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

NORTH-HOLLAND MATHEMATICS STUDIES Editor: Leopoldo NACHBIN Quantum Physics, Relativity, and Complex Spacetime Towords ...

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NORTH-HOLLAND

MATHEMATICS STUDIES Editor: Leopoldo NACHBIN

Quantum Physics, Relativity, and Complex Spacetime Towords a New Synthesis G. KAISER

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 New York, N.Y. 10010, U.S.A.

Library o f Congress Cataloging-in-Publlcation Data

Kaiser. Gerald. Ouantun phystcs, relativity. and complex spacetine : towards a new synthesis / Gerald Kaiser. p. cu. -- (North-Holland matheratics studies : 163) Includes btbliopraphtcal references 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. Series. 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 U.S.A., 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

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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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 1.5. Windowed Fourier transforms ...........................34 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

Cont ent s

x

.

Chapter 4 Complex Spacetime 4.1. Introduction ...........................................155 4.2. Relativity, phase space and quantization ............... 156 4.3. Galilean frames ........................................169 4.4. Relativistic frames .....................................183 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 5.4. Free Klein-Gordon fields .............................. 273 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 References Index

..................................................347

........................................................357

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 "Phase-Space 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 mat hematical 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 oo)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 represent ations 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 IRn to give a function on Cn 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 complexificat ion 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 negative-frequency parts of free fields, so that Wick ordering becomes unnecessary. A functionalintegral representation 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 t o 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 os meus colegas Brasileiros!

xvi

Suggestions to the Reader

The reader primarily interested in the phase-space 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.

Chapter 1 COHERENT-STATE REPRESENTATIONS

1.l. 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. Spacetime and its Dual

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

IR"+'. 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

1. Coherent-State Representations

where xO 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 kx instead of k(x)) can be written as

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 ,

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

-t

IR by

x*(XI) = x

.

XI,

thus giving a map

X*, with

Since g,, is non-degenerate, it also defines a scalar product on X*, whose metric tensor is denoted by gpY.The map x -+ x* establishes an isomorphism between the two spaces, which we use to identify them. If x denotes a set of inertial coordinates in free spacetime, then the scalar product is given by g,, = diag(c2,-1, - 1 , s -

, -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

x . xt = c2tt' - x xt and kx = k . x* = Lot - k . x,

(5)

where x . xt and k . x denote the usual Euclidean inner products in IRS.

2. Fourier ?f-ansforms (which, to avoid The Fourier transform of a function f : X + analytical subtleties for the present, may be assumed to be a Schwartz test function; see Yosida (19711)is a function f:X* + Q given by

where dx dt dsx is Lebesgue measure on X . f can be reconstructed from f by the inverse Fourier transform, denoted by " and given by

where dk = dko dsk denotes Lebesgue measure on X* a X. Note that the presence of the 27r factor in the exponent avoids the usual need for factors of (2~)-('+')I2 or (27r)-"-' in front of the integrals. Physically, k represents a wave vector: ko v is a frequency in cycles per unit time, and k j 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(xl) = exp(-27rikxt) through the spacetime translation x, i.e. 27rk "measures" the radian phase shift. Now in pre-quantum relativity, it was realized

4


that the energy E combines with the moment urn 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 re-interpreted 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, p = 2nfik.

(8')

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: (Ppf)(x) = or, in terms of x*,

Jx*

dkp,e-2""xf(k)

d

= ih-dx f(x),

(9)

5

1.1. Preliminaries

Po= ih-

a

at

and

d axk

Pk= -ah-.

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 (x) satisfies a differential equation, such as the Schrodinger equation or the Klein-Gordon equation, then j(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 Rs. We will write the solution as

where f ( p ) is, by a mild abuse of notation, the LLrestriction"of f to P (actually, lf(p)12 is a density on P ) and dp(p) I p(p)dsp is an appropriate invariant measure on P. For the Schrodinger equation p(p) r 1, whereas for the Klein-Gordon equation, p(p) = I po I Setting t = 0 then shows that f(p) is related to the initial wave function by

-'.

where now " '" denotes the the s-dimensional inverse Fourier transform of the function on P w IRS. We will usually work with "natural units," i.e. physical units so chosen that h = c = 1. However, when considering the non-

i

6


relativistic limit (c + m) or the classical limit ( f i + 0), c or fi will be re-inserted 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 not ation and, hopefully, convince mathematicians of its correctness and value. When applied to coherent-state representat ions, as opposed to representat ions in which the posit ionor 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*:3-1+ G b y

The Riesz represent at ion theorem guarantees that the converse is also true: Each bounded linear functional L : 3-1 + has the form L = f * for a unique f E 3-1. Define the bra (f 1 corresponding to f by

(fl = 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

lg): c + ' H defined by Is)@) = Xg,

AE

a,

(16)

which will be called kets. Thus elements of 'H will be denoted alternatively by g or by )g). We may now consider the composite map bra-ke t (fig):

a 7 a,

given by

(f Is)(A) = f * ( W = Xf*(s) = Yf, g).

(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 (f 1 g) for both the map and the inner product. The reverse composition

Is)(fl: 'H

+

3-1

(19)

may be viewed as acting on kets to produce kets:

Id(f l(lW = 19) ((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 the usual expansion of an arbitrary vector f in

H

H. Then

takes the form

8


from which we have the "resolution of unity"

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

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

L2(IR8),say, are often written as f (x)

= (x ( f

), with

( x I x' ) = 6(x - x'), 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 1 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 ( g n I f ), ( hn I f ), ( x 1 f ) and ( k I f ) merely different "representations" (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 x ; 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.


We begin by recalling the original coherent-state representations (Bargmann [1961],Klauder [1960,1963a, b], Segal [1963a]). Consider a spinless non-relativistic particle in R " (or $13 such particles in R3), whose algebra of observables is generated by the position operators Xk 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 real Lie algebra known as the Heisenberg algebra, which is irreducibly represented on L2(R8)by

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

1. Coherent-State Represent ations

10

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 11 f 11 = 1 (where 11 . 11 denotes the norm in L2(Rs))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 A k is non-Hermitian, it is real in the Schrodinger represent ation. Let

denotes the complex-conjugate of Ak - zkI we have ( 6Ak ) = 0 and

where

Zk

0 _< 116Ak f 112 =

~ k . Then

+ b2Agk- b.

for

6Ak r

(7)

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


11

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

for some real number b (which may actually depend on k) and some z E Cs. But square-integrable solutions exist only for b > 0, and then there is a unique solution (up to normalization) X, for each z E C8. To simplify the notation, we now choose b = 1. Then Ak and A; satisfy the commutation relations

and X, is given by

where the normalization constant is chosen as N = 7r-#I4, SO that llx, 11 = 1for z = 0. Here f is the (complex) inner product of f with itself. Clearly X, is in L2(IR8),and if z = x - ip, then

in the state given by x,. The vectors X, are known as the canonical coherent states. They occur naturally in connection with the harmonic oscillator problem, whose Hamiltonian can be cast in the form

12


with

(thus b = llmw). They have the remarkable property that if the ini) z ( t ) is the orbit tial state is x,, then the state at time t is x , ( ~ where in phase space of the corresponding classical harmonic oscillator with initial data given by s. These states were discovered by Schrodinger himself [1926], at the dawn of modern quantum mechanics. They were further investigated by Fock [I9281 in connection with quantum field theory and by von Neumann [I9311 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 Iclauder 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


13

(next section). The coherent-st ate represent ation is now defined as follows: Let F be the space of all functions

where f runs through L2(IR8). Because the exponential decays rapidly in x', f" is entire in the variable a E Ca. Define an inner product on F by

where z

x

- ip and

Then we have the following theorem relating the inner products in L2(IR8)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 ,J

be the corresponding

Proof. To begin with, assume that f is in the Schwartz space S(IRa) of rapidly decreasing smooth test functions. For z = x - i p , we have X z ( ~= l ) N exp[-r2/2

+ x2/2 - (x' - x)'/2 + ips 1',

14

1. Coherent-Stat e Representations

hence

f ( x - i p ) = N exp[(x2+ p 2 ) / 4 + i p x / 2 ]

ex^[-(XI - ~ ) ~ / 2 )*(PI 1 f (19)

where * denotes the Fourier transform with respect to x t . Thus by Plancherel's theorem (Yosida [1971]),

(20)

Therefore

after exchanging the order of integration. This proves that

for f E S(IR8), hence by continuity also for arbitrary f E L 2 (IRb). B y polarization the result can now be extended from the norms to the inner products. U The relation f t, f can be summarized neatly and economically in terms of Dirac's bra-ket notation. Since


f(z)=

(x. If) = (f

lx*)1

theorem 1 can be restated as

Dropping the bra (f 1 and ket Ig), we have the operator identity

where I is the identity operator on L2(IRs)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 L 2 ( R s )can now be expressed as an integral operator B on F as follows:

Particularly simple represent at ions are obtained for the basic position- and momentumdoperators. We get

*

As will be shown in a more general context in the next section, under favorable conditions the integral actually converges in the strong operator topology.

1. Coherent-St ate Representations

thus

Hence X k and Pkcan be represented as differential rather than integral operators. As promised, the continuous resolution of the identity makes it from its transform jE +: possible to reconstruct f E L2(IRs)

that is,

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

L2(IRs).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(IRs) (and hence also 3)is separable. In particular, the above reconstruction formula can be used to express xZ in terms of all the x,'s:


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 {x,1 z E I?) are already complete since for any function f orthogonal to them all, f"(z) = 0 for all r 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 Axk 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 ti 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 tract abledefinition of uncertainty uses the notion of entropy, which involves all moments rather than just the second moment. See Bialynicki-Birula and Mycielski [I9751 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.

+

18


1.3. Generalized Frames and Resolutions of Unity Let M be a set and p be a measure on M (with an appropriate ualgebra 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) I ( h , I f ) is measurable, and

2. there exist constants 0 < A 5 B such that r

3 - 1 ~is a frame (see Young [I9801 and Daubechies [1988a]) 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

We will henceforth drop the adjective "generalized" and simply speak of "frames." The above case where M is countable will be refered to as a discrete frame. If A = B , the frame 3 - 1 ~is called tight. The coherent states of the last section form a tight frame, with M = d P ( z ) , 12, = xZ and A = B = 1.

as,dp(m)

=

1.3. Generalized I+ames and Resolutions of Unity

19

Given a frame, let T be the map taking vectors in 'H to functions on M defined by

(Tf )(m)

( hm

If)

f(m),

fE

(4)

Then the frame condition states that Tf is square-integrable with respect to dp, so that T defines a map

A Ilf

112

2 IITf ll2.(dP) 5 B Ilf

l12.

(6)

The frame property can now be stated in operator form as

A I 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

IftheintegrdG(N)convergesinthestrong operator topology of ? whenever i N has finite measure, then so does the complete integral representing G = T*T.

Propositioll1.2.

20


Since M is a-finite, we can choose an increasing seProof*. quence { M n ) of sets of finite measure such that M = UnMn. 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). 1 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 L2(dp), 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: (a) Find the range %T

c L2(dp) of the map T .

(b) Find a left inverse S of T, which enables us to reconstruct f from

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

G-I . For let

Then it is easy to see that

*

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

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 ,

for if g = Tf for some f in 3.t, then Pg = PTf = Tf = g , and conversely if for some g we have Pg = g , then g = T(G-lT*g)z T f . Thus ?JtT is a closed subspace of L2(dp)and a function g E L2(dp)is in ?JtT 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 gT.But it differs from the S-function in some important respects. For one thing, it is bounded by


for all m and m'. Furthermore, the "test functions" which K(m, m') reproduces form a Hilbert space and K(m,ml) defines an integral operator, not merely a distribution, on ?RT. 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 9lT 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(m, 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( d p ) 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

23

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

This gives f as a linear combination of the vectors

hm = G-lh,. Note that

therefore the set

is also a frame, with frame constants 0 < B-' 5 A-I. We will call 7iM the frame reciprocal to ?tM. (In Daubechies [1988a], the corresponding discrete object is called the d u d 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 7 i ~'HM , of reciprocal frames:

24


Under the assumptions of proposition 1.2, the Corollary 1.3. above resolutions of unity converge in the strong operator topology of 'H. 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

~ ' H M both reduce to bases of linearly independent, the frames 7 - l and

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 dp is the counting measure on M (re-normalize the hm's if necessary). Then the above relation becomes hr =

hmK(m,k),

and linear independence requires that K be the Kronecker 6: K ( m ,k) = 6;. Thus when the hm7sare linearly independent, 'HM and ' H M reduce to a pair of reciprocal bases for 'H. The resolutions of unity become

and we have the relation


where

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 = L2(dp) 12(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 = A I implies that

so 'HM and 'HM become orthogonal bases. Requiring A = B = 1 means that 'HM = 'HM reduce to a single orthonormal basis. Returning to the general case, we may summarize our findings as follows: EM,'HM, K(m, m') and g(m, m') s ( hm 1 Ghmt ) 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.

1. Coherent-St at e Representations

26

hturning to the general situation, we must still supply a way of computing G-' , 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 BI it follows that 8

Hence letting

6=-

B-A B+A

and

2

c =B+A'

we have

Since 0 5 6 < 1 and c

> 0, we can expand

and the series converges uniformly since

( ( I- cG((5 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


27

If 0 < S 112. 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 = 112W. Our reconstruction formula reduces to

>

+

The smaller the ratio Au/ 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,.(t) = e-2"i"th(t - s).

(1)

The basic window function h(t) was assumed to vanish outside of the interval -7 < t < 0 and to be reasonably smooth with no steep slopes, so that its Fourier transform h(u) was also centered in a small interval about the origin. Of course, since h(t) has compact support, h(u) 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 R(u) decays rapidly outside of a small interval containing the origin. At any rate, the factor exp(-2.rrivt) amounted to a translation of the window in frequency, so that h,,, was a "window" in the time-frequency plane centered about (v, s). 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, h,,, 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 T 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 lal-'I2 is included so that 00

dt I ha,a(t) 1 2 = llhl12.

a

(3)

-00

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 time-frequency 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.ansforrns

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) = 6(t))

and we have

with

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 v P. 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,PI. Then the above expression shows that j(a, s ) depends only on the frequency component of f ( t ) in the band a / a 5 v 5 p/a (if a > 0) or @/a 5 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,

<
0 and v < 0 separately. If v > 0, let

If v < 0, let

t = av.

Then

= -av. Then

giving the same expression. Therefore the frame condition requires that

unless k(v) vanishes in a neighborhood of the origin. Note that the above expression for H ( v ) shows that if both p(a) and R(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(a) and h ( t ) are chosen such that 0 < A 5 H ( v ) 5 B is

48


satisfied. The "metric operator" G and its inverse are given in terms of Fourier transforms by Gf =

( ~ j ) "and

G-'f

= (H-lfl".

(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:

where c+ and c- are non-negative constants (not both zero). Then for v

> 0,

and for v

< 0,

Thus H(v) = A = B requires either that I h ( ~ 1 ) = I h(-f) 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

1.6. Wavelet ~ a n s f o r m s

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 affine 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 !RK is the space of functions (Kf)(a, s ) = ( ha,. 1 f ) I f(a, s) depending on the scale parameter a as well as the time coordinate s. As a + 0, ha,, becomes peaked around t = s and

f

-

lal1I2cf(s)

(23)

h(u)du. The transformed signal j is a smoothed-out version of f and a serves as a resolution parameter. Ultimately, all computations involve a finite number of opera-

where c =

tions, hence as a first step it would be helpful t o 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

1. Coherent-St at e Representations

50

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

= a-"I2 h(a-'t

- nb).

