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

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.