Discrete model of spacetime in terms of inverse spectra of the $T_0$ Alexandroff topological spaces

摘要

The theory of inverse spectra of $T_0$ Alexandroff topological spaces is usedto construct a model of $T_0$-discrete four-dimensional spacetime. The universeevolution is interpreted in terms of a sequence of topology changes in the setof $T_0$-discrete spaces realized as nerves of the canonical partitions ofthree-dimensional compact manifolds. The cosmological time arrow arises beingconnected with the refinement of the canonical partitions, and it is defined bythe action of homomorphisms in the proper inverse spectrum of three-dimensional$T_0$-discrete spaces. A new causal order relation in this spectrum ispostulated having the basic properties of the causal order in thepseudo-Riemannian spacetime however also bearing certain quasi-quantumfeatures. An attempt is made to describe topological changes between compactmanifolds in terms of bifurcations of proper inverse spectra; this led us tothe concept of bispectrum. As a generalization of this concept, inversemultispectra and superspectrum are introduced. The last one enables us tointroduce the discrete superspace, a discrete counterpart of theWheeler--DeWitt superspace.