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.

$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$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.

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 $CCR(V,\omega)$ 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.

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.