Quantum Mechanics in Phase Space: An Overview with Selected Papers

OUANTUM MECHANICS IN PHASE SPACE An Overview with Selected Papers

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World Scientific Series in 20th Century Physics

vol,34

OUANTUM MECHANICS IN PHASE SPACE I

An Overview with Selected Papers

Editors

Cosmas K. Zachos Argonne National Laborato y , USA

David B. Fairlie University of Durham, UK

Thomas L. Curtright University of Miami, USA

wp World Scientific N E W JERSEY

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.

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V

CONTENTS

Preface

Overview of Phase-Space Quantization

vii

1

1 Introduction

1

2 The Wigner Function

3

3 Solving for the Wigner Function

5

4 The Uncertainty Principle

9

5 Ehrenfest’s Theorem

10

6 Illustration: The Harmonic Oscillator

11

7 Time Evolution

13

8 Nondiagonal Wigner Functions

16

9 Stationary Perturbation Theory

17

10 Propagators

18

11 Canonical Transformations

19

12 The Weyl Correspondence

21

13 Alternate Rules of Association

24

14 The Groenewold-van Hove Theorem and the Uniqueness of MBs and *-Products

25

15 Omitted Miscellany

26

16 Selected Papers: Brief Historical Outline

28

References

31

vi

List of Selected Papers

39

Index

43

vii

PREFACE Wigner’s quasi-probability distribution function in phase space is a special (WeylWigner) representation of the density matrix. It has been useful in describing transport in quantum optics; nuclear physics; and quantum computing, decoherence, and chaos. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter century. It furnishes a third, alternative, formulation of quantum mechanics, independent of the conventional Hilbert space, or path integral formulations. In this logically complete and self-standing formulation, one need not choose sides between coordinate and momentum space. It works in full phase-space, accommodating the uncertainty principle, and it offers unique insights into the classical limit of quantum theory. The variables (observables) in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, but are composed together in novel algebraic ways. This volume is a selection of 23 useful papers in the phase-space formulation, with an introductory overview which provides a trail-map to these papers and an extensive bibliography. (Still, the bibliography makes no pretense to exhaustiveness. An up-to-date database on the large literature of the field, with special emphasis on its mathematical and technical aspects, may be found at http://idefix.physik.uni-freiburg.de/Nstar/en/download.html) The overview collects often-used formulas and simple illustrations, suitable for applications to a broad range of physics problems, as well as teaching. It provides supplementary material that may be used for a beginning graduate course in quantum mechanics. D. Morrissey is thanked for the helpful comments and Prof Curtright would also like to express his thanks to Ms Diaz-Heimer. Errata and other updates to the book may be found on-line at ht tp://server.physics.miami.edu/Ncurtright/QMPS

C.K. Zachos, D. B. Fairlie, and T. L. Curtright

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OVERVIEW OF PHASE-SPACE QUANTIZATION 1

Introduction

There are at least three logically autonomous alternative paths to quantization. The first is the standard one utilizing operators in Hilbert space, developed by Heisenberg, Schrodinger, Dirac, and others in the 1920s. The second one relies on path integrals, and was conceived by Dirac [Dir33] and constructed by Feynman. The third one (the bronze medal!) is the phase-space formulation, based on Wigner's (1932) quasi-distribution function [Wig321 and Weyl's (1927) correspondence [Wey27] between quantum-mechanical operators in Hilbert space and ordinary c-number functions in phase space. The crucial composition structure of these functions, which relies on the *product, was fully understood by Groenewold (1946) [Gro46], who, together with Moyal (1949) [Moy49], pulled the entire formulation together. Still, insights into interpretation and a full appreciation of its conceptual autonomy took some time to mature wHith the work of, among others, Takabayasi [Tak54], Baker [Bak58], and Fairlie [Fai64]. This complete formulation is based on the Wigner function (WF), which is a quasiprobability distribution function in phase space: (1)

It is a generating function for all spatial autocorrelation functions of a given quantummechanical wave function $ ( x ) . More important, it is a special representation of the density matrix (in the Weyl correspondence, as detailed in Section 12). Alternatively, in a 2ndimensional phase space, it amounts to (2) where $ ( x ) = ( X I $ )

in the density operator p, (3)

There are several outstanding reviews on the subject: Refs. HOS84, Tak89, Ber80, BJ84, Lit86, deA98, Tat83, Coh95, KN91, Kub64, DeG74, KW90, Ber77, Lee95, DahOl, Sch02, DHSOO, CZ83, Gad95, HH02, Str57, McD88, Leo97, Sny80, Ba163, BFF78. Nevertheless, the central conceit of the present overview is that the above input wave functions may ultimately be bypassed, since the WFs are determined, in principle, as the solutions to suitable functional equations in phase space. Connections to the Hilbert space operator formulation of quantum mechanics may thus be ignored, in principle-even though they are provided in Section 12 for pedagogy and confirmation of the formulation's equivalence. One might then envision an imaginary world in which this formulation of quantum mechanics had preceded the conventional Hilbert-space formulation, and its own techniques and methods had arisen independently, perhaps out of generalizations of classical mechanics and statistical mechanics. 1

2

It is not only wave functions that are missing in this formulation. Beyond the ubiquitous (noncommutative, associative, pseudodifferential) operation, the +product, which encodes the entire quantum-mechanical action, there are no linear operators. Expectations of observables and transition amplitudes are phase-space integrals of c-number functions, weighted by the WF, as in statistical mechanics. Consequently, even though the W F is not positive-semidefinite (it can be, and usually is negative in parts of phase space [Wig32]), the computation of expectations and the associated concepts are evocative of classical probability theory. Still, telltale features of quantum mechanics are reflected in the noncommutative multiplication of such c-number phase-space functions through the *-product, in systematic analogy to operator multiplication in Hilbert space. This formulation of quantum mechanics is useful in describing quantum transport processes in phase space. Such processes are of importance in quantum optics [SchO2, Leo97, SMOO], nuclear and particle physics [BakGO, SP81, MM84, CC03, BJY041, condensed matter [MMP94, DBB02, KKFR89, BP96, KLO1, JBMOS], the study of semiclassical limits of mesoscopic systems [Imr67, OR57, Sch69, Ber77, KW87, OM95, MS95, MOT98, V089, V0781, and the transition to classical statistical mechanics [JD99, Fre87, BD98, Raj83, CV98, SMOO, FZOl, Za1031. It is the natural language to study quantum chaos and decoherence [JN90, ZP94, BC99, KZZ02, KJ99, Zu91, FBA96, Kol96, GH93, CL03, OCO3] (of utility in, e.g. quantum computing [BHP02]), and provides crucial intuition in quantum mechanical interference problems [Wis97], probability flows as negative probability backflows [BM94, FMSOO], and measurements of atomic systems [Smi93, Dun95, Lei96, KPM97, LvoOl, JS02, BHS02, Ber02, Cas911. The intriguing mathematical structure of the formulation is of relevance to Lie Algebras [FFZ89],martingales in turbulence [FanOS],and string field theory [BKMOS]. It has recently been retrofitted into M-theory advances linked to noncommutative geometry [SW99] (for reviews, see Refs. CasOO, HarOl, DNO1, HS02), and matrix models [TayOl, KS021; these apply space-time uncertainty principles [Pei33, Y089, JY98, SSTOO] reliant on the *-product. (Transverse spatial dimensions act formally as momenta, and, analogously to quantum mechanics, their uncertainty is increased or decreased inversely to the uncertainty of a given direction.) As a significant aside, the W F has extensive practical applications in signal processing, filtering, and engineering (time-frequency analysis) , since time and frequency constitute a pair of Fourier-conjugate variables just like the and p pair of phase space.a For simplicity, the formulation will be mostly illustrated for one coordinate and its conjugate momentum, but generalization to arbitrary-sized phase spaces is straightforward [DM86],including infinite-dimensional ones, namely scalar field theory [DitSO, Les84, Na97, CZ99, CPPO1, MM941: the respective WFs are simple products of single-particle WFs. "Thus, time-varying signals are best represented in a W F as time-varying spectrograms, analogously to a music score , i.e. the changing distribution of frequencies is monitored in time [BBL80, Wok97, QC96, MH97, Coh95, GroOl]: even though the description is constrained and redundant, it gives an intuitive picture of the signal that a mere time profile or frequency spectrogram fails to convey. Applications abound [CGBSl, Lou96, MH97] in bioengineering, acoustics, speech analysis, vision processing, turbulence microstructure analysis, radar imaging, seismic data analysis, and the monitoring of internal combustion engine-knocking, failing helicopter component vibrations, and so on.

3

2

The Wigner Function

As already indicated, the quasi-probability measure in phase space is the WF,

f(z,p)=

27r

S d y $* (x

-

5.)

e-iyp$ (z

+ zy). ti

(4)

It is obviously normalized; J d p d z f ( a : , p ) = 1. In the classical limit, ti --+ 0, it would reduce to the probability density in coordinate space x, usually highly localized, multiplied by &functions in momentum: the classical limit is “spiky” and certain! This expression has more z - p symmetry than is apparent, as Fourier transformation to momentum-space wave-functions yields a completely symmetric expression with the roles of z and p reversed, and, upon rescaling of the arguments x and p , a symmetric classical limit. The WF is also manifestly real.b In addition, it is constrained by the Schwarz inequality 5 f ( z , p ) 5 .; Again, this bound disappears in the spiky classical limit. to be bounded, Respectively, p or z-projection leads to marginal probability densities: a spacelike shadow, J d p f(z,p) = p ( x ) , or else a momentum-space shadow, J d z f ( x , p ) = a ( p ) . Either is a bona-fide probability density, being positive-semidefinite. But neither can be conditioned on the other, as the uncertainty principle is fighting back: The W F f ( z , p ) itself can, and most often is negative in some areas of phase space [Wig32, HOS841, as is illustrated below, a hallmark of QM interference in this language. (In fact, the only pure state W F which is non-negative is the Gaussian [Hud74], a state of maximum entropy [Raj83].) The counter-intuitive “negative probability” aspects of this quasi-probability distribution have been explored and interpreted [Bar45, Fey87, BM941 (for a popular review, see LPM98), and negative probability flows amount to legitimate probability backflows in interesting settings [BM94]. Nevertheless, the WF for atomic systems can still be measured in the laboratory, albeit indirectly [Smi93, Dun95, Lei96, KPM97, LvoOl , BAD96, BHS02, Ber02, BRWK991, and reconstructed. Smoothing f by a filter of size larger than ti (e.g. convolving with a phase-space Gaussian) results in a positive-semidefinite function, i.e. it may be thought to have been coarsened to a classical“ distribution [Car76, Ste80, OW81, Raj831. Among real functions, the WFs constitute a rather small, highly constrained set. When is a real function f ( x , p ) a bona-fide Wigner function of the form (4)? Evidently, when its

-;

one space dimension, by virtue of nondegeneracy, $J has the same effect as $J*, and f turns out to be p-even, but this is not a property used here. CThisone is called the Husimi distribution [Tak89, TA991, and sometimes information scientists examine it on account of its non-negative feature. Nevertheless, it comes with a heavy price, as it needs to be “dressed” back to the W F for all practical purposes when expectation values are computed with it, i.e. it does not serve as an immediate quasi-probability distribution with no further measure (see Section 13). The negative feature of the W F is, in the last analysis, an asset, not a liability, and provides an efficient description of “beats” [BBL80, Wok97, QC96, MH97, Coh951; cf. Fig. 1. If, instead, strictly inequivalent (improper) expectation values were taken with the Husimi distribution without the requisite dressing of Section 13, i.e. as though it were a bonaifide probability distribution, such expectation values would reflect loss of quantum information: they would represent classically coarsened observables [WO87]. n

5 c-number functions. Such functions are often classical quantities but, in general, are uniquely associated with suitably ordered operators through Weyl’s correspondence rule [Wey27]. Given an operator ordered in this prescription,

1 (2d2

@ ( r , P ) = -JdrdCdxdp g(z,p)exp[iT(P- p ) + ia(F -

.>I ,

(7)

the corresponding phase-space function g(z,p ) (the “Weyl kernel function of the operator”) is obtained by PHP,

r-5.

(8)

That operator’s expectation value is then a “phase-space average” [Gro46, Moy491,

(@) = /dXdP f(z,PI S(Z,P).

(9)

The kernel function g(z,p ) is often the unmodified classical observable expression, such as a conventional Hamiltonian, H = p2/2m V(z), i.e. the transition from classical mechanics is straightforward. However, it contains h corrections when there are quantummechanical ordering ambiguities, such as in the observable kernel of the square of the angular momentum C C: This contains a term, -3h2/2, introduced by the Weyl ordering [She59, DS82, DS021, beyond the mere classical expression ( L 2 ) ,and accounts for the nontrivial angular momentum of the ground-state Bohr orbit. In such cases (including momentum-dependent potentials), even nontrivial O(h) quantum corrections in the kernel functions (which characterize different operator orderings) can be produced efficiently without direct, cumbersome consideration of operators [CZ02, Hie841. More detailed discussion of the Weyl and alternate correspondences is provided in Sections 12 and 13. In this sense, expectation values of the physical observables specified by kernel functions g ( z , p ) are computed through integration with the WF, in close analogy with classical probability theory, except for the non-positive-definiteness of the distribution function. This operation corresponds to tracing an operator with the density matrix (cf. Section 12).

+

3

Solving for the Wigner Function

Given a specification of observables, the next step is to find the relevant W F for a given Hamiltonian. Can this be done without solving for the Schrodinger wave functions $, i.e. not using Schrodinger’s equation directly? Indeed, the functional equations which f satisfies completely determine it. Firstly, its dynamical evolution is specified by Moyal’s equation. This is the extension of Liouville’s theorem of classical mechanics, for a classical Hamiltonian H ( z ,p), namely atf + {f,N} = 0, to quantum mechanics, in this language [Wig32, Moy491: (10)

5

6

where the *-product [Gro46] is (11)

The right-hand side of (10) is dubbed the “Moyal Bracket” (MB), and the quantum commutator is its Weyl correspondent. It is the essentially unique one-parameter (ti) associative deformation of the Poisson brackets of classical mechanics [Vey75, BFF78, FLS76, Ar83, Fle90, deW83, BCG97, TD971. Expansion in ti around 0 reveals that it consists of the Poisson bracket corrected by terms O(h). The equation (10) also evokes Heisenberg’s equation of motion for operators, except that H and f here are classical functions, and it is the *-product which enforces noncommutativity. This language makes the link between quantum commutators and Poisson brackets more transparent. Since the *-product involves exponentials of derivative operators, it may be evaluated in practice through translation of function arguments ( “Bopp shifts”), (12)

The equivalent Fourier representation of the *-product is [NeuSl, Bak581

(13)

An alternate integral representation of this product is [HOS84] (14)

which readily displays noncommutativity and associativity. * multiplication of c-number phase-space functions is in complete isomorphism to Hilbert-space operator multiplication [Gro46], (15)

The cyclic phase-space trace is directly seen in the representation (14) to reduce to a plain product, if there is only one * involved:

S

dpdx f * g

=

J

dpdx fg =

S

dpdx g * f .

(16)

Moyal’s equation is necessary, but does not suffice to specify the W F for a system. In the conventional formulation of quantum mechanics, systematic solution of time-dependent equations is usually predicated on the spectrum of stationary ones. Time-independent purestate Wigner functions *-commute with H , but clearly not every function *-commuting with H can be a bona-fide W F (e.g. any * function of H will *-commute with H ) . 6

7

Static WFs obey more powerful functional *-genvalue equations [Fai64] (also see Refs. Kun67, Coh76, Dah83):

iti =

+

f ( V )* H ( z , p ) = E f ( z , d

(17)

?

where E is the energy eigenvalue of fj$ = E$. These amount to a complete characterization of the WFs [CFZ98].

Lemma 1 For real functions f ( z , p ) , the Wigner form (4) for pure static eigenstates is equivalent to compliance with the *-genvalue equations (17) (8 and S parts). Proof

(18)

Action of the effective differential operators on $* turns out t o be null. Symmetrically,

.f * H

=E f(Z,P),

(19)

where the action on $ is now trivial. Conversely, the pair of *-eigenvalue equations dictate, for f ( x , p ) = Jdy e-iYpf(z, y)

,

(20)

+

Hence, real solutions of (17) must be of the form f = Jdy e-iyp$*(x - ?y)$(x ;y)/2", such that fj$ = E$. 0 Equation (17) lead to spectral properties for WFs [Fai64, CFZ981, as in the Hilbert space formulation. For instance, projective orthogonality of the * genfunctions follows from associativity, which allows evaluation in two alternate groupings:

f * H * g = E f f * g = E , f*g. 7

(21)

8

Thus, for Eg # E f , it is necessary that (22)

f*g=O.

Moreover, precluding degeneracy (which can be treated separately), choosing f = g above yields

f*H*f=Ef f*f=H*f*f, and hence f

*f

(23)

must be the stargenfunction in question,

f*f.(f.

(24)

Pure state fs then *-project onto their space. In general, it can -e shown [Tak54, CFZ981 that, for a pure state, fa

*f b =

1 Ja,b f a

(25)

'

The normalization matters [Tak54]: despite linearity of the equations, it prevents superposition of solutions. (Quantum mechanical interference works differently here, in comportance with density matrix formalism.) By virtue of (16), for different *-genfunctions, the above dictates that

J

dpdx f g

= 0.

(26)

Consequently, unless there is zero overlap for all such WFs, at least one of the two must go negative someplace to offset the positive overlap [HOS84, Coh951-an illustration of the feature of negative values. This feature is an asset and not a liability. Further, note that integrating (17) yields the expectation of the energy,

s

(27)

H(x,p)f (x,p)dxdp = E

Likewise,d note that integrating the above projective condition yields

s

dxdp f 2

1 =-

h

,

(28)

i.e. the overlap increases to a divergent result in the classical limit, a s the WFs grow increasingly spiky. dThis discussion applies to proper WFs, corresponding to pure states' density matrices. E.g. a sum of two WFs is not a pure state in general, and does not satisfy the condition (6). For such generalizations, the impurity is [Gro46] 1 - h ( f ) = s d x d p (f - h f 2 ) 2 0, where the inequality is only saturated into a n equality for a pure state. For instance, for w (fa fb)/2 with fa * f b = 0, the impurity is nonvanishing, d x d p (w - hw') = 1/2. A pure state affords a maximum of information, while the impurity is a measure of lack of information [Fan57, Tak541-it is the dominant term in the expansion of the quantum entropy around a pure state [Bra94].

s

+

8

9

4

The Uncertainty Principle

In classical (non-negative) probability distribution theory, expectation values of nonnegative functions are likewise non-negative, and thus result in standard constraint inequalities for the constituent pieces of such functions, e.g., moments of the variables. But it was just seen that for WFs which go negative for an arbitrary function g, (1gI2)need not be 2 0. This can be easily seen by choosing the support of g to lie mostly in those regions of phase-space where the W F f is negative. Still, such constraints are not lost for WFs. It turns out they are replaced by:

Lemma 2 (9**9) 1 0.

(29)

In Hilbert space operator formalism, this relation would correspond to the positivity of the norm. This expression is non-negative because it involves a real non-negative integrand for a pure state W F satisfying the above projective Condition", /dpdx(g**g)f = h

dxdp(g**g)(f*f) = h

s

dxdp(f*g*)*(g*f) = h

s

dxdplg*f12. (30) 0

To produce Heisenberg's uncertainty relation [CZOl], one only needs to choose g =a

+ bx + cp,

(31)

for arbitrary complex coefficients a , b, c. The resulting positive semi-definite quadratic form is then a*a+b*b(z*x) +c*c(p*p)

+ (a*b+b*a)(x)+(a*c+c*a)(p)

+c*b(p*x) +b*c(x*p) 2 0 , (32)

for any a, b, c. The eigenvalues of the corresponding matrix are then non-negative, and thus so must be its determinant. Given 2

x*x=x,

p*p=p2,

ih

ih

p*x=px--, 2

x*p=px+-, 2

(33)

and the usual

= ((x - (xc)>2),

@PI2

= ((P- (P)l2>,

(34)

this condition on the 3 x 3 matrix determinant amounts to (35) and hence

ti

AxAp>-. 2

(36)

eSimilarly, if fl and f2 are pure state WFs, the transition probability (I JdzQit; ( Z ) & ( Z ) ~ ~between ) the respective states is also non-negative [OWSl],manifestly by the same argument [CZOl], namely J d p d x f i f2 = ( 2 7 r F ~J) ~d x d p lfl*f2I2 2 0.

9

10

The ti entered into the moments’ constraint through the action of the * product [CZOl]. More general choices of g likewise lead to diverse expectations’ inequalities in phase space; e.g. in six-dimensional phase space, the uncertainty for g = a + bL, + cL, requires 1(1+ 1) 2 m(m l),and hence 1 2 m, etc. [CZOl, CZ021. For a more extensive formal discussion of moments, cf. Ref. N086.

+

5

Ehrenfest’s Theorem

Moyal’s equation (lo), 8.f

(37) at = V J , f ,l serves to prove Ehrenfest’s theorem for expectation values. For any phase-space function k ( x , p ) with no explicit time dependence,

1

= zJdxdp = Jdxdp

(H* f

-

f *H)*k

fgk,Hl= (gk,Hl).

(38)

+

(Any convective time-dependence, J d x d p [?az (f lc) p a,( f k ) ] , amounts to an ignorable surface term, J d z d p [ a z ( k f k )+ a , ( $ f k ) ] , by the x , p equations of motion.) Note the ostensible sign difference between the correspondent to Heisenberg’s equation,

(39) and Moyal’s equation above. The z , p equations of motion reduce to the classical ones of Hamilton, x = a p H , p = -&H. Moyal [Moy49] stressed that his eponymous quantum evolution equation (10) contrasts to Liouville’s theorem for classical phase-space densities, dfcl afcl (40) - =-+;:a,.fcl+papfcl=o. dt at Specifically, unlike its classical counterpart, in general, f does not flow like an incompressible fluid in phase space. For an arbitrary region R about a representative point in phase space,

(41) That is, the phase-space region does not conserve in time the number of points swarming about the representative point: points diffuse away, in general, without maintaining the density of the quantum quasi-probability fluid, and, conversely, they are not prevented from coming together, in contrast to deterministic flow. For infinite R encompassing the entire phase space, both surface terms above vanish to yield a time-invariant normalization 10

11

for the WF. The O(ti2)higher momentum derivatives of the WF present in the MB (but absent in the PB-higher space derivatives probing nonlinearity in the potential) modify the Liouville flow into characteristic quantum configurations [KZZ02, FBA96, ZP941.

6

Illustration: The Harmonic Oscillator

To illustrate the formalism on a simple prototype problem, one may look at the harmonic oscillator. In the spirit of this picture, one can, in fact, eschew solving the Schrodinger problem and plugging the wave functions into (4); instead, one may solve (17) directly for H = (p2 x2)/2 (with m = 1, w = 1):

+

[(x

iti + ,itia,)" + ( p - -8z)2 2 - 2E] f ( ~ , p= ) 0.

(42)

For this Hamiltonian, the equation has collapsed to two simple PDEs. The first one, the Imaginary part,

(43) restricts f to depend on only one variable, the scalar in phase space, z = 4 H / h = 2(x2 p 2 ) / h . Thus the second one, the Real part, is a simple ODE,

+

(44) Setting f ( z ) = exp(-z/2)L(z) yields Laguerre's equation,

1

za;

L

'1

+ (1 - .)az + -t i 2

L ( z ) = 0.

(45)

2

It is solved by Laguerre polynomials, 1 d;(e-"z") n!

L, = -e" for n = E / h - 1/2 = 0 , 1 , 2 , . . . , so the

Lo=1,

L1=l--

* gen-Wigner

4H ti'

,

(46)

functions are [Gro46]

8H2 L2=--h2

8H ti

+ 1,

1..

.

(47)

But for the Gaussian ground state, they all have zeros and go negative. These functions become spiky in the classical limit ti + 0 ; e.g. the ground state Gaussian fo goes to a 6 function. 11

14

evolution operator, a “*-exponential” [BFF78] itH/ti

U,(x,p;t) = e,

-

=1

+ (it/h)H(x,p) + (it2!/ti) ~

(it/ h)3 H * H + 7H * H * H + ..., (54)

for arbitrary Hamiltonians. The solution to Moyal’s equation, given the W F a t t = 0, then, is f ( x , p ; t ) = U?(X,P;t) *f(x,p;O) *U*(x,p;t).

(55)

In general, just like any *-function of H , the *-exponential (54) resolves spectrally [Bon84]:

(56) (Of course, for t = 0, the obvious identity resolution is recovered.) In turn, any particular *-genfunction is projected out formally by (57)

which is manifestly seen to be a *-function. For oscillator *-genfunctions, the *-exponential (56) is directly seen to sum to exp,

(7)

-1

= [cos(;)]

exp [;Htan(;)]

,

(58)

which is to say, a Gaussian [BFF78] in phase space.f For the variables x and p, the evolution equations collapse to mere cZassicaZ trajectories, dx dt

- --

x*H - H*x iii

= apH = p

,

(59)

(60) where the concluding member of these two equations hold for the oscillator only. Thus, for the oscillator, x(t) = a : c o s t + p s i n t ,

p(t) = p c o s t - x s i n t .

(61)

As a consequence, for the oscillator, the functional form of the Wigner function is preserved along classical phase-space trajectories [Gro46]: f ( x , p; t ) = f ( x cos t

- p sin t ,p cos t

+ x sin t ;0).

(62) (62)

fAs an application, note that the celebrated hyperbolic tangent *-composition law of Gaussians follows trivially, since [BFF78]. That these amount to *-exponentials with additive time intervals, exp,(tf) * exp,(Tf) = exp,[(t T)f)], is,

+

14

16

In Dirac’s interaction representation, a more complicated interaction Hamiltonian superposed on the oscillator one leads to shape changes of the W F configurations placed on the above “turntable,” and serves to generalize to scalar field theory [CZ99].

8

Nondiagonal Wigner Functions

More generally, to represent all operators on phase space in a selected basis, one looks at the Weyl-correspondents of arbitrary ) . 1 (bl, referred to as nondiugonal WFs [Gro46]. These enable investigation of interference phenomena and the transition amplitudes in the formulation of quantum-mechanical perturbation theory [BM49, W088, CUZOl]. Both the diagonal and the non-diagonal WFs are represented in (2), by replacing p +

I$a>($bI:

=$a(x)

* J ( P ) * $;(a:)

(63)

1

The representation on the last line is due to Ref. Bra94 and lends itself to a more compact and elegant proof of Lemma 1. Just as pure-state diagonal WFs obey a projection condition, so too the non-diagonals. For wave functions which are orthonormal for discrete state labels, da: $:(a:)$b(z) = dab, the transition amplitude collapses to

J dxdp

fab

(x,P ) = dab

(64)

*

To perform spectral operations analogous to those of Hilbert space, it is useful to note that these WFs are *-orthogonal [Fai64], ( 2 r f i ) fba

* f d c = dbcfda

(65)

7

as well as complete [Moy49] for integrable functions on phase space,

(2.h)

fab

( z l , P l )fba

( z 2 , P 2 ) = 6 (z1 - x 2 )

6(P1 - P 2 )

*

(66)

a,b

For example, for the SHO in one dimension, non-diagonal WFs are

(67) (cf. coherent states [CUZOl, DG80)]. Explicitly, in terms of associated Laguerre polynomials, these are [Gro46, BM49, Fai641 ei(n-k)

arctan(p/z)

(-‘Ik

(68)

7rfi

16

17

The SHO nondiagonal WFs are direct solutions to [Fai64] H*fkn=Enfkn

fkn

3

* H = Ek f k n

-

(69)

i)

i)

The energy *-genvalue conditions are (En - /ti = n, an integer, and (Ek - / h = k, also an integer. The general spectral theory of WFs is covered in Refs. BFF78, FM91, Lie90, BDW99, CUZOl.

9

Stationary Perturbation Theory

Given the spectral properties summarized, the phase-space perturbation formalism is selfcontained: it need not make reference to the Hilbert-space treatment [BM49, W088, CUZO1, SS02, MS961. For a perturbed Hamiltonian,

H (.,P>

+

= Ho(.,P)

Hl(z,P) ,

(70)

seek a formal series solution, 00

03

(71) k=O

k=O

of the left-right-*-genvalue equations (17), H

* f n = Enfn = f n * H .

Matching powers of X in the eigenvalue equation [CUZOl],

(72)

(73) For example, consider all polynomial perturbations of the harmonic oscillator in a unified treatment, by choosing

H I = eYX+6P

* * 0 0

= YX+JP = eYX e,

e6P

e(Y6/2 =

e6P

eY"

e-iY6/2

(74)

to evaluate a generating function for all the first-order corrections to the energies [CUZOl], 00

00

(75) n=O

J

n=O

hence

(76) 17

18

From the spectral resolution (56) and the explicit form of the +exponential of the ti], it follows that oscillator Hamiltonian (58) [with eit --t s and ELo) = ( n

+ i)

(77) and hence

E(*)(s) = -

dxdp eyx+6pexp

7rti(l+ s )

1-s

exp

[fi.4

(72

+ 62)

“‘“1

1-s

I-s (--) ti x‘+p’

lfs

(78)

.

For example, specifically,

(79) and so on. All the first order corrections to the energies are even functions of the parameters-only even functions of x and p can contribute to first-order shifts in the oscillator energies. First-order corrections to the WFs may be similarly calculated using generating functions for nondiagonal WFs. Higher order corrections are straightforward but tedious. Degenerate perturbation theory also has an autonomous formulation in phase-space, equivalent to Hilbert space and path-integral treatments. 10

Propagators

Time evolution of general WFs beyond the above treatment is discussed at length in Refs. BM49, Ber75, GMSO, CUZOl, BR93, wo82, W002, FMO3. A further application of the spectral techniques outlined is the computation of the W F time-evolution operator from the propagator for wave functions, which is given as a bilinear sum of energy eigenfunct ions,

(80) as it may be thought of as an exponentiated effective action. (Henceforth in this section, take h = 1.) This leads directly to a similar bilinear double sum for the W F time-transformation kernel [Moy49],

T ( X , P ; X,P ;t>= 2i.r

C

f b a (5,P>e

ah

18

-i(E,-Eb)t

f a b ( X , p) .

(81)

19

Defining a “big star” operation as a *-product for the upper-case (initial) phase-space variables,

(82) it follows that T ( X , Pi

x,p ;t)*.fdc(X,

p )=

f b c ( z , P ) e-

i (Ec-Eb) t

fdb(X,P) 7

(83)

b

and hence [cf. (55)],

/ d X d P T ( x , p ;X , P ;t ) f d c ( X P , ) = fdc(x,p)e-i(Ec-Ed)t = UL1*fdC(z,p; O)*U, = f d c ( z , p ;t ) .

(84) For example, for a free particle of unit mass in one dimension, H = p 2 / 2 , WFs propagate according to

=

6(z-X-Pt)6(p-P),

(85)

amounting to “classical” motion,

(86) 11

Canonical Transformations

Canonical transformations ( x ,p ) H [ X ( x p, ) , P ( x , p ) ]preserve the phase-space volume (area) element (again, take h = 1) through a trivial Jacobian,

d X d P = dxdp { X ,P }

,

(87)

i.e. they preserve Poisson brackets

{u,v),p { X ,q

= -- -- , axap apax

x p =

1,

{X,P}XP =

(88) 1.

(89)

Upon quantization, the c-number function Hamiltonian transforms “classically,” ‘FI(X,P ) = H ( x ,p ) , like a scalar. Does the *-product remain invariant under this transformation?

Yes, for linear canonical transformations [KLOl], but clearly not for general canonical transformations. Still, things can be put right, by devising general covariant transformation rules for the *-product [CFZ98]: the WF transforms in comportance with Dirac’s quantum canonical transformation theory [Dir33].

20

In conventional quantum mechanics, for classical canonical transformations generated by

K l ( G q ,

(90) the energy eigenfunctions transform in a generalization of the "representation-changing" Fourier transform [Dir33], (91)

(In this expression, the generating function F may contain ti corrections [BCT82] to the classical one, in general-but for several simple quantum-mechanical systems it manages not to [CG92, DG021.) Hence [CFZ98], there is a transformation functional for WFs, I ( x , p ;X, P ) , such that f ( ~ , p= ) J d x d p T(Z,P;

x,P)*FT(X,P ) =

J

d x d I~( Z , P ;

x,P ) F(X,P ) ,

(92)

where

--s -

" I2n 2

(93)

[

dYdy exp -iyp

+ ~ P -Yi F * ( x - -,x Y - -Y >2 + i 2

+

~ ( x Y-, x 2

I'

+ 2)

Moreover, it can be shown that [CFZ98],

H ( w ) * q x ,P ;

x,P ) = 7 ( x ,P; x,P)*

'H(XP).

(94) (94)

That is, if F satisfies a *-genvalue equation, then f satisfies a *-genvalue equation with the same eigenvalue, and vice versa. This proves useful in constructing WFs for simple systems which can be trivialized classically through canonical transformations. A thorough discussion of MB automorphisms may start from Ref. BCW02 . (Also see Refs. Hie82, DKM88, GR94, DV97, Hak99, KL99, DPOl.) Time evolution is a canonical transformation [Dir33], with the generator's role played by the effective action A of the previous section, incorporating quantum corrections to both phases and normalizations; it connects initial wave functions to those at a final time. For example, for the linear potential, with

H=p2+z,

(95) (95)

wave function evolution is determined by the propagator

(96)

T then evaluates to

Tin

(

~

P7 ; X , P ;t )

21 -

- 8n2t

/

[

it

t X.) p%-2t = 6(p+t-P) 6(x-2tp-t2-X) = -2t 16

(

= d(x-X-(p+P)t)

1

i

+ iPY - -2 (y + Y )+ 2t (x - X)(y - y ) 6 (p - - - - .-x>

d Y d y exp -iyp

;

2t

6(P-p-t).

(97)

The 6 functions enforce exactly the classical motion for a mass= 1 / 2 particle subject to a negative constant force of unit magnitude (acceleration = -2). Thus the W F evolves “classically” as

f ( x , p ;t ) = f(x

- 2pt

-t2,p

+ t ;0 ) .

(98)

Note that time independence follows for f ( z , p ; 0) being any function of the energy variable, since x p2 = x - 2pt - t 2 + ( p + t ) 2 . The evolution kernel T propagates an arbitrary W F through just

+

f ( x , p ;t ) =

s

x,

d X d P T ( x , p ; P ;t ) f(X, P ;0 ) .

(99)

The underlying phase-space structure, however, is more evident if one of the wave-function propagators is given in coordinate space, and the other in momentum space. Then the path integral expressions for the two propagators can be combined into a single phase-space path integral. For every time increment, phase space is integrated over to produce the new Wigner function from its immediate ancestor. The result is

T (x,p ;

x,p ;t )

(100)

o>*eiz2P2e-2‘(x--zz)(p-Pz),

= 1 ~ ~ l d p l J d z 2 d p 2 e 2 ’ ( z - z ~ ) ( ~ - m ) e - (~Z-lz; t~\ v x l2 ;0) (pl; tlp2; 7 9

where ( q t lz2;O) and ( p 1 ; t (p2;O)are the path integral expressions in coordinate space, and in momentum space. Blending these x and p path integrals gives a genuine path integral over phase space [Ber80, DK851. For a direct connection of U, to this integral, see Refs. Sha79, Lea68, SamOO.

12

The Weyl Correspondence

This section summarizes the bridge and equivalence of phase-space quantization to the conventional formulation of quantum mechanics in Hilbert space. The Weyl correspondence merely provides a change of representation between phase space and Hilbert space. In itself, it does not map (commutative) classical mechanics to (non-commutative) quantum mechanics, but it makes that deformation map easier to grasp, defined within a common representation, and thus more intuitive. Weyl [Wey27] introduced an association rule mapping invertibly c-number phase-space functions g(x,p ) (called phase-space kernels) to operators 8 in a given ordering prescription.

21

22

Specifically, p

H p,

x H r, and, in general,

/

1 - d r d o d x d p g ( z , p ) exp [ir(p - p) io(F - x)] . (101) (2Tl2 The eponymous ordering prescription requires that an arbitrary operator, regarded as a power series in F and p, be first ordered in a completely symmetrized expression in F and p, by use of Heisenberg's commutation relations, [F,p] = ih. A term with m powers of p and n powers of F is obtained from the coefficient of T ~ O " in the expansion of (rp + which serves as a generating function of Weyl-ordered polynomials [GF91]. It is evident how the map yields a Weyl-ordered operator from a polynomial phase-space kernel. It includes every possible ordering with multiplicity one, e.g.

@(F, p)

=

6p2x2

P2F2

+

+ F2P2 + PFPF + PF2P + FPFP + FP2F

(102)

*

In general [McC32],

(103) Phase-space constants map to the identity in Hilbert space. In this correspondence scheme, then, l"r6 =

s

(104) (104)

dxdpg.

Conversely [Gro46, Kub64, HOS841, the c-number phase-space kernels g ( x ,p) of Weylordered operators @(F, p) are specified by p H p, F H x, or, more precisely, by the "Wigner map,"

(105) since the above trace reduces to

I

dz

e i r u ~ / (zle-iuFe-irp 2

@lz)= 27r

I

d z ( z - hrI@jlz)eiu(rh12--z).

(106)

Thus, the density matrix inserted in this expression [Moy49] yields the hermitean generalization of the Wigner function (63) encountered,

1

=27r / d y e - i Y w z

h

-

ti

p)i&+ ZY)

= fb*.hP)

7

(107)

where the $,(x)s are (ortho)normalized solutions to a Schrodinger problem. (Wigner [Wig321 mainly considered the diagonal elements of the pure-state density matrix, denoted

22

23

above as fm = /mm-) As a consequence, matrix elements of operators, i.e. traces of them with the density matrix, are produced through mere phase-space integrals [Moy49], = / dxdp g(x,p)fmn(x,p),

(108)

and thus expectation values follow for m = n, as utilized throughout in this overview. Hence, {•0m| exp i(cr$ + Tp)\ij>m) = I dxdp fm(x,p) exp i(ax + rp),

(109)

the celebrated moment-generating functional [Moy49] of the Wigner distribution. Products of Weyl-ordered operators are not necessarily Weyl-ordered, but may be easily reordered into Weyl-ordered operators through the degenerate Campbell-Baker-Hausdorff identity. In a study of the uniqueness of the Schrodinger representation, von Neumann [NeuSl] adumbrated the composition rule of kernel functions in such operator products, appreciating that Weyl's correspondence was in fact a homomorphism. (Effectively, he arrived at the Fourier space convolution representation of the star product.) Finally, Groenewold [Gro46] neatly worked out in detail how the kernel functions / and g of two operators # and (3 must compose to yield the kernel of $ dP -m

gegebene *, Denn die Abbildung V . verwandelt 4-J (4) in CrJ

+J;iqip+4 -m

'p

(PIdP = J eiQ sp (P - 4 dT, -m

U, aber in 1Yi.p.

e-

9 (PId F .

-s

Es ist also 'p (PI7 ,

= 'p (P - 4, sp (P>ulr = e--(crp0 (PI*

(53)

III.T e i l . D a s d y n a m i s c h e P r o b l e m . fj 9. D a s O e s e t z d e r z e i t l i c h e n V e r a n d e r u n g . D i e Z e i t g e s a m t h e i t. Die bisherigen Ansatze beanspruchen a.Ugemeine Geltung. Nicht so gunstig steht es mit dem d y n a m i s c h e n P r o b l e m , das eng mit der Frage nach der Rolle zusammenhangt, welche R a u m und Z e i t in der Quantehphysik spielen. In der F e l d t h e o r i e werden Zustandsgr6Sen behandelt, die in Raum und Zeit ausgebreitet sind, die M e c h a n i k im engeren Sinne hat es nur mit der Zeit als der einzigen unabhBngigen Veranderlichen zu tun. Die unabhangigen Veranderlichen sind keine

* Nach einem wichtigen Sakt von P l a n c h e r e l (Rend. Circ. Mat. Palermo 30, 330, 1910) und Titchmarsh [Lond. Math. SOC.Proc. (2) 23, 279, 19241 hat diese TransEormation fur alle absolut quadratisch integrierbaren Funktionen einen klaren Sinn und erhalt (bis auf den Faktor 2 n) das Quadratintegral.

79

36

Quantenmechanik und Gmppentheorie.

gemessenen GrBden, sie sind ein willkiirlich in die Welt hineingetragenes gedachtes Koordinatenspinngewebe. Die Abhhgigkeit einer physikalischen GrgDe von diesen Variablen ist also auch nicht etwas durch Messung zu KontroUierendeS; erst wenn mehrere physikalische Grir Oen vorliegen, kommt man durch Elimination der unabhangigen Vergnderlichen zu Beziehmgen zwischen beobachtbaren GroBen. Es mag sein, daB unter diesen ZustandsgrirDen die Raumkoordinatm eines Elektrons auftreten; gemessener, real markierter Ort und nat,urlich auch real markierte Zeit sind Zustandsgrafien und werden also durch H e r m i t e schen Formen zu reprasentieren sein. Diesem Sachverhalt gegeniiber ist die nicht-relativistische Mechanik in der glucklichen Lage, die Zeit als Zustandsgr8He ignorieren zu kbnnen, wghrend die Relativititsmechanik parallel mit den mellbaren Raumkoordinaten auch die medbaren Zeitkoordinaten der Teilchen benbtigt. Eine vollstiindige Durchfuhrung der Quantentheorie liegt bisher nur in dem Umfang vor, in welchem die Zeit als einzige unabhiingige Variable und die Zeit n u r a l s unabhgngige Variable auftritt. Da die H e r m it e sche Form, welche zu einer physikalischen Gr6De gehort, nichts zu t u n hat mit besonderen Werten, welche die Grolle unter Urnstanden, insbesondere im Laufe der Zeit annimmt, bleibt sie von der Zeit unberiihrt. W a s sich im Laufe der Zeit t iindert, ist allein der reine Fall h (t). Das d y n a m i s c h e G e s e t z gibt die infinitesimale Verschiebung au, die (t) wahrend des Zeitelements d t erftihrt: d h - __. 2ni FE. (54) at h I

Hier ist iE die infinitesimale unitare Abbildung, welche mit der die Energie reprasentierenden Hermiteschen Form E gekoppelt ist, k das Wirkungsquantum. Die mit dem Vorriicken der Zeit um d t verbundene d y ) - A(y) irgend einer E e r m i t e s c h e n Form A (F)ist, Anderung A (y wie man leicht ausrechnet, 2nidt d A = -( E A - A R ) . (W h dE ist = 0. Bringt man die H e r m i t e s c h e Form E der Energie auf Hauptachsen : Z(h) = E,x,X, +E,z,X,+ E,,XnZnl 50 bezeichnen die Nummern 1 bis n die mbglichen Quantenzusttinde, Ei die zugehbrigen Energiestufen, und in den Gleichungen (54) separieren sich die Variablen: dxv 2aiEw

+

0 . -

+

3*

80

36

H.Weyl,

Die Integration l5iDt sich sofort ausfuhren :

(56) Die Hermitesche Form A(.$)= X a , , z p , ist nach Ablauf der Zeit t ubergegangen in

Die Komponenten apv im Hauptachsensystem der Energie fiihren alsa einfache Schwingungen aus mit den B o h r s c h e n F r e q u e n z e n . Nach (56) bleiben nicht nur die Energiestufen E,,widwend der Bewegung erhalten sondern auch die Hhfigkeiten Iz, (t) Iz = zyla, mit denen sie vertreten sind. Das bisher Gesagte gilt fur ein abgeschlossenes System. Wenn man innerhalb eines abgeschlossenen Systems ein Teilsystem ins Auge fa&, das unter dem EinfluB des Restes steht, dessen Ruckwirkung auf den Rest aber vernachlassigt wird, so hat man den Fall der von aullen eingepriigten Kriifte : die Hamiltonsche Funktion hsngt explizite von der Zeit ab. Die Hermiteschen Formen, welche die Energie und andere GroDen CG am System darstellen, sind Funktionen der Zeit: A = A ( t ;r). Das Gesetz der zeitlichen Verschiebung des reinen Falles p(t) bleibt das gleiche. Die Formel (3 1) in 8 6 gestattet die integrale Aneinanderreihung der von Schritt zu Schritt in der Zeit sich vollziehenden infinitesimalen Drehungen (54). So berechne man die Drehung V ( t , , tz), welche von ~ ( t , j zu p(t,) fuhrt. Findet die Einwirkung von auDen nur in dem ZeiL intervall tlt2 statt, wahrend vor t, und nach tz das System abgeschlossen ist, so entnimmt man der Matrix U(t,, tz) insbesondere, wie sich die Wahrscheinlichkeiten fiir die verschiedenen Energiestufen Ep durch die Einwirkung verschoben haben. Darauf bezieht sich die Untersuchung von M. Born iiber das Adiabatenprinzip in der Quantenmechanik*. Wenn die Zeit nicht mel3bare GrBOe, sondern nur unabhhgige Variable ist, haben nur solche Beziehungen konkrete Bedeutung, aus deneo die Zeit eliminiert ist. Tatbestinde von diesem Charakter sind in der Quantenmechanik e k e s abgeschlossenen Systems : der Wertevorrat welchen eine gegebene GrBBe durchlaufen kann, und die zeitlichen 3f it t e l w e r t e der Wahrscheinlichkeiten W (F),mit denen eine gegebena

I

*

ZS. f. Phys. 40, 167, 1927.

81 Quantenmechanik und Gruppentheorie.

37

Gr6fie Werte in gegebenen Grenxen annimmt. Handelt es sich urn den reinen Fall F: sv = c V e ( y v ) (cv 2 0, y. reell), so durchbiuft (t) nach (56)l wenn die Energiestufen nicht speziellen linearen Rationalitatsbeziehungen geniigen, gleichma0ig dicht das gmze

durch

Is*I = c11 lz,l = c,, i 1 5 1= cn defbierte Gebilde @ von reellen Dimensionen. In den Ausnahmefallen reduziert sich die Dimensionszahl *. Zur Berechnung der zeitlichen Yittelwerte ist iiber dieses gleichmSi8ig dicht von der Zeitkurve erfiillte Gebiet Dl die ,,Zeitgesamtheit ', zu integrieren. Ich erinnere noch kurz an die Beziehung der Energie und der H am i 1t o n schen Gleichungen zu den kanonischen Variablen. Hat das mechanische System einen Freiheitsgrad und ist eine Funktion (42) der kanonischen Variablen p , p reprasentiert durch die Matrix (41), so sind - - a

f =fpl d f gemaB unserer Festsetzung die beiden Ableitungen d- = fq dP &l reprgsentiert durch

+-

F, = i I l e ( G P + z & ) . G ~ ( G , z ) d G d z , -OD

+-

Fq = i j I e ( G P + z Q ) . z $ ( G , z ) d G d t , - 0 2

da entsprechendc Fourierentwicklungen fur f p und fq gelten. Wegen (38) ergibt die Kommutatorregel (35), wenn man U(Z) wieder infinitesimal werden lafit, die beiden Gleichungen P . e ( d P z Q ) - e (GP z &) . P = z . e (GP f z Q),

+ &. e(GY f z Q ) - e ( 6 P + z Q ) . @ = - G. e(GP + z Q ) , +

also

Fq = i(PF-FFP). -Fp = i(QF-FQ), Das dynamische Gesetz (54)lg5t sich daher, wenn f ( p , q) die Energiefuriktion ist, nach (55) so fassen:

Daraus sieht man: wenn u und b zwei reelle Zahlen vom Produkt h / 2 1~ sbd, reprleentieren a P und 59 GraDen, welche k a n o n i s c h sind i n

* Vgl. H. W e y l , Ober die Gleichverteilung von Zahlen mod. Eins, Math. Am. 77, 313, 1916.

82

38

H. Weyl,

dem S i n n e , da13 f u r s i e d i e klassischen Bewegungsgleichungen gelten. Auf diese Weise wird in konkreten Beispielen die Bestimmung der Energie als GriiBe im Gruppengebiet durchgefiihrt. Bei solcher Beschreibung kommt das Wirkungsquantum n u r e i n m a l vor: in dem dynamischen Gesetz und nicht in den Vertauschungsrelationen. Sie basiert auf der fiberzeugung, daB die formalen Beziehungen der klassischen Physik als solche zwischen den reprasentierenden Matrizen, nicht zwischen den angenommenen Werten, bestehen bleiben. Will man den gerugten Mange1 des Zeitbegriffs der aiten vorrelativistischen Mechanik aufheben, so werden die meBbaren GrijSen : Zeit t und Energie El als ein weiteres kanonisch konjugiertes Paar auftreten, wie ja bereits das Wirkungsprinzip der analytischen Mechanik erkennen l&Bt; das dynamische Gesetz kommt ganz in Fortfall. Die Behandlung eines Elektrons im elektromagnetischen Felde nach der Relativit'iitstheorie durch S c h r ij d i n g e r u. a. entspricht bereits diesem Standpunkt *. Eine allgemeine Formulierung liegt noch nicht vor. 8 10. K i n e t i s c h e E n e r g i e und Coulombsche K r a f t i n der I:e l a t i v i s tis c h e n Qu a n t e nm e c h a n i k. Innerhalb des Schemas, das die Zeit nur als unabhangige Variable kennt, ist wenigstens eine halbrelativistische Mechanik moglich, welche den richtigen Ausdruck fur die kinetische Energie verwendet, aber die potentielle Energie nach mie vor als eine Funktion der Lagekoordinaten, und das Ireat doch genauer: ihrer s i m u l t a n e n Werte, annimmt. Zur Illustration der Theorie behandle ich den Fall eines oder mehrerer Teilchen, deren Lage durch ihre rechtwinkligen Koordinaten 2,y, B gekennzeichnet wird. Der Ausdruck der kinetischen Energie in den zugehorigen Impulsen u,o, w lautet, wenn m die Masse des Teilchens bedeutet und c die Lichtgeschwindigkeit : c lmaca

+ zca + + wa. v2

Fur die Durchrechnung ist es zweckmIfiig, die Koordinaten und Impulse k bzw. mc zu beziehen; dam des Teilchens auf die Mafieinheiten ~

2xmc

sind sie dimensionslose Gr6Den und zugleich mit der von uns befiirworteten Normierung der kanonischen Koordinaten in Einldang. Es handelt sich darum, die Abbildung oder Hermitesche Form zu konstruieren, welche dieser GroDe entspricht im Raume der Funktionen $(s, y, 2). Als Musterbeispiel dime der eindimensionale Fall. Es id

*

Siehe etwa E. Schrodinger, Abhandlungen zur Wellenmechanik, Leipzig = Ann. d. Phys. (4) 81, 133, 1926.

1927, S. 163,

83 Quantenmechanik nnd Gruppentheorie.

zu ersetzen mit einem kleinen positiven gieren zu lassen. Setzen wir

cd

39

und dann a gegen 0 konver-

oo

1%[ e - a 2 b V 1 X

f a 2 e - ~ l r ~ d= u Gat,$,

" 0

(68)

so is+, die der Grofle (57) korrespondierenda Abbildung +m

@(+4&) =+yvj(2:-~1 G a ( 6 ) d d = J G a ( ~ - - B ~ ( B d(59) & - m

-m

die Herniitesche Form der willkiirlichen Funktion +($) lautet: C m

Urn an der geraden Funktion G,(G) fur 6> 0 den Grenziibergang zu a = 0 zu vollziehen, schlagen Wir in dem Integral, von dem xGa(ci) nach (58) der Realteil ist, den htegrationsweg in die negative imaginiire Halbachse hiniiber : u = - it, indem wir die Singularitat zb = - i nach rechts hin umgehen :

0

Im Limes fur

1

cd

= 0 ist der Realteil also

Uarauaus liest man sofort ab, da13

- G (G) =

1 XG

-r ( G )

ist, wo r f u r G = 0 nur noch logarithmisch unendlich wird. In (59) macht der Grenzubergang zu a = 0 an dem r-Teil keine Schwierigkeit. In (60) ist der erste Summand bei a i d = 0 regular, der zweite hangt eng mit derjenigen Hankelschen Bylinderfunktion erster Ordnung H zusammen, die mit positiv wachsendem G exponentiell zu 0 geht; er ist

+

h

40

H. Weyl,

ngmlich = El(a -k ") a f i G

.

Darum ist bis au€ einen additiv hinzutretenden

Teil, der an der kritischen Stelle a endlich wird:

+ i6 = 0

nur logarithmisch un-

So kommt als Reprasentation der kinetischen Energie die Operation

+-

W++'(d = @ * ( 4 + + J ' ( x - t ) @ ( w t ,

(61)

- 00

(der Einfachheit halber ist @ reell angenommen). Der Grenzubergang ist so zu verstehen, da13 z komplex = 5 i y ist mit positivema Imaginiirteil y und y zu 0 strebt. Dns in der letzten Gleichung hinter dem Zeichen 8 stehende Integral ist das i-€ache der Ableitung derjenigen 0, deren Realteil @ analytischen Funktion in der oberen Halbebeni y auf der reellen Achse mit unserem 9 (z) zusammenfallt. - (x) ist

+

>

+*

demnach die nach der inneren Normale n genommene Ableitung all, dn

dieser Potentiallunktion am Rande. Da das tiber den Rand erstreckte Integral von - 9 dllr - nichts anderes ist als das Dirichletsche Integral dn

D(+)uber die obere Halbebene, haben wir schlieDlich als die der GrtiSe f1 +us zugehbrjge Hermitesche Form: + m

Wenn es sich um ein einzelnes Teilchen handelt und eine (in der Einheit m ca gemessene) potentielle Energie V ( s ) da ist, besteht das Eigenwertproblem darin,

+ J J m- E )

+--I w) ~YWX + m

f m

DW

~

5m ) d

-m

s a

10

+'T

zum Extremum zu machen unter der Nebenbedingung

-m

Extremalwerte A sind die Energiestufen.

= 1. Die

85 Quantenmechanik und Gruppentheorie. ES ist

41

klar, d d die Operation (61), wenn sie zweimal ausgefiihri d r d , zu derjenigen fiihren mud, die 1 ua korrespondiert, d. i. zu

+

Deshalb kann die Schwingungsgleichung fur das einzelne @PJ + !?!. &a

Teilchen auch in der form einer gewohnlichen differentialgleichung angeschrieben Werden:

Aber hier tritt der Eigenwertparameter 1 nicht mehr in h e a r e r Weise auf, und die Halfte der Eigenwerte sind falsche. Auf solchem Wege gelang es S c h r b d i n g e r und P. E p s t e i n , die Energiestufen und Eigenfunktionen des Wasserstoffatoms relativistisch zu berechnen *. Wenn aber mehrere Teilchen im Spiel sind, ist es unmtiglich, durch Iteration zu Differentialgleichungen zu gelangen. Wenn die wirkenden Kriifte Coulombsohe K r g f t e sind, die von &ern festen Kern ausgehen, ist es zweckmgbig, die Komponenten y der reineu Fzille im Hauptachsensystem der Impulskomponenten zu benutzen. Die kinetische Energie ist dann einfach reprgsentiert durch die Multiplikation y + y': y'(u,v,w)= .r/l+Sa.cp(zt,v,w) (SB = ua 2r2 wa).

+ +

Es gilt, die reprasentierende Eermitesche Form fur das Potential l/r (9= sa ya 2) zu finden. Bus Konvergenzgriinden werde 11.

+ +

e-

1~

zuniichst ersetzt durch -, wo 1 eine kleine positive Konstante ist. r Fur das Integral in der Fourierzerlegung dieser Funktion

h d e t man leicht durch Einfuhrung von Polarkoordinaten

* E.Schrodinger, Abbandlnngen zur Wellenmechanik, 1927, S. 164, = Ann.

d. Phys. (4) 81, 134, 1926.

P. S. E p s t e i n , Two Remarks on Schr6dinger's Quantum Theory, Proc. Amer. Nat. Acad. 13, 94, 1927..

86

42

B. Weyl,

Die gesuchte Abbildung ist also diejknige, welche cp (N, v, w) verwandelt in

(62)

In der letzten Gestalt bedeutet M,(cp) den Mittelwert der Funktion cp auf der Eugel vom Radius G um den Punkt (u, v, w) im Impulsraum. Behglt man 2 zunachst noch bei, so tritt im Ausdruck (62) der Summand l9 im Nenner hinzu. Die Funktion, die sich so egibt, ist im vierdimensionalen Raum mit den Roordinaten 240, w,1 diejenige Potentialfunktion F, welche aus der Massenbelegung der ,,Ebene" 1 = 0 mit der Dichte cp entsteht. Pq sind ihre Werte auF der belegten Ebene. Ds offenbar

ist, wo 1, 2, 0 = (spy) drei Punkte im Impulsraum bedeuten und I-*,,, raOlr,a ihre gegenseitigen Abstande, liefert die Wiederholung Pa von P den Prozed, der im dreidimensionalen Impulsraum 'p iiberfiihrt in die durch die Raumbelegung cp erzeugte Potentialfunktion di. Es gilt bekanntlich

m

aaai

dw

&T+du"+--=cpd Wa

Man wird nach Kugelfunktionen zerspalten. Benutzt man die oben erwiihnte vierdimensionale barmonische Funktion F und macbt den Ansatz

F = Y, . F(s, Z), in welchem Y, eine nur von der Richtung u : v : w abhangige Kugel0 funktion n-ter Ordnung sein solt, so geniigt im oberen Halbraum X der nur von s und 1 abhangige Faktor F der Gleichung

>

daF

= n(12 + 1)E;

und die Operation P bedeutet den ubergang von den Rsndwerten ihrer normalen Ableitung zu ihren eigenen Randmerten. Vielleicht ist es

87

43

Quantenmechanik nnd Gruppentheorie.

bequemer, statt F (S, 0 die Funktion s F (s, Z) = F* (s, Z) zu behutzen. Far die Differen tidgleichung 693% daF'': - f i ( f i ds2

+ F-

+ up*,

S2

> 0 einer (s, 2)-Ebene, welche

y* ist eine Funktion in der oberen Helfte 1 bei Spi-egelnng an der 1-Achse nngerade ist. Faktor I/X2,

R2 = (21 - a)a f (w - b)2

+

(W

- Indem

man in (62) den

- r)2= S2 + G2 - 8 S G cos 6,

nacb Rugelfunktionen P,, (cos 8)entwickelt :

angesetzt wird, als Ausdruck der Operation P an solchen Funktionen die

0

+1

-1

Wenn das Einkbrperproblem vorliegt, wird man, auf die Gefahr hin, eine Serie falscher Eigenwerte einzuschmuggeln, P iterieren und dadurch zu einer reinen Differentialgleichung kommen. Fur das nichtrelativistiscbe Wasserstoffatom sind die Eigenfunktionen cpp,(u,w, w), die durch die Fouriersche Transformation aus den Schrbdingerschen Eigenfunktiohen (x,y, z), den Laguerreschen Polynomen, hervorgehen, in meiner Dissertation angegeben *. Sie kbnnen auch sehr schlin direkt auf dem hier skizzierten Wege gewonnen werden. Im Mehrkorperproblem versagt die Iterationsmethode. Coulombsche Krrtfte zwischen m e h r e r e n beweglichen Teilchen. Dem reziproken Abstand l/r19 zweier Teilchen 1 und 2 entspricht im Gebiet der Impulsfmktionen sp (u~, w,, w,; aUp , 'u2, us),wie man auf die gleiche Weise erkennt, die Abbildung

+,,

+m

* Math.

AM. 66, 307-309, 317-324, 1908.

88

44

H. Weyl,

Die Beneichnung soll naturlich nicht ausschlieden, da4 y auch von den Iqpulsen der ubrigen Teilchen abhangt, diese werden aber von der Transformation nicht mit betroffen. Ha t h e m a t ischer Anhang. B e w e i s d e s S a t z e s von d e r . H a u p t a c h s e n t r a n s f o r m a t i o n e i n e r u n i t a r e n Abbildung. 1st die unitire Abbildung A = 11 aik 11 gegeben, so bestimmen wir einen Vektor F' # 0, der durch A in ein Multiplum von sich selber ubergeht: n

(63)

Wahlen wir

E

als eine Wurzel der SPkulargleichung det ( E 1 - A ) = 0,

so existierf tatsschlich ein derartiger Vektor = el. Indem n-ir seinen Betrag zu 1 normieren, erganzen wir ihn durch weitere n - 1 Vektoren e2, .... en zu einem normalen Koordinatensystem. Da in ihm die Gleichungen (63) fur el, d. i. fiir = 1, x9 = 0, .... zn = 0 erfiillt sind, ist jetzt a,, = E , a,, = = a,, = 0. Die Quadratsnmme der absoluten Betrage der ersten Koeffizientenzeile in A mu13 1 sein, darum ist E I = 1. Aber auch die absolute Quadratsumme der Glieder, welche in der ersten Spalte stehen, ist = 1, und das liefert = 1, uZ1 = = an, = 0. I Ia211a ... Das ist der entscheidende Schlu13. Die Matrix A zerfallt nunmehr in der aus dem Schema ersichtlichen Weise:

...

I

+

+

...

E

0 0

O 893

0

. . .0

a93

*

*

.

a39

*

1

a2n a3n

............ a

. . . . . . . . . . . . 0

an2

an3

-

*

*

ann

Durch Induktion in bezug auf die Dimensionszahl 12 ist damit der Beweis vollendet. Lie& die unitire Abbildung A in der Normalform vor, init den Termen at in der Hauptdiagonale, so geniigen der Gleichung (63) offenbar alle und nur diejenigen Vektoren, welche sich a m Grundvektoren ec zusammensetzen, fiir die at z e ist. Daraus geht bervor, da8 die vers c h i e d e n e n Eigenwerte a', a", ... mit ihrer Vielfachheit und die zu-

89

45

Quantenmechanik und Grappentheorie.

geh~rigenTeilraume W (a'), W (a"), eindeutig durch A bestimmt sind.

..., von denen

in

8 1 die Redc

war,

11

Wenn A = 11 aikll, B = btkll v e r t a u s c h b a r e u n i t a r e Mat ri zen s i n d , l a s s e n s i e s i c h s i m u l t a n auf H a u p t a c h s e n t r a n s f o r m i e r e n . Beweis: A kann sogleich in der Normalform angenommen werden, in welcher nur Glieder a,. in der Hauptdiagonale auftreten. Die Vertauschbark&sforderung besagt

(ai - ak)bik = 0.

(64)

Wir teilen die Indizes in Klassen, indem i und k in dieselbe Klasse kommen, wenn ai = ak ist. Die Gleichung (64) zeigt, daS bik = 0 ist, Wenn die Indizes i und k verschiedenen Klassen angehijren; d. h. B zerfillt in der gleichen Weise in Teilmatrizen: B', B", ..., die sich lings der Hauptdiagonale aneinanderreihen, wie sich die ca, in Klassen untereinander gleicher aufteilen: a', a", ... Die Abbildung B l i B t die zu den ... einzeln Eigenwerten a', dr1... gehdrigen Teilrhme 8 (a'), % invariant. Die Normalform von A wird nicht zerstiirt, wenn die Variablen, welche der gleichen Klasse angehijren, untereinander unitar transformiert werden, Durch geeignete Wahl dieser einzelnen unitairen Transformationen in den R h m e n 8 (a'), W (a"), ... konnen aber B', B", ... auf die Normalform gebracht werden. - Das Verfahren ist ohne weiteres auf irgend eine kommutative Gesamtheit von unitiiren Matrizen zu iibertragen. Der Satz von der Hauptachsentransformation der Hermitescben Formen ist ein Grenzfall des soeben bewiesenen, kann aber auch nach der gleichen Methode direkt abgeleitet werden. Der SchluD von aI2 = -..= a,, = 0

aal = ... = a,, = 0 geschieht hier vermijge der Symmetriebedingung aki = auf

Bew eis d e s S a t z e s , d a d e i n e u n i t a r e A b b i l d u n g A n o t wendig = & 1 i s t , wenn sie m i t allen u n i t a r e n Abbildungen eines g e g e b e n e n i r r e d u z i b l e n S y s t e m s U v e r t a u s c h b a r ist. Man fiihre dasjenige normale Koordinatensystem ein, in welchem A mit den Eigenwerten a, zur Diagonalmatrix wird. Sind nicht alle a einander gleich, so zerfallen die samtlichen Matrizen U der vorgegebenen Gesamtheit in der gleichen Weise, wie die a$ in Klassen untereinander gleicher zerfallen; A bewirkt dann einen simultanen Zerfall aller Natrizen des Systems U.

90 H. Weyl, Qnantenmechanik und Gruppentheorie.

46

Den S a t z uber die lineare Transformation einer nichtausgearteten schiefsymmetrischen reellen Bilinearform f

(65)

beweist man so. 'Man fasse das einzelne Zahlsystem (zl,zzl ..., q)als einen Vektor auf und bezeichne (65) als das schiefe Produkt [ g g ] der beiden Vektoren g und 9 = (yi). Man wghle einen Vektor el qk 0. Nach Voraussetzung ist [el r] nicht identisch in r gleich 0; ich kann also einen zweiten Vektor e, so finden, daD [ele,] == 1 ist. Die simultan zu erfiillenden Gleichungen k, TI = 01 [e,rl = 0 haben mindestens f - 2 linear unabhhngige Lasungen e,, . ep Auch zwischen ihnen und el, ea findet keine lineare Relation statt. Denn ist ..)

+6ae, +

$- * " +&re/ = O! so folgt durch Bildung der beiden schiefen Produkte [el?] = lZl e , ? ] = - ,&, dal3 = & = 0 wird. Man kann also el, e,, ..., ef als Koordinatensystem, als Vektorenbasis verwenden. In den darauf bezuglichen Romponenten &, qi der beiden Vektoren g und 9 laute das schiefe

F = 61el

6ae3

Produkt i,

R=

1

GemaB der Bestimmung der Grundvektoren gilt €iir die Koeffizienten Yik

= [eiekl: y,1 = 0, y 1 2 = 1; y , 3 = 0,'", yal = - 1, yza = 0; y23= 0, " ' 1

ylf

= 0,

y2p=

0.

Wegen der schiefen Symmdrie sind infolgedessen auch alle yil, yi2 mit i = 3, .. ., f gleich 0 ; und die Matrix der y i k zerfhllt in das zweireihige Quadrat

1I-Y

und eine (f

- 2) - dimensionale

schiefsym-

metrische Matrix. Durch Induktion i n bezug auf die Dimensionszahl f ergibt sich der behauptete Satz.

91

Die Eiudeutigkeit der Schrodiiigerschen Operatmen. Von

J. v. Neumann

in Berlin.

1. Die sogenannte Vertauschungsrelatioi~ PQ-QP==

.1

ist in der neuen Quantentheorie von fundamentaler Bedeutuiig, sie ist e8, die den ,,Koordinaten-Operator" R und den ,,Impuls-Operator'bP im wesentlichen definiertl). Mathematisch gesprochen, liegt darin die folgende Annahme: Seien P,Q zwei Hermitesche Punktioneloperatoren des Hilbertschen Raumes, dam werden sie durch dis Vertauschungsrelation bis auf eine Drehung des Hilbertschen Raumes, d. i. eine mitiire Transformation U, eindeutig festgelegtg). Es lie& im Wesen der Sache, da13 noch der Zusatz gemacht werden mull: vorausgesetzt, daJ3 P, Q ein irreduzibles System bilden (vgl. weiter unten Anm. 8 ) ) . Wird nun, wie es sich durch die Schrodingersche Bassung der Quantentheorie als besonders giinstig erwies, der Hilbertsche Raum als Funlrtionenraum interpretiert - der Einfachheit halber etwa als Raum aller komplexen Pnnktioneii f ( q ) (- co < q < m )

+

+m

mit endlichem J l f ( p ) ( " d q

-, so gibt

es nach Schrodinger ein besonders

-m

einfaches Losungssystem der Vertauschungsrelation

I) vgl. Born-Heisenberg-Jordan, Zeitschr. f. Phys. 34 (1929, 8. 858-888, ferner Dirac, Proc. Roy. SOC.109 (1925) U. f. Resonders in der letztgenannten Darstellung ist die Rolle dieser Relation fundtxmental. Einen interessanten Versuch zur Begrundung des im folgenden zu dishtierenden Eindeutigkeitssatzes machte Jordan, Zeitsohr. f. Phys. 37 (192G), 8. 383-356. Indemen beruht dieser auf Konvergenzannahmen tibw Potenzreihen unbeschrhkter Operatoren, deren Giiltigkeitsbereioh fraglioh ist. ") Dieselbe bewirkt ein Ersetzen von P,Q durch U P U - I , U Q U-l, woduroh weder der Hermitesohe Char&ter noch das Bostehen der Vertausohungerelation bertihrt w i d 3, vgl. Sohrijdinger, Annalen d. Phys. 79 (192G), 8. 734-756.

92

J. v. Neumenn. Eindeutigkeit der Schodingerschen Operetoren.

57 1

Sind nun dies die im wesentlichen einzigen (irreduziblen) Losungen der Vertauschungsrelation? Indessen ist die Aufgabe in dieser Form nicht geniigend prkzis formuliert. Denn als P,Q sind, wie es die Schrodingerschen LSsungen zeigen, auch unbeschrlinkte, nicht uberall definierte Operatoren ins Auge zu fassen, und fiir diese wird der Operator PQ - Q P nicht iiberall definiert sein, wiihrend es der (auf der anderen Seite der Vertauschungsrelation stehende ) ib

Operator 2rr( 1 ist. Die beiden Seiten konnen also nur gleichgesetzt werden, wenn ihre Definitionsbereiche (d. h. der der linken Seite) naher umschrieben werden. Dieser Schwierigkeit kann man folgendermafien aus dem Wege gehen: Durch formale Operatorenrechnung folgt der Vertauschungsrelation (P(x) analytisch, P'(z) seine Ableitung, vgl. Anm. I))

P-F(&)- P ( & ) P = & P ' ( & ) , 2ni -Ps

und hieraus fiir P ( z )= e

Hieraus folgt wieder formal

e

rni

--2 ; i P Q

--BQ

P ( P )e

= P(P+

h

,8 I),

Ini

-ao

und somit fur P ( x ) = e

Jb

Diese Gleichung ist von Weyl aufgestellt nnd als Ersatz der Vertanschungsrelation vorgeschlagen wordend). Ihr grol3er Vorzug besteht in folgendem: Es ist unter Urntiinden moglich, mit Hilfe der Operatoren P,Q einparaB a6

2376

-aP U ( a )= e h

--8Q

, V(@= ) e

metrige Scharen sind, uncl dem Multiplikationsgesetz

U ( a )U ( P )= q a

+P ) ,

R

zu dehiereii, die nnitiir

V ( a )V ( P )=V(a

+ B>

geniigeii5). Dann stehen auf beiden Seiten der Weylschen Gleichung 9zci

t-ap

U ( a )V ( P ) = e

J(P) U ( 4

Vgl. Weyl, Zeitschr. f. Phya. 46 (1928), Seite 1-46. Vgl. Weyl, Anm. *), ferner Stone, Proo. of Net. Acdemy 1930. Im Sohrijdingeracheii Falle wird, wie man leioht erkennt: 41 6)

2ni

u(4: f ( d - f f ( q t 4 ,

W j : f(a)+e

T P C l

f ( a )-

93

572

J. v. Neumanii.

unitlire, also beschrankte und uberall definierte Operatoren, ao daLl ihr Sinn ein vollig klarer ist. Es bliebe daher zu zeigen, da13 die einzigen irreduziblen Losungen6) der Weylschen Gleichungen die Schrodingerachen (d. h. die am Anm. ") sind. Beweiaansiitze hierfiir gab Stone (vgl. Anm. ") an, jedoch ist bisher ein Beweis auf dieser Grundlage, wie mir Herr Stone freundlichst mitteilte, nicht erbracht worden. Im folgenden sol1 der genannte Eindeutigkeitssatz bewiesen werdeu. Wir werden Rogar alle (auch die reduziblen) Losungen angeben konnen. 2. Sei 8 der Hilbertsche Raum (etwa durch alle Bolgen komplexer m

2 I x, 1' realisiert; oder much durch alle n= 1 em komplexcn Punktionen f (9), -co < g < +m, mit endlichem I f ( q ) I9 d q ) . -cn Wir benutzen die geometrische Terminologie in 8, indem wir das ,,innere Zahlen { x,, x, , .. .} mit endlichem

Betrag"

If l

=

dFn (gleich I/ 2 I x,, l9 bzw. n=l

1xf( I

q ) " d q ) einfiihren';).

--m

Wkc werden ausschIieB&h beschrankt-lineare (iiberdl definierte) Operatoren in 8 betrachten, den transponiert-konjugierten Operator des Operators A nennen wjs A" (er ist durch ( A f ,g) = ( f ,A * g ) , ( f , A g ) = ( A * f ,g) dehiert). Wir erwahnen iioch eius: Wenn der Operator A ( a ) vom Parameter a abhangt, 80 nennen wir diese Abhlingigkeit meflbar, wenn alle Funktionen ( A ( a ) f, g) (dies sind komplexe Zahlenfunktionen der. reellen Zahlenvariablen a, dagegen betrachten wir f , g als Parameter) im Lebesgue schen Sinne in 01 meJ3bar sind8). DaB mit A ( a ) auch a A (a), A ( a ) * und mit A ( a ) , B ( a ) auch A ( a ) B ( a ) meflbar ist, ist klar, aber auch A (a)B (a) ist es. Dies folgt aus den bekannten Regeln der Matrizenmultiplikation, oder auch direkt, ' p l , q9, ... sei ein vollstandiges, normierteu

+

.

0) Ein System von Operatoren A , B , . . (im vorliegenden Falle besteht es ails allen U ( a ) und TI@)) heiBt irredunibel, wenn ee aul3er 0 und dem vollen Hilbertsoheii Raume keine abgescblossene Linearmannigfaltigkeit (d. h. Hyperebene) EJl rnit der folgenden Eigensohaft gibt: mit f gehoren auoh A f , B f , . zu 92. Vgl. auch die Ausfiihrungen im Buoh von Born und Jordan, El~mentareQumtenmechanik. Berlin 1930. ') Vgl. E. Schmidt,, Rend. Ciro, Mat. Pdermo 25 (1908), S. 57-73, ferner die Arbeit des Verf,, Math. Annalen 102 (1930), S. 49-131, an die die Bezeichnungsweise anlehnt. 8) Sei qol, y e , , , . ein vollstidiges, normiertes Orthogonalsystem in 8 . Dann M N W Q1) ist f = 2 ~ , , v , s~=, 2 y Y , v n , also ( A ( a ) f , g ) = limes Z 2:,,,%,(A(~9~,~,,c~,)-

..

n=1

N , N - k c a ?n=1lt=l

9t=l

Somit geniigt die MeBbarkeit cler ( A (a)ynI,q J I )d. , h. der Matrizenelemente von im Koordinatensystem der pl, y 9 , . . .

.

94 Eindeutigkeit der Sohrodingersohen Operatoren.

573

Orth ogonalsystem : ( A ( a ) (a)f, g> = ( B (a)f, A (a)*g j =1 2 4 =

E(B( a ) f, 97,) (qn,~ * ( ag ) E ( A( a ) 9 , V n )( B (4f, V,J.

?&=l

Dasselbe gilt, wem an der Stelle von a mehrere Variable a, b , . . . stehen. Wir kehren nun zu unserem Problem zuriick, ersetzen aber in V ( p ) /? durch =hp . Dann lautet es so: Alle U ( a ) ,V ( p ) seien unitare Operatoren, die meflbar von CI, p abhangen.

Es

gelten die Relationen

U ( a )U(B)= U ( a

+s> 2

+

V ( 4W)= V ( a B ) ,

~ ( 8=) e""P(B)

~ ( a )

~ ( 0 ) .

Alle derartigen Xyeteme sind zu bestimmen. Wenn wir die (von 01, p mefibar abhiingende, unithe) Operatorenschar

x (u,p ) = e-"

idup

~ ( aV )( P )= e'

V ( P )~ ( a )

einfiihren, so konnen wir die obigen Relationen z u + i (aJ-/fv)

+ +

fi(aJb)x(Y, d ) = e f l ( a Y,B 9) zusammenfassen. Infolgedessen ist S (0,O) die Einheit, rind daher X (- a,- p ) zu E ( a , 8) reziprok, also X (01, ,f?)*= X(- a, - /I).Es sollen nun Linearaggregate der X (a,,8) betrachtet werden, diese werden folgendermafien defkiert: Sei a (a,p ) eine iiber die game a, P-Ebene absolut integrierbare Funktion, dann ist wegen der Schwarzschen Ungleichheit

I(&

NG B ) fl.19 I = I fl.1

(a,B> f, 9 ) I I I

9I9

d. h. besohrhkt, also auch das Integral

JS.(.,

B ) (8( a , B ) f, s) dadP

absolut lronvergent. Und zwar ist es, wenn wir c = S j 1 n ( a , p ) I da d p setzen, absolut 5 c . 1 f g Dabei ist es in f linear und in g konjugiertlinear. Daher ist ein Satz von F. Rielj aiiwendbarO), wonach bei festem f ein f * existiert, so da13 dieser Ausdruck fiir jedes g = ( f * ,g) ist, und zwar ist f" 1 6 c I f I. f * ist durch f bestimmt, und zwar ist die Abhangigkeit linear, wir konnen also einen linearen Operator A durch A f = f" definieren, nach der obigen Formel ist A auch beschriinkt. Wir schreiben symbolisch

I I I.

-

I

A = S J a ( ~ , p ) X ( a , Bd ) w L obwohl die Definition eigentlich lautet. a ( a , 16) 0)

( A f ,9 ) = S , f a ( ' J p ) (' heiBe der Kern von A .

f , g) dadP

Vgl. aiich a. a. 0. A m . '), Math. Annalen 102 (1930), S. 94, Anm. as).

95

574

J. v. Neumann.

Wir beweisen einige Rechenregeln fur diese Operatoren. DaB a A den 8 (- a,- /3) den Kern a a(a, p ) hat, ist Mar, A* hat wegen 8 (a,/I)*= Kern a (- a, - p), A S ( u , w) und S (u,w ) A wegen der Multiplikationsregel der S ( a , @ j den Kern e

+i(KV-pU)

a ( a - u,

- w)

bzw. e

-$. i (K W-p

21)

a ( a - u,p -w).

+

a),

Haben A, B die bzw. Kerne a ( a , b ( a , I ) ,so hat A B offenbar u ( a , ,8) I , ( a , b ) , bei A B dagegen ist eine kleine Rechnung notwendig:

+

( A B f >9 ) = ( B f J

= =

ss

(a,B )

(A

SSJS7) ( a , 8) +,

JJ

[JS e+i

A*g)

( a>

B ) f ,9 )

acyp-scv,

p- 6 4

=,fs b(a,p ) ( x ( a >p> f>' * g ) dadP d.B

a(y ---a, d-IY)(X(y,

S> f , g j d a : d B d y d d

- P ) 7) ( a , P ) dad^] ( 8 (Y,6) L s) dy dd. Der Kern von A B ist also (statt y , B schreiben wir wieder a, p, statt 1 i (a - p 5) a, 6 J 9 ) ,fJe2 a ( a - 6,p - 7 ) Z, ( t ,7 ) dE d ~ .(Die absolute ~

(Y

a ( y --

01,s

9)

Integrierbarkeit folgt aus der Deduktion.) Schliel3lich zeigen wir: wenn A verschwindet, 80 ist auch sein Kern (bis auf eine Lebesguesche Nullmenge) gleich 0. Ans A = 0 folgt namlich S (- u, - w)A S ( u , w ) = 0, also, da dieses den Kern ei(a"-8u)GG ( a , /?) hat,

1se w w %a (a,6) ( 8(a,B, f , 9 ) dad/?= 0

'

Somit ist j edenfalls

J , f P ( " , B ) a ( a , B ) ( S ( a , 8f ), 9 ) d ~ d B = O , wenn P(a, p) ein Linearaggregat von endlich vielen e i ( r a + ' P )ist, also fur jedes trigonometrische Polynom mit einer Periode p > 0 in a, ,f?. Da der zweite Yaktor absolut integrierbar ist, und der dritte beschrtinlrt, konnen wir mit dern ersten ( P ( a ,p ) ) Grenziibergange ausfiihren, falls dieser dabei gleichm5Big beschrllllrt bleibt. So konnen wir die Klasse der P( a , ,b) sukzessiv erweitern: 1. zu allen stetigen Funktionen mit einer Periode p > 0 in a,/?,2. zu allen beschriinkten stetigen Bunktionen, 3. zu allen beschriinkten Funktionen der ersten Baireschen Klasse. Wenn also 9? ein beliebiges (endliches) Rechteck in der a , /I-Ebene ist, so konnen Wir P ( a , p ) in % gleich 1 und aul3erhalb = 0 setzen, as wird:

JJa(.,B) ( m a , P 1 > f , q ) d a ~ p = O Dt

fur alle diese %. Daher ist (mit Ausnahme einer a , B-Nullmenge) CL (a, 1)( S (a,,8) f , g ) = 0. Dies gilt bei festem f , g, ist aber nur f fest, wBhrend g ein vollstiindiges normiertes Orthogonalsystem durchlauft, so @t es fur dieses f und alle genannten g auch noch mit Ausnahme einer

96

575

Eindeutigkeit der Sohrodingerschen Operatoren. a , @-Nullmenge,

In diesem Falle ist aber n ( a , ! ) 8 (a,b) f = 0. Da nun fiir f f 0 I s ( a , , B ) f l = I f J > O ist, mu13 dann a(a,B)=O sein -wornit alles bewiesen ist. 3. Die Losung des Eindeutigkeits-Problems wird durch Betrachten des Operators A = sJe-Qa'-*P'S( a , p ) da d p gewonnen. Dieser Operator ist nach unseren bisherigen Resultaten hermitesoh (a. h. A = A " ) und =+ 0, und hat aderdem die bemerkenswerte Eigenschaft, da13 sich A S ( u , v ) A nur urn einen Zahlenfaktor von A unterscheidet. In der Tat hat A den Kern e-*az-'p, also S ( u , u ) A den Kern -&i(av-flu) -&(u-u)2-&(/?-v)l e e : also A S ( u , v ) A

ss e* - ss -9

e -& (a- n c -& v- 9)' e -4i (S1)- u)e -5 (E-W-& (7-TI)' d4 dv rz+(+ a+ Q u-Q i@ -4 i u) 5 +(4 pi-+ Y+ 3i a+ 9 i u)q -& -k p -& ua- & vz

i (a9 -B 5 )

=e =e =e

5%-

a E

-a a%- & pl

-&'u - & v' -&ZL~-*V'

-)u'-*v'

=2ne

-*uz-*u'

71

ss e ss e

e-iu2-&p

e

-*a*-+p

.e

-4 p+ + ( a tu-ip -i V ) -4 -+(E-&

711

+

t d (a+u-iS -iv )71

dE dy

d ldr

(a+u-ib-(v))g-3(,~-~i(a+u-iP-io))B

d 5 dq

dy

JJ e-+"*-+y'dx

-*.1-&p

Hierin ist der erste Faktor konstant und der zweite der Kern von A , also gilt -4t P -$ 3- A . AS(u,v)A=2n;e Wir betrachten nun die Losnngen der Gleichung A f = 2 m f , da A linear-bemhrinkt ist, bilden eie eine abgeschlossene Linearmannigfaltiglreit im Hilberbchen Raume, '$R. Jede von ihnen hat die Form Ag (mit 1

g = L?rld f ) , und umgekehrt gehort jedes Ag zu h e n , da As = 2

n

a

d

ist

(man setze in der obigen Gleichung u = w = 0). Die ZLI 5132 orthogonale abgeschlossene Linewmannigfaltigkeit sei 8,die Elemente f von Zn sind durch Orthogonalitlit zu allen Elementen von 92 gekennzeichnet, d. h, zu allen A g . D.h.: immer ( f , A g )= 0 , oder: immer ( A f , g )= 0 , oder: Af=O. Wenn f , g zu 9 ' 2 gehoren, so kt 1

( X ( a , P ) f y S ( y , b ) g ) =m(s(@Yp)Af,S ( Y , W g ) 1

4 n 2 ( A8 (- y,

L-

-6 ) x ( a ,B ) A f , 9

1 e+i(aa-SY) =4aQ (AS ( @

- Y,B - 8) A f , 9 )

97

576

J. v. Neumann.

.

Sei nun . . ein normiertes Orthogonalsystem, welches in 91 vollsthdig ist (d. h. %Q aufgespannt, die Zahl seiner Elemente ist endlich oder abzahlbar unendlich). Aus ('pm,p ' ,) = d,,, (d. i. 1 fur m = n, 0 sonst) folgt -& (a- UP- a (1-SP+& i (ad -P u) ( # ( a , PI F,,,, 8 ( Y , 6) 9,) = e ~ , nDie durch alle S ( a , ,6) p ' , ( n fest, a, ,6 variieren) aufgespannte abgesohlossene Linearmannigfaltigkeit heil3e p, nach der obigen Formel sind fur m n !$, '@fl zueinander orthogonal, Die p2, . mogen eusammen die abgeschlossene Linearmannigfaltigkeit Q aufspannen, die zu Q komplementare abgeschlossene Linearmannigfaltigkeit sei 5 . Da, jedes 8 ( y , 6) die 8 ( a ,S ) v,, (bis auf Zahlenfaktoren) in ebensolche transformiert, bildet es !&, auf ein TeiI von sich ab; da dasselbe fiir S ( y , S)-'= S (- y , - 6) gilt, ist ?j3, genau invariant. Also ist auch S und S invariant. (5 umfaBt alle p,,,also alle qn, also YR, daher liegt 5 in (57. Somit gilt in Z stets A f = 0. Nun gelten alle msere uber A migestellten Betrachtungen schon in &, d e n die X (a,8) konnen als Operatoren in Z angesehen werden, da dieses ihnen gegeniiber invariant ist. Da nun in Z A = 0 ist, kann nach unserem Beweise in 7X niemals f + 0 sein. Also enthalt S nur die 0, G ist der Hilbertsche Raum, d. h.: Q,,'j&, . . . spannen den vollen Hilbertschen Raum auf. Der Hilbertsche Raum erscheint somit als in eine endliche oder ahzahIbar unendliche ZahI von (paarweise orthogonalen) Unterriiumen Ipl, !&,... zerlegt; jeder derselben ist gegeniiber allen S (g, ,8) invariant, es geniigt also das Verhalten der S ( a , /I)(d. i. der U ( a ) , V (/I)) in einem jeden derselben gesondert zu ermitteln, urn uber sie restlos informiert zu sein. (Im Palle der Irreduzibilitat darf es natiirlich nur ein p,, geben, und dieses ist dann der volle Hilbertsche Raum.) In wissen wir nun uber die 8 (a, ) die f olgenden Tatsachen : Wir nennen Pn 9,S(CX,/~)P),,fa,@. D a m gilt: $2

..

+

und die Linearaggregate endlich vieler fm,@ (die beliebig wahlbar sind!) liegen in 'g iiberall dicht. Wenn wir nun zeigen konnen, daB irgend zwei solche p, in deren jedem eine Schar von unjtaren Operatoren S ( a , 6 ) und Punkten fa,@ mit den obigen Eigenschaften gegeben ist, isomorph sind, so sind wir am Ziele. Isomorphie bedeutet : Existenz einer ein-eindeutigen, linearen und Iangentreuen Abbildung der beiden !'$ aufeinander, die die fa,@ und S(a, @)des ! in dieselben des anderen iiberfiihrt. einen $

98

577

Eindeutigkeit der Schriidingerachen Operatoren.

Unsere Pormel fiir ( f o p , f y 8 ) erlaubt fiir jedes Linearaggregat endlich vieler faa den Absolutwert zu berechnen, wenn wir also die gleichlautenden f -Linearaggregate beider '$ einander zuordnen, so haben wir eine ein?b eindeutige, lineare und langentreue Abbildung dieser Mengen aufeinander. Da sie in den bzw. $' 3 iiberall dicht sind, s h d sie stetig auf die gansen ausdehnbar. Dabei bleiben Linearitat und Langentreue, also auch Eineindeutigkeit, erhalten. Die bzw. fa, entsprechen einander. Wegen der 8 ( y , 6) f@,,-Formeln gehen auch die S ( y , 6) in ihre entsprechenden iiber: wenigstens fiir die fa,,B, aber d a m auch fiir deren Linearaggregate und Damit ist, wenn wir wieder z u den Haufungspunkte - also in ganz U ( a ) ,I7(/?) zuruckkehren, folgendes bewiesen : Bin Bystem unitarer Operatoren U ( a ) , V (8) nebst einem System von Punkten fa, b , d i e zusammen den ganzen Hilbertwhen Raum aufspannen, ist durch die Eigenschajten

bis auf eine unitare Transformationlo) eindeutig festgelegt. Ein System unitarer Operatoren U (a), V (,8) mit den Weylschen Multiplikations- Relationen (vgl. 1.) ist entweder eines der soeben genannten Systeme, oder es entsteht dadurch, dap der Eilbertsche Raum in endlich oder abzahlbar unendlich viele (paarweise orthogonale, Hilbertsche) Unterraume xerfallt, und in jedem derselben ein solches Xystem angenommen wird. D.h. es entsteht durch das Zusammenfiigen dersetben. Die irreduziblen Losungen sind offenbar die ersteren (die Zahl der h e a r unabhgngigen Losungen von A f= 2 n f nimmt fur sie ihren Minimalwert 1an). 4. Zum Schlul3 noch ehige Zusatzbemerkungen. Im Falle der Schrodingerschen Operatoren haben mir U ( a ) ,V ( p ) in 7a

f ( q ) - e e " a @ f ( q(GI ) stattp!), daher

Anm.6) angegeben: f ( q ) - f ( q + a ) , ist &'(a, p ) : f ( q ) - + e iB(g+$) f ( q

$q +m

f(q>

e

-4q'-

sind also die c.e-"',

&4' =

f

+ a ) , und,

( 6 )drj'.

wie man leicht berechnet, A:

Die einzigen Losungen von A f

somit ist cpl=

y,(q) = n-'

-11-42

e

, und

Man verihiert dann leicht maere diesbeziiglichen Formeln.

-

-

= 2 TZ f

99

578

J. v. Neumann. Eindeutigkeit der Schrodingerschen Operatoren.

Bei quantenmeohanischen Problemen mit k (= 1, 2, graden tritt das allgemeine Vertauschungsrelationen-System

. ..)

Preiheits-

auf. Durch Einfuhren von entstehen hieraus die Weylschen Relationen

Auch hier ist der Eindeutigkeitsbeweis mit unseren Methoden aus 2. bis 3.

durchfuhrbar. Wir setzen

Dam konnen wir, genau wie in 3,)

+;-... - au;- 2.;- . . AS(u,, .,., uk,vl, ..., q c ) A = ( 2 n ) " e beweisen, und (durch Untersuchen der Losungeii von A f = (2 n)".f ) genau wie dort a m Ziel kommen. Vom allgemeinen darstellungstheoretischeii Gesichtspunkte aus betrachtet, ist unsere Betrachtungsweise mit der Brobeniusschen Behendluig endlicher Gruppen mittels ihrer ,,charakteristischen Einheiten" verwandt bzw. mit der Weylschen Untersuchung abgeschlossener kontinuierlicher Gruppen mit Hilfe ihrer Gruppenzahlenl'). Die Operatoren A = (c ( a , ,6) S ( a ,/?)d a d ,8 sind niimlich als ,,Gruppenzahlen der S ( a ,S) - Gruppe deutbnr, und A S ( u , w ) A = c,,.A ( G ~ eine , ~ Zahl!) ist der dehierenden Eigenschaft der ,,primitiven" cherakteristischen Einheiten gleichwertig. *

ss

'I)

(1926),

Vgl. Frobenius, Berl. Ber. 1896

11.

f . , Peter und Weyl, Math. Annalen 96

S. 737-755. (Eingegaagen am 31. 8. 1930.)

100 J U N E 1, 1932

PHYSICAL REVIEW

VOLUME 40

On the Quantum Correction For Thermodynamic Equilibrium B y E. WIGNER Department of Physics, Princcton University (Received March 14, 1932) The probability of a configuration is given in classical theory by the Boltzrnann formula exp [ - V / h T ]where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. T h e formula is developed for this correction by means of a probability function and the result discussed.

1 N classical statistical mechanics the relative probability for the range $1 to Pl+d+l; $2 to P z S d P z ; . * . ; $n to $,,+d@, for the momenta and x1 to xI+dx1; x 2 to x 2 + d x z ; . . . ; x, to x , + d x , for the coordinates' is given for statistical equilibrium by the Gibbs-Boltzmann formula

I

P(x1,.

. . , x , ; p l , . . .,pn)dxl. . .dx,dpl. . .dp,

= e-fltdxl.

. .dx,dpl. . .dpn(l)

where E is the sum of the kinetic and potential energy V (2)

and p is the reciprocal temperature T divided by the Boltzmann constant /3 = 1 / k T .

(3)

In quantum theory there does not exist any similar simple expression for the probability, because one cannot ask for the simultaneous probability for the coordinates and momenta. Moreover, it is not possible to derive a simple expression even for the relative probabilities of the coordinates alone-as is given in classical theory by e-flV(? ' . ' =n). One sees this by considering t h a t this expression would give at once the square of the wave function of the lowest state [ +,,(xl . * . x,) when p = to is inserted and on the other hand we know that it is not possible, in general, to derive a closed formula for the latter. The thermodynamics of quantum mechanical systems is in principle, however, given by a formula of Neumann,' who has shown that the mean value of any physical quantity is, (apart from a normalizing constant depending only on temperature), the sum of the diagonal elements of the matrix

l2

Qe-flH

(4)

where Q is the matrix (operator) of the quantity under consideration and H is the Hamiltonian of the system. As the diagonal sum is an invariant under 1

J. von Neumann, Gott. Nachr. p. 273, 1927. 749

101

750

E. WIGNER

transformations, one can choose any matrix or operator-representation for the Q and H. In building the exponential of H o n e must, of course, take into account the non-comrnutability of the different parts of H . 2

I t does not seem to be easy to make explicit calculations with the form (4)of the mean value. One may resort therefore to the following method. If a wave function +(xl . . . x,) is given one may build the following expression2 P(x1,

*

*

. , x n ; PI, =

(')hr,

* *

j

Jm

Pn)

--

* *

.Jdyl.

* '

dyn+(xl+

y l *

. . xn

+

yn>*

+(xl - y l . . . x, - yn)e2i(~,~i+...+~nY,)/h (5) and call i t the probability-function of the simultaneous values of X I . . . xn for the coordinates and $1 . . p , for the momenta. In (5), as throughout this paper, h is the Planck constant divided by 2s and the integration with respect to the y has to be carried out from --OO to 43. Expression ( 5 ) is real, but not everywhere positive. I t has the property, that it gives, when integrated with respect to the p, the correct probabilities I$(xl . . x,) for the different values of the coordinates and also it gives, when integrated with respect to the x , the correct quantum mechanical probabilities

-

Is

1

Jw. . . J $(xl

. . . ~,),-;(~~~:~+...+~n+n)/hdx~ . . . dx,

--m

l2

for the momenta pl, . . , p,. T h e first fact follows simply from the theorem about the Fourier integral and one gets the second by introducing x k + Y k =Uk; Xk-yk=vk into (5). Hence it follows, furthermore, that one may get the correct expectation values of any function of the coordinates or the momenta for the state $ by the normal probability calculation with ( 5 ) . As expectation values are additive this even holds for a sum of a function of the coordinates and a function of the momenta as, e.g., the energy H. I n formulas, i t is --m

--m

(6)

for any $ , j , g, if P is given by (5). 2

This expression was found by L. Szilard and the present author some years ago for another

purpose.

102 751

QUANTUM CORRECTION

Of course P(x1, . . . , x,; PI, . . . , $,) cannot be really interpreted as the simultaneous probability for coordinates and momenta, as is clear from the fact, that i t may take negative values. But of course this must not hinder the use of it in calculations as an auxiliary function which obeys many relations we would expect from such a probability. I t should be noted, furthermore, that ( 5 ) is not the only bilinear expression in 9, which satisfies (6). There must be a great freedom in the expression ( S ) , as it makes from a function 9 of n variables one with 272 variables. I t may be shown, however, that there does not exist any expression P(x1 . ' . x,; p1 . . . p,) which is bilinear in $, satisfies (6) and is everywhere (for all values of x l , . . xn, $11 . . f i n ) positive, so ( 5 ) was chosen from all possible expressions, because it seems to be the simplest. If $(xl, . . , x,) changes according to the second Schrodinger equation 1

(7)

the change of P(x1, .

ap

- = -

at

. , x,,;$1,

+

p k ap c--+c n

k-1

mk a x k

.

*

*

, $,) is given by

axl+. . * + X n v axlXl.

-.

( h / 2i) A

* .+Xn--l

xl! . . . A,!

aXnXn

a A I+ aplkI.

* . .+A,p

. . ap,~n (8)

where the last summation has to be extended over all positive integer values of X I , . . . , A, for which the sum Xl+Xz+ . . * +X, is odd. In fact we get for aP/at by ( 5 ) and (7)

(9)

+ hi

-"v(~l

+

yl,

- V(x1- yl, . . *

*

1

* 1

xn

xn

+ yn)

- yn)]+(x1+

~

1

.,

* * 1

xn

+

Yn)*+(X1-

yl,

* *

'xn

- yn)

1

Here one can replace the differentiations with respect to x k by differentiations with respect to y k and perform in the first two terms one partial integration with respect to y k . In the last term we can develop V(xlfy1, . . . , X n + y n ) and V ( x l - y l , . * . , x,-yn) in a Taylor series with respect to the y and get

103

752

E. WIGNER

(10)

which is identical with (8) if one replaces now the differentiations with respect to y k by differentiations with respect to x k . Of course, ( 8 ) is legitimate only if it is possible to develop the potential energy V in a Taylor series, Eq. (8) shows the close analogy between the probability function of the classical mechanics and our P : indeed the equation of continuity

-a p- -at

ap av aP c---+ c--P k

k

m k

a x k

k a x k

a p k

differs from (8) only in terms of a t least the second power of h and a t least the third derivative of V . Expression (8) is even identical with the classical when V has no third and higher derivatives as, e.g., in a system of oscillators. There is an alternative form for aP/at, which however will not be used later on. I t is

+

. ' . J d j , . . . dj,P(?cl, . ' ' , x,; P1

J W

+ j,,

(11) * *

f

, P,

+ j,)J(x.,,

* * *

, x,;

--m

where J(x1, . . . , x,; j l , , . . , j,) can be interpreted as the probability of a jump in the momenta with the amounts j l , ' . . , j , for the configuration x l , . . , x,. T h e probability of this jump is given by J(x1,

'

'

'

7

xn; j ~ ,'

' 1

jn)

- T/' (

-

,, . . . , xn - y,) ] e-

( 2 i I h ) (Y I i

,+. .S + Y j~ d

(114

t h a t is, by the Fourier expansion coefficients of the potential V(xl, . . , x,,). This form clearly shows the quantum mechanical nature of our P: the momenta change discontinuously by amounts which would be half the momenta of light quanta if the potential were composed of light.z*T o derive (11) one can insert both for P and J their respective values (5) and (lla) on the rigLt hand side of (11). In the first term one can replace f i k e 2 i ( p l M + . ' . + p n y n ) l h by +

28

Cf. F. Bloch, Zeits. f. Physik 52, 555 (1929).