(25)

To see that this is exactly what is desired, suppose k(v) is concentrated on an interval around v = F (i.e., k is a band-pass filter). Then Am. is concentrated around v = F / a m . For given integer rn, the "samples" fmns(hmnIf),

n

~

(26)

z

therefore represent (in discrete "time" n) the behavior of that part of the signal f ( t )with frequencies near F / a m . If m >> 1, f m n will vary slowly with n, and if m 0, so x + ( v ) vanishes for v 0. However, we can choose h(t),a and b such that x + ( v ) satisfies the frame condition for v > 0. Negative frequencies will be taken care of by starting with the complex-conjugate of the original wavelet. We adopt the notation

0, then

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

%gb= {k:,

kkn Im,n E 22).

(44)

The metric operator

is given by

and satisfies the frame condition 0

< A I 5 G 5 B. Since G is no

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

54

1. Coherent-Stat e 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 - 1)b as above and let k be any non-negative integer or k = m. Choose a real-valued function r) E C'(IR) (i.e., r ) is k times continuously differentiable) such that 0 for x 5 0 ?r/2 for x 2 1. (Such functions are easily constructed; they are used in differential geometry, for example, to make partitions of unity; see Warner [I9711.) Define h(t) through its Fourier transform k+(v) by

k+(v) =

sin

u-F/a [r) ( F - F / a ) ]

0s

[r)

(

1,

5 for

Y

(48)

2 F.

Note that k+(v) is Ck since the derivatives of q(x) up to order k all vanish at x = 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 I. = [Flu, Fa]. The width of its support is Wo = ( a - a-l)F, and for each frequency v > 0 there is a unique integer M such that F / a < a M v 5 F, hence also F < aM+lv 5 a F . Therefore, for v > 0,

Thus 0 foru5O

1 forv > 0,

1.6. Wavelet llansforms

55

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

does not vanish for

E'

= E, n1 = n and m' = m 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 Coowith compact support and which are, simultaneously, unconditional bases for all the spaces LP (IR)with

1 < p < oo as well as all the Sobolev spaces and some other popular spaces to boot. Similar bases were constructed in connection with quantum field theory (Battle [1987]) which are only C kfor finite k but, in return, are better localized in the time domain (they have exponential decay). The concept of multiscale analysis (Mallat [1987],Meyer [1986])provided a general met hod for the construction and study of orthonormal bases of wavelets. This was then used by

56


Daubechies \1988b] to construct orthonormal bases of wavelets having compact support and arbitrarily high regularity. The mere existence of. such bases has surprised analysts and made wavelets a hot new topic in current mathematical research. They are also finding important applications in a variety of areas such as signal analysis, computer science and quantum field theory. They are the subject of the next chapter, where a new, algebraic, method is developed for their study.

Chapter 2

WAVELET ALGEBRAS AND COMPLEX STRUCTURES

2.1. Introduction As stated at the end of chapter 1, orthonormal bases of wavelets are finding important applications in mathematics, physics, signal analysis and other areas. In this chapter we present a new treatment of such systems, based on an algebraic approach. This approach was actually discovered while the author was doing work initially unrelated to this book, in preparation for a conference on wavelets (Kaiser [1990a]). But it turned out that orthonormal bases of wavelets are closely associated with the concept of a complex structure, i.e. a linear map J satisfying J2= -I where I is the identity. J unifies certain fundamental operators H and G associated with the wavelets, known as the low-pass and high-pass filters, in much the same way as the unit imaginary i combines the position- and momentum operators in the coherent-state construction. This provides us with yet another example of the central theme of this book, namely that competing (or complementary) quantities can often be reconciled through complexification. For this reason, I decided to include these new results in the book. Furthermore, there may be a direct connection between wavelets and relativistic quantum mechanics (aside from their application to quantum field theory, which is less direct) based on the fact

58

2. Wavelet Algebras and Complex Structures

that relativistic windows (which are like those associated with the windowed Fourier transform but modified so as to be covariant under the Poincark group) behave like wavelets because they undergo a Lorentz contraction in the direction of motion, which is in fact a dilation. This idea is touched on in chapters 4 and 5.

The theory of orthonormal wavelet bases is closely related to multiscale analysis (Mallet [1987], Meyer [1986]),in which functions are decomposed (or filtered) recursively into smoother and smoother functions (having lower and lower frequency spectra) and the remaining high-frequency parts at each stage are stored away. In the limit, the smooth part vanishes (for L2 functions) and the original function can be expressed as the sum of the details drawn off at the various frequency bands. Each recursion involves the application of a "low-frequency filter" H and a "high-frequency filter" G. The entire structure is based on a function 4, called an averaging function, which satisfies a so-called dilation equation. Roughly, 4 may be thought of as representing the shape of a single pixel whose translates and dilates are used to "sample" functions at various locations and scales. Although the operators H and G have very different interpretations, they exhibit a remarkable symmetry whose origin has not been entirely obvious. What is especially striking is that there exists a function $ whose (discrete) translates and dilates span all the highfrequency subspaces. That is, $ is a "basic wavelet" (also called a mother wavelet) in the same sense as that used for the function h(t) in section 1.6, except that now all the translates and dilates of t,b (corresponding to the functions h,,)

form an orthonormal basis. t,b

is related to G in a way formally similar to the way

H.

4 is related

to

2.2. Operational Calculus

59

The complex structures developed in this chapter are orthogonal operators* which relate G to H and t,b to 4, thus explaining the symmetry between these entities in terms of a "complex rotation." The plan of the chapter is as follows. In section 2 we develop an operational calculus for wavelets, which conceptually simplifies the formalism and helps in the search for symmetries. This is used in section 3 to construct the complex structures. These structures, in turn, suggest a new decomposition- and reconstruction algorithm for wavelets, which is considered in section 4. Section 5 consists of an appendix in which we summarize the operational calculus and state how our notation is related to the standard one.


In wavelet analysis (see Daubechies [1988b], Mallat [1987], Meyer [1986], Strang [I9891 and the references therein), one deals with the representation of a function ("signal") at different scales. One begins with a single real-valued function 4 of one real variable which we take, for simplicity, to be continuous with compact support. One assumes that for some T > 0, the translates &(t) = $(t - nT), n E 24, form an orthonormal set in L2(IR) (such functions can be easily constructed). The closure of the span of the vectors 4, in L2(R) forms a subspace V which can be identified with 1 2 ( Z ) since , for a real sequence u {un) we have

*

We assume that the function spaces are real to begin with; if they are complex instead, then the complex structures are unitary,


n

n

We introduce the shift operator

which leaves V invariant and is an orthogonal operator on L2(IR) (we shall be dealing with r e d spaces, unless otherwise stated). A general element of V can be written uniquely as

where u(e'eT) is the square-integrable function on the unit circle (It15 T I T )having {u,) 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 Pq5, 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 95, means that u(S)$ = 0 implies u(S) = 0. Our results could actually be extended to operators u(S) with {u,) E l1(Z) c 12(Z), which also form an algebra since the product u(S-')w(S) corresponds to the convolution of the sequences {un) and {w,). We resist the temptation.


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 from f by cutting off all frequencies ( with I(I > TIT. That is, fo coincides with f for n/T but vanishes outside this interval. The value of fo at t, is then

which is just the Fourier coefficient of the periodic function

obtained from fo(() by identifying ( domain,

+ 2s/T

with f. In the time

This has the same form as u(S)$, if we set un = 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 u, are no longer the values at the sharp times tn but are smeared over $,, since un = ( $n, u(S)$ ). In fact, acts as a filter, i.e. as a convolution operator, since (u(S)$)^(t) = u(eitT)$(t). Roughly speaking, we may think of q5 as giving the shape of a pixel.

+

62


Next, a scaled family of spaces V,, cu E ZZ, is constructed from V as follows. The dilation operator D, defined by

( D f)(t)= 2-'I2f ( t / 2 ) ,

(8)

is orthogonal on L2(lR). It stretches a function by a factor of 2 without altering its norm and is related to S by the commutation rules

Hence D "squares" S while D-' takes its "square root." A repeated application of the above gives

D ~ S = S ~ ~ DC ~~ ,E Z .

(10)

Define the spaces

V, = DaV,

(11)

which are closed in L2(IR) (Vo V ) . An orthonormal basis for V, is given by

-

9E(t) D"Sng(t) = 2-°124 (2-"t - nT) ,

(12)

and V, can also be identified with t 2 ( Z ) The . motivation is that V, 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 cu = - 1) of the form


63

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 h, vanish, so h(S) is a polynomial in S and S-', 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 4,. The coefficients h, uniquely determine 4, up to a sign. For if we iterate D-'h(S) = h ( s ' I 2 ) ~ - ' , we obtain

Since the Fourier transform of

satisfies

we obtain formally N

4 = $(o)Nlim

-+OO

11[2-ll2h (s2-')I6,

where S(t) is the Dirac distribution. The normalization is determined up to a sign by IIcjII = 1. See Daubechies [1988b] for a discussion of the convergence and the regularity of

4.

Note that the singular case 4 = S, discussed above, satisfies the diIation equation with h(S) = 41, where I denotes the identity operator. A more regular solution, related to the Ham basis, is the case where C$ is the indicator function for the interval [0, 1) and h(S) =

(I+ s)/& In general, integration of Dq5 = h(S)$ with respect to t gives

2, Wavelet Algebras and Complex Structures

Also, the regularity of 4 is determined by the order N of the zero which h(S) has at S = -I, i.e.

with L(s) regular at S = -I. For example, N = 0 for N = 1for the Haar system. See Daubechies [1988b].

4 = 6, and

The next step is to introduce a "multiscale analysis" based on the sequence of spaces V,. We shall do this in a basis-independent fashion. Since shifts and dilations are related by D S = S2D, 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 V,+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 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 z . 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


65

H: is interpreted as an operator which interpofrom Va to lates 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,+l. It is the orthogonal complement of the image of H:, i.e. of V,+1, in

v,: - ker H, = V, 8 H:V,+l Wa+l =

= Va 8 V,+i.

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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 --t 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 HE 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 2T) followed by an interpolation (which

66


restores the sampling interval to its original value T). Thus H * is a zoom-in operator! Its adjoint

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,) rather than the functions u(S)$. They are especially useful when considering iterated decomposition- and reconstruction algorithms (section 3). To find the action of H,, it suffices to find the action of H. Note that H*u(S)$ = h(S)u(S2)q5, where u(S2) is even in S. This will be 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,

Define the operator E* (for even) on V by


67

Then E*u(S)d = u(S2)$ =

un)2,.

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n

Also define the operator 0*(for odd) by 0*= SE*,so that o * ~ ( s )= d su(S2)) =

C ~n)ln+l.

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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 F and denote the adjoints of E* and 0* by E and 0. Then

(4

EE* = OO* = I, OE* = EO* = 0,

(note that (a) is a special case with v(S) = I),and

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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 OO* = 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 Cdculus

+ = EE*v+(S)+ EO*v-(S) = v+(S),

Ev(S)E* = E ( v + ( s ~ ) S V - ( s ~ ) ) E * OV(S)O*= E S - ~ V ( S ) S E= * v+(s),

+ = OE*v+(S)+ EE*v-(S) = V - ( S ) , Ev(S)O* = E(v+(s2)+ SV-(S2))sE* = EO*v+(S)+ EE*Sv-(S) = SV-(S). O V ( S E* ) = o(v+(s~) sv-(s2))~*

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Lastly, ( d ) follows from

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 4))

ve= {v(S2)+I v ( S ) E P ) ,

V0 = {Sv(S2)41 v ( S ) E P ) ,

and

This decomposition will play an important role in the sequel.

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70


Proposition 2.2.

The maps H: V -, V and H,: V, -, V,+i are

given by

Proof. Since H* = h(S) E*, it follows that H = E h(S-') and H, D, = D,+' H = DO+'E ~ ( s - ~ ) . 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,+' of V,+l in V,. The orthogonal decomposition V, = V,+l $ 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.' which is related to the spaces W, and the maps G, in a way almost totally symmetric to the way ) is related to V, and Ha. This is not merely a consequence of the orthogonal decomposition but


71

is somehow related to the fact that V,+l is "half" of V,, due to the doubling of the sampling interval upon dilation, as expressed by the commutation relation D S = S 2D. 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 V such that J 2 = -I. a complex structure on V, i.e. a map J : V (To illustrate this concept, consider the complex plane as the real space I R ~ 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

4 s )E P, We further require that J preserve the inner product, i.e. that J* J = I . Combined with J2 = -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.

Note that CM = MC and that C* = C and M* = M, since

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 I-+ -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 e $ V0 is nothing but the spectral decomposition associated with M, since Ve and V0 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,

In terms of C and M ,


73

Proposition 2.3. The conditions J* = -J and J2 = -I hold if and only if c(S) satisfies

Proof. We have

hence J* = -J if and only if E(-S) = -e(S). Assume this to be the case. Then

hence J 2 = -I if and only if E(S-')E(S) = I.

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(e) 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 ~ . (11) More interesting examples can be obtained by enlarging P to a topological algebra, for example allowing u(S) with {u,) E el

(z).

74

2. Wavelet Algebras and Complex Structures , F ,

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 a-an tilinear and is interpreted as Hermitian conj ugat ion. At an arbitrary scale a, define maps J,: V,

-, V,

by naturality, i.e.

which implies that J: = -I, and J: = - J,. J, is related to S by

showing that

S2a J, = -J,s-~*. In particular, note that S1I2J-1 = -J-1 s-ll2, 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,+1 of V,, i.e. the map

Kt:

-, V,

defined by


75

K: is natural with respect to the scale gradation, and its home version K l D = JH*. It will turn out will, as usual, be denoted by K* that its adjoint K is essentially equivalent to the usual filter G (to be introduced below) .but is more natural from the point of view of the complex structure. Define the vector $ E by

where the function

will play a similar role for the high-frequency components as does h ( S ) for the low-frequency components. Namely, g(S) is a "differencing operator," just as h(S) is an averaging operator. [For exarnple, in the simplest case h(S) = (I ~ ) / f i and E(S) = -S, we This gives rise to the Haar system.] For obtain g(S) = (I- s)/& w(S) E P,we have

+

IC*w(S)$ = JH*w(S)$ = Jh(S)w(S2)q5 =g(~)w(~-2= ) $g(S)E*Cw(S)$,

hence

Proposition 2.4. The adjoints of K* and I{: are given by

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76


Proof. Since IC = ECg(S-') and K,Da = DQ+'K, we have

and

K,) Proposition 2.5. The pairs of operators {H, IC) and {Ha, satisfy H H * = E~(s-')~(s)E* = I, ICK* = E~(s-')~(s)E* = I,

HK* = E ~ ( S - ~ ) ~ ( S ) E *=C0, ICH* = C E ~ ( S - ~ ) ~ ( S ) E=*O. and

H,H:

= K,K:

= I,+',

Ha K: = K,H:

= 0.

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Proof. (Note that H,H: = I,+' has already been shown; it is included here for completeness, since it belongs with the other identities.) The first equation follows from H * = H$D and H$Ho = I. The second follows from the first and K * -= JH*,since J* 3 = I. The last two equations follow from lemma 1, since h(S-')g(S) and g(S-')h(S) are odd functions, hence their even parts vanish. This proves the identities for H and K. The other identities follow by naturality. I


77

Proposition 2.6. The pairs { H ,K ) and { H a ,K,) give orthogonal decompositions of V and V,. We have (a)

HV=KV=V

Proof. We have

HV = D - ' H ~ V = D - ~ V= ~ V,

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and since I< = - H J , it follows that KV = HJV = HV = V . Also

KK* = I and H K * = 0 (proposition 2.4) imply that K* is injective and its range is orthogonal to that of H*, i.e. to Vl. Hence K*V C Wl. To show that K * V = W l , let u(S) E P. We need to find v(S), w(S) E P such that

or, equivalently, dropping

4,

Use lemma 1 to decompose this equation into its even and odd parts:


78

which can be written in matrix form as

But proposition 2.5 is precisely the statement that the matrix U ( S ) is unitary i.e., U ( S ) * U ( S = ) I. Multiplying by U ( S ) * ,we obtain the unique solution

h+(S-') h-(S-') w(S-1) = g+(S-') 9-(S-'1 which shows that V Vl $ W l = H*V $ IO.

