Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis 0444884653, 9780444884657, 9780080872742

Table of contents :
Content:
Edited by
Page iii

Copyright page
Page iv

Dadication page
Page v

Unified Field Theory
Page vii

Preface
Pages xi-xv

Suggestions to the Reader
Page xvi

Chapter 1 Coherent-State Representations
Pages 1-56

Chapter 2 Wavelet Algebras and Complex Structures
Pages 57-94

Chapter 3 Frames and Lie Groups
Pages 95-154

Chapter 4 Complex Spacetime
Pages 155-234

Chapter 5 Quantized Fields
Pages 235-319

Chapter 6 Further Developments
Pages 321-356

Index
Pages 357-359

Citation preview

QUANTUM PHYSICS, RELATIVITY, AND COMPLEX SPACETIME Towards a New Synthesis

NORTH-HOLLAND MATHEMATICS STUDIES 163 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.

NORTH-HOLLAND - AMSTERDAM

NEW YORK

OXFORD

TOKYO

QUANTUM PHYSICS, RELATIVITY, AND COMPLEX SPACETIME Towards a New Synthesis

Gerald KAISER Department of Mathematics University of Lowell Lowell, MA, USA

1990

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas NewYork, N.Y. 10010, U.S.A.

Library o f Congress Cataloging-in-Publlcation Data

Kaiser, Gerald. Ouantun physics. relativity. and complex spacetine : towards a new SynthEbiS / Gerald Kaiser. p. cn. -- (North-Holland matheiatlcs studies ; 163) Includes b l b l i O Q r ~ p h i C ~references 1 and index. ISBN 0-444-88465-3 1. Quantum theory. 2. Relativity (Physics) 3. S p a c e and tine. 4. Mathematical physics. I. Title. 11. Serios. QC174.12.K34 1990 530.1'2--dc20 90-7979

CIP

ISBN: 0 444 88465 3 Q ELSEVIER SCIENCE PUBLISHERS B.V., 1990

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. / Physical Sciences and Engineering Division, P.O. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the USA., should be referred to the publisher. NO responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in The Netherlands

To my parents, Bernard and Cesia and to Janusz, Krystyn, Mirek and Renia

This Page Intentionally Left Blank

Vii

UNIFIED FIELD THEORY In the beginning there was Aristotle, And objects at rest tended to remain at rest, And objects in motion tended to come to rest, And soon everything was at rest, And God saw that it was boring. Then God created Newton, And objects at rest tended to remain at rest, But objects in motion tended to remain in motion, And energy was conserved and momentum was conserved and matter was conserved, And God saw that it was conservative. Then God created Einstein, And everything was relative, And fast things became short, And straight things became curved, And the universe was filled with inertial frames, And God saw that it was relatively general, but some of it was especially relative. Then God created Bohr, And there was the principle, And the principle was quantum, And all things were quantized, But some things were still relative, And God saw that it was confusing. Then God was going to create Fergeson, And Fergeson would have unified, And he would have fielded a theory, And all would have been one, But it was the seventh day, And God rested, And objects at rest tend to remain at rest. by Tim Joseph copyright 01978 by The New York Times Company Reprinted by permission.

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ix

CONTENTS

Preface ........................................................

.xi

Suggestions to the Reader ..................................

xvi

.

Chapter 1 Coherent-State Representations 1.1. Preliminaries ............................................ 1 1.2. Canonical coherent states ................................ 9 1.3. Generalized frames and resolutions of unity ............. 18 1.4.Reproducing-kernel Hilbert spaces ...................... 29 34 1.5. Windowed Fourier transforms ........................... 1.6. Wavelet transforms ..................................... 43

.

Chapter 2 Wavelet Algebras and Complex Structures 2.1. Introduction ............................................ 57 2.2. Operational calculus .................................... 59 2.3. Complex structure ...................................... 70 2.4. Complex decomposition and reconstruction ............. 82 2.5. Appendix .............................................. 92

.

Chapter 3 Frames and Lie Groups 3.1. Introduction ............................................ 95 3.2. Klauder’s group-frames .................................95 3.3. Perelomov’s homogeneous G-frames ................... 103 3.4. Onofri’s’s holomorphic G-frames .......................113 3.5. The rotation group .................................... 135 3.6. The harmonic oscillator as a contraction limit ..........145

Contents

X

.

Chapter 4 Complex Spacetime 4.1. Introduction ........................................... 155 4.2. Relativity, phase space and quantization ............... 156 4.3. Galilean frames ........................................ 169 183 4.4. Relativistic frames ..................................... 4.5. Geometry and Probability ............................. 207 4.6.The non-relativistic limit .............................. 225 Notes ...................................................... 230

.

Chapter 5 Quantized Fields 5.1. Introduction ...........................................

235 5.2. The multivariate Analytic-Signal transform ............239 5.3. Axiomatic field theory and particle phase spaces .......249 273 5.4. Free Klein-Gordon fields .............................. 5.5. Free Dirac fields ....................................... 289 5.6. Interpolating particle coherent states ..................302 5.7. Field coherent states and functional integrals .......... 308 Notes ...................................................... 318

.

Chapter 6 Further Developments 6.1. Holomorphic gauge theory .............................

321 6.2. Windowed X-Ray transforms: Wavelets revisited ...... 334

..................................................

347

........................................................

357

References Index

xi

PREFACE The idea of complex spacetime as a unification of spacetime and classical phase space, suitable as a possible geometric basis for the synthesis of Relativity and quantum theory, first occured to me in 1966 while I was a physics graduate student at the University of Wisconsin. In 1971, during a seminar I gave at Carleton University in Canada, it was pointed out to me that the formalism I was developing was related to the coherent-state representation, which was then unknown to me. This turned out to be a fortunate circumstance, since many of the subsequent developments have been inspired by ideas related to coherent states. My main interest at that time was to formulate relativistic coherent states. In 1974, I was struck by the appearance of tube domains in axiomatic quantum field theory. These domains result from the analytic continuation of certain functions (vacuum expectaion values) associated with the theory to complex spacetime, and powerful methods from the theory of several complex variables are then used to prove important properties of these functions in real spacetime. However, the complexified spacetime itself is usually not regarded as having any physical significance. What intrigued me was the possibility that these tube domains may, in fact, have a direct physical interpretation as (extended) classical phase spaces. If so, this would give the idea of complex spacetime a firm physical foundation, since in quantum field theory the complexification is based on solid physical principles. It could also show the way to the construction of relativistic coherent states. These ideas were successfully worked out in 1975-76,culminating in a mathematics thesis in 1977 at the University of Toronto entitled “Phasespace Approach to Relativistic Quantum Mechanics.”

