1 - The self-energy of the electron

Summary

Zeitschrift für Physik, 65: 4–13 (1930). Received 3 August 1930.

The behavior of very fast electrons is investigated, whose energy is large compared with mc² and Mc². Since the rest mass of the electron can be neglected for such motions, a characteristic electron radius plays no role in the question of self-energy. The conditions are investigated under which the self-energy of the electron vanishes.

Introduction

In classical theory, the field strengths E and H become arbitrarily large in the neighborhood of the point-charge e, so that the integral over the energy density 1/8π(E² + H²) diverges. To overcome this difficulty, one therefore assumes a finite radius r₀ for the electron in classical electron theory. This radius is related to the mass m of the electron in the order-of-magnitude relation r₀ ~ e²/mc²; the integral over the energy density is then of the order mc². In quantum theory, not only this radius r₀ but possibly also another length λ₀ = h/mc, which is characteristic of the electron, plays a role in the self-energy. In a superficial consideration in terms of the correspondence principle, one would suspect that the self-energy of the point-like electron must also become infinite in quantum theory.

In fact, Oppenheimer and Waller have indeed shown that a perturbation method which proceeds in powers of e does not yield finite values for the self-energy. At first, it appears as if, in quantum theory also, only the introduction of a finite electron radius can provide a way out of this difficulty.