Graduate Texts in Contemporary Physics

Series Editors:

А. Stephen Berry Joseph L. Birman Jeffrey W. Lynn Mark Р. Silverman Н. Eugene Stanley Mikhail Voloshin

Springer Science+Business Media, LLC

Graduate Texts in Contemporary Physics

S.T. Ali, J.P. Antoine, and J.P. Gazeau: Coherent States, Wavelets and Their Generalizations

A Auerbach: Interacting Electrons and Quantum Magnetism

B. Felsager: Geometry, Particles, and Fields

P. Di Francesco, P. Mathieu, and D. Senechal: Conformal Field Theories

J.H. Hinken: Superconductor Electronics: Fundamentals and Microwave Applications

J. Hladik: Spinors in Physics

Yu.M. Ivanchenko and AA Lisyansky: Physics of Critical Fluctuations

M. Kaku: Introduction to Superstrings and M-Theory, 2nd Edition

M. Kaku: Strings, Conformal Fields, and M-Theory, 2nd Edition

H.V. Klapdor (ed.): Neutrinos

J.W. Lynn (ed.): High-Temperature Superconductivity

H.J. Metcalf and P. van der Straten: Laser Cooling and Trapping

R.N. Mohapatra: Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics, 2nd Edition

H. Oberhummer: Nuclei in the Cosmos

G.D. Phillies: Elementary Lectures in Statistical Mechanics

R.E. Prange and S.M. Girvin (eds.): The Quantum Hall Effect

B.M. Smimov: Clusters and Small Particles: In Gases and Plasmas

M. Stone: The Physics of Quantum Fields

F.T. Vasko and AV. Kuznetsov: Electronic States and Optical Transitions in Semiconductor Heterostructures

(continued following index)

Syed Twareque Ali Jean-Pierre Antoine Jean-Pierre Gazeau

Coherent States, Wavelets and Their Generalizations

With 14 Figures

, Springer

Syed Twareque AIi lean-Pierre Antoine Department of Mathematics and Statistics Concordia University Loyola Campus

Institut de Physique Theorique Universite Catholique de Louvain Chernin du Cyclotron 2

7141 Sherbrooke Street West Montreal, Quebec Н4В IR6 Canada

В-1348 Louvain-Ia-Neuve Belgium

Jean-Pierre Gazeau Laboratoire Physique Theoretique and Matiere Condensee 2, Place lussieu Universite de Paris 7 - Denis Diderot Paris Р-75251 France

Series Editors R. Stephen Белу Department of Chemistry University of Chicago Chicago, IL 60637 USA

Mark Р. Silverman Departmentof Physics Trinity College Hartford, СТ 06106 USA

Joseph L. Бirman Department of Physics City College of CUNY NewYork, NY 10031 USA

Н. Eugene Stanley Center for Polymer Studies Physics Department Боstоп University Боstоп, МА 02215 USA

Library of Congress Саtа1оgiпg-iп-РиbIiсаtiоп Data AIi, S. Тwareque (Syed Twareque)

Jeffrey W. Lynn Department of Physics University of Maryland College Park, MD 20742 USA

Mikhail Voloshin Theoretical Physics Institute Tate Laboratory of Physics Тhe University of Мiпnеsоtа Minneapolis, MN 55455 USA

Coherent states, wavelets and their genera1izations / S. Twareque Ali, Jean-Pierre Antoine, Jean-Pierre Gazeau.

р. ст. - (Graduate texts in contemporary physics) Includes bibIiographical references and index. ISBN 978-1-4612-7065-2 ISBN 978-1-4612-1258-4 (eBook) DOI 10.1007/978-1-4612-1258-4 1. Coherent states. 2. Wavelets (Mathematics) 1. Antoine, Jean­

Pierre. П. Gazeau, Jean-Pierre. Ш. Title. IУ. Series. QC6.4.C65A55 2000 530.12'4--dc21 99-39821

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Preface

Nitya kaaler utshab taba Bishyer-i-dipaalika

Aami shudhu tar-i-mateer pradeep Jaalao tahaar shikhaa 1

- Tagore

Should authors feel compelled to justify the writing of yet another book? In an overpopulated world, should parents feel compelled to justify bringing forth yet another child? Perhaps not! But an act of creation is also an act of love, and a love story can always be happily shared.

