Many posts here turn around the question how exactly spacetime symmetries are represented on (projective) Hilbert spaces in quantum mechanics. The question here is why quantum states should live in (projective) representation spaces of these symmetry groups in the first place.
After all, spacetime symmetries merely reflect a redundant and arbitrary choice we make to describe a system in classical mechanics. Physical objects themselves are of course unaffected by a symmetry transformation, that's the whole point of a symmetry. The prominent role of such "artifacts" of our classical description in the construction of quantum mechanics seems surprising (see, e.g., L. E. Ballentine: Quantum Mechanics, chapter 3, and S. Weinberg, The Quantum Theory of Fields, chapter 2).
- Is there a good a priori reason to expect that quantum states should live in (projective) representation spaces of spacetime symmetries, besides the fact that quantum mechanics "works", as we see a posteriori?
- Is there a formulation of quantum mechanics, or QFT for that matter, which gets rid of the redundancy (spacetime symmetries), instead of carrying it over from classical mechanics, sticking to "bad habits"?
Edit: In what sense do spacetime symmetries merely reflect a redundant and arbitrary choice of how we describe a classical system? By spacetime symmetries, I mean elements of the Galilean group or of the Poincaré group in non-relativistic and relativistic physics, respectively, acting on vector spaces and objects that live on them (tensors). Now, for example, rotating a system actively (including everything with which it interacts) does not have observable consequences. Equivalently, a passive rotation acts on the basis vectors and also transforms tensor components, such that the tensor itself as a geometrical object is invariant.
In this sense, applying spacetime symmetries, whether active or passive, to a system produces different descriptions of the system, all of which are equivalent in that they describe the same physical situation.