= Boundary-induced canonical transformation in Feynman gauge
{title2=$\pi^0=\widetilde\pi^0+\partial_iA_i,\quad\pi^i=\widetilde\pi^i-\partial_iA_0$}
In mostly-minus signature, adding the <boundary term> $\partial_\mu K^\mu$ with $K^\mu=\frac12(A_\nu\partial^\nu A^\mu-A^\mu\partial_\nu A^\nu)$ to the subtracted <Feynman gauge> density changes $\widetilde\pi^\mu=-\dot A^\mu$ to the displayed <canonical momenta>. For fields with vanishing spatial boundary terms, $\int K^0d^3x=\int A_0\partial_iA_i\,d^3x$, whose functional derivatives are these momentum shifts. They depend only on coordinates, so $[A_\mu,\pi^\nu]=i\delta_\mu{}^\nu\delta^3$ is unchanged. The only potentially nonzero mixed momentum bracket is $[\pi^0(x),\pi^i(y)]=i(\partial_{y^i}+\partial_{x^i})\delta^3(x-y)=0$. Thus both density representatives define <canonical commutation relations>, although their momenta must not be equated componentwise.
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