@MISC{_discretespace-time, author = {}, title = {Discrete space-time}, year = {} }

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Abstract

One conceivable hypothesis on the structure of space in the microcosmos, conceived as a collection of disconnected elements in space (points) which cannot be distinguished by observations. An acceptable formalization of discrete space-time can be given in terms of topological spaces in which the connected component of a point is its closure, and, in a Hausdorff space, is this point itself (totally disconnected spaces). Examples of include a discrete topological space, a rational straight line, analytic manifolds, and Lie groups over fields with ultra-metric absolute values. The discrete space-time hypothesis was originally developed as a variation of a finite totally-disconnected space, in models of finite geometries on Galois fields V.A. Ambartsumyan and D.D. Ivanenko (1930) were the first to treat it in the framework of field theory (as a cubic lattice in space). In quantum theory the hypothesis of discrete space-time appeared in models in which the coordinate (momentum, etc.) space, like the spectrum of the-algebra of corresponding operators, is totally disconnected (e.g. like the spectrum of the-algebra of probability measures). It received a serious foundation in the concept of "fundamental lengths " in non-linear generalizations of electrodynamics, mesondynamics and Dirac's spinor theory, in which the constants of field action have the dimension of length, and in quantum field theory, where it is necessary to introduce all kinds of "cut-off " factors. These ideas, later in conjunction