(4)

(Note also that since H is unbounded, no analytic continuation to the upper-half time plane is possible.) The operator e - u H is familiar from two other contexts: it constitutes the evolution semigroup for the heat equation (where u is time), and it is also the unnormalized density matrix for the Gibbs canonical ensemble describing the statistical equilibrium of a quantum system at temperature T (where u = l/kT). To get a feel for our use of this operator, let us be heuristic for a moment and consider what happens when a free classical free particle of mass m is evolved in complex time

T

= t - iu. If its

initial position and momentum are x and p respectively, then its new position will be Z(T)

=x

+ (t - iu)p/m

= (X

+ tp/m) - i(u/m)p

(5)

= x(t) - i(u/m)p.

Since x(t) is just the position evolved in real time t, we see that z ( t ) is, in fact, a complex phase space coordinate of the same type

4.3. Galilean Frames

171

we encountered in the construction of the canonical coherent states! Armed with this intuition, let us now return to quantum mechanics and see if this idea has a quantum mechanical counterpart. The operator e-UH, when applied to any function in L~(R"),gives

If we replace x in the integrand by an arbitrary z E C8,the integral still converges absolutely since the quadratic terni in the exponent dominates the linear term for large /p(.Clearly the resulting function is entire in z (differentiating the integrand with respect to zk still gives an absolutely convergent integral). This shows that the group of Galilean space-time translations,

extends analytically to a semigroup of complex space-time translations U(E, T ) = exp(-i7H

+ i l . P)

(8)

defined over the complex space-time domain

This translation semigroup can be combined with the rotations and boosts to give an analytic semigroup Si extending G2. Let 'Flu be the vector space of all the entire functions f,(z) as f(P) runs through L2(IRs). Then

4. Complex Spacetime

172

f U ( z )= (e:

~i).

where ez(p) = (27r)-a exp[-up2/2m - i p . &] U

= (27r)-' exp my2/2u - -(p 2m

[

-r n ~ / u) ~i p x]

(l1)

are seen to be Gaussian wave packets in momentum space with expected position and momentum given in terms of z EE x - iy by

The e:'s are easily shown to have minimal uncertainty products. The momentum uncertainty can be read off directly from the exponent and is

hence

We now have our prospective coherent states and their label space M = a'. To construct a coherent-state representation, we need a measure on M which will make the e,U's into a frame. Since the er's are Gaussian, the measure in not difficult to find: dpu(z) = ( m / ~ u ) exp ~ ' ~(-my2 /u) d'x dsy. Defining

(15)

4.3. Galilean frames we have T h e o r e m 4.1. (a) ( I . ) 'Hu is an inner product on N u under which Xu is a Hilbert space. (b) The map e+"' is unitary from L 2 ( R s )onto 71.. (c) The e: 's define a resolution of unity on L2(IRd) given by

=

Proof. We prove prove that 11 f ( f 1 f )xu = 11.f 1 s 1 ; The inner product can be recovered by polarization. To begin with, assume that f is in the Schwartz space S ( R s ) of rapidly decreasing smooth test functions. Then

hence by Plancherel's theorem,

and

174


where exchanging the order of integration was justified since the integrals are absolutely convergent. This proves (b), hence also (a), for f E S(IRs). Since the latter space is dense in L2(IRb), the proof extends to f E L~(IR')by continuity. (c) follows by noting that

and dropping ( f

1

and

I 4 ).

I

Using the map e W U Hwe , can transfer any structure from LZ(IR') to 3-1,. In particular, time evolution is given by

fu(z, t) = (e-uH (e

-itH

f ) )(4

where r = t - iu and the wave packets

are obtained from the e i 's by evolving in real time t. They cannot be of minimal uncertainty since the free-particle Schrodinger equation is neccessarily dissipative. Instead, they give the following expectations and uncertainties:

4.3. Galilean Ehmes

Since es,r=e

itH u

e5,

it follows that

thus we have a frame { e,,, 1 z E C8 ) at each complex "inst ant" t - iu, with the corresponding resolution of unity

T

=

The space L2(IR8)carries a representation of the quantum mechanical Galilean group 62. Since the e5,,'s were obtained from the dynamics associated with this group, they transform naturally under its action. A typical element of g2 has the form g = (R,v,xo,to,B), where R is a rotation, v is a boost, xo is a spatial translation, to is a time-translation and 0 is the "phase" parameter associated with the central element M = m/h r m in our representation (see section 4.2). g acts on the complex space-time domain 2) by sending the point (2, T ) to ( T I , el), where


The parameter 8 has no effect on space-time; it only acts on wave functions by multiplying them by a phase factor. The representation of

62 is defined by

(u,f )(z, T ) = e-imef (g-l (z, 7)).

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Thus we have

and the e,,,'s

-

are "projectively covariant" under the action of G2;

if we define e,,,,4

e-"4e,,,,

then this expanded set is invariant

under the action of 6 2 , with 4' = 4-8. Since e,,, and e,,,,,j, represent the same physical state, we won't be fussy and just work with the e,,,'s. Anyway, this anomaly will disappear when we construct the corresponding relativistic coherent st at es. The above representation of G2 on L2(IR") can be transfered to

H ' , using the map e-UH. This map therefore intertwines (see Gelfand, with Graev and Vilenkin [1966])the representations on 62 on L~(IR") the one on Xu. We conclude with some general remarks. 1.

Since the e:'s are spherical and therefore invariant under S O ( n )

(which is, after all, why they describe spinless particles), they can be parametrized by the homogeneous space W = G1/ S O ( n )as long as we keep u fixed (u is a parameter associated with the Hamiltonian, which

4.3. Galilean frames

177

is a generator of G2 but not of GI). The action of W as a subgroup of on the e:'s is preserved in passing to the homogeneous space, hence W acts to translate these vectors in phase space. This explains the similarity between the ei's and the canonical coherent states. On the other hand, dynamics (in imaginary time) is responsible for the parameter u. If we write k G (m/u)y, then e:(p) = ( 2 ~ ) - 'exp G

U u k2 - -(p - k12 - ips x] [2m 2m

(~s).

exp [uk2/2m] e - ' ~ h' ~

The measure dp,(z) is now dpu(x,k) = ( u / ~ r n ) ' lexp(-uk2/m) ~ dsx d'k.

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Hence the exponential factor exp[uk2/2m]in e r ,when squared in the reconstruction formula, precisely cancels the Gaussian weight factor in dpu(x,k), leaving the measure

in phase space. It follows from the above form of ez that 2Aq = J2m/u plays the role of a scale factor in momentum space (as used in the wavelet transforms of chapter I), hence its reciprocal Ax, = acts as a scale factor in configuration space. Thus the Galilean coherent states combine the properties of rigid "windows" with those of wavelets, due to the fact that their analytic semigroup Gi includes both phase-space translations and scaling, the latter due to the heat operator e - u H . However, note that u is constant, though arbitrary, in the resolution of unity and the corresponding reconstruction formula. Since there is an abundance of "wavelets" due

Ju/2m

178


to translations in phase space, only a single scale is needed for reconstruction. (One could, of course, include a range of scales by integrating over u with a weight function, but this seems unnecessary.) In the treatment of relativistic particles, u becomes the time component of a four-vector y = (u, y), hence will no longer be constant. This is because relativistic windows shrink in the direction of motion, due to Lorentz contractions, thus automatically adjusting to the analysis of high-frequency components of the spectrum.

2. Notice that ef is essentially the heat operator e - U H applied to the &function at x, then analytically continued to ii = x iy. The fact that all the ef 's have minimal uncertainties shows that the action of the heat semigroup {U(-iu)) is such that while the position undergoes the normal diffusion, the momentum undergoes the opposite process of refinement, in just such a way that the product of the two variances remains constant. This is reflected in the fact that the operator e - U H , whose inverse in L2(IRs)is unbounded, becomes unitary when the functions in its range get analytically continued, and the reconstruction formula is just a way of inverting e - u H . Hence no information is lost if one looks in phase space rather than configuration space! It seems to me that this way of "inverting" semigroups

+

must be an example of a general process. If such a process exists, I am unaware of it. In our case, at least, it appears to be possible because of analyticity.

3. So far, it seems that coherent-state representations are intimately connected with groups and their representations. However, there is a reasonable chance that coherent-st ate representat ions similar to the above can be constiucted for systems which, unlike free particles, do not possess a great deal of symmetry. Suppose we are

179

4.3. Galilean frames given a system of

9/3

particles in

R~which

interact with one an-

other and/or with an external source through a potential V(x). We assume that V(x) is time-independent, so the system is conservative. (This means that we do have one symmetry, namely under time translations. If, moreover, the potential depends only on the differences xi - x j between individual particles, we also have symmetry with respect to translations of the center of mass of the entire system; but we do not make this assumption here.) This system is then described by a Schrodinger equation with the Hamiltonian operator

+

H = Ho V, where Ho is the free Harniltonian and V is the operator of multiplication by V(x). We need to assume that this (unbounded) operator can be extended to a self-adjoint operator on L2(IR8). How far can the above construction be carried in this case? The key to our method was the positivity of the free Hamiltonian Ho = P2/2m. But a general Hamiltonian must at least satisfy the stability condition:

(S)

The spectrum of H is bounded below.

If H fails to meet this condition, then the system it describes is unstable, and a small perturbation can make it cascade down, giving off an infinite amount of energy. For a stable system, the evolution group U ( t ) = eWitHcan be analytically continued to an analytic semigroup U(T) = e-irH in the lower-half complex time plane as in the free case. Depending on the strength of the potential, the functions f, = U(-iu) f may be continued to some subset of ad. Formally, this corresponds to defining

for an initial function f (x) in L2(RS). As mentioned, this expression is formal since the operator eyePis unbounded and e-irH f may not

180


be in its domain. But it can make sense operating on the range of e-irH , which coincides with the range of e-uH, provided y is not too large. Let yu be the set of all y's for which eJ'" is defined on the range of e-uH and, furthermore, the function exp [iz . P] x exp [ - i r ~f] is sufficiently regular to be evaluated at the origin in R8, no matter which initial f was chosen in L~(Rs). For many potentials, of course, Yu will consist of the origin alone; in that case there are no coherent states. We assume that yu contains at least some open neighborhood of the origin. Intuitively, we may think of y, as the set of all imaginary positions which can be attained by the particle in an imaginary time-interval u, while moving in the potential V. In the free case, yu = Rs and there is no restriction on y provided only that u > 0. This corresponds to the fact that there is no "speed limit" for free non-relativistic free particles, hence a particle can get to any imaginary position in a given positive imaginary time. For relativistic free particles, yUis the open sphere of radius uc, where c is the speed of light. Retuning to our system of interacting particles, define the associated complex space-time domain

This is the set of all complex space-time points which can be reached by the system in the presence of the potential V(x), and it is the label space for our prospective coherent states. These are now defined as evaluation maps on the space of analytically continued solutions:

the inner product being in L 2 ( R 8 ) .Then from the above expression, again formally, we have the dynarnical coherent states

4.3. Galilean fiames

181

for (z, T) in ZH. What is still missing, of course, is the measure dp:. (Since the potential is t-independent, so will be the measure, if it exists.) Finding the measure promises to be equally difficult to finding the propagator for the dynamics. The latter is closely related to the reproducing kernel, H

KH(z,T;zl, t l ) = (e,,,

I e,,,,.H

).

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h'~ depends on T and 7' only through the difference T - ?I, and is the analytic continuation of the propagator to the domain ZH x ZH. It is related to the measure through the reproducing property,

where the integration is carried out over a "phase space" a, in ZH with a fixed value of T = t - iu. A reasonable candidate for dp: (see section 4.4) is

Rather than finding the measure explicitly, a more likely possibility is that its existence can be proved by functional-analytic methods for some classes of potentials and approximation techniques may be used to estimate it or at least derive some of its properties. The theoretical possibility that such a measure exists raises the prospect of an interesting analogy between the quantum mechanics of a single system and a statistical ensemble of corresponding classical systems at

182


equilibrium with a heat reservoir. In the case of a free particle, if we set k = ( m / u ) yas above (see remark 1) and define T by u = 1/2kT where k is Boltzmann's constant, then it so happens that our measure dp, is identical to the Gibbs measure for a classical canonical ensemble (see Thirring [1980]) of s/3 free particles of mass m in

R3,at equilibrium with a heat reservoir at absolute temperature T. Thus, integrating with dp, over phase space is very much like taking the classical thermodynamic average at equilibrium! It remains to be seen, of course, whether this is a mere coincidence or if it has a generalization to interacting systems. There is also a connection between the expectation values of an operator A in the coherent states eET and its thermal average in the Gibbs state,

( A),

2-' Trace

(e-BH A) = 2-' Trace ( e - ~ ~ A l ' e-BH12

),

(41) where Z a Trace ( e - p H ) . Namely, if we have the resolution of unity

then

( A ) , = 2-'

1

dp:(z)

(e:T

where we have used the formula

le

-PHI' Ae-BH12

ecT)

4.4. Relativistic fiames

183

which follows easily from eq. (42). Thus taking the thermal average means shifting the imaginary part u by P/2 in the integral.

4.4. Relativistic Frames We are at last ready to embark on the main theme of this book: A new synthesis of Relativity and quant um mechanics through the geometry of complex spacetime. The main tool for this synthesis will be the physically necessary condition that the energy operator of the total system be non-negative, also known in quantum field theory as the spectral condition. The (unique) relativistically covariant statement of this condition gives rise to a canonical complexfication of spacetime which embodies in its geometry the structure of quantum mechanics as well as that of Special Relativity. The complex spacetime also has the structure of a classical phase space underlying the quantum system under consideration. Quantum physics is developed through the construction of frames labeled by the complex spacetime manifold, which thus forms a natural bridge between the classical and quantum aspects of the system. It is hoped that this marriage, once fully developed, will survive the transition from Special to General Relativity. As mentioned at the beginning of this chapter, the Perelomovtype constructions of chapter 3 do not apply directly to the Poincark group since its time evolution (dynamics) is non-trivial. Pending a generalization of these methods to dynamical groups, we merely use the ideas of chapter 3 for inspiration rather than substance. In fact,

184


it may well be that a closer examination of the construction to be developed here may suggest such a generalization. We begin with the most basic object of relativistic quantum mechanics, the Klein-Gordon equation, which describes a simple relativistic particle in the same way that the Schrodinger equation describes a non-relativistic particle. The spectral condition will enable us to analytically continue the solutions of this equation to complex spacetime, and the evaluation maps on the space of these analytic solutions will be bounded linear functionals, giving rise to a reproducing kernel as in section 1.4. Physically, the evaluation maps are optimal wave packets, or coherent states, and it is this interpretation which establishes the underlying complex manifold as an extension of classical phase space. The next step is to build frames of such coherent states. (Recall from section 1.4 that a frame determines a reproducing kernel, but not vice versa.) The coherent states we are about to construct are covariant under the restricted Poincark group, hence they represent relativistic wave packets . As we have seen, such a covariant family is closely related to a unitary irreducible representation of the appropriate group, in this case %. Such representations are called elementary systems, and correspond roughly to the classical notion of particles, though with a definite quant urn flavor. (For example, physical considerations prohibit them from being localized at a point in space, as will be discussed later.) We will focus on representations corresponding to massive particles. (A phase-space formalism for massless particles would be of great interest, but to my knowledge, no satisfactory formulation exists as yet.) Such representations are characterized by two parameters, the mass m > 0 and the spin j = 0,1/2,1,3/2,. . . of the corresponding particle. w e will specialize to spinless particles ( j = 0) for simplicity. The extension of our construction to particles


185

with spin is not difficult and will be taken up later. Thus we are interested in the (unique, up to equivalence) representation of Po with m > 0 and j = 0. A natural way to construct this representation is to consider the space of solutions of the Klein-Gordon equation (0

+ m2c2)f (2) = 0,

(1)

where

is the Del'Ambertian, or wave operator, A is the usual spatial Laplacian and 8, = d/ax". The function f is to be complex-valued (for spin j, f is valued in c2j+').We set c = 1 except as needed for future reference. If we write f (x) as a Fourier transform,

f (x) = ( ~ T ) - ~ -/R,+l I d5+'p

cii' !(P),

then the Klein-Gordon equation requires that supported on the mass shell

$2,

j@)be a distribution

is a two-sheeted hyperboloid,

where

Jmk u ( p ) on R$.

