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= f(t - TI,
(2)
which leaves V invariant and is an orthogonal operator on L2(R)(we shall be dealing with r e d spaces, unless otherwise stated). A general element of V can be written uniquely as
n
where u(eilT) is the square-integrable function on the unit circle (It15 T / T )having {Un} as its Fourier coefficients and u(S) is defined as an operator on "nice" functions (e.g., Schwartz test functions) f(t) through the Fourier transform, i.e.
For the purpose of developing our operational calculus, we shall consider operators u( s)which are polynomials in S and s-'. These form an abelian algebra P of operators on V . Moreover, it will suffice to restrict our attention to the dense subspace of finite combinations in V , i.e. to P#,since our goal here is to produce an L2 theory and this can be achieved by developing the algebraic (finite) theory and then completing in the L2 norm. Note that the independence of the vectors 4 n means that u(S)# = 0 implies u(S) = 0. Our results could actually be extended to operators u(S) with {un} E ['(Z) c 12(ZZ), which also form an algebra since the product u( S-')w(S) corresponds to the convolution of the sequences {tin} and {wn}. We resist the tempt ation.
2.2. Operational Calculus
61
Let us stop for a moment to discuss the “signal-processing” interpretation of u(S)$, since that is one of the motivations behind wavelet theory. It is natural to think of u(S)$ as an approximation to a function (“signal”) f(t) obtained by sampling f only at t , = nT, n E Z. Let fo denote the band-limited function obtained > T / T . That is, fo from f by cutting off all frequencies ( with coincides with for 5 T / T but vanishes outside this interval. The value of fo at t , is then
3
which is just the Fourier coefficient of the periodic function
obtained from fo(E) by identifying ( domain,
Fo(t) =
c
+ 2n/T with f.
Tfo(nT)s(t - nT).
In the time
(7)
n
This has the same form as u(S)$, if we set t i n = Tfo(nT)and $ ( t ) = 6 ( t ) where S is the Dirac distribution. Hence the usual sampling theory may be regarded as the singular case 4 = 6, and then u(S)$ characterizes the band-limited approximation fo of f . For squareintegrable +, the samples un are no longer the values at the sharp times t n but are smeared over $n, since = { $n, u ( S ) $ ) . In fact, acts as a filter, i.e. as a convolution operator, since (u(S)$)^(Q= u(eEt*)$(t). Roughly speaking, we may think of q5 as giving the shape of a pixel.
+
62
2. Wavelet Algebras and Complex Structures
Next, a scaled family of spaces V,,a E Z, is constructed from V as follows. The dilation operator D , defined by
( D f ) ( t )= 2-'/'f(t/2),
L2(lR). It stretches
is orthogonal on
(8)
a function by a factor of 2
without altering its norm and is related to S by the commutation rules
D S = S'D,
D-'S2 = SD-1.
(9)
Hence D "squares" S while D-l takes its "square root." A repeated application of the above gives
D"S = S'~D,,
CY
E
z.
(10)
Define the spaces
V, = D"V,
(11)
which are closed in L2(IR)(Vo G V ) . An orthonormal basis for V, is given by
4E(t) G D a S n 4 ( t ) = 2-*/'4 (2-at - nT) ,
(12)
and V, can also be identified with t'(Z). The motivation is that Va will consist of functions containing detail only up to the scale of 2,, which correspond to sequences { u z } in t2((;z)representing samples at t , = 2"nT, n E Z. For this to work, we must have Va+l c V, for all a. A necessary condition for this is that 4 must satisfy a functional equation (taking cy = - 1) of the form
n
n
63
2.2. Operational Calculus
for some (unique) set of coefficients h,. Since we assume that 4 has compact support, it follows that all but a finite number of the coefficients hn vanish, so h ( S ) is a polynomial in S and S-l, i.e. h ( S ) E P. This operator averages, while D-' compresses. Hence C$ is a fixed point of this dual action of spreading and compression. The equation Dq5 = h(S)4,called a dilation equation, states that the dilated pixel Dq5 is a linear combination of undilated pixels dn. The coefficients hn uniquely determine 4, up to a sign. For if we iterate D-'h(S) = h(S'/2)D-', we obtain
n N
C$ = [o-'h(S)]
4=
h (S2-=) D - N 4 .
(14)
Ct=l
Since the Fourier transform of D-Ng5 satisfies 2*12 (D-Nq5)n(t) = &2-Nt)
--$
&O)
as N + 00,
(15)
we obtain formally
where b ( t ) is the Dirac distribution. The normalization is determined up to a sign by
11cj11
= 1. See Daubechies [1988b] for a discussion of
the convergence and the regularity of
4.
Note that the singular case 4 = 6, discussed above, satisfies the diIation equation with h(S) = f i r , where I denotes the identity operator. A more regular solution, related to the Ham basis, is the case where
is the indicator function for the interval [0, 1)and h( 5') =
( I + S)/&
In general, integration of Dq5 = h(S)$ with respect to t
gives
64
2, Wavelet Algebras and Complex Structures
Ch,=JZ,
or
h ( ~=) &I.
(17)
n
Also, the regularity of 4 is determined by the order N of the zero which h ( S ) has at S = -I, i.e.
h(S)= ( I
+ S)%(S),
with k(S) regular at S = -I. For example, N = 0 for N = 1for the Haar system. See Daubechies [1988b].
4 = S, and
The next step is to introduce a "multiscde analysis" based on the sequence of spaces V,. We shall do this in a basis-independent fashion. Since shifts and dilations are related by DS = S 2 D ,we have
This defines a map H:: V,+1 + V,, given by
Since the two sides of this equation are actually identical as functions or elements of L2(IR), H: is simply the inclusion map which establishes VO+l c V,. This shows that the relation D4 = h(S)+ is not only necessary but also sufficient for V,+l c V,. Although a vector in Va+l is identical with its image under HE as an element of L2(R),it is useful to distinguish between them since this permits us to use operator theory to define other useful maps, such as the adjoint Ha:V, + V,+l of H:. Since the norm on V, is that of L2(R) and HE is an inclusion, it follows that H , HE = Ia+l, the identity on Va+l. In particular, Ha is onto; it is just the orthogonal projection
2.2. Operational Calculus
65
from Va to Va+l. H: is interpreted as an operator which interpolates a vector in V,+l, representing it as the vector in V, obtained by replacing the “pixel” 4 with the linear combination of compressed pixels D-lh(S)d. The adjoint H , is sometimes called a “low-pass filter” because it smooths out the signal and re-samples it at half the sampling rate, thus cutting the freqency range in half. However, it is not a filter in the traditional sense since it is not a convolution operator, as will be seen below. The kernel of H , is denoted by W,+1. It is the orthogonal complement of the image of H:, i.e. of V,+l, in V, :
Note that HC is “natural” with respect to the scale gradation, i.e.
Our “home space” will be V. All our operators will enjoy the above naturality with respect to scale. Because of this property, it will generally be sufficient to work in V . Define the operator H*: V 4 V by
We will refer to H* as the “home version” of H:. Home versions of operators will generally be denoted without subscripts. Note that while HC preserves the scale (it is an inclusion map!), H* involves a change in scale. It consists of a dilation (which spreads the sample points apart to a distance 257) followed by an interpolation (which
66
2. Wavelet Algebras and Complex Structures
restores the sampling interval to its original value 2’). Thus H* is a zoom-in operator! Its adjoint
H = D-lHo
(23)
consists of a “filtration” by HO (which cuts the density of sample points by a factor of 2 without changing the scale) followed by a compression (which restores it to its previous value). H is, therefore, a zoom-out operator. It is related to H , by
The operators H and H* are essentially identical with those used by Daubechies, except for the fact that hers act on the sequences { u n } rather than the functions u(S)$. They are especially useful when considering iterated decomposition- and reconstruction algorithms (section 3). To find the action of H,, it suffices to find the action of H . Note where u ( S 2 )is even in S. This will be that H*u(S)$= h(S)u(S2)q5, an important observation in what follows, hence we first study the decomposition of V into its even and odd subspaces. An arbitary polynomial u(S) in S,S-’
can be written uniquely
as the sum of its even and odd parts,
n
n
= u+(S2) + Su-(S2).
(25)
Define the operator E* (for even) on V by
E*S = S2E*,
E*$= 4.
(26)
2.2. Operational Calculus
67
Then
E*u(s)$ = u(s2)4 =
C
un42n.
(27)
n
Also define the operator O* (for odd) by O* = SE",so that
o*u(s)$ = su(s2)4 = C Un42n+l.
(28)
n
H* is related to E* by H* = h(S)E*. Hence to obtain H is suffices to find the adjoint E of E*. Lemma 2.1. Let v(S) E by E and 0. Then
P and denote the adjoints of E* and O*
Ov(S)O* = v+(S), 1 -1 Ov(S)E*= v-(S) = -D 2 Ev(S)O* = Sv-(S)
s-1
[v(S) - v(-S)] D
(note that (a) is a special case with v(S) = I ) , and
68
2. Wavelet Algebras and Complex Structures
(4 E*E
+ 0*0= I .
Proof. For u(S),v( S) E P, we have
where the last equality follows from the invariance of the inner product under S H S2, i.e.
Hence EE* = I , so 00*= ES-'SE* = I .
EO* = OE* = 0 follows
from the orthogonality of even and odd functions of S (applied to
4).
This proves (a). To show (b), note that due to the orthogonality of even and odd functions,
where we have used (a). This proves the first equation in (b). The second follows from 0 = ES-I and S-'v(S) = v-(S2)+S-'v+(S2). To prove (c), note that u(S2)E*= E*u(S)and Su(S2)E*= O*u(S), hence
2.2. Operational Calculus
69
+ = EE*v+(S)+ EO*v-(S) = v+(S),
Ev(S)E* = E(v+(S2) Sv-(S2))E* Ov(S)O* = ES-lv(s)SE* = v+(S),
Ow(S)E*= O(v+(S2) + Sv-(S2))E*
(36)
+ Ev(S)O* = E(V+(S2)+ S v - ( P ) ) S E * = EO*v+(S)+ EE*Sv-(S) = Sv-(S). = OE*v+(S) EE*v-(S) = v-(S),
Lastly, (d) follows from
+
+ = v+(S2)$ + Sw-(S2)$b= v(S)$.I
E*Ev(S)$ O*Ov(S)$ = E*v+(S)$ O*v-(S)$
(37)
Remark. The algebraic structure above is characteristic of orthogonal decompositions and will be met again in our discussion of low- and high-frequency filters. E*E and 0'0 are the projection operators to the subspaces of even and odd functions of S (applied to $),
V" = {v(S2)$ I v(S) E P},
V" =- {Sv(S2)bI v(S) E P}, (38)
and
v = V" @ V". This decomposition will play an important role in the sequel.
(39)
70
2. Wavelet Algebras and Complex Structures
Proposition 2.2. given by
The maps H : V
V a n d H,: V,
V,+l are
H u ( S ) b = Eh(S-l)u(S)$
+
= [h+(S--l)u+(S) h-(S-l)u-(S)] 4,
(40)
H,D"u(S)4 = D"+'E h(S-l)u(S)d
+
= D*+l [h+(s-l)u+(S) h - ( S - l ) u - ( s ) ]
4.
Proof. Since H * = h ( S ) E * ,it follows that H = E h ( S - l ) and H,D" = D"+lH = D"+'Eh(,Y1). I
2.3. Complex Structure
Up to this point, it could be argued, nothing extraordinary has happened. We have a filter which, when applied repeatedly, gives rise to a nested sequence of subspaces V,. However, the next step is quite surprising and underlies much of the interest wavelets have generated. It is desirable to record the information lost at each stage of filtering, i.e., that part of the signal residing in the orthogonal complement W,+1 of V,+l in V,. The orthogonal decomposition V, = V,+1 @ W,+l is described by filters H , and G,, where H , is as above and G, extracts high-frequency information. For this reason, H, and G, obey a set of algebraic relations similar to those satisfied by E and 0 above. What is quite remarkable is that there exists a vector 1c, in V.1 which is related to the spaces W, and the maps G, in a way almost totally symmetric to the way 4 is related to V, and H,. This is not merely a consequence of the orthogonal decomposition but
2.3. Complex Structure
71
is somehow related to the fact that Va+l is “half” of Va,due to the doubling of the sampling interval upon dilation, as expressed by the commutation relation DS = S 2 D . However, the precise reason for this symmetry has not been entirely clear. The usual constructions are somewhat involved and do not appear to shed much light on this question. It was this puzzle which motivated the present work. As an answer, we propose the following new construction. Begin by defining a complex structure on V , i.e. a map J : V V such that J 2 = -I. (To illustrate this concept, consider the complex plane as the real space R2. Then multiplication by the unit imaginary i is represented by a real 2 x 2 matrix whose square is -I.) J is defined by giving its commutation rule with respect to the shift and its action on 4:
where e(S) is an as yet undetermined function. It follows that for
4s)E p , Ju(S)$ = E(S)U(--S-~)$.
