In many textbooks introducing to QFT, the definition of elementary particles never implies a projective representation. I don't understand why, and that's the subject of my question.
Context
We know that (quantum) physical states $|\psi\rangle$ correspond to rays in a projective Hilbert space$^{[1]}$. Since our physical theory is naturally described by a projective space, it is useful to look at projective representations$^{[2]}$ of the groups of interest (in particular a group like the Poincaré group or any group over maximally symmetric spaces, I suppose).
The following lemma exists:
$$ \text{PGL}(n, V) = \text{GL}(n, V)/V^* \cong \text{SL}(n, V)/C_n $$
where $C_n$ is the group of $n$th-roots of unity. For example, the projective representation of $\text{SO}^+(1, 3)$ is given by a morphism
$$ \varphi : \text{SO}^+(1, 3) \to \text{SL}(2, \mathbb C) / \{\pm 1\}. $$
To determine $\varphi$ and projective representations, in general, we can apparently use a theorem, Bargmann's theorem, which states that any projective and unitary representation of a group $G$ can be deduced from unitary and irreducible representations of the universal cover $\text{SL}(n, V)/C_n$ (source: Unitary representations of the Poincare group, by Vincent Bouchard). I don't know this theorem, but apparently it exists.
Question
Is the brief explanation I've given the reason we want to study unitary and irreducible representations of the Poincaré group? Most of the time, explanations motivate this with the arguments involved in a previous question and it never invokes projective representations, which strikes me as rather odd given that projective Hilbert spaces are axiomatic (I think).
So, do the need of studying the Poincaré's irreps follows naturally from the result of Bargmann's theorem? If not, why not? If so, what is the link between "classic motivation" (as asked in the shared question above) and the one presented here?
PS: Bargmann's theorem is mentioned several times in the wikipedia page and on the internet, but I can't seem to find a clear statement. Perhaps it would help me to understand or refuted what my question implies.
[1] The states of a physical system correspond (but not necessarily uniquely) to vectors in a Hilbert space $H$. Physically, the $\psi$ state and the $\lambda\psi$ state (with $\lambda\neq0$) represent the same state and the requirement that $\langle\psi|\psi\rangle = 1$ does not completely determine the $\psi$ state (as it can be multiplied by a $\lambda$ of unit norm). Such a $\lambda$ is written $$ \lambda = e^{i\phi} $$ where $\phi$ is a phase. A projective Hilbert space $P(H)$ of a Hilbert space $H$ corresponds to the set of equivalence classes $[\varphi]$ of non-zero $\varphi\in H$ such that $|\varphi\rangle = \lambda|\psi\rangle$. In our cass, the equivalence class of $\psi$ is the ray $$ R_\psi := \left\{ |\varphi\rangle \in H \;|\; \exists\lambda\in\mathbb C_0 \; |\varphi\rangle = \lambda|\psi\rangle \right\}. $$ In other words, each element of such a ray is statistically indistinguishable (according to Born's rule). A ray is more fundamental than a state $|\psi\rangle$ because it takes into account all physically equivalent states.
[2] A projective representation of any group $G$ on a vector space $V$ with field $K$ is a group homomorphism $G \to \text{PGL}(V)$ where $$ \text{PLG}(V) := \text{GL}(V)/K^* $$ where $K^*$ stands for the normal subgroup of non-zero multiples of the identity.
