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 φ in an N-dimensional representation of
a gauge group G, one assigns a φ(x) to
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 φ
S:= βg ∑plaquettes Ep + βm ∑i,j φi Uij φj ,
where β:= 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 U3
U4, 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]; lattice QCD;
QED and modified QED [Schwinger Model];
yang-mills gauge theory [chaos].
@ Electromagnetism, U(1):
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];
Giuliani et al AP(12) [honeycomb lattice gauge theory for graphene];
Kaplan & Stryker a1806 [Hamiltonian formulation, Gauss' law and duality];
Sulejmanpasic & Gattringer NPB(19)-a1901 [θ terms];
> s.a. light [propagating on a lattice].
@ SU(2): Bloch et al NPB(04) [propagators and coupling];
Golterman & Shamir NPPS(05)hl/04 [chiral];
Gutbrod NPB(05) [gauge singularities].
@ SU(n): Bringoltz & Teper PLB(05)hl [bulk thermodynamic properties].
@ Standard model:
Preparata & Xue PLB(91) [electroweak];
Creutz et al PLB(97);
Zubkov PRD(10)-a1008 [electroweak, continuum limit].
@ Supersymmetric:
Bietenholz MPLA(99) [Wess-Zumino];
Fujikawa NPB(02)ht [Leibniz rule];
Kaplan NPPS(03)hl/02,
et al JHEP(03)hl/02;
Itoh et al JHEP(03);
Feo MPLA(04);
Catterall et al PRP(09)-a0903;
Joseph a1409-conf [with matter fields];
Catterall & Schaich JHEP(15)-a1505 [N = 4 supersymmetric Yang-Mills theory];
Bergner & Catterall a1603-IJMPA [rev];
> s.a. supersymmetric theories.
@ Other theories: Wadia pr(79)-a1212 [3D U(N) lattice gauge theory];
Bodwin PRD(96) [chiral gauge theories];
Kawamoto et al NPB(00)ht/99 [BF];
Larsson mp/02,
Wise gq/05 [p-forms];
Wiese AdP(13)-a1305 [and ultracold quantum gases];
Wetterich NPB(13) [scalar lattice gauge theory];
Lipstein & Reid-Edwards JHEP(14)-a1404 [2-form gauge field, lattice gerbe theory];
Brower et al a1904 [beyond the standard model].
> Other: see connection representation
and loop quantum gravity.
References
> s.a. lattice field theory; monopoles;
topological defects; tensor networks;
Wilson Loops.
@ Intros / reviews:
Hasenfratz & Hasenfratz ARNPS(85);
Montvay & Münster 94;
Sharpe hl/98-conf;
Münster & Walzl hl/00-ln;
Wilson NPPS(05)hl/04 [history];
Oeckl 05;
Rothe 12.
@ Loop representation and states:
Brügmann PRD(91);
Aroca et al PRD(96)ht [path integral];
Mathur PLB(06)hl/05,
NPB(07).
@ Quantum simulations: Zohar & Burrello PRD(15)-a1409;
Bender et al NJP(18)-a1804;
Lamm et al a1903;
Bañuls et al a1911;
Halimeh & Hauke a2001.
@ Approches, formulations:
Kogut & Susskind PRD(75) [Hamiltonian formulation];
Loll NPB(92) [variables, constraints];
Milton NPPS(97)hl/96 [alternative approach];
Berges et al PRD(07)hl/06 [real time – Lorentzian];
Vairinhos & de Forcrand JHEP(14)-a1409 [with link variables integrated out];
Delcamp & Dittrich JHEP(18)-a1806 [3+1, dual spin network basis];
Bañuls & Cichy a1910 [new methods];
Haase et al a2006 [resource-efficient protocol].
@ On quantum computers: Martinez et al nat(16)jun-a1605 [on a few-qubit quantum computer];
Brower et al PoS-a2002;
Mathis et al a2005 [scalable].
@ At finite T:
Fodor & Katz PLB(02) [finite chemical potential];
Meyer NPB(08) [sum rules].
@ Continuum limit:
Gross CMP(83) [3D U(1) theory];
McIntosh & Hollenberg PLB(02);
Thiemann CQG(01)ht/00.
@ Simplicial lattice:
Rajeev ht/04-conf [2+1 Yang-Mills];
Christiansen & Halvorsen JMP(12)-a1006 [gauge-invariant discrete action].
@ 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];
Andersen a1210 [Lorentz-covariant lattice graph].
@ Duality: Oeckl & Pfeiffer NPB(01)ht/00 [and spin-foam models];
Grosse & Schlesinger IJTP(01) [categorical methods];
Mathur & Sreeraj PRD(16)-a1604 [SU(N) lattice theory and dual SU(N) spin model];
Riello PRD(18)-a1706 [self-dual phase space].
@ Symmetry-preserving coarse-graining schemes:
Tagliacozzo & Vidal PRB(11)-a1007;
Bahr et al NJP(11)-a1011 [and linearized gravity].
@ Related topics:
Boulatov CMP(97) [deformation];
Ma MPLA(00) [gluon propagator];
Burgio et al NPB(00) [\(\cal H\)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,
NPPS(04)hl/03 [localization];
Silva & Oliveira NPB(04) [Gribov copies];
Stannigel et al PRL(14)-a1308
[generating constrained dynamics via the Zeno effect using engineered classical noise];
Vilela Mendes IJMPA(17)-a1711 [consistent measure];
Knappe et al a1909 [observable algebra, stratified structure];
> s.a. entanglement entropy;
Tomboulis-Yaffe Inequality.
Generalizations
> s.a. spin networks [gauge networks].
* 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.
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