Wilson Loop - An Equation

An Equation

The Wilson line variable (or better Wilson loop variable, since one is always dealing with closed lines) is a quantity defined by the trace of a path-ordered exponential of a gauge field transported along a closed line C:

Here, is a closed curve in space, is the path-ordering operator. Under a gauge transformation

,

where 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 ; is an arbitrary real function of, and are the three Pauli matrices; as usual, a sum over repeated indices is implied.

The invariance of the trace under cyclic permutations guarantees that 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 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, 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 much-debated quark-gluon plasma.