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International Journal of Theoretical Physics

An Actual Discussion of Einstein’s Early Fusion Picture for Photons and Classical Maxwell Fields

Alfred Rieckers

Published Feb 25, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1007/s10773-025-05927-5

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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.
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.
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.
The study of light through the centuries has produced a number of theories and experiments. Newton advanced the corpuscular theory, building on the early Greek view of matter composed of atoms. Then Young essentially proved the wave nature of light through the interference experiments. Maxwell provided the firm theoretical basis for the wave nature. During the late 1800s and early 1900s, further experiments showed predictions based on a continuous wave led to the well-known ultraviolet catastrophe. Plank and Einstein originated and developed the notion of the photon as an elementary quantum of propagating electromagnetic (EM) energy. The first half of the 20th century saw the development of the field quantization and the quantum electrodynamics (QED). Feynmann labeled QED as ‘‘the best theory we have’’ to describe the matter-light interaction. The second half of the 20th century witnessed the development of the coherent and squeezed optical states attributable to Glauber and Yuen. The study of light and electromagnetic fields has a long history. This chapter explores the nature of light in terms of quantum optics.
In this chapter, we address proposals (by Einstein and by Bohr, Kramers and Slater) according to which quantum events are influenced by ‘guiding fields’ in 3-space. These ideas led to a predicted violation of energy-momentum conservation for single events, in contradiction with experiment. The contradiction was resolved only by the introduction of guiding fields in configuration space. All this took place before the fifth Solvay conference, but nevertheless forms an important background to some of the discussions that took place there. Einstein's early attempts to formulate a dynamical theory of light quanta Since the publication of his light-quantum hypothesis in 1905, Einstein had been engaged in a solitary struggle to construct a detailed theory of light quanta, and to understand the relationship between the quanta on the one hand and the electromagnetic field on the other. Einstein's efforts in this direction were never published. We know of them indirectly: they are mentioned in letters, and they are alluded to in Einstein's 1909 lecture in Salzburg. Einstein's published papers on light quanta continued for the most part in the same vein as his 1905 paper: using the theory of fluctuations to make deductions about the nature of radiation, without giving details of a substantial theory. Einstein was essentially alone in his dualistic view of light, in which localised energy fragments coexisted with extended waves, until the work of de Broglie in 1923 – which extended the dualism to all particles, and made considerable progress towards a real theory (see Chapter 2).
Classical electromagnetics in free space leaves no room for accommodation of a light particle. To underline this, it is sufficient to remember that Einstein in order to reach his lightquantum (energy-quantum) hypothesis had to mix classical theoretical physics with a piece of experimental information that defies description in classical terms. In his 1905-paper Einstein concluded [60]:Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of mutually independent energy quanta of magnitude hν. The introduction of light quanta in the Wien regime (the high-frequency part of the Planck blackbody radiation law) is the first step toward the concept of electromagnetic radiation as a Bose gas of photons [180, 39, 68, 69]. Although a photon gas obeys Bose-Einstein statistics for all frequencies the statistical independence assumed by Einstein in 1905 is not true in general. Einstein used Boltzmann statistics when he conjectured the existence of light quanta. In the Wien regime Boltzmann and Bose-Einstein statistics lead to the same conclusion since the mean number of particles (photons) with a given energy is much less than one. The circumstance that the light quanta are not statistically independent in Bose-Einstein counting finds its explanation in quantum mechanics. Hence, the statistical correlation is induced by the requirement of totally symmetric wave functions.
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.