C, P, T and general first and second order partial differential relativistic wave equations

Abstract

The extended Lorentz group, which contains space and time inversions, is extended by the two element group Z 2; so that representations of the resulting group, called the full Lorentz group L , describe integral or half integral spin states. Some of these representations are finite-dimensional, while others have infinite dimensions. The full Poincare group P is constructed as the semiproduct of L with the translational group J . The main hypothesis is that single particle systems form representations of P . First or second order partial differential relativistic wave equations are used to project out those subspaces to whose vectors single physical particle states may be assigned. Under a set of hypotheses originating from heuristic arguments, solutions of wave equations are shown to possess the transformation properties of free particle states under P , so that the assignment of physical states to solutions is meaningful. Since parity and time-inversion are elements of P , wave equations are automatically invariant under them. Parities of particles are found to be ±1 or ± i. A “Poincare invariant” exists so that distinct particle states are orthogonal. In all those equations having particle anti-particle pairs, a charge conjugation operator can always be defined.