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$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$Let $V$ be a real vector space equipped with an antisymmetric bilinear form $\omega$. Recall that there is a $C^\ast$ algebra $$\CCR(V,\omega)$$ (for "canonical commutation relations") freely generated by elements $$\{W(f) \mid f \in V\}$$ subject to the relations

  1. $W(-f) = W(f)^\ast$ and

  2. $W(f)W(g) = e^{i\omega(f,g)} W(g)W(f)$

Let $\pi: \CCR(V,\omega) \to B(H)$ be a representation of $\CCR(V,\omega)$ on a Hilbert space $H$. Then from (1) and (2) we see that $\pi$ induces a projective unitary representation of $V$ on $H$, i.e., a group homomorphism $V \to \PU(H)$, where $V$ is considered as a group under addition.

Question: Let $\phi: V \to \PU(H)$ be a projective representation of the additive group of a real vector space $V$ (perhaps satisfying some continuity conditions?). Then does there exist an antisymmetric form $\omega$ on $V$ such that $\phi$ arises from a representation of $\CCR(V,\omega)$ on $H$ (perhaps with corresponding continuity conditions)?

Background:

My understanding is that in physics, one is interested in the case where $V$ is the underlying real vector space of a Hilbert space which serves as the state space for a single particle of some sort, with $\omega(f,g) = \Im \langle f,g \rangle$; in this case there is an interesting representation of $\CCR(V,\omega)$ on the bosonic Fock space associated to $V$ from which free field operators and creation / annihilation operators are constructed.

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    $\begingroup$ Yes, to an extent. See Baggett-Kleppner (1973) whose Introduction starts: “The classification of the irreducible multiplier representations of a real vector group is an easy consequence of the Stone-von Neumann theorem. With no loss of generality one may assume that the multiplier is the exponential of an alternating bilinear form.” (And they go on to generalize to “quite arbitrary locally compact abelian groups.” That includes discrete [no continuity condition] but not $\infty$-dimensional topological vector groups...) $\endgroup$
    – Francois Ziegler
    Commented Apr 15, 2020 at 19:06
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    $\begingroup$ @FrancoisZiegler, I appreciate your re-posting the edited comment so that I look prophetic. $\endgroup$
    – LSpice
    Commented Apr 15, 2020 at 19:26
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    $\begingroup$ I am very predictable :-) $\endgroup$
    – Francois Ziegler
    Commented Apr 15, 2020 at 19:32
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    $\begingroup$ @FrancoisZiegler Thanks! For the finite-dimensional case, I'm having trouble seeing how this follows from the Stone-von Neumann theorem. Is it as obvious as they indicate? Regarding infinite dimensions, perhaps I need to be a bit more precise in my question, but I am interested in this case. The Stone-von Neumann theorem isn't available here. $\endgroup$
    – Tim Campion
    Commented Apr 15, 2020 at 20:21
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    $\begingroup$ If you have access to the paper, their argument for $\mathbf R^n$ is on p. 314 with reference to Kleppner (1965). $\endgroup$
    – Francois Ziegler
    Commented Apr 15, 2020 at 20:41

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