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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

All labels observed (3)

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

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 in quantum mechanics ⓘ surface form: Born rule for transition probabilities 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 ⓘ

Referenced by (4)

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 symmetry transformations ⓘ

this entity surface form: Wigner’s theorem on degeneracies

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

Kramers degeneracy → formalizedUsing → Wigner’s theorem on symmetry transformations ⓘ

this entity surface form: Wigner’s theorem