Xii

Preface

Up to that point, the theory could only describe free particles. The next goal was to see how interactions could be added. Some progress in this direction was made in 1979-80, when a natural way was found to extend gauge theory to complex spacetime. Further progress came during my sabbatical in 1985-86, when a method was developed for extending quantized fields themselves (rather than their vacuum expectation values) to complex spacetime. These ideas have so far produced no “hard” results, but I believe that they are on the right path. Although much work remains to be done, it seems to me that enough structure is now in place to justify writing a book. I hope that this volume will be of interest to researchers in theoretical and mathematical physics, mathematicians interested in the structure of fundamental physical theories and assorted graduate students searching for new directions. Although the topics are fairly advanced, much effort has gone into making the book self-contained and the subject matter accessible to someone with an understanding of the rudiments of quantum mechanics and functional analysis.

A novel feature of this book, from the point of view of mathematical physics, is the special attention given to “ signal analysis” concepts, especially time-frequency localization and the new idea of wavelets. It turns out that relativistic coherent states are similar to wavelets, since they undergo a Lorentz contraction in the direction of motion. I have learned that engineers struggle with many of the same problems as physicists, and that the interplay between ideas from quantum mechanics and signal analysis can be very helpful to both camps. For that reason, this book may also be of interest to engineers and engineering students. The contents of the book are as follows. In chapter 1 the simplest

Preface

Xiii

examples of coherent states and time-frequency localization are introduced, including the original “canonical” coherent states, windowed Fourier transforms and wavelet transforms. A generalized notion of frames is defined which includes the usual (discrete) one as well as continuous resolutions of unity, and the related concept of a reproducing kernel is discussed. In chapter 2 a new, algebraic approach to orthonormal bases of wavelets is formulated. An operational calculus is developed which simplifierthe formalism considerably and provides insights into its symmetries. This is used to find a complex structure which explains the symmetry between the low- and the high-frequency filters in wavelet theory. In the usual formulation, this symmetry is clearly evident but appears to be accidental. Using this structure, complex wavelet decompositions are considered which are analogous to analytic coherent-state representations. In chapter 3 the concept of generalized coherent states based on Lie groups and their homogeneous spaces is reviewed. Considerable attention is given to holomorphic (analytic) coherent-state representations, which result from the possibility of Lie group complexification. The rotation group provides a simple yet non-trivial proving ground for these ideas, and the resulting construction is known as the “spin coherent states.” It is then shown that the group associated with the Harmonic oscillator is a weak contraction limit (as the spin s 4 00) of the rotation group and, correspondingly, the canonical coherent states are limits of the spin coherent states. This explains why the canonical coherent states transform naturally under the dynamics generated by the harmonic oscillator. In chapter 4,the interactions between phase space, quantum mechanics and Relativity are studied. The main ideas of the phasespace approach to relativistic quantum mechanics are developed for

xiv

Preface

free particles, based on the relativistic coherent-state representations developed in my thesis. It is shown that such representations admit a covariant probabilistic interpretation, a feature absent in the usual spacetime theories. In the non-relativistic limit, the representations are seen to “contract” smoothly to representations of the Galilean group which are closely related to the canonical coherent-state representation. The Gaussian weight functions in the latter are seen to emerge from the geometry of the mass hyperboloid. In chapter 5 , the formalism is extended to quantized fields. The basic tool for this is the Analytic-Signal transform,which can be applied to an arbitrary function on R”to give a function on a!” which, although not in general analytic, is “analyticity-friendly” in a certain sense. It is shown that even the most general fields satisfying the Wightman axioms generate a complexification of spacetime which may be interpreted as an extended classical phase space for certain special states associated with the theory. Coherent-st ate represent ations are developed for free Klein-Gordon and Dirac fields, extending the results of chapter 4. The analytic Wightman two-point functions play the role of reproducing kernels. Complex-spacetime densities of observables such as the energy, momentum, angular momentum and charge current are seen to be regularizations of their counterparts in real spacetime. In particular, Dirac particles do not undergo their usual Zitterbewegung. The extension to complex spacetime separates, or polarizes, the positive- and negativefrequency parts of free fields, so that Wick ordering becomes unnecessary. A functionalintegral represent ation is developed for quantized fields which combines the coherent-state representations for particles (based on a finite number of degrees of freedom) with that for fields (based on an infinite number of degrees of freedom). In chapter 6 we give a brief account of some ongoing work, begin-

Preface

xv

ning with a review of the idea of holomorphic gauge theory. Whereas in real spacetime it is not possible to derive gauge potentials and gauge fields from a (fiber) metric, we show how this can be done in complex spacetime. Consequently, the analogy between General Relativity and gauge theory becomes much closer in complex spacetime than it is in real spacetime. In the “holomorphic” gauge class, the relation between the (non-abelian) Yang-Mills field and its potential becomes linear due to the cancellation of the non-linear part which follows from an integrability condition. Finally, we come full circle by generalizing the Analytic-Signal transform and pointing out that this generalization is a higher-dimensional version of the wavelet transform which is, moreover, closely related to various classical transforms such as the Hilbert, Fourier-Laplace and Radon transforms.

I am deeply grateful to G. Emch for his continued help and encouragement over the past ten years, and to J. R. Klauder and R. F. Streater for having read the manuscript carefully and made many invaluable comments, suggestions and corrections. (Any remaining errors are, of course, entirely my responsibility.) I also thank D. Buchholtz, F. Doria, D. Finch, S. Helgason, I. Kupka, Y. Makovoz, J. E. Marsden, M. O’Carroll, L. Rosen, M. B. Ruskai and R. Schor for miscellaneous important assistance and moral support at various times. Finally, I am indebted to L. Nachbin, who first invited me to write this volume in 1981 (when I was not prepared to do so) and again in 1985 (when I was), and who arranged for a tremendously interesting visit to Brazil in 1982. Quero tarnbkm agradecer a todos 0s meus colegas Brasileiros!

xvi

Suggestions t o the Reader The reader primarily interested in the phasespace approach to relativistic quantum theory may on first reading skip chapters 1-3 and read only chapters 4-6, or even just chapter 4 and either chapters 5 or 6, depending on interest. These chapters form a reasonably self-contained part of the book. Terms defined in the previous chapters, such as “frame,” can be either ignored or looked up using the extensive index. The index also serves partially as a glossary of frequently used symbols. The reader primarily interested in signal analysis, timefrequency localization and wavelets, on the other hand, may read chapters 1 and 2 and skip directly to sections 5.2 and 6.2. The mathematical reader unfamiliar with the ideas of quantum mechanics is urged to begin by reading section 1.1, where some basic notions are developed, including the Dirac notation used throughout the book.

1

Chapter 1 COHERENT-STATE REPRESENTATIONS

1.1. Preliminaries In this section we establish some notation and conventions which will be followed in the rest of the book. We also give a little background on the main concepts and formalism of non-relativistic and relativistic quantum mechanics, which should make this book accessible to nonspecialists. 1. Spacetirne and its Dual

In this book we deal almost exclusively with flat spacetime, though we usually let space be R"instead of R3,so that spacetime becomes X =

IRs+l. The reason for this extension is, first of all, that it involves little cost since most of the ideas to be explored here readily generalize to IRS+l, and furthermore, that it may be useful later. Many models in constructive quantum field theory are based on two- or threedimensional spacetime, and many currently popular attempts to unify physics, such as string theories and Kaluza-Klein theories, involve spacetimes of higher dimensionality than four or (on the string worldsheet) two-dimensional spacetimes. An event x € X has coordinates 2

= ( 2 P ) = (xO,xj),

1 . Coherent-St ate Represent ations

2

where x o

3

t is the time coordinate and x j are the space coordinates.

