Wilson loop
Encyclopedia
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.
In quantum field theory
, the definition of Wilson loop observables as bona fide
operators on Fock space
(actually, Haag's theorem
states that Fock space does not exist for interacting QFTs) 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 the 1970s 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. 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, but there they are superseded by spin network
s, 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.
(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
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 permutation
s 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, 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 much-debated quark-gluon plasma
.
Gauge theory
In physics, 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 and Nobel Prize winner.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 parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the definition of Wilson loop observables as bona fide
Bona Fide
Bona Fide is a studio album from rock band Wishbone Ash. It is the first studio album in six years and is the only studio album to feature guitarist Ben Granfelt...
operators 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...
states that Fock space does not exist for interacting QFTs) is a mathematically delicate problem and requires regularization
Regularization (physics)
-Introduction: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. The two types of hadrons are the mesons and the baryons...
, 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 an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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 existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
and string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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-orderingPath-ordering
In theoretical physics, path-ordering is the procedure of ordering a product of many operators according to the value of one chosen parameter:...
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 (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
under cyclic permutation
Cyclic permutation
A cyclic permutation or circular permutation is a permutation 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 groupLie 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 the 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 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 or quark soup is a phase of quantum chromodynamics which exists at extremely high temperature and/or density. This phase consists of asymptotically free quarks and gluons, which are several of the basic building blocks of matter...
.