Theorem 1.1

Quantum Physics, Relativity, and Complex Spacetime:

Towards a New Synthesis

Gerald Kaiser

Department of Mathematics

University of Massachusetts–Lowell

kaiser@wavelets.com

ABSTRACT

The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube in complex spacetime. For Klein-Gordon particles, we interpret as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps of wave functions on are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved ”probability current” can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where is interpreted as a probability density on all 6D phase spaces in which, when integrated over the ”momentum” variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.

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 rinciple And the principle was uantum And all things were quantized But some things were still elative And God saw that it was confusing. Then God was going to create And would have unified And 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.

Adapted from a poem by Tim Joseph

CONTENTS

Preface iv Suggestions to the Reader vii

Chapter 1. Coherent–State Representations

1.1. Preliminaries 1

1.2. Canonical coherent states 7

1.3. Generalized frames and resolutions of unity 14

1.4. Reproducing–kernel Hilbert spaces 22

1.5. Windowed Fourier transforms 26

1.6. Wavelet transforms 33

Chapter 2. Wavelet Algebras and Complex Structures

This chapter is not essential, hence omitted to save space; the same material is available at http://wavelets.com/pdf/92Siam.pdf

Chapter 3. Frames and Lie Groups

3.1. Introduction 44

3.2. Klauder’s group–frames 44

3.3. Perelomov’s homogeneous –frames 50

3.4. Onofri’s’s holomorphic –frames 57

3.5. The rotation group 74

3.6. The harmonic oscillator as a contraction limit 82

Chapter 4. Complex Spacetime

4.1. Introduction 89

4.2. Relativity, phase space and quantization 90

4.3. Galilean frames 99

4.4. Relativistic frames 110

4.5. Geometry and Probability 128

4.6. The non–relativistic limit 142

Notes 147

Chapter 5. Quantized Fields

5.1. Introduction 151

5.2. The multivariate Analytic–Signal transform 154

5.3. Axiomatic field theory and particle phase spaces 161

5.4. Free Klein–Gordon fields 180

5.5. Free Dirac fields 193

5.6. Interpolating particle coherent states 203

5.7. Field coherent states and functional integrals 208

Notes 216

Chapter 6. Further Developments

6.1. Holomorphic gauge theory 218

6.2. Windowed X–Ray transforms: Wavelets revisited 227

References 237

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.”

Up to that point, the theory could only describe free particles. The next goal was to see how interactions could be added. Some progress in this direction was made in 1979-80, when a natural way was found to extend gauge theory to complex spacetime. Further progress came during my sabbatical in 1985-86, when a method was developed for extending quantized fields themselves (rather than their vacuum expectation values) to complex spacetime. These ideas have so far produced no “hard” results, but I believe that they are on the right path.

Although much work remains to be done, it seems to me that enough structure is now in place to justify writing a book. I hope that this volume will be of interest to researchers in theoretical and mathematical physics, mathematicians interested in the structure of fundamental physical theories and assorted graduate students searching for new directions. Although the topics are fairly advanced, much effort has gone into making the book self–contained and the subject matter accessible to someone with an understanding of the rudiments of quantum mechanics and functional analysis.

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

The contents of the book are as follows. In chapter 1 the simplest 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 reproducing kernels is discussed.

In chapter 2 a new, algebraic approach to orthonormal bases of wavelets is formulated. An operational calculus is developed which simplifies the 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 ) 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 phase–space approach to relativistic quantum mechanics are developed for free particles, based on the relativistic coherent–state representations developed in my thesis. It is shown that such representations admit a covariant probabilistic interpretation, a feature absent in the usual spacetime theories. In the non–relativistic limit, the representations are seen to “contract” smoothly to representations of the Galilean group which are closely related to the canonical coherent–state representation. The Gaussian weight functions in the latter are seen to emerge from the geometry of the mass hyperboloid.

