Wilson loops

In gauge field theory, a Wilson loop or loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop.

In classical theory, the collection of all Wilson loops contains enough information to reconstruct the gauge connection modulo a gauge transformation.

In quantum field theory, the definition of Wilson loop observables as full-fledged operators in Fock space is a delicate mathematical problem and requires regularization, usually by equipping each loop with a framing. The action of the Wilson loop operators has the interpretation of creating an elementary excitation of the quantum field that is located in the loop. In this way, the "flow tubes" of Faraday become elementary excitations of the quantum electromagnetic field.

Wilson loops were introduced in the 1970s in an attempt at a non-perturbative formulation of quantum chromodynamics (QCD), or at least as a convenient collection of variables to describe the tight coupling regime in QCD. Wilson's ties were invented to solve the problem of confinement, which remains unresolved today.

The fact that tightly coupled quantum gauge field theories have nonperturbative elementary excitations that are loops motivated Aleksandr Poliakov to formulate one of the first string theories, describing the propagation of a quantum elementary loop in the space time.

Wilson's loops play an important role in the formulation of loop quantum gravity, but generalized to Spin Lattice (SN).