On the Group Theory of the Polarization States of A Gauge Field

We study the group theoretical structure of the polarization states of the Maxwell field, as a guide to the construction of the polarization states of a higher dimensional generalization. The transformation properties of tensor (and spinor) local fields are defined according to the action of the symmetry group on the indices and on the argument. In Fourier representation, the support of a massless field is invariant under a stabilizing little group with the property that it contains an abelian (“translation”) subgroup. The (finite dimensional) field must then transform trivially under this subgroup. This results in a derivation of the requirement of gauge symmetry and the Gupta-Bleuler condition, as well as an identification of the physical degrees of freedom of the field. We apply this method to the construction of the polarization states of a higher dimensional field in Section 3.