Lattice Gauge Theory - Wikipedia, The Free Encyclopedia

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Quantum field theoryQuantum field theory

(Feynman diagram)

History of...

Gauge theory

Field theory

Poincar; symmetry

Quantum mechanics

Spontaneous symmetry breaking

Crossing

Charge conjugation

Parity

Time reversal

Anomaly

Effective field theory

Expectation value

FaddeevPopov ghosts

Feynman diagram

Lattice gauge theoryLattice gauge theory

LSZ reduction formula

Partition function

Propagator

Quantization

Renormalization

Vacuum state

Wick's theorem

Wightman axioms

Dirac equation

KleinGordon equationProca equations

Lattice gauge theoryLattice gauge theoryFrom Wikipedia, the free encyclopedia

In physics, lattice gauge theorylattice gauge theory is the study ofgauge theories on a spacetime that has beendiscretized into a lattice. Gauge theories are

important in particle physics, and include theprevailing theories of elementary particles:quantum electrodynamics, quantumchromodynamics (QCD) and the Standard Model.Non-perturbative gauge theory calculations incontinuous spacetime formally involveevaluating an infinite-dimensional path integral,which is computationally intractable. By workingon a discrete spacetime, the path integralbecomes finite-dimensional, and can beevaluated by stochastic simulation techniquessuch as the Monte Carlo method. When the sizeof the lattice is taken infinitely large and its sitesinfinitesimally close to each other, the continuumgauge theory is recovered intuitively. Amathematical proof of this fact is lacking.

ContentsContents1 Basics2 YangMills action3 Measurements and calculations4 Other applications5 See also6 Further reading7 External links8 References

BasicsBasics

In lattice gauge theory, the spacetime is Wickrotated into Euclidean space and discretized intoa lattice with sites separated by distance a andconnected by links. In the most commonly-considered cases, such as lattice QCD, fermionfields are defined at lattice sites (which leads tofermion doubling), while the gauge fields aredefined on the links. That is, an element Uof the

BackgroundBackground

SymmetriesSymmetries

ToolsTools

EquationsEquations


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WheelerDeWitt equation

Electroweak interaction

Higgs mechanism

Quantum chromodynamics

Quantum electrodynamics

YangMills theory

Quantum gravity

String theory

Supersymmetry

Technicolor

Theory of everything

Jamal Nazrul Islam Adler Bethe

Bogoliubov Callan Candlin Coleman

DeWitt Dirac Dyson Fermi Feynman

Fierz Frhlich Gell-Mann Goldstone

Gross 't Hooft Jackiw Klein Landau

Lee Lehmann Majorana Nambu Parisi

Polyakov Salam Schwinger Skyrme

Stueckelberg Symanzik Tomonaga

Veltman Weinberg Weisskopf Wilson

Witten Yang Yukawa Hoodbhoy

Zimmermann Zinn-Justin

compact Lie group Gis assigned to each link.Hence to simulate QCD, with Lie group SU(3),there is a 33 special unitary matrix defined oneach link. The link is assigned an orientation, withthe inverse element corresponding to the samelink with the opposite orientation.

YangYangMills actionMills action

The YangMills action is written on the latticeusing Wilson loops (named after Kenneth G.Wilson), so that the limit formallyreproduces the original continuum action.[1]

Given a faithful irreducible representation ofG, the lattice Yang-Mills action is the sum over alllattice sites of the (real component of the) traceover the nlinks e1, ..., en in the Wilson loop,

Here, is the character. If is a real (orpseudoreal) representation, taking the realcomponent is redundant, because even if theorientation of a Wilson loop is flipped, its

contribution to the action remains unchanged.There are many possible lattice Yang-Millsactions, depending on which Wilson loops are used in the action. The simplest "Wilsonaction" uses only the 11 Wilson loop, and differs from the continuum action by "latticeartifacts" proportional to the small lattice spacing a. By using more complicated Wilsonloops to construct "improved actions", lattice artifacts can be reduced to be proportional to

a2, making computations more accurate.

Measurements and calculationsMeasurements and calculations

Quantities such as particle masses are stochastically calculated using techniques such asthe Monte Carlo method. Gauge field configurations are generated with probabilities

proportional to eS, where S is the lattice action and is related to the lattice spacing a.The quantity of interest is calculated for each configuration, and averaged. Calculations areoften repeated at different lattice spacings a so that the result can be extrapolated to thecontinuum, .

