Photons in Fock Space and Beyond

The following sections are included:

• Preface
• Contents

When Einstein introduced the “heuristic principle” of light quanta in 1905, named photons since 1926 (according to a proposal from the physical chemist Gilbert Lewis from Berkeley), he was guided by the analogy between the thermodynamic functions for a cavity filled with black body radiation and a container filled with an ideal gas of material particles. The energy of the light quanta corresponded to the quantized energy differences of the material Planck oscillators (deduced thermodynamically, though being in accordance with the principle of resonance), but Einstein assigned the light quanta an independent existence in free space. Disconnected from the emitting medium, these particle-like excitations should carry energy in discrete portions. Thus it is remarkable, that Einstein applied the analogous idea of quantizing vibrations also to the eigenoscillations of a solid (and gained a deduction of the Third Law of thermodynamics). In the latter case, the vibrational quanta, called phonons, are not at all detached from the medium, rather describe special states of it…

The following sections are included:

• Historical Developments
  • Early Force Equations
  • Electromagnetic Fields
• A First Look on Maxwell's Equations
  • Recognition of the Electrodynamic Laws
  • Hertz Radiation
• Formal Vector Relations and Integrals
• Field-Plus-Matter System
  • Lorentz Force
  • Conservation Quantities
• Field Mediated Interactions
• Special Field Expressions
  • Intensity of Asymptotic Radiation
  • General Plane Waves

The following sections are included:

• Maxwell's Equations in Media
  • Deterministic Clustering
  • Field Systems in Media

The reference theory for a mathematically rigorous treatment of Maxwell's equations is its L²-Hilbert space formulation. By Conclusion 2.6-1 on page 35 that means one inquires on finite energy solutions of the dynamical equations and defers the investigation of typical radiation fields (with unbounded energies) to ulterior considerations…

In the preceding treatment, the Maxwell dynamics with subsidiary conditions has been investigated for the total electric and magnetic fields. The physical meaning of the longitudinal and transversal parts of the E-fields is, however, very different, and in multiply connected regions or cavities Λ there arise also additional types of electric and magnetic fields, which would be both longitudinal and transversal according to the naive classification. Since they characterize the cohomological features of Λ, involving the first and second Betti numbers, we term them “cohomological fields”. They got into focus rather recently, in spite of the following remark of Maxwell: “We are led here to considerations belonging to the Geometry of Position, a subject which, though its importance was pointed out by Leibniz and illustrated by Gauss, has been little studied”…

We have mentioned in the historical introduction that vector potentials have been used already by Franz Neumann and von Helmholtz, even before Maxwell developed his electromagnetic theory. Later on they have been employed as a technical tool for the calculation of force fields, and mostly considered devoid of a direct physical significance. It is, however, not clear what kind of theoretical concepts convey a “direct” physical meaning. (There was a famous controversial discussion on this issue between the young Heisenberg, who considered his matrix elements as especially “physical”, and Einstein who mocked him.) In the context of ED, it has been emphasized, e.g., by G. Ludwig in [Lud74], that potentials are as “physical” as force fields, if they are used to describe the physical facts as concisely as the latter. Formally that means, that one has to combine potentials into equivalence classes or to specify distinguished representatives of these classes. It also means, that the boundary conditions have to be expressed in terms of the potentials as accurately as in terms of the force fields, in spite of not being given by direct physical arguments…

The following sections are included:

• The Standard Lagrangian for General Regions
  • Relativistic Foundation of the Lagrangian in Free Space
  • The Velocity Phase Space
  • Principle of Stationary Action
  • Derivation of the Maxwell Equations
  • On the Existence of Euler–Lagrange Solutions
  • Global Gauge Sections in the Velocity Phase Space
• Transition to Hamiltonian Formulation
  • Hamilton Formalism for Hyper-regular Lagrangians
  • The Canonical Momenta of ED
  • Gauge Dependent Standard Hamiltonians
  • The Temporal Gauge Hamilton Formalism
  • The Coulomb Gauge Hamilton Formalism

The following sections are included:

• Generalized Canonical and Force Fields
  • The Twofold Gelfand Triple Structure
  • The LC-Test Function Spaces and their Gelfand Triples
  • Differentiable Trajectories in the LC-Dual Space
  • Generalized Maxwell and Continuity Equations
  • Dynamics for the Extended Longitudinal Canonical Fields
  • Dynamics of the Extended Cohomological Canonical Fields
  • Dynamics of the Extended Transversal Canonical Fields
  • A Note on Generalized Gauge Transformations
    • The Helmholtz–Hodge Compatible Generalized Potentials
    • Generalized Transversal Vector Potentials
    • Generalized Trajectorial Gauge Transformations
    • A Remark on the Coulomb Gauge Condition
    • Generalized Vector Potential for the Cohomological Magnetic Field
• Poisson Formalism and Phase Space Dynamics
  • Poisson Bracket
  • Hamiltonian Phase Space Flow
• Complex ED on the Complexified Phase Space
  • Classical “Annihilation” and “Creation” Field Functions
  • Complex Conjugation and Decomposition of the Phase Space
• Quadratic Hamiltonians and Symplectic Generators
  • Symplectic One-Parameter Groups
  • Symplectic Generators, Quadratic Hamilton Functions
  • Phase Space Flows by Quadratic Hamiltonians

The diagonalization of one-parametric symplectic groups is a concept we have developed for relating canonical field quantization, which originally concerns only Hermitian fields, with a certain complexification. If it applies, it prepares already on the classical level the particle structure of the quantized fields…

For two classes of spatial domains Λ ⊆ ℝ³, we want to carry out, how appropriate test function spaces, as generally characterized in Sec. 8.1.2 on page 155, may be constructed. For that we consequently apply the method of twofold Gelfand triples, in order to obtain continuous vectorial differential operators and dynamical generators. We specify also the extended families of electromagnetic fields contained in the associated duals of the described test function spaces.

The following sections are included:

• Momentum Maps
• Translation Group and Electromagnetic Momentum
  • Representation of the Translation Group for Transversal Radiation
  • The Translations as a Hamiltonian Flow
• Rotations and Electromagnetic Angular Momentum
  • The Rotation Group
  • General Representation Theory of the Rotation Group
  • The Representation of the Rotation Group for Transversal Light
  • The Rotations as a Hamiltonian Flow

The following sections are included:

• A Word on Statistical Theories
• Overview on Canonical Test Function Spaces
• The C*-Algebra of Classical Observables
• Algebraic Symplectic Geometry
• *-Automorphic Actions
• The Statistical Field States of Classical ED
  • Characteristic Functions, Bauer Simplex of States
  • Regular States and Weak Distributions on Phase Space
• Field Expectations
  • Field Expectations and Moments
  • Field Fluctuations
  • Fluctuation Free States

Here and in Chapter 14 we apply the C*-algebraic frame to canonical ED in vacuo, enclosed in an arbitrary spatial domain Λ ⊆ ℝ³ with perfect conductor boundary…

We continue the C*-Weyl algebraic dynamical theory of the previous Chapter 13 and investigate the radiation in the infinite time limits t → ±∞. The scattering theory, elaborated in the first Sec. 14.1, seems to work for all exterior Λ, since then the dynamical operator has a pure absolutely continuous spectrum…

We treat here the electromagnetic field in a large cavity Λ ⊂ ℝ³ and assume thermal equilibrium. Physically, this radiation state is to be expected if the walls of the cavity have a uniform temperature and absorb all frequency components of the radiation (in this sense being black), subsequently emitting them according to a well defined probability. In order that there be enough time to reach equilibrium the cavity is closed apart from a small hole. The radiation leaving the cavity through the hole is called black body radiation…

The following sections are included:

• Historical Steps to Quantum Algebra
  • From Black Body Radiation to Einstein Coefficients
  • From Dispersion Theory to Quantum Mechanics
  • Wave Functions, Hilbert Space, and Statistics
• Canonical Quantization
  • Canonical Field Quantization
  • Canonical Fields and Weyl Systems
  • Weyl Form of Canonical Quantization

Many important (linear) operators occurring in Physics are necessarily unbounded, leading to many problems and misunderstandings in Theoretical Physics. As a consequence of their unboundedness they are discontinuous and their domains of definition are not all of the underlying Hilbert space. Most notions, such as e.g., the operator sum or product, the self-adjointness or the exponential series, are very hard to handle, if the operators are unbounded. Only a rigorous mathematical investigation of their domains of definition and further properties may then bring the solution. We refer to Chapter 43 on page 1521 for an overview and an introduction to the pertinent mathematical notions and results for Hilbert space operators, emphasizing the treatment of unboundedness…