Greek indices run from 0 to s, while latin indices run from 1 to s. If we think of x as a translation vector, then X is the vector space of all translations in spacetime. Its dual

X* is the set of all linear maps

k : X + IR. By linearity, the action of k on x (which we denote by k x instead of k ( x ) ) can be written as 5

kx =

C k,xp

k,xp,

(2)

,=o

where we adopt the Einstein summation convention of automatically summing over repeated indices. Usually there is no relation between x and k other than the pairing (x,k ) H kx. But suppose we are given a scalar product on X , 5 . 2I

=g,,x

CL x 'V

(3)

where (g,,) is a non-degenerate matrix. Then each x in X defines a linear map x * : X + *: X

---f

IR by

x*(x') = x

. x',

thus giving a map

X * , with ( x * ) , =- 5 , = g , , x Y

(4)

Since g,, is non-degenerate, it also defines a scalar product on X * , '

whose metric tensor is denoted by g"". The map x x* establishes an isomorphism between the two spaces, which we use to identify them. If 2 denotes a set of inertial coordinates in free spacetime, then the scalar product is given by

g,, = diag(c2,-1, -1,

- - - ,-1)

where c is the speed of light. X , together with this scalar product, is called Minkowskian or Lorentzian spacetime.

1.1. Preliminaxies

3

It is often convenient to work in a single space rather than the dual pair X and X * . Boldface letters will denote the spatial parts of vectors in X * . Thus x = ( t , -x), k = (ko, k ) and

where x . x' and k - x denote the usual Euclidean inner products in

IR". 2. Fourier ?f-ansforms The Fourier transform of a function f : X + (E (which, to avoid analytical subtleties for the present, may be assumed to be a Schwartz test function; see Yosida [1971])is a function f:X * + a given by

where dz 3 dt d"x is Lebesgue measure on X . f can be reconstructed from by the inverse Fourier transform, denoted by " and given by

3

f ( ~ =)

d k e-2xikzf ( k ) = (P)"(x),

(7)

where d k = dk0d"k denotes Lebesgue measure on X * M X . Note that the presence of the 27r factor in the exponent avoids the usual ~ (27r)-'-' in front of the integrals. need for factors of ( 2 ~ ) - ( " + ' ) /or Physically, k represents a wave vector: ko v is a frequency in cycles per unit time, and kj is a wave number in cycles per unit length. Then the interpretation of the linear map k: X + 1R is that 27rkx is the total radian phase gained by the plane wave g(x') = exp(-27rikxt) through the spacetime translation x , i.e. 27rk "measures" the radian phase shift. Now in pre-quantum relativity, it was realized

4

I. Coherent-St ate Representations

that the energy E combines with the momentum p to form a vector

p ( p , ) = (E,p) in X * . Perhaps the single most fundamental difference between classical mechanics and quantum mechanics is that in the former, matter is conceived to be made of “dead sets” moving in space while in the latter, its microscopic structure is that of waves descibed by complex-valued wave functions which, roughly speaking, represent its distribution in space in probabilistic terms. One important consequence of this difference is that while in classical mechanics one is free to specify position and momentum independently, in quantum mechanics a complete knowledge of the distribution in space, i.e. the wave function, determines the distribution in momentum space via the Fourier transform. The classical energy is reinterpreted as the frequency of the associated wave by Planck’s Ansatz,

where tL is Planck’s constant, and the classical momentum is reinterpreted as the wave-number vector of the associated wave by De Broglie’s relation,

These two relations are unified in relativistic terms as p , = 27rlik,. Since a general wave function is a superposition of plane waves, each with its own frequency and wave number, the relation of energy and momentum to the the spacetime structure is very different in quantum mechanics from what is was in classical mechanics: They become operators on the space of wave functions:

or, in terms of x*,

5

1.1. Preliminaries

This is, of course, the source of the uncertainty principle. In terms of energy-moment um, we obtain the “quantum-mechanical” Fourier transform and its inverse,

If f(s) satisfies a differential equation, such as t,,e Schrodinger equation or the Klein-Gordon equation, then f ( p ) is supported on an s-dimensional submanifold P of X * (a paraboloid or two-sheeted hyperboloid, respectively) which can be parametrized by p E

R”.

We will write the solution as

where f ( p ) is, by a mild abuse of notation, the “restriction” of

P (actually, l f ( p > l 2 is a density on P ) and d p ( p )

f

to

p ( p ) d ” p is an appropriate invariant measure on P. For the Schrodinger equation p(p) = 1, whereas for the Klein-Gordon equation, p ( p ) = lpo I Setting t = 0 then shows that f(p) is related to the initial wave function by G

-’.

where now

“-”

denotes the the s-dimensional inverse Fourier trans-

form of the function

f

on

P

fi?

IR”.

We will usually work with “natural units,” i.e. physical units so chosen that h = c = 1. However, when considering the non-

6

1. Coherent-State Representations

relativistic limit ( c + 00) or the classical limit (ti + 0), c or ti will be reinserted into the equations.

3. Hilbert Space Inner products in Hilbert space will be linear in the second factor and antilinear in the first factor. Furthermore, we will make some discrete use of Dirac’s very elegant and concise bra-ket notation, favored by physicists and often detested or misunderstood by mathematicians. As this book is aimed at a mixed audience, I will now take a few paragraphs to review this notation and, hopefully, convince mathematicians of its correctness and value. When applied to coherent-state representat ions, as opposed to representat ions in which the positionor momentum operators are diagonal, it is perfectly rigorous. (The bra-ket notation is problematic when dealing with distributions, such as the generalized eigenvectors of position or momentum, since it tries to take the “inner products” of such distributions.) Let ‘FI be an arbitrary complex Hilbert space with inner product (-,-). Each element f E 7-t defines a bounded linear functional

f * : ? t + (Jby

The Riesz represent at ion theorem guarantees that the converse is also true: Each bounded linear functional L : 3-1 + for a unique

f

E 3-1. Define the bra

(fl

(fl

a has the form L = f *

corresponding to f by

= f* : 3-1 +

a.

(14)

Similarly, there is a one-to-one correspondence between vectors g E ‘H and linear maps

7

1.1. Preliminaries

1s):

c + ‘H

defined by

Is)(X) = Xg,

E

a,

(16)

which will be called kets. Thus elements of ‘H will be denoted alternatively by g or by 1s). We may now consider the composite map bra-ke t

given by

(f I 9 ) P ) = f*(W = Xf*(s> = X(f, 9).

