Rudiments of Dual Feynman Rules for Yang-Mills Monopoles in Loop Space

TL;DR

This paper derives dual Feynman rules for Yang-Mills monopoles using loop space variables and the Wu-Yang criterion, revealing a dual symmetry and connecting the dual potential to the original gauge potential.

Contribution

It introduces a novel approach to formulate dual Feynman rules for Yang-Mills monopoles via loop space and Wu-Yang constraints, demonstrating dual symmetry in the theory.

Findings

Dual potential has the same propagator and interaction vertex as the original gauge potential.

In the abelian case, the rules match QED to all orders, confirming dual symmetry.

The approach employs gauge fixing and ghost integration to derive the generating functional.

Abstract

Dual Feynman rules for Dirac monopoles in Yang-Mills fields are obtained by the Wu-Yang (1976) criterion in which dynamics result as a consequence of the constraint defining the monopole as a topological obstruction in the field. The usual path-integral approach is adopted, but using loop space variables of the type introduced by Polyakov (1980). An anti-symmetric tensor potential $L_{\mu\nu}[\xi|s]$ appears as the Lagrange multiplier for the Wu-Yang constraint which has to be gauge-fixed because of the ``magnetic'' $\widetilde{U}$-symmetry of the theory. Two sets of ghosts are thus introduced, which subsequently integrate out and decouple. The generating functional is then calculated to order $g^0$ and expanded in a series in $\widetilde{g}$. It is shown to be expressible in terms of a local ``dual potential'' $\widetilde{A}_\mu (x)$ found earlier, which has the same progagator and the same interaction vertex with the monopole field as those of the ordinary Yang-Mills potential $A_\mu$ with a colour charge, indicating thus a certain degree of dual symmetry in the theory. For the abelian case the Feynman rules obtained here are the same as in QED to all orders in $g$, as expected by dual symmetry.