Residual value

2013-05-14T20:31:56Z

173.161.124.197: made commas in numbers at bottom nice

{{Refimprove|date=February 2011}}
{{Redirect|Wilson line|the Wilson Line shipping company|Thomas Wilson Sons & Co.}}
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]].<ref name="Giles 1981">{{cite journal| first= R.| last= Giles| journal=[[Physical Review D]] | title=Reconstruction of Gauge Potentials from Wilson loops | volume= 24| issue=8| page= 2160| year=1981| doi=10.1103/PhysRevD.24.2160|bibcode = 1981PhRvD..24.2160G }}</ref>

In [[quantum field theory]], the definition of Wilson loop observables as ''[[bona fide]]'' [[operator (mathematics)|operator]]s 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 (physics)|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, [[Michael Faraday|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.<ref name="Wilson 1974">{{cite journal| authorlink=Kenneth G. Wilson| first=K.| last= Wilson| journal=[[Physical Review D]] | title=Confinement of quarks | volume= 10| issue=8| page= 2445| year= 1974| doi=10.1103/PhysRevD.10.2445|bibcode = 1974PhRvD..10.2445W }}</ref> The problem of [[colour confinement|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 Markovich Polyakov|Alexander Polyakov]] to formulate the first [[string theory|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 network]]s, 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 line''' variable <math>W_C</math> (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]] <math>A_\mu</math> transported along a closed line C:

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

Here, <math>C</math> is a closed curve in space, <math>\mathcal{P}</math> is the [[path-ordering]] operator. Under a gauge transformation

:<math>\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)\,</math>,

where <math>x\,</math> 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 <math>g^{\pm 1}(x)\equiv\exp\{\pm i\alpha^j(x)\frac{\sigma^j}{2}\}</math>; <math>\alpha^j(x)</math> is an arbitrary real function of <math>x\,</math>, and <math>\sigma^j</math> are the three Pauli matrices; as usual, a sum over repeated indices is implied.

The invariance of the [[Trace (linear algebra)|trace]] under [[cyclic permutation]]s guarantees that <math>W_C</math> is invariant under [[gauge transformation]]s. Note that the quantity being traced over is an element of the gauge [[Lie group]] and the trace is really the [[character (mathematics)|character]] of this element with respect to one of the infinitely many [[irreducible representation]]s, which implies that the operators <math> A_\mu\,dx^\mu</math> 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 form|connection]] over a [[principal bundle|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 [[color confinement|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]].

==See also==
* [[Stochastic vacuum model]]
* [[Winding number]]

==References==
{{reflist}}

[[Category:Quantum field theory]]
[[Category:Quantum chromodynamics]]

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Residual value