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Wilson loops played an important role in the formulation of [[loop quantum gravity]], but there they are superseded by [[spin network]]s, a certain generalization of Wilson loops.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

Latest revision as of 17:32, 27 December 2014

Succeeding with a home-based business takes a lot of time, dedication and hard work if you want it to pay off. The one thing that all people who have found success with a home-based business is that they had the drive, determination, and effort needed to succeed in a very demanding area. A few savvy people do find success working from home with a variety of businesses. Many people dream of working from home, but for most this dream remains just that, a dream that goes unfulfilled. Working from home can free up your time for family and give you the financial independence you have always dreamed of.

Then consider how this could affect a patients records and you understand why licensing is important. Many work at home opportunities are more to sell training materials and lists of potential employers once you have completed the training. Several of the course may also be taken with less expense at your local community college or trade school.

Make sure the work at home opportunity you choose is something you can feel good about doing and that you do not have unrealistic expectations about the outcome of your endeavor. No opportunity will make you rich overnight, but if you put in the time and work you can be successful.

The convenience factor is also readily visible in the home business sector. The convenience of working from home is an extremely attractive quality. One who works from home has the convenience of not having to commute to and from work, ability to get to work as early as they want or as late as they wish and take breaks during the workday to tend to other important issues.

The home business arena is booming for many reasons but one in particular is that it provides individuals with the opportunity to earn a living working from home and care for their children while doing so. The ability to work and care for the children at home is a highly desirable factor associated with owning a home business.

There will be adjustments and you may have to work harder than you have ever worked before, as you launch your home based business and build it up to something that not only you can be proud of, but it also generates the income you require for the lifestyle that you want to lead.

The company you are joining should have a system which is automated and capable of generating prospects, giving them a view of the business. This company should also give you support, giving you proper training and orientation to know the business

To know more click this link:
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