Loop variables for compact two-dimensional quantum electrodynamics

Variables parametrized by closed and open curves are defined to reformulate compact U(1) quantum electrodynamics in a circle with a massless fermion field. It is found that the gauge-invariant nature of these variables accommodates into a regularization scheme for the Hamiltonian and current operators that is especially well suited for the study of the compact case. The zero-mode energy spectrum, the value of the axial anomaly, and the anomalous commutators this model presents are hence determined in a manifestly gauge-invariant manner. Contrary to the noncompact case, the zero-mode spectrum is not equally spaced and consequently the theory does not lead to the spectrum of a free scalar boson. All the states are invariant under large gauge transformations. In particular, that is the case for the vacuum, and consequently the $\theta$ dependence does not appear.