Editing Wilson loop (section)

===Mandelstam identities===

Gauge groups that admit fundamental representations in terms of <math>N\times N</math> matrices have Wilson loops that satisfy a set of identities called the Mandelstam identities, with these identities reflecting the particular properties of the underlying gauge group.<ref>{{cite journal|last1=Mandelstam|first1=S.|authorlink1=Stanley Mandelstam|date=1968|title=Feynman Rules for Electromagnetic and Yang–Mills Fields from the Gauge-Independent Field-Theoretic Formalism|url=https://link.aps.org/doi/10.1103/PhysRev.175.1580|journal=Phys. Rev.|volume=175|issue=5|pages=1580–1603|doi=10.1103/PhysRev.175.1580|pmid=|arxiv=|bibcode=1968PhRv..175.1580M |s2cid=|access-date=|url-access=subscription}}</ref> The identities apply to loops formed from two or more subloops, with <math>\gamma = \gamma_2 \circ \gamma_1</math> being a loop formed by first going around <math>\gamma_1</math> and then going around <math>\gamma_2</math>.

The Mandelstam identity of the first kind states that <math>W[\gamma_1\circ \gamma_2] = W[\gamma_2 \circ \gamma_1]</math>, with this holding for any gauge group in any dimension. Mandelstam identities of the second kind are acquired by noting that in <math>N</math> dimensions, any object with <math>N+1</math> [[antisymmetric tensor|totally antisymmetric]] indices vanishes, meaning that <math>\delta^{a_1}_{[b_1}\delta^{a_2}_{b_2}\cdots \delta^{a_{N+1}}_{b_{N+1}]} = 0</math>. In the fundamental representation, the holonomies used to form the Wilson loops are <math>N\times N</math> [[matrix representation]]s of the gauge groups. Contracting <math>N+1</math> holonomies with the [[Kronecker delta|delta functions]] yields a set of identities between Wilson loops. These can be written in terms the objects <math>M_K</math> defined iteratively so that <math>M_1[\gamma] = W[\gamma]</math> and

:<math>
(K+1)M_{K+1}[\gamma_1, \dots, \gamma_{K+1}] = W[\gamma_{K+1}]M_K[\gamma_1,\dots, \gamma_K] - M_K[\gamma_1 \circ \gamma_{K+1},\gamma_2, \dots, \gamma_K] -\cdots - M_K[\gamma_1, \gamma_2, \dots, \gamma_K\circ \gamma_{K+1}].
</math>

In this notation the Mandelstam identities of the second kind are<ref>{{cite book|last=Gambini|first=R.|author-link=|date=2008|title=Loops, Knots, Gauge Theories|url=|doi=|location=|publisher=|chapter=3|pages=63–67|isbn=978-0521654753}}</ref>

:<math>
M_{N+1}[\gamma_1, \dots, \gamma_{N+1}] = 0.
</math>

For example, for a <math>\text{U}(1)</math> gauge group this gives <math>W[\gamma_1]W[\gamma_2] = W[\gamma_1\circ \gamma_2]</math>.

If the fundamental representation are matrices of unit [[determinant]], then it also holds that <math>M_N(\gamma, \dots, \gamma)=1</math>. For example, applying this identity to <math>\text{SU}(2)</math> gives

:<math>
W[\gamma_1]W[\gamma_2] = W[\gamma_1\circ \gamma_2^{-1}]+W[\gamma_1\circ \gamma_2].
</math>

Fundamental representations consisting of [[unitary matrix|unitary matrices]] satisfy <math>W[\gamma] = W^*[\gamma^{-1}]</math>. Furthermore, while the equality <math>W[I] = N</math> holds for all gauge groups in the fundamental representations, for unitary groups it moreover holds that <math>|W[\gamma]|\leq N</math>.