(2)
We further require that J preserve the inner product, i.e. that J* J = I . Combined with J 2 = - I , this gives J* = - J . That is, J will behave like multiplication by i also with respect to the inner product, giving it an interpretation as a Hermitian inner product. In order to study J, we first define two simpler operators C and M as follows.
cs = s-lc, M S = -SM,
c4 = 4
M $ = $. Note that C M = M C and that C* = C and M* = M , since
(3)
2. Wavelet Algebras and Complex Structures
72
where u(S)* = u(S-') was used in the second line and the third line follows from the invariance of the inner product under S H -S. Since
C and M are also involutions, i.e.
it follows that they are orthogonal operators. Hence they represent symmetries, which makes them import ant in themselves, especially in the abstract context where one begins with an algebra and constructs a representation (see the remark at the end of section 3). In fact, the orthogonal decomposition V = V" @ V" is nothing but the spectral decomposition associated with M , since V" and V" are the eigenspaces of M with eigenvalues 1 and -1, respectively.
C has
a
simple interpretation as a conjugation operator, since for u(S) E P,
Cu(S)C = u ( S - l ) = u(S)*.
(6)
In terms of C and M ,
J
= E(S)CM.
(7)
2.3. Complex Structure
73
Proposition 2.3. The conditions J* = -J and J2 = -I hold if and only if c(S) satisfies e(-S) = --E(S),
E(S-l)€(S) = 1.
(8)
Proof. We have
J* = MCe(S-l) = M € ( S ) C= €(-S)MC = E(-S)CM, hence J* = -J if and only if case. Then
E(
(9)
-S) = -e(S). Assume this to be the
J2 = c(S)CMe(S)CM= E(S)E(-S-') = --E(S)E(S-'), hence J 2 = -I if and only if E(S-')E(S)= I .
(10)
I
Remarks. 1. J is determined only up to the orthogonal mapping E(S). This corresponds to a similar freedom in the standard approach to wavelet theory, where a factor eix(c) in Fourier space relates the functions H ( [ ) and G(() associated with the operators H and G (Daubechies [1988b], p. 943, where T = 1). The relation between e(S) and A([) is given in the appendix. 2. The simplest examples of a complex structure are given by choosing E(S) = S 2 P + l , P E Z .
(11)
More interesting examples can be obtained by enlarging P to a topological algebra, for example allowing u(S) with { u n } E
P(Z).
74
2. Wavelet Algebras and Complex Structures ,
3. The above proof used the symmetry of the inn-roduct. Later we shall complexify our spaces and the inner product becomes Hermitian. However, this proof easily extends to the complex case (when transposing, also take the complex conjugate). C then becomes C-antilinear and is interpreted as Hermitian conjugation. At an arbitrary scale a , define maps J,: V,
J,D"
4
V, by naturality, i.e.
= D" J,
(12)
which implies that J: = -I, and J: = - J,. J , is related to S by
showing that
S 2 a J , = -J,S-'*. In particular, note that S'I2 J-1 = -J-1S-'/2, hence
We are now in a position to construct the basic wavelet $, the spaces W , and an appropriate set of high-frequency filters in a way which will make the symmetry with 4, V, and H , quite clear. Consider the restriction of J, to the subspace V,+l of V,, i.e. the map I 2s. Then for 0 5 n 5 29,
S i w n = ( n - s)w,.
To see how the generators Si must be scaled, note that the relation between C and z implies that
where
Define
= S d / d g , so that 1i't = (I 0.
(4)
(Note also that since H is unbounded, no analytic continuation to the upper-half time plane is possible.) The operator e-uH is familiar from two other contexts: it constitutes the evolution semigroup for the heat equation (where u is time), and it is also the unnormalized density matrix for the Gibbs canonical ensemble describing the statistical equilibrium of a quantum system at temperature
T (where
u = l/kT). To get afeel for our use of this operator, let us be heuristic for a moment and consider what happens when a free classical free particle of mass m is evolved in complex time
T
=t
- iu. If its
initial position and momentum are x and p respectively, then its new position will be z(7) = x
+ ( t - iu)p/m
= (x
+ @/m)- i(u/m)p
(5)
= x(t) - i(u/m)p.
Since x ( t ) is just the position evolved in real time t , we see that z ( t ) is, in fact, a complex phase space coordinate of the same type
4.3. Galilean F'rames
171
we encountered in the construction of the canonical coherent states! Armed with this intuition, let us now return to quantum mechanics and see if this idea has a quantum mechanical counterpart. The operator e+"', when applied to any function in L2(IR.'), gives
If we replace x in the integrand by an arbitrary z E a?, the integral still converges absolutely since the quadratic terni in the exponent dominates the linear term for large lpl. Clearly the resulting function is entire in z (differentiating the integrand with respect to z k still gives an absolutely convergent integral). This shows that the group of Galilean space-time translations, ~ ( x , t=) e x p ( - i t i ~ + i x . ~ ) ,
(7)
extends analytically to a semigroup of complex sp tions
-tim t ransla-
U ( E 5, ) = exp(-iTH
+ i~ - P)
defined over the complex space-time domain 2, =
{ ( Z , T ) 1z E CS,T E c-}.
(9)
This translation semigroup can be combined with the rotations and boosts to give an analytic semigroup Si extending G2. Let H ' , be the vector space of all the entire functions fu(z) as f ( p ) runs through L2(IRd). Then
172
4. Complex Spacetime
are seen to be Gaussian wave packets in momentum space with expected position and momentum given in terms of z EE x - iy b y
The e:'s are easily shown to have minimal uncertainty products. The momentum uncertainty can be read off directly from the exponent and is
hence
Axk = J-. We now have our prospective coherent states and their label space M = Ca. To construct a coherent-state representation, we need a measure on M which will make the e:'s into a frame. Since the er's are Gaussian, the measure in not difficult to find:
d p u ( z ) = ( m / n ~ ) "exp / ~( - m y 2 / u ) daxd"y. Defining
(15)
4.3. Galilean fiames
173
we have T h e o r e m 4.1. (a) ( I is an inner product on 3.1, under which 'FI, is a Hilbert space. (b) The map e+"' is unitary from L2(IR")onto F ' I,. (c) The e: 's define a resolution of unity on L2(IR")given by
-
e ) ~ , ,
(17)
=
Proof. We prove prove that 11 f (f I f ) ~ = , , I I f I I i 2 . The inner product can be recovered by polarization. To begin with, assume that f" is in the Schwartz space S(R")of rapidly decreasing smooth test functions. Then
hence by Plancherel's theorem,
and
4. Complex Spacetime
174
where exchanging the order of integration was justified since the integrals are absolutely convergent. This proves (b), hence also (a), for f E S(IR").Since the latter space is dense in L2(IRs),the proof extends to
f E L2(Rs)by continuity. (c) follows by noting that
and dropping ( f^ I and
14 ).
I
Using the map e W u Hwe , can transfer any structure from L2(IRs) to 3-1,. In particular, time evolution is given by
where
T
= t - iu and the wave packets
are obtained from the ei's by evolving in real time t. They cannot be of minimal uncertainty since the free-particle Schrodinger equation is neccessarily dissipative. Instead, they give the following expectations and uncertainties:
4.3. Galilean Ehtnes
175
Since
it follows that
thus we have a frame { es,r 1 z E Cs} at each complex “instant” r = t - iu, with the corresponding resolution of unity
The space L2(R”)carries a representation of the quantum mechanical Galilean group 62. Since the e5,,’s were obtained from the dynamics associated with this group, they transform naturally under its action. A typical element of & has the form g = (R,v,xo,to,B), where R is a rotation, v is a boost, xg is a spatial translation, t o is a time-translation and 8 is the “phase” parameter associated with the central element M = m / h m in our representation (see section 4.2). g acts on the complex space-time domain 2, by sending the point (2, T ) to ( T I , e l ) , where
176
4. Complex Spacetime XI
= R x + t v + xo
y’=Ry+uv
t’ = t + t o uI = 21. The parameter 6 has no effect on space-time; it only acts on wave functions by multiplying them by a phase factor. The representation of 92 is defined by
Thus we have
and the e,,,’s are “projectively covariant” under the action of 9 2 ; if we define ee,,,4 e --im4e,,r, then this expanded set is invariant under the action of 92, with 4’ = 4-6. Since e,,, and e,,,,,j, represent the same physical state, we won’t be fussy and just work with the e,,,’s. Anyway, this anomaly will disappear when we construct the corresponding relativistic coherent states. The above representation of & on L2(IR”)can be transfered to
N uusing the map e-uH.
This map therefore intertwines (see Gelfand, Graev and Vilenkin [1966])the representations on 92 on L2(R”)with
the one on Xu. We conclude with some general remarks.
1. Since the e:’s are spherical and therefore invariant under S O ( n ) (which is, after all, why they describe spinless particles), they can be parametrized by the homogeneous space W = Q1 /SO( n) as long as we keep u fixed ( u is a parameter associated with the Hamiltonian, which
177
4.3. Galilean frames
is a generator of 6 2 but not of 61). The action of W as a subgroup of 61 on the e:)s is preserved in passing to the homogeneous space, hence W acts to translate these vectors in phase space. This explains the similarity between the ei's and the canonical coherent states. On the other hand, dynamics (in imaginary time) is responsible for the parameter u. If we write k G ( m / u ) y ,then e:(p)
U
= (Zr)-"exp -k2
[2m
G
21
- -(p
exp [u k2/2m]e--iP'x h
2m
- k)2- ip - x ]
(JS) *
The measure dp,(z) is now
d p u ( x ,k) = ( ~ / n m ) exp(-uk2/rn) "/~ d"xd"k.
(32)
Hence the exponential factor exp[uk2/2m] in e,U, when squared in the reconstruction formula, precisely cancels the Gaussian weight factor in dpu(x,k), leaving the measure
in phase space. It follows from the above form of ez that 2 A q = ,/- plays the role of a scale factor in momentum space (as used in the wavelet transforms of chapter l), hence its reciprocal Axk = ,/* acts as a scale factor in configuration space. Thus the Galilean coherent states combine the properties of rigid "windows" with those of wavelets, due to the fact that their analytic semigroup @ includes j both phase-space translations and scaling, the latter due to the heat operator e - u H . However, note that u is constant, though arbitrary, in the resolution of unity and the corresponding reconstruction formula. Since there is an abundance of "wavelets" due
178
4. Complex Spacetime
to translations in phase space, only a single scale is needed for reconstruction. (One could, of course, include a range of scales by integrating over u with a weight function, but this seems unnecessary.) In the treatment of relativistic particles, u becomes the time component of a four-vector y = ( u ,y ) , hence will no longer be constant. This is because relativistic windows shrink in the direction of motion, due to Lorentz contractions, thus automatically adjusting to the analysis of high-frequency components of the spectrum.
2. Notice that ef is essentially the heat operator e-uH applied to the &function at z, then analytically continued to Z = x i y . The fact that all the ef’s have minimal uncertainties shows that the action of the heat semigroup {U(-iu)} is such that while the position undergoes the normal diffusion, the momentum undergoes the opposite process of refinement, in just such a way that the product of the two variances remains constant. This is reflected in the fact that the is unbounded, becomes unioperator e - u H , whose inverse in L2(IR”) tary when the functions in its range get analytically continued, and the reconstruction formula is just a way of inverting e - - u H . Hence no information is lost if one looks in phase space rather than configuration space! It seems to me that this way of “inverting” semigroups
+
must be an example of a general process. If such a process exists, I am unaware of it. In our case, at least, it appears to be possible because of analyticity.
3. So far, it seems that coherent-state representations are intimately connected with groups and their representations. However, there is a reasonable chance that coherent-state representations similar to the above can be constiucted for systems which, unlike free particles, do not possess a great deal of symmetry. Suppose we are
4.3. Galilean frames
179
given a system of 9/3 particles in R3 which interact with one another and/or with an external source through a potential V(x). We assume that V(x) is timeindependent, so the system is conservative. (This means that we do have one symmetry, namely under time translations. If, moreover, the potential depends only on the differences xi - xj between individual particles, we also have symmetry with respect to translations of the center of mass of the entire system; but we do not make this assumption here.) This system is then described by a Schrodinger equation with the Hamiltonian operator
H = Ho + V, where Ho is the free Hamiltonian and V is the operator of multiplication by V(x). We need to assume that this (unbounded) How operator can be extended to a self-adjoint operator on L2(RB). far can the above construction be carried in this case? The key to our method was the positivity of the free Hamiltonian HO= P2/2m. But a general Hamiltonian must at least satisfy the stability condition:
(S)
The spectrum of H is bounded below.
