= Goldstone current matrix element in a phase-only scalar theory
{c}
{title2=$\langle0|J^\mu(x)|\pi(p)\rangle=-ifp^\mu e^{-ipx}$}
For a canonically normalized massless phase <scalar field> with shift $\pi\mapsto\pi+f\alpha$, the <Noether current> is $J^\mu=f\partial^\mu\pi$. With invariant one-particle normalization, the displayed <matrix> element follows from $\langle0|\pi(x)|\pi(p)\rangle=e^{-ipx}$. Its divergence is $-fp^2e^{-ipx}=0$ on the massless shell, despite a nonzero current <matrix> element. This is the elementary current <coupling> of a <Goldstone boson>; conservation constrains its <momentum> contraction, not every component separately.
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