Symplectic Geometry and Quantum Mechanics

Maurice de Gosson, Daniel Alpay (Editor)

Research output: Book/ReportBookpeer-review

Abstract

This book is devoted to a rather complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a rigorous presentation of the basics of symplectic geometry and of its multiply-oriented extension. Further chaptersconcentrate onLagrangian manifolds, Weyl operators and the Wigner-Moyal transform as well as on metaplectic groups and Maslov indices. Thus the keys for the mathematical description of quantum mechanics in phase space are discussed. They are followed by a rigorous geometrical treatment of the uncertainty principle. Then Hilbert-Schmidt and trace-class operators are exposed in order to treatdensity matrices. In the last chapter the Weyl pseudo-differential calculus is extended to phase space in order to derive a Schrödinger equation in phase space whose solutions are related to those of the usual Schrödinger equation by a wave-packet transform. The text is essentially self-contained and can be used as basis for graduate courses. Many topics areof genuine interest for pure mathematicians working in geometry and topology.
Original languageEnglish
Place of PublicationBasel Switzerland; Boston
PublisherBirkhauser
Number of pages367
ISBN (Print)3764375744, 9783764375744, 9783764375751
DOIs
StatePublished - 2006

Publication series

NameOperator Theory: Advances and Applications
PublisherBirkhauser
Volume166
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Heisenberg group
  • Lie group
  • Weyl calculus
  • phase space
  • quantum mechanics
  • symplectic geometry
  • partial differential equations

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