Loop Variable Inequalities in Gravity and Gauge Theory

TL;DR

This paper highlights the necessity of inequalities for loop variables in gravity and gauge theories to ensure equivalence with traditional local gauge potential formulations.

Contribution

It identifies the incomplete nature of loop variable formulations and establishes the need for inequalities to fully describe gravitational and gauge theories.

Findings

Loop variables must satisfy specific inequalities.

Inequalities are essential for equivalence to local gauge potential descriptions.

Addresses a fundamental gap in the loop variable approach.

Abstract

We point out an incompleteness of formulations of gravitational and gauge theories that use traces of holonomies around closed curves as their basic variables. It is shown that in general such loop variables have to satisfy certain inequalities if they are to give a description equivalent to the usual one in terms of local gauge potentials.