Coherent states, 6j symbols and properties of the next to leading order asymptotic expansions

TL;DR

This paper derives the asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, computing the Hessian determinant and extending the analysis to next-to-leading order terms with new properties and recursion relations.

Contribution

It provides the first complete derivation of the 6j symbol's asymptotic expansion via coherent states, including Hessian computation and next-to-leading order analysis.

Findings

Computed the Hessian determinant for the asymptotic expansion.

Extended the asymptotic formula to next-to-leading order terms.

Derived a recursion relation for the 6j symbol.

Abstract

We present the first complete derivation of the well-known asymptotic expansion of the SU(2) 6j symbol using a coherent state approach, in particular we succeed in computing the determinant of the Hessian matrix. To do so, we smear the coherent states and perform a partial stationary point analysis with respect to the smearing parameters. This allows us to transform the variables from group elements to dihedral angles of a tetrahedron resulting in an effective action, which coincides with the action of first order Regge calculus associated to a tetrahedron. To perform the remaining stationary point analysis, we compute its Hessian matrix and obtain the correct measure factor. Furthermore, we expand the discussion of the asymptotic formula to next to leading order terms, prove some of their properties and derive a recursion relation for the full 6j symbol.