(18)

Therefore the “bra-ket” map is simply the multiplication by the inner product (f, g) (whence it derives its name). Henceforth we will identify these two and write (flg) for both the map and the inner product. The reverse composition

Is)(fl : ‘H 3-1 +

may be viewed as acting on kets to produce kets: IS)(fl(lW = lg)((fIN)*

(20)

To illustrate the utility of this notation, as well as some of its pitfalls, suppose that we have an orthonormal basis {gn} in H. Then the usual expansion of an arbitrary vector f in H takes the form

n

n

8

1. Coherent-State Representations

from which we have the “resolution of unity”

n

where I is the identity on ‘FI and the sum converges in the strong operator topology. If { h,} is a second orthonormal basis, the relation between the expansion coefficients in the two bases is

In physics, vectors such as gn are often written as I n ) , which can be a source of great confusion for mathematicians. Furthermore,

If),

with functions in L2(IRB), say, are often written as f ( x ) = ( x ( x I 2’ ) = 6 ( x - 2 ’ ) ) as though the I x ))s formed an orthonormal basis. This notation is very tempting; for example, the Fourier transform is written as a “change of basis,”

with the “transformation matrix” ( k I x ) = exp(2~ikx). One of the advantages of this notation is that it permits one to think of the Hilbert space as “abstract,” with ( gn

If),

( h,

If

), ( x 1 f ) and ( k I f )

merely different “representat ions” (or “realizations”) of the same vector f . However, even with the help of distribution theory, this use of Dirac notation is unsound, since it attempts to extend the Riesz representation theorem to distributions by allowing inner products of them. (The “vector” ( x I is a distribution which evaluates test func-

tions at the point z; as such, I x‘ ) does not exist within modern-day distribution theory.) We will generally abstain from this use of the bra-ket notation.

1.2. Canonical Coherent States

9

Finally, it should be noted that the term “representation” is used in two distinct ways: (a) In the above sense, where abstract Hilbertspace vectors are represented by functions in various function spaces, and (b) in connection with groups, where the action of a group on a Hilbert space is represented by operators. This notation will be especially useful when discussing frames, of which coherent-state representations are examples.

1.2. Canonical Coherent States

We begin by recalling the original coherent-state representations (Bargmann [1961],Klauder [1960,1963a, b], Segal [1963a]). Consider a spinless non-relativistic particle in IR.” (or 9/3 such particles in IR.~), whose algebra of observables is generated by the position operators XI, and momentum operators Pk,k = 1,2,. . .s. These satisfy the “canonical commutation relations”

where I is the identity operator. The operators -iXk, -iPk and -iI together form a r e d Lie algebra known aa the Heisenberg algebra, which is irreducibly represented on L2(R8)by

the Schrodinger representation. As a consequence of the above commutation relations between xk and Pk, the position and momentum of the particle obey the

10

1. Coherent-St ate Represent ations

Heisenberg uncertainty relations, which can be derived simply as follows. The expected value, upon measurement, of an observable represented by an operator F in the state represented by a wave function f ( x ) with llfll = 1 (where 11 11 denotes the norm in L2(R"))is given

-

by

In particular, the expected position- and momentum coordinates of the particle are ( Xk ) and ( pk ). The uncertainties Ax, and Apk in position and momentum are given by the variances

Choose an arbitrary constant b with units of area (square length) and consider the operators

Notice that although Ak is non-Hermitian, it is real in the Schrodinger representation. Let

where Zk denotes the complex-conjugate of

Zk.

Then for 6Ak =-

Ak - Z k I we have ( 6 & ) = 0 and

The right-hand side is a quadratic in b, hence the inequality for all b demands that the discriminant be non-positive, giving the uncertainty relations

1.2. Canonical Coherent States

AX$AP$L

11

1

-2 '

Equality is attained if and only if 6Akf = 0, which shows that the only minimum-uncertainty states are given by wave functions f(z) satisfying the eigenvalue equations

for some real number b (which may actually depend on k) and some z E a'. But square-integrable solutions exist only for b > 0, and then there is a unique solution (up to normalization) xz for each z E as. To simplify the notation, we now choose b = 1. Then Ak and A*, satisfy the commutation relations

and

xz is given by

where the normalization constant is chosen ELS N = 7rr-s/4, so that llxzll = 1for z = 0. Here f 2 is the (complex) inner product of Z with itself. Clearly xz is in L2(IRd),and if z = z - ip, then

in the state given by

xz. The vectors xz are known as the canon-

ical coherent states. They occur naturally in connection with the

harmonic oscillator problem, whose Hamiltonian can be cast in the form 1 1 sw H = -( P 2+ m 2 u 2 X 2 )= - w 2 A * - A + 2m 2 2

(13)

12

1, Coherent4t a t e Representations

with

(thus b = l / r n w ) . They have the remarkable property that if the initial state is xt,then the state at time t is x t ( q where ~ ( tis)the orbit in phase space of the corresponding classical harmonic oscillator with initial data given by z. These states were discovered by Schrodinger himself [1926], at the dawn of modern quantum mechanics. They were further investigated by Fock [1928] in connection with quantum field theory and by von Neumann [1931]in connection with the quantum measurement problem. Although they span the Hilbert space, they do not form a basis because they possess a high degree of linear dependence, and it is not easy to find complete, linearly independent subsets. For this reason, perhaps, no one seemed to know quite what to do with them until the early 1960’9, when it was discovered that what really mattered was not that they form a basis but what we shall call a generalized frame. This allows them to be used in generating a representation of the Hilbert space by a space of analytic functions, as explained below. The frame property of the coherent states (which will be studied and generalized in the following sections and in chapter 3) was discovered independently at about the same time by Klauder, Bargmann and Segal. Glauber [1963a,b] used these vectors with great effectivenessto extend the concept of optical coherence to the domain of quantum electrodynamics, which was made necessary by the discovery of the laser. He dubbed them “coherent states,” and the name stuck to the point of being generic. (See also Klauder and Sudarshan [1968].) Systems of vectors now called “coherent” may have nothing to do with optical coherence, but there is at least one unifying characteristic, namely their frame property

1.2. Canonical Coherent States

13

(next section). The coherent-state representation is now defined as follows: Let F be the space of all functions

J

=N

J

d'x'

exp[-r2/2

+ z - x' - ~ ' ~ / 2 ] f ( ~ ' )

(15)

where f runs through L2(IR'). Because the exponential decays rapidly in x', f" is entire in the variable I E C'. Define an inner product on F by

where

I

x - ip and

dp(z) = (27r)-' exp( --E

- 2/2) d'x

d'p.

