Lattice Field Theory

Every method so far — Feynman diagrams, the RG, the $1/N$ and coupling expansions — is perturbative or semiclassical. Lattice field theory is the one systematically nonperturbative approach: discretize Euclidean spacetime onto a grid and evaluate the path integral numerically. It is the only ab initio tool for the strongly coupled regime of QCD — confinement, the hadron spectrum, and the QCD phase diagram.

Conventions: Euclidean signature; lattice spacing $a$, so momenta are cut off at $\sim \pi /a$.

The lattice as a regulator

Replace continuous spacetime by a finite hypercubic lattice of spacing $a$. This is a regulator — the shortest wavelength is $2a$, so $a$ provides a hard UV cutoff $\Lambda \sim \pi /a$ — but a special one:

• it is fully nonperturbative, defining the theory without reference to Feynman diagrams;
• it is gauge-invariant (via the Wilson formulation below);
• it makes the Euclidean path integral a finite-dimensional, convergent ordinary integral — computable by a computer.

The physical continuum theory is recovered as $a\to 0$, taken along a line of constant physics dictated by the renormalization group: thanks to asymptotic freedom, $a\to 0$ corresponds to bare coupling $g\to 0$ in a controlled way.

Gauge fields on the lattice: Wilson's construction

The key idea (Wilson, 1974) is to put the gauge field on the links between sites rather than the sites themselves. The dynamical variable is the link variable — the parallel transport across one link,

$$U_{\mu }(x)=e^{iag{A}_{\mu }(x)}\in G,$$

a group element, not an algebra element. Gauge-invariant quantities are traces of link products around closed loops. The simplest is the plaquette (the smallest loop), and the Wilson action is

$$S=\frac{2}{g^2}\sum _{\text{plaquettes}}\mathrm{Re}\mathrm{Tr}(1-U_{\text{plaq}})\stackrel{a\to 0}{\to }\frac{1}{4}\int F_{\mu \nu }^2,$$

reducing to Yang–Mills in the continuum. Because it is built from group elements, exact gauge invariance holds at finite $a$ — no gauge fixing or Faddeev–Popov ghosts are needed, a major advantage over perturbative regulators.

Confinement from the strong-coupling expansion

Wilson's original triumph: in the strong-coupling limit ($g\to \infty$), the lattice gauge theory exhibits confinement analytically. The Wilson loop — the trace of link variables around a large rectangular loop — obeys an area law,

$$⟨W(C)⟩\sim e^{-\sigma \text{Area}(C)},$$

which corresponds to a linearly rising potential $V(r)=\sigma r$ between static quarks: separating them costs energy proportional to distance, so they are permanently confined. This is the cleanest demonstration of confinement in any framework — inaccessible to perturbation theory, which sees only a Coulomb $1/r$.

Monte Carlo evaluation

For realistic couplings the path integral is done numerically. The Euclidean weight $e^{-S_E}$ is a positive probability measure (the payoff of Wick rotation), so observables are computed by importance sampling — generating field configurations with probability $\propto e^{-S_E}$ via Markov-chain Monte Carlo and averaging:

$$⟨O⟩=\frac{1}{Z}\int \mathcal{D}UO{e}^{-S_E}\approx \frac{1}{N}\sum _{\text{configs}}O[U_i].$$

This is literally the statistical-mechanics analogy made computational: the QFT vacuum is sampled like a thermal ensemble. Modern lattice QCD computes the light-hadron spectrum (proton, neutron, pion masses) from first principles to percent accuracy, confirming that most of the proton mass is QCD binding energy.

The fermion doubling problem

Putting fermions on the lattice is subtle. Naively discretizing the Dirac operator produces $2^d=16$ species instead of one — the doubling problem — and the Nielsen–Ninomiya theorem proves this is unavoidable: no lattice fermion action can be simultaneously local, chirally symmetric, and doubler-free. Workarounds each sacrifice something:

• Wilson fermions add a term that lifts the doublers but explicitly breaks chiral symmetry at finite $a$.
• Staggered (Kogut–Susskind) fermions keep a remnant chiral symmetry but scramble flavor.
• Domain-wall / overlap fermions realize exact lattice chiral symmetry at the cost of an extra dimension or a nonlocal operator.

The doubling problem is intimately tied to the chiral anomaly: the lattice cannot represent a single chiral fermion precisely because that would evade the anomaly the continuum insists on.

Limitations

• The sign problem. At finite baryon density or in real (Minkowski) time, $e^{-S_E}$ becomes complex and is no longer a probability — Monte Carlo fails. This blocks lattice access to the dense-QCD phase diagram (neutron-star interiors) and to real-time dynamics. It is a major open problem.
• Cost. Approaching $a\to 0$ and physical quark masses is enormously expensive; lattice QCD is among the largest consumers of supercomputer time in science.

Summary

• Lattice = discretize Euclidean spacetime; the only systematically nonperturbative regulator, exactly gauge invariant via link variables.
• Confinement emerges as a Wilson-loop area law at strong coupling.
• Monte Carlo importance-samples $e^{-S_E}$ — computing the hadron spectrum ab initio.
• Fermion doubling (Nielsen–Ninomiya) forces compromises; the sign problem blocks finite density and real time.

Where this leads

• The theory it solves nonperturbatively: QCD, asymptotic freedom and confinement.
• The Euclidean/statistical foundation: the generating functional.
• Finite-temperature phase transitions on the lattice: finite-temperature QFT.

References

• Wilson, Phys. Rev. D 10, 2445 (1974).
• Montvay & Münster, Quantum Fields on a Lattice.
• Gattringer & Lang, Quantum Chromodynamics on the Lattice.
• Nielsen & Ninomiya, Nucl. Phys. B 185, 20 (1981).