We describe 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, depends on the cosmological constant. The Mandelstam identities are extended to a set of relations which are governed by the Kauffman bracket so that the spin network basis is deformed to a basis of SU(2)q spin networks. Corrections to the actions of operators in non-perturbative quantum gravity may be readily computed using recoupling theory; the example of the area observable is treated here. Finally, eigenstates of the q-deformed Wilson loops are constructed, which may make possible the construction of a q-deformed connection representation through an inverse transform.Comment: 12 pages, many figure
Using a form of modified dispersion relations derived in the context of quantum geometry, we investigate limits set by current observations on potential corrections to Lorentz invariance. We use a phenomological model in which there are separate parameters for photons, leptons, and hadrons. Constraints on these parameters are derived using thresholds for the processes of photon stability, photon absorption, vacuumČerenkov radiation, pion stability, and the GZK cutoff. Although the allowed region in parameter space is tightly constrained, non-vanishing corrections to Lorentz symmetry due to quantum geometry are consistent with current astrophysical observations.Date: January 2002. the authors find that, in an analysis of particle propagation, photon [17] and neutrino [16] dispersion relations are modified. In Section 3 we give a brief overview of threshold calculations before turning to a number of processes including: photon stability, photon non-absorption, vacuumČerenkov radiation for electrons and protons, proton non-absorption, and pion stability. We calculate constraints from the threshold calculations to investigate whether it is possible that observed effects may be accounted for by Lorentz symmetry corrections. In 3.2 we show that asymmetric momentum partitioning, first noticed by Liberati, Jacobson, and Mattingly [13], dramatically affects the constraints on the dispersion relation modifications. The results of Section 3 are summarized in Table 1. Finally in Section 4, we apply the constraints together with current observations to limit the extent of potential Lorentz symmetry corrections. We summarize the constraints in the final section and in Figures 2 and 3. We find that present day observations tightly constrainbut still leave open -the possibility of Lorentz symmetry corrections of this form.Particularly close to the present work is the paper by Jacobson, Liberati, and Mattingly [13] in which many of these results were summarized. For the most part the present work agrees with this paper where the subject overlaps, although this work also includes new threshold calculations and constraints for proton vacuum Cerenkov radiation, the GZK threshold, and pion stability.
Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two "bundles" of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles.
An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally finite partial order. A new topology on causal sets using "thickened antichains" is constructed. This topology is then used to recover the homology of a globally hyperbolic spacetime from a causal set which faithfully embeds into it at sufficiently high sprinkling density. This implies a discrete-continuum correspondence which lends support to the fundamental conjecture or "Hauptvermutung" of causal set theory.
Abstract. Spin networks, essentially labeled graphs, are "good quantum numbers" for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite dimensional systems, gauge theory, and knot theory. Though accessible to undergraduates, spin network techniques are buried in more complicated formulations. In this paper a diagrammatic method, simple but rich, is introduced through an association of 2 × 2 matrices to diagrams. This spin network diagrammatic method offers new perspectives on the quantum mechanics of angular momentum, group theory, knot theory, and even quantum geometry. Examples in each of these areas are discussed.
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