ,ρ k∈D ∗

β,L

and a Grassmann integration

307

∗ k∈Dβ,L

ρ=1,2 σ =↑↓

ˆ + d ˆ− d k,σ,ρ k,σ,ρ

defined as the linear operator on the Grassmann algebra such that, given a monomial ˆ −, ˆ + ) in the variables ˆ ± , its action on Q( ˆ −, ˆ + ) is 0 except in the case Q( k,σ,ρ ρ=1,2 − + − + ˆ ˆ ˆ , ˆ ) = k∈D∗ Q( , up to a permutation of the variables. In β,L

σ =↑↓

k,σ,ρ

k,σ,ρ

this case the value of the integral is determined, by using the anticommuting properties of the variables, by the condition ⎤ ⎡ ρ=1,2 ρ=1,2 ⎥ ˆ− ˆ+ ⎢ − + ˆ ˆ k,σ,ρ k,σ,ρ = 1. d d (3.13) ⎣ k,σ,ρ k,σ,ρ ⎦ ∗ σ =↑↓ k∈Dβ,L

∗ σ =↑↓ k∈Dβ,L

Defining the free propagator matrix gˆk as −1 −ik0 −v ∗ (k) gˆk = χ0 (γ −M |k0 |) −ik0 −v(k)

(3.14)

and the “Gaussian integration” P(d) as ⎤ ⎡ σ =↑↓ 2 2 −M 2 −β || [χ0 (γ |k0 |)] ⎢ + + ˆ k,σ,1 ˆ − d ˆ k,σ,2 ˆ− ⎥ d d d P(d) = ⎣ k,σ,1 k,σ,2 ⎦ 2 2 k0 + |v(k)| ∗ k∈Dβ,L ⎧ ⎫ ⎪ ⎪ σ =↑↓ ⎨ ⎬ + ˆ k,σ,· ˆ− · exp −(β||)−1 gˆk−1 (3.15) k,σ,· ⎪ , ⎪ ⎩ ⎭ k∈D ∗ β,L

it turns out that

" # ˆ− ˆ+ P(d) k1 ,σ1 ,ρ1 k2 ,σ2 ,ρ2 = β||δσ1 ,σ2 δk1 ,k2 gˆ k1 ρ

1 ,ρ2

,

(3.16)

so that, if x − y ∈ β Z × {0},

1 − + e−ik(x−y) gˆk = lim y,σ = S0 (x − y), P(d)x,σ M→∞ β|| M→∞ ∗ lim

(3.17)

k∈Dβ,L

± are defined by where S0 (x − y) was defined in (2.5) and the Grassmann fields x,σ ± x,σ,ρ =

1 ˆ± , e±ikx k,σ,ρ β|| ∗

x ∈ (−β/2, β/2] × .

(3.18)

k∈Dβ,L

Let us now consider the function on the Grassmann algebra, − − + + x,↑,ρ x,↓,ρ x,↓,ρ V () = U dx x,↑,ρ ρ=1,2

=

U (β||)3

ρ=1,2

k,k ,p

+ ˆ k−p,↑,ρ ˆ− ˆ+ ˆ− k,↑,ρ k +p,↓,ρ k ,↓,ρ ,

(3.19)

308


where, in the first line, the symbol dx must be interpreted as β/2 dx = dx0 , −β/2

(3.20)

x ∈

∗ , while the sums over p and, in the second line, the sums over k, k run over the set Dβ,L ∗ : p is the run over the set 2πβ −1 Z × D L (with the constraint that k − p, k + p ∈ Dβ,L transferred momentum). We introduce the following Grassman integrals: e−β||FM,β,L = P(d)e−V () , (3.21) P(d)e−V () xε11,σ1 ,ρ1 · · · xεnn ,σn ,ρn SnM,β, (x1 , σ1 , ε1 , ρ1 ; . . . ; xn , σn , εn , ρn ) = . P(d)e−V () (3.22)

Note that these Grassmann integrals are well defined for any U ; they are indeed polynomials in U , of degree depending on M and L. It is well known that the Grassmann integrals in (3.21) and (3.22) can be used to compute the thermodynamic properties of the model with Hamiltonian (2.1), as ensured by the following proposition: Proposition 1. For any β, L < +∞, assume that there exists U0 independent of β and L M,β, are analytic in the complex domain |U | ≤ U0 , uniformly such that FM,β,L and Sn convergent as M → ∞. Then, if |U | ≤ U0 , 2 1 log Tr{e−β H } = − log 2 + 2 cosh(β|v(k)|) + lim FM,β,L , − M→∞ β|| β|| DL k∈

(3.23) and the Schwinger functions at distinct space-time points, defined in (2.4), can be computed as Snβ, (x1 , σ1 , ε1 , ρ1 ; . . . ; xn , σn , εn , ρn ) = lim SnM,β, (x1 , σ1 , ε1 , ρ1 ; . . . ; xn , σn , εn , ρn ). M→∞

(3.24)

For completeness, the proof of Proposition 1 is reported in Appendix B; its result guarantees that the thermodynamic properties of the model with Hamiltonian (2.1) can be inferred from the analysis of the Grassmann integrals (3.21) and (3.22), provided that the latter satisfy the uniform analyticity properties assumed in Proposition 1. The rest of the paper is devoted to the study of the Grassmann integrals (3.21) and (3.22); our analysis implies, in particular, the uniform analyticity properties assumed in Proposition 1, see Corollary 1 and Sect. 3.4 below. It is important to note that both the Gaussian integration P(d) and the interaction V () are invariant under the action of a number of remarkable symmetry transformations, which will be preserved by the subsequent iterative integration procedure and will guarantee the vanishing of some running coupling constants (see below for details). Let us collect in the following lemma all the symmetry properties we will need in the following:


309

Lemma 1. For any choice of M, β, , both the quadratic Grassmann measure P(d) defined in (3.15) and the quartic Grassmann interaction V () defined in (3.19) are invariant under the following transformations: ˆε ˆε (1) spin exchange: → k,σ,ρ ← k,−σ,ρ ; ε iεα ˆ ε , with ασ ∈ R independent of k; ˆ → e σ (2) global U (1): k,σ,ρ k,σ,ρ ˆε ˆε cos θ sin θ k,↑,ρ k,↑,ρ and θ ∈ T (3) spin S O(2): ˆ ε → Rθ ˆ ε , with Rθ = − sin θ cos θ k,↓,ρ

k,↓,ρ

independent of k; ˆ± (4) discrete spatial rotations:

(k0 ,k),σ,ρ

→

ˆ± e∓i k(δ3 −δ1 )(ρ−1)

, (k0 ,T1 k),σ,ρ

with

de f

T1 x = R2π/3 x ; note that in real space this transformation simply reads ± ± ± ± a(x x),σ → a(x0 ,T1 x ),σ and b(x0 , x),σ → b(x0 ,T1 x ),σ ; 0 , ± ± ˆ ˆ (5) complex conjugation: k,σ,ρ → −k,σ,ρ , c → c∗ , where c is a generic constant appearing in P(d) and/or in V (); ˆ± ˆ± (6.a) horizontal reflections: → (k0 ,k1 ,k2 ),σ,1 ← (k0 ,−k1 ,k2 ),σ,2 ; ± ± ˆ ˆ → ; (6.b) vertical reflections: (k0 ,k1 ,k2 ),σ,ρ (k0 ,k1 ,−k2 ),σ,ρ ˆ∓ → i . (k0 ,k),σ,ρ (k0 ,−k),σ,ρ ± ± ρ ˆ ˆ → i(−1) . (k0 ,k),σ,ρ (−k0 ,k),σ,ρ

ˆ± (7) particle-hole: (8) inversion:

Proof. A moment’s thought shows that the invariance of V () under the above symmetries is obvious, and so is the invariance of P(d) under (1)-(2)-(3). Let us then prove the invariance of P(d) under (4)-(5)-(6.a)-(6.b)-(7)-(8). More precisely, let us consider the term + ˆ k,σ,· ˆ− = gˆk−1 k,σ,· k

−i

+ ˆ k,σ,1 ˆ− − k0 k,σ,1

k

−i

+ ˆ k,σ,1 ˆ− − v ∗ (k) k,σ,2

k

+ ˆ k,σ,2 ˆ− v(k) k,σ,1

k

+ ˆ k,σ,2 ˆ− k0 k,σ,2

(3.25)

k

in (3.15), and let us prove its invariance under the transformations (4)-(5)-(6.a)-(6.b)(7)-(8). Under the transformation (4), the first and fourth term in the second line of (3.25) are obviously invariant, while the sum of the second and third is changed into δ3 −δ1 ) ˆ − +i k( ˆ+ ˆ+ ˆ− − v ∗ (k)e + e−i k(δ3 −δ1 ) v(k) k

=−

(k0 ,T1 k),σ,1

(k0 ,T1 k),σ,2

(k0 ,T1 k),σ,2

(k0 ,T1 k),σ,1

δ1 −δ2 ) ˆ − + + +i k( ˆ k,σ,1 ˆ k,σ,2 ˆ− k,σ,2 + v ∗ (T1−1 k)e e−i k(δ1 −δ2 ) v(T1−1 k) k,σ,1 . (3.26)

k

δ1 −δ2 ) = = ei k( Using that v(T1−1 k) v(k), as it follows by the definition v(k) i k(δi −δ1 ) , we find that the last line of (3.26) is equal to the sum of the seci=1,2,3 e ond and third term in (3.25), as desired.

310


The invariance of (3.25) under the transformation (5) is very simple, if one notes that it follows by the definition of v(k). = v ∗ (k), v(−k) Under the transformation (6.a), the sum of the first and fourth term in the second line of (3.25) is obviously invariant, while the sum of the second and third is changed into + ∗ ˆ− + ˆ− ˆ (k ˆ (k − − ,−k ,k ),σ,2 v (k) ,−k ,k ),σ,1 v(k) 0

k

=−

1

2

(k0 ,−k1 ,k2 ),σ,1

+ ˆ− − ˆ k,σ,2 v ∗ ((−k1 , k2 )) k,σ,1

0

k

1

(k0 ,−k1 ,k2 ),σ,2

2

k + ˆ− . ˆ k,σ,1 v((−k1 , k2 )) k,σ,2

(3.27)

k

Noting that v((−k1 , k2 )) = v ∗ (k), one sees that this is the same as the sum of the second and third term in (3.25), as desired. Similarly, noting that v((k1 , −k2 )) = v(k), one finds that (3.25) is invariant under the transformation (6.b). Under the transformation (7), the sum of the first and fourth term in (3.25) is obviously invariant, while the sum of the second and third term is changed into ˆ+ ˆ+ ˆ− ˆ− + v ∗ (k) + v(k) k

=−

(k0 ,−k),σ,1

k

(k0 ,−k),σ,2

+ ˆ k,σ,2 ˆ− − v ∗ (−k) k,σ,1

k

(k0 ,−k),σ,2

(k0 ,−k),σ,1

+ ˆ k,σ,1 ˆ− . v(−k) k,σ,2

(3.28)

k

we see that the latter sum is the same as the sum = v ∗ (k), Using, again, that v(−k) of the second and third term in (3.25), as desired. Finally, under the transformation (8), all the terms in the right hand side of (3.25) are separately invariant, and the proof of Lemma 1 is concluded. 3.2. Free energy: The ultraviolet integration. We start by studying the partition function M,β,L = e−β||FM,β,L with FM,β,L defined in (3.21). Note that our lattice model has an intrinsic ultraviolet cut-off in the k variables, while the k0 variable is not bounded uniformly in M. A preliminary step to our infrared analysis is the integration of the ultraviolet degrees of freedom corresponding to the large values of k0 . We proceed in the following way. We decompose the free propagator gˆk into a sum of two propagators supported in the regions of k0 “large” and “small”, respectively. The regions of k0 large and small are defined in terms of the smooth support function χ0 (t) introduced at the beginning of Sec. 3.1; the constant a0 entering its definition $ is chosen so that the sup$ − 2 + 2 2 2 and χ0 are disjoint (here | · | is the port of χ0 k0 + |k − p F | k0 + |k − p F | euclidean norm√over R2 /∗ ). In order for this condition to be satisfied, it is enough that 2a0 γ < 4π/(3 3); in the following, for reasons that will become clearer √ later, we shall assume the slightly more restrictive condition 2a0 γ < 4π/3 − 4π/(3 3). We define $ $ − 2 2 + 2 2 f u.v. (k) = 1 − χ0 − χ0 (3.29) k0 + |k − p F | k0 + |k − p F | and f i.r. (k) = 1 − f u.v. (k), so that we can rewrite gˆk as: de f

gˆk = f u.v. (k)gˆk + f i.r. (k)gˆk = gˆ (u.v.) (k) + gˆ (i.r.) (k).

(3.30)


311 (u.v.)±

(i.r.)±

We now introduce two independent sets of Grassmann fields {k,σ,ρ } and {k,σ,ρ }, ∗ , σ =↑↓, ρ = 1, 2, and the Gaussian integrations P(d (u.v.) ) and with k ∈ Dβ,L P(d (i.r.) ) defined by (u.v.) ˆ (u.v.)+ ˆ (u.v.)− (k1 )ρ1 ,ρ2 , P(d (u.v.) ) k1 ,σ1 ,ρ1 k2 ,σ2 ,ρ2 = β||δσ1 ,σ2 δk1 ,k2 gˆ (i.r.) ˆ (i.r.)+ ˆ (u.v.)− (k1 )ρ1 ,ρ2 . (3.31) P(d (i.r.) ) k1 ,σ1 ,ρ1 k2 ,σ2 ,ρ2 = β||δσ1 ,σ2 δk1 ,k2 gˆ Similarly to P(d), the Gaussian integrations P(d (u.v.) ), P(d (i.r.) ) also admit an explicit representation analogous to (3.14), with gˆk replaced by gˆ (u.v.) (k) or gˆ (i.r.) (k) and the sum over k restricted to the values in the support of f u.v. (k) or f i.r. (k), respectively. It easy to verify that the ultraviolet propagator g (u.v.) (x − y) = −1 −ik(x−y) (β||) gˆ (u.v.) (k) satisfies k∈D ∗ e β,L

|g (u.v.) (x − y)| ≤

CN , 1 + ||x − y|| N

(3.32)

$ uniformly in M; here ||x|| = |x0 |2β + | x|2 , with | · |β the distance over the one-dimensional torus of length β and | · | the distance over the periodic lattice . The definition of Grassmann integration implies the following identity (“addition principle”): (i.r.) (u.v.) ) P(d (i.r.) ) P(d (u.v.) )e−V ( + , (3.33) P(d)e−V () = so that we can rewrite the partition function as M,β,L = e−β||FM,L ,β = ≡ e−β||F0,M

⎧ ⎫ ⎨ 1 ⎬ T Eu.v. P(d (i.r.) ) exp (−V ( (i.r.) + ·); n) ⎩ ⎭ n! n≥1

P(d (i.r.) )e−V M

( (i.r.) )

,

(3.34)

T is defined, given any polynomial V ( (u.v.) ) with where the truncated expectation Eu.v. 1 coefficients depending on (i.r.) , as ∂n (u.v.) ) T Eu.v. (V1 (·); n) = n log P(d (u.v.) )eλV1 ( (3.35) , λ=0 ∂λ

and V M is fixed by the condition V M (0) = 0. It can be shown (see Appendix C) that V M can be written as ⎡ ⎤ ∞ n (i.r.)+ (i.r.)− ⎣ ⎦ ˆ ˆ V M () = (β||)−2n k2 j−1 ,σ j ,ρ2 j−1 k2 j ,σ j ,ρ2 j n=1

σ1 ,...,σn =↑↓ ρ1 ,...,ρ2n =1,2 k1 ,...,k2n

j=1

n ·Wˆ M,2n,ρ (k1 , . . . , k2n−1 ) δ( (k2 j−1 − k2 j )), j=1

(3.36)

312


where ρ = (ρ1 , . . . , ρ2n ) and we used the notation = || δ(k) δk,n δ(k) = δ(k)δ(k 0 ), 1 b1 +n 2 b2 ,

δ(k0 ) = βδk0 ,0 , (3.37)

n 1 ,n 2 ∈Z

with b1 , b2 a basis of ∗ . The possibility of representing V M in the form (3.36), with the kernels Wˆ M,2n,ρ independent of the spin indices σi , follows from the symmetries listed in Lemma 1 and from the remark that P(d (u.v.) ) and P(d (i.r.) ) are separately invariant under the same symmetries. The regularity properties of the kernels are summarized in the following lemma, see Appendix C for a proof. Lemma 2. The constant F0,M in (3.34) and the kernels Wˆ M,2n,ρ in (3.36) are given by power series in U , convergent in the complex disc |U | ≤ U0 , for U0 small enough and independent of β, L , M; after Fourier transform, the x-space counterparts of the kernels Wˆ M,2n,ρ satisfy the following bounds: ⎡ ⎤ ||xi − x j ||m i, j ⎦ W M,2n,ρ (x1 , . . . , x2n ) dx1 · · · dx2n ⎣ 1≤i< j≤2n

≤

β||Cmn |U |max{1,n−1} ,

(3.38)

for some constant Cm > 0, where m = 1≤i< j≤2n m i, j . Moreover, the limits F0 = lim M→∞ F0,M and W2n,ρ (x1 , . . . , x2n ) = lim M→∞ W M,2n,ρ (x1 , . . . , x2n ) exist and are reached uniformly in M, so that, in particular, the limiting functions are analytic in the same domain |U | ≤ U0 . Remarks. 1) It is well known that the ultraviolet problem for lattice fermions can be solved in any dimension by a multiscale expansion, see [3,5,4]; for completeness, it will be presented in a self-contained form in Appendix C. Recently, a different proof based on a single scale integration step and using improved bounds on determinants associated to “chronological products” was proposed [20]. 2) Once the ultraviolet degrees of freedom have been integrated out, the remaining infrared problem (i.e., the computation of the Grassmann integral in the second line of (3.34)) is essentially independent of M, given the fact that the limit W2n,ρ of the kernels W M,2n,ρ is reached uniformly and that the limiting kernels are analytic and satisfy the same bounds as (3.38). For this reason, in the infrared integration described in the next two sections, M will not play any essential role and, for this reason, from now on we shall not stress anymore the dependence on M, for notational simplicity. It is important for the incoming discussion to note that the symmetries listed in Lemma 1 also imply some non-trivial invariance properties of the kernels. We will be particularly interested in the invariance properties of the quadratic part Wˆ M,2,(ρ1 ,ρ2 ) (k), which will be used below to show that the structure of the quadratic part of the new effective interaction has the same symmetries as the free integration. The crucial properties that we will need are the following: Lemma 3. Let Wˆ aa (k) ≡ Wˆ M,2,(1,1) (k), Wˆ b b (k) = Wˆ M,2,(2,2) (k), Wˆ ab (k) = Wˆ M,2,(1,2) (k) and Wˆ ba (k) = Wˆ M,2,(2,1) (k). Then the following properties are valid:


313

∗ (k); (i) Waa (k) = Wbb (k) and Wab (k) = Wba ω (ii) as β → ∞, for ω = ±, Waa (0, p F ) = Wab (0, pωF ) = 0; (iii) as β, || → ∞, for ω = ±, % & Re ∂k0 Wˆ aa (0, p Fω ) = 0, ∂k0 Wˆ ab (0, p Fω ) = 0, (3.39) ∂k Wˆ aa (0, p Fω ) = 0, % & % & Re ∂k1 Wˆ ab (0, pFω ) = Im ∂k2 Wˆ ab (0, pFω ) = 0,

i∂k1 Wˆ ab (0, p Fω ) = ω∂k2 Wˆ ab (0, pFω ). Remarks. 1) For simplicity, the properties (ii) and (iii) are spelled out only in the zero temperature limit and in the thermodynamic limit; however, as it will be clear from the proof, those properties all have a finite temperature/volume counterpart. 2) Lemma 3 implies that in the vicinity of the Fermi points the kernel W M,2,(ρ,ρ ) (k) can be rewritten in the form −i z 0 k0 δ0 (ik1 − ωk2 ) W M,2,(ρ,ρ ) (k0 , pωF + k ) , (3.40) δ0 (−ik1 − ωk2 ) −i z 0 k0 ρ,ρ for some real constants z 0 , δ0 , modulo higher order terms in (k0 , k ). Therefore, it is apparent that its structure is the same as the one of Sˆ0 (k), modulo higher order terms in (k0 , k ). Proof. As remarked after (3.37), P(d (u.v.) ) and P(d (i.r.) ) are separately invariant under the symmetry properties listed in Lemma 1. Therefore V() is also invariant under the same symmetries, and so is the quadratic part of V(), that is ˆ (i.r.)− ˆ (i.r.)+ ˆ (i.r.)+ ˆ (i.r.)− (β||)−2 δ(p) k,σ,1 k+p,σ,1 Waa (k) + k,σ,1 k+p,σ,2 Wab (k) σ

k,p

ˆ (i.r.)+ ˆ (i.r.)+ ˆ (i.r.)− ˆ (i.r.)− + k,σ,2 k+p,σ,1 Wba (k) + k,σ,2 k+p,σ,2 Wbb (k)

.

(3.41)

ˆ (i.r.) consists of Recall that, as assumed in the lines preceding (3.29), the support of − + two disjoint regions around p F and p F , respectively; in particular, we assumed that √ 2a0 γ < 4π/3 − 4π/(3 3). Under this condition, it is easy to realize that if both k and ˆ (i.r.) , then |p| < 4π/3. As a consequence, in (3.38), the p + k belong to the support of only non-zero contributions correspond to the terms with p = 0 (in fact, if p is = 0 and belongs to the support of δ(p), then |p| ≥ 4π/3, which means that either k or k + p is ˆ (i.r.) , and the corresponding term in the sum is identically zero). outside the support of This means that the sum (i.r.)+ (i.r.)− ˆ (i.r.)+ ˆ (i.r.)− ˆ ˆ k,σ,1 k,σ,1 Waa (k) + k,σ,1 k,σ,2 Wab (k) σ,k

ˆ (i.r.)+ ˆ (i.r.)− ˆ (i.r.)+ ˆ (i.r.)− + k,σ,2 k,σ,1 Wba (k) + k,σ,2 k,σ,2 Wbb (k)

(3.42)

is invariant under the symmetries (1)–(7) listed in Lemma 1. Invariance under symmetry (4) implies that: = Waa (k0 , T −1 k), Waa (k0 , k) 1 =e Wab (k0 , k)

δ1 −δ2 ) i k(

= Wbb (k0 , T −1 k), Wbb (k0 , k) 1

Wab (k0 , T1−1 k),

=e Wba (k0 , k)

δ1 −δ2 ) −i k(

(3.43) Wab (k0 , T1−1 k);

314


invariance under (5) implies that: Waa (k) = Waa (−k)∗ , Wab (k) = Wab (−k)∗ ,

Wbb (k) = Wbb (−k)∗ , Wba (k) = Wba (−k)∗ ;

(3.44)

invariance under (6.a) implies that: Waa (k0 , k1 , k2 ) = Wbb (k0 , −k1 , k2 ),

Wab (k0 , k1 , k2 ) = Wba (k0 , −k1 , k2 ) ; (3.45)

invariance under (6.b) implies that: Waa (k0 , k1 , k2 ) = Waa (k0 , k1 , −k2 ), Wab (k0 , k1 , k2 ) = Wab (k0 , k1 , −k2 ),

Wbb (k0 , k1 , k2 ) = Wbb (k0 , k1 , −k2 ), (3.46) Wba (k0 , k1 , k2 ) = Wba (k0 , k1 , −k2 ) ;

invariance under (7) implies that: = Waa (k0 , −k), Waa (k0 , k) Wab (k0 , k) = Wba (k0 , −k).

= Wbb (k0 , −k), Wbb (k0 , k)

(3.47)

Finally, invariance under (8) implies that: = −Waa (−k0 , k), = −Wbb (−k0 , k), Wbb (k0 , k) Waa (k0 , k) = Wab (−k0 , k), = Wba (−k0 , k). Wab (k0 , k) Wba (k0 , k)

(3.48)

Now, combining the first of (3.45), the second of (3.46) and the second of (3.47), we find that Waa (k) = Wbb (k). Combining the third of (3.44), the third of (3.47) and the last of (3.48), we find that Wab (k) = Wba (k)∗ . This concludes the proof of item (i). = 0, and this proves, in The first of (3.48) implies that, as β → ∞, Waa (0, k) particular, that Waa (0, pωF ) = 0 and that, in the limit || → ∞, ∂k Waa (0, pωF ) = 0. Using that pωF is invariant under the action of T1 , we see that the third of (3.43) implies ω

ω

that (1 − ei pF (δ1 −δ2 ) )Wab (k0 , pωF ) = 0. Since ei pF (δ1 −δ2 ) = −eiωπ/3 = 1, this identity proves, in particular, that Wab (0, pωF ) = 0, and ∂k0 Wab (0, pωF ) = 0. This concludes the proof of item (ii). Now, combining the first of (3.44) with the first of % (3.47), we find&that Waa (k0 , k) = ω ∗ , which implies, in particular, that Re ∂k0 Wˆ aa (0, p ) = 0. Waa (−k0 , k) F Finally, let Wab (0, pωF + k ) α1ω k1 + α2ω k2 , modulo higher order terms in k . Using √ 3/2 −1/2 √ in the third of (3.43), we find that that T1−1 = − 3/2 −1/2 √ √ α1ω k1 + α2ω k2 = e−iωπ/3 α1ω (k1 /2 − 3k2 /2) + α2ω ( 3k1 /2 + k2 /2) , (3.49)

which implies α1ω = −iωα2ω . Moreover, using the third of (3.44) we find that αiω = −(αi−ω )∗ , and using the third of (3.46) we find that α2ω = −α2−ω . Therefore, α2ω = −α2−ω = −(α2−ω )∗ , and we see that α2ω is real and odd in ω, that is α2ω = ωa, for some real constant a. Therefore, α1ω = −iωα2ω = −ia, and this concludes the proof of item (iii).


315

3.3. Free energy: The infrared integration. Multiscale analysis. In order to compute (3.34) we proceed in an iterative fashion, using standard functional Renormalization Group methods [2,10,17]. As a starting point, it is convenient to decompose the infrared propagator as: ω g (i.r.) (x, y) = e−i pF (x−y) gω(≤0) (x, y), (3.50) ω=±

where, if k = (k0 , k ), gω(≤0) (x, y)

−1 1 −v ∗ (k + p Fω ) −ik0 −ik (x−y) = χ0 (|k |)e , −v(k + p Fω ) −ik0 β|| ω k ∈Dβ,L

(3.51) ω Dβ,L

Dβ∗

DωL ,

Dβ∗

{ nL1 b1

DωL

(γ −M |k

= × with = Dβ ∩ {k0 : χ0 = + and 0 |) > 0} and n2 ω F , 0 ≤ n 1 , n 2 ≤ L − 1}. L b2 − p Correspondingly, we rewrite (i.r.) as a sum of two independent Grassmann fields: ω (i.r.)± (≤0)± = ei pF x x,σ,ρ,ω , (3.52) x,σ,ρ ω=±

and we rewrite (3.34) in the form: M,β,L = e−β||F0,M

Pχ0 ,A0 (d (≤0) )e−V

(0) ( (≤0) )

,

(3.53)

where V (0) ( (≤0) ) is equal to V M ( (i.r.) ), once (i.r.) is rewritten as in (3.52), i.e., V (0) ( (≤0) )

=

∞

(β||)−2n

n=1

ω1 ,...,ω 2n =±

σ1 ,...,σn =↑↓ ρ1 ,...,ρ2n =1,2

⎛

(0)

·Wˆ 2n,ρ,ω (k1 , . . . , k2n−1 ) δ⎝

=

∞

⎡ dx1 · · · dx2n ⎣

n=1 σ ,ρ,ω

2n

k1 ,...,k2n

⎡ ⎣

(3.54) n j=1

⎤

ˆ (≤0)+ ,σ k 2 j−1

j ,ρ2 j−1 ,ω2 j−1

ˆ (≤0)− k ,σ ,ρ 2j

j

⎦

2 j ,ω2 j

⎞

ω

(−1) j (p F j + kj )⎠

j=1 n

⎤

⎦ W (0) (x1 , . . . , x2n ), x(≤0)+ (≤0)− 2n,ρ,ω 2 j−1 ,σ j ,ρ2 j−1 ,ω2 j−1 x2 j ,σ j ,ρ2 j ,ω2 j

j=1

with: 1) ω = (ω1 , . . . , ω2n ), σ = (σ1 , . . . , σn ) and pωF = (0, pFω ); ω ω (0) 2) Wˆ 2n,ρ,ω (k1 , . . . , k2n−1 ) = Wˆ M,2n,ρ (k1 + p F j , . . . , k2n−1 + p F2n−1 ), see (3.36); (0) (x1 , . . . , x2n ) are defined as: 3) the kernels W2n,ρ,ω (0)

W2n,ρ,ω (x1 , . . . , x2n ) = (β||)−2n

k1 ,...,k2n

e

i

2n

j=1 (−1)

⎛ jk

jxj

(0) ) δ⎝ Wˆ 2n,ρ,ω (k1 , . . . , k2n−1

2n

⎞

ωj (−1) j (p F + kj )⎠. j=1

(3.55)

316


Moreover, Pχ0 ,A0 (d (≤0) ) is defined as Pχ0 ,A0 (d (≤0) ) ⎡ ⎤ |)>0 χ0 (|k ⎢ ˆ (≤0)+ ˆ (≤0)− ⎥ = N0 −1 ⎣ d k ,σ,ρ,ω d k ,σ,ρ,ω ⎦ ω k ∈Dβ,L

⎧ ⎪ ⎨

σ,ω,ρ

· exp −(β||)−1 ⎪ ⎩ ω=±,σ =↑↓

χ0 (|k |)>0 ω k ∈Dβ,L

ˆ (≤0)− ˆ (≤0)+ χ0−1 (|k |) k ,σ,·,ω A0,ω (k )k ,σ,·,ω

⎫ ⎪ ⎬ ⎪ ⎭

.

(3.56) where: −v ∗ (k + p Fω ) −ik0 A0,ω (k ) = −v(k + p Fω ) −ik0 c0 (ik1 − ωk2 ) + t0,ω (k ) −iζ0 k0 + s0 (k ) , = c (−ik − ωk ) + t ∗ (k ) −iζ k + s (k ) 0

1

2

0 0

0,ω

0

N0 is chosen in such a way that Pχ0 ,A0 (d (≤0) ) = 1, ζ0 = 1, c0 = 3/2, s0 ≡ 0 and |t0,ω (k )| ≤ C|k |2 . It is apparent that the (≤0) field has zero mass (i.e., its propagator decays polynomially at large distances in x-space). Therefore, its integration requires an infrared multiscale analysis. We consider the scaling parameter γ > 1 introduced above, see the lines preceding (3.29), and we define a sequence of geometrically decreasing momentum scales γ h , h = 0, −1, −2, . . .. Correspondingly we introduce compact support functions f h (k ) = χ0 (γ −h |k |) − χ0 (γ −h+1 |k |) and we rewrite χ0 (|k |) =

0

f h (k ).

(3.57)

h=−∞

The purpose is to perform the integration of (3.53) in an iterative way. We step by step decompose the propagator into a sum of two propagators, the first supported on momenta ∼ γ h , h ≤ 0, the second supported on momenta smaller than γ h . Correspondingly we rewrite the Grassmann field as a sum of two independent fields: (≤h) = (h) + (≤h−1) and we integrate the field (h) . In this way we inductively prove that, for any h ≤ 0, (3.53) can be rewritten as M,β,L = e−β||Fh

Pχh ,Ah (d (≤h) )e−V

(h) ( (≤h) )

,

(3.58)

h where Fh , Ah , V (h) will be defined recursively, χh (|k |) = k=−∞ f k (k ) and (≤h) (≤0) (≤0) Pχh ,Ah (d ) is defined in the same way as Pχ0 ,A0 (d ) with , χ0 , A0,ω , ζ0 , c0 , s0 , t0,ω replaced by (≤h) , χh , Ah,ω , ζh , ch , sh , th,ω , respectively. Moreover


V (h) (0) = 0 and ∞ V (h) () = (β||)−2n

⎡

⎣

σ ,ρ,ω k ,...,k 1 2n

n=1

(h) ·Wˆ 2n,ρ,ω (k1 , . . . , k2n−1 ) δ(

=

n

∞

⎡ dx1 · · · dx2n ⎣

n=1 σ ,ρ,ω

j=1

317

⎤ ˆ (≤h)+ ,σ k 2 j−1

j ,ρ2 j−1 ,ω2 j−1

ˆ (≤h)− k ,σ ,ρ 2j

j

2 j ,ω2 j

⎦

2n ω (−1) j (p F j + kj )) j=1

n

⎤

⎦ x(≤h)+ (≤h)− 2 j−1 ,σ j ,ρ2 j−1 ,ω2 j−1 x2 j ,σ j ,ρ2 j ,ω2 j

j=1

(h) ×W2n,ρ,ω (x1 , . . . , x2n ).

(3.59)

(≤h)

(h)

Note that the field k ,σ,·,ω , whose propagator is given by χh (|k |)[Aω (k )]−1 , has the same support as χh , that is on a neighborood of size γ h around the singularity k = 0 (that, in the original variables, corresponds to the Dirac point k = pωF ). It is important (h) for the following to think Wˆ 2n,ρ,ω , h ≤ 0, as functions of the variables {ζk , ck }h0 ω,σ

k

(≤h)+

ˆ · k ,σ,·,ω

−i z h k0 + σh (k ) δh (ik1 − ωk2 ) + τh,ω (k ) ˆ (≤h)− k ,σ,·,ω . × δ (−ik − ωk ) + τ ∗ (k ) −i z h k0 + σh (k ) h 2 1 h,ω (3.65) Then we include LV (h) in the fermionic integration, so obtaining (h) (≤h) Pχh ,Ah−1 (d (≤h) ) e−RV ( )−β||(Fh +eh ) ,

(3.66)

where eh =

n & 1 (−1)n % (h) Tr χh (k )A−1 (k )W (k ) h,ω 2,ρ,(ω,ω) β|| ω,σ n k

(3.67)

n≥1

is a constant taking into account the change in the normalization factor of the measure and ch−1 (ik1 − ωk2 ) + t h−1,ω (k ) −iζ h−1 k0 + s h−1 (k ) Ah−1,ω (k ) = ch−1 (−ik1 − ωk2 ) + t ∗h−1,ω (k ) −iζ h−1 k0 + s h−1 (k ) (3.68) with: ζ h−1 (k ) = ζh + z h χh (k ), s h−1 (k ) = sh (k ) + σh (k )χh (k ),

ch−1 (k ) = ch + δh χh (k ), t h−1,ω (k ) = th,ω (k ) + τh,ω (k )χh (k ). (3.69)


319

Now we can perform the integration of the (h) field. We rewrite the Grassmann field (≤h) as a sum of two independent Grassmann fields (≤h−1) + (h) and correspondingly we rewrite (3.66) as (h) (≤h−1) + (h) ) e−β||(Fh +eh ) Pχh−1 ,Ah−1 (d (≤h−1) ) P f h ,Ah−1 (d (h) ) e−RV ( , (3.70) where

ch−1 (ik1 − ωk2 ) + th−1,ω (k ) −iζ k0 + sh−1 (k ) Ah−1,ω (k ) = c (−ikh−1 − ωk ) + t ∗ −iζh−1 k0 + sh−1 (k ) h−1 1 2 h−1,ω (k ) (3.71)

with: ζh−1 = ζh + z h , ch−1 = ch + δh , sh−1 (k ) = sh (k ) + σh (k ), th−1,ω (k ) = th,ω (k ) + τh,ω (k ). The single scale propagator is P f h ,Ah−1 (d (h) )x(h)− (h)+ = δσ1 ,σ2 δω1 ,ω2 gω(h) (x1 , x2 ) 1 ,σ1 ,ρ1 ,ω1 x2 ,σ2 ,ρ2 ,ω2

ρ1 ,ρ2

(3.72)

, (3.73)

where gω(h) (x1 , x2 ) =

" #−1 1 e−ik (x1 −x2 ) f h (k ) Ah−1,ω (k ) . β|| ω

(3.74)

k ∈Dβ,L

After the integration of the field on scale h we are left with an integral involving the fields (≤h−1) and the new effective interaction V (h−1) , defined as (h−1) ( (≤h−1) )−e β|| (h) (≤h−1) + (h) ) h e−V = P f h ,Ah−1 (d (h) ) e−RV ( , (3.75) with V (h−1) (0) = 0. It is easy to see that V (h−1) is of the form (3.59) and that Fh−1 = Fh + eh + eh . It is sufficient to use the well known identity 1

(−1)n+1 EhT RV (h) (≤h−1) + (h) ; n , (3.76) eh + V (h−1) ( (≤h−1) ) = n! n≥1

where EhT (X ( (h) ); n) is the truncated expectation of order n w.r.t. the propagator gω(h) , which is the analogue of (3.35) with (u.v.) replaced by (h) and with P(d (u.v.) ) replaced by P f h ,Ah−1 (d (h) ). Note that the above procedure allows us to write the effective renormalizations vh = (ζh , ch ), h ≤ 0, in terms of vk , h < k ≤ 0, namely vh−1 = βh ( vh , . . . , v0 ), where βh is the so–called Beta function. Tree expansion for the effective potentials. An iterative implementation of (3.76) leads to a representation of V (h) ( (≤h) ) in terms of a tree expansion, defined as follows:

320


v r

v0

h

h+1

hv

−1

0

+1

Fig. 1. A tree τ ∈ Th,n with its scale labels

1) Let us consider the family of all trees which can be constructed by joining a point r , the root, with an ordered set of n ≥ 1 points, the endpoints of the unlabeled tree, so that r is not a branching point. n will be called the order of the unlabeled tree and the branching points will be called the non-trivial vertices. The unlabeled trees are partially ordered from the root to the endpoints in the natural way; we shall use the symbol < to denote the partial order. Two unlabeled trees are identified if they can be superposed by a suitable continuous deformation, so that the endpoints with the same index coincide. It is then easy to see that the number of unlabeled trees with n end-points is bounded by 4n . We shall also consider the labelled trees (to be called simply trees in the following); they are defined by associating some labels with the unlabelled trees, as explained in the following items. 2) We associate a label h ≤ −1 with the root and we denote Th,n the corresponding set of labeled trees with n endpoints. Moreover, we introduce a family of vertical lines, labeled by an integer taking values in [h, 1], and we represent any tree τ ∈ Th,n so that, if v is an endpoint or a non trivial vertex, it is contained in a vertical line with index h v > h, to be called the scale of v, while the root r is on the line with index h. In general, the tree will intersect the vertical lines in a set of points different from the root, the endpoints and the branching points; these points will be called trivial vertices. The set of the vertices will be the union of the endpoints, of the trivial vertices and of the non-trivial vertices; note that the root is not a vertex. Every vertex v of a tree will be associated to its scale label h v , defined, as above, as the label of the vertical line to whom v belongs. Note that, if v1 and v2 are two vertices and v1 < v2 , then h v1 < h v2 . 3) There is only one vertex immediately following the root, which will be denoted v0 and cannot be an endpoint; its scale is h + 1. 4) Given a vertex v of τ ∈ Th,n that is not an endpoint, we can consider the subtrees of τ with root v, which correspond to the connected components of the restriction of τ to the vertices w ≥ v. If a subtree with root v contains only v and an endpoint on scale h v + 1, it will be called a trivial subtree. 5) With each endpoint v we associate one of the monomials with four or more Grassmann fields contributing to RV (0) ( (≤h v −1) ), corresponding to the terms with n ≥ 2 in the r.h.s. of (3.54) (with (≤0) replaced by (≤h v −1) ) and a set xv of space-time points (the corresponding integration variables in the x-space representation).


321

6) We introduce a field label f to distinguish the field variables appearing in the terms associated with the endpoints described in item 5); the set of field labels associated with the endpoint v will be called Iv ; note that |Iv | is the order of the monomial contributing to V (0) ( (≤h v −1) ) and associated to v. Analogously, if v is not an endpoint, we shall call Iv the set of field labels associated with the endpoints following the vertex v; x( f ), ε( f ), σ ( f ), ρ( f ) and ω( f ) will denote the space-time point, the ε index, the σ index, the ρ index and the ω index, respectively, of the Grassmann field variable with label f . In terms of these trees, the effective potential V (h) , h ≤ −1, can be written as V (h) ( (≤h) ) + β||ek+1 =

∞

V (h) (τ, (≤h) ),

(3.77)

n=1 τ ∈Th,n

where, if v0 is the first vertex of τ and τ1 , . . . , τs (s = sv0 ) are the subtrees of τ with root v0 , V (h) (τ, (≤h) ) is defined inductively as follows: i) if s > 1, then V (h) (τ, (≤h) ) =

(−1)s+1 T ¯ (h+1) Eh+1 V (τ1 , (≤h+1) ); . . . ; V¯ (h+1) (τs , (≤h+1) ) , s! (3.78)

where V¯ (h+1) (τi , (≤h+1) ) is equal to RV (h+1) (τi , (≤h+1) ) if the subtree τi contains more than one end-point, or if it contains one end-point but it is not a trivial subtree; it is equal to RV (0) (τi , (≤h+1) ) if τi is subtree; " a trivial # T RV (h+1) (τ1 , (≤h+1) ) if τ1 is not ii) if s = 1, then V (h) (τ, (≤h) ) is equal to Eh+1 " # T RV (0) ( (≤h+1) ) − RV (0) ( (≤h) ) if τ1 is a a trivial subtree; it is equal to Eh+1 trivial subtree. Using its inductive definition, the right hand side of (3.77) can be further expanded, and in order to describe the resulting expansion we need some more definitions. We associate with any vertex v of the tree a subset Pv of Iv , the external fields of v. These subsets must satisfy various constraints. First of all, if v is not an endpoint and v1 , . . . , vsv are the sv ≥ 1 vertices immediately following it, then Pv ⊆ ∪i Pvi ; if v is an endpoint, Pv = Iv . If v is not an endpoint, we shall denote by Q vi the intersection of Pv and Pvi ; this definition implies that Pv = ∪i Q vi . The union Iv of the subsets Pvi \Q vi is, by definition, the set of the internal fields of v, and is non empty if sv > 1. Given τ ∈ Th,n , there are many possible choices of the subsets Pv , v ∈ τ , compatible with all the constraints. We shall denote Pτ the family of all these choices and P the elements of Pτ . With these definitions, we can rewrite V (h) (τ, (≤h) ) in the r.h.s. of (3.77) as: V (h) (τ, (≤h) ) = V (h) (τ, P), V (h) (τ, P) =

where + (≤h) (Pv ) =

P∈Pτ (h+1)

+ (≤h) (Pv0 )K dxv0 τ,P (xv0 ), f ∈Pv

(≤h)ε( f )

x( f ),σ ( f ),ρ( f ),ω( f ) ,

(3.79)

(3.80)

322

A. Giuliani, V. Mastropietro (h+1)

and K τ,P (xv0 ) is defined inductively by the equation, valid for any v ∈ τ which is not an endpoint, (h )

v 1 + (h v ) (Pv1 \Q v1 ), . . . , + (h v ) (Pvs \Q vs )], [K v(hi v +1) (xvi )] EhTv [ v v sv !

s

K τ,Pv (xv ) =

i=1

(3.81) + (h v ) (Pvi \Q vi ) has a definition similar to (3.80). Moreover, if vi is an endwhere (h +1) point K vi v (xvi ) is equal to one of the kernels of the monomials contributing to (0) RV ( (≤h v ) ), corresponding to the terms with n ≥ 2 in the r.h.s. of (3.54) (with (h +1) (h v +1) (≤0) replaced by (≤h v ) ); if vi is not an endpoint, K vi v = K τi ,P , where Pi = i {Pw , w ∈ τi }. Equations (3.77)–(3.81) are not the final form of our expansion; we further decompose V (h) (τ, P), by using the following representation of the truncated expectation in the r.h.s. of (3.81). Let us put s = sv , Pi ≡ Pvi \Q vi ; moreover we order in an arbitrary way the sets Pi± ≡ { f ∈ Pi , ε( f ) = ±}, we call f i±j their elements and we define x(i) = ∪ f ∈P − x( f ), y(i) = ∪ f ∈Pi+ x( f ), xi j = x( f i−j ), yi j = x( f i+j ). Note that s s i + − i=1 |Pi | = i=1 |Pi | ≡ n, otherwise the truncated expectation vanishes. A couple l ≡ ( f i−j , f i+ j ) ≡ ( fl− , fl+ ) will be called a line joining the fields with labels f i−j , f i+ j , sector indices ωl− = ω( fl− ), ωl+ = ω( fl+ ), ρ-indices ρl− = ρ( fl− ), ρl+ = ρ( fl+ ), and spin indices σl− = σ ( fl− ), σl+ = σ ( fl+ ), connecting the points xl ≡ xi j and yl ≡ yi j , the endpoints of l. Moreover, if ωl− = ωl+ , we shall put ωl ≡ ωl− = ωl+ . Then, we use the Brydges-Battle-Federbush formula (e.g., see [10,17]) saying that, up to a sign, if s > 1, + (h) (P1 ), . . . , + (h) (Ps )) EhT ( δω− ,ω+ δσ − ,σ + gω(h) (x − y ) = l l l l

l

l

ρl− ,ρl+

l

T l∈T

d PT (t) det G h,T (t), (3.82)

where T is a set of lines forming an anchored tree graph between the clusters of points x(i) ∪ y(i) , that is T is a set of lines, which becomes a tree graph if one identifies all the points in the same cluster. Moreover t = {tii ∈ [0, 1], 1 ≤ i, i ≤ s}, d PT (t) is a probability measure with support on a set of t such that tii = ui · ui for some family of vectors ui ∈ Rs of unit norm. Finally G h,T (t) is a (n − s + 1) × (n − s + 1) matrix, whose elements are given by (h) G ih,T (3.83) j,i j = tii δω− ,ω+ δσ − ,σ + gωl (xi j − yi j ) − + , l

l

l

l

ρl ,ρl

with ( f i−j , f i+ j ) not belonging to

T . In the following we shall use (3.80) even for s = 1, when T is empty, by interpreting the r.h.s. as equal to 1, if |P1 | = 0, otherwise as equal + (h) (P1 )). to det G h = EhT ( Remark. It is crucial to note that G h,T is a Gram matrix, i.e., defining e+ = e↑ = (1, 0) and e− = e↓ = (0, 1), the matrix elements in (3.83) can be written in terms of scalar products: j ) tii δω− ,ω+ δσ − ,σ + gω(h) (x − y (3.84) i j i l l l l l ρl− ,ρl+ = ui ⊗ eω− ⊗ eσ − ⊗ A(xi j − ·), ui ⊗ eωl+ ⊗ eσl+ ⊗ B(xi j − ·) ≡ (fα , gβ ), l

l


323

where A(x) =

1 , e−ik x f h (k ) 1, β|| ω k ∈Dβ,L

" #−1 1 , B(x) = e−ik x f h (k ) Ah−1,ω (k ) . β|| ω

(3.85)

k ∈Dβ,L

The symbol (·, ·) denotes the inner product, i.e.,

ui ⊗ eω ⊗ eσ ⊗ A(x − ·), ui ⊗ eω ⊗ eσ ⊗ B(x − ·) = (ui · ui ) (eω · eω ) (eσ · eσ ) · dz A∗ (x − z)B(x − z),

(3.86)

and the vectors fα , gβ with α, β = 1, . . . , n − s + 1 are implicitly defined by (3.84). The usefulness of the representation(3.84) is that, by the Gram-Hadamard inequality (see, e.g., [10]), | det(fα , gβ )| ≤ α || f α || ||gα ||. In our case, ||fα || ≤ Cγ 3h/2 and ||gα || ≤ Cγ h/2 . Therefore, || f α || ||gα || ≤ Cγ 2h , uniformly in α, so that the Gram determinant can be bounded by C n−s+1 γ 2h(n−s+1) . If we apply the expansion (3.82) in each vertex of τ different from the endpoints, we get an expression of the form (h) + (≤h) (Pv0 )W (h) (xv0 ) ≡ V (τ, P) = V (h) (τ, P, T ), (3.87) dxv0 τ,P,T T ∈T

T ∈T

where T is a special family of graphs on the set of points xv0 , obtained by putting together an anchored tree graph Tv for each non-trivial vertex v. Note that any graph T ∈ T becomes a tree graph on xv0 , if one identifies all the points in the sets xv , with v an endpoint. Given τ ∈ Th,n and the labels P, T , calling vi∗ , . . . , vn∗ the endpoints of τ (h) and putting h i = h vi∗ , the explicit representation of Wτ,P,T (xv0 ) in (3.87) is Wτ,P,T (xv0 ) =

·

⎧ ⎪ ⎨ ⎪ ⎩

v

not e.p.

- n i=1

1 sv !

. (h ) K v ∗ i (xvi∗ ) i

⎡

d PTv (tv ) det G h v ,Tv (tv ) ⎣

l∈Tv

δω− ,ω+ δσ − ,σ + l

l

l

l

⎤⎫ ⎪ ⎬ ⎦ , ρl− ,ρl+ ⎪ ⎭

gω(hl v ) (xl − yl )

(3.88) Analyticity of the effective potentials. The tree expansion described above allows us to express the effective potential V (h) in terms of the running coupling constants ζh , ch and of the renormalization functions σk (k), tk,ω (k). The next goal is the proof of the following result. Theorem 2. There exists a constant U0 > 0, independent of M, β and L, such that the (h) kernels W2l,ρ,ω (x1 , . . . , x2l ) in (3.59), h ≤ −1, are analytic functions of U in the complex domain |U | ≤ U0 , satisfying, for any 0 ≤ θ < 1 and a suitable constant Cθ > 0

324


(independent of M, β, L), the following estimates: 1 (h) dx1 · · · dx2l |W2l,ρ,ω (x1 , . . . , x2l )| ≤ γ h(3−2l+θ) (Cθ |U |)max(1,l−1) . β|| (3.89) Moreover, the constants eh and eh defined by (3.67) and (3.76) are analytic functions of U in the same domain |U | ≤ U0 , and there they satisfy the estimate |eh | + |eh | ≤ Cθ |U |γ h(3+θ) . Proof of Theorem 2. Let us preliminarily assume that, for h ≤ h ≤ −1, and for suitable constants c, cn , the corrections z h , δh , σh (k ) and τh (k ) defined in (3.63) and (3.65), satisfy the following estimates: max {|z h |, |δh |} ≤ c|U |γ θ h ,

sup

γ h −1 ≤|k |≤γ h +1

{||∂kn σh (k )||, ||∂kn τh,ω (k )||} ≤ cn |U |γ 2(h −h) γ (1+θ−n)h . (3.90)

Using (3.90) we inductively see that the running coupling functions ζh , ch , sh (k ) and th (k ) satisfy similar estimates: max {|ζh − 1|, |ch − 3/2|} ≤ c|U |, γh

−1

sup ≤|k |≤γ h

+1

{||∂kn sh (k )||, ||∂kn (th,ω (k ) − t0,ω (k ))||} ≤ cn |U |γ 2(h −h) γ (1+θ−n)h . (3.91) (h)

Now, using the definition of gω , see (3.74) and (3.68), we get, after integration by parts, for any N ≥ 0, (h) γ 2h ≤ C N g (x1 , x2 ) , (3.92) ω ρ,ρ 1 + (γ h ||x1 − x2 ||) N where C N is a suitable constant and ||x1 − x2 || is the distance on the torus, defined after (3.32) Using the tree expansion described above and, in particular, Eqs.(3.77), (3.79), (3.87) and (3.88), we find that the l.h.s. of (3.89) can be bounded from above by - n . (h ) i d(xl − yl ) |K v ∗ (xvi∗ )| n≥1 τ ∈Th,n

⎡

P∈Pτ |Pv0 |=2l

T ∈T

l∈T ∗

i=1

i

⎤ 1 || gω(hl v ) (xl − yl ) ||⎦ , max det G h v ,Tv (tv ) ·⎣ tv s ! v v not e.p.

(3.93)

l∈Tv

where || · || is the spectral norm and where T ∗ is a tree graph obtained from T = ∪v Tv , by adding in a suitable (obvious) way, for each endpoint vi∗ , i = 1, . . . , n, one or more lines connecting the space-time points belonging to xvi∗ .


325

A standard application of the Gram–Hadamard inequality, combined with the dimensional bound on gω(h) (x) given by (3.92), see the remark after (3.83), implies that |detG h v ,Tv (tv )| ≤ c

sv

i=1 |Pvi |−|Pv |−2(sv −1)

· γ hv

sv

i=1 |Pvi |−|Pv |−2(sv −1)

.

(3.94)

(h)

By the decay properties of gω (x) given by (3.92), it also follows that 1 1 γ −h v (sv −1) . d(xl − yl ) ||gω(hl v ) (xl − yl )|| ≤ cn s ! s ! v not e.p. v v not e.p. v

(3.95)

l∈Tv

The bound (3.38) on the kernels produced by the ultraviolet integration implies that

d(xl − yl )

l∈T ∗ \∪v Tv

n

(h )

|K v ∗ i (xvi∗ )| ≤ i

i=1

n

C pi |U |

pi 2

−1

,

(3.96)

i=1

where pi = |Pvi∗ |. Combining the previous bounds, we find that (3.93) can be bounded above by ⎡ ⎤. n

sv 1 pi hv |P |−|P |−3(s −1) n⎣ p −1 v v v i ⎦ i=1 i γ C C |U | 2 . s ! P∈P v not e.p. v n≥1 τ ∈Th,n

τ |Pv0 |=2l

T ∈T

i=1

(3.97) Let us define n(v) = i:v ∗ >v 1 as the number of endpoints following v on τ and v as i the vertex immediately preceding v on τ . Recalling that |Iv | is the number of field labels associated to the endpoints following v on τ (note that |Iv | ≥ 4n(v)) and using that . - s v |Pvi | − |Pv | = |Iv0 | − |Pv0 |, v not e.p.

i=1

(sv − 1) = n − 1,

v not e.p.

(h v − h)

v not e.p.

- s v

.

|Pvi | − |Pv | =

i=1

(h v − h)(sv − 1) =

v not e.p.

(h v − h v )(|Iv | − |Pv |),

v not e.p.

(h v − h v )(n(v) − 1),

(3.98)

v not e.p.

we find that (3.97) can be bounded above by C n γ h(3−|Pv0 |+|Iv0 |−3n) n≥1 τ ∈Th,n

⎡

P∈Pτ |Pv0 |=2l

T ∈T

⎤n 1 (h v −h v )(3−|Pv |+|Iv |−3n(v)) ⎦ ⎣ γ · C pi |U | s ! v v not e.p.

i=1

. pi 2

−1

.

(3.99)

326


Using the identities

γ hn

γ (h v −h v )n(v) =

γ

γ h v ,

v e.p.

v not e.p. h|Iv0 |

γ

(h v −h v )|Iv |

=

γ h v |Iv | ,

(3.100)

v e.p.

v not e.p.

we obtain 1 (h) dx1 · · · dx2l |W2l,ρ,ω (x1 , . . . , x2l )| ≤ β||

n≥1 τ ∈Th,n

P∈Pτ |Pv0 |=2l

C n γ h(3−|Pv0 |)

T ∈T

⎤⎡ ⎤n 1 −(h v −h v )(|Pv |−3) ⎦ ⎣ h v (|Iv |−3) ⎦ ⎣ γ · γ C pi |U | s ! v v e.p. v not e.p. ⎡

. pi 2

−1

. (3.101)

i=1

Note that, if v is not an endpoint, |Pv | − 3 ≥ 1 by the definition of R. Moreover, if v is an endpoint, |Iv | − 3 ≥ 1; in particular, we get γ h v (|Iv |−3) ≤ γ h ∗ −1 , (3.102) v e.p.

with h ∗ the highest scale label of the tree. Now, note that the number of terms in T ∈T n can be bounded by C v not e.p. sv !. Using also that |Pv |−3 ≥ 1 and |Pv |−3 ≥ |Pv |/4, we find that the l.h.s. of (3.101) can be bounded as 1 (h) Cn γ h ∗ −1 dx1 · · · dx2l |W2l,ρ,ω (x1 , . . . , x2l )| ≤ γ h(3−|Pv0 |) β|| n≥1 τ ∈Th,n ⎛ ⎞ ·⎝ γ −θ(h v −h v ) γ −(1−θ)(h v −h v )/2 ⎠ v not e.p.

×

⎛ ⎝

P∈Pτ |Pv0 |=2l

⎞ γ

−(1−θ)|Pv |/8 ⎠

v not e.p.

n

C pi |U |

pi 2

−1

.

(3.103)

i=1

Now, the sum over P can be bounded using the following combinatorial inequality (see for instance A6.1 of [10]): let { pv , v ∈ τ }, with τ ∈ Th,n , a set of integers such Sect. sv that pv ≤ i=1 pvi for all v ∈ τ which are not endpoints; then, if α > 0,

γ −αpv ≤ Cαn .

v not e.p. pv

This implies that P∈Pτ |Pv0 |=2l

⎛ ⎝

v not e.p.

⎞ γ −(1−θ)|Pv |/8 ⎠

n i=1

C pi |U |

pi 2

−1

≤ Cθn |U |n .


Finally, using that γ h ∗

v not e.p. γ

−θ(h v −h v )

327

≤ γ θ h , and that, for 0 < θ < 1,

γ −(1−θ)(h v −h v )/2 ≤ C n ,

τ ∈Th,n v not e.p.

as it follows by the fact that the number of non-trivial vertices in τ is smaller than n − 1 and that the number of trees in Th,n is bounded by const n , and collecting all the previous bounds, we obtain 1 (h) C n |U |n , (3.104) dx1 · · · dx2l |W2l,ρ,ω (x1 , . . . , x2l )| ≤ γ h(3−|Pv0 |+θ) β|| n≥1

which is the desired result. We now need to prove the assumption (3.90). We proceed by induction. The assumption is valid for h = 0, as it follows by (3.38) and by the discussion in Appendix C. Now, assume that (3.90) is valid for all h ≥ k + 1, and let us prove it for k − 1. The functions −i z k k0 + σk (k ) and δk (ik1 − ωk2 ) + τk,ω (k ) admit a representation in terms (k) of W2,ρ,(ω,ω) (x, y). In particular, 1 max{|z k |, |δk |} ≤ β||

(k)

dx1 dx2 ||x − y|| |W2,ρ,(ω,ω) (x, y)|,

(3.105)

and γ

h −1

sup ≤|k |≤γ

Cγ 2h ≤ β||

h +1

{||∂kn σk (k )||, ||∂kn τk,ω (k )||} (k)

dx1 dx2 ||x − y||n+2 |W2,ρ,(ω,ω) (x, y)|.

(3.106)

The same proof leading to (3.104) shows that the r.h.s. of (3.105) can be bounded by the r.h.s. of (3.104) times γ −k (that is the dimensional estimate for ||x−y||), and that the r.h.s. of (3.105) can be bounded by the r.h.s. of (3.104) times γ 2h γ −(n+2)k (where γ −k(n+2) is the dimensional estimate for ||x − y||n+2 ). This concludes the proof of Theorem 2. It remains to prove the estimates on eh , eh . The bound on eh is an immediate corollary of the discussion above, simply because eh can be bounded by (3.93) with l = 0. Finally, (h) remember that eh is given by (3.67): an explicit computation of A−1 h,ω (k )W2,ρ,(ω,ω) (k ) (h) θh and the use of (3.90)–(3.91) imply that ||A−1 h,ω (k )W2,ρ,(ω,ω) (k )|| ≤ C|U |γ , from which: |eh | ≤ C γ 3h n≥1 (C|U |γ θ h )n , as desired.

The existence and analyticity of the specific free energy is a corollary of Theorem 2, see Appendix D for the proof. Corollary 1. The limit f β (U ) = lim L→∞ lim M→∞ FM,β,L , with FM,β,L defined in (3.21), exists and is reached uniformly in U ; in particular, f β (U ) is analytic in |U | ≤ U0 , with U0 the same constant of Theorem 2. Moreover, the limit e(U ) = limβ→∞ f β (U ) exists and is reached uniformly in U ; in particular, e(U ) is analytic in |U | ≤ U0 . Corollary 1 implies the part of the statement of Theorem 1 concerning the free energy and the ground state energy. For the proof of analyticity of the Schwinger functions, see the next section.

328


3.4. The two point Schwinger function. In this section we describe how to modify the expansion for the free energy described in previous sections in order to compute the Schwinger functions at distinct space-time points. For simplicity, we shall restrict our attention to the case of the two point Schwinger function. The general case can be worked out along the same lines. The Schwinger functions can be derived from the generating function defined as + − + − −V (ψ)+ dx φx,σ,ρ x,σ,ρ +x,σ,ρ φx,σ,ρ W(φ) = log P(d)e , (3.107) ε where summation over repeated indices is understood and the variables φx,σ,ρ are Grassε mann variables, anticommuting among themselves and with the variables x,σ,ρ . The de f

M,β,

two–point Schwinger function S(x − y)ρ,ρ = S2 by S(x − y)ρ,ρ =

(x, σ, −, ρ; y, σ, +, ρ ) is given

∂2 W(φ) . − + φ=0 ∂φx,σ,ρ ∂φy,σ,ρ

(3.108)

We start by studying the generating function and, in analogy with the procedure described before, we begin by decomposing the field in an ultraviolet and an infrared compo ω (i.r.)± (≤0)± nent: = (u.v.) + (i.r.) , with x,σ,ρ = ω=± ei pF x x,σ,ρ,ω . After the integration ω (u.v.) ± ± of the variables, and after rewriting φx,σ,ρ = ω=± ei pF x φx,σ,ρ , we get: (≥0) eW (φ) = e−β||F0 +S (φ) Pχ0 ,A0 (d (≤0) ) · ·e

+ (≤0)− (≤0)+ − −V (0) (ψ (≤0) )−B (0) ( (≤0) , φ)+ dx φx,σ,ρ,ω x,σ,ρ,ω +x,σ,ρ,ω φx,σ,ρ,ω

S (≥0) (φ)

,

(3.109)

S (≥0) (0)

where (chosen in such a way that = 0) collects the terms depending on φ but not on (≤0) and B (0) ( (≤0) , φ) the terms depending both on φ and (≤0) generated by the ultraviolet integration. Proceeding as in Sec. 3.3, we inductively show (see below for details) that, if h ≤ 0, eW (φ) can be rewritten as: (≥h) eW (φ) = e−β||Fh +S (φ) Pχh ,Ah (d (≤h) ) ·e

(h+1) (h+1) ˆ (≤h)+ ˆ (≤h)− −V (h) ( (≤h) )−B (h) ( (≤h) , φ)+ dk φˆ k+ ,σ,ρ,ω Qˆ k ,ω,ρ,ρ + φˆ − Qˆ k ,σ,ρ ,ω k ,σ,ρ,ω k ,ω,ρ,ρ k ,σ,ρ ,ω

where

dk must be interpreted as equal to (β||)−1

, (3.110)

ω ; k∈Dβ,L

B (h) ( (≤h) , φ) can be

written as Bφ(h) ( (≤h) ) + W R(h) , with W R(h) containing the terms of third or higher order (h)

in φ and Bφ ( (≤h) ) of the form

-

+ dx φ·,σ,ρ ∗ G (h+1) ω,ρ1 ,ρ2 ∗ 1 ,ω

+ +φ·,σ 1 ,ρ1 ,ω1

∗

G (h+1) ω1 ,ρ1 ,ρ2

∗

∂ V (h) ( (≤h) ) (≤h)+

∂·,σ,ρ2 ,ω

+

∂ V (h) ( (≤h) )

∂2 (≤h)+

(≤h)−

(≤h)−

∂·,σ,ρ1 ,ω

∂·,σ1 ,ρ2 ,ω1 ∂·,σ2 ,ρ3 ,ω2

RV

(h)

(

− ∗ G (h+1) ω,ρ1 ,ρ2 ∗ φ·,σ,ρ2 ,ω

. (≤h)

)∗

G (h+1) ω2 ,ρ3 ,ρ4

− ∗ φ·,σ 2 ,ρ4 ,ω2

,

(3.111)


329

with (h+1) Gˆ ω,ρ,ρ (k ) =

1

(k)

(k)

gˆω,ρ,ρ (k ) Qˆ k ,ω,ρ ,ρ

(3.112)

k=h+1 (h) and Qˆ k ,ω,ρ,ρ defined inductively by the relations (h) (h+1) (h) (h+1) Qˆ k ,ω,ρ,ρ = Qˆ k ,ω,ρ,ρ − W2,ρ,ρ ,(ω,ω) (k )Gˆ ω,ρ ,ρ (k ),

(1)

Q k ,ω,ρ,ρ ≡ δρ,ρ , (3.113)

(h)

(1)

where W2,ρ,ω is the kernel of LV (h) , as defined in (3.62). In (3.112), gˆω is defined as gˆω(1) (k )

= gˆ (k 1 ||k || < ||k + pωF − p−ω F || 1 −ω ω + 1 ||k || = ||k + p F − p F || , 2 (u.v.)

+ pωF )

(h)

de f

where pωF = (0, pωF ). Note that, by the compact support properties of gˆω (k ), if gˆω(h) (k ) = 0, h < 0, then gˆ ( j) (k) = 0 for | j − h| > 1, so that (h+1) ˆ (h+1) ˆ (h) Qˆ (h) k ,ω,ρ,ρ = 1 − W2,ρ,ρ1 ,(ω,ω) (k )gˆ ω,ρ1 ,ρ2 (k ) Q k ,ω,ρ2 ,ρ ,

and, therefore, proceeding by induction, we see that on the support of gˆω(h) (k ) we have (h) || Qˆ k ,ω − 1|| ≤ C|U |γ θ h ,

(h) ||∂kn Qˆ k ,ω || ≤ Cn |U |γ (θ−n)h .

(3.114)

In order to derive (3.114), we used Theorem 2 and the decay bounds (3.92). Using (3.114), the definition (3.112) and the decay bounds (3.92), we find that −(1+ j)h dx |x| j ||G (h) . (3.115) ω (x)|| ≤ C j γ Let us now prove (3.110). We proceed by induction. For h = 0 (3.110) is clearly true (it coincides with (3.109)). Assuming inductively that the representation (3.110) is valid up to a certain value of h < 0, we can show that the same representation is valid for h − 1. In fact, we can rewrite the term V (h) in the exponent of (3.110) as V (h) = LV (h) +RV (h) , as in (3.61), and we can “absorb” LV (h) in the fermionic integration, as explained in Sec. 3.3, see (3.64)–(3.66). Similarly we rewrite ∂ (h) (≤h)∓ (h) (≤h) V ( ) = dy W2,(ρ,ρ ),(ω,ω) (x, y)y,σ,ρ ,ω (≤h)± ∂x,σ,ρ,ω ∂ + (≤h)± RV (h) ( (≤h) ). (3.116) ∂x,σ,ρ,ω This rewriting induces a decomposition of the first line of (3.111) into two pieces, the first proportional to W2(h) , the second identical to the first line of (3.111) itself, with V (h) (h) replaced by RV (h) , that we will call RBφ ( (≤h) ). We choose to “absorb” the term

330

A. Giuliani, V. Mastropietro (h)

proportional to W2 into the definition of Q (h) , and this gives the recursion relation (h) (3.113). Moreover, note that combining RBφ ( (≤h) ) with RV (h) ( (≤h) ) we find: (h)

(h)

RV (h) ( (≤h) ) + RBφ ( (≤h) ) = RV (h) ( (≤h) + G (h+1) ∗ φ) + W R,1 , (3.117) (h)

(h)

(h)

(h)

with W R,1 containing terms of third or higher order in φ. We define W R = W R +W R,1 . After these splittings and redefinitions, we can rewrite (3.110) as (≥h) eW (φ) = e−β||(Fh +eh )+S (φ) Pχh−1 ,Ah−1 (d (≤h−1) ) P f h ,Ah−1 (d (h) ) ·e

(h) (h) ˆ (≤h)− ˆ (≤h)+ ˆ (h) ˆ − −RV (h) ( (≤h) +G (h+1) ∗φ)−W R + dk φˆ k+ Qˆ k +k Q k φk k

.

(3.118)

Integrating the field (h) , we get the analogue of (3.75): P f h ,Ah−1 (d

·e

)e

(h) (h) ˆ (≤h)− ˆ (≤h)+ ˆ (h) ˆ − Q k φk −RV (h) ( (≤h) +G (h+1) ∗φ)−W R + dk φˆ k+ Qˆ k + k k

(h) (h) (h−1) dk φˆ k+ Qˆ k gˆ (h) (k ) Qˆ k φˆ k− −W R,2 (h) ˆ (≤h−1)− ˆ (≤h−1)+ ˆ (h) ˆ − dk φˆ k+ Qˆ k + k Q k φk k

V = e−eh β||−

(h)

(h−1) ( (≤h−1) +G (h) ∗φ)+

·

,

(3.119)

(h−1) with G (h) defined by the recursion relation (3.112) and W R,2 a term of third or higher order in φ. Equation (3.119) can be proved by making use of a formal change of Grassˆ ˆ = ˆ k − gˆ (h) (k )Q (h) mann variables k φk , as described in Ch.4 of [2]. At this point k it is straightforward to check that the final expression for eW (φ) that we end up with is given by the r.h.s. of (3.110), with h replaced by h − 1, and the inductive assumption is proved. From the definitions and the construction above, we get

Sρ,ρ (x − y) =

ω

e−i pF (x−y) Sω,ρ,ρ (x − y) ≡

ω=±

·

1

ω

e−i pF (x−y) ·

ω=± (h)

(h) Q (h) ω,ρ,ρ1 ∗ gω,ρ1 ,ρ2 ∗ Q ω,ρ2 ,ρ (x − y)

h=−∞

(h−1) (h) − G (h) ω,ρ,ρ1 ∗ W2,(ρ1 ,ρ2 ),(ω,ω) ∗ G ω,ρ2 ,ρ (x − y) .

(3.120)

Analyticity of Sρ,ρ (x−y) follows from this representation and the results of Theorem 2. Concerning the representation (2.10), let us take the Fourier transform of Sω,ρ,ρ (x − y). (h) (≤0) If we define h k = min{h : gˆω (k ) ≡ 0}, we get, for k inside the support of k ,σ,ρ,ω , Sˆω,ρ,ρ (k ) =

h k +1

( j)

( j)

( j) Q k ,ω,ρ,ρ1 gω,ρ (k )( j) Q k ,ω,ρ2 ,ρ 1 ,ρ2

j=h k

−

h k +1 j=h k

( j−1)

( j)

j) G (ω,ρ,ρ (k )W2,(ρ1 ,ρ2 ),(ω,ω) (k )G ω,ρ2 ,ρ (k ), 1

(3.121)


331 (h)

which readily implies (2.10): in fact, using the explicit expression of gω and the inductive bounds on Q (h) , see (3.114), it is easy to see that the term in the first line of (3.121) can be written as in (2.10) and that their only singularity is located at k = 0. The contributions from the second line can be bounded using the bounds on W2(h) proved in Theorem 2, and we find that they can be bounded by C|U |γ h k (−1+θ) , which means that they only contribute to the error term appearing in (2.10). This also implies that no other singularity, besides the one at the Fermi points, can be produced by such terms. Finally, if k does not belong to the support of (≤0) , we can write (u.v.) (u.v.) (u.v.) (u.v.) Sˆρ,ρ (k) = Sˆρ,ρ (x − y), (x − y) − gρ,ρ ∗ W ∗ g (k) = g 2,(ρ ,ρ ) 1 2 ρ2 ,ρ 1 (3.122) with W2,ρ defined by (3.36). The bounds discussed in Sec. 3.2 and Appendix C imply (u.v.) that Sρ,ρ (x − y) decays faster than any power, so that no singularity can appear in its Fourier transform. Note that all the bounds discussed in this section are uniform in M, β, L and this fact, in analogy with the results and proofs of Lemma 2 and Corollary 1, implies the β existence of the two-point Schwinger function S2 and of its zero temperature limit S2 , see Appendix D for details. A similar expansion can be obtained for higher order Schwinger functions, but we will not belabor the details here. This concludes the proof of the uniform analyticity properties of 3.22 assumed in Proposition 1 and of Theorem 1.

Appendix A. The Non-interacting Theory In this Appendix we give some details about the computation of the Schwinger functions of the non-interacting theory, i.e., of model (2.1) with U = 0. In this case the Hamiltonian of interest reduces to H0, = − ax+ ,σ b− + bx+ +δ ,σ ax− (A.1) ,σ , x ∈ i=1,2,3

x +δi ,σ

σ =↑↓

i

± with , ax± defined as in items (1)–(4) after (2.1). ,σ , bx +δi ,σ First of all, let us recall that, being H0, quadratic, the 2n-point Schwinger functions satisfy the Wick rule, i.e.,

T{x−1 ,σ1 ,ρ1 · · · x−n ,σn ,ρn y+1 ,σ ,ρ · · · y+n ,σ ,ρ }

1

n

1

G i j = δσi σ j T{x−i ,σi ,ρi y+j ,σ ,ρ }

j

j

β,

n

β,

.

= − det G, (A.2)

β,

Moreover, n every n–point Schwinger function Sn (x1 , ε1 , σ1 , ρ1 ; . . . ; xn , εn , σn , ρn ) with i=1 εi = 0 is identically zero. Therefore, in order to construct the whole set of β, Schwinger functions of H0, , it is enough to compute the 2–point function S0 (x−y) = + − T{x,σ,ρ y,σ,ρ } , and in order to do this, it is convenient to first diagonalize H0, . β, Let us proceed as follows: We identify with the set of vectors in a fundamental cell, and we write = {n 1 a1 + n 2 a2 : 0 ≤ n 1 , n 2 ≤ L − 1},

(A.3)

332


with a1 = 21 (3,

such that ei K x formula:

√

√ 3) and a2 = 21 (3, − 3). The reciprocal lattice ∗ is the set of vectors = 1, if x ∈ . A basis b1 , b2 for ∗ can be obtained by the inversion

b11 b12 b21 b22

−1 a11 a21 = 2π , a12 a22

(A.4)

which gives √ 2π b1 = (1, 3), 3

√ 2π b2 = (1, − 3). 3

(A.5)

We call D L the set of quasi-momenta k of the form m1 m2 k = b1 + b2 , L L

m 1 , m 2 ∈ Z,

(A.6)

identified modulo ∗ ; this means that D L can be identified with the vectors k of the form (2.2) and restricted to the first Brillouin zone: % & m1 m2 (A.7) D L = k = b1 + b2 : 0 ≤ m 1 , m 2 ≤ L − 1 . L L Given a periodic function f : → R, its Fourier transform is defined as 1 i kx ˆ f ( x) = e f (k), ||

(A.8)

DL k∈

which can be inverted into = fˆ(k)

e−i k x f ( x),

k ∈ D L ,

(A.9)

x ∈

where we used the identity

ei k x = ||δk, 0 ,

(A.10)

x ∈

and δ is the periodic Kronecker delta function over ∗ . ± We now associate to the set of creation/annihilation operators ax± the cor ,σ , bx +δi ,σ responding set of operators in momentum space: 1 ±i kx ± 1 ±i kx ˆ ± e aˆ , b± = e (A.11) b . ax± ,σ = || x +δ1 ,σ k,σ k,σ || DL k∈

DL k∈

Note that, using (A.8)–(A.10), we find that ˆ± = aˆ ± = e∓i k x ax± , b e∓i k x b± ,σ k,σ

x ∈

k,σ

x ∈

x +δ1 ,σ

(A.12)

are fermionic creation/annihilation operators satisfying

ε ε {ak,σ k δε,−ε δσ,σ , , ak ,σ } = ||δk,

ε ε {bk,σ k δε,−ε δσ,σ , , bk ,σ } = ||δk,

(A.13)


and {a ε , bε

k ,σ

k,σ

} = 0. With these definitions, we can rewrite

H0, = −

x ∈ i=1,2,3

σ =↑↓

(ax+ ,σ b−

x +δi ,σ

+ bx+ +δ ,σ ax− ,σ ) = − i

+ ˆ− e+i k x e−i k (x+δi −δ1 ) aˆ k,σ × b

k ,σ

k ∈D L k,

=−

333

1 ||2 x ∈ i=1,2,3

σ =↑↓

+ e−i k x e+i k (x+δi −δ1 ) bˆk+ ,σ aˆ −

k,σ

1 ∗ + ˆ− ˆ + aˆ − , vk aˆ k,σ + v b b k k,σ k,σ k,σ ||

(A.14)

D L σ =↑↓ k∈

with vk =

3

e

i(δi −δ1 )k

= 1 + 2e

i=1

−i 23 k1

√ 3 k2 . cos 2

(A.15)

The Hamiltonian H0, can be diagonalized by introducing the fermionic operators v ∗ aˆ k,σ k αˆ k,σ bˆk,σ = √ + √ , 2 2|vk |

v ∗ aˆ k,σ k βˆk,σ bˆk,σ = √ − √ , 2 2|vk |

in terms of which we can rewrite 1 + + ˆ ˆ H0, = −|vk |αˆ k,σ , α ˆ + |v | β β k,σ k k,σ k,σ ||

(A.16)

(A.17)

D L σ =↑↓ k∈

with

|vk | =

2 √ √ 1 + 2 cos(3k1 /2) cos( 3k2 /2) + 4 sin2 (3k1 /2) cos2 ( 3k2 /2), (A.18)

which is vanishing iff k = pωF , ω = ±, with 2π 2π , ω √ ). (A.19) 3 3 3 ±i kx α = ||−1 k∈ ˆ k,σ and βx±,σ = ||−1 DL e

p Fω = (

Now, for x ∈ , we define αx± ,σ

± = e H0, x0 α ± e−H0, x0 and e±i k x αˆ k,σ ) we define αx,σ ; moreover, if x = (x0 , x x ,σ ± e H0, x0 βx ,σ e−H0, x0 . A straightforward computation, see, e.g., Appendix 1 of [2], shows that, if −β < x0 − y0 ≤ β, DL k∈ ± = βx,σ

− + T{αx,σ αy,σ }

β,

. δσ,σ −i k( e(x0 −y0 )|vk | e(x0 −y0 +β)|vk | x−y ) = e − 1 (x0 − y0 ≤ 0) 1 (x0 − y0 > 0) , || 1 + eβ|vk | 1 + eβ|vk | k∈D L

(A.20)

334

A. Giuliani, V. Mastropietro − + T{βx,σ βy,σ }

β,

δ = ||

σ,σ

e

-

x−y ) −i k(

1 (x0 − y0 > 0)

DL k∈

e−(x0 −y0 )|vk | 1 + e−β|vk |

− 1 (x0 − y0 ≤ 0)

e−(x0 −y0 +β)|vk | 1 + e−β|vk |

. ,

(A.21) + − β + }

− and T{αx,σ y,σ β, = T{βx,σ αy,σ } β, = 0. A priori Eq. (A.21) and (A.22) are defined only for −β < x0 − y0 ≤ β, but we can extend them periodically over the whole real axis; the periodic extension of the propagator is continuous in the time variable for x0 − y0 ∈ β Z, and it has jump discontinuities at the points x0 − y0 ∈ β Z. Note that at x0 − y0 = βn, the difference between the right and left limits is equal to (−1)n δx ,y , so For x − y ∈ β Z × 0, we that the propagator is discontinuous only at x − y = β Z × 0. can write − + αy,σ T{αx,σ }

β,

− + T{βx,σ βy,σ }

β,

=

1 δσ,σ −ik(x−y) , e β|| −ik0 − |vk |

(A.22)

1 δσ,σ −ik(x−y) . e β|| −ik0 + |vk |

(A.23)

k∈Dβ,L

=

k∈Dβ,L

Note indeed that for x0 − y0 ∈ β Z the sums over k0 in (A.22) are convergent, uniformly in M; if x0 − y0 = βn and x = y, the r.h.s. of (A.22) is equal to 1 2

lim

x0 −y0 →(βn)


+ β,

− + = T{αx,σ αy,σ }

β,

x0 −y0 =βn

+

lim

x0 −y0 →(βn)


β, −

.

(A.24)

− β + }

A similar remark is valid for T{βx,σ y,σ

β,

± and β ± . If we now re-express αx,σ x,σ

± and b± in terms of ax,σ x+δ1 ,σ , using (A.16), we get (2.6) and (3.17). Note that if, on the

contrary, x = y, then (3.17) is not valid. In fact

1 1 0 −v ∗ (k) . lim gˆk = 2 −v(k) 0 M→∞ β|| k02 + |v(k)| k∈D k∈D ∗

(A.25)

β,L

β,L

In particular, the diagonal part of (A.25) is vanishing, while, using (A.16) and the fact that αˆ + βˆk ,σ

= βˆ+ αˆ k ,σ

= 0, we have that k,σ

β,

k,σ

β,

+ 1,1 = S0 (0− , 0) 2,2 = − 1 α + αx ,σ

S0 (0− , 0) + β β

x ,σ x ,σ x ,σ β, β, 2 β|v | −β|v | k e k 1 e 1 (A.26) =− + =− . β|v | −β|v | k 2|| 2 1+e k 1+e DL k∈


335

Appendix B. Proof of Proposition 1 Let us start by proving (3.23), which is equivalent to 0 / Tr{e−β H } = exp −β|| lim FM,β,L . M→∞ Tr{e−β H0, }

(B.1)

The first key remark is that, if β, L are finite, the left hand side of (B.1) is a priori well defined and analytic on the whole complex plane. In fact, by the Pauli principle, the ± Fock space generated by the fermion operators ax± ∈ , σ =↑↓, is ,σ , b , with x x +δ1 ,σ

finite dimensional. Therefore, writing H = H0 + U V , with H0 and V two bounded operators, we see that Tr{e−β H } is an entire function of U , simply because e−β H converges in norm over the whole complex plane: ||e−β H || ≤

∞ ∞ n βn 0 β k |U |k ||V ||k β n−k ||H0 ||n−k ||H || + |U | ||V || = n! k! (n − k)! n=0

k=0

=e

n≥k

β||H0 ||+β|U | ||V ||

, (B.2) , where the norm || · || is, e.g., the Hilbert-Schmidt norm ||A|| = Tr(A† A). On the other hand, by assumption, FM,β,L is analytic in |U | ≤ U0 , with U0 independent of β, L , M, and uniformly convergent as M → ∞. Hence, by the Weierstrass theorem, the limit Fβ,L = lim M→∞ FM,β,L is analytic in |U | ≤ U0 and its Taylor coefficients coincide with the limits as M → ∞ of the Taylor coefficients of FM,β,L . Moreover, lim M→∞ e−β||FM,β,L = e−β||Fβ,L , again by the Weierstrass theorem. It is well known that the Taylor coefficients of e−β||Fβ,L coincide with the Taylor coefficients of Tr{e−β H }/Tr{e−β H0, }: this can be shown by developing the trace in power series by using Trotter’s product formula; the coefficients of the resulting expansion are expressed in terms of Feynman graphs, which are order by order finite for any fixed β and L (in fact at any fixed order they can be written as a finite combination of integrals over imaginary time and spatial momenta of products of propagators, which are bounded and integrable). Note that, in order to guarantee that the two formal power series are the same, the correct choice of the interaction (3.19) expressed in Grassmann variables does not include terms bilinear in the fields, contrary to the interaction in second quantized form, see (2.1): in fact, with this choice, in both perturbative expansions the "tadpoles" are exactly vanishing, as required by the condition that the system is at half filling, even though the Grassmann propagator at the origin does not coincide with see (A.25) and (A.26). S0 (0− , 0), In conclusion, Tr{e−β H }/Tr{e−β H0, } = e−β||Fβ,L in the complex region |U | ≤ U0 , simply because the l.h.s. is entire in U , the r.h.s. is analytic in |U | ≤ U0 and the Taylor coefficients at the origin of the two sides are the same. Taking logarithms at both sides proves (3.23). Regarding (3.24), we note that, by analyticity, Tr{e−β H }/Tr{e−β H0, } never vanishes on |U | ≤ U0 ; therefore, the same argument used above for the pressure can be now repeated for the Schwinger functions. Appendix C. The Ultraviolet Integration In this Appendix we prove Lemma 2, that is we prove Eq.(3.38) and the existence (and uniformity) of the M → ∞ limit. Note that in order to get (3.38), a simple application of

336


(3.82) and determinant bounds is not enough, because g (u.v.) (x) does not admit a Gram representation, which is a key property needed for the implementation of standard fermionic cluster expansion methods. As mentioned in Sect. 3.2, a way out of this problem is to decompose the ultraviolet propagator into a sum of propagators, each admitting a Gram representation, and performing a simple multiscale analysis of the ultraviolet problem, in analogy with the standard strategy for ultraviolet problems in fermionic Quantum Field Theories [9,14]. This multiscale analysis is very similar to (but much simpler than) the one described in Sect. 3.3; it has been performed in several previous papers [3,5,4] and it is reported here for completeness. Appendix C.1. Proof of (3.38). Let M be the integer introduced at the beginning of Sect. 3.1, let β, L be fixed throughout this Appendix and let us rewrite the Fourier transform of gˆ (u.v.) (k) as g (u.v.) (x) =

M

g (h) (x),

(C.1)

h=1

where g (h) (x) =

1 f u.v. (k)Hh (k0 )e−ikx gˆk , β|| ∗

(C.2)

k∈Dβ,L

with H1 (k0 ) = χ0 (γ −1 |k0 |) and, if h ≥ 2, Hh (k0 ) = χ0 (γ −h |k0 |) − χ0 (γ −h+1 |k0 |). Note that [g (h) (0)]ρρ = 0, ρ = 1, 2, and, for any integer K ≥ 0, g (h) (x) satisfies the bound CK ||g (h) (x)|| ≤ , (C.3) 1 + (γ h |x0 |β + | x| ) K where | · |β is the distance on the one dimensional torus of size β and | · | is the distance on the periodic lattice . Moreover, g (h) (x) admits a Gram representation: g (h) (x − y) = dz A∗h (x − z) · Bh (y − z), with , 1 e−ikx 10 , f u.v. (k)Hh (k0 ) 2 Ah (x) = 2 01 β|| k + |v(k)| k∈Dβ,L

Bh (x) =

0

, 1 ik0 −v ∗ (k) , f u.v. (k)Hh (k0 ) e−ikx −v(k) ik0 β||

(C.4)

k∈Dβ,L

and

||Ah ||2 =

for a suitable constant C. Our goal is to compute e−β||F0 −V (

(i.r ) )

dz|Ah (z)|2 ≤ Cγ −3h ,

||Bh ||2 ≤ Cγ 3h ,

(i.r.) [1,M] ) = lim P(d [1,M] )e V ( + M→∞ 0 / (i.r.) [1,M] ) , = exp lim log P(d [1,M] )e V ( + M→∞

(C.5)

(C.6)


337

where P(d [1,M] ) is the fermionic “Gaussian integration” associated with the propaga M (h) gˆ (k) (i.e., it is the same as P(d (u.v.) )); the fact that the limit M → ∞ can tor h=1 be exchanged with the logarithm in (C.6) follows from the analysis below. We perform the integration of (C.6) in an iterative fashion, analogous to the procedure described in Sec. 3.3 for the infrared integration. We can inductively prove the analogue of (3.58), i.e., (h) (i.r ) (i.r.) [1,h] e−β||F0,M −V M ( ) = e−β||Fh (C.7) P(d [1,h] )eV M ( + ) , where P(d [1,h] ) is the fermionic “Gaussian integration” associated with the propagator h (k) k=1 gˆ (k) and ⎡ ⎤ ∞ n (h) ⎦ x[1,h]+,σ ,ρ x[1,h]− V ( [1,h] ) = dx1 · · · dx2n ⎣ ,σ ,ρ M

2 j−1

n=1 ρ,σ

j

2 j−1

2j

j

2j

j=1

(h)

× W M,2n,ρ (x1 , . . . , x2n ).

(C.8)

In order to inductively prove (C.7)–(C.8) we simply use the addition principle to rewrite (h) (i.r.) [1,h] P(d [1,h] )eV M ( + ) = P(d [1,h−1] ) (h) (i.r.) [1,h−1] + (h) ) × P(d (h) )eV M ( + , (C.9) where P(d (h) ) is the fermionic Gaussian integration with propagator gˆ (h) (k). After the integration of (h) ) we define (h−1) (h) (i.r.) [1,h−1] )−β||e (i.r.) [1,h−1] + (h) ) h = eV M ( + P(d (h) )eV M ( + , (C.10) which proves (C.7). In analogy with (3.76) we have 1

(h) (h−1) (−1)n+1 EhT V M + (h) ; n . eh + V M () = n!

(C.11)

n≥1

As described in Sect. 3.3, the iterative action of EhTi can be conveniently represented in terms of trees τ ∈ T M;h,n , where T M;h,n is a set of labelled trees, completely analogous to the set Th,n described before Eq.(3.77), unless for the following modifications: 1. a tree τ ∈ T M;h,n has vertices v associated with scale labels h + 1 ≤ h v ≤ M + 1, while the root r has scale h; 2. with each end-point v we associate V ( [1,M] ), with V () defined in (3.19). (h)

(0)

In terms of these trees, the effective potential V M , 0 ≤ h ≤ M (with V M ( (i.r.) ) identified with V( (i.r.) )), can be written as (h) V M ( [1,h] ) + β||eh+1

=

∞

V (h) (τ, [1,h] ),

(C.12)

n=1 τ ∈T M;h,n

where, if v0 is the first vertex of τ and τ1 , . . . , τs (s = sv0 ) are the subtrees of τ with root v0 , V (h) (τ, [1,h] ) is defined inductively as follows:

338


v v0

r

h

h+1

hv

M−1 M M+1

Fig. 2. A tree τ ∈ T M;h,n with its scale labels

i) if s > 1, then V (h) (τ, [1,h] ) =

(−1)s+1 T ¯ (h+1) Eh+1 V (τ1 , [1,h+1] ); . . . ; V¯ (h+1) (τs , [1,h+1] ) , s! (C.13)

where V¯ (h+1) (τi , [1,h+1] ) is equal to V (h+1) (τi , [1,h+1] ) if the subtree τi contains more than one end-point, or if it contains one end-point but it is not a trivial subtree; it is equal to V ( [1,h+1] ) if τi is a trivial subtree; " (h+1) # T ii) if s = 1, then V (h) (τ, (≤h) ) is equal to Eh+1 V (τ1 , [1,h+1] ) if τ1 is not a " # T V ( [1,h+1] ) − V ( [1,h] ) if τ1 is a trivial subtree. trivial subtree; it is equal to Eh+1 Note that, with V () defined as in (3.19) and with present choice of the " the[1,h+1] # ultravioT let cutoff (such that [g (h) (0)]ρρ = 0), we get Eh+1 V ( ) − V ( [1,h] ) = 0. This implies that, if v is not an endpoint and n(v) is the number of endpoints following v on τ , and if τ has a vertex v with n(v) = 1, then its value vanishes: therefore, in the sum over the trees, we can freely impose the constraint that n(v) > 1 for all vertices v ∈ τ . From now on we shall assume that the trees in T M;h,n satisfy this constraint. Repeating step by step the discussion leading to (3.79), (3.87) and (3.88), and using analogous definitions, we find that (h) + [1,h] (Pv0 )W (h) (xv0 ) ≡ V (τ, P) = V (h) (τ, P, T ), (C.14) dxv0 τ,P,T T ∈T

T ∈T

where + [1,h] (Pv ) =

f ∈Pv

and Wτ,P,T (xv0 ) = U

⎧ ⎪ ⎨

[1,h]ε( f )

x( f ),σ ( f ),ρ( f )

1 sv !

d PTv (tv ) det G h v ,Tv (tv ) ⎪ ⎩ v not e.p. ⎤⎫ ⎡ ⎬ ⎣ δσ − ,σ + g (h v ) (xl − yl ) − + ⎦ . l l ⎭ ρl ,ρl

n

l∈Tv

(C.15)

(C.16)


339

Moreover, G h v ,Tv (tv ) is a matrix, analogous to (3.83), with δω+ ,ω− replaced by 1 and l

l

(h) . Note that W (h) gω(h) τ,P,T and, therefore, V (τ, P) do not depend on M: l replaced by g (h) V M () depends on M only through the choice of the scale labels (i.e., the dependence on M is all encoded in T M;h,n ). As in the proof of Theorem 2, we get the bound

1 β|| ⎡ ·⎣

(h)

dx1 · · · dx2l |W M,2l,ρ (x1 , . . . , x2l )| ≤

n≥1

v not e.p.

|U |n

τ ∈T M;h,n

⎤ 1 h v ,Tv (h v ) || g (xl − yl ) ||⎦ (tv ) max det G sv ! tv

P∈Pτ |Pv0 |=2l

T ∈T

d(xl − yl )

l∈T

(C.17)

l∈Tv

and, using the analogues of the estimates (3.94), (3.95) and (3.96), taking into account the new scaling of the propagator, we find that (C.17) can be bounded above by ⎡ ⎤ 1 γ −h v (sv −1) ⎦ . (C.18) C n |U |n ⎣ s ! v P∈P v not e.p. n≥1 τ ∈T M;h,n

τ |Pv0 |=2l

T ∈T

Using (3.98) we find that the latter expression can be rewritten as ⎡ ⎤ 1 γ −(h v −h v )(n(v)−1) ⎦ , (C.19) C n |U |n γ −h(n−1) ⎣ s ! v P∈P v not e.p. n≥1 τ ∈T M;h,n

τ |Pv0 |=2l

T ∈T

where we remind the reader that n(v) > 1 for any τ ∈ T M;h,n . Performing the sums over T, P and τ as in the proof of Theorem 2, we finally find 1 (h) (C.20) dx1 · · · dx2l |W M,2l,ρ (x1 , . . . , x2l )| ≤ C|U |max{1,n−1} , β|| which (3.38) with m = 0. The proof of the general case, m ≥ 0, is completely analogous. By the uniformity of the constant C with respect to M, β, L, the bounds above imply analyticity of the kernels in |U | ≤ U0 , for a suitable U0 independent of M, β, L. Appendix C.2. The M → ∞ limit. In this subsection we prove that, if M ≥ M, 1 (0) (0) dx1 · · · dx2l W M ,2l,ρ (x1 , . . . , x2l ) − W M,2l,ρ (x1 , . . . , x2l ) β|| ≤ C1 |U |max{1,n−1} γ −M/2 ,

(C.21)

which readily implies the last statement of Lemma 2. In fact, (C.21) implies that (0) {Wˆ k,2l,ρ }k∈N is a Cauchy sequence, uniformly in U for |U | ≤ U0 . Since the kernels (0) are analytic in U in the same domain, by the Weierstrass theorem the kernels Wˆ M,2l,ρ (0) admit a limit Wˆ 2l,ρ as M → ∞; the limit is analytic in |U | ≤ U0 and its Taylor (0) coefficients are the limits of the coefficients of Wˆ M,2l,ρ .

340


Using the same representation leading to (C.17) and following the steps leading to (C.18) and (C.19), we see that the l.h.s. of (C.21) can be bounded as 1 (0) (0) dx1 · · · dx2l W M ,2l,ρ (x1 , . . . , x2l ) − W M,2l,ρ (x1 , . . . , x2l ) β|| ⎡ ⎤ 1 ≤ γ −(h v −h v )(n(v)−1) ⎦ . C n |U |n ⎣ s ! v P∈P v not e.p. n≥1 τ ∈T M ; 0,n \T M; 0,n

τ |Pv0 |=2l

T ∈T

(C.22) Note that the set of trees over which we are summing is T M ;h,n \T M;h,n , i.e., the trees (0) (0) contributing to the difference W M ,2l,ρ − W M,2l,ρ must have at least one endpoint on

scale M < h ∗ ≤ M + 1. By using the fact that n(v) ≥ 2, we can bound the r.h.s. of (C.22) from above by ⎡ ⎤ 1 M γ −(h v −h v )(n(v)−3/2) ⎦ γ− 2 C n |U |n ⎣ s ! v P∈P v not e.p. n≥1 τ ∈T M ; 0,n \T M; 0,n

τ |Pv0 |=2l

T ∈T

≤ C1 |U |max{1,n−1} γ −M/2 ,

(C.23)

which proves (C.21). Appendix D. The Thermodynamic and Zero Temperature Limits In this Appendix we first prove Corollary 1, discussing the existence (and uniformity) of the thermodynamic limit and of the zero temperature limit of the free energy. Finally, we discuss the existence of the thermodynamic and zero temperature limits for the Schwinger functions. Let us start by studying the thermodynamic limit of the free energy. The discussion in Appendix C implies that lim M→∞ Fβ,L = F0 + 0h=h β (eh +eh ), where F0 was defined in Lemma 2 and eh and eh were defined in (3.67) and (3.76), respectively 1 . Note that both F0 and eh , eh depend on L and β, through the propagators (which depend on β, L) and through the definition of the integration interval and of the sum over the scale labels. In order to make this dependence apparent, let us rename them as F0,β,L , eh,β,L and eh,β,L . Similarly, when needed, we shall attach extra labels β, L to the kernels of the effective potentials, to the propagators and to the Gram determinants, to make their dependence on β, L apparent. We already know that F0,β,L , eh,β,L and eh,β,L are analytic in the uniform domain |U | ≤ U0 , where they satisfy bounds of the form: |F0,β,L | ≤ C|U |, |eh | + |eh | ≤ C|U |γ h(3+θ) , 0 ≤ θ < 1. Our first goal is to prove that, for any β < +∞ and for any 0 < K < 4, |F0,β,L − F0,β | +

0

C K |U | |eh,β,L − eh,β | + |eh,β,L − eh,β | ≤ , LK

(D.1)

h=h β

1 With some abuse of notation, we are denoting by the same symbols both the functions e and e comh h puted at finite M, and their limits as M → ∞ (which exist, by Lemma 2, Theorem 2 and an application of the Weierstrass theorem: note in fact that eh and eh in (3.67) and (3.76) have a very weak dependence on M, (0) induced by the kernels of V M , that is essentially irrelevant, as proved in Appendix C).


341

for suitable functions F0,β , eh,β , eh,β , analytic in |U | ≤ U0 , and a suitable constant C K . Let us start by considering 1 (0) F0,β,L = (D.2) dxv0 Wτ,P,T,β,L (xv0 ), β|| ∗ P∈P n≥1 M≥1 τ ∈T0,n (M)

τ |Pv0 |=0

T ∈T

∗ (M) is the set of trees with root on scale 0, n endpoints and the highest scale where T0,n label equal to M + 1. We observe that, by using translation invariance, we can fix one variable at the origin: remember that xv0 = {xv : v is an e.p. of τ }, so that, if v ∗ is one arbitrarily chosen endpoint of τ , (0) d x¯ v0 Wτ,P,T,β,L (¯xv0 ), (D.3) F0,β,L = ∗ (M) n≥1 M≥1 τ ∈T0,n

P∈Pτ |Pv0 |=0

T ∈T (β,)

where x¯ v0 = {xv − xv ∗ : v is an e.p. of τ } and ⎡ ⎤ β/2 ⎣ ⎦. d x¯ v0 = dx0,v (β,)

v e.p. v =v ∗

−β/2

(D.4)

x v ∈

We want to estimate F0,β,L − F0,β =

∗ (M) n≥1 M≥1 τ ∈T0,n

P∈Pτ |Pv0 |=0

T ∈T

(0)

(β,)

d x¯ v0 Wτ,P,T,β,L (¯xv0 ) −

(β,B)

(0) d x¯ v0 Wτ,P,T,β (¯xv0 ) ,

(D.5) (0)

(0)

where B is the infinite triangular lattice and Wτ,P,T,β the kernel obtained from Wτ,P,T,β,L (h)

(h)

by replacing all the propagators gβ,L (x) by their infinite volume limits gβ (x) = (h)

lim L→∞ gβ,L (x). Let us fix 0 < δ < 1/4 and let us define δ = {n 1 a1 + n 2 a2 : |n 1 |, |n 2 | ≤ δL}. We rewrite (D.5) as F0,β,L − F0,β =

∗ (M) n≥1 M≥1 τ ∈T0,n

P∈Pτ |Pv0 |=0

T ∈T

(D.6)

(1) (2) (3) Rτ,P,T,β,L + Rτ,P,T,β,L + Rτ,P,T,β,L , (D.7)

where (1) Rτ,P,T,β,L (2) Rτ,P,T,β,L (3)

=

(β,)

=

Rτ,P,T,β,L =

(0) d x¯ v0 Wτ,P,T,β,L (¯xv0 ) −

(β,δ )

(β,δ )

(0) d x¯ v0 Wτ,P,T,β (¯xv0 ) −

(0)

(β,δ )

(β,B)

d x¯ v0 Wτ,P,T,β,L (¯xv0 ), (0)

d x¯ v0 Wτ,P,T,β (¯xv0 ),

(0) (0) d x¯ v0 Wτ,P,T,β,L (¯xv0 ) − Wτ,P,T,β (¯xv0 ) .

(D.8)

342

A. Giuliani, V. Mastropietro (1)

(2)

The contributions to F0,β,L − F0,β associated to Rτ,P,T,β,L and Rτ,P,T,β,L , in analogy with (C.17), can be bounded from above by ∗ |U |n d(xl − yl ) ∗ (M) M≥1 τ ∈Th,n

n≥1

⎡

P∈Pτ |Pv0 |=0

T ∈T

l∈T

⎤ 1 (h ) h ,T v v || gβ,Lv l (xl − yl ) ||⎦ , max det G β,L ·⎣ (tv ) v tv s ! v v not e.p.

(D.9)

l∈Tv

∗ means that the integrawhere L v and L l can only assume the values L or +∞, and ˜ tion region satisfies the following constraint: there exists a subtree T ⊆ T such that | l∈T˜ (yl − xl )| ≥ δL. Using this constraint and (C.3), we get the following improved version of the analogue of (3.95): 1 1 cnK (h v ) γ −h v (sv −1) . d(xl − yl ) ||gβ,L (xl − yl )|| ≤ K l s ! 1 + (δL) s ! v v v not e.p. v not e.p. l∈Tv

(D.10) This implies that the contributions to F0,β,L − F0,β associated to (2) Rτ,P,T,β,L

are bounded by C K

|U |L −K ,

(1) Rτ,P,T,β,L

and

as desired. (3)

Let us now look at the contributions to F0,β,L − F0,β associated to Rτ,P,T,β,L . Using (4)

(5)

(C.16) we can bound it from above by Rβ,L + Rβ,L , with (4)

Rβ,L =

n≥1

|U |n

∗ (M) M≥1 τ ∈Th,n

P∈Pτ |Pv0 |=0

T ∈T

⎧⎛ ⎪ ⎨ 1 d(xl − yl ) ⎝ ⎪ (β,δ ) l∈T ⎩ v not e.p. sv !

⎫ ⎞ ⎪ ⎬ ⎠ (h l ) (h l ) (h l ) h v ,Tv || gβ,L (xl − yl ) || ||gβ,L (xl − yl ) − gβ (xl − yl )|| · max det G β,L v (tv ) · l ⎪ tv ⎭ l ∈T l∈T l =l

and

⎧ ⎪ ⎪ ⎨ (5) n g (h l ) (xl − yl ) Rβ,L = |U | d(xl − yl ) β,L l ⎪ ⎪ ∗ (M) P∈Pτ T ∈T (β,δ ) l∈T ⎩ l∈T n≥1 M≥1 τ ∈Th,n |Pv0 |=0 ⎞⎫ ⎛ ⎪ ⎪ ⎜ 1 ⎟⎬ 1 h w ,Tw h w ,Tw h v ,Tv ⎟ ⎜ max det G β,L (tw ) − det G β max det G β,L (tv ) ⎠ · (tw )| · ⎝ v ⎪ s ! tw s ! tv ⎪ v not e.p. v w not e.p. w ⎭

v =w

where, again, L l and L v can only assume the values L or +∞. By Poisson’s summation (h) (h) (h) formula, gβ,L (x) − gβ (x) = n=0 gβ (x0 , x + a1 n 1 L + a2 n 2 L), with a1,2 the two basis’ vectors of the triangular lattice B; therefore, C K γ −hl (h ) (h ) d(xl − yl )||gβ,Ll (xl − yl ) − gβ l (xl − yl )|| ≤ , (D.11) LK (β,δ )


343

which, combined with the same bounds leading from (C.17) to (C.20), implies that (4) (5) Rβ,L ≤ C K |U |L −K . Now, in order to get a bound on Rβ,L , if G h w ,Tw is an s × s matrix, we rewrite h w ,Tw det G β,L (tw ) − det G βh w ,Tw (tw ) (D.12) (h) (h) (h) (h) (−1)p (gβ,L )1, p(1) · · · (gβ,L )s, p(s) − (gβ )1, p(1) · · · (gβ )s, p(s) = p

=

(−1)p

p

s

(h) (h) · · (gβ,L )1, p(1) · · · (gβ,L ) j−1, p( j−1)

j=1

(h) (h) (h) (h) (gβ,L ) j, p( j) − (gβ ) j, p( j) (gβ ) j+1, p( j+1) · · · (gβ )s, p(s) , where p = ( p(1), . . . , p(s)) is a permutation of the indices in the (unordered) set J = {1, . . . , s}. We rewrite the two sums over p and j in the following way: s p

s ∗∗ s ∗ = ,

(D.13)

j=1 k=1 J1 ,J2 p

j=1

where the ∗ on the second sum means that the (unordered) sets J1 and J2 are s.t. (J1 , J2 ) is a partition of J \ {k}; the ∗∗ on the third sum means that p(1), . . . , p( j − 1) belong to J1 , p( j) = k and p( j + 1), . . . , p(s) belong to J2 . Using (D.13), we rewrite (D.12) as s s (h) (h) h w ,Tw det G β,L (gβ,L ) j,k − (gβ ) j,k (tw ) − det G βh w ,Tw (tw ) = ·

∗

(−1)

J1 ,J2

π

⎛ (−1)

p1 ,p2

p1 +p2

j=1 k=1

⎞⎛

⎞

(h) (h) ⎝ (gβ,L )i, p1 (i) ⎠ ⎝ (gβ )i , p2 (i ) ⎠ , i∈J1 i ∈J2

(D.14)

where: (−1)π is the sign of the permutation leading from the ordering (1, . . . , s) to the ordering ( f, J¯1 , J¯2 ), with J¯i a fixed (arbitrary) reordering of Ji ; pi , i = 1, 2 is a permutation of J¯i and (−1)pi is its sign. In conclusion, using the obvious notation, h w ,Tw det G β,L (tw ) − det G βh w ,Tw (tw ) s ∗ s (h) (h) hw (gβ,L ) j,k − (gβ ) j,k = (−1)π det G β,L (J1 ) · det G βh w (J2 ), J1 ,J2

j=1 k=1

(D.15) hw hw (J1 ) and G β,L (J1 ) are two G β,L ∗ is equal to 2s . By sum J1 .J2

where in the summation formula, we get

Gram matrices. Note that the number of terms the Gram-Hadamard inequality and Poisson’s

cs h w ,Tw (tw ) − det G βh w ,Tw (tw )| ≤ KK , max det G β,L tw L

(D.16)

which, combined with the same bounds leading from (C.17) to (C.20), implies the desired (5) bound, Rβ,L ≤ C K |U |L −K .

344


We now need to prove that 0h=h β |eh,β,L − eh,β | + |eh,β,L − eh,β | ≤ C|U |L −K . The quantity |eh,β,L − eh,β | can be bounded by following a strategy completely analogous to the one used to bound |F0,β,L − F0,β |, the only difference being that now the trees involved in the expansions are the infrared ones (with root on scale h and highest scale ≤ 1); therefore, the analogue of (D.10) is changed into 1 (h ) d(xl − yl ) ||gωl v,β,L l (xl − yl )|| s ! v not e.p. v l∈Tv

cnK ≤ 1 + (γ h δL) K

1 −h v (sv −1) ; γ s ! v not e.p. v

(D.17)

the analogue of (D.11) is changed into (β,δ )

) d(xl − yl )||gω(hl l,β,L (xl − yl ) − gω(hl l,β) (xl − yl )|| ≤

C K γ −hl ; (γ hl L) K

(D.18)

the analogue of (D.16) is changed into cs γ 2h w s h w ,Tw (tw ) − det G βh w ,Tw (tw )| ≤ Kh . max det G β,L tw (γ w L) K

(D.19)

These estimates imply that, for any 0 ≤ θ < 1 and any K > 0, |eh,β,L − eh,β | ≤ (3+θ)h (γ h L)−K ; therefore, for any K < 3 + θ , we get the desired bound, C K ,θ |U |γ −K . A similar estimate is valid for |e h≥h β h,β,L − e h,β | ≤ C K |U |L h≥h β |eh,β,L − eh,β |, but we will not belabor the details here. This concludes the proof of the first claim of Corollary 1, concerning the thermodynamic limit of the free energy. We are now left with discussing the zero temperature limit limβ→∞ f β (U ). More precisely, we need to prove that, for any β > β and 0 < K < 4, 0 0 C K |U | |F0,β − F0,β | + (eh,β + eh,β ) − (eh,β + eh,β ) ≤ . (D.20) βK h=h β

h=h β

If we follow step by step the discussion above, leading to the estimate |F0,β,L − F0,β | ≤ C K |U |L −K , we find that, similarly, |F0,β − F0,β | ≤ C K |U |β −K , K > 0; the proof of this bound is based on a decomposition of the difference F0,β − F0,β into a sum of terms involving either integrals over constrained regions (such that l∈T |xl −yl | ≥ δβ) or dif (h) ferences of propagators |gβ (x) − g (h) (x)| ≤ n=0 |g (h) (x0 + nβ, x )| ≤ C K (γ h β)−K ; the technical details are similar to those discussed above for the thermodynamic limit and will not be repeated here. Let us now consider the difference 0h=h β eh,β − 0h=h eh,β : its absolute value β can be bounded from above by h ≤h 0, even for the Anderson model, see [10,11] (a resulting lower bound for the localization length is proven in [5]). In fact, our technique employs results of the type proven in these references as an ingredient of the proof; see Sect. 4.3. We might add that we expect that the model treated in the present paper can also be analyzed using operator-theoretic techniques introduced for the study of return to equilibrium in open quantum systems, see e.g. [2,14], and we are currently working on such a formulation. The technique used in the present paper is largely based on [20].

Quantum Brownian Motion in a Simple Model System

367

1.4. Outline. In Sect. 2, we introduce our model, making precise the description in the Introduction. Then, in Sect. 3, we state our assumptions and main results with as few divagations as possible. Section 4 contains the main ideas of the paper and the plan of the proof. The technical parts of the proof are postponed to Sect. 5, which contains the proof of Theorem 4.4, and Sect. 6, where one finds the proof of Theorem 4.5. 2. Model 2.1. Conventions and notation. Given a Hilbert space E , we use the standard notation B p (E ) := S ∈ B(E ), Tr (S ∗ S) p/2 < ∞ , 1 ≤ p ≤ ∞, (2.1) with B∞ (E ) ≡ B(E ) the bounded operators on E , and 1/ p S p := Tr (S ∗ S) p/2 ,

S := S∞ .

(2.2)

For bounded operators acting on B p (E ), i.e. elements of B(B p (E )), we use in general the calligraphic font: V, W, T , . . .. An operator X ∈ B(E ) determines an operator ad(X ) ∈ B(B p (E )) by ad(X )S := [X, S] = X S − S X,

S ∈ B p (E ).

(2.3)

We will mainly use the case p = 2. The norm of operators in B(B2 (E )) is defined by W :=

W(S)2 . S2 S∈B2 (E ) sup

(2.4)

For vectors κ ∈ Cd , we let κ, κ denote the vectors (κ 1 , . . . , κ d ) and (κ 1 , . . . , √ κ d ), respectively. The scalar product on Cd is written as (·, ·) and the norm as |κ| := (κ, κ). The scalar product on an infinite-dimensional Hilbert space E is written as ·, ·, or, occasionally, as ·, ·E . All scalar products are defined to be linear in the second argument and anti-linear in the first one. We write s (E ) for the symmetric (bosonic) Fock space over the Hilbert space E and we refer to [7] for definitions and discussion. If ω is a self-adjoint operator on E , then its (self-adjoint) second quantization, ds (ω), is defined by ds (ω)Sym(φ1 ⊗ · · · ⊗ φn ) :=

n

Sym(φ1 ⊗ · · · ⊗ ωφi ⊗ · · · ⊗ φn ),

(2.5)

i=1

where Sym projects on the symmetric subspace and φ1 , . . . , φn ∈ E . 2.2. The particle. We set HS = l 2 (Zd ) (the subscript S refers to ‘system’, as is customary in system-reservoir models). We define the one-dimensional projector 1x on HS by (1x f )(x ) := δx,x f (x ),

x, x ∈ Zd , f ∈ l 2 (Zd ).

(2.6)

368

W. De Roeck, J. Fröhlich, A. Pizzo

We will often consider the space HS in its dual representation, i.e. as L 2 (Td , dk), where Td is the d-dimensional torus, which is identified with L 2 ([−π, π ]d ). We define the ‘momentum’ operator P as multiplication by k ∈ Td , i.e., (Pθ )(k) := kθ (k),

θ ∈ L 2 (Td , dk).

(2.7)

Although P is well-defined as a bounded operator, it does not have nice properties; e.g., it is not true that [X i , P j ] = iδi, j . Throughout the paper, we only use operators f (P), where f is periodic on Rd with period 2π , i.e. a function on Td . We choose a periodic function ε to be the dispersion law of the system. Although this is not essential, we require ε to have inversion symmetry, i.e., ε(k) = ε(−k),

k ∈ Td .

(2.8)

The Hamiltonian of our particle is given by HS := ε(P).

(2.9)

Our first assumption ensures that HS is sufficiently regular. Assumption 2.1 (Analyticity of system dynamics). The function ε, defined originally on Td , extends to an analytic function in a strip of width δε > 0. That is, when viewed as a periodic function on Rd , ε is analytic in (R + i[−δε , δε ])d . Moreover, we assume that the function Td k → (υ, ∇ε(k)) does not vanish identically for any vector υ ∈ Rd , υ = 0.

d The most natural choice for ε satisfying Assumption 2.1 is ε(k) = j=1 (2 − j 2 cos(k )), which corresponds to −HS being the discrete Laplacian. 2.3. The reservoirs. 2.3.1. Reservoir spaces. We consider an array of independent reservoirs. With each site x ∈ Zd we associate a one-particle Hilbert space hx (one can imagine that hx = L 2 (Rd )) with a positive one-particle Hamiltonian ωx . The reservoir at x is now described by the Fock space s (hx ) with Hamiltonian ds (ωx ). The full reservoir space is HR := s (⊕x∈Zd hx ) with Hamiltonian HR := ds (ωx ). (2.10) x∈Zd

We choose the different reservoir one-particle spaces to be isomorphic copies of a fixed space h so that ϕ ∈ hx is naturally identified with an element of hx that is also denoted by ϕ without further warning. Likewise, ωx is naturally identified with ωx . Hence, if no confusion is possible we simply write h and ω to denote the (one-particle) one-site space and the Hamiltonian, respectively. For ϕ ∈ h, the operators ax∗ (ϕ)/ax (ϕ) stand for the creation/annihilation operators on the Fock space s (hx ). By the embedding of hx into ⊕ y∈Zd h y , these creation/annihilation operators act on HR in a natural way. They satisfy the commutation relations [ax (ϕ), ax∗ (ϕ )] = δx,x ϕ, ϕ h, where a # stands for either a ∗ or a.

[ax# (ϕ), ax# (ϕ )] = 0,

(2.11)


369

2.3.2. Interaction and initial reservoir state. We pick a ‘structure factor’ φ ∈ h and we choose the interaction between the system and the reservoir at site x to be given by 1x ⊗ x (φ), where x (φ) = ax (φ) + ax∗ (φ).

(2.12)

So far, we have not made any assumptions concerning ω and φ, but their form will be restricted by Assumption 2.2 in (2.20). The particle interacts with all reservoirs in a translation invariant way. Hence the total interaction Hamiltonian is given by HSR :=

1x ⊗ x (φ) on HS ⊗ HR .

(2.13)

x∈Zd

Next, we put the tools in place to describe the positive temperature reservoirs. Let C be the ∗-algebra consisting of polynomials in ax (ϕ), ax∗ (ϕ ), with ϕ, ϕ ∈ h, x, x ∈ Zd . We introduce the positive operator Tβ = (eβω − 1)−1 on h; β should be thought of as the inverse temperature. β We let ρR be a quasi-free state defined on C . It is fully specified1 by 1) Gauge-invariance β β ρR ax∗ (ϕ) = ρR [ax (ϕ)] = 0. 2) Two-point correlation functions β β ρR ax∗ (ϕ)ax (ϕ ) ρR ax∗ (ϕ)ax∗ (ϕ ) β β ρR ax (ϕ)ax (ϕ ) ρR ax (ϕ)ax∗ (ϕ )

0 ϕ |Tβ ϕ . = δx,x 0 ϕ|(1 + Tβ )ϕ )

(2.14)

(2.15)

3) Quasi-freeness, i.e. , the higher-point correlation functions are expressed in terms of the two-point function by β β ρR ax#1 (ϕ1 ) . . . ax#2n (ϕ2n ) = ρR ax#r (ϕr )ax#s (ϕs ) , (2.16) pairings π (r,s)∈π

β # ρR ax#1 (ϕ1 ) . . . a2n+1 (ϕ2n+1 ) = 0,

(2.17)

where a pairing π is a partition of {1, . . . , 2n} into n pairs and the product is over these pairs (r, s). A quantity that will play an important role in our analysis is the on-site-reservoir correlation function defined by β ˆ ψ(t) := ρR x (eitω φ)x (φ) = φ, Tβ eitω φ + φ, (1 + Tβ )e−itω φ.

(2.18)

1 The reason why, in models like ours, it is enough to know the state on C , has been explained in many places, e.g. [1,3,8,13].

370


ˆ It is useful to introduce ψ, the inverse Fourier transform of ψ, 1 ˆ ψ(t) =√ dξ eiξ t ψ(ξ ). 2π

(2.19)

R

As is explained in Appendix A, ψ is the (squared norm of) the effective structure factor. In particular, ψ(ξ ) ≥ 0. The following assumption requires the reservoir to have exponential decay of correlations. Assumption 2.2. There is a decay rate gR > 0 such that gR |t| ˆ < ∞. sup |ψ(t)|e t∈R

(2.20)

We assume that ψˆ ≡ 0, or equivalently ψ ≡ 0. The assumption that ψˆ ≡ 0 ensures that the particle interacts effectively with the fields describing the reservoirs. In Appendix A, we discuss examples of reservoirs that satisfy Assumption 2.2, provided that β < ∞. 2.4. The dynamics. Consider the zero-temperature Hilbert space HS ⊗ HR . The Hamiltonian (with coupling constant λ) is formally defined by Hλ := HS + HR + λHSR .

(2.21)

This operator generates the zero-temperature dynamics. However, we need to consider the dynamics at positive temperature. In particular, we must understand the reduced positive-temperature dynamics of the system S after the reservoir degrees of freedom have been traced out. β By a slight abuse of notation, we use ρR to denote the conditional expectation from B(HS ⊗ C ) to B(HS ) given by β

β

ρR (S ⊗ R) := SρR (R),

S ∈ B(HS ), R ∈ C ,

(2.22)

β

where ρR (R) is defined through (2.15–2.17). Formally, the reduced dynamics in the Heisenberg picture is given by β Ztλ,∗ (S) := ρR eit Hλ (S ⊗ 1) e−it Hλ

(2.23)

whenever the RHS is well-defined. A mathematically precise definition of the reduced dynamics is the subject of the next lemma. Lemma 2.3. Suppose that Assumption 2.2 (see (2.20)) holds and define HSR (t) := 1x (t) ⊗ x (eitω φ) with 1x (t) := eit HS 1x e−it HS . x∈Zd

(2.24)


371

The series2 Ztλ,∗ (S)

:=

n∈Z+

β

(iλ)

dt1 . . . dtn ρR

n 0≤t1 ≤···≤tn ≤t

ad(HSR (t1 )) . . . ad(HSR (tn )) eitad(HS ) (S ⊗ 1)

(2.25)

is well-defined for any λ, t ∈ R and arbitrary S ∈ B(HS ), i.e., the RHS converges absolutely in the norm of B(HS ), and Ztλ,∗ has the expected properties, namely Ztλ,∗ (1) = 1,

Ztλ,∗ (S) ≤ S.

(2.26)

One can prove this lemma (under less restrictive conditions than those in Assumption 2.2) by direct estimates of the RHS of (2.25). For this purpose, the estimates given in the present paper amply suffice. However, one can also define the system-reservoir dynamics as a dynamical system on a Von Neumann algebra through the Araki-Woods representation. This is the usual approach in the mathematical physics literature; see e.g. [8,13,14]. Finally, we define Ztλ : B1 (HS ) → B1 (HS ), the reduced dynamics in the Schrödinger picture, by duality, i.e., Tr[ρS Ztλ,∗ (S)] = Tr[Ztλ (ρS )S],

S ∈ B(HS ), ρS ∈ B1 (HS ).

(2.27)

We could also have started by defining the full initial state ρSR of the total system consisting of the particle and reservoirs as the positive, normalized functional β

ρSR := ρS ⊗ ρR

on B(HS ) ⊗ C ,

(2.28)

where we abuse notation by employing the same symbol ρS for both the density operator (a positive element of B1 (HS )) and the state it determines on B(HS ), i.e., ρS [S] := Tr[ρS S],

S ∈ B(HS ).

(2.29)

Then, ρSR eit Hλ (S ⊗ 1) e−it Hλ = Tr[Ztλ (ρS )S].

(2.30)

In what follows, we simply write ρ for ρS . For convenience, we treat ρ as an element of the Hilbert space B2 (HS ), which is justified since B1 (HS ) ⊂ B2 (HS ). 2 In fact, one needs to do things more carefully, since H (t) ∈ / C . A possible solution is to define the SR

cut-off interaction HS−R, (t) = x∈ 1x (t) ⊗ x (eitω φ), for some finite subset ⊂ Zd , and to show that one can take the limit Zd in the expression analogous to (2.25).

372


3. Result We now state our main results. Recall that the position operator X on l 2 (Zd ) is given by (X f )(x) = x f (x), For κ ∈

Cd ,

x ∈ Zd , f ∈ l 2 (Zd ).

(3.1)

we define i

i

Jκ S := e− 2 (κ,X ) S e− 2 (κ,X ) ,

S ∈ B(HS ).

(3.2)

Note that Jκ is unbounded if κ ∈ / Rd . We choose an initial state ρ ∈ B1 (HS ) satisfying ρ > 0,

Tr[ρ] = 1

Jκ ρ2 < ∞,

(3.3)

Cd .

for κ in some open neighborhood of 0 ∈ Our first result says that the momentum distribution of the particle tends to a stationary distribution exponentially fast. Theorem 3.1 [Equipartition Theorem]. Suppose that Assumption 2.1 (see Sect. 2.2) and Assumption 2.2 (see (2.20)) hold, and let ρ satisfy condition (3.3). There are positive constants λ0 > 0 and g > 0 such that for 0 < |λ| ≤ λ0 , there is a function ζλ0 ∈ L 2 (Td ) satisfying Tr[θ (P)Ztλ (ρ)] = θ, ζλ0 L 2 (Td ) + O(θ 2 e−λ

2 gt

∞

),

as t ∞,

for any θ = θ ∈ L (T ), d

(3.4)

and e−βε(k) + O(λ2 ), −βε(k) Td dk e

ζλ0 (k) =

λ 0.

(3.5)

The decay rate λ2 g is strictly smaller than gR , introduced in (2.20). Define a probability density µλt depending on the initial state ρ ∈ B1 (HS ) by µλt (x) := Tr 1x Ztλ (ρ) . (3.6) It is easy to see that µλt (x) ≥ 0,

µλt (x) = Tr[ρ] = 1.

(3.7)

x∈Zd

We claim that the particle exhibits a diffusive motion. This is the content of the next result. Theorem 3.2 [Diffusion] Under the same assumptions as in Theorem 3.1, the following holds. Let the initial state ρ satisfy condition (3.3) and let µλt be as defined in (3.6). There is a positive constant λ0 such that, for 0 < |λ| ≤ λ0 , 1 − √i (q,x) µλt (x)e t −→ e− 2 (q,Dλ q) , q ∈ Rd , (3.8) x∈Zd

t∞

where the diffusion matrix Dλ is positive-definite (i.e., has strictly positive eigenvalues), and λ 0, (3.9) Dλ = λ−2 Dkin + O(λ2 ) , with Dkin a λ-independent positive-definite matrix introduced in Sect. 4.3.


373

We refer to Sect. 1 for an explanation of the connection between this result and diffusion in the physicists’ sense. We close this section with some remarks concerning possible extensions of our results. Remark 3.3. Our proof of Theorem 3.2 actually gives a stronger result. Assume the n th moments of the initial distribution are bounded, or, equivalently, q → µλ0 (x)e−i(q,x) is n times differentiable. (3.10) x∈Zd

Then the rescaled n th moments converge to the n th moments of the limiting distribution, or equivalently, the derivatives of n th order of − √i (q,x) q → µλt (x)e t (3.11) x∈Zd

converge, as t ∞, to the derivatives of e−(q,Dλ q) . For n = 2, this implies (1.19). Note that the condition (3.10) is a weaker assumption than (3.3); in fact, (3.3) implies that (3.10) is a real-analytic function. Remark 3.4. By the same technique as employed in our proofs, one can show that correlations decay rapidly in time. As explained in the Introduction, this rapid decay provides an intuitive explanation why the particle motion is diffusive. Define the particle velocity operator by V (t) := ieit Hλ [Hλ , X ]e−it Hλ ,

(3.12)

V (0) = i[Hλ , X ] = i[HS , X ] = (∇ε)(P).

(3.13)

and observe that

Suppose that Assumptions 2.1 and 2.2 hold and let ρ = ρS satisfy condition (3.3). By reasoning similar to that in Lemma 2.3, one can define the velocity-velocity correlation function ρSR [V (t1 )V (t2 )]. Let the coupling strength λ and the positive constant g be as in Theorem 3.1. Then, for all 0 ≤ t1 , t2 < ∞, |ρSR [V (t1 )V (t2 )]| ≤ c e−λ

2 g|t

2 −t1 |

,

for some c < ∞.

(3.14)

Remark 3.5. The condition that the particle dispersion satisfies ε(k) = ε(−k) is not really necessary for our results to hold. If one did not impose this condition, the particle could have a drift velocity vdr given by vdr := ∇ε, ζλ0 ,

(3.15)

and the particle motion would still be diffusive, but one would now consider the “random variable” √1t (xt − vdr t), instead of √1t xt . In other words, in (3.8), one would have to replace − √i (q,x) − √i (q,(x−vdr t)) µλt (x)e t by µλt (x)e t . (3.16) x∈Zd

x∈Zd

Similarly, in Eq. (3.14), one would have to replace V (t) by V (t) − vdr .

374


4. Discussion and Outline of the Proof 4.1. Translation invariance. Consider the space of Hilbert-Schmidt operators B2 (HS ) ∼ B2 (l 2 (Zd )) ∼ L 2 (Td × Td , dk1 dk2 ), and define ˆ 1 , k2 ) := 1 S(k S(x1 , x2 )e−i(x1 ,k1 )+i(x2 ,k2 ) , S ∈ B2 (l 2 (Zd )). (4.1) (2π )d d x1 ,x2 ∈Z

ˆ To deal conveniently with the translation In what follows, we simply write S for S. invariance in our model, we make the change of variables k=

k1 + k2 , 2

p = k1 − k2 ,

(4.2)

and, for a.e. p ∈ Td , we obtain a well-defined function S p ∈ L 2 (Td ) by putting (S p )(k) := S(k +

p p , k − ). 2 2

(4.3)

This follows from the fact that the Hilbert space B2 (HS ) ∼ L 2 (Td × Td , dk1 dk2 ) can be represented as a direct integral p dp H , S= d p Sp, (4.4) B2 (HS ) = ⊕Td

where each ‘fiber space’ H lattice translation

p

⊕Td

is naturally identified with L 2 (Td ). Let Tz , z ∈ Zd , be the

(Tz S)(x1 , x2 ) := S(x1 + z, x2 + z),

S ∈ B(HS ),

(4.5)

or, equivalently, (Tz S) p (k) = ei( p,z) S p ,

S ∈ B(HS ).

(4.6)

β

Since Hλ and ρR are translation invariant, it follows that T−z Ztλ Tz = Ztλ .

(4.7)

Let W ∈ B(B2 (HS )) be translation invariant in the sense of Eq. (4.7), i.e., T−z WTz = W. Then it follows that, in the representation defined by (4.4), W acts diagonally in p, i.e. (W S) p depends only on S p , and we define W p by (W S) p = W p S p .

(4.8)

For the sake of clarity, we give an explicit expression for W p . Define the kernel W(x, y; x , y ) by (W S)(x , y ) = W(x, y; x , y )S(x, y), x , y ∈ Zd . (4.9) x,y∈Zd

Translation invariance is expressed by W(x, y; x , y ) = W(x + z, y + z; x + z, y + z),

z ∈ Zd ,

(4.10)


375

and, as an integral kernel, W p ∈ B(L 2 (Td )) is given by W p (k , k) =

i

ei(k,x−y)−i(k ,x −y ) e 2 ( p,(x +y )−(x+y)) W(x, y; x , y ).

x, y, x , y ∈ Zd x =0

(4.11)

Next, we state an easy lemma. Lemma 4.1. Let S ∈ B1 (HS ). Then, S p , as defined in (4.3), is well-defined as a function in L 1 (Td ) for every p, and e−i px S(x, x) = 1, S p , (4.12) Tr[J p S] = x∈Zd

where 1 ∈ L 2 (Td ) ∩ L ∞ (Td ) is the constant function with value 1(k) = 1. Assume, moreover, that there is a constant δ > 0 such that Jκ S2 < ∞

for

|κ| < δ,

(4.13)

then the function p → S p ∈ L 2 (Td ) has a bounded-analytic extension to the strip | p| < δ. The first statement of the lemma follows from the singular-value decomposition for trace-class operators and standard properties of the Fourier transform. In fact, the correct statement asserts that one can choose S p such that (4.12) holds. Indeed, one can change the value of the kernel S(k1 , k2 ) on the line k1 − k2 = p without changing the operator S, and hence S p in (4.12) can not be defined via (4.3) for all p, but only for almost all p. The second statement of Lemma 4.1 is the well-known relation between exponential decay of functions and analyticity of their Fourier transforms. Since we will always demand the initial density matrix ρ0 to be such that Jκ ρ0 2 is finite for κ in a complex domain, we will mainly need the second statement of Lemma 4.1.

4.2. Return to equilibrium inside the fibers. The main idea of our proof is that the reduced evolution in the ‘low momentum fibers’, (Ztλ ) p , for p near 0, has an invariant state to which every well-localized initial state relaxes exponentially fast. Recalling that HS = ε(P) and that the system is weakly coupled to a heat bath at inverse temperature β, we expect that, in an appropriate sense, and for arbitrary initial states ρ ∈ B1 (HS ), Ztλ (ρ) “−→ t↑∞

1 −βε(P) e + o(λ0 ), Z (β)

λ 0.

(4.14)

We observe that e−βε(P) ∈ / B1 (HS ), hence (4.14) cannot hold in norm (in other words, Z (β) = ∞). One way to interpret (4.14) is that it gives the correct asymptotic expectation value of functions of the momentum, and that is exactly what Theorem 3.1 states. For every ρ satisfying (3.3), we have that Tr[θ(P)Ztλ (ρ)] = θ, (Ztλ ρ)0 ,

θ ∈ L ∞ (Td ),

(4.15)

376


by applying Lemma 4.1 with S := θ(P)Ztλ (ρ). Hence, we should apparently attempt to prove ‘return to equilibrium’ for the evolution (Ztλ )0 on L 2 (Td ). The dynamics in the fibers corresponding to small values of p provides information on the diffusive character of the system. The probability density µλt (x) corresponding to some initial state ρ is defined as in (3.6). By Lemma 4.1, µλt (x)e−i( p,x) = (Ztλ ρ)(x, x)e−i( p,x) x∈Zd

x∈Zd

dk(Ztλ ρ)(k +

= Td

p p , k − ) = 1, (Ztλ ρ) p . 2 2

(4.16)

To establish diffusion, it suffices to show that, for λ fixed and for p in a neighborhood of 0 ∈ Td , 1

1, (Ztλ ρ) p = et (− 2 ( p,Dλ p)+o( p )) (1 + o(t 0 ) + o( p 0 )), 2

t ∞, p 0,

(4.17)

for some positive-definite matrix Dλ . Indeed, by (4.16), Theorem 3.2 follows from (4.17) by taking p = √qt . Thus, in order to prove Theorem 3.2, we are led to study the long-time

asymptotics of the evolution (Ztλ ) p , for small p. However, as our approach is perturbative in λ, expression (4.17) is not a good starting point, since ( p, Dλ p) = O(λ−2 ), for fixed p (as can be seen from the statement of Theorem 3.2), and hence one cannot perturb around ( p, Dλ p)λ=0 . The way out of this difficulty is to set up the perturbation on a scale where the diffusion constant is finite (this will turn out to be the kinetic scale), or, in other words, to take the p-neighborhood in (4.17) to shrink, as λ 0. Since λ approaches 0, one must wait a time of order λ−2 , before one sees the effect of the interaction. Since, between collisions, the velocity of the free particle is unaffected, it travels a distance of order λ−2 . This means that when both space and time are measured in units of λ−2 ; x = λ−2 x˜λ , we expect a diffusion constant D˜ λ ∼ x2 t

(x˜λ )2 t˜λ

is of order λ−2 . The limit

fact that Dλ ∼ limit, as outlined in the next section.

t = λ−2 t˜λ ,

(4.18)

of order O(1). This is consistent with the D˜ λ0 is the diffusion constant in the kinetic

4.3. The kinetic limit. To control the asymptotics of the effective time-evolution (Ztλ ) p , we compare it with the corresponding evolution in the kinetic limit, which is the limit approached when microscopic space and time are taken to be λ−2 x, λ−2 t, respectively, and the coupling strength λ → 0; as announced in the previous section. It has been proven in [9] (for models with only one thermal reservoir) that, in this limit, the dynamics is described by a linear Boltzmann equation. Our variant of this result is described below. 4.3.1. Convergence to a linear Boltzmann equation. The effective reservoir structure factor ψ has been defined in (2.18–2.19). For convenience, we introduce a positive function r (·, ·), with r (k, k ) := ψ[ε(k ) − ε(k)] ≥ 0.

(4.19)


377

For κ ∈ Rd , we define a bounded linear operator, M κ , on L 2 (Td ) by (M κ θ )(k) := i(κ, ∇ε)(k)θ (k) + dk r (k , k)θ (k ) − r (k, k )θ (k) ,

θ ∈ L 2 (Td ), (4.20)

Td

where (κ, ∇ε)(k) stands for the scalar product in Cd of κ and ∇ε(k). The operator M κ has a straightforward interpretation: Consider a classical particle whose states are specified by a position x ∈ Rd and a ‘momentum’ k ∈ Td . The momentum k evolves according to a Poisson process with a rate r (k, k ) for the transition from state k to k . Between two momentum jumps, the particle moves freely, with speed given by (∇ε)(k). The translation of this picture into a mathematical statement is as follows: The state-space distribution of the classicalparticle at time t is given by a probability density νt (·, ·) on Rd × Td ; (ν(x, k) ≥ 0 and dxdk νt (x, k) = 1). Then ∂ νt (x, k) = (∇k ε, ∇x νt )(x, k)+ dk r (k , k)νt (x, k )−r (k, k )νt (x, k) . (4.21) ∂t Td

One checks that νˆ tκ (k) := (2π )−d/2

dx e−i(κ,x) νt (x, k)

(4.22)

Rd

satisfies an evolution equation generated by M κ ; ∂ κ νˆ = M κ νˆ tκ . ∂t t

(4.23)

We claim that the rates r (k, k ) satisfy the identity

r (k, k ) = r (k , k)e−β(ε(k )−ε(k)) ,

(4.24)

known as the detailed balance condition in the context of Markov processes. It is a direct consequence of the KMS-condition for the reservoirs. In our context, it is easily derived from (2.15). The detailed balance condition implies that 0 M 0 ζkin = 0,

e−βε(k) . −βε(k) Td dke

0 where ζkin (k) =

(4.25)

0 is a stationary state. In the language of Markov processes, ζkin κ The relevance of M is that it describes the evolution Zλλ−2 t in the fiber indexed by λ2 κ in the limit λ 0. Moreover, the convergence of the fiber dynamics (Zλλ−2 t )λ2 κ holds even after analytic continuation to complex κ. One can prove the following result

Proposition 4.2. Assume Assumptions 2.1 and 2.2. Then, for |κ| sufficiently small and 0 < t < ∞, κ λ (4.26) (Zλ−2 t )λ2 κ − et M −→ 0, λ0

where the norm is the operator norm on L 2 (Td ).

378


We do not prove this proposition (which is not needed for the proof of our results). In fact, the proof is based on the same reasoning as in Sect. 6. Of course, one can also express Proposition 4.2 in terms of the rescaled Wigner function, as is done in [9,12]. Indeed, setting κ κ αˆ tκ (k) := lim Zλλ−2 t ρ (k + λ2 , k − λ2 ) = lim (Zλλ−2 t ρ)λ2 κ (k), (4.27) λ0 λ0 2 2 one obtains from Proposition 4.2 that αˆ tκ (k) satisfies the evolution equation (4.23). (It would thus be justified to call αˆ tκ (k) simply νˆ tκ (k)). Its inverse Fourier transform −d/2 dκ ei(κ,x) αˆ tκ (k) (4.28) αt (x, k) = (2π ) Rd

is a probability density on Rd × Td and satisfies (4.21) with initial condition α0 (x, k) = δ(x)ρ(k, k). We state another useful consequence of Proposition 4.2. Recall that the probability density µt (·) has been defined in (3.6), for any initial state ρ. Taking the scalar product with 1 ∈ L 2 (Td ) on both sides of (4.27) and using (4.16), we obtain that −iλ2 (κ,x) λ e µλ−2 t (x) −→ dk αˆ tκ (k). (4.29) x∈Zd

λ0

Td

As outlined in Sect. 4.2, the t ∞ asymptotics of the LHS of (4.29) contains information on the diffusive behavior of the particle. In the next section we discuss the t ∞ asymptotics of the RHS of (4.29). 4.3.2. Diffusive behavior of solutions of the Boltzmann equation. To realize that the Boltzmann equation describes diffusion, one studies the spectral properties of M κ , for small κ. We state a crucial result, Theorem 4.3, and we refer the reader to [6] for complete proofs and a more extended discussion of quantum dissipative evolutions. Theorem 4.3. Suppose that Assumptions 2.1 and 2.2 hold, and let M κ ∈ B(L 2 (Td )) be defined as in (4.20). Then there is a positive constant δkin such that the operator M κ , with |κ| ≤ δkin , has a simple eigenvalue, f kin (κ), separated from the rest of the spectrum by a gap, dist( f kin (κ), ) =: gkin > 0, where

:= ∪|κ| sup spT consists of isolated eigenvalues of finite multiplicity. From the pointwise positivity of r (·, ·), the Perron-Frobenius theorem and from the fact that M 0 generates a contractive semigroup on L 1 (Td ) we then conclude that the eigenvalue 0 of M 0 is simple and that it is separated 0 is explicitly given by a gap from the rest of the spectrum. The spectral projection Pkin by 0 0 θ = 1, θ ζkin , Pkin

θ ∈ L 2 (Td )

(4.36)

0 as in (4.25). The analyticity of f (κ) and ζ κ is proven with the help of with ζkin kin kin analytic perturbation theory. Using the assumption that ε(k) = ε(−k), we check that 0 0 Pkin ∇ε Pkin = 0.

(4.37)

Employing explicit expressions of second order perturbation theory, we obtain formula (4.32) as a consequence of the fact that M κ − M 0 = i(κ, ∇ε) and (4.37). Since f kin (κ) = f kin (−κ), it follows that the matrix Dkin has real entries. The positive-definiteness of Dkin is established as follows. Consider the bounded operator 1

(W θ )(k) = e 2 βε(k) θ (k),

θ ∈ L 2 (Td ),

(4.38)

and notice that M˜ := W −1 M 0 W is a self-adjoint operator on L 2 (Td ), in particular 0 = W −1 1, (i.e., the left and right eigenvector corresponding to the eigenζ˜ := W ζkin value 0 are identical). For κ ∈ Rd , we can rewrite (4.33) as −1 ˜ ˜ ˜ κ, ∇ε ζ . κ, Dkin κ = − κ, ∇ε ζ , M (4.39) By Assumption 2.1, the function k → (κ, ∇ε(k)) does not vanish identically on Td ˜ expression (4.39) is strictly (for κ = 0). Hence, by the spectral theorem applied to M, positive. ! "

380


Let νˆ tκ (k) be a solution of the evolution equation (4.23) for κ in some neighborhood of 0 in Cd . Using Theorem 4.3 and reasoning similar to that in Sect. 4.2, it follows that 1 dk νˆ tκ (k) −→ e− 2 (q,Dkin q) , q ∈ Rd . (4.40) q κ= √ , t∞ t

Td

Hence a solution νt (x, k) of the Boltzmann equation (4.21) behaves diffusively, with diffusion tensor Dkin . 4.4. Perturbation around the kinetic limit. Up to now, we have seen that, in the kinetic limit, the particle motion is described by a linear Boltzmann equation. Since solutions of the linear Boltzmann equation behave diffusively for large times (as is essentially stated in Theorem 4.3), we can associate a diffusion constant to our model. Indeed, by (4.29) and (4.40), lim lim

t∞ λ0

x∈Zd

µλλ−2 t (x)e

2

λ −i √ (q,x) t

1

= e− 2 (q,Dkin q) .

(4.41)

However, (4.41) does not give information on the long-time asymptotics of our system for small, but fixed |λ| > 0. The least one would wish for is to be able to exchange the order of limits in (4.41), and, indeed, Theorem 3.2 states that one can do so without affecting the RHS. We stress this point, because it is an improvement of our paper when compared to most earlier results on diffusion. κ Since we have learned that (Zλλ−2 t )λ2 κ has a well-defined limit, et M , as λ 0, (see Proposition 4.2), it is natural to expand (Zλλ−2 t )λ2 κ around this limit, in such a way that we can take t ∞. We perform the expansion on the Laplace transform of Ztλ , Rλ (z) := dt e−t z Ztλ . (4.42) R+

Theorem 4.4 below summarizes the result of our expansion. Loosely speaking, a key consequence of this theorem is the fact that, in the fibers indexed by λ2 κ, one has that (Rλ (z))λ2 κ = (z − λ2 M κ − A(z, λ, κ))−1 ,

(4.43)

where the operator A(z, λ, κ) is “small” compared to λ2 M κ . Theorem 4.4. Suppose that Assumptions 2.1 and 2.2 in Sect. 2 hold. Then, there are operators L(z) and Rex λ (z) in B(B2 (HS )) such that the following statements hold: 1) For (z, λ) ∈ C × R satisfying z > λ2 L(z) + Rex λ (z), −1 Rλ (z) = (z − ad(iHS ) − λ2 L(z) − Rex λ (z)) .

(4.44)

2) The operators L(z) and Rex λ (z) have the following properties: There are positive constants δ1 , δ2 , g > 0 such that Jκ L(z)J−κ ,

Jκ Rex λ (z)J−κ

(4.45)


381

are analytic in the variables (z, κ) ∈ C × Cd in the region defined by |κ| ≤ δ1 , z > −g , |λ| ≤ δ2 . Moreover, sup

Jκ L(z)J−κ = O(1),

λ 0,

(4.46)

4 Jκ Rex λ (z)J−κ = O(λ ),

λ 0,

(4.47)

|κ|≤δ1 ,z>−g

sup

|κ|≤δ1 ,z>−g

where · refers to the operator norm on B(B2 (HS )) (as in (2.4)). 3) Let M κ be defined as in Sect. 4.3. Then λ2 κ 0, λ 0. ad(iHS ) + λ2 L(0) 2 −λ2 M κ = O(λ4 κ 2 )+ O(λ4 κ), λ κ

(4.48) The proof of Theorem 4.4 is the subject of Sect. 5. From that proof, it becomes clear that g can be chosen to be any fraction of gR by making δ1 and δ2 small enough. From Theorem 4.4, one obtains our main result by using Theorem 4.3 and standard analytic perturbation theory. More precisely, we prove the following theorem. Theorem 4.5. Suppose that Assumptions 2.1 and 2.2 in Sect. 2 hold. Then, there are positive constants δ1 , δ2 , g > 0 such that, for (λ, κ) ∈ R × Cd and |κ| ≤ δ1 , 0 < |λ| ≤ δ2 , there is a rank 1 operator P λ,κ and a function f (λ, κ) satisfying (Ztλ )λ2 κ − et f (λ,κ) P λ,κ = O(et ( f (λ,κ)−λ

2 g)

),

t ∞

(4.49)

and κ P λ,κ − Pkin = O(λ2 ),

| f (λ, κ) − λ2 f kin (κ)| = O(λ4 ),

λ 0 (4.50)

Moreover, P λ,κ and f (λ, κ) are analytic in κ ∈ Cd in the region defined by |κ| ≤ δ1 , |λ| ≤ δ2 . By making δ2 small enough, the constant g can be chosen to be any fraction of gkin and δ1 can be chosen to be given by δkin , with gkin , δkin as in Theorem 4.3. Theorems 3.1 (Equipartition Theorem) and 3.2 (Diffusion ) then follow as discussed in Sect. 4.2. We briefly recapitulate our reasoning. Proof of Theorems 3.1 and 3.2. We first prove Theorem 3.1. Using (4.15) and Theorem 4.5, we write, for θ = θ ∈ L ∞ (Td ), Tr[θ (P)Ztλ ρ] = θ, (Ztλ ρ)0 = θ, et f (λ,0) P λ,0 ρ0 + O(et ( f (λ,0)−λ

2 g)

). (4.51)

Since Ztλ ρ has trace 1 (it is a density matrix) for all t ≥ 0, we deduce that f (λ, 0) = 0 and, setting θ = 1, 1, P λ,0 ρ0 = 1.

(4.52)

The fact that P λ,0 is a rank 1 operator (by Theorem 4.5) implies, together with (4.52), that, P λ,0 η = ζλ0 1, η,

for any η ∈ L 2 (Td ),

for some ζλ0 ∈ L 2 (Td ) which satisfies 1, ζλ0 = 1. Theorem 3.1 follows.

(4.53)

382


We define the diffusion matrix by (Dλ )i, j := −λ−4

∂2 f (λ, κ)κ=0 , ∂κ i ∂κ j

i, j = 1, . . . d.

(4.54)

From (4.12), with S := Ztλ ρ, we conclude that f (λ, κ) = f (λ, −κ), and hence the matrix Dλ has real entries. Positive-definiteness of Dλ follows then from positivedefiniteness of Dkin , for λ small enough. Using Theorem 4.5, we find that

e

− √i (q,x) λ t µt (x)

= 1, (Ztλ ρ) √q

(4.55)

t

x∈Zd

q with κ = λ−2 √ , q ∈ Rd t

= 1, (Ztλ ρ)λ2 κ ,

= 1, et f (λ,κ) P λ,κ ρλ2 κ (1 + O(e−gt )), −t (λ4 21 (κ,Dλ κ)+O(κ 3 ))

= 1, e as κ 0 1

P

λ,0

as t ∞

ρ0 (1 + O(κ))(1 + O(e−gt )),

= 1, e− 2 (q,Dλ q)+O(tκ ) P λ,0 ρ0 (1 + O(κ))(1 + O(e−gt )), 3

which proves Theorem 3.2 upon using 1, P λ,0 ρ0 = 1 and κ = λ−2 √qt .

" !

Remark 3.3 follows by standard reasoning, using the following facts: 1) The family of operators (Ztλ )λ2 κ − et f (λ,κ) P λ,κ

(4.56)

is analytic in κ in a neighborhood of 0 ∈ Cd and bounded by a constant independent of κ and t. 2) The function f (λ, κ) and the rank 1 operator P λ,κ are analytic in κ in a neighborhood of 0 ∈ Cd . This is related to the general fact that the central limit theorem follows from the existence and analyticity of the large deviation generating function, as described in [4]. Indeed, κ → f (λ, κ) can be viewed as the large deviation generating function corresponding to the family of random variables xt , t > 0, as defined in (1.17). 5. Dyson Expansion and Proof of Theorem 4.4 To construct a “polymer model”, we first write a Dyson expansion for Ztλ . 5.1. Dyson expansion. In this section, we set up a convenient notation to handle the Dyson expansion, which has been introduced in Lemma 2.3. Define the unitary group Ut on B2 (HS ) by Ut S := e−it HS Seit HS ,

S ∈ B2 (HS ),

(5.1)


383

Fig. 5.1. Graphical representation of a term contributing to the RHS of (5.3) with π = {(1, 3), (2, 4), (5, 8), (6, 10), (7, 11), (9, 12)} ∈ P6 . The times ti correspond to the position of the points on the horizontal axis Starting from this graphical representation, we can reconstruct the corresponding term in (5.3) - an operator on B2 (HS ))- as follows: • • •

To each straight line between the points (ti , xi , li ) and (ti+1 , xi+1 , li+1 ), one associates the operators Uti+1 −ti . To each point (ti , xi , li ), one associates the operator λ2 Ixi ,li , defined in (5.2). To each curved line between the points (tr , xr , lr ) and (ts , xs , ls ), with r < s, we associate the factor ) ˆ s − tr ) lr = L ψ(t δxr ,xs ˆ − t )) ψ(−(t s r lr = R.

Rules like these are commonly called “Feynman rules” by physicists.

and the operators Ix,l , with x ∈ Zd and l ∈ {L , R} (L , R stand for “left” and “right”), as if l = L i 1x S Ix,l S := (5.2) if l = R. −i S1x Let Pn be the set of partitions π of the integers 1, . . . , 2n into n pairs. We write (r, s) ∈ π if (r, s) is one of these pairs, with the convention that r < s. Note that the same notation was already used in (1.22) and in (2.16). Elements in R2n , (Zd )2n , {L , R}2n are denoted by t, x, l, with ti , xi , li their respective components, for i = 1, . . . , 2n. We evaluate (2.25) by using (2.15) and (2.16)-(2.17): Ztλ=

n∈Z+

λ2n

2n ( dti ) ζπ (t, x, l) Ut−t2n Ix2n ,l2n . . . Ix2 ,l2 Ut2 −t1 Ix1 ,l1 Ut1 ,

0≤t1 ...≤t2n ≤t i=1

x,l π∈Pn

(5.3) where ζπ (t, x, l) :=

(r,s)∈π

) δxr ,xs

ˆ s − tr ) ψ(t lr = L , ˆ ψ(−(ts − tr )) lr = R,

(5.4)

and, for n = 0, the integral in (5.3) is meant to be equal to Ut . We introduce some more terminology, extending the above definition of pairings. It will be helpful in classifying the pairings. Definition 5.1. 1) Let n be the set of sets of n pairs of (distinct) natural numbers. More concretely, for each σ ∈ n , we can write σ = {(r1 , s1 ), . . . , (rn , sn )} , ri , si ∈ N,

(5.5)

384


... Fig. 5.2. Graphical representation of a pairing π ∈ P9 . The pair (r, s) belongs to π whenever the natural numbers r, s are connected by an arc. This type of diagrams differs from those of Fig. 5.1 in that we don’t keep track of the ti -coordinates, but only of the topological structure of the pairings. Below is the decomposition of π into irreducible components

for natural numbers ri , si , i = 1, . . . , n which are all distinct. By convention, ri < si , i = 1, . . . , n and ri < ri+1 , i = 1, . . . , n − 1. If σ1 ∈ n 1 and σ2 ∈ n 2 , we write σ1 < σ2 whenever all elements of the pairs (ri1 , si1 ) in σ1 are smaller than all elements of the pairs (r 2j , s 2j ) in σ2 , i.e., si1 < r 2j ,

i = 1, . . . , n 1 , j = 1, . . . , n 2 .

(5.6)

2) Recall the definition of Pn , the set of pairings with n pairs. Obviously Pn ⊂ n , n {r , s } = {1, . . . , 2n}. Further, with any and σ ∈ n belongs to Pn whenever ∪i=1 i i σ ∈ n , we associate the unique pairing π ∈ Pn for which there is a monotone increasing function q on {1, . . . , 2n} such that (i, j) ∈ π ⇔ (q(i), q( j)) ∈ σ.

(5.7)

3) We set P := ∪n≥1 Pn and write |π | = n whenever π ∈ Pn . 4) We call σ ∈ n irreducible (Notation: irr. ) whenever there are no two sets σ1 ∈ n 1 , σ2 ∈ n 2 , n 1 +n 2 = n such that σ = σ1 ∪σ2 and σ1 < σ2 . For any σ ∈ n that m σ = σ) is not irreducible, we can thus find partitioning subsets σ1 , . . . , σm (∪i=1 i such that σi=1,...,m are irreducible and σi < σi+1 for i = 1, . . . , m − 1. 5) Consider some π ∈ P and its partitioning into irreducible subsets σ1 , . . . , σm , as defined above. By (5.7), we can associate to each of the σi a unique πi in P. We call the set (π1 , . . . , πm ) of pairings, obtained in this way the decomposition of π into irreducible components. 6) For each n ∈ N, we define a distinguished pairing π ∈ Pn , which is called the minimally irreducible pairing (Notation: min.irr. ). For n > 2, this minimally irreducible pairing is given by (r1 , s1 ) = (1, 3), (rn , sn ) = (2n − 2, 2n), (ri+1 , si+1 ) = (2i, 2i + 3), for i = 1, . . . , n − 2. (5.8) For n = 1 and n = 2, the minimally irreducible pairing is defined to be (1, 2) and {(1, 3), (2, 4)} respectively. Intuitively, the minimally irreducible pairing in Pn is characterized by the fact that if one removes any pair, other than the pair with r = 1 or s = 2n, the resulting pairing is no longer irreducible. For an irreducible pairing π ∈ Pn , we introduce (using the same conventions as in (5.3), (5.4)), Vt (π ) :=

(

2n−1

0=t1 ≤...≤t2n =t i=2

dti )

ζπ (t, x, l) Ix2n ,l2n Ut−t2n−1 . . . Ix2 ,l2 Ut2 −t1 Ix1 ,l1 .

x,l

(5.9)


π1

π2

385

π4

π3

Fig. 5.3. The irreducible components π1 , π2 , π3 , π4 . Explicitly, π1 = π2 = {(1, 2)}, π3 = {(1, 6), (2, 3), (4, 7), (5, 8)} and π4 = {(1, 3), (2, 5), (4, 6)}. The pairings π1 , π2 and π4 are minimally irreducible, whereas π3 is not. Indeed, one can remove the pair (4, 7) from π3 without destroying the irreducibility

We can now rewrite (5.3) as a sum over collections of irreducible pairings; Ztλ = dt1 . . . dt2m m∈Z+ 0≤t ...≤t ≤t 1 2m

m (2 i=1 |πi |)

λ

Ut−t2m Vt2m −t2m−1 (πm ) . . . Ut3 −t2 Vt2 −t1 (π1 )Ut1 .

(5.10)

π1 , . . . , πm ∈ P π1 , . . . , πm irr.

To obtain this last expression, we decompose each pairing π in (5.3) into its irreducible components π1 , . . . , πm , and we made use of a simple factorization property of (5.3). The term on the RHS of (5.10) corresponding to m = 0 is understood to be equal to Ut . In expression (5.10), we view the pairings πi with |πi | ≥ 2 as excitations. If |πi | = 1, for all i = 1, . . . , m, the corresponding term in (5.10) is called a ladder diagram. These ladder diagrams provide the leading contribution to the dynamics, and they are the only terms that survive in the kinetic limit. We define separately the Laplace transforms of the irreducible “excitation” diagrams (Rex λ ) and the irreducible “ladder” diagram (L): Rex dt e−t z λ2|π | Vt (π ), (5.11) λ (z) := |π | ≥ 2 π irr.