104 753

QUANTUM CORRECTION

( k / 2 i )(d/dyk)e2i(pl%+*.. +PnYn)/* and then perform a partial integration with respect to Y k . Then one can replace the differentiation with respect to y by differentiationwith respect to x , upon which some terms cancel and the rest goes over t o h c-J.. . J d Y l . 2im k

* . . d y n [ ~ 2 + ( x l + ~ ~ , * , ~ n + y n )+(x1-~1, .. axk2 *

*

xn-yn)

(12)

which is just what we need for the left side of (11). By integrating the second term on the right side of (11)

- V(x1 - 81 . . . xn - zn) 3 e - 2 i ( z

.

ir+.

,+gn

id / h

with respect to z and j one gets because of the Fourier theoremS

J.

+

(i/h) * . J d y l * * . dyn+(xl+ y l ’ . . xn yn)*+(xl- y l . * . xn - yn> e2i(piyl+...+~n~n)’h. [v(xl y1 . . . xn yn) - ~ ( 3 ~ 1yl * * . xn - yn)] (12a)

+

+

and this gives the second part of the left side of (11).

3 So far we have defined only a probability function for pure states, which gives us the correct expectation values for quantities f(P1 * . * Pn)+ g(x1 . . If, however, we have a mixtureI4e.g., the pure states $1, $z, $3, * ‘ with the respective probabilities w1, w2, wS, . . * (with wl+wz+wa+ * . . = 1) the normal probability calculation suggests a probability function (13) P(x1, * * ’ 9 xn, 91, * . * , Pn) = C wx Px(x~, * * - 7 xn, * * Pn) A

where PAis the probability function for $A. This probability function gives obviously the correct expectation values for all quantities, for which (5) gives correct expectation values and therefore will be adopted. For a system in statistical equilibrium at the temperature T = l/kP the relative probability of a stationary state +A is e-oEA where E Ais the energy of $x. Therefore the probability function is a part from a constant Cf. e. g., R. Courant und D. Hilbert, Methoden der mathematischen Physik I. Berlin 1924. p. 62, Eq. (29). J. v. Neumann, Gott Nachr. 245, 1927. L. Landau, Zeits. f. Physik 45,430 (1927).

’

754

E. WIGNER

e-BEh#(xl

Now

C fiA

(

~

* 1* *

- yl . . . xn - yn)e2'(pi"~+...+P"~")/h, ~ , ) * j( E A ) fix

(211 *

*

(14)

. Vn)

A

is that matrix element of the operatorf(H), ( H is the energy operator) which is in the u 1 * . un row and V I * . . v n column. Therefore (14) may be written as

P(x1 *

-

* *

x n ; PI

J:. .

S

dy,

' '

Pn)

. . . ~ y , e ~ [ ( z i + U ~ ) P i + . ~ ~ + ( ~ n + ~ n ) P n l ~ ~.,.=, [ e -+un;z,-ul...zn-vn ~~~~~+~~

-m ,e - i [ ( z , - U , ) P , + .

so that we have under the integral sign the x ~ + y I *

. . + ( ~ n - ~ n ) ~ n l/ h . (15) . . x,+y,; x l - y ~ . *

xn-yn element of the matrix e - B f l H transformed by the diagonal matrix e ' ( p l z l + * . . + p n Z n ) l h . Instead of transforming e-PH we can transform H first and then take the exponential with the transformed expression. By transforming H we get the operator (the p are numbers, not operators!)

which is equal to (16)

where c =

2 -+ V(x1, Pk2

k=l

'

, x,).

(17)

2mk

(18)

By calculating the mean value of a quantity Q =f(P1, . . . , Pn)fg(x1, . . . , x n ) by (18) one has to obtain the same result as by using the original expression (4) of Neumann. If we are dealing with a system, the behavior of which in statistical equilibrium is nearly correctly given by the classical theory, we can expand (18) into a power of h and keep the first few terms only. The term with the zero power of h is -P)'e'/r! Now r' is the operator of multiplication with the r

cp(

106

755

QUANTUM CORRECTION

power of (17). I t s xl+yl, sequen tly

+ yl, . .

xn

*

. , xn+yn; x l - y ~ , . . . , x,-y,, element is con-

*

+ yn)'6(xl+

yl, x1

- yl)

' ' '

&(xn

+ yn, xn -

yn).

a", . . . ) only depends on the difference of its two arguments, one can write 6 (-2y1) . . . 6(-2yn) for the last factors and perform the integration by introducing - 2 ~ 1 , . . , -2yn as new variables. T h e terms with the zero power of h, arising from the first part of (16) only, give thus

As 6 (also 6',

(1/2")

C ( - ~ ) r e ( x l , . , x,,)r/r! = e-b6/2*

(19) (19)

*

r

which is just the classical expression. T h e higher approximations of the probability function can be calculated in a very similar way. The terms of e - f l , involving the first power of the second part of B only, are

(20) By replacing all operators by symbolic integral-kernels one gets for the x l + y l , . . . , x,+y,; xl-yy,, . . . , x n - y n element of the operator (20)

h2

- -6(-

2y1)

2mk

. . . 6',(-

2Yk)

'

. . 6 ( - 2y,)

1

€(XI

- y1,.

* *

, xn -

y,)r-p.

Now

so that the summation over p and r can be performed in (21). By introducing again new variables w l , . . . , w , for - 2yl, . . . , - 2y, and performing the integration one has

1 2"

mk

dWk

2mk

dWk2 g-8f(Sl....

e(x1,

,lk-Wk/2,..*

,zn)

- e-bt(Zk,."

. . , X!+ - qwk, ' . . , x,) - €(XI, '

* *

,Zk+Wk/2.".

. , Xk

+

vZ?t)

+Wk,

. . , x,) *

where wk = O must be inserted after differentiation. T h e first differential quotient vanishes at wk=O, as the expression to be differentiated is an even function of ' w k . T h e second part gives

107 756

E. WIGNER

In principle it is possible to calculate in the same way the terms involving the higher powers of the second part of H also, the summation over r and the quantities corresponding to our p can always be performed in a very similar way. In practice, however, the computation becomes too laborious. Still i t is clear, t h a t if we develop our probability function for thermal equilibrium in a power series of h

XI, . * , xn; P I , . . . , p,,)

= e-pr

+ h/i + h2jz + . .

(22) (we can omit the factor 1 / 2 " before e-oc, as we are dealing with relative probabilities anyway) all terms will be quite definite functions of the ;b, V and t h e different partial derivatives of t h e latter. Furthermore it is easy t o see, t h a t f k will not involve higher derivatives of V than the k-th nor higher powers of p than the k-th. These facts enable us to calculate t h e higher terms of (22) in a somewhat simpler way, than the direct expansion of (18) would be. T h e state ( 2 2 ) is certainly stationary, so t h a t i t would give identically aP/at=O when inserted into (8). B y equating the coefficients of the different powers of h in aP/at to zero one gets the following equations : *

(23, 0) (23,1)

av a j 2 x---h2 a%-@ c - _ _ _+ c--h2 - z-.-----0 8 P k

k

d3V

af2

m k dXk

k f i

k

dxk

k

a p k

a3v

a3e-Pt

aXk2dXi

apk'api

aXk3

24

apk3

(23, ( 2 3 , 2) 2)

and so on. T h e first of these equations is an identity because of (17), as i t must be; ( 2 3 , a ) , ( 2 3 , 2 ) , . . . will determine f1, fi, . . . respectively. All Eqs. ( 2 3 , a ) are linear inhomogeneous partial differential equations for the unknown f. From one solution fa of ( 2 3 , a ) one obtains the general solution b y adding to i t the general solution F of the homogeneous part of ( 2 3 , a ) , which is always

av z---+C--=O. k

p k

dF

mk

3%

aF

k

axk a p k

This equation in turn is t h e classical equation for the stationary character of the probability distribution F(x1, . . . , x n ; @ I , . . . , p,). I t has in general only one solution which contains only a finite number of derivatives of V , namely ~ ( x l* ,*

, x m ; P I , . . ' , pn)

= F(

c k

In fact, if it had other integrals, like

Pk2 I _

2mk

108

then all mechanical problems would have in addition to the energy-integral further integrals of the form (24) which, of course, is not true. One solution of ( 2 3 , l ) isfi = O and the most general we have to consider is therefore f l = F ( E ) .We have to take however F ( E )=O as fl has to vanish for a constant V . So we get f i = 0, as we know it already from the direct expansion of (18). Thc same holds consequently forf3,f6, . . . , as the inhomogeneous part of the equation for f 3 only contains fl, the inhomogeneous part of the equation for fs only f l and f 3 , and so on. For f2 one easily gets

(25) as a solution of (23, 2) and it is also clear, that this is the solution we need. The first two terms of fz we have already directly computed ( Z l ) , the third arises from terms with the second power of the second part of i?. Similarly f4 is for one degree of freedom ( n = 1) 64m2P-2 eo‘jd = H 4 ( q )[p2V“2/72- pV/”’/120]

+ Hz(q) [/33V’21‘”/18- 2p2V”z/15 - p2Vv’V”’/lS + pV1”’/15] + Ho(q) [p4Vt4/18 - 2Zp31/”2V’”/45 + 2p2V“2/5 + 8/”’Y’’’/15 ( 2 6 ) - 4/3V””/15]

where H , is the r-th Hermitean polynomial and 4 =p11zp/(Zm)1/2. I t does not seem to be easy to get a simple closed expression forfk, but it is quite possible to calculate all of them successively. A discussion of Eqs. (23) shows, t h a t the g in P(x1, .

. . , x,; P I , . . . , pn)

=

e-af(l

+ k2gz + k4g4 + . . . )

(27)

are rational expressions in the derivatives of V only (do not contain V itself) and all terms of g k contain k differentiations and as functions of the p are polynomials of not higher than the k-th degree. The first term in (27) with the zero power of h is the only one, which occurs in classical theory. There is no term with the first power, so that if one can develop a property in a power series with respect to h, the deviation from the classical theory goes a t least with the second power of k in thermal equilibrium. One familiar example for this is the inner energy of the oscillator, where the term with the first power of h vanishes just in consequence of the zero point energy. The second term can be interpreted as meaning that a quick variation of the probability function with the coordinates is unlikely, as i t would mean a quick variation, a short wave-length, in the wave functions. This however would have the consequence of a high kinetic energy. T h e quantum mechanical probability is therefore something like the integral of the classical expression e-ot over a finite range of coordinates of the magnitude -h/p where p is the mean momentum ~ ( k T r n ) The ~ / ~correction . terms of (27) have, among other effects,

109

758

E. WIGNER

the consequence t h a t the probability for a particle being in a narrow hole is smaller than would be in classical statistics. From now on we will keep only the first two terms of (27). 4 From (25) one easily calculates the relative probabilities of the different configurations by integration with respect to the p :

s

*

..

Sdpl

..

'

dp,P(zl

. . . x,; p 1 . . . p,,) (28)

Hence the mean potential energy is

(29) I-

'

24

where dx is written for dxl . dx, and t h e higher power terms of h are omitted, Similarly the mean value of the kinetic energy is I

(30) J

This formula also is correct only within the second power of h ; in order to derive i t one has to perform again some partial integrations with respect to the x. Eqs. (28), (29), (30) have a strict quantum mechanical meaning and i t should be possible to derive them also from (4).One sees that the kinetic energy is in all cases larger than the classical expression +nkT. 5

One fact still needs to be mentioned. We assumed t h a t the probability of a state with the energy E is given by e--BE. This is not true in general, since t h e P a d principle forbids some states altogether. The corrections thus introduced by the Bose or Fermi statistics even give terms with the first power of h, so t h a t i t seems, t h a t as long a s one cannot take the Bose of Fermi statistics into account, Eq. (25) cannot be applied to a n assembly of identical par-

110 Q U A N T U M CORRECTION

759

ticles, as, e.g., a gas. There is reason to believe however, t h a t because of the large radii of the atoms this is not true and the corrections due to Fermi and Bose statistics may be neglected for moderately low temperatures. T h e second virial coefficient was first calculated i n quantum mechanics by F. London on the basis of his theory of inneratomic force^.^ He also pointed out that quantum effects should be taken into account at lower temperatures. Slater and Kirkwooda gave a more exact expression for the inneratomic potential of He and Kirkwood and Keyes' calculated on this basis the classical part of the second virial coefficient of He. H. Margenaus and Kirkwoodg performed the calculations for the quantum-correction. T h e present author also tried to calculate it by the method just outlined. H e got results, which differ from those of Margenau and Kirkwood in some cases by more than 100 percent.1° I t does not seem however to be easy to compare these results with experiment, as the classical part of the second virial coefficient is a t low temperatures so sensitive to small variations of the parameters occurring in the expression of the interatomic potential, t h a t i t changes by more than 20 percent if the parameter in the exponential (2.43) is changed by 4 percent and i t does not seem to be possible to determine the latter within this accuracy.

F. London, Zeits. f . Physik63,245 (1930). J. C. Slater and J . G. Kirkwood, Phys. Rev. 37,682 (1931). 7 J . G. Kirkwood and F. G. Keyes, Phys. Rev. 38,516 (1931). H. Margenau, Proc. Nat. Acad. 18, 56, 230 (1932). Cf. also J. C. Slater, Phys. Rev. 38, 237 (1931). 9 J. G. Kirkwood, Phys. Zeits. 33,39 (1932). 10 I am very much indebted to V. Rojansky for his kind assistance with these calculations. The reason for the disagreement between our results and those of Margenau and Kirkwood may be the fact that they did not apply any corrections for the continuous part of the spectrum. I n a paper which appeared recently in the Zeits. f. Physik (74, 295 (1932)) F. Bloch gets results which are somewhat similar to those of the present paper. (Note added at proof.) 6

111 October 1946

Physica XII, no 7

ON THE PRINCIPLES O F ELEMENTARY QUANTUM MECHANICS by H. J. GROENEWOLD Natuurkundig Laboratorium der Rijks-Universiteit te Groningen

Summary Our problems are about a t h e correspondence a +--+ a between physical quantities a and quant u m operators a (quantization) a n d p t h e possibility of understanding t h e statistical character of q u a n t u m mechanics b y averaging over uniquely determined processes as in classical statistical mechanics (interpretation). a and p are closely connected. Their meaning depends on the notioneof observability. We have tried t o p u t these problems in a form which is fit fcr discussion. We could not bring them t o a n issue. (We are inclined t o restrict the meaning of CL t o t h e trivial correspondence a + a (for lim fi -+0) a n h to deny t h e possibility suggested in p). Meanwhile special attention has been paid t o t h e measuring process (coupling, entanglement; ignoration, infringement; selection, measurement). For t h e sake of simplicity the discussion has been confined t o element a r y non-relativistic quantum mechanics of scalar (spinless) systems with one linear degree of freedom without exchange. Exact mathematical rigour has not been aimed a t .

1. Statistics and correspondence. 1.01 Meaning. When poring over M the correspondence a +-+ a between observables n and the operators a,b y which they are represented in elementary quantum mechanics, 9 the statistical character of elementary quantum mechanics (we need 0: for p), we run a continuous risk of lapsing into meaningiess problems. One should keep in mind the meaning of the conceptions and statements used. We only consider - 405

Physica XTI

26 *

112

406

H.J . GROENEWOLD

M , : observational m e a n i n g , determined by the relation with what is (in a certain connection) understood as observation,

M,: formal m e a n i n g , determined with respect to the mathematical formalism without regard t o observation. Only hlo is of physical interest, M , is only of academic interest. Dealing with hl, may sometimes suggest ideas, fruitful in the sense of M,, but may often lead one astray. 1.02 Quantization. Very simple systems suffice for demonstrating the essential features of cc and p. I n elementary classical point mechanics a system is described b y the coordinates q of the particles and the conjugate momenta p . We only write down a single set p,q, corresponding to one degree of freedom. Any other measurable quantity (observable) a of the system is a function a(p,q) of p and q (and possibly of the time t). The equations of motion can be expressed in terms of P o i s s o n brackets

(1.01) When the same system is treated in elementary quantum mechanics, the (real) quantities a are replaced by (H e r m i t i a n) operators a, which now represent the observables. I n the equations of motion the P o i s s o n brackets (1.01) are replaced b y the operator brackets

i

[a,b] =-(ab-ba) F,

h (h=--, h P l a n c k ’ s constantof action). (1.02) 2x

Problem ct1 is to find the correspondence a --+ a (other problems are stated further on). 1.03 Statistical character. The statements of quantum mechanics on observations are in general of statistical character. Problem g is whether the statistical quantum processes could be described by a statistical average over uniquely determined processes (statistical description of the 1st kind, type S’)or not (statistical description of the 2nd kind, type S2).The observability of the uniquely determined processes may be required (proper statistical description, type So)or not (formal statistical description, type S,). (Classical statistical mechanics, e.g. are properly of the 1st kind, type Si). 1.04 Transition ofierator, Before going on we have to deal for a moment with the operators and the wave functions.

u

113 ON T H E PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

407

The H e r m i t i a n operators a form a non-commutative ring. The normalized elements (wave function$) of (generalized) H i 1b e r t space on which they act from the left are denoted by 'pp, the adjoint elements on which they act from the right are denoted by .9; Unless otherwise stated the inner product of p': and 'pv is simply written cpL'p,,. The outer product of 'pf and 'pv defines the transition operator

kvp = cpy'p:> kip = kpv. Take a complete system of orthonormal wave functions orthonormality is expressed by cp:cpv

= 8,VP

the completeness b y

2 cp,cpf

= 1.

(1.03) 'pv.

The

(1..04)

(1.05)

P

I n continuous regions of the parameter p the W e i e r s t r a s z 8-symbol must be replaced b y the D i r a c %function and the sum by an integral. (1.04) and (1.05) show that every (normalizable) function cp can be expanded into 'p = C f P p ', CL

with f , = p': cp.

(1.06)

k, and k& transform 'ppI and y;, according to ( 1.07) k",'p$ = 'pv$L,, and ' p : q L = 8,'pTv t (that is why they are called transition operators). (1.04) gives k,vkv,p,= kp,&,,,. (1.08) In particular k,, and k, are for p # v orthogonal projection operators (belonging to 'pp and yv respectively). The trace of an operator a (resulting when a acts towards the right upon itself from the left, or opposite; when it bites its tail) is (according t o (1.05))defined b y (1.09) Tra = C p': a p'., P

(Because the right hand member is invariant under unitary transformations of the ? ,, this definition is independent of the special choice of the complete orthonormal system of p,). This gives

Wkvpa) = 9: a 'pv. (1.04) and (1.05) can be written Trk, = a,, C k,, = 1 P

(1.10) (1.11)

(1.12)

114

4C8

H. J . GROENEWOLD

and further imply

Tr(k,ukv#p*) = 6p#,6,*, C k,Tr(k,,a)

= a (for every

(1.13) a).

(1.14)

(1.13) and (1.14) show t h a t every operator a (with adjoint at) can be expanded into a = C xvpkPuwith a,,,= Tr(k,,a). (1.15) P.”

a”, is the matrix element (1.10) of a with respect to (py and yp. It follows further that if Tr(ac) = 0 for every a, then c = 0 and therefore (1.14) is equivalent t o

CI Tr(k,b) Tr(k,,a)

=

Tr(ab) (for every a and b). (1.16)

Further

Tr(ab) = Tr(ba).

(1.17)

When a is a H e r m i t i a n operator

a+ = a, ,:a = a,,,

(1.18)

(the asterik denotes the complex conjugate), the system of eigenfunctions 9, with eigenvalues up

acp,

= a,cp,

(1.19)

can serve as reference system. I n this representation (1.15)takes the diagonal form a = C upk,,. (1.20) P

1.05 Statistical operutor l). The quantum state of a system is said to be pure, if it is represented by a wave function rpp. The statistical operator of the state is defined b y the projection operator k,, of cpp. We will see that the part of the statistical operator is much similar to that of a statistical distribution function. The most general quant u m state of the system is a statistical mixture of (not necessarily orthogonal) pure states with projection operators k,, and nonnegative weights K,, which are normalized by

X k p = 1.

(1.21)

P

(In some cases the sum diverges and the right member actually should symbollically be written as a &function). The statistical operator cf the mixture is (in the same way as it would be done for

115 ON T H E PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

409

a distribution function) defined by

k = Z: k,kp,

(1.22)

P

and because of (1.21) normalized by

T r k = 1.

(1.23)

(we will always write 1 for the right member, though in some cases it actually should be written as a %function). For brevity we often speak of the state (or mixture) k. An arbitrary non-negative definite normalized H e r m i t i a n operator k (Trk = 1) has non-negative eigenvalues k,, for which Z k,= 1 CL

and corresponding eigenstates with projection operators k,. Therefore k can according to (1.20) be expanded in the form (1.22) and represents a mixture of its (orthogonal) eigenstates with weights given by the eigenvalues. The statistical operator k,, of a pure state is from the nature of the case idempotent (k2 = k,,). If on the other hand an idempotent Y normalized H e r m i t 1 a n operator k is expanded with respect to its eigenstates k,, with eigenvalues .k,, we get

k2 = k, k t

= k,;

Trk

=

1, C k , = 1 , P

/

(1.24)

so that one eigenvalue k, is 1 , all other are 0. Then k is the projection operator of the pure state 3,

k = k,”.

(1.25)

Therefore pure states and only these have idempotent statistical operators. Suppose the normalized statistical operator k of an arbitrary quantum state is expanded in some way into other normalized (but not necessarily orthogonal) statistical operators k, with non-negative weights k, k = C k,k,; k , > 0. (1.26) I

‘This gives

k - k2 = X k, (k, - k:)

+ Q C K,k,(k, - k,)2.

(1.27)

1,s

1

If we expand with respect to pure states k, (kp = k,), (1.27) becomes k - k2 = C k,k,(k, - k,)2 (1.28)

+

116

410

H.J. GROENEWOLD

This shows that k - k2 is a non-negative definite operator. If the given state is pure (k2= k) all terms at the right hand side of (1.27) (which are non-negative definite) must vanish separately. For the terms of the first sum this means that all states k with non-vanishing weight (k, > 0) must be pure, for the terms of the second sum it means further that all these states must be identical with each other and therefore also with the given state (k, = k). The given state is then said to be indivisible. If the given state is a mixture, k - k2 must be positive definite. Then at least one term at the right hand side of (1.28) must be different from zero. This means that at least two different states k, and k, (k,# k,) must have non-vanishing weight (k, > 0, k, > 0). The given state is then said to be divisible. Thus pure states and only these are indivisible. This has been proved in a more exact way by v o n N e u m a n n l). 1.06 Observation. In order to establish the observational meaning Mo, one must accept a definite notion of observation. We deal with 3 different notions: 0,: the classical notion: all observables a@,q) can be measured without fundamental restrictions and without disturbing the system, 0,: the quantum notion (elucidated in 2): measurement of an observable, which is represented by an operator a, gives as the value of the observable one of the eigenvalues up of a and leaves the system in the corresponding eigenstate k (cf. (1.20)); if beforehand the P? system was in a state k, the probability of this particular measuring result is Tr(kk,,). Suppose for a moment that the statistical description of quantum mechanics had been proven to be formally of the 1st kind St, but with respect to 0, properly of the 2nd kind S:,. Then (if any) the only notion, which could give a proper sense to the formal description, would be 0,: the utopian notion: the uniquely determined processes are observable by methods, hitherto unknown, consistent with and complementary to the methods of 0,. With respect to quantum theory classical theory is incorrect, though for many purposes it is quite a suitable approximation (for Zivvt A --f 0).With regard to the utopian conception quantum theory would be correct, but incomplete. In this a description is called correct if none of its statements is in contradiction with observational data. It. is called complete if another correct description,

117 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS 41 1

which leads to observable statements not contained in the given description, is impossible. This need not imply that all possible observational statements can be derived from a complete theory. 1.07 The fundamental controversy. Problem p intends t o state certain aspects of the well known controversy about the statistical character of quantum mechanics in a form fit for a reasonable discussion. Such a discussion is only possible as long as the theory is accepted as essentially correct (or rejected and replaced by a more correct theory). The completeness of the theory may be questioned. The physical reasonings of B o h r a.0. and the mathematical proof of v o n N e u m a n n 1) (reproduced in 1.08) have shown that (with respect to 0,) the statistical description of quantum mechanics is properly of the 2nd kind Szq (problem PI). Yet many of the opponents did not throw up the sponge, some because they did not grasp the point,others because they perceived a gap in the reasoning. I t seems t h a t a great many of the escapes (as far as they consider quantum mechanics as essentially correct) debouch (if anywhere) into an expectation, which either is already contented with a fopm'al statistical description of the lste kind Sj, or moreover hopes to give such a description a proper sense of type .Sk by proclaiming the utopian notion of observation 0,. The examination of this conqption is problem p2. Even if one did (we could not satisfactorily) succeed in proving the formal impossibility of type .Sj (and consequently of type Sh), many of the opponents would not yet strike the flag. We have already gone to meet them in trying to formulize some of their most important objections in a form fit for fruitful discussion. I t would be like flogging a dead horse in trying to do so with all vague objections they might possibly raise. Actually that is their own task. If they succeed in doing so, we try to prove the impossibility, they try to find the realization of their (formal or proper) expectations. Formal expectations can be realized by a formal construction, proper ones also require the realization of the type of observations from which they draw their observational meaning. As soon as the opponents succeed in finding a realization, we will (formally or properly) be converted (but not a minute before). As often as we succeed in proving the impossibility, some of the opponents may formulize (if anything) new objections for ever. At best they might be compelIed to retreat step by step, they could never be finally vanquished. I t

118

412

H . J . GROENEWOLD

may also happen that nobody succeeds in going further. Thus because of running on an infinite track or into a dead one, the controversy may be left undecided. Meanwhile we expect that in $n infinite regression the opponents objections will lose more and more interest after every retreat. 1.08 v o n N e u m a n n’s proof. The only states with a meaning M,,with respect to quantum observations 0, are quantum states (pure states or mixtures). Therefore in a statistical description of the 1st kind .SAq a quantum state should be described as a statistical ensemble of quantum states. This is impossible for a pure state, because such a state is indivisible (cf. 1.05).Then the statistical description of quantum mechanics must (with respect to quantum observations) be of the 2nd kind S:,. This is in our present mode of expression the point of v o n N e u m a n n’s proof 1). It should be noted that in 1.05 the admission of non-negative probabilities only (non-negative weights and non-negative definite statistical operators) is an essential (and natural) feature of the proof. Now before going into the details of problem pz, we first turn to problem u (we need a5 for pz). 1.09 Correspondence afp,y) t3 a. I n passing from classical to quantum mechanics, the coordinate and momentum q and p , for which (PJY) = (1.29) ‘

J

are replaced by coordinate and momentum operators q and p, for which ( 1.30)

p and q are the generating elements of the commutative ring of i-,!assical quantities n(q,p), p and q the generating elements of the non-commutative ring of quantum operators a. The non-commutability (1.30) of p and q entails t h a t the quantities a($,q) cannot unambiguously be replaced by a(p,q). The ambiguity is of the order of fi. The classical quantities a(p,q) can be regarded as approsimations to the quantum operators a for 2im F, -+ 0. The former can serve as guides to get on the track of the latter. Problem u1 asks for a rule of correspondence a@,q) -+ a, by which the quantum operators a can be uniquely determined from the classical quantities a(p,q). In practical problems no fundamental difficulties seem to occur

119 ON T H E PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS 4 13

in finding the appropriate form of the required operators a. This suggests the problem (not further discussed here) whether all or only a certain simple class of operators a occur in quantum mechanics. Suppose for a moment that all relevant quantum operators a had been fixed in one or other way. Then one might ask for a rule a +- a($,q), by which the corresponding classical quantities a($, q) are uniquely determined (problem a2).Problem a2 would be easily solved in zero order of A, ambiguities might arise in higher order. Now (with respect to 0,) the classical quantities have only a meaning as approximations to the quantum operators for Zim A -+ 0. Therefore, whereas in zero order of A it is hardly a problem, in higher order problem u2 has no observational meaning M , (with respect to 0,). Problems a1 and a2 could be combined into problem a3,asking for a rule of one-to-one correspondence a(fi,q) t-+ a between the classical quantities a($,q) and the quantum operators a. Beyond the trivial zero order stage in A, problem a3 can (with respect to 0,) only have an observational meaning M,, as a guiding principle for detecting the appropriate form of the quantum operators (i.e. as problem al).A formal solution of problem a3 has been proposed by W e y 1 z, (cf. 4.03). We incidentallycome back to problem a3 in 1.18. 1.10 Quantum observables. In this section a will not denote a classical quantity a(fi,q),but it will stand as a symbol for the observable, which (with regard to 0,) is represented b y the quantum operator a. According to 0, two or more observables a,b , . . . . can be simultaneously measured or not, according as the corresponding operators a, b , . . . . respectively do or do not commute i.e. as they have all eigenstates in common or not. Problem a4 deals with the (one-to-one) correspondence a +-3a between the symbols a and the operarors a. Problem z4has no sense as long as the symbols a are undefined. They may, however, be implicitely defined just by putting a rule of correspondence. (When the symbols a are identified with the classical quantities a(p,q), problem g4 becomes identical with problem a3). 1' o n N e u m a n n l) has proposed the rules

I

/(a), if a t+ a, then /(a)t-+

+

I1 a and b t+ b, then a b t+ a + b. /(a) is defined as the operator, which has the same eigenstates as a with eigenvalues /(a,), where ap are those of a. Then I seems to be obvious. The observable /(a)can be measured simultaneously with if u

t+

120

414

H . J. GROENEWOLD

a, its value is /(a,), where a, is that of a. When a and b commute, a b has the same eigenstates as a and b with eigenvalues a, b,, where a, and b, are those of a and b. Then I1 seems also to be obvious. a b can be measured simultaneously with a and b, its value b,, where a, and b, are t h e values of a and b. When a and b is a, do not commute, I1 is proposed with some hesitation. Because according to 0, the probability of finding a value a p for a in a state k is Tr(kk,,) (and because of 1.20)), the expectation value (average value) of a in this state is

+

+

+

+

E x ( a ) = 2 Tr(kk,,)a, = Tr(ka)

(1.31)

P

and similar for b. If one requires t h a t for a certain pair of observables a and b always E x ( a b) = E x ( a ) E x ( b ) , ( 1.32)

+

+

one must, because of

+ b)) = Tr(ka)+ Tr(kb), E x ( a + b) = Tr(k(a + b)).

Tr(k(a have that

(1.33) (1.34)

Because this has to hold for all states k, a and b have to satisfy rule 11. When I1 is given up for certain pairs a,b, the additivity of the expectatiori values of these pairs has also t o be given up. I n 4.01 it will be shown that, if I and I1 shall be generally valid, the symbols a have to be isomorphic with the operators a. But then there is no reason to introduce the former, their task (if any) can be left t o the latter. Accordingly for the sake of brevity we shall henceforth speak of the (quantum) observable a. When on the other hand, the symbols a are intended as real commuting quantities, the general validity of I and I1 cannot be maintained. As long as the symbols a are not further defined, problem c ( ~ comes to searching for a one-to-one correspondence a +-+a between the commutative ring of real symbols a and the non-commutative ring of H e r m i t i a n operators a. There may be no, one or more solutions. After the pleas for I and for 11, one might be inclined to maintain 1 and to restrict 11. In 1.13 we meet with a particular case (problem as! for which I1 has to be maintained and therefore I has to be restricted. Because we are further exclusively interested in problem as, we will not examine the possibility of solutions for which II is restricted.

121 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

415

1.1 1 Hidden fiarameters. We try to trace the conditions for the assumption that the statistical description of quantum mechanics is (at least formally) of the 1st kind S' (problem p). A statistical description S' must be obtained by statistical averaging over uniquely determined processes. The averaging must be described by integration or summation over a statistical distribution with respect to certain parameters. Unless they are further specified, we denote all parameters by a single symbol 5 and integration (including a possible density function) and summation over continuous and discrete parameters b y / & Parameters, which are in no way observable with respect to O,,are called hidden parameters. (We exclude their occurence in 1,15). As a pure superstate we define a state for which all parameters (inclusive the hidden ones) have a definite value. 1.12 Distributions. A quantum state must be described as an ensemble of pure superstates. The statistical operator k of the quantum state must correspond to at least one (non-negative definite) distribution function k(c) for the superstates. For each definite value of 6 all k(E) must have definite values and therefore must commute. k(6)must be normalized b y / d t k(6) = 1, so that with ( 1 2 3 )

Trk = f d6 A(().

(1.35)

Further the correspondence must be linear if kl t-+ kl(6) and k2c3 k,([), then kl

+ k, ++ k,(E)+k,(E). (1.36)

The observable (with respect to 0,) represented by the statistical operator k,, of a pure quantum state has the eigenvalue 1 in this quantum state and 0 in all orthogonal states. The probability of measuring in a system, which is originally in a quantum state k, the value 1 (and leaving the system in the pure quantum state kI"") is Tr(kk,,). I n a description of type S' this probability must be interpreted as the probability that any superstate belonging to the ensemble with distribution function k(EJ corresponding to k also belongs to the ensemble with distribution function K,(E) corresponding to k",. The latter probability is J dc K(E)k,,(E). Therefore the correspondence k t-+ A ( [ ) must be so that always

Tr(k1kz) = J d E

h(E)b(E).

(1.37)

For two orthogonal states k l and k2 this expression is zero, which guarantees that the distribution functions A,( tl (2.49) performed on the same system 1. The first measurement measures the states cpl,(t) or a corresponding observable al(t),the second one measures the states cp;,(t) or a corresponding observable a;@). If the first measuring result indicates the final pure quantum state cpl,(t) (tl < t < th), the statistical operator at the beginning t;l of the second measurement is klpP(t;), which is expanded according to

k1,Jth) = c X;,,,”*,G> with

l4,,”M

,‘,”’

4pp,v~p4d = Tr(k;y,,.(t~)kl,,(t~).

(2.50)

The statistical probability, that, after the first measuring result has indicated the pure quantum state cpl,(t) (tl < t Gth), the second measuring result will indicate the pure quantum state &,.(t) (t ti) is

4,,,”43

= T%,VG)

klpp(&))

=

I ma cpl,(th) 12*

(2.5 1)

This conditional probability is actually the most elementary expression contained in the formalism, which denotes an observable connection and which has a directly observable statistical meaning. When the functions &(t) coincide with the cp(tlp),i.e. when a;(t) and al(t)commute, (2.51) becomes 4 p p , v , J ( t ; ) ) = SJp

(2.52)

and the second measuring result can be predicted with certainty from the first. I n this case we have essentially the repetition of a measurement. (2.52) expresses the reproducibility of the measuring result. 2.09 Intevrningled states. The entangled state of two object systems 1 and 2 after a coupling of the type described above is of the kind

(2.53) The probability of finding system 1 in a state kl and 2 in a state k2 is

T~(k12kikd= C xvp Tr(ki,,ki) Tr(kzPyk2).

(2.54)

PlV

When kl and k2 coincide with the projection operators kl, and k2””,

143 ON T H E P R I N C I P L E S OF ELEMENTARY QUANTUM MECHANICS

437

(2.54) becomes equal to xPPaPV This might (wrongly) suggest that (after the coupling and before the measurement) the state of 1 and 2 is the mixture

k;2 =

Xppkl,,,

k2,p

(2.55)

P

instead of the state (2.53). I n this way the correlation between 1 and 2 would be partially destroyed by the omission of the non-diagonal terms. In the mixture (2.55) the expectation value of the states kl and k2 would be (2.56)

instead of f2.54). It has been emphasized by F u r r y 3) (*in a somewhat different form and particularly against our common opponents, cf. 2.1 1) that only if neither kl nor k2 coincides with any of the klPP or k2, respectively, (2.56) can be different from (2.54). Because the latter case hardly occurs in the relevant applications, one is apt to make the mistake of replacing (2.53) by (2.55) (and to draw unjustified conclusions whenever this case does occur). If 1 and 2 had been coupled with one or more further systems 3,. . . . according t o

k123.... =

xvp k l p k2p

bP”.. *

*

(2.57)

PL.V

and these further systems had been ignored afterwards, the infringed state of 1 and 2 would correctly be given b y (2.55) indeed. This infringed state is quite distinct from the entangled state (2.53). 2.10 Multilateral correlation. I n (2.53) the transition operators klPvand kZPV belong to two systems of orthonormal wave functions y I pand y 2 p ,which span the (generalized) H i 1b e r t subspaces R1 and R2. An interesting case 4) is that for which k12can similar to (2.53) also be expanded with respect to the transition operators llpaand lzPu belonging to any two systems of wave functions + l p and +2p in R1 and R2, when one system is chosen arbitrarily variable but orthonormal and complete, the other system suitably to the first (2.58)

A necessary and sufficient condition 4) for the occurrence of this case is that the xvP are of the form (2.59)

144

438

H. J. GROENEWOLD

The factorization of x , , ~means that klz is a pure quantum state of the combined systems 1 and 2 with wave function 'PI2

=

(2.60)

xp 'Plp 'Pzp. P

The unimodular coefficients x,/x could even be included in cplp or YZp.

The special case under discussion can easily be generalized to the following case. The functions 'plP and ' p z p are taken together in groups (plP1, (plw,. . . . and 'pzp,, (pzp,, . . . . which span the (generalized) H i 1b e r t subspaces Rll,R12,.. . . and RZ1,RZ2,. . . . respectively (R1 = R11 Rl2 . . . . and R2= Rzl RZ2 . . .). In these subspaces we take any two sets of systems + l f i , + l p , l . . . . and +zp, JcZp,, . . . ., of which one set is chosen arbitrarily variable but orthonormal and complete, the other suitably to the first. It is easily seen that the last part of condition (2.59) then has to be replaced by I x- I = xp. In I-dimensional subspaces R l p and R2p all 1-representations are essentially the same. An equivalent formulation of the generalized case is obtained by taking instead of any two systems of wave functions +lp.and +2p, as in the special case, two definite systems of which one is chosen arbitrarily fixed but orthonormal and complete, the other suitably to the first. R11,R12,.. . . or Rzl,Rz2,.. . . are then determined by the sharpest division of R1 or R2 into subspaces, which span linearly independent groups of 'pip and + l p or ' p z p and JC2, at the same time. We restrict ourselves to the special case. First we show the necessity of (2.59). With (1.13) it follows from (2.58) that

+

+

Xvp

+

+ .

Tr1F l p u 1lup) = AupTr2(k2,12pu)

(2.61)

xupTr2(k2pv12pu) = AupWklup 1lpu) * I t follows directly that +u%p

TY1(klpul l u p ) = ,A hpu Tr1 (klpvllup) (1 = 1 2 ) *

(2.62)

so that (with xtLY= x$, Apu = A&)

I xpv j2

= 1 hpu l2 or Trl (klpv11,)

= 0 ( I = 1 and 2).

(2.63)

Because one of the systems llpa or I., is arbitrarily variable and complete in R1 or Rz the latter alternative is excluded and we must have = ppul =2=h2(x=h> 0). (2.64)

145 ON T H E PRIXCIPLES O F ELEMENTARY QUANTUM MECHANICS

439

LVith (1.13)it further follows from (2.58) that x v p 1 (kl, 1lup) kzpu = hup 1,J P*V

2 xvp7-r2(k,

hup) klpv

(2.65)

= ,A 11,.

PIV

These relations connect the arbitrarily and the suitably chosen systems and establish the orthonormality and completeness of the latter. With (1.08) we derive from (2.65) (2.66)

and (2.67)

and similarly for interchanged indices 1 and 2. (2.66) and (2.67) must be identical according t o (1.08). Because one of the systems l l p uor l,, is arbitrarily variable and complete in R1 or R2, we must have (remembering (2.64)) xpIvxvp = x 2 x p # p ;hp&p=h~hp’p (. = A

> 0).

(2.68)

Then xvp and A, must have the form

= x ; ~ u p = x ~ p J l ~ p l = v .(2.69)

x v p-- X*y x p , [ x p I

This shows the necessity of (2.59). The sufficiency can be shown in the following way. Choose, say in R 1 , a complete system of orthonormal wave functions + l p and choose for each p a constant Ap with I hp 1 = A = x . Then take the functions 1

h p=

A c x p ( 4 f p Yip)

(P2/.47

(2.70)

P P

which are orthonormal and complete in R2.From (2.70) it follows that (2.7 1) The indices 1 and 2 could equally well have been interchanged. For the transition operators we get 12/30

=

1

A PL.” c xvp Trl(kl,v1lup) k z p v , UP

1

(2.72)

146

440

H. J. GROENEWOLD

and x v p T 7 1 P l p v llup)

(2.73)

= AupTr2 (k2vp f 2 p A *

Therefore

c xvp k l p v kzp” = p,uXvp Tr1 @lpv 11,) l l p o k 2 p v = c ,A Trz(k2vp12up) 11pukZpv = ~ u p l l p u 1 2 p u * pJJ;p, ptv;

P*V

(2.74)

P*O

0

This shows the sufficiency of (2.59). I t is of importance for the discussion of the measuring process, multilateral cort h a t (contrary t o the expectation of R u a r k relation between more than two systems is impossible. We first show this impossibility for the case of 3 systems. Suppose we would have the expansions k123 =

xvp klpv kZpvk3pv

PIV

=

(2.75)

hup11pu12pu13pu. PIU

With (1.13) it follows from (2.75) that X v p 1 P l p v llup) Tr,(k,v

12up)

= hupTr3 &,up 13pu)

(CYCJ.)

I

(2.76)

XUpTr35k3p” 13up) = ~ u p T r 1 ( k 1 ,11,u)Trz(k2vp 12pu) ( C Y C J . ) .

In the same way as before it follows that

I xpv j 2

= I hpu 1’

or TrJ(kJpv 11,) = 0

(I = 1, 2 and 3). (2.77)

Because one of the systems I,,, must be arbitrarily variable and complete in R,, we must have IxpvI= IhpoI=xZ=h2 (x=A>O).

(2.78)

I t further follows from (2.76) that (2.79) Then we must have ~ ~ l ( k l j I v 1 l u= p ~~r2(k2,”12,p)

= Tr3(k3pv13op) = 1 or 0. (2.80)

This would mean that the systems of llpu,12p0 and l,, should (but for a simultaneous change of enumeration of the Greek indices of the three corresponding operators and but for unimodular constants) , and k3pv.This is against the be identical with those of k l p ukZpv assumption. Multilateral correlation between the states of 1, 2 and 3 is therefore impossible.

147 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

441

For more systems 1, 2, 3,. . . . the impossibility of multilateral correlation can easier be shown in the following way. Suppose we would have the expansions

k123....=

x ~ v p k l p v k 2 p v k 3 p v -*

*

=

Aupllpo12p013pu*

P.0

PPV

- - - (2.81) *

Then (2.82)

Similar to (2.61) and (2.62) we get xppTrl

( k l p p 'lpp)

xpp7-rdk2,p

'2pp)

= AppTr2(k2pp '2pP)

J

(2.83)

= ~ p p W k 1 p hpp p )

and

hPP)= h;pTrdkrpp

x;p%(klPP

Ilpp)

(I = 12%

(2.84)

so that x P p = & A,

or Tri(klpPllp,,) = 0 (I = 1 and 2).

(2.85)

Because one of the systems lip,,is arbitrarily variable the latter alternative is excluded and because the traces in (2.83) are non-negative we must have XPP = hpp. (2.86) Further we have similar to (2.65) 7.Yl(klpp11pp)k2pp

= I2pp

(2.87) from which we derive llpp1luu

=

c~

~ l ~ ~ l ~ ~ ~ l p p kzpp ~ ~ ~ l ~ ~ (2.88) l p p ~ l u u ~

P

and 1lpp

a,,

Tr1 W l p p

= P

*

lpp)

$xJ

k2pp

(2.89)

and similarly for interchanged indices 1 and 2. Because (2.88) and (2.89) have to be identical according to (1.08) we must have T r l ( k 1 p p I l p p ) T ~ l ( k l p p ~ l u a= ) TrlPlpphpp~qxr.

(2.90)

This would require Wklpp11pp)

= 6pu

(2.91)

for every p, p and cr, which is impossible. Multilateral correlation cannot extend over more than two systems.

148

442

H. J. GROENEWOLD

The proofs given for the special case of multilaterial Correlation in the entire spaces R1, R,, . . . can easily be generalized to the general case of multilateral correlation in the subspaces RI1,Rzl,.. . . ; Rl2,RZ2,.. . . ;. . . . only. Now we see that also in the measuring process multilateral correlation (in the special or in the generalized sense) cannot be transmitted through the chain of systems of the measuring instrument. The correlation (2.28) is uniquely determined. This excludes the possibility of surpassing in the measurement conclusion the maximum inference discussed in 2.04 by the application of multilateral correlation. 2.1 1 E i n s t e i n’s fiaradox. We return to the two object systems 1 and 2 in the multilateral correlated state (2.58). If the state of one of the systems, say 2, is entirely ignored, the infringed state of 1 becomes (2.92)

.

The sums (which are identical) denote the projection operator of the In the mixture (2.92)all states (generalized) H i 1b e r t subspace RI. in R , have the same probability x2 = A2. If R1 coincides with the entire (generalized) H i 1 b e r t space of wave functions of 1, the infringed state (2.92) becomes entirely undetermined. If in dealing with the entangled state (2.58) one would make the mistake pointed out by F u r r y (cf. 2.09), one would get x2

C

kl,

kz,,

= A2

C h p p I,,,.

(2.93)

P

P

In dealing with (2.82) we have seen that (2.93) cannot hold. (2.85) does not express a correlation between pure quantum states of 1 and pure quantum states of 2 (in the way a member of (2.93) would do). If, however, (after the interaction between 1 and 2, which establishes the state (2.58)) one of the systems, say 2, interacts with a measuring instrument, which measures the states 12pp, the infringed state of 1 and 2 together after the latter interaction 1s 12 c 11pp12pp. (2.94) P

This mixture is different for different types of measurements, i.e. for different systems l,,,,. (2.94) does express a correlation between peur states of 1 and pure states of 2. This correlation is of unilateral

149 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

443

type. When the measuring result selects for 2 the state l,, the state of 1 is 1lPP. After the interaction between 1 and 2 has taken place, an observable bl of 1 with eigenstates 11, can be measured in two different ways: either by a direct measurement on 1, or by measuring an observable b2 of 2 with eigenstates 12,, (corresponding to llpP) by a direct measurement on 2 (then 2 can be conceived as a part of the measuring chain). At a first glance it might seem surprising and perhaps even paradoxical that it is still possible to decide which observable of 1 will be measured by a measurement on 2 after all interaction with 1 has been abolished 6, and that it is possible to measure independently two incommensurable observables al and bl([al,bl]# 0) by applying the two measuring methods side by side ') ". (Of course one should care for not making the mistake of (2.93), which would naturally lead to paradoxical results). When the eigenstates of a, are kl, and those of bl are 11,, a measurement of al selects a state out of the left member, a measurement of bl selects a state out of the right member of the expression (2.92) for the infringed state of 1. The probability that one measure ment selects the state kl,, if the other selects the state llpp(or opposite) is according to (2.51) (2.95a)

no matter whether al and bl are both (successively)measured directly on 1 or (no matter whether successively or simultaneously) one of them on 1 and the other one on 2. When both are directly measured on 1 , the state in which 1 is left after the succeeding measurements is kl, if the final measurement was that of a,, it is I+,, if the final measurement was that of b l . A paradoxical situation seems to arise if one asks in which state 1 is left after a; has been measured on 1 and bl on 2 (or opposite). We have to remember (cf. 2.08) that all observational statements bear on connections between measurements. The state in which 1 is left has only an observational meaning with regard to a succeeding measurement of an observable of 1, say c1 with eigenstates mln. When the measurement of a, has selected the state kl,, the probability that the measurement of c1 will select the state mlrris (2.95b)

150

4 44

H. J. GROENEWOLD

When the measurement of bl has selected the state llPp,the probability that the measurement of c1 will select the state mlrris (2.95~) Thus we get two different probabilities for the same event. This is not unfamiliar in statistics, because the probabilities are (always) conditional. They have onIy a meaning for a great number of combined measurements of al, bl and cl. The probability of finding a state kl, is x2,the probability of finding a state 11, is h2,the probability of finding a state ml,is then according to (2.95b) or (2.95~) x2 X Tr(kl,, rnlT7)or 1’ X Ty(llP,, fi

ml,).

(2.96)

P

OnIy these sums have to be identical and they are so according to (2.92).The correlations between the measuring results for a l , bl and c1 are described by (2.95). Let us consider once more the measurement of al and of bl, one of them directly on 1 and the other directly on 2. The latter measurement can also be conceived as a direct measurement on 1 (the system 2 is then regarded as a part of the measuring chain), which preceedes the first mentioned measurement. The only pecularity of the present case is that after the coupling between the object system 1 and the first system 2 of the measuring chain of the earliest measurement has been abolished (and even after the succeeding measurement has been performed) one can thanks to the multilateral correlation between 1 and 2 still decide which observable will be measured by this earliest measurement. But when we pay due regard to the correlations between the various measuring results, this leads to no paradox. An illustrative example, which has been discussed by E i ns t e i n a.0. ’) 4) and by B o h r a.0. 8 , 3, 6), is that of two particles (each with one linear degree of freedom) in an entangled state for which the wave function reads in q-representation (2.97) This state can be realized by two particles 1 and 2 directly after passing through two parallel slits at a distance Q in a diaphragm. (2.97) describes the motion in the direction perpendicular to the slits, parallel to the diaphragm. The total momentum P can be determined from the total momentum directly before the passage

15 1 ON T H E PRINCIPLES O F ELEMENTARY QUANTUM MECHANICS

445

through the diaphragm and the change of momentum of the diaphragm. The slits can b e taken so far apart, that exchange effects can be neglected. (2.97) is of the form (2.60)with (2.59), as can be seen by expanding (2.97) with respect t o e.g. coordinate or momentum eigenfunctions of 1 and 2 1 i[ P Q 6 (qz - - 2 ‘p 12 = .\/h dE e 8 q1-E 2)

/”

(

”)

+

(2.98)

Ri coincides with the entire (generalized) H i 1 b e r t space of wave functions of 1. The infringed state of 1 is entirely undertermined. After a measuring result q2 = qZpor p 2 = p Z p 1 is “left” in the state (2.99) or PI = p - p 2 p respectively. I n this way the and q1 = q + - Q coordinate or momentum of 1 is measured by the coordinate or momentum of 2 after the interaction between 1 and 2. We come back to this example in 5.06.

3. Operator relations. 3.01 Exponentials. I n the ring of operators a generated by two non-commuting ,He T m i t i a n basic operators p and q, for which [p,q] = 1, i.e. pq - qp

h

>

= 7- (tt a

(3.01)

0),

we are going to derive a F o u r i e r expansion similar t o that in a commutative ring of functions a($,q) of two real basic variables 9 and q. For this purpose we need some exponential relations. I t should be remembered that we still have a rather specialized case, because the commutator (3.01) of p and q commutes with p and q. With (3.01) one has 2, ,h (P+@ 1 i - lim ( 1 -- (p q))% = lim (( n-tcr, n h n--foo

+

+

i

i

i

- e ~ P e ~ q e -(3.02) ~ .

152

446

H.J. GROENEWOLD

With (xp

+ yq) and (x’p + y’q) instead of p and q we get for (3.01)

+ Yq),

UXP

(X’P

+ Y’@I

(3.03)

= xyf -P’

and for (3.02) i

i - ,T (xp+yq)

e-((x+X’)P+o)+YWl) n

i

i

eT(X’P+YW e- 5 W Y ’ Y z ’ )

.

(3.04)

(Important special cases are y = x’ = 0 or x = y‘ = 0). Further

.-x i

(tP+9q)

i i ex h + Y d ex ( t P + T d -

i

(xp+yq)

i

(.?j-Yt)

.

(3.05)

Analogous to the (symbolical) relation i

Lh2/ j d q dq ex

(xP+rd

-

-

’b)J

(3.06)

(3.05)gives the operator relation

J- jjdE drl e-

i

b (tP+W)

h2

exi (m+yq)exi (Ip+Tlq) - F(x) ah).

(3.07)

Further analogous to 3.08)

we have

i

=//dc

d-q e-x

i

a

(tp+~q)

(tp+vq)

a(6) s(-q) = a.

(3.09)

In the same way as (3.08) and (3.06) show that every (normalizablej function a(+,q) can be expanded into a F o u r i e r integral

(3.10)

(3.09)and (3.07) show that every operator a (with adjoint at) can be expanded into

153 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS 447

(3.11)

This is already the F o u r i e r expansion, but the coefficients a(x,y) can still be expressed in a more simple form. 3.02 The trace. When U is a unitary operator

UtU = 1,

(3.12)

the unitary transformation

a‘ = UtaU; 9’

= Uy, ‘p’t = qtut

(3.13)

leaves all operator relations invariant. Therefore the latter can be derived in a suitably chosen representation. The eigenvalues q of q and of p are assumed to run continuously between -00 and 00. In q-representation the operators q and p can be taken in the form

+

+

(3.14)

(a&

is meant to operate to the left). With (3.04) we can write (3.15)

Expressing occasionally the inner product explicitely by an integral, we get with (1.09), (3.15) and (1.05)

(3.16)

The result is independent of the chosen representation. Comparing (3.16) with (3.07) and remembering the linear expansion (3.11) of a, we see that Tra can invariantly be represented by the operator relation i i 1 Tr a = -//dE 1 dq e- (5Pfrlq) a ex ( t P f q q ) . (3.17) h hZ 3.03 F o u Y i e Y expansion. Rewriting (3.07), (3.09) and (3.11) with the help of (3.17) we get

- T yeF( a f y q ) - 6(x)6 ( Y ) , h

(3.18)

154

448

H . J. GROENEWOLD

(3.19)

and

(3.20)

(3.18), (3.19) and (3.20) are entirely analogous to (1.13), (1.14) and (1.15). (3.18) and (3.19) respectively express the orthonormality and the completeness of the systems of operators i 1 p(.P+Yq)

(with variable x and y ) . dh (1.15) and (3.20) are the two ways we use for the expansions of operators.

4. Correspondence. 4.01 v o n N e u m a n n’s rules. We now examine the rules of correspondence I, 11, 111, 11’ and V’. First I and 11. We show that if between the elements a of one ring and the elements a of another ring there is a one-to-one correspondence a c-+ a, which satisfies v o n N e u m a n n’s rules (cf. 1 .lo) if a t+ a, then / ( a ) +-+ if a

a and b

t-+

t+

b, then a

/(a),

+ b +-+a + b,

I I1

the two rings are isomorphous. We get using I and I1 (a

+ b)2 - a2 - b2 = ab + ba +-+ab + ba

(4.01)

and also using (4.01) a(ab

+ ba) + (ab + ba)a - a2b - ba2 = 2nba

2aba

t-+

(4.02)

and further ‘using (4.02) (ab

+ b a y - b(2aba)- (2aba)b = _-

(ab - ba)2 t+ - (ab - ba)2. (4.03)

Therefore we have ab - - a

t+

f (ab-ba),

(4.04)

155 ON THE PRIKCIPLES OF ELEMENTARY QUANTUM MECHANICS

449

and with (4.01)

ab +-+ ab (for all a and b) or ab

t+

ba (for all u and b). (4.05)

This means that the rings are isomorphous. It follows that, if one ring is commutative and the other not, I and I1 are inconsistent ". (When the commutators are of the order of Ti, the discrepancy is according t o (4.03)of the order of A2). 4.02 Bracket expressions. Then V'. For the correspondence a +-+ a between the commutative ring with generating elements p and q and the non-commutative ring with generating elements p and q with commutator (3.01) (fi t+ p and q t--f q) we show that the rule (cf. 1.18)

if 4 p , q )

a and b(P,q) +-+b, then (a(p,q),b($,g))

+-+

++

[a,bl V'

is self contradictory. With

p2 ++ x1, q2 ++ x,; p 3 we find from

4 (P2,q) = p 4 (p2,p)= 0

-

++

Yl,

4"

+-+

(4.06)

Y2

Q [x,,ql = Pt

f-+

$ [Xl,Pl

(4.07)

=0

(and similar relations for q2 and x,) that

p 2 f+

p2

+ c1, q2

++

q2

+

(4.08)

c2

and from (4.09) (and similar relations for q3 and y2) that

fi3 +-+p3 + 3Clp

+

dl,

q3 ++ q3 4-3C2q f d, (4.10)

(cl, c2; d,, d2 are undetermined constants). Further we get

+(P3,q2)=pZqt-f~[(P3+3c,P+d1),(q2+c2)1=4(P2q+qP2) fclq@* 1) (4.11)

Pq2

3(pq2+q2p)+C2P

+-+

and

f (p3,q3)= p2q2 = B (P2$

++

Physica XI1

3(p3

+ 3clp + dl), (s3+ 3c2q + &)I

+ q2p2)+ 4h2 + c1q2 + C,P2 +

ClCZ.

(4.12) 29

156

45 0

H . J. GROENEWOLD

With (4.1 1) we get

+ [(*(P2cl+ 9P2) + clq), (* (PQ2+ q2P)+ + (P2Q2+ q2p2)+ Q fi2 -c1q2- -+

a @2q,pq2)= P2q2 =

+-+

C2P2

ClC2.

C2P)l

(4.13)

(4.12) and (4.13) can only be identical for c1 = c2 = 0 and A = 0.

Therefore V' is self inconsistent (the deficiency is of the order of A2). 4.03 W e y 1's correspondence. And finally I11 and IV with parameters p and q (i.e. for the same rings as in 4.02). We denote the density function by p(p,q). The rules (cf. 1.13) 1 +--+ 1, I11 if a(p,q) ++ a and b(p,q) +-+b, then f j d p ' q P(fiJ4)a(#Jq) b(p,q) = Tr(ab) IV

can be satisfied b y (1.55)

a =lf ' q P(#,q) m(+,q) a@Jq) (4'14) with a transformation nucleus m@,q), which satisfies (1.57), (1.58) ; a(fiJq) = Tr(m(fi,q)a),

(1.59), (1.60)

Trm@,q)= 1, (4.15) (4.16) fJ@ a4 P(Pd m@4)= 1; Tr(m@,q) m(?',q')) = P-'(ib,q) ' ( p ' ( q -q')J (4'17) / l d p dq P(P,q) Tr(m(pJq)a) Tr(m@,q) b, = = Tr(ab) (for every a and b), (4.18) When we replace in (1.56) the complete orthonormal systems Kz,(p,q) of (1.54) and k,, of (1.15) by the complete orthonormal systems i 1 -2 -e A ( x P I - 9 4 ) of (3.10) and eT(Xp+yn) of (3.20), h

we find a solution

m(p,q)=

i

dy e t ( z p + y q ) , - ~ ( z f i + ~ q )

(4.19)

of (4.15), (4.16); (4.17), (4.18) with the density function

P(PJq) =

1 '

(4.20)

Then we get for (4.14) (4.21)

157 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

451

With the F o u r i e r expansions (3.10) and (3.20) this correspondence reads (4.22)

which is W e y 1’s correspondence 2). I1 is a consequence of IV and is therefore satisfied by the correspondence (4.21). We will see what is left of I and V’. If a ++ a and b t-+ b according to (4.21) we find with (3.04)

ab =

‘I/. . . .

dy dx‘ dy’ d p dq dp’ dq’

/”dx

h4

i

. e -h ((x+x’)p+(r+r’)s)

i

i

@Y’-YX‘)

- ++Yq+x‘P’+Y’q’)

e

With the variables [=x+x’,

q=y+y‘,

,

x-x’

5’ = ___ 2

)

Yi

.

=-y -y‘,

2

a= 6’ =

P

+ P‘ 2

p - p‘)

alP,q) b(P,q). (4.23)

’

7

7 1

+ q‘

4=-

2

= q - 41,

’ (4.24)

this becomes

(4.25) The expressions in brackets at the end are a symbolical representation of T a y 1 o r expansion. With the substitution i4.26) F-. x, y1-+ y , a+ p , T+ q we get by partial integration

(4.27)

158

452

H. J . GROENEWOLD

This gives for the H e r m i t i a n operators

. (ab - ba)

i 4 (ab + ba) and 2 ’

the correspondence

b($,q)

t+

8 (ab + ba), (4.28)

i b($,q) t+ - (ab - ba). (4.29) 2

To the neglect of terms of order of h2 and higher (4.28) and (4.29) would read a(P,q) b(fiJq) e+ Q (ab f ba), (4.30)

--)

b(fi,q) t+ i (ab - ba). (4.31)

(4.30) would lead to I , (4.31) is equivalent to V’. We examine which functions /(a)satisfy 1. From (4.28) we see that the correspondence

if a

t+

a, then an t+ a* (for every integer n)

(4.32)

only holds if

(4.33) First take for n a homogeneous polynomial in p and q of degree n. An elementary calculation shows that the condition (4.34) or

(4.35)

+

is only satisfied if a is of the form (@ yq)”. Then it follows that any polynomial in p and q can only satisfy (4.33) if it is a polyyq. This finally means that I can only be satisfied nomial in xp if a is a function of a certain linear combination xp yq of 9 and q. IVith the help of the F o u r i e r expansion (4.22) it is easily seen that every (normalizable) function of xfi yq does satisfy I. Therefore the least restricted form of.1, which is consistent with the correspondence (4.21) is f ( x P Y4) t-+ i ( X P 3- Y9). (4.36)

+

+

+

+

159 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS 453

As to V‘, we see from (4.31) that for the correspondence (4.21) the bracket expression ( ( a ( p , q )b, ( p , q ) ) ) (cf. 1-14) defined by if a(p,q)+ + - aand Q,q) ++b, then ( ( u ( p , q ) ,b(P,q)))+-+[a,b] (4.37) is given by

(4.38) If a@,q) or b(j5,q) is a polynomial in j5 and q of a t most 2nd degree, we have a special case for which the bracket expressions ((a,b))and (a,b) coincide. The correspondence (4.21) is a solution of 111 and IV. We have not investigated the possibility of other solutions with the same parameters P and q.

5. Quasi-distributions. 5.01 Proper and improper representations. With W e y 1’s correspondence (4.22) as a special solution of

1t+1 if k t+ K(p,q) and a

I11 t+

a(P,q), IV

(with parameters p and q and density function p(p,q) = l / h ) , we obtain a special case of a transformation between a representation in terms of operators k and a and a representation in terms of functions K(p,q) and a(f,q). Quantum statistics are usually represented in terms of operators, classical statistics in terms of functions. We assert that the usual description is also the proper one. The statistical operator k of the quantum representation and the statistical distribution function k(p,q) of the classical representation are non-negative definite, but in general the quantum @,q) and the classical k are not. This makes that for orthogonal states, for which

Wklk2)

=

l I^Sdp d q k,(P,q) k,(P,q) = 0, h,

(5.01)

the product klk2or K,(p,q)K,(p,q) vanishes in the proper representation, but in the improper representation it need not. The equations

160

454

H.

J. GROENEWOLD

of motion of the quantum k are described by infinitesimal unitary transformations, those of the classical @,q) by infinitesimal canonical transformations (contact transformations), but the equations of motion of the classical k and the quantum k($,q) are in general not of these types. Because the improper representation is formally equivalent to the proper one, it is (provided it is not misinterpreted) a correct description, though it is in general a rather impracticable one. In spite of its deficiences, or rather because of them, we discuss some aspects of the improper representation of quantum mechanics in terms of k(P,q) and a(p,q),i.e. the quasi-statistical description of the 1st kind Q’ (cf. 1.19). It more or less illustrates the ways along which some opponents might hope to escape B o h r’s reasonings and v o n N e u m a n n ’ s proof and the places where they are dangerously near breaking their necks. 5.02 Transition functions. For the transition functions k,,(#,q) corresponding to the transition operators (1.03) according to (4.21) we find with the help of the q-representation (occasionally expressing the inner product explicitely by an integral) similar to (3.16)

(5.02)

Because the wave functions cpp are only determined but for a factor eiIRYP (y real), the K,,(p,q) are only determined but for a factor ei’ (Yp-Yv). The distribution functions, which are thus obtained with W e y 1’s correspondence 2) become identical to those given by W i g n e r 10). 5.03 Proper value. In a distribution k or @,q) a quantity a or a($,q) can be regarded to have a proper value if the condition (2.10)

Tr(kf(a)) = f(Tr(ka))

(5.03)

or

(5.04)

161 ~~

~

ON THE PRINCIPLES O F ELEMENTARY QUANTUM MECHANICS

455

is satisfied for every f . Whereas the validity of (5.04) is for a proper (non-negative definite) K(+,q) already guaranteed by the validity of the special case /(a) = a2, it is not for a proper k or an improper K(9,q).For a proper k the validity of (5.03) or (2.1 1) requires that a is of the form a(xP Y q ) (5.05)

+

and k an eigenstate of a. For any H$,q) the validity of (5.04) requires that K(9,q) is of the form

Ya($,q) -

(5.06

which is a proper (i.e. non-negative definite) one. Because (5.03)and (5.04) are identical, the conditions (5.03 and (5.06) are equivalent. This means that the eigenstates of the operators a(xp yq, and of no other operators correspond with proper (and orthonormal and therefore non-overlapping) distributions of the form (5.06), in which up is the corresponding eigenvalue. This case would be rather encouraging for a statistical description of the 1st kind S ' , if it were not just an exceptional case. The eigenfunctions of a ( x p y q ) are in q-representation

+

+

5.07) Qp(d

=

-Y(P)

dY V Y q - P) e h

for x

= 0.

(y(p) real arbitrary). The corresponding eigenvalues are aip)

a(xP f yq)(Pp = 4 P ) Y p .

(5.08)

+

which is the eigenvalue of xp y q (for arbitrary fixed x and y), runs between - 00 and 00. The domain of eigenvalues of a(xp yq) is therefore the same as that of the functions n(z) (- 00 < z Q m).This means that the domain of the proper values of observables, which have such, are unrestricted by quantum conditions. Inserting the eigenfunctions (5.07) in (5.02) we get p,

+

+

(5.09)

(The expression in brackets in the exponent in (5.09) is a canonical

I62

456

H. J. GKOENEWOLD

+

conjugate of x p yq). The k,,(p,q) are actually of the form (5.06). 5.04 The harmonic oscillator. After we have treated in 5.03 a special case for which the k(p,q) are of proper type themselves, we now deal with a case for which their equations of motion are of proper type. According to (1.43) and condition V’ they are if ( ( H ( p , q ),k(P,q))) coincides with (H(P,q), k ( p , q ) ) and according to (4.38) this is the case for every K(p,q) if H@,g) is a polynomial in p and q of at most 2nd degree. This condition is satisfied for the harmonic oscillator, for which H($,q) coincides with the classical Hamiltonian

(5.10) is the mass, w the classical circular frequency of the binding. We consider 9’ and q’ as new canonical coordinates and omit the dash. In q-representation the normalized stationary solutions of the wave equation

iiz

(5.11) (5.11) are

~ ~ (=4 )

1

-__

_ _1

e

2h

Q’H, __

e

(TZ=

0,1,2,. . . .). (5.12)

4/2“n!

The H e r m i t i a n polynomials function

H,

(&)have the generating (5.13)

(5.02) becomes with (5.12)

(5.14)

ON T H E PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

457

With (5.13) we get

(5.15)

This gives

(x 2

LLL;(z;

(P2+4'))

ei (tn--n)

(arc tan

2 --wi) . (5.16)

The Lf') are associated L e g e n d r e polynomials. k,,(p,q) is separated into a product of functions of the canonical conjugates $(p2 q2>and arc tan @/q). The K,,(p,q) actually form a complete orthonormal system. For the distribution function km,,($,q) of the nzlh eigenstate of $(p2 q2), the average value of $(p2 q2) is (m+&)A,but it is not a proper value. With (5.10) the transformation (1.47) gives the contact transformation determined b y

+

+

@

_ _ --

at

+

4

- wq, = wp, at

(5.17)

with solutions

p

= a cos (ot-

x),q = a sin (at -x ) .

(5.18)

164

458

H . J. GROENEWOLD

The representative point in the phase space of a superstate rotates uniformly about the origin with constant radius and circular frequency a.The rotation of the entire distribution k,,,,(P,q) with this circular frequency o produces according t o the last factor of (5.16) a periodicity with circular frequency (m-n)o (like a rotating wheel with 1 m - n I spokes). Also this would have a hopeful aspect for a description of type S’, if it were not one out of a few exceptional cases. 5.05 The scale system. We shortly return t o the measuring process. We start with the most favourable case for a description of the 1st kind S’ and consider a system 1 in the measuring chain, for which the ,, distributions klpp(@J,qI)do not overlap. The corresponding k are then eigenstates of a n operator of the form xpl yg, (cf. 5.03). The scale system is a special case ( x = 0), which shows all essential features. According t o (5.09) we have

+

(5.19) By ignoration of one or more systems of the measuring chain the non-diagonal functions (p # v) are dropped and only the diagonal functions remain. Instead of (5.19) we get (5.20)

(The latter &-function is actually a remainder of the ignored distribution functions). The effect on (5.19) of ignoration of other systems is formally the same as that of integration over fi with density function 1 / h . This illustrates even more plainly than before (cf. 2.07) how the correlation between pl and other observables is completely destroyed by the reading of ql. So far there is no difficulty with an interpretation of the 1st kind. We are only concerned with the value of ql, which is a proper value and uniquely determines the distribution (5.20). The value of f l is indifferent. As soon as inference is made about other systems in the chain with overlapping k,+(p,q), correct results are only obtained after the integration over pl (with density function l/h) has been performed (cf. 1.19). In a description of the 1st kind this integration could only be interpreted as an averaging over a great number of measurements. But the integration has already to be performed in a single reading and therefore an interpretation of the 1st kind is excluded.

165 ON THE PRINCIPLES OF ELEMENTARY QUANTUM MECHANICS

459

5.06 E in s t e i n's fiaradox. The multilateral correlated state (2.97) has according t o (5.02) the distribution

6(Pi -!?2 -k Q ) ~ ( $ I -k $2 -PI. (5.21) This shows clearly the correlation between q1 and q2 and between p1 and $,.The similarity to a genuine distribution of the 1st kind is very tempting. Because (5.21) is highly singular we also consider the distribution ~

WPQ Q( ~ , ! ? ; I #2,!?2)=

(5.22) (properly instead of \5.21) we should use eigendifferentials). The infringed distribution after a measurement of q2 or f 2 can be found from. (5.22) by integration over f12 or q2 respectively with density function I/h. This gives

(5.23) or

(2.24) respectively. For the distribution (5.21)this becomes

(5.25) The correlation between f i l and f i 2 or q1 and q2 respectively has entirely disappeared. If the state of 2 is entirely ignored, the distribution of the infringed state of 1 can be found from (5.22) by integration over p 2 and q2 with density function l/h. This gives h

(5.26)

For the distribution (5.21) the result is llh, the infringed state is entirely undetermined (the normalization can be understood from (5.26)). A measuring result q2 = qZpor p 2 = p2,, selects from (5.23)

166

460 ON T H E P R I N C I P L E S O F E L E M E N T A R Y QUANTUM MECHANICS

or (5.24) for 1 the distribution

(5.27) or

(5.28) For (5.25) this gives 1

-S(P1 k

+

P2,-P).

(5.29)

Also in this example, in which all distribution functions derived from (5.21) are non-negative definite, it is already the particular part of the immediate integration over half of the parameters even in a single measurement, which does not fit into an interpretation of the 1st kind. These few attempts and failures to carry through a genuine statistical description of the 1st kind S' may suffice to illustrate the intention and troubles of such a conception.

REFERENCES

J. v. N e u ni a n 11, Matheniatische Grundlagen der Quantenmechanik, Berlin 1932; iVew York 1943. H. W e y l , Z.Phys. 4 6 , 1, 1927; Gruppentheorie und Quantenmechanik, Leipzig 1928. W. H. I: u r r y, Phys. Rev. (2)4 9 , 393, 476, 1936. E. S c h r o d i n g e I, Proc. Camb. Phil. SOC.31,555, 1935;32,446, 1936;Naturw. 23, 807, 823, 844, 1935. A. E. R u a r k , Phys. Rev. (2) 48,446, 1935. L. F. v. W e i z s a c k e r, Z. Phys. i0,114, 1931. A. E i n s t e i n , B. P o d o l s k y and iV. R o s e n , Phys.Rev.(2)47,777,1935. N. B o h r, Phys. Rev. (2)40,466,1935. G. T e m p l e , Nature 136, 957, 1935; H. F r o h l i c h and E. G u t h , G. T e m p 1 e, Nature 136, 179, 1935; R. P e i e r 1 s, Nature 130,395, 1935. 10) E. W i g n e r, Phys. Rev. (2)4 0 , 749, 1932.

167

I: 99 1 QUANTUM MECHANICS AS A STATISTICAL THEORY

BY J. E. MOYAL Communicated by M. S. BARTLETT

Received 12 November 1947 1. INTRODUCTION Statistical concepts play an ambiguous role in quantum theory. The critique of acts of observation, leading to Heisenberg’s ‘principle of uncertainty’ and to the necessity

for considering dynamical parameters as statistical variates, not only for large aggregates, as in classical kinetic theory, but also for isolated atomic systems, is quite fundamental in justifying the basic principles of quantum theory; yet paradoxically, the expression of the latter in terms of operations in an abstract space of ‘state’ vectors is essentially independent of any statistical ideas. These are only introduced as apost hoc interpretation, the accepted one being that the probability of a state is equal to the square of the modulus of the vector representing it; other and less satisfactory statistical interpretations have also been suggested (cf. Dirac (I)). One is led t o wonder whether this formalism does not disguise what is an essentially statistical theory, and whether a reformulation of the principles of quantum mechanics in purely statistical terms would not be worth while in affording us a deeper insight into the meaning of the theory. From this point of view, the fundamental entities would be the statistical variates representing the dynamical parameters of each system; the operators, matrices and wave functions of quantum theory would no longer be considered as having an intrinsic meaning, but would appear rather as aids to the calculation of statistical averages and distributions. Yet there are serious difficulties in effecting such a reformulation. Classical statistical mechanics is a ‘crypto-deterministic ’ theory, where each element of the probability distribution of the dynamical variables specifying a given system evolves with time according to deterministic laws of motion; the whole uncertainty is contained in the form of the initial distribu. tions. A theory based on such concepts could not give a satisfactory account of such non-deterministic effects as raclioa,ctive decay or spontaneous emission (cf. Whittaker ( 2 ) ) . Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. I n the general case, there is the possibihty of ‘dEusion’ of the probability ‘fluid’, so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics.

I. QUA.NTUM KDTEMATICS 2. THE EXISTENCE OF PHASE-SPAUE DISTRIBUTIONS I N QUANTUM THEORY

In the accepted statistical interpretation of quantum theory, the possible values of a dynamical variable s are the eigenvalues siof the corresponding operator (observable ) 7-2

168

100

J. E. MOYAL

S in the Hilbert space of the state vectors. The probability of finding sc in a state @ is then equal to the square of the modulus I a$l2 of the projection a, of @ on the corresponding eigenvector 1C.i. A complete or irreduciblerepresentation for a given mechanical system is given by a set of commuting observables s such that their eigenvectors pi span the whole space, i.e. such that any ?b, = Ca,$i. Hence we obtain directly from i

II.the joint distribution of the variables s. It is known, however, that these s are not sufficientin themselves to specify the system completely; we need, in addition, another complementary set, say r, which does not in general commute with s; for example, a complete representation is given by either the Cartesian coordinates q or their conjugate momenta p, but the complete dynamical specification of the system requires both q’s and p’s. Hence, the phase-space distributions of complete sets of dynamical variables, which are required for a statistical theory, are not given directly by y?. It has been argued(3) that such distributions do not exise, because of the impossibility of measuring non-commuting observables simultaneously. This argument is not conclusive for two reasons; one is that the impossibility of physical measurements does not preclude us from considering the proposition that there exists a well-defined probability for the two variables t o take specsed values or sets of values; in fact, the theory of probability is introduced to deal with suoh situations where exact measurement c ir impossible (see Jeffreys ( 4 ) ) .The other reason is that it is possible in principle to form operators G corresponding to functions #(r, s) of non-commuting observables; the expectation value of G in a state 9 is then given by the scalar product ($, GP). But the j o h t distribution of r and s can be reconstructed from a set of such expectation values, e.g. the values of all the joint moments ykSn. The formalism of quantum theory allows us therefore to derive the phase-space distributions indirectly if a theory of functions of non-commuting observables is specified and conversely. There are serious difficdtie’s to be met, however, in defining these distributions unambiguously. This may be seen, for example, in the case of the harmonic oscillator. The energy eigenvalues form a discrete set E,, = (n 8)hv, The corresponding eigenfunctions uu,(q),un(p)are sets of Hermite functions, continuous in p and q. Hence any joint distribution for p and g in a state consistent with the individual distributions

+

must extend continuously over the whole ( p , q ) plane, while any joint distribution for the energy H = +(p2/m+27rmvq2)and the phase angle 0 = tan-lp/q consistent with probabilities an@ for En,wiU be concentrated on a set of ellipses +(p2/mi27rmvq2)= (ni-9) hv.

We are thus forced to the conclusion that phase-space distributions are not unique for a givert state, but depend 0% the variables one is going to measure. In Heisenberg’s words (61, ‘the statistical predictions of quantum theory are thus significant only when combined with experiments which are actually capable of observing the phenomena treated by the statistics’. Since the introduction of statistical concepts in atomic theory is justsed by an analysis of the interaction between observed system and observer, it is perhaps not surprising that different distributions should arise according to the

169

Quantum mechunics us u statistical theoyy

101

experimentd set-up. For example, measurement of the spectra of a4 atom corresponds to a distribution with discrete values for the energy and angular momenta. Direct transformation of this distribution t o ( p ,q ) space, corresponding to a distribution concentratedon discrete orbits, would not be appropriate for the treatment of collisions of the same atom with a beam of electrons; the appropriate distribution in the latter case arises from wave functions filling the whole space continuously, and is incompatible with discrete orbits. The statistical interpretation of quantum kinematics will thus have to give methods for setting up the appropriate phase-spacedistributions of each basic system of dynamical variables in terms of the wave vector^, and for transforming such distribution into one mother. 3. PJXASE-SPAUE DISTRIBUTIONS IN TERMS OF WAVE VEUTORS

We denote by r a set of commuting observables or operators giving a complete representation, s the complementary set, such that s do not commute with r and that r and s together form a basic set of dynamical variables, characterizing a given system; r and s me their possible values or eigenvalues (these are, of course, ordinary commuting variables). The most natural way of obtaining the phase-space distribution P(r,s) is to look for its Bourier inverse, i.e. the mean of exp{i(mi-Bs)} (known in statistical terminology as the characteristicfunction). On forming the corresponding in operator (3.1) M(7,B) = exp {i(m+ 8s)) = 2 --J (71-+ B s ) ~ , n

the characteristic function in a state $ is given by the scalar product M(T,8 ) = ($, e f ( ~ ~ + @ $ ) .

(3.2) From well-knownformulae for Fourier inversion, the phase-spacedistribution function is then

(3.3)

for continuous eigenvaluesf, and

(3.4) for discrete eigenvalues ri , 8k (Cram& ( 6 ) ) $ . The operator (3.1) takes a specially simple form for canonically conjugate coordinates and momenta q, p (pq-qp = &/i), M(7,0) = e ~ < k Oe i 8 4 e h P = e-i%7PeiOCe?IiTP (3.6) (cf. Kermack and McCrea (7)).From the second expression for M,we find (3.6)

-

co t o + a. $ The term distribution function is used in this paper t o denote the probability density of conthous eigenvalues, and the finite probability of discrete eigenvalues.

t when no f i t s are specified, all integrals are t o be taken as from

170

J. E. MOYAL

102 and hence by Fourier inversion F ( P ,P) =

,s$*(s 1

- &) eiT9 @ + liw (d7, !I

(3.7)

an expression h s t given by Wigner ( 8 ) . From the &st operator form of M in (3.6), and by expressing $(q) in terms of the momentum wave function $ ( p ) $(q) = h”J$(P) f+lfidP,

(3.8)

we find, by a series of partial integrations,

M ( T , ~=) JL-) = h-f

ss

[@*(q)$(p) &‘@/fi]

e+fi7i7eeib+W)pdq

fs

ef(Wa*/a~ &[@-*(q)#(p) e { ~ / f ei(TP-k&)dpdq, i]

and hence the alternative expression for the phase-space distribution p ( p ,4) = h-k eWWazhaQ[$*(q)$(P)ed~dfi],

(3.9) (3.10)

It is shown in Appendix 1 that the Heisenberg inequality Ap Aq 2 ?$follows directly from the expression for P ( p ,q) given above. In this sense, the expression of the phasespace distributions in terms of the wave vectors may be considered as a more complete formulation of the uncertainty principle than that given by the inequalities, since it should contain all possible restrictions on the probabilities and expectation values of non-commuting observables. This choice of expression for the phase-Bpace distributions constitutes a new hypothesis, not already included in the basic postulates of quantum theory a8 they are usually formulated. The discussion of certain difficulties associated with this choice, in particular the appearance of ‘negative probabilities ’ for certain states, is made clearer by further developments of the theory, and will therefore be deferred to $15. 0ther possible choices and the possibilities of experimental verification are discussed briefly in Q 17. 4.

PHASE-SPACD EIQENPUNCTIONS

If we insert the expansion of the wave vector $ in terms of an orthonormal set of eigenvectors

@ = Z%@z 1

(4.1)

in the expression (3.3) for F(Y,s),we find for the latter the expansion

(4.2) (4.3) (4.4) (4.5)

171

Quuntum mechuks as a stutistical theory 3.03 where (4.4)refers to the case of continuous eigenvalues r , a and (4.6)to that of discrete eigenvalues ra, s8. The functionsf,(r, s) form a complete orthogonal set in the Hilbert Bpaceof the phase-space functions F(r,8),satisfying the relations?

(4.6) (4.7)

(4.8) (4.9)

In the general case, this follows from the fact that (4.3)and (4.4) or (4.5) form a unitary transformation from a vector, say @lk, of oomponents ?,41*,@k in the product space of the vectors ?,h*with the vectors @, to ,f& The veotors @lk form a complete orthogonal (and self-orthogonal) set, and these properties are invariant under a unitary transformation. Furthermore, it is easily seen from their definition that the fir, form a Hermitian matrix with respect to their subscripts I, k fik(r,

s, = ff21(r,

(4-10)

We shall see later (557 and 8) that the& can be interpreted as the eigenfunctions of characteristic equations for the phase-space distribution functions, corresponding t o the eigenvalue equations of the 9 ’ s ; we therefore call them piwcse-spaceeigenfunctions. In the case of the canonical coordinates and momenta q and p , relations (46)-(4.9) can be proved by elementary methods (cf. Appendix 2), and the f z A ( pq) , have the explicit expressions, corresponding to (3.7) and (3.8)’ 1 t i k ( p ,9) = % J h ( q - iw + ~ p $lc(q 467)a7,

(4.1 1)

fik(p,q) = 7t-* e*(fi/t)aa/aalJn aalJg[@r(q) & ( p )e @ g / a ] .

(4.12)

+

Substituting the eigenfunctions @&)

= h-* e4P’*Ifi in a p-representation, we find

(4.13)

(4.14)

t Integration must be replaced by summation h what follows when the eigenvahes of r, s are discrete.

172

J. E. MOYAL

104 6 . MBANVALUES,

OPBRATORS AND MATRIUES OF FUNCTIONS OF TBB: DYNA.MIUAL VAFUBLEB

The mean value of an ordinary function #(r,s) taken with respect to the phase-space distribution P(r,s) is

B = JJ#(Y,

$1 P(r,s)dras

= [/[[Q(r,

s ) ($,ei(*+OQ$)e47r+0@drdsd~d6

(5.1) where y(7,6) is the ordinary Fourier inverse of G(r, s) y(7, 8) = S f a c r , s)e-q7?+")drc~3.

(5.2)

0 is thus the mean of the operator G = / / y ( ~ 8) , ef17r+o@d7d6,

(5.3)

which is thus the operator corresponding t o the ordinary function G(r,s) in our theory. It now ~OUQWS that the matrix Gl,c of G in any representation of eigenvectors $-1 can be obtained by integration of the ordinary function G(r,s) with respect to the corresponding pkse-space eigenfunctionflrc(r,s)

u ~=, [ J o ( r ,s)fzk(r,s)drds = / J / [ a ( r , s) (pz,eqTr+")pIc) drdsd7do =W

Z > G$k)*

(5.4)

Since file is a Hermitian matrix with respect to I and k, we see at once from (5.4)that Glkwill be Hermitian if Q(r,s) is real. The operators and matrices corresponding to any function of the basic variables r , s are thus uniquely defined by the phase-space distributions. I n other words, our theory of phase-space distributions is equivalent t o a theory of functions of noncommuting operators. Inversely, this theory of functions defines the phase-space distributions uniquely. In the special case of functions U(p,q) of canonically conjugate coordinates and momenta, (6-3) coincides with an expression derived by Weyl(9)on group-theoretical for P ( p ,q) is considerations.A n alternative expression corresponding t o (3-10)

G

e ~ ( n / i ) a ~ / aGo(q,p), ~aa

(5.5)

where G,(q, p) is obtained directly from the ordinary function G ( p ,q) by writing all the operators p to the right (e.g.cpp'n), and this order is maintained when applying the operator e$(fi/i)@/aPagto Go(cf. Appendix 3 for the proof; see also McCoy(10)).The form of the usual operators of quantum theory: energy, angular momenta, radial momenta, etc., are not changed when they are derived by this method from the correspon&g classical functions of p and q.

173

Quantum mechanics as

CL

statistical theory

105

II. QUANTUM DYNAMICS 6 . THBLAWS OB MOTION OF GENEBAL DYNAMIUAL STOUHASTIC

PROUESSES

We now come to the statistical interpretation of quantum dynamics. What we have t o do for this purpose is to find the temporal transformation laws of the phase-space distributions of quantum theory corresponding to the quantum equations of motion. mentioned in 5 1, thia cannot be done within the framework of clasaioal statistical meohanics, which is a ' crypto-deterministic' theory, but appears rather as a special cage in the general theory of dynamical stochastic processea. We start therefore with a brief survey of the integral and differential relations through whioh laws of motion can be expressed for such processes. The theory will be developed for Cartesian and momenta only. The fundamental integral relation connecting the probability distributions P ( p ,q; t ) and Fo(po,qo;to)at times t and tofor a given mechanical system is

(6.1) where R is the distribution of p , q at t conditional in p,, qo at to. X is therefore the temporal transformation function, and must express the laws of motion of the system. While Faand P depend on the initial and h a 1 states of the system, K must be independent of these states, and depend on the inherent dynumical properties of the system. Hence the assumption that K is homogeneous, i.e. invariant for a translation of the origin in t, and dependent only on the interval t-to (as long as there are no external time-dependent forces acting on the system). K gives the transformation for fhite intervals. We now derive the corresponding infinii;esimal transformation. The characteristic function A for the differences q - 6, p - 7 conditional in 5 , is~

A(T,8 I 7,E; t - t o )

=

ss

e'[o(q--8+T@-'l)lK(p, q 1 7, E; t -to) dpdq.

(6.2)

We make the second assumption that in the stochastic processes of physics, the probability of a transition from [, 7 t o q [, p + 7 in a small interval t - tois of the order of t-io, For t = to, obviously K = 6 ( p - ~ ) 6 ( q - [ ) and A = 1. Hence (A.-l)/(t-to) tends to a finite limit L when t +to

+

(6.3) We shall call L the derivate characteristic function. If M(r,8; to)is the characteristic function at to

M(T,8; to)=

fs

ei(Tq+@) Po(7, [;to)dTd6,

(6.4)

then the characteristic function at t is

M(T,8; t ) = /Jei(Tv+60A(.r,0 I 7, 6;t -

t o )Fo(7,E;

$0) d

~dt.

174

J. E. MOYAL

106 Hence

(6.5)

(6.6) (first suggested to the author by Prof. M. S. Bartlett). ( 6 6 ) and (6.6) express the idnitesimal transformation corresponding to (6.1) in terms of characteristic functions; they can be inverted to express this transformation directly in terms of distribution functions. This may be achieved in two ways; if L admits a Fourier inverse

~ ( pq I ,7,&)=

jJ~(7, B I7 , t )

e([7($--23)+0(5--4)3d7d~,

(6.7)

we obtain for J' the integro-differential equation

a

$(P,

q; t ) =

p(P, I

q 7,5) P(7,5; t )d7dk.

(6.8)

If, on the other hand, it is possible to expand L in the form

(6.9) (where the anr(r,t)are called the derivate moments of the system), then the differentid equation of infinite order

_F1

satisfies

(6.10) This reduces to an equation of finite order if the expansion (6.9) for L terminates, i.e. if the derivate moments vanish above given powers o f p and q. 7. EQUATIONS OF THP

MOTION FOR THE PHASB-SPAUI DISTRIBUTIONS OF QUANTUM THLORY

In order t o derive the equations of motion for the quantum phase-space distributions, we look for the time derivatives of their cheracteristic functions. We find from the Poisson-bracket form of the quantum equations of motion -=JP*(s)[M,WlP(s)dq aiv =~~~*(p)[MH-HMlP(q)dl, at

(74

where M(7,O) is the characteristic funotion operator (3.5), and H the Hamiltonian operator, expressed from (5.3)by

(7.2)

175

Quantum mechanics as u statistical theory

507 ~ ( c Tp,) being the Fourier inverse of the aorresponding classical Hamiltonian B ( p ,q). Hence, using expression (3-5), we obtain

(7.3) (7.4) where a/apH, a/aqHin the right hand of (7.4) operate only on H and a/apR, a/aqF on F . The comparison of (7.3) with (6.5) gives the derivate characteristic function L(T,O

I Ip,d = ~ [ H ( p + g n s , q - ~ ) i ~ ) - - , Y ( p - - t n e , ( l + ~ ) i 7 ) ] .

(7.5) (7.5)

If L possesses a Fourier transform a ( p , q I 7, 6 ) = i{/[H(T -I-&fie,

6-

@) -B(y -@6, $4- @7)] eirT(lr-ll)+s(s-a)ld7d6, (7.6)

then P ( p ,q; t ) satisfies an integro-differential equation of form (6.8)

(7.7)

(7.8) which is easily shown equivalent to (6-10)with derivate moments

(7.9) Inversely, the quantum equations of motion, and in particular the Schrodinger equation, may ba derived from the equations above for P(p,q;t ) (cf. Appendix 4). There is thus complete equivalence between the two. Pinally, we may notice the analogy between the right-hand side of (7.8) and the olassical Poisson bracket. This may be generalized in the following way. It may be shown by a method similar t o that leading t o (7*8),that the commutator ifi[RG - GR] of two operators R, G obtained (e.g. by (5.3) or (55)) from the ordinary functions R ( p ,q), @ ( p ,q) is identical with the operator corresponding (by the same rules) to (7.10)

In other words, (7.10) is the analogue of the classical Poisson bracket when the laws of quantum mechanics are expressed in phase-space, and the commutator is the

176

108

J. E. MOYAL

corresponding operator in a q- or ^-representation. It is also seen from this that operators whose classical analogue is 0 may correspond to non-vanishing phase-space functions in the present theoryf-

8. THE GHARAOTEKISTIO EQUATIONS OF PHASE-SPACE EIGENETOTOTIONS

The expansion of the distributions F( p, q; £) of a conservative system in' terms of its energy phase-space eigenfunctions/.^.^, q) is, from (4-2), , q) e«*H**»W.

( 8-1)

Substituting in (7-7) and identifying term-by-term, we see that the/#c are the eigenfunctions of the homogeneous integral equation (8-2)

The kernel 8 can therefore be expanded in terms of the/iA. ).

(8-3)

i, le

Similar characteristic equations can be found for the eigenfunctions girc(p,q) of any operator G corresponding to the classical function G(p,q). Let ji be the eigenvalues of G Gu g) = ^). (84) i(

7i

Calculating the mean of the commutator [G, M] from the two sides of (8-4) J(g) [GM - MG] uk(q] dq = (y? - 7,,) j*J e«^> gik(p, q) dp dq

we find the characteristic equations for gik *%

/* /*

> q) - rr-73 Sa(P> I^' rk~7iJJ

)N

'

where the kernel

= 2^i,Ss (r* - r*) «T«(J f This question was raised by the referee.

(8/ 0 q21Pl,ql; t2-tl)E(pIJq1(p0,qO; tl-tO)dpldql'

(9'8)

The transformation (9-1) form therefore a continuous unitary group. Stochastic processes satisfying the iteration relations (9.8) are known as Markoff processes (cf. Hostinsky (11);see also Jeffreys (12)). The energy eigenfunctions f $ k ( p ,q ) of a conservative system are easily seen from (9.4) and (9.6) to satisfy the homogeneous integral equation f , k ( p , q) = ~ - w $ - W f - ~ o ) / f i

(9-9)

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J. E. MOYAL

110

The transformation function of a conservative system therefore forms a kernel symmetrical in p , 4, antisymmetrical in t

K(P,Q I 230, 4 0 ; t-to)

= h CfdlhA9) f&(Po, PO) ei(ErEkl(i-fo)’fi k

=K

(Po,40 I PI 4; to -t)-

(9.10)

An alternative expression for K can be given in terms of the transformation wave function (9.11) Substituting expression (4-11)for fik in (9*10),we find

%J,4 Ip o ,qo; t - to)= 10. THE

ss

- I& I qo- $670) $(q+

e-c(Tp--70po)$*(q

+

457 I q, +h0) d7a~~. (9.12)

RELATION BBTWEEN INFIMTESIMrlL AND FINITE TRANSFORMATIONS:

APPLICATION TO WAVE PACKET AND COLLISION PROBLEMS AND TO THE CALCULATION OF TRANSITION PROBABILITIES

It is seen from the expansions (8.3) and (9.10) of S and K in terms of the energy eigenfunctions that

&A

4 IPo,4 0 ) =

a

IL‘(P,PI P0,qO; f-t,).

(10-1)

Inversely, K can be expanded in terms of S

(10.2) S(rln,h I Po,Qo) d71d5, * * + dT,,dl,,. This is easily verified by substituting from (8.3)for Sand comparing with (9.10)for K . Binoe S has a simple expression, obtained directly from the Hamiltonian, (10.2) supplies also a convenient method of approximation for K when the energy eigenfunctions are not known exactly. We have thus a new method of solving problems in *.*

qwntum mechunics, without having to solve the Schrodinger equation. The distribution P ( p ,4; t ) of a wave packet at any time t is obtained by the transformation (9.1) from the initial distribution Po(po,qo; to)at to.We can apply tkis to the solution of collision problems by introducing a suitable initial distribution $0 describing the motion of the two particles before the collision, and calculating the trensformation function K by (10.2) and (7.6)from the Hamiltonian for the colliding particles. These methods can also be applied to the calculation of transition probabilities. , be the energy eigenfmctions corresponding to the unperturbed HamilLet f l k ( p q) tonian H,. We can approximate for K from (10*2),using the kernel S corresponding to the complete Hamiltonian H = Ho+Hl (where Rlis the perturbing term). Taking a single diagonal eigenfunction fkk(po,qO)as the initial distribution at t = 0, the expansion of the transformed distribution Fk(p,q ; t ) at time d in terms of thef,, is (10.3) F I C h g; t>= z: 4*CCkmfnrn(P,4). n, 111.

179

Quantum rnechmics as n statistical theory

111

The transition probabilities ckn(t) from state Ic t o state n are the diagonaI coefficients Cbn c&akn whose expression in terms of R will clearly be I

///k(P, I

q pO?40;t ) f k k ( p O , %)fntL(p,q)dpOdqUdPdq* (Ioe4)

= aZnaktb

11. THEPROBLEM OF DETERMINISM IN QUANTUM MEUH~MUS

The present theory should help to elucidate the question whether quantum mechanics is deterministic in the classical kinetic theory sense?, since it permits a direct comparison between the two. The infinitesimal time transformation of quantum phasespace distributions (7.8) may be written in the form (11.1)

where @lap,3/84] is the phase-space differential operator giving the classical Poisson bracket. The corresponding transformation of classical kinetic theory is given by (11.2)

Its deterministic character may be seen from the fact that the characteristics of this first order partial differential equation are simply the classical paths in phase-space. Alternatively, we may say that P is an integral invariant of the transformation generated by the operator {a/@,2/84); an element Soof phase-space will transform to 8, in the interval t, and r

r

(11.3)

This no longer holds in the case of quantum theory; the transformation generated by the operator (2/7i)sin +Ji{a/ap,a/i3q}is equivalent to {a/ap,a/Q}when applied t o H p , Hq, but not in general when applied to HP,so that while Sowill transform into X,exactly as for the corresponding classical system, yet generally n

(11.4) Hence the present theory leads t o the conclusion that quantum theory is not generally deterministic in the classical sense. In the correspondence principle limit, when h+O, the quantum equation (11.1) is seon t o reduce t o the classical equation (11.2); this will equally well be the case if the Hamiltonian H ( p , q ) is a second degree polynomial in q and p , leading t o the amprising conclusion that systems such as a free or uniformly accelerated particle, or a harmonic oscillator, are deterministic in quantum theory: this should not be taken too seriously, since even small perturbations or non-linear terms would, according t o ( l l s l ) , destroy this deterministic character. The phase-space transformations with time of quantum theory form a continuous Unitary group, which reduces therefore to the group of contact transformation of

t

Cf. in this connexion Whittaker (2), Jeffreys (12) and SO Reichenbech (25).

