# Current algebra > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Current_algebra > Markdown URL: https://mediated.wiki/source/Current_algebra.md > Source: https://en.wikipedia.org/wiki/Current_algebra > Source revision: 1296601578 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Infinite dimensional Lie algebra occurring in quantum field theory}} Certain [commutation relation](/source/commutation_relation)s among the current density operators in [quantum field theories](/source/quantum_field_theory) define an infinite-dimensional [Lie algebra](/source/Lie_algebra) called a '''current algebra'''.{{harvnb|Goldin|2006}} Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra.{{cite book |last=Kac |first=Victor |date=1983 |title=Infinite Dimensional Lie Algebras |publisher=Springer |page=x |isbn=978-1475713848}} ==History== The original current algebra, proposed in 1964 by [Murray Gell-Mann](/source/Murray_Gell-Mann), described weak and electromagnetic currents of the strongly interacting particles, [hadrons](/source/hadrons), leading to the '''Adler–Weisberger formula''' and other important physical results. The basic concept, in the era just preceding [quantum chromodynamics](/source/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.{{harvnb|Gell-Mann|Ne'eman|1964}} The commutators involved in current algebra amount to an infinite-dimensional extension of the [Jordan map](/source/Jordan_map), where the quantum fields represent infinite arrays of oscillators. Current algebraic techniques are still part of the shared background of [particle physics](/source/particle_physics) when analyzing symmetries and indispensable in discussions of the [Goldstone theorem](/source/Goldstone_boson). ==Example== In a [non-Abelian](/source/Non-abelian_group) [Yang–Mills](/source/Yang%E2%80%93Mills) 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{{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 }}{{harvnb|Treiman|Jackiw|Gross|1972}} : \bigl[\ V^a(\vec{x}),\ V^b(\vec{y})\ \bigr] = i\ f^{ab}_c\ \delta(\vec{x}-\vec{y})\ V^c(\vec{x})\ , and : \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}) ~, where {{mvar|f}} are the structure constants of the [Lie algebra](/source/Lie_algebra). To get meaningful expressions, these must be [normal order](/source/normal_order)ed. The algebra resolves to a direct sum of two algebras, {{mvar|L}} and {{mvar|R}}, upon defining : 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)\ , whereupon \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})~. ==Conformal field theory== For the case where space is a one-dimensional circle, current algebras arise naturally as a [central extension](/source/Lie_algebra_extension) of the [loop algebra](/source/loop_algebra), known as [Kac–Moody algebra](/source/Kac%E2%80%93Moody_algebra)s or, more specifically, [affine Lie algebra](/source/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](/source/complex_plane), thus avoiding some of the formal divergence difficulties commonly encountered in quantum field theory. When the [Killing form](/source/Killing_form) of the Lie algebra is contracted with the current commutator, one obtains the [energy–momentum tensor](/source/energy%E2%80%93momentum_tensor) of a [two-dimensional conformal field theory](/source/two-dimensional_conformal_field_theory). When this tensor is expanded as a [Laurent series](/source/Laurent_series), the resulting algebra is called the [Virasoro algebra](/source/Virasoro_algebra).{{citation|first=Jurgen|last= Fuchs|title=Affine Lie Algebras and Quantum Groups|year=1992|publisher=Cambridge University Press|isbn=0-521-48412-X}} This calculation is known as the [Sugawara construction](/source/Sugawara_construction). The general case is formalized as the [vertex operator algebra](/source/vertex_operator_algebra). == See also == * [Affine Lie algebra](/source/Affine_Lie_algebra) *[Chiral model](/source/Chiral_model) *[Jordan map](/source/Jordan_map) * [Virasoro algebra](/source/Virasoro_algebra) * [Vertex operator algebra](/source/Vertex_operator_algebra) *[Kac–Moody algebra](/source/Kac%E2%80%93Moody_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](/source/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](/source/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](/source/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.] {{Authority control}} Category:Quantum field theory Category:Lie algebras Category:Murray Gell-Mann {{quantum-stub}} --- Adapted from the Wikipedia article [Current algebra](https://en.wikipedia.org/wiki/Current_algebra) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Current_algebra?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.