(17)

Then we have the following theorem relating the inner products in L2(IR') and F. T h e o r e m 1.1. Let f,g E L2(IRa) and let entire functions in F. Then

f", ij

be the corresponding

Proof. To begin with, assume that f is in the Schwartz space S(IR') of rapidly decreasing smooth test functions. For z = x - ip, we have

xl(x') = N exp[-E2/2

+ x2/2 - (5' - ~

+

) ~ / i2p * 4,

14

1. Coherent-State Representations

hence

where denotes the Fourier transform with respect to x'. Thus by Plancherel's theorem (Yosida [1971]),

(27r)-'

/

d'p exp(-p2/2)lf"(z - ip)12 = N 2 exp(x2/2)

J

d'z' exp[-(z' - x ) ~lf(x')l2. ] (20)

Therefore

=N2

/ 1 d'z

)'.(fI

d a d exp[-(z' - x ) ~ ]

1'

(21)

after exchanging the order of integration. This proves that

11311:

= Ilf

IIL

(22)

for f E S(IR"),hence by continuity also for arbitrary f E L2(IRb). By polarization the result can now be extended from the norms to the inner products. I The relation f H f' can be summarized neatly and economically in terms of Dirac's bra-ket notation. Since

1.2. Canonicd Coherent States

15

theorem 1 can be restated as

Dropping the bra

(fl

and ket

Is), we have the operator identity

where I is the identity operator on L2(IRB) and the integral converges at least in the sense of the weak operator topology,* i.e. as a quadratic form. In Klauder’s terminology, this is a continuous resolution of unity. A general operator B on L2(RB) can now be expressed as an integral operator B on F as follows:

Particularly simple representations are obtained for the basic position- and momentum’ operators. We get

*

As will be shown in a more general context in the next sec-

tion, under favorable conditions the integral actually converges in the strong operator topology.

1. Coherent-State Representations

16

thus

Hence xk and Pk can be represented as differential rather than integral operat ors. As promised, the continuous resolution of the identity makes it possible to reconstruct f E L2(IR")fr lm its transform f" E E

that is,

"

Thus in many respects the coherent states behave like a basis for

L2(IR").But they differ from a basis i s at least one important

respect: They cannot all be linearly independent, since there are uncountably many of them and L2(IR")(and hence also F)is separable. In particular, the above reconstruction formula can be used to express x I in terms of all the xw's:

1.2. Canonical Coherent States

17

In fact, since entire functions are determined by their values on some discrete subsets I? of Cs,we conclude that the corresponding subsets of coherent states {xZ I z E I?} are already complete since for any function f orthogonal to them all, f(z) = 0 for all z E r and hence f = 0, which implies f = 0 a.e. For example, if I’is a regular lattice, a necessary and sufficient condition for completesness is that r contain at least one point in each Planck cell (Bargmann et al., [1971]), in the sense that the spacings Azk and Apk of the lattice coordinates zk = X k ipk satisfy AxkApk _< 27rh 21r. It is no accident that this looks like the uncertainty principle but with the inequality going “the wrong way.” The exact coefficient of h is somewhat arbitrary and depends on one’s definition of uncertainty; it is possible to define measures of uncertainty other than the standard deviation. (In fact, a preferable-but less tractabledefinition of uncertainty uses the notion of entropy, which involves all moments rather than just the second moment. See Bialynicki-Birula and Mycielski [1975] and Zakai [1960].) The intuitive explanation is that if f gets “sampled” at least once in every Planck cell, then it is uniquely determined since the uncertainty principle limits the amount of variation which can take place within such a cell. Hence the set of all coherent states is overcomplete. We will see later that reconstruction formulas exist for some discrete subsystems of coherent states, which makes them as useful as the continuum of such states. This ability to synthesize continuous and discrete methods in a single representation, as well as to bridge quantum and classical concepts, is one more aspect of the a.ppea1 and mystery of these systems. I

+

18

1. Coherent-State Representations

1.3. Generalized Frames and Resolutions of Unity Let M be a set and p be a measure on M (with an appropriate aalgebra of measurable subsets) such that { M ,p } is a a-finite measure space. Let 3-1 be a Hilbert space and hm E 3-1 be a family of vectors indexed by m E M .

Definition.

The set

is a generalized frame in 3-1 if 1. the map h: m H hm is weakly measurable, i.e. for each f E 3.1 the function f ( m ) ( hm I f ) is measurable, and

2. there exist constants 0 < A I B such that r

3 - 1 ~is a frame (see Young [1980] and Daubechies [198Sa]) in the special case when M is countable and p is the counting measure on M (i.e., it assigns to each subset of M the number of elements

contained in it). In that case, the above condition becomes

C

A I I ~ I I ~ S I(hmIf)12 ( m (>hm

If )

.f(m),

Then the frame condition states that

f E 3.t.

(4)

Tf is square-integrable with

respect to dp, so that T defines a map

T : ‘H + L 2 ( d p ) ,

(5)

witb

The frame property can now be stated in operator form as

AI 5 T*T 5 B I ,

(7)

where I is the identity on ‘H. In bra-ket notation,

where the integral is to be interpreted, initially, as converging in the weak operator topology, i.e. as a quadratic form. For a measurable subset N of M , write

Proposition 1.2. I f the integral G ( N ) converges in the strong operator topology of ‘H whenever N has finite measure, then so does the complete integral representing G = T*T.

20

1. Coherent-State Representations

Proof*. Since M is a-finite, we can choose an increasing sequence { M n } of sets of finite measure such that M = U,Mn. Then the corresponding sequence of integrals G , forms a bounded (by G) increasing sequence of Hermitian operators, hence converges to G in the strong operator topology (see Halmos [1967], problem 94). I If the frame is tight, then G = A1 and the above gives a resolution of unity after dividing by A. For non-tight frames, one generally has to do some work to obtain a resolution of unity. The frame condition means that G has a bounded inverse, with

Given a function g(m) in L 2 ( d p ) ,we are interested in answering the following two questions: (a) Is g = Tf for some f E 'H? (b) If so, then what is f ? In other words, we want to:

c L 2 ( d p ) of the map T. (b) Find a left inverse S of T,which enables us to reconstruct f from (a) Find the range %T

Tf by f = S T f . Both questions will be answered if we can explicitly compute

G-l. For let

P

= TG-I T*:L2(dp) -+

L2(dp).

Then it is easy to see that

*

I thank M. B. Ruskai for suggesting this proof.

(11)

1.3. Generalized fiames and Resolutions of Unity

21

( a ) P* = P

( b ) P2 = P (c)

PT

= T.

