Massless Particle Fields, with Momentum Matrices

Nontrivial translation matrices occur for spin (A,B) ⊕ (C,D) with | A − C | = | B−D | = 1/2, necessarily associating a (C,D) field with a spin (A,B) field. Including translation matrices in covariant non-unitary Poincaré representations also introduces new gauge terms in the construction of massless particle fields from canonical unitary fields. In the usual procedure without spacetime translation matrices, gauge terms arise from ‘translations’ of the massless little group; the little group combines spacetime rotations and boosts making a group isomorphic with the Euclidean group E2, including E2 translations. The usual remedy is to invoke gauge invariance. But here, the spacetime translation gauge terms can cancel the little group gauge terms, trading the need for gauge invariance with the need to specify displacements and to freeze two little group degrees of freedom that are not wanted anyway. The cancelation process restricts the helicity to A − B − 1 for A − C = −(B − D) = 1/2 and A − B + 1 for A − C = −(B − D) = −1/2. However, the cancelation only works for the little group standard momentum and specific transformations and, in general, gauge invariance is still needed to obtain massless particle fields. Expressions for massless particle fields for each spin type are found.