I'm trying to gain some intuition behind the definition that states a particle is an irreducible unitary representation of the restricted Poincare group (or more specifically, its double cover).
Let's say I have some Hilbert space of states and a priori no definition of what a particle is. I then decide to define a particle as a subset of the Hilbert space whose elements are indiscernible to an observer under any Poincare transformation he can make. Is there anything that says this subset is a vector space? If I accept that projective Hilbert spaces are the actual thing we are interested in, then it makes sense scalar multiples should also be in the subset. I'm more confused about the closure under addition.
I'm guessing that this probably has to do with the general structure of any quantum theory which states that true-false statements do not correspond to Borel subsets, but to closed linear subspaces. I could further my question by then asking, given that our quantum system's true-false statements don't form a Boolean algebra, what are the experimental facts that force the alternative to use closed linear subspaces.