Limitations of describing QED interactions in the Coulomb gauge? When we work with the S-matrix operator to describe interactions between the quantized Maxwell field and a classical source or a Dirac field, are there any limitations one needs to keep in mind when working with the Coulomb instead of the Lorentz gauge?
Some thoughts:

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*The gauge condition should not affect physical predictions because it represents a mathematical (not physical) degree of freedom.

*The Coulomb gauge has a residual degree of freedom which is usually removed via $A_0=0$ but this is only correct if there are no static sources present.

*The Coulomb gauge condition has to be imposed for every reference frame but interactions might depend on the reference frame, see Unruh effect?

*Although, only transverse polarizations are physical, longitudinal and scalar polarizations participate as virtual particles in interactions.

 A: Having thought more about this question and reading more about the differences in the canonical quantization of Lorentz and Coulomb gauge in Field Quantization by W. Greiner and J. Reinhardt, I arrived at the following conclusion:
Calculating QED interactions in the Lorentz gauge is most general and should reduce to the results obtained in the Coulomb gauge after selecting a particular reference frame.
That said, calculating QED interactions in the Coulomb gauge is equally valid but limited to a particular reference frame.
I don't believe that doing QED in the Coulomb gauge "misses" specific interactions - at least it won't miss interactions we could measure in our particular reference frame. The reason being that QED in the Coulomb gauge is not manifest Lorentz-covariant BUT still Lorentz-invariant - just as Maxwell equations are Lorentz invariant but not manifest Lorentz-covariant (I mean the "standard vector version" of Maxwell equations). Performing a Lorentz-boost with Maxwell equations is possible although the transformations become very tedious.
Why should one use the Coulomb instead of the Lorentz gauge anyway if the later is more general? In the Gupta-Bleuler quantization, we only remove the unphysical scalar and longitudinal degrees of freedom at the very end (by projecting them out of the Hilbert space, I believe). Until then, we need to carry them over when calculating interactions.
The answer might not be very satisfying, so I will only keep it here until some more qualified answer is posted.
