Finite canonical representation interface

Assume the Axiom of Choice. Let $n\in {\mathbb{N}}_0$, let $V$ be a real vector space of dimension $2n$, and let $\omega :V\times V\to \mathbb{R}$ be bilinear and alternating, meaning $\omega (v,v)=0$ for every $v\in V$, and nondegenerate, meaning that $\omega (v,w)=0$ for every $w\in V$ implies $v=0$. Let $\lambda \in \mathbb{R}\setminus \{0\}$, and let $\mathcal{H}\mathrm{\ne }\{0\}$ be a complex Hilbert space. A unitary operator on $\mathcal{H}$ means a bijective complex-linear isometry. Suppose $W:V\to U(\mathcal{H})$ satisfies $W(0)=I$ and

$W(v)W(w)=e^{-i\lambda \omega (v,w)\mathrm{/}2}W(v+w)(v,w\in V),$

is regular in the sense that $t\mapsto W(tv)\xi$ is norm-continuous for every $v\in V$ and $\xi \in \mathcal{H}$, and is irreducible in the sense that its only closed subspaces $M\subseteq \mathcal{H}$ satisfying $W(v)M\subseteq M$ for every $v\in V$ are $\{0\}$ and $\mathcal{H}$. Choose a symplectic basis identifying $(V,\omega )$ with ${\mathbb{R}}^n\oplus {\mathbb{R}}^n$ and $\omega ((q,p),(q^{\mathrm{\prime }},p^{\mathrm{\prime }}))=q\cdot p^{\mathrm{\prime }}-p\cdot q^{\mathrm{\prime }}$. Then $W$ is unitarily equivalent, meaning that there is a unitary map $U:\mathcal{H}\to L^2({\mathbb{R}}^n)$ intertwining the two Weyl actions, to the Schrödinger system

$(W_{\lambda }(q,p)f)(x)=e^{i\lambda p\cdot (x-q\mathrm{/}2)}f(x-q)$

on $L^2({\mathbb{R}}^n)$. Consequently, there is exactly one unitary-equivalence class of nonzero irreducible regular Weyl systems with this fixed finite-dimensional phase space and parameter. No conclusion is asserted for infinite-dimensional phase spaces, nonregular systems, or zero central parameter.

Stone–von Neumann uniqueness for finite-dimensional regular Weyl systems

The Axiom of Choice