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.

Aliases (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 NERFINISHED →
relatedConcept	Gleason’s theorem NERFINISHED → Stone’s theorem on one-parameter unitary groups NERFINISHED → projective Hilbert space → quantum state space as rays →
relatesTo	Born rule for transition probabilities NERFINISHED → 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 (2)

Subject (surface form when different)	Predicate
Eugene Wigner →	knownFor
Longuet-Higgins theorem in molecular symmetry ("Wigner’s theorem on degeneracies") →	relatedTo