2001/02/22 by A. Mallios, I. Raptis
Physics and Astronomy · #gr-qc
published as Int.J.Theor.Phys. 40 (2001) 1885-1928 · This is a shortened version of the abstract. The paper is submitted to the International Journal of Theoretical Physics
arxiv created 2001/02/22 · arxiv updated 2009/11/30
A locally finite, causal and quantal substitute for a locally Minkowskian principal fiber bundle \calP of modules of Cartan differential forms \omg over a bounded region X of a curved C∞-smooth differential manifold spacetime M with structure group \bf G that of orthochronous Lorentz transformations L+:=SO(1,3)\uparrow, is presented. \calP is the structure on which classical Lorentzian gravity, regarded as a Yang-Mills type of gauge theory of a sl(2,\com)-valued connection 1-form \calA, is usually formulated. The mathematical structure employed to model this replacement of \calP is a principal finitary spacetime sheaf \calPn of quantum causal sets \amgn with structure group \bf Gn, which is a finitary version of the group \bf G of local symmetries of General Relativity, and a finitary Lie algebra \bf gn-valued connection 1-form \calAn on it, which is a section of its sub-sheaf \amg1n. \calAn is physically interpreted as the dynamical field of a locally finite quantum causality, while its associated curvature \calFn, as some sort of `finitary Lorentzian quantum gravity.