Lattice Gauge Theory
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, lattice gauge theory is the study of
gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
on a spacetime that has been
discretized In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
into a lattice. Gauge theories are important in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and ...
, and include the prevailing theories of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiq ...
s:
quantum electrodynamics In particle physics, quantum electrodynamics (QED) 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 spec ...
,
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ty ...
(QCD) and particle physics'
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. I ...
.
Non-perturbative In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function : f(x) = e^, which does not have a Taylor series at ''x'' = 0. Every coefficient of the Tayl ...
gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional path integral, which is computationally intractable. By working on a discrete
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...
, the path integral becomes finite-dimensional, and can be evaluated by stochastic simulation techniques such as the
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
. When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.


Basics

In lattice gauge theory, the spacetime is
Wick rotated In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that sub ...
into
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
and discretized into a lattice with sites separated by distance a and connected by links. In the most commonly considered cases, such as
lattice QCD Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the ...
,
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
fields are defined at lattice sites (which leads to fermion doubling), while the gauge fields are defined on the links. That is, an element ''U'' of the
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
''G'' (not
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
) is assigned to each link. Hence, to simulate QCD with Lie group
SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
, a 3×3
unitary matrix In linear algebra, a Complex number, complex Matrix (mathematics), square matrix is unitary if its conjugate transpose is also its Invertible matrix, inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, esp ...
is defined on each link. The link is assigned an orientation, with the
inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
corresponding to the same link with the opposite orientation. And each node is given a value in \mathbb^3 (a color 3-vector, the space on which the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defi ...
of SU(3) acts), a bispinor (Dirac 4-spinor), an ''nf'' vector, and a
Grassmann variable In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...
. Thus, the composition of links' SU(3) elements along a path (i.e. the ordered multiplication of their matrices) approximates a
path-ordered exponential The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential ...
(geometric integral), from which Wilson loop values can be calculated for closed paths.


Yang–Mills action

The Yang–Mills action is written on the lattice using Wilson loops (named after Kenneth G. Wilson), so that the limit a \to 0 formally reproduces the original continuum action. Given a faithful irreducible representation ρ of ''G'', the lattice Yang–Mills action, known as the Wilson action, is the sum over all lattice sites of the (real component of the) trace over the ''n'' links ''e''1, ..., ''e''n in the Wilson loop, :S=\sum_F -\Re\. Here, χ is the
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
. If ρ is a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
(or
pseudoreal In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space ''V'' with an invariant quaternionic structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies ...
) 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 Wilson 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 a. By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to a^2, 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 methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...
. Gauge field configurations are generated with
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
proportional to e^, where S is the lattice action and \beta is related to the lattice spacing a. The quantity of interest is calculated for each configuration, and averaged. Calculations are often repeated at different lattice spacings a so that the result can be extrapolated to the continuum, a \to 0. Such calculations are often extremely computationally intensive, and can require the use of the largest available supercomputers. To reduce the computational burden, the so-called quenched approximation 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 (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
or
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assumed to be isolated in the sense that it canno ...
algorithms. The results of lattice QCD computations show e.g. that in a meson not only the particles (quarks and antiquarks), but also the " fluxtubes" of the gluon fields are important.


Quantum triviality

Lattice gauge theory is also important for the study of
quantum triviality In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. ...
by the real-space
renormalization group In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the ...
. The most important information in the RG flow are what's called the ''fixed points''. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be ''trivial'' or noninteracting. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question. Triviality has yet to be proven rigorously, but lattice computations have provided strong evidence for this. This fact is important as quantum triviality can be used to bound or even predict parameters such as the mass of
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the St ...
.


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, who worked in the field of phase transitions. When only 1×1 Wilson loops appear in the action, lattice gauge theory can be shown to be exactly dual to
spin foam In physics, the topological structure of spinfoam or spin foam consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structur ...
models.


See also

*
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. The Hamiltonian is then re-expressed as a function of degrees ...
* Lattice field theory *
Lattice QCD Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time. When the size of the ...
*
Quantum triviality In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. ...
* Wilson action


References


Further reading

* Creutz, M., ''Quarks, gluons and lattices'', Cambridge University Press, Cambridge, (1985). * Montvay, I., Münster, G.,
Quantum Fields on a Lattice
', Cambridge University Press, Cambridge, (1997). * Makeenko, Y., ''Methods of contemporary gauge theory'', Cambridge University Press, Cambridge, (2002). . * Smit, J., ''Introduction to Quantum Fields on a Lattice'', Cambridge University Press, Cambridge, (2002). * Rothe, H., ''Lattice Gauge Theories, An Introduction'', World Scientific, Singapore, (2005). * DeGrand, T., DeTar, C.,
Lattice Methods for Quantum Chromodynamics
', World Scientific, Singapore, (2006). * Gattringer, C., Lang, C. B., ''Quantum Chromodynamics on the Lattice'', Springer, (2010). * Knechtli, F., Günther, M., Peardon, M., ''Lattice Quantum Chromodynamics: Practical Essentials'', Springer, (2016). * {{cite journal , author = Weisz Peter, Majumdar Pushan , year = 2012 , title = Lattice gauge theories , journal = Scholarpedia , volume = 7 , issue = 4, page = 8615 , doi = 10.4249/scholarpedia.8615 , bibcode = 2012SchpJ...7.8615W , doi-access = free


External links


The FermiQCD Library for Lattice Field theory

US Lattice Quantum Chromodynamics Software Libraries
Lattice field theory