PO = k Taking

(6)


for some function a(p) on

a,,

and using

6(p2 - m2) = 6 ((pa - w)(po + w))

we get

where

is the unique (up to a constant factor) Lorentz-invariant measure (The factor w corrects for Lorentz contraction in frames on 52,. at momentum p.) For physical particles, we must require that the energy be positive, i.e. that a(p) = 0 on 52;. Hence the physical

-'

states are given as positive-energy solutions,

The function a(p) can now be related to the initial data by setting xO t = 0, which shows that fo(x) = f (x, 0) =

n / $,

dp" eBxaPa(p)

4.4. Relativistic frames

a@)

= ~ ( w , P=) %.io(p),

(13)

where denotes the spatial Fourier transform. In particular, f (x) is determined by its values on the Cauchy surface t = 0. For general solutions of the Klein-Gordon equation, we would also need to specify 8f /at on that surface, but restricting ourselves to positive-energy solutions means that f(x) actually satisfies the first-order pseudodifferential non-local equation A

--.

(which implies the Klein-Gordon equation), hence only f (x,0)is necessary to determine f . (We will see that when analytically continued to complex spacetime, positive-energy solutions have a local characterization.) The inner product on the space of positive-energy solutions is defined using the Poincarkinvariant norm in momentum space,

We will refer to the Hilbert space

as the space of positive-energy solutions in the momentum representation. It carries a unitary irreducible representation of Po defined as follows. The natural action of Po on spacetime is (b, A)x = Ax

+ b,

(17)

where A is a resticted Lorentz transformation (A E Lo ) and b is a spacetime translation. Since the Klein-Gordon equation is invariant


188

under %, the induced action on functions over spacetime transforms solutions to solutions. Since the positivity of the energy is also invariant under Po,the subspace of positiveenergy solutions is also left invariant. Po acts on solutions by (U(b, 4 f ) ( 4 = f (A-I ( 1 - b)) .

(18)

The invariance of the inner product on ~ : ( d j )then implies that the induced action on that space (which we denote by the same operator)

(U(b, A) a) (p) = e i b p a ( A - ' ~ )

.

(19)

The invariance of the measure dj3 then shows that U(b, A) is unit ary, thus (b, A) H U(b, A) is a unitary representation of 'Po. It can be shown that it is, furthermore, irreducible. Neither of the "function" spaces {f(x)) and ~:(d$) are reproducing-kernel Hilbert spaces, since the evaluation maps f H f (x) and a H a(p) are unbounded. To obtain a space with bounded evaluation maps, we proceed as in the last section. Due to the positivity of the energy, solutions can be continued analytically to the lowerhalf time plane:

f ( x t - iu) =

/nt

dfi exp (-;ti - w

+ ix

p) ~ ( p ) .

(20)

where u > 0. As in the non-relativistic case, the factor exp(-uw) decays rapidly as I p 1 + oo,which permits an analytic continuation of the solution to complex spatial coordinates z = x - iy. But since w(p) r is no longer quadratic in 1 p 1, y cannot be arbitrarily large. Rather, we must require that the four-vector (u, y) satisfy the condition


In covariant notation, setting

189

u, we must have

so that the complex exponential exp [-i(x - iy )p] remains bounded as p varies over R;. This implies that yp > 0 for all p E V+, where

is the open forward light cone in momentum space. In general, we need to consider the closure of V+ , i.e. the cone

which contains the light cone { p 2 = 0 I po > 0) (corresponding to massless particles) and the point { p = 0) (corresponding in quantum field theory to the vacuum state). The set of all y's with yp > 0 is i.e., called the dual cone of

v+,

It is easily seen that y belongs to V; if and only if I y I < cyO. Note contracts to the non-negative pa-axis while V; that as c + m, expands to the half-space {(u,y) 1 u > 0, y E Eta) which we have encountered in the last section. V. coincides with V+ when c = 1, but it is important to distinguish between them since they "live" in different spaces (see section 1.I). Thus for y E V;, setting z = x - iy, we define

v+

190


The integral converges absolutely for any a E L:(d@) and defines a function on the forward tube

7+E { x - i y E as+'I y E v;}, also known as the future tube and, in the mat hematical literature, as the tube over V;. Differentiation with respect to z p under the integral sign leaves the integral absolutely convergent, hence the function f ( 2 ) is holomorphic, or analytic, in 7+.As y + 0 in V;, f ( z ) + f ( x ) in the sense of L $ ( @ ) . Thus f (x) is a boundary vdue of f ( 2 ) . Clearly f ( 2 ) is a solution of the Klein-Gordon equation in either of the variables z or x. Let K be the space of all such holomorphic solutions: = If ( 4l a E ~ : ( d @ ) l .

(28)

Then the map a ( p ) H f ( 2 ) is one-to-one from L$(d@)onto IC. Hence we can make K into a Hilbert space by defining

where the inner product on the right-hand side is understood to be that of L$(d@). We now show that IC is a reproducing-kernel Hilbert space. Its evaluation maps are given by

where

4.4. Relativistic Frames

Lemma 4.2. 1. For each z E I+, e, belongs to L:(dfi),

with

where v = (s - 1)/2,

and Ir', is a modified Bessel function (Abramowitz and Stegun [1964];the speed of light has been inserted for future reference.) 2. In particular, the evaluation maps on

K

are bounded, with

I f (4I 5 IlezII Ilf II.

(34)

Proof. Set c = 1. (To recover c, replace m by mc in the end.) Then

Since G(y) is Lorentz-invariant and y E V l , we can evaluate the integral in a Lorentz frame in which y = (A, 0 ) :

G(y) = G(A, 0)=

LL

d@e-Z*w

d8P

Set p = mq. Then

(36) exp [ - 2 ~ J 2 7 7 ] .

192


G(y) = (27r)-sms-'

mS-l

-

Jm]

exp [ - 2 ~ m

/2$&,

I"

dt sinhs-' t exp [-2Xm cosh t ]

(4~)s/~rr(~/2)

The reproducing kernel can be obtained by analytic continuation from llez112:

where

q

,/= [(y'

+ Y ) -~(1' - x ) +~ 2i(y1+ ~ ) ( X I x)] -

112

(39)

is defined by analytic continuation from z' = z (when 77 = 2X) as follows: The square-root function is defined on the complex plane cut along the negative real axis. Since y and y' both belong to V'., so does y'

+ y. Now the argument of the square root is real if and only

+ y)(xl - x ) = 0, and this can happen only when (x' - x ) 5~ 0. (0therwise, either x' - x or x - x' belongs to V; , hence ( y' +y)(xl - x)

if (y'

is positive or negative, respectively.) But then,

4.4. Relativistic n a m e s

193

Thus for z', z E 7+,the quantity -(zl - z ) cannot ~ belong to the negative real axis, so 77 is well-defined. The reproducing kernel is closely related to the analytically continued (Wightman) 2-point function for the scalar quantum field of mass m (Streater and Wightman [1964]):

We will encounter this and other 2-point functions again in the next chapter, in connection with quantum field theory. Note: We will be interested in the behavior of ary of 7+,i.e. when X that

N

11 e, 11

near the bound-

0. From the properties of Ii', it follows

when X

N

0.

In particular, the evaluation maps are no longer bounded when X = 0.

Po acts on 7+by a complex extension of its action on real spacetime, i.e., z'

= (b, A) z = Az + b.

This means that x' = Ax

+b

(43)

as before, and y1 = Ay. (This is consistent with the phase-space interpret ation of 7+to be established below.) The induced action on K: is therefore


This implies that the wave packets e , transform covariantly under

Po, i.e.

We have now established that the space

K of holomorphic po-

sitive-energy solutions is a reproducing-kernel Hilbert space. Recall that picking out the positive-energy part of f ( x ) was a non-local operation in real spacetime, involving the pseudodifferential operator

d m .

However, when extended t o ' I such + functions , mag be

characterized locally, as simultaneaous solutions of the Klein-Gordon equation and the Cauchy-Riemann equations, since the negativeenergy part of f (x) does not have an analytic continuation to 7+. We now show that 7+may, in fact, be interpreted as an extended phase space for the underlying classical relativistic particles. Clearly, x '8 are the spacetime coordinates. Their relation to the expectation values of the relativistic (Newton-Wigner) position operators will be discussed below. We now wish t o investigate the relation of the ima,ginary coordinates

yp

-3 z p to the energy-momentum

vector. The bridge between the "classical" coordinates yp and the quantum-mechanical observables Pp will be, as usual, the (future) coherent states e,. Before getting involved in computations, let us take a closer look at these wave packets in order to get a qualitative picture. Since y p is Lorentz-invariant, it can be evaluated in a reference frame where p = (mc2,0 ) . Thus


195

with equality if and only if y = 0, i.e. when y and p are parallel. This is a kind of reverse Schwarz inequality which holds in V . x V + under the pairing provided by the Lorentzian scalar product. Thus for fixed y E V; and variable p E R,: we have

the maximum being attained when and only when p = (mc/X)y Hence we expect, roughly, that

py.

Therefore the vector y, while itself not an energy-momentum, acts as a control vector for the energy-momentum by filtering out p's which are "far" from pv. The larger we take the parameter A, the finer the filter. The expected energy-momentum can be computed exactly by noting that

where G(y) = lie, 112 as before. Since G depends on y only through the invaria,nt quantity A, we have

where we have used the recurrence relation (Abramowitz and Stegun [I9641)


a (X-VI m, which can be seen as follows. For all p, p' E 02 we have the "reverse Schwarz inequality" pp' 2 m2, with equality if and only if p = p'. Hence


197

This is a kind of "renormalization effect" due to the uncertainty, or fluctuation, of the energy-momentum in the state e,. It appears to go in the "wrong direction" (i.e., ( P ) 2 > ( P2)) for the same reason as does the inequality pp' 2 m2, namely because of the Lorentz metric. Thus ( P, ) is proportional, by a y-dependent but Po-invariant factor, to yp. We may therefore consider the 9,'s as homogeneous coodinates for the direction of motion of the classical particle in (real) spacetime. Alternatively, the expectation of the velocity operator

PIPo can

+

be shown to be y/yo. Thus of the s 1 coordinates y,, only s have a "classical" interpretation. It is important to understand that the parameter X has no relation to the mass; it can be chosen to be an arbitrary positive number and has the physical dimensions of length. It is the relativistic counterpart of the parameter u encountered in connection with the non-relativistic coherent states, and its significance will be studied later. At this point we simply note that X measures the invariant distance of z from the boundary of 7+.The larger A, the more smeared out are the spatial features and the more refined are the features in momentum space. (Recall that the imaginary part u of the time played a similar role in the non-relativistic theory. )

Because the vector y is so fundamental to our approach, it deserves a name of its own. We will call it the temper vector. The name is motivated in part by the smoothing effect which y has on spacetime quantities, and also by the fact that y plays a role similar to that played by the inverse teperature P = l / k T in statistical mechanics; the latter controls the energy.

From the asymptotic properties of the I 0 and s(z) depends only on x. Then the sdimensional manifold

is a potential generalized configuration space, and a has the "product" form

+

a = S - i R A a {x-iy € I + xES [ , ~ R:E }.

(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 a is a phase space if and only if S is a (generalized) configuration space.

Theorem 4.8.

Let a = S - iR: symplectic if and only if

be as above. Then (a,rr.) is

that is, if and only if S is nowhere timelike.

Proof. On a, we have

8s

{s, h ) = 2yp # 0, axr (16) and we may assume {s, h) > 0 without loss. For fixed x E S , the above inequality must hold for all y E a:, hence for all y E V'.. This

implies that the vector ds/dx"s

in the dual V+ of V i . I

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

Po.Note that

213

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 ad,defines a positive measure on a,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 do = (s!A x )-1 a,a.

(17)

do is the restriction to a of a 2s-form defined on d l of 7+,which we also denote by da. (This is a mild abuse of notation; in particular, the "8' here must not be confused with exterior differentiation!) By eq. (7), we have

We now derive a concrete expression for do. Since s obeys eq. (15) and ds # 0 on a, we can solve ds = 0 (satisfied by the restriction This (and a similar of ds to a ) for dxO and substitute this into procedure for y) gives

zi.

on a. Hence


We identify o with IFt2' by solving s(x) = 0 for x0 = t (x) and mapping

(t(XI - id=,

&

x - iy) u (x, y).

(21)

zo

We further identify OA with the Lebesgue measure d8y d8x on I R ~ ' (this amounts to choosing a non-st andard orientation of IFt2'). Thus we obtain an expression for do as a measure on IR*'. s(x) = 0 on a implies that

Now

which can be substituted into the above expression to give

=A ';

(1- V t . (Y/yo)) d8y d8x.

But eq. (15) implies that I Vt (x) 1

< 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 do becomes "asymptotically" degenerate as ( y I + w in the direction of V t (x). That is, if a is lightlike at ( t (x), x), then do becomes small as the velocity y / yo approaches the speed of light in the direction of V t (x). This means that functions in L2(da) (and, in particular, as we shall see, in K ) are allowed high

215


velocities in the direction Vt (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.) a )Hilbert space of all complexFor a E C, denote by ~ ~ ( dthe valued, measurable functions on a with

we restrict it to a and define 11 f 11 as If f is a CM function on 7+, above. Our goal is to show that llflla = Ilfllx: for every f E 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 ut,r of the last section to an arbitrary a = S without changing the norm. For f E K ,define

in:

where 0: has the orientation defined by

& O , so that JO

((s)

is posi-

tive. Then

where S has the orientation defined by to S does not vanish since I Vt (x) I Theorem 4.9.

go.(The restriction of zo

5 1.)

Let f(p) be Cm with compact support. Then Jp(x)

is Coo and satisfies the continuity equation


aJ'- 0.

ax'

Proof. By eq. ( 1 9 ) ,

h

where dg r d y O / y o . The function

is in L1(dg x dfi x dq"),hence by Fubini's theorem,

.

+

.

where, setting k m p q, 7 I @ and using the recurrence relation for the ICu's given by eq. ( 5 1 ) in section 4.4, we compute

k'H(7). H ( 7 ) is a bounded, continuous function of 7 for 7

> 2m,and


J p ( x ) = A;'

d ~ d ~4 X [P~ x ( P- q ) ~f(p) kq) (P" + q')

~ ( r ) ) .

(33) Since f(p) has compact support, differentiation under the integral sign to any order in x gives an absolutely convergent integral, proving that J p is Coo.Differentiation with respect to x" brings down the factor i(pp - qp) from the exponent, hence the continuity equation follows from p2 = q2 = rn2. I

Remark. The continuity equation also follows from a more intuitive, geometric argument. Let

oriented such that

(the outward normal on 8 ~ points : "down," whereas "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 dy,'s except for dyP, we have h

218


where dy is Lebesgue measure on B:. To justify the use of Stokes' theorem, it must be shown that the contribution from 1 y 1 + oo 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 a x r ay,

= 0,

which follows from the Klein-Gordon equation combined with analyticity, since

Incident ally, this shows that

is a "microlocal," spacetime-conserved probability current for each fixed y E V;, so the scalar function

I f(z) 1

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


219

red spacetime. The latter (Itzykson and Zuber [1980]) is given by

which leads to a conceptual problem since the time component, which should serve as a probability density, can become negative even for positive-energy solutions (Gerlach, Gomes and Petzold [1967], Barut and Malin [1968]). By contrast, eq. (36) shows that JO(x)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 o = S - in: E C and f E K

.