It follows that P is the orthogonal projection onto the range of T,

Tf = g, and conversely if for some g we have Pg = g , then g = T(G-'T*g) z Tf.

for if g =

Tf for some f in 3.t, then Pg = P T f

=

Thus 8~is a closed subspace of L2(dp) and a function g E L 2 ( d p ) is in 8~if and only if

The function

therefore has a property similar to the Dirac &-functionwith respect to the measure dp, in that it reproduces functions in %T. But it differs from the S-function in some important respects. For one thing, it is bounded by

1. Coherent-St ate Representations

22

IK(m,m')l = l ( h m l G - l h ~ ) l

5 IIG-' II IIhmII IIhml II I A-'lIhml\ Ilhm)II < 00

(16)

for all m and m'. Furthermore, the "test functions" which K ( m ,m') reproduces form a Hilbert space and K(m,m') defines an integral operator, not merely a distribution, on %T. In the applications to relativistic quantum theory to be developed later, M will be a complexification of spacetime and K(m,m') will be holomorphic in m and antiholomorphic in m'. The

Hilbert space %T and the associated function K ( m ,m') are an example of an important structure called a reproducing-kernel Hilbert space (see Meschkowski [1962]), which is reviewed briefly in the next section. K(rn,m') is called a reproducing kernel for SRT. We can thus summarize our answer to the first question by saying that a function g E L2(dp) belongs to the range of T if and only if it satisfies the consistency condition

Of course, this condition is only useful to the extent that we have information about the kernel K(m,m') or, equivalently, about the operator G-l. The answer to our second question also depends on the knowledge of G-l. For once we know that g = Tf for some

f

E 3.1, then

Thus the operator

1.3. Generalized Fkaxnes and Resolutions of Unity

S = G-lT*: L2(dp)+ 'FI

23

(19)

is a left inverse of T and we can reconstruct f by

f = Sg = G-'T*Tf

This gives f as a linear combination

Note that

d p ( m ) I h" ) ( hm I = G-lGG-'

= G-I,

/M

therefore the set

3-IM

3

(h" Im E M )

(23)

is also a frame, with frame constants 0 < B-' 5 A-I. We will call ' H M the frame reciprocal to 3 - 1 ~ . (In Daubechies [ISSSa], the corresponding discrete object is called the dual frame, but as we shall see below, it is actually a generalization of the concept of reciprocal basis; since the term "dual basis" has an entirely different meaning, we prefer "reciprocal frame" to avoid confusion.) The above reconstruction formula is equivalent to the resolutions of unity in terms of the pair ' H M , E M of reciprocal frames:

24

1. Coherent-State Representations

Corollary 1.3. Under the assumptions of proposition 1.2, the above resolutions of unity converge in the strong operator topology of ‘FI. The proof is similar to that of proposition 1.2 and will not be given. The strong convergence of the resolutions of unity is important, since it means that the reconstruction formula is valid within

3-1 rather

than just weakly. Application to f = hk for a fixed k E M gives

which shows that the frame vectors hm are in general not linearly independent. The consistency condition can be understood as requiring the proposed function g(m) to respect the linear dependence of the frame vectors. In the special case when the frame vectors are linearly independent, the frames 7 - l and ~ ‘FIM both reduce to bases of

7-l. If 7-l is separable (which we assume it is), it follows that M must be countable, and without loss in generality we may assume that d p is the counting measure on M (renormalize the hm’s if necessary). Then the above relation becomes

mEM

and linear independence requires that K be the Kronecker 6: K ( m ,k ) = 6;. Thus when the hm’s are linearly independent, 3-1121 and ‘FIM reduce to a pair of reciprocal bases for ‘FI. The resolutions of unity become

GEM and we have the relation

1.3. Generalized fiames and Resolutions of Unity

25

(28) where gmk

I

(hm hk) =

(h" I G h k )

(29)

is an infinite-dimensional version of the metric tensor, which mediates between covariant and contravariant vectors. (The operator G plays the role of a metric operator.) In this case, %T = L 2 ( d p ) 1 2 ( M )and the consistency condition reduces to an identity. The reconstruction formula becomes the usual expression for f as a linear combination of the (reciprocal) basis vectors. If we further specialize to the case of a tight frame, then G = AI implies that

(h,(hk)=A6:

and

(hmJhk)=A-'6P,

(30)

so 'HM and 'HM become orthogonal bases. Requiring A = B = 1

means that ' H M = 'KM reduce to a single orthonormal basis. Returning to the general case, we may summarize our findings as follows: E M ,N M ,K(m,m') and g(m,m') (h" I Ghml) are generalizations of the concepts of basis, reciprocal basis, Kronecker delta and metric tensor to the infinite-dimensional case where, in addition, the requirement of linear independence is dropped. The point is that the all-important reconstruction formula, which allows us to express any vector as a linear combination of the frame vectors, survives under the additional (and obviously necessary) restriction that the consistency condition be obeyed. The useful concepts of orthogonal and orthonormal bases generalize to tight frames and frames with A = B = 1, respectively. We will call frames with A = B = 1normal. Thus normal frames are nothing but resolutions of unity.

=

26

1. Coherent-State Representations hturning to the general situation, we must still supply a way

of computing G - l , on which the entire construction above depends. In some of the examples to follow, G is actually a multiplication operator, so G-' is easy to compute. If no such easy way exist, the following procedure may be used. From AI 5 G 5 B I it follows that 8

1 1 - - ( B - A)I 5 G - - ( B 2 2

1 + A ) I 5 -(B - A)I. 2

Hence letting

6=-

B-A B+A

and

c=-

2 B+A'

we have

-61 5 I - CG 5 6I. Since 0

(33)

5 6 < 1 and c > 0, we can expand 00

-'= c C(I - c G ) ~

G-' = c [ I - ( I - cG)]

(34)

k=O

and the series converges uniformly since

( ( I- ~ G l 5 l 6 < 1. The smaller 6, the faster the convergence. For a tight frame,

(35)

S=0

and cG = I , so the series collapses to a single term G-' = c. Then the consistency condition becomes

and the reconstruction formula simplifies to

1.3. Generalized fiames and Resolutions of Unity

27

If 0 < S 1/2. The above condition on Av therefore implies that T > 1/2W. It would thus seem that we could get away with a slightly larger sampling interval T than the Nyquist interval TN = 1/2W. Our reconstruction formula reduces to

+

f(t)

X

g(t)-'

c

h(t - nT)f"(0, nT).

(24)

n

The smaller the ratio A v / W , the better this approximation is likely

1.6. Wavelet ?Eansforms

43

to be. But a small Av means a large r , hence the samples f(0, nT) are smeared over a large time interval.

1.G. Wavelet Transforms The frame vectors for windowed Fourier transforms were the wave packets

h&)

= e-2Ti”%(t

- 3).

(1)

The basic window function h ( t ) was assumed to vanish outside of the interval -T < t < 0 and to be reasonably smooth with no steep slopes, so that its Fourier transform k ( v ) was also centered in a small interval about the origin. Of course, since h ( t ) has compact support, f ( v )is the restriction to IR of an entire function and hence cannot vanish on any interval, much less be of compact support. The above statement simply means that A(.) decays rapidly outside of a small interval containing the origin. At any rate, the factor exp(-2rivt) amounted to a translation of the window in frequency, so that hV,# was a “window” in the time-frequency plane centered about ( v , ~ ) .

