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The BKS Theory and the Light-Quantum Hypothesis: Virtual Entities and Transitions to Intermediate States, But in Different Conceptual Frameworks (1923–1925)

Markus Ehberger

Published Jan 1, 2026• 1 citation
Publisher

Abstract

Abstract Prior to the development of matrix and wave mechanics, virtual entities figured prominently in the so-called BKS theory. According to some historians, the virtual oscillators of the BKS theory and their mathematical representations are directly connected to the concepts of virtual transitions/states. I will analyse the meaning of the term “virtual” in BKS, discuss the theory’s most important conceptual building blocks and its impact on dispersion theoretic accounts, most importantly the Kramers-Heisenberg paper of 1925. Then I will turn to a development that has nearly faded from sight: Shortly before the development of quantum mechanics, but from a light quantum point of view, physicists developed the idea that scattering and dispersion is composed of successive emission and absorption processes through “intermediate states”. The discussion of both of these strands, BKS and the light-quantum viewpoint, will serve as basis for the further development and for an important question to be answered in the following: Was there a generative connection between virtual oscillators and what came to be known as virtual transitions?

DOI
10.1007/978-3-032-09188-8_4

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Abstract Prior to the development of matrix and wave mechanics, virtual entities figured prominently in the so-called BKS theory. According to some historians, the virtual oscillators of the BKS theory and their mathematical representations are directly connected to the concepts of virtual transitions/states. I will analyse the meaning of the term “virtual” in BKS, discuss the theory’s most important conceptual building blocks and its impact on dispersion theoretic accounts, most importantly the Kramers-Heisenberg paper of 1925. Then I will turn to a development that has nearly faded from sight: Shortly before the development of quantum mechanics, but from a light quantum point of view, physicists developed the idea that scattering and dispersion is composed of successive emission and absorption processes through “intermediate states”. The discussion of both of these strands, BKS and the light-quantum viewpoint, will serve as basis for the further development and for an important question to be answered in the following: Was there a generative connection between virtual oscillators and what came to be known as virtual transitions?
Abstract This paper studies the notion of virtuality in the Bohr-Kramers-Slater theory of 1924. We situate the virtual entities of BKS within the tradition of the correspondence principle and the radiation theory of the Bohr model. We show how, in this context, virtual oscillators emerged as classical substitute radiators and were used to describe the otherwise elusive quantum transitions. They played an effective role in the quantum theory of radiation while remaining categorically distinct and ontologically separated from the quantum world of the Bohr model. The notion of virtuality thus differs markedly from its counterpart in quantum mechanics or QFT.
The report by Born and Heisenberg on ‘quantum mechanics’ may seem surprisingly difficult to the modern reader. This is partly because Born and Heisenberg are describing various stages of development of the theory that are quite different from today's quantum mechanics. Among these, it should be noted in particular that the theory developed by Heisenberg, Born and Jordan in the years 1925–6 and known today as matrix mechanics (Heisenberg 1925b [1], Born and Jordan 1925 [2], Born, Heisenberg and Jordan 1926 [4]) a differs from standard quantum mechanics in several important respects. At the same time, the interpretation of the theory (the topic of Section II of the report) also appears to have undergone important modifications, in particular regarding the notion of the state of a system. Initially, Born and Heisenberg insist on the notion that a system is always in a stationary state (performing quantum jumps between different stationary states). Then the notion of the wave function is introduced and related to probabilities for the stationary states. At a later stage, probabilistic notions (in particular, what one now calls transition probabilities) are extended to arbitrary observables, but it remains somewhat unclear whether the wave function itself should be regarded as a fundamental entity or merely as an effective one. This may reflect the different routes followed by Born and by Heisenberg in the development of their ideas. The common position presented by Born and Heisenberg emphasises the probabilistic aspect of the theory as fundamental, and the conclusion of the report expresses strong confidence in the resulting picture.
This work examines the physical and conceptual evolution of the light quantum from Planck's blackbody theory to the theoretical and experimental developments that led to the quantization of the electromagnetic field. The present study shows that the decisive transition occurred in Einstein's quantum theory of radiation (1916-1917). Absorption, stimulated emission, and spontaneous emission were formulated as elementary probabilistic mechanisms whose statistical balance alone reproduces the blackbody spectrum. In particular, spontaneous emission requires the emission of a light quantum, thereby implicitly proving its physical necessity before its theoretical status was clarified. At the same time, the already existing term photon began to acquire a stable usage following Lewis's 1926 proposal and became increasingly associated with Einstein's light quantum. By the mid-1920s, the central problem had shifted from whether light quanta were physically required to how radiation could be incorporated into the emerging quantum-mechanical formalism. This transition marks a key stage, illustrating how initial debates about the existence of light quanta gave way to their integration into a comprehensive theoretical structure. The resulting asymmetry between the novel quantum description of matter and the still-classical description of radiation, called into question by the phenomenon of spontaneous emission, identifies the physical problem that led to the quantization of the electromagnetic field.
Abstract The new quantum mechanics could at first be used to answer questions concerning radiation only through analogies with the classical theory. In Heisenberg’s original matrix theory, for instance, it is assumed that the matrix elements of the polarisation of an atom determine the emission and absorption of radiation analogously to the Fourier components in the classical theory. In more recent theories a certain expression for the electric density obtained from the quantum mechanics is used to determine the emitted radiation by the same formulæ as in the classical theory. These methods give satisfactory results in many cases, but cannot even be applied to problems where the classical analogies are obscure or non-existent, such as resonance radiation and the breadths of spectral lines. A theory of radiation has been given by the author which rests on a more definite basis. It appears that one can treat a field of radiation as a dynamical system, whose interaction with an ordinary atomic system may be described by a Hamiltonian function. The dynamical variables specifying the field are the energies and phases of its various harmonic conrponents, each of which is effectively a simple harmonic oscillator. One must, of course, in the quantum theory take these variables to be q-numbers satisfying the proper quantum conditions. One finds then that the Hamiltonian for the interaction of the field with an atom is of the same form as that for the interaction of an assembly of light-quanta with the atom. There is thus a complete formal reconciliation between the wave and light-quantum points of view.
Abstract This chapter focuses on Paul Dirac’s introduction of a verbally explicit description of scattering in terms of two successive emission and absorption processes, each coupled to a transition of the atom, not obeying energy conservation and located on the level of probability amplitudes. Historical actors called these energy non-conserving transitions “virtual transition” later on. I will contextualize Dirac’s introduction of this verbal model in reference to his research at the time, his interpretation of quantum theory and his rather pragmatic to flexible approach to mostly philosophical aspects of the interpretation of quantum theory. According to my argument, one of the most important reasons for Dirac to come up with this description was its usefulness in the practice of theory. A comparison between Dirac’s concepts and the concepts of the old quantum theory shows that neither the “intermediate states” of the light quantum point of view nor the “virtual” entities of the BKS theory had an important influence on Dirac.