The phrase "harmonic analysis in phase space" is a concise if somewhatinadequate name for the area of analysis on Rn that involves the Heisenberggroup, quantization, the Weyl operational calculus, the metaplectic representation, wave packets, and related concepts: it is meant to suggest analysis on theconfiguration space Rn done by working in the phase space Rn×Rn. The ideasthat fall under this rubric have originated in several different fields--Fourieranalysis, partial differential equations, mathematical physics, representationtheory, and number theory, among others. As a result, although these ideas areindividually well known to workers in such fields, their close kinship and thecross-fertilization they can provide have often been insufficiently appreciated.One of the principal objectives of this monograph is to give a coherent accountof this material, comprising not just an efficient tour of the major avenues butalso an exploration of some picturesque byways.

Preface

Prologue. Some Matters of Notation

CHAPTER 1. THE HEISENBERG GROUP AND ITS REPRESENTATIONS

 1. Background from physics

Hamiltonian mechanics, 10. Quantum mechanics, 12. Quantization, 15.

 2. The Heisenberg group

The automorphisms of the Heisenberg group, 19.

 3. The Schr~Sdinger representation

The integrated representation, 23. Twisted convolution, 25.

The uncertainty principle, 27.

 4. The Fourier-Wigner transform

Radar ambiguity functions, 33.

 5. The Stone-yon Neumann theorem

The group Fourier transform, 37.

 6. The Fock-Bargmann representation

Some motivation and history, 47.

 7. Hermite functions

 8. The Wigner transform

 9. The Laguerre connection

 10. The nilmanifold representation

 11. Postscripts

CHAPTER 2. QUANTIZATION AND PSEUDODIFFERENTIAL OPERATORS

 1. The Weyl correspondence

Covariance properties, 83. Symbol classes, 86. Miscellaneous remarks

and examples, 90.

 2. The Kohn-Nirenberg correspondence

 3. The product formula

 4. Basic pseudodifferential theory

Wave front sets, 118.

 5. The Calder6n-VaiUancourt theorems

 6. The sharp G~rding inequality

 7. The Wick and anti-Wick correspondences

CHAPTER 3. WAVE PACKETS AND WAVE FRONTS

 1. Wave packet expansions

 2. A characterization of wave front sets

 3. Analyticity and the FBI transform

 4. Gabor expansions

CHAPTER 4. THE METAPLECTIC REPRESENTATION

 1. Symplectic linear algebra

 2. Construction of the metaplectic representation

The Fock model, 180.

 3. The infinitesimal representation

 4. Other aspects of the metaplectic representation

Integral formulas, 191. Irreducible subspaces, 194. Dependence on

Planck's constant, 195. The extended metaplectic representation, 196.

The Groenewold-van Hove theorems, 197. Some applications, 199.

 5. Gaussians and the symmetric space

Characterizations of Gaussians, 206.

 6. The disc model

 7. Variants and analogues

Restrictions of the metaplectic representation, 216. U(n,n) as a complex

symplectic group, 217. The spin representation, 220.

CHAPTER 5. THE OSCILLATOR SEMIGROUP

 1. The Schrodinger model

The extended oscillator semigroup, 234.

 2. The Hermite semigroup

 3. Normalization and the Cayley transform

 4. The Fock model

Appendix A. Gaussian Integrals and a Lemma on Determinants

Appendix B. Some Hilbert Space Results

Bibliography

Index