180

J. E. MOYAL

112

classical mechanics in the correspondence principle limit and for the ‘deterministic’ quantum systems whose Hamiltonian is a second degreepolynomial;the transformation function K of 3 9, which is the probability distribution of p and q at time t conditional in po,qo, a t time to, degenerates in the classical limit t o a singular distribution, with complete concentration of the probability ‘mass’ on the classical path in phase-space issuing from p,, qo; K may then be expressed as a product of delta functions

K = 43 --P(PO, 4 0 , t - toll @ - d P 0 , qo, t - to112 where p and q are the classical solutions as functiona of the initial values po, qo and the interval t-to. The phase-space distributions P at t, will be obtained from Poa t t, by substituting the classical solutions for p and q. This has been shown directly by Prof, M. S. Bartlett and the author in the ‘deterministic’ cases of the free and uniformly accelerated particle and the harmonic oscillator. Owing t o the fact that the transformation is unitary, the eigenvalues of the integral equations (9-8), (9.9) ar8 all of modulus 1; in fact, of the form = eKj%-Ed(t-to)/fi.

I n the theory of discrete Markoff processes (where the random variables have only a discrete and finite set of possible values) characteristic roots of modulus 1 for the transformation matrix correspond to deterministicprocesses (non-degenerateprocesses involving roots of the form [ e-N-fo)l < 1).Yet we saw abovethat the quantum mechanical process is not deterministic in the classical sense. The explanation of this discrepancy must await the further study of unitary-Markoff processes of this type.

III. QUANTUM STATISTICS 12. (XIBBS’S

ZWSIZMBLESAND PHASE-SPAUE DISTRIBUTIONS

A possible field of application for the statistical approach t o quantum mechanics lies in the kinetic theories of matter, where the joint distributions of coordinates.and momenta are required. As a first step in this direction, we shall study the equilibrium distributions in large assemblies of similar systems. The notion of Gibbs’s ensemble is translated into the quantum theory of statistical assemblies by introducing ‘mixed’ states, where the assembly has a probability P, t o be in a state y?n and the average of any dynamical variable G is given by the diagonal sum = c ($It, G$,)P, (12.1) .n

(Dirac(13));the introduction of Gibbs’s ensembles in quantum theory is due to von Neumann. The phase-space distribution corresponding to a Gibbs’s ensemble may be found in accordance with the method of 0 3, by calculating the mean of e t ~ ~ ( ~ u r ~ from (12.1) (yU, 9, being the dynamical variables characterizing the assembly), and taking its Fourier inverse, For the Cartesian coordinates and momenta of an assembly of N degrees of freedom

(12.2)

181

Qaantzcm mechanics as a statistical theory

113

and the phase-spacedistribution p is a s u m of diagonal eigenfunctions pn (see$9 3 and 4) (12-3) A P u , 510) = Pn(Pu,qu)Pn,

x n

= JL-~N e+(wozuanm,auu[7kt(qu) #,(Pu) e ~ z u ~ u ~ ~ n ~ ,

(12.4)

where $n(qu), q5n(p,,)are the eigenfunctions in qu, pu representations respectively. Since each term ,on in the right-hand side of (12.3) is a solution of the phase-space equation of the motion (7.8), the transformation with time of p will be governed by the game equation, which now appears as a generalization of Liouville's theorem for the probability densities in phase-space of statistical assemblies. Introducing the phaaespace differential operator of a Poisson bracket (12.5) we have symbolically

(12.6)

It has been held that the existence of Gibbs's ensembles 'is rather surprising in view of the fact that phase-space has no meaning in quantum mechanics' (Dirac(13)). This apparent paradox is removed by the statistical approach to quantum theory, which leads, as seen above, t o an interpretation of ensembles closely analogous t o that of classical statistical mechanics. 13. PEASE-SPACB DISTRIBUTIONS OF ONE MBMBER OF A STATI9TIU& ASSEMBLY We consider now an assembly of similar particles in weak interaction. For a given energy - - En of the whole assembly, we find complexions anwith a, particles of energy k

lC

1

1

el, u2 of energy b2, ...,ak of energy E , ~ , N = I;ai,and .En =

aiei. Assume at first that

the energy eigenstates of individual particles are non-degenerate. The eigenfunctions corresponding to a, are M.B. case: Pan= %(qJ 4%) - * - %(GI) %(%zl+l) * " 2c,(PN)>

B.E. case:

A,, = ( W - t 2 pr%(ql) Ul(ClZ) P

F.D. case:

=

* . - %7&)I,

(N!)-t2 k P[u,(q,) ul(q,) ... zlk(qN)], P

1

(13.1)

where M.B. refers to a Maxwell-Boltzmann, B.E.to a Bose-Einstein (symmetrical), 8ndF.D. to a Fermi-Dirac (antisymmetrical),assembly, P denotes all the permutations of the qc, and the + or - signs in the F.D. case refer t o even or odd permutations. The numbers of distinct wave functions for each energy En are " M.B. case: 0,. = a,!u,! ...uk!'

B.E. case:

C,, = I for all an,

F.D. case:

C~ =

(01

(13.2)

when all ad = 0 or 1,

ifanyui>l. 8

182

J. E.MOYAL

114

The phase-space distribution p(pu,qu) and eigenfunctionspdn( p g ,qr) for the assembly are obtained by substituting from (13.1) in (12.3), (124). It is easily seen that in the M.B. case pa, is a product of diagonal eigenfunotionsfii(p,q ) of the individual particles only, while in the B.E. and F.D. cases, non-diagonal eigenfutlctions occur too. The phase-space distribution for one particle is obtained by integrating over the coordinates and momenta of the remaining particles f(pJ.2ql) =

f

‘**

2(N-1)

/ d p u , 4~7)drpZ&?Zd~3d!?3

...@ I V ~ ~ “

(13.3)

Owing to this integration, all terms in plrninvolving non-diagonal eigenfimction cancel,

becauseJJf&pdq eigenfunctions

= 8i,c.Hence in all three casesf(pl,ql) appears as a sum of diagonal

f (Pl,Q1) = z %h(Pl,

Sl)*

(13.4)

.1

It is easily shown that the nc are simply the average frequencies of the occupation numbers a, of (13.1). Introducing a canonical ensemble, where the Pn of (12.3) are proportional to e-Ed6T, we obtain (13.6)

By substituting from (13.2) for the C, .the n$ can be calculated by the method of ‘sums-over-states’ (Schrodinger(u)), leading to the well-known expressions 1 nc = (l/f es4kT-y’

(13.6)

(13.7) M.B. case: y = 0; B.E. case: y = 1; F.D. case: y = - 1, which can be substituted in (13.4) to give an explicit expression for the phase-space distribution of one member of an assembly, As usual in equilibrium theory, all results are independent of the type of ensemble provided that the dispersion of the total energy is sufficiently small. The effect of degeneracy of the individual energy eigenstates i8 to introduce nondiagonal terms in (13.4). As a result, the lzi must be multiplied by the corresponding order of degeneracy wi,while the fic must each be replaced by (13.8)

whers the indices k, 1 refer to the degenerate phase-space eigenfunctions at the ith level, supposed orthogonal. The foregoing may be used to justify the introduction of ensembles in quantum theory. If we think of an ensemble as an assembly of similar assemblies, then the distribution of one assembly will have the diagonal expansion (12.3) for the same reason that the distribution of one particle in an assembly has the diagonal expansion (13.4), even if the ensemble is in a pure state. If the ensemble consists of an infinite number of distinguishable assemblies, then the coefficients P, of the expansion must be M.B. factors e-E.”T(-Q being now the energy of one whole assembly) rtnd we thus have a canonical ensemble.

Quantum mechanics as a statistical theory

P 15

We may compare averaging over an ensemble t o averaging over time. If an assembly in a pure state, non-diagonal terms in the expansion of its distribution function P(Pu, 4b; t, = a?akPik((Pa, %) e'(E'-Edt'n (13.9) 2, k

cancelin a time average, leaving a diagonal expansionsimilar to (12.3).Thisis analogous to the ergodic principle of classical theory. 14. JOINT PHASR-SPACE DISTRIBUTION FOR TWO MEMBERS OF BN ASSEMBLY The distribution function for two particles is obtained by integrating p over the coordinates and momenta of the remaining particles. f ( p 1 , 41, p2,q 2 )

=

f

"*

2(N-2)

/P(PcJ q g ) h

d q 3 *.'

(14.1)

*NdqN*

In the M.B. case, the integration of each eigenfmction pa, yieIds onIy products of diagonal eigenfunctions of the form ,fi{(pl,Q )fiFe(p2, q2). In the other two cases, it is seen that if i p k, there will be in addition non-diagonalterms (obtained by permuting the two particles) fik(pl,ql)fw(p2,q2), preceded by a + sign in the B.E. case, a - sign in the F.D.case. Other non-diagonal terms in pm cancel by integration as in the case of a single particle. Hence we can write for all three cases

f($kIll, P 2 , q 2 ) = i,2kn i k f i i ( P 1 , ql)f k k ( p 2 , 9 2 ) + y i z k n i k

f i k ( p 1 , 4 1 ) fk.l($b

qa),

(l4.2)

where y has the same meaning as in (13.7). The coefficients of this expansion are easily found t o be for a canonical ensemble

(14.3) Carrying out the summations in (14.3) by the ' sum-over-states' method, we find that the non-diagonal coefficients (i Ic) are

+

nik

(14.4)

= n(nk,

where the niare the average frequencies of the a$,as given in (13.6),while the diagonal coefficients are (14.6) M.B. case: mi$= n!, B.E. case: niC= 2 4 F.D. case: nii = 0.t The last (F.D. case) is of course a result of the exclusion principle. Substituting in (144)we have (14*6) f k i r 41,Pz,4 2 ) = 2: %nkfii(%k41)f k k ( p Z , & ) +Y n i n l c f i k ( p l , nl) f k k ( p 2 , 4, 16

4, k

which may be written, after comparison with (13*4),

(14.7)

t Strictly speaking, the right-hand sides of (14.4) and (14.6) should be multiplied by a normalizing factor (1+yx.n:). i

8-2

184

J. E.MOYAL

116

We see thus that symmetry (or antispmetry) conditions introduce a probability dependence between any two particles in B.E. (or P.D.) assemblies even in the absence of any energy interaction. For example the coordinates and momenta of the two particles will be correlated, with covariance I

6,Ic

where

Qnk,

(14.8)

Pn, are the matrices of the individual q’s and p’s,

It is this dependence which gives rise to the ‘exchange energy’ between the particles when they interact. 15. LIMITATIONS OF THE STATISTIUAL APPROAOH

TO QUANTUM THEORY

The results obtained so far seem t o offer a, fairly complete scheme for treating quantum mechanics as a form of statistical dynamics. It is important now to return to the difficulties mentioned a t the beginning of this paper, and discuss the limitations of this approach. First, we notice that phase-space eigenfunctions must generally take negative as well as positive values, since they are orthogonal. Only one eigenfimction (generafly the ground state one) may possibly be non-negative for all values of the dynamical variables, except for singular eigenfunctions involving delta functions, such as the momenta eigenfunctions (4.13). Hence, on taking for example Cartesian coordinates and momentap, q as the basic system, the phase-space distribution in the nth energy eigenstate formed according to the method of 6 3 would be the diagonal eigedunction, fnn(p,q), which can be negative, and is therefore not a true probability. This is not really surprising, because we have seen in 9 that the dynamical equations are those of a, Markoff process. The existence of eigenfunction solutions for the fundamental equstions (9-8), (9.9) of Markoff processes is well known (see Hostinsky(rl)),and it is also hiown, that these eigenfunctions are not generally probabilities by themselves. Probability distributions are expressed as non-negative linear combinations of these eigenfunctions. In the language of quantum theory, we may say that true probabilitg distributiolzs of any given set of non-commuting variables do not exist for every state; the physical interpretation would be that where the distribution, as calculated by the method of § 3, can take negative values, it is not an observable quantity, This is a restatement of the necessity, already discussed in 5 2, for postulating tho existence of Werent phase-space distributions according to the basic set of dynamical variables. Take, for example, a system composed of one proton and one electron. The distribution P(p,q)corresponding to the @(q)of a Gaussian wave-packet is positive for allp and q, and is hence an observable quantity. On the other hand, there would be no observable ( p , q )

Quantum mechanics as a statisticd theory

I17

distributions for the energy eigenstates of a hydrogen atom, though an observable distribution may exist for some other set of variables, It is usually accepted that a dynamical variable G is exactly equal to its eigenvalue g, when the system is in the corresponding eigenstate. This means that the operator W corresponding to the function T(U) should be equal to the function W of the operator G, W = W(G),since if U is exactly equal to gn the mean of W is FV = W(g,), and hence (15.1) = (1C.,,W,J = (@n,W(G)@n) = (Pw W(gJ II.,> =~(QJNow it is easily seen (Appendix 5) that according to the theory of functions of 3 5 this condition is fulfilled only when B is a function of some linear combination of the basic variables r , s: G(ar -I-bs). This again i s connected with the necessity for phase-space distributions adapted to the experimental situation; if the latter involves observation of B, then the distributions must be set up for 8ome set of variables r , s such that B = G(ar + bs). In order for the scheme to be consistent, it should be possible to prove that if a state $,I admits a non-negative phase-space distribution F at the time t = 0, then F wiU be non-negative at any time t. This is easily seen for isolated systems possessing at least one cyclic coordinate 8.Suppose that 0 and its conjugate g are obtained by a canonical transformationfrom the original system q4,pi,and let Qd, pi be the other (transformed) ooordinates and momenta, B(g,0,4,&$) the transformed Hamiltonian. Then

aH

-=

ae

0,

aH - constant = w . -

ag The transformed equationof the motion (7.8) can be written

-+o-+-sinaF at aF ae %

Y-a -a 1

EIP

2 ae2aQs

= 2ip

= 0.

( p constant),

= &@+0/4~

(15.2)

(18.3)

(15.4)

I

Comparing with the expansion of P in energy eigenfunctions, we see that it must be of the form ~ ( g8 ,, 8 , Q ~t>;= 2 a? qCQa(g, 4, e~((Ei-Ed(l+elw)}/T. (16.5) i,k

Hence, if P>O for all 0 a t t = 0, it must be non-negative for all t. This proof was suggested t o the author by Prof. M. S. Bartlett. Binally, we may discuss the meaning in the present theory of observables having no classical analogue. gS2-5 on quantum kinematics are framed SO as to apply t o such observables as well as to those having a classical analogue. The phase-space distributions represent for both types the joint distributions of eigenvalues for non-oommuting Bets, and are subject t o the same restrictions. The quantum equations of motion in phase-space, on the other hand, were expressed only for Cartesian coordinates and momenta, so as t o bring out the relationship with the theory of general stoahastic

186

118 J. E. MOYAL processes. It is clear, however, that they can be extended to general quantum observables, say r and s. If P(r,6 , t ) is their joint distribution, then as in $ 7 , aP/at is obtained (16.6)

16. PRAUl’IUfi

AF’PLIOATIONS OF THE THEORY

The foregoing restrictions are necessary a0 long as we require probabilities in phasespace. They may be relaxed in practical applications of the theory, where we introduce phase-space distributions as aids to calculation, and where the observable quantities we wish t o calculate are necessarily non-negative, independently of whether the phasespace distribution takes negative values or not. It is not difEcult to see that the phasespace distributions and eigenfunctions obtained by the rules of $$ 3 and 4, though not necessarily non-negative, obey the other fundamental rules of probability theory, i.e. the addition and multiplication laws. Bartlett (15) has discussed the introduction of such ‘negative probabilities’ as aids to calculation, and has shown that they can be manipulated according to the rules of the calculus of probabilities (with suitable precautions) provided we combine them in the end to give true (non-negative)probabilities. He remarks that ‘where negative probabilities have appeared spontaneously in quantum theory, it is due t o the mathematical segregation of systems or states which physically only exist in combination ’. Now this relaxation will be possible in practical applications, because the phasespace distributions oontsin more information than is generally required for comparisonwith observations. For example, ifwe wish t o calculate the way the distribution in space p(q;b) of a wave-packet varies with time, we may use the method of $10,

s

becausep(q;t) = P(rp,q;t ) d p = +(q;t)$-*(p;t)willneverbenegative,evenifP(p,q;t ) can be negative. Similarly, transition probabilities calculated by the method outlined in the same paragraph will always be non-negative, whether P takes negative values or not. Finally, we may use the methods of sg12-14 t o calculate the phase-space distributions of members of an assembly even if the phase-space distribution for the whole assembly can be negative. We conclude that in applications of the theory, we need not be concerned whether the phase-space distributions are true probabilities, provided that the final results, expressed either as linear combinations of these distributions or as integrals over part of their range, are necessarily true, non-negative probabilities. 17. UNIQUENESS OF TRPJTHEORY AND POSSTSILITIES OF EXPERTMENTAL VERIFIOATION

The statistical approach to quantum theory involves the introduction of an additional postulate on the form of the phase-space distribution, which is equivalent to a theory of functions of non-commuting observables. The choice of this postulate is not unique. Dirac (16) has given a, theory of functions of non-commuting observables which differs from the one obtained in 5 of this paper; it has the advantage of being

187

Quuntum mechanics

a statistical theory

119 bdependent of the basic set of variables, but, as might be expected from the foregoing discussion, it leads to complex quantities for the phase-space distributions which can never be interpreted as probabiliti6s.t It is natural to ask therefore whether any experimental evidence is obtainable on this subject. In so far as observable results calculated by such theories are equivalent to those obtained by orthodox methods, e.g. transition probabilities, or distributions of coordinates only, this is obviously impossible. However, though the simultaneous measurement of coordinates and momenta is not possible for single particles, there is Borne hope that experiments on large number of particles might be devised to verify the phase-space distributions predicted by the theory. Alternatively, one might hope t o verify the corresponding theory of functions of non-commuting observables if experimental evidence became available on 8ome Hamiltonian involving products of and P. GGS

APPENDICES Appendix 1, #pace-conditionul averages of the momenta and the uncertainty relations The space-conditional moments are the means of pn when 4 is given, They may be obtained either from expression (4.14)for P(p,q)

F$

(A 1.1)

J

where p ( q ) = P ( p )q ) dp = @(a)@p+(q),or from the characteristic function X(7 I a) of

p conditional in'g (see Bartlett (17)) which is seen, from (3.77, t o be 1 psm,

*w

M(7 1 4r) = 4) eZ7g%J = @*(a$(q+ *W/**(q) @(d* On writing $(q) = p"q) eis(Q)lfi the logarithm of M(7 I 9) or 'cumulant' function (KendaU(18))

(A 1.2) (A 1-3)

I

K(7 14) = logM(7 q ) = ~ l o g p ( p + ~ ~ ~ ) ~ ( s - ~ ~ 7 ) - l o g p ( p ) f g r S ( 4 + 3 K ~ ) - S ( S - ~ ~ 7 ) 1 (A 1.4) leads t o a, simple expression for the 'oumulants' %(a) (coefficientsof ( i ~ ) ~in / nthe ! Taylor expansion of K )

(A 1.5)

t Note added in proof. Reference should ako be made to a recent paper by Feynman (26) giving an alternative approach. $ The double bar denotes EL conditional moment, while t b single bar - denotes a mean over the di8tribution of bokh p and q.

-

188

J. E. MOYAL

120

The Zn bear simple relations to the moments fist moment by both methods is %(9) =

(Kendall(18)). In particular, the

as a) ='

(A 1.0)

leading to the interpretation of the argument of the wave-function @(q) as the potential #(q) of the apace-conditional mean F(q). The conditional mean-square deviation ist

(A 1.7) We note also that the asymmetry of a distribution depends only on its odd c u d a n t s ; hence the asymmetry of the conditional distribution of p depends entirely on S(q). Pormulae (A 1.6) and (A 1-7)lead directly to Heisenberg's inequality for the meansquare deviations of p and q. Let a , ,8 be any two random variables with zero means. We have the well-known Schwarz inequality

I

--

(a2/q

= rauflB

PI.

(A 1.8)

Now take a = $(q), where we suppose to become random when we allow q to vary; take also p = q. Then from (A 1.8) above, and assuming (as can be done without loss of generality) that B = = 0, we obtain

(A 1.9)

Hence, fiom (A 1+8),

~ , e s 6 1 q P & 7 >ag2.

(A 1-10) (A 1.11)

Since

the sum of the two inequalities (A 1.9) and (A 1.10) gives Heisenberg's inequality r;.;

z (p4)2+

agz.

(A 1.12)

This derivation of Heisenberg's inequality was pointed out to the author by Prof. M. 8. Bartlett.

t The fact that 41, can be negative according to (A 1.7) results from the possibfity of th0 formal expression for P ( p , 9) being negative in certain states. The restrictions thus imposed on the interpretation of P ( p , q ) as a probability me discussed in 16.

Quantum mechunics as a, statistical theory

121

Appendix 2. Orthogonality and completeness of the phase-space eige.rtf.unctionsfw canon&xzlly conjugate variables The orthogonality relations of the phase-space eigenfunctions for canonically conjugate variables can be seen quite simply. We have, from ( 4 ~ 1 1 ) ~

\jj&J 4)fi%p2 q)dPdq = (.2n)-2j///UT(9.= ~ - 1 J / ~ f ( z%(%) )

467)Uk(q+ @7) 'Uz,(p-

@7') W$((I

uk(y)Uzd(y) axdv = WS~S,,

f $87') e--2(7-7')gdTdT'dfldq

(A 2.1)

(thesecond Line following from the change of variables 2 = 4 - i?iry y = q -+ @T), md

A 2.2)

(A 2.3)

(A 2.4) Appendix 3. Operators corresponding to functions of canonically conjugate variables The proof of (6.15)follows from expression (3.10) for the phase-space distribution

(A 3.1) (A 3-2)

190

J. E. MOYAL

122

The operator corresponding t o a function '(PI

4) = I;@ n k ) P n n

(A 3-31

is obtained very simply from (A 1.1).We have

(A 3.4) (A 3.5) This could also be derived from (A 3.2) (cf. McCoy (lo)). Appendix 4. Transport equations and the Xchrodinyer equation The 'transport' equation of any quantity g(p,q,t) is defined as the equation governing the time variation of the mean z(q,t) at every point q (space-conditional mean). It is obtaineclfrom (7.7) or (7.8) by integrating over themomentap and making use of the expressions in Appendix 1 for the conditional moments of p . In the case of a particle of mass nz, charge e in an electromagneticfield, whose classical Hamiltonianis

(A 4.1) (A&, t ) being the vector, V(q,, t ) the scalar, potentials) integration of (7.8) and substitution of = aS/aq, from (A 1-6)lead to the continuity equation (A 4.2) where p(qt) is the distribution function of the coordinates. Multiplying (7.8) by P k and integrating gives the transport equation for (A 4.3) Substituting in the above from (A 1.6) and (A 1.7), and combining with (A 4-2), we h d (A 4-41 Hence the quantum-mechanical equivalent of the classical Hamilton-Jacobi equation

(A 4.5) Substituting p = $@* and S = &/2ilog(+/$*) mcl adding and subtracting (A 4.2) and (A 4.5) we fmd the Schrodinger equation of a charged particle in the field (A 4.6)

191

Quantum mechcmics

us u stutistica2 theory

I23

Appendix 6. Operators cmresponding to functions of linear combinations of the basic variables According to (5.2) and ( 5 ~ 3 the ) ~ operator corresponding to Q(ar+bs), where a and b are constants, is

(A 5.1) Changing to the variables

6 = ar + bs,

A=-+-

2a

2b'

7 8 111 = --2a 2b'

(A 5.2) 5.2)

(A 5.3) I should like t o acknowledge my indebtedness to Profs. P. A. M. Dirac, H. Jeffreys and the late R. H. Fowler for their criticisms, suggestions and encouragement in carrying out this work, and my gratitude to Prof. M. S. Bartlett for many invaluable discussions and the communication of his various results referred to in the text. M. J. Bass and Dr H. J. Groenewold have studied the same subject independently (cf. Bass (19)(201, Groenewold (21)),and I have benefited from discussions and correspondence with them. The papers of Powell (22),Stueckelberg(23).Dedebant (24) and Reichenbach's book (25) also have a bearing on the questions disoussed in the present paper (I am indebted to Prof. Bartlett for these last references). SUMMARY

An attempt is made t o interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the distribution functions of the complete set of clynamical variables specifying a mechanical system (phase-space distributions), which are fundamental in any form of statistical dynamics, are expressed in terms of the wave vectors of quantum theory. This is shown to be equivalent to speclfylng a theory of functions of non-commuting operators, and may hence be considered as an interpretation of quantum Icinernatics. In the second part, the laws governing the transformation with time of these phase-space distributions are derived from the equations of motion of puantztm dynamics and found t o be of the required form for a dynamical stochastic process. It is shown that these phase-space transformation equations can be used as an alternative t o the Schrodinger equation in the solution of quantum mechanical problems, such as the evolution with time of wave packets, collision problems and the calcuhtion of transition probabilities in perturbed systems; an approximation method is derived for this purpose. The third part, quantum statistics, deals with the phase-space distribution of members of large assemblies, with a view to applications of quantum mechanics to kinetic theories of matter. Finally, the limitations of the theory, its uniqueness and the possibilities of experimental verxcation are discussed.

192

J. E. M o v a

124

REFEREN CE S

DIRAO,P. A. M. C o m m n . DubEn Imt. Adv. Stud. A, 1 (1943), 11. WHITTAKER, E. T. Proc. Phys. Soc., London, 55 (1943), 469-70. (3) DIBAO,P. A. M. The principles of quantum mechanics (2nd ed., Cambridge, 1935), p. 48. (4) JEFX-REYS, H. Phd. Mag. 32 (194l), 195. ( 5 ) HEISENBERG, W. The physical principles of the quuntum theory (Cambridge, 1930), p. 34. (6) CRAMS~R, H. Random variables and probability distributions (Cambridge, 1937), p. 24. (7) KERMACK, W. 0. and MOCREA,W. H. Proc. Edinbwgh Math. Soo. 2, seriea 2 (1931), 224. (8) WIUNER,E. Phys. Rev. 40 (1932), 749. ( 9 ) WEYL, H. T h e theory of groups and quantum mechanics (London,1931), p. 274. (10) McCoy, N. H. Proc. Nat. Acud. Sci., Vash., 18 (1932), 674. (11) HOSTINSKY, M.B. Mdthodes gdndrales d u calcul dea probabititb (Park, 1931), p. 47. (12) JEPFREYS, H. Phil. Mag. 33 (1942), 816. (13) DIRAO, P. A. M. Theprinciples of quantum mechanics (2nd ed., Cambridge, 1936), pp. 140-2. (14) SOJXROD~UER, E. fltatistical thermodynamics (Cambridge, 1946), p. 43. (15) B A R ~ E TM. T , S. Proc. Cambridge Phil. SOC.41 (1944), 71. (16) DIRAO, P. A. M. Rev. Mod. Phys. 17 (1946), 195-9. (17) BARTLETT, M. S. J. London Math. 800. 13 (1938), 62-7. (18) KENDALL, M.a. The advanced theory of statistics, 1 (London,1943), 60, 90. (19) BAss, J. Oompte8 Rendw, 221 (1946), 46-9. (20) B~86, J. Rev. 8&ent. 3240 (1945), 11-18. (21) CROENEWOLD, H. J. Physica, 12 (1946), 406-60. F. C. Proc. Cambridge Phil. Soc. 41 (1945), 57. (22) POWELL, E. U. G. Helv. Php. Acta, 18 (1945), 195. (23) STUEOIIELBERG, (24) DIDEBANT,G. PortugaEiae Pltysica, 2 (1946), 149. H. Philoscphic foundations of quantum mecbnics (Berkeleyand Los Angelea, (25) REIOBENBAOH, (1) (2)

1946). (26)

FEY",

R. P. Rev. Mod. Phys. 20 (1948), 367-87.

DEPARTMENT OF MATHEMATICAL PHYSICS T m QUEEN'SUNIVERSITY, BELFAST

THE EXACT TRANSITION PROBABILITIES O F QUANTUMMECHANICAL OSCILLATORS CALCULATED BY THE PHASE-SPACE METHOD BY M. S. EARTLETT AND J. E. MOYAL Received 14 September 1948

INTRODUUTION The calculation by the usual perturbation methodg of transition probabilities between 1.

the unpedurbed states of a quantum-mechanical system yields approximate results, valid only for small perturbations. The object of this paper is to calculate the ezuct transition probabilities between the unperturbed states of quantum osciuators, valid for lalye as well as sinall perturbations, by using the 'phase-space' method developed by one of the authors (Moyal(I), referred to henceforth as (I)). We give first the main results of (I)required in this paper. The probability distribution in phase-space of a system in a state clescribed by the wave-function $(q) in p-space ist 1

~ ( pq ),= %J+:k(q

- +jiT) e - c ~+(q +417i7) ciT.

(1.1)

Corresponding to the expansion of @(q;t ) in terms of energy eigenfunctionsUJQ)

(1.2) we have an expansion for P ( p ,4; t )

(1.3) in terms of the energy phase-space eigenfunctions

(1.4) These functions form a complete orthogonal system in phase-space which is also 'selforthogonal' and hermitian with respect to the indices 7c, n, i.e.

ss

f k , J ~ ~ m dq ~ d ~ ~ - l ~ k ~ Jlfkndpdp ~ ~ ~ n ~= ~ &, k w

f7m = f ; k s

(1.5)

Furthermore, the matrix GIlcn corresponding to an ordinary function Q ( p ,q) is given by Gkn =

/k(p'

q)fkn(p,9)

(1.6)

The transformation with time of P ( p ,4; t) corresponding to the quantum equations of the motion is given symbolically by

(1.7) -f When no limits are specified, all integrals are t o be taken

from

- co t o + m.

194

546

M. S. BARTLETT AND J. E. MOYAL

where alapH, a/aqH operate only on the classical Hamiltonian H ( p ,q ) of the system; a/8pP,a/aq, on P ( p ,q, t). This is seen t o be an extension of Liouville’s theorem

(1.8)_ and reduces to the latter in the correspondence principle limit (h+O) and for systems whose Hamiltonian is a polynomial of the 2nd degree or less inp andq. For suchsysteme (and they include the free and the uniformly accelerated particle, and the oscillator) the transformation with time of P ( p ,q; t ) is of the ‘deterministic’ type of classical kinetic theory, each element of the distribution transforming in phase-space according t o the laws of classical mechanics. A direct veriiication of the deterministic character of these systems in given in Appendix 1 and in 5 2.t Equation (1.7) specifies the infinitesimal transformation with time of P ( p ,q; t). The transformation over a finite interval t - tocan be given in terms of a transformation function’K(p,ql po,qo;t--to);

(1.9) K is interpreted as the probability of p , q a t t conditional in p o , qo at to, and may be expressed either in terms of the phase-space eigenfuimtions by an expansion similar t o (1.3) (1.10) fic7~(220, ‘lo) fzn(l.’, 4)ei(Ek-EdcL-‘o)’fi, (1.10) q I Po, 4 0 ; t - t o ) = k, n

or in terms of the wave transformation-function

I

$(q qo; t - to) =

u,(qo) tP((q) n e-~~~(~-Lo)/~~,

(1.11) (1.11)

n

by an integral similar to (1.1)

(1.12) The phase-space theory of quantum mechanics may be applied to calculate the transition probabilities of a perturbed system. If K is luzown for the perturbed Hamiltoiuan H , and we wish to calculate the transition probabilities from the lcth unpertu’.bed state in an interval t, we take as initial distribution the lcth diagonal phase-space eigenFo(po,qo) = fL?(l)o, go). function corresponding to the unperturbed Hamiltonian So): The transformed distribution after an interval t is then from (1.9) and (1.3)

(1.13) The transition probability from state lc to state n is then given exactly by$ (1.14)

t cf.,in this comexion. Coulson and Rushbrooke(7). $ It may be shown that the transition probabilities obt&inedin this way are in fact ide11tio8’ with those of ordinary quantum theory; the proof of this statement is given in Appendix 2-

Transition probabilities of quantum-mechcmicul millators

547

In the particular case of detcmninistic systems the transformation function K must reduce to a product of delta-functions expressing the contact transformation of classical mechanics (this is verified in 9 2 for the harmonic oscillator).The distribution at t follows simply by substituting the classical solutions in the initial distribution: F(k)(it),4 ; t ) = fi%it)g(p, SI, t),4&, q, t)J. 2. PHASE-SPACE THEORY OF THE

EARMONIC OSCILLATOR

We now develop the phase-space theory of the one-dimensionaloscillator of mass m, angular frequency w , coordinate Q and momentum P.In terms of the reduced variables q = (mw/?i)*(2, p = (mu$)-* P,its Hamiltonian is H = $(P2/m+rnw2&2) = * ( I p 2 + q 2 ) f i ' d , (2.1) The energy eigenfunctions in p - and q-space are un(q),un(p),where un(q) = ( - 1)n(21~7~*n!)+ e*aa

(e-Q') = (2nnfn!)-te-~~'H,(q),

(2.2)

(2.3)

or, on introducing tho variables x = g + ip,x:k = q - ip

a a (@) (z) IC

fIc&,

z * ) = ( - ~ / * ) ? L + / c (nzn!ii!)-+ezz*

?L,

(e-?.

(2.4)

A inoye convenient expression is obtained by substituting the variables w = 222:':= 2 ( p 2 + q 2 )= 4H/lio, 0 = tan-l(p/q). The phase-space eigenfunctions then break up into the products of an associated Laguerre function of w sncl a trigonometric function OP 0

(&Y

fic,L(w,0 )dwcZ0 = +( - l)k (/c!n!)-+w6~1c-)~)e*~

(W?Le-W>

dw (2n)-l ei(7c-W do

= +( - 1)"(k!/n.!)a,-~(r-n)e-tw~~-k(W)dw (27r)-lei(k-?@d0.

(2.5)

Tho diagonal eigenfunctions &resimple Laguerre functions fnn(w, 8) dud8 = $(

- 1)"

(%!)-I

e*w

(aY -

(W?& e-w)

dw d8/2n

= &(- 1 ) n e - h L , ( W ) ~ W ~ B / ~ T . An expression for these eigenfunctionshas been found independently by Dr EL wold (2) I

(2.6)

I. Qroene-

196

M. S. BARTLETT AND J. E. MOYAL

548

The determinism of the oscillator may be verified by calculating the phase-space transformation function over a finite interval of time t, on using equation (1.12). The wave transformation function over this interval is given by the well-known MehIer expansion for Hermite polynomials (3)

I 40; t ) = r,n u,(!lo) 4 x 4 ) ei(n*)ol = (2n-ishut)-* exp {i(2sinwt)-l

[(q2-+qi)coswt - 24q0]}.

(2.7)

The phase-space transformation function is then

-w,Y I P O ?

40;

t)

exp [i{7,p0-7p + (sinwt)-1[(7q+70q,)coswt-~q,,-~oql}ld n h 0 PI-

J exp{i(sinwt)-l[~(q~~s~~t-pssinot-q,)

= (it+sinwt)--1J

+7"(y0cosot + p o sin o ~-t q)])d

~ d ~ ,

= sinoJt8(qcoswt-psino~t-q,,) 8(q,~oswt+13~sino~t-q), (2.8) a product of delta functions expressingtlie contact transformation of classicalmechanics eigenfunctiom for the oscillator. The expansion (1.10) of K in terms of the l~hasc-spaoe gives the following interesting formula for tlie associated Laguerre functions (32n)-1 r, (k! I % ! ) W;f(k--r~) e-al/)uL;;-lc(w w--8(1c--1L) e-B~t~L?~-l~ $(7+ic)(o-uu-d) 0) 1c 15, n

=~

- w0)&(a-0,- ~ +t 2 ~ 7 ~ )(2.9) )

( L U

where the 8-functions are norinalized over the ranges of zu and 0 , r being an integer such that 0 < 8, -+ wt - 2 m < 27r. In Appendix 3, the above results are applied to derive the equilibrium phase-space distribution of members of a statistical assembly of oscillators. 3. TRANSITION PROBABILITIES OF

A PERTURBED OSCILLATOR

We shall now apply the method outlined in $ 1to calculate the exact transition probabilities of a perturbed oscillator for a perturbing potential of the forin V = p%)~ where &(t) is an arbitrary function of the time. On using equation (1.14), the transition probability from state k to state n is given in terms of the variables w = 2(p2 + q2) and 8 = t a r 1 ( p / q )used in $ 2 by 4)

pkTL(t)= 87i/

0

1

2n

0

~ f k k ( ~eo) 0 JL??'~,(~J

)'

dw,d60J

(3.1)

where wo,8, are the initial values a t t = 0, w and 0 those at time t obtained from the classical solution for the perturbed oscillator

q = qocoswt+posinwt-w

Hence where

8(7)sinw(t-7)d7,

(3.9)

197

Trunsition pobubili ties of ~ua;?ttzcrn-mecfLcicc61 oscillators

649

It is easily seen that e = &?hai s the non-fluotuating part of the work done by the perturbing force, while q5 is the phase change of the oscillator. We now substitute these

solutions in (3.1) and also introduce two auxiliary variables y , t which we shall equate to 1after all the indicated integrations and differentiations have been carried out. This yields a closed expression for the transition probability

where Io(s)is the modified zero-order Bessel function of the first kind (I&) = Jo(is)). The integration is easily carried out, for example, by substituting for I&) i b Taylor expansion and integrating term by term, whence

(3.3) Carrying out the indicated differentiation on y”, [ n ) one finds that

(3.4) With the use of the e-qression for the Laguerre polynomials generating function (3)

(3.5) ‘probability generating function’ (p.g.f.) may be calculated for the p G(r,0) = C 2 1 , OrcrlL ~~

a

--

(3.8)

1- r O

The coefficient of 816 in the Taylor expansion of G ( r ,0) in powers of 0 only, will be the p.g.f. CrlC(r) for transitions froin the lcth state; that of B I c P in the expansion in powers of both 0 and T will be pkn(t). E’or n. 2 k

(12-k)!(ri+v)!(-2)~

). (3-7)

198

550

11.,S. BARTLETT AND J. E. MOYAL

Since, as is obvious from (3-3)or ( 3 4 ) , the pkn satisfy the principle of detailed balance, i.e. pkn = Pnk, the probability of transitions t o states n < Ic is simply obtained by interchanging the indices n and k: in the right-hand side of (3-7). The p.g.f. Cr0(7)and the probabilities pOn(t)for transitions from the gromcl state are simply those of a Poisson distribution

(3.8) These exact results may now be compared with the approximate ones obtained by the standard perturbation method. The Taylor expansion of the approximate expression for the p.g.f. QL(T,l9)

--

E

(1-7)(1-0)

gives approximate values for the

1

(3.9)

(3.10)

correct to the first power in E , whose expression is identical with the first approximation o j Me perturbation melhod. Expression (3.7) shows that in general the probability of a 2nth-pole transition for small E is of the order of (e/fiu)’L,i.e. only dipole trailsitions have an appreciable probability. The perturbation method equates t o 1the exponentid factor exp { - E / ~ J } in the exact expressioii (3.7). This procedure is justified only for small B ; as the perturbation eneryzJ increases, however, multipole transitions become progressively moreprobable. In order to find the most probable oiies from the ground state, let us substitute the continuous variable x for n in (3.8)

The most probable transition is therefore t o the state n given by

-

where the logaritlmic derivative $(x) of the factorial function x! is an iiicreain& log x,and hence diu = En - 30 6. function of x (Jalullre-Emde(4)). I?or large x,I,@) Similar consideratioiis apply to the general case: it will be seen from (3.7) that the Dost probable transitions are those from states k t o states n such that (n- h)&w = En - 31, Hence the physically plausible result that for large perturbations the most probable transitions will be those for which the c7mnge in energy i s approximateJy equal to the work, done by the perturbing forces. APPENDIX

(1) Pree particle and paytick under constant force

The way in which the phase-space method may be used to solve wave-paoket problem is easily exemplifiedin the deterministic cases of the free particle and the particle under

199

TmwitiorL probabilities of yziuntzLrrL-)nechc~.1aicalosciElaior,5

551

force. I n both these cases, the Schrodhger equation is most easily solved in Domenturn space. We describe the particle at t = 0 by a Gaussian wave-packet

ww

(A. 1.1) correspondingto the minimum uncertainty uo,so = 4% on its position and momentum (of. Kennard (5)), so and uobeing respectively the mean square deviations of p o and qo &Ddtalring the origin at the mean of p o and qo. The corresponding phase-apace distribution is by (1.1) $ o ( ~ o )=

~ O ( P 0 , Y O= )

~ X [P-P~/*S;I

,1 ~ ~ ~ ( ~ o - ~ ~ ~ ) ~ b-fi1-lexp - B(Pt/44-48/4)1.

(A. 1.2) The solution is then obtained simply by substituting in (A. 1-2)the classical solutions for p and p in terms of po, qo, t. In the case of the free particle, this gives for the distribution at t R P ,4; f) = ( T WexP [ - 8(P1so)2- & @ / g o -P~1mf10’o)21. (A. 1.3) It is easily shown that this corresponds by (1.1)to the wave-function (A. 1-4) $ ( p ; t ) = (2nst)-fexp [ - (;u/2~,)~+ip~t/2mKj, which is the solution of the Schrodinger equation =

(A. 1-6) with 4 0 ( p o )as initial. wave-function a t t = 0 (cf. Darwin (6)). Similarly, for the particle under constant force my we find for the same initial dis-

(A. 1.6) which again corresponcls t o the solution

(A. 1.7) of the Schrodinger equation

p2 -$+-2na

mg?i,a$ $84 = --z 211 i at‘

(A. 1.8)

( 2 ) Equivalence between the trunsition probabilities calculated by the phase-space method and those of standard quantum theory

It is convenient in the calculations that follow to introcluce Dirac’s notation: (sl I ql) for the coorclinate eigenfmction corresponding to state sl, at time t,, (s2 I q2) for state s2 at t2, (ql I qZ) for the wave transition function from 4, at tl t o q2 at t,, (sl I s2) for the transition matrix from s, at 4 to s2 at t,; the corresponding transition probability in the standard theory is then I (a1 \ s2) 12. The transition probability p,,,,, calculated by the phase-space method, is from (1.13) and (1.14)

(A. 2.1)

200

562

M, S. BARTLETT AND J. E. MOYAL

Substituting from expressions (1.4) for fss(p,q) in terms of (8 I q ) and (1-12) for .K in terms of (ql I q2), we have

where the cliaiige of variables x1 = q1 + iZkrl, z;

= q1- *fir1, etc., has been

made. Heiice

(A. 2.2) P S L S 2 = I (81 I 6 2 ) 15 i.e. the two expressions for the transition probabilities are formally equivalent. (3) YJime-spacedistribution of a memboy of u statistical assembly of oscillators The equilibrium distribution of the coordinates and momenta of one member of a statistical assembly of similar particles is expressed in I, p. 114, as a sum of cliagond phase-space energy eigenfunctions relative t o the inciividud particles

f(13,q) = c c nlLf,clc(lJ,q ) ,

(A. 3.1)

16

where C is a normalizing constant and the u,, = e-c7Jkz' in the case of a Maxwell-Boltzmann assembly. On substituting from (2.6) for an assenibly of oscillators, this becomes f(w, 8) = (47r)-1(1 - e--JLw/'c'l' ) c ( - 1)ne-!lr-'lRo/kT~ 1E.(w) 72

=

(47r)-l tsnh (6wl2lcT)exp { - &[zu taiih (&u/2l;.T)]},

(A. 3.2)

where the last line follows from (3.5). Transforming back from w and 0 to p and 4, we finally find EL Gaussian distribution for the coordinates and momenta of a member of an assembly of oscillstors

f ( P >q ) = (274-l (fi4Jv exP { - 8[(P2+ q2)(fiu/-QI), (A. 3.3) where E is the mean energy E = @w coth (fiwl2kT)= ?i,w(efl*/162'- 1)-1+ &. (A. 3.4) We may note that the mean-square deviation of the energy cdculated from the ~. above distribution is & = E2 instead of the usual expression u& = E 2 - ( 3 5 ~ )TGS follows from the fact that the phase-spacetheory of quantum mechanics yields different distributions according to the basic system of variables chosen (see I, p. 100 for a fuller discussion). If the energy is one of these variables, then it is quantized, with possible values Ek = (k+ 8)fiw, and its distribution is simply (A. 3-81 P ( E ~ )= 2 sinh (?L/fLlcT) e-6@2', yielding the second value CT= ; Ea- ( + $ u )for ~ the m.5. deviation of the eiiergY- If

p and q are chosen as the basic system, then tho energy distribution becomes continuous, with a m.s. deviation of (+%J)~ for each of the energy eigeifunctions, leading thus to the first value cg = E2.

20 1

T v a i d i o i a pobubilities of quaiztum-mecha?ziculoscil1aiol.s

553

SUMMARY

The ‘phase-space’ method in quantum theory is used to derive exact expressions for the transition probabilities of a perturbed oscillator. Comparisonwith the approximate results obtained by perturbation methods shows that the latter must be multiplied by an exponential factor exp ( - e/%w),where E is the non-fluctuating part of the work done by the perturbing forces; mi long 8s 6 is small, exp (- E/&w) 1 and only dipole transitions have a11 appreciable probability. As the perturbation energy increases, however, this is no longer true, and multipole transitions become progressively more probable, the most probable ones being those for which the change in energy is approximately equal to the work done by the perturbing forces. N

REFXRENCES

J. E. Proc. Uambvidge Phil. Xoc. 45 (1949), 99-124. GROBNBWOLD, H.J. P?l3/siccc, 12 (1946), 406-60.

( I ) Mouffi, (2)

(3) Sz~a6,G. Orthogonal polynomials (New York, 1939), pp. 97, 102. (4) JAHNXRP, E. and EMDE, F. Tables offunctions (3rd ed.) (Leipzig, 1938), p. 19. ( 5 ) I ~ N N AE. R H. D , 8. Phps. 4 4 (1927), 326. ( 6 ) DARWIN, S. (2. PTOC. ROY.XOC.A, 117 (1928), 258-93. (7)COULSON,C. A. and RUSHBROOKB, G. S. Proc. Cambridge Phil. 800.42 (1946),286-91.

DEPARTMENTOF M~THEMATICS UNIVERSITY OF MANOHESTER

202 311 Progress of Theoretical Physics, Vol. 11, Nos. 4-5,

April-May

1954

The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space’’ Takehiko TAKABAYASI

(Received November 16, 1953)

T h e formulation of non-relatiristic quantum mechanics in terms of ensemble in phase space is established by clarifying the subsidiary conditions for the phase space ensemble to represent a pure state, and thereby the equivalent correspofidence between this formulation and the alternative formulation in terms of quantum potential previously developed is exhibited.

§ 1. Introduction and summary The ordinary formulation of quantum mechanics, as established by the fusion of Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics, is certainly the most fundamental and powerful one, having its own ‘ picture ’ in a broad sense* essentially nonclassical. Nevertheless we may consider another consistent formulation of quantum mechanics with its associated picture, for instance, path integral formulation by Feynman”. Generally such a new formulation and picture would reveal new aspects of physical and mathematical construction of quantum mechanics, and might serve to suggest new clues to future progress of quantum theory itself **, apart from its usefulness for practical applications to specified class of problems. From such viewpoint we have examined in detail a certain formulation of quantum mechanics in previous paper^^'^' (see 5 (a)): T h e method is based on the transformation of customary Schrodinger equation into simultaneous equations for the phase and amplitude of the wave function, which are found to be of the form of Hamilton-Jacobi-like equation or Euler’s equation of motion for velocity potential, and the equation of continuity. According to this expression we have the representation of quantum mechanical motion in terms of an ensemble of trajectories in configuration space subject to some additional force (so-called ‘ quantum force ’), or equivalently in terms of an irrotational flow of perfect fluid with peculiar internal stress (‘ quantum stress ’). W e shall call this method the method of the c o ~ j g u m t i o nspnrc. crzscmblt (abbreviated as cs. en.) . hTow the Schrodinger equation can be transformed into a form describable in classical

*) Extending “ the meaning of the word ‘ picture ’ t o include any way of looking at the fundamental laws which makes their self-consistency obvious ”, according to Dirac?). **> In this paper, however, we shall not try any such suggestion, confined merely in the reformulation

of the present quantum theory.

203

312

T. Takabayasi

languages in still another way : That is, we can transform the Schrodinger equation into Liouville-like equation for a distribution function in phase space which is produced as a certain fourier transform of a bilinear form of the wave function, leading to the picture of certain Markoff-like process of an ensemble in phase space for quantum-mechanical motion. This method, which we will refer to as the method of the p h s c S ~ O C L ’c’7tscmbh (abThe purpose of breviated as ps. en.), was initiated by Wignerfi’ and later by Moyal”. the present paper is to develop this method into a consistent formulation of quantum mechanics by establishing the srLbsidiafy conditions for a ps. en. to represent a ~ Z W P state, and also to prove thereby the equivalence of this formulation with the cs. en. formulation formerly mentioned. I n the phase space formulation the knowledge involved in the phase of the original wave function is reflected upon the momentum distribution in such a manner that the phase space distribution function (abbreviated as ps. df.) implies the representation of a state symmetrical in coordinates and momenta. But the manifold of the ps. df. covers wider possibilities than that of the original wave functions. Now, according to our prescripwith positive coefficients of relevant tion, a mixirig of states corresponds to a sz~~,~7~positioiz distribution functions which as well satisfies the same Liouville-like equation, because the latter is linear in the df. Accordingly a ps. df. in general would be the representative of a mixed state, in so far as it satisfies certain ‘ positivity condition ’. Thus in this formulation of quantum mechanics it is an essential problem to obtain the subsidiary conditions*’ that a ps. df. should particularly correspond to a pure state. W e explicitly obtain these conditions, which must be of some non-linear relations (§ 4). This is made tractable by first replacing the usual pure state condition for the density matrix (4.2) by Zocal n~lntions (4.4). Transforming the latter we acquire the pure state conditions on the ps. df., which consist of the condition of irrotationality of mean momentum field, (4.19), and the condition (4.6) which we call the ‘ quantum condition ’. T h e latter will further be transformed into a series of relations between distribution moments in respect to momentum components of successively higher orders. Now, in virtue of these pure state conditions, we can prove the equivalence and correspondence between the cs. en. formulation and the ps. en. formulation (S 5 ) . For instance, the quantum potential in the former may be looked upon as an apparent force appearing as a result of ‘projecting’ the ps. en. onto the configuration space. The CS. en. formulation and ps. en. formulation, though they are equivalent and transmutable to each other, are of very different characters. The effect of quantum fluctuations is represented with fluctuations of continuous trajectories5’ due to quantum potential in the former, while in the latter with Markoff-like transitions, the properties of which we shall examine in detail ( § 3 ) . The ps. en. seems to be one step superior to the cs. en. in that it correctly yields quantum-mechanical expectation values as the mean values over the ensemble for wider class of dynamical quantities, yet it must be emphasized that *) Moyal’) unnoticed the presence of these conditions, while other authors have been unable to obtain their expression.

204

it cannot do so for aZL hermitian quantities and that we must in return allow of ~ccgntive 2). The ps. en. formulation is formally consistent within its limited range of applicability and accompanied with the picture working along classical lines, but we cannot take the picture too realistically, just as in the case of the cs. en. formulation4’. Instead, these formulations provide coiicrrtc nm@scs us to thc drgm in which thc stntisticnZ pivprutics

pobnbiZtties

(s

qunntum mi~chnriic~ mu bc uirdcrstood along- n i y stntisticnl scheme bnsrd o n some hid&n vni-inbks. The ps. df. is a 7 ~ a Zquantity produced as a bidiircnr- form of the wave function, as is needed for the representation in terms of it to have classical pictures, but that would just mean greater complications in mathematical treatments usually. It is well known that We the method of ps. df. is useful for the treatment of quantum statistical shall, however, show how this method of ps. en. can effectively be applied to pure state problems for a few elementary examples (§ 6). I n the last section we consider the positivity condition and also express the pure state condition in an alternative form.

§ 2. Phase space distributions and mean values For simplicity, we shall confine ourselves to the simplest case of a single non-relativistic particle without spin throughout*. A quantum-mechanical mixed state can be specified by a density matrixRa) or ( x l p l X ’ ) = p ( X , x’)in coordinate representation, which must

c,

satisfy the conditions to be hermitian :

p(x, x’)={)*(XI,X),

(2.1)

normalizable :

s p p= (rn,p(s,

(2 2 )

and positive definite :

s p (f

X)dX=l,

432 0,

for any hermitian operator Conversely, any function p ( x ,

-

x’) satisfying

A. -

(2 * 3)

**

these conditions can be expanded in a form

with where

#,,(x)and

nci, mean eigen-functions and eigen-values of -,O respectively, which fact

indicates that the p ( ~ x’) , corresponds to a mixture of pure states of wave functions $ f i , n ( ~ ) with respective weights zun. The density matrix

[I

gives the quantum-mechanical expectation value of any quantity

*) The region of applicability of our formulation is rather limited. When a vector potential is acting, the Markoff-like picture to be stated in $ 3 requires certain extension. **I In this paper we indicate an abstract operator by attaching an underline.

205

T. Takabayasi

341

@>,u

=Sp(_P

4).

Now, from p we define, according to Wigner'), a function

(2.5)

f(x,p ) by

the transfor-

mation, (2.6)"

and will use the latter for the specification of the mixed state. written in the rrcijvocnb form :

The function can also be

(2.7)*

where p(p, p') is the momentum representation of the density matrix. Eqs. ( 2 . 6 ) and ( 2 . 7 ) show that the function f(x,p ) is the fourier transform of ,o(x, x') or p ( p , p') along its antidiagonal ; in other words, if we consider the density matrix to be a function of ' mean coordinate ' and ' relative coordinate ', as ?(XI

v) = p ( x - ! / / 2 ,

x+v/2),

f

is the fourier transform of r) in respect to the relative coordinate. The function f ( x , p ) should be, corresponding to ( 2 . I ) , f(x,p )=real, (though not necessarily positive), and satisfy, corresponding to ( 2 2 ) ,

-

( f ( x ,p ) d x d p = 1 .

(2.8)

(2.9)

W e may thus imagine, corresponding to a mixed state, an ' ensemble ' with the ' probability distribution ' in phsr spncr, f ( x , 21) , though we must then allow of negative probabilities. The function f(x,p ) , which we call the phase space distribution function (ps. df.), can be regarded as the Y L ~r'smtativcz I of statc .gwi?m*iricti//'Ti cooidimtc niicr' mo?nnrrituiir. From ( 2 . 6 ) or ( 2 . 7 ) we get ( 2 * lo) (2.11)

that is, our ps. df.

f ( x , p ) leads to the positional distribution and momentum distribu-

*) In this paper integral usually means a definite integral over whole coordinate space or whole momentum space. In such case we shall hereafter omit to note the boundaries =tx.

206

The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space 347 tion of quantum-mechanical particle correctly.

(A),,,, =Sp(pA(:z, provided -A (- x_,

23)

22)) ={

. ~ . ~.

I

Accordingly, we have further

A(x,p > f ( ~~ ), ) d ~ & ,

(2.12a)

belongs to the quantities that separate as

A(%, - p- ) =Al(:x) +A&)).

(2.12b)

-

This means that our phase space ensemble (ps. en.) correctly gives the quantum-mechanical expectation values as the average values over the ensemble for any quantity of type (2.12b)”. However, for a more general function A(%, 1)) involving cross terms of x and p there exist ambiguities in the definition of the corresponding quantum-mechanical function A x., p- ) of non-commutable operators x and p , so that in such a case our ps. df. may - (-

A‘\”’(%, - p) function A ( x ,$3).

give the mean value correctly only for a special quantity

(x ,p ) corresponding to the same c-number ties A ~-

among the quantiSuch A‘\”’ _ (x, _ p)

is the same with that defined by Wey1’2) procedure, i.e., A(’0) - (x,

p .- ) =

(Q,

(2.13)

Lir,

r ) ei(’StT19

where n(a, r ) is the classical fourier coefficient of A ( % ,p ) : A ( X , $3) = j (T(6,T ) L , i ! o - l ’ )

n‘o dr.

(2.14)

The above mentioned validity of the relation,

(A””’(x, - - -p ) ) q , , =( A(%, p > f ( x P) , dx

49,

(2.15)

is clear from another expression of the ps. df.:

dQ dr.

( 2 . 1 6 j7)

Thus it may readily be found that the ps. df. does not give mean values correctly for quantities such as, e.g., the commutator [ -f i r , x,], the square of the energy N‘ (except fot the case of free particle), or the magnitude* of angular momentum Z‘.

I n other words

our ps. df. does not reflect quantum-mechanical probability distributions correctly for H or -ld (a component of angular momentum), in contrast to the case quantities such as of

_A,(?)

or

A_@)

*

By the way, our ps. df. can also be written as f ( z , p ) =exp(ft/2i.PP

(2.17)

V.J

in terms of

(2.18)

g(x,p ) = (pI{@)(xlp).

Here the latter distribution function g ( ~p,) , though not real, gives the mean values correctly for the ‘ well-ordered ’ functions’” of the form, A(%, p“’. _ _ 1)) _ = Cn,&,,, n,

*)

1”

For the components of angular momentum, the ps. df. yields expectation values correctly.

207

T. Takabayasi

346

Now we must consider the condition ( 2 . 3 ) .

p -

Taking a function of z - alone, or of

alone, ( 2 ' 3 ) requires

i

->-

Sp(p +,(Xy)= jA,(x)~P(r;>d32 0 , SP(CA ? ( ~ ) ) ' ) = S A * ( P ) ' Q ( ~ ) ) ~ P ~ O .

These relations hold for any real function

Z'(X)~O

A,(%) or A , ( p ) ,

SO

that we must have

Q(p)ZO.

and

(2.19)

This is a necessary condition but not sufficient for ( 2 . 3 ) (see § 7). We could represent a general quantum-mechanical mixed state with a phase space ensemble, whose distribution function f(x,p ) is not itself necessarily positive everywhere, though required to produce a positive configuration-space df. P ( x ) and positive momentumspace df. Q ( p ) . Conversely, any ps. df. f(x,P) satisfying ( 2 . 8 ) , ( Z a g ) , and certain ' positivity condition ' including ( 2 . 1 9 ) , is a permissible one corresponding to a mixed state. The non-positive-definiteness of our general ps. df., which stems from the same propetty of the ps. df. for pure states, discloses the physically unreal nature of our ps. en. It is a general characteristic of quantum-mechanical probability distributions for non-commuting quantities that they cannot be derived from a single statistical ensemble based on hidden variables, at least without admitting of negative probabilities. When we are given the configuration and momentum distribution functions P ( x ) and Q ( p ) independently, except for the normalization condition,

s P ( X >d32 = s a (P)4!J=1, we can construct many ps. dfs. which are compatible with those given Indeed the ps. df. fo

( G P ) =J'@)

Q @)

(2 * 2 0 )

P(x) atid Qb). (2.21)

is clearly such one, and moreover any ps, df.

a

+A (GP)

(2.22)

SA(x9 P ) d P = I f , ( % , P)dx=O,

(2-23)

P>=l'(x) in which

f,is

an arbitrary function staisfying

also leads to the given P ( x ) and Q ( p ). Such ps. df., (2 2 1) or ( 2 . 2 2 ) , is a permissible one, corresponding generally to a mixed state, provided it satisfies the positivity condition. The ps. df. that factorizes as (2.21) is a particular one having no correlation at all between particle position and momentum*). I n this connection, it is further to be

*) Such propetty cannot genetally be conserved with the passage of time in classical as well as in quantum ps. en. In the former, however, this property persists in the special case of stationary canonical ensemble, while in the latter it is not so. This is due to the difference of the ' Liouville ' equations for ps. dls. in both cases (see 4 3 ) .

208

_ The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space 347 remarked that there can exist many distribution functions in phase space other than/~(03, p) , which satisfy (2-8) and ( 2 - 9 ) and give expectation values correctly for any quantity of type (2-12b). Examples are /.(«5. 2>)=cos(«/2-r p P,) f(/dt=Hp_—pH,

The equa-

(3-1)

is written, in ^-representation, as

(3-2) since we take hamiltonian, ff=p~/2m+ V(af) , corresponding to a particle in a scalar potential V(OC). For the intermediary function

?(a3, y)=c(x-y/2, x+y/2) = \f(x, p^-'™'* dp,

(3-3)

(3 • 2) is written as

which, by fourier transformation, leads to the equation of motion for our df.: d//dt+p/m-rf=A[f\,

(3-5)

with 4/1= !/(*• P-PO/to 2>'> */.

(3 • 6)

(3 7)

'

*) /(I,

It is given by 7?) = («^)-

209 T. Takabayasi

348

(3.8) where

(3-9) The appearance of the fourier component, V ( 2 p ) , may more easily be understood by starting from the momentum representation of ( 3 . 1 ) ,

and transforming it by use of (2.7). By expansion] may also be written as

(3.10) where & ~ t ' / 4 and , the integral operator

1 ' 1

means the summation over positive odd integers. Using this form, is expressed as an infinite series of differential operators"'):

(3.11) (3 12)

with the understanding that P c in ( 3 . 1 2 ) operates on V ( X )alone. Expressions ( 3 . 1 0 ) and ( 3 . 1 1 ) may be regarded as expansions in ascending powers of &, of which first terms are p ( T , 1.))= j 7 v . P l , ; ( p ) ,

If

.I.,

I*'(%)

11'""f]=DI.'.P,

f.

-

(3 13)

is a polynomial in or below second power, only the first terms ( 3 % 13) in the

*) T o next page : T h e circumstance that in case of potential quadratic (including below quadratic) in the ps. en. moves purely classically is closely connected with the fact that in this case, speaking with the

language of matrix mechanics, the Hamilton's or Newton's rguntfou of uzotioit for y-number coordinates is l h e o ~ ' ~ ) . Generally in quantum mechanics formeily the same equation of motion as in classical theory holds for y-number quantities, but, if this equaticn of motion be linear, there does not appear anyj./.oduci o f y - n u / A - , / x , the rule on which specifies the essential diserence of quantum mecbanics from classical theory, and theiefore each matrix element changes like clzssical quantity. I n this case the diserence of quantum mechanics from classical theory can only prescnt itself in the dehr.ition of the initial c o d c i o n s which is repiesented by the commutation relation in matrix mechanics, and by the subsidiary conditions for pure state (to be stated in § 4) in out formulation. It is t o be noted that the above case includes a generaiized/bfZ.:,(f 'oscillator ' with the hamiitonian, I;r=a(t)&l-

B(452

210

The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space 349 series survive, and then (3.5) reduces to the form identical with the Liouville equation for the classical df. Therefore in such a case the time development of our ps. en. can be regarded as i s produced by the process that each point of the ensemble moves along its purely classical continuous trajectory, in precisely the same manner as in the classicalstat istical ensemble.* Such a picture, however, fails in general cases, where A[f] is an integral operator in momentum space. Yet, looking upon the second and higher order terms in the expansion (3 e l l ) as quantum-mechanical corrections, we call ( 3 5 ) ‘ quantum-mechanical Liouville equation ’. The formal interpretation of this equation leads to the following sfochnstic picture for the time development of the ensemble : The coordinate x of each particle of the epsemble changes continuously with the velocity p / m , while the value of its momentum ( J ( x ,p ) d meaning ~ the probability with which the jumps with a ‘transition probability’ -1 momentum jumps in unit time by an amount p--p+dl> at point I.). Now this stochastic picture has further following features : (i) J is an odd function of 23 ; therefore it takes negative as well as positive values, and also

-

!](I.,

p’)n‘r’=o,

that is, this transition probability is not normalizable, though (3.14) the distribution probability :

(3 * 14) serves to ensure

(3.15) (ii) The transition probability depends upon the amount of jump alone, irrespective of the value of momentum before or after the jump. (iii) The external field V ( X )acts so as to induce indeterministic transitions in particle momentum, but the transition probability] itself is perfectly determined by this potential. (iv) As is seen in (3.8) J ( x , p ) is of a form of a perfect sinusoidal wave in x-space with no damping in far away irrespective of the form of V ( X ) . The amplitude of this wave is the fourier component** V(2p);while its wave-length is Z/2$, so that for larger jumps of momentum the Xspace oscillation of J is more rapid, having therefore less effects when we consider the Since space average over the ensemble. (v)

jpJ(x, p)dy,= -v

v,

(3 * 16)

the rate of average change of momentum agrees with the classical value. In each individual case, however, particle R uctuates, performing jumps in momentum, which may well be far greater than the mean value, and may thus, for instance, penetrate a potential barrier. (vi)

The various

‘ transition

moments

’ of

momentum components are found to be

**) The circumstance that the probablity of momentum transition from 270 to 1, is detetmined by b 7 ( 2 ( p-2’0)) in out stochsstic picture is somewhat similar to the usual perturbation-theoretical result of quantum mechanics, where the probability of transition from the state of momentum p o to that of 2, is proportional to

I~~l~-PcI)~?.

21 1

T. Takabayasi

350

(3.17)

-

including ( 3 . 1 4 ) and (3 16) as special cases.

O n account of the odd character of

%f,

transition moments of even orders vanish while those of odd orders survive, in contrast to the usual Brownian processes. In the latter, the transition moments in and above third orders are assumed to vanish, resulting in the differential equation of Fokker-Plan~k-Kramers’~’ type for the df.; the diffusion takes place in a manner essentially determined by the second order moment, the distribution always diffusing monotonously and irreversibly. O n the other hand in our case the moments of third and higher odd orders give rise to a quite The non-vanishing of third or higher order moments different type of ‘ diffusion ’*I**’***). means that in our process the probability that the value of momentum changes by a finite amount in a small time interval cannot be regarded as small. W e have thus obtained a stochastic picture of distinctive features for a quantummechanical change of state. Though this picture cannot be taken as a real one, for instance, on account of (i), it may be said to represent quantum fluctuations in a picturesque manner.

(b)

Transition probability for a finite time interval

OUKdf. develops with time according to the linear integro-differential equation (3 - 5 ) which is of the first degree in t , and therefore, given the initial distribution f(x,, po,t o ) , later distributions will be uniquely determined. T h e time development off may thus be written in an integral form,

f i x , P,4= s T(xPlxoPo~0lf(% P O ) tO)dX,

dP”,

(3 * 18)

where the kernel T ( x p t / qpo , t o ) embodies the temporal development law of the distribution independently of the initial condition, and implies the tramitinn p1-06nhiCity in phasespace for finite time interval, i.e., the distribution at t conditional in x,,p , at to. Naturally this function is closely connected with the kernel (propagation function) for the time development of the wave function +(z, d ) , which is written as

p (x,I ) = j zqxt Ixot0) + (X,k,) &,.

(3-19)

*) Cf. S 6 (a) and appendix A. **) Such a result corresponds to the circumstance that the wave function (the probability nm$ln(iti/J~~) satisfies a diffusion-type diiferential equation with imaginary time coefficient, 2nd therefore, when we transform this equation to that [or a real quantity interpretable as a ptobability dist~i6ii!io12 by an iteration procedure, the latter equation can no longer be of a diffusion type. ***) In conventional Brownian processes, lurthermote, there acts a frictional force which is propottional to particle velocity and makes it tend to the equilibrium distribution. In our process, however, such effects do not occur and it is impossible that any distribution should converge with the passage oi time to some equilibrium distribution that would correspond to a stationary state [see 5 6 (b) iii)].

212 The Formulation of Quantum Mechanics in terms of Ensemble in Phjase Space 351 This kernel K(xtlxoto)is nothing but the transformation function which transforms the x,, (particle position at t o ) diagonal to the one making x, (posirepresentation making tion at t) diagonal, i.e., (3-20)

Hence it satisfies, as is well known, the iteration law :

K(xt I%to) = pqxt I x14)K(x,t,Ix,t,)dx,

(3 * 2 1 )

9

and the unitarity condition : and

j K(#t Ixoto)K( xoto IX’Z) Cdi, =b (x-x’) ,

(3.22)

K(xt,J3C,to)=i;(X-xo).

(3 *23)

SO

As K ‘ ( x / ~ z also ~ ~ satisfies ~) the Schrodinger equation, it can be determined by solving that equation under the initial condition ( 3 . 2 3 ) . For a conservative system K may be written as

K ( x t INot,) = ( x I E - ‘ - - l o ) ’ f i l

x,),

(3.24)

which is a function of t-to alone in respect to the time, and satisfies

K( xll%,to) =K * (xoto 1xt).

(3 * 25)

Now, corresponding to ( 3 * 1 9 ) , the kernel for the time development of the density , t) is given by matrix p ( ~ x’,

j K(at Ixoto) K * (X’t I%;to) dx&lx;, and accordingly the relation of our transformation function T with

(3.26)

K is found

to be

(3.27) From the properties of K, (3.21)-(3.25), we can find the corresponding properties of T. First, T is real, satisfies the iteration law :

T ( ~ p t l ~ o , o t o ) = 5 T ( ~ p d lt~i )i T p (, x , ~ tilxo~?oto)dxl i d~i,

(3.28)

and the unitarity :

T ( x pt lx,p , to)T(x,p,t

o pp’ t ) d X 0 dpo= 8 (x-x’) 6(P-P’),

(3 * 29)

and so we also have

T(xpt,~x@,dO) = ~ ( x - x o ) ~ ~ P - P* o )

(3’30)

The T-function itself satisfies the quantum Liouville equation :

(3.31)

213

T.

352

Takabayasi

and we may determine T by solving (3 - 3 1 ) under the initial condition ( 3 . 3 0 ) . For a coiiservative system, T(xp tl:rop0 r,) depends on t-to alone in respect to the time, and is symmetric :

~ ( ~ ~ ~ l ~ o P , ~ o ) = ~ 7 1 ( ~ 0 1 3 o ~ o l ~ ~(3~* 32) ) , and normalized :

j T ( x pt \ ~ , , pt0)dx , dp= 1.

(3 * 33)

Eq. ( 3 . 2 8 )

with ( 3 . 3 0 ) and ( 3 . 3 3 ) shows that our stochastic process is a sort of & f a y h f $ m c r ~ s in phase-space with the transition probability T, though we must allow of negative probabilities, since T as well as f is not necessarily positive. T h e law of composition ( 3 . 2 8 ) may be regarded as the integral equation (" Smoluchowski equation ") for the transition probability function T, which may be reduced to the ' differential form ' ( 3 31) with (3 * 1 0 ) . T h e latter equation, however, is of a type quite different from the Fokker-Kramers-Kolmogoroffequation for usual Brownian processes, since, in our process, the differential coefficients of Y i n and above third degree, in so far as they do not vanish, play the roles of higher transition moments for momentum components. Eq. (3.31) with (3.30) is written as

(3.34)

which may be solved by iteration as a power series in d - t o :

1, TI(%]> I z o p o ) = -6(p-pa)V.6(z-zo) //2

+6(~~-z")-~(~,~)-.z)o),

1

Y'? ( X I ) I z , 7 p o ) = -711-,. 6 (P- 2)o) ( 1 ) V ) ( 2 h V O ) 6 (x- xn) p +/70 /I2

J7zlY(z-x:o) y ( x ! 2)-2)0)

iS(:X:

--X:o)

1J ( z p-p') , J(z,'p/-.p0)dp/.

When the potential t'(x) is, in particular, a polynomial in or below szcond power, 1 ' is the solution of the classical Liouviile equation with the initial condition (3.30), and hence is given by

T ( z pI

z,,)

where (z,, ticle we have

zoP0'0)

=6(x--2,(zoPof0))6(P--Pt( Z 0 2 ) 0 ~ " ) ) ,

is the solution of the classical Newtonian equation of motion.

Y'CZ~HI x o 2 , o t 0 ) =6(2)-j,o)6(zx;-

-ieo-2)0/~1.

(3.35)

For example, for a free par-

(t.--!,J),

(3.36)

which is quite difFeferent fiom the kernel for the usual diffusion, while the kernel for probability amplitude is, as i s well known, of diffusion type with imaginary time coefficient : (3.37) But the fouriet transCotmation of (3.36), taking account of the relation (3.27), gives

214

The Formulation of Quantum Mechanics in terms of Ensemble in Phase Space 347

(3.38)

from which we can derive (3.37).

§ 4. Subsidiary conditions for pure state (a) As was stated in $ 2, generally a ps. df. f ( x , p ) , in so far as it satisfies the positivity condition, corresponds to a mixed state, and the procedure of mixing of states corresponds to a superposition (with positive coefficients) of relevant distribution functions. Therefore, various relations thus far stated which are valid for any mixed states (including pure states as their special cases) must be linear ones (except for the positivity condition). In fact, the expression of mean values (2.12), and the time development equation (3.5) or (3.18) are all linear in f. Therefore the subsidiary condition that must be imposed upon a df. in order that the df. should in particular correspond to a pure state, must be some non-linear relation. It must be further of such character as to restrict the functional space of the df. which is one real function of six independent variables to that of two real functions of three independent variables, as a pure state correspmds to a complex function $(x) or d ( p ) . Since our df. f correctly gives expectation values of dynamical quantities at least of type (2.12b) as the mean values over f,the df. for pure state must satisfy the ‘ uncettainty relation ’ as the relation between the mean deviations of x and p :

(4.1) This relation, being non-linear in f, is a necessary condition for the correspondence of f to a pure state, but not a sufficient one. The condition that a mixed state should in particular fall into a pure state can be expressed, in terms of the density matrix, as po==p, - i.e.,

\ p (x,x”)p (X”,x’)dx”=p (x,x’).

(4 * 2 )

Transforming this relation according to ( 2 . 6 ) we may immediately obtain the pure state condition for but then the result is not of a convenient form (see § 7)*; so we will prefer another way. Now, if (4 2) is satisfied, there exists a suitable complex function S ” ( X ) that makes p(x, x;’) written as

p (n, x’)=sl, (X)+* (x’) ,

(4 * 3)

on account of ( 2 . 1 ) , ( 2 . 2 ) , and ( 2 - 3 ) . It is because ( 4 . 2 ) with ( 2 . 3 ) requires ZU,=J,,~(with certain fixed ? i n ) , when p ( ~ x’) , is put in the form (2.4). But if

( 4 . 3 ) be satisfied, we have clearly “) But there the condition is expressed in a form symmetrical in

Y

and

p.

215

__ 354

T. Takabayasi

dfl 3,> &,> d(>I —l -- '— — i) -- £- = — [i— 3xt dxk'

dxtdxk'

d .log! / a .

--

dxt' dXi

\

logo 1 = 0 ;

\dxt'

/

(4-4)

(*, k=l, 2,3) and conversely, if ( 4 - 4 ) be satisfied for every t and k, we get (4-3) by integrating (4-4) and by use of (2-1), (2-2). We can therefore adopt flip local relation (4-4), in place of the integral relation (4-2), as the pure state condition, under the premises (2-1), (2-2), and (2-3). Eq. (4 • 4) is of course compatible* with the equation of motion for (>, which determines f>(oc, OC1, f) from its initial value f>(oc, X1 , ta) uniquely, so it is sufficient to impose the pure state condition at a certain instant. Now we can get the pure state condition for the ps. A{.f(ac, f t ) satisfying the general conditions (2-8), (2-9) by transforming (4-4) into the relation as to/" through (2-6). For the intermediary function (/(oc, y) of (3-3), (4-4) is written as

9 'd \-)«•( ( 1----9 , 19-V }p I/ I---\2 dx dJ1 \2 dx j1 J_ + J_V,

dx* dyk r

(4-5)

which, by the fourier transformation, separates into the following real and imaginary part equations :

where the notation such as /i*/"2 means the convolution with respect to p, i.e.,

, p") 3(p-p'-p"}dp'Jp".

(4 • 8)

We have thus obtained the subsidiary conditions for the correspondence to a pure state, which have following properties :

(i) The condition consists of six relations (4 • 6) symmetrical in i and k and three real anti-symmetrical ones (4 • 7) . Since f is a. scalar, (4 -6) is a tensor equation and (4 • 7) a vector equation, both covariant to coordinate transformations. (ii) They are ' kinematical ' relations independent of the dynamical characteristics of the system. Planck's constant which appeared in the time development equation (3 • 5) with (3-11) also enters the first condition (4-6), both in the form £=/52/4. Thus in our *)

See the foot-note of appendix B.

216

The Formulation of Quantum Mechanics in terms of Eusemble in Phase Spcae 355 formulation of quantum mechanics Planck’s constant has to play such ~ z u q h d dr-odcs. We may call ( 4 . 6 ) the ‘quantum condition’** for the ps. en. formulation, and ( 4 . 7 ) the ‘ irrotationality condition ’ (see below). (iii) T h e conditions are surely not linear in f, but are again integral equations quadratic satisfying both (4 - 6 ) and ( 4 . 7 ) , the disin f. If we have two distributions f,and f.. tribution obtained by their superposition, (which corresponds to mixing of states), no longer satisfies these conditions, as it should not. W e could adopt, in place of ( 4 . 4 ) , the similar relation for p ( p , 2)’) :

(4.9) which, by fourier transformation using (2 . 7 ) , leads to relations :

(4 * 10) (4.11)

where the convolution is to be taken in respect to x in place o f p . This set of relations is an alternative form of the pure state condition, being equivalent to the set ( 4 . 6 ) and (4.7).

-

(b) Our next task is to re-express the conditions (4 * 6) and (4 7) in another forms. For that purpose, first, we integrate ( 4 6) and ( 4 7) throughout over the 23-space, which procedure we shall call the ‘$rq>ctioiz of the relations onto the coordinate space ’, and employ the factorization formula for the convolution :

-

5

(f1*f*)@=

SfidP,Sf?dP.

(4.12)

Then we get from ( 4 - 6 )

(ai=apxi)

P‘ Pik =& (Pdsak1’- sip. a k P ) ,

(4.13)

and from ( 4 . 7 ) sip. pk-

a$&

= p(ajpk-

akc’ 0 and ap 5 1, where afl = (fi/2)( ~ ~ c r J - ~ . Integrating by dp and changing variables so that x = q - $tfi, y = q we obtain

+ tth,

(4) f(u)

=

y*(u) exp [u' (-4473 - 1/4pA)

+ u (aq'/Zi + ip'/A)]

and y = (1/2h) (1/b - a), we can write [for well behavedf(u)]

(7rha)f

--

2 X W

m

exp [ - ~ x q ' ~ / f i ] yn/n!

f ' * ( y ) yn dy. ( 5 )

n=O

When a, ,6 > 0 and ap 5 1, we have y 2 0. Thus W, (p', 4') 2 0, since each term in the sum is of the form y"/n! c;c,,, and is hence non-negative.

252 212

N.D. CARTWRLGHT

References E. P. Wigner, Phys. Rev. 40 (1932) 749. J.VIieger, P.Mazur and S.R. de Groot, Physica 27 (1961j 353, 957 and 974. H.Mori, 1.Oppenheim and J.Ross, in Studies in Statistical Mechanics, Vol. I, J. de Boer and G. E.Uhlenbeck, eds. (North-Holland, Amsterdam, 1962). 1.Bialynicki-Birula and Z.Bialynicki-Birula, Phys. Rev. A8 (1973) 3146. R. J. Glauber, Phys. Rev. 130 (1963) 2529. C.L.Mehta and E.Wolf, Phys. Rev. 134 (1964) A1149. M.Lax and W.H.Louisel1, I.E.E.E. J. of Q. Electronics QE-3 (1967) 47. J. P. Hansen, Phys. Rev. A 8 (1973) 309k E.L.Pollack and J.P.Nansen, Phys. Rev. A 8 (1973) 3110. L. J. Roszman and C. F.Hooper, Phys. Rev. A7 (1973) 2121. 5 ) H.Mori, 1.Oppenheim and J.Ross, loc. cit. n2.

253 PHYSICAL REVIEW A

VOLUME 15, NUMBER 2

FEBRUARY 1977

Wigner function as the expectation value of a parity operator Antoine Royer Centre de Recherches MathCmatiques, UniversitC de MontrCal, Montrkal H3C 357, Canada (Received 30 August 1976)

It is pointed out that the Wigner function f ( r , p ) is 2 / h times the expectation value of the panty operator that performs reflections about the phase-space point r, p. Thus f( r, p) is propor&ioualto the overlap of the wave function Jc with its mirror image about r, p; this is clearly a measure of how much Jc is centered about r, p . and the Wigner distribution function now appears phyeically more meaningful and natural than it did previously.

(5)

In 1932, Wigner’ associated with the quantum wave function +(r)a phase-space quasiprobability distribution function

f(r,p ) =

aJ

d s e-zips/$(r- s)*+(r+s) ,

(5')

(1)

o r , in terms of the momentum representative &p) = h-l’zJcir e+r”+(v),

(2) This Wigner “representation” has proved useful for studying the passage from quantum to classical mechanics and establishing quantum corrections to classical results, and generally it enhances understanding by favoring the use of classical intuition in quantum problems.2 At first sight the constructions (1)and (2) seem rather a d hoc and devoid of any deep physical o r mathematical significance. A somewhat more meaningful expression for f(r,p ) was provided by Moyal: namely

(3) where 6 and 9 are the position and m2mTntum operators, respectively, satisfying [R,PI =iti. The fofma(3) is conspicuous to statisticians: et(kR+sP) I$) appears as a “characteristic function,” being the expectation of the operator that corresponds to the function ei(h+sp)in Weyl’s rule of a s s ~ c i a t i o n . ~ Here we wish to point out t h a t f ( r , p ) has a much more direct physical meaning, in that it is the expectation value of the parity operator about the phase-space point r ,p. To show this, let us first rewrite

($1

f(r,p)=(2/h)(+)n,,l+),

(5'') ( 5’’) where lr) and Ip) a r e eigenstates of fi and p, respectively. Let u s now consider the special case r = 0, p = 0, and denote IIr=o,Bsp II; we have

(6) (6') (6') From (6) or (6’) it is immediately apparent that lT is the parity operator (about the origin): it changes $(r)into +( -r) and &p) into $( -p), o r equivalently (note that n-’=n),

nrin=-8, nr;n=-P. (7) We now observe that n, may be obtained from n by a unitary transformation

nrp=D ( r , p ) n D ( r , p ) ” ; ~(~,p)~~i(~iP-rB)/n

though related, type of phase-space representation of quantum mechanics, the coherent-state representation. We have the actions ~(r,p)-~fi~(r,p)=Ei+r, D(r,p)-’f’D(r,p)= $ + p , and more generally

15 -

( 9)

is a phase-space displacement operator, introduced by Glauber5 in connection with a different,

(4) (4)

where the operator Itr, has the following three equivalent expressions in view of (1)-(3):

( 8)

here

449

(10)

254 450

ANTOINE ROYER

D(Y, p)-l~(Ei,$ ) ~ ( r , p )= q R + Y , P + p )

(11)

that is, II,# reflects about the phase-space point r , p and is thus the parity operator about that point. Note that

(nJ=

1.

(12)

The Wigner function, Eq. (4), is thus 2/h times the expectation value of the parity operator about Y , p. Alternatively, f ( r ,p) is proportional to the overlap of $ with its mirror image about r , p , which is clearly a measure of how much $ is “centered” about Y,p. Let u s now discuss some simple implications of the preceding considerations. We first observe that rI, has eigenvalues f1 [in view of (12)], and its eigenfunctions @;#, satisfying

n,#l@;#)=*l@;p)2

(13)

a r e functions that a r e either symmetric o r antisymmetric about Y ,p. They may be obtained by displacing in phase-space functions of the same symmetry about the origin, i.e.,

I@$P)

I

= W r ,P) @*)9

(14)

1

1

where @+ and @- satisfy n @*>=* @*), or equiva$*(-PI= *$*(p). lently @*(-r)=*@*(r), Let us define projectors PrPand Pip on the spaces of functions symmetric and functions antisymmet r i c about r , p , respectively: P;

=h(1f nrp)

,

(P:J2=

[ F ( k P) , being a power series in f? and $1, whence Eq. (8), in view of (5”) and (6”) [and noting that D ( r , p Y = D(-Y, -PI]. Using (71, (8)) and (lo), we readily verify that

nr@(R - r)II,,=-(I? - r) , rI,,(P - p)rI,,= -(P - p) ;

15 -

(15)

N1,

= D ( r , P)P*D(Y,

*

where P*=+(l n ) projects on the space of functions symmetric (antisymmetric) about the origin. We have

IE. P. Wigner, Phys. Rev. g, 749 (1932). *See, for instance, R. Balescu, Equilibrium and NonEquilibrium Statistical Mechanics (Wiley, New York, 1975); S. R. de Groot and L. G. Suttorp, Foundations of Electrodynamics (North-Holland, Amsterdam, 1972); E. A. Remler, Ann. Phys. (N.Y.) 95, 455 (1975); B. Leaf, J. Math. Phys. 2, 65, 769 (1968). 3J. E. Moyal, Proc. Cambr. Phil. SOC.45, 99 (1949).

(16)

p*l,+P;#=l,

(17)

Pip- Pi*= nrp.

(18)

Let us now separate $ into components symmetric and antisymmetric about r,p:

+=

+:p+

$;p

(19)

7

where

I

I

$:p) e p $) . By (16) we have

(20)

I1 $@: II * .

($k# I$) = ($:@ I$:p) Then by (4),

(la),

(21)

and (21),

II - II$#; II9 .

f(r,P) = (Z/h)( II@;$

(22)

That is, the Wigner function equals 2/h times the difference of the squared norms of the symmetric and antisymmetric (about r , p ) parts of $. By (17) and (21) we further have ($ 14) = 1=

II~:@1I2 I G@112 11 $:# 1 C 1, implying in turn, +

’

(23)

in view This implies of (22), t h a t f ( r , p ) is bounded by the values -2/h and 2/h: -2/h such that the components of A in the charts are constant. Such a chart is called a natural chart for the manifold.

r

(e) A Poisson connection (resp. symplectic connection) on the Poisson (resp. symplectic) manifold ( W , A) is a linear connection without torsion such that VA = 0, where G is the operator of covariant differentiation defined by If I‘;k are the usual coefficients of a connection I’ in a natural chart {xi),introduce the quantities Pk= AjzAkmT;m;I‘is a Poisson connection if and only if the P k ’ s are completely symmetric for each natural chart. It is easy to see that a regular Poisson manifold admits an infinity of Poisson connections; for a symplectic manifold the difference between two symplectic connections is deduced from a completely symmetric contravariant tensor of rank 3.

r.

2. EXAMPLEOF POISSON MANIFOLD: COADJOINT REPRESENTATION OF A LIE

ALGEBRA

(a) Let sfbe a Lie algebra of dimension m over 88, d*the dual vector space of d , , ) the bilinear duality form. Denote by {LA}( A = 1,..., m) a basis for &‘ and by (A,) the dual basis for d*; then


0

L'), 11')

(3-3)

262

68

BAYEN ET AL.

and 0)

u

(u * A

10)

=

C At C

(ap,/r! s!) P'(u, P*(v, IV)).

r+a=t

t-0

We search for functions f such that (3-3) is satisfied formally, that is

T,(u,U, W) = &(u,

U, W)

(t = 0, 1,..-)

(3-4)

where we have defined:

Tt(u,u, w) =

C

(ap,/r!s!)

PT(PS(U, u),

w),

rf8-t

V,(u, u, w) =

C

( u p s / r !s!) Pr(u, Ps(u, w)).

r+a-t

We obtain, taking into account the properties of the connection: pr@,P # ( ~w)) , = Aflil ... A f r i v A x l L I ... Akaz#vil...i,u

x

vjl...j,,Rl...L,u

(r') SI-0

vj,,+l...j,ll .z, M'.

(3-5)

(3-6)

ul-,u, = a,,a,,

For z

= 0,

(r'

+ s < t; t = 0,1,2 ,...).

(3-7)

1, we obtain only identities; the coefficient a, is arbitrary and we can take For t = 2, (3-7) gives a, = 1. We deduce

a, = 1 by a linear change of the variable A.

263 69

DEFORMATION THEORY AND QUANTIZATION, I

from (3-7) by induction that necessarily at = 1 for each t. Conversely, if such is the case, (3-3) is satisfied. We have thus obtained:

THEOREM 3. r f (W, A) is afrat Poisson manifold, there is onb one formal function of the Poisson bracket (up to a constant factor and a linear chmge of variable) that generates a formal deformation of the associative aIgebra defined on N by the usual product: it is the exponentialfunction. Symbolically, we can write this deformation, for instance,

u *A v

=

exp(hP)(u, 0).

(3-8)

Since the composition law (3-8) is associative, the bracket defined by (u * A v - v *A u) satisfiesthe Jacobi identity. We are led to consider the bracket: (2A)-’(exp(hP) - exp(-AP)}(u, u), that is: X-l

sinh(AP)(u, v )

(3-9)

which can be written: OD

[u, t:]” =

C (vr/(2r-+ I)!)

(3-10)

P2rL1(u,v )

r-0

where we h u e set Y = h2. The first term of (3-10) is P(u, u). We see that (3-9) or (3-10) defines a formal deformation of the Poisson Lie algebra N. (b) Let (W, A) be any regular Poisson manifold. Consider the Chevalley cohomology of the corresponding Poisson Lie algebra N ; it is the cohomology, with values in Nitself, defined by the adjoint representation: ap-cochain is an alternate p-linear map of N p in N, the O-cochains being identified with the elements of N. The coboundary of the p-cochain C is the ( p 1)achain 2C given by

+

ac(uLl * * - - >

up>

= (1/p!)

‘ $ ~ ~ > ~ ~ A .9

c(uA, ,.--,UA)]

- (1/2(p - I)!) c$:;’C((ulo

3

UA,]t UAz

?*--*

uAp)

(3-1

where uA,E N. We note that the space of the I-wcycles of N is the space of the derications of N,the space of the exact I-cocycles is the space of the inner derivations. A pcochain C is called local (support preserving) if, for each ul E N, such that u1 :V = 0 on a domain U,we have C(u, ,..., u,) Iu = 0. If C is local, aC is local. We obtain thus the so-called “diagonal complex” in the terminology of GelfandFuks. A p a h a i n C is called d-differentiable (d 2 1) if the cochain is local and if its restriction to each domain U of Wis a d-differentiablep-mchainof N(U) = C*(V; R) in a clear sense. Such a m h a i n is &fined by a multidifferential operator of maximum

264

70

BAYEN ET AL.

order d. If C is d-differentiable, aC is also d-differentiable. We have proved [I23 the following proposition (see Section 6b for the symplectic case): PROPOSITION 4. Let (W, A) be a regular Poisson manifold. If C is a local I-cochain of N such that aC is a d-diferentiable 2-cochain, then C is itselfd-differentiable. We note that if (W, A ) is symplectic and if aC is null on the constants, we have C(1) = constant. (c) We come back to a fiat Poisson manifold (W;A). Given a formal series f ( z ) with constant coefficients, without constant term, we substitute P r for z+ in the development of A-lf(Az), where P r is defined by (3-1). We obtain a bilinear map N x N + E(N; A) which is skew symmetrical only iff is odd. If D

f(z) =

2 (ay/(2r+ I)!) +=o

Z*~+I

(ao = I )

we have: 0:

Pv(u, t;) = P(u, v ;

Y)

=

C v*(aY/(2r+ I)!)

PPr+I(u,V )

(V

= A').

(3-12)

r-0

We search for functions f such that (3-12) determines a formal deformation of the Lie algebra N; (3-12) satisfies formally the Jacobi identity if and only if, for t = 1, 2,..,: Dt

=

2

aPsEr.a= 0

r+b-t;r.r)O

where, for u, V , w

E

N,

Introduce the local Chevalley cohomology of N. For t = 1, D1= 0 expresses only that Psis a 2-cocycle for this cohomology and the coefficient a, is arbitrary. We can take a, = 1 by a linear change of the variable Y. We obtain by an argument similar to the argument of Section 3a that Dt= 0 if and only if a, = 1 for each t. We have thus determined the required formal deformations of the Poisson Lie algebra N. Consider the 2-cocycle P3 which corresponds to the term linear in Y. If this cocycle were exact, it would be the coboundary of a local I-cochain which would be necessarily 3-dilTerentiable, according to Proposition 4. But it is easy to see that such a coboundary has no term of bidifferential type (3,3). More generally, it is possible to prove in the same way that for a flat Poisson manifold, the second space of local cohomology of N has the dimension I ; P3 defines a cohomology 2-class which is a generator for this space. We have proved:

THEOREM 5. If (W, A) is spat Poisson manifold, there is only one formal function of the Poisson bracket (up to a constant factor and a linear change of variable) that

265 71

DEFORMATION THEORY AND QUANTIZATION, I

generates a formaI deformation of the Poisson Lie algebra N of the manifold it is the sinhfunction. The corresponding deformation is nontrivial evenfor the order 1 . Thus we obtain only the deformation M(u, v; v = A2) given by (3-9), which is therefore nontrivial. It is remarkable that for h = %/2, we obtain a bracket given many years ago by Moyal [5]. We often suppress the mention of the variable v and call M(u, v ) the Moyal-Vey bracket of the flat Poisson manifold (W, A, I'); this bracket depends on the choice of the flat connection F. (For W = R2 and = 0, the unicity of the sine-bracket as a function of P has been noticed in 1964 by C . L. Mehta).

r

(d) Consider a general symplectic manifold ( W,A ) and a linear connection such that V A = 0. We introduce the bidifferential operators P ' defined by:

P'(u, r);"

=

AQY1'.' A'r'r VZI ...2,u

V2L...,,~ (u, c E N )

(3- 13)

and set n

u

zFI t'

=

1 ( P / r ! )Pr(u, u).

(3- 14)

r-0

If (3-14) is limited to the order 2, we see that the associative property (3-3) is satisfied up to the order 3 if and only if Vil,zuis always symmetric, that is if r i s without torsion. If (3-14) is limited to the order 3, then (3-3) is satisfied up to the order 4 if and only ~ t l , , i , u is completely symmetric, that is if I ' is without forsion and curvature (flat symplectic connection). In this case (3-14) satisfies (3-3) to all orders. 4. DEFORMATIONS OF THE POISSON LIEALGEBRA OF A SYMPLECTIC MANIFOLD

A) with an arbitrary symplectic con(a) Consider a symplectic manifold (W, nection I' (without torsion, but with curvature). We denote by F the closed 2-form of the manifold. If {I'&} are the usual coefficients of the connection in a naturaI chart {x~],we introduce the quantities rijk = FiLT;k which are completely symmetric. If u E N, we denote by Xu,for simplicity, the corresponding Hamiltonian vector field CX,,= p-Ydu)); 9 ( X , ) I ' is the completely symmetric covariant 3-tensor defined by means of the Lie derivative of the connection by the vector field Xu. We have IocalIy for a natural chart:

r

r

(4-1) If T is a completely symmetric covariant 3-tensor, we have:

(4-2) Consider the 2-cochains Sr3 defined by:

(4-3)

266

72

BAYEN ET AL.

In the special case when Pis flat, Sr3 coincides with P3. In the general case, we deduce from *„ = *• SSF\P(u, v), W) = On the other hand, we deduce from the properties of the Lie derivative by Xu :

It follows that:

SP(Sr\u, v), H.) = and thus SSr*(P(u, v), w) + SP(Sr3(u, v), w) = 0. According to (3-11), this means that dSr* = 0 and Srs is a 2-cocycle for the considered cohomology of N. The same argument as for the flat case shows that the 2-cocycle 5V3 is nonexact. (b) Let T be a completely symmetric covariant 3-tensor. We can associate to the 3-tensor T and to the symplectic connection J" the second-order differential operator AT given by: (v 6 N)

and the third-order differential operator BT given by: B,(p) = AViA^A'^TtMVe&J r)iliti,

(v e N).

(44)

The coboundary 8C of a 1-cochain C is given by: 8C(u, v) = J2?(X-J C(o) - 2(Xj C(u) - C({u, v})

(u, v e N).

We deduce from (4-5):

&AT(u, v) = 2Ai^A^Ai'i^(Xv) T)w, ((XJ T)hMt

and: (JS?(JT.) r)Wj

(4-5)

267 DEFORMATION TKEORY AND QUAiNTIZATION, I

73

PROPOSITION 6 . The cohomology 2-class defined by the 2-cocycle Sr3 does not depend on the choice of the symplectic connection

r. Consider two symplectic connections r and r' and take for T the 3-tensor

Proof. defined by the difference between the two connections. We obtain:

(s",t - S,",(U,

u) = A i1j1Ai2kA '"'"{(P(xu) T)ilirir ( y ( X u > Ojz,da

+ (g(xu>

(p(xv)Thl~z,sl

r)f1i2i3

+ Ai1j~Aiz'~~/li~f3(.k9(X,) T)ilizil(P(Xu) T),,,ds. Therefore we have:

s,,- s,"

=

a(&

+ $A?-)

(4-7)

and the cohomology class of Sr3 is independent of the choice of I'.

defined by a bidire7. Let Q3 be a 2-cocycle belonging to /I, (c) PROPOSITION rential operator of maximum order 3 on each argument, null on the constants. There exists a unique symplectic connection such that

r

(4-8) where K is a differential operator of order 1) are not known.

5. DERIVATIONS INDEPENDENT

OF v OF A

VEY LIE ALGEBRA

The Lie algebra of the injinitesimal automorphisms of an arbitrary Lie algebra is given by the Lie algebra of the derivations. (a) Concerning the Poisson Lie algebra N , we recall that a derivation D is an endomorphism D : N -+ N such that, for any u, v E N, we have:

{Du, V}

+ {u,Dv} - D{u,

U} = 0

(5-1)

that is, 2D = 0. We note that we have for D no locality, continuity, or differentiability assumption. A vector Z defines a symplectic infinitesimal transformation (i.t.) of (W,F) if U ( Z )F = 0; we denote by L the Lie algebra of the symplectic i.t.; 2 defines a con-

269 DEFORMATION THEORY AND QUANTIZATION, I

75

formal symplectic i.t. if 9 ( Z )F = aF, where a is a scalar; if dim W > 2, a is necessarily a constant. In all cases, we denote by Lc the Lie algebra of vector fields Z such that there exists a constant k ( Z ) for which: 2 ( Z )F

(5-2)

+ k(Z)F = 0

or

(5-3)

9 ( Z )A = k(Z)(I.