In chapter 5, the formalism is extended to quantized fields. The basic tool for this is the Analytic–Signal transform, which can be applied to an arbitrary function on to give a function on which, although not in general analytic, is “analyticity–friendly” in a certain sense. It is shown that even the most general fields satisfying the Wightman axioms generate a complexification of spacetime which may be interpreted as an extended classical phase space for certain special states associated with the theory. Coherent–state representations 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 functional–integral 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, beginning with a review of the idea of holomorphic gauge theory. Whereas in real spacetime it is not possible to derive gauge potentials and gauge fields from a (fiber) metric, we show how this can be done in complex spacetime. Consequently, the analogy between General Relativity and gauge theory becomes much closer in complex spacetime than it is in real spacetime. In the “holomorphic” gauge class, the relation between the (non–abelian) Yang–Mills field and its potential becomes linear due to the cancellation of the non–linear part which follows from an integrability condition. Finally, we come full circle by generalizing the Analytic–Signal transform and pointing out that this generalization is a higher–dimensional version of the wavelet transform which is, moreover, closely related to various classical transforms such as the Hilbert, Fourier–Laplace and Radon transforms.

I am deeply grateful to G. Emch for his continued help and encouragement over the past ten years, and to John Klauder and Ray 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 também agradecer a todos os meus colegas Brasileiros!

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, time–frequency 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

1.1. Preliminaries

In this section we establish some notation and conventions which will be followed in the rest of the book. We also give a little background on the main concepts and formalism of non–relativistic and relativistic quantum mechanics, which should make this book accessible to non–specialists.

1. Spacetime and its Dual

In this book we deal almost exclusively with flat spacetime, though we usually let space be instead of , so that spacetime becomes . 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 , and furthermore, that it may be useful later. Many models in constructive quantum field theory are based on two– or three– dimensional 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 world–sheet) two–dimensional spacetimes. An event has coordinates

 x=(xμ)=(x0,xj),

where is the time coordinate and are the space coordinates. Greek indices run from 0 to , while latin indices run from 1 to . If we think of as a translation vector, then is the vector space of all translations in spacetime. Its dual is the set of all linear maps . By linearity, the action of on (which we denote by instead of ) can be written as

 kx=s∑μ=0kμxμ≡kμxμ,

where we adopt the Einstein summation convention of automatically summing over repeated indices. Usually there is no relation between and other than the pairing . But suppose we are given a scalar product on ,

 x⋅x′=gμνxμx′ν

where is a non–degenerate matrix. Then each in defines a linear map by , thus giving a map , with

 (x∗)ν≡xν=gμνxμ.

Since is non–degenerate, it also defines a scalar product on , whose metric tensor is denoted by . The map establishes an isomorphism between the two spaces, which we use to identify them. If denotes a set of inertial coordinates in free spacetime, then the scalar product is given by

 gμν=diag(c2,−1,−1,⋯,−1)

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

It is often convenient to work in a single space rather than the dual pair and . Boldface letters will denote the spatial parts of vectors in . Thus and

 x⋅x′=c2tt′−x⋅x′  and  kx=k⋅x∗=k0t−k⋅x,

where and denote the usual Euclidean inner products in .

2. Fourier Transforms

The Fourier transform of a function (which, to avoid analytical subtleties for the present, may be assumed to be a Schwartz test function; see Yosida [1971]) is a function given by

 ^f(k)=∫Xdxe2πikxf(x)

where is Lebesgue measure on . can be reconstructed from by the inverse Fourier transform, denoted by and given by

 f(x)=∫X∗dke−2πikx^f(k)≡(^f)ˇ(x),

where denotes Lebesgue measure on . Note that the presence of the factor in the exponent avoids the usual need for factors of or in front of the integrals. Physically, represents a wave vector: is a frequency in cycles per unit time, and is a wave number in cycles per unit length. Then the interpretation of the linear map is that is the total radian phase gained by the plane wave through the spacetime translation , i.e. “measures” the radian phase shift. Now in pre–quantum relativity, it was realized that the energy combines with the momentum p to form a vector in . 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,

 p0≡E=2πℏν

where is Planck’s constant, and the classical momentum is re–interpreted as the wave–number vector of the associated wave by De Broglie’s relation,

 p=2πℏk.

These two relations are unified in relativistic terms as . 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:

 (Pμf)(x)=∫X∗dkpμe−2πikx^f(k)=iℏ∂∂xμf(x),

or, in terms of ,

 P0=iℏ∂∂t  and  Pk=−iℏ∂∂xk.

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

 ^f(p)=∫Xdxeipx/ℏf(x)f(x)=(2πℏ)−s−1∫X∗dpe−ipx/ℏ^f(p).

