Editing Lattice gauge theory

{{Quantum field theory}}
{{short description|Theory of quantum gauge fields on a lattice}}

In [[physics]], '''lattice gauge theory''' is the study of [[Gauge theory|gauge theories]] on a spacetime that has been [[Discretization|discretized]] into a [[lattice (group)|lattice]].

Gauge theories are important in [[particle physics]], and include the prevailing theories of [[elementary particle]]s: [[quantum electrodynamics]], [[quantum chromodynamics]] (QCD) and particle physics' [[Standard Model]]. [[Non-perturbative]] gauge theory calculations in continuous spacetime formally involve evaluating an infinite-dimensional [[Path integral formulation|path integral]], which is computationally intractable. By working on a discrete [[spacetime]], the [[Functional integration|path integral]] becomes finite-dimensional, and can be evaluated by [[stochastic simulation]] techniques such as the [[Monte Carlo method]].  When the size of the lattice is taken infinitely large and its sites infinitesimally close to each other, the continuum gauge theory is recovered.<ref name="wilson">{{cite journal | authorlink=Kenneth G. Wilson | first=K. | last= Wilson | journal=[[Physical Review D]]| volume=10 | issue=8 | page=2445 | title=Confinement of quarks | year= 1974 | doi=10.1103/PhysRevD.10.2445|bibcode = 1974PhRvD..10.2445W }}</ref>

==Basics==
In lattice gauge theory, the spacetime is [[Wick rotated]] into [[Euclidean space]] and discretized into a lattice with sites separated by distance <math>a</math> and connected by links. In the most commonly considered cases, such as [[lattice QCD]], [[fermion]] fields are defined at lattice sites (which leads to [[fermion doubling]]), while the [[Gauge boson|gauge fields]] are defined on the links.  That is, an element ''U'' of the [[compact group|compact]] [[Lie group]] ''G'' (not [[Lie algebra|algebra]]) is assigned to each link.  Hence, to simulate QCD with Lie group [[Special unitary group|SU(3)]], a 3×3 [[unitary matrix]] is defined on each link.  The link is assigned an orientation, with the [[inverse element]] corresponding to the same link with the opposite orientation. And each node is given a value in <math>\mathbb{C}^3</math> (a color 3-vector, the space on which the [[fundamental representation]] of SU(3) acts), a [[bispinor]] (Dirac 4-spinor), an ''n<sub>f</sub>'' vector, and a [[Grassmann number|Grassmann variable]].

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]] values can be calculated for closed paths.

==Yang–Mills action==
The [[Yang–Mills theory|Yang–Mills]] action is written on the lattice using [[Wilson loop]]s (named after [[Kenneth G. Wilson]]), so that the limit <math>a \to 0</math> formally reproduces the original continuum action.<ref name="wilson" />  Given a [[faithful representation|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 (matrix)|trace]] over the ''n'' links ''e''<sub>1</sub>, ..., ''e''<sub>n</sub> in the Wilson loop,

:<math>S=\sum_F -\Re\{\chi^{(\rho)}(U(e_1)\cdots U(e_n))\}.</math>

Here, χ is the [[character (mathematics)|character]].  If ρ is a [[real representation|real]] (or [[pseudoreal representation|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 <math>a</math>.  By using more complicated Wilson loops to construct "improved actions", lattice artifacts can be reduced to be proportional to <math>a^2</math>, making computations more accurate.

