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Classical electrodynamics as a distribution theory

Published online by Cambridge University Press:  24 October 2008

J. G. Taylor
J. G. Taylor
Affiliation:
Christ's CollegeCambridge
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Abstract

In this paper a precise distribution-theoretic formalism for the description of point charges interacting through a classical electromagnetic field is given. The distribution solutions of the Maxwell equations are shown to reduce to the Lienard–Wiechert fields, which are summable over the whole of space-time. It is not possible to obtain directly a unique field intensity, on the world lines, from these distribution solutions. A brief analysis shows that analytic continuation methods also do not give a unique field intensity on the world lines. The energy tensor for the field is a distribution which can only be rigorously defined in terms of the motions of the charges. Thus only the value of the field energy or momentum residing in any finite region of space-time can be given any meaning. Equations of motion for the charges may be derived from the field energy tensor, using the principle of conservation of total energy and momentum for the system. These equations agree with those obtained previously by Dirac (2). Also any canonical formalism in which the field potentials and the particle variables enter in an equivalent manner is not possible. The difficulty of the definition of products of distributions in the stress tensor does not occur.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 1 , January 1956 , pp. 119 - 134
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

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