2001/01/19 by I. Raptis, Ioannis Raptis, Raptis, Ioannis
Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/0101082
65 pages, LaTeX2e,typos corrected
openalex publication_date 2001/01/19 · arxiv created 2001/02/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A quantum causal topology is presented. This is modeled after a non-commutative scheme type of theory for the curved finitary spacetime sheaves of the non-abelian 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 `gauge-independent'. 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 non-commutative scheme-theoretic curved quantum causal topology with some recent C*-quantale models for non-abelian generalizations of classical commutative topological spaces or locales, as well as with some relevant recent results obtained from applying sheaf and topos-theoretic ideas to quantum logic proper. Motivated by the latter, we organize our finitary spacetime primitive algebra schemes of curved quantum causal sets into a topos-like 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 time-asymmetry 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 topos-theoretic terms.