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Complementarity relation between the U(p,q) and U(n) lie groups and some applications to atomic physics

par Quesne, Christiane
Référence Journal of Physics A: Mathematical and General, 19, 14, page (2689-2706), 009
Publication Publié, 1986

Article révisé par les pairs

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Résumé :

The complementarity relation between the unitary groups U(d) and U(n) within the symmetrical irreducible representations of the larger unitary group U(dn) is extended to non-compact groups. It is proved that the pseudo-unitary group U(p,q) is complementary with respect to U(n) within some positive discrete series irreducible representations of the larger pseudo-unitary group U(pn,qn). The latter arise when reducing the metaplectic irreducible representations ((1/2)dn) and ((1/2)dn-13/2) of the real symplectic group Sp(2dn, R), where d=p+q, and they are characterised by a single label, the eigenvalue of the first order Casimir operator. Some applications of the U(p,q)-U(n) complementarity to atomic physics are outlined. For such purposes, the isomorphism between the Lie algebras of SU(2,2) and SO(4,2) is used extensively. The mathematical framework underlying the Kibler-Negadi approach (1983,4) of the hydrogen atom dynamical group is extended to the independent-electron dynamical group of intrashell many-electron states as well as to the correlated electron dynamical group of intrashell doubly excited states.