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

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.

31 pages (+ 24 pages appendices), 13 figures

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• The Ponzano-Regge model
• Area-angle variables for general relativity
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• Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
• Hessian in the spinfoam models with cosmological constant