Symplectic geometry and quantum mechanics

Title:

Personal Author:

Gosson, Maurice de

Series:

Operator theory, advances and applications ; 166

Publication Information:

Basel : Birkhauser Verlag, 2006

ISBN:

9783764375744

Subject Term:

Symplectic geometry

Quantum theory

Boundary value problems -- Weyl theory

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010113505	QA665 G67 2006	Open Access Book	Book	Searching... Unknown

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.

Preface	p. xiii
Introduction	p. xiii
Organization	p. xiv
Prerequisites	p. xv
Bibliography	p. xv
Acknowledgements	p. xv
About the Author	p. xvi
Notation	p. xvii
Number sets	p. xvii
Classical matrix groups	p. xvii
Vector calculus	p. xviii
Function spaces and multi-index notation	p. xix
Combinatorial notation	p. xx
Part I Symplectic Geometry
1 Symplectic Spaces and Lagrangian Planes
1.1 Symplectic Vector Spaces	p. 3
1.1.1 Generalities	p. 3
1.1.2 Symplectic bases	p. 7
1.1.3 Differential interpretation of ¿	p. 9
1.2 Skew-Orthogonality	p. 11
1.2.1 Isotropic and Lagrangian subspaces	p. 11
1.2.2 The symplectic Gram-Schmidt theorem	p. 12
1.3 The Lagrangian Grassmannian	p. 15
1.3.1 Lagrangian planes	p. 15
1.3.2 The action of Sp(n) on Lag(n)	p. 18
1.4 The Signature of a Triple of Lagrangian Planes	p. 19
1.4.1 First properties	p. 20
1.4.2 The cocycle property of ¿	p. 23
1.4.3 Topological properties of ¿	p. 24
2 The Symplectic Group
2.1 The Standard Symplectic Group	p. 27
2.1.1 Symplectic matrices	p. 29
2.1.2 The unitary group U(n)	p. 33
2.1.3 The symplectic algebra	p. 36
2.2 Factorization Results in Sp(n)	p. 38
2.2.1 Polar and Cartan decomposition in Sp(n)	p. 38
2.2.2 The "pre-Iwasawa" factorization	p. 42
2.2.3 Free symplectic matrices	p. 45
2.3 Hamiltonian Mechanics	p. 50
2.3.1 Hamiltonian flows	p. 51
2.3.2 The variational equation	p. 55
2.3.3 The group Ham(n)	p. 58
2.3.4 Hamiltonian periodic orbits	p. 61
3 Multi-Oriented Symplectic Geometry
3.1 Souriau Mapping and Maslov Index	p. 66
3.1.1 The Souriau mapping	p. 66
3.1.2 Definition of the Maslov index	p. 70
3.1.3 Properties of the Maslov index	p. 72
3.1.4 The Maslov index on Sp(n)	p. 73
3.2 The Arnol'd-Leray-Maslov Index	p. 74
3.2.1 The problem	p. 75
3.2.2 The Maslov bundle	p. 79
3.2.3 Explicit construction of the ALM index	p. 80
3.3 q-Symplectic Geometry	p. 84
3.3.1 The identification {{\rm Lag}}_{{\infin}}(n) = {{\rm Lag}}(n) \times \op {{Z}}	p. 85
3.3.2 The universal covering Sp &infty; (n)	p. 87
3.3.3 The action of Sp q (n) on Lag 2q (n)	p. 91
4 Intersection Indices in Lag(n) and Sp(n)
4.1 Lagrangian Paths	p. 95
4.1.1 The strata of Lag(n)	p. 95
4.1.2 The Lagrangian intersection index	p. 96
4.1.3 Explicit construction of a Lagrangian intersection index	p. 98
4.2 Symplectic Intersection Indices	p. 100
4.2.1 The strata of Sp(n)	p. 100
4.2.2 Construction of a symplectic intersection index	p. 101
4.2.3 Example: spectral flows	p. 102
4.3 The Conley-Zehnder Index	p. 104
4.3.1 Definition of the Conley-Zehnder index	p. 104
4.3.2 The symplectic Cayley transform	p. 106
4.3.3 Definition and properties of ¿(S &infty; )	p. 108
4.3.4 Relation between ¿ and \mu_{{l_{{P}}}}	p. 112
Part II Heisenberg Group, Weyl Calculus, and Metaplectic Representation
5 Lagrangian Manifolds and Quantization
5.1 Lagrangian Manifolds and Phase	p. 123
5.1.1 Definition and examples	p. 124
5.1.2 The phase of a Lagrangian manifold	p. 125
5.1.3 The local expression of a phase	p. 129
5.2 Hamiltonian Motions and Phase	p. 130
5.2.1 The Poincaré-Cartan Invariant	p. 130
5.2.2 Hamilton-Jacobi theory	p. 133
5.2.3 The Hamiltonian phase	p. 136
5.3 Integrable Systems and Lagrangian Tori	p. 139
5.3.1 Poisson brackets	p. 139
5.3.2 Angle-action variables	p. 141
5.3.3 Lagrangian tori	p. 143
5.4 Quantization of Lagrangian Manifolds	p. 145
5.4.1 The Keller-Maslov quantization conditions	p. 145
5.4.2 The case of q-oriented Lagrangian manifolds	p. 147
5.4.3 Waveforms on a Lagrangian Manifold	p. 149
5.5 Heisenberg-Weyl and Grossmann-Royer Operators	p. 152
5.5.1 Definition of the Heisenberg-Weyl operators	p. 152
5.5.2 First properties of the operators \widehat {{T}}(z)	p. 154