Hence the frequency components of f ( t ) were picked out by means of rigidly translating the basic window in both time and frequency. (It is for this reason that the windowed Fourier transform is associated to the Weyl-Heisenberg group, which is exactly the group of all translations in phase space amended with the multiplication by phase factors necessary to close the Lie algebra, as explained in chapter 3.) Consequently, hV,#has the same width r for all frequencies, and the number of wavelengths admitted for analysis is ur. For low frequencies with vr > 1, too many wavelengths are admitted. For such waves, a time-interval of duration r seems infinite, thus negating the sense of “locality” which the windowed Fourier transform was designed to achieve in the first place. This deficiency is remedied by the wavelet transform. The window h ( t ) , called the basic wavelet, is now scaled to accomodate waves of different frequencies. That is, for a # 0 let

The factor ]al-1/2 is included so that Ilha,,l12

=J

00

-00

dt

I ha,&) I 2

= llhl12.

(3)

The necessity of using negative as well as positive values of a will become clear as we go along. It will also turn out that h will need to satisfy a technical condition. Again, we think of both h and f as real but allow them to be complex. The wavelet transform is now defined by

Before proceeding any further, let us see how the wavelet transform localizes signals in the timefrequency plane. The localization in time is clear: If we assume that h ( t ) is concentrated near t = 0 (though it will no longer be convenient to assume that h has compact support), then f ( a , s) is a weighted average of f(t) around t = s (though the weight function need not be positive, and in general may even be complex). To analyze the frequency localization, we again

1.6. Wavelet ?f.ansforms

45

want to express f in terms of the Fourier transforms of h and f . This is possible because, like the windowed Fourier transform, the wavelet transform involves rigid time-translations of the window, resulting in a convolution-like expression. The ‘‘impulse response” is now (setting f ( t ) = b ( t ) ) ga(s) = lal-1/2h(-s/a),

(5)

and we have

with

J-00

Later we will see that discrete tight frames can be obtained with certain choices of h ( t ) whose Fourier transforms have compact support in a frequency interval interval a 5 v 5 @. Such functions (or, rather, the operations of convolutiong with them) are called bandpass filters in communication theory, since the only frequency components in f(t) to survive are those in the “band” [a,@].Then the above expression shows that f(a,s) depends only on the frequency component of f ( t ) in the band a / a 5 v 5 @ / a (if a > 0) or @ / a5 v 5 a / a (if a < 0). Thus frequency localization is achieved by dilations rather than translations in frequency space, in contrast to the windowed Fourier transform. At least from the point of view of audio signals, this actually seems preferable since it appears to be frequency ratios, rat her than frequency differences, which carry meaning. For example, going up an octave is achieved by doubling the frequency. (However,

46

1. Coherent-State Representations

frequency differences do play a role in connection with beats and also in certain non-linear phenomena such as difference tones; see Roederer [1975].)Let us now try to make a continuous frame out of the vectors h,,,. This time the index set is

M = { ( a ,s) I a

# 0, s E lR} !a IR*x IR,

(8)

IR*denotes the group of non-zero real numbers under multiplication. M is the fine group of translations and dilations of the

where

+

real line, t’ = at s, and this fact will be recognized as being very important in chapter 3. But for the present we use a more pedestrian approach to obtain the central results. This will make the power and elegance of the group-theoretic approach to be introduced later stand out and be appreciated all the more. At this point we only make the safe assumption that the measure dp on M is invariant under time translations, i.e. that dp(a, s) = p(a)da ds

where p ( a ) is an as yet undetermined density on Plancherel’s theorem, r

(9)

lR*. Then, using

1.6. Wavelet Tkansforms

47

The frame condition is therefore A L. H ( v )

5 B for some positive

where

constants A and B. To see its implications, we analyze the cases

v > 0 and v < 0 separately. If v

If v < 0, let

> 0, let t = av. Then

= -uv. Then

giving the same expression. Therefore the frame condition requires that P ( t / V > = O ( ( t / y r 2 > as

unless

t/v

4

fw,

(14)

vanishes in a neighborhood of the origin. Note that the

&(I/)

above expression for H ( v ) shows that if both p(a) and &v) vanish for negative arguments then H ( v )

0 and no frame exists. Hence to support general complex-valued windows (such as bandpass filters for a positive-frequency band), it is necessary to include negative as well as positive scale factors a. The general case, therefore, is that we get a (generalized) frame whenever p(u) and h ( t ) are chosen such that 0 < A 5 H ( v ) 5 B is

1. Coherent-St ate Represent ations

48

satisfied. The “metric operator” G and its inverse are given in terms of Fourier transforms by

Gf = (Hf”)” and

G-’f = ( H - l f ) ” .

(15)

Since G is no longer a multiplication operator in the time domain (as it was in the case of the discrete frame we constructed from the windowed Fourier transforms), the action of G-’ is more complicated. It is preferable, therefore, to specialize to tight frames. This requires that H ( v ) be constant, so the asymptotic conditions on p reduce to the requirement that p be piecewise continuous: c + / a 2 , if a c - / a 2 , if a

>o 0,

and for u

< 0,

k(-t)

Thus H ( v ) = A = B requires either that I k ( ( ) I = I I (which holds if h ( t ) is real) or that c+ = c - . Since we want to accomodate complex wavelets, we assume the latter condition. Then we have

We have therefore arrived at the measure dp(a, s ) = c+da d s / a 2

(20)

1.6. Wavelet lhnsforms

49

for tight frames, which coincides with the measure suggested by group theory (see chapter 3). In addition, we have found that the basic wavelet h must have the property that

In that case, h ( t ) is said to be admissible. This condition is also a special case of a grouptheoretic result, namely that we are dealing with a square-integrable representation of the appropriate group (in this case, the f i n e group IR* x IR). To summarize, we have constructed a continuous tight frame of wavelets ha,, provided the basic wavelet is admissible. The corresponding resolution of unity is

The associated reproducing-kernel Hilbert space !Rzh: is the space of

=

, = ( ha,, I f ) f(a, s) depending on the scale functions ( K f ) ( a s) parameter a as well as the time coordinate s. As a + 0, ha,, becomes peaked around t = s and

where c = Sh(u)du. The transformed signal f" is a smoothed-out version of f and a serves as a resolution parameter. Ultimately, all computations involve a finite number of operations, hence as a first step it would be helpful to construct a discrete subframe of our continuous frame. Toward this end, choose a fundamental scale parameter a > 1 and a fundamental time shift b > 0. We will consider the discrete subset of dilations and translations

D

= {(a",na"b) Im,n E

Z}c IR* x IR.

(24)

50

1. Coherent-State Representations

Note that since am > 0 for all m, only positive dilations are included in D,contrary to the lesson we have learned above. This will be remedied later by considering h ( t ) along with h ( t ) . Also, D is not a subgroup of R*x R,as can be easily checked. The wavelets parametrized by D are

hmn =

h

(t

)

- numb am = u-m/2h(a-mt - nb).

(25)

To see that this is exactly what is desired, suppose k ( u ) is concentrated on an interval around u = F (i.e., k is a band-pass filter). Then jLmn is concentrated around u = F l u m . For given integer m, the “samples” fmns((hmnIf),

~

E

Z

(26)

therefore represent (in discrete “time” n ) the behavior of that part of the signal f ( t ) with frequencies near Flu”. If m >> 1, f m n will vary slowly with n, and if m .

