A Set-Matrix Duality Principle for the Dirac Equation

TL;DR

This paper introduces a set-matrix duality principle for the Dirac equation, providing a new mathematical framework to understand spontaneous mirror symmetry violation and the structure of matter fields involving matrices and sets.

Contribution

It proposes a novel set-matrix duality principle that offers exact definitions and axioms, advancing the mathematical understanding of parity violation in elementary particle theories.

Findings

Defines internally disclosed and undisclosed matrices.

Establishes axioms and lemmas for set-matrix duality.

Unifies matrix and set operations in a new framework.

Abstract

Spontaneous mirror symmetry violation is carried out in nature as the transition between the usual left (right)-handed and the mirror right (left)-handed spaces, in each of which the usual and mirror particles have the different lifetimes. As a consequence, all equations of motion in a unified field theory of elementary particles include the mass, energy and momentum as the matrices expressing the ideas of the left- and right-handed neutrinos are of long- and short-lived objects, respectively. These ideas require in principle to go away from the chiral definitions of the structure of matter fields taking into account that the Dirac matrices are, in the Weyl presentation, reduced to the matrices indicating to the absence in nature of a place for parity conservation but not allowing to follow the dynamical origination of its spontaneous violation. We discuss a theory in which a set comes forward at the new level, namely, at the level of set-matrix duality principle as a criterion for matrices. This connection gives the exact mathematical definitions of internally disclosed and undisclosed matrices, allowing to formulate three more definitions, three lemmas and two pairs of axioms. Thereby, it involves that there is no single matrix, for which an absolutely empty matrix would not exist. The sets of matrix elements and the matrices of set elements thus found unite all of matrix and set operations necessary for deciding the problems in a unified whole.