Scattering charge selection rule from global gauge symmetry ($[Q,S]=0⟹(Q_f-Q_i)⟨f|S|i⟩=0$)

For a conserved charge $Q$ and invariant vacuum, its global symmetry commutes with the S-matrix. Matrix elements between charge eigenstates give the displayed selection rule. Constant electromagnetic phases cancel at every QED vertex, and nonzero physical amplitudes conserve total charge. Local gauge invariance strengthens this global rule to the Ward-Takahashi identity.