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Symplectic Geometry and Quantum Mechanics

  • Book
  • © 2006

Overview

Authors:
  1. Maurice Gosson

    1. Institut für Mathematik, Universität Potsdam, Potsdam, Germany

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  • Complete theory of the Maslov index and its variants
  • Discussion of the metaplectic group and the Conley-Zehnder index
  • Rigorous mathematical treatment of the Schrödinger equation in phase space
  • Includes supplementary material: sn.pub/extras

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

Part of the book sub series: Advances in Partial Differential Equations (APDE)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xx
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  3. Heisenberg Group, Weyl Calculus, and Metaplectic Representation

    1. Front Matter

      Pages 121-121
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    2. Lagrangian Manifolds and Quantization

      Pages 123-157

    3. Heisenberg Group and Weyl Operators

      Pages 159-193

    4. The Metaplectic Group

      Pages 195-233

  4. Quantum Mechanics in Phase Space

    1. Front Matter

      Pages 235-235
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    2. The Uncertainty Principle

      Pages 237-269

    3. The Density Operator

      Pages 271-302

    4. A Phase Space Weyl Calculus

      Pages 303-332

  5. Back Matter

    Pages 333-368
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Keywords

About this book

Introduction We have been experiencing since the 1970s a process of “symplectization” of S- ence especially since it has been realized that symplectic geometry is the natural language of both classical mechanics in its Hamiltonian formulation, and of its re?nement,quantum mechanics. The purposeof this bookis to providecorema- rial in the symplectic treatment of quantum mechanics, in both its semi-classical and in its “full-blown” operator-theoretical formulation, with a special emphasis on so-called phase-space techniques. It is also intended to be a work of reference for the reading of more advanced texts in the rapidly expanding areas of sympl- tic geometry and topology, where the prerequisites are too often assumed to be “well-known”bythe reader. Thisbookwillthereforebeusefulforbothpurema- ematicians and mathematical physicists. My dearest wish is that the somewhat novel presentation of some well-established topics (for example the uncertainty principle and Schrod ¨ inger’s equation) will perhaps shed some new light on the fascinating subject of quantization and may open new perspectives for future - terdisciplinary research. I have tried to present a balanced account of topics playing a central role in the “symplectization of quantum mechanics” but of course this book in great part represents my own tastes. Some important topics are lacking (or are only alluded to): for instance Kirillov theory, coadjoint orbits, or spectral theory. We will moreover almost exclusively be working in ?at symplectic space: the slight loss in generality is, from my point of view, compensated by the fact that simple things are not hidden behind complicated “intrinsic” notation.

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)

Authors and Affiliations

  • Institut für Mathematik, Universität Potsdam, Potsdam, Germany

    Maurice Gosson

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