Canonical Commutation Relations: A quick proof of the Stone-von Neumann theorem and an extension to general rings
Bachir Bekka
TL;DR
The article introduces canonical commutation relations (CCR) relative to a unitary character $\lambda$ on a locally compact ring $R$ and identifies a Schrödinger pair as a canonical model. It proves a quick Stone–von Neumann theorem for local fields and extends it to general locally compact rings by analyzing inflations $U^{(\infty)}, V^{(\infty)}$ and employing Fourier analysis and Fell's absorption principle to show (approximate) equivalence to Schrödinger inflations. The results hinge on symmetry and duality conditions (Sym), (Iso), and (Dens); the paper demonstrates a spectrum of examples, including adèles, matrix rings, and finite/profinite rings, and culminates in a Heisenberg-group reformulation: CCR representations of $R^d$ correspond to unitary representations of $H_{2d+1}(R)$ with central character $\lambda$, with the Schrödinger representation serving as the canonical model. These findings provide a unifying, representation-theoretic view of CCR across a broad class of rings and expose a deep connection between Stone–von Neumann-type uniqueness and Fourier-analysis on noncommutative rings.
Abstract
Let $R$ be a (not necessary commutative) ring with unit, $d\geq 1$ an integer, and $λ$ 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 $λ$) if $U(a) V(b)= λ(a\cdot b)V(b) U(a)$ 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 $λ$). Provided that $λ$ 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 $(Φ_n)_n$ of unitary isomorphisms $Φ_n: \mathcal{H}_1^{(\infty)}\to \mathcal{H}_2^{(\infty)}$ such that $\lim_{n} \Vert U_2^{(\infty)}(a) - Φ_n U_1^{(\infty)}(a) Φ_n^{*}\Vert=0$ and $\lim_{n} \Vert V_2^{(\infty)}(b) - Φ_n V_1^{(\infty)}(b) Φ_n^{*}\Vert=0,$ uniformly on compact subsets of $R^d.$