Namespaces
Variants
Actions

Difference between revisions of "Chiral anomaly"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
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]]).
+
{{TEX|done}}
 +
One of the quantum-field theoretic manifestations of chiral dissymmetry or chiral asymmetry. Chiral anomaly in $2$-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]] $g$, then the whole model possesses symmetries, whose generators belong to the Borel–Lie anti-commutative central extension of the double $g+g$ (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.
 
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====
 
====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>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Juriev,  "Noncommutative geometry, chiral anomaly in the quantum projective $sl(2,C)$-invariant field theory and $jl(2,C)$-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>

Latest revision as of 11:13, 5 October 2014

One of the quantum-field theoretic manifestations of chiral dissymmetry or chiral asymmetry. Chiral anomaly in $2$-dimensional 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 and the theory of anti-commutative algebras (cf. Anti-commutative algebra), which are not Lie algebras [a1], [a2]. Namely, if the chiral sectors admit symmetries described by a semi-simple Lie algebra $g$, then the whole model possesses symmetries, whose generators belong to the Borel–Lie anti-commutative central extension of the double $g+g$ (an anti-commutative algebra is called a Borel–Lie algebra (or BL-algebra) if every solvable subalgebra of it is a 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

[a1] D. Juriev, "Noncommutative geometry, chiral anomaly in the quantum projective $sl(2,C)$-invariant field theory and $jl(2,C)$-invariance" J. Math. Phys. , 33 (1992) pp. 2819–2822
[a2] D. Juriev, "Erratum" J. Math. Phys. , 34 (1993) pp. 1615
How to Cite This Entry:
Chiral anomaly. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chiral_anomaly&oldid=16702
This article was adapted from an original article by D.V. Juriev (D.V. Yur'ev) (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article