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This is indeed the same problem, from different viewpoints. The deep source of the problem is that the number of $A_\mu$ fields is higher than the actual physical degrees of freedom. Here are a few remarks: The operator you are trying to invert does not have an inverse because it is just a projection operator(you can check this by direct computation, the ...


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Just add a source term of the form $\int d^4 x J(x) \phi(x)$, or $\int d^4 x J^{\mu}(x) A_{\mu}(x)$ for a scalar field or $U(1)$ Maxwell field. $J$ is a function that you are free to specify.



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