Deformation principle as foundation of physical geometry and its application to spacetime geometry 2004jbu  Rylov Yuri A.
Physical geometry studies mutual disposition of geometrical objects and
points in space, or spacetime, which is described by the distance function $%
d$, or by the world function $\sigma =d^{2}/2$. One suggests a new general
method of the physical geometry construction. The proper Euclidean geometry is described in terms of its world function $\sigma _{\mathrm{E}}$. Any physical geometry $\mathcal{G}$ is obtained from the Euclidean geometry as a result of replacement of the Euclidean world function $\sigma _{\mathrm{E}}$ by the world function $\sigma $ of $\mathcal{G}$. This method is very simple and effective. It introduces a new geometric property: nondegeneracy of geometry. Using this method, one can construct deterministic spacetime geometries with primordially stochastic motion of free particles and
geometrized particle mass. Such a spacetime geometry defined properly (with
quantum constant as an attribute of geometry) allows one to explain quantum
effects as a result of the statistical description of the stochastic
particle motion (without a use of quantum principles).
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