Synthesis of Covariant Particle Equations

Abstract

By combining two irreducible representations of the proper inhomogeneous Lorentz group, certain irreducible unitary representations of the complete Lorentz group including space and time inversion are obtained, together with a Schrödinger equation whose solutions constitute the representation space for these representations. The representations thus define a "canonical" form for covariant particle theories, in which not only the wave equations but the manner in which the wave functions transform under Lorentz transformations is prescribed. It is shown that by a suitable choice of representation, the Dirac, Klein-Gordon, and Proca equations can all be reduced to this canonical form. It is further shown that in the representation space provided, several possibilities exist for the identification of the transformations to be associated with space inversion, time inversion, and charge conjugation, thus suggesting the existence of several distinct relativistic theories for particles of any given spin. Conjectures are made as to the physical significance of these different possibilities when the equations are second-quantized. It is shown that each of the conventional theories employs only one of the available possibilities for these transformations, the choices being different for integral and half-integral spin theories.