Lattice gauge theory

# Lattice gauge theory

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, lattice gauge theory is the study of gauge theories
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

on a spacetime that has been discretized into a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

. Gauge theories are important in particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, and include the prevailing theories of elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

s: quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

(QCD) and the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

. Non-perturbative gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional path integral
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...

, which is computationally intractable. By working on a discrete spacetime, the path integral becomes finite-dimensional, and can be evaluated by stochastic simulation
Stochastic simulation
Stochastic simulation algorithms and methods were initially developed to analyse chemical reactions involving large numbers of species with complex reaction kinetics. The first algorithm, the Gillespie algorithm was proposed by Dan Gillespie in 1977...

techniques such as the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered intuitively. A mathematical proof of this fact is lacking.

## Basics

In lattice gauge theory, the spacetime is Wick rotated into Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

and discretized into a lattice with sites separated by distance and connected by links. In the most commonly-considered cases, such as lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....

, fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

fields are defined at lattice sites (which leads to fermion doubling
Fermion doubling
The fermion doubling problem is a problem that is encountered when naively trying to put fermionic fields on a lattice. It consists in the appearance of spurious states, such that one ends up having 2d fermionic particles for each original fermion...

), while the gauge fields
Gauge boson
In particle physics, gauge bosons are bosonic particles that act as carriers of the fundamental forces of nature. More specifically, elementary particles whose interactions are described by gauge theory exert forces on each other by the exchange of gauge bosons, usually as virtual particles.-...

are defined on the links. That is, an element U of the compact
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

G is assigned to each link. Hence to simulate QCD, with Lie group SU(3)
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

, there is a 3×3 special unitary matrix defined on each link. The link is assigned an orientation, with the inverse element
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...

corresponding to the same link with the opposite orientation.

## Yang–Mills action

The Yang–Mills action is written on the lattice using Wilson loop
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...

s (named after Kenneth G. Wilson
Kenneth G. Wilson
Kenneth Geddes Wilson is an American theoretical physicist and Nobel Prize winner.As an undergraduate at Harvard, he was a Putnam Fellow. He earned his PhD from Caltech in 1961, studying under Murray Gell-Mann....

), so that the limit formally reproduces the original continuum action. Given a faithful
Faithful representation
In mathematics, especially in the area of abstract algebra known as representation theory, 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.In more abstract language, this means...

irreducible representation ρ of G, the lattice Yang-Mills action is the sum over all lattice sites of the (real component of the) trace over the n links e1, ..., en in the Wilson loop,

Here, χ is the character
Character (mathematics)
In mathematics, a character is a special kind of function from a group to a field . There are at least two distinct, but overlapping meanings...

. If ρ is a real
Real representation
In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...

(or pseudoreal) representation, taking the real component is redundant, because 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 loops are used in the action. The simplest "Wilson action" uses only the 1×1 Wilson loop, and differs from the continuum action by "lattice artifacts" proportional to the small lattice spacing . By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to , making computations more accurate.

## Measurements and calculations

Quantities such as particle masses are stochastically calculated using techniques such as the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

. Gauge field configurations are generated with probabilities
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

proportional to , where is the lattice action and is related to the lattice spacing . The quantity of interest is calculated for each configuration, and averaged. Calculations are often repeated at different lattice spacings so that the result can be extrapolated
Extrapolation
In mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...

to the continuum, .

Such calculations are often extremely computationally intensive, and can require the use of the largest available supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s. To reduce the computational burden, the so-called quenched approximation
Quenched approximation
In particle physics, the quenched approximation is an approximation often used in lattice gauge theory in which the quantum loops of fermions in Feynman diagrams are neglected. Equivalently, the corresponding one-loop determinants are set to one...

can be used, in which the fermionic fields are treated as non-dynamic "frozen" variables. While this was common in early lattice QCD calculations, "dynamical" fermions are now standard. These simulations typically utilize algorithms based upon molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...

or microcanonical ensemble
Microcanonical ensemble
In statistical physics, the microcanonical ensemble is a theoretical tool used to describe the thermodynamic properties of an isolated system. In such a system, the possible macrostates of the system all have the same energy and the probability for the system to be in any given microstate is the same...

algorithms.

The results of lattice QCD computations show e.g. that in a meson not only the particles (quarks and antiquarks), but also the "fluxtube
Flux tube
A flux tube is a generally tube-like region of space containing a magnetic field, such that the field at the side surfaces is parallel to those surfaces...

s" of the gluon fields are important.

## Other applications

Originally, solvable two-dimensional lattice gauge theories had already been introduced in 1971 as models with interesting statistical properties by the theorist Franz Wegner
Franz Wegner
Franz Wegner is emeritus professor for theoretical physics at the University of Heidelberg. Franz Wegner attained a doctorate in 1968 with thesis advisor Wilhelm Brenig at Technical University Munich with the thesis "Zum Heisenberg-Modell im paramagnetischen Bereich und am kritischen Punkt" ...

, who worked in the field of phase transitions.

Lattice gauge theory has been shown to be exactly dual to spin foam
Spin foam
In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral description of quantum gravity...

models provided that only 1×1 Wilson loops appear in the action.

• Hamiltonian lattice gauge theory
Hamiltonian lattice gauge theory
In physics, 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...

• Lattice field theory
Lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a spacetime that has been discretized onto a lattice. Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by...

• Lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....

• Quantum triviality
Quantum triviality
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. Ifthe only allowed value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting...