Wilson loop

In gauge theory, a Wilson loop (named after Kenneth G. Wilson) is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. 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 loop 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.[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Wilson loops played an important role in the formulation of loop quantum gravity, but there they are superseded by spin networks (and, later, spinfoams), a certain generalization of Wilson loops.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.

An equation

The Wilson loop variable is a quantity defined by the trace of a path-ordered exponential of a gauge field \( A_\mu \) transported along a closed line C:

\( W_C := \mathrm{Tr}\,(\, \mathcal{P}\exp i \oint_C A_\mu dx^\mu \,)\,. \)

Here, C is a closed curve 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 \( x\, \) corresponds to the initial (and end) point of the loop (only initial and end point of a line contribute, whereas gauge transformations in between cancel each other). For SU(2) gauges, for example, one has \( g^{\pm 1}(x)\equiv\exp\{\pm i\alpha^j(x)\frac{\sigma^j}{2}\} ; \alpha^j(x) \) is an arbitrary real function of \( x\,, \) and \( \sigma^j \) are the three Pauli matrices; as usual, a sum over repeated indices is implied.

The invariance of the trace under cyclic permutations guarantees that \( W_C \) 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 one of the infinitely many irreducible representations, which implies that the operators \( A_\mu\,dx^\mu \) don't need to be restricted to the "trace class" (thus with purely discrete spectrum), but can be generally hermitian (or mathematically: self-adjoint) as usual. Precisely because we're finally 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.

At T=0, where T corresponds to temperature, the Wilson loop variable characterizes the confinement or deconfinement of a gauge-invariant quantum-field theory, namely according to whether the variable increases with the area, or alternatively with the circumference of the loop ("area law", or alternatively "circumferential law" also known as "perimeter law").

In finite-temperature QCD, the thermal expectation value of the Wilson line distinguishes between the confined "hadronic" phase, and the deconfined state of the field, e.g., the quark–gluon plasma.

22. Computation of the Wilson Loop