If F is nonexact (in particular if W is compact), Lc coincides with L. I f F is exact, L is. the commutator ideal of LG and dim Lc/L = 1. Avez and Lichnerowicz [I31 have proved the following: THEOREM 9. Let ( W ,F ) be a symplectic manifold and N its Poisson Lie algebra. r f W is noncompact, each derivation D of N is given by U(Z) k(Z), where Z E Lc. r f W is compact, each derivation D of N is given by:

+

Du

= 9 ( Z )u

4c “W

u~

where Z E L, c E W, and 71 is the symplectic volume element; rhese are nonlocal derications (for c # 0). We suppose now that W is noncompact. The results concerning the compact case are similar, since it is possible to prove that the nonlocal derivations do not appear in the following study. (b) Consider an infinitesimal Vey deformation of the Poisson Lie algebra of the symplectic manifold ( W, A )

(5-4) where Q3 is a 2cocycle satisfying Proposition 7; we have

(5-5)

r

where is a symplectic connection and where Kis a differential operator of order G2. An infinitesimal automorphism D-independent of v-of the bracket (5-4)is defined by an endomorphism D:N + N such that, for every u, v E Nand v E C, we have:

(5-6) The space of these infinitesimal automorphisms admits, for the natural bracket of the endomorphisms, a structure of Lie algebra. For v = 0, we obtain aD = 0;

270

76

BAYEN ET AL.

that is, D is a derivation of the Poisson Lie algebra. It follows from the above theorem that necessarily

D

= 9(Z)

-+ k ( 2 )

(ZE Lc)

(5-7)

Q3(Du, u)

+ Q3(u,Du) - DQ3(u,U) = 0.

(5-8)

and (5-6)is then reduced to a,D(u,

U)

E

(c) Consider the differential operator K of order 2 introduced in (5-5). This operator is defined in terms of the connection by a symmetric contravariant 2-tensor H, a vector J, and a constant c so that, on the domain U of a chart, we have:

r

K(u) Iu = Hk'Vk,u

+ JkV,u + CU.

(5-9)

Here c is a constant since aK is null on the constants (as noticed at the end of Section 3b). We obtain easily according to the Ricci identity

~ K ( uv)( , u = di5 ViHkZ(VktuV ~ U V~LVV ~ U-) A"HkZ(VkiuV i j ~- V k i ~VljU)

- A"HkzRl,rk(V,u V ~ V V,V V ~ U )

+ (A" ViJk - A'k VJ') V ~ V,V U + cA'~V,U V ~ U

(5-10)

r.

where we have introduced the curvature tensor of We are led to evaluate, for D = O(2) k ( 2 ) (with Z E Lc):

+

E ~ ( uU) ,

~ K ( D uV, ) f ~ K ( uDu) , - DaK(u, v);

that is:

(5-11) We have the following lemma:

LEMMA 10. The bidzflerential operator Ez dejined by (5-11), where ZELc, is of maximum order 2 on each argument. Proof. Computing (5-1 1) with the help of (5-10) and the identity

(P(9 VkV, - V , V , - W ) )

2.4 =

(%TI

D k J

v,u

we see immediately that all terms in Ez of order >2 in each argument cancel out, and the lemma is proved.

27 1

77

DEFORMATION THEORY AND QUANTIZATION, I

(d) We have

(5-12) Introduce the symmetric covariant 3-tensor T = U ( Z )r a n d the differential operator Br corresponding to and T defined in (4-4).It follows from (5-12) that:

r

a3D

= -2BT

(5-13)

- 4k(Z) Sr3 + Ez

(5-14) Introduce the symmetric covariant 3-tensor T = U(Z) rand the differential operator We have at this point:

where we can choose arbitrarily the symmetric 3-tensor 7 related, according to (4-1), to the third derivatives of u at x. Lemma 10 gives &(u, v)(x) = 0 and we deduce from (5-14):

+

79i1fzfa 2k(ZX9(X,)

Ti1i2fa[(Y(x*)

r),,f,f,l(X)

= 0.

It follows that necessarily, for every u E N

9(XJ T

+ 4k(Z) T(XJ I' = 0.

(5- 15)

If we take for u an element of N admitting a null 2-jet at x, we obtain k(2) = 0. Therefore 2 belongs necessariIy t o the Lie aZgebra L; (5-15) gives the result

.Y(X,)T = 0 for every u E N. Consider an element v e N such that (a,,u)(x) According to (4-2), we have at x: (Wi,i,iJ(x)

(5-16) = 0, (a,u)(x)

being arbitrary.

=0

and the components TtIi,,,of T a r e constant on U. We shall raise indices in T with the help of A. If u is now a function such that at x : (awuXx) = 7 x 2

272 78

BAYEN ET AL.

where T is an arbitrary symmetric 2-tensor, (5-16) gives at x : (5-17) Choose for 7 a tensor which has as only nonvanishing component rI1= 1; take il = 1 and successively i, = i3 = 1; is = 1, i3 # 1; i2 # 1 , i3 # 1. We obtain T:2i3= 0 for arbitrary indices i2, i3 ; 1 being an arbitrary index, it follows T = 0; that is,

qz)r = 0.

r.

(5-18)

In particular Z preserves separately Sr3 and EK. A similar argument shows that if Z preserves (1, and also aK,we have with the representation (5-9) 2 preserves the connection

r,

U ( Z )H = 0

(5-19)

[ 9 ( Z )J , A ] = 0.

(5-20)

and The vector field 9 ( Z ) J defines a symplectic i.t. We note that the same argument proves also the uniqueness of the connection defined by Proposition 7. We denote by Ls,, the symplectic subalgebra of the Lie algebra of the affine infinitesimal transformations corresponding to the connection Let L(Q3 be the Lie algebra of infinitesimal automorphisms (independent of v) of the bracket (5-4) (Vey infinitesimal deformation). We have proved:

r.

THEOREM 11. The Lie algebra L(Ql) of infinitesimal automorphisms of the bracket

Q1which do not depend on v is a subalgebra of the algebra LSAof the symplectic affine infinitesimal transformationsof the symplectic connection r associated with the 2-cocycle Q3. Therefore L(QJ is finite dimensional.

We note that if Q3 = Sr3, we have L(Ql)= LSA. For a Vey Lie algebra, with the bracket Q given by (4-lO),it follows from Theorem 11 that: COROLLARY 12. The Lie algebra L(Q) of the derivations independent of v of a formal Vey Lie algebra is finite dimensional. If ( W, A, is a pat symplectic manifold and M the natural Moyal-Vey bracket given by the sine function, we have L(M) = LSA.

r>

(e) Remarks. Invariance of a Vey deformation. (i) The Lie algebra of vector fields Z (infinitesimal geometric transformations of the manifold W )which do not depend on v and preserve the bracket Q, is also L(QJ [9]. (We note that in this case necessarily 2 E L.) (ii) If we now look for vector fields 2, = 2, vZ,,where 2, and 2, are

+

273

79

DEFORMATION THEORY AND QUANTIZATIOh', I

vector fields independent of Y, which preserve an infinitesimal Vey deformation Ql given by (5-4),we see immediately that ZoE L(QJ and the invariance of Q1 (in our notations) reduces to:

i([A,[Z, , J ]

+ Z,])(du

A

dv) = 0:

PROPOSITION 13. A vector field 2, + vZ,preserves an infinitesimal Vey deformation Q,

if and only i f 2, E L(QJ and [Zo, 31 + 2, E L.

(iii) More generally. if we look for a formal series *

Z" =

1

(5-2 1)

V'Z,

r=0

preserving formally a Vey deformation Q given by (+lo), we need more information on the 2-cochains Q2r+1.In the case of afrat symplectic manifold, (5-21) will preserve formally the natural Moyal-Vey bracket given by (3-10) if and only if, for every r, -Y(Z,) A = 0 = U(ZJ that is 2, E Lsa . For instance, in the case W = R2" with the usual connection and the usual Moyal bracket, 2;belongs to the algebra of affine symplectic transformations with coefficients in the field of formal series in Y. (iv) Still more generally for a flat Poisson manifold W, if U, E N has an inverse (uA*j-l with respect to the * A product (3-8), v ++ U, *,, u * A (uA*)-' is a n automorphism of N( W )depending on A. This leads us to the next section.

r,

DEPENDING ON Y OF 6. DERIVATIONS Consider linear maps D,: N

---f

A VEY

LIE ALGEBRA

E(N; V)defined by means of the formal series: "-

I

D,=

1 V'D, 8-0

(6-1)

where D, is an endomorphism of N independent of v. Let us look for derivations D, of a formal Vey Lie algebra corresponding to a bracket defined by (4-10); we will have formally, for every u, o E N :

(6-2) (a) Taking the term of (6-2) independent of Y, we see first that Do is a derivation of the Poisson Lie algebra. There is a Z, = Z E Lc such that

Do For the term linear in

Y,

= g(Z)

+ k(ZJ

we obtain with the above notations: i 3 0 ,+ (1/3!) &Do = 0.

(6-3)

274

80

BAYEN ET AL.

If we set T = 9(Z)

r, we deduce from (5-13) that (6-3) can be written: aD, - (1/3!)(aB,

+ 4k(Z)Sr3 - Ez) = 0,

(6-4)

where E, is a bidserential operator of maximum order 2. It follows from (6-4) that aD, is a bidifferential operator of maximum order 3. Therefore D , is necessarily a differential operator of maximum order 3 and aD, has no terms of bidifferential type (3,3). This fact implies k ( 2 ) = 0. We obtain

LEMMA14. We have necessarily Do = Y(Zo),where 2, E L. (b) If D is an endomorphism of N , we set: (82r+lD)(~, V ) E Q2++'(Du,U )

+ QZr+'(~,Dv) - DQ2'+'(u,

0).

(6-5)

Let 2 be an element of L. If U is a contractible domain of W, there is a function w u E N( U), defined up to an additive constant, such that:

(6-6) Since Q2r+l is null on the constants, we can define a differential operator order (2r 1) by

+

Q$''(U)~ u = Q Z r + ' ( ~ vu, I u )

(U E

N).

QZ+'

of

(6-7) (6-7)

I t is easy to verify that, if Z E L satisfies (6-6) on U,we have:

(6-8) For r = 1, aQa = 0 and (6-3) can be written with the above notations: a(D, - (1/3!) Q",) = 0. We thus obtain:

PROPOSITION 15. The derivations Dv = Do

+ vD, of Q up to the order 2 are given

by: Do = y(ZJ,

Di = (1/3!)

Q"Z,

+ p(zi)+ k(Z3

where 2,E L and 2, E Lc. (c) Q being a deformation of the Poisson Lie algebra, we have:

(6-9)

275 81

DEFORMATION THEORY AND QUANTIZATION, I

where S is the summation after cyclic permutation on u, u, w. According to the defination of QF+', we have on the domain U,for Z E L satisfying (6-6):

9 ' SQ"+l(Q"+l(~ I , y , li ),I

9 '(l$,.+,Q$'+l)(~,li)l

wU) =

(u,v E A'),

(6-10)

where Y is the symmetrizer in (r,s). It follows from (6-8), (6-9), (6-10) that for Z EL: 1 (2t i l ) !

rz=t

1 (2r + I)! (Zs + I)! a2r+lQy+r. (6-1 1)

+ aQY1) = -

(&+1-W)

r. s>l

We deduce from (6-1 1) that 0,defined by (6-1) satisfies formally (6-2) if and only if for each t = 1,2,-..:

(6- I 2)

(d) Consider a derivation of Q: 1-1

D, =

V'D, 5-0

up to the order t . We shall prove by induction that we have for s

= A(@, dg). Every a E N defines an infinitesimal canonical transformation given by X,f = { a , f } that preserves the Poisson bracket (Jacobi identity):

This is an inner derivation of the Poisson Lie algebra N . Let a *-product on W (a bilinear, associative, internal composition law on N) be given and let a E N. We say that the *-product is a-invariant if X,preserves the *-product: xa(f * 8) = (xaf)* g f * (Xag)*

+

In other words

@,f* 8) = hf)* g + f

* (4 g),

tJf, g E N-

(9-1)

If the *-product is a-invariant and b-invariant then it is {a,b}-invariant. The set of all a E N for which the *-product is a-invariant is thus a Poisson Lie algebra of derivations of N (with the *-product structure).

283 DEFORMATION THEORY AND QUANTIZATION, I

89

(b) Conversely, let & be any finite-dimensional subalgebra of the Poisson Lie algebra N. We pose the problem of constructing a *-product on Wthat is &-invariant; that is, a-invariant for every a E d.This, in our view, is a central problem of quantization, a problem that is discussed at length in the companion article [lo]. RecalI that t h e Moyal *-product is invariant under the group of affine symplectic transformations, and that conventional quantization realizes this group as a group of unitary operators. In general, the C“ functions on W that belong to d C N will be taken to have significance as a preferred set of physical observables; hence we require that the Lie algebra .d be preserved by the deformation; precisely (henceforth we write * for * A ) a * b - b * a = iti{a, b),

Va, b E d

(9-2)

where ih/2 = X is the deformation parameter. In addition, it will be supposed that the set of observables belonging to d is “sufficiently large.” Let {LA}( A = I,...y m) be a basis for .dand let J be the map of Winto d* given by x ++ [(x) where t A ( x )= LA(x) [17]. The action of the vector field X , in W is now transferred to the image of J by dJ and this is just the coadjoint action ad*d of d in d*. The image M of W by the map J is therefore an ad*&-invariant submanifold of .a‘*. The algebra s2 is “sufficiently large” if this manifold is diffeomorphic to the manifold of actual interest; this happens if J W + M is locally bijective (if W is the manifold of interest) or more generally if we are interested in constraint manifolds with constraints in Ker J. In any case we now direct the attention to MC&* and formulate a precise mathematical problem. (c) Let d be a Lie algebra of dimension m over R and d* its real dual. We identify d with the space .d**of linear functionals over d*;in particular this means that a basis { L A ) ( A = m) for a2 is also a set of coordinates for d*.A polynomial over d* is a polynomial in L.l,...Lm y with coefficients in @. We denote by U(32) the enveloping algebra and by S ( d ) the symmetric algebra of the complex extension of sd, and by 9 the algebra of polynomials over d * ,naturally isomorphic to S ( d ) [IS]. The natural Poisson structure on d* was described in Section 2. Let M be an algebraic submanifold of d * ,invariant under ad*d ,and let I7 denote the projection of 8,B -+ 9’/9 where 4 is the ideal of polynomials in B vanishing on M . Thus Ker I 7 = 9 = { fS;fl ~ M = 0} and l7(9‘)= 9/Kerl7. Commutativity of the diagram

defines the Poisson bracket { f‘, g’) on Zi’(9)and induces on M the structure of a Poisson manifold. This is well defined because Ker Zi’is addinvanant: f E Ker 17 * kf) E Kern, V a E d .

284

90

BAYEN ET A t .

DEFINITION 21. An invariant *-product on M is an internal composition law on

n(.9), writtenf, g ++ f *g, associative and distributive ouer C, such that

k*f=f*k=kf, Vk E C ; a, b ~ d , a * b - b * a = %{a, b}, vj; g E { a , f * s> = { a , f >* g + f * {a, g),

n(q.

(9-4)

Here and below we still denote by a, b the restrictions to M of a, b E d ,considered as linear functions on A*. The problem that we are interested in is the construction of all invariant *-products on M. Concerning the motivation, we may add that the restriction of at'* to M sometimes has a direct physical interpretation. In fact, any element q in the center of the Lie algebra 9 (with Poisson bracket) is a canonical invariant and is expected to remain so after quantization. Fixing q in 88 defines an invariant submanifold M of @'*; see also Section llb.

10. EXAMPLES OF

INVARIANT *-PRODUCTS

We shall make use of the formula (3-8) to obtain some examples of invariant *-products. (a) Let (W,F, I') be a connected, paracompact, symplectic C" manifold of dimension 2n, fundamental 2-form F, and connection It will be assumed throughout that the covariant derivatilie V of F vanishes. Let {xi> (i,j = 1 ,..., 2n) be a local chart with domain U.For u, v E P(W, C) define

r.

(10-1)

with A

=

ifi/2 and

(10-2) Recall from Section 3d that this product is associative if and only if the torsion and the curvature ofl' vanish and suppose that this is the case. Denote by Inv(F, r ) the subalgebra of the Poisson Lie algebra N = C'-(W,C) that consists of all (I EN, independent of A, such that (10-1) is a-invariant (Section 9a). This algebra is ;SOmorphic to-and, if I' = 0, equal to-the semidirect product of Sp(2n, R) and the Heisenberg Lie algebra H , . Of course, its representation by vector fields (considered in Section 5 ) is the semidirect product of Sp(2n, R) and the Abelian algebra R2,.

r)

(b) Let .dbe any subalgebra of Inv(F, and J the map of W into @'* defined in Section 9c. Let M = Im(J) C @'* and denote by J* the map of Ca(M, C) into N defined by J * : f + + f J. (10-3) 0

285

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DEFORMATION THEORY AND QUANTIZATION, I

Finally, let No = Im(J*) C N a n d assume that No is a *-subalgebra of N. In that case a *-product on M is defined by (f * g ) 5 J We give some examples, taking ( W ,F, structure.

=

cj-0

J)

*

( 10-4)

( g c J).

r)to be Rn %

Rn with the ordinary symplectic

EXAMPLE 22. Let a2 = H , with basis xl,..., x2,, xZn+l= e; M = Im(J) is determined by e = 1 and is a single 2n-dimensional orbit of ad*& in &*. Moreover, ad*& can be identified with W,J is the identity map, and No = N. EXAMPLE 23. .Let srl = Sp(2n, W) with basis (xixJ}(i,j = I ,..., 2n); M is a single, 2n-dimensional orbit; No consists of all even functions of xl,..., xZnand is a *-subalgebra of N. EXAMPLE 24. Let x*,..., x2n = 41,...,4”; p‘, ... p”, and a? = SO(n) with basis y‘pj - qjpi (i,j = I,,.., n), The cornmutant 93 of d in Inv(F, I‘) has the basis gigi, 2 q%i, x p i p i . The cornmutant NB of in N consists of functions of Lij and therefore it coincides with N o . It follows (see the Lemma below) that No is a *-subalgebra of N . In this case M contains many orbits. Lij

This last example has a number of easy generalizations; thus one may introduce

a symmetric, nondegenerate form g on R“ and take for 9 the span of g(q, q), g(q, p ) , g ( p , p ) , obtaining an invariant *-product on ME&*, where d is the pseudoorthogonal algebra defined by g. The success of these constructions depends on LEMMA25. Let 9 be any subalgebra of Inv(F, r)and N g the cornmutant of 3 in N; then iVa is a *-subalgebra of N .

Proof: If u, v E N9 andb E so that

I(

*V

*

* + u*{b, u> = 0

C Inv(F, r),then (b, u v ) = (6, u) v

E N ~ .

These ideas have interesting applications to systems with constraints, especially in connection with the problem of quantization. Such applications, where one takes for the algebra of first and second class constraints, are discussed in Section 8 and in Part I1 [lo]. (c) We return to a flat symplectic (W,F, I‘) with structure tensor For x E U,let = uA(x)/axi

L/(X)

= p-l(n.

(10-5)

be the components of dl l2 .Any contravariant tensor S(x)with components P 1 - - - f f ( ~ ) Iz S(x) at 5 = J(x), with components

at x determines a contravariant tensor $.$ = dJ S”I...“I(S)

= L;yX) ... 1

q(x) g””””(x),

(1 0-6)

286 92

BAYEN ET AL.

and a tensor field S on W, projectable by J, determines a tensor field on M = Im(J). In particular, J A B = LfALjBAU, (10-7) and A describes the restriction A , to M of the structure 2-tensor of .a‘* introduced in Section 2a. A tensor defined by (10-6) vanishes if any argument is in Ker(dJ 13; therefore by restriction to T,M a J-projectable tensor field S on W defines by (10-6) a tensor field SMon M . Note that any tensor field SMon M can be defined by components with respect to a basis in &*. From now on it will be supposed that the map J: W - t M is locally bijective. Pf J is locally bijective except on a subset of lower dimension one removes this subset from W.] In this case M is an orbit of ad*& in d* and AMdetermines a symplectic structure on M also Ker(dJ) = Ker A. We stress that this limitation is a matter of convenience, since a *-product is induced on M under the weaker condition that No be a *-subalgebra of N . The image f = d J ( r ) is a “pseudoconnection” and its components can be defined by

p . c = L&B((Lik/p’L,C,

- p’kLk=)

( 10-8)

where Lz2 = a2Lc/axkaxL. We can set RADfiC= so that the components f g B of the “pseudoconnection” are determined (and need to be determined) only to the extent that they are defined by this identity. The “contravariant derivative” of the field (10-6) is the image by dl of the contravariant derivative of S. Since s vanishes on Ker d,its components are well defined by fABsC,

(10-9) The *-product f * g for ft g E C“(M, 6 ) was defined by (10-4); direct calculation give

f*g =

a

(hk/k!)fA’”’Akg,,...,k

(10-10)

k-0

where fAi*-.Ak

- {LA”; =

{.-;

{LA’,f} -..}

(10-11)

and where the “covariant components” g A are defined (and need to be defined) only to the extent that they are determined by the equation g B l ” . B k = RBIAl..RBkAkgA,.-.A,

.

(d) Conversely, we may pose the question of finding necessary and sufficient conditions on 2” in order that (10-lo), (10-11) define an associative, invariant

287

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DEFORMATION THEORY AND QUANTIZATION, I

*-product. Since the covariant derivative of A vanishes, and an elementary calculation gives

is completely symmetric (10-12)

(10-13) The last equation is the condition for (LA;dsc) to vanish; hence (M,AYf) is a symplectic manifold with fundamental 2-tensor A , and “pseudoconnection” fi such that the contravariant derivative of vanishes. Moreover, r is d-invariant by hypothesis and therefore f is also d-invariant, which is expressed by { L A , f”C.”}

- ,;BpC.D

- C;CfrBE.D

- ,jD,C.E

= 0.

(10-14)

Finally, we express the fact that R(T)vanishes. Let X be a vector field on d* 1 M that vanishes on K e r A and whose components can therefore be expressed as as X B = RABPB. Then one finds {LA;{ L B ;

XC}}

-{LB; {LA;

XC}} =

A”MylD

where the contravariant “curvature tensor”

(10-105) PROPOSITION 26. The *-product dhfined 6y (10-lo), (10-11) is d-inaarianr and associative if and only ifthe ‘‘pseudo connection” f satisfies (10-12)-(10-15).

(e) Equations (10-12)-(10-1s) imply that

f ABC

= {LC;{ L B ; {LA,f}}} = 0

iffis linear,

(10-16)

which means that the series that defines L A * g ends with the third term. If g is a polynomial in L1,...,Lm (more precisely, g ~n(9’), in the terminology of Section 9), LA

* g = LAg + A{LA, g } + $ h y f = w C

where KAB

+ KADaD)g

= -Lc D AL CDB.

(10-17)

(10-18 )

In particular,

LA

L B

= LALB f

AAAB

Concerning (10-18) we find by direct calcuIation:

+ iA2KAB.

(10-19)

288

94

BAYEN ET AL.

PROPOSITION 27. (1) K A B is constant on each orbit; that is, {LA,K E C } 3-tensor KADCiCis completely skew symmetric [19].

=

0. (2) The

The *-product on M is determined by (10-17); precisely:

PROPOSITION 28. If f satisfies (10-12)-(10-15), then (10-17) is compatible with a unique invariant *-product on M ynamely, that given by (10-101, (10-1 1). EXAMPLE 29. Let W = IW2n, and of A = p-'(F) are related by

= 0,

A"Fkj

and d = Sp(2n, R). The components of F =

82.

(10-20)

Lower case Latin indices are raised and lowered as follows

x,= XjF,,.

xi = LPX5 ;

(10-21)

A basis for d is given by real, second-order polynomials LA = &Z;.X'X.', A = I, ..., rn

= n(2n

+ 1)

where Z$ = Zfi. Define gABby m

zy = - B-1 c gA BzjBi and normalize so that z$zBklgAB= 8ikc8j2 + 82i 8:, Z$Bi5

= 2gAB.

Here ( g A B ) is the inverse matrix of ( gAB).The zA's may be chosen so that

(10-22) (10-23) (10-24)

289 95

DEFORMATION THEORY AND QUANTIZATION, I

The case n = 1: Sp(2, R) is locally isomorphic to SO(2, 1) and is of interest in connection with quantization of the harmonic oscillator. In this case we take L1 = Hx'x'

+ x'x'),

Lz= X'X',

L3 = +(x'x'

- x'x'),

(10-25) (10-26) ((10-27)

The centers of 9 and of the *-algebra are generated by (10-28) (10-29)

For future reference we list the lowest order symmetrized *-monomials: (1/2!)

c

L A

Perm

* LB = LALB + P g " B ,

(10-30) (10-30) (10-31)

11. CONSTRUCTION OF INVARIANT *-PRODUCTS

(a) We begin by studying invariant *-products on d*.Let US call *-polynomial an expression involving elements of C and of d C B and the operations of addition and *-multiplication, and formar *-product akebra (*-algebra for brevity) the algebra of *-polynomials with the identifications (9-2). This algebra is obviously isomorphic (by rescaling in d)to U(sa). A *-product on d*maps the *-polynomials into 9;precisely: PRoPosInoN 30. An invariant *-product on d* defines a C-htear map @: U ( 4 + Bsuch that, Va E d:

@(I) = 1, @(a) = a/%, gZ 0 ad&) = add(a) @, Ker @ = bilateral ideal of V ( d ) . 0

(11-1) (11-1)

Comersely, any C-linear map 0 of U ( d )onto B that satisfies (11-1) definesa *-product on d* given by

f * g = @O(@,Yf)

@Ol(g)),

tlf, g E 9

where CDo is rhe byection U(sB)lKer CD + B induced by @.

(11-2)

290

96

BAYEN !3 AL.

EXAMPLE 31. The canonical bijection between B and U ( 4 ,together with a change of scale, gives a linear map defined by @(an)= (a@)", or a * * Q = an, Va E d . This is the quantization rule proposed by Abellanas and Martinez-AIonso [20].

.--

EXAMPLE 32. More generally, for k = 2, 3,..., let (11-3)

where ck E C and T is a tensor of rank k whose components are polynomials of order less than k. This defines a bijection if ck # 0, k = 2,3, .... Example 31 is recovered bytakingck = 1 and T = 0.

A *-product on a'* will be called regular if @ is onto. More generally, let M be any algebraic ad*&-invariant submanifold of d*,and fl(9') the associated projection of 9. To any invariant *-product on M we associate a map Y (with properties analogous to (11-1)). DEFINITION 33. An invariant *-product on M is called regular if the map

Y: U ( d ) -+ 17(9)associated with it is onto [21]. The *-product (11-3) is regular if ck # 0, k

= 2,

3,....

(b) Next, consider the possibility of inducing an invariant *-product on M by restriction from d*. More generally, let M C M' E d*, where M and M' are ad*d-invariant algebraic submanifolds, and let IT be the projection of polynomials induced by the restriction of M to M.

DEHNKION 34. An invariant *-product on M' is said t o be compatible with IT if Ker 17 is *-invanant; that is, iff E Ker 17 implies that a * f E Ker 17,Va E d. If an invariant *-product on M' is compatible with D, then commutativity of the following diagram induces an invariant *-product on N :

Let Z(P) denote the center of the Poisson Lie algebra 9' and let Q denote the set of homomorphisms from Z(P) to C that reduce to the identity on the scalars. For any a E %',the family of equations q - r ( q ) = 0, Vq E Z ( q , defines an ad*, invariant submanifold M,. If riff is the projection of B defined by M,, we have fl, I Z(@)= v . We now restrict ourselves to projections of this type.

DEFINITION 35. Let R and R' be commutative rings and $: R -+ R' a ring homomorphism. Let A be an R-module, A' an R' module and #: A -+ A' any map that

29 1

97

DEFORMATION THEORY AND QUANTIZATION, I

+

preserves addition: Y(a, a2) = Y(a,) fY(a,>. Then Y is called [22] semilinear (relatively to $), in short $-linear, if Y ( r a ) = $(r) Y(a), for all r E R and a E A .

Let Z(d) be the center of U ( d ) and consider U ( d ) as a Z(emodu1e and n,,(P) as a C-module. PROPOSITTON 36. Let @: U(s4) B be the map associated with any *-product is on 9,compatible with l7,, , and q5 the restriction of @ to Z ( d ) . Then $ = TT M algebra homomorphism and y/ = 17 @ is $-linear. ---f

Q

Q

Proof. Let q E Z(P) and f E 8 ; q - p(q) E K e r n , . Since the *-product is compatible with l7, we have (q - dq)) *f E Kern, and thus Un(q*f) = .rr(q)nn(flThus for any 4 E Z ( d ) and U(ss), Y’(4f) = f l n 0 (43) = f l n ( # 4 ) @(!)I =

*

3.

*(@

WP).

A converse is also true; it will be formulated below. Since, for every q E Z ( @ there is some T E Q such that v(q) # 0, we have COROLLARY 37. Let a regular *-producton d*be given, and let SP be the associated map of U ( d ) onto 8. Then each of the folIowing statements implies the other two:

n,,

, VTTE V ; (i) The *-product is compatible with (ii) @ is +linear [+ zz 0 I Z ( d ) ] ; (iii) q * f = qf, V q E Z(P), Vf E 9.

(1 1-5)

DEFINITION 38. A regular, invariant *-product on .d* that satisfies any one (and therefore all) of the conditions (1 1-5) is called normal. The associated map gZ will also be called normal in this case; hence any map ds: U ( d ) 9’ is normal if it is onto, +-linear, and satisfies (11-1).

-

(c) Let an invariant *-product on M, be given and consider the question of its “lifting” to d * ;the following gives a partial answer that covers the most interesting

cases:

WEOREM 39. Let an invariant, regular *-product on M, be given, with the associated supposed +linearporn the Z(d)-moduIe lJ(sa>to the @-module map F U ( 4 l7=(P) n,(q ($ = ?P1 Z ( 4 ) . Suppose that either a’= H,, , the Heisenberg algebra, or that .d is semisimple and that the ideal Ker Y is generated by Ker $. Then there exists a m r d *-product on A?* with associated map 0:U ( d ) -+ 9 (boective in the semisimple case) such that 17,o @ = Y. --f

Prmf. In both cases U ( d ) and B are free modules over Z ( d ) and Z(9) (respectively); Z ( d ) is a polynomial algebra in a finite number of “Casimir” elements g= E Z ( d )1181. We denote by @, the “canonical‘’ bijection U ( d ) 9’obtained from the canonical bijedon U(@ + S(d) by rescaling, and by its restriction to Z(d). we give a proof for the semisimple case.

-

292

98

BAYEN ET AL.

+

(i) Construction of = @ I Z(4. We need an algebra isomorphism Z(P) such that 7r 0 4 = 4. To show that one exists let +o(&) = qe and fl4J = qu - dq,) #(&) = qa h, = q i with X, in @. We extend 4 as a (polynomiaI) ring isomorphism Z ( d ) = @[gal Z P )= C[q,‘l.

4: ZW)

--.f

+

+

-

(i) Construction of bases. We can write 9 = 02=oBn as a graded algebra and 9 = (J:d9’)n as a filtered algebra. The adjoint action of d in B preserves the grading and filtration and is fully reducible in each component (since d is semisimple). The ideal 4 = K e r n , is add invariant and filtered into 9 = u& ,4, = .# n 9,. This allows us to construct by induction a (graded) supplementary subspace B to Y in 9,invariant under a d d , and therefore a basis {xf}(i = 1, 2, ...) of 9 as a Z ( q module with each x* in 8.(These constructions are similar to those of [18, Sect. 8.21.) Let ( x i ) (i = 1,2, ...) be a basis for B as a Z(9) module obtained in this way, and let i, = @;Vxj); then (Zi} is a basis for U ( d ) as a Z ( d ) module and the IT,(xi) form a basis for n.C.9’(as ) a C-module). We have Y(Q = CjC,fl,,(x,) (finite sum), with C, in C. (iii) Construction of @. We take @(id)= Ct C,,X, and extend by +linearity: any i in U ( d ) can be written in a unique way as 2 = Qt2, , with Q,E Z ( d ) ; we define @ );( = &Qf)@GJ= Cis,flQf)Cijx,. Then @ is +linear (by construction) and commutes with add (because Y/ does). Furthermore, @(I) = 1 and auica)= u/ih for a in d . Finally, the @ that is constructed in this way is bijective (because Kcr Yis generated by Ker # and Ker 9 = (0)). The Heisenberg case is straightforward. The fact that Ker @ is a bilateral ideal can be Seen using the unique decomposition of any X in U ( d ) into X = K (ek - fl8))Y, Yo with Yr in @(@ ’; ) ( j = 0, I,..., K).

xi

-+

EXAMPLE40. Let d = H,,with basisp, q, e and structure (11-6)

Let S mean symmetrization with respect to the order of factors and take, for k , I , m = 0 , 1 , 2 ,...: (11-7)

where {Ck*mj are polynomials in p , q, e, with Coo0 = 1, C1W = p/ifi, Cola = q/8, Coo’= el%. This *-product is &-invariant if and only if the following recursion relation s hold : (1 1-8)

The center of U W )is generated by e; hence (1 1-7) is +linear if and only if e *f = ef, Vf E 3;that is, Cklm= (elifz)m Cklwhere Ckzis a polynomial in p , 4. The necessary

293 DEFORMATION THEORY AND QUANTIZATION, I

99

and sufficient conditions for (1 1-7) to be both invariant and +linear can be expressed by 1231 (11-9)

where C(a, is a formal power series in a,/3 and the equality is to be interpreted in the sense of equality of formal power series in a, p. Comparing the constant and linear terms in (11-9) one sees that C(a,j?) must have constant term = 1 and no linear terms, so that

+

1 where the unwritten terms are monomials inp, q of order .

=

The result (12-6) is easily generalized to a large class of functions, including all polynomials on d** The essential point here is that the functions (U(o1,p, Y ) ) form ~ a group under *-multiplication that is isomorphic to the operator group (12-3). We now investigate invariant *-products on any Lie algebra from this point of view. (b) Consider the formal group [24] with Lie algebra d.For a E d,let {exp(tu)} t’) a)-

-

( t E R) be the one-parameter formal group exp(ta) exp(t’a) = exp((t For u, b E d let c = c(a, b) be defined by the Campbell-Hausdoe formula:

-

exp(u) exp(b) = exp(c).

+

(12-7)

Let 8’ be the extension of B to the algebra of formal power series over &*.

DEFINITION 43. We call *-exponential the function Exp: d defined by m

--f

9,u ~ + Exp(a),

i

(12-8) (a * )” = a *

--- * a

(n factors).

(12-9)

PROPOSITION 44. The function Exp enjoys the foIlowing properties:

(i) Exp(u) * Exp(b) = Exp(c)

(12-10)

where c = c(a, b) is defined by (12-7). (ii) (add(a) Exp)(b) = {a, Exp(b)} = ad*d(a) Exp(b)

(12-11)

where add is the extension of the adjoint action of I in I toformalpower series over d. (iii) Exp(u) = 1

+ a/% + .-(12-12)

where T(,,) is an n-linear map of d into 9. Proof. (i) This follows from (12-7) and Eq. (9-2):

* - b * a)/% = {u, b},

(a 6

Va, b E d.

(12-13)

297

103

DEFORMATION THEORY AND QUANTIZAlTON, I

(ii) From (12-10) we extract the terms linear in a and obtain

*

(l/iti>(a Exp(b) - Exp(b) * a) = (a, Exp(b)};

(12-14)

the Ieft-hand side is (add(a) Exp)(b). (iii) This follows immediately from (12-8). Conversely, we have

PROPOSITTON 45. Let f: d

---f

9' be any function such that

(12-15) m

r

(12-16)

where T(n)is an n-linear map of A? into 9. Then there is a unique @-linear map @: U ( d )-+B, given by (a * >"

=

T(,,l(a,..., a).

(12-17)

r f Ker @ is a biiateral ideal of U ( 4 , then @ defines a unique invariant *-product on d* and f is the function Exp f o r this *-product.

Proof. From (12-15) and (12-16) we conclude that 4j commutes with add and that @(l) = 1, @(a)= a/iti; therefore, if Ker @ is a bilateral ideal of V ( d ) , we have an invariant *-product by Proposition 30. Substituting (12-17) into (12-16), we see that f coincides with the function Exp defined by (12-8). Remarks. (1) The condition that Ker @ is a bilateral ideal of U ( d ) is necessary as well as sufficient. (2) Any function f: d 3 9" that satisfies the conditions of Proposition 45, with the exception of (12-15), defines an associative *-product on B.

E ~ ~ ~ 46. P L Let E d be. the Heisenberg algebra with basis XI..., , Jczn+l = ql,...,qn; pl,-.-,pn;e with [ q i , p j ] = W e , and the other commutators equal to zero. Let g1,...,p" + Ql, ...,Pn = y l ,...,y2n be a canonical transformation and define Exp(oriq* + pipi + ye) = e(a,V'+BtP'+ve)/i~.

Then the associated *-product is given by (10-1) and (10-2), with X

= ih/2 and

it is &-invariant if and only if yr - xi:are constants and then it is an invanant *-product on d*.

298

104

BAYEN ET AL.

(c) Next, consider *-products on an invariant submanifold M of d*and the restrictions on Exp implied by compatibility of the *-product with the associated projection 17. Recall from Section Ilc, Theorem 39, and Corollary 37, that many regular, invariant *-products on M,,,rr E V, can be obtained by projection from a regular, invariant *-product on d* with the property 4 * f = qL

v q E z ( q , VfEP.

(12-18)

Of course, (12-18) remains valid forfE 9".In particular,

* Exp(b) = q Exp(b),

Vq E Z ( q , Vb E d .

(12-19)

We shall show that the structure equations (12-10) allow us to reduce these equations to partial differential equations for Exp. First, an example.

EXAMPLE 47. Suppose that d has a nontrivial center do;then any rr E V defines, by restriction, a linear map cr: do+ @. [This map defines a submanifold Muof d*; the following applies to the problem of construction of a *-product on either M, or Mu.]Let a normal *-product on d* be given, and let Exp be the associated *-exponential. Then (12-18) holds and implies that U,

* Exp(b) = a, Exp(b),

VU,, E 54, , Vb E d .

Using (12-8) and (12-10) it is now easy to see that Exp(b

+ ao)= Exp(b) e%"lL,

Vu,

E

do, Vb E at'.

(12-20)

In particular, Exp reduces to an ordinary exponential if d is Abelian. The calculations that follow will give analogous results for the general case.

Let t~ R and ct = c(ra, b), where C(Q, b) is defined as in (12-7). From Exp(ru) * Exp(b) = Exp(cJ we get by formal differentiation of formal power series with respect to t at t = 0: a

* Exp(b) = ifia, Exp(cJ It4

= ifi77" Exp(b)

(12-21)

where 7" denotes the vector field on d defined by 9" I D = ate,

It4

-

[The tangent space at any point b E Jaz is canonically identified with d.1 We have ' 9

ID

=a

+ t[aa,b1 f

i%

[[a,b1, b] f--'

299 105

DEFORMATION THEORY AND QUANTIZATION, I

where x is the linear operator xa = -add(b)a and the coefficientsB,, are the Bernoulli numbers. The identity B,xR = x/(l - e-.) between formal power series can be used to evaluate qa. For example, if add@) is diagonable and A' = @A EAis the eigenspace decomposition and (PA}the associated projection operators, then we have x = C AP, and

zz4

-

(12-22)

Let {LA)(A, B = l,...,m ) be a basis for d, and a = aALA,7" =

then

1 ~is a~ family ) of vector fields over d and we have

LA * Exp = ihf Exp.

(12-23)

Let Q be any element of the center of the formal *-product algebra of the form

Q = Q AI...A,LA'*

*--

* LA-

where the coefficient tensor is completely symmetric. The value of denoted Q.Then

Q * EXP = (iti)" Q A , ...A',

A1

... .qAn Exp.

(12-24) in Z(P) is also (12-25)

PROPOSITION 48. Let a normal *-product on A'* be given, and let Exp be the associated *-exponential. Let Q be any element of the center of the formal *-product algebra of theform (12-24). Then Exp satisfies thefollowingpartial dyerential equation: (12-26)

Proof. Replace q +Q in (12-19) andcompare with (12-25). Combining this with Proposition 45 one has

THEOREM 49. L e t f : A'

(i) W) (iii) (iv) form

--f

8'be any function such that:

f ( b ) = (a,f(b)). a/ih i- = (I/n!)(ih)-nT(,,)(a,..., a). The C-linear map @: U ( d ) + B defined by (a *)* = T(,,)(a,...7a) is bijective. For every element Q of the center of the formal *-product algebra of the ad..&)

f(u) = I

p=

& A x...A.LA1

+

1 . -

zy=,,

* .--* L A - ,

Then @ is normal and defines a unique invariant *-product on A'*, compatible with D=for every z E %,' and f is the associatedfunction Exp.

300 106

BAYEN ET AL.

. EXAMPLE 50. Let d = SO(2,l) and adopt the notations of Example 41. Let b = bALAE d and let 6denote the matrix of -ada(b). We have

bAB= C;'bc ,

bZA*= 4(bAbB- t28AB),

g(b, b) = b,2 - bI2 - b2'.

t2

If t 2 # 0, then 5 is diagonable, with projectors satisfying

Thus we get from (12-22) (and the definition 7' = aAqA):

.

where aB denotes differentiation with respect to bB The center of U ( d ) is generated by the second-order Casimir operator and the set (10-26) of differential equations reduces to

(ih)' gABqAqB EXP = Q EXP

(12-27)

or

This equation is of course add invariant and can be separated by introducing pseudcspherical coordinates: t , z = cosh 8 = b(t2Q)-112 and a polar angle. The invariance property of Exp, namely, Eq. (12-1 l), means that Exp does not depend on the polar angle and can be expanded in terms of Legendre functions; formally Exp(b) = CaC',(f)P,(z). Now Exp(b) must be interpreted as a formal power series in { b ~ ) as well as in (LA>;we therefore restrict the summation to values of a E C such that each term is a formal power series; that is (12-28)

Here P,(b) is the "solid spherical polynomial" defined by (11-13) and (11-14); it diEers from the usual Legendre functions by a factor t a p 1 2 and is a polynomial in both {bA]and {LA].Of course, (12-28) must be interpreted as an equality of formal power series. For C&) we find pt2

2 +tan t at

sin2 t

')

-

(i?i)-Z

Q] tac,,(t) = 0.

(12-29)

301

107

DEFORMATION THEORY AND QUANTIZATION, I

Consider the limit t2 -+ 0 with b # 0, then

P,(N

(2a 3

- l)!! ba.

(12-30)

a!

Therefore, the only acceptable solution of (12-29) is the one that is regular at t = 0: (12-31)

where c, is a constant and the parameter I is defined by

Q = (2fi)Z Z(I

+ 1).

(12-32)

+

Equating equal powers of {bA) in (12-28) we get (b *)" = c&" a polynomial of order n in {LA) that vanishes when either Q or t Evanishes. The constants c, in (12-31) can therefore be identified with the constants c, in (11-15'); that is, Eqs. (12-28) and (12.31) give the functionExp for the *-product definedby Eq. (11-15). Note that c0 and cl must both be equal to 1. I n particular, if we specialize once again to the interesting special case Q = 0, = d(ifi)E(i.e., I = -2) and c, = 1, then

e

(12-33)

In Part TI [lo] we shall rediscover this formula and obtain from it the spectrum of the harmonic oscillator. I n order to show the connection between (12-28) and (11-15) in more detail we give explicitly all terms up to the third order in {bA}in Eq. (12-28):

from which one recovers Eq. (11-16). (d) We believe that the function E x p can be a useful tool for representation theory. Consider the formula (12-34)

where f is some function on d,di; is the Lebesgue measure on d , and Exp,(a) is

302

108

BAYEN ET AL.

the value of Exp(a) at f E sd*. When d is Abelian and the *-product is normal, then this reduces to the definition of the ordinary Fourier transform. In general, one has a novel and possibly interesting notion of Fourier transform that relates f on d to f on d*.We shall not approach the problem of determining the precise conditions under which (12-34) makes sense, but limit ourselves to some formal observations. One has

Here and below c stands for the Campbell-HausdoriTfunction C(Q, 6). If the *-product is normal we can restrictfand g to M,, and integrate to obtain a type of Plancherel formula: (12-35)

where dp is the Liouville measure on M,, and

-

In the case of the Heisenberg algebra, the support of K reduces to {c

jf* g dp(0

J& @(a); this shows that If*

= 0)

and

g dp(6) = J f g dp(6) in this case.

Another interesting possibility (verified in some cases) is that Exp(ta), for t~ R and a E d ,have a Fourier-Dirichlet expansion: Exp(ta) =

J ~ ~ e ” ”dA.”

(1 2-36) (12-36)

One can define the “spectrum” 2 of a to be the support of mA and the “multiplicity” of a discrete point h E 2 by the integral of mAover phase space. This idea finds support in several particular instances that are investigated in Part I1 [lo]. Let a faithful, unitary representation of the group SO(2, 1) be given, together with a map of the type of the Weyl correspondence, that maps operators to B’: A F+Aw . A *-product is defined on 9’ by A * Bw = (A& . Let H be the operator that represents the generator t l of the compact subgroup of SO(2, I), so that Hw = L1and (etH’’pw= Exp(tL1). Now, since the spectrum of H/2fi consists of integers, we must have Exp(mL1) = 1 for this *-product. Examination of (12-31) reveals that the period of each term in (12-28) is 27m, where n is the smallest integer such that 2nZ is integer. This indicates that the *-products that were found for SO(2, 1) are related to faithful representations of SO(2,l) or to coverings of SO(2, 1). In particular, it is known that (12-33) is related to a representation of the fourfold (metaplectic) covering of SO(2, 1); that is, the twofold covering of SL(2, R).

303 DEFORMATION THEORY AND QUANTIZAnON, I

109

ACKNOWLEDGMENTS We thank Professor J. Schwinger for helpful advice and Dr. D. Amal and Professor R. Ra& for interesting discussions. One of the authors (M.F.)thanks the Physics Department, UCLA. for hospitality. Another author (C.F.)thanks l’universitk de Dijon for hospitality received during the early part of this collaboration.

REFERENCES 1. E. P. WIGNER,Phys. Rev. 40 (1932), 749. 2. H. WEn, “Gruppentheorie und Quantenmechanik,” Hirzel-Verlag, Leipzig, 1928; 2. P h y ~ 46 (1927), 1. 3. G. S . AGARWAL AND E. WOLF, Phys. Rev. D 2 (1970), 2161. 4. K. CHILnr, J. Math. Phys. 17 (1976), 859. 5. J. E. MOYAL, Proc. Cambridge PhiI. Soc. 45 (1949), 99. See also E. A. Render, Ann. Phys. (N.Y.) 95 (1975), 455. 6. M. FLATO, A. LICHNEROWICZ, AND D. STERNHEIMER, (a) C. R. Acad. Sci. Pans Ser. A 279 (1974), 877; (b) Compositio Math. 31 (1975). 47; (c) J. Math. Phys. 17 (1976), 1754. Ann. Math. 79 (1964), 59. 7. M. GERSTENHABER, 8. J. VEY, Comment. Math. Helvet. 50 (1979, 421. 9. M. FLATO, A. LrcHNeRowxz, AND D. STERNHEIMER, C. R. Acad. Sci. Paris Ser. A 283 (1976), 19. 10. F. BAYEN,M. FLATO,C. FRONSDAL, A. LICHNEROWICZ. AND D. STERNHEIMER, Deformation theory and quantization. 11. Physical applications, Ann. Phys. (N.Y.)111 (1978), 111. 11. A. NUENHLJIS, Indag. Math. 17 (1955). 390. 12. A. LICHNEROWICZ, Les Varittts de Poisson et leurs Algkbres de Lie associks. J. Differential Geometry, to appear. 13. A. Avez AND A. LICHNEROWICZ, C. R. Acad. Sci. Paris Ser. A 275 (1972). 113; A. AVEZ, A. LICHNEROWICZ, AND A. DIAZ-MIRANDA, J. DifferentialGeometry 9 (1974), I. 14. F. BAYEN, M. FLATO,C. FRONSDAL, A. LICHNEROWICZ, AND D. STERNHEIMER, Lett. Math. Phys. I, 521 (1977). 15. P. A. M. DIRAC, “Lectures on Quantum Mechanics,” Belfer Graduate School of Sciences Monograph Series No. 2, Yeshiva Univ., New York, 1964. 16. A. LICHNEROWICZ, (a) C. R. Acad. Sci. Paris Ser. R 280 (1976), 523. (b) Leu. Mafh. Phys. 2(2) (1977). “Structures des Systkmes Dynarniques,” Duncd. 17. This map was introduced by J. M. SOURIAU, Paris. 1970. 18. For informationabout enveloping algebras, see J. Dixmier, ”Algebres Enveloppantes,” GauthierV a h , Paris, 1974. 19. In the examples of *-products for semisimple algebras that have been calculated so far, K turned Out to be a multiple of the Killing form. 20. L.ABELLANM AND L. MARTINEZ-ALONSO, J. Math. Phys. 17 (1976), 1363. See also M. Vergne. Bull. Soc. Math. France 100 (1972), 301. When at = H, (with basis q1 - p I , e), the “canonical” bijection between B and U(& defines a bijection Yobetween the quotient of U ( 4 by the ideal generated by (e - 1)-a Weyl algebra [18]-and the quotient of 9’by the corresponding ideal; in the Heisenberg representation of d , Y;*gives the Weyl quantization mle. Other bijections, delined with different vector space bases of U ( 4 , correspond to different ordexings [3] (standard. no-4 etc.). This is one of the motivations for the restriction made from the next subsection onward to projections defined by characters of the center of the Poisson Lie algebra 9. 21. We have found some invariant *-products that are not regular; they are related to h i t d i m e n sional representations of the algebra and are of interest on compact symplectic manifolds. 22. N.W A“‘Algtibre,” K I , Chap. II, Sect. 1, No. 13,3rd ed., 1962 or new edition, 1970, Hermann. Paris.

304 110

BAYEN ET AL.

23. The general form (11-9) for the Exp of an Hl-invariant, normal *-product corresponds exactly to the general “Bassociation 1aw”considered by Agarwal and Wolf [3]. The result is easily extended to H,. 24. E. B. DY“, Amer. Math. SOC.Truml. 97 (1953). pspehi Mat. Nauk (N.S.) 5, No. 1 (35) (1950),135]. See also N. JacobSon,‘‘Lie Algebras,” Interscience, New York, 1962; G. Hochschild, “The Structure of Lie Groups,” Holden-Day, San Francisco, 1966.

305 ANNALS OF PHYSICS

110, 111-151 (1978)

Deformation Theory and Quantization. * II. Physical Applications F. BAYEN DPpartement de Mathkmatiques, Universitk de Paris 6, 75230 Paris Cedex 05, France

M. FLATO~ AND C. FRONSDAL University of California, Los Angeles, California 90024 AND

A. LICHNEROWICZ AND D.

STERNHEIMER

Physique Mathkmatique, College de France, 75231 Paris Cedex 05, France

Received May 19,1977

In the preceding paper general deformations of the structures based on the classical symplectic manifolds were examined. Quantization can be understood as a deformation of the algebra of observables without any need for introducing a Hilbert space. By a slight but crucial restatement of the usual interpretation of classical mechanics we find a framework for the description of both classical and quantum mechanics, within which the continuity of the quantization process is brought out. The spectra of some important physical observables are determined by direct phase space methods; this helps support the belief that a complete and autonomous theory, equivalent to ordinary quantum mechanics in special e, but capable of wide generalization, can be constructed.

A. GENERAL PKysrcAL CONSIDERATIONS 1. Contractions and Deformations in Physics

(a) Nonrelativistic physics (more precisely: Galilei-relativistic physics) can be viewed as a “contraction” of relativity theory (Lorentz- or Poincari-relativistictheory). The word contraction is used here in the sense of Segal-Wigner-Inonii contraction of groups [I]; the Poincark group contracts to the Galilei group in the limit c + 03 (contraction parameter c = velocity of light). A deformation [2] is a sort of inverse

* This work was supposted in part by the National Science Foundation. *Permanent address: Physique Mathkmetique; College de France, 75231 Paris Cedex 05 and uniVexsik5 de Bjon, 21000 Dijon, France. 111 0003-4916/78/1111-0111~5.00/0 Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

306 112

BAYEN ET AL.

to contraction: one determines, in a precise mathematical sense, all groups that are “close” to the Galilei group and finds the PoincarC group (among a small number of possibilities). In this sense, relativity theory is a deformation of nonrelativistic physics. Another instance of contraction is the passage from quantum theory to the classical limit fi + 0 (contraction parameter fi = Planck‘s constant). Here the inverse process of deformation is nothing but the general problem of quantization. The radical change in interpretation that accompanies the passage to quantum theory might discourage attempts to apply the concepts and techniques of deformation theory; nevertheless it is our aim to show that it can be done. (b) The basic mathematical structures of classical mechanics are the symplectic structures attached to phase space: the algebra ( N ) of Cwfunctions on phase space (W) under ordinary multiplication of functions, and the Lie algebra structure induced on N by the Poisson bracket that is defined by the symplectic form ( F ) on W. In the preceding paper [3] we have examined the formal differentiable deformations of these algebras; in particular, we have introduced an associative algebra (*-product algebra or *-algebra for brevity) that is a deformation of the algebra N of functions with ordinary multiplication. For ft g E N = CO-(W7C) we write the new (deformed) composition law on N as g) ~ f g. The * corresponding Lie algebra defined by (Ag) I+ [f*g] = (f*g - g * f ) / i h is a deformation of the Poisson Lie algebra. A particular instance of this type of deformation of classical mechanics is familiar and is known as the Moyal product and associated Moyal bracket. It is our intention to demonstrate that quantum mechanics can be replaced by a deformation of classical mechanics: a description of quantum phenomena in terms of ordinary functions on phase space, including a complete and autonomous physical interpretation. Naturally, this alternative formulation of quantum theory will include some features that are not usually associated with phase space. In order t o know what to expect it is worth while to recall the elements of the theory of Weyl [4], Wigner [5], and Moyal [6]. See also the preceding paper [3]. (I, Section 12a).

u,

2. The Weyl- Wigner-Moyal Formalism

(a) Consider the Schrodinger quantum mechanical description of a particle interacting with a potential. Let A be a linear operator in Hilbert space; the following can be made rigorous provided A belongs to a large class of operators including those that are Hilbert-Schmidt. A unique function on phase space (here W = R3 x R3} is defined in terms of the configuration space matrix elements of A (integral kernel associated with A ) by the formula [5] (2-1)

This defines a one-one correspondence-the

Weyl correspondence-between a large class of operators in Hilbert space and a large family JV of functions or distri-

307

113

DEFORMATION THEORY AND QUANTIZATION. I1

butions on phase space 171. The map that sends Aw to A is called the Weyiappfication and denoted Q; one has Q: f +f =

If@, 3) -.-.

++

e(f'P+n.O)Iifi

d35 d31)

(2-2)

where f is the inverse Fourier transform of the function or distribution f, and P, 0 are the Schrodinger operators that correspond to j,4'. An associative *-product (Moyal *-product) is induced in N in a natural manner. Let f,g E M and 3, the corresponding operators, so that f = f w ,f = f), etc.; then f * g = (3i)wEN(2-3)

a(

That is, the composition law ( L g ) w f + g is defined by the commutativity of the diagram

Direct calculation gives (2-4)

where Pn is the nth power of the Poisson bracket (interpreted as a bidifferential operator acting on the couple Cr, g ) and is defined by (throughout part A of this paper, we take for P what is denoted by -P in part B and in the preceding paper): (2-5)

The Moyal bracket [6] is M g ) M

= v*g1= C f * g - g * f ) / i f i .

(2-6) (2-6)

It is this bracket, and not the Poisson bracket, that corresponds to the quantum , = (i/i+i)[f,ilW. commutator: ~ fgjM The fact that one has to do with a formal deformation of classical mechanics, with Planck's constant +i playing the role of deformation parameter, is brought out b y Es. (2-4). Other rules of association (other orderings) between functions and operators have been considered in the literature (see e.g., Agarwal and Wolf, Ref. 7). They are of the type (2.2), with the difference that the measure d5 dv is multiplied by a weight function

c @T&*r, co

-n, = 1 -t

+-1

308

114

BAYEN ET AL.

where upris a homogeneous polynomial; we shall call them Q,-laws. In the orderings usually considered (standard, normal, etc.) Ol= exp(tiw,), with o2a homogeneous quadratic polynomial. Its Fourier transform will thus be a formal series a

where the T, are differential operators. If the functions f and g give the operators under the O,-law, we shall denote by f * ’g the inverse Q1-image o f f i . Then we have T,f = fw and therefore T,( f * ‘g) = Tfif * Trig. Thus T,,realizes the cequivalence (in the sense of Definition 20 of I) between the Moyal bracket and the O,-bracket [f* ’g], and similarly between the two associative algebra deformations. All orderings correspond therefore to cohomologically equivalent deformations, and any treatment done in one ordering can be translated into another ordering via this equivalence. In this sense the two formalisms are equivalent, though the same classical function will in general correspond to operators of different form in the different orderings.

f and j

(b) Many problems of quantum physics have already been translated into the Moyal idiom [8]. Our point of view is different: we want to make the “classical” formulation autonomous in order to open up a vast field of generalizations with all kinds of interesting applications (see Section 5). For this reason it is necessary to deal with some problems of interpretation. We begin with an examination of the Moyal equations of motion. Letf be any operator without explicit time dependence, A the Hamiltonian operator, and f, H the corresponding functions on phase space. The Schrodinger equation of motion for f is translated by the WeyI correspondence into the Moyal equations of motion forf, namely, (2-7)

The Moyal bracket defines a derivation of the *-product algebra; that is, (2-8)

whence d

dt

d d cr * g ) = (z f) * g + f * (x g).

(2-9)

However, unlessfis a polynomial of order ,P d3P d39.

EXAMPLE 1. Take H = p2/2m,f = q2; Eq. (2-12) gives the trajectory in N given + tjj/mj2.With p = S3( j - j’)S3(q‘ - B’) one gets ( f ) t = (T + rF/m)2,

byf(t) = (4‘

This interpretation lends itself to statistical generalizations though it remains a m pletely deterministic as long as the first-order equation of motion (2-12) is retained, Determinism, in the usual narrow sense, is lost when (2-12) is replaced by the nonlocal equation of motion (2-7). It is then that phenomena such as “spreading of the wave packet” appear [9]. We return now to the discussion of (2-10). In classical mechanics we are dealing with the algebra N of functions on phase space with ordinary multiplication of functions. The Poisson bracket { H ,f} defines a derivation of this algebra and therefore so does the time development determined by the equations of motion. In addition, it is clear that (fg)(t) must perforce be equal to f ( t ) g(t) as long as one is dealing with a single trajectory in phase space and f ( t ) , g(r), Cfgj(t) are identified with their numerical values at the point on the trajectory labeled by t. In Moyal dynamics, on the other hand, the relevant structure on N is the *-product; the equation of motion shows that d / d is a derivation of this new structure and f ( t ) * g ( t ) = Cf*g)(t). The “paradox” of (2-11) means that trajectories in phase space have no invariant meaning: if jj7 (q‘ -+ P, 0 is a canonical transformation, then the trajectory j ( t ) ,q(t) does not necessarily coincide with the trajectory P ( t ) , &(t). Of course, this is directly related to the fact that the system (j

- p”)

*p

= 0 = (4‘

- q) * p ,

(P I ,

6)) E I w 6

(2-16)

does not possess solutions when fi # 0. Cc) The analog of the formula (2-15) for the measured value ( f ) t of the observablefe Jy at time t for the state p E M , in Moyal dynamics, is

(2-17)

This can be derived from quentum mechanics by means of the Weyl correspondence 12-1) with the following interpretations: j3 = Q(p) is a density matrix, f ( t ) = Q ( f ( t ) ) 7 ’ f > t = tr(fj3). If j3 is pure, 9 = \ $)(+I, then p = (1 $>(# is called the Wigner function for the state I.#)Remark. Equation (2-17) appears in the literature in the equivalent form (2-15). The equality g P p d3q = Jfg d3p d3q was discussed in {3] (I, Sect. 12d); it is

If*

311 DEFORMATION THEORY AND QUANTIZATION. I1

117

valid for the Moyal *-product (and equivalent orderings related to the Heisenberg algebras Ifn),but not for *-products in general. We shall adopt (2-17) in the general case. 3. General *-Products and the Notion of Preferred Coordinates

(a) The general notion of deformation of syrnplectic structures studied in the preceding paper provides the means for a wide generalization of the conventional process of quantization. Given any classical mechanical system, is it possible to identify a more or less unique *-product and thus the “correct” quantization scheme? The question is not well posed, since the answer must depend on the extent to which the classical system is well defined and endowed with a complete physical interpretation. We shall suggest a partial answer based on the idea of the existence of a set of “distinguished observables” or “preferred coordinates.” (b) It is curious to notice that the concept of “observable” seems to be more fully developed in quantum theory than in classical mechanics. Indeed, a fairly accurate definition in quantum theories without superselection rules is to call observable every self-&joint operator. In classical mechanics there has been a tendency to call observable every function on phase space. To avoid semantic difficulties we shall call “good observables” functions of a more restricted class. Ifpis a self-adjoint operator in Hilbert space, then one has a I-parameter group G of unitary operators (eittj ( t E R). Let f be the function on phase space related to f by the Weyl correspondence; then f is real and defines an infinitesimal canonical transformation; that is, a derivation g H {A g) of N, and a globally Hamiltonian vector field on phase space. By the usual exponentiation of vector fields one obtains a n action of G in W. This motivates the following

DEFINITION 2. A function F E N is a good observable if it generates, by the Poisson bracket, a group of symplectic diffeomorphisms of W. For a E N we have (I, Sect. 9) called a-invariant any *-product such that (3-1)

k t z2 be the Lie algebra of all a E N such that the *-product is &-invariant. In the case of a Vey deformation (defined by the differentiablecochains of increasing order), we have Seen that d is always finite dimensional. [It is likely, though not proved, that d is finite dimensional for more general deformations as well.] Therefore, euery Vey qzuznrization scheme (one formalizable as a Vey deformation) distinguishes ajinite subalgebra of N. This algebra is a subalgebra of the algebra Inv(F, r)that generates the infinitesimal symplectic transformations, affine for a connection T on W. The Moyal *-product is invariant under Inv(F, 0, where r is the usual Aat connection on W’.If we take the natural coordinates on R2’ such that the components

3 12

118

BAYEN ET AL..

Tik of I‘ vanish, then the elements of Inv(F, I‘) are the polynomials of order , for Va E d, Vf E N, where it? is the deformation parameter. The elements of d will be called the distinguished observables or the preferred coordinates for the quantization. [By associating quantum observables with self-adjoint operators we have deliberately, for simplicity, restricted ourselves to the narrow framework of conventional wisdom. As we know by certain examples in quantum mechanics, and also from studying the question of integrability of finite-dimensional Lie algebras to Lie groups (both in geometry and in analysis), it may very well be so that special circumstances require a more liberal definition of observables. Consequently, it may sometimes be necessary to extend our definition of good observables to include certain types of derivations that do not exponentiate to Lie groups of symplectomorphisms.] According to Definition 4, the problem of quantization of any physical system may be approached as follows: (i) identify a finite algebra of distinguished observables; (ii) replace W by the image M of the map J W-. d * ;(iii) select an invariant *-product on M. This last problem was solved, in principle, under fairly general conditions, in the preceding paper. We are led to the crux of the problem of quantization: To quantize a physical system, one must identifv a finite algebra d of distinguished observables. That the algebra A of distinguished observables plays a very special role in the

313 DEFORMATION THEORY AND QUANTIZATION. I1

119

deformed (quantized) theory is evident. LA a ~ and d consider the differentia1 equation (d/dt)f = [a * f]. This coincides with (d/dt)f = {a,f}. For the Poisson bracket we have the ordinary derivation rule; therefore this differential equation defines a group of geometric transformations of phase space. In particular, i f the Hamiltonian H belongs to d,then the time development of the system can be associated with a classical trajectory through phase space. The harmonic oscillator is a wellknown example. (c) The specification of the distinguished observables for a classical physical system is an important part of the interpretation of it. Since it is impossible to treat this problem in general we shall discuss only one particular case. The classical Kepler problem is characterized by the following Hamiltonian function: H = $p2 (I/r). (3-2)

-

[Units have been chosen so that both physical parameters of the problem are equal to unity.] Here p2

=@

- 9,

r

I -

(q'

*

4)1P

> 0,

(3-3)

and (q, fi) E (R3- (0)) x R3. The singularity of (3-2) at the origin must be removed; this will be done by a method that goes back to Poincark and even to Kepler himself. The equation of motion forfE P ( W , C), namely, (d/dt)f = - { H , f } ,

(3-4)

together with the subsequent restriction to an energy surface:

H-EEO,

EER

(If - E vanishes weakly in Dirac's sense Ill J' ) can be replaced by (d/ds)f = {r(E - H ) , f )

and

r(E - H)M 0.

(3-5)

Here s is the mean anomaly introduced by Kepler; comparison of (3-4) and (3-5) shows that dsldt = l/r. In this way the singular Hamiltonian (3-2) is replaced by the smooth function r(E - H) = Er - J.rp2- 1.

However, singularities are introduced if one evaluates (dlds) p'

= {r(E

- H, p} = (E - &p2)q/r;

therefore the choice of the pi as preferred coordinates leaves something to be desired. No singularities appear in (dlds)4' = {r(E - H), 4') = rp'

3 14 120

BAYEN ET AL.

or in higher derivatives such as (3-7)

This strongly suggests that 4‘ and r j be taken as distinguished coordinates. These six functions generate, by the Poisson bracket, a 10-dimensional algebra isomorphic to the Poincard algebra, with the basis

{LA}= {q3,r; qipj - q j p i , rpi}

( A = 1,..., 10).

(3-8)

This algebra would be a reasonable choice of distinguished observables for many systems with a I/r potential. The special case of the Kepler problem, however, is characterized by a special symmetry-the SO(4) symmetry associated with the RungeLenz constants of the motion [12]. The generators of SO(4) are not included in the Poincart algebra spanned by (3-8) and this suggests that the set of distinguished observables be enlarged. That this can be done is well known; the easiest construction is found by asking whether the function (3-6) can be included. One finds a 15-dimensional algebra isomorphic to so(4, 2), with the basis (3-8) and

-

{LA}= {rp*,p q, (q p ) j - &p2?j}

( A = 11,..., 15).

(3-9)

The larger algebra includes the so(4) symmetry algebra of the Hamiltonian. Although the Poincard algebra spanned by (3-8) is sufficiently large, there are advantages to including (3-9) and taking so(4,2) as the algebra of distinguished observables. Because the function r(E - H) is an element of so(4,2) (for E fixed in W), one finds that every element of so(4,2) has simple harmonic s-dependence, which trivializes the solution of the equations of motion [13]. In addition, the arbitrariness in the choice of *-product is always reduced when the algebra of invariance is enlarged. The map J defined by (q,p) H hALA(j,if) ( A = 1,..., 15) sends W onto a sixdimensional submanifold of the dual &* of & = s0(4,2). This is an exceptional orbit of the coadjoint action ad*d of d in d* (most orbits are 12-dimensional; see I, Sect. lld). The construction of invariant *-products on such orbits is a very interesting problem. 4. Spectral Theory, a Beginning (a) For f E N and t E R consider the formal series (see I, Sect. 12) (4-1)

Assume that there exists a p > 0 such that for I t I < p the power series in t converges to a distribution on W.Suppose also that, for t fixed in some complex neighborhood

315 DEFORMATION THEORY AND QUANTIZATION. II

121

of the origin, ExpCft) considered as a distribution on W has a Fourier-Dirichlet expansion ExpCft)

=

1 TAentlm

(4-2)

AEI

where I is a sequence in C and T,,E N for h E I. In .the special case of the Moyal *-product, if Q(f)is a normal operator in Hilbert space, then Zis the spectrum of this operator and the are the projectors for the spectral decomposition. This motivates the following definition (we return to the general case):

DEFINITION 5. Iff€ N satisfies (4-2) we call I the spectrum off, any h E Z an eigenvalue off, and r Athe projector associated with A. From (4-1) one obtains

f * Exput) = ifi(d/dt)Expuf)

(4-3)

and from this it follows easily that

(4-4)

* T A ~ = 8AA'wA.

TA

More generally, we may consider the Fourier transform (in a generalized-function sense); formally ExpCft) = @Iin dp(A). (4-5)

I

In general, the support of dp(A) will be referred to as the spectrum off. It is the (Fourier) spectrum of ExpCft) as a distribution in t , in the sense of L. Schwartz. (b) Let a *-product position law

(Lg)

f * g on N = Ca(W,C) be given; then the com-

H

(L g) - f

-

8

=2 *3

(f = complex conjugate o f f ) defines another *-product on N. In particular, let a2 be a real Lie algebra, d*its real dual, and f * g an invariant *-product on some ad*&invariant submanifold M of d* (see I, Sects. 9-11); then f i g is another invariant *-product on M. DEFINITION 6. A *-product - is called symmetric if, for f, g E N, we have = f T g , that is, iff * g = g * J From now on, we limit ourselves to symmetric *-products (the Moyal *-product is symmetric).

f *g

(c) Iff in (4-2) is real, then I is symmetric (that is, X E I =- E I> and +A = q

-

316

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BAYEN ET AL.

Moreover, if g * g 2 0 for any real g E N (the expansion (4-2) being supposed unique), then I and T,, are real. We suppose now that this is the case, and define NAby ( N AE R if the integral converges, otherwise we shall say that NAis infinite): (4-6)

where dp(x) is the Liouville measure on W (with some normalization to be chosen later). Assuming N , is finite we have a normalized state p,, = nA/N,. Now let g be another function with an expansion of the type (4-2), with real spectrum I' a n d projectors n,', and define N,' = sw7r,I dp(x), p: = 7TK)/N,'. Measuring the observable nAin the state p,' and the observable nK'in the state pA one obtains

The principle of detailed balance thus tells us that NAand N,' are (up to a common factor) the multiplicities of the states pA and p i ; that is, the multiplicities of the eigenvalues h off and K of g. Henceforth, we shall take the Liouville measure dp(x) on W to be so normalized that (4-6)gives precisely the multiplicity of the eigenvalue h off. In the case of the Moyal *-product on RZn one verifies easily that NA = tr 9(nA)if one takes (4-7)

5. Suggestions

(a) It is apparent that we have tried to avoid dealing with distributions (generalized functions) on phase space beyond those that can be defined in some sense by formal power series. Attempts to generalize lead to interesting questions; for example, we need to know under what conditions associativity is retained. Our "classical" spectral theory is very far from being developed; one may ask, for example, under what conditions a distributionfhas a real spectrum for a given quantization. Equations (4-2) and (4-5) pose a problem that seems to have received scant notice up to now: given a distribution, find the support of its Fourier-Dirichlet series (or Fourier transform). This question replaces the much more familiar but hardly easier problem of the determination of spectra of operators. In fact, it seems that all aspects of operator theory now find analogs in classical analysis. (b) A promising aspect of the approach to quantization by means of deformation theory is its generality. Symplectic manifolds with nontrivial homology groups are of interest-sometimes for obvious reasons and sometimes as a result of the globalization of phase space that is realized by the choice of distinguished observables. The states are not necessarily one-valued on phase space and this gives rise to cohomologically nontrivial factors and projective representations of groups of

3 17 DEFORMATION THEORY AND QUANTIZATION. 11

I23

invariance. The Bohm-Aharonov effect should provide a very interesting example [14]. Applications to open systems are also possible. One-differentiable rigorous deformations of the Poisson Lie algebra have been shown to apply to classical systems with friction or viscosity [15]. [The losses occur as a consequence of giving up the derivation rule.] Perhaps this type of deformation could be combined with the quantization to create a coherent description of decaying systems. (c) In classical field theory one has a Poisson bracket (in terms of functional derivatives) [I 61. Formally it is straightforward to write a corresponding Moyal bracket [ 171. It would be worthwhile to develop techniques (on infinite-dimensional symplectic manifolds) for giving a rigorous meaning to deformations of the Poisson bracket of classical field theory. An analog of the *-product for infinite-dimensional symplectic manifolds would also be of interest. I t would be natural to begin with an investigation of semiclassical approximations, including WKB and tree approximations, in field theories subject to strictly canonical quantization. For want of a rigorous treatment one could study quantizations at some order of fi. Even more exciting is the possibility of discovering new quantization schemes for field theories, such as quantum electrodynamics, Yang-Mills theories with or without a Higgs-Kibble mechanism, theories with soliton or monopole solutions, for which the conventional canonical quantization is clearly inappropriate. Also, “quarks” may be “confined” because quark fields are quantized so as never to create any free particle states in the first place [18]. (d) Anti-Poisson brackets on Grassmann algebras have been studied recently [19], after the invention of supersymmetry in dual models [20] and in quantum field theory [21]. Let N be the Grassmann algebra generated by (O‘} (a, b = I,,.., n) and A a real, constant symmetric 2-tensor; that is a A = (Ad} with Aab= AbaE 88. Define the anti-Poisson bracket for f, g E N by (5-1)

The simplest deformation of N is given by the anti-Moyal *-product dehed for f,gENbY f * g = fiAFg (A = ifi/2). (5-2) With this product N is a Clifford algebra. The corresponding deformation of the anti-Poisson algebra is given by

This bracket has the correct symmetry properties and satisfies the (generalized) Jacobi identity.

318 124

BAYEN ET AL.

Generalizations are immediate. In the first place, A need not be constant. The antiPoisson bracket (5-1) satisfies the Jacobi identsy if and only if the (generalized) Schouten bracket [A,A ] vanishes. The powers of P that appear in (5-2) must be understood in terms of covariant derivatives and conditions are imposed on the connection by associativity. Applications include quantization of classical (Grassmann algebra valued) Fermi fields. (e) From the general considerations and the particular examples of invariant *-products developed in I, from the remarks made about our “classical” spectral theory in this paper, and from the partial success of the Kostant-Souriau geometrical approach to group representations, it seems more than evident that *-products have a promising future in representation theory. (f) One of the most important theorems of classical mechanics-it can be taken as the starting point for a formulation of statistical mechanics--is the Liouville theorem. One form of this theorem states that the generalized velocity vector in phase space is divergenceless. In ordinary classical mechanics (phase space W = W2” with {xi} = {ql .*. p“}) this reads -div f = (a/8xi){H,xt} = 0. The validity of this formula follows easily from the equations of motion. In Moyal mechanics the Liouville theorem continues to hold. The reason for this is that [H* a]/% = {H, a} for every first-order polynomial a = aixfin 9,..., xZn. [This statement remains valid for all the generalizations of the Weyl correspondence considered in the literature.] It is interesting to study the validity of the Liouville theorem for general deformations and invariant *-products. (g) The Weyl application establishes a correspondence between functions on phase space and linear operators in Hilbert space. To the product of operators corresponds the *-product of functions and to the commutator of operators conesponds the *-commutator of functions (Moyal bracket). One may ask: What is the image of the Poisson bracket of functions in the ring of operators ? That is, what is the contraction limit fi + 0 of the ring of operators of quantum mechanics ? In other words, we are interested in the empty box in Fig. 1. The application indicated by a dashed line is defined by commutativity of the diagram. The physics described by the mathematical structure in the empty box applies to certain situations in quantum optics in the coherent states formalism.

1-

Weyl

1

application Deformations

[Mcyolbracket]

.

m 1

Contractions Weyl appl icot ion

FIGURE 1

1-

,

I

I

319 DEFORMATION THEORY AND QUANTIZATION. I1

125

(h) Finally, it should be noted that our suggestion to regard Moyal mechanics and its generalizations as autonomous physical theories raises the question of whether or not all such theories can be cast in a form that fits the general axiomatic formulation of quantum mechanics.

B. CALCULATION

OF SPECTRA

6. Harmonic Oscillator In this section the energy levels of an I-dimensional harmonic oscillator will be computed with the help of the Moyal product. This will be achieved by considering the function Exp(Ht) and its Fourier decomposition as explained in the previous section. Though some of the calculations that will appear may be more or less well known, it does not seem that such a “classical” treatment exists in the literature. Here by “classical” we mean that we deal only with functions defined in phase space and that no operator of the quantum mechanical Hilbert space corresponding to the problem appears. (a) One-dimensional case. We denote the Hamiltonian function by H; for the harmonic oscillator

H(P,4 ) = H P 2

+ 43.

Suppose f: R -+ C is a C“ function. Then f(H) is a C“ function defined in phase space and a straightforward computation gives:

H * f ( H ) = Hf(H)- (ti2/4)f ’(H) - (ti2/4) Hf”(H).

(6-1)

Equation (6-1) proves that in this case (H*)” is a function of H only. We can there fore write (H*)” = K,,(K) (n = 0,1,...) where K,,is a function of one variable. According to (6-1) we have: Kn+I(H) = HK,(H) - (75”4) L ‘ ( H )

- (h2/4) HKXH);

(6-2)

Kn is therefore a polynomial of degree n and the same parity as n. We are now in position to formulate:

PROPOSITION 1. For any fixed ( p , q) E W, the power series in t: (6-3)

has a radius of convergenceequal to m. For 1 t 1

< m one has (6-4)

320

126

BAYEN ET AL.

Proof. Consider the function

which is analytic in the following open set U of C2:

U

= {(t,H) E C2 I t # (2k

Write the Taylor expansion of

and remark that

+ 1)n,k E Z}.

4 with respect to t around the origin:

4 satisfies the following partial differential equation in U

It is obvious that the coefficients C,,satisfy the recursion relation (6-2) with C, = 1. It follows that for any n, C, = K,,. Compare I, Eq. (12-33). This pointwise convergence property of the series EL,, (I/n!)((Ht/ih)*),, allows us to prove the convergence of (6-3) in the usual Y(W) space of distributions in the variables p and q.

PROPOSITION 2. For $xed t E ]-7r, n[ (or [ t converges in 9'(W) for rhe weak topology to

1 < n and t E C) the series (6-3) (6-5)

Proof. In fact with respect to the variable H EC the series converges uniformly on compact sets in the complex plane. It follows that when His replaced by $( p 2 qz) the series converges uniformly on compact sets in the ( p , q) plane. Now the map t -l 1 t t w (cos T ) exp (p2 q 2 ) tan

+

[x +

7-

from the disk I r [ < n into 9'(R2) is weakly analytic and has a (weak) analytic continuation in the open set U' = C - {(2k 1) T I k E Z}. We therefore define Exp(Hr) as the distribution (6-5)in the variables (p , q)for$xed t E Li'. One easily checks 0, the distribution Exp(Ht) E Y l ( W ) ,i.e., is tempered. that if t E U', Im t In order to define the Fourier expansion of Exp(Ht) we remark that for fixed (pyq) E R2 the function (6-5) is periodic in r with period h.However, this function does not belong to L1(O, h).We therefore consider (6-5) as a distribution in the variable r E R. In fact we have:

+


2 will cancel. Hence we get for such an 1":

We are now in position to prove:

LEMMA4. In the open set U we have ( Y * ) ~= g2- [(31- 4)/4]fi2 and more generally ( y * I n = g-lGn+l(g)

if

n 2 1.

(7-7)

One has thus shown that the nth Moyal powers of the function y( p, q ) = g( p , q ) -

[(I - 2)/4Jfi2g(p,q ) - I are identical to the (n + 1)th Moyalpowers of the Hamiltonian H of the I - 2-dimensional harmonic oscillator divided by g (after replacing H by g ) and that (y * )2 = C [(I - 2)"4] fi2. It is easy to check that the function g-l E L~,,(BB2z).Thus for any integer n > 0, (y *)" E W(Rzz)and these distributions are tempered. It is now trivial to prove:

+

t 29 DEFORMATION THEORY AND QUANTIZATION. I1

135

PROPOSITION 1. For any ( p , q) E U,the power series in s (7-8)

has a radius of convergence equal to TT. rf I sI

< T one has (7-9)

Moreover the series (7-8) converges in 9j'(!R2')for the weak topology to the right-hand side of (7-9) i f 1 s I < r r and s is fixed. Proof. Equations (7-2) and (7-7) imply that in U

if I s I < T. Suppose now that s is fixed and I s J < T. It is clear that with respect to x the series C ( l / n ! ) ( ~ / i Gn+,(x) f i ) ~ converges uniformly on compact sets in R. Thus for any b > 0 and any E > 0 there exists No = No(€,b) such that for any n 2 No and any x E [-b, b], 1 Xm,' (l/m!)(s/ifi)mGm+,(x)I I R,(x)l < E . Let d~E 9(R2'), K = SUPP4, b = S U P ( D . ~ ) E K g ( p , 41, and = JK g ( P , q)-' dp dq- For any n 2 N,(EM-~I\ 4 ,';I\ 6 ) we have

If I s 1

< i-r and s is fixed we can now define g)) EXP(ys) = a/as(~fig-l#(s,

= %(2g)-l(cos s/2)-l[(l- 2) sin(sl2) cos(s/2)

x exp((2g/ih) tan s/2).

+ 2g(ih)-l] (7-10)

This is an element of 9'( R2z)and the process of weak analytic continuation allows, as in Section 6, to define Exp(ys) by the same formula if s E U' = C -((2k -I- I)TT I k E Z}. Tt is obvious that if s E U',Im s < 0, the distribution Exp(ys) is tempered. Expansion (6-12) may now be used to expand Exp(ys) in a Fourier-Dirichlet series.

330 136

BAYEN ET AL.

PROPOSITION 2. Forfixed s E C with Im s

< 0, one has (7-1 1)

(7-12)

the series converging in Y’(UP)for the weak topology. Prooj: To simplify the proof we replace g/fi by g and write e-fs = 7,IT,, = g-la,(g). With the help of the generating function of the generalized Laguerre polynomials it is straightforward to prove that for any s E C with Im s < 0 one has

Exp(ys) = g-1e--*+Wz

c ataw

rn.

n>o

Moreover, Eq. (7-10) allows one to make an estimate of the remainder of the previous series: for any 7 E C such that I 7 I r < p < 1 where r and p are arbitrary, any g 2 0 and any n 2 0, the following inequality holds:


, 2 ) and let r

+ +

and attempt to get its spectrum by *-product algorithms. One may at first think of (at least) three approaches, all of which meet with difficulties if we keep the usual Moyal product in W E . (i) Direct exponentiation of H as in Sections 6 and 7. This seems for the moment hopeless: successive *-powers (H*)* of H are not in general functions of H only; these. powers become more and more singular as n increases and no simple recursion formula seems to hold among them. For example, when I = 3 one finds that

Besides, H * y5 has in general an infinite number of terms and a direct resolution of the equations H * y5 = 4 * H = A$ seems difficuIt.

332

138

BAYEN ET AL.

+

(ii) Direct transcription of the well-known so(2, 1) C so(l 1,2) dynamica1 method. In our case this amounts to linearizing, with the help of the generators of an so(2, 1) Lie algebra, the function U = r * (If - E ) or, if one wants to deal with real functions, Y = r112* ( H - E ) * r1I2.In the first case, the functions that generate a representation of the Lie algebra so(2, 1) with the Moyal bracket are

r, = -21 * ( p + 1 ) = -2I

r(p2

ifi p q + 1) - -- (Z - 1 ) h2 2 r 8r '

ifi p . q - ( I - 1 j ti2 r4= -21 r * ( p 2 - 1) = -21 r(p2 - 1) - 2 r 8r ' S =p - q

(8-1)

- iA/2 = T - 312.

The Casimir is c=(r,*j2-((r4*)2-(s*)2=g2-~(z-

I)#

and one has

u = &(r, + r4)- ~ ( -r r,) , -I as expected. Let E < 0 and write ea = ( -2E)ll2. The function Exp(Ts) may be easily computed with the help of the results of Section 6. One verifies that, iff is a polynomial in the variables p and q, ExP(--J)

* f( P,4) = W

P ,e - w * EXP(- Ts).

Thus this equation will hold on formal series and hence on all (generalized) functions for which the above *-products are convergent. This allows us to write EXP(-T~) * f(P , 4 ) * EXP(TJ) = f (espye-'qj.

In the case of U one gets ExP(-Ts)

* U * Exp(Ts) = 4e"(ro + I'J - Ee-d(T,,- r4)- 1 = (-2,9112

r,,- I.

An analogous manipulation is possible in the case E > 0 (I-', replacing To).However, though the spectrum of the Casimir is known (Section 7), this is not the case for To. Here also a direct exponentiation of Toseems difficult: the *-powers of Toare not functions of Toonly, and no simple recursion formula seems to hold among them. However, here the even *-powers of Toare polynomials in p , q and thus regular. In the case where one considers V, the generators of the Lie algebra $ 4 2 , 1) are r-112 * To* 1112 3 yo', r-112 * * rIl2 3/41, T = r-lP * S * r1/2 = pq and an identical treatment is a priori possible. Though here the generators are real valued, the difficultiesare the same and furthermore all *-powers of the compact generator are singular.

r,

333

139

DEFORMATION THEORY AND QUANTIZATION. I1


f l i k = q < P k - q k p j @j,l+l = 641 * ( P 2 ) - P I * q k * p k ) - 6qj Bj,1+2= f i ~ Z + 3~ q3 , M,+l,z+2= S (1 < j , k 0. (iii) A third approach, based on the SO(Z+ 1) symmetry group of the and Mj,l+l = A, = problem, is the following. The functions MIk = q5pk - t ( p 2 - 1) qj ( p * q ) p j ( j = 1,...,I ) define with Moyal or Poisson brackets a representation of so(l I), the first Casimir of which is

+

Y

+


2 we get P q g , +)

+ (1 - 2) n(n - 1) pn-yg-1,

4) = 0

(A-3)

in the case 4 = Q(g2). We now show that the previous equality holds for any = F(g). Let n > 2 be a fixed integer. It is easy to prove that there exists a sequence of polynomials (Pk)k>l such that the polynomials Rk defined by &(x) = pk(xz) have the following property: on any compact subset of R+*, the polynomials & , R'k ,..., RE) converge uniformly (respectively) to F, F',..., F(n)as k 4 co. Let us write R k ( g ( p ,4)) = $k( p , 4). On any compact subset of U the expression Pn(g, +k) ( I - 2) n(n - 1) Pn-2(g-1, 4 k ) y which is 0, converges uniformly to P ( g , 4) (I - 2) n(n - 1) Pn-2(g-1, 4) as k 4 00. Hence this last expression is identically 0 on U.

+ +

LEMMA 4. Let g(x) = x - [(I - 2)/4x]P.One immediately checks that (Ag)(x) = x2 - [(31- 4)/4]iizy which proves the lemma for ( Y * ) ~ . We prove by induction rhe end of the lemma. One has glx) = x-lGZ(x).Suppose that (7-7) holds for n and that x-lG,+l(x) = R,(x) is a polynomial in fi. Lemma 3 implies: (p)n+l

= Y * ("/In

= (ARn)(g) = Rn+l(g) = g-lGn+z(g)

and obviously g-1Gn+2(g)is a polynomial in ri. Remark. Even powers of y are polynomials in the variable g2 and odd powers of y are quotients of polynomials in the variable g2 by g .

REFERENCES 1.

I. E. SEGAL,Duke Math. J. 18 (1951), 221; E. I N ~ N UAND E. P. WIGNER,Proc. Nat. Acad. Sci.

U.S.A. 39 (19531, 510, and 40 (1954), 119. 2. M. GERSTENHABER, Ann. Math. 79 (1964), 59.

345 DEFORMATION THEORY AND QUANTIZATION. I1

151

3. F. BAYEN,M. FLATO, C. FRONSDAL, A. LICHNEROWICZ, AND D. STERNHEIMER, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys. (N.Y.) 111 (1978), 61 ; henceforth referred to as I. 4. H.WEYL,“The Theory of Groups and Quantum Mechanics,” Dover, New York, 1931. 5. E. P. WIGNER, Phys. Rev. 40 (1932), 749. Physica 6. J. E. MOYAL,Proc. Cambridge Phil. Soc. 45 (1949), 99. See also H. J. GROENEWOLD, 12 (1946), 405. 7. The domain of the Weyl correspondence has been studied by many authors. In particular, it e.g., K. CHI LIU, J. Math. Phys. 17 (1976), 859 and references quoted there. See also G. S. AGARWAL AND E. WOLF,Phys. Rev. D2(1970), 2161. 8. See, e.g., E. REMLER, Ann. Phys. (N.Y.) 95 (1973,455. See also S. R. DE GRWTAND L. G. S U ~ O R P“Foundations . of electrodynamics.” North-Holland, Amsterdam, 1972; AND S. R. DE GROOT.“La transformation de Weyl et la fonction de Wigner: une forme alternative de la mkanique quantique.” Presses Universitaires de Montrtal, 1974. 9. The nonlocality of the equations of motion provides the basis for the introduction of fuzzy phase space, in the sense of E. Prugovdki, J. Math. Phys. 17 (1976), 517, 1673; S. T. ALI AND E. PRUGOVEEKI, J. Math. Phys. 18 (1977), 219. :O. The map J is Souriau’s “moment”; J. M. SOURIAU,“Structure des Systemes Dynamiques,” p. 105, Dunod, Paris, 1970;See also C.-M. Marle, in “Differential Geometry and Relativity,” volume dedicated to A. Lichnerowicz (M. Cahen and M. Flato, Eds.), Reidel, Dordrecht, 1977. 11. P. A. M. DIRAC,“Lectures on Quantum Mechanics,” Belfer Graduate School of Sciences Monograph Series No. 2, Yeshiva Univ., New York, 1964. Nuovo Cimento A 53 (1968), 717. 12. See, e.g., G. GYORGYI, 13. See, e.g., C. FRONSDAL AND R. W. HUFF,Phys. Rev. D 7 (1973), 3609. L4. See C. MARTIN,Lett. Math. Phys. 1 (1976), 155. 15. M. FLATO, A. LICHNEROWICZ, AND D. STERNHEIMER, J. Math. Phys. 17 (1976), 1754. 16. I. E. SEGAL,“Symposia Math.,” Vol. XIV, pp. 99-117, Academic Press, New York/London, 1974, and earlier references quoted therein. 17. One simply substitutes in (2-4) formal powers of the field-theoretical Poisson bracket (using higher-order functional derivatives). This has of course to be justified. See also G . S. AGARWAL AND E. WOLF,Phys. Rev. D 2 (1970), 2187, 2206; E. REMLER, preprint, College of William and Mary, 1977. 18. M.GUENIN,Phys. Left., 62B (1976),81 and Ref. 8. M. M. FLATO AND M. GUENIN, Helv. Phys. Acta, 50 (1977),117. CERN preprint TH. 21 39 (March 1976). K. DRUHL,Max-Planck-Institute 19. R. CASALBUONI, preprint, Starnberg, December 1976. A. SUDBERY, Univ. of York preprint, October 1975. 20. J. L. GERVALS AND B. SAKITA, Nucl. Phys. 834 (1971). 21. J. WESSAND B. ZUMINO,Nucl. Phys. B 70 (1974),39.See also J. Schwinger, Phys. Rev. 92 (1953), 1283 and Proc. Nut. Acad. Sci. U.S.A. 48 (1962), 603; M. FLATO AND P. HILLION,Phys. Rev. D 1 (1970), 1667. 22. A. ERDBLYI(Ed.),“Higher Transcendental Functions,” Vol. 2, McGraw-Hill, New York, 1953.

346 J. Phys. A: Math. Gen., 13 (1980) L185-Ll88. Printed in Great Britain

LETTER TO THE EDITOR

Wigner distribution functions and the representation of canonical transformations in quantum mechanics G Garcia-Calder6n and M Moshinskyt Instituto de Fisica, UNAM, Apdo Postal 20-364, Mexico 20, DF Received 10 March 1980

Abstract. In this Letter we show how for classical canonical transformations we can pass, with the help of Wigner distribution functions, from their representation U in the configurational Hilbert space to a kernel K in phase space. The latter is a much more transparent way of looking at representations of canonical transformations, as the classical limit is reached when h + O and the successive quantum corrections are related with the power of P, n = I, 2, . , . .

In recent publications one of the authors (MM) and his collaborators have discussed extensively the representation in quantum mechanics of non-linear and non-bijective canonical transformations (Mello and Moshinsky 1975, Kramer et a1 1978, Moshinsky and Seligman 1978, 1979a, b). The representations, to be denoted by U, are given in definite Hilbert spaces like, for example, the one associated with coordinate q ; thus the matrix elements (4'1Ul4")related with specific canonical transformations were derived explicitly. It is not easy though to see from these matrix elements the quantum modifications to the canonical transformations, as the latter are formulated in phase space rather than in Hilbert space. Thus it is interesting to discuss the representation of canonical transformations in the phase space version of quantum mechanics that was developed originally by Wigner (1932), with the help of the distribution functions that now bear his name. We shall do this in the present Letter, illustrating the analysis with the representations of some simple examples of canonical transformations. We begin by recalling the definition of Wigner's distribution function f(q,p ) for a given wavefunction +(q),i.e. m

f ( 4 ,~ ) = ( . r r W - ' I m ( $ 1 4 + y ) ( 4 - y l $ ) e y p ( ~ ) dy,

(1)

where we use Dirac's notation (419)= 9 ( 4 ) ,($14) = 9*(4),and restrict ourselves to a single degree of freedom. As is well known (Wigner 1932), the integration of f ( q , p ) with respect to p or 4 gives the probability density for the state I$) in configuration or momentum space respectively. We consider now a canonical transformation (2) t Member of the Instituto Nacional de Investigaciones Nucleares and El Colegio Nacional.

0305-4470/80/060185

-+ 04$01.50

@ 1980 The Institute of Physics

L185

347

L186

Letter to the Editor

under which a classical distribution function f(q, p) would of course transform into F(q, P ) given by

F(4,P ) = f [ Q ( 4 ,P ) , P(q,PI].

(3)

I$) transforms into (Mello and Moshinsky 1975, Kramer et a1 1978, Moshinsky and Seligman 1978, 1979a, b)

In quantum mechanics though, the state

(4)

(5)

*

Writing z* = q’ y ’ when it is associated with 9,and za= q’* j j ’ when it is associated with U, and integrating over q’, y’, y’, y, with the extra factor 6(y’-

jq = ( v h ) - l

1-m

exp(2iP’(Y’-9’)) h dp’,

we immediately arrive at the relation m

m,P )

=

11

dq’ dP’fk.7’9P’Nq’P’lIClqP),

(6)

--03

in which the kernel K is given by m

Wp‘lKJqp)= 2(.nh)-’

J jdy dy’ (q ’ + y ‘ l U tl q + y ) -m

x(4 -

(7)

where from (3) we expect that

22 ( ~ ‘ P ’ I K I ~ P ) = S [ ~ ’ - PQ)(I~S, [ P ‘ - P (PI]. ~,

(8)

To obtain K we must known U which, in principle (Dirac 1947), is determined by the equations (Mello and Moshinsky 1975, Kramer et a1 1978, Moshinsky and Seligman 1978, 1979a, b) Q(q, P ) = W U t , P(4, P) = UPU’, (9) where q,p are now quantum mechanical operators. As U t U =I,we can pass U’ to the left-hand side, and taking matrix elements between a bra (4’1 and a ket 14”)obtain the

348 Letter to the Editor

L187

equations (Mello and Moshinsky 1975, Rramer et al 1978, Moshinsky and Seligman l978,1979a, b) Q(q , h ~a) ( ~ f l ~ 1 4 3 = ~ f ’ ( q ’ l U / ~ ’ ‘ ) ,

(10a) (10b)

Of course these equations only make sense when Q, P are well defined operators; otherwise, more sophisticated procedures need to be used (Moshinsky and Seligman 1978, 1979a, b). We shall now consider two simple examples of canonical transformations. The first will be the linear one

Q = aq + bp, P = cq + dp; a d - b c = l , b>0, where the constants are all real. We have then (Moshinsky and Quesne 1971) ( q ’ l ~ l q ’= ’ ) (~!.rrb)-’/~ exp[(-i/~b)(aq’~ - 2qrq“+dqrf2)3,

(11)

(12)

which satisfies equations (10) if we note from (11)that c = (ad - l ) / h Introducing (12) in (7) and using the relation (q’1Utlq”)= (q”IUlq‘)*we immediately obtain (q’p‘lKlqp)=

w -(as + bP)lS[P‘- (cq + dpll.

(13)

Thus for the linear canonical transformation the kernel coincides with its classical limit (8), in agreement with the fact that for this type of transformation Poisson and Moyal (1949) brackets coincide. In the second example we take Q as the Hamiltonian of a linear potential (Landau and Lifshitz 1958), and thus we have the canonical transformation

Q = (p2/2m)--Foq,

f’ = -p/Fo,

(14)

where m is the mass, Foa constant of the dimension of force, and {Q, P } = 1. Equation (10a) leads then to an Airy function (Landau and Lifshitz 1958), and we also satisfy (106) and get a normalised (Landau and Lifshitz 1958) unitary representation if we write (4’1UIq“) =AW-5),

(15~)

5 = [q’+ ( ~ ” / F o ) I ( ~ ~ F o / ~ ~ ) ” ~ , A = (2m)1/3.rr-1/2~;l/621-213,

(15b) (15~) (15d)

Substituting (15a) into (7) and making use of (15d)we can show straightforwardly that for the canonical transformation (14) the kernel K becomes (16)

We note first that when h + 0 the function becomes (Landau and Lifshitz 1958) either very small or very rapidly oscillating except when q‘ = (p2/2rn)- Foq. Furthermore, with the help of ( 1 5 d ) we easily see that v-1/2 @(x)dx = 1. Thus the expression in

349

L188

Letter to the Editor

{ } in (16) tends to a S function in the limit h+0, so that the kernel K goes into its classical limit (8),where Q and P are given by (14). To see what the quantum corrections are, it is best to apply the K of (16) to a smooth distribution function f(q,p ) , rather than study it directly. We choose

f ( 4 ,p ) = (Tab)-' exp[- ( q 2 / a 2-(p2/b2)1, )

(17)

where from (1)we will have the relation b = h / a iff is obtained from a Gaussian state in configuration space, Again using (15d) we obtain for the new distribution function F ( 4 ,p ) the expression

(18) 3 k - : even

where Q, P are given by (14). As indicated in (3),f ( Q ,P)is thaclassical change in the distribution function due to the canonical transformation, and it will be the only one remaining in (18) if h+0. Thus the terms associated with the higher powers of R2 indicate the successive quantum corrections to the distribution function when we perform the canonical transformation. The examples discussed in this Letter are very specialised, but they clearly indicate the procedure to be followed in general. Among the more interesting cases where this formalism can be applied are those of non-bijective (Kramer eta1 1978, Moshinsky and Seligman l978,1979a, b) canonical transformations. The concepts of ambiguity group and ambiguity spin used in the derivation of the representation U can then give interesting insights into the structure of phase space as a carrier of canonical transformations, as will be discussed in future publications. The authors are indebted to Professor E P Wigner for a stimulating presentation of his distribution function formalism and helpful discussions, during his recent stay in Mexico.

