Reduction of Relativistic Wavefunctions to the Irreducible Representations of the Inhomogeneous Lorentz Group. I. Nonzero Mass Components

Abstract

When the space‐time coordinates of a relativistic system undergo the transformations of the proper, orthochronous, inhomogeneous Lorentz group, the wavefunction of the system undergoes transformations which may be considered to constitute a representation of the group. We give a simple algorithm for reducing this representation to the irreducible unitary ray representations if we assume that only nonzero mass representations occur. The extension to cases in which zero mass representations occur will be given in a later paper. The form in which the reduction is given is an expansion of the wavefunction as given in configuration space in terms of a basis such that the coefficients transform in accordance with the Foldy‐Shirokov realization of the irreducible representations. Any wave equation which the wavefunction satisfies and any auxiliary conditions, such as the Lorentz condition or reality conditions, eliminate or relate in a simple way some of the representations which can appear. As examples, we reduce the scalar wavefunction, the four‐vector with and without the Lorentz condition, the Dirac wavefunction, the wavefunction which transforms like the electromagnetic field, and a wavefunction which transforms as a generalization of the Dirac wavefunction. In these examples it is also shown that if one replaces the amplitudes associated with the irreducible representations by annihilation and creation operators in a suitable manner, one obtains the usual canonical formalism for second quantization in configuration space. The reduction technique given herein is a simple application of the results of an earlier paper by the author and J. S. Lomont in which is shown how to reduce any unitary ray representation of the inhomogeneous Lorentz group.