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Physics - Theoretical, Mathematical & Computational Physics | Mathematical Topics Between Classical and Quantum Mechanics

Mathematical Topics Between Classical and Quantum Mechanics

Landsman, Nicholas P.

1998, XIX, 529 p.

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  • About this book

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.

Content Level » Research

Keywords » Mathematica - classical mechanics - field theory - functional analysis - gauge theories - mechanics - quantization - quantum field - quantum field theory - quantum mechanics - quantum theory

Related subjects » Theoretical, Mathematical & Computational Physics

Table of contents 

Introductory Overview.- I. Observables and Pure States.- Observables.- Pure States.- From Pure States to Observables.- II. Quantization and the Classical Limit.- Foundations.- Quantization on Flat Space.- Quantization on Riemannian Manifolds.- III. Groups, Bundles, and Groupoids.- Lie Groups and Lie Algebras.- Internal Symmetries and External Gauge Fields.- Lie Groupoids and Lie Algebroids.- IV. Reduction and Induction.- Reduction.- Induction.- Applications in Relativistic Quantum Theory.- I Observables and Pure States.- 1 The Structure of Algebras of Observables.- 1.1 Jordan-Lie Algebras and C*-Algebras.- 1.2 Spectrum and Commutative C*-Algebras.- 1.3 Positivity, Order, and Morphisms.- 1.4 States.- 1.5 Representations and the GNS-Construction.- 1.6 Examples of C*-Algebras and State Spaces.- 1.7 Von Neumann Algebras.- 2 The Structure of Pure State Spaces.- 2.1 Pure States and Compact Convex Sets.- 2.2 Pure States and Irreducible Representations.- 2.3 Poisson Manifolds.- 2.4 The Symplectic Decomposition of a Poisson Manifold.- 2.5 (Projective) Hilbert Spaces as Symplectic Manifolds..- 2.6 Representations of Poisson Algebras.- 2.7 Transition Probability Spaces.- 2.8 Pure State Spaces as Transition Probability Spaces.- 3 From Pure States to Observables.- 3.1 Poisson Spaces with a Transition Probability.- 3.2 Identification of the Algebra of Observables.- 3.3 Spectral Theorem and Jordan Product.- 3.4 Unitarity and Leibniz Rule.- 3.5 Orthomodular Lattices.- 3.6 Lattices Associated with States and Observables.- 3.7 The Two-Sphere Property in a Pure State Space.- 3.8 The Poisson Structure on the Pure State Space.- 3.9 Axioms for the Pure State Space of a C*-Algebra.- II Quantization and the Classical Limit.- 1 Foundations.- 1.1 Strict Quantization of Observables.- 1.2 Continuous Fields of C*-Algebras.- 1.3 Coherent States and Berezin Quantization.- 1.4 Complete Positivity.- 1.5 Coherent States and Reproducing Kernels.- 2 Quantization on Flat Space.- 2.1 The Heisenberg Group and its Representations.- 2.2 The Metaplectic Representation.- 2.3 Berezin Quantization on Flat Space.- 2.4 Properties of Berezin Quantization on Flat Space.- 2.5 Weyl Quantization on Flat Space.- 2.6 Strict Quantization and Continuous Fields on Flat Space.- 2.7 The Classical Limit of the Dynamics.- 3 Quantization on Riemannian Manifolds.- 3.1 Some Affine Geometry.- 3.2 Some Riemannian Geometry.- 3.3 Hamiltonian Riemannian Geometry.- 3.4 Weyl Quantization on Riemannian Manifolds.- 3.5 Proof of Strictness.- 3.6 Commutation Relations on Riemannian Manifolds.- 3.7 The Quantum Hamiltonian and its Classical Limit.- III Groups, Bundles, and Groupoids.- 1 Lie Groups and Lie Algebras.- 1.1 Lie Algebra Actions and the Momentum Map.- 1.2 Hamiltonian Group Actions.- 1.3 Multipliers and Central Extensions.- 1.4 The (Twisted) Lie-Poisson Structure.- 1.5 Projective Representations.- 1.6 The Twisted Enveloping Algebra.- 1.7 Group C*-Algebras.- 1.8 A Generalized Peter-Weyl Theorem.- 1.9 The Group C* Algebra as a Strict Quantization.- 1.10 Representation Theory of Compact Lie Groups.- 1.11 Berezin Quantization of Coadjoint Orbits.- 2 Internal Symmetries and External Gauge Fields.- 2.1 Bundles.- 2.2 Connections.- 2.3 Cotangent Bundle Reduction.- 2.4 Bundle Automorphisms and the Gauge Group.- 2.5 Construction of Classical Observables.- 2.6 The Classical Wong Equations.- 2.7 The H-Connection.- 2.8 The Quantum Algebra of Observables.- 2.9 Induced Group Representations.- 2.10 The Quantum Wong Hamiltonian.- 2.11 From the Quantum to the Classical Wong Equations.- 2.12 The Dirac Monopole.- 3 Lie Groupoids and Lie Algebroids.- 3.1 Groupoids.- 3.2 Half-Densities on Lie Groupoids.- 3.3 The Convolution Algebra of a Lie Groupoid.- 3.4 Action *-Algebras.- 3.5 Representations of Groupoids.- 3.6 The C*-Algebra of a Lie Groupoid.- 3.7 Examples of Lie Groupoid C*-Algebras.- 3.8 Lie Algebroids.- 3.9 The Poisson Algebra of a Lie Algebroid.- 3.10 A Generalized Exponential Map.- 3.11 The Groupoid C*-Algebra as a Strict Quantization.- 3.12 The Normal Groupoid of a Lie Groupoid.- IV Reduction and Induction.- 1 Reduction.- 1.1 Basics of Constraints and Reduction.- 1.2 Special Symplectic Reduction.- 1.3 Classical Dual Pairs.- 1.4 The Classical Imprimitivity Theorem.- 1.5 Marsden-Weinstein Reduction.- 1.6 Kazhdan-Kostant-Sternberg Reduction.- 1.7 Proof of the Classical Transitive Imprimitivity Theorem.- 1.8 Reduction in Stages.- 1.9 Coadjoint Orbits of Nilpotent Groups.- 1.10 Coadjoint Orbits of Semidirect Products.- 1.11 Singular Marsden-Weinstein Reduction.- 2 Induction.- 2.1 Hilbert C*-Modules.- 2.2 Rieffel Induction.- 2.3 The C*-Algebra of a Hilbert C*-Module.- 2.4 The Quantum Imprimitivity Theorem.- 2.5 Quantum Marsden-Weinstein Reduction.- 2.6 Induction in Stages.- 2.7 The Imprimitivity Theorem for Gauge Groupoids.- 2.8 Covariant Quantization.- 2.9 The Quantization of Constrained Systems.- 2.10 Quantization of Singular Reduction.- 3 Applications in Relativistic Quantum Theory.- 3.1 Coadjoint Orbits of the Poincaré Group.- 3.2 Orbits from Covariant Reduction.- 3.3 Representations of the Poincaré Group.- 3.4 The Origin of Gauge Invariance.- 3.5 Quantum Field Theory of Photons.- 3.6 Classical Yang-Mills Theory on a Circle.- 3.7 Quantum Yang-Mills Theory on a Circle.- 3.8 Induction in Quantum Yang-Mills Theory on a Circle.- 3.9 Vacuum Angles in Constrained Quantization.- Notes.- I.- II.- III.- IV.- References.

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