Nonlinear Poisson brackets: geometry and quantization
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Providence, R.I.
American Mathematical Society
1993
|
| Series: | Translations of mathematical monographs
119 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XI, 366 S. graph. Darst. |
| ISBN: | 0821845969 |
Staff View
MARC
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| 245 | 1 | 0 | |a Nonlinear Poisson brackets |b geometry and quantization |c M. V. Karasev ; V. P. Maslov |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 1993 | |
| 300 | |a XI, 366 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Translations of mathematical monographs |v 119 | |
| 500 | |a Aus d. Russ. übers. | ||
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| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 7 | |a Systèmes hamiltoniens |2 ram | |
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Record in the Search Index
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| adam_text | Contents
Preface ix
Introduction 1
CHAPTER I. Poisson Manifolds 7
§ 1. Poisson brackets related to Lie groups 7
1.1. Symplectic leaves and the Darboux theorem (8)
1.2. Linear brackets. Phase space over a Lie group (14)
1.3. Brackets generated by 1 forms. Cocycles of Lie bialgebras (18)
1.4. Examples of compatible brackets. The Yang Baxter equation in
Lie algebras (23)
§2. Reduction and deformation of brackets 27
2.1. Lagrangian and coisotropic submanifolds. Hamiltonian flows (27)
2.2. Bifibrations and brackets on their bases (30)
2.3. Lie Cartan reduction. Action angle variables (35)
2.4. Examples of reduced brackets (38)
2.5. Brackets generated by 2 forms. The Dirac bracket (44)
§3. Perturbations and cohomology of Poisson brackets 53
3.1. The infinitesimal deformation problem. Examples (53)
3.2. Structure of the Poisson manifold near nondegenerate leaves (56)
3.3. Free brackets. Nonisotropic deformations (61)
3.4. Anomalies in the Jacobi identity (67)
3.5. Tower of obstructions. General outline for the calculation of
tensor cohomology, cocyles, and coboundaries (71)
CHAPTER II. Analog of the Group Operation for Nonlinear Poisson
Brackets 75
§1. Phase space over a Poisson manifold 75
1.1. Symplectic groupoids (75)
1.2. Analogs of direct Lie theorems (78)
1.3. System of Lie equations (81)
1.4. Gluing of the phase space. An analog of the third inverse Lie
theorem (83)
1.5. Multiplication in phase space. Analogs of the 1st and 2nd inverse
Lie theorems (87)
§2. Examples of symplectic goupoids 91
2.1. Actions of groupoids and bifibrations (91)
2.2. Polar groupoid (94)
V
vi CONTENTS
2.3. Nilpotent and solvable brackets (96)
2.4. The Cartan structure (99)
2.5. The groupoid for the Cartan structure. Affine brackets (103)
§3. Finite dimensional pseudogroups and connections on Poisson
manifolds 106
3.1. Actions of finite dimensional pseudogroups (108)
3.2. Reconstruction of a pseudogroup from canonical vector fields
and structure functions (112)
3.3. Canonical actions on symplectic manifolds (115)
3.4. Linear connections and basis of the pseudoalgebra (117)
3.5. Poisson brackets on groups and pseudogroups compatible with
them (121)
3.6. Adjoint almost brackets and almost Poisson actions (126)
3.7. Local vanishing of torsion and non Hamiltonian actions (130)
3.8. The symplectic groupoid generated by a pseudogroup (133)
CHAPTER III. Poisson Brackets in R2 and Semiclassical
Approximation 139
§ 1. Lagrangian submanifolds as fronts of wave packets 139
1.1. Quantum density of a packet (140)
1.2. Gaussian and oscillating packets (143)
1.3. Theorem on the Lagrangian property of fronts (146)
1.4. Functorial properties of density (149)
1.5. Localization of wave packets (153)
1.6. Holography (154)
§2. The correspondence principle in the language of Lagrangian
geometry 157
2.1. Intertwining of classical and quantum variables (157)
2.2. One dimensional obstructions. Path index (162)
2.3. Formulas for the intertwining operator (168)
2.4. Quantization of solutions to Hamiltonian systems. The
eigenvalue problem (173)
2.5. The Cauchy problem. The oscillator and 90° rotations (177)
CHAPTER IV. Asymptotic Quantization 185
§1. Review of general approaches to quantization 185
1.1. General ideas and notation (185)
1.2. Quantization of symplectic manifolds (188)
1.3. Quantization of degenerate Poisson brackets (190)
§2. Sheaf of wave packets over a symplectic manifold 192
2.1. Action of Poisson mappings on wave packets (192)
2.2. Nonlocal cocycle over the groupoid of Poisson mappings (197)
2.3. Two dimensional obstructions to gluing a sheaf. Global
? product of symbols (203)
2.4. Relationship with the theory of geometric quantization (211)
2.5. Torus, sphere, and sphere with horns (214)
CONTENTS vii
§3. Quantization of two dimensional surfaces 226
3.1. Index of two dimensional surfaces (226)
