Every irreducible strongly continuous representation of the Weyl relations with fixed nonzero central character is unitarily equivalent to the translation-modulation representation on . The Gaussian projection in a Weyl representation supplies a cyclic vector whose Weyl-orbit inner products agree with those of a Gaussian in the standard representation. Matching those orbit vectors gives the unitary intertwiner, with no appeal to a classification assertion in place of the proof.
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The Stone–von Neumann theorem is a significant result in the theory of representations of groups, particularly in the context of quantum mechanics and mathematical physics. It pertains to the representation of the canonical commutation relations (CCR) associated with the position and momentum operators in quantum mechanics. **Statement of the Theorem:** The Stone–von Neumann theorem states that, under certain conditions, any two irreducible representations of the canonical commutation relations are unitarily equivalent.