Ward Identity & Noether's Theorem

U(1) gauge symmetry → current conservation ∂_μ J^μ = 0. Ward-Takahashi identity: k_μ Γ^μ = G^{-1}(p+k) − G^{-1}(p).

Phase θ: 0.0

Coupling e: 1.0

Gauge Transformation ψ → e^{iθ}ψ

Noether Current J^μ

Ward Identity k_μΓ^μ = G^{-1}(p+k)−G^{-1}(p)