NonCommutative Topology for Curved Quantum Causality
Abstract
A quantum causal topology is presented. This is modeled after a noncommutative scheme type of theory for the curved finitary spacetime sheaves of the nonabelian incidence Rota algebras that represent `gravitational quantum causal sets'. The finitary spacetime primitive algebra scheme structures for quantum causal sets proposed here are interpreted as the kinematics of a curved and reticular local quantum causality. Dynamics for quantum causal sets is then represented by appropriate scheme morphisms, thus it has a purely categorical description that is manifestly `gaugeindependent'. Hence, a schematic version of the Principle of General Covariance of General Relativity is formulated for the dynamically variable quantum causal sets. We compare our noncommutative schemetheoretic curved quantum causal topology with some recent $C^{*}$quantale models for nonabelian generalizations of classical commutative topological spaces or locales, as well as with some relevant recent results obtained from applying sheaf and topostheoretic ideas to quantum logic proper. Motivated by the latter, we organize our finitary spacetime primitive algebra schemes of curved quantum causal sets into a toposlike structure, coined `quantum topos', and argue that it is a sound model of a structure that Selesnick has anticipated to underlie Finkelstein's reticular and curved quantum causal net. At the end we conjecture that the fundamental quantum timeasymmetry that Penrose has expected to be the main characteristic of the elusive `true quantum gravity' is possibly of a kinematical or structural rather than of a dynamical character, and we also discuss the possibility of a unified description of quantum logic and quantum gravity in quantum topostheoretic terms.
 Publication:

arXiv eprints
 Pub Date:
 January 2001
 arXiv:
 arXiv:grqc/0101082
 Bibcode:
 2001gr.qc.....1082R
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 65 pages, LaTeX2e,typos corrected