Stone’s theorem on one-parameter unitary groups

E443147

mathematical theorem result in operator theory result in quantum mechanics theorem in functional analysis

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.

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Stone’s theorem on one-parameter unitary groups canonical	1

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This entity first appeared as the object of triple T4461421 — 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’s theorem on one-parameter unitary groups
Context triple: [Wigner’s theorem on symmetry transformations, relatedConcept, Stone’s theorem on one-parameter unitary groups]

A. 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.
B. 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).
C. Faddeev’s axioms
Faddeev’s axioms are a set of conditions characterizing Shannon entropy in information theory, providing an alternative but equivalent axiomatization to the Shannon–Khinchin framework.
D. Theory of Linear Operations
Theory of Linear Operations is a foundational 1932 monograph by Stefan Banach that systematically developed functional analysis and the theory of Banach spaces.
E. Mathematical Foundations of Quantum Mechanics
Mathematical Foundations of Quantum Mechanics is John von Neumann’s landmark 1932 treatise that rigorously formulates quantum theory using functional analysis and operator theory on Hilbert spaces.
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’s theorem on one-parameter unitary groups
Target entity description: 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.

A. 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.
B. 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).
C. Faddeev’s axioms
Faddeev’s axioms are a set of conditions characterizing Shannon entropy in information theory, providing an alternative but equivalent axiomatization to the Shannon–Khinchin framework.
D. Theory of Linear Operations
Theory of Linear Operations is a foundational 1932 monograph by Stefan Banach that systematically developed functional analysis and the theory of Banach spaces.
E. Mathematical Foundations of Quantum Mechanics
Mathematical Foundations of Quantum Mechanics is John von Neumann’s landmark 1932 treatise that rigorously formulates quantum theory using functional analysis and operator theory on Hilbert spaces.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in operator theory ⓘ result in quantum mechanics ⓘ theorem in functional analysis ⓘ
appliesTo	self-adjoint operators on a Hilbert space ⓘ strongly continuous one-parameter unitary groups on a Hilbert space ⓘ
assumes	real parameter t in R for the one-parameter group ⓘ
characterizes	self-adjoint operators as generators of unitary groups ⓘ strongly continuous one-parameter unitary groups ⓘ
classificationRole	classifies strongly continuous one-parameter unitary groups by their self-adjoint generators ⓘ
concerns	densely defined self-adjoint operator A ⓘ one-parameter unitary group {U(t)}_{t in R} ⓘ
converseStatement	If A is a self-adjoint operator on a Hilbert space H, then U(t)=exp(itA) defines a strongly continuous one-parameter unitary group ⓘ
domain	Hilbert spaces NERFINISHED ⓘ
ensuresProperty	the generator of a strongly continuous one-parameter unitary group is closed ⓘ the generator of a strongly continuous one-parameter unitary group is densely defined ⓘ
field	functional analysis ⓘ mathematical physics ⓘ operator theory ⓘ quantum mechanics ⓘ
formalStatement	If {U(t)}_{t in R} is a strongly continuous one-parameter unitary group on a Hilbert space H, then there exists a unique self-adjoint operator A such that U(t)=exp(itA) for all real t ⓘ
generalizationOf	finite-dimensional diagonalization of normal matrices to continuous unitary flows ⓘ
generatorDefinition	the generator A is defined by Aψ = i lim_{t→0} (U(t)ψ − ψ)/t on its domain ⓘ
historicalPeriod	20th century ⓘ
implies	the generator of a strongly continuous one-parameter unitary group is self-adjoint ⓘ the time evolution in quantum mechanics is governed by a self-adjoint Hamiltonian ⓘ
importance	links self-adjointness with physical observables in quantum mechanics ⓘ provides rigorous foundation for Schrödinger time evolution ⓘ
mathematicalArea	C*-algebras and unitary representations of groups ⓘ
namedAfter	Marshall Harvey Stone NERFINISHED ⓘ
relatedTo	Hille–Yosida theorem NERFINISHED ⓘ Stone–von Neumann theorem NERFINISHED ⓘ spectral theorem NERFINISHED ⓘ
relatesConcept	one-parameter group ⓘ self-adjoint operator ⓘ strong continuity ⓘ unitary group ⓘ
requiresCondition	U(0)=I, the identity operator ⓘ group property U(t+s)=U(t)U(s) ⓘ strong continuity of the unitary group ⓘ
statesThat	every self-adjoint operator generates a strongly continuous one-parameter unitary group ⓘ every strongly continuous one-parameter unitary group is generated by a unique self-adjoint operator ⓘ
typicalNotation	U(t)=e^{itA} ⓘ
usedIn	construction of unitary groups from self-adjoint Hamiltonians ⓘ description of time evolution of quantum states ⓘ mathematical formulation of quantum mechanics ⓘ
usesConcept	spectral theorem for self-adjoint operators ⓘ strong operator topology ⓘ

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Wigner’s theorem on symmetry transformations → relatedConcept → Stone’s theorem on one-parameter unitary groups ⓘ

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