If satisfies a differential equation, such as the Schrödinger equation or the Klein–Gordon equation, then is supported on an –dimensional submanifold of (a paraboloid or two–sheeted hyperboloid, respectively) which can be parametrized by . We will write the solution as

 f(x)=(2πℏ)−s∫Pdμ(p)e−ipx/ℏ^f(p),

where is, by a mild abuse of notation, the “restriction” of to (actually, is a density on ) and is an appropriate invariant measure on . For the Schrödinger equation , whereas for the Klein–Gordon equation, . Setting then shows that is related to the initial wave function by

 f(x,0)=(ρ(p)^f)ˇ(x),

where now “” denotes the the –dimensional inverse Fourier transform of the function on .

We will usually work with “natural units,” i.e. physical units so chosen that . However, when considering the non–relativistic limit () or the classical limit (), or 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 notation and, hopefully, convince mathematicians of its correctness and value. When applied to coherent–state representations, as opposed to representations in which the position–or 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 be an arbitrary complex Hilbert space with inner product . Each element defines a bounded linear functional by

 f∗(g)=⟨f,g⟩.

The Riesz representation theorem guarantees that the converse is also true: Each bounded linear functional has the form for a unique . Define the bra corresponding to by

 ⟨f|=f∗:H→Cl.

Similarly, there is a one-to-one correspondence between vectors and linear maps

 |g⟩: Cl→H

defined by

 |g⟩(λ)=λg,λ∈Cl,

which will be called kets. Thus elements of will be denoted alternatively by or by . We may now consider the composite map bra–ket

given by

 ⟨f|g⟩(λ)=f∗(λg)=λf∗(g)=λ⟨f,g⟩.

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

 |g⟩⟨f|:H→H

may be viewed as acting on kets to produce kets:

 |g⟩⟨f|(|h⟩)=|g⟩(⟨f|h⟩).

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

 f=∑ngn⟨gn|f⟩≡∑n|gn⟩⟨gn|f⟩,

from which we have the “resolution of unity”

 ∑n|gn⟩⟨gn|=I

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

 ⟨hn|f⟩=⟨hn|If⟩=∑m⟨hn|gm⟩⟨gm|f⟩.

In physics, vectors such as are often written as , which can be a source of great confusion for mathematicians. Furthermore, functions in , say, are often written as , with , as though the ’s formed an orthonormal basis. This notation is very tempting; for example, the Fourier transform is written as a “change of basis,”

 ⟨k|f⟩=∫dx⟨k|x⟩⟨x|f⟩

with the “transformation matrix” . One of the advantages of this notation is that it permits one to think of the Hilbert space as “abstract,” with and merely different “representations” (or “realizations”) of the same vector . 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” is a distribution which evaluates test functions at the point ; as such, does not exist within modern–day distribution theory.) We will generally abstain from this use of the bra–ket notation.

Finally, it should be noted that the term “representation” is used in two distinct ways: (a) In the above sense, where abstract Hilbert–space vectors are represented by functions in various function spaces, and (b) in connection with groups, where the action of a group on a Hilbert space is represented by operators.

This notation will be especially useful when discussing frames, of which coherent–state representations are examples.

1.2. Canonical Coherent States

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

 [Xk,Xl]=0,[Pk,Pl]=0,[Xk,Pl]=iδklI,

where is the identity operator. The operators , and together form a real Lie algebra known as the Heisenberg algebra, which is irreducibly represented on by

 Xkf(x)=xkf(x),Pkf(x)=−i∂∂xkf(x),

the Schrödinger representation.

As a consequence of the above commutation relations between and , the position and momentum of the particle obey the Heisenberg uncertainty relations, which can be derived simply as follows. The expected value, upon measurement, of an observable represented by an operator in the state represented by a wave function with (where denotes the norm in ) is given by

 ⟨F⟩=⟨f|Ff⟩.

In particular, the expected position– and momentum coordinates of the particle are and . The uncertainties and in position and momentum are given by the variances

 Δ2Xk=⟨(Xk−⟨Xk⟩)2⟩=⟨X2k⟩−⟨Xk⟩2Δ2Pk=⟨(Pk−⟨Pk⟩)2⟩=⟨P2k⟩−⟨Pk⟩2.

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

 Ak=Xk+ibPk=xk+b∂∂xk.

