't Hooft's Order-Disorder Parameters and the Dual Potential

TL;DR

This paper demonstrates that a dual potential operator satisfies the correct commutation relations with Wilson operators, confirming its role as an order-disorder parameter in gauge theories.

Contribution

It introduces and verifies a dual potential operator that fulfills the 't Hooft order-disorder criteria using the recently derived dual potential.

Findings

The operator $B(C)$ satisfies the correct commutation relations with Wilson operators.

The dual potential $ ilde{A}(x)$ is consistent with 't Hooft's order-disorder parameters.

The coupling $ ilde{g}$ relates to $g$ via Dirac quantization, ensuring proper quantum behavior.

Abstract

It is shown that the operator $B(C) = Tr [P \exp i \tilde{g} \oint \tilde{A}_i(x) dx^i]$ constructed with the recently derived dual potential $\tilde{A}(x)$ and a coupling $\tilde{g}$ related to $g$ by the Dirac quantization condition satisfies the correct commutation relation with the Wilson operator $Tr [P \exp ig \oint A_i(x) dx^i]$ as required by 't Hooft for his order-disorder parameters.