Stone–von Neumann Theorem

Full treatment. Read the Stone–von Neumann Theorem for the imprimitivity proof architecture, multiplicity form, and exact taxonomy of excluded representations.

Statement

For finitely many canonical pairs, every strongly continuous irreducible Weyl system with fixed nonzero central character $\hbar$ is unitarily equivalent to the Schrödinger representation on $L^2(\mathbb R^n)$.

With

$$T(q)=e^{-iq\cdot P/\hbar}, \qquad M(p)=e^{ip\cdot Q/\hbar},$$

the convention is

$$T(q)M(p) = e^{-iq\cdot p/\hbar}M(p)T(q).$$

Check every hypothesis

• the number $n$ of canonical pairs is finite;
• the Weyl subgroups are strongly continuous (regularity);
• the joint representation is irreducible;
• the same nonzero central character is fixed;
• the statement concerns the global Weyl relation, not only a formal dense-domain commutator.

A regular reducible representation can contain a multiplicity space, $L^2(\mathbb R^n)\otimes\mathcal K$. Infinite systems, nonregular representations, compact configuration spaces, and $\hbar=0$ are outside the irreducible uniqueness theorem.

References

• G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989.
• J. von Neumann, “Die Eindeutigkeit der Schrödingerschen Operatoren,” Mathematische Annalen 104, 570–578, 1931, doi:10.1007/BF01457956.