Poisson reduction and quantization for the n +1 photon

Source abstract

For a dynamical system in which the constraints are given by the vanishing of a singular momentum map J, reduction in the usual group-theoretic sense may not be possible. Nonetheless, one may still ‘‘reduce’’ J−1(0), at least on the level of Poisson algebras. An example of such a singular constrained system is the ‘‘n+1 photon,’’ that is, a massless, spinless particle in (n+1)-dimensional Minkowski space-time. We apply the generalized reduction procedure to the n+1 photon, explicitly constructing the Poisson algebra of gauge invariant observables. This technique also enables us to completely analyze the effects of the singularities in J−1(0) on the system. We then quantize, obtaining results which are in agreement with a quantization of the extended phase space and the subsequent imposition of the constraint.