Wilson Loops
In standard model physics, by the time one arrives at QCD, one has already hidden away a lot of abstract algebra, which means that at some point, topological objects become a good way to cheat the math, especially since most of them will disappear in integrals anyway. This makes solving matrix or tensor models much easier in practice, once the model has been expressed in terms of Wilson loops. Because of where these tools enter the model, all of the ideas here have the underlying gauge transformations. Keeping this in mind, the propagators should be fixed first, before the loop will make sense. The partial derivative is of course replaced by the standard covariant derivative. The propagator in an external electromagnetic field is expressed along some set axis and normalized time τ.
Transverse components the electromagnetic field describe photons, and longitudinal components are related to gauging the phase of a wave function. Since the wave-function phase is itself unobservable, it needs to be made visible using the phase differences (via interference phenomena), which depends on the value of the phase factor for some path along which the parallel transport occurs. Using the Aharonov-Bohm experiment, the explicit dependence of the interference picture from an external electric current is made visible.
where the closed contours Γ is decomposed into the positive and negative components, though it doesn't depend on their shapes, but rather on the magnetic flux through the solenoid. The Aharanov-Bohm effect can be used as a generalized approach to effects of external fields. Assuming the usual gauge invariance for Lie groups, the non-Abelian phase factors are where Wilson loops become relevant. The Abelian extension is characterized by path-ordering, and unitarity. Due to the traits of parallel transporters, it follows that the matrix representation U is gauge invariant.
In placing a Wilson loop into a space, it's often convenient to place it onto a lattice, and use the lattice points for reference. Matter/quark fields are attributed to those lattice sites, so a continuous field is approximated by moving the lattice sites closer together. This eventually moves us toward the coarse-graining method which is something for a different time. The gauge field is attributed to the links of the lattice. The link itself is characterized by a coordinate and a direction. This makes sense, since the in the path-ordered integral depend heavily on the orientation. The lattice phase factors then are analogous to Wilson loops, as is the process of using them to construct the contours from the lattice links. The phase factor for the simplest closed contour on a lattice is the oriented boundary of a plaquette.
The simplest Wilson action on a lattice then is
The summation is over all the elementary plaquettes of a lattice, regardless of orientation. Using the gauge-theory-associated partition function, the measure is left to be identified. Preserving the gauge invariance at finite lattice spacing, the integration is over the "Haar measure" which is an invariant group measure under multiplication by an arbitrary group element from either side. For SU(2), that is
The lattice phase factors are associated with paths drawn on the lattice. For some contour C = {x; μᵢ}, the lattice phase factor U(C) is given first for the links with negative direction, and for closed contours Σμᵢ = 0, the gauge invariant trace of the phase factor give rise to the formal definition for Wilson loops via its average.
Exponential dependence of the Wilson loop average on the area of the minimal surface is customarily assumed to hold for loops of large area in pure SU(3) gauge theory. The quarks then are confined. There then is no need for physical in/out-quark states. This makes the Wilson's confinement criterion, meaning that physical amplitudes don't have quark singularities when the Wilson criterion is satisfied. For a linear potential, it means that lines of force between static quarks are depicted for linear and Coulomb interaction potentials. E(R) = KR, with the coefficient K is called the string tension, contracting the gluon field between quarks into a tube/string with energy proportional to its length. This string stretches with the distance between quarks and prevents them from moving apart to macroscopic distances.
The asymptotic scaling is - as it usually is in QFT - a different sort of approximation for small coupling, and handled mostly numerically (discounting any involvement of Feynman diagrams). A different approach for the 1/N-expansion is through a latticed variant of Feynman diagrams called index/ribbon-graphs, which introduces the direction of vertices to the standard Feynman diagram, as per construction via lattice plaquettes.