nLab Stone-von Neumann theorem

Idea

The Stone–von Neumann theorem — due to Stone 1930 and von Neumann 1931/32 — 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, the Weyl relations, the canonical commutation relations.

This was originally motivated from and has special impact in quantum mechanics, where it says that phase spaces which are symplectic vector spaces, hence of the form $\mathbb{R}^{2n}$ equipped with its canonical symplectic form (such as describe for instance non-relativistic particles propagating on Euclidean space $\mathbb{R}^n$), have an essentially unique quantization (disregarding the choice of quantization of further observables such as Hamiltonians), namely with half of the canonical coordinates on $\mathbb{R}^{2n}$ represented as multiplication operators and the remaining canonical momenta represented as partial derivatives on square integrable functions.

Curiously, the analogous uniqueness statement does not hold for infinitely many generators which generically appear in quantum field theory: this is the statement of Haag's theorem.

It was observed by Mackey 1949 that the Stone-von Neumann theorem naturally generalizes from the usual Heisenberg groups based on the additive group of real numbers to analogous Heisenberg groups based on any locally compact abelian group, whence some authors speak of the Stone-von Neumann-Mackey theorem (e.g. Prasad 2011).

For instance, the phase space of abelian Chern-Simons theory (a quantum field theory, but “topological”, expected to describe fractional quantum Hall systems at certain filling fractions) over the torus has as algebra of observables the integer Heisenberg group which is based on the discrete integer subgroup $\mathbb{Z} \hookrightarrow \mathbb{R}$. In this case, Mackey’s generalization of the Stone-von Neumann theorem applies and characterizes the finite dimensional Hilbert space of states of the theory (cf. Gelca & Uribe 2010 Thm 2.4, Gelca & Hamilton 2012/15).

Statement

The form of the theorem closest to traditional discussion in quantum mechanics textbooks is in terms of representations of the Lie algebra or Weyl algebra of “canonical commutation relations”:

• For canonical commutation remations

But mathematically the theorem has been and is most naturally expressed in terms of the corresponding “exponentiated” operators representing a Heisenberg group.

In this form it is natural to generalize the underlying abelian group $\mathbb{R}$ to any locally compact abelian group:

• For general Heisenberg groups

For canonical commutation relations

The canonical commutation relations on two generators (canonical coordinate $q$ and canonical momentum $p$) in the form

$$[q,p] = \mathrm{i} \hbar$$

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

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

where on the right we have the partial 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)

For general Heisenberg groups of abelian groups

Consider

• $A$ a locally compact topological abelian group,

• $\mathbb{R}/\mathbb{Z}$ the circle group,

• $\widehat A$ the Pontrjagin dual group of continuous group homomorphisms $A \xrightarrow{\;} \mathbb{R}/\mathbb{Z}$.

Definition

(Generalized Heisenberg group)
The Heisenberg group $H(A)$ is that generated by

• $W_{(a,b)}$ for $a \in A$, $b \in \widehat{A}$

• $\zeta^t$ for $t \in \mathbb{R}/\mathbb{Z}$

subject to the group commutator relations

$$\big[W_{(a,0))},\, W_{(a',0)}\big] \;=\; \mathrm{e} \,, \;\;\;\; \big[W_{(0,b)},\, W_{(0,b')} \big] \;=\; \mathrm{e} \,, \;\;\;\; [\zeta^t, -] = \mathrm{e}$$

and

$$W_{(0,b)} W_{(a,0)} \;=\; e^{2 \pi \mathrm{i} b(a)} \, W_{(a,0)} W_{(0,b)} \,.$$

Theorem

(Mackey’s generalization of Stone-vonNeumann)
On the Hilbert space $L^2(A)$ of square integrable functions (with values in $\mathbb{C}$) over $A$, the following formulas define a unitary linear representation of $H(A)$ (Def. ):

$$\begin{array}{ccccccl} H(A) &\xrightarrow{\phantom{---}}& \mathrm{U}\big(L^2(A)\big) \\ W_{(0,b)} &\mapsto& \widehat{W}_{(0,b)} &\colon& f &\mapsto& \big( x \mapsto e^{2 \pi \mathrm{i} b(x) }\, f(x) \big) \\ W_{(a,0)} &\mapsto& \widehat{W}_{(a,0)} &\colon& f &\mapsto& \big( x \mapsto f( x - a ) \big) \\ \zeta^t &\mapsto& \widehat{\zeta}^t &\colon& f &\mapsto& \big( x \mapsto e^{2\pi \mathrm{i} t} \, f(x) \big) \end{array}$$

which is irreducible in that $L^2(A)$ does not have a closed linear proper subspace invariant under this action.

