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Published online by Cambridge University Press: 24 October 2008
It is shown in this paper and the preceding one that two separate forms of theory can be developed in which a “finite size” is attributed to a charged particle by means of its interaction with the radiation field. The region attributed in this way to the particle is four dimensional and is determined in such a manner that the usual difficulties with relativistic invariance do not arise.
The advantage of such a theory becomes clear when the theory is applied to those problems in which the usual calculations give infinite results. The problem of the method of successive approximations is considered and satisfactory results are obtained provided that the space dimensions of the finite region are of the order of the classical radius of the electron, when the electron is at rest.
It may be noted explicitly that the difficulty that has been associated with the emission of low energy quanta by “Bremsstrahlung” will not arise in the present formulation of the electromagnetic interaction between field and particles. This case is interesting since an infinity arises here which is not analogous to the self energy infinities, but occurs in the direct calculation of a physical process and not in a virtual transition.
The theory seems satisfactory so far as low energy processes (< 137 mc2) are concerned and the real test of its applicability may be expected to arise in discussing processes of high energy. It is hoped to treat these in a later paper.
† The relativistic invariance follows as above now that the time derivatives of the commutation relations are zero.
† In this case − e times the particle density is the charge density.
† R is the length occurring in the definition of the Z-regions for the rest system.
‡ There will also be a contribution for | k | ˜ | p0 | which may be roughly of the same order of magnitude, e 2/8πħc. But this need not be considered since this type of term will occur in the nth approximation with a coefficient ˜ (e 2/ħc)n/8π. That is, there is no question of divergencies arising from this type of term.
† The distinction introduced above between self energy terms and light quanta and electrostatic terms implies that in the self-energy interaction the phases of the time integrals occurring are never zero, whereas for the light quanta and electrostatic interactions the phases are integrated through their zero values.