Wilson loop

Last updated

In gauge theory, a Wilson loop (named after Kenneth G. Wilson) is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. In the classical theory, the collection of all Wilson loops contains sufficient information to reconstruct the gauge connection, up to gauge transformation. [1]



In quantum field theory, the definition of Wilson loop observables as bona fide operators on Fock spaces is a mathematically delicate problem and requires regularization, usually by equipping each loop with a framing. The action of Wilson loop operators has the interpretation of creating an elementary excitation of the quantum field which is localized on the loop. In this way, Faraday's "flux tubes" become elementary excitations of the quantum electromagnetic field.

Wilson loops were introduced in 1974 in an attempt at a nonperturbative formulation of quantum chromodynamics (QCD), or at least as a convenient collection of variables for dealing with the strongly interacting regime of QCD. [2] The problem of confinement, which Wilson loops were designed to solve, remains unsolved to this day.

The fact that strongly coupled quantum gauge field theories have elementary nonperturbative excitations which are loops motivated Alexander Polyakov to formulate the first string theories, which described the propagation of an elementary quantum loop in spacetime.

Wilson loops played an important role in the formulation of loop quantum gravity, but there they are superseded by spin networks (and, later, spinfoams), a certain generalization of Wilson loops.

In particle physics and string theory, Wilson loops are often called Wilson lines, especially Wilson loops around non-contractible loops of a compact manifold.

An equation

The Wilson loop variable is a quantity defined by the trace of a path-ordered exponential of a gauge field transported along a closed line C:

Here, is a closed curve in space, is the path-ordering operator. Under a gauge transformation


where corresponds to the initial (and end) point of the loop (only initial and end point of a line contribute, whereas gauge transformations in between cancel each other). For SU(2) gauges, for example, one has ; is an arbitrary real function of , and are the three Pauli matrices; as usual, a sum over repeated indices is implied.

The invariance of the trace under cyclic permutations guarantees that is invariant under gauge transformations. Note that the quantity being traced over is an element of the gauge Lie group and the trace is really the character of this element with respect to one of the infinitely many irreducible representations, which implies that the operators don't need to be restricted to the "trace class" (thus with purely discrete spectrum), but can be generally hermitian (or mathematically: self-adjoint) as usual. Precisely because we're finally looking at the trace, it doesn't matter which point on the loop is chosen as the initial point. They all give the same value.

Actually, if A is viewed as a connection over a principal G-bundle, the equation above really ought to be "read" as the parallel transport of the identity around the loop which would give an element of the Lie group G.

Note that a path-ordered exponential is a convenient shorthand notation common in physics which conceals a fair number of mathematical operations. A mathematician would refer to the path-ordered exponential of the connection as "the holonomy of the connection" and characterize it by the parallel-transport differential equation that it satisfies.

At T=0, where T corresponds to temperature, the Wilson loop variable characterizes the confinement or deconfinement of a gauge-invariant quantum-field theory, namely according to whether the variable increases with the area, or alternatively with the circumference of the loop ("area law", or alternatively "circumferential law" also known as "perimeter law").

In finite-temperature QCD, the thermal expectation value of the Wilson line distinguishes between the confined "hadronic" phase, and the deconfined state of the field, e.g., the quark–gluon plasma.

See also

Related Research Articles

Quantum chromodynamics Theory of the strong nuclear interactions

In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years.

Quantum field theory Theoretical framework combining classical field theory, special relativity, and quantum mechanics

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics, but not general relativity's description of gravity. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.

Color charge property of quarks and gluons that is related to the particles strong interactions in the theory of quantum chromodynamics

Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).

Linking number

In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. The linking number is always an integer, but may be positive or negative depending on the orientation of the two curves.

Lattice gauge theory gauge theory implemented as a lattice field theory

In physics, lattice gauge theory is the study of gauge theories on a spacetime that has been discretized into a lattice.

Yang–Mills theory Physical theory unifying the electromagnetic, weak and strong interactions

Yang–Mills theory is a gauge theory based on a special unitary group SU(N), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces as well as quantum chromodynamics, the theory of the strong force. Thus it forms the basis of our understanding of the Standard Model of particle physics.

Propagator Function in quantum field theory showing probability amplitudes of moving particles

In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions.

Wheeler–DeWitt equation A field equation, part of a theory that attempts to combine quantum mechanics and general relativity

The Wheeler–DeWitt equation is a field equation. It is part of a theory that attempts to combine mathematically the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group".

In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of representatives is a submanifold and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish. The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity.

Mathematical formulation of the Standard Model The mathematics of a particle physics model

This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as containing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs particle.

In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion.

In theoretical physics Pohlmeyer charge, named for Klaus Pohlmeyer, is a conserved charge invariant under the Virasoro algebra or its generalization. It can be obtained by expanding the holonomies

Canonical quantum gravity

In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity. It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.

In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.

In theoretical physics, the BRST formalism, or BRST quantization denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.

Gauge theory Physical theory with fields invariant under the action of local "gauge" Lie groups

In physics, a gauge theory is a type of field theory in which the Lagrangian does not change under local transformations from certain Lie groups.

Gluon field strength tensor second order tensor field characterizing the gluon interaction between quarks

In theoretical particle physics, the gluon field strength tensor is a second order tensor field characterizing the gluon interaction between quarks.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

Gluon field four vector field characterizing the propagation of gluons in the strong interaction between quarks

In theoretical particle physics, the gluon field is a four vector field characterizing the propagation of gluons in the strong interaction between quarks. It plays the same role in quantum chromodynamics as the electromagnetic four-potential in quantum electrodynamics – the gluon field constructs the gluon field strength tensor.

Loop representation in gauge theories and quantum gravity

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. 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. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.


  1. Giles, R. (1981). "Reconstruction of Gauge Potentials from Wilson loops". Physical Review D . 24 (8): 2160. Bibcode:1981PhRvD..24.2160G. doi:10.1103/PhysRevD.24.2160.
  2. Wilson, K. (1974). "Confinement of quarks". Physical Review D . 10 (8): 2445. Bibcode:1974PhRvD..10.2445W. doi:10.1103/PhysRevD.10.2445.