Asymptotics of classical spin networks
Abstract
A spin network is a cubic ribbon graph labeled by representations of $\mathrm{SU}(2)$. Spin networks are important in various areas of Mathematics (3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin networks (using the theory of $G$-functions), (b) a rationality property of the generating series of all evaluations with a fixed underlying graph (using the combinatorics of the chromatic evaluation of a spin network), (c) rigorous effective computations of our results for some $6j$-symbols using the Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube $12j$ spin network (including a non-rigorous guess of its Stokes constants), in the appendix.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2009
- DOI:
- 10.48550/arXiv.0902.3113
- arXiv:
- arXiv:0902.3113
- Bibcode:
- 2009arXiv0902.3113G
- Keywords:
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- Mathematics - Geometric Topology;
- General Relativity and Quantum Cosmology;
- High Energy Physics - Phenomenology;
- Mathematics - Quantum Algebra
- E-Print:
- 24 pages, 32 figures