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