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Lattice Gauge Theory (Wilson 1974): To regularize gauge theories non-perturbatively,
fields are placed on a spacetime lattice. Link variables U_μ(x) ∈ U(1) (here: e^{iθ})
carry the gauge field; plaquettes (elementary squares) encode the field strength.
The Wilson action is S = β·Σ_□ (1 − Re[U_□]) where U_□ = product of links around a plaquette.
The Wilson loop W(R,T) = ⟨Tr U around R×T rectangle⟩ is the order parameter:
in the confined phase (small β, strong coupling), W ~ e^{−σRT} (area law) with string tension σ;
in the deconfined phase (large β, weak coupling), W ~ e^{−c(R+T)} (perimeter law).
Colors show link phases θ; hot colors = excited (disordered), cool = ordered. A Monte Carlo
Metropolis update thermalizes the system at temperature T=1/β.