If H fails to meet this condition, then the system it describes is unstable, and a small perturbation can make it cascade down, giving off an infinite amount of energy. For a stable system, the evolution group ~ ( t=) eWitH can be analytically continued to an analytic semigroup U ( T ) = e-irH in the lower-half complex time plane as in the free case. Depending on the strength of the potential, the functions fu = U ( - i u ) f may be continued to some subset of C8. Formally, this corresponds to defining
for an initial function f(x) in L2(RS).As mentioned, this expression is formal since the operator is unbounded and e - i r H f may not
180
4. Complex Spacetime
be in its domain. But it can make sense operating on the range of e - i r H , which coincides with the range of e--rH, provided y is not too large. Let y, be the set of all y's for which e"J' is defined on the range of e-rH and, furthermore, the function exp[iz P] x exp [ - i 7 H ] f is sufficiently regular to be evaluated at the origin in IR", no matter which initial f was chosen in L2(IR"). For many potentials, of course, y , will consist of the origin alone; in that case there are no coherent states. We assume that Y, contains at least some open neighborhood of the origin. Intuitively, we may think of y , as the set of all imaginary positions which can be attained by the particle in an imaginary time-interval u, while moving in the potential V . In the free case, y, = R"and there is no restriction on y provided only that u > 0. This corresponds to the fact that there is no "speed limit" for free non-relativistic free particles, hence a particle can get to any imaginary position in a given positive imaginary time. For relativistic free particles, Y , is the open sphere of radius uc, where c is the speed of light. Returning to our system of interacting particles, define the associated complex space-time domain
-
This is the set of all complex space-time points which can be reached by the system in the presence of the potential V(x), and it is the label space for our prospective coherent states. These are now defined as evaluation maps on the space of analytically continued solutions:
the inner product being in L2(Rs).Then from the above expression, again formally, we have the dynarnical coherent states
4.3. Galilean fiames
181
for ( z , ~ in ) ZH. What is still missing, of course, is the measure d p f . (Since the potential is t-independent, so will be the measure, if it exists.) Finding the measure promises to be equally difficult to finding the propagator for the dynamics. The latter is closely related to the reproducing kernel,
K H ( z T, ; z',
H
I
H
7 ' ) = ( e5,7 e5f,rt).
(38)
h ' ~depends on T and 7' only through the difference T - ?', and is the analytic continuation of the propagator to the domain ZH x ZH. It is related to the measure through the reproducing property,
J..
dp,H(z) K H ( z 'T, ' ; E , 7 ) K H ( z ,T ; B", 7")
(39) = K H ( d , 7';z",?"),
where the integration is carried out over a "phase space" ur in ZH with a fixed value of T = t - iu. A reasonable candidate for dp: (see section 4.4)is
Rather than finding the measure explicitly, a more likely possibility is that its existence can be proved by functional-analytic methods for some classes of potentials and approximation techniques may be used to estimate it or at least derive some of its properties. The theoretical possibility that such a measure exists raises the prospect of an interesting analogy between the quantum mechanics of a single system and a statistical ensemble of corresponding classical systems at
182
4. Complex Spacetime
equilibrium with a heat reservoir. In the case of a free particle, if we set k = ( m / u ) y as above (see remark 1) and define T by u = 1/2kT where k is Boltzmann’s constant, then it so happens that our measure dpu is identical to the Gibbs measure for a classical canonical ensemble (see Thirring [1980]) of s/3 free particles of mass m in
IR3, at equilibrium with a heat reservoir at absolute temperature T. Thus, integrating with dpu over phase space is very much like taking the classical thermodynamic average at equilibrium! It remains to be seen, of course, whether this is a mere coincidence or if it has a generalization to interacting systems. There is also a connection between the expectation values of an operator A in the coherent states e,H[, and its thermal average in the Gibbs state,
(A)P where 2
2-l Trace ( e - P H A ) = 2-’ Trace ( e - P H / 2 A e - B H / 2
),
= Trace (e-PH>.
(41) Namely, if we have the resolution of unity
then
= 2-’ S,,
dp,H(z) A ( 2 , T
where we have used the formula
- iP/2),
4.4.Relativistic fiames
TraceA=
J..
dpf(z)(e&IAe&),
183
(44)
which follows easily from eq. (42). Thus taking the thermal average means shifting the imaginary part u by p / 2 in the integral.
4.4. Relativistic Frames We are at last ready to embark on the main theme of this book: A new synthesis of Relativity and quantum mechanics through the geometry of complex spacetime. The main tool for this synthesis will be the physically necessary condition that the energy operator of the total system be non-negative, also known in quantum field theory as the spectral condition. The (unique) relativistically covariant statement of this condition gives rise to a canonical complexfication of spacetime which embodies in its geometry the structure of quantum mechanics as well as that of Special Relativity. The complex spacetime also has the structure of a classical phase space underlying the quant um system under consideration. Quantum physics is developed through the construction of frames labeled by the complex spacetime manifold, which thus forms a natural bridge between the classical and quantum aspects of the system. It is hoped that this marriage, once fully developed, will survive the transition from Special to General Relativity. As mentioned at the beginning of this chapter, the Perelomovtype constructions of chapter 3 do not apply directly to the Poincark group since its time evolution (dynamics) is non-trivial. Pending a generalization of these methods to dynamical groups, we merely use the ideas of chapter 3 for inspiration rather than substance. In fact,
184
4. Complex Spacetime
it may well be that a closer examination of the construction to be developed here may suggest such a generalization. We begin with the most basic object of relativistic quantum mechanics, the Klein-Gordon equation, which describes a simple relativistic particle in the same way that the Schrodinger equation describes a non-relativistic particle. The spectral condition will enable us to analytically continue the solutions of this equation to complex spacetime, and the evaluation maps on the space of these analytic solutions will be bounded linear functionals, giving rise to a reproducing kernel as in section 1.4. Physically, the evaluation maps are optimal wave packets, or coherent states, and it is this interpretation which establishes the underlying complex manifold as an extension of classical phase space. The next step is to build frames of such coherent states. (Recall from section 1.4 that a frame determines a reproducing kernel, but not vice versa.) The coherent states we are about to construct are covariant under the restricted Poincark group, hence they represent relativistic wave packets
. As we have seen, such a covariant family is closely re-
lated to a unitary irreducible representation of the appropriate group, in this case
Po. Such representations
are called elementary systems,
and correspond roughly to the classical notion of particles, though with a definite quantum flavor. (For example, physical considerations prohibit them from being localized at a point in space, as will be discussed later.) We will focus on representations corresponding to massive particles. (A phase-space formalism for massless particles would be of great interest, but to my knowledge, no satisfactory formulation exists as yet.) Such representations are characterized by two parameters, the mass m > 0 and the spin j = 0,1/2,1,3/2,. . . of the corresponding particle. w e will specialize to spinless particles ( j = 0) for simplicity. The extension of our construction to particles
4.4. Relativistic fiames
185
with spin is not difficult and will be taken up later. Thus we are interested in the (unique, up to equivalence) representation of POwith m > 0 and j = 0. A natural way to construct this representation is to consider the space of solutions of the Klein-Gordon equation
where
= a%$
is the Del'Ambertian, or wave operator, A is the usual spatial Lapla-
cian and a,, = d/Bz". The function f is to be complex-valued (for spin j , f is valued in C2j+'). We set c = 1 except as needed for future reference. If we write f(z) as a Fourier transform,
then the Klein-Gordon equation requires that f ( p ) be a distribution supported on the mass shell
Qm
is a two-sheeted hyperboloid,
where
4. Complex Spacetime
186
f(p) = 2 n S(p2 - m 2 )a @ ) for some function a ( p ) on
(7)
am,and using
6(P2 - m2)= 6 ((Po - w)(po + w ) ) 1
=2w P ( p 0
- w ) + 6(po + 41,
we get
where
is the unique (up to a constant factor) Lorentz-invariant measure (The factor w - l corrects for Lorentz contraction in frames on 52,. at momentum p . ) For physical particles, we must require that the energy be positive, i.e. that a ( p ) = 0 on 52;. states are given as positive-energy solutions,
f(z) =
LA
d@ e-izp a ( p ) .
Hence the physical
(11)
The function a ( p ) can now be related to the initial data by setting zo
so
t = 0, which shows that
4.4. Relativistic frames
187
4 4 = 4 4 P ) = %fo(P),
(13)
where denotes the spatial Fourier transform. In particular, f(z) is determined by its values on the Cauchy surface t = 0. For general solutions of the Klein-Gordon equation, we would also need to specify on that surface, but restricting ourselves to positive-energy solutions means that f(x) actually satisfies the first-order pseudodifferential non-local equation ---. A
af/&
(which implies the Klein-Gordon equation), hence only f(x,0) is necessary to determine f. (We will see that when analytically continued to complex spacetime, positiveenergy solutions have a local characterization.) The inner product on the space of positive-energy solutions is defined using the Poincarkinvariant norm in momentum space,
We will refer to the Hilbert space
L:(dp")
= {a E L2(dp")I a ( p ) = 0 on a,}
as the space of positive-energy solutions in the momentum representation. It carries a unitary irreducible representation of Po defined as follows. The natural action of Po on spacetime is
(b, A)a: = Ax
+ b,
(17)
where A is a resticted Lorentz transformation (A E Lo ) and b is a spacetime translation. Since the Klein-Gordon equation is invariant
188
4. Complex Spacetime
under ’PO, the induced action on functions over spacetime transforms solutions to solutions. Since the positivity of the energy is also invariant under ’Po, the subspace of positiveenergy solutions is also left invariant.
Po
acts on solutions by
( U ( 4 4 . f ) (4= f (A%
- b)) *
(18)
The invariance of the inner product on L:(dj) then implies that the induced action on that space (which we denote by the same operator) is
( U ( b ,A) a ) ( p ) = eibpa (A-’p)
.
(19)
The invariance of the measure dj3 then shows that V (b, A) is unitary, thus (b,A) H U(b,A) is a unitary representation of ’Po. It can be shown that it is, furthermore, irreducible. Neither of the “function” spaces {f(z)} and L?(d$) are reproducing-kernel Hilbert spaces, since the evaluation maps f H f(x) and a H a ( p ) are unbounded. To obtain a space with bounded evaluation maps, we proceed as in the last section. Due to the positivity of the energy, solutions can be continued analytically to the lowerhalf time plane:
where u > 0. As in the non-relativistic case, the factor exp(-uw) decays rapidly as I p I + 00, which permits an analytic continuation of the solution to complex spatial coordinates z = x - iy. But since
d
m
w(p) is no longer quadratic in I p I, y cannot be arbitrarily large. Rather, we must require that the four-vector ( u , y ) satisfy the condition
4.4.Relativistic f i m e s
uw - y * p
>0
In covariant notation, setting yo
V(w,p) E a+,. u , we
(21)
must have
VPEfl+,,
YP>O
189
(22)
so that the complex exponential exp [ - i ( x - iy)p] remains bounded as p varies over fl;. This implies that yp > 0 for all p E V+,where
is the open forward light cone in momentum space. In general, we need to consider the closure of V+,i.e. the cone
-
v+ = {P E IRa+l I lPl I P O / C } ,
(24)
which contains the light cone { p 2 = 0 I PO > 0) (corresponding to massless particles) and the point { p = 0) (corresponding in quantum field theory to the vacuum state). The set of all y’s with yp > 0 is called the d u d cone of i.e.,
v+,
v; 3 {y E R”+lI yp > 0 v p E V+}.
(25)
It is easily seen that y belongs to V; if and only if I y I < cyo. Note contracts to the non-negative Po-axis while V; that as c + 00, expands to the half-space { ( u ,y) I u > 0, y E R’} which we have encountered in the last section. V . coincides with V+ when c = 1, but it is important to distinguish between them since they “live” in different spaces (see section 1.1). Thus for y E V;, setting z = x - iy, we define
v+
f(z) =
J’,+ d@e-;*’ rn
a@).
