The spin 1 $A^\mu$ field transforms under the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ representation of the Lorentz field. When restricted to the $SO(3)$ subgroup, it decomposes into the $0 \oplus 1$ representation of $SO(3)$. However, when using the massive Proca action, the spin $0$ degree of freedom is "killed off" by the equations of motion. (If the particle is massless, an extra degree of freedom is killed off, let's ignore that.)
What then was the point of using the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ representation to begin with? Why didn't we just use the $(1, 0)$ representation from the beginning?
Similarly, when we want to describe a spin 2 particle, we use the $(1, 1)$ representation and kill off degrees of freedom in the Lagrangian just like we did we the Proca action. What's the point? Why not just start with the $(0, 2)$ representation from the get-go?
To elaborate, there is a difference between the representations of the fields and the representations of the particles. However, which particle representations are actually present and "physical" are determined by the Lagrangian. The massive Proca Lagrangian kills of the spin $0$ particle. Is there some general way to construct Lagrangians that only give you the particles you want? And what's the point of keeping these un-physical representations around?
