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(Background) In the scenario where a gauge symmetry is spontaneously broken and the gauge field eats the Goldstone boson to acquire a mass, the massive gauge field acquires a longitudinal polarization. I have seen the argument that the massless field has only two physical polarizations, and I have heard the hand-waving argument that now we can boost into the frame of the massive gauge boson, so I think I understand conceptually what has happened. However, I still don't understand how to show mathematically and rigorously that the new polarization has appeared, and I can't find any really decent explanation. This raises the following question

Given only the Lagrangian of some QFT, and assuming that one does not already know anything about the answer from more elementary arguments, how does one (mathematically, rigorously) find the polarizations of a given field appearing therein?

The proof that a massless vector field has only two physical polarizations (at least the one from my QFT I class) assumes the Lorentz gauge condition; this suggests the question

Can one find the polarizations of a field in general gauge?

Thank you in advance.

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