Wilson loops in the Hamiltonian formalism

TL;DR

This paper explores the role of Wilson and 't Hooft loops in gauge theories within the Hamiltonian formalism, clarifying their effects on the physical Hilbert space and their relation to test charges at finite temperature.

Contribution

It provides a detailed analysis of how Wilson and 't Hooft loops modify the Hilbert space and connect to test charges and propagators in gauge theories at nonzero temperature.

Findings

Wilson loops enlarge the Hilbert space by adding electric test charges.

Polyakov loops are related to the propagator of a test charge at finite temperature.

't Hooft loops do not change the physical Hilbert space, representing magnetic test charges.

Abstract

In a gauge theory, the gauge invariant Hilbert space is unchanged by the coupling to arbitrary local operators. In the presence of Wilson loops, though, the physical Hilbert space must be enlarged by adding test electric charges along the loop. I discuss how at nonzero temperature Polyakov loops are naturally related to the propagator of a test charge. 't Hooft loops represent the propagation of a test magnetic charge, and so do not alter the physical Hilbert space.