(29)

We now assume that k(v) = k(v) vanishes outside the interval

where F

> 0 is some fixed frequency to be determined below. The

width of the “band” I0 is Wo = ( a - u-’)F. Therefore the function k ( a ” v ) f ” ( v )is supported on the compact interval

I , = [F/am,F/am-l]

(31)

of width W, = WO/am,and we can expand it in a Fourier series in that interval:

n

where

cmn = Wi’

I

00

du exp(-27rinv/Wm) k(amu)f ( u ) .

(33)

Comparing this with 00

fmn

dv aml2 exp(-27rivnbam) k ( u m u ) f ( u )

= J-00

suggests that we choose F so that Wm = l/amb, which gives

(34)

1. Coherent-State Representations

52

U

F= (a2

(35)

- I) b

and

Th Fourier series representation above only holds in th interval I m , since the left-hand side vanishes outside this interval while the right-hand side is periodic. To get equality for all frequencies and reconstruct f(t), multiply both sides by k( am v) and sum over m:

m,n

(37) To have a frame we would need the series on the left-hand side to converge to a function x+( v) with 0

< A 5 x+(Y)

C I k ( u m v )l 2 5 B

(38)

m

for some constants A and B. But this is a priori impossible, since

R(V) is supported on an interval of positive frequencies and am > 0, 5 0. However, we can choose h ( t ) ,a and b such that x + ( v )satisfies the frame condition for v > 0. Negative freso x + ( v ) vanishes for v

quencies will be taken care of by starting with the complex-conjugate of the original wavelet. We adopt the notation

h+(t) = h(t), h-(t)

G

h(t).

Then the Fourier transforms k* of hf are related by

(39)

1.6. Wavelet Zlansforms

53

k-(v) = k+(-v),

(40)

hence k - ( v ) is supported on -I0 = [-Fa, ment to the above gives

-Flu].A similar argu-

and

x-(Y)

C I k-(amv) I

= X+(-V).

(42)

m

Hence if

x+

satisfies the frame condition for u > 0, then

for all u # 0. Since (0) has zero measure in frequency space, the frame condition is satisfied by the joint set of vectors

7igb= {kf,,

k,;

Im,n E 22).

(44)

The metric operator

is given by

(Gf>(t>=

(x' + x-)!)

and satisfies the frame condition 0

"(t)

(46)

< A1 5 G 5 B. Since G is no

longer a multiplication operator in the time domain (as was the case

54

1. Coherent-State Representations

with the discrete frame connected to the windowed Fourier transform), the recovery of signals would be greatly simplified if the frame was tight. The following construction is borrowed from Daubechies [1988a]. Let F = a/(a2 - l ) b as above and let k be any non-negative integer or k = 00. Choose a real-valued function 77 E C'(lR,) (i.e., 77 is k times continuously differentiable) such that 0 for x 5 0 n / 2 for x 2 1.

(47)

(Such functions are easily constructed; they are used in differential geometry, for example, to make partitions of unity; see Warner [1971].) Define h ( t ) through its Fourier transform k+(v)by

Note that k + ( v ) is C ' since the derivatives of q(z) up to order k all vanish at IC = 0 and x = n/2. This means that the wavelets in the frame we are about to construct are all Ck.Also, k+ vanishes outside the interval 10 = [ F / a ,Fa]. The width of its support is Wo = ( a - a - l ) F , and for each frequency Y > 0 there is a unique integer M such that F / a < u M v 5 F , hence also F < a M + l v 5 aF. Therefore, for v > 0,

Thus 0 for Y 5 0 1 forv > 0,

55

1.6. Wavelet Tkansforms

i.e., x + ( Y ) is the indicator function for the set of positive numbers. It follows that x - ( v ) is the indicator function for the negative reals, and

-

This choice of k+ and k- = k+ gives us a tight frame,

This frame is not a basis; if it were, it would have to be an orthonormal basis since it is a normal frame, hence the reproducing kernel would have to be diagonal. But

K ( Ern, , n; E', m', n') = ( kk,,I k$,, does not vanish for

E'

)

(53)

= E , n' = n and rn' = rn f 1, due to the overlap

of wavelets with adjacent scales. However, it is possible to construct orthonormal bases of wavelets which, in addition, have some other surprising and remarkable properties. For example, such bases have been found (Meyer [1985], Lemarie and Meyer [1986])whose Fourier transforms, like those above, are C" with compact support and which are, simultaneously, unconditional bases for all the spaces Lp (IR)with

1

= 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 0 by

Such sets are not covariant, but a covariant extension will be found in the next section. As for the measure, a Gaussian weight function (such as exp(-my2/u), which occured in dpu(y)) is no longer satisfactory since it cannot be covariant. It turns out that we do not need a weight function at all! This can be seen as follows: In the non-relativistic case, the shift to complex time was performed once and for all by the operator e-uH. For fixed u > 0, the weight function served to correct for the non-unitary translation from the real point x in space to the complex point z = x - iy. However, if we restrict ourselves to the subset u,then a translation to imaginary space is necessarily accompanied by a translation in imaginary time. The analog of the above translationis (t-iA,x)

H

(t-i,/m,x-iy).

The increase in yo, it turns out, precisely compensates for the shift to complex space! This follows from the fact that the operator e-Yp, which affects the total shift to complex spacetime, is relativistically invariant, hence the point y = 0 no longer plays a special role. We will show later that in the non-relativistic limit, we recover the weight function naturally. Hence the Gaussian weight function associated with the Galilean coherent states (which, as we have seen, is closely related to that associated with the canonical coherent states) has its origin in the geometry of the relativistic phase space, i.e. in the curvature of the hyperboloid (y2= A'}.

For

CT

= ot,X as above and f E Ic, define

4.4. Relativistic names

where we parametrize

Q

203

by (x,y) E R2" and the measure do is given

with

=

(z)'-

m G(A).

Then we have the folowing result.

Theorem 4.3.

For all f E K ,

Proof. Assume, to begin with, that a(p) a(w,p) belongs to the Schwartz space S(IRd)of rapidly decreasing test functions. Then

Hence, by Plancherel's theorem,

204

4. Complex Spacetime

Exchanging the order of integration in the integral representing \If[ :) we obtain

We now evaluate J(p)= as follows: Consider all s

J

dSye-2yp

(76)

+ 1 components of p as independent and

define m(p)f @. From the integral computed earlier, i.e.

J

G(y) 3 ( 2 ~ ) - ~d"p(2u)-1e-2yp

(77)

4nX we obtain by exchanging p and y (as well as rn and A):

Taking the partial derivative with respect to p o on both sides gives

d [t-"Kv(t)] dPO

= -(2nX2)y-

(79)

where t ( p ) 2Xrn(p). Using again the recurrence relation for the 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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