Wilson loop

In gauge theory

Gauge theory

Gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

, a Wilson loop (named after Kenneth G. Wilson

Kenneth G. Wilson

Kenneth Geddes Wilson is an American theoretical physicist.As an undergraduate at Harvard, he was a Putnam Fellow. He earned his PhD from Caltech in 1961, studying under Murray Gell-Mann....

) is a gauge-invariant observable obtained from the holonomy

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...

 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.

In quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...

, the definition of Wilson loops observables as bona fide operator

Operator

In mathematics, an operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the...

s on Fock space

Fock space

The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...

 (actually, Haag's theorem

Haag's theorem

Rudolf Haag postulatedthat the interaction picture does not exist in an interacting, relativistic quantum field theory, something now commonly known as Haag's Theorem. The theorem was subsequently proved by a number of different authors...

 states that Fock space does not exist for interacting QFTs) is a mathematically delicate problem and requires regularization

Regularization (physics)

In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...

, usually by equipping each loop with a framing.

In gauge theory

Gauge theory

Gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

, a Wilson loop (named after Kenneth G. Wilson

Kenneth G. Wilson

Kenneth Geddes Wilson is an American theoretical physicist.As an undergraduate at Harvard, he was a Putnam Fellow. He earned his PhD from Caltech in 1961, studying under Murray Gell-Mann....

) is a gauge-invariant observable obtained from the holonomy

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...

 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.

In quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...

, the definition of Wilson loops observables as bona fide operator

Operator

In mathematics, an operator is a type of function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the...

s on Fock space

Fock space

The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...

 (actually, Haag's theorem

Haag's theorem

Rudolf Haag postulatedthat the interaction picture does not exist in an interacting, relativistic quantum field theory, something now commonly known as Haag's Theorem. The theorem was subsequently proved by a number of different authors...

 states that Fock space does not exist for interacting QFTs) is a mathematically delicate problem and requires regularization

Regularization (physics)

In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...

, 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

Michael Faraday

Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....

's "flux tubes" become elementary excitations of the quantum electromagnetic field.

Wilson loops were introduced in the 1970s in an attempt at a nonperturbative formulation of quantum chromodynamics

Quantum chromodynamics

In theoretical physics, Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

 (QCD), or at least as a convenient collection of variables for dealing with the strongly-interacting regime of QCD. The problem of confinement

Colour confinement

Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons...

, 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

String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...

, which described the propagation of an elementary quantum loop in spacetime.

Wilson loops played an important role in the formulation of loop quantum gravity

Loop quantum gravity

Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...

, but there they are superseded by spin network

Spin network

In physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in quantum mechanics. From a mathematical perspective, the diagrams are a concise way to represent multilinear functions and functions between representations of...

s, a certain generalization of Wilson loops.

In particle physics

Particle physics

Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called high energy physics, because many elementary particles do not occur under normal circumstances in nature, but can be created and detected...

 and string theory

String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...

, Wilson loops are often called Wilson lines, especially Wilson loops around non-contractible loops of a compact manifold.

An equation

The Wilson line variable (or better Wilson loop variable, since one is always dealing with closed lines) 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

Path-ordering

In theoretical physics, path-ordering is the procedure of ordering a product of many operators according to the value of one chosen parameter:Here is a permutation that orders the parameters:For example:- Examples :...

 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

Trace

Trace may refer to:Mathematics, computing and electronics:* Trace of a square matrix or a linear transformation* Trace of a surgery on a manifold* Trace class, a certain set of operators in a Hilbert space...

 under cyclic permutation

Cyclic permutation

A cyclic permutation is built from one or more sets of elements in cyclic order.The notion cyclic permutation is used in different, but related ways:- Definition 1 :right|mapping of permutation...

s guarantees that is invariant under gauge transformations. Note that the quantity being traced over is an element of the gauge Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

 and the trace is really the character

Character (mathematics)

In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...

 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

Connection form

In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....

 over a principal G-bundle

Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G of a space X with a group G...

, the equation above really ought to be "read" as the parallel transport

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...

 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, the Wilson loop variable characterizes the confinement

Confinement

Confinement may refer to either* civil confinement for psychiatric patients* color confinement, the physical principle explaining the non-observation of color charged particles like free quarks* solitary confinement, a strict form of imprisonment...

 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 much-debated quark-gluon plasma

Quark-gluon plasma

A quark–gluon plasma is a phase of quantum chromodynamics which exists at extremely high temperature and/or density. This phase consists of free quarks and gluons, which are the basic building blocks of matter...

.