Spin in Special and General Relativity

Abstract

Spin is the ultimate gyroscope. The smallest possible amount of angular momentum is ħ/Π- that possessed by a spin 1/2 particle. When the day comes that it becomes realistic to theorise about and to measure the precession of a spin 1/2 particle it will be necessary to have to hand the relevant theoretical tools; in other words, to be able to give a description of spin one-half particles which is consistent with Special Relativity, and to generalise that description to General Relativity. In general terms, then, this is an exercise in relativistic quantum mechanics, and in the case of General Relativity, in quantum mechanics in a curved space. This latter is, of course, different from quantum gravity. Quantum gravity is a theory describing the quantum nature of the gravitational field itself, for example in terms of gravitons. For our purposes the gravitational field is treated classically, as a curved space-time. The only thing to be quantised is the spin 1/2 particle.

It may be thought that this exercise has already been performed, since the Dirac equation is nothing other than a relativistic equation for spin 1/2 particles. It turns out, however, that the Dirac equation itself does not automatically yield a relativistic spin operator. The problems connected with finding such an operator were already identified in 1950b y Foldy and Wouthuysen. After outlining these problems, we shall describe how this operator is constructed. The paper concludes with some remarks about the extent to which it makes sense to talk about spin in the context of general relativity. In particular it will be pointed out that spin precession is inevitable in GR; there is no such thing as conserved spin in curved spacetime.