3.2. Rule of quantization (231)
3.3. Intertwining operators in quantized symplectic manifolds (233)
3.4. Example. Asymmetric SO(3) top (234)
3.5. Quantization of Poisson mappings. Lifting of asymptotics from
reduced spaces (237)
§4. Nonlinear commutation relations in semiclassical
approximation 245
4.1. Quadratic relations with a small parameter (246)
4.2. Quantum corrections to Poisson brackets (247)
4.3. Generators of the * product on oscillating symbols (249)
4.4. Representation of commutation relations by ft pseudodifferential
operators (255)
4.5. Convolution corresponding to nonlinear Poisson brackets (258)
Appendix I. Formulas of Noncommutative Analysis 265
1.1. Ordered functions of operators and Weyl functions (265)
1.2. Formulas of differentiation and disentangling (271)
1.3. Permutation of operators. Commutation with the exponent (277)
1.4. Functions of functions of operators (284)
1.5. Reduction to normal form (287)
1.6. Paradoxes of formal calculations with functions of operators
(292)
Appendix II. Calculus of Symbols and Commutation Relations 297
2.1. Generalized Jacobi conditions and Poincare Birkhoff Witt
property (297)
2.2. Change of order and * product over the Heisenberg algebra (306)
2.3. Semilinear commutation relations (309)
2.4. Strongly nonlinear and solvable relations (313)
2.5. Quantum Yang Baxter equation (321)
2.6. Reduction to triangular form (324)
2.7. Spectrum and cospectrum of quadratic linear relations (328)
2.8. Transformation of scale and structure constants (335)
2.9. Algebras equivalent to Lie algebras (346)
References 353
|
| any_adam_object | 1 |
| author | Karasev, Michail V. Maslov, Viktor P. 1930- |
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| indexdate | 2024-07-09T17:13:43Z |
| institution | BVB |
| isbn | 0821845969 |
| language | English Russian |
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| physical | XI, 366 S. graph. Darst. |
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| series2 | Translations of mathematical monographs |
| spelling | Karasev, Michail V. Verfasser aut Nelinejnye skobki Puassona Nonlinear Poisson brackets geometry and quantization M. V. Karasev ; V. P. Maslov Providence, R.I. American Mathematical Society 1993 XI, 366 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 119 Aus d. Russ. übers. Globale analyse gtt Manifolds gtt Poisson, Parenthèses de Poisson, Variétés de Poisson, parenthèses de ram Poisson, variétés de ram Quantification géométrique Systèmes hamiltoniens Systèmes hamiltoniens ram Hamiltonian systems Poisson brackets Poisson manifolds Poisson-Klammer (DE-588)4332536-1 gnd rswk-swf Poisson-Klammer (DE-588)4332536-1 s DE-604 Maslov, Viktor P. 1930- Verfasser (DE-588)123340853 aut Translations of mathematical monographs 119 (DE-604)BV000002394 119 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005305915&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Karasev, Michail V. Maslov, Viktor P. 1930- Nonlinear Poisson brackets geometry and quantization Translations of mathematical monographs Globale analyse gtt Manifolds gtt Poisson, Parenthèses de Poisson, Variétés de Poisson, parenthèses de ram Poisson, variétés de ram Quantification géométrique Systèmes hamiltoniens Systèmes hamiltoniens ram Hamiltonian systems Poisson brackets Poisson manifolds Poisson-Klammer (DE-588)4332536-1 gnd |
| subject_GND | (DE-588)4332536-1 |
| title | Nonlinear Poisson brackets geometry and quantization |
| title_alt | Nelinejnye skobki Puassona |
| title_auth | Nonlinear Poisson brackets geometry and quantization |
| title_exact_search | Nonlinear Poisson brackets geometry and quantization |
| title_full | Nonlinear Poisson brackets geometry and quantization M. V. Karasev ; V. P. Maslov |
| title_fullStr | Nonlinear Poisson brackets geometry and quantization M. V. Karasev ; V. P. Maslov |
| title_full_unstemmed | Nonlinear Poisson brackets geometry and quantization M. V. Karasev ; V. P. Maslov |
| title_short | Nonlinear Poisson brackets |
| title_sort | nonlinear poisson brackets geometry and quantization |
| title_sub | geometry and quantization |
| topic | Globale analyse gtt Manifolds gtt Poisson, Parenthèses de Poisson, Variétés de Poisson, parenthèses de ram Poisson, variétés de ram Quantification géométrique Systèmes hamiltoniens Systèmes hamiltoniens ram Hamiltonian systems Poisson brackets Poisson manifolds Poisson-Klammer (DE-588)4332536-1 gnd |
| topic_facet | Globale analyse Manifolds Poisson, Parenthèses de Poisson, Variétés de Poisson, parenthèses de Poisson, variétés de Quantification géométrique Systèmes hamiltoniens Hamiltonian systems Poisson brackets Poisson manifolds Poisson-Klammer |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005305915&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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