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Nonlinear Poisson Brackets: Geometry and Quantization
M. V. Karasev and V. P. Maslov
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Translations of Mathematical Monographs
1993; 366 pp; softcover
Volume: 119
ISBN-10: 0-8218-8796-3
ISBN-13: 978-0-8218-8796-7
List Price: US$115
Member Price: US$92
Order Code: MMONO/119.S
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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.

Table of Contents

  • Poisson manifolds
  • Analog of the group operation for nonlinear Poisson brackets
  • Poisson brackets in \(\mathbb R^{2n}\) and semiclassical approximation
  • Asymptotic quantization
  • Appendix I: Formulas of noncommutative analysis
  • Appendix II: Calculus of symbols and commutation relations
  • References
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