Point Form Electrodynamics and the Gupta-Bleuler Formalism

The Gupta-Bleuler formalism for photons is generalized by choosing the representation of the little group for massless particles, the two dimensional Euclidian group, to be the four dimensional nonunitary representation obtained by restricting elements of the Lorentz group to the Euclidian group. Though the little group representation is nonunitary, it is shown that the representation of the Poincaré group is unitary. Under Lorentz transformations photon creation and annihilation operators transform as irreducible representations of massless particles, and not as four-vectors. As a consequence the polarization vector, which connects the four-vector potential with creation and annihilation operators, is given in terms of boosts, coset representatives of the Lorentz group with respect to the Euclidian group. Several polarization vectors (boost choices) are worked out, including a front form polarization vector. The different boost choices are shown to be related by the analogue of Melosh rotations, namely Euclidian group transformations1.