N.P. Landsman

Real- and imaginary-time field theory at finite temperature and density

Abstract This report gives a detailed account of relativistic quantum field theory in the grand canonical ensemble. Three approaches are discussed: traditional Euclidean Matsubara, and two recently developed real-time methods, namely, Minkowskian time-path and thermo field dynamics. The first two formulations are derived in a unified manner from the path-integral representation for the contour-ordered generating functional. Fields with spin and gauge fields, in particular, are included. Zero-temperature renormalizability id shown to imply UV finiteness at any temperature and density. Thermo field dynamics, which is basically an operator theory, is presented in a C∗-algebraic context. Relevant parts of the HHW formalism of quantum statistical mechanics, and the Tomita-Takesaki theory are explained. The next chapter contains an analysis of the structure and analytic properties of the self-energy and its relation to the full propagator. In this connection the concept of a statistical quasiparticle is briefly described. This is followed by a discussion of thermal WT identities, Results are applied to discuss transversality of the SU(N) gluon polarization tensor. The final chapter deals with the diagrammatic rules for evaluating the pressure and energy density. The energy-momentum tensor is analyzed as a composite operator, and a renormalized virial theorem is established to provide the link with the thermodynamic potential. The pressure of the SU(N) chromoplasma is calculated up to third order.

Mathematical topics between classical and quantum mechanics

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 Poincare Group.- 3.2 Orbits from Covariant Reduction.- 3.3 Representations of the Poincare 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.

A Topos for Algebraic Quantum Theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos

Rieffel induction as generalized quantum Marsden-Weinstein reduction

{\mathcal{T}(A)}

Strict deformation quantization of a particle in external gravitational and Yang-Mills fields

in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra

The geometry of inequivalent quantizations

{\underline{A}}