Extremal properties of Singe's world function and discrete geometry

Yu.A. Rylov

Institute for Problems in Mechanics, Academy of Sciences of the USSR, Prospect Vernadskogo 101-1 , Moscow 117526, USSR

email: rylov@ipmnet.ru

Properties of sigma space [a set of points P with real function sigma(P,P') given on this set] are investigated. A continuity of the set is not necessary and, generally, geometry is discrete. The properties of the world function sigma are investigated. At certain (extremal) world function properties the sigma space is shown to be a subset of points of Euclidean space or Riemannian space. The presented approach has the pecularity that no operation other than the world function is given on sigma space. In particular, all such operations as linear operation over vectors, constructing lines and planes, and dimension of the space are expressed through the world function and only through it (if it is extremal). A violation of the sigma-space extremality leads to going out beyond the frames of Riemannian geometry (lines are substituted by tubes of lines, etc.). The presented approach can be useful in quantum gravitation, string models, and other problems, where the properties of the event space at small distances are important.

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