Consider $\hat{r} = (\hat{x}\_1,\hat{p}\_1,\ldots \hat{x}\_n,\hat{p}\_n)^\intercal$ with commutation relation
\begin{equation}
[\hat{r},\hat{r}^\intercal] = i\Omega.
\end{equation}


I want a simple statement like:



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> Any two irreducible representations of the canonical commutation relations are unitarily equivalent.
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so that I can prove that $\hat{r}$ and $S\hat{r}$ are unitarily equivalent when $S \in \operatorname{Sp}(2n,\mathbb{R})$. I seek out the Stone von Neumann theorem! But [here](http://www.math.umd.edu/~jmr/StoneVNart.pdf) I am told that



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> There is substantial literature on solutions to the commutation relations that do not necessarily integrate to unitary groups and for these the Stone von Neumann theorem breaks down completely...We bypass these difficulties by going to the Weyl integrated form.
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which provides a reason that the theorem is always stated along these lines:



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> Let $(V,\Omega)$ be a finite dimensional symplectic vector space. Let $(\mathcal{H},\hat{W}(y))$ and $(\mathcal{H}',\hat{W}(y))$ be strongly continuous, irreducible, unitary representations of the Weyl relations. Then $(\mathcal{H},\hat{W}(y))$ and $(\mathcal{H}',\hat{W}(y))$ are unitarily equivalent.
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Can I say something like: 'I don't want to consider the horrible cases', rigorously? Something like



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> Any two irreducible representations of the canonical commutation relations ''that are nice'' are unitarily equivalent
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The Stone-von Neumann theorem without exponentials?
From a [random review](http://www.ams.org/mathscinet-getitem?mr=217618):



> 
> There are many theorems that answer this question, due to [Rellich](http://www.ams.org/mathscinet-getitem?mr=22310), [Dixmier](http://www.ams.org/mathscinet-getitem?mr=101478), [Tillmann](http://www.ams.org/mathscinet-getitem?mr=169595), [Foiaş-Gehér-Sz.-Nagy](http://www.ams.org/mathscinet-getitem?mr=113147), [Kristensen-Mejlbo-Poulsen](http://www.ams.org/mathscinet-getitem?mr=165369), and [the reviewer](http://www.ams.org/mathscinet-getitem?mr=151867). (...) Some of the proofs obtain the result directly, mostly via considering the operator $A=2^{−1/2}(Q+iP)$ and its adjoint $A^\*$ which satisfy the formal relation $AA^\*=A^\*A+I$. Others deduce the Weyl-von Neumann relation. {It seems to the reviewer that none of these theorems is quite satisfactory.}
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