nLab Stone-von Neumann theorem



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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 algebra, Weyl relations, 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 qq and canonical momentum pp) in the form

[q,p]=i [q,p] = i \hbar

may be represented as unbounded operators on the Hilbert space of square integrable functions L 2()L^2(\mathbb{R}) on the real line by defining them on the dense subspace of smooth functions ψ:\psi \colon \mathbb{R} \to \mathbb{C} as

(qψ)(x)xψ(x)AAAA(pψ)(x)ixψ(x), (q \psi)(x) \coloneqq x \psi(x) \phantom{AAAA} (p \psi)(x) \coloneqq -i \hbar \frac{\partial}{\partial x} \psi(x) \,,

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, cf. eg. Redei (to be distinguished from “Schrödinger picture” which is a related but different concept).


The original articles:


  • Miklós Rédei, Von Neumann’s proof of Uniqueness of Schrödinger representation of Heisenberg’s commutation relation (pdf)

Further discussion:

See also

Last revised on December 2, 2023 at 17:58:56. See the history of this page for a list of all contributions to it.