Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity

Quantum gravity is studied nonperturbatively in the case in which space has a boundary with finite area. A natural set of boundary conditions is studied in the Euclidean signature theory, in which the pullback of the curvature to the boundary is self-dual (with a cosmological constant). A Hilbert space which describes all the information accessible by measuring the metric and connection induced in the boundary is constructed and is found to be the direct sum of the state spaces of all $SU(2)$ Chern-Simon theories defined by all choices of punctures and representations on the spatial boundary $\cal S$. The integer level $k$ of Chern-Simons theory is found to be given by $k= 6\pi /G^2 \Lambda + \alpha$, where $\Lambda$ is the cosmological constant and $\alpha$ is a $CP$ breaking phase. Using these results, expectation values of observables which are functions of fields on the boundary may be evaluated in closed form. The Beckenstein bound and 't Hooft-Susskind holographic hypothesis are confirmed, (in the limit of large area and small cosmological constant) in the sense that once the two metric of the boundary has been measured, the subspace of the physical state space that describes the further information that the observer on the boundary may obtain about the interior has finite dimension equal to the exponent of the area of the boundary, in Planck units, times a fixed constant. Finally,the construction of the state space for quantum gravity in a region from that of all Chern-Simon theories defined on its boundary confirms the categorical-theoretic ``ladder of dimensions picture" of Crane.Comment: TEX File, Minor Changes Made, 59 page

"So what will you do if string theory is wrong?"

I briefly discuss the accomplishments of string theory that would survive a complete falsification of the theory as a model of nature and argue the possibility that such a survival may necessarily mean that string theory would become its own discipline, independently of both physics and mathematics

The Bekenstein bound, topological quantum field theory and pluralistic quantum cosmology

An approach to quantum gravity and cosmology is proposed based on a synthesis of four elements: 1) the Bekenstein bound and the related holographic hypothesis of 't Hooft and Susskind, 2) topological quantum field theory, 3) a new approach to the interpretational issues of quantum cosmology and 4) the loop representation formulation of non-perturbative quantum gravity. A set of postulates are described, which define a {\it pluralistic quantum cosmological theory.} These incorporates a statistical and relational approach to the interpretation problem, following proposals of Crane and Rovelli, in which there is a Hilbert space associated to each timelike boundary, dividing the universe into two parts. A quantum state of the universe is an assignment of a statistical state into each of these Hilbert spaces, subject to certain conditions of consistency which come from an analysis of the measurement problem. A proposal for a concrete realization of these postulates is described, which is based on certain results in the loop representation and topological quantum field theory, and in particular on the fact that spin networks and punctured surfaces appear in both contexts. The Capovilla-Dell-Jacobson solution of the constraints of quantum gravity are expressed quantum mechanically in the language of Chern-Simons theory, in a way that leads also to the satisfaction of the Bekenstein bound

The linearization of the Kodama state

We study the question of whether the linearization of the Kodama state around classical deSitter spacetime is normalizable in the inner product of the theory of linearized gravitons on deSitter spacetime. We find the answer is no in the Lorentzian theory. However, in the Euclidean theory the corresponding linearized Kodama state is delta-functional normalizable. We discuss whether this result invalidates the conjecture that the full Kodama state is a good physical state for quantum gravity with positive cosmological constant.Comment: 14 pages, statement on the corresponding Yang-Mills case correcte

Self-organized critical behavior: the evolution of frozen spin networks model in quantum gravity

In quantum gravity, we study the evolution of a two-dimensional planar open frozen spin network, in which the color (i.e. the twice spin of an edge) labeling edge changes but the underlying graph remains fixed. The mainly considered evolution rule, the random edge model, is depending on choosing an edge randomly and changing the color of it by an even integer. Since the change of color generally violate the gauge invariance conditions imposed on the system, detailed propagation rule is needed and it can be defined in many ways. Here, we provided one new propagation rule, in which the involved even integer is not a constant one as in previous works, but changeable with certain probability. In random edge model, we do find the evolution of the system under the propagation rule exhibits power-law behavior, which is suggestive of the self-organized criticality (SOC), and it is the first time to verify the SOC behavior in such evolution model for the frozen spin network. Furthermore, the increase of the average color of the spin network in time can show the nature of inflation for the universe.Comment: 5 pages, 5 figure