The two most important developments in twentieth century physics remain unreconciled. Not only are general relativity and quantum theory are not understood as coming from one theory, but the core principles of the two theories clash. Given the lack of experimental work, the fledgling field of quantum gravity is rich and varied. This dissertation explores one approach to this theory known as nonperturbative canonical quantization. The focus is on a deformation of quantum geometry based on the new variables, q-Quantum gravity. Though far removed from everyday experiments, this theory might have observable astronomical phenomenon. There are hints and theoretical results which point to the final form. In the Introduction, these are reviewed with an emphasis on a canonical approach.

A brief survey of this field is presented in Chapter 2. Beginning with Einstein- Hilbert action, a self-dual action suitable for canonical quantization is derived. Included is a discussion of the recent ``Immirzi ambiguity.'' The review of nonperturbative quantum gravity continues in Chapter 3 with a discussion on the origins of one of the key structures in nonfigurative quantum gravity, spin networks. These were introduced as a combinatorial basis for spacetime. One of the central results, the Spin Geometry Theorem, shows that such a discrete model for spacetime can give a continuum of angles. More recently, spin networks have found an application as the state space for quantum geometry.

The connection between the classical theory of gravity presented in Chapter 2 and spin networks is discussed in Chapter 4. In both the loop and connection representations, spin networks are introduced as a space of states. An inner product is introduced. It is shown how they form a basis (including a discussion of higher valence vertices) for diffeomorphism invariant states. The chapter concludes with a detailed look at the regularization of the area operator.

Once the review is completed, an analysis of gravity in a bounded region is presented in Chapter 5. By requiring a well-defined variational principle, local boundary conditions are identified, surface observables derived, and their algebra computed. The observables arise as induced surface terms, which contribute to a non-vanishing Hamiltonian. A new approach is given which satisfies the needs of functional differentiability and which couples surface and interior degrees of freedom.

The heart of this work, a new formulation of quantum gravity, is presented in the last four chapters. q-Quantum gravity is a deformation of the observable algebra of quantum gravity in which the loop algebra is extended to framed loops. This allows an alternative nonperturbative quantization which is suitable for describing a phase of quantum gravity characterized by states which are normalizable in the measure of Chern-Simons theory. The deformation parameter, q, is related to the cosmological constant. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory. The results include a new definition of framed graphs, the use of q-spin networks as a representation of this new algebra, eigenstates of one of the algebra elements, and treatment of the area and volume observables. As Chern-Simons theory is the effective description of the quantum Hall effect. This same deformation plays a role in a gauge invariant description of this physical system and is described in Chapter 7. The dissertation concludes with some speculation on the physical content of the deformation in Chapter 9.

The whole document in zipped postscript.

© S. Major 1993-2004 Last modified 11 April 2004