I am trying to do the canonical quantization of an abelian vector field $A_\mu(x)$ in the $R_\xi$ gauge. So the gauge-fixing Lagrangian is given by $$ \mathscr{L}_{gf} = -\frac{1}{2\xi} (\partial_\mu A^\mu)^2. $$
Like in Coulomb gauge, should the commutation relations imposed on the field be modified such that there is explicit dependence on $\xi$? Or we still impose the canonical commutation relations given by $$ [A_\mu(t, \vec{x}), \pi_\nu(t, \vec{y})] = \iota\hbar \, \eta_{\mu \nu} \delta^{(3)}(\vec{x} - \vec{y}), $$ and the information of $\xi$ will reflect in an updated commutation relation for the oscillator modes derived from the above relation?
Moreover, do the completeness relation for the polarization vectors, $\epsilon_\mu(\vec{k}, \lambda)$, remain unchanged, given by $$ \sum_{\lambda, \lambda'} \eta_{\lambda \lambda'} \epsilon_\mu(\vec{k}, \lambda) \epsilon_\nu(\vec{k}, \lambda')^* = \eta_{\mu \nu}, $$ or it must be changed to include a $\xi$-dependence? We also need modifications to the completeness relation in Coulomb gauge (the details can be found in David Tong's QFT notes, at https://www.damtp.cam.ac.uk/user/tong/qft.html, on page 130). What is the intuition for the completeness relation among the polarization vectors?