In writing this book, it has been our feeling that, in all of the wealth of material on coherent states and wavelets, there exists a lack of a discern­able, unifying mathematical perspective. The use of wavelets in research and technology has witnessed explosive growth in recent years, while the use of coherent states in numerous areas of theoretical and experimental physics has been an established trend for decades. Yet it is not at all un­common to find practitioners in either one of the two disciplines who are hardly aware of one discipline's links to the other. Currently, many books are on the market that treat the subject of wavelets from a wide range of perspectives and with windows on one or several areas of a large spectrum

IThine is an eternal celebration '" A cosmic Festival of Lights! '" Therein I am a mere flicker of a wicker lamp, , . 0 kindle its flame (my Master!),

vi Preface

of possible applications. On the theory of coherent states, likewise, sev­eral excellent monographs, edited collections of papers, synthetic reviews, or specialized articles exist. The emphasis in most of these works is usu­ally on physical applications. In the more mathematical texts, the focus is usually on specific properties - arising either from group theory, holomor­phic function theory, and, more recently, differential geometry. The point of view put forward in this book is that both the theory of wavelets and the theory of coherent states can be subsumed into certain broad functional analytic structures, namely, positive operator-valued measures on a Hilbert space and reproducing kernel Hilbert spaces. The specific context in which these structures arise, to generate particular families of coherent states or wavelet transforms, could of course be very diverse, but typically they em­anate either from the property of square integrability of certain unitary group representations on Hilbert spaces or from holomorphic structures as­sociated to certain differential manifolds. In talking about square integrable representations, a broad generalization of the concept has been introduced here, moving from the well-known notion of square integrability with re­spect to the whole group to one based on some of its homogeneous spaces. This generalization, while often implicit in the past in physical discussions of coherent states (notably, in the works of Klauder, Barut and Girardello, or even in the case of the time-honored canonical coherent states, as dis­covered by Schrodinger), was not readily recognized in the mathematical literature until the work of Gilmore and Perelomov. In this book, this gener­alization is taken even further, with the result that the classes of coherent states that can be constructed and usefully employed extends to a vast array of physically pertinent groups. Similarly, it is generally known and recognized that wavelets are coherent states arising from the affine group of the real line. Using coherent states of other groups to generate higher di­mensional wavelets, or alternative wavelet-like transforms, however, is not a common preoccupation among practitioners of the trade (an exception being the recent book by Torresani). About a third of the present book is devoted to looking at wavelets from precisely this point of view, thereby displaying the richness of possibilities that exist in this domain. Consider­able attention has also been paid in the book to the discretization problem, in particular, with the discussion of r-wavelets. The interplay between dis­crete and continuous wavelets is a rich aspect of the theory, which does not seem to have been exploited sufficiently in the past.

In presenting this unifying backdrop, for the understanding of a wide sweep of mathematical and physical structures, it is our hope that the re­lationship between the two disciplines - wavelets and coherent states -will have been made more transparent, thereby aiding the process of cross­fertilization as well. For graduate students or research workers approaching the disciplines for the first time, such an overall perspective should also make the subject matter easier to assimilate, with the book acting as a dovetailed introduction to both subjects - unfortunately, a frustrating in-

Preface vii

coherence blurs the existing literature on coherence! Besides being a primer for instruction, the book, of course, is also meant to be a source material for a wide range of very recent results, both in the theory of wavelets and of coherent states. The emphasis is decidedly on the mathematical aspect of the theory, although enough physical examples have been introduced, from time to time, to illustrate the material. While the book is aimed mainly at graduate students and entering research workers in physics and math­ematical physics, it is nevertheless hoped that professional physicists and mathematicians will also find it interesting reading, being an area of math­ematical physics in which the intermingling of theory and practice is most thorough. Prerequisite for an understanding of most of the material in the book is a familiarity with standard Hilbert space operator techniques and group representation theory, such as every physicist would acknowlegde from the days of graduate quantum mechanics and angular momentum. For the more specialized topics, an attempt has been made to make the treatment self-contained, and, indeed, a large part of the book is devoted to the development of the mathematical formalism.

If a book such as this can make any claims to originality at all, it can mainly be in the manner of its presentation. Beyond that, we believe that a body of material is also presented here (for example, in the use of POV measures or in dealing with the discretization problem) that has not ap­peared in book form before. An attempt has been made throughout to cite as many references to the original literature as were known - omissions should therefore be attributed to our collective ignorance and we would like to extend our unconditional apologies for any resulting oversight.

This book has grown out of many years of shared research interest and, indeed, camaraderie, between us. Almost all of the material presented here has been touched on in courses, lectures, and seminars given to students and among colleagues at various institutions in Europe, America, Asia, and Africa - notably, in graduate courses and research workshops, given at different times by all of us, in Louvain-Ia-Neuve, Montreal, Paris, Porto­Novo, Bialystok, Dhaka, Fukuoka, Havana, and Prague. One is tempted to say that the geographical diversity here rivals the mathematical menagerie!