R+

L(z) :=

dt e R+

−t z

|π |=1

Vt (π ) =

dt e−t z Vt ({(1, 2)}).

(5.12)

R+

Here and in what follows, we omit the specification π ∈ P under the summation symbol. We observe that, in (5.12), the only element of P1 is the set containing the single pair (1, 2). The operators Rex λ (z) and L(z) have already appeared in Theorem 4.4. We will prove Theorem 4.4 in Sect. 5.3. First, we establish some helpful estimates. 5.2. Estimates on the Dyson expansion. 5.2.1. A priori estimates. The following Lemma 5.1 is a useful a-priori estimate. Its main assertion, Statement 2), i.e., Eq. (5.14), gives a bound on Vt (π ), the contribution of the irreducible pairing π to the dynamics, in terms of the temporal coordinates t. In particular, the sum over the other coordinates, x and l is already performed. This is possible because the matrix elements of the free dynamics (e−it HS )(0, x) decay exponentially in space, for fixed t; (see Statement 1 of Lemma 5.1, or Eq. (5.17)). Equation (5.18) tells us that one can sum over x at the cost of introducing an exponential growth in time. This exponential growth in time is also visible in (5.14), in the factor e2tcε (γ1 ) . However, this exponential growth is harmless, because the reservoir correlation functions ψˆ on the

386


RHS of (5.14) are exponentially decaying in time, by Assumption 2.2, and the growth constant cε (γ1 ) can be chosen arbitrarily small. In particular, it can be chosen smaller than the reservoir decay rate gR , and this fact will be exploited in Sect. 5.3.2. Lemma 5.1. Suppose that Assumption 2.1 holds (with some δε > 0) and define cε (δ) := supk∈Td sup|κ|≤δ |ε(k + κ)|, (cε (δ) < ∞, for 0 < δ < δε ),

bd (δ) := x∈Zd e−δ|x| , (bd (δ) < ∞, for 0 < δ). Then the following statements hold true: 1) For any κ ∈ Cd with |κ| ≤ γ1 , for some γ1 < δε , ei(κ,X ) e−itε(P) e−i(κ,X ) ≤ etcε (γ1 ) ,

t ≥ 0.

(5.13)

2) Let π ∈ Pn , and choose constants 0 < γ < γ1 < δε . For any κ ∈ Cd satisfying |κ| ≤ γ1 − γ , ⎧ ⎪ b (2γ ) [bd (γ1 − γ − |κ|)]2n 22n e2tcε (γ1 ) ⎪ ⎨ d 2n−1

Jκ Vt (π ) J−κ ≤ (5.14) * * ⎪ × dti |ψ(ts − tr )|. ⎪ ⎩ 0=t1 ≤...≤t2n =t

i=2

(r,s)∈π

We recall that · in (5.14) refers to the operator norm on B(B2 (HS )). Proof. Statement 1). Recall that HS = ε(P). By analytic continuation from κ = 0 to |κ| ≤ δε , one has that ei(κ,X ) e−itε(P) e−i(κ,X ) = e−itε(P−κ) .

(5.15)

Since, for |κ| ≤ γ1 , e−itε(P−κ) ≤ etε(P−κ) ≤ etcε (γ1 ) ,

t ≥ 0,

(5.16)

the claim (5.13) is proven. We observe that (5.13) implies

|(e−it HS )(x, x )| ≤ etcε (γ1 ) e−γ1 |x −x| ,

for any 0 < γ1 < δε ,

t ≥ 0,

(5.17)

and hence eγ |x −x| |(e−it HS )(x, x )| ≤ etcε (γ1 ) bd (γ1 − γ ), for any 0 < γ < γ1 < δε , x ∈Zd

t ≥ 0.

(5.18)

Statement 2). To estimate the integrand in (5.9), we choose 0 < γ < γ1 < δε and find that ⎛ ⎞ eγ (|y −y|+|z −z|) ⎝ ⎠ ζπ (t, x, l)Ix2n ,l2n . . . Ix2 ,l2 Ut2 −t1 Ix1 ,l1 (y, z; y , z ) x,l y ,z ≤ (sup |ζπ (t, x, l)|) x,l

l

e2tcε (γ1 ) (bd (γ1 − γ ))2n ,

(5.19)


387

where we can replace “ l ” by 22n , the number of terms in the sum. The bound (5.19) is obtained by applying (5.18) 2n times. For clarity, we illustrate this with an example: Take n = 4 and (l1 , l2 , l3 , l4 , l5 , l6 , l7 , l8 ) = (L , R, L , L , R, L , R, R). First, we notice that Ix ,l . . . Ix ,l Ut −t Ix ,l (y, z; y , z ) (5.20) 8 8 2 2 2 1 1 1 vanishes unless x1 = y and x8 = z , and that it is bounded by )

w(t3 − t1 , x3 − x1 ) × w(t4 − t3 , x4 − x3 ) × w(t6 − t4 , x6 − x4 ) × w(t − t6 , y − x6 ) × w(t2 − 0, x2 − z) × w(t5 − t2 , x5 − x2 ) × w(t7 − t5 , x7 − x5 ) × w(t8 − t7 , x8 − x7 ),

(5.21) where w(u, x) := |(e−iu HS )(0, x)|, t1 = 0, t8 = t. We use the decomposition (recall that x1 = y and x8 = z) |y − y| ≤ |x3 − x1 | + |x4 − x3 | + |x6 − x4 | + |y − x6 |, |z − z| ≤ |x2 − z| + |x5 − x2 | + |x7 − x5 | + |x8 − x7 |, and (5.21) to factorize the sum over y , z , x on the LHS of (5.19). Those sums can then be carried out with the help of (5.18), yielding the bound exp (cε (γ1 ) [(t8 − t6 ) + (t6 − t4 ) + (t4 − t3 ) + (t3 − t1 )]) (bd (γ1 − γ ))8 × exp (cε (γ1 ) [(t8 − t7 ) + (t7 − t5 ) + (t5 − t2 ) + (t2 − 0)]) = (bd (γ1 − γ ))8 e2tcε (γ1 ) .

(5.22)

Note that this bound only depends on |π | and t, and not on t, l, or π . Hence it can be applied for all l, which yields the factor 22n in (5.19). For a linear operator W on l 2 (Zd × Zd ), a straightforward application of the Cauchy-Schwarz inequality yields ⎛ ⎞ W ≤ bd (2δ) ⎝ sup |W(y, z; y , z )|eδ(|y −y|+|z −z|) ⎠. (5.23) y,z∈Zd y ,z ∈Zd

Starting from the explicit definition of Jκ Vt (π ) J−κ (as in (3.2) and (5.9)) , one uses (5.23) and (5.19) with γ := γ + |κ|. This yields Statement 2). ! " 5.2.2. A combinatorial estimate. In the next step of our analysis of the Dyson series, we show that one can perform the integration over all pairings π and temporal coordinates t contributing to (5.11). The following lemma is purely combinatorial, i.e., it only employs notions introduced in Definition 5.1. Lemma 5.2. Consider a positive function h on R+ and a pairing π ∈ P. We define χt (π ) :=

(

2n−1

0=t1 ≤...≤t2n =t i=2

dti )

(r,s)∈π

h(ts − tr ),

with n = |π |.

(5.24)

388


Fig. 5.4. This figure illustrates the change of variables from (π, t), with π ∈ P3 and t1 < . . . < t6 , to (u i , vi )i=1,2,3 , with u i ≤ vi and u i ≤ u i+1

Then

χt (π ) ≤

π irr.

⎛

χt (π ) × exp ⎝t

π min. irr.

⎞ dwh(w)⎠,

(5.25)

R+

and, if π is the minimally irreducible pairing in Pn and z ∈ R, ⎛ ⎞ ⎛ ⎞n−1 dt e−t z χt (π ) ≤ ⎝ dwh(w)e−wz ⎠ × ⎝ dy dw h(y + w)e−wz ⎠ . (5.26) R+

R+

R+

R+

Proof. Given π ∈ Pn , we can relabel the times t1 , . . . , t2n by setting u i = tri ,

vi = tsi

for i = 1, . . . , n.

(5.27)

Using our conventions for the labels of the pairs (ri , si ), it follows that 0 ≤ u i ≤ vi ≤ t,

0 ≤ u i ≤ u i+1 ≤ t,

0 = u 1 , t = max{vi }.

(5.28)

Conversely, a set of n pairs of times (u i , vi ), i = 1, . . . , n, satisfying (5.28) uniquely determines a pairing π ∈ Pn and corresponding times 0 = t1 ≤ . . . ≤ t2n = t. Consider an irreducible pairing π ∈ Pn . It is easy to see that we can always find a subset j1 , . . . , jn of {1, . . . , n }, for some n ≤ n , such that 1) the pairs (r ji , s ji ), i = 1, . . . , n determine a minimally irreducible pairing π ∈ Pn ; 2) these pairs contain the boundary points, i.e. r j1 = 1 and maxi {s ji } = 2n . We write π → π whenever π and π are related in this way; (note, however, that π is not uniquely determined). It follows that χt (π ) ≤ χt (π ). (5.29) |π | = n π irr.

|π | ≤ n π min.irr.

|π | = n π → π

For n ≥ 2, the inequality is strict, since π is not necessarily uniquely determined by π , and hence the same irreducible π can appear more than once on the RHS of (5.29). Using the change of variables (5.27), one can convince oneself that, for all m := n − n ≥ 0, m χt (π ) = χt (π ) dudv h(vi − u i ), (5.30) |π | = n π → π

0 ≤ u1 ≤ . . . ≤ um ≤ t 0 ≤ u i ≤ vi ≤ t

i=1


389

Fig. 5.5. Illustration of (5.30). Three pairings in P5 contributing to the LHS of (5.30). We have chosen π to be the minimally irreducible pairing (1, 3), (2, 5), (4, 6) in P3 , as in Fig. 5.4. For each of these 3 pairings in P5 , the five pairs (u i , vi )i=1,...,5 contain a subset of three pairs identified with π . We have only shown the two other pairs, relabeling them as (u i , vi )i=1,2 . The same strategy is used to prove (5.30) in general

where π is the minimally irreducible pairing in Pn , and where we have abbreviated du := du 1 . . . du n and dv := dv1 . . . dvn . The relation (5.30) expresses the fact that one can add any set of pairs, corresponding to times u, v satisfying the first two conditions of (5.28), to a minimally irreducible π , thus obtaining a new irreducible pairing (see also Fig. 5.5). By explicit computation, ⎞m ⎛ m dudv h(vi − u i ) ≤ du ⎝ dwh(w)⎠ m∈Z+ 0 ≤ u ≤ . . . ≤ u ≤ t m 1

m∈Z+ 0≤u ≤...≤u m ≤t 1

i=1

0 ≤ u i ≤ vi ≤ t

⎛ ≤ exp ⎝t

R+

⎞ dwh(w)⎠,

(5.31)

R+

which proves the bound (5.25) starting from (5.29) and (5.30). When we perform the change of variables (5.27) for a minimally irreducible pairing π , the variables u, v satisfy the constraint u i+1 ≤ vi ≤ u i+2 in addition to the constraints 0 ≤ u i ≤ u i+1 ≤ t and 0 ≤ u i ≤ vi ≤ t. Let π be the minimally irreducible pairing in Pn . Then (u 1 = 0 is a dummy variable) dte

−t z

∞ χt (π ) =

R+

dv1 h(v1 − u 1 )e

−z(v1 −u 1 )

0 vn−4

... vn−5

dvn−2 . . .

vn−3 vn−1

∞

du n vn−2

∞

vn−3

∞

du n−1

dv2 . . .

du 2 0

vn−2

∞

du n−2

v1

v1

dvn−1 e−z(vn−1 −vn−2 ) h(vn−1 − u n−1 )

vn−2

dvn e−z(vn −vn−1 ) h(vn − u n ).

(5.32)

vn−1

Performing the change of variables wi = vi − vi−1 and yi = vi−1 − u i (for i > 1) and extending the range of integration of yi to R, the above expression factorizes and one obtains the bound (5.26). ! " 5.3. Proof of Theorem 4.4. In this section, we prove Theorem 4.4. Statement 2) is proven separately for L(z) and Rex λ (z) in Sects. 5.3.1 and 5.3.2, respectively. Statement 3) is proven in Sect. 5.3.1 and Statement 1) in Sect. 5.3.3.

390


It is mainly in Sect. 5.3.2 that we use the preparatory work summarized in Lemma 5.1 and Lemma 5.2, in order to obtain a bound on Rex λ (z). 5.3.1. Properties of L(z). We compute L(z) starting from (5.9) and (5.12): ⎧ ˆ ⎨ ψ(t) l=L . Il ,x Ut Il,x L(z) = dt e−t z ⎩ ψ(−t) ˆ l = R d l,l ∈{L ,R} x∈ Z + R To display the result, we introduce the functions ψ+ , ψ− as it z it z ˆ ˆ ψ+ (z) = dt ψ(t)e , ψ− (z) = dt ψ(t)e ,

z ∈ C,

(5.33)

(5.34)

R−

R+

ˆ with ψˆ as defined in Sect. 2.3.2; (we recall that ψ(u) decays exponentially). Since ˆ ˆ ψ(−u) = ψ(u) (as follows from (2.18)), one has that ψ+ (z) = ψ− (¯z ),

ψ(z) = ψ+ (z) + ψ− (z),

with |z| < gR .

(5.35)

Using (5.33), we calculate L(z)S, for S ∈ L 2 (Td × Td ), p p (L(z)S)(k + , k − ) 2 2 p p p p = dk ψ+ ε k − − ε k+ +iz + ψ− ε k + −ε k − −iz 2 2 2 2 Td

p p (5.36) ×S k + , k − 2 2 p p p p − dk ψ+ ε k − −ε k + +iz) + ψ− ε k + −ε(k − )−iz 2 2 2 2 Td

p p . ×S k + , k − 2 2 The claim about L(z) in Statement 2) of Theorem 4.4 follows by noticing that the above expression can be analytically continued in z and p. This follows from the analyticity of ε (Assumption 2.1) and ψ+ , ψ− (consequences of Assumption 2.2). To prove Statement 3), we first check that (L(0))0 = M 0 by setting p = 0 and z = 0 in (5.36), and using (5.35). It remains to verify that λ2 (M κ − M 0 ) = iλ2 (κ, ∇ε) = (ad(iHS ))λ2 κ + O((λ2 κ)2 )

(5.37)

as operators on L 2 (Td ), where (κ, ∇ε) is the multiplication operator given by the function k → (κ, ∇ε)(k). Equation (5.37) follows by writing explicitly p p ((ad(iHS )) p θ )(k) = i ε(k + ) − ε(k − ) θ (k), θ ∈ L 2 (Td , dk), (5.38) 2 2 expanding in powers of p and putting p = λ2 κ.


391

5.3.2. Properties of Rex λ (z). Choose positive constants γ1 > γ > 0, as in Lemma 5.1, and define the quantity χt (π ) as in Lemma 5.2, with h given by ˆ h(t) := 22 bd (γ1 − γ − |κ|)2 λ2 |ψ(t)|.

(5.39)

It follows from Statement 2) of Lemma 5.1 and Eqs. (5.9), (5.11) that ⎞

⎛ Jκ Rex λ (z)J−κ ≤ bd (2γ ) R+

⎟ ⎜ ⎟ ⎜ dt e2cε (γ1 )t e−zt ⎜ χt (π )⎟, ⎠ ⎝

(5.40)

|π | > 1 π irr.

and hence, using Lemma 5.2, that Jκ Rex λ (z)J−κ ≤ bd (2γ )

dte−(z−a)t

+

χt (π ),

with a := 2cε (γ1 )

π min. irr.

R+

dw h(w)

R+

⎛

≤ bd (2γ ) ⎝ ⎛ ×F ⎝

⎞

dw h(w)e−w(z−a) ⎠

R+

dy R+

⎞ dw h(y + w)e−w(z−a) ⎠,

R+

x where F(x) := 1−x , provided that |x| < 1. To prove the first inequality above, we use (5.40) and (5.25), and, for the second inequality, we use (5.26) and sum the geometric series. Statement 2) of Theorem 4.4 now follows by fixing the constants and using the ˆ For example, choose γ1 , γ such that exponential decay of ψ.

2cε (γ1 ) ≤

1 gR , 4

γ :=

1 γ1 , 2

(5.41)

and δ2 small enough such that for |λ| ≤ δ2 , dw h(w) ≤ R+

1 gR , 4

dy

R+

1

dw h(y + w)e−w(− 4 gR −a) ≤ 1.

R+

Then (4.47) is satisfied with δ1 := 14 γ1 , g := 41 gR and δ2 as determined above.

(5.42)

392


5.3.3. Proof of Equation (4.44) in Statement 1) of Theorem 4.4. To simplify the following calculations, we abbreviate ex 2 Rirr λ (z) := Rλ (z) + λ L(z),

RS (z) := (z − ad(iHS ))−1 .

(5.43)

By the self-adjointness of ad(HS ), one has that RS (z) < |z|−1 . We choose λ and z irr −1 such that z > 0 and Rirr λ (z)RS (z) ≤ |z| Rλ (z) < 1. Then Rλ (z) := dt e−t z Ztλ R+

=

n∈Z+

n RS (z) Rirr λ (z)RS (z)

−1 = RS (z) 1 − Rirr λ (z)RS (z) −1 = z − ad(iHS ) − Rirr λ (z) −1 = z − ad(iHS ) − λ2 L(z) − Rex , λ (z)

(5.44)

where the second equality follows by Laplace transforming (5.10), and the third equality represents the sum of a geometric series. Hence, Statement 1) of Theorem 4.4 is proven. 6. Proof of Theorem 4.5 In this section we prove Theorem 4.5. Our reasoning is based on a standard application of analytic perturbation theory and the inverse Laplace transform. We abbreviate A(z, λ, κ) := ad(iHS ) + λ2 L(z) + Rex (z) 2 − λ2 M κ (6.1) λ κ

and we define ⎫ ⎧ + ⎨ ⎬ δ 1 , G := (z, λ, κ) ∈ C×R×Cd z > −g , |κ| < δkin , |λ| < min δ2 , ⎩ δkin ⎭

(6.2)

where g , δ1 , δ2 are as described in Theorem 4.4 and δkin is as described in Theorem 4.3. Theorem 4.4 implies that, on the domain G, the function λ2 M κ + A(z, λ, κ) is analytic in the variables (z, κ) and, for z large enough, (Rλ (z))λ2 κ = (z − λ2 M κ − A(z, λ, κ))−1 .

(6.3)

We may extend the (operator-valued) function z → (Rλ (z))λ2 κ into the region z > −g . This will be useful, because, at the end of this section, we calculate the reduced evolution (Ztλ )λ2 κ from the inverse Laplace transform of (Rλ (z))λ2 κ . From (6.3) we see that any singular point of the function z → (Rλ (z))λ2 κ must satisfy z ∈ sp(λ2 M κ + A(z, λ, κ)).

(6.4)


393

Recall that by Theorem 4.3, M κ has a simple isolated eigenvalue f kin (κ), and let ⊂ C be as defined in (4.30), i.e., spM κ \ { f kin (κ)} . (6.5) := ∪ |κ| 0, for all m > 0, such that sup

z∈C, dist(z,spM κ )≥r (m)

(z − M κ )−1 ≤ m.

(6.8)

Choose m −1 := sup(z,λ,κ)∈G λ−2 A(z, λ, κ) (by (6.7), m −1 = O(λ0 )). Using the Neumann series for (z−λ2 M κ − A(z, λ, κ))−1 , it follows that, if dist(z, λ2 spM κ ) ≥ λ2 r (m), then z cannot satisfy (6.4). If, however, dist(z, λ2 spM κ ) ≤ λ2 r (m), then A(z, λ, κ) = O(λ4 ), as λ 0; (this follows from (6.7) and the analyticity of L(z)). The claim now follows from analytic perturbation theory, using that λ2 f kin (κ) is an isolated simple eigenvalue. ! " Lemma 6.2. For sufficiently small |λ|, there is a unique z =: z˜ at a distance O(λ4 ) from λ2 f kin (κ) satisfying (6.4). Let P λ,κ be the residue of (z − λ2 M κ − A(z, λ, κ))−1 at z = z˜ . It follows that P λ,κ is a rank one-operator and κ = O(λ2 ) P λ,κ − Pkin κ Pkin

the one-dimensional spectral projection of with simple eigenvalue f kin (κ), as in Theorem 4.3.

Mκ

(6.9) corresponding to the isolated

Proof. By analytic perturbation theory, the operator λ2 M κ + A(z, λ, κ) has at most one eigenvalue at a distance O(λ4 ) of f kin (κ). This means that (6.4) has at most one solution at a distance O(λ4 ) of f kin (κ). We now prove that there is at least one solution. Indeed, if no such solution existed, we could choose a contour Cκ,a = {z ∈ C | |z − f kin (κ)| = a},

a > 0,

(6.10)

with a small enough such that Cκ,a stays away from . We then calculate κ − 0) = dz(z − λ2 M κ )−1 − dz(z − λ2 M κ − A(z, λ, κ))−1 2π i(Pkin λ2 Cκ,a

= λ2 Cκ,a

λ2 Cκ,a

dz(z − λ2 M κ )−1 1 − (1 − A(z, λ, κ)(z − λ2 M κ )−1 )−1

≤ (2πa) b(a, κ) 1 −

1 , 1 − b(a, κ)O(λ2 )

(6.11)

394


where b(a, κ) := sup (z − M κ )−1 , z∈Cκ,a

and, here and in what follows, the contour integrals are meant to be oriented clockwise. Since the last line of (6.11) is of order λ2 , we arrive at a contradiction to the fact that κ = 0. Pkin The claim about the residue is most easily seen in an abstract setting: Let F(z) be a Banach-space valued analytic function in some open domain containing 0, and such that 0 ∈ spF(0) is an isolated eigenvalue. We have hence the Taylor expansion F(z) =

zn Fn , n!

Fn := F (n) (0),

0 ∈ spF0 .

(6.12)

n≥0

If F1 − 1 is small enough, then also F1−1 F0 has 0 as an isolated eigenvalue. We denote the corresponding spectral projection by 10 (F1−1 F0 ) and we calculate Res(F(z)−1 ) = Res(F0 + z F1 )−1 = Res(F1−1 F0 + z)−1 F1−1 = 10 (F1−1 F0 )F1−1 . (6.13) The last expression is clearly a rank-one operator. In the case at hand, F1−1 = 1 + O(λ2 ), as λ 0, which yields (6.9). ! " We set f (λ, κ) := z˜ and we define P λ,κ as the residue of (z − λ2 M κ − A(z, λ, κ))−1 at z = z˜ . It is clear that f (λ, κ) and P λ,κ enjoy the analyticity properties claimed in Theorem 4.5. We define the horizontal contours := {z ∈ C |z = l + iR},

:= {z ∈ C |z = −(g − ) + iR},

(6.14)

with l large enough such that all singular points of z → (Rλ (z))λ2 κ lie below , and > 0 small enough such that all singular points with z > −g lie above (the notions ‘below’ and ‘above’ are meant as in Fig. 6.1). These contours are oriented from left to right. By Theorem 4.3, we can construct a contour C which encircles and such that f kin (κ) is separated by a gap g from this contour: g := inf f kin (κ) − sup C > 0. |κ|≤δkin

By performing an inverse Laplace transform we find that 1 (Ztλ )λ2 κ = dz et z (z − λ2 M κ − A(z, λ, κ))−1 . 2π i

(6.15)

(6.16)

For λ small enough, Lemma 6.1 ensures that one can deform contours and obtain = + + . (6.17)

λ2 Ca,κ

λ2 C


395

z

Γ λ2 Cκ,a

l 2

λ C

i z λ2 g

O(λ2 )

g −

Γ

Fig. 6.1. The (rotated) complex plane. The black dots and thick black line indicate the spectrum of λ2 M 0 : The upper dot is the eigenvalue 0 and the thick vertical line is the continuous spectrum. In the picture, we have drawn only one other eigenvalue, but, in general, there can be more than one (or none) further eigenvalues. The function λ2 M κ + A(z, λ, κ) is analytic above the lowest gray (rectangular) region. The other gray regions contain the singularities of the function (Rλ (z))λ2 κ for (z, λ, κ) ∈ G. The integration contours , and λ2 Cκ,a , λ2 C are drawn in dashed lines. In this picture, the contour λ2 Cκ,a encircles λ2 f (λ, κ), for all (λ, κ), (i.e., such that (z, λ, κ) ∈ G), which can be achieved by choosing a large enough

The first term on the RHS of (6.17) equals etλ f (κ,λ) P λ,κ ; this follows from Lemma 6.2. The second term is dominated by −1 d|z| 2 (z −λ2 M κ )−1 1−(1− A(z, λ, κ)(z −λ2 M κ )−1 eλ t (sup(C )) . 2π 2

λ2 C

(6.18)

396


By the choice of Cλ and the bound (6.7), the integral on the RHS is bounded by a constant, for λ small enough. The third term of the RHS of (6.17) is split as tz 2 κ −1 dz e (z − λ M − A(z, λ, κ)) = dz et z (z − λ2 M κ )−1

+

dz et z (z − λ2 M κ )−1 A(z, λ, κ)

×(z − λ2 M κ − A(z, λ, κ))−1 .

(6.19)

The first integral can be closed in the lower half-plane and equals 0, the second integral has an integrand of order z −2 for large z, and hence its contribution is bounded by a constant times e−t (g −) . It follows that the crucial estimate (4.49) holds with δ1 := δkin and g as in (6.15). APPENDIX A Here we consider the effective structure factor, which, in Sect. 2.3, has been introduced as the Fourier transform of the reservoir correlation function. We use the spectral theorem to represent the positive operatorω as multiplication by ξ ∈ R+ . There are Hilbert spaces hξ for ξ ∈ R+ such that h = ⊕R+ dξ hξ , and for all ϕ ∈ h, there are ϕξ ∈ hξ such that ϕ= dξ ϕξ , ωϕ = dξ ξ ϕξ . (A-1) ⊕R+

⊕R+

The structure factor φ ∈ h has been introduced in Sect. 2.3. We construct an effective form factor φ β as an element of h ⊕ h. We choose h−ξ to be isomorphic to hξ , and we β define φ β = ⊕R φξ as an element of h ⊕ h ∼ ⊕R hξ by setting ) 1 √ φξ , ξ > 0, β eβξ −1 (A-2) φξ := 1 √ φ−ξ , ξ < 0. βξ 1−e

The function φ β plays the role of the form factor if one constructs the positive-temperature dynamical system. We just note that β

ψ(ξ ) = φξ 2hξ .

(A-3)

Assume that the on-site one-particle space is given by h = L 2 (Rd ), and the one-particle Hamiltonian acts by multiplication with a function ξ(r ), where r := |q|, for q ∈ Rd . We also assume that r → ξ(r ) is differentiable and monotonically increasing. Hence we can define the inverse function ξ → r (ξ ). The form factor φ ∈ L 2 (Rd ) is taken to be spherically symmetric, φ(q) ≡ φ(r ). Then the Hilbert spaces hξ are naturally identified with L 2 (Sd−1 ), and ) βξ

(e − 1)−1/2 φ(r (ξ )), ξ > 0, d−1 ∂r (|ξ |) −1/2 β (A-4) 1Sd−1 φξ = r (|ξ |) 2 βξ −1/2 ∂|ξ | (1 − e ) φ(r (−ξ )), ξ < 0,


397

where 1Sd−1 ∈ L 2 (Sd−1 ) is the constant function on Sd−1 with 1Sd−1 = 1. Next, we return to Assumption 2.2. By properties of the Fourier transform, e.g. Th. IX.14 of [19], this assumption is equivalent to the assumption that ψ extends to an analytic function in the strip |ξ | < gR , and sup dx |ψ(x + iy)| < ∞. (A-5) −gR t and t− < t, or sets G of the form G = int(J + (x) ∩ J − (y)), where y lies in the open interior of J + (x). Now we collect some well-known results (well-known mainly in the context of the quantized Dirac field on curved spacetimes) on the existence and uniqueness of advanced and retarded fundamental solutions for the Dirac operator DV . Proposition 1 ([3,17]). (a) DV f, h = f, DV h ( f, h ∈ C0∞ (Rn , C N )). (b) There is a unique pair of linear maps R V± : C0∞ (Rn , C N ) → C ∞ (Rn , C N ) having the properties DV R V± f = f = R V± DV f and supp R V± f ⊂ J ± (supp f ) ( f ∈ C0∞ (Rn , C N )). R V± is called advanced(+)/retarded(-) fundamental solution of DV . (c) C R V± = R V± C. (d) Writing R V = R V+ − R V− , the form ( f, h)V = f, i R V h is a sesquilinear form on C0∞ (Rn , C N ), and C is a conjugation for this form: (C f, Ch)V = (h, f )V = ( f, h)V , ( f, h ∈ C0∞ (Rn , C N )).

(8)

406

M. Borris, R. Verch

(e) For each t ∈ R it holds that ( f, h)V = (Pt R V f, Pt R V h)D , ( f, h ∈ C0∞ (Rn , C N )), where Pt : C ∞ (Rn , C N ) → C ∞ (t , C N ) is the map given by Pt : ϕ → ϕ(t, ·) for ϕ : (x 0 , x) → ϕ(x 0 , x) in C ∞ (Rn , C N ), x = (x 1 , . . . , x s ). Hence, (·, ·)V is positive-semidefinite on C0∞ (Rn , C N ). (f) The Cauchy-problem for the Dirac-equation DV ϕ = 0 is well-posed: Given any Cauchy-hyperplane t and Cauchy-data w ∈ S (t , C N ), there is a unique ϕ ∈ C ∞ (Rn , C N ) such that DV ϕ = 0 and Pt ϕ = ϕ|t = w. Furthermore, the solution ϕ fulfills the causal propagation property in the sense that supp ϕ ⊂ J (supp w). (g) Let E V be the subspace of all f ∈ C0∞ (Rn , C N ) so that ( f, f )V = 0, and let KV be the Hilbert space arising as completion of C0∞ (Rn , C N )/E V with respect to the scalar product induced by (·, ·)V (which will be denoted by the same symbol). The quotient map C0∞ (Rn , C N ) → C0∞ (Rn , C N )/E V will be written f → [ f ]V . Then for each t ∈ R, the map Q V,t : [ f ]V → Pt R V f

(9)

extends to a unitary map from KV onto Dt . (h) Let G be a hyperbolic subset of n dimensional Minkowski spacetime, and suppose that V1 and V2 are real-valued, C ∞ , and that V1 = V2 on G. Then R V±1 f = R V±2 f on G for all f ∈ C0∞ (G, C N ).

(10)

Proof (sketch). (a) This is a straightforward calculation. (b) This is proved using the same argument as for Theorem 2.1 in [17], which applies also in the presence of a real scalar potential V , together with the existence and uniqueness result for fundamental solutions of hyperbolic wave operators, which can be found (in far greater generality than needed here) in [3]. (c) This is a consequence of the uniqueness of the R V± together with C DV = DV C. (d) The only non-obvious part (C f, Ch)V = (h, f )V of the claim follows easily from (c), Eq. (7) and the relation R V f, h = − f, R V h, which is shown in the proof of Theorem 2.1 in [17]. (e) The argument is the same as for Proposition 2.4 (d) in [17].

Dirac Field on Moyal-Minkowski Spacetime and Non-commutative Potential Scattering

407

(f) This statement is proved analogously to Thm. 2.3 in [17]. It is proved there for the case that the Cauchy-data are C0∞ . However, existence and uniqueness of a distributional solution is proved in Prop. 2.4 in [17] for distributional Cauchy-data. The smoothness of the solution in case of Cauchy-data that are of Schwartz type can be proved by making use of the causal propagation property of the solutions (i.e. supp ϕ ⊂ J (supp w)) in combination with a partition of unity argument. (g) In view of (e), what remains to be checked is the surjectivity of Q V,t . To see this, let w ∈ C0∞ (t , C N ), and let ϕ ∈ C ∞ (Rn , C N ) be the solution of DV ϕ = 0 having Cauchy-data w on t , i.e. Pt ϕ = w. We will construct some f ∈ C0∞ (Rn , C N ) so that Pt R V f = w. To this end, we take two further Cauchy-hyperplanes, ± , with ± = {x = (x 0 , . . . , x s ) : x 0 = t ± 1}. Then we can consider the open sets G ± = int(J ± (∓ )) = {x = (x 0 , . . . , x s ) : ±x 0 > t ∓ 1}. The sets G ± form an open covering of Rn . Let χ± be a C ∞ partition of unity of Rn subordinate to the covering. It is easy to see that the functions χ± can be chosen in such a way that they depend only on x 0 , and we will assume that this choice has been made (although this is not relevant at this point; see however the proof of Prop. 5 (g) later). Then one has DV (χ+ ϕ) = −DV (χ− ϕ), and owing to the support properties of χ± , one concludes that both DV (χ± ϕ) have support contained in G + ∩ G − = {(x 0 , . . . , x s ) : t + 1 ≥ x 0 ≥ t − 1}. On the other hand, since we also have supp ϕ ⊂ J (supp w) and since supp w was assumed to be compact, this implies that both DV (χ± ϕ) are C0∞ . Setting now f = DV (χ+ ϕ), it holds that f ∈ C0∞ (Rn , C N ), and moreover, we see that DV (R V f − ϕ) = 0. However, we also have that R V f = R V+ f − R V− f , and owing the support properties of R V± , on Rn \G − = {(x 0 , . . . , x s ) : x 0 > t + 1} it holds that R V f = R V+ f = χ+ ϕ = ϕ. This means that Pτ (R V f − ϕ) = 0 for all real τ > t + 1 and hence, since (R V f − ϕ) is a C ∞ solution of the Dirac equation with C0∞ Cauchy-data, one actually concludes that (R V f − ϕ) = 0 on all of Rn . Hence we have shown that there is some f ∈ C0∞ (Rn , C N ) with R V f = ϕ, implying Pt R V f = Pt ϕ. This shows that the range of Q V,t is dense, and by its isometric property, Q V,t is actually surjective. (h) The spacetime region G, endowed with the Minkowski metric (and standard spin structure) is a globally hyperbolic spacetime. Given a smooth real-valued V : G → R as a potential function, one can define the “intrinsic” Dirac operator of G, DV |G : C ∞ (G, C N ) → C ∞ (G, C N ) by DV |G f = DV f , f ∈ C ∞ (G, C N ), which is nothing but the canonical restriction of DV onto G. According to [17] (cf. also [3]), there are unique advanced/retarded fundamental solutions RV±|G : C0∞ (G, C N ) → C ∞ (G, C N ) for DV |G . Now, if V1 = V2 = V on G, then the appropriate restrictions of R V±1 and R V±2 onto C0∞ (G, C N ) (more precisely, the maps f → (R V±j ) , f ∈ C0∞ (G, C N ), j = 1, 2) have the same properties G

as the map R V±|G . Hence, by the uniqueness statement, these restrictions must be equal to R V±|G .

408

M. Borris, R. Verch

Starting from (KV , C), the Hilbert space KV with conjugation C, one can form, following [2], the corresponding self-dual CAR-algebra F(KV , C). It is defined as follows: One introduces a ∗-algebra generated by symbols B(ξ ) = BV (ξ ), ξ ∈ KV , subject to the relations B(ξ )∗ = B(Cξ ), B(ξ1 ) B(ξ2 ) + B(ξ2 )B(ξ1 )∗ = 2(ξ1 , ξ2 )V 1l, ξ → B(ξ ) is complex linear, ∗

where 1l is an algebraic unit. One can show that the resulting ∗-algebra admits a unique C ∗ -norm, and F(KV , C) is the completion of that ∗-algebra with respect to the C ∗ - norm. Therefore, F(KV , C) is a C ∗ -algebra. Writing ( f ) = V ( f ) = BV ([ f ]V ) for f ∈ C0∞ (Rn , C N ), F(KV , C) is generated by “abstract field operators” ( f ), which are C-linear and obey the relations ( f )∗ = (C f ), {( f )∗ , (h)} = 2( f, h)V 1l, (DV f ) = 0. The construction of F(KV , C) can also be carried out, in an analogous manner, for “local subspaces” of KV . For this purpose, let G be a hyperbolic subset of n dimensional Minkowski spacetime. For f ∈ C0∞ (G, C N ), we introduce the equivalence class ∞ N [ f ]G V = { f + h : h ∈ C 0 (G, C ), R V h = 0}.

As before, the space of the [ f ]G V carries a scalar product (·, ·)V in the same fashion as C0∞ (Rn , C N )/E V (denoted by the same symbol as there is no danger of confusion). The resulting Hilbert-space completion will be denoted by KVG . Again, the charge conjugation C induces a conjugation on KVG as well. Whence, we can form the self-dual CARG algebra F(KVG , C), which is the C ∗ -algebra generated by symbols BVG ([ f ]G V ), [ f ]V ∈ KVG , obeying relations akin to those fulfilled by the B([ f ]V ) above. Lemma 1. Suppose that G is a hyperbolic neighbourhood of a Cauchy hyperplane in n dimensional Minkowski spacetime. Moreover, suppose that V1 and V2 are two smooth, real-valued potentials which coincide on the region G. Then (a) The map G uG V1 ,V2 : [ f ]V1 → [ f ]V2 ,

f ∈ C0∞ (G, C N )

extends to a unitary between KVG1 and KV2 commuting with the charge conjugation C. (b) There is a ∗-algebra isomorphism αVG1 ,V2 : F(KVG1 , C) → F(KV2 , C) induced by αVG1 ,V2 BVG1 ([ f ]G V1 ) = BV2 ([ f ]V2 ),

f ∈ C0∞ (G, C N ).


409

Proof. (a) In view of (h) of Proposition 1, R VG1 f = R V2 f on G for all f ∈ C0∞ (G, C N ). Using the definition of (·, ·)V , this implies that the map u G V1 ,V2 is isometric. To show that the map is surjective, let h ∈ C0∞ (Rn , C N ). Since G is an open neighbourhood of a Cauchy surface, there is some f ∈ C0∞ (G, C N ) such that R V2 ( f − h) = 0 ([3]) and hence [ f ]V2 = [h]V2 . (b) This is a straightforward consequence of the fact that u G V1 ,V2 is a unitary intertwining the action of C, see [1] (or [7,8]). We will now make use of the lemma. Suppose that a smooth scalar (real) potential V is given on n dimensional Minkowski spacetime, having support contained in the time-slice {(x 0 , x 1 , . . . , x s ) : λ− < x 0 < λ+ } for some real numbers λ− < λ+ . Then one can consider the regions 1 G + = {(x 0 , x 1 , . . . , x s ) : x 0 > λ+ + } and 2 1 G − = {(x 0 , x 1 , . . . , x s ) : x 0 < λ− − }. 2 They form hyperbolic neighbourhoods of the Cauchy hyperplanes + = {(x 0 , x 1 , . . . , x s ) : x 0 = λ+ + 1} and − = {(x 0 , x 1 , . . . , x s ) : x 0 = λ− − 1}, respectively. Lemma 1 then warrants the following C ∗ -algebraic isomorphisms: G

G

α0± = α0,0± : F(K0 ± , C) → F(K0 , C) G

G

B0 ± ([ f ]0 ± ) → B0 ([ f ]0 ), G

f ∈ C0∞ (G ± , C N ),

G

αV ± = α0,V± : F(K0 ± , C) → F(KV , C) G

G

B0 ± ([ f ]0 ± ) → BV ([ f ]V ),

f ∈ C0∞ (G ± , C N ).

Since these maps are isomorphisms, they can be combined into an automorphism βV : F(K0 , C) → F(K0 , C), −1 −1 ◦ αV,+ ◦ α0,+ . βV = α0,− ◦ αV,−

(11)

This isomorphism is reminiscent of a similar object defined in Sec. 4 of [9], and it has similar properties. Its significance is that it describes the scattering of the quantized Dirac field by the classical potential V at the level of a C ∗ -algebraic Bogoliubov transformation. In order to see this more clearly, we will discuss how βV relates to the perhaps more familiar scattering formalism in terms of time-evolution on the Cauchy data. For this purpose, let us first revisit βV . We have βV (B0 ([ f ]0 )) = B0 (UV [ f ]0 ), where UV is the unitary given by −1 UV = u 0,− ◦ u −1 V,− ◦ u V,+ ◦ u 0,+ ,

(12)

410

M. Borris, R. Verch

and where, similarly as for the isomorphisms above, we have used the abbreviations G

G

± . u 0,± = u 0,0± , u V,± = u 0,V

The action of the succession of unitaries on the right hand side of the defining equation of UV can be described as follows: [ f ]0

u −1 0,+

/ [ f G + ]G + 0

u V,+

/ [ f G + ]V

u −1 V,−

u / [ f G − ]G − 0,− / [ f G − ]0 . 0

(13)

In this chain of mappings, f G + is any element in C0∞ (G + , C N ) such that R0 ( f − f G + ) = 0, and f G − is any element in C0∞ (G − , C N ) such that R V ( f G + − f G − ) = 0. Turning to the description of the quantized Dirac field in terms of its Cauchy data, we recall that D0 = L 2 (0 , d s x), where 0 is the x 0 = 0 Cauchy hyperplane. We have also seen that the charge conjugation C acts as a complex conjugation on D0 . Hence one can associate to D0 and C the CAR-algebra F(D0 , C) with generators BD0 (v), v ∈ D0 , linear in v, and with the relations BD0 (v)∗ = BD0 (Cv), {BD0 (v)∗ , BD0 (w)} = 2(v, w)D 1l.

(14)

Writing Q 0 for Q V,0 in the case of V = 0, Q 0 : K0 → D0 , [ f ]0 → P0 R0 f is a unitary intertwining the actions of C on the respective Hilbert spaces. Consequently ([2]), there is a canonical isomorphism : F(K0 , C) → F(D0 , C) of CAR-algebras induced by (B0 ([ f ]0 )) = BD0 (Q 0 ([ f ]0 )).

(15)

On F(D0 , C), we can introduce two types of time evolutions, one corresponding to a vanishing potential V = 0 in the Dirac equation (the “free” dynamics), and another corresponding to a nonvanishing C ∞ potential term V in the Dirac equation (the “interacting” dynamics). These dynamical evolutions will be defined on the Cauchy-data space D0 . To this end, we define on C0∞ (Rs , C N ) the operators ∂ (H0 f )(x 1 , . . . , x s ) = iγ 0 γ k k + γ 0 m f (x 1 , . . . , x s ), ∂x (16) ∂ (HV (t) f )(x 1 , . . . , x s ) = iγ 0 γ k k + γ 0 m + γ 0 V (t) f (x 1 , . . . , x s ), ∂x where f (x 1 , . . . , x s ) is regarded as a column vector on which the γ - matrices act by matrix multiplication. These operators are symmetric with respect to the scalar product (·, ·)D , and even essentially selfadjoint under very general conditions on (the realvalued) V (t) (e.g. see Theorem 1.1 of [43], and Theorem X.69 of [35]). Moreover, it is easy to check that the operators anti-commute with the charge conjugation C, C H0 = −H0 C, C HV (t) = −HV (t)C. There is hence a continuous unitary group Tt , t ∈ R, on D0 such that 1 d Tt v = H0 v, v ∈ C0∞ (Rs , C N ). i dt t=0

(17)


411

(V )

There is also a continuous family of unitarities Ts,t , s, t ∈ R, so that (V )

(V )

(V )

(V ) ◦ Ts,t = Tr,t , Tt,t = 1l and Tr,s 1 d (V ) Ts,t v = HV (t)v, v ∈ C0∞ (Rs , C N ). i ds

(18)

s=t

(V ) Let us indicate that the existence of the family Ts,t with said properties is implied by the well-posedness of the Cauchy problem for the Dirac equation: For each v ∈ C0∞ (Rs , C N ) ⊂ Dt there is a unique solution ϕ ∈ C ∞ (Rn , C N ) to the Dirac equation

DV ϕ = 0 having initial data v on t , i.e. Pt ϕ = ϕ|t = v. The solution property is equivalent to 1 d Pt ϕ = HV (t)Pt ϕ. i dt

(19)

(V ) On the other hand, the uniqueness statement implies that there is a map Tt,t : Pt ϕ → (V ) (V ) (V ) (V ) (V ) 1 d Pt ϕ with the properties T ◦ T = T and Tt,t = 1l. And Ts,t = HV (t) t,t

t ,t

i ds s=t

t,t

(V ) on C0∞ (Rs , C N ) then follows from (19). The unitarity of Tt,t is implied by Proposition 1 (e). We note also that (V )

(V )

C Tt = Tt C and C Tt,t = Tt,t C

(20)

(V ) on account of (17). Therefore, Tt and Tt,t give rise to CAR-algebra automorphisms τt

(V ) and τt,t of F(D0 , C) induced by

τt (BD0 (v)) (V ) τt,t (BD0 (v))

= BD0 (Tt v), (V )

= BD0 (Tt,t v).

As before, we will now assume that the potential V ∈ C ∞ (Rn , R) has support contained in the set {(x 0 , x 1 , . . . , x s ) : λ− < x 0 < λ+ } for some real numbers λ− < λ+ . The scattering operator for the Dirac equation at the level of Cauchy data on 0 is the operator (V ) Tsc(V ) = lim Tt−1 ◦ Tt,t ◦ Tt t →∞ t→−∞

(21)

on D0 . The restriction on the time-support of V implies that the limit (21) is reached as soon as t > λ+ and t < λ− , so that (V )

Tsc(V ) = Tt−1 ◦ Tt,t ◦ Tt , for t > λ+ , t < λ− .

(22)

(V )

We denote by τsc the corresponding scattering morphism on F(D0 , C) given by (V ) (BD0 (v)) = BD0 (Tsc(V ) v). τsc

(23)

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Now we want to demonstrate that (V ) ◦ . ◦ βV = τsc

(24)

Q 0 ◦ UV = Tsc(V ) ◦ Q 0 .

(25)

Thus we aim at showing

(V )

To prove this, we write the action of Q −1 0 ◦ Tsc following form: [ f ]0

/ P0 R0 f

Tt

/ P RV h G + t / P R h G− t 0

Q0

(∗2)

(∗4)

◦ Q 0 on an element [ f ]0 ∈ K0 in the

/ Pt R0 f

(∗1)

/ P R0 h G + t

Tt,t

/ P R h G+ t V

(∗3)

/ P R h G− t V

Tt−1

/ P R h G− 0 0

Q −1 0

(V )

/ [h G − ] . 0 (26)

In this succession of maps, at (∗1) an element h G + ∈ C0∞ (G + , C N ) is chosen so that R0 h G + = R0 f . At (∗2), it is used that R V h G + = R0 h G + on G + because of the support properties of the functions V and h G + , cf. Proposition 1 (h). At (∗3), an element h G − ∈ C0∞ (G − , C N ) is chosen so that R V h G + = R V h G − . Then at (∗4), it is again used that R0 h G − = R V h G − on G − owing to the support properties of V and h G − . Comparing (13) and (26), one can see that the specifications of f G ± and h G ± are such that one can may even choose (starting from the same given f ) f G ± = h G ± , and this then proves the relation (25). Summarizing, we have proved: Lemma 2. The morphism βV of F(K0 , C) defined in (11) and the scattering morphism (V ) describing the potential scattering of the quantized Dirac field at the level of the τsc Cauchy-data CAR-algebra F(D0 , C) are intertwined by the CAR-algebra isomorphism : F(K0 , C) → F(D0 , C) defined in (15), i.e. it holds that (V ) ◦ βV = τsc ◦ .

One advantage of working with βV is that it can be associated to localization in spacetime: It acts trivially outside of J (supp V ). This is our next assertion. Proposition 2. Let f ∈ C0∞ (Rn , C N ) have support causally disjoint from supp V , i.e. supp f ∩ J (supp V ) = ∅. Then βV (0 ( f )) = 0 ( f ). Proof. According to Proposition 1 (h), if supp f ∩ J (supp V ) = ∅, then R0 f = R V f on Rn \ J (supp V ), since Rn \ J (supp V ) is a hyperbolic region in n dimensional Minkowski spacetime. On the other hand, supp f ∩ J (supp V ) = ∅ is equivalent to J (supp f ) ∩ supp V = ∅. Now, R V f is a solution to (D + V )R V f = 0, and supp R V f ⊂ J (supp f ), thus supp R V f ∩ supp V = ∅, implying that D R V f = 0. Consequently, R0 f and R V f are both solutions to the Dirac equation with vanishing potential V = 0, and coincide in the neighbourhood of a Cauchy surface for n dimensional Minkowski spacetime (which


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is implied by R0 f = R V f on Rn \ J (supp V ) and supp R0 f ∪supp R V f ⊂ J (supp f )). This implies that R0 f = R V f on all of n dimensional Minkowski spacetime. Now consider the map U V : [ f ]0

/ [ f G + ]G + 0

/ [ f G + ]V

/ [ f G − ]G − 0

/ [ f G − ]0 . (27)

In this succession of mappings, f G + is any element in C0∞ (G + , C N ) so that R0 ( f − f G + ) = 0, and f G − is any element in C0∞ (G − , C N ) so that R V ( f G + − f G − ) = 0. However, since R0 f = R V f , it holds that R V ( f − f G + ) = R0 f − R V f G + = R0 f − R0 f G + = 0 on G + , hence R V ( f − f G + ) = 0 on G + , and hence R V ( f − f G + ) = 0 everywhere on n dimensional Minkowski spacetime. Furthermore, R V f G − = R V f G + , from which R V ( f − f G − ) = 0 obtains. On the other hand, we also have R V f G − = R0 f G − on G − , and R V f = R0 f , thus R0 f = R0 f G − on G − , and hence everywhere on n dimensional Minkowski spacetime. This shows that [ f ]0 = [ f G − ]0 and therefore, UV [ f ]0 = [ f ]0 . In view of (12), this yields the claimed proposition. 3. Scattering of the Dirac Field in the Vacuum Representation and Implementability of the Scattering Transformation The Hamilton operator H0 defined in (16) is essentially selfadjoint on C0∞ (Rs , C N ) ⊂ L 2 (Rs , C N ) (see Theorem 1.1 of [43]). Therefore, its selfadjoint extension, again denoted by H0 , possesses a spectral decomposition, and we denote by p+ the spectral projection of H0 corresponding to the spectral interval (0, ∞). Since the mass term m in the Dirac equation has been assumed to be strictly greater than 0, p+ projects in fact on the spectral values in [m, ∞) and the orthogonal projector p− = 1l − p+ coincides with the spectral projector of the spectral interval (−∞, −m]. Owing to C Tt = Tt C for all t ∈ R, it holds that C p+ = p− C. Thus, p+ is a basis projection in the sense of [2]. To this basis projection one can associate a pure, quasifree state ω p+ on F(D0 , C) whose two-point function is given by ω2 + (BD0 (u)∗ BD0 (w)) = (u, p+ w)D , u, w ∈ D0 . p

The state can be pulled back by to a pure, quasifree state ωvac = ω p+ ◦ ↑ on F(K0 , C). This state is actually just the usual (P˜ + (n)-invariant) vacuum state on F(K0 , C). Writing

e+ = Q −1 0 p+ Q 0 , its GNS-representation (Hvac , π vac , vac ) can be realized as follows: Hvac = FF (e+ (K0 )) is the Fermionic Fock space over the one-particle Hilbert space e+ (K0 ) (e+ projects on the “positive frequency” solutions of the Dirac equation), vac = (1, 0, 0, . . .) the Fock vacuum vector, π vac (0 ( f )) = A(e+ C[ f ]0 ) + A+ (e+ [ f ]0 ),

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where A(χ ) and A+ (χ ) denote, respectively, the Fermionic annihilation and creation operators of a χ in the one-particle Hilbert space. We will sometimes use the notation ψ( f ) = π vac (0 ( f )),

f ∈ C0∞ (Rn , C N ),

for the field operators of the quantized Dirac field in the vacuum representation. For several reasons, it is important to investigate the question of unitary implementability of the scattering transformation in the vacuum representation. In the situation at hand, this is the question if there exists a unitary operator SV : Hvac → Hvac such that SV π vac (0 ( f ))SV−1 = π vac (βV (0 ( f ))),

f ∈ C0∞ (Rn , C N ).

(28)

This issue has been investigated for the Dirac field on Minkowski spacetime by several authors in various publications that have appeared over the last decades. The result is that there is such an operator, or “S-matrix”, provided that the potential V is sufficiently regular and sufficiently fast decaying. A sufficient condition to this end, which is convenient for comparison with developments presented later in this article, is the following Proposition 3. If V is in S (Rn , R) (the class of Schwartz functions) and if V has compact support with respect to the time-coordinate x 0 , then there is a unitary operator SV on Hvac implementing the potential scattering morphism βV in the vacuum representation, i.e. relation (28) holds. This is, however, a very specialized version of results which have been obtained previously. We make no attempt to review these results here, but mention the following. It is quite obvious that one may generalize the result by dropping the compact support of V in time, relaxing the smoothness requirement and replacing the rapid decay conditions by suitable conditions of integrability. Furthermore, one can generalize V to a matrixvalued function as long as the resulting Hamilton operator HV (t) remains essentially selfadjoint and still fulfills C HV (t) = −HV (t)C. Generalizations of this type have been considered by Palmer [32], and he has found that the S-matrix SV implementing the scattering transformation exists, if ∂tα Vˆ (t, ·) L q (Rs ) is integrable over t ∈ R for all 1 ≤ q < 2 + ε and for all 0 ≤ α < s/2 + ε. Vˆ denotes the Fourier transform of V with respect to the spatial variables x 1 , . . . , x s . We refer to [32] for further details, and also for references to related, earlier work. 4. Moyal Minkowski Spacetime As it is usually introduced, n = 1 + s dimensional Minkowski spacetime gets Moyal-deformed if one postulates the following commutation relations between the coordinates: [xµ , xν ] = iθµν (µ, ν = 0, . . . , s)

(29)

with θ being some antisymmetric, real (n × n)-matrix. Of course, this stems from the idea of generalizing the behaviour of the quantum mechanical position operators xµ originating in motivations like [18] (restricting event localization by incorporating the uncertainty principle in general relativity) and [42] (string theory). Alternatively one


415

can implement the relations (29) by changing the product structure on the spacetime manifold such that [xµ , xν ] = xµ xν − xν xµ = iθµν (µ, ν = 0, . . . , s) is fulfilled between the coordinate chart functions xµ of the manifold. Thereby is the non-commutative Moyal product. But let us make this last point more precise now. Let q, p ∈ N0 , with p = 2l for l ∈ N0 , and let θ > 0. Then we define the (q + p) × (q + p)-matrix ⎤ ⎡ θ M = Mθ = 2

⎢ 0q×q ⎢ ⎢ ⎢ ⎢ ⎣ 0 p×q

0q× p 0l×l −1ll×l

⎥ ⎥ ⎥ ⎥ 1ll×l ⎥ ⎦

(30)

0l×l

having the 2l × 2l-dimensional standard symplectic matrix in the lower right corner, and zeros everywhere else. With this notation, we introduce the Moyal product 1 c (q, p) g(x) = c(x − Mu)g(x + v)e−iu·v d q+ p ud q+ p v, x ∈ Rq+ p , (2π )q+ p for (complex-valued) Schwartz functions c, g ∈ S (Rq+ p ). By u · v we denote the standard Euclidean scalar product of vectors u, v ∈ Rq+ p . One can show, either directly or by adapting the arguments of [20], that c (q, p) g is again in S (Rq+ p ) and that the product c (q, p) g is jointly continuous in c and g with respect to the usual test-function topology on S (Rq+ p ). In the case that q = 0, M = Mθ is invertible, and then one has 1 −1 c(x − u)g(x + v)e−iu·M v d p ud p v, c (0, p) g(x) = (π θ ) p which is the usual Moyal product investigated in several references (see [20,21]). In the other extreme case, p = 0, one finds 1 c (q,0) g(x) = c(x)g(x + v)e−iu·v d q ud q v = c(x)g(x), (2π )q i.e. the product c (q,0) g coincides with the usual pointwise product of functions. In the general case, it is straightforward to check that (c ⊗ ϕ) (q, p) (g ⊗ ξ ) = (c (q,0) g) ⊗ (ϕ (0, p) ξ )

(31)

for c, g ∈ S (Rq ) and ϕ, ξ ∈ S (R p ). Together with the continuity of · (q, p) · in both entries and the fact that S (Rq+ p ) = S (Rq ) ⊗ S (R p ) topologically, this shows that the product (q, p) is associative and furnishes an algebra product on S (Rq+ p ), because these properties are known for (q,0) and (0, p) . Furthermore, the standard complex conjugation induces a ∗-involution on S (Rq+ p ) with respect to the product (q, p) . We denote this by c → c∗ = c. ¯ As a ∗-involution, it has the property c∗ (q, p) g ∗ = (g (q, p) c)∗ .

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With the algebra product (q, p) and the complex conjugation as a ∗- involution, S (Rq+ p ) . By (31), we have is turned into a ∗-algebra which we denote by SM (q, p) SM = SM ⊗ SM , (q, p) (q,0) (0, p)

(32)

which holds also in the topological sense. One can adapt the arguments in [20] to observe that the product (q, p) can be extended to much larger spaces of functions and even distributions. An important case is that one factor in c (q, p) g is in S (Rq+ p ) and the other is in L 2 (Rq+ p ). Again we consider this situation first for q = 0. Using Lemma 2.12 of [20] (resp. reference [43] therein, which is [21] here), it holds that c (0, p) g is in L 2 (R p ) if both c and g are in L 2 (R p ). One can thus also define the operator of left Moyal multiplication on L 2 (R p ), Lc : g → c (0, p) g, g ∈ L 2 (R p ), for c ∈ L 2 (R p ). It is proved in [21] that this operator is bounded, more precisely, that Lc g L 2 ≤

1 c L 2 g L 2 . (2π θ ) p/2

(33)

The same estimate holds then also for the operator of right multiplication by c ∈ L 2 (R p ) given by Rc : g → g (0, p) c, g ∈ L 2 (R p ), since Rc g L 2 = Lc¯ g ¯ L 2 and g ¯ L 2 = g L 2 , where the overlining denotes complex conjugation. For p = 0, as c (q,0) g = c · g = g ·c = g (q,0) c is just the usual pointwise product of functions, one has c (q,0) g L 2 = g (q,0) c L 2 ≤ c∞ g L 2 , where · ∞ is the supremum norm. This entails that for c = cq ⊗ c p with cq ∈ S (Rq ) and c p ∈ S (R p ), the operators Lc : g → c (q, p) g, and Rc : g → g (q, p) c, g ∈ L 2 (Rq+ p ), are bounded operators whose operator norms are not greater than (2π θ )− p/2 cq ∞ c p L 2 . Since each c ∈ S (Rq+ p ) can be approximated by a sequence Nj=1 cq, j ⊗ c p, j as N → ∞, so that for all of the Schwartz norms ·s there holds ∞ j=1 cq, j ⊗c p, j s < ∞, it follows that Lc and Rc are bounded operators on L 2 (Rq+ p ) for all c ∈ S (Rq+ p ). Furthermore, we put on record here the following hermiticity property of Lc and Rc . Lemma 3. Let c ∈ SM and let ϕ, ψ ∈ L 2 (Rq+ p ). Then (q, p) (c (q, p) ϕ, ψ) L 2 = (ϕ, c∗ (q, p) ψ) L 2 , (ϕ (q, p) c, ψ) L 2 = (ϕ, ψ (q, p) c∗ ) L 2 .

(34) (35)


Proof. Consider first the case q = 0. Then 1 −1 c(w)ϕ(v)e−i(x−w)·M (x−v) ψ(x)d p wd p vd p x, (c (q, p) ϕ, ψ) L 2 = (π θ ) p 1 −1 (ϕ, c∗ (q, p) ψ) L 2 = ϕ(x)c(y)ψ(z)ei(x−y)·M (x−z) d p zd p yd p x. (π θ ) p

417

(36) (37)

Carrying out the substitution (w, v, x) → (y, x, z), the right hand side of (36) becomes 1 −1 c(y)ϕ(x)e−i(z−y)·M (z−x) ψ(z)d p yd p xd p z. (38) (π θ ) p Thus one can see that (38) coincides with (37) upon noticing that, using the antisymmetry of M −1 , (z − x) · M −1 (z − y) = −x · M −1 z + x · M −1 y − z · M −1 y coincides with (x − y) · M −1 (x − z) = −x · M −1 z + y · M −1 z − y · M −1 x. This proves (34) in the case q = 0, and (35) is proved analogously. Then we notice that (34) and (35) are obviously correct for p = 0. Therefore we obtain, using the tensor product decomposition of (q, p) as in (31), ϕq ⊗ ϕ p , (cq ⊗ c p )∗ (q, p) (ψq ⊗ ψ p ) L 2 = ϕq ⊗ ϕ p , (cq∗ (q,0) ψq ) ⊗ (c∗p (0, p) ψ p ) 2 L = (cq (q,0) ϕq ) ⊗ (c p (0, p) ϕ p ), ψq ⊗ ψ p L 2 = (cq ⊗ c p ) (q, p) (ϕq ⊗ ϕ p ), ψq ⊗ ψ p L 2 , whenever cq ∈ SM , c p ∈ SM and ϕq , ψq ∈ L 2 (Rq ), ϕ p , ψ p ∈ L 2 (R p ). This (q,0) (0, p) implies (34). Relation (35) is proved analogously. 5. The Dirac Field on Moyal-Deformed Minkowski Spacetime as a Lorentzian Spectral Geometry — General Discussion We will now embark on a — rather informal — discussion on the setting in which we wish to view the quantized Dirac field on Moyal-deformed Minkowski spacetime. Assume that n ≥ 2, n = 1 + s, and assume the restrictions on n made before in (2). Let q + p = n where p is even. Let C ∞ (Rn , C N ), N = N (n) as in (1), denote the space of smooth spinor fields on flat Minkowski spacetime Rn = R1+s as introduced in Sec. 2. We can introduce a scalar product on the spinors given by ψ¯ A (x)δ AB η B (x)d n x (ψ, η) = (39) Rn

for ψ = (ψ A ) NA=1 , η = (η A ) NA=1 in L 2 (Rn ) ⊗ C N . Let H = Hn denote the Hilbert space of square-integrable spinors L 2 (Rn ) ⊗ C N , carrying the scalar product (39). Then

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S (Rn , C N ) ∼ can act from = S (Rn ) ⊗ C N is a dense subspace of H. The algebra SM (q, p) the left or the right on H; an explicit representation of the left action is (Lc ψ) A = c (q, p) ψ A

(40)

for ψ = (ψ A ) NA=1 in H. We denote by A M the represented algebra LSM AM ,

(q, p)

. Thus, we

have a ∗-algebra of bounded linear operators, acting on H, (cf. last section), and if p = 0, then this algebra is non-commutative. Furthermore, we have the usual Dirac operator D defined in (4), whereas we set D = D0 for potential V = 0 here, acting on a dense domain in H; for convenience, we shall take this domain to be C0∞ (R) ⊗ S (Rs ) ⊗ C N . Said data A M , H, D are reminiscent of the data of a spectral triple in the spectral triple approach to non-commutative geometry by Connes [12,13 and 14] (cf. also [36]), and in fact, this is how we would like to think of them. There are, however, a few technical obstructions to doing so, since the original spectral geometry approach generalizes compact Riemannian spin geometries, while in our case A M is a non-commutative deformation of an algebra of functions over the non-compact Rn , and D is the Dirac operator of a metric of Lorentzian signature. This means that a modified structure needs to be provided in order to attain a spectral geometry generalization of non-compact Lorentzian spin geometries of comparable strength as in the compact, Riemannian case. This endeavour will be carried out elsewhere [33], we report here only about some of the important ingredients in a rather non-technical manner, and largely tailored to our Moyal spacetime case at hand. We begin by noting that structural elements in addition to A M , H, D are needed already in the Riemannian spectral geometry framework. What is required is an anti-unitary involution C on H, playing the role of a charge conjugation, and in our Moyal-setting, C will in fact be defined as in (3). Additionally, one needs an operator γ on H which induces an orientation, and in our Moyal-case at hand, γ = γ0 γ1 . . . γs is the product of Dirac matrices, acting on L 2 -spinors in H by matrix multiplication from the left. Supposing for a moment (for the purpose of comparison) that A M , H, D, C, γ were describing a compact (non-commutative) Riemannian spin geometry in the framework of spectral geometry — which actually is not the case — then the just listed items would be required to fulfill important structural properties, such as: (i) A M is a unital pre-C ∗ -algebra, (ii) D is hermitian and elliptic, and (D − λ1l)−n is in a suitable Schatten class for λ ∈ spec D, (iii) a series of (anti-)commutation relations between A M , D, C and γ , (iv) certain “regularity” conditions on A M and D (including domain conditions). (See [23] for a detailed exposition of the required properties.) Now in the present case, where A M , H, D, C, γ actually derive from Moyal-deformed Minkowski spacetime, several of these properties, in particular (5) and (5), no longer hold, but need to be replaced by suitable generalizations. We won’t discuss here the appropriateness of the generalizations envisaged (see [33]), but only give a few indications of their nature. A M is not a unital algebra, so one needs, as a further datum, a unitization A M,I ⊃ A M , where A M,I is a unital pre-C ∗ -algebra. The work [20] contains an extended discussion on the best choice of A M,I in the Riemannian Moyal-algebra case (actually, for q = 0), and since this discussion concerns mainly topological aspects of the non-commutative space as opposed to its metric structure, the results of this apply here as well.


419

In [20], A M,I is constructed as follows. Let c be a C ∞ function on R p which is bounded together with all of its derivatives. Then define the operator (cf. (40)) Lc : ψ → Lc ψ for all ψ ∈ H = L 2 (R p ) ⊗ C N . This is a bounded operator with respect to the operator norm on L 2 (R p ) ⊗ C N . The ∗-algebra generated by these operators is taken as A M,I . Note that Lc1 Lc2 ψ = Lc1 (0, p) c2 ψ when c1 (0, p) c2 is defined, and likewise Lc∗ = Lc∗ . One can opt for this choice of A M,I also in the case of (q, p) . Another modification is needed for (ii). Already in the non-compact Riemannian case, (D − λ1l)−n is not compact for resolvent values λ of D, but this can be remedied by requiring that a(D − λ1l)−n is in a suitable Schatten class for a ∈ A M . However, in the Lorentzian case, D is not elliptic, and thus a(D − λ1l)−n is non-compact. Moreover, D is not hermitian with respect to the L 2 scalar product. A way to get around this difficulty is to introduce another element of structure in the form of a further linear, bounded operator β : H → H. This operator carries the information of a “time-like” direction and thereby encodes the Lorentzian metric signature; in our case, β = γ0 , acting as (matrix) multiplication operator on the spinors. The characteristic properties of β, besides β 2 = 1 and suitable Clifford relations with C and γ , are β D = D ∗ β on the C ∞ -domain of D, and that

D =

1 ∗ (D D + D D ∗ ) 2

is an elliptic operator so that a(D − λ1l)−n is in a suitable Schatten class for resolvent values λ of D and a ∈ A M . (The adjoint D ∗ is defined with respect to the scalar product of H.) Whence, the collection of objects A M,I ⊃ A M , H, D, β, C, γ in combination with a list of relations and conditions that will be discussed in detail in [33], can be viewed as a “Lorentzian spectral triple” (LOST), i.e. the generalization of spectral geometry from Riemannian to Lorentzian signature. As we have outlined, Moyal-deformed Minkowski spacetime can be fit into this setting. If one now contends that non-commutative Lorentzian spacetimes are described in terms of LOSTs with data A M,I ⊃ A M , H, D, β, C, γ , one is faced with the question as to what a quantum field theory on a LOST should be, and how such quantum field theories can, on one hand, be constructed, and on the other hand, be interpreted. A fairly immediate idea is this: Since a Hilbert space H with a Dirac-operator D and a charge conjugation C acting in it are part of the data describing a LOST, one may define the Dirac field on a LOST as an abstract CAR algebra corresponding to these data. One must remember, however, that the Hilbert space H does not play the role of the Hilbert space K (= KV , V = 0) in Proposition 1, describing the space of equivalence classes of smooth, compactly supported elements in L 2 (Rn ) ⊗ C N modulo the kernel of

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the operator R = R + − R − (where R ± are the advanced/retarded fundamental solutions of D). Nevertheless, the Hilbert space structure of H = L 2 (Rn ) ⊗ C N is used to obtain a Hilbert space structure on the set of equivalence classes. In the case of a general LOST, it is at present not clear how to characterize advanced and retarded fundamental solutions of D. One of the difficulties is caused by the circumstance that “advanced” and “retarded” refer to localization properties which are notoriously difficult to capture in non-commutative geometry. This notwithstanding let us, for the time being, suppose that we have a LOST where advanced and retarded fundamental solutions of D are given as quadratic forms on a suitable domain D contained in the joint C ∞ -domain of D and D ∗ . Abusing notation, we will denote these quadratic forms by f, h → ( f, R ± h),

f, h ∈ D.

The fundamental solution property amounts to the condition (D ∗ f, R ± h) = ( f, h) = ( f, R ± Dh) for all f, h ∈ D. Guided by the example of the Dirac field on commutative Minkowski spacetime, one is led to the assumption that ( f, h)(R) = eiδ ( β f, R + h) − ( β f, R − h) defines, upon choice of a suitable phase δ, a scalar product on D/ ker(·, ·)(R) . [At present it is not clear if such a property can actually be proved under suitable additional “regularity” conditions on LOSTS, or if this is genuinely an extra assumption; but in our Moyal spacetime example in the next section we will see that this property is fulfilled.] With this assumption, one can define the Hilbert space K(R) as the completion of D/ ker(·, ·)(R) with respect to (·, ·)(R) . Under these circumstances, the conjugation C on D induces a conjugation C on K(R) via C[ f ](R) = [C f ](R) . Thus, one has a Hilbert space K(R) with a conjugation C on it. One can therefore define the associated CAR-algebra F(K(R) , C) in a manner completely analogous to the example of the free Dirac field on Minkowski spacetime, cf. Sec. 2. That is, F(K(R) , C) is generated by B([ f ](R) ), f ∈ D, which are linear in [ f ](R) , and subject to the relations B([ f ](R) )∗ = B(C[ f ](R) ), {B([ f ](R) ) , B([h](R) )} = 2([ f ](R) , [h](R) )(R) 1l, B([D f ](R) ) = 0. ∗

At this stage, one has constructed abstractly a quantum field theory on a non-commutative geometry described by a LOST and some additional structure. The quantum field theory was then essentially obtained by second quantization. The question arises how such a quantum field theory should be interpreted. Regarding this point, let us specialize to the case that A M is the Moyal- deformed algebra of functions on Minkowski spacetime, and H = L 2 (Rn ) ⊗ C N , D = C0∞ (R) ⊗ S (Rs ) ⊗ C N , with the Dirac operator as in (5). This means that H and D are the same as in the case of commutative, “undeformed” Minkowski spacetime, just the domain D has changed, but this does not lead to a significant modification. As will be explained in the next section, there will again be uniquely determined advanced and retarded fundamental solutions R ± of D. The CAR-algebra F(K(R) , C) one obtains in this case coincides with F(K, C) defined in Sec. 2, except that K(R) is larger than K owing to


421

the fact that D is taken larger than it was in the case of commutative spacetime. This difference would, however, disappear in the vacuum representation of the Dirac field (defined with respect to the time-translations) upon passing to von Neumann algebras in that representation. Thus, the von Neumann algebras of the CAR- algebras of the Dirac field, in vacuum representation, constructed either for classical Minkowski spacetime, or for Moyal-Minkowski spacetime, both coincide. It is therefore worth contemplating if the sketched way of “abstract” quantization of the LOST corresponding to Moyal-deformed Minkowski spacetime leads to anything different from the usual quantized Dirac field on usual Minkowski spacetime. We argue that this is indeed the case. One must remember that, in operational terms, a quantum field theory — on a classical spacetime — is described by an assignment of observables to spacetime regions and that the physical content of the theory lies mainly in the localization properties of the observables (and their algebraic relations) relative to each other, see [26,27] and discussion further below. We must, in the case of MoyalMinkowski spacetime, specify the observables of the quantum field theory we have defined, and study their localization properties in connection with the algebraic structure of the Moyal-Minkowski-algebra A M . In the vacuum representation (Hvac , π vac , vac ) of F(K0 , C), we have defined the field operators ψ( f ) = π vac (0 ( f )),

f ∈ C0∞ (Rn , C N ).

These operators do not correspond directly to observable quantities since they fulfill anticommutativity upon spacelike separation of the test-spinors f . Therefore, one needs to build operators corresponding to observables from the ψ( f ). A common choice is to take operators of the form ψ( f 1 )∗ ψ( f 2 ) as building blocks for observables. Then ψ( f 1 )∗ ψ( f 2 ) commutes with ψ(h 1 )∗ ψ(h 2 ) if the supports of f 1 and f 2 are spacelike separated from the supports of h 1 and h 2 . Certain operators arising as limits of linear combinations of such operators have interesting properties. Among them is the Wick-product : ψ + ψ : (c) which is indexed by scalar testing functions c ∈ C0∞ (Rn , R). One may define : ψ + ψ : (c) as follows. Take two finite families of spinors, eµ and ηµ (µ = 1, . . . , L) in C N , with the property L that µ=1 e A µ ηµB = 14 γ0 AB (the matrix entries of γ0 ). Then define, for q1 and q2 in C0∞ (Rn , R), the operator ψ ψ(q1 ⊗ q2 ) = +

L

ψ(q1 eµ )∗ ψ(q2 ηµ ).

µ=1

The map q1 ⊗q2 → ψ + ψ(q1 ⊗q2 ) defines a real-linear operator-valued distribution and thus extends to C0∞ (Rn × Rn , R). Let j be a family of real-valued functions in C0∞ (Rn ) approaching the δ-measure peaked at 0 for → 0, and set, for q1 , q2 ∈ C0∞ (Rn , R), F (x, y) = q1 (x)q2 (y) j (x − y) (x, y ∈ Rn ). Moreover, denote by W ⊂ Hvac the dense subspace generated by Pvac , where P ranges over all polynomials in the ψ( f ) with f ∈ C0∞ (Rn , C N ) (including the case that P has degree zero, i.e. P is a multiple of 1l). With these conventions, we define : ψ + ψ : (c)χ = lim ψ + ψ(F )χ − (vac , ψ + ψ(F )vac )χ →0

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M. Borris, R. Verch

for all χ ∈ W and c(x) = q1 (x)q2 (x) (x ∈ Rn ). It turns out (see Sec. 7 and Appendix A) that : ψ + ψ : (c) is an essentially selfadjoint operator on W which furthermore turns out to be independent of the choices made for eµ and ηµ (µ = 1, . . . , L). The : ψ + ψ : (c) are local operators in the sense that : ψ + ψ : (c1 ) commutes with : ψ + ψ : (c2 ) if the supports of c1 and c2 are spacelike separated. For c ≥ 0, : ψ + ψ : (c) can be interpreted as the observable of (squared) field strength density weighted with the function c. An interesting property of : ψ + ψ : (c), proven in Appendix A, is [: ψ + ψ : (c), ψ( f )] = −iψ(c R0 f ) C0∞ (Rn ) and all

(41)

C0∞ (R) ⊗ S (Rs ) ⊗ C N . On the other hand, we will also

for all c ∈ f ∈ show in Sec. 8 that, identifying c with the scalar potential in the discussion of potential scattering in Sec. 6, there holds d d −1 vac π (βλc (0 ( f ))) = Sλc ψ( f )Sλc = ψ(c R0 f ) (42) dλ dλ λ=0

λ=0

C0∞ (Rn )

C0∞ (Rn , C N ).

and f ∈ for all c ∈ d S In view of (41) and (42), the observables : ψ ∗ ψ : (c) are identified as −i dλ λ=0 λc where Sc is the scattering matrix corresponding to the localized scattering potential c. This connection between localized observables and the derivative of the scattering matrix of a localized interaction with respect to the interaction strength is, of course, long known, especially in the context of perturbative interacting quantum field theory, and often goes by the name “Bogoliubov’s formula” [6]. We now wish to point out that one can obtain in a similar manner observables for the quantized Dirac field on Moyal-deformed Minkowski spacetime employing Bogoliubov’s formula. The precise mathematical discussion of the considerations we present here will be given in the next section. In the case of the Dirac field on usual Minkowski spacetime, the scattering matrix SV ≡ Sc was constructed for the Dirac operator DV = D + V , where the potential term was V f = c f , c f meaning the usual pointwise (and component-wise) multiplication of a scalar function c with a spinorfield f . We should now recall that classical Minkowski spacetime is also described by the structure of a LOST. The data for the LOST corresponding to classical Minkowski spacetime coincide with the data for the LOST of Moyal-Minkowski spacetime, except that instead of the non-commutative algebra A M = SM we have the commu(q, p) ∞ Min n = C0 (R ) of scalar functions on spacetime. The map c → cϕ, tative algebra A ϕ ∈ H = L 2 (Rn , C N ) produces a faithful representation of AMin on the Hilbert-space of square-integrable spinor fields. For the case of Moyal-Minkowski spacetime, one can regard the potential term V in a similar light, and define, for ϕ ∈ H, for instance V ϕ = Lc ϕ + Rc ϕ = c ϕ + ϕ c

(43)

= Moyal-deformed Minkowski spacetime with commutative time coordinate (corresponding to q = 1 and p even in the notation of (30)) there is a Bogoliubov-transformation βVM on the CAR-algebra F(K = K(R0 ) , C) describing scattering by the non-commutative potential V given in (43). (This needs mild further assumptions on c, see Sec. 6 for details.) Furthermore, we will show that this scattering transformation is unitarily implementable in the vacuum-representation (Hvac , π vac , vac ), so that there is a unitary operator SVM with with real-valued c in A M

SM . In the next section we will show that in the case of a (q, p)

SVM π vac (0 ( f ))(SVM )−1 = π vac (βVM (0 ( f )))


423

for all f ∈ C0∞ (R)⊗S (Rs )⊗C N . Consequently, one can formally define the derivative d SM (44) (c) = −i dλ λ=0 λV which, following the ideas underlying Bogoliubov’s formula alluded to just before, would correspond to an observable quantity. In Sec. 7 we will in fact show that d M −1 S M ψ( f )SλV = [i(c), ψ( f )] = ψ(V R0 f ) (45) dλ λ=0 λV for all f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N with an essentially selfadjoint operator (c) on W. d M , in the sense that One may therefore identify (c) with the derivative −i dλ λ=0 SλV (45) holds. In the case of usual Minkowski spacetime, the assignment c →: ψ + ψ : (c), where c is a scalar C0∞ test-function on spacetime, has the typical properties of an observable quantum field of Wightman type [40]. The support of the test-function c limits the localization of the observable : ψ + ψ : (c), which is reflected by the relations (41) and (42) and the fact that the changes of states which βλc induces are localized in the support of c. In the algebraic approach to quantum field theory [26,27], one therefore considers the ∗-algebras R(O) generated by all observable quantum field operators : ψ + ψ : (c), where the support of c is contained in the spacetime region O.1 Then one obtains an assignment O → R(O) of spacetime regions to operator algebras with the two characteristic properties of Isotony: O1 ⊂ O2 ⇒ R(O1 ) ⊂ R(O2 ), Locality: O1 ⊥ O2 ⇒ [F1 , F2 ] = 0 for F j ∈ R(O j ) ( j = 1, 2), where O1 ⊥ O2 means that the spacetime regions are causally separated, i.e. there is no causal curve joining them. According to the algebraic approach to quantum field theory, a quantum field theoretical model is basically characterized by a map O → R(O) with these properties (see [26,27,38]), describing especially the localization of observables of the quantum system under consideration on a “classical” spacetime with commutative coordinate functions. Let us now discuss some, however vague, ideas how this may be generalized to quantum field theories on non-commutative spacetimes, where again we stay at the level of the Dirac field on Moyal-Minkowski spacetime. The scattering by a noncommutative potential furnishes the assignment c → (c) of (44). We interpret (c) as an observable, and hence we have an assignment of elements c in the non-commutative algebra A M to (unbounded) operators in Hvac . The c now carries the information about the spacetime localization of (c), but due to the non-commutativity of A M , this is subject to uncertainties. In particular, in general (c1 ) and (c2 ) won’t commute anymore if the supports of c1 and c2 , viewed as test-functions, are causally separated. Therefore, if one defines the algebras R M (O) as being generated by the (c), where c has support in O, then the assignment O → R M (O) is clearly different from O → R(O) as defined 1 Two things should be noted here. (1) Actually, R(O) would have to be defined as algebraically generated by all observable quantum field operators smeared with test-functions supported in O; we use : ψ + ψ : as a placeholder for any observable quantum field at this point. (2) In the algebraic approach to quantum field theory it is customary to define R(O) as algebraically generated by the bounded functions of quantum field operators smeared with test-functions supported in O; here, our R(O) are algebras of unbounded operators.

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above for usual Minkowski spacetime, and thus we see that we derive indeed a different system of observables from the scattering morphisms via Bogoliubov’s formula in the non-commutative case, without an obvious locality structure. Nevertheless, one may attempt to mimic the algebraic approach to quantum field theory in a generalized form, upon forming algebras of observables R M (P) labelled by subsets P of A M , understanding that R M (P) be generated by the (c) with c ∈ P. It is not clear at this stage what structure these subsets should have, e.g. if they should be subalgebras of A M . In comparison to the classical case, what seems to be required is a partial ordering on the collection of chosen P, and a concept of causal separation [38]. An idea could be to choose the P as sets of (approximate) projections, inspired by the situation on classical spacetime, where a subset O may be identified with its characteristic function, which is a projection in the commutative algebra of coordinate functions. The ordering relation may then be taken as operator ordering. It is more difficult to capture the concept of causal separation. In the case of classical Minkowski spacetime, the supports of two C0∞ test-functions c1 and c2 are causally separated if and only if ic1 f, Rm c2 h = 0 for all spinor fields f and h, where Rm is the causal propagator for the Dirac equation for any mass term m (corresponding to R V for V = m, cf. Eq. (8)). By assumption, the causal propagator is available in our non-commutative setup as the quadratic form (., . )(R) on the domain D ⊂ H, and thus one may characterize the causal disjointness of two subsets P1 and P2 of A M with the help of this quadratic form for any mass term. Ideally, one might want to define P1 ⊥ P2 to be equivalent to (c1 f, c2 h)(R) = 0 for all c j ∈ P j and all f, h ∈ D, provided this is compatible with the ordering relation. If that cannot be had, the second best option would be to define P1 ⊥ P2 as meaning that (c1 ., c2 . )(R) is, in a suitable sense, “small” compared to (., . )(R) — a sort of “infinitesimal” quantity in the sense of spectral geometry. Supposing that suitable forms of a partial ordering relation P1 ≤ P2 and a causal separation relation P1 ⊥ P2 have been found for suitably chosen subsets P of A M , it seems well possible that the generalized version of a quantum field theory on noncommutative spacetime in the operator algebraic setting may take the shape of an assignment P → R M (P), where the R M (P) are operator algebras, subject to the relations of Isotony: P1 ≤ P2 ⇒ R M (P1 ) ⊂ R M (P2 ), Locality: P1 ⊥ P2 ⇒ [F1 , F2 ] = 0 for F j ∈ R M (P j ) ( j = 1, 2). Actually, it could happen that the condition of locality ought to be relaxed requiring only that [F1 , F2 ] is in a suitable sense “small” compared to F1 and F2 if P1 ⊥ P2 , similar in spirit to the possibly generalized condition of causal separation. Admittedly, this is at present all speculation, and a careful study of examples is required before a clear picture of the basic structure of quantum field theory on (Lorentzian) non-commutative spacetime will emerge. 6. The Dirac Field on Moyal-Deformed Minkowski Spacetime – The Model Now our intention is to follow the lines of Sec. 2 under the modifications of using n = 1 + s = q + p dimensional Moyal-deformed Minkowski spacetime and a suitable different potential term for the Dirac operator. The spacetime of interest (with dimension n = 1 + s = q + p) will be described as in Sec. 4, with the exception that we restrict ourselves to Moyal matrices M of the more


specialized form

425

⎡

⎤ ··· 0 0 ⎢ ⎥ M = ⎣ ... M ⎦ (q+ p−1)×(q+ p−1) 0 (q+ p)×(q+ p)

(46)

i.e. the first row and the first column shall vanish, q ≥ 1. Nothing is changed (cf. Sec. 2) in the manner of how we define the algebra of Dirac matrices (γ0 , γ1 , . . . , γs ) and the charge conjugation C. Again the Dirac operator (m > 0 constant) acting on C0∞ (R) ⊗ S (Rs ) ⊗ C N is denoted by DV = (−i ∂ + m) + V. But now the “potential term” operator V acting on f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N is not just the multiplication operator multiplying f with a scalar function, but one of the following operators: (i) (V f ) A (x) = (V(i) f ) A (x) = (c (q, p) f A )(x) + ( f A (q, p) c)(x),

(47)

(ii) (V f ) (x) = (V(ii) f ) (x) = (c (q, p) f

(48)

A

where c ∈

A

C0∞ (R, R) ⊗ S (Rs , R)

A

(q, p) c)(x),

is a function of the form

c(x) = a(t)b(x),

(49)

with a ∈ C0∞ (R, R), b ∈ S (Rs , R), t = x 0 , x = (x 1 , . . . , x s ). We aim at presenting an analogue of Proposition 1 for the potential operators V = V(i) or V = V(ii) which describe “scattering by a time-dependent, spatially non-commutative potential”. Remark 1. The definitions of (i) and (ii) are two possibilities to generalize a scalar potential to the non-commutative setting. Note that both a left and right action of c with respect to the Moyal product must occur to ensure that the potential is invariant under charge conjugation. Variant (ii) is more regular than (i). It is useful, at this point, to consider first the Cauchy-data version of the dynamical problem. As in (16), we have the free Hamiltonian 0 k ∂ 0 (H0 f )(x) = iγ γ + γ m f (x) (50) ∂xk and the Hamiltonian with time-dependent interaction term, ∂ (HV (t) f )(x) = iγ 0 γ k k + γ 0 m + γ 0 V (t) f (x), ∂x

(51)

acting on f ∈ S (Rs ) ⊗ C N . Here V (t) stands for the operators V(i) (t) : f → V(i) (t) f,

f ∈ S (Rs ) ⊗ C N ,

(V(i) (t) f ) (x) = a(t)(b (q−1, p) f (x) + f A

A

A

(q−1, p) b(x))

(52) (53)

or V(ii) (t) : f → V(ii) (t) f,

f ∈ S (Rs ) ⊗ C N ,

(V(ii) (t) f ) A (x) = a(t)2 (b (q−1, p) f A (q−1, p) b(x)).

(54)

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M. Borris, R. Verch

By the assumptions made on a and b above, V (t) = V(i) (t) and V (t) = V(ii) (t) are bounded operators on L 2 (Rs , C N ). As in the case of a scalar potential, we have again that C HV (t) = −HV (t)C, which is a consequence of the easily checked equation CRc = Lc C,

(55)

obviously in the case of V(i) and under additional usage of the associativity of the Moyal product in the case of V(ii) . And HV (t) is a symmetric (in fact essentially selfadjoint) operator on S (Rs , C N ) ⊂ L 2 (Rs , C N ). As in the case considered before in Sec. 2, a smooth function ϕ ∈ C ∞ (Rs , C N ) is a solution of DV ϕ = 0

(56)

if and only if 1 d Pt ϕ = HV (t)Pt ϕ i dt with Pt ϕ = ϕ|t . Establishing existence and uniqueness of the Cauchy problem for the Dirac equation (56) is therefore equivalent to proving existence and uniqueness of solutions for the initial value problem 1 d vt = HV (t)vt , i dt

vt |t=0 = w.

This will be our next auxiliary result. (V ) 2 s N Proposition 4. (a) There is a unique family of unitaries Tt,t on L (R , C ), strongly continuous in t and t , so that (V )

(V )

(V )

(V )

Tt,t ◦ Tt ,s = Tt,s , Tt,t = 1l

(57)

1 d (V ) T w = HV (t)w i dt t,0

(58)

and

(V )

for all w ∈ L 2 (Rs , C N ). Moreover, Tt,t maps S (Rs , C N ) into itself. (b) Given w ∈ S (Rs , C N ), the map (V ) N ((t, t , x) ∈ R × R × Rs ) (t, t , x) → Tt,t w(x) ∈ C

is jointly C ∞ in all variables. (c) The Cauchy-problem for the Dirac-equation DV ϕ = 0 with the potential term V = V(i) or V = V(ii) is well-posed in the following sense. For any given w ∈ S (Rs , C N ) and t ∈ R there is a unique ϕ ∈ C ∞ (R) ⊗ S (Rs ) ⊗ C N such that DV ϕ = 0 and Pt ϕ = w.


427 (V )

Proof. Part (a). We shall work in the interaction picture, i.e. we obtain Tt,t as (V )

(V )

Tt,t = e−it H0 T˜t,t eit

H 0

,

(59)

(V ) where T˜t,t is the Dyson series for

U˜ (t) = eit H0 γ 0 V (t)e−it H0 , meaning that (V ) T˜t,t =

∞

Tñ(V ) (t, t ),

n=0 (V ) where Tñ (t, t ) is iteratively defined by

1 (V ) T˜0 (t, t ) = 1l, Tñ(V ) (t, t ) = i

t t

(V ) U˜ (r )Tñ−1 (r, t )dr.

Since the operators V (t) and γ 0 V (t) are bounded operators (with uniform bound in t) on L 2 (Rs , C N ), and H0 is essentially selfadjoint, one can rely on Theorem X.69 in [35] (V ) (V ) to see that Tt,t is a family of unitaries with the required properties, provided that Tt,t maps S (Rs , C N ) into itself. To show this, we note that (cf. [43], Theorem 1.2 and Appendix 1.D) (eit H0 f )(x) ds p (k)|t) − (γ 0 γ k pk + iγ 0 m) sin (| H (k)|t) fˆ( p) , = eik·x cos (| H (k)| (2π )s |H (k)| = where fˆ is the Fourier transform of f , p = ( p1 , . . . , ps ) ∈ Rs , and | H it H s N |k|2 + m 2 . This shows that e 0 f is in S (R , C ) (t ∈ R) for f ∈ S (Rs , C N ) and that, moreover, eit H0 f is C ∞ in t with respect to the S -topology. In the next step, we note that n t tn t2 1 (V ) ˜ ··· Tn (t , t ) = U˜ (tn ) · · · U˜ (t1 )dt1 · · · dtn . i t t t We set v f = v(i) f = γ 0 (b (q−1, p) f + f (q−1, p) b), V = V(i) ,

(60)

v f = v(ii) f = γ (b (q−1, p) f (q−1, p) b), V = V(ii) .

(61)

0

Then V (t) f = a(t)v f , and n t tn t2 1 (V ) ˜ Tn (t , t ) f = ··· a(tn ) · · · a(t1 ) f (n) (t (n) )dt1 · · · dtn , i t t t where f (n) (t (n) ) = eitn H0 vei(tn−1 −tn )H0 v · · · ei(t1 −t2 )H0 ve−it1 H0 f.

428

M. Borris, R. Verch

This implies that, given any pair of multi-indices α, β ∈ Ns0 , we obtain an estimate of the form2 Ca (t , t )2n α β ˜ (V ) α β (n) (n) D f (t ) (62) max x D Tn (t , t ) f 2 ≤ x 2 L L n! t j ∈[t ,t ] with the constant Ca (t , t ) = max |a(t)||t − t |. t∈R

Now we will show that there is a constant Cb,α,β (t , t ) and a constant m(α, β) such that α β (n) (n) x max D f (t ) sup D γ f L 2 . (63) 2 ≤ Cb,α,β (t , t )n t j ∈[t ,t ]

L

|γ |≤2m(α,β)

The proof will be given by induction on n, and we will only treat the case v = v(i) (corresponding to V = V(i) ) as the other case v = v(ii) is completely analogous. The inductive proof will only be needed for the special case x α = 1 (i.e. all α j = 0) which makes it more transparent, and the result will be used in the proof of the general case. Let β ∈ Ns0 be a multi-index and define B|β| = sup|γ |≤|β| D γ b L 2 . We want to prove by induction that there is a constant F > 0 with β (n) (n) n sup D γ f L 2 . (64) D f (t ) 2 ≤ 2n · 2|β|n F n B|β| L

|γ |≤|β|

Let us show that this is correct for n = 1: f (1) (t1 ) = eit1 H0 ve−it1 H0 f = eit1 H0 γ 0 b (q−1, p) (e−it1 H0 f ) + (e−it1 H0 f ) (q−1, p) b . The Leibniz rule for coordinate derivatives applies with respect to the Moyal product = (q−1, p) : ∂ ∂ ∂ (h g) = h g + h g , h, g ∈ S (Rs ). (65) ∂x j ∂x j ∂x j ∂ ∂x j

Since the coordinate derivatives |β|

β

D f

(1)

(t1 ) =

2

commute with eit H0 , we hence obtain

eit1 H0 γ 0 (D β (k) b) (D β (k) e−it1 H0 f )

k=1

+(D β (k) e−it1 H0 f ) (D β

(k)

b) ,

(66)

where β (k) and β (k) are suitable multi-indices3 with β j (k) + β j (k) = β j . Using the fact that h g L 2 ≤ Fh L 2 g L 2 for h, g ∈ S (Rs ), with F = (2π θ )− p/2 , one deduces β (1) (67) D f (t1 ) 2 ≤ 2 · 2|β| F B|β| sup D γ f L 2 . L

|γ |≤|β|

2 For the remainder of this proof, D β = (−i∂/∂ x 1 )β1 · · · (−i∂/∂ x s )βs ; here, D is not to be confused with the Dirac operator. 3 The case β (k ) = β (k ) and β (k ) = β (k ) for some k = k typically occurs in our sum decom1 2 1 2 1 2 position of multiple derivatives of a product. Usually, this is written as a sum over fewer terms, occurring with a multiplicity expressed by binomial coefficients.


429

In order to conclude that the validity of (64) for some n ∈ N implies the relation (64) with n + 1 in place of n, we note that f (n+1) (t (n+1) ) = eitn+1 H0 ve−itn+1 H0 f (n) (t (n) ) = eitn+1 H0 γ 0 b (e−itn+1 H0 f (n) (t (n) )) +(e−itn+1 H0 f (n) (t (n) )) b . Hence, relation (66) continues to hold under the simultaneous replacements f (1) (t1 ) → f (n+1) (t (n+1) ), e±it1 H0 → e±itn+1 H0 and f → f (n) (t (n) ). Therefore, (67) also holds when making these replacements, leading to the estimate β (n+1) (n+1) (t ) 2 ≤ 2 · 2|β| F B|β| sup D γ f (n) (t (n) ) L 2 D f L

|γ |≤|β|

≤2

n+1

·2

|β|(n+1)

n+1 F n+1 B|β| sup D γ f L 2 , |γ |≤|β|

where the induction hypothesis (64) was used in the second inequality. This proves by induction that (64) holds for all n ∈ N. Turning to the general case, the first observation is that, given multi-indices α, β ∈ Ns0 and a finite real interval [t , t ], there are constants αβ (t , t ) > 0 and m(α, β) > 0 such that α β it H0 ψ 2 ≤ αβ (t , t ) x ρ D δ ψ L 2 x D e L

|ρ|,|δ|≤m(α,β)

[t , t ],

S (Rs , C N )

holds for all ψ ∈ and all t ∈ where ρ, δ are multi-indices. The second observation is that the action of multiplication by a coordinate function x j on a Moyal product can be decomposed as follows (cf. [21]): There are numbers ε( j) which may take the values 0, 1 or −1, and for each coordinate x j there is a coordinate x ι( j) , such that iε( j)θ ∂h x j (h g) = h (x j g) + g 2 ∂ x ι( j) ∂g iε( j)θ h ι( j) = (x j h) g − (68) 2 ∂x for all h, g ∈ S (Rs ). Now we define: Mα,β =

sup

|γ |,|σ |≤m(α,β)

Cb,α,β (t , t ) = 2 · 2

2m(α,β)

x γ D σ b L 2 ,

|θ | m(α,β) Fαβ (t , t )m(α, β) Mα,β 1 + , 2

2

and we will show that (63) holds with these definitions for all n ∈ N. It holds that α β (n+1) (n+1) (t ) 2 x D f L = x α D β eitn+1 H0 ve−itn+1 H0 f (n) (t (n) ) 2 L ρ δ −itn+1 H0 (n) (n) ≤ αβ (t , t ) x D ve f (t ) L 2 (69) |ρ|,|δ|≤m(α,β)

430

M. Borris, R. Verch

for all tn+1 ∈ [t , t ] and all n ∈ N0 (with f (0) (t (0) ) = f ). Now we use (65) and (68) to conclude that we can write x ρ D δ ve−itn+1 H0 f (n) (t (n) ) |ρ|

=

|δ|

2 2

µ(l, k)γ 0 D ρ (l) D δ (k) e−itn+1 H0 f (n) (t (n) ) x ρ (l) D δ (k) b

l=1 k=1

+ν(l, k)γ 0 x ρ (l) D δ (k) b D ρ (l) D δ (k) e−itn+1 H0 f (n) (t (n) ) ,

(70)

where µ(l, k), ν(l, k) are complex numbers and ρ (l), ρ (l), δ (k), δ (k) are suitable multi-indices, where δ j (k) + δ j (k) = δ j ; |ρ (l)|, |ρ (l)| ≤ |ρ|, and |µ(l, k)|, |ν(l, k)| ≤ |ρ| 1 + |θ| . Thus we obtain for the sum on the right hand side of (70) the L 2 -norm 2 bound 2·2

|ρ|

·2

|δ|

|θ | |ρ| F sup x γ D σ b L 2 · sup D γ f L 2 , 1+ 2 |γ |≤|ρ|, |σ |≤|δ| |γ |≤|ρ|+|δ|

using (64) again; inserting this into (69) yields α β (n+1) (n+1) (t ) x D f

L2

|θ | m(α,β) ≤ αβ (t , t )m(α, β)2 · 2 · 22m(α,β) 1 + F 2 · sup x γ D σ b L 2 · sup D γ f L 2 |γ |,|σ |≤m(α,β)

n+1

≤ Cb,α,β (t , t )

|γ |≤2m(α,β)

sup

|γ |≤2m(α,β)

D γ f L 2

with the above definitions. This proves that (63) holds for all n ∈ N. In combination with (62), we have thus proved that for each given f ∈ S (Rs , C N ), ˜ (V ) the Dyson series ∞ n=0 Tn (t , t ) f converges in all Schwartz norms and thus yields (V ) s N again an element in S (Rs , C N ). Therefore, Tt,t also maps S (R , C ) into itself and thence has (as mentioned, by Thm. X.69 in [35]) the properties claimed in statement (a) of the lemma. Part (b). The arguments showing the claimed property are quite standard in view of the estimates given to establish part (a), so we will mainly sketch them. Let µ be any C-valued C0∞ -function on R × R and denote by Y the function (V )

(t, t , x) → µ(t, t )Tt,t w(x) ≡ Y (t, t , x). (V )

Since (t, t ) → Tt,t w ∈ L 2 (Rs , C N ) is continuous, Y is in L 2 (R × R × Rs , C N ). Thus we need only show that, if s+2 denotes the Laplacian in s +2 dimensions, (1−s+2 ) J Y is again in L 2 (R × R × Rs , C N ) for all J ∈ N; the claimed statement on smoothness then follows by Sobolev’s Lemma (cf. Thm. IX.24 in [35]). In turn, the required property


431

(V )

follows from the fact that Tt,t maps S (Rs , R N ) into itself and that, as established in the proof of (a) or following immediately thereof, ∂ (V ) (V ) T w = i HV (t)Tt,t w, ∂t t,t ∂ (V ) (V ) T w = −i Tt,t HV (t )w, ∂t t,t (ii) x α D β HV (t)x α D β is a continuous operator on (i)

S (Rs , C N ) uniformly in t ranging over compact intervals. Part (c). It follows from parts (a) and (b) that there exists for any given Cauchy-datum w ∈ S (Rs , C N ) a solution ϕ ∈ C ∞ (R) ⊗ S (Rs ) ⊗ C N of DV ϕ = 0 with Pt ϕ = w. Recalling the definition δ AB v A (x)w b (x) d s x (v, w)D = Rs

for v, w ∈ L 2 (Rs , C N ), one finds d (Pt ϕ, Pt ψ)D = (i HV (t)Pt ϕ, Pt ψ)D + (Pt ϕ, i HV (t)Pt ψ)D = 0 dt for all ϕ, ψ ∈ C ∞ (R) ⊗ S (Rs ) ⊗ C N which are solutions of the equations DV ϕ = 0 and DV ψ = 0. Hence, in particular, if for two solutions ϕ und ψ there holds (Pt (ϕ − ψ), Pt (ϕ −ψ))D = 0 for some real t , then it follows that (Pt (ϕ −ψ), Pt (ϕ −ψ))D = 0 for all real t. This shows that, if ϕ and ψ have the same Cauchy-datum on some Cauchy-hyperplane t , then actually ϕ = ψ. On C0∞ (R) ⊗ S (Rs ) ⊗ C N we can introduce the sesquilinear form γ0 AB ( f¯ B (q, p) h A )(x)d n x f, h = Rn = γ0 AB f¯ B (x)h A (x)d n x, Rn

where the last equality follows from the tracial property of the Moyal product for q = 0 (Lemma 2.1 (v) in [20]) and its obvious generalization to arbitrary (q, p) due to the trivial case p = 0 and the tensor product structure (32). Therefore we still have C f, Ch = −h, f ( f, h ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ). We recall the definitions t = {x = (x 0 , x 1 , . . . , x s ) ∈ Rn : x 0 = t}, Dt = L 2 (t , C N ), (v, w)D = δ AB (v¯ A (q, p) w B )(x)d s x t = v¯ A (x)δ AB w B (x)d s x (v, w ∈ Dt ), t

and the property (Cv, Cw)D = (w, v)D of a conjugation C induced on each Dt by the charge conjugation C (same symbol).

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Fig. 1. Sketch of the regions T ± (K )

For a subset G of n dimensional Moyal-deformed Minkowski spacetime the sets J ± (G) and the notion of hyperbolicity are defined in exactly the same way as in Sec. 2. Now we transfer the results collected in Proposition 1 to our new setting. For this purpose the following definition is needed. Definition 1. Let K be a non-empty subset of Rn . Then let κ− (K ) = inf{x 0 : x = (x 0 , x) ∈ K }, κ+ (K ) = sup{x 0 : x = (x 0 , x) ∈ K }, and define T + (K ) = {(x 0 , x) ∈ Rn : x 0 ≥ κ− (K )}, T − (K ) = {(x 0 , x) ∈ Rn : x 0 ≤ κ+ (K )}. Proposition 5. (a) DV f, h = f, DV h ( f, h ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ). (b) There is a unique pair of continuous linear maps R V± : C0∞ (R) ⊗ S (Rs ) ⊗ C N → C ∞ (R) ⊗ S (Rs ) ⊗ C N having the properties DV R V± f = f = R V± DV f and supp R V± f ⊂ T ± (supp f ) ( f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ). (c) C R V± = R V± C. (d) Writing R V = R V+ − R V− , the form ( f, h)V = f, i R V h is a sesquilinear form on C0∞ (R) ⊗ S (Rs ) ⊗ C N , and C is a conjugation for this form: (C f, Ch)V = (h, f )V = ( f, h)V ( f, h ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ). (e) For each t ∈ R it holds that ( f, h)V = (Pt R V f, Pt R V h)D , ( f, h ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ), where Pt : C ∞ (R) ⊗ S (Rs ) ⊗ C N → S (t , C N ) is the map given by Pt : ϕ → ϕ(t, ·) for ϕ : (x 0 , x) → ϕ(x 0 , x) in C ∞ (R) ⊗ S (Rs ) ⊗ C N , x = (x 1 , . . . , x s ). Hence, (·, ·)V is positive-semidefinite on C0∞ (R) ⊗ S (Rs ) ⊗ C N .


433

(f) Let E V be the subspace of all f ∈ C0∞ (R)⊗S (Rs )⊗C N so that ( f, f )V = 0, and let KV be the Hilbert space arising as completion of (C0∞ (R) ⊗ S (Rs ) ⊗ C N )/E V with respect to the scalar product induced by (·, ·)V (which will be denoted by the same symbol). The quotient map C0∞ (R) ⊗ S (Rs ) ⊗ C N → (C0∞ (R) ⊗ S (Rs ) ⊗ C N )/E V will be written f → [ f ]V . Then for each t ∈ R, the map Q V,t : [ f ]V → Pt R V f extends to a unitary map from KV onto Dt . (g) Let G be an open time-slice of n dimensional Moyal-deformed Minkowski spacetime, i.e. G = {(x 0 , x 1 , . . . , x s ) : λ1 < x 0 < λ2 } for some real numbers (or infinite) λ1 < λ2 , and suppose that V1 and V2 are potentials of the form described by (47), (49), and that V1 = V2 on G. Then R V±1 f = R V±2 f on G for all f ∈ C0∞ ((λ1 , λ2 )) ⊗ S (Rs ) ⊗ C N . Proof (sketch). (a) Clearly the only difference compared to Proposition 1 (a) is the partial claim V f, h = f, V h. To show this, calculate (only for the case V = V(i) ; the other one is completely analogous) V f, h = γ0 AB (V f ) B (x)h A (x)d n x = γ0 AB ((V f ) B , h A ) L 2 Rn

= γ0 AB (c (q, p) f B + f B (q, p) c, h A ) L 2 = γ0 AB ( f B , c (q, p) h A + h A (q, p) c) L 2 = γ0 AB f¯ B (x)(V h) A (x)d n x = f, V h, Rn

since c is real-valued, using Lemma 3. (b) To prove this, the fundamental solutions will be constructed explicitly. Using the notation f t (. ) = f (t , . ), define for f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N , (V ) ± 0 (R V f )(t, x) = ±iγ θ (±(t − t ))Tt ,t f t (x) dt , where θ is the Heaviside step function. Note that the integral is well-defined since f t (x) has compact support in t . By Lemma 4 it follows that (t, x) → (R V± f )(t, x) is C ∞ , and of Schwartz type with respect to x. Using standard arguments, and ) ± exploiting the properties of Tt(V ,t given in Lemma 4, one proves that D V R V f = ± ± ± f = R V DV f . The next step consists in showing that supp(R V f ) ⊂ T (supp f ). / T + (supp f ). Then x 0 < κ− (supp f ) and To this end, suppose that (x 0 , x) ∈ therefore (V )

θ (x 0 − t )Tt ,t f t (x) = 0 for all values of t, t and x. To see this, note that if t ≥ x 0 , then θ (x 0 − t ) = 0, and if t < x 0 < κ− (supp f ), then f t (. ) = 0 by the definition of κ− (supp f ).

434

(c) (d) (e) (f) (g)

M. Borris, R. Verch

Hence, it holds that (R V+ f )(x 0 , x) = 0 if (x 0 , x) ∈ / T + (supp f ), implying that + + supp(R V f ) ⊂ T (supp f ). The inclusion supp(R V− f ) ⊂ T − (supp f ) can be shown in an analogous manner. The uniqueness property of the fundamental solutions follows by a standard argument owing to the well-posedness of the Cauchyproblem for the Dirac-equation DV ϕ = 0. This is a consequence of the uniqueness of the R V± together with C DV = DV C. Analogous to Proposition 1; the crucial point is the validity of R V h, f = −h, R V f , for which the argument is again similar to the proof of Thm. 2.1 in [17]. The argument is the same as in Proposition 1. The proof is identical to the corresponding statement (g) of Proposition 1. The modification lies in the generalized class of Cauchy data. The choice of a partition of unity χ± depending only on the time-coordinate x 0 is needed at this point. Apart from the modified assumption on the subset G, providing an adjusted timedirection behaviour for the non-commutative case, this can obviously be proved the same way as Proposition 1, (h).

Analogous to Sec. 2 the self-dual CAR-algebra F(KV , C) is generated by the C-linear “abstract field operators” ( f ) = V ( f ) = BV ([ f ]V ), for f ∈ C0∞ (R)⊗S (Rs )⊗C N , obeying the relations ( f )∗ = (C f ), {( f ) , (h)} = 2( f, h)V 1l, (DV f ) = 0. ∗

Note that because of the trivial action of V with respect to the first coordinate (the time) it holds that DV f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N . Again this construction can be carried out for “local subspaces” of KV as well (cf. Sec. 2), and we get F(KVG , C) generated by BVG ([ f ]G V ), but this time only for an open time-slice G of n dimensional Moyal-deformed Minkowski spacetime and no longer for arbitrary hyperbolic subsets. Being mainly a consequence of Proposition 1 (h), Lemma 1 can be transferred almost unchanged. Lemma 4. Suppose that G is a hyperbolic neighbourhood of a Cauchy hyperplane in n dimensional Moyal-deformed Minkowski spacetime, of the form as in Proposition 5 (g). Moreover, suppose that V1 and V2 are two potentials of type (47), (49), which coincide on the region G. Then (a) The map G uG V1 ,V2 : [ f ]V1 → [ f ]V2 ,

f ∈ C0∞ ((λ1 , λ2 )) ⊗ S (Rs ) ⊗ C N

extends to a unitary between KVG1 and KV2 commuting with the charge conjugation C. (b) There is a ∗-algebra isomorphism αVG1 ,V2 : F(KVG1 , C) → F(KV2 , C) induced by αVG1 ,V2 BVG1 ([ f ]G V1 ) = BV2 ([ f ]V2 ),

f ∈ C0∞ ((λ1 , λ2 )) ⊗ S (Rs ) ⊗ C N .


435

Since our potential operator V (see (47),(49)) was chosen to be a compactly supported multiplication operator with respect to the time coordinate, we can maintain exactly the same geometrical setting as in Sec. 2 involving the same time-slice {(x 0 , x 1 , . . . , x s ) : λ− < x 0 < λ+ } for some real numbers λ− < λ+ and the same regions G + , G − being hyperbolic neighbourhoods of the Cauchy hyperplanes + , − . As a result we again arrive at an automorphism βV : F(K0 , C) → F(K0 , C), −1 −1 ◦ αV,+ ◦ α0,+ . βV = α0,− ◦ αV,−

(71)

(V )

As in Sec. 2 we can again define Tsc as the scattering transformation on D0 N s 0 , C ) (0 R ), the space of Cauchy data for the Dirac equation at coordinatetime t = 0, by setting L 2 (

(V )

Tsc(V ) = Tt−1 ◦ Tt,t ◦ Tt for t > λ+ , t < λ− (recall that the interval [λ− , λ+ ] is the time-support of the potential term V ). As before in Sec. 2, Tt denotes the “free” evolution of the Dirac equation with(V ) . In consequence one obtains, exactly as in out potential term, coinciding with Tt,0 (V )

V =0

Eq. (23), an induced automorphism τsc on the CAR-algebra F(D0 , C), given by (V ) (BD0 (v)) = BD0 (Tsc(V ) v), v ∈ D0 , τsc

where the BD0 (v) are the generators of F(D0 , C). Again, there is a canonical identification between the CAR algebras F(K0 , C) and F(D0 , C), (B0 ([ f ]0 )) = BD0 (Q 0 ([ f ]0 )). In the next section we will study the problem of unitary implementability of βV in the Fock-vacuum-representation of F(K0 , C), and the following Lemma, which is the counterpart of Lemma 2 for the case of potential V defined as in (47) and (48), guarantees (V ) that unitary implementability of τsc in the vacuum representation is just the equivalent problem. Lemma 5. The morphism βV of F(K0 , C) defined in (71) and the scattering morphism (V ) τsc (defined like (23)) describing the potential scattering of the quantized Dirac field at the level of the Cauchy-data CAR-algebra F(D0 , C) (with D0 = L 2 (0 , d s x)) are intertwined by the CAR-algebra isomorphism : F(K0 , C) → F(D0 , C) defined in (15), i.e. it holds that (V ) ◦ βV = τsc ◦ .

(72)

7. Moyal-Deformed Minkowski Spacetime: Scattering of the Dirac Field in the Vacuum Representation and Implementability of the Scattering Transformation In the present section we will prove unitary implementability of the scattering transformation βV on F(K0 , C) for the “Moyal-Minkowski-potentials” V defined in (47) and (48), in the Fock-vacuum representation (Hvac , π vac , vac ) of the vacuum state

436

M. Borris, R. Verch

ωvac on F(K0 , C). Owing to Lemma 5, this is equivalent to the problem of unitary (V ) implementability of the scattering transformation τsc on F(D0 , C) in the Fock-vacuum p p p p vac representation (H + , π + , + ) of ω + (where ω = ω p+ ◦ ), the pure, quasifree ground state on F(D0 , C) with respect to the time-evolution induced by the Hamiltonian H0 of (50) on the domain S (Rs , C N ) ⊂ L 2 (Rs , C N ). Recall that p+ is the spectral projection of H0 corresponding to the spectral interval [m, ∞), and that the conjugation C intertwines p+ and 1 − p+ , and hence p+ is a basis projection on (D0 , C) according (V ) to [2]. A well-established criterion for unitary implementability of τsc in the Fock(V ) vacuum representation is that [ p+ , Tsc ] is Hilbert-Schmidt [2,39]. If and only if this is the case, then there is a unitary operator Sτ (V ) on H p+ such that sc

p+ (V ) Sτ (V ) π p+ (BD0 (v))S −1 τsc (BD0 (v)) = π p+ (BD0 (Tsc(V ) v)), v ∈ D0 , (73) (V ) = π sc

τsc

(V ) which is just what it means to say that τsc is unitarily implementable. The condition (v) (V ) that [ p+ , Tsc ] is Hilbert-Schmidt is equivalent to the condition that [ε, Tsc ] is Hil2 s N bert-Schmidt as an operator on L (R , C ), where ε = sign(H0 ) = H0 /|H0 | is the sign function of H0 in the sense of the functional calculus, since p+ = (1l + ε)/2. In an interesting work, Langmann and Mickelsson [30] have shown that certain conditions (V ) on the potential term V (t) (cf. Eq. (51)) are sufficient to conclude that [ε, Tsc ] is Hilbert-Schmidt. Their argument is interesting as it involves a non-local regularization of the interacting dynamics which nevertheless leads to the same scattering transformation (V ) Tsc . We refer to [30] for details and present only the relevant conditions, adapted to our notation. Let the interaction potential W (t) = γ 0 V (t) in the Hamiltonian HV (t) of Eq. (51) have the following properties:

(I) W (t) is a bounded operator on L 2 (Rs , C N ) for each t ∈ R, such that t → W (t) is C ∞ . (II) There is a core for H0 , contained in the C ∞ -domain of H0 , which is left invariant by all W (t) and (∂t )k W (t) (k ∈ N) and by all C ∞ functions of H0 which, together with all their derivatives, are polynomially bounded. (III) There is some ν ∈ N0 so that |H0 |−ν W (t) and |H0 |−ν (∂t )k W (t), k = 1, . . . , ν, are Hilbert-Schmidt operators for all t. n (W (t)) and δ n ((∂ )k W (t)), (IV) Defining δ|H0 | (A) = [|H0 |, A], it holds that δ|H t |H0 | 0| 2 s k = 1, . . . , ν, are bounded operators on L (R , C N ) for all n ∈ N (t ∈ R). n (W (t)) and |H |−ν δ n ((∂ )k W (t)), k = 1, . . . , ν, are Hilbert-Schmidt (V) |H0 |−ν δ|H 0 t |H0 | 0| operators for all n ∈ N (t ∈ R). (VI) W (t) = 0 if t < λ− and if t > λ+ for some real numbers λ− < λ+ . We cite the result relevant for our purposes. Theorem 1. (Langmann and Mickelsson [30]). If the interaction term W (t) = γ 0 V (t) (V ) in HV (t) (cf. (51)) satisfies Conditions (I) . . . (VI), then [ε, Tsc ] is Hilbert-Schmidt, (V ) and hence τsc is implementable in the vacuum-representation (H p+ , π p+ , p+ ) of the Dirac field. Consequently, what we will now set out to demonstrate is


437

Proposition 6. Let V (t) be any of the V(i) (t) or V(ii) (t) defined in (53) and (54), with a ∈ C0∞ (R, R) and b ∈ S (Rs , R). Then W (t) = γ 0 V (t) fulfills the criteria (I) . . . (VI) (V ) above. Therefore, τsc is unitarily implementable in the vacuum representation, so that there is a unitary Sτ (V ) on H p+ such that (73) holds. sc

Proof. Observing that (cf. (53),(54)) W (t) f = a(t)v ˜ f ( f ∈ L 2 (Rs , C N )) ˜ = a(t)2 for V = V(ii) , the time-dependence of with a(t) ˜ = a(t) for V = V(i) and a(t) W (t) is trivial in the context of Conditions (I). . .(VI), and they need only be checked for the time-independent operator v, which is v f = v(i) f = γ 0 (Lb f + Rb f ) or v f = v(ii) f = γ 0 (Lb Rb f ). We note also that the multiplication with γ 0 is irrelevant for checking Conditions (I). . .(VI) since γ 0 commutes with |H0 |. Thus, the conditions need only be checked for Lb , Rb and Lb Rb . A quick inspection shows that the conditions are algebraic in the sense that, if they hold for Lb and Rb , then they hold also for the operator product Lb Rb . As will become clear from the Fourier-representations of Lb and Rb (see below), checking the conditions for Rb is completely analogous to the case of Lb , so it is sufficient to show that the conditions are fulfilled for the operator Lb . Let, for g ∈ L 2 (Rs , C), the Fourier-transform be defined by 1 (Fg)(k) = g(k) ˆ = g(y)e−i y·k d s y ; (74) (2π )s/2 Rs this definition is extended componentwise to elements in L 2 (Rs , C N ). It is easy to see that 1 ˆ − u)eiu·Mk g(u) FLb F −1 g(k) ˆ = ˆ d s u, (75) b(k (2π )s/2 Rs 1 ˆ − u)e−iu·Mk g(u) ˆ = ˆ d s u, (76) b(k FRb F −1 g(k) (2π )s/2 Rs where M is the block-entry M(q+ p−1)×(q+ p−1) in the matrix (46). Moreover, one finds (k)g(k), F H0 F −1 g(k) ˆ =H ˆ κ −1 (k)|κ g(k) ˆ = |H ˆ (κ ∈ R), F|H0 | F g(k)

(77)

(k)| = (|k|2 + m 2 )1/2 (k ∈ Rs ). (k) = −γ 0 γ j k j + γ 0 m, | H H

(78)

with

This implies n Fδ|H (Lb )F −1 g(k) ˆ = 0|

1 ˆ − u)eiu·Mk g(u) 0 (k)| − | H 0 (u)|)n b(k (| H ˆ d s u (79) (2π )s/2 Rs

438

M. Borris, R. Verch

for all n ∈ N and all gˆ ∈ S (Rs , C N ), and similarly n (Lb )F −1 g(k) ˆ F|H0 |−ν δ|H 0| 0 (k)| − | H 0 (u)|)n (| H 1 ˆ − u)eiu·Mk g(u) = ˆ d s u. b(k s/2 0 (k)|ν (2π ) |H Rs

(80)

The discussion in Sec. 4 shows that Lb is bounded. Moreover, we see from the Fourier-representation of H0 that S (Rs , C N ) is a core with the properties demanded in (II). In view of what we observed previously, (IV) is proved once we have shown that n (L ) is a bounded operator for all n ∈ N. We use that, given n, there are constants δ|H b 0| α, β > 0 such that 0 (k)| − | H 0 (u)| |n ≤ α|k − u|2n + β (k, u ∈ Rs ). | |H

(81)

Now the integral kernel in (79) is actually a matrix, and owing to (81), each of its entries has a modulus which can be bounded by n b)(k − u)| + β| α|(− b(k − u)|,

(82)

n (L )F −1 has an operator where denotes the Laplace operator. This shows that Fδ|H b 0| norm which can be dominated by a constant times 1/2 n b)(k − u)| + β| [α|(− b(k − u)|]2 d s k (83) sup u

Rs

which is finite since b is of Schwartz type. It remains to check Conditions (III) and (V). To this end, we observe that the integral kernel in (80) is a matrix where each entry has a modulus which, for given n and ν, can be estimated by a constant times 1 n b(k − u)| + β| [α| − b(k − u)|]. (|k|2 + m 2 )ν/2

(84)

One can obviously choose ν large enough so that this expression is, for each n, square integrable over (k, u) ∈ Rs × Rs , using again that b is a Schwartz function. This shows n (L )F −1 is Hilbertthat a number ν can be chosen large enough so that F|H0 |−ν δ|H b 0| −ν −1 Schmidt for all n (including n = 0, i.e. F|H0 | Lb F ). Finally we remark that Condition (VI) is clearly fulfilled since it was assumed that a ∈ C0∞ (R, R). We will also show in this chapter that the generator of the S-matrix Sτ (V ) with respect sc to variations of V exists as an essentially selfadjoint operator in the sense of derivations. Using the fact that SVM and Sτ (V ) are intertwined by a unitary establishing the equivalence sc between π vac and π p+ ◦ , this will allow the conclusion that also the generator of the S-matrix SVM with respect to variations of V exists as an essentially selfadjoint operator on a suitable domain. In preparing the proof of the assertion we aim to establish, we need an auxiliary result. Proposition 7. Let V be any of the operators V(0) , V(i) or V(ii) , where V(0) f (x) = c(x) f (x) ( f ∈ S (Rn , C N ), x ∈ Rn ),


439

and V(i) and V(ii) are defined as (47) and (48), with c(x) = a(x 0 )b(x) for any a ∈ C0∞ (R, R) and b ∈ S (Rs , R), x = (x 0 , x) ∈ R1+s = Rn . Then (a) Defining d dTsc(V ) v = −i Tsc(λV ) v (v ∈ S (Rs , C N )), (85) dλ λ=0

it holds that dTsc(V ) v(x) = −

∞ −∞

i H0 t −i H0 t a(t)e ˜ ve v(x) dt (x ∈ Rs ),

(86)

where v is either of the operators v(0) , v(i) or v(ii) , with v(i) and v(ii) defined in (60) and (61) and v(0) v(x) = γ 0 b(x)v(x), and with a(t) ˜ = a(t) in the cases V = V(0) , V(i) , while a(t) ˜ = a(t)2 in case V = V(ii) . (V ) 2 The operator dTsc is bounded and selfadjoint on L (Rs , C N ). (b) The commutator [dTsc(V ) , p+ ] is a Hilbert-Schmidt operator on L 2 (Rs , C N ). Here, p+ denotes again the spectral projection of H0 corresponding to the spectral interval [m, ∞). Before we give the proof of that proposition (see towards the end of this section), we explain how this result allows it to conclude the statements made in Sec. 5, which re-appear below in Eqs. (89), (90) and (92). Recall that H p+ = FF ( p+ L 2 (Rs , C N )) is the Fermionic Fock-space with one-particle space p+ L 2 (Rs , C N ). For v ∈ L 2 (Rs , C N ), define the field operators ψ(v) = A( p+ Cv) + A+ ( p+ v), v ∈ D0 ≡ L 2 (Rs , C N ), where A and A+ denote the Fermionic annihilation and creation operators. In other words, ψ(v) = π p+ (BD0 (v)).

(87)

By F we denote the ∗-algebra generated by all ψ(v) and the unit operator, and we 2 s N set W = F p+ . Now consider orthonormal bases {χ ± j } j∈N of p± L (R , C ). If [dTsc(V ) , p+ ] is Hilbert-Schmidt, we can form the operator

: G(dTsc(V ) ) := lim Gk (dTsc(V ) ) − ( p+ , Gk (dTsc(V ) ) p+ ) k→∞

upon defining Gk (dTsc(V ) ) =

k − ∗ ψ(dTsc(V ) χ +j )∗ ψ(χ +j ) + ψ(dTsc(V ) χ − ) ψ(χ ) . j j j=1 (V )

According to Sec. 10 in [43] (cf. also [11]), : G(dTsc ) : defines an essentially selfadjoint operator on W, and it holds that [: G(dTsc(V ) ) :, ψ(v)] = ψ(dTsc(V ) v)

(88)

440

M. Borris, R. Verch (V )

for all v ∈ L 2 (Rs , C N ). Moreover, : G(dTsc ) : is independent of the choice of (V ) {χ ± j } j∈N . (Note that in the notation of [43 and 11], : G(dTsc ) : would be written : dTsc(V ) ψ ∗ ψ :.) On the other hand we have, by Eq. (73) and owing to (87), the relation d d −1 (λV ) S (λV ) ψ(v)S (λV ) = ψ T v = ψ(idTsc(V ) v) τsc dλ λ=0 τsc dλ λ=0 sc for all v ∈ L 2 (Rs , C N ), resulting in d (V ) S (λV ) ψ(v)S −1 (λV ) = [i : G(dTsc ) :, ψ(v)] τsc dλ λ=0 τsc

(89)

(90)

for all v ∈ L 2 (Rs , C N ). In view of ωvac = ω p+ ◦ with the morphism in (15) and (72), there is a canonical unitary operator υ : Hvac → H p+ so that

υ π vac (A)υ −1 = π p+ ◦ (A)

(A ∈ F(K0 , C))

and

υ vac = p+ .

(91)

It is easy to see that υ W = W, where W has been introduced as the domain for : ψ + ψ : (c) in Sec. 5. Furthermore, defining SVM = υ −1 Sτ (V ) υ , sc

and using (72), one can see that (73), which was proven in Prop. 6, is equivalent to SVM π vac (A)(SVM )−1 = π vac (βV (A)) (A ∈ F(K0 , C)). Setting (c) = υ −1 : G(dTsc(V ) ) : υ , it holds that (c) is essentially selfadjoint on W, and by Eq. (89), there holds d M −1 S M ψ( f )SλV = [i(c), ψ( f )] = ψ(V R0 f ) dλ λ=0 λV

(92)

for all f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N . Thus we have established the statements announced in Sec. 5. A further remark is in order here. We do not directly establish the relation d M implementing the scattering trans−i dλ S M = (c). Notice that the unitary SλV λ=0 λV M is formation is not uniquely determined, but only determined up to a phase, i.e. if S˜λV M M −1 ˜ another choice of unitary implementer of the scattering matrix, then SλV (SλV ) = eir (λ) M is indeed differentiable at with some real-valued function r (λ). However, if λ → SλV λ = 0 (upon a suitable choice of λ → r (λ)), then it follows that its derivative with respect to λ at λ = 0 in fact equals the above defined (c) up to an additive multiple of the unit operator. This is a consequence of (92) together with the fact that the ∗-algebra generated by 1l and the ψ( f ) acts irreducibly in the vacuum representation. The additive constant may be compensated for by a re-definition of the phase function r (λ).


441

(V )

Proof (of Proposition 7). In order to show that [dTsc , p+ ] is Hilbert-Schmidt, it is (V ) sufficient to prove that p+ dTsc p− is Hilbert-Schmidt. It holds that ∞ (V ) i H0 t a(t)e ˜ p+ v p− e−i H0 t dt, p+ dTsc p− = − −∞

implying F p+ dTsc(V ) p− F −1 ∞ −i H0 t =− a(t)Fe ˜ p+ F −1 FvF −1 F p− ei H0 t F −1 dt, −∞

where F is the Fourier-transform as in (74). In view of the Fourier-expressions for H0 (cf. (77),(78)), one has ˆ

ˆ = ei H0 (k)t pˆ + (k)g(k), ˆ Fei H0 t p+ F −1 g(k) where pˆ + (k) is an N × N -matrix valued projection. By the properties of H0 and the resulting Hˆ 0 (k), it follows that there is a smooth family of unitary matrices U(k) diagonalizing Hˆ 0 (k) and with the property that 1l 0 00 −1 −1 ≡ P+ , U(k) pˆ − (k)U(k) = ≡ P− , U(k) pˆ + (k)U(k) = 00 0 1l where 1l denotes the N /2 × N /2-unit matrix (recall that N is even). Since U(k) Hˆ 0 (k)U(k)−1 is diagonal and Hˆ 0 (k)∗ Hˆ 0 (k) = | Hˆ 0 (k)|2 1l N ×N is a multiple of the N × N -unit matrix, the eigenvalues of U(k) Hˆ 0 (k)U(k)−1 are ±| Hˆ 0 (k)|, and one obtains ˆ

ˆ

ˆ

ˆ

U(k)ei H0 (k)t pˆ + (k)U(k)−1 = e−i| H0 (k)|t P+ , U(k)e−i H0 (k)t pˆ − (k)U(k)−1 = e−i| H0 (k)|t P− . Using the Fourier-representations of Rb and Lb given in (75) and (76), it furthermore follows that −1 FvF g(k) ˆ = vˆ (k, )g( ˆ ) d s (g ∈ L 2 (Rs , C N )) with a smooth, bounded, N × N -matrix valued function vˆ (k, ). Taking together all these observations, we find F p+ dTsc(V ) p− F −1 g(k) ˆ ∞ −it (| Hˆ 0 (k)|+| Hˆ 0 ( )|) a(t)e ˜ pˆ + (k)vˆ (k, ) pˆ − ( )g( ˆ ) d s dt =− −∞ Rs √ (F a)(| ˜ Hˆ 0 (k)| + | Hˆ 0 ( )|) pˆ + (k)vˆ (k, ) pˆ − ( )g( ˆ ) ds . = − 2π Rs

The Fourier-transform (F a) ˜ of a˜ is in the Schwartz-class while the modulus of pˆ + (k)vˆ (k, ) pˆ − ( ) is continuous and uniformly bounded; therefore the integral kernel (V ) of the last integral is clearly L 2 in the (k, ) variables, proving that F p+ dTsc p− F −1 (V ) and hence p+ dTsc p− is Hilbert-Schmidt.

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M. Borris, R. Verch

8. Bogoliubov’s Formula In this section, we will derive the expressions for d/dλ|λ=0 βλV that we alluded to in Sec. 5 (cf. Eqs. (42) and (45)). We proceed using the geometrical setting from the last part of Sec. 6 (which has also been investigated already in the commutative case in Sec. 2). Under consideration is, respectively, one of the following “potential” operators: (0) (V f ) A (x) = (V(0) f ) A (x) = c(x) f A (x),

(93)

(i) (V f ) (x) = (V(i) f ) (x) = (c (q, p) f )(x) + ( f A

A

(ii) (V f ) (x) = (V(ii) f ) (x) = (c (q, p) f A

A

A

A

A

(q, p) c)(x),

(q, p) c)(x),

(94) (95)

where c ∈ C0∞ (R, R) ⊗ S (Rs , R) is a function of the form c(x) = a(t)b(x),

(96)

with a ∈ C0∞ (R, R), b ∈ S (Rs , R), t = x 0 , x = (x 1 , . . . , x s ). These operators act on f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N and (q + p) × (q + p) is the dimension of the matrix M, which still shall be of the form ⎡ ⎤ 0 ··· 0 ⎢ ⎥ M = ⎣ ... M . ⎦ (q+ p−1)×(q+ p−1)

0

(q+ p)×(q+ p)

These potentials act non-trivially only inside the time-slice {(x 0 , x 1 , . . . , x s ) : λ− < x 0 < λ+ } for some real numbers λ− < λ+ . We recall the definitions of the regions G+, G−, 1 , G + = (x 0 , x 1 , . . . , x s ) : x 0 > λ+ + 2 1 , G − = (x 0 , x 1 , . . . , x s ) : x 0 < λ− − 2 forming hyperbolic neighbourhoods of the Cauchy hyperplanes + = {(x 0 , x 1 , . . . , x s ) : x 0 = λ+ + 1}, − = {(x 0 , x 1 , . . . , x s ) : x 0 = λ− − 1} respectively. With these assumptions, we obtain the following result, the proof of which makes use of Proposition 1, Lemma 1 (commutative case) and Proposition 5, Lemma 4 (noncommutative case). Theorem 2. It holds that

⎧ ⎪ ⎨0 (c R0 f ) βλV (0 ( f )) = 0 (c (q, p) R0 f + (R0 f ) (q, p) c) ⎪ ⎩ (c λ=0 0 (q, p) (R0 f ) (q, p) c),

d dλ

λ being a real parameter, for the respective choices of the operator V = V(0) , V(i) , V(ii) .


443

Proof. The priority of this proof lies on the non-commutative cases. The commutative case can be carried out along the same lines. We recall the automorphism βλV from (71), βλV : F(K0 , C) → F(K0 , C),

−1 −1 βλV = α0,− ◦ αλV ,− ◦ αλV ,+ ◦ α0,+ ,

together with βλV (B0 ([ f ]0 )) = B0 (UλV [ f ]0 ), where UλV is the unitary given by −1 UλV = u 0,− ◦ u −1 λV ,− ◦ u λV ,+ ◦ u 0,+ ,

and where, similarly as for the isomorphisms above, we have used the abbreviations G

G

± . u 0,± = u 0,0± , u λV ,± = u 0,λV

These equations arise from Lemma 4. The proof relies now on exactly the same strategy as the one for the very similar Theorem 4.3 of [9]. With that in mind we start by finding more explicit expressions for the inverses in the chain of mappings UλV : [ f ]0

u −1 0,+

−1

u u u / [ f G + ]G + λV ,+ / [ f G + ]λV λV ,− / [ f G − ]G − 0,− / [ f G − ]0 , 0 0

where f G + is any element in C0∞ ((λ+ , ∞)) ⊗ S (Rs ) ⊗ C N such that R0 ( f − f G + ) = 0, and f G − is any element in C0∞ ((−∞, λ− )) ⊗ S (Rs ) ⊗ C N such that RλV ( f G + − f G − ) = 0. According to [9], we choose f G + = −D0 χ+ret R0 f,

f G − = −DλV χ−ret RλV f G + ,

(97)

where χ±ret are defined as follows: It has been demanded that the open regions G ± contain Cauchy hyperplanes ± . Then there are two pairs of further Cauchy surfaces in G ± , adv in the timelike future of and ret in the timelike past of . Thus namely ± ± ± ± adv ret ◦ O± = [J − (± ) ∩ J + (± )]

are open neighbourhoods of ± and O± ⊆ G ± . Now a partition of unity {χ±adv , χ±ret } is ret ) and χ ret = 0 on J + ( adv ). introduced on Rn with χ±adv = 0 on J − (± ± ± The crucial point lies in having to ensure that f G ± both lie in the domain of RλV in view of the weakened support properties of the fundamental solutions R V± (cf. Proposition 5(b)) compared to the situation in [9]. Obviously it holds DλV χ−adv RλV f = −DλV χ−ret RλV f,

(98)

since DλV RλV f = 0 and χ−adv + χ−ret = 1. The left hand side of (98) vanishes on ret ) and the right hand side on J + ( adv ). Thus we can conclude from (98) that J − (− − both DλV χ−adv RλV f and −DλV χ−ret RλV f lie in C0∞ (R) ⊗ S (Rs ) and, hence, in the domain of Rλ V for any λ . This shows immediately that f G + lies in the domain of any Rλ V , and, iterating the argument, the same holds for f G − .

444

M. Borris, R. Verch

Putting the definitions of (97) into the chain of mappings composing UλV results in UλV [ f ]0 = [DλV χ−ret RλV D0 χ+ret R0 f ]0 . In the following we would like to abbreviate formally “δ =

d dλ λ=0 ”.

We calculate

δUλV [ f ]0 = δ[DλV χ−ret RλV D0 χ+ret R0 f ]0 = −[δ DλV χ−ret R0 f ]0 + [D0 χ−ret (δ RλV )D0 χ+ret R0 f ]0 , since R0 D0 χ+ret ϕ = −ϕ. It is easy to see that δ DλV and χ−ret have disjoint supports, and thus δUλV [ f ]0 = [D0 χ−ret δ RλV D0 χ+ret R0 f ]0 . + − R − implies RλV = RλV λV − + D0 χ+ret ϕ − χ−ret RλV D0 χ+ret ϕ, χ−ret RλV D0 χ+ret ϕ = χ−ret RλV

whereof the first term on the right hand side vanishes, since supp χ−ret ⊆ J − (G − ), + D χ ret ⊆ T + (G ) and T + (G ) ∩ J − (G ) = ∅. Hence supp RλV 0 + + + − − D0 χ+ret R0 f ]0 . δUλV [ f ]0 = [−D0 χ−ret δ RλV

And this equals [D0 χ−ret R0− δ DλV R0− D0 χ+ret R0 f ]0 , because of the following deduction: − − RλV DλV = 1l ⇒ (δ RλV )D0 + R0− (δ DλV ) = 0

− ⇒ δ RλV D0 R0− = −R0− δ DλV R0−

− ⇒ δ RλV = −R0− δ DλV R0− .

Support arguments lead to χ−ret R0+ δ DλV = 0 and δ DλV R0+ D0 χ+ret ϕ = 0 and therefore δUλV [ f ]0 = [D0 χ−ret R0 δ DλV R0 D0 χ+ret R0 f ]0 = [δ DλV R0 f ]0 , since R0 D0 χ±ret = −1l. Obviously δ DλV = V , which is just one of the three choices of a potential operator. Hence δβλV (0 ( f )) = δβλV (B0 ([ f ]0 )) = δ B0 (UλV [ f ]0 ) = B0 ([V R0 f ]0 ) = 0 (V R0 f ).


445

9. Conclusion and Outlook We have given a brief sketch of how a simple quantum field theory on Moyal-Minkowski spacetime can be constructed in such a way as providing a model for the construction of quantum field theories on more general Lorentzian non-commutative spacetimes in a setting inspired by spectral geometry. For this quantum field theory — which is the quantized Dirac field — we have seen that a construction of observable field operators labelled by elements of the deformed function algebra of Moyal-Minkowski space can be derived, via Bogoliubov’s formula, from the S-matrix describing scattering of the usual Dirac field on Minkowski spacetime by a non-commutative scalar potential. Again, it is feasible that this procedure can be generalized to more general Lorentzian non-commutative spacetimes. However, it is certainly inappropriate, at this stage, to judge the generality of the method. Moyal-Minkowski spacetime with commutative time is a very simple and very special non-commutative geometry whose physical relevance is not compelling, to say the least. On the other hand, due to the quite unusual and counter-intuitive properties of non-commutative geometries, one surely needs examples as one’s guidance towards developing physical theories in non-commutative geometries, such as quantum field theory. This clearly shows a dilemma: the examples for non-commutative geometries that are manageable are probably un-physical and may therefore do a very poor job as a guidance when attempting to find some central principles, while without such principles, it is hard to judge which non-commutative geometries are related to physics. Nevertheless, one can probably do better, and try and investigate our method of construction of quantum field theories and their observables for non-commutative spacetimes that have a greater physical appeal, such as developed in [18 and 5], for example. Even if this appears to be a considerably more difficult task, we think it is worth an attempt. A. The Action of the Wick Square In this Appendix we will prove that the Wick-square acts as a derivation on the Dirac field operators in the same way as the derivative of the S-matrix with respect to the scalar scattering potential. The assumptions, wherever not spelled out in detail, are those stated in Sec. 5. Proposition 8. Let eµ and ηµ (µ = 1, . . . , L) be elements in C N , chosen such that L 1 AB A B . Define for q1 , q2 ∈ C0∞ (Rn , R), µ=1 eµ ηµ = 4 γ0 ψ + ψ(q1 ⊗ q2 ) =

L

ψ(q1 eµ )∗ ψ(q2 ηµ )

µ=1

and whence, : ψ + ψ : (c) = lim ψ + ψ(F ) − (vac , ψ + ψ(F )vac )1l →0

N on W with F (x, y) = q1 (x)q2 (y) j (x − y) and c(x) = q1 (x)q2 (x) (x, y ∈ R ), where lim→0 q(x) j (x − y) d n x = q(y) (q ∈ C0∞ (Rn )). Then

[: ψ + ψ : (c), ψ( f )] = −iψ(c R0 f ) ( f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ).

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M. Borris, R. Verch

Moreover, : ψ + ψ : (c) is independent of the choice of families eµ , ηµ ∈ C N fulfilling 1 AB A B . µ eµ ηµ = 4 γ0 Proof. Using the relations of the generators of the CAR algebra, it follows that (χ1 , [ψ(q1 eµ )∗ ψ(q2 ηµ ), ψ( f )]χ2 ) µ

=2

(χ1 , ψ(Cq1 eµ )χ2 )(Cq2 ηµ , f )0 − (χ1 , ψ(q2 ηµ )χ2 )(q1 eµ , f )0

!

(99)

µ

holds for all vectors χ1 , χ2 in the dense domain W ⊂ Hvac and for all f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N . We recall here the definition ( f, h)0 = i γ0 AB f B (x)(R0 h) A (x) d n x (for V = 0, see Prop. 5 or respectively 1 and Eq. (6)). One can show either directly, or by falling back onto general arguments [19], that for each pair of vectors χ1 , χ2 ∈ W there are smooth functions ξ A on Rn (A = 1, . . . , N ) such that (χ1 , ψ( f )χ2 ) = ξ A (y) f A (y) d n y ( f ∈ C0∞ (R) ⊗ S (Rs ) ⊗ C N ). (100) With this notation, the right-hand side of (99) assumes the form q 1 (y)q2 (x)ξ A (y)(Ceµ ) A (y)γ0B D Cηµ D (R0 f ) B (x) d n x d n y 2i µ

−2i

µ

q 1 (x)q2 (y)ξ A (y)ηµA γ0D B eµ B (R0 f ) D (x) d n x d n y.

(101)

Using the defining property µ eµ A ηµB = 41 γ0 AB and γ0 2 = 1l, the second integral of (101) simplifies to i − q 1 (x)q2 (y)ξ A (y)(R0 f ) A (x) d n x d n y. (102) 2 In a similar manner, using also the relations C 2 = 1l and (7), one can check that the first integral in (101) simplifies to i − q 1 (y)q2 (x)ξ A (y)(R0 f ) A (x) d n x d n y, (103) 2 so that (101) becomes equal to i − (q1 (y)q2 (x) + q2 (y)q1 (x))ξ A (y)(R0 f ) A (x) d n x d n y, 2

(104)

observing that q1 and q2 are real-valued. Replacing here q1 ⊗ q2 by F and taking the limit → 0 turns the last expression into (105) − i ξ A (x)c(x)(R0 f ) A (x) d n x.


447

On using (100), we have therefore proved the first claim of the proposition. To see the independence of the definition of : ψ + ψ : (c) of the mentioned choices, we note that the commutator formula for : ψ + ψ : (c), together with the fact that the ∗-algebra generated by 1l and all the ψ( f ) acts irreducibly, fixes : ψ + ψ : (c) up to addition of a multiple of the unit operator 1l. On the other hand, by construction we have (vac , : ψ + ψ : (c)vac ) = 0, so that the scalar multiple in question must vanish in the vacuum state, which implies that it is zero. This demonstrates the claimed independence of the definition of : ψ + ψ : (c) of the possible choices for eµ and ηµ . Acknowledgement. The second named author would like to thank Mario Paschke for many discussions on the topics presented here and for continuing collaboration on Lorentzian spectral geometry of which some ideas have been sketched in the text. Thanks are also extended to Sergio Doplicher, Klaus Fredenhagen and Raimar Wulkenhaar for discussions and comments. The first named author gratefully acknowledges financial support by the German Research Foundation (DFG).

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23. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Basel: Birkhäuser, 2000 24. Grosse, H., Lechner, G.: Noncommutative deformations of Wightman quantum field theories. JHEP 0809, 131 (2008) 25. Grosse, H., Wulkenhaar, R.: Renormalization of φ 4 theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) 26. Haag, R.: Local Quantum Physics. 2nd ed., Berlin: Springer-Verlag, 1996 27. Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964) 28. Hawkins, E.: Hamiltonian gravity and noncommutative geometry. Commun. Math. Phys. 187, 471 (1997) 29. Kopf, T., Paschke, M.: A spectral quadruple for de Sitter space. J. Math. Phys. 43, 818 (2002) 30. Langmann, E., Mickelsson, J.: Scattering matrix in external field problems. J. Math. Phys. 37, 3933 (1996) 31. Moretti, V.: Aspects of noncommutative Lorentzian geometry for globally hyperbolic space-times. Commun. Math. Phys. 232, 189 (2003) 32. Palmer, J.: Scattering automorphisms of the Dirac field. J. Math. Anal. Appl. 64, 189 (1978) 33. Paschke, M., Rennie, A., Verch, R.: Lorentzian spectral triples. In preparation 34. Paschke, M., Verch, R.: Local covariant quantum field theory over spectral geometries. Class. Quant. Grav. 21, 5299 (2004) 35. Reed, M., Simon, B.: Fourier Analysis, Self-Adjointness; Methods of Modern Mathematical Physics II. New York: Academic Press, 1975 36. Rennie, A., Varilly, J.C.: Reconstruction of manifolds in noncommutative geometry. http://arxiv.org/abs/ math.OA/0610418, 2006 37. Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative φ 4 -theory by multi-scale analysis. Commun. Math. Phys. 262, 565 (2006) 38. Roberts, J.E.: More Lectures on Algebraic Quantum Field Theory. Noncommutative Geometry, 263–342, Lecture Notes in Math. 1831, Berlin: Springer-Verlag, 2004, pp. 263–342 39. Shale, D., Stinespring, W.F.: Spinor representations of infinite orthogonal groups. J. Math. & Mech. 14, 315–322 (1965) 40. Streater, R.F., Wightman, A.S.: PCT, Spin and Statistics, and All That. New York: Benjamin, 1968 41. Strohmaier, A.: On noncommutative and semi-Riemannian geometry. J. Geom. Phys. 56, 175 (2006) 42. Szabo, R.: Quantum field theory on noncommutative spaces. Phys. Rep. 387, 207 (2003) 43. Thaller, B.: The Dirac Equation. Berlin: Springer-Verlag, 1992 Communicated by Y. Kawahigashi


Communications in


A Universal Inequality for Axisymmetric and Stationary Black Holes with Surrounding Matter in the Einstein-Maxwell Theory Jörg Hennig1 , Carla Cederbaum1 , Marcus Ansorg1,2 1 Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476 Golm, Germany.

E-mail: [email protected], [email protected]

2 Institute of Biomathematics and Biometry, Helmholtz Zentrum München, Ingolstädter Landstr. 1, D-85764

Neuherberg, Germany. E-mail: [email protected] Received: 15 December 2008 / Accepted: 29 April 2009 Published online: 1 September 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: We prove that in Einstein-Maxwell theory the inequality (8π J )2 +(4π Q 2 )2 < A2 holds for any sub-extremal axisymmetric and stationary black hole with arbitrary surrounding matter. Here J, Q, and A are angular momentum, electric charge, and horizon area of the black hole, respectively. 1. Introduction For a single rotating, electrically charged, axisymmetric and stationary black hole in vacuum (described by the Kerr-Newman family of solutions), the angular momentum J , the electric charge Q, and the horizon area A are restricted by the inequality p 2J + p 2Q ≤ 1 with p J :=

8π J , A

p Q :=

4π Q 2 . A

(1)

Equality in (1) holds if and only if the Kerr-Newman black hole is extremal. That is to say, p 2J + p 2Q < 1

(2)

holds for any non-extremal Kerr-Newman black hole. As was shown in [1], the equality p 2J + p 2Q = 1 holds more generally in EinsteinMaxwell theory for axisymmetric and stationary degenerate1 black holes with surrounding matter. Moreover, it was conjectured in [1] that inequality (1) is still valid if the black hole is surrounded by matter (i.e. if it is not a member of the Kerr-Newman family). Inequality (2) was proved in [7] for axisymmetric and stationary black holes with surrounding matter in pure Einsteinian gravity (without Maxwell field). In that article, emphasis was put on “physically relevant” configurations by assuming the black hole to 1 Degeneracy of an axisymmetric and stationary black hole is defined by vanishing surface gravity κ.

450

J. Hennig, C. Cederbaum, M. Ansorg

be sub-extremal. This condition requires the existence of trapped surfaces (i.e. surfaces with a negative expansion of outgoing null geodesics) in every sufficiently small interior vicinity of the event horizon, see [3]. Here, we consider again sub-extremal axisymmetric and stationary black holes with arbitrary surrounding matter, but provide a proof of (2) which is valid in the full Einstein-Maxwell theory. The idea of the proof relies on showing that a black hole cannot be sub-extremal for p 2J + p 2Q ≥ 1. In order to prove this, we study the Einstein-Maxwell equations in a vicinity of the black hole horizon. It turns out that a reformulation can be found which states that an appropriate functional I (to be defined below) must always be greater than or equal to 1. In this way, we encounter a variational problem, and the corresponding solution provides a proof of inequality (2). As will be shown below, this variational problem can be treated with methods from the calculus of variations. This paper is organized as follows. In Sec. 2, we introduce appropriate coordinates which are adapted to the subsequent analysis. Moreover, we list the Einstein-Maxwell equations and the corresponding boundary and regularity conditions in these coordinates. In Sec. 3, we express the ingredients p J and p Q , which appear in the inequality (2), in terms of metric and electromagnetic potentials. We formulate the variational problem mentioned above in Sec. 4 and solve it in Sec. 5. Finally, we conclude this paper with a discussion on physical implications of inequality (2), see Sec. 6. In an appendix, we establish a connection to degenerate black holes. 2. Coordinate Systems and Einstein Equations Following Bardeen [2], we describe an exterior electrovacuum vicinity of the black hole2 in spherical coordinates (R, θ, ϕ, t) in terms of a Boyer-Lindquist-type3 line element d R2 4 2 2 + dθ + uˆ sin2 θ (dϕ − ωdt)2 − (R 2 − rh2 )dt 2 , (3) ds = µˆ 2 2 uˆ R − rh where the metric potentials µ, ˆ uˆ and ω are functions of R and θ alone and where in addition µˆ and uˆ are positive functions. The event horizon H is located at R = rh , rh = constant > 0. The electromagnetic field gives rise to an energy momentum tensor 1 1 Ti j = (4) Fki F kj − gi j Fkl F kl , 4π 4 where, using Lorenz gauge, the electromagnetic field tensor Fi j can be written in terms of a potential (Ai ) = (0, 0, Aϕ , At ), Fi j = Ai, j − A j,i .

(5)

Note that, like the metric quantities, Aϕ and At also depend on R and θ only. 2 For a stationary spacetime, the immediate vicinity of a black hole event horizon must be electrovacuum, see [5 and 2]. 3 In the special case without any exterior matter, i.e. for the Kerr-Newman black hole, we obtain BoyerLindquist coordinates (r, θ, ϕ, t) through a linear transformation r = 2R + M, where M is the black hole mass.

A Universal Inequality for Black Holes with Surrounding Matter

451

In the Boyer-Lindquist-type coordinates, the Einstein-Maxwell equations in electrovacuum are given by4 (R 2 − rh2 )u˜ ,R R + 2R u˜ ,R + u˜ ,θθ + cot θ u˜ ,θ 2 ω,θ uˆ 2 1 2 2 2 2 2 2 =1− − r )A + A (R − sin θ ω,R + 2 ϕ,θ h ϕ,R 8 R − rh2 uˆ sin2 θ (,θ − Aϕ ω,θ )2 uˆ − , (,R − Aϕ ω,R )2 + 4 R 2 − rh2

(6)

(R 2 − rh2 )µ˜ ,R R + R µ˜ ,R + µ˜ ,θθ 2 ω,θ uˆ 2 2 2 sin θ ω,R + 2 = − (R 2 − rh2 )u˜ 2,R + R u˜ ,R − u˜ ,θ (u˜ ,θ + cot θ ), (7) 16 R − rh2 (R 2 − rh2 )(ω,R R + 4ω,R u˜ ,R ) + ω,θθ + ω,θ (3 cot θ + 4u˜ ,θ ) 4 2 = (R − rh2 )Aϕ,R (,R − Aϕ ω,R ) + Aϕ,θ (,θ − Aϕ ω,θ ) , 2 uˆ sin θ

2 (R − rh2 ) ,R R − Aϕ ω,R R + 2u˜ ,R (,R − Aϕ ω,R ) − Aϕ,R ω,R + ,θθ − Aϕ ω,θθ + (2u˜ ,θ + cot θ )(,θ − Aϕ ω,θ ) − Aϕ,θ ω,θ = 0,

2 (R − rh2 ) Aϕ,R R − 2u˜ ,R A,ϕ,R + 2R Aϕ,R + Aϕ,θθ − (2u˜ ,θ + cot θ )Aϕ,θ ,θ − Aϕ ω,θ uˆ 2 2 sin θ (,R − Aϕ ω,R )ω,R + = ω,θ . 4 R 2 − rh2

(8)

(9)

(10)

Here, we have used the dimensionless quantities u˜ :=

uˆ µˆ 1 1 ln , µ˜ := ln , 2 uˆ N 2 uˆ N

where uˆ N is the the north pole value of u, ˆ uˆ N := u(R ˆ = rh , θ = 0). Moreover, we have replaced At by the comoving electric potential = At + ω Aϕ . At the horizon, the metric potentials obey the boundary conditions (cf. [2]) H:

ω = constant = ωh ,

2rh = constant = κ, = constant = h , (11) µˆ uˆ

where ωh , κ, and h denote the angular velocity of the horizon, the surface gravity, and the value of the comoving electric potential at the horizon, respectively. 4 Throughout this paper we consider a vanishing cosmological constant, = 0. (Note that inequality (1) can be violated for = 0. An example is the Kerr-(A)dS family of black holes, see [3].)

452


On the horizon’s north and south pole (R = rh and sin θ = 0), the following regularity conditions hold5 : µ(R ˆ = rh , θ = 0) = u(R ˆ = rh , θ = 0) = µ(R ˆ = rh , θ = π ) = u(R ˆ = rh , θ = π ) = uˆ N = Aϕ (R, θ = 0) = Aϕ (R, θ = π ) = 0.

2rh , κ

(12) (13)

In the forthcoming calculations we need relations between metric and electromagnetic quantities at the black hole horizon H. These are provided by an investigation of Eqs. (6) and (7). For the evaluation of these equations in the limit R → rh , we introduce the regular horizon potentials (cf. [1]) ωˆ :=

ω − ωh − h ˆ := , , R − rh R − rh

(14)

from which it follows that lim

R→rh

lim

R→rh

2 ω,θ

R 2 − rh2

= lim

R→rh

R − rh 2 ωˆ R + rh ,θ

= 0,

(,θ − Aϕ ω,θ )2 R − rh 2 ˆ ,θ − Aϕ ωˆ ,θ )2 = 0. = lim ( R→rh R + rh R 2 − rh2

Using these relations, we obtain for (6) and (7) in the limit R → rh , 2

2rh u˜ ,R + u˜ ,θθ + cot θ u˜ ,θ = 1 − rh µ˜ ,R + µ˜ ,θθ =

Aϕ,R uˆ 2 uˆ 2 sin2 θ ω,R − − (,R − Aϕ ω,R )2 , 8 4 uˆ sin2 θ

uˆ 2 2 sin2 θ ω,R + rh u˜ ,R − u˜ ,θ (u˜ ,θ + cot θ ). 16

(15) (16)

3. Calculation of p J and p Q In order to find suitable expressions for p J and p Q , we introduce the following functions which are defined as follows in terms of the metric and electromagnetic quantities at the black hole horizon H: 1 uˆ 1 ln , V (x) := uˆ ω,R H , 2 uˆ N H 4 Aϕ uˆ S(x) := (,R − Aϕ ω,R )H , T (x) := , 2 uˆ N uˆ N H

U (x) :=

where x := cos θ . 5 Note that (13) holds on the entire rotation axis.

