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

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.