I have a few questions about projective representations on Hilbert spaces and Wigner theorem. I would like to make a bit more precise and rigorous my intuition behind it. The initial motivation is quantum physics but my questions are of a more mathematical nature. These are probably rather standard questions but I could not really find definitive 100% rigorous answers where I looked.
Let $H$ be a complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and $R(H)$ the space of rays of $H$, i.e. the set of unit vectors in $H$ modulo multiplication by $e^{i\alpha}$, $\alpha\in\mathbb{R}$. For a unit vector $\Psi\in H$, I will denote by $\underline{\Psi}$ its class in $R(H)$. It is then clear that for any $\underline{\Psi},\underline{\Phi}\in R(H)$, $(\underline{\Psi},\underline{\Phi})=|\langle\Psi,\Phi\rangle|^2$ does not depend on the choice of representatives $\Psi,\Phi\in H$ and thus defines a well defined map from $R(H)\times R(H) \to \mathbb{R}^+$. I then call a symmetry transformation a map $T:R(H)\to R(H)$ which preserves $(\cdot,\cdot)$. Wigner theorem then tells us that $T$ is induced on $R(H)$ by a unitary or anti-unitary map $T':H\to H$, which I call a lift of $T$.
I am more interested in the case not of a single symmetry transformation but of a group $G$ acting on $R(H)$ by symmetry transformations $T_g : R(H) \to R(H)$, $g\in G$. From now on let's say that $G$ is a connected finite-dimensional real Lie group. Through Wigner theorem, the maps $T_g$ of $R(H)$ ($g\in G$) "lift" to transformations $T'_g: H \to H$ (I take a choice of such $T'_g$ here). My first question is the following.
- As I supposed that $G$ is connected, I would have the impression that the $T'_g$ are unitary (and not anti-unitary). More generaly, if $G$ is not connected, I would imagine that if $g$ and $h$ belong to the same connected component of $G$ then $T'_g$ and $T'_h$ are either both unitary or both anti-unitary. Is this indeed the case in full generality? Does one have to assume anything further on $G$ or the action on $R(H)$ (if possible I would like to have assumptions direclty on the maps $T_g$ acting on $R(H)$ and not on the maps $T'_g$ acting on $H$)?
Let me now assume that all the $T'_g$, $g\in G$, are unitary. My other questions concern the structure that these $T'_g$ form. They satisfy: $T'_g T'_h = e^{i\omega(g,h)} T'_{gh}$, where $\omega$ is a 2-cocycle of $G$. This then defines a projective representation of $G$. From my understanding, the presence of this cocycle can have two main incarnations:
i) The first one is of a more algebraic nature and is typically that at the infinitesimal level, the mpas $T'_g$ induce a representation of a Lie algebra which is a central extension of $\mathfrak{g}=Lie(G)$ (then classified by the cohomology group $H^2(\mathfrak{g})$).
ii) The second one is of a more topological nature and expresses the fact that one can pass to a cover of the Lie group, like typically $SO(3)$ to $SU(2)$ with spinorial representations.
- My second question is basically if this is rigorously always true. More precisely, do we have something like that:
Let $G$ be a connected finite-dimensional real Lie group. Any central extension of $Lie(G)$ lifts to a unique connected and simply-connected Lie group: we denote by $E(G)$ the set of such groups. Then there is a 1-to-1 correspondence between unitary projective actions of $G$ and unitary representations of groups in $E(G)$.
If not do we have something similar with additional assumptions on $G$ or the actions on $R(H)$?
Do we know additional things on these unitary representations? Like (weak/strong) continuity, ...?
What happens if $G$ is infinite dimensional? If it is just a topological group and not necessarily a Lie group?
Thank you in advance for any answer, even partial, to these questions or any reference.