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Semiclassical Analysis of SU(2) Spin Networks


In the current pursuit of a quantum theory of gravity, the lack of experimental data at the Planck scale has forced physicists to rely on checking the classical limits of their proposed theories to select viable candidates. In loop quantum gravity, the best candidates are the spin foam models in four dimensions, where the vertex amplitudes of a 4-simplex are formulated in terms of the Wigner 15j-symbols. Historically, the asymptotic expression of the Wigner 6j-symbol has played an essential role in the classical limit of the spin foam model in three dimensions. Thus, we believe the asymptotic expression of the 15j-symbol will play an equally important role in the classical limits of the spin foam models in four dimensions. In this thesis, we employ a wide range of techniques, including stationary phase approximation, WKB theory, classical mechanics, symplectic reduction, symbol correspondence, star products, and the Born-Oppenheimer approximation, to derive old and new asymptotic formulas for the Wigner 3j-, 6j-, 9j-, 12j-, and 15j-symbols, which are examples of SU(2) spin networks.

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