(17) (18)


453

In terms of these quantities we obtain for angular momentum J , charge Q and horizon area A (cf. [1]): 6 1 J = (m i; j + 2m k Ak F i j )dSi j 8π H

1 π uˆ 2 ω,R sin θ − Aϕ (,R − Aϕ ω,R ) sin θ dθ =− uˆ H 4 0 4 uˆ N 1 =− [V e2U (1 − x 2 ) − 2ST ]dx, (19) 4 −1 1 1 π Q=− F i j dSi j = − u( ˆ ,R − Aϕ ω,R )H sin θ dθ (20) 4π 4 0 H uˆ N 1 =− S dx (21) 2 −1 π A = 2π µˆ uˆ H sin θ dθ = 4π uˆ N . (22) 0

Here, we have used conditions (11) and (12). Finally, we arrive at pJ ≡

1 8π J =− A 2

pQ ≡

4π Q 2 1 = A 4

1

−1

[V e2U (1 − x 2 ) − 2ST ]dx,

(23)

2

1

S dx

−1

.

(24)

4. Reformulation in terms of a Variational Problem As a first step towards the proof of the inequality (2) for sub-extremal black holes, we consider the following lemma. Lemma 1 (Characterization of sub-extremal black holes). A necessary condition for the existence of trapped surfaces in the interior vicinity of the event horizon of an axisymmetric and stationary charged black hole is

π 0

(µˆ u) ˆ ,R H sin θ dθ > 0.

(25)

This lemma was originally derived in the setting of pure Einsteinian gravity (without Maxwell-field), see [7]. As the corresponding proof presented in [7] carries over to the full Einstein-Maxwell theory, we may use the lemma in the forthcoming investigation. The proof of (2) relies on showing that for p 2J + p 2Q ≥ 1 inequality (25) is violated, which implies by virtue of Lemma 1 a violation of the sub-extremality condition: p 2J + p 2Q ≥ 1

⇒ 0

π

(µˆ u) ˆ ,R H sin θ dθ ≤ 0.

6 Note that m i denotes the Killing vector with respect to axisymmetry.

(26)

454


Using Einstein equations (15) and (16) together with the boundary conditions (11), we may rewrite the integrand in (25) as 2 ) 2( u ˆ uω ˆ ,R 2 2 N sin θ − u˜ ,θ (u˜ ,θ + 2 cot θ ) (µˆ u) ˆ ,R H = 1− rh 4 A2ϕ,θ uˆ 2 − (27) − (,R − Aϕ ω,R ) . 4 uˆ sin2 θ Hence we can express (25) in terms of S, T , U , and V : 1 1 2 (V + U 2 )(1 − x 2 ) − 2xU + (S 2 + T 2 )e−2U dx < 1, 2 −1

(28)

where := d/dx. With the expressions for p J and p Q [see (23), (24)] we can thus write the implication in (26) as follows:

1 −1

⇒

Ve 1 2

2U

1

−1

(1 − x ) − 2ST dx 2

2

1 + 4

4

1

−1

Sdx

≥4

(V 2 + U 2 )(1 − x 2 ) − 2xU + (S 2 + T 2 )e−2U dx ≥ 1.

(29)

In the following we show that this implication holds for all sufficiently regular functions7 S, T, U, V : [−1, 1] → R which satisfy the boundary conditions U (±1) = T (±1) = 0.

(30)

The conditions in (30) follow from (17), (18), (12), (13). In the next step we formulate a variational problem which is a sufficient criterion for the validity of the implication in (29). Applying the Cauchy-Schwarz inequality to the first inequality in (29), we obtain ⎛ ⎞ 1 2 1 4 1 1 1 ⎝ V 2 (1 − x 2 )dx e4U (1 − x 2 )dx + 2 S T dx ⎠ + Sdx ≥ 4. 4 −1 −1 −1 −1 With the abbreviations 1 4U 2 e (1 − x )dx, c2 := c1 := −1

1

1 ST dx, c3 := √ 2 −1

this inequality leads to the estimate 1 V 2 (1 − x 2 )dx ≥ M22 , −1

7 A precise statement about the required regularity properties follows below.

1 −1

Sdx,

(31)

(32)


where

√ M1 − 2|c2 | . M2 := max 0, c1

M1 := max 0, 4 − c34 , Using (32) in order to replace the term it follows immediately that 1 I [S, T, U ] := 2

1 −1

455

(33)

V 2 (1 − x 2 ) dx in the second inequality in (29),

[U 2 (1 − x 2 ) − 2xU (x) + (S 2 + T 2 )e−2U ]dx +

M22 ≥1 2

(34)

is a sufficient condition for the validity of the implication in (26). We summarize this result in the following lemma. Lemma 2 (Variational problem). The inequality p 2J + p 2Q < 1 holds for any sub-extremal axisymmetric and stationary charged black hole with surrounding matter provided that the inequality I [S, T, U ] ≥ 1

(35)

is satisfied for all S ∈ L 2 (−1, 1), T, U ∈ W01,2 (−1, 1). Remark. The Lebesgue and Sobolev spaces L 2 and W01,2 contain all functions S and T , U , respectively, that arise in the physical situation above. With this lemma, we have reduced inequality (2) to the variational problem of calculating the minimum of I [S, T, U ] and showing that this is greater than or equal to 1. In the next section, we solve this problem with methods from the calculus of variations. 5. Solution of the Variational Problem 5.1. An approximating family of functionals. Analyzing the functional I proves difficult as the factor 1 − x 2 is singular at the boundary x = ±1, cf. definition of I in (34). We therefore approximate it by a family of slightly modified functionals Iε which are conducive to analysis using techniques of the calculus of variations. We work on the Hilbert space X := (L 2 × W01,2 × W01,2 )(−1, 1)

(36)

endowed with the inner product ˜ T˜ , U˜ ) := (S, T, U ), ( S,

1 −1

S S˜ + T T˜ + U U˜ (1 + ε − x 2 ) dx

depending on a fixed ε > 0. Recall that this inner product is equivalent to the ordinary one by the fundamental theorem of calculus. Moreover, we have Proposition 1 (Theorem 2.2 in Buttazzo-Giaquinta-Hildebrandt [4]). On any bounded interval J ⊆ R, W 1,2 (J ) → C 0 (J ) compactly. Moreover, the fundamental theorem of calculus holds in W 1,2 (J ).

456


For ε ≥ 0, we consider the functional Iε : X → R given by 1 Iε [S, T, U ] := 2

1

M ε [S, T, U ]2 U 2 (1 + ε − x 2 )−2xU + (S 2 + T 2 )e−2U dx + 2 , 2 (37)

−1

where the auxiliary functionals M2ε , M1 , c1ε , c2 , c3 : X → R are defined by c1ε [S, T, U ]

:=

1

−1

1 c3 [S, T, U ] := √ 2

e4U (1 + ε 1

−1

S dx,

−

x 2 )dx,

c2 [S, T, U ] :=

1 −1

ST dx,

M1 [S, T, U ] := max 0, 4 − (c3 [S, T, U ])4 ,

(38) (39)

and M2ε [S, T, U ]

√

:= max 0,

M1 − 2|c2 | [S, T, U ] , c1ε

(40)

respectively. All of these functionals can easily be seen to be well-defined, and all auxiliary functionals are weakly continuous on X by Poincaré’s inequality. Also, c1ε is positive and both M1 , M2ε are non-negative. We now show that for ε > 0 there exists a global minimizer (S, T, U ) ∈ X for Iε and study its value Iε [S, T, U ]. Following this investigation, we take the limit ε → 0 and see that the claim of Lemma 2 follows. 5.2. Existence and characterization of the minimizer. Now let ε > 0 be fixed. Iε then has the following properties: 2 √ = (i) Iε is bounded from below. Using 0 ≤ √ x 2 − U (x) 1 + ε − x 2 1+ε−x

x2 1+ε−x 2

− 2xU (x) + U 2 (x)(1 + ε − x 2 ) we conclude that Iε [S, T, U ] ≥ −

1 2

1 −1

x2 dx =: C(ε) > −∞ 1 + ε − x2

for any (S, T, U ) ∈ X . (ii) Iε is coercive with respect to the weak topology on X . Indeed, applying the 1 Cauchy-Schwarz inequality to −1 xU (x)dx, we obtain that Iε [S, T, U ] ≥

1 U 2 − C(ε) U 2

for any (S, T, U ) ∈ X with C(ε) > 0. Hence, for every P ∈ R there exists Q P ∈ R such that Iε [S, T, U ] ≥ P whenever (S, T, U ) ≥ Q P . This is equivalent to coercivity of the functional Iε with respect to the weak topology on X , where both the norm · and the weak topology refer to the inner product defined above.


457

(iii) The functional Iε is sequentially lower semi-continuous (lsc) with respect to the weak topology on X . To see this, recall that lower semi-continuity is additive and that the first terms can be dealt with by standard theory (see e.g. [9]), and use Proposition 1 as well as the Lipschitz continuity of exp on bounded intervals. For the last term, the weak continuity of the auxiliary functionals yields the claim. We are now in a position to show existence of a global minimizer for Iε : As we have seen in (i), Iε is bounded from below on X . We can hence choose a minimizing sequence (Sk , Tk , Uk ) ∈ X which must be bounded by coercivity (ii) and thus has a weakly converging subsequence by Hilbert space techniques (theorem of EberleinShmulyan [9]) tending to a limit (S ∗ , T ∗ , U ∗ ). Lower semicontinuity as in (iii) then gives us Iε [S ∗ , T ∗ , U ∗ ] = inf{Iε [S, T, U ] | (S, T, U ) ∈ X } and thus asserts that (S ∗ , T ∗ , U ∗ ) is a global minimizer. However, Iε is not Fréchet-differentiable at (S, T, U ) ∈ X with c2 [S, T, U ] = 0 and c34 [S, T, U ] = 4 due to the maximum-terms in the definitions of Miε (i = 1, 2). It is consequently impossible to derive Euler-Lagrange equations for Iε directly. To circumvent this problem, we introduce the constraints ciε = constant (i = 1, 2, 3) and use the method of Lagrange multipliers to minimize Iε under these constraints. This leads to a Fréchet-differentiable functional on every class K with fixed values of ciε . Moreover, the asserted global minimizer (S ∗ , T ∗ , U ∗ ) also minimizes Iε in its class K∗ which induces conditions on the constants specifying K∗ and explicit expressions for the related Lagrange multipliers. 5.3. The Euler-Lagrange equations. Setting ci∗ := ciε [S ∗ , T ∗ , U ∗ ] and M ∗j := M εj [S ∗ , T ∗ , U ∗ ], (i = 1, 2, 3; j = 1, 2), the class K∗ containing the global minimizer (S ∗ , T ∗ , U ∗ ) is characterized by K∗ := (S, T, U ) ∈ X | ciε [S, T, U ] = ci∗ (i = 1, 2, 3) . In this class, Iε can be evaluated as follows: (M2∗ )2 1 1 2 Iε [S, T, U ] = . U (1 + ε − x 2 ) − 2xU + (S 2 + T 2 )e−2U dx + 2 −1 2 By the theory of Lagrange multipliers, for each minimizer (S, T, U ) of Iε in the class K∗ , there is (λ1 , λ2 , λ3 ) ∈ R3 such that (S, T, U, λ1 , λ2 , λ3 ) ∈ X × R3 is a critical point of the functional Jε∗ : X × R3 → R given by 1 1 2 U (1 + ε − x 2 ) − 2xU + (S 2 + T 2 )e−2U dx Jε∗ [S, T, U, λ1 , λ2 , λ3 ] := 2 −1 ! " + λ1 (c1ε [S, T, U ])2 − (c1∗ )2 + λ2 c2 [S, T, U ] − c2∗ √ ! " + 2 λ3 c3 [S, T, U ] − c3∗ , which is well-defined and indeed sufficiently smooth by Proposition 1. In other words, there is (λ∗1 , λ∗2 , λ∗3 ) ∈ R3 such that (S ∗ , T ∗ , U ∗ ) satisfies 1 1 2 0= U ψ (1 + ε − x ) − xψ dx + Sρ + T ϕ − (S 2 + T 2 )ψ e−2U dx −1

+ 4λ∗1

1 −1

e4U ψ(1 + ε − x 2 ) dx + λ∗2

−1 1

−1

(Sϕ + Tρ) dx + λ∗3

1

−1

ρ dx

(41)

458


for all (ρ, ϕ, ψ) ∈ X . This can be restated by saying that (S ∗ , T ∗ , U ∗ ) is a weak solution of 0 = −U (1 + ε − x 2 ) + 2xU + 1 − (S 2 + T 2 )e−2U + 4λ∗1 (1 + ε − x 2 ) e4U , (42) 0 = −T + 2U T + λ∗2 Se2U , + (λ∗2 T

(43)

+ λ∗3 )e2U ,

0=S 0 = T (±1) = U (±1)

(44)

on (−1, 1). Any weak solution (S, T, U ) ∈ X of the system (42), (43), and (44) can be shown to be smooth and to satisfy the equations strongly via a bootstrap argument: For all (ρ, ϕ, ψ) ∈ X , we can rewrite (41) as 1 x 0= U (1 + ε − x 2 ) − x + (S 2 + T 2 ) e−2U dt −1 −1

x − 4λ∗1 e4U (1 + ε − t 2 )dt ψ dx, 1

−1 1

0= 0=

−1

−1

T e−2U − λ∗2

x −1

S dt ϕ dx,

Se−2U + λ∗2 T + λ∗3 ρ dx,

where we used integration by parts and Proposition 1. By the fundamental lemma of the calculus of variations, there are constants a, b ∈ R such that the equations x (S 2 + T 2 ) e−2U dt a = U (x)(1 + ε − x 2 ) − x + −1 x e4U (1 + ε − t 2 ) dt, (45) − 4λ∗1 −1 x S dt, (46) b = T (x) e−2U (x) − λ∗2 0 = S(x) e

−2U (x)

+ λ∗2

−1

T (x) + λ∗3

(47) T , U , and

hold almost everywhere on (−1, 1). Solving iteratively for S, we deduce the respective smoothness of S, T , and U up to the boundary by a bootstrap argument (similar to p. 462 in [6]) using Propostion 1 in every step. Differentiating Eqs. (45) and (46), we get validity of (42) and (43) in the strong sense. In particular, (S ∗ , T ∗ , U ∗ ) is a smooth classical solution of the Euler-Lagrange equations of Jε∗ with (λ1 , λ2 , λ3 ) = (λ∗1 , λ∗2 , λ∗3 ). 5.4. Solution of the Euler-Lagrange equations. Let us now determine the minimizer (S, T, U ) := (S ∗ , T ∗ , U ∗ ) explicitly, dropping the asterisk in what follows for ease of notation. S can obviously be expressed as S(x) = −[λ3 + λ2 T (x)] e2U (x)

(48)

by Eq. (44). Inserting this expression into (43), we get the equation 0 = T − 2U T + λ2 (λ3 + λ2 T ) e4U ,

(49)


459

a linear ODE of second order for T for given U . To solve (49), consider two separate cases: (i) Assume λ2 = 0. Then (49) reduces to T = 2U T which has the general solution x T (x) = a −1 e2U (t) dt + b with a, b ∈ R, so that T (±1) = 0 induces T ≡ 0. (ii) Assume now λ2 = 0. In this case, (49) has the general solution x x

λ3 T (x) = a sin λ2 e2U (t) dt + b cos λ2 e2U (t) dt − 1 (50) λ2 −1 −1 with a, b ∈ R. For λ3 = 0, inserting the boundary values T (±1) = 0 gives us b = 1 and # $ 1 $ 1 − cos λ2 1 e2U (t) dt 1 − cos λ2 −1 e2U (t) dt −1 $ = ±% . (51) a= 1 2U (t) 1 sin λ2 −1 e dt 1 + cos λ2 −1 e2U (t) dt The task of determining U remains to be completed. To this end, set γ := − S(x)2 + T (x)2 e−4U (x) ≤ 0

(52)

and observe that dγ /dx = 0, so that γ is a non-positive constant. Moreover, from the explicit expressions obtained for S and T , we see that γ = −λ23 (1 + a 2 ) where, as defined above, a = 0 if λ2 = 0 and a is as in (51) otherwise. Recall that (S, T, U ) is a global minimizer of Iε . Although Iε is not globally Fréchetdifferentiable w.r.t. S and T , it can straightforwardly be shown that it is continuously Fréchet-differentiable w.r.t. U . We thus deduce via integration by parts and by the fundamental lemma of the calculus of variations that 0 = −U (1 + ε − x 2 ) + 2xU + 1 − (S 2 + T 2 )e−2U −

2M22 c12

(1 + ε − x 2 ) e4U .

Comparing this equation with (42), we obtain the explicit expression λ1 = −

M22 2c12

≤ 0.

(53)

Moreover, the Euler-Lagrange equation (42) for U , which can now be written as 0 = −U (1 + ε − x 2 ) + 2xU + 1 + γ e2U + 4λ1 (1 + ε − x 2 ) e4U , has an integrating factor and leads to the first order ODE F := −(1 + ε − x 2 )2 U 2 + 2x(1 + ε − x 2 )U + 2λ1 e4U (1 + ε − x 2 )2 −x 2 + γ (1 + ε − x 2 ) e2U ≡ constant, because

F (x) = 2[(1 − x 2 )U (x) − x] −U (1 + ε − x 2 ) + 2xU + 1 + γ e2U + 4λ1 (1 + ε − x 2 ) e4U = 0.

(54)

460


We now proceed to calculate U . Substituting W (x) := (1 + ε − x 2 ) e2U (x) > 0 on [−1, 1], Eq. (54) can be reformulated to say W 2λ1 W 2 + γ W − F =± , (55) 2W 1 + ε − x2 which in particular implies F ≤ 0 as both λ1 , γ ≤ 0 and W > 0 by definition. We would like to divide by the square root on the right hand side and integrate the equation. We must first find out where the zeros of W can lie, if they exist at all. A careful discussion of the ODE (55) referring to (42), the boundary values W (±1) = ε, and using the fact that we are discussing the class K∗ containing the global minimizer, shows that W has exactly one zero x˜ ∈ (−1, 1) and that W (x) ˜ < 0. Moreover, from this discussion we obtain F < 0 and the fact that λ1 and γ cannot vanish simultaneously. Integrating (55) on both [−1, x) ˜ and (x, ˜ 1] and using W (x) ˜ < 0 to determine the correct sign on each interval we obtain √ 2F 2 −F x ± C, W± (x) = artanh √ , y± (x) := √ 1+ε 1+ε γ − γ 2 + 8λ1 F cosh y± (x) where √

2 −F 1 C = −√ artanh √ 1+ε 1+ε

⎛

⎞ ε2

2λ1 + γ ε − F ⎠ + artanh ⎝ √ , ε −F + 2√γ−F

W− : [−1, x) ˜ → R, W+ : (x, ˜ 1] → R. As the solution W we are looking for is smooth by the above and agrees with W± where they exist, W− and W+ must smoothly fit together at x. ˜ Also, the induced functions U− , U+ both smoothly extend to [−1, 1] and must agree at x˜ to all orders. Moreover, they both solve Eq. (42). Thus, Picard’s uniqueness theorem (cf. p. 9 in [8]) tells us they agree on the whole interval [−1, 1]. From W− (−x) = W+ (x) we deduce symmetry of W , x˜ = 0, and C = 0. Altogether, we know that W has the following form: √ 2F x 2 −F . (56) W (x) = artanh √ , y(x) := √ 1+ε 1+ε γ − γ 2 + 8λ1 F cosh y(x) 5.5. Estimating the minimal value of Iε . In order to estimate the value of Iε at its global minimizer, we use the fact that (54) allows us to simplify our expression for Iε . Using (53), we obtain √ 1 F Iε [S, T, U ] = 1 − artanh √ 1+ε+ √ . (57) 1+ε 1+ε We now intend to estimate F from above via F ≤ −(1 + ε) 1 −

2+ε 2−ε (2+ε)2 ε

2 ln ln

2 ,

(58)

which allows us to conclude that lim inf ε→0 Iε ≥ 1, see Subsect. 5.8. We prepare this estimate with the study of two auxiliary functions f and g, see below. We then use these


461

functions to obtain (58) in the cases c2 = 0 and c2 = 0 (and several subcases), see Subsecs. 5.6 and 5.7. We define f (α) :=

1 1 ε M [(1 + α)S, T, U ]2 , g(α) := M2ε [S, (1 + α)T, U ]2 . 2 2 2

The function g : R → R can be seen to be differentiable at α = 0 and we obtain g (0) = −

2 |c2 | M2 . c1

(59)

As (S, T, U ) simultaneously is a minimizer of Iε and a critical point of Jε , it follows from (41) on the other hand that 1 1 0= T 2 e−2U dx + λ2 c2 = T 2 e−2U dx + g (0). (60) −1

−1

We also find that f : R → R is differentiable at α = 0 unless both c2 = 0 and c34 = 4. Recall that this singular case also led us to the introduction of Lagrange multipliers. We then have & 2 M c4 g (0) − c √2M3 if M1 = 0 1 1 f (0) = , (61) 0 if M1 = 0 unless both c2 = 0 and c34 = 4. In addition, it follows as above that 0=

1 −1

S 2 e−2U dx + λ2 c2 +

√

2 λ3 c3 =

1 −1

S 2 e−2U dx + f (0),

(62)

or equivalently 0 = −γ = −γ

1 −1 1 −1

e

2U

1

−1 1

dx −

e2U dx −

−1

T 2 e−2U dx + λ2 c2 + T 2 e−2U dx + f (0).

√ 2 λ3 c3 (63)

5.6. Estimating the minimal value of Iε : the case c2 = 0. The explicit expression (59) for g (0) suggests separate treatment of the cases c2 = 0 and c2 = 0. We begin with c2 = 0. Four different subcases arise, namely (a) (b) (c) (d)

c34 c3 c3 c3

= 4, = 0, = 0, M1 = 0, = 0, M1 = 0.

We will find that the last two cases cannot occur in the minimizing class K∗ . In the first two cases, we will indeed arrive at estimate (58). Let us discuss the singular case (a) first. Here, (60) implies T ≡ 0, c3 = 0 assures S = 0 so that we can deduce λ2 = 0 from (43). Recall M1 = M2 = λ1 = 0, γ = 0.

462


λ3 1 2U Then (48) implies S = −λ3 e2U so that c3 = − √ e dx. Let us proceed to calcu2 −1 1 2U 2F late −1 e dx. The boundary condition W (±1) = ε implies γ = ε(1+cosh y1 ) and we are in a position to calculate

1

−1

e

2U (x)

dx =

1 −1

√ ε sinh(y1 ) W (x) −F y1 2 y dy = √ 1 − tanh dx = − . 2 1+ε−x 2γ 2 −F −y1

! " Recalling γ = −λ23 , we deduce 2 = c32 = 2ε sinh2 y21 so that cosh y1 = 2+ε ε , whence by definition of y1 , F = −(1 + ε) in accordance with (58). We now proceed to a discussion of case (b). From (60) and (62), we get T ≡ 0 and S ≡ 0, respectively. This implies γ = 0 so that λ1 = 0 by the above. We therefore obtain √ −F W (x) = √ −2λ1 cosh y(x) so that the boundary condition W (±1) = ε leads to λ1 = c12

=

1 −1

F . 2ε2 cosh2 y1

We calculate

√ W2 −F dx = − tanh y1 . 2 1+ε−x 2λ1

Recall that in this particular case also λ1 = − c24 by (53) so that y1 = arsinh 2ε and we 1

arrive at estimate (58) using arsinh x = ln(x +

x 2 + 1) and artanh x =

1 1+x ln . 2 1−x

Let us continue with case (c). From (60) we get T ≡ 0, whereas M1 = 0 implies M2 = 0 and thus λ1 = 0 by (53). On the other hand, we get f (0) = 0 from (61) so that by (63) we have γ = 0, a contradiction, because we have seen in the previous subsection that λ1 and γ cannot vanish simultaneously. Finally, we discuss case (d). As before, we get T ≡ 0 and thus by (43) λ2 = 0 as √ 2c3

c3 = 0 ensures S ≡ 0. Equation (62) then leads to λ3 = − c2 3 . From this, we obtain 1 6 1 2U c12 2 so that γ = − 2c3 . e dx = , where we used (48) and c = 0. Also, γ = −λ 3 2 4 3 −1 c c 3

In particular, Iε [S, T = 0, U ] = Iε [0, 0, U ] + [S, T = 0, U ] being a global minimizer of Iε .

c34 2c12

1

> Iε [0, 0, U ]. This contradicts

5.7. Estimating the minimal value of Iε : the case c2 = 0. Finally let c2 = 0. If λ1 = 0 were possible, then by (53) M2 = 0 so that g (0) = 0 and thus T ≡ 0 follow from (59), (60). Equation (61) then tells us that f (0) = 0 and whence S ≡ 0, so that also γ = 0, in contradiction to the above exclusion of λ1 = γ = 0. Thus, λ1 = 0 which implies


both M2 = 0 and M1 = 0. Using again (60), (62), and (53), we obtain: ' 2 4 4 − c3 − 2|c2 | λ1 = − , 2c14 ' 4 − c34 − 2|c2 | 2 λ2 = − sign(c2 ), c12 √ ' 2 4 − c34 − 2|c2 | c33 ' λ3 = − . 4 − c34 c12 x Rewrite S, T in terms of A(x) := λ2 0 e2U (t) dt, A1 := A(1) and use W (±1) equation (52), and our definition of y1 as well as symmetry of U to obtain

λ3 cos A(x) T (x) = −1 , λ2 cos A1 cos A(x) 2U (x) S(x) = −λ3 e , cos A1 λ2 γ = − 23 , cos A1 √ −F γ 2 + 8λ1 F sinh y1 2 c1 = − 2λ1 γ 2 + 8λ1 F cosh y1 − γ γ + γ 2 + 8λ1 F y1 2γ arctan tanh + √ , √ 2 8λ1 F 8λ1 F λ23 A1 − sin A1 cos A1 , cos2 A1 λ22 √ λ3 c3 = − 2 tan A1 , λ2 γ + γ 2 + 8λ1 F y1 λ2 A1 = √ tanh arctan , √ 2 −2λ1 8λ1 F c2 = −

ε=

2F

γ− + 8λ1 F cosh y1 √ 2 −F 1 y1 = √ artanh √ . 1+ε 1+ε Now set φ := arccos √

γ2

−γ γ 2 +8λ1 F

,

463

(64)

(65)

(66) = ε, (67) (68) (69)

(70) (71) (72) (73) (74) (75)

∈ (0, π2 ) which is well-defined as λ1 · F > 0. Using

this new constant, Eqs. (67) through (75) take on a simpler form. In particular, these equations lead to 0 < |A1 | ≤ φ < π2 and c12 =

4 tan φ cos A1 | sin A1 − A1 cos A1 | , c3 = ± 2 cos A1 . √ 4 −F sin A1

464


For the Lagrange multipliers, we get F sin4 A1 , 8 tan2 φ cos2 A1 4(sin A1 − A1 cos A1 ) λ2 = , c12 sin2 A1 3 | sin A1 − A1 cos A1 | 4 λ3 = ∓ 2 cos 2 A1 . c1 sin3 A1 λ1 =

With the above expressions, we obtain ε=

2 cos A1 sin2 φ . sin2 A1 cos φ (cosh y1 + cos φ)

! π" 2x As sin cos x is monotonically increasing on 0, 2 and |A1 | ≤ φ, we have ε ≥ in other words 2 −1 . y1 ≥ arcosh ε

2 cosh y1 +1

or

This implies √ " ! 1 + ε arcosh 2ε − 1

√ −F ≥

2 artanh √ 1

,

1+ε

√ where we have used (75). Recall arcosh x = ln x + x 2 − 1 to deduce (58) also in the discussed case c2 = 0. 5.8. The limit ε → 0. We conclude as promised that for c2 = 0 cases (c) and (d) cannot apply for the minimizer (S, T, U ), whereas in the remaining cases (a) and (b), as well as for c2 = 0, we can estimate using (57) and (58) that Iε [S, T, U ] ≥ 1 +

√

⎡ 1+ε ⎣ 1−

2+ε 2−ε (2+ε)2 ε

2 ln ln

√ 2+ε . ≥ 1 − 2 1 + ε ln 2−ε

2

⎤ − 1⎦ artanh √

1 1+ε (76)

We now study the limit ε → 0. For any (S, T, U ) ∈ X , c2 [S, T, U ], c3 [S, T, U ] and thus M1 [S, T, U ] are independent of ε, limε→0 c1ε [S, T, U ] = c10 [S, T, U ] and thus limε→0 M2ε [S, T, U ] = M20 [S, T, U ] so that ε |I [S, T, U ] − Iε [S, T, U ]| ≤ 2

M [S, T, U ]2 M2ε [S, T, U ]2 2 − U (x) dx + →0 2 2 −1

1

2


465

as ε → 0, i.e. Iε [S, T, U ] is continuous at ε = 0 for fixed (S, T, U ). This finally leads us to an esimate of the original functional I . We obtain I [S, T, U ] = lim Iε [S, T, U ] ε→0

≥ lim inf Iε [S ∗ , T ∗ , U ∗ ] ε→0 √ 2+ε ≥ lim inf 1 − 2 1 + ε ln ε→0 2−ε = 1, where (S ∗ , T ∗ , U ∗ ) denotes the global minimizer of Iε . This proves the claim of Lemma 2 and therefore the inequality (2).

Finally, after we have seen that the functional I has a lower bound of 1, we can ask the question of whether there exist functions S, T , and U for which I takes on this value. The investigation of this question together with a discussion of the meaning of I in the context of degenerate black holes can be found in Appendix A. 6. Discussion With techniques from the calculus of variations, we have shown that the inequality p 2J + p 2Q < 1 holds for axisymmetric and stationary sub-extremal black holes with surrounding matter in full Einstein-Maxwell theory. In particular, we have proved the inequality I [S, T, U ] ≥ 1 for the functional I defined in (34). As I could not directly be seen to have a local minimizer, we introduced a family of approximating functionals Iε which could be shown to have one. Together with a theorem for degenerate black holes in [1], we can deduce the following. Theorem 1. Consider Einstein-Maxwell spacetimes with vanishing cosmological constant. Then, for every axisymmetric and stationary sub-extremal black hole with arbitrary surrounding matter we have the inequality (8π J )2 + (4π Q 2 )2 < A2 . If the axisymmetric and stationary black hole is degenerate, the equation (8π J )2 + (4π Q 2 )2 = A2 holds. Observe that the assumptions for the result in [1] which has been used here have been weakened, see Appendix A. Theorem 1 provides a remarkable relation between the geometrical concept of the existence of trapped surfaces and the physical black hole properties described by rotation rate p J and charge rate p Q . We see that “physically reasonable” (sub-extremal) black holes cannot rotate “too fast” and cannot be charged “too strongly”. Finally, our results shed new light on the notions of sub-extremality and extremality of axisymmetric and stationary black holes. Any sub-extremal black hole in the sense of [3] (the notion of which we have used throughout this paper) is also sub-extremal in the sense that p 2J + p 2Q < 1. In fact, p 2J + p 2Q = 1 holds in the degenerate limit, for which reason we may call these black holes “extremal”.

466


Acknowledgement. We would like to thank Herbert Pfister for many valuable discussions and John Head for commenting on the manuscript. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre SFB/TR7 “Gravitational wave astronomy” and by the International Max Planck Research School for “Geometric Analysis, Gravitation and String Theory”. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

A. Remarks on Degenerate Black Holes In order to discuss extremal black holes (as done in [1]) and to get an idea of the meaning of the appearing functional I , one can apply similar techniques as used in Sec. 5 to I itself to derive Euler-Lagrange equations and a complete characterization of the minimizers of I . As minimizers of I need not be limits of minimizers of Iε , this renewed analysis is necessary. It turns out that the Euler-Lagrange equations for S and T are just as before, cf. (43) and (44). Moreover, there again exists an integrating factor for the Euler-Lagrange equation for U leading to −1 = −(1 − x 2 )2 U 2 + 2x(1 − x 2 )U + 2λ1 e4U (1 − x 2 )2 − x 2 + γ (1 − x 2 ) e2U , where γ is defined as in (52). Introducing W (x) := (1 − x 2 )e2U (x) on the interior (−1, 1), we find that Eq. (55) holds on (−1, 1) with ε = 0 and F = −1. Discussing the radicand in (55) as before, we see that it vanishes at at most one inner point. Assuming non-vanishing of the radicand and integrating the equation on (−1 + δ, 1 − δ) for some δ > 0 leads to a contradiction as the unique solution U derived from W diverges as δ → 0 while we know that the smooth solution U exists on the whole interval by the same bootstrap argument as sketched above. Thus we know that there exists exactly one interior zero of the radicand and we can integrate the equation as before to obtain e2U (x) =

2 , (1 + γ )x 2 + 1 − γ

(77)

and the consistency condition γ 2 − 8λ1 = 1, using the boundary values for U , U (±1) = 0. In other words, U belongs to a family parametrized by γ ∈ [−1, 0]. Proceeding as above, we find that S and T are given by

' 3 S(x) = ± (−γ ) 2 + 1 − γ 2 |T (x)| e2U , ' T (x) = ± −γ (1 − γ 2 )

1 − x2 1 − γ + (1 + γ )x 2

with γ ∈ [−1, 0]. The signs of S and T can be chosen independently of each other. It can a posteriori be seen that all functions S, T , U of this form with γ ∈ [−1, 0] in fact satisfy I [S, T, U ] = 1


467

so that we have identified all minimizers of I . Moreover, one can show that the Lagrange parameters λ1 , λ2 , λ3 can explicitly be expressed as ' 3 1 (78) λ1 = − (1 − γ 2 ), λ2 = ± 1 − γ 2 , λ3 = ±(−γ ) 2 , 8 where again the signs are not correlated. By comparison with [1], one finds that these are exactly the functions S, T , and U arising in the context of arbitrary degenerate black holes with surrounding matter.8 Moreover, the differential equations characterizing S, T , and U in [1] are exactly the Euler-Lagrange equations of I derived in this paper where the Lagrange parameters in (78) correspond to the constants appearing in [1]. We arrive at two conclusions: First, our analysis dispenses with additional assumptions made in [1], namely equatorial symmetry and the existence of a continuous sequence of spacetimes, leading from the Kerr-Newman solution in electrovacuum to the discussed black hole solution. The latter was necessary to assure uniqueness (up to a parameter) of the solution to the horizon equations in [1]. As a matter of fact, any smooth solution of the equations in [1] is a minimizer of I as can be seen by solving the equations as done above and using the relations in (78) between γ and the Lagrange parameters. Thus, any solution of these equations is automatically equatorially symmetric and of the form assumed in [1]. Hence, the unnecessary assumptions of [1] can be dropped. Secondly, we see that the functional I plays the role of a “primitive” of the Einstein equations on the event horizon of degenerate black holes: Remarkably, the Euler-Lagrange equations corresponding to I lead uniquely to the electromagnetic and metric potentials S, T , and U belonging to degenerate black holes. References 1. Ansorg, M., Pfister, H.: A universal constraint between charge and rotation rate for degenerate black holes surrounded by matter. Class. Quantum Grav. 25, 035009 (2008) 2. Bardeen, J.M.: Rapidly rotating stars, disks, and black holes. In: Black holes (Les Houches), deWitt, C., deWitt, B.S., ed., London: Gordon and Breach, 1973, pp. 241–289 3. Booth, I., Fairhurst, S.: Extremality conditions for isolated and dynamical horizons. Phys. Rev. D 77, 084005 (2008) 4. Buttazzo, G., Giaquinta, M., Hildebrandt, S.: One-dimensional variational problems. Oxford: Clarendon Press, 1998 5. Carter, B.: Black hole equilibrium states. In: Black Holes (Les Houches), deWitt, C., deWitt, B.S., ed., London: Gordon and Breach, 1973, pp. 57–214 6. Evans, L.C.: Partial Differential Equations. Providence, RI: Amer. Math. Soc. 2002 7. Hennig, J., Ansorg, M., Cederbaum, C.: A universal inequality between the angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter. Class. Quantum Grav. 25, 162002 (2008) 8. Rauch, J.: Partial Differential Equations. Berlin: Springer, 1991 9. Yosida, K.: Functional Analysis. Berlin: Springer, 1995 Communicated by P.T. Chru´sciel

8 The parameter α ∈ [−1, 1] in [1] is related to γ via γ = −(1 − α 2 )/(1 + α 2 ).


Communications in


Non-Equilibrium Dynamics of Dyson’s Model with an Infinite Number of Particles Makoto Katori1 , Hideki Tanemura2 1 Department of Physics, Faculty of Science and Engineering,

Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan. E-mail: [email protected] 2 Department of Mathematics and Informatics, Faculty of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan. E-mail: [email protected] Received: 19 December 2008 / Accepted: 17 June 2009 Published online: 10 September 2009 – © Springer-Verlag 2009

Abstract: Dyson’s model is a one-dimensional system of Brownian motions with longrange repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson’s model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of Z is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel. 1. Introduction In order to understand the statistics of eigenvalues of random matrix ensembles as distributions of particle positions in the one-dimensional Coulomb gas systems with log-potentials, Dyson introduced stochastic models of particles in R, which obey the stochastic differential equations (SDEs), d X j (t) = d B j (t) +

β 2

1≤k≤N ,k= j

dt , 1 ≤ j ≤ N , t ∈ [0, ∞), X j (t) − X k (t) (1.1)

where B j (t)’s are independent one-dimensional standard Brownian motions [3]. The Gaussian orthogonal ensemble (GOE), the Gaussian unitary ensemble (GUE), and the Gaussian symplectic ensemble (GSE) of random matrices correspond to the SDEs (1.1) with β = 1, 2 and 4, respectively [14]. Spohn [20] has considered an infinite particle system obtained by taking the N → ∞ limit of (1.1) with β = 2 and called the system Dyson’s model. He studied the equilibrium dynamics with respect to the determinantal

470

M. Katori, H. Tanemura

(Fermion) point process µsin , in which any spatial correlation function ρm is given by a determinant with the sine kernel [18,19], 1 sin{π(y − x)} , x, y ∈ R, (1.2) dk eik(y−x) = K sin (y − x) = 2π |k|≤π π(y − x) √ where i = −1. By the Dirichlet form approach Osada [16] constructed the infinite particle system represented by a diffusion process, which has µsin as a reversible measure. Recently he proved that this system satisfies the SDEs (1.1) with N = ∞ [17]. On the other hand, it was shown by Eynard and Mehta [4] that multitime correlation functions for the process (1.1) are generally given by determinants, if the process starts from µGUE , the eigenvalue distribution of GUE with variance σ 2 . Nagao and Forrester N ,σ 2 [15] evaluated the bulk scaling limit σ 2 = 2N /π 2 → ∞ and derived the so-called extended sine kernel with density 1, 1 2 Ksin (t − s, y − x) = dk ek (t−s)/2+ik(y−x) − 1(s > t) p(s − t, x|y) 2π |k|≤π ⎧ 1 2 2 ⎪ ⎪ du eπ u (t−s)/2 cos{π u(y − x)} if t > s ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ (1.3) = K sin (y − x) if t = s ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ 2 2 ⎪− ⎩ du eπ u (t−s)/2 cos{π u(y − x)} if t < s, 1

s, t ≥ 0, x, y ∈ R, where 1(ω) is the indicator function of condition ω, and p(t, y|x) is the heat kernel 2 e−(y−x) /2t 1 2 p(t, y|x) = √ dk e−k t/2+ik(y−x) , t > 0. (1.4) = 2π R 2π t Since lim N →∞ µGUE = µsin , the process, whose multitime correlation functions N ,2N /π 2 are given by determinants with the extended sine kernel (1.3), is expected to be identified with the infinite-dimensional equilibrium dynamics of Spohn and Osada. This equivalence is, however, not yet proved. Fritz [5] established the theory of non-equilibrium dynamics of infinite particle systems with a finite-range smooth potential. Here we study the non-equilibrium dynamics of infinite-particle Dyson’s model with a long-range log-potential, in which the force acting on each particle is singular both for short and long distances (see (1.1)). We denote by M the space of nonnegative integer-valued Radon measures on R, which is a Polish space with the vague topology: we say ξn , n ∈ N ≡ {1, 2, . . . } converges to ξ vaguely, if limn→∞ R ϕ(x)ξn (d x) = R ϕ(x)ξ(d x) for any ϕ ∈ C0 (R), where C0 (R) is the set of all continuous real-valued functions with compact supports. Any element ξ of M can be represented as ξ(·) = j∈ δx j (·) with a sequence of points in R, x = (x j ) j∈ satisfying ξ(K ) = { j ∈ : x j ∈ K } < ∞ for any compact subset K ⊂ R. The index set is N or a finite set. We call an element ξ of M an unlabeled configuration, and a sequence x a labeled configuration. For A ⊂ R, we write the restriction of ξ on A as (ξ ∩ A)(·) = j∈:x j ∈A δx j (·). As an M-valued process (P, (t), t ∈ [0, ∞)), we consider the system such that, for any integer M ≥ 1, f m ∈ C0 (R), θm ∈ R, 1 ≤ m ≤ M, 0 < t1 < · · · < t M < ∞,

Non-Equilibrium Dynamics of Dyson’s Model with Infinite Number of Particles

the expectation of exp

M

m=1 θm R f m (x)(tm , d x)

471

can be expanded with χm (x) =

eθm fm (x) − 1, 1 ≤ m ≤ M as G ξ [χ ] ≡

···

N1 ≥0

×

M N M ≥0 m=1

Nm M

NM N1 1 d x (1) · · · d x (M) j j NM N m ! R N1 R j=1 j=1

(m) (1) (M) ρ t1 , x N 1 ; . . . ; t M , x N M , χm x j

m=1 j=1 (m) (m) where x (m) Nm denotes (x 1 , . . . , x Nm ), 1 ≤ m ≤ M. Here ρ’s are locally integrable functions, which are symmetric in the sense that (m)

(m)

(m)

(m)

(m)

ρ(. . . ; tm , σ (x Nm ); . . . ) = ρ(. . . ; tm , x Nm ; . . . ) with σ (x Nm ) ≡ (xσ (1) , . . . , xσ (Nm ) ) (M) for any permutation σ ∈ S Nm , 1 ≤ ∀m ≤ M. In such a system ρ(t1 , x (1) N1 ; . . . ; t M , x N M ) is called the (N1 , . . . , N M )-multitime correlation function and G ξ [χ ] the generating function of multitime correlation functions. There are no multiple points with probability one for t > 0. Then we assume that there is a function K(s, x; t, y), which is continuous with respect to (x, y) ∈ R2 for any fixed (s, t) ∈ [0, ∞)2 , such that

(1) (M) (m) (n) det K(tm , x j ; tn , xk ) ρ t1 , x N 1 ; . . . ; t M , x N M = 1≤ j≤Nm ,1≤k≤Nn 1≤m,n≤M

M of positive integers, and any time for any integer M ≥ 1, any sequence (Nm )m=1 sequence 0 < t1 < · · · < t M < ∞. That is, the finite dimensional distributions of the process are determined by the function K. Let T = {t1 , . . . , t M }. We note that T = t∈T δt ⊗(t) is a determinantal (Fermion) point process on T×R with an opera tor K given by K f (s, x) = t∈T R dy K(s, x; t, y) f (t, y) for f (t, ·) ∈ C0 (R), t ∈ T. The process is then said to be determinantal with the correlation kernel K. When K is symmetric, Soshnikov [19] and Shirai and Takahashi [18] gave sufficient conditions for K to be a correlation kernel of a determinantal point process. Though such conditions are not known for asymmetric cases, a variety of processes, which are determinantal with asymmetric correlation kernels, have been studied. As mentioned above the pro cess (t) = Nj=1 δ X j (t) with the SDEs (1.1) with β = 2 starting from its equilibrium is an example [4]. The infinite particle system of Nagao and Forrester measure µGUE N ,σ 2 [15] is also determinantal with the extended sine kernel, which is asymmetric as shown by (1.3). (For other examples, see, for instance, [10,21].) In the present paper we first show that, for any fixed configuration ξ N ∈ M with N ξ (R) = N , Dyson’s model starting from ξ N is determinantal and its correlation kernel N Kξ is given by using the multiple Hermite polynomials [2,7,8] (Proposition 2.1). For ξ ∈ M, when Kξ ∩[−L ,L] converges to a continuous function as L → ∞, the limit is written as Kξ . If Pξ ∩[−L ,L] converges to a probability measure Pξ on M[0,∞) , which is determinantal with the correlation kernel Kξ , weakly in the sense of finite dimensional distributions as L → ∞ in the vague topology, we say that the process (Pξ , (t), t ∈ [0, ∞)) is well defined with the correlation kernel Kξ . (The regularity of the sample paths of (t) will be discussed elsewhere [11].) In the case ξ(R) = ∞, the process (Pξ , (t), t ∈ [0, ∞)) is Dyson’s model with an infinite number of particles.

472


For ξ ∈ M with ξ({x}) ≤ 1, ∀x ∈ R, we give sufficient conditions so that the process (Pξ , (t), t ∈ [0, ∞)) is well defined, in which the correlation kernel is generally expressed using a double integral with the heat kernels of an entire function represented by an infinite product (Theorem 2.2). The configuration in which every point of Z is occupied by one particle, ξ Z (·) ≡ ∈Z δ (·), satisfies the conditions and we will show that Dyson’s model starting from ξ Z is determinantal with the kernel Z

Kξ (s, x; t, y) = Ksin (t − s, y − x) 1 2 + dk ek (t−s)/2+ik(y−x) {ϑ3 (x − iks, 2πis) − 1} (1.5) 2π |k|≤π = Ksin (t − s, y − x) 1 2 2 2 2 + e2πi x−2π s du eπ u (t−s)/2 cos [π u{(y −x)−2πis}] , 0

∈Z\{0}

s, t ≥ 0, x, y ∈ R, where ϑ3 is a version of the Jacobi theta function defined by 2 ϑ3 (v, τ ) = e2πiv+πiτ , τ > 0.

(1.6)

∈Z ξZ

Z

The lattice structure K (s, x + n; t, y + n) = Kξ (s, x; t, y), ∀n ∈ Z, s, t ≥ 0 is clear in (1.5) by the periodicity of ϑ3 , ϑ3 (v + n, τ ) = ϑ3 (v, τ ), ∀n ∈ Z. We can prove Z

lim Kξ (u + s, x; u + t, y) = Ksin (t − s, y − x),

u→∞

(1.7)

which implies that µsin is an attractor of Dyson’s model and ξ Z is in its basin. We are interested in the continuity of the process with respect to initial configuration. For Dyson’s model with finite particles, the weak convergence of the processes (PξnN , (t), t ∈ [0, ∞)) → (Pξ N , (t), t ∈ [0, ∞)) as n → ∞ is guaranteed by the vague convergence of the initial configurations ξnN → ξ N as n → ∞, where ξnN , ξ N ∈ M with ξnN (R) = ξ N (R) = N < ∞, n ∈ N. Based on this continuity, Dyson’s model can be defined for any initial configurations with finite particles, which can have multiple points (see Proposition 2.1). On the other hand, we have found that, if ξ(R) = ∞, the weak convergence of processes in the sense of finite dimensional distributions cannot be concluded from the convergence of initial configurations in the vague topology. In the present paper we consider a stronger topology for infinite-particle configurations (Definition 2.3). We introduce the spaces Yκm , κ ∈ (1/2, 1), m ∈ N of initial configurations such that the convergence of processes is guaranteed by that of the initial configurations in this new topology (Theorem 2.4). Note that the union of the spaces Y = κ∈(1/2,1) m∈N Yκm is large enough to carry the Poisson point processes, Gibbs states with regular conditions, µsin , as well as infiniteparticle configurations with multiple points. In particular, using the fact µsin (Y) = 1 and the continuity with respect to the initial configurations, we can prove that the process (Psin , (t), t ∈ [0, ∞)) of Nagao and Forrester, which is determinantal with the extended sine kernel (1.3), is Markovian [11]. The paper is organized as follows. In Sect. 2 preliminaries and main results are given. In Sect. 3 the definitions of some special functions used in the present paper are given and their basic properties are summarized. Sect. 4 is devoted to proofs of results.


473

2. Preliminaries and Main Results For ξ(·) = j∈ δx j (·) ∈ M, we introduce the following operations: δx j +u (·), (shift) For u ∈ R, τu ξ(·) = j∈

(dilatation) For c > 0, c ◦ ξ(·) = (square) ξ 2 (·) =

δcx j (·),

j∈

δx 2 (·). j

j∈

We use the convention such that f (x) = exp ξ(d x) log f (x) = R

x∈ξ

f (x)ξ({x})

x∈supp ξ

for ξ ∈ M and a function f on R, where supp ξ = {x ∈ R : ξ({x}) > 0}. For a multivariate symmetric function g we write g((x)x∈ξ ) for g((x j ) j∈ ). For s, t ∈ [0, ∞), x, y ∈ R and ξ N ∈ M with ξ N (R) = N ∈ N, we set 1 N Kξ (s, x; t, y) = dz p(s, x|z) dy p(t, −i y|y ) 2πi (ξ N ) R 1 iy − z × 1− iy − z N x −z x ∈ξ

−1(s > t) p(s − t, x|y),

(2.1)

where (ξ N ) is a closed contour on the complex plane C encircling the points in supp ξ N on the real line R once in the positive direction. Proposition 2.1. Dyson’s model (Pξ N , (t), t ∈ [0, ∞)), starting from any fixed configuration ξ N ∈ M with ξ N (R) = N < ∞, is determinantal with the correlation kernel N Kξ given by (2.1). We put M0 = {ξ ∈ M : ξ({x}) ≤ 1 for any x ∈ R}. Since any element ξ of M0 is determined uniquely by its support, it is identified with a countable subset {x j } j∈ of R. For ξ N ∈ M0 , a ∈ C, we introduce an entire function of z ∈ C, z−a 1− , (ξ N , a, z) = x −a N c x∈ξ ∩{a}

whose zero set is supp (ξ N ∩ {a}c ) (see, for instance, [12]). Then, if ξ N ∈ M0 , (2.1) is written as N Kξ (s, x; t, y) = ξ N (d x ) p(s, x|x ) dy p(t, −i y|y )(ξ N , x , i y ) R

R

−1(s > t) p(s − t, x|y).

(2.2)

474


For L > 0, α > 0 and ξ ∈ M we put ξ(d x) ξ(d x) 1/α M(ξ, L) = , Mα (ξ, L) = , α x [−L ,L]\{0} [−L ,L]\{0} |x| and M(ξ ) = lim M(ξ, L), L→∞

Mα (ξ ) = lim Mα (ξ, L), L→∞

if the limits finitely exist. We introduce the following conditions: (C.1) There exists C0 > 0 such that |M(ξ )| < C0 , (C.2) (i) There exist α ∈ (1, 2) and C1 > 0 such that Mα (ξ ) ≤ C1 , (ii) There exist β > 0 and C2 > 0 such that M1 (τ−a 2 ξ 2 ) ≤ C2 (|a| ∨ 1)−β ∀a ∈ supp ξ. We denote by X the set of configurations ξ satisfying the conditions (C.1) and (C.2), and put X0 = X ∩ M0 . For ξ ∈ X0 , a ∈ R and z ∈ C we define (ξ, a, z) = lim (ξ ∩ [a − L , a + L], a, z). L→∞

We note that |(ξ, a, z)| < ∞ and (ξ, a, ·) ≡ 0, if |M(τ−a ξ )| < ∞ and M2 (τ−a ξ ) < ∞. Theorem 2.2. If ξ ∈ X0 , the process (Pξ , (t), t ∈ [0, ∞)) is well defined with the correlation kernel given by Kξ (s, x; t, y) = ξ(d x ) p(s, x|x ) dy p(t, −i y|y )(ξ, x , i y ) R

R

−1(s > t) p(s − t, x|y).

(2.3)

In case ξ(R) = ∞, Theorem 2.2 gives Dyson’s model with an infinite number of particles starting from the configuration ξ ∈ X0 . From (2.3) it is easy to check that Kξ (t, x; t, y)Kξ (t, y; t, x)d xd y → ξ(d x)1(x = y), t → 0 in the vague topology. An interesting and important example is obtained for the initial configuration, in which every point in Z is occupied by one particle, ξ Z (·) ≡ ∈Z δ (·). In this case Z ξ Z (·) ∈ X0 and we can show that the correlation kernel Kξ is given by (1.5). The process (Psin , (t), t ∈ [0, ∞)) is reversible with respect to µsin . The result (1.7) implies that the process (Pξ Z , (u + t), t ∈ [0, ∞)) converges to (Psin , (t), t ∈ [0, ∞)), as u → ∞, weakly in the sense of finite dimensional distributions. In other words, (Pξ Z , (t), t ∈ [0, ∞)) is the relaxation process from an initial configuration ξ Z to the invariant measure µsin , which is determinantal, and this non-equilibrium dynamics is completely determined via the temporally inhomogeneous correlation kernel (1.5). (See Remark in Sect. 4.3.) For κ > 0, we put g κ (x) = sgn(x)|x|κ , x ∈ R, and ηκ (·) = δgκ () (·). ∈Z

Since g κ is an odd function, ηκ satisfies (C.1) for any κ > 0. For any κ > 1/2 we can show by simple calculation that ηκ satisfies (C.2)(i) with any α ∈ (1/κ, 2) and some


475

C1 = C1 (α) > 0 depending on α, and does (C.2)(ii) with any β ∈ (0, 2κ − 1) and some C2 = C2 (β) > 0 depending on β. This implies that ηκ is an element of X0 in any case κ > 1/2. Note that η1 = ξ Z . If there exists β < (β − 1) ∧ (β/2) for ξ ∈ M0 such that {x ∈ ξ : ξ([x − β |x| , x + |x|β ]) ≥ 2} = ∞, then ξ does not satisfy the condition (C.2) (ii). In order to include such initial configurations as well as those with multiple points in our study of Dyson’s model with an infinite number of particles, we introduce another condition for configurations: (C.3) there exists κ ∈ (1/2, 1) and m ∈ N such that m(ξ, κ) ≡ max ξ [g κ (k), g κ (k + 1)] ≤ m. k∈Z

We denote by Yκm

the set of configurations ξ satisfying (C.1) and (C.3) with κ ∈ (1/2, 1) and m ∈ N, and put Y= Yκm . κ∈(1/2,1) m∈N

Noting that the set {ξ ∈ M : m(ξ, κ) ≤ m} is relatively compact for each κ ∈ (1/2, 1) and m ∈ N, we see that Y is locally compact. We introduce the following topology on Y. Definition 2.3. Suppose that ξ, ξn ∈ Y, n ∈ N. We say that ξn converges -moderately to ξ , if lim (ξn , i, ·) = (ξ, i, ·) uniformly on any compact set of C.

n→∞

(2.4)

It is easy to see that (2.4) is satisfied, if ξn converges to ξ vaguely and the following two conditions hold: ξn (d x) = 0, (2.5) lim sup L→∞ n>0 [−L ,L]c x 2 ξn (d x) lim sup = 0. (2.6) L→∞ n>0 [−L ,L]c x Note that for any a ∈ R and z ∈ C, lim (ξn , a, z) = (ξ, a, z),

n→∞

(2.7)

/ supp ξ . if ξn converges -moderately to ξ and a ∈ Then the second theorem of the present paper is the following. Theorem 2.4. (i) If ξ ∈ Y, (Pξ , (t), t ∈ [0, ∞)) is well defined with a correlation kernel Kξ . In particular, when ξ ∈ Y0 ≡ Y ∩ M0 , Kξ is given by (2.3). (ii) Suppose that ξ, ξn ∈ Yκm , n ∈ N for some κ ∈ (1/2, 1) and m ∈ N. If ξn converges -moderately to ξ , then the process (Pξn , (t), t ∈ [0, ∞)) converges to the process (Pξ , (t), t ∈ [0, ∞)) weakly in the sense of finite dimensional distributions as n → ∞ in the vague topology. In the proof of this theorem given in Section 4.4, we will give an expression (4.26) to Kξ , which is valid for any ξ ∈ Y. There we will use special functions such as the Hermite polynomials, Hk , k ∈ N0 ≡ N ∪ {0}, the complete symmetric functions h k , k ∈ N0 , and the Schur functions s(k|) , k, ∈ N0 .

476


3. Special Functions 3.1. Multivariate symmetric functions. For n ∈ N, let λ = (λ1 , λ2 , . . . , λn ) be a partition of length less than or equal to n, and δ = (n − 1, n − 2, . . . , 1, 0). For x = (x1 , x2 , . . . , xn ) consider the skew-symmetric polynomial x λj k +n−k . aλ+δ (x) = det 1≤ j,k≤n

If λ = ∅, it is the Vandermonde determinant, which is given by the product of difference of variables: x n−k = aδ (x) = det (x j − xk ). j 1≤ j,k≤n

1≤ j 0 such that ξ ([x0 − ε, x0 + ε]) = 0. We see that for fixed z ∈ C, z−x 1− (ξ, x, z) = u−x u∈ξ −δx z − x0 1 1− = u − x0 1 − (x − x0 )/(u − x0 ) u∈ξ u∈ξ −δx 1 = (ξ, x0 , z) hr (x − x0 )r , u − x0 u∈ξ r ∈N0

where (3.3) has been used. Then (ξ, x, z) is a smooth function of x on [x0 − ε, x0 + ε]. 3.2. Multiple Hermite polynomials. For any ξ ∈ M with ξ(R) < ∞, the multiple Hermite polynomial of type II, Pξ is defined as the monic polynomial of degree ξ(R) that satisfies for any x ∈ supp ξ , 2 dy Pξ (y)y j e−(y−x) /2 = 0, j = 0, . . . , ξ({x}) − 1. (3.4) R

The multiple Hermite polynomials of type I consist of a set of polynomials Aξ ( · , x) : x ∈ supp ξ, deg Aξ (·, x) = ξ({x}) − 1

(3.5)

such that the function

Q ξ (y) =

Aξ (y, x)e−(y−x)

2 /2

(3.6)

x∈supp ξ

satisfies

dy Q ξ (y)y = j

R

0, j = 0, . . . , ξ(R) − 2 1, j = ξ(R) − 1.

(3.7)

The polynomials {Aξ (·, x)} are uniquely determined by the degree requirements (3.5) and the orthogonality relations (3.7) [8]. The multiple Hermite polynomial of type II, Pξ and the function Q ξ defined by (3.6) have the following integration representations [2]:

2 e−(y +i y) /2 Pξ (y) = dy (i y − x), √ 2π R x∈ξ 2 1 e−(z−y) /2 1 ! . dz √ Q ξ (y) = 2πi (ξ ) (z − x) 2π x∈ξ

(3.8)

478


Now we fix ξ N ∈ M with ξ N (R) = N ∈ N. We write ξ N (·) = Nj=1 δx j (·) with a labeled configuration x = (x j ) Nj=1 such that x1 ≤ x2 ≤ · · · ≤ x N . Then we define ξ0N (·) ≡ 0 and ξ jN (·) =

j

δxk (·), 1 ≤ j ≤ N .

k=1 N ({x}), ∀x ∈ R, 0 ≤ j ≤ By definition ξ jN (R) = j, 0 ≤ j ≤ N and ξ jN ({x}) ≤ ξ j+1 N − 1. We define (−)

Hj

(y; ξ N ) = Pξ N (y), j

(+)

H j (y; ξ N ) = Q ξ N (y), 0 ≤ j ≤ N − 1. j+1

(3.9)

By the orthogonality relations (3.4), (3.7) and the above definitions, we can prove the biorthonormality [2], (−) (+) dy H j (y; ξ N )Hk (y; ξ N ) = δ jk , 0 ≤ j, k ≤ N − 1. (3.10) R

For N ∈ N, let W N = {x ∈ R N : x1 < x2 < · · · < x N }, the Weyl chamber of type A N −1 . Lemma 3.1. Let y = (y j ) Nj=1 ∈ W N . For any ξ N (·) = Nj=1 δx j (·) ∈ M with a labeled configuration x = (x j ) Nj=1 such that x1 ≤ x2 ≤ · · · ≤ x N , 1 2 (+) det e−(yk −x j ) /2 = (−1) N (N −1)/2 (2π ) N /2 det H j−1 (yk ; ξ N ) . 1≤ j,k≤N aδ (x) 1≤ j,k≤N (3.11) Here when some of the x j ’s coincide, we interpret the LHS using l’Hôpital’s rule. ! ! j−1 Proof. First we assume ξ N ∈ M0 . Since aδ (x) = (−1) N (N −1)/2 Nj=2 m=1 (x j −xm ), by the multilinearity of determinant 1 2 det e−(yk −x j ) /2 aδ (x) 1≤ j,k≤N # " −(yk −x j )2 /2 e 1 = (−1) N (N −1)/2 (2π ) N /2 det √ ! j−1 1≤ j,k≤N 2π m=1 (x j − x m ) ⎡ ⎤ j −(yk −x )2 /2 e 1 ⎦. ! = (−1) N (N −1)/2 (2π ) N /2 det ⎣ √ 1≤ j,k≤N 2π 1≤m≤ j,m= (x − x m ) =1

By definition (3.9) with (3.8), if ξ N ∈ M0 , ξ N (R) = N , 2 1 e−(yk −z) /2 1 (+) ! dz H j−1 (yk ; ξ N ) = √ 2πi (ξ jN ) 2π x∈ξ jN (z − x) 2 1 e−(yk −z) /2 1 = dz √ ! j 2πi (ξ jN ) 2π =1 (z − x ) =

2 j e−(yk −x ) /2 1 ! , 1 ≤ j ≤ N. √ 2π 1≤m≤ j,m= (x − x m ) =1

(3.12)


479

Then (3.11) is proved for ξ N ∈ M0 . When some of the x j ’s coincide, the LHS of (3.11) is (+) interpreted using l’Hôpital’s rule and in the RHS of (3.11) H j−1 (yk ; ξ N ) should be given by the first expression of (3.12). Then (3.11) is valid for any ξ N ∈ M, ξ N (R) = N . Lemma 3.2. Let N ∈ N, ξ N ∈ M with ξ N (R) = N . For 0 ≤ s ≤ t, x, y ∈ R, 0 ≤ j ≤ N − 1, s j/2 y x 1 1 dy H j(−) √ ; √ ◦ ξ N p(t − s, y|x) = H j(−) √ ; √ ◦ ξ N , t s s t t R (3.13) ( j+1)/2 x s y 1 1 d x p(t − s, y|x)H j(+) √ ; √ ◦ ξ N = H j(+) √ ; √ ◦ ξ N , t s s t t R (3.14) where p is the heat kernel (1.4). Proof. Consider the integral y 1 (−) N p(t − s, y|x) dy H j √ ;√ ◦ξ t t R √ 2 1 1 x 2 iy − √ =√ dy dy e−(y−x) /{2(t−s)}−(y +i y/ t) /2 √ 2π(t − s) 2π R t R N ( =

x∈ξ j

√ 2 t 1 x i y − √ e−t (y +i x/ t) /(2s) . dy √ s 2π R t N

x∈ξ j

√ Change the integral variable y → y t/s to obtain the equality (3.13). Similar calculation gives (3.14). When ξ N (·) = N δ0 (·),

√ H j(−) (y; N δ0 ) = 2− j/2 H j (y/ 2),

√ 2− j/2 2 √ H j (y/ 2)e−y /2 , 0 ≤ j ≤ N − 1, j! 2π where H j (x) is the Hermite polynomial of degree j, (+)

H j (y; N δ0 ) =

H j (x) = j!

[ j/2]

(−1)k

k=0

(2x) j−2k k!( j − 2k)!

√ e−y /2 dy √ (i y + 2x) j 2π R 2 j! e2zx−z = dz . 2πi (δ0 ) z j+1 2

= 2 j/2

(3.15)

The last expression (3.15) implies that the generating function of the Hermite polynomials is given by zj 2 e2zx−z = (3.16) H j (x). j! j∈N0

480


4. Proofs of Results 4.1. Proof of Proposition 2.1. For x, y ∈ W N and t > 0, consider the Karlin-McGregor determinant of the heat kernel (1.4) [9], f N (t, y|x) =

det

)

1≤ j,k≤N

* p(t, y j |xk ) .

If ξ N ∈ M0 with ξ N (R) = N ∈ N, ξ N can be identified with a set x ∈ W N . For any M ≥ 1 and any time sequence 0 < t1 < · · · < t M < ∞, the multitime probability density of Dyson’s model is given by [6,10] pξ

t1 , ξ (1) ; . . . ; t M , ξ (M)

N

= aδ (x (M) )

M−1

f N (tm+1 − tm , x (m+1) |x (m) ) f N (t1 , x (1) |x)

m=1

where ξ (m) (·) =

N

j=1 δx (m) (·), j

1 , aδ (x)

1 ≤ m ≤ M.

Define x 1 √ ; √ ◦ ξN , t t x 1 N −( j+1)/2 (+) N , φ (+) (t, x; ξ ) ≡ t H ; ◦ ξ √ √ j j t t (−)

(−)

φ j (t, x; ξ N ) ≡ t j/2 H j

0 ≤ j ≤ N − 1, t > 0, x ∈ R. From the biorthonormality (3.10) of the multiple Hermite polynomials and Lemma 3.2, the following relations are derived: Lemma 4.1. For ξ N ∈ M with ξ N (R) = N ∈ N, 0 ≤ t1 ≤ t2 ,

R

(−) N N d x2 φ (−) j (t2 , x 2 ; ξ ) p(t2 − t1 , x 2 |x 1 ) = φ j (t1 , x 1 ; ξ ), 0 ≤ j ≤ N − 1, (+)

(+)

d x1 p(t2 − t1 , x2 |x1 )φ j (t1 , x1 ; ξ N ) = φ j (t2 , x2 ; ξ N ), 0 ≤ j ≤ N − 1, (+) N N d x1 d x2 φ (−) j (t2 , x 2 ; ξ ) p(t2 − t1 , x 2 |x 1 )φk (t1 , x 1 ; ξ ) = δ jk , R R

R

0 ≤ j, k ≤ N − 1. Put µ(±) (t, x; ξ N ) = (−)

det

1≤ j,k≤N

(±) φ j−1 (t, xk ; ξ N ) .

Since H j is a monic polynomial of degree j, µ(−) (t, x; ξ N ) = (−1) N (N −1)/2 aδ (x). By Lemma 3.1, f N (t1 , x (1) |x)/aδ (x) will be replaced by (−1) N (N −1)/2 µ(+) (t1 , x (1) ; ξ N )


481

to extend the expression to the case ξ N ∈ M. Then the multitime probability density of Dyson’s model is expressed as

N p ξ t1 , ξ (1) ; . . . ; t M , ξ (M) = µ(−) (t M , x (M) ; ξ N )

M−1

f N (tm+1 − tm ; x (m+1) |x (m) )µ(+) (t1 , x (1) ; ξ N )

m=1

(4.1) for ξ N ∈ M with ξ N (R) = N ∈ N. For x = (x1 , . . . , x N ) with ξ(·) = Nj=1 δx j (·) and M N ∈ {1, 2, . . . , N }, we put x N = (x1 , . . . , x N ). For a sequence (Nm )m=1 of positive integers less than or equal to N , we obtain the (N1 , . . . , N M )-multitime correlation function by

ξN (1) (M) ρ N t1 , x N 1 ; . . . ; t M , x N M =

!M m=1

R N −Nm

pξ

N

t1 , ξ (1) ; . . . ; t M , ξ (M)

M

m=1

1 (N − Nm )!

N

(m)

dx j .

j=Nm +1

(4.2) For f = ( f 1 , · · · , f M ) ∈ C0 (R) M , and θ = (θ1 , · · · , θ M ) ∈ R M , the generating function for multitime correlation functions is given as ⎧ ⎫⎤ ⎡ N M ⎨ ⎬ N G ξ [χ ] = Eξ N ⎣exp θm f m (X jm (tm )) ⎦ ⎩ ⎭ m=1

=

N N1 =0

×

···

N

M

N M =0 m=1

Nm M

jm =1

NM N1 1 (1) (M) dx j · · · dx j N N m ! R N1 M R j=1 j=1

N

(m) (1) (M) ρ ξ t1 , x N 1 ; . . . ; t M , x N M , χm x j

m=1 j=1

where χm (x) = eθm fm (x) − 1,

1 ≤ m ≤ M.

By the argument given in Sect. 4.2 in [10], the expression (4.1) with Lemma 4.1 leads to the Fredholm determinantal expression for the generating function, N S m,n (x, y; ξ N )χn (y) , G ξ [χ ] = Det δmn δ(x − y) + + where + S m,n (x, y; ξ N ) = S m,n (x, y; ξ N ) − 1(m > n) p(tm − tn , x|y) with

482


S m,n (x, y; ξ N ) =

N −1

N (−) N φ (+) j (tm , x; ξ )φ j (tn , y; ξ )

j=0

1 =√ tm

N −1 j=0

tn tm

j/2

(+) Hj

x 1 √ ; √ ◦ ξN tm tm

(−) Hj

y 1 N . √ ;√ ◦ξ tn tn

Here the Fredholm determinant is expanded as Det δmn δ(x − y) + + S m,n (x, y; ξ N )χn (y) =

N N1 =0

×

N M

···

N M =0 m=1

Nm M

NM N1 1 (1) (M) dx j · · · dx j N N m ! R N1 M R j=1 j=1

(m) χm x j

m=1 j=1

det

1≤ j≤Nm ,1≤k≤Nn 1≤m,n≤M

(m) (n) + S m,n (x j , xk ; ξ N ) .

Proof of Proposition 2.1. Inserting the integral formulas for H j(±) , the kernel S m,n is written as √ 2 √ −(y +i y/ tn )2 /2 1 1 e−(z−x/ tm ) /2 m,n N e S (x, y; ξ ) = √ dz dy √ √ tm 2πi (tm−1/2 ◦ξ N ) 2π 2π R √ k/2 !k N −1 tn =1 (i y − x / tn ) × !k+1 √ tm =1 (z − x / tm ) k=0 √ 2 √ −(y +i y/ tn )2 /2 e−(z−x/ tm ) /2 1 e dz dy = √ √ 2πi (tm−1/2 ◦ξ N ) 2π 2π R √ N −1 !k =1 (i tn y − x ) . × !k+1 √ =1 ( tm z − x ) k=0 For z 1 , z 2 ∈ C with z 1 ∈ / {x1 , . . . , x N }, the following identity holds: ! N −1 k =1 (z 2 − x ) !k+1 =1 (z 1 − x ) k=0 (z 2 − x1 )(z 2 − x2 ) · · · (z 2 − x N −1 ) 1 z 2 − x1 + ··· + + z 1 − x1 (z 1 − x1 )(z 1 − x2 ) (z 1 − x1 )(z 1 − x2 ) · · · (z 1 − x N ) N z 2 − x 1 −1 . = z 1 − x z2 − z1

=

=1

By this identity, we have S

m,n

√ 2 √ −(y +i y/ tn )2 /2 e−(z−x/ tm ) /2 e dz dy √ √ −1/2 2π 2π (tm ◦ξ N ) R √ N i tn y − x 1 × −1 √ √ . √ tm z − x i tn y − tm z

1 (x, y; ξ ) = 2πi N

=1


483

Note that √ 2 √ −(y +i y/ tn )2 /2 1 e−(z−x/ tm ) /2 1 e dz dy √ √ √ √ −1/2 2πi (tm ◦ξ N ) i tn y − tm z 2π 2π R √ 2 ( j +i y/√t )2 /2 −(z−x/ t ) /2 −(y m n tm z 1 e 1 e = dz dy √ √ √ 2πi (tm−1/2 ◦ξ N ) i tn y tn i y 2π 2π R j∈N0

= 0. By changing the integral variables appropriately, we find that + S m,n (x, y; ξ N ) is equal to (2.1) with s = tm , t = tn . This completes the proof. 4.2. Proof of Theorem 2.2. In this subsection we give a proof of Theorem 2.2. First we prove some lemmas. Lemma 4.2. If Mα (ξ ) < ∞ for some α ∈ (1, 2), then α

M1 (ξ, L)α/(α−1) 2 ≤ Mα (ξ )α /(α−1) . L(L + 1)α

L∈N

Proof. By Hölder’s inequality we have ξ(d x) M1 (ξ, L) = ≤ Mα (ξ )ξ ([−L , L] \ {0})(α−1)/α . |x| 0 L δ1 . Then ∞ αδ/(α−1) M1 (ξ, L)α/(α−1) L1 >α α L(L + 1)α L(L + 1)α L=L 1 L∈N ∞ αδ/(α−1) > αL 1 dy y −(α+1) L 1 +1

αδ/(α−1)

= L1

(L 1 + 1)−α =

L1 L1 + 1

α

α(δ−α+1)/(α−1)

L1

.

From Lemma 4.2 we have α L1 2 α(δ−α+1)/(α−1) L1 ≤ Mα (ξ )α /(α−1) . L1 + 1 Hence L1
0 and θ ∈ (α ∨ (2 − β), 2) such that * ) ∀y ∈ R, ∀a ∈ supp ξ. |(ξ, a, i y)| ≤ exp C3 (|y|θ ∨ 1) + (|a|θ ∨ 1) Proof. First we estimate the entire function (ξ, a, z), z ∈ C, in the case that a = 0 ∈ supp ξ . In case 2|z| < |x|, by using the expansion z (−1)k−1 z k = log 1 + , x k x k∈N

we have (−1)k−1 z k z = ξ(d x) log 1 + ξ(d x) x k x 2|z| 0 such that Mα τ−a (ξ − Ca ) ≤ C4 (κ, m)(|a| ∨ 1)(1−κ)/κ ∀a ∈ supp ξ,

(4.14)

and (C.2) (i) holds, that is, there exists C1 = C1 (α, ξ ) such that Mα (ξ ) ≤ C1 .

(4.15)

4a satisfies (C.2) (ii) ∀a ∈ supp ξ , (ii) Suppose that β ∈ (0, 2κ − 1). Then ξ − −C 4a = C−k in case Ca = Ck . That is, there exists C2 (κ, m) > 0 such that where C

4a )2 ≤ C2 (κ, m)(|a| ∨ 1)−β ∀a ∈ supp ξ. (4.16) M1 τ−a 2 (ξ − Ca − C Ca

492


Proof. By simple calculations we see that there exists a positive constant C(κ) such that Mα (τ−a ηκ ) ≤ C(κ)(|a| ∨ 1)(1−κ)/κ ∀a ∈ supp ηκ .

(4.17)

Suppose that Ca = Ck , k ∈ Z. Then ξ − Ca = ξ ∩ [bk−1 , bk ]c . We divide the set [bk−1 , bk ]c into the following four sets: * A1 = −∞, g κ (k − 2) , A2 = g κ (k − 2), bk−1 , ) A4 = g κ (k + 2), −∞ . A3 = bk , g κ (k + 2) , Then we have

(ξ − Ca )(d x) |x − a|α R

From (4.7) and (4.8), we have ξ(d x) α A1 |x − a| ξ(d x) α A2 |x − a| ξ(d x) |x − a|α A 3 ξ(d x) |x − a|α A4

1/α ≤

4 j=1

Aj

ξ(d x) |x − a|α

≤m

−∞ 0 are satisfied. Then there exists C6 = C6 (α, β, C1 , C5 ) > 0 such that |M(τ−a ξ ) − M(ξ )| ≤ C6 |a|δ1 , where δ1 = α(1 + γ ) − 1.


493

Proof. From Lemma 4.3 and the fact that M1 (τ−a ξ , L) is increasing in L, we see that max M1 (τ−a ξ , L) = M1 (τ−a ξ , L 0 ) ≤ (2Mα (τ−a ξ ))αδ1 /(δ1 −α+1) ≤ C|a|δ1

0≤L≤L 0

from (4.18) with a constant C > 0. Combining this estimate with Lemma 4.3, we have M1 (τ−a ξ , L) ≤ C|a|δ1 ∨ L δ1 .

(4.19)

We assume a = 0. By the definitions of M(ξ ) and M(τ−a ξ ), ξ(d x) 1 + ξ({0}) |M(τ−a ξ ) − M(ξ )| ≤ + |a| . c |a| {a,0} |x(x − a)| We divide the set {a, 0}c into the three disjoint subsets {x : 0 < |x| < 2|a|, 2|a − x| > |a|}, {x : |x| ≥ 2|a|} and {x : 0 < |x| < 2|a|, 0 < 2|a − x| ≤ |a|}. By simple calculation, we see 2 2 ξ(d x) ξ(d x) ≤ = M1 (ξ, 2|a|). |a| 0 t) p(s − t, x|y) ξ ∩ [−L , L](d x ) p(s, x|x ) dy p(t, −i y|y )(ξ ∩ [−L , L], x , i y) = lim L→∞ R R = lim p(s, x|ck ) dy p(t, −i y|y )k (t, ξ ∩ [−L , L], i y , x).

L→∞

L→∞

k∈Z

R

496


By Lemma 4.8, we can apply Lebesgue’s convergence theorem to show that the limit is p(s, x|ck ) dy p(t, −i y|y )k (t, ξ, i y , x). R

k∈Z

We can repeat the argument in the proof of Theorem 2.2 given at the end of Sect. 4.2. Then if ξ ∈ Y0 , (Pξ , (t), t ∈ [0, ∞)) is well-defined with the correlation kernel ξ K (s, x; t, y) = p(s, x|ck ) dy p(t, −i y|y )k (t, ξ, i y , x) R

k∈Z

−1(s > t) p(s − t, x|y).

(4.26)

It is equal to (2.3) of Theorem 2.2 by the equality (4.25). When ξ ∈ Y \ Y0 , (4.25) is not valid. For any L > 0, however, the equality Kξ ∩[−L ,L] (s, x; t, y) + 1(s > t) p(s − t, x|y) = p(s, x|ck ) dy p(t, −i y|y )k (t, ξ ∩ [−L , L], i y , x) R

k∈Z

holds by the continuity with respect to the initial configuration for Dyson’s model with finite particles. Then, again by Lemma 4.8 with Lebesgue’s convergence theorem, we will obtain the result (4.26). (ii) By the fact (2.7) and the definition of k , we see that for any k ∈ N, t ≥ 0, and x, y ∈ R, lim k (t, ξn , i y , x) = k (t, ξ, i y , x).

n→∞

By using Lemma 4.8 we see that, for fixed t ≥ 0, x ∈ R, there exist θ ∈ (1, 2), 57 = C 57 (t, x) > 0 such that C7 = C7 (t, x) > 0, and C * ) 57 exp C7 |y |θ + |ck |θ , k ∈ Z, y ∈ R, n ∈ N. |k (t, ξn , i y , x)| ≤ C Therefore, by applying Lebesgue’s convergence theorem, we obtain the theorem.

Acknowledgement. The present authors would like to thank T. Shirai and H. Spohn for useful comments on the manuscript. A part of the present work was done during the participation of M.K. in the ESI program “Combinatorics and Statistical Physics” (March and May in 2008). M.K. expresses his gratitude for the hospitality of the Erwin Schrödinger Institute (ESI) in Vienna and for the well organized program by M. Drmota and C. Krattenthaler. M.K. is supported in part by the Grant-in-Aid for Scientific Research (C) (No.21540397) of Japan Society for the Promotion of Science. H.T. is supported in part by the Grant-in-Aid for Scientific Research (KIBAN-C, No.19540114) of Japan Society for the Promotion of Science.

References 1. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge: Cambridge University Press, 1999 2. Bleher, P.M., Kuijlaars, A.B.: Integral representations for multiple Hermite and multiple Laguerre polynomials. Ann. Inst. Fourier 55, 2001–2014 (2005) 3. Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962) 4. Eynard, B., Mehta, M.L.: Matrices coupled in a chain: I. Eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998) 5. Fritz, J.: Gradient dynamics of infinite point systems. Ann. Probab. 15, 478–514 (1987)


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6. Grabiner, D.J.: Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. Henri Poincaré, Probab. Stat. 35, 177–204 (1999) 7. Imamura, T., Sasamoto, T.: Polynuclear growth model with external source and random matrix model with deterministic source. Phys. Rev. E 71, 041606/1-12 (2005) 8. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge: Cambridge University Press, 2005 9. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 9, 1141–1164 (1959) 10. Katori, M., Tanemura, H.: Noncolliding Brownian motion and determinantal processes. J. Stat. Phys. 129, 1233–1277 (2007) 11. Katori, M., Tanemura, H.: In preparation 12. Levin, B.Ya.: Lectures on Entire Functions. Translations of Mathematical Monographs, 150, Providence R. I.: Amer. Math. Soc., 1996 13. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. 2nd edition, Oxford: Oxford Univ. Press, 1995 14. Mehta, M.L.: Random Matrices. 3rd edition, Amsterdam: Elsevier, 2004 15. Nagao, T., Forrester, P.J.: Multilevel dynamical correlation functions for Dyson’s Brownian motion model of random matrices. Phys. Lett. A247, 42–46 (1998) 16. Osada, H.: Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions. Commun. Math. Phys. 176, 117–131 (1996) 17. Osada, H.: Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. http://arXiv.org/abs/0902.3561v1[math.PR], 2009 18. Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point process. J. Funct. Anal. 205, 414–463 (2003) 19. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000) 20. Spohn, H.: Interacting Brownian particles: a study of Dyson’s model. In: Hydrodynamic Behavior and Interacting Particle Systems, Papanicolaou, G. (ed), IMA Volumes in Mathematics and its Applications, 9, Berlin: Springer-Verlag, 1987, pp. 151–179 21. Tracy, C.A., Widom, H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7–41 (2004) Communicated by H. Spohn


Communications in


Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains Li Yuxiang1,2, , Philippe Souplet2 1 Department of Mathematics, Southeast University, Nanjing 210096,

P. R. China. E-mail: [email protected]

2 CNRS UMR 7539, Laboratoire Analyse Géométrie et Applications, Université Paris 13,

99, Avenue J.B. Clément, 93430 Villetaneuse, France. E-mail: [email protected] Received: 22 December 2008 / Accepted: 17 August 2009 Published online: 13 October 2009 – © Springer-Verlag 2009

Abstract: Consider the diffusive Hamilton-Jacobi equation u t = u + |∇u| p , p > 2, on a bounded domain with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ∇u may blow up only on the boundary ∂. In this paper, under suitable assumptions on ⊂ R2 and on the initial data, we show that the gradient blow-up singularity occurs only at a single point x0 ∈ ∂. This is the first result of this kind in the study of problems involving gradient blow-up phenomena. In general domains of Rn , we also obtain results on nondegeneracy and localization of blow-up points. 1. Introduction In certain physical models, for instance of ballistic deposition processes, the evolution of the profile of a growing interface is described by diffusive Hamilton-Jacobi type equations. In [27], the Kardar-Parisi-Zhang (KPZ) equation u t = υu + λ|∇u|2 + η(x, t)

(1.1)

is proposed, where u(x, t) represents the height of the interface profile (surface). Here the growth term λ|∇u|2 accounts for the deposition of new particles on the surface, υu corresponds to the relaxation of the interface by a surface tension and η(x, t) is a noise term. Krug and Spohn [28] generalized the (deterministic) KPZ equation to u t = υu + λ|∇u| p

(1.2)

with p ≥ 1 in order to study the effect of the form of the nonlinearity. An extreme case of the generalized KPZ equation (1.2) is the singular equation u t = u + ln |∇u|,

(1.3)

Supported in part by National Natural Science Foundation of China 10601012 and Southeast University Award Program for Outstanding Young Teachers 2005.

500

Y. Li, Ph. Souplet

which describes interface growth in the Complex Directed Polymer problem CDP [46]. For more extensive discussion of the nonlinear interface growth equations, we refer to [24]. We would like to point out that Eq. (1.2) is also the viscosity approximation of Hamilton-Jacobi type equations from stochastic control theory [35]. As a typical model-case in the theory of parabolic PDEs, Eq. (1.2) is the simplest example of a parabolic equation with a nonlinearity depending on the first-order spatial derivatives of u and is considered as an analogue of the extensively studied reactiondiffusion equation with zero order nonlinearity u t − u = u p ,

p > 1.

(1.4)

Our interest in this paper lies in the singularity formation in the interface growth process described by the generalized KPZ equation (1.2). By a dimensionless procedure, we can set υ = λ = 1. Let p > 0 and let ⊂ Rn be a smoothly bounded domain, namely of class C 2+α . We consider the problem u t − u = |∇u| p , u(x, t) = 0, u(x, 0) = u 0 (x),

x ∈ , t > 0, x ∈ ∂, t > 0, x ∈ .

(1.5) (1.6) (1.7)

We assume that u 0 ∈ X + := {v ∈ C 1 (); v ≥ 0, v|∂ = 0}. Then it follows from [16, Theorem 10, p. 206] that problem (1.5)–(1.7) admits a unique maximal, nonnegative classical solution u ∈ C 2,1 ( × (0, T )) ∩ C 1,0 ( × [0, T )), where T = T (u 0 ) is the maximal existence time. The nature of the singularity formation is very different between problems (1.5)–(1.7) and (1.4). Equation (1.4) is well-known to exhibit L ∞ blow-up whenever p > 1. On the contrary, for problem (1.5)–(1.7), we have u(t)∞ ≤ u 0 ∞ , 0 < t < T, by the maximum principle. Since (1.5)–(1.7) is well posed in X + , it follows that, if T < ∞, then lim ∇u(t)∞ = ∞.

t→T

This phenomenon of ∇u blowing up with u remaining uniformly bounded is known as gradient blow-up. It actually occurs if and only if p > 2. For p > 2 this happens for suitably large initial data (cf. [2,26,40,42] and see also [1,13,15,34] for related results). For 0 < p ≤ 2 it is well-known that all solutions are global (cf. [31] and see also [7,34,43]). Blow-up analysis, including blow-up rate, blow-up set, blow-up profile, continuation after blow-up, etc., for Eq. (1.4) has been studied extensively in the past twenty years, see e.g. [17,20,25,37–39,44,45] and the references therein. Comparing with Eq. (1.4), gradient blow-up analysis of problem (1.5)–(1.7) is far from complete. From [42, Theorem 3.2], it follows that ∇u blows up only on the boundary ∂. More precisely, it is proved there that |∇u| ≤ C1 δ −1/( p−1) (x) + C2 in × [0, T ),

(1.8)

where C1 = C1 (n, p) > 0 and C2 = C2 ( p, , u 0 C 1 ) > 0. Here and in the rest of the paper we denote δ(x) = dist(x, ∂).

Diffusive Hamilton-Jacobi Equations

501

The estimate (1.8) is known to be optimal for n = 1 due to the results in [2,4]. For the related one-dimensional problem u t − u x x = |u x | p , x ∈ (0, 1), t > 0, u(0, t) = 0, u(1, t) = M > 0, t > 0, the precise gradient blow-up rate estimate C3 (T − t)−1/( p−2) ≤ u x (t)∞ ≤ C4 (T − t)−1/( p−2) , T /2 < t < T,

(1.9)

is known for monotone in time nonglobal solutions. (Note that the existence of such solutions requires M to be larger than some minimal positive value.) The upper estimate in (1.9) was recently proved in [23]. The lower estimate, which is true actually without monotonicity assumption, is due to [12] (see also [23] for extensions to higher dimensions). Estimate (1.9) is very interesting since it shows that the actual blow-up rate does not coincide with that suggested by the self-similar invariance of the equation. Some results on continuation after gradient blow-up for problem (1.5)–(1.7) and its variants can be found in [6,14,29]. For results on interior gradient blow-up in other related equations, see [3,5,19]. As far as we know, the question of the location of gradient blow-up points within the boundary for problem (1.5)–(1.7) with n ≥ 2 has not been addressed so far. The gradient blow-up set of u is defined by G BU S(u 0 ) = x0 ∈ ∂; ∇u is unbounded in ( ∩ Bη (x0 )) × (T − η, T ) for any η > 0 . Note that by definition, G BU S(u 0 ) is compact. The main goal of this paper is to show that under some assumptions on ⊂ R2 and u 0 , the gradient blow-up set G BU S(u 0 ) contains only one point. In the context of KPZ-type models, a possible physical interpretation is that the surface tension (diffusion) forces the steep region to become more and more concentrated near a single boundary point (for suitably localized and monotone initial data – see below). We shall consider two types of domains: disks and the following class of locally flat symmetric domains, defined by the assumptions: is symmetric with respect to the line x = 0, (−ρ, ρ) × {0} ⊂ ∂ for some ρ > 0, νx ≥ 0 on ∂∩{x > 0}, where ν = (νx , ν y ) is the unit outward normal vector.

(1.10) (1.11) (1.12)

Our main results are the following theorems. Theorem 1.1. Let p > 2 and ⊂ R2 be a smoothly bounded domain satisfying (1.10)– (1.12). (i) There exist initial data u 0 ∈ X + such that T (u 0 ) < ∞ and u 0 is symmetric with respect to the line x = 0, u 0,x ≤ 0 in + = {(x, y) ∈ ; x > 0}, G BU S(u 0 ) ⊂ (−ρ/2, ρ/2) × {0}. (ii) For any such u 0 , G BU S(u 0 ) contains only the origin.

(1.13) (1.14) (1.15)

502

Y. Li, Ph. Souplet

We now consider the case = B1 (0) ⊂ R2 . We shall denote by (r, θ ) ∈ [0, 1] × [−π, π ] the polar coordinates and write u˜ 0 (r, θ ) = u 0 (r cos θ, r sin θ ). Theorem 1.2. Let p > 2 and = B1 (0) ⊂ R2 . (i) There exist initial data u 0 ∈ X + such that T (u 0 ) < ∞ and u 0 is symmetric with respect to the line y = 0, u˜ 0,θ ≤ 0 in B+ = {(r, θ ); 0 < r < 1, 0 < θ < π }, G BU S(u 0 ) = ∂.

(1.16) (1.17) (1.18)

(ii) For any such u 0 , G BU S(u 0 ) contains only the point (x, y) = (1, 0). Remark 1.1. (a) Based on Theorems 1.1 and 1.2, we conjecture that single-point gradient blow-up should be true in general symmetric domains without the flatness assumption (1.11) (for suitable initial data). However we have been unable to show this up to now. (b) It should be possible to generalize Theorems 1.1 and 1.2 to higher dimensions, for suitable axisymmetric initial data and domains, at the expense of further technical complication. However, we have refrained from doing this, in order to keep the presentation of the new phenomenon of single-point gradient blow-up as simple as possible. Moreover, the main motivating physical problem (interface growth [27,28]) is two-dimensional. (c) Assumption (1.18) in Theorem 1.2 is of course essential, since one easily produces examples such that G BU S(u 0 ) = ∂ by considering radially symmetric initial data. (d) Examples of initial data satisfying the conditions of Theorem 1.1 are given by k −1 2 2 u 0 (x) = Cε ϕ ε (1.19) x + (y − ε) , where k = ( p − 2)/( p − 1), C ≥ C0 ( p) > 0, ε > 0 is sufficiently small, and ϕ ∈ C ∞ ([0, ∞)) satisfies ϕ ≤ 0,

ϕ(s) = 1, s ≤ 1/4,

ϕ(s) = 0, s ≥ 1/2.

(1.20)

Note that these functions have support concentrated near the boundary point (0, 0), with large derivative in some sense, but that u 0 ∞ can be made arbitrarily small. The same example works in Theorem 1.2 after replacing (x, y) with (y, 1 − x). Single-point gradient blow-up is not expected to hold for general initial data nor general domains. However, in any smooth domain and any dimension, we have the following result which shows that, for suitable initial data, the gradient blow-up set can be localized in an arbitrarily small neighborhood of a given boundary point. Theorem 1.3. Let p > 2 and ⊂ Rn be any smoothly bounded domain. Let x0 ∈ ∂ and ρ > 0. There exist initial data u 0 ∈ X + such that T (u 0 ) < ∞ and G BU S(u 0 ) ⊂ Bρ (x0 ) ∩ ∂.

(1.21)

As a by-product of our analysis, we also obtain the following nondegeneracy property for gradient blow-up points, of independent interest. It implies in particular that the upper gradient bound (1.8) is optimal near blow-up points and is achieved in the normal direction. On the other hand, the precise behavior of the tangential derivatives near an isolated gradient blow-up point still remains an open problem.


503

Proposition 1.1. Let p > 2 and ⊂ Rn be any smoothly bounded domain. Let u 0 ∈ X + be such that T = T (u 0 ) < ∞ and let x0 ∈ ∂ be a gradient blow-up point. Then lim sup δ −( p−2)/( p−1) (x) u(x, t) ≥ c0 ( p) > 0.

x→x0 , t→T

(1.22)

Denote by ν˜ = ν˜ (x) the normal vector at the boundary point which is closest to x (this point is unique for δ(x) small enough). Then we have in particular ∂u lim sup δ 1/( p−1) (x) (x, t) ≥ c1 ( p) > 0. ∂ ν˜ x→x0 , t→T The proof of Theorems 1.1(ii) and 1.2(ii) is delicate. It is based on a combination of a nontrivial modification of the Friedman-McLeod method [17], the spatial estimate (1.8) on ∇u, and the nondegeneracy property (1.22). One of the difficulties is that, whereas the method of [17] is designed for L ∞ blow-up and one-dimensional (or radial) problems, we here have to deal with gradient blow-up and a genuinely two-dimensional situation. The proof of the nondegeneracy property (1.22) relies on the constuction of suitable supersolutions, along with a local boundary control property for the gradient (Lemma 2.1). The proof of the latter is based on Bernstein-type arguments. As for Theorem 1.3 (and assertion (i) of Theorems 1.1 and 1.2), the localization of the blow-up set is achieved by combining additional supersolution arguments with the above local boundary control property, while the global nonexistence follows from a dilation argument. Finally, let us mention some work concerning other aspects of problem (1.5)–(1.7). Existence, uniqueness and regularity of weak solutions of problem (1.5)–(1.7) with irregular initial data are studied in [1,8,11,22] and references therein. Viscosity solutions are considered in [6]. Asymptotic behavior of global solutions of (1.5)–(1.7) and of the corresponding Cauchy problem is investigated in [2,4,9,10,21,30,33,41,42] and references therein. The rest of the paper is organized as follows. In Sect. 2, we first show the local boundary control property (Lemma 2.1) and we next prove Proposition 1.1 and Theorem 1.3. The proofs of Theorems 1.1 and 1.2 are then given in Sects. 3 and 4. 2. Nondegeneracy and Localization of Gradient Blow-up Points Let ⊂ Rn be a smoothly bounded domain with n ≥ 1. The unit outward normal vector is denoted by ν. Let x0 ∈ ∂. Let u be a gradient blow-up solution of problem (1.5)–(1.7) with finite blow-up time T = T (u 0 ) < ∞. We start with the local boundary control property for the gradient. Its proof is based on Bernstein-type arguments similar to [42, Theorem 3.2]. Lemma 2.1. If there exist M, R > 0 such that |u ν | ≤ M on (B R (x0 ) ∩ ∂) × [0, T ),

(2.1)

then x0 is not a gradient blow-up point. Proof. We give the brief proof. For more details on the computation, we refer to [42, Theorem 3.2]. Put w = |∇u|2 , then Lw = −2|D 2 u|2 ,

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Y. Li, Ph. Souplet

where |D 2 u|2 = i, j (u xi x j )2 and Lw = wt − w − p|∇u| p−2 ∇u · ∇w. Let a ∈ (0, 1). We select a cut-off function η ∈ C 2 (B R (x0 )), with η = 0 for |x − x0 | = R, 0 < η ≤ 1 for |x − x0 | < R and such that |∇η| ≤ C R −1 ηa , |η| + 4η−1 |∇η|2 ≤ C R −2 ηa , for |x − x0 | < R, with C = C(a) > 0. Such a function η is given for instance in the proof of [42, Theorem 3.2]. Put z = ηw, (x, t) ∈ Q := (B R (x0 ) ∩ ) × (0, T ). Then Lz = ηLw + wLη − 2∇η · ∇w in Q. Since 2|∇η · ∇w| ≤ 4η−1 |∇η|2 w + η|D 2 u|2 , it follows that Lz + η|D 2 u|2 ≤ w(|η| + 4η−1 |∇η|2 ) + pw ( p+1)/2 |∇η| ≤ C R −2 ηa w + C p R −1 ηa w ( p+1)/2 . √ Using |w p/2 − u t | = |u| ≤ n|D 2 u|2 , hence w p /(2n) ≤ |D 2 u|2 + |u t |2 , we get 1 ηw p ≤ C R −2 ηa w + C p R −1 ηa w ( p+1)/2 + |u t |2 . 2n Taking a = ( p + 1)/(2 p) and using Young’s inequality, we obtain Lz +

1 ηw p ≤ C R −2 p/( p+1) + |u t |2 . 4n On the other hand, by local theory [36], there exists t0 ∈ (0, T ) such that Lz +

u(t0 )C 2 () + sup u(t)C 1 () ≤ C(u 0 C 1 () , p, ). t∈[0,t0 ]

By [42, Prop.2.4], it follows that p

|u t | ≤ u(t0 )∞ + ∇u(t0 )∞ ≤ C(u 0 C 1 () , p, ) in × [t0 , T ). So we obtain Lz ≤ −

1 p z + A in (B R (x0 ) ∩ ) × (t0 , T ), 4n

with A = C(n, p)R −2 p/( p−1) + C(u 0 C 1 () , p, ). It follows from the maximum principle (cf. e.g. [42, Prop. 2.2]) that sup z(x, t) ≤ max max z(x, t0 ), max z(x, t), (4n A)1/ p .

(B R (x0 )∩)×(t0 ,T )

B R (x0 )∩

(B R (x0 )∩∂)×(t0 ,T )

So we get |∇u| ≤ C in (B R/2 (x0 ) ∩ ) × (t0 , T ), where C = C(u 0 C 1 () , M, p, , ρ). The proof is complete.


505

We now turn to the proof of Proposition 1.1, which amounts to showing that condition (2.1) can be relaxed to allow weak singularity of |∇u| in a neighborhood of x0 ∈ ∂. Weak singularity means |∇u| ≤ εδ −1/( p−1) (x) in (Bρ (x0 ) ∩ ) × [0, T ),

(2.2)

with suitably small ε > 0, comparing with the spatial gradient estimate (1.8). Note that (2.2) implies u≤

p−1 p−2

εδ ( p−2)/( p−1) (x) in (Bρ˜ (x0 ) ∩ ) × [0, T ),

(2.3)

for some ρ˜ > 0, but that (2.3) is weaker than (2.2). Proposition 1.1 is actually an immediate consequence of the following nondegeneracy lemma. Lemma 2.2. Let ⊂ Rn be a smoothly bounded domain and x0 ∈ ∂. There exists c0 = c0 ( p) > 0 such that, if u ≤ c0 δ ( p−2)/( p−1) (x) in (Bρ (x0 ) ∩ ) × [0, T ),

(2.4)

for some ρ > 0, then x0 is not a gradient blow-up point. Proof. We will complete the proof by showing that (2.4) implies (2.1). Without loss of generality, decreasing ρ if necessary, we assume that x0 is the origin and the hyperplane xn = 0 is tangent to ∂ at the origin, (2.5) ∂ can be locally represented as xn = f (x ), for x = (x , xn ), |x | < ρ, where f is smooth. (2.6) Denote K = f L ∞ (Bρ ) . For some small constants r ∈ (0, ρ) and 0 < d < (1 − β)/(4 pK ),

β = 1/( p − 1) < 1,

(2.7)

we can assume that D := {x ∈ Rn ; |x | < r, f (x ) < xn < f (x ) + d} ⊂ , with |∇ f | ≤ 1 for |x | ≤ r,

(2.8)

νn > 1/2 on ∂ ∩ D = {x ∈ Rn ; |x | ≤ r, xn = f (x )}

(2.9)

δ(x) ≤ 2 xn − f (x ) in D.

(2.10)

and

Let us define the comparison function

v = v(x , xn , t) = ε xn − f (x ) V −β in D × [0, T ), with V = xn − f (x ) + η(r 2 − |x |2 )t,

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Y. Li, Ph. Souplet

where ε, η > 0 are to be determined. Our aim is to show that u ≤ v in D × (0, T ),

(2.11)

which implies, due to (2.9), that −β 1 ∂u |u ν | = ≤ 2ε η(r 2 − |x |2 )T ≤ M on (Br/2 (x0 ) ∩ ∂) × (0, T ), νn ∂ xn for some constant M > 0. Then this lemma will follow from Lemma 2.1. For (x, t) ∈ D × (0, T ), we compute vt − v = −εβη(xn − f )(r 2 − |x |2 )V −β−1 + 2εβV −β−1 |∇ f |2 + 1 + 2ηt x · ∇ f + εV −β f − εβ(xn − f )V −β−1 ( f + 2(n − 1)ηt) −εβ(β + 1)(xn − f )V −β−2 1 + |∇ f + 2ηt x |2 and

|∇v|2 = ε2 V −2β |1 − β(xn − f )V −1 |2 + | − ∇ f + β(xn − f )V −1 (∇ f + 2ηt x )|2 . Using (2.7), (2.8), the fact that 0 ≤ β(xn − f )V −1 ≤ 1 and taking η = η( p, n, d, r, T ) > 0 small enough, we obtain vt − v ≥ εβV −β−1 −ηdr 2 + 2|∇ f |2 + 2 + 4ηt x · ∇ f − ( p − 1)K (d + ηr 2 T ) −d(K + 2(n − 1)ηT ) − (β + 1) 1 + |∇ f |2 + 4ηt x · ∇ f + 4η2 T 2 |x |2 ≥ εβV −β−1 (1 − β) − dpK

−η dr 2 + ( p − 1)r 2 T + 2d(n − 1)T + 4βr T + 4(β + 1)ηr 2 T 2 ≥ε and

β(1 − β) −β−1 V 2 |∇v|2 ≤ ε2 V −2β 1 + (2 + 2ηr T )2 ≤ 6ε2 V −2β .

Since pβ = β + 1, if ε = ε0 ( p) > 0 is small enough, we get vt − v ≥ |∇v| p in D × (0, T ). On D × {0}, using (2.10) and choosing c0 = ε0 /2, we get

1−β u ≤ c0 δ 1−β (x) ≤ c0 21−β xn − f (x ) ≤ v.

(2.12)

(2.13)

Moreover, (2.13) holds also on the lateral boundary part {x ∈ Rn ; |x | = r, f (x ) ≤ xn ≤ f (x ) + d} × (0, T ). On the surface {x ∈ Rn ; |x | ≤ r, xn = f (x )} ⊂ ∂, we have, for 0 < t < T , u(·, ·, t) = v(·, ·, t) = 0.


507

On the remaining part {x ∈ Rn ; |x | ≤ r, xn = f (x ) + d} × (0, T ), assuming in addition that η ≤ dr −2 T −1 , we obtain u ≤ c0 (2d)1−β ≤ ε0 d(d + ηr 2 T )−β ≤ v. By the comparison principle, we deduce (2.11). The proof is complete.

Based on Lemma 2.1, we can show the following lemma, which immediately implies Theorem 1.3. Lemma 2.3. Assume (2.5)–(2.6). Let u 0 be given by u 0 (x) = Cεk ϕ ε−1 |x |2 + (xn − ε)2 , where ϕ ∈ C ∞ ([0, ∞)) satisfies (1.20)), k = ( p − 2)/( p − 1), C ≥ C0 (n, p) > 0, and ε > 0 is sufficiently small. Then u 0 ∈ X + and u 0 satisfies (1.21). Proof. We divide the proof into two steps. Step 1. ∇u blows up for C ≥ C0 (n, p) and small ε. Since is smooth, there exists ε0 > 0 such that, for any ε ∈ (0, ε0 ), Bε := {(x , xn ); |x |2 + (xn − ε)2 < ε} ⊂ and ∂ Bε is tangent to ∂ at the origin. Moreover, u 0 ∈ C0∞ (Bε ). Consider the following problem: vt − v = |∇v| p , x ∈ B1 (0), t > 0, v(x, t) = 0, x ∈ ∂ B1 (0), t > 0, v(x, 0) = φ(x) := Cϕ(|x|), x ∈ B1 (0).

(2.14) (2.15) (2.16)

By [39, Thm.40.2] (see also [42, Prop.7.1]), there exists C0 = C0 (n, p) > 0 such that, if φ1 ≥ C0 , then T (φ) < ∞. Therefore, we have T (φ) < ∞ whenever C is greater than some C˜ 0 (n, p) > 0. We now use the scale invariance of the equation. Namely we consider the rescaled functions vε (x , xn , t) = εk v ε−1 x , ε−1 (xn − ε), ε−2 t . Then vε solves (2.14)–(2.16) in Bε with initial data u 0 . By the comparison principle, we get u ≥ vε in Bε × (0, T˜ ), where T˜ = min(T (u 0 ), Tε ) and Tε = ε2 T (φ). In view of u(0, 0, t) = vε (0, 0, t) = 0, we deduce that ∂u ∂vε − (0, 0, t) ≥ − (0, 0, t), 0 < t < T˜ . ∂ xn ∂ xn On the other hand, as a consequence of the maximum principle applied to ∇v (see e.g. [39, Prop.40.3]), we know that ∂v − , 0 < τ < T (φ). max ∇v(·, t)∞ = max ∇v(·, 0)∞ , max t∈[0,τ ] ∂ B0 (1)×[0,τ ] ∂ν Since v is radially symmetric it follows that lim sup t→Tε

hence T (u 0 ) ≤ Tε < ∞.

∂vε (0, 0, t) = ∞, ∂ xn

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Step 2. G BU S(u 0 ) ⊂ Bρ (x0 ) ∩ ∂ for small ε. Let h ≥ 0 be a smooth function in such that 1, x ∈ ∂ ∩ Bρ/4 (x0 ) h(x) = 0, x ∈ ∂ \ Bρ/2 (x0 ). Consider the elliptic problem −ψ = 1, ψ = h,

x ∈ , x ∈ ∂.

Then −(cψ) = c ≥ |∇(cψ)| p , x ∈ , − p/( p−1)

if 0 < c < ∇ψ L ∞ () small, we have

. According to the continuity of ψ in , if ε is sufficiently cψ ≥

c ≥ Cεk ≥ u 0 in Bε , 2

hence cψ ≥ u 0 in , due to ψ > 0. By the comparison principle, it follows that u ≤ cψ in × (0, T (u 0 )). Therefore,

0 ≤ −u ν ≤ −cψν ≤ M on ∂ \ Bρ/2 (x0 ) × (0, T (u 0 )). The conclusion now follows from Lemma 2.1. 3. Single-Point Gradient Blow-up for Locally Flat Symmetric Domains Note that, due to (1.11)–(1.12), we can assume that lies on only one side of the line y = 0, e.g. ⊂ R2+ = {(x, y) ∈ R2 : y > 0}. Proof of Theorem 1.1. (i) Let u 0 be as in Lemma 2.3. Then u 0 satisfies (1.15) and one easily checks that properties (1.13) and (1.14) are also verified. (ii) We divide the proof into three steps. Step 1. Preparations. By (1.10)–(1.12), there exist x1 ∈ (ρ/2, ρ) and y1 ∈ (0, 1) such that the rectangle (0, x1 ) × (0, 2y1 ) ⊂ . Denote D = (0, x1 ) × (0, y1 ). Let us fix a positive constant σ < 1/(2( p − 1)). We set F(u) = u q , q > 1 close to 1, d(y) = y −γ , γ = (1 − 2σ )(q − 1), c(x) = kx, k > 0 small,

(3.1) (3.2) (3.3)

and we consider the auxiliary function J (x, y, t) = u x + c(x)d(y)F(u) in D × (0, T ). Observe that, for each T < T , we have u ≤ C y in D × [0, T ]

(3.4)


509

for some C = C(T ) > 0. Since γ < q, we have in particular J ∈ C(D × [0, T )) ∩ C 2,1 (D × (0, T )).

(3.5)

Our aim is to use the maximum principle to prove that J ≤ 0 in D × (t0 , T )

(3.6)

with some t0 < T , for sufficiently small q − 1 and k. If (3.6) is proved, then integrating (3.6) over (0, x) for 0 < x < x1 , we get u ≤ C x −2/(q−1) y 1−2σ in D × (t0 , T ), with C = C(q, k) > 0. Then, since 1 − 2σ > ( p − 2)/( p − 1), the theorem will immediately follow from Lemma 2.2. Step 2. Derivation of a parabolic inequality for J . Let us compute Jt = u xt + cd F u t , J = u x + cd F u + cd F |∇u|2 + 2c d F u x + 2cd F u y + cd F. Then we obtain Jt − J = (|∇u| p )x + cd F |∇u| p − 2c d F u x − 2cd F u y − cd F |∇u|2 − cd F. Using u x = J − cd F, we write (|∇u| p )x = p|∇u| p−2 ∇u · ∇u x = p|∇u| p−2 ∇u · ∇ J − p|∇u| p−2 ∇u · ∇(cd F) = p|∇u| p−2 ∇u · ∇ J − p|∇u| p−2 (cd F |∇u|2 + u x c d F + u y cd F), hence (|∇u| p )x = p|∇u| p−2 ∇u · ∇ J − pcd F |∇u| p − pc d F|∇u| p−2 J + pcc d 2 F 2 |∇u| p−2 − pcd F|∇u| p−2 u y . We also have −2c d F u x = −2c d F J + 2cc d 2 F F . So we get Jt − J = a J + b · ∇ J −( p − 1)cd F |∇u| p + pcc d 2 F 2 |∇u| p−2 − pcd F|∇u| p−2 u y + 2cc d 2 F F − 2cd F u y − cd F |∇u|2 − cd F,

(3.7)

a = − pc d F|∇u| p−2 − 2c d F and b = p|∇u| p−2 ∇u.

(3.8)

where

We note that, in view of (3.4) and γ < q − 1, a is bounded in D × (0, T ) for each T < T .

(3.9)

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Y. Li, Ph. Souplet

Let P J = Jt − J − a J − b · ∇ J. From (3.1)–(3.3) and (3.7), we obtain PJ |∇u| p u q |∇u| p−2 u q−1 |∇u| p−1 ≤ −( p − 1)q + kp + 2kq + pγ cd F u yγ y yγ 2 2qγ |∇u| γ (γ + 1) |∇u| + . − q(q − 1) 2 − y u u y2

(3.10)

By Young’s inequality, we have 2qγ |∇u| qγ 2 1 |∇u|2 ≤ q(q − 1) 2 + y u u q − 1 y2 and pγ

σγ p2 γ |∇u| p−1 ≤ 2+ |∇u|2 p−2 , y 2y 2σ

hence 2qγ |∇u| |∇u|2 γ (γ + 1) − q(q − 1) 2 − ≤ y u u y2 and |∇u| p−1 |∇u| p + pγ ≤ − ( p − 1)q u y

qγ 2 − γ (γ + 1) q −1

p2 γ u|∇u| p−2 − ( p − 1)q 2σ

1 2σ γ =− 2 2 y y (3.11)

|∇u| p σ γ + 2. u 2y (3.12)

On the other hand, by (1.8), we have |∇u| ≤ C y −1/( p−1) and u ≤ C y ( p−2)/( p−1) in D × [0, T ), where C = C( p, , u 0 C 1 ) > 0. Consequently, 1 u q |∇u| p−2 (q−1) p−2 p−1 −γ = C q+ p−2 y (q−1)(2σ − p−1 ) , ≤ C q+ p−2 y γ y 1 u q−1 (q−1) p−2 p−1 −γ = C q−1 y (q−1)(2σ − p−1 ) , ≤ C q−1 y γ y

u|∇u| p−2 ≤ C q+ p−1 . Combining (3.10)–(3.15), we obtain 1 PJ 3σ γ 1 (q−1)(2σ − p−1 )+2 q+ p−2 q−1 ≤ k pC y + 2qC − cd F 2 y2 2 p p γ q+ p−1 |∇u| C . + − ( p − 1)q 2σ u

(3.13) (3.14) (3.15)

(3.16)


511

Now, we may choose q > 1 close enough to 1, thus γ = (1 − 2σ )(q − 1) small enough, such that p 2 γ q+ p−1 1 C + 2 > 1. (3.17) < p − 1 and (q − 1) 2σ − 2σ p−1 Next recalling that y1 < 1 and assuming k > 0 to be so small that k( pC q+ p−2 + 2qC q−1 ) −

3σ γ ≤ 0, 2

we then deduce from (3.16) that P J ≤ 0 in D × (0, T ).

(3.18)

(Note, at this point, that q is fixed and that any small k > 0 is allowable.) Step 3. Initial and boundary conditions for J . Since u 0 ≥ 0 in , by the maximum principle we have u > 0 in × (0, T ). So by Hopf’s lemma, we get u ν < 0 on ∂ × (0, T ). Therefore by (1.12), we have u x = νx u ν ≤ 0 on {(x, y) ∈ ∂; x > 0} × (0, T ). By symmetry of u 0 , we also have u x = 0 on ( ∩ {x = 0}) × (0, T ). Since w = u x satisfies wt − w = p|∇u| p−2 ∇u · ∇w in + × (0, T ),

(3.19)

then the maximum principle implies that u x < 0 in + × (0, T ).

(3.20)

By (1.15), there exists a constant M such that |∇u| ≤ M in (\{(−ρ/2, ρ/2) × (0, y1 /4)}) × (0, T ). Consequently, by parabolic estimates,

u can be extended to a function u ∈ C 2,1 (Q) with Q = \{(−x2 , x2 ) × (0, y1 /2)} × (0, T ] and ρ/2 < x2 < x1 . Therefore by (3.19), (3.20), (1.5) and Hopf’s Lemma, there exist constants c1 , c2 > 0 such that u x ≤ −c1 y on {x1 } × (0, y1 ) × (T /2, T ), u x ≤ −c1 x on (0, x1 ) × {y1 } × (T /2, T ), u ≤ c2 y on {x1 } × (0, y1 ) × (0, T ). Hence on the lateral boundary ∂ D × (T /2, T ), we have J (x, 0, t) = 0 on (0, x1 ) × {0} × (T /2, T ), J (0, y, t) = 0 on {0} × (0, y1 ) × (T /2, T ), −γ

(3.21) (3.22)

q

J (x, y1 , t) ≤ −c1 x + kx y1 u 0 ∞ ≤ 0 on (0, x1 ) × {y1 } × (T /2, T ), (3.23) q

J (x1 , y, t) ≤ −c1 y + kx1 c2 y q−γ ≤ 0 on {x1 } × (0, y1 ) × (T /2, T ), if k is sufficiently small since q > γ + 1.

(3.24)

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Y. Li, Ph. Souplet

We now claim that, for a fixed t0 ∈ (T /2, 3T /4), there exists c3 > 0 such that u x (x, y, t0 ) ≤ −c3 x y in D.

(3.25)

This follows from a parabolic version of “Serrin’s corner Lemma” (cf. [18, Lemma S and p. 238-242] for a general result in the elliptic case). Here we give a simple proof based on the following comparison argument. Let K = sup×(T /2,3T /4) p|∇u| p−1 and 0 < < min(x1 , y1 ). Fix a nontrivial smooth function φ ≥ 0 such that w = u x satisfies − w(x, y, T /2) ≥ φ(x)φ(y), (x, y) ∈ (0, )2 .

(3.26)

Let v be the (global) solution of vt − vx x = −K |vx |, x ∈ (0, ), t > T /2, v(0, t) = v(, t) = 0, t > T /2, v(x, T /2) = φ(x), x ∈ (0, ). By Hopf’s Lemma, we have v(x, t0 ) ≥ cx on (0, /2) for some c > 0. Let then z(x, y, t) = v(x, t)v(y, t). An immediate computation yields z t − z = −K |z x | − K |z y |, (x, y) ∈ (0, )2 , t > T /2.

(3.27)

By (3.19), (3.27) and the comparison principle, we deduce that −w(x, y, t0 ) ≥ v(x, t0 )v(y, t0 ) ≥ c2 x y in (0, /2)2 . By a further use of Hopf’s Lemma, one easily extends this inequality to the whole region D to yield (3.25). Now (3.25) implies q

J (x, y, t0 ) ≤ −c3 x y + kxc2 y q−γ ≤ 0 in D

(3.28)

if k is sufficiently small since q > γ + 1. Then (3.6) follows from (3.18), (3.21)–(3.24), (3.28) and the maximum principle. Note that the use of the maximum principle is justified in view of (3.5) (or, alternatively, of the fact that a < 0). The proof is complete. 4. Single-Point Gradient Blow-up for Disks The main idea of the proof is similar to that of Theorem 1.1. However since we use polar coordinates for disks, there are significant differences in the computation. Therefore we prefer to give enough details for the convenience of the reader. Proof of Theorem 1.2. (i) It is similar to that of Theorem 1.1(i). (ii) We also divide the proof into three steps. Step 1. Preparations. Let x = r cos θ, y = r sin θ . We have sin θ ∂ f ∂ f cos θ ∂ f ∂f ∇ f = ( f x , f y ) = cos θ − , sin θ + , ∂r r ∂θ ∂r r ∂θ 1 1 f = f x x + f yy = frr + fr + 2 f θθ , r r 1 ∇ f · ∇g = f x gx + f y g y = fr gr + 2 f θ gθ . r

(4.1) (4.2) (4.3)


513

Differentiating Eq. (1.5) with respect to θ , we get wt − w = p|∇u| p−2 (u r wr +

1 u θ wθ ) = p|∇u| p−2 ∇u · ∇w, r2

(4.4)

where w = u θ . It follows from (1.16) that u(·, t) is symmetric with respect to the line y = 0 for 0 < t < T . Therefore u θ (·, t) = 0 on the line y = 0. In view of (1.17), we have u θ (·, t) = 0 on the parabolic boundary of B+ × (0, T ) and it follows from the strong maximum principle that u θ < 0 in B+ × (0, T ).

(4.5)

Since G BU S(u 0 ) is compact, by (1.18) and (4.5), there exists 0 < θ0 < π such that G BU S(u 0 ) does not contain the arc {(r, θ ); r = 1, θ0 < θ < 2π − θ0 }.

(4.6)

Denote D = (r0 , 1) × (0, θ0 ) with 1/2 < r0 < 1. Fix a positive constant σ < 1/ (2( p − 1)) and denote F(u) = u q , q > 1 close to 1, d(r ) = (1 − r )−γ , γ = (1 − 2σ )(q − 1), c(θ ) = kθ, 0 < k < 1 small,

(4.7) (4.8) (4.9)

and J (r, θ, t) = u θ + c(θ )d(r )F(u) in D × (0, T ). Observe that, for each T < T , we have u ≤ C(1 − r ) in D × [0, T ]

(4.10)

for some C = C(T ) > 0. Since γ < q, we have in particular J ∈ C(D × [0, T )) ∩ C 2,1 (D × (0, T )). Our aim is to prove that J ≤ 0 in D × (t0 , T ),

(4.11)

with some t0 < T , for sufficiently small q − 1, k and 1 − r0 . If inequality (4.11) is proved, then integrating (4.11) over (0, θ ) for 0 < θ < θ0 , we get u ≤ Cθ −2/(q−1) (1 − r )1−2σ in D × (t0 , T ), with C = C(q, k) > 0. Then, since 1 − 2σ > ( p − 2)/( p − 1), the theorem will immediately follow from Lemma 2.2. Step 2. Derivation of a parabolic inequality for J . Let LJ = Jt − J − r −2 a J − b · ∇ J,

514

Y. Li, Ph. Souplet

where a, b are given by (3.8). Clearly, in view of (4.10) and γ < q − 1, r −2 a is bounded in D × (0, T ) for each T < T . Using (4.1)–(4.3) and arguing similarly as in the proof of (3.7), we obtain 1 pcc d 2 F 2 |∇u| p−2 − pcd F|∇u| p−2 u r r2 1 1 (4.12) + 2 2cc d 2 F F − 2cd F u r − cd F |∇u|2 − c d + d F. r r

LJ = −( p − 1)cd F |∇u| p +

From (4.7)–(4.9) and (4.12), we deduce LJ |∇u| p u q |∇u| p−2 u q−1 |∇u| p−1 ≤ −( p − 1)q + 4kp + 8kq + pγ cd F u (1 − r )γ 1−r (1 − r )γ 2 |∇u| 2qγ |∇u| γ (γ + 1) 2γ − q(q − 1) 2 − , (4.13) + + 1−r u u (1 − r )2 1 − r where we also used r > 1/2. Using estimates (3.11)–(3.15) with y replaced by 1 − r , we get 1 LJ 1 (q−1)(2σ − p−1 )+2 ≤ 4 pC q+ p−2 + 2qC q−1 (1 − r ) − σγ cd F (1 − r )2 2 σ γ p γ q+ p−1 |∇u| p C + 2(1 − r ) − + − ( p − 1)q , (4.14) 2σ u 2 (1 − r )2 where we used k < 1 in the first term. Now, fix again q > 1 close enough to 1, so that (3.17) holds, and next choose r0 ∈ (1/2, 1) so close to 1 that σ 4 pC q+ p−2 + 4qC q−1 (1 − r0 ) − σ γ < 0 and 2(1 − r0 ) − < 0. 2 Then by (4.14) and (3.17) we finally get LJ ≤ 0 in D × (0, T ).

(4.15)

(Note, at this point, that q, r0 are fixed and that any k ∈ (0, 1) is allowable.) Step 3. Initial and boundary conditions for J . Similarly as in Step 3 of the proof of Theorem 1.1, there exist constants c1 , c2 > 0 such that u θ ≤ −c1 (1 − r ) on (r0 , 1) × {θ0 } × (T /2, T ), u θ ≤ −c1 θ on {r0 } × (0, θ0 ) × (T /2, T ), u ≤ c2 (1 − r ) on (r0 , 1) × {θ0 } × (0, T ).

(4.16)

Hence on the parabolic boundary ∂ D × (0, T ), we have J (1, θ, t) = 0 on {1} × (0, θ0 ) × (0, T ), (4.17) J (r, 0, t) = 0 on (0, r0 ) × {0} × (0, T ), (4.18) q −γ J (r0 , θ, t) ≤ −c1 θ + kθ (1 − r0 ) u 0 ∞ ≤ 0 on {r0 } × (0, θ0 ) × (T /2, T ), (4.19) q q−γ J (r, θ0 , t) ≤ −c1 (1 − r ) + kθ0 c2 (1 − r ) ≤ 0 on (r0 , 1) × {θ0 } × (T /2, T ). (4.20)


515

We now claim that, for a fixed t0 ∈ (T /2, 3T /4), there exists c3 > 0 such that u θ (r, θ, t0 ) ≤ −c3 (1 − r )θ in D.

(4.21)

To prove (4.21), we modify the proof of (3.25) as follows. Let K = r0−1 1 +

sup

×(T /2,3T /4)

p|∇u| p−1 .

(4.22)

Fix a nontrivial smooth function φ ≥ 0 on [r0 , 1], with supp(φ) ⊂ (r0 , 1), and a smooth function ψ on [0, θ0 ] such that ψ = 0 on [0, θ0 /2], ψ(θ0 ) = 1 and ψ , ψ ≥ 0. Let v, V be the respective (global) solutions of vt − vrr = −K |vr |, r ∈ (r0 , 1), t > T /2, v(r0 , t) = v(1, t) = 0, t > T /2, v(r, T /2) = φ(r ), r ∈ [r0 , 1], and Vt − Vθθ = −K Vθ , θ ∈ (0, θ0 ), t > T /2, V (0, t) = 0, V (θ0 , t) = 1, t > T /2, V (θ, T /2) = ψ(θ ), θ ∈ [0, θ0 ].

(4.23)

By the maximum principle we have 0 ≤ V ≤ 1 and Vθ ≥ 0. By (4.23), we deduce that Vθθ (θ, t) ≥ 0 for θ ∈ {0, θ0 } and t > T /2. Since ψ ≥ 0, it follows from the maximum principle that Vθθ ≥ 0. Moreover, by Hopf’s Lemma, we have v(r, t0 ) ≥ c(1 − r ) in (r1 , 1),

V (θ, t0 ) ≥ cθ in (0, θ0 ),

(4.24)

with r1 = (1 + r0 )/2 and some c > 0. Let then z(r, θ, t) = v(r, t)V (θ, t). We compute z t − zrr − r −2 z θθ = V (vt − vrr ) + v(Vt − r −2 Vθθ ) ≤ V (vt − vrr ) + v(Vt − Vθθ ) = −K |zr | − K |z θ |, where we used Vθ , Vθθ ≥ 0, hence z t − z ≤ −(K − r0−1 )|zr | − K |z θ |, (r, θ ) ∈ (r0 , 1) × (0, θ0 ), t > T /2. (4.25) Next, for µ > 0 sufficiently small, due to (4.5) and supp(φ) ⊂ (r0 , 1), we have −u θ (·, T /2) ≥ µφ(r )ψ(θ ) = µz(r, θ, T /2) in D. Moreover, for µ possibly smaller, using (4.16), we see that −u θ (r, θ0 , t) ≥ c1 (1 − r ) ≥ µv(r, t) = µz(r, θ0 , t), r ∈ (r0 , 1), t ∈ (T /2, 3T /4). Since z = 0 on the rest of the lateral boundary (i.e., for r ∈ {r0 , 1} or θ = 0), it follows from (4.4), (4.25), (4.22), the comparison principle and (4.24) that −u θ (x, y, t0 ) ≥ µv(r, t0 )V (θ, t0 ) ≥ c2 (1 − r )θ in (r1 , 1) × (0, θ0 ). By a further use of Hopf’s Lemma, one easily extends this inequality to the whole region D to yield (4.21). Finally, (4.21) implies q

J (r, θ, t0 ) ≤ −c3 θ (1 − r ) + kθ c2 (1 − r )q−γ ≤ 0 in D

(4.26)

if k is sufficiently small since q > γ + 1. Then (4.11) follows from (4.15), (4.17)–(4.20), (4.26) and the maximum principle. The proof is complete.

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Acknowledgements. This research was performed while Li Yuxiang was a visiting scholar at the Laboratoire Analyse Géométrie et Applications in Université Paris-Nord. He is grateful to this institution for its hospitality and stimulating atmosphere.

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28. Krug, J., Spohn, H.: Universality classes for deterministic surface growth. Phys. Rev. A 38, 4271– 4283 (1988) 29. Kutev, N.: Gradient blow-ups and global solvability after the blow-up time for nonlinear parabolic equations. In: Evolution Equations, Control Theory, and Biomathematics (Han sur Lesse, 1991). Lecture Notes in Pure and Appl. Math. 155, New York:Dekker, pp.301–306, 1994 30. Laurençot, Ph.: Convergence to steady states for a one-dimensional viscous Hamilton-Jacobi equation with Dirichlet boundary conditions. Pacific J. Math. 230(2), 347–364 (2007) 31. Ladyzhenskaya, O., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Providence, RI: Amer. Math. Soc., 1967 32. Laurençot, Ph., Souplet, Ph.: On the growth of mass for a viscous Hamilton-Jacobi equation. J. Anal. Math. 89, 367–383 (2003) 33. Li, Y.-X.: Stabilization towards the steady-state for a viscous Hamilton-Jacobi equation. Comm. Pure Appl. Anal. 8(6), 1917–1924 (2009) 34. Lieberman, G.M.: The first initial-boundary value problem for quasilinear second order parabolic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 13(3), 347–387 (1986) 35. Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations. Research Notes in Mathematics, 69. Advanced Publishing Program. Boston, MA.-London:Pitman, 1982 36. Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications, 16. Basel, Birkhauser Verlag, 1995 37. Matano, H., Merle, F.: On nonexistence of type II blowup for a supercritical nonlinear heat equation. Comm. Pure Appl. Math. 57(11), 1494–1541 (2004) 38. Merle, F., Zaag, H.: Stability of the blow-up profile for equations of the type u = u + |u| p−1 u. Duke Math. J. 86, 143–195 (1997) 39. Quittner, P., Souplet, Ph.: Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basel Textbooks, Basel, Birkhäuser Verlag, 2007 40. Souplet, Ph.: Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions. Diff. Int. Eq. 15(2), 237–256 (2002) 41. Souplet, Ph., Vázquez, J.L.: Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Disc. Cont. Dyn. Syst. 14(1), 221–234 (2006) 42. Souplet, Ph., Zhang, Q.S.: Global solutions of inhomogeneous Hamilton-Jacobi equations. J. Anal. Math. 99, 355–396 (2006) 43. Tersenov, Al., Tersenov, Ar.: Global solvability for a class of quasilinear parabolic problems. Indiana Univ. Math. J. 50, 1899–1913 (2001) 44. Velázquez, J.J.L.: Estimates on the (n − 1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J. 42, 445–476 (1993) 45. Weissler, F.B.: Single point blow-up for a semilinear initial value problem. J. Diff. Eq. 55, 204–224 (1984) 46. Zhang, Y.-C.: Singular dynamic interface equation from complex directed polymers. J. Phys. I France 2, 2175–2180 (1992) Communicated by P. Constantin

Commun. Math. Phys. 293, 519–543 (2010) Digital Object Identifier (DOI) 10.1007/s00220-009-0916-z

Communications in


Entire Solutions of Hydrodynamical Equations with Exponential Dissipation Claude Bardos1 , Uriel Frisch2 , Walter Pauls3 , Samriddhi Sankar Ray4 , Edriss S. Titi5,6 1 Université Denis Diderot and Laboratoire J.L. Lions, Université Pierre et Marie Curie,

Paris, France. E-mail: [email protected]

2 UNS, CNRS, Laboratoire Cassiopée, OCA, BP 4229, 06304 Nice cedex 4,

France. E-mail: [email protected]

3 Max Planck Institute for Dynamics and Self-Organization,

Göttingen, Germany. E-mail: [email protected]

4 Center for Condensed Matter Theory, Department of Physics, Indian Institute of Science,

Bangalore, India. E-mail: [email protected]

5 Department of Mathematics and Department of Mechanical and Aerospace Engineering,

University of Irvine, Irvine, CA 92697-3875, USA. E-mail: [email protected]

6 Department of Computer Science and Applied Mathematics,

Weizmann Institute of Science, Rehovot 76100, Israel Received: 31 December 2008 / Accepted: 18 June 2009 Published online: 10 September 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: We consider a modification of the three-dimensional Navier–Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e|k|/kd at high wavenumbers |k|. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e−C(k/kd ) ln(|k|/kd ) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C = 1/ ln 2. The same behavior with a universal constant C is conjectured for the Navier–Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier–Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed. 1. Introduction More than a quarter of a millenium after the introduction by Leonhard Euler of the equations of incompressible fluid dynamics, the question of their well-posedness in three dimensions (3D) with sufficiently smooth initial data is still moot [1–4] (see also many papers in [5] and references therein). Even more vexing is the fact that switching to viscous flow for the solution of the Navier–Stokes equations (NSE) barely improves the situation in 3D [6–10]. Finite-time blow up of the solution to the NSE can thus not be ruled out, but there is no numerical evidence that this happens. In contrast, there is strong numerical evidence that for analytic spatially periodic initial data both the 3D Euler and NSE have complex space singularities. Indeed, when

520

C. Bardos, U. Frisch, W. Pauls, S. S. Ray, E. S. Titi

such equations are solved by (pseudo-)spectral techniques the Fourier transforms of the solution display an exponential decrease at high wavenumbers, which is a signature of complex singularities [11]. This behavior was already conjectured by von Neumann [12] who pointed out on p. 461 that the solution should be analytic with an exponentially decreasing spectrum. Recently Li and Sinai used a Renormalization Group method to prove that for certain complex-valued initial data the 3D NSE display finite-time blow up in the real domain (and, as a trivial corollary, also in the complex domain) [13]. For some PDEs in lower space dimensions explicit information about the position and type of complex singularities may be available. For example, complex singularities can sometimes be related to poles of elliptic functions in connection with the reaction diffusion equation [14] and 2D incompressible Euler equations in Lagrangian coordinates [15]. The best understood case is that of the 1D Burgers equation with ordinary (Laplacian) dissipation:1 its singularities are poles located at the zeroes of the solutions of the heat equation to which it can be mapped by using the Hopf–Cole transformation (see, e.g., [16,17] and references therein). We now return to the 3D NSE with real analytic data. It is known that blow up in the real domain can be avoided altogether by modifying the dissipative operator, whose Fourier-space symbol is µ|k|2 , to a higher power of the Laplacian with symbol µ|k|2α (α > 5/4) [6,18]. The numerical evidence is however that complex singularities cannot be avoided by this “hyperviscous” procedure, frequently used in geophysical simulations (see, for example, [19]). Actually, we are unaware of any instance of a nonlinear space-time PDE, with the property that the Cauchy problem is well posed in the complex space domain for at least some time and which is guaranteed never to have any complex-space singularities at a finite distance from the real domain. In other words the solution stays or becomes entire for all t > 0. Here we shall show that solutions of the Cauchy problem are entire for a fairly large class of pseudo-differential nonlinear equations, encompassing variants of the 3D NSE, which possess “exponential dissipation”, that is dissipation with a symbol growing exponentially as e|k|/kd with the ratio of the wavenumber |k| to a reference wavenumber kd . The paper is organized as follows. In Sect. 2 we consider the forced 3D incompressible NSE in a periodic domain with exponential dissipation. The initial conditions are assumed just to have finite energy. The main theorem is established using classes of analytic functions whose norms contain exponentially growing weights in the Fourier space [20,21]. In Sect. 3 we show that the Fourier transform of the solution decays at ˜ for any C < 1/(2 ln 2). Here, k˜ := |k|/kd high wavenumbers faster than exp (−C k˜ ln k) is the nondimensionalised wavenumber. In Sect. 4 we briefly present extensions of the result to other instances: different space dimensions and dissipation rates, problems formulated in the whole space and on a sphere, and different equations. In Sect. 5 we then turn to the 1D Burgers equation with a dissipation growing exponentially at high wavenumbers, for which the same bounds hold as for the 3D Navier–Stokes case. However in the Burgers case, simple heuristic considerations (Sect. 5.1) and very accurate numerical simulations performed by two different techniques (Sects. 5.2 and 5.3), indicate that ˜ We observe that the leading-order asymptotic decay is precisely exp ((−1/ ln 2)k˜ ln k). the heuristic approach, which involves a dominant balance argument applied in spatial Fourier space, is also applicable to the 3D Navier–Stokes case with exactly the same 1 The case of the Burgers equation with modified dissipation will be considered in Sect. 5.

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

521

prediction regarding the asymptotic decay. In the concluding Sect. 6 we discuss open problems and a possible application. 2. Proof that the Solution is Entire We consider the following 3D spatially periodic Navier–Stokes equations with an exponential dissipation (expNSE) ∂u + u · ∇u = −∇ p − µDu + f, ∂t u(x, 0) = u 0 (x).

∇ · u = 0,

(2.1) (2.2)

Here, D is the (pseudo-differential) operator whose Fourier space symbol is e2σ |k| , that is a dissipation rate varying exponentially with the wavenumber |k|, u 0 is the initial condition, f is a prescribed driving force and µ and σ are prescribed positive coefficients. The problem is formulated in a periodic domain (for simplicity of notation we take = [0, 2π ]3 ). The driving force is assumed to be a divergence-free trigonometric polynomial in the spatial coordinates. For technical convenience we use σ in the statements and proofs of mathematical results, while the use of the reference wavenumber kd = 1/(2σ ) is preferred when discussing the results. The initial condition is taken to be a divergence-free periodic vector field with a finite L 2 norm (finite energy). As usual the problem is rewritten as an abstract ordinary differential equation in a suitable function space, namely du 1/2 + µe2σ A u + B(u, u) = f ; dt u(0) = u 0 ;

(2.3) (2.4)

where A := −∇ 2 and B(u, u) is a suitable quadratic form which takes into account the nonlinear term, the pressure term and the incompressibility constraint (see, e.g. [6,7,9]). Note that the Fourier symbol of A1/2 is |k|. The problem is formulated in the space H := {ϕ ∈ (L 2 ())3 : ϕ is periodic, 1/2 ϕ d x = 0, ∇ · ϕ = 0}. Here, for any λ ≥ 0, the Fourier symbol of the operator eλA is given by eλ|k| , where k ∈ Z3 \{(0, 0, 0)}. To prove the entire character, with respect to the spatial variables, of the solution u(t) of expNSE for t > 0, it suffices to show that its Fourier coefficients decrease faster than exponentially with the wavenumber |k|. This will be done by showing that, for any 1/2 λ > 0, the L 2 norm of eλA u, the solution with an exponential weight in Fourier space, is finite.As usual, we here denote the L 2 norm of a real space-periodic function f by m 2 2 | f | := [0,2π ]3 | f (x)| d x. Moreover, H will be the usual L Sobolev space of index m (i.e., functions which have up to m space derivatives in L 2 ). The main result (Theorem 2.1) will make use of the following proposition which was inspired by [20] (see also [21]) Proposition 2.1. Let α ≥ 0, β > 0, κ > 3 and ϕ ∈ dom (e(α+β)A ). Then 1/2 2−a(κ) 1/2 (α+β)A1/2 a(κ) αA B(ϕ, ϕ) ≤ CA (lκ (β))a(κ) eα A ϕ ϕ , e e 1/2

(2.5)

522


where CA is a universal constant and κ −βx

lκ (β) := sup x e 0≤x 0 is a universal constant. By using (2.10), (2.13) and the fact that |W | = |w|, we obtain 1/2 1 1 1/2 2 3 1/2 2 α A1/2 B(ϕ, ϕ), w) ≤ CA eα A ϕ Aeα A ϕ A 2 eα A ϕ |w|. (2.14) (e And by using the interpolation inequality between L 2 and H κ , where κ > 3, we obtain2 1/2 2−a(κ) κ α A1/2 2 α A1/2 a(κ) B(ϕ, ϕ), w) ≤ CA eα A ϕ ϕ |w|. (2.15) (e A e Now, to obtain the inequality in Proposition 2.1, we just need to estimate the L(H ) operator norm κ κ κ 2 −β A1/2 κ −βx e ≤ sup x e = e−κ = lκ (β). (2.16) A L(H ) β 0≤x 0. This is why in estimating L 2 norms of the solution with exponential weights we have to stay clear of t = 0. Theorem 2.1. Let u 0 ∈ H , fix T > 0 and let f (t) = f (. , t) be an entire function with respect to the spatial variable x. Then for every n = 0, 1, 2 . . . there exist constants Cn , C¯ n , K n and K¯ n which depend on |u 0 |, µ, T, σ and on the norm T 2 (n−1)σ A1/2 f (s) ds, (2.17) e 0

moreover there exist integers pn , qn ≥ 1 such that 2 Kn nσ A1/2 u(t) ≤ p + Cn , for all t ∈ (0, T ] e t n and

T

|e(n+1)σ A

1/2

u(s)|2 ds ≤

t

K¯ n ¯ + Cn , for all t ∈ (0, T ], t qn

(2.18)

(2.19)

where u(t) is the solution of (2.3)–(2.4). Corollary 2.1. Let u 0 ∈ H, T > 0 and let f (x, t) be an entire function with respect to 2 T 1/2 the spatial variable x such that for every M ≥ 0 we have 0 e Mσ A f (s) ds < ∞. Then, the solution u(t) of (2.3)–(2.4) is an entire function with respect to the spatial variable for all t ∈ (0, T ], and satisfies the estimates (2.18) and (2.19) in Theorem 2.1 for any n = 1, 2, . . .. 2 The simplest formulation is obtained for κ = 5 but the optimization of the bound for the law of decay in Sect. 3 requires using arbitrary κ.

524


Proof of Corollary 2.1. Consider the Fourier series representation ei k·x u(k, ˆ t). u(x, t) =

(2.20)

k

From (2.18) and Parseval’s theorem, we have, for any n = 0, 1, 2 . . ., 2 ˆ t) < ∞, e2nσ |k| u(k,

(2.21)

k

for t > 0. In (2.20) we change x to a complex location z = x + i y and obtain u(x + i y, t) = ei k·(x+i y) u(k, ˆ t)

(2.22)

k

=

ei k·x e−|k| e−k·y+|k| u(k, ˆ t) .

(2.23)

k

For any n = 1, 2, . . ., the series (2.23) of complex analytic functions converges uniformly in the strip |y| + 1 ≤ σ n. This is because the sum in (2.23) is shown to be bounded, for any y, by use of the Cauchy–Schwarz inequality applied to the two bracketed expressions and use of (2.21) with |y| + 1 ≤ σ n. Hence the Fourier series representation converges in the whole complex domain. This concludes the proof of the entire character of the solution with respect to the spatial variables. Remark. This corollary just expresses the most obvious part of the Paley–Wiener Theorem. Proof of Theorem 2.1. The proof of the theorem proceeds by mathematical induction. Step n = 0. We prove the statement of the theorem for n = 0. We take the inner product of (2.3) with u and use the fact that (B(u, u), u) = 0 to obtain (when there is no ambiguity we shall henceforth frequently denote u(t) by u) 1 d 1/2 2 1/2 1/2 |u|2 + µ eσ A u = ( f, u) = (e−σ A f, eσ A u) (2.24) 2 dt 1/2 1/2 ≤ e−σ A f eσ A u |e−σ A f |2 µ σ A1/2 2 ≤ + |e u| , 2µ 2 1/2

(2.25)

where Young’s inequality has been used to obtain the third line. Therefore −σ A1/2 f |2 d σ A1/2 2 |e 2 |u| + µ e . u ≤ dt µ Integrating the above from 0 to T , we obtain T 2 1 T σ A1/2 2 e |u(t)|2 + µ u(s) ds ≤ C := |u | + 0 0 µ 0 0

2 −σ A1/2 f (s) ds. e

(2.26)

(2.27)

Hence |u(t)|2 ≤ C0 ,

(2.28)


and

T

µ 0

2 σ A1/2 u(s) ds ≤ C0 . e

525

(2.29)

From (2.28) and (2.29) we obtain (2.18) and (2.19) for the case n = 0. Here C0 is given by (2.27), K 0 = 0, K¯ 0 = 0 and C¯ 0 = C0 . Notice that since K 0 = K¯ 0 = 0 there is no need to determine the integers p0 and q0 ; however, for the sake of initializing the induction process we chose p0 = q0 = 1. Step n → n + 1. Assume that (2.18) and (2.19) are true up to n = m and we would like 1/2 to prove them for n = m + 1. Let us take the inner product of (2.3) with e2(m+1)σ A u and obtain 1 d (m+1)σ A1/2 2 1/2 2 u + µ e(m+2)σ A u e 2 dt 1/2 1/2 ≤ ( f, e2(m+1)σ A u) + (B(u, u), e2(m+1)σ A u) 1/2 1/2 1/2 1/2 ≤ emσ A f e(m+2)σ A u + emσ A B(u, u) e(m+2)σ A u . Now we use Proposition 2.1 to majorize the previous expression by 1/2 1/2 ≤ emσ A f e(m+2)σ A u + CA (lκ (β))a(κ) 1/2 2−a(κ) (m+2)σ A1/2 1+a(κ) × emσ A u u . e By Young’s inequality we have 1/2 2−a(κ) (m+2)σ A1/2 1+a(κ) CA [lκ (β)]a(κ) emσ A u u e 2κ + 5 d(κ) 2κ − 5 −d(κ) 2f(κ) 2a(κ)f(κ) µ ≤ CA (lκ (β)) 4κ κ mσ A1/2 2e(κ) µ (m+2)σ A1/2 2 × e u + e u . 4 It follows that 2 d (m+1)σ A1/2 2 1/2 2 1/2 2 u + µ e(m+2)σ A u ≤ emσ A f e dt µ 2κ − 5 −d(κ) 2f(κ) µ + CA (lκ (β))2a(κ)f(κ) 2κ 2κ + 5 d(κ) mσ A1/2 2e(κ) × u . e κ

(2.30)

(2.31)

(2.32)

Now we integrate this inequality on the interval (s, t) ⊂ (0, T ), obtaining t 2 (m+2)σ A1/2 2 (m+1)σ A1/2 u(t) + µ u(s ) ds e e s t 2 2 t 2 1/2 mσ A1/2 mσ A1/2 2e(κ) ≤ e(m+1)σ A u(s) + f (s ) ds + C u(s ) ds e e µ s s t 2 2 T 2 1/2 mσ A1/2 mσ A1/2 2e(κ) ≤ e(m+1)σ A u(s) + f (s ) ds + C u(s ) ds , e e µ 0 s (2.33)

526


where we have set for brevity 2κ − 5 −d(κ) 2f(κ) µ CA (lκ (β))2a(κ)f(κ) 2κ

C = C(µ, β, κ) :=

2κ + 5 κ

d(κ) ,

(2.34)

and where lκ (β) is given by (2.6). Now we come to the point where we use the actual induction assumptions. We use (2.18) and the midpoint convexity to estimate the integrand in the last integral: e(κ) e(κ) 2e(κ) K m mσ A1/2 f(κ) K m u(t) ≤ + Cm ≤2 + 2f(κ) Cme(κ) . (2.35) e t pm t pm Whence it follows that t mσ A1/2 2e(κ) C u(s ) ds e s

≤2

f(κ)

1 C K e(κ) e(κ) pm − 1 m

1

1

− e(κ) p −1 m s e(κ) pm −1 t 1 K e(κ) +2f(κ) C Cme(κ) (t − s) ≤ 2f(κ) C e(κ) pm − 1 m 1 × e(κ) p −1 + 2f(κ) C Cme(κ) (t − s). (2.36) m s 2 t 1/2 Discarding the positive term µ s e(m+2)σ A u(s ) ds in (2.33), we obtain from (2.33) and (2.36) 2 2 2 T 2 1/2 mσ A1/2 (m+1)σ A1/2 u(t) ≤ e(m+1)σ A u(s) + f (s ) ds e e µ 0 1 1 +2f(κ) C (2.37) K me(κ) e(κ) p −1 + 2f(κ) C Cme(κ) (t − s). m e(κ) pm − 1 s Integrating this inequality with respect to s over (t/2, t) we get 2 2 t 2 2 2 T mσ A1/2 (m+1)σ A1/2 (m+1)σ A1/2 u(t) ≤ u(s) ds + f (s ) ds e e e t t/2 µ 0 e(κ) pm −1 e(κ) 2 t 2f(κ) C K m (2.38) + + 2f(κ) C Cme(κ) . (e(κ) pm − 1)(e(κ) pm − 2) t 4 Note that pm ≥ 1 implies that e(κ) pm − 2 > 0.

(2.39)

By using (2.19), we have qm +1 2 2 2 2 2 T mσ A1/2 (m+1)σ A1/2 + u(t) ≤ K + C f (s ) ds e e m m t t µ 0 e(κ) pm −1 e(κ) 2 T 2f(κ) C K m + + 2f(κ) C Cme(κ) . (e(κ) pm − 1)(e(κ) pm − 2) t 4

(2.40)


527

From this relation follows that (2.18) holds for m + 1 with pm+1 = max {e(κ) pm − 1, qm + 1} , qm+1 = max {e(κ) pm − 1, qm + 1} .

(2.41) (2.42)

By the induction assumption we use (2.33) to estimate

2 2qm K¯ m ¯ (m+1)σ A1/2 u(s) ds ≤ + Cm . e t qm t/2 T

(2.43)

From this estimate and the above we conclude the existence of the constants K m+1 , Cm+1 and the integer pm+1 such that (2.18) holds for n = m +1. Using the estimate that we have just established in (2.18) for n = m + 1, and substituting this in (2.33), we immediately obtain the estimate (2.19) for n = m + 1. This concludes the proof of Theorem 2.1. 3. Rate of Decay of the Fourier Coefficients The purpose of this section is to specify the behavior of various constants appearing in the preceding section to obtain the rate of decay with the wavenumber of the Fourier coefficients u(k, ˆ t) for t > 0. We again consider the 3D case in the periodic domain. Since the decay may depend on the rate of decay of the Fourier transform of the forcing term f (x, t), for simplicity we assume zero external forcing, which we expect to behave as the case with sufficiently rapidly decaying forcing. The adaptation to sufficiently regular forced cases, for example a trigonometric polynomial, is similar but more technical.3 Furthermore, it is enough to prove the decay result up to a time T such that 1/N := T U/L < 1, where L and U are a typical length scale and velocity of the initial data. Extending the results to later times is easy (by propagation of regularity). We shall show that the bound for the square of the L 2 norm of the velocity weighted 1/2 by enσ A is a double exponential in n. Specifically, we have Theorem 3.1. Let u(t) be the solution of (2.3)–(2.4) in [0, T ] with f = 0 and 0 < T < L/U . Then for every κ > 3 and δ > 0, there exists a number , depending on δ and κ, such that, for all integers n ≥ 0,

t

T

2 L an nσ A1/2 u(t) ≤ , t ∈ (0, T ], e Ut 2 L an (n+1)σ A1/2 u(s) ds ≤ , t ∈ (0, T ], e Ut 4κ − 5 n . where an = (1 + δ) 2κ − 5

(3.1) (3.2) (3.3)

Corollary 3.1. For any t > 0 the function u(t) of (2.3)–(2.4) is an entire function in the space variable and its (spatial) Fourier coefficients tend to zero in the following 3 It is conceivable that the results can be extended to forces entire in the space variables whose Fourier transforms decrease faster than e−C|k| ln |k| with sufficiently large C.

528


faster-than-exponential way: there exists a constant such that, for any 0 < ε < 1, we have

|u(k, ˆ t)| ≤ e

− σ β(1−ε) |k| ln |k| κ,δ

,

for all |k| ≥

L Ut

βκ,δ εσ

,

(3.4)

where 4κ − 5 . βκ,δ = ln (1 + δ) 2κ − 5

(3.5)

Proof of Corollary 3.1. Since we are dealing with a Fourier series, the modulus of any 1/2 Fourier coefficient of the function e(n+1)σ A u(t) cannot exceed its L 2 norm, hence it is bounded by (3.1). Thus, discarding a factor (2π )−3/2 < 1, we have for all k and n, |u(k, ˆ t)| ≤ e

−nσ |k|

L Ut

((1+δ) e(κ))n

= exp ln

L nβκ,δ e − nσ |k| , Ut

(3.6)

where e(κ) is defined in (2.7). Now choosing 1 βκ,δ ln ln |k| ≥ ε σ

L , Ut

(3.7)

we obtain with n = ln |k|/βκ,δ the following estimate: √ σ βκ,δ ln L/U t − σ (1−ε) |k| ln |k| ≤ e βκ,δ |u(k, ˆ t)| ≤ exp − |k| ln |k| 1 − . βκ,δ σ ln |k|

(3.8)

Remark 3.1. Since ε and δ can be chosen arbitrarily small and κ arbitrarily large, Corollary 3.1 implies that, in terms of the dimensionless wavenumber k˜ = 2σ k, the Fourier ˜ of the form e−C k˜ ln k˜ for any C < 1/(2 ln 2). amplitude has a bound (at high enough k) We shall see that the upper bound for the constant C can probably be improved to 1/ ln 2. Proof of Theorem 3.1. We proceed again by induction. We assume that the following inequalities hold: 2 Kn nσ A1/2 u(t) ≤ a , e t n

(3.9)

and

T t

2 Kn (n+1)σ A1/2 u(s) ds ≤ a , e t n

(3.10)

where K n and an ≥ 1 are still to be determined. Starting from expNSE (2.3), we take 1/2 the inner product with e2(n+1)σ A u. Then we obtain from Proposition 2.1 with α = nσ


529

and β = 2σ , 1 d (n+1)σ A1/2 2 1/2 2 1/2 u + µ e(n+2)σ A u ≤ B(u, u), e2(n+1)σ A u e 2 dt 1/2 1/2 ≤ enσ A B(u, u) e(n+2)σ A u 1/2 2−a(κ) (n+2)σ A1/2 1+a(κ) ≤ CA (lκ (β))a(κ) enσ A u u e ≤

2κ − 5 −d(κ) 2f(κ) µ CA (lκ (β))2a(κ)f(κ) 4κ µ 1/2 2e(κ) 1/2 2 × (1 + a(κ))d(κ) enσ A u + e(n+2)σ A u . 2

(3.11)

Then it follows that d (n+1)σ A1/2 2 1/2 2 u + µ e(n+2)σ A u e dt 2κ + 5 d(κ) nσ A1/2 2e(κ) 2κ − 5 −d(κ) 2f(κ) ≤ CA u . µ (lκ (β))2a(κ)f(κ) e 2κ 2κ

(3.12)

By using the induction assumption we obtain e(κ) Kn d (n+1)σ A1/2 2 (n+2)σ A1/2 2 u + µ e u ≤ C , e dt t an

(3.13)

where we have set C =

2κ − 5 −d(κ) 2f(κ) µ CA (lκ (β))2a(κ)f(κ) 2κ

2κ + 5 2κ

d(κ) .

(3.14)

Renaming the time variable in (3.13) from t to s and integrating over s from s to t (with 0 < s < t ≤ T ) we obtain t 2 (n+1)σ A1/2 (n+2)σ A1/2 2 u(t) + µ u(s ) ds e e s 2 1 1 1 (n+1)σ A1/2 e(κ) C Kn + ≤ − u(s) e an e(κ) − 1 s an e(κ)−1 t an e(κ)−1 1 1 1/2 2 C K ne(κ) a e(κ)−1 + e(n+1)σ A u(s) . ≤ an e(κ) − 1 s n

(3.15)

Omitting the positive integral term on the left-hand side of the inequality we obtain 2 (n+1)σ A1/2 u(t) ≤ e

2 1 1 1/2 C K ne(κ) a e(κ)−1 + e(n+1)σ A u(s) . an e(κ) − 1 s n

(3.16)

530


Choosing 1 < γ ≤ N δ = (L/U T )δ and integrating over s from t/γ to t we obtain 2 t 1/2 (γ − 1) e(n+1)σ A u(t) γ γ an e(κ)−2 1 an e(κ)−2 1 1 e(κ) C Kn ≤ − an e(κ) − 1 an e(κ) − 2 t t t γ an e(κ)−2 2 1 1 (n+1)σ A1/2 C K ne(κ) + u(s) ds ≤ e an e(κ) − 1 an e(κ) − 2 t t/γ γ an +K n , (3.17) t where we have used the induction assumption (3.10). We obtain thus the following estimate: 2 γ an e(κ)−1 1 1 1 (n+1)σ A1/2 C K ne(κ) u(t) ≤ e γ − 1 an e(κ) − 1 an e(κ) − 2 t γ an +1 1 Kn + , (3.18) γ −1 t which holds for every 0 < t ≤ T . 2 T 1/2 To estimate t e(n+2)σ A u(s) ds we integrate (3.13) from t to T : 2 (n+1)σ A1/2 u(T ) + µ e

2 (n+2)σ A1/2 u(s) ds e t 2 1 1 (n+1)σ A1/2 e(κ) e C Kn + u(t) ≤C . an e(κ) − 1 t an e(κ)−1 T

Omitting the first term on the right-hand side and using (3.18) we obtain T 2 1 1 (n+2)σ A1/2 C K ne(κ) a e(κ)−1 µ u(s) ds ≤ C e n a e(κ) − 1 t n t γ an e(κ)−1 1 1 1 C K ne(κ) + γ − 1 an e(κ) − 1 an e(κ) − 2 t γ an +1 1 Kn + . γ −1 t We conclude that since an ≥ 1 and

an + 1 ≤ an 2 +

5 2κ − 5

T t

(3.20)

= an e(κ),

for a suitable constant E > 0 we have 2 γ an e(κ) (n+1)σ A1/2 u(t) ≤ E K ne(κ) e t and

(3.19)

2 γ an e(κ) (n+2)σ A1/2 u(s) ≤ E K ne(κ) . e t

(3.21)

(3.22)

(3.23)


531

Since, for t ≤ T , γ Nδ ≤ = t t

L UT

δ

1 ≤ t

L Ut

1+δ

U , L

it follows that an (1+δ)e(κ) an e(κ) 2 L U (n+1)σ A1/2 e(κ) e u(t) ≤ E K n Ut L

(3.24)

and

T t

an (1+δ)e(κ) an e(κ) 2 L U (n+2)σ A1/2 e(κ) u(s) ≤ E K n . e Ut L

(3.25)

This finishes the induction step. From the above follows that we can take an+1 = an (1 + δ)

4κ − 5 , 2κ − 5

K n+1 = E K ne(κ) .

