View source for Chiral anomaly

One of the quantum-field theoretic manifestations of chiral dissymmetry or chiral asymmetry. Chiral anomaly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110190/c1101901.png" />-dimensional [[Quantum field theory|quantum field theory]] means that the quantum field observables from the left and the right sectors of a field model do not commute. Chiral anomaly is deeply related to non-commutative [[Geometry|geometry]] and the theory of anti-commutative algebras (cf. [[Anti-commutative algebra|Anti-commutative algebra]]), which are not Lie algebras [[#References|[a1]]], [[#References|[a2]]]. Namely, if the chiral sectors admit symmetries described by a semi-simple [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110190/c1101902.png" />, then the whole model possesses symmetries, whose generators belong to the Borel–Lie anti-commutative central extension of the double <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110190/c1101903.png" /> (an anti-commutative algebra is called a Borel–Lie algebra (or BL-algebra) if every solvable subalgebra of it is a [[Lie algebra|Lie algebra]]).

Field models with chiral anomaly are efficiently used for anomalous stereo-synthesis (e.g., octonionic stereo-synthesis) in real-time interactive binocular video-systems.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Juriev,   "Noncommutative geometry, chiral anomaly in the quantum projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110190/c1101904.png" />-invariant field theory and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110190/c1101905.png" />-invariance"  ''J. Math. Phys.'' , '''33'''  (1992)  pp. 2819–2822</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Juriev,   "Erratum"  ''J. Math. Phys.'' , '''34'''  (1993)  pp. 1615</TD></TR></table>