- Lattice gauge theory
In

physics ,**lattice gauge theory**is the study of gauge theories on a spacetime that has been discretized onto a lattice. Although most lattice gauge theories are notexactly 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, one finally will be able to recover the behaviour of the continuum theory.In lattice gauge theory, the spacetime is

Wick rotated intoEuclidean 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 withfermion doubling , though. SeeWilson-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 theparallel 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 3special unitary matrix defined on each link. The faces of the lattice are called plaquettes. TheYang-Mills action is rewritten usingWilson loop s over plaquettes (it's simply a character evaluated over the composition of link variables around the plaquette) in such a way that the limit $a\; o\; 0$ formally gives the original continuous action.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 acompact Lie group G to each edge given an orientation for it called U. Basically, the assignment for an edge in a given orientation is thegroup 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 faithfulirreducible representation ρ of G, the latticeYang-Mills action is:$S=sum\_F\; -Re\{chi^\{(\; ho)\}(U(e\_1)cdots\; U(e\_n))\}$

(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 . e_{1}, ..., 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.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 $a^2$, 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 theMonte Carlo method is used to estimate the quantity. Random configurations (values of the gauge fields) are generated with probabilities proportional to $e^\{-eta\; S\}$, where $S$ is the lattice action for that configuration and $eta$ 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 $a=0$.Lattice gauge theory is a particularly important tool for

quantum chromodynamics (QCD). The discretized version of QCD is calledLattice QCD . QCD confinement has been shown inMonte Carlo simulation s. Deconfinement at high temperature leads to the formation of aquark-gluon plasma .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.**Other Applications**Originally, solvable two-dimensional lattice gauge theories had already been introduced in 1969 as models with interesting statistical properties by the theorist Franz Wegner, who worked in the field of phase transitions.

**ee also***

Lattice field theory andlattice QCD

*Hamiltonian lattice gauge theory

*Path integral formulation

*quenched approximation **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"

* [*http://usqcd.jlab.org/usqcd-docs/chroma/ The Chroma Library for Lattice Field Theory*]

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### Look at other dictionaries:

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