Extended Hamiltonian Formalism of the Pure Space-Like Axial Gauge Schwinger Model

We demonstrate that pure space-like axial gauge quantizations of gauge fields can be constructed in ways which are free from infrared divergences. We begin by constructing an axial gauge formulation in auxiliary coordinates: $x^{+}=x^0\sin \theta +x^1\cos \theta ,x^{-}=x^0\cos \theta -x^1\sin \theta$. For \theta less than \pi\over 4 we can take $x^{-}$ as the evolution parameter and construct a traditional canonical formulation of the temporal gauge Schwinger model in which residual gauge fields dependent only on $x^{+}$ are static canonical variables. Then we extrapolate the temporal gauge operator solution into the axial region, \theta > \pi \over 4, where $x^{+}$ is taken as the evolution parameter. In the axial region we find that we have to change representations of the residual gauge fields from one realizing the PV prescription to one realizing the ML prescription in order for the infrared divergences resulting from $({\mathrm{\partial }}_{-}{)}^{-1}$ to be canceled by corresponding ones resulting from the inverse of the hyperbolic Laplace operator. Finally, by taking the limit $\theta \to \frac{\pi }{2}-0$ we obtain an operator solution and the Hamiltonian of the axial gauge (Coulomb gauge )Schwinger model in ordinary coordinates. That solution includes auxiliary fields and the representation space is of indefinite metric, providing further evidence that physical'' gauges are no more physical than unphysical’’ gauges.