Editing Current algebra

{{Short description|Infinite dimensional Lie algebra occurring in quantum field theory}}
Certain [[commutation relation]]s among the current density operators in [[quantum field theory|quantum field theories]] define an infinite-dimensional [[Lie algebra]] called a '''current algebra'''.<ref>{{harvnb|Goldin|2006}}</ref> Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra.<ref>{{cite book |last=Kac |first=Victor |date=1983 |title=Infinite Dimensional Lie Algebras |publisher=Springer |page=x |isbn=978-1475713848}}</ref>

==History==
The original current algebra, proposed in 1964 by [[Murray Gell-Mann]], described weak and electromagnetic currents of the strongly interacting particles, [[hadrons]], leading to the '''Adler–Weisberger formula''' and other important physical results. The basic concept, in the era just preceding [[quantum chromodynamics]], was that even without knowing the Lagrangian governing hadron dynamics in detail, exact kinematical information – the local symmetry – could still be encoded in an algebra of
currents.<ref>{{harvnb|Gell-Mann|Ne'eman|1964}}</ref>

The commutators involved in current algebra amount to an infinite-dimensional extension of the [[Jordan map]], where the quantum fields represent infinite arrays of oscillators.

Current algebraic techniques are still part of the shared background of particle physics when analyzing symmetries and indispensable in discussions of the [[Goldstone boson|Goldstone theorem]].

==Example==
In a [[Non-abelian group|non-Abelian]] [[Yang–Mills]] symmetry, where {{mvar|V}} and {{mvar|A}}   are flavor-current and axial-current 0th components (charge densities),  respectively, the  paradigm of a current algebra is<ref>{{cite journal |last1=Gell-Mann |first1=M. |year=1964 |title=The Symmetry group of vector and axial vector currents |journal=Physics |volume=1 |issue=1 |page=63 |doi=10.1103/PhysicsPhysiqueFizika.1.63|pmid=17836376 |doi-access=free }}</ref><ref>{{harvnb|Treiman|Jackiw|Gross|1972}}</ref>
:<math> \bigl[\ V^a(\vec{x}),\ V^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ V^c(\vec{x})\ , </math> and 
:<math>
\bigl[\ V^a(\vec{x}),\ A^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x} - \vec{y})\ A^c(\vec{x})\ ,\qquad
\bigl[\ A^a(\vec{x}),\ A^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x} - \vec{y})\ V^c(\vec{x}) ~,</math>
where {{mvar|f}} are the structure constants of the [[Lie algebra]]. To get meaningful expressions, these must be [[normal order]]ed.

The algebra resolves to a direct sum of two algebras, {{mvar|L}} and {{mvar|R}}, upon defining
:<math> L^a(\vec{x})\equiv \tfrac{1}{2}\bigl(\ V^a(\vec{x}) - A^a(\vec{x})\ \bigr)\ , \qquad R^a(\vec{x}) \equiv  \tfrac{1}{2}\bigl(\ V^a(\vec{x}) + A^a(\vec{x})\ \bigr)\ ,</math>
whereupon
<math> \bigl[\ L^a(\vec{x}),\ L^b(\vec{y})\ \bigr]= i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ L^c(\vec{x})\ ,\quad
\bigl[\ L^a(\vec{x}),\ R^b(\vec{y})\ \bigr] = 0, \quad
\bigl[\ R^a(\vec{x}),\ R^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ R^c(\vec{x})~.
</math>

==Conformal field theory==
For the case where space is a one-dimensional circle, current algebras arise naturally as a [[Lie algebra extension#Central|central extension]] of the [[loop algebra]], known as [[Kac–Moody algebra]]s or, more specifically, [[affine Lie algebra]]s. In this case, the commutator and normal ordering can be given a very precise mathematical definition in terms of integration contours on the complex plane, thus avoiding some of the formal divergence difficulties commonly encountered in quantum field theory.

When the [[Killing form]] of the Lie algebra is contracted with the current commutator, one obtains the [[energy–momentum tensor]] of a [[two-dimensional conformal field theory]].  When this tensor is expanded as a [[Laurent series]], the resulting algebra is called the [[Virasoro algebra]].<ref>{{citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|isbn=0-521-48412-X}}</ref>  This calculation is known as the [[Sugawara construction]].

The general case is formalized as the [[vertex operator algebra]].

== See also ==
* [[Affine Lie algebra]]
*[[Chiral model]]
*[[Jordan map]]
* [[Virasoro algebra]]
* [[Vertex operator algebra]]
*[[Kac–Moody algebra]]

==Notes==
{{Reflist}}

== References ==
*{{cite journal |last1=Gell-Mann |first1=M. |year=1962 |title=Symmetries of baryons and mesons |journal=Physical Review |volume=125 |issue=3 |pages=1067–84 |doi=10.1103/PhysRev.125.1067 |bibcode=1962PhRv..125.1067G |doi-access=free }}
*{{cite book |editor1-first=M. |editor1-last=Gell-Mann|editor-link1=Murray Gell-Mann|editor2-first=Y. |editor2-last=Ne'eman|editor-link2=Yuval Ne'eman|year=1964|title=The Eightfold Way|url=http://bookzz.org/book/1271076/8ff905|publisher=[[W. A. Benjamin]]|lccn=65013009}}
*{{cite book|last=Goldin|first=G.A.|editor-first1=J-P.|editor-last1=Françoise|editor-first2=G. L.|editor-last2=Naber|editor-first3=T. S.|editor-last3=Tsun |title=Encyclopedia of Mathematical Physics|at=Current Algebra|isbn=978-0-12-512666-3|year=2006}}
*{{cite book |first1=S. B.|last1=Treiman|authorlink1=Sam Treiman|first2=R.|last2=Jackiw|authorlink2=Roman Jackiw|first3=D.J.|last3=Gross|authorlink3=David J. Gross|title=Lectures on current algebra and its applications|series=Princeton Series in Physics|publisher=[[Princeton University Press]]|location=Princeton, N.J.|year=2015|orig-year=1972|isbn=978-1-4008-7150-6|doi=10.1515/9781400871506|via=[[De Gruyter]]|url=http://www.degruyter.com/view/product/459870|url-access=subscription |ref={{harvid|Treiman|Jackiw|Gross|1972}}}}  [https://books.google.com/books&id=ZP99BgAAQBAJ&pg=PA3 Sample.]

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[[Category:Quantum field theory]]
[[Category:Lie algebras]]
[[Category:Murray Gell-Mann]]
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