==Measurements and calculations==
[[File:Fluxtube_meson.png|thumb|150px|This result of a [[Lattice QCD]] computation shows a [[meson]], composed out of a quark and an antiquark. (After M. Cardoso et al.<ref>{{cite journal | last1=Cardoso | first1=M. | last2=Cardoso | first2=N. | last3=Bicudo | first3=P. | title=Lattice QCD computation of the color fields for the static hybrid quark-gluon-antiquark system, and microscopic study of the Casimir scaling | journal=Physical Review D | volume=81 | issue=3 | date=2010-02-03 | issn=1550-7998 | doi=10.1103/physrevd.81.034504 | page=034504|arxiv=0912.3181| bibcode=2010PhRvD..81c4504C | s2cid=119216789 }}</ref>)]]

Quantities such as particle masses are stochastically calculated using techniques such as the [[Monte Carlo method]].  Gauge field configurations are generated with [[probability|probabilities]] proportional to <math>e^{-\beta S}</math>, where <math>S</math> is the lattice action and <math>\beta</math> is related to the lattice spacing <math>a</math>.  The quantity of interest is calculated for each configuration, and averaged.  Calculations are often repeated at different lattice spacings <math>a</math> so that the result can be [[extrapolation|extrapolated]] to the continuum, <math>a \to 0</math>.

Such calculations are often extremely computationally intensive, and can require the use of the largest available [[supercomputer]]s.  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.<ref>{{cite journal | author=A. Bazavov| title=Nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks | journal=Reviews of Modern Physics | volume=82 | issue=2 | year=2010 | pages=1349–1417 | doi=10.1103/RevModPhys.82.1349 | arxiv=0903.3598 | bibcode=2010RvMP...82.1349B| s2cid=119259340 |display-authors=etal}}</ref>  These simulations typically utilize algorithms based upon [[molecular dynamics]] or [[microcanonical ensemble]] algorithms.<ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Microcanonical Ensemble Formulation of Lattice Gauge Theory | journal=Physical Review Letters | volume=49 | year=1982 | issue=9 |pages=613–616 | doi=10.1103/PhysRevLett.49.613 | bibcode=1982PhRvL..49..613C}}</ref><ref>{{cite journal | author=[[David Callaway|David J. E. Callaway]] and [[Aneesur Rahman]] | title=Lattice gauge theory in the microcanonical ensemble | journal=Physical Review | volume=D28 |year=1983 | issue=6 | pages=1506–1514 | doi=10.1103/PhysRevD.28.1506|bibcode = 1983PhRvD..28.1506C | url=https://cds.cern.ch/record/144746/files/PhysRevD.28.1506.pdf }}</ref> An alternative method could be simulations on [[Quantum computing|quantum computers]].<ref>{{Cite journal |last=Meth |first=Michael |last2=Zhang |first2=Jinglei |last3=Haase |first3=Jan F. |last4=Edmunds |first4=Claire |last5=Postler |first5=Lukas |last6=Jena |first6=Andrew J. |last7=Steiner |first7=Alex |last8=Dellantonio |first8=Luca |last9=Blatt |first9=Rainer |last10=Zoller |first10=Peter |last11=Monz |first11=Thomas |last12=Schindler |first12=Philipp |last13=Muschik |first13=Christine |last14=Ringbauer |first14=Martin |date=2025-03-25 |title=Simulating two-dimensional lattice gauge theories on a qudit quantum computer |url=https://www.nature.com/articles/s41567-025-02797-w |journal=Nature Physics |language=en |pages=1–7 |doi=10.1038/s41567-025-02797-w |issn=1745-2481|doi-access=free |pmc=11999872 }}</ref>

The results of lattice QCD computations show e.g. that in a meson not only the particles (quarks and antiquarks), but also the "[[Flux tube|fluxtube]]s" of the gluon fields are important.{{citation needed|date=April 2025}}

==Quantum triviality==
Lattice gauge theory is also important for the study of [[quantum triviality]] by the real-space [[renormalization group]].<ref>{{cite journal | last=Wilson | first=Kenneth G. |author-link=Kenneth G. Wilson| title=The renormalization group: Critical phenomena and the Kondo problem | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=47 | issue=4 | date=1975-10-01 | issn=0034-6861 | doi=10.1103/revmodphys.47.773 | pages=773–840| bibcode=1975RvMP...47..773W }}</ref> 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.<ref>{{cite journal
 | author=[[David J E Callaway|D. J. E. Callaway]]
 | year=1988
 | title=Triviality Pursuit: Can Elementary Scalar Particles Exist?
 | journal=[[Physics Reports]]
 | volume=167
 | issue=5 | pages=241–320
 | doi=10.1016/0370-1573(88)90008-7
|bibcode = 1988PhR...167..241C }}</ref>

