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Current algebra

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Infinite dimensional Lie algebra occurring in quantum field theory

Certaincommutation relations among the current density operators inquantum field theories define an infinite-dimensionalLie algebra called acurrent algebra.^[1] Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra.^[2]

History

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The original current algebra, proposed in 1964 byMurray Gell-Mann, described weak and electromagnetic currents of the strongly interacting particles,hadrons, leading to theAdler–Weisberger formula and other important physical results. The basic concept, in the era just precedingquantum 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 ofcurrents.^[3]

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

Current algebraic techniques are still part of the shared background ofparticle physics when analyzing symmetries and indispensable in discussions of theGoldstone theorem.

Example

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In anon-Abelian Yang–Mills symmetry, whereV andA are flavor-current and axial-current 0th components (charge densities), respectively, the paradigm of a current algebra is^[4]^[5]

{\bigl [}\ V^{a}({\vec {x}}),\ V^{b}({\vec {y}})\ {\bigr ]}=i\ f_{c}^{ab}\ \delta ({\vec {x}}-{\vec {y}})\ V^{c}({\vec {x}})\ ,

and

{\bigl [}\ V^{a}({\vec {x}}),\ A^{b}({\vec {y}})\ {\bigr ]}=i\ f_{c}^{ab}\ \delta ({\vec {x}}-{\vec {y}})\ A^{c}({\vec {x}})\ ,\qquad {\bigl [}\ A^{a}({\vec {x}}),\ A^{b}({\vec {y}})\ {\bigr ]}=i\ f_{c}^{ab}\ \delta ({\vec {x}}-{\vec {y}})\ V^{c}({\vec {x}})~,

wheref are the structure constants of theLie algebra. To get meaningful expressions, these must benormal ordered.

The algebra resolves to a direct sum of two algebras,L andR, 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 ${\bigl [}\ L^{a}({\vec {x}}),\ L^{b}({\vec {y}})\ {\bigr ]}=i\ f_{c}^{ab}\ \delta ({\vec {x}}-{\vec {y}})\ L^{c}({\vec {x}})\ ,\quad {\bigl [}\ L^{a}({\vec {x}}),\ R^{b}({\vec {y}})\ {\bigr ]}=0,\quad {\bigl [}\ R^{a}({\vec {x}}),\ R^{b}({\vec {y}})\ {\bigr ]}=i\ f_{c}^{ab}\ \delta ({\vec {x}}-{\vec {y}})\ R^{c}({\vec {x}})~.$

Conformal field theory

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For the case where space is a one-dimensional circle, current algebras arise naturally as acentral extension of theloop algebra, known asKac–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 thecomplex plane, thus avoiding some of the formal divergence difficulties commonly encountered in quantum field theory.

When theKilling form of the Lie algebra is contracted with the current commutator, one obtains theenergy–momentum tensor of atwo-dimensional conformal field theory. When this tensor is expanded as aLaurent series, the resulting algebra is called theVirasoro algebra.^[6] This calculation is known as theSugawara construction.

The general case is formalized as thevertex operator algebra.