Page 92 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

The Real Generators of the Unitary Group

P. CASSAM-CHENAI
Equipe d'Astrochimie Quantique, Laboratoire de Radioastronomie
EMS., 24 rue Lhomond, F-75231 Paris Cedex 05, France

This note is dedicated to G. Berthier who has always emphasized the importance of a
rigorous use of the language in scientific papers. I would like to expose here an "abus de
langage" regarding "the generators of the unitary group U(n) ", usually denoted by
which dates back to their introduction in quantum chemistry [1]. As a matter of fact, in
the original paper, the author concedes that they are not the generators of U(n) but those
of the linear group ; however, as far as I am aware, none of his followers has
ever mentioned this point.

The generators which are chosen such that:

are Hermitian only for They generate, using complex numbers, the Lie algebra of
. This algebra contains the Lie algebra of U(n) , but it is indeed much larger.
The Lie algebra of U(n) can be generated more specifically, using real numbers, with
Hermitian generators denoted

The generators , and the generators
are related in the same way as the angular moment operators" and

where is the Kronecker symbol, and
It is convenient to extend these relations to all couples (i,j), and to write compactly :

The structure constants for Hermitian generators are purely imaginary :

Y. Ellinger and M. Defranceschi (eds.). Strategies and Applications in Quantum Chemistry, 77–78.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.