Editing Anomaly (physics) (section)

{{Short description|Asymmetry of classical and quantum action}}
{{Quantum field theory|cTopic=Tools}}
{{distinguish|Anomaly (natural sciences)}}
In [[quantum physics]] an '''anomaly''' or '''quantum anomaly''' is the failure of a [[symmetry]] of a theory's classical [[action (physics)|action]] to be a symmetry of any [[regularization (physics)|regularization]] of the full quantum theory.<ref>
{{cite journal
 |last=Bardeen |first=William |year=1969
 |title=Anomalous Ward identities in spinor field theories
 |journal=[[Physical Review]]
 |volume=184
 |issue=5 |pages=1848–1859
 |doi=10.1103/physrev.184.1848|bibcode = 1969PhRv..184.1848B }}</ref><ref>{{cite book
 |last1=Cheng |first1=T.P.|last2=Li |first2=L.F. |date=1984 |title=Gauge Theory of Elementary Particle Physics|publisher=Oxford Science Publications }}</ref>  
In [[classical physics]], a '''classical anomaly''' is the failure of a symmetry to be restored in the limit in which the symmetry-breaking parameter goes to zero. Perhaps the first known anomaly was the dissipative anomaly<ref>{{cite web|title=Dissipative Anomalies in Singular Euler Flows|url=https://www-n.oca.eu/etc7/EE250/presentations/Eyink.pdf}}</ref> in [[turbulence]]: time-reversibility remains broken (and energy dissipation rate finite) at the limit of vanishing [[viscosity]]. 

In quantum theory, the first anomaly discovered was the [[Adler–Bell–Jackiw anomaly]], wherein the [[Chiral_anomaly|axial vector current]] is conserved as a classical symmetry of [[electrodynamics]], but is broken by the quantized theory. The relationship of this anomaly to the [[Atiyah–Singer index theorem]] was one of the celebrated achievements of the theory. Technically, an anomalous symmetry in a quantum theory is a symmetry of the [[action (physics)|action]], but not of the [[measure (physics)|measure]], and so not of the [[partition function (quantum field theory)|partition function]] as a whole.