5.5.3 The Grossmann-Royer operators	p. 156
6 Heisenberg Group and Weyl Operators
6.1 Heisenberg Group and Schrödinger Representation	p. 160
6.1.1 The Heisenberg algebra and group	p. 160
6.1.2 The Schrödinger representation of H n	p. 163
6.2 Weyl Operators	p. 166
6.2.1 Basic definitions and properties	p. 167
6.2.2 Relation with ordinary pseudo-differential calculus	p. 170
6.3 Continuity and Composition	p. 174
6.3.1 Continuity properties of Weyl operators	p. 174
6.3.2 Composition of Weyl operators	p. 179
6.3.3 Quantization versus dequantization	p. 183
6.4 The Wigner-Moyal Transform	p. 185
6.4.1 Definition and first properties	p. 186
6.4.2 Wigner transform and probability	p. 189
6.4.3 On the range of the Wigner transform	p. 192
7 The Metaplectic Group
7.1 Definition and Properties of Mp(n)	p. 196
7.1.1 Quadratic Fourier transforms	p. 196
7.1.2 The projection \pi^{{{{\rm Mp}}}} : {{\rm Mp}}(n) \rarr Sp(n)	p. 199
7.1.3 Metaplectic covariance of Weyl calculus	p. 204
7.2 The Metaplectic Algebra	p. 208
7.2.1 Quadratic Hamiltonians	p. 208
7.2.2 The Schrödinger equation	p. 209
7.2.3 The action of Mp(n) on Gaussians: dynamical approach	p. 212
7.3 Maslov Indices on Mp(n)	p. 214
7.3.1 The Maslov index \widehat \mu (\widehat {{S}})	p. 215
7.3.2 The Maslov indices \widehat {{\mu}}_l(\widehat {{S}})	p. 220
7.4 The Weyl Symbol of a Metaplectic Operator	p. 222
7.4.1 The operators \widehat {{R}}_{{\nu}}(S)	p. 223
7.4.2 Relation with the Conley-Zehnder index	p. 227
Part III Quantum Mechanics in Phase Space
8 The Uncertainty Principle
8.1 States and Observables	p. 238
8.1.1 Classical mechanics	p. 238
8.1.2 Quantum mechanics	p. 239
8.2 The Quantum Mechanical Covariance Matrix	p. 239
8.2.1 Covariance matrices	p. 240
8.2.2 The uncertainty principle	p. 240
8.3 Symplectic Spectrum and Williamson's Theorem	p. 244
8.3.1 Williamson normal form	p. 244
8.3.2 The symplectic spectrum	p. 246
8.3.3 The notion of symplectic capacity	p. 248
8.3.4 Admissible covariance matrices	p. 252
8.4 Wigner Ellipsoids	p. 253
8.4.1 Phase space ellipsoids	p. 253
8.4.2 Wigner ellipsoids and quantum blobs	p. 255
8.4.3 Wigner ellipsoids of subsystems	p. 258
8.4.4 Uncertainty and symplectic capacity	p. 261
8.5 Gaussian States	p. 262
8.5.1 The Wigner transform of a Gaussian	p. 263
8.5.2 Gaussians and quantum blobs	p. 265
8.5.3 Averaging over quantum blobs	p. 266
9 The Density Operator
9.1 Trace-Class and Hilbert-Schmidt Operators	p. 272
9.1.1 Trace-class operators	p. 272
9.1.2 Hilbert-Schmidt operators	p. 279
9.2 Integral Operators	p. 282
9.2.1 Operators with L 2 kernels	p. 282
9.2.2 Integral trace-class operators	p. 285
9.2.3 Integral Hilbert-Schmidt operators	p. 288
9.3 The Density Operator of a Quantum State	p. 291
9.3.1 Pure and mixed quantum states	p. 291
9.3.2 Time-evolution of the density operator	p. 296
9.3.3 Gaussian mixed states	p. 298
10 A Phase Space Weyl Calculus
10.1 Introduction and Discussion	p. 304
10.1.1 Discussion of Schrödinger's argument	p. 304
10.1.2 The Heisenberg group revisited	p. 307
10.1.3 The Stone-von Neumann theorem	p. 309
10.2 The Wigner Wave-Packet Transform	p. 310
10.2.1 Definition of U ¿	p. 310
10.2.2 The range of U ¿	p. 314
10.3 Phase-Space Weyl Operators	p. 317
10.3.1 Useful intertwining formulae	p. 317
10.3.2 Properties of phase-space Weyl operators	p. 319
10.3.3 Metaplectic covariance	p. 321
10.4 Schrödinger Equation in Phase Space	p. 324
10.4.1 Derivation of the equation (10.39)	p. 324
10.4.2 The case of quadratic Hamiltonians	p. 325
10.4.3 Probabilistic interpretation	p. 327
10.5 Conclusion	p. 331
A Classical Lie Groups
A.1 General Properties	p. 333
A.2 The Baker-Campbell-Hausdorff Formula	p. 335
A.3 One-parameter Subgroups of GL(m, \op {{R}} )	p. 335
B Covering Spaces and Groups
C Pseudo-Differential Operators
C.1 The Classes S_{{\rho, \delta}}^m, L_{{\rho, \delta}}^{{m}}	p. 342
C.2 Composition and Adjoint	p. 342
D Basics of Probability Theory
D.1 Elementary Concepts	p. 345
D.2 Gaussian Densities	p. 347
Solutions to Selected Exercises	p. 349
Bibliography	p. 355
Index	p. 365

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