Triviality has yet to be proven rigorously, but lattice computations 
have provided strong evidence for this.<ref>[[Kenneth G. Wilson|K.G. Wilson]](1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. '''47''', 4, 773.</ref><ref>{{Cite journal|last1=Callaway|first1=D.J.E.|last2=Petronzio|first2=R.|doi=10.1016/0550-3213(87)90657-2|title=Is the standard model Higgs mass predictable?|journal=[[Nuclear Physics B]]|volume=292|pages=497–526|year=1987|bibcode=1987NuPhB.292..497C|url=https://cds.cern.ch/record/172532}}{{cite journal| last=Heller| first=Urs| author2=Markus Klomfass |author3=Herbert Neuberger |author4=Pavols Vranas | s2cid=7146602|date=1993-09-20|journal=[[Nuclear Physics B]]| volume=405| doi=10.1016/0550-3213(93)90559-8| pages=555–573|arxiv = hep-ph/9303215 |bibcode = 1993NuPhB.405..555H|title=Numerical analysis of the Higgs mass triviality bound|issue=2–3 }}, which suggests {{math|''M<sub>H</sub>'' < 710 GeV}}.</ref><ref>{{Cite journal | doi = 10.1016/0550-3213(84)90246-3 | title = Monte Carlo renormalization group study of {{math|''φ''<sup>4</sup>}} field theory| journal = Nuclear Physics B| volume = 240| issue = 4| pages = 577| year = 1984| last1 = Callaway | first1 = D. J. E. | last2 = Petronzio | first2 = R. |bibcode = 1984NuPhB.240..577C | url = https://cds.cern.ch/record/150964}}</ref><ref>{{cite journal| last1=Gies| first1=Holger|last2=Jaeckel| first2=Joerg| s2cid=222197| title=Renormalization Flow of QED| journal=[[Physical Review Letters]]| date=2004-09-09| volume=93| doi=10.1103/PhysRevLett.93.110405 | page=110405| bibcode=2004PhRvL..93k0405G|arxiv = hep-ph/0405183|issue=11| pmid=15447325}}</ref><ref>{{cite journal | doi = 10.1016/0550-3213(86)90431-1 | title =Can elementary scalar particles exist?: (II). Scalar electrodynamics | journal = Nuclear Physics B| volume = 277| issue = 1| pages = 50–66| year = 1986| last1 = Callaway | first1 = D. J. E. | last2 = Petronzio | first2 = R. |bibcode = 1986NuPhB.277...50C | url =https://cds.cern.ch/record/167168 }}</ref><ref>{{cite journal |last1=Göckeler|first1=M. |last2=Horsley|first2=R. |last3=Linke|first3=V. |last4=Rakow|first4=P. |last5=Schierholz|first5=G. |last6=Stüben|first6=H. |s2cid=119494925 |year=1998|title=Is There a Landau Pole Problem in QED?|journal=[[Physical Review Letters]]|volume=80|doi=10.1103/PhysRevLett.80.4119| pages=4119–4122| bibcode=1998PhRvL..80.4119G|arxiv = hep-th/9712244|issue=19 }}</ref>   This fact is important as quantum triviality can be used to bound or even predict parameters such as the mass of [[Higgs boson]].  Lattice calculations have been useful in this context.<ref>For example, {{Cite journal|last1=Callaway|first1=D.J.E.|last2=Petronzio|first2=R.|doi=10.1016/0550-3213(87)90657-2|title=Is the standard model Higgs mass predictable?|journal=[[Nuclear Physics B]]|volume=292|pages=497–526|year=1987|bibcode=1987NuPhB.292..497C|url=https://cds.cern.ch/record/172532}}{{cite journal| last=Heller| first=Urs| author2=Markus Klomfass |author3=Herbert Neuberger |author4=Pavols Vranas | s2cid=7146602|date=1993-09-20|journal=[[Nuclear Physics B]]| volume=405| doi=10.1016/0550-3213(93)90559-8| pages=555–573|arxiv = hep-ph/9303215 |bibcode = 1993NuPhB.405..555H|title=Numerical analysis of the Higgs mass triviality bound|issue=2–3 }}, which suggests {{math|''M<sub>H</sub>'' < 710 GeV}}.</ref>

