3 research outputs found
Semiclassical Quantum Gravity: Obtaining Manifolds from Graphs
We address the "inverse problem" for discrete geometry, which consists in
determining whether, given a discrete structure of a type that does not in
general imply geometrical information or even a topology, one can associate
with it a unique manifold in an appropriate sense, and constructing the
manifold when it exists. This problem arises in a variety of approaches to
quantum gravity that assume a discrete structure at the fundamental level; the
present work is motivated by the semiclassical sector of loop quantum gravity,
so we will take the discrete structure to be a graph and the manifold to be a
spatial slice in spacetime. We identify a class of graphs, those whose vertices
have a fixed valence, for which such a construction can be specified. We define
a procedure designed to produce a cell complex from a graph and show that, for
graphs with which it can be carried out to completion, the resulting cell
complex is in fact a PL-manifold. Graphs of our class for which the procedure
cannot be completed either do not arise as edge graphs of manifold cell
decompositions, or can be seen as cell decompositions of manifolds with
structure at small scales (in terms of the cell spacing). We also comment
briefly on how one can extend our procedure to more general graphs.Comment: 16 pages, 5 figure
Background Independent Quantum Gravity: A Status Report
The goal of this article is to present an introduction to loop quantum
gravity -a background independent, non-perturbative approach to the problem of
unification of general relativity and quantum physics, based on a quantum
theory of geometry. Our presentation is pedagogical. Thus, in addition to
providing a bird's eye view of the present status of the subject, the article
should also serve as a vehicle to enter the field and explore it in detail. To
aid non-experts, very little is assumed beyond elements of general relativity,
gauge theories and quantum field theory. While the article is essentially
self-contained, the emphasis is on communicating the underlying ideas and the
significance of results rather than on presenting systematic derivations and
detailed proofs. (These can be found in the listed references.) The subject can
be approached in different ways. We have chosen one which is deeply rooted in
well established physics and also has sufficient mathematical precision to
ensure that there are no hidden infinities. In order to keep the article to a
reasonable size, and to avoid overwhelming non-experts, we have had to leave
out several interesting topics, results and viewpoints; this is meant to be an
introduction to the subject rather than an exhaustive review of it.Comment: 125 pages, 5 figures (eps format), the final version published in CQ