Analytical complements to the parity-independent Racah-Wigner calculus for the superalgebra osp(1| 2). Part II (9 -j S symbols)
Lionel Bréhamet · Il Nuovo Cimento B · 2006
An original method for defining 9-j S symbols for osp(1|2), directly in terms of parity-independent 6-j S symbols is developed, from both osp(1|2) Racah and Biedenharn-Elliott sum rules only. The important property of invariance under transposition is preserved. This method exhibits at the same time their definition and their permutational symmetries. Formulas of the six different 9-j S symbols with one argument zero are listed. In addition to the pseudo-orthogonality relation, various sum rules up to triple summations, analogous to those known for su(2), are also derived, the phase factors of which are given with the most compact form as possible, mainly in terms of integral parts. osp(1|2) analog of the famous Innes-Ufford identity is written down. Definition of mixed tensor operators is given, followed by the formula of its associated osp(1|2) tensorial permutation identity. Expressions for their reduced matrix elements in coupled basis are derived and explicited in terms of 9-j S symbols. After a detailed approach of the concept of a total super angular momentum J(1 ⊗ 2), examples of calculus with products of mixed tensor operators are presented, showing clearly the occurrence of 9-j S symbols in the study of reduced matrix elements problems.