Canonical Commutation Relations: A quick proof of the Stone-von Neumann theorem and an extension to general rings

Abstract

Let $R$ be a (not necessary commutative) ring with unit, $d\ge 1$ an integer, and $\lambda$ a unitary character of the additive group $(R,+).$ A pair $(U,V)$ of unitary representations $U$ and $V$ of $R^d$ on a Hilbert space $\mathcal{H}$ is said to satisfy the canonical commutation relations (relative to $\lambda$) if $U(a)V(b)=\lambda (a\cdot b)V(b)U(a)$ for all $a=(a_1,\dots ,a_d),b=(b_1,\dots ,b_d)\in R^d$, where $a\cdot b={\sum }_{k=1}^d{a}_k{b}_k.$ We give a new and quick proof of the classical Stone von Neumann Theorem about the essential uniqueness of such a pair in the case where $R$ is a local field (e.g. $R=\mathbf{R}$). Our methods allow us to give the following extension of this result to a general locally compact ring $R$. For a unitary representation $U$ of $R^d$ on a Hilbert space $\mathcal{H},$ define the inflation $U^{(\infty )}$ of $U$ as the (countably) infinite multiple of $U$ on ${\mathcal{H}}^{(\infty )}={\oplus }_{i\in \mathbf{N}}\mathcal{H}$. Let $(U_1,V_1),(U_2,V_2)$ be two pairs of unitary representations of $R^d$ on corresponding Hilbert spaces ${\mathcal{H}}_1,{\mathcal{H}}_2$ satisfying the canonical commutation relations (relative to $\lambda$). Provided that $\lambda$ satisfies a mild faithful condition, we show that the inflations $(U_1^{(\infty )},V_1^{(\infty )}),(U_2^{(\infty )},V_2^{(\infty )})$ are approximately equivalent, that is, there exists a sequence $({\Phi }_n{)}_n$ of unitary isomorphisms ${\Phi }_n:{\mathcal{H}}_1^{(\infty )}\to {\mathcal{H}}_2^{(\infty )}$ such that ${\lim }_n\Vert U_2^{(\infty )}(a)-{\Phi }_n{U}_1^{(\infty )}(a){\Phi }_n^{\ast }\Vert =0$ and ${\lim }_n\Vert V_2^{(\infty )}(b)-{\Phi }_n{V}_1^{(\infty )}(b){\Phi }_n^{\ast }\Vert =0,$ uniformly on compact subsets of $R^d.$

14 pages