Lattice gauge theory

A lattice gauge theory is a gauge theory defined on a discrete space-time. Lattice gauge theories are among the few ways to perform non- perturbative calculations in quantum field theories.

The basic idea is to regularize by introducing a minimum distance in space and time the theory, so that no divergences occur at high energies. This minimum distance corresponds to a cut-off energy (English Cut -Off) in momentum space. A constant reduction of the minimum lattice spacing corresponds to the transition to the original theory in continuous space by removing the highest energies in momentum space.

In order to enable simulations of lattice gauge theories on computers, is additionally performed a Wick rotation generally, which one goes to the Euclidean space. Then, a relationship exists for statistical physics, and it may be the powerful tool of the Monte Carlo simulation are used.

Of special importance was the method in the framework of quantum chromodynamics ( QCD), so that one, unless otherwise stated, usually the lattice QCD has in mind. Because the lattice regularization is a non- perturbative regularization, can be carried out in lattice gauge theories and calculations for low energies, which are not accessible to perturbation theory. This makes it possible, inter alia, the masses of hadrons, d quark states bound h, examined by thermodynamic quantities or of important topological excitations ( monopoles, instantons and solitons ).

In addition to the QCD and other gauge theories and spin systems are studied on the lattice, in particular those with non- Abelian gauge group ( general Yang-Mills theories similar to QCD).


Introduced by Kenneth Wilson in 1974 lattice theory of quantum chromodynamics discretized the effect of QCD on a four-dimensional cubic lattice with lattice spacing. An important principle in the construction Wilson grating theory is that the effect is explicitly gauge invariant even when finite lattice spacing. Is further selected Wilson effect that results in the effect in the continuum threshold. Usually one considers the formulation for the gauge sector separately from that for fermions, since the transfer of the chiral symmetry of the Fermionfelder on the grid is a different problem area.

Pure gauge theory

For the discretization of the Yang-Mills action, which describes the dynamics of the gauge bosons, we define so-called link variables connecting adjacent grid points,

Here, the gauge fields are elements of the adjoint representation of the algebra of the gauge group of QCD, SU (3). The link variables themselves elements of the gauge group, ie SU (3) matrices, each of which connects two adjacent grid points. In terms of the differential geometry, they can be regarded as a finite parallel transport.

The gauge field part of the effect can now be represented as the trace over closed loops of link variables. Each track of such Wilson loops is gauge invariant. A simple calibration effect can therefore be written as follows:

Here, ( the so-called plaques variables) are defined as the smallest to the closed rectangular loop corresponding sizes

Analogously to the geometry of a square that is indexed, for example, with a positive sense of rotation by the four numbers 1 2 3 and 4.

Instead of the coupling constant one often uses the inverse gauge coupling

Since the shape of the effect is determined only by the continuum limit, the above mentioned calibration effect, the so-called Wilson or Plaque - effect, not unique, but can by terms which vanish in the continuum limit, are modified. This observation is used to construct improved effects at a faster continuum approach.

Fermions on the lattice

While the link variables connecting two lattice points Fermionfelder are defined on these points. This allows gauge-invariant combinations of the form with form that can be used as building blocks of the discretized covariant derivative.

Substituting now the derivatives in the Dirac effect by finite differences, we obtain a naive discretization of the theory, which not only describes a single fermion, but sixteen ( with the number of dimensions). This phenomenon is known as the Doppler and problem related to the realization of the chiral symmetry on the lattice. In fact, says the Nielsen - Ninomiya theorem, that at the same time on the grid no Dirac operator with the correct continuum limit Doppler- free, local, translationally invariant and can be chiral symmetry. To take into account the physically correct Doppler problem, various kinds of Fermiondiskretisierung be used which are described below.

Wilson fermions

To eliminate the Doppler one can incorporate additional terms in the action which give the unphysical Fermionmoden an additional mass. In forming the continuum limit de-couple the resulting Doppler modes of the theory, since their mass diverges. This is the approach of Wilson Fermionwirkung:

Which can be freely chosen as a prefactor in front of the newly introduced Wilson term f denotes the flavor degree of freedom of the fermions and r. For r = 0 we obtain the original discretized naive fermions with doublers, while for the usual choice r = 1 are eliminated Doppler as described above.

