Gell-Mann-Dashen algebra

An infinite-dimensional Lie algebra occurring in quantum field theory. Let $ {\widetilde{\mathfrak g} } $ be a finite-dimensional Lie algebra and $ {\mathcal S} ( \mathbf R ^ {n} ) $ the space of Schwartz test functions (cf. Generalized functions, space of). The Lie algebra $ \mathfrak g = {\mathcal S} ( \mathbf R ^ {n} ) \otimes {\widetilde{\mathfrak g} } $ is defined by

$$ [ f \otimes X,g \otimes Y ] = fg \otimes [ X,Y ] $$

and can be interpreted as the Lie algebra of the group of gauge transformations (cf. Gauge transformation) [a1]. Representations of $ \mathfrak g $ are called current algebras in quantum field theory. Let $ J : \mathfrak g \rightarrow \mathfrak h $ be a homomorphism of Lie algebras and let $ ( A _ \alpha ) $ be a basis of $ {\widetilde{\mathfrak g} } $ with structure constants $ c _ {\alpha \beta \gamma } $. The mapping $ {\mathcal S} ( \mathbf R ^ {n} ) \ni f \mapsto J ( f \otimes A _ \alpha ) \in \mathfrak h $ defines an $ \mathfrak h $- valued distribution $ J _ \alpha ( x ) \in {\mathcal S} ^ \prime ( \mathbf R ^ {n} ) \otimes \mathfrak h $ and it is true that

$$ [ J _ \alpha ( x ) ,J _ \beta ( x ^ \prime ) ] = \delta ( x - x ^ \prime ) \sum _ \gamma c _ {\alpha \beta \gamma } J _ \gamma ( x ) . $$

Passing to the Fourier image one sets $ { {J _ \alpha } hat } ( k ) = J ( e ^ {ik \cdot x } \otimes A _ \alpha ) $ for $ k \in \mathbf R ^ {n} $; then

$$ [ { {J _ \alpha } hat } ( k ) , { {J _ \beta } hat } ( k ^ \prime ) ] = \sum _ \gamma c _ {\alpha \beta \gamma } { {J _ \gamma } hat } ( k + k ^ \prime ) . $$

R. Dashen and M. Gell-Mann (1966) studied and applied the latter commutation relations in the particular case when $ {\widetilde{\mathfrak g} } = \mathfrak s \mathfrak u ( 3 ) \oplus \mathfrak s \mathfrak u ( 3 ) $, [a2].

General references for current algebras are [a3], [a4].

References

[a1]	R. Hermann, "Lie algebras and quantum mechanics" , Benjamin (1970)
[a2]	R. Dashen, M. Gell-Mann, "Representation of local current algebra at infinite momentum" Phys. Rev. Lett. , 17 (1966) pp. 340–343
[a3]	S.L. Adler, R. Dashen, "Current algebras" , Benjamin (1968)
[a4]	B. Renner, "Current algebras and their applications" , Pergamon (1968)