4. Complex Spacetime
190
The integral converges absolutely for any a E L:(dp") and defines a function on the forward tube
also known as the future tube and, in the mathematical literature, as the tube over V;. Differentiation with respect to z p under the integral sign leaves the integral absolutely convergent, hence the function
f(z) is holomorphic, or analytic, in 7+.As y -+ 0 in V;, f ( z ) + f ( z ) in the sense of Li(dj5). Thus f( 2) is a boundary value of f( z). Clearly f(z) is a solution of the Klein-Gordon equation in either of the variables z or z. Let
K
be the space of all such holomorphic solutions:
Then the map a ( p ) H f(z) is one-to-one from L$(dp")onto Ic. Hence we can make
K
into a Hilbert space by defining
where the inner product on the right-hand side is understood to be that of L$(d@).We now show that Ic is a reproducing-kernel Hilbert space. Its evaluation maps are given by
E , ( f ) = f(z) = where
JCk dp" e-"fpa(p)
E (e,
Ia),
(30)
191
4.4. Relativistic Frames
Lemma 4.2. 1. For each z E I+, e, belongs to Li(dfi), with
where v = (s - I)/&
and I 0 and rn #
-
0, and shows that
206
4. Complex Spacetime The vectors e, belong t o L$(dfi), but correspond to vectors E,
in
Ic defined by
( f I f ) - on K provides us with an interpretaThe norm llfll: tion of I f ( z ) I as a probability density with respect to the measure dax on the phase space Q. Within this interpretation, the wave packets e , have the following optimality property: For fixed z E 7+let
Proposition 4.5. Up to a constant phase factor, the function g Z is the unique solution to the following variational problem: Find f E K: such that llfll = 1 and I f ( z ) I is a maximum.
Proof. This follows at once from the Schwarz inequality and theorem 4.3,since by eq. (26),
If(z)l
= I(ezla)l=
I(e"zIf>l
5 I l 4 I llfll = l l e z l l Ilfll, with equality if and only if f is a constant multiple of e",.
I
According to our probability interpretation of I f(z) I ', this means that the normalized wave packet i, maximizes the probability of finding the particle at z . Note: Unlike the non-relativistic coherent states of the last section, the e,'s do not have minimum uncertainty products. In fact, since the uncertainty product is not a Lorentz-invariant notion, it is a
4.5. Geometry and Probability
207
priori impossible to have relativistic coherent states with minimum uncertainty products. The above optimality, which is invariant, may be regarded as a reasonable substitute. Actually, there are better ways to measure uncertainty than the standard one used in quantum
mechanics, which is just the variance. From a statistical point of view, the variance is just the second moment of the probability distribution. Perhaps the best definition of uncertainty, which includes all moments, is in terms of entropy (Bialynicki-Birula and Mycielski
[ 19751) Zakai [19601). Being necessarily non-linear, however, makes this definition less tractable.
4.5. Geometry and Probability
The formalism of the last section was based on the phase space and the measure do, neither of which is invariant under the action of Po on 7+.Yet, the resulting inner product ( I .)o is ot,~ o
-
clearly invariant. It is therefore reasonable to expect that 0 and do merely represent one choice out of many. Our purpose here is to construct a large natural class of such phase spaces and associated measures to which our previous results can be extended. This class will include u and will be invariant under Fo. In this way our formalism is freed from its dependence on u and becomes manifestly covariant. As a byproduct, we find that positiveenergy solutions of the Klein-Gordon equation give rise to a conserved probability current, so the probabilistic interpretation becomes entirely compatible with the spacetime geometry. As is well-known, no such compatibility is possible in the usual approach to Klein-Gordon theory. We begin by regarding 7+as an extended phase space (symplec-
208
4. Complex Spacetime
tic manifold) on which Po acts by canonical transformations. Candidates for phase space are 2s-dimensional symplectic submanifolds
c
7+,and Po maps different u’s into one another by canonical transformations. A submanifold of the “product” form IY = S - iQ:, where S (interpreted as a generalized configuration space) is an su
submanifold of (real) spacetime IRS+l, turns out to be symplectic if and only if S is given by xo = t ( x ) with
lot1 5
1, that is, if and only if S is nowhere timelike. (This is slightly larger than the class of all spacelike configuration spaces admissible in the standard theory.) The original ut,A corresponds to t (x)=constant. The results of the last section are extended to all such phase spaces of product form. The action of ’POon 7+is not transitive but leaves each of the
(2s
+ 1)-dimensional submanifolds 7 -= {x - iy E 7+I y2 = P}
invariant. Each 7’
(1)
is a homogeneous space of Po, with isotropy
subgroup SO(s), hence
Thus 7’ corresponds to the homogeneous space C of section 4.2 (where we had specialized to s = 3). In view of the considerations in sections 4.2 and 4.4, each 7 ; can be interpreted as the product of spacetime with ‘(momentumspace”. Phase spaces u will be obtained by taking slices to eliminate the time variable. On the other hand, we also need a covariantly assigned measure for each u. The most natural way this can be accomplished is to begin with a single Po-invariant symplectic form on 7+and require that its restriction to each u be symplectic. This will make each u a
4.5. Geometry and Probability
209
symplectic manifold (which, in any case, it must be to be interpreted as a classical phase space) and thus provide it with a canonical (Liouville) measure. Thus we look for the most general 2-form a on 7+ such that (a) a is closed, i.e., da = 0;
(b) a is non-degenerate, i.e., the (29+2)-fonn as+1 c y A a A . - - A a vanishes nowhere; (c) for every g = (.,A) E PO,g*a = a,where g*a denotes the pullback of a under g (see Abraham and Marsden [1978]). Since every Po-invariant function on 7+ depends on z only through y2, the most general invariant 2-form is given by
Now the restriction (pullback) of the second term to 7 ;
vanishes,
since it contains the factor ypdyp = d(y2)/2. Furthermore, the coefficient $(y2) of the first term is constant on 7:. confine our attention to the form
a = dy,, A dx” without any essential loss of generality.
Hence we may
(4)
This form is symplectic as
well as invariant, hence it fulfills all of the above conditions. 7+, together with a,is a symplectic manifold, and invariance means that each g E Po maps 7+into itself by a canonical transformation.
A general 2s-dimensional submanifold c of 7+will be a potential phase space only if the restriction, or pullback, of cy to t~ is a symplectic form. We denote this restriction by a,. Let c be given bY
210
4. Complex Spacetime
0
= {Z E ' I I+ S(Z) = h(z) =0},
(5)
where s ( z ) and h ( z ) are two real-valued, C" (or at least C') functions on 7+such that d s A d h # 0 on u. For example, Q ~ , X can be obtained from S ( Z ) = xo - t and h ( z ) = y2 - X2. The pullback au depends only on the submanifold 6,not on the particular choice of s and h.
Proposition 4.6. Poisson bracket
The forrn a , is symplectic if and only if the
{s, h } everywhere on
ds dh dh as -- -- # O ax, dy, ax, dy,
Q.
Proof. a , is closed since a is closed and d ( a u ) = (da),. Hence a , is symplectic if and only if it is non-degenerate, i.e. if and only if its s-th exterior power a" vanishes nowhere on Q. Now (a,)" equals the pullback of a" to Q, and a straightforward computation gives h
h
a" = S! d y " A d z , ,
(7)
where h
dy' = (-1)' dyo A dyl A h
--
d x , = (-1)' dx" A dx"-l A
*
A dy,-l
- - - A dx""
A dy,+l
A
--
*
A dxP-' A
A dy, *
- - A dx'.
(8)
(z z,
and are essentially the Hodge duals (Warner [1971]) of dy, and dx,, respectively, with respect to the Minkowski metric.) Let ( ~ 1 , .. . , ~ 2 ~ , v 1 , v 2be} a basis for the tangent space of 7+ at z E Q, with ( 2 1 1 , . . . ,u2"} a bas& for the tangent space Q, of Q. Since ds and d h vanish on the vectors u j ,
4.5. Geometry and Probability
211
By assumption, ( d s A d h ) ( v l ,u2) # 0. Therefore a, is non-degenerate at z if and only if a' A ds A d h # 0 at z. But by eq. (7))
a' A d s A d h = s! { s ,h } d y A d x ,
(10)
where
Hence a:
# 0 at
z if and only if {s, h } # 0 at z . I
Let us denote the family of all such symplectic submanifolds a
by Co.
Proposition 4.7. Let a E CO and g E PO.Then ga E CO and the restriction g : a + g a is a canonical transformation from (a,a,,)to
(so,Q g 4 . Proof. Let
g* denote the pullback map defined by g, taking forms
on ga to forms on u. Then the invariance of
Q
implies that
Hence agUis non-degenerate. It is automatically closed since closed. Thus ga E
CO.To say that
Q
is
g: a + g a is a canonical trans-
formation means precisely that a, and agaare related as above. I
212
4. Complex Spacetime
We will be interested mainly in the special case where h ( z ) = y2 - X2 for some X > 0 and s ( z ) depends only on x. Then the sdimensional manifold
s = (5 E IR”l I s ( x ) = 0)
(13)
is a potential generalized configuration space, and u has the “product” form u = s - iR,+
{x - iy E T+ I x E S, y E Q:).
(14)
The following result is physically significant in that it relates the pseudeEuclidean geometry of spacetime and the symplectic geometry of classical phase space. It says that u is a phase space if and only if S is a (generalized) configuration space.
Theorem 4.8.
Let u = S - i Q i be as above. Then (a,a,) is
symplectic if and only if
that is, if and only if S is nowhere timelike.
Proof. On
6 ,we
have {s,
h } = 2-
89 yp # 0,
dXP
and we may assume { s , h } > 0 without loss. For fixed x E S, the above inequality must hold for all y E R i , hence for all y E V’.. This of V i . I implies that the vector a s / a x ~is in the dual
v+
We denote the family of all a ’ s as above (i.e., with S nowhere timelike) by C. It is a subfamily of CO and is clearly invariant under
4.5. Geometry and Probability
213
PO.Note that C admits lightlike as well
as spacelike configuration spaces, whereas the standard theory only allows spacelike ones.
We will now generalize the results of the last section to all a E C. The 2s-form CY; defines a positive measure on u , once we choose an orientation (Warner [1971]) for a. (This can be done, for example, by choosing an ordered set of vector fields on a which span the tangent space at each point; the order of such a basis is a generalization of the idea of a “right-handed” coordinate system in three dimensions.) The appropriate measure generalizing da of the last section is now defined as da = ( s ! A x )-1 a,8,.
(17)
do is the restriction to u of a 2s-form defined on all of 7+,which we also denote by do. (This is a mild abuse of notation; in particular, the ‘(8’here must not be confused with exterior differentiation!) By
eq. (7), we have
-
h
da = A ,-1 d y p Adz,.
We now derive a concrete expression for da. Since s obeys eq. (15) and ds # 0 on u, we can solve ds = 0 (satisfied by the restriction n of ds to a) for dzo and substitute this into dzk. This (and a similar procedure for y) gives
(2) -1
n
dz, =
on u . Hence
as
-dxo ax,
4. Complex Spacetime
214
We identify a with mapping
R2"by solving s ( x )
( t ( x >-
id=,
= 0 for zo = t ( x ) and
x - iy) H (x,y).
(21)
&
We further identify AZ o with the Lebesgue measure d"y d'x on R2"(this amounts to choosing a non-standard orientation of R2"). Thus we obtain an expression for da as a measure on R'". Now s(z) = 0 on a implies that
which can be substituted into the above expression to give
-
= AX1 (1 - V t (y/yo)) d"y d " ~ .
But eq. (15) implies that I Vt ( x ) I
5 1, hence for y E V i ,
and da is a positive measure as claimed. The above also shows that if I Vt (x)I = 1for some x,then da becomes "asymptotically" degenerate as y I + 00 in the direction of V t ( x ) . That is, if (r is lightlike at ( t (x),x), then da becomes small as the velocity y/yo approaches the speed of light in the direction of V t ( x ) . This means that functions in L 2 ( d a )(and, in particular, as we shall see, in K ) are allowed high
I
215
4.5. Geometry and Probability
velocities in the direction V t ( x ) at ( t ( x ) , x )E S. This argument is an example of the kind of microlocal analysis which is possible in the phase-space formalism. (In the usual spacetime framework, one cannot say anything about the velocity distribution of a function at a given point in spacetime, since this would require taking the Fourier transform and hence losing the spatial information.) For u E C, denote by L2(da)the Hilbert space of all complexvalued, measurable functions on 0 with
llfll2,= J do l f I 2 < 00.
(25)
0
If f is a C” function on 7+, we restrict it to u and define above. Our goal is to show that
llflld
llfllo
= Ilfllx: for every f E
as
K. To
do this, we first prove that each f E K defines a conserved current in spacetime, which, by Stokes’ theorem, makes it possible to deform the EC phase space at,^ of the last section to an arbitrary u = S without changing the norm. For f E K , define
in:
where 52:
has the orientation defined by
&
O,
so that Jo(z)is posi-
tive. Then
where S has the orientation defined by to S does not vanish since I V t (x)I
go. (The restriction of 30
5 1.)
Let f(p) be C” with compact support. Then J ” ( z ) is C” and satisfies the continuity equation Theorem 4.9.
4. Complex Spacetime
216
8 JP ax@
- 0.
