The wave equation in conformal space

Abstract

In a recent paper Dirac has shown that by passing from the ordinary Euclidean space to a four-dimensional conformal space, some of the equations of physics can be written in a tensor form, the indices of which take on six values. Those equations which can be written in this form are then invariant under conformal transformations of the Euclidean space. Among the equations of physics which have this more general invariance are the Maxwell equations, as was proved by a direct transformation a long time ago by Cunningham, and Bateman, so that Dirac's paper provides an alternative and more general proof of this result. Certain errors in Dirac's paper, however, necessitate a reformulation of the proof. Before we do this in section 2, we briefly recapitulate in section 1 some of the general results derived there. In section 3 we investigate further the conformal invariance of the wave equation for an electron in the presence of a general electromagnetic field.