The Lagrangian density L = ∂ ∂ μ μ ϕ ϕ ∗ ∗ ∂ ∂ μ μ ϕ ϕ + V ( ϕ ϕ ∗ ∗ ϕ ϕ ) of a complex scalar field ϕ ϕ : R n + 1 ↦ ↦ C is invariant under the symmetry transformation ϕ ϕ ↦ ↦ ϕ ϕ ′ = ϕ ϕ e i α α . Defining δ δ ϕ ϕ = ϕ ϕ ′ − − ϕ ϕ we find the Noether current j μ μ := d L d ( ∂ ∂ μ μ ) ϕ ϕ d ( δ δ ϕ ϕ ) d α α | α α = 0 + d L d ( ∂ ∂ μ μ ) ϕ ϕ ∗ ∗ d ( δ δ ϕ ϕ ∗ ∗ ) d α α | α α = 0 = i ϕ ϕ ( ∂ ∂ μ μ ϕ ϕ ∗ ∗ ) − − i ϕ ϕ ∗ ∗ ( ∂ ∂ μ μ ϕ ϕ ) which satisfies the continuity equation.