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Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity

  • Author(s): Haggard, Hal Mayi
  • Advisor(s): Littlejohn, Robert G
  • et al.

This work initiates a study of the semiclassical limit of quantum gravity using a geometrical formulation of WKB theory and the Hamilton-Jacobi equation. Few conceptual principles are available to guide physicists in the construction of a quantum theory of gravity. Experimentally accessible signals are notoriously difficult to extract from existing proposals and one of the few reasonable constraints that we can impose is that the proposals agree with general relativity in the classical limit. Because general relativity is such a rich classical theory this is a non-trivial condition, one that has yet to be quantitatively achieved by any theory of quantum gravity. The main focus of the dissertation is on the semiclassics of $SU(2)$ spin networks. Spin networks play an important role in the loop gravity approach to quantum gravity, where they furnish a convenient and geometrically meaningful basis for the Hilbert space. Previous work on the semiclassics and asymptotics of spin networks have focused on a coherent state approach. Here we provide alternative methods based on geometrical Lagrangian manifolds. This new perspective is complementary; for example, calculation of amplitudes is very straightforward, and should open new research avenues.

The thesis consists of two parts. In the first part, Foundations, we review the geometrical formulation of WKB theory and introduce the theory of spin networks from the beginning. These chapters make the tools and applications covered in this thesis readily accessible to new researchers and open the door to further cross-fertilization between researchers in semiclassics and loop gravity. In the second part, Applications, we focus on two applications of semiclassical theory to objects arising in loop gravity. In the loop approach to quantum gravity the geometry of space becomes discretized. Our first application is a derivation of the semiclassical spectrum and wavefunctions of the volume operator of a tetrahedral grain of space. A comparison of this spectrum with that found in loop gravity shows excellent agreement. This provides a simplified derivation of the quantization of space that strengthens earlier proposals along these lines. The second application is an asymptotic formula for the 9j-symbol including its amplitude, phase, and all of the phase adjustments. The 9j-symbol is a more complex spin network than has been treated at this level of detail before and arises as part of the vertex amplitude in spin foams, the loop gravity analog of the path integral approach to quantum gravity. More broadly this quantitative result provides further motivation for developing the asymptotics of higher 3nj-symbols; in the long term these asymptotics, which are accurate even for small quantum numbers, may furnish an effective computational tool for bridging loop gravity predictions to testable experiments.

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