Proof. By eq. (19),
= d y ‘/yo. h
where dg
The function
is in L 1 ( d g x d??;x dg), hence by Fubini’s theorem,
(31) where, setting k p q, 7 @ and using the recurrence relation for the ICv’sgiven by eq. (51) in section 4.4,we compute
+
E
kC”H(7).
H ( 7 ) is a bounded, continuous function of 7 for 7 2 2m, and
4.5. Geometry md Probability
217
dPd4 exp b ( P - dl 3 ( P ) P(d (P” + 4‘9H(rl).
J p ( x )= AX1
(33) Since f(p) has compact support, differentiation under the integral sign to any order in 5 gives an absolutely convergent integral, proving that J p is C”. Differentiation with respect to x p brings down the factor i ( p p - q p ) from the exponent, hence the continuity equation follows from p 2 = q2 = rn2. I
Remark. The continuity equation also follows from a more intuitive, geometric argument. Let
oriented such that
(the outward normal on aB: points “down,” whereas f-2: “up”). Then by Stokes’ theorem,
is oriented
Here, d represents exterior differentiation with respect to y, and since the s-form dy p contains all the dyy’s except for dya, we have h
Jp(x) = -AT1
d
- iY) I 2 ,
(37)
218
4. Complex Spacetime
where dy is Lebesgue measure on B z . To justify the use of Stokes’ theorem, it must be shown that the contribution from l y l + 00 to the first integral vanishes. This depends on the behavior of f(z), which is why we have given the previous analytic proof using the Fourier transform. Then the continuity equation is obtained by differentiating under the integral sign (which must also be justified) and using
a2If l 2 axray,
= 0,
(38)
which follows from the Klein-Gordon equation combined with analyticity, since
Incident ally, this shows that j”(2)
= --d I f ( Z ) I (40)
ay, =i
[f(t)a,f(.)
- d,f(
is a “microlocal,” spacetimeconserved probability current for each fixed y E V;, so the scalar function I f(z) I is a potential for the probability current. We shall see that this is a general trend in the holomorphic formalism: many vector and tensor quantities can be derived from scalar potentials. Eqs. (37) and (40) also show that our probability current is a regularized version of the usual current associated with solutions in
4.5. Geometry and Probability
219
real spacetime. The latter (Itzykson and Zuber [1980]) is given by
which leads t o a conceptual problem since the time component, which should serve as a probability density, can become negative even for positiveenergy solutions (Gerlach, Gomes and Petzold [1967], Barut and Malin [1968]).By contrast, eq. (36) shows that Jo(s) is stricly non-negative. The tendency of quantities in complex spacetime to give regularizations of their counterparts in real spacetime is further discussed in chapter 5. We can now prove the main result of this section.
Theorem 4.10.
Let u = S - in: E C and f E X:
.
Then
llflld = IlfllK. Proof. We will prove the theorem for ](p) in the space D ( R 8 )of C" functions with compact support, which implies it for arbitrary f" E L:(d$) by continuity. Let S be given by zo = t(x), and for R > 0 let
I ER = {x E R"+'I SOR = {Z E R'+l I SR = {X E ]Rd+' I D R = {a: E Rd+'
1x1 < R,':a E [O,t(x)]}, 1x1 = R,':a E [O,t (x)]},
1x1 < R,ZO= o},
(41)
1x1 < R,'5 = t(X)}, where [0, t (x)]means [t(x),01 if t (x)< 0. We orient SORand SR by dxo , E R by the "outward normal" h
220
4. Complex Spacetime
+
and D R so that ~ D = R SR - SOR ER. Now let f(p) E D(IR3). Then J P ( x ) is C", hence by Stokes' theorem,
J
S R -SOR+ER
J p ( x )&,, =
kR(
d Jp
= (-1)"
J
DR
z,,) dJC"
dx -= 0.
(43)
ax'
We will show that
A ( R ) = J,, J",,
+
o as R + 00
(44)
(i.e., there is no leakage to 1x1 + oo),which implies that
= Ilfl:o,
= llfll:,
by theorem 1 of section 4.4. To prove that A(R) + 0, note L a t on h
ER,dxo = 0 and h
h
h
dxk = xk- dx 1 = xk- d52 = ' ' ' = X k - dX3 9
h
21
22
5 3
each form being defined except on a set of measure zero; hence h
i.=R-.d X 1 21
(47)
221
4.5. Geometry and Probability
=s
LR
J o i. = a(R).
Now by eqs. (31) and (32),
J o ( x )=
k2,
d"pd"q eiz(p-q)$(p, q),
where
D=jZ*V,, where 2 = x/R, and observe that for x E ER,
=-
i t(x,p) ~ eizp,
where v = p/po. Since q5 has compact support, there exists a constant a < 1 such that IvI 5 a and lv'l 5 CY for all (p,p') in the support
4. Furthermore, Ixo I < R(1+ e) for
of
IE(x,p)l
since (Vt(z)I 5 1, given any E > 0 we have E ER for R sufficiently large; hence
1 1 - a(1+
E)
for z E ER and p E supp
4.
(54)
4. Complex Spacetime
222
Choose 0 < E < 0-l - 1, substitute
into the expression for Jo(x) and integrate by parts:
This procedure can be continued, giving (for x E E R )
where
r-')"
Now ( D o is a partial differential operator in p whose coefficients are polynomials in D'((E-l) with Ic = 0,1, ... ,n. We will show that for x E ER with R sufficientlylarge, there are constants b k such that
ID' (t-') I< bk which implies that
Ic = 0,1,-,
(59)
4.5. Geometry and Probability
223
for some constants cn, so that by eqs (49) and (57),
a(R) = s
LR
Jot
if we choose n > s. To prove eq. (59)) note that it holds for k = 0 by eq. (54) and let u = 2 v. Then
-
and if for some k pk(u) Dku = Pt
where pk is a constant-coefficient polynomial, then
-
pk+1( U ) p;+l
hence eq. (63) holds for k = 1,2,
which implies
'
- - by induction. Thus
224
4. Complex Spacetime
But D k ( t - ' ) is a polynomial in
[-'and DE,D 2 [ ,- - - ,Ilk[;hence eq.
(59) follows from eqs. (54) and (66).
I
The following is an immediate consequence of the above theorem.
Corollary 4.11. (a) For every o E C, the form
defines a Po-invariant inner product on
K , under
which
K is a
Hilbert space.
(b)
=
The transformations (V,f)(z) f(g-'z), g E PO,form a unitary irreducible representation of PO under the above inner prod-
f^
f from L$(dfi) to K: intertwines this representation with the u s u d one on L:(d@). (c) For each t~ E C, we have the resolution of unity uct, and the map
H
on L:(d@) (or, equivalently, on
K
if e, is replaced by e",. ) I
Note: As in section 4.4,all the above results extend by continuity to the case X = 0. #
4.6. The Non-Relativistic Limit
225
4.6. The Non-Relativistic Limit
We now show that in the non-relativistic limit c + 00, the foregoing coherent-state representation of Po reduces to the representation of 62 derived in section 4.3, in a certain sense to be made precise. As a by-product, we discover that the Gaussian weight function associated with the latter representation (hence also the closely related weight function associated with the canonical coherent states) has its origin in the geometry of the relativistic (dual) “momentum space” That is, for large lyl the solutions in K: are dampened by the factor in momentum space, which in the non-relativistic exp[limit amounts to having a Gaussian weight function in phase space.
Qi.
d w u ]
In considering the non-relativistic limit, we make all dependence on c explicit but set h = 1. Also, it is convenient to choose a coordinate system in which the spacetime metric is g = diag(1, -1,. . . ,-I), so that yo = yo = and po = po = Fix u > 0 and let X = uc. Then
d
m
d w .
n
= umc2
++ 2m up‘
n
my“ ~(c-~). 2u
+
Working heuristically at first, we expect that for large c, holomorphic solutions of the Klein-Gordon equation can be approximated by
226
4. Complex Spacetime
where T = t - iu and fNR is the corresponding holomorphic solution of the Schrodinger equation defined in section 4.3. Note that the Gaussian factor exp[-my2/2u] is the square root of the weight function for the Galilean coherent states, hence if we choose f(p) E L2(IR")c L$(dfi), then
We now rigorously justify the above heuristic argument. Let f ( z ) be the function in K: corresponding to f(p) and denote by fc its restriction to z o = t and y2 = u2c2,for fixed u > 0. Theorem 4.12. Let u
> 0 and f(p)
E L2(IRs).Then
Proof. Without loss of generality, we set u = m = 1 and t = 0 to simplify the notation. Note first of all that
4.6. The Non-Relativistic Limit
where A,
Ax (A
G
227
uc = c). But
Choose a , y such that 1/2 < y < a < 1. Then
-+
0 as c + 00,
I
where xc is the indicator function of the set {p IpI > c ~ - ~ }Define . 6 and d by IyI = csinh6 and IpI = csinh4. Then yo = cosh8 and w = c2 cosh 6, hence
228
4. Complex Spacetime
Thus for arbitrary a 2 0,
Let a = sinh-'(c-7).
Then for IpI
"'2po B(p0pb) 6(p2 - m 2 )6 ( ~ -' p~ 2 ) S(p - p') = (27r)8+22poq p o p ; ) 6(p2 - m 2 )6(pb2 - p i ) S(p - p') = sign(p0) ( 2 ~ ) 6(p2 ~ + ~ m 2 )~ ( -pp')
(13) for arbitrary p,p' E mented by
IRS+l.
For charged fields, this must be supple-
Recall that in the general case we had
(
q I @pi
) = 0 ( p 2 )(2T)"+l S(p - p')
(15)
v+,
for p,p' E where a ( p 2 )is the spectral density for the two-point function (sec. 5.3, eq. (33)). For the free field now under consideration we have
(
q I 'pp") = ( Qo I 6(P)i(P')* Qo ) I [W &)')*I
) = (27r)5+26(p2 - m 2 )6(p - p'), = ( Qo
?
Qo
(16)
which shows that the spectral density for the free field is 0 ( p 2 )= 27r6(p2 - m2).
The spectral condition implies that
(17)
5. Quantized Fields
278
since otherwise these would be states of energy-momentum -p. Hence the particle coherent states defined in the last section are now given by
where the vectors 6: are generalized eigenvectors of energy-momenturn p E
with the normalization
-+ I @+,p
(@p
I 42-4.(p')*Qo 1
) =( =(9
0
I
[+)l
.(P')*IQO )
(20)
= 2w(27r)s S(p - p').
The wave packets e$ span the one-particle subspace 'FI1 of 'FI and have the momenturn represent at ion ._ ( aP -+ I e+, ) = carp.
(21)
A dense subspace of 'HI is obtained by applying the smeared operators
5.4. B e e Klein-Gordon Fields where
279
fo is the restriction of f” to 522. Hence the functions
exactly the holomorphic positive-energy solutions of the KleinGordon equation discussed in section 4.4,with e;f corresponding to the evaluation maps e,. The space K: of these solutions can thus be are
identified with 3-11, and the orthogonal projection from 7f to given by 4
is
Consequently, the resolution of unity developed in chapter 4 can now be restated as a resolution of Ill:
where a+,earlier denoted by a,is a particle phase space, i.e. has the form a+ = { x - i y I z E
s,
y
E 52;)
(27)
for some X > 0 and some spacelike or, more generally, nowhere timelike (see section 4.5) submanifold S of real spacetime. As in section 4.5, the measure do is given in terms of the Poincar&invariant symplectic form Q! = dy, A dx” by restricting as a A . A a to a+ and choosing an orient at ion:
--
h
h
da = (s!Ax)-1 a s = AX1 dy” A d x , .
