Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F\_{\mu\nu}F^{\mu\nu}\right),\text{ where }F\_{\mu\nu} = \partial\_\mu A\_\nu - \partial\_\nu A\_\mu.$$The Lagrangian for $A\_\mu$, including a gauge fixing term, is$$\mathcal{L} = -{1\over4}F^2 - {\lambda\over2}(\partial\_\mu A^\mu)^2.$$Could anyone show or supply me a reference for the computation of the equal time commutators $[\dot{A}\_\mu(\textbf{x}), A\_\nu(\textbf{y})]$ and $[\dot{A}\_\mu(\textbf{x}), \dot{A}\_\nu(\textbf{y})]$ for general $\lambda$ and the fact that they simplify for $\lambda = 1$? I need this result, but I am not a physicist by training.



Gauge field quantization, electromagnetism
the answer is on page 189 of [Field Quantization](https://books.google.nl/books?id=VvBAvf0wSrIC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) by Greiner & Reinhardt (their $\zeta$ is your $\lambda$):


![](https://ilorentz.org/beenakker/MO/commutator.png)