==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 transition]]s.<ref>F. Wegner, "Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter", ''J. Math. Phys.'' '''12''' (1971) 2259-2272. Reprinted in [[Claudio Rebbi]] (ed.), ''Lattice Gauge Theories and Monte-Carlo-Simulations'', World Scientific, Singapore (1983), p. 60-73. [http://www.tphys.uni-heidelberg.de/~wegner/Abstracts.html#12 Abstract]</ref>

When only 1×1 Wilson loops appear in the action, lattice gauge theory can be shown to be exactly dual to [[spin foam]] models.<ref>{{cite journal |author1=R. Oeckl |author2=H. Pfeiffer |year=2001 |title=The dual of pure non-Abelian lattice gauge theory as a spin foam model |arxiv=hep-th/0008095 |doi=10.1016/S0550-3213(00)00770-7 |volume=598 |issue=1–2 |journal=Nuclear Physics B |pages=400–426|bibcode=2001NuPhB.598..400O |s2cid=3606117 }}</ref>

==See also==
*[[Hamiltonian lattice gauge theory]]
*[[Lattice field theory]]
*[[Lattice QCD]]
*[[Quantum triviality]]
*[[Wilson action]]

==References==
{{reflist|2}}

==Further reading==
* Creutz, M., ''Quarks, gluons and lattices'', Cambridge University Press, Cambridge, (1985). {{ISBN|978-0521315357}}
* Montvay, I., Münster, G., ''[https://books.google.com/books?id=NHZshmEBXhcC&q=%22Lattice+gauge+theory%22 Quantum Fields on a Lattice]'', Cambridge University Press, Cambridge, (1997). {{ISBN|978-0521599177}}
* Makeenko, Y., ''Methods of contemporary gauge theory'', Cambridge University Press, Cambridge, (2002). {{ISBN|0-521-80911-8}}.
* [[Jan Smit (physicist)|Smit, J.]], ''Introduction to Quantum Fields on a Lattice'', Cambridge University Press, Cambridge, (2002). {{ISBN|978-0521890519}}
* Rothe, H., ''Lattice Gauge Theories, An Introduction'', World Scientific, Singapore, (2005). {{ISBN|978-9814365857}}
* DeGrand, T., DeTar, C., ''[https://books.google.com/books?id=r8bICgAAQBAJ&q=%22Lattice+gauge+theory%22 Lattice Methods for Quantum Chromodynamics]'', World Scientific, Singapore, (2006). {{ISBN|978-9812567277}}
* Gattringer, C., Lang, C. B., ''Quantum Chromodynamics on the Lattice'', Springer, (2010). {{ISBN|978-3642018497}}
* Knechtli, F., Günther, M., Peardon, M., ''Lattice Quantum Chromodynamics: Practical Essentials'', Springer, (2016). {{ISBN|978-9402409970}}
* {{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==
* [https://web.archive.org/web/20050304091436/http://www.fermiqcd.net/ The FermiQCD Library for Lattice Field theory]
* [http://usqcd.jlab.org/usqcd-software/ US Lattice Quantum Chromodynamics Software Libraries]

[[Category:Lattice field theory]]