To all of our colleagues and students who have participated in these dis­cussions, we would like to extend our heartfelt thanks. In particular, a few colleagues graciously volunteered to critically read parts of the manuscript and to offer numerous suggestions for improvement and clarity. Among them, one ought to especially mention J. Hilgert, G. G. Emch, S. De Bievre, and J. Renaud. In addition, the figures would not exist without the programming skills of A. Coron, L. Jacques, and P. Vandergheynst (Louvain-Ia-Neuve), and we thank them all for their gracious help. During the writing of the book, we made numerous reciprocal visits to one another's institutions. To Concordia University, Montreal, the U niversite Catholique de Louvain, Louvain-Ia-Neuve, and the Universite Paris 7 - Denis Diderot, Paris, we would like to express our appreciation for hospitality and colle-

viii Preface

giality. The editor from Springer-Verlag, Thomas von Foerster, deserves a special vote of thanks for his cooperation and for the exemplary patience he displayed, even as the event horizon for the completion of the manuscript kept receeding further and further! It goes without saying, however, that all responsibility for errors, imperfections, and residual or outright mistakes is shared jointly by all of us.

Syed Twareque Ali Jean-Pierre Antoine Jean-Pierre Gazeau

Montreal, Canada Louvain-la-Neuve, Belgium

Paris, France

March 1999

Contents

Preface

1 Introduction

2 Canonical Coherent States 2.1 Minimal uncertainty states ..... 2.2 The group-theoretical backdrop .. 2.3 Some functional analytic properties 2.4 A complex analytic viewpoint (.to)

2.5 Some geometrical considerations. 2.6 Outlook ............. . 2.7 Two illustrative examples. . . . .

2.7.1 A quantization problem (.to)

2.7.2 An application to atomic physics (.to)

3 Positive Operator-Valued Measures and Frames 3.1 Definitions and main properties .... . 3.2 The case of a tight frame ........ . 3.3 Example: A commutative POV measure 3.4 Discrete frames . . . . . . . . . . . . . .

4 Some Group Theory 4.1 Homogeneous spaces, quasi-invariant, and

invariant measures ............ .

v

1

13 13 17 20 24 26 27 27 27 29

33 34 40 41 42

47

47

x Contents

4.2 Induced representations and systems of covariance . 54 4.2.1 Vector coherent states ........... 59 4.2.2 Discrete series representations of SU(l, 1) 61 4.2.3 The regular representations of a group . . 67

4.3 An extended Schur's lemma . . . . . . . . . . . . 68 4.4 Harmonic analysis on locally compact abelian groups 70

4.4.1 Basic notions ...... 70 4.4.2 Lattices in LCA groups. . . . . . 72 4.4.3 Sampling in LCA groups . . . . . 73

4.5 Lie groups and Lie algebras: A reminder 75 4.5.1 Lie algebras . . . . . . . . . . . . 75 4.5.2 Lie groups . . . . . . . . . . . . . 77 4.5.3 Extensions of Lie algebras and Lie groups 82 4.5.4 Contraction of Lie algebras and Lie groups . 85

5 Hilbert Spaces with Reproducing Kernels and Coherent States 89 5.1 A motivating example. . . . . . . . . 89 5.2 Measurable fields and direct integrals 92 5.3 Reproducing kernel Hilbert spaces. . 94

5.3.1 Positive-definite kernels and evaluation maps. 94 5.3.2 Coherent states and POV functions. . . . . 99 5.3.3 Some isomorphisms, bases, and II-selections 101 5.3.4 A reconstruction problem: Example of a

holomorphic map (eft) . . . . . • . . . . . . . 104 5.4 Some properties of reproducing kernel Hilbert spaces 106

6 Square Integrable and Holomorphic Kernels 109 6.1 Square integrable kernels . . . . . . . . 109 6.2 Holomorphic kernels ............. 112 6.3 Coherent states: The holomorphic case . . . 116

6.3.1 An example of a holomorphic frame (eft) • 119 6.3.2 A nonholomorphic excursion (eft) . 122

7 Covariant Coherent States 127 7.1 Covariant coherent states. . . . . . . . . . . . . . . 128

7.1.1 A general definition. . . . . . . . . . . . . . 128 7.1.2 The Gilmore-Perelomov CS and vector CS . 129 7.1.3 A geometrical setting. . . . . . . . . . . . 133

7.2 Example: The classical theory of coherent states 136 7.2.1 CS of compact semisimple Lie groups. . . 136 7.2.2 CS of noncompact semisimple Lie groups. 139 7.2.3 CS of non-semisimple Lie groups ..... 141

7.3 Square integrable covariant CS: The general case 141 7.3.1 Some further generalizations. . . . . . . . 144

Contents

8 Coherent States from Square Integrable Representations 8.1 Square integrable group representations. 8.2 Orthogonality relations . . . . . . . . . . . . . 8.3 The Wigner map ..... . . . . . . . . . . . 8.4 Modular structures and statistical mechanics .

