On the magnetic moment of the electron

Abstract

1. In a recent papers the present writer discussed Dirac's new formulation of the quantum mechanics of the electron from the point of view of the wave theory. In the course of that work formulæ were obtained for the electric density and current associated with any type of electron wave. These formulæ link the electron with the æther, so that to a first approximation (that is, excluding such questions as the reaction of its own emitted radiation on the electron) we have a complete expression of the state of affairs, which is suitable for use in the classical electromagnetic theory. But there are certain problems in which it is convenient to attribute part of the magnetic field of a moving electron to an electric convection current and part to its intrinsic magnetisation; for example, this would be so in discussing the Stern-Ger1ach effect for free electrons. In the earlier paper a separation of this kind was carried out for an electron in an atom and a short account was given of the corresponding case for an electron moving freely. As the paper was intended to cover a much wider field, these matters were only reviewed rather briefly, and the purpose of the present note is to elaborate them somewhat more fully. The process is one of pure classical theory, for the establishment of current and density is the only appeal that need be made to the quantum theory. We shall exhibit directly the magnetic moment of the electron, working it out in the first instance for slow motions, as it is easy to generalise this case by relativity principles. It would no doubt be possible to develop the whole result in one step, but the purely classical problem of the fields of a swiftly moving magnet is quite troublesome, even though the principles have been fully mastered, so that it would be an unnecessary and profitless exercise to follow this procedure. It may be recalled that Dirac's equations were easily shown to be invariant for any space or relativity transformation, even though their form is quite unsymmetrical. The associated density-current functions naturally have the property of covariance, though also quite unsymmetrical in appearance. We shall here meet other tensors, also quadratic in the Ψ’s, and having the same unsymmetrical appearance. Their construction is by no means obvious, and the only process seems to be a direct application of all the transformations of the group to each component—a straightforward but laborious method. Before Dirac’s equations were found one would have said without hesitation that the correct procedure in such a case would be to reduce the equations themselves to tensor form, for then the associated tensors should become obvious. It is not hard to throw the equations into space-vector form, but if the relativity transformation is to included, they obstinately refuse to go into any but a very clumsy form and nothing is gained; this being so, it seems not worth while to use the space-venctor either. I have the hope that this subject may interest some analytical geometer and may tempt him to investigate these curiously unsymmetrical invariant properties. As we shall see, they appear to be connected with the stereographic projection of a sphere and perhaps with the homographic transformations of a complex variable, subjects with which I am not very familiar.