For finite a, however, the chiral symmetry is given by the Wilson term explicitly broken and is only produced in the continuum limit again. A practical consequence is that the lattice artifacts differently than for other effects already occur in linear order of the lattice spacing. To resolve this problem, in numerical simulations almost exclusively so-called enhanced effects are used. The most widely used in this connection are the so-called clover fermions, for a further term is added to the effect of, the free parameters may be selected so that the leading grid artifacts are eliminated. In addition, look for Wilson fermions with a modified mass term under the name twisted mass fermions use.

Staggered fermions

In addition to the Wilson fermions in particular the so-called staggered fermions (English staggered fermions ) can be used. These use a Spindiagonalisierung by the number of Doppler to reduce by a factor of 4. To describe a theory with exactly one Fermionart, has a theoretically controversial process, known as rooting, are applied.

Chiral fermions

In the continuum theory of the Dirac operator D of a chirally symmetric theory satisfies the relationship with γ5, as well as D, are to be provided from the Dirac theory as known. The Wilson term explicitly breaks this symmetry. This can be circumvented by a weaker definition of chiral symmetry on the lattice,


Wherein R is a local grid operator. ~ A behavior leads to an effective smoothing of the interfering terms ~ 4 / a of Wilson's functional.

By this substitution to obtain the so-called Overlapoperator and from the Wilson equation arises Ginzparg the Wilson equation. In addition to exact solutions, there are also a number of common fermions whose Dirac operator only approximately satisfies the Ginsparg -Wilson equation. The best known are the domain -wall fermions, which (in the case of infinite expansion of a fifth dimension ) correspond to the Overlapoperator. In practical simulations, however, this dimension is always finite.

Alternative formulations

The discretizations mentioned above are the methods most commonly used to treat fermions on the lattice. There are also other, such as the fermions with minimal doubling ( minimally doubled fermions ), which achieve a minimization of the Doppler problem a modification of the lattice geometry. Another variant is the breaking of translational invariance by introducing an additional dimension, as happens in the domain -wall fermions.

Context of physical quantities with simulation parameters

In a lattice QCD simulation, you can set a number of parameters: the number of grid points in spatial and temporal direction, the grating coupling and possibly quark mass parameter and parameters which lead to the improvement of the theoretical continuum behavior. To translate such a " Setup" dimensionless numbers in physical units, ie by dimensional measures, such as the lattice spacing ( in fm ) or hadron masses to obtain ( in MeV/c2 ), have selected physical properties ( such as mass or decay constant of the pion ) are fixed to the reduction of the scale. All other calculated quantities are then predictions of lattice QCD to the given parameters.

The computer power required increases with decreasing mass, so that the achievement of physical quark masses without further extrapolation still requires a huge effort and not has been reached with all Fermiondiskretisierungen. In addition, it is important to keep the systematic effects that are due to the extrapolation to vanishing lattice spacing and infinite volume under control.

The asymptotic freedom of QCD ensures with its fixed point in the flow of the coupling constant, that the continuum limit of vanishing coupling (or ) is reached.


As actual birth of lattice QCD is now publishing the work of the physicist Kenneth Wilson in 1974, which very soon became one of the core areas of the then state of research and triggered a rapid development of the method.

The method of lattice QCD is analogous to special spin models, which were set up in 1971 in solid-state theoretical context by Franz Wegner. This lattice - spin models are, like in QCD, by a local gauge invariance and by an analog to the gauge field energy term. Nowadays, however, the solid-state theoretical context, the theory largely forgotten (but see the paper by J. Kogut ).

Although quantum chromodynamics is a main field of application of lattice gauge theory, there are even in high -energy physics experiments with multigrid methods, which go beyond the QCD, such as the Higgs mechanism.

Selected Results

One advantage of simulations of lattice gauge theories is that particularly gauge-invariant quantities are available. This led to a calculation of all meson and baryon ground states containing up, down or strange quarks. A typical result (see the graph) shows that not only the particles, quarks and antiquarks, but also the " flux tubes " of the gluon fields are important in a meson.

Such calculations also collective effects can be studied, which could be related to the confinement phenomenon. This can, for example, be topological excitations, such as instantons, monopoles and solitons, or Perkolationseffekte the center of the gauge group. These and other tasks are partly supported by the German Research Foundation and / or the EU by about regional funding.

It can also calculations of QCD at high temperatures lead to study the transition to the quark -gluon plasma, which was measured in experiments at particle accelerators above about 1.2 x1012 Kelvin.