nLab
lattice gauge theory

Context

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

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 3,1\mathbb{R}^{3,1} to Euclidean field theory on a lattice inside 3×S 1\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 4T^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 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 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-simulated 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)).

Properties

Sign problem

See at sign problem in lattice QCD.

References

General

The concept was introduced in

Introduction and review:

Rigorous discussion in view of the mass gap problem:

Visualization:

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

See also

Computer simulations

Introduction:

Account of computer simulation results in lattice QCD:

See also

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

reviewed in

Discussion specifically of numerical computation of form factors:

Relation to tensor networks:

Renormalization

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

reviewed in

Topological effects and instantons

Discussion of instantons in lattice QCd

For super Yang-Mills theories

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

General

Compactification to D=1D = 1

The BFSS matrix model:

Including the BMN matrix model:

Compactification to D=0D= 0

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