nLab lattice gauge theory

Idea

Lattice gauge theory (introduced in Wegner 71, 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 us 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 simulations provide, besides actual experiment, key insights into the non-perturbative nature of the theory, such as its instanton sea (Gruber 13) and notably the phenomenon 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-simulated experiment”. While it allows us to see the 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)).

Properties

Sign problem

See at sign problem in lattice QCD.

Related concepts

• Symanzik effective field theory

• lattice model

• non-perturbative effect

• lattice renormalization

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

• QCD cosmology

• AdS/QCD correspondence

• Euclidean field theory

• string bit model

References

General

The concept was introduced in

• Franz Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, Journal of Mathematical Physics 12, 2259 (1971) (doi:10.1063/1.1665530)

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

(Wilson 1974 envisioned to take the continuum limit by way of block spin transformations?, but Creutz 1979 successfully applied Monte Carlo methods, and that has been the method of choice ever since.)

Introduction and review:

• 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)

• Brambilla et al., Section 3.3.1 of: QCD and strongly coupled gauge theories - challenges and perspectives, Eur Phys J C Part Fields. 2014; 74(10): 2981 (arXiv:1404.3723, doi:10.1140/epjc/s10052-014-2981-5)

• Mattia Dalla Brida, Past, present, and future of precision determinations of the QCD parameters from lattice QCD (arXiv:2012.01232)

• Franz Gross, Eberhard Klempt et al., Chapter 4 of: 50 Years of Quantum Chromodynamics, EJPC [arXiv:2212.11107]

Rigorous discussion in view of the mass gap problem:

• Sourav Chatterjee, A probabilistic mechanism for quark confinement, Comm. Math. Phys. 2020 (arXiv:2006.16229)

Visualization:

• James Biddle et al., Publicising Lattice Field Theory through Visualisation [arXiv:1903.08308]

See also:

• Wikipedia, Lattice gauge theory

• Wikipedia, Lattice QCD

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 A Vol. 31, No. 22, 1643006 (2016) (arXiv:1604.05421)

With strong electromagnetic background fields:

• Gergely Endrodi, QCD with background electromagnetic fields on the lattice: a review [arXiv:2406.19780]

For topological phases of matter via higher lattice gauge theory:

• Joe Huxford, The higher lattice gauge theory model for topological phases of matter, PhD thesis, Oxford (2022) [uuid:e789bf29-179b-4b79-9168-605bf8c035ba, pdf]

• Joe Huxford, Steven H. Simon, Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview [arXiv:2202.08294]

Further on higher lattice gauge theory:

• Juan Orendain, José A. Zapata, Higher homotopy and lattice gauge fields [arXiv:2311.02363]

Comparison to holographic QCD:

• Nicolas Kovensky, Andreas Schmitt: Thermal pion condensation: holography meets lattice QCD [arXiv:2407.12641]

Computer simulations

Introduction:

• Masanori Hanada, Markov Chain Monte Carlo for Dummies (arXiv:1808.08490)

• Anosh Joseph, Markov Chain Monte Carlo Methods in Quantum Field Theories: A Modern Primer, Springer Briefs in Physics 2020 (arXiv:1912.10997)

Account of computer simulation results in lattice QCD:

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

The idea of using Monte Carlo method for lattice gauge theory originates with:

• Michael Creutz, Confinement and the Critical Dimensionality of Space-Time, Phys. Rev. Lett. 43 (1979) 553 [doi:10.1103/PhysRevLett.43.553]

• Michael Creutz, Monte Carlo study of quantized $SU(2)$ gauge theory, Phys. Rev. D 21 (1980) 2308-2315 [doi:10.1103/PhysRevD.21.2308, pdf]

• Michael Creutz, Monte Carlo study of renormalization in lattice gauge theory Phys.Rev. D 23 (1981) 1815 [doi:10.1103/PhysRevD.23.1815, pdf&rbrack

• Michael Creutz, Laurence Jacobs, Claudio Rebbi, Monte Carlo computations in lattice gauge theories, Physics Reports 95 4 (1983) 201-282 [doi:10.1016/0370-1573(83)90016-9, 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

Discussion specifically of numerical computation of form factors:

• B.B. Brandt, S. Capitani, M. Della Morte, D. Djukanovic, J. Gegelia, G. von Hippel, A. Juttner, B. Knippschild, H.B. Meyer, H. Wittig, Form factors in lattice QCD, Eur. Phys. J. ST 198:79-94, 2011 (arXiv:1106.1554)

Relation to tensor networks:

• Luca Tagliacozzo, Alessio Celi, Maciej Lewenstein, Tensor Networks for Lattice Gauge Theories with continuous groups, Phys. Rev. X 4, 041024 (2014) (arXiv:1405.4811)

• M.C. Bañuls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C.A. Muschik, B. Reznik, E. Rico, Luca Tagliacozzo, K. Van Acoleyen, Frank Verstraete, U.-J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller:

  Simulating Lattice Gauge Theories within Quantum Technologies

  (arXiv:1911.00003)

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)

and via the homotopy theory of bundle gerbes:

• Jing-Yuan Chen, Instanton Density Operator in Lattice QCD from Higher Category Theory [arXiv:2406.06673]

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)

Including the BMN matrix model:

• Hrant Gharibyan, Masanori Hanada, Masazumi Honda, Junyu Liu, Toward simulating Superstring/M-theory on a quantum computer (arXiv:2011.06573)

• Georg Bergner, Norbert Bodendorfer, Masanori Hanada, Stratos Pateloudis, Enrico Rinaldi, Andreas Schäfer, Pavlos Vranas, Hiromasa Watanabe, Confinement/deconfinement transition in the D0-brane matrix model – A signature of M-theory?, JHEP 05 (2022) 096 [arXiv:2110.01312]

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)