Polar analysis of the Dirac equation through dimensions

Abstract

We consider the polar form of the spinor field equation in an n-dimensional space-time, studying the way in which the space-time dimension influences the number of the independent field equations and the number of the degrees of freedom of the spinor field written in the polar form: we will find that this polar form is the clearest tool to make manifest the fact that the degrees of freedom of the spinor field and the independent field equations match in dimensions 4 and 8 alone while in all the other instances there will be problems of general under-determination or over-determination.