Lattice gauge theory
   HOME

TheInfoList



OR:

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 r ...
, 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 into a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
. 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) an ...
, 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, an ...
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 type ...
(QCD) and particle physics'
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
.
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 Taylor ...
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 differen ...
, the path integral becomes finite-dimensional, and can be evaluated by
stochastic simulation A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities.DLOUHÝ, M.; FÁBRY, J.; KUNCOVÁ, M.. Simulace pro ekonomy. Praha : VŠE, 2005. Realizations of these ...
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 determi ...
. 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 into
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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 an ...
fields are defined at lattice sites (which leads to
fermion doubling In lattice field theory, fermion doubling occurs when naively putting fermionic fields on a lattice, resulting in more fermionic states than expected. For the naively discretized Dirac fermions in d Euclidean dimensions, each fermionic field res ...
), 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 additio ...
''G'' (not
algebra Algebra () is one of the 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 mathematics. Elementary a ...
) 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 special ...
, a 3×3
unitary matrix In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is ...
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 is ...
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 defini ...
of SU(3) acts), a
bispinor In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons. It is a specific embodiment of a spinor, speci ...
(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 (geometric integral), from which
Wilson loop In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representat ...
values can be calculated for closed paths.


Yang–Mills action

The Yang–Mills action is written on the lattice using
Wilson loop In quantum field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge information of the theory, allowing for the construction of loop representat ...
s (named after
Kenneth G. Wilson Kenneth Geddes "Ken" Wilson (June 8, 1936 – June 15, 2013) was an American theoretical physicist and a pioneer in leveraging computers for studying particle physics. He was awarded the 1982 Nobel Prize in Physics for his work on phase ...
), so that the limit a \to 0 formally reproduces the original continuum action. Given a faithful
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
ρ of ''G'', the lattice Yang–Mills action, known as the
Wilson action In lattice field theory, the Wilson action is a discrete formulation of the Yang–Mills action, forming the foundation of lattice gauge theory. Rather than using Lie algebra valued gauge fields as the fundamental parameters of the theory, grou ...
, is the sum over all lattice sites of the (real component of the)
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''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) (2010) ...
(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 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 determi ...
. Gauge field configurations are generated with probabilities 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 In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between kn ...
to the continuum, a \to 0. Such calculations are often extremely computationally intensive, and can require the use of the largest available
supercomputer A supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second ( FLOPS) instead of million instructions ...
s. To reduce the computational burden, the so-called
quenched approximation In lattice field theory, 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 ...
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 the ...
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 Stand ...
.


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 Joachim Wegner (born 15 June 1940) is emeritus professor for theoretical physics at the University of Heidelberg. Education Franz Wegner attained a doctorate in 1968 with thesis advisor Wilhelm Brenig at the Technical University Munich with ...
, 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 o ...
*
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 space or spacetime that has been discretised onto a lattice. Details Although most lattice field theories are not exactly sol ...
*
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 In lattice field theory, the Wilson action is a discrete formulation of the Yang–Mills action, forming the foundation of lattice gauge theory. Rather than using Lie algebra valued gauge fields as the fundamental parameters of the theory, grou ...


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