References Dirac P A M 1947 The Principles of Quantum Mechanics (Oxford: UP) Kramer P, Moshinsky M and Seligman T H 1978 J. Math. Phys. 19 683 Landau L D and Lifshitz E M 1958 Quanrum Mechanics (London: Pergamon) pp 73-4 Mello P A and Moshinsky M 1975 J. Math. Phys. 16 2017 Moshinsky M and Quesne C 1971 J. Murh. Phys. 1 2 1772 Moshinsky M and Seligman T H 1978 Ann. Phys., NY 114 243 -1979a Ann. Phys., NY 120 402 -1979b J. Phys. A: Math. Gen. 12 L135 45 99 Moyal J E 1949 Proc. Camb. Phil. SOC. Wigner E P 1932 Phys. Rev. 40 749

350 MOLECULAR PHYSICS,1982, VOL. 47, No. 5, 1001-1019

Wigner’s phase space function and atomic structure I. The hydrogen atom ground state by JENS PEDER DAHL and MICHAEL SPRINGBORG Department of Chemical Physics, Technical University of Denmark, DTH 301, DK-2800 Lyngby, Denmark (Receiwed 25 May 1982 ; accepted 29 ’June 1982)

We have constructed the Wigner function for the ground state of the hydrogen atom and analysed its variation over phase space. By means of the Weyl correspondence between operators and phase space functions we have then studied the description of angular momentum and resolved a dilemma in the comparison with early quantum mechanics. Finally we have discussed the introduction of local energy densities in coordinate space and demonstrated the validity of a local virial theorem.

1. INTRODUCTION This is the first of a series of papers devoted to the phase space description of atomic and molecular systems. Phase space representations of quantum mechanics have been extensively discussed since the classical works by Weyl [l], Wigner [2], Groenewold [3] and Moyal [4]. They have been applied in quantum statistical studies of transport processes and radiation (see, for example, [S] and [ 6 ] ) , and in treatments of molecular and nuclear dynamics (for example, [7-101). They have, however, not yet been used in such detailed theories as the theories of atomic and molecular electronic structure. I n this and forthcoming papers we shall investigate the possibility of extending the application of phase space representations to such theories as well. Very accurate wavefunctions are now available for all atoms and for a large class of molecules. These wavefunctions have always been generated in coordinate space, but there has been a considerable interest in their momentum space representatives as well. The use of phase space representations allows one to include the coordinate and momentum characteristics in a single picture, and hence it may serve to improve our understanding of the dynamical behaviour of electrons in atoms and molecules. The phase space formulation of quantum mechanics treats states and transitions in an equivalent manner. Thus, there is a phase space function associated with every quantum state and with every quantum transition as well. This function is the Wigner function. I n the present paper we shall only consider Wigner functions associated with states. Operationally, such functions play the rBle of probability densities in phase space. The values of the functions are, however, not restricted to being positive or zero, although they are always real. Hence, one may not interpret the functions as probability densities. Such an interpretation would of course also be inconsistent with the uncertainty principle. 0026-8976/82/4705 1001 804.00 (B 1982 Taylor & Francis Ltd

M.P.

2K

35 1

J. P. Dahl and M. Springborg

1002

If a Wigner function cannot be interpreted as a probability density, how may it then be interpreted ? This question, which has attracted considerable attention, was discussed in a recent article by one of the authors [Il]. The conclusion is that one must reoonsider the r61e played by a point in phase space. The significance of such a point is not, as in classical mechanics, that it defines a simultaneous position and momentum of a particle. It is instead that it defines an inversion operator, the so-called Wigner operator [12, 131. The properties of the Wigner operator and the group theoretical basis it lends to the phase space representation of quantum mechanics has been thoroughly discussed by one of us [14], but since only one-dimensional motion was considered we shall here list a few of the relevant expressions for a particle in three dimensions. The inversion operator defined by the phase space point (r, p ) is 1

n(r, p ) = ( s )

3

f

du dv exp

[$o.

u + p . v)

1 (1)

with the caret (‘ ) denoting operators. normalized state vector [ #) is

The Wigner function associated with a

(2)

It is normalized such that

Jf

(3)

f(r, P) dr dP = 1.

If +(r) and +(p) are the coordinate and momentum wavefunctions, respectively, that is (4) $(r)= , (4) (5)

+(PI= (PI$>,

then we may also write f(r, p) =

(i)’ f

dr’$( r - r’)*#(r

+ r’) exp (

-: P r’)

(6)

and (7)

Further we have the relations

and

s dP f(r, P) = $(r>+Krh s dr f(r, P) = +(P)#+(Ph ($lSl$>= s s dr dP f(r, P M r , PI,

(8) (9)

(10)

where a(r, p) is the Weyl transform [l] of the operator 8, as discussed in the Appendix.

352

Wigner function for the hydrogen atom

1003

It is the relations ( 3 ) and @)-(lo) that show how the Wigner function plays the r81e of a probability density in an operational sense. Let us also note that the eigenvalues of R(r, p) must be 5 1, since it is an inversion operator, Hence [13] we get that (11)

and consequently that f(r, P) must have support in a volume not smaller than

(h/2)3.

With the value of f at the point (r, p) being equal to 2/h times the overlap between l # ) and its mirror image with respect to (r, p), we may say that f(r, p) is a measure of the way the point ( r , p) supports the given quantum state. Similarly, we may talk about the way a certain region or a certain trajectory in phase space‘supports a state. A proper use of this kind of language leads to an integrated description of the wave and particle characteristics of quantum states, It is the purpose of the present paper to show how this kind of description works for the ground state of the hydrogen atom.

2. THEWIGNER FUNCTION FOR IS ORBITALS In what follows we shall use atomic units, and hence put m,e and ti equal to one, m being the electron’s mass and - e its charge. We shall consider nuclear masses as infinite and exclude spin and relativistic effects. The hamiltonian for a hydrogen-like atom with nuclear charge 2 is then (12)

T h e coordinate wavefunction for the ground state has the well known form (13)

and the corresponding momentum wavefunction is [15] (14)

Thus we obtain the following equivalent expressions for the Wigner function by using equations ( 6 ) and (7)

23

fls(r, p ) = - ~ j dr’ exp ( - Zlr

- r’l)

exp ( - Z l r + r ’ l ) exp (-2ip. r’)

(15)

+ + Zz]-2exp (2ir . p’).

(16)

and fls(r,

8Z5

p) =yrl

j

dp‘ [(p - p’)’+ Za]-2[(p p’)

Neither of the integrals involved can be evaluated in a closed analytical form and the Wigner function is consequently not expressible in terms of standard functions, This is in accordance with the fact that it satisfies a differential equation of infinitely high order [16], while the usual standard functions satisfy differential equations of the first or second order. 2K2

353

J. P. Dahl and M. Springborg

1004

The integrals (15) and (16) are, however, not newcomers in the theory of electronic structures. T h e integral (15) is, for a fixed value of r, the Fourier transform of the product of two 1s orbitals centred around the points - r and r respectively. Thus it is recognized as a generalized scattering factor in the theory of X-ray diffraction of molecular crystals and as a standard integral in those band theories of solids that are based on Fourier transform methods. Several procedures have, accordingly, been devised for the evaluation of this integral by methods of approximation. Thus, McWeeny [17] and Silverstone [18] have studied methods in which one orbital is expanded in an infinite series about the origin of the other. Such methods are only rapidly convergent for small values of r, and hence they are not applicable in the present context where all values of r must be considered. Other methods implying the summation of an infinite series or the numerical evaluation of an integral have been suggested by several authors ((19-211 and references therein). Although very powerful for single values of r where the accuracy can be readily assessed, these methods are again not easily applied when r is allowed to vary freely. The natural procedure to follow in the present context is to approximate the function (13) by a finite series of gaussians and insert this series in (15). The resulting integrals can then be evaluated analytically. This method is capable of giving a good representation of the Wigner function for all values of r and p, and it can easily be extended to other orbitals than the 1s orbital. Generalized scattering factors have been calculated along these lines by McWeeny [22] and Stewart [23]. We write accordingly M

(17)

with Xi(r) being a normalized gaussian Xi(r) =

This . gives ,

fle(M)(r, P) =

(

?)3’4

M

C i=1

exp ( - c q P ) .

(18)

M

ci2 p d r , P)+

C i>j=l

w#’dr,

~ > - t P d rPI) ,

where we have used the definition 1 Pi,(r, p) =- j dr’ xi(r - r‘) Xj( r ir‘) exp ( - 2ip r‘). 7r3

.

(19)

(20)

A straightforward integration results in (21)

in which (22)

and (23))

354 Wignkr function for the hydrogen atom

1005

Hence we get that

(24)

In the following sections we shall study this expression for the hydrogen atom, using an M = 10 representation of the 1s orbital determined by Duijneveldt [24]. This is the extremely good approximation to the true 1s orbital, leading for instance to an energy which only deviates 0*00015 per cent from the true value. The parameters defining the expansion are reproduced in table 1. Table 1. Gaussian approximation to the 1s hydrogen orbital [24].

0.062157 0.1 38046 0.304802 0.710716 1.794924 4.91 5078 15.018344 54.698039 254.017712 1776.775559

1

2 3 4 5 6 7 8 9 10

0.107330 0.339658 0.352349 0,213239 0.090342 0.030540 0.008863 0-002094 0*000372 0~000044

We shall also make certain comparisons with the variationally determined

M = 1 representation of the hydrogen Is orbital. As is well known, this approximation is obtained for a = 8/9n= 0.282942 3. A CHANGE OF VARIABLES The Wigner function is a function in six-dimensional phase space. It is, however, obvious from (24)that f18(r, p) only depends upon the three quantities r , p and u, with u being the angle between the vectors r and p. Let us therefore define new phase space variables instead of = (XI, x2, x 3 ) (25) and (26) P = (Pll P 2 l P 3 ) by introducing three mutually orthogonal unit vectors [25] ,)!-(=le 1 2 stnu12 r p

1

e2=-

('+").

1 2 cosu/2 r p

1 e3

=w

u

r x p = el x e2.

}

J

(27)

355 1006

J. P. Dahl and M. Springborg

These vectors define a right handed coordinate system S in ordinary threespace. The orientation of S with respect to a laboratory system So may be specified by three Euler angles a:, /I, y such that S is obtained from Soby (1) a rotation about the third axis of Sothrough the angle a, (2) a rotation aboilt the new second axis through the angle j, (3) a rotation about the new third axis through the angle y. The following relations are then valid :

(281 x g = r sin /I sin y - -

(

3 (29)

p,=p sin sin ( y + d and

dr dp = r2 dr p2 dp sin u du sin 1d p dct dy.

(30) and p

The quantities r , p , u, a,/3, y are our new phase space variables ; Y range from 0 to co, u and jl from 0 to T , a and y from 0 to 27~. We shall refer to the plane defined by r and p as the dynamical plane. Its normal, which is e,, has the spherical polar coordinates (,9, a). The angle y will be calIed the dynamical angle. With these designations we may express the fact that f l a is independent of a:, ,8 and y by saying, that all dynamical planes and all dynamical angles are equivalent. By displaying the dependence on r , p and u we obtain a complete picture of the 1s-state in the phase space representation. Let us first consider the picture obtained in the M = 1 approximation.

4. THESINGLE GAUSSIAN APPROXIMATION TO THE Is STATE This simple approximation corresponds to a coordinate wavefunction of the form (18), that is (31))

with a: = 8/9n= 0,282942 a,-a.

The corresponding momentum wavefunction is (32)

356 Wigner function for the hydrogen atom

1007

and the Wigner function (24) becomes fls(l)( r,

1

p) =7ra - exp ( - 2ar2)exp

(-$),

(33)

This function is not only independent of a, j? and y . It is independent of u as well. By integrating it over a, /3, y , u and multiplying it with r 2 p 2we obtain the function

(34) which is normalized such that m m

J J

0

C

.5

F18(1)(Y,

0

1.5

1.0

p ) dr dp = 1.

2.0

2.5

3.0

(35) 3.5

1

4.0

4.0

p/0;dh

3.5

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.5

I .5

1.0

1.0

.5

.5

0.0 0.0

.S

1.5

1.0

2.0

2.5

3.0

0.0 3

3.5 c/oo

) . function attains its maximum Figure 1. Contour map of the function F ~ ( ~ ) ( r , pThe value, 0.6893 ti-1, at the point ( y o , Po)= (1.3293 ao, 0.7523 ao-I ti). Starting from the maximum,contours have been drawn at 0.6, 0.3, 0.1, 0.06, 0.03, 0.01, 0.006 ?F1.

T h e function FI8(1)(r,p)is displayed in figure 1 through a contour map. As is evident from (34) it is everywhere non-negative. It has a maximum at the point (yo,

p o )=

(m, 1

+!a))

= (1.3293 aol, 0.7523 a0-l ti)

(36)

the maximum value being 16

Fls(l)(~o, po)=-= 0,6893 kl. re2

(37)

357

1008

J. P. Dahl and M. Springborg

It is obvious that FI8(l)(y,p)gives us complete information about the 1s state in the single gaussian approximation. Thus, integration over p gives the radial density in configuration space (with maximum at y o ) and integration over Y gives the radial density in momentum space (with maximum at po). Actually is nothing but the product of these two radial densities. As far as the function fu(l)( r, p) itself is concerned, we note that it attains its maximum at the origin (0, 0). The 1s state is symmetric with respect to inversion in this point, and the corresponding value of is accordingly the largest possible one, which from the relation (11) is known to be (2/h)a,that is, (1 8-3 in atomic units.

5. THEM=10 DESCRIPTION OF THE Is STATE This description, whose parameters are listed in table 1, is as previously mentioned an exceedingly accurate one. T h e Wigner function, for which we have the expression (24), is no longer independent of u and hence we cannot display all its features by means of a single contour map of the type shown in figure 1. A complete picture requires the drawing of a map for each value of u in the interval 0 6 u < 4 2 . As is obvious from the expression (24), we obtain the same maps for u and rr - u. It is still expedient to integrate the expression (24) over a,/3 and y and to multiply with y2 p2. Hence we obtain the function F*&Y,p , u )= 8ray Z p 2 fU(lo)(r, p).

(38)

Figure 2. Contour map of the function FIs(r,p, u) for u=O. Starting from the nodal curves (dashed lines, contour value (0.0)) contours have been drawn at 0.01, 0.03, 0.1, 0.2 k1(solid lines), -0.01, -0.03 ?V1(dotted lines).

358 Wigner function for the hydrogen atom

1009

0

1

4.0

4.0

3.5

3.5

3.0

3.0

2.5

2.5

2.0

2.0

1.5

1.5

1.0

1.0

.5

.5

p/a;’h

0.0 0

.5

1.0

1.5

2.0

2.5

3.0

0.0 I

3.5 rho

Figure 3.

As figure 2 except u= 7r/4,

Figure 4. Contour map of the function F&,p, u) for u = s / 2 . Starting from the maximum (yo, p,)=(1*405 a,, 0.759 a,-l ti), contours have been drawn at 0.3, 0.1, 0.06, 0.03,0.01, 0.006, 0.003 f i - l .

359

1010

J. P. Dahl and M. Springborg ”

This function is displayed in figures 2, 3 and 4 for u equal to 0, 7114 and 712 respectively. For u = 4 2 it is non-negative everywhere, but for all other values of u it has negative as well as positive regions. For several purposes it is sufficient to know the function obtained from (38) by integrating over u. We shall call this function the radial phase space function and designate it FIF,,(~, p ) . It satisfies the normalization condition (35). From (24) it is found to be 16

FlS(y,p ) = - r2 p 2 7r

M

C

ci2 exp ( - Z a i y 2 ) exp

i=l

32

M

7r

i>j-l

(-f) (39))

where (40)

is a spherical Bessel function.

The function attains i t s maximum Contour map of the function K & , p ) . value, 0.5617 at the point (yo, p,) = (1.30a,, 0.68 oo-l ti). Starting from the nodal curves (dashed lines, contour value OaO), contours have been drawn at 0.01, 0.02, 0.05, 0.1,0.2, 0.5 ti-’ (solid lines), -0.01 ti-’ (dotted lines).

Figure 5 .

360

Wigner function for the hydrogen atom

1011

The function Fl#(r,p)is displayed in figure 5. It is characterized by a dominant region in which it is positive, and by an oscillatory behaviour outside this region. The amplitudes of the oscillations are, however, fairly small (see table 2), but they are definitely not due to lack of accuracy in the wavefunction used. Table 2. Selected values of the function F&, p ) for Y =p. (Y/UOI

=(p/ao-l fiI

F d r , p1lh-l

0.0 0.3 0.6 0.9 1.2

0.0 0.0326 0.2676 0.4648 0.301 5 0.0473 -0.0047 0.0081 - 0.0031 0.0014 - 0*00063 0~00021

1.5 1.8

2.1 2.4 2-7 3.0 3.3

When the product of p and r is large, a regular albeit weak damped wave is disclosed by figure 5. The presence of such a wave is readily understandable from the expression (39). When both Y and p are large, the dominant terms in (39) will be cross terms for which one 01 is large and the other a is small, since it is only for such terms that both yu and l / ( q + a I ) become small and hence lead to slowly decaying exponentials. Since (23) shows that IT$,( z 1 when one a value is much larger than the other, we find that the relevant Bessel functions in (39) approach jo(p?),and this leads to a damped wave as observed. Applying a similar argument to the expression (24) shows that the contour maps for the functions Fls(r,p , u ) must disclose damped oscillations in cos (Xp r) when both p and r are large. That this is in fact the case is apparent from figures 2 and 3. I n figure 4, cos (2p r) equals 1 for all r and p (the wavelength becomes infinite), and the phase space function is accordingly nonnegative everywhere. The damped oscillations which we have discussed will, of course, have their counterparts in the theory of generalized scattering factors (cf. $2). T h e appearance of the oscillations in that context has been noticed and discussed by Avery [26], on the basis of arguments quite different from ours. I n closing this section it is worthwhile drawing attention to the complexity of the exact phase space function, as compared to the simplicity of the approximate phase space function discussed in the previous section, and a natural question presents itself. What must a wavefunction look like in order that the associated Wigner function be non-negative everywhere ? Hudson [27] has given a mathematical answer to this question for one-dimensional motion. His analysis showed that the wavefunction must have the form

.

.

-

+ +

$(x) = exp [ i(ax2 26% c)],

(41 1

36 1

1012

J. P. Dahl and M. Springborg

where u, 6 are arbitrary complex numbers with Re a > 0 and the complex number c is chosen so as to ensure correct normalization. When Im u is zero, this wavefunction describes a minimum uncertainty state [28] in one dimension. The conclusions of Hudson may probably be extended to three dimensions in a straightforward manner. That the wavefunction (31) describes a minimum uncertainty state in three dimensions is in accordance with this assumption. 6. THEDOMINANT SUBSPACE AND THE CLASSICAL SUBSPACE The function F18(r,p)of (39) and figure 5 was obtained from the function Fls(r,p , u ) of (38) by integrating over u. It is interesting to integrate over I and p instead to obtain a function Fls(u),normalized such that n

S FIa(u)sin u du = 1.

(42))

0

The expression for this function is found to be 10

Fdu)=3 with

c

10

-227,~’

Ci2 $-

i= 1

C$p~~’4(Uij

COS2 U ) / ( U < j + Tij’ COS2 U ) 5 ’ 2

(43)

i>j-l

as defined by (23) and

T ~ ,

q j=

4LYicij (mi+ a j y

(44)

I

Figure 6 shows the functions Fls(u) and Fls(u)sin u. Both functions have a sharp maximum at u = rr/Z.

Thus, the condition u = 4 2 defines a dominant subspace in which the Wigner function finds it maximum support. As already seen (figure 4), the function is everywhere non-negative in this subspace.

/ - . 0.0

-4

.I

.2

.3

.4

.5

.6

.7

.E

.9

.

1.0

u/n Figure 6 . The functions K a ( u )(upper curve) and Fl,(u) sin u (lower curve) as a function of uln.

362 Wigner function for the hydrogen atom

1013

T h e dominant subspace is five-dimensional. It contains a three-dimensional subspace of particular interest, namely the subspace obtained by putting r = 1 a, and$= 1 a,-l ti. I t is represented by the point (1, 1) in figure 4. This is the subspace to which the ground state motion was restricted in early quantum mechanics, since a Bohr orbit (in ordinary space) was just a circle with radius Y = 1 a,, in which the electron was sipposed to move with the constant momentum p = 1 A ao-l. Hence we shall call the subspace in question the classical subspace. T h e following important statement may now be made, on the basis of the present section and figures 2 and 3. The Wigner function attains a large, positive and constant value in the classical subspace. It is also large and positive in a large region surrounding this subspace. I n particular, it is everywhere non-negative in the dominant subspace. The regions in which the Wigner function becomes negative are well separated from the classical subspace.

7. THEANGULAR MOMENTUM DILEMMA I n this section we shall comment on the angular momentum of the 1s state, as calculated by the expression on the right hand side of (10). a(r, p) is, as mentioned in 5 1, the Weyl transform of the operator 6. The Weyl correspondence is discussed in the Appendix and there it is shown that if ri is a component of the angular momentum vector operator, say

e, =

- a&$,

(45)

then a(r, p ) is the same component of the classical angular momentum vector, that is, (46) 1, = xiP2 - XaPi. As is well known, the left hand side of (10) is zero when I#} is the 1s state and a" is jl, or j3. Hence the right hand side must also be zero. That this is actually the case is easily seen by remembering that f(r, p) is independent of the Euler angles a, j3 and y . This makes all directions of the vector e3 in (27) equivalent. But the direction of e3 is also seen to be the direction of the angular momentum vector and thus each component of this vector does in fact have a zero mean value. As regards the length of the angular momentum vector, it is shown in the Appendix that the classical function

e2

A2 = I1Z

+ 1,= + 1:

(471

is the Weyl transform of the operator (48)

Evaluating the right hand side of (10) with a( r, p) = Xa will accordingly produce the value #Aa. This interesting result allows us to resolve a pedagogical dilemma which has bothered writers of elementary textbooks [29]. How does one bring the fact that the angular momentum in a Bohr orbit is ti into accordance with the fact that the angular momentum in the SchrBdinger picture is zero ?

363 1014

J. P. Dahl and M. Springborg

The dilemma is obviously resolved by remarking that the operator that occurs in the SchrBdinger picture, viz.

+ i2z +

12 = 2e,

(49) is different from the operator that corresponds to the classical function X2 of (47). Averaging A2 over the classical subspace described in the previous section does in fact give the value P, as in the Bohr description. la2

8. LOCALDENSITIES An advantage of the phase space formulation of quantum mechanics is that it leads to a natural definition of a local density in coordinate space for a given operator and a given state. Thus (10) suggests that we consider the quantity

41)"

I

dP f(r9 p)a(r, P)

as being the local density associated with the operator a* and the state Integration over the spatial coordinate gives the expectation value of d

=s

dr 4 r ) .

(50)

I$). (51)

When a(r, p ) =a(r), a function of r alone, we may use (8) to get 4 r ) = dr)lb(r)"$(r).

(52)

As an example, the potential energy operator

13,

L

: Y

(53)

defines the local potential energy density 2

d r )=

-; +(r)*+(r)*

(54)

For the ground state of the hydrogen atom this becomes, by observing (13), Ep(r)=

--n1 exp r( - 2 r )

(55)

As an important example of an operator that is not a function of P we consider the kinetic energy operator

rf=-

2'

(56)

The local kinetic energy density becomes (57))

It may be evaluated when f(r, p) is known, but as shown by Ziff et al. [30] and by Cohen [31] it may also be evaluated directly from the coordinate wavefunction by using ( 6 ) . T h e result is d r )=

iH IW2-kV21#I2)

(58)

364

Wigner function for the hydrogen atom

1015

or equivalently

It is seen that where

EK(r)=*(*lv#/2-)#yv2

16-i)#v29").

'Idr)=!d'RB(r) + EKC(r)), EKB(r)

and

=*I V#12

+

eKc(r) = - t(##V2 $ $V2 #*). When # is real, eKC(r) becomes equal to eKA(r) =

- +,PO2 #.

(59)

(60)

(61) (62) (63 1

The expressions EKE, eKc and cK were all discussed by Cohen 1311 (who used the designations KA,KB,K c and K,, respectively). He showed that each expression could be derived from phase space descriptions discussed earlier by him [32]. For the ground state of the hydrogen atom we obtain the following explicit expressions

(;-&) exP ( - Z y ) ,

1 1 eKA(r)=eKC(r)=G 1 €KB(r)=g

exp (-2y),

1 eK(r)=-exp(-2r). 27rr

(64) (65 1

(66)

These will be discussed further in the following section. With reference to Cohen's work [31, 321, we want to make the comment that, although a whole set of mathematically consistent phase space representations of quantum mechanics exists, there are compelling reasons why one should consider the Weyl-Wigner representation used in the present work as the canonical one. These reasons were discussed at length in a previous paper

P119. A

LOCAL VIRIAL THEOREM

Let (67) (68) (69)

be,the exuectation values of total enerb, respectively, for a stationary state of the-hydrogen atom. As for any Coulomb system we then have the well-known virial theorem

Ex= -aEp

(70)

which expresses a global balance between the kinetic and potential energy. Due to the obvious relation E = E K i Ep (71)

365 1016

J. P. Dahl and M. Springborg

we may also write

-E,

(72) Bader and his co-workers have shown that for a molecular system it is possible to perform a spatial partitioning of the charge distribution in such a manner that the kinetic and potential energies of the resulting fragments obey the virial theorem (see [33]). For the ground state of the hydrogen atom they also noted [33] that E K B ( ~ ) = -E141z, (73) which is a local virial theorem if the right hand side is identified as the local counterpart of the total energy. Such an identification is, however, not quite satisfactory, since the natural definition of this local energy density would be EK=

= ‘%B ( r, -k EP(r) (74) in accordance with (71) and this quantity is different from the right hand side of (73). As a consequence EItn(r) and rp(r) do not satisfy a local analogue of (70)It is, on the other hand, easily seen that if we work with EK(r), defined through the Weyl-Wigner correspondence, then we obtain a completely satisfactory local form of the virial theorem

(75) (76) (77)

It must be stressed, however, that this remarkable result only holds for the ground state of a hydrogen-like atom. In forthcoming publications we plan to study the local balance between the potential and kinetic energy for other atomic and molecular systems. 10. DISCUSSION A N D CONCLUSIONS The phase space description of a quantum mechanical system is an alternative to the description based on wavefunctions. It is from several points of view a more complicated description. From other points of view it is a richer description. The state of a system is described by a Wigner function, I n the present paper we have constructed this function for the ground state of the hydrogen atom and analysed its properties. T h e analysis was facilitated by the introduction of concepts like the dynamical plane and the dynamical angle, the classical subspace and the dominant subspace. The orientation of the dynamical plane is closely related to the angular momentum vector ; the classical subspace is that part of phase space to which the ground state motion was confined in early quantum mechanics. The Wigner function is found to be independent of the orientation of the dyanamical plane. It attains a large, positive and constant value in the classical subspace and it remains large and positive in an extended region containing that subspace. Outside this region the Wigner function shows a rich structure

366 WigneY function for the hydrogen atom

1017

which includes oscillations of a de Broglie wave-like character for large values of Y andp. We have illustrated the general variation over phase space by means of a series of contour maps and a graph which shows that the Wigner function finds its maximum support in the subspace where r p = 0, the so-called dominant subspace. A dynamical variable is represented by an operator in a description based on wavefunctions. In the phase space description it is represented by an ordinary function of r and p. T h e connection between the two representations is given by the so-called Weyl correspondence which we have described in the Appendix. I n 5 7 we have discussed the Weyl transformation of the angular momentum and thus resolved a pedagogical dilemma which previously obscured the comparison between early quantum mechanics and the Schrodinger description. I n 5 s 8 and 9 we have shown how an integration over the momentum coordinates leads to a local configuration space description of all dynamical quantities. We have then studied the local kinetic and potential energies and shown that the virial theorem is locally satisfied for the ground state of the hydrogen atom. I n forthcoming papers we shall extend the present study by including excited states, and atoms and molecules with more than one electron.

.

We are very grateful to Dr. Sten Rettrup for his kind assistance at an early stage of this work. We also want to thank Dr. Helge Johansen for letting us use his density contour programs and Dr. Kurt Nielsen for interesting discussions. APPENDIX

The Weyl correspondence To every operator a" in spin-free Hilbert space there corresponds a phase space function u(r, p), such that (10) is valid. The relation between d and a(r, p ) is given by the so-called Weyl correspondence [l], which we prefer to write in the form [ll] (A-1)

where fr(r, p) is the inversion operator (1). a^ and a(r, p) are said to be mutual Weyl transforms. Whenever a(r, p) is a function of r or p alone, then d is the same function of the vector operator P or b. In the general case one obtains the operator a" from the function a(r, p) by the replacement of r with P and p with 8, followed by a proper symmetrization of products of non-commuting operators. This symmetrization is such that, if x, and p , are corresponding components of r and p, then the operator associated with the function ~~~p~~ is

(A 2)

These expressions were first derived by McCoy [34].

367

J. P. Dahl and M. Springborg

1018

As an important example let us consider the classical angular momentum vector l = r x P = ( x ~ A - x ~ PxaPi-xiP3, ~, *de-*di)* (A3) Direct substitution does not introduce products of non-commuting operators. The Weyl transform of I is accordingly

i= p

(A 4) which is just the ordinary quantum mechanical angular momentum vector. For the square of an angular momentum component, l3 say, we obtain fi=(22$3-d3$2,

d3$1-%$8,

z32 = X12 $22

+

4$2-’2$1)

- 2Xd,X,P2.

(A 5 ) Symmetrization according to (A 2) gives the corresponding operator which we denote Az2 X22 $12

52=$12 bZ2+ 222$12- Wlbl - t b l 4 ) ( % $ 2

(A 6)

+$232).

Squaring the third component of 1gives, on the other hand, the operator

+

e3z = .El2$22 $22 Using the commutation relation

$12

- 4$1$2i32

[$k, b k l

twice shows that

=ifi

+p i 2 = + a%?, fa= + x2z + i 3 s iz = + + X32 = 132

and hence that

12

where

I2

A12

and

e,2

e22

e2 ,.

-$14d2$2.

(A 7) (A 8)

(A () (A 10) (A 11) (A 12)

Thus, there is a difference of $ti2 between the Weyl transform of l2 and the ordinary quantum mechanical operator P. This difference was apparently first noticed by Shewell [35] in connection with a general discussion of correspondence and symmetrization rules.

REFERENCES [l] WEYL,H., 1931, The Theory of Groups and Quantum Mechanics (Dover). [2] WIGNER, E., 1932, Phys. Rev., 40, 749. [3] GROENEWOLD, H. J,, 1946, Physica, 12,405. [4] MOYAL, E. J., 1949, Proc. Camb. Phil. SOC.math. phys. Sn‘., 45, 99. [ 5 ] BALESCU, R., 1975, Equilibrium and Nonequilibrium Statistical Mechanics (John W i b

& Sons).

161 LOUISELL, W. H., 1973, Quantum Statistical Properties of Radiation (John Wiley Lk Sons). [7] BILLING,G. D., 1978, Thesis, University of Copenhagen. [8] BROWN, R.‘C., and HELLER, E. J., 1981, J. chem. Phys., 75, 186. [9] HUTCHINSON, J, S., and WYATT,R. E., 1981, Phys. Rev. A, 23, 1567. [lo] ESBENSEN, H., WINTHER,A., BROGLIA, R. A., and DASSO, C. H., 1978, Phys. Rea. Lett., 41, 296. [ll] DAHL,J. P . , 1982, Physica A, 112, 439. [12] GROSSMANN, A., 1976, Commun. Math. Phys., 48, 191.

368

Wiper function for the hydrogen atom

1019

[13] ROYER, A., 1977,Phys. Rev. A, 15, 449. [13]DAHL,J . P., 1982,Physica scripta, 25, 499. [15] PODOLSKY, B., and PAULING, L., 1929,Phys. Rev., 34, 109. [16] DAHL,J . P.,1982,Molecular Structure and Energy Scrambling, edited by J . Hinze (Plenum). [17] MCWEENY, R., 1952,Acta crystallogr., 5, 463. [18] SILVERSTONE, H.J., 1968,J . chem. Phys., 48, 4098. [19] LAURENZI, B.J., 1969,Int. J . quant. Chem., 3, 489. [20] MONKHORST, H.J., and HARRIS, F. E., 1972,Int.J. quant. Chem., 6, 601. C., ARRIGHINI, G. P., and MARINELLI, F., 1979,Theor. chim. Acta, 53, 165. [21] GUIDOTTI, [22]MCWEENY, R., 1953,Acta crystallogr., 6 , 631. R. F., 1969,J.chem. Phys., 51, 4569. [23] STEWART, F.B., 1971,Gaussian Basis Sets f o r the Atoms H-Ne for Use in [24] VANDUIJNJWELDT, Molecular Calculations, IBM Technical Report RJ 945. [25] DAHL,J . P., Mat. fys. Meddr., 39, No. 12. [26] AVERY,J., 1978,Acta crystallogr. A, 34, 582. R.L., 1974,Rep. Math. Phys. (Poland), 6 , 249. [27] HUDSON, L.I., 1955,Quantum Mechanics (McGraw-Hill). [28] SCMIFF, L., 1970,General Chemistry, 3rd ed. (W. H. Freeman & Co.), p. 125. [29] PAULING, G. E., and KAC,M., 1977,Phys. Reg., C 32, 1970. [30] ZIFF, R. M., UHLENBECK, [31] COHBN,L.,1979,J. chem. Phys., 70, 788. [32] COHEN,L., 1966,J. math. Phys., 7, 781. P. M., 1972,J. chem. Phys., 56, 3320. [33] BADER,R.F. W., and BEDDALL, 1932,Proc. natn. Acad. Sci. U.S.A., 18,674. [34] McCoy, N.H., J . R., 1959,Am. J . Phys., 27,16. [35] SHEWELL,

369

DISTRIBUTION FUNCTIONS IN PHYSICS: FUNDAMENTALS M. HILLERY Institute for Modern Optics,

!iversity of New L x i c o , Albuquerque, Nh, 8713 U.S.A. and

Max-Planck Znstitut fur Quantenoptik, 0-8046 Garching bei Munchen, West Germany

R.F. O’CONNELL Depamnent of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

M.O. SCULLY Max-Planck Xnstitut fur Quantenoptik, 0-8046 Gatching bei Munchen, West Germany and

Institute for Modern Optics, University of New Mexico, Albuquerque, NM 87131, U.S.A.

E.P. WIGNER

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, L A 70803, U.S.A.

NORTH-HOUND PHYSICS PUBLISHING-AMSTERDAM

370 PHYSICS REPORTS (Review Section of Physics Letters) 106. No. 3 (1984) 121-167. North-Holland, Amsterdam

DISTRIBUTION FUNCTIONS IN PHYSICS: FUNDAMENTALS M . HILLERY Inslitule for Modem Oplics, Uniwrsiry of New Mexico, Albuquerque, NM 87131, U.S.A.

and Max-Planck h.stitu~fur Quantenoptik, 0-8046 Garching bei Munchen, West Germany

R.F. O’CONNELL Department of Physics and Ashonomy. Louisiana Slate Uniuersiiy, Baton Rouge. LA 7W3,U.S.A.

M.O. SCULLY Max-Planrk Imtitut fur Quanienoptik. 0-8046Garching bei Munchen. West Germany

and Institute for Modern Optics. University of New Mexico, Albuquerque. NM 87131, U.S.A.

E.P. WIGNER* Department of Physics and Astronomy, Louisiana Slate Uniwrsity, Baton Rouge, LA 70803, U.S.A.

Received December 1983

Contents:

I . Introduction 2. Wigner distribution 2.1. Properties 2.2. Associated operator ordering 2.3. Dynamics 2.4. An example 2.5. Statistics and second-quantized notation 3. Other distribution functions 4. Distribution functions in terms of creation and annihilation

123 126 126 132 135 142

I46

Normal ordering Symmetric ordering Anti-normal ordering Examples Distribution functionson four-dimensionalphase space 5. Conclusion References 4. I . 4.2. 4.3. 4.4. 4.5.

1%

158 161

162 163 166 166

150

152

operators

* Permanenl address: Department of Physics, Joseph Henry Laboratory, Princeton University, Princeton, NJ 08540, U.S.A. ~~~~

Single orders for this issue

PHYSICS REPORTS (Review Section of Physics Letters) 106, No. 3 (1984) 121-167 Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 29.00, postage included.

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37 1 M.Hillery el al., Distribution functions in physics: Fundamentals

123

Ahslracf:

This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us @PW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.

1. Introduction

It is well known that the uncertainty principle makes the concept of phase space in quantum mechanics problematic. Because a particle cannot simultaneously have a well defined position arid momentum, one cannot define a probability that a particle has a position q and a momentum p, i.e. one cannot define a true phase space probability distribution for a quantum mechanical particle. Nonetheless, functions which bear some resemblance to phase space distribution functions, “quasiprobability distribution functions”, have proven to be of great use in the study of quantum mechanical systems. They are useful not only as calculational tools but can also provide insights into the connections between classical and quantum mechanics. The reason for this latter point is that quasiprobability distributions allow one to express quantum mechanical averages in a form which is very similar to that for classical averages. As a specific example let us consider a particle in one dimension with its position denoted by 4 and its momentum by p. Classically, the particle is described by a phase space distribution Pcl(q,p). The average of a function of the position and momentum A(q,p) can then be expressed as

(1.1) The integrations in this equation are from --03 to +w. This will be the case with all integrations in this paper unless otherwise indicated. A quantum mechanical particle is described by a density matrix p (we will designate all operators by a *) and the average of a function of the position and momentum operators, A(q, p ) is

(1.2) (Tr Q means the trace of the operator 6). It must be admitted that, given a classical expression A(q,p), the corresponding self-adjoint operator A is not uniquely defined-and it is not quite clear what the purpose of such a definition is. The use of a quasiprobability distribution, P&, p ) , however, does give such a definition by expressing the quantum mechanical average as

(1.3) where the function A(q, p) can be derived from the operator A(d,$)by a well defined correspondence rule. This allows one to cast quantum mechanical results into a form in which they resemble classical ones. The first of these quasiprobability distributions was introduced by Wigner [1932a] to study quantum corrections to classical statistical mechanics. This particular distribution has come to be known as the

372 I 3

M. Hillen, er d ,Disrrihuiionfuncrions in physics: Fundanientals

Wigner distribution.? and we will designate it as P,. This is, and was meant to be. a reformulation of Schrodinger’s quantum mechanics which describes states by functions in configuration space. It is non-relativistic in nature because it is not invariant under the Lorentz group; also, configuration space quantum mechanics for more than one particle would be difficult to formulate relativistically. However. it has found many applications primarily in statistical mechanics but also in areas such as quantum chemistry and quantum optics. In the case where Pa in eq. (1.3) is chosen to be P,, then the correspondence between A(9,p ) and A is that proposed by Weyl [1927], as was first demonstrated by Moyal [ 19491. Quantum optics has given rise to a number of quasiprobability distributions, the most well-known being the P representation of Glauber [1963a] and Sudarshan [1963], which have also found extensive use. As far as the description of the electromagnetic field is concerned, these do exhibit (special) relativistic invariance. Other distribution functions have also been proposed (Husimi [ 19401; Margenau and Hill [1961]; Cohen [1966]) but have found more limited use, although, more recently, extensive use has been made of the generalized P representations by Drummond, Gardiner and Walls [1980, 19811. In this paper we will discuss the basic formalism of these quasiprobability distributions and illustrate them with a few simple examples. We will defer any detailed consideration of applications to a later paper. We now proceed to the basic problem: how do we go about constructing a quantum mechanical analogue of a phase space density? Let us again consider, for simplicity, a one particle system in one dimension which is described by a density matrix 6. In this paper we will work, for simplicity, in one dimension; the generalization to higher dimensions will be given in a few cases but is in most circumstances obvious. It is possible to express the position and momentum distributions of the particle as

(1.4a) (1.4b) where S(9 - 4) is the operator which transforms 19’) as follows:

(1.5) and similarly for 6 ( p - P ) , We introduce the function p(q’, 9”) defined by

(1.6) where wA is the probability of the system being in the state

$A,

and the {$A} form a complete set. Then

h1.7a) and Pmo,(p)

=(2h)-’

1dx \ dx’ p ( x , x‘) exp{ip(x’

J

J

- x)/h}.

(1.7b)

t We use this designation here and throughout the paper despite the strenuous objections of one of us since the majority of us feel we should adhere to what is now common nomenclature.

373 M.Hillery el al., Distribution functions in physics: Fundamentals

I25

To show that this corresponds to the usual definition we will examine Pp(q). We have that, in the Dirac bracket notation,

(1.8) which is a more conventional expression for the position density. A first guess for some kind of a phase space density might then be

(1.9) On the other hand, we might choose instead

(1.10) But these expressions are not equal and although either of them, or a combination of both, could be used to evaluate expectation values of functions of 4 and 0 (provided the operators are ordered properly, the ordering for PI being different than that for &), they do not possess what we regard as desirable properties (see section 2). In fact, they are, in general, not real. The association of distribution functions with operator ordering rules (or, equivalently, the association of operators with classical expressions) is one which will recur throughout this paper. Each of the distribution functions which we will discuss can be used to evaluate expectation values of products of operators ordered according to a certain rule. We will consider distribution functions which can be used to compute expectation values of products of the position and momentum operators 4 and 0,and also distribution functions which can be used to compute expectation values of prclducts of the creation and annihilation operators, d’ and d. The latter are useful in problems involving electromagnetic fields. Because the creation and annihilation operators are simply related to 4 and rj there is a relation between these two types of distribution functions. The Wigner distribution, for example, has proved useful in both the 6, 6’ and i, 4 contexts. The basic criterion for the choice of a distribution function for a particular problem is convenience. In the next two sections we will continue to examine distribution functions expressed in terms of both the position and momentum variables. The Wigner function, Pw, will be discussed first in section 2 for not only was it the first quantum mechanical phase space distribution to be considered, but also it satisfies a number of properties which make it quite useful in applications. First of all, we will discuss its properties and then show that Wigner’s distribution function gives the same expectation value for every function of p and q as does the corresponding operator, as proposed by Weyl [1927], for the density matrix which describes the same state to which the distribution function corresponds. As was mentioned before, this was first observed by Moyal [1949]. Next we derive an equation, in many different forms, for the time dependence of P,. Finally, we apply the formalism we have developed to the calculation of P, for the eigenstates of the harmonic oscillator and also for the case of a canonical ensemble of harmonic oscillators at temperature T. In section 3 we discuss distribution functions other than P, which correspond to operator ordering

374 126

M. Hillery ef ul.. Disfrihution func~ionsin physics: Fundumenrals

schemes different from that of Weyl-Wigner. Then in section 4 we treat distribution functions in terms of creation and annihilation operators, with emphasis on normal, symmetric and anti-normal ordering. In particular, we emphasize the normal ordering from which arises the well-known P distribution of quantum optics. We also discuss the generalized P representations. Finally, in section 5 we present our conclusions. Applications will be treated in a future paper but we would be remiss not to mention the recent extensive review of quantum collision theory using phase space distributions (Carruthers and Zachariasan [1983]) and the work on relativistic kinetic theory- in addition to extensive discussions on the Wigner-Weyl correspondence - by the Amsterdam group (Suttorp and de Groot [1970]; Suttorp [1972]; de Groot [1974]; de Groot, van Leeuwen and van Weert [1980]). Also, a brief overview of some applications is presented in O’Connell [2983a,bl. 2. Wigner distribution 2.1. Properties

In a 1932 paper (Wigner [1932a]) the distribution

2.1) was proposed to represent a system in a mixed state represented by a density matrix ii In the case of a pure state, $, it follows from eq. (1.6) that p ( q ‘ , 4”)= $(q‘) $*(q”)and hence (2.2a)

The latter result refers to one dimension. In the case of more than one dimension, the n h must be replaced by (d)-”, where n is the number of the variables of $ (or the number of variables of the rows or columns of 8) and 9, y and p are n-dimensional vectors, with py the scalar product of the two. The integration is then over all components of y. Explicitly, eq. (2.2a) generalizes to

(2.2b) It was mentioned that this choice for a distribution function was by no means unique and that this particular choice was made because it seemed to be the simplest of those for which each Galilei transformation corresponds to the same Galilei transformation of the quantum mechanical wave functions. In later work Wigner [1979] returned to this issue by considering properties which one would want such a distribution to satisfy. He then showed that the distribution given by eq. (2.1) was the only

hhj M.Hillery er al.. Dishibution functions in physics: Fundamentals

127

one which satisfied these properties. A subsequent paper by O’Connell and Wigner [1981a] considered a somewhat different list of properties and showed that these, too, led to the expression in eq. (2.1). The properties for a distribution function, P(q, p), which were considered of special interest, for the case of a pure state (generalization to the case of a mixed state is straightforward), are as follows (O’Connell and Wigner [1981al): (i) P ( q , p ) should be a Hermitean form of the state vector $(q), i.e. P is given by

(2.3) where a ( 4 , p ) is a self-adjoint operator depending on p and q. Therefore, P(q, p) is real. (ii)

(2.4) (2.5) (2.6) (iii) P ( q , p ) should be Galilei invariant, i.e. if $(q)+$(q+ a ) then P ( q , p ) + P ( q + a , p ) and if $h)+exp(ip’qlh) $(q) then P(q, p ) + P ( q , P - P‘). (iv) P(q, p ) should be invariant with respect to space and time reflections, i.e. if $(q)+ $(-q) then P(q, p ) + P ( - q , -P) and if $(4)+$*(q) then P(q,p)+P(q, --PI. It should be admitted, however, that neither of these transformations is relativistic and also that they do not yet involve the spin variable. (v) In the force-free case the equation of motion is the classical one (2.7)

(vi) If P,(q, p ) and P+(q,p) are the distributions corresponding to the states $(q) and 4(q) respectively then

2.8) Property (vi) has two interesting consequences. If we set 444)= $(q) we get

(2.9) and, in the case of a mixed state, the right-hand side of eq. (2.9) is multiplied by ZBw’k where the wk are the probabilities for the different states (the characteristic values of b). This implies that P,(q, p ) is not too highly peaked and rules out such distributions as P,(4, p) = 6(q - 4) S ( p - p) which would be

376 12H

M. Hillery el nl., Distribution funcfions in physics: Fundamentals

possible classically. We can also choose 4 and i,b so that they are orthogonal. We then have that (2.lo)

which implies that P(9, p ) cannot be everywhere positive. This conclusion is actually rather general. Wigner I19791 has shown that any distribution function as long as it satisfies properties (i) and (ii) assumes also negative values for some p and q. (vii)

(2.11) where A(9, p ) is the classical function corresponding to the quantum operator to Wigner’s prescription, by A(q, p ) =

I

dz eip2’*( q - fzl&

A.and is given, according

+ $2)

(2.12)

so that d9 dpA(q, p ) = 2.rrh Tr(A). A similar relation exists between B(9, p ) and 6. The proof of eq. (2.11) will be shown below to follow as a particular case of a more general relation (eq. (2.23)) for F(9,p), in terms of A(9,p) and B(q,p),where E = Ab. From eq. (2.12), it is at once is real if is self-adjoint evident that the phase space description A(q,p) of the operator (Hermitean) and is imaginary if A is skew Hermitean. Since in neither case does A(9,p) vanish, it is evident that if it is real, its operator A is self-adjoint, if it is imaginary A is skew symmetric. It is also evident that the phase space description of the Hermitean adjoint of A is the complex conjugate of the similar description of A. Similarly, if the phase space descriptions of two operators are complex conjugates of each other, then the operators are Hermitean adjoints of each other. By comparison of eqs. (2.1) and (2.12), it is clear that P ( q , p ) , derived from the density matrix, is (2?rh)-’ times the phase space operator which corresponds to the same matrix. Also, for A = 6 and B equal to the unit matrix, eq. (2.6) immediately follows from eq. (2.11). Furthermore, for B = 6, eq. (2.11) reduces to

A

A

a+

(2.13) which is equivalent to eqs. (1.2) and (1.3). This result was originally obtained (Wigner [1932]) for the special case of A being the sum of a function of j j only and a function of 4 only but Moyal [1949] showed it was actually true in the case where A is any function of 9 and jj, if A(9,jj) is the Weyl operator (discussed below in section 2.2) for A(4,p ) . In addition, if we take = B = p^ in eq. (2.11) and use the fact that, if b represents a pure state, Tr@)2= Tr b = 1, we obtain eq. (2.9) again. (viii) If we define the Fourier transform of the wave function

a

d ( p )= (2&)-’

I

d9 $(q) e-iqp’h I

(2.14)

377 M.Hillcry et al., Distribution functions in physics: Fundamenfals

129

then eq. (2.2a) can be re-written in the form P(q, p ) = (?rh)-’

dp’ 4 * ( p + p’) 4 ( p - p’) e-ziw’lh,

(2.15)

exhibiting the basic symmetry under the interchange q t)p. It may be worth observing also that the contraction of the distribution function from n to n - 1 variables

(2.16) gives the distribution function which corresponds to the properly contracted p (in square brackets). Actually, this is true also for the other distribution functions which will be considered in section 3. Wigner in his 1971 paper also showed that properties (i)-(v) determined the distribution function uniquely. OConnell and Wigner [1981a] showed that properties (i)-(iv) and (vi) also accomplish this. In both cases the distribution function was that given by eq. (2.1). Finally, we draw attention to two restrictions on the distribution function discussed above. First of all, as already mentioned, it is non-relativistic. Secondly, not all functions P(q, p) are allowed, as we will now demonstrate by turning to the question of the admissability of P and asking what condition is necessary so that P implies the existence of the density function b, the expectation values of which are, naturally, positive or zero. Our starting-point is eq. (2.2a) from which it follows that dp e-zipy’hP(q, p ) = p(q - Y,q + y) .

(2.17)

Hence, changing variables to u = 4 + y and u = q - y , we obtain p(u, u ) =

I

dp e-ip(u-u)l*P(%u + v), P) .

(2.18)

We remark that since p on the right-side of eq. (2.18) is a dummy variable it is clear that it could be replaced by q. Now the condition for P(q,p) to be a permissible distribution function is that the corresponding

378 130

M.Hi//ery ef a/., Dishburion functions in physics: Fundamenlafs

density matrix be positive definite, i.e.

II II dx

dx' $ * ( x ) p ( x , x') $(x') 1 0

(2.19a)

for all $. Using eq. (2.18) and eq. (2.19a), it follows that the condition that P(9, p) be permissible is that dq

dpP(q, P ) P'(q, P ) 2 0

(2.19b)

for any P'(q, p) which corresponds to a pure state. This is evident already from eq. (2.8). It also follows Eq. (2.19b) holds, of course, for any P' which is itself from eq. (2.11) and the fact that Tr($)?O. permissible but the permissibility of P follows already if it is valid for all P' which correspond to a pure state. Eight properties of the distribution function were discussed above, eqs. (2.3) to (2.16), with the emphasis on the use of this function to form another description of a quantum mechanical state, i.e. be a substitute for the density matrix. Just as eq. (2.1) permits one to give a phase space formulation to the density matrix b, we emphasize that eq. (2.12) permits one also to give a phase space formulation to any matrix -or operator - and it may be useful to consider the properties of eq. (2.12). In particular, we wish to derive an expression for the function F(4, p) which corresponds to the product F' = ad of two operators and B to which the q, p functions A(q,p) and B(q, p) correspond. We assume that the operators and B are matrices, the rows and columns of which can be characterized by a single variable, but the generalization to a many-dimensional configuration space is obvious. We can write, therefore

a a

P(x, x") =

I

a(x, x') f i ( x ' , x") dx' .

(2.20)

Analogous to eqs. (2.17) and (2.18), eq. (2.20) can be written as (taking h

I

dpl F($(x + x"), pl)e-im(x"-x) = (272)-'

II

=

1 for this derivation)

dx' dp' dp" A(&x t x'), p') e-ip'(x'-x)

B ( $ ( ~+' y),p ~ !e-ip"(x"-x') )

(2.21)

Substituting x = q + q', X" = q - q', multiplying with e-zi9'pand integrating over q' one obtains

F(q, p) = 2 (272)-'

I\ I

dq' dx' dp' dp" A($(q + q' + x'), p') B(i(q - q' f x'), p")

x exp{-iq'(2p - pr - p") - i(p"- p') (q - x')}

Introducing finally new variables y

x exp{-4i y'(p =

16 (27r)-'

II1I

-p )

= $(q

.

(2.22)

+ x'), y' = iq', p' = p - p', p" = p + p', one obtains

- 4i p'(q - y)}

dy dy' dp dp' A(q f y f y', p f p - p') B ( q + y - Y'. p + p -t p ' ) e4i@y'+yp') (2.23)

379 M. Hillery ei al., Dbnibulion funcfiom in physics: Fundamenrals

131

This expression for F(q,p), which is a new result, also shows the similarity of the roles of p and q in Hamiltonian mechanics. In the next subsection, another expression (eq. (2.59)) for F(q, p) will be presented . If we integrate F(q, p) in eq. (2.23) over q and p, we obtain

x exp(4iy'pt 4ip'y) (4.rrz/16)6(y') a@') .

(2.24)

Hence

(2.25) Since the 1eft:hand side of this equation is the same as (2n-h) Tr(fi), it is clear that eq. (2.25) is the same as eq. (2.11). In the case of n dimensions, the nth power of (4/.rr*) appears in the expression corresponding to eq. (2.23). Eq. (2.23) provides also a means to ascertain, in terms of the phase space descriptions of and B, whether these two operators commute, Naturally, the condition for the commutative nature is

a

(2.26) Since this is valid for all p and q, the integration over the variables which are their fa!tors i? the exponent (i.e. y' and p') can be omitted. This gives as condition for the commutability of A and B (we replace y, y' by q, q' and p, p' by p, p'):

x exp{4i(q'p + p'q)} = o ,

(2.27)

a somewhat unexpected expression. The last quantum mechanical relation that will be translated into phase space language is the equation Ap^= Ab specifying that the wave functions of which b consists are characteristic functions (eigenfunctions) of with the characteristic value A. Whether p^ contains only one or more such characteristic functions depends whether or not its phase space representation, Pw, satisfies eq. (2.9), i.e. whether its square integral is equal to or smaller than (217fi)-'. The @ , = Ab relation, with represented by Pw, reads, according to eq. (2.23), in phase space language:

A

380 132

M. Hillery el a/.. Distribution funcriom in phyrirs: Fundamenfals

=

(2.28)

A Pw(q, p ) ’

In order to simplify this, one can multiply with exp{4i(q’p t p‘q)} and integrate over p and 9 to obtain, substituting also 9 and p for the integration variables y and p.

(2.29) Both eqs. (2.27) and (2.29) are a good deal more complicated than the quantum mechanical equations for which they substitute. It is questionable whether they are really useful. We thought that they should be derived in spite of this because the final form is considerably simpler than the original one and because they clearly demonstrate the essential phase space equivalence of 9 and p . It may be worth remarking finally that in the case of several dimensions all variables should be considered as vectors, and products like 9’p or p‘9 should be replaced by scalar products of these vectors.

2.2. Associated operator ordering We will now discuss the connection between a classical function of 9 and p and a quantum mechanical operator which is supposed to correspond to it, The result of the measurement of a quantum mechanical operator is well defined: it is supposed to transfer the state of the system on which the measurement is carried out into one of the characteristic vectors of the operator in question, and the probabilities with which the different characteristic vectors would result from the measurement are also well defined. They are the squares of the scalar products of the normalized initial state of the system on which the measurement is carried out and of the operator’s normalized characteristic vector into which the state of the system is transformed. It must be admitted, even in this case, that, given an arbitrary operator, it is in many cases difficult, in others impossible, to construct an apparatus which can carry out the measurement, i.e. the desired change of the state of the system on which the measurement is to be carried out. But as far as the measurement of a classical function of p and 9 is concerned, no similar postulate exists which can be formulated in classical terms. But Weyl did propose the association of a quantum mechanical operator to every function of 9 and p and defined the measurement of the classical quantity as being identical with the above described quantum mechanical measurement of the operator which he associated to the classical function of q and p. This association will be described below. What is remarkable, however, and what has been first pointed out by Moyal [1949], is the close connection between Weyl’s proposal and the distribution function as defined above. In particular, the expectation which Weyl associates with the classical value of the result of the measurement of the operator function A(4, p ) if carried out on a system in the state +,

a,

(2.30)

381 M.Hillery er a/.. Disfribuion functions in physics: Fundarnenrols

133

a

is equal to the expectation value of the classical function A(q,p) to which corresponds assuming that the system is described by the distribution function Pw(4,p) which corresponds to 6. This is the content of eq. (2.30) and it is valid, as will be demonstrated below, for every state vector (I and also €or any density matrix 6

(2.31) Actually eq. (2.31) is an easy consequence of eq. (2.30) and only the latter will be proved below. In order to prove eq. (2.30), we start with Weyl's expansion of A(4,p) into a Fourier integral (taking h = 1 for the purpose of this proof): A(4, p) =

du

(2.32)

d.rcr(u, 7) ei(Oq+'p)

Weyl then defines the operator which corresponds to the exponential in the integrand on the right-hand side of eq. (2.32) as exp(i(mj t $)}. The operator which corresponds to A(q,p) is then given by

&,j)

=

J du J d7 a(u, T ) exp(i(a4 + $)} .

(2.33)

a

If we substitute this result for into the left-hand side of eq. (2.30) and replace A(4,p) on the right-hand side by the right-hand side of eq. (2.32), it becomes evident that all we have to prove is that

(2.34) The integration over p gives 27-r S(y t T ) and hence the right-hand side of eq. (2.34) becomes

In order to evaluate the left-hand side of (2;34) we note that according to-the Baker-Hausdodl theorem (Messiah [1961]), if the commutator d. = [A, B ] commutes with and B then

a

eA+B = eA

es e-m2,

(2.35a)

It then follows that ei(p4+ri3)

= eiud ei'P

eim/12

Hence, the left-hand side of eq. (2.34) becomes

(2.35b)

382 I34

M.Hillery et nl., Distributionjuncrions in physics:

Fundamenrnl

Next, using the fact that

(2.36) and transferring the eid to the left-hand side, this becomes @(x)I@(x+ T ) ) = eim/2(e-iux

@*W@(x+ 7)

1

(2.37)

which is equal to the expression obtained above for the right-hand side of eq. (2.34). Thus, we have proved eq. (2.34) and hence also eq. (2.30). In summary, if a classical function (2.38)

goes over to the quantum operator (2.39)

then the relation between A(q,p) and that if, for all A(p, q )

a is that given by Wigner in eq. (2.12). Furthermore, it is clear (2.40)

then P' is identical with P. In addition, we mention that under the Weyl correspondence the classical quantity qnp" becomes

(2.41) as can be seen by considering the d""" coefficient in (cr4 + TO)"+'". Finally, we would like to mention the role played by the characteristic function. This is a description of the state by means of a function of two new variables, cr and T, C(a,T) = Tr@ &(u,T)) ,

(2.42)

(2.43)

383 M.Hillery CI a/.. Distribution functions in physics: Fundamentals

135

Here we are following the nomenclature of Moyal which has now become standard in describing this quantity as a "characteristic function". This description stems from statistical terminology, and, in particular, should not be confused with the sometime usage of "characteristic function" in quantum mechanics to denote an eigenfunction. C(a,7)is just the Fourier transform of P(q, p). To see this we note that the function corresponding to t ( u ,7)is just exp{(i/fi)(uq + 7p)). Making use of eq. (2.11) gives

(2.44)

so that (2.45)

We can use the characteristic function to compute expectation values of Weyl-ordered products of p and q. We have that (2.46)

the right-hand side of which is just the average of the Weyl-ordered product q"p". 2.3. Dynamics

We would now like to derive equations for the time-dependence of P,. As before, our detailed considerations will be confined to one dimension but some results will also be quoted for the multi-dimensional case. The time-dependence of P, may be decomposed into two parts (Wigner [1932a1) (2.47) the first part resulting from the (ih/2rn) a2/aq2part, the second from the potential energy V/ih part of the expression for a+/at. From the definition of Pw, given by eq. (2.2a), it follows that (2.48) where we have taken advantage of the functional dependence of $ to replace dZ/aqZby a21dyz. Next we perform one partial integration with respect to y to obtain (2.49)

384 136

M. Hillery el a/.. Dishiburion functions in physics: Fundamentals

since the boundary term does not contribute. Switching back from a/ay to a/aq, we finally obtain (2.50) This is identical with the classical (Liouville) equation for the corresponding part of dP/at, as was mentioned at eq. (2.7). We next calculate

(2.5I) Assuming that V can be expanded in a Taylor series, we write V(q + Y) =

25

W 9 )

(2.52)

A =O

where V(")(q)= aAMasA. It follows that

/

-- 2i at

?rh2

dy A

5

V(')(q) $*(q + y) $(4 - y) ezipy'*,

(2.53)

where now the summation over A is restricted to all odd positive integers. It is clear that in the powers y A in the integrand we can replace y by (h/2i)(a/ap). It then follows that (2.54) A again being restricted to odd integers. An alternative form for &PW/at is given by

(2.55) where

(2.55a) is the probability of a jump in the momentum by an amount j if the positional coordinate is 9. The first part of eq. (2.55a) may be verified by inserting the Fourier expansion, with respect to y, of V(q + y) -

385 M. Hi/lery el al., Distribution functions in physics: Fundamentals

137

V(q - y ) into eq. (2.51). The second part is obtained by replacing the exponential by c o s t i sin and noting that the expansion in the square bracket is odd so that the integral of the cos part vanishes. In the multi-dimensional case where P, = P,(ql, . . . 4,; pl, . . .pn), the corresponding results are

(2.56) where the last summation has to be extended over all positive integer values of Al, . . . An for which the sum A 1 t A,+ * t A, is odd. The lowest term of eq. (2.56) in which only one A is 1 and the others vanish, and which has no h factors, is identical with the corresponding term of Liouville's equation. Hence eq. (2.56) reproduces the classical (but non-relativistic) equation if h is set equal to zero. The h 2 terms give the quantum correction if this is very small. We will obtain a somewhat similar equation for the 1/T dependence of the distribution function of the canonical ensemble, which also is useful if the temperature T is not too low so that the quantum correction is small. Eq. (2.56) is the generalization of eq. (2.50) and eq. (2.54) for an n-dimensional configuration space. The same generalization of eq. (2.50) with eq. (2.55) is

-

(2.57) where J(ql,.. . q,; jl, . . .In) can be interpreted as the probability of a jump in the momenta with the amounts jl,. . .j, for the configuration q l , . . . 4.. The probability of this jump is given by

(2.58) that is, by the Fourier expansion coefficientsof the potential V(ql,.. . 4,). From eq. (2.56) it is clear that the equation of motion is the same as the classical equation of motion when V has no third and higher derivatives as, for example, in the case of a uniform electric field or for a system of oscillators. However, there is still a subtle difference in that the possible initial conditions are restricted. This comes about because not all P(q, p ) are permissible (see eq. (2.19b)). While we consider that the above form for the equations of motion (Wigner [1932]) are the simplest to use in practice, we will now discuss some other forms which occur frequently in the literature. Before doing so it is useful to take note of another relation, ip addition to that given by eq. (2.23), which expresses the Weyl function corresponding to an operator F = AB in terms of the Weyl functions corresponding to and B. This relation was first derived by Groenewold [1946] and was also discussed by Imre, Ozizmir, Rosenbaum and Zweifel [1967]. They find that the function corresponding to fi is

a

(2.59)

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M.Hillery el al.. Distribution Junrfions in physics: Fundamentals

where

(2.60) and the arrows indicate in which direction the derivatives act. Also (8/8p)(d89) is considered as the multi-dimensional scalar product of dldp and 8/89, or, in other words, it is equal to (8/dpi)(8/89,), where i = ( l , , . . n) and n denotes the number of dimensions and, as usual, repeated indices denotes summation. To derive this result we first note that

(2.61) where a is defined by eq. (2.32). This result follows from eq. (2.33) by taking the matrix element of both sides. A similar result follows for (q@(q') except that a is replaced by p, the Fourier transform of B(% P): (2.62)

We can now calculate F(9, p). We have from eq. (2.12) that

(2.63) We now define two new variables of integration r = 9' - 9 + (z/2) and r'

=9

- 9' t (z/2) so that (2.64)

It is possible to replace the exponential factor exp{(i/h) (U'T- mf)/2} by exp(hA/2i) so that eq. (2.64) becomes

(2.65) i.e. just the first expression appearing on the right-hand side of eq. (2.59). The second expression also follows readily from eq. (2.64).

387 M. Hillery ef al., Dicbibuh'on funcfionr in physics: Fundamentals

139

We can also make use of eq. (2.64) to find an alternative expression for F(q,p) involving the Bopp operators (Bopp [1961] and Kubo [19641)

Q=q--2i ap '

h a 2i aq '

(2.66)

We first note that (2.67)

so that (2.68) Using this result in eq. (2.64) we then have that (2.69)

(2.70)

is just the Weyl-ordered operator A(4, j)with 9 + Q and jj P. A(Q, P) is also an operator but not on is an operator; it operates on functions in phase space. We can, the Hilbert space on which therefore, express F(q, p) as

&,a)

F(q,p ) = A(Q, P)B(q, P) .

(2.71)

In a similar manner one can show that

(2.72) where

h a

Q * = q t -2i- ,ap

h a P * = P - - 2i - ilq *

(2.73)

It is now possible to make use of the fact that the Wigner distribution is the function which is associated with (1/27rh)i. The equation of motion for C is just

ih ab/a~ = [fi,81.

(2.74)

388 141)

M. Hillery er ul., Disrrihurion functions in physics: Fundainenla1.s

This implies that we have for the Wigner function

or

h aP,/df= -2 H(q,p ) sin(hAl2) Pw(4,p ) ,

(2.75)

where H(q, p ) is the function corresponding to the Hamiltonian operator for the system, fi. Actually, this is an abbreviated form of eq. (2.56) as can be verified by expanding the sin into a power series. Note that if we take the h .-+ 0 limit of this equation we obtain the classical Liouville equation

df :/at + { P.: H}= 0 .

(2.76)

where { } denote Poisson brackets and the superscript c on P, indicates the classical limit. For an H ( q , p ) which is at most quadratic in 4 and p , e.g. a free particle or an harmonic oscillator, eqs. (2.75) and (2.76) coincide. In these systems, then, the difference between a classical and a quantum ensemble is the restriction on the initial conditions in the case of latter (cf. eq. (2.19)). We also want to quote two alternate forms of eq. (2.75). The first follows immediately from our discussion of the Bopp operators. We have, using eqs. (2.65), (2.71), (2.72) and (2.75), that ih aP,/at

=

[H(Q, P ) - f i ( ~ *P *. ) ]~

~ (p )9 ,

(2.77)

a result first obtained by Bopp 119611. Analogous to the definition of A ( 0 , P ) ? given by eq. (2.70). Q and .-+ P, where 0 and P are defined in eq. (2.66). These equations do not exhaust the possible formulations of the dynamics of the Wigner function. One can also make use of propagation kernels. This approach is discussed by Moyal [1949] and Mori, Oppenheim and Ross [1962]. We turn now to a consideration of a canonical ensemble. If /? = l/kT where k is Boltzmann‘s constant and T is the temperature, then the density matrix of the canonical ensemble is

R(Q,P ) is the Weyl-ordered operator with 4

(2.78)

and Z ( p ) = Tr(e-OH). The unnormalized density matrix, d!, then satisfies the equation

ahlap = -Hh = -hH,

(2.79)

subject to the initial condition fi@? = 0) = i where f is the identity operator. Eq. (2.79) is referred to as the Bloch [1932] equation for the density matrix of a canonical ensemble. Using the product rule given by eq. (2.59) we have that

(2.80) A being given by eq. (2.60) so that

389 M.H i l h y el a/., Disfribufionjuncfions in physics: Fundarnenfals

141

(2.81) This is the Wigner translation of the Bloch equation, which was entensively studied by many authors and was first derived in this form by Oppenheim and Ross [1957]. It is useful in the calculation of quantum mechanical corrections to classical statistical mech?nics. The initial condition for this equation is just the Wigner function corresponding to h(p = 0)= I. Inserting I in eq. (2.12) we find that the initial condition is just Jl(q, p)la-o = 1. It is also worth noting that P,(q,p) does not satisfy the Wigner translation of the Bloch equation simply because of the fact that it must be multiplied by the P-dependent factor ( 2 h ) Z ( p ) in order to obtain O(q,p). Finally, we emphasize that all equations from eq. (2.59) onwards hold in the multi-dimensional case, where we simply interpret (4,p) to be (ql, . . . q,; pl, . , .p,) and the simple products in the exponents as scalar products. The solution of eq. (2.81) in the multi-dimensional case, is to order hZ(Wigner [1932a]),

(2.82) Actually, the Wigner translation of the Bloch equation, eq. (2.18) above, can be simplified further into a form, analogous to that of eq. (2.56), which is more convenient for applications. This is achieved by writing the cos term as the real part of the operator

(2.83) where we have used the explicit form for A given in eq. (2.60), again noting that the arrows indicate in which direction the derivatives act and that the gradient operators are 3N-dimensional. Next we decompose 8 by means of the Baker-Hausdorff theorem (eq. (2.35a)), and using the fact that

(2.84) it follows that we may write

(2.85) where we have neglected terms which do not contribute in the present context. Again because of eq. (2.y), and also using the fact that we are only interested in the real part, it follows that the only terms in 0 which contribute are (2.86) where i, j = 1, . . . n (and as usual, it is understood that ( d a q ) ( d d p ) stands for (a/aqi)(slap,)). From henceforth, we wit1 assume that we are dealing with a system of ( 4 3 ) identical particles of mass m.

390 142

M. Hillery el al.. Distriburionfunctions in physics: Fundamentals

Hence, since H = (p2/2m)+ V, it follows that

H O a = exp----ifi a’ a’ V---th Z a’’

{ [

2 39 apl

+?3

2m 892 2m

(2.87)

where it is to be understood that the (&’as)term in the exponential operates only on V and not on R (whereas the p/a9’ term operates on 0).Also, the 8 a p term has no effect on V and thus operates only on 0. Since all arrows now operate to the right, they will be omitted from henceforth so that we finally obtain (2.88a)

(2.88b) where the 8/39 term in the cos and sin terms is to be understood as operating only on V. Such a form was given for the first time by Alastuey and Jancovici [1980] and, in fact, their result also takes account of the presence of a magnetic field. We recall that (3/3p)(8/89) is considered as the multi-dimensional scalar product of 8/8p and 439, or, in other words it is equal to (a/@,) (8/89,)where i goes from 1 to n and n denotes the number of dimensions. Hence, the explicit form of eq. (2.88a) is (2.89) where the last summation is to be extended over all positive integer values, as well as zero values, of A ,, A’, . . . A,,, for which the sum A I t A 2 t * * t An is even. This form for the Wigner translation of the Bloch equation is the most convenient from the point of view of applications. One of the earliest applications of these results was to the quantum corrections of the classical equations of state and to similar corrections to chemical reaction rates (Wigner [1932b, 19381) and they have been extensively used in statistical mechanics (Oppenheim and Ross [1957]; Mori, Oppenheim and Ross [1962]; Nienhuis [ 19701, for example). However, we will defer a detailed discussion of applications to Part I1 of our review, to be published at a later date. 1

2.4. An example

We would now like to use some of the formalism which we have developed to actually calculate some distribution functions. The system which we will consider is the harmonic oscillator and we will consider both pure and mixed states. We will find the Wigner functions corresponding to the eigenstates of the harmonic oscillator and also the function corresponding to a canonical ensemble of harmonic oscillators at temperature T. The eigenstates of the harmonic oscillator are (Landau and Lifshitz [1965])

(2.90)

39 1 M. Hillery er al., Distributionfunctions in physics: Fundamenfah

143

where H n is the nth Hermite polynomial and a = (mw/h)lR. Substituting this expression into the definition of the distribution function, eq. (2.2a), we find that a2

u*.(q+Y) un(q-Y)=

1/2

1

(7) , , , e ~ ~ { - ~ * t ( q + Y ) ~ + ( q - y ) ~ 1 / 2 } ~ n ( a ( q + ~ ) ) . ~ . ( a ( q - y ) ) (2.91)

so that

(2.92) We now note that

a2y2- 2ipylh = a'(y - ipla'h)'

+ p2/aZh2

(2.93)

and define a new variable z = a(y - ipla'h).

(2.94)

We then have that (2.95) where /3 = iplah. Noting the H,(-x)= (-l)"H,(x) we find (2.96) The above integral can be done (Gradshteyn and Ryzhik [1980]) and is

J' dz e-" H,(z + /3 t aq) H,(z t p - aq) = 2" d; n! L,(2(azqz - p')) ,

(2.97)

where L, is the nth Laguerre polynomial. Re-expressing a and p in terms of q and p we have (2.98)

so that (Groenewold [1946]; Takabayaski [1954]; Dahl [1982]) (2.99) Before discussing this result we will first calculate the distribution for an ensemble of oscillators at temperature T (Imre, Ozizmir, Rosenbaum and Zweifel[1967]). Here we proceed by way of the Wigner

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translation of the Bloch equation (eq. (2.88b)) which for this system results in (2.100)

Because V is quadratic in q2 it is clear that only the leading order term in the sin2 expansion will contribute, and since d2V/dq2= mw2, it follows that the Wigner translation of the Bloch equation for the oscillator reduces to

(2.10 1) To solve this equation we make the Ansatz

R(q,P) = exp{-A(P) H t

W)}

(2.102)

where A(0) = B(0)= 0, and H = (p2/2m)t imo’q2. Substituting this into eq. (2.101) gives us

(2.103) This equation can be re-expressed in the form (2.104) Because this equation must hold for all q and p, and the terms in the brackets are independent of q and

p, they must vanish independently, i.e. (2.105)

(2.106) Eq. (2.105) can be integrated directly. One has that dA -1dp 1 - ( h ~ / 2 A’ ) ~-

(2.107)

or

P

=

hw In

[(1 t

A ) / (1 -

A

)I

.

(2.108)

393 M.Hiky et a/., Dishiburion funcfions in physics: Fundamentals

145

Inverting this equation gives us that A@) = (2/hw) tanh(hwp/2).

(2.109)

This can now be substituted into eq. (2.106) to give B

d#l' t a n h ( y ) = -In c o s h ( y )

B @ )= -

.

(2.110)

0

Therefore, we have

L?(q,p) = sech(hwp/2)exp[-(2/hw) tanh(hwpl2) H(q, P)] .

(2.111)

To complete our derivation we need to normalize the above expression. As was noted before the Wigner function is the function which corresponds to the operator @/2?rf)).From eq. (2.78) we then have

. . (2.112)

p ) is just the function corresponding to e-@'. We also have from eq. (2.11) (setting as and B = I)

Z(B) = Tr(e-@') =

a ' I dqI dp O(q,p)

,

a = e-@ (2.113)

Substituting eq. (2.111) into eq. (2.113) we find

Z@)= f[sinh(hwp/2)]-' .

(2.114)

Finally we obtain for Pw(q,p), from eqs. (2.111), (2.112) and (2.114),

Pw(q,P ) = ( l / d ) tanh(hwp/2) exp[-(2/hw) tanh(hwlY2) H(q, p ) ] .

(2.115)

We now want to compare the two expressions (eq. (2.99) and eq. (2.115)) for P, for the pure and mixed states, respectively. Examining the first few Laguerre polynomials

L&)

=1

L,(x)= 1 - x

(2.116)

L,(x) = 1 - 2x + x2

we see that for the ground state of the oscillator Pw(q,p ) > 0 while for excited states P,(q, p) can assume negative values. The result for the canonical ensemble, however, is always positive. It does not have the oscillatory structure which is present in the expressions given by eq. (2.99). The incoherence induced by a finite temperature leads to a much smoother distribution function.

M. Hillery el a/., Distribution functions in physics: Fundamenlals

146

2.5. Statistics and second-quantized noration (Klimintovich i1958); Brittin and Chappell [1962]; Imre, Ozizmir, Rosenbaum and Zweifel [1967]) When one is dealing with more than one particle one has to include the effects of quantum statistics. To illustrate how these effects come in to the Wigner function we will first consider an example. We will then show how the Wigner function can be expressed in second-quantized notation. In this form it is easier to take the effects of statistics into account, but two of us have an article in preparation (O’Connell and Wigner [1983]) which not only will take the effect of statistics into account, but will also include spin effects. Let us consider two identical particles in one dimension in a harmonic potential well. We will further assume that the particles are bosons. The Hamiltonian for the system is

(2.117) Suppose that we want to find the Wigner distribution for a canonical ensemble of these systems at a temperature T. We would again like to use the Wigner translation of the Bloch equation but now we must be more careful; the initial condition is no longer so simple. To see this we first find the density matrix for the system. The eigenstates of the Hamiltonian given by eq. (2.117) are

(2.118) (2.118)

(2.119) (2.119)

(2.120) (2.120)

d~

= 0) =

C

I4nim)

(4ninJ

(2.121) (2.121)

nisn2

Taking matrix elements we find

(2.122) We can now make use of the identity

395 hi.Hillrry el 41.. Distribution f u d m in physics: Fundomenlrrlr

147

(2.123) (2.123) to give

(qL4iIfM = 0)hh q 2 ) = @(q; - 41)

w;-

q2)+

6(q;

-

42)

S@-

9111,

as was to be expected. If we operate on an arbitrary two particle state, I$), with

(2.124) (2.124)

b@= 0) we have that

(2.125 If t+h is symmetric the result on the right-hand side of eq. (2.125) is +, if

+ is anti-symmetric the result is

0.

Therefore, h(j3= 0) is just the projection operator, pssay, onto the state of symmetrictwo-particle wave functions. This result is also true for an arbitrary number ofparticles, N. Our result that d(g = 0) is was derived for bosons. Similarly, if the particles are fermions O(p = 0) is PA,the projection onto the space of anti-symmetric N-particle wave functions, but in this case, the spin variable should also be included. Returning now to our example we want to find the initial condition for the Wigner translation of Bloch equation, i.e. we must find the function corresponding to ts. Making use of the two-particle extension of eq. (2.12) we find

(2.126) The corresponding result for fermions has a minus sign in front of the second term. This initial condition is considerably more complicated than the initial condition, R(q,p) = 1, which was obtained in the one-dimensional case. The situation rapidly becomes worse with larger numbers of particles. Second-quantized notation provides, in principle, a convenient way to deal with the problems imposed by quantum statistics, We will consider a Fock space an! designat! the vacuum state of this space by lo), and the quan$zed field operators at the point r by $+(r) and $(r). The interpretation of the field operators is that $'(r) adds a particle at point r to the system whereas $(r) removes a particle at point r. They are defined as (2.127a) (2.12%)

396 148

M. H i h y et a / . , Distribution funclions in physics: Fundamentab

where the so-called annihilation and creation operators, Cip and id, respectively (discussed in detail in section 4), act to remove or create a particle of momentum p in a box of volume V. For bosons these operators obey the commutation relation (2.128a) (2.128b) and for fermions the anti-commutation relation (2.129a) (2.129b) To every N-particle state IPN)in the Fock space corresponds an N-particle wave function given by (Schweber [1961])

(2.130) then, is given by The distribution function for the state I PN)

(2.131)

(2.132) where, in the case of a pure state,

(2.133) with I@,) denoting the N-particle ket basis vector. An N-particle density matrix has the property that if QN, and QN- are "-particle and N"-particle states respectively, then (@N"(bNl@N.) = 0 unless N ' = N " = N. Therefore, eq. (2.132) can be expressed as

397 M. Hillcry et al., Distribution functions in physics: Fundamentoh

149

(2.134) This is the desired expression for the Wigner function in second-quantized form (Brittin and Chappell [1962]; Imre, Ozizmir, Rosenbaum and Zweifel [1967]). It is also possible to derive expressions for the reduced distribution functions in terms of the quantized field operators (Brittin and Chappell [1%2]; Imre, Ozizmir, Rosenbaum and Zweifel [1967]). The distribution function of order N, reduced to the jth order, is defined as

(2.135) and this definition will be used for the rest of this section. This can also be expressed, by making use of eq. (2.134), as

(2.136)

d3r Ct(r)C(r)= fi,

(2.137)

where N is just the number operator. We then have that, for both bosons and fermions A] =

.

(2.138)

Therefore, (2.139)

and

398 ~

~~

M.Hillery el al., Distribution junctions in physics: Fundamentals

150

(2.140)

x Tr(bN$(rl

+ iyl). . . $(rj

(N- 1) (N- N t j + 1) $(rj - iy,)

t iyi) N

* * *

*

*

*

(2.141)

Because bN is an N-particle density matrix we have that

A(A - 1) . (A- N t j t 1) &rj

. $(rl - 4~1)).

- fyj). . * &(rl- +yl)bN = ( N - j ) ! &rj

- :yj). . &(r,- $yl)fiN,

so that our final expression for the reduced Wigner function is

(2.142)

(2.143) It is now possible to formulate the dynamics of this theory in a way which is independent of the number of particles. We first go to the Heisenberg picture in which the field operators become time dependent. We then consider the operators 1

8(rl, . . . rj;p i , . . . p i ) = (s) J’ d3yl 3J

*

x

.J’ d3yjexp(i(p, - y lt . . t pi -yj)/h}

it(,-, t iy,; t ) . . . ijt(rj + +yj;t ) . . . J(rj - iyj; t ) . .

*

J ( r l - tyl; t ) .

(2.144) (2.144)

The distribution functions for an N-particle theory are then just

(2.145)

6

We see that in this formulation all of the dynamical information is contained in the operators which contain no reference to a specific particle number and also contain the information about the statistics of the particles. Thus, in principle, the second-quantized formalism should be a useful starting-point for the incorporation of statistics into problems involving a system of identical particles. However, it must be admitted that - to our knowledge - no application has been made along these lines.

3. Other distribution functions We now want to examine certain other distributions besides the one considered so far. These may arise out of a desire to make use of an operator ordering scheme other than that proposed by Weyl or a desire to have a distribution function with certain properties. For example, we may want to make use of

399 M.Hillery cf a/., Disbibution funcfionr in physics: Fundamentals

151

symmetric ordering qnPn

+I(2

P

q - m A n+

pqn),

(3.1)

in which case we would use the distribution function (Margenau and Hill [1%1]; Mehta [1964])

(3.2) On the other hand, we may want to consider a distribution which is always greater than or equal to zero. We will discuss a distribution which has this property shortly. A scheme for generating distribution functions was proposed by Cohen [1966] and further examined by Summerfield and Zweifel [1%9]. They give the rather general expression

(3.a) (3.3b) for the distribution function of the pure state $(4), where

(3.4) Thus the function P is simply the original function P,,,smeared with another function g. The basic requirement which leads to eq. (3.3) is that P transform correctly with respect to space displacement, +(q)+ $(q - a), and transition to a uniformly moving coordinate system, $(q) + exp(-imuq) +(q). These requirements were formulated in giving the form eq. (3.3a) to P,-and the satisfaction of the requirements can easily be verified; eq. (3.3b) then follows. Cohen also pointed out that it is possible to obtain distributions whose dependence upon the wave function of the system is other than bilinear simply by choosing g(a, T) to depend upon $(q). For example, one can choose

(3.5) where qo is an arbitrary value of q. This choice for g(a,T) satisfies g(0, T) = g(a,0) = 1 so that the correct marginal distributions are obtained. On the other hand, we now have the rather awkward situation that the function-operator correspondencedepends upon the wave function. An even simpler choice is, of course

(3.6) where d(p),the Fourier transform of $(q) is defined by eq. (2.14). The conditions on g(q, p) which must be satisfied so that the correct marginal distributions are obtained are

400 152

M. Hillery ef al., Drshiburion functions in physics: Fundamenfals

(3.7a) (3.7b) (3.7b) One choice of g(q, p ) which does not satisfy eqs. (3.7) but which is interesting nonetheless is given by

(3.8) The use of this smearing function was first proposed by Husimi [1940] and has been investigated by a number of authors since (Bopp [1956]; Kano [1965]; McKenna and Frisch [1966]; Cartwright [1976]; Prugovecki [1978]; O’Connell and Wigner [1981b]). It leads to a distribution function, pH(4, p ) , where the subscript H denotes Husimi, which is non-negative for all p and q. One can see this by noting that g(q - 9’, p - p’; a ) is just the Wigner distribution function which one obtains from the displaced (in both position and momentum) harmonic oscillator ground state wave function

(3.9) which we will call Pq,p(O’Connell and Wigner [1981b]). If the Wigner distribution in question, P4, corresponds to a wave function 4(9) we have (3.10) (3.10) where we have used eq. (2.8).Note that in order to get a positive distribution function we had to violate condition (ii) on our list of properties of the Wigner function. Property (vi) is also violated as was shown by Prugovecki I19781 and by O’Connell and Wigner [1981a]. We will encounter PH(9,p ) again in the next section in a somewhat different form. It is the “0”or “anti-normally-ordered” distribution function of quantum optics. It is one of a number of distributions which are useful in the description of harmonic oscillators, and, hence, modes of the electromagnetic field. We now proceed to examine these distribution functions.

4. Distribution functions in terms of creation and annihilation operators The harmonic oscillator is a system that is ubiquitous in physics, so that it is not surprising that quantum distribution functions have been developed which are tailored to its description. It is in the description of the modes of the electromagnetic field that these distribution functions have found their widest application. It should be emphasized that many problems in quantum optics require a fully quantized treatment not only of the atoms but also of the field. For example, an analysis of experiments dealing with photon counting or a derivation of the fluctuations in intensity of a laser near threshold both require the quantum theory of radiation (Scully and Lamb [1967]; De Giorgio and Scully [1970]; Graham and Haken

40 1 M.Hillery er 01.. Dishibufionfwctiom in physics: Fundamentals

153

[19701). The latter is developed within the framework of annihilation and creation operators for bosons (see below) but it is then possible to go to a description in terms of c-numbers (while fully retaining the quantum aspects of the situation) by means of distribution functions. In most cases, this greatly facilitates the calculation while, at the same time, it contributes to a better understanding of the connection between the quantum and classical descriptions of the electromagnetic field. A number of studies of these distribution functions have been done (Mehta and Sudarshan [1965]; Lax and Louisell [1%7); Lax [1968]; Cahill and Glauber [19691; Agarwal and Wolf [1970); Louisell [1973]). We will rely most heavily upon the papers by Cahill and Glauber [1969] in our treatment. Their discussion considers a continuum of possible operator ordering schemes, and hence distributions (an even larger class is considered in Agarwal and Wolf [1970]) but we will consider only three of these. A final section will discuss distributions defined on a Cdimensional, rather than a 2-dimensional, phase space. We will describe the system in terms of its annihilation and creation operators (4.la) (4.lb) satisfying

(4.2) As mentioned before, it is assumed that the field operators we consider obey Bose statistics. Each pair of Ci, d' refers to a certain function of position. These functions form an orthonormal set which is countable if the basic domain is assumed to be finite, and continuous if infinite. We deal with a very large, but finite, system so that the system is only approximately relativistically invariant (exact invariance is achieved for an infinitely large system, but this would make the calculation in other ways difficult). The various functions of d and d' are investigated individually because the corresponding 6 and 2' do not interact with the d and d' of another member of the set. They interact with the matter which is in the basic domain. Thus, for example, when we apply this formalism to the case of the electromagnetic field, we investigate each mode (corresponding to a definite momentum and definite direction of polarization) separately, and the operators associated with different modes commute (no interaction between modes). In addition, there will be a distribution function corresponding to each mode. The ci and d' operators act on the basis vectors In),the so-called "particle number states", and have the properties: d In) = d; In - 1)

(4.2a) (4.2b)

d'a In) = n In)

(4.2~)

d 10) = 0 .

(4.2d)

In addition, one can prove that

402 M. Hillery er a/.. Disfribufionfunctions in physics: Fundamenrah

I54

[d. (;+y] = n(h+y-I .

(4.2e)

and if we are If we are considering an oscillator of mass m and angular frequency w we take A = considering a mode of the electromagnetic field of angular frequency w we set A = (h '"w/c). We also want to consider a special class of states known as coherent states (Schrodinger [1926]; Glauber [1963a]; Glauber [1963b]; Sudarshan [1963]; Glauber [1965]). To define these we first define for each complex number a the unitary displacement operator: &)

= e(m5+-m-h) - e-la12/z

ea6+ e-u'h

(4.3)

where the last expression is obtained by use of the Baker-Hausdorff theorem (eq. (2.35a)) and the commutation relation given by eq. (4.2). The operator d(a)has the property that

P ( a )8 B ( a )= 8 t a

(4.4a)

&'(a) 8' D(a)= 8' t a*

(4.4b)

The proof of eq. (4.4) readily follows from eqs. (4.2e) and (4.3). We now define the coherent state (Glauber [1963a]; Glauber [1963b]; Sudarshan [1963]), which we denote by la), as

(4.5) where 10) is the ground state of the oscillator. This state has the property that it is an eigenstate of the annihilation operators with eigenvalue a. Again, this can be verified by using eq. (4.2e). Perhaps it should be emphasized that the symbol a always refers to a complex eigenvalue whereas la) always denotes a state, just as n denotes a real eigenvalue and In) a state, the so-called "number state". Also, just as In) refers to a definite state of excitation of a system of one mode, (a)also refers to a state of one mode. The la) states are not orthogonal but they are complete (in fact overcomplete). Explicitly,

(4.6) which follows immediately from eq. (4.5) and the fact that the number states are orthonormal. Furthermore, it is possible to express the identity operator as (4.7a) where d2a = d(Re a ) d(Im a)= 4da da*. The proof of eq. (4.7a) follows by setting a = r e", so that d2a = r dr do, and then using eq. (4.5) to get

(4.7b)

403 M.H i h y CI a/., Distriburion functions in physics: Fundomenfals

155

where we have used the fact that the angular integral simply equals 27rS,,,,. The latter radial integral equals n!/2, so that using the fact that Z, In) (nl = 1, eq. (4.7a) readily follows. A direct consequence of eq. (4.7a) is that the trace of any operator d is just

1 Tr(d) = - d2a ( a l a l a ) ,

(4.8)

7T

It is also of use to compare the expression for the displacement operat9r &a) to our*previousresults and use this comparison to derive an expansion for a general operator A in terms of D-’(a).This will be of use later. First we note that if we set a = (2h)-“2 (AT t iA-’a) then (see eq. (4.1))

d(a)= exp((i/h)(ad t $1)

=

&,

71,

(4.9)

where was defined earlier by eq. (2.43). Thus from eqs. (2.42) and (4.9) it is clear that the characteristic function is the expectation value of the displacement operator. This in conjunction with eqs. ( 4 3 , (4.6) and (4.8) gives

T&a)

8-’(/3)) = P S0)(a- p ) ,

(4.10)

where the 6 function here is a’(() = S(Re 6 ) 6(Im 6). Suppose that we can expand the operator a(d, a+) as (4.11) Using eq. (4.10) we find that

g(6) = Tr(d

m

*

(4.12)

It can be shown (Cahill and Glauber [1969]) that if d is HilbertSchmidt (i.e. Tr(a+a)< a) then the function g(5) is square integrable. The three types of ordering of the operators a^ and 4’ which we wish to consider are defined as follows: (i)Normal ordering - A product of rn annihilation operators and n creation operators is normally ordered if all of the annihilation operators are on the right, i.e. if it is in the form (ci+)” ci”. (ii) Symmetric ordering - A product of m annihilation operators and n creation operators can be ordered in (n t rn)!/n!m! ways. The symmetrically ordered product of these operators, denoted by {(ci’)” d ” } , is just the average of all of these differently ordered products. For example (4.13a) (4.13b) (4.13~) (iii) Anti-normal ordering - A product of m annihilation operators and n creation operators is

404 I Sh

M.Hillery

er ol.. Disrribution funcfiom in physics: Fundamenrols

anti-normally ordered if all of the annihilation operators are on the left, i.e. if it is of the form i”’(6+)”. For each operator ordering we have a rule which associates a function of a and a * with a given operator. The rule is as follows: for any operator ordering scheme the product of m annihilation and n creation operators, ordered according to that scheme, is associated with the function (a*)”am.For example, if we are considering normal ordering the product ( d ” ) ” i m is associated with ( , * ) “ a m ;if anti-normal ordering is being considered then i m ( d + > , is associated with a m ( a * ) ”We , will now make the meaning of our rule more explicit by considering each of these orderings and its associated distribution function. 4.1. Normal ordering

Let us suppose that we can expand a given operator

a(&,d ’ ) in a normally ordered power series (4.14)

Let us further suppose that we can express the density matrix as (4. IS)

where P(a) is a c-number and the state la) is given by eq. (4.5). P(a) is called the P-representation of the density matrix (or the distribution function representing the density matrix) of the particular mode under study. It should be emphasized that both the real and imaginary parts of a are used as the variables of the distribution function. Also, it is probably worthwhile mentioning again that our discussion is restricted to a system of bosons and thus the distribution functions under study are not applicable to, for instance, a gas of neutrinos. Also, we are dealing with a very large but countable set since we assumed that the basic domain is finite. From eqs. (4.10) and (4.7a) and because ( ~ l h ’ ~ 6 “ ’ l=a a*na”‘, ) it follows that

=

I

d2aP(a)

[2

n.m = O

c,,(a*)”a”

1=

d2aP(a)AN(a,a * ) ,

(4.16)

with (4.17) Therefore, we associate the operator A(Ci,ci’) with the function AN(a,a*) in the evaluation of expectation values with the P-representation, We now want to derive two expressions for P ( a ) in terms of the density matrix. It is not always possible to find a useful representation of b of the form given by eq. (4.15). For some density matrices P ( a ) would have to be so singular that it would not even be a tempered distribution (Cahill [1965]; Klauder and Sudarshan [1968]). This difficulty will be apparent in our formal expression for P(a).

405 157

M.Hilkry ef al., Dism'bution functions in physics: Fundamentals

Let us now choose for the operator A

a=

e-owa

(4.18)

The corresponding function is then AN(a,a*)= exp([a* - [*a).Inserting these expressions into eq. (4.16) we find that

x&) = TrG eta+e-

d2a P ( a )eb"'-t'" .

(4.19)

The function ,yN([) is known as the normally ordered characteristic function. The right-hand side of eq. (4.19) is just a Fourier transform in a somewhat disguised form. In fact one has that if

(4.20a) then

f ( [ )=

d2a e f u ' - f ' u f ( a ) ,

(4.20b)

and vice versa. Therefore, we have for P ( a )

4

P ( a )= P

I

d2[ e"t'-"'fx,(~).

(4.21)

The problem with this expression is that xN([) can grow rather rapidly. In fact we have that because exp([d+ - [*S) is unitary

IxN(()I= elf12n1Tr@efa+-f*d)I5 e1f12/2,

(4.22)

which suggests the type of behavior which is possible. For example, if p = In)(nl, where In) is the eigenstate of the number operator with eigenvalue n, then for large ( 1 we have IxN([)l ([I2". This representation, then, is not appropriate for all density matrices, but, nonetheless, is useful in many of the cases of interest. Finally, we will derive an expression for P ( a ) in terms of a series expansion for the density matrix. Let us suppose that we can express the density matrix as an anti-normally ordered series

-

m

p=

2

pnrndrn(d+)n.

(4.23)

n.m - 0

If we again consider the expression for

A(& Ci+)

given by eq. (4.14) we find that (4.24)

The trace in eq. (4.24) can be expressed as

406 158

M.Hillery

er a/.. Dismbution functions in physics: Fundamenlals

so that

c m

1 Tr(&d)=;Id%

m

C pnmcrsa*nama*rcxr.

(4.25)

n.m=O r,s=O

Comparing this with eq. (4.16) we see that I

(4.26) n.m =O

The difficulties which we had when considering eq. (4.21) suggest that we will have similar problems with eq. (4.26). In fact the problem goes back to eq. (4.23). The class of operators for which a meaningful anti-normally ordered expansion exists is highly restricted. One can see this by considering the representation for an operator given by eq. (4.11). Expand &I(() = exp((*d) exp(-@+) exp($l[l’) in an anti-normally ordered power series and insert it back into eq. (4.11). This gives us an anti-normally-ordered power series for A :

a= c m

dn,dm(i+)n,

(4.27)

n.m=O

with the coefficients given by dnm= -n ! m ! 7~

d26Tr(AD(6)) e1612/2(-~)”(~*)m

(4.28)

For these coefficients to exist Tr(Ah(6)) must be a very rapidly decreasing function of 161. Our previous remarks indicate that this will not be true in general for Hilbert-Schmidt opTr!tors and, in fact, will not be true in general for operators of trace class (operators, A,for which Tr([AtA]”’) < m) such as density matrices. It should be mentioned that normally-ordered power series expansions are far better behaved. A derivation similar to the one above gives for the coefficients cnm in eq. (4.14)

c,,

=

liL n! m . IT

d2[ Tr(Ab(6)) e-lft2/’(-6)”

(r*)”

(4.29)

This clearly exists for a much wider class of operators than does dnm.The cnm’s exist, in fact, for all Hilbert-Schmidt operators and the series converges in the sense that if one takes its matrix element between two coherent states, (a1 on the left and Ip) on the right, the resulting series converges to ( aIAIP).

4.2. Symmetric ordering Before proceeding with a discussion of the distribution function for this case we would like to consider a few properties of the ordering scheme itself. We first note that

407 M.Hillery et a/., Distribution junctions in physics: Fundamentals

(&ti+t (zci)"

=

2

5Y-I)

5:

1-0

(;) {(6+)n-lCi'},

159

(4.30)

which implies that (4.31) Our operator-function correspondence is now done in a way analogous to that of the preceding section. Expand an operator a(6,4') in a symmetrically ordered power series

a=

bn, ((6')" d " } .

(4.32)

n.m =O

The function corresponding to the operator is then m

brim (a*)"a m .

A&, a * )=

(4.33)

n.m = O

Under this correspondence we see from eq. (4.30) that the function b(6)goes to

(4.34) Comparison with eq. (4.9) shows us that this is nothing other than Weyl ordering expressed in a different form. The distribution function, therefore, should be the Wigner function. As before we define this as the Fourier transform of the characteristic function x(T) (see eqs. (2.42)-(2.45)) and we use the real and imaginary parts of a = a, t iai as the variables of the distribution function, so that, analogous to eq. (4.1), (Y = (2h)-'" (Aq t (i/A)p), where A = (rno)ln.Thus

(4.35) where (4.36) It may be verified, using eqs. (2.42), (2.45) and (4.35) that

(4.37)

408 IMI

M. Hillery et al.. Distribution functions in physics: Fundamentals

where, in the derivation of the last line from the previous line, we have used eq. (2.90) and where

p = (rnw/h)'" and (nl@lrn)= $:$,,. Examination of eqs. (4.11) and (4.34) shows us that the function A , which corresponds, by eqs. (4.32) and (4.33), to the operator 4+) can also be represented as

a(&

d2t Tr(Ab(5))eE'a-cu.'.

(4.38)

We would now like to use this to show that Tr@) = -!- d2a A,(a, a *) W ( a )

(4.39)

IT

Evaluating the right-hand side we see that IT

I

'I I

d2aA.(a, a * ) W ( a )= 7 IT d2a d2t Tr(ab(5)) ef'a-co' W ( a )'

(4.40)

Making use of the relation

(4.41)

(4.42) We also have from eqs. (4.11) and (4.36)

(4.43) so that

-1 1

7r

d2a As(qa * ) W ( a )=

;I 1

d25Tr(a&))X(-t)

= Tr(&),

(4.44)

which proves eq. (4.39) and shows that Ar(a,a * ) and W ( a )can be used to calculate the expectation values of symmetrically ordered operators. We would also like to say a word about symmetrically ordered power series. Comparison of eqs. (4.11) and (4.31) allows us to calculate the coefficients appearing in eq. (4.32)

I

bnm= dZ[ Tr(ab(6)) (-5)" ([*)"' . n!rn!v

(4.45)

409 161

M Hillery el al., Distribution functions in physics: Fundarnenfals

These coefficients, then, will exist for all operators which have the property that all moments of Tr(&)) are finite. While this behavior is not as good as that for a normally ordered power series it is certainly better than that of anti-normally ordered series. It is also of interest to examine the behavior of W(a).First we note that

(4.46) so that ~ ( 5 is ) a square integrable function. As W ( a ) is just the Fourier transform of ~ ( 5 )it too is square integrable. Therefore, W ( a ) is far better behaved than P ( a ) and will exist for all density matrices. It is also possible to express the Wigner distribution in terms of the P representation. If we can represent the density matrix as in eq. (4.15) we then have that

(4.47)

) us, with the use of eqs. (4.35) and (4.37), Taking the Fourier transform of ~ ( 5 gives

(4.48)

4.3. Anti-normal ordering

Let us suppose that we have an operator given by an anti-normally ordered power series as in eq. (4.27). The function corresponding to the of eq. (4.27) is then

a

A,(a, a*)=

A m

a"' (a*)" .

(4.49) (4.49)

n.m - 0

By analogy with our discussion of the P representation (eq. (4.26)) we can then express &, 4') as

a(6,d') =

$

d2aA,(&, a*)la) (a1.

(4.50) (4.50)

We then have that (4.51)

410 162

M.Hillery et al.. Distribution functions in physics: Fundamentals

where we have set (Kano [ 19651) (4.52)

This distribution can also be expressed in terms of a characteristic function

(4.53) We have that (4.54)

so that

(4.55) Again by considering our derivation of the P representation we can derive an alternate expression for A*(&,a*).Examining eq. (4.21) we see that

(4.56) The "function" A.(a, a*)has, of course, all of the singularity problems of the P representation. The distribution function, Q(a),has, on the other hand, no singularity problems at all. It exists for all density matrixes, is bounded, and is even greater than or equal to zero for all a. The problems in this ordering scheme arise in the representation of the operators. As a final remark, we note that all of the distribution functions can be written in terms of the Wigner distribution function (McKenna and Frisch [1966]; Agarwal and Wolf [1970]; Haken [1975]; O'Connell [1983b]), by use of integrals or derivatives. 4.4. Examples

We would now like to calculate Q ( a ) and P ( a ) for a single mode of the radiation field of angular frequency w . The system which we will consider will be a canonical ensemble at temperature T = (kp)-'. Our discussion will follow that given in Nussenzveig [1973]. The density matrix for this system is We first consider the anti-normal distribution function a@). (4.57) "=O

41 1 163

M. Hillery et a/.. Distribution functions in physics: Fundamentals

For Q ( a ) we than have from eq. (4.52)

1

e-BnO)exp[-(aIZ (1 - e-Bmw)] .

= -(I?T

To obtain P ( a ) we make use of our result for Q(a). We first find ~ s = (1- e-@-),

f = x t iy,

(Y

=r

(4.58)

~ (from 6 )eq. (4.54). If we set

t ik

(4.59)

then

x&)

=

/ '\ \

dZae e * - e * u dr

=P

To calculate x&),

e-slolz

/

=7T dr

dk exp(-s(r - iy/s)'-

dk exp{-2i(kx - r y ) - s(rz t k2)}

s(k t ix/s)2}exp{-(x2 t y2)/s} = e-1eI2/s.

(4.60)

given by eq. (4.19), we now use the general relation

(4.61) Therefore, we see that

x&)

(4.62)

= exp(-lfl2(1- s)/s}.

If we set A = (1- s)/s = (8*w - l)-' then from eq. (4.21) we have 1

P ( Q )= 7 =

jdzf e"f""'e

1

e-b12/A

?TA

e-A1e12 = IT2

/ / dx

1 = -(grw- 1)exp[-IaI2 7T

dy exp{2i(kx - ry) - A(x2t y'))

(8"- I)] .

(4.63)

For this system P(a) is a well-behaved function, a Gaussian in fact, and has no singularities. It is even positive definite. Q(a) is also well behaved, but this comes as no surprise. Our general discussion had ensured that this would be the case.

4.5. Distribution functions on four-dimensional phase space We would now like to briefly discuss some distribution functions which are functions on a fourrather than a two-dimensional phase space. The first of these, the R representation, was discussed by

412 164

M.Hillery el a/., Distribution functions in physics: Fundamentals

Glauber in his 1963 paper. It is very well behaved but has found little use in applications. More recently a new class of these distributions, the generalized P representations, has been used to study the photon statistics of various non-linear optical devices [Walls, Drummond and McNeil [ 19811; Drummond and Gardiner [1980]; Drummond, Gardiner and Walls [1981]). The R representation of the density matrix is obtained by using the coherent state resolution of the identity twice. One has

(4.64) where la) is defined in eq. (4.5) and

IP) has the corresponding meaning, and (4.65)

This representation has no singularity problems, Also it exists and is unique for all density matrices provided that R(a*,/3) is an analytic function of a* and P (Glauber [1963b]). It can be used to evaluate normally ordered products. One has 1

((8')" 4") = Tr[b (6')" Ci"] = 7 d2a

d2P exp{-(la12 t ]PI') t p * a } R ( a * ,p ) a m(p*)". (4.66)

The generalized P representations (Drummond and Gardiner [ 19801; Drummond, Gardiner and Walls [198l]) are again functions of two complex variables but are not necessarily defined for all values of these variables. To define these representations we define the operator

(4.67) and an integration measure dp(a, p). It is the choice of this measure which determines the distribution function. We will consider two different choices. The density matrix is then (4.68)

where D is the domain of integration. Normally ordered products are then given by

(4.69) Our first integration measure is dp(a, /3) = d a dp where a and P are to be integrated on some contours C and C' respectively. This gives rise to what is called the complex P representation. Let us consider the case in which C and C' are contours which enclose the origin. One can then show (Drummond and Gardiner [1980]) that if the density matrix is of the form (4.70)

413 M.Hillery et al., Distributionfunctions in physics: Fundamentals

165

where both sums are finite then P ( a , p ) exists and is analytic when neither a nor /3 is 0. Whether P ( a , p ) exists for a general density matrix is not known. The complex P representation is also not unique; if one complex P representation exists for a given density matrix, then an infinite number of representation exist. The second measure which we wish to consider is dp(a, /3) = da2dp2. Because the coherent states are linearly dependent such a representation is not unique. In fact we have encountered one representation of this type already, the R representation. It is possible to choose P(a,P)so that it is real and non-negative (Drummond and Gardiner [1980]), i.e. P(a, p ) = (1/4.rr2)exp{-fla - p*12)(%a + p*)IBI&~ t P*)>.

(4.71)

This representation, the positive P representation, is defined for all density matrices. These two distributions have been used in problems in which non-classical photon states (states which are more like number states than coherent states) are produced. Under these conditions the above defined generalized P representations are better behaved than the original P representation. For example, the P representation corresponding to a density matrix = In) (nl contains derivatives of delta functions up to order 2n. On the other hand, the complex P representation for this state (again defined on two contours C and C’ encircling the origin) is just (Drummond and Gardiner [1980]) P(a, p ) = -(1/4aZ) n! e’@ (llap)””

(4.72)

while the positive P representation is, from eq. (4.70)

P(a,p ) = (1/4.rr2)(l/n!) exd-fla - /3*(’} exp(-f)a t p*I2}If(at p*)P.

(4.73)

Both of these functions are far less singular than the original P representation. The original motivation for the introduction of these generalized P distributions was connected with their practical applicability to the solution of quantum mechanical master equations (Drummond and Gardiner [1980]; Drummond, Gardiner and Walls [1981]). In general, using a coherent state basis, it is possible to develop phase-space Fokker-Planck equations that correspond to quantum master equations for the density operator (Haken [1970]; Louise11 [1973]). From this equation observables are obtained in terms of moments of the P function. However, for various problems, as for example the analysis of recent experiments on atomic fluorescence (Kimble, Dagenais and Mandelt19781) where we are dealing with non-classical photon statistics (Carmichael and Walls [1976]), the Glauber-Sudarshan P function is singular whereas the generalized P function discussed above is not. Also, use of the latter leads to Fokker-Planck equations with positive semi-definite diffusion coefficients whereas the former gives rise to non-positive-definite diffusion coefficients. In particular, the generalized P representations were applied successfully to non-linear problems in quantum optics (two-photon absorption; dispersive bistability; degenerate parametric amplifier) and chemical reaction theory (Drummond and Gardiner [1980]; Drummond, Gardiner and Walls [1981]; Walls and Milburn [1982]). On the other hand, the usefulness of the Wigner distribution in quantum optics has been demonstrated in a paper by Lugiato, Casagrande and Pizzuto [1982] who consider a system of N two-level atoms interacting with a resonant mode radiation field and coupled to suitable reservoirs. The presence of an external CW coherent field injected into the cavity is also included, which allows for the possibility of treating optical bistability (which occurs when a non-linear optical medium, interacting with a coherent driving field, has more than one stable steady state) as well as a laser with injected signal.

414 166

M. Hillery et al.. Distribution functions in physics: Fundamentals

5. Conclusion We have given what we hope is a useful summary of some of the formalism surrounding the use of quantum mechanical quasiprobability distribution functions. To be of use, however, the formalism should either provide insight or convenient methods of calculation. In our next paper dealing with applications we hope to show that this particular formalism does both in that it has proven to be a tool of great effectiveness in many areas of physics.

Acknowledgments R.F.O’C. acknowledges support from the Dept. of Energy, Division of Material Sciences, under Contract No. DE-AS05-79ER10459. He would also like to thank the Max-Planck Institute for Quantum Optics for hospitality, during the summers of 1981 and 1982, at which time part of this work was carried out. M.O.S. acknowledges support from the Office of Scientific Research, under Contract No. AFOSR81-0128-A,

References Aganval, G.S. and E. Wolf, 1970. Phys. Rev. D10,2161. 2187,2206. Alastuey, A. and B. Jancovici, 1980.Physica 102A. 327. Bloch, F.. 1932, Zeits. f. Physik 74,295. Bopp. F., 1956,Ann. Inst. H. Poincare 15. 81. Bopp, F.. 1%1, Werner Heisenberg und die Physik unserer Zeit (Vieweg. Braunschweig) p. 128. Brittin, W.E. and W.R. Chappell, 1962,Rev. Mod. Phys. 34.620. Cahill. K.E.. 1%5, Phys. Rev. 138, 81566. Cahill, K.E. and R.J. Glauber. 1969. Phys. Rev. 177, 1857, 1883. Carmichael, H.J. and D.F. Walls, 1976,J. Phys. B9. 1199. Carruthers, P. and F. Zachariasan, 1983,Rev. Mod Phys. 55, 245. Cartwright, N.D., 1976, Physica 83A, 210. Cohen. L., 1966, J. Math. Phys. 7.781. Dahl, J.P., 1982,Physica Scripta 25,499. De Giorgio, V. and M.O. Scully, 1970,Phys. Rev. A2, 1170. De Groot. S.R.and L.G.Suttorp, 1972,Foundations of Electrodynamics(North-Holland. Amsterdam). De Groot. S.R., 1974, La transformation de Weyl et la fonction de Wigner: une forme alternative de la mkcanique quantique (Les Presses Universitaires de Montrtal, Montreal). De Groot, S.R.,W.A. van Leeuwen and C.G. van Weert. 1980, Relativistic Kinetic Theory (North-Holland, Amsterdam). Drummond, P.D. and C.W. Gardiner. 1980. J. Phys. A13, 2353. Drummond. P.D., C.W. Gardiner and D.F. Walls, 1981,Phys. Rev. A24.914. Glauber, R.J.. 1%3a, Phys. Rev. Lett. 10, 84. Glauber, R.J.,1%3b. Phys. Rev. 131,2766. Glauber, R.J.. 1%5, in: Quantum Optics and Electronics, eds. C. DeWitt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach. New York). Gradshteyn, IS. and M. Ryzhik, 1980, Table of Integrals. Series and Products (Academic Press, New York) p. 838. Graham, R. and H. Haken, 1970, Z. Physik 237,31. Groenewold. H.J., 1946, Physica 12,405. Haken, H., 1970, Handbuch der Physik, Vol. XXV/2c (Springer, Berlin). Haken, H.. 1975. Rev. Mod. Phys. 47.67. Husimi, K.. 1940. Proc. Phys. Math. Soc. Japan 22, 264. Imre, K., E. Ozizmir, M.Rosenbaum and P. Zweifel. 1%7. J. Math. Phys. 8. 1097. Kano. Y.,1965,J. Math. Phys. 6, 1913. Kimble, H.J., M.Dagenais and L. Mandel, 1978, Phys. Rev. A M . 201. Klauder J.R. and E.C.G. Sudarshan, 1968. Quantum Optics (Benjamin, New York) p. 178. Klimintovich, I.L., 1958,Sov. Phys. JETP 6,753.

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Kubo, R.. 1964, J. Phys. Soc. Japan 19,2127. Landau, L.D. and E.M. Lifshitz, 1965. Quantum Mechanics (Pergamon, oxford). Lax, M.,1968, Phys. Rev. 172, 350. Lax, M. and W.H. Louisell, 1%7, 1. Quant. Electron. QE3.47. Louisell, W.H..1973. Quantum Statistical Roperties of Radiation (Wiley. New York). Lugiato. L.A., F. Cassagrande and L. Piuuto, 1982, Phys. Rev. A%, 34%. Margenau, H.and R.N. Hill, 1%1, Prog. Theoret. Phys. (Kyoto) 26.722. McKenna, J. and H.L. Frisch, 1966, Phys. Rev. 145,93. Mehta, C.L., 1964, J. Math. Phys. 5, 677. Mehta, C.L. and E.C.G. Sudarshan. 1%5, Phys. Rev. 138, B274. Messiah, A.. 1%1, Quantum Mechanics (North-Holland. Amsterdam) Vol. 1, p. 442. Mon. H., I. Oppenheim and J. Ross, 1%5, in: Studies in Statistical Mechanics, eds. J. de Boer and G.E. Uhlenheck (North-Holland, Amsterdam) Vol. 1, p. 213. Moyal, J.E., 1949, Proc. Cambridge Phil. Soc. 45.99. Nienhuis, G.,1970. J. Math. Phys. 11, 239. Nussenzveig, H.M., 1973, Introduction to Quantum Optics (Gordon and Breach, New York) p. 71. OConnell. R.F., 1983a. Found. Phys. 13, 83. OConnell, R.F., 1983b, in: Proc. Third New Zealand Symp. on Laser Physics. eds. J.D. Harvey and D.F. Walls (Springer-Verlag. Berlin and New York). Lecture Notes in Physics No. 182. OConnell. R.F.and E.P. Wigner, 1981a. Phys. Lett. @A, 145. OConnell, R.F. and E.P. Wigner, 1981b, Phys. Lett. 85A, 121. OConnell, R.F. and E.P. Wigner. 1983, in preparation. Oppenheim, I. and J. Ross. 1957. Phys. Rev. lW,28. hgovecki. E.. 1978, Ann. Phys. (NY) 110. 102. SchrGdinger, E., 1926. Natunvissenschaften 14, 664. Schweber, S.S..1%1. An Introduction to Relativistic Quantum Field Theory (Harper and Row, New York) pp. 133, 140. Scully, M.O.and W.E. Lamb Jr., 1967, Phys. Rev. 159,208. Sudarshan, E.C.G., 1963, Phys. Rev. Lett. 10,277. Summerfield, G.C. and P.F. Zweifel. 1969, J. Math. Phys. 10,233. Suttorp. L.G. and S.R. de Groot, 1970, Nuovo Cimento 6SA, 245. Takabayashi. T., 1954, Pmg. Theor. Phys. 11.341. Walls,D.F., P.D. Drummond and K.J.McNeil, 1981. in: Optical Bistability. 4 s . C.M.Bowden, M.Ciftan and H.R. Rob1 (Plenum, New York). Walls, D.F. and G.J. Milbum, 1982, in: Proc. NATO AS1 in Bad Windsheim. West Germany, ed. P. Meystre (Plenum, New York). Weyl, H., 1927. Z. Phys. 46,1. Wigner. E.,1932a. Phys. Rev. 40,749. Wigner, E., 1932b. Z. Phys. Chem. 819,203. Wigner, E., 1938. Trans. Faraday Soc. 34.29. Wigner, E.P., 1979, in: Perspectives in Quantum Theory, eds. W. Yourgrau and A. van der Menve (Dover, New York) p. 25.

416

Canonical transformation in quantum mechanics Y. S . Kim Department of Physics and Astronomy. University of Maryland, College Park, Maryland 20742

E. P. Wigner Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544

(Received 23 November 1988; accepted for publication 22 June 1989) The phase-space picture of quantum mechanics and some examples illustrating it are presented. Since the position and momentum are c numbers in this picture, it is possible to introduce the concept of phase space in quantum mechanics. The uncertainty relation is stated in terms of an area element in phase space, whose minimum size is Planck's constant. Area-preserving canonical transformations in phase space are therefore uncertainty-preserving transformations. The wavepacket spread, coherent-state representation, and squeezed states of light are discussed as illustrative examples.

I. INTRODUCTION The present organization of the first-year graduate course in quantum mechanics is largely based on the Schrodinger picture and its applications to atomic and nuclear physics. The first widely accepted textbook on this subject 439

Am. J. Phys. 58 (5), May 1990

was Schiff 's book entitled Quantum Mechanics, whose first edition was published in 1949.' There are now many excellent textbooks, but their basic organization is not significantly different from that of Schiff s first edition. These days, due to many new physical applications, we are led to consider adding to the physics curriculum representations 0 1990 American Association of Physics Teachers

439

417 of quantum mechanics other than the Heisenberg or Schrodinger picture. The phase-space picture of quantum mechanics is a case in Starting from the Shrodinger wavefunction, it is possible to construct a distribution function in phase space in terms of the c-number position and momentum variables. Using this distribution function, we can perform canonical transformations in quantum mechanics. The earliest application of the phase-space distribution function was made in quantum corrections to thermodynamics in 1932.4 Now, this phase-space approach is an important scientific language for many branches of physics, and there are a number of review articles.'-' From the pedagogical point of view, the phase-space picture of quantum mechanics is a useful tool for demonstrating the transition from classical to quantum mechanics. In this picture, it is possible to perform canonical transformations, just as in classical mechanics.',' Canonical transformations in quantum mechanics lead to a more precise picture of the uncertainty relation, particularly for the spreading wave packet." It also allows us to define the uncertainty relation in a Lorentz-invariant manner. ' I In addition, the phase-space picture of quantum mechanics is a practical research tool in modern optics. Coherent and squeezed states are of current interest,"-" and they deserve to be included in the existing quantum mechanics curriculum in their own right. The phase-space picture of quantum mechanics is the simplest language for these states." The group theory of linear canonical transformations in phase space is a very important theoretical tool in many branches of physics.'"*' In particular, the group of homogeneous linear canonical transformations is locally isomorphic to the ( 2 1 )-dimensional Lorentz group." This allows us to study Lorentz transformations while performing canonical transformations in phase space that correspond to physical processes in optics laboratories. I' The purpose of this article is to present some of the features of the phase-space picture that can be easily accommodated in the existing quantum mechanics curriculum. In Sec. 11, we introduce the phase-space distribution function and its general properties. In Secs. 111and IV, we discuss linear canonical transformations in classical rnechanics and in the phase-space picture of quantum mechanics. In Sec. V, the wave-packet spread is shown to be an example of canonical transformations in phase space. In Sec. VI, coherent and squeezed states of light are introduced as minimum-uncertainty states. In Secs. VII and VIII, the coherent and squeezed states are discussed in terms of canonical transformations. It is pointed out in Sec. IX that we can study the Lorentz kinematics of the Thomas precession in terms of canonical transformations of squeezed states.

+

The parameters x andp are c numbers. Therefore, this form is a distribution function defined over the two-dimensional phase space of x and p. We shall call this function the phase-space distribution function or PSD function. The PSD function is real, but not always positive. The PSD function is not a probability distribution function. Nevertheless, it is possible to derive from this function many useful relations in quantum mechanics. It is not unlike the case of the partition function in statistical mechanics. It is very difficult, if not impossible, to give a physical interpretation to the partition function, but it is possible to derive many useful physical quantities from it. We can derive from the PSD function the positive probability distributions in the position and momentum coordinates4-h.22. p ( x , t ) = l$(X,t)12 =

a ( p , t ) = IX(P,t) l 2

=

s

I

W(x,p,t)dp, W(x,p,t)dx,

(3)

wherex(p) is the momentum wavefunction. It is also possible to compute the absolute square of the inner product of two wavefunctions. Let us next consider two wavefunctions $ ( x , t ) and d ( x , t ) . If W,(x,p,t) and W , ( x , p , t ) are the PSD functions for $ ( x ) and d ( x ) , respectively, then4-6.?3 W , ( x , p , t )W, (x,p,t)dxdp = (1/237) l ( d ( x , t ) , $ ( x , t ) ) 1 2 . (4) This expression is nonnegative, but can become zero if the two wavefunctions are orthogonal to each other, indicating that the PSD functions are not positive everywhere in phase space. For instance, the one-dimensional harmonic oscillator wavefunction takes the form

(5) for the ground and first-excited states, respectively. Then, the corresponding PSD functions are

~ , ( x , p= ) (1/r)e-'"'+P'', W , ( x , p )= ( 2 / r " ~ ~ + p ~ - t ) e - ' ~ ' + ~ ' ' . (6) These examples confirm the properties of the PSD function given in Eqs. ( 3 ) and (4). The time-dependent Schrodinger equation leads to the differential equation4"

11. BASIC PROPERTIES OF T H E PHASE-SPACE DISTRIBUTION FUNCTION If t j ( x , t ) is a solution of the Schrodinger equation

. a $ ( x , t ) = -I at 2m

(y $( + V ( x ) $ ( x ) , ax x,t)

(7) (1)

we can construct a function of x , p , and t defined as44

'I

W ( x , p , t )= -

$ * ( x + y , t ) t j ( x - y , t ) e'"dy.

?T

440

Am. J. Phys., Vol. 58, No. 5 , May 1990

(2)

where m is the mass of the particle, and V ( x )is the potential. If the particle is free, the above differential equation becomes

a

- W ( x , p , t )=

at

-

(9L -

-W ( x , p , t ) .

Y . S. Kim and E.P.Wigner

(8) 440

418 In the case of the harmonic oscillator with V ( x ) = Kx2/2, the differential equation becomes

a W(x,p,t). + KX -

We are interested in linear canonical transformations of the form X=a,ix+aig+bi, P = a 2 ~ x a 2 9 b,. (13) The parameters b , and b, are for translations, which are area-preserving canonical transformations. If we do not consider this transformation by setting 6 , = 6, = 0, the above equations represent homogeneous linear transformations. The most familiar linear transformation is the rotation around the origin:

+

(9)

aP There are many other interesting properties of the PSD function, and they are extensively discussed in the literat ~ r e . ~ -In " this article, we are interested only in the fact that canonical transformations are possible in quantum mechanics through the use of the PSD function. The probability density functions in Eq. ( 3 ) clearly indicate that x andp are the position and momentum variables in quantum mechanics. On the other hand, they are c numbers, allowing us to define a function over the phase space of x andp. Does this mean that we can determine the x andp variables simultaneously?The answer to this question is no. The uncertainty principle does not allow us to determine a point in phase space. However, it does not forbid the area element that satisfies (Ax) ( Ap) > 1. Since canonical transformations are area-preserving transformations, they preserve the uncertainty relation. Indeed, we may achieve a deeper understanding of the uncertainty principle through canonical transformations in phase space, which are possible in the phase-space picture of quantum mechanics.

+

- sin(B/2))

cos(B/2)

(x)

We use the angle B/2 instead of B for later convenience. Another area-preserving linear transformation is the squeeze along the x axis:

(15) We are now ready to formulate the group linear canonical transformations in phase space. This group consists of translations, rotations, and squeezes in phase space. The coordinate transformation representing translations X = x + b , , P=p+b,, can be written as

[ U'V ]X P

auav auav ax ap ap ax

Under the canonical transformation X=X(x,p), P = P(X,P),

(11)

the Poisson bracket remains invariant. For one pair of canonical variables, this leads to the condition that the Jacobian determinant be 1:

I- ax I

(16) (16)

(17)

111. LINEAR CANONICAL TRANSFORMATIONS IN CLASSICAL MECHANICS Classical mechanics can be formulated in terms of the position and momentum variables.If Uand Vare two functions of x andp, then their Poisson bracket is defined asx

(14)

The matrix performing the rotation around the origin by 8 / 2 takes the form (cos(:/2) -sin(B/2) 0 (18) R ( 8 ) = sin(B/2) cos(B/2) 0). (18) 0 1 It is possible to define the olar variables r and 4 in phase space, with r = ( x 2 +p2)IRand 4 = tan-'(p/x). The x direction in this case is the 4 = 0 direction. We can thus . write S, (7) = S(4 = 0 , ~ )Then,

I

(19)

aP -

I

aP aP This means that the area element in phase space is preserved under canonical transformations.

+

lcosh(qd2) (cos 4)sinh(v/2) (sin #)sinh(v/2) 0

The elongation along the x axis is necessarily the contraction along thep axis. The squeeze along the direction that makes an angle 4/2 with the x axis is

(sin #)sinh(v/2) cosh (7/2) - (cosd)sinh(7/2) 0

o\

I

0 . 1

(20)

r Since a canonical transformation followed by another one is a canonical transformation, the most general form of 441

Am. J. Phys., Vol. 58, No. 5 , May 1990

the transformation matrix is a product of the above three forms of matrices. We can simplify this mathematics by Y . S . Kim and E.P. Wigner

441

419 using the generators of the transformation matrices. If we use T(bl,bz)for the translation matrix given in Eq. (17), it can be - i ( b , N ,+ b.N.1 (21 1 T(u,v) ~

where

O O i N,O 0 0 ( 0 0 3 .

0 0 0 N2= 0 0 i ( 0 0 3

: :),

Since the PSD function is real and defined over the phase space ofx andp, we can perform area-preserving canonical transformations as in the case of classicalmechanics. Let us consider first linear canonical transformations applicable to a function of x and p. This is then a matter of converting the matrix generators given in Sec. 111, into differential forms. The generators of translations are

a

N I-i-

These generators commute with each other:

[NiJfzI = 0. The rotation matrix is generated by 0 -i/2 0 L=(if

IV. LINEAR CANONICAL TRANSFORMATIONS IN QUANTUM MECHANICS

(22)

N2=

. a

-I--,

aP while rotations around the origin are generated by ax'

(30)

(31) (23)

The squeezes along the x axis and along the direction that makes an angle of 45" are generated by

and

R(0)=,-IeL. (24) These generators satisfy the following commutation relations with N , and N2: [ N , , L1 = ( i / 2 ) N 2 , [ N , , L 1 = ( - i / 2 ) N 1 . (25) Indeed, L, N , , and Nz satisfy the closed commutation relations. They generate the two-dimensional Euclidean group consisting of rotations and translations in two-dimensional spa~e.~~.~~ The squeeze matrix of Eq. (19) can be written as s(0,~) = e-'''Km, where i/2 0 K,=(:

(26)

-?

3.

In addition, if we introduce the matrix K z defined as i/2 0 Kz=(i!2 (27)

: ;),

[ K , , LI =

\K,,L ] = iK,. (28) This set of commutation relations is identical to that for the generators of the ( 2 + 1)-dimensional Lorentz group. 19,2425 The group generated by the above three operators is known also as the symplectic group Sp(2 ),Icz1 and its connection with the Lorentz group has been extensively discussed in the literature. If we take into account the translation operators, the commutation relations become ] ( - i/2)N2, [ K , , N l l= ( i / 2 ) N l , [ K I , N 2= [ K z , N l l= ( i / 2 ) N 2 , [K,,N21 = (i/2)Nl. (29) These commutators together with those of Eq. ( 2 8 ) form the set ofclosed commutation relations (or Lie algebra) of the group of canonical transformations. This group is the inhomogeneous symplectic group in two-dimensional space which is often called ISp(2).'h.Z".2'.26 442

- iL,

a ax

h

N I -- - i - - ,

A

N2=x, (33)

which generates the squeeze along the direction that makes a 45" angle with the x axis, then the matrices L, K , , and K 2 satisfy the following commutation relations: [KI,K21 =

(32) respectively.These operators satisfy the commutation relations for the generators of the group of linear canonical transformations given in Sec. 111. Therefore, we can continue using the matrix formalism of classical mechanics in the phase-space picture of quantum mechanics. Next, let us see why canonical transformations are not discussed in the Schrodinger picture of quantum mechanics. In the Schrodinger picture, the wavefunction is a function ofx orp, but not both. On the other hand, the transformations corresponding to those applicable to the PSD function are possible on the Schrodinger wavefunction. Indeed, the following transformations on $ ( x , t ) , through the definition given in Eq. ( 2 ) ,lead to the transformations given in Eqs. ( 3 0 ) - ( 32) 16:

- iK,,

Am. J . Phys., Val. 58, No. 5, May 1990

h

These operators are all Hermitian in the Schrodinger picture. Therefore, linear canonical transformations in phase space correspond to unitary transformations in the Schrodinger picture of quantum mechanics. The question then is whether these operators satisfy the commutation relations for the generators of the group of linear canonical transformations. The answer to this question is no, but almost yes. They satisfy ail the conputation relations except one. The operators N , and N2 do not commute with each other, while N , and Nz do: A

h

[ N , , N , ]= - i, while [ N , , N , ]= 0. (34) This causes a factor of modulus unity when the translation alongp is commuted with the translation along the x direction, and the group of linear canonical transformations in the Schrodinger picture is not the same as that in classical mechanics. Let us next look at the subgroups. The group generated Y . S. Kim and E. P. Wigner

442

420 by N , and N2 is the translation subgroup of the group of linear canonical transformations in phase space. xhis is an Abelian group. On the other hand, the operators N , and N2 do not form a closed Lie algebra. They have to be supplemented by an identity operator to form a group. This is known as the Heisenberg group. This means that the Heisenberg group is represented in phase space as the translation group. The operators N , , N,, and L form thn txodimexsional Euclidean group. On the other hand, N , , N,, and L need the identity operator to form a group. The operatorsly,, K,, and L generate the group of homogeneous linear transformations. The coxnterpaxts in the Schrodinger picture, unlike the case with N , and N2,satisfy the same set of commutation relations given in Eq. (28). Furthermore, the Lie algebra of Eq. (28) is identical to that of the (2 + 1)-dimensional Lorentz group, where K , and K2are the generators ofboosts along the x andy directions, respectively,while L generates rotations around the z axis. This correspondence allows us to study the group of Lorentz transformations using the phase-space picture of quantum mechanics.

V. WAVE-PACKET SPREADS IN TERMS OF CANONICAL TRANSFORMATIONS Let us consider the wave-packet spread of a free particle with mass m. If the initial momentum distribution is of the form

g(k) =( b / ~ r ) ~ ’ ~ e - ’ ~ ’ ’ ~ , (35) at t = 0, the time-dependent Schrodinger wavefunction becomes tl(x,t) = (b/.n)”4[1/(b+it/m)1”2 - x ’ / 2 ( b + t:/mI

(36) In order to obtain the PSD function, we solve the differential equation for W ( x , p , t )given in Eq. (28) for the free particle. The solution is

IP

Fig. 1. Wave-packet spread in phase space. If the uncertainty is defined in terms of the volume of the error box, the spreading wave packet preserves the uncertainty. On the other hand, in the Schrbdinger picture of quantum mechanics, Ap remains constant, while Ax increases as time progresses. This is commonly known as the wave-packet spread. The phasespace picture gives a more precise description of the uncertainty relation for spreading wave packets. 443

Am. J. Phys., Vol. 58, No. 5, May 1990

W(x,p,t)= W x - pr /m,p,O). (37) If the initial momentum distribution is Gaussian as is given in Eq. (35),

W(x,p,t)= (I/n-)exp{-

[(~-pt/m)~/b+bp~]).

(38) This distribution is concentrated within the region where the exponent is less than 1 in magnitude. This is the region of uncertainty, which these days is called the error We can choose the coordinate system in which b = I . Then, the above PSD function is a circle at t = 0. As time progresses, the circle becomes a tilted ellipse while preserving its area,”.“ as is described in Fig. 1. In the Schrodinger picture of quantum mechanics, as time progresses, the momentum distribution u ( p , t ) remains unchanged, while the spatial distribution p ( ~be) comes widespread. This is called the wave-packet spread. On the other hand, in the phase-space picture of quantum mechanics, the uncertainty is defined in terms of the error box. The volume of the error box, which measures the uncertainty, remains constant. Thus we achieve a deeper understanding of the wave-packet spread in the phase-space picture. The above-mentioned elliptic deformation is a canonical transformation. Indeed, the transformation matrix is

(39) This transformation is generated by G = ( “ i, 0 0 ’ which leads to

(40) (41)

The transformation property in phase space of this shear is illustrated in Fig. 1. The question then is whether the above transformation can be derived from the generators of linear canonical transformations given in Sec. 111. The simplest answer to this question is to note that the generator G can be written as G = K z - L. It is also possible to prove this using explicit matrices. With t / m = 2[sinh(77/2)], the above matrix can be written as

1 2 [ si n h ( ~ / 2 ) ] )- (cos(a/2) - sin(a/2)) (0 1 sin(a/2) cos(a/2) cosh(q/2) sinh(v/2) sinh(qV2) cosh(v/2) (cos(a/2) - sin(a/2)) x sin(a/2) . cos(a/2) ’ (42) where sin a = tanh( 77/2). On the right-hand side, each of the three matrices is a canonical transformation derivable from the generators given in Sec. 111. Thus the left-hand side is a canonical transformation matrix.

VI. MINIMUM-UNCERTAINTY STATES Because the variables x andp are c numbers, it is possible to perform even nonlinear canonical transformations in the phase-space picture of quantum mechanics. Let us considY. S. Kim and E. P. Wigner

443

42 1 er the transformation from the Cartesian coordinate system of q andp into a polar coordinate system, as is indicated in Fig. 2,and the canonical transformation n = i ( q 2+ p 2 - I ) , 4 = tan-'(p/q), (43) with

(alNla) = aa*. (49) In order to see the uncertainty relation associated with the photon number, we note that each number state can be represented by a harmonic oscillator wavefunction:

(44)

where the variable q is not the x variable for spatial displacement. This variable, together with its conjugate p , is related to the photon number n through the canonical transformation of Eq. (43).If we write a and at as

This means that the uncertainty ( h n ) ( A # ) (Ref. 28) is equal to ( Aq) ( A p ) . Thus, in order to study the minimum uncertainty in 4 and n, it is sufficient to study the uncertainty for the q andp variables. In the real world, the relation ( A n ) ( A 4 ) appears as the phase-intensity or phase-number uncertainty relation for nonlocalizable light waves.I' Let us consider the coherentstate representation. If In) is the n-photon state, the coherent state is defined

(45) If a and at are the annihilation and creation operators, respectively, with [a,at] = 1, (46) from this commutation relation, it is possible to define the number operator N = ata, where

N l n ) = nln). (47) The coherent state la) is not an eigensate of this number operator, but is an eigenstate of the annihilation operator, satisfying the eigenvalue equation ala) = ala). (48) The probability of the coherent state la) being in the nphoton stateis (aa*)"/[n! exp(aa*)], whichmeans that the number of photons in the coherent state has a Poisson distribution. The expectation value of the number operator is

IP

(aln) = [ l/(fi2"n!)] "'Ha (q)exp( - q 2 / 2 ) ,

(50)

(51) then Eq. (48) is a first-order differential equation in q, and its normalized solution is12

(qla) = ( 1 / ~ ) 1 / 4 ~Ilrn(a)l'e(q-{Za)'/2. (52) The parameter a is a complex number that can be represented in the two-dimensional complex plane. This wavefunction of the Gaussian form gives the minimum-uncertainty product.'.'' It is possible to change the width of the above Gaussian form without changing the minimality in uncertainty. l4.I5 While it is posible to write the coherent-state representation of Eq. (45)as la) = exp(aat - a*o)lo), (53) we can consider the state I W = S(0la),

(54)

with

S(6)=expC:[6(atat) - 4 * ( a a ) l ) . If 6 is a real number 7,S ( 5 ) can be written as (55) h

The generator of the above form is K ,of Eq. (33) with x replaced by q. This operator expands the width of the Gaussian wavefunction of Eq. (52)by e9". Consequently, the width of the momentum distribution is contracted by e - 9'2. This is an operation of squeeze. What happens if 6 becomes complex? This question can most conveniently be answered in the phase-space picture of quantum mechanics, as we shall see in the following sections.

VII. CANONICALTRANSFORMATIONS OF COHERENT STATES It is straightforward to evaluate the PSD function for the coherent state given in Eq. (52).The result is5-7311.'2

W(q,p)= (l/a)exp[ - ( q - a)* - ( p - b)*I,

Fig. 2. Polar coordinate system in phase space. Since both q and p are c numbers, it is possible to make the canonical transformation given in Eq. (43). The minimum uncertaintv in n and 1 means the same uncertaintv in q and p The coherent or squeeied state is a minimum-uncertainty stite. 444

Am. J. Phys., Vol. 58, No. 5, May 1990

(56)

where a = Re( a) and b = 4 Im(a). This function is concentrated within a circular region described by the equation ( q - a ) * + ( p - b)* = 1. (57) circle is ifcx = 0, then both and b vanish, and the centered around the Origin: 8 +p' = 1. (58) Y.S.Kim and E. P. Wigner

444

422 This is the vacuum or zero-photon state. We can obtain the circle of Eq. (57) by translating the above circle to (a,b). This translation is a canonical transformation. If we multiply a by e””, the circle of Eq. (57) becomes rotated around the origin, and the resulting equation is ( q - a’)2 (p - b ’ ) 2 = 1, (59) where cos(8/2) - sin(8/2))

+

(3

The group of transformations consisting of rotations and translations in the two-dimensional space is called the twodimensional Euclidean g r o ~ p .The ~ ~above . ~ ~rotation preceded by the translation to (a,b) from the origin is represented by cos(8/2) -sin(8/2) CI) (1 0 a ) 0 0 b , (60) (sin(:R) cos(6/2) 0 1 0 0 1 applicable to the column vector (x,p,1) ,as was discussed in Sec. 111. The above form can also be written as

(61) 0

where a’ and b ’ are given in Eq. ( 5 9 ) . However, the circle centered at the origin is invariant under rotations. Therefore, the net effect is the translation

(62) Indeed, every coherent state is a translation of the vacuum state. This two-dimensional Euclidean group is isomorphic to the internal space-time symmetry group of massless particles and has been discussed extensively in the literature.24*25.29 It is interesting to note that we can study this group in terms of canonical transformations of coherent states of light.

VIII. SQUEEZED STATES Squeezed states of light are of current intere~t.’~.’~.~’ They are often called generalized coherent states and are, like coherent states, represented by infinite series.26On the other hand, as is seen in Sec. VII, the coherent state takes an unusually simple form in the phase-space picture of quantum mechanics. We shall therefore continue using this representation for squeezed states. In Sec. VII, the coherent state has a Gaussian distribution in phase space that can be described by a circle. The squeezed state also has a Gaussian distribution in phase space. Unlike the case of coherent states, the distribution for a squeezed state is elliptic. The circle in phase space for the coherent state is linearly deformed in such a manner that the area is preserved. The squeeze along the q direction means that q andp are replaced by ( e - ” / ’ ) q and (e7I2)p, respectively, in the equation for a circle. The result is that the circle in Eq. (57) becomes an ellipse specified by e - n ( q - e e ” ’ 2 a ) 2 + e n ( p - e e - ” ~ 2 b )1.2 = (63) This is a canonical transformation. The squeeze can also be made along an arbitrary direction. The squeeze along the 4/2 direction is given in the matrix form in Eq. (20). If we perform the rotation R ( 8 ) on the circle of Eq. (58) centered around the origin, it remains invariant. If the same rotation is applied to the circle of Eq. (57), which is not centered around the origin, its effect is ( q - a’)’

+ ( p - b’)’ = 1,

(64)

where

t’)

cos(8/2)

= (sin (

12)

- sin(8/2))

cos(e /2 )

(3

As was noted in Sec. 111, if there are no translations, we can represent the transformations using 2 X 2 matrices. Under the squeeze S(B,A), the circle of Eq. (57) becomes a tilted ellipse''^": e - A [ ( q - a ” ) c o s ( ~ / 2 ) (p-b”))sin(Q/2)I2 + e A [ ( q - a ” ) s i n ( 8 / 2 ) - ( p - b ” ) c o s ( 8 / 2 ) I 2 = 1, (65) with

+

t”.)(

cosh(A/2) + (cos B)sinh(A/2) (sin B)sinh(A /2) cosh(A /2) - (cos B)sinh(A /2) (sin B)sinh(A /2) This is of course an area-preserving transformation. Let us next consider repeated squeezes. If we apply S(8,A) after S(0,v)on the circle centered around the origin with (I = b = 0, the net effect is another tilted ellipse which can be obtained from the operation of S(4,g) on the circle, with cash{= (coshT)coshA (sinhT)(sinhA)cosB, =

+ +

(sinO)[sinhA (tanhv)(coshA- l ) c o s e ] (sinhA)cosB + ( t a n h v ) [ l + (coshA- I ) ( c o s ~ ) ~ ]

This means that the resulting ellipse is that of Eq. (65), where 0 and A are replaced by 4 and 6, respectively, with a” = b ” = 0. Thus we are tempted to conclude which leads to that S(B,A)S(O,T) = S ( # , l ) , [S(4,{)]-’S(O,A) S(0,v)= 2 x 2 identity matrix. This is 445

Am. J. Phys., Vol. 58, No.5, May 1990

(66)

notcorrect! The matrix multiplication ofthe left-hand side gives [S(#,O1 ‘S(8,A )S(O,v)= R

(a1,

(67)

where Y.S . Kim and E. P.Wigner

445

423 R

(sin8)[tanh(A/2)][tanh(7/2)]

tan - = 2 1 + [tanh(R/2)][tanh(~/2)](cos8) ’ Since the circle centered at the origin is invariant under rotations, the effect of R ( R ) is the same as the identity transformation. Indeed, if we apply the above three successive squeezes on the circle of Eq. (57) not centered at the origin, the net effect is ( q - ” f ) z + ( P - g ) z = 1, (68) where cos(R/2) - sin(W2))

();

This is clearly a rotation.” Since the group of canonical transformation discussed in this section is locally isomorphic to the (2 1)-dimensional Lorentz group, it is possible to study rotations associated with successive Lorentz boosts, including the Thomas p r e c e ~ s i o n . ~We ’ . ~ shall ~ illustrate this in terms of a simple Lorentz kinematics in Sec. IX.

sented by 2 X 2 matrices, satisfy the same set of commutation relations as that for the (2 + 1)-dimensional Lorentz group applicable to the space of (x,y,t). Therefore, it is possible to carry out calculations of Lorentz transformations using the 2 x 2 matrix representation of Sp(2). 19*34 In fact, a detailed calculation of the Thomas rotation angle has been carried out within the framework of this 2 x 2 repre~entation.~’ In this section, we shall discuss a simpler Lorentz kinematics that contains all the essential features of the Thomas prece~sion.’~ The (2 1)-dimensional Lorentz group consists of boosts along the x and y directions and rotations around the z axis. Indeed, the 2 x 2 rotation matrix R ( 8 ) of Eq. (14) corresponds to the rotation matrix applicable to

+

(x,y,t):

+

IX. KINEMATICS OF THE THOMAS PRECESSION As was noted in Secs. I11 and IV, the generators of ho-

mogeneouscanonical transformations, which can be repre-

I + (cosh/l-l)cosz$ $(1 -coshA)sin(2$) (sinhA)sin4

cos 8 R(e)=(si;e

;),

-sin 8 0

(69)

and there is a correspondence between the squeeze matrix S(A,B) of Eq. (20) and the boost matrix applicable to (x,y,t):

:(l-cosh/l)sin(2$) (sinh/l)cos$ 1 + (coshA- l ) s in Z 4 (sinhA)sin$ cosh A (sinh A)cos $

(70)

, matrix represents a boost along the If 4 = 0, the above matrix becomes a boost matrix along thex direction. If$ = ~ / 2 this y axis. It is indeed possible to establish the relation given in Eq. (67) using this matrix representation. This correspondence allows us to study Lorentz transformations using squeezed light.’6*31.35 As an illustrative example, let us consider the following Lorentz transformations on a particle at rest with the fourmomentum (O,O,rn). We are interested in transforming this particle to a state with the momentum along the direction that makes a 45” angle with the x axis. First, we boost along the x direction with S, (7)= S(0,v). The resulting four-momentum will be rn(sinh v,O,cosh 7). We then boost this four-momentum along they direction with S,,(A) = S(d 2 , A ) . Then, the four-momentum becomes rn[sinh v,(sinhA)cosh 7,(cosh A)sinh 71. In order that this momentum have the same x and y components, A and 7 should satisfy the relation:

(71)

sinh A = tanh 7. The resulting transformation matrix is cosh 9

(72)

Wecanreversetheordertoget thesameresultonthemo[S,(R)Sy(7/)]-’SY(/z)S,(7) =R(fl), mentum by performing the transformation S, (A)S,(,7) on the particle at rest. However, are the transformation with matrices identical to each other? The answer to this question is no: sin R = (tanh 7)’. 446

Am. J . Phys., Vol. 58, No. 5, May 1990

Y . S. Kim and E. P. Wigner

(73)

446

424 It would be very difficult, if not impossible, to carry out this experiment in special relativity. On the other hand, this experiment is possible in optics laboratories. For squeezed states, we can use the following 2 x 2 matrices for S, (7) andS,(A):

S,(A)

cosh(A /2) =(sinh(A . /2)

sinh(A /2) cosh(A/2)

(74)

Thus S,, ( A ) S , (7)should be

Sy(A)Sx(7) e7/* cosh(A/2) e-”” sinh(A /2) e7”sinh(A/2) e-q/2cosh(A/2) Then these matrices satisfy Eq. (73) with

=(

’Y. S.Kim and E. P. Wigner, “Covariant phase space representation for localized light waves,” Phys. Rev. A 36,1293-1297 (1987). This paper contains an extensive list ofthe paperson applicationsofthe phase-space picture ofquantum mechanics to statistical mechanics, nuclear physics, elementary particle physics, condensed matter physics, atomic and molecular physics, chemical physics, and quantum optics. ‘Herbert Goldstein, Classical Mechanics ( Addison-Wesley, Reading, MA, 1980), 2nd ed. 9N. L. Balazs. “Weyl’s association, Wigner’s function and a6ine geometry,’’Physica A 102,236-254 (1980) and “On the Weyl association of infinitesimal unitary and canonical transformations,” Physica A 109, 3 17-327 ( 198 1). See also M. Moshmsky, “The structure of phase space and quantum mechanics,” in The Proceedings of the First Internatianal Conferenceon the Physics ofSpace, College Park, Maryland, 1986, edited by Y.S. Kim and W. W. Zachary (Springer-Verlag. Berlin, 1987), pp. 314-318.

)

‘

(75)

(76) Each of these 2 X 2 matrices represents a concrete measurable operation in modern optics laboratories. To a coherent state with a = ( a + ib)/,h, we can apply two repeated squeezes, namely, S, (A)S,(7)and S, (A)S,(?). The difference between these two operations is the rotation R ( a),which represents a phase change in a according to Eq. ( 5 6 ) . In principle, this angle can be measured in optics laboratories. If the present pace of development continues in optical t e c h n ~ l o g y ,it~ may ~ . ~ ~be possible in the near future to include this or a similar experiment” in an advanced laboratory course in the physics curriculum.

‘‘Jerzy Kijowski, “On the time operator in quantum mechanics and the Heisenberg uncertainty relation for energy and time,” Rep. Mach. Phys. 6,361-386 (1974); Hai-Wmng Lee, “Spreadingof afree wave packet,” Am. I. Phys. 50,438440 ( 1982); Antoine Royer, “Squeezed states and their Wigner functions,’’ in The Pmceedings of the First International Conferencean the PhysicsofPhaseSpace, CollegePark, Maryland, 1986, edited by Y. S. Kim and W. W. Zachary (Springer-Verlag, Berlin. 1987), pp. 253-257. “Y. S. Kim and E. P. Wigner, “Covariant phase space representation for harmonic oscillators,” Phys. Rev. A 38, 1159-1 167 (1988). ”Edwin Goldin, Wavesand Photons (Wiley, New York, 1982). “John R. Klauder and Bo-Sture S. Skagerstam, Coherent States (World Scientific, Singapore, 1985). I4David Stoler. “Equivalence classes of minimum uncertainty product,” Phys. Rev. D 1, 3217-3219 (1970); Horace P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 222C2243 (1976); D. F. Walls, “Squeezed states of light,” Nature 306, 141-146 (1983).

”For a pedagogical paper on squeezed states, see Richard W. Henry and Sharon C. Glotzer, “A squeezed states primer;’ Am. 1. Phys. 56, 318328 (1988).

“‘D. Han, Y. S. Kim, and Marilyn E. Noz, “Linear canonical transformations of coherent and squeezed states in the Wigner phase space,” Phys. Rev. A37,807-814 (1988). “Vladmir 1. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, 1978). This book is an English translation by K. Vogtman and A. Weinstein of Arnold’s original b w k in Russian, Matematischeskie Mefody Klassicheskoi Mekheniki (Nauka, Moscow, 1974).

‘Leonard I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1949).

‘We would like to thank Professor Bertrand I. Halperin for informing us that he discussed the phase-space distribution function in his quantum mechanics class at Harvard University in 1988. ‘There are textbooks on statistical mechanics containing discussions of the phase-space distribution function. See, for example, R. P. Feynman, Statistical Mechanics (Benjamin/Cummings, Reading, MA, 1972). ‘E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40,749-759 (1932). ’M. Hillery, R. F. OConnell, M. 0.Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Phys. Rep. 106, 121-167 ( 1984); G. S. Agarwal, “Wiper-function description of quantum noise in interferometers,” J. Mod. Opt. 34, 909-921 (1987); D. Han, Y. S. Kim, and Marilyn E. Noz, “Linear canonical transformations in the Wigner phase space 11. Quantitative analysis,” Phys. Rev. A 40, 902912 (1989).

hEugene P. Wigner, ‘*Quantummechanical distribution functions revisited,” in Perspective in Quantum Theory, edited by W. Yourgrau and A. van der Merwe (MIT, Cambridge, MA, 1971 ), pp. 25-36; R. F. OConnell, “The Wigner distribution function-50th birthday,” Found. Phys. 13, 83-93 (1983); P. Carruthers and F. Zachariassen, “Quantum collision theory with phase-space distributions,’’ Rev. Mod. Phys. 55, 245285 (1983); N. L. Balazs and B. K. Jennings, “Wigner function and other distribution functions in mock phase space,” Phys. Rep. 104,347-

391 (1984). 447

Am. I. Phys., Vol. 58, No. 5, May 1990

“Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics (BenjamidCummings, Reading, MA, 1978), 2nd ed. ” Y . S. Kim and Marilyn E. Noz, “Illustrative examples of the symplectic group,” Am. I. Phys. 51,368-375 (1983). ’“Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics (Cambridge U.P., London, 1984). ”Robert G. Littlejohn, “The semiclassical evolution of wave packets,” Phys. Rep. 138, 193-291 (1986). ‘>For the uniqueness condition for W(x,p,t)givingp(x,l) or o ( p , t ) after integration, see R. F. OConnell and E. P. Wigner, *‘Quantummechanical distribution functions: Conditions for uniqueness,” Phys. Lett. 83A, 145-148 (1981).

”For physical applications of this formula, see W. Schleich, D. F. Walls, and J. A. Wheeler, “Area of overlap and interference in phase space versus Wigner pseudoprobabilities,” Phys. Rev. A 38, 1177-1 186 (1988); Y. S. Kim and E. P. Wigner, “Covariant phase space representation and overlapping distribution functions,” Phys. Rev. A 39, 2829 (1989).

24E.Wigner, “On unitary representations of the inhomogeneous Lorentz group,” Ann. Math. 40,149-204 ( 1939); Eugene P. Wigner, “Relativistic invariance and quantum phenomena,” Rev. Mod. Phys. 29,255-268 (1957).

“Y. S. Kim and Marilyn E. Noz, TheoryandApplicatiansof the Poincari. Group (Reidel, Dordrecht, The Netherlands, 1986). “A. M. Perelomov, Generalized Coherent States and Their Applications (Springer, Heidelberg, 1986). Y. S. Kim and E. P. Wigner

447

425 ”Carlton M. Caves, Kip S. Thorne, Ronald W. P. Drever, Vernon D. Sandberg, and Mark Zimmerman, “On the requirement of a weak classical force coupled to a quantum-mechanical oscillator,” Rev. Mod. Phys. 52,341-392 (1980). *‘R. Dirl, P. Kasperkovitz, and M. Moshinsky, “Wigner distribution functions and the representation of a non-bijective canonical transformation in quantum mechanics.” I. Phys. A 21, 1835-1846 (1988). Whilt ‘his nonlinear transformation is possible in the phase-space picture of quantum mechanics in which the variables q, p. n, and are c numbers, it is not possible to construct Hermitian operators for n and 4. See W. F. Louisell, “Amplitude and phase uncertainty relations,’’ Phys. Lett. 7, 60-61 (1963); Leonard Susskind and Jonathan Glogower, “Quantum mechanical phase and time operator,” Physics 1, 49-61 (1964). 291twas shown in Ref. 24 that the internal space-time symmetry group for massless particles is locally isomorphic to the two-dimensional Euclidean group. See D. Han and Y. S. Kim, “The little group for photons and gauge transformations,” Am. J. Phys. 49,348-351 (1981 ); D. Han, Y. S. Kim, and Marilyn E. Noz, “Internal space-time symmetries of massive and massless particles,” Am. J. Phys. 52, 1037-1043 (1984); Y. S. Kim and E. P. Wigner, “Cylindrical group and massless particles,” I. Math. Phys. 28,1175-1179 (1987). “‘For papers on the generation of squeezed states, see Horace P. Yuen and Jeffrey H. Shapiro, “Generation and detection of two-photon coherent states in degenerate four-wave mixing,” Opt. Lett. 4, 334 (1979); R. E. Slusher, L. W. Hollenberg, B. Yurke, J. C Metz, and 1 F Valley, “Observation ofsqueezed states generated by four-wave mixing in an optical cavity,”Phys. Rev. Lett. 55, 2409 (1995); R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691 (1986); Ling-An Wu, H. 1. Kimble, 1. L. Hall, and Huifa Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57,2520 (1986); Mari W. Maeda, Prem Kumar, and Jeffrey H. Shapiro, “Squeezing experiments in sodium vapor,” J.

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Opt.Soc.Am.84, 1501-1513 (1987);SasumuMachidaandYoshihisa Yamamoto, ‘,‘Ultrabroad amplitude squeezing in a semiconductor laser,” Phys. Rev. Lett. 60, 792 (1988). ”D. Han, E. E. Hardekopf, and Y. S. Kim, “Thomas precession and squeezed states of light,” Phys. Rev. A 39, 1269-1276 (1989). ”This rotation is sometimes called the Wigner rotation in the literature. See Refs. 17 and 31. For earlier papers dealing with this rotation in the ( 3 1 )-dimensional Lorentz group, see V. I. Ritus, “Transformations of the inhomogeneous Lorentz group and the relativistic kinematics of polarized states,” Sov. Phys. IETP 13,240-248 ( 1961);A. Chakrabarti, “Applications of the Lorentz transformation properties of canonical spin tensors,”J. Math. Phys. 5, 1747-1755 (1964); D. Han, Y. S. Kim, and D. Son, “Eulerian parametrization of Wigner’s little groups and gauge transformations in terms of rotations in two-component spinors,” J. Math. Phys. 27,2228-2235 (1986) ”D. Han, Y. S. Kim, and D. Son, “Thomas precession, Wigner rotations, and gauge transformations,” Class. Quantum Grav. 4, 1777-1783 (1987). 14Thissituation is quite analogous to the case where the calculations of the three-dimensional rotation matrices can be carried out in the regime of the 2 x 2 Pauli matrices. ”Raymond Y. Chiao and Thomas F. Jordan, “Lorentz-group Berry phase in squeezed light;’ Phys. Lett. A 132, 77-80 (1988). See also Raymond Chiao, ”Lorentz-group Berry phases in squeezed light,” in The Proceedings of the InternafionalSymposium on Spacetime Symmetries in Commemoration of the 500th Anniversary of Eugene Paul Wigner’s Fundamental Paper on the Inhomogeneous Lorentz Group, College Park, Maryland, 1988, edited by Y. S. Kim and W. W. Zachary (North-Holland, Amsterdam, 1989), pp. 327-223. ”Richard E. Slusher and Bernard Yurke, “Squeezed light,” Sci. Am. 285(5), 50-56 (1988). ”M. Kitano and T. Yabuzaki, “On the observation of Lorentz-group Berry phases in polarization optics,” Phys. Lett. A 142, 321-324 (1989).

+

FEYNMAN ON THE MYSTERY OF QUANTUM MECHANICS The question is, how does [the two-slit experiment] really work? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation than I have given... They can give you a wider explanation, in the sense that they can do more examples to show how it is impossible to tell which hole the electron goes through and not at the same time destroy the interference pattern ... But that is just repeating the same thing to drive it in. It is not any deeper; it is only wider. The mathematics can be made more precise; you can mention that they are complex numbers instead of real numbers, and a couple of other minor points which have nothing to do with the main idea. But the deep mystery is what I have described, and no one can go any deeper today.

Richard P. Feynman, The Characferof Physical Law (MIT, Cambridge, 1967)

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Am. J. Phys., Vol. 58, No. 5 , May 1990

Y. S. Kim and E. P. Wigner

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426

Negative probability Richard P. Feynman California Institute of Technology

Some twenty years ago one problem we theoretical physicists had was that if we combined the principles of quantum mechanics and those of relativity plus certain tacit assumptions, we seemed only able to produce theories (the quantum field theories) which gave infinity for the answer to certain questions. These infinities are kept in abeyance (and now possibly eliminated altogether) by the awkward process of renormalization. In an attempt to understand all this better, and perhaps to make a theory which would give only finite answers from the start, I looked into the ‘tacit assumptions’ to see if they could be altered. One of the assumptions was that the probability for an event must always be a positive number. Trying to think of negative probabilities gave me a cultural shock at first, but when I finally got easy with the concept I wrote myself a note so I wouldn’t forget my thoughts. I think that Prof. Bohm has just the combination of imagination and boldness to find them interesting and amusing. I am delighted to have this opportunity to publish them in such an appropriate place. f have taken the opportunity to add some further, more recent, thoughts about applications to two-state systems. Unfortunately I never did find out how to use the frecdorn of allowing probabilities to be negative to solve the original problem of infinities in quantum field theory! It is usual to suppose that, since the probabilities of events must be positive, a theory which gives negative numbers for such quantities must be absurd. I should show here how negative probabilities might be interpreted. A negative number, say of apples, seems like an absurdity. A man starting a day with five apples who gives away ten and is given eight during the day has three left. I can calculate this in two

235

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Richard P. Feynman

steps: 5 - 10 = - 5 ; and -5 + 8 = 3. The final answer is satisfactorily positive and correct, although in the intermediate steps of calculation negative numbers appear. In the real situation there must be special limitations of the time in which the various apples are received and given since he never really has a negative number, yet the use of negative numbers as an abstract calculation permits us freedom to do our mathematical calculations in any order, simplifying the analysis enormously and permitting us to disregard inessential details. The idea of negative numbers is an exceedingly fruitful mathematical invention. Today a person who balks at making a calculation in this way is considered backward or ignorant, or to have some kind of mental block. It is the purpose of this paper to point out that we have a similar strong block against negative probabilities. By discussing a number of examples, I hope to show that they are entirely rational of course, and that their use simplifies calculations and thought in a number of applications in physics. First let us consider a simple probability problem, and how we usually calculate things, and then see what would happen if we allowed some of our normal probabilities in the calculations to be negative. Let us imagine a roulette wheel with, for simplicity, just three numbers: 1, 2, 3. Suppose, however, the operator, by control of a switch under the table, can put the wheel into one of two conditions, A, B, in each of which the probability of 1, 2, 3 are different. If the wheel is in condition A, the probabilities of 1, p I A= 0.3 say, of 2 is p l A = 0.6, of 3 is p 3 A = 0.1. But if the wheel is in condition B, these probabilities are p l B = 0.1, pZB= 0.4, p3B= 0.5, say, as in Table 13.1. Table 13.1 Probability table for roulette wheel with tv'o conditions -~

1

2 3

-

Condition A

Condition B

0.3 0.6 0.1

0.1 0.4 0.5

We use the table in this way: suppose the operator puts the wheel into condition A 7/10 of the time and into B the other 3/10 of the time at random (that is, the probability of condition A, PA = 0.7, and of B, P, = 0.3.),then the probability of getting 1 is Prob. 1 = 0.7 (0.3) 0.3 (0.1) = 0.24, etc. In general, of course, if a are conditions and pi, is a conditional probability (the probability of getting the result i if the condition a holds), we have (pi,= Prob (if a then i)):

+

[1]

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Negative probability

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where Pa are the probabilities that the conditions a obtain, and Piis the consequent probability of the result i. Since some result must occur in any condition, we have:

[2] where the sum is that over all possible independent results i. If the system is surely in some one of the conditions, so if: a

then:

c31

xipi= 1

meaning we surely have some result, in virtue of [23. Now, however, suppose that some of the conditional probabilities are negative; suppose the table reads so that, as we shall say, if the system is in condition B the probability of getting 1 is -0.4 (see Table 13.2). This sounds absurd, but we must say it this way if we wish that our way of thought and language be precisely the same whether the actual quantities pi. in our calculations are positive or negative. That is the essence of the mathematical use of negative numbers - to permit an efficiency in reasoning so that various cases can be considered together by the same line of reasoning, being assured that intermediary steps which are not readily interpreted (like - 5 apples) will not lead to absurd results. Let us see what plB= -0.4 ‘means’ by seeing how we calculate with it. Table 13.2 Probability table with negative probability

1

2 3

Condition A

Condition B

0.3 0.6 0.1

-0.4 1.2

0.2

We have arranged the numbers in the table so that plB p Z B p3a = 1, in accordance with equation [2]. For example, if the condition A has probability 0.7 and B has probability 0.3, we have for the probability of result 1:

+

+

p1 = 0.7 (0.3) + 0.3 (-0.4) = 0.09

which would be all right. We have also allowed pZBto exceed unity. A

429 238

Richard P. Feynman

probability greater than unity presents no problem different from that of negative probabilities, for it represents a negative probability that the event will not occur. Thus the probability of result 2 is, in the same way: p 2 = 0.7 (0.6)

+ 0.3 (1.2) = 0.78

Finally, the probability of result 3 presents no problem for: p 3 = 0.7 (0.1)

+ 0.3 (0.2) = 0.13

The sum of these is 1.00 as required, and they are all positive and can have their usual interpretation. The obvious question is what happens if the probability of being in condition B is larger; for example, if condition B has probability 0.6, the probability of result 1 is negative 0.4(0.3) + 0.6 (-0.4) = -0.12. But suppose nature is so constructed that you can never be sure the system is in condition B. Suppose there must always be a limit of a kind to the knowledge of the situation that you can attain. And such is the limitation that you can never know for sure that condition B occurs. You can only know that it may occur with a limited probability (in this case less than 3/7, say). Then no contradiction will occur, in the sense that a result 1 or 2 or 3 will have a negative probability of occurrence. Another possibility of interpretation is that results 1, 2, 3 are not directly observable but one can only verify by a final observation that the result had been 1, 2 or 3 with certain probabilities. For example, suppose the truly physically verifiable observations can only distinguish two classes of final events. Either the result was 3 or else it was in the class of being either 1 or 2. This class has the probability p 1 + p 2 , which is always positive for any positive P,,P,. This case corresponds to the situation that 1, 2, 3 are not the finally observed results, but only intermediaries in a calculation. Notice that the probabilities of conditions A and B might themselves be negative (for example, PA = 1.3, PB = -0.3) while the probabilities of the results 1, 2, 3 still remain positive. It is not my intention here to contend that the final probability of a verifiable physical event can be negative. On the other hand, conditional probabilities and probabilities of imagined intermediary states may be negative in a calculation of probabilities of physical events or states. If a physical theory for calculating probabilities yields a negative probability for a given situation under certain assumed conditions, we need not conclude the theory is incorrect. Two other possibilities of interpretation exist. One is that the conditions (for example, initial conditions) may not be capable of being realized in the physical world. The other possibility is that the situation for which the probability appears to be negative is not one that can be verified directly. A

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Negative probability

239

combination of these two, limitation of verifiability and freedom in initial conditions, may also be a solution to the apparent difficulty. The rest of this paper illustrates these points with a number of examples drawn from physics which are less artificial than our roulette wheel. Since the result must ultimately have a positive probability, the question may be asked: Why not rearrange the calculation so that the probabilities are positive in all the intermediate states? The same question might be asked of an accountant who subtracts the total disbursements before adding the total receipts. He stands a chance of going through an intermediary negative sum. Why not rearrange the calculation? Why bother? There is nothing mathematically wrong with this method of calculating and it frees the mind to think clearly and simply in a situation otherwise quite complicated. An analysis in terms of various states or conditions may simplify a calculation at the expense of requiring negative probabilities for these states. It is not really much expense. Our first physical example is one in which one usually uses negative probabilities without noticing it, It is not a very profound example and is practically the same in content as our previous example. A particle diffusing in one dimension in a rod has a probability of being at x at time t of ~ ( x , tsatisfying ) aP(x,t)/dt = - a 2 p ( x , ~ ) / d xSuppose 2. at x = 0 and x = 71 the rod has absorbers at both ends so that P(x,l) = 0 there. Let the probability of being at x at t = 0 be given as P(x,O) = f ( x ) . What is P(x,t) thereafter? It is: ~ ( x , t= )

2p,, sin nx exp( -

n2r)

c41

n=l

where p,, is given by: W

f(x) =

1p n sin nx

c51

n=l

or:

CSl The easiest way of analyzing this (and the way used if P(x,t) is a temperature, for example) is to say that there are certain distributions that behave in an especially simple way. If , f ( x ) starts as sin n x it will remain that shape, simply decreasing with time as P-"*'. Any distributionf(x) can be thought of as a superposition of such sine waves. But f'(x) cannot be sin nx if f ( x ) is a probability and probabilities must always be positive. Yet the analysis is so simple this way that no one has really objected for long. To make the relation to our previous analysis more clcar, the

431 240 Richard P. Frvnman various conditions c1 are the conditions n (that is, the index ct is replaced by n). The a priori probabilities are the numbers pn. The conditions i are the positions x (the index i is replaced by x) and the conditional probabilities (these do not satisfy equation [2], for we have particles ‘lost’ off the end of the rod, and the state of being off the rod is not included among the possibilities i) (if n then s at time t ) are: pi#-+ px,n = e-”lt sin nx

Equation [4] is then precisely equation [f], for the probabilities pi of having result n is now what we call P(x,t). Thus equation [4] is easily interpreted as saying that if the system is in condition n, the chance of finding it at x is exp ( - n 2 t ) sin nx, and the chance of finding it in condition n is pn. No objection should be made to the negative values of these probabilities. However, a natural question is: What are the restrictions which ensure that the final probability for the event (finding a particle at x at time r) are always positive? In this case they are simple. It is that the a priori probabilities, although possibly negative, are restricted by certain conditions. The condition is that they must be such that they could come from the Fourier analysis of an everywhere positive function. This condition is independent of what value of x one wishes to observe at time t . In this example, the restrictions to ensure positive probabilities can be stated once and for all in a form that does not depend on which state we measure. They are all positive simultaneously. Another possibility presents itself. It can best be understood by returning to our roulette example. It may be that the restrictions on the conditions A, B which yield a positive probability may depend on what question you ask. In an extreme example, there may be no choice for the p a that simultaneously make all pi positive at once. Thus, although certain restrictions may make probability of result 1 positive, result 3 under these circumstances would have a negative probability. Likewise, conditions ensuring that p 3 is positive might leave p 1 or p 2 negative. In such a physical world, you would have such statements as: ‘If you measure 1 you cannot be sure to more than a certain degrce that the condition is A; on the other hand it will be all right to think that it is certainly in condition A, provided you are only going to ask for the chance that the result is 3.’ For such a circumstance to be a viable theory, there would have to be certain limitations on verification experiments. Any method to determine that the result was 3 would automatically exclude that at the same time you could determine whether the result was 1. This is reminiscent of the situation in quantum mechanics in relation to the uncertainty principle. A particle can have definite momentum, or a definite position in the sense that an experiment may be devised to measure either one. But no experiment can be devised to decide what the momentum is, to error of

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Negative probability

24 1

order Ap, which at the same time can determine that the position x is within Ax unless Ax > h/Ap. It is possible, therefore, that a closer study of the relation of classical and quantum theory might involve us in negative Probabilities, and so it does. In classical theory, we may have a distribution function F(x,p) which gives the probability that a particle has a position x and a momentum p in dx and dp (we take a simple particle moving in one dimension for simplicity to illustrate the ideas). As Wigner has shown, the nearest thing to this in quantum mechanics is a function (the density matrix in a certain representation) which for a particle in a state with wave function $(x) is:

r c71 (If the state is statistically uncertain we simply average F for the various possible wave functions with their probabilities.) In common with the classical expression, we have these properties. 1 F(x,p) is real. 2 Its integral with respect to p gives the probability that the particle is at x: JF(X3P)dPlW = $*(x)$(x)

3

I31

Its integral with respect to x gives the probability that the momentum is p: JF(-w)dx = cp*(P)cp(P) where p(p) is the usual Fourier transform of cp(p) = je-iPx$(x)dx. The average value of a physical quantity M is given by:

c91 I&).

[10] where wM is a weight function depending upon the character of the physicai quantity. The only property it does not share is that in the classical theory F(x,p) is positive everywhere, for in quantum theory it may have negative values for some regions of x,p. That we still have a viable physical theory is ensured by the uncertainty principle that no measurement can be made of momentum and position simultaneously beyond a certain accuracy. The restriction this time which ensures positive probabilities is that the weight functions w,(x,p) are restricted to a certain class - namely, those that belong to hermitian operators. Mathematically, a positive probability will result if wp is of the form:

+

w ( x , ~= ) JX(x - Y / ~ ) ~ + “ ” X * ( XY/2)dY

c113

where X i s any function and X * is its complex conjugate. Generally, if w(x,p) is the weight for the question ‘What is the probability that the

433

242 Richard P. Fe ynman physical quantity M has numerical value m?’, w must be of the form equation [11] or the sum of such forms with positive weights. With this limitation, final probabilities are positive. To make the analogy closer to those previously used, we can take two systems a, b, in interaction, such that measurements on b can provide predictions of probabilities for u. Thus, using the onedimensional case again, we have a two-point correlation function F(x,,p,; xb,pb) defined via an obvious generalization of equation [7] to two variables. This corresponds to the conditional probability p,=. Then if a quantity A4 is measured in 6, the a pfiori probabilities for various Xb,Pb are given by an appropriate w,(x),,&,) (the analogue of Pa in equation [I]). The probability that system ‘a’ has position and momentum xa,puis (the analogue of Pi), then: p(xa,Pa)

=

jF(xa,pa; xb,Pb)w,(xb,Pb)dxbdPb

the analogue of equation [l]. As an example, we may take the strong correlation possible arising from the two-particle wave function 6(xa - xb) which is: F(xa#a;

Xb@b)

= 6(P,

+ Pb)d(Xa - xb)

which means that the particles a, b, have the same position and opposite momenta so that a measurement of b’s position would permit a determination of a’s and a measurement of b’s momentum would determine u’s (to be the opposite). This particular F is entirely positive and classical in its behaviour, so that letting w,(xb,pb) be 6(xa - b) x 6 ( p a - Q ) would not lead to negative probabilities directly, for equation [l] gives P(x,,p,) = 6(x, - b)6(Pa + Q ) in this case, but further use of such a P in subsequent interactions has the danger of producing negative probabilities. We have become quite used to the rules of thought and limitations of an experiment, which ensures that they never arise in quantum mechanics. It is not our intention to claim that quantum mechanics is best understood by going back to classical mechanical concepts and allowing negative probabilities (for the equations for the development of F in time are more complicated and inconvenient than those of $). (The classical equations for F for a particle moving in a potential are: aqx,p,t)lat = - p / m . aFpX

+ vyx)aF/ap

while the quantum equations are: aF(x,t)/at = - p / m . aF/dx

+ JG(x,Q)F(x,p+ Q)dQ

so instead of the momentum changing infinitesimally during an infinitesimal time, A f , it may jump by an amount Q with probability when it is at x:

AtG(x,Q) = Af . 2 Im JeiQ.YV(x+ Y/2)d Y

434 Negutive psobabilitv

243

which is a real. but possibly negative probability.) Rather we should likc t o emphasize the idea that negative probabilities in a physical theory does not exclude that theory, providing special conditions are put on what is known or verified. But how are we to find and state these special conditions if we have a new theory of this kind? It is that a situation for which a negative probability is calculated is impossible, not in the sense that the chance for it happening is zero, but rather in the sense that the assumed conditions of preparation or verification are experimentally unattainable. We may give one more example. In the quantum theory of electrodynamics, the free photon moving in the z direction is supposed to have only two directions of polarization transverse to its motion x,y, When this field is quantized, an additional interaction, the instantaneous Coulomb interaction, must be added to the virtual transverse photon exchange to produce the usual simple:

[12] virtual interaction between two currents, j and j'. I t is obviously relativistically invariant with the usual symmetry of the space j,,jy,jzand time .j, components of the current (in units where the velocity of light is c = I ) . The original starting Hamiltonian with only transverse components does not look invariant. Innumerable papers have discussed this point from various points of view but perhaps the simplest is this. Let the photon have,fotir directions of polarization of a vector .~,y,z,f, no matter which way it is going. Couple the time component with ie instead or e so that the virtual contribution for it will be negative, as required by relativity in equation [12]. For real photons, then, the probability of a t-photon emission is negativc, proportional to -l(fkJi}l2 the square of the matrix element of j , between initial and final states, just as the probability to emit an x photon is + ~(f~,~i)~2. The total probability of emitting any sort of photon is the algebraic sum of the probabilities for the four possibilities:

c131 It is always positive, for by the conservation of current there is a relation of .if and the space Components of j, k, j , = 0 if k, is the fourvector of the photon. For example, if k is in the z direction, kz = w, and k , = k,, = 0 so j , = j , and we see equation [13] is equal to the usual result where we add only the transverse emissions. The probability to cmit a photon of definite polarization e, is (assume e, is not a null vector): - I 0), and another t photon is emitted say independently with probability -p(p > 0), the chance of emitting both is positive (-a)( - 8) = ap > 0. Should we not expect then to see physical emission of two such photons? Yes, but (if these photons are moving in the z direction) there is a probability to emit z photons u and /3 also, and there are four emission states: two t photons with probability +up; two z photons with probability +up; the first L’ and second c probability (+ a)( -p) = - afi and the first t second z with probabilities -ap so again, for total emission rate only the transverse photons contribute. Although it is true that a negative probability for some situations in a theory means that that situation is unattainable or unverifiable, the contrary is not true; namely, a positive probability for a situation does not mean that that situation is directly verifiable. We have no technique for detecting t photons which is not similarly sensitive to i photons, so that we can only always respond to a combination of them. Likewise, no direct test can be made that the two t photons are indeed present without including the additional probabilities of having z photons. The fact for example, that F(x,p) is everywhere positive:

p’lm (exp( -

+ mw’x 2Rw

for the ground state of an oscillator does not mean that for that state we can indeed measure both x and p simultaneously. As another example we will give an analogue of the Wigner function for a spin half system, or other two-state system. Just as the Wigner function is a function of x and p , twice as many variables as in the wave function, here we will give a ‘probability’ for two conditions at once. We choose spin along the z-axis and spin along the x-axis. Thus let f++ represent the ‘probability’ that our system has spin up along the z-axis and up along the x-axis simultaneously. We shall define the quantity f++ for a pure state to be the expectation of $(l cr= a, cry), where ax,cry, and oz are the Pauli matrices. For a mixed state we take an average over the pure state values. Likewise f+- is the expectation of $(I + az - o.r - a,), ,f-+ is the expectation of a(l - az + ox - ay) and f-- is the expectation of 1 z(1 - a= - ax cry). Understanding that this ‘probability’ can be negative, we shall train

+ + +

+

436

Negative probability

245

ourselves to deal with it otherwise as a real probability and thus dispense with the warning quotes hereafter. Analogously f+- is the probability that the spin is up along the z-axis and down along the xaxis (that is pointing in the negative x direction). Likewise f-+ and f - - give the probability that the spin is along the negative z-axis and along the x-axis in the positive or negative sense, respectively. These are all the possible conditions so we have f + + + f+- + f - + f-- = 1. As an example, we might have f + + = 0.6, f+-= -0.1, f-+ = 0.3 and f-- = 0.2. Now the probability that the spin is up along z is simply the sum of the probability that it is up along z and up along x, and the other possibility, that it is up along z but down along x; that is simply j ++ + f +- or 0.6 (-0.1) = 0.5 in our example. The probability the spin is down along z is f-+ + f--,also 0.5. In the same way the probability that the spin is along the positive x-axis, independent of its value along z is f++ f-+ or 0.9. We, of course, cannot measure simultaneously the spin in the z and in the x direction, so we cannot directly determine f +- and there is no difficulty with its negative value. These four numbers give a complete expression of the state of the system, and the probability for any other question you can ask experimentally is some linear combination of them. For example, the probability that a measurement of spin along the y-axis gives ‘up’ is j + ++ f-- or 0.8, and that it gives ‘down’ is f + -+ f-+ or 0.2. In fact, for a two-state system any question is equivalent to the question ‘Is the spin up along an axis in some direction?’ If that direction is defined by the unit vector V with components V,,V,,,V, then we can say the probability that the spin is up along this direction if the condition of the electron is + + is p + + ( V ) = gl + V, + V, + V,). For the other conditions we have p + - ( V ) = 31 + V, - V, - V,), p - + ( V ) = 3(1 - V , V, - V,), and p - - ( V ) = 31 - Vz - V, V,). In the general case then where the fs give the a priori probabilities of each condition the probability of finding the spin up along V is the sum on a of p , ( V ) f , or 1 2((1 + V z + Y x + V J f + + + (1 + V z - v x - V,>f++ (1 - V , + V , - V,)f-+ + (1 - V, - V , + V , ) f - - ) . In order that this always gives positive results, in addition to the condition that the sum of the f s is unity, there is the restriction that the sum of the squares of the four fs be less than 3.It equals f for a pure state. If there are two electrons in a problem we can use classical logic, considering each of them as being in one of the four states, + -, - +, - -. Thus suppose we have two electrons, correlated so their total spin is zero, moving into two detectors, one set to determine if the spin of the first electron is in the direction Vand the other set to measure whether the second electron has its spin in the direction U. The probability that both detectors respond is gl - U.V). Thus if one IS found up along any axis, the other is surely down along the

+ +

+

+

+

+ +,

437

246 Richard P.Feynman same axis. This situation usually causes difficulty to a hidden variable view of nature. Suppose the electron can be in one of a number of conditions a, for each of which the chance of being found to be spinning up along the V-axis is p a ( V ) . If the second electron is in condition 6, its probability of being found along U is ph(U ) . Suppose now that the chance of finding the two electrons in conditions a,b, respectively, is Pab.This depends on how the electrons were prepared by the source. Then the chance of finding them along the V and U axes is ~ , a P a , , pV)pb( a ( U) which is equal to +(1 - U.Y ) . This js well known to be impossible if all the ‘probabilities’ P a b and p are positive. But everything works fine if we permit negative probabilities and use for a our four states with the p,(V) as defined previously. The probabilities for the correlated states in the case that the total spin is zero are Pab equal Q if a and b are different states, and -Q if they are the same. For another example of a two-state system, consider an electron going through a screen with two small holes to arrive at a second screen (see Figure 13.1). We can say there are four ways or conditions by which the electron can go through the holes, corresponding to the + +, + -, - +, and - - conditions. If we take up spin to correspond to going through hole number 1 and down spin to represent going through hole 2, then the other variable corresponding to spin i n the x direction means going through the two holes equally in phase Ordinarily we cannot say which hole it goes through and what the phase relation is (just as ordinarily we do not say which way the zspin is and which way the x-spin is) but now we can and do. For example, f-- gives the probability of going through hole 2 but 180 degrees out of phase (whatever that could mean). For each of these conditions we can calculate what the chance is that the electron arrives at a point x along the screen. For example, P + + ( x ) , the probability for arrival at x for the condition (through 1 in phase) and P + - ( x ) , the probability for + - (through 1 but out of phase) are sketched roughly in Figure 13.1 as the curves (b) and (c) respectively. The independent probabilities are negative for some values of x. The functions through hole 2 are these reflected in x; P- +(x) = P+ +( - x) and P - - ( x ) = P + - ( - x ) . The total chance to go through hole 1, P, + + P+ -, the sum of the two irregular curves shown in the figure, is just the smooth bump, the solid line at (a),with its maximum under hole 1, not showing interference effects. But the total probability to P- -, shows the typical interarrive with holes out of phase, P+ ference pattern at the bottom of the figure at (6). Obviously the particular choice we used for the two-state system is arbitrary, and other choices may have some advantages. One way that generalizes to any number of holes or of states, finite or otherwise, is this. Suppose an event can happen in more than one way, say ways A , B, C, etc., with amplitudes a, 6, c, respectively, so that the probability

++

+

438

Negative urobabilitv 247

ure 13.1 3 A two-state system in which an electron goes through a screene e n hith

two small holes to a a second screen. (a). Te t;a; cjam

through hole 1. (b) The probability of going through hole 1 in phase. (c) The probability of going through hole 1 but out of phase. (d) The total probability to arrive with holes out of phase.

+ + +

of occurring is the absolute square of a b c . . . This can be described by saying the event can happen in two ways a t once. For example we can say that the event happens by ‘coming’ in way A and ‘going’ in way B (or, if you prefer, by ‘looping’ via A and B) with a ‘probability’ P(A,B) = gl i)a*b $(l - @*a, where a* stands for the complex conjugate of a. The probability of ‘coming’ and ‘going’ by the same way A is P(A,A) = a*u and is the conventional positive probability that the event would occur if way A only

+

+

439

248

Richard P. Feynman

were available to it. The total probability is the sum of these P for every pair of ways. If the two ways in P,‘coming’ and ‘going’ are not the same, P is as likely to be negative as positive. The density matrix, pij, if the states are i is then represented instead by saying a system has a probability to be found in each of a set of conditions. These conditions are defined by an ordered pair of states ‘coming’ in i and ‘going’ in j , with ‘probability’ p(ij? equal to the real part of (1 + i)pii. The condition that all physical probabilities remain positive is that the square of p ( i j ) not exceed the product p(i,i)p( j , j ) (equality is reached for pure states). Finally, suppose that, because of the passage of time, or other interaction, or simply a change in basis, the state i has an amplitude Smiof appearing as state m, where S is a unitary matrix (so the new density matrix p’ is given by S ’ p S ) . We then discover we can find the new probabilities p’(rn,n) by summing all alternatives ij of p(i,j) times a factor that can be interpreted as the probability that the state ‘coming’ in i, ‘going’ in j turns into the state ‘coming’ in m, ‘going’ in n. This ‘probability’ is: h Z(SimSjn * sj*nsim) +j<sj*,s, - Si*,Sj,)

+

With such formulas all the results of quantum statistics can lie described in classical probability language, with states replaced by ‘conditions’ defined by a pair of states (or other variables), provided we accept negative values for these probabilities. This is interesting, but whether it is useful is problematical, for the equations with amplitudes are simpler and one can get used to thinking with them just as well. My interest in this subject arose from many attempts to quantize electrodynamics or other field theories with cut-offs or using advanced potentials, in which work apparently negative probabilities often arose. It may have applications to help in the study of the consequences of a theory of this kind by Lee and Wick.

440

EXISTENCE O F STAR-PRODUCTS A N D OF FORMAL DEFORb4ATIONS OF THE POISSON LIE ALGEBRA O F ARBITRARY SYMPLECTIC MANIFOLDS

MARC DE WILDE and PIERRE B.A. LECOMTE

Universite'de Lit?ge, Institut de Mathhatique, 15 Avenue des Tilleuls, 84000, Li2ge, Belgium

ABSTRACT. We prove the existence of star-products and of formal deformations of the Poisson

Lie algebra of an arbitrary symplectic manifold. Moreover, all the obstructions encountered in the step-wise construction of formal deformations are vanishing.

0. INTRODUCTION In the Hamiltonian formulation of classical mechanics, a phase space is nothing else but a symplectic manifold. Passing to quantum theory in the classical way implies a fundamental change in the nature of observables and makes the interpretation of the classical theory as a limit of the new one uneasy in many respects. An important aspect of quantization is its relation to deformations of classical theories. In that spirit, Hato, Lichnerowicz and Sternheimer have proposed building up quantum mechanics on an ordinary phase space in such a way that quantization manifests itself in a deformation of the algebra of observables. The value of the parameter of deformation is closely related to the Planck constant and letting it tend to zero gives back classical mechanics as a limit case. An account of the deformation approach to quantization can be found in (1,6]. The algebra of observables is the space of smooth functions over a symplectic manifold with its natural structure of associative algebra and the appropriate deformations of this structure are called the star-products. The first star-product appeared as the inverse Weyl transform of the product of operators (Moyal [9]).It was rediscovered by Vey [lo] who also proved the existence of nontrivial deformations of the Poisson Lie algebra structure for a symplectic manifold with a vanishing third De Rham cohomology group. The result was extended to associative deformations by Neroslavsky and Vlassov [8] under the same assumption. In the mean time, various classes of manifolds where this assumption is not necessary have been exhibited [4]. The cohomological obstructions appear as follows: a star-product or a deformation of the Poisson bracket are usually constructed step by step. In passing from step k to step k t 1, one encounters a Chevalley or a Hochschild cocycle which should be a coboundary to allow the construction to continue. The work of Vey and Neroslavsky and Vlassov consists of confining this cocycle in the De Rham cohomology. We show in this paper that there exists no obstruction at all: each formal deformation of order 487 Letters in Mathematical Physics 7 (1983) 487-496.0377-9017/83/0076-0487 0 I983 by D. Reidel Pllblishing Company.

$01.50.

44 1 k of the Poisson bracket extends to a formal deformation and a similar result holds true for starproducts. The basic tools are, first, cohomological properties of the Nijenhuis-Richardson bracket, showing in particular that the bracket of a one-differentiablecocycle with an arbitrary cocycle is always exact. Secondly, if $. is a conformal nonsymplectic vector field for the symplectic formF of M,homogeneity with respect to allows us to avoid the obstructions. This was fust observed in an analytic setting in [2]. An algebraic interpretation led to the proof of the existence of starproducts for exact symplectic manifolds [4,5].A further refinement combined with gluing allows us to use this type of argument for nonexact F.

1. N O T A T I O N S A N D D E F I N I T I O N S

We will mainly use the notations and defMtions of IS]. Some of them have just to be pre'cised. Let Mbe a smooth connnected Hausdorff second countable manifold equipped with a symplectic form F. We suppose dim M > 2. We denote by A(M) the space of smooth forms on M and by X(M) the space of smooth vector fields on M. As usual, we set N = Ao(M> and L x denotes the Lie derivative in the direction of X E J€(Wacting on A(M). If V and V' are vector spaces, A p ( V. V') is the space of (p + 1)-linear alternating maps from V p + ' into V' and A ( V, V ' )is the direct sum of theAP(V. V'))s (p 2 -1). For simplicity, we set A"( v)= A p ( V , V ) and A( V ) = A( V , V). It is known that (A( V), [ ] ), where [ J is the NijenhuisRichardson bracket, is a graded Lie algebra, the degree of C E A P ( v > being p . If 9 is a Lie algebra structure on V, the Chevalley coboundary operator of the adjoint representation of (V, 9 )is up to +1 the adjoint action of B onA(V). We denote byAI,@(M), Aq(M)) [re~p.A~,,,~,(N)]the space of all C€A(JCO, Aq(W)[resp.A(N)I which are local [resp. local and vanishing on the constants]. We set also A,(N) =E(A(N), u] and Av,ioc, ncQ = E(A lot, nc(N)r V ) ,where E( V , v) denotes the space of formal power series in v with coefficients in V. The mapping p: K(M) + A' (M): X + -i(X)F induces an isomorphism between the spaces of contravariant and covariant tensor fields on M. One sets A = p-'F and, for u EN, Xu = p - l du. For C €AP(X(M), 1 \ 4 0 ) , set p*C(uo, ..., u p )= C(X,, , ..., X ) and, if q = 2, p'C = (A, p*Cj. In particular, P = p*F is t h e Poisson bracket ofM (A(M) being identified once and for all with a subspace ofA(X(M), N) in the natural way). Denote by Z&nc(N) the space of Chevalley twococycles of (N, P).Then a being the coboundary operabr and r a symplectic connection of (M, F), each C EZ&, nc is of the form C = rS: + p*sZ + aE (r E IR, R E Az (M) closed and E EA&.,(N));S; is a cocycle of the form p'ar,where aFisa cocycle of the Chevalley cohomology of the representation (A(M), X +Lx) of K(M). Further details may be found in [3,5,71* PROPOSITION. 1.1. Thereexistsalinearmap r : A~,,,.,(N)~A:,(JC(M),N)such t h a t p * o r =

1 on A' (M), @$) = ( A, 4r ) and r 0.&4Lc,.,(N) C im a' (ll denotes the appropriate identical map; a' is the coboundaty operator of the cohomology of the representation (A(WY 7 0 p* =

x

--*

Lx)of Jc(M).1

h o f : It is easily seen that p*:Afoc(JC(lC(M),N)+Af,,n,(N) position 488

is surjective. Take then a decom-

n,

442

where

Observe that p*: A2(M) -+A{oc,n,(N)is injective. Choose then any right inverse of p*, r1 on El and r2 and E2 and set rS: = ( A , aPr ), o LL*~,,~(M) = 1,r o a = a' 0 r1 on El and r = r2 on E z . This defines a linear map T : Afoc,nc(N) +A&,(K(M). N), which obviously verifies the first three required properties. For the last property, one notes that by 15, prop. 2.21, p*A(M) n aA&c,nc(N>= ap*A1(M, and that onA'(M), 7 0 a 0 p = 7 0 p* 0 a'= a'.

h this paper, we fur once and for all a r such as in Proposition 1.1. LEMMA 1.2. LetRiEAIoc,n,(~,SiEA~~,,nc(N)andaclosed C2EA2(M) begiven. Letalso Rf, S,!, T'EAlo,(3C(M);N) besuch thatRi=p*Ri, Si=p*Sfand Ci[Ri,Si] =p*T'. There exisfs T E A lo, .,(M such that

(1) whenever 52 = di(X)F on an open subset U CM. Proof: If 521 = di (X')F and U n U' # @, then X ' - Xis a symplectic vector field on U n U'.It is, thus, locally of the form Xu.Since p*i(X,) = i(u)p*, the right-hand side of (1) vanishes for X = Xu by the graded Jacobi identity.

2. ONE-DIFFERENTIABLE DEFORMATIONS OF P Recall that a formal deformation of order k of P is an element ovkCk of A:,lm,nc(w such that C, = Pand such that IL,, L,I vanishes at order k (i.e., the components [LV,L"] I of [L,, L,] are vanishing for 1 Q k). A f o m a l deformation o f P is an element L, E A :,lot, such that Co = P and that IL, L,r = 0.

nc(w

THEOREM 2.1. Let Lv be a formal deformation

(i) There exists a sequence

of P and let C2 E Az(M) be closed. = X, and that (k E IN)of elements O ~ A , , ~,, , ~(,* such that

(2) whenever 52 = di(X)F'on the open subset LI C M. (ii) JL; is a cocycle for L, and its f v s t term is p*R. More generally, 489

443

is a formal deformation in p of L, (iii) In particular, for each t 2 1,

is a formal deformation of Psuch that EL = E, + vtp*51 at order t. (iv) I f E, is one-differentiable at orders (i.e.pC, is one-differentiable f o r 1 Q s), then L t ( k E IN) and LL are onedifferentiable at orders. (If it is needed to recall L, and C2, we sometimes will .write = IL: 3 (L,; 51)). Proof: Assume the existence of solutions ( I < k ) of (2) such that

(3) From Lemma 1.2, we obtain the existence of a globally defined IL; verifying (2) for 1 = k. Then applying (2) and the graded Jacobi identity:

in view of (3). Hence, (i), (ii) and (iii). For (iv), one has simply to note that the NijenhuisRichardson bracket of one-differentiable cochains is one-differentiable. Let us say that a formal deformation L, of order k is a driver of a formal deformation 1: if &, = Ev at order k. THEOREM 2.2. Every one-differentiable formal deformation of order k of P is a driver of a onedifferentiable formal deformation of P. In particular, there exist one-differentiable formal deformations o f P with driver P + vc(*52, where is an arbitrary closed two-formof M. Pro05 Let L, be a one-dimensional formal deformation of order k of P. Let 0 Q 1< k be @en. Suppose that there exists a one-differentiable formal deformation l; of P such that C,!= Ci for i < 1. Then C:+, - C1+,is a one-differentiable cocycle. It is thus of the form p*C2 for some closed two-form 51. By theorem 2.1

490

444 is a one-differentiable formal deformation of P such that result follows by induction on 1.

= Ci for i

< 1 + 1 . Taking L;

= P,the

3. F O R M A L DEFORMATIONS O F P WITH DRIVERP + vrs;

Let U be open and suppose that Flu = di(t)F for some t E JC(U). It has been seen in [6] that

In addition, one may state LEMMA 3.2. F o r p # 2 , 3 , L E + p i - a o p * o i ( i ) o ris~linearproofbijectionfrornZ~,,,~,(~ into itself: Proof: Recall that C EZ&.&N) admits a decomposition C = rS; + p*i2 + aE where i2 E A2(M> is closed, r E IR.Moreover, Cis exact if and only if r = 0 and i2 is exact.

By Proposition 1.l, C' = p*R

+ aE = ~ * T ( C 'where ) T(C')is a cocycle for a'.

Using Lemma 3.1,

(4) Thus L E + p l - a o p* o i(t) o T is surjective. Moreover, if the right-hand side of (4)is vanishing, r = 0 and C' = 0. Hence, the injectivity. Let ll €Ao(E(N v)) be defmed by

and set 0 = 211 - 1.As can easily be seen, for every Tv EA,(N),

THEOREM 3.3. Let r E & be given. There exists a unique formal deformation L, of P of driver

P + urS$ such that K..#,;

F) + a,o = 0.

((5) 5) 49 1

445

(a, is the coboundary operator associated to the adjoint representation of (E(N, u). Xu)). hoof: (i) If U is open and Flu = di(t)F for some E E WU), after its definition in Theorem 2.1, iLv(L; F)reads

Thus, the kth component of the left-hand side of (5) is

Observe, moreover, that L t = p*i(.$)F = p*i(t)T(Co).Hence, the above expression can be written, for k > 0,

(ii) Let 8, and LL have driver P + vrS$ and satisfy (5). Assume that k > 1 and that C;=Ci for i < k. Then

(&

+ (2k t i)n

- ap*i(E)T)(c;, -

c,) = 0.

It follows from Lemma 3.2 that c; = ck on U.Thus c; = ck, hence, the uniqueness by induction on k. (iii) Observe that (5) means that for every open subset U on which F = di ($)F, gU = c(*i($)r(&)+@ is a derivation of Xu,,. Set C, = P,C1 = rS:, L: = C, + YC, and 9 = L Et p*i(&-(S$) + 0. One has

and

b y Lemma 3.1. Let now k > 1 and suppose that the Ct's EA&c,n,(N)(i < k ) have been constructed such that L: - I = & 0,

L; = l L v ; ( L y ; T(C; - C y ) ) . It is easily seen that L; = 1, at orders. If s = k, the proof is achieved. Suppose then that s < k and that we have found a formal deformation f : of P such that 2: =L, at order I , with s < 1 < k. Then the cocycle C;+l - Cr+l is of the form r'SF + p*R + aT for some r' E IR,some closed R E A2(M> and some E E A ; , ,,.(N).Replacing .Cb by Ad(exp $+'QE>, we may assume that T = 0. On the other hand, set

Then, Cif- C; is one-differentiable for i Q 1 + 1 and vanishes for i Q I - s. Define formal deformations ~ ~ - -..., s2';' + ' ofPinductively , by $-"" = L," a n d f o r j > l - s ,

494

448

T h e n z r ’ is a formal deformation o f P which coincides with L, at order 1 + 1. Hence, the result by induction on 1 < k.

