Photons in Fock Space and Beyond

The following sections are included:

• Preface and Overview III
• Contents

The present chapter gives an introduction to the used notions and results for Hilbert space operators, especially for unbounded operators. The main difficulties, when dealing with unbounded operators, arise from their discontinuity and the fact that their domains of definition in general cannot be the entire Hilbert space (cf. the Hellinger–Töplitz result in Proposition 43.1-2 below). Unfortunately, many important operators used in physics are unbounded, and many problems (and misunderstandings) in theoretical physics struggle, in fact, with this attribute. A rigorous mathematical investigation of the domains may then bring the solution…

A concise formulation of electrodynamics requires the introduction of vectorial differential operators, such as gradient, divergence, curl (also called rotation), curlcurl, and Laplacian, as densely defined, closed, respectively self-adjoint operators acting between L²-Hilbert spaces. The precise handling of these operators, respectively of the associated forms, makes extensive use of Sobolev spaces, which are chosen according to the appropriate boundary conditions…

The theory of C*-algebras arose mathematically as an abstraction of the norm closed *-algebras of bounded operators in Hilbert space. The latter, in which the *-operation is realized as the Hermitian adjointing, are sometimes called “concrete C*-algebras”. C*-algebras generalize also the “rings of operators”, earlier developed by J. von Neumann and now named after him. These are closed even in a weaker-than-norm topology and provided the basic connections between algebraic and topological notions…

The following sections are included:

• Basics of von Neumann Algebras
  • Locally Convex Topologies on ℒ(ℋ)
  • Definition and Some Elementary Properties of General von Neumann Algebras
  • Predual and Normal States
• Spectral and Classificatory Notions
  • Arveson Spectrum and Borchers–Arveson Theorem
  • Quasiequivalence, Disjointness, and Folia
  • Faces, Projections, Supports, and Equivalence
• Modular Theory and Thermal Fields
  • Standard von Neumann Algebras
  • Standard Implementation of Automorphisms

Convex sets turn up in our book mainly as state spaces and distinguished subsets of them. The first section of this overview treats basic notions of geometric convexity. In the second section, we outline some general features of state spaces and characterize quantum mechanical state spaces by their 3-ball property. For deeper insight into quantum mechanical notions, we mention in the third Section state spaces with n-ball property. For n > 3, they are even more non-classical than quantum mechanical state spaces. This leads in following sections to a rather general formulation and interpretation of “properties”, coherent superpositions of states, and of transition probabilities…

The following sections are included:

• Orthogonal Measures
  • Basic Notions from Measure Theory
  • Choquet Theory
  • Measures on the State Space
• Spatial Decomposition Theory
  • Measurable Families of Hilbert Spaces
  • Direct Integrals of von Neumann Algebras
  • Direct Integrals of Representations
  • Superselection Sectors and Rules
• Ergodic Averages
  • Invariant Quantities and Automorphism Groups
  • Group Averages
  • Ergodic Decompositions
  • Asymptotic Abelian Systems
  • Quasi-Free Ergodic Boson States
• Algebraic Transition Probabilities
  • Generalities
  • Transition Probabilities between Finite Products of States
  • Infinite Tensor Product Spaces and their Operators
    • Infinite Products of c-Numbers
    • (Infinite) Tensor Products of Hilbert Spaces
  • Transition Probabilities between Infinite Product States
  • Integral Decompositions

The following sections are included:

• Systems of Semi-Norms
• Sobolev Chains for (Q)ED
  • Motivation for Test Function Topologies
  • Polynormed F-Spaces
  • Operator Restrictions, Part 1
  • Operator Restrictions, Part 2
• Twofold Gelfand Triples
  • LC-Continuous Operators in LC-Gelfand Triples
  • The Structure of Twofold Gelfand Triples

Besides for our own applications to classical and quantum mechanical states in field theory, the integration theory over function spaces has far reaching importance for several fields of mathematical physics. It arose in the theory of random processes, especially for the Brownian motion. In this connection one has to mention the Wiener integrals, which were generalized by A. N. Kolmogorov…

In the present and next two chapters we supplement mathematical details of the electron–photon perturbation theory in a somewhat generalized version.

We continue our elaboration on perturbation theory from the previous chapter. Here, however, the special situation of commuting matter interaction parts is treated.

The material system on the Hilbert space ℋ be as before. But for the Boson field we generalize our ansatz, which we prepare in algebraic terms. Instead of using the Fock representation, we describe now the Boson observables by the C*-Weyl algebra

The following sections are included:

• Differentiable Mappings on CLC-Spaces
• Differentiable Manifolds and Fiber Bundles
  • Differentiable Manifolds
  • Fiber Bundles
• Geometric Bundles and Equivalence of Bundles
  • Strict Equivalence and Geometric Bundles
  • Equivalence and (Non-)Triviality
• (Co-)Tangent Vectors and Differential Forms
  • Tangent and Cotangent Vectors
  • Differential Forms
• Lie Groups, Principal Bundles, and Connections
  • Lie Groups
  • Principal Fiber Bundles
  • Connections in Principal Bundles
• Associated Bundles

The following sections are included:

• Bundles above Space–Time
  • General Remarks
  • Gauge Bundles above General Space-Time
  • Gauge Bundles for Fixed Time
  • Construction of Non-relativistic Gauge Bundles
  • Phases in Associated Line Bundles
  • Transition to Non–Smooth Boundaries and Sections
• Bundles above General Force-Field Trajectories
  • Trajectorial Gauge Groups with Split Gauge Functions
  • Tentative Extension of the Trajectorial Gauge Group
  • Gauge Bundles for Both Cohomological Vector Potentials
  • Reduction to the Generalized Coulomb Gauge by Time Fixing
• Gauges, Wave Equations and Causality in Free Space
  • The Setup
  • The Helmholtz–Hodge Decomposition in Free Space
  • Helmholtz–Hodge Decomposition of the Maxwell Equations
    • Vacuum Maxwell Equations in Free Space
    • The Dynamical Part of the Vacuum Maxwell Equations
    • Helmholtz–Hodge Decomposition of the Maxwell Equations
    • The Longitudinal Electric Part of the Maxwell Equations
    • The Transversal Electromagnetic Part of the Maxwell Equations
  • The Inhomogeneous WE for the Force Fields
    • Wave Equations for the Electric and Magnetic Fields
    • Helmholtz–Hodge Decomposition of the Wave Equations
  • Charge Conservation is Basic for WE
  • Causality in Wave Solutions
    • Solution of the IVP for the Inhomogeneous Wave Equation
    • Instantaneous Versus Propagating Electrodynamic Fields
  • Potentials and Causality

The following sections are included:

• Bibliography
• Index