Consistency questions in the light cone gauge based on equal time commutation rules

Abstract

We investigate whether time displacement invariant propagators are compatible canonical formulations in the light cone gauge based on equal time commutation rules. We conclude that, in the light cone gauge, time displacement invariant propagators are not consistent with the requirement that, in canonical formulations of gauge theories, only transversely polarized, massless gauge field excitations (photons, or gluons in perturbative QCD), can contribute to the transverse part of a time displacement invariant propagator. When the time displacement invariant light cone gauge propagator is represented as a four-dimensional momentum space Fourier integral the following is observed: Transverse parts of the propagator obtain time displacement invariant contributions from the ( k 3− k 0) pole, as well as from the ( | k| 2−k 0 2 ) pole. But since, in the Schrödinger picture (i.e. at t=0), the divergence-free part of the gauge field consists of transversely polarized gauge field excitations only, the transverse part of the propagator either can have time displacement invariant time dependence determined by the ( | k| 2−k 0 2 ) pole; or, if any part of the transverse propagator has time dependence determined by the ( k 3− k 0) pole, it cannot be time displacement invariant.