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 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 ≈ 10 −66+2n , where 10 −n is the measuring scale and the Planck scale ∼ 10 −33 cm is assumed. The second one is also too small for the conventional temperatures, that is those much below the Plancks. 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 * Fax: (+375) 172 326075; e-mail: a.shalyt@mail.ru;alexm@hep.by 1 (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.

Entropy in the Present and Early Universe: New Small Parameters and Dark Energy Problem

“…In conclusion it should be noted that in a series of the author's works [75][76][77][78][79][80][81][82][83][84] a minimal α-deformation of QFT has been formed. By "minimal" it is meant that no space-time noncommutativity was required, i.e., there was no requirement for noncommutative operators associated with different spatial coordinates…”

“…Besides, QMFL has been framed first using the deformed density matrix and then in the produced corresponding Heisenberg algebra deformation [75][76][77][78][79][80][81][82][83][84], the density matrix deformation ρ(α) in QMFL being a starting object called the density pro-matrix and the deformation parameter (additional parameter) α = l 2 min /x 2 , where x is the measuring scale and l min ∼ l p . As indicated in this paper, the deformation parameter α varies within the limits 0 < α ≤ 1/4.…”

“…where < B > is the average in well-known QFT [80,81] denoted as QF T α . All the variables associated with the considered α-deformed quantum field theory are hereinafter marked with the superscript α .…”

“…At the same time, there is no place for such a parameter in the conventional QFT, whereas in QFT with the fundamental length, specifically in QF T α , it is quite natural [75,76,78,80,81,83].…”

“…Moreover, in QF T α there is no need in renormalization as from the start we are concerned with the ultraviolet-finiteness. Moreover, a series expansion of (59) in terms of α is a complete analogue of the expansion in terms of the same parameter, re-determining the measuring procedure in QM F L α [76,78,80,83]:…”