Notice that although is non–Hermitian, it is real in the Schrödinger representation. Let

 ¯zk≡⟨f|Akf⟩=⟨Xk⟩+ib⟨Pk⟩,

where denotes the complex–conjugate of . Then for we have and

 0≤∥δAkf∥2=Δ2Xk+b2Δ2Pk−b.

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

 ΔXkΔPk≥12.

Equality is attained if and only if , which shows that the only minimum–uncertainty states are given by wave functions satisfying the eigenvalue equations

 Akf=(xk+b∂xk)f=¯zkf

for some real number (which may actually depend on ) and some . But square–integrable solutions exist only for , and then there is a unique solution (up to normalization) for each . To simplify the notation, we now choose . Then and satisfy the commutation relations

 [Ak,Al]=0,[Ak,A∗l]=2δklI,

and is given by

 χz(x′)=Nexp[−¯z2/4+¯z⋅x′−x′2/2],

where the normalization constant is chosen as , so that for . Here is the (complex) inner product of with itself. Clearly is in and if , then

 ⟨Xk⟩=xk and ⟨Pk⟩=pk

in the state given by . The vectors 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

 H=12m(P2+m2ω2X2)=12mω2A∗⋅A+sω2

with

 Ak=Xk+iPk/mω

(thus ). They have the remarkable property that if the initial state is , then the state at time is where is the orbit in phase space of the corresponding classical harmonic oscillator with initial data given by . These states were discovered by Schrödinger himself [1926], at the dawn of modern quantum mechanics. They were further investigated by Fock [1928] in connection with quantum field theory and by von Neumann [1931] in connection with the quantum measurement problem. Although they span the Hilbert space, they do not form a basis because they possess a high degree of linear dependence, and it is not easy to find complete, linearly independent subsets. For this reason, perhaps, no one seemed to know quite what to do with them until the early 1960’s, 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 effectiveness to extend the concept of optical coherence to the domain of quantum electrodynamics, which was made necessary by the discovery of the laser. He dubbed them “coherent states,” and the name stuck to the point of being generic. (See also Klauder and Sudarshan [1968].) Systems of vectors now called “coherent” may have nothing to do with optical coherence, but there is at least one unifying characteristic, namely their frame property (next section).

The coherent-state representation is now defined as follows: Let be the space of all functions

 ~f(z)≡⟨χz|f⟩=∫dsx′¯¯¯¯¯¯¯¯¯¯¯¯¯¯χz(x′)f(x′)=N∫dsx′exp[−z2/2+z⋅x′−x′2/2]f(x′)

where runs through . Because the exponential decays rapidly in , is entire in the variable Define an inner product on by

 ⟨~f|~g⟩F≡∫lCsdμ(z)¯¯¯¯¯¯¯¯¯¯¯~f(z)~g(z),

where and

 dμ(z)=(2π)−sexp(−¯z⋅z/2)dsxdsp.

Then we have the following theorem relating the inner products in and .

###### Theorem 1.1

Let and let be the corresponding entire functions in . Then

 ⟨~f|~g⟩F=⟨f|g⟩L2.

Proof. To begin with, assume that is in the Schwartz space of rapidly decreasing smooth test functions. For , we have

 χz(x′)=Nexp[−¯z2/2+x2/2−(x′−x)2/2+ip⋅x′],

hence

 ~f(x−ip)=Nexp[(x2+p2)/4+ipx/2](exp[−(x′−x)2/2]f)^(p)

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

 (2π)−s∫dspexp(−p2/2)|~f(x−ip)|2=N2exp(x2/2)∫dsx′exp[−(x′−x)2]|f(x′)|2.

Therefore

 ∫dμ(z)|~f(z)|2=N2∫dsx∫dsx′exp[−(x′−x)2]|f(x′)|2=∫dsx′|f(x′)|2

after exchanging the order of integration. This proves that

 ∥~f∥2F=∥f∥2L2

for , hence by continuity also for arbitrary . By polarization the result can now be extended from the norms to the inner products.

The relation can be summarized neatly and economically in terms of Dirac’s bra-ket notation. Since

 ¯¯¯¯¯¯¯¯¯¯¯~f(z)=¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨χz|f⟩=⟨f|χz⟩,

theorem 1 can be restated as

 ∫ lCsdμ(z)⟨f|χz⟩⟨χz|g⟩=⟨f|g⟩.

