In physics, lattice gauge theory is the study of the behaviour of lattice model gauge theories. That is, it is the study of gauge theories on a spacetime that has been discretized onto a lattice. Although most lattice gauge theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, that one will be able to recover the behaviour of the continuum theory. Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ... In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... See lattice for other meanings of this term, both within and without mathematics. ... In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable therefore implies solvablility. ...

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. 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 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. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In lattice theories, fermion fields experience (at least) a doubling of the number of particle types in a lattice. ... 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 n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... 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... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... 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. Unsolved problems in physics: What causes anything to have mass? Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to. ... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations. ... Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. ...

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. ... Colour confinement (often just confinement) is the physics phenomenon that color charged particles (such as quarks) cannot be isolated. ... 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 QGP is formed at the collision point of two relativistically accelerated gold ions in the center of the STAR detector at the relativistic heavy ion collider at the Brookhaven national laboratory. ...

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. ... This article or section is in need of attention from an expert on the subject. ... 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
• I. Montvay and G. Münster, Quantum Fields on a Lattice
• H. Rothe, Lattice Gauge Theories, An Introduction
• J. Smit, Introduction to Quantum Fields on a Lattice
• The Chroma Library for Lattice Field Theory