Editing Wilson loop (section)

==Order operator==

Since temporal Wilson lines correspond to the configuration created by infinitely heavy stationary quarks, Wilson loop associated with a rectangular loop <math>\gamma</math> with two temporal components of length <math>T</math> and two spatial components of length <math>r</math>, can be interpreted as a [[quark]]-antiquark pair at fixed separation. Over large times the [[vacuum expectation value]] of the Wilson loop projects out the state with the [[ground state|minimum energy]], which is the [[potential energy|potential]] <math>V(r)</math> between the quarks.<ref>{{cite book|last=Rothe|first=H.J.|author-link=|date=2005|title=Lattice Gauge Theories: An Introduction|series=World Scientific Lecture Notes in Physics: Volume 43|url=https://library.oapen.org/handle/20.500.12657/50492|doi=10.1142/8229|location=|publisher=World Scientific Publishing|chapter=7|volume=82 |pages=95–108|isbn=978-9814365857}}</ref> The [[excited state]]s with energy <math>V(r)+\Delta E</math> are exponentially suppressed with time and so the expectation value goes as

:<math>
\langle W[\gamma]\rangle \sim e^{-TV(r)}(1+\mathcal O(e^{-T\Delta E})),
</math>

making the Wilson loop useful for calculating the potential between quark pairs. This potential must necessarily be a [[monotonic function|monotonically increasing]] and [[concave function]] of the quark separation.<ref>{{cite journal|last1=Seiler|first1=E.|authorlink1=|date=1978|title=Upper bound on the color-confining potential|url=https://link.aps.org/doi/10.1103/PhysRevD.18.482|journal=Phys. Rev. D|volume=18|issue=2|pages=482–483|doi=10.1103/PhysRevD.18.482|pmid=|arxiv=|bibcode=1978PhRvD..18..482S |s2cid=|access-date=|url-access=subscription}}</ref><ref>{{cite journal|last1=Bachas|first1=C.|authorlink1=|date=1986|title=Concavity of the quarkonium potential|url=https://link.aps.org/doi/10.1103/PhysRevD.33.2723|journal=Phys. Rev. D|volume=33|issue=9|pages=2723–2725|doi=10.1103/PhysRevD.33.2723|pmid=9956963|arxiv=|bibcode=1986PhRvD..33.2723B |s2cid=|access-date=|url-access=subscription}}</ref> Since spacelike Wilson loops are not fundamentally different from the temporal ones, the quark potential is really directly related to the pure Yang–Mills theory structure and is a phenomenon independent of the matter content.<ref>{{cite book|last=Greensite|first=J.|author-link=|date=2020|title=An Introduction to the Confinement Problem|edition=2|url=|doi=|location=|publisher=Springer|chapter=4|pages=37–40|isbn=978-3030515621}}</ref>

[[Elitzur's theorem]] ensures that local non-gauge invariant operators cannot have a non-zero expectation values. Instead one must use non-local gauge invariant operators as order parameters for confinement. The Wilson loop is exactly such an order parameter in pure [[Yang–Mills theory]], where in the confining [[phase (matter)|phase]] its expectation value follows the area law<ref>{{cite book|last=Makeenko|first=Y.|date=2002|title=Methods of Contemporary Gauge Theory|series=Cambridge Monographs on Mathematical Physics|url=|doi=10.1017/CBO9780511535147|location=Cambridge|publisher=Cambridge University Press|chapter=6|pages=117–118|isbn=978-0521809115}}</ref>

:<math>
\langle W[\gamma]\rangle \sim e^{-aA[\gamma]}
</math>

for a loop that encloses an area <math>A[\gamma]</math>. This is motivated from the potential between infinitely heavy test quarks which in the confinement phase is expected to grow linearly <math>V(r) \sim \sigma r</math> where <math>\sigma</math> is known as the string tension. Meanwhile, in the [[Higgs phase]] the expectation value follows the perimeter law

:<math>
\langle W[\gamma]\rangle \sim e^{-bL[\gamma]},
</math>

where <math>L[\gamma]</math> is the perimeter length of the loop and <math>b</math> is some constant. The area law of Wilson loops can be used to demonstrate confinement in certain low dimensional theories directly, such as for the [[Schwinger model]] whose confinement is driven by [[instanton]]s.<ref>{{cite book|last=Paranjape|first=M.|author-link=|date=2017|title=The Theory and Applications of Instanton Calculations|url=|doi=|location=|publisher=Cambridge University Press|chapter=9|page=168|isbn=978-1107155473}}</ref>