Such calculations are often extremely computationally intensive, and can require the use of

the largest available supercomputers. To reduce the computational burden, the so-calledquenched 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,

Standard ModelStandard Model

Incomplete theoriesIncomplete theories

ScientistsScientists


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This result of a Lattice

QCD computation

shows a meson,composed out of a

quark and an

antiquark. (After M.

Cardoso et al. [2])

"dynamical" fermions are now standard.[3] These simulationstypically utilize algorithms based upon molecular dynamics or

microcanonical ensemble algorithms.[4][5]

The results of lattice QCD computations show e.g. that in a mesonnot only the particles (quarks and antiquarks), but also the

"fluxtubes" of the gluon fields are important.

Other applicationsOther applications

Originally, solvable two-dimensional lattice gauge theories hadalready been introduced in 1971 as models with interestingstatistical properties by the theorist Franz Wegner, who worked in

the field of phase transitions.[6]

Lattice gauge theory has been shown to be exactly dual to spinfoam models provided that only 11 Wilson loops appear in theaction.

See alsoSee also

Hamiltonian lattice gauge theoryLattice field theoryLattice QCDQuantum triviality

Further readingFurther reading

M. Creutz, Quarks, gluons and lattices, Cambridge University Press 1985.I. Montvay and G. Mnster, Quantum Fields on a Lattice, Cambridge University Press1997.Y. Makeenko, Methods of contemporary gauge theory, Cambridge University Press2002, ISBN 0-521-80911-8.J. Smit, Introduction to Quantum Fields on a Lattice, Cambridge University Press2002.

H. Rothe, Lattice Gauge Theories, An Introduction, World Scientific 2005.T. DeGrand and C. DeTar, Lattice Methods for Quantum Chromodynamics, WorldScientific 2006.C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice, Springer2010.

External linksExternal links

The FermiQCD Library for Lattice Field theory (http://www.fermiqcd.net)

US Lattice Quantum Chromodynamics Software Libraries(http://usqcd.jlab.org/usqcd-software/)


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ReferencesReferences

1. ^ Wilson, K. (1974). "Confinement of quarks". Physical Review D1010 (8): 2445. Bibcode1974PhRvD..10.2445W (http://adsabs.harvard.edu/abs/1974PhRvD..10.2445W) .doi:10.1103/PhysRevD.10.2445 (http://dx.doi.org/10.1103%2FPhysRevD.10.2445) .

2. ^ M. Cardoso et al., Lattice QCD computation of the colour fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimir scaling, Phys. Rev. D 81, 034504 (2010) ).

3. ^ A. Bazavov et al. (2010). "Nonperturbative QCD simulations with 2+1 flavors of improvedstaggered quarks". Reviews of Modern Physics8282 (2): 13491417. arXiv:0903.3598(http://arxiv.org/abs/0903.3598) . Bibcode 2010RvMP...82.1349B(http://adsabs.harvard.edu/abs/2010RvMP...82.1349B) . doi:10.1103/RevModPhys.82.1349(http://dx.doi.org/10.1103%2FRevModPhys.82.1349) .

4. ^ David J. E. Callaway and Aneesur Rahman (1982). "Microcanonical Ensemble Formulation ofLattice Gauge Theory". Physical Review Letters4949 (9): 613616. Bibcode 1982PhRvL..49..613C(http://adsabs.harvard.edu/abs/1982PhRvL..49..613C) . doi:10.1103/PhysRevLett.49.613(http://dx.doi.org/10.1103%2FPhysRevLett.49.613) .

5. ^ David J. E. Callaway and Aneesur Rahman (1983). "Lattice gauge theory in the microcanonicalensemble". Physical ReviewD2 8D28 (6): 15061514. Bibcode 1983PhRvD..28.1506C

(http://adsabs.harvard.edu/abs/1983PhRvD..28.1506C) . doi:10.1103/PhysRevD.28.1506(http://dx.doi.org/10.1103%2FPhysRevD.28.1506) .

6. ^ F. Wegner, "Duality in Generalized Ising Models and Phase Transitions without Local OrderParameter", J. Math. Phys. 1212 (1971) 2259-2272. Reprinted in Claudio Rebbi (ed.), Lattice GaugeTheories and Monte-Carlo-Simulations, World Scientific, Singapore (1983), p. 60-73. Abstract(http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html#12)

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