Institute for Problems in Mechanics, Russian
Academy of Sciences
101-1 ,Vernadskii Ave., Moscow, 119526, Russia
Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm
or mirror Web site: http://gasdyn-ipm.ipmnet.ru/~rylov/yrylov.htm
The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via the Euclidean world function \sigma _E and declared to be concepts and objects of any physical geometry, provided the Euclidean world function \sigma _E is replaced by the world function \sigma of the physical geometry in question. The set of physical geometries is more powerful, than the set of Riemannian geometries, and one needs to choose a true space-time geometry. In general, the physical geometry is multivariant (there are many vectors which are equivalent to a given vector, but are not equivalent between themselves). The multivariance admits one to describe quantum effects as geometric effects and to consider existence of elementary particles as a geometrical problem, when the possibility of the physical existence of an elementary geometric object in the form of a physical body is determined by the space-time geometry. Multivariance admits one to describe discrete and continuous geometries, using the same technique. A use of physical geometry admits one to realize the geometrical approach to the quantum theory and to the theory of elementary particles.