This book deals with two old mathematical problems. The first is the problem of constructing an analog of a Lie group for general nonlinear Poisson brackets. The second is the quantization problem for such brackets in the semiclassical approximation (which is the problem of exact quantization for the simplest classes of brackets). These problems are progressively coming to the fore in the modern theory of differential equations and quantum theory, since the approach based on constructions of algebras and Lie groups seems, in a certain sense, to be exhausted. The authors' main goal is to describe in detail the new objects that appear in the solution of these problems. Many ideas of algebra, modern differential geometry, algebraic topology, and operator theory are synthesized here. The authors prove all statements in detail, thus making the book accessible to graduate students.
Readership: Graduate students and researchers
Author(s): M. V. Karasev and V. P. Maslov
Series: Translations of Mathematical Monographs, Vol. 119
Publisher: American Mathematical Society
Year: 1993
Language: English
Pages: C+xii+366+B
Cover
Nonlinear Poisson Brackets: Geometry and Quantization
Copyright ®1993 by the American Mathematical Society.
ISBN 0-8218-4596-9
QA614.83.K3713 1993 514'.74-dc20
LCCN 92-42061 CIP
Contents
Preface
Introduction
CHAPTER I Poisson Manifolds
§1. Poisson brackets related to Lie groups
1.1. Symplectic leaves and the Darboux theorem
1.2. Linear brackets. Phase space over a Lie group
1.3. Brackets generated by 1-forms. Cocycles of Lie bialgebras
1.4. Examples of compatible brackets. The Yang-Baxter equation in Lie algebras
§2. Reduction and deformation of brackets
2.1. Lagrangian and coisotropic submanifolds. Hamiltonian flows
2.2. Bifibrations and brackets on their bases
2.3. Lie-Cartan reduction. Action-angle variables
2.4. Examples of reduced brackets
2.5. Brackets generated by 2-forms. The Dirac bracket
§3. Perturbations and cohomology of Poisson brackets
3.1. The infinitesimal deformation problem. Examples
3.2. Structure of the Poisson manifold near nondegenerate leaves
3.3. Free brackets. Nonisotropic deformations
3.4. Anomalies in the Jacobi identity
3.5. Tower of obstructions. General outline for the calculation of tensor cohomology, cocycles, and coboundaries
CHAPTER II Analog of the Group Operation for Nonlinear Poisson Brackets
§1. Phase space over a Poisson manifold
1.1. Symplectic groupoids
1.2. Analogs of direct Lie theorems
1.3. System of Lie equations
1.4. Gluing of the phase space. An analog of the third inverse Lie theorem
1.5. Multiplication in phase space. Analogs of the 1st and 2nd inverse Lie theorems.
§2. Examples of symplectic groupoids
2.1. Actions of groupoids and bifibrations
2.2. Polar groupoid
2.3. Nilpotent and solvable brackets
2.4. The Cartan structure
2.5. The groupoid for the Cartan structure. Affine brackets
§3. Finite-dimensional pseudogroups and connections on Poisson manifolds
3.1. Actions of finite-dimensional pseudogroups
3.2. Reconstruction of a pseudogroup from canonical vector fields and structure functions
3.3. Canonical actions on symplectic manifolds
3.4. Linear connections and basis of the pseudoalgebra
3.5. Poisson brackets on groups and pseudogroups compatible with them.
3.6. Adjoint almost brackets and almost Poisson actions.
3.7. Local vanishing of torsion and non-Hamiltonian actions
3.8. The symplectic groupoid generated by a pseudogroup
CHAPTER III Poisson Brackets in R^2n and Semiclassical Approximation
§1. Lagrangian submanifolds as fronts of wave packets
1.1. Quantum density of a packet.
1.2. Gaussian and oscillating packets
1.3. Theorem on the Lagrangian property of fronts
1.4. Functorial properties of density
1.5. Localization of wave packets
1.6. Holography
§2. The correspondence principle in the language of Lagrangian geometry
2.1. Intertwining of classical and quantum variables
2.2. One-dimensional obstructions. Path index
2.3. Formulas for the intertwining operator
2.4. Quantization of solutions to Hamiltonian systems. The eigenvalue problem.
2.5. The Cauchy problem. The oscillator and 90° rotations
CHAPTER IV Asymptotic Quantization
§1. Review of general approaches to quantization
1.1. General ideas and notation
1.2. Quantization of symplectic manifolds
1.3. Quantization of degenerate Poisson brackets
§2. Sheaf of wave packets over a symplectic manifold
2.1. Action of Poisson mappings on wave packets
2.2. Nonlocal cocycle over the groupoid of Poisson mapping
2.3. Two-dimensional obstructions to gluing a sheaf. Global *-product of symbols
2.4. Relationship with the theory of geometric quantization
2.5. Torus, sphere, and sphere with horns
§3. Quantization of two-dimensional surfaces
3.1. Index of two-dimensional surfaces
3.2. Rule of quantization
3.3. Intertwining operators in quantized symplectic manifolds
3.4. Example. Asymmetric SO(3)-top
3.5. Quantization of Poisson mappings. Lifting of asymptotics from reduced spaces.
§4. Nonlinear commutation relations in semiclassical approximation
4.1. Quadratic relations with a small parameter
4.2. Quantum corrections to Poisson bracket
4.3. Generators of the *-product on oscillating symbols.
4.4. Representation of commutation relations by h-pseudodifferential operators.
4.5. Convolution corresponding to nonlinear Poisson brackets.
APPENDIX I Formulas of Noncommutative Analysis
1.1. Ordered functions of operators and Weyl functions.
1.2. Formulas of differentiation and disentangling
1.3. Permutation of operators. Commutation with the exponent
1.4. Functions of functions of operators
1.5. Reduction to normal form.
1.6. Paradoxes of formal calculations with functions of operators
APPENDIX II Calculus of Symbols and Commutation Relations
2.1. Generalized Jacobi conditions and Poincare-Birkhof Witt property
2.2. Change of order and *-product over the Heisenberg algebra.
2.3. Semilinear commutation relations.
2.4. Strongly nonlinear and solvable relations
2.5. Quantum Yang-Baxter equation.
2.6. Reduction to triangular fo
2.7. Spectrum and cospectrum of quadratic-linear relations
2.8. Transformation of scale and structure constants
2.9. Algebras equivalent to Lie algebras
References
Back Cover