(3.26)

Note that in the induction step we use the assumption that an ≥ 1. This fixes the value of a0 = 1. The solution of the recursion relations is given by 4κ − 5 n an − 1 an = (1 + δ) , ln K n = ln E (3.27) + an ln K 0 . 4κ−5 2κ − 5 (1 + δ) 2κ−5 −1 Finally, choosing a sufficiently large number we get the desired estimates (3.1) and (3.2). This concludes the proof of Theorem 3.1. 4. Remarks and Extensions for the Main Results Although our main theorems are stated for the case of the 3D expNSE, their statements and proofs are easily extended mutatis mutandis to arbitrary space dimensions d: with exponential dissipation for any d the solution is entire in the space variables and the decay ˜ ln |k|) ˜ for any C < C = 1/(2 ln 2). of Fourier coefficients is bounded by exp(−C|k| Some of the intermediate steps in the proof, such as the formulation of Agmon’s inequality, change with d but not the result about the constant 1/(2 ln 2). We can also easily change the functional form ofαthe dissipation.4 One instance is a dissipation operator D with a Fourier symbol e2σ |k| with 0 < α < 1. One can prove that the solution in this case satisfies 2 Kn α ˆ t) ≤ p + Cn , e2n|k| u(k, (4.1) t n k=0

for all t ∈ (0, T ] and for all n. Hence the solution in this case belongs to C ∞ but is not necessarily an entire function. In fact it belongs to the Gevrey class G 1/α . Gevrey 4 Note that the proof of Proposition 2.1 and Theorem 2.1 holds mutatis mutandis if we replace, in the argument of the exponential, |k| by a subadditive function of |k| subject to some mild conditions, such as |k|α with 0 < α < 1.

532


regularity with 0 < α < 1 does not even imply analyticity.5 Actually, with such a dissipation, the solutions are analytic even when α < 1. We shall return to this case of dual Gevrey regularity and analyticity in Sect. 5.1. Next, consider the case α > 1. The dissipation has a lower bound of the ordinary exponential type, so that the entire character of the solution is easily established. However, for α > 1 the bound exp(−C|k| ln |k|) can be improved in its functional form, as we shall see in Sect. 5.1. Obviously, the results of Sects. 2 and 3 do still hold if we change the functional form of the Fourier symbol of the dissipation at low wavenumbers |k| while keeping its exponential growth at high wavenumber. One particularly interesting instance, to which we shall come back in the next section on the Burgers equation and in the conclusion, is “cosh dissipation”, namely a Fourier symbol −µ(1 − cosh(k/kd )) with µ > 0. The dissipation rate at wavenumber much smaller than kd is then ν|k|2 with ν = µ/(2kd 2 ), just as for the ordinary Navier–Stokes equation. It is worth mentioning that the key results of Sects. 2 and 3 still hold when the problem is formulated in the whole space Rd rather than with periodicity conditions. Similarly they should hold on the sphere S 2 , a case for which spherical harmonics can be used (see [23]). Of course the result on the entire character of the solution, when exponential dissipation is assumed, holds for a large class of partial differential equations. Besides the exponential modification of the Navier–Stokes equations it applies to similar modifications, for example, of the magnetohydrodynamical equations and of the complex Ginzburg–Landau equation ∂u ∂ 2u − α 2 + βu + γ |u|2 u = 0, ∂t ∂x

(4.2)

Re α > 0, Re γ > 0.

(4.3)

where

The main idea would be in proving the analogue of Proposition 2.1 for the corresponding nonlinear terms in the underlying equations following our proof combined with ideas presented in [21 and 24]. 5. The Case of the 1D Burgers Equation The (unforced) one-dimensional Burgers equation with modified dissipation reads: ∂u ∂u +u = −µDu, ∂t ∂x

(5.1)

u(x, 0) = u 0 (x).

(5.2)

We shall mostly consider the case of the cosh Burgers equation when D has the Fourier symbol −µ(1 − cosh(k/kd )). Since the cosh Burgers equation is much simpler than expNSE we can expect to obtain stronger results or, at least, good evidence in favor of stronger conjectures. 5 The special class when α = 1 of analytic functions is considered by some authors as one of the Gevrey classes [20,21].


533

Let us observe that the cosh Burgers equation can be rewritten in the complexified space of analytic functions of z := x + i y as ∂u(z, t) µ ∂u(z, t) + u(z, t) = − [u(z + i /kd , t) + u(z − i /kd , t) − 2u(z, t)] . ∂t ∂x 2

(5.3)

This is the ordinary Burgers equation with the dissipative Laplacian replaced by its centered second-order finite difference approximation, differences being taken in the pure imaginary direction with a mesh 1/kd . As already stated, Corollary 2.1 on the entire character of the solution and Corol˜ ln |k|) ˜ lary 3.1 on the bound of the modulus of the Fourier coefficients by exp(−C|k| for any C < 1/(2 ln 2) hold in the same form as for the expNSE. Of course, if the finite differences were taken in the real rather than in the pure imaginary direction, the solution would not be entire. Actually, (5.3) relates the values of the velocity on lines parallel to the real axis shifted by ±1/kd in the imaginary direction. It thereby provides a kind of Jacob’s Ladder allowing us to climb to complex infinity in the imaginary direction. This can be used to show, at least heuristically, that the complexified velocity grows with the imaginary coordinate y as exp C2|y|kd . Such a heuristic derivation turns out to be equivalent to another derivation by dominant balance which can be done on the Fourier-transformed equation, the latter being not limited to cosh dissipation. Section 5.1 is devoted to Fourier space heuristics for different forms of the dissipation. For exponential and cosh dissipation this suggests a leading-order behavior of the Fourier coefficients for large wavenumber of the form exp(−C |k| ln |k|) with C = 1/ ln 2, a substantial improvement over the rigorous bound. Various numerical and semi-numerical results, discussed in Sects. 5.2 and 5.3, support this improved result. 5.1. Heuristics: A dominant balance approach. We want to handle dissipation operators D with an arbitrary positive Fourier symbol, taken here to be eG(k) , where G(k) is a real even function of the wavenumber k ∈ Z which is increasing without bound for k > 0. It is then best to rewrite the Burgers equation in terms of the Fourier coefficients. We set u(x, t) = ei kx u(k, ˆ t), (5.4) k∈Z

and obtain from (5.1), ∂ u(k, ˆ t) i k + u( ˆ p, t)u(q, ˆ t) = −µeG(k) u(k, ˆ t). ∂t 2

(5.5)

p+q=k

This is the place where we begin our heuristic analysis of the large-wavenumber asymptotics. First, we drop the time derivative term since it will turn out not to be relevant to leading order. (A suitable Galilean change of frame may be needed before this becomes true.) For simplicity we now drop the time variable completely. The next heuristic step is to balance the moduli of the two remaining terms, taking |u(k)| ˆ ∼ e−F(k) ,

(5.6)

where F(k) is still to be determined but assumed sufficiently smooth and the symbol ∼ is used here to connect two functions “asymptotically equal up to constants and algebraic

534


prefactors” (in other words, asymptotic equality of the logarithms). The convolution in (5.5) can be approximated for large wavenumbers by a continuous wavenumber integral ∼ e−F( p)−F(k− p) dp. Next we evaluate the integral by steepest descent, assuming that the leading order comes from the critical point p = k/2, where the p-derivative of F( p) + F(k − p) obviously vanishes. This will require that this point be truly a minimum of F( p) + F(k − p). Balancing the logarithms of the nonlinear term and of the dissipative term, we obtain the following simple equation for the function F(k): k 2F = F(k) − G(k). (5.7) 2 This is a linear first order finite difference equation (in the variable ln k) which is easily solved for values of the wavenumber of the form k = 2n : G(2) G(4) G(2n ) n n . (5.8) F(2 ) = 2 F(1) + + + ··· + 2 4 2n For exponential dissipation (and for cosh dissipation when |k|/kd 1), we have G(k) 2σ |k| and we obtain from (5.8), to leading order for large positive k, F(k)

1 ˜ ˜ ˜ k k ln k; k := 2σ k = . ln 2 kd

(5.9)

If this heuristic result is correct – and the supporting numerical evidence is strong as we shall see in Sects. 5.2 and 5.3 – the estimate given by Corollary 3.1 (adapted to the Burgers ˜ ˜ ˜ and any C < C = 1/(2 ln 2) case) that |u(k)| ˆ < e−C|k| ln |k| for sufficiently large |k| still leaves room for improvement as to the value of C . It can be shown that this dominant balance argument remains unchanged if we reinsert the time-derivative term, since its contribution is easily checked to be subdominant. Actually, the conjecture that the solution of NSE is entire with exponential or cosh dissipation was based on precisely this kind of dominant balance argument, which suggests a faster-than-exponential decay of the Fourier coefficients. When G(k) = 2σ |k|α with α > 1 we obtain to leading order F(k)

2σ |k|α . 1 − 21−α

(5.10)

This is an even faster decay of the Fourier coefficients than in the exponential case (2.3).6 It is easily checked that for α ≥ 1 the condition of having a minimum of F( p) + F(k − p) at p = k/2 is satisfied. If however we were to use (5.10) for 0 < α < 1 the condition would not be satisfied. In this case it is easily shown for the Burgers equation and the NSE, by using a variant of the theory presented in Sect. 2, that the solution is in the Gevrey class G 1/α in the whole space Rd . It is actually not difficult to show that the solution is also analytic when 0 < α < 1, in a finite strip in Cd about the real space Rd . For this it suffices to adapt to the proof of analyticity given for the ordinary NSE under the condition of some mild regularity. Such regularity is trivially satisfied with the much stronger dissipation assumed here [20].7 We also found strong numerical evidence for analyticity. 6 Actually, one can show, for the Burgers equation and the NSE that when α > 1 the Fourier coefficients of the solution decay faster than exp(−C|k|α− ) for any > 0. 7 The first results on analyticity, derived in the more complex setting of flow with boundaries, were obtained in [25].


535

It is of interest to point out that, although analyticity is a stronger regularity than Gevrey when 0 < α < 1, the Gevrey result implies a decay of the form exp(−C|k|α ln |k|), independently of the viscosity coefficient µ, whereas analyticity in a finite strip gives a decay of the form exp(−η|k|), where η depends on µ [26]. 5.2. Spectral simulation for the Burgers case. Here we begin our numerical tests on the 1D Burgers equation. So far we have a significant gap in the value of the constant C ˜ ˜ appearing in the e−C|k| ln |k| estimate of the Fourier coefficient, between the bounds and a heuristic derivation of the asymptotic behavior. In this section we shall exclusively consider the case of the unforced Burgers equation with initial condition u 0 (x) = − sin x and dissipation with a rate 1 − cosh k. (Thus, µ = 1 and kd = 1.) The numerical method is however very easily extended to other functional forms of the dissipation and other initial conditions. The spectral method is actually quite versatile. Its main drawback will be discussed at the end of this section. The standard way of obtaining a high-orders scheme when numerically integrating PDE’s with (spatial) periodic boundary conditions is by the (pseudo)-spectral technique with the 2/3 rule of alias removal [27]. The usual reason this is more precise than finite differences is that the truncation errors resulting from the use of a finite number N of collocation points (and thus a finite number N /3 of Fourier modes) decreases exponentially with N if the solution is analytic in a strip of width δ around the real axis. Indeed this implies a bound for the Fourier coefficients at high |k| of the form |u(k)| ˆ < e−C|k| for any C < δ. In the present case, the solution being entire, the bound is even better. There are of course sources of error other than spatial Fourier truncation, namely rounding errors and temporal discretization errors. Temporal discretization is a nontrivial problem here because the dissipation grows exponentially with |k| and thus the characteristic time scale of high-|k| modes can become exceedingly small. Fortunately, these modes are basically slaved to the input stemming from nonlinear interaction of lower-lying modes. It is possible to take advantage of this to use a slaving technique which bypasses the stiffness of the equation (a simple instance of this phenomenon is described in Appendix B of [28]). We use here the slaved scheme Exponential Time Difference Runge Kutta 4 (ETDRK4) of [29] with a time step of 10−3 .8 As to the rounding noise, it is essential to use at least double precision since otherwise the faster-than-exponential decrease of the Fourier coefficients would be swamped by rounding noise beyond a rather modest wavenumber. Even with double precision, rounding noise problems start around wavenumber 17, as we shall see. Hence it makes no sense to use more than, say, 64 collocation points, as we have done. Figure. 1 shows the discrepancy Discr (k) := −

ln |u(k, ˆ 1)| 1 − , |k| ln |k| ln 2

(5.11)

which, according to heuristic asymptotic theory (5.6)–(5.9), should converge to 0 as |k| → ∞. It is seen that the discrepancy falls to about 3.5% of the nominal value 1/ ln 2 before getting swamped by rounding noise around wavenumber 17. 8 This is far larger than would have have been permitted without the slaving. Actually it can still be increased somewhat to 5 × 10−3 without affecting the results.

536


0.4 0.3 0.2

Discr (k)

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 1

3

5

7

9

11

13

15

17

19

21

k

Fig. 1. Discrepancy Discr, as predicted by (5.11), vs wavenumber k for data obtained by spectral simulation in double precision; rounding noise becomes significant beyond wavenumber 17

It is actually possible to significantly decrease the discrepancy by using a better processing of the numerical output, called asymptotic extrapolation, developed recently by van der Hoeven [30] and which is related to the theory of transseries [31,32]. The basic idea is to perform on the data a sequence of transformations which successively strip off the leading and subleading terms in the asymptotic expansion (here for large |k|). Eventually, the transformed data allow a very simple interpolation (mostly by a constant). The procedure can be carried out until the transformed data become swamped by rounding noise or display lack of asymptoticity, whichever occurs first. After the interpolation stage, the successive transformations are undone. This determines the asymptotic expansion of the data up to a certain order of subdominant terms. An elementary introduction to this method may be found in [33], from which we shall also borrow the notation for the various transformations: I for “inverse”, R for “ratio”, SR for “second ratio”, D for “difference” and Log for “logarithm”. The choice of the successive transformations is dictated by various tests which roughly allow to find into which broad asymptotic class the data and their transformed versions fall. In the present case, the appropriate sequence of transformations is: Log, D, D, I, D. Because of the relatively low precision of the data it is not possible to perform more than five transformations, so that the method gives us access only to the leading-order asymptotic behavior, namely |u(k, ˆ 1)| e−C |k| ln |k| . (5) (5) It may be shown that the constant C = −1/u , where u is the constant value of the high-|k| interpolation u (5) (|k|) after the 5th stage of transformation. Figure. 2 shows the discrepancy u (5) (|k|) + ln 2. The absolute value of the discrepancy lingers around 0.002 to 0.005 before being swamped by rounding noise at wavenumber 17. Thus with asymptotic extrapolation the discrepancy does not exceed 0.7% of the nominal value ln 2. The accuracy of the determination has thus improved by about a factor 5, compared to the naive method without asymptotic extrapolation. To improve further on this result and get some indication as to the type of subdominant corrections present in the large-wavenumber expansion of the Fourier coefficients, it would not suffice to increase the spatial resolution, since rounding noise would still swamp the signal beyond a wavenumber of roughly 17. Higher precision spectral


537

0.55 0.45 0.35

Discr (k)

0.25 0.15 0.05 0 -0.05 -0.15 -0.25 5

7

9

11

13

15

17

19

21

k

Fig. 2. Discrepancy Discr vs wavenumber k for the same data as in Fig. 1, but processed by a 5-stage asymptotic extrapolation method

calculations are doable but not very simple because high-precision fast Fourier transform packages are still in the experimental phase. In the next section we shall present an alternative method, significantly less versatile as to the choice of the initial condition because it exploits the algebraic structure of a certain special class of solutions, but which also allows to work easily in arbitrary precision and thus to make better use of asymptotic extrapolation for determining the constant C .

5.3. Half-space (Fourier) supported initial conditions. So far we have limited ourselves to initial conditions that are real entire functions. Hence the Fourier coefficients had Hermitian symmetry: uˆ 0 (−k) = uˆ ∗0 (k), where the star denotes complex conjugation. With complex initial data there are no analytical results when the dissipation is exponential, even when the initial conditions are entire because the energy conservation relation – now about a complex-valued quantity – ceases to give L 2 -type bounds. Actually, as already pointed out, Li and Sinai [13] showed that the 3D NSE can display finite-time blow up with suitable complex initial data. It is however straightforward to adapt to complex solutions the heuristic argument of Sect. 5.1 and to predict a high-wavenumber leading-order term exactly of the same form as for real solutions. This is of interest since we shall see that there is a class of periodic complex initial conditions for which, provided the Burgers equation is written in terms as the Fourier coefficients as in [34] and [13], any given Fourier coefficient can be calculated at arbitrary times t with a finite number of operations, most easily performed on a computer, by using either symbolic manipulations or arbitrary-high precision floating point calculations. For the case of the Burgers equation, this class consists of initial conditions having the Fourier coefficients supported in the half line k > 0.9 We shall refer to this class of initial data as “half-space (Fourier) supported”. 9 If the coefficient for wavenumber k = 0 is non-vanishing a simple Galilean transformation can be used to make it vanish.

538


Because of the convolution structure of the nonlinearity when written in terms of Fourier coefficients, it is obvious that with an initial condition supported in the k > 0 half line, the solution will also be supported in this half line. A similar idea has been used in three dimensions for studying the singularities for complex solutions of the 3D Euler equations [35]. Specifically, we consider again the 1D Burgers equation (5.1)–(5.2) with 2π -periodic boundary conditions for t ≥ 0, rewritten as (5.5), in terms of the Fourier coefficients u(k, ˆ t), assumed here to exist. The k-dependent real, even, non-negative dissipation coefficient µeG(k) is denoted by ρ(k). The initial conditions uˆ 0 (k) (k = 1, 2, . . .) are chosen arbitrarily, real or complex. We then have the following proposition, which is of purely algebraic nature: Proposition 5.1. Equation (5.5) with the initial conditions u(k, ˆ 0) = 0 (k ≤ 0) and u(k, ˆ 0) = uˆ 0 (k) (k = 1, 2, . . .) defines, for all k > 0 and t > 0, u(k, ˆ t) as a polynomial functions of the set of uˆ 0 ( p) with 0 < p ≤ k. Proof. From (5.5), after integration of the dissipative term, we obtain, for k > 0 and t > 0, u(k, ˆ t) = e−tρ(k) uˆ 0 (k) −

ik 2

t

ds e−(t−s)ρ(k)

0

k−1

u( ˆ p, s)u(k ˆ − p, s).

(5.12)

p=1

Observe that u(k, ˆ t), given by (5.12), involves u(k, ˆ 0) (linearly) and the set of Fourier coefficients u( ˆ p, s) for 1 ≤ p ≤ k − 1 and 0 ≤ s < t (quadratically). The proof follows by recursive use of this property for k, k − 1, . . . , 1.10 Note that the solution can be obtained without any truncation error on a computer, using symbolic manipulation. Alternatively it can be calculated in arbitrary highprecision floating point arithmetic. Now we specialize the initial condition even further, by assuming that the only Fourier harmonic present in the initial condition has k = 1.11 We take u 0 (x) = i Aei x ,

(5.13)

for which uˆ 0 (1) = i A, while all the other coefficients vanish. Setting, for k > 0, v(k, ˆ t) :=

u(k, ˆ t) , i Ak

(5.14)

we obtain from (5.12) by working out the power series to the second order the following fully explicit expressions of the first two Fourier coefficient at any time t > 0: v(1, ˆ t) = e−ρ(1)t ,

v(2, ˆ t) =

e−ρ(2)t e−2ρ(1)t − . ρ(2) − 2ρ(1) ρ(2) − 2ρ(1)

(5.15)

10 This proposition has an obvious counterpart for the NSE in any dimension when the the Fourier coefficients of the initial condition are compactly supported in a product of half-spaces. 11 What follows can be easily extended to the case of a finite number of non-vanishing initial Fourier harmonics.


539

0.025 0.02 0.015

Discr (k)

0.01 0.005 0 -0.005 -0.01 -0.015

8

10

12

14

16

18

20

22

24

k

Fig. 3. Fifth stage of asymptotic extrapolation showing the discrepancy with respect to the heuristic prediction C = 1/ ln 2. Note the decaying oscillatory behavior

For the one-mode initial condition with A = 1 and ρ(k) = e|k| (exponential dissipation with µ = 1 and (kd = 1), we have calculated the Fourier coefficients v(k, ˆ 1) for k = 1, 2, . . . , 24 using Maple symbolic calculation with a forty-digit accuracy. The data have then been processed using asymptotic extrapolation as in Sect. 5.2 with the same five transformations Log, D, D, I, D. Figure 3 shows the discrepancy vˆ (5) (k, 1) + ln 2 between the 5th stage of interpolation and the prediction from the dominant balance argument. It is seen that the discrepancy drops to −2.3 × 10−3 . Thus the relative error is about 3 × 10−3 . The oscillations in the discrepancy, if they continue to higher wavenumbers would indicate that the first subdominant correction to the ˜ ˜ asymptotic behavior of the Fourier coefficient e−(1/ ln 2)|k ln |k| is a prefactor involving a complex power of the wavenumber. Finally, we address the issue of what kind of solution we have constructed by this Fourier-based algebraic method. Is it a “classical” sufficiently smooth global-in-time solution of the Burgers equations (5.1)–(5.2) written in the physical space? We have here obtained strong numerical evidence that the Fourier coefficients decrease faster than exponentially with the wavenumber and thus define a classical solution which is an entire function of the space variable. This is however just a conjecture. The tools used in Sect. 2 to prove the entire character of the solution rely heavily on the definite positive character of the energy, a property lost with complex solutions.12 6. Conclusions In this paper we have proved that for a large class of evolution PDE’s, including the 3D NSE, exponential or faster-growing dissipation implies that the solution becomes and remains an entire function in the space variables at all times. Exponential growth constiα tutes a threshold: subexponential growth with a Fourier symbol e|k| , where 0 < α < 1, makes the solution analytic (but not entire) as is the case in 2D (and generally conjectured 12 It is however not difficult to prove, for short times, that our solution is also a classical entire solution.

540


in 3D). Furthermore, for the 3D NSE and the 1D Burgers equation with a dissipation having the Fourier symbol µe|k|/kd , we have shown that the amplitude of the Fourier ˜ ˜ coefficients is bounded by e−C|k| ln |k| (where k˜ := k/kd ) for any C < 1/(2 ln 2). For the case of the 1D Burgers equation we have good evidence that this can be improved to C < C = 1/ ln 2 since the high-|k| asymptotics seems to have a leading term precisely ˜ ˜ of the form e−C |k| ln |k| ; the evidence comes both from a heuristic dominant balance argument and from high-precision simulations. The heuristic argument can actually be carried over somewhat loosely to the expNSE in any dimension: again the dominant nonlinear interaction contributing to wave vector k comes from the wave vectors p = q = k/2; actually the condition of incompressibility kills nonlinear interactions between exactly parallel wave vectors but this is only expected to modify algebraic prefactors in front of the exponential term. We thus conjecture that the C = 1/ ln 2 also holds for expNSE in any space dimension d ≥ 2. Of course there is a substantial gap between the bound and the conjectured asymptotic behavior. It seems that such a gap is hard to avoid when using L 2 -type norms. For proving the entire character of the solution such norms were appropriate. Beyond this, it appears more advisable to try bounding directly the moduli of Fourier coefficients by using the power series method [13,34]. A first step in this direction would be to prove that C = 1/ ln 2 for initial conditions whose Fourier coefficients are compactly supported in a product of half-spaces, of the kind considered in Sect. 5.3.13 Exponential dissipation differs from ordinary dissipation (with a Laplacian or a power thereof) not only by giving a faster decay of the Fourier coefficients but by doing so in a universal way: with ordinary dissipation the decay of the Fourier coefficient is generally conjectured to be, to leading order, of the form e−η|k| , where η depends on the viscosity ν and on the energy input or on the size of the initial velocity; with exponential ˜ ˜ dissipation the decay is e−C |k| ln |k| , where C = 1/ ln 2 and thus depends neither on the coefficient µ which plays the role of the viscosity nor on the initial data.14 As a consequence, it is expected that exponential dissipation will not exhibit the phenomenon of dissipation-range intermittency, which for the usual dissipation can be traced back either to the fluctuations of η [36] or to complex singularities of a velocity field that is analytic but not entire [37]. Finally some comments on the practical relevance of modified dissipation. First, let us comment on “hyperviscosity”, the replacement of the (negative) Laplacian by its power of order α > 1. Of course we know that specialists of PDE’s have traditionally been interested in the hyperviscous 3D NSE, perhaps to overcome the frustration of not being able to prove much about the ordinary 3D NSE. But scientists doing numerical simulations of the NSE, say, for engineering, astrophysical or geophysical applications, have also been using hyperviscosity because it is often believed to allow effectively higher Reynolds numbers without the need to increase spatial resolution. Recently, three of us (UF, WP, SSR) and other coauthors have shown that when using a high power α of the Laplacian in the dissipative term for 3D NSE or 1D Burgers, one comes very close to a Galerkin truncation of Euler or inviscid Burgers, respectively [38]. This produces a range of nearly thermalized modes which shows up in large-Reynolds number spectral simulations as a huge bottleneck in the Fourier amplitudes between the inertial range and the far dissipation range. Since the bottleneck generates a fairly large eddy viscos13 Progress on this issue has been made and will be reported elsewhere. 14 It does however depend on the type of nonlinearity. For example with a cubic nonlinearity the same kind

of heuristics as presented in Sect. 5.1 predicts a constant C = 1/ ln 3.


541

ity, the hyperviscosity procedure with large α actually decreases the effective Reynolds number. Next, consider exponential dissipation. In 1996 Achim Wirth noticed that when used in the 1D Burgers equation, cosh dissipation produces almost no bottleneck although it grows much faster than a power of the wavenumber at high wavenumbers [39]. It is now clear that such a dissipation will produce a faster-than-exponential decay at the highest wavenumbers. But at wavenumbers such that |k| kd a dissipation rate −µ(1 − cosh k/kd ) reduces to µ|k|2 /(2kd 2 ), to leading order, which is the ordinary (Laplacian) dissipation. With the ordinary 1D Burgers equation it may be shown analytically that there is no bottleneck. For the ordinary 3D NSE, experimental and numerical results show the presence of a rather modest bottleneck (for example the “compensated” three-dimensional energy spectrum |k|+5/3 E(|k|) overshoots by about 20%.). If in a simulation with cosh dissipation µ and kd are adjusted in such a way that dissipation starts acting at wavenumbers slightly smaller than kd , the beginning of the dissipation range will be mostly as with an ordinary Laplacian, that is with no or little bottleneck.15 At higher wavenumbers, where the exponential growth of the dissipation rate is felt, faster than exponential decay will be observed. In principle this can be used to avoid wasting resolution without developing a seri 2 ous bottleneck. Faster than exponentially growing dissipation, e.g. µ e(|k|/kd ) − 1 , may be even better because the prediction is that the Fourier coefficients will display Gaussian decay.16 Testing the advantages and drawbacks of different types of faster-than-algebraically growing dissipations for numerical simulations is left for future work. Acknowledgements. We thank J.-Z. Zhu and A. Wirth for important input and M. Blank, K. Khanin, B. Khesin and V. Zheligovsky for many remarks. CB acknowledges the warm hospitality of the Weizmann Institute and SSR that of the Observatoire de la Côte d’Azur, places where parts of this work were carried out. The work of EST was supported in part by the NSF grant No. DMS-0708832 and the ISF grant No. 120/06. SSR thanks R. Pandit, D. Mitra and P. Perlekar for useful discussions and acknowledges DST and UGC (India) for support and SERC (IISc) for computational resources. UF, WP and SSR were partially supported by ANR “OTARIE” BLAN07-2_183172 and used the Mésocentre de calcul of the Observatoire de la Côte d’Azur for computations. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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6. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Paris: GauthierVillars, 1969 7. Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics. Chicago: University of Chicago Press, 1988 8. Fefferman, C.: Existence & smoothness of the Navier–Stokes equation. Millenium problems of the Clay Mathematics Institute (2000). Available at www.claymath.org/millennium/Navier-Stokes_Equations/ Official_Problem_Description.pdf 9. Temam, R.: Navier–Stokes Equations. Theory and numerical analysis. Revised edition. With an appendix by F. Thomasset. Published by AMS Bookstore. Providence, RI: Amer. Math. Soc., 2001 10. Sohr, H.: The Navier–Stokes Equations. Basel: Birkhäuser, 2001 11. Brachet, M.-E., Meiron, D.I., Orszag, S.A., Nickel, B.G., Morf, R.H., Frisch, U.: Small-scale structure of the Taylor-Green vortex, J. Fluid Mech. 130, 411–452 (1983) 12. von Neumann, J.: Recent theories of turbulence (1949), In: Collected works (1949–1963) 6, ed. A.H. Taub, New York: Pergamon Press, 1963, pp. 37–472 13. Li, D., Sinai, Ya.G.: Blow-ups of complex solutions of the 3D Navier–Stokes system and renormalization group method. J. Eur. Math. Soc. 10, 267–313 (2008) 14. Oliver, M., Titi, E.S.: On the domain of analyticity for solutions of second order analytic nonlinear differential equations. J. Diff. Eq. 174, 55–74 (2001) 15. Pauls, W., Matsumoto, T.: Lagrangian singularities of steady two-dimensional flow. Geophys. Astrophys. Fluid. Dyn. 99, 61–75 (2005) 16. Senouf, D., Caflisch, R., Ercolani, N.: Pole dynamics and oscillation for the complex Burgers equation in the small-dispersion limit. Nonlinearity 9, 1671–1702 (1996) 17. Poláˇcik, P., Šverák, V.: Zeros of complex caloric functions and singularities of complex viscous Burgers equation. Preprint. 2008, http://arXiv.org/abs/math/0612506v1 [math.AP], 2006 18. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow (1st ed.). New York: Gordon and Breach, 1963 19. Holloway, G.: Representing topographic stress for large-scale ocean models. J. Phys. Oceanogr. 22, 1033–1046 (1992) 20. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369 (1989) 21. Ferrari, A., Titi, E.S.: Gevrey regularity for nonlinear analytic parabolic equations. Commun. Part. Diff. Eq. 23, 1–16 (1998) 22. Agmon, S.: Lectures on Elliptic Boundary Value Problems. Mathematical Studies, Princeton, NJ: Van Nostrand, 1965 23. Cao, C., Rammaha, M., Titi, E.S.: The Navier–Stokes equations on the rotating 2 − D sphere: Gevrey regularity and asymptotic degrees of freedom. Zeits. Ange. Math. Phys. (ZAMP) 50, 341–360 (1999) 24. Doelman, A., Titi, E.S.: Regularity of solutions and the convergence of the Galerkin method in the Ginzburg–Landau equation. Num. Funct. Anal. Optim. 14, 299–321 (1993) 25. Masuda, K.: On the analyticity and the unique continuation theorem for solutions of the Navier–Stokes equation. Proc. Japan Acad. 43, 827–832 (1967) 26. Doering, C.R., Titi, E.S.: Exponential decay rate of the power spectrum for solutions of the Navier–Stokes equations. Phys. Fluids 7, 1384–1390 (1995) 27. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods. Philadelphia: SIAM, 1977 28. Frisch, U., She, Z.S., Thual, O.: Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model. J. Fluid Mech. 168, 221–240 (1986) 29. Cox, C.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 76, 430–455 (2002) 30. van der Hoeven, J.: On asymptotic extrapolation. J. Symb. Comput. 44, 1000–1016 (2009) 31. Ecalle, J.: Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac. Actualités mathématiques. Paris: Hermann, 1992 32. van der Hoeven, J.: Transseries and Real Differential Algebra. Lecture Notes in Math. 1888, Berlin: Springer, 2006 33. Pauls, W., Frisch, U.: A Borel transform method for locating singularities of Taylor and Fourier series. J. Stat. Phys. 127, 1095–1119 (2007) 34. Sinai, Ya.G.: Diagrammatic approach to the 3D Navier-Stokes system. Russ. Math. Surv. 60, 849–873 (2005) 35. Caflisch, R.E.: Singularity formation for complex solutions of the 3D incompressible Euler equations. Physica D 67, 1–18 (1993) 36. Kraichnan, R.H.: Intermittency in the very small scales of turbulence. Phys. Fluids 10, 2080–2082 (1967) 37. Frisch, U., Morf, R.: Intermittency in nonlinear dynamics and singularities at complex times. Phys. Rev. A 10, 2673–2705 (1981)


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38. Frisch, U., Kurien, S., Pandit, R., Pauls, W., Ray, S.S., Wirth, A., Zhu, J.-Z.: Hyperviscosity, Galerkin truncation and bottlenecks in turbulence. Phys. Rev. Lett. 101, 144501 (2008) 39. Wirth, A. Private communication, 1996 40. Zhu, J.-Z. Private communication, 2008 41. Orszag, S.A. Private communication, 1979 Communicated by A. Kupiainen


Communications in


Matryoshka of Special Democratic Forms Chandrashekar Devchand1 , Jean Nuyts2 , Gregor Weingart3 1 Institut für Mathematik der Universität Potsdam, Am Neuen Palais 10,

D-14469 Potsdam, Germany. E-mail: [email protected]

2 Physique Théorique et Mathématique, Université de Mons-Hainaut,

20 Place du Parc, B-7000 Mons, Belgium. E-mail: [email protected]

3 Instituto de Matemáticas, Universidad Nacional Autónoma de México,

62210 Cuernavaca, Morelos, Mexico. E-mail: [email protected] Received: 6 January 2009 / Accepted: 3 August 2009 Published online: 2 October 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract: Special p-forms are forms which have components ϕµ1 ...µ p equal to +1, −1 or 0 in some orthonormal basis. A p-form ϕ ∈ p Rd is called democratic if the set of nonzero components {ϕµ1 ...µ p } is symmetric under the transitive action of a subgroup of O(d, Z) on the indices {1, . . . , d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P ≥ p and D ≥ d. In particular, we display a remarkable nested structure of special forms including a U(3)-invariant 2-form in six dimensions, a G2 -invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form. 1. Introduction Special holonomy plays an important role in field theories. For instance, supersymmetry often requires that target manifolds have special holonomy. This property is also important for Yang-Mills theories. In dimensions greater than four, special holonomy offers the possibility of constructing solutions of the Yang-Mills equations satisfying the generalised self-duality equations first introduced for flat Euclidean spaces in [1] (see also [2–4]); 1 2 Tmnpq F pq

= λFmn , m, n, · · · = 1, . . . , d.

(1.1)

Here, Fmn are components of the curvature of a Yang-Mills connection ∇ taking values in the Lie algebra of the gauge group and Tmnpq are components of a 4-form T . This 4-form acts as an endomorphism on the space of 2-forms. The curvature F restricted to eigenspaces of T corresponding to nonzero eigenvalues λ satisfy the Yang-Mills equations ∇m Fmn = 0 in virtue of the Bianchi identities ∇[m Fnp] ≡ 0. Interesting examples

546

C. Devchand, J. Nuyts, G. Weingart

are the 4-forms invariant under (Sp(n)⊗Sp(1))/Z2 , Spin(7) and G2 corresponding to Yang-Mills equations on quaternionic Kähler and exceptional holonomy manifolds (see e.g. [5–10]). Further examples of special holonomy structures are the U(n) invariant Kähler 2-forms in 2n (real) dimensions and the G2 invariant Cayley 3-form in seven dimensions. It turns out that the latter forms are not only related to each other, but also to interesting higher rank forms in higher dimensions. Recall that a constant p-form ϕ in a d-dimensional Euclidean space is a calibration if for any p-dimensional subspace spanned by a set of orthonormalised vectors e1 , . . . , e p , (ϕ(e1 , . . . , e p ))2 ≤ 1 ,

(1.2)

where equality holds for at least one subspace. Constant p-forms can always be rescaled to be calibrations. Many of the interesting calibrations which characterise special holonomy manifolds can be presented as special forms, all of whose nonzero components saturate the bound (1.2) (see Definition 1). In this article, we wish to highlight relationships between special p-forms in d dimensions and certain special P-forms (P ≥ p) in D dimensions (D ≥ d), governed by discrete symmetries. Symmetric ways of embedding the d-dimensional space in the D-dimensional space leads us to a notion of democratic forms. We study examples, focusing our attention on specially interesting structures in dimensions seven, eight and ten. A remarkable nested structure, reminiscent of a matryoshka1 , emerges in successively higher dimensions. In particular, this structure provides new examples of selfdualities. 2. Special Forms, Symmetries and Democracy We concentrate on what we call special forms. Let (e1 , . . . , ed ) denote an orthonormal basis of Rd . Definition 1. A special p-form ϕ is a p-form ϕ ∈ p Rd on d–dimensional Euclidean space Rd in the orbit under the special orthogonal group SO(d, R) of ϕ = ϕµ1 ···µ p eµ1 ∧ eµ2 ∧ · · · ∧ eµ p (2.1) 1≤µ1 d as follows:

(i1 ) ... (id ) (d+1) ... (D) = i1 ... id ,

= id .

(4.1)

Clearly, the d-dimensional completely antisymmetric tensor i1 ... id is fully democratic, the bisymmetry group Sd acting transitively on its indices. 4.2. u(n)-invariant forms. Consider the u(n)-invariant 2-form ω on R2n , with non-zero components ω12 = ω34 = ω56 = · · · = ω(2n−1)(2n) = 1.

(4.2)

The vertex space P2 (2n) consists of n points. The permutation symmetry group of the 2-form ω is G n! = Sn , the group of permutations of the n ordered pairs of indices, {1, 2}, {3, 4}, {5, 6}, . . . , {(2n − 1), (2n)}, generated by σ1 := ( 1 3 )( 2 4 )( 5 )( 6 ) · · · ( 2n ), σ2 := ( 1 3 5 · · · 2n−1 )( 2 4 6 · · · 2n ).

(4.3)

The number of permutation antisymmetries is also n!, generated by the composition of the generators in (4.3) with, for example, the permutation τ1 := ( 1 2 )( 3 4 )( 5 6 ) . . . ( 2n−1 2n ).

(4.4)

Matryoshka of Special Democratic Forms

551

There are 2n n! orthogonal symmetries and 2n n! orthogonal antisymmetries. These are either permutation symmetries or permutation antisymmetries multiplied by 2n possible sign factors. The 2-form ω, as well as the 2k-forms ϕ = k!1 ωk , k ≤ n, are democratic and special. For example, the permutation symmetry group of the 4-form 21 ω2 is the permutation bisymmetry group of ω and it has no antisymmetries. Example 4.2.1. Starting from any one non-zero component, say ω12 , the other components in (4.2) can be generated by the subgroup Hn ⊂ G n! generated by σ2 . Hence a presentation is given by P[1; 1](ω) = {ω12 ; σ2 }. For n = 3 the invariant subgroup H3 is the commutator subgroup (the closure of the set of elements of the form b−1 a −1 ba ∀a, b ∈ G 6 ). Example 4.2.2. The 2-form ω ∈ 2 R2n may be constructed from the 2-form in two dimensions, ∈ 2 R2 , with non-zero component 12 = 1 thus: ω (a) (b) = ab , a, b = 1, 2, ∈ {σ2m | m = 1, . . . , n} = Hn ⊂ G n! ,

(4.5)

where Hn is the subgroup generated by σ2 in (4.3). 4.3. The g2 -invariant form. Consider the G2 invariant special 3-form ψ on R7 , with non-zero components ψabc given by any choice of the structure constants of the imaginary octonions. Let {ea , a = 1, . . . 7} denote the standard basis for Im(O) R7 , with ea eb = ψabc ec − δab . A choice of the structure constants ψabc is given by ψ127 = ψ163 = ψ154 = ψ253 = ψ246 = ψ347 = ψ567 = 1.

(4.6)

The vertex space P3 (7) consists of 7 points. The 7-valent graph connecting these vertices has all edges labeled by distance 2. The permutation symmetry group of ψ (and naturally of its 4-form dual ψ) is a group of order twenty-one, G 21 , generated by the permutations σ3 := (1 2 5 4 6 7 3), σ4 := (1 3 5)(2 4 6)(7).

(4.7) (4.8)

The permutation σ3 generates the commutator subgroup H7 ⊂ G 21 . Since the permutation symmetry group includes an order 7 permutation, the form ψ is manifestly democratic. There are no permutation antisymmetries τ . Using Maple we have determined that the number of orthogonal symmetries is 672, with an order 168 commutator subgroup generated by either {(1 5 7)(2 4 6)(3), (1 2 6 5)(4 7)(3), (1 4 7)(2 5 3)(6)}

(4.9)

or, alternatively, by {(1 4)(2 3)(5)(6)(7), (1 2 4 7 6 3 5), (1 3 4)(5 6 7)(2)}.

(4.10)

The number of orthogonal antisymmetries is also 672, obtained from the orthogonal symmetries by multiplying all the ηi by −1.

552


Example 4.3.1. The component choice (4.6) can be generated from any non-zero component in (4.6) by the iterated action of σ3 . Thus a presentation is given by, for instance, P[1; 1](ψ) = {ψ127 ; σ3 }. Alternatively, a less economical presentation is given by P[3; 1](ψ) = {ψ127 , ψ136 , ψ246 ; σ4 }. Example 4.3.2. The components of the 4-form dual ψ ∈ 4 R7 can be obtained in the following way: ψ (1) (2) (3) (4) = 1234 , ∈ H7 ⊂ S7

(4.11)

where H7 is the group generated by the permutation σ3 = (1 2 5 4 6 7 3). Example 4.3.3. The 2-form ω in R6 given by (4.2) for n=3 affords an extension to the 3-form ψ thus: ψ (i) ( j) (7) = ωi j , i, j = 1, . . . , 6 , ∈ H ⊂ S7 ,

(4.12)

where there are three types of ‘minimal’ choices of H , having only one generator: • H = H7 generated by σ3 = (1 2 5 4 6 7 3), • H = H3 generated by (1)(2 6 5)(3 4 7), • H = H3 generated by (2)(1 3 6)(4 7 5). Composing the mappings in Eqs. (4.5) and (4.12) immediately yields: Example 4.3.4. Consider the two dimensional form ∈ 2 R2 , 12 = 1. It yields the components (4.6) of the 3-form ψ in seven dimensions: ψ (i) ( j) (7) = i j , i, j = 1, 2, ∈ H7 ⊂ S7 .

(4.13)

Choosing H7 to be the group generated by σ3 = (1 2 5 4 6 7 3) again gives the set of ψ’s in (4.6). In fact, choosing H7 to be the group generated by any seven-cycle of the form (1 2 ∗ ∗ ∗ 7 ∗) or (1 2 ∗ 7 ∗ ∗ ∗) provides equivalent choices of components of this 3-form. This example yields a simple mnemonical construction of the structure constants of the imaginary octonions. 4.4. The spin(7)-invariant form. Consider the Spin(7)-invariant self-dual 4-form φ in d = 8 [1] with non-zero components 1 = φ1234 = φ1256 = φ1278 = φ1357 = φ1386 = φ1485 = φ1476 = φ2385 = φ2376 = φ2475 = φ2468 = φ3456 = φ3478 = φ5678 .

(4.14)

We note that each pair (i, j) of indices occurs precisely thrice and the contraction with any 2-plane spanned by {ei , e j } yields the u(3)-invariant 2-form (4.2) with φ(ei , e j , · , ·)|R6 = ω. The vertex space P4 (8) consists of 14 points. In the corresponding graph, every vertex is connected to 12 others by edges of distance 2 and to one antipodal point at distance 4. This form is democratic and has a permutation symmetry group G 168 generated by σ5 := (1 2)(3 6 7 4 5 8), σ6 := (8)(1 2 5 4 6 7 3).

(4.15)


553

The permutation σ6 has a 11 71 cycle decomposition. Its powers, apart from the identity, generate six independent permutations having the same cycle decomposition. There are 8 such permutations, corresponding to all 8 choices of the 1-cycle. In all they generate 48 permutations in the class 11 71 . The permutation σ5 has a 21 61 cycle decomposition. Clearly, its inverse, σ55 , also. Further, σ52 and σ54 have 12 32 cycles. There are 28 permutations of each of these four types, corresponding to all choices of the 2-cycle in σ5 . So, these generate 56 permutations in each of the classes 21 61 and 12 32 . The third power, σ53 , generates a 24 cycle. There are seven such permutations. Including the identity, we therefore have the 168 = 48 + 56 + 56 + 7 + 1 elements of G 168 . The permutation σ5 clearly decomposes into the product of the order 3 and order 2 permutations σ7 := (1)(2)(3 5 7)(4 6 8), σ8 := (1 2)(3 4)(5 6)(7 8),

(4.16)

and a presentation for G 168 is given by σ6 and σ7 . The commutator subgroup of G 168 is the order 56 group generated by σ6 and σ8 . It contains the 48 elements in the class 11 71 , the seven elements in the class 24 and the identity. The orthogonal symmetries of (4.14) total 10752 elements, with the commutator subgroup being the order 1344 group generated by (7)(1 3 2 8 4 5 6) and (6)(1 5 7 2 8 3 4). The form φ has no antisymmetries. Example 4.4.1. The order 12 subgroup H12 ⊂ G 168 leaving the component φ1234 invariant is generated by σ8 and σ9 := (1)(6)(2 3 4)(5 8 7).

(4.17)

It has 14 left-cosets corresponding to the 14 components of φi jkl , more precisely the action of the 14 cosets on φ1234 generates the 14 non-zero components of φ. The 4-form φ in (4.14) may be constructed in various ways from the so(n)-, su(3) ⊕ u(1)-, and g2 -invariant forms discussed above. Example 4.4.2. Starting from the 4-form in R4 we can generate the 4-form φ in eight dimensions with components (4.14) thus: φσ (1)σ (2)σ (3)σ (4) = 1234 , σ ∈ H12 ⊂ S8 ,

(4.18)

where H12 is the group generated by σ8 and σ9 . Example 4.4.3. From the components (4.6) of the g2 -invariant form ψ ∈ 3 R7 , we may obtain the Spin(7)-invariant 4-form φ: φσ (i)σ ( j)σ (k)σ (8) = ψi jk , i, j, k = 1, . . . , 7, σ ∈ H ⊂ S8 .

(4.19)

By choosing H appropriately, we obtain the components (4.14). Two possibilities are a) H = H6 generated by σ5 = (1 2)(3 6 7 4 5 8), b) H = H8 generated by the set {(1 2)(3 4)(5 6)(7 8), (1 3)(2 4)(5 7)(6 8), (1 5)(2 6)(3 7)(4 8)}.

554


The φmnpq obtained this way satisfy the well-known relations (e.g. [1]), φi jk8 = ψi jk , φi jkl = 16 i jklmnp ψmnp , i, . . . , p = 1, . . . , 7.

(4.20)

Example 4.4.4. Analogously to (4.13), we may directly obtain the Spin(7)-invariant set of φ’s in (4.14) from i j , 12 = 1, in two dimensions: φσ (i)σ ( j)σ (7)σ (8) = i j , i, j = 1, . . . , 2, σ ∈ H21 ⊂ S8 ,

(4.21)

where H = H21 is generated by (3)(1264758) and (1)(6)(234)(587). 5. A D = 10 Structure from a Spin(7) Structure in d = 8 5.1. Construction of a 6-form in D = 10 from a Spin(7)-invariant 4-form in d = 8. Consider the Spin(7)-invariant self-dual 4-form φmnpq in d = 8 given in (4.14). Its discrete symmetry group G 168 generated by the permutations (4.15) include the Z2 × Z2 transformations generated by the 24 cycles ρ1 := (18)(27)(36)(45), ρ2 := (14)(23)(58)(67).

(5.1)

Define the permutation σ of d indices, for d even, σ := (135 . . . d−1)(246 . . . d) ,

(5.2)

which acts on the set of ordered pairs {1, 2}, {3, 4}, {5, 6}, . . .. We see that for d = 8, this mapping squared, σ 2 = ρ1 · ρ2 . We want to embed the form φ (4.20) into a form in R10 with orthonormal basis (en )n=1,...,10 . For the components, we shall denote the 10th index by a 0. Clearly, a 6-form 0 in ten dimensions which reduces to the above 4-form in eight dimensions may be defined in a trivial fashion by requiring the non-zero components to be given by 0mnpq 9 0 = φmnpq , m, n, p, q = 1, . . . , 8,

(5.3)

i.e. the 6-form 0 contracted with the volume form on the 9–10 plane yields the Spin(7)-invariant tensor (4.20). However, there is a less trivial possibility. Consider a 6-form 1 in D = 10 with non-zero components 1σ (m) σ (n) σ ( p) σ (q) 1 2 = φmnpq , m, n, p, q = 1, . . . , 8.

(5.4)

We note that the components of 1 are compatible with the components of 0 , in that 1mnpq 9 0 = 0mnpq 1 2 , m, n, p, q = 3, . . . , 8.

(5.5)

Similarly, the further 6-form having non-zero components 2σ 2 (m) σ 2 (n) σ 2 ( p) σ 2 (q) 3 4 = φmnpq , m, n, p, q ∈ {1, . . . , 8}\{3, 4}

(5.6)

is consistent with both 0 and 1 , i.e. 0mnpq 3 4 = 2mnpq 1 2 , m, n, p, q = 5, . . . , 10, 1mnpq 3 4 = 2mnpq 9 0 , m, n, p, q = 1, 2, 5, . . . , 8.

(5.7)


555

Fig. 1. The 6-form is symmetric under 2π/5 rotations generating a Z 5 symmetry. It is antisymmetric under reflections in the dotted line

In fact the five 6-forms σNN (m) σ N (n) σ N ( p) σ N (q) σ N (9) σ N (0) , N = 0, . . . , 4 are all compatible, allowing the definition of a 6-form in ten dimensions manifestly invariant under the Z5 transformations between the five ordered pairs of indices in Fig. 1 generated by σ = (13579)(24680), i.e. {1, 2} → {3, 4} → {5, 6} → {7, 8} → {9, 10} → {1, 2}.

(5.8)

This Z5 -invariant 6-form has components given by σ N (m) σ N (n) σ N ( p) σ N (q) σ N (9) σ N (0) = φmnpq , N = 0, . . . , 4, m, n, p, q = 1, . . . , 8.

(5.9)

Explicitly, for the choice (4.14), these are given by the 50 non-zero elements 123456 = 123478 = 123490 = 123579 = −123580 =−123670 =−123689 =−124570 =−124589 =−124679 = 124680 = 125678 = 125690 = 127890 =−134579 = 134580 = 134670 = 134689 = 135679 =−135680 =−135789 = 135790 = 136780 =−136890 =−145670 =−145689 = 145780 =− 145890 = 146789 =−146790 = 234570 = 234589 = 234679 =−234680 =−235670 =−235689 = 235780 =−235890 = 236789 =−236790 =−245679 = 245680 = 245789 =−245790 =−246780 = 246890 = 345678 = 345690 = 347890 = 567890 = 1.

(5.10)

The vertex space P6 (10) consists of 50 points, corresponding to a 49-valent graph having two types of vertices: Type A vertices connected to 30 vertices at distance 2, 16 vertices at distance 3 and 3 vertices at distance 4 and Type B vertices connected to 4 vertices at distance 1, 24 vertices at distance 2, 16 vertices at distance 3 and 5 vertices at distance 4. There are 10 vertices of Type A and 40 of Type B. The symmetries of this democratic form are as follows. The permutation symmetry group is the order 60 alternating group A5 of five elements, the five ordered pairs in (5.8).

556


The number of permutation antisymmetries is also 60, obtained from the elements of the permutation symmetry group by multiplication by, for example, the reflection in the vertical axis of Fig. 1: τ := (12)(03)(94)(85)(76).

(5.11)

There are 120 orthogonal symmetries and 120 orthogonal antiymmetries. The orthogonal symmetries which are not permutation symmetries have all their respective ηi = −1. In the eight dimensional subspaces orthogonal to nonexceptional planes {a, b}, not in the set (5.8), this 6-form reduces to an SU(2)-invariant 4-form, which we discuss further in Sect. 5.3. 5.2. Self-duality. The six-form given by (5.9) defines skew-symmetric endomorphisms on the space of 3-forms, yielding generalised self-duality equations analogous to (1.1), 1 6

g m 4 n 1 g m 5 n 2 g m 6 n 3 m 1 m 2 m 3 m 4 m 5 m 6 G n 1 n 2 n 3 = λG m 1 m 2 m 3 .

(5.12)

Its 4-form dual = defines a symmetric endomorphism on the space of 2-forms, satisfying equations of the form (1.1). To find the eigenvalues of a 2k-form on k R D , we identify the components of the k-forms G in D dimensions, {G m 1 m 2 ...m k , 1 ≤ m 1 < m 2 < · · · < m k ≤ D} with the components of a vector in the Dk -dimensional space k R D thus: G A = G m 1 m 2 ...m k where A = 1 +

k m

i −1

i

= 1, . . . ,

D k .

(5.13)

(5.14)

i=1

On this vector the 2k-form may be represented as a AB = m 1 ...m k m k+1 ...m 2k ,

A := 1 +

k m

i −1

i

,

i=1

D k

×

D

matrix,

k

B := 1 +

2k m i −1 . i

(5.15)

i=k+1

In this notation, self-duality equations like (5.12) and (1.1) take the form of matrix equations allowing direct evaluation of the eigenvalues and eigenvectors using an algebraic computation programme like Maple or Reduce. We find the characteristic polynomials for the 6-form (5.9) to be 4 6 35 λ6 + 51λ4 + 699λ2 + 1369 λ4 + 42λ2 + 361 λ2 + 1 λ2 + 9 (5.16) and that of its dual 4-form to be 6 (λ + 4) (λ + 1)8 (λ − 1)24 λ2 + 2λ − 19 .

(5.17)

We have checked that the stability group H ⊂ S O(10) of (or equivalently = ) has dimension 16 and is the direct product H = SU (4) ⊗ U (1).

(5.18)


557

Under this stabilty group the D = 10 dimensional vector module V and the 45, 120 and 210 dimensional spaces of the two- three- and four-forms, respectively, decompose as V = 10 = 40 + 40 + 1+1 + 1−1 , V = 45 = 150 + 60 + 60 + 4+1 + 4−1 + 4+1 + 4−1 + 10 + 10 , 2

3 V = 120 = 200 + 200 + 15+1 + 15−1 + (40 )3 + (40 )3 + (6+1 )2 + (6−1 )2 + 1+1 + 1−1 ,

(5.19)

4 V = 210 = 200 + 20+1 + 20−1 + 20+1 + 20−1 + 150 + 150 + 100 + 100 + (60 )4 + (4+1 )2 + (4−1 )2 + (4+1 )2 + (4−1 )2 + (10 )4 , where the exponent denotes the multiplicity and the subscript the U(1) charge. We identify the eigenspaces in 2 V of the 4-form as follows: λ = +1 ⇔ 15 + 4 + 4 + 1, λ = −1 ⇔ 4 + 4, λ = −4 √ ⇔ 1, λ = −1 − 2√ 5 ⇔ 6, λ = −1 + 2 5 ⇔ 6.

(5.20)

The λ = 1 eigenspace is the most interesting, satisfying a set of 21 equations amongst the 45 components of 2 V . The equations for the other eigenspaces are rather overdetermined. For the six-form, the roots of the 6th order polynomial in (5.16) correspond to the 4’s and 4’s, the roots of the quartic correspond to the four 6’s, the eigenspaces with λ = i and −i transform √ as (20 + 15) and (20 + 15), respectively, and the two singlets have eigenvalues ±i 3. 5.3. Reduction to nonexceptional planes. As we have seen, in the five exceptional eight dimensional embeddings in ten dimensions, the 6-form (5.10) reduces to the Spin(7)-invariant 4-form (4.14). Contracted with the bivectors spanning all the other planes, i.e. for {a, b} not in the set of planes in (5.8), we obtain a 4-form with 17 nonzero components T pqr s . For example in the space orthogonal to the {1, 10} plane we have T1256 = −T1678 = −T2356 = −T1357 = −T3467 = −T1458 = T2457 = −T2347 = T1247 = −T2567 = T3568 = −T1238 = T2578 = T3456 = −T2468 = T3478 = T1346 = 1.

(5.21)

This 4-form in the eight dimensional space orthogonal to the {1, 10} plane is invariant under an SU(2) subgroup of SO(8). Under this SU(2) the 8-dimensional vector module of SO(8) decomposes as: 8 = 4[0] ⊕ 2[ 21 ],

(5.22)

i.e. four spin 0 modules and two spin 21 modules. Here [s] denotes the 2s + 1 dimensional module of SU(2). The infinitesimal generators of su(2) are the following 8 × 8 matrices acting on the subspace V 2−9 spanned by the basis vectors e2 , . . . , e9 :

558


⎛

0 ⎜0 ⎜ ⎜0 1 ⎜ ⎜0 T1 = √ ⎜ 0 2 3⎜ ⎜0 ⎜ ⎝0 0 ⎛

0 ⎜0 ⎜ ⎜0 1 ⎜ ⎜0 T2 = √ ⎜ 0 2 3⎜ ⎜0 ⎜ ⎝0 0 ⎛ 0 ⎜0 ⎜ ⎜0 1⎜ ⎜0 T3 = ⎜ 6 ⎜0 ⎜0 ⎜ ⎝0 0

0 0 0 −1 0 1 0 0

0 0 0 0 −1 0 1 0

0 1 0 0 0 −1 0 0

0 0 1 0 0 0 −1 0

0 −1 0 1 0 0 0 0

0 0 −1 0 1 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

0 0 0 0 −1 0 1 0

0 0 0 1 0 −1 0 0

0 0 −1 0 1 0 0 0

0 1 0 −1 0 0 0 0

0 0 1 0 0 0 −1 0

0 −1 0 0 0 1 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ 0⎠ 0

0 0 −2 0 1 0 1 0

0 2 0 −1 0 −1 0 0

0 0 1 0 1 0 −2 0

0 −1 0 −1 0 2 0 0

0 0 1 0 −2 0 1 0

0 −1 0 2 0 −1 0 0

(5.23)

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

satisfying the standard commutation relations [Ti , T j ] = i jk Tk . These Ti ’s clearly have nontrivial action on the subspace V 3−8 spanned by the basis vectors e3 , . . . , e8 . In the eight dimensional space V 2−9 the vectors (1, 0, . . . , 0), (0, . . . , 0, 1), (0, 1, 0, 1, 0, 1, 0, 0) and (0, 0, 1, 0, 1, 0, 1, 0) are the four invariant vectors under the su(2) action. We find that the two spinor modules in (5.22) are spanned by the basis vectors √ √ √ √ {b1 =(0, 1 + 3i, 0, 1 − 3i, 0, −2, 0, 0), b2 =(0, 0, 3 + i, 0, − 3 + i, 0, −2i, 0)} (5.24) and

√ √ √ √ c1 = 0, 0, 1 + 3i, 0, 1 − 3i, 0, −2, 0 , c2 = 0, 3 − i, 0, 2i, 0, − 3 − i, 0, 0 .

(5.25) b1 and c1 are the eigenvectors of T1 with eigenvalue −i/2, whereas b2 and c2 are its eigenvectors with eigenvalue i/2. The above decomposition of the 8-dimensional vector module leads immediately to the following decomposition of the 28-dimensional space of 2-forms 28 = 9[0] ⊕ 8[ 21 ] ⊕ [1].

(5.26)

With the 4-form T defined in (5.21), the characteristic polynomial of the self-duality equation (1.1) is λ4 (λ2 − 5)(λ4 − 8λ2 + 3)4 (λ6 − 14λ4 + 33λ2 − 12).

(5.27)


559

Let us now discuss the association of the roots of (5.27) with the decomposition of the space of 2-forms (5.26) under su(2): • We identify the eight roots of (λ2 − 5)(λ6 − 14λ4 + 33λ2 − 12), which are all distinct, with eight of the nine spin [0] states in (5.26). • The four zero eigenvalues correspond to the ninth spin [0] state together with the spin [1] state. • The remaining 16 eigenvalues, the roots of (λ4 − 8λ2 + 3)4 , correspond to the spin [ 21 ] states in the decomposition (5.26). Since there are only four distinct eigenvalues ±λi , i = 1, 2 the corresponding eigenspaces transform as the four dimensional [ 21 ] ⊕ [ 21 ] representation. 6. SU(4)⊗ U(1)-Invariant 4-Forms in Ten Dimensions As we see from the decomposition of the 210 = 4 (40 + 40 + 1+1 + 1−1 ), SU(4)⊗ U(1) has four singlets. One of them arises from the tensor product 4 ⊗ 4 ⊗ 1 ⊗ 1 and the other three singlets have their origin in 4 (40 + 40 ) and correspond to the three SU(4)-invariant 4-forms discussed in Appendix B of [1]. Choosing complex coordinates z 1 = x1 + i x2 , z 2 = x3 + i x4 , z 3 = x5 + i x6 , z 4 = x7 + i x8 , z 5 = x9 + i x10 , the four SU(4)⊗ U(1)-invariant forms may be expressed thus: A = mnpqzi z¯i z j z¯ j z k z¯ k , (6.1) mnpq 1≤i< j 0 such that for all x ∈ K and all |t| T , one has 0t (x) ∈ / K. (iv) The sets G(E) ∩ supp(H − H0 ) and G 0 (E) ∩ supp(H − H0 ) are compact. In concrete situations, Assumption 2.1(iv) can be considerably relaxed; we do not pursue this direction. 2.2. Wave operators and the scattering map. For x ∈ 0E , let W± (x) = lim −t ◦ 0t (x). t→±∞

(2.2)

If K 0 ⊂ 0E is a compact, then, taking K = supp(H − H0 ) ∪ K 0 in Assumption 2.1(iii), we see that the above limits exist and are attained at finite values of t: W+ (x) = −t ◦ 0t (x), W− (x) = t ◦ 0−t (x), ∀x ∈ K 0 , ∀t T.

(2.3)

Since H0 (resp. H ) is constant on the orbits of 0 (resp. ), we get H (W+ (x)) = H (−t ◦ 0t (x)) = H (0t (x)) = H0 (0t (x)) = H0 (x), ∀x ∈ K 0 , ∀t T and in the same way, H (W− (x)) = H0 (x). It follows that W± ( 0E ) ⊂ E . However, it is easy to construct examples such that W± ( 0E ) = E . Thus, we make an additional assumption: W+ ( 0E ) = W− ( 0E ) = E .

(2.4)

566

V. Buslaev, A. Pushnitski

This is analogous to the completeness assumption in quantum scattering, see (A.1). Since t and 0t are symplectic diffeomorphisms of N for each t, it follows from (2.3) that W± : 0E → E are diffeomorphisms and that W±∗ (ω| E ) = ω| 0 . E Next, since the definition of W± can also be written as W± (x) = lims→±∞ −t−s ◦ 0t+s (x) for any t ∈ R, we get the intertwining property W± ◦ 0t = t ◦ W± , ∀t ∈ R.

(2.5)

The completeness assumption (2.4), together with the intertwining property (2.5), ensures that the non-trapping Assumption 2.1(iii) holds true also for the flow t on E . Assuming completeness (2.4), we can define the scattering map S E = W+−1 ◦ W− .

(2.6)

S E (x) = 0−t ◦ 2t ◦ 0−t (x), ∀x ∈ K 0 , ∀t, |t| T.

(2.7)

By (2.3), one can write

It follows that S E : 0E → 0E is a diffeomorphism onto 0E and S E∗ (ω| 0 ) = ω| 0 . E

E

(2.8)

From (2.5) (or directly from (2.7)) it follows that S E ◦ 0t = 0t ◦ S E , ∀t ∈ R.

(2.9)

The scattering map is often defined initially on the whole of N (or for some range of energies) and then restricted onto 0E . The above constructions are very well known; see e.g. [13,14,24,30]. The fact that the wave operators and the scattering map are symplectic transformations is particularly emphasized in the works by W. Thirring, see [30 or 22,29].

2.3. Symplectic reduction and the Poincaré scattering map. One can consider the set of 0 . Then all orbits of 0t on the constant energy surface 0E as a symplectic manifold E 0 0 → which is sometimes called the the map S E generates a quotient map SE : E E Poincaré scattering map; it is not difficult to see that S E is a symplectic diffeomorphism. Let us discuss the details of this construction. By Assumption 2.1(i)–(iii), the action of the group 0 on 0E is smooth, proper and free, and therefore (see [1, Prop. 4.1.23]) the orbit space admits a smooth manifold 0 is a submersion. It is easy to construct structure and the quotient map π0 : 0E → E 0 charts on E by choosing sufficiently small (2n − 2)-dimensional submanifolds of 0E such that X 0 is non-tangential to these manifolds; see the proof of Lemma 5.1 below. If 0 can be identified 0 , then the tangent space Ty x ∈ 0E is a point of an orbit y ∈ E E 0 with the quotient space Tx E / span{X 0 (x)}. There exists a unique symplectic form ω0 0 such that π ∗ on ω = ω| ; see e.g. [1, Theorem 4.3.1 and Example 4.3.4(ii)]. It 0 0 E 0 E is not difficult to prove that if N is exact (i.e. there exists a 1-form α on N such that 0 is also exact, see Lemma 5.1 below. ω = dα), then E

Scattering Matrix in Hamiltonian Mechanics

567

If f : 0E → R is a smooth function such that f ◦ 0t = f for all t ∈ R, then f 0 → R such that generates a smooth function f : f ◦ π0 = f . In a similar way, by E (2.9), the scattering map S E generates the diffeomorphism 0E → 0E , π0 ◦ S E = SE : S E ◦ π0 .

(2.10)

From (2.8) and (2.10) one easily obtains π0∗ ( S E∗ ω0 − ω0 ) = 0. S E∗ ω0 = ω0 , i.e. S E is a symplectic diffeomorSince π0 is a surjection, it follows that phism. It is interesting to note that S E does not have to be homotopic to the identity map, see Example 4.3 below. Since the action of on E is also free, smooth and proper, one can consider the sym E of the orbits of on E , with the natural projection π : E → E plectic manifold and a symplectic form ω on E . By the intertwining property (2.5), there exist symplectic diffeomorphisms ± : ± ◦ π0 = π ◦ W± . 0E → E , W W

(2.11)

In the case n = 1 the above reduction produces a “manifold” of dimension zero, i.e. a discrete set of orbits. The Poincaré scattering map becomes just a permutation map on the set of these orbits. In this case, integration over the “volume forms” ωn−1 , ω0n−1 , n = 1 will be understood simply as summation over this set of orbits. 0 , E , we Since we are going to discuss integration of forms over N , 0E , E , E should fix orientation on these manifolds. Orientation on N is fixed in such a way that the form ωn is positive on a positively oriented basis. In the same way, orientation on 0 is fixed in such a way that the form ω0n−1 is positive on a positively oriented basis. E 0 Orientation on E is fixed such that if (e1 , . . . , e2n−1 ) is a positively oriented basis in Tx 0E and ξ ∈ Tx N is such that dx H0 (ξ ) > 0, then (ξ, e1 , . . . , e2n−1 ) is a positively oriented basis in Tx N . In other words, 0E is considered as a boundary of G 0 with E and E is fixed in a similar way to 0 , 0 . induced orientation. Orientation on E E 2.4. Poincaré section. The above procedure of symplectic reduction looks particularly simple if one makes Assumption 2.2. There exists a smooth submanifold ⊂ 0E of dimension 2n − 2 such that: / Tx for all x ∈ ; (a) X 0 (x) ∈ (b) for all x ∈ 0E , there exists a unique z = z(x) ∈ and a unique t = t (x) such that x = 0t (z). In this case, the elements x ∈ 0E can be considered as pairs (z, t) ∈ × R such that x = t (z). It is easy to see that Assumption 2.1(i),(ii) + Assumption 2.2 ⇒ Assumption 2.1(iii). 0 is a diffeoLet i : → 0E be the natural embedding. Then γ0 = π0 ◦ i : → E ∗ morphism and γ0 ω0 = ω| . Thus, can be considered as a “concrete” realisation of 0 . the “abstract” manifold E

568


Using the above identification of 0E and ×R, the free dynamics can be represented as 0s : (z, t) → (z, t + s) and the scattering map as S E : (z, t) → (z , t ),

z = s E (z), t = t − τ E (z),

(2.12)

S E ◦ γ0 : → is a symplectic diffeomorphism, and τ E : → R where s E = γ0−1 ◦ is a smooth function. The map τ E is often called time delay, or sojourn time. We note that the definition (2.12) of τ E depends on the choice of ; there is no invariant way of 0 . defining a time delay function on E The manifold is, of course, the well known Poincaré section; see e.g. [1, Sect. 7.1]. The map s E : → in concrete cases appeared before in physics literature in [11, 15,27] and in mathematical literature in [3,16]. The connection of s E with the “quantum” scattering matrix in the framework of semiclassical analysis has been discussed in physics literature, see e.g. [27] and in mathematical literature in [3]. One of the earliest rigourous works which established the connection between the “quantum” scattering matrix and the analogous classical objects was [31].

2.5. Total time delay. Although the definition of time delay τ E above depends on the choice of , the total (or average) time delay TE can be defined (see (2.15) below) in an invariant way. Let 1 ⊂ 2 ⊂ · · · ⊂ N be a sequence of open pre-compact sets such that 0 0 ∪∞ k=1 k = N . Let us define the functions u k : E → R and u k : E → R by u 0k (x) = u k (x) =

∞

−∞ χk

◦ 0t (x)dt, x ∈ 0E ,

(2.13)

−∞ χk

◦ t (x)dt, x ∈ E .

(2.14)

∞

It is straightforward to see that u 0k ◦0t = u 0k and u k ◦t = u k for all t ∈ R and therefore 0 → R, E → R such that u 0k : uk : u 0k ◦ π0 = u 0k and u 0k , u k generate functions E uk ◦ π = uk . Theorem 2.3. Suppose Assumption 2.1 and completeness (2.4) hold true, and let 1 ⊂ 2 ⊂ · · · ⊂ N and u 0k , u k be as defined above. Then the limit TE = lim

k→∞

ωn−1 (y) − u k (y) (n − 1)! E

0 E

ωn−1 (y) u k0 (y) 0 (n − 1)!

(2.15)

exists and is independent of the choice of the sequence {k }. If, in addition, Assumption 2.2 holds true and τ E : → R is as defined by (2.12), then TE =

τ E (x)

ωn−1 (x) . (n − 1)!

(2.16)

TE is the total time delay. The above statement (in various concrete forms) is well known; for completeness, we give a proof in Sect. 7. See [10] for a survey of time delay and [25] for a rigourous discussion of semiclassical aspects.


569

− : 0 → E (see (2.11)) is a symplectic diffeomorphism, the Remark 2.4. Since W E difference of the integrals in (2.15) can be rewritten as

ωn−1 (y) − u k (y) (n − 1)! E

0 E

ωn−1 (y) u k0 (y) 0 (n − 1)!

=

0 E

− (y) − ( uk ◦ W u k0 (y))

ω0n−1 (y) . (n − 1)! (2.17)

− (y) − The quantity uk ◦ W u k0 (y) and its limit − (y) − uk ◦ W u k0 (y)) lim (

k→∞

(2.18)

− ) is often interpreted as the time delay related + instead of W (or similar objects with W to the orbit y. Note, however, that the limit (2.18) may not exist unless the sequence k is chosen in a special way; see [12,28] for a discussion of this issue.

2.6. Regularised phase space volume. Suppose that Assumption 2.1(iv) holds true for some E. Let us denote ωn (2.19) (χG 0 (E) − χG(E) ) . ξ(E) = n! N It is interesting to note that if ±(H (x) − H0 (x)) 0 for all x ∈ N , then ±ξ(E) 0. The “quantum” analogue of ξ(E) is the spectral shift function. See the survey [26] for an extensive discussion of this analogy in semiclassical context. 3. Main Results We will use some notation and terminology from symplectic topology. We collect the required material in Appendix B; for the details, see [20]. In particular, we use the notion of the Calabi invariant. Let M be an exact non-compact symplectic manifold; the Calabi invariant CAL is a functional on a certain subset (which we denote by Dom(CAL, M)) of the set of symplectic diffeomorphisms of M. We note that our sign conventions and normalisation of the Calabi invariant are different from those of [20]. In Appendix B (see Sect. B.3) we explain the analogy between CAL and the functional −Im log det on the set of unitary operators. 3.1. CAL( S E ) and the regularised phase space volume. Theorem 3.1. Let n 2; suppose that Assumption 2.1 and completeness (2.4) hold true 0 is non-compact. Then the for some E ∈ R. Assume also that N is exact and that E 0 → 0 belongs to Dom(CAL, 0 ) and the identity map SE : E E E CAL( S E ) = ξ(E) holds true.

(3.1)

570


This should be compared to the Birman-Krein formula (A.3) in “quantum” scattering theory, bearing in mind the analogy between the Calabi invariant and the logarithm of determinant, see Sect. B.3. The proof of Theorem 3.1 is given in Sects. 5 and 6 entirely in the framework of Hamiltonian mechanics. The proof can probably be obtained, at least in some particular cases, by analysing the semiclassical asymptotics of the Birman-Krein formula. However, this route is likely to be much more technically difficult. Similarly to the Birman-Krein formula, the relation (3.1) can be applied to a wide variety of concrete situations. In Sect. 4 we discuss applications to scattering of classical particles.

3.2. The scattering matrix and the total time delay. Theorem 3.2. Suppose that Assumption 2.1 and completeness (2.4) hold true for some E ∈ R. Suppose also that Assumption 2.1(iv) holds true for some E 1 > E. Then the derivative ddξE (E) exists and the identity TE = −

dξ (E) dE

(3.2)

holds true. This result in various concrete forms appeared before in physics literature; see e.g. [6,19,22]. For completeness, we give a proof in Sect. 7. Combining Theorem 3.2 with Theorem 3.1, we get d CAL( S E ) = −TE . dE

(3.3)

This should be compared to the Eisenbud-Wigner formula (A.4) in “quantum” scattering. A related result was obtained in [5] in the framework of Hilbert space classical scattering.

4. Application to Classical Mechanics 2n 4.1. The ngeneral case. Let N = R , n 2, with the2nstandard symplectic form ω = dp ∧ dqi , (q1 . . . qn , p1 . . . pn ) = x ∈ R . Of course, N is exact, i=1 i ω = d(− i qi dpi ). We denote by ·, · the usual inner product in Rn , and |q|2 = q, q. Let

H0 (q, p) =

1 2 | p| + v0 (q), 2

where v0 ∈ C ∞ (Rn ) satisfies the following assumptions: 1 sup (v0 (q) + q, ∇v0 (q)) < ∞, 2 n q∈R

(4.1)

inf v0 (q) > −∞.

(4.2)

q∈Rn


571

The quantity v0 + 21 q, ∇v0 in (4.1) is known as the virial. Next, let H (q, p) = H0 (q, p) + v(q), where v ∈ C0∞ (Rn ). Let E ∈ R be such that E is not a critical value of v0 or v0 + v; 1 E > sup (v0 (q) + q, ∇v0 (q)). 2 q∈Rn

(4.3) (4.4)

Fix any R ∈ R and consider

= {(q, p) | h 0 (q, p) = E, q, p = R} ⊂ 0E .

(4.5)

Lemma 4.1. Assume (4.1) through (4.4). Then Assumption 2.1 and Assumption 2.2 hold true with as in (4.5). Proof. Assumption 2.1(i) follows from (4.3). Assumption 2.1(ii) follows from (4.2), since q˙ = p and | p|2 = 2(E − v0 ) C < ∞. Let us check Assumption 2.1(iii). For any trajectory (q(t), p(t)) = 0t (x), we have d d |q(t)|2 = 2 q(t), p(t) = 2| p(t)|2 − 2q(t), ∇v0 (q(t)) C > 0 dt dt by (4.4), and so |q(t)| → ∞ as t → ±∞, as required. Let us check Assumption 2.1(iv). We have

(4.6)

1 G 0 (E) ∩ supp(H − H0 ) = {(q, p) | q ∈ supp v, | p|2 E − v0 (q)}; 2 using (4.2) we see that this set is compact. In the same way, G(E) ∩ supp(H − H0 ) is compact. Let us check Assumption 2.2. In order to check that is a smooth manifold in 0E , it suffices to verify that d H0 and dq, p are linearly independent on . Suppose that d H0 = λdq, p at some point (q, p) ∈ . Then ∇v0 (q) = λp and p = λq, and so E = 21 | p|2 + v0 (q) = 21 ∇v0 (q), q + v0 (q) < E by (4.4) — contradiction. Next, we need to check that X 0 is non-tangential to . We have X 0 (q, p) = ( p, −∇v0 (q)). The tangent space T(q, p) consists of vectors (ξ, η) such that ξ, p + η, q = 0 and ξ, ∇v0 (q) + η, p = 0. For (ξ, η) = X 0 (q, p) we have ξ, p + η, q = | p|2 − q, ∇v0 (q) = 2(E − v0 (q)) − q, ∇v0 (q) > 0 by (4.4), and so X 0 (q, p) is not in T(q, p) . d Finally, by (4.6), for any trajectory (q(t), p(t)) = 0t (x), we have dt q(t), p(t) C > 0 and so there exists a unique t ∈ R such that q(t), p(t) = R. Thus, (4.1)–(4.4) ensure that the wave operators W± : 0E → E exist. In order to ensure that completeness (2.4) holds true, we have to make more specific assumptions. Let us assume that E > sup (v0 (q) + v(q) + 21 q, ∇v0 (q) + 21 q, ∇v(q)). q∈Rn

(4.7)

Then by the same argument as in the proof of Lemma 4.1 (see (4.6)), we obtain that all trajectories of t leave supp v for large |t| and therefore coincide with some trajectories of the free dynamics for large ±t > 0. From here we get completeness (2.4). To summarise: Suppose that (4.1)–(4.4), (4.7) hold true. Then Assumptions 2.1, (2.4) and Assumption 2.2 hold true and therefore Theorems 3.1 and 3.2 hold true. We note without proof that under the above assumptions one can check that 0 ). S E ∈ Hamc ( E

572


4.2. The case v0 ≡ 0. In the case v0 ≡ 0 it is easy to give explicit formulas for the phase space volume, time delay, and the Calabi invariant of SE . The formula for the regularised phase space volume (2.19) is very well known (see e.g. [26] for a comprehensive discussion) and easy to compute: n/2 ξ(E) = κn 2n/2 E − (E − v(q))+ d n q, E > 0, (4.8) Rn

π where κn = (1+n/2) is the volume of a unit ball in Rn , d n q is the Lebesgue measure in n R , and (a)+ = (a + |a|)/2. Next, the formula for the time delay is also well known; see e.g. [21]. Let x− = − (q , p − ) ∈ and let x+ = (q + , p + ) = S E (x− ). Then the time delay function τ E from (2.12) can be expressed as n/2

τ E (x− ) =

1 (q − , p − − q + , p + ), 2E

and the total time delay is TE =

τ E (x− )

×

ωn−1 (x− ) = (2E)(n−3)/2 d n−1 pˆ − (n − 1)! Sn−1

q − , p − =R

d n−1 q − (q − , p − − q + , p + ),

where pˆ − = p − /| p − |, and d n−1 pˆ − is the Lebesgue measure on the unit sphere in Rn . Let us display formula (3.2) in the cases n = 2, 3: TE = 0,

√ TE = 6π 2

R3

x−

1/2

E − (E − v(q))+

(n = 2), d 3 q,

(n = 3).

Finally, let us give a formula for CAL( S E ). As above, we have S E : 0E → 0E , − − + + = (q , p ) → (q , p ); define −

ρ(x ) =

n i, j=1

qi− q +j

∂ p +j ∂qi−

(q − , p − ).

It is straightforward to see that this definition is independent of the choice of the coordinate system in Rn . Lemma 4.2. Under the assumptions (4.1)–(4.4), (4.7), one has ρ ◦ 0t = ρ for all t ∈ R and 1 ωn−1 (x) CAL( SE ) = − ρ(x) (4.9) n(n − 1)

(n − 1)! (2E)(n−1)/2 n−1 d pˆ d n−1 q ρ(q, p). (4.10) =− n(n − 1) Sn−1 q, p=R


573

0 → R such that Remark. (1) Since ρ ◦ 0t = ρ, there exists a function ρ : E ρ ◦ π0 = ρ. Then (4.9) can be rewritten in an invariant form as ωn−1 (z) 1 . ρ (z) 0 CAL( S E ) = − n(n − 1) (n − 1)! 0 E (2) It is possible to prove Lemma 4.2 by a direct calculation in local coordinates. Instead, below we give a proof which clarifies the main idea behind the definition of ρ and can be adapted to more general situations. n Proof. 1. Fix α = − i=1 qi dpi . Let us prove that the identity 1 (S E∗ α) ∧ α ∧ ωn−1 = − ρωn n

(4.11)

holds true in some open neighbourhood of 0E . Here S E should be understood as an extension of the map S E : 0E → 0E into a neighbourhood of 0E . More precisely, if the hypothesis of the lemma is fulfilled for the energy E, then it is clearly also fulfilled for an open interval of energies (E 1 , E 2 ) E. Then S E can be extended to {x | E 1 < H0 (x) < E 2 } by using the same construction. In order to check (4.11), we note that S E∗ α = −

n

qi+ dpi+ = −

i=1

n i,k=1

qi+

∂ pi+

n

∂qk

i,k=1

− − dqk −

qi+

∂ pi+

∂ pk−

dpk− ,

and therefore (S E∗ α) ∧ α ∧ ωn−1 ⎛ ⎞ ⎛ ⎞ n n n + + ∂ p ∂ p −⎠ i i =⎝ qi+ − dqk− + qi+ − dpk− ⎠ ∧ ⎝ q− ∧ ωn−1 j dp j ∂q ∂ p k k k,i=1 k,i=1 j=1 ⎛ ⎞ ⎛ ⎞ n n ∂ pi+ − −⎠ =⎝ qi+ − dqk ⎠ ∧ ⎝ q− ∧ ωn−1 j dp j ∂q k k,i=1 j=1 =

n 1 + − ∂ pi+ − qi q j dq j ∧ dp −j ∧ ωn−1 n ∂q − j

=−

i, j=1 n

1 n

i, j=1

qi+ q − j

∂ pi+

ω ∂q − j

n−1

1 = − ρωn , n

as required. 2. By inspection, α(X 0 ) = 0 and (t0 )∗ α = α.

(4.12)

This is crucial for our proof. Since S commutes with 0t (see (2.9)), it follows that (0t )∗ S ∗ α = S ∗ (0t )∗ α = S ∗ α. Using these relations and applying (0t )∗ to both sides of (4.11), we get ρ ◦ 0t = ρ.

574


3. Let us prove that (S E∗ α) ∧ α ∧ ωn−2 = −

1 ρωn−1 on 0E . n−1

(4.13)

Choose a vector field Y in the neighbourhood of 0E such that d H0 (Y ) = −1. Then i(X 0 )ω = −d H0 = 0 on 0E and i(Y )i(X 0 )ω = ω(X 0 , Y ) = −d H0 (Y ) = 1, i(Y )i(X 0 )ω = (n − 1)(i(Y )i(X 0 )ω)) ∧ ωn−2 = (n − 1)ωn−2 on 0E . By the same argument, one obtains i(Y )i(X 0 )ωn = nωn−1 on 0E .

(4.14)

Using these identities and (4.12), we get i(Y )i(X 0 )(S ∗ α) ∧ α ∧ ωn−1 = i(Y )((S ∗ α) ∧ α ∧ (i(X 0 )ωn−1 ) = (S ∗ α) ∧ α ∧ i(Y )i(X 0 )ωn−1 = (n − 1)(S ∗ α) ∧ α ∧ ωn−2 on 0E . (4.15) Applying i(Y )i(X 0 ) to both sides of (4.11) and using (4.14), (4.15), one obtains (4.13). 4. By the definition (2.12) of s E , we have S E (z) = 0−τ (z) ◦ s E (z), z ∈ . It follows that dz S E (ξ ) = dz (0t ◦ s E )(ξ )|t=−τ (z) − X 0 (S E (z))dz τ (dz s E (ξ )), for any ξ ∈ Tz . By (4.12), we get α(dz S E (ξ )) = α(dz (0t ◦ s E )(ξ ))|t=−τ (z) = α(dz s E (ξ )). It follows that s E∗ α = (S E∗ α)| .

(4.16)

5. According to formula (B.4) for CAL, we have 1 CAL( S E ) = CAL( sE ) = n!

(˜s E∗ α) ∧ α ∧ ωn−2 .

From here, using (4.16) and (4.13), one obtains (4.9). Formula (4.10) is just (4.9) with ωn−1 expanded.


575

4.3. The case of rotationally symmetric v. Let v0 ≡ 0 and let v be rotationally symmetric, v(x) = v1 (|x|). Then formula (4.9) can be recast in terms of the usual variables of scattering theory: the impact parameter s and the scattering angle φ (see [17, Sect. 18]). For (q, p) ∈ R2n , p = 0, the impact parameter s > 0 is defined by q, p p . s = q − | p|2 Due to the conservation of angular momentum, the impact parameter is the integral of motion for both dynamics 0 , . If (q + , p + ) = S E (q − , p − ), then the scattering angle φ is defined such that cos φ =

p+ , p− . | p + |2

(4.17)

Of course, this does not yet define φ uniquely. In order to fix φ, let us note that due to the rotational symmetry, φ depends only on E and s. Thus, for a fixed energy E let us define φ as a continuous function of s ∈ (0, ∞) such that (4.17) holds true and sin φ =

p+ , q − , | p + ||q − |

where q − is chosen such that q − , p − = 0. In other words, φ(s) 0 for small repulsive potentials (v 0) and φ(s) 0 for small attractive potentials (v 0). Formula for φ is well known (see e.g. [17, Sect. 18]): −1/2 v1 (r ) s 2 sdr 1 − − , 2 E r2 rmin r v1 (r ) s 2 − 2 =0 . rmin (s) = max r : 1 − E r

∞

φ(s) = π − 2

(4.18)

It is easy to compute that ρ = ρ(s) = and therefore CAL( S E ) = −κn κn−1 (2E)n/2

∞

s 0

n dφ

ds

√

2Es 2

dφ , ds

ds = nκn κn−1 (2E)

n/2

∞

s n−1 φ(s)ds.

0

(4.19) Substituting (4.19) into (4.19), and using (4.8), it is not difficult to check directly the validity of Theorem 3.1 in this case. Example 4.3. Let us give an example where the Poincaré scattering map is not homotopic to the identity map. Let n = 2, v0 = 0 and v be of the form v(q) = v1 (|q|), v1 0, v1 (r ) = 0 for r 1, v1 (r ) < 0 for all r ∈ (0, 1). Let E ∈ (0, v1 (0)). Under these assumptions, (4.7) may or may not hold true. However, using the separation of variables, one can directly check that the completeness condition (2.4) holds true.

576


In order to make our notation more succinct, let us identify R2 with C in the usual way. Then the trajectories q = q(t), p = p(t) of the free dynamics 0 can be parameterised by θ ∈ [0, 2π ) and σ ∈ R so that √ √ q(t) = iσ eiθ + t 2Eeiθ , p(t) = p(0) = 2Eeiθ . Of course, |σ | is the impact √ parameter. 0 , and so 0 can be It is easy to see that σ 2E, θ are symplectic coordinates on E E ∗ 1 identified with a cylinder T S (as in Example B.3). Due to the conservation of angular momentum, the Poincaré scattering map has the form S E : (σ, θ ) → (σ, θ + ϕ(σ )), where ϕ(σ ) = 0 for σ −1 and ϕ(σ ) = 2π for σ 1. Here ϕ is the scattering angle with a different normalisation. Since the map S E “twists” the cylinder T ∗ S1 , it is easy to see that S E is not homotopic to the identity map: SE ∈ / Sympc0 (T ∗ S1 ). 5. Auxiliary Statements 0 . Recall that N is called exact if there exists a 1-form α on N such 5.1. Exactness of E 0 . In the course that dα = ω. Here we prove that exactness of N implies exactness of E 0 . of the proof, we construct an atlas on E Lemma 5.1. Let Assumption 2.1 (i)–(iii) hold true and suppose that N is exact. Then the 1-form α on N such that dα = ω can be chosen in such a way that i(X 0 )α = 0 on 0E . Moreover, there exists a 1-form α on

0 E

such that d α= ω0 and

(5.1) π0∗ α=

α| 0 . E

0 . Fix y0 ∈ 0 and choose x0 ∈ y0 . Using Proof. 1. First let us construct an atlas on E E local coordinates around x0 , it is easy to construct a manifold 0 ⊂ 0E of dimension 2n − 2 such that x0 ∈ 0 and X 0 (x) ∈ / Tx 0 for all x ∈ 0 . By Assumption 2.1(iii), there exists T > 0 such that for all |t| T , one has 0t ( 0 ) ∩ 0 = ∅; here 0 is the closure of 0 in 0E . It follows that the orbit 0t (x0 ) can intersect 0 only finitely many times. By reducing 0 if necessary, we can ensure that 0t (x0 ) ∈ / 0 for all t = 0. Next, since we may assume 0 to be pre-compact, there exists > 0 such that for all 0 < |t| < , one has 0t ( 0 ) ∩ 0 = ∅. Since the closed set K = {0t (x0 ) | |t| T } has empty intersection with 0 , there exists an open neighbourhood of K which has empty intersection with 0 . It follows that one can choose a subset

⊂ 0 (which is itself a manifold of dimension 2n −2) such that 0t ( )∩ = ∅ for all t = 0. Thus, we have constructed a “local Poincaré section”, i.e. parameterises {0t (x) | t ∈ R, x ∈ } ⊂ 0E rather than the whole manifold 0E . 0 ; then the correspondSuch a manifold can be constructed for any point y0 ∈ E 0 . Let us choose a locally finite subcover ing sets π0 ( ) form an open cover of E 0 subordinate to {π0 ( j )} of this cover and a smooth partition of unity { ζ j } on E this subcover (i.e. for each j, supp ζ j lies entirely within one of the sets π0 ( j )). ζ j ◦ π0 . Clearly, we have an associated partition of unity {ζ j } on 0E , ζ j = Of course, if Assumption 2.2 holds true, then the above atlas can be chosen to consist of just one map.


577

2. Since N is exact, there exists a 1-form β on N such that dβ = ω. Let us construct F ∈ C ∞ (N ) such that α = β + d F has the required properties. We will construct F using the atlas described above. For each j, let us construct a function F j on j = {0t (x) | t ∈ R, x ∈ j } such that β(X 0 ) + d F j (X 0 ) = 0 on j . This can be done by setting F j = 0 on j and then extending F j onto j by integrating the differential equation d F j ◦ 0t (x) = −β(X 0 ◦ 0t (x)) dt

(5.2)

along the orbits of 0 . Now let us define F = j ζ j F j on 0E and extend F onto the whole of N as a smooth function. Then on 0E we have i(X 0 )α = i(X 0 )β + i(X 0 )d F =

ζ j (i(X 0 )β + i(X 0 )d F j ) +

j

F j i(X 0 )dζ j .

j

(5.3) The first sum in the r.h.s. of (5.3) vanishes by the construction of F j . Next, by the construction of ζ j we have 0=

d ζ j ◦ 0t (x)|t=0 = dζ j (X 0 (x)), dt

and so the second sum in the r.h.s. of (5.3) also vanishes. Thus, α satisfies (5.1). 0 such that π ∗ 3. Let us prove that there exists a 1-form α on 0 α = α. First note that, E by Cartan’s formula for Lie derivative, L X 0 α = di(X 0 )α + i(X 0 )dα = 0 + i(X 0 )ω = −d H0 = 0 on 0E , where we have used (5.1). It follows that (0t )∗ α| 0 = α| 0 E

E

for all t ∈ R.

(5.4)

Next, let x, y ∈ 0E and let ξ ∈ Tx 0E , η ∈ Ty 0E be such that π0 (x) = π0 (y) and dx π0 (ξ ) = dy π0 (η). This means that for some t, c ∈ R, one has y = 0t (x) and η = dx 0t (ξ ) + cX 0 (y). Then, using (5.1) and (5.4), we obtain αy (η) = αy (dx 0t (ξ ) + cX 0 (y)) = αy (dx 0t (ξ )) = αx (ξ ). Thus, α : T 0E → R is constant on the pre-images of any point under the map 0 . This shows that one can define a smooth 1-form 0 dπ0 : T 0E → T α on E E ∗ such that π0 α = α. 4. Let us prove that d α= ω0 . We have π0∗ ω0 = ω = dα = dπ0∗ α = π0∗ d α, and so π0∗ ( ω0 − d α ) = 0. Since π0 is a surjection, it follows that d α= ω0 .

578


5.2. Separation of variables in integrals over 0E . We will need the following version of Fubini’s theorem: Lemma 5.2. Let µ be a 1-form on 0E and ⊂ 0E be an open pre-compact set. Define ∞ δ(x) = χ ◦ 0t (x) (i(X 0 )µ) ◦ 0t (x)dt, x ∈ 0E , −∞

0 → R be the corresponding function such that δ ◦ π0 = δ. Then and let δ: E δ ω0n−1 . µ ∧ ωn−1 = 0 E

We note that this lemma, with obvious modifications, can (and will) also be applied E instead of 0 , 0 . to integrals over E and E E Proof. 1. Let ζ j , ζ j , j be as in the proof of Lemma 5.1. It suffices to prove that ζj δ ω0n−1 . ζ j µ ∧ ωn−1 = (5.5) 0 E

0 which is a symplectic diffeoNext, as in Sect. 2.4, we have a map γ j : j → E morphism onto its range. Using this map, we can rewrite the integral in the r.h.s. of (5.5) as the integral over j . Thus, it suffices to prove that ∞ n−1 n−1 ζjµ ∧ ω = ζ j (x)ω (x) χ ◦ 0t (x) (i(X 0 )µ) ◦ 0t (x)dt.

j

−∞

(5.6) 2. Let us prove (5.6). Consider the map φ : R × j → 0E , φ(t, x) = 0t (x). We have n−1 ζjµ ∧ ω = (ζ j ◦ φ)(χ ◦ φ)(φ ∗ µ) ∧ (φ ∗ ω)n−1 . (5.7) R× j

Since 0t is a symplectic map, we get for any ξ, η ∈ Tx j : (φ ∗ ω)(t,x) (ξ, η) = ω0 (x) (dx 0t (ξ ), dx 0t (η)) = ωx (ξ, η). t

∂ Further, let (t, x1 , . . . , x2n−2 ) be local coordinates on R× j and let ∂t∂ , ∂x∂ 1 , . . . , ∂x2n−2 be the corresponding basis in the tangent space T(t,x) (R × j ). For any ξ ∈ Tx j , we have ∂ ∗ , ξ = ωx (X 0 (x), dx 0t (ξ )) = −dx H0 (dx 0t (ξ )) = 0, (φ ω)(t,x) ∂t

and so

∂ ∂ ∂ ((φ µ) ∧ (φ ω) )(t,x) , ,..., ∂t ∂x1 ∂x2n−2 ∂ ∂ 0 n−1 . = µ0 (x) (X 0 (t (x)))ωx ,..., t ∂x1 ∂x2n−2 ∗

∗

n−1

It follows that we can separate integration over j and over R in (5.7), which yields the required identity (5.6).


579

6. Proof of Theorem 3.1 6.1. A generating function of S E . Here we use the notion of a generating function of a symplectic map; see Appendix B. Let α be a 1-form on N as in Lemma 5.1. Let us define a function on E by ∞ (x) = − (i(X )α) ◦ t (x)dt, x ∈ E . (6.1) −∞

Note that by (5.1), for all sufficiently large |t| one has (i(X )α) ◦ t (x) = (i(X 0 )α) ◦ t (x) = 0 and so the integration in (6.1) is actually performed over a bounded set of t. It follows that ∈ C ∞ ( E ). It follows directly from the definition that is constant on the orbits of t and ∈ C ∞ ( E ) such that ◦ π = . Since E ∩ supp(H − H0 ) therefore there exists is compact and so ∈ C ∞ ( E ). is compact, it follows that supp 0 0 ) Lemma 6.1. Under the assumptions of Theorem 3.1, we have S E ∈ Dom(CAL, E and ωn−1 (x) (x) n CAL( S E ) = . (6.2) (n − 1)! E Proof. 1. First recall the well known formula for the derivative of reduced action. Let T > 0 and T f T (x) = (i(X )α) ◦ t (x)dt, x ∈ E ; 0

then d f T = ∗T α − α + T d H.

(6.3)

On E , we have H = E and so the last term in the r.h.s. of (6.3) vanishes. 2. Let us check that the function 0 = ◦ W− ∈ C ∞ ( 0E ) satisfies the identity α − S E∗ α = d0

on 0E .

(6.4)

Let K ⊂ 0E be a compact set. We can choose T > 0 sufficiently large so that for all x ∈ K , S E (x) = 0−T ◦ 2T ◦ 0−T (x) and

(x) = −

T −T

(i(X )α) ◦ t (x)dt = − f 2T ◦ −T (x).

Thus, using (5.4) and (6.3), we have on K : S E∗ α = (0−T )∗ (2T )∗ (0−T )∗ α = (0−T )∗ (d f 2T + α) = d( f 2T ◦ 0−T ) + α = d( f 2T ◦ −T ◦ W− ) + α = −d( ◦ W− ) + α = −d0 + α, which proves (6.4).

580


0 ∈ C ∞ ( 0 ) be a function such that 0 ◦ π0 = 0 . Using the formula 3. Let 0 E ∗ α = π0 α and (2.10), we can rewrite (6.4) as 0 . π0∗ ( α − S E∗ α ) = π0∗ d Since π0 is a surjection, it follows that 0 α − S E∗ α = d

0E , on

(6.5)

0 is an 0 ) and, accordi.e. α -generating function of S E . Thus, S E ∈ Dom(CAL, E ing to the definition (B.3), ωn−1 (x) 0 (x) 0 n CAL( SE ) = . (6.6) (n − 1)! 0 E − be as in (2.11). Since W −∗ 4. Let W ω= ω0 , by a change of variable in the integral (6.6) we obtain (6.2). 6.2. Application of Stokes’ formula. Below we apply Stokes’ formula to rewrite the integral (2.19) in the definition of ξ . The following statement is used in the proof of Theorem 3.1 but it might be of an independent interest. Proposition 6.2. Let α be a 1-form on N as in Lemma 5.1; then the identity 1 ξ(E) = − α ∧ ωn−1 n! E

(6.7)

holds true. Proof. 1. We first note that α ∧ ωn−1 | 0 = 0.

(6.8)

E

Indeed, i(X 0 )α| 0 = 0 by (5.1), and i(X 0 )ω = −d H0 = 0 on 0E since 0E is E

a constant energy surface. Thus, i(X 0 )(α ∧ ωn−1 )| 0 = 0; since α ∧ ωn−1 has a E

maximal rank on 0E , it follows that (6.8) holds true. 2. Let ⊂ N be a compact set with a smooth boundary such that supp(H − H0 ) ∩ G(E) ⊂ and supp(H − H0 ) ∩ G 0 (E) ⊂ . Since E \ = 0E \, by (6.8) the integrand in (6.7) vanishes outside . Thus, the integration in (6.7) is in fact performed over E ∩ and so the r.h.s. is finite. 3. Writing G = G(E) and G 0 = G 0 (E) for brevity, we get ωn ωn ωn ωn ξ(E) = (χG 0 − χG ) (χG 0 − χG ) = = − . n! n! N N ∩ G 0 ∩ n! G∩ n! We have d(α ∧ ωn−1 ) = ωn and therefore, by the Stokes’ formula, n−1 n!ξ(E) = α∧ω − α ∧ ωn−1 ∂(G 0 ∩) ∂(G∩) n−1 n−1 = α∧ω − α∧ω =− α ∧ ωn−1 0E ∩

E ∩

since the integrals over G 0 ∩ ∂ and G ∩ ∂ cancel out.

E


581

6.3. The rest of the proof of Theorem 3.1. By (6.2) and (6.7), it remains to prove that n−1 (x) ωn−1 (x). − α∧ω = E

E

This follows from Lemma 5.2 with µ = α, = E . 7. Time delay: Proof of Theorems 2.3 and 3.2 7.1. Proof of Theorem 2.3. 1. Our first aim is to prove that the r.h.s. of (2.17) is independent of k for all sufficiently 0 = π0 (K 0 ). It is easy to see that large k. Denote K 0 = supp(H − H0 ) ∩ 0E and K − − 0 and therefore the integration in the r.h.s. of (2.17) is supp( uk ◦ W u 0k ) ⊂ K 0 . actually performed over K By Assumption 2.1(iii), there exists T > 0 such that for all |t| T one has 0t (K 0 ) ∩ K 0 = ∅.

(7.1)

Then we have t ◦ W− (x) = 0t (x), t −T, ∀x ∈ K 0 , t ◦ W− (x) = 0t ◦ S E (x), t T ∀x ∈ K 0 . Let us choose sufficiently large so that for all |t| T , we have 0t (K 0 ) ⊂ , t ◦ W− (K 0 ) ⊂ . Then for all x ∈ K 0 and all k : ∞ 0 u k ◦ W− (x) − u k (x) = (χk ◦ 0t ◦ S E (x) − χk ◦ 0t (x))dt.

(7.2)

(7.3)

0

Next, let us define

vk (x) = 0

∞

χk \ ◦ 0t (x)dt, x ∈ K 0 , k .

(7.4)

Comparing (7.3) and (7.4), we get (u k ◦ W− (x) − u 0k (x)) − (u ◦ W− (x) − u 0 (x))=vk ◦ S E (x) − vk (x), x ∈ K 0 . (7.5) From (7.2) it follows that vk ◦ 0t (x) = vk (x) for all x ∈ K 0 and |t| T . Let us 0 → R such that vk ◦ π0 (x) = vk (x), x ∈ K 0 . Then from (7.5) it define vk : K follows that ωn−1 (y) − (y) − − (y) − ( uk ◦ W u 0k (y)) − ( u ◦ W u 0 (y)) 0 (n − 1)! 0 K ωn−1 (y) . (7.6) ( vk ◦ S E (y) − vk (y)) 0 = (n − 1)! 0 K 0 , 0 → 0 is a symplectic diffeomorphism and Since SE : S E (x) = x for x ∈ /K E E we see that the r.h.s. of (7.6) vanishes. This proves that the r.h.s. of (2.17) is independent of k . Thus, the limit (2.15) exists.

582


2. Let us prove that the limit in (2.15) is independent of the choice of the sequence {k }. First note that the limit (2.15) can be calculated over any subsequence of {k }. Next, let {k } be another sequence of sets with the same properties as {k }. Then it is easy to construct sequences of indices p1 < p2 < · · · and q1 < q2 < · · · such that p1 ⊂ q 1 ⊂ p2 ⊂ q 2 ⊂ · · · .

(7.7)

Then the sequence { pk } is a subsequence of both the sequence (7.7) and the sequence {k }. It follows that the limits (2.15) over the sequence (7.7) and over the sequence {k } coincide. In the same way, the limits (2.15) over the sequence (7.7) and over the sequence {k } coincide. 3. Let us prove (2.16). Since supp(H − H0 ) ∩ 0E is compact, there exists a compact set 0 ⊂ such that for all z ∈ \ 0 and all t ∈ R, one has t (z) = 0t (z) (for example, one can take 0 = ∩ K 0 ). Then we have supp τ E ⊂ 0 and also ωn−1 (z) TE = , (u k ◦ W− (z) − u 0k (z)) (n − 1)!

0 for all sufficiently large k. Let be sufficiently large so that (7.2) holds true and also assume (by increasing if necessary) that for all k and all z ∈ 0 , one has 0t (z) ∈ k for |t| |τ (z)|. Using (7.3) and the representation (2.12) for the scattering map, we get for all z ∈ 0 : ∞ 0 u k ◦ W− (z) − u k (z) = {χk ◦ 0t ◦ 0−τ (z) ◦ s E (z) − χk ◦ 0t (z)}dt 0

∞

= 0

0 {χk ◦ 0t ◦ s E (z) − χk ◦ 0t (z)}dt + χk ◦ 0t ◦ s E (z)dt −τ (z) ∞ {χk ◦ 0t ◦ s E (z) − χk ◦ 0t (z)}dt + τ (z). = 0

(7.8) Next, since s E : → is a symplectic diffeomorphism, we get χk ◦ 0t ◦ s E (z)ωn−1 (z) = χk ◦ 0t (z)ωn−1 (z)

0

0

(7.9)

for all t ∈ R. Finally, integrating (7.8) over 0 with respect to the symplectic volume ωn−1 form (n−1)! and using (7.9), we arrive at (2.16). 7.2. Proof of Theorem 3.2. 1. Let 1 ⊂ 2 ⊂ · · · ⊂ N be a sequence of open pre-compact sets such that ∪∞ k=1 k = N . Choose sufficiently large so that G(E 1 ) ∩ supp(H − H0 ) ⊂ Then for all k , dξ d (E) = dE dE

and G 0 (E 1 ) ∩ supp(H − H0 ) ⊂ .

G 0 (E)∩k

ωn − n!

G(E)∩k

ωn . n!


583

2. Let Y0 be a vector field defined in a neighbourhood of 0E ∩ k such that d H0 (Y0 )| 0 = 1. Similarly, let Y be a vector field defined in a neighbourhood E of E ∩ k such that d H (Y )| E = 1. Then dξ 1 1 (E) = i(Y0 )ωn − i(Y )ωn dE n! 0E ∩k n! E ∩k 1 1 = (i(Y0 )ω) ∧ ωn−1 − (i(Y )ω) ∧ ωn−1 . (n − 1)! 0E ∩k (n − 1)! E ∩k 3. Let us apply Lemma 5.2 to the integrals in the r.h.s. of the last identity. We will take µ = i(Y0 )ω for the first integral and µ = i(Y )ω for the second integral. Note that i(X )i(Y )ω = i(Y )d H = 1 by our choice of Y , and in the same way i(X 0 )i(Y0 )ω = 1. Application of Lemma 5.2 yields dξ (E) = dE

0 E

u k0 (y)

ω0n−1 (y) − (n − 1)!

E

u k (y)

ωn−1 (y) (n − 1)!

with u k0 , u k as in Sect. 2.5. By (2.15), the r.h.s. is (−TE ), as required. Appendix A: Key Formulas in Quantum Scattering Here we collect those definitions and formulas in quantum scattering theory which are relevant to the rest of the paper. We do not make any attempt at being rigourous or even precise about the required assumptions. Our cavalier approach will probably horrify experts in quantum scattering but this collection of formulas might be useful for the purposes of comparison of “classical” and “quantum” cases. Let H0 and H be self-adjoint operators in a Hilbert space H. In order to simplify our discussion, let us assume that both H0 and H have purely absolutely continuous spectrum (see e.g. [23, Sect. VII.2]). The wave operators W± : H → H are defined by W± ψ = lim eit H e−it H0 ψ, ψ ∈ H, t→±∞

whenever these limits exist. The wave operators are easily seen to be isometric and intertwine H and H0 : W± H0 = H W± . Completeness of the wave operators is the relation Ran W+ = Ran W− = H.

(A.1)

If H has a non-empty pure point or singular continuous spectrum, then the definition of the wave operators has to be modified and H in the right-hand side of the last relation has to be replaced by the absolutely continuous subspace of H . See [32, Sects. 2.1, 2.3] for the details. If the wave operators exist and are complete, one defines the scattering operator S = W+−1 W− . The scattering operator is unitary and commutes with H0 : S H0 = H0 S. It follows that S is diagonalised by the direct integral of the spectral decomposition of H0 : ⊕ H= h(E)d E, (H0 f )(E) = E f (E), (S f )(E) = S E f (E). (A.2) Spec(H0 )

584


Here S E is a unitary operator in the fibre space h(E); S E is called the scattering matrix. See [32, Sect. 2.4] for the details. Let P (k) be a family of operators in H such that P (k) ψ → ψ as k → ∞ for any ψ ∈ H. Let T (k) be the operator defined by ∞ ∞ P (k) e−it H W− ψ2 dt − P (k) e−it H0 ψ2 dt, (T (k) ψ, ψ) = −∞

−∞

assuming that these integrals exist. Then T (k) commutes with H0 and therefore is diagonal with respect to the direct integral decomposition (A.2). Let T E(k) : h(E) → h(E) be (k) the corresponding fibre operator. Then the limit T E = limk→∞ T E , whenever it exists, is called the global time delay operator, and TE = Tr T E is called the global time delay. See [25] for the details. The spectral shift function ξ(E) is defined by the relation ξ(E) = Tr(χ(−∞,E) (H0 ) − χ(−∞,E) (H )) which has to be understood in a certain regularised sense; see [32, Sect. 8.2] for the details. The existence of the spectral shift function and the global time delay requires some trace class assumptions on H and H0 , such as H − H0 ∈ Trace class or (H + a I )−m − (H0 + a I )−m ∈ Trace class for some appropriate a and m. The Birman-Krein formula reads det S E = e−2πiξ(E) ,

(A.3)

for a.e. E in the absolutely continuous spectrum of H0 . This formula first appeared in [8,18] for the one-dimensional Schrödinger operator and was established in [2] in the general case; see e.g. [4] for the details and historical references. In concrete applications this formula is sometimes stated in the form d dξ(E) d SE log det S E = Tr S E∗ = −2πi ; dE dE dE see e.g. [7]. The Eisenbud-Wigner formula reads d Im log det S E = TE ; dE

(A.4)

see e.g. [9,25] for the details and references. Appendix B: Symplectic Diffeomorphisms and the Calabi Invariant B.1. Symplectomorphisms. We recall some notation and preliminaries from symplectic topology; see e.g. [20] for the details. Let (M, ω) be a non-compact 2m-dimensional symplectic manifold, possibly with boundary. We need the following notation: Symp(M) is the group of all symplectomorphisms (=symplectic diffeomorphisms) on M. Sympc (M) is the group of all symplectomorphisms of M with compact support, supp ψ = Clos{x | ψ(x) = x}. Sympc0 (M) is the path connected component of the identity map in Sympc (M).


585

Hamc (M) is the set of all compactly supported Hamiltonian symplectomorphisms of M. This means that ψ ∈ Hamc (M) can be constructed as a time one flow of a family of time-dependent compactly supported Hamiltonians. More precisely, ψ ∈ Hamc (M) means that there exists a smooth family h t , t ∈ [0, 1] of Hamiltonians on M such that ∪t∈[0,1] supp h t lies in a compact set and if ψt is the corresponding flow and X t the corresponding vector field, d ψt = X t ◦ ψt , i(X t )ω = −dh t , ψ0 = id, dt

(B.1)

then ψ = ψ1 . It is easy to see (cf. [20, Sect. 10]) that Hamc (M) ⊂ Sympc0 (M) ⊂ Sympc (M).

(B.2)

B.2. The Calabi invariant. Let us assume that M is exact, i.e. there is a 1-form α such that ω = dα. We recall the definition of the Calabi invariant CAL; for the details, see [20]. For our purposes we need to define CAL on a wider set of symplectomorphisms than is usually done. Let Dom(CAL, M) be the set of all ψ ∈ Sympc (M) such that there exists a 1-form α and f ∈ C0∞ (M) with dα = ω and α − ψ ∗ α = d f . In this case we will say that f is an α-generating function of ψ. If ψ ∈ Dom(CAL, M) and f is an α-generating function of ψ, let us define 1 ωm (x) CAL(ψ) = . f (x) m+1 M m!

(B.3)

Note that our sign conventions and normalisation differ from those of [20]. Since the choice of α above is not unique, a symplectomorphism can have many generating functions. However, we have Proposition B.1. (i) CAL(ψ) is independent of the choice of α. (ii) If f is an α-generating function of ψ, then CAL(ψ) can be calculated as 1 CAL(ψ) = (ψ ∗ α) ∧ α ∧ ωm−1 . (m + 1)! M

(B.4)

(iii) Suppose ψ ∈ Hamc (M) is generated by a family of Hamiltonians {h t }, see (B.1). Then ψ ∈ Dom(CAL, M) and CAL(ψ) =

dt 0

Proof.

1

M

h t (x)

ωm (x) . m!

(i) This is a well known calculation, see e.g. [20, Lemma 10.27]. Suppose that α j − ψ ∗ α j = d f j , f j ∈ C0∞ (M), dα j = ω, j = 1, 2. Denote β = α2 − α1 , g = f 2 − f 1 ; then we have dβ = 0, β − ψ ∗ β = dg. We need to prove that m M gω = 0.

586


As in [20, Lemma 10.27], we have m m−1 gω = g dα1 ∧ ω = M

M

M

=−

M

=

=

M

(ψ β) ∧ α1 ∧ ω

m−1

−

M

M

(ψ ∗ β − β) ∧ α1 ∧ ωm−1

ψ ∗ (β ∧ α1 ∧ ωm−1 )

ψ ∗ β ∧ (α1 − ψ ∗ α1 ) ∧ ωm−1 =

=−

(dg) ∧ α1 ∧ ωm−1 = ∗

M

(d(gα1 ) − (dg) ∧ α1 ) ∧ ωm−1

M

M

ψ ∗ β ∧ d f 1 ∧ ωm−1

d( f 1 ψ ∗ β ∧ ωm−1 ) = 0,

as required. (ii) Similarly to the previous calculation, using Stokes formula, we have (m + 1)! CAL(ψ) = f (dα) ∧ ωm−1 = f d(α ∧ ωm−1 ) M M m−1 =− df ∧α ∧ω = (ψ ∗ α − α) ∧ α ∧ ωm−1 M M = ψ ∗ α ∧ α ∧ ωm−1 , M

since α ∧ α = 0. (iii) Is well known; see [20, Lemma 10.27].

Remark B.2. The Calabi invariant is usually defined as a map Hamc (M) → R, in which case it is a homomorphism; see [20]. On the domain Dom(CAL, M) this is in general not the case: it is not difficult to show that Dom(CAL, M) is, in general, not a subgroup of Sympc (M). Example B.3 below shows that in general neither of the two sets Sympc0 (M), Dom(CAL, M) is a subset of the other one. Example B.3. Let (M, ω) be T ∗ S1 with the canonical symplectic structure of the cotangent bundle (see e.g. [20, Sect. 3.1]). Let (s, θ ) ∈ R×[0, 2π ] be the coordinates in T ∗ S1 . All possible one-forms α such that dα = ω can be described as α = sdθ + γ dθ + dg, where γ ∈ R and g ∈ C ∞ (T ∗ S1 ). Consider the symplectic diffeomorphism ψ of T ∗ S1 , defined by ψ : (s, θ ) → (s, θ + φ(s)), where φ ∈ C ∞ (R). Consider the following cases:

(i) Suppose φ(s) = 0 for s −1 and φ(s) = 2π for s 1, and R sφ (s)ds = 0. Then ψ is not homotopic to identity, but ψ ∗ (sdθ ) − sdθ = sφ (s)ds and so ψ ∈ Dom(CAL, T ∗ S1 ). Thus, Dom(CAL, T ∗ S1 ) is not a subset of Sympc0 (T ∗ S1 ). ∞ (ii) Suppose φ ∈ C0 (R), and R sφ (s)ds = 0. Then it is easy to see that ψ ∈ Sympc0 (T ∗ S1 ) but ψ ∈ Dom(CAL, T ∗ S1 ).


587

B.3. log det and CAL. Here, without any attempt at being rigourous, we point out an analogy between the Calabi invariant of a symplectic map and the logarithm of the determinant of a unitary operator. This analogy helps to understand the relation between Theorems 3.1, 3.2 and their “quantum” counterparts (A.3), (A.4). In order to make our discussion concrete, suppose M = R2n and let us use the Weyl quantisation procedure. That is, for a real valued function h ∈ C0∞ (R2n ), let us define the self-adjoint operator n n (H u)(q) = eiq−q , p h( q+q 2 , p)u(q )d q d p. R2n

Then, clearly, Tr H =

h(q, p) d q d p = n

R2n

n

R2n

h(x)

ωn (x) . n!

(B.5)

Further, let U be the unitary operator obtained from H by means of exponentiation: U = exp(−i H ). U can be regarded as a time one map corresponding to the differential d equation i dt U (t) = HU (t). The “classical” analogue of this procedure is taking the time one map of the Hamiltonian flow t generated by h. We have ωn = exp(−i CAL(1 )). det U = exp(−i Tr H ) = exp −i h n! In other words, we have a diagram h ⏐ classical ⏐ dynamics

Quantisation

−−−−−−−−→

H ⏐ ⏐ quantum dynamics

1 ⏐ ⏐

e−i H ⏐ ⏐

exp(−i CAL(1 ))

det e−i H

This to some extent explains the analogy between −Im log det and CAL. Acknowledgements. Research was partially supported by the London Mathematical Society. A.P. is grateful to H. Dullin, A. Gorodetski, M. Hitrik, and A. Strohmaier for useful discussions and references to the literature and to N. Filonov for reading the manuscript and making a number of very helpful remarks.

References 1. Abraham, R., Marsden, J.E.: Foundations of Mechanics. Second edition. Reading, MA: Benjamin/ Cummings Publishing Co., 1978 2. Birman, M.Sh., Krein, M.G.: On the theory of wave operators and scattering operators. Soviet Math. Dokl. 3, 740–744 (1962) 3. Alexandrova, I.: Structure of the short range amplitude for general scattering relations. Asymptot. Anal. 50(1–2), 13–30 (2006) 4. Birman, M.Sh., Yafaev, D.R.: The spectral shift function. The work of M. G. Krein and its further development. St. Petersburg Math. J. 4(5), 833–870 (1993)

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5. Bollé, D.: On classical time delay. In: New Developments in Mathematical Physics, Edited by H. Mitter and L. Pittner, Berlin-Heidelberg-New York: Springer, 1981 6. Bollé, D., Osborn, T.A.: Sum rules in classical scattering. J. Math. Phys. 22(4), 883–892 (1981) 7. Buslaev, V.S.: The trace formulae and certain asymptotic estimates of the kernel of the resolvent for the Schrödinger operator in three- dimensional space. (in Russian). Probl. Math. Phys., No. I, Spectral Theory and Wave Processes, Leningrad: Izdat. Leningrad. Univ., 1966, pp. 82–101 8. Buslaev, V.S., Faddeev, L.D.: Formulas for traces for a singular Sturm-Liouville differential operator (in Russian). Soviet Math. Dokl. 1, 451–454 (1960) 9. Jensen, A.: Time-delay in potential scattering theory. Commun. Math. Phys. 82, 435–456 (1981) 10. de Carvalho, C.A.A., Nussenzveig, H.M.: Time delay. Phys. Rep. 364, 83–174 (2002) 11. Doron, E., Smilansky, U.: A scattering theory approach to semiclassical quantization. In: Lecture notes, Summer School “Mesoscopic systems and chaos: a novel approach”, 3–6 August 1993, Adratico Res. Conf., Singapore: World Scientific, 1995 12. Gérard, C., Tiedra de Aldecoa, R.: Generalized definition of time delay in scattering theory. J. Math. Phys. 48(12), 122101 (2007) 13. Herbst, I.: Classical scattering with long range forces. Commun. Math. Phys. 35, 193–214 (1974) 14. Hunziker, W.: The S-Matrix in Classical Mechanics. Commun. Math. Phys. 8(4), 282–299 (1968) 15. Jung, C.: Poincaré map for scattering states. J. Phys. A 19, 1345–1353 (1986) 16. Knauf, A.: Qualitative aspects of classical potential scattering. Regul. Chaotic Dyn. 4(1), 3–22 (1999) 17. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics. Vol. 1. Mechanics. Third edition. OxfordNew York-Toronto: Pergamon Press, 1976 18. Lifshits, I.M.: On the problem of scattering of particles by a centrally symmetric field in quantum mechanics (in Russian). Khar’Kov Gos. Univ. Uchen. Zap. 27, 105–107 (1948) 19. Lewenkopf, C.H., Vallejos, R.O.: On the classical-quantum correspondence for the scattering dwell time. Phys. Rev. E 70, 036214 (2004) 20. McDuff, D., Salamon, D.: Introduction to Symplectic Topology, Oxford: Oxford University Press, 1998 21. Narnhofer, H.: Another definition for time delay. Phys. Rev. D 22(10), 2387–2390 (1980) 22. Narnhofer, H., Thirring, W.: Canonical scattering transformation in classical mechanics. Phys. Rev. A 23(4), 1688–1697 (1981) 23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol 1: Functional analysis. New York: Academic Press, 1972 24. Simon, B.: Wave operators for classical particle scattering. Commun. Math. Phys. 23, 37–48 (1971) 25. Robert, D.: Relative time-delay for perturbations of elliptic operators and semiclassical asymptotics. J. Funct. Anal. 126(1), 36–82 (1994) 26. Robert, D.: Semiclassical asymptotics for the spectral shift function. In: Differential Operators and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, 189, Providence, RI: Amer. Math. Soc., 1999, pp. 187–203 27. Rouvinez, C., Smilansky, U.: A Scattering Approach to the quantization of Hamiltonians in 2 Dimensions — Application to the Wedge Billiard. J. Phys. A: Math. Gen. 28, 77–104 (1995) 28. Sassoli de Bianchi, M., Martin, Ph.A.: On the definition of time delay in scattering theory. Helv. Phys. Acta 65(8), 1119–1126 (1992) 29. Thirring, W.: Classical Scattering Theory, In: New Developments in Mathematical Physics, Edited by H. Mitter, L. Pittner, Berlin-Heidelberg-New York: Springer, 1981 30. Thirring, W.: Classical Mathematical Physics. Dynamical Systems and Field Theories. Third edition. New York: Springer-Verlag, 1997 31. Va˘ınberg, B.R.: Asymptotic Methods in Equations of Mathematical Physics. New York: Gordon & Breach, 1989 32. Yafaev, D.R.: Mathematical Scattering Theory. General theory. Providence, RI: Amer. Math. Soc., 1992 Communicated by I. M. Sigal


Communications in


Marginally Trapped Tubes Generated from Nonlinear Scalar Field Initial Data Catherine Williams Department of Mathematics, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected] Received: 20 January 2009 / Accepted: 14 July 2009 Published online: 10 September 2009 – © Springer-Verlag 2009

Abstract: We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal, spherically symmetric marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the 1 initial data is sufficiently small and decays like O(r − 2 ), and the potential function V is nonnegative with bounded second derivative. This result can be loosely interpreted as a statement about the stability of ‘nice’ asymptotic behavior of marginally trapped tubes under certain small perturbations of Schwarzschild. 1. Introduction Black holes lie at the core of many current efforts to further our understanding of gravitation. Questions surrounding their existence and properties are central to some of the most significant open problems in mathematical relativity, while a major focus in numerical relativity is locating evolving black holes in simulations, and considerable research in physics communities is dedicated to carrying over concepts from quantum mechanics to black hole regimes. In all of these contexts, certain spacetime hypersurfaces known as marginally trapped tubes (MTTs) play an important role. (Marginally trapped tubes of certain causal characters are often referred to as dynamical or isolated horizons in the physics literature.) On one hand, mathematically, these hypersurfaces generally lie inside of black holes and can be thought of as forming boundaries between the regions of weak and strong gravitational fields. Understanding their behavior thus sheds some light on the portions of black holes’ interiors in which singularities and/or Cauchy horizons may form. On the other hand, numerical relativists and physicists, e.g. those developing loop quantum gravity, have largely set aside the traditionally-defined event horizon and instead use MTTs as models of surfaces of black holes [3,12]; the advantage of the latter is primarily that they are defined quasi-locally, whereas the former notion requires global information. In any case, whether interpreting MTTs as black hole surfaces or just

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interesting structures inside them, it is useful to characterize their long term behavior and its relationship with classical event horizons. There is a general expectation that MTTs that form during gravitational collapse become spacelike or null and asymptotically approach the event horizon. Essentially nothing is currently known about the asymptotic behavior of MTTs in general, however, i.e. without symmetry. Indeed, it follows from the existence results of Andersson, Mars, and Simon [1,2] that a general black hole spacetime may contain uncountably many distinct MTTs, and in fact, there may be open sets of points having the property that each point lies on uncountably many distinct MTTs. It is an open problem whether any one of these MTTs need be asymptotic to the event horizon, much less all of them. Imposing spherical symmetry, however, the problem becomes simpler, since each point can lie in at most one spherically symmetric MTT, i.e. one foliated by round two-spheres. We can then ask whether this particular MTT has the expected “nice” asymptotic behavior. Henceforth we shall focus our attention solely on such spherically symmetric MTTs. Even in spherical symmetry, a major problem in trying to compare the asymptotic behavior of MTTs with classical event horizons is that the latter are difficult to locate in general, since their definition requires global information. One can start by examining known exact black hole solutions to Einstein’s equations, but unfortunately this list of spacetimes is rather quickly exhausted. The exact spherically symmetric black hole solutions are the Schwarzschild, Reisner-Nordström, and Vaidya spacetimes, whose matter models are vacuum, electro-vacuum, and ingoing null dust, respectively. In all three of these, the (spherically symmetric) MTTs do exhibit the nice asymptotic behavior just described. In fact, in Schwarzschild and Reisner-Nordström, the MTTs coincide exactly with the black hole event horizons. In Vaidya, provided the dominant energy condition holds, the MTT is achronal and is either asymptotic to or eventually coincides with the event horizon [3,13]. General spherically symmetric black hole spacetimes satisfying only the dominant energy condition were considered in [13], and it was shown there that if four particular inequalities involving the metric and stress-energy tensor are satisfied in a certain small region near future timelike infinity i + , the future limit point of the event horizon, then the black hole contains an MTT which eventually becomes achronal and asymptotically approaches the event horizon. It was also shown that these inequalities are satisfied in two families of self-gravitating Higgs field black holes. The Higgs field matter model is generated by a scalar field φ satisfying the nonlinear (coupled) wave equation g φ = V (φ), where V is some given potential function; by Higgs field black holes, we mean spacetimes which are assumed a priori to contain black holes. In particular, the hypotheses of the general theorem were shown to hold provided that certain quantities involving the metric, φ, and V (φ) either: one, decay at specified rates along the event horizon, or two, satisfy several rather restrictive monotonicity properties along the event horizon at late times. Given a particular matter model, an alternate, more physical strategy is to recast the Einstein equations as an initial value problem and generate spacetimes from initial data, then locate any black holes and look for MTTs inside them. This program has been successfully carried out for several matter models. In particular, the maximal development of spherically symmetric asymptotically flat initial data for the Einstein equations coupled with a scalar field, the Maxwell equations and a real scalar field, or the Vlasov equation (describing a collisionless gas) does indeed contain an MTT which is asymptotic to the event horizon [4,6,9,10]. Furthermore, in the scalar field cases, the MTT is necessarily achronal.

MTTs Generated from Nonlinear Scalar Field Data

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In this paper we reconsider the problem for Higgs field spacetimes, this time taking the latter approach and beginning with asymptotically flat spherically symmetric Higgs field initial data. In order to circumvent the difficulties associated with the global existence problem for the Einstein equations, we rely on the general results of [8] to tell us that as long as our data contains a trapped surface, contains no weakly anti-trapped surfaces, and generates a spacetime with nonempty future null infinity, I + = ∅, then the maximal development of the data will contain a black hole. We make two main addi1 tional assumptions: one, that the scalar field decays to some limiting value like O(r − 2 ), where r is a radial coordinate which tends to infinity on the asymptotically flat end; and two, that the scalar field and the mass flux are sufficiently small outside of the outermost marginally trapped sphere. We then show that the black hole generated by these initial data contains an achronal, spherically symmetric MTT which is asymptotic to the event horizon, and moreover, this MTT is smooth and connected, intersecting the initial hypersurface at precisely the outermost marginally trapped surface. A few remarks are in order. First, this result is considerably stronger than those of [13], since the Higgs field black hole generated under the assumptions just described need not satisfy either set of conditions along the event horizon mentioned previously; the latter are much more restrictive. Secondly, it should be emphasized that, unlike previous results, here we locate the whole (connected) future development of the MTT emanating from the Cauchy surface, rather than just a small portion of it near i + . And thirdly, this result can be loosely interpreted as a statement about certain small perturbations of Schwarzschild. Suppose we take a spherically symmetric, spacelike slice of a Schwarzschild spacetime which extends from an inner boundary inside the black hole region out to spacelike infinity i 0 , and suppose we perturb the vacuum metric on this slice to one given by a very small, decaying Higgs field, say with compact support. Then the result of this paper says that, while the perturbed MTT may not coincide exactly with the event horizon as it does in Schwarzschild, it will still be achronal and connected and will asymptotically approach the event horizon from inside the black hole. 2. Preliminaries The spacetimes we wish to consider are characterized by the following four properties: their matter is described by a Higgs field matter model satisfying the dominant energy condition; they are spherically symmetric; they arise evolutionarily, i.e. as maximal future developments of asymptotically flat initial Cauchy data prescribed on spacelike hypersurfaces; and the Cauchy data from which they arise is physically reasonable black hole data. We describe the details of each of these requirements below. It should be noted that we do not deal with the Cauchy problem and the issue of specifying initial data per se. Instead we posit those conditions which are necessary to address the situation at hand, i.e. the evolution of a spherically symmetric black hole, and then in the statement of the main theorem identify further, more restrictive conditions on the data which guarantee the existence and desired behavior of a unique spherically symmetric marginally trapped tube inside the black hole. 2.1. Higgs field matter model & dominant energy condition. The Higgs field matter model on a spacetime (M, g) consists of a scalar function φ ∈ C 2 (M) and a potential function V (φ), V ∈ C 2 (R), such that Tαβ

Rαβ − 21 Rgαβ = 2Tαβ , = φ;α φ;β − 21 φ;γ φ ;γ + V (φ) gαβ ,

(1) (2)

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and g φ = g αβ φ;αβ = V (φ).

(3)

The dominant energy condition stipulates that, for a vector field ξ α on M, −Tβα ξ β is future causal wherever ξ α is future causal. Using the spherical symmetry of the metric described below, namely the decomposition of the metric given by (4), one readily computes that the dominant energy condition is satisfied if and only if V (φ) ≥ 0 everywhere on M. We therefore assume a priori that the potential function V is an everywhere nonnegative function of its argument. 2.2. Spherical symmetry. A self-gravitating Higgs field spacetime (or Cauchy surface) is said to be spherically symmetric if the Lie group S O(3) acts on it by isometries under which φ remains invariant, with orbits which are either spacelike two-spheres or fixed points. We shall in fact make a slightly stronger assumption, that in addition to M admitting such an S O(3)-action, the quotient Q = M/S O(3) inherits from M the structure of a 1+1-dimensional Lorentzian manifold, possibly with boundary, with metric g such that g = g + r 2γ .

(4)

Here γ is the usual round metric on S 2 , and r is a smooth nonnegative function on Q, called the area-radius, whose value at each point is proportional to the square root of the area of the corresponding two-sphere upstairs in M. Since it is preserved by the S O(3)action, the function φ descends to a function Q. The advantage of such an assumption on the set Q is that such features of (M, g) as black holes, event horizons, and spherically symmetric marginally trapped tubes are preserved and may be studied at this quotient level. We further assume that Q admits a conformal embedding into a subset of 2dimensional Minkowski space M1+1 . Such an embedding preserves causal structure, so identifying Q with its image under this embedding, we make use of the usual global double-null coordinates u, v on M1+1 to write g = −2 dudv, where = (u, v) > 0 on Q. (In our Penrose diagrams, we will always depict the positive u- and v-axes at 135◦ and 45◦ from the usual positive x-axis, respectively.) Since now φ = φ(u, v) and r = r (u, v) as well, we can rewrite (1)-(3) as a system of pointwise equations on Q: ∂u (−2 ∂u r ) = −r −2 (∂u φ)2 , ∂v (−2 ∂v r ) = −r −2 (∂v φ)2 , ∂u m = r 2 V (φ)∂u r − 2−2 (∂u φ)2 ∂v r , ∂v m = r 2 V (φ)∂v r − 2−2 (∂v φ)2 ∂u r , and

2 V (φ) = −4−2 ∂uv φ + ∂u φ (∂v log r ) + ∂v φ (∂u log r ) ,

(5) (6) (7) (8)

(9)


593

where m = m(u, v) =

r r (1 − g (∇r, ∇r )) = (1 + 4−2 ∂u r ∂v r ) 2 2

(10)

is the Hawking mass. Note that the null constraints (5) and (6) are just Raychaudhuri’s equation applied to each of the two null directions in Q. Since we can pass back and forth between (M, g, φ) and (Q, , r, φ) without losing information, we may work directly on Q without any loss of generality. 2.3. Evolution from asymptotically flat initial data. Next we assume that our spacetime arises as the future evolution of initial data for the Einstein-Higgs system (1)-(3). In particular, we require that (M, g) be the maximal future Cauchy development of initial data prescribed on a spherically symmetric spacelike hypersurface ⊂ M. Since is preserved under the S O(3)-action, it descends to a spacelike curve S ⊂ Q, and our assumption upstairs implies that on the quotient level we have Q = D + (S); in particular, S is the past boundary of Q. The remaining assumptions on initial data may be formulated directly on S. We further assume that the initial hypersurface S has at least one asymptotically flat end, and for simplicity, we focus our attention on a single such end. Here asymptotic flatness means first of all that S is a connected spacelike curve along which r → ∞ in one direction, say the direction of increasing v; the other end may or may not have boundary. Without loss of generality we assume that r is strictly positive on S. Moreover, we require that the metric and stress-energy tensor approach the Euclidean and vacuum ones, respectively, as r → ∞ along S. By inspection of (2), it follows that for Higgs initial data, ∇φ and V (φ) → 0 along S. We therefore assume that φ → φ+ along S and that V (φ+ ) = 0, where the limiting value φ+ is some finite constant. From our assumption in Sect. 2.1 that V ≥ 0 uniformly, it then follows that V (φ+ ) = 0 as well. Lastly, we assume that the Hawking mass m is uniformly bounded along S. We remark that the requirement that φ have a finite limit along S results in a slight loss of generality, since we are excluding possibilities such as φ ∼ log r with limx→∞ V (x) = 0. However, this drawback is outweighed by the advantage of being able to estimate V (φ) and V (φ) in terms of φ using the mean value theorem. For definiteness let us assume that Q ⊂ [0, u 0 ] × [v0 , ∞) ⊂ M1+1 , some u 0 , v0 > 0, and that in particular v → ∞ along S. We abuse notation somewhat to allow Q to include points along the “ray” [0, u 0 ] × {∞}, so as to be able to refer to points “at infinity,” e.g. spacelike infinity i 0 , future timelike infinity i + , and future null infinity I + . (We could achieve the same end more rigorously by requiring our embedding Q → M1+1 to be a conformal compactification, but the certain core elements of the proof of the theorem are clearer with v scaled to have infinite range on Q.) Constant-u curves are said to be outgoing and constant-v ones ingoing. 2.4. Physically reasonable black hole data. Finally we suppose that the asymptotically flat initial hypersurface S is equipped with physically reasonable black hole initial data. By this we mean three things: one, that S should contain at least one (spherically symmetric) closed trapped surface; two, that S should not contain any (spherically symmetric) weakly anti-trapped surfaces; and three, that the data should decay sufficiently rapidly toward the asymptotically flat end to insure that future null infinity I + = ∅, where I + is defined as in [8]. We introduce relevant definitions and discuss the first two of these requirements in greater detail in the next section. As for the third assumption, we do not address here the issue of precisely what rate of decay is sufficiently rapid to insure

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Fig. 1. A spacetime generated by “physically reasonable asymptotically flat black hole initial data” as described in Sects. 2.3 and 2.4. The existence of the black hole B = Q \ J − (I + ) is guaranteed by [8], given that S contains a trapped surface and that I + = ∅; H is its event horizon

that future null infinity I + = ∅, we simply assume our initial data has this property. Alternately, we could avoid all discussion of decay by taking φ and ∇φ to be compactly supported on S and assuming V (0) = 0; Birkhoff’s theorem [11] would then imply that our spacetime contains a region isometric to an exterior region of the Schwarzschild spacetime and would thus guarantee that I + = ∅. With these assumptions in place, it now follows from [8] that Q contains a black hole region. In particular, I + is complete with future limit point i + and past limit point i0 , the black hole region is B := Q\ J − (I + ), and its event horizon is H := ∂ Q \ J − (I + ) ∩Q. (Here and henceforth, set boundaries and closures are to be taken with respect to the topology of M1+1 rather than the relative topology of Q.) See Fig. 1 for a representative Penrose diagram. The event horizon necessarily has the property that sup p∈H r ( p) = r+ < ∞. 2.5. Regions R, T , and A. The characterization of a closed spacelike two-surface S in M as trapped, marginally trapped, weakly anti-trapped, etc., depends on the signs of the inner and outer future null expansions, θ+ and θ− , on S. These expansions are defined as follows: if + and − are two future-directed null vector fields normal to S, with + pointing towards the asymptotically flat region and − away from it, then at a point p ∈ S, θ± = div S ± = h αβ ± β;α , where h αβ is the Riemannian metric on S induced from g. Rescaling + or − by a positive factor rescales the corresponding expansion but does not change its sign, so the signs of θ± are well-defined given M’s time-orientation and asymptotically flat end. We then say that S is trapped if both θ+ < 0 and θ− < 0 at all points in S, marginally trapped if θ+ = 0 and θ− < 0 at all points in S, and weakly anti-trapped if θ− ≥ 0 at all points in S (no restriction on θ+ ). By contrast, closed spacelike two-surfaces in flat or nearly flat space will always have θ+ > 0 and θ− < 0 everywhere on S. Because of the spherical symmetry, θ+ and θ− are constant on round two-spheres in M and therefore descend to pointwise functions on Q. Since u is the ingoing and v the outgoing coordinate on Q, one computes that θ+ is proportional to ∂v r and θ− is proportional to ∂u r . Thus the first two of the “physically reasonable black hole initial data” assumptions listed above are satisfied provided that ∂u r < 0 everywhere on S and that there exists q ∈ S such that ∂v r (q) < 0 as well. We note also that since r tends to infinity in the direction of increasing v, ∂v r must eventually become positive along S in the direction of the flat end, and there must therefore exist an outermost marginally trapped surface p∗ along S. (Here and henceforth all trapped or marginally trapped surfaces should be understood to be spherically symmetric, whether or not this


595

is made explicit.) This surface p∗ plays a large role in the statement and proof of the main theorem. Now, the object of this paper is to locate and elucidate the causal and asymptotic behavior of any (spherically symmetric) marginally trapped tubes in the spacetime. In general, a marginally trapped tube in the spacetime M is a smooth hypersurface, of any causal character, which is foliated by closed, marginally trapped two-surfaces. In our spherically symmetric setting, we restrict our attention to those marginally trapped tubes foliated by round two-spheres, in which case the tubes descend to curves in the quotient Q. In fact, we define three subsets of interest in Q: the regular region R = {(u, v) ∈ Q : ∂v r > 0 and ∂u r < 0}, the trapped region T = {(u, v) ∈ Q : ∂v r < 0 and ∂u r < 0}, and the marginally trapped tube, A = {(u, v) ∈ Q : ∂v r = 0 and ∂u r < 0}. Note that A is a smooth hypersurface in Q wherever 0 is a regular value of ∂v r . It is clear that Q = R ∪ A ∪ T if and only if Q contains no weakly anti-trapped surfaces. We have assumed that S contains no such surfaces, and it turns out that this is sufficient to guarantee that none evolve: Proposition 1 [5,8]. If ∂u r < 0 along S, then ∂u r < 0 everywhere in Q. Proof. Let (u, v) be any point in Q. Suppose the ingoing null ray to the past of (u, v) intersects S at the point (u , v). Then integrating Raychaudhuri’s equation (5) along this ray, we obtain u −2 −2 ( ∂u r )(u, v) = ( ∂u r )(u , v) − r −2 (∂u φ)2 (u, ¯ v) d u. ¯ u

By assumption ∂u r (u , v) < 0, so the right-hand side of this equation is strictly negative, and hence so is the left-hand side. Integrating the other Raychaudhuri equation (6) yields a slightly different but equally useful result: Proposition 2 [5,8]. If (u, v) ∈ T ∪ A, then (u, v ) ∈ T ∪ A for all v > v. Similarly, if (u, v) ∈ T , then (u, v ) ∈ T for all v > v. Proof. Integrating (6) along the null ray to the future of a point (u, v) ∈ Q yields (−2 ∂v r )(u, v ) = (−2 ∂v r )(u, v) −

v v

r −2 (∂v φ)2 (u, v) ¯ d v¯

for v > v. The right-hand side of this equation is nonpositive if ∂v r (u, v) ≤ 0, and strictly negative if ∂v r (u, v) < 0; both statements of the proposition follow.

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Fig. 2. The regions R and T , shaded light and dark, respectively, are separated by marginally trapped tube A. The point p∗ represents the outermost marginally trapped sphere along S; the dashed lines inside the black hole indicate portions of the spacetime boundary, Q \ Q. As depicted here, A need be neither achronal nor connected, but it must lie inside the black hole and comply with Proposition 2

Since I + is characterized by the property that r has infinite supremum along any outgoing null ray with a limit point on I + , an immediate consequence of Proposition 2 is that all trapped and marginally trapped surfaces must lie inside the black hole, T ∪ A ⊂ B. We shall be concerned only with the connected component of R containing the exterior of the black hole J − (I + ), so let us assume for convenience that R ∩ S is connected, i.e. that points on S interior to p∗ are trapped or marginally trapped. Indeed, since we made no assumptions concerning the inner boundary of S, we can cut off any other components of S ∩ R without affecting our hypotheses. Proposition 2 now guarantees that R is connected in Q. Figure 2 provides a Penrose diagram indicating a possible configuration of R, T , and A in Q. With notation in place, we now give a brief overview of the proof of the main theorem. Logically speaking, the proof comprises three main parts, though they are presented somewhat out of order in the actual proof given in Sect. 3. The first and most involved piece is a bootstrap argument whose purpose is to establish positive lower bounds for two particular quantities in the region R ∩ {r ≤ R} for a carefully chosen constant R. The two quantities in question are called κ and α (defined by (22) and (51), respectively), and the constant R is chosen in such a way that R ∩ {r < R} is an open neighborhood of H but R is not too large. The primary tools used in closing the bootstrap are the smallness of initial data stipulated by the theorem and the energy estimates from Sect. 2.6 below. The second part of the proof shows that the component of A containing p∗ must be achronal and that it must extend all the way out to v = ∞, i.e. either to i + or the Cauchy horizon. The key ingredients here are the lower bound for α obtained from the bootstrap and the extension principle of [8], which is known to hold for the Higgs field matter model. The last step of the proof is to show that if A terminates at the Cauchy horizon, leaving a gap between itself and H, then we can derive a contradiction essentially by 2 r twice in the region formed by this gap. The contradiction arises directly integrating ∂uv from the lower bounds on α and κ obtained from the bootstrap argument in addition to bounds on the radial function r . 2.6. Monotonicity & energy estimates. From Proposition 1 and our assumption of no weakly anti-trapped surfaces on S, we have ∂u r < 0

everywhere in Q,

(11)

everywhere in R ∪ A.

(12)

and by definition, ∂v r ≥ 0


597

These inequalities, together with Eqs. (7), (8), and the dominant energy condition V (x) ≥ 0, imply that everywhere in R ∪ A we have ∂u m ≤ 0,

(13)

∂v m ≥ 0.

(14)

and

Set M := sup m(q),

(15)

m 1 := m( p∗ ),

(16)

r1 := r ( p∗ ),

(17)

q∈S

and where p∗ corresponds to the outermost marginally trapped sphere along S as described in the previous section. Since S is spacelike, (11), (12) and (13), (14) imply that r is increasing and m is non-decreasing, respectively, along S ∩R toward the asymptotically flat end. Thus we have m1 =

inf

m(q)

(18)

inf r (q).

(19)

q∈S ∩R

and r1 =

q∈S ∩R

Note also that M < ∞ by hypothesis (Sect. 2.3), and m 1 , r1 > 0, since (10) and the fact that ∂v r ( p∗ ) = 0 together yield r1 = 2m 1 , and we assumed in Sect. 2.3 that r > 0 on S. Proposition 3. For all q ∈ R, m 1 ≤ m(q) ≤ M

(20)

r1 ≤ r (q).

(21)

and Proof. Given q ∈ R, the outgoing null ray to the past of q must also lie in R by Proposition 2. Then (21) and the left-hand inequality of (20) follow immediately from (14) and (18), or (12) and (19), respectively. The ingoing null ray to the past of a point q ∈ R may or may not lie entirely in R. If it does, then the right-hand inequality of (20) is immediate from (13) and (15). If not, then suppose q = (u , v ), so that the ingoing null ray in question is given by {(u, v) : v = v }, and suppose (u , v ) is the point at which this ray intersects S. Further, let u 1 and u 2 be the minimum and maximum, respectively, of the set {u ∈ [u , u ] : (u, v ) ∈ T ∪ A}. It follows that (u 1 , v ), (u 2 , v ) ∈ A, so ∂v r (u 1 , v ) = ∂v r (u 2 , v ) = 0. Then applying inequalities (11), (13), and Eq. (10), we have r r m(q) ≤ m(u 2 , v ) = (u 2 , v ) ≤ (u 1 , v ) = m(u 1 , v ) ≤ m(u , v ) ≤ M, 2 2 so the right-hand inequality holds as well.

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Let κ denote the quantity κ := −

2 ∂v r = , 4∂u r 1 − 2m r

(22)

and observe that (11) implies that κ > 0 everywhere in Q. We now have the necessary tools to derive the two energy estimates for φ in R which are crucial for the main theorem. Proposition 4. For any interval [u 1 , u 2 ] × {v} ⊂ R, u2 r ∂u φ 2 1 2m − 1 − ∂u r (u, ¯ v) d u¯ ≤ M − m 1 , 2 r ∂u r

(23)

and for any interval {u} × [v1 , v2 ] ⊂ R, v2 1 2 −1 2 ¯ v¯ ≤ M − m 1 . 2 r κ (∂v φ) (u, v)d

(24)

u1

v1

Proof. To obtain the first of these estimates, we integrate Eq. (7) along [u 1 , u 2 ] × {v} and apply (11), the dominant energy condition, and Proposition 3 to get u2 r 2 V (φ)∂u r − 2−2 (∂u φ)2 ∂v r (u, ¯ v)d u¯ m1 − M ≤ u 1u 2 − 2r 2 −2 (∂u φ)2 ∂v r (u, ¯ v)d u. ¯ ≤ u1

Using (22) and rearranging the factors of the integrand, this yields (23). Analogously, for the second estimate, we integrate Eq. (8) along {u} × [v1 , v2 ] and apply (12), the dominant energy condition, and Proposition 3, arriving at v2 M − m1 ≥ r 2 V (φ)∂v r − 2−2 (∂v φ)2 ∂u r (u, v)d ¯ v¯ v1 v2 ≥ − 2r 2 −2 (∂v φ)2 ∂u r (u, v)d ¯ v. ¯ v1

Again, making use of (22) and rearranging yields (24).

3. Main Result Theorem. Consider spacelike spherically symmetric initial data for the Einstein-Higgs field equations with one asymptotically flat end. Assume that the Higgs potential function is such that the dominant energy condition is satisfied, i.e. V (x) ≥ 0 for all x ∈ R, and assume further that |V (x)| ≤ B < ∞ for all x ∈ R.


599

Fig. 3. The result of the theorem is that the marginally trapped tube A emanating from the outermost marginally trapped sphere p∗ is in fact connected, achronal, and asymptotic to the event horizon H, as shown here

As described in Sect. 2, suppose (Q, , r, φ) is the 2-dimensional Lorentzian quotient of the future Cauchy development of the initial hypersurface S equipped with asymptotically flat physically reasonable black hole initial data, with φ → φ+ along S for some φ+ ∈ R, and with global coordinates (u, v) obtained from identifying Q with its image under a conformal embedding into M1+1 . Let p∗ ∈ Q be the point corresponding to the outermost marginally trapped sphere along S, and let H denote the event horizon of the black hole region in Q. Define m 1 , M, and κ as in (16), (15) and (22), respectively, and assume that there exists a constant κ0 such that 0 < κ0 ≤ κ(q) ≤ 1 for all q ∈ S. Fix a constant C such that r ∂u φ (q) : q ∈ S, r (q) ≤ 2M . C > sup ∂u r

(25)

(26)

Then there exist constants ε > 0 and ρ ∈ (0, 1), depending only on M, B, κ0 , and C, such that if ε |φ − φ+ | ≤ √ r

on S

(27)

and ρ≤

m1 , M

(28)

then Q contains a connected, achronal, marginally trapped tube A which intersects S at p∗ and is asymptotic to the event horizon H. See Fig. 3 for a representative Penrose diagram. Remarks. Assumption (25) amounts to fixing a gauge along S, while (26) poses no restriction on the initial data. The constants ε and ρ should both be understood as smallness parameters. In particular, ρ should be thought of as being very close to 1, forcing m 1 to be close to M, and hence allowing very little matter flux along the asymptotically flat end.

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C. Williams

The possibility that A coincides with part or all of H is included in our interpretation of the term “asymptotic.” Indeed, there are two different, reasonable definitions of what it means for A to be “asymptotic to” H. We discuss these in the course of the proof and show that A is asymptotic to H in both senses. Finally, although we do not pursue the matter in the present paper, it seems plausible that one could remove the requirement that V have a uniformly bounded second derivative on all of R. Instead, one would carefully track the range of φ, showing that it lies in some compact interval which can be identified a priori, depending only on the initial data, and then replace the uniform bound B with the maximum of |V | over this interval. Proof. To begin, note that since (5)-(9) depend only on derivatives of φ except through the potential V (φ), by translating φ and shifting the domain of V , we may assume without loss of generality that φ+ = 0 and V (0) = V (0) = 0 (cf. Sect. 2.3). Such a modification has no real effect except to simplify the presentation of proof. The proof comprises four parts. First we establish the necessary constants, including ε and ρ and a host of auxiliary parameters. Second, we set up a bootstrap region V in R in which certain nice estimates hold. Third, we show how the statement of the theorem follows from retrieval of the bootstrap estimates in V ∩ R. And finally, we show that V = ∅ and retrieve each of the bootstrap conditions in V ∩ R. First fix ε to be any value such that 1 . ε 1 sufficiently close to 1 that

log k < min 21 −2 log 2, log 23 , and the following inequality is satisfied: 2

ε 1 + 21 log 2 log k < . k3 √ 16M 2 B σ 2M Note that such a choice is possible by (30).

(30)

(31)

(32)

(33)

(34)


601

Finally, choose λ such that 2 < λ < 2k and, recalling (26), such that r ∂u φ (q) : q ∈ S, r (q) ≤ λM ≤ C, sup ∂u r and note that (33) and (35) imply that λ < 3. We now fix ρ such that λ 2 max , 1−L 1− , σ 2 < ρ < 1. 2k λ

(35)

(36)

(37)

Such a choice is possible because (35) and the fact that σ < 1 together imply that the maximum on the left-hand side of (37) is strictly less than 1. It is helpful to note that (28) now implies not only that m 1 ≥ ρ M, but also r1 ≥ 2ρ M, since 2m 1 = 2m( p∗ ) = r ( p∗ ) = r1 . λ Two other useful consequences of this definition (37) are that 2ρ < k and 1 − ρ < L. Next we define all the constants needed for the bootstrap argument. First, for convenience, set R := λM, 1−ρ := . 1 − λ2 Note that (37) implies that < L. Finally, define log 2 C1 := , λ 2 log( 2ρ )

:= √ε + 21 log 2 log k, C σ 2M √ B , C := 4M κ1 := 21 κ0 · e−2/λ , α1 :=

1 2 ρ M.

(38) (39)

(40) (41) (42) (43) (44)

Now we have the necessary constants in place to define our bootstrap region. Let V be the set of all points (u, v) in {r ≤ R} ∩ Q such that the following six inequalities are satisfied for all (u, ˜ v) ˜ ∈ J − (u, v) ∩ {r ≤ R}: r ∂u φ < C1 , ( u, ˜ v) ˜ (45) ∂ r u |φ(u, ˜ v))| ˜ < C, (46) |V (φ(u, ˜ v))| ˜ < C, α(u, ˜ v) ˜ > α1 , κ(u, ˜ v) ˜ > κ1 , ∂v r (u, ˜ v) ˜ > 0,

(47) (48) (49) (50)

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C. Williams

where the function α(u, v) is defined as α := m − r 3 V (φ).

(51)

The first step of the proof is to show that inequalities (45)-(50) hold along the curve So , where So := S ∩ R ∩ {r ≤ R} = S ∩ {r1 < r ≤ R}; then V contains an open neighborhood of So in Q. Since V is a past set in {r ≤ R}, i.e. J − (V) ∩ {r ≤ R} ⊂ V, its future boundary ∂ + V := V \ V must be achronal. (Recall that set closures are taken here with respect to the topology of the underlying Minkowski space, so in particular, ∂ + V need not a priori lie entirely in Q.) Next consider a point p ∈ V ∩R. By definition of V, p has the property that non-strict versions of inequalities (45)-(50) hold for all q ∈ J − ( p) ∩ {r ≤ R}. Using this property, we shall show below that in fact strict inequalities (45)-(50) hold at p. Since strict inequalities (45)-(50) must then also hold in a neighborhood of p and ∂ + V is achronal, we must have p ∈ V. That is, V is closed in R, and hence in R ∩ {r ≤ R} as well. Clearly V is also open in R ∩ {r ≤ R}. Thus by continuity we have V = R ∩ {r ≤ R}, and it follows that ∂ + V ∩ Q ⊂ A. Setting aside for the moment the matter of retrieving the bootstrap inequalities in V ∩ R, we now show how the statement of the theorem follows from these assertions, i.e. that V = R ∩ {r ≤ R} and ∂ + V ∩ Q ⊂ A. It was shown in [13] that if the inequality Tuv −2 < 4r12 holds at every point of A, then each connected component of A is a smooth curve in Q which is everywhere spacelike or outgoing-null. Since Tuv = 21 2 V (φ) here, the inequality becomes 2r 2 V (φ) < 1 in our setting. Now from inequality (48), we have that α ≥ α1 > 0 everywhere in V, so in particular at points in ∂ + V ∩ Q ⊂ A we have 0 < α = m − r 3 V (φ) =

r r − r 3 V (φ) = 1 − 2r 2 V (φ) . 2 2

Thus each connected component of ∂ + V ∩ Q is indeed a smooth curve which is everywhere spacelike or outgoing-null. Consider the connected component of ∂ + V ∩ Q containing the outermost marginally trapped sphere p∗ . Then p∗ must constitute the inner endpoint of this curve segment; let q∗ = (u ∗ , v∗ ) denote the segment’s other, outer endpoint. As described in Sect. 2.3, we abuse notation slightly to allow the possibility that v∗ = ∞. Since ∂ + V ∩ Q is smooth / Q. Thus q∗ ∈ Q \ Q, and either and consequently non-degenerate at every point, q∗ ∈ v∗ < ∞ or v∗ = ∞. We wish to show that v∗ must be infinite, so suppose by way of contradiction that v∗ < ∞. Since V is a past set in {r ≤ R} and q∗ ∈ V, we must have I − (q∗ ) ∩ J + (S) ⊂ V ∪ {r ≥ R} ⊂ R. Let u , v be such that (u ∗ , v ), (u , v∗ ) ∈ S. Now, q∗ is the outer endpoint of a smooth, nonempty, achronal curve in Q, so the global hyperbolicity of Q implies that the outgoing null ray from S to q∗ , i.e. {u ∗ } × [v , v∗ ), must lie entirely in Q and hence in R ∪ A. The ingoing null ray from S to q∗ need not lie entirely in Q, however, so let

u ∗∗ = sup u ∈ [u , u ∗ ] : [u , u) × {v∗ } ⊂ Q , and set q∗∗ = (u ∗∗ , v∗ ). (Note that q∗ may equal q∗∗ .) Then I − (q∗∗ ) ∩ J + (S) ⊂ I − (q∗ ) ∩ J + (S) ⊂ R, but since Q \ S is open, q∗∗ ∈ / Q.


