Loop Variables and Gauge Invariance in (Open) Bosonic String Theory

TL;DR

This paper refines the loop variable approach for gauge-invariant equations in open string theory, demonstrating gauge invariance to all orders and connecting these equations with Wilsonian RG and on-shell amplitudes.

Contribution

It provides a simplified, more complete formulation of gauge-invariant equations using loop variables, including a proof of gauge invariance to all orders and their relation to Wilsonian RG equations.

Findings

Loop variable equations resemble free equations with an extra parameter.

Gauge invariance holds to all orders in the formulation.

Equations match on-shell scattering amplitudes in the infrared limit.

Abstract

We give a simplified and more complete description of the loop variable approach for writing down gauge invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite world-sheet cutoff and that in the infrared limit, equivalence with the Callan-Symanzik $\beta$-functions should ensure that they reproduce the on-shell scattering amplitudes in string theory. It is applied to the tachyon-photon system and the general arguments for gauge invariance can be easily checked to the order calculated. One can see that when there is a finite world sheet cutoff in place, even the U(1) invariance of the equations for the photon, involves massive mode contributions. A field redefinition involving the tachyon is required to get the gauge transformations of the photon into standard form.