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The Stone–von Neumann theorem (due to Marshall Stone and John von Neumann) says that there is – up to isomorphism – a unique irreducible unitary representation of the Heisenberg group on finitely many generators (equivalently: of the Weyl relations of the canonical commutation relations).
The analogous statement does not hold for infinitely many generators (as they appear in quantum field theory); this is Haag's theorem.
Explicitly, the canonical commutation relations on two generators (canonical coordinate $q$ and canonical momentum $p$) in the form
may be represented as unbounded operators on the Hilbert space of square integrable functions $L^2(\mathbb{R})$ on the real line by defining them on the dense subspace of smooth functions $\psi \colon \mathbb{R} \to \mathbb{C}$ as
where on the right we have the derivative along the canonical coordinate function on $\mathbb{R}$.
This is often called the Schrödinger representation (after Erwin Schrödinger, e.g. Redei), to be distinguished from “Schrödinger picture” which is a related but different concept.
The original articles are
John von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren , Mathematische Annalen (Springer Berlin / Heidelberg) 104: 570–578,
John von Neumann, Über Einen Satz Von Herrn M. H. Stone (in German), Annals of Mathematics, Second Series 33 (3): 567–573, ISSN 0003-486X
Marc Rieffel, On the uniqueness of the Heisenberg commutation relations (pdf)
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See also
Last revised on October 26, 2019 at 18:30:04. See the history of this page for a list of all contributions to it.