Dropping the bra and ket , we have the operator identity

 ∫ lCsdμ(z)|χz⟩⟨χz|=I,

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

 (~B~f)(z)≡(Bf)~(z)=⟨χz|Bf⟩=⟨χz|BIf⟩=∫ lCsdμ(w)⟨χz|Bχw⟩⟨χw|f⟩≡∫ lCsdμ(w)~B(z,¯w)~f(w).

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

 Xkχz(x′)≡x′kχz(x′)=(∂∂¯zk+¯zk2)χz(x′)Pkχz=−i(Ak−Xk)χz=i(∂∂¯zk−¯zk2)χz,

thus

 ~Xk~f=(∂∂zk+zk2)~f~Pk~f=−i(∂∂zk−zk2)~f.

Hence and can be represented as differential rather than integral operators.

As promised, the continuous resolution of the identity makes it possible to reconstruct from its transform :

 f=If=∫lCsdμ(z)|χz⟩⟨χz|f⟩,

that is,

 f(x′)=∫lCsdμ(z)χz(x′)~f(z).

Thus in many respects the coherent states behave like a basis for . But they differ from a basis in at least one important respect: They cannot all be linearly independent, since there are uncountably many of them and (and hence also ) is separable. In particular, the above reconstruction formula can be used to express in terms of all the ’s:

 χz=∫lCsdμ(w)|χw⟩⟨χw|χz⟩.

In fact, since entire functions are determined by their values on some discrete subsets of , we conclude that the corresponding subsets of coherent states are already complete since for any function orthogonal to them all, for all and hence , which implies a.e. For example, if is a regular lattice, a necessary and sufficient condition for completesness is that contain at least one point in each Planck cell (Bargmann et al., [1971]), in the sense that the spacings and of the lattice coordinates satisfy . It is no accident that this looks like the uncertainty principle but with the inequality going “the wrong way.” The exact coefficient of 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 tractable—definition of uncertainty uses the notion of entropy, which involves all moments rather than just the second moment. See Bialynicki–Birula and Mycielski [1975] and Zakai [1960].) The intuitive explanation is that if 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 appeal and mystery of these systems.

1.3. Generalized Frames and Resolutions of Unity

Let be a set and be a measure on (with an appropriate –algebra of measurable subsets) such that is a –finite measure space. Let be a Hilbert space and be a family of vectors indexed by . Definition.  The set

 HM≡{hm|m∈M}

is a generalized frame in if 1. the map is weakly measurable, i.e. for each the function is measurable, and 2. there exist constants such that

 A∥f∥2≤∫Mdμ(m)|⟨hm|f⟩|2≤B∥f∥2∀f∈H.

is a frame (see Young [1980] and Daubechies [1988a]) in the special case when is countable and is the counting measure on (i.e., it assigns to each subset of the number of elements contained in it). In that case, the above condition becomes

 A∥f∥2≤∑m∈M|⟨hm|f⟩|2≤B∥f∥2∀f∈H.

We will henceforth drop the adjective “generalized” and simply speak of “frames.” The above case where is countable will be refered to as a discrete frame.

If , the frame is called tight. The coherent states of the last section form a tight frame, with and .

Given a frame, let be the map taking vectors in to functions on defined by

 (Tf)(m)≡⟨hm|f⟩≡~f(m),f∈H.

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

 T: H→L2(dμ),

with

 A∥f∥2≤∥Tf∥2L2(dμ)≤B∥f∥2.

The frame property can now be stated in operator form as

 AI≤T∗T≤BI,

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

 G≡T∗T=∫Mdμ(m)|hm⟩⟨hm|,

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 of , write

 G(N)≡∫Ndμ(m)|hm⟩⟨hm|.

Proposition 1.2.   If the integral converges in the strong operator topology of whenever has finite measure, then so does the complete integral representing . Proof**I thank M. B. Ruskai for suggesting this proof..   Since is –finite, we can choose an increasing sequence of sets of finite measure such that . Then the corresponding sequence of integrals forms a bounded (by ) increasing sequence of Hermitian operators, hence converges to in the strong operator topology (see Halmos [1967], problem 94).

If the frame is tight, then and the above gives a resolution of unity after dividing by . For non-tight frames, one generally has to do some work to obtain a resolution of unity. The frame condition means that has a bounded inverse, with

 B−1I≤G−1≤A−1I.

