Nicholas P. Landsman

Mathematical Topics Between Classical and Quantum Mechanics (eBook, PDF)

• Format: PDF

Produktbeschreibung

This monograph draws on two traditions: the algebraic formulation of quantum mechanics as well as quantum field theory, and the geometric theory of classical mechanics. These are combined in a unified treatment of the theory of Poisson algebras of observables and pure state spaces with a transition probability, which leads on to a discussion of the theory of quantization and the classical limit from this perspective. A prototype of quantization comes from the analogy between the C*- algebra of a Lie groupoid and the Poisson algebra of the corresponding Lie algebroid. The parallel between reduction of symplectic manifolds in classical mechanics and induced representations of groups and C*- algebras in quantum mechanics plays an equally important role. Examples from physics include constrained quantization, curved spaces, magnetic monopoles, gauge theories, massless particles, and $theta$- vacua. Accessible to mathematicians with some prior knowledge of classical and quantum mechanics, and to mathematical physicists and theoretical physicists with some background in functional analysis.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 529
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461216803
• Artikelnr.: 44058185

Inhaltsangabe

Introductory Overview.- I. Observables and Pure States.- II. Quantization and the Classical Limit.- III. Groups, Bundles, and Groupoids.- IV. Reduction and Induction.- I Observables and Pure States.- 1 The Structure of Algebras of Observables.- 2 The Structure of Pure State Spaces.- 3 From Pure States to Observables.- II Quantization and the Classical Limit.- 1 Foundations.- 2 Quantization on Flat Space.- 3 Quantization on Riemannian Manifolds.- III Groups, Bundles, and Groupoids.- 1 Lie Groups and Lie Algebras.- 2 Internal Symmetries and External Gauge Fields.- 3 Lie Groupoids and Lie Algebroids.- IV Reduction and Induction.- 1 Reduction.- 2 Induction.- 3 Applications in Relativistic Quantum Theory.- Notes.- I.- II.- III.- IV.- References.