We already introduced, in the context of canonical Hilbert space quantization, the notion of a Weyl system (W_𝒦,ℋ_𝒦) over the real, pre-symplectic space (E, ħσ) (see Sec. 16.2 on page 353). The unitaries W_𝒦(f), f ∈ E, acting on ℋ_𝒦 are demanded to satisfy the Weyl relations, that is, the mapping E ∋ f ↦ W_𝒦(f) constitutes a projective, unitary representation of the vector group E on the Hilbert space ℋ_𝒦. Since the W_𝒦(f) are bounded operators, it is possible to abstract from the Hilbert space ℋ_𝒦 and to formulate the Weyl relations within a C*-algebraic setup leading to the C*-Weyl algebra 𝒲(E, ħσ). It incorporates the basic algebraic structure of the canonical quantization of a system with possibly infinitely many degrees of freedom. In the case of an infinite-dimensional test function space, 𝒲(E, ħσ) represents the basic (quasilocal) observables of infinitely many material Bosons or of a quantized classical field (displaying not necessarily a physical particle structure)…

In the present chapter, we are again concerned with an arbitrary real pre-symplectic space (E, σ), which describes — as the so-called “test function space” — the degrees of freedom of a physical system. The number of degrees of freedom is the cardinality of a (real) basis of E, divided by two. Let us emphasize once more that a degree of freedom of motion is a universal concept, independent of how the motion is described theoretically within kinematics…

In Sec. 20.1 we take up and elaborate an ansatz of the Segal school, namely that a positive diagonalized form of an implementing (renormalized) Hamiltonian be not only of practical but also of basic conceptual importance, providing a particle interpretation of the quantized field system. We characterize those representations which allow for a “physical particle structure”, and give a condition, under which there is a unique one…

For formulating a microscopic model of charged particles in interaction with the quantized electromagnetic field one most frequently works in the Coulomb gauge. We also perform our analysis under the Coulomb gauge condition only and neglect transformations to other gauges. Otherwise, there would arise extreme mathematical difficulties, which e.g., result from the fact that the quantum Weyl algebra for the canonical longitudinal field remains commutative in the Coulomb gauge but gets non-commutative in other gauges, like the temporal gauge. This feature would also prevent in non-Coulomb gauges the usual picture of an atom or molecule, stably bound by the classical Coulomb potential…

The asymptotic correspondence between classical and quantum physics is intimately connected with the name of N. Bohr. But it was P.A.M. Dirac, who first shaped this general principle into a limiting equality of the scaled quantum commutator with the classical Poisson bracket (see Sec. 16.1). The various concise mathematical formulations, however, for such a classical correspondence limit ħ → 0 developed much later…

Projective representations of groups arise in many fields of physics, not only in all branches of quantum theory, but also in classical theory (as, e.g., in the representation theory of the Galilei group). It is well known that for fixed multiplier the projective representations of a locally compact group E are in 1:1-correspondence with the representations of the twisted group Banach algebra respectively C*-algebra. The theory of twisted group algebras is traditional, and has been especially initiated by the mathematical foundations of quantum mechanics. There are intimate connections to C*-algebraic systems and C*-algebraic structure theory. Twisted group C*-algebras acquire also increasing importance in the developments of Poisson manifolds and momentum maps, but also in the theory of strict deformation quantization, and in many other deformation strategies (cf. the previous chapter, and references therein)…

Since the creation of quantum mechanics, most investigations on quantization, especially the numerous treatments on (strict) deformation quantization, are concerned with the algebras of observables. And we were so, too, in the previous two chapters…

Throughout the present chapter, let (E, σ) be an arbitrary real pre-symplectic space, with σ a degenerate or non-degenerate symplectic form. We fix an arbitrary Planck parameter ħ ∈ ℝ and get another (pre-) symplectic form ħσ, where even ħ = 0 is allowed and represents the classical case.

In (Q)ED, optical coherence is investigated only for the transversal part of the electromagnetic field, in both the classical regime ħ = 0 and quantum regime ħ > 0. Translated into the present context, this leads to the point of view that optical coherence concerns states on the Weyl algebra 𝒲(E^Τ, ħIm(.|.)), built on the transversal complex test function space E^Τ after diagonalization. Let us recall (from Sec. 9.3.5 on page 206) that E^Τ is a complex subspace of ℋ^Τ ⊂ L²(Λ,ℂ³), equipped with the complex inner product (.|.) inherited from L²(Λ,ℂ³). The symplectic form is proportional to the imaginary part of this scalar product…

The following sections are included:

• Bibliography
• Index