Difference between revisions of "Gell-Mann-Dashen algebra"
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Revision as of 18:52, 24 March 2012
An infinite-dimensional Lie algebra occurring in quantum field theory. Let be a finite-dimensional Lie algebra and
the space of Schwartz test functions (cf. Generalized functions, space of). The Lie algebra
is defined by
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and can be interpreted as the Lie algebra of the group of gauge transformations (cf. Gauge transformation) [a1]. Representations of are called current algebras in quantum field theory. Let
be a homomorphism of Lie algebras and let
be a basis of
with structure constants
. The mapping
defines an
-valued distribution
and it is true that
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Passing to the Fourier image one sets for
; then
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R. Dashen and M. Gell-Mann (1966) studied and applied the latter commutation relations in the particular case when , [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=22503