Massive Particle Fields, with Momentum Matrices

Including translation matrices in covariant non-unitary Poincaré representations alters the construction of massive particle fields from canonical unitary fields. The conventional procedure without spacetime translation matrices determines covariant fields that transform by matrix representations of the homogeneous Lorentz group combined with a differential operator representation of the Poincaré group that acts on spacetime coordinates. The differential operator part generates translations nontrivially with the momentum operator proportional to the gradient, but the finite dimensional matrix part represents translations trivially, i.e. the momentum matrices vanish. This paper generalizes the construction of a massive particle field to produce a field that also transforms according to a finite dimensional non-unitary representation of translations generated by nonzero momentum matrices. The more general field evaluated at spacetime coordinates x results when the translation matrix for a displacement x is applied to the conventional field evaluated at x. ∗affiliation and mailing address: Department of Applied Mathematics and Sciences, Wentworth Institute of Technology, 550 Huntington Avenue, Boston, MA, USA, ZIP 02115, telephone number: (617) 989-4338, fax number: (617) 989-4591 , e-mail address: [email protected]