Stone–von Neumann theorem

E860124

mathematical theorem theorem in mathematical physics theorem in representation theory

The Stone–von Neumann theorem is a fundamental result in functional analysis and quantum mechanics that classifies all irreducible unitary representations of the canonical commutation relations as being unitarily equivalent.

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All labels observed (2)

Label	Occurrences
Stone–von Neumann theorem canonical	6
Weyl–Stone–von Neumann theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T10389306 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Stone–von Neumann theorem
Context triple: [Weil representation, constructedVia, Stone–von Neumann theorem]

A. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
B. Stone’s theorem on one-parameter unitary groups
Stone’s theorem on one-parameter unitary groups is a fundamental result in functional analysis and quantum mechanics that characterizes strongly continuous one-parameter unitary groups as being generated by unique self-adjoint operators.
C. Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
D. Wigner’s theorem on symmetry transformations
Wigner’s theorem on symmetry transformations is a fundamental result in quantum mechanics stating that any symmetry of transition probabilities is represented by either a unitary or antiunitary operator on the system’s Hilbert space.
E. Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Stone–von Neumann theorem
Target entity description: The Stone–von Neumann theorem is a fundamental result in functional analysis and quantum mechanics that classifies all irreducible unitary representations of the canonical commutation relations as being unitarily equivalent.

A. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
B. Stone’s theorem on one-parameter unitary groups
Stone’s theorem on one-parameter unitary groups is a fundamental result in functional analysis and quantum mechanics that characterizes strongly continuous one-parameter unitary groups as being generated by unique self-adjoint operators.
C. Peter–Weyl theorem
The Peter–Weyl theorem is a fundamental result in representation theory and harmonic analysis that decomposes square-integrable functions on a compact topological group into a direct sum of finite-dimensional irreducible unitary representations.
D. Wigner’s theorem on symmetry transformations
Wigner’s theorem on symmetry transformations is a fundamental result in quantum mechanics stating that any symmetry of transition probabilities is represented by either a unitary or antiunitary operator on the system’s Hilbert space.
E. Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in mathematical physics ⓘ theorem in representation theory ⓘ
appliesInContext	finite number of degrees of freedom ⓘ
appliesTo	Weyl form of the canonical commutation relations ⓘ canonical commutation relations ⓘ irreducible unitary representations ⓘ
asserts	all irreducible unitary representations of the canonical commutation relations are unitarily equivalent ⓘ there is essentially a unique irreducible unitary representation of the canonical commutation relations up to unitary equivalence ⓘ
characterizes	irreducible unitary representations of the canonical commutation relations ⓘ
concerns	Heisenberg commutation relations ⓘ linked to: Heisenberg Lie algebra position and momentum operators ⓘ self-adjoint operators satisfying canonical commutation relations ⓘ
doesNotExtendTo	quantum field theory with infinitely many degrees of freedom ⓘ
field	functional analysis ⓘ mathematical physics ⓘ operator algebras ⓘ quantum mechanics ⓘ representation theory ⓘ
formalizes	equivalence of all irreducible regular representations of the Weyl relations ⓘ
guarantees	uniqueness of the canonical quantization of finite-dimensional symplectic vector spaces ⓘ
hasConsequence	equivalence of different realizations of the same quantum system ⓘ uniqueness of quantization for finite-dimensional canonical systems ⓘ
hasVersion	Weyl–Stone–von Neumann theorem ⓘ linked to: Stone–von Neumann theorem
historicalPeriod	20th century ⓘ
implies	uniqueness of the Schrödinger representation up to unitary equivalence ⓘ
namedAfter	John von Neumann ⓘ Marshall H. Stone ⓘ
relatedTo	C*-algebras ⓘ CCR algebra ⓘ Fock space representations ⓘ Heisenberg uncertainty principle ⓘ linked to: uncertainty principle Weyl quantization ⓘ von Neumann algebras ⓘ
requires	irreducibility of the representation ⓘ regularity conditions on the representation ⓘ separability of the Hilbert space ⓘ
subjectOf	Heisenberg group ⓘ linked to: Heisenberg Lie algebra Weyl relations ⓘ canonical commutation relations ⓘ
usedIn	classification of representations of the Heisenberg group ⓘ construction of the Schrödinger representation ⓘ mathematical foundations of quantum mechanics ⓘ
uses	Weyl form of the canonical commutation relations ⓘ strong continuity of one-parameter unitary groups ⓘ

How these facts were elicited

Referenced by (7)

Full triples — surface form annotated when it differs from this entity's canonical label.

Weil representation → constructedVia → Stone–von Neumann theorem ⓘ

Stone’s theorem on one-parameter unitary groups → relatedTo → Stone–von Neumann theorem ⓘ

Heisenberg Lie algebra → playsRoleIn → Stone–von Neumann theorem ⓘ

Marshall H. Stone → notableWork → Stone–von Neumann theorem ⓘ

metaplectic group → isConstructedVia → Stone–von Neumann theorem ⓘ

Stone–von Neumann theorem → hasVersion → Weyl–Stone–von Neumann theorem ⓘ

linked to: Stone–von Neumann theorem

Sur certains groupes d’opérateurs unitaires → relatedTo → Stone–von Neumann theorem ⓘ

subject linked to: Weil’s "Sur certains groupes d’opérateurs unitaires"

All labels observed (2)

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Statements (45)

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Referenced by (7)

How this entity was disambiguated

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