603

Fig. 4. If q∗ does not have infinite v-coordinate, then we can find a rectangle J − (q∗∗ )∩ J + ( p)\{q∗∗ } ⊂ R∪A as shown and derive a contradiction to the extension principle of [8] − − + Finally, fixa point p ∈ S ∩ I (q∗∗ ). By construction, J (q∗∗ ) ∩ J ( p) \ {q∗∗ } lies in V ∪ {r ≥ R} ∩Q and hence in R∪A as well. (See Fig. 4 for a Penrose diagram depicting these points and regions.) But since the bound (21) of Proposition 3 guarantees that q∗∗ lies away from the center of symmetry of Q, the extension principle formulated in [8], which was shown in [7] to hold for self-gravitating Higgs fields with V (x) bounded below, implies that q∗∗ ∈ R∪A ⊂ Q, a contradiction. So indeed we must have v∗ = ∞. We have now shown the existence of a smooth, connected marginally trapped tube A = ∂ + V, which is everywhere spacelike or outgoing-null, intersects S at p∗ , and exists for all v ∈ [v( p∗ ), ∞). The final piece of the proof is to show that A is in fact asymptotic to the event horizon H. Now, H is an outgoing null ray {U } × [V, ∞), where {(U, V )} = H ∩ S. Recall that r+ := sup p∈H r ( p); in fact, since ∂v r ≥ 0 on H, we have r+ = limv→∞ r (U, v). Furthermore, Lemma 3 of [8] shows that r ≤ 2M f on H, where M f is the final Bondi mass. Then since M f ≤ M and λ > 2, we have r+ ≤ 2M f ≤ 2M < λM = R. In particular, this implies that the timelike curve {r = R} lies in the exterior of the black hole B and tends to i + , never intersecting H. Since H ⊂ R ∪ A and V = R ∩ {r ≤ R}, it follows that H ⊂ V. Also, since ∂u r < 0 in Q, we have r ≤ r+ everywhere in J + (H) ∩ Q. We showed above that the outer “endpoint” of the marginally trapped tube A is q∗ = (u ∗ , ∞). One way of interpreting the idea that A should be asymptotic to H in this regime is to require that u ∗ = U ; then A and H are asymptotic as curves in the underlying M1+1 . Alternately, perhaps more geometrically, one might instead require that lim p→q∗ , p ∈A (r ( p)) = r+ . (One readily computes that r is non-decreasing along A toward q∗ , and since r ≤ r+ on A, this limit must exist.) Note that both of these interpretations of “asymptotic” allow the possibility that A coincides with H for all v ≥ v, some v ≥ V . We shall show that A is indeed asymptotic to H in both of these senses; both follow directly from the bootstrap estimates in V and Eq. (52) below. For the first statement, we argue by contradiction: suppose u ∗ > U , and fix a value U < u < u ∗ . By construction, the infinite rectangle [U, u] × [V, ∞) ⊂ V. Now, differentiating (10) with respect to u, combining the result with Eqs. (5) and (7), and finally 2 r , we obtain solving for ∂uv

2 ∂uv r = − 21 2 r −2 m − r 3 V (φ) = 2κr −2 (∂u r )α.

(52)

Since r (u, v) ≤ r+ in [U, u] × [V, ∞) ⊂ V, rearranging (52) and using this bound together with the fact that inequalities (49) and (48) hold in V, we have ∂v log(−∂u r ) = 2κr −2 α

> 2κ1r+−2 α1

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C. Williams

everywhere in [U, u] × [V, ∞). For any U ≤ u ≤ u, integrating along an outgoing null segment {u} × [V, v] yields −2

−∂u r (u, v) > −∂u r (u, V ) e2κ1 r+

α1 (v−V )

.

Since [U, u] × {V } is compact, we may assume −∂u r (u, V ) ≥ b for all U ≤ u ≤ u, some b > 0, so for all (u, v) ∈ [U, u] × [V, ∞), we have −2

−∂u r (u, v) > be2κ1 r+

α1 (v−V )

.

Now integrating this along an ingoing null ray segment [U, u] × {v} for any v ≥ V , we arrive at −2

r (u, v) < r (U, v) − be2κ1 r+

α1 (v−V )

(u − U ).

But since u − U > 0 and r (U, v) →v→∞ r+ < ∞, the right-hand side tends to −∞ as v → ∞, while the left-hand side remains positive, a contradiction. So in fact u ∗ = U , and A and H are asymptotic as curves in M1+1 . To show that A and H are asymptotic in the second sense described above, we again argue by contradiction: parametrize A by {(u(v), v)}, v( p∗ ) ≤ v < ∞ — this is possible since A is everywhere spacelike or outgoing-null — and suppose that r (u(v), v) → r+ −δ as v → ∞, some constant δ > 0. Then we again apply bounds (49) and (48), integrate (52) along the ingoing null segment [U, u(v)]×{v}, and use the fact that ∂v r (u(v), v) ≡ 0 to obtain the inequality ∂v r (U, v) > 2κ1r+−2 α1 (−r (u(v), v) + r (U, v)) for all v ≥ V . Taking the lim inf of both sides as v → ∞, we have lim inf (∂v r (U, v)) > 2κ1r+−2 α1 (−(r+ − δ) + r+ ) = 2κ1r+−2 α1 δ > 0, v→∞

a contradiction to the fact that limv→∞ (r (U, v)) = r+ < ∞. Thus A and H do indeed tend to the same limiting radius as v → ∞ as well. We have now completed the proof of the theorem except for the bootstrap itself. That is, it remains to show, one, that strict inequalities (45)-(50) hold along So = S ∩ {r1 ≤ r ≤ R}, and two, that if non-strict versions of (45)-(50) hold for all points within {r ≤ R} to the causal past of a point (u, v) ∈ R ∩ {r ≤ R}, then in fact strict versions of the six inequalities must hold at p. First, from (26), (32), (33), (37), and (40), we have C1 > C, so it follows from (36) that inequality (45) is satisfied on So . To see that (46) holds on So , we apply (27), (19), (28), (37), and (41): ε ε ε ε ε |φ| ≤ √ ≤ √ = √ < C. ≤√ ρ M − λ3 M 3 B 16M 2 Bk 3 3 1 λ = ρM − M 2 2k > ρ M − 21 ρ 3 M > ρ M − 21 ρ M = α1 , where in addition to (46) we have used (37) and the fact that ρ > ρ 3 (since ρ < 1). Also, |V (φ)| ≤ B|φ| ≤ BC 1 1. Thus (47) and (48) hold on So , since we have already shown that (46) does. Since the six inequalities (45)-(50) are all satisfied along So , we have So ⊂ V, and V = ∅. It remains to retrieve (45)-(50) in V ∩ R. Clearly (50) is automatically satisfied by definition of R, so the five nontrivial inequalities to consider are (45)-(49).

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C. Williams

We begin with (49). To retrieve the desired inequality in V ∩ R, we estimate κ in two pieces, first using an energy estimate to get a stronger bound on So ∪ {r = R}, then integrating in u and using the bootstrap assumptions to obtain the desired one in V. For the bound on {r = R}, first recall that by hypothesis, we have κ ≥ κ0 on all of S. Also observe that in the exterior region {r ≥ R = λM}, we have 1−

2m 2M 2 ≥1− = 1 − > 0. r R λ

(54)

Now, rearranging Eq. (5), we obtain ∂u κ =

r ∂u φ ∂u r

2

∂u r r

κ,

(55)

and integrating along the ingoing null ray [u , u] × {v}, where the point (u, v) ∈ {r = R = λM} and (u , v) ∈ S, we apply (54), the energy estimate (23), as well as (28), (11), and (39), to get 2m r ∂u φ 2 1 κ(u, v) = κ(u , v) exp 1− ˜ v)d u˜ ∂u r · (u, r ∂u r 1 − 2m u r r u 1 2m r ∂u φ 2 ≥ κ0 exp 1− ∂u r (u, ˜ v)d u˜ r ∂u r (λ − 2) M u 2 1 − mM1 ≥ κ0 exp − (λ − 2)

u

≥ κ0 · e−2/λ . Since κ ≥ κ0 on S and e−2/λ < 1, we thus have κ ≥ κ0 · e−2/λ everywhere on So ∪ {r = R}. Now, given any point (u, v) ∈ V ∩ R, the ingoing null ray to the past of (u, v) must intersect S o ∪ {r = R}, say at the point (u , v). Again we integrate (55), this time applying the bootstrap inequality (45) along with (11) and (40) to obtain 2 r ∂ ∂ φ r u u κ(u, v) = κ(u , v) exp (u, ¯ v)d u¯ ∂u r r u r (u, v) ≥ κ0 · e−2/λ exp C12 log r (u , v) 2ρ ≥ κ0 · e−2/λ exp C12 log λ log 2 2ρ −2/λ = κ0 · e exp log λ λ 2 log( 2ρ )

1 = κ0 · e−2/λ · √ 2 > κ1 .

u


607

Thus we have retrieved and in fact improved (49) in V ∩ R. Separately, note also that it follows from (55) that ∂u κ < 0 everywhere in Q (since κ > 0 by (11)), and thus our assumption that κ ≤ 1 on S implies that κ ≤ 1 in all of Q. Next, as was shown previously, in order to retrieve (46), (47), and (48) in V ∩ R, it suffices to retrieve (46). As with (49), retrieving (46) requires two steps: we first estimate |φ| on So ∪ {r = R}, again utilizing an energy estimate, then use that bound to obtain in V ∩ R. |φ| < C √ From (53), we know already that |φ| < ε/ 2σ 2 M on So . To see that the same bound holds along {r = R}, we integrate |∂u φ| along an ingoing null ray emanating from S. Suppose (u, v ) ∈ {r = R} and {(u , v )} = S ∩ {v = v }. Then since the ingoing null ray segment [u , u] × {v } ⊂ {r ≥ R = λM} ⊂ R, integrating ∂u φ along it and applying (27), Cauchy-Schwarz, (54), the energy estimate (23), as well as (39), (31), (30) and (35), yields u |φ(u, v)| ≤ |φ(u , v)| + |∂u φ(u, ˜ v)|d u˜ u

− 12

≤ ε · r (u , v) u r ∂u φ 2 2m + − 1− ∂u r (u, ˜ v)d u˜ r ∂u r u u ∂u r − ˜ v)d u˜ · (u, 2 1 − 2m u r r u 1 1 ∂u r −2 − 2 (u, ˜ v)d u˜ ≤ ε · r (u , v) + 2(M − m 1 ) 2 r 1− λ u 1 1 − 12 · < ε · r (u , v) + 2(M − m 1 ) 2 1− λ R 2(1 − mM1 ) ε + ≤ √ λ−2 λM ε 2 + ≤ √ λ λM 1 √ ε + 2L M < √ λM ε . < √ σ 2M ε everywhere along So ∪ {r = R}. Thus we have |φ| < √ σ 2M Now consider a point (u, v) ∈ V ∩ R with past ingoing null ray intersecting S o ∪ {r = R} at the point (u , v). Then: u |φ(u, v)| ≤ φ(u , v) + |∂u φ(u, ¯ v)|d u¯ u u r ∂u φ ∂u r ε ¯ v)d u¯ + < √ r (u, σ 2M u ∂u r

608

C. Williams

ε r (u , v) + C1 log √ r (u, v) σ 2M ε R ≤ √ + C1 log r1 σ 2M log 2 ε λ · log , + = √ λ 2ρ σ 2M 2 log 2ρ ε λ + 21 log 2 log 2ρ , = √ σ 2M

ε + 21 log 2 log k < √ σ 2M = C, ≤

where we have used (45), (40), (21) and (37), in addition to our bound for |φ| on So ∪ {r = R}. Thus (46) holds everywhere in V ∩ R, and hence so do (47) and (48). Finally, it remains only to retrieve (45). Combining Eqs. (9) and (52) and rearranging terms, we derive r ∂u φ 2κα r ∂u φ ∂v =− 2 + r κ V (φ) − ∂v φ. (56) ∂u r r ∂u r Now, the past boundary of V lies in (S ∩ {r1 ≤ r ≤ R}) ∪ {r = R}, and since {r = R} is everywhere timelike, the outgoing null segment to the past of a point (u, v) ∈ V must intersect So , say at the point (u, v ). Then integrating (56) along the null segment {u} × [v , v] ⊂ V, we have v 2κα r ∂u φ ˜ v˜ r ∂u φ ≤ e v − r 2 (u,v)d (u, v) (u, v ) ∂ r ∂ r u u v v − 2κα (u,v)d ˜ v˜ ¯ v¯ + r κ V (φ) − ∂v φ (u, v)d e v¯ r 2 v v 2κ α − 1 1 (v−v) ¯ ≤C+ RC + |∂v φ| (u, v)d e R2 ¯ v¯ v 2κ α C R3 − 1 1 (v−v ) 1 − e R2 ≤C+ 2κ1 α1 v

+

v

e

−

4κ1 α1 (v−v) ¯ R2

r −2 κ(u, v)d ¯ v¯

v

v

(∂v φ)2 r 2 κ −1 (u, v)d ¯ v¯

C R3 ≤C+ 2κ α 1 1

4κ α R2 − 1 1 (v−v ) 1 − e R2 · 2(M − m 1 ) 4κ1 α1 C R3 R 2 (M − m 1 ) ≤C+ + , 2κ1 α1 2r12 κ1 α1 + r1−2 ·

(57)


609

where we have used (36), the bootstrap inequalities (49), (48), and (47), CauchySchwarz, (21), the energy estimate (24), and the fact that κ ≤ 1. Now, the second term here is √ C R3 B (λM)3 = · 1 2κ1 α1 4M 2( 2 κ0 e−2/λ )( 21 ρ M) √ 3 λ BM = −2/λ 2ρ κ0 e √ 2L/λ B Me kλ2 < κ0 √ 9 B Me L < , κ0 where we have applied (42), (38), (43), (44), (37), and the facts that < L, λ > 2, k < 1, and λ < 3; and the third one is R 2 (M − m 1 ) (λM)2 (M − ρ M) ≤ 2 2r1 κ1 α1 2(2ρ M)2 ( 21 κ0 e−2/λ )( 21 ρ M) λ 2(1 − ρ)e2/λ = · 2ρ ρκ0 λ L/2 2(1 − ρ) e < 2ρ ρκ0 2L < ke L/2 σ 2 κ0 L 3e L/2 < , σ 2κ0 where this time we used (38), (43), (44), < L, λ > 2, various consequences of (37), and k < 23 . Thus, resuming from (57), we have √ L L/2 r ∂u φ 9 B Me L 3e + ∂ r (u, v) < C + κ0 σ 2κ0 u = log 2 2 log k log 2 0 is uniformly translation bounded and hence precompact finite measures vol(B r) in the vague topology by [32, Prop. 2.2]. One can thus always select converging subsequences to define an autocorrelation (which then depends on the sequence of averaging sets). As long as balls are used, one speaks of natural autocorrelations. More generally, one may work with any averaging sequence A = {An | n ∈ N} of relatively compact, open sets An ⊂ R d that satisfy An ⊂ An+1 for all n ∈ N together with n∈N An = R d . Again, for translation bounded measures ω, the corresponding limit in (6) exists, at least along suitable subsequences. An important further ingredient is the concept of a van Hove sequence, which is an averaging sequence with a restricted ‘surface to volume’ ratio. To formalise this, let K , C ⊂ R d be compact and define ∂ K C := (C + K )\C ◦ ∪ ( R d \C − K ) ∩ C , (7) which may be viewed as a K -thickened boundary of C. Then, A is called van Hove when, for every compact K ⊂ R d , vol(∂ K An ) = 0. n→∞ vol(An ) lim

(8)

Now, the comparison of limits taken along different averaging sequences makes sense, and becomes independent of A for ergodic systems; compare [55, Lemma 1.1]. Also, as follows from [55, Lemma 1.2], translation bounded measures satisfy the relation lim

n→∞

ωn ∗ ω n ωn ∗ ω = lim , n→∞ vol(An ) vol(An )

(9)

Diffraction of Stochastic Point Sets

617

provided that A is van Hove and one of the limits exists. Here, ωn = ω| An , and ωn ∗ ω is well-defined by [13, Prop. 1.13]. This freedom will be used several times below. The general situation for a translation bounded measure ω is as follows. The van Hove property of A implies that |ω|(An ) ≤ c vol(An ) with a constant c > 0. An obvious modification of [32, Prop. 2.2] in conjunction with [55, Lemma 1.2] then gives the following result. Lemma 3. Let ω be a translation bounded measure, and A a van Hove averaging ωn ∗ ω ωn ∗ ω n sequence. With γn := vol(A and γn;mod := vol(A , the families {γn | n ∈ N} and n) n) {γn;mod | n ∈ N} are uniformly translation bounded and hence precompact in the vague topology. Any accumulation point of either family, of which there is at least one, is also an accumulation point of the other family, and a translation bounded, positive definite measure.

Lemma 1 applies to any autocorrelation measure, and the corresponding measure γ is then a positive, translation bounded measure. It is called the diffraction measure of ω, relative to the averaging sequence A. In ergodic situations, we have no dependence on A and thus suppress it. Then, the diffraction measure is also related to the Bartlett spectrum known from stochastic geometry, though there are important differences to be discussed later; see Remark 15 below. In general, an interesting initial question concerns the spectral type of γ , which follows from the spectral decomposition γ = ( γ )pp + ( γ )sc + ( γ )ac

(10)

of γ into its pure point, singular continuous andabsolutely continuous parts relative to λ, the latter being the Haar measure on R d with λ [0, 1]d = 1. Lattices and regular model sets [8,55] are examples with γ = ( γ )pp , while the Thue-Morse and the Rudin-Shapiro sequence show singular continuous and absolutely continuous components, respectively; compare [35] and references given there. Absolutely continuous components appearing as a result of stochastic influence are the main theme below. 3. Renewal Processes in One Dimension An illustrative class of examples is provided by the classical renewal process on the real line, defined by a probability measure on R + = {x > 0} of finite mean as follows. Starting from some initial point, at an arbitrary position, a machine moves to the right with constant speed and drops a point on the line with a random waiting time that is distributed according to . When this happens, the clock is reset and the process resumes. In what follows, we assume that both the velocity of the machine and the expectation value of are 1, so that we end up (in the limit that we let the initial point move to −∞) with realisations that are almost surely point sets in R of density 1. Clearly, the process just described defines a stationary process. It can thus be analysed by considering all realisations which contain the point 0. Moreover, there is a clear (distributional) symmetry around this point, so that we can determine the autocorrelation (in the sense of (6)) of almost all realisations from studying what happens to the right of 0 (we will make this approach rigorous in Proposition 1 below). Indeed, if we want to know the frequency per unit length of the occurrence of two points with distance x (or the corresponding density), we need to sum the contributions that x is the first point

618

M. Baake, M. Birkner, R. V. Moody

after 0, the second point, the third, and so on. In other words, we almost surely obtain the autocorrelation γ = δ0 + ν + ν (11) ∗n and with ν = + ∗ + ∗ ∗ + · · · = ∞ ν as defined above, provided that n=1 the sum in Eq. (11) converges properly. Note that the point measure at 0 simply reflects that the almost sure density of the resulting point set is 1. In the slightly more general case of a probability measure on R + ∪ {0}, one has the following convergence result. It is essentially a measure theoretic reformulation of the main lemma in [25, Sec. XI.1], but we prefer to give a complete proof that is adjusted to our setting. Lemma 4. Let be a probability measure on R + ∪ {0}, with (R + ) > 0. Then, νn := + ∗ + · · · + ∗n with n ∈ N defines a sequence of positive measures that converges towards a translation bounded measure ν in the vague topology. Proof. Note that the condition (R + ) > 0 implies 0 ≤ ({0})< 1, hence excludes the case = δ0 . When = δa for some a > 0, one has νn = nm=1 δma by a simple convolution calculation, and the claim is obvious. In all remaining cases, it is possible to choose some a ∈ R + with ({a}) = 0 and 0 < ([0, a)) = p < 1, so that also ([a, ∞)) = 1 − p < 1. Since the sequence νn is monotonically increasing, the claimed vague convergence follows from showing that lim supn→∞ νn ([0, x)) is bounded by C1 + C2 x for some constants Ci . As there are at most countably many points y with ({y}) > 0, it is sufficient to show these for all x ∈ R + with ({x}) = 0. In a estimates ∗n ([b, b + x)) is bounded by 1 + C + C x, second step, we then demonstrate that ∞ 1 2 n=1 independently of b, which establishes translation boundedness. If (X i )i∈N denotes a family of i.i.d. random variables, with common distribution according to (and thus values in R + ∪ {0}), one has P (X 1 + · · · + X m < x) = ∗m ([0, x)). On the other hand, for the a chosen above, one has the inequality P(X 1 + · · · + X m < x) ≤ P (card{1 ≤ i ≤ m | X i ≥ a} ≤ x/a) [x/a] m (1 − p) p m− , = =0

where

m

= 0 whenever > m. Observing

∞ m m=1

∞

m=1

(1 − p) p m− = (1 − p)

p m = p/(1 − p) and

∞ 1 d m 1 p = ! d p 1− p m=0

for all ≥ 1, the previous inequality implies, for arbitrary n ∈ N, νn ([0, x)) ≤

∞ [x/a] m m=1 =0

(1 − p) p m− =

which establishes the first claim.

p + [x/a] p 1 ≤ + x, 1− p 1 − p a(1 − p)


619

For the second estimate, we choose b ≥ 0, x > 0 and observe ∞

∗n ([b, b + x)) =

n=1

= ≤ =

∞

P (b ≤ X 1 + · · · + X n < b + x)

n=1 ∞ n n=1 k=1 n ∞

P (X 1 + · · · + X k−1 ≤ b ≤ X 1 + · · · + X k and b ≤ X 1 + · · · + X n < b + x) P (X 1 + · · · + X k−1 ≤ b ≤ X 1 + · · · + X k ) P (X k+1 + · · · + X n < x)

n=1 k=1 ∞

∞

k=1

n=k

P (X 1 + · · · + X k−1 ≤ b ≤ X 1 + · · · + X k )

=1+

∞

P (X k+1 + · · · + X n < x)

P (X 1 + · · · + X m < x) ,

m=1

with the convention to treat empty sums of random variables as 0. The last step used the i.i.d. property of the random variables together with P(0 ≤ X ) = 1 and ∞

P (X 1 + · · · + X k−1 ≤ b ≤ X 1 + · · · + X k ) = 1.

k=1

In conjunction with our previous estimate, this completes the proof.

When ({0}) > 0, we are outside the realm of (renewal) point processes, and formula (11) for the autocorrelation no longer applies. This case might nevertheless be analysed with the methods of Secs. 4 and 5, see Example 7 and Corollary 1 in particular. For the remainder of this section, we assume ({0}) = 0, so that is a measure on R + ; see Remark 14 below for an alternative approach via random counting measures, or [25, Ch. XI.9]. Proposition 1. Consider a renewal process on the real line, defined by a probability measure on R + with mean 1. This defines a stationary stochastic process, whose realisations are point sets that almost surely possess the autocorrelation measure γ = δ0 + ν + ν of (11). ∞ ∗n Here, ν = is a translation bounded positive measure. It satisfies the n=1 renewal equations ν = +∗ν

and

(1 − ) ν = ,

where is a uniformly continuous function on R. In this setting, the measure γ is both positive and positive definite. Proof. The renewal process is a classic stochastic process on the real line which is known to be stationary and ergodic; compare [25, Ch. VI.6] for details. Consequently, the measure of occurrence of a pair of points at distance x + dx (or the corresponding density) can be calculated by fixing one point at 0 (due to stationarity) and then determining the ensemble average for another point at x + dx (due to ergodicity). This is the justification for the heuristic reasoning given above, prior to Eq. (11).

620


By Lemma 4, ν is a translation bounded measure, so that the convolution ∗ ν is well defined by [13, Prop. 1.13]. The first renewal identity is then clear from the structure of ν as a limit, while the second follows by Fourier transform and the convolution theorem. The autocorrelation is a positive definite measure by construction, though this is not immediate here on the basis of its form as a sum, see [1] for a related discussion.

Let us now consider the spectral type of the resulting diffraction measure for the class of point sets generated by a renewal process. This requires a distinction on the basis of the support of . To this end, the second identity of Proposition 1 is helpful, because one has ν(k) =

(k) 1 − (k)

(12)

at all positions k with (k) = 1. This is in line with summing ν as a geometric series, which gives the same formula for ν(k) for all k with | (k)| < 1 and has (12) as the unique continuous extension to all k with | (k)| = 1 = (k). In fact, one sees that ν(k) is a continuous function on the complement of the set {k ∈ R | (k) = 1}. For most , the latter set happens to be the singleton set {0}. In general, a probability measure µ on R is called lattice-like when its support is a subset of a translate of a lattice, see [27] for details. We need a slightly stronger property here, and call µ strictly lattice-like (called arithmetic in [25]) when its support is a subset of a lattice. So, the difference is that we do not allow any translates here; see [2] for related results. Lemma 5. If µ is a probability measure on R, its Fourier transform, µ(k), is a uniformly continuous and positive definite function on R, with | µ(k)| ≤ µ(0) = 1. Moreover, the following three properties are equivalent: (i) card{k ∈ R | µ(k) = 1} > 1; (ii) card{k ∈ R | µ(k) = 1} = ∞; (iii) supp(µ) is strictly lattice like. Proof. One has µ(k) = R e−2πikx dµ(x), whence the first claims are standard consequences of Fourier analysis; compare [48, Prop. 5.2.1] and [52, Sec. 1.3.3]. If µ = x∈Γ p(x)δx for a lattice Γ ⊂ R, with p(x) ≥ 0 and x∈Γ p(x) = 1, one has µ(k) = p(x) e−2πikx , x∈Γ

µ(k) = 1}, so that we so that µ(k) = 1 for any k ∈ Γ ∗ . In particular, Γ ∗ ⊂ {k ∈ R | have the implications (iii) ⇒ (ii) ⇒ (i). −2πikx Conversely, if µ(k) = 1 for some k = 0, one has R e dµ(x) = 1 and hence (13) (1 − cos(2π kx)) dµ(x) = (1 − cos(2π kx)) dµ(x) = 0, R

supp(µ)

where supp(µ), the support of the probability measure µ, is a closed subset of R and measurable. The integrand is a continuous non-negative function that, due to k = 0, vanishes precisely on the set k1 Z, which is a lattice.


621

˙ as a disjoint union of measurable sets, with A = supp(µ)∩ 1 Z Write supp(µ) = A∪B k and B = supp(µ) ∩ (R\ k1 Z). We can now split the second integral in (13) into an integral over A, which vanishes because the integrand does, and one over the set B, which would give a positive contribution by standard arguments, unless B = ∅. But this means supp(µ) = A ⊂ k1 Z, so that (i) ⇒ (iii), which establishes the result.

At this point, we can state the main result of this section, the diffraction properties of renewal processes; compare [19, Ex. 8.2(b)] for a special case. Theorem 1. Let be a probability measure on R + with mean 1, and assume that is not strictly lattice-like. Assume further that a moment of of order 1 + ε exists for some ε > 0. Then, the point sets obtained from the stationary renewal process based on almost surely have a diffraction measure of the form γ = δ0 + ( γ )ac with γ )ac = (

1 − | (k)|2 λ = (1 − h) λ, |1 − (k)|2

where h is a continuous function on R\{0} that is locally integrable. It is given by 2 | (k)|2 − Re( (k)) h(k) = |1 − (k)|2 and measures the difference from a constant background as described by λ. When is strictly lattice-like, the pure point part becomes a lattice Dirac comb, and the behaviour of h at 0 repeats at each point of the underlying lattice, see Remark 3 for details. Proof. The process has a well-defined autocorrelation γ , by an application of Proposition 1, in the sense that almost every realisation of the process is a point set Λ with this autocorrelation. Since γ is a positive definite measure, it is Fourier transformable by Lemma 1(ii), with γ being a positive measure on R. The point measure at 0 with intensity 1 reflects the fact that the resulting point set Λ almost surely has density 1. To see this, define gn = n1 1[− n2 , n2 ] and h n = gn ∗ gn . Here, h n is a positive definite, tent-shaped function with support [−n, n] and maximal value sin(π kn) 2 1 n at 0. It has (inverse) Fourier transform h n (k) = ( π kn ) , which is a non-negative function (with maximum value 1 at k = 0) that concentrates around 0 as n → ∞. Let ωr := δΛ∩[−r,r ] . Using (6) together with (ωr ∗ ωr ) (gn ∗ gn ) ≥ 0, it is not difficult to

n→∞

see that γ (h n ) −−−−→ (dens(Λ))2 , which is almost surely 1 (for this, assume first that r n 1, then take the limit r → ∞ followed by the limit n → ∞). On the other hand, one has n→∞

γ (h n ) = γ ( h n ) −−−−→ γ ({0}) , γ (h n ) =

due to the concentration property of h n (in particular, for all ε > 0, one verifies the relation γ (Bε (0)) ≥ (dens(Λ))2 > 0, which proves the existence of a point measure at 0). Due to the assumption that supp() is not contained in a lattice, we may invoke Lemma 5 to see that (k) = 1 whenever k = 0, so that we have pointwise convergence n→∞

νn (k) −−−→ ν(k) =

(k) 1 − (k)

622


on R\{0}, and similarly for ν. Since is uniformly continuous on R and (k) = 1 on R\{0}, both ν and ν are represented, on R\{0}, by continuous Radon-Nikodym densities. Writing (δ0 + ν + ν ) = (1 − h)λ, hence (ν + ν ) = −h λ, the formula for h follows from ν = ν. It remains to show that 1 − h is locally integrable near 0. Let X be a random variable with distribution . Since the latter has mean 1 and our assumption guarantees that ∞ X 1+ε = 0 x1+ε d(x) < ∞, we have the Taylor series expansion (k) = 1 − 2πik + O(|k|1+ε ), as |k| → 0, by an application of [61, Thm. 1.5.4]. Inserting this into the expression for h results in h(k) = 2 + O(k −1+ε ), as |k| → 0, which establishes integrability around 0, and thus absolute continuity of the measure (1 − h)λ. As the contribution to the central peak is already completely accounted for by the term δ0 , the claim follows.

Remark 2. Asymptotic behaviour of h. When, in the setting of Theorem 1, the second moment of exists, one obtains from [61, Thm. 1.5.3] the slightly stronger expansion (k) = 1 − 2πik − 2π 2 X 2 k 2 + O(|k|2 ), as |k| → 0. This leads to the asymptotic behaviour h(k) = 2 − X 2 + O(1), as |k| → 0, which implies that h is bounded and can continuously be extended to h(0) = 2−X 2 = 1 − σ 2 , where σ 2 is the variance of . Clearly, the existence of higher moments implies stronger smoothness properties. Remark 3. Complement of Theorem 1. When happens to be strictly lattice like, the Z-span of the finite or uniformly discrete1 set supp() is a lattice of the form Γ = bZ, where b > 0 is unique (in other words, Γ is the coarsest lattice in R that contains supp()). Then, one finds the diffraction γ = δZ/b + (1 − h)λ, with the function h from Theorem 1. Note that h is well-defined (and continuous) on R\Γ ∗ , with Γ ∗ = Z/b being the dual lattice of Γ . Moreover, it is locally integrable around all points of Γ ∗ , so that (1 − h)λ is again an absolutely continuous measure. Note that, since the underlying point set is always a subset of Γ , the diffraction measure is periodic, with Γ ∗ as its lattice of periods; compare [2] for general results in this direction. When supp() is a finite set, one is in the situation of a random tiling with finitely many prototiles. A more detailed discussion, together with an explicit calculation of h for this case, is given in [4, Thm. 2]; see also Example 5 and Remark 7. 1 Recall that a set S ⊂ R d is called uniformly discrete when there is a number s > 0 such that the distance between any two distinct points of S is at least s.


623

Let us turn to some examples, for which we employ the Heavyside function, ⎧ ⎪ ⎨1, if x > 0, Θ(x) := 21 , if x = 0, ⎪ ⎩0, if x < 0.

(14)

This formulation of Θ has some advantage for formal calculations around generalised functions and their Fourier transforms. Example 1. Poisson Process on the Real Line. The probably best-known stochastic process is the classical (homogeneous) Poisson process on the line, with intensity 1, where = f λ is given by the density f (x) = e−x Θ(x). It is easy to check that the convolution of n + 1 copies of this function yields e−x xn Θ(x)/n!, which gives ν = Θλ. As the intensity is 1, this results in the autocorrelation ν = δ0 + λ, γ = δ0 + ν + and thus in the diffraction γ = γ , compare Eq. (4). Remark 4. Characterisation of Poisson Processes. Let N denote a homogeneous Poisson process on the real line, so that, for any measurable A ⊂ R, N (A) is the number of renewal points that fall into A. It is well-known that N (A) is then Poisson(λ(A))-distributed, which means that P(N (A) = k) =

e−λ(A) (λ(A))k k!

with k ∈ N0 , and that, for any collection of pairwise disjoint sets A1 , A2 , . . . , Am , the random numbers N (A1 ), . . . , N (Am ) are independent. In fact, this property characterises the Poisson process (compare [19, Ch. 2.1]), and it can serve as a definition in higher dimensions or in more general measure spaces, to which the renewal process cannot be extended. Example 2. Renewal Process with Repulsion. A perhaps more interesting example in this spirit is given by the density f (x) = 4x e−2x Θ(x). It is normalised and has mean 1, as in Example 1, but models a repulsion of points for small distances. Note that this distribution can be realised out of Example 1 by taking only every second point, followed by a rescaling of time. By induction (or by using well-known properties of the gamma distributions, compare [25, Sec. II.2]), one checks that f ∗n (x) =

4n (2n−1)!

x2n−1 e−2x Θ(x),

which finally results in the autocorrelation γ = δ0 + (1 − e−4|x| ) λ = δ0 + λ − e−4|x| λ

624


1.4 1.2 1 0.8 0.6 0.4 0.2 -3

-1

-2

1

2

3

Fig. 1. Absolutely continuous part of the diffraction measure from Example 3, for α = 0.7 (top curve), α = 1 (horizontal line, which also represents Example 1), α = 2 (see also Example 2) and α = 8 (overshooting curve)

and in the diffraction measure γ = δ0 +

2 + (π k)2 2λ λ = δ0 + λ − . 2 4 + (π k) 4 + (π k)2

This is illustrated in Fig. 1. The ‘dip’ in the absolutely continuous part around 0, and thus the deviation from the previous example, reflects the effectively repulsive nature of the stochastic process when viewed from the perspective of neighbouring points. Example 3. Renewal Process with Gamma Law of Mean 1. The previous two examples are special cases of the gamma family of measures. For fixed mean 1, they are parametrised by a real number α > 0 via α = f α λ and the density f α (x) :=

α α α−1 −αx x e Θ(x).

(α)

(15)

While α = 1 is the ‘interaction-free’ Poisson process of Example 1, the density implies an effectively attractive (repulsive) nature of the process for 0 < α < 1 (for α > 1). When α = k ∈ N, the process can also be interpreted as a modified Poisson process where one keeps only every k th point (followed by an appropriate rescaling). α nα Observing f α∗n (x) = (nα) xnα−1 e−αx Θ(x) for n ∈ N, this leads to the measure να = gα Θ λ

with

gα (x) = α e−αx

∞ (αx)nα−1 n=1

(nα)

.

(16)

Note that, for fixed α, one has limx→∞ gα (x) = 1. The calculations result in the autocorrelation γα = δ0 + gα (|x|) λ


625

and in the diffraction γα = δ0 + (1 − h α ) λ, where h α is the symmetric function defined by h α (k) =

2 (1 − Re((1 + 2πik/α)α )) . |1 − (1 + 2πik/α)α |2

The latter follows from the general form of h in Theorem 1, together with the observation that f α (k) = (1 + 2πik/α)−α . It is easy to see that limk→±∞ h α (k) = 0, for any fixed α > 0, which makes the role of h α as the deviation from the Poisson process diffraction more transparent, where α = 1 and h 1 ≡ 0. Note also that limα→∞ γα = δZ in the vague topology, in line with the limits mentioned before. This can nicely be studied in a series of plots of the diffraction with growing value of the parameter α. Figure 1 shows some initial cases. Remark 5. Construction of Delone Sets. Of particular interest in the applications are Delone sets (which are point sets that are both uniformly discrete and relatively dense), because points (representing atoms, say) should neither be too close nor too far apart. Such sets can also arise from a renewal process. In fact, if one considers a probability measure on R + , the resulting point sets are always Delone sets when supp() ⊂ [a, b] with 0 < a ≤ b < ∞, and conversely. This equivalence does not depend on the nature of on [a, b], while the local complexity of the resulting point sets does. In particular, if is absolutely continuous, the point sets will not have finite local complexity (see below for a definition). It is clear that no absolutely continuous is lattice-like, hence certainly not strictly lattice-like, so that all these examples match Theorem 1. Probability measures with supp() contained in a lattice are covered by Remark 3. They are of interest because they form a link to point sets and tilings of finite local complexity, which have only finitely many patches of a given size (up to translations). Let us consider some examples. Example 4. Deterministic Lattice Case. The simplest case is = δ1 . From δ1 ∗δ1 = δ2 , one sees that ν = δN and hence γ = δ0 + δN + δ−N = δZ , which is a lattice Dirac comb, with Fourier transform γ = δZ according to the Poisson summation formula (1). This is the deterministic case of the integer lattice Z, covered in this setting. Remark 6. Deterministic Limit of Example 3. The last example can also be seen as a limiting case of the measure α defined by Eq. (15). In particular, one has lim α→∞ α = δ1 and limα→∞ να = δN , with να as in (16) and both limits to be understood in the vague topology. This can also be seen by means of the strong law of large numbers. For each n ∈ N, by well-known divisibility properties of the family of Gamma distributions, n is the distribution of n 1 Xi , n i=1

where the X i are independent and exponentially distributed random variables with √ mean 1. This sum then concentrates around 1, with a standard deviation of order 1/ n.

626


Example 5. Random Tilings with Finitely Many Prototiles. Consider the measure = αδa + (1−α)δb , with α ∈ (0, 1) and a, b > 0, subject to the restriction αa + (1 − α)b = 1 to ensure density 1. Each realisation of the corresponding renewal process results in a point set that can also be viewed as a random tiling on the line with two prototiles, of lengths a and b. As before, place a normalised point measure at each point of the realisation. Then, the diffraction (almost surely) has a pure point and an absolutely continuous part, but no singular continuous one. The pure point part can be just δ0 (when b/a is irrational) or a lattice comb (see Remark 3); details are given in [4], including an explicit formula for the AC part. This has a straightforward generalisation to any finite number of prototiles, with a similar result. Also in this case, there is an explicit formula for the diffraction measure, which was derived in [4] by a direct method, without using the renewal process. Remark 7. Continuous Diffraction with ‘Needles’. Looking back at Lemma 5, one realises that Example 5 revolves around the lattice condition in an interesting way. Namely, even if is not strictly lattice-like, supp() for a random tiling example with finitely many prototiles is a finite set, and thus a subset of a Meyer set (which is a relatively dense set Λ whose difference set Λ − Λ is uniformly discrete). We then know from the harmonic analysis of Meyer sets, compare [44] and references therein, that (k) will come ε-close to 1 with bounded gaps in k. This means that the diffraction measure, though it is absolutely continuous apart from the central peak at k = 0, will develop sharp ‘needles’ that are close to point measures in the vague topology — a phenomenon that was also observed in [4] on the basis of the explicit solution. 4. Arbitrary Dimensions: Elementary Approach Let us now develop some intuition for the influence of randomness on the diffraction of point sets and certain structures derived from them in Euclidean spaces of arbitrary dimension. In this section, our point of view is from a single point set Λ ⊂ R d that is being modified randomly, by replacing each point by a complex, finite, random cluster. This is still relatively easy as long as Λ is sufficiently ‘nice’. In Sec. 5, we revisit this situation from the point of view of a stationary ergodic point process, which treats almost all of its realisations at once and permits a larger generality for the sets Λ, though the clusters will then be restricted to positive or signed measures. Let Λ ⊂ R d be a fixed point set, which we assume to be of finite local complexity (FLC). By definition, this means that there are only finitely many distinct patches of any given size (up to translations) in Λ. This property is equivalent to the difference set Λ − Λ being locally finite [55], the latter saying that K ∩ (Λ − Λ) is a finite set for all compact K ⊂ R d . In particular, since 0 is then isolated in Λ − Λ, the set Λ itself is uniformly discrete; see Remark 11 for a possible extension. Attached to Λ is its Dirac comb δΛ = x∈Λ δx , which is a translation bounded measure, as a consequence of the FLC property. We associate to δΛ the autocorrelation and the diffraction measure as explained in Sec. 2, for a suitably chosen averaging sequence A = {An | n ∈ N} of van Hove type. A natural choice is An = Brn (0), with Br (0) denoting the open ball of n→∞ radius r around 0, for a non-decreasing series of radii with rn −−−→ ∞ (alternatively, nested cubes are also quite common).


627

Set Λn = Λ ∩ An (so that Λn Λ in the obvious local topology [55]) and consider γΛ,n :=

δΛn ∗ δ 1 Λn δx−y . = vol(An ) vol(An ) x,y∈Λn

We now make the assumption that the limit lim γ n→∞ Λ,n

=: γΛ

(17)

exists in the vague topology, which is then the autocorrelation measure of the set Λ relative to the averaging sequence A. Remark 8. Accumulation Points versus Limits. Due to translation boundedness of δΛ , the sequence of measures γΛ,n always has points of accumulation; see [32, Prop. 2.2] and Lemma 3. Consequently, one can always select a subsequence of A for which the assumption (17) is satisfied. This remains true even if we relax the nesting condition for A. In this sense, when the autocorrelation is not unique (as in the example of the visible lattice points without nesting [9]), we simply select one of the possible autocorrelations by a suitable choice of A. Our results below apply to any autocorrelation of this kind separately. In this sense, the assumption made in (17) is not restrictive. As briefly explained in Sec. 2, see Lemma 3, the van Hove property of A in the context of (17) implies that one also has δΛn ∗ δ Λ = γΛ , n→∞ vol(An )

lim γΛ,n;mod := lim

n→∞

(18)

the difference between the two approximating measures in (17) and (18) being a ‘surface term’ that vanishes in the infinite volume limit n → ∞. Equation (17) explicitly shows that the measure γΛ is positive definite (hence transformable by Lemma 1), while (18) is easier to work with for (pointwise) calculations in the presence of random modifications as introduced below. Since Λ − Λ is locally finite by assumption, Eq. (18) is equivalent to the existence of all the pointwise limits lim η (z) n→∞ n

=: η(z),

(19)

with the approximating coefficients ηn (z) =

card{x ∈ Λn | x − z ∈ Λ} , vol(An )

where η(z) = 0 for any z ∈ Λ − Λ. Clearly, the measure γΛ as well as the coefficients η(z) may (and generally will) depend on the averaging sequence; compare Remark 8. The next step consists in modifying Λ by a random process in a local way. To come to a reasonably general formulation that includes several notions of randomness known from lattice theory, compare [30,63], we employ a formulation with finite, random, complex measures. Let Ω denote a measure-valued random variable, and Q the corresponding law, which is itself a probability measure on Mbd = Mbd (R d ), the space of finite complex measures on R d . To keep the notation compact, we use the symbol E Q for the various expectation values that arise in connection with (Ω, Q). In particular, we write E Q (Ω) = Mbd ω dQ(ω), where ω refers to the realisations of Ω as usual. Note

628


that we also refer via the index Q to the underlying law for one random variable for more complicated expectation values, rather than using the underlying (though hidden) probability space. This will be explained in more detail in Sec. 5 below. To proceed, we need a version of the strong law of large numbers (SLLN) for measures. Lemma 6. Let (Ωi )i∈N be a sequence of integrable, finite, i.i.d. random measures, with common law Q. Then, with probability 1, one has n 1 n→∞ Ωi −−−→ E Q (Ω1 ) n i=1

in the vague topology. Proof. By definition, integrability means that E Q (|Ωi |), which is independent of i ∈ N, is a finite measure. As the space of continuous functions Cc (R d ) is separable, the almost n sure convergence of the measures follows from the almost sure convergence of n1 i=1 Ωi (ϕ) for an arbitrary (but fixed) bounded, continuous function ϕ. This, in turn, follows from the conventional SLLN [24], possibly after splitting the sums into their real and imaginary parts and applying the SLLN twice.

Recall that ω is the measure defined by ω(ϕ) = ω( ϕ ). Let Ω and Ω be two independent random measures, with the same law Q, and such that E Q (|Ω|) is a finite measure, and also assume the second moment condition E Q (|Ω|(R d ))2 < ∞. Then, the convolution Ω ∗Ω is well defined, and one obtains from elementary calculations the important relations = E ) = E Q (Ω) ∗ E E Q (Ω) Q (Ω) and E Q (Ω ∗ Ω Q (Ω ),

(20)

the second due to the assumed independence. Let us now fix an FLC set Λ, which is assumed to possess the autocorrelation measure γΛ relative to the van Hove averaging sequence A chosen, and consider the family (Ωx )x∈Λ of integrable, complex, i.i.d. random measures, with common law Q and subject to the moment conditions mentioned above. When Ω is any representative of these random measures, E Q (|Ω|) is a finite measure by assumption, and the measure-valued exist (note that also E Q (|Ω∗Ω|) is a finite measure, expectations E Q (Ω) and E Q (Ω∗Ω) due to the condition on the second moment). We are now interested in the random object (Ω) Ωx ∗ δx , (21) δΛ = x∈Λ

which is almost surely a locally finite measure (though not necessarily translation bounded). d To see this, we observe that, for any bounded Borel set B ⊂ R , the random sum x∈Λ |(Ωx ∗ δx )(B)| converges almost surely, since E Q (|(Ωx ∗ δx )(B)|) = E Q (|Ω(B − x)|) ≤ E Q (|Ω|) (B − x) and the convolution δΛ ∗ E Q (|Ω|) is a well-defined locally finite measure due to the translation boundedness of δΛ (note that the summands in x∈Λ |Ωx ∗ δx |(B) are non-negative, hence convergence of the means implies almost sure convergence). As


629

the Borel σ -algebra on R d is countably generated, we can find a set of Q-measure 1 on which the sum (21) converges (absolutely) for each Borel set B, and the limit is a measure. Let A be fixed and assume for simplicity that each An is invariant under x → −x. (Ω) The (n th ) approximating autocorrelation of δΛ reads (Ω)

1 (Ω) (Ω) δ ∗δ vol(An ) Λ | An Λ | An 1 y ∗ δ−y Ω = (Ωx ∗ δx )| ∗ | An . An vol(An )

γΛ,n =

x∈Λ

(22)

y∈Λ

For certain pointwise calculations and arguments, it will be more convenient below to consider the modified approximating autocorrelation ⎞ ⎛ ⎞ ⎛ 1 (Ω) ⎝ y ∗ δ−y ⎠ . γΛ,n;mod := Ω Ωx ∗ δx ⎠ ∗ ⎝ (23) vol(An ) x∈Λn

y∈Λ

To this end, we need a probabilistic analogue of Eq. (9). (Ω)

(Ω)

Proposition 2. Almost surely, γΛ,n of (22) and γΛ,n;mod of (23) define sequences of locally finite random measures. Moreover, we can choose a strictly increasing subsequence (n k )k∈N such that, in the vague topology, we almost surely have (Ω)

(Ω)

k→∞

γΛ,n k − γΛ,n k ;mod −−−→ 0. (Ω)

(Ω)

In particular, if γΛ,n or γΛ,n;mod almost surely converges to γ along A or along a subsequence of it, we can choose a subsequence A of A so that both sequences almost surely converge to γ along A . Proof. We abbreviate µ(·) := E (|Ω|(·)), ν(·) := E |Ω|(R d ) |Ω|(·) . Due to the assumption E (|Ω|(R d ))2 < ∞, both µ and ν are finite, positive measures. Consequently, since Λ is uniformly discrete, µ ∗ δΛ and ν ∗ δΛ are translation bounded by [13, Prop. 1.13] (and thus certainly locally finite). Let us first verify that the expression in (23) almost surely defines a locally finite measure (the estimate for (22) is completely analogous, with the same upper bound). If B ⊂ R d is a bounded Borel set, we have (Ω) vol(An ) E Q |γΛ,n;mod |(B) y | ∗ δ−y (v) EQ 1 B (u + v) d (|x | ∗ δx ) (u) d | ≤ R d ×R d

x∈Λn y∈Λ

≤

R d ×R d x∈Λn y∈Λ\{x}

+

x∈Λn

≤

1 B (u + v) d(µ ∗ δx )(u) d( µ ∗ δ−y )(v)

EQ

R d ×R d

x |(v) 1 B (u + v) d|x |(u) d|

∗ δΛ (B) + ν(R d ) card(Λn ) < ∞. µ ∗ δΛn ∗ µ

630


Thus, by arguments analogous to those used before, the sum on the right hand side of (23) almost surely converges absolutely when applied to any bounded Borel set B. Again, since the Borel σ -algebra on R d is countably generated, this suffices to ensure (Ω) that γΛ,n;mod is a locally finite, random measure. Furthermore, again for a bounded Borel set B, one has (Ω) (Ω) vol(An ) E Q |γΛ,n − γΛ,n;mod |(B) y | ∗ δ−y (v) ≤ EQ 1 B (u +v) 1An (u)1An (v) − 1An (x) d (|x | ∗ δx ) (u) d | R d ×R d

x,y∈Λ

=

1 B (u + v) 1An (u)1An (v) − 1An (x) d(µ ∗ δx )(u) d( µ ∗ δ−y )(v)

d d x,y∈Λ R ×R

x=y

+

EQ

x∈Λ

(24)

x | ∗ δ−x (v) . 1 B (u + v) 1An (u)1An (v) − 1An (x) d (|x | ∗ δx ) (u) d |

R d ×R d

We estimate the term in the third line of (24) as follows: d d x,y∈Λ R ×R x=y

=

1 B (u + v) 1An (u)1An (v) − 1 d(µ ∗ δx )(u) d( µ ∗ δ−y )(v)

d d x∈Λn y∈Λ R ×R y=x

+ ≤

1 B (u + v) 1An (u)1An (v) − 1An (x) d(µ ∗ δx )(u) d( µ ∗ δ−y )(v)

1 B (u + v) 1An (u)1An (v) d(µ ∗ δx )(u) d( µ ∗ δ−y )(v)

d d x∈Λ\Λn y∈Λ R ×R y=x

d d R ×R

1 B (u + v) 1An (u)1An (v) − 1 d µ ∗ δΛn (u) d µ ∗ δΛ (v)

∗ δΛ (v) 1 B (u + v) 1An (u)1An (v) d µ ∗ δΛ\Λn (u) d µ + R d ×R d (B) ≤ µ ∗ δΛn | ∗ δΛ | ∗ µ ∗ δΛ (B) + µ ∗ δΛn ∗ µ R d \An R d \An + µ ∗ δΛ\Λn | ∗ µ (B), ∗ δΛ | An An where, in the first inequality, we have used the fact that removing the restriction x = y in the summation only adds positive terms (note that µ is a positive measure by definition), and employed the estimate 1An (u)1An (v) − 1 ≤ 1R d \A (u) + 1R d \A (v) for the second n n inequality. There is a constant c B that depends on B (as well as on µ and the averaging sequence) and a sequence dn → 0 that is independent of B such that the sum in the last two lines above is bounded by dn c B vol(An ). This comes from the fact that there are only contributions from ‘surface terms’; compare the arguments in [55].


631

By way of example, we verify that, for any R > 0, µ ∗ δΛn | ∗ µ ∗ δΛ (B) R d \An ≤ µ(R d ) {x ∈ Λn | d(x, R d \An ) ≤ R} +µ(R d \B R )|Λn | sup µ ∗ δΛ (B +x). x∈R d

Note that this (together with analogous statements for the two other summands above) yields the claim by the van Hove property of the averaging sequence A. Observe next that, for a (possibly) complex measure ξ with |ξ |(R d ) < ∞, a translation bounded measure ν and a bounded Borel set B, we have (by an application of [32, Prop. 2.2] and its proof) the estimate |(ξ ∗ ν)(B)| ≤ |ξ |(R d ) supx∈R d |ν|(B + x) < ∞. Finally, note that µ ∗ δΛ is translation bounded by [13, Prop. 1.13], and that µ ∗ δΛn | µ| ∗ δx (R d \An ) (R d ) = BR R d \A n

x∈Λn

+

µ| ∗ δx (R d \An ) R d \B R

x∈Λn

≤ µ(R d ) {x ∈ Λn | d(x, R d \An ) ≤ R} + µ(R d \B R )|Λn |. Similarly, the term in the fourth line of (24) is bounded from above by

EQ

x∈Λn

R d ×R d

x | ∗ δ−x (v) 1 B (u + v) 1R d \A (u) + 1R d \A (v) d (|x | ∗ δx ) (u) d |

EQ

x | ∗ δ−x (v) 1 B (u + v)1An (u)1An (v) d (|x | ∗ δx ) (u) d |

+

x∈Λ\Λn

≤

n

R d ×R d

d x (R ) E Q Ω

x∈Λn

Rd

+

x∈Λ\Λn

1R d \A (u) d (|x | ∗ δx ) (u) n

x | ∗ δ−x (v) 1R d \A (v) d | n d R d x (R ) E Q Ω 1An (u) d (|x | ∗ δx ) (u)

Rd

∗ δΛn (R d \An ) + ν ∗ δΛ\Λn (An ) ≤ dn vol(An ), = ν ∗ δΛn (R d \An ) + ν

E Q |Ωx | (R d ) + x∈Λn

n

with a sequence dn −→ 0. Combining the above estimates, we obtain ⎛ (Ω) ⎞ (Ω) γΛ,n − γΛ,n;mod (B) ⎠ ≤ (c B + 1) dn EQ ⎝ vol(An ) for a sequence dn −→ 0; hence, for ε > 0, ⎞ ⎛ (Ω) (Ω) γΛ,n − γΛ,n;mod (B) c + 1 > ε⎠ ≤ B dn P⎝ vol(An ) ε

(25)

(26)

(27)

632


by Markov’s inequality. If we choose (n k )k∈N such that k dnk < ∞, we obtain from (27) and the (first) Borel-Cantelli lemma that (Ω) (Ω) γΛ,n k − γΛ,n k ;mod (B) −→ 0 , almost surely as k → ∞. (28) vol(An k ) By (27), we may choose the subsequence (n k )k∈N independently of B in such a way that, for each bounded Borel set B, (28) holds almost surely. Finally, since the Borel σ -algebra on R d is countably generated, this implies the main claim of the lemma. The last statement is then obvious.

()

Let us now resume our study of γΛ . Invoking Proposition 2 and replacing the averaging sequence A = (An )n∈N by the subsequence chosen there, we may use the modified (Ω) measures γΛ,n;mod as our approximating measures. Observing

(Ω) y ∗ δ−y , Ω δΛ = y∈Λ

the modified autocorrelation approximant reads ⎞ ⎛ ⎞ ⎛ 1 (Ω) ⎝ y ∗ δ−y ⎠ Ω γΛ,n;mod = Ωx ∗ δx ⎠ ∗ ⎝ vol(An ) x∈Λn y∈Λ 1 (Ω) = Ωx ∗ Ωx−z ∗ δz =: ζz,n ∗ δz , (29) vol(An ) z∈Λ−Λ

x∈Λn x−z∈Λ

z∈Λ−Λ

(Ω)

where we now need to analyse the behaviour of the random measures ζz,n . Let us first look at z = 0, where we obtain (Ω)

ζ0,n =

1 card(Λn ) n→∞ x − Ωx ∗ Ω −−→ dens(Λ) · E Q (Ω ∗ Ω) vol(An ) card(Λn )

(a.s.)

x∈Λn

(30) by an application of Lemma 6. Note that dens(Λ) = η(0) as introduced in Eq. (19). (Ω) Next, assume z ∈ Λ − Λ with z = 0. Then, we split ζz,n into two sums, (1) (0) 1 (Ω) x−z + y−z , (31) = Ωx ∗ Ω Ωy ∗ Ω ζz,n vol(An ) x∈Λn x−z∈Λ

y ∈Λn y−z∈Λ

where the upper index stands for the following additional restriction: Given z, our point set Λ is the disjoint union of countably many maximal linear chains of the form {. . . , x + 2z, x + z, x, x − z, x − 2z, . . .} with all points lying in Λ, and x being chosen as its representative. Such a chain may be finite or infinite, but has no gaps by construction. For each of these chains, the random x+(m−1)z , are identically distributed, but not independent (due to the measures Ωx+mz ∗ Ω


633

index overlap). However, those with m even (type (0)) are mutually independent, as are those with m odd (type (1)). This way, each element of Λ inherits the type as a label, and the terms in (31) are distributed to the two sums according to their type. This approach (Ω) (Ω) guarantees that the terms for ζz,n which already showed up in ζz,n−1 end up in the same sum as before, no matter what the detailed structure of the (nested) averaging sequence might be. We also split the number of terms card{x ∈ Λn | x − z ∈ Λ} = Nn(0) + Nn(1) accordingly. We can now rewrite our previous expression in the form (0) (1) (0) (1) card{x ∈ Λn | x − z ∈ Λ} Nn Nn (Ω) + (0) ζz,n = , (0) (1) (0) (0) (1) vol(An ) Nn + Nn Nn Nn + Nn Nn (32) (0) where the term in brackets is a convex combination of two random measures (0)/Nn (1) (1) and /Nn . By (19), the factor in front of the bracket converges to η(z). When this n→∞ limit is non-zero, we know that Nn(i) −−−→ ∞ for i ∈ {0, 1}, so that Lemma 6 and Eq. (20) imply 1 (i) (i) Nn

n→∞ −−−→ E Q (Ω) ∗ E Q (Ω)

(a.s.).

(33)

Although we do not know whether the rational prefactors in (32) converge, we have a convex combination of two sequences that each almost surely converge to the same limit, which must then also be the limit of their convex combination. Put together, this gives (Ω) ζz,n −−−→ η(z) · E Q (Ω) ∗ E Q (Ω) n→∞

(a.s.)

(34)

for all z ∈ Λ − Λ with z = 0. These considerations will be sufficient when the random measures almost surely have a (deterministic) compact support. To formulate the main result of this section in greater generality, we need one further technical property. For brevity, we write Brc = R d \Br (0). Lemma 7. If M is a set of uniformly translation bounded, positive measures and ν a finite, positive measure on R d , there is a sequence Rk ∞ such that ∞ k=1

sup ω|

ω∈M

B Rc

∗ ν (K ) < ∞ k

holds for any compact set K ⊂ R d . Proof. Since ν is a finite, positive measure and ν(Brc ) a decreasing function that tends c to 0 as r → ∞, we can choose radii Rk with ν(B Rc k ) < 1/k 2 , so that ∞ ν(B k=1 Rk ) < π 2 /6 < ∞. Moreover, we may do this in such a way that the differences between consecutive radii do not decrease, meaning that R2 ≥ 2 R1 and Rk+2 − Rk+1 ≥ Rk+1 − Rk for all k ∈ N.

634


Uniform translation boundedness of M means that, for any compact K ⊂ R d , there is a positive constant α K with ω(x + K ) ≤ α K , simultaneously for all x ∈ R d and all ω ∈ M . If K ⊂ R d is compact, we have K ⊂ Br (0) for some r > 0. If R > r , one has ω| c ∗ ν (K ) = 1 K (x + y) 1 B c (x) dω(x) dν(y) BR R d d R ×R c ω (K − y) ∩ B Rc dν(y) ≤ α K ν(B R−r ), = Rd

where the last step follows because ω (K − y) ∩ B Rc vanishes whenever |y| ≤ R − r . For radii 0 ≤ R ≤ r , the bound is simply given by α K ν(R d ), which is finite. Now, for some m ∈ N, we have Rm − r ≥ R1 , with the sequence of radii chosen before. The additional difference property of the radii makes sure that the radii Rm+k with k ∈ N give a summable contribution, while the remaining terms are finite by construction. Since this argument is uniform in ω ∈ M and holds for all compact K ⊂ R d , our claim follows.

Theorem 2. Let Λ ⊂ R d be an FLC point set such that its Dirac comb δΛ possesses the autocorrelation measure γΛ of (17), relative to the fixed averaging sequence A, and thus the diffraction measure γΛ . Let (Ωx )x∈Λ be a family of integrable, complex, i.i.d. random measures with common law Q and finite second moment measure, with Ω being (Ω) of (21). any representative of this family, and consider the random measure δΛ Then, possibly after replacing A by a suitable subsequence A , the sequence of (Ω) approximating measures γΛ,n of (22) almost surely converges, as n → ∞, to the positive definite, translation bounded autocorrelation measure − E Q (Ω) ∗ E γΛ,Q = E Q (Ω) ∗ E Q (Ω) ∗ γΛ + dens(Λ) E Q (Ω ∗ Ω) Q (Ω) ∗ δ0 . This measure has the Fourier transform 2 − E Q (Ω) ∗ E γΛ + dens(Λ) E Q (Ω ∗ Ω) γΛ,Q = E Q (Ω) · Q (Ω) · λ, (Ω) relative to A . which is the almost sure diffraction measure of the random measure δΛ

Proof. The previous calculations establish the individual almost sure convergence of (Ω) the (countably many) measures ζz,n , with the limits as given in Eqs. (30) and (34). Our assumptions on Ω ensure that E Q (Ω) ∗ E Q (Ω) in (34) is a finite, positive definite mea d r →∞ sure. It is concentrated at 0 in the sense that E Q (Ω) ∗ E −−−→ 0, Q (Ω) R \Br (0) − d g ) > 0 for all 0 = g ∈ Cc (R ). In view of (30) and (34), while E Q (Ω) ∗ E Q (Ω) (g ∗ our (claimed) almost sure limit γΛ,Q inherits translation boundedness from γΛ . The (deterministic) measure γΛ,Q is positive definite, hence transformable by Lemma 1. Its Fourier transform has the form claimed as a result of the convolution theorem [13, Excs. 4.18]. The latter is applicable here because all expectation measures involved are finite measures, so that their Fourier transforms are represented by uniformly continuous functions on R d . It remains to establish the limit property. Let us first assume that there is a (determinis(Ω) tic) compact set C so that supp(Ωx ) ⊂ C almost surely. This implies supp(ζz,n ) ⊂ C−C


635 (Ω)

for all n and z, so that only terms from finitely many z ∈ Λ − Λ contribute to γΛ,n;mod on any compact K ⊂ R d . In this case, we may use an elementary pointwise calculation (Ω) to see that γΛ,n;mod tends to the claimed limit, and Proposition 2 gives the assertion. In the general case, this simple argument is not conclusive, and we need to estimate (Ω) putative contributions from distant points z ∈ Λ−Λ to γΛ,n;mod . For bounded K , B 0, with µ(·) := E (|Ω|(·)) as above, we have 1 EQ Ωx ∗ Ωx−z ∗δz (K ) vol(An ) x∈Λn x−z∈Λ

z∈Λ−Λ z∈ B

≤

z∈Λ−Λ z∈ B

=

z∈Λ−Λ z∈ B

! 1 x−z | ∗δz (K ) E Q |Ωx | ∗|Ω vol(An ) x∈Λn x−z∈Λ

1 µ ∗δz ) (K ) (µ ∗ vol(An )

= µ ∗ µ∗

x∈Λn x−z∈Λ

1 vol(An )

= µ ∗ µ ∗(γΛ,n;mod |

δz

(K )

z∈Λ−Λ, x∈Λn x−z∈Λ, z∈ B

R d \B

) (K ) =: φn (K , B).

(35)

Since {γΛ,n;mod | n ∈ N} are uniformly translation bounded by Lemma 3, we can choose a sequence of radii Rk ∞ according to Lemma 7 such that, for all compact K ⊂ R d , ∞

sup φn K , B Rk (0) < ∞.

k=1 n∈N

(36)

(Ω) (Ω) We have proved above that, for each z ∈ Λ − Λ, ζz,n −−−−→ ζz,∞ , almost surely in (Ω) (Ω) and ζz,∞ the vague topology, where ζ0,∞ = dens(Λ) · E Q (Ω ∗ Ω) = η(z) · E Q (Ω) ∗ E Q (Ω) for z = 0. Possibly after passing to another subsequence, we may now assume that the convergence is so fast that, for any g ∈ Cc (R d ), n→∞ (Ω) (Ω) ζz,n − ζz,∞ ∗ δz (g) −−−−→ 0 (a.s.). (37) n→∞

z∈Λ−Λ |z| 0 that φn K , B Rn (0) (Ω) , (38) P ζz,n ∗ δz (K ) ≥ ε ≤ ε z∈Λ−Λ |z|≥Rn

which is summable by (36). Hence, by Borel-Cantelli, n→∞ (Ω) ζz,n ∗ δz −−−−→ 0 (a.s.). z∈Λ−Λ |z|≥Rn

(39)

636


Combining this with (37) shows that the limit is the expected one (from the pointwise calculation) also in this case, which yields the claim.

Note that our argument is based on the potential selection of a subsequence of the original (deterministic) A. However, it also shows that the limit derived in Theorem 2 is the only point of accumulation along any deterministic subsequence of A. Remark 9. Randomisation of Meyer Sets. A particularly relevant class of point sets in the theory of aperiodic order are Meyer sets, which are relatively dense sets Λ such that Λ − Λ is uniformly discrete. Such sets always have a diffraction measure with a non-trivial pure point part, with a relatively dense supporting set [2,59], despite the fact that Meyer sets can have entropy2 . If modified by a family of random measures according to Theorem 2, the resulting diffraction still shows the original diffraction with 2 its non-trivial pure point component, modulated by the function |E Q (Ω)| , in addition to the diffuse background originating from the added randomness. Let us look at consequences of Theorem 2 in terms of some examples. Example 6. Deterministic Clusters. Let S ⊂ R d be a finite point set, and consider Ω ≡ δS = = x∈S δx . Clearly, this completely deterministic case gives E Q (|Ω|) = δ S ∗ δS , so that Theorem 2 gives γ (Ω) = δ S ∗ δS ∗ γ E Q (Ω) = δ S and E Q (Ω ∗Ω) Λ Λ (Ω) and γΛ = |δS |2 · γΛ , which is always true (rather than almost always) in this case. A particularly simple instance of this emerges from S = {a}, which effectively means (Ω) (Ω) a global translation by a. This leads to the relations γΛ = γΛ and γΛ = γΛ , as it must. Example 7. Random Weight Model. Here, we consider Ω = H δ0 , where H is a complex-valued random variable with a law µ that satisfies E µ (|H |2 ) < ∞ (hence also = E µ (|H |2 ) δ0 , E µ (|H |) < ∞). Clearly, this gives E Q (Ω) = E µ (H ) δ0 and E Q (Ω∗Ω) so that Theorem 2 results in the diffraction formula (Ω) (a.s.). γΛ + dens(Λ) E µ (|H |2 ) − |E µ (H )|2 · λ γΛ = |E µ (H )|2 · The autocorrelation is clear from Theorem 2. Remark 10. Interpretation as Particle Gas. A widely used special case of Example 7 is the random occupation model, or ‘Λ-gas’. Here, Ω may take the value δ0 (with probability p, for ‘occupied’) or 0 (with probability 1 − p, for ‘empty’). This gives the diffraction (Ω)

γΛ

= p2 · γΛ + dens(Λ) · p(1 − p) · λ

(a.s.),

which was derived in a similar setting in [7], and later generalised to Bernoulli and Markov systems [4], to systems with finite range Gibbs measures [11], and beyond [40,41]. The results of Examples 6 and 7 can be extended in many ways, some of which will be met later on. One further possibility consists in replacing a point by a ‘profile’, as described by an integrable function, or by a finite collection of such profiles, which could represent different types of atoms. The corresponding formulas for the autocorrelation and the diffraction are then easy analogues of the ones given so far. 2 The binary random tilings of Example 5 produce Meyer sets whenever b/a ∈ Q.


637

Example 8. Random Displacement Model. Consider the random measure Ω = δ X , where X is an R d -valued random variable with law ν. If A ⊂ R d is a Borel set, one has δx (A) dν(x) = 1 A (x) dν(x) = ν(A), E Q (Ω) (A) = Rd

Rd

= ν(R d ) δ0 = δ0 . Theorem 2 which shows that E Q (Ω) = ν. One also finds E Q (Ω ∗Ω) now results in the equations (Ω)

γΛ

(Ω) γΛ

= (ν ∗ ν ) ∗ γΛ + dens(Λ) (δ0 − ν ∗ ν)

(a.s.),

= | ν|2 · γΛ + dens(Λ) (1 − | ν|2 ) · λ

(a.s.),

which recovers Hof’s result on the diffraction at high temperature [33]. In comparison, Hof’s approach to the random displacement model [33] also uses the SLLN, but does not require the FLC property. Instead, he needs an ergodicity assumption on the underlying point set; compare also [42]. Remark 11. Extension of Theorem 2. The argument above is shown for FLC sets in a pointwise fashion, to make the result more transparent. However, it is clear that one does not need the FLC property itself. Indeed, it is sufficient to assume that the fixed point set Λ, relative to a chosen van Hove averaging sequence A, possesses an autocorrelation that is a pure point measure of the form γ = z∈F η(z)δz with F a locally finite point set. An argument with local test functions will then still connect to the SLLN and thus avoid the need for ergodic assumptions on the underlying set Λ. With hindsight, it is rather clear that the formulas of Theorem 2 are robust, and should also hold for other point sets, such as those coming from a homogeneous Poisson process or from a model set based particle gas, as introduced in [7]. So, to complement our approach of this section, let us now consider ergodic point processes instead, meaning that also the set Λ becomes part of the random structure. 5. Arbitrary Dimensions: Point Process Approach Here, we are interested in the diffraction of certain random subsets of Rd , where we restrict ourselves to the situation that these subsets are self-averaging in a suitable way. This will be guaranteed by the ergodicity of the underlying stochastic process. One further benefit is that we are freed from details of the averaging sequence and the potential selection of subsequences thereof. It is convenient to start by putting ourselves in the context of random counting measures, which we now summarise in a way that is tailored to diffraction theory. 5.1. Random measures and point processes. Let M+ denote the set of all locally finite, positive measures φ on R d (where we mean to include the 0 measure). That φ is locally finite (some authors say φ is boundedly finite or that φ is a Radon measure) means that, for all bounded Borel sets A, φ(A) < ∞. The space M+ is closed in the topology of vague convergence of measures (in fact, M+ is a complete separable metric space, see [19, A 2.6]3 ). We let ΣM+ denote the σ -algebra of Borel sets of M+ . The latter can be 3 We refer to the second edition of this work throughout, which comes in two volumes [19,20]. All results we need are also contained in the original one volume edition [18], sometimes with a slightly different numbering.

638


described as the σ -algebra of subsets of M+ generated by the requirement that, for all Borel sets A ⊂ R d , the mapping φ → φ(A) is measurable; compare [37, Chs. 1.1 and 1.2] for background. A random measure on R d is a random variable Φ from a probability space (Θ, F, π ) into (M+ , ΣM+ ). Let us write P(M+ ) for the convex set of probability measures on M+ . The distribution of a random measure Φ is the probability measure P = PΦ ∈ P(M+ ), defined by P = π ◦ Φ −1 . In other words, P is the law of Φ, written as L(Φ) = P. Note that, as soon as P is given or determined, one can usually ignore the underlying probability space. For each t ∈ R d , let Tt denote the translation operator on R d , as defined by the mapping x → t +x. Clearly, one has Tt Ts = Tt+s , and the inverse of Tt is given by Tt−1 = T−t . For functions f on R d , the corresponding translation action is defined via Tt f = f ◦T−t , so that Tt f (x) = f (x − t). Similarly, for φ ∈ M+ , let Tx φ := φ ◦ T−x be the image measure under the translation, so that (Tx φ)(A) = φ(T−x (A)) = φ(A −x) for any measurable subset A ⊂ Rd , and (Tx φ)( f ) = R d f (y) d(Tx φ)(y) = R d f (x + z) dφ(z) = φ(T−x f ) for functions. This means that there is a translation action of R d on M+ . Finally, we also have a translation action on P(M+ ), via (Tx Q)(A) = Q(T−x A) for any measurable A ⊂ M+ . Our primary interest is in random counting measures. A measure φ on R d is called a counting measure if φ(A) ∈ N 0 for all bounded Borel sets A. These are positive, integer-valued measures of the form φ = i∈I δxi , where the index set I is (at most) countable and the support of φ is a locally finite subset of R d . The (positive) counting measures form a subset N + ⊂ M+ . We can repeat the above discussion of M+ by restricting everything to N + . The vague topology on N + is the same as its topology inherited from M+ , and its σ -algebra of Borel sets ΣN + consists of the intersections of the elements of ΣM+ with N + . The concepts of the law of a random measure and the translation action by R d carry over. In particular, for x ∈ supp(φ) with φ ∈ N + , T−x φ corresponds to the counting measure obtained from φ by translating its support so that x is shifted to the origin. A point process on R d is a random variable Φ from a probability space (Θ, F, π ) into (N + , ΣN + ). Alternatively, a point process is a random measure for which π -almost all θ ∈ Θ are counting measures. Furthermore, it is called simple when, for π -almost all θ ∈ Θ, the atoms of φ = Φ(θ ) have weight (or multiplicity) 1. In many instances, the point processes we are dealing with are simple. Whenever this happens, we feel free to identify point measures with their supports. In this case, the measures almost surely are Dirac combs of the form φ = δ S with S ⊂ R d locally finite. Later on, we create compound processes in which an underlying point process is decorated with a random finite measure, and this will take us from N + to M+ , which is also the reason why we introduced random measures above. For a random measure (or a point process) Φ with law P, the expectation measure E P (Φ) is defined by φ(A) d P(φ), for A ⊂ Rd Borel. (40) (E P (Φ)) (A) = E P (Φ(A)) = N+

It is a measure on Rd which gives the expected mass (or number of points) that Φ has in A. More precisely, in terms of the underlying probability space (Θ, F, π ), one writes E P (Φ(A)) = Φ(θ )(A) dπ(θ ) = Φ(A) d P(Φ). Θ

N+


639

It is common in the probability literature (and we adopt this slight abuse of notation here, too) to suppress the explicit dependence on (Θ, F, π ) by simply writing Φ for the general instance Φ(θ ) of the process Φ. The latter is called stationary when its law P is translation invariant, which means that Tt P = P ◦ T−t = P holds for all t ∈ R d . Remark 12. Intensity of a Process. If P is stationary, we have Tt E P (Φ) = E P (Φ) for all t ∈ R d , whence E P (Φ) must be a multiple of Lebesgue measure (the latter being Haar measure on R d ). Consequently, I P (Φ) = E P (Φ) = ρ λ, where ρ ∈ [0, ∞] is usually called the intensity of P. Unfortunately, this term is already in use for the positive weights of Bragg peaks in diffraction theory. In the setting of simple point processes, ρ also has the meaning of a point density, averaged over all realisations of the process. In the ergodic case (see below for a definition), it is then almost surely the density of a given realisation in the usual sense. We thus often prefer to call ρ the point density of the simple point process or the density of the random measure. From now on, we always assume that ρ is finite. Let Φ : (Θ, F, π ) −→ (X , ΣX ) be a stationary random measure (where X = M+ ) or point process (X = N + ), with law P. Then, (X , ΣX , P) is a probability space with translation invariant probability measure P. In fact, we will usually simply assume that (X , ΣX , P) is itself the probability space (or basic process) we are dealing with. In general, there will be several different spaces, and to keep track of the processes, we use the law of the basic process as an index. Let us recall that the random measure or point process Φ is called ergodic when (X , ΣX , P) is ergodic as a dynamical system [62] under the translation action of R d , see below for more. In particular, we do not refer to strict ergodicity this way. 5.2. Palm distribution and autocorrelation. Let P ∈ P(N + ) be stationary with finite density 0 < ρ < ∞. The assumption ρ > 0 is no restriction, since it is easy to see that a realisation of a stationary point process with intensity ρ = 0 almost surely is the zero measure. Let 1B , as usual, denote the characteristic function of the set B ⊂ N + , and choose a Borel set A ⊂ Rd with 0 < λ(A) < ∞. The Palm distribution P0 is the probability measure on N + that satisfies 1 P0 (B) = Φ({x}) 1B (T−x Φ) d P(Φ) (41) E P (Φ(A)) N + x∈A∩supp(Φ)

for any B ∈ ΣN + , compare [57, Ch. 4.4] or [38, Ch. 3] for background. Due to stationarity, Remark 12 applies to E P (Φ(A)), whence the prefactor simplifies to (ρ λ(A))−1 . Note that the sum under the integral runs over at most countably many points. Moreover, the definition does not depend on the actual choice of A. Intuitively, P0 describes the configuration Φ as seen from a typical point in supp(Φ), with that point translated to the origin. Alternatively, in the case of simple point processes, one can think of P0 as the distribution of Φ, conditioned on having a point measure at 0. This actually amounts to condition properly on an event of probability 0, which might need some further explanation. The first point of view can be made precise, at least in the ergodic case, as a limit, via sampling points in Φ over larger and larger balls, see [38, Thm. 3.6.6] or [20, Prop. 13.4.I

640


and Prop. 13.4.IV] as well as Eq. (43) below. The second interpretation can be corroborated by conditioning Φ to have a point in a small ball around 0 and then again taking a limit, see [20, Thm. 13.3.IV]. In more precise terms, P0 would be called the Palm distribution with respect to 0 ∈ Rd , compare [36, Ch. 10] or [20, Ch. 13.1]. Since we will mostly be dealing with the stationary scenario, we refrain from spelling out the full name. There is an alternative approach to the Palm distribution, which also applies to the random measure case, compare [20, Chs. 13.1 and 13.2]. Let Φ : (Θ, F, π ) −→ (M+ , ΣM+ ) be a stationary random measure with law P and finite mean density ρ < ∞. Then, the Palm distribution is the unique probability measure P0 on M+ that satisfies EP g(x, Φ) dΦ(x) = ρ g(x, Tx ψ) d P0 (ψ) dx (42) Rd

Rd

M+

for all non-negative functions g on R d × M+ for which R d M+ g(x, φ) dφ(x) d P(φ) is finite. When dealing with point processes, all this reduces to N + by simply replacing M+ with N + throughout Eq. (42), compare [20, Ch. 13.2 and Thm. 13.2.III]. If Φ is an ergodic, stationary random measure, an application of the ergodic theorem implies 1 n→∞ F(T−x Φ) dΦ(x) −−−→ ρ F(Ψ ) d P0 (Ψ ) (a.s.), (43) λ(Bn ) Bn M+ for any non-negative measurable function F : M+ → R, see [20, Prop. 13.4.I] or the proof of [38, Thm. 3.6.6]. Here and below, we write Bn for Bn (0) and λ(Bn ) for vol(Bn (0)). In the literature, the probability measure P0 is usually called the Palm distribution of P (with respect to 0), while the term Palm measure is also in use for the unnormalised version ρ P0 , a convention we adopt here. The first moment measure of the latter coincides with the autocorrelation measure of the underlying process and is denoted (Φ) by γ P . This is motivated by the following result on the autocorrelation γ P of a given realisation, which is somewhat implicit in the literature. Its importance in our present context was first emphasised by Goueré in [28]; see also [43] for complementary aspects. Theorem 3. Let Φ be a stationary, ergodic, positive random measure with distribution P. Assume that P has finite density ρ, and that P has locally finite second moments in the sense that E P (Φ(A)2 ) < ∞ for any bounded A ⊂ Rd (this follows for instance from the condition E P (Φ(Br (x))2 ) < ∞ for all x ∈ R d and some fixed radius r ). Let Φn := Φ| Bn (0) denote the restriction of Φ to the centred ball of radius n. Then, the natural autocorrelation γ P(Φ) of Φ, defined via an averaging sequence of centred nested balls, almost surely exists and satisfies (Φ)

γP

:= lim

n→∞

n Φn ∗ Φ Φn ∗ Φ = lim = ρ IP = γ P , 0 n→∞ vol(Bn (0)) vol(Bn (0))

where the limit refers to the vague topology on M+ . Here, I P0 is the first moment measure of the Palm distribution, Ψ (A) d P0 (Ψ ), for A ⊂ Rd Borel. I P0 (A) = M+


641

Proof. As test function, fix a non-negative continuous function g : Rd → [0, ∞) with compact support. With Bnc := R d \Bn , we have 1 1 n (x) = g(x) d Φn ∗ Φ g(x − y) dΦ(x) dΦ(y) λ(Bn ) R d λ(Bn ) Bn ×Bn 1 = g(x − y) dΦ(y) − g(x − y) dΦ(y) dΦ(x) λ(Bn ) Bn Rd Bnc 1 = Fg (T−x Φ) dΦ(x) − Rn (g) λ(Bn ) Bn (note that both integrals inside the big brackets in the second line are finite because g has compact support), where φ → Fg (φ) = R d g(−z) dφ(z) defines a measurable function, and the remainder is given by 1 Rn (g) = g(x − y) dΦ(y) dΦ(x). λ(Bn ) Bn Bnc Note that Rn , which is a random measure, is precisely the difference between the elements of the two approximating sequences of random measures in the claim. In view of (43), it thus remains to show that limn→∞ Rn = 0 almost surely. Choose k so that g(x) = 0 for |x| > k, and fix some ε > 0. We then have, for n > k/ε, g∞ g∞ Rn (g) ≤ Φ Bnc ∩ (x + Bk ) dΦ(x) ≤ Φ(x + Bk ) dΦ(x) , λ(Bn ) Bn λ(Bn ) Bn \B(1−ε)n where φ → G(φ) := φ(Bk ) is again measurable. Hence we obtain g∞ Rn (g) ≤ G (T−x Φ) dΦ(x) λ(Bn ) Bn λ(B(1−ε)n ) g∞ − G T−y Φ dΦ(y) λ(Bn ) λ(B(1−ε)n ) B(1−ε)n n→∞ −−−→ 1 − (1 − ε)d g∞ ρ G(Ψ ) d P0 (Ψ ) M+ = 1 − (1 − ε)d g∞ ρ IP (Bk ), 0

almost surely by (43). Now take ε 0 to conclude.

Continuing with the hypotheses of Theorem 3, our assumptions guarantee that the second moment measure µ(2) of P, defined on cylinder sets A × A ⊂ R d × R d via (2) µ (A × A ) = N + Φ(A) Φ(A ) d P(Φ), is locally finite. This is a necessary and sufficient condition for the existence of the first moment measure of the Palm distribution (as a locally finite measure). In fact, in the stationary scenario, the autocorrelation of (2) (2) the process, denoted by γ P , satisfies γ P = µred , where µred is the so-called reduced second moment measure of P, and this, in turn, is the same as the intensity of the Palm measure. We offer a brief explanation of this (for more details, see [20, Prop. 13.2.VI] or [57, Ch. 4.5]).

642

M. Baake, M. Birkner, R. V. Moody (2)

First, µred is obtained from µ(2) by disintegration, via factoring out the translation (2) invariance. More precisely, following [20], µred is the unique positive measure on Rd such that (2) h(x, y) dµ(2) (x, y) = h(u, u + v) dµred (v) dλ(u) , (44) R d ×R d

Rd

Rd

for all (real) functions h ∈ Cc (R d × R d ). When h = f ⊗ g is a product function (meaning that h(x, y) := f (x)g(y)), one finds the relation ˜ µ(2) ( f ⊗ g) = µ(2) red ( f ∗ g)

(45) (2)

via Fubini’s theorem. Choosing g = f , it is clear that the measure µred is positive definite. More generally, when dealing with complex-valued functions, one has to consider (2) µ(2) ( f¯ ⊗ g) = µred ( f˜ ∗ g),

which leads to some technical complications later on. Since we consider real-valued component processes only, we can stick to the simpler case of real-valued functions. The connection of the reduced second moment to the intensity measure of the Palm measure comes through applying (42) to a function on R d × M+ defined by (x, φ) → g(x)

Rd

Tx h(y) dφ(y) ,

(46)

where g, h are arbitrary, but fixed, non-negative measurable functions on R d . The left hand side of (42) then reads

g(x) h(y − x) dΦ(y) dΦ(x) d Rd R g(x)h(y − x) dΦ(y) dΦ(x) = EP Rd Rd (2) g(x)h(y − x) dµ(2) (x, y) = λ(g) · µred (h), =

EP

R d ×R d

where we employed Fubini’s theorem and (44), while the right hand side reads ρ

R d M+

=ρ

Rd

g(x)

M+

(Tx h)(y) d(Tx φ)(y) d P0 (φ) dλ(x)

R d

g(x)

Rd

h(y) dφ(y) d P0 (φ) dλ(x) = λ(g) · ρ I P0 (h).

Here, we used the notation of the intensity of the Palm measure for its first moment. Comparing these two calculations gives (2)

µred = ρ I P0 = γ P .

(47)


643 (2)

Remark 13. Equivalent Definitions of µred . There are several different ways to define a reduced measure via disintegration. In particular, one could employ h(u, u ± v) as well as h(u ± v, u) in Eq. (44). Using translation invariance of Lebesgue measure, this boils down to just two different possibilities, the one with h(u, u + v) introduced above and the one with h(u, u − v), which is used in [37, Prop. I.60]. Observing the relation f ∗ g˜ = f˜ ∗ g together with µ(2) = µ(2) , one can check that both versions define the same measure, as the process is restricted to positive (and thus real) random measures, so that no complex conjugation shows up in the . -operation. Alternatively, one can use commutativity of the convolution together with the symmetry of µ(2) , which implies µ(2) ( f ⊗ g) = µ(2) (g ⊗ f ). Remark 14. Renewal Process Revisited. Consider a stationary renewal process Φ on the real line, viewed as a random counting measure, with inter-arrival law . The latter is assumed to be a probability measure on R + , with expectation R + x d(x) = 1. It is well known (compare [20, Thm. 13.3.I and Ex. 13.3(a)]) that the Palm distribution P0 , with respect to the origin, of (the law of) Φ equals the law of δ0 +

i∈Z

δ Si ,

(48)

where S0 = 0, " Si =

X1 + · · · + Xi , if i ≥ 1, X i+1 + · · · + X 0 , if i ≤ −1,

and (X i )i∈Z are i.i.d. with law . We see immediately from (48) that the first moment measure I P of P0 is given by formula (11), see also Proposition 1, and thus recover 0 Theorem 1 as well as Remark 3 by specialising Theorem 3 to the case of a renewal process on the line. With the interpretation as a random counting measure, we are no longer restricted to laws on R + . Indeed, when is any probability measure on R with expectation 1 (which prevents the process from being recurrent), the random counting measure almost surely leads to the autocorrelation given in (11), and thus avoids the complications mentioned after Lemma 4; see also [25, Ch. XI.9] for a systematic exposition, and [19, Ex. 8.2(b)] for comparison. Remark 15. Bartlett The diffraction measure γ of a stationary random Spectrum. measure Φ (with E |Φ(A)|2 < ∞ for all compact A ⊂ R d , say) is closely related to (2) the so-called Bartlett spectrum Γ := (cred ) of Φ as follows; compare [19, Ch. 8.2]. Recall that the covariance measure c(2) is defined on cylinder sets via c(2) (A × A ) = Φ(A) Φ(A ) d P(Φ) − Φ(A) d P(Φ) Φ(A ) d P(Φ) = µ(2) (A × A ) − ρ 2 (λ ⊗ λ) (A × A ),

644


where ρ is the density of Φ, compare [20, Eq. (9.5.12)]. Consequently, the relation between the reduced second moment measure µ(2) red and the reduced covariance measure (2) cred of Φ is (2)

(2)

cred = µred − ρ 2 λ.

(49)

(2) Since γ is the Fourier transform of µ(2) red and Γ that of cred , Eq. (49) translates into

Γ = γ − ρ 2 δ0 .

(50)

From our perspective, the positive definite autocorrelation measure γ is a slightly more γ cornatural and universal object than the inverse Fourier transform Γ of Γ , since responds to a physically observable quantity, namely diffraction. Equation (50) gives

Γ = γ − ρ 2 λ, which is no longer positive definite. Rather, it is tailored to situations where 0 supports the only atom of γ , as in the homogeneous Poisson process; compare Example 9 below and the discussion in [19, p. 305], and see [19, Sec. 8.2] for further explicitly computable examples. To formulate the standard Poisson process in this setting, let us start with an intuitive picture. Imagine independently putting single points on the sites of εZd ⊂ R d , each with probability ρ εd , and imagine a process that arises from this construction in the limit ε → 0. For a rigorous construction, one can start from a tiling of Rd with translates of [0, 1)d and then proceed, independently for each cell, as follows: Put a Poisson-(ρ) distributed number of points in the cell, with their locations independently and uniformly distributed over the given cell, see [57, Sec. 2.4.1] for details. Such a more elaborate approach is needed when d > 1, as there is no analogue of the renewal process we used for d = 1. Example 9. Homogeneous Poisson Process. This process on Rd , with (point) density ρ (compare Remark 4), is a random counting measure Φ (with distribution P) such that Φ(A) is Poisson-(ρ λ(A))-distributed for any measurable A ⊂ Rd and that the random variables Φ(A1 ), . . . , Φ(Am ) are independent for any collection of pairwise disjoint A1 , . . . , Am ⊂ Rd . With this setting, the expectation measure of the process is given by E P (Φ) = ρ λ. It is well-known that, under the Palm distribution, a Poisson process looks like the same Poisson process augmented by an additional point at 0, so that P0 (B) = 1B (Φ + δ0 ) d P(Φ), for B ⊂ N (51) (alternatively, write L(Φ + δ0 ) = P0 , or P ∗ δδ0 = P0 ), by a theorem of Slivnyak, compare [57, Ex. 4.3]. This is intuitively obvious from the approximation via independent coin flips on εZd and the idea of obtaining the Palm distribution via conditioning on the presence of a point at 0. Here, this gives in I P0 = δ0 + I P = δ0 + ρ λ. Since homogeneous Poisson processes are stationary and ergodic for the translation action of R d , we can now apply Theorem 3. Consequently, for almost all realisations Φ of a homogeneous Poisson process of density ρ, the autocorrelation measure and the diffraction measure are given by γ P = ρ δ0 + ρ 2 λ

and

γ P = ρ 2 δ0 + ρ λ,

(52)


645

by an application of Eq. (4). This also extends Example 1 to arbitrary finite values of the point density ρ; compare also [19, Ex. 8.2(a)]. Remark 16. Matérn’s Hard-Core Process. One drawback of the Poisson process (of point density ρ > 0 in R d , with d ≥ 2 say) for applications in physics is the missing uniform discreteness. The latter was introduced by Matérn through a hard-core condition, realised via a local thinning process applied to each realisation, see [58] and references therein. Informally, for some fixed radius R > 0, his construction works as follows. Each point of a realisation of a Poisson process is equipped with an independent mark that is drawn uniformly at random from [0, 1] (technically, this is a marked Poisson process). Then, only those points x are kept for which there exists no point with a smaller mark in B R (x). The autocorrelation of the modified process is still radially symmetric. Moreover, if R is the radius of the hard-core condition and B R = B R (0) as before, the autocorrelation for distances r ≥ 2R is that of a homogeneous stationary Poisson process with a new, effective point density ρeff =

1 − e−ρ vol(B R ) . vol(B R )

2 λ − ν, where In fact, the new autocorrelation almost surely has the form γ = ρeff δ0 + ρeff ν is a radially symmetric measure with support B2R , as follows from [58, Thm. 1]. In 2 1 fact, ν is absolutely continuous with density ρeff B − g, where g is a smooth function R

on B2R \B R . This density has a jump (with sign change) at |x| = R, but tends smoothly to 0 as |x| 2R. The diffraction of (this version of) the Matérn model is thus given by 2 δ0 + ρeff − h λ (a.s), γ = ρeff where h is a smooth (even analytic) function with h(k) = O |k|−(d+1)/2 as |k| → ±∞. Indeed, one has d/2 R 1 B (k) = Jd/2 (2π |k| R), R |k| which is responsible for the above estimate via the exact asymptotic behaviour of the Bessel function Jd/2 for large arguments. The remaining contribution, using the explicit expression of [58, Eq. (3.2)] and the reduction of the radially symmetric Fourier transform to a one-dimensional Hankel transform, gives another term. It can also be computed explicitly, though the resulting formula is too lengthy to write it down here. Its decay is as for the previous term, by an application of the (refined) Paley-Wiener theorem. At this point, let us recall from [7] one process that is of particular interest in the study of aperiodic solids. Unlike the Poisson process and most of its siblings, it shows a substantial amount of point spectrum. It is related to Remark 10, but based on the cut and project formalism, for which we refer the reader to [44]. Example 10. Model Set Based Particle Gas. Let Λ ⊂ R d be a regular model set, for simplicity with internal space R m . Let L be the lattice in R d × R m that is needed for the cut and project scheme, with projection image L in R d . We denote the corresponding star map (from L into R m ) by , so that (up to a translation) Λ = {x ∈ L | x ∈ W },

646


where W ⊂ R m is the window of Λ. The latter is assumed to be a compact set with non-empty interior and a boundary of measure 0. This guarantees Λ to be a Meyer set. Let f now be a continuous function on W , and consider the weighted Dirac comb f (x ) δx , ω = x∈Λ

which is pure point diffractive by the model set theorem [7,32,55] with diffraction measure 2 γω = f (−k ) δk . k∈M

Here, M is the projection of the dual lattice L∗ into R d , on which the star map is well-defined, too. It is known as the Fourier module of the model set Λ. Assume now that 0 ≤ f (y) ≤ 1 on W , and define a family of independent binary random variables (Ux )x∈Λ , each taking values 0 or 1 with P ({Ux = 1}) = f (x ). The stochastic counterpart ωs of ω is U x δx , ωs = x∈Λ

which can be interpreted as a particle gas on Λ. By [7, Thm. 2 and Eq. (58)], it almost surely has diffraction # γωs = γω + dens(Λ) V λ.

1 Here, V = vol(W ) W f (y) (1 − f (y)) dy is the mean variance of the random variables, averaged over Λ, which is a consequence of the uniform distribution result for model sets [45,54]. One can also calculate the entropy of this system [7]. Moreover, it is not difficult to restrict the process to produce Meyer sets – one simply has to choose a function f that is 1 on a subset of W with non-empty interior. 5.3. Compound processes. Let us now go one step further by adding random clusters to the picture. To this end, let a stationary, ergodic, point process Φ be given, with law P, density ρ, and locally finite expectation measure E P (Φ). This is called the centre process from now on. Moreover, let Ψ ∈ M+bd be a positive random measure with law Q, subject to the condition that both its expected total mass, m := E Q Ψ (Rd ) > 0, and 2 the second moment of the total mass, E Q Ψ (R d ) , are finite. This is the component process. We will also consider signed component processes Ψ with values in Mbd , in which case we assume that the second moment of the total variation measure is finite; see the Appendix for some details on the required notions and modifications for signed measures. A combined cluster process, or cluster process for short, is a combination of a centre process and a component process of cluster type, and is obtained by replacing each point x ∈ supp(Φ) by an independent copy of Ψ , translated to that point x. We denote such a process by the pair (Φ P , Ψ Q ). As before, we restrict ourselves to finite clusters here. Formally, let Ψ1 , Ψ2 , . . . be independent copies of Ψ (these are the individual clusters). When Φ = i δ X i , we put TX i Ψi = δ X i ∗ Ψi , Φcl := i

i


647

and denote the resulting law by Pcl = R. Note that, when Ψ ≡ δ0 is deterministic and concentrated to one point, we simply obtain L(Φ P , Ψ Q ) = L(Φ), and the cluster process coincides with the centre process. If Ψ is a counting measure, the cluster process (Φ P , Ψ Q ) is again stationary and ergodic, and its expected density is given by mρ, by [20, Prop. 12.3.IX]. This property actually holds in larger generality, which we need later on. Proposition 3. Let Φ be a stationary and ergodic point process with law P, finite density ρ and locally finite second moments. Let Ψ be a (signed) random measure with law Q, finite mean and finite second moment. Then, the combined cluster process, which is a random measure, is again ergodic. Sketch of proof. If the component process is a (positive) point process as well, this result is stated and proved in [20, Sec. 12.3]. The necessary modifications for an extension to a (possibly signed) random measure as component process, which seem to be well-known but which we could not explicitly trace in the literature, are provided in the Appendix.

The second moment measures of the three processes are connected in a way that permits an explicit calculation of the autocorrelation γ R in terms of γ P and various expectation measures of the component process with law Q. To employ this powerful connection, we recall another disintegration formula, this time for any random variable Ξ of the cluster process, E R (Ξ ) = E R (E R (Ξ | given the centres)) = E P E Q (Ξ | given the centres) , (53) which (with obvious meaning) follows from the standard theorems on conditional expectation. We are now in the position to use Eq. (44) in conjunction with Theorem 3 and Eq. (53) (2) to calculate µ(2) cl = µ R , and thus the autocorrelation of almost all realisations of the cluster process, where we first concentrate on positive random measures. The extension to signed measures follows in Sec. 5.4. We first need some technical results. Lemma 8. Let λ be Lebesgue measure on R d , as before, and µ a finite Borel measure. Then, one has µ ∗ λ = cλ with c = µ(R d ). Proof. Let g be an arbitrary continuous function on R d , with compact support. For all x ∈ R d , we have λ(T−x g) = (Tx λ)(g) = λ(g) due to translation invariance of λ. The convolution µ ∗ λ is well-defined as µ is finite while λ is translation bounded [13, Prop. 1.13]. One thus has g(x + y) dλ(y) dµ(x) = λ(T−x g) dµ(x) (µ ∗ λ) (g) = d d Rd R × R λ(g) dµ(x) = µ(R d ) λ(g). = Rd

Since g was arbitrary, the claim follows.

Given a measure µ ∈ M+ and a continuous function g on R d with compact support (possibly complex-valued), we define a new function gµ on R d via gµ (x) := (Tx µ)(g),

(54)

which is certainly measurable. It is easy to check that gµ satisfies gµ = g˜µ˜ .

(55)

648


Lemma 9. Let µ ∈ M+bd and let γ be a positive, translation bounded measure on R d . For arbitrary (possibly complex-valued) f, g ∈ Cc (R d ), one has the identity ˜ = γ ( f µ ∗ gµ ). (µ ∗ µ˜ ∗ γ ) ( f ∗ g) This identity also holds when both µ and γ are signed measures. Proof. Let f and g be µ-measurable functions such that f ∗ g˜ is a continuous function with compact support, which includes the situation of the claim. One then finds, with Fubini, that ˜ = f (x + z + ξ ) dµ(x) g(y ˜ − ξ ) dµ(y) ˜ dλ(ξ ) dγ (z) (µ ∗ µ˜ ∗ γ ) ( f ∗ g) = Tz+ξ µ ( f ) T−ξ µ˜ (g) ˜ dλ(ξ ) dγ (z) = f µ (z + ξ ) g˜µ˜ (−ξ ) dλ(ξ ) dγ (z) = γ ( f µ ∗ gµ ) , where all integrals are over R d and (55) was used in the last step.

Specialising Lemma 9 to γ = δ0 gives the relation ˜ ( f ∗ g) ˜ = f µ ∗ gµ (0) = λ( f µ gµ ), (µ ∗ µ)

(56)

which simplifies our further discussion. Remark 17. Test Functions for Measures. Recall that two measures µ, ν ∈ M(R d ) are equal when µ(h) = ν(h) for all h ∈ Cc (R d ). When the measures are positive or signed (but not complex), real-valued functions suffice. In the latter case, it will be particularly helpful to restrict to functions of the form h = f ∗ g with f, g ∈ Cc (R d ). Since the space Cc (R d ) contains an approximate unit for convolution, the linear combinations of such functions are dense in Cc (R d ), so that they suffice to assess equality of measures. Lemma 10. Under our general assumptions on the component process, one has ) ( f ∗ g) E Q (Ψ ∗ Ψ ˜ = λ E Q ( f Ψ g Ψ ) and E Q (Ψ )∗ E ˜ = λ f E Q (Ψ ) g E Q (Ψ ) , Q (Ψ ) ( f ∗ g) where f, g ∈ Cc (R d ), possibly complex-valued. Proof. Let f and g be chosen as in the previous proof, with complex-valued functions permitted. For the first claim, observing that each realisation of Ψ is a finite measure, a direct calculation with Eq. (56) gives ( f ∗ g) ) ( f ∗ g) E Q (Ψ ∗ Ψ ˜ = E Q λ( f Ψ gΨ ) = λ E Q ( f Ψ g Ψ ) , ˜ = EQ Ψ ∗ Ψ while the second identity simply is Eq. (56) with µ = E Q (Ψ ), which is a finite measure by assumption.


649

Recall that the covariance of two real-valued random variables X and Y related to the law Q (using our general notation as explained above) is defined as covQ (X, Y ) := E Q (X Y ) − E Q (X ) E Q (Y ) ,

(57)

where the index Q highlights the reference to the underlying law Q. Proposition 4. Let (Ξ, R) be a combined cluster process with stationary centre point process (Φ, P) and real component process (Ψ, Q), both with the usual assumptions on means and second moments ! as used above. Then, Ξ is locally square integrable in the sense that E R (Ξ (B))2 < ∞ for any bounded Borel set B, and we have the reduction formula (2) (2) µ R ( f ⊗ g) = µ P f E Q (Ψ ) ⊗ g E Q (Ψ ) + ρ λ covQ ( f Ψ , g Ψ ) , where ρ is the density of the centre process, f, g are continuous with compact support, and the covariance is defined as in (57). ! Proof. In order to check that E R (Ξ (B))2 < ∞ for bounded, Borel measurable B ⊂ R d , one can trace through the steps below, replacing f and g by 1 B (the corresponding integrals then involve only positive terms and are finite by Fubini’s theorem). In general, by assumption and the disintegration formula (53), one finds (2) µ R ( f ⊗ g) = Ξ ( f ) Ξ (g) d R(Ξ ) M+ = EQ Ψx (T−x f ) Ψy (T−y g) d P(Φ) , N+

x,y∈supp(Φ)

where Ψx denotes the random measure at centre x. Since Ψx and Ψy are independent for x = y, the double sum over the support is split into a sum over the diagonal (x = y) and a sum over all remaining terms (x = y). Using the linearity of the expectation operator, the integrand can now be rewritten as a sum over two contributions, namely

E Q (Ψ (T−x f )) E Q Ψ (T−y g)

and

x,y

E Q (Ψ (T−x f ) Ψ (T−x g)) − E Q (Ψ (T−x f )) E Q (Ψ (T−x g)) . x

Inserting the first term into the previous calculation leads to the contribution (2) (2) µ P E Q ( f Ψ ) ⊗ E Q (g Ψ ) = µ P f E Q (Ψ ) ⊗ g E Q (Ψ ) , while the second results in E P (Φ) covQ ( f Ψ , g Ψ ) = ρ λ covQ ( f Ψ , g Ψ ) , where the last step follows from the stationarity of (Φ, P).

650


Theorem 4. Let Φ be a stationary and ergodic point process with law P, finite density ρ and locally finite second moments. Let Ψ be a random measure with law Q, finite expectation measure and finite second moments. If (Ξ, R) denotes the combined cluster process built from the centre process (Φ, P) and the component process (Ψ, Q), it is also stationary and ergodic. Moreover, the autocorrelation of the combined process satisfies ) − E Q (Ψ ) ∗ E γ R = E Q (Ψ ) ∗ E Q (Ψ ) ∗ γ P + ρ E Q (Ψ ∗ Ψ Q (Ψ ) , and this is almost surely the natural autocorrelation of a given realisation of the cluster process. Proof. Choose two measurable functions f and g such that f ∗ g˜ ∈ Cc (R d ). Then, in line with Remark 17 and Eq. (45), one finds via Proposition 4 that (2) (2) γ R ( f ∗ g) ˜ = µ R ( f ⊗ g) = µ P f E Q (Ψ ) ⊗ g E Q (Ψ ) + ρ λ covQ ( f Ψ , g Ψ ) + ρ E = γ P f E Q (Ψ ) ∗ g (Ψ ∗ Ψ ) − E (Ψ ) ∗ E (Ψ ) ( f ∗ g) ˜ , Q Q Q E Q (Ψ ) where E P (Φ) = ρ λ due to stationarity of (Φ, P). The second step makes use of Lemma 10. The formula for the autocorrelation now follows from the observation that = E ( f ∗ g), ˜ γ P f E Q (Ψ ) ∗ g (Ψ ) ∗ E (Ψ ) ∗ γ Q Q P E Q (Ψ ) which is an application of Lemma 9. The remaining claims are clear due to the assumed ergodicity, via an application of Proposition 3.

An application of the convolution theorem gives the following consequence, where ) was used to highlight the structure of the result. also the identity E Q (Ψ ) = E Q (Ψ Corollary 1. Under the assumption of Theorem 4, the diffraction measure of the combined cluster process is given by )2 · |2 ) − |E Q (Ψ )|2 λ, γ R = E Q (Ψ γ P + ρ E Q (|Ψ which is then almost surely also the diffraction measure of a given realisation.

The result parallels our previous formulas, as was to be expected. Nevertheless, it does not follow from Theorem 2 in general, because realisations of stationary point processes in R d generically fail to be FLC sets. Before we discuss possible generalisations beyond the case of positive random measures, let us look at some examples. Example 11. Poisson Cluster Process. An important special case emerges when the centre process is the homogeneous Poisson process of Example 9, with point density ρ. Let γ P and γ P be the corresponding measures. If we couple a cluster component process Ψ to it, with law Q and m := E Q (Ψ )(R d ) its expected number of points, our general formula for the compound process (Φ P , Ψ Q ) applies. With Lemma 8, the convolution formula can be simplified, and the result reads as follows:


651

For almost all realisations of a Poisson cluster process (Φ P , Ψ Q ), the natural autocorrelation measure exists and is given by ) , γ R = γ P,Q = (mρ)2 λ + ρ E Q (Ψ ∗ Ψ ) is a finite positive measure (of expected total mass ≥ m 2 ), due to our where E Q (Ψ ∗ Ψ general assumption that E Q (Ψ (R d ))2 is finite. Consequently, the diffraction measure is almost surely given by ) · λ, γ R = (mρ)2 δ0 + ρ E Q (Ψ ∗ Ψ ) is a uniformly continuous Radon-Nikodym density for Lebesgue where E Q (Ψ ∗ Ψ measure. These formulas include the case of deterministic clusters; compare Example 6. Remark 18. Random Displacement of Poisson Processes. An interesting pair of processes is the combination of the homogeneous Poisson process from Example 9 with Hof’s random displacement model from Example 8. A simple calculation shows that (ν)

γ R = γ P = γ P and γR = γP in this case (and, in fact, R and P have the same law here). From a physical point of view, this is in line with the behaviour of an ideal gas at high temperatures. When the Poisson process is a good model for the gas, and random displacement one for the disorder due to high temperature, compare the discussion in [33], the combination should still be an ideal gas – and this is precisely what happens, as reflected by the two identities. Remark 19. Particle Gas Cluster Process. It is clear that the particle gas of Example 10 satisfies all requirements for a centre process, so that we can apply the cluster process machinery to it, too. This produces physically interesting and relevant examples with a substantial amount of point spectrum. This observation remains true for more complicated particle gas models with interactions, under certain conditions on the potential of the underlying Gibbs measure, say; compare [11] for further details and examples. Example 12. Neyman-Scott Processes. Let K be a non-negative random integer with law L(K ) = µ, mean m := E µ (K ) and finite second moment, E µ (K 2 ) < ∞. Now, let Y1 , Y2 , . . . be a family of Rd -valued i.i.d. random variables with common distribution ν, and independent of K . Define the cluster distribution via Ψ :=

K

δY j ,

j=1

i.e., a cluster has a random size K , while the positions of its atoms are independently drawn from the probability distribution ν. The induced distribution for Ψ is again called Q. With a calculation similar to the one in Example 8, one finds K K E Q (Ψ )(A) = E Q 1 A (X i ) = E µ 1 A (xi ) dν(xi ) i=1

= E µ (K · ν(A)) = m ν(A)

i=1

Rd

652


) = m 2 (ν ∗ for A ⊂ R d Borel, so that E Q (Ψ ) = m ν and E Q (Ψ ) ∗ E Q (Ψ ν ). Moreover, one has ⎛ ⎞ K )(A) = E Q ⎝ E Q (Ψ ∗ Ψ 1 A (X k − X )⎠ = m δ0 (A) + E µ (K (K − 1)) (ν ∗ ν )(A), k,=1

) = m δ0 + E µ (K (K − 1)) (ν ∗ ν), so that the general formulas which gives E Q (Ψ ∗ Ψ from Theorem 2 can now be applied again. Note that E µ (K (K − 1)) = E µ (K 2 ) − m. If the centre process is once more the homogeneous Poisson process with mean (point) density ρ, Lemma 8 gives similar simplifications as in Example 11. Consequently, for the resulting law R, the autocorrelation is almost surely given by γ R = (mρ)2 λ + mρ δ0 + ρ E µ (K 2 ) − m (ν ∗ ν) , whence the corresponding diffraction measure is given by ν|2 λ , γ R = (mρ)2 δ0 + ρ m + (E µ (K 2 ) − m)| which is an interesting extension of the Poisson process; compare [19, Ex. 8.2(f)] for a circularly symmetric case in R 2 . 5.4. Autocorrelation for signed (ergodic) processes. It is intuitively clear that the results of this section are not really restricted to point processes or positive measures for the clusters. Here, we sketch how they can be adapted to the situation of signed random measures. Consider a stationary, possibly signed, random measure Ψ (with law Q and ‘finite second moments’, meaning that E Q (|Ψ |(A))2 < ∞ holds for any bounded A ⊂ R d ), with second moment measure µ(2) , defined as before via bounded f of compact support as f (x, y) dµ(2) (x, y) = E Q f (x, y) dΨ (x) dΨ (y) . R d ×R d

Rd ×Rd

(2)

The reduced second moment measure µred on R d with the property ˜ = µ(2) ( f ⊗ g) µ(2) red ( f ∗ g)

(58)

is defined in complete analogy to the positive case. The analogue of Theorem 3 is: Theorem 5. Let Φ be a stationary and ergodic, random, signed measure with distribu tion P. Assume that Φ has finite second moments in the sense that E P (|Φ|(A))2 < ∞ for any bounded measurable set A ⊂ Rd (which follows, for example, from E P (|Φ|(Br (x)))2 < ∞ for all x ∈ R d and some open ball Br ). Let Φn := Φ| Bn denote the restriction of Φ to the ball of radius n around 0. Then, the natural autocorrelation of Φ, which is defined with an averaging sequence of nested, centred balls, almost surely exists and satisfies (Φ)

γP

:= lim

n→∞

n Φn ∗ Φ Φn ∗ Φ (2) = lim = µred = γ P , n→∞ λ(Bn ) λ(Bn ) (2)

where the limit refers to the vague topology on M. Here, µred is the reduced second moment measure of P according to (58).


653

Proof. The proof is a variation of that of Theorem 3. Fix a continuous function h : R d → R with compact support. We have to check that 1 n→∞ (2) n (h) −− Φn ∗ Φ −−→ µred (h) λ(Bn )

(a.s.).

(59)

Let Φ be an ergodic, random, signed measure as above and F an ergodic random function on R d , the latter with the property that |F(x)| d|Φ|(x) < ∞ (60) EP A

for any bounded measurable A ⊂ R d . We can then define an additive covariant spatial process X A in the sense of [46], indexed by bounded measurable subsets A, via X A := F(x) dΦ(x). A

Note that ergodicity of Φ and F implies that (X A , R d ) is again ergodic, meaning that the shift-invariant σ -field is trivial. Now, [46, Cor. 4.9] yields 1 1 (a.s.). X Bn = E P X B1 lim n→∞ λ(Bn ) λ(B1 ) Applying this to Φ as in the theorem, together with F(x) := R d h(x − y) dΦ(y), yields 1 1 (h) Φn ∗ Φ lim F(x) dΦ(x) = lim n→∞ λ(Bn ) B n→∞ λ(Bn ) n 1 = EP h(x − y) dΦ(y) dΦ(x) λ(B1 ) B1 R d 1 1 B (x) h(x − y) dµ(2) (x, y) = λ(B1 ) R d × R d 1 1 (2) (2) 1 B1 (x) h(z) dµred (z) dx = h dµred = λ(B1 ) R d R d Rd and Φn ∗ Φ n almost surely, which is almost the claim. The difference between Φn ∗ Φ can be treated as in the proof of Theorem 3.

Combining Proposition 5 and Theorem 5, and observing that the calculations in the proof of Proposition 4 carry over literally to the signed case, we obtain Corollary 2. The statements of Theorem 4 and Corollary 1 remain true for cluster processes with signed clusters.

Example 13. Signed Poisson Process. If we combine the homogeneous Poisson process of Example 9 with the random weight model of Example 7, and choose weights 1 and −1 with equal probability, Corollary 2 implies the almost sure diffraction γ = ρ λ. In particular, one has γ = λ for density ρ = 1, which makes this signed Poisson point set, on the level of the 2-point correlations, indistinguishable from the signed Bernoulli sequence (or process) on Zd . This is remarkable in view of the rather different geometric structure and demonstrates the intrinsic difficulty of the corresponding inverse problem.

654


5.5. Equilibria of critical branching Brownian motions in d ≥ 3. Consider a system of particles performing independent Brownian motions in Rd , d ≥ 3 (for ease of comparison with the cited literature, we assume that the variance parameter is σ 2 = 2). Additionally, each particle, after an exponentially distributed lifetime with parameter V , either doubles or dies, where each possibility occurs with probability 1/2. In the situation of a birth event, the daughter particles appear at the position of the mother. Note that if we start with a finite number of particles, the expected number of particles is preserved for all time, as the expected number of offspring equals 1. This is what ‘critical’ in the name refers to. Imagine we start such a system from a homogeneous Poisson process of density ρ, denote by Φt the random configuration observed at time t ≥ 0, and its distribution by Pt . Here, Pt is stationary with density ρ, see [31] and the references given there for background. It follows from [31, Thm. 2.3] that the first moment measure of the Palm distribution of Pt is given by I(Pt )0 = δ0 + (ρ + f t )λ , where

(61)

t

V 2t f t (x) = V ps (0, y) ps (y, x) dy ds = pu (0, x) du , 2 0 0 Rd with pt (x, y) = (4π t)−d/2 exp −|x − y|2 /(4t) the d-dimensional Brownian transition density (with variance parameter 2). As explained in [31], there is a genealogical interpretation behind (61): In view of the interpretation of the Palm distribution as the configuration around a typical individual, δ0 is the contribution of this individual, f t λ that from its relatives in the family decomposition of the branching process, and ρ λ is the contribution from unrelated individuals. Furthermore, by [31, Thm. 2.2], Pt converges (vaguely) towards P∞ , which is the unique, ergodic, equilibrium distribution of density ρ (cf. [15] for uniqueness), and the limit t → ∞ can be taken in (61) to obtain I(P∞ )0 = δ0 + (ρ + f ∞ )λ ,

where V f ∞ (x) = 2

0

∞

1 V d−2 2 pu (0, x) du = d/2 2 4π |x|d−2

is (up to the prefactor V /2) the Green function of Brownian motion. Thus, using Lemma 2, we have Theorem 6. Let Φ∞ be a realisation of the critical branching Brownian motion, from the equilibrium distribution P∞ . The autocorrelation is then almost surely given by γ = ρ δ0 + ρ (ρ + f ∞ )λ , while

1 V λ γ = ρ 2 δ0 + ρ 1 + 2 4π 2 |k|2

is the corresponding diffraction measure.


655

Remark 20. Extension of Theorem 6. One can also consider the scenario where, instead of Brownian motion, particles move during their lifetime according to a symmetric, stable process of index α ∈ (0, 2] in Rd (α = 2 corresponds to Brownian motion). Such processes have discontinuous paths, and their transition density (α) (α) pt (x, y) = pt (0, y − x) satisfies (α) eik·x pt (0, x) dx = exp(−t|k|α ) Rd

(in general, no explicit form of pt(α) is known). By [31, Thm. 2.2], non-trivial equilibria exist if the spatial dimension d satisfies d > α. In this case, a reasoning analogous to (α) that above yields the following: The autocorrelation of a realisation Φ∞ of the equilibrium of a system of critical, branching, symmetric α-stable processes (with density ρ) is almost surely given by (α) γ = ρ δ0 + ρ (ρ + f ∞ )λ ,

where (α) f∞ (x)

V = 2

0

∞

pu(α) (0, x) du =

1 V ((d − α)/2) 2 2α π d/2 (α/2) |x|d−α

(for the form of the Green function of the symmetric α-stable process, see [14, Ex. 1.7]). Hence, the diffraction measure is almost surely given by 1 V 2 λ, γ = ρ δ0 + ρ 1 + 2 (2π )α |k|α by another application of Lemma 2. Note that, due to the independence properties of the branching mechanism, these equilibria can also be considered as Poisson cluster processes. In contrast to the scenario considered above, clusters in Φ∞ are infinite, and the spatial correlation decays only algebraically (without being integrable). 6. Outlook This article demonstrates that various aspects of mathematical diffraction theory for random point sets and measures can be approached systematically with methods from point process theory, as was originally suggested in [28]. At the same time, the approach is sufficiently concrete to allow for many explicitly computable examples, several of which were presented above. They comprise many formulas from the somewhat scattered literature on this subject in a unified setting. There are, of course, many more examples, but we hope that the probabilistic platform advertised here will prove useful for them as well. The next step in this development needs to consider point processes and random measures with interactions, such as those governed by Gibbs measures. First steps are contained in [4,10,11,21,28,33,40,41] and indicate that both qualitative and quantitative results are possible, though some further development of the theory is needed. A continuation along this path would also make the results more suitable for real applications in physics and crystallography, though it is largely unclear at the moment what surprises the corresponding inverse problem might have to offer here.

656


Appendix: Ergodicity for Cluster Processes with Signed Random Measures Let M = M(R d ) be the space of (locally finite) real or signed measures on R d , equipped with the topology of vague convergence, with M+ = M+ (R d ) denoting the subspace of positive measures. Let ΣM denote the Borel σ -algebra of R d . Note that the latter is also generated by the mappings M µ → µ(A), for bounded and measurable sets A ⊂ R d . Recall that any µ ∈ M admits a unique Hahn-Jordan decomposition µ = µ+ − µ− , with µ+ , µ− ∈ M+ mutually singular. The mappings µ → µ+ and µ → µ− are ΣM -measurable. We write |µ| := µ+ + µ− ∈ M+ for the total variation measure of µ. A random signed measure Φ is a random variable with values in (M, ΣM ). In the context of signed random measures, it is convenient to work with the characteristic functional ! (62) ϕΦ (h) := E exp i h dΦ , which is defined for any h : R d → R that is bounded and measurable with compact support. Here and below, we suppress R d as the integration region. In analogy to the Laplace functional for positive random measures, the distribution of Φ is determined by ϕΦ . Here, we are interested in signed cluster processes: Let Φ be a stationary counting process with finite intensity ρ, and Ψ j (with j ∈ N) independent (and independent from Φ), identically distributed, random, signed measures such that E [|Ψ1 |] is a finite measure. Then, given a realisation Φ = j δ X j , where X j are the positions of the atoms of Φ (in some enumeration), the cluster process is defined as Ξ := TX j Ψ j . (63) j

Note that for any bounded B ⊂ R d , |Ψ j |(B − X j ) E [|Ξ (B)|] ≤ E =ρ

j

Rd

dE [|Ψ1 |] dx = ρ (E [|Ψ1 |] ∗λ) (B) < ∞ , B−x

so that (63) is indeed well-defined. Lemma 11. Let Ψ be a signed random measure on R d . The following are equivalent: (1) Ψ is ergodic. (2) For any U, V ∈ ΣM , 1 lim (P (Ψ ∈ U ∩ Tx V ) − P(Ψ ∈ U ) P(Ψ ∈ V )) dx = 0. n→∞ λ(Bn ) B n (3) For any g, h : R d → R measurable with compact support, 1 ϕΨ (g + Tx h) − ϕΨ (g) ϕΨ (h) dx = 0. lim n→∞ λ(Bn ) B n


657

Furthermore, it suffices to restrict to U, V a semiring which generates ΣM in (2), and it suffices to restrict to continuous g, h with compact support in (3). Proof. This is a straightforward adaptation of the proofs of Propositions 12.3.III and 12.3.VI and Lemma 12.3.II of [20] to the signed case.

The following result is an analogue [20, Prop. 12.3.IX] for the signed measure case. Since we have not been able to find a proof in the literature, we provide a sketch. Proposition 5. Let Φ, Ψ j , and Ξ := j TX j Ψ j be as above. If Φ is ergodic, then Ξ is ergodic as well. Sketch of proof. We verify condition (3) from Lemma 11. Observe that for any f : R d → R with compact support and any ε > 0, we can find R < ∞ such that ⎞ ⎛ f d(TX Ψ j ) ≥ ε⎠ ≤ ε. (64) P⎝ j j : |X j |≥R

To check (64), let R be large enough so that supp( f ) ⊂ [−R , R ]d , and note that for R > R , the left-hand side of (64) is bounded by ⎛ ⎞ ε ⎠ P⎝ |Ψ j | [−R , R ]d + X j ≥ || f ||∞ j : |X j |∞ ≥R ⎡ ⎤ || f ||∞ ⎣ E ≤ |TX j Ψ j | [−R , R ]d ⎦. ε j : |X j |∞ ≥R

The expectation on the right-hand side above equals ρ 1[−R ,R ]d (x − y) dE [|Ψ1 |] (y) dx R d \[−R,R]d R d

≤ ρ(2R )d E [|Ψ1 |] R d \[−(R − R ), (R − R )]d ,

which converges to 0 as R → ∞ because E [|Ψ1 |] is a finite measure. Let g, h : R d → R continuous with compact support and define ! G(Φ) := E exp i g dΞ | Φ] , H (Φ) := E exp i h dΞ Φ . Decompose

(g + Tx h) dΞ =

TX j g dΨ j +

j : X j ∈[−R,R]d

+ j :Xj

∈[−R,R]d −x

TX j g dΨ j

j : X j ∈[−R,R]d

TX j +x h dΨ j + j :Xj

∈[−R,R]d −x

TX j +x h dΨ j ,

658


and choose R so large that (64) is fulfilled for f = g and f = h. Recall that, for any real-valued random variables X , Y with P(|Y | ≥ ε) ≤ ε, we have ) ( i(X +Y ) − E ei X ≤ E ei(X +Y ) − ei X ≤ E ei X eiY − 1 ≤ ε + P(|Y | ≥ ε) ≤ 2ε. E e For A ⊂ R d , write Ξ A :=

TX j Ψ j for the random measure d R \[−2R, 2R]d , we then have

j : X j ∈A

which consists of

clusters with centres in A. For x ∈ ! E exp i (g + Tx h) dΞ − E [G(Φ)H (Tx Φ)] ! ! ≤ E exp i (g + Tx h) dΞ − E exp i g dΞ[−R,R]d + i Tx h dΞ[−R,R]d −x !! Φ − E [G(Φ)H (Tx Φ)] . + E E exp i g dΞ Tx h dΞ d +i d [−R,R]

[−R,R] −x

The first term on the right-hand side is bounded by 2ε. Observing that the conditional expectation in the second term is in fact a product because clusters with centres in disjoint regions are (conditionally) independent, we can bound the second term from above by ! ! ! E E exp i g dΞ [−R,R]d Φ E exp i Tx h dΞ[−R,R]d −x Φ − H (Tx Φ) ! ! + E E exp i g dΞ[−R,R]d Φ − G(Φ) H (Tx Φ) ≤ E exp i Tx h dΞ[−R,R]d −x − exp i Tx h dΞ + E exp i g dΞ g dΞ , d − exp i [−R,R]

which is not more than 2ε. !! Thus, using the relation E E exp i (g + Tx h) dΞ | Φ = ϕΞ (g + Tx h) together with E [G(Φ)] = ϕΞ (g) and E [H (Φ)] = E [H (Tx Φ)] = ϕΞ (h), we obtain 1 (65) ϕΨ (g + Tx h) − ϕΨ (g) ϕΨ (h) dx lim sup n→∞ λ(Bn ) Bn 1 + 4ε = 4ε ≤ lim sup Φ)] − E E (Φ)]) dx (T [G(Φ)] [H [G(Φ)H (E x n→∞ λ(Bn ) Bn by ergodicity of Φ (in order to deduce this literally from statement (2) in Lemma 11, one can for instance discretise the support of g and h and approximate G(Φ), H (Φ) with functions depending only on the random vector (Φ(ci ))1≤i≤N , where {ci | 1 ≤ i ≤ N } is a collection of disjoint (small) cubes). Finally, take ε → 0 to conclude.

Acknowledgements. This work was supported by the German Research Council (DFG), within the CRC 701, by the Natural Sciences and Engineering Research Council of Canada (NSERC), and by the RiP program at Oberwolfach. We thank the referees for their thorough analysis of the paper and for making useful suggestions that have helped to improve it.

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Communications in


The Nekrasov Conjecture for Toric Surfaces Elizabeth Gasparim1 , Chiu-Chu Melissa Liu2 1 School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building,

The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland. E-mail: [email protected]

2 Department of Mathematics, Columbia University, 2990 Broadway,

New York, NY 10027, USA. E-mail: [email protected] Received: 8 September 2008 / Accepted: 18 September 2009 Published online: 19 November 2009 – © Springer-Verlag 2009

Abstract: The Nekrasov conjecture predicts a relation between the partition function for N = 2 supersymmetric Yang–Mills theory and the Seiberg-Witten prepotential. For instantons on R4 , the conjecture was proved, independently and using different methods, by Nekrasov-Okounkov and Nakajima-Yoshioka. We prove a generalized version of the conjecture for instantons on noncompact toric surfaces. Contents 1.

2. 3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Partition functions for instantons on noncompact toric surfaces 1.3 Seiberg-Witten prepotential . . . . . . . . . . . . . . . . . . . 1.4 Nekrasov conjecture . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . Moduli Spaces of Framed Bundles on Surfaces . . . . . . . . . . . 2.1 Dimension of the moduli space . . . . . . . . . . . . . . . . . 2.2 The natural bundle . . . . . . . . . . . . . . . . . . . . . . . Torus Action and Fixed Points . . . . . . . . . . . . . . . . . . . . 3.1 Torus action on the surface . . . . . . . . . . . . . . . . . . . 3.2 Torus action on moduli spaces . . . . . . . . . . . . . . . . . 3.3 Torus fixed points in moduli spaces . . . . . . . . . . . . . . . Gauge Theory Partition Functions . . . . . . . . . . . . . . . . . . 4.1 Equivariant parameters . . . . . . . . . . . . . . . . . . . . . 4.2 Multiplicative classes of the tangent and natural bundles . . . . 4.3 4d pure gauge theory . . . . . . . . . . . . . . . . . . . . . . 4.4 4d gauge theory with N f fundamental matter hypermultiplets . 4.5 4d gauge theory with one adjoint matter hypermultiplet . . . . 4.6 5d gauge theory compactified on a circle of circumference β .

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662 662 662 663 664 665 665 665 669 669 670 670 670 671 671 672 673 673 674 674

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E. Gasparim, C.-C. M. Liu

4.7 Hirzebruch χy genus . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Elliptic genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The Instanton Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The tangent bundle: adjoint representation . . . . . . . . . . . . . . 5.2 The natural bundle: fundamental representation . . . . . . . . . . . 5.3 Formula for instanton partition functions . . . . . . . . . . . . . . . 5.4 Nekrasov conjecture for C2 : instanton part . . . . . . . . . . . . . . 5.5 Nekrasov conjecture for toric surfaces: instanton part . . . . . . . . 6. The Perturbative Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The virtual tangent bundle of Mr,d,n (X 0 ) . . . . . . . . . . . . . . 6.2 The natural virtual bundle . . . . . . . . . . . . . . . . . . . . . . . 6.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Nekrasov conjecture: perturbative part . . . . . . . . . . . . . . . . Appendix A: Kobayashi–Hitchin Correspondence and Existence of Instantons Appendix B: Equivariant Cohomology . . . . . . . . . . . . . . . . . . . . . B.1 Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Seiberg-Witten Prepotential . . . . . . . . . . . . . . . . . . . C.1 SU (2) case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Higher rank case . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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675 675 676 676 680 681 683 684 687 687 688 689 690 693 694 694 695 695 696 696 697 699

1. Introduction 1.1. Background. The Nekrasov conjecture [Ne2] predicts a surprising relation between two seemingly unrelated quantities: the partition function for N = 2 supersymmetric Yang–Mills theory, defined in terms of instantons on R4 , and the Seiberg-Witten prepotential [SW], defined in terms of period integrals of a family of hyperelliptic curves. For gauge group U (r ), Nekrasov and Okounkov proved the conjecture for a list of gauge theories (4d pure gauge theory, 4d gauge theory with matter, 5d theory compactified on a circle) [NO], Nakajima and Yoshioka proved the conjecture for 4d pure gauge theory [NY1] and for 5d theory compactified on a circle [NY2] (see also Göttsche-NakajimaYoshioka [GNY2]). Braverman and Etingof studied 4d pure gauge theory with arbitrary gauge groups [Br,BrE]. In this paper we prove a generalized version of the conjecture for instantons on noncompact toric surfaces. Instantons on toric surfaces have been studied in [Ne3,GNY1, GNY2]. In field theory terms, Nekrasov’s insight involves a comparison of the infrared and ultraviolet limits of the SUSY gauge theories, as follows. The vacuum expectation value of their observables is not sensitive to the energy scale. In the ultraviolet, the theory is weakly coupled and dominated by instantons; whereas in the infrared, there appears a relation to the prepotential of the effective theory. In this instance, the physical argument is accompanied by completely rigorous mathematical definitions, thus allowing us to prove the conjecture. 1.2. Partition functions for instantons on noncompact toric surfaces. Let X 0 = X \ ∞ be an open toric surface that can be compactified to a non-singular projective toric surface X by adding a line at infinity ∞ ∼ = P1 with positive self-intersection number, so

The Nekrasov Conjecture for Toric Surfaces

663

that Tt = (C∗ )2 acts on X 0 and on X . Let Mr,d,n (X, ∞ ) denote the moduli space of rank r torsion free sheaves over X having Chern classes c1 = d and c2 = n, and framed over ∞ . Then Mr,d,n (X, ∞ ) is a smooth variety over C, and it admits a Tt × Te -action with isolated fixed points, where Te ∼ = (C∗ )r is the maximal torus of the complex gauge group G L(r, C) which acts on framings. We define 1 Mr,d,n (X,∞ )

by formally applying the Atiyah-Bott localization formula. The above integral is a rational function in equivariant parameters 1 , 2 ∈ HT2t (pt) and a1 , . . . , ar ∈ HT2e (pt). The Nekrasov partition function for supersymmetric SU (r ) instantons on X 0 is defined as def (1−r )d·d 2r n inst Z X 0 ,d (1 , 2 , a ; ) = 1, Mr,d,n (X,∞ )

n≥0

where is a formal variable. It lies in the ring Q(1 , 2 , a1 , . . . , ar )[[]]. In further generality, given two multiplicative classes A, B we define def (1−r )d·d 2r n Z inst ( , , a ; ) = A T˜ (TM)BT˜ (V ), 1 2 X 0 ,A,B,d Mr,d,n (X,∞ )

n≥0

where TM is the tangent bundle and V is the natural bundle on Mr,d,n (X, ∞ ) (see Definition 2.9).

1.3. Seiberg-Witten prepotential. We briefly recall the definition of the SeibergWitten prepotential for 4d pure SU (r ) gauge theory. Appendix C contains a more detailed discussion and definitions for other gauge theories. Consider the family of hyperelliptic curves parametrized by and u = (u 2 , u 3 , . . . , u r ): 1 = P(z) = z r + u 2 z r −2 + u 3 z r −3 + · · · + u r . Cu : r w + w The parameter space for u is called the u-plane. The Seiberg-Witten differential dS =

dw 2π −1 w 1 √

z

def is a meromorphic differential defined on the total space of this family such that ω p = ∂ ∂u p (d S) | p = 2, . . . , r is a basis of holomorphic differentials on the genus (r − 1) curve Cu . Choose a symplectic basis {Aα , Bβ | α, β = 2, . . . , r } of H1 (Cu , Z), and define √ D aα = d S, aβ = 2π −1 d S. Aα

Bβ

664


Then the 1-form rα=2 aαD daα is closed, so there exists a locally defined function, the Seiberg-Witten prepotential F0 , such that r α=2

aαD daα = dF0 , i.e., aαD =

∂F0 . ∂aα

The above definitions of d S, aα , aαD are the same as those in [NO], but are the corresponding definitions in [NY,NY1].

√

−1 times

1.4. Nekrasov conjecture. Let q0 , q1 be the two Tt fixed points in ∞ ⊂ X , and let u, v ∈ Z1 ⊕ Z2 be the weights of the Tt -action on (N∞ / X )q0 , (N∞ / X )q1 , respectively, where N∞ / X is the normal bundle of ∞ in X . If w is the weight of Tt -action on Tq0 ∞ and k = ∞ · ∞ > 0, then v = u − kw. Define def

(1 , 2 , a ; ) = −u(u − kw) log Z inst ; ). F Xinst X 0 ,A,B,d (1 , 2 , a 0 ,A,B,d We now state the prototype statement of the conjecture for toric surfaces, which will have 8 incarnations. Main Theorem (Nekrasov conjecture for toric surfaces: prototype statement). (a) F X···0 ,A,B,d (1 , 2 , a , m; ) is analytic in 1 , 2 near 1 = 2 = 0. (b) lim F X···0 ,A,B,d (1 , 2 , a ; ) = kF0··· ( a , ), where F0··· ( a , ) is the · · ·part of the 1 ,2 →0

Seiberg-Witten prepotential of matter case A, B, m, and k = ∞ · ∞ > 0 is the self intersection number of ∞ .

The 8 cases we prove are • Instanton part: Theorem 5.21. With the ··· replaced by inst , we prove the following cases of the conjecture: (1) 4d pure gauge theory: A = B = 1, m = ∅. (2) 4d gauge theory with N f fundamental matter hypermultiplets: A = 1, B = (E m )(V ) is the Tm -equivariant Euler class of V ⊗ M, where V is the natural bundle over the moduli space, M is the fundamental representation of U (N f ), Tm is the maximal torus of U (N f ), m = (m 1 , . . . , m N f ). (3) 4d gauge theory with one adjoint matter hypermultiplet: A = E m (TM) is the equivariant Euler class of the tangent bundle of the moduli space, B = 1, m = m. (4) 5d gauge theory compactified on a circle: A = Aˆ β (TM) is the Aˆ β genus of the tangent bundle (the usual Aˆ genus being the case β = 1), B = 1, m = ∅ but F depends on the additional parameter β. • Perturbative part: Theorem 6.8. With the ··· replaced by pert , we derive 4 more cases of the conjecture, with the same restrictions as in the first part: (1) 4d pure gauge theory. (2) 4d gauge theory with N f fundamental matter hypermultiplets. (3) 4d gauge theory with one adjoint matter hypermultiplet. (4) 5d gauge theory compactified on a circle of circumference β. The instanton part follows by localization, from known results in the C2 case. Indeed, localization calculations yield an expression of the instanton partition function Z inst X 0 ,A,B,d over X 0 in terms of contributions from vertices (Tt fixed points in X 0 ) and from legs


665

(Tt invariant P1 ’s in X 0 ). Each vertex contributes one copy of the instanton partition function of C2 , for which the singularity along 1 = 2 = 0 is already known. The contribution from legs does not introduce more poles along 1 = 2 = 0. A priori, the tangent weights at all Tt fixed points in X 0 appear in the denominator, but an argument similar to that in [Ne3, Sect. 6.1] shows that these poles mostly cancel out, and we are left with the two normal weights u, u − kw at the Tt fixed points on ∞ . The perturbative part is fairly straightforward. 1.5. Outline of the paper. In Sect. 2, we describe properties of the instanton moduli spaces. In Sect. 3, we study torus actions on these moduli spaces and the fixed point sets. In Sect. 4, we introduce a general instanton partition function depending on two multiplicative classes A, B for noncompact toric surfaces; different choices of A, B give partition functions of different gauge theories. Section 5 contains localization computations on instanton moduli spaces, and the proof of the instanton part of the conjecture. Section 6 contains definitions of the perturbative part of the partition function, and the proof of the perturbative part of the conjecture. 2. Moduli Spaces of Framed Bundles on Surfaces We work over C. Let X be a non-singular projective surface. Let ∞ ⊂ X be a smooth divisor. In this section, we introduce moduli spaces of framed bundles on X , and describe basic properties of these moduli spaces, generalizing the discussion in [NY1, Sect. 2] on the case X = P2 . The framed moduli spaces were constructed in much more general setting by Huybrechts-Lehn [HL]. Given a positive integer r , an integer n, and a cohomology class d ∈ H 2 (X ; Z), let Mr,d,n (X, ∞ ) be the moduli space which parametrizes isomorphism classes of pairs (E, ) such that (1) E is a torsion free sheaf on X which is locally free in a neighborhood of ∞ . (2) rank(E) = r , c1 (E) = d and X c2 (E) = n. ∼

(3) : E|∞ → O⊕r is an isomorphism called “framing at infinity”. ∞ Note that (1) and (2) imply ∞ d = 0.

2.1. Dimension of the moduli space. Given a divisor D ⊂ X , let E(−D) = E ⊗ O X (−D). Proposition 2.1. Suppose that ∞ · ∞ > 0. (a) For any (E, ) ∈ Mr,d,n (X, ∞ ) we have Ext 0O X (E, E(−∞ )) = 0. (b) Assume in addition that ∞ ∼ = P1 , then for any (E, ) ∈ Mr,d,n (X, ∞ ) we have Ext 0O X (E, E(−∞ )) = Ext 2O X (E, E(−∞ )) = 0. Remark 2.2. If X is a non-singular projective surface which contains a smooth divisor ∞ ∼ = P1 such that k = ∞ · ∞ > 0. Then TX ∞ ∼ = OP1 (k) ⊕ OP1 (2), so X is rationally connected, or equivalently, X is a rational surface. The arithmetic genus of X is pa (X ) = χ (O X ) − 1 = 0.

666


Proof of Proposition 2.1. (a) Assuming that ∞ · ∞ > 0, we will show that HomO X (E, E(−∞ )) = 0. Let s be a section of O X (∞ ) such that its zero locus is ∞ . The exact sequence s·

0 → E(−(m + 1)∞ ) → E(−m∞ ) → E(−m∞ ) ⊗ O D → 0 induces a long exact sequence 0 → HomO X (E, E(−(m + 1)∞ )) → HomO X (E, E(−m∞ )) → HomO X (E, E(−m∞ ) ⊗ O∞ ) → Ext 1O X (E, E(−(m + 1)∞ ) → Ext 1O X (E, E(−m∞ )) → · · · , where HomO X (E, E(−m∞ ) ⊗ O∞ ) ∼ = H 0 (∞ , O X (−m∞ )|∞ )⊕r , 2

since E|∞ is trivial. Let k = ∞ · ∞ > 0. Then H 0 (∞ , O X (−m∞ )|∞ ) ∼ = H 0 (P1 , OP1 (−mk)) = 0 when m > 0. So, for any positive integer m, HomO X (E, E(−(m + 1)∞ )) → HomO X (E, E(−m∞ )) is an isomorphism, and Ext 1O X (E, E(−(m + 1)∞ )) → Ext 1O X (E, E(−m∞ )) is injective. As a consequence, any element in HomO X (E, E(−∞ )) restricts to zero in a formal neighborhood of ∞ in X . So HomO X (E, E(−∞ )) = 0. (b) We now assume that ∞ ·∞ > 0 and ∞ ∼ = P1 . By Serre duality, Ext 2O X (E, E(−∞ )) is dual to HomO X (E, E(K X + ∞ )). We will show that HomO X (E, E(K X + ∞ )) = 0. The exact sequence s·

0 → E(K X −m∞ ) → E(K X +(1 − m)∞ ) → E(K X + (1 − m)∞ ) ⊗ O D → 0 induces a long exact sequence 0 → HomO X (E, E(K X − m∞ )) → HomO X (E, E(K X + (1 − m)∞ )) → HomO X (E, E(K X + (1 − m)∞ ) ⊗ O∞ ) → Ext 1O X (E, E(K X − m∞ ) → Ext 1O X (E, E(K X + (1 − m)∞ )) → · · · . E|∞ is trivial and K ∞ = (K X + ∞ )|∞ , so HomO X (E, E(K X + (1 − m)∞ ) ⊗ O∞ ) ∼ = H 0 (∞ , O∞ (K ∞ ) ⊗ O X (−m∞ )|∞ )⊕r . 2


667

Note that H 0 (∞ , O∞ (K ∞ ) ⊗ O X (−m∞ )|∞ ) ∼ = H 0 (P1 , OP1 (−2 − mk)) = 0 for all m ≥ 0. So, for any nonnegative integer m, HomO X (E, E(K X − m∞ ) → HomO X (E, E(K X + (1 − m)∞ )) is an isomorphism, and Ext1O X (E, E(K X − m∞ )) → Ext 1O X (E, E(K X + (1 − m)∞ )) is injective. As a consequence, any element in HomO X (E, E(K X + ∞ )) restricts to zero in a formal neighborhood of ∞ in X , and we conclude that HomO X (E, E(K X + ∞ )) = 0.

Corollary 2.3. Let X be a non-singular projective surface, and let ∞ be a smooth divisor of X such that ∞ · ∞ > 0. Then for any (E, ) in Mr,d,n (X, ∞ ), dimC Ext 1O X (E, E(−∞ )) − dimC Ext 2O X (E, E(−∞ )) = 2r n + (1 − r )d · d − r 2 ( pa (X ) + pa (∞ )), where d · d = genus of ∞ .

X

d 2 , pa (X ) is the arithmetic genus of X , and pa (∞ ) is the arithmetic

Proof. Let (E, ) ∈ Mr,d,n (X, ∞ ) be locally free. By Proposition 2.1 (a), dimC Ext 1O X (E, E(−∞ ))−dimC Ext 2O X (E, E(−∞ )) = −χ (End(E) ⊗ O X (−∞ )). Let ν ∈ H 4 (X ; Z) be the Poincaré dual of [pt] ∈ H0 (X ; Z), and let e ∈ H 2 (X ; Z) be the Poincaré dual of [∞ ] ∈ H2 (X ; Z). By Hirzebruch-Riemann-Roch, χ (End(E) ⊗ O X (−∞ )) = deg (ch(End(E))ch(O X (−∞ ))td(TX )) , where ch(End(E)) = ch(E)ch(E ∨ ) = r 2 + (r − 1)d 2 − 2r nν, k e2 = 1 − e + ν for k = ∞ · ∞ > 0, ch(O X (−∞ )) = 1 − e + 2 2 1 1 td(TX ) = 1 + c1 (X ) + (c1 (X )2 + c2 (X )). 2 12 Let N∞ / X be the normal bundle of ∞ in X . Then

c1 (∞ ) + c1 (N∞ / X ) = 2 − 2 pa (∞ ) + k. ec1 (X ) = X

∞

668


Consequently, deg (ch(End(E))ch(O X (−∞ ))td(TX )) 2 r r2 kr 2 = (c1 (X )2 + c2 (X )) − ec1 (X ) + (r − 1)d 2 + ( − 2r n)ν 2 2 X 12 2 2 r r kr 2 = − 2r n (c1 (X )2 + c2 (X )) − (k + 2 − 2 pa (∞ )) + (r − 1) d2 + 12 X 2 2 X = −2r n + (r − 1) d 2 + r 2 ( pa (X ) + pa (∞ )). X

Corollary 2.4. Let X be a non-singular projective rational surface, and let ∞ be a divisor of X such that ∞ ∼ = P1 and ∞ · ∞ > 0. Then Mr,d,n (X, ∞ ) is smooth of (complex) dimension where d · d =

2r n + (1 − r )d · d,

X

d 2.

Example 2.5. Let X = P2 , and let ∞ = {[Z 0 , Z 1 , Z 2 ] ∈ P2 | Z 0 = 0} ∼ = P1 .

Then ∞ ·∞ = 1 > 0. The moduli space Mr,d,n (P2 , ∞ ) is nonempty only if ∞ d = 0, which implies d = 0. By Corollary 2.4, the moduli space Mr,0,n (P2 , ∞ ) is smooth of complex dimension 2r n. (See [NY1, Cor. 2.2]). def

Example 2.6. Let X = Fk = P(OP1 (−k) ⊕ OP1 ) be the k th Hirzebruch surface, where k is a positive integer. Let 0 = P(0 ⊕ OP1 ) ∼ = P1 , ∞ = P(OP1 (−k) ⊕ 0) ∼ = P1 . Then 0 · 0 = −k < 0 and ∞ · ∞ = k > 0. The moduli space Mr,d,n (Fk , ∞ ) is nonempty only if ∞ d = 0, which implies d = m0 for some m ∈ Z. By Corollary 2.4, the moduli space Mr,m0 ,n (Fk , ∞ ) is smooth of complex dimension 2r n + (r − 1)km 2 . Example 2.7. Let ⊂ P2 be a curve of degree 1, and let p1 , . . . , pk be k generic points in P2 which are disjoint from . Let π : Bk → P2 be the blowup of P2 at p1 , . . . , pk . Let ∞ = π −1 () ∼ = P1 , and let i = π −1 ( pi ) be the exceptional divi2 sors. Let e∞ , e1 , . . . , ek ∈ H (Bk ; Z) be the Poincaré duals of [∞ ], [1 ], . . . , [k ], respectively. Then H 2 (Bk ; Z) = Ze∞ ⊕ Ze1 ⊕ · · · Zek . The moduli space Mr,d,n (Bk , ∞ ) is nonempty only if ∞ d = 0, which implies d = m 1 e1 + · · · + m k ek , m i ∈ Z. By Corollary 2.4, the moduli space Mr,m 1 e1 +···+m k ek ,n (Bk , ∞ ) is smooth of complex dimension 2r n + (r − 1)(m 21 + · · · + m 2k ).


669

2.2. The natural bundle. In this subsection, X is a non-singular projective rational surface, and ∞ is a smooth rational curve in X such that ∞ · ∞ > 0. The proof of the following proposition is very similar to that of Proposition 2.1. Proposition 2.8. H 0 (X, E(−∞ )) = H 2 (X, E(−∞ )) = 0. Let E → X × Mr,d,n (X, ∞ ) be the universal sheaf. Let p1 : X × Mr,d,n (X, ∞ ) → X and p2 : X × Mr,d,n (X, ∞ ) → Mr,d,n (X, ∞ ) be the projections to the two factors. Definition 2.9. The natural bundle over Mr,d,n (X, ∞ ) is V = (R 1 p2 )∗ (E ⊗ p1∗ (O X (−∞ ))). def

Corollary 2.10. V is a vector bundle of rank 1 n − (d · d + c1 (X ) · d) 2 over Mr,d,n (X, ∞ ). Proof. We use the notation in the proof of Corollary 2.4. Let (E, ) ∈ Mr,d,n (X, ∞ ) be locally free. The rank of V is given by −χ (E(−∞ )). By Hirzebruch-Riemann-Roch, χ (E(−∞ )) = deg (ch(E)ch(O X (−∞ ))td(TX )) , where d2 k e2 − nν), ch(O X (−∞ )) = 1 − e + = 1 − e + ν, 2 2 2 1 1 2 td(TX ) = 1 + c1 (X ) + (c1 (X ) + c2 (X )). 2 12 ch(E) = r + d + (

Consequently, deg (ch(E)ch(O X (−∞ ))td(TX )) r kr 1 d2 2 = (c1 (X ) + c2 (X )) + (d − r e)c1 (X ) + + ( − n)ν 2 2 2 X 12 1 r r kr −n = (c1 (X )2 + c2 (X )) − (k + 2) + (d 2 + c1 (X )d) + 12 X 2 2 X 2 1 = −n + (d 2 + c1 (X )d) + r pa (X ), 2 X where pa (X ) = 0 since X is a rational surface.

3. Torus Action and Fixed Points def

In this section, X is a non-singular projective toric surface. Therefore Tt = (C∗ )2 acts on X . We use notation similar to that in [NY1, Sect. 2, 3].

670


3.1. Torus action on the surface. We assume that ∞ is a Tt -invariant P1 in X , and ∞ · ∞ = k > 0. Then X 0 = X \ ∞ is a non-singular, quasi-projective toric surface. Let be a graph such that the vertices of are in one-to-one correspondence with the Tt fixed points in X 0 , and two vertices are connected by an edge if and only if the corresponding fixed points are connected by a Tt -invariant P1 . Then is a chain, so #V ( ) − #E( ) = 1, and χ (X 0 ) = #V ( ) = χ (X ) − 2, where E( ) is the set of edges in and V ( ) is the set of vertices in . Let pv be the Tt fixed point in X 0 which corresponds to v ∈ V ( ), and let e be the Tt -invariant P1 which corresponds to e ∈ E( ). Any Tt -invariant divisor D in X disjoint from ∞ is of the form D= m e e ∼ = H2 (X 0 ; Z), e∈E( )

where m e ∈ Z. 3.2. Torus action on moduli spaces. Let Te be the maximal torus of G L(r, C) consisting of diagonal matrices, and let T˜ = Tt × Te . We define an action of T˜ on Mr,d,n (X, ∞ ) as follows: for (t1 , t2 ) ∈ Tt , let Ft1 ,t2 be the automorphism of X defined by Ft1 ,t2 (x) = (t1 , t2 ) · x. Given e = diag(e1 , . . . , er ) ∈ Te , let G e denote the isomorphism of O⊕r ∞ given by (s1 , . . . , sr ) → (e1 s1 , . . . , er sr ). For (E, ) ∈ Mr,d,n (X, ∞ ), we define

∗ (t1 , t2 , e) · (E, ) = (Ft−1 , ) E,

,t 1 2 where is the composite of homomorphisms (Ft−1 )∗

1 ,t2

φt1 ,t2

G e

)∗ E|∞ −−−−−→ (Ft−1 )∗ O⊕r −−−→ O⊕r −→ O⊕r . (Ft−1 1 ,t2 1 ,t2 ∞ ∞ ∞ Here φt1 ,t2 is the homomorphism given by the action. ˜

3.3. Torus fixed points in moduli spaces. The fixed points set Mr,d,n (X, ∞ )T consists of (E, ) = (I1 (D1 ), 1 ) ⊕ · · · ⊕ (I2 (Dr ), r ) such that (1) Iα (Dα ) is a tensor product Iα ⊗ O X (Dα ), where Dα is a Tt -invariant divisor which does not intersect ∞ , and Iα is the ideal sheaf of a 0-dimensional subscheme Q α contained in X 0 . (2) α is an isomorphism from (Iα )∞ to the α th factor of O⊕r . ∞ (3) Iα is fixed by the action of Tt . The support of Q α must be contained in X 0Tt , the Tt fixed points set of X 0 . Thus Q α is a union of {Q vα | v ∈ V ( )}, where Q vα is a subscheme supported at the Tt - fixed point pv ∈ X 0 . If we take a coordinate system (x, y) around pv , then the ideal of Q vα is generated by monomials xi y j , So Q vα corresponds to a Young diagram Yαv . Therefore the fixed point set is parametrized by 2r -tuples (D, Y) = (D1 , Y1 , . . . , Dr , Yr ),


671

where Dα ∈

Ze ∼ = H2 (X 0 ; Z), Yα = {Yαv | v ∈ V ( )},

e∈E( )

and each Yαv is a Young diagram. Let |Yα | =

v∈V ( )

|Yαv |.

Let d ∨ ∈ H2 (X ; Z) be the Poincaré dual of d ∈ H 2 (X ; Z). Then ∨ d ∈ e∈E( ) Z[e ]. The constraints are α r

2r

r α=1

α 2, but a semistrict Lie n-algebra is just an n-term chain complex equipped with the structure of an L ∞ -algebra. So, it would be easiest to start by considering a generalization of the semi-bracket, and see if this can be used to construct a semistrict Lie n-algebra. – Does the Lie 2-algebra of observables in 2-plectic geometry extend to something like a Poisson algebra? It is far from clear how to define a product for Hamiltonian 1-forms, and the usual product of a Hamiltonian 1-form and a smooth function is not Hamiltonian. – The based loop space X of a manifold X equipped with a closed (n + 1)-form ω is an infinite-dimensional manifold equipped with a closed n-form η defined ‘by transgression’ as follows: 2π η(v1 , . . . , vn ) = ω(γ (σ ), v1 (γ (σ )), . . . , vn (γ (σ )) dσ, 0

where vi are tangent vectors at the loop γ ∈ X and vi (γ (σ )) are the corresponding tangent vectors at the point γ (σ ) ∈ X . Even when ω is n-plectic, η is rarely (n − 1)-plectic. However when X = G is a compact simple Lie group equipped with the 2-plectic structure of Example 1, η becomes symplectic after adding an exact form. The interplay between the 2-plectic structure on G and the symplectic structure on G plays an important role in the theory relating the Wess–Zumino– Witten model, central extensions of the loop group G, gerbes on G and the string 2-groups Stringk (G) [6]. It would be nice to have a more general theory whereby the loop space of an n-plectic manifold became an (n − 1)-plectic manifold. – When a symplectic structure ω on a manifold X defines an integral class in H 2 (X, R), there is a U(1) bundle over X equipped with a connection whose curvature is ω. As mentioned in the Introduction, this plays a fundamental role in the geometric quantization of X . Similarly, when a 2-plectic structure ω on a manifold X defines an integral class in H 3 (X, R), there is a U(1) gerbe over X equipped with a connection whose curvature is ω [11]. Is there an analogue of geometric quantization that applies in this case? Following the ideas of Freed [17], we might hope that geometrically quantizing this gerbe will give a ‘2-Hilbert space’ of states. However, Freed’s work only treats Schrödinger quantization, and that only in the special case where the resulting 2-Hilbert space is finite-dimensional. Finite-dimensional 2-Hilbert spaces are by

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J. C. Baez, A. E. Hoffnung, C. L. Rogers

now well-understood [3], but the infinite-dimensional ones are still being developed [4,36]. Geometric quantization for gerbes is an even greater challenge. However, we expect the problem of geometrically quantizing a U(1) gerbe on X to be closely related to the better-understood problem of geometrically quantizing the corresponding U(1) bundle on the loop space of X . Acknowledgements. We thank Urs Schreiber, Allen Knutson and Dmitry Roytenberg for corrections and helpful conversations. This work was partially supported by a grant from the Foundational Questions Institute. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Communications in


Q-Systems, Heaps, Paths and Cluster Positivity Philippe Di Francesco1 , Rinat Kedem2 1 Institut de Physique Théorique du Commissariat à l’Energie Atomique,

Unité de Recherche Associée du CNRS, CEA Saclay/IPhT/Bat 774, F-91191 Gif sur Yvette Cedex, France. E-mail: [email protected]

2 Department of Mathematics, University of Illinois MC-382, Urbana,

IL 61821, U.S.A. E-mail: [email protected]; [email protected] Received: 28 November 2008 / Revised: 28 July 2009 / Accepted: 16 September 2009 Published online: 6 November 2009 – © Springer-Verlag 2009

Abstract: We consider the cluster algebra associated to the Q-system for Ar as a tool for relating Q-system solutions to all possible sets of initial data. Considered as a discrete integrable dynamical system, we show that the conserved quantities are partition functions of hard particles on certain weighted graphs determined by the choice of initial data. This allows us to interpret the solutions of the system as partition functions of Viennot’s heaps on these graphs, or as partition functions of weighted paths on dual graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the Q-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the Ar Q-system. We also give the relation to domino tilings of deformed Aztec diamonds with defects. Contents 1.

2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Q-system . . . . . . . . . . . . . . . . . . . . . . . 1.2 Q-systems as cluster algebras . . . . . . . . . . . . . . . 1.3 Outline of the paper . . . . . . . . . . . . . . . . . . . . Properties of the Q-System . . . . . . . . . . . . . . . . . . 2.1 The fundamental domain of seeds . . . . . . . . . . . . 2.2 Discrete Wronskians . . . . . . . . . . . . . . . . . . . Conserved Quantities and Hard Particles . . . . . . . . . . . 3.1 Conserved quantities of the Q-system . . . . . . . . . . 3.2 Recursion relations for discrete Wronskians with defects 3.3 Conserved quantities as hard particle partition functions . 3.4 Generating functions . . . . . . . . . . . . . . . . . . .

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4.

