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In this article, I talk about a model which is known and often used to unifyboth massless and massive vector free-field theory, discuss the importance ofauxiliary scalar field introduced in the Lagrangian density that we consider inthis theory which turns out to be unphysical in massive vector field case whichI also show. But for massless theory in 3+1-dimensional case, I show that thisauxiliary field makes the physical Hilbert space non-trivial in terms ofintroducing an extra physical degree of freedom or an extra non-trivial kind ofpolarized state which is absent in Maxwell's electrodynamics theory(free-massless vector field theory). And this state is also shown to be presentin 1+1-dimensional theory whereas there is no on-shell photon in Maxwell'stheory of electromagnetism, which is a massless vector field theory.

A unified theory for both massless, massive vector fields and non-triviality of physical Hilbert space of this theory