Current algebra

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 of Gell-Mann of 1964 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 densities, respectively, the paradigm of a current algebra is

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 ${\displaystyle [L^{a}({\vec {x}}),L^{b}({\vec {y}})]=if_{c}^{ab}\delta ({\vec {x}}-{\vec {y}})L^{c}({\vec {x}}),\quad [L^{a}({\vec {x}}),R^{b}({\vec {y}})]=0,\quad [R^{a}({\vec {x}}),R^{b}({\vec {y}})]=if_{c}^{ab}\delta ({\vec {x}}-{\vec {y}})R^{c}({\vec {x}})~.}$

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