Hamiltonian Solution of the Schwinger Model with Compact U(1)

The complete exact solution of the Schwinger model with compact gauge group U(1), in the Hamiltonian approach, is presented . The compactification is imposed by demanding that the only surviving true electromagnetic degree of freedom has angular character. Not surprinsingly, this topological condition defines a version of the Schwinger model which is different from the standard one, where $c$ takes values on the line . The main consequences are: the spectra of the zero modes is not degenerated and does not correspond to the equally spaced harmonic oscillator, both the electric charge and a modified gauge invariant chiral charge are conserved (nevertheless, the axial-current anomaly is still present) and, finally, there is no need to introduce a $\theta$-vacuum. A comparison with the results of the standard Schwinger model is pointed out along the text.