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Über dieses Buch

Subject Matter The original title of this book was Tractatus Classico-Quantummechanicus, but it was pointed out to the author that this was rather grandiloquent. In any case, the book discusses certain topics in the interface between classical and quantum mechanics. Mathematically, one looks for similarities between Poisson algebras and symplectic geometry on the classical side, and operator algebras and Hilbert spaces on the quantum side. Physically, one tries to understand how a given quan­ tum system is related to its alleged classical counterpart (the classical limit), and vice versa (quantization). This monograph draws on two traditions: The algebraic formulation of quan­ tum mechanics and quantum field theory, and the geometric theory of classical mechanics. Since the former includes the geometry of state spaces, and even at the operator-algebraic level more and more submerges itself into noncommutative geometry, while the latter is formally part of the theory of Poisson algebras, one should take the words "algebraic" and "geometric" with a grain of salt! There are three central themes. The first is the relation between constructions involving observables on one side, and pure states on the other. Thus the reader will find a unified treatment of certain aspects of the theory of Poisson algebras, oper­ ator algebras, and their state spaces, which is based on this relationship.

Inhaltsverzeichnis

Frontmatter

Introductory Overview

Abstract
The aim of the first chapter is to give two descriptions of classical and quantum mechanics, each of which enables one to see in a different way what their common properties as well as their striking differences are. The first description focuses on the observables of the theory, whereas the second one is based on the pure states.
N. P. Landsman

Chapter I. Observables and Pure States

Abstract
In this section we specify the key algebraic and functional-analytic structures relevant to classical and quantum mechanics. Our main aim is to look at a C*- algebra from the point of view of its self-adjoint part. From this perspective the relationship between the respective algebraic structures of classical and quantum mechanics is particularly transparent.
N. P. Landsman

Chapter II. Quantization and the Classical Limit

Abstract
The aim of quantization theory as presented in this book is to relate Poisson algebras or Poisson manifolds to C*-algebras or their pure state spaces. A slightly awkward feature of the first relationship is that usually Poisson algebras are not Banach spaces; a nonzero Poisson bracket on some Poisson subalgebra \(\widetilde{\mathfrak{A}}_{\mathbb{R}}^{0}ofC_{b}^{\infty }(P,\mathbb{R}{\text{)}}\) cannot be extended to the closure \(\mathfrak{A}_{\mathbb{R}}^{0}of\widetilde{\mathfrak{A}}_{\mathbb{R}}^{0}\) in the sup-norm.
N. P. Landsman

Chapter III. Groups, Bundles, and Groupoids

Abstract
This section describes the main class of examples of Poisson manifolds that are not symplectic. Here G is a Lie group, g its Lie algebra, and g* is the dual of g.
N. P. Landsman

Chapter IV. Reduction and Induction

Abstract
We start with a geometric description of symplectic reduction in a rather general form, and subsequently relate this to the notion of a constraint.
N. P. Landsman

Backmatter

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