Wigner’s theorem on symmetry transformations

E98262

result in mathematical physics theorem in quantum mechanics

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.

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All labels observed (4)

Label	Occurrences
Wigner’s theorem on symmetry transformations canonical	2
Wigner’s symmetry classification	1
Wigner’s theorem	1
Wigner’s theorem on degeneracies	1

How this entity was disambiguated

This entity first appeared as the object of triple T818202 — 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: Wigner’s theorem on symmetry transformations
Context triple: [Eugene Wigner, knownFor, Wigner’s theorem on symmetry transformations]

A. 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.
B. 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.
C. Osterwalder–Schrader axioms
The Osterwalder–Schrader axioms are a set of mathematical conditions that characterize Euclidean quantum field theories in a way that allows them to be rigorously continued to physically meaningful relativistic quantum field theories.
D. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
E. Brillouin–Wigner perturbation theory
Brillouin–Wigner perturbation theory is a formulation of quantum mechanical perturbation theory that uses an energy-dependent effective Hamiltonian to obtain improved approximations to eigenvalues and eigenstates.
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: Wigner’s theorem on symmetry transformations
Target entity description: 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.

A. 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.
B. 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.
C. Osterwalder–Schrader axioms
The Osterwalder–Schrader axioms are a set of mathematical conditions that characterize Euclidean quantum field theories in a way that allows them to be rigorously continued to physically meaningful relativistic quantum field theories.
D. Noether's theorem
Noether's theorem is a fundamental result in theoretical physics and mathematics that links continuous symmetries of a physical system to corresponding conservation laws, such as energy or momentum conservation.
E. Brillouin–Wigner perturbation theory
Brillouin–Wigner perturbation theory is a formulation of quantum mechanical perturbation theory that uses an energy-dependent effective Hamiltonian to obtain improved approximations to eigenvalues and eigenstates.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in mathematical physics ⓘ theorem in quantum mechanics ⓘ
appliesTo	pure states in quantum mechanics ⓘ rays in Hilbert space ⓘ
assumes	symmetry preserves transition probabilities ⓘ
characterizes	automorphisms of the projective Hilbert space preserving transition probabilities ⓘ projective symmetries of Hilbert space ⓘ
clarifies	why quantum symmetries are represented by unitary or antiunitary operators ⓘ
concerns	Hilbert space structure of quantum states ⓘ symmetry transformations in quantum mechanics ⓘ transition probabilities in quantum theory ⓘ
domain	Hilbert space of a quantum system ⓘ
ensures	symmetry transformations preserve absolute values of inner products ⓘ symmetry transformations preserve transition probabilities between pure states ⓘ
field	functional analysis ⓘ mathematical physics ⓘ quantum mechanics ⓘ
formalizes	connection between physical symmetries and linear operators on Hilbert space ⓘ
hasConsequence	internal symmetries are represented by unitary operators ⓘ spatial rotations are represented by unitary operators ⓘ time-reversal symmetry is represented by an antiunitary operator in many systems ⓘ
historicalPeriod	20th century ⓘ
implies	symmetry transformations act by unitary or antiunitary operators on Hilbert space ⓘ symmetry transformations are isometries of projective Hilbert space ⓘ
influenced	axiomatic approaches to quantum mechanics ⓘ modern representation theory of quantum symmetries ⓘ quantum information theory treatments of symmetry ⓘ
language	mathematical physics terminology ⓘ
mathematicalFormulation	bijections of the projective Hilbert space preserving transition probabilities are induced by unitary or antiunitary operators ⓘ
namedAfter	Eugene Wigner ⓘ
relatedConcept	Gleason’s theorem ⓘ Stone’s theorem on one-parameter unitary groups ⓘ projective Hilbert space ⓘ quantum state space as rays ⓘ
relatesTo	Born rule for transition probabilities ⓘ linked to: Born rule in quantum mechanics antiunitary operators ⓘ projective representations of groups ⓘ ray representations of symmetry groups ⓘ unitary operators ⓘ
requires	complex Hilbert space structure ⓘ
statesThat	any symmetry of transition probabilities is implemented by a unitary or antiunitary operator ⓘ
typeOf	structure theorem for symmetry transformations ⓘ
usedIn	analysis of parity and charge-conjugation symmetries ⓘ analysis of time-reversal symmetry ⓘ classification of quantum symmetries ⓘ derivation of projective unitary representations of symmetry groups ⓘ foundations of quantum theory ⓘ

How these facts were elicited

Referenced by (5)

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

Eugene Wigner → knownFor → Wigner’s theorem on symmetry transformations ⓘ

Longuet-Higgins theorem in molecular symmetry → relatedTo → Wigner’s theorem on degeneracies ⓘ

linked to: Wigner’s theorem on symmetry transformations

Wigner Jenő Pál → knownFor → Wigner’s theorem on symmetry transformations ⓘ

Kramers degeneracy → formalizedUsing → Wigner’s theorem ⓘ

linked to: Wigner’s theorem on symmetry transformations

Dyson index β → generalizationOf → Wigner’s symmetry classification ⓘ

linked to: Wigner’s theorem on symmetry transformations

All labels observed (4)

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Statements (47)

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Referenced by (5)

How this entity was disambiguated

How these facts were elicited