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.

Spin networks arise in many areas of physics and are a key component in both the canonical formulation (loop quantum gravity) and the path-integral formulation (spin-foam gravity) of quantum gravity. In loop quantum gravity the spin networks are used to construct a countable basis for the physical Hilbert space of gravity. The basis states may be interpreted as gauge-invariant wavefunctionals of the connection. Evaluating the wavefunctional on a specific classical connection involves embedding the spin network into a spacelike hypersurface and finding the holonomy around the network. This is equivalent to evaluating a ``g-inserted'' spin network (a spin network with a group action acting on all of the edges of the network). The spin-foam approach to quantum gravity is a path-integral formulation of loop quantum gravity in which the paths are world-histories of embedded spin networks. Depending on the spin-foam model under consideration the vertex amplitude (the contribution a spin-foam vertex makes to the transition amplitude) may be represented by a specific simple closed spin network. The most important examples use the 6j-symbol, the 15j-symbol, and the Riemannian 10j-symbol. The semiclassical treatment of spin networks is the main theme of this dissertation.

To show that classical solutions of general relativity emerge in the appropriate limits of loop quantum gravity or spin-foam gravity requires knowledge of the semiclassical limits of spin networks. This involves interpreting the spin networks as inner products and then treating the inner products semiclassically using the WKB method and the stationary phase approximation. For any given spin network there are many possible inner product models which correspond to how the spin network is ``split up'' into pieces. For example the 6j-symbol has been studied in both a model involving four angular momenta (Aquilanti et al 2012) and a model involving twelve angular momenta (Roberts 1999). Each of these models offers advantages and disadvantages when performing semiclassical analyses. Since the amplitude of the stationary phase approximation relies on determinants they are easiest to calculate in phase spaces with the fewest dimensions. The phase, on the other hand, is easiest to compute in cases where all angular momenta are treated on an equal footing, requiring a larger phase space.

Surprisingly, the different inner product models are not related by symplectic reduction (the removal of a symmetry from a Hamiltonian system). There is a connection between the models, however. On the level of linear algebra the connection is made by considering first not inner products but matrix elements of linear operators. A given matrix element can then be interpreted as an inner product in two different Hilbert spaces. We call the connection between these two inner product models the ``remodeling of an inner product.'' The semiclassical version of an inner product remodeling is a generalization of the idea that the phase space manifold that supports the semiclassical approximation of a unitary operator may be considered the graph of a symplectomorphism. We use the manifold that supports the semiclassical approximation of the linear map to ``transport'' features from one space to another. Using this transport procedure we can show that the amplitude and phase calculations in the phase spaces for the two models are identical. The asymptotics of a complicated spin network, and thus the fundamental amplitudes of loop quantum gravity and spin-foam gravity, may be computed by first setting up an inner product remodeling and then picking and choosing which features of the calculation to perform in which space.

In this dissertation we first introduce the remodeling of an inner product and the semiclassical features of the remodeling. We then apply the remodeling to the well-studied cases of the 3j-symbol and the 6j-symbol. Finally we explore how the remodel procedure applies to more complicated spin networks such as the 15j-symbol and the g-inserted spin networks of loop quantum gravity.

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.