Is there a reason why the subset of our Hilbert space that corresponds to a particle is a vector subspace?

Question

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.

Answer 1

I think you can understand the statement if you consider a more general formulation of a quantum system, and states, in terms of $C^*$-algebras.

Given a $C^*$-algebra (e.g. the group algebra of the Poincaré group), then you can construct a $*$-representation of it in some Hilbert space by the Gelfand-Naimark-Segal construction (actually there is a $*$-isomorphism between any $C^*$ algebra and an algebra of bounded operators on a Hilbert space).

The quantum states are defined as the positive linear functionals of norm one on the $C^*$-algebra. Since there is also a $1-1$ correspondence between the non-degenerate $*$-representations of the $C^*$ group algebra (of a locally compact group) and the strongly continuous unitary representations of the group itself, you see that the linear functionals of the $C^*$ Poincaré algebra correspond to the states of any irreducible representation of the Poincaré group.

The structure of the space of states is inherited from the structure of the $C^*$-algebra since it is a subset of the topological dual (you can consider many topologies on it, and it is a convex set). In particular, the structure of vector space is naturally defined, but only the convex sum of two states is again a state, because of the norm one condition.

Answer 2

I think that Wigner's kinematic definition of particles is strongly inspired by the results of representation theory of the special Poincaré group $P=SL(2,\mathbb C)\ltimes\mathbb R^4$. It turns out that this group is of type I, and therefore every representation of $P$ decomposes in a direct sum/integral of irreducible representations. This is generally true for compact groups (see Peter-Weyl theorem), but fails in general for non-compact groups, and $P$ is known to be locally compact, non-compact. Therefore one can restrict the analysis of the representations of $P$ to just the irreducible ones. By Schur's lemma, the centre of such representations contains only multiples of the identity. Examples of such operators are the Casimir operators $p^\mu p_\mu$ and $W_\mu W^\mu$, where $p^\mu$ is the 4-momentum operator and $W^\mu$ is the Pauli-Lubanski pseudovector. Since they commute with everything in the representation, and they are self-adjoint, there are real numbers $m$ and $j$ such that $p_\mu p^\mu = m^2I$ and $W_\mu W^\mu = j^2I$. These two numbers have been interpreted by Wigner as the mass of a particle and its spin (here the definition of particle is purely kinematic, in the sense the even a state of bounded particles, like an atom, is considered a particle).

Given now a generic representation of the group $P$ on some Hilbert space $H$, one has a decomposition of $H$ into invariant subspaces (because $P$ is of type I), which can be indexed by pairs $(m,j)$ and a possible multiplicity, and it seems quite natural to interpret each of these subspaces as those associated to states of particles with mass $m$ and spin $j$.