Lattice
Gauge Theories| Lattice
Gauge Theories |
In General > s.a. non-commutative
field theory; quantum gauge theory.
* History: First proposed by K Wilson in 1973 in order to explain
quark confinement.
* Idea: Given a matter
field 
in
an N-dimensional representation of a
gauge group G, one assigns a (x)
to each lattice site and a U(x,x')
in
G to each
(arbitrarily) oriented link; U corresponds to the holonomy exp( ie
xx' Aa dxa).
* Action: The sum of a Wilson action Sg for U and
a term
Sm for the field
S:= 
g 
plaquettes
Ep + m i,j i Uij j ,
where :=
2N/g2; The energy for a
plaquette (the smallest possible Wilson loop) along directions i, k is
EP:=
1 – (1/2N) tr (UP +
UP
) → (g2/N)
l a4 (Fikl)2 (continuum
limit) ,
UP:= U1 U2 U3U4,
and l labels the Lie algebra generators.
* Uses: Used to "show" confinement
in non-abelian gauge theories, although there are
difficulties
coming from the fact that one encounters phase transitions going
to
the weak coupling limit (which corresponds to the continuum).
Specific Theories > s.a. graphs
in physics [QED];
QED and modified
QED [Schwinger Model]; yang-mills
gauge theory [chaos].
@ Electromagnetism: Aroca & Fort PLB(94),
et al
PLB(94)
[Lagrangian loop formulation]; Orthuber qp/03;
He & Teixeira PLA(05)
[polyhedral complex, degrees of freedom], PLA(06)
[geometric, Galerkin duality]; Di Bartolo et al MPLA(05)
[Wilson loops]; > s.a. light [propagating
on a lattice].
@ SU(2): Bloch et al NPB(04)
[propagators and coupling]; Golterman
hl/04-in
[chiral]; Gutbrod NPB(05)
[gauge singularities].
@ SU(n): Bringoltz & Teper PLB(05)hl [bulk
thermodynamic properties].
@ QCD, rev:
Weingarten SA(96)feb;
Wilczek NPPS(03)hl/02-in
[rev]; DeTar & Gottlieb PT(04)feb;
DeGrand IJMPA(04);
DeGrand & DeTar 06; Davies pw(06)dec;
Schierholz IJMPA(07);
Onogi IJMPA(09) [realistic unquenched simulations].
@ QCD, other phenomenology: Shipsey hl/04-in
[and experiments]; Beane et al PRL(06)
+ sr(06)jul
[N-N scattering].
@ QCD, masses: Montvay RMP(87)
[hadrons]; Davies PRL(04),
Allison et al PRL(05)
+ pn(05)may
+ pw(05)jul
[mesons].
@ QCD, observable
algebra:
Jarvis et al JPA(05)ht/04;
Kijowski & Rudolph RPMP(05).
@ QCD, other topics: Wilson PRD(74);
Kogut RMP(83);
de Forcrand et al NPPS(98)hl/97 [topology];
Kijowski & Rudolph JMP(02)
[charge and flux]; Charzynski et al JGP(05)ht/04 [stratified
configuration space]; NPPS(05)140
issue; Lepage AP(05)
[high-precision, rev]; Narayanan & Neuberger hl/05-in
[large N,
fermionic sector]; Charzynski et al JGP(08)ht/05 [reduced
classical configuration space]; Di Pierro IJMPA(06)
[with fermions, lattice]; Del Debbio et al JHEP(06)
[stability]; Meurice ht/06-in
[series truncation]; Sommer NPPS(06)
[fundamental parameters]; Bowman et al NPPS(06)
[QCD propagators]; Walker ht/07 [failure
of cluster decomposition method]; Miller PRP(07)
[equation of state].
@ Standard model: Preparata & Xue PLB(91) [electroweak]; Creutz et
al PLB(97).
@ Supersymmetric: Bietenholz MPLA(99)
[Wess-Zumino]; Fujikawa NPB(02)ht [Leibniz
rule]; Kaplan hl/02-in,
et al JHEP(03)hl/02;
Itoh et al JHEP(03);
Feo MPLA(04);
Catterall et al PRP(09)-a0903.
@ Other theories: Bodwin PRD(96)
[chiral gauge theories]; Kawamoto et al NPB(00)ht/99 [BF];
Larsson mp/02,
Wise gq/05 [p-forms].
> Other: see connection representation and loop
quantum gravity.
References > s.a. lattice
field theory; topological defects.
@ Intros / reviews: Hasenfratz & Hasenfratz ARNPS(85);
Sharpe hl/98-in;
Münster & Walzl hl/00-ln;
Montvay & Münster 97; Wilson hl/04-in
[history];
Oeckl 05; Rothe 05.
@ Loop representation and states: Brügmann PRD(91);
Aroca et al PRD(96)ht [path
integral];
Mathur PLB(06)hl/05,
NPB(07).
@ Finite T: Fodor & Katz PLB(02)
[finite chemical potential].
@ Continuum limit: Gross CMP(83)
[3D U(1) theory]; McIntosh & Hollenberg PLB(02);
Thiemann CQG(01)ht/00.
@ Simplicial lattice: Rajeev ht/04-in
[2+1 Yang-Mills].
@ Random lattice: Christ et al NPB(82), NPB(82),
NPB(82);
Itzykson in(84), & Drouffe
89;
Burda et al PRD(99)hl,
NPPS(00)hl/99 [fermions].
@ Other lattices: Chodos
PRD(78) [dynamical structure].
@ Duality: Oeckl & Pfeiffer NPB(01)ht/00 [and
spin-foam models]; Grosse & Schlesinger IJTP(01)
[categorical methods].
@ Related topics: Kogut & Susskind PRD(75)
[Hamiltonian formulation]; Loll NPB(92)
[variables, constraints]; Milton NPPS(97)hl/96 [alternative
approach]; Boulatov CMP(97)
[deformation]; Ma MPLA(00)
[gluon propagator];
Burgio et al NPB(00)
[
phys];
Caselle
IJMPA(00)
[and AdS-cft]; Adams NPB(02)hl/01 [fermionic
topological
charge],
NPB(02)
[space of lattice fields]; de Forcrand & Jahn NPB(03)
[SO(3)
vs SU(2)]; Golterman & Shamir PRD(03)hl,
hl/03-in
[localization]; Silva & Oliveira
NPB(04)
[Gribov copies]; Berges et al PRD(07)hl/06 [real
time – Lorentzian]; Meyer NPB(08)
[sum rules, finite T]; > s.a. Tomboulis-Yaffe
Inequality; Wilson
Loops.
Generalizations
* Tensor categories: The role of the gauge group is played by a tensor
category, the admissible type (spherical, ribbon, symmetric) depending
on
the dimension of the underlying manifold (3, 4, any); Ordinary LGT is recovered
if the category is the (symmetric) category of representations of a
compact
Lie group.
@ Tensor categories: Oeckl JGP(03)ht/01.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 12
nov 2009