Abstract

Relevance. This study is relevant because it addresses the long-standing problem of unifying general relativity (GR) and quantum field theory (QFT) by focusing on the essential concepts of discreteness and continuity.

Purpose. The purpose of this work is to introduce and explore the concept of a quantum set (q-set), which provides a framework for defining quantum manifolds (QM) that are equivalent to infinite hyperalgebras.

Methodology. The approach involves defining a specific symmetry for hyper-algebras, which restricts the solution space to a unique hyperalgebra. This hyperalgebra combines both matter and spacetime in a unified structure, allowing for the derivation of observable physical phenomena.

Results. The application of the q-set framework leads to several significant findings. It predetermines spacetime to be locally Minkowski without relying on the assumptions of special relativity. The framework successfully generates the fundamental fermionic and bosonic fields of the Standard Model. The introduction of 0-generation particles, which supplement the three known generations, provides a possible explanation for the nature of Dark Matter. At the Planck scale, the framework describes gravity as being mediated by ten-gauge vector fields, with three of these fields being massive and short-range, while one represents a repulsive force. The study also gives a possibility of the consideration of the Pre-Big Bang Universe structure.

Conclusions. The concept of QM within the framework of a q-set provides a unified description of both matter and spacetime. This new approach successfully connects the Standard Model of particle physics with a novel view of gravity and the universe’s early structure.

Keywords: minimal length; quantum set; hypercomplex algebra; Standard Model; Dark Matter; Big Bang

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