Moreover, every other continuous unitary representation $\widehat{(-)} \,\colon\, H(A) \xrightarrow{\;} \mathscr{H}$ on some separable complex Hilbert space $\mathscr{H}$, for which $\widehat{\zeta}^t \,=\, e^{2 \pi \mathrm{i}t}\, \mathrm{id}$, is isomorphic to an orthogonal direct sum of copies of this representation.

The original proofs are due to Mackey 1949 and Rieffel 1972. The above formulation follows Prasad 2011 §4.1,

Related concepts

• Weyl relation, Weyl algebra

• canonical commutation relation

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

References

General

The original articles:

• Marshall H. Stone: Linear transformations in Hilbert space. III. Operational methods and group theory, Proceedings of the National Academy of Sciences of the United States of America, 16 (1930) 172–175 [jstor:85485, doi:10.1073/pnas.16.2.172]

• 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]

The original generalization to Heisenberg groups of locally compact abelian groups:

• George W. Mackey: A theorem of Stone and von Neumann, Duke Mathematical Journal, 16 (1949) 313–326 [doi:10.1215/S0012-7094-49-01631-2]

• 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]

On the history of the theorem:

• Jonathan Rosenberg: A Selective History of the Stone-von Neumann Theorem, in: Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John von Neumann and Marshall H. Stone, Contemporary Mathematics 365, AMS (2004) [doi:10.1090/conm/365, pdf]

Review:

in mathematical physics:

• Michael Reed, Barry Simon, Thm. VIII.14 in: Functional Analysis, Volume I of: Methods of Modern Mathematical Physics, Academic Press (1978) [ISBN:9780125850506, doi:10.1016/B978-0-12-585001-8.X5001-6, pdf]

in representation theory:

• Gérard Lion, Michèle Vergne, §1.3 in: The Weil representation, Maslov index and Theta series, Progress in Mathematics 6, Birkhäuser (1980) [doi:10.1007/978-1-4684-9154-8]

• Amritanshu Prasad: An easy proof of the Stone-von Neumann-Mackey theorem, Expositiones Mathematicae 29 (2011) 110-118 [arXiv:0912.0574, doi:10.1016/j.exmath.2010.06.001]

• Miklós Rédei: Von Neumann’s proof of Uniqueness of Schrödinger representation of Heisenberg’s commutation relation [pdf]

in Lie theory:

• Karl-Hermann Neeb: The Stone–von Neumann-Mackey Theorem, Appendix VIII in: Holomorphy and Convexity in Lie Theory, De Gruyter (2000) [doi:10.1515/9783110808148.742]

See also:

• Wikipedia, Stone-von Neumann theorem

For the integer Heisenberg group

Discussion for the integer Heisenberg group in view of abelian Chern-Simons theory:

• Răzvan Gelca, Alejandro Uribe, Thm. 2.4 in: From classical theta functions to topological quantum field theory, in: The Influence of Solomon Lefschetz in Geometry and Topology: 50 Years of Mathematics at CINVESTAV, Contemporary Mathematics 621, AMS (2014) 35-68 [arXiv:1006.3252, doi;10.1090/conm/621, ams:conm-621, slides pdf, pdf]

• Răzvan Gelca, Alastair Hamilton: Classical theta functions from a quantum group perspective, New York J. Math. 21 (2015) 93–127 [arXiv:1209.1135, nyjm:j/2015/21-4]

• Răzvan Gelca, Alastair Hamilton: The topological quantum field theory of Riemann’s theta functions, Journal of Geometry and Physics 98 (2015) 242-261 [doi:10.1016/j.geomphys.2015.08.008, arXiv:1406.4269]