nLab
lattice gauge theory

Idea

Lattice gauge theory (introduced in Wilson 74) is gauge theory (Yang-Mills theory, such as quantum chromodynamics) where continuum spacetime is replaced by a discrete lattice, hence a lattice model for gauge field theory.

Usually this is considered after Wick rotation from Minkowski spacetime $\mathbb{R}^{3,1}$ to Euclidean field theory on a lattice inside $\mathbb{R}^3 \times S^1$, and typically one further identifies the spatial directions periodically to arrive at Euclidean gauge field theory on a lattice inside the 4-torus $T^4$.

This discretization and further compactification has the effect that the would-be path integral of the theory becomes an ordinary finite- (albeit high-)dimensional integral, hence well defined and in principle amenable to explicit computation.

This allows to consider (Wick-rotated) path integral quantization at fixed lattice spacing, this being, in principle, a non-perturbative quantization, in contrast to perturbative quantum field theory in terms of a Feynman perturbation series. On the other hand, much of the subtlety of the latter now appears in issues of taking the continuum limit where the the lattice spacing is sent to zero. In particular, different choices of discretizing the path integral over the lattice correspond to the renormalization-freedom seen in perturbative quantum field theory.

Hence lattice gauge theory lends itself to brute-force simulation of quantum field theory on electronic computers, and the term is often understood by default in this sense. See Fodor-Hoelbling 12 for a good account.

Since the explicit non-perturbative formulation of Yang-Mills theories such as QCD is presently wide open (see the references at mass gap and at quantization of Yang-Mills theory) these numerical simulation provide, besides actual experiment, key insight into the non-perturbative nature of the theory, such as its instanton sea (Gruber 13) and notably the phenomenonon of confinement/mass gap and explicit computation of hadron masses (Durr et al. 09, see Fodor-Hoelbling 12, section V)

Despite the word “theory”, lattice gauge theory is more like “computer-simpulated experiment”. While it allows to see phenomena of QCD, it usually cannot provide a conceptual explanation, and of course not a mathematical derivation of problems such as confinement/mass gap. Lattice gauge theory is to the confinement/mass gap-problems as explicit computation of zeros of the Riemann zeta-function is to the Riemann hypothesis (see there)).

Related concepts

• lattice model

• non-perturbative effect

• lattice renormalization

• Discussion of QCD instantons in LGT includes (Moore 03, section 7, Gruber 13)

• AdS/QCD correspondence

• Euclidean field theory

• string bit model

References

The concept was introduced in

• Kenneth Wilson, Confinement of quarks, Phys. Rev. D10, 2445, 1974 (doi:10.1103/PhysRevD.10.2445)

General

• S. Gupta, Introduction to lattice field theory, March 2011, (pdf)

• G. Münster, M. Walzl, Lattice Gauge Theory - A short Primer (arXiv:hep-lat/0012005)

• Kenneth Wilson, The Origins of Lattice Gauge Theory, (arXiv:hep-lat/0412043)

• Guy Moore, Informal lectures on lattice gauge theory, 2003 (pdf)

• Kasper Peeters, Marija Zamaklar, section 5 of Euclidean Field Theory, Lecture notes 2009-2011 (web, pdf)

Visualization:

• James Biddle et al. Publicising Lattice Field Theory through Visualisation (arXiv:1903.08308)

Relation to string theory/M-theory (such as via BFSS matrix model) in view AdS-CFT duality:

• Masanori Hanada, What lattice theorists can do for superstring/M-theory, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) (arXiv:1604.05421)

See also

• Wikipedia, Lattice gauge theory

• Wikipedia, Lattice QCD

Computer simulations

A good general account of computer simulation of lattice QCD is in

• Zoltan Fodor, Christian Hoelbling, sections II-IV of Light Hadron Masses from Lattice QCD, Rev. Mod. Phys. 84, 449 (2012) (arXiv:1203.4789)

See also

• Michael Creutz, Monte Carlo study of quantized SU(2) gauge theory Phys. Rev. D21 (1980) 2308-2315 (journal, pdf)

• Michael Creutz, Monte Carlo study of renormalization in lattice gauge theory Phys.Rev. D23 (1981) 1815 (pdf)

• Michael Creutz, Laurence Jacobs, Claudio Rebbi, Monte Carlo computations in lattice gauge theories, Volume 95, Issue 4, April 1983, Pages 201–282 (pdf)

Specifically computation of hadron-masses (see mass gap problem) in lattice QCD is reported here:

• S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, S.D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K.K. Szabo, G. Vulvert,

  Ab-initio Determination of Light Hadron Masses,

  Science 322:1224-1227,2008 (arXiv:0906.3599)

reviewed in

• Fodor-Hoelbling 12, section V of

Renormalization

A proposal for a rigorous formulation of renormalization in lattice gauge theory is due to

• Tadeusz Balaban, Renormalization group approach to lattice gauge field theories: I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions, Communications in Mathematical Physics, Volume 109, Issue 2, pp.249-301 (web)

• …

reviewed in

• Jonathan Dimock, The renormalization group according to Balaban, I. Small fields, Rev. Math. Phys., 25, 1330010 (2013) (doi:10.1142/S0129055X13300100)

• …

Topological effects and instantons

Discussion of instantons in lattice QCd

• Florian Gruber, Topology in dynamical Lattice QCD simulations, 2013 (web, pdf)

For super Yang-Mills theories

Lattice simulation of torus-KK-compactifications of 10d super Yang-Mills theory and numerical test of AdS/CFT:

General

• Anosh Joseph, Review of Lattice Supersymmetry and Gauge-Gravity Duality (arXiv:1509.01440)

• Masanori Hanada, What lattice theorists can do for superstring/M-theory, International Journal of Modern Physics AVol. 31, No. 22, 1643006 (2016) (arXiv:1604.05421)

Compactification to $D = 1$

The BFSS matrix model:

• Veselin G. Filev, Denjoe O’Connor, The BFSS model on the lattice, JHEP 1605 (2016) 167 (arXiv:1506.01366)

• Masanori Hanada, Paul Romatschke, Lattice Simulations of 10d Yang-Mills toroidally compactified to 1d, 2d and 4d (arXiv:1612.06395)

Compactification to $D= 0$

The IKKT matrix model and claims that it predicts spontaneous KK-compactification of the $D = 10$ non-perturbative type IIB string theory/F-theory to $D = 3+1$ macrocopic spacetime dimensions:

• S.-W. Kim, J. Nishimura, and A. Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions, Phys. Rev. Lett. 108, 011601 (2012), (arXiv:1108.1540).

• S.-W. Kim, J. Nishimura, and A. Tsuchiya, Late time behaviors of the expanding universe in the IIB matrix model, JHEP 10, 147 (2012), (arXiv:1208.0711).

• Yuta Ito, Jun Nishimura, Asato Tsuchiya, Large-scale computation of the exponentially expanding universe in a simplified Lorentzian type IIB matrix model (arXiv:1512.01923)

• Toshihiro Aoki, Mitsuaki Hirasawa, Yuta Ito, Jun Nishimura, Asato Tsuchiya, On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model (arXiv:1904.05914)