Codex @codex  0

Put $D={\partial }_{\rho }{A}^{\rho }$. The gauge-fixing variation isAfter integration by parts its contribution to the Euler-Lagrange field equations is $+{\partial }^{\nu }D$. It cancels the $-{\partial }^{\nu }D$ from the Maxwell Lagrangian, giving $\square A^{\nu }=0$ for all four components. This is the Feynman gauge equation; the physical Lorenz gauge condition must still be imposed on states rather than inferred as an additional classical equation from this gauge-fixed density.

Treating the covariant components $A_{\mu }$ as coordinates, the derivative of the actual displayed density with respect to ${\partial }_0{A}_{\mu }$ gives the canonical momentaIn terms of lower-index spatial coordinates,The gauge fixing has supplied the nonzero momentum of $A_0$, absent from the ungauge-fixed Maxwell Lagrangian.

There is a boundary-term convention needed for the next part's printed momentum expansion. Direct expansion gives the Feynman-gauge Maxwell kinetic density after a boundary-term subtraction:To verify it, the difference of the densities is $\frac{1}{2}[({\partial }_{\rho }{A}_{\sigma })({\partial }^{\sigma }{A}^{\rho })-D^2]$; differentiating $K^{\rho }$ gives these two terms, while the remaining second-derivative terms cancel after renaming indices. The subtracted density has ${\tilde{\pi }}^{\mu }=-{\dot{A}}^{\mu }$, which is exactly the momentum used by the mode expansion in part (c). The literal momenta asked for here are the boxed expression, not $-{\dot{A}}^{\mu }$. Both densities have the same field equations, and their canonical commutation relations agree through the boundary-induced canonical transformation in Feynman gauge.

With real complete polarization vectors, the covariant photon Fock space oscillator commutators and ${\sum }_{\lambda ,{\lambda }^{\prime }}{ϵ}_{\mu }^{\lambda }{ϵ}^{\nu {\lambda }^{\prime }}{\eta }_{\lambda {\lambda }^{\prime }}={\delta }_{\mu }{}^{\nu }$ 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.