P. Di Francesco, R. Kedem

Positivity: A Heap Interpretation . . . . . . . . . . . . . . . . . . . . . . . 4.1 Heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Positivity from heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Continued fraction expressions for generating functions . . . . . . . . . 4.4 Continued fraction rearrangements . . . . . . . . . . . . . . . . . . . . 5. Path Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . 5.1 From heaps on G r to paths on G 5.2 Motzkin paths and path graphs . . . . . . . . . . . . . . . . . . . . . . 5.3 From Motzkin paths to heap graphs . . . . . . . . . . . . . . . . . . . . 6. Path Interpretation of R1,n and Cluster Mutations . . . . . . . . . . . . . . . 6.1 Path partition functions in terms of transfer matrices . . . . . . . . . . . 6.2 Reduction and continued fractions . . . . . . . . . . . . . . . . . . . . 6.3 Mutations as fraction rearrangements . . . . . . . . . . . . . . . . . . . 6.4 Weights and the mutation matrix B . . . . . . . . . . . . . . . . . . . . 6.5 Positivity of the cluster variables R1,n . . . . . . . . . . . . . . . . . . 7. Strongly Non-Intersecting Path Interpretation of Rα,n . . . . . . . . . . . . 7.1 Ascending Motzkin paths: Paths on trees T2r +2 (I ) . . . . . . . . . . . . 7.2 Strongly non-intersecting paths on M . . . . . . . . . . . . . . . . . . 7.3 Positivity of Rα,n for all α, n . . . . . . . . . . . . . . . . . . . . . . . 8. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The limit A∞/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Numbers of configurations . . . . . . . . . . . . . . . . . . . . . . . . 9. The Relation to Domino Tilings . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Paths and matchings of the Aztec diamond . . . . . . . . . . . . . . . . 9.2 Lattice paths and domino tilings . . . . . . . . . . . . . . . . . . . . . 9.3 Q-system solutions as functions of x0 and domain tilings . . . . . . . . 9.4 Solutions of the Q-system as functions of xM and tilings of domains with defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. The Case of A3 in the Heap Formulation . . . . . . . . . . . . . . . Appendix B. The Case of A3 in the Path Formulation . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The Ar Q-system is a recursion relation satisfied by characters of special irreducible finite-dimensional modules of the Lie algebra Ar [15]. It is a discrete integrable dynamical system. On the other hand, the relations of the Q-system are mutations in a cluster algebra defined in [14]. One of the goals of this paper is to prove the positivity property of the corresponding cluster variables, by using the integrability property. We do this by solving the discrete integrable system, which can be mapped to several different types of statistical models: path models, heaps on graphs, or domino tilings. The choice initial conditions for the recursion relations determines the specific model, one for each choice of initial data. The Boltzmann weights are Laurent monomials in the initial data. Our construction gives an explicit solution of the Q-system as a function of any possible set of initial conditions. It is a conjecture of Fomin and Zelevinsky [9] that the cluster variables at any seed x of a cluster algebra, expressed as a function of the cluster variables of any other seed y of the algebra, are Laurent polynomials with non-negative coefficients. In the language

Q-Systems, Heaps, Paths and Cluster Positivity

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of dynamical systems, the choice of seed variables y corresponds to the choice of initial conditions. The methods used in this paper appear to be new in the context of the positivity conjecture. They have the advantage that they give explicit solutions, and extend immediately to T -systems [6] and to certain integrable non-commutative cluster algebras introduced by Kontsevich [7,16].

1.1. The Q-system. First, let us recall some definitions. Let Ir = {1, . . . , r }, and consider the family of commutative variables {Q α,n : α ∈ Ir , n ∈ Z}, related by a recursion relation of the form: Q α,n+1 Q α,n−1 = Q 2α,n − Q α+1,n Q α−1,n ,

Q 0,n = Q r +1,n = 1, (n ∈ Z, α ∈ Ir ). (1.1)

A solution of this system is specified by giving a set of boundary conditions. The original Ar Q-system [15] is the recursion relation (1.1) (where n ≥ 1), together with the boundary conditions Q α,0 = 1,

Q α,1 = chV (ωα ), α ∈ Ir .

(1.2)

Here, V (ωα ) is one of the r fundamental representations of the Lie algebra slr +1 . The solutions, written as functions of the initial variables, are the characters of the Ar Kirillov-Reshetikhin modules [15]: Q α,n = chV (nωα ). The boundary conditions (1.2) are singular, in that Q α,−1 = 0. All solutions are therefore polynomials in {Q α,1 : α ∈ Ir } (Lemma 4.2 of [4]). In this paper, we relax this boundary condition. Moreover, as we are interested in the positivity property of cluster algebras, we renormalize the variables in (1.1).1 Let Rα,n = α Q α,n ,

α = eiπ

r

β=1 (C

−1 )

α,β

= eiπ α(r +1−α)/2 ,

(1.3)

where C is the Cartan matrix of Ar . Then 2 + Rα+1,n Rα−1,n , Rα,n+1 Rα,n−1 = Rα,n

R0,n = Rr +1,n = 1, (α ∈ Ir , n ∈ Z). (1.4)

In the rest of the paper we work only with the renormalized variables in (1.3), and henceforth we refer to Eq. 1.4 as the Ar Q-system. 1 Alternatively, we can introduce coefficients in the second term (see Appendix A in [4]). The two approaches are equivalent in this case.

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1.2. Q-systems as cluster algebras. Instead of specifying the boundary conditions of the form (1.2), we may specify much more general boundary conditions by picking a set of 2r variables in a consistent manner, and setting them to be formal variables. Any solution is then a function of these formal variables. In order to explain what we mean by “a consistent manner”, we can use the formulation of these recursion relations as mutations in a cluster algebra. It was shown in [14] that the system (1.4) can be expressed in terms of a cluster algebra [9]: Theorem 1.1 [14]. Equations (1.4) for each α and n are mutations in the cluster algebra without coefficients, of rank 2r , defined by the seed (x0 , B0 ), where x0 is a cluster variable and B0 is an exchange matrix, with 0 −C x0 = (R1,0 , . . . , Rr,0 ; R1,1 , . . . , Rr,1 ), B0 = , (1.5) C 0 where C is the Cartan matrix of Ar . A cluster algebra of rank n without coefficients is defined as follows [9]. Consider an n-regular tree Tn , with nodes connected via labeled edges. Each node is attached to n edges with distinct labels 1, . . . , n. To each node t, we associate a seed consisting of a cluster variable and an exchange matrix. The cluster variable xt has n components, xt = (xt,0 , . . . , xt,n ), and the exchange matrix Bt is an n × n skew-symmetric matrix. Cluster variables at connected nodes are related by mutations given by the exchange matrix. Suppose node t is connected to node t by an edge labeled k, then the seeds at these nodes are related by a mutation µk . The effect of this mutation is as follows: [(Bt )i j ]+ n [−(Bt )i j ]+ n ,k= j xt,−1j x + x i=1 t,i i=1 t,i µk (xt, j ) = xt , j = xt, j otherwise. Here, [n]+ = max(0, n). The matrices Bt and Bt are related via the mutation µk as follows. if i = k or j = k, −(Bt )i j µk ((Bt ))i j = (Bt )i j = (Bt )i j + sign((Bt )ik )[(Bt )ik (Bt )k j ]+ otherwise. The cluster algebra is the commutative algebra over Q of the cluster variables. In the case of the cluster algebra defined in Theorem 1.1, there is a subgraph Gr of T2r , which includes the node with seed (1.5), and is the maximal subgraph with the property that the mutation of the cluster variables along each edge of the graph is one of Eqs. (1.4). The union of the cluster variables over all nodes of the graph is the set {Rα,n : α ∈ Ir , n ∈ Z}.2 To describe the seeds in Gr , note that the mutation of the variable Rα,n−1 into Rα,n+1 uses the variables {Rα,n−1 , Rα−1,n , Rα,n , Rα+1,n }. If these are entries of the cluster variable xt at node t, then it is possible to apply the mutation t → t which sends Rα,n−1 → Rα,n+1 . With our indexing convention, this is the mutation µα if n is odd, µα+r if n is even. Such a mutation will be called a “forward” mutation, as it increases the value of the index n − 1 → n + 1. (Cluster mutations are involutions, hence we also have “backward” mutations which send n + 1 → n − 1.) 2 Note that this graph is different from the graphs described in [4,14], which were minimal rather than maximal.


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Fig. 1. The subgraph of Q-system mutations for A3 . The blue dot stands for the node 0

We see that any cluster variable in Gr has the properties: (i) The variable consists of pairs of the form (Rα,m α , Rα,m α +1 ), α ∈ Ir and (ii) m α−1 − 1 ≤ m α ≤ m α−1 + 1 for all α ∈ Ir . Define a path on the two-dimensional lattice in the (n, α)-plane, by connecting the points M = {(m α , α)}rα=1 . Then condition (ii) implies that M is a Motzkin path, with steps of type (1, 1), (−1, 1) and (0, 1). Therefore we have Lemma 1.2. The nodes of Gr are in bijection with Motzkin paths on the square lattice with r − 1 steps of the form (1, 1), (−1, 1) or (0, 1), connecting vertices with integer coordinates (n, α) such that 1 ≤ α ≤ r . We define xM to be the seed with the variables {Rα,m α +i : i = 0, 1, α ∈ Ir }. Let M0 denote the Motzkin path with m α = 0, α ∈ Ir . Then x0 = xM0 . Forward mutations µα and µα+r act on Motzkin paths by increasing or decreasing one of the indices m α . If the resulting path is also a Motzkin path, such mutations are guaranteed to be of the type (1.4). The graph Gr is an infinite strip, and we display a section of it for r = 3 in Fig. 1, where we have identified the nodes sharing the same cluster variables. Most of the results of this paper can be reduced to working with the fundamental domain of this strip. 1.3. Outline of the paper. The Laurent property [10] guarantees that the cluster variables are Laurent polynomials when expressed as functions of the cluster variables of any seed in the cluster algebra. It is a general conjecture [9] that the coefficients of the monomials in these Laurent polynomials are non-negative integers. In this paper, we prove positivity for the cluster variables in Gr . The Q-system is a way of relating the solutions of (1.4) to all possible sets of initial data. Mutations of the cluster algebra allow us to move within the set of possible initial data, as long as this initial data is of the form xM . Our aim is to give an explicit combinatorial description of Rα,n , for each choice of initial data xM within this set, as the partition function for a statistical model with positive Boltzmann weights. We proceed using the following steps: Step 1. We give a combinatorial description of {R1,n , n ∈ Z} as functions of the initial data x0 . All other variables {Rα,n , α > 1} are discrete Wronskians of {R1,n+ j , 1 − α ≤ j ≤ α − 1}. The variables {R1,n , n ∈ Z} satisfy a linear recursion relation with constant

732


coefficients, which are the conserved quantities of the Q-system. Here, the integrability of the system plays a crucial role. The conserved quantities are partition functions of hard particles on a weighted graph G r , with weights which are Laurent monomials in the cluster variables x0 . Therefore (r ) n the generating function F1 (t) = n≥0 t R1,n is equal to the partition function of (r )

Viennot’s heaps on G r . Thus, there is a simple expression for F1 (t) as a finite continued fraction. (r ) Elementary rearrangements of the continued fraction allow us to express F1 (t) as the generating function for heaps on different graphs with different weights. Our goal is then to prove that these rearrangements are mutations of the initial data xM . Step 2. We use a heap-path correspondence to re-express {R1,n (x0 )}n∈Z+ as partition r . functions for weighted paths on a dual graph G Step 3. For each Motzkin path M, we construct a graph M and edge weights ye (M) which are Laurent monomials in the cluster variables xM . The variables {R1,n }n∈Z+ are expressed as the generating functions for weighted paths on M . This proves the positivity conjecture for R1,n and the claim that continued fraction rearrangements are cluster mutations of the initial data. Step 4. We use the discrete Wronskian expression for Rα,n to interpret these variables as partition functions for families of strongly non-intersecting paths on M . This is done by a generalization of the Lindström-Gessel-Viennot [11,19] determinant formula for the counting of non-intersecting lattice paths. This implies the positivity property for Rα,n as functions of xM . The paper is organized as follows. In Sect. 2, we show that the Q-system amounts to a discrete Wronskian equation for the R1,n ’s, and that Rα,n with α > 1 are expressed as α × α discrete Wronskians of R1,n . We deduce that R1,n satisfies a linear recursion relation with constant coefficients, interpreted as the conserved quantities of the Q-system. In Sect. 3, the conserved quantities are interpreted as partition functions for hard particles on a certain target graph G r , with vertex weights which depend on the initial data x0 of (1.5). We rephrase the linear recursion relation for R1,n in terms of generating functions which are simple rational functions. In Sect. 4 we re-interpret these rational functions as partition functions for Viennot’s heaps [24] on the same target graph G r . This gives a first proof of the positivity of R1,n as a Laurent polynomial of the initial data x0 (1.5). We also give an explicit expression for the generating function of R1,n ’s, as a finite continued fraction. We show how a simple rearrangement lemma for fractions allows, by iterative use, to rewrite the partition function for heaps on G r as a partition function for heaps on other graphs, with weights accordingly transformed. The proof that these rearrangements correspond to mutations of the cluster variables appears in Sect. 6. In Sect. 5 we reformulate the heap partition function in terms of weighted paths on “dual” target graphs, the language of which is more amenable to mutations. For each Motzkin path, we construct a corresponding graph with edge weights which are monomials in xM . Partition functions for paths on these graphs are the cluster variables R1,n . For completeness, we also explain how to construct the dual graphs for the heap models from these graphs. In Sect. 6 we prove that R1,n is the partition function for weighted paths on a target graph M with weights determined by the initial data xM . Using the transfer matrix


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formulation, we prove the statement that fraction rearrangements correspond to mutations of the initial data. This leads to the main positivity theorem for the cluster variables R1,n , in terms of any initial data xM . The extension of this result to Rα,n , α > 1 is presented in Sect. 7, where Rα,n is interpreted à la Lindström-Gessel-Viennot as the partition function for families of strongly non-intersecting weighted paths on the same target graph as for R1,n . This completes the statistical-mechanical interpretation of all the solutions of the Q-system, and proves the positivity of all the cluster variables of the subgraph corresponding to the Q-system, when expressed in terms of cluster variables at any of its nodes. In Sect. 8 we discuss the limiting case of A∞/2 and present various exact and asymptotic path enumeration results, which correspond to picking initial data xM with entries all equal to 1. Finally, since the Q-system is a limit of the T -system, aka the octahedron equation, in Sect. 9 we give the relation of our results to the known results on the octahedron equation. We show how to relate weighted domino tilings of the Aztec diamond to our weighted non-intersecting families of lattice paths, and to interpret the result of cluster mutations in that language as weighted tilings of suitably deformed Aztec diamonds, by means of dominos and also pairs of square “defects”. The appendices include explicit examples of our constructions for the case A3 . 2. Properties of the Q-System 2.1. The fundamental domain of seeds. Our goal is to give explicit expressions for Rα,n as a function of any initial seed data xM . First, we use the symmetries of the Q-system to enable us to restrict our attention to a finite fundamental domain of initial seeds, parametrized by Motzkin paths which have a minimum node at 0. There are three obvious symmetries of the system. A symmetry σ : {Rα,n } → {Rα,n } of (1.4) is a map with the property that Rα,n = f (x), then σ (Rα,n ) = f (σ (x)). Equation (1.4) is invariant under σ (Rα,n ) = Rα,−n+1 . Therefore, Lemma 2.1. “Time reversal”: if Rα,n = f (R1,0 , R2,0 , . . . , Rr,0 ; R1,1 , R2,1 , . . . , Rr,1 ), then Rα,−n+1 = f (R1,1 , R2,1 , . . . , Rr,1 ; R1,0 , R2,0 , . . . , Rr,0 ),

(2.1)

for all n ∈ Z and α ∈ Ir . We also have the reflection symmetry of (1.4), σ (Rα,n ) = Rr +1−α,n . Therefore, Lemma 2.2. if Rα,n = f (R1,0 , R2,0 , . . . , Rr,0 ; R1,1 , R2,1 , . . . , Rr,1 ), then Rr +1−α,n = f (Rr,0 , Rr −1,0 , . . . , R1,0 ; Rr,1 , Rr −1,1 , . . . , R1,1 ),

(2.2)

for all n ∈ Z and α ∈ Ir . Finally, we have the translational invariance, σ (Rα,n ) = Rα,n+k : Lemma 2.3. if Rα,n = f (R1,0 , R2,0 , . . . , Rr,0 ; R1,1 , R2,1 , . . . , Rr,1 ), then Rα,n+k = f (R1,k , R2,k , . . . , Rr,k ; R1,k+1 , R2,k+1 , . . . , Rr,k+1 ), for all n, k ∈ Z and α ∈ Ir .

(2.3)

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Fig. 2. The Motzkin paths for the 9 seeds in the fundamental domain for the A3 case, and the mutations relating them. The leftmost vertices of each Motzkin path lie on the vertical axis x = 0

This lemma may also be viewed as a special case of the substitution property of deformed Q-systems defined in [3]. (M) For any Motzkin path M = (m 1 , . . . , m r ), let f α,n (x) denote the function of x such that (M) Rα,n = f α,n (xM ).

Let M + k = (m 1 + k, . . . , m r + k). Then Eq. (2.3) can be written as (M )

(M0 ) 0 Rα,n = f α,n (xM0 ) = f α,n−k (xM0 +k ).

(2.4)

(M) (M0 ) (M0 ) (M) Rα,n+k = f α,n+k (xM ) = f α,n+k (xM0 ) = f α,n (xM0 +k ) = f α,n (xM+k ).

(2.5)

More generally,

Therefore, (M)

(M)

Theorem 2.4. Let Rα,n = f α,n (xM ), where f α,n is a positive Laurent polynomial of xM , α ∈ Ir , n ∈ Z. Then as a function of xM+k , k ∈ Z, Rα,n is also a positive Laurent (M+k) polynomial, Rα,n = f α,n (xM+k ). We can thus restrict our attention to the fundamental domain: Definition 2.5. The fundamental domain Fr for the Ar Q-system is indexed by the Motzkin paths with r − 1 steps of the form {(m α , α)}α∈Ir , with Minα∈Ir (m α ) = 0. There are exactly 3r −1 Motzkin paths in the fundamental domain. As an example, Fig. 2 shows the 9 Motzkin paths of the fundamental domain for A3 and the mutations relating them, c.f. the graph G3 in Fig. 1.


735 j ,..., j

2.2. Discrete Wronskians. For any matrix M, let Mi11,...,ikm be the matrix obtained by it by removing rows i 1 , . . . , rk and columns j1 , . . . , jm . 2.2.1. Plücker relations Let T be an n × (n + k)-matrix. The Plücker relations for the minors of T are |T a1 ,...,ak | |T b1 ,...,bk | =

k

|T b p ,a2 ,...,ak | |T b1 ,...,b p−1 ,a1 ,b p+1 ,...,bk |.

(2.6)

p=1

In particular, when k = 2, |T a1 ,a2 | |T b1 ,b2 | = |T b1 ,a2 | |T a1 ,b2 | + |T b2 ,a2 | |T b1 ,a1 |.

(2.7)

Let n = r + 1, a2 = j1 , b2 = j2 , and (T )i,a1 = δi,i1 , (T )i,b1 = δi,i2 . Then Eq. (2.7) gives the Desnanot-Jacobi formula for the minors of the matrix M = T a1 ,b1 of size (r + 1) × (r + 1): j ,j

j

j

j

j

|M| |Mi11,i22 | = |Mi11 | |Mi22 | − |Mi12 | |Mi21 |.

(2.8)

2.2.2. Wronskian formula for R1,n Using (1.4), it is possible to eliminate the variables {Rα,n }α>1 in favor of {R1,n }. The remaining equations determine {R1,n } in terms of the initial data. As a consequence, R1,n satisfies a linear recursion relation with constant coefficients. This can then be extended trivially to all Rα,n . Define the α × α matrix Mα,n with (Mα,n )i, j = R1,n+i+ j−1−α . That is, ⎞ ⎛ R1,n R1,n−α+1 R1,n−α+2 · · · R1,n+1 ⎟ ⎜ R1,n−α+2 R1,n−α+3 · · · ⎟, (2.9) Mα,n = ⎜ . . .. . ⎠ ⎝ .. .. .. . R1,n

R1,n+1

···

R1,n+α−1

and define the discrete Wronskian determinant to be Wα,n = |Mα,n |. Lemma 2.6. We have Rα,n = Wα,n . Proof. Applying the Desnanot-Jacobi formula (2.8) to the (α + 1) × (α + 1) matrix M with entries (M)i, j = R1,n+i+ j−α−2 with the choice of rows i 1 = 1, i 2 = α + 1, and columns j1 = 1, j2 = α + 1, we have 2 + Wα+1,n Wα−1,n Wα,n+1 Wα,n−1 = Wα,n

(2.10)

for any sequence R1,n , and any α ≥ 1, with the convention that W0,n = 1 for all n. The sequence Wα,n is the unique solution to eq.(2.10) such that W0,n = 1 and W1,n = R1,n . Comparing this to the Q-system (1.4), we deduce that Rα,n = Wα,n , α = 1, . . . , r , and the lemma follows. The boundary condition Rr +1,n = 1 yields the following polynomial relation for R1,n : Corollary 2.7. R1,n−r +1 · · · R1,n R1,n−r R1,n−r +1 R1,n−r +2 · · · R1,n+1 (2.11) Wr +1,n = .. .. .. = 1. .. . . . . R R1,n+1 · · · R1,n+r 1,n

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2.2.3. Integrals of motion The determinant Wr +1,n is a discrete version of the Wronskian ( j−1) determinant W ( f 1 , . . . , fr ) = deti, j ( f i ). In the theory of linear differential equations, the Wronskian of r linearly independent solutions to an r th order linear differential equation is a constant. This is proved by differentiating the Wronskian and noting that a linear combination of its columns vanishes, due to the differential equation. Conversely, if the Wronskian is a (non-zero) constant (so that its columns are linearly independent), there exists a vanishing linear combination between the column vectors of its derivative, −1 ( j−1) (r ) namely f i = rj=1 a j fi , where the f ’s are a linearly independent set of solutions of these equations. Theorem 2.8. The variables {R1,n }n∈Z satisfy a linear recursion relation involving r + 2 terms: r +1

(−1)m cr +1−m R1,n+m = 0, n ∈ Z,

(2.12)

m=0

with the coefficients c0 = cr +1 = 1, and with c1 , c2 , . . . , cr some constant (independent of n) coefficients determined by the initial conditions. Proof. In analogy with the continuous situation, consider the discrete derivative Wr +1,n+1 − Wr +1,n = 0. Since Wr +1,n and Wr +1,n+1 have r identical columns ((Wr +1,n+1 )i, j = (Wr +1,n )i+1, j , i ∈ Ir ), R1,n+i+ j−r −1 − (−1)r δ j,r +1 R1,n+i−r −1 = 0. (2.13) Wr +1,n+1 − Wr +1,n = det 1≤i, j≤r +1

As a consequence, there exists a non-trivial linear combination of the columns of this difference which vanishes. From the form of the entries of these columns (in which the indices are shifted by −1 relative to each other) the coefficients of this linear combination are independent of n. 3. Conserved Quantities and Hard Particles 3.1. Conserved quantities of the Q-system. Since Wr +1,n = 1, it is a conserved quantity, i.e. it is independent of n. More generally, we claim that there are r + 1 linearly independent conserved quantities, and therefore the Q-system is a discrete integrable system in the Liouville sense. Theorem 3.1. The following polynomials ci−1 = |(Mr +2,n )rr +2−i +2 |, i = 0, . . . , r, where cr +1 = c0 = 1, are independent of n, and c0 , . . . , cr are the linearly independent conserved quantities of the Ar Q-system. Proof. This follows from the fact that Wr +2,n = 0, as a consequence of the boundary condition Wr +1,n = 1 and the Q-system relation. The conserved quantities are the minors of the expansion of this determinant with respect to the last row, as in Eq. (2.12). We get only r + 1 linearly independent minors, since c0 = Wr +1,k−1 = 1 = Wr +1,k = cr +1 .


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Example 3.2. For r = 1, we have R R1,1 1 R1,0 R c1 = 1,k−2 1,k = + + . R1,k−1 R1,k+1 R1,0 R1,0 R1,1 R1,1 2 Using the Q-system for A1 to eliminate R1,k+2 = (R1,k+1 + 1)/R1,k−1 and R1,k−2 = 2 (R1,k+1 + 1)/R1,k−1 , we get the conservation law:

c1 =

R1,k 1 R1,k−1 R1,1 1 R1,0 + + = + + . R1,k−1 R1,k−1 R1,k R1,k R1,0 R1,0 R1,1 R1,1

This is a two-term recursion relation in k, whereas the Q-system is a three-term recursion. The former is an explicit discrete “first integral” of the latter. Another way of understanding the conserved quantities of Theorem 3.1 is via the translational invariance of the cluster algebra, expressed in Lemma 2.3. We get the following immediate Corollary 3.3. The quantities ci , expressed in terms of the seed x0 , are conserved, namely: ci (R1,0 , . . . , Rr,0 ; R1,1 , . . . , Rr,1 ) = ci (R1,k , . . . , Rr,k ; R1,k+1 , . . . , Rr,k+1 ) (3.1) for all k ∈ Z and for i = 0, 1, 2, . . . , r . 3.2. Recursion relations for discrete Wronskians with defects. We now derive explicit relations between the ci and the initial data. It is useful to work in the context of the A∞/2 Q-system, which is obtained from the Ar system by relaxing the boundary condition Rr +1,n = 1: 2 rα,n+1rα,n−1 = rα,n + rα+1,n rα−1,n , r0,n = 1, n ∈ Z, α ≥ 1.

(3.2)

Again, as in Lemma 2.6, we have rα,n :=

det (r1,n+i+ j−1−α ).

1≤i, j≤α

Define the α × α Wronskians with a “defect” at position α − m (0 ≤ m ≤ α, n ∈ Z): j if j ≤ m sm ( j) = cα,m,n = det r1,n+i+sα−m ( j)−α−1 , , (3.3) j + 1 if j > m 1≤i, j≤α where c0,0,n = 1 for all n. Note that cα,0,n = rα,n and cα,α,n = rα,n+1 . In addition, ci = cr +1,i,k if rα,n = Rα,n , that is, when we impose the condition rr +1,n = 1. Lemma 3.4. The cα,m,n satisfy the following recursion relation: rα−1,n rα−1,n+1 cα,m,n = rα,n rα−1,n+1 cα−1,m,n + rα−1,n rα,n+1 cα−1,m−1,n + rα,n rα,n+1 cα−2,m−1,n .

(3.4)

738


Proof. Applying (2.8) to the α × α matrix M with entries Mi, j = r1,n+i+ j−α , i, j ∈ {1, . . . , α}, with i 1 = 1, i 2 = α, j1 = α − m and j2 = α, rα,n+1 cα−2,m−1,n + rα−1,n+1 cα−1,m,n = rα−1,n cα−1,m,n+1 .

(3.5)

Using this to simplify the sum of the first and last terms on the r.h.s. of (3.4), must prove that rα−1,n+1 cα,m,n = rα,n+1 cα−1,m−1,n + rα,n cα−1,m,n+1 .

(3.6)

Multiplying (3.6) by rα,n+1 , and using (3.2), rα,n+1 rα−1,n+1 cα,m,n −(rα,n+2 rα,n −rα+1,n+1 rα−1,n+1 )cα−1,m−1,n −rα,n+1 rα,n cα−1,m,n+1 = rα−1,n+1 (rα,n+1 cα,m,n +rα+1,n+1 cα−1,m−1,n )−rα,n (rα,n+2 cα−1,m−1,n +rα,n+1 cα−1,m,n+1 ) = rα,n (rα−1,n+1 cα,m,n+1 − rα,n+2 cα−1,m−1,n − rα,n+1 cα−1,m,n+1 ) = 0,

where we have used again (3.5), with α → α + 1, to simplify the second line. The last equation follows from (2.7), using the α × α + 2 matrix T with entries Ti,1 = δi,α and Ti, j = r1,n+i+ j−α−1 (2 ≤ j ≤ α + 2 and 1 ≤ i ≤ α), with a1 = 1, a2 = 2, b1 = α + 2 − m and b2 = α + 2. Then Eq. (2.7) becomes rα,n+2 cα−1,m−1,n = cα,m,n+1 rα−1,n+1 − rα,n+1 cα−1,m,n+1

which is the desired relation. Define vα,n =

rα,n , α = 1, 2, . . . , n ∈ Z. rα−1,n

Equation (3.4) can be written as cα,m,n = vα,n cα−1,m,n + vα,n+1 cα−1,m−1,n + vα,n vα,n+1 cα−2,m−1,n .

(3.7)

Together with the initial conditions c0,0,n = 1 and c1,0,n = v1,n , c1,1,n = v1,n+1 , (3.7) gives cα,m,n as a polynomial in the variables {vk,n , vk,n+1 }1≤k≤α , of total degree α. In particular, we have cα,0,n = v1,n v2,n · · · vα,n , cα,α,n = v1,n+1 v2,n+1 · · · vα,n+1 .

(3.8)

Next we introduce the quantities which we call weights, for reasons which will become clear below: y2α−1,n =

vα,n+1 rα−1,n rα,n+1 = , vα,n rα,n rα−1,n+1

y2α,n =

vα+1,n+1 rα−1,n rα+1,n+1 = , α ≥ 1. vα,n rα,n rα,n+1 (3.9)

We define Cα,m,n = Then (3.7) becomes

cα,m,n . v1,n v2,n · · · vα,n

(3.10)


739

Fig. 3. The graph G r , with 2r + 1 vertices labeled i = 1, 2, . . . , 2r + 1

Theorem 3.5. The quantities Cα,m,n of (3.10) satisfy Cα,m,n = Cα−1,m,n + y2α−1,n Cα−1,m−1,n + y2α−2,n Cα−2,m−1,n ,

(3.11)

with the y’s as in Eq. (3.9). Together with the initial condition Cα,0,n = 1, this gives {Cα,m,n } as polynomials of homogeneous degree m in yk,n ’s, with 1 ≤ k ≤ 2α − 1. Example 3.6. The first few C’s for α = 0, 1, 2, 3 read: C0,0,n C2,0,n C3,0,n C3,2,n

= = = =

1, C1,0,n = 1, C1,1,n = y1,n , 1, C2,1,n = y1,n + y2,n + y3,n , C2,2,n = y1,n y3,n , 1, C3,1,n = y1,n + y2,n + y3,n + y4,n + y5,n , y1,n y3,n + y1,n y4,n + y1,n y5,n + y2,n y5,n + y3,n y5,n , C3,3,n = y1,n y3,n y5,n .

We apply the above results to the conserved quantities of the Q-system of Theorem 3.1. We identify Rα,n = rα,n by imposing the boundary condition rr +1,n = 1 for all n. Then vr +1,n = 1/rr,n and v1,n v2,n . . . vr +1,n = 1.

(3.12)

ci = cr +1,i,k = Cr +1,i,k = Cr +1,i,0 .

(3.13)

Therefore,

In particular, we recover c0 = cr +1 = 1 from the explicit expressions for cr +1,0,n and cr +1,r +1,n of Eq. (3.8), together with (3.12). We note that Cr +1,i,k = Cr +1,i,0 are independent of k for all j = 0, 1, . . . , r + 1, in other words we have the conservation laws: c j ({yα,k }) = c j ({yα,0 }) for all k ∈ Z. The identification (3.13) gives an expression for ci in terms of the initial data x0 : By iterative use of the recursion relations of Theorem 3.5 for n = 0 and 1 ≤ α ≤ r + 1, we get expressions for Cr +1,m,0 as polynomials of homogeneous degree m in the weights yα,0 , 1 ≤ α ≤ 2r + 1. These involve only the entries of x0 . 3.3. Conserved quantities as hard particle partition functions. The recursion relations of the previous section lead directly to an interpretation of the quantities ci as partition functions of hard particles on a graph, with weights which depend only on x0 . Let G r be the graph of Fig. 3. When r = 1, G 1 is the chain with 3 vertices. To each vertex labeled i in the graph, we assign the weight yi . Let G be a graph with vertices labeled by the index set I . Definition 3.7. A hard particle configuration C on G is a subset of I such that i, j ∈ C only if there is no edge connecting vertices i and j in G.

740


Let HP(G) be the set of all hard particle configurations on G. If we assign a weight yi to each vertex i ∈ I , the weight of a configuration C is w(C) = i∈C yi . The partition function of hard particles on G is

w(C). (3.14) Z (G) ({yi }) = C∈HP(G)

If we limit the summation to the set of configurations fixed cardinality j, we have the j-particle partition function Z (G) j . In the particular case G = G r of Fig. 3, we have the partition function of j hard r) r) := Z (G (y1 , . . . , y2r +1 ). These satisfy recursion particles on G r , denoted by Z (G j j relations, coming from the structure of G r . (G r )

Theorem 3.8. The partition functions Z j (G r )

Zj

(G r −1 )

= Zj

satisfy the following recursion relations: (G

)

(G

)

+ y2r +1 Z j−1r −1 + y2r Z j−1r −2 .

(3.15)

Proof. Depending on the occupation numbers of the last two vertices, three situations may occur: (1) Vertices 2r and 2r + 1 are both vacant. This is a configuration of j hard particles on G r −1 , obtained by erasing these two vertices and their adjacent edges. (2) Vertex 2r + 1 is occupied, and hence the vertex 2r is empty. Such configurations are those of j − 1 hard particles on G r −1 . (3) The vertex 2r is occupied, and hence vertices 2r + 1, 2r − 1 and 2r − 2 are empty. Such configurations are those of j − 1 hard particles on G r −2 obtained by erasing the vertices 2r − 2, 2r − 1, 2r, 2r + 1 and their incident edges. Each of these occupation states gives rise to one of the terms on the right–hand side of the equation. Example 3.9. For r = 2, the hard particle model on G 2 has the partition function Z (G 2 ) (y1 , y2 , y3 , y4 , y5 ) = 1 + (y1 + y2 + y3 + y4 + y5 ) +(y1 y3 + y1 y4 + y2 y5 + y3 y5 + y1 y5 ) + (y1 y3 y5 ), (3.16) where the various terms correspond to the configurations depicted in Fig. 4. Theorem 3.10. The j th conserved quantity c j of the Q-system for Ar is equal to the (G ) partition function Z j r (y1,n , . . . , y2r +1,n ) for j hard particles on the graph G r , with vertex weights yi ≡ yi,n defined in Eq. (3.9), for i = 1, 2, . . . , 2r + 1, and for any choice of n ∈ Z. Proof. Equations (3.11) and (3.15) are identical upon taking yi = yi,n . Moreover the r) = initial conditions are also identical, as Z 0(G r ) = Cr +1,0,n = 1. We deduce that Z (G j Cr +1, j,n = c j , independently of n. Corollary 3.11. The conserved quantities c j can be expressed in terms x0 as the partition functions for j hard particles on G r , with vertex weights y2α−1 =

Rα−1,0 Rα,1 , 1 ≤ α ≤ r + 1, Rα,0 Rα−1,1

y2α =

Rα−1,0 Rα+1,1 , 1 ≤ α ≤ r. (3.17) Rα,0 Rα,1


741

Fig. 4. The configurations of hard particles on G 2 , with, from left to right, 0, 1, 2 or 3 particles

Example 3.12. In the case G 2 of Example 3.9, we have (with y1 y3 y5 = 1): c0 = c3 = 1, and R2,0 R1,1 R1,0 R2,1 R2,1 R1,0 + + + + , R2,1 R1,0 R2,0 R1,1 R1,0 R1,1 R2,0 R2,1 R1,0 R2,1 R2,0 R1,1 R1,1 R2,0 c2 = + + + + . R1,1 R2,0 R1,0 R2,1 R2,0 R2,1 R1,0 R1,1 c1 =

(3.18) (3.19)

The two integrals of motion of the A2 Q-system correspond to writing c1 and c2 with the substitutions Rα,i → Rα,i+n−1 , i = 0, 1, α = 1, 2. These yield a system of recursion relations involving only indices n and n − 1, as opposed to the original Q-system, which involves the indices n − 1, n and n + 1.

3.4. Generating functions. 3.4.1. A generating function for R1,n It is useful to introduce generating functions. Define

F1(r ) (t) = R1,n t n . (3.20) n≥0

Theorem 3.13. We have the relation r j j j=0 (−1) d j t (r ) F1 (t) = r +1 , j j j=0 (−1) c j t

dj =

j

R1, j−i (−1) j−i ci .

(3.21)

i=0

r +1 ∞ i j c t j . Then all terms R t (−1) Proof. Consider the product of series j i=0 1,i j=0 of order r + 1 or higher in t vanish, due to Theorem 2.8. We are left with the terms of order 0, 1, . . . , r , the term of order j being exactly (−1) j d j .

742


Example 3.14. For r = 1, we have c1 = d0 = R1,0 , d1 = c1 R1,0 − R1,1 =

2 +1 R1,0 R1,1

(1)

F1 (t) = R1,0

1−

R1,1 R1,0

+

1 R1,0 R1,1

R1,0 R1,1

+

from Example 3.2, and

, hence

1− R1,1 R1,0

R1,0 R1,1

+

+

1 R1,0 R1,1

1 R1,0 R1,1

+

R1,0 R1,1

t

t + t2

.

(3.22)

3.4.2. Generating function and hard particles Theorem 3.15. F1(r ) (t) = R1,0

Z (G r ) (0, −t y2 , −t y3 , . . . , −t y2r +1 ) Z (G r ) (−t y1 , −t y2 , −t y3 , . . . , −t y2r +1 )

(3.23)

with yi as in (3.17). Proof. In the expression (3.21), the denominator is the partition function (r ) Z (G r ) (−t y1 , . . . , −t y2r +1 ), according to Corollary 3.11. The numerator of F1 (t)/R1,0 r is j=0 (−t) j d j /R1,0 , where

dj R1, j−i = (−1) j−i ci . R1,0 R1,0 j

i=0

We proceed as for the c j . First, we relax the condition that Rr +1,n = 1, hence work with the rα,n , the solutions of (3.2). Define Dα,m,n =

m

r1,m−i (−1)m−i Cα,i,n . r1,0

(3.24)

i=0

Then d j /r1,0 = Dr +1, j,n = Dr +1, j,0 , independently of n when we impose the condition rr +1,n = 1, due to (3.13). Substituting the recursion relations (3.11) into this expression, we obtain an analogous recursion relation for Dα,m,0 : Dα,m,0 =

m

i=0

(−1)m−i

r1,m−i (y2α−1 Cα−1,i−1,0 + y2α−2 Cα−2,i−1,0 + Cα−1,i,0 ) r1,0

= y2α−1 Dα−1,m−1,0 + y2α−2 Dα−2,m−1,0 + Dα−1,m,0 , with Cα,−1,0 = 0 and yk := yk,0 . The initial values of D for α = 1 are D1,0,0 = C1,0,0 = 1 and D1,1,0 = C1,1,0 − y1 C1,0,0 = y2 + y3 . Both coincide with the values of C1,0,0 and C1,1,0 , respectively, when restricted to y1 = 0. As the recursion relation for Dα,m,0 is identical to that for Cα,m,0 | y1 =0 , we deduce that Dα,m,0 = Cα,m,0 | y1 =0 for all α, m. This relation remains true after imposing the condition (3.12). Therefore d j = R1,0 Dr +1, j,0 = R1,0 Cr +1, j,0 (0, y2 , . . . , y2r +1 ) with the y’s as in Corollary 3.11. We deduce that the numerator of F1(r ) (t)/R1,0 is equal to the denominator of (3.21), restricted to the value y1 = 0.


743

Fig. 5. A heap on the graph G 3 . Solid discs are piled up above each vertex of G 3 . Diameters are such that only adjacent vertex discs may overlap

3.4.3. Translational invariance From the translational invariance property of Lemma (r ) 2.3, we may easily deduce an invariance property for the generating function F1 (t). (r ) (r ) Let us first write F1 (t) as an explicit expression F1 (t) = ((Rα,0 )rα=1 ; (Rα,1 )rα=1 ; t) involving only the initial data x0 . Theorem 3.16. The generating function satisfies the following translation property: (R1,0 , . . . , Rr,0 ; R1,1 , . . . , Rr,1 ; t) =

k−1

R1,n t n + t k (R1,k , . . . , Rr,k ;

n=0

×R1,k+1 , . . . , Rr,k+1 ; t) for all k ≥ 0. (r ) n k n Proof. We write F1 (t) = k−1 n≥0 R1,n+k t , and apply Eq. (2.4) to all n=0 R1,n t + t R1,n+k in the second term. 4. Positivity: A Heap Interpretation We now have an expression for the generating function of R1,n (n ≥ 0) in terms of the ratio of two partition functions of hard particles. Positivity of the terms R1,n , when expressed in terms of the fundamental cluster variables x0 , follows from a theorem relating this ratio to the partition function of heaps. This interpretation also allows us to find an explicit formula for the generating function of cluster variables. 4.1. Heaps. Given a graph G, heaps on G are defined as follows (see [24] and the beautiful expository article [18]). The graph G is represented in the x y plane in R3 , and we attach half-lines parallel to the positive z-axis, originating at each vertex of G. (See Fig. 5.) The vertices of G are endowed with a partial ordering. On each half-line above vertex i, we stack an arbitrary number of discs of radius Ri and thickness a, with a hole at their center, so that they can freely slide along the

744


half-lines (gravity points in the negative z-direction). The disc radii Ri are such that the distance between any pair of adjacent vertices i, j of G is < Ri + R j , while the distance between any pair of non-adjacent vertices i, j is > Ri + R j . Thus, the order in which the various discs are stacked matters only on neighboring half-lines (i.e. with connected projections on G), but not on distant ones. For a given stack of discs, its foreground is the set of discs that touch the x y-plane. A stack is said to be admissible if it is empty or if its foreground is reduced to one disc, positioned at a vertex of smallest order. (Such configurations are also called “pyramids” in the heap jargon [24]) We will call such admissible stacks of discs heaps on G. To each heap h on G, we associate a weight W (h) = d∈h W (d), where the product extends over all discs d of the heap, and where the weight W (d) = z i if d is stacked above vertex i. The partition function for heaps on G is

(G) ({z i }) = W (h). (4.1) h heap on G

Theorem 4.1. (G) ({z i }) =

Z (G) (0, −z 2 , −z 3 , . . .) . Z (G) (−z 1 , −z 2 , −z 3 , . . .)

(4.2)

Proof. This follows from the general theory of heaps [18,24]. We write

W (h) w(c) = Z (G) (0, −z 2 , −z 3 , . . .) (G) ({z i }) Z (G) (−z 1 , −z 2 , −z 3 , . . .) = (h,c)

=

w(c ),

(4.3)

c with vertex 1 empty

where the sums extend over pairs (h, c) made of a heap h and a configuration c of hard particles on G, and configurations c of hard particles on G such that the vertex 1 remains unoccupied. We define an involution ϕ between pairs (h, c) which reverses the sign of W (h) w(c). Let c◦h be the heap obtained by replacing the particles of c by discs, and by adding those discs on top of h. We define the background of any heap h to be the foreground of the heap obtained by flipping the configuration upside-down. In particular, the background of c ◦ h contains c. Let d be the disc in the background of c ◦ h that has the smallest vertex index i. Then if i is occupied in c, we form c by removing the particle at i, and h by adding a disc on top of the i vertex. If i is unoccupied in c, then we form c by adding a particle at i, and h by removing the top disc at i. Finally if h is empty and the vertex 1 is unoccupied in c, then we leave the pair unchanged. These rules define an involution ϕ(h, c) = (h , c ). By construction, we have W (h) w(c) = −W (h ) w(c ), if (h , c ) = (h, c). Hence the distinct pairs ((h, c), (h , c )) image of one-another under ϕ contribute zero to the sum on the l.h.s. of Eq. (4.3), and we are left only with the contribution of the fixed points of the involution. The latter correspond to the situation where h = ∅ and c has no particle at the vertex 1, producing the r.h.s. of (4.3). 4.2. Positivity from heaps. Applying Theorem 4.1 to the case of G = G r , and comparing the expressions (3.23) and (4.2), we arrive at the main theorem of this section, which allows to interpret the R1,n as partition functions for heaps.


745

Theorem 4.2. The solution R1,n to the Q-system for n ≥ 0 is, up to a multiplicative factor R1,0 , the partition function for configurations of heaps of n discs on G r with weights z i = t yi , with yi as in (3.17). (r )

Proof. Using Theorem 3.15, we may rewrite F1 (t)/R1,0 exactly in the form of the (r ) r.h.s. of (4.2), with G = G r and the weights z i = t yi . We deduce that F1 (t)/R1,0 is the generating function for heaps on G r with weight t yi per disc above the vertex i. The coefficient of t n in the corresponding series corresponds to heap configurations with exactly n discs. Corollary 4.3. For all n ∈ Z, R1,n is a positive Laurent polynomial of the initial seed x0 . Proof. By Theorem 4.2, R1,n /R1,0 for n ≥ 0 is a sum over heap configurations, each contributing a weight made of a product of yi ’s. The resulting sum is therefore a manifestly positive polynomial of the yi ’s, themselves products of ratios of initial data. For n < 0, the result follows by applying Lemma 2.1.

4.3. Continued fraction expressions for generating functions. The heap interpretation allows us to write an explicit expression for F1(r ) (t) as a rational function of t. Theorem 4.4. R1,0

(r )

F1 (t) =

1−t

y1 1−t

1−t y3 −t

(4.4) y2

1−t y5 −t

y4

..

y6

. 1−t y

2r +1

where yi are defined in (3.17). Proof. Define a partial ordering on the vertices of G r by their geodesic distance from vertex 1. Any nonempty heap on G r is constructed by repeating the following two steps: (1) Stack one disc above vertex 1. (2) Construct a heap on the graph G r = G r \ {1} (the graph G r without the vertex 1 and its incident edges). (r )

Let c(t) be the generating function for heaps on G r , then clearly F1 (t)/R1,0 = 1/(1 − t y1 c(t)). Similarly, c(t) is found via a similar construction of heaps on G r . Clearly, c(t) = 1/(1 − t y2 d(t)), where d(t) is the generating function for heaps on the graph G r = G r \ {2}. To find d(t), we note that G r has two minimal vertices, but they are connected, so that a heap on G r may have as its foreground either node 3 or node 4 but not both. Thus, d(t) = 1/(1 − t y3 − t y4 e(t)), where e(t) is the generating function for heaps on G r \ {3, 4}. This procedure is iterated (see Fig. 6), resulting in Eq. (4.4). When expanded as a power series in t, (4.4) has manifestly positive coefficients which are Laurent polynomials in the yi ’s.

746


Fig. 6. The graph G r , arranged into a hierarchical structure. The depth of the vertices is indicated on the (r ) center, and the corresponding terms of the continued fraction for the generating function F1 (t) are displayed on the right

4.4. Continued fraction rearrangements. One can rewrite the continued fraction expres(r ) sion for F1 (t) in various ways using two simple lemmas: Lemma 4.5. For all a, b there is an identity of power series of t: a 1 , =1+t a 1 − t 1−tb 1 − ta − tb

(4.5)

and Lemma 4.6. For all a, b, c, d such that c = 0 and a + b = 0, a+

a b = , c 1 − t 1−td 1 − t 1−tcb −td

(4.6)

where a = a + b, b =

a c bc ac a b , c = , a= , b = , c = b + c . a+b a+b b + c b + c (4.7)

For example, applying Lemma 4.5 to the expression (4.4) with a = y1 and b = y2 /(1 − t y3 · · · ), we have ⎛ ⎞ ⎜ ⎜ ⎜ ⎜ (r ) F1 (t) = R1,0 ⎜1 + t ⎜ 1 − t y1 − ⎜ ⎝ where yi are defined in (3.17).

y1 t y2 1−t y3 −

..

.

t y4

ty 1−t y2r −1 − 1−t y2r 2r +1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(4.8)


747

Fig. 7. The local transformation of Lemma 4.6

Example 4.7. For r = 1, we have ⎛ ⎜ ⎜ (1) F1 (t) = R1,0 ⎜1 + ⎝

⎞ R

1,1 t R1,0

1−t

R1,1 R1,0

−

t R1,0 R1,1 R 1−t R1,0 1,1

⎟ ⎟ ⎟= ⎠

R1,0 1−t 1−t

R1,1 R1,0 1 R1,0 R1,1 R 1−t R1,0 1,1

.

(4.9)

Figure 7 is a graphical interpretation of Lemma 4.6. It shows that the generating function of heaps on G can be rewritten as a generating function for heaps on G with new weights. Note that a, b, c, d and the associated nodes might stand for composite generating functions and the associated subgraphs. Starting from the initial seed x0 and its associated graph G r , repeated application of Lemmas 4.5 and 4.6 on the expression (4.8) produces new expressions for F1(r ) (t) with manifestly positive series expansions in t. (r ) Writing the expression in (4.8) as F1 (t) ≡ FM0 (x0 ; t), let x M be another seed in the fundamental domain. Then we claim that there is a sequence of applications (r ) Lemmas 4.5 and 4.6, such that (F1 (t)) = FM (xM ). It turns out that we can generate in this way all the mutations of x0 within the fundamental domain Fr . The proof of this statement appears in Sect. 6. Here, we illustrate this result with the example of A2 . The example of A3 is given in Appendix Appendix A. (2)

Example 4.8. Consider the case of A2 . We present all the rearrangements of F1 (t) which correspond to mutations of the weights within the fundamental domain. (1) FM0 (x0 ) with x0 = (R1,0 , R2,0 ; R1,1 R2,1 ): Equation 4.8 is R1,1

(2)

F1 (t) = R1,0 + t

1 − t y1 −

(4.10)

t y2 ty 1−t y3 − 1−t4y

5

with y1 =

R1,1 R2,1 R1,0 R2,1 R1,0 R2,0 , y2 = , y3 = , y4 = , y5 = . R1,0 R1,0 R1,1 R2,0 R1,1 R2,0 R2,1 R2,1 (4.11)

(2) Fµ2 (M0 ) (µ2 (x0 )), where µ2 (x0 ) = (R1,0 , R2,2 ; R1,1 R2,1 ): Use Lemma 4.6 with a = y3 , b = y4 , c = y5 and d = 0. Then (2)

F1 (t) = R1,0 + t

R1,1 1 − t x1 −

1−

t x2

t x3 tx 1− 1−t4x 5

=

R1,0 1−

t x1

1−

t x2 t x3 1− tx 1− 1−t4x 5

(4.12)

748


(a)

(b) (r )

Fig. 8. The graph heap formulation of the application of Lemma 4.6 on the generating function F1 (t)/R1,0

(r ) (a), and on the generating function (t) such that F1 (t) = R1,0 + t R1,1 (t) (b). The circled vertices

correspond to a, b, c in the Lemma 4.6

where the second expression follows from application of Lemma 4.5 with a = x1 . Here x1 = y1 =

R1,1 R2,1 , x2 = y2 = , R1,0 R1,0 R1,1

x3 = a = a + b = y3 + y4 =

2 +R ) R1,0 (R2,1 1,1

=

R2,1 R1,1 R2,0 y bc y R R1,1 4 5 1,0 x 4 = b = = = 2 = , a y3 + y4 R2,1 R2,2 R2,1 a ac y3 y5 R1,0 R2,1 x 5 = c = = = = .. a y3 + y4 R1,1 a R2,2

R1,0 R2,2 , R2,1 R1,1

(4.13)

by use of the Q-system. This is an expression for the generating function in terms of (R1,0 , R2,2 ; R1,1 R2,1 ) = µ2 (x0 ). It has a manifestly positive series expansion in t. Figure 8 (a) illustrates this transformation. The generating function is interpreted in terms of that for heaps on H2 : F1(2) (t) = R1,0

Z (H2 ) (0, −t x2 , −t x3 , −t x4 , −t x5 ) . 1 , −t x 2 , −t x 3 , −t x 4 , −t x 5 )

Z (H2 ) (−t x

(3) Fµ1 (M0 ) (µ1 (m0 )), where µ1 (x0 ) = (R1,2 , R2,0 ; R1,1 R2,1 ): Use Lemma 4.6 on 2 R (4.10), with a = y1 , b = y2 , c = y3 + y4 /(1 − t y5 ) = R2,0 R1,0 R2,1 + R1,1 / 2,1 R1,1 R2,0 (1 − t R2,1 ) , and d = 0 (see Fig. 8 (b)): F1(2) (t) = R1,0 + t

R1,1 1− 1−t

t z1 z z 2 + 1−t6z 5 z 1−t z 3 −t 1−t4z 5

(4.14)


749

Fig. 9. The graphs encoding the R1,n ’s for the case A2 , and the corresponding mutations of cluster variables

with 2 +R R1,1 R1,2 2,1 z 1 = a = a + b = y1 + y2 = = , R1,0 R1,1 R1,1 z6 bc y4 = y2 y3 + z2 + = b = 1 − t z5 a+b 1 − t y5 ⎛ ⎞ 1 R 1,1 2 ⎝ R2,1 ⎠, = + R R2,0 R1,2 R1,1 1 − t R2,0 2,1 z4 ac y 4 = y1 y3 + z3 + = c = 1 − t z5 a+b 1 − t y5 ⎛ ⎞ R1,1 R 1,1 2 ⎝ R2,1 ⎠, = + (4.15) R R2,0 R2,1 R1,2 1 − t 2,0 R2,1

where we have used the Q-system. This yields positivity of all R1,n in terms of 2 + R )/R . (R1,2 , R2,0 ; R1,1 R2,1 ) = µ1 (x0 ), by noting that R1,0 = (R1,1 2,1 1,2 Figure 8 (b) illustrates this transformation. Equation (4.14) may be rewritten in terms of the generating function of heaps on K 2 as: F1(2) (t) = R1,0 + R1,1 t

Z (K 2 ) (0, −t z 2 , −t z 3 , −t z 4 , −t z 5 , −t z 6 ) . Z (K 2 ) (−t z 1 , −t z 2 , −t z 3 , −t z 4 , −t z 5 , −t z 6 )

Note that the graph K 2 has one more node than G 2 and H2 , but the weights depend on the same number (4) of free parameters, since z 1 z 3 z 5 = 1, and z 2 z 4 = z 3 z 6 . (2)

The rearranged expressions for F1 (t) considered above have been related to mutations of cluster variables in the fundamental domain F2 , as well as to configurations of heaps on particular graphs, see Fig. 9. 5. Path Generating Functions In order to prove the claim of the last section, it is simpler to work with a path interpretation. There exist certain bijections between the partition function of heaps on a graph

750


Fig. 10. A heap on G 3 with 8 discs, and the corresponding lattice path of length 16. The heap has small discs above vertices 3 and 5 of G 3 , and large discs above the other vertices. The large discs are in bijection with the descending steps, and the small discs are in bijection with the horizontal steps. This path has weight y12 y2 y3 y4 y5 y6 y7 . On the left is the path graph G˜ 3 , with labeled vertices (in black) and labeled edges (in gray)

Such G, and the partition function of paths on an associated weighted rooted graph G. bijections are standard in the theory of heaps [24]. We will establish this bijection in the case of G r . Then, we construct bijections between Motzkin paths representing cluster variables in the fundamental domain, and weighted graphs, on which the path partition function gives the generating function (r ) F1 (t), in terms of the new cluster variables. For completeness, we give the bijection with the related graphs for heaps at the end of this section. r . We start with the graph G r defined in Fig. 3. 5.1. From heaps on G r to paths on G r is a vertical chain of r +3 vertices labeled 0, 1, 2, . . . , r +2, Definition 5.1. The graph G with r +2 vertical edges (i, i +1) (0 ≤ i ≤ r +1), together with r −1 vertices 2 , 3 , . . . , r and r − 1 horizontal edges (i, i ) (2 ≤ i ≤ r ). It is rooted at its bottom vertex 0. (See the left of Fig. 10 for the r = 3 example.) r is “dual” to G r , in the sense that its edges are in bijection with the The graph G vertices of G r . We may denote the edges by the same labels, i = 1, 2, . . . , 2r + 1. r are defined as follows. There is a weight yi for each Definition 5.2. The weights of G step along the edge i which goes towards the root vertex 0, and a weight 1 to all others. That is, the step i → i has weight y2i−1 , the step i → i − 1 has weight y2i if 1 < i < r + 1, step 1 → 0 has weight y1 , and step r + 2 → r + 1 has weight y2r +1 . r , from the root to the root, also referred to as G r -paths below, can be Paths along G represented on a two-dimensional lattice as follows. Paths of length 2n start at the point (0, 0), end at the point (2n, 0), and cannot go below 0 or above r + 2. They may contain steps of the type (i, j) → (i + 1, j ± 1) (if j ± 1 is in the range 0, . . . , r + 2), and steps (i, j) → (i + 2, j) (2 ≤ j ≤ r ). r -paths of length Theorem 5.3. The heaps on G r with n discs are in bijection with the G 2n, and their partition functions are equal, with weights as in Definition 5.2. Proof. Given a heap on the graph G r , we associate a large disc to each vertex along the “backbone” {1, 2, 4, . . . , 2r − 2, 2r, 2r + 1}. We associate a small disc to the other vertices, {3, 5, . . . , 2r − 1}. The overlap between discs is given by the graph G r , namely


751

Fig. 11. The bijection between heaps on K 2 and paths on the “dual” graph K˜ 2 . The discs of the heap give rise to polygons, whose top right edges are exactly the descending steps of the path. We have indicated the weights z i of the various discs, that are transferred to the descending steps of the path. We have indicated the vertex labels on K 2 , and the corresponding dual edge labels on K˜ 2

two discs overlap if and only if the corresponding vertices of G r are connected via an edge. These discs are represented by bars in the plane in Fig. 10. We draw the bars on the faces of a tilted square lattice, in such a way that they are in bijection with the descending (for a large disc) or horizontal (for a small disc) steps of r -path. In this picture, the descending steps the two-dimensional representation of a G of the path are the north-east edges of the square faces in which the large discs sit, while the horizontal steps are the horizontal diagonals of the square faces in which the small discs sit. There is a unique way of placing the discs with this constraint. We decompose the heap into successive shells. Each shell has faces containing the discs placed from the bottom to the top and from left to right, the spaces in-between being covered with only up steps. In this way, each large disc corresponds to a descending step and each small one to a double horizontal step of the path. The correspondence of weights is clear: descending steps receive the weights of the corresponding large discs, while the horizontal steps receive the weights of the corresponding small discs. r -path, we associate bijectively a large disc to each descending Conversely, given a G step and a small disc to each horizontal step. The resulting configuration is a heap over G r , as the path always ends with a descending step corresponding to a disc over vertex 1 of G r . Corollary 5.4. For n ≥ 0, the solution R1,n of (1.4), expressed in terms of the variables r -paths of length 2n, with weights as in x0 , is R1,0 times the partition function of G Definition 5.2. The partition function for heaps F1(r ) on the other graphs corresponding to fraction rearrangements can also be written in terms of paths on graphs. For example, it is easy r corresponding to the heap graph Hr (the chain with 2r + 1 to see that the path graph H r correspondvertices) is a vertical chain with 2r + 2 vertices, with edge (i, i + 1) of H ing to vertex i + 1 in Hr . The edge weights for descending edges are the same as the corresponding vertex. Ascending edges have weight 1. Example 5.5. Consider the case of A2 (see Fig. 9). We have a path formulation of the 2 as a function of x0 . Paths on the graph H 2 partition function in terms of paths on G correspond to the generating function in terms of µ2 (x0 ). The path formulation for the cluster variable µ1 (x0 ), corresponding to heaps on K 2 , is the partition function for 2 in Fig. 11. Note that the edge labeled 6 is an oriented edge (with graphs on the graph K weight z 6 ).

752


Fig. 12. The graphs (k) corresponding to strictly descending Motzkin paths with k = 1, 2, 3, 4 vertices. We have indicated the extra edge labels (in black) and the vertex labels (gray) along the vertical chain for the case when i = j = 0

In general, it is quite complicated to work out the direct bijection between G and Instead, we introduce a bijection between Motzkin paths in the fundamental domain G. and path graphs, and a bijection between the same Motzkin paths and heap graphs. This establishes an identification between path and heap partition functions in the cases of interest.

5.2. Motzkin paths and path graphs. 5.2.1. The graphs M For any Motzkin path M in the fundamental domain, we associate a graph M by (1) decomposing M into “strictly descending” pieces; (2) associating a subgraph to each descending piece and (3) gluing the subgraphs. (1) Decomposition of M: Each path M, consists of strictly descending pieces m1 , m2 , . . . , m p , where mi = {(x, y), (x − 1, y + 1), . . .}. (We consider a single vertex to be a descending piece). These are separated by p − 1 ascending steps of type (i) the step (1,1) in the plane, or (ii) the step (0,1) in the plane. (2) Graphs for mi : For each strictly descending piece m with k vertices, we define a (k) := m as follows. When k = 1, m is a single vertex, and (1) is a chain of four vertices, as shown in the top left of Fig. 12. When m contains k ≥ 2 vertices, the graph m = (k) consists of a “skeleton” k , plus extra oriented edges j → i if j − i > 1, j < k + 2 and i > 0. There is a G total of k(k − 1)/2 extra oriented edges in (k), which we label by the vertices they connect. See Fig. 12. Each (k) has four distinguished vertices denoted by (t, t , b , b) = (k + 2, k + 1, 1, 0). (3) Gluing the subgraphs: Ascending steps of M are of type (i) (1, 1) or (ii) (0, 1). The graph M is obtained by gluing the graphs mi , i = 1, 2, . . . , p as follows.


753

Fig. 13. The graphs mi−1 and mi are glued together according to whether the pieces mi−1 and mi are separated by a step of type (i) or (ii)

Denote the edge (t , t) of mi by (ti , ti ), and so forth. We then make the following identifications of vertices and edges (See Fig. 13): • Type (i): Identify ti−1 with bi and ti−1 with bi . The common edge is represented vertically. • Type (ii): Identify ti−1 with bi and ti−1 with bi . The common edge is represented horizontally. We denote this gluing procedure by the symbol “|”. Thus, M = m1 |m2 | · · · |mk .

(5.1)

Finally, all vertices are renumbered sequentially from bottom to top. Example 5.6. Consider the case of an ascending Motzkin path containing steps of type (i) and (ii) only. It decomposes into r isolated vertices, corresponding to mi = (1). r (see Fig. 14). For the path M0 , with steps only of type (ii), the result is M0 = G More generally, M is obtained by gluing r chains (1) vertically or horizontally. The path M is composed of p vertical chains of lengths l1 , . . . , l p ≥ 1, with steps p of type (ii) only with i=1 li = r . These are separated by p − 1 steps of type (i). Each li , which are glued according to rule (i). This vertical chain corresponds to a graph G results in a tree which has a vertical chain of length 2 + i (li + 1), and consecutive sequences of li − 1 horizontal edges, separated by pairs of vertices. The top two and bottom two vertices have no horizontal edge attached. (see Fig. 15 for r = 4).

754


Fig. 14. The graph M0 corresponding to the vertical Motzkin path M0 with r vertices. The path is decomposed into k isolated vertices, each corresponding to a chain of type (1), glued as indicated. The resulting graph is the tree G˜ r

Fig. 15. The 8 trees T2r +2 (i 1 , i 2 , . . . , i s ) for r = 4. We have indicated the values of the i’s under each tree, and the labeling of the edges (gray) and vertices (black)

The vertices i j to which the horizontal edges are attached 1 < i 1 < i 2 < · · · < i s < 2r − s, where s = r − p, have the property that i 1 − 1 is odd, and i j+1 − i j is odd for all j < s. In fact, {i 1 , i 2 , . . . , i s } = {0, 1, 2, . . . , 2r − s + 1}\ ∪0≤ j≤ p { j +

j

i=1

li , j + 1 +

j

li }.

i=1

Definition 5.7. The trees obtained in the previous example are denoted by T2r +2 (i 1 , i 2 , . . . , i s ). The 2r +1 edges of such trees are ordered from bottom to top and labeled 1, 2, . . . , 2r +1, including the horizontal edges, which have labels i 1 + 1, i 2 + 2, . . . , i s + s.


755

Noting that the sequence ja = (i a +a −1)/2 satisfies1 ≤ j1 < j2 < · · · < js ≤ r −1 without further constraint, we see that there are exactly r −1 s such trees for r and s fixed, r −1 when we sum over s = 0, 1, 2, . . . , r − 1. Note that when s = 0, hence a total of 2 T2r +2 () = H˜ r , the “dual” of the chain of 2r + 1 vertices Hr introduced in Sect. 5.1. This example is important, as it illustrates all 2r −1 possible skeleton trees any M in the fundamental domain can have: Definition 5.8. The skeleton tree associated to M is the tree M , where M is obtained from M by replacing all the steps (−1, 1) by vertical steps (0, 1). In other words, M is obtained from M by removing all its extra down-pointing edges. 5.2.2. Weights on M We will compute the partition function of paths on M , for which purpose, we assign a weight to each oriented edge of M . An unoriented edge, in this context, is considered to be a pair of edges oriented in opposite directions. The edge (v, v ) is considered to be an ascending edge if the distance of v from the vertex 0 is greater than that of v. Otherwise, it is a descending edge. We assign weight 1 to all ascending edges, and weight ye (M) to each descending edge e. Edges are labeled as follows. Given M , consider the skeleton tree M = T2r +2 (i 1 , i 2 , . . . , i s ) of Definition 5.7. Its edges have labels 1, 2, . . . , 2r + 1, which we retain for the graph M , and we assign to the descending skeleton edge i the weights yi (these are independent variables, not necessarily related to the weights encountered earlier). The extra descending edges of M are labeled by the pairs (i, j) of vertices they connect, with 1 < j + 1 < i ≤ 2r − s (see Fig. 12). The weights yi, j corresponding to down-pointing edges e = (i, j) where i − j > 1 can be expressed in terms of the skeleton weights. In view of the gluing procedure, we restrict our attention to strictly descending Motzkin paths m with k vertices (see Fig. 12). There are k(k − 1)/2 descending edges of type (i → j), 2 ≤ j + 1 < i ≤ k + 1. Then (y2 j , y j+2, j , . . . , yk+1, j ) ∝ (y2 j+1 , y2 j+2 , y j+3, j+1 , . . . , yk+1, j+1 ) for j = 1, 2, . . . , k − 1.

(5.2) The proportionality is via overall non-vanishing scalar factors, so these relations allow to express all the weights yi, j with i > j + 1 in terms of those of the skeleton tree, y1 , y2 , . . . , y2k+1 . In fact, denote yi+1,i := y2i ,

and yi,i := yi ,i = y2i−1 .

(5.3)

yi, j ym, = yi, ym, j for i > m and j > .

(5.4)

Then Eqs. (5.2) are equivalent to

That is, i−1 yi, j =

= j y +1, i−1 = j+1 y ,

i−1

= j

= i−1

y2

= j+1 y2 −1

.

(5.5)

These relations between edge weights will play a crucial role in Sect. 7 below, when we discuss the path interpretation of Rα,n . They will also become clear when we express the effect of cluster mutations on the weighted graphs.

756


Fig. 16. The path-heap correspondence. To each descending step (t, i) → (t + 1, j) of any path on M , we associate the parallelogram indicated. The corresponding heap disc is a segment on the vertical diagonal of the parallelogram. To each double horizontal step we associate the triangle indicated. The corresponding heap disc is on the vertical median of the triangle

5.3. From Motzkin paths to heap graphs. For completeness, we give the bijection between the path graphs M and the heap graphs dual to them. This subsection is not necessary for further computations in this paper. We construct a heap graph G M for each Motzkin path M via a generalized path-heap correspondence. Given a graph M , we represent paths on the graph from 0 to 0 on the two-dimensional lattice as before. We advance by one step in the “time” (x) direction for each step in the path, and record the height of the vertex visited in the vertical coordinate. Associate to each descending step (t, i) → (t + 1, j) in this picture the parallellogram with vertices {(t, i), (t + 1, j)(t, j − 1)(t − 1, i − 1)}, and represent a vertical segment {(t, x), x ∈ [ j − 1 + , i − ]} for > 0 sufficiently small (see Fig. 16). For double horizontal steps (steps of the form (t − 1, i) → (t, i) → (t + 1, i)), we draw the half-diamond with vertices {(t − 1, i), (t + 1, i), (t, i − 1)}, and the segment {(t, x), x ∈ [i − 1 + , i − ]} for > 0 sufficiently small. These segments represent the discs of the heap. The heap graph G M encodes the overlap between these segments. The vertices of G M are in bijection with the descending steps on M , and the edges connect any pair of steps such that the associated discs cannot freely slide horizontally without touching each-other. Weights on the vertices are assigned according to edge weights. To construct G M directly from M, we proceed as for M . We associate a graph G mi to each strictly decending piece mi of M. We then glue these pieces according to the type of separating step between them. Let m be a strictly descending Motzkin path with k vertices. Let G m = G(k). Its vertices are indexed by the descending steps on (k). The descending steps on (k) are indexed by (i, j) for i = j = 2, 3, . . . , k, i = j + 1 = 1, 2, . . . , k + 2 and k +1 ≥ i > j +1 ≥ 2, hence G(k) has a total of k −1+k +2+k(k −1)/2 = (k +1)(k +2)/2 vertices. The descending steps k + 2 → k + 1 and 1 → 0 are singled out, and form the top and bottom vertices of G(k), denoted by t and b. The descending step (i, j) (i ≥ j) overlaps with any descending step (m, ) (m ≥ ) such that m or ∈ [ j, i] (or both). The set of all these overlapping descending steps (i, j) − (m, ) forms the edges of G(k). We have represented the graphs G(k) for k = 1, 2, 3 in Fig. 17, together with the path graphs (k) of Fig. 12 for k = 1, 2, 3. The graphs G mi−1 and G mi are glued in two possible ways, according to whether mi−1 and mi are separated by a step (1, 1) (type (i)) or (0, 1) (type (ii)) (see Fig. 18):


757

Fig. 17. The heap graphs G(k) corresponding to the strictly descending Motzkin paths with k = 1, 2, 3 vertices, together with the corresponding path target graphs (k). The vertices of G(k) are indexed by the descending steps on (k)

Fig. 18. The heap graphs G mi−1 and G mi corresponding to the strictly descending Motzkin paths mi−1 and mi are glued in two possible ways, according to whether the Motzkin paths are separated by (i) a step (1, 1) or (ii) a step (0, 1). Both involve identifying the top vertex of G mi−1 with the bottom vertex of G mi , but in the case (ii) all the vertices connected to the bottom vertex of G mi must be connected to all the vertices connected to the top vertex of G mi−1

758


• Type (i): The top vertex ti−1 of G mi−1 is identified with the bottom vertex bi of G mi . • Type (ii): The top vertex ti−1 of G mi−1 is identified with the bottom vertex bi of G mi , and all the vertices connected to ti−1 are connected to all the vertices connected to bi , via additional edges. Example 5.9. Consider µ1 (G 3 ) of Fig. 36 of Appendix Appendix A (top of second column from left). The corresponding Motzkin path is {(1, 1), (0, 2), (0, 3)}, and has two strictly descending pieces m1 = {(1, 1), (0, 2)} and m2 = {(0, 3)}. The graphs G m1 and G m2 correspond to the vertices denoted {1, 2, 3, 4, 5, 8} and {5, 6, 7} respectively in Fig. 36. As they are glued according to the rule (ii), all the vertices connected to 5 in G m1 , namely 4 and 8, are connected to all the vertices connected to 5 in G m2 , namely 6. Remark 5.10. Let us define the overlap between descending edges of M as follows: the descending edge (i, j) (i ≥ j) overlaps the descending edge (m, ) (m ≥ ) if and only if m or ∈ [ j, i] (or both). We may now bypass the above gluing procedure by directly associating to M the graph G M as follows: (i) the vertices of G M are in bijection with the descending edges on M and (ii) the edges of G M connect all pairs of vertices such that the corresponding descending edges overlap. The bijection between paths on M and heaps on G M follows as before from decomposing any heap into shells of successive foregrounds, and the fact that there is a unique way of arranging the discs (and their surrounding parallelograms) on the square lattice from bottom to top and left to right in each shell, and filling the spaces in-between with ascending steps of the path. 6. Path Interpretation of R1,n and Cluster Mutations Path partition functions give a combinatorial interpretation for {R1,n }, expressed in terms of seed in the fundamental domain. For each Motzkin path M, F1(r ) (t) = any cluster n n≥0 t R1,n is a path partition function on a graph M with positive weights depending on xM . This results in a positivity theorem for all {R1,n }n∈Z , as well as explicit expressions for the generating function. 6.1. Path partition functions in terms of transfer matrices. (n) 6.1.1. Path transfer matrix Let PM (a, b) be the set of paths on the graph M starting at vertex a and ending at vertex b with n descending steps. The partition function is

(n) Z M (a, b) = ye (M). (6.1) (n)

p∈PM (a,b) e∈ p

(Recall that only descending edges have non-trivial weights.) Define the generating function

(n) Z M (a, b) = t n ZM (a, b). (6.2) n≥0

This can be computed by use of the transfer matrix TM , the weighted incidence matrix of M , with an additional factor t per descending step. Its rows and columns are indexed


759

by the vertices of M (the ordering on vertices is such that i < i < i + 1), with non-vanishing entries 1 if j → i is an ascending edge, (TM )i, j = t y j,i if j → i is a descending edge, where y j,i is the weight of the oriented edge j → i. We have Z M (a, b) = (I − TM )−1 ,

(6.3)

b,a

where I is the identity matrix. Let FM (t) := Z M (0, 0). The entry ((I − TM )−1 )0,0 is computed by row-reduction, using upper unitriangular row operations to obtain a lower triangular matrix, then taking the (0, 0) entry of the latter. To be precise, we iterate the following procedure: (1) Let t be the last row in the matrix A. Define the matrix A with entries Ai,t At, j , i < t. At,t

Ai, j = Ai, j −

= 0 for all i < t. Then Ai,t (2) Truncate A by deleting its last row and column.

Repeat this until the result is a 1 × 1 matrix, which is 1/(A−1 )0,0 . For our particular set of graphs, the reduction procedure has a graphical interpretation. It gives the partition function in terms of a pruned graph with the top part removed, and replaced by a single “loop” with a different weight, corresponding to the partition function on the pruned branch. Example 6.1. The transfer matrix of the graph K˜ 2 of Fig. 12 is ⎛ ⎞ 0

TK˜ 2

t y1 0 1 0 0 0

⎜1 ⎜0 = ⎜0 ⎝ 0 0

0 t y2 0 1 1 0

0 0 t y3 0 0 0

0 t y3,1 t y4 0 0 1

0 0 ⎟ 0 ⎟ 0 ⎟ ⎠ t y5 0

(6.4)

with vertex order (0, 1, 2, 2 , 3, 4). Note that y3,1 = y2 y4 /y3 due to (5.2). The reduction of A = I − TK˜ 2 is I − TK˜ 2

⎞ ⎛ 1 −t y ⎞ ⎛ 1 −t y 0 0 0 0 0 1 1 t y3,1 −1 1 −t y2 0 −t y3,1 ⎟ ⎜−1 1 −t y2 − 1−t y5 0 ⎟ ⎜ → ⎝ 0 −1 1 −t y3 −t y4 ⎠ → ⎝ ⎠ ty −t y3 0 −1 1 − 1−t4y 0 0 −1 1 0 5 ⎛ →

0

0

−1

0

1 − t y5

1 −t y1 0 ty ⎝−1 1 −t y2 − 1−t3,1y5 ty 0 −1 1 − t y3 − 1−t4y

⎞

⎛

0

−t y1

⎠→⎝

−1 1 − t

→ 1−

y

1−t

5 ty 1−t y3 − 1−t4y 5

= (I − TK˜ 2 )−1 )

0,0

.

⎞ ⎠

5 ty

1−t y3 − 1−t4y

1

3,1 y2 + 1−t y

y

3,1 y2 + 1−t y

5

5

t y1

−1

0

1

1

760


Fig. 19. Construction of the transfer matrix TM in the case of (i) vertical and (ii) horizontal gluing of the . (a) Is the motzkin path, (b) is the graph gluing and (c) the gluing of the two subgraphs m and m i−1 i diagonal blocks T and T . We have shaded the graph i in (b) and the corresponding matrix elements i−1

i

of the block T in (c). The added element y stands for the weight t yb,a in both cases (i) and (ii) i

This is identical to the inverse of the factor of t R1,1 in the second term of Eq. (4.14), upon identifying z i = yi , i = 1, 2, 3, 4, 5 and z 6 = y3,1 . 6.1.2. Block structure To see how this works in general, we now describe the structure of TM . It consists of blocks which are put together according to the gluing procedure. be the Recall the decomposition of M into strictly descending pieces mi . Let m i graph with its bottom and top vertices and edges removed. Define Tmi to be the transfer . Let also and matrix of m m0 m p+1 denote the bottom and top vertices, glued via i and . We write vertical edges to m M = m0 ||m1 || · · · ||m p+1 to denote the mp 1 gluing procedure. (Gluing via the operation || consists of adding a vertical edge or a horizontal edge and vertex.) The matrix TM is obtained by gluing the diagonal blocks Ti according to whether the corresponding graph gluing is via a (i) vertical or (ii) horizontal edge (see Fig. 19): • Type (i): The blocks Ti−1 and Ti occupy successive diagonal blocks in the matrix TM (vertex ti−1 of i−1 is followed by vertex bi of Ti ). They are “glued” by the ,b = y = t yb ,t . addition of two matrix elements, (TM )bi ,ti−1 = 1 and (TM )ti−1 i i i−1 • Type (ii): We place the two blocks Ti−1 and Ti in the matrix TM in such a way that ) coincide with the first row and col the last row and column of Ti−1 (with index ti−1 umn of Ti (with index bi ). We then insert a new row and column labeled bi following , with nonvanishing entries (T ) ,b = y = t yb ,t . ti−1 = 1 and (TM )ti−1 M bi ,ti−1 i i i−1


761

6.2. Reduction and continued fractions. 6.2.1. General reduction process Now consider the reduction procedure performed on the matrix A = I − TM . (1) Let (t , t) denote the last two row indices of A. The bottom right 2 × 2 submatrix of 1 −y I − TM has the form with y = t yt,t = t y2r +1 . Reduction erases the row −1 1 and column t of A, and replaces At ,t = 1 → 1 − y. (2) Inductive step: After reduction of all rows with index j > ti , the resulting matrix T ||···|| differs from the original only in the element (ti , ti ), denoted by γi , where m0

i

(1 − γi )−1 = (I − T{t

i

) } ||i+1 ||···|| p+1

−1

t i ,t i

.

We then reduce all indices j > ti−1 + 1. This results in the matrix A . Its lower right 2 × 2 matrix, indexed by the vertices a, b, has the form: 1 −y , where y = t yb,a and (1−c)−1 = (I − Ti ||···|| p+1 )−1 Case (i): . −1 1 − c b,b One more reduction step eliminates row and column b and gives the new entry y Aa,a → 1 − 1−c . 1 − c −y Case (ii): . One more step in the reduction gives Aa,a → 1 − y − c. −1 1 In both cases, the net result is a modification of A in which the bottom right element is 1 − γi−1 , where . (1 − γi−1 )−1 = (I − T{t } ||i ||···|| p+1 )−1 t

i−1

i−1 ,t i−1

The bottom right element of the reduced transfer matrix is γi−1 equal to (i) y/(1−c) or (ii) y + c. 6.2.2. The case of a strictly descending Motzkin path Consider the transfer matrix corresponding to (k) (Fig. 12)). For later use, we use a more general matrix T (k), which has T (k)k+1,k+1 = c instead of 0. With vertex order (1, 2, 2 , 3, 3 , . . . , k, k , k + 1), this matrix is ⎞ ⎛ 0 ty 0 ty 0 ty 0 · · · ty 0 ty 2

⎜1 ⎜0 ⎜0 ⎜ ⎜0 T (k) = ⎜ ⎜0 ⎜ .. ⎜. ⎜0 ⎝

0 1 1 0 0 . .. 0 0 0 0 0

3,1

t y3 0 0 0 0 . .. 0 0 0

t y4 0 0 1 1 . .. 0 0 0

4,1

0 0 t y5 0 0 . .. 0 0 0

t y4,2 0 t y6 0 0 . .. 0 0 0

0 0 0 0 t y7 . .. 0 0 0

··· ··· ··· ··· ··· .. . ··· ··· ···

k,1

t yk,2 0 t yk,3 0 t yk,4 . .. 0 1 1

k+1,1

0 0 0 0 0 . .. t y2k−1 0 0

t yk+1,2 ⎟ 0 ⎟ t yk+1,3 ⎟ ⎟ 0 ⎟ t yk+1,4 ⎟ ⎟. . ⎟ .. ⎟ ⎟ t y2k ⎠ 0 c

(6.5)

Reduction of the last index in A = (I − T (k)) results in ⎧ yk+1,i in row i < k − 1; ⎨ yk,i → yk,i + 1−c y y2k−2 → y2k−2 + k+1,k−1 1−c in row k − 1; ⎩ y2k Akk → 1 − t 1−c in row k, y2k whereas reduction of the next index changes Akk = 1−t y2k /(1−c) → 1−t y2k−1 −t 1−c .

762


Let

ϕk (y2 , . . . , y2k ; y3,1 , y4,1 , . . . , yk+1,k−1 ; c) = (I − T (k))−1

1,1

.

(6.6)

As a result of the two reduction steps, ϕk is replaced by ϕk−1 , with suitable substitutions of variables. Lemma 6.2. The function ϕk =

−1 (I − T(k) )

1,1

is determined by the recursion

relation ϕk ({y j }2k j=2 ; {y j,i }2≤i+1< j≤k+1 ; c) yk+1,k−1 2k−2 yk+1,i y2k } }2≤i+1< j≤k ; t y2k−1 + t ), ; {y j,i + δ j,k = ϕk−1 ({y j + δ j,2k−2 1 − c j=2 1−c 1−c with −1 1 1 −t y2 ϕ1 (y2 ; ; c) = = . (6.7) −1 1 − c 1 − t y2 1,1

1−c

Example 6.3. For k = 2, we have y3,1 t y4 ϕ2 (y2 , y3 , y4 ; y3,1 ; c) = ϕ1 y2 + ; ; t y3 + = 1−c 1−c

1 1−

t y3,1 1−c t y4 1−t y3 − 1−c

t y2 +

.

For k = 3, we have:

y4,2 y4,1 t y6 ϕ3 (y2 , y3 , y4 , y5 , y6 ; y3,1 , y4,1 , y4,2 ; c) = ϕ2 y2 , y3 , y4 + ; y3,1 + ; t y5 + 1−c 1−c 1−c 1 . = t y4,1 t y3,1 + 1−c t y6 1−t y5 − 1−c t y4,2 t y4 + 1−c 1−t y3 − y6 1−t y5 −t 1−c

t y2 +

1−

Thus, ϕk is a finite, multiply branched continued fraction. Its power series expansion in t has coefficients which are polynomials in the weights with non-negative integer coefficients. This expression is still valid if we relax the proportionality conditions (5.2), but if we use these for j = 1, we see that 1 ϕk = (6.8) y2 t y3 +t y4 wk , 1 − y3 1−t y3 −t y4 wk where wk does not depend on y2 , y3 , y4 . 6.3. Mutations as fraction rearrangements. 6.3.1. Mutations and Motzkin paths One can associate a unique sequence of mutations to each Motzkin path M using the following procedure. The path M = {(m α , α)}α∈Ir is considered on the lattice, and each lattice point (x, α) with 1 ≤ x ≤ m α corresponds to a mutation µα (µα+r ) if x is odd (even), respectively. The compound mutation µ such that µ(x0 ) = x M is the product of these mutations read from bottom to top, and from right to left. For example, in Fig. 20 (a), µ = µ11 µ12 µ15 µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ9 .


(a)

763

(b)

Fig. 20. A path M with r = 9 (a), and the product of mutations (circled) yielding M when applied on M0 . In (b), the two generic moves (i) and (ii) corresponding to the actions of the mutation are indicated, together with their “boundary” version, (iii) and (iv) when α = 1, and (v) when α = r

This restricted set of mutations acting on M0 yields any path M in the fundamental domain. We need to use only the two elementary moves shown in Fig. 20 (b) (i)–(ii), and their “boundary” versions (iii)–(v). Without loss of generality we may therefore restrict ourselves to this subset of mutations. To see how this set of mutations acts on the partition function, we compare the graphs and weights (M , y(M)) and (M , y(M )), where M = µα (M) or µα+r (M). 6.3.2. Bulk mutations of type (a) and (b) Consider first the two bulk moves (i)-(ii) of Fig. 20 (b). Their effect on the corresponding target graphs is shown in Fig. 21 (a) and (b). The path M is decomposed into a vertex labeled α, and bottom and top pieces m1 and m2 , corresponding to graphs (1), 1 and 2 respectively. In Fig. 21 (a), the edge labels of the piece (1) corresponding to the central isolated vertex α of M are identified; they are 2α − 1, 2α and 2α + 1. We find that all the independent edge weights in M and M are identical, except for the three weights , y y2α−1 , y2α , y2α+1 of M . These transform to y2α−1 , y2α 2α+1 in M . As we shall see below, the new weights correspond to the mutation of cluster variables. The matrix AM = 1 − TM , after the reduction of the block corresponding to 2 , becomes: ⎛ ⎞ 0

0

. . ⎜ ⎟ ⎛ ⎞ . . . . ⎜ I − T1 ⎟ 0 0 0 0 ⎜ ⎟ 0 0 . . . ⎜ ⎜ ⎟ ⎟ . . . . . . −t y2α−1 −t y2α ⎜ I − T1 ⎜ ⎟ ⎟ ⎜ ⎜ 0 ··· 0 − 1 ⎟ ⎟ 0 0 0 1 0 0 · · · 0 ⎜ ⎟ → ⎜ ⎟ −t y2α−1 −t y2α 0 ⎜ ⎜ 0 ··· 0 − 1 ⎟ 0 1 −t y2α+1 0 · · · 0 ⎟ ⎜ ⎜ ⎟ ⎟ 1 0 0 1 0 ⎜ ⎟ ⎝ 0 ··· 0 − 1 ⎠ 0 ··· 0 − 1 0 1 −t y2α+1 ⎜ ⎟ 0 0 ⎜ ⎟ 0 · · · 0 0 0 −1 1 − c 0 I − T2 ⎠ .. .. ⎝ . 0

. 0

(6.9)

764


(a)

(b)

Fig. 21. The two bulk moves of Fig. 20 (b) acting on the paths and graphs. The edges involved in the transformation are indicated. The mutation (a) involves the three edges of the center piece (1), and is independent of the structure of m1 and m2 . Mutation (b) involves four edges, and depends on the length of the strictly descending subpath above vertex α +1, increasing its length by 1. On the graph it transforms the block (k −1) into (k), creating k new descending edges

with (1 − c )−1 = (I − T2 )−1 b ,b (b2 is the bottom vertex of 2 ). Three more 2 2 iterations of reduction replace the bottom right element of I − T1 with f = 1 − t y2α−1 −

t y2α 1−

t y2α+1 1−c

.

(6.10)

Similarly, reduction of I − TM yields (after reducing the 2 part) ⎞ ⎛ 0 0 0 I − TM

I −T

. . . ⎜ . . . ⎜ 1 ⎜ 0 0 → ⎜ −t y2α−1 0 ⎜ ⎜ 0 0 ··· 0 − 1 1 −t y2α ⎝ 0 0 ··· 0 0 0 ··· 0

0 0

−1 0

. ⎟ . . ⎟ ⎟ 0 ⎟ 0 ⎟ ⎟ 0 ⎠ 1 − c −t y2α+1 −1 1

(6.11)

with c as above. Three more reduction iterations result in the updated bottom right element of I − T1 , f = 1−

t y2α−1

1−

t y2α 1−t y2α+1 −c

.

(6.12)

The partition functions of paths on M and M must coincide, they are the same partition function expressed in terms of different variables. They coincide if and only


765

, y if the weights y2α−1 , y2α 2α+1 are such that f = f (all the other weights are equal). Application of the rearrangement Lemma 4.6, with a = y2α−1 , b = y2α , c = y2α+1 and d = c , gives f = f if and only if: y2α−1 = y2α−1 + y2α ,

y2α =

y2α y2α+1 , y2α−1 + y2α

y2α+1 =

y2α−1 y2α+1 . y2α−1 + y2α

(6.13)

We interpret this transformation as the effect of the mutation µα or µα+r on the skeleton weights of M , resulting in a rearrangement of the continued fraction FM (t) into FM (t). The mutation in Fig. 21 (b) is slightly more subtle, because it depends on the length of the strictly descending subpath in m above α, whose length is increased by 1. All independent edge weights are unaffected by the mutation, except those of edges 2α−1, . . . , 2α+2 (recall that the other edge weights are ratios of skeleton weights y1 , y2 , . . . , y2r +1 ). The transfer matrix is ⎞ ⎛ 0 0

TM

.. ⎜ T . ⎜ 1 ⎜ 0 ⎜ t y2α−1 ⎜ =⎜ ⎜0 0 ··· 0 1 0 ⎜ ⎜ 0 ⎜ .. ⎝

0

. 0

.. . 0 t y2α 0

0

⎟ ⎟ ⎟ ⎟ ⎟ ··· 0 ⎟, ⎟ ⎟ ⎟ T y2α+1 , y2α+2 (k−1)|2 ⎟ ⎠

(6.14)

where y2α+1 ,y2α+2 (k − 1) is the circled graph in M , on the bottom of Fig. 21 (b), for which we indicate the values of the weights of its first two lowest edges. After reduction, we are left with (I − T1 ), with the bottom diagonal element replaced by f = 1 − t y2α−1 −

t y2α , 1 − c(y2α+1 , y2α+2 )

(6.15)

where (1 − c(y2α+1 , y2α+2 ))−1 = (I − T y2α+1 ,y2α+2 (k−1)|2 )−1

b ,b

the next-to-bottom vertex of (k − 1). On the other hand, ⎛

0 . ⎜ . . ⎜ 1 ⎜ 0 ⎜ t y2α−1 0 ⎜ ⎜ 0 0 · · · 0 1 0 t y 0 ∗ 0 ∗ 0 · · · 0 ∗ 2α ⎜ ⎜ 1 ⎜ 0 ⎜ ⎝ .. y2α+1 , y2α+2 . 0

T

TM =

0 . .. 0

0

0

T

, where b denotes

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ (k−1)|2 ⎠

(6.16)

where the entries ∗ stand for matrix elements of the form t y j,i , where the vector , (t y )) is proportional to the vector (t y (t y2α 2α+1 , t y2α+2 , (t y j,i )), which are the weights j,i

766


appearing in the row below it. This fact can be used to eliminate the ∗ entries using the row below. Then the reduction gives the matrix ⎛ ⎞ 0 0 .. . 0 0

I −T

. ⎜ .. ⎜ 1 ⎜ 0 ⎜ t y2α−1 ⎜ ⎜ ⎝ 0 0 · · · 0 − 1 1 + y 2α y 0 0 ··· 0

−

2α+1

y2α

y2α+1

1 − c(y2α+1 , y2α+2 )

−1

0

⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

−1 where 1 − c(y2α+1 , y2α+2 ) = (I − T y

(k−1)|2 2α+1 ,y2α+2

)−1

(6.17)

b ,b

, with c as in (6.15).

Two further reduction steps replace the bottom right element of (I − T1 ) by f = 1−

t y2α−1

1−

y2α c(y2α+1 ,y2α+2 ) y2α+1 1−c(y2α+1 ,y2α+2 )

.

(6.18)

The partition functions for graphs on M and M must be equal, so we look for , y the transformation (y2α−1 , y2α , y2α+1 , y2α+2 ) → (y2α−1 , y2α 2α+1 , y2α+2 ) such that f = f . The function c = c(y2α+1 , y2α+2 ) is identified by row-reducing (I − T(k)|2 ). Reducing the 2 part results in replacing the bottom right element of (I − T(k) ) with y c 1 − d , where (1 − d )−1 = (I − T2 )−1 b ,b . This yields the relation 1 − y 2α 1−c = 2

2

2α+1

, y ϕk−1 (y2α 2α+1 , y2α+2 , y2α+3 , . . . , y2α+2k−2 ; {y j,i }; d ), with ϕk as in Lemma 6.2, and we get c = y2α+1 +wk y2α+2 , by comparison with the general expression (6.8). Therefore

f = 1 − t y2α−1 −

t y2α = f = 1− 1 − t y2α+1 − t y2α+2 wk 1−

y2α y2α+1

t y2α−1

t y2α+1 +t y2α+2 wk 1−t y2α+1 +t y2α+2 wk

.

(6.19) Applying the rearrangement Lemma 4.6 with a = y2α−1 , b = y2α and c = y2α+1 + y2α+2 wk , we conclude that (6.19) is satisfied if and only if y2α−1 = y2α−1 + y2α ,

= y2α

y2α y2α+1 , y2α−1 + y2α

y2α+1 =

y2α−1 y2α+1 , y2α−1 + y2α

y2α+2 =

y2α+2 y2α−1 , y2α−1 + y2α

(6.20) yi

while all other weights are equal to yi . This is interpreted as the effect of the mutation µα or µr +α on the graph weights. 6.3.3. Boundary mutations We consider the mutations of Fig. 20 (b) (iii-v). • Case (v). We have α = r , which is just case (a) of Fig. 21, with m2 = ∅ and 2 reduced to a vertex. The transformation of weights is (6.13). • Cases (iii-iv). Here, α = 1. We take m1 = ∅ and 1 a single vertex in both cases (a) and (b) of Fig. 21. However, the action of µ1 or µr +1 introduces a new feature, which we call re-rooting. This is because the effect of the mutation is an application of the rearrangement Lemma 4.6 on the corresponding weighted path partition function FM (t). This is possible only if the graph M is rooted at vertex b = 1 instead of (t), where F (t) = (I − T )−1 vertex b = 0. We write FM (t) = 1 + t y1 (M)FM , M M 0,0 (t) = (I − T )−1 . while FM M 1,1


767

This re-rooting must take place whenever we act via the moves (iii-iv) as a direct consequence of the cases (a) and (b) of Fig. 21. Indeed, 1 is reduced to a vertex. The lower edge 2α − 1 = 1 is horizontal, and the vertex common with 1 is b = 1, rather than b = 0, so that 1 = {b }. The weight y1 of the horizontal edge is associated with the step 0 → 1 in the re-rooted formulation. In general, the weighted path partition function FM (t), corresponding to the Motzkin −1 (r ) path M, is related to the initial generating function FM0 (t) = R1,0 F1 (t), via the following sequence of re-rootings. We start from the canonical sequence of Motzkin paths leading from M0 to M = (m 1 , . . .) via our restricted set of mutations. Within this sequence, the paths M(i ) , = 1, 2, . . . , m 1 are those which differ from their predecessor only via the action of the “boundary” mutations µ1 or µr +1 . Note that m 1 (M ) = . Only the boundary mutations µ1 and µr +1 imply a re-rooting (otherwise FM(i−1) = FM(i) ). Thus, −1 (r ) Lemma 6.4. The partition function FM (t) is obtained from FM0 (t) = R1,0 F1 (t) via the sequence of re-rootings:

FM −1 (t) = 1 + t y1 (M −1 )FM (t), = 1, 2, . . . , m 1 with FM = FMm 1 , where m 1 = m 1 (M) is the first entry of M = (m α , α)rα=1 . This allows to interpret R1,n+m 1 (xM ), expressed as a function of the cluster variable xM , in terms of the partition function FM (t). Let us denote by F(t) n the coefficient of t

t n in the series F(t). Lemma 6.5.

R1,n+m 1 = R1,0 FM0 (t) n+m = R1,0 y1 (M0 )y1 (M1 ) . . . y1 (Mm 1 ) FM (t) n . t

1

t

Proof. We use Lemma 6.4. The prefactor is obtained by collecting the successive multiplicative weights of each re-rooting. 6.3.4. Main theorem Let us summarize the results of this section in the following: Theorem 6.6. The mutation of cluster variables µα or µα+r : xM → xM is equivalent to a rearrangement relating the continued fractions FM → FM that generate weighted paths on the rooted target graphs M and M . The edge weights of the corresponding skeleton trees, y(M) = {y1 (M), . . . , y2r +1 (M)} and y = y(M ) are related through either of the two transformations (6.13) or (6.20), while all other weights remain the same. 6.4. Weights and the mutation matrix B. There is a simple expression for the edge weights of M in terms of the cluster variables xM and the mutation matrix B(M) at the seed M. To specify all the edge weights M , one need only specify y1 (M), y2 (M), . . . , y2r +1 (M), due to the relations (5.2) for the other weights. Theorem 6.7. The weights y1 (M), y2 (M), . . . , y2r +1 (M) of the skeleton tree M are the following Laurent monomials of the cluster variable xM : y2α−1 (M) =

λα,m α

λα−1,m α−1

, (α = 1, 2, . . . , r + 1),

(6.21)

768


y2α (M) =

µα+1,m α +1 λα+1,m α+1 1 − δm α ,m α+1 +1 (1 − ) (6.22) µα,m α λα+1,m α λα−1,m α ) , (α = 1, 2, . . . , r ), × 1 − δm α−1 ,m α +1 (1 − λα−1,m α−1

where λα,n =

Rα,n+1 , Rα,n

µα,n =

Rα,n . Rα−1,n

(6.23)

Proof. Weights are determined by their initial values (3.17) at seed M0 and by the recursion relations (6.13) (in the case m α−1 = m α < m α+1 ) and (6.20) (in the case m α−1 = m α = m α+1 ) for each mutation in our restricted set, all other weights being invariant. Using the Q-system (1.4), one checks that the relations (6.13) and (6.20) are satisfied by the weights (6.21) and (6.22). Note that Eqs. (6.21)-(6.22) involve only the data (Rα,m α , Rα,m α+1 )rα=1 , the cluster variables at the seed M. Example 6.8. Consider ascending Motzkin paths as in Example 5.6. The weights from Theorem 6.7 are y2α−1 (M) =

λα,m α , λα−1,m α−1

y2α (M) =

µα+1,m α +1 , µα,m α

(6.24)

since only mutations of type (a) of the previous subsection are used. In the particular case of M = M0 , Eqs. (6.24) reduce to (3.17). In the case of the strictly ascending path Mmax = (0, 1, . . . , r − 1), with cluster variables (Rα,α−1 , Rα,α )α∈Ir , x2α−1 := y2α−1 (Mmax ) =

Rα−1,α−2 Rα,α Rα−1,α−1 Rα+1,α , x2α := y2α (Mmax ) = . Rα−1,α−1 Rα,α−1 Rα,α−1 Rα,α (6.25)

The graph Mmax = H˜ r , the chain of 2r + 2 vertices. The partition function is (r )

FMmax (t) = F1 (t) =

t n R1,n =

n≥0

R1,0 1−t

x1 1−t

1−t

.

(6.26)

x2 x3

..

.

x 1−t 1−t x2r 2r +1

Remark 6.9. Recalling the discrete Wronskian expression Rα,n = det 1≤i, j≤α (R1,n+i+ j−α−1 ), we may view the result (6.25)–(6.26) as a particular case of a theorem of Stieltjes [23] (see also [1]), which expresses the formal power series expansion F(λ) = k≥0 (−1)k ak /λk+1 around ∞ for any sequence ak , k ∈ Z≥0 , as the continued fraction F(λ) =

1 β1 λ +

1 1 β2 + β λ+··· 3

,


769

where β2k = (0k )2 /(1k 1k−1 ), β2k+1 = (1k )2 /(0k 0k+1 ), and m k =

det (am+i+ j−2 )

(6.27)

1≤i, j≤k

are Hankel determinants involving the sequence ak . Indeed, writing λ = −1/t and taking ak = R1,k , we easily identify the two continued fraction expressions F(λ) and F1(r ) (t) of Eq. (6.26) upon taking xk = βk β1k+1 , while 1/β1 = a0 = R1,0 yields the overall prefactor. Note that the continued fraction is actually finite, as 1/β2r +3 ∝ r0+2 = 0. The weights computed in Theorem 6.7 are simply related to the exchange matrix B(M) of the cluster algebra at the seed M. Theorem 6.10. The exchange matrix B(M) at the point M = {(m α , α)}rα=1 reads, for 1 ≤ i, j ≤ r : m j +1 mj mi B(M)i, j = (−1) 2 (−1) 2 − (−1) 2 δ|i− j|,1 , B(M)i, j+r = (−1) B(M)i+r, j = (−1) B(M)i+r, j+r = (−1)

mj m i +1 2 + 2

+1

Ci, j ,

m +1 m 2i + j2 m 2i

Ci, j , m j +1 m i +1 (−1) 2 − (−1) 2 δ|i− j|,1 ,

where C is the Cartan matrix of Ar .

(6.28)

0 −C t . Proof. By induction. The theorem is true when M = M0 , where BM0 = C 0 Assuming it is true for some M with m α−1 = m α ≤ m α+1 , let us prove it for M , with m β = m β + δβ,α . We distinguish according to the parity of m α : if m α is even, we use the mutation µα , otherwise we use µα+r . Assume m α even. Under the mutation µα , the matrix elements B(M)i,α and B(M)α, j

are negated. This is compatible with Eq. (6.28) by noting that m +1 α2

m α 2

= m2α and that

= m α2+1 + 1, which gives the expected extra minus sign. For the other entries of B(M), we use (1.6): B(M )i, j = B(M)i, j + sgn(B(M)i,α )[B(M)i,α B(M)α, j ]+ m

with i, j = α. Assume j ≤ r . Then B(M)α, j = 0 only if 2j = m2α ± 1, while |i − α| = | j − α| = 1. Due to the Motzkin path condition, this is only possible if m j = m α − 1, but is impossible, as we must have m α+1 , m α−1 ≥ m α . Therefore Bi, j (M) = Bi, j (M ). Similarly, if i ≤ r , we have the same conclusion. If i, j > r , we write i = i + r and j = j + r , with i , j ≤ r . Then B(M)i +r,α B(M)α, j +r = −Ci ,α Cα, j (−1)

mi 2

+

m j 2

is positive only if i = α and j = α ±1, in which case (i) m j = m α or (ii) j = α +1 and m j = m α+1 = m α +1, (or with i and j interchanged). Then B(M)i +r,α B(M)α, j +r = 2. In the case (i), we have Bα+r, j +r (M) = 0, hence Bα+r, j +r (M ) = 0 + 2 = 2, compatible with Eq. (6.28), as

m α +1 2

= m α2+1 + 1. In the case (ii), we have Bα+r,α+1+r (M) = −2,

770


hence Bα+r,α+1+r (M ) = −2 + 2 = 0, also in agreement with Eq. (6.28), as mα 2

m α +1

+ 1 = 2 . The case of m α odd is treated analogously.

m α+1 +1 2

=

Remark 6.11. It is interesting to note that the B-matrices of Theorem 6.10 only have entries in {0, ±1, ±2}. This is true only for the cluster subgraph Gr , as the entries grow arbitrarily in the complete cluster graph T2r . We also note the remarkable property, that the sum of the four blocks r × r of the B-matrices always vanishes, namely that Bi, j + Bi+r, j + Bi, j+r + Bi+r, j+r = 0 for i, j = 1, 2, . . . , r . The sequence (Rα,m α )rα=1 , (Rα,m α +1 )rα=1 and the sequence of cluster variables 2r (where the R xM = (Reven , Rodd ) ≡ (xi (M))i=1 α,m are listed first for the even entries in m and then for the odd ones) are related via a permutation σM : ⎧ if m α is even ⎨ α σM (α) = α + r if m α is odd, and α ≤ r . (6.29) ⎩α − r if m is odd, and α > r α We have the following expression: Theorem 6.12. The weights computed in Theorem 6.7 are related to the exchange matrix of the cluster algebra B(M): y2α−1 (M) = xi (M) B(M)i,σM (α) , y2α (M) 2r

i=1

y2α (M) = xi (M) B(M)i,σM (α+r ) , (α ∈ Ir ). y2α+1 (M) 2r

i=1

(6.30) 6.5. Positivity of the cluster variables R1,n . The variables R1,n can now be shown to be positive Laurent polynomials of any possible set of cluster variables xM . Note that the graphs M are invariant under translations of M, but the same is not true for the weights y(M). Theorem 6.13. For any Motzkin path M = {(m α , α)}rα=1 with r vertices and any n ≥ 0, the solution R1,n+m 1 (xM ) of the Ar Q-system, expressed as a function of xM , is equal to R1,m 1 times the generating function for weighted paths on M , starting and ending at the root, with n down steps, and with weights ye (M) given by Theorem 6.7, namely: R1,n+m 1 (xM ) = R1,m 1 FM (t) n . t

Moreover, R1,n+m 1 is expressed as an explicit Laurent polynomial of the cluster variable xM , with non-negative integer coefficients, for all n ∈ Z. Proof. By Theorem 2.4, we can restrict ourselves to the Motzkin paths of the fundamental domain. The first statement of the theorem follows from Lemma 6.5, with the prefactor R1,0 y1 (M0 ) . . . y1 (M p ) = R1,m 1 , by use of (6.21) for α = 1. The second statement is clear for n ≥ 0. It is a direct consequence of the reinterpretation of R1,n+m 1 as the generating function for weighted paths with n down steps, for all n ≥ 0, by also noting that the weights ye (M) are all positive Laurent monomials of the initial data, as a consequence of Theorem 6.7.


771

For n < 0, we simply use an enhanced version of the reflection property of Lemma 2.1. Indeed, noting that the Q-system equations are invariant under the reflection n → −n, we deduce that the quantity R1,n+m 1 , expressed as a function of the initial data xM , is the same as R1,−n−m 1 expressed as a function of the reflected initial data xMt , where the reflected Motzkin path Mt = {(m tα , α)} satisfies m tα = −(m α + 1) for α = 1, 2, . . . , r . In other words, if R1,n+m 1 = f (xM ), then R1,−n−m 1 = f (xMt ). Let us now use the translation invariance of the Q-system in the form of Eq. (2.5). We first translate the reflected Motzkin path Mt back to the fundamental domain, namely so that its lowest entry m tα is zero. This is done by considering m t = Minα=1,2,...,r (m tα ), and the translate M = Mt − m t , with entries m α = m tα − m t for all α. Then for n < 0, the quantity R1,−n−m 1 +m t is still given by the same expression as before in terms of the shifted Motzkin path M , namely R1,−n−m 1 +m t = f (xM ). But now the first part of the theorem applies, as m 1 = m 1 (M ) = −m 1 − 1 − m t . Indeed, as −n + 1 ≥ 0, we deduce that R1,−n+1+m 1 = f (xM ) is a positive Laurent polynomial of the reflected-translated data xM . As R1,n+m 1 = f (xM ) via the same function f , we conclude that R1,n+m 1 is also a positive Laurent polynomial of the initial data xM , for all n < 0. This completes the proof of the theorem. In view of the correspondence between path partition functions on M and heap partition functions on G M , we have also Corollary 6.14. The weighted heap graph G M associated to the Motzkin path M is constructed as above. The heaps on G M with n discs are in bijection with the weighted paths on M with n descending steps, starting and ending at the root. For any Motzkin path M and any n ≥ 0, the quantity R1,n+m 1 is expressed in terms of the cluster variable xM as R1,m 1 times the partition function for weighted heaps with n discs on G M . 7. Strongly Non-Intersecting Path Interpretation of Rα,n We now provide a combinatorial interpretation of the determinant expressions for Rα,n (α > 1) as partition functions of families of strongly non-intersecting paths on M . For this, we need to generalize the usual notion of non-intersecting lattice paths. As a result we obtain a proof of the positivity of the determinant expression for Rα,n with α > 1. 7.1. Ascending Motzkin paths: Paths on trees T2r +2 (I ). First consider the case of the ascending Motzkin paths. The graphs M are skeleton trees, as in Sect. 5.6, denoted by T2r +2 (I ), where I = {i 1 , i 2 , . . . , i s } with 1 < i 1 < i 2 < · · · < i s < 2r − s and i 1 − 1, i 2 − i 1 ,…, i s − i s−1 odd. The variables R1,n are partition functions of paths on these trees, starting and ending at vertex 0. Paths on T2r +2 (I ) are equivalent to paths on the two-dimensional lattice, with the y-coordinate restricted to 0 ≤ y ≤ 2r + 1 − s, with the following possible steps: (i) (x, y) → (x + 1, y + 1); (ii) (x, y) → (x + 1, y − 1) and (iii) (x, y) → (x + 2, y) (y ∈ I ). We say that two such paths intersect if they share a vertex. Theorem 7.1. Let n ≥ α−1. Then Rα,n is (R1,0 )α times the partition function of families of α non-intersecting paths on T2r +2 (I ), starting at the points (0, 0), (2, 0), . . . , (2α − 2, 0) and ending at the points (2n, 0), (2n + 2, 0), . . . , (2n + 2α − 2, 0), with weights as in Example 6.8.

772


Fig. 22. The eight pairs of non-intersecting G˜ r -paths of 8 and 4 time-steps for r ≥ 3. The top box contains 2 the non-intersecting H˜r - (Dyck-) paths. There are six pairs of paths on G

Proof. We apply the Lindström-Gessel-Viennot (LGV) theorem [11,19]. Consider the ,...,E α partition function Z AE11,...,A of non-intersecting families of α weighted paths from the α initial points A1 , A2 , . . . , Aα to endpoints E 1 , E 2 , . . . , E α . We assume that if i < j and k < l, then a path Ai → El must intersect any path A j → E k . Let Z Ai →E j be the partition function for weighted paths from Ai to E j , then ,...,E α = det Z Ai →E j . (7.1) Z AE11,...,A α 1≤i, j≤α

Now, R1,n is R1,0 times the partition function of paths on T2r +2 (I ) from (0, 0) to (2n, 0). Let Ai = (2i − 2, 0) and E i = (2n + 2α − 2i, 0) (i = 1, 2, . . . , α). Since Z Ai →E j is the partition function for paths of 2n + 2α − 2i − 2 j + 2 time-steps, it may be identified with R1,n+α−i− j+1 /R1,0 . We conclude that ,...,E α = Z AE11,...,A α

1

det

(R1,0 )α 1≤i, j≤α

where we have used Lemma (2.6).

R1,n+α−i− j+1 =

Rα,n , (R1,0 )α

(7.2)

Example 7.2. For α = 2, n = 2 and r ≥ 3, R2,3 = R1,4 R1,2 − (R1,3 )2 = x12 x2 x3 (x1 x3 + x1 x4 + x2 x4 ) = y12 y2 (y1 y32 + 2y1 y3 y4 + y1 y42 + (y1 + y2 )y4 (y5 + y6 )) with xi as in (6.25) and yi as in (3.17). The first expression corresponds to the three pairs of paths in the top of Fig. 22. The second expression corresponds to the eight pairs of non-intersecting G˜ r -paths of 8 and 4 time-steps depicted in Fig. 22. We also indicated in this figure the six pairs of non-intersecting paths on G˜ 2 , for which no horizontal step at height 3 is allowed, eliminating the two configurations of the bottom row. 7.2. Strongly non-intersecting paths on M . Theorem 7.1 can be generalized to paths on M even when M is not a tree. To do this, we must generalize the notion of nonintersecting paths, using a representation of M -paths on the two-dimensional lattice which preserves the property that the number of descending steps is half the horizontal length of the path. (Note that this representation is different from the one used in Fig. 16.)


773

Fig. 23. The two-dimensional path representation of a path on M with M = (2, 1, 0) (see Fig. 12 with r = 3). Descents of height 2 are vertical (horizontal displacement 2 − h = 0), while descents of height 3 go back one step horizontally (2 − h = −1). With this choice, the horizontal distance between the starting and ending point is twice the number of descents (16 = 2 × 8 here)

7.2.1. Two-dimensional representation of M -paths Some of the descending steps on M may have height h > 1, in case there is an edge j → i with j − i > 1. When h = 0 or 1, each descending step is in bijection with an ascending one, but this is not true if h > 1. We require that the total horizontal displacement in a path i → i + 1 → · · · → j − 1 → j → i should be independent of the height h = j − i of descending step used. Therefore, on the two-dimensional lattice, we draw a descent along an edge of length h as a line from (x, y) to (x − h + 2, y − h). Then a path which goes up h single steps, then down one step of height h has horizontal length 2, independently of h. With this convention, the horizontal distance between the start and end of a path from the 0 to 0 on M is twice the number of its descending steps (see Fig. 23 for an illustration). Definition 7.3. The two-dimensional representation of a path with n descents on M is a path in (Z+ )2 , starting at (0, 0) and ending at (2n, 0), with the possible steps: • (x, y) → (x + 1, y + 1) whenever there is an edge y → y + 1 in M . • (x, y) → (x + 2, y) whenever there is an edge y → y → y in M . • (x, y) → (x + 2 − h, y − h), h ≥ 1 whenever there is an edge y → y − h in M . Note that these steps all preserve the parity of x + y, which is even, so the path is on the even sublattice in Z2 , with even x + y. 7.2.2. Intersections of paths on the lattice The determinant Rα,n = det 1≤i, j≤α (R1,n+α+1−i− j ) may be interpreted using a generalization of the LGV formula. Intersections of the paths in Definition 7.3 are of a more general type than the usual case. Paths may intersect along edges, and not just at a vertex. It is clear that edge intersections can occur only along descending edges. The possible crossing types can be catalogued as follows.

774


Fig. 24. The possible intersections of descending steps of two paths. The first descent is (h − 2, h) → (0, 0), and the second is of type (i + 2 − k, i + k) → (i, i) with 0 ≤ k < h. There are h − k − 1 possible values of i

Lemma 7.4. Consider a step of path P1 of type (h − 2, h) → (0, 0). Then a descending step of a second path P2 may cross this step only via the following h(h − 1)/2 possible steps: (i + k − 2, i + k) → (i, i),

for 0 ≤ i ≤ h − k − 1, 0 ≤ k ≤ h − 1.

A generic case is represented in Fig. 24. 7.2.3. A weight preserving involution on families of paths Given the list of edge intersections in Lemma 7.4, we define a weight-preserving involution on families of paths which we call flipping. Suppose two paths, P1 and P2 , intersect along an edge e1 = (u + h − 2, v + h) → (u, v) of P1 and e2 = (u + i + k − 2, v + i + k) → (u + i, v + i) of P2 . Suppose Pi = L i ∪ ei ∪ Ri (i = 1, 2), where L i is the subpath of Pi before the vertex (u + h − 2, v + h), and so forth. Definition 7.5. The flipping of P1 and P2 w.r.t. the intersection along the edges e1 and e2 the configuration of two new paths P1 , P2 such that Pi = L i ∪ ei ∪ Ri , where e2 = (u + i + k − 2, v + i + k) → (u, v) and e1 = (u + h − 2, v + h) → (u + i, v + i). Flipping is illustrated in Fig. 25. The weight of two paths (P1 , P2 ) is equal to the product of the weights of P1 and P2 . Lemma 7.6. The weight of the two paths (P1 , P2 ) is equal to that of the flipped paths (P1 , P2 ) of Definition 7.5. Proof. w(e1 )w(e2 ) = yh+v,v yi+k+v,i+v = yh+v,i+v yi+k+v,v = w(ei )w(e2 ), by virtue of Eq. (5.4). The rest of the weights of the path configurations are unchanged by the flipping operation, so the lemma follows. For the graph M , there is a finite list of all possible results of the flipping procedure. Lemma 7.7. Given M , the results of flipping an intersection between a pair of paths is a pair of paths which include pairs of steps of the form ((u + h − 2, v + h) → (u + i, v + i), (u + j − 2, v + j) → (u, v)) for any 1 ≤ i ≤ j < h, where there is a descending edge v + h → v in M .


(a)

775

(b)

Fig. 25. A typical edge intersection of M -paths (a) and the result of the flipping operation on it (b). We have the weights of the steps. We have the identity of path weights, since yh+v,v yi+k+v,i+v = yh+v,i+v yi+k+v,v . The paths in (b) are said to be “too close” to each other

Proof. By construction of M (see Sect. 5.2.1), if there exists a descending edge v +h → v on M , then for all 1 ≤ i ≤ j < h, the edges v + j → v + i exist on M , as well as v + h → v + i and v + j → v. The list of the definition therefore exhausts all possible cases of flippings of intersections along edges. Definition 7.8. Two paths obtained as the result of the flipping of an intersection are called “too close” to each other. For example, the paths in Fig. 25 (b) are too close. Conversely, we may define the flipping of a pair of paths which are too close to each other (with respect to the pair of edges that are too close) as the inverse of the transformation of Definition 7.5. The flipping thus defined is an involution. Definition 7.9. Two paths are said to be strongly non-intersecting if (i) they do not intersect and (ii) they are not too close. 7.2.4. Generalization of LGV We have the formula R1,n+m 1 +α+1−i− j Rα,n+m 1 , = det 1≤i, j≤α (R1,m 1 )α R1,m 1

(7.3)

where n + m 1 ≥ α − 1. Using Theorem 6.13 and the path presentation above, R1,n+m 1 +α+1−i− j /R1,m 1 is the sum over weighted M -paths from Ai = (2i − 2, 0) to E j = (2n + 2α − 2 j, 0). The proof of the LGV formula (7.1) uses an involution on configurations of paths, which leaves their weight invariant up to a sign. This cancels various pairs of intersecting paths in the determinant expansion. This involution is defined as follows: it leaves nonintersecting configurations invariant, otherwise, it interchanges the beginnings (until the first intersection) of the two leftmost paths that intersect first. The only terms not cancelled in the determinant correspond to the non-intersecting path configurations. α The determinant (7.3) is also as a sum over families of paths from {Ai }i=1 to the α {E j } j=1 , with their weight multiplied by the signature of the permutation σ such that Ai is connected to E σ (i) .

776


We define an involution as follows. If a configuration has at least one intersection or edges which are too close to each other, then “flips” the beginnings of the two leftmost paths which intersect or are too-close, whichever comes first. That is, we either interchange the beginnings of paths if there is an intersection along a common vertex, or we apply the flipping procedure described above. Configurations which have no intersections nor edges which are too close to each other are invariant under . The map is clearly an involution. Applying Lemma 7.6, the total weight of a flipped configuration is invariant. As for the ordinary case, pairs up terms in the expansion of the determinant which cancel each other, and we are left only with the -invariant configurations. Therefore the determinant (7.3) is equal to the partition function of strongly non-intersecting paths of Definition 7.9. Theorem 7.10. For any Motzkin path M, and for n + m 1 ≥ α − 1, Rα,n+m 1 /(R1,m 1 )α is the partition function of configurations of α strongly non-intersecting paths on M starting at the points Ai = (2i − 2, 0), i = 1, 2, . . . , α and ending at the points E j = (2n + 2α − 2 j, 0), j = 1, 2, . . . , α. The weights are expressed in terms of the Motzkin path as in Theorem 6.7, and (5.5). 7.3. Positivity of Rα,n for all α, n. Theorem 7.10 implies that Rα,n+m 1 is a positive Laurent polynomial of any initial data, provided n + m 1 ≥ α − 1, since all the weights in Theorem 6.7 and (5.5) are positive Laurent monomials of the initial data xM . We now consider the case when n + m 1 < α − 1. The determinant in Eq. (7.3) involves some quantities R1,m with indices m < 0, for which we have no interpretation as partition functions for paths. In order to generalize the result for arbitrary n ∈ Z, we relate the expressions Rα,n+m 1 in the range n +m 1 < α −1 to some expressions Rα ,n +m 1 with n + m 1 ≥ α − 1. Consider the case when m α + 1 < n + m 1 < α − 1. Due to the structure of the Q-system, each Rβ,n is inductively obtained from the values Rβ,n−1 , Rβ,n−2 , Rβ+1,n−1 and Rβ−1,n−1 . Consequently, Rα,n+m 1 is a function only of the initial data that are contained in a “light-cone” of values of (β, m), such that α − n − m 1 + k ≤ β ≤ α + n + m 1 − k, for k < n + m 1 (see Fig. 26). In fact, Rα,n+m 1 depends only on the initial data xM corresponding to a subset M = {(m α , α)}rα=a of the Motzkin path M, namely such that (a, m a + 1) lies on the boundary of the light-cone, i.e. a = α − n − m 1 + k and m a + 1 = k. For the sake of this calculation we may freely modify the values of Ra−1,m and set them to 1, as they are not involved in the expression of Rα,n+m 1 . This has the effect of transforming the problem into one for Ar , with r = r − a + 1. More precisely, Rα,n+m 1 , as a function of a subset of the initial data xM , may be reinterpreted as the solution Rα ,n +m of the Ar Q-system, expressed in terms of the 1

initial data xM , with m β = m a+β−1 , β = 1, 2, . . . , r − a + 1. In this new expression, we have n + m 1 = n + m 1 = α − a + m a + 1 = α + m a ≥ α . Note that the minimum m of m β on the interval [a, r ] may be strictly positive. In that case, we must use the translational invariance property of Lemma 2.3 (see also Eq. (2.4)) to first translate both the Motzkin path M and the index n by −m . We get n − m + m 1 = α + m 1 − m ≥ α , and Theorem 7.10 can be applied to conclude that Rα ,n −m +m is a positive Laurent polynomial of the translated initial data xM corre1

sponding to m β = m β − m for all β. By Lemma 2.3, we find that Rα ,n +m is a positive 1 Laurent polynomial of the un-translated data xM .


777

Fig. 26. The generic situation m α + 1 < n < α − 1 in the (m, α) plane. We have represented the light-cone of values determining Rα,n+m 1 (shaded area inside the wedge on the left of the point (α, n + m 1 )), as well as the initial data, in the form of two parallel Motzkin paths M = {(m β , β)}rβ=1 and {(m β + 1, β)}rβ=1 . The light-cone cuts out a portion M = {(m β , β)}rβ=a of M so that Rα,n+m 1 only depends on the corresponding initial data. We may therefore truncate the picture by taking as a new origin the (dashed) line β = a − 1, and interpret Rα,n+m 1 as the solution Rα ,n +m a of the Ar −a+1 Q-system, with initial data indexed by M , and with α = α − a + 1 and n = n + m 1 − m a

We deduce that Rα,n is a positive Laurent polynomial of the initial data xM for all values of n > m α + 1. Finally, the positivity result is extended to n < m α (including negative values of n) by use of the corresponding invariance n → −n of the Q-system (see Lemma 2.1), and the same trick as in the proof of Theorem 6.13, involving the reflected-translated Motzkin path M . We deduce the final: Theorem 7.11. The solution Rα,n of the Q-system for Ar is a Laurent polynomial with non-negative integer coefficients of any initial data xM indexed by any Motzkin path M, for all n ∈ Z. 8. Asymptotics In this section, we consider two limiting cases of the results: Solutions in the the limit r → ∞, corresponding to the Q-sytem of A∞/2 , and solutions R1,n when n → ∞. In the latter case, we are interested in the asymptotic behavior of the number of paths contributing to the partition function R1,n as n → ∞. 8.1. The limit A∞/2 . In Eq. (1.4), retaining only the boundary condition at α = 0, R0,n = 1 (n ∈ Z), but dropping the condition at α = r + 1, gives us the solutions for the algebra A∞/2 . All of the results of the previous sections generalize in a straightforward way.

778


The initial seed is an infinite sequence, (Rα,m α , Rα,m α +1 )α∈Z+ indexed by semi-infinite Motzkin paths M. For example, for M0 = (0, 0, . . .), and the corresponding heap graph is G ∞/2 , generalizing Fig. 6. Then R1,n as a function of xM0 is R1,0 times the generating function for weighted heaps on G ∞/2 , F1 (t) obtained as the limit of (4.4): F1 (t) = 1 + t

y1 1 − t y1 − t

y2 1−t y3 −t

. y4

1−t y5 −t

(8.1)

y6

..

.

1−t y2α−1 −t

y2α

..

.

Using the rearrangement Lemmas 4.5 and 4.6 we can write the expressions for the generating function in terms of other initial seeds. For instance, the generating function corresponding to the “maximal” Motzkin path Mmax with m α = α − 1 is the continued fraction: 1 1−t

.

y1

1−t

(8.2)

y2 y3 1−t y4 1−t y5 1−t y6 1−t

..

1−t

.

yα

..

.

The limit of the generating function corresponding to the strictly descending Motzkin path with r vertices, of the form 1/(1 − t y1 ϕr ), with ϕr as in Lemma 6.2 is a “continued fraction” with infinitely many branchings, a sort of self-similar object.

8.2. Numbers of configurations. We consider the number of configurations contributing to R1,n in general. For this purpose, set Rα,m α = Rα,m α+1 = 1 (α = 1, 2, . . . , r ). This implies that all the edge weights are ye = 1. Therefore, Rα,n is a non-negative integer equal to the numbers of configurations of the related statistical model. For example, when M = M0 , Lemma 8.1. For the initial data Rα,0 = Rα,1 = 1, α = 1, 2, . . . , r , the generating (r ) function F1 (t) = n≥0 R1,n t n reads: √ m 1 Pr (t) (r ) 2 (8.3) F1 (t) = 1 + t , with Pm (t) = t Um √ − t , Pr +1 (t) t where Um are the Chebyshev polynomials of the second kind, with Um (2 cos θ ) = sin(m+1)θ sin θ . The corresponding limit r → ∞ for A∞/2 reads: √ 3 − t − 1 − 6t +t 2 (∞) F1 (t) = 1+t z(t) = = 1+t +2t 2 +6t 3 +22t 4 +90t 5 +394t 6 +· · · , 2 (8.4) where z(t) =

√ 1−t− 1−6t+t 2 2t

is the generating function of the large Schroeder numbers.


779

Proof. We use the expression (4.4) with all weights equal to 1. Then there is a recursion relation, Pr +1 (t) = (1 − t)Pr (t) − t Pr −1 (t), with P0 (t) = 1 and P1 (t) = 1 − t. The limit z(t) := limr →∞ Pr (t)/Pr +1 (t) therefore satisfies z = 1 − t + zt . It also coincides with the continued fraction (8.1) with yi = 1 for all i. The rate of growth of R1,n , considered as the number of paths of length 2n on M0 , can also be analyzed. The radius of convergence of the series F1(r ) (t) is given by the √ smallest root of the denominator of the fraction, which is Ur +1 √1t − t . As the zeros k of the Chebyshev polynomial Um are 2 cos π m+1 , k = 1, 2, . . . , m, when n → ∞,

R1,n

2n π π 2 + cos +1 ∼ Cr × cos r +2 r +2

(8.5)

for some constant Cr . The number of paths on Mmax is also simple to compute, as it is the number of Dyck paths of length 2n, which are limited by height 2r + 1: U2r +1 √1t 1 (r ) . F1 (t) = √ t U2r +2 √1 t

(8.6)

In the limit r → ∞, (∞)

F1

(t) =

1−

√ 1 − 4t = 1 + t + 2t 2 + 5t 3 + 14t 4 + 42t 5 + 429t 6 + · · · 2t

(8.7)

1 2n is the generating function of the Catalan numbers cn = n+1 n . As a result of the correspondence to domino tilings in the next section, this function counts the number of domino tilings of the (possibly truncated) halved Aztec diamond with d tiles missing. The function Rα,n counts the number of its indented versions. 2n The large n behavior is R1,n ∼ Cr 2 cos( 2rπ+3 ) , for some constant Cr . Now consider the expression as a function of the initial seed corresponding to M2 , with m α = r − α (the maximal descending Motzkin path), where R1,n+r −1 /R1,r −1 is as in Lemma 6.2. (r )

F1 (t) =

r −2

R1, j t j + t r −1 R1,r −1 r (t; y),

r (t; y) =

j=0

1 , 1 − t y1 ϕr ({y j }; {yi, j }; c)

where ϕr is defined in Lemma 6.2, and the arguments are the weights yi = yi (M2 ), i = 2, 3, . . . , 2r , yi, j = yi, j (M2 ), 2 ≤ j + 1 < i ≤ r + 1, and c = t y2r +1 (M2 ). Lemma 8.2. Setting Rα,r −α = Rα,r −α+1 = 1 (α ∈ Ir ), r (t) = 1 + t

√ Vr (t) , Vm (t) = (−1)m U2m ( t). Vr +1 (t)

(8.8)

780


Proof. Setting the weights y’s to 1 in 1/(1 − tϕr ), using the recursive definition of ϕr of Lemma 6.2 with ϕm ({y j }; {yi, j }; t y2m+1 )|y=1 = 1−θ1m (t) (m ≥ 0), ϕ0 = 1 and θ0 = 0,

t+θm−1 . Let 1 − t − θm = VVm+1 , then we have a three-term recursion we have θm = 1−t−θ m m−1 relation for Vm : Vm+1 = (2 − t)Vm − Vm−1 , where V0 = √ 1, V1 = 1 − t (or equivalently V−1 = V0 = 1). The solution is Vm (t) = (−1)m U2m ( t), by iterating the Chebyshev recursion relation Um+1 (x) = xUm (x) −Um−1 (x), to get Un+1 = (x 2 − 2)Un−1 −Un−3 , and identifying the initial conditions.

The asymptotic behavior of R1,n with √ these boundary conditions is found from the smallest root of the denominator U2r +2 ( t). Then R1,n ∼ Cr ×

2 sin

1 π 2(2r +3)

2n

(8.9)

as n → ∞, for some constant Cr . In this case, the limit r → ∞ of r is ill-defined. The Motzkin path M2 has m 1 = r − 1 → ∞, hence does not have a good limit in our picture. More importantly, the graph M2 has an infinite number of incoming edges at each vertex 1, 2, 3 . . . .. Hence, as we count paths according to their numbers n of down steps, we get an infinite number of paths from 0 to 0 as soon as n ≥ 2. This can also be seen in the fact that the rate of growth in Eq. (8.9) diverges as 2r/π . Example 8.3. In the cases r = 1, 2, 3, we have 1 (t) = 2 (t) =

1 1−

t t 1− 1−t

1 1−

t

1−

3 (t) =

√ 1−t U2 ( t) =1+t =1−t √ , 1 − 3t + t 2 U4 ( t)

t t+ 1−t t 1−t− 1−t

√ 1 − 3t + t 2 U4 ( t) =1+t =1−t √ , 1 − 6t + 5t 2 − t 3 U6 ( t)

1 1−

t

t t+ 1−t t 1−t− 1−t 1− t t+ 1−t 1−t− t 1−t− 1−t

√ 1 − 6t + 5t 2 − t 3 U6 ( t) =1+t =1−t √ . 1 − 10t + 15t 2 − 7t 3 + t 4 U8 ( t)

t+

More generally, for arbitrary choices of M and the associated initial conditions (r ) Rα,m α = Rα,m α +1 = 1, we have F1 (t) = L r (t)/K r +1 (t), where L m (t), K m (t) are polynomials of degree m with integer coefficients depending on M, and K m (0) = 1 and K m (t) ∼ (−t)m for large t. Indeed, from the results of Sect. 3, we know that K r +1 (t) is the generating function for hard particles on G M , with weight −t per particle. So the empty configuration contributes K r +1 (0) = 1. Due to connectivity of G M , the maximally occupied hard-particle configuration corresponds to all odd vertices 2i + 1, i = 0, 1, . . . , r being occupied by a particle (these are the duals of all the skeleton-tree edges of M , with odd labels). The corresponding weight is therefore (−t)r +1 . (r ) For r large, it is always possible to reexpress F1 (t) for any Motzkin path M obtained from M0 via finitely many mutations in the form of a rational fraction of the variables


781

t and Pr −1 (t)/Pr (t). Indeed, the branchings of the continued fraction expression for F1(r ) (t) all include some “tails” of the form 1/(1 − t ya − t (yb /(1 − t yc . . .))) which, when we set all yi = 1, reduce to some ratio Pm (t)/Pm+1 (t). By use of the 3-term recursion relation for Pm , this can always be rewritten as a rational fraction of t and Pr −1 /Pr . Then, in the case of A∞/2 , upon taking the limit r → ∞, we see that, as √ (r ) Pr −1 /Pr → z(t), F1 (t) takes the general form A(t) + B(t) 1 − 6t + t 2 , where A and B are rational fractions of t with integer coefficients. For instance let us take M = µ1 M0 . Then we have t R1,1 (r ) F1 (t) = R1,0 + y1 1−t y3,1 y=1 y2 +

1−t

= 2+

y 1−t y5 −t 1−t y6 −··· 7 y4 1−t y3 −t y 1−t y5 −t 1−t y6 −··· 7

t 1−

P t (t) P (t) 1−t 1+ Pr −2 (t) rP−1 r (t)

=2+t +

t2 2−t −

r −1

Pr −1 (t) Pr (t)

,

R 2 +R2,1

where we have used the Q-system to rewrite R1,0 = 1,1R1,2 = 2, and the recursion relation for P to eliminate Pr −2 . In the limit r → ∞, this yields for the A∞/2 Q-system with initial conditions R1,1 = R1,2 = 1 and Rα,0 = Rα,1 = 1 for all α ≥ 2: √ 2 2 t2 (∞) 2 1 − 5t + 2t + 1 − 6t + t F1 (t) = 2 + t + . =2+t +t 2 − t − z(t) 1 − 7t + 5t 2 − t 3 9. The Relation to Domino Tilings The solutions of the A∞ T -system, also known as the octahedron equation [10,17,21], Ti, j,k+1 Ti, j,k−1 = Ti, j+1,k Ti, j−1,k − Ti+1, j,k Ti−1, j,k

(9.1)

were given in [22] in terms of the partition function for domino tilings of the Aztec diamond and its generalizations. The Ar T -system (the same equation, with additional boundary conditions) is the fusion relation satisfied by the transfer matrix of the generalized, inhomogeneous Heisenberg spin chain [15]. The Q-system (1.1) is a limit of the T -system, with the index j dropped, whereas the renormalized Q-system (1.4) is obtained by formally dropping the index i. Therefore one should be able to recover the solutions of the Q-system from those of the T -system. Here, we give the explicit connection between the path formulation of Rα,n and domino tilings. In particular, we express Rα,n (xM ) for any Motzkin path M, as partition functions for tilings of certain domains of the square lattice by means of 2 × 1 and 1 × 2 dominos, and of rigid “defect” pairs of square tiles 1 × 1. 9.1. Paths and matchings of the Aztec diamond. In this subsection, we consider the solutions Rα,n as functions of x0 . For a restricted subset of the indices (α, n) there is an alternative combinatorial interpretation of Rα,n , related to the results of [22] on the solutions of the octahedron equation. This relation holds only for a restricted subset, because the

782


boundary conditions of [22] are incompatible with our truncation3 R0,n = Rr +1,n = 1. The connection is therefore valid only sufficiently far away from these boundaries. Consider a system of equations with indices (α, n) ∈ Z × Z (the A∞ Q-system): 2 + ρα+1,n ρα−1,n , α, n ∈ Z. ρα,n+1 ρα,n−1 = ρα,n

(9.2)

To write ρα,n as a function of {ρα,0 , ρα,1 }α∈Z we use the recursion relation ρα,n+1 =

2 +ρ ρα,n α+1,n ρα−1,n . ρα,n−1

By induction, ρα,n depends only on the subset of the initial data {ρβ,0 , ρβ,1 } with α − n + 1 ≤ β ≤ α + n − 1. Hence, if we identify ρα,i = Rα,i (i = 0, 1), then ρα,n = Rα,n if n ≤ Min(α, r + 1 − α).

(9.3)

We will show that Rα,n , with indices in the set, is the generating function for positively weighted matchings of the Aztec diamond. Recall the interpretation of ρα,n in terms of partition functions of matchings of Aztec diamonds [22]. The Aztec diamond centered at (α, 0) of radius n is the set Aα,n = (β, γ ) ∈ Z2 , | |α − β| + |γ | < n with boundary Bα,n = {(β, γ ) ∈ Z2 , | |α − β| + |γ | = n}. We denote by Aα,n and Bα,n the subsets of the square lattice with vertices in Z2 + ( 21 , 21 ) made of squares centered on the points of Aα,n and Bα,n respectively. Consider the matchings, or compact dimer coverings, of Aα,n . These are configurations of occupation of edges (including their vertices) of Aα,n by dimers, where each vertex is covered by exactly one dimer. Each square (β, γ ) ∈ Aα,n has either 2,1, or 0 edges covered, and we define (β, γ ) = −1, 0, 1, respectively. Each square (β, γ ) ∈ Bα,n has either 0 or 1 edge covered, and we define (β, γ ) = 0, 1 respectively. Let 0 if α + β + γ + n = 0 mod 2 θα,n (β, γ ) = . 1 otherwise Then the generating function for matchings of Aα,n is

(β,γ ) Mα,n = ρβ,θα,n (β,γ ) .

(9.4)

matchings of Aα,n (β,γ )∈Aα,n ∪Bα,n

The following is proved in [22]: Theorem 9.1. The solution ρα,n of the system (9.2), expressed as a function of {ρα,0 , ρα,1 }α∈Z , is equal to the generating function for matchings of Aα,n (9.4). Example 9.2. Figure 27 shows, for n = 3, (a) the eight matchings of the Aztec diamond Aα,3 , centered at (α, 0) after rotation by π/4. The corresponding weights of eq.(9.4) are (we denote by ai the weight of the configuration labeled i in Fig. 27): a1 = a4 =

2 2 2 Rα−1,1 Rα−1,1 Rα+1,1 Rα+2,1 Rα−2,1 Rα,1 Rα+2,1 , a2 = , a = , 3 2 Rα−1,0 Rα+1,0 Rα−1,0 Rα+1,0 Rα,0 Rα,1 2 Rα−2,1 Rα+1,1

3 2 2 Rα,1 Rα−1,1 Rα+1,1 Rα−1,1 Rα,1 Rα+1,1 , a5 = 2 , a6 = , a7 = a8 = , 2 Rα−1,0 Rα+1,0 Rα−1,0 Rα,0 Rα+1,0 Rα,0 Rα,0

3 This truncation is also different from the truncation considered in [12] for the so-called bounded octahedron recurrence.


783

(a)

(b) Fig. 27. (a) The 8 matchings of Aα,3 and (b) the non-intersecting G r -paths from (0, 0), (2, 0), (4, 0) to (6, 0), (8, 0), (10, 0), where r ≥ 4. The labeling corresponds to the weights ai , i = 1, 2, . . . , 8

8 and we have Rα,3 = i=1 ai . Note that this expression is valid only for α = 2, 3, . . . , r − 1, r ≥ 3, and provided we set R0,n = Rr +1,n = 1 for all n. This can be compared to our previous result for Rα,3 as solution of the Ar Q-system, obtained by cutting out the initial data at β = α − 3, hence working instead with Ar , r = r − α + 3. This allows to for A reinterpret Rα,3 for Ar as R3,3 r −α+3 , with initial data Rα ,i = Rα +α−3,i , i = 0, 1 )3 ) times and α = 1, 2, . . . , r − α + 3. As such it is interpreted as (Rα−2,0 )3 (= (R1,0 the generating function for triples of non-intersecting G r -paths (for any r ≥ 4) from the points (0, 0), (2, 0), (4, 0) to the points (6, 0), (8, 0), (10, 0), but with the weights R Rβ+α−4,0 Rα−2,1 = Rβ+α−4,0 Rβ+α−2,1 for and y2β−1 = Rβ+α−3,1 for β ≥ 2 and y2β y1 = Rα−2,0 Rβ+α−3,0 Rβ+α−3,1 β+α−3,0 Rβ+α−4,1 3 β ≥ 1. There are exactly 8 such triples, and their weights (multiplied by Rα−2,0 ) match one by one the weights ai above. They are represented in Fig. 27 (b).

9.2. Lattice paths and domino tilings. There is a standard bijection between domino r lattice paths. (see e.g. [13]). Consider the dominos as tiling a chessboard. tilings and G Then there are four types of 2 × 1 or 1 × 2 dominos (see Fig. 28). These are in bijection with the four path steps: (1, 1), (1, −1), (2, 0) = (1, 0) + (1, 0) or no step at all. Thus, a domino tiling of a domain D can be rephrased in terms of a configuration of ∞ non-intersecting paths connecting points on the boundary ∂D. Conversely, consider G lattice paths in the non-negative integer quadrant Z2+ from (0, 0) to (2n, 0) of length 2n. (We call these Aztec paths in this section.)

784


(a)

(b)

(c)

(d)

Fig. 28. A bijection between lattice paths and domino tilings. The dual square lattice is bicolored, so there are 4 possible domino tiles, labeled a, b, c, d. Each corresponds to a step of path as indicated: a : (1, 1), b : (1, −1), c : no step, and d : (2, 0) = (1, 0) + (1, 0), double horizontal step

Fig. 29. The bijection between paths of length 2n steps from (0, 0) to (2n, 0) and domino tilings of a “halved” Aztec diamond H An , of width 2n and height n + 1

Lemma 9.3. Aztec paths are in bijection with domino tilings of the “halved” Aztec diamond H An represented in Fig. 29, of width 2n and height n + 1, with a floor at height h = −1/2, a ceiling at height h = n + 1/2, and a white face on the bottom left. Proof. Any Aztec path may be obtained from the “maximal” Aztec path p0 (n up steps followed by n down steps) using the procedure shown in Fig. 30. Given the path p0 , use the tiles a, b, d of Fig. 28 to tile the region traversed by the path. One must then tile H An−1 , the half Aztec diamond below this region, with width 2n − 2 and with the bottom row removed. The bottom left and right corner tiles are frozen, and are of type c, as are all the tiles touching the left and right boundaries of H An−1 . By induction, the entire domain H An−1 must be tiled with c tiles, a unique configuration. Therefore there is a unique tiling of H An associated to the maximal path p0 . For any other path p, we use the sequence of moves in Fig. 30, from p0 to p, and apply the corresponding transformations on the associated tiling by use of the bijection of Fig. 28. This produces a unique tiling of H An for each Aztec path p. Conversely, given any tiling of H An , there is a unique path associated to it via the map in Fig. 28. A path can only enter the domain from its lower left end (1 × 2 domino of type a or 2 × 1 domino of type d), as all faces touching the left boundary are black, hence the tile d cannot be used. Similarly, the path can only exit via the lower right end


785

Fig. 30. The local moves (i-ii) which give any Aztec path p from the maximal path p0 , and an example of such a transformation. First one goes from p0 to the Dyck path pˆ closest to p by a sequence of “box removals” (i), before getting to p via “half-box removals” (ii)

(1 × 2 domino of type b or 2 × 1 domino of type d). We deduce that there is a unique path associated to the tiling, that goes from (0, 0) to (0, 2n). We assign weights to the steps of Aztec paths according to the height h at which the step starts: • Weight 1 for steps of type (1, 1), irrespectively of h. • Weight z 2h−2 for a step (1, −1) from height h to h − 1, h ≥ 2; weight z 1 for a step (1, −1) from h = 1 to h = 0. • Weight z 2h−1 for a step (2, 0) at height h, h ≥ 2; weights w0 , w1 for steps (2, 0) at heights h = 0 and h = 1, respectively. Domino tilings receive weights according to the bijection of Fig. 28, with domino c having weight 1. We will refer to those as weighted domino tilings in the following. 9.3. Q-system solutions as functions of x0 and domain tilings. Recall the interpretation of R1,n as the partition function of weighted paths on G˜ r . These are weighted Aztec paths with the following identification of weights: z 2r +1 = w0 = w1 = z j = 0 ( j ≥ 2r + 3), z i = yi (i = 1, 2, . . . , 2r ) and z 2r +2 = y2r +1 , with y j as in (3.17). The condition z j = 0 for j ≥ 2r + 4 corresponds to truncating the tiled domain to the inside of a strip of height r + 3, with a ceiling at height h = r + 5/2, with floor at height h = −1/2. The conditions w0 = w1 = z 2r +1 = z 2r +3 = 0 forbid the use of the tile d in the two bottom (−1/2 ≤ h ≤ 3/2) and top (r + 1/2 ≤ h ≤ r + 5/2) rows. Lemma 9.4. The families of α non-intersecting Aztec paths on Z2+ from (0, 0), (2, 0), . . . , (2α − 2, 0) to (2n, 0), (2n + 2, 0), . . . , 2n + 2α − 2, 0), with steps (1, 1), (1, −1) and double steps (2, 0) = (1, 0) + (1, 0), are in bijection with domino tilings of the “indented halved” Aztec diamond I H An,α represented in Fig. 31. Proof. Along the same lines as for the α = 1 case of Lemma 9.3, we first associate a tiling to any configuration of α non-intersecting Aztec paths. We simply use the fact that

786


Fig. 31. The bijection between families of α Aztec paths on Z2+ from (0, 0), (2, 0), . . . , (2α − 2, 0) to (2n, 0), (2n + 2, 0), . . . , 2n + 2α − 2, 0) and the domino tilings of the indented halved Aztec diamond I H An,α of width 2n + 2α − 2 and height n + α. Here, α = 3

any configuration of α non-intersecting Aztec paths may be obtained from the configuration where all paths are maximal via successive applications of the local moves of Fig. 30. We apply the construction of the proof of Lemma 9.3 to the bottom-most path, then to the next bottom-most, etc. Conversely, to construct the path configurations from the tilings, we note that the two successions of α − 1 indentations of the bottom boundary (left and right) impose the presence of α − 1 tiles a and one a or d on the left and α − 1 tiles b and one b or d on the right, and that paths can enter or exit nowhere else on the boundary, due to coloring constraints. This corresponds to exactly α paths touching the boundary of the domain, entirely determined by the tiling. Given that Rα,n is the partition function of α non-intersecting Aztec paths, with the identification of weights as above, this gives the relation to domino tilings. For sufficiently large r , the restriction of the weights effectively reduces the domain to be tiled to the shaded region in Fig. 31, as the tiling of the rest of the domain is entirely fixed (only dominos a on the left, and b on the right). The case α = 1 is also depicted with the same convention in Fig. 29. In the particular case when α = n, and r ≥ 2n − 1, the shaded domain to be tiled is exactly the Aztec diamond itself (with width and height 2n − 2), with no constraints. It is interesting to compare the partition function Dn (y) for the weighted tilings of this domain to the partition function for weighted Aztec diamond matchings (9.4). On one hand, we know that it is equal to Dn (y1 , y2 , . . . , y2r +1 ) =

Rn,n n y n y n−1 y n−2 · · · y R1,0 2n−2 1 2 4

,

(9.5)

n as the generating function for where we have used the interpretation of Rn,n /R1,0 G˜ r -paths, applied the path-tiling bijection of Sect. 9.2, and removed the contributions of the tiles outside of the shaded domain. As discussed in Sects. 7.3 and 9.1, the solution ρα,n of the A∞ Q-system for pairs α, n , the solution of the A Q-system for any r ≥ 2n −1, obeying (9.3) is identified with Rn,n r = Rα−n+β,1 for β = 1, 2, . . . , r , namely and with initial data Rβ,0 = Rα−n+β,0 and Rβ,1


787

with the weights (see also Example 9.2): Rβ+α−n,1 Rβ+α−n−1,0 for β = 1, 2, . . . , r + 1, Rβ+α−n,0 Rβ+α−n−1,1 Rβ+α−n−1,0 Rβ+α−n+1,1 = for β = 1, 2, . . . , r. Rβ+α−n,0 Rβ+α−n,1

y2β−1 = y2β

So we get two different ways of computing the same partition function ρα,n for Aztec . diamond matchings, one via the expression (9.4), the other as Rn,n As is apparent from Example 9.2, the weights leading to expression (9.4) and those of the families of non-intersecting paths are in bijection, but are incompatible with our weighted bijection. Indeed, if we rotate all the matchings of Fig. 27 (a) clockwise by a quarter-turn, we find that the corresponding dual domino tilings match the configurations of non-intersecting paths of Fig. 27 (b) via our bijection only in the cases 2, 3, 4, 7, 8, while some permutation of 1, 5, 6 must be applied. This suggest perhaps that other natural bijections should exist.

9.4. Solutions of the Q-system as functions of xM and tilings of domains with defects. We now ask for a tiling interpretation for the weighted M -paths, describing Rα,n as a function of the seed data xM . Consider first Mmax , with m α = α −1, where R1,n is a partition function of weighted Dyck paths on the strip 0 ≤ y ≤ 2r + 1 from (0, 0) to (2n, 0). Comparing this with the situation described in Lemma 9.3, we can make a direct connection by forbidding the steps (2, 0), or by using tiles a, b, c only. Thus, R1,n /R1,0 is the generating function for tilings of the domain H An by means of tiles a, b, c only, and with weights 1 per tiles a or c, and yi per tile b with center at height i − 1/2, for i = 1, 2, . . . , 2r + 1. More generally, let M = T2r +2 (i 1 , . . . , i s ), a skeleton tree as in Definition 5.7. As before we choose the weights w0 = w1 = 0, z j = 0 for j ≥ 2(2r + 1 − s) − 1, z 2(2r −s) = y2r +1 (M), and z 2 j−1 = 0 for all indices j ∈ [2, 2r − s − 1] \ {i 1 , i 2 , . . . , i s }, corresponding to missing horizontal edges at height j in M . The remaining z i ’s are identified with y j (M), in increasing order of indices. Again, the vanishing conditions impose some truncation to a strip h ∈ [−1/2, 2r − s + 3/2], and forbid the use of tiles d except at heights i 1 , i 2 , . . . , i s . Next, consider the case of the Motzkin path M(k) := µ1 µ2 . . . µk (M0 ), with m 1 = m 2 = · · · = m k = 1, and m j = 0 for j > k. The graph M(k) is G˜ r with one additional down-pointing edge k + 2 → k, assuming that r ≥ k + 1. We define modified Aztec paths (AM(k) -paths) as the corresponding M(k) -paths in the Z2+ quadrant (without the usual restriction due to r being finite). It has an extra step (0, −2) only from height h = k + 2 to height h = k. The weighting is as in Sect. 9.2, and the extra step receives a weight z k+2,k . Paths from (0, 0) to (2n, 0) are in bijection with tilings of the domain depicted in Fig. 32, by means of the domino tiles of Fig. 28, plus rigid pairs of 1 × 1 defects tiles (see the medallion in Fig. 32), with centers at positions (t − 1/2, k) and (t + 1/2, k + 2) with k + 2 ≤ t ≤ 2n − k, t integer. The “maximal” path p0 now involves a succession of n + 1 up steps (1, 1), followed by n − k down steps (1, −1), then one down step (0, −2) and finally k down steps (1, −1). The local moves (i) and (ii) of Fig. 30 must now be supplemented by new moves expressing the reduction of the special down steps (0, −2).

788


Fig. 32. The bijection between modified Aztec AM(k) -paths from (0, 0) to (2n, 0) and the domino tilings of a domain made of the half Aztec diamond (dashed lines), enhanced as indicated, tiled by means of dominos and rigid pairs of 1 × 1 squares with centers at height k + 2 and k (depicted in the medallion)

(n)

Fig. 33. The local rules for constructing the domain DM out of the maximal AM -path

The partition function for weighted AM(k) -paths is identified with R1,n expressed as a function of xM(k) (up to a multiplicative factor of R1,0 ), choosing the weights w0 = w1 = z 2k+1 = z 2k+3 = 0, z j = 0 for j ≥ 2r + 4, z 2k+2 = y2k+1 , z j = y j for j = 1, 2, . . . , 2k, and z k+2,k = yk+2,k = y2k y2k+2 /y2k+1 , where the y j ≡ y j (M(k)). Equivalently, R1,n /R1,0 is the partition function for weighted tilings of the domain of Fig. 32, delimited by the tiles corresponding to the maximal path p0 , and by the floor at height h = −1/2. For an arbitrary Motzkin path M, the AM Aztec paths are the M -paths without restriction due to r . (n)

Definition 9.5. The domain DM is defined as follows. Starting from M , remove any down-pointing edges i → j if M has an edge i → j with [ j, i] ⊂ [ j , i ] (strict inclusion). This leaves us with a set of “maximal” down-pointing edges. Define the “maximal” path from (0, 0) to (2n, 0), which has no horizontal step, and goes as far up and to the right as possible, namely starts with a maximal number of steps (1, 1), (n) descends via n maximal descending steps. The domain DM is constructed by associating boundary pieces to each step, as shown in Fig. 33. Finally, the domain is completed by a horizontal line at height h = −1/2. Lemma 9.6. The AM -paths from (0, 0) to (2n, 0) are in bijection with tilings of the (n) domain DM by means of the usual 2 × 1 and 1 × 2 tiles, plus rigid pairs of square 1 × 1


789

(n)

Fig. 34. A typical domain DM , with its maximal AM -path (in red), the target graph M and the Motzkin path M. We have indicated the two maximal edges, along which the maximal path takes its maximal descents

(n)

Fig. 35. The indented domain DM , serving for the representation of Rα,n in terms of tilings, and the target graph M (here α = 3). Note that the tiling involves defect pairs corresponding to descents of M , and that horizontal edges (tiles d) are forbidden at heights where M has no horizontal edges. Outside of the shaded blue domain, all the tiles are fixed by the indentations and the absence of d tile in the bottom row

tiles centered at points of the form (t − i− 2j−1 , j) and (t + i− 2j−1 , i) for all the pairs i, j of vertices connected by down-pointing edges on M . We refer to Fig. 34 for an example. We can now say that R1,n+m 1 /R1,m 1 is the partition (n) function of tilings of DM as described above. Finally, recall that Rα,n+m 1 /(R1,m 1 )n is the generating function for families of α strongly non-intersecting M -paths, starting at (0, 0), . . . , (2α − 2, 0) and ending at (2n, 0), . . . , (2n + 2α − 2, 0). These paths can be represented as tilings of the domain (n) DM with α −1 additional indentations on the bottom left and bottom right boundary (see Fig. 35), by means of dominos and rigid defect pairs corresponding to down-pointing

790


edges of M . The strong non-intersection of the paths imposes extra constraints on the tilings, by forbidding some local configurations.