9 Some Examples and Generalizations 9.1 A class of semidirect product groups

9.1.1 Three concrete examples (") . 9.1.2 A broader setting ...... .

9.2 A generalization: a- and V-admissibility 9.2.1 Example of the Galilei group (") 9.2.2 CS of the isochronous Galilei group (") 9.2.3 Atomic coherent states ...... .

10 CS of General Semidirect Product Groups 10.1 Squeezed states (") ........... . 10.2 Geometry of semidirect product groups ..

10.2.1 A special class of orbits ...... . 10.2.2 The coadjoint orbit structure of r . 10.2.3 Measures on r . . . . . . . . . . . . 10.2.4 Induced representations of semidirect products.

10.3 CS of semidirect products ... . 10.3.1 Admissible affine sections ............ .

11 CS of the Relativity Grouf.s 11.1 The Poincare groups 'P +(1,3) and 'Pt(l, 1)

11.1.1 The Poincare group in 1+3 dimensions, 'P~(1, 3) (") ............... .

11.1.2 The Poincare group in 1+1 dimensions, 'Pt(1,1) (") .............. .

11.1.3 Poincare CS: The massless case ... . 11.2 The Galilei groups Q(l, 1) and Q == Q(l, 3) (") 11.3 The anti-de Sitter group 800 (1,2) and

its contraction(s) (") ............. .

12 Wavelets 12.1 A word of motivation .............. . 12.2 Derivation and properties of the 1-D continuous

wavelet transform (") . . . . . . . . . . . . . . . 12.3 A mathematical aside: Extension to distributions 12.4 Interpretation of the continuous wavelet transform.

12.4.1 The CWT as phase space representation .. 12.4.2 Localization properties and physical interpretation

of the CWT .................... .

xi

147 148 155 160 164

171 171 176 179 181 186 190 196

199 200 204 204 206 209 213 214 220

225 225

225

235 240 242

245

257 257

261 266 272 272

273

xii Contents

12.5 Discretization of the continuous WT: Discrete frames 274 12.6 Ridges and skeletons 276 12.7 Applications....................... 278

13 Discrete Wavelet Transforms 283 13.1 The discrete time or dyadic WT . . . . . . . . . . 283

13.1.1 Multiresolution analysis and orthonormal wavelet bases .. . . . . . . . . 284

13.1.2 Connection with filters and the subband coding scheme . 285

13.1.3 Generalizations . . . . . . . . . 287 13.1.4 Applications. . . . . . . . . . . 288

13.2 Towards a fast CWT: Continuous wavelet packets. 289 13.3 Wavelets on the finite field Zp (") . . . . 290 13.4 Algebraic wavelets ............ 292

13.4.1 r-wavelets of Haar on the line (") 292 13.4.2 Pisot wavelets, etc. (") . . . . . . 303

14 Multidimensional Wavelets 307 14.1 Going to higher dimensions. 307 14.2 Mathematical analysis (") . 308 14.3 The 2-D case. . . . . . . . . 316

14.3.1 Minimality properties. 316 14.3.2 Interpretation, visualization problems,

and calibration ............ 318 14.3.3 Practical applications of the CWT in

two dimensions . . . . . . . . . . . . 322 14.3.4 The discrete WT in two dimensions. 327 14.3.5 Continuous wavelet packets in two dimensions 330

15 Wavelets Related to Other Groups 331 15.1 Wavelets on the sphere and similar manifolds 331

15.1.1 The two-sphere (") . . . . . . . . 331 15.1.2 Generalization to other manifolds . . . 336

15.2 The affine Weyl-Heisenberg group (") . . . . . 338 15.3 The affine or similitude groups of space-time. 342

15.3.1 Kinematical wavelets (") . . . . . . . . 342 15.3.2 The affine Galilei group (") ...... 344 15.3.3 The (restricted) Schrodinger group (") 347 15.3.4 The affine Poincare group (") ..... 349

16 The Discretization Problem: Frames, Sampling, and All That 353 16.1 The Weyl-Heisenberg group or canonical CS . 354 16.2 Wavelet frames ................. 357

Contents xiii

16.3 Frames for affine semidirect products . . . 359 16.3.1 The affine Weyl-Heisenberg group. 359 16.3.2 The affine Poincare groups (10) • . • 360 16.3.3 Discrete frames for general semidirect products 362

16.4 Groups without dilations: The Poincare groups (10) 364

16.5 A group-theoretical approach to discrete wavelet transforms ......... 369 16.5.1 Generalities on sampling . . . . . 369 16.5.2 Wavelets on Zp revisited (10) . • • 370 16.5.3 Wavelets on a discrete abelian group 372

16.6 Conclusion................... 387

Conclusion and Outlook 389

References 393

Index 415