Difference between revisions of "Gell-Mann-Dashen algebra"
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| − | + | An infinite-dimensional [[Lie algebra|Lie algebra]] occurring in [[Quantum field theory|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|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|Gauge transformation]]) [[#References|[a1]]]. Representations of $ \mathfrak g $ | |
| + | are called current algebras in [[Quantum field theory|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 | ||
| − | + | $$ | |
| + | [ J _ \alpha ( x ) ,J _ \beta ( x ^ \prime ) ] = \delta ( x - x ^ \prime ) \sum _ \gamma c _ {\alpha \beta \gamma } J _ \gamma ( x ) . | ||
| + | $$ | ||
| − | R. Dashen and M. Gell-Mann (1966) studied and applied the latter commutation relations in the particular case when | + | 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 ) $, | ||
| + | [[#References|[a2]]]. | ||
General references for current algebras are [[#References|[a3]]], [[#References|[a4]]]. | General references for current algebras are [[#References|[a3]]], [[#References|[a4]]]. | ||
Latest revision as of 19:41, 5 June 2020
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) |
Gell-Mann-Dashen algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gell-Mann-Dashen_algebra&oldid=47060