The Poincaré Group

Abstract

In Chapter 1 the Dirac equation was defined in such a way that it automatically satisfies the energy-momentum relation as required by classical relativistic mechanics — at least in a particular inertial frame. According to the principles of relativity, different inertial frames are related by Poincaré transformations (Sect. 2.1). In this chapter we are going to prove that the Dirac equation — and the quantum theory developed in Chapter 1 — is invariant under Poincaré transformations. This is perhaps obscured by the fact that the position is represented by an operator, whereas time remains a parameter. After all, this is not quite in the spirit of the theory of relativity, where space and time coordinates are mixed up by Lorentz transformations.