On Covariant Derivatives and Gauge Invariance in the Proper Time Formalism for String Theory

TL;DR

This paper demonstrates how to incorporate gauge invariance and covariant derivatives into the proper time formalism of string theory, extending it to higher spin modes and illustrating the approach with explicit examples.

Contribution

It introduces a method to implement gauge invariance in the proper time formalism for string theory, including higher spin modes, using loop variables.

Findings

Covariant derivatives can be implemented in the proper time formalism.

Gauge invariance for higher spin modes can be made manifest.

Explicit examples illustrate the formalism's effectiveness.

Abstract

It is shown that the idea of ``minimal'' coupling to gauge fields can be conveniently implemented in the proper time formalism by identifying the equivalent of a ``covariant derivative''. This captures some of the geometric notion of the gauge field as a connection. The proper time equation is also generalized so that the gauge invariances associated with higher spin massive modes can be made manifest, at the free level, using loop variables. Some explicit examples are worked out illustrating these ideas.