(28)
Similarly, the antiparticle coherent states for the free field are given by
5. Quantized Fields
280
Since for p E !2$ and n E 7- we have
. it follows that e;
has exactly the same spacetime behavior as e t ,
confirming the interpretation of an antiparticle as a particle moving backward in time. An antiparticle phase space is defined as a submanifold of 7- given by
where S is as above. The resolution of
n-1
is then given by
Many-particle or -antiparticle coherent states and their corresponding phase spaces can be defined similarly, and the commutation relations imply that such states are symmetric with respect to permutations of the particles' complex coordinates. For example,
since 4(z1) and 4(z2) commute. In this way, a phase-space formalism can be buit for an indefinite number of particles (or charges), analogous to the grand-canonical ensemble in classical statistical mechanics. This idea will not be further pursued here. Instead, we
5.4. R e e Klein-Gordon Fields
281
now embark on option (b) above, i.e. the construction of global, conserved field observables as integrals over particle and antiparticle phase spaces. The particle number and antiparticle number operators are given by
N+ and N - generalize the harmonic-oscillator Hamiltonian A'A to the infinite number of degrees of freedom possessed by the field, where normal modes of vibration are labeled by p E $22for particles and p E 0; for antiparticles. The total charge operator is Q = e (N+ - N - ) , as can be seen from its commutation relations with u ( p ) and b(p). But the resolution of unity derived in chapter 4 can now be restated as
for p,p' E 02,where the second identity follows from the first by replacing z with Z and o+ with g-. It follows that N h can be expressed as phase-space integrals of the extended field q5(z):
282
5. Quantized Fields
Hence the charge is given by
The two integrals can be combined into one as follows: Define the total phase space as u = a+ - u-, where the minus sign means that u- enters with the opposite (“negative”) orientation to that of u+, in the sense of chains (Warner [1971]).The reason for this choice of orientation is that B; and By are both open sets of IR8+’, hence must have the same orientation, and we orient 0: and 0, so that
Since the outward normal of B: points “down” and that of B, points ((up,”this means that 0, must have the opposite orientation to that By and 0 x G 0: - a, we have of 0;. Thus, setting Bx B:
=
+
This gives u- the orientation opposite to that of u+, and we have
Next, define the Wick-ordered product (or normal product) by
This coincides with the usual definition, since in 7+, $* is a creation operator and 4 is an annihilation operator, while in 7-these roles are reversed. The charge can now be written in the compact form
5.4. R e e Klein-Gordon Fields
283
We may therefore interpret the operator p(z)
f€
: 4(z)* 4(z) :
(42)
as a scalar phase-space charge density with respect to the measure
da.
The Wick ordering can be viewed as a special case of imaginarytime ordering, if we define 4 * ( z ) = $(Z)*: : +(z')* 4 ( z ) :=: 4 * ( ~ ' $) ( z ) := T i [d*(Z')
4(~)],
(43)
where
and
when z and z' are in the same half of 7,whereas if they are in opposite halves of 7, the sign of 3(zk - zo) is invariant. Note: For the extended fields, the Wick ordering is not a necessity but a mere convenience, allowing us to combine the integrals over a+ and 0- into a single integral. Each of these integrals is already in normal order, since the extension to complex spacetime polarizes the free field into its positiveand negativefrequency parts. Also,
5. Quantized Fields
284
the extended fields are operator-valued functions rather than distributions, hence products such as 4(z)*c$( z ) are well-defined, which is not the case in the usual formalism. A similar situation will occur in the expressions for the other observables (energy-momentum, angular momentum, etc.) as phasespace integrals. Hence the phasespace formalism resolves the problem of zero-point energies without the need to subtract infinite terms “by hand”! In this connection, see the remarks on p. 21 of Henley and Thirring [1962]. # The above expression for the charge can be related to the usual one in the spacetime formalism, which is ~~~~d = ia
G
J, z,, : +*-dx, d4 - %* - qqx): ax, (47)
kz,,Jc”(x),
by using Rx = -dBx and applying Stokes’ theorem:
where j”(.)
= --d
dY,
p(z)
is the phase-space current density. Using the notation
(49)
5.4. n e e Klein-Gordon Fields
285
we have
--d - i(d” - 8”). Hence, by the holomorphy of
j”(2)
4, d
=-
E G
.$*4: *
(a. - a”) : f$* f$ : = i&: 4* - &4* - 4 : = i&: 4*- 34 - %* 4 : . = i&
axr
dx”
Our expression for the charge is therefore
where
is seen to be a regulmized version of the usual current density J”(z) obtained by first extending it to 7 and then integrating it over Bx.
The vector field fixed y E V’, since
jP(z)
is conserved in real spacetime for each
5. Quantized Fields
286
by virtue of the Iclein-Gordon equation combined with the holomorphy of 4 in 7. This implies that JfLx,(x) is also conserved, hence the charge does not depend on the choice of S or u.
Note: In using Stokes’ theorem above, we have assumed that the contribution from lyol + 00 vanishes. (This was implicit in writing the non-compact manifold Rx as -dBx.) This is indeed the case, as has been shown rigorously in the context of the one-particle theory in chapter 4 (theorem 4.10). Also, we see another example of the pattern, mentioned before, that in the phase-space formalism vectorand tensor fields can often be derived from scalar potentials. Here, p ( z ) acts as a potential for j p ( z ) . Note also that the Klein-Gordon equation can be written in the form
which is manifestly gauge-invariant.
#
&call now that $ ( z ) is a “root vector” of the charge with root value -e:
Substituting our expression for Q, we obtain the identity
287
5.4. n e e Klein-Gordon Fields
K is
a distribution on (I?+1 x
(59) which is piecewise analytic in
7 x 7, with
K(Z',2 ) =
{
y ( Z '
- Z; m ) ,
Z',Z
Z'
E 7-
E T+,ZE 7-
(60)
The two-point functions -iA+ and iA- are analytic in 7+and 7-, respectively, and act as reproducing kernels for the subspaces with charge e and -6. Because of the above property, it is reasonable to call K ( z ' , Z) a reproducing kernel for the field + ( z ) , though this differs somewhat from the standard usage of the term as applied to Hilbert spaces (see chapter 1). Note that K propagates positivefrequency components of the field into the forward ("future") tube
5. Quantized Fields
288
and negative-frequency components into the backward (“past”) tube. This is somewhat reminiscent of the Feynman propagator, but K is a solution of the homogeneous Klein-Gordon equation in the real spacetime variables rather than a Green function. The energy-momentum and angular momentum operators may be likewise expressed as conserved phase-space integrals of the extended field:
Pp = i l d o : $*a,$: r
Mpv = i
d o : $*(xcdv - x V a p ) $ : .
Like Q, these may be displayed as regularizations of the usual, more complicated expressions in real spacetime. Note first that Pp can be rewritten as
The angular momentum can be recast similarly as
5.5. n e e Dirac Fields
289
Using s2x = -aBx and applying Stokes' theorem, we therefore have
where
is a regularized energy-momentum density tensor which, inciclzntally, is automatically symmetric. Similarly,
where
is a regularized angular momentum density tensor.
5.5. Free Dirac Fields
For simplicity, we specialize in this section (only) to the physical
case of three spatial dimensions, s = 3. The proper Lorentz group
290
5. Quantized Fields
Lo is then S0(3,1)+, where the plus sign indicates that At > 0, so that A preserves the orientations of space and time separately. Its universal covering group can be identified with SL(2, C) as follows (Streater and Wightman [1964]): An event x E R4 is identified with the Hermitian 2 x 2 matrix
x =x b p=
xo + x3 x1 +ix2
-
x1 ix32) xo - x
where QO = I (2 x 2 identity) and b k (k = 1,2,3) are the Pauli spin matrices. Note that det X = x2 x x. The action of SL(2,C) on
-
Hermitian 2 x 2 matrices given by
X' = AXA*,
A E SL(2,C),
(2)
induces a linear transformation on IR4 which we denote by ?r(A):
x' = n(A) x
(3)
From
it follows that 7r(A) is a Lorentz transformation, and it can easily be seen to be proper. Hence
7r
7r:
defines a map
SL(2,C) + Lo,
(5)
which is readily seen to be a group homomorphism. Clearly, ?r( -A) =
.rr(A),and it can be shown that if 7r(A) = ?r(B),then A = fB.Since SL(2, C)is simply connected, it follows that SL(2, a) is the universal covering group of LO,the correspondence being two-to-one. The relativistic transformation law as stated in section 5.3,
5.5. n e e Dirac Fields
291
applies to scalar fields, i.e. fields without any intrinsic orientation or spin. To generalize it to fields with spin, note first of all that since the representing operator U ( a ,A) occurs quadratically, the law is invariant under U + 4. This means that U could, in fact, be a representation, not of PO,but of the inhomogeneous version of
SL(2,a!),
which acts on R4 by ( u , A ) s= ~ ( A ) a : + a .
(8)
P2 is the two-fold universal covering group of PO. A field $(a:) of arbitrary spin is a distribution taking its values in the tensor product L(7-l)8 Y of the operator algebra of the quantum Hilbert space 3.c with some finite-dimensional representation space V of SL(2, a!). The transformation law is
where S is a given representation of SL(2, a!) in V. S determines the spin of the field, which can take the values j = 0,1/2,1,3/2,2,. . .. The locality condition for the scalar field (axiom 4) can be extended to non-scalar fields as
[$a(a:),$,9(a:c')]
=o
if (a: - z ' ) ~< o
292
5. Quantized Fields
where $a are the components of $. Now it follows from the axioms that if the field has half-integral spin ( j = 1/2,3/2,. . .), then the 0. above locality condition implies that it is trivial, i.e. that $(z) A non-trivial field of hal-integral spin can be obtained, however, if we modify the locality condition by replacing commutators with ant icommutat ors:
Replacing the commutators with anticommutators means that changing the order in which $(z) and $(d)are applied to a state vector in Hilbert space merely changes the sign of the vector, which has no observable effect. Hence the physical interpretation that events at spacelike separations cannot influence one another is still valid. Similarly, for fields of integral spin ( j = 0,1,. . .), the locality condition with anticommutators gives a trivial theory, whereas a nontrivial theory can exist using commutators. The choice of commutators or anticommutators in the locality condition does, however, have an important physical consequence. For we have seen that the free asymptotic fields can be written as sums of creation and destruction operators for particles and antiparticles. If xl,2 2 , - - - z, are n distinct points in the hyperplane zo = 0 and $+(z) denotes the positive-frequency part of the field (which can be obtained from $(a: - iy) by taking y --+ 0 in V;), then
is a state with n particles located at these points. Since any two of these points are separated by a spacelike distance, the locality condition implies that this state is symmetric with respect to the exchange
5.5. fiee Dirac Fields
293
of any two particles if commutators are used and antisymmetric with respect to the exchange if anticommutators axe used. Particles whose states are symmetric under exchange are called Bosons, and ones antisymmetric under exchange are called Fermions. The choice of symmetry or antisymmetry crucially affects the large-scale statistical behavior of the particles. For example, no two Fermions can occupy the same state due to the antisymmetry under exchange; this is the Pauli
exclusion principle. Hence the choice of commutators or anticommutators is known as the choice of statistics, and the above theorem correlating this choice with the spin is known as the Spin-Statistics theorem of quantum field theory (Streater and Wightman [1964]). This theorem is fully supported by experiment, and represents one of the successes of the theory. The Dirac field is a quantized field with spin 1/2 whose associated particles and antiparticles are typically taken to be electrons and positrons, though it is also used (albeit less accurately) to model neutrons and protons. Our treatment follows the notation used in Itzykson and Zuber [1980],with minor modifications. The free Dirac field is a solution of the Dirac equation
where
is the Dirac operator and the 7’s are a set of 4 x 4 Dirac matrices, meaning they satisfy the Clifford condition with respect to the Minkowski metric:
5. Quantized Fields
294
{ y p , 7')
= ypyv + 7 " 7 p = 2gp'.
(15)
The components of $ satisfy the Klein-Gordon equation, since q 2 = 0 , and the solutions can be written as
where u" and v" are positive- and negative-frequency four-component spinors and summation over the polarization index a = 1 , 2 is implied. b, and d, are operators satisfying the "canonical anticommutation relations"
G y p ) u"p) = -fi"(p) v q p ) = 6"B
where a summation on cr is implied in the last two equations and the adjoint spinors are defined by
5.5. B e e Dirac Fields
295
ba(p) and d Q ( p ) are interpreted as annihilation operators for particles and antiparticles, respectively, while their adjoints are creation operators. The adjoint field is defined by
and satisfies
The particle- and antiparticle number operators are now
(23)
r
and the charge operator is
Q = &(N+- N - ) .
(24)
As for the Klein-Gordon field, we wish to give a phasespace repusing the resentation of Q. The first step is to extend $(z) to Analytic-Signal transform, which gives
a?
Again, the extended field is analytic in 7,with the parts in 7+and 7containing only positive and negative frequncies, respectively. Using the above orthogonality relations, as well as
296
5. Quantized Fields
for p , q E ni, we obtain the following expressions for the particleand antiparticle number operators as phase-space integrals:
J u-
where the fields in the first integral are already in normal order and the second integral involves two changes of sign: one due to the normal ordering, and another due to the orthogonality relation for the P’s.The charge operator can therefore be given the following compact expression as a phasespace integral over the oriented phase space CT = a+ - a-:
Q = a / da : $lC,
:= e
U
J dap(z),
(28)
U
where p = : $$ : is the scalar phase-space charge density. The usual expression for the charge as an integral over a configuration space S is
To compare these two expressions, we again use Rx = -dBx and invoke Stokes’ theorem:
Define the phase-space current density
-
j p ( z ) E 2ma : + ( z ) $~( z~)
:,
297
5.5. n e e Dirac Fields
where the factor 2m is included to give j p the correct physical dimensions, given our normalization. Note that
jp(z)
is conserved in
spacetime, i.e.
