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© 2006

Symplectic Geometry and Quantum Mechanics

Book

Part of the Operator Theory: Advances and Applications book series (OT, volume 166)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Symplectic Geometry

  3. Heisenberg Group, Weyl Calculus, and Metaplectic Representation

    1. Front Matter
      Pages 121-121
    2. Pages 195-233
  4. Quantum Mechanics in Phase Space

    1. Front Matter
      Pages 235-235
    2. Pages 271-302
  5. Back Matter
    Pages 333-368

About this book

Introduction

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 chapters concentrate on Lagrangian 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 treat density 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 are of genuine interest for pure mathematicians working in geometry and topology.

Keywords

Heisenberg group Lie group Weyl calculus phase space quantum mechanics symplectic geometry

Authors and affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

Bibliographic information

Reviews

From the reviews:

"De Gosson’s book is an exhaustive and clear description of almost all the more recent results obtained in connected areas of research like symplectiv geometry, the combinatorial theory of the Maslov index, the theory of the metaplectic group and so on. It fills an important niche in the literature." -Mircea Crâsmareanu, Analele Stiintifice

"This book concerns certain aspects of symplectic geometry and their application to quantum mechanics. … This book seems best suited to someone who already has a solid background in quantum theory and wants to learn more about the symplectic geometric techniques used in quantization. … the book contains useful information about various important topics." (Brian C. Hall, Mathematical Reviews, Issue 2007 e)

“This book covers … symplectic geometry and their applications in quantum mechanics with an emphasis on phase space methods. … The exposition is very detailed and complete proofs are given. … the book takes a particularly fresh point of view on some of the topics and contains a lot of useful information for readers with some background in quantum theory and an interest in the use of symplectic techniques.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 155 (1), September, 2008)