Editing Wilson loop (section)

===Makeenko–Migdal loop equation===

Similarly to the [[functional derivative]] which acts on [[functional (mathematics)|functions of functions]], functions of loops admit two types of [[derivative]]s called the area derivative and the perimeter derivative. To define the former, consider a contour <math>\gamma</math> and another contour <math>\gamma_{\delta \sigma_{\mu\nu}}</math> which is the same contour but with an extra small loop at <math>x</math> in the <math>\mu</math>-<math>\nu</math> plane with area <math>\delta \sigma_{\mu\nu}=dx_\mu \wedge dx_\nu</math>. Then the area derivative of the loop functional <math>F[\gamma]</math> is defined through the same idea as the usual derivative, as the normalized difference between the functional of the two loops<ref>{{cite journal|last1=Migdal|first1=A.A.|authorlink1=Alexander Arkadyevich Migdal|date=1983|title=Loop Equations and 1/N Expansion|url=|journal=Phys. Rep.|volume=102|issue=4|pages=199–290|doi=10.1016/0370-1573(83)90076-5|pmid=|arxiv=|s2cid=|access-date=}}</ref>

:<math>
\frac{\delta F[\gamma]}{\delta \sigma_{\mu\nu}(x)} = \frac{1}{\delta \sigma_{\mu\nu}(x)}[F[\gamma_{\delta \sigma_{\mu\nu}}]-F[\gamma]].
</math>

The perimeter derivative is similarly defined whereby now <math>\gamma_{\delta x_\mu}</math> is a slight deformation of the contour <math>\gamma</math> which at position <math>x</math> has a small extruding loop of length <math>\delta x_\mu</math> in the <math>\mu</math> direction and of zero area. The perimeter derivative <math>\partial_\mu^x</math> of the loop functional is then defined as

:<math>
\partial_\mu^x F[\gamma] = \frac{1}{\delta x_\mu}[F[\gamma_{\delta x_\mu}]-F[\gamma]].
</math>

In the [[1/N expansion|large N-limit]], the Wilson loop vacuum expectation value satisfies a closed functional form equation called the Makeenko–Migdal equation<ref>{{cite journal|last1=Makeenko|first1=Y.M.|authorlink1=|last2=Migdal|first2=A.A.|authorlink2=Alexander Arkadyevich Migdal|date=1979|title=Exact Equation for the Loop Average in Multicolor QCD|url=|journal=Phys. Lett. B|volume=88|issue=1–2|pages=135–137|doi=10.1016/0370-2693(79)90131-X|pmid=|arxiv=|bibcode=1979PhLB...88..135M |s2cid=|access-date=}}</ref>

:<math>
\partial^x_\mu \frac{\delta}{\delta \sigma_{\mu\nu}(x)}\langle W[\gamma]\rangle = g^2 N \oint_\gamma dy_\nu \delta^{(D)}(x-y) \langle W[\gamma_{yx}]\rangle \langle W[\gamma_{xy}]\rangle.
</math>

Here <math>\gamma = \gamma_{xy}\cup \gamma_{yx}</math> with <math>\gamma_{xy}</math> being a line that does not close from <math>x</math> to <math>y</math>, with the two points however close to each other. The equation can also be written for finite <math>N</math>, but in this case it does not factorize and instead leads to expectation values of products of Wilson loops, rather than the product of their expectation values.<ref>{{cite book|last=Năstase|first=H.|author-link=Horațiu Năstase|date=2019|title=Introduction to Quantum Field Theory|url=|doi=|location=|publisher=Cambridge University Press|chapter=50|pages=469–472|isbn=978-1108493994}}</ref> This gives rise to an infinite chain of coupled equations for different Wilson loop expectation values, analogous to the [[Schwinger–Dyson equation]]s. The Makeenko–Migdal equation has been solved exactly in two dimensional <math>\text{U}(\infty)</math> theory.<ref>{{cite journal|last1=Kazakov|first1=V.A.|authorlink1=|last2=Kostov|first2=I.K.|authorlink2=|date=1980|title=Non-linear strings in two-dimensional U(∞) gauge theory|url=https://dx.doi.org/10.1016/0550-3213%2880%2990072-3|journal=Nuclear Physics B|volume=176|issue=1|pages=199–215|doi=10.1016/0550-3213(80)90072-3|pmid=|arxiv=|bibcode=1980NuPhB.176..199K |s2cid=|access-date=|url-access=subscription}}</ref>