Why is it interesting?

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In the algebraic approach, the fundamental structure of a quantum theory is given by an abstract algebra of the canonical commutation relations, which can be given various different representations in terms of subalgebras of the bounded operators on a Hilbert space. Two such representations, each consisting of a Hilbert space $H$ and the set of bounded operators $\hat O$ defined on it, are unitarily equivalent if there is a (one-to-one, linear, norm preserving) map $U:G\to H'$ such that $(\forall i)(U^{-1}\hat{O}'_iU=\hat{O}_i)$. The Stone–von Neumann theorem guarantees that in the finite-dimensional case all irreducible representations of the abstract algebra are unitarily equivalent, but in the infinite-dimensional case there are unitarily inequivalent representations of the algebra.http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf

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Examples

Example1

Example2:

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History

• A Selective History of the Stone-von Neumann Theorem by Jonathan Rosenberg