\"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 representa-tion, wave packets, and related concepts: it is meant to suggest analysis on theconfiguration space Rn done by working in the phase space Rn x Rn. The ideasthat fall under this rubric have originated in several different fidds??Fourieranalysis, partial differential equations, mathematical physics, representationtheory, and number theory, among others.\"

\"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 SchrSdinger representation The integrated representation, 23. Twisted convolution, 25. The uncertainty principle, 27. 4. The Fourier-Wigner transform Radar ambiguity functions, 33. 5. The Stone-von 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 CalderSn-Vaillancourt theorems 6. The sharp Garding 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 CHAPPTER 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 meta\"