10. Conclusion In this paper, we found a simple structural explanation for the cluster mutations, which allow for sweeping the set of possible initial data for the Q-system, in terms of simple local rearrangements of the continued fractions that generate the R1,n ’s. This local move allowing for generating mutations is reminiscent of the local “Yang-Baxter”-like relation used in [8] in the context of total positivity of the Grassmannian, as expressed through local positive transfer matrices for networks. We do not yet fully undertand this relation, although a partial explanation is found in [5]. In view of Example 6.8, we can think of the constructions of this paper as generalizations of the Stieltjes theorem [23], that now allow to re-express the series F(λ) = k≥0 (−1)k ak /λk+1 in different ways as (mutated) possibly multiply branching continued fractions, whose coefficients are particular combinations of Hankel determinants involving the sequence ak , each such rewriting corresponding to a Motzkin path. To make contact with our results, we simply have to take t = −1/λ, ak = R1,k and to identify Rα,n = n−α+1 with the Hankel determinants of Eq. (6.27). For each Motzkin α path M, we find a new continued fraction expression for F(λ) involving only the Hankel determinants corresponding to the cluster variable xM , via the weights yi (M). Equations called Q-systems exist for all simple Lie algebras. We have checked that in these cases, R1,n also satisfies linear recursion relations with constant coefficients. We expect the constructions of the present paper to generalize to all these cases. In particular, we expect cluster positivity to be a consequence of the LGV formula applied to counting possibly interacting families of non-intersecting paths. The Q-system is a specialization of the T -sytem, a discrete integrable system with one additional parameter. It was shown in [4] that T -systems can be considered as cluster algebras, in general, of infinite rank. The statistical models in this paper are particularly well-suited to the solution of the T -system [6]. In the case of the cluster variables corresponding to seeds of the cluster algebra outside the subgraph Gr , we do not have an interpretation in terms of a statistical model. Outside this subgraph, the exchange matrix B has entries which grow arbitrarily. This seems to suggest that the evolution in directions leading outside Gr is not integrable. Appendix A. The Case of A3 in the Heap Formulation In this appendix, we detail the heap interpretation of the solution R1,n of the A3 Q-system, as expressed in terms of the various cluster variables corresponding to the 9 seeds of the fundamental domain. We apply the same systematic method as in Example 4.8. For simplicity, we have just depicted in Fig. 36 the heap graphs, together with the mutations relating them, for each cluster variable of the fundamental domain. The corresponding (3) rearrangements of the generating function F1 (t) and new weights are given below, with the labeling of vertices indicated in Fig. 36. These are straightforwardly generated by carefully following the successive applications of Lemmas 4.5 and 4.6. Initial cluster variable: G 3 . We start from the graph G 3 and the associated heap gener(3) ating function F1 (t) = 1 + t y1 (G 3 ) (t), with


791

Fig. 36. The heap graphs corresponding to the R1,n ’s for the case A3 , and the corresponding mutations of seeds

(G 3 ) (t) =

1 1 − t y1 − t

1−t y3 −t

y2 y4 y 1−t y5 −t 1−t6y 7

with the y’s as in (3.17), namely R1,1 R2,1 R1,0 R2,1 R1,0 R3,1 , y2 = , y3 = , y4 = , R1,0 R1,0 R1,1 R2,0 R1,1 R2,0 R2,1 R2,0 R3,1 R2,0 R3,0 y5 = , y6 = , y7 = . R3,0 R2,1 R3,0 R3,1 R3,1

y1 =

Note that the weights are related via y1 y3 y5 y7 = 1. Mutation µ1 (G 3 ). We apply Lemma 4.6 to vertices 1, 2 and the structure attached to 3. (3) This gives F1 (t) = 1 + t y1 (µ1 (G 3 )) (t), with (µ1 (G 3 )) (t) =

1 z1

1−t

z8 z 1−t z 5 −t 1−t6z 7 z4 1−t z 3 −t z 1−t z 5 −t 1−t6z z2 +

1−t

7

with z 1 = y1 + y2 , z 2 =

y2 y3 y2 y4 y1 y3 y1 y4 , z8 = , z3 = , z4 = , z 5 = y5 , z 6 = y6 , z 7 = y7 y1 + y2 y1 + y2 y1 + y2 y1 + y2

792


2 + R : namely, upon using the Q-system relation R1,0 R1,2 = R1,1 2,1 2 2 R R2,1 R1,1 R1,2 R1,1 R2,1 3,1 , z2 = , z3 = , z4 = , R1,1 R2,0 R1,1 R1,2 R2,0 R1,2 R2,0 R2,1 R1,2 R2,0 R3,1 R2,0 R3,0 R3,1 z5 = , z6 = , z7 = , z8 = . R3,0 R2,1 R3,0 R3,1 R3,1 R2,0 R1,2

z1 =

Note that there is one more z than y’s, but that we now have two relations z 1 z 3 z 5 z 7 = 1 and z 3 z 8 = z 2 z 4 . Mutation µ2 (G 3 ). We apply Lemma 4.6 to vertices 3, 4 and the descendent structure attached to 4. This gives F1(3) (t) = 1 + t y1 (µ2 (G 3 )) (t), with (µ2 (G 3 )) (t) =

1 1 − t x1 − t

x2

1−t

x3 x x4 + 1−t8x 7 1−t x6 1−t x5 −t 1−t x 7

with x1 = y1 , x2 = y2 , x3 = y3 + y4 , x4 = x5 =

y4 y5 y4 y6 , x8 = , y3 + y4 y3 + y4

y3 y5 y3 y6 , x6 = , x7 = y7 , y3 + y4 y3 + y4

2 +R namely, upon using the Q-system relation R2,0 R2,2 = R2,1 1,1 R3,1 : 2

x1 =

R1,1 R3,1 R1,1 R2,1 R1,0 R2,2 , x2 = , x3 = , x4 = , R1,0 R1,0 R1,1 R1,1 R2,1 R3,0 R2,1 R2,2

x5 =

R2,1 R2,1 R3,1 R3,0 R1,1 , x6 = , x7 = , x8 = . R3,0 R2,2 R3,0 R3,1 R2,2 R3,1 R3,0 R2,2

2

As for the z’s above, there is one more x than y’s, but they obey two relations: x1 x3 x5 x7 = 1 and x5 x8 = x4 x6 . Mutation µ3 (G 3 ). This is part of the sequence of graphs of Example 5.6. It is obtained by applying Lemma 4.6 to the vertices 5, 6, 7 of G 3 . This gives F1(3) (t) = 1+t y1 (µ3 (G 3 )) (t), with (µ3 G 3 ) (t) =

1 1 − t t1 − t

,

t2 1−t t3 −t

1−t

t4

t5 t 1−t 1−t6 t

7

with t1 = y1 , t2 = y2 , t3 = y3 , t4 = y4 , t5 = y5 + y6 , t6 =

y6 y7 y5 y7 , t7 = . y5 + y6 y5 + y6


793

2 +R : namely, upon using the Q-system relation R3,0 R3,2 = R3,1 2,1

R1,1 R2,1 R1,0 R2,1 R1,0 R3,1 , t2 = , t3 = , t4 = , R1,0 R1,0 R1,1 R2,0 R1,1 R2,0 R2,1 R2,0 R3,2 R2,1 R3,1 t5 = , t6 = , t7 = . R2,1 R3,1 R3,1 R3,2 R3,2

t1 =

Note that the t satisfy the relation t1 t3 t5 t7 = 1. Mutation µ1 µ2 (G 3 ). We apply Lemma 4.6 to the vertices 1, 2, and the substructure attached to 3 in the graph µ2 (G 3 ) (see Fig. 36). This gives (µ1 µ2 (G 3 )) (t) = (µ2 (G 3 )) (t), with (µ1 µ2 (G 3 )) (t) =

1 1−t

(A.1)

u1 1−t

u2

u8 u 4 + 1−tu 7 1−tu 3 −t u6 1−tu 5 −t 1−tu 7

with u 1 = x1 + x2 , u 2 =

x2 x3 x1 x3 , u3 = , u 4 = x4 , u 5 = x5 , u 6 = x6 , u 7 = x7 , u 8 = x8 , x1 + x2 x1 + x2


u1 =

2 R1,1 R3,1 R1,2 R2,2 R1,1 R2,2 , u2 = , u3 = , u4 = , R1,1 R1,1 R1,2 R1,2 R2,1 R3,0 R2,1 R2,2

u5 =

2 R2,1 R2,1 R3,1 R3,0 R1,1 , u6 = , u7 = , u8 = . R3,0 R2,2 R3,0 R3,1 R2,2 R3,1 R3,0 R2,2

Note that the u’s satisfy the two relations u 1 u 3 u 5 u 7 = 1 and u 5 u 8 = u 4 u 6 . It is instructive to see how to arrive at the same result from the sequence of mutations µ2 µ1 (G 3 ) = µ1 µ2 (G 3 ). We now start from the graph µ1 (G 3 ), and apply Lemma 4.6 with (i) the vertex 2 and its substructure, (ii) the vertex 8 and its substructure common with that of vertex 2 (iii) its attached substructure not common with that of the vertex 2, namely with z2 a= , z4 1 − t z3 − t z6 1−t z 5 −t 1−t z

7

z8 b= 1 − t z3 − t

z4 z 1−t z 5 −t 1−t6z

c = z5 +

, 7

z6 . 1 − t z7

The result is (µ2 µ1 (G 3 )) (t) =

1

(A.2)

u 1

1−t

u 2

1−t

⎞

⎛ z4 ⎝1−t z 3 −t z 1−t z 5 −t 1−t6z

7

⎛

⎜ ⎜ ⎠⎜1−t ⎜ ⎝

u 8 1−tu 7 u 6 1−tu 5 −t 1−tu 7 u 4 +

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

794


with y3 + y4 x2 x3 = = u2, y1 + y2 x1 + x2 z8 z5 y4 y5 z8 z6 y4 y6 u 4 = = = x4 = u 4 , u 8 = = = x8 = u 8 , z2 + z8 y3 + y4 z2 + z8 y3 + y4 z2 z5 z2 z6 y3 y6 u 5 = = x5 = u 5 , u 6 = = = x6 = u 6 , u 7 = z 7 = x7 = u 7 , z2 + z8 z2 + z8 y3 + y4

u 1 = z 1 = y1 + y2 = u 1 , u 2 = z 2 + z 8 = y2

where we have used the expressions of the z’s and the x’s in terms of the y’s. Applying again Lemma 4.6 to the first factor under u 2 , with a = z 3 , b = z 4 , c = z 5 + z 6 /(1 − t z 7 ), we may rewrite: z3 +

z4 = z6 1 − t z 5 − t 1−t z7 1−t

u 3

u 4

1−tu 5 −t

(A.3) u 6

1−tu 7

with y3 + y4 x1 x3 z4 z5 z8 z5 = = u 3 , u 4 = = = u 4 = u 4 , y1 + y2 x1 + x2 z3 + z4 z2 + z8 z3 z5 z2 z5 u 5 = = = u 5 = u 5 , u 7 = z 7 = u 7 , z3 + z4 z2 + z8

u 3 = z 3 + z 4 = y1

where we have used the relation z 3 /z 4 = z 2 /z 8 between the z’s. Substituting this into (A.2), we finally recover Eq. (A.1). This case is made particularly cumbersome by the fact that the vertex 2 contains a substructure (which has not yet been the case so far), encoded in the fraction (A.3). From this example, we learn that the mutation µ2 consists actually of two simultaneous applications of Lemma 4.6, one “usual” and one within the substructure attached to the initial vertex 2. Mutation µ1 µ3 (G 3 ). We apply Lemma 4.6 to the vertices 1, 2, and the substructure attached to 2 in the graph µ3 (G 3 ) (see Fig. 36). This gives (µ1 µ3 (G 3 )) (t) = (µ3 (G 3 )) (t), with (µ1 µ3 (G 3 )) (t) =

1 1−t

⎛

w1

w8 w2 + w5 1−t ⎜ w6 1−t 1−tw ⎜ 7 1−t ⎜ w ⎜ 1−tw −t 4 w5 3 ⎝ 1−t w6 1−t 1−tw 7

⎞

(A.4)

⎟ ⎟ ⎟ ⎟ ⎠

with w1 = t1 + t2 , w2 =

t2 t3 t2 t4 t1 t3 t1 t4 , w8 = , w3 = , w4 = , w5 = t5 , w6 = t6 , w7 = t7 , t1 + t2 t1 + t2 t1 + t2 t1 + t2

2 + R : namely, upon using the Q-system relation R1,0 R1,2 = R1,1 2,1 2 2 R R2,1 R1,1 R1,2 R3,1 3,1 , w2 = , w3 = , w4 = , R1,1 R2,0 R1,1 R1,2 R1,2 R2,0 R2,0 R2,1 R1,2 R2,0 R3,2 R2,1 R3,1 R3,1 w5 = , w6 = , w7 = , w8 = . R2,1 R3,1 R3,1 R3,2 R3,2 R2,0 R1,2

w1 =


795

Note the relations w1 w3 w5 w7 = 1 and w2 w4 = w3 w8 . For completeness, let us compute the function (µ3 µ1 (G 3 )) = (µ1 (G 3 )) , and check that indeed µ1 µ3 (G 3 ) = µ3 µ1 (G 3 ) yield the same result. We now apply Lemma 4.6 with the vertices 5, 6 and 7 of the graph µ1 (G 3 ), with respectively a = z 5 , b = z 6 and c = z 7 , while d = 0. This gives (µ3 µ1 (G 3 )) =

1 w1

1−t

w2 + 1−t

(A.5) w8 w5

w6 1−tw7 w4 1−tw3 −t w5 1−t w6 1−t 1−tw7 1−t

1−t

with y2 y3 t2 t3 = = w2 , y1 + y2 t1 + t2 y2 y4 t2 t4 y1 y3 t1 t3 w8 = z 8 = = = w8 , w3 = z 3 = = = w3 , y1 + y2 t1 + t2 y1 + y2 t1 + t2 y1 y4 t1 t4 w4 = z 4 = = = w4 , w5 = z 5 + z 6 = y5 + y6 = t5 = w5 , y1 + y2 t1 + t2 z6 z7 y6 y7 z5 z7 y5 y7 w6 = = = t6 = w6 , w7 = = = t7 = w 7 , z5 + z6 y5 + y6 z5 + z6 y5 + y6

w1 = z 1 = y1 + y2 = t1 + t2 = w1 , w2 = z 2 =

where the various identifications are made by use of the expressions of the z’s and t’s in terms of the y’s. We deduce that the expressions (A.4) and (A.5) are identical. Mutation µ2 µ3 (G 3 ). This is part of the sequence of graphs of Example 5.6. It is obtained by applying Lemma 4.6 to the vertices 3, 4 and the structure attached to 5 of the graph µ3 (G 3 ) (see Fig. 36). This gives (µ2 µ3 (G 3 )) (t) = (µ3 (G 3 )) (t), with (µ2 µ3 (G 3 )) (t) =

1 1 − ts1 − t

s2

1−t

(A.6)

s3 s4 1−t s6 1−ts5 −t 1−ts 7

with s 1 = t1 , s 2 = t2 , s 3 = t3 + t 4 , s 4 =

t4 t5 t3 t5 , s5 = , s 6 = t6 , s 7 = t7 , t3 + t4 t3 + t4

2 +R namely, upon using the Q-system relation R2,0 R2,2 = R2,1 1,1 R3,1 :

R1,1 R2,1 R1,0 R2,2 R1,1 R3,2 , s2 = , s3 = , s4 = , R1,0 R1,0 R1,1 R1,1 R2,1 R2,1 R2,2 R2,1 R3,2 R2,1 R3,1 s5 = , s6 = , s7 = . R3,1 R2,2 R3,1 R3,2 R3,2

s1 =

Note the relation s1 s3 s5 s7 = 1. For completeness, let us compute the function (µ3 µ2 (G 3 )) = (µ2 (G 3 )) , and check that indeed µ2 µ3 (G 3 ) = µ3 µ2 (G 3 ) yield the same

796


result. We now apply Lemma 4.6 on the graph µ2 (G 3 ), with (i) the vertex 4 and its substructure (ii) the vertex 8 and its substructure common to vertex 4, and the substructure of vertex 8 not common to vertex 4. This amounts to taking a = x4 /(1−t x5 −t x6 /(1−t x7 )), b = x8 /(1 − t x5 − t x6 /(1 − t x7 )) and c = x7 , while d = 0. This yields (µ3 µ2 (G 3 )) =

1 1 − ts1

s2

−t 1−t 1−t

(A.7) s3 s4

x 1−t x5 −t 1−t6x

7

1−t

s6 1−ts7

with s1 = x1 = y1 = t1 = s1 , s2 = x2 = y2 = t2 = s2 , s3 = x3 = y3 + y4 = t3 + t4 = s3 , y5 + y6 t4 t5 x7 x8 y6 y7 s4 = x4 + x8 = y4 = = s4 , s6 = = = t6 = s 6 , y3 + y4 t3 + t4 x4 + x8 y5 + y6 x7 x4 y5 y7 s7 = = = t7 = s 7 , x4 + x8 y5 + y6 where identifications are made by expressing the x’s, s’s and t’s in terms of the y’s. As in the case of µ2 µ1 (G 3 ), we see that we must still apply Lemma 4.6 to the first factor under s4 in (A.7), which corresponds to the substructure attached to the initial vertex 4. This allows to rewrite x5 +

s5 x6 = s6 1 − t x7 1 − t 1−ts 7

with y5 + y6 t3 t5 x6 x7 y6 y7 = = s5 , s6 = = = t6 = s 6 , y3 + y4 t3 + t4 x5 + x6 y5 + y6 x5 x7 y5 y7 s7 = = = t7 = s 7 . x5 + x6 y5 + y6

s5 = x5 + x6 = y3

Substituting these into (A.7) allows to identify it with (A.6). Mutation µ6 µ2 µ3 (G 3 ) = H3 . This is part of the sequence of graphs of Example 5.6. It is obtained by applying Lemma 4.6 to the vertices 5, 6, 7 of the graph µ2 µ3 (G 3 ) (see Fig. 36). This gives (µ6 µ2 µ3 (G 3 )) (t) = (µ2 µ3 (G 3 )) (t), with (µ6 µ2 µ3 (G 3 )) (t) =

1 1 − tr1 − t

r2 1−t 1−t

r3 r4 r5 1−t r6 1−t 1−tr 7

with r 1 = s1 , r 2 = s2 , r 3 = s3 , r 4 = s4 , r 5 = s5 + s6 , r 6 =

s6 s7 s5 s7 , r7 = , s5 + s6 s5 + s6


797


R1,1 R2,1 R1,0 R2,2 R1,1 R3,2 , r2 = , r3 = , r4 = , R1,0 R1,0 R1,1 R1,1 R2,1 R2,1 R2,2 R2,1 R3,3 R2,2 R3,2 r5 = , r6 = , r7 = . R2,2 R3,2 R3,2 R3,3 R3,3

r1 =

Note the relation r1r3r5r7 = 1. Mutation µ4 µ1 µ2 (G 3 ). It is obtained by applying Lemma 4.6 to the vertices 1, 2 and the structure attached to 2 of the graph µ1 µ2 (G 3 ) (see Fig. 36). Writing (µ1 µ2 (G 3 )) (t) = 1 + tu 1 (µ1 µ2 (G 3 )) (t), we have (µ4 µ1 µ2 (G 3 )) (t) = (µ1 µ2 (G 3 )) (t), with (µ4 µ1 µ2 (G 3 )) (t) =

1 1−t

v1

⎞ v10 v9 + 1−tv 7 ⎟ ⎜ v2 + v6 ⎟ ⎜ 1−tv5 −t 1−tv ⎜ 7 ⎞⎟ 1−t ⎜ ⎛ ⎟ v8 v4 + 1−tv ⎟ ⎜ 7 ⎠ ⎝ 1−t ⎝v3 + v6 ⎠ 1−tv5 −t 1−tv 7 ⎛

(A.8)

with u2u3 u2u4 u2u8 u1u3 , v9 = , v10 = , v3 = , u1 + u2 u1 + u2 u1 + u2 u1 + u2 u1u4 u1u8 v4 = , v5 = u 5 , v6 = u 6 , v7 = u 7 , v8 = , u1 + u2 u1 + u2

v1 = u 1 + u 2 , v2 =


v1 =

2 2 R2 R2,2 R1,2 R1,3 R1,2 R2,2 3,1 , v2 = , v3 = , v4 = , R1,2 R2,1 R1,2 R1,3 R2,1 R1,3 R3,0 R2,1 R2,2 R1,3

v5 =

2 2 R2,1 R1,2 R2,1 R3,1 R3,0 , v6 = , v7 = , v8 = , R3,0 R2,2 R3,0 R3,1 R2,2 R3,1 R3,0 R2,2 R1,3

v9 =

2 R3,1

R3,0 R2,1 R1,3

, v10 =

1 . R3,0 R1,3

Note that as we have 3 more v’s than y’s, we expect four relations between them. These read: v1 v3 v5 v7 = 1, v2 v4 = v3 v9 , v8 v9 = v4 v10 and v2 v8 = v3 v10 . We conclude that for r = 3, R1,n is a positive Laurent polynomial of all the mutations of the initial data x0 in the fundamental domain F3 . Appendix B. The Case of A3 in the Path Formulation In this appendix, we detail the path interpretation of the solution R1,n of the A3 Q-system, as expressed in terms of the various initial data corresponding to the 9 cluster variables of the fundamental domain F3 , labelled by the 9 Motzkin paths of Fig. 2. As a direct illustration of the constructions of Sect. 6, we list below for each of the variables the corresponding transfer matrix T , and the generating function nine cluster (I − T )−1 0,0 , in terms of dummy weights yi and yi, j , that are actually shorthand

798


Fig. 37. The target graphs M for the nine seeds of the case A3 (labelled by the 9 Motzkin paths of Fig. 2), and the corresponding mutations. We have indicated vertex and edge labels

notations for the weights yi (M) and their redundant counterparts, to be extracted from Theorem 6.7 and eq. (5.5). The corresponding graphs are represented in Fig. 37, together with the mutations relating the associated cluster variables. For M0 = {(0, 1), (0, 2), (0, 3)}. The corresponding target graph is M0 = G˜ 3 . The transfer matrix and the associated generating function are:

TM0

⎛ 0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜0 ⎜ ⎝0 0

t y1 0 1 0 0 0 0 0

0 t y2 0 1 1 0 0 0

0 0 t y3 0 0 0 0 0

0 0 t y4 0 0 1 1 0

0 0 0 0 t y5 0 0 0

0 0 0 0 t y6 0 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0


(I − TM0 )−1

0,0

=

799

1 1−t

.

y1 1−t

y2 1−t y3 −t

y4 y6 1−t y5 −t 1−t y 7

For M1 = µ1 M0 = {(1, 1), (0, 2), (0, 3)}, with y3,1 = y2 y4 /y3 : ⎛ 0 t y1 0 0 0 0 ⎜1 0 t y2 0 t y 0 3,1 ⎜ ⎜0 1 0 t y3 t y4 0 ⎜ 1 0 0 0 ⎜0 0 TM1 = ⎜ 1 0 0 t y5 ⎜0 0 ⎜0 0 0 0 1 0 ⎜ ⎝0 0 0 0 1 0 0 0 0 0 0 0 1 (I − TM1 )−1 = . y1 0,0 1−t y3,1

0 0 0 0 t y6 0 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0

y2 +

1−t

y 1−t y5 −t 1−t6y 7 y4 1−t y3 −t y6 1−t y5 −t 1−t y 7

For M2 = µ2 M0 = {(0, 1), (1, 2), (0, 3)}, with y5,3 = y4 y6 /y5 : ⎛ 0 t y1 0 0 0 0 0 ⎜1 0 t y2 0 0 0 0 ⎜ ⎜0 1 0 t y3 0 0 0 ⎜ 1 0 t y4 0 t y5,3 ⎜0 0 TM2 = ⎜ 0 1 0 t y5 t y6 ⎜0 0 ⎜0 0 0 0 1 0 0 ⎜ ⎝0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 (I − TM2 )−1 = . y1 0,0 1−t y2 1−t

1−t

y3 y5,3 y4 + 1−t y7 1−t y 1−t y5 −t 1−t6y 7

For M3 = µ3 M0 = {(0, 1), (0, 2), (1, 3)}: ⎛ 0 t y1 0 0 ⎜1 0 t y2 0 ⎜ ⎜0 1 0 t y3 ⎜ 1 0 ⎜0 0 TM3 = ⎜ 1 0 ⎜0 0 ⎜0 0 0 0 ⎜ ⎝0 0 0 0 0 0 0 0 1 (I − TM3 )−1 = y1 0,0 1−t y2 1−t

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎟ ⎟ t y7 ⎠ 0

1−t y3 −t

1−t

0 0 t y4 0 0 1 0 0

y4 y5 y 1−t 1−t6y 7

0 0 0 0 t y5 0 1 0 .

0 0 0 0 0 t y6 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0

800


For M2,1 = µ2 µ1 M0 = {(1, 1), (1, 2), (0, 3)}: With y4,2 = y4 y6 /y5 , ⎛

TM2,1

(I − TM2,1 )−1

0,0

0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜0 ⎜ ⎝0 0 =

1−t

t y1 0 1 0 0 0 0 0

0 t y2 0 1 1 0 0 0 1

0 0 t y3 0 0 0 0 0

0 0 t y4 0 0 1 1 0

0 0 t y4,2 0 t y6 0 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0

.

y1

1−t

0 0 0 0 t y5 0 0 0

y2 y4,2 y4 + 1−t y7 1−t y3 −t y 1−t y5 −t 1−t6y 7

For M3,1 = µ3 µ1 M0 = {(1, 1), (0, 2), (1, 3)}: With y3,1 = y2 y4 /y3 , ⎛

TM3,1

(I − TM3,1 )−1

0,0

0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 = ⎜ ⎜0 ⎜0 ⎜ ⎝0 0 =

t y1 0 1 0 0 0 0 0

0 0 t y2 0 0 t y3 1 0 1 0 0 0 0 0 0 0 1

0 t y3,1 t y4 0 0 1 0 0

0 0 0 0 0 t y6 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0

.

y1

1−t

0 0 0 0 t y5 0 1 0

y3,1 y5 1−t y 1−t 1−t6y 7 y4 1−t y3 −t y5 1−t y 1−t 1−t6y 7 y2 +

1−t

For M3,2 = µ3 µ2 M0 = {(0, 1), (1, 2), (1, 3)}:

TM3,2

(I − TM3,2 )−1

0,0

⎛ 0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜0 ⎜ ⎝0 0 =

1−t

t y1 0 1 0 0 0 0 0

0 t y2 0 1 0 0 0 0 1

0 0 t y3 0 1 0 0 0 y1

1−t

0 0 0 t y4 0 1 1 0

y2 y3 1−t y4 1−t y 1−t y5 −t 1−t6y 7

0 0 0 0 t y5 0 0 0 .

0 0 0 0 t y6 0 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0


801

For M4,2,1 = µ4 µ2 µ1 M0 = {(2, 1), (1, 2), (0, 3)}: With y3,1 = y2 y4 /y3 , y4,2 = y4 y6 /y5 , y4,1 = y2 y4 y6 /(y3 y5 ),

TM4,2,1

(I − TM4,2,1 )−1

0,0

⎛ 0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜0 ⎜ ⎝0 0 =

t y1 0 1 0 0 0 0 0

0 t y2 0 1 1 0 0 0 1

0 0 t y3 0 0 0 0 0

0 t y3,1 t y4 0 0 1 1 0

1−t

0 t y4,1 t y4,2 0 t y6 0 0 1

0 0 0 0 t y5 0 1 0

0 0 0 0 0 t y6 0 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ 0⎟ ⎟ t y7 ⎠ 0

.

y1

1−t

0 0 0 0 t y5 0 0 0

y4,1 y3,1 + 1−t y7 y2 + y 1−t y5 −t 1−t6y 7 y4,2 y4 + 1−t y7 1−t y3 −t y 1−t y5 −t 1−t6y 7

For M6,3,2 = µ6 µ3 µ2 M0 = {(0, 1), (1, 2), (2, 3)}: ⎛

TM6,3,2

(I − TM6,3,2 )−1

0,0

0 ⎜1 ⎜ ⎜0 ⎜ ⎜0 =⎜ ⎜0 ⎜0 ⎜ ⎝0 0 =

1−t

t y1 0 1 0 0 0 0 0

0 t y2 0 1 0 0 0 0 1

0 0 t y3 0 1 0 0 0 y1

1−t

0 0 0 t y4 0 1 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ , 0⎟ ⎟ ⎟ 0⎟ t y7 ⎠ 0

.

y2 y3 1−t y4 1−t y5 1−t y 1−t 1−t6y 7

Note finally that the graphs in Figs. 36 and 37 are duals of each other. This duality is best seen by expressing a bijection between the paths (represented with their time extension) and the corresponding heaps, by associating a disc of the heap to each descent of the path. The heap graph is simply the graph whose vertices stand for the discs and whose edges indicate the pairs of overlapping discs. Acknowledgements. We thank M. Bergvelt, C. Krattenthaler, A. Postnikov, H. Thomas, and particularly S. Fomin for many interesting discussions. We thank the organizers of the semester “Combinatorial Representation Theory” and the Mathematical Science Research Institute, Berkeley, CA, USA for hospitality, as well as the organizers of the program “Combinatorics and Statistical Physics” and the Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria. R.K. acknowledges the hospitality of the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France. P. D.F. acknowledges the support of the ENIGMA research training network MRTN-CT-2004-5652, the ANR program GIMP, and the ESF program MISGAM. R. K.’s research is supported by NSF grants DMS 0500759 and 0802511.

802


References 1. Beals, R., Sattinger, D.H., Szmigielski, J.: Continued fractions and integrable systems. J. Comput. Appl. Math. 153(1-2), 47–60 (2003) 2. Caldero, P., Reineke, M.: On the quiver Grassmannian in the acyclic case. J. Pure Appl. Algebra 212(11), 2369–2380 (2008) 3. Di Francesco, P., Kedem, R.: Proof of the combinatorial Kirillov-Reshetikhin conjecture. Int. Math. Res. Notices 2008, rnn006–57 (2008) 4. Di Francesco, P., Kedem, R.: Q-systems as cluster algebras II. http://arXiv.org/abs/0803.0362v2[math. RT], 2008 5. Di Francesco, P., Kedem, R.: Q-systems cluster algebras, paths and total positivity. http://arXiv.org/abs/ 0906.3421v2[math.CO], 2009 6. Di Francesco P., Kedem, R.: Positivity of the T -system cluster algebra, preprint 7. Di Francesco, P., Kedem, R.: In progress 8. Fomin, S., Zelevinsky, A.: Total positivity: tests and parameterizations. Math. Intelligencer 22, 23–33 (2000) 9. Fomin, S., Zelevinsky, A.: Cluster algebras I. J. Amer. Math. Soc. 15(2), 497–529 (2002) 10. Fomin, S., Zelevinsky, A.: The Laurent phenomenon. Adv. in Appl. Math. 28(2), 119–144 (2002) 11. Gessel, I.M., Viennot, X.: Binomial determinants, paths and hook formulae. Adv. Math. 58, 300–321 (1985) 12. Henriquès, A.: A periodicity theorem for the octahedron recurrence. J. Alg. Comb. 26(1), 1–26 (2007) 13. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Prob. Th. Rel. Fields 123(2), 225–280 (2002) 14. Kedem, R.: Q-systems as cluster algebras. J. Phys. A: Math. Theor. 41, 194011 (2008) 15. Kirillov, A.N., Reshetikhin, N.Yu.: Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math. 52, 3156–3164 (1990) 16. Kontsevich, M.: Private communication 17. Knutson, A., Tao, T., Woodward, C.: A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. Electr. J. Combin. 11, RP 61 (2004) 18. Krattenthaler, C.: The theory of heaps and the Cartier-Foata monoid. Appendix to the electronic republication of Problèmes combinatoires de commutation et réarrangements, by P. Cartier and D. Foata, available at http://mathnet.preprints.org/EMIS/journals/SLC/books/cartfao.pdf 19. Lindström, B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5, 85–90 (1973) 20. Postnikov, A.: Total positivity, Grassmannians, and networks. http://arXiv.org/abs/math/ 0609764v1[math.CO], 2006 21. Robbins, D., Rumsey, H.: Determinants and alternating sign matrices. Adv. in Math. 62, 169–184 (1986) 22. Speyer, D.: Perfect matchings and the octahedron recurrence. J. Alg. Comb. 25(3), 309–348 (2007) 23. Stieltjes, T.J.: Recherches sur les fractions continues. In: Oeuvres complètes de Thomas Jan Stieltjes. Vol. II, No. LXXXI, Groningen: P. Nordhoff, 1918 pp. 402–566 (see in particular Eq. (1), p 402 and Eq. (7) p 427) 24. Viennot, X.: Heaps of pieces I: Basic definitions and combinatorial lemmas. In: Combinatoire énumérative, eds Labelle, G., Leroux, P. Lecture Notes in Math. 1234, Berlin: Springer-Verlag, 1986, pp. 321–325 Communicated by L. Takhtajan


Communications in


Remarks on Chern–Simons Invariants Alberto S. Cattaneo1 , Pavel Mnëv2, 1 Institut für Mathematik, Universität Zürich–Irchel, Winterthurerstrasse 190,

CH-8057 Zürich, Switzerland. E-mail: [email protected]

2 Petersburg Department of V. A. Steklov Institute of Mathematics, Fontanka 27,

191023 St. Petersburg, Russia. E-mail: [email protected] Received: 18 December 2008 / Accepted: 31 August 2009 Published online: 17 November 2009 – © Springer-Verlag 2009

Abstract: The perturbative Chern–Simons theory is studied in a finite-dimensional version or assuming that the propagator satisfies certain properties (as is the case, e.g., with the propagator defined by Axelrod and Singer). It turns out that the effective BV action is a function on cohomology (with shifted degrees) that solves the quantum master equation and is defined modulo certain canonical transformations that can be characterized completely. Out of it one obtains invariants. Contents 1. 2. 3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective BV Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toy Model for Effective Chern–Simons Theory on Zero-Modes: Effective BV Action on Cohomology of dg Frobenius Algebras . . . . . . . . . . . . . . 3.1 Abstract Chern–Simons action from a dg Frobenius algebra . . . . . . . 3.2 Effective action on cohomology . . . . . . . . . . . . . . . . . . . . . 3.3 Dependence of the effective action on cohomology on induction data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Comments on relaxing the condition K 2 = 0 for chain homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Invariants from the relaxed effective action . . . . . . . . . . . . . 3.6 Examples of dg Frobenius algebras . . . . . . . . . . . . . . . . . . . . Three-Manifold Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Induction data and the propagator . . . . . . . . . . . . . . . . . . . . .

804 805 809 810 811 816 817 821 823 826 828 828

This work has been partially supported by SNF Grant 20-113439, by the European Union through the FP6 Marie Curie RTN ENIGMA (contract number MRTN-CT-2004-5652), and by the European Science Foundation through the MISGAM program. The second author was also supported by RFBR 08-01-00638 and RFBR 09-01-12150-ofi_m grants.

804

A. S. Cattaneo, P. Mnëv

4.2 The improved propagator . . . . . . . . . . . . . . . . . 4.3 On property P5 . . . . . . . . . . . . . . . . . . . . . . 4.4 The construction of the invariant by a framed propagator 4.5 The unframed propagator . . . . . . . . . . . . . . . . . Appendix A. The Chern–Simons Manifold Invariant . . . . . . . . Appendix B. Regular Forms . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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830 831 832 833 834 836 836 836

1. Introduction Since its proposal in Witten’s paper [18], Chern–Simons theory has been a source of fruitful constructions for 3-manifold invariants. In the perturbative framework one would like to get the invariants from the Feynman diagrams of the theory. These may be shown to be finite, see [2]. However, as in every gauge theory, one has to fix a gauge and then one has to show that the result, the invariant, is independent of the gauge fixing. In the case when one works around an acyclic connection, this was proved in [2], but this assumption rules out the trivial connection. Gauge-fixing independence for perturbation theory around the trivial connection of a rational homology sphere was proved in [1] and, for more general definitions for the propagator, in [11] and in [3]. The flexibility in the choice of propagator allows one to show that the invariant is of finite type [13]. The case of general 3-manifolds was not treated in detail, even though the propagators described in [2,3,11] are defined in general. The main point is the presence of zero modes, namely—working around the trivial connection—elements of de Rham cohomology (with shifted degree) of the manifold tensor the given Lie algebra. Out of formal properties of the BV formalism, it is however clear [7,6,14,15] what the invariant in the general case (of a compact manifold) should be: a solution of the quantum master equation on the space of zero modes modulo certain BV canonical transformations. This effective action has already been studied—but only modulo constants—in [7], which has been a source of inspiration for us. In the first part of this note, we make this precise and mathematically rigorous, working with a finite-dimensional version [17] of Chern–Simons theory, where the algebra of smooth functions on the manifold is replaced by an arbitrary finite-dimensional dg Frobenius algebra (of appropriate degrees). We are able to produce the solution to the quantum master equation on cohomology and to describe the BV canonical transformations that occur. Out of this we are able to describe the invariant in the case of cohomology concentrated in degree zero and three (the algebraic version of a rational homology sphere) and to extract invariants in case the first Betti number is one or, more generally, when the Frobenius algebra is formal. In the second part we revert to the infinite-dimensional case and show that whatever we did in the finite-dimensional case actually goes through if the propagator satisfies certain properties. It is good news that the propagator introduced by Axelrod and Singer in [2] does indeed satisfy them. In particular, we get an invariant for framed 3-manifolds as described in Theorem 1 in Subsect. 4.5. The problems with this scheme are that there is little flexibility in the choice of propagator and that the invariants are defined up to a universal constant that is very difficult to compute. (Notice that this constant is the same that appears anyway in the case of rational homology spheres in [2,3]). For the general case, that is of a propagator as in [11 or 3], we are able to show that all properties but one can easily be achieved. We

Remarks on Chern–Simons Invariants

805

reduce the last property to Conjecture 1 in Subsect. 4.3, which we hope to be able to prove in a forthcoming paper. During the preparation of this note, we have become aware of independent work by Iacovino [10] on the same topic. 2. Effective BV Action Let1 (F, σ ) be a finite-dimensional graded vector space endowed with an odd symplectic form σ ∈ 2 F ∗ of degree -1, which means σ (u, v) = 0 ⇒ |u| + |v| = 1 for u, v ∈ F. The space of polynomial functions Fun(F) := S • F ∗ is a BV algebra with anti-bracket {•, •} and BV Laplacian2 generated by the odd symplectic form σ . In coordinates: let {u i } be a basis in F and {x i } be the dual basis in F ∗ . Let us denote σi j = σ (u i , u j ). Then σ =

i (−1)gh(x ) σi j δx i ∧ δx j , i, j

1 ∂ ∂ i (−1)gh(x ) (σ −1 )i j i f, 2 ∂x ∂x j i, j ⎞ ⎛ ← − − → ∂ ∂ ⎠ g. { f, g} = f ⎝ (σ −1 )i j i ∂x ∂x j f =

i, j

In our convention σi j = −σ ji and we call grading on Fun(F) the “ghost number”: gh(x i ) = −|u i |. Suppose (F , σ ) is another odd symplectic vector space and ι : F → F is an embedding (injective linear map of degree 0) that agrees with the odd symplectic structure: σ (u , v ) = σ (ι(u ), ι(v )) for u , v ∈ F . Then F can be represented as F = ι(F ) ⊕ F ,

(1)

where F := ι(F )⊥ is the symplectic complement of the image of ι in F with respect to σ . Hence the algebra of functions on F factors Fun(F) ∼ = Fun(F ) ⊗ Fun(F ) (the isomorphism depends on the embedding ι). Since (1) is an (orthogonal) decomposition of odd symplectic vector spaces, the BV Laplacian also splits: = + .

(2)

1 This section is an adaptation of Section 4.2 from [16]. 2 In the general case, on an odd-symplectic graded manifold, the BV Laplacian is constructed from the

symplectic form and a consistent measure (the “S P-structure”). But here we treat only the linear case, and an odd-symplectic graded vector space (F , σ ) is automatically an S P-manifold with Lebesgue measure (the constant Berezinian) µF .

806


Here is the BV Laplacian on Fun(F ), associated to the odd symplectic form σ , and is the BV Laplacian on F associated to the restricted odd symplectic form σ |F . Let S ∈ Fun(F)[[]] be a solution to the quantum master equation (QME): e S/ = 0

1 {S, S} + S = 0 2

⇔

(the BV action on F). Let also L ⊂ F be a Lagrangian subspace in F . We define the effective (or “induced”) BV action S ∈ Fun(F )[[]] by the fiber BV integral e S (x )/ = e S(ι(x )+x )/µL , (3) L

where µL is the Lebesgue measure on L and we use the notation x = i u i x i for the canonical element of F ⊗ Fun(F), corresponding to the identity map F → F, and analogously for x ∈ F ⊗ Fun(F ), x ∈ F ⊗ Fun(F ). Proposition 1. The Effective action S defined by (3) satisfies the QME on F , i.e.

e S / = 0. Proof. This is a direct consequence of (2) and the BV-Stokes theorem (integrals over Lagrangian submanifolds of BV coboundaries vanish): e S (x )/ = e S(ι(x )+x )/µL = ( − )e S(ι(x )+x )/µL L L S(ι(x )+x )/ = e µL − e S(ι(x )+x )/µL = 0. L

L

In the last line the first term vanishes due to QME for S, while the second term is zero due to the BV-Stokes theorem. The BV integral (3) depends on a choice of embedding ι : F → F and Lagrangian subspace L ⊂ F (notice that F = ι(F )⊥ itself depends on ι). We call the pair (ι, L) the “induction data” in this setting. We are interested in the dependence of the effective BV action S on deformations of induction data. Recall that a generic small Lagrangian deformation L of a Lagrangian submanifold L ⊂ F is given by a gauge fixing fermion ∈ Fun(L) of ghost number -1. If (q a , pa ) are Darboux coordinates on F such that L is given by p = 0, then L is ∂ L = (q, p) : pa = − a ⊂ F . ∂q In our case we are interested in linear Lagrangian subspaces and thus only allow quadratic gauge fixing fermions. A general small deformation of induction data (ι, L) from (F, σ ) to (F , σ ) can be written as ι → ι + δι⊥ + δι|| , L → (idF − ι ◦ (δι⊥ )T )L ,

(4)

where the “perpendicular part” of the deformation of the embedding δι⊥ : F → F is a linear map of degree 0 satisfying δι⊥ (F ) ⊂ F , while the “parallel part” is of the form δι|| = ι ◦ δφ|| with δφ|| : F → F a linear map of degree 0 satisfying


807

(δφ|| )T = −δφ|| (i.e. δφ|| lies in the Lie algebra of the group of linear symplectomorphisms δφ|| ∈ sp(F , σ )). We use superscript T to denote transposition w.r.t. symplectic structure. Thus it is reasonable to classify small deformations into the three following types: • Type I: small Lagrangian deformations of Lagrangian subspace L → L leaving the embedding ι intact. • Type II: small perpendicular deformations of the embedding ι → ι+δι⊥ accompanied by an associated deformation3 of the Lagrangian subspace L → (idF − ι ◦ (δι⊥ )T )L

(5)

(this is necessary since here we deform the splitting (1) and L is supposed to be a subspace of the deformed F ). • Type III: small parallel deformations of the embedding ι → ι + δι|| = ι ◦ (idF + δφ|| ) leaving L intact. A general small deformation (4) is the sum of deformations of Types I, II, III. We call the following transformation of the action S → S˜ = S + {S, R} + R

(6)

(regarded in first order in R) the infinitesimal canonical transformation of the action with (infinitesimal) generator R ∈ Fun(F)[[]] of ghost number -1. Equivalently ˜

e S/ → e S/ = e S/ + (e S/ R).

(7)

The transformed action also solves the QME (in first order in R). Infinitesimal canonical transformations generate the equivalence relation on solutions of QME. Lemma 1. If the action S is changed by an infinitesimal canonical transformation S → S + {S, R} + R, then the effective BV action is also changed by an infinitesimal canonical transformation S → S + {S , R } + R with generator

R

∈

(8)

Fun(F )[[]]

given by the fiber BV integral −S / R =e e S/ R µL . L

(9)

Proof. This follows straightforwardly from (2), the BV-Stokes theorem and the exponential form of canonical transformations (7): S / S˜ / e → e = (e S/ + (e S/ R))µL L S / =e + e S/ R µL + (e S/ R) µL = e S / + (e S / R ) L

L

0

with

R

given by (9).

3 Formula (5) is explained as follows: one can express the deformation of the embedding as ι + δι = ⊥ T ∈ sp(F , σ ), so that id + δ is an infinitesimal symplecto(idF + δ) ◦ ι, where δ = δι⊥ ◦ ιT − ι ◦ δι⊥ F morphism of the space F , accounting for the deformation of ι. Then (5) just means L → (idF + δ) ◦ L.

808


Proposition 2. Under general infinitesimal deformation of the induction data (ι, L) as in (4) the effective BV action S is transformed canonically (up to constant shift): S → S + {S , R } + (R − R I I I )

(10)

e S/( + σ (x , δι⊥ x )) µL . R = σ (x , δφ|| x ) +e−S /

L

(11)

with generator

R I I I

Proof. The deformation (4) can be represented in the form ι → (idF + δ) ◦ ι , L → (idF + δ)L, where δ ∈ sp(F, σ ) is the infinitesimal symplectomorphism given by T ) + δι|| ◦ ιT . δ = {•, } + (δι⊥ ◦ ιT − ι ◦ δι⊥

Here {•, } = exp({•, }) − idF is understood as the (infinitesimal) flow generated by the Hamiltonian vector field {•, } in unit time. The pull-back (idF +δ)∗ : Fun(F) → Fun(F) acts on functions as canonical transformation f → f + { f, R}

(12)

with generator given by R=

1 1 σ (x, δ x) = + σ (x, (δι⊥ ◦ ιT )x) + σ (x, (δι|| ◦ ιT )x) .

2

2

RI

RI I

RI I I

It is important to note that only the third term (the effect of Type III deformation) contributes to R: R = R = R I I I =

1 Str F δφ|| 2

(Str F denotes supertrace over F ) and R = 0. The latter implies that (idF + δ)∗ µL = µL . Now we can compute the transformation of the effective BV action S due to infinitesimal change of induction data: S / S˜ / e → e = (idF + δ)∗ e S/µL L −Str F δφ|| =e e(S+{S,R}+R)/µL = e−StrF δφ|| e(S +{S ,R } + R )/. (13) L

Here we used Lemma 1 and the generator is given by (9): R = e−S / e S/ RµL L 1 −S / =e e S/( + σ (x, (δι⊥ ◦ ιT )x) + σ (x, (δι|| ◦ ιT )x))µL 2 L which yields (11). At last note that 21 Str F δφ|| = 21 σ (x , δφ|| x ) = R I I I which explains the constant shift in (10).


809

Remark 1. If we were treating BV actions as log-half-densities (meaning that e S/ is a half-density), we would write an honest canonical transformation (8) instead of (10), with no − R I I I shift. This is because the pull-back (idF + δ)∗ would then be acting on S by transformation (6) instead of (12). But in practice one works with effective BV actions defined by a normalized BV integral: if the initial BV action is of the form S = S0 + Sint with S0 quadratic in fields (the “free part” of BV action) and Sint the “interaction part”, one usually defines 1 e S (x )/ = e S(ι(x )+x )/µL N L with the normalization factor N = L e S0 (x )/µL . The effective action defined via a normalized BV integral is indeed a function rather than log-half-density, and transforms according to (10) under change of induction data. 3. Toy Model for Effective Chern–Simons Theory on Zero-Modes: Effective BV Action on Cohomology of dg Frobenius Algebras By a dg Frobenius algebra (C, d, m, π ) we mean a unital differential graded commutative algebra C with differential d : C • → C •+1 and (super-commutative, associative) product m : S 2 C → C, endowed in addition with a non-degenerate pairing π : S 2 C → R of degree −k (which means π(a, b) = 0 ⇒ |a| + |b| = k; and k is some fixed integer), satisfying the following consistency conditions: π(da, b) + (−1)|a| π(a, db) = 0, π(a, m(b, c)) = π(m(a, b), c)

(14) (15)

for a, b, c ∈ C. By a dg Frobenius–Lie4 algebra (A, d, l, π ) we mean a differential graded Lie algebra A with differential d : A• → A•+1 and Lie bracket l : ∧2 A → A, endowed with a non-degenerate pairing π : S 2 A → R of degree −k satisfying the conditions π(A, l(B, C)) = π(l(A, B), C), π(d A, B) + (−1)|A| π(A, d B) = 0

(16) (17)

for A, B, C ∈ A. If g is an (ordinary) Lie algebra with non-degenerate ad-invariant inner product πg : S 2 g → R (one calls such Lie algebras “quadratic”) and (C, d, m, π ) is a dg Frobenius algebra, then (g ⊗ C, d, l, πg⊗C ) is a dg Frobenius–Lie algebra. Here one defines d(X ⊗ a) := X ⊗ da, l(X ⊗ a, Y ⊗ b) := [X, Y ] ⊗ m(a, b), πg⊗C (X ⊗ a, Y ⊗ b) := πg(X, Y ) π(a, b). We will usually write π instead of πg⊗C . Example 1. If M is a closed (compact, without boundary) orientable smooth manifold of dimension D, then the de Rham algebra • (M) is adg Frobenius algebra with de Rham differential, wedge product and Poincaré pairing M • ∧ • of degree −D. If g is a finite-dimensional Lie algebra with invariant non-degenerate trace tr, then the algebra

• (M, g) = g ⊗ • (M) of g-valued differential forms on M is a dg Frobenius–Lie algebra with pairing tr M • ∧ • of degree −D. 4 Alternatively one could use the term “cyclic dg Lie algebra”.

810


3.1. Abstract Chern–Simons action from a dg Frobenius algebra. Let (C, d, m, π ) be a finite dimensional non-negatively graded dg Frobenius algebra with pairing π of degree -3, C = C 0 ⊕ C 1 [−1] ⊕ C 2 [−2] ⊕ C 3 [−3]. We denote by Bi = dim H i (C) the Betti numbers. Due to non-degeneracy of π , there is an isomorphism π : C • ∼ = (C 3−• )∗ (the Poincaré duality). Induced pairing on cohomology is also automatically non-degenerate, and so Poincare duality descends to cohomology: π : H • (C) ∼ = (H 3−• (C))∗ . Hence B0 = B3 , B1 = B2 . We will suppose in addition that B0 = B3 = 1 (so that C models the de Rham algebra of a connected manifold). Let g be a finite dimensional quadratic Lie algebra of coefficients and let (g ⊗ C, d, l, π ) be the corresponding dg Frobenius–Lie structure on g ⊗ C. Then we can construct an odd symplectic space of BV fields F = g ⊗ C[1] with odd symplectic structure of degree -1 given by σ (s A, s B) = (−1)|A| π(A, B).

(18)

Here s : g ⊗ C → g ⊗ C[1] is the suspension map. Let us also introduce the notation ω for the canonical element of (g ⊗ C) ⊗ Fun(F) corresponding to the desuspension map s −1 : F → g ⊗ C. If {e I } is a basis in C and {Ta } is an orthonormal basis in g, then we can write Ta e I ω I a , ω= I,a

where {ω I a } are the corresponding coordinates on F. By abuse of terminology we call ω the “BV field”. Let us introduce notations for the structure constants: π I J = π(e I , e J ), m I J K = π(e I , m(e J , e K )), f abc = πg(Ta , [Tb , Tc ]), d I J = π(e I , de J ). We will also use the shorthand notation for degrees |I | = |e I |. In terms of π the BV Laplacian and the anti-bracket are 1 −1 I J ∂ ∂ f = (π ) f, I a 2 ∂ω ∂ω J a I,J,a ⎛ ⎞ ← − − → ∂ ∂ ⎠ g. { f, g} = f ⎝ (−1)|I |+1 (π −1 ) I J ∂ω I a ∂ω J a I,J,a

Proposition 3. The action S ∈ Fun(F) defined as 1 1 π(ω, dω) + π(ω, l(ω, ω)) 2 6 1 1 |I |+1 = (−1) dI J ω I a ω J a + 2 6

S :=

I,J,a

(−1)|J | (|K |+1) f abc m I J K ω I a ω J b ω K c

I,J,K ,a,b,c

(19) satisfies the QME with BV Laplacian defined by the odd symplectic structure (18) on F.


811

Proof. Indeed, let us check the CME: 1 1 1 {S, S} = {π(ω, dω), π(ω, dω)} + {π(ω, dω), π(ω, l(ω, ω))} 2 8 12 1 + {π(ω, l(ω, ω)), π(ω, l(ω, ω))} 72 1 1 1 = − π(ω, d 2 ω) − π(ω, dl(ω, ω)) − π(ω, l(ω, l(ω, ω))) = 0. 2 2 8 The first term vanishes due to d 2 = 0, the second — due to the Leibniz identity for g ⊗ C, since property (16) implies 1 1 π(ω, dl(ω, ω)) = π (ω, dl(ω, ω) − l(dω, ω) + l(ω, dω)), 2 6 and the third term is zero due to the Jacobi identity for g ⊗ C. Next, check the quantum part of the QME: S = −

1 1 Str g⊗C d − Str g⊗C l(ω, •) = 0. 2 2

Here the first term vanishes since d raises degree and the second term vanishes due to unimodularity of Lie algebra g. The BV action (19) can be viewed as an abstract model (or toy model, since C is finite dimensional) for Chern–Simons theory on a connected closed orientable 3-manifold. We associate such a model to any finite dimensional non-negatively graded dg Frobenius algebra C with pairing of degree −3 and B0 = B3 = 1, and arbitrary finite dimensional quadratic Lie algebra of coefficients g. We are interested in the effective BV action for (19) induced on cohomology F = H • (C, g)[1]. We will now specialize the general induction procedure sketched in Sect. 2 to this case. 3.2. Effective action on cohomology. Let ι : H • (C) → C • be an embedding of cohomology into C. Note that ι is not just an arbitrary chain map between two fixed complexes, but is also subject to condition ι([a]) = a + d(. . .) for any cocycle a ∈ C. This implies in particular that the only allowed deformations of ι are of the form ι → ι + d δ I , where δ I : H • (C) → C •−1 is an arbitrary degree -1 linear map. This is indeed a Type II deformation (in the terminology of Sect. 2), while Type III deformations are prohibited in this setting. Let also K : C • → C •−1 be a symmetric chain homotopy retracting C • to H • (C), that is a degree -1 linear map satisfying d K + K d = idC − P , π(K a, b) + (−1)|a| π(a, K b) = 0, K ◦ ι = 0,

(20) (21) (22)

where P = ι ◦ ιT : C → C is the orthogonal (w.r.t. π ) projection to the representatives of cohomology in C. We require the additional property K2 = 0

(23)

812


Remark 2 (cf. [9]). An arbitrary linear map K 0 : C • → C •−1 satisfying just (20) can be transformed into a chain homotopy K with all the properties (20,21,22,23) via a chain of transformations K 0 → K 1 → K 2 → K 3 = K , where 1 (K 0 − K 0T ), 2 K 2 = (idC − P ) K 1 (idC − P ), K3 = K2 d K2. K1 =

(24) (25)

Having ι and K we can define a Hodge decomposition for C into representatives of cohomology, d-exact part and K -exact part: C = im(ι) ⊕ Cd−ex ⊕ C K −ex .

im(d)

(26)

im(K )

Properties (21), (22), (23) and skew-symmetry of differential (14) imply the orthogonality properties for Hodge decomposition (26): im(ι)⊥ = Cd−ex ⊕ C K −ex , (Cd−ex )⊥ = im(ι) ⊕ Cd−ex , (C K −ex )⊥ = im(ι) ⊕ C K −ex . In terms of Hodge decomposition (26) the splitting (1) of the space of BV fields F = g ⊗ C[1] is given by g ⊗ C[1] = ι(H • (C, g)[1]) ⊕ g ⊗ Cd−ex [1] ⊕ g ⊗ C K −ex [1] .

F

ι(F )

F

and we choose the Lagrangian subspace L K = g ⊗ C K −ex [1] ⊂ F .

(27)

We define the “effective BV action on cohomology” (or “on zero-modes”) W ∈ Fun(F )[[]] for an abstract Chern–Simons action (19) by a normalized fiber BV integral 1 W (α)/ e = e S(ι(α)+ω )/µL K , (28) N LK where N=

LK

e S0 (ω

)/

µL K

(29)

is the normalization factor and S0 (ω ) =

1 π(ω , dω ) 2

is the free part of the action S. To lighten somewhat the notation, we denoted the effec• (C, g)[1] by α tive action by W instead of S and the BV field associated to F = H instead of ω . Let {e p } be a basis of the cohomology H • (C). Then α = a, p Ta e p α pa ,


813

where α pa are coordinates on F with ghost numbers gh(α pa ) = 1 − |e p |. We have the following decomposition of S(ι(α) + ω ): S(ι(α) + ω ) =

1 1 π(ι(α), l(ι(α), ι(α))) + π(ω , dω )

2

6 Wprod (α)

S0 (ω )

1 1 1 + π(ω , l(ι(α), ι(α))) + π(ι(α), l(ω , ω )) + π(ω , l(ω , ω )) . 2 6

2 Sint (α,ω )

(30) The perturbation expansion for (28) is obtained in a standard way and can be written as

1 −1 ∂ ∂ W (α) = Wprod (α) + log e− 2 π ( ∂ω ,K ∂ω ) |ω =0 ◦ e Sint (α,ω )/ =

∞ l=0

l

∞ n=0 ∈G l,n

1 W (α), |Aut()|

(31)

where G l,n denotes the set of connected non-oriented Feynman graphs with vertices of valence 1 and 3 (we would like to understand them as trivalent graphs with “leaves” allowed, i.e. external edges), with l loops and n leaves. The contribution W (α) of each Feynman graph ∈ G l,n is a homogeneous polynomial of degree n in {α pa } and of ghost number 0 obtained by decorating each leaf of by ι Ip α pa , each trivalent vertex by f abc m I J K and each (internal) edge by δ ab K I J , and taking contraction of all indices, corresponding to incidence of vertices and edges in . One should also take into account signs for contributions, which can be obtained from the exponential formula for perturbation series (31). The cubic term 1 π(ι(α), l(ι(α), ι(α))) 6 1 = (−1)|eq | (|er |+1) f abc µ pqr α pa α qb αr c 6 p,q,r

Wprod (α) =

(32)

a,b,c

is the contribution of the simplest Feynman diagram 0,3 , the only element of G 0,3 . Here µ pqr = π(ι(e p ), m(ι(eq ), ι(er ))) are structure constants of the induced associative product on H • (C) (hence the notation Wprod ). Remark 3. The perturbative expansion (31) is related to homological perturbation theory 1 (HPT) in the following way. Denote by cl,n (α) = n! ∈G l,n |Aut()| W (α) the total contribution of Feynman graphs with l loops and n leaves to the effective action (31), with additional factor n!. Then each cl,n ∈ S n (F ∗ ) ∼ = Hom(∧n (H • (C, g)), R) can be understood as a (super-)anti-symmetric n-ary operation on cohomology H • (C, g), taking values in numbers. Now suppose ln ∈ Hom(∧n (H • (C, g)), H • (C, g)) are the L ∞ operations on cohomology, induced from dg Lie algebra g ⊗ C (by means of HPT). Then it is easy to see that c0,n+1 (α0 , . . . , αn ) = π (α0 , ln (α1 , . . . , αn )) and trees for HPT (the

814


Lie version of trees from [12], cf. also [9]) are obtained from Feynman trees for W (α) by assigning one leaf as a root and inserting the inverse of pairing5 (π )−1 there, or vice versa: Feynman trees are obtained from trees of HPT by reverting the root6 with π and forgetting the orientation of edges (cf. Sect. 7.2.1 of [16]). Thus we can loosely say that the BV integral (28) defines a sort of “loop enhancement” of HPT for a cyclic dg Lie algebra g ⊗ C. Also, in this language (due to A. Losev), using the BV-Stokes theorem to prove that the effective action W satisfies the quantum master equation can be viewed as the loop-enhanced version of using the HPT machinery to prove the system of quadratic relations (homotopy Jacobi identities) on induced L ∞ operations ln . Let us introduce a Darboux basis in H • (C). Namely, let e(0) = [1] be the basis vector in H 0 (C), the cohomology class of unit 1 ∈ C 0 (recall that we assume B0 = dim H 0 (C) = 1) and let e(3) be the basis vector in H 3 (C), satisfying π (e(0) , e(3) ) = 1 (i.e. e(3) is represented by some top-degree element v = ι(e(3) ) ∈ C 3 , normalized by the i } the dual basis condition π(1, v) = 1). Let also {e(1)i } be some basis in H 1 (C) and {e(2) in H 2 (C), so that π (e(1)i , e(2) ) = δi . The BV field α is then represented as a ia i a a α= e(0) Ta α(0) + e(1)i Ta α(1) + e(2) Ta α(2)i + e(3) Ta α(3) . j

a

j

a,i

a

a,i

ia , α a , α a } the BV Laplacian on F is In Darboux coordinates {α0a , α(1) (2)i (3)

=

a

∂ ∂ ∂ ∂ + . a a ia ∂α a ∂α(0) ∂α(3) ∂α(1) (2)i a,i

It is also convenient to introduce g-valued coordinates on F : a i ia a a Ta α(0) , α(1) = Ta α(1) , α(2)i = Ta α(2)i , α(3) = Ta α(3) . α(0) = a

a

a

a

i ,α The ghost numbers of α(0) , α(1) (2)i , α(3) are 1, 0, −1, −2 respectively. In terms of this Darboux basis, the trivial part (32) of the effective action is 1 a b c Wprod (α) = f abc α(0) α(0) α(3) 2

−

a,b,c

a ib c f abc α(0) α(1) α(2)i +

a,b,c i

1 ia jb kc f abc µi jk α(1) α(1) α(1) 6 a,b,c i, j,k

1 1 j i k = πg (α(3) , [α(0) , α(0) ]) − πg (α(0) , [α1i , α(2)i ]) + µi jk πg (α(1) , [α(1) , α(1) ]) . 2 6

i i, j,k

W 003 prod

012 Wprod

111 Wprod

5 We use notation π = π(ι(•), ι(•)) : C ⊗ C → R for the induced pairing on cohomology H • (C). By a slight abuse of notation, we also use π to denote the pairing on H • (C, g) induced from πg⊗C . 6 This means the following: let T be a binary rooted tree with n leaves, oriented towards the root; let T¯ be the non-oriented (and non-rooted) tree with n +1 leaves, obtained from T by forgetting the orientation (and treating the root as an additional leaf). Then the weight WT¯ (α) of T¯ as a Feynman graph and the contribution l T of tree T (without the symmetry factor) to the induced L ∞ operation ln are related by WT¯ (α) = π (α, l T (α, . . . , α )).

n

Pictorially this is represented by inserting a bivalent vertex (associated to the operation π (•, •)) at the root of T . Both edges incident to this vertex are incoming, thus we say that the root becomes reverted.


815

Here µi jk is the totally antisymmetric tensor of structure constants of multiplication of 1-cohomologies: µi jk = π(ι(e(1)i ), m(ι(e(1) j ), ι(e(1)k ))). Proposition 4. The effective BV action W , induced from the abstract Chern–Simons action (19) on F = H • (C, g)[1] has the form B1 1 W (α) = Wprod (α) + F(α(1) , . . . , α(1) ; ),

(33)

g where F ∈ Fun(g B1 )[[]] is some -dependent function on H 1 (C, g)[1] ∼ = g B1 , invariant under the diagonal adjoint action of g, i.e. B1 B1 B1 1 1 1 F(α(1) + [X, α(1) ], . . . , α(1) + [X, α(1) ]; ) = F(α(1) , . . . , α(1) ; )

mod X 2

(34)

at the first order in X ∈ g. Proof. Ansatz (33) follows from the observation that the values of individual Feynman i }). The graphs = 0,3 in (31) depend only on the 1-cohomology: W = W ({α(1) argument is as follows: suppose not all leaves of are decorated with insertions of 1-cohomology. Then, since gh(W ) = 0, there is at least one leaf decorated with insertion of 0-cohomology. Since ι(H 0 (C)) = R · 1, the value of will contain one of the expressions K 2 (· · · ), K (ι(· · · )). Hence Feynman diagrams with insertion of cohomology of degree = 1 vanish due to (22), (23). This proves (33). By construction, W has to satisfy the QME (Proposition 1). The QME for an action satifying ansatz (33) is equivalent to ad-invariance of F (34): 1 ∂ 012 a ib {W, W } + W = {Wprod , F} = f abc α(0) α(1) F ic 2 ∂α(1) i a,b,c ∂ i ], i F, = [α(0) , α(1) ∂α(1) i g

where < •, • >g denotes the canonical pairing between g and g∗ . Another explanation of (34) is the following: if g is the Lie algebra of the Lie group G, then the original abstract Chern–Simons action (19) is invariant under the adjoint action of G, i.e. ω → gωg −1 for g ∈ G. The embedding ι and the choice of the Lagrangian subspace L ⊂ F are also compatible with this symmetry. Hence W (α) is also invariant under the adjoint action of G: α → gαg −1 , and (34) is the infinitesimal form of this symmetry. Remark 4. There is another argument for ansatz (33) that can be formulated on the level of the BV integral (28) itself, rather than on the level of Feynman diagrams. Namely, the restriction of Sint (α, ω ) (we refer to decomposition (30)) to the Lagrangian subspace L K does not depend on α(0) . This means that the only term depending on α(0) in (30) is the trivial one Wprod (α), constant on L K . Hence the non-trivial part of the effective action W (α) − Wprod (α) does not depend on α(0) . Since it also has to be of ghost number zero, it can only depend on α(1) .

816


3.3. Dependence of the effective action on cohomology on induction data. The effective action W (α) depends on induction data (ι, K ), and we are interested in describing how W (α) changes due to the deformation of (ι, K ). In the terminology of Sect. 2, the Type I deformations of induction data are of the form • ι → ι, K → K +[d, δκ], where δκ : Cd−ex → C K•−2 −ex is a skew-symmetric linear map of degree -2. The corresponding deformation of the Lagrangian subspace L K is described by the gauge fixing fermion = 21 π(ω, d δκ d ω). Type II deformations change embedding as ι → ι + d δ I , where δ I : H • (C) → C K•−1 −ex , and change chain homotopy in a minimal way (so as not to spoil properties (22), (20)): K → K + ι δ I T − δ I ιT . Type III transformations are forbidden in this setting, as discussed above (we have a canonical surjective map from ker d ⊂ C to H • (C) that sends the cocycle α to its cohomology class [α]). Due to Proposition 2, an infinitesimal deformation of (ι, K ) induces an infinitesimal canonical transformation of W (α), W → W + {W, R } + R with generator given by the fiber BV integral R (α) = e−W (α)/

LK

e S(ι(α)+ω

)/

1 π(ω , d δκ dω ) + π(ω , d δ I α) µL K . 2 (35)

This integral is evaluated perturbatively in analogy with (31): 1 −1 ∂ ∂ R (α) = e−W (α)/ · e− 2 π ( ∂ω ,K ∂ω ) ◦ ω =0 1 ◦ e Sint (α,ω )/ · ( π(ω , d δκ dω ) + π(ω , d δ I α)) 2 ∞ ∞ 1 R (α). = l M )| M |Aut( M M l=0

(36)

n=0 ∈G l,n

M is the set of connected nonHere the superscript M stands for “marked edge”, G l,n oriented trivalent graphs with l loops and n leaves and either one leaf or one internal edge marked. Values of Feynman graphs R M (α) are now homogeneous polynomials of degree n and ghost number -1 on F , obtained by the same Feynman rules as for W (α), supplemented with a Feynman rule for the marked edge: we decorate the marked leaf with δ I pI α pa and marked internal edge with δ ab δκ I J .

Proposition 5. The generator of the infinitesimal canonical transformation induced on the effective BV action W (α) by the infinitesimal change of induction data (ι, K ) has the following form: R (α) =

a,i

B1 a 1 α(2)i G ia (α(1) , . . . , α(1) ; ),

(37)


where G =

a,i

G ia

∂ ia ∂α(1)

817

∈ (Vect(g B1 )[[]])g is some -dependent vector field on

H 1 (C, g)[1] ∼ = g B1 , equivariant under the diagonal adjoint action of g, i.e. B1 B1 B1 1 1 1 G i (α(1) + [X, α(1) ], . . . , α(1) + [X, α(1) ]; ) = [X, G i (α(1) , . . . , α(1) ; )]

mod X 2 (38)

at first order in X ∈ g, for all i = 1, . . . , B1 (and we set G i := canonical transformation with generator (37) in terms of ansatz (33) is 111 + F) + div(G). F → F + G ◦ (Wprod

a

Ta G ia ). The (39)

Proof. The argument for ansatz (37) is pretty much the same as for (33): the value M is linear in α R M (α) of each Feynman graph M ∈ G l,n (2) and does not depend on α(0) , α(3) for the following reason: Unless we decorate one leaf of M by α(2) and all other leaves by α(1) , some leaf has to be decorated by α(0) (since the total ghost number of R M (α) has to be -1). Then the contribution of this decoration of M vanishes due to ι(H 0 (C)) = R · 1 and the vanishing of the expressions δ I (e(0) ), K 2 , K δκ, δκ K , K ι, δκ ι, one of which necessarily appears as a contribution of a neighborhood of the place of insertion of 0-cohomology on the Feynman graph. This proves ansatz (37). The equivariance of G (38) is equivalent to the fact that a canonical transformation with generator (37) preserves ansatz (33) for W (α). Indeed, if G were not equivariant, the 012 , R } would produce α -dependence for the canonically transformed effecterm {Wprod (0) tive action. The other explanation is that the equivariance of G is due to the invariance of R under the adjoint action of the group G, which is due to the fact that a deformation of (ι, K ) is trivial in g-coefficients and hence consistent with the adjoint G-action. Rewriting the canonical transformation of the effective action (33) with generator (37) as (39) is straightforward. Remark 5. Analogously to Proposition 4, one can prove ansatz (37) on the level of the BV integral (35) instead of using Feynman diagrams. Namely, expressions S(ι(α) + ω ) − Wprod (α), W (α) − Wprod (α) and the expression in parentheses in (35) all do not depend on α(0) . Hence R does not depend on α(0) . But it also has to be of ghost number -1, which can only be achieved if it is of form (37). 3.4. Invariants. We are interested in describing the effective action W (α) on cohomology modulo changes of induction data (ι, K ). Due to Propositions 4, 5, we can give a complete solution (i.e. describe the complete invariant) in case B1 = 0, and find some partial solution (i.e. describe some, probably incomplete, invariant) for the case of a formal Frobenius algebra C, meaning that we can find representatives for cohomology ι : H • (C) → C closed under multiplication. In particular, in case B1 = 1 the algebra C is necessarily formal. Proposition 6. If B1 = 0, the effective action on cohomology is 003 (α) + F(), W (α) = Wprod

(40)

003 (α) = 1 π (α , [α , α ]) and F() is an -dependent constant, invariwhere Wprod (0) (0) 2 g (3) ant under deformations of induction data (ι, K ).

818


Proof. Ansatz (40) is a restriction of (33) to the case B1 = 0. Due to (37) and B1 = 0, the generator of induced canonical transformation necessarily vanishes R = 0. Hence F() is invariant under deformation of (ι, K ). So F() is the complete invariant of W (α) for the B1 = 0 case (which is an abstract model for Chern–Simons theory on a rational homology sphere) and is given by 1 −1 ∂ ∂ 1 1 F() = log e− 2 π ( ∂ω ,K ∂ω ) |ω =0 ◦ e 6 π(ω ,l(ω ,ω )) =

∞ l=2

l

vac ∈G l,0

1 F vac , |Aut( vac )|

(41)

where we sum over trivalent connected non-oriented graphs without leaves vac (the “vacuum loops”). The contribution of a Feynman graph F vac ∈ R is a number, computed by the same Feynman rules as for (31), just without the insertions of α. Example 2. (Chevalley–Eilenberg complex of su(2)) We obtain an interesting example of abstract Chern-Simons theory with B1 = 0 if we choose C = S • (su(2)∗ [−1]) = R ⊕ su(2)∗ [−1] ⊕ ∧2 su(2)∗ [−2] ⊕ ∧3 su(2)∗ [−3] — the Chevalley-Eilenberg complex of the Lie algebra su(2). This C is naturally a dg Frobenius algebra with super-commutative wedge product, Chevalley–Eilenberg differential d:

e1 → e2 e3 , e2 → e3 e1 , e3 → e1 e2 ,

and pairing π(1, e1 e2 e3 ) = π(e1 , e2 e3 ) = π(e2 , e3 e1 ) = π(e3 , e1 e2 ) = 1. Here {e1 , e2 , e3 } is the basis in su(2)∗ , dual to the basis {− 2i σ1 , − 2i σ2 , − 2i σ3 } in su(2), where {σi } are the Pauli matrices; 1 is the unit in C 0 = R. This C can be understood as the algebra of left-invariant differential forms on the Lie group SU (2) ∼ S 3 which is indeed quasi-isomorphic (as a dg algebra) to the whole de Rham algebra of the sphere S 3 ; thus the abstract Chern–Simons theory associated to this C is in a sense a toy model for true Chern–Simons theory on S 3 . The Hodge decomposition (26) for C is unique: C = R ⊕ ∧3 su(2)∗ [−3] ⊕ su(2)∗ [−1] ⊕ ∧2 su(2)∗ [−2],

C K −ex

H • (C )

Cd−ex

and the BV field ω is ω = α(0) 1 + α(3) e1 e2 e3 + ω1 e1 + ω2 e2 + ω3 e3 + ω23 e2 e3 + ω31 e3 e1 + ω12 e1 e2 .

α

ωK −ex

ωd−ex

Here α(0) , α(3) are g-valued coordinates on F = g ⊗ H • (C)[1] and {ω I }, {ω I J } are g-valued coordinates on F ⊂ g⊗C[1]. The Lagrangian subspace (27) is L = g⊗su(2)∗ .


819

The effective action on cohomology, as defined by the integral (28), satisfies the ansatz (40) with F() given by

1

g⊗su(2)∗ e F() = log

3 1 2

I =1 πg (ω 1 1

g⊗su(2)∗

e·2

I ,ω I )+π

3

I =1 πg (ω

The perturbative expansion (41) now reads − 21 3I =1 πg−1 ( ∂ I , ∂ I ) ∂ω ∂ω F() = log e =

∞

(−1)l+1 l

vac ∈G l,0

l=2

g (ω

1 ,[ω2 ,ω3 ])

I ,ω I )

ω I =0

dω1 dω2 dω3

dω1 dω2 dω3

◦e

1

πg (ω1 ,[ω2 ,ω3 ])

.

(42)

1 g (2) · L vac . L suvac |Aut( vac )|

(43)

g

Here L vac denotes the “Lie algebra graph” (or the “Jacobi graph”), i.e. the number7 obtained by decorating vertices of vac with the structure constants f abc of the Lie su(2) algebra g and taking contraction of indices over edges of vac . In particular, for L vac vertices are decorated with structure constants of su(2) — the Levi-Civita symbol8 I J K ∈ {±1, 0}. First terms in the series (43) are: 1 · 6 · f abc f abc 12 1 1 +3 · 12 · f acd f bcd f ae f f be f + · 6 · f abc f ade f be f f c f d + · · · . 16 24

F() = −2

In particular, for g = su(N ) we have 1 7 23 F() = −2 (N 3 − N ) + 3 (N 4 − N 2 ) − 4 (N 5 − N 3 ) + · · · . 2 8 8 su(2)

As a side note, L vac can be interpreted combinatorially as the number of ways to decorate edges of vac with 3 colors, such that in each vertex edges of all 3 colors meet (these decorations should be counted with signs, determined by the cyclic order of colors in each vertex). Thus for g = su(2) the invariant F() is given by F() =

∞ l=2

(−1)l+1 l

vac ∈G l,0

1 (2) 2 (L suvac ) , vac |Aut( )|

and it can be viewed as a generating function for certain interesting combinatorial quantities (and on the other hand, there is an “explicit” integral formula (42) for F() in terms of a 9-dimensional Airy-type integral). Proposition 7. Suppose C is formal and the embedding ι : H • (C) → C is an algebra homomorphism, then: 7 Strictly speaking, one also has to choose a cyclic ordering of half-edges for each vertex of vac , and g g this choice affects the total sign of L vac . But this ambiguity is cancelled in (43) since the factors L vac and

su(2)

L vac change their signs simultaneously if we change the cyclic ordering of half-edges in any vertex of vac . 8 We assume that the inner product on su(2) is normalized as π su(2) (x, y) = −2 tr(x y) (in the fundamental representation of su(2)), so that the structure constants for the orthonormal basis are really IJK .

820


• The tree part of the effective action on cohomology contains only the trivial Wprod term, i.e. B1 1 , . . . , α(1) )+ W (α) = Wprod (α) + F (1) (α(1)

∞

B1 1 l F (l) (α(1) , . . . , α(1) ),

(44)

l=2

where F (l) ∈ (Fun(g B1 ))g for l = 1, 2, . . .. • The 1-loop part of effective action can be written as F (1) (α(1) ) =

1 Str g⊗C log 1 + K ◦ l(ι(α(1) ), •) . 2

(45)

• Restriction of the 1-loop part of the effective action F (1) |MC to the Maurer-Cartan set is invariant under deformations of (ι, K ) (preserving the homomorphism condition for ι). Here MC ⊂ H 1 (C, g)[1] is given by j k µi jk [α(1) , α(1) ] = 0. j,k

Proof. The fact that ι is a homomorphism implies K ◦ l(ι(α), ι(α)) = 0.

(46)

This means that all Feynman diagrams for W (α) that contain a vertex adjacent to two leaves and one internal edge, vanish. In particular, all tree diagrams except 0,3 vanish. Together with Proposition 4 this implies (44). Also (46) implies that among 1-loop diagrams only “wheels” survive, and they are summed up to form (45) in the standard way. The next observation is that the tree part R 0 of the generator of the canonical transformation (36) induced by changing (ι, K ) vanishes. This is a consequence of properties (46), δκ ◦ l(ι(α), ι(α)) = 0 and π(δ I (α), l(ι(α), ι(α))) = 0 (which all follow from the ∞fact lthat(l)ι is a homomorphism). In terms of the vector field G (37), we have G = l=1 G and the l-loop part of effective action is transformed according to (39) as F

(l)

→ F

(l)

+G

(l)

◦

111 Wprod

+

l−1

G (l−i) ◦ F (i) + div G (l−1) .

i=1

In particular, the 1-loop part is transformed as 111 F (1) → F (1) + G (1) ◦ Wprod . 111 , the Since the Maurer-Cartan set is precisely the locus of stationary points of Wprod (1) restriction F |MC is invariant. This finishes the proof.

Special case of formal C is the case B1 = 1. Here the Maurer-Cartan set is MC = H 1 (C, g)[1] ∼ = g, so the 1-loop part of effective action F (1) ∈ Fun(g)g is invariant (without any restriction).


821

3.5. Comments on relaxing the condition K 2 = 0 for chain homotopy. Let us introduce the notation 1 −1 ∂ ∂ ¯ ¯ Indι,K ( S)(α) := log e− 2 π ( ∂ω ,K ∂ω ) |ω =0 ◦ e( S(ι(α)+ω )−S0 (ω ))/ ∈ Fun(F )[[]]

(47)

for the “effective action” for some (not necessarily abstract Chern–Simons) action S¯ ∈ Fun(F)[[]], defined by perturbation series, rather than by the BV integral itself. Expression (47) is indeed the perturbation series generated by the BV integral ¯

eIndι,K ( S)(α)/ =

1 ¯ e S(ι(α)+ω )/µL K N LK

(48)

with normalization N as before (29). In particular for abstract Chern–Simons action S¯ = S we recover the definition of W : Indι,K (S)(α) = W (α). The important observation with which we are concerned here is that definition (47) makes sense for a chain homotopy K not necessarily satisfying property K 2 = 0 (we assume that the other properties we demanded (20,21,22) hold), while the definition via the BV integral (48) is less general and uses essentially the K 2 = 0 property for construction of Lagrangian subspace L K . To avoid confusion we will denote by Kˆ the chain homotopy without property (23) and reserve notation K for the “honest” chain homotopy with property (23). We will call the effective action defined via (47) with Kˆ as chain homotopy the “relaxed” effective action, while for an honest chain homotopy K we call the effective action (defined equivalently by (47) or by the BV integral (48)) “strict”. Proposition 8. Let Kˆ : C • → C •−1 be a chain homotopy satisfying properties (20,21,22), but with Kˆ 2 = 0, and let K = Kˆ d Kˆ be the construction (25) applied to Kˆ (i.e. K satisfies all the properties (20,21,22,23)). Then the relaxed effective action Indι, Kˆ (S) is equivalent (i.e. connected by a canonical transformation) to the strict effective action Indι,K (S). Proof. The first observation is that since K = Kˆ d Kˆ , we can write the relaxed chain homotopy as Kˆ = K + dd,

(49)

where = Kˆ 3 : C 3 → C 0 . In fact, formula (49), with arbitrary skew-symmetric, degree -3 linear map : C 3 → C 0 , gives a general (finite) deformation of the honest chain homotopy K , preserving properties (20,21,22), but violating (23) and satisfying in addition K = Kˆ d Kˆ . In other words, deformation (49) is the inverse of projection (25). Second, we interpret the Feynman diagram decomposition for Indι, Kˆ with propagator ˆ K given by (49) as the sum over graphs with edges decorated either by K or by dd, and then raise the Feynman subgraphs with edges decorated only by dd into action. I.e. we obtain Indι, Kˆ (S) = Indι,K (S + ),

(50)

822


where 1 1 π(l(ω, ω), dd l(ω, ω)) + π(l(ω, ω), dd l(ω, dd l(ω, ω))) 8 8 1 + π(l(ω, ω), dd l(ω, dd l(ω, dd l(ω, ω)))) + · · · 8 1 1 + · Str F dd l(ω, dd l(ω, •)) 2 2 1 1 + · Str F dd l(ω, dd l(ω, dd l(ω, •))) + · · · . (51) 2 3 Only the simplest trees (“branches”) and only the simplest one-loop diagrams (“wheels”) contribute to , because any diagram with 3 incident internal edges decorated by dd automatically vanishes due to =

π(dd(· · · ), l(dd(· · · ), dd(· · · ))) = 0 (which is implied by Leibniz identity in g ⊗ C, by d 2 = 0 and skew-symmetry of d). Third, we notice that there is a canonical transformation from S to S + . A convenient way to describe a finite canonical transformation is to present a “homotopy” S (t, dt) = S (t) + dt · R (t) ∈ Fun(F)[[]] ⊗ • ([0, 1]) — a differential form on the interval [0, 1] with values in functions on F (t ∈ [0, 1] is a coordinate on the interval), satisfying the QME on the extended space F ⊕ T [1]([0, 1]): 1 (dt + )S (t, dt) + {S (t, dt), S (t, dt)} = 0 2

(52)

(where dt = dt ∂t∂ is the de Rham differential on the interval), and satisfying boundary conditions S (t)|t=0 = S, S (t)|t=1 = S + . The extended QME is equivalent to the fact that S (t) is a solution to the QME on F for any given t ∈ [0, 1] plus the fact that S (t + δt) is obtained from S (t) by the infinitesimal canonical transformation with generator δt · R (t). In our case it is a straightforward exercise to present the desired homotopy between S and S + : 1 S (t, dt) = S + t + dt Rt , t where t is defined by (51) with rescaled by t, and R is given by 1 1 π(dd ω, l(ω, ω)) + π(dd ω, l(ω, dd l(ω, ω))) 4 4 1 + π(dd ω, l(ω, dd l(ω, dd l(ω, ω)))) + · · · . 4 We showed that S and S + are connected by a homotopy. Due to Proposition 1 and Lemma 1 the effective actions Indι,K (S) and Indι,K (S + ) are also connected by a homotopy defined by 1 e S (t,dt)(ι(α)+ω )/µL K . (53) e W (t,dt)(α)/ = N LK R =

Together with (50) this finishes the proof.


823

Remark 6. An immediate consequence of Proposition 8 is that the relaxed effective action Indι, Kˆ (S) satisfies the QME on F . Remark 7. Expression (51) suggests that it is itself the value of a certain Gaussian integral. Namely, the restriction of S + to the subspace ι(F )⊕L K ⊂ F can be written as 1 1 a fiber integral over fibers g ⊗ Cd−ex [1] of vector bundle ι(F ) ⊕ L K ⊕ g ⊗ Cd−ex [1] → ι(F ) ⊕ L K (which is a sub-bundle of F → ι(F ) ⊕ L K ): 1 −1 e(S+ )/ = e(S+ 2 π(ω,K K ω))/µg⊗C 1 [1] ι(F )⊕L K

1 g⊗Cd−ex [1]

d−ex

(by µ(··· ) we always mean the Lebesgue measure on the vector space). Here we assume for simplicity that : C 3 → C 0 is an isomorphism, and we denote −1 : C 0 → C 3 its inverse. Now we can write the relaxed effective action as 1 −1 Indι, Kˆ (S)(α)/ = e S(ι(α)+ω )/ e 2 π(ω ,K K ω ) µL K ⊕g⊗C 1 [1] , e 1 L K ⊕g⊗Cd−ex [1]

d−ex

(54) 1 [1] ⊂ F instead of where we integrate over the coisotropic subspace L K ⊕ g ⊗ Cd−ex 1

just the Lagrangian subspace L K ⊂ F , with measure e 2 π(ω,K

−1 K ω)

µL K ⊕g⊗C 1

d−ex [1]

1 that is constant in the direction of L K and is Gaussian in the direction of g ⊗ Cd−ex [1]. So expression (54) gives an elegant interpretation of the relaxed effective action via a “thick” fiber BV integral (over a Gaussian-smeared Lagrangian subspace). In the case 1 of of general rank (not necessarily an isomorphism), we should replace Cd−ex by 1 im(d) ⊂ Cd−ex in this discussion, which leads to a thick fiber BV integral over a smaller coisotropic L K ⊕ g ⊗ im(d)[1] ⊂ F .

3.5.1. Invariants from the relaxed effective action We are interested in describing the invariants of the relaxed effective action Indι, Kˆ (S) modulo deformations of the “relaxed induction data” (ι, Kˆ ). Since we have a general description (49) for a non-strict chain homotopy, a general (infinitesimal) deformation of the relaxed induction data (ι, Kˆ ) can be described as a deformation of the underlying strict induction data (ι, K ) studied in the beginning of Sect. 3.3, plus a deformation of . Now we can restate some weak version of Proposition 6 for the case of relaxed induction, where we make a special choice for the Lie algebra of coefficients: g = su(2) (we generalize this in the Remark afterwards). We are able to recover only the two-loop part of the complete invariant F() of the strict effective action in this discussion. Proposition 9. If B1 = 0 and g = su(2), the relaxed effective action on cohomology has the form 003 Indι, Kˆ (S)(α) = A() · Wprod (α) + B(),

(55)

F (2) = 3A(1) + B (2)

(56)

a b c 003 = 1 (1) 2 (2) where Wprod a,b,c abc α(0) α(0) α(3) , and A() = 1+A + A +· · ·, B() = 2 2 B (2) + 3 B (3) + · · · are some -dependent constants. The number is invariant under deformations of the relaxed induction data (ι, Kˆ ).

824


Proof. Ansatz (55) follows from the following argument. Since the relaxed effective action Wˆ = Indι, Kˆ (S) ∈ Fun (g[1] ⊕ g[−2]) [[]] is a function of ghost number zero and g = su(2) is 3-dimensional (and hence an at most cubic dependence on α(0) is possible), it has to be of the form 1 a b c Wˆ (α) = Aabc ()α(0) α(0) α(3) + B(). 2 a,b,c

Since Wˆ also inherits the invariance under adjoint action of G = SU (2): α → gαg −1 for g ∈ G from the ad-invariance of the original abstract Chern–Simons action S, the tensor Aabc () has to be of the form Aabc () = A()abc . This proves (55). The fact that the series for A() starts as A() = 1 + O() is due to the vanishing of all tree diagrams = 0,3 which follows from formality of C (it is automatic for the B1 = 0 case) and Kˆ ◦ ι = 0. The series for B() starts with an O(2 )-term just because G l,0 is empty for n = 0, 1. Next, we know from Proposition 8 that there is a homotopy W (t, dt)(α) connecting 003 (α) + F() (ansatz (40)) and the the strict effective action W = Indι,K (S) = Wprod relaxed one Wˆ = Ind ˆ (S). The general ansatz for W (t, dt), taking into account that ι, K

g = su(2) and that construction (53) is compatible with ad-invariance α → gαg −1 , is the following: 003 (α) + B(; t) W (t, dt)(α) = A(; t)Wprod ⎛ ⎞ 1 a b c d e a b ⎠ +dt · ⎝C(; t) abc δde α(0) α(0) α(0) α(3) α(3) + D(; t) δab α(0) α(3) , 12 a,b,c,d,e

a,b

(57) where A, B, C, D are some functions of and the homotopy parameter t ∈ [0, 1]. The boundary conditions are: A(; 0) = 1, A(; 1) = A(), B(; 0) = F(), B(; 1) = B(). The extended QME (52) for homotopy W (t, dt) is equivalent to the system ∂ A(; t) = C(; t) − A(; t) · D(; t), ∂t ∂ B(; t) = 3 D(; t). ∂t

(58) (59)

Next, the argument of vanishing of non-trivial trees (due to formality of C, K ◦ ι = 0 and d ◦ ι = 0) applies again to construction (53). Hence, we have A(; t) = 1 + A(1) (t) + O(2 ), C(; t) = C (1) (t) + O(2 ), D(; t) = D (1) (t) + O(2 ). And due to G 0,0 = G 1,0 = ∅, we again have B(; t) = 2 B (2) (t) + O(3 ). Equation (58) in order O() together with (59) in order O(2 ) yield ∂ (1) A (t) = −D (1) (t), ∂t

∂ (2) B (t) = 3D (1) (t). ∂t


825

Hence the expression 3A(1) (t)+ B (2) (t) does not depend on t. For t = 0 it is the two-loop part of the invariant F() from (40), while for t = 1 it is the right hand side of (56). This concludes the proof. Remark 8 (Generalization). The generalization of Proposition 9 to an arbitrary (quadratic) g is straightforward. We no longer have ansatz (55) for Wˆ , since there might be more ad-invariant functions on g[1] ⊕ g[−2], but we still can write 003 Wˆ (α) = (1 + A(1) )Wprod (α) + 2 B (2) + O(3 + 2 (α(0) )2 α(3) + (α(0) )4 (α(3) )2 )

(we could prescribe weight 1 to and weight 1/3 to α(0) and α(3) , then we write explicitly terms with weight ≤ 2). The fact that O(α(0) α(0) α(3) )-contribution is propor003 (in principle there could be some other invariant tensor of rank 3) is tional to Wprod explained by the fact that it is given by a single Feynman diagram ∈ G 1,3 — the wheel with 3 leaves — and the Lie algebra part of this diagram is described by contraction d,e, f f ade f be f f c f d ∝ f abc . Following the proof of Proposition 9, for the homotopy we have 003 W (t, dt)(α) = (1+ A(1) (t)) Wprod (α)+ 2 B (2) (t)+ O(3 + 2 (α(0) )2 α(3) + (α(0) )4 (α(3) )2 ) 2 (1) a a 3 2 α(0) α(3) + O α(0) α(3) + (α(0) ) (α(3) ) +dt · D (t) . a

The extended QME at order O(dt (α(0) )2 α(3) + dt 2 ) yields ∂ (1) A (t) = −D (1) (t), ∂t

∂ (2) B (t) = dim g · D (1) (t). ∂t

Hence the two-loop part of the invariant F() is expressed in terms of coefficients of the relaxed effective action Wˆ as F (2) = dim g · A(1) + B (2) .

(60)

The other case discussed in Sect. 3.4, the case of formal C with general B1 , is translated straightforwardly into the setting of relaxed effective actions. Proposition 10. Suppose C is formal and ι : H • (C) → C is an algebra homomorphism. Then the relaxed effective action has the form Indι, Kˆ (S)(α) = Wprod (α) + Fˆ (1) (α) + O(2 ),

(61)

where the one-loop part can be expressed as a super-trace: 1 Fˆ (1) (α) = Str g⊗C log 1 + Kˆ ◦ l(ι(α), •) . 2

(62)

The restriction of the one-loop relaxed effective action to the Maurer-Cartan set Fˆ (1) |MC is invariant under deformations of (ι, Kˆ ), preserving the homomorphism property of ι.

826


Proof. Analogously to the case of a strict chain homotopy K (Proposition 7), formality of C together with Kˆ ◦ ι = 0 imply vanishing of non-trivial trees (hence the ansatz (61)) and that the only possibly non-vanishing one-loop diagrams are wheels (hence the super-trace formula (62)). Second, we know that there is a homotopy W (t, dt)(α) connecting the strict effective action W (α) = Wprod (α) + F (1) (α(1) ) + · · · to the relaxed one Wˆ (α) = Wprod (α) + Fˆ (1) (α) + · · ·: (1) W (t, dt)(α) = Wprod (α) + F (1) (t)(α) + O(2 ) + dt · R (t)(α) + O(2 ) (here we again exploit the vanishing of trees implied by construction (53), formality of C and K ◦ ι = d ◦ ι = 0). The extended QME for the homotopy at order O(dt · ) is ∂ (1) (1) F (t)(α) = {Wprod (α), R (t)(α)} . ∂t Hence the restriction F (1) (t)|MC does not depend on t, where MC = {α| l(ι(α), ι(α)) = 0} ⊂ F is the set of critical points of Wprod (the “non-homogeneous Maurer-Cartan set”). As MC ⊂ MC, the restriction F (1) (t)|MC also does not depend on t, hence Fˆ (1) |MC = F (1) |MC . As the right-hand side is invariant (Proposition 7), so is the left hand side. This concludes the proof. Obviously, going from the restriction to MC to the restriction to MC, we do not lose any invariant information, since F (1) depends only on α(1) and not on other components of α, which implies Fˆ (1) |MC = Fˆ (1) |MC . Note also that in Proposition 9 we managed to reconstruct only the two-loop part of the invariant F() (Proposition 6) in the relaxed setting. On the other hand, we can completely reconstruct the invariant F (1) |MC of Proposition 7 in the relaxed setting. 3.6. Examples of dg Frobenius algebras. In this section we will provide some examples of non-negatively graded dg Frobenius algebras with pairing of degree −3 and zeroth Betti number B0 = 1, i.e. algebras suitable for constructing abstract Chern–Simons actions. Example 1: Minimal dg Frobenius algebra. Let V be a vector space and µ ∈ ∧3 V ∗ an arbitrary exterior 3-form on V . Then we construct the dg Frobenius algebra C from this data as C := R · 1 ⊕ V [−1] ⊕ V ∗ [−2] ⊕ R · v, where v is a degree 3 element. The pairing π is defined to be the canonical pairing between V [−1] and V ∗ [−2], and also we set π(1, v) := 1. We define the product m as m(1, x) = x, m(x(1) , y(1) ) =< µ, x¯(1) ∧ y¯(1) >, m(x(1) , z (2) ) =< x¯(1) , z¯ (2) >,


827

where x ∈ C (element of arbitrary degree), x(1) , y(1) ∈ V [−1], z (2) ∈ V ∗ [−2] and bar means shifting an element to degree zero, e.g. x¯(1) = sx(1) ∈ V , z¯ (2) = s 2 z (2) ∈ V ∗ ; < •, • > denotes the canonical pairing. The differential d is set to zero. This construction gives the most general minimal (i.e. with zero differential) dg Frobenius algebra (non-negatively graded, with pairing of degree -3 and B0 = 1). Abstract Chern–Simons action associated to a minimal algebra is a purely cubic polynomial in the fields, with coefficients being the components of µ. Inducing the effective action on cohomology is the identity operation, since here C = H • (C). Example 2: Differential concentrated in degree C 1 → C 2 . Let V be a vector space, µ ∈ ∧3 V ∗ be some 3-form on V and δ : S 2 V → R some symmetric pairing on V (not necessarily non-degenerate). We define C, the pairing π and product m as in Example 1, but now we construct the differential d : V [−1] → V ∗ [−2] from δ as x(1) → s −2 δ(x¯(1) , •). The two other components of the differential R · 1 → V [−1] and V ∗ [−2] → R · v are set to zero as before. This construction gives the general dg Frobenius algebra with differential concentrated in degree C 1 [−1] → C 2 [−2] only. Here the Hodge decomposition C = ι(H • (C)) ⊕ Cd−ex ⊕ C K −ex is defined by a choice of projection p : V → ker δ or equivalently by choosing a complement V of ker δ ⊂ V in V (here we understand δ as the self-dual map V → V ∗ ). We set H • (C) = R · 1 ⊕ (ker δ)[−1] ⊕ (ker δ)∗ [−2] ⊕ R · v, Cd−ex = (imδ)[−2], C K −ex = V [−1]. The embedding ι : H • (C) → C is canonical in degrees 0,1,3 and given by p ∗ : (ker δ)∗ → V ∗ in degree 2. The (non-vanishing part of the) chain homotopy K is the inverse map for the isomorphism d : V [−1] → (imδ)[−2]. So the induction data (ι, K ) is completely determined by the choice of p. This means in particular that only the deformations of induction data of Type II are possible here. Also there are no relaxed chain homotopies Kˆ (other than the strict one described above), which is obvious from (49). Despite these simplifications, the effective action on cohomology for such C is in general non-trivial. A particular example here is the case C = S • (su(2)∗ [−1]) discussed in Sect. 3.4. Example 3: “Doubled” commutative dga. Let V = V 0 ⊕ V 1 [−1] be a unital commutative associative dg algebra, concentrated in degrees 0 and 1, with differential dV and multiplication m V , and satisfying dim H 0 (V) = 1. Then we set C := V ⊕ V ∗ [−3] = V 0 ⊕ V 1 [−1] ⊕ (V 1 )∗ [−2] ⊕ (V 0 )∗ [−3] with pairing π generated by the canonical pairing between V and V ∗ . The component of differential V 0 → V 1 [−1] is given by dV , the component (V 1 )∗ [−2] → (V 0 )∗ [−3] — by the dual map (dV )∗ , and the component V 1 [−1] → (V 1 )∗ [−2] is set to zero. Multiplication for elements of C of degrees 0 and 1 is given by m V and is extended to other degrees by the cyclicity property (15): m(x(0,1) , y(0,1) ) := m V (x(0,1) , y(0,1) ), m(x(0,1) , z (2,3) ) :=< m V (•, x(0,1) ), z (2,3) > for x(0,1) , y(0,1) ∈ V and z (2,3) ∈ V ∗ [−3]. In particular, the product of elements of degree 1 in C is zero. Hodge decomposition for C is fixed by choosing an embedding ιV : H • (V) → V (it is canonical in degree zero since H 0 (V) = R · 1, but non-canonical in degree one) and a retraction rV : V → H • (V ). Equivalently, we choose a splitting of V into representatives of cohomology and the complement V = ιV (H • (V)) ⊕ V , i.e.

828


V 0 = R · 1 ⊕ V 0 , V 1 = ιV (H 1 (V)) ⊕ V 1 . Then the Hodge decomposition for C is C = ι(H • (C)) ⊕ Cd−ex ⊕ C K −ex with H • (C) = R ⊕ H 1 (V)[−1] ⊕ (H 1 (V))∗ [−2] ⊕ R · v,

Cd−ex = V 1 [−1] ⊕ (V 0 )∗ [−3], C K −ex = V 0 ⊕ (V 1 )∗ [−2]. Here v ∈ (V 0 )∗ [−3] is the element defined by the component of retraction rV : V 0 → R · 1. The embedding ι : H • (C) → C is given by ιV in degrees 0,1 and by (rV )∗ in degrees 2,3. As in Example 2, only Type II deformations of the induction data (ι, K ) are possible here. However, one can introduce the relaxed chain homotopy (49) here with : (V 0 )∗ [−3] → V 0 . The BV integral (28) is Gaussian in this example since the cyclic product π(•, m(•, •)) vanishes on (C K −ex )⊗3 (for trivial degree reasons). Hence, the only Feynman graphs contributing to W (α) are wheels and the trivial tree 0,3 (“branches” could also contribute, but they vanish since the component of the multiplication ι(H 1 (C)) ⊗ ι(H 1 (C)) → C 2 vanishes and the non-trivial part of W (α) depends only on α(1) due to Proposition 4). The effective action can be written as 003 012 W (α) = Wprod (α) + Wprod (α) + 2 ·

1 tr g⊗V 0 log 1 + K ◦ l(ι(α(1) ), •) 2

(the factor 2 accounts for the contribution of the trace over g ⊗ (V 1 )∗ — the other half of the Lagrangian subspace L K ). 4. Three-Manifold Invariants We now wish to extend the results of the previous sections to the Frobenius algebra

• (M) of differential forms on a smooth compact 3-manifold M (with de Rham differential, wedge product and integration pairing). This will provide us with the effective action (around the trivial connection) of Chern–Simons theory [18]. The invariant of this Frobenius algebra will also constitute an invariant of 3-manifolds modulo diffeomorphisms. The discussion of the previous sections, however, does not go through automatically since • (M) is infinite dimensional. As in previous works [2,3,11] the way out is to restrict oneself to a special class of chain homotopies K for which the finite dimensional arguments are simply replaced by the application of Stokes’ theorem. However, at some point of the construction we have to choose a framing and hence we get invariants of framed 3-manifolds. 4.1. Induction data and the propagator. As in the finite-dimensional case we fix an embedding ι : H • (M) → • (M). By χ ∈ 3 (M × M) we will denote the represenPoincaré dual tative of the of the diagonal determined by this embedding. Namely, if 1, {αi , β i }i=1,...,B1 , v is a basis of ι(H • (M)) (with 1 the constant function, αi ∈ 1 , j β i ∈ 2 , v ∈ 3 and M αi β j = δi , M v = 1), then χ = v2 − αi,1 β2i + β1i αi,2 − v1 , where we have used Einstein’s convention over repeated indices and for any form γ ∈ • (M) we write γi for πi∗ γ with π1 and π2 the two projections M × M → M.


829

The chain homotopy K is assumed to be determined by a smooth integral kernel η. ˆ Namely, let C20 (M) := M × M \ = {(x1 , x2 ) ∈ M × M : x1 = x2 } be the open configuration space of two points in M and let C2 (M) be its Fulton–MacPherson– Axelrod–Singer (FMAS) compactification [8,2] obtained by replacing the diagonal with its unit normal bundle. Let π1,2 be the extensions to the compactification of the projection maps πi (x1 , x2 ) = xi . Then ηˆ is a smooth 2-form on C2 (M) and the chain homotopy K is defined by K α = −π2∗ (ηˆ π1∗ α),

α ∈ • (M),

where a lower ∗ denotes integration along the fiber. For K to be a symmetric chain homotopy satisfying (20), ηˆ must satisfy the following properties (see [3,5] for more details): P1 d ηˆ = π ∗ χ , with π the extension to C2 (M) of the inclusion of C20 (M) into M × M. P2 ∂C2 (M) ηˆ = −1. P3 T ∗ ηˆ = −η, ˆ where T is the extension to C2 (M) of the involution (x1 , x2 ) → (x2 , x1 ) of C20 (M). Observe that ∂C2 (M) is canonically diffeomorphic to the sphere tangent bundle ST M of M (ST M is the quotient of T M by the action (x, v) → (x, λv), λ ∈ R+∗ ). Condition P2 may then be refined to P2’ ι∗∂ ηˆ = −η, where ι∂ is the inclusion map ST M = ∂C2 (M) → C2 (M) and η is a given odd global angular form on ST M. Recall that a global angular form on a sphere bundle S → M is a differential form on the total space S whose restriction to each fiber generates its cohomology and whose differential is minus the pullback of a representative of the Euler class (in our case zero). These two properties are consistent with the restriction to the boundary of P1 and P2. Odd here means T ∗ η = −η, where T is the antipodal map on each fiber, and this is compatible with the restrictions to the boundary of P3. Global angular forms always exist. Remark 9 (Kontsevich [11]). Since ST M is trivial for every 3-manifold, there is a simple choice for η only depending on a choice of framing or, equivalently, a choice of trivi∼ alization f : ST M → M × S 2 . One simply sets η = f ∗ ω, where ω is the normalized S O(3)-invariant volume form on S 2 (tensor 1 ∈ 0 (M)). Remark 10 ([3]). It is also possible to construct an odd global angular form depending on the choice of a connection but not on a framing. Namely, realize ST M as the S 2 -bundle associated to the frame bundle F(M): ST M = F(M) × S O(3) S 2 (we reduce the structure group of the frame bundle to S O(3) by picking a Riemannian metric). By choosing a metric connection θ (e.g., the Levi-Civita connection), one defines η¯ := ω + d(θi x i )/(2π ); here the x i s, i = 1, 2, 3, are the homogeneous coordinates on S 2 , while the θi s are the coefficients of the connection in the basis {ξi } of so(3) given by (ξi ) jk = i jk . It is then easy to show that η¯ is a global angular form on F(M) × S 2 and that it is basic. Hence it defines a global angular form η on ST M. Lemma 2 ([3]). For any given odd global angular form η on ST M, there exists a propagator ηˆ ∈ 2 (C2 (M)) satisfying P1, P2’, P3.

830


Proof. The complete proof in the case of a rational homology sphere is contained in [3]. (The general case, whose proof is a straightforward generalization of the previous case, is spelled out in [5].) For completeness, we briefly recall the idea of the proof. One chooses a tubular neighborhhood U of ∂C2 (M) and a subneighborhood V . One picks a compactly supported function ρ which is constant and equal to −1 on V and is even under the action of T . Let p be the projection U → ∂C2 (M). The differential of the form ρp ∗ η is a representative of the Thom class of the normal bundle of . The form ρp ∗ η may be extended by zero to the whole of C2 (M) and its differential is then a representative of the Poincaré dual of . Thus, its difference from the given representative χ will be exact and actually regular on the diagonal (see Appendix B for the definition). Namely, dρp ∗ η = π ∗ (χ + dα), α ∈ 3 (M × M). Since both ρp ∗ η and χ are odd under the action of T , one may also choose α to be odd. Finally, one sets ηˆ := ρp ∗ η − π ∗ α. 4.2. The improved propagator. In the previous sections it was also important to assume the conditions K ◦ ι = 0 and K 2 = 0. In terms of the propagator ηˆ they correspond to additional conditions that are easily expressed by using the Definition 1 (Compactification of configuration spaces). The FMAS compactification Cn (M) of the configuration space Cn0 (M) := {(x1 , . . . , xn ) ∈ M n : xi = x j ∀i = j} is obtained by taking the closure Bl(M S , S ), Cn (M) := Cn0 (M) ⊂ M n × S⊂{1,...,n}:|S|≥2

where Bl(M S , S ) denotes the differential-geometric blowup obtained by replacing the principal diagonal S in M S with its unit normal bundle N ( S )/R+∗ . See [2,4]. We recall that the Cn (M) are smooth manifolds with corners. Notation 1. Given a differential form γ on M, we write γi := πi∗ γ , where πi is the extension to Cn (M) of the projection Cn0 (M) → M, (x1 , . . . , xn ) → xi . We also set ηˆ i j := πi∗j η, ˆ where πi j is the extension to the compactifications of the projection Cn0 (M) → C20 (M), (x1 , . . . , xn ) → (xi , x j ). Given a product of such forms on a compactified configuration space and a set of indices i, j, . . ., we denote by i, j,... the fiber integral over the points xi , x j , . . .. This notation also takes care of orientation. Namely, if i, j, . . . are not ordered, the integral carries the sign of the permutation to order them. We are finally ready to list the two simple properties corresponding to K ◦ ι = 0 and K 2 = 0: P4 2 ηˆ 12 γ2 = 0 for all γ ∈ ι(H • (M)). P5 2 ηˆ 12 ηˆ 23 = 0. Lemma 3. For any given odd global angular form η on ST M, there exists a propagator ηˆ ∈ 2 (C2 (M)) satisfying P1, P2’, P3, P4. Proof. The idea is to apply transformation (24) to a propagator as the one constructed in Lemma 2. This must however be reformulated in terms of integral kernels. Namely, let ηˆ satisfy P1,P2’,P3. Set


831

λ12 :=

η13 v3 − η23 v3 + η13 α3,i β2i − β1i η23 α3,i + η13 β3i α2,i + 3 3 3 3 3 j i i i + α1,i η23 β3 − α1,i β3 η34 β4 α2, j − β1 α3,i η34 v4 − v3 η34 α4,i β2i . 3

3,4

3,4

3,4

By construction the new propagator ηˆ − λ satisfies P3 and P4. It is not difficult to check that λ is closed, so P1 is still satisfied. Finally observe that the integrals on one argument simply produce a form on M, while the integrals on two arguments simply produce a number. As a consequence, λ is (the pullback by π of) a form on M × M. As it is also T -odd, its restriction to the boundary vanishes. Thus, P2’ still holds. 4.3. On property P5. Lemma 4. For any given odd global angular form η on ST M, there exists a propagator ηˆ ∈ 2 (C2 (M)) satisfying P1, P2, P3, P4, P5. Proof. We apply transformation (25) in the equivalent form K 3 = K 2 + [d, dK 23 ] to the propagator ηˆ constructed in Lemma 3. In terms of integral kernels, the new propagator is ηˆ + γ with γ12 := dd1 f 12 and ηˆ 13 ηˆ 34 ηˆ 42 . f 12 := 3,4

Properties P1 and P2 are obviously satisfied as we have changed the propagator by an exact term. As for Property P3, observe that equivalently γ may be written as d2 d1 f 12 and that f is even under the action of T . Finally Property P4 is easily checked by integration. It would be very useful to prove the following Conjecture 1. The propagator constructed in Lemma 4 also satisfies Property P2’. Observe that since γ is T -odd, it would suffice to show that it is regular on the diagonal (i.e., a pullback from M × M). For this it would suffice to show that f or at least d f has this property. By its definition f looks rather regular, but at the moment we have no complete proof of this fact. Remark 11 (The Riemannian propagator). The physicists’ treatment of Chern–Simons theory would simply be to choose a Riemannian metric g and use it to impose the Lorentz gauge-fixing. Out of this one gets the propagator d∗ ◦ G, where G is the Green function for +P , where is the Laplace operator and P is the projection to harmonic forms in the Hodge decomposition. The integral kernel of this propagator is a smooth two-form ηˆ on C2 (M) satisfying Properties P1, P2’, P3, P4, P5 with odd angular form on the boundary of the type described in Remark 10. In this case, the metric connection is actually the Levi-Civita connection for the chosen Riemannian metric. See [2] and [3, Remark 3.6].

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4.4. The construction of the invariant by a framed propagator. We now fix the boundary value of the propagator as in Remark 9 for a given choice f of framing. We also choose an embedding ι of cohomology and denote by Pι, f the space of propagators satisfying Properties P1,P2’,P3. Observe that, by Lemma 3, Pι, f is not empty. For ηˆ ∈ Pι, f we define Indι,ηˆ ∈ Fun(F )[[]], F = H • (M, g)[1] = H • (M)[1] ⊗ g, analogously to Indι,K as at the beginning of Subsect. 3.5 by the following obvious changes of notations: (1) Every chain homotopy K is replaced by a propagator η; ˆ (2) Every vertex is replaced by a point in the compactified configuration space over which we eventually integrate. Signs may be taken care of by choosing an ordering of vertices and of half edges at each vertex (see, e.g., [3]). All computations in Sect. 3 go through as they are simply replaced by Stokes theorem: n d = (−1) d+ . Cn (M)

Cn (M)

∂Cn (M)

Among the codimension-one boundary components of Cn (M) we distinguish between principal and hidden faces: the former correspond to the collapse of exactly two points, the latter to the collapse of more than two points. Principal faces contribute by the same combinatorics as in Sect. 3, whereas hidden faces do not contribute by our choice of propagators because of Kontsevich’s Vanishing Lemmata [11]. As a result we conclude that Indι,ηˆ satisfies the quantum master equation for every choice of induction data as above. For more details, we refer to Appendix A. By the same reasoning and by the arguments of Subsect. 3.5, we may prove that Indι,ηˆ is canonically equivalent to a strict effective action. This allows us to recover at least the two-loop part of the complete invariant of a rational homology sphere as discussed in Subsect. 3.5.1. Remark 12. If Conjecture 1 were true, the space Pι, f of propagators satisfying Properties P1,P2’,P3,P4,P5 would not be empty. We could then repeat the above construction using Pι, f instead of Pι, f and get a strict effective action directly. The discussions of Subsects. 3.2, 3.3 and 3.4 would then go through and, in particular, Propositions 4, 5, 6, 7 would hold. Remark 13. In [3] an invariant for framed rational homology spheres was introduced. The boundary condition for the propagator was different (see the next subsection), but this is immaterial for the present discussion. Namely, choose a propagator in Pι, f and define η˜ 123 := ηˆ 12 + ηˆ 23 + ηˆ 31 . If M is a rational homology sphere, η˜ 123 is closed. Now take the graphs appearing in the constant part of the strict effective action and reinterpret them as follows: (1) Each vertex is replaced by a point in the compactified configuration space; 3 (2) An extra point x0 is added on which one puts the representative v ∈ ι(H (M)) with v = 1; (3) Each chain homotopy is replaced by η˜ (more precisely, the chain homotopy between vertices i and j is replaced by η˜ i j0 ). It is now possible to show that this produces an invariant of (M, f ). This is a different way of getting the invariant corresponding to the constant part of the strict effective action for a choice of propagator not necessarily satisfying Property P5. We do not have


833

a direct proof that this invariant is the same. The indirect proof consists of showing that both invariants are finite type with the same normalizations along the lines of [13]. If Conjecture 1 were true, then it would immediately follow that this invariant is exactly the constant part of the strict effective action. In fact, for a propagator as in the conjecture, it is not difficult to show that only the term ηˆ i j in each η˜ i j0 would contribute (and the integration over x0 would then decouple). Since the induced action is constant on Pι, f , by restriction to Pι, f ⊂ Pι, f we would prove the claim. 4.5. The unframed propagator. Instead of using Kontsevich’s propagator, one may proceed as in [3] and define the propagator by choosing the global angular form on ∂C2 (M) as in Remark 10. Recall that in this case no choice of framing is required. On the other hand, one needs to specify a Riemannian metric g and a metric connection θ . We denote by Pι,g,θ the space of propagators corresponding to these choices. We proceed exactly as in the previous subsection to define the effective action (see Appendix A for more details). In particular we want to check independence on the induction data; so we choose a path in Pι,g,θ and consider the effective action W as a function on the shifted cohomology tensor the differential forms on [0, 1] and check whether the extended QME (d + )W + {W, W }/2 = 0 holds. The only difference with respect to the previous subsection is that there is an extra set of boundary components of the configuration spaces that may appear: namely, the most degenerate faces corresponding to the collapse of all points. These faces may be treated exactly as in [2,3] and one shows that their contribution is a multiple of the first Pontryagin form − tr Fθ2 /(8π 2 ), where Fθ is the curvature of θ . The important point is that the coefficient depends only on the graph involved but not on the 3-manifold M. As a result the effective action might not satisfy the extended quantum master equation. However, one may easily compensate for this by adding to it the integral over M of the Chern–Simons 3-form of the connection θ pulled back from F(M) to M by choosing a section f (i.e., a framing). The framing now appears because of this correction but is not present in the propagator. The main disadvantage of this approach is that one does not know how to compute the universal coefficients. (It is known that the coefficients vanish for graphs with an odd number of loops, while for the graph with two loops one may compute the coefficient explicitly and see that it is not zero.) The advantage is that Pι,g,θ contains a subspace of propagators satisfying also Property P5: These are the integral kernels constructed in [2], see Remark 11. With these choices, and the addition of the frame-dependent constant as in the previous paragraph, one gets an induced effective action satisfying all properties stated in Subsects. 3.2, 3.3 and 3.4. More precisely, let 1 2 3 ∗ CS(M, θ, f ) := − 2 f Tr θ dθ + θ 8π M 3 be the Chern–Simons integral for a connection θ on the frame bundle F(M) of M and a framing f (regarded here as a section of F(M)). Theorem 1. Let M be a compact 3-manifold and g a quadratic Lie algebra. Then (1) For every choice of Riemannian metric g on M, the effective action W constructed using the Riemannian propagator of Remark 11 is a function on H • (M, g)[1], solves the quantum master equation and has the properties described in Proposition 4. In addition it has the form given in (40) in case B1 (M) = 0 and in (44) in case M is formal.

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(2) There is a universal element φ ∈ 2 R[[2 ]], depending only on the choice of Lie algebra g, such that the modified effective action (M, g, f ) := W (M, g) + φ CS(M, θg , f ), W where θg is the Levi-Civita connection for g, solves the QME and is independent of g modulo canonical transformations as in Proposition 5. In particular we get invariants for the framed 3-manifold (M, f ) as in Propositions 6 and 7. Remark 14. The leading contribution to φ may be explicitly computed and yields φ = C2 (g)2 /48 + O(4 ) with C2 (g) = f abc f abc , where the f abc are the structure constants of g in an orthonormal basis. It is not known whether there are nonvanishing higher order corrections. Appendix A. The Chern–Simons Manifold Invariant In this Appendix we give more details on the construction outlined in Subsect. 4.5. To a graph with || vertices we associate an element ω of • (C|| (M)) ⊗ Fun(F ) as follows:9 • to each edge we associate the pullback of a propagator by the corresponding projection from C|| (M) to C2 (M); • to each leaf associate the pullback (by the corresponding projection from C|| (M) we a γ µ e , where {γ µ } is the chosen basis of H • (M), {e } is an orthonormal to M) of z µ a a a }s are the corresponding coordinate functions on F ; basis of g, and the {z µ • on each vertex we put a structure constant in the orthonormal basis chosen above. Then we take the wedge product of the differential forms and sum over Lie algebra indices for each edge. The result does not depend on the choice of orthonormal basis for g but depends on a choice of numbering of the vertices and of orientation of the edges. If we however make the same choice also to orient C|| (M), then ω ∈ Fun(F ) C|| (M)

is well defined. We define Z (the exponential of W ) to be the sum of ω|| /| aut | over all trivalent graphs, where | aut | is the order or the group of automorphisms of . Proving the QME for W is equivalent to proving Z = 0. The main observation is that Property P1 of the propagator and the same combinatorics as in the toy model imply that Z is obtained by replacing the ω s by dω s one by one. We then use Stokes theorem. The contributions of principal faces (i.e., boundary faces of configuration spaces corresponding to the collapse of two vertices) sum up to zero thanks to the Lie algebra contributions (this is also the same combinatorics as in the toy model). Hidden faces (i.e., the other boundary faces) may in principle contribute. Let γ be the subgraph corresponding to a hidden face (i.e., the vertices of γ are those that collapse and its edges are the edges between such vertices). By simple dimensional reasons, the hidden faces corresponding to γ vanishes if γ has a univalent vertex; if 9 Here and in the rest of the Appendix, the symbol ⊗ is understood as the completed tensor product: i.e., the space of functions on the Cartesian product of the corresponding supermanifolds.


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γ has a bivalent vertex, its contribution also vanishes by Kontsevich’s lemma thanks to Property P3. Since we only consider trivalent graphs, we are left with contributions coming from the collapse of all vertices of a connected component of s with no leaves. These are the graphs that contribute to the constant part of Z and thus of W . The latter contributions also vanish by a simple dimensional argument. More generally, to keep track of the choices involved in the propagator, we consider a one parameter family of choices with parameter t ∈ I := [0, 1] and show Z˜ = 0 with := + dt dtd and Z˜ ∈ Fun(F ) ⊗ • (I ) constructed as follows. Let ηˆ be a one-parameter family of propagators regarded as an element of 2 (C2 (M) × I ) related, at every t ∈ I , by Property P1 to the one parameter family {γ µ } of bases of • (M). 2 µ • Let η˙µ∈ ν (C2 (M) × I ) and γ˙ ∈ (M × I ) be their t-derivatives. We may assume M γ γ˙ = 0, ∀µ, ν, for more general choices may be compensated by a linear transformation of H • (M). Observe that by Property P2’, the restriction of ηˆ to the boundary is fixed, so the restriction of η˙ vanishes. Actually, by construction η˙ vanishes in a whole neighborhood of the boundary, so it is a regular form. Let λ13 :=

2 ηˆ 12 η˙ 23

− η˙ 12 ηˆ 23 , 2

µ µ which is regular by Lemma 5 in Appendix B. Also set ξ1 := ( ηˆ 12 γ˙2 ). We define η˜ := ηˆ − λ dt, γ˜ µ := γ µ + ξ µ dt, χ˜ := gµν γ˜ µ γ˜ ν = χ + O(dt), where gµν is the metric on H • (M) in the given basis. A simple computation then shows D η˜ = χ˜ and D γ˜ µ = 0 with D := d + dt dtd . Finally, we define Z˜ as above by using η˜ and γ˜ instead of ηˆ and γ , respectively. We now observe that applying to Z˜ is the same as applying D to the propagators. Reasoning as above by Stokes theorem, we see that the only possible nonvanishing contributions come from hidden faces corresponding to the collapse of all vertices of a connected component with no leaves.10 These contributions also vanish if one uses a framed propagator, whereas the choice of an unframed propagator yields some constant times the integral of the Pontryagin form on M as a one-form on I , see [3]. By writing Z˜ = Z + ζ dt, we may see that the equivalences are produced by ζ . These are very particular kinds of BV equivalences as ζ consists of graphs decorated by propagators and generators of cohomology classes with the exception of one edge that is decorated by λ or one leaf that is decorated by ξ . ˜ If Property P5 holds, drastically. By construction we the computation of Z simplifies µ obtain 2 ηˆ 12 λ23 = 2 λ12 ηˆ 23 = 0 and 2 η12 ξ2 = 0. As a result, whenever a vertex is decorated by 1 ∈ 0 (M) the corresponding integral vanishes. Thus, Z˜ , as a function on H • (M, g)[1] is independent of the coordinates in degree 1 apart from the trivial classical term. Since Z is of degree zero, it will only depend on the coordinates of degree zero. Since ζ is of degree −1, it will be linear in the coordinates of degree −1 with coefficients depending on the coordinates of degree zero. 10 Observe that, since λ is regular, only η will appear in the boundary computations.

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Appendix B. Regular Forms Let : Cn (M) → M n be the extension to the compactification of the inclusion Cn (M)0 → M n . We call a form on Cn (M) regular if it is a pullback by . Recall the maps πi s and πi j s defined in Notation 1 in Subsect. 4.2. one of them be Lemma 5. Let α and β be differential forms on C2 (M) and let∗at least ∗ β) is regular. regular. Then their convolution α ∗ β := 2 α12 β23 := π13,∗ (π12 α π23 Proof. Suppose that, e.g., α is regular; i.e., α = ∗ α , α ∈ • (M × M). Define γ = (pr 1 ×π2 )∗ (pr 1 ×π1 )∗ α pr ∗2 β ∈ • (M × M), where pr 1 and pr 2 are the projections from M ×C2 (M) to the two factors. It then follows that α ∗ β = ∗ γ .

Acknowledgements. We are grateful to K. Costello and D. Sinha for insightful discussions. We also thank C. Rossi and J. Stasheff for useful remarks.

References 1. Adams, D.H.: A note on the Faddeev–Popov determinant and Chern–Simons perturbation theory. Lett. Math. Phys. 42, 205–214 (1997) 2. Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. In: Proceedings of the XXth DGM Conference, ed. S. Catto, A. Rocha, Singapore: World Scientific, 1992, pp. 3–45; “Chern–Simons perturbation theory. II.” J. Diff. Geom. 39, 173–213 (1994) 3. Bott, R., Cattaneo, A.S.: Integral invariants of 3-manifolds. J. Diff. Geom. 48, 91–133 (1998) 4. Bott, R., Taubes, C.: On the self-linking of knots. J. Math. Phys. 35, 5247–5287 (1994) 5. Cattaneo, A.S.: Configuration space integrals and invariants for 3-manifolds and knots. In: Low Dimensional Topology, ed. H. Nencka, Cont. Math. 233, Providence, RI: Amer. Math. Soc., 1999, pp. 153–165 6. Cattaneo, A.S., Felder, G.: Effective Batalin–Vilkovisky theories, equivariant configuration spaces and cyclic chains. http://arXiv.org/abs/0802.1706v1[math-ph], 2008 7. Costello, K.J.: Renormalisation and the Batalin–Vilkovisky formalism. http://arXiv.org/abs/0706. 1533v3[math.QA], 2007 8. Fulton, W., MacPherson, R.: A compactification of configuration spaces. Ann. Math. 139, 183–225 (1994) 9. Gugenheim, V.K.A.M., Lambe, L.A.: Perturbation theory in differential homological algebra I. Illinois J. Math. 33(4), 566–582 (1989) 10. Iacovino, V.: Master equation and perturbative Chern–Simons theory. http://arXiv.org/abs/0811. 2181v1[math.DG], 2008 11. Kontsevich, M.: Feynman diagrams and low-dimensional topology. First European Congress of Mathematics, Paris 1992, Volume II, Progress in Mathematics 120, Basel-Boston: Birkhäuser, 1994, pp. 97–121 12. Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry, Seoul 2000, River Edge, NJ: World Sci. Publ., 2001, pp. 203–263 13. Kuperberg, G., Thurston, D.P.: Perturbative 3-manifold invariants by cut-and-paste topology. http:// arXiv.org/abs/math/9912167v2[math.GT], 2000 14. Losev, A.: BV Formalism and Quantum Homotopical Structures. Lectures at GAP3, Perugia, 2006 15. Mnev, P.: Notes on simplicial BF theory. Moscow Math. J. 9(2), 371–410 (2009) 16. Mnev, P.: Discrete BF Theory. http://arXiv.org/abs/0809.1160v2[hep-th], 2008 17. Schwarz, A.: A-model and generalized Chern–Simons theory. Phys. Lett. B 620, 180–186 (2005) 18. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989) Communicated by L. Takhtajan


Communications in


Back to Balls in Billiards Françoise Pène1,2, , Benoît Saussol1,2 1 Université Européenne de Bretagne, Brest, France 2 Université de Brest, Laboratoire de Mathématiques, CNRS UMR 6205,

6 avenue Victor Le Gorgeu, F-29238 Brest, France. E-mail: [email protected]; [email protected] Received: 19 December 2008 / Accepted: 15 June 2009 Published online: 19 November 2009 – © Springer-Verlag 2009

Abstract: We consider a billiard in the plane with periodic configuration of convex scatterers. This system is recurrent, in the sense that almost every orbit comes back arbitrarily close to the initial point. In this paper we study the time needed to get back in an ε-ball about the initial point, in the phase space and also for the position, in the limit when ε → 0. We establish the existence of an almost sure convergence rate, and prove a convergence in distribution for the rescaled return times. 1. Introduction 1.1. Periodic Lorentz gas. We consider a planar billiard with periodic configuration of scatterers. Such a model is also called a Lorentz process. The motion of a free point particle bouncing on the scatterers according to Descartes’ reflection law defines a flow. The flow conserves the initial speed, so that without loss of generality we will assume that the particle moves with unit speed. This is a Hamiltonian flow which preserves a Liouville measure. Observe that the phase space is spatially extended and thus the measure is infinite. We will suppose that the horizon is finite, i.e. the time between two consecutive reflections is uniformly bounded. We are interested in the quantitative aspect of Poincaré’s recurrence for the billiard flow. It is known that this system is recurrent, in particular almost every orbit comes back arbitrarily close to the initial point. In this paper, our goal is to study the return time in balls, in the limit when the radius goes to zero. Our main result is that (i) the time Z ε to get back ε-close to the initial point in the phase space is of order exp( ε12 ) for Lebesgue almost all initial conditions, (i’) the time Zε to get back ε-close to the initial position is of order exp( 1ε ) for Lebesgue almost all initial conditions, Françoise Pène is partially supported by the ANR project TEMI (Théorie Ergodique en mesure infinie).

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F. Pène, B. Saussol

Fig. 1.1. Motion of a point particule in the Lorentz process

(ii) we determine the fluctuations of ε2 log Z ε and of ε log Zε by proving a convergence in distribution to a simple law. This subject has been well studied recently in the setting of finite measure preserving transformations and typical behavior has been proved in a variety of chaotic systems: exponential statistics of return time, Poisson law, relation between recurrence rate and dimensions (see e.g. [1] for a state of the art in a probabilistic setting; also [2,8]). The present work differs by two points from the existing literature. First, the system in question has continuous time; second, the main novelty is that its natural invariant measure is σ -finite. Very few works have appeared on the topic in this situation [3,12,23]. A first reduction of the dynamics at the time of collisions with the scatterers (Poincaré section) and a second reduction by periodicity defines the praised billiard map. This map belongs to the class of hyperbolic systems with singularities. Since the work of Sinaï [27] establishing the ergodicity of the billiard map, it has been studied by many authors (let us mention [4–7,13]) giving: Bernoulli property, central limit theorem. In the past ten years, the new approach of L.-S. Young [29] has been exploited to get new significant results for the billiard map. Among them, let us mention the exponential decay of correlations [29], a new proof of the central limit theorem [29] and the local limit theorem proved by Szász and Varjú [28]. Conze [9] and Schmidt [25] proved that recurrence of the Lorentz process follows from some central limit theorem for the billiard map. Szász and Varjú [28] used their local limit theorem to give another proof of the recurrence. As proved by Simányi [26] and the first named author [20], once its recurrence is proved, it is not difficult to prove the total ergodicity of the Lorentz process. More recently, estimates on the first return time in the initial cell have been established by Dolgopyat, Szász and Varjú in [10] and an analogous estimate for the return time in the initial obstacle follows from a paper of the first named author [22]. 1.2. Precise description of the model and statement of the results. We now precisely define the billiard flow Φt . Let (Oi )i∈I be a finite number of open, convex subsets of

Back to Balls in Billiards

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O1+(0,1)

O1+(−1,1)

O2+(−1,0)

O1 +(1,1)

O2+(1,0)

O2

O1+(−1,0)

O2+(−1,−1)

O2 +(1,1)

O 2 +(0,1)

O2+(−1,1)

O1

O2+(−1,0)

O1+(1,0)

O2 +(1,−1)

Fig. 1.2. Labeling of the obstacles

v nq

v Fig. 1.3. Elastic reflection

R2 with C 3 boundaries and non-null curvature. We let Q = R2 \

+ Oi be the

i∈I,∈Z2

billiard domain in the plane. We suppose that the sets + Oi in this union have pairwise disjoint closure. The flow is given by the motion of a point particle with position q ∈ Q and velocity v ∈ S 1 . Namely, the motion is ballistic if there are no collisions with an obstacle in the time interval [0, t]: Φt (q, v) = (q + tv, v). At the time of a collision the velocity changes according to reflection law v → v : If n q denotes the normal to the boundary of the obstacle at the point of collision q ∈ ∂ Q, pointing inside the domain (i.e. outside the obstacle) then the angle ∠(n q , v ) = π − ∠(n q , v); see Fig. 1.3. We assume that the billiard has finite horizon, in the sense that the time between two consecutive collisions is uniformly bounded. We endow the space X = Q × S 1 with the product metric d((q, v), (q , v )) = max(d(q, q ), d(v, v )), where for simplicity we denote all the distances by d. The flow preserves the Lebesgue measure on Q × S 1 ; it is σ -finite but nevertheless the system is well known to be recurrent [9,25,28].

840


For x ∈ X and ε > 0 we define the minimal time to get back ε-close to the initial point by Z ε (x) := inf {t > ε : d(Φt (x), x) < ε} .

(1.1)

The quantity Z ε (x) is well defined and finite for, at least, Lebesgue a.e. x. We denote by Π Q : X = Q × S 1 → Q the canonical projection. We also define the minimal time to get back ε-close to the initial position by (1.2) Zε (x) := inf t > ε : d(Π Q (Φt (x)), Π Q (x)) < ε . In the paper we give a precise asymptotic analysis of the return times Z ε and Zε expressed by our main theorem. We say that a random variable Yε defined on X converges in the strong distribution sense to a random variable Y if for any probability P Leb, Yε → Y in distribution under P. Theorem 1.1. The billiard flow satisfies log log Z ε (x) = 2; − log ε 2 (ii) the random variable ε log Z ε converges as ε → 0 in the strong distribution sense to a random variable Y0 with distribution P(Y0 > t) = 1 +1β0 t for some constant β0 > 0; log log Zε (x) = 1; (iii) for Lebesgue a.e. x ∈ X we have lim ε→0 − log ε (iv) the random variable ε log Zε converges as ε → 0 in the strong distribution sense to a random variable Y1 with distribution P(Y1 > t) = 1 +1β1 t for some constant β1 > 0. (i) for Lebesgue a.e. x ∈ X we have lim

ε→0

Remark 1.2. The constant β0 is equal to

2β , i∈I |∂ Oi |

with β =

where Σ 2 is the asymptotic covariance matrix of the cell shift function κ for the billiard map (T¯ , µ) ¯ . defined by (A.1); See Sect. 4 for precision. The constant β1 is equal to 2πβ |∂ O | √1 , 2π det Σ 2

i∈I

i

In Sect. 2 we define the billiard maps associated to our billiard flow. In Sect. 3 we investigate the behavior of return times for the billiard map. In Sect. 4 we pursue this analysis for the extended billiard map, and building on the previous section we prove some preparatory results. Section 5 is then devoted to the proof of the part of Theorem 1.1 relative to returns in the phase space. Finally, in Sect. 6 we prove the part relative to returns for the position. 2. Billiard Maps 2.1. Discrete time dynamics and new coordinates. In order to study the statistical properties of the billiard flow, it is classical to make a Poincaré section at collisions times, i.e. when Φt (q, v) ∈ ∂ Q × S 1 . For definiteness, when q ∈ ∂ Q we choose the velocity v pointing outside the obstacle, that is right after the collision. Denote for such a q ∈ ∂ Q and v ∈ S 1 by τ (q, v) the time before the next collision: τ (q, v) = min{t > 0 : Φt (q, v) ∈ ∂ Q × S 1 }. Let φ be the Poincaré map: φ(q, v) = Φτ (q,v) (q, v) = (q , v ) (see Fig. 2.1).


841

ví q’

q

v nq

τ (q,v)

Fig. 2.1. The Poincaré section at collisions times

Next, we make a change of coordinates for the base map. For each obstacle Oi we choose an arbitrary origin and parametrize its boundary ∂ Oi by counter-clockwise arclength. The position q ∈ ∂ Q is represented by (, i, r ) if q ∈ + ∂ Oi and r is the parametrization of the point q. The normal of the boundary at each point q is denoted by n q and the velocity v is represented by its angle ϕ ∈ [− π2 , π2 ] with n q . Let π π {(, i)} × R/|∂ Oi |Z × − , M= 2 2 2 ∈Z i∈I

endowed with the product metric. Denote by ψ : M → ∂ Q × S 1 the change of coordinate, such that ψ(, i, r, ϕ) = (q, v). The extended billiard map T : M → M is the Poincaré map φ in these new coordinates: T = ψ −1 ◦ φ ◦ ψ. The flow Φt is conjugated to the special flow Ψt defined over the map T under the free flight function τ ◦ ψ. Let Mτ = {(m, s) ∈ M ×R : 0 ≤ s < τ (ψ(m))}. We denote by π : Mτ → M the projection onto the base defined by π(m, s) = m and extends the conjugation ψ to Mτ by setting ψ(m, s) = Φs (ψ(m)). Let M¯ be the subset of M corresponding to the cell = 0. We define the billiard map T¯ : M¯ → M¯ corresponding to the quotient map of T by Z2 ; this is well defined by Z2 -periodicity of the obstacles. The cell shift function κ : M → Z2 is defined by κ(, i, r, ϕ) = − if T (, i, r, ϕ) = ( , i , r , ϕ ). During the proof of our theorems on the billiard flow we will prove a version of the local limit theorem for the billiard map suitable for our purpose, as well as a property of recurrence called exponential law for the return time statistics. 2.2. Different quantities related to recurrence. The notion of recurrence in these billiard maps gives rise to the definition of the following different quantities. Let m ∈ M and ¯ m¯ ∈ M. Let W A (m) be the first iterate n ≥ 1 such that T n m ∈ A for some subset A ⊂ M. ¯ Let W¯ B (m) ¯ be the first iterate n ≥ 1 such that T¯ n m¯ ∈ B for some subset B ⊂ M. Let Wε (m) be the first iterate n ≥ 1 such that d(T n m, m) < ε for some ε > 0. Let W¯ ε (m) ¯ be the first iterate n ≥ 1 such that d(T¯ n m, ¯ m) ¯ < ε for some ε > 0. 3. Recurrence for the Billiard Map Recall that the billiard map T¯ preserves a probability measure µ¯ equivalent to the ¯ whose density is given by Lebesgue measure on M, 1 cos ϕ, where Γ := |∂ Oi |. ρ(, i, r, ϕ) = 2Γ i∈I

842


¯ T¯ ) is two dimensional with one negative and one positive The billiard system ( M, Lyapunov exponent and the singularities are not too wild, therefore the result on recurrence rate [24] applies. Theorem 3.1 ([24]). The recurrence rate of the billiard map is equal to the dimension: log W¯ ε = 2 µ¯ a.e. ε→0 − log ε lim

¯ for all c1 > 0, c2 > 0, α > 0 and for all Lemma 3.2. For µ-almost ¯ every m ∈ M, family (Dε )ε of sets containing m such that ¯ ε ) ≥ c1 (diam(Dε ))2 , Dε ⊆ B(m, c2 ε) and µ(D we have µ( ¯ W¯ B(m,ε) ≤ ε−2 + α |Dε ) → 0. Proof. Let α > 0, c1 > 0 and c2 > 0. Choose some a ∈ (0, α) and set for some ε0 > 0, log W¯ ε (m) Fa = {m ∈ M¯ : ∀ε ≤ ε0 , ≥ 2 − a}. − log ε By Theorem 3.1 we have µ(F ¯ a ) → 1 as ε0 → 0. There exists ε1 > 0 such that, for any ε < ε1 we have the inclusions Dε ∩ {W¯ B(m,ε) ≤ ε−2 + α } ⊂ Dε ∩ {W¯ (1+c2 )ε ≤ ε−2 + α } ⊂ Dε ∩ Fac . Thus for any density point m of the set Fa relative to the Lebesgue basis given by (B(·, ε))ε we obtain ¯ ac |Dε ) µ( ¯ W¯ B(m,ε) ≤ ε−2 + α |Dε ) ≤ µ(F ≤ µ(F ¯ ac |B(m, diam Dε )) as ε → 0.

µ(B(m, ¯ diam Dε )) →0 µ(D ¯ ε)

We call non-sticky a point m satisfying the conclusion of Lemma 3.2 and we denote by N S the set of non-sticky points. We emphasize that µ(N ¯ S) = 1. The next theorem says that the return times and entrance times in balls are exponentially distributed for the billiard map. Theorem 3.3. Let m ∈ N S be a non-sticky point. We have µ( ¯ µ(B(m, ¯ ε))W¯ B(m,ε) (·) > t|B(m, ε)) → e−t , µ( ¯ µ(B(m, ¯ ε))W¯ B(m,ε) (·) > t) → e−t , uniformly in t ≥ 0, as ε → 0. We denote by A[η] the η-neighborhood of a set A.


843

Proof. We use an approximation by cylinders, the exponential mixing and the method developed in [15] for exponential return times and entrance times. We write A = B(m, ε) for convenience. According to Theorem 2.1 in [15], it suffices to show that

¯ W¯ A > n|A) − µ( ¯ W¯ A > n) = oε (1), sup µ( n

since it will imply that the limiting distributions exist and are both exponential. Let c3 > 0 be such that µ(∂ A[η] ) ≤ c3 η independently of ε. Let k be an integer such that δ k ≈ ε3 . Let g be an integer such that θ g−2k ≈ ε3 , where θ is the constant appearing in Theorem A.3. If m is a non-sticky point, observing that g is logarithmic in ε, we have for any integer n,

µ( ¯ W¯ A > n|A) − µ( ¯ W¯ A ≤ g|A) = oε (1). ¯ W¯ A ◦ T¯ g > n − g|A) ≤ µ( Set E = {W¯ A > n − g}. We approach A and E by a union of cylinder sets: Let A be the union of all the cylinders (see Appendix A.1 for the precise definition) k such that Z ⊂ A. We have A ⊂ A and A\A ⊂ ∂ A[c0 δ k ] by Lemma A.1. Thus Z ∈ ξ−k ) ≤ c c δk . we get µ(A\A ¯ 3 0 Let

E =

n−g

T¯ − j (∪ Z ∈ξ k+ j

−k− j ,Z ∩A =∅

j=1

Z )c .

We have E ⊂ E and by Lemma A.1 again E\E ⊂ (∂ A)[c0 δ ] ∪ k

n−g

k+ j T¯ − j (∂ A)[c0 δ ] .

j=1 ) ≤ c c δ . Using the decay of correlations Thus by the invariance of µ¯ we get µ(E\E ¯ 3 0 1−δ (for cylinders, see Theorem A.3 in Appendix A.1) we get that

µ(A ¯ ∩ T¯ −g E ) − µ(A ¯ )µ(E ¯ ) ≤ Cθ g−2k = o(µ(A)). ¯ k

Furthermore,

≤ µ(

µ( ¯ W¯ A ≤ g) ≤ g µ(A) ¯ W¯ A > n) − µ(E) ¯ ¯ = o(1).

Putting together all these estimates gives

µ( ¯ W¯ A > n|A) − µ( ¯ W¯ A > n) = o(1), uniformly in n ∈ N.

Next, using the mixing property again we can condition on a smaller set and still get the same limiting law. Proposition 3.4. For any m ∈ N S there exists a function f m , lim f m (ε) = 0, and such that the following holds: For any ε > 0 and any balls Dε , Aε of M¯ such that

ε→0

844


(i) m ∈ Dε ⊂ Aε = B(m, ε), (ii) µ(D ¯ ε ) ≥ ε2.25 , we have for any n,

¯ ¯ ε )

¯ W Aε (·) > n|Dε ) − e−n µ(A

µ(

≤ f m (ε). Proof. We approximate the sets D and E = {W¯ A > n} from the inside by sets D and E as we approximated the sets A and E in the proof of Theorem 3.3. With the same g we get

µ( ¯ W¯ A ≤ g|D) = o(1) ¯ W¯ A > n|D) − µ( ¯ W¯ A ◦ T¯ g > n − g|D) ≤ µ( for non-sticky points. Using the exponential decay of correlations for cylinders given by Theorem A.3 we get that ¯ W¯ A ◦ T¯ g > n − g) + o(1) µ( ¯ W¯ A ◦ T¯ g > n − g|D) = µ( ¯ = e−n µ(A) + o(1)

by Theorem 3.3.

The following result of independent interest will not be used in the sequel and can be derived from Proposition 3.4 as Proposition 4.7 would be derived from Proposition 4.6. Therefore we omit its proof. Proposition 3.5. The random variable 4ε2 ρ(·)W¯ ε (·) converges, in the strong distribution sense, to the exponential law with parameter one. The random variable ε2 W¯ ε (·) converges, under the law of µ, ¯ to a random variable Y which is a continuous mixture of exponentials. More precisely Y has distribution P(Y > t) = e−4tρ d µ. ¯ M¯

4. Recurrence for the Extended Billiard Map Recall that the extended billiard map (M, T ) preserves the σ -finite measure µ equivalent to the Lebesgue measure on M, which is the image of the Lebesgue measure on Q × S 1 , whose density is equal to cos ϕ. Note that ¯ µ| M¯ = 2Γ µ.

(4.1)

4.1. Preliminary results on the extended billiard map. We will use the following extension of Szász and Varjú’s local limit theorem [28]. We denote by Sn κ =

n−1

κ ◦ T¯ k

k =0

the cell shift after n reflections. We refer to the beginning of Appendix A.1 for the precise definition of the dynamically defined partitions ξkk12 . For simplicity we use the notation µ(A ¯ 1 ; . . . ; An ) = µ(A ¯ 1 ∩ · · · ∩ An ).


845

Proposition 4.1. Let p > 1. There exists c > 0 such that, for any k ≥ 1, if A ⊂ M¯ is k and B ⊂ M ¯ is a union of ξ ∞ then for any n > 2k and a union of components of ξ−k −k 2 ∈Z ,

1 1 2 −1

p ck µ(B) ¯ βe− 2(n−2k) (Σ ) ·

−n ¯ ¯ Sn κ = ; T (B)) − , µ(A) ¯ µ(B) ¯

µ(A;

≤

(n − 2k) 23 (n − 2k) where β =

√1 . 2π det Σ 2

The proof of Proposition 4.1 is in Appendix A.2. Proposition 4.2. Let c1 , c2 , c3 and c4 be some positive constants. For any m ∈ N S there exists a function f m such that limε→0 f m (ε) = 0 and such that the following holds: For any ε > 0 and any subsets Dε , Aε of M¯ such that (i) (ii) (iii) (iv)

m ∈ Dε ⊂ A ε , c1 ε2 ≤ µ(A ¯ ε ) and Aε ⊂ B(m, c2 ε), [η] [η] ¯ Dε ) ≤ c3 η, for any η > 0, µ(∂ ¯ Aε ) ≤ c3 η, and also µ(∂ 2 2.25 µ(D ¯ ε ) ≥ c1 (diam(Dε )) and µ(D ¯ ε ) ≥ c4 ε , 1

uniformly in N ∈ (elog ε , e ε2.5 ) we have 2

µ(W ¯ Aε (·) > N |Aε ) =

1 + oε (1) 1 + log(N )µ(A ¯ ε )β

and µ(W ¯ Aε (·) > N |Dε ) =

1 + oε (1), 1 + log(N )µ(A ¯ ε )β

where the error terms oε (1) are bounded by f m (ε). Lemma 4.3. Under the hypothesis of Proposition 4.2, for all m ∈ M¯ (even those not belonging to N S), we have µ(W ¯ A > N |D) + β log(N )µ(A) ¯ µ(W ¯ A > N |A) ≤ 1 + oε (1), where the error term only depends on the positive constants ci . Proof. As used by Dvoretzky and Erdös in [11], a partition of D with respect to the last entrance time q into the set A in the time interval [0, . . . , N ] gives µ(D) ¯ =

N

µ(D; ¯ Sq κ = 0; T¯ −q (A ∩ {W A > N − q}))

q =0

≥

N

µ(D; ¯ Sq κ = 0; T¯ −q (E))

q =0

with E = A ∩ {W A > N }. Let k be such that δ k ≈ ε3 . We approach D and E by cylindrical sets:

(4.2)

846

F. Pène, B. Saussol k such that Z ⊂ D. We have D ⊂ D and Let D be the union of cylinders Z ∈ ξ−k

) ≤ c c δk . D\D ⊂ ∂ D [c0 δ ] by Lemma A.1, thus by the hypothesis (iii) we get µ(D\D ¯ 3 0 Let A be the corresponding cylindrical approximation for A and set ⎛ ⎞ N {S j κ = 0} ∪ T¯ − j (∪ Z ∈ξ k+ j ,Z ∩A=∅ Z )c ⎠ . E = A ∩ ⎝ k

−k− j

j=1

We have E ⊂ E and by Lemma A.1, E\E ⊂ (∂ A)[c0 δ ] ∪ k

N

k+ j T¯ − j (∂ A)[c0 δ ] .

j=1 ) ≤ c c δ . Thus by the hypothesis (iii) and the invariance of µ¯ we get µ(E\E ¯ 3 0 1−δ −a Set p0 ≈ ε with a = 4.6 > 2 × 2.25. By (4.2) and the inclusions we get k

µ(D) ¯ ≥ µ(D ¯ ∩ E) +

N

µ(D ¯ ; Sq κ = 0; T¯ −q E ).

q= p0

It follows from Proposition 4.1 that µ(D) ¯ ≥ µ(D ¯ ∩ E) +

N q= p0

The error term is bounded by log p0 = o(log N ),

√ ck p0 −2k

β

N ¯ ) ck µ(D ¯ )µ(E . − 3 q − 2k q= p0 (q − 2k) 2

= O(log(ε)εa/2 ) c4 ε2.25 ≤ µ(D). ¯ Thus, since

¯ ) ≤ µ(D)(1 ¯ + o(1)). µ(D ¯ ∩ E) + β log(N )µ(D ¯ )µ(E ) ≤ c c δ k = c c ε 3 c ε 2.25 − c c ε 3 ≤ µ(D ¯ ), we get Therefore, using µ(D\D ¯ 3 0 3 0 4 3 0

¯ + o(1)). µ(D ¯ ∩ E) + β log(N )µ(D) ¯ µ(E ¯ ) ≤ µ(D)(1 ) log N ≤ Notice that µ(E\E ¯

c3 c0 k 1−δ δ

log N = o(1), from which it follows that

µ(D ¯ ∩ E) + β log(N )µ(D) ¯ µ(E) ¯ ≤ µ(D)(1 ¯ + o(1)). A division by µ(D) ¯ yields, since E = A ∩ {W A > N } and D ⊂ A, ¯ µ(W ¯ A > N |A) ≤ 1 + o(1). µ(W ¯ A > N |D) + β log(N )µ(A) Lemma 4.4. Under the hypotheses of Proposition 4.2 we have ¯ µ(W ¯ A > N |A) = 1 + oε (1). µ(W ¯ A > N |D) + β log(N )µ(A)


847

Proof. Let α ∈ (0, 0.25) and set Mε = ε2(−1+α) . We use the same decomposition as in Eq. (4.2) again, with n N = N log(N ) and m N = n N − N : nN

µ(D) ¯ =

µ(D; ¯ Sq κ = 0; T¯ −q (A ∩ {W A > n N − q})).

q =0

We divide this sum into four blocks: S0 is the term for q = 0, S1 is the sum for q in the range 1, . . . , Mε , S2 in the range Mε + 1, . . . , m N and S3 in the range m N + 1, . . . , n N . The value of S0 is simply S0 = µ(D; ¯ W A > n N ) ≤ µ(D; ¯ W A > N ). By assumption (conclusion of Lemma 3.2), we have S1 = µ(D; ¯ W A ≤ Mε ) ≤ µ(D; ¯ W¯ B(m,c2 ε) ≤ Mε ) = o(µ(D)). ¯ When q ≤ m N we have n N − q ≥ N , therefore we have mN

S2 ≤

µ(D; ¯ Sq κ = 0; T¯ −q (E))

q=Mε +1

with E = A ∩ {W A > N }. Let k be such that δ k ≈ ε3 . We approximate the sets D and k such that Z ∩ D = ∅. Let A E by cylinders: let D be the union of cylinders Z ∈ ξ−k be the corresponding enlargement for A and let N {S j κ = 0} ∪ T¯ − j (∪ Z ∈ξ k+ j E =A ∩

−k− j ,Z ⊆A

j=1

Z)

c

.

We have D ⊂ D and by Lemma A.1, D \D ⊂ (∂ D)[c0 δ ] . Thus by Hypothesis (iii) we get that µ(D ¯ \D) ≤ c3 c0 δ k . Similarly, E ⊂ E and k

E \E ⊂ (∂ A)[c0 δ ] ∪ k

N

k+ j T¯ − j (∂ A)[c0 δ ] .

j=1 δ Thus by hypothesis (iii) we get that µ(E ¯ \E) ≤ c3 c0 1−δ which implies that log(m N ) µ(E ¯ \E) = o(1). By Proposition 4.1 with p such that 1 + 2p > 2.5, we obtain k

S2 ≤

mN q=Mε +1

≤

mN q=Mε +1

µ(D ¯ ; Sq κ = 0; T¯ −q (E ))

1

¯ ) p ¯ ) ck µ(E µ(D ¯ )µ(E + β 3 q − 2k (q − 2k) 2

1

ck µ(A ¯ ) p ≤ log(m N )β µ(D ¯ )µ(E ¯ )+ √ Mε − 2k

≤ log(m N )β µ(D) ¯ µ(E)(1 ¯ + o(1)) + o(µ(D)) ¯ + O(log(ε)ε1−α ε2/ p ).

848


The last error term is o(µ(D)) ¯ provided 1 − α + 2/ p > 2.25. In addition we have log m N ∼ log N , hence S2 ≤ β log(N )µ(D) ¯ µ(E)(1 ¯ + o(1)) + o(µ(D)). ¯ Finally, by Proposition 4.1 we get S3 ≤

nN q=m N nN

µ(D ¯ ; Sq κ = 0; T¯ −q A )

ck ¯ ) µ(D ¯ )µ(A + ≤ β 3 q − 2k (q − 2k) 2 q=m N 2ck nN µ(D) ¯ µ(A)(1 ¯ + o(1)) + √ . ≤ β log mN m N − 2k Moreover we have log( mn NN ) = o(1), and the last error term is again o(µ(D)) ¯ since mN ≥ N. We conclude that (1 + o(1))µ(D) ¯ ≤ µ(D ¯ ∩ E) + β log(N )µ(D) ¯ µ(E). ¯ A division by µ(D) ¯ yields, since E = A ∩ {W A > N } and D ⊂ A, µ(W ¯ A > N |D) + β log(N )µ(A) ¯ µ(W ¯ A > N |A) ≥ 1 − o(1). The reverse inequality also holds by Lemma 4.3, finishing the proof.

Proof of Proposition 4.2. Lemma 4.4 with D = A gives us µ(W ¯ A > N |A) =

1 + o(1) . 1 + β log(N )µ(A) ¯

(4.3)

This proves the proposition in the special case D = A. We turn now to the general case. Applying Lemma 4.4 again, together with (4.3) we get, µ(W ¯ A > N |D) + β log(N )µ(A) ¯ which proves the proposition.

1 + o(1) = 1 + o(1), 1 + β log(N )µ(A) ¯

4.2. Recurrence results for the extended billiard map. Proposition 4.5. The recurrence rate for the extended billiard map is given by log log Wε = 2 µ-a.e. ε→0 − log ε lim


849

¯ Proof. Note that by Z2 -periodicity it suffices to prove the statement µ¯ a.e. in M. Upper bound: Let δ > 0 and set M¯ δ = {m ∈ N S : ρ(m) > δ and sup f m (ε) ≤ 1}, ε≤δ

where the function f m (ε) appears in Proposition 4.2. Let us notice that there exist constants ci for which the hypotheses of Proposition 4.2 are satisfied for any Dε = Aε = B(m, ε/2), with m ∈ M¯ δ . Let α ∈ (0, 21 ), n ≥ 1 and set εn = log−α n. Take a cover of M¯ δ by some sets B(m, εn /2), m ∈ Pn ⊂ M¯ δ such that #Pn = O(εn−2 ). According to Proposition 4.2, we have

εn εn

¯ δ) ≤ µ¯ W B(m, εn ) ≥ n B(m, ) µ¯ B(m, ) µ({W ¯ εn ≥ n} ∩ M 2 2 2 m∈Pn

≤ O((1 + βc1 log1−2α n)−1 ). Now, by taking n k = exp(k 2/(1−α) ) and according to the Borel-Cantelli lemma, we get that, for almost all m in M¯ δ , there exists Nm such that, for any k ≥ Nm , Wεnk (m) < n k , and hence log log Wεnk (m)

lim

− log εn k

k→+∞

≤

1 . α

Since log εn k ∼ log εn k+1 , we get that µ-a.e. ¯ on M¯ δ , 1 log log Wε ≤ . ε→0 − log ε α lim

¯ we have We conclude that almost everywhere in M, lim

ε→0

log log Wε ≤ 2. − log ε

Lower bound: Let α > 1/2. Let n ≥ 1 and εn = log−α n. We consider a cover of M¯ by balls B(m, εn ) for m ∈ Pn such that #Pn = O(εn−2 ). Let k be such that δ k ≈ εn5 . For each m ∈ Pn we consider the sets Bm and Cm constructed from B(m, εn ) and B(m, 2εn ) (respectively) like A was constructed from A in the proof of Lemma 4.4. Applying Proposition 4.1 we get µ(d(·, ¯ T n (·)) < εn ) ≤

µ(B(m, ¯ εn ) ∩ {Sn = 0} ∩ T¯ −n (B(m, 2εn )))

m∈Pn

≤

µ(B ¯ m ∩ {Sn = 0} ∩ T¯ −n (Cm )) + O(δ k )

m

≤

µ(B ¯ )µ(C ¯ ) m

m

+

n − 2k ≤ O n −1 log−2α n . m

ck + O(δ k ) (n − 2k)3/2

850


¯ there exists Hence, according to the first Borel Cantelli lemma, for almost every m ∈ M, Nm such that, for all n ≥ Nm , we have d(m, T n (m)) ≥ εn . Let u = min(d(m, T n (m)), n = 1, . . . , Nm ). Note that u > 0, otherwise we would have m = T p (m) for some p and hence m = T n (m) infinitely often, which would contradict d(m, T n (m)) ≥ εn . For all n ≥ Nm such that εn < u we have Wεn (m) ≥ n. Hence log log Wεn (m) 1 lim ≥ . − log ε α n→+∞ n 1 log log Wε ¯ Therefore ≥ almost everywhere on M. α ε→0 − log ε

Since log εn ∼ log εn+1 we get lim log log Wε ≥ 2 µ-a.e. ¯ ε→0 − log ε lim

Proposition 4.6. For a.e. m ∈ M¯ and any sequences of sets (Aε ) and (Dε ) such that the hypotheses (i)–(iv) of Proposition 4.2 are satisfied we have µ(W Aε > exp(

t 1 )|Dε ) → as ε → 0. µ(A ¯ ε) 1 + tβ

t Proof. Proposition 4.2 with N = exp( µ(A ¯ ε ) ) immediately gives the result.