It follows from Theorem 3.4 that [C, c] is exact for each two-cocycle C. Moreover, - diff, nc(h‘) denoting the cohomology of one-differentiablenc cochains, it can be shown that

4.

STAR-PRODUCTS

Recall that a star-product of (M,fl is a formal deformation Ak= X;;”=ohkCkof the associative algebra (N A),with driver M . t XP, where for k > 0, the Ck’s are local, vanishing on the constants and such that ck(v, u ) = (-l)kCk(U, v ) for all u , v EN. THEOREM 4.1. Every star-product of order 2k is a driver of a star-product. I n particular. e v e v symplectic manifold admits a star-product. hoof: Let = X i < 2 k i i C i be a star-product of order 2k ( k > 0). Then

(8) is a formal deformation of order k - 1 of P. In the proof of the Neroslavsky-Vlassov theorem [8], which asserts the existence of a starproduct when the third De Rham cohomology space of M vanishes, the key steps are the following. = MA+ VZk+’C2k+l is a star-product O f (i) There exists CZk+l €Akc,nc(w such that order UC t 1; C2kt1 is determined up to an arbitrary one-differentiable element ofAk,,nc(N). extends to a star-product of order 2k + 2 if and only if 8, f V k C Z k + 1 is a (ii) formal deformation of order k of P. (iii)

[CZP+l’czq+11

(9)

p+q= k

is a one-differentiable Chevalley cocycle. By Theorem 3.5,(8) is a driver for a formal deformation of P. If vkC is its kth component,

then a ( C 2 k + l - C ) is one-differentiable. n u s C z k + l - c - p * a is a Chevalley cocycle for a suitable E A2(M). Then replacing C,,,, by C2k+l -p*fl, (9) vanishes and &k extends to a by induction star-product of order 2k + 2. Hence, the existence of a star-product with driver on k. To conclude, observe now that each symplectic manifold admits a star-product of order 2 [8 J . 495

449 ~~

REFERENCES

1. Arnal, D., Cortet, J.C., Flato, M.,and Stedeimer, D., ‘Star-Products: Quantization and Representation Without Operators’, in E. Tirapegui (ed.), Field Theoly Quantization and StatisticalPhysics, Reidel, Dordrecht, 1981, pp. 85-1 11. 2. Cahen, M. and Gutt, S., Lett. Math. Phys. 6, 395-404 (1982). 3. De Wilde, M., Gutt, S., and Lecomte, P., ‘A propos des deuxieme et troisi6me espaces de cohomologie de l’algibre de Lie de Poisson d’une van& symplectique’, Ann. Inst. Poincar6, to appear. 4. De Wilde, M. and Lecomte, P.,Letr. Math. Phys. 7,235-241 (1983). 5. De Wilde, M. and Lecomte, P., ‘Existence of Star-Products on Exact Symplectic Manifolds’, to appear. 6. Flato, M., and Sternheimer, D., ‘Deformationof Poisson Brackets’, in J. Wolf et al. (eds.),Hamonic Analysis and Representation of SemiSimple Lie Croup, Reidel, Dordrecht, 1980, pp. 385-448. 7. Gutt, S., Ann. Inst. Poincare‘33, 1-31 (1981). 8. Lichnerowicz, A., Ann. Znst. Fourier 32, 157-209 (1982). 9. Moyal, J.E., Proc. Cambridge Phil. SOC.4599-124 (1949). 10. Vey, J., Comm Math. Helv. 50,421-454 (1975).

(Received July 28.J 983)

496

450 J. DIFFERENTIAL GEOMDRY 40 ( 1994) 2 13-238

A SIMPLE GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION BORIS V. FEDOSOV Abstract A construction, providing a canonical star-product associated with any symplectic connection on symplectic manifold, is considered. An action of symplectomorphismsby automorphisms of star-algebra is introduced, as well as a trace construction. Generalizationsfor regular Poisson manifolds and for coefficients in the bundle Hom(E, E ) are given.

1. Introduction

A manifold M is called a Poisson manifold, if for any two functions u , v E C ” ( M ) , a Poisson bracket is defined by ij

au a v

{ u , v} = t -ax‘ ax’ The bracket is a bilinear skew-symmetric operation, satisfying the Jacobi identity

{ u , { ~ , ~ } } + { ~ , { w , ~()u } , v+) ){=wO,. An important particular case is a symplectic manifold. In this case the matrix t” has maximal rank 2n equal to the manifold dimension. The inverse matrix o i j defines the exterior 2-form o = $a,, d x ’ A dx’ which is closed in virtue of Jacobi identity. In [ 11 it has been proved that, if the tensor t” has constant rank 2n > d i m M , there exists a symplectic foliation of the manifold M , a Poisson manifold with this property being said to be regular. The leaves F of this foliation locally are symplectic manifolds, and a Poisson bracket is defined by the symplectic form o (closed 2-form of the rank 2n = dim F) defined on the leaves. In the same paper [I] the question of deformation quantization of Poisson and in particular symplectic manifolds is considered. The problem is to define an associative multiplication operation * , depending on parameter h (Planck constant), of two functions so that the space C”(M) with Received November 1 1 , 1992 and, in revised form, April 27, 1993.

45 1 214

B. V. FEDOSOV

usual linear operators and *-product would be a formal deformation of commutative algebra of functions with a Poisson bracket. More exactly it means the following. Let 2 be the linear space, the elements of which are formal series

c 00

a = a(x , h ) =

hka,(x),

k=O

where ak(x) E C""(M). Further for any a , b E 2 let an associative product operation k=O

be defined with the following properties: (i) ck are polynomials in a, , b, and their derivatives; 01) C O ( 4 = ao(x)b,(x); , where dots mean the (iii) [ a , b] = a * b - b * a = -ih{ao , b,} terms of higher orders. The algebra 2 is called the algebra of quantum observables. Property (i) means the locality of *-product, property (ii) means that algebra 2 is a deformation of the commutative algebra of C" functions, property (iii) is the so-called correspondence principle. The question of the existence of such a product for symplectic manifolds has been completely solved in [2]. Subsequently, an equivariant generalization of this construction [7] for symplectic manifolds was obtained, as well as a generalization for regular Poisson manifolds [6]. The constructions considered in these works are based on the analysis of Hochschild cohomologies. In [3] the author, being unaware of the results of [2], proposed another construction of *-product for a symplectic manifold. This construction admits straightforward generalizations for both the equivariant case and the case of a regular Poisson manifold. In subsequent papers [4], [S] the author studied the action of symplectic diffeomorphisms, proposed a trace construction in algebra 2 , introduced the concept of index, generalizing the index of elliptic operators and obtained an index formula. Unfortunately, work [3] was published in a local issue of Moscow Institute of Physics and Technology in very few copies, so it remained unknown to most mathematicians. The purpose of the present article, containing the extended exposition of some results of [3], [4], is to introduce the results to broader mathematical circles. Let us briefly describe the contents of subsequent sections. In $2 we consider the Weyl algebras bundle W , W-valued differential forms, and a

+

452 GEOMETRICAL CONSTRUCIlON OF DEFORMATION QUANTIZATION

215

connection in the bundle W . These notions give us a basic machinery. In 93 we introduce the notion of Abelian connection and prove the existence of such connections. The sections of the Weyl algebras bundle, which are flat with respect to a fixed Abelian connection, form an associative algebra. We prove that these sections are in one to one correspondence with the functions from 2 . This allows us to transfer the associative algebra structure to the set 2 and thus to define a *-product. The next sections are concerned with the notion of trace in the algebra of flat sections. First of all we construct isomorphisms of this algebra, corresponding to any symplectic diffeomorphism of symplectic manifolds. This construction, introduced in $4, is used in 555 and 6 to define a trace by means of localization and reduction to the case of standard symplectic space It2". In 97 two generalizations are exposed. The first one gives the construction of deformation quantization and the trace for the case where the coefficients are homomorphisms of a vector bundle over M . The second one deals with a generalization of the results obtained in 352 and 3 for the case of regular Poisson manifolds. A few years ago there appeared a paper [8], in which quantization is based on the idea of identifying functions on a symplectic manifold with the sections of the Weyl bundle. We use a similar approach. But their means of such an identlfication is much more complicated than ours. This article has been written during a visit to MIT. Taking this opportunity, I would like to express my profound gratitude to the Department of Mathematics at MIT and to Professor Guillemin for the kind invitation. I would also like to thank Professor A. Weinstein for his useful corrections and remarks. 2. Weyl algebras bundle

Let (M, o)be a symplectic manifold of dimension 2n. The form w determines a symplectic structure in each tangent space T'M . Definition 2.1. The formal Weyl algebra W x , corresponding to the symplectic space TxM , is the associative algebra over C with a unit, its elements being formal series 2k+l>O

. .

where h is a formal parameter, y = (y 1 , , y2") E T,M is a tangent vector, and i+, i,...i,are covariant tensors. The degrees 1 and 2 are prescribed for the variables y' and h respectively. The product of elements

453 B. V. FEDOSOV

216

a , b E W, is determined by the Weyl rule

It is easily seen that the multiplication (2.2) does not depend on the choice of a basis in T,M and is associative. Taking the union of the algebras W, , x E M , we obtain the bundle of formal Weyl algebras whose sections are "functions" (2.3) 2k+1>0

where uk , ..i, are symmetric covariant tensor fields on M . The set of sections also forms an associative algebra with respect to the fiberwise multiplication (2.2). The unit in this algebra is the "function" identically equal to 1 . To simplify notation we shall also denote the algebra of the sections by W (instead of the pedantic C"(M, W)) , which, to our mind, should not cause any confusion. It is easy to see that the center of W consists of the sections not containing y 's. Thus the central sections are defined by the series of form (1.2), and consequently the center of W may be identified as a linear space with the space 2 mentioned in the introduction. There is a filtration in the algebra W : W 2 Wl 2 W2 2 . . with respect to the total degree 2k + I of the terms of the series (2.3). We shall also need differential forms on M with values in W . A differential q-form is defined by the series I,,

-

(2.4)

whose coefficients are covariant tensor fields symmetric with respect to indices i, , . - , ip and antisymmetric with respect to J , , . . .Jq . The differential forms constitute an algebra W @ A = @:to( W @A), in which the multiplication is defined by means of the exterior product of differentials dx' and Weyl product (2.2) of polynomials in y' (dx' commute with y ' ) . The product of two forms will be denoted by the same symbol a o b , such as the product sections of W . A filtration W @ A 2 Wl@ A3 W2@A2 - is introduced with respect to the total degree 2k + p corresponding to the variables h , y' .

-

454 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

217

Let us introduce the commutator of two forms a E W @ A'!, b E W @ Aq2 defined by [ a , b] = a 0 b - (- 1)Iq2bo a . A form a is said to be central, if for any b E W @ A the commutator of a and b vanishes. It is clear that the central forms are just the ones not containing y 's, i.e., 2 @ A is the center. Define two projections of the form a = a ( x , y , dx , h) onto the center: a. = a ( x , 0 , d x , h ) and a, = a ( x , 0 , 0, h) . In the particular case where a = a ( x , y , h ) E W , we shall use the notation a ( a ) for a. = a ( x , 0 h ) and call a(a) the symbol of the section a . Consider two important operators on forms:

6 a = d xk

(2.5)

A-

aa

ayk '

6 * a = yk i ( G a) a ,

where i ( a / t l x k ) means the contraction of the vector field a / a x k and the form, multiplication by y k being the usual commutative product of functions. The operator 6: Wp@ A' --t Wp-l @ A'+1 which reduces the filtration by 1 is similar to the exterior derivation. The operator 6* : Wp@ A' 4 @A'-' raises the filtration by 1 . In other words the operator 6 acts on the monomial y11y12. . . Y I P dxJl A dxJz A . . . A dXJq (2.6) by replacing one by one the variables y'l , y J 2,. . - , y" by dx'l , dx12 , . . . , dx'p respectively; the operator 6* acts on (2.6) by replacing d x J i, d x J 2, . . . , dxjq by y'1 , -yJ2 . . . (- 1)'yJq respectively. Lemma 2.2. The operators S and 6" do not depend on the choice of local coordinates and have the following properties : (i) 62 = ( s * ) = ~ 0, (ii) for monomial (2.6) we have 66' + 6*6 = ( p q ) id. The lemma is easily proved by a direct check. q.e.d. Note that 6 is an antiderivation, i.e., for a E W@Aql and b E W @ h q 2 we have

+

+

6(a o b ) = (6a)0 b (-1)'lu 0 b , (2.7) ( 6' does not possess this property). Define the operator 6-' acting on the monomial (2.6) by 6-' = s*/@ 4) for p + q > 0 , and 6-l = o for p + q = 0. BY Lemma 2.2 it can be derived that any form a E W @ A has the representation

+

(2.8)

a = 6S-'a

+~

' 6 +aa m ,

which is similar to the Hodge-De Rham decomposition.

455 B. V. FEDOSOV

218

Definition 2.3. A symplectic connection is a torsion-free connection preserving tensor w . . i.e., dimjk = 0 , di being a covariant derivative "i' with respect to d / d x . In Darboux local coordinates the coefficients riik= of the symplectic connection are completely symmetric with respect to indices i j k . The symplectic connection always exists but is not unique, unlike the Riemannian connection [ 11. Two symplectic connections differ by a . completely symmetric tensor Arijk Let a be a symplectic connection on the manifold M . Using the covariant derivation of tensor fields, which are coefficients in (2.4), define a connection in the bundle W as an operator a : W @ A q-t W @ h q + 'by

wi,,,q/

aa = dx' A d,a .

(2.9)

Definition 2.3 implies the following properties of the connection d in the bundle W 8 A : (i) a ( ~ o b ) = d a o b + ( - l ) ~ ~ a ofor abU E W@Aql. (ii) For any scalar form q E Aq , d(p A a) = d p A a + (- l)qpA d a . In Darboux local coordinates the connection d can be written in the form (2.10)

aa = da

+ [ ( i / h ) T ,a ] ,

where I'= tI'ijkyly'dxk is a local 1-form with values in W , d = dx' A a / d x ' being the exterior differential with respect to x . We shall consider more general connections D in the bundle W , namely, connections of the form (2.11)

~a = aa

+ [ ( i / h ) y, a3 = da + [ ( i / h ) (+r

,4,

where y is a globally determined 1-form on M with values in W (i.e., section of W @ A') . Note that the operator 6 , introduced above, may be written in the form (2.12)

Lemma 2.4.

6a = -[(i//z)w,,y'dx', a ] . Let d be a symplectic connection. Then

(2.13)

dSa + 6da = 0 ,

(2.14)

a2a = d ( a a ) = [(Z/h)R,a ] ,

where

R = i R i j k r iyy jd x k A d x I ,

Rijkl= mimR>, being the curvature tensor of the symplectic connection.

456 GEOMETRICAL. CONSTRUCTION OF DEFORMATION QUANTIZATION

219

Proof: Identities (2.13),(2.14) are obvious consequences of equations (2. lo), (2.12). Note that (2.14) is a compact form of the Ricci identity. Definition 2.5. Let D be a connection in the bundle W of the form (2.1 1) with (2.15)

yo = 0.

We shall call the 2-form

(2.16) the curvature of the connection D . Lemma 2.6. For any section a E W 8 A we have (2.17)

D2a = [(i/h)52,a].

The proof is straightforward. Remark. Note that the form y in (2.11) is determined by the connection D not uniquely but up to a central 1-form, because it appears in commutators. For the uniqueness of y and therefore of the curvature (2.16) some normalizing condition is required. We assume that this condition has the form (2.15) and call it Weyl normalizing condition. The corresponding curvature (2.16) is called Weyl curvature. 3. Abelian connections and quantization

Definition 3.1. A connection D in the bundle W is said to be Abelian if for any section a E W 8 A (34

D2a = [ ( i / h ) i 2 ,a ] = 0.

By (3.1) we can show that the curvature of Abelian connection is a central

form and conversely. In this section we prove the existence of an Abelian connection of the form D = - a + a + [ i r , , ] = a + [ f ( w i j y di x j + r ) , ,

-1

d being a fixed symplectic connection, and r E W38 A' being a globally defined 1-form, satisfying Weyl normalizing condition ro = 0 . Calculating the curvature of this connection and using Lemma 2.4, we get (3.2) R = -$aijdx' A dx' + R - Sr + a r + ( i / h ) r 2 . The Abelian property will be fulfilled, provided (3.3)

2

Sr = R + a r + ( i / h ) r .

457

B. V. FEDOSOV

220

Then for the curvature we shall have $2= --o , so that SZ will really be a scalar form. Theorem 3.2. Equation (3.3) has a unique solution, satisfiiing the con-

dition 6% = 0.

(3.4)

(Note that (3.4) implies ro = 0). Proof. Let r E 5 8 A’ satisfy (3.3), (3.4). The decomposition (2.8) for the form r becomes r = F ’ 6 r as 66-’r = 0 by (3.4) and roo = 0 , since r is a 1-form. Applying the operator 6-’ to (3.3) we get (3.5)

r = 6-l R

+ 6-l

( a r + ( i / h ) r 2 ).

The operator a preserves the filtration and 6-’ raises it by 1 , so iteration of equation(3.5) shows that it has a unique solution. Conversely, we will show that the solution of equation (3.5) satisfies -1 2 (3.3), (3.4). The condition (3.4) is evidently fulfilled because of (6 ) = 0 . Let A = ~r - R - a r - ( i / h ) r 2 be the difference between the left-hand and the right-hand sides of (3.3), r being the solution of (3.5). Show that A satisfies the equation

6 A = a A + ( i / h ) [ r, A] (3.6) and the “initial” condition (3.7)

6 - ’ A = 0.

From (3.6) and (3.7) it follows that A vanishes. Indeed, applying 6-l to both sides of equation (3.6) and using (3.7) we shall get similar to (3.5) A = 6-’ ( a A

+ ( i / h ) [ r ,A ] ) ,

from which by iterations it follows that A = 0 . For checking (3.7) we have S - ’ A = s - ’ s ~- 6-I (R+ a r + ( i / h ) r 2 )= ~ - ‘ 6 r r

= 0.

Here we have used (3.5), condition (3.4), and the Hodge-De Rham decomposition. For checking (3.6) by taking into account that 6 S r = 0 , we obtain S A = -6R - S ( a r )

+ [ ( i / h ) r 6, r ] ,

since

SR = $ R i J k ,‘yd x J A d x k A d x ‘ ,

45 8 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

22 I

which is equal to 0 because of the relation

+

R i j k r+ R i k l j Riijk= 0 for the curvature tensor. Further, S ( a r ) = - a ( S r ) according to Lemma 2.4. Thus,

S A = a(sr>+[ ( i / h ) r ,R + d r + ( i / h ) r 2 1 . (3.8) We have d R = 0 according to the Bianchi identity for the curvature tensor, d d r = [ ( i / h ) R ,r ] invirtue of Lemma 2.4 (Ricci identity), a ( i / h ) r 2= [ a r , ( i / h ) r ] .Taking into account that [ ( i / h ) r ,( i / h ) r 2 ]= 0 , we get that the last two terms in (3.8) would equal to 0 , and this proves equality (3.6). q.e.d. Note that iterating equation (3.5) we can effectively construct the form r and, consequently Abelian connection D . The first two terms are i j k

r = $R,,,y y y d x

I

+ &,,Rijkiy

i j k m

y y y dx

I

+.(.,

dm being a covariant derivative with respect to the vector field a / d x m. Fuaher terms would contain not only y 's but also powers of h because of the term ( i / h ) r 2 in (3.5). Introduce now the main object: the subalgebra W ' c W , consisting of flat sections, i.e., such that Da = 0 . Theorem 3.3. For any a, E Z there exists a unique section a E WD S U C ~that O ( U ) = a,. Recall that, for the section a(x , y , h ) E W , a(a) means the projection onto the center, i.e., a(a) = a ( x , 0 ,h ) . Prouf The equation Da = 0 can be written in the form Sa = d a

(3.9)

+ [ ( i / h ) r ,a ] .

Applying the operator 6-' and using Hodge-De Rham decomposition yield

(3.10)

a = a.

+ ~ - ' ( a a+ [ ( i / h ) r ,a ] ) ,

wherefrom by iterations we should get that equation (3.10) has a unique solution because S- increases the filtration. Conversely, let a be the solution of (3.10). Then evidently we have o(a) = a, since the result of applying 6-' contains only positive powers of y 's. Further, using reasoning similar to the proof of Theorem 3.2, we can show that the difference

'

A = 6a - d a - [ ( i / h ) r ,a ] = Da

459 B. V. FEDOSOV

222

between the left-hand and the right-hand sides of (3.9) satisfy the equation

6 A = a A + [ ( i / h ) r ,A]

(3.11)

.

and the trivial "initial" condition

6 - ' A = 0.

(3.12)

Equation (3.11) is fulfilled, since it means that DA = 0 , so taking into account that A = Da we shall have D A = D(Da) = 0 because D is an Abelian connection. Further, 6 - l ~ =s

+

- ' ~ u- ~ ' ( a a[ ( i / h ) r ,a ] ) = K ' S U

- a + a,

according to equation (3. lo). The last expression is equal to 0 by HodgeDe Rham decomposition, since 6-'a = 0. q.e.d. It is easily seen that for any a ( y , h) E W with fixed xo E M there XO exists a flat section a(x y , h ) E W (not unique, of course) such that 4 a(x, y , h ) = a(y , h ) . This fact implies that the centralizer of WD in W coincides with the center Z of W . In other words, if a section b E W commutes with any flat section a E W-, , then b E 2 . Similarly, the centralizer of WD in W 8 A is 2 @ A . Iterating equation( 3.10) we can effectively construct the section a E WD by its symbol a, = a ( a ) : a = a,

+ aia,y i + @iajaoyiyJ+ '6ai0 j. ak aOy i y j y k

-L 24 R i j k [ o l m a ~ a O Y i Y j Y k +

, a .

If the curvature tensor is equal to 0 , iterations would give the explicit expansion

It is clear that, provided Abelian connection D is fixed, flat sections form a subalgebra WD with respect to fiberwise Weyl multiplication 0 in the algebra W . Theorem 3.3 states that the map Q : WD 2 is bijective. WD has been defined. Thus, the inverse map 0 - l : 2 Now we can explain the construction of *-product in the space Z . Namely, by using the bijections Q and Q - ' associative product o in the algebra W is transferred to the set 2 , i.e., we assume for a , b E Z -+

-+

(3.14)

CL

* b = a(a-'(a)0 2 ( b ) ) .

Using (3.13) it is easily checked that such defined *-product satisfies all the

460 GEOMETRICALCONSTRUCTION OF DEFORMATION QUANTIZATION

223

conditions (i)-(iii), formulated in the introduction. However, as will be seen later, it is more convenient to use the subalgebra WD with 0-product than the *-product. Therefore later on we shall not mention *-product at all. The subalgebra WD will be called an algebra of quantum observables.

4. An action of symplectomorphisms

Let M , O , W , a , D , WD denote, similar to those in the previous sections, a symplectic manifold, a symplectic form, the Weyl algebras bundle (and the algebra of its sections), a symplectic connection on A4 , the Abelian connection in the bundle W and the subalgebra of flat sections. For a symplectic diffeomorphism f:M + M the pullbacks f* of these objects are evidently defined ($*w = o since f is a symplectic map). For example, for a section a ( x , y , h ) E W we assume Y

f'

4.m)

( f a > ( x> Y h ) = > f Y h )> being a differential of the map f. Since f is a diffeomorphism, both Y

pullbacks f* and pushforwards S, = (f-')*are defined for all geometric objects. Consider in more detail the action of symplectomorphisms f:M -+ M on connections. Let 8 be a symplectic connection considered as the connection in the bundle W according to (2.7), and D be the Abelian connection corresponding to D by Theorem 3.2. Since f* and f, are evidently automorphisms of the algebra W , we can define the connections 6 = L a , fi = f , D by the formulas Ba = f , ( a ( f * a ) ) ,

Zia = f , ( ~ ( f * a > > .

It is clear that 5 is also an Abelian connection. The operators 6 , 6-' and Weyl normalizing condition are invariant under diffeomorphisms. (Because of the uniqueness of the solution of (3.3), (3.4), fi corresponds to 8 , i.e., 5 is obtained from 8 by Theorem 3.2. Theorem 4.1. The autornorphism f,: W + W isomorphically maps

the subalgebra WD onto the subalgebra W z . Besides, if the symplectic connection a is invariant under f , i.e., if 0 = f,a = d , then f , defines the autornorphism of the algebra WD. Proof: The proof directly follows from the definitions. If a E W, then f,a E W E ,since Ef,a = f , ( D a ) = 0 . Further on, if

8=d

the property of the uniqueness of Theorem 3.2 implies that 5 = D , WE = WD, i.e., 1; is an automorphism of the algebra of quantum observables WD. q.e.d.

46 1 B. V. FEDOSOV

224

In particular, if G is a group of symplectomorphisms of M , and a is a G-invariant symplectic connection, then the corresponding Abelian connection is also G-invariant and the group G acts by automorphisms on the algebra of quantum observables WD. In a general case, when 8 is not invariant with respect to f , it is nevertheless possible to define automorphism Af , corresponding to f , by using the fiberwise conjugation automorphisms. To do so introduce an extension W' of the algebra W as follows: (i) Elements U E W + are given by the series (2.1), but the powers of h can be both positive and negative. (ii) The total degree 2k + I of any term of the series is nonnegative. (iii) There exists a finite number of terms with a given nonnegative total degree. It is clear that W + is also an algebra with respect to Weyl fiberwise multiplication, and the connections a and D act on sections a E Wt . Lemma 4.2, Let a E W + and Da = 0 . Then a does not contain negative powers of h , i.e., a E WDc W c W'. ProoJ Let a ( a )= a ( x , 0 , h ) . Nonnegativeness of the total degree of series terms implies a(a) E 2 , i.e., it does not contain negative powers of h . According to Theorem 3.3 a flat section is uniquely defined by its symbol a(a) E 2 and thus belongs to WD. q.e.d. Like W the algebra W + has the filtration with respect to the total degree 2k I of series terms (2.3). Introduce a group, consisting of invertible elements of the algebra W + with the leading term 1 having the form

+

(4.1)

where H E W,. It follows from the Campbell-HausdorR formula that such elements form a group. It is clear that the map (4.2)

(the commutator is taken k times) is an automorphism of W + , which maps the algebra W onto itself. It is also clear that this map preserves the filtration but not the degrees of the series terms. Let D be the Abelian connection in the bundle W of the form (2.1 l), y satisfying Weyl normalizing condition yo = 0 . Automorphism (4.2) defines a new Abelian connection 5 by the rule

462 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

(4.3)

Eja

=U

o

D(U-'o a o U ) 0 U - '

= Da

-

[DU o U-' , a ] .

From (4.3) it follows that the form 9 , corresponding to the connection and satisfying normalizing condition, has the form (4.4)

3 =y+Ay

=7

- D U O U-'

+ (DU

22s

o

fi

U-'),,.

Hence for the curvature we shall have

(f-A? ) + ( ; y ) 2 = - ; c o + d ( D ( ' o C ' -

-iQ- = - SiZ + D h h

1 ),.

The last equality is obtained by using the relation

-D(DU

0

U - l ) + (DU 0

u- 1 )2 = 0.

The scalar form (DU o U-l)o belongs to W28 A' n Z , i.e., begins with the first power of h . Thus we obtain that the curvatures of these two connections D and 6 differ by an exact 2-form belonging to ( W .@ A 2n) 2.

Theorem 4.3. Let d , 8 be two symplectic connections, and D , 6 be the Abelian connections corresponding to d , 8 by Theorem 3.2. Then there exists a section U E W + of the f i r m (4.1) such that

?j=D-[Duou-

1

,.I.

5 can be written in the form 5 = D + [(i/h)Ay, -1,

Prooj The connection where

Ay = f

-

I-+

? - r E W2@ A '

satisfies Weyl normalizing condition (Ay), = 0 . Hence for the curvature ( i / h ) f i of the connection 5,

Since this expression is to be equal to (i/h)S2,we have (4.5)

D(Ay)+ (i/h)(Ay)2= 0 .

Find the section U E W + as a solution of the equation

DU which is equivalent to

o

U-' = -(i/h)Ay ,

463 B. V. FEDOSOV

226

DU = -(i/h)Ay

(4.6)

o

U.

Condition (4.5)is necessary for the solvability of equation (4.6) in W + . Indeed, applying operator D ,we get 0 on the left-hand side, since D2U = 0 . Thus 0 = - ( i / h ) D ( A y ) o U + ( i / h ) A yo D U . Substituting (4.6)for DU in the above equation, we obtain

o = -{( which is fulfilled according to (4.5). Let us show that condition (4.5) is also sufficientfor the solvability of equation (4.6). Rewrite (4.6)in the form

SU

+ S ) U + ( i / h ) A ~U

= (D

0

and apply the operator S-' to both sides of the equation. Taking Uo = 1 and using the Hodge-De Rham decomposition, we get

U = 1 + S-'{(D

(4.7)

+ S)U + ( i / h ) A yo U } .

+

Since the operator D + 6 = d [ ( i / h ) r ,-1 does not change the filtration, multiplication by ( i / h ) A y in W l 8 A' does not change the filtration either, and 6-' raises the filtration by 1 , the iterations of equation (4.7) give a unique solution. The resulting solution is an invertible element of the algebra W f , since its leading term is equal to 1 . Conversely, let us show that the solution of equation (4.7)satisfies (4.6). Let A = DU ( i / h ) A yo U . Then

+

6-'A

(4.8)

=0

according to (4.7).Further we have

i

i

D A = - ( D A Y ) 0 U - -A? h h

'

0

DU

1

=

h { ( D A Y )+ ;(A?)'}

0

i U - -Ay h

0

{ DU + i A y

o

U},

Hence, in consequence of (4.9,A satisfies the equation

(4.9)

D A + ( i / h ) A y o A=O.

So, using reasoning similar to Theorem 3.2, we get A tion (4.9) together with condition (4.8)gives

= 0.

Indeed, equa-

464 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

227

+

A = 6 - ' ( ( D 6 ) A + ( i / h ) A yo A ) , Thus the iterations yield a trivial solution. The solution of equation (4.6) is not unique. Let V be another solution. Then for U-' o V we get D(U-'

0

=

U-'

o

+ u-'

u-'

0

V

( i / h ) A yo V

-

U-'

V )= -u- 1 ODUO

0

ODV

( i / h ) A yo V

= 0.

Consequently, U-l 0 V = C E W,, and the two solutions of equation (4.6) differ by the factor C , which is an invertible flat section. Let us finally show that the solution can always be chosen in the form (4.1) with H E W3. Introduce the section O0

(4.10)

H = -ihlnU = -ihln(l+(U-1))

=-

(_l)k+'

ihx

k

(U-Ilk

7

k= 1

where powers are understood with respect to the multiplication 0 . Since U - 1 E Wl , the series converges with respect to filtration in W + and defines the section H E W: . Multiplying U by the proper factor C = ea to the right ( a E W, and the exponent is calculated in the algebra W,) we can always achieve Ho = 0. Indeed, if Ho E W4 n 2 is not equal to 0 , then by Theorem 3.3, we can construct a E W, n W4 , such that a ( a ) = Ho. Taking in (4.10) U o e-' instead of U , we get a new section H ,for which Ho E W,nZ . Repeating this procedure, we shall get sections H with Ho having higher and higher degree so that in the limit we obtain H with Ho = 0 . Let us show that this section belongs not only to W: but also to W3 as well. Indeed, derivating the exponent U = exp((i/h)H) and substituting into equation (4.6) we obtain exp(ad((i/h)H))- 1 i i DUO u-' = -DH = --Ay, h ad((i/h)H) h where a d ( ( i / h ) H )= [ ( i / h ) H, - 3 . This gives an equation for H which can be written in the form

6H =

ad((i'h)H) Ay exp(ad((i/h)H))- 1

+ (D + 6)H.

Applying 6-' and using Ho = 0 , we shall get, according to HodgeDeRham decomposition,

H = 6-l

Since Ay

E

(exp(ad((i/h)H)) ad((i'h)Hi) Ay+(D+6)H) . -1

W2@A' ,all the iterations will give elements of

% , so that the

section H belongs to W 3 . Hence the theorem has been proved. q.e.d.

465

B. V. FEDOSOV

228

Now if we are given a symplectic diffeomorphism f: M

+ M , then A / : W + W , mapping the subalgebra WD onto itself, an automorphism can be associated with f in the following way. Let a be a symplectic connection on M , and D be the Abelian connection in the bundle W , corresponding to a by Theorem 3.2. Let, further, 8 = f,a and 5 = f , D be pushforwards of d and D under diffeomorphism f. Since Weyl curvatures of D and 5 both are equal to - ( i / h ) w according to Theorem 3.2, by Theorem 4.3 there exists the section U E Wf of the form (4.1) such that the connection D goes to the connection D under conjugation autornorphism (4.2). So automorphism A / defined by the relation

A/:a

(4.1 1)

w

U O

( ~ ao U-' )

maps the sections of WD to the sections belonging to WD. Generally speaking, these automorphisms do not satisfy natural cocycle condition

(4.12)

=id,

if f,f2f3= id. However, we can state that the left-hand side of (4.12) is the conjugation automorphism by the section U = Ul o f,,(U, o f2*U3)E W + of the form (4.1). Lemma 4.4. Let conjugation automorphism (4.2) map the subalgebra WD onto itselj Then locally there exists a function a, such that U e p E WD. Proof: For any U E WD we have Aa = U o a o U-' E WD. Then

o = D ( U o a o u-')= [DU o U-' , u o a o u-']+ u o Da o u-' , wherefrom it follows that

[DUOu-',u o a 0 u-']= 0. Thus, the form v/ = DU 0 U - ' commutes with any section U o a o U-' E WD, i.e., it is the central form. This form is closed since dv

=D(DUo

u- 1 ) = (DUOu- 1)2 = W A

= 0.

So, locally y = dq , and

D(Ue-')e'

o

U - ' = DU

o

U-' - dv, = 0,

which means that the section Ue-' is flat in W + and then, according to Lemma 4.2, it automatically belongs to W .

466 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

Corollary 4.5. Automorphism Al, A,,.. A& , where an inner automorphism of the algebra W .

229

fifif3 = id, locally is

5. A trace in the algebra of quantum observables on R2"

In this section we shall consider the case M = R2" with the standard symplectic structure. The standard symplectic form has constant coefficients, therefore we can take the operator of exterior derivation as a symplectic connection. The Abelian connection D in the bundle W = W(R2"), corresponding to d by Theorem 3.2, has the form D = -6 + D . In this case the isomorphism a-' and the multiplication * have a very simple explicit form

a

-1

:a(x,h) H a ( x + y , h)=

" 1 -a(a)a(x, h ) y a , a!!

lffl=O

(5.1)

Definition 5.1. The trace in the algebra WD(R2")is the linear functional defined on the ideal WrmP(R2"),consisting of the flat sections with compact support by formula (5.2)

Thus the trace has values in Laurent formal series in h with negative powers of h not greater than n , i.e., M

k=O

Lemma 5.2.

The trace has the property tr a o b = tr b o a ,

(5.3) where a E W',mP(R2n),b E W(R2n). Prm$ Since a ( a o b) = a(a)* a ( b ) ,according to (5.1) it is sufficient to check the equality dkb aka akb aka i z n axil . . . axik a x j l . . . a j k con = L 2 n axjl . . . a x j k axil . . . a X l k which is easily verified by integrating by parts. q.e.d. 9

467

B. V. FEDOSOV

230

We shall prove that property (5.3) implies the invariance of the trace under isomorphisms A,, considered in $4. For an open contractible set 0 c R2" we shall denote the algebra of flat sections with support in 0 by Wr"'(0). Let f be a symplectic diffeomorphism, defined on 0 and mapping it onto the open set f (0),and let A : Wrmp(0)-+ Wrmp(f (0)) be an isomorphism corresponding to f by formula (4.1 1). Theorem 5.3. For any a E WrmP(0), tra = tr(Afa).

(5.4)

Proof: For the proof we construct a family a ( t ), t E [ 0 , 1 3 , of flat sections with compact support such that a(0) = a , a( 1) = A f a , satisfying the Heisenberg equation

W )= ( i / h ) [ H ( t )401

(5.5)

9

7

with the Hamiltonian H ( t ) E WD(R2") . Then according to (5.3) we have

d i dt h wherefrom it follows that tr a( t ) = const. Lemma 5.4. Let ft be a family of symplectic difleomorphisms of the open set 0 and let - tra(t) = - t r [ H ( t ) ,a ( t ) ]= 0 ,

A , : W:mp(0)

+

c W,""'(R2")

W;"'(ft(O))

be the corresponding family of isomorphisms A,a

(5.6)

=

Uf0 (A*.)

0

u-' .

Then a ( t ) = A,a satisfies equation (5.5) with the Hamiltonian H ( t ) E WD(R2n).

Proof of the Lemma. We have (5.7)

ir(t) =

Ol 0 u!- 0 a ( t ) - a ( t >o Ol 0 ut-'+ ufo (&,a) o q-'. 1

Denoting the map, inverse to

ft

by gt , we shall have

Since the section a is flat, aa/ax' = a a / a y ' , so that the last expression can be written in the form

468 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

23 1

Here we have used the fact that for a symplectic map f the expression o~jui(afk/ax')(ag~/ax')v' is symmetric with respect to u , E R2" . Substituting into (5.7), we get equation ( 5 . 5 ) , where H ( t ) belongs to W,+ but is, generally speaking, not a flat section. We will show that it is possible Applying the operator D to the both sides of to pick H ( t ) E WD(R2"). (5.5) and taking into account that Da(t) = 0 we get ( i / h ) [ D H ( t,) A,a] = 0 . Being fulfilled for any a E Wrmp(0), this equation means that D H ( t) is a central (i.e., scalar) 1-form ly , which is closed because

d y =Dry = D(DH) = 0 . Hence, y = d p , where qt = q p , ( xh) , is a scalar function, which is uniquely defined if subjected to the normalizing condition q , ( J ; ( x o )= ) 0, where xo E 0 is any fixed point. Replacing H ( t ) by H ( t ) - qt we do not change equation ( 5 4 , as p, belongs to the center, and, on the other hand, D ( H ( t )- pt) = 0 , i.e., H ( t ) = pt is a flat section. q.e.d. Let us proceed to prove the theorem. According to the lemma, it is sufficient to construct a family of symplectic diffeomorphisms fi , so that = id , and = f . Besides, we may confine ourselves to a sufficiently small neighborhood 0., of an arbitrary fixed point xo E 0 . A general case would be obtained by using a partition of unit subordinated to a sufficiently fine covering of a compact set supp a . The desired deformation fi is constructed in two steps. At the first step, consider the linear part L f of the map f at the point xo given by

fo

fi

Since the group of linear symplectic transformations is connected, there exists a deformation, connecting the identity map with L/.. At the second step, consider a nonlinear map LT1f(x) . In a sufficiently small neighborhood of x,, it is close to the identity map, so it may be given by a generating function S(z ) according to the formulas

(5.8)

i

i

ijaS(z)

x = z +o

-,

aZJ

f'(X)=Z

i

--o

ij

-.as(z ) aZ J

469 232

B. V. FEDOSOV

From ( 5 . 8 ) we get (5.9)

Z=

+

f(x)

dS(z) =

1 pi’( f q x ) - x’) dz’ .

2 ’ It is easy to see that the 1-form on the right-hand side of the second equation of (5.9) is exact. Indeed, its exterior differential is equal to

o i j ( d f ‘- d x ’ )A (df’

+ dx’) = w i j d f iA df’ - wijdx‘ A dx’ + o i j d f A dx’ - oijd x i A df’ .

The first two summands give 0 , since f is a symplectic diffeomorphism. The second two summands also give 0 , because i

oijd x A df’ = -aijdf’ A dx’ = ojidf’ A dx‘ .

Thus, (5.9) determine the generating function S ( z ), provided the first equation of (5.9) determines a diffeomorphism x H z , which is just so in a sufficiently small neighborhood O., . Besides, we have S ( z ) = O ( [ Z ~ Replacing the function S ( z ) in (5.8) by the functions t S ( z ), t E [0, 11, we shall have the desired deformation f , ( x ) in the sufficiently small neighborhood O., by formulas (5.8).

f is a linear transformation f ’ ( x ) = Afix’ with a symplectic matrix A:. , formulas (5.8), (5.9) give the Cayley transformation. Remark. If

6. A localization and a trace

In this section we construct a trace in the algebra W,(M) on an arbitrary symplectic manifold M . The basic tool is a localization, i.e., a representation of the algebra W’(M) by a compatible family of the algebras of quantum observables in R2“ . We shall denote the standard symplectic form on R2” by oo and the Abelian connection -6 d in W(RZ”) by Do. Let { 01}be a locally finite covering of the manifold A4 by local Darboux charts, { p i ( x ) }be a partition of unity subordinated to this covering, and : Ol 4R2” be coordinate maps. For a given symplectic connection a and the corresponding Abelian connection D in the bundle W = W ( M ) consider the algebra W,(M) of flat sections determined on M and its subalgebra Wzmp(Ui), consisting of flat sections with supports in Oi . Using the constructions of $4 we may define isomorphisms

+

x,

A i : W;mP(Oi)

---t

WDOc o m p ( ~ i ( ~ i ) )

470 GEOMETRtCAL CONSTRUCTION OF DEFORMATION QUANTIZATION

233

of the form (4.1 1). More precisely, for a E W ~ m p ( O ,we ) take its pushforward Xi,a, which is a flat section in Wcomp(R2n)with respect to the comp connection Di = x , * D , i.e., Xi*a E W- ( ~ $ 0 , )After ) . that we pass N

Di

from the connection Zi to the connection Do using the conjugation automorphism. Finally we get

A ~ w;""(o~) : 3 a ++ U,o (xj,u) o ui-' E

w~~~~ Do (x;( 0 ; ) ) '

We shall call A , coordinate isomorphisms. For the algebra Wrmp(Oin O j ) we have two coordinate isomorphisms A , and A j , and thus transition isomorphisms are defined as follows: 1

w comp

A . .= A.AT : (x,(o,n oj))-+ w ~ ~ n ~oil). ~ (6.2) IJ I J Do From (6.2) it immediately follows that A i j satisfies a cocycle condition

A i j A j k A k i= id

(6.3)

n O, n o k ) .

in the algebra w'(Oi

Using Theorem 3.3 we can construct the flat sections Di = a - ' ( p i ) E WFmP(Oi) , which form a partition of unity in the algebra WD( M ) . Indeed

Cjj,= C a - l ( p J = 6' (p)= fJ

-I

i

(1) = 1 .

i

So, we obtain a set of flat sections (6.4)

ui = A,(Di o U ) E W ~ ~ ( x , ( Oc, WDo(R2") ))

corresponding to the flat section a E W D ( O i )We . shall call this set a local representation of the section a , or shorter a localization. It is clear that a = C i A i 1 a1 ' , Definition 6.1. A trace in the algebra W,(M) is a linear functional defined on an ideal Wrmp(M)with values in Laurent formal series, containing negative powers of h not greater than n = $ dim M . For any a E W,"""(M) and b E W D ( M )the equality t r a o b = tr b o a

(6-5)

must be fulfilled. Theorem 6.2. A trace in the algebra WD(M)does exist. PrmJ For a given coordinate covering {Oiland a partition of unity { P i W ) take

tra=Ctrai =CtrAi(jioa), i

1

47 1

B. V. FEDOSOV

234

where a E W,(M) and the traces tra, are given by formula (5.2) in Wcomp(R2n).We should check the correctness of the definition, i.e., indeDO pendence of the choice of a covering, a partition of unity, and coordinate isomorphisms A; , and then prove property (6.5). Let us prove the independence of the choice of coordinate isomorbe a flat section with a support in the cophisms. Let a E W,””*(O) ordinate neighborhood 0 c M , x and x’ be two coordinate diffeomorphisms 0 --,R 2 ” , and A and A’ be the corresponding coordinate isomorphisms, mapping W,””’(O)onto W,”o”p(x(O)), W T ( x ’ ( 0 ) )respectively. Then the symplectic map f = x ‘ x - ’ : ~ ( 0 +) ~‘(0) and the are decorresponding isomorphism Af : Wcomp(x(0))-+ W:mp(x’(0)) DO fined. According to Theorem 5.3 we have tr AfAa = tr Aa .

(6.7)

Generally speahng, the automorphisms AfA and A’: W;”’(O) -+ W“””(~‘(0))do not coincide. However they differ by an inner automorDO phism of the algebra W T ( x ’ ( 0 ) )(see Corollary 4.5), i.e., there exists a section S E Wtmp(x’(0)), such that A,Aa = S o (A’a)0 S-’ . Hence, according to property (5.3) of the trace in the algebra WD0(R2“) we get tr AfAa = tr S o (A’a)o S-’ = tr A’a ,

as desired. The independence of a covering and a partition of unity is now proved in a standard way, i.e., by passing to a refined covering { Oin O j } and the corresponding partition of unity { p i p j } . Let us prove the equality (6.5). We have a0b =

i ,;

=trboa, which proves the theorem.

C(pi a ) 0

0

( p j 0 b ),

472 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

235

7. Generalizations The above constructions of the deformation quantization and the trace in the algebra of quantum observables allow different generalizations. Quantization with coefficients in Hom(E , E) . An evident generalization consists in considering matrix coefficients. No change is necessary, except that in the definition of the trace (5.2) a matrix trace under integral sign must be taken. A less evident generalization is obtained if we admit that the coefficients ak , i, ...i ,j , ...j , ( x ) of series (2.3), (3.4) take values in a bundle Hom(E , E) , wheri E is a vector bundle over M . Let us consider this case in more detail. Let as be a symplectic connection on M , and aE be a connection in the vector bundle E . Then d = as8 1 1 c3 dE defines the connection in Weyl algebras bundle W 8 Hom(E , E) ; we will denote this bundle by W as before for short. We shall look for Abelian connection D in the bundle W in the same form as in (3.1). The same equation (3.3) is obtained for r , R being now equal to

+

(h/2)R: dx’

A

dx’

+ :RijklyiyJdxk A dx’ ,

where the first term is the curvature of dE ,and the second one is the same as in (2.1 1); the superscript s means “symplectic”. Theorems 3.2 and 3.3 are completely valid in this case. As to the action of symplectic diffeomorphisms, the results of $4 are also valid with some modifications. Let &: M --+ A4 be a symplectic diffeomorphism of M , and f , * E = (&-‘)*E be a pushforward of the bundle E under 4 , i.e., (&-’)* is an induced bundle. Let a fiberwise homomorphism a, : ( f , , E ) + E be given as well. Then the formula (7.1)

defines a lifting of the map f’ onto a bundle space Hom(E, E) . We define the pushforward of a section a ( x , y , h) E W 8 Hom(E, E) , by assuming (7.2)

So, if lifting (7.1) is given, the pushforwards and pullbacks of the sections and the connections a and D are defined as in 54. In the case of coefficientsin Hom(E , E) localization, considered in $6, is constructed as before with some modifications. More exactly, not only

413 B. V. FEDOSOV

236

a coordinate mapping x, : 0. R2nis to be given, but the trivialization of the bundle E as well. The trivialization defines a lifting x of x, , so that sections of Hom(E , E) over 0 go to matrix-valued functions on ~ ~ and allow us to define coordinate isomorphisms

and then a trace can be defined as before by formulas (6.6), (5.2) with the matrix trace under integral sign in (5.2). Deformation quantization of regular Poisson manifolds. As mentioned in the introduction, a regular Poisson manifold has a symplectic foliation. It means that it is possible to introduce local coordinates (Darboux coordinates) I

2

2n

x ,x , ..* > x , x

2n+l

,

m

,x ;

2n = rank(tij), m = d i m M ,

in which the components of Poisson tensor t” have the form i+n

- 1,

ti+n, i

- -1;

i = 1 ,2 ,

,n ,

and the rest of its components are equal to 0 . The leaves F of the foliation are locally defined by equations x k = const , k = 2n + 1 , - m7 dx’ A dxn+i defines a symplectic structure on the and the form o = leaves. Thus the regular Poisson manifold can be locally considered as a rn family of symplectic manifolds depending on parameters x 2n+ 1 , ,x . The quantization construction, given in 552, 3, smoothly depends on parameters and is local, so it is evidently valid for the case of regular Poisson manifolds. More precisely, the construction looks as follows. We consider a tangent bundle T F along the leaves and the exterior algebra A, = A(T*F). A homomorphism i* : T * M .+ 7°F is defined, induced by a local embedding of the leaf i: P --,M . In Darboux local coordinates we introduce the natural basis of vector fields e k = a / 8 x k ( k = 1, - , 2 n ) tangent to the leaves and the dual basis 8 = (8’, . . . , 1 9 ~in~ )T*F . Instead of series (2.4) we will now consider the series

x:=l

--

Y

* * .

(7.3)

where the terms uk , p , are the sections of Sp(TF) @ Aq(T * F ), x E M , the range of the indices i, , j , is from 1 to 2n. Such series form an

474 GEOMETRICAL CONSTRUCTION OF DEFORMATION QUANTIZATION

237

algebra with respect to the fiberwise multiplication o (the exterior product 8 and Weyl product (2.2) for monomials in y 's). The algebra of of ' such series will be denoted by W 8 A, . The operators 6 , S* , 6-' are introduced similar to (2.5) as follows: k .

6*a = v z(e, ) a .

(7.4)

and have the same properties, including Hodge-De Rham decomposition. We shall also need a Poisson connection along the leaves a : C" ( T F ) .-+ C"(TF 8 A:) . For such a connection, its local restriction on each leaf gives a symplectic connection on the leaf. For the sake of completeness let us give the construction of such a connection. Let V be an arbitrary connection in the bundle T F over M . Let us denote the indices ranging from 1 to 2n by Roman letters and those ranging from 1 to rn by Greek letters. In Darboux local coordinates we have i j i Ve = e rjad x " . Restricting it to the vectors tangent to the leaves, we get a connection 0, along the foliation (7.5)

V,e

i

=e

j

i

rjke

k

,

the Jacobian matrix of the transition diffeomorphism between two Darboux local charts $ = ax'"/axS has a triangular form, because a*'"/axi=O,

a=2n+l,..* m, 2

(4)

and its upper left block gives a transition function of the bundle T F . Hence, a skew-symmetric part of the connection coefficients defines i a tensor rjjk) in the bundle TF (a torsion tensor), since afi!/axk=

a2fi/Llx'dxk are symmetric with respect to j , k . Thus, symmetrizing the coefficients r;k in ( 7 4 ,we get a new torsion-free connection V F along the foliation. which is symmetric in lower indices and Finally we find a tensor m;.k, k such that the connection 8 = V F +r:,k8 preserves the tensor oijinverse to t i J . We have

475 238

B. V. FEDOSOV

wherefrom we obtain the equations (7.6) A r i j kfor A r i j k = w IP. ArpJk

In Darboux local coordinates

' f k being the coefficients of given by

V F . A partial solution of

system (7.6) is

(7-7)

(the general solution is obtained by adding to (7.7) any completely symmetric 3-tensor). Thus, we obtain the connection a along the leaves in the bundle T F such that its restriction to any leaf gives a symplectic connection on the leaf. According to (7.7) it smoothly depends on the coordinates x 2n+l , ...

9

x m , which are parameters, defining the leaf. Theorems 3.2 and 3.3 give a

smooth dependence on these parameters and thus define quantization for regular Poisson manifolds. As for the results of $84, 5, 6 it is not quite clear whether a reasonable generalization of these results for regular Poisson manifolds could be made.

References [ l ] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerovicz & D. Stemheimer, Deformation fheory and quanfizafion,Ann. Phys. 111 (1978) 61-151. [ 21 M. De Wilde & P. B. A. Lecomte, Existence of star-product and of formal deformations in Poisson Lie algebra of arbitrary symplectic manifold, Lett. Math. Phys. 7 (1983) 487-496. [3] B. Fedosov, Formal quantization, Some Topics of Modem Math. and Their Appl. to Problems of Math. Phys., Moscow, 1985, pp. 129-1 36. [4] -, Quantization and index, Dokl. Akad. Nauk. SSSR 291 (1986) 82-86, English transl. in Soviet Phys. DOH.31 (1986) 877-878. [ S ] -, An index theorem in the algebra of quantum observables, Dokl. Akad. Nauk SSSR 305 (1989) 835-839, English transl. in Soviet Phys. DOH.34 (1989) 318-321. [6] A. Masmoudi, Ph.D. Thesis, Univ. de Metz (1992). [7] D. Melotte, Invariant deformations of the Poisson Lie algebra of a symplectic mani-

fufd and star-products,Deformation Theory of Algebras and Structures and Applications, Ser. C: Math. and Phys. Sci., Vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, 961-972. 181 H. Omori, Y. Macda & A. Yoshioka, Weyl manifolds and deformation quantization, Advances in Math. (China) 85 (1991) 224-255.

Moscow INSTITUTE OF PHYSICS AND TECHNOLOGY

476 PHYSICAL REVIEW D, VOLUME 58, 025002

Features of time-independent Wigner functions Thomas Curtright* Department of Physics, University of Miami. Box 248046, Coral Gables, Florida 33124

David Fairlie’ Department of Mathematical Sciences, Universily of Durham, Durham, DHl3LE. United Kingdom

Cosmas Zachod High Energy Physics Division, Argonne National Laboratory, Argonne. Illinois 60439-4815 (Received 26 November 1997; published 8 June 1998)

The Wigner phase-space distribution function provides the basis for Moyal’s deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (“star”) eigenvalue equations they satisfy: their projective orthogonality spectral properties; their Darboux (“supersymmetric”) isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Poschl-Teller potential, and the Liouville potential. [S0556-2821(98)00714-0] PACS number(s): 11.15.Tk,03.65.Db, 04.20.Fy, 05.30.-d I. INTRODUCTION

Wigner functions have been receiving increasing attention in quantum optics, dynamical systems, and the algebraic structures of M theory [l]. They were invented by Wigner and Szilard [2], and serve as a phase-space distribution alternative to the density matrix, to whose matrix elements they are related by Fourier transformation. The diagonal, hence, real, time-independent pure-state Wigner function f ( x , p ) corresponding to the eigenfunction JI of HJI= EJI, is

f(x,p)=&

dyJI*(x-iy)e-’”PJI(x+iy).

(1)

These functions are not quite probability distribution functions, as they are not necessarily positive-this is illustrated below. However, upon integration over p or x , they yield bona fide positive probability distributions, in x or p . respectively. Wigner functions underlie Moyal’s formulation of quantum mechanics [3], through the unique [4,5] one-parameter (ti) associative deformation of the Poisson-brackets structure of classical mechanics. Expectation values can be computed on the basis of phase-space c-number functions: given an operator A(x,p), the corresponding phase-space function A ( x , p ) obtained by p e p , XHX yields that operator’s expectation value through

(2)

assuming the usual normalization J d x d pf ( x . p )= 1 and further assuming Weyl ordering, as addressed by Moyal, who took matrix elements of all such operators:

X exp[ i T( p - p )

.

(3)

Wigner functions are c numbers, but they compose with each other nonlocally. The properties of these compositions were explored in, e.g., [6,7], and were codified in an elegant system in [5]: to parallel operator multiplication, the Wigner functions compose with each other through the associative star product

(4) Recalling the action of a translation operator exp(aa,)h(x) =h(x+a), it is evident that the * product induces simple “Bopp” shifts:

(5) etc., where a’ and a’ here act on the arguments o f f and g, respectively. This intricate convolution samples the Wigner function over the entire phase space, and thus provides an alternative to operator multiplication in Hilbert space. Antisymmetrizing and symmetrizing the star product, yields the Moyal (sine) brackets [3]

*Electronic address: [email protected] ‘Electronic address: [email protected] *Electronic address: [email protected] 0556-2821/98/58(2)/025002( 14)/$15.00

+ ia(x-x)]

(6) 58 025002-1

0 1998 The American Physical Society

477 ~~~

PHYSICAL REVIEW D 58 025002

THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS

holds for the time-independent pure-state Wigner functions (lemma l), and amounts to a complete characterization of them (lemma 2). We will explore the features of this *-genvalue equation, and illustrate its utility on a number of solvable potentials, including both the harmonic oscillator and the linear one. The * multiplications of Wigner functions will be seen to parallel Hilbert-space operations in marked detail. The Poschl-Teller potential will reveal how the hierarchy of factorizable Hamiltonians familiar from supersymmetric quantum mechanics finds its full analogue in * space. We determine the Wigner function's transformation properties under (phase-space volume-preserving) canonical transformations, which we finally elaborate in the context of the Liouville potential.

and Baker's [6] cosine brackets

(7) respectively. Note [7,8] that

I

dpdxf*g=] dpdxfg.

(8) (8)

Further note the Wigner distribution has a *-factorizable integrand:

(9) In general, a systematic specification of time-dependent Wigner functions is predicated on the eigenvalue spectrum of the time-independent problem. For pure-state static distributions, Wigner and, more explicitly, Moyal showed that

11. *-GENVALUE EQUATION

Lemma 1. Static, pure-state Wigner functions obey the *-genvalue equation

(10) (10)

{ { H ( x 4 J ) . f ( x 7 p )= } }0;

i.e., H and f * commute. However, there is a more powerful functional equation, the "star-genvalue" equation, which

H ( x , p ) * f ( x , p )= Ef(x,p).

Without essential loss of generality, consider H ( x , p ) =p2/2m+ V ( x ) ,

(x- 5 Y )

I'= I'=

*(

2 a

d y [ ( p - i 5 2 x ) 2 / 2,+V(x+5y)]e-'"**(x-5y)*[x+5y)

2 a

d y e C t Y p I i( e y + i s 2 x ) ? /

=1

x+ 5 Y )

,m+V(x+5.)]**(x-sy)*(x+syi

j

dye-IYp$*(x- i y ) E + ( x + s y ) = E f ( x , p ) ,

(12)

2a since the action of the effective differential operators on f *H= 2 n

] dye-I'p[ ( 3)-

**

(1 1 )

turns out to be null, and, likewise,

i 2,) * /

2m

+ V ( x-

5 ) ] 5 ) +( y

$* ( x - y

x+ i y )

(13)

= Ef(X,P). I

Thus, both of the above relations (10) and lemma 1 obtain. This time-independent equation was introduced in Ref. [7], such that the expectation of the energy H ( x , p ) in a pure state time-independent Wigner function f(x,p) is given by

(14) On account of the integration property of the star product, Eq. (8). the left-hand side of this amounts to J d x d p H ( x , p )* f ( x , p ) . Implicitly, this equation could have

been inferred from the Bloch equation of the temperatureand time-dependent Wigner function, in the early work of [9]. *-genvalue equations are discussed in some depth in the second reference of Ref. [5] and in [ l o ] . By virtue of this equation, Fairlie also derived the general *-orthogonality and spectral projection properties of static Wigner functions [7]. His results were later formalized in the spectral theory of the second of Ref. [5] [e.g., Eq. (4.4)]. Consider g corresponding to the (normalized) eigenfunction $g corresponding to energy E , . By lemma 1 and the associativity of the * product,

02:im2-2

478 FEATURES OF TIME-INDEPENDENTWIGNER FUNCTIONS

(15)

f*H*g=Eff*g=E8f*g. Then, if E , f E f , this is only satisfied by

N.B. The integrated version is familiar from Wigner’s paper, d x d pf *g =

duced by Fairlie appears local, but is, of course, highly nonlocal, by virtue of the convolving action of the * product. Precluding degeneracy, for f = g , f * H *f = E,f

(16)

f*g=O.

I

PHYSICAL REVIEW D 58 025002

I

d x d pf g

= 0,

*f = H *f *f ,

(18)

which leads, by virtue of associativity, to the normalization relation [6]

(17) f*faf.

and demonstrates that all overlapping Wigner functions cannot be everywhere positive. The unintegrated relation intro-

(19)

Both relations (16) and (19) can be checked directly:

(20) I

The second integral factor is 0 or l / h , depending on f Z g or f = g , respectively, specifying the normalization f * f = f l h in Eq. (19). In conclusion, Corollary 1. f,*fb= llh6,,bf0. These spectral properties are summoned up by their own necessity; much of their meaning, nevertheless, resides in their margins: For nonnormalizable wave functions, the above second integral factor may diverge, as illustrated below for the linear potential, but the orthogonality properties still hold. Thus, e.g., for an arbitrary function(a1) F ( z ) ,

(21) and, for

* genfunctions of lemma

1,

(22 Baker’s converse construction extends to a full converse of lemma 1 , namely, the following lemma. Lemma 2. Real solutions of H ( x , p ) * f ( x , p ) = E f ( x , p ) [=f ( x , p ) * H ( x , p ) ] must be of the Wigner form, f =Jdye-’YPJI*[x- ( h l 2 ) y ] J I [ x +( h / 2 ) y ] / 2 ~ such , that HJI= EJI. As seen above, the pair of *-eigenvalue equations dictate, for f ( x .p ) = J d y e - ‘ Y P Y ( x ,y ) ,

(23) This constrains T ( x , y ) to consist of bilinears JI*[x - ( h l 2 ) y l $ [ x + ( h l 2 ) y l of unnormalized eigenfunctions $ ( x ) corresponding to the same eigenvalue E in the Schrodinger equation with potential V. W These two lemmata then amount to the statement that, for real functions f ( x . p ) , the Wignerform is equivalent to compliance with the *-genvalue equation (real and imaginary part). 111. EXAMPLE: THE SIMPLE HARMONIC OSCILLATOR

The eigenvalue equation of lemma 1 may be solved directly to produce the Wigner functions for specific potentials, without first solving the corresponding Schrodinger problem (as in, e.g., [l 11). Following [7],for the harmonic oscillator, H = ( p 2 + x 2 ) / 2 (with h= 1, m= l ) , the resulting equation is

025002-3

479 THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS By virtue of its imaginary part (nd,-pd,)f=O, f is seen to depend on only one variable, z = 4 H = 2 ( x 2 + p z ) , and so the equation reduces to a simple ordinary differential equation;

(25)

Moreover, setting f(z) =exp(-r/2)L(z), this yields

(

2l l L ( z ) = O ,

zd:+(l-z)d,+E-

PHYSICAL REVIEW D 58 025002

These states are real, like the Gaussian ground state, and are thus left-right symmetric * genstates. They are also transparently * orthogonal for different eigenvalues, and they project to themselves, as they should, since the Gaussian ground state does, f o * f o a f o . It will be seen below that even the generalization of this factorization method for isospectral potential pairs goes through without difficulty.

IV. FURTHER EXAMPLE: THE LINEAR POTENTIAL

(26)

For simplicity, take m = 1/2, A = 1. Recall [12] that the problem readily reduces to a free particle: H ( x , p ) = p 2 + x - H f r e e = P is accomplished by canonically transforming through the generating function F ( x , X ) = - fX3-xX. The energy eigenfunctions are Airy functions,

which is the equation satisfied by Laguerre polynomials L, =e'd"(e-'z"), for n=E-1/2=0,1,2,. . . , so that the unnormalized Wigner eigenfunctions are

$E(x)=

2;;

I+-m

dXe'F(X-X)e'EX=Ai(x-E). (33)

The *-genvalue equation in this case is

f, = e-2HL,(4H), Lo= 1, L , = l - 4 H ,

L2=16H2- 16H+2,. . . .

(34)

(27)

Note that the eigenfunctions are not positive definite, and are the only ones satisfying the boundary conditions,f(O) finite andf(z)+O, as z+m. In fact, Dirac's Hamiltonian factorization method for algebraic solution caries through (cf. [5]) intact in * space, Indeed,

1 H = -2( x - i p ) * ( x + i p ) +

1

v?€

= f ( * + P 2 ) = f ( H ) . The real Part of the equation is then an ordinary second-order equation, just as in the above harmonic oscillator case. Moreover, here the real part Of the *-genValUe equation is essentially the same as the usual energy eigenvalue equation:

(35)

1

-, 2

(28) where z = x + p 2 . Hence, the Wigner function is again an Airy function, like the above wave functions, except that the argument has a different scale and shift:'

motivating the definition of a= - ( x + i p ) ,

whose imaginary Part ( i d ~ - p d x ) f ( x J ' ) = o gives f ( x - p )

1 at= v?€ ( x - iP).

22~3

(29)

Thus, noting that a*nt-at*a=

1

1

a*fo= - ( x + y9

ip)* e - ( x 2 + p 2 ) = 0

(31)

provides a *-Fock vacuum, it is evident that associativity of the * product permits the entire ladder spectrum generation to go through as usual. The * genstates of the Hamiltonian, such that H* f =f * H , are thus f n m ( a t*)"fa( * a ) " .

(36)

(30)

and also that, by the above,

(32)

22~3

f ( x , p ) = %A' I ( ~ ~ ~ ( z - E )Z;;~i(22J3(X+p2)= E))

The Airy functions are not square integrable, so that the conventional normalization f*f= ( 1/2~r)f does not strictly apply. On the other hand, the energy eigenfunctions are nondegenerate, and the general corollary 1 projection relations f n * f b a 6,,bf, still hold for the continuous spectrum:

'This case is similar to the Gaussian wave function, i.e., the harmonic oscillator ground state encountered above, whose Wigner function is also a Gaussian, but of different width. S. Habib kindly informed us that this solution is also given in Ref. [13], Eq. (29).

025002-4

480 FEATURES OF TIME-INDEPENDENT WIGNER FUNCTIONS

by virtue of the direct definition (36).

PHYSICAL REVIEW D 58 025002

i.e., the one with a partner potential

V. DARBOUX CONSTRUCTION OF WIGNER FUNCTION

(44)

RECURSIONS

Analogous ladder operators for eigenstates corresponding to “essentially isospectral” pairs of partner potentials [14] [familiar from supersymmetric quantum mechanics (SSQM)] can also be defined muratis murandis for Wigner functions and * products. They faithfully parallel the differential equation structures. Consider a positive semidefinite Hamiltonian H = p 2 / 2 m + V(x).

This can be written as a

has Wigner function of H . Specifically,

H*f=Q**Q*f=

are

* genfunctions

H ’ * ( Q *f*Q * ) = Q * Q * * Q *f*Q * = E ( Q *f*Q * ) , (46)

* product of two operators,

provided

(45)

f*Q**Q=Ef

implies that the real functions Q*f*Q* of H ’ with the same eigenvalue E ,

(38)

(39)

* genstates of the same energy as those

unless f is the Wigner function corresponding to t,bo, since Q *fa = 0. In consequence, EA = E , + for n 5 0. Conversely, for g * genfunctions of H ’ , Q * * g * Q are * genfunctions of H with the same eigenvalues. l/$o will be an invalid zero mode eigenMoreover, function of H’, as seen from the sign flip in Eqs. (41) and (44).Consequently, an unnormalized, runaway zero-energy solution of the Schrodinger equation with V‘(x) will invert to the legitimate ground state of H and will permit construction of V given V’. For example, starting from the trivial potential with a continuous (unnormalizable) spectrum,

@A=

(40) This Riccati equation, familiar from SSQM, can be Darboux transformed by changing variable for the “superpotential” Wx).

(41)

V’ = 1,

(47)

and the solution which reduces the condition to the Schrodinger equation for a zero eigenvalue: ti2 - 2m _ a,’*o+ v(x)*o=o.

(42)

Also note Q* f o = 0 for the corresponding Wigner function. It is easy to generalize this by adding a constant to H to shift the ground state eigenvalue from zero. By virtue of associativity, it is evident that the partner Hamiltonian 275 H I = Q* Q* = H + -ax

&

w,

(43)

(48) results via Eq. (40) in the symmetric, reflectionless PoschlConTeller potential [15], V= 1 - 2/cosh2[(&&/fi)]. versely, starting from this potential,

2 V(x) = 1 -

7) ’

coshz( &x

there is a single bound state (normalizable to J” I,@=2),

025002-5

(49)

48 1 PHYSICAL REVIEW D 58 025002

THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS

(50 so that

(51)

V’= I.

Thus, the Wigner function ground state (for m = 1/2) is 1

- iyp

%

fo(x’p)’

f o ( N ; x , p ) = [ s e c h ( n / R ) * ] N - ~ fl;x,p)[* o( sech(x/h)]’”-l. (58)

dy 2 cosh(x/fi-y/2)cosh(x/h+y/2)

COS(YP) dycosh(2x/fi)+cosh(y)

-

(57) where the integral only need be evaluated from the above fo( 1 ;x,p). Alternatively,

sin( 2xplh) sinh(2x/R)sinh( v p ) ’

(52)

The (unnormalized) state above the ground state at E =-(N-l)’ is [ ( ~ I \ / ~ ~ ) ~ ’ , - W ( N ) ] I , O O ( N and - ~ ) , its corresponding Wigner function (setting m= 1/2) is found recursively from the ground state of H ( N - I ) , through Q*(N)*fo(N- 1)*Q(N).

I

:”

Q*fo= p--d

i;l ; 11

- i tanh - + - a p

1

!

[N.B. It is not positive definite or a function of just N(x,p).] It may be verified directly that

p*fo(N- l ) + i N tanh - * f o ( N - 1) *Q(N)

fo(x,p)=O.

(53)

2N-1

=(

This appendage of bound states to a potential generalizes [ 161 to the hierarchy associated with the Korteweg-de Vries (KdV) equation. Specifically,

W ( n ) = n tanhi

9)

m)p*fo(N-

(59)

l)*P.

by virtue of

Q(N-l)*fo(N-I)=O=fo(N-l)*Q*(N-l). (60)

(54) (54)

The state above that, at E = -(N-2)’, sively through

connects the reflectionless Poschl-Teller potential

Q*(N)*Q*(N- l)*fo(N-2)*Q(N-

is found recur1)*Q(N), (61)

V’(x)=n’-n(n-

and so forth. Thus, the entire Wigner *-genfunction spectrum of H ( N ) is obtained with hardly any reliance on Schrodinger eigenfunctions.

l)/cosh2

to its contiguous

VI. CANONICAL TRANSFORMATION OF THE WIGNER FUNCTION

(55) which has one more bound state (shape invariance). Recursively, then, one may go in N steps, with the suitable shifts of the potential by 2n- 1 in each step, from the constant potential to

For notational simplicity, take ti= 1 in this section. The area element in phase space is preserved by canonical transformations (x,P)H(x(x,P).p(x,P))

which yield trivial Jacobians (dXdP =dxdp{X,P}) by preserving the Poisson brackets

( 7).

V ( N ; x ) = N Z - N ( N + I)/co~h’ -

(62)

(56)

au {U,V},p=-----

Shifting this potential down by N Z assigns the energy E = -AJ2 to the corresponding ground State $o(N)=SechN(x) (unnormalized), which is the null state of (hl&)d, + W ( N ) . The corresponding (unnormalized) Wigner function is the *-null state of Q(N).

av

ax ap

au a, ap ax‘

(63)

They thus preserve the “canonical invariants” of their functions:

025002-6

{ X , P } , , = 1 and hence { ~ , p } ~1. ~ = (64)

482 FEATURES OF TIME-INDEPENDENT WIGNER FUNCTIONS

PHYSICAL REVIEW D 58 025002

Equivalently,

(66) (65)

{x.pI= { X . P ) ,

in any basis. Motion being a canonical transformation, Hamilton's classical equations of motion are preserved, for ' H ( X , P ) = H ( x , p ) , as well [17]. What happens upon quantization? Since, in deformation quantization, the Hamiltonian is a c-number function, and so transforms "classically," ' H ( X , P ) = H ( x , p ) , the effects of a canonical transformation on the quantum *-genvalue equation of lemma 1 will be carried by a suitably transformed Wigner function. Predictably, the answer can be deduced from Dirac's quantum transformation theory. Consider the canonical transformations generated by F ( x , X ) :

Following Dirac's celebrated exponentiation [18] of such a generator, in the implementation of [ 12,191, the energy eigenfunctions transform canonically through a generalization of the "representation-changing" Fourier transform:

(67) Thus,

(68) The pair of Wigner functions in the respective canonical variables, f ( x , p ) and

'I i

3(X,P)=2 a

dYq* X--Y

")

e-jyPq x+-Y 2

,

(69)

are connected by a transformation functional T ( x , p ; X , P ) ,

(70) where 0 is with respect to the variables X and P . To find this functional, let X = $ ( X I+ X 2 ) and Y = X , - X I , so that J d X l J d X 2 = J d X J d Y . Noting that

(:ii

'I!* X - - Y

'I! X + - Y

=

dPeiYP3(X,P),

(71)

it follows that Eq. (68) reduces to

(72)

483 PHYSICAL REVIEW D 58 025002

THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS

which leads to the following lemma. Lemma 3. ~ ( n ,;x, p P ) = ( I ~ I ~ / 2 . r r ) J d ~exp[ d y - iyp +iPY-iF*(x-y/2$Y/2) iF(x+yR,X+ Y/2)]. Corollary 2. This phase-space transformation functional obeys the “two-star” equation

+

H ( x , p ) * Z ( x , p ; X , P ) = ~ ( x , p ; X , P ) O 7 - I ( X , P ) ,(73)

as follows from H(x, - id,)exp[iF(x,X)] =li(X,idx)exp[iF(x,X)]. If F satisfies a O-genvalue equation, then f satisfies a *-genvalue equation with the same eigenvalue, and vice versa. Note that, by virtue of the spectral projection feature (16), (19), this equation is also solved by any representationchanging equal-energy bilinear in real Wigner * genfunctions of H and H,

(74) for arbitrary real g ( E ) . Such a bilinear transformation functional is nonsingular (invertible) if and only if g ( E ) has no zeros on the spectrum of either Hamiltonian? As an example, consider the linear potential again, which transforms into a free particle (‘H=P) through

1 F = - -X3- x x * p = - x ,

x=P-X2.

(75)

By direct computation,

(76) Note N,= l/Gfor the free-particle energy eigenfunction normalization choice .U,(X) = (2 n)-In exp(iEX). Thus, indeed, the free-particle Wigner function F,(X,P) = 6 ( E - P ) / ( 2 ~ r )transforms into

‘I

f(x,p)=G

dPdXZ6(E-P)

2 ~ 3

= -Ai(22’3(x+p2-~)), 2n

(77)

as it should, and Eq. (73) is seen to be satisfied directly, by virtue of the linearity of the respective Hamiltonians in the variables P , x , conjugate to those of the arguments of S ( p +X). The structure of the result in Eq. (76) underscores that the linear potential is as “close to classical” as one can get, in simple quantum mechanics. It has been noted before [12] that the transformation functional for linear potential wave functions is exactly the exponential of the classical generating function for the canonical transformation to a free particle, and that this is not the case for any other potential. The present result for the transformation functional for Wigner functions is further evidence for this “close to classical” behavior. The delta function 6(p+X) in Eq. (76) is halfof the classical story. Were the Airy function also a delta function of its argument, we would have an exact implementation of the X,P-x,p classical correspondence. As it is, there is some typically quantum mechanical spread around the classical constraint x X z - P = 0, in the form of oscillations of the Airy function, and, in consequence, the Wigner functions of the free particle do not retain their delta-function form under the canonical transformation to the linear potential Wigner functions. Reinstating fi into Eq. (36); and taking the limit fi-0 converts the Airy function to a delta function, 8(x + X 2 - P ) . thereupon producing the complete classical correspondence between the two sets of phase space variables, in that limit. As already seen, there is substantial nonuniqueness in the choice of transformation functional. For example, for the linear potential again, Eq. (73),

+

’In general, if the transformation functional effects a map to a free particle, the P integration is trivial in Eq. (70), and the result for the Wigner function of the x,p theory is just an average over X of the transformation functional. That is, if F(X,P)= S(P - k ( E ) ) , where k ( E ) is the momentum-energy relation for the free particle theory in question:

II

I

f(x,p)= dX dPZ(x,p;X,P)F(X,P)= dXIT(x,p;X,k(E)). One might then be tempted to wonder if just I(x,p;X,P)= $:(x - ~ X 1 2 ) e ~ ’ x p ~ ~ ( x + ~ XB(x,p;X,P). 1 2 ) 1 2 ~ ~However, what determines the allowed range for P? It is always possible to embed any real energy spectrum into the real line, but knowing this does not help at all to determine what points are to be embedded. From the point of view of this paper, even when the spectrum is obvious, such a choice for the transformation functional in general does not satisfy the two-* equation (73). Rather, the equation fails by total derivatives that vary contingent on particularities of the case. E.g., for free-particle plane waves, @&)=exp(iEx), so that p*@ -@@P=d,@. This choice for I , then, does not yield useful information on the Wigner functions.

(n+p’)*B(x,p;X,P)=B(x,p;X,P)OP

(78)

is also satisfied by a different (and somewhat simpler) choice:

(79) ’The exponent of the integrand turns into i y ( E - x - p 2

- h’yZl12).

025002-8

484 FEATURES OF TIME-INDEPENDENT WIGNER FUNCTIONS

PHYSICAL REVIEW D 58 025002

This transformation functional also converts the free-particle Wigner function F,(X,P)= 6 ( E - P ) / 2 r into an Airy function (as above) after integrating over the free-particle phase space, JdXdP.

Actually, it is not necessary to integrate over the phase space. In general, * multiplying a delta function spreads it out, and yields a Fourier transform with respect to the conjugate variable. Thus, for the example considered,

(80) Hence,

I! &'

dpe'[(-2/3)X3-2(X+PZ-P)Xl*

(81)

J ( p - E ) = 22Rr Ai(2m(, +$- E ) ) ,

Compare this to the action of the above Z(x,p;X,P),

'I

+

[Ai(22/3(x X2 - P )) 6 ( p + X)]* 6( P - E ) = e21X(P-E) r

+

+

dZe - 2 i Z ( P - E ) Ai(2u3(x 2'- P ) ) S ( p 2 )

(82) I

Aside from innocuous normalizations. the difference in the two transformation functionals acting on the free-particle Wigner function is just the phase factor e*ip(P-E)and the argument of the Airy function, where E has been replaced by P . Indeed, the phase factor precisely compensates for the different energy eigenvalue occurring in the argument of Ai, when acted upon by ( x + p 2 ) * . Such simple phase factors may be used to shift a * genvalue whenever the Hamiltonian is linear in any variable.

*E(x)=

$m

K , d e x ) ,

(85)

which are normalized such that J?",x $ , f , ( x ) $,,(x) = 6(E l - E , ) . There is no solution [20] for E = 0. For completeness, consider the Fourier transform (including a convergence factor, necessary for x-+--m to control plane wave behavior, but not for x+m)

VII. ILLUSTRATIONS USING LIOUVILLE QUANTUM MECHANICS

A summary illustration of all the above, in particular the canonical transformation effects on Wigner functions, is provided by the Liouville model [20]. Our conventions for the model [which are essentially those of [21], with their m = 1/(4r) and their g= 11 are given by

xr

(

+

.[

- i ( p + i t . ) i fi) - i ( p + it.) - i fi 2

2 (86)

(83)

This follows, e.g., from a result in [22], Vol. 11, p 51, Eq. (27):

The energy eigenfunctions are then solutions of

(87)

(84) The solutions are Kelvin (modified Bessel) K functions, for O<E<m,

valid for %(l + p ? v)>O (i.e., the previous transform is valid for E > O ) . The right-hand side of this last relation clearly displays the symmetry Y+ - Y , which just amounts

025002-9

485 THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS

to the physical statement that the energy eigenfunctions are nondegenerate for the transmissionless exponential potential of the Liouville model. Further note the effect on @E(p+iE) of shifting p+p +2i, using T(l+z)=zI'(z),

+

aE(p 2i + i E ) = 4

PHYSICAL REVIEW D 58 025002

Many, if not all, properties of the Liouville wave functions may be understood from the following integral representation "241, Chap. VI, Sec. 6.22, IZq. (lo)]. Explicitly emphasizing the abovementioned nondegeneracy,

(-i(p+ie)+ifi) 2

(90) = [ E - ( p f i ~ ) ~ ] @+~i e( )p.

(88)

So, as 6-0, QE(p+2i)=(E-p2)OE(p).But this simple difference equation is just the Liouville energy eigenvalue equation in the momentum basis, ( p 2 - ~ ) @ ~ +e2'dr@,(p) (p) =o.

(89)

Such first-order difference equations invariably lead to gamma functions [23]. Below, it turns out that the Wigner functions also satisfy momentum difference equations, but of second order.

(Also see [25], Eq. 9.6.22.)This integral representation may be effectively regarded as the canonical transformation of a free-particle energy eigenfunction e i k Xthrough use of the generating function F ( x , X ) = e x sinh X. Classically, p = d F l d x = ex sinh X and P = - d F l d X = - e x cosh X , and so Pz-pz= e2'. That is, HLiouuille=XfHf,,,=P2under the classical effects of the canonical transformation. The quantum effects are detailed below, by * acting with the Liouville and free Hamiltonians on the suitable transformation functional. The Liouville Wigner function may be obtained from the definition (1) in terms of known higher transcendental functions:

(91) The following

K transform was utilized to express this result in closed form:

(92) The right-hand side involves a special case of Meijer's C function,

(93) (cf. [22], Sec. 5.3). which is fully symmetric in the parameter subsets {al ,...,a,,}, {a,+,

,...,u p } , {bl ,.... b,,,}, and

{b,,,+] ,..., b q } . It is possible to reexpress the result as a linear combination of generalized hypergeometric functions of type , F 3 , but there is little reason to do so here. This transform is valid for %>O, and is taken from [26], p. 711, Eq. (55); The

transform is complementary to [27], Sec. 10.3, JZq. (49), in an obvious way, a K transform which appears in perturbative computations of certain Liouville correlation functions [21]. The result (91) may be written in slightly different alternate forms f(x,p)=

4 7r'

- sinh(7rG) gT3

[ e4'l i&-ip -i&-ip GGx 2 ' 2

1- 2 i f i - 2 i p 4 i&+ip

-i

1+2i&+2ip 4

~

1 -2i&+2ip 4

-

(94)

4There is an error in this result as it appears in [27], Vol. 11, Sec. 10.3, Eq. (58), where the formula has a2z2/4instead of a2zZ/16as the argument of the G function. The latter argument is correct, and appears in Meijer's original paper cited here. 025002- 10

486 FEATURES OF TIME-INDEPENDENT WIGNER FUNCTIONS

PHYSICAL REVIEW D 58 025002

by making use of the parameter translation identity for the G function “221, Sec. 5.3.1, Eq. (9)]:

(95) Yet another way to express the result utilizes the Fourier transform of the wave function, Q. (86). in terms of which the Wigner function reads, in general,

(96) The specific result (86) then gives, as e+O,

(97) However, this is a contour integral representation of the particular G function given above. Because of the E prescription, the contour in the variable z=k/2+ie runs parallel to the real axis, but slightly above the poles of the r functions located on the real axis at z = p - fi,z = p + fi,z = - p + fi,and z= - p - fi.Changing variables to s = i i z yields

(98)

+

where the contour C in the s plane runs from - i m to im, just to the left of the four poles on the imaginary s axis at i ( p + f i ) / 2 , i ( p - f i ) / 2 , i ( - p + f i ) / 2 , and i ( - p - f i ) / 2 . This is recognized as the Mellin-Barnes-type inte(: function [cf. [22],Sec. 5.3, Eq. (l)] gral definition of the 3 in agreement with the second result above, Eq. (94). The translation identity (95) is seen to hold by virtue of Eq. (98). through simply shifting the variable of integration, s . Moreover, deforming the contour in Eq. (98) to enclose the four sequences of poles s, = n i( 2 p 2 $)/2 reveals the equivalence of this particular G function to a linear combination of four functions, one for each of the sequences of poles. Evaluating the integral by the method of residues for all these poles produces the standard hypergeometric series. It should now be straightforward to directly check that the explicit result forf(x,p) is indeed a solution to the Liouville *-genvalue equation,

+

ffLiouuille*f(x,P)

1

jpz-E-zd:+s’~

COS

1

dp f(x,p)=O.

(101)

The first of these is a first-order differential-difference equation relating the x and p dependence: 1 e - 2 ” a ~ ( x , p=) 2ip I f ( x , p + i ) - f ( x , p - i) I .

(102)

Similarly, the real part of the *-genvalue equation is a second-order differential-difference equation: e - 2 x ( p z -E -

;

d:)fW

1

+ y [ f ( x , p + i ) + f ( x , p - i ) ] = 0. (103)

The previous first-order equation may now be substituted (twice) into this last second-order equation, to convert it from a differential-difference equation into a second-order difference-only equation in the momentum variable, with nonconstant coefficients:

(99) For real E and real f ( x . p ) . the imaginary part of this genvalue equation is (-pd,+e2’

sin d , ) f ( x , p ) = O ,

*-

( 100)

(104)

while the real part is 025002-11

487 THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS

PHYSICAL REVIEW D 58 025002

We leave it as an exercise for the reader to exploit the recursive properties of the Meijer G function and show that this difference equation is indeed obeyed by the result (91). Rather than pursue this in detail, we turn our attention to the transformation functional which connects the above result for f to a free-particle Wigner function. Given Eq. (90). it follows that

(105) and hence NE=[4rrfieTJE sinh(rrfi)]ln/2rr, if we choose a S ( E , - E , ) normalization for the free-particle plane waves as well as for the Liouville eigenfunctions. Therefore, lemma 3 yields

IN”

l ( x , p ; X , P ) = - dYdy exp[-iyp+iP Y - ~ F * ( X - ~ / ~ X - Y / ~ ) + ~ F ( X + ~ / ~ , X + Y / ~ ) ] 2rr 1 = 7 [ 4 7 r f i e n E s i n h ( r f i ) ] \ dYdy exp - i y p + i P Y-iex-yn sinh (2n) 1 =7 [ 4 n f i e T E

4rr

sinh(rrfi)]

(106)

(107)

(108)

(109)

(110)

(111) (112) (113) (114) 025002-12

488

(115) I

T h e s u m o f E q s . (110),(111),and(115)showsthatEq. (109) is, indeed, satisfied. Integrating over X and P the product of Z ( x , p ; X , P ) and the free-particle Wigner function, as given here by (4w4?)-'6(P - fi),yields another expression for the Liouville Wigner function which checks against the previous result, Eq. (91). Using Eq. (92) and the parameter translation identity for the G function, this other expression is just Eq.

(94). Supersymmetric Liouville quantum mechanics is obtained by carrying through the Darboux construction detailed above (with fi = 1 = 2m), for the choice W ( x )= e x .

(116)

The conventions used essentially follow [28]. The first Hamiltonian of the essentially isospectral pair is then H=p2+t?-eX,

The second Hamiltonian of the pair i s

H' = p Z + e Z x + e x ,

( 120)

and the allowed spectrum is O<E<m, excluding zero en erg^.^ The E>O eigenfunctions are then r

.

1 1"

x [ i K l n - , d e " ) - i K ~ z + , d e ~ ) l , (121) and may be obtained from the previous E>O eigenfunctions, as c ~ . ~ ( x ) = ( ~ / f i ) ( ~ ~ + ~ ) c ~ . ~ ( x ) . For both Hamiltonians, the Wigner functions are straightforward to construct directly, once again leading to the K transform (92) and particular Meijer G functions. W e find it sufficient here to consider only the ground state for H ,

(117)

and the allowed spectrum is O<E<m, including zero energy, for which there is a bounded wave function normalized as part of the continuum,

(118)

(122) a single modified Bessel function. It smoothly [29] satisfies [p - i W ( x ) ] *f o = 0 and, hence, the *-genvalue equation H*fo=O. ACKNOWLEDGMENTS

The other, E>O, eigenfunctions are

W e thank Y. Hosotani for helpful discussions. This work was supported in part by NSF Grant No. PHY 9507829 and by the U S . Department of Energy, Division of High Energy Physics, Contract No. W-31-109-ENG-38.

(119)

6

'The candidate @&x) = l/$o(x) = exp(e? solves the Schrodinger equation, but is obviously unbounded, as expected.

[I] M. Hillery, R. O'Conell, M. Scully, and E. Wigner, Phys. Rep. 106, 121 (1984); H.-W. Lee, ibid. 259, 147 (1995); N. Balasz and B. Jennings, ibid. 104,347 (1984); R. Littlejohn, ibid. 138, 193 (1986); T. Curtright, D. Fairlie, and C. Zachos, Phys. Lett. B 405, 37 (1997); D. Fairlie, Mod. Phys. Lett. A 13, 263 (1998). [2] E. Wigner, Phys. Rev. 40, 749 (1932). [3] J. Moyal, Proc. Cambridge Philos. SOC.45, 99 (1949). [4] J. Vey, Comments Math. Helvet. 50. 412 (1975); M. Flato, A. Lichnerowicz, and D. Stemheimer, J . Math. Phys. 17, 1754

(1976); M. de Wilde and P. Lecomte, Lett. Math. Phys. 7,487 (1983); P. Fletcher, Phys. Lett. B 248, 323 (1990). [51 F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Stemheimer, Ann. Phys. (N.Y.) 111, 61 (1978); 111, 111 (1978). [6] G. Baker, Phys. Rev. 109, 2198 (1958). [71 D. Fairlie, Proc. Cambridge Philos. SOC. 60, 581 (1964); D.

Fairlie and C. Manogue, J. Phys. A 24, 3807 (1991). [81 F. Hansen, Publ. RIMS Kyoto Univ. 26, 885 (1990). [9]I. Oppenheim and J. Ross, Phys. Rev. 107,28 (1957); also see

025002-13

489 THOMAS CURTRIGHT, DAVID FAIRLIE, AND COSMAS ZACHOS

K. Imre et al., J. Math. Phys. 8, 1097 (1967). [lo] K. Takahashi, Prog. Theor. Phys. Suppl. 98, 109 (1989); J. Dahl, in Energy Storage and Redistribution, edited by J. Hinze (Plenum, New York, 1983), pp. 557-571; L. Cohen, J. Math. Phys. 17, 1863 (1976); W. Kundt, 2. Naturforsch. 22A, 1333 (1967). [ I l l H. Groenewold, Physica (Amsterdam) 12, 405 (1946); this work may well be the first to define the t product, under a different name. M. Bartlett and J. Moyal, Proc. Cambridge Philos. SOC.45, 545 (1949). [I21 T. Curtright and G. Ghandour, in Quantum Field Theory, Sfatistical Mechanics, Quantum Groups, and Topology, edited by T. Curtright, L. Mezincescu, and R. Nepomechie (World Scientific, Singapore, 1992), pp. 333-335 [hep-th/9503080]. [I31 S . Habib, Phys. Rev. D 42, 2566 (1990); also see N. Balms and G. Zipfel, Ann. Phys. (N.Y.) 77, 139 (1973). [I41 G. Darboux, C. R. Hebd. Seances Acad. Sci. 94, 1456 (1882), cited E. L. Ince, Ordinary Dtfferential Equations (Dover, New York, 1926), p. 132; M. M. Crum, Q.J. Math. 6, 121 (1955); P. Deift, Duke Math. J. 45, 267 (1978); E. Witten, Nucl. Phys. B188, 513 (1981); reviewed in F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995). [I51 G. Poschl and E. Teller, 2. Phys. 83, 143 (1933). [I61 W. Kwong and J. Rosner, Prog. Theor. Phys. Suppl. 86, 366 (1986). [I71 For a review of the relevant features, consider Appendix A in

PHYSICAL REVIEW D 58 025002

T. Curtright, T. Uematsu, and C. Zachos, Nucl. Phys. B469, 488 (1996) [IS] P. Dirac, Phys. 2. Sowjetunion 3, 64 (1933). [I91 T. Curtright and C. Zachos, Phys. Rev. D 49, 5408 (1994). [ZO] E. DHoker and R. Jackiw, Phys. Rev. D 26, 3517 (1982). [21] E. Braaten, T. Curtright, G. Ghandour, and C. Thorn, Ann. Phys. (N.Y.) 153, 147 (1984). [22] A. Erddyi el aL, Higher Transcendental Functions, Bateman Manuscript Project (McGraw-Hill, New York, 1953). [23] C. Bender and S . Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Chap. 5. [241 G. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, England, 1966). [25] Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun, Natl. Bur. Stand. Appl. Math. Ser. No. 55 ( U S . GPO, Washington, D.C., 1965). [26] C. Meijer, Proc. K. Ned. Akad. Wet. 43, 702 (1940). [27] A. Erddyi et al., Tables of Integral Transforms, Bateman Manuscript Project (McGraw-Hill, New York, 1954). [28] T. Curtright and G. Ghandour, Phys. Lett. 136B, 50 (1984); also see E. D’Hoker, Phys. Rev. D 28, 1346 (1983). [29] “To think intellectually is a wonderfully human trait. My dog has no interest in the Associative Law of Multiplication.” The Recollections of Eugene P. Wigner, as told to Andrew Szanton (Plenum, New York, 1992), p. 307.

025002-14

490 Modern Physics Letters A, Vol. 16, No. 37 (2001) 2381-2385 @ World Scientific Publishing Company

NEGATIVE PROBABILITY AND UNCERTAINTY RELATIONS

THOMAS CURTRIGHT Department of Physics, University of Miami, Box 248046, Coral Gables, F L 33124, USA [email protected]. edu COSMAS ZACHOS High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439-481 5, USA zachosOhep. anl.gov Received 31 October 2001 A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl’s correspondence, or to marginal distributions of x and p . Keywords: Uncertainty relations; star product; Wigner functions; phase-space quantization; negative probability. PACS Nos.: 03.65.Ta, 03.65.Db, 03.65.Fd, 11.15.K~

Phase-space quantization is the third autonomous and logically complete formulation of quantum mechanics beyond the conventional ones based on operators in Hilbert space or path integral~.l-~ It is free of operators and wave functions: observables and matrix elements are computed through phase-space integrals of c-number functions (“classical kernels”) weighted by a Wigner function (WF).334 This is a phase-space distribution function which is not positive semi-definite, and constitutes the Weyl correspondent‘ of the density matrix in the conventional formulation,

(1) Operators of the conventional formulation, when properly ordered ( e g Weylordered), correspond uniquely to phase-space classical kernel functions, while operator products correspond to *-products6 of these classical kernels, the *-product being a noncommutative and associative operation encoding quantum mechanical action. The above wave functions, however, may be forfeited, since the WFs are determined, in principle, as the solutions of the celebrated *-genvalue functional equation^.^-'^ Connections to the original, operator, formulation of quantum mechanics may thus be ignored. 2381

49 1 2382

T. Curtright & C. Zachos

Recent M-theory advances linked to noncommutative geometry and matrix models" apply space-time uncertainty principles" reliant on phase-space quantization and the *-product. Transverse spatial dimensions act formally as momenta, and, analogously to quantum mechanics, their uncertainty is increased or decreased inversely t o the uncertainty of a given direction. For classical (non-negative) probability distributions, expectation values of nonnegative functions are likewise non-negative, and thus result in standard constraint inequalities for the constituent pieces of such functions. On the other hand, in phase-space quantization, the distribution functions are non-positiwe-definite,such as, in general, the quasi-probability WF: it was interpreted early on by Bartlett,13 and later by Feynman,14 as a "negative probability function", with the proper nonnegative marginal probabilities upon projection to either x or p space. Hence, a frequent first question in phase-space quantization is how Heisenberg's standard quantum mechanical uncertainty relation arises for moments of such distributions. To be sure, Moyal derived these uncertainty relations, in his original formulation of quantum mechanics in phase space, by careful analysis of conditioned and marginal probabilities. Nevertheless, plain evaluations of expectation values of the c-number variables ( x ' ) , (p'), etc., do not evince constraints; and the student of deformation quantization is left wondering how ii enters the constraint of such expectation values of (c-number) observables when the variables 2,p do not contain h. How do their moments manage to constrain each other by extracting ti out of the Wigner function? The answer lies in Groenewold's associative *-product ,6

(2) which is the cornerstone of phase-space quantization. Its mechanics is reviewed in Refs. 7, 10 and 15. An alternate, integral, representation of this product i d 6

f*g =

s

dudvdwdz f ( X + u , p + v ) g ( x + w , p + z )

(3) which readily displays associativity. The phase-space trace is directly seen in this representation to obey

Idpdxf*g =

s

dpdx f g

=

s

dpdxg* f .

(4) (4)

The WF spectral propertied are reviewed and illustrated in Refs. 8 and 10. For example the trace-normalization condition,

(5) (5)

492 Negative Probability and Uncertainty Relations

2383

and the spectral orthogonality condition^,^ f m n * f k l = Sml f k n / 2 7 r h . Given (4),it follows that J d~ dpfmn(Z,p)f&(x,p) = S m l d n k / 2 7 r h . For complete sets of input wave functions, it also follows that

(6) An arbitrary phase-space function cp(x,p) can thus be expanded as cp(z,p) = c m n f m n ( Z ,P ) . Here, a concise proof of all uncertainty relations is provided completely within the autonomous framework of phase-space quantization, unlike extant discussions of such correlation inequalities, which rely on the operator formulation of quantum mechanics. It is stressed that, in the following, no operators occur, only the *-product operation, and x and p are c-numbers. The controlling fact is that expectation values of arbitrary real *-squares are positive semi-definite, even though the Wigner distribution f(x,p ) itself is not. Specifically, for any complex phase-space function g(x,p ) , and any (real) Wigner function f(x,p ) representing a pure state, the following inequality holds:

Em,,

*

(9* 9 ) =

1

dP W g * * g ) f 2 0 .

(7)

The * is absolutely crucial here, and its removal leads to violation of the inequality, as can easily be arranged by choosing the support of g to lie mostly in those regions of phase-space where the Wigner function is negative. (The only pure state W F which is non-negative is the G a u s ~ i a n . ~In- ~Hilbert ~ ~ ~ ) space operator formalism, this relation (7) would correspond to the positivity of the norm. By (4), J d p d z ( g * * g ) f = J d p dz(g* * g) * f,i.e. inside a phase-space integral an ordinary product can be extended to a *-product, provided it not be part of a longer string. For example the one *-product of the left-hand side cannot be eliminated, because of the extra ordinary product with f . To prove the inequality (7), it suffices to recognize that, for a pure state, its (real) Wigner function can be expanded in a complete basis of Wigner *-genfunctions of a convenient Hamiltonian," f = ctcnfmn, for complex coefficients c,,s.t. (cnI2= 1 , to satisfy (5). Then, it follows thatl8>l9

En

En,,

f *f

=f/h.

