Let $\mathfrak h$ be the Heisenberg algebra in dimension 1, generated by vectors $P$, $Q$ and $I$ satisfying $[P,Q] = I$, $[P,I] = [Q,I] = 0$. A representation of $\mathfrak h$ on a Hilbert space $X$ is a lie algebra homomorphism from $\mathfrak h$ to the set of linear operators on $X$ (with the commutator bracket). Such a representation is called integrable if there is a corresponding representation $\rho$ of the Heisenberg group $H$ such that $P = \frac{d}{dt}|_{t=0} \rho(\exp(tP))$ and $Q = \frac{d}{dt}|_{t=0} \rho(\exp(tQ))$, where $U$ and $V$ are the one-parameter unitary groups generated by the generators of the Heisenberg group. I am looking for an example of a representation of $\mathfrak h$ which is not integrable. Does anyone have one?


Let me define more accurately what a representation of $\mathfrak h$ is: $X\subseteq H$ is a dense vector subspace of a Hilbert space, $P,Q,I$ are linear operators on $X$ satisfying the commutation relations and are formally skew-adjoint, that is $\langle P\varphi,\psi\rangle=-\langle \varphi,P\psi\rangle,\dots$ for all $\varphi,\psi\in X.$

Take $H=L^2(0,1),$ $X=C_0^\infty(0,1),\ P=d/dt,\ Q=it,\ I=i.$ This representation is not integrable because $iP$ is symmetric and not essentially self-adjoint, thus $P$ is not a generator of a $1$-parameter group. But this representation can be extended to an integrable representation, namely Schrödinger representation on $\mathcal S(\mathbb R)\subseteq L^2(\mathbb R).$

The existence of representations which are not integrable and not extendable is mentioned in the paper Woronowicz "The quantum problem of moments. I." Rep. Mathematical Phys. 1 1970/1971 for the operator relation $AA^*-A^*A=Id.$ Setting $P=(A-A^*)/\sqrt 2,$ $\ Q=i(A+A^*)/\sqrt 2,$ $I=iId$ you get a non-extendable representation of $\mathfrak h.$

Explicit examples of non-extendable representations are contained in: K. Schmüdgen, "On the Heisenberg commutation relation. I." J. Funct. Anal. 50 (1983), no. 1, 8–49.


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