= Photon oscillator completeness and canonical brackets
{title2=$[a^\lambda_{\mathbf p},a^{\lambda'\dagger}_{\mathbf q}]=-\eta^{\lambda\lambda'}(2\pi)^3\delta^3(\mathbf p-\mathbf q)$}
With real complete <polarization vectors>, the <covariant photon Fock space> oscillator <commutators> and $\sum_{\lambda,\lambda'}\epsilon_\mu^\lambda\epsilon^{\nu\lambda'}\eta_{\lambda\lambda'}=\delta_\mu{}^\nu$ imply the equal-time <canonical commutation relations> $[A_\mu(\mathbf x),-\dot A^\nu(\mathbf y)]=i\delta_\mu{}^\nu\delta^3(\mathbf x-\mathbf y)$. The two mixed creation-annihilation terms add; the two coordinate-coordinate or momentum-momentum terms cancel after their polarization sums. The sign of the time-component oscillator is essential to the identity. <Boundary-induced canonical transformation in Feynman gauge> explains the momenta for the unsubtracted density.
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