This question has been bugging me for a long time now. So the field equation for the electromagnetic field operator in QED is, brushing aside concerns about gauge fixing, Maxwell's equations. Similarly, in non-relativistic QM, the momentum operator obeys Newton's second law. In both these cases, there are situations - in the "classical limit" - where real physical experiments can be well-approximated by just doing classical mechanics with these differential equations: classical particle mechanics and classical electromagnetism work fairly well to describe a large number of physical situations.
Now, when you do QFT, the situation for the electron field looks very similar to the EM field: you have a field equation (i.e. the Dirac equation) for the field operator. However, I have never seen anyone try and use the Dirac equation as a classical field equation. I've seen full QFT calculations performed with the aid of "classical solutions" (thinking of instantons used for Euclidian Green's functions), but that's a bit different to just doing classical mechanics with the Dirac equation. Overall, while in the case of non-relativistic particle mechanics and the EM field the "classical limit" seems to be a really important, physical phenomenon, with the Dirac field it seems to be a highly formal mathematical oddity.
So my question is, does there exist a principle why the "classical limit" of the Dirac equation can never actually be reached, or is it a contingent issue to do with the sorts of physical situations we are actually interested in/can be produced in a laboratory?
