Multimass Fields, Spin, and Statistics

Abstract

The most general free field which transforms locally under Lorentz transformations according to a unitary representation of the homogeneous Lorentz group is constructed. Such a field is a linear combination of annihilation and creation operators for particles belonging to an infinite tower of irreducible multiplets of the Poincaré group. Different spin multiplets in the tower may have different masses, and appear with arbitrary spin-dependent and mass-dependent weight factors in the field expansion. It is shown that the requirement of causal (anti-)commutation relations for these fields can be satisfied simply if one assumes Bose statistics for the particles in the towers for both integer and half-odd-integer spin. However, by a judicious use of the Gelfand matrices (generalized Dirac matrices) the causality condition can also be satisfied using Fermi statistics. Such Fermi fields are constructed for two particular unitary representations—one of integer and one of half-odd-integer spin. These generalized Fermi fields still do not enable one to construct an index-invariant theory consistent with the unitarity of the $S$ matrix which incorporates particles satisfying Fermi statistics. Thus the incompatibility for an index-invariant causal theory between unitarity and Fermi statistics, which was established in a previous paper under more restrictive assumptions, remains valid for these more general fields.