Wilson loop

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

In the classical theory, the collection of all Wilson loops contains sufficient information to reconstruct the gauge connection, up to gauge transformation.^[1]

In quantum field theory, the definition of Wilson loops observables as bona fide operators on Fock space (actually, Haag's theorem states that Fock space does not exist for interacting QFTs) 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.^[2] 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 Alexander 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Wilson loops played an important role in the formulation of loop quantum gravity, but there they are superseded by spin networks, a certain generalization of Wilson loops.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 a compact manifold.

[edit] An equation

A Wilson line $W C$ is a quantity defined by a path-ordered exponential of a gauge field $A μ$

$W_C = \mathrm{Tr}\, \mathcal{P}\exp i \oint_C A_\mu dx^\mu$

Here, $C$ is a contour in space, $\mathcal{P}$ is the path-ordering operator. Under a gauge transformation

$\mathcal{P}e^{i \oint_C A_\mu dx^\mu} \to g(x) \mathcal{P}e^{i \oint_C A_\mu dx^\mu} g^{-1}(x)\,$ ,

where $x\,$ is the initial point of the loop. The invariance of the trace under cyclic permutations guarantees that the operator is invariant under gauge transformations. Note that the quantity being traced over is an element of the gauge Lie group and the trace is really the character of this element with respect to an irreducible representation, which means there are infinitely many traces, one for each irrep. Precisely because we're looking at the trace, it doesn't matter which point on the loop is chosen as the initial point. They all give the same value.

Actually, if A is viewed as a connection over a principal G-bundle, the equation above really ought to be "read" as the parallel transport of the identity around the loop which would give an element of the Lie group G.

Note that a path-ordered exponential is a convenient shorthand notation common in physics which conceals a fair number of mathematical operations. A mathematician would refer to the path-ordered exponential of the connection as "the holonomy of the connection" and characterize it by the parallel-transport differential equation that it satisfies.

In finite temperature QCD, the expectation value of the Wilson line distinguishes between the confined phase and the deconfined phase of the theory.