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Would you please advise me what does gauge fixing term do (physical point of view) ?

Does it eliminate unnecessary spin components from lagrangian for example:

Vector particle has two (massless case) or three (massive case) degrees of freedom.

Vector itself has four, and a vector (four component object) can handle spin-0 and spin 1 mode so adding a gauge fixing term does it project out spin-0 component ?

Or absence of spin-0 part is due to specific choice of kinetic term (F_mu_nu)....

Similarly in gravity... summetric tensor has 10 independent components and it can handle spin-0 spin-1 spin-2 modes... so does employment of harmonic gauge eliminate unnecessary modes or what role has it in this business?

Thanks