Given a function in , we are interested in answering the following two questions: (a)  Is for some ? (b)  If so, then what is ? In other words, we want to: (a) Find the range of the map . (b) Find a left inverse of , which enables us to reconstruct from by . Both questions will be answered if we can explicitly compute . For let

 P=TG−1T∗: L2(dμ)→L2(dμ).

Then it is easy to see that

 (a) P∗=P(b) P2=P(c) PT=T.

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

 P: L2(dμ)→RT,

for if for some in , then , and conversely if for some we have , then . Thus is a closed subspace of and a function is in if and only if

 g(m)=(Pg)(m)=(TG−1T∗g)(m)=⟨hm|G−1T∗g⟩H=⟨TG−1hm|g⟩L2(dμ)=∫Mdμ(m′)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(TG−1hm)(m′)g(m′)=∫Mdμ(m′)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨hm′|G−1hm⟩g(m′)=∫Mdμ(m′)⟨hm|G−1hm′⟩g(m′).

The function

 K(m,m′)≡⟨hm|G−1hm′⟩

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

 |K(m,m′)|=|⟨hm|G−1h′m⟩|≤∥G−1∥∥hm∥∥hm′∥≤A−1∥hm∥∥hm′∥<∞

for all and . Furthermore, the “test functions” which reproduces form a Hilbert space and defines an integral operator, not merely a distribution, on . In the applications to relativistic quantum theory to be developed later, will be a complexification of spacetime and will be holomorphic in and antiholomorphic in .

The Hilbert space and the associated function are an example of an important structure called a reproducing–kernel Hilbert space (see Meschkowski [1962]), which is reviewed briefly in the next section. is called a reproducing kernel for .

We can thus summarize our answer to the first question by saying that a function belongs to the range of if and only if it satisfies the consistency condition

 g(m)=∫Mdμ(m′)K(m,m′)g(m′).

Of course, this condition is only useful to the extent that we have information about the kernel or, equivalently, about the operator . The answer to our second question also depends on the knowledge of . For once we know that for some , then

 f=G−1Gf=G−1T∗(Tf)=G−1T∗g.

Thus the operator

 S=G−1T∗: L2(dμ)→H

is a left inverse of and we can reconstruct by

 f=Sg=G−1T∗Tf=G−1∫Mdμ(m)|hm⟩⟨hm|f⟩=∫Mdμ(m)G−1|hm⟩g(m).

This gives as a linear combination of the vectors

 hm≡G−1hm.

Note that

 ∫Mdμ(m)|hm⟩⟨hm|=G−1GG−1=G−1,

therefore the set

 HM≡{hm|m∈M}

is also a frame, with frame constants . We will call the frame reciprocal to . (In Daubechies [1988a], the corresponding discrete object is called the dual frame, but as we shall see below, it is actually a generalization of the concept of reciprocal basis; since the term “dual basis” has an entirely different meaning, we prefer “reciprocal frame” to avoid confusion.)

The above reconstruction formula is equivalent to the resolutions of unity in terms of the pair of reciprocal frames:

 ∫Mdμ(m)|hm⟩⟨hm|=I=∫Mdμ(m)|hm⟩⟨hm|.

Corollary 1.3. Under the assumptions of proposition 1.2, the above resolutions of unity converge in the strong operator topology of . 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 rather than just weakly. Application to for a fixed gives

 hk=∫Mdμ(m)hmK(m,k),

which shows that the frame vectors are in general not linearly independent. The consistency condition can be understood as requiring the proposed function to respect the linear dependence of the frame vectors. In the special case when the frame vectors are linearly independent, the frames and both reduce to bases of . If is separable (which we assume it is), it follows that must be countable, and without loss in generality we may assume that is the counting measure on (re–normalize the ’s if necessary). Then the above relation becomes

 hk=∑m∈MhmK(m,k),

and linear independence requires that be the Kronecker : . Thus when the ’s are linearly independent, and reduce to a pair of reciprocal bases for . The resolutions of unity become

 ∑m∈M|hm⟩⟨hm|=I=∑m∈M|hm⟩⟨hm|,

and we have the relation

 hk=∑m∈Mhm⟨hm|hk⟩≡∑m∈Mhmgmk

where

 gmk≡⟨hm|hk⟩=⟨hm|Ghk⟩

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

 ⟨hm|hk⟩=Aδkmand⟨hm|hk⟩=A−1δmk,

so and