=O by the Dirac equation combined with the analyticity of 1c, in 7.The same combination also implies
where
are the spin matrices. The real part of this equation gives a phasespace version of the Gordon identity
The two terms are conserved separately, since
298
5. Quantized Fields
and the second term, which is due to spin, does not contribute to the total charge since it is a pure divergence with respect to x. Thus
where
is a “regularized” spacetime current.
Note: The Dirac equation can also be written in the manifestly gauge-invariant form
Again, $ is a “root vector” of the charge operator, since it removes a charge E from any state to which it is applied:
[$(z’),Q]= E$(z’)
VZ’ E C4.
(40)
Substituting for Q the above phase-space integral and using the commut ator identity
[ A ,BC]= { A ,B}C - B { A , C}
(41)
and the canonical anticommutation relations, we obtain
where the “reproducing kernel” for the Dirac field is a matrix-valued distribution on C4 x C4 given by
5.5. fiee Dirac Fields
299
Here, K is the reproducing kernel for the Klein-Gordon field and 8' is the Dirac operator with respect to the real part x' of z'. Like K, ICo is piecewise holomorphic in z' - 2 for z', z E 7.Another form of the reproducing relation can be obtained by substituting the more complicated expression for Q given by eq. (37) into eq. (40): $ ( z ' ) = 2mAi1
(44)
This form is closer to the usual relation. The energy-moment um and angular momentum operators for the Dirac field can likewise be represented by phase-space integrals as
Pp = l d u : $ia,,$:
+
iwPV= J,du : s(ix,av - ixvaP + a P v ) : .~
(45)
More generally, let $ ( z ) represent either a KleinTGordon field (in which case II) will mean $*) or a Dirac field, and let Ta be the local generators of an arbitrary internal or external symmetry group, so that the infinitesimal change in $ ( z ) is given by
5. Quantized Fields
300
For example, T, is multiplication by 6 for U(1) gauge symmetry, Tp = ia, for spacetime translations (where the derivative is with respect to zp), etc. (In case the theory has an internal symmetry higher than U(1),of course, 1c, must have extra indices since it must be valued in a representation space of the corresponding Lie algebra.) The generators satisfy the Lie relations
where Cib are the structure constants. Then we claim that the conserved global field observable corresponding to T, is
Q , = l d c :$T,$:. For this implies [$(z'),
&,I
=
J. d o K D ( ~3'),Ta$(z),
(49)
where K D is replaced by K if $ is a Klein-Gordon field. Since T, generates a symmetry, it follows that Ta$(z) is also a solution of the appropriate wave equation, hence it is reproduced by K D :
J.
du KD(z',Z)Ta$(z) = Talc,(z').
(50)
Therefore Q , has the required property
It can furthermore be checked that
[&, &a]
=
1
do : $ [Ta, Tb]$ := CZb QC,
4
hence the mapping
T, H Qa is a Lie algebra homomorphism.
(52)
5.5. B e e Dirac Fields
301
Finally, we show that due to the separation of positive and negative frequencies in 7, the interference effect known &s Zitterbewegung does not occur for Fermions in the phasespace formalism. Let St be the configuration space defined by xo = t. Then the components of the “regularized” three-current at time t are
and a straightforward computation gives
The right-hand side is independent of t , hence no Zitterbewegung occurs. In real spacetime, Zitterbewegung is the result of the inevitable interference between the positive- and negativefrequency components of qb. Its absence in complex spacetime is due to the polarization of the positive and negative frequencies of II, into 7+and 7-, respectively. In the usual theory, Zitterbewegung is shown to occur in the singleparticle theory; the above computation can be repeated for the classical (i.e., “first-quantized”) Dirac field, with an identical result except for a change in sign in the second term due to the commutation of dz and d,. Alternatively, the above argument also implies the absence of Zitterbewegung for the one-particle and oneantiparticle states of the Dirac field.
302
5. Quantized Fields
5.6. Interpolating Particle Coherent States We now return to the interpolating charged scalar field 4. The asymptotic fields satisfy the Klein-Gordon equation,
and have the same vacuum expectation values as the free KleinGordon field discussed in section 5.4. Hence, by Wightman’s reconstruction theorem (Streater and Wightman [1964]),these three fields are unitarily related. We identify the free field of section 5.4 with &n. Then there is a unitary operator S such that
S is known as the scattering operator. Define the source field j ( x ) by j(.)
3
(n + m2)4(.).
(3)
It is a measure of the extent of the interaction at x, and by axiom 5,
o
j(x)
(weakly) as x o
-, foe.
(4)
Note that we are not making any additional assumptions about j . If j is a known function (i.e., if it is a multiple of the identity on ‘H for each x), then it acts as an external source for 4. If, on the other hand, j is a local function of such as : 43:, it represents a self-interaction of 4. In any case, the above equations can be “solved” using the Green functions of the Klein-Gordon operator, which satisfy
+
+
(0, m2)G ( x )= 6(x).
(5)
5.6. Interpolating Particle Coherent States
303
In general, we have formally
4(x) = 4o(x)
+ J dx' G(x - .')j(.'),
(6)
where 4 0 is a free field determined by the initial or boundary conditions at infinity used .to determine G. The retarded Green function (we are back to s spatial dimensions) is defined as
where
with
E
> 0 and the limit E 5 0 is taken after the integral is evaluated.
Gret propagates both positive and negative frequencies forward in time, which means that it is causal, i.e. vanishes when xo < 0. Since it is also Lorentz-invariant, it follows that
Gret(x - z') is interpreted as the causal effect at x due to a unit disturbance at x'. The corresponding choice of free field 4 0 is q5jn, hence
If j is a known external source, this gives a complete solution for q5(x). If j is a known function of 4, it merely gives an integral equation which
4 must satisfy.
Similarly, the advanced Green function is defined by
5. Quantized Fields
304
=
with p (po - ie,p) and E: 3. 0, and propagates both positive and negative frequencies backward in time, which means it is anticausal. The corresponding free field is q50ut, hence
4(.)
= $out(z)
+
1
dx'
Gadv(2
- x')j(.')-
(12)
Let us now apply the Analytic-Signal transform to both of these equations:
where (with z = z - iy)
and
Since the Analytic-Signal transform involves an integration over the entire line x(r) = x - r y , the effect of Gret(z - 2') is no longer
5.6. Interpolating Particle Coherent States
305
causal when regarded as a function of z and 5’. Rather, it might be interpreted as the causal effect of a unit disturbance at x’ on the line parametrized by 2. (Note that only those values of r for which z - r y - z’ E contribute to the integral.) A similar statement goes for Gadv( z - 2’). Whereas djn(z) and dout(z) are holomorphic in 7,4 ( z ) is not (unless j (z) = 0), since Gret( z -z‘) and G d v( z -5 ’ ) are not holomorphic. This breakdown of holomorphy in the presence of interactions is by now expected. Of course 4, Gret and Gad” are all holomorphic along the vector field y, as are all Analytic-Signal transforms.
Qt,
In Wightman field theory, the vacua Qytand QO of the in-, out- and interpolating fields all coincide (the theory is “alreadyrenormalized”). Let us define the asymptotic particle coherent states by
as the interpolating particle coherent states. By eq.
and
(13),
5. Quantized Fields
306
From the definitions it follows that
Gadv(z - 2') = Gret(z' - z), hence eq. (18) can be rewritten as
Eqs. (19) and (21) display the interpolating character of e$. Note that when j ( ~is) an external source, then the interpolating particle coherent states differ from the asymptotic ones by a multiple of the vacuum.
As in the case of the free theory, a general state with a single positive charge
E
can be written in the form
q = 4*(f)Qo.
(22)
For interacting fields, this may, in general, no longer be interpreted as a one-particle state, since no particle-number operator exists.* But
*
If the spectrum C contains an isolated mass shell
is concentrated around St$, then
Qi
and f ( p ) is, in fact, a one-particle state.
This is the starting point of the' Haag-Ruelle scattering theory (Jost [1965]).I thank R. F. Streater for this remark.
5.6. Interpolating Particle Coherent States
307
the charge operator does exist since charge (unlike particlenumber) is conserved in general, due to gauge invariance; hence Qf makes sense as an eigenvector of charge with eigenvalue 6. !PF can be expressed in terms of particle coherent states as
f(z ) satisfies the inhomogeneous equations
where the last equation is a definition of 6 ( z - 3') as the AnalyticSignal transform with respect to x of b(x - z'). The above is easily seen to reduce to
308
5. Quantized Fields
where j ( z ) is the Analytic-Signal transform of j ( s ) . Equivalently, eq. (3) can be extended to by applying the Analytic-Signal transform, giving
(02
+ m2)J(z) = ( q o I (0,+ m 2 )+ ( z ) I Q:
) = ( Qo I j ( z ) Q; ). (28)
For a known external source, this is a “perturbed” Klein-Gordon equation for j ( z ) ; if j depends on 4, it appears to be of little value.
5.7. Field Coherent States and Functional Integrals
So far, all our coherent states have been states with a single particle or antiparticle. In this section, we construct coherent states in which the entire field participates, involving an indefinite number of particles. We do so first for a neutral free Klein-Gordon field (or a generalized free field; see section 5.3), then for a free charged scalar field. A similar construction works for Dirac fields, but the “functions” labeling the coherent states must then anticommute instead of being “classical” functions and a generalized type of functional integral must be used (Berezin [1966], Segal [1956b, 19651). We also indulge in some speculation on generalizing the construction to interpolating fields. An extended neutral free Klein-Gordon field satisfies the canonical commutation relations
5.7. Field Coherent States and finctiond Integrals
309
for all z,z‘ E 7+,as well as the reality condition +(z)* = +(Z). The basic idea is that since all the operators 4 ( z ) ( z E 7+) commute, it may be possible to find a total set of simultaneous eigenvectors for them. This is not guaranteed, since + ( z ) is not self-adjoint (it is not even normal, by eq. (1))and, in any case, it is unbounded and thus may present us with domain problems. However, this hope is realized by explicitly constructing such eigenvectors. This construction mimics that of the canonical coherent states in section 3.4, which used the lowering and raising operators A and A*. As in the case of finitely many degrees of freedom, the canonical commutation relations mean that +* acts as a generator of translations in the space in which is “diagonal.” The construction proceeds as follows: Let f(p) be a function on IR”, which will also be regarded as a function To simplify the analysis, we assume to begin with that f on is a (complex-valued) Schwartz test function, although this will be relaxed later. f determines a holomorphic positive-energy solution of the Klein-Gordon equation,
+
i22.
Define
where cr+ is any particle phase space and the second equality follows from theorem 4.10 and its corollary. (Note: this is not the same as the smeared field in real spacetime, since the latter would involve an integration over time, which diverges when f is itself a solution rather than a test function in spacetime.) The canonical commutation relations imply that for z E 7+,
5. Quantized Fields
310
and for n 2 1,
We now define the field coherent states of $ by the formal expression
Then if z E ' I so + that $ () z ) 90= 0, eq. ( 5 ) implies that $(z)
Ef = [ $ ( z ) , ,4*(f)] Qo (7)
= f(z)Ef.
Hence Ef is a common eigenvector of all the operators $ ( z ) , z E 7+. This eigenvalue equation implies that the state corresponding to
Ef
is left unchanged by the removal of a single particle, which requires that Ef be a superposition of states with 0,1,2,. . . particles. Indeed,
Ef =
c+
n=O
n.
$*(f)"
9 0 .
The projection of Ef to the one-particle subspace can be obtained by using the particle coherent states e,:
=
where the last equality follows from d(f) 90 ($*(f))*90= 0. More generally, the n-particle component of the n-particle coherent state
Ef is given by
projecting to
5.7. Field Coherent States and Ftrnctiond Integrals
so all particles are in the same state
31 1
f and the entire system of par-
ticles is coherent! Similar states have been found to be very useful in the analysis of the phenomenon of coherence in quantum optics Klauder and Sudarshan [1968]),where the name “co(Glauber [1963], herent states” in fact originated. In the usual treatment, the positivefrequency components have to be separated out “by hand” using their Fourier representation, since one is dealing with the fields in real spacetime. For us, this separation occured automatically though the use of the Analytic-Signal transform, i.e. $*(f) can be defined directly as an integral of f ( z ) over a+. (This would remain true even if f had a negative-frequency component, since the integration over CT+ would still restrict f to positive frequencies.) The inner product of two field coherent states can be computed as follows. Note first that if g(z) is another positiveenergy solution, then
Ja+ L
where, by theorem 4.10,
5. Quantized Fields
312
Hence
Thus Ef belongs to 'FI (i.e., is normalizable) if and only if f(p) belongs to L:(dfi) or, equivalently, f(z) belongs to the oneparticle space Ic of holomorphic positive-energy solutions. If we suppose this to be the case for the time being, then the field coherent states Ef are parametrized by the vectors f" E Li(dfi) or f E K. Next, we look for a resolution of unity in 'H in terms of the Ef's. The standard procedure (section 1.3)would be to look for an appropriate measure dp(f) on Ic. Actually, it turns out that due to the infinite dimensionality of Ic, a larger space Kb 3 Ic will be needed to support dp. Thus, for the time being, we leave the domain of integration unspecified and write formally (15)
where dp is to be found. Taking the matrix element of this equation between the states Eh and E g , we obtain
With h = -g this gives /'dp(f)e(fIg)-(glf)
= e-(gIg)
s[g].
