THE 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, gaug e invariance is maintained off-mass-shell. However, at the second orde r in the gauge field interaction, one finds that [U(1)] gauge invarian ce is violated due to the finite cutoff. Interestingly, we find, to th e lowest order, that by adding a massive mode with appropriate gauge t ransformation laws to the sigma model background, we can restore gauge invariance. The gauge transformation law is found to be consistent, t o the order calculated, with what one expects from the interacting equ ation of motion of the massive field. We also extend some previous dis cussion on applying the proper time formalism for propagating gauge pa rticles, to the interacting (i.e. Yang-Mills) case.