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

TL;DR

This paper explores 1-D non-local field theories with specific interactions and gauge fields, demonstrating that at fixed points they satisfy reparametrization invariance Ward identities, with implications for dissipative quantum mechanics and string theory.

Contribution

It establishes the presence of reparametrization invariance Ward identities in certain non-local 1-D field theories and shows their solutions' stability to all orders in potential strength.

Findings

Fixed points satisfy infinite Ward identities.

Potential solutions remain valid to all orders for special gauge fields.

Applications to dissipative quantum mechanics and string theory.

Abstract

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.