Reduction of Relativistic Wavefunctions to the Irreducible Representations of the Inhomogeneous Lorentz Group. Part II. Zero-Mass Components

Abstract

In a previous paper, we showed how wavefunctions which transform in a relativistic manner in configuration space can be expanded in terms of amplitudes, which for nonzero mass transform like the wavefunctions for irreducible representations of the proper, orthochronous, inhomogeneous Lorentz group. A simple algorithm was given to obtain the expansion. In the present paper, we extend the results to include zero-mass amplitudes. It is shown that for wavefunctions which are required to transform under the homogeneous Lorentz group such that the matrices which involve the spinor indices are finite dimensional, the zero-mass amplitudes transform under nonunitary representations of the inhomogeneous Lorentz group. However, it is possible to split up each such nonunitary representation into a part which corresponds to a unitary representation for finite spin and into a part which corresponds to an unphysical change of wavefunction. As examples of the technique, we consider wavefunctions which transform as an antisymmetric real tensor (i.e., as an electromagnetic field) as a four-vector with and without the Lorentz condition, and as a Dirac spinor. The results offer interesting contrasts with the reductions of Part I where only nonzero-mass components were considered. It is shown that the expansion of the present paper, when applied to the solution of Maxwell's equations, leads to an expansion in terms of photon wavefunctions and that the unphysical change of wavefunction is zero. For a real vector potential with the Lorentz condition (i.e., the electromagnetic vector potential), the expansion corresponds to the sum of an expansion in terms of photon wavefunctions and a wavefunction which sets the gauge of the vector potential. The nonphysical part of the transformation of the electromagnetic vector potential is merely a gauge change. Finally, solutions of the massless Dirac equation are expanded in terms of wavefunctions for massless particles of spin ½ for which the nonphysical part of the change is zero. In the present paper, we also show how invariant inner products are to be introduced, how negative-energy representations can be replaced by positive-energy representations (''antiparticles''), and show the connection with the usual canonical formalism. Finally, second quantization of the theory is given.