Wilson loop

In gauge field theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. It is named for Kenneth Wilson.

In the classical theory, the collection of all Wilson loops contains sufficient information to reconstruct the gauge connection, up to gauge transformation.

In quantum field theory, the definition of Wilson loops observables as bona fide operators on Fock space is a mathematically delicate problem and requires regularization, usually by equipping each loop with a framing. The action of Wilson loop operators has the interpretation of creating an elementary excitation of the quantum field which is localized on the loop. In this way, Faraday's "flux tubes" become elementary excitations of the quantum electromagnetic field.

Wilson loops were introduced in the 1970s in an attempt at a nonperturbative formulation of quantum chromodynamics (QCD), or at least as a convenient collection of variables for dealing with the strongly-interacting regime of QCD. The problem of confinement, which Wilson loops were designed to solve, remains unsolved to this day.

The fact that strongly-coupled quantum gauge field theories have elementary nonperturbative excitations which are loops motivated Maxim Polyakov to formulate the first string theories, which described the propagation of an elementary quantum loop in spacetime.

Wilson loops played an important role in the formulation of loop quantum gravity, but there they are superseded by spin networks, a certain generalization of Wilson loops.

In particle physics and string theory, Wilson loops are often called Wilson lines, especially Wilson loops around non-contractible loops of the compact manifold.

An equation

Wilson line $W_{C}$ is a quantity defined by a path-ordered exponential of a gauge field $A_{\mu }$

$W_{C}=\mathrm {Tr} \,{\mathcal {P}}\exp \left(\oint _{C}i\,dx^{\mu }A_{\mu }\right)$

Here, $C$ is a contour in space, ${\mathcal {P}}$ is the path-ordering operator, and the trace Tr guarantees that the operator is invariant under gauge transformations.

Note that a path-ordered exponential is a convenient shorthand notation common in physics which conceals a fair amount of mathematical operations.