Then

llf llu = Ilf l l ~ . Proof. We will prove the theorem for in the space D(D(R8) of Coofunctions with compact support, which implies it for arbitrary f E L:(d$) by continuity. Let S be given by x0 = t (x), and for R > 0 let DR = {x E IRa+l

ER = {x E IRS+l

SOR = {x E IRS+'

I I I I

1x1 < R, x0 E [O,t(x)] }, 1x1 = R, x0 E [0,t (x)] }, 1x1 < R,xO=o},

(41)

SR= {x E IR"' 1x1 < R, x0 = t (x)}, where [0,t (x)] means [t (x), 01 if t (x) < 0. We orient SoRand SRby dxO,ER by the "outward normal" h

220


+

and DR so that 8 D R = S R - SOR ER. NOW let Then J I ( x ) is COO, hence b y Stokes' theorem,

/

J Y ( ~Z, ) =

SR-SOR+ER

kR

d (J'

= (-I)'/

Zp) dx-

DR

f(P) E V ( I R s ) .

dJC"

ax'

= 0.

(43)

We will show that

A(R)=

kR

J p Z , + 0 as R + oo

(44)

(i.e., there is no leakage to 1x1 + oo), which implies that

=

llf

1I:OA

= llf

1;

by theorem 1 of section 4.4. To prove that A(R) + 0 , note that on h

ER,dxo = 0 and

each form being defined except on a set of measure zero; hence

BY eq. (29),

hence


=s

LR

J0 i

-

o ( ~ ) .

Now by eqs. (31) and (32),

where

+(P, q) = j ( ~ j(q) ) $(P, q), and $ E c ~ ( R . ~ Hence ~).

+ E 0(IR2').

Let

D=jZ.Vp, where i = x/R, and observe that for x E ER,

where v = p/po. Since q5 has compact support, there exists a constant a < 1 such that Ivl 5 a and Ivtl 5 a for all ( p , p t ) in the support of 4. Furthermore, since (Vt(x)] 5 1, given any E > 0 we have IxOI < R ( l e) for E ER for R sufficiently large; hence

+

IE(x,p)l 1 1- a(1

+ E ) for x E ER and p E

supp 6.

(54)

222 Choose 0


< E. < a-'

- 1, substitute

into the expression for JO(x)and integrate by parts:

( i )

/

.

.

d'pdaqeix(p-q)q5:(p, q). 28

This procedure can be continued, giving (for x E ER)

where

Now (D o (-')" is a partial differential operator in p whose coefficients are polynomials in D~([-') with k = 0,1,. . . ,n. We will show that for x E ER with R sufficientlylarge, there are constants bk such that

which implies that

4.5. Geometry and Probability for some constants c,, so that by eqs (49) and (57),

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 ( 4 Dku = -

P: where

Pk is a constant-coefficient polynomial, then

hence eq. (63) holds for k = 1,2, .

which implies

. by induction.

Thus


But D k ( t - ' ) is a polynomial in [-' and DE, D2E,.. . ,D"; hence eq.

( 5 9 ) follows from eqs. (54) and (66).

1

The following is an immediate consequence of the above theorem.

Corollary 4.11. (a) For every a E C, the form

defines a %-invariant inner product on

K , under

which

K

is a

Hilbert space.

=

(b) The transformations (Ug f ) ( z ) f ( g - l ~ ) , g E PO,form a unitary irreducible representation of Po under the above inner product, and the map

j

I-+

f from ~ $ ( d f i )to K intertwines this

representation with the usual one on L:(d@). (c) For each a E C, we have the resolution of unity

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 thecaseX=O. #

4.6. The Non-Relativistic Limit 4.6. The Non-Relativistic Limit

We now show that in the non-relativistic limit c + oo, the foregoing coherent-state representation of Po reduces to the representat ion of G2 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" R:. That is, for large JyJ the solutions in K: are dampened by the factor exp[- J ~ w in momentum ] space, which in the non-relativistic limit amounts to having a Gaussian weight function in phase space.

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 let X = uc. Then

andpo =p. =

Jw. > Fix u

0

Working heuristically at first, we expect that for large c, holomorphic solutions of the Klein-Gordon equation can be approximated by

226


w

J(

""

2 ~ 2mc ) ~

exp [-umc2 w

exp [-it (mc2

+ &) + i x . p]

x

(2)

-2m - 2~ + y S p ] j ( p )

(2mc)-' exp [-irmc2 - m y 2 / 2 u ] fNR(x - i y , T ) ,

where T = t - i u and fNRis the corresponding holomorphic solution of the Schrodinger equation defined in section 4.3. Note that the Gaussian factor e x p [ - m y 2 / 2 u ] is the square root of the weight function for the Galilean coherent states, hence if we choose j ( p ) E L 2 ( R s )c L:(@), then

We now rigorously justify the above heuristic argument. Let f (r) be the function in IC corresponding to f ( p ) and denote by fc its restriction to x 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 r uc = c). But Ac = *"K,+~(2c2) =

(6)

[I + 0(~-2)]

and

I~IIZ:(~,-)5 ( 2 ~ ) - ~ ( 2 ~ ) - ~ 1 i 1 1 2 ~ ~ ~ . ) .(7) Thus

112ce~~fc11~2~tp) 5 (4*)-s1211/112z(~9 [1

+ 0 ( ~ - ~ ,) ]

showing that 2cec% approaches a limit in L~((CS) as c

Choose a,y such that 1/2

(8)

m. Now

< y < a < 1. Then

I

where xCis the indicator function of the set {p lpl > cl-a}. Define 0 and 4 by lyl = csinh0 and Ipl = csinh4. Then yo = cosh8 and w = c2 cosh 4, hence

228


Thus for arbitrary a 2 0,

J

Ga(p)

day e

2 ~ Z - 2 ~

lyl>ccosh a

:(Ti; la

dB sinhs-I B cosh Be-' dB e("-')' (es

cl-7

and

Finally,

where

1

< -IY2 2c2

- P21

5

+~

;

(c-27

- 2 ~ )

- C-~Y. < We have used the estimate

JT;;~-JS~=~/' "" , / U

G

f

5 l / l v x d x ~= v 2 -u21. Hence for sufficiently large c and Ipl < cl-",

(20)


-

h(c)

-,0

as c -, oo.

Thus

which proves that J(c) -, 0 as c + oo.

I

Notes

This chapter represents the main body of the author's mathematics thesis at the University of Toronto (Kaiser 11977~1).All the theorems, corollaries, lemmas and propositions (labeled 4.1-4.12) have appeared in the literature (Kaiser [1977b, 1978a1). In 1966, when the idea of complex spacetime as a unification of spacetime and phase space first occurred to me, I had found a kind of frame in which both the bras and the kets were holomorphic in 3 and the resolution of unity was obtained by a contour integral, using Cauchy's theorem. During a seminar I gave in 1971 at Carleton University in Ottawa (where I

Notes

231

was then a post-doctoral fellow in physics), L. Resnick pointed out to me that this "wave-packet representation" appeared to be related to the coherent-state representation, which was at that time unknown to me. The kets were identical to the canonical coherent states, but the bras were not their Riesz duals; in the language of chapter 1, they belonged to a (generalized) frame reciprocal to that of the kets, and the resolution of unity was of the type given by eq. (24) in section 1.3, which may be called a continuous version of biorthogonality. A version of this result was reported at a conference in Marseille (Kaiser [1974]). I was later informed by J. R. Klauder that a similar representation had been developed by Dirac in connection with quantum electrodynamics (Dirac [1943, 19461). The original idea of complex spacetime as phase space was to consider a complex combination of the (symmetric) Lorentzian metric with the (antisymmetric) symplectic structure of phase space, obtaining a hermitian metric on the complex spacetime parametrized by local coordinates of the type x ibp. (I have since learned that this structure, augmented by some technical conditions, is known as a K a l e r metric; see Wells [1980].) The above "wavepacket representation" indicated that this combination may in fact be interesting, but so far it was ad hoc and lacked a physical basis. Also, the representation was non-relativistic, and it was not at all clear how to extend it to the relativistic domain, as pointed out to me by V. Bargmann in 1975. The standard method of arriving at canonical coherent states is to use an integral transform with a Gaussian kernel in the configuration-space representation, and there is no obvious relativistic candidate for such a kernel. The more general methods described in chapter 3 do not work, since the representations of interest are not squareintegrable (section 4.3). An important clue came in 1974 from the study of axiomatic quantum field theory, where I

+

232


was fascinated by the appearance of tube domains. These domains occur in connection with the analytic continuation of vacuum expectation values of products of fields, and are therefore extensions of such products to complex spcetime. However, the complexified spacetimes themselves are not taken seriously as possible arenas for physics. They are merely used to justify the application of powerful methods from the theory of several complex variables, in order to obt ain results concerning the restrict ions of vacuum expectaion values to real spacetime. (However, the restrictions to Euclidean spacetime do have import ant consequences for st at istical mechanics; see Glimm and JaiTe [I9811.) I felt that if these tube domains could somehow be given a physical interpretation as extended classical phase spaces, this would give the phase-space formulation of relativistic quantum mechanics a firm physical foundation, since in quantum field theory the extension to complex spacetime is based on solid physical principles such as the spectral condition. This idea was first worked out at the level of non-relativistic quantum mechanics, leading to the representation of the Galilean group given in section 4.3. That amounted to a reformulation of the canonical coherent-state representation in which the Gaussian kernel appears naturally in the momentum representation, as a result of the analytic continuation of solutions of the Schrodinger equation. This "explained" the combination x ibp (section 4.3, eq. (5)) and gave the coherent-state representation a dynamical significance. It also cleared the way to the construction of relativistic coherent states, since now the Gaussian kernel merely had to be replaced with the analytic Fourier kernel e-'"p on the mass shell. An important tool was the use of groups to compute certain invariant integrals, which I learned from a lecture by E. Stein on Hardy spaces in 1975. The construction of the relativistic coherent states given in sections 4.4 and 4.5 was carried out in 1975-76, culminating

+

Notes

233

in the 1977 thesis. Related results were announced at a conference in 1976 (Kaiser [1977a]) and at two conferences in 1977 (Kaiser [1977d, 1978b1). To my knowledge, this was the first successful formulation of relativistic coherent states, which have since then gained some popularity (see De Bikvre [1989], Ali and Antoine [1989]). An earlier attempt to formulate such states was made by PrugoveEki [1976], but this was shown to be inadequate since the proposed states were merely the Gaussian canonical coherent states in disguise, hence not covariant under the Poincark group (Kaiser [1977c], remark 4 in sec. 11.5 and addendum, p. 133.) After the results of the thesis appeared in the literature, PrugoveEki [1978; see also 19841 discovered that they can be generalized by replacing the invariant functions e-w with arbitrary (sufficiently regular) invariant functions. The price of this generalization is that solutions of the Klein-Gordon equation are no longer represented by holomorphic functions and the close connection with quantum field theory (chapter 5) appears to be lost. The relation between the two formalisms and their history was discussed at a conference in Boulder in 1983 (Kaiser [1984b]),where an inconsistency in PrugoveEki's formalism was also pointed out. The classical limit of solutions of the Klein-Gordon equation in the coherent-state representation was studied in Kaiser [1979]. In an effort to underst and interactions, the notion of holomorphic gauge theory was introduced (Kaiser [1980a, 19811). This is reviewed in section 6.1. An early attempt was also made to extend the theory to the framework of interacting quantum fields (Kaiser [1980b]), but that was soon abandoned as unsatisfactory. A more promising approach was developed later (Kaiser [1987a]) and is presented in the next chapter. Note that our phase spaces a are not unique, since the configura-

234


tion space S can be chosen arbitrarily as long as it is nowhere timelike and X > 0 can be chosen arbitrarily. The freedom in S is, in fact, related to the probability-current conservation, while the freedom in A, combined with holomorphy, allowed us to express the probability current as a regularization of the usual current by the use of Stokes' theorem (section 4.5, eqs. (37) and (40)). By contrast, the phase spaces obtained by De Bikvre [I9891 are unique. They are "coadjoint orbits" of the Poincark group, related to "geometric quantization" theory (Kirillov [1976], Kostant [1970], Souriau [1970]). Although this uniqueness seems attractive, it involves a high cost: the dynamics must be factored out. This means that the ensuing theory is no longer "local in time." Since one of the attractions of coherent-state representations is their "pseud~locality"in both space and momentum, and since in a relativistic theory time ought to be treated like space, it seems to me an advantage to retain time in the theory. Perhaps a more persuasive argument for this comes from holomorphic gauge theory (section 6.1), where a theory describing a free particle can, in principle, be "perturbed" by introducing a non-trivial fiber metric to obtain a theory describing a particle in an electromagnetic (or Yang-Mills) field. This cannot be done in a natural way once time has been factored out. Some very interesting work done recently by Unt erberger [I9881 uses coherent states which are essentially equivalent to ours to develop a pseudodifferential calculus based upon the Poincarh group as an alternative to the usual Weyl calculus, which is based on the Weyl-Heisenberg group. Since the Poincard group contracts to a group containing the Weyl-Heisenberg group in the non-relativistic limit (section 4.2), Unterberger's "Iclein-Gordon calculus" similarly contracts to the Weyl calculus.

Chapter 5 QUANTIZED FIELDS

5.1. Introduction We have regarded solutions of the Klein-Gordon equation as the quantum states of a relativistic particle. But such solutions also possess another interpretation: they can be viewed as classical fields, something like the electromagnetic field (whose components, in fact, satisfy the wave equation, which is the Klein-Gordon equation with zero mass). This interpretation is the basis for quantum field theory. The general idea is that just as the finite number of degrees of freedom of a system of classical particles was quantized to give ordinary ("point") quantum mechanics, a similar prescription can be used to quantize the infinite number of degrees of freedom of a classical field. It turns out that the resulting theory implies the existence of particles. In fact, the asymptotic free in- and out-fields are represented by operators which create and destroy particles and antiparticles, in agreement with the fact that such creation and destruction processes occur in nature. These particles and antiparticles are represented by posit ive-energy solutions of the asymptotic free wave equation, e.g. the Iclein-Gordon or Dirac equation. Thus the formalism of relativistic quantum mechanics appears to be, at least partially, absorbed into quant um field theory.

236

5. Quantized Fields

In regarding solutions of the Klein-Gordon equation as the physical states of a relativistic particle, it was appropriate to restrict our attention to functions having only posit ive-frequency Fourier components, since the energy of the particle must be positive. Even a small negative energy can be made arbitrarily large and negative by a Lorentz transformation, leading to instability. When the solutions are regarded as classical fields, however, no such restriction on the frequency is necessary or even justifiable. For example, in the case of a neutral field (i.e., one not carrying any electric charge), the solutions must be real-valued, hence their Fourier transforms must contain negative- as well as positive-frequency components. On the other hand, the analytic extension of the solutions to complex spacetime appeared to depend crucially on the positivity of the energy. We must therefore ask whether an extension is still possible for fields, or if it is even desirable from a physical standpoint, since the connection between solutions and particles is not as immediate as it was earlier. In this chapter we find an affirmative answer to both of these questions. A natural method, which we call the Analytic-Signal transform, will be developed to extend arbitrary functions from IR'+' to p + 1 , and when the functions represent physical fields, the double tube 7 = 7+U 7-in C8+' will be shown to have a direct physical significance as an extended classical phase space, not for the fields themselves but for certain "particlen- and "antiparticle" coherent states e$ associated with them. These states are related directly to the dynainical (interpolating) fields, not their asymptotic free in- and out-fields. To be precise, they should be called charge coherent states rather than particle coherent states, since they have a well-defined charge whereas, in general, the concept of individual particles does not make sense while interactions are present. If the given fields satisfy some (possibly non-linear) equations, the coherent states satisfy

5.1. Introduction

237

a Klein-Gordon equation with a source term. Hence they represent dynamical rather than "bare" particles. For free fields, e z reduces to the state e, defined in the last chapter and e; to its complex conjugate, which is a negative-energy solution of the Klein-Gordon equation holomorphic in I-. Complex tube domains also appear in the contexts of axiomatic and constructive quantum field theory, and our results suggest that those domains, too, may have interpretations related to classical phase space, a point of view which, to my knowledge, has not been explored heretofore.* While our extended fields are not analytic in general, they are "analyticity-friendly," i.e. have certain features which yield various analytic objects under different circumstances. For example, their two-point functions are piecewise analytic, and the pieces agree with the analytic Wightman functions. In the special case when the given fields are free, the extended fields themselves are analytic in 7. f i r thermore, the fields in general possess a directional analyticity which looks like a covariant version of analyticity in time. Since the latter forms the basis for the continuation of the theory (in the form of vacuum expectaion values) from Lorentzian to Euclidean spacetime (see Nelson [1973a,b] and Glirnrn and J&e [1981]), it may be that our extended fields, when restricted to the Euclidean region, bear some relation to the corresponding Euclidean fields. The formalism we are about to develop for fields is a natural

*

R. F. Streater has recently told me that G. Kiillkn was informally

advocating the interpretation of the holomorphic Wightman twopoint function as a correlation function in phase space around 1957. Nothing appears to have been published on this, however.

