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

Related concepts

• Weyl relation, Weyl algebra

• canonical commutation relation

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

References

The original articles:

• John von Neumann, Die Eindeutigkeit der Schrödingerschen Operatoren, Mathematische Annalen 104 (1931) 570–578 [doi:10.1007/BF01457956]

• John von Neumann, Über Einen Satz Von Herrn M. H. Stone, Annals of Mathematics, Second Series 33 3 (1932) 567-573 [doi:10.2307/1968535, jstor:1968535]

Review:

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

Further discussion:

• Marc Rieffel, On the uniqueness of the Heisenberg commutation relations, Duke Math. J. 39 4 (1972) 745-752 [doi:10.1215/S0012-7094-72-03982-8]

See also

• Wikipedia, Stone-von Neumann theorem