NationMaster - Encyclopedia: Wilson loop

In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. It is named after Kenneth Wilson. Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Kenneth Geddes Wilson (born June 8, 1936) is an American physicist. ...

In the classical theory, the collection of all Wilson loops contains sufficient information to reconstruct the gauge connection, up to gauge transformation. Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

In quantum field theory, the definition of Wilson loops 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. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... This article is about operators in mathematics, for other kinds of operators see operator (disambiguation). ... The Fock space is an algebraic system (Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of particles. ... Rudolf Haag showed in 1955 that the interaction picture cannot be rigorously defined in quantum field theory, a result now commonly cited as Haags Theorem. ... The mathematical term regularization has two main meanings, both associated with making a function more `regular or smooth. ... Michael Faraday Michael Faraday (September 22, 1791 - August 25, 1867) was a British scientist (a physicist and chemist) who contributed significantly to the fields of electromagnetism and electrochemistry. ...

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. Events and trends Although in the United States and in many other Western societies the 1970s are often seen as a period of transition between the turbulent 1960s and the more conservative 1980s and 1990s, many of the trends that are associated widely with the Sixties, from the Sexual Revolution... Quantum chromodynamics (QCD) is the physical theory describing one of the fundamental forces, the strong interaction. ... This article refers to a particle physics phenomenon. ...

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. Alexander M. Polyakov is a physicist, formerly at the Landau Institute in Moscow, currently at Princeton University. ... String theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ...

Wilson loops played an important role in the formulation of loop quantum gravity, but there they are superseded by spin networks, a certain generalization of Wilson loops. This article needs to be cleaned up to conform to a higher standard of article quality. ... A spin network is a graph whose edges are associated with representations of a Lie group, G and vertices are associated with intertwiners of the edge reps adjacent to it. ...

In particle physics and string theory, Wilson loops are often called Wilson lines, especially Wilson loops around non-contractible loops of a compact manifold. Particles explode from the collision point of two relativistic velocity (100 GeV) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... String theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. ...

An equation

A Wilson line $W C$ is a quantity defined by a path-ordered exponential of a gauge field $A μ$ In theoretical physics, path-ordering is the procedure (or a meta-operator ) of ordering a product of many operators according to the value of one chosen parameter: Here is a permutation that orders the parameters: Examples If an operator is not simply expressed as a product, but as a function... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

Here, $C$ is a contour in space, is the path-ordering operator, and the trace Tr guarantees that the operator 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 an irreducible representation, which means there are infinitely many traces, one for each irrep. The Comet Nucleus Tour (CONTOUR) was a Discovery-class space mission. ... In theoretical physics, path-ordering is the procedure (or a meta-operator ) of ordering a product of many operators according to the value of one chosen parameter: Here is a permutation that orders the parameters: Examples If an operator is not simply expressed as a product, but as a function... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, the term irreducible is used in several ways. ...

Precisely because we're 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. In differential geometry, the connection form describes connection on principal bundles (or vector bundles). ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

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.

In finite temperature QCD, the expectation value of the Wilson line distinguishes between the confined phase and the deconfined phase of the theory. In particle physics, a hadron is a subatomic particle which experiences the strong nuclear force. ... Quark gluon plasma is a phase of Quantum Chromodynamics (QCD) which exists at extremely high temperature and density. ...

See also