5. Quantized Fields

238

extension of the one constructed for particles in the last chapter. Like its predecessor, it posesses a degree of regularity not found in the usual spacetime formalism. Some examples of this regularity are: (a) The extended fields $ ( z ) are, under reasonable assumptions, operator-valued functions (rather than distributions, as usual) when restricted to 7. (b) The theory contains a natural, covariant ultraviolet damping which is a permanent feature of the theory. This comes from the possibility of working directly in phase space, away from real spacetime. From the point of view of the usual (real spacetime) theory, our formalism looks like a LLregularization".From our point of view, however, no regularization is necessary since, it is suggested, reality takes place in complex spacetime! In other words, this "regularization" is permanent and is not to be regarded as a kind of trick, used to obtain finite quantities, which must later be removed from the theory. (c) In the case of free fields, the formalism automatically avoids zero-point energies without normal ordering, due to a polarization of the positive- and negative frequency components into the forward and backward tubes, respectively. Observables such as charge, energy-momentum and angular momentum are obtained as conserved integrals of bilinear expressions in the fields over phase spaces a c 7.These expressions, which are densities for the corresponding observables, look like regularizations of the corresponding expressions in the usual spacetime theory. The analytic (Wightman) two-point function acts as a reproducing kernel for the fields, much as it did for the wave functions in chapter 4.

(d) The particles and antiparticles associated with the free Dirac

5.2. The Multivariate Analytic-Signal lI-andorm

239

field do not undergo the random motion known as Zitterbewegung (Messiah [1963]),again because of the aforementioned polarization.

5.2. T h e Multivariate Analytic-Signal Transform

As mentioned above, in dealing with physical fields such as the electromagnetic field, rather than quantum states, we can no longer justify the restriction that frequencies must be positive. For one thing, as we shall see, in the presence of interactions there is no longer a covariant way to eliminate negative frequencies. Hence the method used in chapter 4 to analytically continue solutions of the KleinGordon equation to complex spacetime will no longer work directly. In this section we devise a method for extending arbitrary functions from IR"~ to W1.When the given functions are positive-energy solutions of the Klein-Gordon or the Wave equation, this method reduces to the analytic extension used in chapter 4. But it is much more general, and will enable us to extend quantized fields, whether Bose or Fermi, interacting or free, to complex spacetime. We begin by formulating the method for functions of one variable, where it is closely related to the concept of analytic signals. For motivational purposes, we think of the variable as time (s = 0). In this chapter, Fourier transforms will usually be with respect to spacetime (R"') rather than just space (IRs). Hence we will denote them by f, reserving f for the spatial Fourier transform, as done so far. Suppose we are given a "time-signal," i.e. a real- or complexvalued function of a single real variable x. To begin with, assume that f is a Schwartz test function, although most of our considerations will

5. Quantized Fields

240

extend to certain kinds of distributions. Consider the positive- and negative- frequency parts of f , defined by

f+(x)

I( 2 ~ ) - '

1" 1"

dpe-'"~ f(p)

0

f-(x)

(27r)-l

(1)

dP e-izp f ( P ) .

Then f+ and f- extend analytically to the lower-half and upper-half complex planes, respectively, i.e.

f

-)

=(

2 )

I" 0

f- ( x - i y ) = ( 2 ~ ) - '

dp e-'("-'p)p

f@),

Y

d p e - i ( x - i y ) p f"( P ) ,

>0 (2)

Y 0), then f*(z) = 1 e x p ( ~ i a z )so , the boundary values are fk(x) = $ e x p ( ~ i a x ) . In order to extend the concept of analytic signals to more than one dimension, let us first of all unify the definitions of f+ and f- by defining f ( X - iy) E (2n)-l

I"

dp g(yp) e - ' ( ~ - ' ~ )f~( ~ )

(7)

J-00

for arbitrary x - i y E

a, where 8 is the unit step function, defined by

Then we have

(9) f-(z),

YO

Qo = {P I p2 = 0, P # 0) Q o o = (0)

Qim={plp2=-m2),

m>O,

which are, in fact, the various orbits of the full Lorentz group L. Greenberg [I9621 has shown that q5 is a generalized free field (i.e., a sum or integral of free fields of varying masses m 2 0) if vanishes

6

on any of the following types of sets:

5.3. Axiomatic Field Theory and Particle Phase Spaces

257

He has also shown, by giving counter-examples, that this conclusion cannot be drawn if vanishes on sets of the type

4

(See also Dell'Antonio [I9611 and Robinson [1962].) Note: Up to now, we have not assumed that the field satisfies the canonical commutation relations (section 5.4), hence our conclusions are quite general and should hold for an arbitrary (system of mutually) interacting field(s). The "Lie algebra" generated by the fields (obtained by including, along with the fields, their commutators [$(p), $(P')] as we11 as higher-order commutators) has a very interesting formal structure, although it is not a Lie algebra in the usual sense. (For one thing, it is uncountably infinite-dimensional rather than finite-dimensional.) Namely, the above relations suggest that the operators Pp be regarded as belonging to a Cartan subalgebra and that d(p) (with p in the support of

4) is a root vector with as-

sociated root -p. The spectrum C is therefore reminiscent of a set

A+ of positive roots. In general, the Cartan subalgebra consists of a maximal set of commuting observables. When considering charged fields, the charge will also belong to the Cartan subalgebra, with root values 0, fc, f2c, . . ., where E is a fundamental unit of charge. The vacuum is a vector (not in the Lie algebra but in an associated representation space) of "highest" (or lowest) weight which, as in the finite-dimensional case, generates a representation of the algebra because of its cyclic property. To my knowledge, this important analogy between the structures of general quantized fields (i.e., apart from the canonical commutation relations or any particular models of interac-

258

5. Quantized Fields

tions) and Lie algebras has not been explored, although the methods of Lie-algebra theory could add a powerful new tool to the study of quantized fields. (In a somewhat different context, the structures of quantized fields and infinitedimensional Lie algebras are united in string theory; see Green, Schwarz and Witten [I9871.) # Let us now formally extend the quantized field #(x) to C8+', using the Analytic-Signal transform developed in section 5.2. Recall that this transform was originally defined for Schwartz test functions. In principle, we would like to define 4(z) by using its distributional Fourier transform $(p):

This presents us with a technical problem, as already noted in the last section, since 0-"P is not a Schwartz test function in p. One way out is to smear 4(z) with a test function f ( z ) over C8+l. Although this is the safest solution, it is not very interesting since not much appears to have been gained by extending the field to complex spacetime: the new field is still an operat or-valued distribution. However, we shall see that there are reasons to expect 4(z) to be more regular than a "generic" Analytic-Signal transform, due in part to the fact that 4(x) satisfies the Wightman axioms. When 4 is a (generalized) free field, the restriction of $ ( z ) to the double tube 7 turns out to be a holomorphic operator-valued function. We will see that even for general Wightman fields, 7 is an important subset of C8+'. In the presence of interactions, holomorphy is lost but some regularity in 7 is expected to remain. We now proceed to find conditions which do not force # to be a generalized free field but still allow $(z) to


259

be an operator-valued function on 7.The arguments given below have no pretense to rigor; they are only meant to serve as a possible framework for a more precise analysis in the future. All statements and conditions concerning convergence, integrability and decay of operator-valued expressions are meant to hold in the weak sense, i.e. for matrix elements between fixed vectors. Since the operators involved are unbounded, we must furthermore assume that the vectors used to form the matrix elements are in their (form) domains. For a fixed timelike "temper" vector y, 8-"p fails to be a Schwartz test function in two distinct ways: (a) It has a discontinuity on the spacelike hyperplane Ny = {p I yp = 01,and (b) it has a constant modulus on hyperplanes parallel to N9, hence cannot decay there. On the other hand, by relativistic covariance, the support of must be smeared over the orbits of L,given by eq. (16). This gives a "stratification" of as a sum of tempered distributions

4

4

with support properties

Although fl+ and f2- contain

no,the distributions $+ and $-

have no contributions from p2 = 0. (For example, the distribution

1+6(p) on IR has a decomposition TI+To where TI has support IR and

260

5. Quantized Fields

To has support (0) c IR, but the p = 0 contribution to TIvanishes.) Jo has no contribution from Similarly, although Ro contains aO0, p = 0. Corresponding to the above decomposition of 6, we have, formally, 9%~= ) 4+(4

+ 4o(z) + $oo(z) + 4-(z).

(22)

We will show that 1. $+(z) and $0 (z) are holomorphic operator-valued functions in 7; 2. c,hoo(z)must be a constant field, to be physically reasonable; and 3. under certain (hopefully not too restrictive) conditions, 4- (z), though not holomorphic, is an operator-valued function on 7.

First of all, we claim that each of the fields 4,) ( a = f, 0,OO) is still covariant under Po.* To see this, take the Fourier transform of eq. (1) and use the invariance of Lebesgue measure d p under Lo. This leads to &p) = eiapU(a, A) $ ( A ' ~ )U(a, A)',

(23)

where A' is the transpose of A E Lo. Since the different components are "essentially" supported on disjoint subsets and these subsets are invariant under Lo, we conclude that eq. (23) holds for each To show that d+(z) and 40(z) are holomorphic in 7+,let z E 7+. Since O-'+p vanishes for p E V-\{O), and since and have no contribution from p = 0, we have

4,.

4+

*

However, 4, need not satisfy other Wightman axioms such as locality; for example, q5+ need not be a generalized free field.


261

V+

Now e-"P may be regarded as the restriction to of a Schwartz test function fz(p) which is of compact support in p for each fixed po and vanishes when po < -E, for some E > 0. Thus $+(z) and $o(z) make sense as operator-valued functions on 7+, and they are cleary holomorphic there. (This will be shown explicitly below.) A similar analysis shows that the same can be done for z E 7-. Next, we consider boo(z). Since 4 0 0 is supported on { p = 01, it must have the form P ( d ) 6(p), where P(d) is a partial differential operator. In the x-domain (i.e., in real spacetime), this corresponds to P(-ix), for which the Analytic-Signal transform is not well-defined, although it is possible that a regularization procedure would cure this. But in any case, non-constant polynomials in x do not appear to be of physical interest since they correspond to unbounded fields even in the classical sense (as functions of x). Hence we assume that (a)

doe@) = 2AS(p), where A is a constant operator.

This corresponds to a constant field $(x) = 2A and, correspondingly, 4oo(z) A (see eq. (15) in section 5.2). In order that +(z) be an operator-valued function on 7, it therefore remains only for 4-(z) to be one. Note that so far, the only assumption we needed to make, in addition to the Wightman axioms, was (a). To make &(z) a function, we now make our second assumption: (b)

6-(p) is integrable on all spacelike hyperplanes. 4

=

the integral of over the hyperplane HY,, grows at most polynomially in v .

firthemore, { p I yp = v ) ( y E V')

It is not clear what specific minimal conditions on

4-

produce this

property. The integral occurring in (b) is known as the Radon trans-

262 form ( ~ d ) (v ~ ) of,

5. Quantized Fields

6 when y is a (Euclidean) unit vector, and will

be further discussed in section 6.2. (See also Helgason [1984].) Unlike the Fourier transform, the Radon transform does not readily generalize to tempered distributions (which were, after all, designed specifically for the Fourier transform). However, it does extend to distributions of compact support and can be further generalized to distributions with only mild decay. Also, the relation of assumptions

(a), (b) (or their future replacements, if any) to the Wightman axioms needs to be investigated. In order to compute d-(z) for z E 7 it suffices, by covariance, to do so for x = 0 and y = (u, 0), for all u # 0. The analyses for u > 0 and u < 0 are similar, so we restrict ourselves to u > 0. Eq. (19) then gives

For fixed po 2 0, condition (b) implies that the integral over p converges, giving an operator-valued function F(po) which is of at most polynomial growth in po . 4-( -iu, 0) is then the Laplace transform of F(po), which is indeed well-defined. Note: The behaviors of d(z) and ?(p) exhibit a certain duality which reflects the dual nature of y E R'+' and p E (lR8+')* (section 1.1). We have just seen that when d(p) behaves reasonably for spacelike p, then $ ( z ) behaves reasonably for timelike y. In the trivial case when $(p) 0 for p2 < 0 and (a) holds, d(z) is holomorphic for y2 > 0. In fact, 4 is then a generalized free field, hence may be said to be "trivial." This dual behavior also extends to p2 2 0 and any non-constant field, d(p) is non-trivial for

y, the hyperplane

p2

y2

< 0: For

2 0; for spacelike

H y," (which contains timelike as well as spacelike


263

directions) therefore intersects the support of 6 in a non-compact set and we do not expect $(z) to make sense as an operator-valued function outside of 7. # No claims of analyticity can be made for d(z) in general. In fact, Greenberg's results show that d(z) may not be analytic anywhere in CS+l unless 4 is a generalized free field. For as in the classical case, formal differentiation with respect to 2, gives

hence 47ri8, 4 is the inverse Fourier transform of p,J in the hyperplane Ny. If $ is not a generalized free field, then, according to Greenberg, the support of contains sets of timelike as well as sets of spacelike p's with positive Lebesgue measure. Hence, for any nonzero y E IR'", the intersection of the support of 6 with N , has positive measure in N y ,so 4 will not be holomorphic at x - iy in general. As in the classical case, however, the above equation for 44 implies that $(z) is holomorphic along the vector field y, i.e.,

4

This is a covariant condition which, when specialized to y = (yo,0), states that $(z) is holomorphic in the complex time-direction. As we have seen, this result simply follows from the nature of the AnalyticSignal transform. A similar situation forms the basis of Euclidean quantum field theory. However, there one is dealing not directly with the field but with its vacuum expectation values, and the mathematical reason for the analyticity is the spectral condition, which would appear to have little in common with the Analytic-Signal transform.

264

5. Quantized Fields

Incidentally, eq. (26) provides a simple formal proof that 4+(z) is holomorphic in 7. The support of d+(p) is contained in V , hence for any y in V', its intersection with N , is either empty or equal to Roo. But the contribution from p = 0 vanishes, hence eq. (26) shows that %4+(z) = 0 in 7. The same argument also shows that free fields and generalized free fields are holomorphic in 7.In the next two sections we shall study free Klein-Gordon and Dirac fields in more detail. Although d ( z ) is not holomorphic in general, we will be able to establish for it one essential ingredient of the foregoing phasespace formalism, namely the interpretation of the double tube 7 as an extended classical phase space for certain "particles" and "antiparticles" associated with the quantized field 4. First, let us expand the above considerations to include charged fields by allowing $(x) to be non-Hermitian (i.e., a non-Hermitian operator-valued distribution). Then the extended field $(z) need not . charge Q is defined satisfy the reflection condition )(z)* = r j ( ~ ) The as a self-adjoint operator which generates overall phase translations of the field, i.e.,

for real a , where e is a fundamental unit of charge. This implies

showing that )(x) and &) remove a unit E of charge from the field, while their adjoints add a unit of charge. We assume that phase translations commute with Poincark transformations, and in particular with spacetime translations. Thus


265

so charge is conserved. Q can be included in the "Cartan subalgebra" containing the P, 's, and the above commutation relations show that d(p) and &p)* are still "root vectors," with Q-root values -E and E , respectively. We also assume that the vacuum is neutral, i.e. Q\ko = 0. Repeated applications of r$ and to Bo show that the spectrum of Q is (0, fE , f2 ~ .,. .).

