Symplectic Geometry and Quantum Mechanics

1. Auflage 2006
ISBN: 978-3-7643-7575-1
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 368 Seiten

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.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Contents;6
2;Preface;12
3;Notation;16
4;I Symplectic Geometry;20
4.1;1 Symplectic Spaces and Lagrangian Planes;21
4.1.1;1.1 Symplectic Vector Spaces;21
4.1.2;1.2 Skew-Orthogonality;29
4.1.3;1.3 The Lagrangian Grassmannian;33
4.1.4;1.4 The Signature of a Triple of Lagrangian Planes;37
4.2;2 The Symplectic Group;45
4.2.1;2.1 The Standard Symplectic Group;45
4.2.2;2.2 Factorization Results in;56
4.2.3;2.3 Hamiltonian Mechanics;68
4.3;3 Multi-Oriented Symplectic Geometry;82
4.3.1;3.1 Souriau Mapping and Maslov Index;83
4.3.2;3.2 The Arnol’d–Leray–Maslov Index;91
4.3.3;3.3;101
4.3.4;Symplectic Geometry;101
4.4;4 Intersection Indices in Lag(n) and Sp(n);112
4.4.1;4.1 Lagrangian Paths;112
4.4.2;4.2 Symplectic Intersection Indices;117
4.4.3;4.3 The Conley–Zehnder Index;121
5;II Heisenberg Group, Weyl Calculus, and Metaplectic Representation;137
5.1;5 Lagrangian Manifolds and Quantization;138
5.1.1;5.1 Lagrangian Manifolds and Phase;138
5.1.2;5.2 Hamiltonian Motions and Phase;145
5.1.3;5.3 Integrable Systems and Lagrangian Tori;154
5.1.4;5.4 Quantization of Lagrangian Manifolds;160
5.1.5;5.5 Heisenberg–Weyl and Grossmann–Royer Operators;167
5.2;6 Heisenberg Group and Weyl Operators;173
5.2.1;6.1 Heisenberg Group and Schr¨ odinger Representation;174
5.2.2;6.2 Weyl Operators;180
5.2.3;6.3 Continuity and Composition;188
5.2.4;6.4 The Wigner–Moyal Transform;199
5.3;7 The Metaplectic Group;208
5.3.1;7.1 De.nition and Properties of;209
5.3.2;7.2 The Metaplectic Algebra;221
5.3.3;7.3 Maslov Indices on;227
5.3.4;7.4 The Weyl Symbol of a Metaplectic Operator;235
6;III Quantum Mechanics in Phase Space;247
6.1;8 The Uncertainty Principle;248
6.1.1;8.1 States and Observables;249
6.1.2;8.2 The Quantum Mechanical Covariance Matrix;250
6.1.3;8.3 Symplectic Spectrum and Williamson’s Theorem;255
6.1.4;8.4 Wigner Ellipsoids;264
6.1.5;8.5 Gaussian States;273
6.2;9 The Density Operator;281
6.2.1;9.1 Trace-Class and Hilbert–Schmidt Operators;282
6.2.2;9.2 Integral Operators;292
6.2.3;9.3 The Density Operator of a Quantum State;301
6.3;10 A Phase Space Weyl Calculus;313
6.3.1;10.1 Introduction and Discussion;314
6.3.2;10.2 The Wigner Wave-Packet Transform;320
6.3.3;10.3 Phase-Space Weyl Operators;327
6.3.4;10.4 Schrödinger Equation in Phase Space;334
6.3.5;10.5 Conclusion;341
7;A Classical Lie Groups;343
7.1;A.1 General Properties;343
7.2;A.2 The Baker–Campbell–Hausdor. Formula;345
7.3;A.3 One-parameter Subgroups of;345
8;B Covering Spaces and Groups;348
9;C Pseudo-Di.erential Operators;350
9.1;C.1 The Classes;351
9.2;C.2 Composition and Adjoint;351
10;D Basics of Probability Theory;353
10.1;D.1 Elementary Concepts;353
10.2;D.2 Gaussian Densities;355
11;Solutions to Selected Exercises;357
12;Bibliography;363
13;Index;372