Discretize a scalar field φ on a 1D lattice and take the continuum limit a→0. Watch the dispersion relation, propagator, and Green's function converge as lattice spacing decreases.
Lattice action: S = Σ_i [½(φ_{i+1}-φ_i)²/a² + ½m²φ_i²]a. Lattice dispersion: ω²(k) = (2/a²)(1-cos(ka)) + m² → k²+m² as a→0. Propagator: G(k) = 1/(ω²(k)). Lattice artifact: species doubling at k=π/a. Fermion doubling is a fundamental challenge in lattice QCD (Nielsen-Ninomiya no-go theorem 1981). Wilson term adds r·a·ψ̄∂²ψ to remove doublers.