6'

Recall that the commutation relation between J(p) and P, imremoves an energy-mometum p from the field. Sirniplied that j@) larly, its adjoint relation

shows that &p)* adds an energy-momentum p to the field. In place of the generalized eigenvectors Q p of energy-momentum which we had for the Hermitian field, we can now define two eigenvectors,

Qi a $(p)* Qo 9;

= &-P)

Qo,

for each p E El. For a non-Hermitian field, these vectors are independent. They are states of charge e and -&, respectively. We may think of them as particles and antiparticles, although they do not have a well-defined mass since p2 will be variable on El, unless 4 is a free field. Each p # 0 in El belongs to the continuous spectrum of the P,'s, since it can be changed continuously by Lorentz transformations. Hence the "vectors" 9: are non-normalizable. Since the and nd; belong to different eigenP,'s are self-adjoint, and since values of the charge operator (which is also self-adjoint), we have

5. Quantized Fields

266

(with the usual abuse of Dirac notation, where "inner products" of distributions are taken)

where a, a distribution with support in El,depends only on p2 by Lorentz invariance. (Charge symmetry requires that a be the same for particles as for antiparticles.) If 4 is the free field of mass m > 0,

- e (PO)2n6(p2 - m2) = 2n6(p2 - m2) in V+\{O). a(p2 ) Now define the particle coherent states by

Like the Q:'s, these do not have a well-defined mass; in addition, they are wave packets, i.e. have a smeared energy-momentum, but they still have a definite charge e . Their spectral components are given by

If z belongs to the backward tube 7-,then yp < 0 on V+\{O), hence e$ = 0. If z belongs to the forward tube 7+, then yp > 0 and the vector e$ is weakly holomorphic in 2. For the free field, it reduces to the coherent state e, defined in chapter 4. Similarly, define the coherent antiparticle states by


267

These are wave packets of charge -e for which

Thus e l vanishes in the forward tube and is weakly holomorphic in the backward tube. In the usual formulation of quantum field theory, particles are associated not directly with the interacting, or interpolating, field

4

but with its asymptotic fields &, and q5,,t, which are free. (We will construct such free-particle coherent states in the next two sections.) However, the coherent states e$ are directly associated with the interpolating field. We shall refer to them as interpolating particle coherent states (section 5.6). We are now ready to establish the phase-space interpretation of

7 in the general case. We will show that 7+and 7-are extended phase spaces associated with the particle- and antiparticle coherent states e$ and e; , respectively, in the sense that they parametrize the classical states of these particles. We first discuss x as a "position" coordinate. In the case of interacting fields there is no hope of finding even a "bad" version of position operators. Recall that position operators were in trouble even in the case of a one-particle theory without interactions! In the general case of interacting fields, this problem becomes even more

5. Quantized Fields

268

serious, since one is dealing with an indefinite number of particles which may be dynamically created and destroyed. (As argued in section 4.2, the generators MOkof Lorentz boosts qualify as a natural, albeit non-commutative, set of center-of-mass operators; although I believe this idea has merit, it will not be discussed here.) Since no position operators are expected to exist, we must not think of x as eigenvalues or even expect ation values of anything, but rat her simply as spacetime parameters or labels. On the other hand, y will now be shown to be related to the expectations of the energy-momentum operators (which do survive the transition to quantum field theory, as we have seen). For z, z' E 7+,we have

where we have set m2

= P2 and used

(27r)-"-' dp = (27r)-'-' with

-

d@

-

[z(27r)'

dpo dSp = (27r)-' dm2 d@,

d m ]-' dsp

(39)

(40)

the Lorentz-invariant measure on 52h. A+(w; m) is the two-point function for the free Klein-Gordon field of mass m, analytically continued to w z' - f E I+. In the limit y, y' -+ 0, this gives the


269

KdGn-Lehmann representation (Itzykson and Zuber [1980]) for the usual two-point function,

( @o I$(.')

d(~)**o) =

dm2 a(m2)~ + ( s-' I; m ) , (41)

which is a distribution. In Wightman field theory, such vacuum expect at ion values are analytically continued using the spectral condition, and conclusions are drawn from these analytic functions about the field in real spacetime. In our case, we have first extended the field (albeit non-analytically), then taken its vacuum expectation values (which, due to the spectral condition, are seen to be analytic functions, not mere distributions). The fact that we arrived at the same result (i.e., that "the diagram commutes") indicates that our approach is not unrelated to Wightman's. However, there is a fundamental difference: The thesis underlying our work is that the "red" physics actually takes place in complex spacetime, and that there is no need to work with the singular limits y -+ 0. The norm of ef is given by

where G(y; m), computed in section 4.4, is given by

fl,

Recall that X E v E (s - 1)/2 and I{. is a modified Bessel function. We assume that e$ is normalizable, which means that the spectral density function a(m2) sat i d e s the regularity condition

5. Quantized Fields

(This condition is automatically satisfied for Wightman fields, where it follows from the assumption that 4 is a tempered distribution; however, it is also satisfied by more singular fields since I{, decays exponentially.) It follows that

Using the recursion relation (Abramowitz and Stegun [1964])

we find that the state e$ has an expected energy-momentum

where

We call mx the effective mass of the particle coherent states; it generalizes the corresponding quantity for Klein-Gordon particles (section 4.4). The name derives from the relation


271

It is import ant to keep in mind that in quantum field theory, the natural picture is the Heisenberg picture, where operators evolve in spacetime and states are fixed. Recall that for a free Klein-Gordon particle (chapter 4), we interpreted e, as a wave packet focused about the event x E Rz and moving with an expected energy-momentum (mx/X)y. This suggests that the above states e> be given a similar interpretation. Thus z becomes simply a set of labels parametrizing the classical states of the particles. This establishes the interpretation of 7+as an extended classical phase space associated with the "particle" states e.:

A similar

computation shows that 7. acts as an extended classical phase space for the "antiparticle" states e,

, whose expected energy-momentum

is

The expected angular momentum in the states ef can be computed similarly. The angular momentum operator Mpu is the generator of rotations in the p-v plane, hence

This implies for the Fourier transform

Since the vacuum is Lorentz-invariant, we have Mpu\ko = 0, hence for z E 7+,

272

5. Quantized Fields

provided that c$(~)vanishes on the boundary of

V + (this excludes

massless fields). The expectation of M,, is therefore related to that of P, by

Similarly, in e, with z E 7-,

This section can be summarized by saying that the vector y plays a similar role for general quantized fields as it did for positive-energy solutions of the Klein-Gordon equation, namely it acts as a control vector for the energy-momentum. In other words, the function 8(yp) e-yp acts as a window in momentum space, filtering out from each mass shell R,

momenta which are not approximately parallel

to y. The step function 8(yp) makes certain that only parallel components of p pass through this filter by eliminating the antiparallel ones (which would make the integrals diverge). Thus we may think of B(yp) e-vp as a kind of "ray filter" in

v,when y E V'. We continue

to refer to y as a temper vector (section 4.4).

5.4. n e e Klein-Gordon Fields Note: The regularity condition given by eq. (44) for a ( m 2 ) ,i.e. the requirement that ef be normalizable, shows that X acts as an effective ultraviolet cutoff, since KV(2Xm)decays exponentially as m --, w, giving finite values to mx and other quantities associated with the field.

#

5.4. Free Klein-Gordon Fields In the context of general quantum field theory, we were able to show that 7 plays the role of an extended phase space for certain "particle" states of the fields. The question arises whether the phase-space formalism of chapter 4 can be generalized to quantized fields. There, we saw that all free-particle states in the Hilbert space could be reconstructed from the values of their wave function on any phase space a c 7+. There are two possible ways in which this result might extend to quantized fields: (a) The vectors e$ belong the subspaces Wkl with charge fE , hence we may try to get continuous resolutions, not of the identity on 3-1 but of the orthogonal projection operators to in terms of these vectors. (This can then be generalized to the resolution of the projection operator

71, with charge nc, n E

II, to the subspace

z:)(b) The global observables of the the-

ory, such as the energy-momentum, the angular momentum and the charge operators, are usually expressed as conserved integrals of the field operators and their derivatives over an arbitrary configuration space S in spacetime (i.e., an s-dimensional spacelike submanifold of IFtJS1); our approach would be to express them as integrals of the extended fields over 2s-dimensional phase spaces a in 7, much as the inner products of positive-energy solutions were expressed as

274

5. Quantized Fields

such integrals. In this section we do both of these things for the free Klein-Gordon field of mass m > 0, which is a quantized solution of

We consider classical solutions at first. The Fourier transform &p) has the form

for some complex-valued function a(p) defined on the two-sheeted mass hyperboloid Q, = R$ U 0;. Write

If the field is neutral, then #(x) is real-valued and b ( p ) r a(p). For charged fields, a(p) and b(p) are independent. At this point, we keep both options open. Then

The extension of #(x) to comp~exspacetime given by the AnalyticSignal transform is

5.4. Ree Klein-Gordon Fields

If y E V;, then yp > 0 for all p E Qk, hence

is analytic in 7+,containing only the positive-frequency part of the and field. Similarly, when y E VL, then yp < 0 for all p E

Thus + ( z ) is also analytic in 7-, where it contains only the negativefrequency part of the field. However, note that the two domains of analyticity 7+and 7- do not intersect, hence the corresponding restrictions of + ( z ) need not be analytic continuations of one another. We are now ready to quantize $(z). This will be done by first quantizing +(x) and then using the Analytic-Signal transform to extend it to complex spacetime. We assume, to begin with, that $ is a neutral field, so b(p) a(p). According to the standard rules (Itzykson and.Zuber [1980]) of field quantization, $(x) becomes an operator on a Hilbert space 'FI such that at any fixed time xo, the field "configuration" operators $(so, x) and their conjugate "momenta" do $(so, x) obey the equal-time commutation relations

5. Quantized Fields

[r(x),$("')Iz;=zo = 0 [4(x),~04(x')lz,=zo= q x - XI).

(8)

This is an extension to infinite degrees of freedom of the canonical commutation relations obeyed by the quantum-mechanical positionand momentum operators. Note that since time evolution is to be implemented by a unitary operator, the same commutation relations will then hold at any other time as well. For the non-Hermitian operators a(p), the corresponding commutation relations are

PI, a(p1)l = PO POP^) ( 2 ~ )&(P ' + P') for p, p' E R,.

(9)

Using the neutrality condition a(-p) = a(p)*, these

can be rewritten in their conventional form [a(p),a(pt)l = 0 [u(P),a(p1)*1= 2~ ( 2 ~ ) &(P ' - P') where now p, p' E R i .

A charged field can be built up from a pair of neutral fields as

where the two fields d2 each obey the equal-time commutation relations and commute with one another. Equivalently, the operators a(p) and b(p) become independent and satisfy

for p, p' E R i . The canonical commutation relations for both neutral and charged fields can be put in the manifestly covariant form

5.4. n e e Klein-Gordon Fields

277

= sign(po)( 2 ~ ) * 6(p2 + ~ - ma) 6(p - p')

(13) for arbitrary p,pl E 1 ~ " ~ For . charged fields, this must be supplemented by

Recall that in the general case we had

v+,

where o(p2) is the spectral density for the two-point for p,pl E function (sec. 5.3, eq. (33)). For the free field now under consideration we have

( a:

I a:

) = ( Q o 1 &P) I(P')* Qo ) = ( *o =

I [d(p),&p')*I 90) 6(p2 - m2)6@ - p'),

which shows that the spectral density for the free field is

The spectral condition implies that

(16)

278

5. Quantized Fields

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 8: are generalized eigenvectors of energy-momenturn p E R i with the normalization

-+ I @+,p

( a,

) = ( *o =(9

0

I

a(pl)**o

l [a(p)1a(p1)*1Qo )

(20)

= %(27r)' 6(p - p').

The wave packets e$ span the one-particle subspace 'FI1 of 'FI and have the momenturn represent at ion .( -+ a pIe,+ ) = elq.

(21)

A dense subspace of 'HI is obtained by applying the smeared operators

4*(f)

= m(j)*I

J dx

)(XI*

f ( x ) = (2.)-"-'

d p &P)*

j@) (22)

to the vacuum, where f is a test function. This gives

5.4. B e e Klein-Gordon Fields where

f is the restriction of j

to

S22. Hence the functions

are exactly the holomorphic positive-energy solutions of the Klein-

Gordon equation discussed in section 4.4, with e;f corresponding to the evaluation maps e,. The space K: of these solutions can thus be identified with 'HI, and the orthogonal projection from 7f to ?ll is given by 4

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

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 da is given in terms of the Poincar&invariant symplectic form a! = dy, A dx, by restricting as a A . A a to a+ and choosing an orientat ion: h

h

do = (s!AA)-1 as = A;' dy A dx,. Similarly, the antiparticle coherent states for the free field are given by

5. Quantized Fields

Since for p E R$ and n E 7- we have

. it follows that e;

(B;~e;)=e'q=

( Q- p+

1 % + ),

(30)

has exactly the same spacetime behavior as ef

,

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 I L l 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(zl ) and 4(z2) commute. In this way, a phase-space forrnalism can be buit for an indefinite number of particles (or charges), analogous to the grand-canonicil ensemble in classical statistical mechanics. This idea will not be further pursued here. Instead, we

5.4. Ree 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 a(p) and b(p). But the resolution of unity derived in chapter 4 can now be restated

do exp(iip - izp') = ( 2 ~ ) %(p) ' 6(p - p') = ( Q"+ p I &+, )

1-

do exp(izp - iip') = ( 2 ~%(p) ) ~ 6(p - p') = ( 6 ;

I 62 )

(34)

for p,pl E 02,where the second identity follows from the first by replacing z with i and a+ with g-. It follows that Nh can be expressed as phase-space integrals of the extended field q5(r):

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 a = a+ - a- , where the minus sign means that a- enters with the opposite ("negative") orientation to that of a+, in the sense of chains (Warner [1971]). The reason for this choice of orientation is that B: and By are both open sets of IR3+', hence must have the same orientation, and we orient R: and 0 , so that

Since the outward normal of B: points "down" and that of BL points "up," this means that must have the opposite orientation to that of 0:. Thus, setting Bx = B: By and RA = R: - R i , we have

+

This gives a- the orientation opposite to that of a+,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 We may therefore interpret the operator

as a scalar phase-space charge density with respect to the measure do. The Wick ordering can be viewed as a special case of imaginarytime ordering, if we define 4* ( z ) = $ ( f ) * :

where

for z, z' E 7. This definition is Lorentz-invariant , since

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 a- 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 positive-and negativefrequency parts. Also,

284

5. Quantized Fields

the extended fields are operator-valued functions rather than distri~ ) well-defined, which butions, hence products such as 4 ( ~ ) * 4 (are 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 phase-space integrals. Hence the phasespace formalism resolves the problem of zero-poin t 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

by using Rx = -dBx and applying Stokes' theorem:

where

is the phase-space current density. Using the notation

5.4. n e e Klein-Gordon Fields we have

Hence, by the holomorphy of

4,

Our expression for the charge is therefore

where

is seen to be a regulmized version of the usual current density J p ( x ) obtained by first extending it to 7 and then integrating it over BA. The vector field fixed y E V', since

jyz) is conserved in real

spacetime for each

5. Quantized Fields

by virtue of the Iclein-Gordon equation combined with the holomorphy of 4 in 7.This implies that Jf;)(x) is also conserved, hence the charge does not depend on the choice of S or a.