Note that in particular the proposition applies to the sequence of balls Aε = Dε = B(m, ε). This is the corresponding result to that of Theorem 3.3 in the case of the extended billiard map. Proposition 4.7. The random variable 4ε2 ρ(·) log Wε (·) converges in the strong distri1 bution sense, to a random variable with law P(Y > t) = 1+βt . Proof. The proof is similar to that of Theorem 1.1-(ii), without the flow direction; see Sect. 5 for details. Since it is an obvious modification of it and since this result will not be used in the sequel, we omit its proof. 5. Proof of the Main Theorem: Recurrence in the Phase Space We prove in this section Theorem 1.1-(i)and (ii) about the return times in the phase space Z ε defined by (1.1). 5.1. Almost sure convergence: the first statement. By Z2 -periodicity it is sufficient to ¯ Let m ∈ M¯ be a point which is not on a singular orbit of T and prove the result on M. such that Wε (m) follows the limit given by Proposition 4.5. By regularity of the change of variable ψ (away from the singular set) there exist two constants 0 < a < b such that, for any 0 ≤ s ≤ τ (m), we have (min τ )(Wbε (m) − 1) ≤ Z ε (Φs ψ(m)) ≤ (max τ )Waε (m),

(5.1)

since the free flight function τ is bounded from above and from below. This implies the result for all the points Φs ψ(m). By Fubini’s theorem this concerns a.e. points in Q × S 1 , which proves the first statement.


851

5.2. Convergence in distribution: the second statement. Unfortunately we cannot exploit the relation (5.1) above anymore. The problem is not with the multiplicative factor coming from τ , but the fluctuations are sensible to the constants a and b and a direct method could only lead to rough bounds in terms of these constants. The following lemma gives the measure of the projection of a ball B(x, ε) onto M. Lemma 5.1. For any x ∈ X and ε > 0 such that the ball B(x, ε) does not intersect the boundary ∂ Q × S 1 , we have µ(π ψ −1 B(x, ε)) = 4ε2 . Proof. Let x = (q0 , v0 ) ∈ X . We consider the ball B(q0 , ε) as a new obstacle added in our billiard domain. Let ∆ε := (q, v) ∈ Q × S 1 : q ∈ ∂ B(q0 , ε), |∠(v0 , v)| < ε, n q , v > 0 . Since the billiard map preserves the measure cos ϕdr dϕ, we have µ(π ψ −1 B(x, ε)) = cos ∠(n q , v) dqdv. ∆ε

For any v such that |∠(v0 , v)| < ε a classical computation gives cos ∠(n q , v) dq = 2ε, {q:(q,v)∈∆ε }

whence the result.

Let P = hdL be the probability measure on X under which we will compute the ¯ By Z2 -periodicity, Z ε has the same distribution under law of Z ε . Let X¯ = ψ(π −1 M). ¯ ¯ P as under P¯ = hdL, where h(·) = ∈Z2 h(· + )1 X¯ . Therefore we suppose that supp h ⊂ X¯ . Assume for the moment that the density h is continuous and compactly supported in the set X¯ = X¯ \(ψ( M¯ × {0} ∪ π −1 R0 )), where R0 = {ϕ = ± π2 }. Then for any r > 0 sufficiently small we have ¯ r < d(m, R0 ), r ≤ s ≤ τ (m) − r }. supp h ⊂ X¯ r := {Φs (ψ(m)) : m ∈ M, Let K ⊂ N S be a set of points where the convergence in Proposition 4.6 is uniform and such that P({Φs (ψ(m)) : m ∈ K , 0 ≤ s < τ (m)}) > 1 − r. For any ε ∈ (0, r ) sufficiently small, the ε-neighborhood of X¯ r is contained in X¯ . Let νε = ε5/4 . Choose a family of pairwise disjoint open balls of radius νε in M¯ such that their union has µ-measure ¯ larger than 1 − 4νε . We drop all the balls not intersecting K and call {Di } the remaining family. For each i we choose a point m i ∈ Di ∩ K . For each i, we take the family of times si j = jνε ∈ (0, min Di τ ). Let Pi j = {Φs ψ(Di ) : si j ≤ s ≤ si j + νε }.

852


We finally drop the Pi j ’s not intersecting X¯ ∩ ψπ −1 K . Set yi j = Φsi j (ψ(m i )). We have t t ≈ ±r + ; P P Z > exp P Z ε > exp ε ij 4ε2 4ε2 i, j t ; Pi j h(yi j )L Z ε > exp ≈ ±r + 4ε2

(5.2)

i, j

by uniform continuity of h. Denote the projection onto the base of the balls by Ai±j = m ∈ M¯ : ∃0 ≤ s < τ (m) s.t. Φs (ψ(m)) ∈ B(yi j , ε ± νε ) = π ψ −1 B(yi j , ε ± νε ). Let τ− = min τ and τ+ = max τ . For any x ∈ Pi j , setting m = π ψ −1 x ∈ M¯ its projection, we have (τ− )(W Ai+j (m) − 1) ≤ Z ε (x) ≤ (τ+ )W A− (m).

(5.3)

ij

Hence we have for any real t > 0, νε µ((τ− )(W Ai+j − 1) > t; Di ) ≤ L(Z ε > t; Pi j ) ≤ νε µ((τ+ )W A− > t; Di ). (5.4) ij

Using the regularity of the projection π on X r , we see that the sets Ai±j fulfill the hypotheses of Proposition 4.6 with uniform constants. Moreover, by Lemma 5.1 and the relation (4.1), we have µ(A ¯ i±j ) =

4(ε ± νε )2 . 2Γ

Therefore by our choice of the m i ’s, the difference

µ (τ∓ )W ± > exp tΓ

Di − 1

Ai j

2 2ε 1 + βt

tends to zero uniformly as ε → 0. Putting it together with (5.4) in the computation (5.2) yields

1

tΓ ≤ r. − lim

P Z ε > exp ε→0 2ε2 1 + βt

Letting r → 0 gives the conclusion for a continuous density compactly supported on X¯ . The conclusion follows by an approximation argument, since any density h ∈ L 1 ( X¯ , L) may be approximated by a sequence h n of such densities.


853

6. Proof of the Main Theorem: Recurrence for the Position In this section we prove Theorem 1.1-(iii) and (iv) about the return times Zε defined by (1.2). The proof follows the scheme of the previous section but has additional arguments. We will detail the differences and indicate the common points. We recall that Π Q is the canonical projection from X = Q × S 1 onto Q. We will use the first return time Z ε in the ε-neighborhood of the initial position modulo Z2 defined by ⎧ ⎫ ⎨ ⎬ Z ε (x) = min t > ε : Φt (x) ∈ B(Π Q (x) + , ε) × S 1 . ⎩ ⎭ 2 ∈Z

For any q in Q and any ε > 0, we define the backward projection of Bε (q) × S 1 on M and on M¯ by Aε (q) = m ∈ M : ∃s ∈ [0, τ (ψ(m))), Φs (ψ(m)) ∈ B(q, ε) × S 1 , ⎧ ⎫ ⎨ ⎬ A¯ ε (q) = m ∈ M¯ : ∃s ∈ [0, τ (ψ(m))), Φs (ψ(m)) ∈ B(q + , ε) × S 1 . ⎩ ⎭ 2 ∈Z

Lemma 6.1. For any q ∈ Q and any ε ∈ (0, d(q, ∂ Q)), we have µ(Aε (q)) = 4π ε and ε so µ( ¯ A¯ ε (q)) = 2π Γ . Proof. Indeed, since the measure cos(ϕ)dr dϕ is preserved by billiard maps, µ(Aε (q)) is equal to the measure of the outgoing vectors based on ∂ Bε (q) (for the measure cos(ϕ)dr dϕ), which is equal to 2 × 2π ε. The second assertion follows from (4.1). We first need a result similar to Theorem 3.1. log Z ε ≥ 1. − log ε ε→0

Lemma 6.2. Lebesgue almost everywhere we have lim

¯ of points in X with previous reflection Proof. We consider again the set X¯ = ψ(π −1 M) ¯ in M. Let α > 0 and set X¯ α = {x = (q, v) ∈ X¯ : d(q, ∂ Q) > α}. Let n ≥ 1 be an integer and set rn :=

1 . n(log n)2

We define the set G n of points in X¯ α

coming back (modulo Z2 ) in the rn -neighborhood of the initial position between the n th and the (n + 1)th reflections by G n := {x ∈ X¯ α : T n−1 (Φτ (x) (x)) ∈ Arn (Π Q (x) + )}. ∈Z2

We take a family of pairwise disjoint open balls Di ⊂ M¯ of radius rn such that their union has µ-measure ¯ larger than 1 − 4rn . As in Sect. 5, we then construct the family Pi j following the same procedure. We drop those Pi j ’s not intersecting X¯ α . For each i, j

854


we fix a point yi j ∈ Pi j ∩ X¯ α . There exists L 0 > 0 such that for all x ∈ X α we have A¯rn (Π Q (x)) ⊂ A¯ L 0 rn (Π Q (y)) whenever d(x, y) < rn . Thus Leb(G n ) ≤

i, j

≤

Leb x ∈ Pi j : T¯ n−1 (Φτ (x) (x)) ∈ A¯rn (Π Q (x))

rn µ¯ Di ∩ T¯ −n A¯ L 0 rn (Π Q (yi j )) .

i, j

Now, we approximate the indicator function of Di by the Lipschitz function i) f i = max(1 − d(·,D rn , 0). We approximate in the same way the indicator function of the set A¯ L 0 rn (Π Q (yi j )) by a Lipschitz function gi j . Using the exponential decay of covariance for Lipschitz functions (Theorem A.3) we get −n ¯ n −2 ¯ µ(D ¯ i ∩ T A L 0 rn (Π Q (yi j ))) ≤ Cθ rn + f i d µ¯ gi j d µ. ¯ Therefore Leb(G n ) ≤ Cθ n rn−5 +

rn 4µ(D ¯ i )µ( ¯ A¯ L 1 rn (Π Q (yi j ))),

i, j

for some constant L 1 (since Π Q ◦ Φs ◦ ψ is Lipschitz for any 0 ≤ s ≤ τ+ ). According to Lemma 6.1 we get n≥1 Leb(G n ) < +∞. Therefore, by the first Borel-Can¯ telli lemma, for almost ! every x ∈ X α , there exists N x such that, for all n ≥ N x , n−1 T (Φτ (x) (x)) ∈ ∈Z2 Arn (Π Q (x) + ). Let ε0 = min{d(Π Q (Φs (x)), Π Q (x) + Z2 ) : s ∈ [α, N x τ+ ]}. We admit temporarily the following result: Sub-Lemma 6.3. The set {x ∈ X : ∃s > 0, Π Q (Φs (x)) − Π Q (x) ∈ Z2 } has zero Lebesgue measure. Hence ε0 is almost surely non-null. Therefore, for almost every point x in X¯ α , for all n ≥ N x such that rn < ε0 , and all k = 0, . . . , n, the point T k−1 (Φτ (x) (x)) ∈ Arn (Π Q (x) + ), ∈Z2

log Z rn (x) ≥ 1. Since log rn ∼ log rn+1 , we n→+∞ − log rn

and so Z rn (x) ≥ (n − 1)τ− . Hence lim

log Z ε ≥ 1 µ-a.e. on X α . The conclusion follows from µ(X α ) → 1 − log ε ε→0 as α → 0. end up with lim

Proof of Sub-lemma 6.3. Let x be a point in X such that, for some s > 0, we have Π Q (Φs (x)) − Π Q (x) ∈ Z2 . Then either s < τ (x) which implies that x has a rational direction, or there exists n ≥ 1 such that a particle with configuration T n−1 (Φτ (x) (x)) will visit Π Q (x) + Z2 before the next reflection. We have to prove that the set C of points


855

x satisfying the second condition has zero Lebesgue measure. For any q in Q\∂ Q, we denote by Cq the set of points of C with position q. We have f q (r )dr, Leb X (C|Π Q = q) = Leb Q (Cq ) = T¯ (A0 (q))∩T¯ −(n−1) (A0 (q))

(for some positive measurable function f q ) where A0 (q) is the set of points m ∈ M¯ that visits q + Z2 before the next reflection. The set T (A0 (q)) is a finite union of curves γ1 given by ϕ = ϕ1 (r ). Analogously, the set T¯ −(n−1) (A0 (q)) is a finite union of curves γ−(n−1) given by ϕ = ϕ−(n−1) (r ). Moreover, each γ1 is transversal to each γ−(n−1) (ϕ1 is strictly increasing and ϕ−(n−1) is strictly decreasing). Hence the intersection of T¯ (A0 (q)) and of T¯ −(n−1) (A0 (q)) is finite. Lemma 6.2 enables us to prove the following lemma analogous to Lemma 3.2. We call M¯ τ := {(m, s) ∈ M¯ × R : 0 ≤ s < τ (ψ(m))}. Lemma 6.4. For µ-almost ¯ every (m, s) ∈ M¯ τ the following holds: For any families (qε )ε of Q, (Dε )ε of subsets of M¯ such that (qε ), (i) m ∈ Dε ⊂ A¯ ε! (ii) Φs (ψ(m)) ∈ ∈Z2 B(qε + , ε) × S 1 , (iii) Dε is either a ball or the set A¯ ε (qε ), we have for all α > 0, µ( ¯ W¯ A¯ ε (qε ) ≤ ε−1 + α |Dε ) → 0 as ε → 0. Proof. We do not detail the proof when Dε is a ball since it is a direct adaptation of the proof of Lemma 3.2 with the use of Lemma 6.2 instead of Theorem 3.1. We suppose that Dε = A¯ ε (qε ). The idea is to consider the billiard flow modulo Z2 and to adapt the proof of Lemma 3.2 thanks to the Fubini theorem. Let α > 0 and let a ∈ (0, α). Let η > 0 and ε0 > 0. We set for all q in Q, # " log Z ε (q , v) 1 0, lim Leb Q ((Q ∩ [0; 1)2 )\Fη (ε0 )) = 0.

ε0 →0

856


Therefore, for a.e. (m, s) and any η > 0 there exists a choice of ε0 such that (6.1) holds. Let ⎛ ⎞ Hε := (B(qε , 2ε) × S 1 ) ∩ Φ−s ⎝ B(qε + , 2ε) × S 1 ⎠ . s∈(6ε(τ+ )ε−1 + α )

∈Z2

There exists ε1 ∈ (0, ε0 ) such that, for all ε ∈ (0, ε1 ), we have Hε ⊂ (B(qε , 2ε) × S 1 ) ∩ {Z 4ε ≤ τ+ ε−1 + α } ⊂ {(q , v) ∈ B(qε , 2ε) × S 1 : v ∈ Bad(q )}. Therefore −1 −1 Leb X (Hε ) = Leb X (Π Q (Fη (ε0 )) ∩ Hε ) + Leb X (Hε \Π Q (Fη (ε0 )))

≤ ηLeb Q (B(qε , 2ε)) + 2π Leb Q (B(qε , 2ε)\Fη (ε0 )). This together with (6.1) yields lim Leb X (Hε |B(qε , 2ε) × S 1 ) ≤

ε→0

η . 2π

Since η > 0 is arbitrary, for almost every (m, s), we get lim Leb X (Hε |B(qε , 2ε) × S 1 ) = 0.

ε→0

Hence Leb X (Hε ∩ (B(qε , 2ε) × S 1 )) = o(ε2 ). Moreover, setting Is (m) = length{s ∈ (0; τ (m)) : Φs (m) ∈ B(qε , 2ε) × S 1 } and using the representation of Φs as a special flow over T gives 1 Is (m) dµ(m) Leb X (Hε ∩ (B(qε , 2ε) × S )) ≥ ≥

A¯ 2ε (qε )∩{W¯ A¯

2ε (qε )

≤ε−1 + α }

Is (m) dµ(m) A¯ ε (qε )∩{W¯ A¯ ε (qε ) ≤ε−1 + α }

≥ εµ(A¯ ε (qε ) ∩ {W¯ A¯ ε (qε ) ≤ ε−1 + α }). This finally gives µ( ¯ A¯ ε (qε ) ∩ {W¯ A¯ ε (qε ) ≤ ε−1 + α }) = o(ε) = o(µ( ¯ A¯ ε (qε ))). We denote by N S the set of couples (m, s) ∈ M¯ τ satisfying the conclusion of Lemma 6.4. This is essential for the following lemma analogous to Proposition 4.2: Lemma 6.5. For all (m, s) ∈ N S , there exists a function f m,s , lim f m,s (ε) = 0, and ε→0

such that, for any families (qε )ε of Q and (Dε )ε of subsets of M such that: (i) m ∈ Dε ⊂ Aε (qε );


857

(ii) Φs ψ(m) ∈ B(qε , ε) × S 1 ; (iii) Dε is a ball of radius larger than ε1.2 or is the set Aε (qε ); 1

for all N ∈ (elog ε , e ε2.5 ), we have:

µ(WA (q ) (·) > N |Dε ) − ε ε

2

≤ f m,s (ε) ¯ 1 + log(N )µ( ¯ Aε (qε ))β

1

and µ(WAε (qε ) (·) > N |Aε (qε )) =

1 + oε (1) , 1 + log(N )µ( ¯ A¯ ε (qε ))β

where the error term oε (1) is bounded by f m,s (ε). Proof. To simplify the proof, we use the notations A = Aε (qε ) and A¯ = A¯ ε (qε ). First step. We adapt the proof of Lemma 4.3 to prove that ¯ µ(W A > N |D) + β log(N )µ( ¯ A)µ(W A > N |A) ≤ 1 + oε (1). A slight difficulty comes from the fact that the set A can be divided into several cells. More precisely, there exist pairwise disjoint subsets A of M¯ such that (with obvious notations) (A + ) and A¯ = A . A= ||≤τ+

||≤τ+

Analogously, there exist pairwise disjoint subsets D of M¯ such that D= (D + ). ||≤τ+

Hence, we have µ(D) =

N

µ(D; T −q (A; W A > N − q))

q =0

≥ µ(D; W A > N ) +

N

µ(D; T −q (A; W A > N ))

q= p0

≥ µ(D; W A > N ) +

N

µ(D + ; T −q (A + ; W A > N ))

q= p0 ,

≥ µ(D; W A > N ) +

N

µ(D ; Sq κ = − ; T¯ −q (A ; W A− > N )).

q= p0 ,

This together with (4.1), as in the proof of Lemma 4.3, give µ(D) ≥ µ(D; W A > N ) + β log(N )

µ(D) µ(A; W A > N ) + o(µ(D)), 2Γ

858


and so ¯ ¯ A)µ(W 1 ≥ µ(W A > N |D) + β log(N )µ( A > N |A) + o(1). Second step. To prove the following lower bound: ¯ ¯ A)µ(W µ(W A > N |D) + β log(N )µ( A > N |A) ≥ 1 + oε (1), we use the notations m N and n N of the proof of Lemma 4.4 and we write nN

µ(D) =

µ(D; T −q (A; W A > n N − q))

q =0

= µ(D; W A > N ) +

nN q=1

µ(D ; Sq κ = − ; T¯ −q (A ; W A− > N )).

,

A first difference with the proof of Lemma 4.4 is that we work with D and A instead of considering directly D and A. We approximate D by a set D and A by a set A as we approximate D by D in the proof of Lemma 4.4. We fix α ∈ (0, 0.5) and we follow the scheme of the proof of Lemma 4.4 for the estimate of S0 and S3 (using D and A ). We take Mε = ε−1 + α instead of Mε = ε2(−1+α) . According to Lemma 6.4, this choice of Mε gives the correct estimate of S1 . We introduce Mε = ε−6 . We decompose S2 in two blocks: S2 is the sum for q in the range Mε + 1, . . . , Mε and S2 in the range Mε + 1, . . . , m N . To estimate S2 and S2 , we approximate E := A ∩ {W A− > N } by a set E as we approximate E by E in the proof of Lemma 4.4. We estimate S2 as we estimate S2 in the proof of Lemma 4.4 with Mε instead of Mε : S2

mN ≤ log Mε

1

p µ(D)µ(E) ck µ(A) ¯ β (1 + o(1)) + $ + o(µ(D)), 2Γ Mε − 2k

and the error term is in O(log(ε)ε1/ p ε3 ) = o(µ(D)) provided 3 + 1/ p > 2.4. To estimate S2 , we use the symmetry π0 on M with respect to the normal n given by: π0 (ψ(, i, r, ϕ)) = π0 (ψ(, i, r, −ϕ)). Let us notice that π0 preserves the measure µ. ¯ Using this symmetry and applying Proposition 4.1 with p such that 1/4 > 2.4(1 − 1/ p), we get

S2 ≤ 2Γ

Mε q=Mε

,

µ(D ¯ , Sq κ = − ; T¯ −q (A ))

≤ 2Γ

Mε q=Mε ,

µ(π ¯ 0 (A ); Sq κ = − ; T¯ −q (π0 (D )))

Mε β µ(A ¯ )1/ p ¯ ) ck µ(D ¯ )µ(D + ≤ 2Γ q − 2k (q − 2k)3/2 q=Mε , c kµ(D)1/ p (1 + o(1)) Mε ¯ β µ( ¯ A)µ(D)(1 + o(1)) + ≤ log , √ Mε Mε − 2k


859

1/ p ε (1−α)/2 = o(µ(D)) ¯ since we have the last error term being in O log(ε)µ(D) ¯ (1 − α)/2 > 2.4(1 − 1/ p). Hence, we have proved that, under the assumptions of Lemma 6.5, we have ¯ ¯ A)µ(W µ(W A > N |D) + β log(N )µ( A > N |A) = 1 + o(1).

(6.2)

In the special case D = A, we conclude that µ(W A > N |A) =

1 + o(1) . ¯ 1 + β log(N )µ( ¯ A)

(6.3)

We turn now to the general case. Applying Eqs. (6.2) and (6.3) we get µ(W A > N |D) =

1 + o(1). ¯ 1 + β log(N )µ( ¯ A)

Proof of Theorem 1.1-(iii). Upper bound. Let X¯ 0 be a set of points of X with previous reflection in M¯ and on which the estimate of Lemma 6.5 is uniform. Let α ∈ (0, 1) and εn = log−α n. Take a cover of X¯ 0 by some balls B(qn , ε2n ) × S 1 for qn ∈ Qn ⊆ Q such that #Qn = O(εn−2 ). We have Leb( X¯ 0 ; Z εn ≥ nτ+ ) ≤ 2

Leb(B(qn ,

qn

≤

qn

≤ εn

εn ); Z εn ≥ nτ+ ) 2 2

εn µ(WA εn (qn ) > n; A εn (qn )) 2

2

qn

µ WA εn (qn ) > n A εn (qn ) µ(A εn (qn )) 2 2 2

≤ O((1 + βc log(n)εn )−1 ) ≤ O((1 + βc log1−α (n))−1 ), 2/(1−α) ) and accordwith c = 2π Γ (according to Lemma 6.1). Now, by taking n k = exp(k ing to the Borel-Cantelli lemma, we get that, for almost all x in X¯ 0 , there exists N x such that, for any k ≥ N x , Z εnk (x) < n k τ+ and hence 2

lim

log log Zεnk (x)

k→+∞

− log εn k

≤

1 . α

Since log εn k ∼ log εn k+1 , we conclude that almost everywhere in X¯ 0 , we have 1 log log Zε ≤ . ε→0 − log ε α lim

Therefore, almost everywhere in X , we have log log Zε ≤ 1. ε→0 − log ε lim

860


¯ Let α > 1. Lower bound. Let X¯ be the set of points of X with previous reflection in M. For all n ≥ 1, we take εn = log−α n and we denote by K n the set of points x ∈ X¯ whose orbit (by the billiard flow) comes back to the εn -neighbourhood for the position between the n th and the (n + 1)th reflections: K n = x ∈ X¯ : ∃s ∈ In (x), d(Π Q (x), Π Q (ψ(T n (π ψ −1 (x)), s))) < εn , with In (x) := [0; τ (T n (π ψ −1 (x)))). We consider a cover of X¯ by sets Cεn (q) = B(q, εn ) × S 1 for q ∈ Qn ⊆ Q such that #Qn = O(εn−2 ). Let n ≥ 1. For any q ∈ Qn , there exist two families of pairwise disjoint subsets (A1, (q)) and (A2, (q)) of M¯ such that A1, (q) + and A2εn (q) = A2, (q) + . Aεn (q) = Let k be such that δ k ≈ εn5 . Let A1, (q) (resp. A2, (q)) be the union of all the cylinders k intersecting A (q) (resp. A (q)). We have Z ∈ Z−k 1, 2, Leb(x ∈ Cεn (q) : T n (π ψ −1 (x)) ∈ A2εn (q)) Leb(K n ) ≤ q

≤ 2εn

,

q

≤ 2εn

µ(A1, (q); Sn κ = − ; T¯ −n (A2, (q)))

,

q

≤ 4εn Γ

µ(A1, (q) + ; T −n (A2, (q) + ))

q

,

µ(A ¯ 1, (q); Sn κ = − ; T¯ −n (A2, (q)))

ck + ≤ 4εn Γ β n − 2k (n − 2k)3/2 q , εn µ(Aεn (q))µ(A2εn (q)) ≤ β (1 + o(1)) + O(εn n −1 ) Γ q n − 2k

µ(A ¯ 1, (q))µ(A ¯ 2, (q))

≤ O(εn n −1 ) = O(n −1 log−α n). Hence, according to the first Borel Cantelli lemma, for almost every x ∈ X¯ , there exists N x such that, for all n ≥ N x , for every s ∈ In (x), we have d(Π Q (x), Π Q (ψ(T n (π ψ −1 (x)), s))) ≥ εn . According to Lemma 6.3, u := min d(Π Q (x), Π Q (ψ(T n (π ψ −1 (x)), s)), n = 1, . . . , N x , s ∈ In (x) is almost surely non-null. Therefore, for almost every point x in X¯ , for all n ≥ N x such that εn < u, Zεn (x) ≥ (n − 1)τ− . Hence, almost everywhere in X , we have log log Z εn ≥ α −1 . n→+∞ − log εn lim


861

Since log εn ∼ log εn+1 , we have limε→0 in X , we have

log log Zε − log ε

≥ α −1 . Therefore, almost everywhere

log log Zε ≥ 1. ε→0 − log ε lim

Sketch of proof of Theorem 1.1-(iv). This result is obtained by following the same scheme as in the proof of Theorem 1.1-(ii) in Sect. 5. We list the differences: – The set K ⊂ N S ⊂ M¯ is replaced by a set K ⊂ N S ⊂ M¯ τ such that the convergence in Lemma 6.5 is uniform and such that P(ψ(K)) > 1 − r. – The family Pi j . We first take a family of pairwise disjoint balls Di of M¯ of radius νε such that their union has µ-measure ¯ larger than 1 − 4νε . We construct the Pi j ’s exactly as in Sect. 5. Finally we drop the Pi j ’s not intersecting K ∩ X¯ r . We choose yi j ∈ Pi j ∩ ψ(N S ). – The sets Ai±j are replaced by Ai±j := Aε±νε (Π Q (yi j )). – We use the formula for the measure of the Ai±j given by Lemma 6.1. A. Transfer Operator and Local Limit Theorem A.1. Hyperbolicity, Young towers and spectral properties of the transfer operator. We do not repeat the construction of stable and unstable manifolds but only emphasize the hyperbolic estimate that is used throughout the proofs. Recall that the pre-singularity ¯ For any integers k1 ≤ k2 , let ξ k2 be the partition of set is R0 = {ϕ = ± π2 } ⊂ M. k1 !k 2 ¯ − j (R0 ) into connected components. With a slight abuse of language we ¯ T M\ j=k1 % j ∞ = ξ−k . Notice that each will call cylinders the elements of ξkk12 . For k ≥ 0 we let ξ−k stable manifold is contained in an element of ξ0∞ .

j≥0

Lemma A.1. There exist some constants c0 and δ > 0 such that for every integer k, k has a diameter diam Z ≤ c δ k . every set Z ∈ ξ−k 0 Proof. We recall that there exists C0 > 0 and Λ0 > 1 such that, for any increasing curve contained in the same connected component of ξ0k , T n γ is an increasing curve satisfying length(T n γ ) ≥ C0 Λn0 length(γ )2 and such that, for any decreasing curve contained in the same connected component of 0 , T −n γ is a decreasing curve satisfying ξ−k length(T −n γ ) ≥ C0 Λn0 length(γ )2 . k and be composed of points based on the same obstacle O . The set Let Z be in ξ−k i Z is delimitated by two increasing curves and two decreasing curves. Let m and m be two points in Z . These two points can be joined by a monotonous curve γ in Z .

862


If the curve γ is increasing we have & & length(T n γ ) π + |∂ Oi | ≤ . length(γ ) ≤ C0 Λn0 C0 Λn0 If the curve γ is decreasing, considering T¯ −n γ , we get length(γ ) ≤

'

π +|∂ Oi | . C0 Λn0

We do not repeat the construction of the tower but only briefly recall its property and then introduce the Banach space suitable for the study of the transfer operator. Young ˜ T˜ , µ) ˆ Tˆ , µ) constructed in [29] two dynamical systems ( M, ˜ and ( M, ˆ such that there ˜ ¯ ˜ ˆ exist two measurable functions π˜ : M → M and πˆ : M → M such that π˜ ◦ T˜ = T¯ ◦ π, ˜ ¯ πˆ ◦ T˜ = Tˆ ◦ πˆ , πˆ ∗ µ˜ = µ. ˆ π˜ ∗ µ˜ = µ, These dynamical systems are towers and are such that for any measurable f : M¯ → C constant on each stable manifold there exists fˆ : Mˆ → C such that fˆ ◦ πˆ = f ◦ π. ˜ For ˆ This -floor is partitioned in each ≥ 0, we denote by ∆ˆ the th floor of the tower M. {∆ˆ , j : j = 1, . . . , j }. The partition D = {∆ˆ , j : ≥ 0, j = 1, . . . , j } is Markov. For any x, y belonging to the same atom of D, we define s(x, y) := max{n ≥ 0 : ∀i ≤ n, D(Tˆ i x) = D(Tˆ i y)}. For any such x, y, the sets π˜ πˆ −1 {x} and π˜ πˆ −1 {y} are contained in the same connected !s(x,y) −k ¯ R0 . ¯ component of M\ k=0 T Let p > 1 and set q such that 1p + q1 = 1. Let ε > 0 and β ∈ (0, 1) be well chosen. q ˆ µ), ˆ Young defines for fˆ ∈ L ( M, C

fˆ = sup fˆ|∆ˆ ∞ e−ε + sup esssupx,y∈∆ˆ , j

, j

| fˆ(x) − fˆ(y)| −ε e . β s(x,y)

q ˆ µ) ˆ : fˆ < ∞}. This defines a Banach space (V, · ), such that Let V = { fˆ ∈ L C ( M, · q ≤ · . Let P be the Perron-Frobenius operator on L q defined as the adjoint of the composition by Tˆ on L p . This operator P is quasicompact on V. The construction of the tower can be adapted in such a way that its dominating eigenvalue on V is 1 and is simple. This choice will be convenient for our proof and we will adopt it, although it is not essential. The cell shift function κ is centered in the sense that κd µ¯ = 0,

and its asymptotic covariance matrix, Σ 2 := lim

n→∞

1 Covµ¯ (Sn κ), n

(A.1)

is well defined and non-degenerated. Since κ : M¯ → Z2 is constant on the local stable manifolds, there exists κˆ : Mˆ → Z2 such that κˆ ◦ πˆ = κ ◦ π˜ .


863

We define the complex perturbation of the Perron-Frobenius operator by Pu ( fˆ) = P(eiu·κˆ fˆ), for any u ∈ R2 . This method introduced by Nagaev [18,19] and developed by Guivarc’h and Hardy [14] and many other authors has been applied in this context by Szász and Varjú [28] (see also [21]). They have established the following key result: Proposition A.2. There exist a real a ∈ (0, π ), a C 3 family of complex numbers (λu )u∈[−a,a]2 , two C 3 families of linear operators on V: (Πu )u∈[−a,a]2 and (Nu )u∈[−a,a]2 such that ( (i) for all u ∈ [−a, a]2 we have Pun = λnu Πu + Nun ; moreover Π0 fˆ = Mˆ fˆ d µˆ for any fˆ ∈ L q ; (ii) there exists ν ∈ (0, 1) such that sup u∈[−a,a]2

|Nun | = O(ν n ) and

sup u∈[−π,π ]2 \[−a,a]2

(iii) we have λu = 1 − 21 Σ 2 u · u = O(|u|3 );

|Pun | = O(ν n );

1

(iv) there exists σ > 0 such that, for any u ∈ [−a, a]2 , |λu | ≤ e−σ |u| and e− 2 Σ 2 e−σ |u| . 2

2 u·u

≤

Note that by taking u = 0 in the proposition we recover the estimate on the rate of decay of correlations below. We state it here in a form suitable for our purpose, in particular to prove the results of Sect. 3. Theorem A.3 ([29]). There exist some constants C > 0 and θ ∈ (0, 1) such that for all Lipschitz functions f and g from M¯ to R, n f ◦ T gd µ¯ − f d µ¯ gd µ¯ ≤ Cθ n f Li p g Li p . (A.2) k and g is the Moreover, if f is the indicator function of a union of components of ξ−k ∞ indicator function of a union of components of ξ−k , then the covariance in (A.2) is simply bounded by Cθ n−2k .

However this information is not sufficient to control the recurrence for the extended billiard map T , therefore we need a finer version. A.2. Conditional uniform local limit theorem. Here we prove the local limit theorem, Proposition 4.1, concerning the billiard map T¯ and its Z2 -cocycle Sn κ. Proposition 4.1. Let p > 1. There exists c > 0 such that, for any k ≥ 1, if A ⊂ M¯ is k and B ⊂ M ¯ is a union of ξ ∞ , then for any n > 2k and a union of components of ξ−k −k ∈ Z2 ,

1 1 − 2(n−2k) (Σ 2 )−1 ·

p ck µ(B) ¯ βe

−n ¯ ∩ {Sn κ = } ∩ T¯ (B)) − µ(A) ¯ µ(B) ¯ ,

≤

µ(A

(n − 2k) 23

(n − 2k) where β =

√1 . 2π det Σ 2

864


Proof. The set T¯ −k A is a union of components of ξ02k and T¯ −k B is a union of components of ξ0∞ . Let Aˆ = πˆ (π˜ −1 T¯ −k A) and Bˆ = π( ˆ π˜ −1 T¯ −k B). Note that π˜ −1 T¯ −k A = πˆ −1 Aˆ −1 −k −1 ¯ ˆ and π˜ T B = πˆ B. Setting ¯ Sn κ = ; T¯ −n B), Cn (A, B, ) := µ(A; we have

Cn (A, B, ) =

Mˆ

1 Aˆ 1{Sn κ=} ◦ Tˆ k 1 Bˆ ◦ Tˆ n d µˆ ˆ

P k (1 Aˆ )1{Sn κ=} 1 Bˆ ◦ Tˆ n−k d µˆ ˆ 1 −iu· = e P k (1 Aˆ )eiu·Sn κ 1 Bˆ ◦ Tˆ n−k d µˆ du. (2π )2 [−π,π ]2 ˆ *+ , )M =

Mˆ

a(u)

By introducing the perturbed operator we get a(u) = Pun (P k (1 Aˆ )1 Bˆ ◦ Tˆ n−k ) d µˆ Mˆ = Puk (1 Bˆ Pun−k P k (1 Aˆ )) d µˆ Mˆ = Puk (1 Bˆ Pun−2k (buk )) d µ, ˆ Mˆ

with buk := Puk P k (1 Aˆ ). Set a1 (u) :=

Mˆ

P k (1 Bˆ Pun−2k (buk )) d µ. ˆ

We have, since |Puk − P k | L 1 →L 1 ≤ |u|kκ∞ , |a(u) − a1 (u)| ≤ |Puk − P k | L 1 →L 1

Mˆ

1 Bˆ |Pun−2k (buk )| d µˆ

ˆ 1/ p ≤ κ∞ k|u|Pun−2k buk ˆν ( B) by the Hölder inequality and since the norm · dominates the L q norm. Let us notice that by the Markov property supu∈[−π,π ]2 buk = O(1), uniformly in A and k. We have by Proposition A.2 (i) and (ii), 1 n−2k |u||Pu |du = |u||λu |n−2k du + O(ν n−2k ). 2 (2π )2 [−π,π ]2 [−a,a] In addition, by Proposition A.2 (iv) we have |u||λu |

[−a,a]2

n−2k

du ≤

-

1 3

(n − 2k) 2 R2

|v|e

−σ |v|2

dv = O

.

1 3

(n − 2k) 2

,

(A.3)


865

with the change of variable v = 1 Cn (A, B, ) = (2π )2

√

[−π,π ]2

n − 2ku. Therefore

−iu·

e

-

Mˆ

1 Bˆ Pun−2k (buk ) d µˆ du

+O

.

1

p k µ(B) ¯

.

3

(n − 2k) 2

By the Hölder inequality and since the norm · dominates the L q norm and according to points (i) and (ii) of Proposition A.2, we have 1 . p k µ(B) ¯ 1 −iu· n−2k k Cn (A, B, ) = e 1 Bˆ λu Πu (bu ) d µˆ du + O . 3 2 (2π ) [−a,a]2 Mˆ (n − 2k) 2 We will use here and thereafter the notation f u = O(gu ) to mean that there exists some constant c∗ such that for all u ∈ [−a, a]2 , we have | f u | ≤ c∗ |gu |. The differentiability of u → Πu gives |Πu − Π0 | = O(|u|). Hence using formula (A.3), we get 1 . p k µ(B) ¯ 1 −iu· n−2k k ˆ Cn (A, B, ) = e λ du µ( ˆ B) b d µ ˆ + O . u u 3 (2π )2 [−a,a]2 Mˆ (n − 2k) 2 For any u ∈ [−a, a]2 , we have k iu·Sk κˆ k ˆ + O(|u|). bu d µˆ = e P (1 Aˆ ) d µˆ = eiu·Sk κˆ ◦ Tˆ k 1 Aˆ d µˆ = µ( ˆ A) Mˆ

Mˆ

Mˆ

Again, using formula (A.3) we have 1 Cn (A, B, ) = (2π )2

-

[−a,a]2

e−iu· λn−2k u

du µ(B) ¯ µ(A) ¯ +O

.

1

p k µ(B) ¯ 3

.

1

.

(n − 2k) 2

According to the point (iii) of Proposition A.2, we have

n−2k 2 2

n−2k − e− 2 Σ u·u ≤ c∗ (n − 2k)e−σ |u| (n−2k−1) O(|u|3 ).

λ u Hence, proceeding similarly as in formula (A.3), we get µ(B) ¯ µ(A) ¯ Cn (A, B, ) = (2π )2

[−a,a]2

e−iu· e

2 − n−2k 2 Σ u·u

du + O

p k µ(B) ¯ 3

(n − 2k) 2

1 . p 1 2 k µ(B) ¯ µ(B) ¯ µ(A) ¯ −i √ v· n−2k e − 2 Σ v·v dv + O e = 3 (2π )2 (n − 2k) R2 (n − 2k) 2 √ with the change of variable v = u n − 2k. Finally, using the formula of the characteristic function of a gaussian, we get 1 . 2 )−1 · $ p k µ(B) ¯ µ(A) ¯ µ(B) ¯ − (Σ2(n−2k) 2 −1 2π det(Σ ) e Cn (A, B, ) = +O , 3 2 (2π ) (n − 2k) (n − 2k) 2 which proves the result after obvious simplifications.

866


References 1. Abadi, M., Galves, A.: Inequalities for the occurrence times of rare events in mixing processes. The state of art. Markov Process. Related Fields 7, 97–112 (2001) 2. Barreira, L., Saussol, B.: Hausdorff dimension of measures via Poincaré recurrence. Commun. Math. Phys. 219, 443–464 (2001) 3. Bressaud, X., Zweimüller, R.: Non exponential law of entrance times in asymptotically rare events for intermittent maps with infinite invariant measure. Ann. I.H.P. Phys. Th. 2, 1–12 (2001) 4. Bunimovich, L., Sinai, Y.: Markov partitions for dispersed billiards. Commun. Math. Phys. 78, 247– 280 (1980) 5. Bunimovich, L., Sinai, Y.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1981) 6. Bunimovich, L., Chernov, N., Sinai, Y.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45(3), 105–152 (1990); translation from Usp. Mat. Nauk 45(3), 97–134 (273) (1990) 7. Bunimovich, L., Chernov, N., Sinai, Y.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46(4), 47–106 (1991); translation from Usp. Mat. Nauk 46(4), 43–92(280) (1991) 8. Collet, P., Galves, A., Schmitt, B.: Repetition time for gibsiann source. Nonlinearity 12, 1225–1237 (1999) 9. Conze, J.-P.: Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications. Erg. Th. Dyn. Syst. 19(5), 1233–1245 (1999) 10. Dolgopyat, D., Szász, D., Varjú, T.: Recurrence properties of planar Lorentz process. Duke Math. J. 142, 241–281 (2008) 11. Dvoretzky, A., Erdös, P.: Some problems on random walk in space. Proc. Second Berkeley Sympos. Math. Statist. Probab., Berkeley, CA: Univ. California Press, 1951, pp. 353–367 12. Galatolo, S., Kim, D.-H., Park, K.: The recurrence time for ergodic systems with infinite invariant measures. Nonlinearity 19, 2567–2580 (2006) 13. Gallavotti, G., Ornstein, D.S.: Billiards and Bernoulli schemes. Commun. Math. Phys. 38, 83–101 (1974) 14. Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. Inst. H. Poincaré (B), Probabilité Et Statistiques 24, 73–98 (1988) 15. Hirata, M., Saussol, B., Vaienti, S.: Statistics of return times: a general framework and new applications. Commun. Math. Phys. 206, 33–55 (1999) 16. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge Univ. Press, 1995 17. Katok, A., Strelcyn, J.-M. (with Ledrappier, F., Przytycki, F.): Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Springer Lecture Notes in Mathematics 1222, Berlin-Heidelberg-NewYork: Springer, 1986 18. Nagaev, S.V.: Some limit theorems for stationary Markov chains. Theor. Probab. Appl. 2, 378–406 (1957); translation from Teor. Veroyatn. Primen. 2, 389–416 (1958) 19. Nagaev, S.V.: More exact statement of limit theorems for homogeneous Markov chains. Theor. Probab. Appl. 6, 62–81 (1961); translation from Teor. Veroyatn. Primen 6, 67–86 (1961) 20. Pène, F.: Applications des propriétés stochastiques du billard dispersif. C. R. Acad. Sci., Paris, Sér. I, Math. 330(12), 1103–1106 (2000) 21. Pène, F.: Planar Lorentz process in a random scenery. Ann. I.H.P. Prob. Stat. 45(3), 818–839 (2009) 22. Pène, F.: Asymptotic of the number of obstacles visited by the planar Lorentz process. Disc. Cont. Dyn. Sys. A 24(2), 567–587 (2009) 23. Pène, F., Saussol, B.: Quantitative recurrence in two-dimensional extended processes. Ann. I.H.P. Prob. Stat., to appear, http://www.imstat.org/aihp/pdf/AIHP195.pdf, 2009 24. Saussol, B.: Recurrence rate in rapidly mixing dynamical systems. Disc. and Cont. Dyn. Sys. 15, 259– 267 (2006) 25. Schmidt, K.: On joint recurrence. C. R. Acad. Sci., Paris, Sér. I, Math. 327(9), 837–842 (1998) 26. Simányi, N.: Towards a proof of recurrence for the Lorentz process. Dyn. sys. and erg. th., 28th Sem. St. Banach Int. Math. Cent., (Warsaw/Pol. 1986), Banach Cent. Publ. 23, 265–276 (1989) 27. Sinai, Y.: Dynamical systems with elastic reflections. Russ. Math. Surv. 25(2), 137–189 (1970) 28. Szász, D., Varjú, T.: Local limit theorem for the Lorentz process and its recurrence in the plane. Erg. Th. Dyn. Syst. 24(1), 257–278 (2004) 29. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147, 585–650 (1998) Communicated by G. Gallavotti


Communications in


The Uncanny Precision of the Spectral Action Ali H. Chamseddine1,3 , Alain Connes2,3,4 1 Physics Department, American University of Beirut, Beirut, Lebanon.


2 College de France, 3 rue Ulm, F75005 Paris, France 3 I.H.E.S., F-91440 Bures-sur-Yvette, France.


4 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

Received: 26 December 2008 / Accepted: 14 August 2009 Published online: 17 November 2009 – © Springer-Verlag 2009

Abstract: Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as an action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round sphere S 3 . We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of S 3 × S 1 , where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms a2n vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function. 1. An Overview Our experimental information on the nature of space-time is based on two sources: • High energy physics based on cosmic ray information and particle accelerator experiments, whose results are encapsulated in the Standard Model of particle physics. • Cosmology based on astronomical observations. The large scale global picture is well described in terms of Riemannian geometry and general relativity, but this picture breaks down at high energy where the quantum

868

A. H. Chamseddine, A. Connes

effects take over. It is thus natural to look for a paradigm of geometry which starts from the quantum framework, where the role of real variables is played by self-adjoint operators in Hilbert space. Such a framework for geometry has been slowly emerging under the name of noncommutative geometry. One of its key features, besides the ability to handle spaces for which coordinates no longer commute with each other, is that this new geometry is spectral. This is in agreement with physics in which most of the data we have, either about the far distant parts of the universe or about high energy physics, are also of spectral nature. The red shifted spectra of distant galaxies or the momentum eigenstates of outgoing particles in high energy experiments both point towards a prevalence of spectral information. In the same vein the existing unit of time (length) is also of spectral nature. From the mathematical standpoint it takes some doing to obtain a purely spectral (Hilbert space theoretical) counterpart of Riemannian geometry. One reason for the difficulty of this task is that, as is well known since the examples of J. Milnor [1], non-isometric Riemannian spaces exist which have the same spectra (for the Dirac or Laplacian operators). Another reason is that the conditions for a (compact) space to be a smooth manifold are given in terms of the local charts, whose existence and compatibility is assumed, but whose intrinsic meaning is more elusive. The paradigm of noncommutative geometry is that of spectral triple. As its name indicates it is of spectral nature. By definition a spectral triple is a unitary Hilbert space representation of “something”. This something is an equipment that allows one to manipulate algebraically coordinates and to measure distances. The algebra of the coordinates is denoted by A and is an involutive algebra, with involution a → a ∗ . The equipment needed to measure distances is the inverse line element D which is unbounded and fulfills D = D ∗ . Altogether these data fulfill some algebraic relations, e.g. if we talk about the simplest geometric space, i.e. the circle S 1 , the relation between the complex unitary coordinate U and the inverse line element D is just [D, U ] = U , which is in the vein of the Heisenberg commutation relations. Thus, a geometry is given as a Hilbert space representation of the pair (A, D) and can be encoded by the spectral triple (A, H, D), where H is the Hilbert space in which both the algebra A and the inverse line element D are now concretely represented, the latter as an unbounded self-adjoint operator. This picture shares with the Wigner paradigm for a particle as an (irreducible) representation of the Poincaré group the feature that it separates the kinematical relations from the choice of the Hilbert space representation. It is only when the latter is chosen that actual measurements of distances between points x and y can be performed by formulas such as

Distance (x, y) = sup | f (x) − f (y)| , f ∈ A , [D, f ] ≤ 1,

where indeed the norm [D, f ] is the operator norm in Hilbert space and depends on the specific choice of the representation. We now have at our disposal a reconstruction theorem (cf. [2]) which shows that ordinary Riemannian spaces are neatly characterized among spectral triples by the following kinematical relations: • The algebra A is commutative. • The commutator [[D, a], b] = 0 for any a, b ∈ A.

The Uncanny Precision of the Spectral Action

869

• The following “Heisenberg type” relation1 holds2 , for some a αj ∈ A: α

a0α [[D, a1α ], [D, a2α ], . . . , [D, anα ]] = 1,

(1)

together with the following spectral requirements: • The k th characteristic value of the resolvent of D is O(k −1/n ). • Regularity. • Absolute continuity. We refer to [2] for the precise statement. The meaning of (1) is that the determinant of the metric g µν does not vanish, and more precisely that its square root multiplied by the volume form α a0α da1α ∧ da2α ∧ · · · danα gives 1. The reason for the last two spectral requirements is technical and allows one to specify the regularity (C ∞ , real analytic...) of the space and to control the spectral measures. The first of the spectral requirements is crucial in that it bounds the “effective dimension” of the spectrum of the space in the representation. There are good physics reasons to consider that the apparent dimension, equal to four, of space-time is governed by the asymptotic behavior of the eigenvalues of the line element, which is the Euclidean propagator. Moreover this spectral dimension is not restricted to be an integer a priori and can model fractal dimension easily. The above reconstruction theorem shows furthermore that the operator D in the spectral triple is a Dirac type operator, i.e. an order one operator with symbol given by a representation of the Clifford algebra. The restriction to spin manifolds is obtained by requiring a real structure, i.e. an antilinear unitary operator J acting in H which plays the same role and has the same algebraic properties as the charge conjugation operator in physics. When the dimension n involved in the reconstruction theorem is even (rather than odd) the right-hand side of (1) is now replaced by the chirality operator γ which is just a Z/2-grading in mathematical terms. It fulfills the rules γ 2 = 1 , [γ , a] = 0,

a ∈ A.

(2)

The following further relations hold for D, J and γ J 2 = ε , D J = ε J D,

J γ = ε γ J,

Dγ = −γ D,

(3)

where ε, ε , ε ∈ {−1, 1}. The values of the three signs ε, ε , ε depend only, in the classical case of spin manifolds, upon the value of the dimension n modulo 8 and are given in the following table [3]: n ε ε ε

0 1 1 1

1 1 −1

2 −1 1 −1

3 −1 1

4 −1 1 1

5 −1 −1

6 1 1 −1

1 Here the multiple commutator is defined as

[T1 , T2 , . . . , Tn ] =

σ

(σ ) Tσ (1) Tσ (2) · · · Tσ (n) .

2 We assume for simplicity that the dimension n is odd.

7 1 1

870


In the classical case of spin manifolds there is thus a relation between the metric (or spectral) dimension given by the rate of growth of the spectrum of D and the integer modulo 8 which appears in the above table. For more general spaces however the two notions of dimension (the dimension modulo 8 is called the K O-dimension because of its origin in K -theory) become independent since there are spaces F of metric dimension 0 but of arbitrary K O-dimension. More precisely, starting with an ordinary spin geometry M of dimension n and taking the product M × F, one obtains a space whose metric dimension is still n but whose K O-dimension is the sum of n with the K O-dimension of F, which as explained can take any value modulo 8. Thus, one now has the freedom to shift the K O-dimension at very little expense, i.e. in a way which does not alter the plain metric dimension. As it turns out the Standard Model with neutrino mixing favors the shift of dimension from the 4 of our familiar space-time picture to 10 = 4 + 6 = 2 modulo 8 [4,5]. The shift from 4 to 10 is a recurrent idea in string theory compactifications, where the 6 is the dimension of the Calabi-Yau manifold used to “compactify”. Effectively the dimension 10 is related to the existence of Majorana-Weyl fermions. The difference between this approach and ours is that, in the string compactifications, the metric dimension of the full space-time is now 10 which can only be reconciled with what we experience by requiring that the Calabi-Yau fiber remains unnaturally small. In order to learn how to perform the above shift of dimension using a 0-dimensional space F, it is important to classify such spaces. This was done in [6,7]. There, we classified the finite spaces F of given K O-dimension. A space F is finite when the algebra A F of coordinates on F is finite dimensional. We no longer require that this algebra is commutative. The first key advantage of dropping the commutativity can be seen in the simplest case where the finite space F is given by A = Mk (C) , H = Mk (C) , D = 0 , J ξ = ξ ∗ ,

ξ ∈ H,

(4)

where the algebra A = Mk (C) is acting by left multiplication in H = Mk (C). We have shown in [8] that the study of pure gravity on the space M × F yields Einstein gravity on M minimally coupled with Yang-Mills theory for the gauge group SU(k). The Yang-Mills gauge potential appears as the inner part of the metric, in the same way as the group of gauge transformations (for the gauge group SU(k)) appears as the group of inner diffeomorphisms. One can see in this Einstein-Yang-Mills example that the finite geometry fulfills a nice substitute of commutativity (of A) namely [a, b0 ] = 0 , ∀ a, b ∈ A,

(5)

where for any operator a in H, a 0 = J a ∗ J −1 . This is called the order zero condition. Moreover the representation of A and J in H is irreducible. This example is (taking γ = 1) of K O-dimension equal to 0. In [6] we classified the irreducible (A, H, J ) and found out that the solutions fall into two classes. Let AC be the complex linear space generated by A in L(H), the algebra of operators in H. By construction AC is a complex algebra and one only has two cases: (1) The center Z (AC ) is C, in which case AC = Mk (C) for some k. (2) The center Z (AC ) is C ⊕ C and AC = Mk (C) ⊕ Mk (C) for some k. Moreover the knowledge of AC = Mk (C) shows that A is either Mk (C) (unitary case), Mk (R) (real case) or, when k = 2 is even, M (H), where H is the field of quaternions (symplectic case). This first case is a minor variant of the Einstein-Yang-Mills case described above. It turns out by studying their Z/2 gradings γ , that these cases are incompatible with K O-dimension 6 which is only possible in case (2). If one assumes


871

that one is in the “symplectic–unitary” case and that the grading is given by a grading of the vector space over H, one can show that the dimension of H which is 2k 2 in case (2) is at least 2 × 16 while the simplest solution is given by the algebra A = M2 (H) ⊕ M4 (C). This is an important variant of the Einstein-Yang-Mills case because, as the center Z (AC ) is C ⊕ C, the product of this finite geometry F by a manifold M appears, from the commutative standpoint, as two distinct copies of M. We showed in [6] that requiring that these two copies of M stay a finite distance apart reduces the symmetries from the group SU(2) × SU(2) × SU(4) of inner automorphisms3 to the symmetries U (1) × SU(2) × SU(3) of the Standard Model. This reduction of the gauge symmetry occurs because of the second kinematical condition [[D, a], b] = 0 which in the general case becomes: [[D, a], b0 ] = 0 , ∀ a, b ∈ A.

(6)

Thus the noncommutative space singles out 42 = 16 as the number of physical fermions, the symmetries of the standard model emerge, and moreover, as shown in [9], the model predicts the existence of right-handed neutrinos, as well as the see-saw mechanism. In the above Einstein-Yang-Mills case, the Yang-Mills fields appeared as the inner part of the metric in the same way as the group of gauge transformations (for the gauge group SU(k)) appeared as the group of inner diffeomorphisms. But in that case all fields remained massless. It is the existence of a non-zero D for the finite space F that generates the Higgs fields and the masses of the Fermions and the W and Z fields through the Higgs mechanism. The new fields are computed from the kinematics but the action functional, the spectral action, uses in a crucial manner the representation in Hilbert space. In order to explain the conceptual meaning of this spectral action functional it is important to understand in which way it encodes gravity in the commutative case. As explained above the spectrum of the Dirac operator (or similarly of the Laplacian) does not suffice to encode an ordinary Riemannian geometry. However the Einstein-Hilbert action functional, given by the integral of the scalar curvature multiplied by the volume form, appears from the heat expansion of the Dirac operator. More generally it appears as the coefficient of 2 in the asymptotic expansion for large of the trace Tr( f (D/ )) ∼ 2 4 f 4 a0 + 2 2 f 2 a2 + f 0 a4 + · · · + −2k f −2k a4+2k + · · ·

(7)

when the Riemannian geometry M is of dimension 4, and where f is a smooth even function with fast decay atinfinity. The choice of the function f only enters in the ∞ ∞ multiplicative factors f 4 = 0 f (u)u 3 du, f 2 = 0 f (u)udu, f 0 = f (0) and f −2k = k k! (2k) (0), i.e. the derivatives of even order at 0, for k ≥ 0. Thus, when f is a (−1) (2k)! f “cutoff” function (cf. Fig. 1) it has vanishing Taylor expansion at 0 and the asymptotic expansion (7) only has three terms: Tr( f (D/ )) ∼ 2 4 f 4 a0 + 2 2 f 2 a2 + f (0)a4 .

(8)

The term in 4 is a cosmological term, the term in 2 is the Einstein-Hilbert action functional, and the constant term a4 gives the integral over M of curvature invariants such as the square of the Weyl curvature and topological terms such as the Gauss-Bonnet, with numerical coefficients of order one. It is thus natural to take the expression Tr( f (D/ )) as a natural spectral formulation of gravity. We are working in the Euclidean formulation 3 of the even part of the algebra.

872


i.e. with a signature (+, +, +, +) and the Euclidean space-time manifold is taken to be compact for simplicity. In the non-compact case we have shown in [10] how to replace the simple counting of eigenvalues of |D| of size < given4 by (7), by a localized counting. This simply introduces a dilaton field. We also tested this idea of taking the expression Tr( f (D/ )) as a natural spectral formulation of gravity by computing this expression in the case of manifolds with boundary, and we found [11] that it reproduces exactly the Hawking-Gibbons [12] additional boundary terms which they introduced in order to restore consistency and obtain Einstein equations as the equations of motion in the case of manifolds with boundary. Further, Ashtekar et al [13] have recently shown that the use of the Dirac operator in a first order formalism, which is natural in the noncommutative setting, avoids the tuning and subtraction of a constant term. One may be worried by the large cosmological term 4 f 4 a4 that appears in the spectral action. It is large because the value of the cutoff scale is dictated, roughly speaking, by the Planck scale since the √ term 2 f 2 a2 is the gravitational action 16π1 G R gd 4 x. Thus it seems at first sight that the huge cosmological term 4 f 4 a4 overrides the more subtle Einstein term 2 f 2 a2 . There is, however, and even at the classical level to which the present discussion applies a simple manner to overcome this difficulty. Indeed the kinematical relation (1) in fact fixes the Riemannian volume form to be5 √ 4 gd x = a0α da1α ∧ da2α ∧ da3α ∧ da4α . (9) α

Thus, if we vary the metric with this constraint we are in the context of unimodular gravity [14], and the cosmological term cancels out in the computation of the conditional probability of a gravitational configuration with total volume V held fixed. The remaining unknown, then, is the distribution of volumes dµ(V ), which is just a distribution on the half-line R+ V . The striking conceptual advantages of the spectral action are • Simplicity: when f is a cutoff function, the spectral action is just counting the number of eigenstates of D in the range [− , ]. • Positivity: when f ≥ 0 (which is the case for a cutoff function) the action Tr( f (D/ )) ≥ 0 has the correct sign for a Euclidean action. • Invariance: one is used to the diffeomorphism invariance of the gravitational action but the functional Tr( f (D/ )) has a much stronger invariance group, the unitary group of the Hilbert space H. One price to pay is that, as such, the action functional Tr( f (D/ )) is not local. It only becomes so when it is replaced by the asymptotic expansion (8). This suggests that one should at least compute the next term in the asymptotic expansion (even though this term appears multiplied by the second derivative f (0) = 0 when f is a cutoff function) just to get some idea of the size of the remainder. In fact both D and have the physical dimension of a mass, and there is no absolute scale on which they can be measured. The ratio D/ is dimensionless and the dimensionless number that governs the quality of the approximation (8) can be chosen to just be the number N ( ) of eigenvalues λ of D whose size is less than , i.e. |λ| ≤ . When f is a cutoff function the size of the error term in (8) should be O(N −k ) for any positive k, using the flatness of the Taylor expansion of f at 0. In the case of interest, where M is the Euclidean space-time, a rough 4 for f a cutoff function. 5 up to a numerical factor.


873

1.0

0.8

0.6

0.4

0.2

3

2

1

1

2

3

Fig. 1. Cutoff function f

estimate of the size of N is the 4-dimensional volume of M in Planck units i.e. an order of magnitude6 of N ∼ 10214 (at the present radius, see Sect. Two for details). Thus, even without the vanishing of f (0), the rough error term N −1/2 ∼ 10−107 is quite small in the approximation of the spectral action by its local version (8). We shall in fact show that a much better estimate holds in the simplified model of Euclidean space-time given by the product Sa3 × Sβ1 . Another advantage of the above spectral description of the gravitational action is that one can now use the same action Tr( f (D/ )) for spaces which are not Riemannian. The simplest case is the product of a Riemannian geometry M (of dimension 4) by the finite space F of (4). The only new term that appears is the Yang-Mills action functional of the SU(k) gauge fields which form the inner part of the metric. This new term appears as an additional term in the coefficient a4 of 0 , and with the positive sign. In other words gravity on the slightly noncommutative space M × F gives ordinary gravity minimally coupled with SU(k)-Yang-Mills gauge theory. The latter theory is massless and the fermions are in the adjoint representation. The fermionic part of the action is easy to write since one has the operator D whose inner fluctuations are D A = D + A + J A J −1 , A =

a j [D, b j ] , a j , b j ∈ A , A = A∗ .

(10)

In the Einstein-Yang-Mills system so obtained, all fields involved are massless. We now consider the product M × F of a Riemannian geometry M (of dimension 4) by the finite space F of K O-dimension 6 which was determined above. The computation shows that (cf. [9]) • The inner fluctuations of the metric give an U (1) × SU(2) × SU(3) gauge field and a complex Higgs doublet scalar field. 6 Using the age of the universe in Planck units to estimate the spatial Euclidean directions and the inverse temperature β = 1/kT also in Planck units, to set the size of the imaginary time component of the Euclidean M.

874


• The spectral action Tr( f (D/ )) plus the antisymmetric bilinear form J ξ, D A η on chiral fermions, gives the Standard Model minimally coupled to gravity, with the Majorana mass terms and see-saw mechanism. • The gauge couplings fulfill the unification constraint, the Yukawa couplings fulfill Y2 = 4g 2 , where Y2 is defined in Eq. (11), and the Higgs quartic coupling also fulfills a unification constraint. Most of the new terms occur in the a4 term of the expansion (8). This is the case for the minimal coupling of the Higgs field as well as its quartic self-interaction. The terms a0 and a2 get new contributions from the Majorana masses (cf. [9]), but the main new term in a2 has the form of a mass term for the Higgs field with the coefficient − 2 . This immediately raises the question of the meaning of the specific values of the couplings in the above action functional. Unlike the above massless Einstein-Yang-Mills system, we can no longer take the above action simply as a classical action, because of the unification of the three gauge couplings, which does not hold at low scale. The basic idea proposed in [8] is to consider the above action as an effective action valid at the unification scale and use the Wilsonian approach of integrating the high frequency modes to show that one obtains a realistic picture after “running down” from the unification scale to the energies at which observations are done. This approach is closely related to the approach of Reuter [15], Dou and Percacci [16], [17]. The coarse graining uses a much lower scale ρ which can be understood physically as the resolution with which the system is observed. The modes with momenta larger than ρ cannot be directly observed and their effect is averaged out by the functional integral. In fact the way the renormalization group is computed in [16] shows that the derivative ρ∂ρ ρ of the effective action is expressed as a trace of an operator function of the propagators and is thus of a similar nature as the spectral action itself, though the trace involves all fields and not just the spin 21 fields as in the spectral action. It is an open question to compute the renormalization group flow for the spectral action in the context of spectral triples. One expects, as explained above, that new terms involvδ2 ing traces of functions of the bosonic propagator7 δ Dδ D Tr( f (D/ )) will be generated. The idea of taking the spectral action as a boundary condition of the renormalization group at unification scale generates a number of severe tests. The first ones involve the dimensionless couplings. These include (1) the three gauge couplings, (2) the Yukawa couplings, (3) the Higgs quartic coupling. As is well known, the gauge couplings do not unify in the Standard Model but the meeting of g2 and g3 specifies a “unification” scale of ∼ 1017 GeV. For the Yukawa couplings the boundary condition gives Y2 = 4 g 2 , Y2 =

σ

(yνσ )2 + (yeσ )2 + 3 (yuσ )2 + 3 (ydσ )2 .

(11)

This yields a value of the top mass which is 1.04 times the observed value when neglecting8 the Yukawa couplings of the bottom quarks, etc...and is hence compatible with 7 We thank John Iliopoulos for discussions on this point. 8 See [9] for the precise satement.


875

experiment. The Higgs quartic coupling (scattering parameter) has the boundary condition of the form: λ˜ ( ) = g32

b ∼ g32 . a2

The numerical solution to the RG equations with the boundary value λ0 = 0.356 at = 1017 GeV gives λ(M Z ) ∼ 0.241 and a Higgs mass of the order of 170 GeV. This value now seems to be ruled out experimentally but this might simply be a clear indication of the presence of some new physics, instead of the “big desert” which is assumed here in the huge range of energies between 102 GeV and 1017 GeV. To be more precise the above “prediction” of the Higgs mass is in perfect agreement with the one of the Standard Model, when one assumes the “big desert” (cf. [19]). In a forthcoming paper [18] we show that the choice of the spectral function f could play an important role, even when it varies slightly from the cutoff function. This is related to the fact that the vev of the Higgs field is proportional to the scale and thus higher order corrections do contribute. This will cause the relation between the gauge coupling constants to be modified and to change the Higgs potential. Such gravitational corrections are known to cause sizable changes to the Higgs mass [20]. The next tests involve the dimensionful couplings. These include (1) (2) (3) (4)

the inverse Newton constant Z g = 1/G, the mass term of the Higgs, the Majorana mass terms, the cosmological constant.

Since our action functional combines gravity and the Standard Model, the analysis of [16] applies, and the running of the couplings Z which have the physical dimension of the square of a mass is well approximated by β Z = a1 k 2 , where the parameter k is fixing the cutoff scale but is considered itself as one of the couplings, while the coefficient a1 is a dimensionless number of order one. For the inverse Z g of the Newton constant, one gets the solution: 1 k2 ¯ Z g = Z g 1 + a1 (12) 2 Z¯ g which behaves like a constant and shows that the change in Z g is moderate between the low energy value Z¯ g at k = 0 and its value at k = m P the Planck scale, for which k2 = 1. We have shown in [9] that a relation between the moments of the cutoff function Z¯ g f involved in the spectral action, of the form f 2 ∼ 5 f 0 suffices to give a realistic value of the Newton constant, provided one applies the spectral action at the unification scale ∼ 1017 GeV. The above discussion of the running of Z g shows that this yields a reasonable low energy value of the Newton constant G. The form β Z = a1 k 2 of the running of a coupling with mass2 dimension implies that, as a rule, even if this coupling happens to be small at low scale, it will necessarily be of the order of 2 at unification scale. For the Majorana mass terms, we explained in [9] why they are of the order of 2 at unification and their role in the see-saw mechanism shows that one should not expect them to be small at small scale, thus a running like (12) is realistic. Things are quite different for the mass term of the Higgs. The spectral action delivers a huge mass term of the form − 2 H 2 and one can check that it is consistent with the sign and order of magnitude of the quadratic divergence of the self-energy of

876


this scalar field. However though this shows compatibility with a small low energy value it does by no means allow one to justify such a small value. Giving the term − 2 H 2 at unification scale and hoping to get a small value when running the theory down to low energies by applying the renormalization group, one is facing a huge fine tuning problem. Thus one should rather try to find a physical principle to explain why one obtains such a small value at low scale. In the noncommutative geometry model M × F of space-time the size of the finite space F is governed by the inverse of the Higgs mass. Thus the above problem has a simple geometric interpretation: Why is the space F so large9 in Planck units? There is a striking similarity between this problem and the problem of the large size of space in Planck units. This suggests that it would be very worthwhile to develop cosmology in the context of the noncommutative geometry model of space-time, with in particular the preliminary step of the Lorentzian formulation of the spectral action. This also brings us to the important role played by the dilaton field which determines the scale in the theory. The spectral action is taken to be a function of the twisted Dirac operator so that D 2 is replaced with e−φ D 2 e−φ . In [10] we have shown that the spectral action is scale invariant, except for the dilaton kinetic energy. Moreover, one can show that after rescaling the physical fields, the scalar potential of the theory will be independent of the dilaton at the classical level. At the quantum level, the dilaton acquires a Coleman-Weinberg potential [21] and will have a vev of the order of the Planck mass [22]. The fact that the Higgs mass is damped by a factor of e−2φ , can be the basis of an explanation of the hierarchy problem. In this paper we investigate the accuracy of the approximation of the spectral action by the first terms of its asymptotic expansion. We consider the concrete example given by the four-dimensional geometry Sa3 × Sβ1 , where Sa3 is the round sphere of radius a as a model of space, while Sβ1 is a circle of radius β viewed as a model of imaginary periodic time at inverse temperature β. We compute directly the spectral action and compare it with the sum of the first terms of the asymptotic expansion. In Sect. Two we start with the round sphere Sa3 and use the known spectrum of the Dirac operator together with the Poisson summation formula, to estimate the remainder when using a smooth test function. This is then applied to the four-dimensional space Sa3 × Sβ1 , where it is shown that, for natural test functions, the spectral action is completely determined by 2 the first two terms, with an error of the order of 10−σ , where σ is the inner diameter µ, µ = inf(a, β) in units of the cutoff . Thus for instance an inner diameter of 10 in cutoff units yields the accuracy of the first hundred decimal places, while an inner diameter of 1031 corresponding to the visible universe at inverse temperature of 3 Kelvin and a cutoff at Planck scale10 , yields an astronomical precision of 1062 accurate decimal places. This is then extended in the presence of Higgs fields. The above direct computation allows one to double check coefficients in the spectral action. It also implies, for Sa3 × Sβ1 , the vanishing of all the Seeley-De Witt coefficients a2n , n ≥ 2, in the heat expansion of the square of the Dirac operator. This is confirmed in Section Three, by a local computation of the heat kernel expansion, where it is shown that a4 and a6 vanish due to subtle cancelations.

9 by a factor of 1016 . 10 while the age of the universe in Planck units gives a ∼ 1061 .


877

2. Estimate of the Asymptotics The number N ( ) of eigenvalues of |D| which are ≤ , N ( ) = # eigenvalues of D in [− , ],

(13)

is a step function N ( ) which jumps by the integer multiplicity of an eigenvalue whenever belongs to the spectrum of |D|. This integer valued function is the superposition of two terms, N ( ) = N ( ) + Nosc ( ). The oscillatory part Nosc ( ) is generically the same as for a random matrix. The average part N ( ) is computed by a semiclassical approximation from local expressions involving the familiar heat equation expansion and will now be carefully defined assuming an expansion of the form11 Trace (e−t ) ∼ aα t α (t → 0) (14) for the positive operator = D 2 . One has, ∞ 1 −s/2 = s e−t t s/2−1 dt, 2 0

(15)

and the relation between the asymptotic expansion (14) and the ζ function, ζ D (s) = Trace (−s/2 )

(16)

is given by12 • α < 0 gives a pole at −2α for ζ D with 2 aα , (−α)

Ress=−2α ζ D (s) =

(17)

• α = 0 (no log t term) gives regularity at 0 for ζ D with ζ D (0) = a0 .

(18)

For simple superpositions of exponentials, as Laplace transforms, ∞ f (u) = e−su h(s) ds,

(19)

0

we can write formally

f (t) =

∞

e−st h(s) ds

(20)

0

and Trace ( f (t)) ∼

aα t α

∞

s α h(s) ds.

0

11 The a defined here is equal to the Seeley-de Witt coefficients a α n+2α in dimension n. 12 One adds dim(Ker(D)) to ζ (0) when D is not injective. D

(21)

878


For α < 0 one has sα = and

∞ 0

1 (−α)

∞

e−sv v −α−1 dv

0

1 s h(s) ds = (−α) α

so that Trace ( f (t)) ∼

∞

f (v) v −α−1 dv,

0

∞ Ress=−2α ζ D (s) 0 f (v) v −α−1 dv t α ∞ +ζ D (0) f (0) + α>0 aα t α 0 s α h(s) ds.

1 α 0 for which aα = 0 are integers and note that ∞ s n h(s) ds = (−1)n f (n) (0) ,

(22)

(23)

0

so that all the terms aα for α > 0 have vanishing coefficients when f is a cutoff function which is constant equal to 1 in a neighborhood of 0. To define the average part we consider the limit case f (v) = 1 for |v| ≤ 1 and 0 elsewhere and get for the coefficients of (22), ∞ 1 tα , (24) f (v) v −α−1 dv t α = 2 0 (−2α) which, with t = −2 , gives the following definition for the average part: k Ress=k ζ D (s) + ζ D (0). N ( ) := k

(25)

k>0

To get familiar with this definition we shall work out its meaning in a simple case: Proposition 1. Assume that Spec kD ⊂ Z and that the total multiplicity of {±n} is P (n) for a polynomial P(x) = ck x . Then one has N ( ) = P (u) du + c , c = ck ζ (−k) , 0

where ζ is the Riemann zeta function.

Proof. One has by construction, with P(x) = ck xk , P(n) n −s = ck ζ (s − k). ζ D (s) = Thus Ress=k ζ D (s) = ck−1 and N ( ) :=

k ck−1 + ζ D (0). k k>0

The constant ζ D (0) is given by and is independent of .

ck ζ (−k)


879

2.1. The sphere S 4 . We check the hypothesis of Proposition 1 for a round even sphere. We recall ([26]) that the spectrum of the Dirac operator for the round sphere S n of unit radius is given by n + k | k ∈ Z, k ≥ 0 , (26) Spec(D) = ± 2 n where the multiplicity of ( n2 + k) is equal to 2[ 2 ] k+n−1 . Thus for n = 4 one gets that k the spectrum consistsof the relative integers, except for {−1, 0, 1}. The multiplicity of for k + 2 = m which gives, for the total multiplicity of ±m, the eigenvalue m is 4 k+3 k P(m) =

4 4 (m + 1)m(m − 1) = (m 3 − m), 3 3

which shows that one gets the correct minus sign for the scalar curvature term after integration using Proposition 1. Thus one gets (up to the normalization factor 43 ) Tr(|D|−s ) = ζ (s − 3) − ζ (s − 1).

(27)

This function has a value at s = 0 given by ζ (−3) − ζ (−1) =

1 1 11 + = 120 12 120

which, taking into account the factor 43 from normalization, matches the coefficient 11 360 × 4 which appears in the spectral action in front of the Gauss-Bonnet term, as will be shown in § 3. 2.2. The sphere S 3 . We now want to look at the case of S 3 and determine how good the approximation of (22) is for test functions. In order to estimate the remainder of (22) in this special case we shall use the Poisson summation formula ˆ ˆ h(n) = h(u)e−2πixu du (28) h(n) , h(x) = Z

R

Z

+ Z in the odd case, the variant

1 = g n+ (−1)n g(n) ˆ 2

or rather, since the spectrum is

1 2

Z

(29)

Z

(obtained from (28) using h(u) = g(u + 21 )). In the case of the three sphere, the eigenvalues are ±( 23 + k), for k ≥ 0 with the mul 1 tiplicity 2 k+2 k . Thus n + 2 has multiplicity n(n + 1). This holds not only for n ≥ 0 but also for n ∈ Z since the multiplicity of −(n + 21 ) is n(n + 1) = m(m + 1) for m = −n − 1. In particular ± 21 is not in the spectrum. Thus when we evaluate Tr( f (D/ )), with f an even function, we get the following sum:

1 / . (30) Tr( f (D/ )) = n(n + 1) f n+ 2 Z

880


We apply (29) with g(u) = (u 2 − 14 ) f (u/ ). The Fourier transform of g is 1 2 u − f (u/ )e−2πixu du g(u)e du = g(x) ˆ = 4 R R 1 v 2 f (v)e−2πi xv dv − f (v)e−2π i xv dv. = 3 4 R R

−2πixu

We introduce the function fˆ(2) which is the Fourier transform of v 2 f (v) and we thus get from (29), Tr( f (D/ )) = 3

1 (−1)n fˆ(2) ( n) − (−1)n fˆ( n). 4 Z

(31)

Z

If we take the function f in the Schwartz space S(R), then both fˆ and fˆ(2) have rapid decay and we can estimate the sums

| fˆ( n)| ≤ Ck −k ,

n=0

| fˆ(2) ( n)| ≤ Ck −k

n=0

which gives, for any given k, an estimate for a sphere of radius a of the form: Tr( f (D/ )) = ( a)

3

1 v f (v)dv − ( a) f (v)dv + O(( a)−k ). 4 R R 2

(32)

The radius simply rescales D and enters in such a way as to make the product a dimenµ D sionless. This can be seen by noting that the ratio contains the term 1 eα γ α ∂µ and µ 1 the radius enters as a in the inverse dreibein eα . Note that, provided that k > 1, one controls the constant in front of ( a)−k from the constants c j with |xk fˆ(x)| ≤ c1 , |xk fˆ(2) (x)| ≤ c2 . To get an estimate of these constants c j , say for k = 2, one can use the L 1 -norms of the functions f (v) and (v 2 f (v)), where = −∂v2 is the Laplacian. If we take for f a smooth cutoff function we thus get that the c j are of order one. In fact we shall soon get a much better estimate (Corollary 4 below) which will show that, for suitable test functions, a size of N in cutoff units, a ∼ N , already ensures a 2 precision of the order of e−N . We shall work directly with the physically more relevant model consisting of the product S 3 ×S 1 viewed as a model of the imaginary time periodic compactification of space-time at a given temperature. Our estimates will work well for a size in cutoff units as small as N ∼ 10 and will give the result with an astronomical precision for larger values. These correspond to later times since both the radius of space and the inverse temperature are increasing functions of time in this simple model.


881

2.3. The product S 3 × S 1 . We now want to move to the 4-dimensional Euclidean case obtained by taking the product M = S 3 × S 1 of S 3 by a small circle. We take the product geometry of a three dimensional geometry with Dirac operator D3 by the one dimensional circle geometry with Dirac D1 =

1 i ∇θ , β

(33)

so that the spectrum of D1 is β1 (Z + 21 ). Lemma 2. Let D be the Dirac operator of the product geometry

0 D3 ⊗ 1 + i ⊗ D1 . D= D3 ⊗ 1 − i ⊗ D1 0

(34)

The asymptotic expansion for → ∞ of the spectral action of D is given by Tr(h(D 2 / 2 )) ∼ 2 β Tr(k(D32 / 2 )) ,

(35)

where the function k is given by k(x) =

∞

x

(u − x)−1/2 h(u) du.

(36)

Proof. By linearity of both sides in the function h (using the linearity of the transformation (36)) it is enough to prove the result for the function h(x) = e−bx . One has

2 0 D3 ⊗ 1 + 1 ⊗ D12 2 D = 0 D32 ⊗ 1 + 1 ⊗ D12 and Tr(e−b D

2 / 2

) = 2 Tr(e−b D1 / ) Tr(e−b D3 / ). 2

Moreover by (33) the spectrum of D1 is and b, one has for all k > 0, Tr(e−b D1 / ) ∼ 2

2

1 β (Z

2

2

2

+ 21 ) so that, using (29), and for fixed β

√ π β b−1/2 + O( −k ).

Thus Tr(e−b D

2 / 2

√ 2 2 ) = 2 β Tr( π b−1/2 e−b D3 / ) + O( −k+3 ),

and the equality (35) follows from ∞ √ (u − x)−1/2 e−bu du = π b−1/2 e−bx x

−bx by the linear transformation which shows that √ the function k associated to h(x) = e (36) is k(x) = π b−1/2 e−bx .

882


One can write (36) in the form

∞

k(x) =

v −1/2 h(x + v) dv ,

(37)

0

which shows that k has right support contained in the right support of h, i.e. that if h vanishes identically on [a, ∞[ so does k. It also gives a good estimate of the derivatives of k since ∞ ∂xn k(x) = v −1/2 ∂xn h(x + v) dv. 0

In fact, in order to estimate the size of the remainder in the asymptotic expansion of the spectral action for the product M = S 3 × S 1 , we shall now use the two dimensional form of (29), 1 1 = g n + ,m + (−1)n+m g(n, ˆ m), (38) 2 2 2 2 Z

Z

where the Fourier transform is given by g(x, ˆ y) = g(u, v)e−2πi(xu+yv) dudv. R2

(39)

For the operator D of (34), and taking for D3 the Dirac operator of the 3-sphere Sa3 of radius a, the eigenvalues of D 2 / 2 are obtained by collecting the following:

2 2

1 1 + n ( a)−2 + + m ( β)−2 , n, m ∈ Z 2 2 with the multiplicity 2n(n + 1) for each n, m ∈ Z. Thus, more precisely 2 2

1 1 2 2 −2 −2 + n ( a) + + m ( β) 2n(n + 1)h Tr(h(D / )) = 2 2 2 Z

which is of the form:

1 1 Tr(h(D / )) = , g n + ,m + 2 2 2

(40)

1

g(u, v) = 2(u 2 − )h u 2 ( a)−2 + v 2 ( β)−2 4

(41)

2

2

Z

where

One has g(0, ˆ 0) =

R2

g(u, v)dudv = 2

= 2 ( a) ( β)

1 h u 2 ( a)−2 + v 2 ( β)−2 dudv u − 4 2

R2

1 h(x2 + y 2 )dxdy ( a)2 x2 − 4 R2


883

using u = x ( a) and v = y ( β) . Thus we get: ∞ h(ρ 2 )ρ 3 dρ − π ( β) ( a) g(0, ˆ 0) = 2π ( β) ( a)3 0

∞

h(ρ 2 )ρdρ.

(42)

0

To estimate the remainder, given by the sum (−1)n+m g(n, ˆ m), (n,m)=(0,0)

we treat separately the Fourier transforms of g1 (u, v) = u 2 h(u 2 ( a)−2 + v 2 (β )−2 ) , g2 (u, v) = h(u 2 ( a)−2 + v 2 ( β)−2 ). One has

gˆ2 (n, m) =

= βa 2

R2

R2

g2 (u, v)e−2πi(nu+mv) dudv

h(x2 + y 2 )e−2πi(n ax+m βy) dxdy = 2 βaκ2 (n a, m β),

where the function of two variables κ2 (u, v) is the Fourier transform, h(x2 + y 2 )e−2πi(ux+vy) dxdy = κ(u 2 + v 2 ). κ2 (u, v) =

(43)

The function κ is related to the function k(x) defined by (36), and one has κ(u 2 ) = k(x2 )e−2πiux dx,

(44)

R2

R

so that κ(u 2 ) is the Fourier transform of k(x2 ). For g1 one has, similarly, gˆ1 (n, m) = g1 (u, v)e−2πi(nu+mv) dudv R2

= 4 βa 3

R2

x2 h(x2 + y 2 )e−2πi(n ax+m βy) dxdy = 4 βa 3 κ1 (n a, m β),

where the function of two variables κ1 (u, v) is the Fourier transform, x2 h(x2 + y 2 )e−2πi(ux+vy) dxdy κ1 (u, v) = R2

which is given in terms of (43) by 1 κ1 (u, v) = −π −2 (u 2 κ (u 2 + v 2 ) + κ (u 2 + v 2 )). 2

(45)

884


Now for any test function h in the Schwartz space S(R), the function x2 h(x2 + y 2 ) is in the Schwartz space S(R2 ) and thus we have for its Fourier transform, and any k > 0, an estimate of the form |κ1 (u, v)| ≤ Ck (u 2 + v 2 )−k . We thus get, for k > 2, | (−1)n+m gˆ1 (n, m)| ≤ (n,m)=(0,0)

= 4 βa 3

(46)

|gˆ1 (n, m)|

(n,m)= (0,0)

|κ1 (n a, m β)| ≤ Ck 4 βa 3

(n,m)=(0,0)

((n a)2 + (m β)2 )−k

(n,m)=(0,0)

≤ Ck 4 βa 3 ( µ)−2k

(n 2 + m 2 )−k , µ = inf(a, β).

(n,m)=(0,0)

We thus get, using a similar estimate for gˆ2 , Theorem 3. Consider the product geometry Sa3 × Sβ1 . Then one has, for any test function h in the Schwartz space S(R), the equality ∞ ∞ h(ρ 2 )ρ 3 dρ − π 2 βa h(ρ 2 )ρdρ + ( ), (47) Tr(h(D 2 / 2 )) = 2π 4 βa 3 0

0

where ( ) = O( −k ) for any k is majorized by |( )| ≤ 2 4 βa 3

(n,m)=(0,0)

1 |κ1 (n a, m β)| + 2 βa 2

|κ2 (n a, m β)|

(n,m)=(0,0)

with κ j defined in (43) and (45). This implies that all the Seeley coefficients a2n vanish for n ≥ 2, and we shall check this directly for a4 and a6 in § 3. This vanishing of the Seeley coefficients does not hold for the 4 sphere and it is worth understanding why one cannot expect to use the Poisson summation in the same way for the 4 sphere. The problem when one tries to use the Poisson formula as above is that, 2 e.g. for the heat kernel, one is dealing with a function like |x|e−tx which is not smooth and whose Fourier transform does not have rapid decay at ∞. 2.4. Specific test functions. We shall now concretely evaluate the remainder in Theorem 3 for analytic test functions of the form h(x) = P(π x)e−π x ,

(48)

where P is a polynomial of degree d. The Fourier transforms of the functions of two variables x2 h(x2 + y 2 ) and h(x2 + y 2 ) are of the form κ j (u, v) = P j (u, v)e−π(u

2 +v 2 )

,


885

where the P j are polynomials. More precisely, since the Fourier transform of e−λπ(x

2 +y 2 )

is λ1 e−π

(u 2 +v 2 ) λ

one obtains the formula for P2 by differentiation at λ = 1 and get (u 2 +v 2 ) 1 κ2 (u, v) = P(−∂λ )λ=1 e−π λ λ

which is of the form κ2 (u, v) = Q(π(u 2 + v 2 ))e−π(u

2 +v 2 )

,

where Q is a polynomial of degree d. The transformation P → Q = T (P) is given by 1 z Q(z) = P(−∂λ )λ=1 e− λ . λ

(49)

Moreover one then gets κ1 (u, v) = −(2π )−2 ∂u2 κ2 (u, v) = (u 2 Z 1 (π(u 2 + v 2 )) + Z 2 (π(u 2 + v 2 )))e−π(u

2 +v 2 )

,

where Z 1 = −Q + 2Q − Q , Z 2 =

1 (Q − Q ). 2π

(50)

We let C P be the sum of the absolute values of the coefficients of Q = T (P). Corollary 4. Consider the product geometry Sa3 × Sβ1 . Let µ = inf(a, β). Then one has, with h any test function of the form (48), the equality ∞ ∞ Tr(h(D 2 / 2 )) = 2π 4 βa 3 h(ρ 2 )ρ 3 dρ − π 2 βa h(ρ 2 )ρdρ + ( ), (51) 0

where, assuming µ ≥

0

d(1 + log d) and µ ≥ 1, π

|( )| ≤ Ce− 2 (µ ) , C = 4 βa 3 C P (8 + 6d + 2d 2 ). 2

(52)

Proof. One has xk e−x/2 ≤ 1 , ∀x ≥ 3k(1 + log k). Thus, for (n, m) = (0, 0) one has π

|κ2 (n a, m β)| ≤ C P e− 2 ((n a)

2 +(m β)2 )

, π

since π((n a)2 + (m β)2 ) ≥ 3d(1 + log d). Moreover, since e− 2 (µ ) ≤ 14 , one gets (n,m)=(0,0)

π

e− 2 ((n a)

2 +(m β)2 )

π

≤ 8e− 2 (µ )

2

2

886

and


π

|κ2 (n a, m β)| ≤ 8 C P e− 2 (µ ) . 2

(n,m)=(0,0)

A similar estimate using (50) yields π 2 |κ1 (n a, m β)| ≤ (2 + 3d + d 2 ) C P e− 2 (µ ) . (n,m)=(0,0)

Thus by Theorem 3, the inequality (52) holds for C = C P (2 4 βa 3 (2 + 3d + d 2 ) + 4 2 βa). One then uses the hypothesis µ ≥ 1 to simplify C. The meaning of Corollary 4 is that the accuracy of the asymptotic expansion is at least π 2 of the order of e− 2 (µ ) . Indeed the term 4 βa 3 in (52) is the dominant volume term in the spectral action and the other terms in the formula for C are of order one. Thus for instance for a size µ ∼ 100 one gets that the asymptotic expansion accurately delivers the first 6820 decimal places of the spectral action. Note that some test functions of the form (48) give excellent approximations to cutoff functions, in particular h n (x) =

n (π x)k

k!

0

e−π x .

(53)

The graph of h n (x2 ) is shown in Fig. 1 for n = 20. For h = h 20 the computation gives C P (8 + 6d + 2d 2 ) ≤ 2 × 106 so that this constant only interferes with the last six decimal places in the above accuracy. In our simplified physical model we test the approximation of the spectral action by its asymptotic expansion for the Euclidean model 3 1 E(t) = Sa(t) × Sβ(t) ,

where space at a given time t is given by a sphere with radius a(t) and β(t) is a uniform value of inverse temperature. One can then easily see that the above approximation to the spectral action is fantastically accurate, going backwards in time all the way up to one order lower than the Planck energy. In doing so the radius a(t) varies between at −34 m), while the temperature least ∼ 1061 Planck unitsand 10 ◦ Planck units (i.e. 10 ◦ 31 varies between 2.7 K and 10 K . It is for an inner size less than 10 in Planck units that the approximation does break down. Remark 5. For later purpose it is important to estimate the constant C P in terms of the coefficients of the polynomial P. Let then P(z) = z n . One has h(x) = (π x)n e−π x and the function k(x) associated to h by (36) is n n n−k 2 2 n −π x x h(x + y )dy = π e y 2k e−π y dy k(x) = k R R 0

= π −1/2 e−π x

n n 0

1 ( + k)(π x)n−k . k 2


887

To obtain Q = T (P) one then needs to compute the Fourier transform κ(u 2 ) of the 2 function k(x2 ) as in (44). The Fourier transform of (π x2 )m e−π x is m (u) = (−4π )−m ∂u2m e−π u = L m (π u 2 )e−π u 2

2

and one checks, using the relation L m+1 (z) = 1/2((1 − 2z)L m (z) + (−1 + 4z)L m (z) − 2z L m (z) that the sign of the coefficient of z k in L m (z) is (−1)k . Thus the sum of the absolute values of the coefficients of L m is equal to L m (−1) = m (iπ −1/2 )/e. Thus since the above sum giving k(x) has positive coefficients we get that, for P(z) = z n , the constant 2 2 C P is given by Q(π(u 2 + v 2 ))e−π(u +v ) /e for (u, v) = (iπ −1/2 , 0), which gives 2 2 √ CP = π n (y 2 + x2 )n e−π y −π x +2 π x−1 dxdy. R2

One then gets CP ≤ 2

∞

u 2n+1 e−(u−1) du = O(λn n!) , ∀λ > 1. 2

(54)

0

Thus, for an arbitrary polynomial P(z) = d0 ak z k one has ∞ 2 CP ≤ 2 |P|(u 2 )e−(u−1) udu , |P|(z) = |ak |z k .

(55)

0

2.5. The Higgs potential. We now look at what happens if one performs the following replacement on the operator D 2 → D 2 + H 2 , where H is a constant. This amounts with the above notations to the replacement h(u) → h(u + H 2 / 2 ).

(56)

As long as H 2 / 2 is of order one, we can trust the asymptotic expansion and we just need to understand the effect of this shift on the two terms of (47). We look at the first contribution, i.e. ∞ ∞ 2π 4 βa 3 h(ρ 2 )ρ 3 dρ = π 4 βa 3 uh(u)du. 0

0

We let x = H 2 / 2 , and get, after the above replacement, ∞ ∞ x ∞ uh(u + x)du = (v − x)h(v)dv = (v − x)h(v)dv − (v − x)h(v)dv x

0

∞

= 0

0

∞

vh(v)dv − x 0

0

x

h(v)dv − 0

(v − x)h(v)dv.

888


∞ The first term corresponds to the initial contribution of π 4 βa 3 0 uh(u)du. The second term gives ∞ ∞ h(v)dv = −π 2 βa 3 H 2 h(v)dv (57) − π 4 βa 3 x 0

0

which is the expected Higgs mass term from the Seeley–de Witt coefficient a2 . To understand the last term we assume that h is a cutoff function. Lemma 6. If h is a smooth function constant on the interval [0, c], then for x = H 2 / 2 ≤ c the new terms arising from the replacement (56) are given by ∞ 1 1 2 3 − π βa h(v)dv H 2 + πβah(0) H 2 + πβa 3 h(0) H 4 . (58) 2 2 0 4 βa 3 ∞ uh(u)du, besides (57), we just need to comProof. For the perturbation of π 0 x pute the last term − 0 (v − x)h(v)dv, and one has x x 1 − (v − x)h(v)dv = h(0) (x − v)dv = h(0)x2 , 2 0 0 since h is constant on the interval [0, x]. We then look at the effect on the second contribution, i.e. ∞ ∞ 1 −π 2 βa h(ρ 2 )ρdρ = − π 2 βa h(u)du. 2 0 0 We let, as above, x = H 2 / 2 , and get ∞ ∞ h(u + x)du = h(v)dv = 0

x

∞

x

h(v)dv −

0

h(v)dv. 0

Thus the perturbation, under the hypothesis of Lemma 6 is 1 1 − π 2 βa(−xh(0)) = πβah(0) H 2 . 2 2 The three terms in formula (58) correspond to the following new terms for the spectral action: • The Higgs mass term coming from the Seeley–de Witt coefficient a2 , • The R H 2 term coming from the Seeley–de Witt coefficient a4 , • The Higgs potential term in H 4 coming from the Seeley–de Witt coefficient a4 . We can now state the analogue of Theorem 3 as follows: Theorem 7. Consider the product geometry Sa3 × Sβ1 . Let µ = inf(a, β). Then one has, with h any test function of the form (48), the equality ∞ ∞ 2 2 2 4 3 2 3 2 Tr(h((D + H )/ )) = 2π βa h(ρ )ρ dρ − π βa h(ρ 2 )ρdρ 0

0

1 + π 4 βa 3 V (H 2 / 2 ) + π 2 βa W (H 2 / 2 ) + ( ), 2


where

V (x) =

∞

889

u(h(u + x) − h(u))du , W (x) =

0

and, assuming µ ≥

x

h(u)du

(59)

0

d(1 + log d), µ ≥ 1, and H 2 −2 ≤ c/π , π

|( )| ≤ Ce− 2 (µ ) , C = 4 βa 3 C P (8 + 6d + 2d 2 ) where, with P(z) = d0 ak z k one has ∞ 2 C P = 4 |P|(u 2 + c)e−(u−1) udu , |P|(z) = |ak |z k . 2

(60)

0

Proof. The new terms simply express the replacement (56) in the formula of Theorem 3. The new function h˜ thus obtained is still of the form (48) since it is obtained from h by a translation. It thus only remains to estimate C P˜ , where P˜ is the polynomial such that ˜ ˜ u)e−π u . For P(z) = z n the constant C ˜ for a translation u → u + x, x ≥ 0 h(u) = P(π P of the variable, is less than the constant C Px for the polynomial n Px (π u) = P(π(u + x)) = (π x)n−k (π u)k . k Thus, by Remark 5, (55), the constant C Px is estimated by ∞ 2 C Px ≤ 2 (u 2 + π x)n e−(u−1) udu, 0

which is an increasing function of x and thus only needs to be controlled for x = c/π in our case. For instance, for h = h 20 the computation gives C P (8 + 6d + 2d 2 ) ≤ 3 × 107 for c = 1, so that this constant only interferes with the last seven decimal places in the accuracy which is the same as in Corollary 4. Moreover as shown in Lemma 6, when h is close to a true cutoff function 1 π 4 βa 3 V (H 2 / 2 ) + π 2 βa W (H 2 / 2 ) 2 ∞ 1 1 h(ρ 2 )ρdρ H 2 + πβah(0) H 2 + πβa 3 h(0) H 4 + δ, = −2π 2 βa 3 2 2 0

(61)

where the remainder δ is estimated from the Taylor expansion of h at 0. For instance for the functions h n of (53), one has by construction 0 ≤ h n (x) ≤ 1 for all x and since a(n, k)(π x)n+k+1 , h n (x) = 1 − a(n, k) = (−1)k /((n + k + 1)n!k!) one gets, for h = h n the estimate |δ| ≤ π 4 βa 3

(π x)n+3 (π x)n+2 + π 2 βa , x = H 2 / 2 . (n + 3)(n + 1)! 2(n + 2)!

While the function W is by construction the primitive of h, and is increasing for h ≥ 0 one has, under the hypothesis of positivity of h,

890


Lemma 8. The function V (x) is decreasing with derivative given by ∞ V (x) = − h(v)dv. x

The second derivative of V (x) is equal to h(x). Proof. One has V (x) =

0

∞

uh (u + x)du = [uh(u + x)]∞ 0 −

∞

h(u + x)du

0

which gives the required results.

3. Seeley–De Witt Coefficients and Spectral Action on S3 × S1 In this section we shall compute the asymptotic expansion of the spectral action on the background geometry of S 3 × S 1 using heat kernel methods. This will enable us to check independently the accuracy of the estimates derived in the last section. This background is physically relevant since it can be connected with simple cosmological models. We refer to [8,23] for the formulas and the method of the computation. The general method we use is also explained in great detail in a forthcoming paper [18]. We start by computing a0 : Tr(1) √ 4 1 3 3 gd x = 4π 2 gd x dx a0 = 16π 2 =

1 4π 2

2π 2 a

3

S3

(2πβ) =

S1 πβa 3 ,

where β is the radius of Sβ1 and the volume of the three sphere Sa3 of radius a is 2π 2 a 3 [25]. Next we calculate a2

1 1 4 √ a2 = d x gTr E + R , 16π 2 6 where E is defined from the relation D 2 = − g µν ∇µ ∇ν + E , where for pure gravity we have 1 E =− R 4 so that (using Tr(1) = 4 ) a2 =

1 4π 2

R √ − d 4x g 12


891

since the curvature is constant. The curvature tensor13 is, using the coordinates of [25] for the three sphere Sa3 with labels i, j, k, l and the label 4 for the coordinate in Sβ1 , Ri jkl = −a −2 gik g jl − gil g jk , i, j, k, l = 1, · · · 3, Ri jk4 = 0, Ri4 j4 = 0,

where gi j is the metric on the three sphere as in [25]. The Ricci tensor is given, following the sign convention of [24] which introduces a minus sign in passing from the curvature tensor to the Ricci tensor, by Ri j = −g kl Rik jl = 2a −2 gi j , Ri4 = 0, R44 = 0. Thus the scalar curvature is R = g i j Ri j =

6 , a2

and the a2 term in the heat expansion simplifies to

1 . a2 = −πβa 2 Next for a4 we have a4 =

1 1 2 16π 360

√ µ d 4 x g T r 12R;µ + 5R 2 − 2Rµν R µν

M

µ + 2Rµνρσ R µνρσ + 60R E + 180E 2 + 60E ;µ + 30 µν µν , where for the pure gravitational theory, we have 1 E = − R, 4

µν =

1 R ab γab . 4 µν

In this case it was shown in [8] that a4 reduces to

1 1 4 √ 2 2 2 a4 = x g 5R − 8R − 7R d µν µνρσ 4π 2 360

1 1 √ 2 = + 11R ∗ R ∗ , d 4 x g −18Cµνρσ 2 4π 360 which is obviously scale invariant. The Weyl tensor Cµνρσ is defined by 1 Rµρ gνσ − Rνρ gµσ − Rµσ gνρ + Rνσ gµρ 2 1 − gµρ gνσ − gνρ gµσ R 6

Cµνρσ = Rµνρσ +

13 The sign convention for this tensor is the same as in [23].

(62)

892


This tensor vanishes on S 3 × S 1 as can be seen by evaluating the components Ci jkl = a −2 −(gik g jl − gil g jk ) + 2(gik g jl − gil g jk ) − (gik g jl − gil g jk ) = 0, Ci jk4 = 0, Ci4k4 = 0. Similarly the Gauss-Bonnet term 1 µνρσ αβγ δ Rµναβ Rρσγ δ 4

γδ αβ = i jk4 αβγ δ Ri j Rk4

R∗ R∗ =

=0 The next step of calculating a6 is in general extremely complicated, but for spaces of constant curvature the expression simplifies as all covariant derivatives of the curvature tensor, Riemann tensor and scalar curvature vanish. The non-vanishing terms are, using Theorem 4.8.16 of [23] and the above sign convention for the Ricci tensor Rµν and the scalar curvature,

1 1

4 √ 2 2 a6 = 35R 3 − 42R Rµν x gTr + 42R Rµνρσ d 16π 2 9 · 7! − 208Rµν Rµρ Rνρ − 192Rµρ Rνσ Rµνρσ − 48Rµν Rµρσ κ Rνρσ κ − 44Rµνρσ Rµνκλ Rρσ κλ − 80Rµνρσ Rµκρλ Rνκσ λ 1

−12µν νρ ρµ − 6Rµνρσ µν ρσ − 4Rµν µρ νρ + 5R2µν + 360 2 2 + 60E 3 + 30E2µν + 30R E 2 + 5R 2 E − 2Rµν E + 2Rµνρσ E

We can now compute each of the above eighteen terms. These are listed in an appendix. Collecting these terms we obtain that the integrand is 4a −6 −35 · 63 + 42 · 72 − 42 · 72 + 208 · 24 − 192 · 24 + 48 · 24 − 44 · 24 − 80 · 6 9 · 7!

4a −6 15 · 27 5 · 27 − 9 + 18 − 12 + 45 + − − 15 · 27 + 10 · 27 − 36 + 36 360 2 2

2 2 − = 0, = a −6 3 3

−

implying that a6 = 0 , which shows that the cancelation is highly non-trivial. We conclude that the spectral action, up to terms of order 14 is given by

∞

S = 4 0

xh (x) dx πβa 3 − 2

∞ 0

1 + O −4 . h (x) dx πβa 2


893

After making the change of variables x = ρ 2 we get 3 S = (πβ ) 2 ( a)

∞

2 ρ h ρ dρ − ( a) 3

0

∞

ρh ρ

2

dρ + O −4 .

0

This confirms Eq. (47) and shows that, to a very high degree of accuracy, the spectral action on S 3 × S 1 is given by the first two terms. Remark 9. It is worth noting that one can also check the value of the Gauss-Bonnet term on S 4 and show that it agrees with the value obtained in (27). To see this note that the Riemann tensor in this case is given by ([25]) Rµνρσ = −a −2 gµρ gνσ − gµσ gνρ which implies14 that Cµνρσ = 0, R ∗ R ∗ = 6a −4 , and thus a4 =

1 11 −4 a 4π 2 60

√ d 4 x g. S4

The volume of S 4 is V4 =

5

8π 2 4 2π 2 √ a , d 4x g = a4 = 3 S4 25

and this implies that a4 =

11 × 4, 360

which agrees exactly with the calculation of (27) based on zeta functions.

Appendix In this appendix we compute the eighteen non-vanishing terms that appear in the a6 term of the heat kernel expansion. Using the properties 2 Rµν = Ri2j = 12a −4 , 2 Rµνρσ = Ri2jkl = 12a −4 , 14 One can double check the value of R ∗ R ∗ using the Gauss–Bonnet Theorem.

894


35R 3 = 35(6)3 a −6 , 2 −42R Rµν = −42 (6) (12) a −6 ,

−208Rµν Rµρ Rνρ = −208 (2)3 gi j gik g jk = 208 (2)3 (3) a −6 , −192Rµρ Rνσ Rµνρσ = −192Rik R jl Ri jkl , = 192 (2)2 gik g jl gik g jl − gil g jk a −6 , = 192 (24) a −6 ,

−48Rµν Rµρσ κ Rνρσ κ = −48 (2) gi j (gik glm − gil gkm ) g jk glm − g jl gkm a −6 , = −48 (4) gi j 2gi j a −6 = −48 (24) a −6 , −44Rµνρσ Rµνκλ Rρσ κλ = 44 gik g jl − gil g jk gi p g jq − giq g j p gkp glq − glq glp a −6 , −80Rµνρσ Rµκρλ Rνκσ λ

= 44 (4) (6) a −6 , = 80 gik g jl − gil g jk gik g pq − giq g pk g jl g pq − g jq g pl a −6 , = 80 3g jl g pq − g pq g jl − gl j g pq + glq g j p g jl g pq − g jq g pl a −6 = 80 (9 − 3 + 3 − 3) a −6 = 80 (6) a −6 .

Collecting the first set of terms we get 4a −6 −35 · 63 + 42 · 72 − 42 · 72 + 208 · 24 − 192 · 24 + 48 · 24 − 44 · 24 − 80 · 6 9 · 7! 2 = a −6 . 3

−

Now we continue with the second set of terms

3 1 ef −12Tr µν νρ ρµ = −12 Tr γab γcd γe f Rµνab Rνρcd Rρµ 4

3 1 = Tr (1) 12 (8) Rµνab Rνρbc Rρµac 4 3 = − gik g jl −gil g jk g jl g pq −g j p glq g pk giq −g pi gkq a −6 Tr (1) 2 3 = − 3gik g pq − gik g pq − gik g pq + giq g pk g pk giq − g pi gkq 4a −6 2 3 = − (3 − 3 + 9 − 3) 4a −6 2 = −9 · 4a −6 , 6 −6Rµνρσ Tr µν ρσ = − 2 Rµνρσ Tr (γab γcd ) Rµνab Rρσcd 4 12 Rµνρσ Rµνab Rρσ ab Tr (1) = 16 3 = − gik g jl − gil g jk gi p g jq − giq g j p gkp glq − gkq glp 4a −6 4 = −3 (9 − 3) 4a −6 = −18 · 4a −6 ,


895

1 −4Rµν Tr µρ νρ = − Rµν Tr (γab γcd ) Rµρab Rνρ cd 4 1 = Rµν Rµρab Rνρab Tr (1) 2 1 = (2) gi j gi p gmq − giq gmp g j p gmq − g jq gmp 4a −6 2 = 2 (9 − 3) 4a −6 = 12 · 4a −6 ,

5 5RTr 2µν = RTr (γab γcd ) Rµνab Rµνcd 16 5 2 = − R Rµνρσ Tr (1) 8 5 = − (6) (12) 4a −6 8 = −45 · 4a −6 ,

1 3 3 60Tr E 3 = 60 − R Tr (1) 4

1 3 3 = 60 − (6) · 4a −6 4 1 = − (15 · 27) · 4a −6 , 2

30Tr

E2µν

1 2 −2 2 3 (−2) − = 30 − a Rµνρσ Tr (1) 2 4 90 (12)4a −6 = 16 1 = (5 · 27) · 4a −6 , 2 30 3 R Tr (1) 16

15 = (6)3 4a −6 8

30R E 2 Tr (1) =

= (15 · 27) · 4a −6 , 5 5R 2 ETr (1) = − R 3 · 4 4 5 = − (6)3 4a −6 4 = − (10 · 27) · 4a −6 ,

896


R 2 2 − 4 −2Rµν ETr (1) = −2Rµν 4

3 4a −6 = −2 (12) − 2 = 36 · 4a −6 , 2 2Rµνρσ ETr (1)

3 4a −6 = 2 (12) − 2 = −36 · 4a −6 .

Collecting the second set of terms we get

4a −6 15 · 27 5 · 27 − − − 15 · 27 + 10 · 27 − 36 + 36 9 + 18 − 12 + 45 + 360 2 2 2 = − a −6 . 3 Thus the sum of all the terms in a6 is zero. Acknowledgements. The research of A. H. C. is supported in part by the Arab Fund for Social and Economic Development and the National Science Foundation under Grant No. Phys-0601213.

References 1. Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA. 51(4), 542 (1964) 2. Connes, A.: On the spectral characterization of manifolds. To appear, available at http://arxiv.org/abs/ 0810.2088v1[math.OA], 2008 3. Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996) 4. Barrett, J.: The Lorentzian version of the noncommutative geometry model of particle physics. J. Math. Phys. 48, 012303 (2007) 5. Connes, A.: Noncommutative geometry and the standard model with neutrino mixing. JHEP 0611, 081 (2006) 6. Chamseddine, A., Connes, A.: Why the Standard Model. J. Geom. Phys. 58, 38–47 (2008) 7. Chamseddine, A., Connes, A.: Conceptual explanation for the algebra in the noncommutative approach to the standard model. Phys. Rev. Lett. 99, 191601 (2007) 8. Chamseddine, A., Connes, A.: The Spectral action principle. Commun. Math. Phys. 186, 731–750 (1997) 9. Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. 11, 991–1090 (2007) 10. Chamseddine, A., Connes, A.: Scale invariance in the spectral action. J. Math. Phys. 47, 063504 (2006) 11. Chamseddine, A., Connes, A.: Quantum gravity boundary terms from the spectral action of noncommutative space. Phys. Rev. Lett. 99, 071302 (2007) 12. Gibbons, G., Hawking, S.: Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752 (1977) 13. Ashtekar, A., Engle, J., Sloan, D.: Asymptotics and Hamiltonians in a first order formalism. Class. Quant. Grav. 25, 095020 (2008) 14. van der Bij, J., van Dam, H., Ng, Y.: Physica A 116, 307 (1982); Wilczek, F., Zee, A.: In: High Energy Physics, ed. Mintz, S., Perlmutter, A.: New York: Plenum, 1985; S. Weinberg, Rev. Mod. Phys. 61, 1 (1989) 15. Reuter, M.: Nonperturbative evolution equation for quantum gravity. Phys. Rev. D 57, 971 (1998) 16. Dou, D., Percacci, R.: The running gravitational couplings. Class. Quant. Grav. 15, 3449 (1998)


897

17. Percacci, R.: Renormalization group, systems of units and the hierarchy problem. J. Phys. A40, 4895 (2007) 18. Chamseddine, A., Connes, A.: Uncovering the noncommutative geometry of space-time: a user manual for physicists. To appear 19. Hambye, T., Riesselmann, K.: Matching conditions and upper bounds for Higgs masses revisited. Phys. Rev. D55, 7255 (1997) 20. Isidori, G., Rychkov, V., Sturmia, A., Tetradis, N.: Gravitational corrections to the standard model vacuum decay. Phys. Rev. D77, 025034 (2008) 21. Coleman, S., Weinberg, E.: Radiative corrections as the origin of spontaneous symmetry breaking. Phys. Rev. D7, 1888 (1973) 22. Buchmüller, W., Busch, C.: Symmetry breaking and mass bounds in the standard model with hidden scale invariance. Nucl. Phys. B349, 71 (1991) 23. Gilkey, P.: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. Wilmington, DE: Publish or Perish, 1984 24. Lawson, H.B., Michelsohn, M-L.: Spin Geometry, Princeton Mathematical Series 38, Princeton, NJ: Princeton University Press, 1989 25. Weinberg, S.: Gravitation and Cosmology. New York: J. Wiley, 1972, pp. 389–390 26. Trautman, A.: Spin structures on hypersurfaces and spectrum of Dirac ooperators on spheres. In: Spinors, Twistors, Clifford Algebras and Quantum Deformations. Dordrecht: Kluver Academic Publishers, 1993 Communicated by A. Kapustin


Communications in


Trivializing Maps, the Wilson Flow and the HMC Algorithm Martin Lüscher CERN, Physics Department, 1211 Geneva 23, Switzerland. E-mail: [email protected] Received: 9 August 2009 / Accepted: 24 September 2009 Published online: 24 November 2009 – © Springer-Verlag 2009

Abstract: In lattice gauge theory, there exist field transformations that map the theory to the trivial one, where the basic field variables are completely decoupled from one another. Such maps can be constructed systematically by integrating certain flow equations in field space. The construction is worked out in some detail and it is proposed to combine the Wilson flow (which generates approximately trivializing maps for the Wilson gauge action) with the HMC simulation algorithm in order to improve the efficiency of lattice QCD simulations. 1. Introduction The Nicolai map transforms interacting supersymmetric theories to non-interacting ones [1]. Supersymmetry is considered to be essential for the existence of these field transformations in view of the fact that their Jacobian is exactly cancelled by the fermion partition function. In lattice gauge theory, a natural question to ask is whether there are field transformations that map the theory to its strong-coupling limit. In particular, if there are no matter fields, one is looking for substitutions U = F(V )

(1.1)

of the gauge field U in the functional integral whose Jacobian cancels the gauge-field action. Similarly to the Nicolai map, this kind of transformation maps the theory to a solvable one, but supersymmetry is not required and the Jacobian plays a different rôle. On a finite lattice, and if the gauge group is compact and connected, the existence of such trivializing maps is guaranteed by a general theorem on volume forms on compact manifolds (see ref. [2], Theorem 1.26, for example). One may be inclined to assume that these transformations are too complicated to be of any use. However, as explained later in this paper, it is possible to build up trivializing maps by integrating flows in field space, whose generators satisfy certain partial differential equations. The latter are

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M. Lüscher

U

F −1

V

HMC

V’

F

U’

Fig. 1. The proposed simulation algorithm for lattice QCD updates the gauge field U in three steps, following the arrows in this diagram, where the Hamilton function used in the HMC step has the standard kinetic term and includes the Jacobian of the field transformation F (cf. Subsect. 2.4)

quite tractable and can, to some extent, be solved analytically in the pure gauge theory. An application of trivializing maps, which can then be envisaged, is the acceleration of lattice QCD simulations. The fact that the efficiency of the available simulation algorithms is unpredictable has always been a weakness of numerical lattice QCD. Already a while ago, empirical studies of the SU(3) gauge theory by Del Debbio, Panagopoulos and Vicari [3] showed that the autocorrelation times of observables related to the topological charge of the gauge field tend to be large and appear to grow exponentially with the inverse of the lattice spacing. Moreover, Schaefer, Sommer and Virotta [4] recently found that the situation is, in this respect, essentially unchanged when the sea quarks are included in the simulations. The rapid slowdown of the simulations at small lattice spacings may conceivably be overcome by combining approximately trivializing maps with the HMC simulation algorithm [5] (see Fig. 1). Since the transformation moves the theory closer to the strong-coupling limit, where the HMC algorithm is known to be highly efficient, the autocorrelation times are, in general, expected to be reduced in this way. Evidently, for the combined algorithm to work out in practice, approximately trivializing maps must be found which are fairly simple and programmable. One of the goals of the present paper is thus to provide a solution to this problem. 2. Field Transformations Most concepts developed in this paper are expected to be widely applicable, but as explained above, the case of immediate interest is lattice QCD. In the following, the gauge group is therefore taken to be SU(3). Since the quarks will play a spectator rôle, they will be added to the theory only in Sect. 6, where the proposed simulation algorithm for lattice QCD is discussed. 2.1. Field space. The lattice theory is set up on a finite hypercubic lattice with periodic boundary conditions. For notational convenience, the lattice spacing is set to unity. As usual, the gauge field variables U (x, µ) ∈ SU(3) are assumed to reside on the links (x, µ) of the lattice (where x ∈ and µ = 0, . . . , 3). The expectation value of any observable O(U ) is then given by the functional integral 1 O = Z D[U ] O(U ) e−S(U ) (2.1) over the space of gauge fields. In this expression, S(U ) denotes the gauge-field action, Z the partition function and D[U ] the product of the normalized SU(3)-invariant integration measures of the link variables U (x, µ). From a purely mathematical point of view, the space of lattice gauge fields is a power of SU(3) and therefore a compact connected manifold. Field transformations are invertible maps of this manifold onto itself. Such transformations will always be required to

Trivializing Maps, the Wilson Flow and the HMC Algorithm

901

be differentiable in both directions and orientation-preserving (here and below, “differentiable” means “infinitely often continuously differentiable”). 2.2. Right-invariant differential operators. Since the link variables U (x, µ) take values in a Lie group, it is natural to express differentiations with respect to them through a basis a , a = 1, . . . , 8, of differential operators that are invariant under the right-action of ∂x,µ the group. The action of these operators on a differentiable function f (U ) of the gauge field is given by tT a e U (x, µ) if (y, ν) = (x, µ), a d (2.2) ∂x,µ f (U ) = dt f (Ut ) t=0 , Ut (y, ν) = U (y, ν) otherwise, a where T a are the SU(3) generators (see Appendix A). In particular, ∂x,µ transforms according to the adjoint representation under the left-action of SU(3). a go along with a basis of 1-forms on the field manifold, The operators ∂x,µ a θx,µ = −2 tr{dU (x, µ)U (x, µ)−1 T a },

such that d f (U ) =

a a θx,µ ∂x,µ f (U )

(2.3)

(2.4)

x,µ

for all functions f (U ). 2.3. Jacobian matrix. For any given gauge field V (y, ν), the transformation (1.1) produces another field U (x, µ) = [F(V )](x, µ). When considering such transformations, one needs to distinguish differentiations with respect to V from those with respect U . The associated 1-forms must also be distinguished. In the following, all symbols carrying a hat represent quantities and operations referring to V . The Jacobian matrix b [F∗ (V )](x, µ; y, ν)ab = −2 tr{∂ˆ y,ν U (x, µ)U (x, µ)−1 T a }

(2.5)

can be considered to be the kernel of a linear operator acting on link fields with values in su(3). In particular, a b θx,µ = [F∗ (V )](x, µ; y, ν)ab θˆy,ν . (2.6) y,ν

Since the functional integration measure D[U ] is proportional to the maximal product of these 1-forms, it follows that D[U ] = D[V ] det F∗ (V ).

(2.7)

The Jacobian of the map (1.1) is thus det F∗ (V ). If the transformation satisfies S(F(V )) − ln det F∗ (V ) = constant,

(2.8)

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M. Lüscher

the substitution U → V of the integration variables in the functional integral maps the theory to the trivial one where the link variables are completely decoupled from one another. The expectation values (2.1) are then given by O = D[V ] O(F(V )). (2.9) Such trivializing maps thus contain the entire dynamics of the theory. Although the remark is likely to remain an academic one, an intriguing observation is that the integral (2.9) can be simulated simply by generating uniformly distributed random gauge fields. Subsequent field configurations are uncorrelated in this case and there are therefore no autocorrelations in the data series for the observables O. 2.4. Transformation behaviour of the HMC algorithm. The HMC algorithm [5] operates on the phase space associated to the field manifold. In particular, the transition V → V in Fig. 1 requires the equations of motion derived from the Hamilton function Hˆ (πˆ , V ) = 21 (πˆ , πˆ ) + S(F(V )) − ln det F∗ (V )

(2.10)

to be integrated, where π(x, ˆ µ) ∈ su(3) is the canonical momentum of V (x, µ). Although the complete update algorithm for the field U described by Fig. 1 looks different, it is in fact equivalent to the HMC algorithm with a non-standard Hamilton function. The equivalence can be established by noting that the transformation from V to U preserves the symplectic 2-form a ˆ = d{πˆ a (x, µ)θˆx,µ }, (2.11) x,µ

ˆ if the momenta of the fields are transformed according to i.e. = , [F∗ (V )](x, µ; y, ν)ab π a (x, µ). πˆ b (y, ν) =

(2.12)

x,µ

The evolution of the transformed fields is then governed by the Hamilton function H (π, U ) = 21 (π, K (U )π ) + S(U ) − ln det F∗ (V ),

(2.13)

where K (U ) = F∗ (V )T F∗ (V ),

V = F −1 (U ).

(2.14)

Note that the Jacobian in Eq. (2.13) cancels when the momenta are integrated over in the functional integral. The algorithm outlined in Fig. 1 thus amounts to applying the HMC algorithm with a modified Hamilton function of the kind considered long ago by Duane et al. [6]. 3. Transformations Generated by Flow Equations Flows in field space build up field transformations from infinitesimal transformations. The latter are generally easier to work with than integral transformations, because they refer to the current field only. Moreover, the differentiability and invertibility of the generated transformations is automatically guaranteed.


903

3.1. Flows in field space. An infinitesimal field transformation, U → U + Z (U )U + O( 2 ),

(3.1)

is generated by a link field [Z (U )](x, µ) with values in su(3). The continuous composition of such transformations amounts to integrating a flow equation U˙ t = Z t (Ut )Ut

(3.2)

with respect to a ficticious time t. A simple choice for the generator of the flow is a [Z t (U )]a (x, µ) = ∂x,µ W0 (U ), tr{U (x, µ, ν)}, W0 (U ) =

(3.3) (3.4)

x,µ=ν

where U (x, µ, ν) denotes the plaquette loop in the (µ, ν)-directions at the point x. In the following, this flow will be referred to as the “Wilson flow”. If Z t (U ) is a differentiable function of t and U , the flow equation (3.2) has a unique solution Ut for any specified initial value U0 = V and all t ∈ (−∞, ∞). Moreover, the solution is differentiable with respect to t and V (for a proof of these statements, see ref. [7], for example). It should be emphasized that the existence of the solution for all times is non-trivial and can only be guaranteed, without further assumptions, because the field manifold is compact.

3.2. Integrated transformations. At fixed t, the field Ut is a well-defined function of the initial field V . Through the integration of the flow equation, one thus obtains a differentiable transformation, V → Ut = Ft (V ), of the field space. The transformation is invertible and its inverse is differentiable, because the flow equation can be integrated backwards from t to 0. Moreover, since the Jacobian det Ft,∗ (V ) is equal to unity at t = 0 and does not pass through zero at any time, the transformation is also orientationpreserving and thus fulfills all requirements for an acceptable map of field space. There is a useful compact expression for the Jacobian which is obtained starting from the equations ln det Ft,∗ (V ) = Tr{F˙ t,∗ (V )Ft,∗ (V )−1 }, (3.5) b {[Z t (Ut )](x, µ)Ut (x, µ)}Ut (x, µ)−1 T a [F˙ t,∗ (V )](x, µ; y, ν)ab = −2 tr ∂ˆ y,ν b − ∂ˆ y,ν Ut (x, µ)Ut (x, µ)−1 [Z t (Ut )](x, µ)T a . (3.6) d dt

Noting b = ∂ˆ y,ν

a [F∗ (V )](x, µ; y, ν)ab ∂x,µ ,

(3.7)

x,µ

a few lines of algebra then lead to the formula t a ln det Ft,∗ (V ) = ∂x,µ ds [Z s (U )]a (x, µ) U =U . 0

x,µ

s

(3.8)

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M. Lüscher

In the case of the Wilson flow, for example, the contribution of the Jacobian to the action of the field V , t 16 ln det Ft,∗ (V ) = − 3 ds W0 (Us ), (3.9) 0

is proportional to the integral of the Wilson plaquette action along the flow. 4. Trivializing Maps Somewhat surprisingly, trivializing maps can, to some extent, be constructed explicitly in the pure gauge theory. The construction is explained in this section, assuming that the gauge action S(U ) is a sum of Wilson loops (plaquettes, rectangles, etc.). 4.1. Trivializing flows. If the generator Z t (U ) of the flow (3.2) is such that t a ds [Z s (U )]a (x, µ) U =U = t S(Ut ) + Ct , ∂x,µ 0

s

x,µ

(4.1)

where Ct may depend on t but not on the fields, the associated integrated transformations satisfy S(Ft (V )) − ln det Ft,∗ (V ) = (1 − t)S(Ft (V )) − Ct .

(4.2)

In particular, the transformation at t = 1 is then a trivializing map. Equation (4.1) is a rather implicit condition on the generator of the flow. However, when differentiated with respect to t, it assumes a more tractable form, a a ∂x,µ [Z t (U )]a (x, µ) − t∂x,µ S(U )[Z t (U )]a (x, µ) = S(U ) + C˙t , (4.3) x,µ

which involves the generator at time t only. Note that the differential condition (4.3) and the flow equation (3.2) imply Eq. (4.1), i.e. it suffices to find a generator Z t (U ) that satisfies Eq. (4.3).

4.2. Existence of trivializing flows. Equation (4.3) is an inhomogeneous linear partial differential equation for the generator Z t (U ). Since it is a scalar equation, one expects that there are many solutions. In the following, the solution will be obtained in the form a [Z t (U )]a (x, µ) = −∂x,µ St (U ),

(4.4)

where the action St (U ) is to be determined. When inserted in Eq. (4.3), the ansatz (4.4) leads to the Laplace equation Lt St = S + C˙t ,

a a a a −∂x,µ ∂x,µ + t ∂x,µ S ∂x,µ Lt = x,µ

(4.5) (4.6)


905

(for simplicity, the argument U is now often omitted). The operator Lt is elliptic and symmetric with respect to the scalar product (φ, ψ) = D[U ] e−t S(U ) φ(U )∗ ψ(U ). (4.7) Lt has therefore a complete set of differentiable eigenfunctions and a purely discrete spectrum with no accumulation points (see ref. [8], Sect. 1.6, for example). Moreover, since

a a (φ, Lt φ) = ∂x,µ φ, ∂x,µ φ ≥ 0, (4.8) x,µ

the function φ(U ) = 1 is the only zero mode of Lt and all other eigenfunctions have eigenvalues separated from the origin by a strictly positive spectral gap. Now if one chooses Ct to be such that C˙ t = −(1, S)/(1, 1),

(4.9)

the zero-mode component is removed from the right-hand side of Eq. (4.5) and ˙ St = L−1 t (S + C t )

(4.10)

is then a well-defined expression that solves the equation. The differentiability of St with respect to t and U essentially follows from the ellipticity of Lt (Appendix E). A constructive proof of the existence of trivializing flows has thus been given. 4.3. Expansion in powers of t. The solution (4.10) is well defined but still quite implicit since it involves the inverse of an operator acting on functions of the gauge field. In this and the next subsection, the solution is worked out analytically in powers of t. When the series St =

∞

tk S (k)

(4.11)

k=0

is inserted in Eq. (4.5), the matching of the terms of a given order in t leads to the recursion L0 S (0) = S + C˙ (0) , a a (k−1) ∂x,µ S ∂x,µ + C˙ (k) , L0 S (k) = − S

(4.12) k = 1, 2, . . . ,

(4.13)

x,µ

for the actions S (k) . The Laplacian L0 coincides with the colour-electric part of the Hamilton operator in lattice gauge theory in 4 + 1 dimensions. In particular, its eigenfunctions are products of SU(3) representation functions of the link variables. Sums of Wilson loops and products of Wilson loops, for example, are eigenfunctions of L0 or can easily (algebraically) be decomposed into eigenfunctions. The solution of the recursion, S (0) = L−1 0 S, S (k) = −L−1 0

(4.14) x,µ

a a (k−1) ∂x,µ S ∂x,µ , S

k = 1, 2, . . . ,

(4.15)

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M. Lüscher

(1)

(2)

(3)

(5)

(6)

(7)

(4)

Fig. 2. Classes of loops and pairs of loops contributing to S (1) in the Wilson theory. The loops 5 − 7 reside on a single plaquette of the lattice. All other loops occupy two plaquettes which can lie in a plane or be at right angles in three dimensions

is thus obtained in the form of sums of Wilson loops and products of Wilson loops. Note that, as already mentioned in Subsect. 4.2, the constant function is the only zero mode of L0 . Since the smallest non-zero eigenvalue of L0 is 43 , the right-hand sides of Eqs. (4.14), (4.15) are therefore unambiguously determined up to an irrelevant additive constant. 4.4. Calculation of S (0) and S (1) in the Wilson theory. For illustration, the first two terms of the series (4.11) are now worked out explicitly for the plaquette action [9] Sw (U ) = − 16 β W0 (U )

(4.16)

(where β = 6/g02 denotes the inverse gauge coupling). A short calculation, using the completeness relation (A.5), shows that the leading term is 1 βW0 = S (0) = − 32

3 16 Sw .

(4.17)

To this order and up to a rescaling of the time parameter t, the trivializing flow in the Wilson theory thus coincides with the Wilson flow. The expression to be worked out at the next order is a a 1 2 −1 β L0 ∂x,µ W0 ∂x,µ W0 . (4.18) S (1) = − 192 x,µ

The Wilson loops and products of Wilson loops that can occur at this point derive from the contractions of two plaquette loops with a common link. Altogether there are then seven classes Ci , i = 1, . . . , 7, of loops and pairs of loops to consider (see Fig. 2). By summing the traces of the associated Wilson loops, each class Ci defines an action tr{U (C)} if i = 1, 2, 5, (4.19) Wi = C∈Ci

Wi =

{C,C }∈Ci

tr{U (C)}tr{U (C )} if i = 3, 4, 6, 7,

(4.20)


907

where U (C) denotes the ordered product of the link variables along the loop C. The sums in these equations extend over all possible positions of the loops on the lattice. Loops with opposite orientation are considered to be different and are both included in the sums. Using the identity (A.5) again, some algebra now yields

a a ∂x,µ W0 ∂x,µ W0 = W1 − W2 − 13 W3 + 13 W4 − 2W5 + 23 W6 − 43 W7 (4.21)

x,µ

up to an additive constant. Furthermore, L0 W1 = 8W1 , L0 W2 =

31 3 W2

(4.22) + W4 ,

(4.23)

L0 W3 = 11W3 − W1 ,

(4.24)

L0 W4 =

31 3 W4

+ W2 ,

(4.25)

L0 W5 =

28 3 W5

+ 4W6 ,

(4.26)

L0 W6 =

28 3 W6

+ 4W5 ,

(4.27)

L0 W7 = 12W7 + constant.

(4.28)

In the subspace of these functions, the operator L0 can be easily inverted and the result 1 2 4 12 1 5 3 β − 33 W1 + 119 W2 + 33 W3 − 119 W4 + 10 W5 S (1) = 192 (4.29) − 15 W6 + 19 W7 is thus obtained. Note that the smallness of the numerical coefficients in this formula is balanced, to some extent, by the number of loops per lattice point in the classes Ci (which are equal to 120, 12 and 6 for i = 1, . . . , 4, i = 5, 6 and i = 7 respectively).

4.5. Miscellaneous remarks. (a) Higher orders. The actions S (k) , k ≥ 2, can be computed algebraically following the steps taken in the previous subsection. Since all loops generated by contracting a plaquette loop with the loops at order k − 1 must be considered, the work required for the calculation is however rapidly growing with k. (b) Locality and convergence. The series (4.11) is an expansion in local terms whose footprint on the lattice increases proportionally to the order k. At the values of t, where the expansion converges, the action St is then guaranteed to be local as well. The norm estimates in Appendix E imply a lower bound on the convergence radius of the series, but this bound is rather poor and vanishes in the infinite-volume limit. It seems nevertheless plausible that the series has a non-zero convergence radius in this limit, because the inverse of the operator Lt in Eq. (4.10) is likely to remain bounded in a complex neighbourhood of t = 0. An analysis that takes the locality properties of Lt into account will however be required to show this.

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M. Lüscher

(c) Truncation of the expansion. If all terms in the series (4.11) of order k ≥ n are dropped, one obtains an approximately trivializing flow that satisfies Eq. (4.2) up to an additive correction of order t n+1 . An at least partial cancellation of the action is achieved in this case as long as t is sufficiently small for the correction to be strongly suppressed. (d) Smoothing property. The Wilson flow satisfies d dt Sw (Ut )

3 = − 16

a a ∂x,µ Sw (U )∂x,µ Sw (U ) U =U ≤ 0 x,µ

t

(4.30)

and therefore lowers the Wilson action as t increases. To leading order in t, the trivializing flow constructed in this section for the Wilson theory thus has a smoothing effect on the gauge field. On the other hand, if the flow is followed in the reverse direction, the gauge field tends to become rougher. (e) Topological charge sectors. In lattice QCD, the topological (instanton) sectors are not a property of the field manifold alone, but are expected to emerge dynamically when the continuum limit is approached. The fact that trivializing maps completely “straighten out” the sectors is therefore not in conflict with the topological properties of the field space. (f) Renormalization group. By composing the trivializing map U = F1 (V ) in the Wilson theory with its inverse at another value of the gauge coupling, one obtains a group of transformations whose only effect on the action is a shift of the coupling. The locality properties of these transformations are not transparent, however, and could be quite different from the ones of a Wilsonian “block spin” transformation.

5. Numerical Integration of the Wilson Flow The discussion in Sect. 1 now suggests to combine the HMC algorithm with the field transformations generated by the trivializing flow constructed in the previous section. In particular, if the Wilson gauge action is used, the transformations generated by the Wilson flow may lead to an algorithm with improved sampling efficiency.

5.1. Forward integration. There is a wide range of numerical integration methods that can in principle be used to integrate the Wilson flow (see ref. [10], for example). The Euler scheme discussed in the following performs the integration in time steps of size and updates the link variables one after another according to U (x, µ) → U (x, µ) = e[Z (U )](x,µ) U (x, µ),

(5.1)

a [Z (U )](x, µ) = T a ∂x,µ W0 (U ).

(5.2)

where

Starting from the gauge field Ut at time t, the field at time t + is thus obtained by running through all links (x, µ) on the lattice and updating the link variable residing there. Note that the ordering of the links matters, since the old value of U (x, µ) is replaced by the new one before going to the next link.


909

The generator of the flow is explicitly given by [Z (U )](x, µ) = − P U (x, µ)U (x + µ, ˆ ν)U (x + νˆ , µ)−1 U (x, ν)−1 ν=µ

+ U (x, µ)U (x + µˆ − νˆ , ν)−1 U (x − νˆ , µ)−1 U (x − νˆ , ν) ,

(5.3)

where µˆ denotes the unit vector in direction µ and P{M} = 21 (M − M † ) − 16 tr(M − M † )

(5.4)

projects any 3 × 3 matrix M to su(3). The Euler integration of the Wilson flow thus amounts to applying a number of “stout smearing” steps [11], except that the link variables are here updated one by one rather than all at once. 5.2. Backward integration. The application of n Euler sweeps maps the initial field V = U0 to the field U = Un at time t = n. If t is held fixed and n is taken to infinity, this map converges to the transformation obtained by integrating the Wilson flow exactly. However, the HMC algorithm may potentially be combined with the map defined by the Euler integrator at fixed n and . For this proposition to be a viable option, the transformation must be invertible, i.e. one must be able to trace back the Euler integration by inverting the link update steps one by one in the reverse order. The question is thus whether Eq. (5.1) has a unique solution U (x, µ) given U (x, µ) (and keeping all other link variables fixed). As explained in Appendix D, the answer is affirmative, for arbitrary values of the field variables, if || < 18 .

(5.5)

Moreover, in this range of , the solution U (x, µ) = e− X ∗ U (x, µ) can be obtained through the fixed-point iteration X 0 = 0, X n+1 = {[Z (U )](x, µ)}U (x,µ)=e− X n U (x,µ) ,

(5.6) n = 0, 1, 2, . . . ,

(5.7)

which converges to X ∗ at an exponential rate. In Appendix D it is also shown that the Jacobian of the transformation (5.1) is strictly positive in the range (5.5). The field transformations obtained through the Euler integration of the Wilson flow are therefore orientation-preserving diffeomorphisms of the field manifold, as required for acceptable maps of field space. 5.3. Jacobian matrix of the Euler integrator. The Euler step (5.1) amounts to applying the field transformation [Z (U )](x,µ) e U (x, µ) if (y, ν) = (x, µ), (5.8) [Ex,µ (U )](y, ν) = U (y, ν) otherwise, to the current gauge field. An Euler sweep is then the composition product of these transformations over all links (x, µ). It is straightforward to show that the Jacobian matrix of a composed transformation is the product of the Jacobian matrices of the factors. The

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M. Lüscher

Jacobian matrix of the Euler integrator is therefore an ordered product of the Jacobian matrices c (5.9) [Ex,µ,∗ (U )](y, ν; z, ρ)ac = −2 tr ∂z,ρ [Ex,µ (U )](y, ν)[Ex,µ (U )](y, ν)−1 T a of the one-link transformations (5.8). The matrix (5.9) can be expressed through the derivative c [Z ∗ (U )](y, ν; z, ρ)bc = ∂z,ρ [Z (U )]b (y, ν)

of the generator of the Wilson flow. Explicitly, one finds that [Ex,µ,∗ (U )](y, ν; z, ρ)ac = δ ac δ yz δνρ + δx y δµν (eAd X − 1)ac δ yz δνρ + J (−X )ab [Z ∗ (U )](y, ν; z, ρ)bc

X =[Z (U )](x,µ)

(5.10)

, (5.11)

where use was made of the SU(3) formulae listed in Appendices A and B.

5.4. Jacobian of the integrated transformations. The Euler integrator generates a sequence of fields V = U0 → U → U2 → . . . → Un = U

(5.12)

by sweeping through the lattice n times and updating the link variables one by one in a specified order. In the following, the intermediate field configurations obtained starting from Uk and updating the link variables on all links that come before (x, µ) will be denoted by Uk,[x,µ] . In particular, Uk,[x,µ] = Uk if (x, µ) is the first link and Uk,[y,ν] = Ex,µ (Uk,[x,µ] )

(5.13)

if (y, ν) follows (x, µ) in the chosen link order. Since the transformation V → U = Fn (V ) is a composition product of one-link update steps, its Jacobian det Fn,∗ (V ) =

n−1

det Ex,µ,∗ (Uk,[x,µ] )

(5.14)

k=0 x,µ

factorizes into the product of the Jacobians of the steps. The latter coincide with the determinants of certain real 8 × 8 matrices given explicitly in Appendix C. 6. Proposed Simulation Algorithm for Lattice QCD With respect to the QCD simulation algorithms used to date, the combination of the transformations obtained through the Euler integration of the Wilson flow and the HMC algorithm is expected to sample the topological sectors more quickly and to be generally more efficient. The use of the Wilson flow is suggested if the gauge action coincides with the Wilson plaquette action. For other actions, the appropriate flow can be easily constructed following the lines of Sect. 4.


911

V = U0

Uε U2ε U = U 3ε Fig. 3. The proposed algorithm evolves the field V using the standard HMC algorithm (thick line). The force that drives the molecular-dynamics evolution is obtained by forward integration of the Wilson flow and subsequent backward propagation of the derivatives of the action and the Jacobian of the flow (n = 3 in this figure)

6.1. Choice of the parameters. The parameters of the Euler integrator are the integration step size and the number n of Euler sweeps that are applied. One also needs to choose a definite ordering of the links of the lattice. The step size should be positive and not larger than, say, 1/16 so that the invertibility of the Euler integrator is guaranteed within a safe margin. Some tuning of the integration time n will certainly be required in order to maximize the efficiency of the algorithm. Note that the unit of time differs from the one used in Subsect. 4.4, i.e. setting t = 1 there corresponds to an integration time n = β/32. The ordering of the links can be chosen arbitrarily. One may, for example, first visit the links (x, 0) on all even points x, then the links (x, 0) on all odd points, then the links (x, 1) on all even points, and so on. This ordering is well suited for parallel processing, since the link variables in a given direction on the even (odd) sites are decoupled from one another and can therefore be updated in parallel.

6.2. Force calculation. At the beginning of an update cycle, the current gauge field U −1 (U ) by applying n backward Euler sweeps to U . is transformed to the field V = Fn The force that drives the molecular-dynamics evolution of V is then given by c S(Fn (V )) − ln det Fn,∗ (V ) , F(z, ρ)c = ∂ˆ z,ρ

(6.1)

where the action S(U ) now includes the usual sea-quark pseudo-fermion actions (for simplicity, the dependence on the pseudo-fermion fields is suppressed). Each time the force is to be calculated, the current field V must be transformed to U again by applying n forward Euler sweeps (see Fig. 3). The fields U0 , U , . . . , Un generated in this process should be stored in memory so that they will be available when the force is propagated from U to V . Note that the intermediate fields Uk,[x,µ] (y, ν) = are then also available.

Uk (y, ν) Uk+ (y, ν)

if (y, ν) ≥ (x, µ), otherwise,

(6.2)

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M. Lüscher

The factorization (5.14) of the Jacobian implies c S(Un ) − F(z, ρ)c = ∂ˆ z,ρ

n−1

c ln det Ex,µ,∗ (Uk,[x,µ] ). ∂ˆ z,ρ

(6.3)

k=0 x,µ

Moreover, c ∂ˆ z,ρ S(Un ) =

y,ν

a ∂ y,ν S(U )

U =Un

Fn,∗ (y, ν; z, ρ)ac ,

(6.4)

and there is a similar formula for the other terms in Eq. (6.3) involving the Jacobian matrices of the transformations V → Uk,[x,µ] . All these matrices, as well as the one in Eq. (6.4), are products of the Jacobian matrices of the one-link transformations (5.8). The force can therefore be computed recursively as follows: c S(U ). 1. Set U = Un and F(z, ρ)c = ∂z,ρ 2. For k from n − 1 to 0, run through all links (x, µ) in reverse order, set U = Uk,[x,µ] and update the force according to c F(z, ρ)c → F(y, ν)a [Ex,µ,∗ (U )](y, ν; z, ρ)ac − ∂z,ρ ln det Ex,µ,∗ (U ). (6.5) y,ν

Note that the field U backtracks the forward integration of the Wilson flow in the course of the recursion. Since the Jacobian matrix Ex,µ,∗ (U ) differs from unity only on the links sharing a plaquette with (x, µ), the total computational effort required for the propagation (6.5) of the force is expected to be similar to the one required for n applications of a nearest-neighbour gauge-covariant difference operator to the force field. 6.3. Domain-decomposed algorithm. Field transformations can also be combined with the DD-HMC algorithm [12]. Only the so-called active link variables are transformed in this case, but the transformations may depend on the inactive field components. In the pure gauge theory, it is then again possible to construct trivializing flows. They operate on the active link variables and contract the gauge action in each domain to a constant (i.e. an expression depending on the inactive link variables only). These flows are not the same as the ones constructed in Sect. 4, but can be obtained by solving the differential equations derived there. In particular, the trivializing flow for the Wilson action is, to leading order, a slightly modified Wilson flow, where the plaquettes containing ν active links are given the weight 4/ν. 7. Concluding Remarks Trivializing maps and flows in field space have here been discussed with a particular application in mind. The underlying concepts are fairly general, however, and may have other uses in rigorous constructive work, numerical perturbation theory or in connection with renormalizable smoothing techniques, for example. Whether the proposed combination of the Wilson flow and the HMC algorithm does indeed sample the topological sectors in lattice QCD more efficiently than the simulation algorithms used so far remains to be determined. As the lattice spacing is taken to smaller and smaller values, the quark fields may eventually have to be included in the flow and


913

perhaps also the next-to-leading order correction discussed in Sect. 4. Trivializing maps in the presence of matter fields is, in any case, a subject that deserves to be studied in its own right. Acknowledgements. In the course of this work, I profited from many discussions with Filippo Palombi and Stefan Schaefer of various questions related to the slow topology-switching in current lattice QCD simulations. I also wish to thank Stefan Schaefer, Rainer Sommer and Francesco Virotta for sharing some of their simulation results before publication.

Appendix A. SU(3) Notation A.1. Group generators. The Lie algebra su(3) of SU(3) may be identified with the space of all anti-hermitian traceless 3 × 3 matrices. With respect to a basis T a , a = 1, . . . , 8, of such matrices, the elements X ∈ su(3) are given by X = XaT a,

(A.1)

where (X 1 , . . . , X 8 ) ∈ R8 (repeated group indices are automatically summed over). The generators T a are assumed to satisfy the normalization condition tr{T a T b } = − 21 δ ab .

(A.2)

The structure of the Lie algebra is then encoded in the commutators [T a , T b ] = f abc T c ,

(A.3)

while the completeness of the generators implies {T a , T b } = − 13 δ ab + id abc T c , a a Tαβ Tγ δ = − 21 δαδ δβγ − 13 δαβ δγ δ .

(A.4) (A.5)

It follows from these equations that the structure constants f abc and the tensor d abc are both real. Moreover, f abc is totally anti-symmetric in the indices and d abc totally symmetric and traceless. A.2. Adjoint representation. The representation space of the adjoint representation of su(3) is the Lie algebra itself, i.e. the elements X of su(3) are represented by linear transformations Ad X : su(3) → su(3),

(A.6)

Ad X · Y = [X, Y ] for all Y ∈ su(3).

(A.7)

The action of Ad X on the group generators is given by Ad X · T b = T a (Ad X )ab ,

(A.8)

(Ad X )ab = − f abc X c

(A.9)

where

is a real antisymmetric 8 × 8 matrix.

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M. Lüscher

A.3. Matrix norms. The natural scalar product in su(3) is (X, Y ) = X a Y a = −2 tr{X Y }.

(A.10)

is a possible definition of the norm of X ∈ su(3). In particular, X = (X, Another useful matrix norm derives from the square norm X )1/2

v 2 = {|v1 |2 + |v2 |2 + |v3 |2 }1/2

(A.11)

of complex colour vectors v. If A is any complex 3 × 3 matrix, one defines A 2 = max Av 2 .

(A.12)

v 2 =1

This norm satisfies A + B 2 ≤ A 2 + B 2 ,

AB 2 ≤ A 2 B 2 ,

(A.13)

for all matrices A, B. Moreover, if A is hermitian or antihermitian, A 2 is equal to the maximum of the absolute values of its eigenvalues. Appendix B. Properties of the SU(3) Exponential Function B.1. Lipschitz bound. For any X, Y ∈ su(3), the relation e X − eY 2 = 1 − e−X eY 2 is unitary. Using the identity 1 1 − e−X eY = ds e−s X (X − Y )esY

follows from the fact that

(B.1)

eX

(B.2)

0

and the subadditivity (A.13) of the norm, the Lipschitz bound e X − eY 2 ≤ X − Y 2

(B.3)

is then obtained. B.2. Differential of the exponential map. Let X be an element of the Lie algebra su(3). A linear mapping J (X ) : su(3) → su(3) is then defined by for all Y ∈ su(3). (B.4) J (X ) · Y = e−X dtd e X +tY t=0

J (X ) is referred to as the differential of the exponential map. Scaling the exponents by a parameter s, as above, one obtains the representation 1 J (X ) · Y = ds e−s X Y es X (B.5) 0

and thus the expansion J (X ) = 1 +

∞

(−1)k (k+1)! (Ad

k=1

which is absolutely convergent for any X ∈ su(3).

X )k

(B.6)


915

The action of J (X ) on the group generators T a is given by J (X ) · T b = T a J (X )ab ,

(B.7)

where J (X )ab is a real 8 × 8 matrix. Note that J (X )Ad X = 1 − e−Ad X .

J (X )T = J (−X ) = eAd X J (X ),

(B.8)

Moreover, Eq. (B.5) implies J (X ) · Y 2 ≤ Y 2

(B.9)

for all X, Y ∈ su(3). Appendix C. Jacobian of the Euler Step The Jacobian matrix (5.11) of the Euler step is equal to unity except for some non-zero elements along the row (y, ν) = (x, µ). Its determinant therefore coincides with the determinant of the (x, µ; x, µ)-element Aac = (eAd X )ac + J (−X )ab [Z ∗ (U )](x, µ; x, µ)bc

X =[Z (U )](x,µ)

(C.1)

of the matrix. The derivative [Z ∗ (U )](x, µ; x, µ)bc can be worked out explicitly in terms of the plaquette sum M=

U (x, µ)U (x + µ, ˆ ν)U (x + νˆ , µ)−1 U (x, ν)−1 ν=µ

+ U (x, µ)U (x + µˆ − νˆ , ν)−1 U (x − νˆ , µ)−1 U (x − νˆ , ν) .

(C.2)

A few lines of algebra then lead to the expression Aac = B ac + 21 C ab id bcd tr{T d (M + M † )} − 13 δ bc tr{M + M † } ,

(C.3)

where B = 21 (eAd X + 1),

(C.4)

C = J (−X ),

(C.5)

X = −P{M}.

In particular, det A = 1 − 43 tr{M + M † } + O( 2 ), as expected from Eq. (3.9).

(C.6)

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M. Lüscher

Appendix D. Inversion of the Euler Step D.1. Basic norm bounds. For any complex 3 × 3 matrix M, the inequality P{M} 2 ≤ 43 M 2

(D.1)

can be established in a few lines. One first observes that 1 2 (M

− M † ) = AD A−1 ,

A ∈ SU(3),

(D.2)

where D is a diagonal matrix with diagonal elements λ1 , λ2 , λ3 . Setting λ¯ =

1 3

3

λk ,

(D.3)

k=1

the estimates

P{M} 2 = max λk − λ¯ k

≤

4 3

max |λk | = 23 M − M † 2 ≤ 43 M 2 k

(D.4)

then show that the inequality (D.1) holds for all matrices M. An immediate consequence of the bound (D.1) and the Lipschitz bound (B.3) is that P{A(e X − eY )B} 2 ≤ 43 X − Y 2

(D.5)

for all A, B ∈ SU(3) and X, Y ∈ su(3). D.2. Solution of Eq. (5.1). For a given link (x, µ) and any fixed U (x, µ) ∈ SU(3), the function f (X ) = {[Z (U )](x, µ)}U (x,µ)=e− X U (x,µ)

(D.6)

maps X ∈ su(3) back to su(3). Recalling Eq. (5.3), the inequality (D.5) immediately implies that f (X ) − f (Y ) 2 ≤ k X − Y 2 ,

k = 8||.

(D.7)

The function f (X ) is therefore a strict contraction if the integration step size is in the range (5.5) (which is assumed to be the case from now on). It is not difficult to prove that strict contractions in a complete metric space have a unique fixed point (see ref. [13], Theorem V.18, for example). In the present case, the fixed point X ∗ can be computed by noting that the sequence X 0 = 0, X 1 , X 2 , . . . generated through the recursion X n+1 = f (X n ) satisfies X n − X ∗ 2 ≤ k X n−1 − X ∗ 2 ,

(D.8)

and therefore rapidly converges to X ∗ . The matrix U (x, µ) = e− X ∗ U (x, µ)

(D.9)

then provides a solution of Eq. (5.1). Moreover, there is no other solution, because the fixed point X ∗ of f (X ) is unique.


917

D.3. Positivity of the Jacobian. In order to prove that the determinant of the Jacobian matrix (C.1) is positive at all in the range (5.5), it suffices to show that the matrix has no zero mode, i.e. that Aac Y c = 0

(D.10)

implies Y = 0. To this end, Eq. (D.10) is first written in the form Y a = − J (X )ab W b ,

(D.11)

where X = [Z (U )](x, µ),

W b = [Z ∗ (U )](x, µ; x, µ)bc Y c .

Recalling Eq. (5.10), the formula W = lim 1t [Z (U )](x, µ)|U (x,µ)→etY U (x,µ) − [Z (U )](x, µ) t→0

(D.12)

(D.13)

may then be derived from which one infers that W 2 ≤ 8 Y 2 .

(D.14)

The inequality (D.5) has here been used again and also the fact that the norm is a continuous map from su(3) to R. The combination of Eq. (D.11) and the bounds (B.9) and (D.14) now implies Y 2 ≤ k Y 2 and thus Y = 0. Appendix E. Differentiability of St The action St solves an elliptic partial differential equation with smooth coefficients and is therefore guaranteed (by elliptic regularity) to be a differentiable function of the gauge field U . In this Appendix, the simultaneous differentiability in the time t is established by expanding St in powers of t − t0 around any fixed time t0 . Using Sobolev norms, the expansion can be shown to converge if t is sufficiently close to t0 . The pointwise uniform convergence of the series and its derivatives, and therefore the differentiability of St , then follows from Sobolev’s lemma. E.1. Sobolev spaces on the field manifold. The definition of the Sobolev spaces on a compact manifold is quite involved and will not be reviewed here. An introduction to the subject and the proof of all statements made in this subsection is given in the first three sections of ref. [8], for example. Let C ∞ be the space of differentiable functions on the field manifold and Hk the associated Sobolev space of order k ∈ Z. The latter is the completion of C ∞ with respect to a certain norm · k . A characteristic feature of these norms is that the bounds Dφ k ≤ ck φ k+ p

(E.1)

hold for all φ ∈ C ∞ and any differential operator D of order p with smooth coefficients, where the constants ck depend on D but not on φ. Such differential operators thus extend to bounded linear operators from Hk+ p to Hk . Moreover, φ k ≤ φ l if k < l and therefore Hl ⊂ Hk .

918

M. Lüscher

A fairly concrete description of the Sobolev space Hk can be given when k is even and non-negative. For any t ∈ R and j = 0, 1, 2, . . ., another norm φ t, j of φ ∈ C ∞ may be defined through j

φ t, j = φ + Lt φ ,

(E.2)

where · is the norm associated to the scalar product (4.7). The fact that Lt is a second-order elliptic differential operator then implies φ 2 j ≤ at, j φ t, j ,

φ t, j ≤ bt, j φ 2 j

(E.3)

for some constants at, j , bt, j . In particular, H2 j is the completion of C ∞ with respect to the norm · t, j . E.2. Properties of the inverse of Lt . Let Pt be the orthogonal projector to the zero mode of Lt in the Hilbert space with scalar product (4.7). Its action on any function φ ∈ C ∞ is given by Pt φ = (1, φ)/(1, 1).

(E.4)

Note that Pt projects φ to a constant, but the constant depends on the definition of the scalar product (· , ·) and therefore on t. The action of the inverse Gt of Lt on φ is then determined by the equations Lt Gt φ = (1 − Pt )φ,

Pt Gt φ = 0.

(E.5)

As already mentioned in Subsect. 4.2, the ellipticity of Lt and the absence of further zero modes imply that these equations have, for any fixed t, a unique differentiable solution Gt φ. Evidently, St = Gt S. Since Lt has a spectral gap in the subspace orthogonal to the zero mode, Gt is a bounded operator with respect to the norm · . As a consequence, there exists a constant gt such that Gt φ t, j ≤ gt φ t, j−1

(E.6)

for all j = 1, 2, . . . and all φ ∈ C ∞ . Moreover, recalling the inequalities (E.3), one concludes that Gt φ 2 j ≤ gt, j φ 2 j−2

(E.7)

for some other constants gt, j . E.3. Expansion of St . For any fixed time t0 , the operator Lt may be decomposed according to

a a Lt = Lt0 + (t − t0 ) L , ∂x,µ L = S ∂x,µ . (E.8) x,µ

Each term in the power series ψt =

∞ n=0

n (t0 − t)n Gt0 L Gt0 S

(E.9)


919

is then a well-defined differentiable function of both t and U . Moreover, since Gt0 L φ 2 j ≤ gt0 , j L φ 2 j−2 ≤ d j gt0 , j φ 2 j−1 ≤ d j gt0 , j φ 2 j ,

(E.10)

all j = 1, 2, . . . and some constants d j , it is clear that the series and all its derivatives with respect to t converge uniformly in the norm · 2 j if t is sufficiently close to t0 . Sobolev’s lemma (statement (c) in Lemma 1.3.5 of ref. [8]) then implies that the series converges pointwise and uniformly, together with all its derivatives in t and its derivatives in U up to some order proportional to j. If t is in a neighbourhood of t0 , where the convergence of the series and its derivatives up to order l ≥ 2 is guaranteed, the action of the operator Lt and the summation may be interchanged and one finds that Lt ψt = S −

∞

n (t0 − t)n Pt0 L Gt0 S = (1 − Pt ) S,

(E.11)

n=0

the second equality being implied by the identities Pt Lt ψt = 0 and Pt Pt0 = Pt0 . As a consequence, St , (1 − Pt ) ψt = Gt S =

(E.12)

which shows that St is, in the specified neighbourhood of t0 , l times continuously differentiable with respect to t and U . Since t0 and l can be chosen arbitrarily, the simultaneous differentiability of St is thus guaranteed at all times t and to all orders. References 1. Nicolai, H.: Phys. Lett. 89B, 341 (1980); Nucl. Phys. B176, 419 (1980) 2. Moser, J., Zehnder, E.J.: Notes on Dynamical Systems. Courant Lecture Notes, Vol. 12, Providence, RI: Amer. Math. Soc. 2005 3. Del Debbio, L., Panagopoulos, H., Vicari, E.: JHEP 0208, 044 (2002) 4. Schaefer, S., Sommer, R., Virotta, F.: Investigating the critical slowing down of QCD simulations. Talk given at the XXVII International Symposium on Lattice Field Theory, Beijing, China (July 2009), arXiv:0910.1465 [hep-lat], to appear in the Proceedings 5. Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Phys. Lett. B195, 216 (1987) 6. Duane, S., Kenway, R., Pendleton, B.J., Roweth, D.: Phys. Lett. B176, 143 (1986); Duane, S., Pendleton, B.J.: Phys. Lett. B206, 101 (1988) 7. Arnold, V.I.: Ordinary Differential Equations. 3rd ed., Berlin: Springer-Verlag, 2008 8. Gilkey, P.B.: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. 2nd ed. Boca Raton, FL: CRC Press, 1995 9. Wilson, K.G.: Phys. Rev. D10, 2445 (1974) 10. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. 2nd ed., Berlin: Springer, 2006 11. Morningstar, C., Peardon, M.: Phys. Rev. D69, 054501 (2004) 12. Lüscher, M.: Comput. Phys. Commun. 165, 199 (2005) 13. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. I, New York: Academic Press, 1972 Communicated by M. Salmhofer

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