Statistical electrodynamics

Abstract

The random pure radiation field postulated in an earlier paper is set up in a relativistically invariant manner. The requirement that this field be isotropic in three dimensions makes much of the formalism identical with the theory of isotropic turbulence, as has been noted by a previous author. It is found, however, that the more stringent requirement of invariance under the Lorentz group, together with the fact that the field components satisfy Maxwell's equations, mean that, to within a multiplicative constant, the entire random process is uniquely specified.In accordance with results obtained previously, the constant is effectively identified as Planck's constant. There then results a striking resemblance between the formalism of the present theory, and that of the quantized pure radiation field, with random variables taking the place of operators, and correlations between these random variables taking the place of commutator brackets.The infinite field energy, which arises also in quantum field theory, where, however, it is usually considered to be purely formal, remains a central difficulty of the random theory. It can be shown to be a natural consequence of assuming Maxwell's equations, and it therefore becomes necessary to consider non-linear modifications of these.The connexion between the present results and the inconsistencies of classical and quantum electrodynamics is discussed. Also some new information is obtained concerning the behaviour of thermodynamic quantities under a Lorentz transformation.1. Introduction. In a previous paper ( (7)), it was shown that, in the presence of a random electromagnetic field, certain charged classical systems, namely, the free particle and the harmonic oscillator, behave essentially like their quantum-mechanical counterparts. The energy spectrum of the field needed to produce this behaviour was obtained, in the non-relativistic limit, and was found to be the familiar Planck spectrum superposed on a certain fixed spectrum. The latter was therefore identified as a zero-temperature spectrum, and it was suggested that such a random radiation field might be capable of explaining other quantum effects in material systems.In the present paper we investigate the requirements which special relativity places on such a random theory. Attention is confined to the pure radiation field, and the question of how non-linear material systems behave in such a field is left for a later paper.As in quantum electrodynamics, we find that a crucial decision has to be made, namely, 'What are the quantities which are observable?' It is necessary to frame

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