Quantization of the space-time based on a formless finite fundamental element

TL;DR

This paper introduces a novel model of space-time based on formless finite fundamental elements (FFFE), providing a framework that bridges discrete and continuous space-time and explores implications for quantum fields and string theory.

Contribution

It proposes a new space-time model using FFFEs, unifies various discrete space-time constructions, and connects to string theory and supersymmetry.

Findings

FFFE space-time can be described as coverings of continuous space.

The model recovers continuous space in the limit of zero fundamental length.

Connections to string theory and supersymmetry are established.

Abstract

The concept of the space (space-time) of the formless finite fundamental elements (FFFE) is suggested. This space can be defined as a set of coverings of the continual space by non-overlapping simply connected regions of any form and arbitrary sizes with some probability measure. The average sizes of each fundamental element are equal to the fundamental length. This definition enables to describe correctly the passage from the space of the formless finite fundamental elements to the continual space in the limit of zero value of the fundamental length. FFFE space-time functional integral construction is suggested. A wave function of a separate FFFE and the overall wave function of a manifold are introduced. It is shown that many other constructions of the discrete space-time (the Regge coverings, the lattice space-time etc.) are the special cases of this space-time. A vacuum action problem is analyzed. One term of this action is proportional to the volume of a fundamental element. It is possible to direct the way for this term to yield the Nambu-Goto action in consideration the string as one-dimensional excitation of a number of FFFEs. Fermionic and bosonic fields in the space-time of FFFEs are excited states of elements. Space-time supersymmetry leads to supposition that the maximal possible number of fermionic excitations at one FFFE is equal to the number of elements in all space-time. The compactification in this space-time means the condition of the neighbour elements absence in compactificated dimensions.