(8)

Consequently, given the relations (4),(g * f ) * = f*g*, and the associativity of the *-product,

=h/dxdp(f *g*)*(g*

f)=h

J

dxdp)g* fI2

(9)

493 2384

T. Curtright €4 C. Zachos

This expression, then, involves a real non-negative integrand and is itself positive semi-definite. (Similarly, if f l and f 2 are pure state WFs, the transition probability between the respective states3 is also manifestly non-negative by the same argument: J d p d x f l f 2 = (27rti)2Jdxdplfl* f 2 I 2 . ) Given ( 7 ), correlations of observables follow conventionally from specific choices of g ( x ,p ) . For example, to produce Heisenberg's uncertainty relation, one only need to choose g =a

+ bx + cp ,

(10)

for arbitrary complex coefficients a , b, c. The resulting positive semi-definite quadratic form is then

a*a+b*b(x*x) + c * c ( p * p )

+ (a*b+b*a)(x)

+ (a*c+ c*a)(p)+ c*b(p* x ) + b*c(x* p ) 2 0 ,

(11)

for any a , b, c. The eigenvalues of the corresponding matrix are then non-negative,

and thus so must be its determinant. Given

x*x=x2,

p*x=px-iti/2,

p*p=p2,

x*p=px+ih/2,

(12)

and the usual

( A x ) 2= ( ( x- (xu2)7

(API23 ( ( P - ( P H 2 )

1

(13)

this condition on the 3 x 3 matrix determinant amounts to

(14) and hence

(15)

A x A p 2 h/2

The inequality is saturated for a vanishing original integrand g * f = 0, for suitable a, b, c, and when the last term of (14) vanishes: x , p statistical independence, such as in a Gaussian ground state W F , f00 = 2hexp(-(x2 + p 2 ) / h ) . More general choices of g will likewise constrain as many observables as this function has terms (-1, if there is a constant term). For instance, for more general (real) observables u ( x , p ) v, ( x , p ) ,the resulting inequality is

AUAV2

1

- J ~ ( U * W -w

2

+

* u ) ~( ~( u- ( t i ) ) * ( .- (v))

+ ( W - ( w ) ) * ( u- ( u ) ) ) ~ . (16) (16)

The minimum uncertainty is realized at (u* v specific coefficients, i.e.

+ w * u ) = 2(u)( w ) , with g * f = 0 for

(Au(w- ( v ) )- k i A ~ ( u ( u ) ) *) f = 0 , where k is the sign of i(u* v Refs. 8-10 and 15.

-

(17) (17)

v * u).Solving such *-equations is elaborated in

494 Negative Probability and Uncertainty Relations 2385

Acknowledgment This work was supported in part by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38, and the NSF Award 0073390. References 1. J . Moyal, Proc. Camb. Phil. SOC.45,99 (1949). 2. F. Bayen, M. Flato, C. F’ronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. 111,61; 111 (1978). 3. For reviews, see M. Hillery, R. O’Connell, M. Scully and E. Wigner, Phys. Rep. 106, 121 (1984); H.-W. Lee, ibid. 259,147 (1995); N. Balasz and B. Jennings, ibid. 104, 347 (1984); R. Littlejohn, ibid. 138, 193 (1986); M. Berry, Philos. Trans. R. SOC. London A287, 237 (1977); M. Gadella, Fortschr. Phys. 43, 229 (1995); L. Cohen, Time-Frequency Analyszs (Prentice-Hall, 1995); F. Berezin, Sou. Phys. Usp. 23,763 (1980); P. Carruthers and F. Zachariasen, Rev. Mod. Phys. 55, 245 (1983); A. M. Ozorio de Almeida, Phys. Rep. 295,265 (1998). 4. E. Wigner, Phys. Rev. 40,749 (1932). 5. H. Weyl, 2. Phys. 46,1 (1927); also reviewed in H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, 1931). 6. H. Groenewold, Physica 12,405 (1946). 7. D. Fairlie, Proc. Camb. Phil. SOC.60,581 (1964). 8. T. Curtright, D. Fairlie and C. Zachos, Phys. Rev. D58,025002 (1998). 9. M. Hug, C. Menke and W. Schleich, Phys. Rev. A57,3188 (1998). 10. T. Curtright, T. Uematsu and C. Zachos, J. Math. Phys. 42, 2396 (2001), hepth/0011137. 11. N . Seiberg and E. Witten, JHEP 9909,032 (1999); for a review, see L. Castellani, Class. Quantum Grav. 17,3377 (2000), hep-th/0005210. 12. T. Yoneya, Mod. Phys. Lett. A4, 1587 (1989); A. Jevicki and T. Yoneya, Nucl. Phys. B535,335 (1998), hep-th/9805069; N. Seiberg, L. Susskind and N. Toumbas, JHEP 0006,044 (2000), hep-th/0005015. 13. M. Bartlett, Proc. Camb. Phil. SOC.41,71 (1945). 14. R. Feynman, “Negative probability”, in Essays in Honor of David Bohm, eds. B. Hiley and F. Peat (Routledge and Kegan Paul, 1987); for a popular review, see D. Leibfried, T. Pfau and C. Monroe, Physics Today, 22 (April 1998). 15. F. Hansen, Rep. Math. Phys. 19,361 (1984); C. Roger and V. Ovsienko, Russ. Math. Surv. 47,135 (1992); C.Zachos, J . Math. Phys. 41,5129 (2000), hep-th/9912238. 16. J. von Neumann, Math. Ann. 104,570 (1931). 17. R. Hudson, Rep. Math. Phys. 6,249 (1974). 18. T. Takabayasi, Prog. Theor. Phys. 11,341 (1954). 19. G. Baker, Phys. Rev. 109,2198 (1958).

495 JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 42, NUMBER 6

JUNE 2001

Generating all Wigner functions Thomas Curtrighta) Department of Physics, University of Miami, P.O. Box 248046, Coral Gables, Florida 33124

Tsuneo Uematsub) Depanment of Fundamental Sciences, FIHS. Kyoto University, Kyoto 606-8501, Japan

Cosmas Zachosc) High Energy Physics Division, Argonne National Laboratory, Argonne, Illinois 60439-4815 (Received 15 January 2001; accepted for publication 26 February 2001) In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasiprobability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space. 0 2001 American Institute of Physics. [DOI: 10.1063/1.1366327]

1. INTRODUCTION General phase-space functions f(x,p) and g ( x , p ) compose noncommutatively through Groenewold's *-product,' which is the unique associative pseudodifferential deformation* of ordinary products:

(1) This product is the cornerstone of deformation (phase-space) quantization^-' as well as applications of matrix models and noncommutative geometry ideas in M-physics.6 Its mechanics, however, is not always straightforward. The practical Fourier representation of this product as an integral kernel has been utilized widely since Baker's' early work, 1 f*g=

dp' d p " d r ' dx"f(x',p')g(x",p")

(2) The determinantal nature of the star product controls the properties of the phase-space "Electronic mail [email protected] b)Electronicmail: [email protected] "Electronic mail: [email protected] 0022-2488/2001/42(6)/2396/20/$18.00

2396

0 2001 American Institute of Physics

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496 J.

Math. Phys., Vol. 42, No. 6, June 2001

Generating all Wigner functions

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(3) The above-mentioned *-product and phase-space integrals provide the multiplication law and, respectively, the trace in phase-space q~antization,~ the third autonomous and logically complete formulation of quantum mechanics beyond the conventional formulations based on operators in Hilbert space or path integrals. (This formulation is reviewed in Refs. 2 and 5.) Properly ordered operators (e.g., Weyl-ordered) correspond uniquely to phase-space c-number functions (referred to as “classical kernels” of the operators in question); operator products correspond to *-products of their classical kernels; and operator matrix elements, conventionally consisting of traces thereof with the density matrix, correspond to phase-space integrals of the classical kernels with the ” ~ celebrated Wigner function (WF), the Weyl correspondent of the density m a t r i ~ . ~The * -genvalue functional equations determining the Wigner functions’’” and their spectral properties (e.g., projective orthogonality”) are reviewed and illustrated in Ref. 4. The functions introduced by Wigner” and Szilard correspond to diagonal elements of the density matrix, but quantum mechanical applications (such as perturbation theory), as well as applications in noncommutative soliton problemsL3often require the evaluation of off-diagonal matrix elements; they therefore utilize the complete set of diagonal and off-diagonal generalized Wigner functions introduced by MoyaL3 For instance, in noncommutative soliton theory, the diagonal WFs are only complete for radial phase-space functions (functions *-commuting with the harmonic oscillator Hamiltonian-the radius squared), whereas deviations from radial symmetry necessitate the complete off diagonal set. As for any representation problem, the particular features of the *-equations under consideration frequently favor an optimal basis of WFs; but, even in the case of the oscillator, the equations are technically demanding. It is pointed out here, however, that suitable generating functions for them, acting as a transform of these basis sets, often result in substantially simpler and more compact objects, which are much easier to use, manipulate, and intuit. In the following, after some elementary overview of the Weyl correspondence formalism (Sec. II), we illustrate such functions for the harmonic oscillator (Sec. III), which serves as the archetype of WF bases indexed discretely; it turns out that these generating functions amount to the phase-space coherent states for WFs, and also the WFs of coherent state wave functions (Appendix A). Direct applications to first-order perturbation theory are illustrated in Appendix B. For sets indexed continuously, the generating function may range from a mere Fourier transform, illustrated by the linear potential (Sec. IV), to a less trivial continuous transform we provide for the Liouville potential problem (Sec. V), where the advantage of the transform method comes to cogent evidence. Throughout our discussion, we provide the typical *-composition laws of such generating functions, as well as applications such as the evaluation of * -exponentials of phase-space functions (Appendix C), or *-versions of modified Bessel functions (technical aspects of integral transforms of which are detailed in Appendix D). Appendix E provides the operator (Weyl-) correspondent to the generating function for the Liouville diagonal WF introduced in Sec V. II. OVERVIEW OF GENERAL RELATIONS IN THE WEYL REPRESENTATION Without loss of generality, we review basic concepts in two-dimensional phase space, ( x , p ) , as the extension to higher dimensions is straightforward. In addition, we first address discrete spectra, E n , n =0,1,2,3,. . . , and will only later generalize to continuous spectra. In the Weyl corre~pondence,’~ c-number phase-space kernels a ( x , p ) of suitably ordered operators d(X,P) are defined by

(4) Conversely, the ordering of these operators is specified through

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497 2398

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Curtright, Uernatsu, and Zachos

(5) An operator product then corresponds to a star-composition of these kernels,'

(6) Moyal' appreciated that the density matrix in this phase-space representation is a Hermitean generalization of the Wigner function:

(7) where the $m(n)'sare (ortho-)normalized solutions of a SchrGdinger problem. (Wigner" mainly considered the diagonal elements of the density matrix (pure states), usually denoted as f, =f,, .) As a consequence, matrix elements of operators are produced by mere phase-space integra~s,~

(8) The standard machinery of density matrices then is readily transcribed in this language, e.g., the trace relation?

(9)

(10)

(11) For complete sets of input wave functions, it also follows that3

(12)

(13) the coefficients being specified through (1l),

(14)

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498 J. Math. Phys., Vol. 42, No. 6, June 2001 Further note the resolution of the

Generating all Wigner functions

2399

id en tit^,^

E f,,(x,p)= n

1 -=

1

2nfi

(15)

-.h

For instance, for eigenfunctions of the Hamiltonian ‘H(X,P)with eigenvalues E n , the corresponding WFs satisfy the following star-genvalue equations’ (also see Refs. 1 1 and 4). with H ( x , p ) , the phase-space kernel of ‘H(X,P): H*fmn=En.fmn

7

fmn*H=Emfmn.

(16)

The time dependence of a pure state WF is given by Moyal’s dynamical e q ~ a t i o n : ~

a ifi x f ( x , p ; t ) = H * f ( x . p ; t ) - f ( x , p ; t ) * H . By virtue of the *-unitary evolution operator (a “*-exponential”

(17)

2).

(18) the time-evolved WF is obtained formally in terms of the WF at t = 0, (19)

f(x,p;t)= u;’(x,p;t)*f(x,p;o)*u,(n,p;t).

(These associative combinatoric operations completely parallel those of operators in the conventional formulation of quantum mechanics in Hilbert space.”) Just like any star-function of H , this *-exponential can be computed,’6 exp,(itH/fi)= exp,( i t H / h ) * 1 = exp,( i t H / f i ) * 2 7 r f i E f n n = 2 7 r h x e i r E n l f i f n n . (20)

”

n

(Of course, for t = 0, the obvious identity resolution is recovered.) For continuous spectra, the sums in the above-mentioned relations extend to integrals over a continuous parameter (the energy), and the Kronecker Smn’s into &functions (these last ones reflecting the infinite normalizations of unnormalizable states). For example, Eqs. (9) and (11) extend to

I

dx d p f E l E 2 ( x , P )= ~

( E I-

~ z )

7

(21) (22)

Completeness (12) extends to

(23) More generally, (10) extends to

1 fElE2*fE;E;=

-E ; ) f E ;

E2’

(24)

Finally, Eq. (15) extends to

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499 J. Math. Phys., Vol. 42, No. 6, June 2001

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Curtright, Uematsu, and Zachos

(25) and hence (20) extends to

(26) 111. GENERATING FUNCTIONS FOR THE HARMONIC OSCILLATOR Consider the harmonic oscillator,

(27) where, without loss of generality, parameters have been absorbed in the phase space variables: rn = 1, w = 1 . Further recall that the normalized eigenfunctions of the corresponding operator Hamiltonian 3-1 are ~ ~ ( ~ ) = ( ~ 2 " n ! ) - " ~ e - ( " ~ for ) " the ~ Heigenvalues ~ ( x ) , E,=fi(n+ 112). Define a radial and an angular variable,

(28) so that

(29) Groenewold,' as well as Bartlett and Moyal,17 have worked out the complete sets of solutions to Moyal's time-evolution equation (17), which are all linear combinations of terms exp(it(m -n))fmn . They solved that equation indirectly, by evaluating the integrals (7) for time-dependent Hermite wave functions, which yield generalized Laguerre polynomial-based functions. More directly, Fairlie' dramatically simplified the derivation of the solution by relying on his fundamental equation (16). He thus confirmed Groenewold's WFs,'.17

(30) The special case of diagonal elements,

(31) constitutes the time-independent "*-genfunctions" of the oscillator hamiltonian kernel4 [i.e., the complete set of solutions of the time-independent Moyal equation H*f -f * H = O , where H * f , = E d , . Incidentally, (10) restricted to diagonal WFs closes them under *-multiplication,'2 fm*f,= SmJm /(2rrfi).]That is to say, "radially symmetric" phase-space functions, i.e., functions that only depend on z but not 0, can be expanded in terms of merely these diagonal elements-unlike the most general functions in phase space which require the entire set of offdiagonal f,, above for a complete basis. Note, however, that all *-products of such radially symmetric functions are commutative, since, manifestly,

(32)

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500 J. Math. Phys., Vol. 42, No. 6, June 2001

Generating all Wigner functions

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Moreover, the *-exponential (20) for this set of *-genfunctions is directly seen to amount to

(33) which is, to say, a Gaussian in phase space.' As an application, note that the hyperbolic tangent *-composition law of Gaussians follows trivially, since these amount to *-exponentials with additive time intervals, exp,(rf)*exp,(Tf )=exp,((t+T)f)?

(34) We now introduce the following generating function for the entire set of generalized Wigner functions:

(35) Utilizing the identity" 8.975.2,

(36)

(37) Thus,

(38) (38)

&(ae-"+pe")-ap-

Since pel') = v ~ ( r y + p )n\ j z i p ( a -

p).

(39) (39)

one can re-express:

(40) As the name implies, from G( a , P ; x , p ) , the f m n ' s are generated by

(41) These functions *-compose as

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501 2402

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e G ( ~ , P ) * G ( E . O ~= G ( E , P ) .

(42)

dxdpG(a,P)=eap.

(43)

The phase-space trace is

By (16). the action of the Hamiltonian kernel on this function is

(44) and

(45) Consequently,

I

dxdpH*G(a,P)=h

(46)

The spectrum then follows by operating on both sides of this equation,

(47) In general, matrix elements of operators may be summarized compactly through this generating function in phase space. This generating function could be interpreted as a phase-space coherent state, or the offdiagonal WF of coherent states, as discussed in Appendix A,I9 G( a.P;x,p) = exp,(pa t)foexp,( a a ) ,

(48)

(49)

and hence Eqs. (44) and (45) amount to

(50)

This formalism finds application in, e.g., perturbation theory in phase space, cf. Appendix B.

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502 J. Math. Phys., Vol. 42, No. 6, June 2001

Generating all Wigner functions

2403

IV. GENERATING FUNCTIONS FOR THE LINEAR POTENTIAL

The linear potential in phase space has been addressed" (also see Refs. 19 and 4). We shall adopt the simplified conventions of Ref. 4,i.e., rn = 1/2, fi = 1. The Hamiltonian kernel is then H(x,p) = p2+x,

(51)

and the eigenfunctions of 'H are Airy functions,

(52) indexed by the continuous energy E . The spectrum being continuous, the Airy functions are not square integrable, but have continuum normalization, Jdx I(Ii,(x) i,bE2(x) = S(El - E 2 ) , instead. Thus, (21) et seq. are now operative. The generalized WFs are"

(53) The *-exponential (26) then is again a plain exponential of the shifted Hamiltonian kernel, exp,( i t ( x + p 2 ) ) = 2 a

I

x+p2- 2

= exp(it(x+p2+ t2/12)).

(54)

(Thk could also be derived directly, as the CBH expansion simplifies dramatically in this case, cf. Appendix C.) As before, the *-composition law for plain exponentials of the hamiltonian kernel function follows,

+

exp(a(x+p2))*exp( b(x+p2)) = exp( ( a b)(x + p 2 - $ a b ) ) .

(55)

Since the complete basis Wigner functions are now indexed continuously, a generating function for them must rely on an integral instead of an infinite sum. The simplest transform is possibly a double Fourier transform with respect to the energy indices [but note the transform factors exp(iEIX), exp(-iE2Y) may also be regarded as plane waves]. Suitably normalized,

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503 2404

J. Math. Phys., Vol. 42, No. 6, June 2001

Curtright, Uematsu, and Zachos

(56) The phase-space trace is

I

dxdpG(X,Y;x,p)=2.rr6(X-Y),

( 5 7 ) (57)

= 6 ( E ,- E ; ) ~ E ; Ethe ~ ,*-composition and, given (24) for these functions, ~ E ~ E * * ~ E ; E (;1/27r)

law for these G's is C(X,Y;x,p)*G( W,Z;x,p)= S(X-Z)C( W,Y;x,p).

(58)

V. GENERATING FUNCTIONS FOR THE LlOUVlLLE POTENTIAL

A less trivial system with a continuous spectrum is the Hamiltonian with the Liouville potential.2032'In the conventions of Ref. 4 (fi= 1, m = 1/2), the Hamiltonian kernel is H=p2+eZx,

(59)

and the eigenfunctions of the corresponding 7-l are (60)

with continuum normalizations Jdx $ l , ( x ) $ E 2 ( x )= 6 ( E l - E z ) . The modified Bessel function (Ref. 22, Chap. VI, Sec. 6.22) can be written in the Heine-Schlafli form, K j p ( e x ) = ; I:-dXexp(

-excoshX+iXp)=K-jp(ex).

(61)

The nondiagonal WF is then

7,/ 1

f ~ , ~ ~ ( x , p ) = dy e - 2 i P y ~ ~ K ~ ~ e ~ - Y ) ~ ~ K ,(62) ~ ( e x ' Y ) . This Wigner function amounts to Meijer's G function, 1 ~E~E~(X9 , P ) =&inh(.rr\jEl)sinh(n\jE2)

(63)

Alternatively, the WF may be written as a double integral representation, 1

dsinh(m6)sinh(rrJE(q))

f ~ ( k E) ( , ) ( ~ , P ) =

X I

[

cosh Y ' p dXdY eJkXe'qYcash x) K2,,(eXJ4 cash X cash U,

(64)

where E ( k ) - k 2 , E ( q ) = q 2 . This is an inverse integral transform, as in Sec. IV, of a generating function

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504 J. Math. Phys., Vol. 42, No. 6, June 2001

Generating all Wigner functions

Kzi,(e'J4 coshX cosh Y ) = G * ( Y,X;x,p).

2405

(65)

The form and construction of this G are consequences of (61), as detailed in Appendix D. However, the *-composition law of this particular generating function is not so straightforward. It is singular, as a consequence of the general relation (24) and the behavior of the integrand in (65) as k,q-+O. The singularity may be controlled by regulating the *-product through imaginary shifts in the momenta,

It follows that one derivative with respect to either of X or Z suffices to eliminate the divergence at E = O ,

Unlike the situation in (58), here the right-hand side vanishes at X = Z . More symmetrically,

[

6-10

-tanhX-

cosh X + cosh Z

By some contrast to the above, Eq. (65), an alternate generating function for just the diagonal WFs, f E E = f E , could be defined through the spectral resolution of the * - K function,

(66)

This can be evaluated by reliance on Macdonald's trilinear

(67)

then is obtained by replacing x + x + Y

and y - + x - Y ,

and Fourier transforming by

(UT)J dY e -liPY,

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505 2406

J. Math. Phys., Vol. 42, No. 6,June 2001

Curtright, Uematsu, and Zachos

(68) Finally, simplifying the right-hand side gives

(69) As a side check of this expression, (69), note that it must satisfy ~ * G ( z ; x , p=) G ( z ; x , p ) * ~ =( -

a;+ e Z z ) G ( z ; x , p ) ,

((70) 70)

which follows from the spectral resolution evident in (66). Indeed, since e-'d,KIp(e') = i p e - Z K , p ( e Z ) - K , p + l ( e z )and , ( - d ~ + e 2 Z ) K , , ( e L ) = p 2 K , , ( e zthese ) , relations are satisfied,

(71) Parenthetically, as an alternative to the ordinary product form in (69), the phase-space kernel (Note: Do not shift the integration parameter y by the phase-space variable n before the star products are evaluated.),

Li may also be represented as an integral either of a *-exponential or of a single *-product

*exp(iyp-eZcoshy).

(72)

i:

(73)

This follows from the identities (cf. Appendix C) *exp(iyp)=exp

- -eZX-I+iyp

The ordinary product form in (69) and the *-exponential form in (72) reveal that G(z;x,p) = G ( z ; x , - p ) , so one may replace exp(iyp) by cos(yp) in the second line of (72). Given these, there are several ways to verify (70). These relations and the star-product expressions for the kernel in (72) are isomorphic to those of the corresponding operators, as discussed in Appendix E. The *-composition law of these generating functions follows from (24) and Macdonald's identity,

'I i :

G ( u ; x , p ) * ~ ( u ; x , p )2=

dw exp - - ( e u ~ u ~ w + e u ~ " ~ w +)jG(w;x.P). e ~ u ~ u ~ w (74)

This also follows directly from the explicit form (69). Again, this is isomorphic to the corresponding operator composition law given in Appendix E. From the orthogonality of the &'r, :he diagonal WFs may be recovered by inverse transformation,

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506 J.

Math. Phys., Vol. 42, No. 6, June 2001

Generating all Wigner functions

2407

(75)

This representation and the specific factorized x,p-dependence of B can be of considerable use, e.g., in systematically computing diagonal matrix elements in phase space. In illustration of the general pattern, consider the first-order energy shift effected by a perturbation Hamiltonian kernel H I . It is, cf. Appendix B; Eq. (Bll), (76)

Choosing H , =e 2 n x e i s ~ R ,

(77)

yields (78)

Now. (79)

and hence (Ref. 18, 6.576.4, a = b),

xr(n-if+iplr(

n-if-ip)

(80)

Thus,

xr[

n+i&-ip 2

n-i&+ip 2

)r[

n-ifi-ip 2

)r(

1.

(81)

Finally (Ref. 18, 6.422.19).

(82)

To sum up, the perturbed energy shift is a Meijer function,

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507 2408

Curtright, Uematsu, and Zachos

J. Math. Phys., Vol. 42,No. 6,June 2001

I

12-n

+ i&

2 -n -

i&\

(83)

In principle, any polynomial perturbation in either x or p can be obtained from this, by differentiation with respect to n and s. (Retaining a bit of exponential in x would be helpful to suppress the region of large negative x). ACKNOWLEDGMENTS

We wish to thank D. Fairlie and T. Hakioglu for helpful conversations. This work was supported in part by the US Department of Energy, Division of High Energy Physics, Contract No. W-31-109-ENG-38; NSF Award No. 0073390; and by the Grant-in-Aid for Priority Area No. 707 of the Japanese Ministry of Education. T. U.and T. C. thank Argonne National Laboratory for its hospitality in the summer of 2000. APPENDIX A: *-FOCK SPACE AND COHERENT STATES

Dirac's Hamiltonian factorization method for algebraic solution of the harmonic oscillator carries through (cf. Ref. 2) intact in *-space. Indeed,

1 2

H=-(x-ip)*(x+ip)+

n

(A1)

-, 2

motivating definition of (A2)

Thus, noting (A3)

and also that, by above, (AS4)

provides a *-Fock vacuum, it is evident that associativity of the *-product permits the entire ladder spectrum generation to go through as usual. The * -genstates of the Hamiltonian, such that H * f = f * H , are thus (A5)

These states are real, like the Gaussian ground state, and are thus left-right symmetric *-genstates. They are also transparently *-orthogonal for different eigenvalues; and they project to themselves, as they should, since the Gaussian ground state does, fo*fo=fo/2nfi. The complete set of generalized WFs can thus be written as 1 f In"

=-

'

( a *)"fo(* a ) m , m , n = 0,1,2,3;

JiG

..

(A6)

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508 J. Math. Phys., Vol. 42, No. 6,

June 2001

Generating all Wigner functions

2409

The standard combinatoric features of conventional Fock space apply separately to left and right (its adjoint) *-multiplication:

(A7)

(A8)

Furthermore, a lefuright (non-self-adjoint) coherent state is naturally ~ ( a , P ) = e x p * ( a a t ) f o e x p , ( p a ) ,a * @ ( a , P ) = a @ ( a , P ) , @ ( a . ~ ) * a " = ~ @ ( a $ ) . ((A91 A9)

Up to a factor of e~p((lal~+IP1~)/2), this is also the WF of coherent states la) and (PI.= As indicated in the text, this coherent state is identifiable with the generating function G for the harmonic oscillator. APPENDIX B: STATIONARY PERTURBATION THEORY

Perturbation theory could be carried out in Hilbert space and its resulting wave functions utilized to evaluate the corresponding W F integrals. However, in the spirit of logical autonomy of Moyal's formulation of quantum mechanics in phase space, the perturbed Wigner functions may also be computed ab initio in phase space,"*25without reference to the conventional Hilbert space formulation. The basics are summarized in the following. As usual, the Hamiltonian kernel decomposes into free and perturbed parts, H=H,+hH,.

(B 1)

Fairlie's stationary, real, * -genvalue equationssB4for the full Hamiltonian, (b2)

are solved upon expansion of their components E and f in powers of A, the perturbation strength, (B3) (B4)

Note the superscripts on E and f are order indices and not exponents. Resolution into individual powers of A yields the real equations: (B5) (B6) (B7)

Left multiplication of (B6) by

E* yields (B8)

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509 2410

J. Math. Phys., Vol. 42, No. 6, June 2001

Curtright, Uematsu, and Zachos

and, by (B5).

E * H *E= E: E *A; 1

(B9)

by (1 l), (10). and the cyclicity of the trace (3),

(B10)

(B11) the diagonal element of the perturbation. For the off-diagonal elements, similarly left-*-multiply (B6) by f i

.

J", *H,*ff, By completeness,

+.tt*H,

*t= E Y m * f ! , + E f f m * E ,

((BIZ) B12)

fk , i f 0, resolves to (B13)

the reality condition dictating

(b14)

(B15) and hence

(B16)

(B17) so that

(B18) Finally, use of (1 l), yields

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5 10 J. Math. Phys., Vol. 42, No. 6,June 2001

Generating all Wigner functions

~ ~ , ~ , , , = ( 2 7 r hd x) ~d p1 Pml*Pm*Hl

241 1

*t

~g-~lj,

(B19)

We also have the similar equation for ZZn. Consequently, is proportional to the matrix element of the perturbation, and it vanishes unless 1 or m is equal to n . [Note: This differs from Ref. 25, Eq. (45).] To sum up,

((B20) B20)

By (8). it can be seen that the same result may also follow from evaluation of the WF integrals of perturbed wave functions obtained in standard perturbation theory in Hilbert space. For example, consider H I = v2 x = a +at . It follows that EA = 0, and

-

-

4,

(E,O- E:)

11

dxdP(Jm+lPm+.l,n+mPm,"+l)

=a n , l ~ J m + l ~ r n + l , n -

for m # n , and the ( m

Jn';+la r n , n + ] ) ,

((B21) b21)

1 ) expression for 1# n . Hence,

fA= &(t1,nft.n-

1)-

J n ? i t t , n + l +t+l,n).

((B22) B22)

APPENDIX C: COMBINATORIC DERIVATION OF IDENTITIES (54) AND (73)

The *-exponential (54) of the Hamiltonian kernel for the linear potential is also easy to work out directly, since the combinatorics in *-space are identical to the combinatorics of any associative algebra. In particular, the Campbell-Baker-Hausdorff expansion also holds for *-exponentials,

exp,(A)*exp,(B)=exp,(A+B+

+[A,Bl,+ h[A,[A,Bl*l*+ h[[A,BI,,Bl*+C),

(C1)

where C represents a sum of triple or more nested *-commutators (Moyal Brackets, [A,B], -2it'p =A*B-B*A). Now, choosing A = i t x and B = i r p 2 + i t 2 p + $ t 3 , yields [A$],= - i t 3 , [A,[A,B]*]*=2it3, [[A,B],,B],=O, and hence C=O. Consequently,

+

exp,( i t x ) *exp,( itp2 i t 2 p+ i t 3 )= exp,( i t x + itp') .

(C2)

But further note exp,(ux)=exp(a), and also exp,(bp2+cp+d)=exp(bp2+cp+d). This reduces the *-product to a mere translation,

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511 2412

J. Math. Phys., Vol. 42, No. 6, June 2001

Curtright, Uernatsu, and Zachos

exp,(ax)*exp,(bp2+ c p + d ) = exp(ax)*exp( b p 2 + c p + d ) = exp(ax

+ $ iad,)exp(

= exp(ax+ b ( p

+

bpz cp + d )

+ t ial2+ c ( p+

i a )+ d )

+

=exp(ax+ bp2+ ( c i n b ) p + d - a a z b + ) i a c ) .

(C3)

+

(C4)

Consequently,

+

exp,( i t x ) *exp,( i t p 2 i t 2 p + j it3) = exp( it( x p2 + t 2 / 1 2 ) ) , and the identity

+ +

exp, ( i t ( x +p z ) ) = exp( it (x pz tz/l2))

(54')

follows. The proof of (73) is similar. Choosing now A = - ) e y - ' e Z x and B = i y p , it follows that [A&],= -2yA, so that only those multiple Moyal commutators survive which are linear in A. This means, then, that in the Hausdorff expansionz6 for Z ( A ,B)=In,(exp,(A)*exp,(B)), only B and terms linear in A survive. Hence, Z reduces to merely (c5) The Hadamard expansion in B ] , means successive right *-commutation with respect to B as many times as the regular power expansion of the function in the parenthesis dictates. Consequently,

(C6) On the other hand, exp( - +eY-ze2x)*exp(iyp)=exp( - +eY-z+2x)exp(iy(p+ i i x / 2 ) )= exp( - + e Z x - z + iyp), (C7) and the identity is proven. APPENDIX D: CONSTRUCTION OF THE GENERATING FUNCTION FOR THE LlOUVlLLE WFS

From (60) and (61), it is evident that the Liouville wave functions can be generated by dk

exp( - e x cosh X) =

I - d z z z cikX

$Edx).

(D1)

where E ( k ) = k 2 . Therefore, the usual wave function bilinears appearing in the WFs are generated by (recalling that the Q s are real)

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512 J. Math. Phys., Vol. 42,No. 6,June 2001

Generating all Wigner functions

2413

dk

exp( -ex-? cosh X)exp( - ex+Y cosh Y)=

x

-ikX-

iqY

4!'E(k)(X--Y)4!'E(q)(x+Y).

(D2)

Consequently, Fourier transforming this produces a generating function for WFs, dy e-2ipyexp(-

coshX)exp( - ex+Ycosh Y)

(D3)

Evaluation of this expression yields just a factor multiplying a modified Bessel function, dy e-2ipyexp( -eX-YcoshX-ex+Ycosh Y )

= /ymdy exp( - 2 i p ( y f Zln(coshX/cosh 1 Y)

exp(-(eXJ4 coshXcosh Y)coshy)

(

coshY jP =2 coshX) K2ip(e"J4coshXcosh Y).

(D4)

Thus, a generating function for the complete set of Liouville Wigner functions is

(-)

2 cosh Y T coshX

dk

jP

K z i p (e'44 cosh X cosh Y ) = -i k X -

iqY

fE(k) E(q)(X,P),

((65') 65')

as in the text. APPENDIX E: OPERATOR ORDERING AND EQ. (69)

Given the factorized phase-space generating function G(z;x,p)=exp(- )e2x-z)Kip(eL).

((69') 69')

what is the operator corresponding to it? According to Weyl's prescription, Eq. ( 5 ) , the associated operator is d T d u dx

dp B( z;x,p)exp( i T( P- p ) + ia(X - x ) )

1 (El) The integrals over x and p may be evaluated separately, if the u contour is first shifted slightly above the real axis, u-+c+ i c , thereby suppressing contributions to the x-integral as x-+ --oo. Now s=)eZx-' gives

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5 13 2414

J. Math. Phys., Vol. 42, No. 6, June 2001

Curtright, Uematsu, and Zachos

-I

-2e

I

-i(z+lnZ)u/Zr

( - i( a+ ie)/2).

dX e-ezCoShX 2774X-

dp K,,(e')exp( - iTp) =

T)= 77e-e2CoSh r.

(E2)

(E3)

so

'I

B(z ;X,P ) = - d~ d a e-i(z+'n2)u'2 ( - i( a+ i e)12) e-e'

'Osh

877

'exp( i TP+irk').

((E4) E4)

r

The shifted a contour avoids the pole in at the origin. Ordering with all P ' s to the right, thereby departing from Weyl ordering, yields exp(iTP + iaX, = exp(iaX,exp(ian'2)exp(i~P). Performing the a integration before the T integration, permits taking the limit e-10 to obtain

'I

B(z;X, P ) = - d T 8n

-ez

1

(I

cosh r

d a r ( - i( a+ i e)/;?)exp( i a X + i a d 2 - i a ( z +In 2)/2)

exp( i TP)

d7(4 exp( - e Z X + r - ( z C l n2)))e-e'

=-

8T

COsh

exp( i ~ p )

(E5) This is the operator correspondent to (72); it reflects the Weyl correspondence through which it was originally defined (although, technically, it was taken out of Weyl ordering above, merely as a matter of convenience, not a bona-fide change of representation). This form leads to a more intuitive Hilbert space representation. Acting to the right of a position eigen-bra, (xlX= (xlx, while the subsequent exponential of the momentum operator just translates, (xlexp(i.rP) = (x + So the full right-operation of 6 is

TI.

(E6) Inserting 1 =I& Ix)(xI gives B(z;X,P)=Jdr Ix)(xl@(z;X,P), and leads to a coordinate space realization of the operator involving an x,y -symmetric kernel,

B(z;X,P)=

7

2

1 1 dx dy Ix)(yl exp( - - e X + y - z - - ex-y+z2 2

I 2

-e-x+y+z

).

037)

This operator is diagonal on energy states: by Macdonald's identity (67), and the reality and orthogonality of the wave functions,

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514 J. Math. Phys., Vol. 42, No. 6, June 2001

Generating all Wigner functions

2415

(E8) This is in agreement with the corresponding phase-space expression, (66). The composition law of this operator also parallels its phase-space isomorph, (74),

(E9) 'H. Groenewold. Physica (Amsterdam) 12, 405 (1946). 'F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Stemheimer, Ann. Phys. (N.Y.) 111, 61 (1978); 111. 1 1 1 (1978); Lett.Math. Phys. 1, 521 (1977). 'J. Moyal, Roc. Cambridge Philos. SOC.45, 99 (1949). 'T. Curtright, D. Fairlie, and C. Zachos, Pbys. Rev. D 58, 025002 (1998). 'For reviews, see M. Hillery, R. OConell, M. Scully, and E. Wigner. Phys. Rep. 106, 121 (1984); H.-W. Lee, ibid. 259, 147 (1995): N. Balasz and B. Jennings, ibid. 104, 347 (1984); R. Littlejohn. ibid. 138. 193 (1986); M. Berry, Philos. Trans. R. Soc. London, Ser. A 287, 237 (1977); M. Gadella, Fortschr. Phys. 43, 229 (1995); P. Carmthers and F. Zachariasen, Rev. Mod. Phys. 55, 245 (1983); L. Cohen, Time-FrequencyAnalysis (Prentice-Hall, Englewocd Cliffs, NJ, 1995): F. Berezin, Sov. Phys. Usp. 23,763 (1980); A. M. Ozorio de Ahneida, Phys. Rep. 295, 265 (1998). 6N. Seiberg and E. Witten, 1. High Energy Phys. 9909,032 (1999); for a review, see L. Castellani, Class. Quantum Grav. 17, 3377 (2000) [hepW0005210]. 7 G . Baker, Phys. Rev. 109, 2198 (1958); formally underlain by J. v Neumann, Math. Ann. 104, 570 (1931). 'D. Fairlie, Proc. Cambridge Ph~los.SOC.60, 581 (1964). 9F. Hansen, Rev. Mod Phys. 19, 361 (1984); C . Roger and V Ovsienko, Russ Math. Surv. 47, 135 (1992); C. Zachos, 1. Math. Phys. 41, 5129 (2000) [hep-tW9912238]. "E. Wigner, Phys. Rev. 40,749 (1932). "I. Dahl, in Energy Storage and Redistribution, edited by J. Hinze (Plenum, New York, 1983). pp, 557-571, see also L. Cohen, I. Math. Phys. 17, 1863 (1976); W. Kundt, Z. Naturforsch. A 22, 1333 (1967). "T. Takabayasi, Prog. Theor. Phys. 11,341 (1954). 13R. Gopakumar, S. Minwalla, and A. Strominger, J. High Energy Phys. 0005,020 (2000) [hep-W0003160]; J. Harvey, P. Kraus, F. Larsen, and E. Martinec, ibid. 0007,042 (2000) [hep-th/0005031]; K. Dasgupta, S. Mukhi, and G. Rajesh, ibid. 0006, 022 (2000) [hep-W0005006]; A. Polychronakos. Phys. Lett. B 495, 407 (2000) [hep-th/0007043]; J. Harvey, hep-WO 102076. I4H. Weyl, Z. Phys. 46. 1 (1927); also reviewed in H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931). "T. Curhight and C. Zachos, J. Phys. A 32, 771 (1999). 16J. Gracia-Bonda, Phys. Rev. A 30, 691 (1984). I7MvI.Bartlett and J. Moyal, Proc. Cambridge Philos. SOC.45, 545 (1949). "I. Gradshteyn and 1. Ryzhik, Table oflntegrals. Series, and Products, 5th ed. (Academic, New York, 1994). I9V. Dodonov and V. Man'ko, Physica A 137, 306 (1986). "E. D'Hoker and R. Jackiw. Phys. Rev. D 26, 3517 (1982); E. Braaten, T. Curtright, G. Ghandour, and C. Thorn, Ann. Phys. (N.Y.) 153, 147 (1984); T. Curtright and G. Ghandour, Phys. Len. B 136, 50 (1984); E. D'Hoker, Phys. Rev. D 28, 1346 (1983). "C. Grosche and F. Steiner, Phys. Lett. A 123, 319 (1987). 22G. Watson, A Treatise on the Theory ofBessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1995). 23T.Curtright, quant-pN0011101. 24V.Han, Y. Kim, and M. Noz, Phys. Rev. A 40, 902 (1989); M. Hennings. T. Smith, and D. Dubin, J. Phys. A 28,6779 (1995); 28, 6809 (1995); J. Klauder and 8. Skagerstam, Coherent States (World Scientific, Singapore, 1985). "L. Wang and R. O'Connell, Found. Phys. 18, 1023 (1988). 26W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954).

Downloaded 16 Oct 2002 to 146.139.180.65. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp

515 VOLUME 57, NUMBER 5

PHYSICAL REVIEW A

MAY 1998

Modified spectral method in phase space: Calculation of the Wigner function. I. Fundamentals M. Hug,‘ C. Menke? and W. P. Schleichl ‘Abfeilungfur Quantenphysik, Universitut Ulm, 0-89069 Ulm, Germany 2Abteilung Numerik. Universitui Ulm, 0-89069 Urn, Germany (Received 30 October 1997) We present a method for the direct computation of the Wigner function by solving a coupled system of linear partial differential equations. Our procedure is applicable to arbitrary binding potentials. We introduce a modified spectral tau method that uses Chebyshev polynomials as shape functions to approximate the solution. Since two differential equations are solved simultaneously, the resulting linear equation system is overdetermined. We approximate its solution by a least-squares method. We prove the stability and convergence of our scheme. As an application, we compute numerically the Wigner function for the harmonic oscillator. Our calculations show excellent agreement with known analytic results. [S1050-2947(98)04704-0]

PACS number(s): 03.65.Bz, 02.70.Hm I. INTRODUCTION

An alternative formulation of standard quantum mechanics i la Schrodinger or Heisenberg is the phase-space approach pioneered by Wigner. Here the standard definition of a Wigner function [l] is based on the density matrix. Hence once we know the density matrix we find the Wigner function by performing a Fourier integral. However, one might ask if phase space alone defines the Wigner function T, Does there exist a set of real partial differential equations in the phase-space variables 9 and p which determine VI? This would avoid the detour through the density operator and the Fourier integral. Indeed, a set of two real linear partial differential equations [2,3] defines the Wigner function. This set is rarely investigated in the literature [4]. although it offers new insights into the Wigner function. To derive solutions for these phase-space equations is the topic of these two papers. According to Heisenberg’s uncertainty principle, position and momentum cannot be measured simultaneously and precisely. So, strictly speaking, one cannot define a phase-space probability distribution in quantum mechanics. However, several quasiprobability distributions [1,5,6] have been proposed and widely used that carry some of the key features of classical phase-space densities. The most famous quasiprobability distribution is the Wigner function [7]. This function has found a wide range of applications (in the fundamentals of quantum mechanics, quantum optics, nuclear physics, and plasma physics, to name just a few areas). The Wigner function shows many features of a classical phase-space density. In particular, this quasiprobability has quantummechanically correct marginal distributions [8]. However, because of the uncertainty principle the Wigner function differs in two important points from a classical phase-space distribution [9]. One is a required symmetrical ordering of observables [l]. Second, the Wigner function may contain areas of negative “probabilities.” These negativities have been considered prominent features of nonclassical behavior [lo]. Quite typically, the negative areas occur as oscillations arising from interference effects [I 1,121. In this paper we introduce a spectral tau method [I31 to compute the Wigner function directly for energy eigenstates. We apply proposals of Fairlie [2(a)] and Kundt [2(b)]. They 1050-2947/98/57(5)/3188( I8)/$15.OO

57 -

derived two real linear partial differential equations for the Wigner function of a bound energy eigenstate. To our knowledge, the present work is the first one that describes how to solve these equations numerically for an arbitrary one-dimensional binding potential. We approximate their solution as a finite sum of Chebyshev polynomials in the two phase-space variables position and momentum. Since we simultaneously solve two differential equations, the resulting linear system of equations is overdetermined. We approximate its solution by a least-squares method. We prove the stability and convergence of our method. Traditionally the Wigner function is calculated by first finding the stationary wave function (solving the Schrodinger equation) and then performing a Fourier transformation. Our method avoids the detour of calculating the wave function. We believe that this procedure may provide new insight into the fascinating properties of quasiprobability distributions, and that our method will become a valuable tool for numerical computations of the Wigner function. We note that our approach can be generalized to any similar system of partial differential equations. In Sec. I1 we present the differential equations for the Wigner function. Section 111 introduces the fundamentals of spectral methods. The truncated system of differential equations that we use in this paper to outline our method is given in Sec. IV. In Sec. V we perform the required calculations to implement our modified spectral method. Aspects of stability, convergence, and error estimates are discussed in Sec. VI. A numerical example of the harmonic oscillator is given in Sec. VII. 11. FORMULATION OF THE PROBLEM

In the present section we briefly summarize the two phase-space equations defining the Wigner function of an energy eigenstate of a one-dimensional binding potential. We introduce dimensionless variables on a quadratic domain of phase space which will make it convenient to apply our method of solving these equations. In Appendix A we show that the Wigner function

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0 1998 The American Physical Society

516 51

MODIFIED SPECTRAL METHOD I N . . . , I.

of an eigenstate I t+&q)) in position representation of energy B i n a one-dimensional potential U = U ( q ) follows from the equations

...

1 x'-[QQo

3189

$ (Qi+Q2)],

where

Qo(1)

k (Qz-Qi).

With the potential

and

V(x)=v(Qox+

i(Qi+Qz)>

we arrive at the two coupled equations

(2) in the phase-space variables' positions q and momentum p , respectively. Here M and fi denote the mass of the particle in the potential and Planck's constant, respectively. In general, these equations are of infinite order in p . However, in the case of a polynomial potential U only a finite number of derivatives contributes. In any case the solution V ( q , p ) is symmetric in the momentum variable p . that is

( 4 .- P) = W q , p ) .

L

and 1

and

(3)

When we introduce the dimensionless variables Q = q / a o and P = ( a o / h ) p with the appropriate characteristic length a o . we find

P2

((4) 4)

((55)) This is the set of equations we solve using our modified tau method. In the present paper we truncate this set after the second derivative of the potential in order to outline our modified spectral method. In paper I1 [14] we extend this technique to treat arbitrary high orders. We conclude this section by noting that indeed these definitions limit the range of the variables x and y to the phase-space square [ - 1.11 x [ - I,l].

d2

-+ V( Q) - E - - 2 2 8 aQ

where V ( Q ) = ( M a i / h 2 ) U ( q ) and E = M ( a i / h 2 ) denote ~ the corresponding dimensionless potential and the energy eigenvalue, respectively. To apply our tau ansatz, it is convenient to transform the phase-space domain [-Po,P,]X[Q,,Qz]with Po>O and Q , < Q 2 , onto the square [ - 1,1] X [ - 1,1]. This is the range for which the Chebyshev polynomials used in our method are defined. We can easily invert this linear mapping once we have found the solution. We accomplish this transformation by introducing the new momentum variable

P ysPO and the new position variable

111. SPECTRAL AND TAU METHODS: A BRTEF REVIEW

In the present section we first briefly review the spectral and tau methods for finding solutions of partial differential equations. Detailed descriptions of these methods can be found in the mathematical literature, e.g., Refs. [13,15], and the references therein. We then adopt these techniques to the specific problem of the two coupled equations (4) and (5). Spectral methods belong to the class of weighted residual methods and address differential equations of the form

L*=f,

((6) 6)

where L is an arbitrary differential operator, f an inhomogeneity, and CC, the solution of that problem in a domain D . The spectral method starts from the approximate solution N

((7) 7) of Eq. (6), that is a linear superposition of N known trial or shape functions 4,. Here the coefficients a , are the constant

517 M. HUG, C. MENKE, AND W. P. SCHLEICH

3190

coefficients to be determined. When we substitute the approximate solution (cIN into Eq. (6),the residual

(8) of the equation is in general not equal to zero. The aim of the method is to find $ N , i.e., the unknown coefficients a i , such that R is small. We achieve this goal by requiring that the integral Rq,=O jD

(9)

of the weighted residuals over D vanishes. Here the quantities cpv with v = 1, ...,N are called weight or test functions. From Eq. (9) we find, with Eq. (8), the N equations for linear operators L,

for the N unknown coefficients a i . When we define the components

of the N X N matrix

r, and the components

of the N vector b, this problem can be written as the linear system of equations

ra= b

(10)

for the vector ( a ) i = a i containing the N unknown coefficients a i of the spectral ansatz (7). The spectral method uses the same orthogonal functions 4i= ‘pias trial and weight functions leading to the relation

j p ” X = 4”const. whereby 15,” and x denote the Kronecker symbol Eq. (CI) and a positive weight function depending on the particular choice of the lp, , respectively. Making use of these orthogonality relations only pairs lpiqvwith i= v give a nonvanishing contribution to the system of equations (10). Usually the trial functions satisfy the same boundary conditions as required for the solution $. When this is not the case the method is called the tau method. Then certain equations of system (10) are discarded and replaced by the boundary constraints. It is worth mentioning that for sufficiently smooth solutions the spectral tau method yields results that are much more accurate than any finite-difference scheme with the same number of unknowns, see Ref. [16]. Since in our specific problem there are no boundary conditions, the tau method is the basis of our approach. In contrast to usual applications of tau methods we have to solve

57 -

two real partial differential equations simultaneously corresponding to the two operators Lev,, and Ldd. This leads to an overdetermined linear system consisting of 2 N equations for N unknowns. Here we have two choices: Either we find a way to reduce the system to a N X N system of full rank, or we keep all the equations with more than one nonvanishing coefficient, and apply a least-squares method to the full system. The second approach is valid, provided that the error introduced by the least-squares approximation is not larger than the spectral error. That means that in practice we do not require that the weighted residuals Eq. (9) vanish, but we minimize them in a least-squares sense. Our numerical experiments show that this approach does indeed yield solutions of very high accuracy for our model problem, where the exact solution is known and can be used to check the numerical solution. IV. TRUNCATION OF THE SYSTEM

In the preceding sections we have laid the foundations for both papers. To bring out most clearly the essential ingredients of our modified tau method we now illustrate this technique for the truncated system of equations. This is the set of equations which we actually solve in this paper. The generalization to infinite order-that is the full Eqs. (4) and (5)-is the topic of paper I1 [14]. In general, Eqs. (1) and (2) are of infinite order in the momentum p . However, in the case of a polynomial potential only a finite number of derivatives contributes. For example, the equations for the harmonic oscillator with U ( q ) = i m o Z q Zare of second order. In this case analytic solutions of the two coupled equations exist [4]. The ultimate goal of the present work is to solve this set of equations including all derivatives. However, to demonstrate our modified spectral tau method and to illustrate the structure of the resulting linear equation system, we neglect in a first approach terms of higher than second order in p . Hence we here consider the simplified equations

(1) and

(12) This is the set of equations we use throughout this paper. We note that Eq. (11) is closely related [7] with the Liouville equation of classical statistical mechanics. V. IMPLEMENTATION OF OUR METHOD

In this section we present an implementation of our modified spectral tau method and proceed in three steps: First we formulate our spectral ansatz Eq. (7) for the two coupled differential equations (11) and (12). Since our problem involves two differential equations we have to evaluate two

518

57

MODIFIED SPECTRAL METHOD I N . . . . I. . . .

residuals Eq. (8) in the second step. The last step is to build up the linear system of equations Eq. (10). We emphasize that the set of equations (11) and (12) can be solved analytically [4] for the harmonic oscillator V(Q) = Q2/2, and thus provides a good test of our numerical results. When we include higher-order terms of the differential equations (1) and (2), the procedure remains the same. Each additional order leads to one additional term in the matrix coefficients of system (10). Since this does not increase the number of equations or the number of unknowns, it is computationally not more expensive to include higherorder terms (see Ref. [14]).

3191

(15) where the coefficients for k = 0.1,. .. read NY

a;.Zk=E s = k 2(2s+1)aj,&+l

9

(16)

(17)

A. Establishing the spectral ansatz

We now present our modified tau method for the secondorder differential equations (11) and (12) for the Wigner function. We follow Ref. [17], and use Chebyshev polynomials in our approach. They are recommended as the best choice whenever a nonperiodic problem occurs or whenever one works with general functions that do not allow us to use special geometries. Since we seek a universal method valid for an arbitrary potential, we face just such a situation. Chebyshev polynomials give good results, and converge fast enough under almost all circumstances. Furthermore, they are easy to handle, because all terms, products, and derivatives can be reexpressed in terms of Chebyshev polynomials. We assume a solution of the form

To satisfy Eq. (15) for all x the products a!,,Tk(O) have to vanish for all k . Since Chebyshev polynomials T z k + , ( y )of odd order vanish at the origin the condition ay,,=O has to be satisfied. According to Eq. (16) this translates itself into the requirement aj,zk+l =o

(18)

for all k . Hence in our specific problem of Eqs. (1 1) and (12) the number of unknowns is reduced by a factor of 2. Since in contrast to the usual spectral method no boundary conditions are present, Eq. (18) is the only information we can extract from the boundaries.

(13) which is a special case of the ansatz Eq. (7). Here we denote the Chebyshev polynomial of order j by Tj, and refer for their properties to Ref. [18] and Appendix B. As usual in Chebyshev series, the primes indicate that the first term in each sum is multiplied by the factor ). To reduce the number of unknowns we take advantage of the symmetry of Eqs. (1 1) and (12) in y . following from Eq. (3). For any approximation l v N of the Wigner function that has a continuous derivative this symmetry is equivalent to von Neumann boundary conditions

(14) on the x axis. When we substitute ansatz (13) into condition (14) we find-xpressing with the help of Eq. (B2) the derivative of a Chebyshev series in another Chebyshev seriesthe constraint

B. Evaluating the residuals

We now insert our spectral ansatz Eq. (13) for V into the two coupled differential equations (11) and (12) for the Wigner function and evaluate the two residuals of these equations. For the sake of simplicity we here only present the main ideas and results and refer to Appendix B for the calculation. Since we have represented the Wigner function T by a Chebyshev series, it is convenient also to express the potential

(19) as a Chebyshev series. For polynomial potentials this representation is exact, while for other smooth potentials the approximation can be made as accurate as desired. This we achieve by an appropriate choice of the upper limit N u . When, according to Appendix B, we express all the terms in Eqs. (11) and (12) by Chebyshev polynomials, for the residual of Eq. (1 1) we obtain the expression

I

(20) whereas the residual of Eq. (12) reads

519 3192

57 -

M. HUG, C. MENKE, AND W. P. SCHLEICH

(21) \

I

2kSNy

The coefficients of the series follow from Eq. (B3), and are given by N*

a;,2k=

X 2 ( j + 1 +2 s ) a , +

2kSNY-2

I

ISN,-2

These simple properties of Chebyshev polynomials bring out most clearly the structure of the residuals. C. Setting up the linear system of equations

1+ 2 s . 2 k .

S=O

NY

++,=2 2(2s+2)aJ&+2 .1

\

=k

Now we are ready to set up the system of equations (10) for the coefficients aJ,k. We require that the integral of the weighted residuals (9) over the computational domain C i s [ - l , l ] X [ - 1 , 1 ] vanishes. With the Chebyshev weight

(22)

and this yields the conditions

NY

r=k

c

(23)

NY

4(2t+1)(2~+2)a,,&+~.

S=I

We obtain the latter relations by applying Eq. (B2) twice. Note that, due to Eq. (18). a;>+ = L Z ; ~I ~=+0. To bring out clearly the variable x or y of the partial derivatives, we have introduced a superscript in the coefficients of the differentiated series. Also, the coefficients of second derivatives carry a double superscript and are double sums given by Eq. (B6). For a detailed discussion of these points we refer to Appendix B. These residuals seem to be very complicated, but their structure can easily be understood. Indeed every term in R , and R , corresponds to a term in Eqs. ( 1 1 ) and (12). Any derivative of a Chebyshev series is again a Chebyshev series. However each derivative decreases the order of the sum by 1 , since a Chebyshev series is a polynomial. The upper limits N,- 1 , N,,- 1, and Nu- 1 in the multiple sums of R , arise from the differentiation of the corresponding Chebyshev series. Equivalently the upper limits N,- 2, N , - 2, and N , - 2 in the multiple sums of R2 occur, where the corresponding quantities have been differentiated twice. Furthermore one has to perform products of Chebyshev polynomials. As shown in Appendix B this leads to symmetrical shifts in the indices of the Chebyshev polynomials, which can also be easily identified in the expressions Eqs. (20) and (21). At last one has to multiply an ordinary polynomial, y 2 . with Chebyshev polynomials, which gives a symmetrical structure of three Chebyshev polynomials T 2 k - 2 , T2, , and T 2 k + 2 .

for both residuals, that is for a = 1 and a = 2 and all pairs v , with ~ v=0,1,2 ,...,N, and p = 0 , 1 , 2 ,...,N y . We substitute Eqs. (20) and (21) for the residuals R , and Rz into Eq. (23). and note that due to the orthogonality

of the Chebyshev polynomials only the integrals

contribute. To satisfy Eq. (23) the prefactors of these integrals, that is, the prefactors belonging to the product T , ( x ) T , ( y ) , must vanish. This condition leads to the homogeneous system of equations

(24) kSN,

We note that this system corresponds to the system Eq. (10) with b=O. We identify the coefficients rJ;[ according to Appendix C, and find

520 57

MODIFIED SPECTRAL METHOD I N . . . . I. . . .

3193

if (25) _L

^

I

fi,Qn

1/JL V

l

h

"=.?

-

and

2/u + 1 = k - 1 - 2n n€ iN

and

if

and

2 W €{0.2

V,}

and

=

if

- 2 - 2n

(26)

and

and 2n€{0,2..,;Vj,-2}

Here the terms Vf and Vf* follow from Eq. (B4). We note that the coefficients Tffi*1 (odd in /z) result from the residual /?i whereas the coefficients rjjf'* (even in /i) originate from /? 2 . The linear system (24) obviously has the trivial solution. To find nontrivial solutions, we need at least one additional equation. We do this by setting the value of the solution at an arbitrary point in the computational domain, such as the origin, to a nonzero value, for example unity. We then obtain from Eq. (13), recalling the properties r 2 y+i(0) = 0 and 72/0) = (- 1V, the additional equation

(27)

Since the original equations (11) and (12) are linear, we can arbitrarily choose the value of the Wigner function and the solution can be normalized after the computation of the unknown coefficients. We approximate the solution of the overdetermined linear system (24) by a standard least-squares algorithm [19]. This introduces an additional error to our solution since then the

52 1 3194

51 -

M. HUG, C. MENKE, AND W. P. SCHLEICH

residuals are not exactly zero, but are minimized in a leastsquares sense using the L2 norm. In general, the norm of the residuals decreases with increasing N , and N , , and therefore can be used as a measure to check the accuracy of the approximation. In Sec. VII we present numerical calculations for our model problem. As the results are of high accuracy for appropriate N , and N , , the least-squares approach can be justified. VI. CONVERGENCE OF OUR METHOD

In this section we analyze the convergence of our integration scheme for the set of differential equations (11) and (12). Throughout the section we make heavy use of results of Canuto et al. [16]. We discuss error estimates, stability, and consistency. For an introduction into the elements of functional analysis and definitions such as stability, consistency, and well-posed problems, we refer to Appendix D. In the present section we just discuss the results qualitatively. The convergence properties and the errors of a Chebyshev series approximating a given function have been studied extensively [ 161. We make use of these results in our approach to control the error of the approximation to the potential V ( x ) by a Chebyshev series. However, these properties are not applicable to the approximation of the solution of Eqs. (11) and (12), since here the approximated function is not given explicitly but implicitly as a solution of differential equations. In order to apply the convergence theorems of Chebyshev series we have to ensure that the approximation converges toward the solution of the differential equation. Hence we have to show that the differential equation has a unique solution. Furthermore we have to investigate whether our approximation converges and represents this solution in the limit N + m . Therefore the question of convergence of a numerical approximation to the exact solution is central to our scheme. According to the Lax-Richtmyer theorem [20], stability and consistency imply the convergence of a well-posed problem. The most straightforward technique for establishing the stability as well as the consistency of the spectral schemes is the so-called energy method [ 161. To apply this technique in our specific problem, we have to prove the inequalities

and the H i norm additionally contains all the first derivatives, that is,

Inequality (28) states that L is a positive operator, which is called coercive over H i , while Eq. (29) is a continuity condition for L in the sense that (.L$,F)xdepends continuously on $ and g. If these inequalities hold there exists a unique solution of the problem L$=f according to a theorem in Ref. [16]. The application of this result to the Chebyshev polynomials of degree N yields, furthermore, the stability and consistency of our spectral scheme in the sense of the energy method, if N is sufficiently large. Under this hypothesis convergence is a consequence of the Lax-Richtmyer theorem. If the inequalities required by the energy method are fulfilled, then the corresponding error estimate for the approximate solution TNreads

(30) for each ( c r H;(R). ~ For the second-order equations we have m = 2 , and hence the error is proportional to 1/N. The second term on the right-hand side contains all derivatives up to second order. This has the consequence that other parameters such as the size of the integration domain D can also influence the error estimate. We will see this in Sec. VII when we investigate the harmonic oscillator. To prove the stability of our scheme, we simply have to follow these instructions with one modification. Usually one has to solve only a single equation with boundary conditions. In the context of Wigner function, however, we have to actually solve two equations simultaneously. The first-order equation is hyperbolic and the second-order equation is elliptic provided d 2 V / d x 2 > 0 . It is easy to see that a hyperbolic differential operator does not fulfill inequality (28). Nevertheless we can apply the energy method to the system of equations (1 1) and (12). To see that, we add Eq. (1 1) to Eq. (12). and arrive at the equivalent system

(28)

i

LzV= -+V(x)-E and

(29) where C , and Cz denote positive constants. These inequalities have to be valid for all functions ( I . E D ( L ) and $ E H i , where D ( L ) is the domain of the differential operator L . For a definition of the Sobolev spaces H i and H i , we refer to Appendix D. Here it will be sufficient to note that the is simply the scalar product norm 11

and

+ V (x) - E -

Po Qo

-y

d

-

1 dV(x) d +PoQo

dx

JY

X

(32) (32) Indeed, both equations are now elliptic for d 2 V ( x ) l d x 2 > 0 .

522 MODIFIED SPECTRAL METHOD I N . . . . I. . . .

57

Now we can start to prove stability, because both equations lead to the same coercivity evaluation as we show in Appendix E. In addition we also prove that inequality (28) holds under certain conditions for an appropriate function q ( x , y ) which is even in y . According to Appendix E, for positive d 2 V / d x Zwe obtain the estimate

i dzvj IIW +/: 1’

3195

o.2

32Qo2 min

(Lzvv,w,~

0

1

( T + V ( x ) - E ‘Pzx d x d y

1

-1

+ B,

(33)

where B represents the boundary terms, given explicitly in Appendix E. Here we used the weighted scalar product Eq. (El). Hence to achieve stability according to Eq. (28), w e have to require that the sum of the integral and the boundary terms on the right-hand side of Eq. (33) is greater than a constant times /l\Y112. For this purpose it is sufficient that the sum of these terms is not negative. W e denote the phaseThis space domain, where { P i y 2 / 2 - [ E - V ( x ) ] } a Oby area corresponds to the “nonclassical” phase-space region outside the classical trajectory E = P/2+ V ( Q ) . Let CL,=n-n,, the complement of C L , , then

-0.2

a,.

4

P

FIG. 1. Wigner function (top) and zero phase-space contour lines (bottom) of the eighth energy eigenstate of the harmonic oscillator.

-/

/a2[[E-V(x)]-

(34)

2

Whenever the integral over a, in Eq. (34) is greater than the integral over CL2. the right-hand side is positive. Then it can be written as a positive constant times llT/12by applying a mean value theorem for integrals. Note that the boundary terms of Eq. (33) are zero when the Wigner function at the borders of the computational domain is zero or at least neg-

ligibly small. In this case inequality (33) is valid. This argumentation holds if the domain CL is chosen in such a way that most of it lies outside the classical region. Hence the choice of 0,i.e., P o , Q , , and Q 2 , influences the stability. For other choices of the computational domain we have to require that the right-hand side of Eq. (33) does not become negative. Our proof of stability might fail, if d 2 V / d x Z becomes negative. Since we have only binding potentials, there is always a phase-space region, where the curvature is convex and therefore d 2 V / d x 2 is positive. In this case we can decompose all integrals containing d 2 V / d x 2 into positive and negative parts in exactly the same way as in Eq. (34), and we

TABLE I.Mean and maximal absolute errors for the Wigner function approximation of the eighth energy eigenstate of the harmonic oscillator depending on the number N,,= N,= N , of Chebyshev polynomials in our ansatz (13). We applied our modified spectral method on the computational phase space domains, D I = [ - 4.5,4.5] X [ - 4.5,4.5], D, = [ - 4.0,4.0] X [ - 4.0,4.0], and D, = [ - 3.5,3.5] X [ - 3.5,3.5]. I n the left part of the table we compare the mean absolute errors with increasing N,, and on the right part the maximal absolute enors which are located in the edges as shown in Table 11. Note that to find an error of the same order of magnitude we have to increase the number of Chebyshev polynomials N,, with increasing size of the computational doman. Maximal absolute error

Mean absolute error N,=N, 30 40 50 60 70

Dl

0 2

1.4X10’0 4.3X10-’ l . l ~ l O + ~ 3.3x10-’ 4.1X10-’ 6.7X10-’ 4.1X10-* 1.7X10-10 3.7X1O-l4 9.3X10-’2

D3

1.6X10-2 2.6~10-~ 6.0X10-9 4.1X10-’3 2.2X10-15

Dl 6.6X1OfZ 5.4~10’~ 2.OX1OZo l.0X10-5 1.7X10-9

D2

9.8X10” 6.4x10-1 9.2X10-6 1.4X10-9 6.6X10-”

D3

3.2X10-1 1.7~10-~ 4.6X10-’ 3.0X10-12 1.3X1O-l4

523 M. HUG, C. MENKE, AND W. P. SCHLEICH

3196

of TABLE 11. Dependence of the absolute errors (times the Wigner function approximation Eq. (13) of the eighth energy eigenstate of the harmonic oscillator on the location ( Q , P )in the phase-space domain D ,with N,,=50. We note the dramatic error located in the comer (Q,P)=(4.5,4.S).

30.4 3.7 1.7 1.9

0.9

0.0

0.9

1.8

2.7

3.6

4.5

Q

57

TABLE 111. Dependence of the absolute errors (times lo-'') of the Wigner function approximation Eq. (13) of the eighth energy eigenstate of the harmonic oscillator on the location ( Q , P )in the phase-space domain D , with N,,=70. Note that in contrast to Table 11, due to the larger number of polynomials the error decreases enormously and the error in the comer is much less dramatic. It will completely disappear for any higher N,. 4.5 3.6 2.7 1.8 0.9 0.0

1.5 10.8

0.0

3.7 12.9 4.0

0.3 8.3 5.9 9.8 3.7 19.7

0.0

0.9

1.8

2.6 10.3 0.2 19.7

4.0

4.0

0.2

0.2

9.1 0.1 1.7 8.3 10.8 10.3

1675.0 9.1 0.2 0.3 1.5 2.6

2.7

3.6

4.5

1.7 10.3 5.9

4.0

Q have to require that the negative parts do not dominate. Again this is influenced by the choice of Q, and Q2. Thus one has to be very careful in choosing a proper domain. To obtain stability with the energy method, one has to choose a domain where the positive parts of d 2 V / d x 2 outweigh the negative parts, and where the additional negative terms do not dominate. To complete our proof of stability, we have to show the inequality (29). For each term of (L2'P,$), this can be done in a straightforward way by using the triangle inequality, partial integrations, the Cauchy-Schwartz inequality, and Hardy's inequality. With that estimation, stability and consistency, and hereby convergence of our method, are achieved, if the coordinate range is chosen carefully. The error estimate (30) holds. We conclude this section by noting that higher-order terms lead to more complicated investigations because more cases occur. Nevertheless the convergence and stability results remain valid under conditions analogous to Eq. (33).

VII. NUMERICAL RESULTS As a first application and a test of our method we choose the harmonic-oscillatof potential V ( Q) = Q2/2. This is motivated by the fact that an analytic expression for the Wigner function of an energy eigenstate is known in this case. Indeed, the Wigner function of the nth energy eigenstate with energy E=n + of the harmonic oscillator is [21]

(35) where L, is a Laguerre polynomial and

R 2 = Q2+ P2. In Fig. 1 we show the Wigner function of the eighth eigenstate together with the contour lines of q ( Q , P ) , According to Eq. (35) the latter are circles centered at the origin of phase space. Multiplying our numerical solution based on the method of Sec. V with a normalization factor enables us to compare it with the exact solution and to calculate the absolute errors.

Table I displays the mean errors' of our approximation of the Wigner function for various numbers N,,= N , = N , of Chebyshev polynomials in the variables x and y. Here we chose E = 8.5 and calculated the approximation in the three computational domains Dl-[ -4.5SQS4.5]X[ -4.5GP G4.51, D,- [ - 4 s QG4] X [ - 4 s PG4], and D 3 = [ - 3.5 G Q s 3 . 5 ] X [ -3.5GPS3.51 of phase space. The smallest domain D, is chosen in such a way that it contains all oscillations of the Wigner function and thus all relevant physical information. The largest domain D , contains the whole region in which the Wigner function is nonzero. For a constant number of Chebyshev polynomials in our ansatz Eq. (13). the errors increase with the size of the chosen domain. In each domain the errors decrease with increasing number of Chebyshev polynomials. The consequence is that for larger computational domains we have to incorporate more Chebyshev polynomials to keep the error constant. Therefore it is important to find the balance to properly choose a computational domain which is large enough to obtain properties of the Wigner function resolved, but not too large, since then the system of equation is very large when we want to arrive at a distinct error. The numerical results for Ne,=N,=N,=50 in Table I1 show that the maximum of the absolute error occurs in the edges of the chosen domain (here 0 ' ) .Far away from the edges the errors are much smaller. The data in Table I11 illustrate that this effect is compressed for a higher number of Chebyshev polynomials Neq=N,=N,=70. In this case the absolute errors are nearly constant. Only in the comers they are three orders of magnitude larger but still of order Therefore the Wigner function is now approximated very well in the whole computational domain. In the regions where the Wigner function is strongly oscillating the absolute errors are of order for the domain D 3= [ - 3.5,3.5]X [ - 3.5,3.5] and of order lo-' for the domain 02 = [ - 4.0.4.01X [ - 4.0.4.01, with N,,= 50. In the larger domain D I = [ - 4.5,4.5] X[-4.5,4.5] the errors are of order lo-, but when we in-

'In order to compare our approximation to the exact solution we evaluate our spectral solution on an equidistant grid consisting of IOOXlOO points. At each grid point we calculate the absolute difference between the approximate and the exact solution. The mean value of these differences we call mean absolute error and their maximum we call maximal absolute error.

524

57

MODIFIED SPECTRAL METHOD I N . . . . I. . . ,

crease N,, only by 10 we obtain errors of order lo-*. The errors for N,,=50 are all still small enough to obtain an accurate impression of the Wigner function, since the error can hardly be seen on the scale of the Wigner function. Our numerical results confirm that our method is a valuable tool for the computation of the solution of partial differential equations of forms (1) and (2).

3197

a

of any operator through an integration in phase space. In this sense the Wigner function acts like a classical phasespace distribution. Here

(A2)

VIII. CONCLUSION

We have presented an alternative approach for the calculation of the Wigner function. Our method avoids the solution of the Schrodinger equation for the wave function. Instead, we apply a result from Fairlie and Kundt [2] and others [3] and solve two coupled real linear partial differential equations. Our numerical method is a modification of the spectral tau method, which uses Chebyshev polynomials as shape functions. In contrast to usual applications, no boundary conditions are available for the equations. Since two differential equations have to be solved simultaneously, the resulting linear equation system is overdetermined. Its solution is approximated by a least-squares method. We could prove stability and convergence of our scheme by making use of results of Ref. [ 161. Generalizations of our approach to any similar system of partial differential equations are straightforward. Our numerical results for the harmonic oscillator are in excellent agreement with the known analytic solution. The mean absolute error is of order for an appropriate choice of the domain and the approximation order N,,= 70. Our modified spectral method is equally applicable to any potential V, and can be extended to the higher-order differential equations. This will be the subject of the following paper. Our results encourage further research on potentials for which no analytic solution can be calculated.

a.

denotes the Weyl-Wigner correspondence of the operator Therefore the Wigner function itself is the Weyl-Wigner correspondence of the operator I;. We can interpret the two coupled partial differential equations (1) and (2) as the phase-space correspondence of the time-independent Schrodinger equation

fiI;=Ei for the density operator which in this case equals the projeccorresponding to the energy tion operator p= I &)( eigenstate I + F) with energy eigenvalue ? !. and the Hamiltonian H = p2/2M+U(q). When we perform the WeylWigner correspondence Eq. (A2) of both sides of this operator equation, with Eq. (Al) we find the relation

$4

(Hpl(q,p)=W'(q,p).

We recall the Weyl-Wigner correspondence 221

for products of operators ab, and arrive at

ACKNOWLEDGMENTS

We thank R. Baltin, M. V. Berry, J. P. Dahl, U. Leonhardt, H. Leschke, S . Schneider, and R. Seydel for many fruitful discussions. This work was partially supported by the Deutsche Forschungsgemeinschaft. M. H. is most grateful to the Studienstiftung des deutschen Volkes for its continuous support. APPENDIX A: A DERIVATION OF THE PHASE-SPACE EQUATIONS

In this appendix we briefly summarize the central ideas of the derivation of the phase-space equations (1) and (2) for the Wigner function. Here we follow Ref. [3(c)] closely. One of the most important properties of the Wigner function

(A1) of the density operator is that it yields the correct quantummechanical marginal probability distributions of position and momentum. Moreover, it enables us to evaluate the expectation value

or

When we expand the potential (I into a Taylor series and take the real and imaginary parts, we arrive at the two real coupled partial differential equations (1) and (2) for the Wigner function in phase space APPENDIX B: PROPERTIES OF CHEBYSHEV POLYNOMIALS AND EVALUATION OF RESIDUALS

In this appendix we first briefly review some useful properties of Chebyshev polynomials, and then apply them to express all terms of the coupled differential equations (11) and (12) in terms of Chebyshev polynomials. Since the operators L , and L , , are linear, this can be done term by term. Here we demonstrate this procedure for one term only, and then outline how to proceed to evaluate the residuals. The

525 51 -

M. HUG, C. MENKE. AND W. P. SCHLEICH

3198

application of the same technique to all the other terms is straightforward, and therefore not shown here.

c.

N" N"

Vy=4x

( l + l + 2 t ) ( l + 2 + 2 ~ ) V f + 2 + 2 ~ .(84)

1=0 s = t

1. Properties of Chebyshev polynomials

Chebyshev polynomials were investigated in great detail in Ref. [18]. Here we only specify those properties that we use in the derivation of our modified spectral method. Products T , ( x ) T , ( x ) of Chebyshev polynomials with positive integers s and r can he written as

Here we have applied Eq. (B3) twice. When we insert our spectral ansatz Eq. (13) into ( d 2 / d y 2 ) V r ,we obtain, from Eq. (B2).

kSNy

(B1) where we defined T - , ( x ) = T S ( x ) . Products of the form n'T,(x) are given by

kGNy-2

where it is now convenient to decompose the coefficients of the differentiated Chebyshev series (B3) into even and odd parts in order to use the symmetry relation (18). The coefficients for the even terms read NY aYY

2(2r+

I,Zk

The derivative of a finite Chebyshev series of degree n can be expressed as

1)a:,2~+I

3

with NY ay,21 + 1 = I = I

(B2) that is a Chebyshev series of degree n - 1 with coefficients

2(2s +

)'J.&+2

9

and for the odd terms, N.

n

a:=x 2 ( i + 1 + 2 s ) ~ ~ + ~ + ~ ,

(B3)

S=O

for i=O,l, ... and with aj=O for i>n. Hence the differentiated Chebyshev series is a sum containing the coefficients a j of the original Chebyshev series. Note that, for higher derivatives, Eq. (B3) is a recurrence relation that connects the coefficients of the n-times differentiated Chebyshev series with the coefficients of the ( n - 1)-times differentiated series. Finally these coefficients of the n-times differentiated series consist of n-fold sums containing the coefficients ai of the original series.

with NY

X

a;,21+2=s = 1 + l 2 ( 2 s + l ) a l . 2 r + l = ~ ; that is, a%+ 1 = O

Here we exploited the symmetry relation Eq. (18). Hence we arrive at

2. Evaluation of residuals

We now demonstrate our technique and express the term [ d 2 V ( x ) / d x Z ] ( d 2 ~ / d yappearing Z) in Eq. (12), in terms of Chebyshev polynomials. The second derivative d 2 V ( x ) / d x 2 of the potential Eq. (19) reads

g*=x ' a2

c.

where

N"

NY

- 4 t x= k

5=1

(2r+1)(2s+2)~,,,+~.

(B6)

Now we can perform the product

N"

v;I= t = O

(B5)

where

'r From Eq. (B2), we find

ar,!2kT~(x)T2k(y),

JGN, 2kSNy-2

2( I

+ 1 + 2t) v;+ + Z f , ' $ V;'ar~k[T~+l(x)+T~-f(x)lT2k(y)

I G N ~ 2kGNy-2 fSN,-2

and hence

of these two terms, where we used Eq. (Bl). Note that T J - f ( x ) = T ~ J - l i ( xFor ) . all the other terms of the coupled

h 51

MODIFIED SPECTRAL METHOD I N . . . . I. . . .

differential equations (1 1 ) and (12) we can proceed in exactly the same way, and find the residuals R , and R , presented in Eqs. (20) and (21). APPENDIX C: DERIVATION OF THE LINEAR SYSTEM

In this appendix we calculate the coefficients ry;j' of the linear system Eq. (24). As mentioned in Sec. V, the system is defined by the vanishing prefactors of the integrals in Eq. (23).We have to compute explicitly the prefactors belonging to a given product T , ( x ) T , ( y ) in their dependence on v and p. Because of the symmetry Eq. (14) in y , the residual R 1 only yields odd coefficients 2 p + 1 in y . and the residual R 2

3199

only yields even coefficients 2 p in y . The problem now is to find, for any fixed pair of numbers v and p,those pairs j and k for which the corresponding product T , ( x ) T , ( y ) of Chebyshev polynomials occurs. For that purpose we introduce the Kronecker delta

sJ'y=

I

1 j=v

0

otherwise,

(c1)

which we use in two dimensions as a product of two Kronecker deltas. With this notation, for the residual R , , Eq. (20), we find

(c2)

(c3) Here the constant c 2 = l / d when both coefficients v and p are positive, c,=2/~r' when one of them is zero, and c2=4/7rZ when both are zero. Equations ('2%)and (C3) represent system (10) of equations, r a = 0. Note, however, that it contains the unknown coeffiexplicitly, we have to express the coefficients of the cients a,,k only implicitly. To obtain the matrix coefficients To do so we have to change the order of summation etc., in terms of the unknowns differentiated Chebyshev series in the double sums in Eq. (B6), and the equivalent equation for J2q/dx2. We start from

ry;!

and

Here J= Ij-ll or J=j + l , and we have used Eq. (B3) twice. For convenience, we have shifted the upper limit of the sum for a?u.zp by k . This does not change the sum since u , , ~ = O for k > N , . When we interchange the order of summation we obtain

and

527 M. HUG, C. MENKE, AND W. P. SCHLEICH

3200

Ny+k s=k

l=k

4(2t+1)(2s+2)~,~+2.

The inner sums over t can be written as products J

2 ( j + 1+ 2 t ) = ( s + I ) ( j + l ) + S ( S +

I)=($+ I)(j+s+I )

f=O

and

Thus the equations of the weighted residuals [Eqs. (C2) and (C3)] take on the forms

and

When we shift some indices of the first equation by T=j+ 1 + 2 s and E=2k+2+2s, it finally has the form

(c4) Similarly we shift some indices of the second equation by T = j + 2 + 2 s and E = 2 k + 2 + 2 s . With u=T-2-2s 2 p = R-2-2s, we can furthermore collect the prefactors

and 4( s This finally yields

+ 1)(2k + s + 1) = ( F- 2@)(K+ 2 p )= ( P - 4p’).

and

528

51

MODIFIED SPECTRAL METHOD I N . . . . I. . . ,

.

3201

N,

(C5)

From Eqs. (C4) and (C5), we immediately can read off the matrix elements ry;l of the system of equations for a j , k .We finally arrive at Eqs. (25) and (26). APPENDIX D: ELEMENTS OF FUNCTIONAL ANALYSIS

In this appendix we briefly review fundamental definitions of the numerical functional analysis and the energy method. This summary provides the theoretical background for the convergence analysis of our spectral scheme (Sec. VI) .

11 Q N f

-fNll
2 appearing in A2m-l and A,, [Eqs. (1) and (2)]. Taking the derivative d l m ~ N ( x , y ) l d f " for any m>O of our spectral ansatz Eq. (3). we obtain

where now the superscript m of a;' denotes the coefficient of the mth derivative with respect to y of the Chebyshev series (3) which is again a Chebyshev series [2]. Note that in contrast to paper I, we do not have to indicate the variable of the differentiation since for any term of higher order than the second all the derivatives of Vr are with respect to y . The coefficient uTk is determined by a recurrence relation that connects the coefficients of the m-times differentiated Chebyshev series with the coefficients of the ( m - 1)-times differentiated series. As shown in Appendix A, we can use this recurrence relation to express these coefficients by the coefficients aj,k of the original Chebyshev series (3) via an ( m - 1)-fold sum. Surprisingly we can factorize this multiple sum and therefore finally this reads for odd indices 2k+ 1, N.

m-l

(6) and for the even ones,

111. ANALYSIS OF THE HIGHER-ORDER TERMS

In this section we concentrate on the additional aspects we have to consider, when we do not truncate the system after the second derivative of the potential but take all terms into account. As there are no changes in the terms correlated to the truncated equations Eqs. (1 1) and (12) of paper I [ 11, the analysis presented there remains valid. For each higherorder differential quotient which we additionally take into account, we obtain one further summand in the equation for

"-1

(7) Due to the symmetry relation "(x, - y ) = ' P ( x , y ) following =0, and confrom Eqs. (1) and (2), the coefficients aj.2K+1 are zero. sequently a;,$+, and

535 3208

51 -

M. HUG, C.MENKE, AND W.P. SCHLEICH

The remarkable consequence of formulas (6) and (7) for the implementation of our method is that we do not have to calculate the multiple sums arising from the recurrence relation, since they can be factorized. Instead we only have to evaluate simple products. Therefore we are able to treat the derivative d m V / d y m for arbitrary m . We give an explicit expression for this sum factorization formula in Appendix A, and prove it in Appendix B.

In the same way we can calculate the expression dmV V;"T/(X). dnm I S N U - m

--

C'

Note that again we can use a recurrence relation and apply the sum factorization formula and arrive at

(8) For the evaluation of the higher-order terms of Eqs. (1) and (2). we have to calculate the product

(9) /GN,-m

where V;" is determined by Eq. (8), and the coefficients u r k by Eqs. (6) and (7). Here we used

(10) m )Eqs. ( 1 ) or (2) contribute to the residual where s and r are integers. Therefore the terms [ d m V ( x ) / d x m ] ( a m V / a y of

(11) where Eq. (9) generates the summands

/ G N U- ( 2 m- I )

In an analogous way, we obtain the residual

(12)

with the summands

2kSNy-2m / G N U- Zm

536 51

MODIFIED SPECTRAL METHOD IN

.. .

. 11.

...

3209

These expressions are for the summation index m= 1 the very same as in paper I, and are regularly continued for higher orders m> 1. Therefore, the residual structure does not become more complicated, when we add arbitrary high orders to the calculation.

(0,

jfk

we conclude that only the integrals

B. Setting up the linear system of equations

We follow the next step of our scheme and set up the linear system of equations which determines the unknown coefficients a j , k . We do this by requiring l : , R , , T , ( n ) T , ( y ) x ( x . y ) d n d y =O.

(13)

which means that the weighted residuals over the domain n=[-l,l]X[-l,l] must vanish for a = l and 2 and all pairs v,p with v=0,1,2 ,...,N , and p=0,1,2 ,...,N , . The abbreviation x ( x , y ) denotes the Chebyshev weight

All terms of the residuals R 1 and R 2 [Eqs. (1 1) and (12)] are expressed in Eq. (9) by Chebyshev polynomials. Because the orthogonality relation for Chebyshev polynomials reads

contribute. Therefore we only have to identify the terms in Eq. (13) containing this specific combination of Chebyshev polynomials. For this combination the integral does not vanish. This means that the prefactors of these terms have to vanish in order to satisfy Eq. (13), yielding a system of linear equations for the unknown coefficients a j , k . Since the product ( d m / d x m )V ( x ) ( a m / d y m ) in Eq. (9) produces only shifts j + l and j - 1 , which are exactly the same shifts that are present in the truncated equations, the technical procedure of finding out these coefficients is the very same as we have done in Appendix C of paper I [l]. When we use these results, the weighted residual equations (13) now read

(15) and

(16)

Here the constant c I = l / a 2 when v is positive, c 1= 2/7r2 when v=O, c 2 = l / d when both coefficients v and p are positive, c 2 = 2 / r 2 when one of them is zero, and c z = 4 / a 2 when both are zero. We obtain all Chebyshev coefficients with a superscript from Eqs. (6), (7), and (8). Then Eqs. (15) and (16) yield the overdetermined system (5) of linear equations

jsNx kSNy

r y : /a j, k =o ,

where we identify the coefficients

ry;/ with

(17)

537 M. HUG, C. MENKE, AND W. P. SCHLEICH

u

if

57

2p

= j - 1 - 2n

+ 1 = [lc - 11

n~lN U € { O , l , ...,N ,

- 1)

'Zii+l€{l

.....N,}

(18)

+

t

f Vi

if

and

2p = k ZPE{O.Z

.....IVY)

(19) if

I

Here, for convenience, we have defined the modified prefactor

(3). recalling the properties T 2 j + l ( 0 ) = 0 and TZj(O) =( - 1)'. the additional equation V(O,O)=l=

.C' ( - l ) j + k a 2 j , 2 k .

F N x ksN,

where the terms V; follow from Eq. (8). This scheme gives the rule about how we have to set up the matrix of the system of equations. We approximate the solution of the overdetermined system Eq. (17) by a standard least-squares algorithm [3]. To avoid calculating only the trivial solution, we have to add at least one further equation. We do this by setting the value of the solution at an arbitrary point in the computational domain, such as the origin to a nonzero value, for example unity. Thus we obtain from Eq.

Since the original equations (1) and (2) are linear, we can arbitrarily choose the value of the Wigner function at the origin and the solution can be normalized after the computation of the unknown coefficients. We note that the coefficients rr;:p+' (odd in p) result from the residual R , , whereas the coefficients r?? (even in p) originate from R 2 . Thus the original two differential equations still occur separately in the resulting system of equations (17). We further note that this equation system contains all orders of the differential equations (1) and (2).

538 57

MODIFIED SPECTRAL METHOD I N . . . . 11. . . .

Nevertheless it consists of ( N , + 1)(NY/2+1 ) unknowns; that is, the number of Chebyshev polynomials in our ansatz without odd ones in the y direction. Therefore the higherorder terms of the derivatives of Eqs. (1) and (2) do not contribute to the numerical cost that is determined by the number of unknowns. The only effect is that the matrix has more nonzero entries, because there are, due to Eqs. (18) and (19). more cases for which a term has to be written into the matrix. Note that the stability analysis would follow the same path that we outlined in paper I. We would obtain more condition equations which have to be fulfilled to prove stability with the energy method. Instead of deriving these conditions, we prefer to show examples in Sec. IV to accentuate the practical benefit of our method. To conclude this section, we note that solving the system of equations (17) by a least-squares algorithm introduces an additional error to our solution, since then the residuals are not exactly zero. This algorithm minimizes the residuals in a least-squares sense using the Euclidean norm. Here the weighted residuals are no longer exactly zero but minimized. In general, the norm of the residuals decreases with increasing N , and N , , and therefore can be used as a measure to check the accuracy of the approximation.

r

321 1

in most examples we did not use more than 50 Chebyshev polynomials in each direction. Due to this relatively low number of polynomials, we obtain all numerical results with few computational cost (-4 min on a Hp9000/K260 workstation). The Morse oscillator potential [4] is given by

V(Q)=~h(l-e-Q’a)z,

(20)

and its energy eigenstates by

IV.APPLICATIONS In this section we apply our algorithm to two potentials: a Morse oscillator and an asymmetric double-well potential that is given by a fourth order polynomial. We use these two potentials to investigate different aspects of our method. To obtain satisfactory results for the Morse potential we have to include higher-order terms of Eqs. (1) and (2). In order to obtain a reference solution, we use an analytical expression for the wave functions of the energy eigenstates, and numerically perform the Fourier transform defining the Wigner function using this expression. This allows us to obtain the Wigner function with high accuracy. Thus we can estimate the errors of our method together with those errors that are due to the truncation of higher-order terms in Eqs. (1) and (2). For the asymmetric double-well potential no analytic solutions are known, and even semiclassical approximations fail for low excited states. Nevertheless these states are most interesting since they are most affected by the barrier of the potential. By qualitative arguments we verify that our method leads to a very good approximation even in cases where other approximations are not valid. A. Morse oscillator

For the Wigner function of the Morse oscillator we present a sequence of examples where we increase the maximum order of the derivatives included in Eqs. (1) and (2) while we keep the number of Chebyshev polynomials constant. This illustrates how many terms we need to calculate a satisfactory approximation for the Wigner function in this particular example. We present results for two different computational domains, since according to the stability considerations of paper I the appropriate choice of the domain is important to obtain good results. Furthermore we show for one specific set of parameters how the error decreases with increasing number N,, of Chebyshev polynomials. Note that

0.2 0

-0.2

P

2

FIG.1. Exact Wigner functions of all energy eigenstates of the Morse oscillator potential [Eq.(ZO)] with A =4. The figure on the bottom shows the ground state, above the first, second and third, i.e., the highest possible energy eigenstate of the Morse oscillator with this specific choice of the parameter A. The potential (thin line) with the energy levels (dashed line) of the energy eigenstates and the position probability distribution (thick line) of each particular state are shown projected on the left-hand wall for each energy eigenstate. To keep the scaling consistent we compressed the potential by a factor of 7 and the position probability distribution by 3.5. Below the three-dimensional representation of the Wigner function of each eigenstate, we show its contour lines. The thicker contour lines denote the zero level, the thin solid contours denote positive values in steps of 0.04, and the dashed contours the negative values, respectively. Note that the position probability distribution is obtained by integrating the Wigner function over y.

539 57 -

M. HUG, C. MENKE, AND W. P. SCHLEICH

3212

TABLE I. Mean absolute errors for a series of Wigner function approximations of the second excited energy eigenstate of the Morse oscillator with constant number of Chebyshev polynomials N,= N,= N,=SO calculated on the computational domains D , and D , . The series index m denotes the truncation order of the Eqs. (I) and (2). Mean absolute error

N,= SO m

Dl

2 3 4

7.4x 10-2 7.0X 2.8X 3.1X lo-* 1.ox 10-2 3 . 4 10-3 ~ 1 . 7 10-3 ~ 1 . 3 10-3 ~

5 6 7 8

9 10

D2

8.9X 2.9X lo-, 2.7X lo-* 3.8X 8 . 9 10-3 ~ 2.9X 10-3 6.8X 3 . 9 I~ O - ~ 1 . 1 ~

2 . 5 10-4 ~

with the abbreviation

and the normalization

J

(2X-2n-l)r.(n+l)

"l)= &r.(2A-n) L r ) denotes a generalized Laguerre polynomial. The energy eigenvalues read TABLE 11. Mean absolute errors for a series of Wigner function approximations of the ground state and the first excited energy eigenstate of the Morse oscillator with constant number of Chebyshev polynomials N,= N,= N,= SO calculated on the computational domains D , and D4 . The series index m on the left denotes the truncation order of the Eqs. (I) and (2). For the ground state and the first excited state we chose the same computational domain D3 to compare the errors. Mean absolute error SO

N,= m

Ground state 0 3

2 3

s.sx

4 5

LOX

6 7

8 9 10

3.3X lo-'

10-3

2 . 6 1~0 - 4 1 . 7 10-4 ~ 2 . 1 ~ 2 . 6 10-5 ~ 2 . 6 10-4 ~ 2.1x10-~

first excited state 0 3

4.5I ~O - ~ 4.0X 3 . 3 10-2 ~ l.lXIO-l

1.4~ 4.9xi0-~ 1.1x10-~ 3.9~ 5.ix10-~

D4 4.5~ 3 . S X lo-'

9 . 7 10-3 ~ 2.8X10-3 9.4~ 2 . 5 lo-' ~ 7.9xi0-~ 2 . 3 1~ 0 - ~ S.OXIO-~

TABLE 111. Mean absolute errors for the Wigner function approximation of the second excited energy eigenstate of the Morse oscillator with constant truncation order m = 10 for various numbers of Chebyshev polynomials N,,= N,= N , . Mean absolute error Nea

Dl

D2

20 30

2.1x 10-2 3.1x 10-3 2.6X 10-4 2.sx

9.4x 10-2

40

so

En=

(,+

;)& (,+ -

1.ox S.6X

I . I X 10-4

y,

with n=O,l, ...,A - 1. In our specific case we choose the dimensionless parameter A = 4.This parameter designates the relation between the height and width of the potential, and therefore the number of bounded states. This potential and all its squared eigenfunctions are sketched on the left walls in Fig. 1, where we also show the corresponding exact Wigner function. We approximated the solutions for the Wigner function of the second excited state shown in Fig. 1 with our modified spectral method on the two computational domains D , = [ -2.O=sQe8.0] X [ - 2 . 5 C P S 2 . 5 ] and D 2 = [ -4.0 CQC8.0]X[-3.0CP/3. With Eq. (C3), we can rewrite the left-hand side of Eq. ( C l ) , N+2

( ~ + 1 )

C

n

e=z

I ~ . . . L ~ + ~

(2tN+4-e+2k+e) \

1 N+2

f u rI

(C4) where we have written the term u = 2 separately. We now rename the variables in each summand in such a way that the separately written factor t ~ + 4 - , , is labeled with the same index t N + 2 in each summand. To accomplish this we shift, all indices t , with v > N + 4 - u by 1, that is, tN+4-e is renamed by t N + 4 - ( e + lif) e G U , which is the case in the first product. Hence Eq. (C4)now reads 1 N+Z

\

x ( 2 t N + 2+ 2k - N - 2 + 2 u)

(C5)

Now we interchange the order of summation of the double sum

and shift the index

e=e + 1 of the first product. Thus Eq. (C5) reads

f u rI

i N+2

\

N+2

(2t~+2+2k-N-2+2U),

where we now are able to combine the first and second products. The factors containing the indices t 2 ,. . . . t N +I are common to all summands = 2,...,N + 2 . W e therefore can factor them out. Hence the right-hand side of the last equation reads

e

N+2

(2tN+4-,+2k+e)

550 MODIFIED SPECTRAL METHOD I N .

51

. . . 11.

3223

,, ,

When we shift FN+Z=tN+2+((T-2) by a - 2 , this reads

We can finally combine the sums

and end up with i N+2

\

It+N

which is indeed the right-hand side of Eq. ( C l ) . APPENDIX D: PROOF OF EQ. (C3)

In this appendix we finally show the formula Eq. (C3) which we used in Appendix C. We carry out our proof by mathematical induction. Equation (C3) formulated for any integer N reads N+Z

(N+ 1)

II ( 2 t N + 4 - e + 2 k + e )

e=2

z2[g2

N+2

-

u-I

[2tN+4-,+2k+(e+l)l

1

(2tN+,-,+2k-N-2+2a)

[ II e:::

i

(2fN+4-~+2k+e) , I

(D1) (D1)

where we recall that H g = m .= 1 for a l p , The starting point of the induction is to verify Eq. ( D l ) for N=O, that is, 2

+

+

( 0 1 )( 2 t 2 2k + 2 ) =

E ( 2 t2 + 2k - 0 - 2 + 2 a ) . u=2

After this trivial step we start for the further proof from Eq. ( D l ) , which is our induction hypothesis. We substitute N+ 1 for N in Eq. ( D l ) and obtain N+3

(N+2)

e=2

(2t~+,-,+2k+e)

(D2) We now have to reduce the left-hand side of Eq. (D2) to the left-hand side of Eq. ( D l ) . We do this by rewriting the product

( N + 2 ) (2tN+3 + 2k + 2 ) = ( N + 1 ) (2tN+3 + 2 k + 3 ) + ( 2 f N +3 + 2 k - N + 1 ) . When we use this relation and shift the index of the product

(::: II

Ni-3

JJ

( ~ + 2 ) (2tN+,-,+2k+e)=

F= e-

(2tN+,-,+2k+e)

e=z

1

1 by 1, Eq. (D2) changes to

(N+2)(2tN+3+2k+2)

N+2

=

n

-

( 2 t N + 4 - c + 2 k + e + 1)[(N+ 1 ) ( 2 t , + 3 + 2 k + 3 ) + ( 2 t N + 3 + 2 k - N +

I)].

~ = 2

When we define

F= k + 5 we now can replace the left-hand side of the hypothesis Eq. ( D l ) by the right-hand side, and obtain

55 1 3224

M. HUG, C. MENKE, AND W. P. SCHLEICH / N+2

/

N+2

\

I N+2

57 \

\

Since we can include the term ( 2 t N + , + 2 F + 2 ) into the first product, and since the last summand is the term u= 1, we can combine this to

With

C?= a+ 1, we arrive at

which is the right-hand side of our formula Eq. (D2) for N+ 1 after resubstitution of

[l] M. Hug, C. Menke, and W. P. Schleich, preceding paper, Phys. Rev. A, 57, 3188 (1998). [2] L. FOXand I. B. Parker, Chebyshev Polynomials in N u m e r i d Analysis, Oxford Mathematical Handbooks (Oxford University Press, London, 1968). [3] E. Anderson, z.~ a i C. , ~ i ~ ~j, hDemel, ~ f , J , Dongma, J , Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S.

F and F by e and k , respectively.

Ostrouchov, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1992). [43 P. M. Morse, Phys. Rev. 34, 57 (1929). [51 M. D. Felt, I. A. Fleck, It., and A. Steiger, 1. Comput. Phys. 47, 412 (1982), and references therein. [61 For a nice presentation and resolution of this problem, see M. V. Berry, Philos. Trans. R. SOC. London, Ser. A 287, 237 (1977).