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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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 NERFINISHED ⓘ 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 NERFINISHED ⓘ
historicalPeriod	20th century ⓘ
implies	uniqueness of the Schrödinger representation up to unitary equivalence ⓘ
namedAfter	John von Neumann NERFINISHED ⓘ Marshall H. Stone NERFINISHED ⓘ
relatedTo	C*-algebras ⓘ CCR algebra NERFINISHED ⓘ Fock space representations ⓘ Heisenberg uncertainty principle NERFINISHED ⓘ Weyl quantization NERFINISHED ⓘ von Neumann algebras NERFINISHED ⓘ
requires	irreducibility of the representation ⓘ regularity conditions on the representation ⓘ separability of the Hilbert space ⓘ
subjectOf	Heisenberg group NERFINISHED ⓘ Weyl relations NERFINISHED ⓘ 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 ⓘ

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Weil representation → constructedVia → Stone–von Neumann theorem ⓘ