This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO.

I've been reading Singer's Linearity, Symmetry, and Prediction in the Hydrogen Atom and am trying to come to terms with the main physical (as opposed to mathematical) argument of the text. The argument posits, if I understand it correctly, that a quantum system described by a Hilbert space $H$ on which a group $G$ of symmetries acts by unitary transformations should have the property that its "elementary states" "are" irreducible subrepresentations of the representation of $G$ on $H$. She begins this argument in section 5.1:

Invariant subspaces are the only physically natural subspaces. Recall from Section 4.5 that in a quantum system with symmetry, there is a natural representation $(G, V, \rho)$. Any physically natural object must appear the same to all observers. In particular, if a subspace has physical significance, all equivalent observers must agree on the question of a particular state's membership in that subspace.

and continues it in section 6.3:

We know from numerous experiments that every quantum system has *elementary states*. An elementary state of a quantum system should be **observer-independent**. In other words, any observer should be able (in theory) to recognize that state experimentally, and the observations should all agree. Secondly, an elementary state should be indivisible. That is, one should not be able to think of the elementary state as a superposition of two or more "more elementary" states. If we accept the model that every recognizable state corresponds to a vector subspace of the state space of the system, then we can conclude that elementary states correspond to irreducible representations. The independence of the choice of observer compels the subspace to be invariant under the representation. The indivisible nature of the subspace requires the subspace to be irreducible. So elementary states correspond to irreducible representations. More specifically, if a vector $w$ represents an elementary state, then $w$ should lie in an *irreducible* invariant subspace $W$, that is, a subspace whose only invariant subspaces are itself and $0$. In fact, every vector in $W$ represents a state "indistinguishable" from $w$, as a consequence of Exercise 6.6.

(For people who actually know their quantum, Singer is ignoring the distinction between representations and projective representations until later in the book.)

My first problem with this argument is that Singer never gives a precise definition of "elementary state." My second problem is that I'm not sure what physical principle is at work when she posits that physically natural subspaces and elementary states should be observer-independent (i.e. invariant under the action of $G$). What underlying assumption of quantum mechanics, or whatever, is at work here? Why should a mathematician without significant training in physics find this reasonable? (I have the same question about the identification of elementary particles with irreducible representations of the "symmetry group of the universe," so any comments about this physical argument are also welcome.)

Singer goes on to use this assumption to deduce the number of electrons that fill various electron orbitals, and I won't be able to convince myself that this makes sense until I understand the physical assumption that allows us to use irreducible representations to do this.

numberof particles in a given state is observer-dependent! This does not happen in relativistic quantum field theory (i.e., in Minkowski spacetime), of course. $\endgroup$2more comments