On the Clebsch–Gordan problem for SU(1,1): Coupling nonstandard representations

Abstract

The Clebsch–Gordan coefficients coupling two unitary, irreducible, positive discrete series representations of SU(1,1), are constructed. In contrast to the Clebsch–Gordan coefficients obtained a long time ago by Holman and Biedenharn [Ann. Phys. (N.Y.) 39, 1 (1966)], the derived coefficients are valid even when coupling nonstandard representations such as those for which the corresponding Bargmann indices k may be k=14 and/or 34, values associated with the “two-photon” realization of the su(1,1) Lie algebra, the corresponding representations covering the even and odd number states, respectively, of the single-mode boson system. These nonstandard cases are actually representations associated with the covering group SU¯(1,1). The results are extended to the coupling of three positive discrete series and the corresponding SU(1,1) Racah coefficients are obtained.