nLab Stone-von Neumann theorem

Idea

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

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

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

$$(q \psi)(x) \coloneqq x \psi(x) \phantom{AAAA} (p \psi)(x) \coloneqq -i \hbar \frac{\partial}{\partial x} \psi(x) \,,$$

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, e.g. Redei), to be distinguished from “Schrödinger picture” which is a related but different concept.

Related concepts

• For the Schrödinger representation obtained via geometric quantization see there.

References

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)

Review includes

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

See also

• Wikipedia, Stone-von Neumann theorem