Lattice QCD: Wilson Loops & Confinement

U(1) gauge theory on a discrete spacetime lattice — area law signals quark confinement

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avg plaquette
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Wilson loop W(R,T)
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string tension σ
1.50
2
2
Wilson loop: W(R,T) = ⟨Tr P exp(i∮A·dx)⟩ around an R×T rectangle.
Area law: W(R,T) ~ exp(−σ·R·T) → confinement (large β = weak coupling = deconfinement).
Perimeter law: W ~ exp(−μ·P) signals a free phase. βc ≈ 1.0 for U(1) in 2D.
Links Uμ(x) = exp(iθ), θ∈[−π,π]. Plaquette: U□ = U₁U₂U₃†U₄†. Action: S = β Σ(1−Re U□).