· 2012
Nonlinear poisson brackets: geometry and quantization
- 265citations
- 2012year
Short summary
This book introduces new mathematical objects to construct Lie group analogs for nonlinear Poisson brackets and address quantization problems in semiclassical approximations.
AI-generated from the title and abstract; the full text is not read.
Key points
- Develops Lie group analogs for nonlinear Poisson brackets.
- Addresses quantization problems for nonlinear Poisson brackets in semiclassical approximation.
- Introduces and details new mathematical objects relevant to these problems.
- Synthesizes concepts from algebra, differential geometry, algebraic topology, and operator theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source., 2012 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics