Reparametrization Invariance in Some Non-Local 1-D Field Theories

In this paper we consider 1-D non-local field theories with a particular $1/r^2$ interaction, a constant gauge field and an arbitrary scalar potential. We show that any such theory that is at a renormalization group fixed point also satisfies an infinite set of reparametrization invariance Ward identities. We also prove that, for special values of the gauge field, the value of the potential that satisfies the Ward identities to first order in the potential strength remains a solution to all orders in the potential strength, summed over all loops. These theories are of interest because they describe dissipative quantum mechanics with an arbitrary potential and a constant magnetic field. They also give solutions to open string theory in the presence of a uniform gauge field and an arbitrary tachyon field.