As far as I understand from posts such as this and this, when determining what is possible in a relativistic theory, Wigner's theorem tells us that we care about objects transforming under projective representations of the Lorentz group, because two state vectors differing by a phase are really equivalent.
However, I don't see how this is primarily relevant for QFT. Isn't it more important that the Lagrangian density is invariant under Lorentz transformations, rather than anything to do with the properties of the states?
