On the group theory of the polarization states of a massless field

It is shown that the theory of representations of the PoincarC group applied to the vector potentials of a massless field yields in a simple and direct way (with- out assuming that the field is a gauge field, and without detailed assumption on the form of the equation of motion), the structure of the polarizations, the Gupta-Bleuler condition, and gauge invariance of the theory. The method is shown to apply to the Maxwell field and to a five-dimensional generalization of the Maxwell field (which properly contains the Maxwell theory) associated with manifestly covariant dynamics. In this paper we study the group theoretical structure of the Maxwell radiation field as a guide to the construction of the polarization states of a higher-dimensional generalization of the Maxwell field. Such a generalization is required to describe the electromagnetic interactions of the manifestly covariant mechanics of Stueckelberg' and Horwitz and Piron. This manifestly covariant relativistic classical and quantum mechanics has for its fundamental dynamical constituents, as for Feynman's formulation of quantum electrodynamics,3 objects defined lo- cally in space-time. The states of such a system evolve with an invariant parameter r, which we take to be a universal time, as that of Newton. Since the conserved current of the Maxwell theory is an integral over the world line of local contributions:5 e.g., in the classical case,