THE UNIVERSE AS A NONUNIFORM LATTICE IN FINITE-VOLUME HYPERCUBE I: FUNDAMENTAL DEFINITIONS AND PARTICULAR FEATURES

Abstract

In this paper a new small parameter associated with the density matrix deformation (density pro-matrix) studied in previous works of the author is introduced into the Generalized Quantum Mechanics (GQM), i.e. quantum mechanics involving description of the Early Universe. It is noted that this parameter has its counterpart in the Generalized Statistical Mechanics. Both parameters offer a number of merits: they are dimensionless, varying over the interval from 0 to 1/4, and assuming in this interval a discrete series of values. Besides, their definitions contain all the fundamental constants. These parameters are very small for the conventional scales and temperatures, e.g. the value of the first parameter is on the order of ≈10-66+2n, where 10-n is the measuring scale and the Planck scale ~ 10-33cm is assumed. The second one is also too small for the conventional temperatures, that is those much below the Planck temperature. It is demonstrated that relative to the first of these parameters the Universe may be considered as a nonuniform lattice in the four-dimensional hypercube with dimensionless finite-length (1/4) edges. And the time variable is also described by one of the above-mentioned dimensions due to the second parameter and Generalized Uncertainty Relation in thermodynamics. In this context the lattice is understood as a deformation rather than approximation.