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Coherent states, 6

*j* 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.

© 2013 AIP Publishing LLC

Received 08 October 2013
Accepted 03 December 2013
Published online 26 December 2013

Acknowledgments:
The authors would like to thank Frank Hellmann, Jerzy Lewandowski, and Krzysztof Meissner for fruitful discussions, and especially Bianca Dittrich for a lot of valuable comments and for showing us the paper.^{11} We would also like to acknowledge Matteo Smerlak who focused our attention to the NLO expansion problem. W.K. acknowledges partial support by the grant “Maestro” of Polish Narodowe Centrum Nauki nr 2011/02/A/ST2/00300 and the grant of Polish Narodowe Centrum Nauki No. 501/11-02-00/66-4162. S.St. gratefully acknowledges a stipend by the DAAD (German Academic Exchange Service). This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Article outline:

I. INTRODUCTION
A. Coherent states and integration kernels
B. Relation to discrete gravity
C. Problem of the next to leading order (NLO) and complete asymptotic expansion
D. Organization of the paper
II. MODIFIED COHERENT STATES, SPIN-NETWORK EVALUATIONS AND SYMMETRIES
A. Intertwiners from modified coherent states
1. Prescription of the modifiers
2. The spin network evaluation
B. The action
C. Symmetry transformations of the action
D. Groups generated by symmetry transformations
E. Geometric lemma
F. Normal vectors to the faces
G. Interpretation of planar (spherical) spin-networks as polyhedra
III. VARIABLE TRANSFORMATION AND FINAL FORM OF THE INTEGRAL
A. Partial integration over ϕ and the new action
1. Stationary points for *S* _{ e }
2. Geometric interpretation of the angle
B. Partial integration over ϕ
1. Expansion around stationary points
2. The total expansion of the edge integral
C. New form of the action
1. *c* transformation as parity transformation
D. Normalization - “Theta” graph
1. Theta graph for integer spins and ∑ *j* even
E. Final formula
IV. ANALYSIS OF 6J SYMBOL AND FIRST ORDER REGGE CALCULUS
A. Stationary point analysis
B. Propagator and Hessian
1. Propagator
2. Hessian
C. Final result
D. First order Regge calculus
V. PROPERTIES OF THE NEXT TO LEADING ORDER AND COMPLETE ASYMPTOTIC EXPANSION
A. Properties of the Dupuis-Livine form
B. Partial integration over ϕ
1. Weak equivalence
2. Equivalences and recursion relations
3. Total expansion and DL property
4. The total expansion of the original integral
C. Different forms of intertwiners and DL property
D. Leading order expansion and a recursion relation for the 6*j* symbol
1. Recursion relation for 6*j* symbols
VI. DISCUSSION AND OUTLOOK

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2013-12-26

2016-05-29

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