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Gell-Mann formula

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A prescription for constructing anti-Hermitian representations of a symmetric Lie algebra (over the real numbers) from representations of an Inönü–Wigner contraction . One assumes that is a direct sum of vector spaces and

Then and there exists an isomorphism of vector spaces such that , and for all , . In addition, one has .

The best studied examples concern the (pseudo-) orthogonal algebras, when or and [a1], [a2]. Then is an inhomogeneous Lie algebra with . Let be the quadratic -invariant element from the symmetric algebra of . If is an anti-Hermitian representation of such that is a multiple of the unit operator, then the formula for the representation of reads: for all , and, for all ,

where is the second-degree Casimir element from the universal enveloping algebra of while and are parameters. Here, is real and arbitrary and is pure imaginary and depends on .

References

[a1] E. Weimar, "The range of validity of the Gell-Mann formula" Nuovo Cim. Lett. , 4 (1972) pp. 43–50
[a2] R. Hermann, "Lie groups for physicists" , Benjamin (1966)
How to Cite This Entry:
Gell-Mann formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gell-Mann_formula&oldid=47062
This article was adapted from an original article by P. Stovicek (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article