Free and Interacting 2-D Maxwell Field Theory on Conformally Flat Space Times

TL;DR

This paper quantizes the 2D Maxwell field on conformally flat space, showing its classical equivalence to a biharmonic string theory, and computes propagators and Wilson loops, with initial insights into interactions via the Schwinger model.

Contribution

It establishes a classical equivalence between 2D Maxwell theory and biharmonic string theory on conformally flat spaces and computes key correlators and Wilson loops.

Findings

Propagator of Maxwell field on conformally flat space constructed

Wilson loop expectation values confirmed previous results

Lowest order correlators of Schwinger model derived on Riemann surfaces

Abstract

The free Maxwell field theory is quantized in the Lorentz gauge on a two dimensional manifold $M$ with conformally flat background metric. It is shown that in this gauge the theory is equivalent, at least at the classical level, to a biharmonic version of the bosonic string theory. This equivalence is exploited in order to construct in details the propagator of the Maxwell field theory on $M$. The expectation values of the Wilson loops are computed. A trivial result is obtained confirming in the Lorentz gauge previous calculations. Finally the interacting case is briefly discussed taking the Schwinger model as an example. The two and three point functions of the Schwinger model are explicitly derived at the lowest order on a Riemann surface.