It has been suggested that this article or section be merged into Lattice field theory. (Discuss)

Lattice gauge theory is a method to deal with gauge theory that is useful for computer-assisted calculations. In lattice gauge theory, the spacetime is Wick rotated into Euclidean space, discretized and replaced by a lattice with lattice spacing equal to a. The quark fields are only defined at the elements of the lattice. There are problems with fermion doubling, though. See Wilson-Ginsparg action. Wikipedia does not have an article with this exact name. ... It has been suggested that Lattice gauge theory be merged into this article or section. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... See lattice for other meanings of this term, both within and without mathematics. ... In lattice theories, fermion fields experience (at least) a doubling of the number of particle types in a lattice. ...

Instead of a vector potential as in the continuum case, the gauge fields variables are defined on the links of the lattice and correspond to the parallel transport along the edge which takes on values in the Lie group. Hence to simulate QCD, for which the Lie group is SU(3), there is a 3 by 3 special unitary matrix defined on each link. The faces of the lattice are called plaquettes. The Yang-Mills action is rewritten using Wilson loops over plaquettes (it's simply a character evaluated over the composition of link variables around the plaquette) in such a way that the limit formally gives the original continuous action. 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. ... The initialism QCD can mean: Quantum chromodynamics Quintessential Player, formerly known as Quintessential CD Quality, Cost, Delivery, A three-letter acronym used in lean manufacturing This page concerning a three-letter acronym or abbreviation is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field of complex numbers, with the group operation that of matrix multiplication. ... In mathematics, a unitary matrix is a n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In gauge theory, a Wilson loop is a gauge-invariant observable obtained from the holonomy of the gauge connection around a given loop. ...

More precisely, we have a lattice with vertices, edges and faces. In lattice theory, the alternative terminology sites, links and plaquettes for vertices, edges and faces is often used. This reflects the origin of the field in solid state physics. While each edge happens to have no intrinsic orientation, to define the gauge variables, we assign an element of a compact Lie group G to each edge given an orientation for it called U. Basically, the assignment for an edge in a given orientation is the group inverse of the assignment to the same edge in the opposite orientation. Likewise, the plaquettes have no intrinsic orientations, but have to be temporarily given an orientation for computational purposes. Given a faithful irreducible representation ρ of G, the lattice Yang-Mills action is In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... This article just presents the basic definitions. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... In mathematics, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ(g). ... In mathematics, the term irreducible is used in several ways. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

(the sum over all lattice sites of the (real component of the) Wilson loop). Here, χ is the character (trace) and the real component is redundant if ρ happens to be a real or pseudoreal representation. e₁, ..., e_n are the n edges of the Wilson loop in sequence. The nice thing about being real is even if the orientation of a Wilson loop is flipped, its contribution to the action remains unchanged. 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 (the diagonal from the upper left to the lower right) of A, i. ... In mathematics and theoretical physics, a real representation is a group representation that is equivalent to its complex conjugate and that also allows the matrices representing the group elements to be real — unlike a pseudoreal representation (symplectic representation). ... In mathematics and theoretical physics, a pseudoreal representation is a group representation that is equivalent to its complex conjugate, but that is not a real representation. ...

There are many possible lattice Yang-Mills actions, depending on which Wilson loop is used in the above formula. The simplest is the Wilson action, in which the Wilson loop is just a plaquette. A disadvantage of the Wilson action is that the difference between it and the continuous action is proportional to the lattice spacing a. It is possible to use more complicated Wilson loops to form actions where this difference is proportional to a², thus making computations more accurate. These are known as improved actions.

To calculate a quantity (such as the mass of a particle) in lattice gauge theory, it should be calculated for every possible value of the gauge field on each link, and then averaged. In practice this is impossible. Instead the Monte Carlo method is used to estimate the quantity. Random configurations (values of the gauge fields) are generated with probabilities proportional to e ^{− βS}, where S is the lattice action for that configuration and β is related to the lattice spacing a. The quantity is calculated for each configuration. The true value of the quantity is then found by taking the average of the value from a large number of configurations. To find the value of the quantity in the continuous theory this is repeated for various values of a and extrapolated to a = 0. Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. ... Monte Carlo methods are a class of computational algorithms for simulating the behavior of various physical and mathematical systems. ... The word probability derives from the Latin probare (to prove, or to test). ... In mathematics, extrapolation is a type of interpolation. ...

Lattice gauge theory is a particularly important tool for quantum chromodynamics (QCD). The discretized version of QCD is called Lattice QCD. QCD confinement has been shown in Monte Carlo simulations. Deconfinement at high temperature leads to the formation of a quark-gluon plasma. Quantum chromodynamics (QCD) is the theory of the strong interaction, a fundamental force describing the interactions of the quarks and gluons found in nucleons (such as the proton and neutron). ... It has been suggested that lattice field theory be merged into this article or section. ... This article is about a particle physics phenomenon. ... Monte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. ... A Quark-gluon plasma is a phase of quantum chromodynamics (QCD) which exists at extremely high temperature and density. ...

Lattice gauge theory has been shown to be exactly dual to spin foam models provided that the only Wilson loops appearing in the action are over plaquettes. In physics, a spin foam is a four-dimensional graph made out of two-dimensional faces that represents one of the configurations that must be summed to obtain Feynmans path integral (functional integration) describing the alternative formulation of quantum gravity known as loop gravity or loop quantum gravity. ...

See also

• Lattice field theory and lattice QCD
• Hamiltonian lattice gauge theory
• Path integral formulation
• quenched approximation

It has been suggested that Lattice gauge theory be merged into this article or section. ... It has been suggested that lattice field theory be merged into this article or section. ... Hamiltonian lattice gauge theory is a calculational approach to gauge theory and a special case of lattice gauge theory in which the space is discretized but time is not. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In particle physics, quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermions in Feynman diagrams are neglected. ...

References and external links

• M. Creutz, Quarks, gluons and lattices
• G. Munster and I. Montvay, Lattice Gauge Theory
• J. Smit, Lattice Gauge Theory
• A standard library of algorithms for lattice gauge theory
• The Chroma Library for Lattice Field Theory