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International Journal of Theoretical Physics

On Sectorial Photon Theories Reconstructing Einstein’s Early Fusion Picture for Light

Alfred Rieckers

Published Feb 25, 2025• 2 citations
Publisher

Abstract

Abstract This is the second part of a pair of companion papers devoted to an analysis of Einstein’s fusion picture for light. While in the foregoing paper “An Actual Discusion of Einstein’s Early Fusion Picture for Photons and Classical Maxwell Fields” (referred to as Rieckers 2025) our conclusions ended with the historical ab initio motivation and mathematical elaboration of a multi-photon theory in Fock space, we now start from the non–separable C*-Weyl algebra gained by abstracting the Fock–represented Weyl algebra. To the rich state space of this antiliminary observable algebra we apply notions of a convex state space approach to identify sub–theories that cover classical field states (including the vacuum with classical zero–field), that are possibly decorated by photonic quantum noise. The main emphasis is laid on direct integrals over irreducible, disjoint representations of the C*-Weyl algebra, decomposing the effective photon theory into sectors. Certain limits of smeared central field operators suggest a mathematical realization of Einstein’s early photon notion consisting of an energy point surrounded by a local force field (respective wave function). Macroscopic classical Maxwell fields are gained by superposing the expectation values of products of those composite photon operators. The ontological status of the explicated photon notion is discussed. Connections to Einstein’s historic photon theory, summarized in Rieckers (2025), as well as to quantum optical applications are hinted at throughout the paper.

