Concept of a veritable osp(1$|$2) super-triangle sum rule with 6-$j^S$ symbols from intrinsic operator techniques: an open problem

Abstract

Efficiency of intrinsic operator techniques (using only products and ranks of tensor operators) is first evidenced by condensed proofs of already known $\bigtriangledown$-triangle sum rules of su(2)/su$_q$(2). {\em A new compact} su$_q$(2)-{\em expression} is found, using a $q$-series $\Phi$, with $\Phi(n)_{| q=1}=1$. This success comes from an ultimate identification process over monomials like $(c_0)^p$. For osp(1$|$2), analogous principles of calculation are transposed, involving a second parameter $d_0$. Ultimate identification process then must be done over binomials like ${(c_{0}+{d_{0}}^{2})}^{\Omega -m} ({d_{0}}^{2})^{m}$. {\em Unknown} polynomials ${\cal P}$ are introduced as well as their expansion coefficients, $x$, over the binomials. It is clearly shown that a hypothetical super-triangle sum rule requires super-triangles $\bigtriangleup^{S}$, instead of $\bigtriangledown$ for su(2)/su$_q$(2). Coefficients $x$ are integers ({\em conjecture 1}). Massive unknown advances are done for intermediate steps of calculation. Among other, are proved {\em two theorems} on tensor operators, "zero" by construction. However, the ultimate identification seems to lead to a dead end, due to analytical apparent complexities. Up today, except for a few of coefficients $x$, no general formula is really available.