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

TL;DR

This paper explores intrinsic operator techniques to derive sum rules for osp(1|2) superalgebra, introduces new polynomial structures, and discusses the challenges in establishing a super-triangle sum rule due to complex analytical issues.

Contribution

It extends intrinsic operator techniques to osp(1|2), introduces new polynomial expansions, and highlights the difficulties in formulating a super-triangle sum rule.

Findings

New compact su_q(2) expression using q-series.

Introduction of unknown polynomials and their coefficients.

Proved two tensor operator theorems that are zero by construction.

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.