Jump to content
 







Main menu
   


Navigation  



Main page
Contents
Current events
Random article
About Wikipedia
Contact us
Donate
 




Contribute  



Help
Learn to edit
Community portal
Recent changes
Upload file
 








Search  

































Create account

Log in
 









Create account
 Log in
 




Pages for logged out editors learn more  



Contributions
Talk
 



















Contents

   



(Top)
 


1 History  





2 Example  





3 Conformal field theory  





4 See also  





5 Notes  





6 References  














Current algebra






العربية
Deutsch
 

Edit links
 









Article
Talk
 

















Read
Edit
View history
 








Tools
   


Actions  



Read
Edit
View history
 




General  



What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Wikidata item
 




Print/export  



Download as PDF
Printable version
 
















Appearance
   

 






From Wikipedia, the free encyclopedia
 


Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra.[1] Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra.[2]

History[edit]

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.[3]

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 theorem.

Example[edit]

In a non-Abelian Yang–Mills symmetry, where V and A are flavor-current and axial-current 0th components (charge densities), respectively, the paradigm of a current algebra is[4][5]

and

where f are the structure constants of the Lie algebra. To get meaningful expressions, these must be normal ordered.

The algebra resolves to a direct sum of two algebras, L and R, upon defining

whereupon

Conformal field theory[edit]

For the case where space is a one-dimensional circle, current algebras arise naturally as a central extension of the loop algebra, known as Kac–Moody algebras or, more specifically, affine Lie algebras. 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.[6] This calculation is known as the Sugawara construction.

The general case is formalized as the vertex operator algebra.

See also[edit]

Notes[edit]

  • ^ Kac, Victor (1983). Infinite Dimensional Lie Algebras. Springer. p. x. ISBN 978-1475713848.
  • ^ Gell-Mann & Ne'eman 1964
  • ^ Gell-Mann, M. (1964). "The Symmetry group of vector and axial vector currents". Physics. 1 (1): 63. doi:10.1103/PhysicsPhysiqueFizika.1.63. PMID 17836376.
  • ^ Treiman, Jackiw & Gross 1972
  • ^ Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521-48412-X
  • References[edit]

  • t
  • e

  • Retrieved from "https://en.wikipedia.org/w/index.php?title=Current_algebra&oldid=1165299962"

    Categories: 
    Quantum field theory
    Lie algebras
    Quantum physics stubs
    Hidden categories: 
    Articles with short description
    Short description matches Wikidata
    Articles with FAST identifiers
    Articles with BNF identifiers
    Articles with BNFdata identifiers
    Articles with GND identifiers
    Articles with J9U identifiers
    Articles with LCCN identifiers
    Articles with SUDOC identifiers
    All stub articles
     



    This page was last edited on 14 July 2023, at 08:54 (UTC).

    Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.



    Privacy policy

    About Wikipedia

    Disclaimers

    Contact Wikipedia

    Code of Conduct

    Developers

    Statistics

    Cookie statement

    Mobile view



    Wikimedia Foundation
    Powered by MediaWiki