(17)
The left-hand side is an infinitedimensional version of the Fourier transform of dp, as becomes apparent if we decompose f and g into their real and imaginary parts. The Fourier transform of a measure is called its characteristic function. Hence we conclude that a
5.7. Field Coherent States and finctional Integrals
313
necessary condition for the existence of dp is that its characteristic function be S[g]. In turn, a function must satisfy certain conditions in order to be the characteistic function of a measure. In the finite-dimensional case, Bochner's theorem (Yosida [1971]) guarantees the existence of the measure if these conditions are satisfied. If the idnite-dimensional space of f's is replaced by (En,the above relation would uniquely determine dp as a Gaussian measure. For the identity
det A-'d2"( exp[-?r(( - At)* A-'(( - At)] = 1,
(18)
where A is a positive-definite matrix, implies
with dp((') = det A-' exp[-~('*A-'(] d2"C.
The integral in eq. (19) is entire in the variables hence it can be analytically continued to
+(()
If ( = (Y + ip and
(20)
< and t* separately,
I* + -t*, giving
e7r( 0,
Thus if h ( t ) decays rapidly, say if
Xh(Xt) + 0 then we expect fh(z,y/X)
+0
as
X + 00,
as X + 00.
(24)
6. f i t h e r Developments
340
Since eq. (11) holds for admissible h, we can now allow f E L2(IRn). We would like to characterize the range %T of the map 2’: f H fh from L2(IR”)to L2(dp). The relation
shows that h,,, acts like an evaluation map taking fh E L2(dp) to its “value” at (5,y). These linear maps on !RT are, however, not bounded if n > 1, since then h,,, is not square-integrable. (In general, the “value” of fh at a point may be undefined.) Hence %T is not a reproducing-kernel Hilbert space (chapter 1). But in any case, the distributional kernel
represents the orthogonal projection from L2(dp) onto RT. Thus a given function in L2(dp) belongs to !RT if and only if it satisfies the consistency condition
where the integral is the symbolic representation of the action of K as a distribution. Remarks. 1. For n = 1, the reconstruction formula is identical with the one for the continuous one-dimensional wavelet transform W f , since by eq. (21,
6.2 Windowed X-Ray Transforms: Wavelets Revisited
341
2. In deriving the resolution of unity and the related reconstruction formula, we have tacitly identified Rnas a Euclidean space, i.e. we have equipped it with the Euclidean metric and identified the pairing px in the Fourier transform as the inner product. The exact place where this assumption entered was in using the rotation group plus dilations to obtain IRr from the single vector q, since rotations presume a metric. Having established fh as a generalization of the one-dimensional wavelet transform, let us now investigate it in its own right. First, note that for n = 1 there were only two simple types of candidates for generalized frames of wavelets: (a) all continuous translations and dilations of the basic wavelet, or (b) a discrete subset thereof. For n > 1, any choice of a discrete subset of vectors h,,, spoils the invariance under continuous symmetries such as rotations, and it is therefore not obvious how to use the above grouptheoretic method to find discrete subframes. In fact, the discrete subsets { ( a m ,namb)} which gave frames of wavelets in section 1.6 and chapter 2 do not form subgroups of the &ne group. One of the advantages of using tensor products of one-dimensional wavelets is that they do generate discrete frames for n > 1, though sacrificing symmetry. However, other options exist for choosing generalized (continuous) subframes when n > 1, and one may adapt one’s choice to the problem at hand. Such choices fall between the two extremes of using all the vectors h,,, and merely summing over a discrete subset, as seen in the examples below. 1. The X-Ray fiansfonn The usual X-Ray transform is obtained by choosing h(t)
1, which
342
6. f i r t h e r Developments
is not admissible in the above sense; hence the above “wavelet” reconstruction fails. The reason is easy to see: Note that now fh has the following symmetries: fh(2,ay) = laI-lfh(z,y)
vaE
fh(z -k sy, 9) = fh(z,y) v s E
m*
IR.
(29)
Together, these equations state that fh depends only on the line of integration and not on the way it is parametrized. The first equation shows that integration over all y
# 0 is unnecessary as well as unde-
sira.ble, and it suffices to integrate over the unit sphere Iyl = 1. The second equation shows that for a given y, it is (again) unnecessary and undesirable to integrate over all z, and it suffices to integrate over the hyperplane orthogonal to y. The set of all such (z, y) does, in fact, correspond to the set of all lines in IR”,and the corresponding ~ a continuous frame which gjves the usual reconset of h z , y ’forms struction formula for the X-Ray transform (Helgason [1984]). The moral of the story is that sometimes, inadmissibility in the “wavelet” sense carries a message: Reduce the size of the frame. 2. The Radon llansform Next, choose v E IR and
Like the previous function, this one is inadmissible, hence the “wavelet” reconstruction fails. Again, this can be corrected by understanding the reason for inadmissibility. Eq. (6) now gives fh, (z, y)
=
J d”P e-2xipxqpy - v ) f”(P).
6.2 Windowed X-Ray Dansforms: Wavelets Revisited For any a
343
# 0, we have
where w = v / a . Hence it suffices to restrict the y-integration to the unit sphere, provided we also integrate over v E R. Also, for any 7
E IR,
Fixing x = 0, the function
is called the Radon transform of f” (Helgason [1984]). It may be regarded as being defined on the set of all hyperplanes in the Fourier and f” can be reconstructed by integrating over the set space (R”)*, of these hyperplanes.
3. The Fourier-Laplace Dansform Now consider
which gives rise to the Analytic-Signal transform. (We have adopted a slightly different sign convention than is sec. 5.2. Also, note that we have reinserted a factor of 27r in the exponent in the Fourier transform, which simplifies the notation.) Then fi is the exponential step function (sec. 5.2)
6. Further Developments
344
and eq. (6) reads
J
(37)
> 0). This is the Fourier-Laplace transform of f in My.For n = 1 and y > 0, it reduces to the usual
where My is the half-space {p Ipy
Fourier-Laplace transform. This h, too, is not admissible. f(z) can be recovered simply by letting y + 0, and fh(z,y) may be regarded as a regularization of A
f(z). If the support of f is contained in some closed convex cone r*C (Rn)*, then fh(2,y) f(z - i y ) is holomorphic in the tube 7r over the cone r dual to r*,i.e.
r = {y E IR"~ p y> o 7i
= { z - i y E C:"ly E
vp E
r*) (38)
r).
(Note that no metric has been assumed.) In that case, f(z) is a
boundary value of f( z -iy). This forms the background for the theory of Hardy spaces (Stein and Weiss [1971]). We have encountered a similar situation when R" was spacetime (n = s l),r*= and f(z) was a positive-energy solution of the Klein-Gordon equation;
+
v+,
then I' = V' and 5 = 7+.But in that case, f(z) was not in L2(RSs1) due to the conservation of probability. There it was unnecessary and undesirable to integrate over all of 7+since it was determined by its values on any phase space a+ C 7+,and reconstruction was then achieved by integrating over a+ (chapter 4).
6.2 Windowed X-Ray llansforms: Wavelets Revisited
345
As seen from these examples, the windowed X-Ray transform has the remarkable feature of being related to most of the “classical” integral transforms: The X-Ray, Radon and Fourier-Laplace transforms. Since the Analytic-Signal transform is a close relative of the multivariate Hilbert transform H , (sec. 5.2), we may also add Hgto this collection.
This Page Intentionally Left Blank
347
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357
INDEX admissible, 49, 100, 112, 339 affine group, 46 analytic signal, 239 -transform, 243ff, 335, 343 analytic vector, 116 anticommutators, 292 averaging operator h ( S ) , 62ff, 75 Bosom, 293 bra-ket not ation , 6ff canonical anticommutation relations, 294 commutation relations, 9, 11, 162, 276 canonical transformation, 211 Cartan subalgebra, 117 central extension, 168 characteristic function , 3 12 charge, 264ff, 282, 296 -current jfi,284, 296 density p, 283, 296 coherent state, canonical, 9ff, 115 field-, 308ff Galilean e i , 172ff holomorphic, 130 interpolating, 302ff particle-, 266 relativistic e,, 19Off -representation, 13ff spin-, 135 complex line bundle, 323 complex manifold, 129 complex structure J , 70ff complex vector bundle, 328 configuration space S, 208, 212 conjugation operator C, 72
connection, 322ff -form, 324 Riemannian, 323 type (1, O), 333 consistency condition, 22, 340 contraction limit, 121, 145ff, 161 control vector y, 195, 272 coset spaces, 109 covariant derivative Dx , 32M curvature 0,331 De Broglie’s relation, 4 decomposition, 82ff complex, 87ff differencing operator g(S), 75 dilation, 45, 58, 336 -equation, 63 -operator D, 62 Dirac equation, 393 -field 4, 289ff directional holomorphy, 247 electromagnetic field F, 326 -potential A, 325 energy, 4 evaluation maps, 32, 190 exponential step function Oc , 241 Fermions, 293 fiber, 323 -metric h, 325, 329 filter, bandpass, 45 complex 2,2, 85ff high-pass Gal 78ff lOW-pasS Ha,65 Fourier transform, 3, 5 windowed, 34, 36
358 Fourier-Laplace transform, 240, 244, 344 frame, l8ff, 33, 37, 49 discrete, 38-40, 55 group-, 95ff holomorphic, 113ff homogeneous, 103ff frame bundle, 159 functional integral, 316 gauge group, 327 -symmetry, 300 gauge transformation, 324 holomorphic-, 329 Galilean group 6,163ff General hlativity, 321 Gordon identity, 297 Haar basis, 63, 75 harmonic oscillator, 11, 145ff Heisenberg algebra, 9 -picture, 198 Hilbert transform H , 243 H,, 247, 345 home versions, 65 -space V, 65 homogeneous space, 110 interpolation operator H: , 65 Killing form, 119 Klein-Gordon equation, 185ff field 4, 273ff light cones V+,V . , 189 Lorentz group L,157 -metric, 2 Lorentzian spacetime, 2 mass rn, 162 -shell Om, 185 metric, 322ff minimal coupling, 326 Mobius transformation, 140
Index momentum, 4 multiscale analysis, 58, 64 natural units, 5 naturality, 65 non-relativistic limit, 225ff number operators N* , 281 , 295 orientation, 2 13 oversampling, 41 Pauli exclusion principle, 293 phase space, 36, 156 b, 202, 207ff, 279ff Planck’s Ansatz, 4 Poincarh group P,157ff polarization (of frequencies), 301 position operators, 162, 198 probability -current, 207ff, 215, 219 -density, relativistic, 206 quantization, 162-163 Radon transform, 342 reconstruction, 38, 40, 82ff, 33Gff complex, 87ff regularized current, 285, 298 representation of group, 33, 96, 124 of vector space, 8 project,ive, 112 Schrodinger, 9, 105 square-integrable, 49, 112 reproducing kernel, 22, 29ff, 49, 193, 287, 298 resolution of unity, 8, 15, 18ff, 23, 24, 37, 49, 205ff root, 118 -subspace, 118 -vector, 118 sampling -rate, 50 time-frequency, 40
Index
359
Schrodinger equation, 169 section (of bundle), 109 shift operator S,60 signal, 34 state space C, 159 statistics, 293 stereographic projection, 145 symplect ic form, 135,208ff geometry, 155 temper vector, 197, 272 tube 7, 190,249 two-point functions Ah, 193,268,287 uncertainty principle, 5, 10, 163 vector bundle, 328 vector potential A,, 326 wave number vector, 4
wavelet, 44 mother (basic) $, 44,58,75,79 -transform, 44ff,334ff weight, 126 Weyl-Heisenberg group W ,38, 104, 159ff Wick ordering, 282ff Wightman axioms,252ff window, 35 relativistic, 58, 178,334 X-ray transform, 245, 341 windowed, 334ff Yang-Mills field, 331-332 -potential, 331 -theory, 327 Zitterbewegung , 301 zoom operators H, H ' , 66,84
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