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In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops.They encode all gauge information of the theory, allowing for the construction of loop representations which fully describe gauge theories in terms of these loops.
The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states (Gauss gauge invariant states).
A lattice gauge action can be constructed from any discretized Wilson loop. As long as the loops are suitably averaged over orientations and translations in spacetime to give rise to the correct symmetries, the action will reduce back down to the continuum result. The advantage of using plaquettes is its simplicity and that the action lends ...
This is the case of the Wilson loop, which is defined as a path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection. The parameter σ that determines the ordering is a parameter describing the contour , and because the contour is closed, the Wilson loop must be defined as a trace in order to be ...
The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge ...
In quantum field theory, the Polyakov loop is the thermal analogue of the Wilson loop, acting as an order parameter for confinement in pure gauge theories at nonzero temperatures. In particular, it is a Wilson loop that winds around the compactified Euclidean temporal direction of a thermal quantum field theory .
With Wilson loops as a basis, any Gauss gauge invariant function expands as, [] = [] []. This is called the loop transform and is analogous to the momentum representation in quantum mechanics (see Position and momentum space).
The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The action integral (path integral) of the field theory in physics is viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on M.