Note: In using Stokes' theorem above, we have assumed that the contribution from (yo1 + oo 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 jp(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

5.4. n e e Klein-Gordon Fields

where, by the canonical commutation relations,

I< is a distribution on 7 x 7, with

as+' x ad+'which is piecewise analytic in

( -iA+(zt - Z;m),

K(zt,2 ) =

{ if-("- Z; m),

zt, z E 7+ zt, z E 7zt E 7+,z E 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(zt,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

288

5. Quantized Fields

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 rat her than a Green function. The energy-momentum and angular momentum operators may be likewise expressed as conserved phase-space integrals of the extended field:

Like Q, these may be displayed as regularizations of the usual, more complicated expressions in real spacetime. Note first that re-written as

The angular momentum can be recast similarly as

P, can be

5.5. n e e Dirac Fields Using

289

= -aBx and applying Stokes' theorem, we therefore have

where

is a regularized energy-momentum density tensor which, incidentally, is automatically symmetric. Similarly,

where

1 2

@(*I (x) 1 -A*' Ccvr

a2

1

- x u aypayr :4*+: (67)

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 SO(3, I)+, 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 Et4 is identified with the Hermitian 2 x 2 matrix

where a0 = I (2 x 2 identity) and O k ( k = 1,2,3) are the Pauli spin matrices. Note that det X = x2 1 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 a(A):

From

it follows that 7r(A) is a Lorentz transformation, and it can easily be seen to be proper. Hence 7r defines a map 7r:

SL(2, C) -,Lo,

(5)

which is readily seen to be a group homomorphism. Clearly, T(-A) = .rr(A),and it can be shown that if 7r(A) = T(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,

291

5.5. n e e Dirac Fields

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 -, -U. 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 IR~ by (a, A) x = T(A) x

+ a.

(8)

P2 is the two-fold universal covering group of Po. A field $(x) of arbitrary spin is a distribution taking its values in the tensor product L(7-l)8 V 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 U(a, A) $(x) U(a, A)' = S(A-l) $(.lr(A)x

+ a),

(9)

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 [

(( x ) ] = 0

if (x - 3')'

0 and the limit E 5 0 is taken after the integral is evaluated. Gr,t 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- x') = O

unless x - x' E

y.

(9)

Gret(x - x') is interpreted as the causal effect at x due to a unit disturbance at x'. The corresponding choice of free field 4o is h,, 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

=

with p(po - ie,p) and e 10, and propagates both positive and negative frequencies backward in time, which means it is anticausal. The corresponding free field is q5,,t, hence

Let us now apply the Analytic-Signal transform to both of these equations:

4(z) = C u t (2)

+ / dx' Gadv

(Z

- X I ) j(x'),

where (with z = x - iy)

and

Since the Analytic-Signal transform involves an integration over the entire line x(7) = x - ry, the effect of Gret(z - x') is no longer

5.6. Interpolating Particle Coherent States

305

causal when regarded as a function of z and st. Rather, it might be interpreted as the causal effect of a unit disturbance at x' on the line parametrized by z . (Note that only those values of T for which x - r y - z' E contribute to the integral.) A similar statement goes for Gadv( Z- 2'). Whereas di,(z) and dOut(z)are holomorphic in 7,4(z) is not (unless j (x) 0), since Gret(z -3') and Gdv (z - X I ) 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.

-

In Wightman field theory, the vacua Yt, Yyt and Qo of the in-, out- and interpolating fields all coincide (the theory is "alreadyrenormalized" ). Let us define the asymptotic particle coherent states by

We will refer to

as the interpolating particle coherent states. By eq. (13), e$ = e;,, = eft,,

and

J +J

+

dx' Gret(z - XI)j(xt)* Yo

(18) dx' Gadv(~- zt)j(x')* Bo

5. Quantized Fields

From the definitions it follows that

hence eq. (18) can be rewritten as

Eqs. (19) and (21) display the interpolating character of e$. Note that when j(x) 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

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

R2,

and j(p) then Q: 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 particle-number) is conserved in general, due to gauge invariance; hence Q f makes sense as an eigenvector of charge with eigenvalue E . !Pf can be expressed in terms of particle coherent states as

j ( z ) satisfies the inhomogeneous equations

But from the definitions it follows that

where the last equation is a definition of 6(z - 3') as the AnalyticSignal transform with respect to x of 6(x - XI). The above is easily seen to reduce to

308

5. Quantized Fields

where j (z) is the Analytic-Signal transform of j(x). Equivalently, eq. (3) can be extended to transform, giving (ox

as+' by

applying the Analytic-Signal

+ m2)+(z) = j ( 4 ,

hence ( n x + m 2 ) . f ( z ) = ( q ~ I ( n , + m ~ ) + IQ:) ( ~ ) = (%lj(z)q:).

(28)

For a known external source, this is a "perturbed" Klein-Gordon equation for .f(z); if j depends on (, 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 L'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 ' I as + well as, the reality condition 4(z)* = 4 ( ~ ) The . basic idea is that since all the operators +(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 q5 is "diagonal." The construction proceeds as follows: Let j(p) be a function on IR', which will also be regarded as a function on 02. To simplify the analysis, we assume to begin with that j is a (complex-valued) Schwartz test function, although this will be relaxed later. j determines a holomorphic positive-energy solution of the Klein-Gordon equation,

Define

where a+ 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

and for n 2 1,

We now define the field coherent states of

4 by the formal expression

I so + that $ , ( z ) !Po = 0, eq. (5) implies that Then if z E '

Hence ~f is a common eigenvector of all the operators 4 ( z ) , z E 7+. This eigenvalue equation implies that the state corresponding to E f is left unchanged by the removal of a single particle, which requires that E f be a superposition of states with 0,1,2, . . . particles. Indeed,

The projection of E f to the one-particle subspace can be obtained by using the particle coherent states e,:

(e.IEf)= (Qo 14(z)Ef) =f

*a I E f ) = f (4,

(9)

where thelast equalityfollowsfrom d(f) 9oE ( $ * ( f ) ) * Jlo = 0. More generally, the n-particle component of ~f is given by projecting to the n-particle coherent state

5.7. Field Coherent States and Ftrnctiond Integrals

ez~z~--z,,

~ ( Z I ) * ~ ( Z' 'Z' $(zn)* )* @o,

31 1

(lo)

which gives

so all particles are in the same state f and the entire system of particles is coherent! Similar states have been found to be very useful in the analysis of the phenomenon of coherence in quantum optics (Glauber [1963],Klauder and Sudarshan [1968]),where the name "coherent states" in fact originated. In t he 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 a+ 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 positive-energy solution, then

where, by theorem 4.10,

312

5. Quantized Fields

Hence

Thus ~f belongs to H (i.e., is normalizable) if and only if f(p) belongs to L:(dfi) or, equivalently, f(z) belongs to the one-particle 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 ~:(d@)or f E IC. Next, we look for ' in terms of the Ef's. The standard procea resolution of unity in H dure (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 ICb > IC will be needed to support dp. Thus, for the time being, we leave the domain of integration unspecified and write formally

where dp is to be found. Taking the matrix element of this equation between the states E~ and E g , we obtain

With h = -g this gives

The left-hand side is an infinite-dimensional 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 infinite-dimensional space of f's is replaced by (En, the above relation would uniquely determine dp as a Gaussian measure. For the identity

C - At)* A-'(C - At)] = 1, det ~ - l d ~ "exp[-n(C

(18)

where A is a positive-definite matrix, implies n(C*€+€*C) = e ~ € * A €

(19)

with dp(C) = det A-' e x p [ - n ~ * ~ -(1' d2"C.

(20)

= ( f ( X I , f ( 4 )= f ( x ) h ( 4 f ( 4

(11)

We require that p be invariant under transport. This means that (f, f ) changes only due to its dependence on x , i.e.

It follows that

which constrains the real part of I' but leaves the imaginary part arbitrary. Writing I? = R + i A, where R and A are real 1-forms, we have

2R = dlog h.

(14)

The real part R of the connection can be transformed away by d d n ing f = h'I2 f and & = 1, which gives P = p and

-

Since p = f f , we may as well assume from the outset that p = 1f l2 and I? = iA is purely imaginary.

6. h t h e r Developments

326

-

Note: The mapping f

H

j is

not a gauge transformation in the

usual sense; in the standard gauge theory the metric is assumed to

I), hence only phase translations are allowed. This corresponds to having already transformed away R. It turns out that in phase space, it will be natural to admit non-constant metrics.

be constant (h(x)

To make the Klein-Gordon equation invariant under gauge transformations, we now replace d, by D p = d,

+ iA,.

The result is

This equation was known (even before gauge theory) to be a relativistically covariant description of a Iclein-Gordon particle in the presence of the electromagnetic field determined by the vector potential A, (x). Hence t he connection, which describes a geometric property of the the complex line bundle, acquires a physical significance with respect to electrodynamics, just as did the connection I? with respect to gravitation. When f is differentiated in the usual way, it is unconsciously assumed that the connection vanishes. Coupling to an electromagnetic field then has to be put in "by hand," through the substitution d,

+

8,

+ iA,,

which is known as the minimal

coupling prescription. Gauge theory gives this ad-hoc prescription a geometric interpretation. But note that in this case, the fiber metric h(x) did not determine the connection. This is due to the complex structure: iAf cancels in the inner product because it is imaginary. The electromagnetic field generated by the potential A is given by the 2-form

6.1. Holomorphic Gauge theory

327

which in fact measures the non-triviality of the connection form A: if F = 0, then A is closed and therefore (locally) exact, i.e. it is due purely to a choice of gauge. This is analogous in Relativity to choosing an accelerating coordinate system, which gives the illusion of gravity. Note the complementary nature of the two theories: in Relativity, the skew part of the connection, which is the torsion, is assumed to vanish. In gauge theory, the inner product becomes Hermitian, and the symmetric and antisymmetric parts correspond to its real and imaginary parts, respectively. It is the real part of the connection which is assumed to vanish in gauge theory. Were r required to be real, it could in fact be transformed away as above, giving rise to no gauge field. That is, only the trivial part of the connection is determined by the metric. The non-trivial (imaginary) part is arbitrary. We will see that when the theory is extended to complex spacetime, the metric does determine a non-trivial connection. Gauge theory usually begins with a Lagrangian invariant under phase translations f H eidf , which form the group U(1). The equations satisfied by f and A,, are derived using variational principles (see Bleeker [1981]). There is a natural generalization where f ( x ) is an n-dimensional complex vector. In that case, the group of phase tanslations is replaced by a non-abelian group G, usually a subgroup of U(n). G is called the gauge group, and the correponding gauge theory is said to be non-abelian or of the Yang-Mills type. Such theories have in recent years been applied with great success to the two remaining known interactions (aside from gravity and electromagnetism), namely the weak and the strong forces, which involve nuclear matter (Appelquist et al. [1987]).

6. f i r t h e r Developments

328

Let us now see how non-abelian gauge theory may be extended to complex spacetime. (This will include the abelian case of electrodynamics when n = 1.) Consider a field f on complex spacetime, say on the double tube 7, whose values are n-dimensional complex vectors. The set of all possible values at z E 7 is a complex vector space F, x called the fiber at z. The collection of all fibers is called a vector bundle. We assume that this bundle is holomorphic

an

(Wells [1980]), so that holomorphic sections z H f(z) E Fz,represented locally by holomorphic vector-valued functions, make sense. Upon transport along a curve z ( t ) having the complex tangent vector Z, f changes by

sf = e(z)f st,

(18)

where 8(Z) is a linear map on each fiber. The total differential change is

If z = x - iy, then

where

Since a general tangent vector has the form

6.1. Holomorphic Gauge theory we have

~(a,,).

where 8, = @(a,) and OF = Let us now try again to derive the connection from a fiber metric, as we have failed to in the case of real spacetime. A positivedefinite metric on the fibers F, must have the form

where h(z) is a positivedefinite matrix. Again, it will suffice to consider the (squared) fiber norm

We define a holomorphic gauge transformation to be a map of the form

where ~ ( zis) an invertible n x n matrix-valued holomorphic function. Clearly p is invariant under holomorphic gauge transformations. The corresponding gauge group acting on a single fiber Fz is the general linear group G = GL(n, a), which includes the usual gauge group U ( n ) . However, analyticity correlates the values of ~ ( z at ) different fibers. Invariance of the inner product under transport gives

from which we obtain the matrix equation 8*h

+ he = dh = 8h + dh.

330

6. f i r t h e r Developments

As in the case of real spacetime, this only determines the Hermitian part (relative to the metric) of 9. But if we make the Ansatz (29)

h9 = ah,

then the resulting connection 9 = h-'ah satisfies the above constraint. We now show that in general, the connection 9 is non-trivial, i.e. cannot be transformed away by a holomorphic gauge transformation. Under such a transformation, 9 becomes

9' = (x*hx)-'i3(x* hx) = X-l h-l ahX =x

-

'ex

+ x-l ax

+ x-lax,

since ax* = 0 by analyticity. It follows that D'f' E (d

+ 9')f1 = X - l ~ ( X f ' ) ,

(31)

hence I 2

(D )

f

1

= x-

1

2

D (xfl).

(32)

If 8 were trivial, then for some gauge we would have 9' = 0, hence = d2 = 0, SO D2 = 0. But

where the 2-form


331

is the curvature form of the connection 0, analogous t o the Riemann curvature tensor in Relativity. Using the matrix equation

dh-1 = -h-'dh. and

h-'

a2 = a2 = aa+aa= 0, we find

a8+8A8=

(-h-'ah-h-'

) A ah + (h-'ah)

A (h-'ah) = 0.

(36) This is an integrability condition for 0, being a consequence of the fact that 6 can be derived from h. One could say that h is a "potential" for 8. Therefore the quadratic term cancels in eq. (34) and the curvature form reduces to

Hence if h is such that 8 (h-'ah) # 0, then the connection is nontrivial. The form e is the complex spacetime version of a Yang-Mills field, and t9 corresponds to the Yang-Mills potential. For n = 1, 8 and O are the complex spacetime versions of the field, respectively. electromagnetic potential and the ele~troma~gnetic Since h ( z ) is a positive function, it may be writ ten as

where 4 ( z ) is real. Then

To relate 8 and O to the electromagnetic potential A and the electromagnetic field F, one performs a non-holomorphic gauge transformation simi1a.r to that in eq.(15): let

6. Further Developments

Then the transformed potential becomes purely imaginary,

giving

for the complex spacetime version of the electromagnetic potential. Note that although A,(z) is a pure gradient in the y-direction, the corresponding electromagnetic field is not trivial, since

need not vanish, in general. Incident ally, there is an intriguing similarity between the inner product using the fiber metric,

and that in Onofri's holomorphic coherent states represent ation (sec.

3.4, eq. (62)). Note that our O coincides with Onofri's symplectic form -w. This possible connection remains to be explored. Remarks. 1. The relation between the Yang-mills potential A and the YangMills field F in real spacetime is F = d A A A A, which is

+


333

quadratic in the non-abelian case (n > I), since the wedge product A A A involves matrix multiplication. In our case, however, the connection satides the integrability condition given by eq. (36), hence the quadratic term cancels and the relation becomes linear, just as it is normally in the abelian case. The non-holomorphic gauge transformation f o f in eq. (40) can be generalized to the non-abelian case as follows: Since h(z) is a postive matrix, it can be written as h(z) = k(z)*k(z), where k(z) is an n x n matrix. (k(z) need not be Hermitian; the holomorphic case 8 k = 0 corresponds to a "pure gauge" field, i.e. @ = 0.) Setting j(z) = k(z)f ( z ) and &(I) I 1 brings us to the unitary gauge, where the new gauge transformations are given by unitary matices j ( z ) cr u ( z ) ~ ( z ) .This amounts to a reduction of the gauge group from GL(n, C) to U ( n ) . In the unitary gauge, the relation between the connection and the curvature becomes non-linear, as it is in the usual Yang-Mills theory. See I

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

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

Quantum physics, relativity, and complex spacetime: Towards a new synthesis