Stone–von Neumann theorem - Imported Knowlpedia Content

Statement

Let $H_n$ be the real Heisenberg group with center $Z=\{(0,0,z):z\in\mathbb R\}$ , and fix $\lambda\in\mathbb R\setminus\{0\}$ . The Stone–von Neumann theorem states that every irreducible continuous unitary representation $\pi$ of $H_n$ satisfying

\pi(0,0,z)=e^{i\lambda z}I

is unitarily equivalent to the Schrödinger representation on $L^2(\mathbb R^n)$ . Thus fixing a nontrivial character of the center fixes the irreducible representation up to unitary equivalence; among representations with nontrivial central action, varying the character gives the family indexed by $\lambda\ne0$ .

Schrödinger model

For the convention

(x,y,z)(x',y',z')=(x+x',y+y', z+z'+\tfrac12(x\cdot y'-y\cdot x')),

one Schrödinger model is

[\pi_\lambda(x,y,z)f](u) =e^{i\lambda(z+y\cdot u+\frac12x\cdot y)}f(u+x).

Translations and modulations therefore fail to commute by exactly the central phase encoded by the Heisenberg group law. The uniqueness theorem and this model are developed in Folland, §1.5.

Why the hypotheses matter

Irreducibility is essential: arbitrary representations with the same central character can be direct sums or direct integrals of copies of the Schrödinger model. Nontriviality of the central character is also essential. When $\lambda=0$ , the representation factors through the abelian quotient $H_n/Z$ , which has many one-dimensional characters rather than one distinguished irreducible representation.

Weyl-relations formulation

An equivalent formulation starts with strongly continuous one-parameter unitary groups satisfying the Weyl commutation relations and an irreducibility condition. The theorem says that such a finite-dimensional system of canonical commutation relations is unitarily equivalent to the standard position and momentum model. This exponentiated formulation avoids the domain ambiguities of writing commutators of unbounded operators Hall, Chapter 14.

References

Gerald B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122, Princeton University Press, 1989. DOI record for the electronic edition. Relevant: §1.5 on the Stone–von Neumann theorem.
Brian C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013. DOI record. Relevant: Chapter 14 on the canonical commutation relations and Stone–von Neumann theorem.