In quantum electrodynamics problems involving background fields, most text books usually makes the following substitution
$$A^\mu \rightarrow A^{'\mu}= A_0^{\mu}+A_{c}^{\mu}$$
Where $A_{c}$ is classical field satisfying Maxwell equations.
In the path integral formalism this seems reasonable to do since, $A_0^{\mu}$ and $A_{c}^{\mu}$ are $c$ numbers but in the canonical formalism $A_0^{\mu}$ are an operator and $A_{c}^{\mu}$ a $c$ number.
Is this a reasonable thing to do?
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