The Universe as a Nonuniform Lattice in Finite-Volume Hypercube. I.Fundamental Definitions and Particular Features

TL;DR

This paper introduces a small, dimensionless parameter in generalized quantum mechanics related to the early universe, revealing that the universe can be modeled as a nonuniform lattice within a finite hypercube, with implications for quantum and thermodynamic descriptions.

Contribution

It proposes a new small parameter in generalized quantum mechanics and demonstrates the universe's structure as a nonuniform lattice in a finite hypercube based on this parameter.

Findings

Universe modeled as a nonuniform lattice in a finite hypercube

Parameters contain all fundamental constants and vary discretely

Lattice deformation relates to generalized uncertainty relations

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 discreet 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 $\approx 10^{-66+2n}$, where $10^{-n}$ is the measuring scale and the Planck scale $\sim 10^{-33}cm$ is assumed. The second one is also too small. 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 relations in thermodynamics. In this context the lattice is understood as a deformation rather than approximation.