DOI
10.1007/s10773-025-05928-4

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Abstract This is the second part of a pair of companion papers devoted to an analysis of Einstein’s fusion picture for light. While in the foregoing paper “An Actual Discusion of Einstein’s Early Fusion Picture for Photons and Classical Maxwell Fields” (referred to as Rieckers 2025) our conclusions ended with the historical ab initio motivation and mathematical elaboration of a multi-photon theory in Fock space, we now start from the non–separable C*-Weyl algebra gained by abstracting the Fock–represented Weyl algebra. To the rich state space of this antiliminary observable algebra we apply notions of a convex state space approach to identify sub–theories that cover classical field states (including the vacuum with classical zero–field), that are possibly decorated by photonic quantum noise. The main emphasis is laid on direct integrals over irreducible, disjoint representations of the C*-Weyl algebra, decomposing the effective photon theory into sectors. Certain limits of smeared central field operators suggest a mathematical realization of Einstein’s early photon notion consisting of an energy point surrounded by a local force field (respective wave function). Macroscopic classical Maxwell fields are gained by superposing the expectation values of products of those composite photon operators. The ontological status of the explicated photon notion is discussed. Connections to Einstein’s historic photon theory, summarized in Rieckers (2025), as well as to quantum optical applications are hinted at throughout the paper.
Abstract This is the first part of a pair of companion papers devoted to an analysis of Einstein’s fusion picture for light-quanta (subsequently also referred to as “photons”) with classical electromagnetic fields that he formulated in 1909 via a purely verbal description. There, the overlap of many of the hypothetical local force fields, surrounding the photonic energy points, should produce the well–known classical Maxwell fields. We claim that, alone from such a logical connection, that apparently did not receive sufficient attention in the literature, one may draw essential conclusions on the photon states. In the present paper, we give a historical overview of Einstein’s photon conceptions until 1927. We extract hints from his formulations, that lead from his local force fields to Schrödinger wave functions, which reproduce the conserved integrals for energy, momentum, and angular momentum of Maxwell theory in form of Hilbert space expectations. Also a preliminary multi–photon theory, including Glauber states, is shown to be motivated by Einstein’s composite photons with possibly (incoherently!) overlapping force fields. In the resulting Fock space theory, gained by symmetrizing the many–photon states the goal of a fusion theory is, however, not realizable. In the subsequent companion paper “On sectorial photon theories reconstructing Einstein’s early fusion picture for light” (referred to as Rieckers 2025) our conclusions are extended to photon states that surpass Fock space theory by describing actual infinitely many photons. These are shown to cover also classical light states in terms of collective structures. Appealing to the more recent mathematical tools of convex state space approach and algebraic quantum field theory, we suggest there mathematical realizations of Einstein’s fusion program.
The deduction starts with the (non-relativistic) one-photon Hilbert space \mathcal{H} , equipped with the one-photon Hamiltonian and basic symmetry generators, as the only input information. We recall in the functorially associated boson Fock space the multi-photon dynamics and symmetry transformations, as well as the field operator (as the scaled self-adjoint part of the creation operator) and the q(uasi)-classical states. There is no reference to a presupposed classical Maxwell theory. By abstraction, we go over to the algebraic formulation of the multi-photon theory in terms of a C*-Weyl algebra. Its test function space E\subset \mathcal{H} is constructed as a nuclear Fréchet space, in which – via infrared damping – the dynamics and symmetries are nuclear continuous and their generators bounded. Each w*-closed, singular subspace of the continuous dual E' determines non-Fock coherent states and their mixtures lead to a representation von Neumann algebra with non-trivial center. The symmetry generators restricted to the center can be transformed into the Maxwell form by means of a symplectic transformation and involve the well-known conservation quantities of electrodynamics. This identifies the central part of the represented photon field operator as composed of the two classical canonical electrodynamic field components. We have obtained, therefore, in free space a kind of fusion of the multi-photon theory and the Maxwell theory of transverse electrodynamic fields, where the latter arise as derived quantities. By means of a Bogoliubov transformation one also gets a fusion of the quantized with the classical Maxwell theory, deduced from the photon concept. A sketch of non-relativistic gauging is added in the appendix to gain longitudinal, cohomological, and scalar potentials.
Abstract Through the classic work of Planck, Einstein, and others the particle nature of the photon has been established. In the previous chapter we considered the wave field of the photon. In this chapter we will emphasize more its particle nature and discuss the extent to which a wave function with a probabilistic interpretation is an appropriate description. Once we have the concept of a particle it is natural to ask how localized the state can be. The degree of localizability of photons is of concern in the interpretation of experiments in quantum optics involving weak light sources and non-linear electromagnetic media. If a photon is not to have the structure of a composite particle its quantum states should be describable by translations, rotations, and Lorentz boosts of a particular state and possibly super positions of such states. This is equivalent to a photon being associated with an irreducible representation of the Poincaré group. A brief overview of the general theory of group representations is given in Appendix 3. We will be able to give a complete discussion of the representations of the Poincaré group pertinent to the photon. Our discussion of localizability will be quite rigorous but we will gloss over measure-theoretic niceties. For more mathematical discussions the reader can refer to the works of Mackey (1978) and Wightman (1962). We will finish the chapter by examining the implication of Poincaré invariance for the metric on the space of quantum states of the photon.
Abstract This book is unique in covering phenomena in photon- matter interactions in a unified way over a range of many orders in energy. The quantum field theoretic approach to the fully relativistic theory of quantum electrodynamics (QED) is presented together with the non- relativistic theory in both confined and unconfined geometries. The predictions of QED have been verified to a greater accuracy than any other physical theory. Moreover QED is a paradigm for other gauge theories and is presented in such a way that the generalisation to other gauge theories is natural. Gauge and Poincare symmetry properties and the non-existence of a photon wave function are thoroughly discussed. Starting from the Dirac equation the non-relativistic interaction of the electron with the electromagnetic field is derived as an effective Hamiltonian of multipole expansions. Much of quantum optics is based on the lowest order dipole approximation. From this point on the treatment of fully relativistic QED and quantum optics is done in parallel. Applications of perturbation theory such as Compton and Moller scattering and the theory photdetection are given. After the impressive successes of QED, the limitation of the theory and the necessity of electroweak theory and quantumchromodynamics are discussed. The remaining chapters are devoted to quantum optics inside cavaties. Various approaches to open systems such as master equations are discussed within the context of active systems (e.g. the laser) and passive systems. Semi- classical approximarions are shown to imply a rich non- linear dynamics including chaos for certain parameter regimes. The effect of fluctuations on such non-linear dynamics is also studied. The final chapter is devoted to highly non- classical states of the light field such as photon number, squeezed and two photon entangled states. The latter are studied for the important system of parametric down conversion and the localisation properties of photons are characterised in terms of asympotic tails in photodetection probabilities as a function of time delay. The range of the book has wider benefits. Workers in quantum optics will gain a deeper understanding of the foundations of their subject and field theorists will see concrete examples of open systems, which are beginning to impinge on fundamental theories.
This chapter begins with a review of the semi-classical model of the interaction of charged particles with light, and then proceeds to the full quantum theory by replacing the classical fields with the corresponding field operators. A presentation of the quantum Maxwell equations and their behavior under parity and time reversal transformations is followed by a discussion of stationary density operators. The notion of positive- and negative-frequency parts of field operators is extended to interacting fields and used to define multi-time correlation functions that describe experimental results. The perturbation expansion is formulated by means of the interaction picture and combined with the dipole approximation to calculate the Einstein A- and B-coefficients. The chapter ends with a discussion of spontaneous emission in a cavity and Raman scattering.