Proper Time Formalism, Gauge Invariance and the Effects of a Finite World Sheet Cutoff in String Theory

TL;DR

This paper explores off-shell effects in string theory using proper time formalism with a finite world sheet cutoff, addressing gauge invariance issues and proposing a method to restore it with massive modes.

Contribution

It introduces a covariant off-shell Klein-Gordon interaction in string theory and demonstrates how to maintain gauge invariance by incorporating massive modes.

Findings

Finite cutoff causes gauge invariance violation at second order.

Adding massive modes restores gauge invariance.

Consistent gauge transformation laws align with equations of motion.

Abstract

We discuss the issue of going off-shell in the proper time formalism. This is done by keeping a finite world sheet cutoff. We construct one example of an off-shell covariant Klein Gordon type interaction. For a suitable choice of the gauge transformation of the scalar field, gauge invariance is maintained off mass shell. However at second order in the gauge field interaction, one finds that (U(1)) gauge invariance is violated due to the finite cutoff. Interestingly, we find, to lowest order, that by adding a massive mode with appropriate gauge transformation laws to the sigma model background, one can restore gauge invariance. The gauge transformation law is found to be consistent, to the order calculated, with what one expects from the interacting equation of motion of the massive field. We also extend some previous discussion on applying the proper time formalism for propagating gauge particles, to the interacting (i.e. Yang Mills) case.