Algebraic Quarks from the Tangent Bundle: Methodology

TL;DR

This paper develops a methodology for algebraic solutions related to quark-like components using tangent bundle mathematics, aiming to bridge algebraic structures with particle phenomenology, and provides formulas for particles and decays.

Contribution

It introduces a new algebraic framework for modeling particle components and interactions, inspired by solutions to inhomogeneous equations involving Kähler's angular momentum.

Findings

Algebraic solutions resemble quark components.

Formulas for hypothetical neutrinos and Z0 particles.

Representation of particles using idempotent algebraic structures.

Abstract

In a previous paper, we developed a table of components of algebraic solutions of a system of equations generated by an inhomogeneous proper-value equation involving K\"ahler's total angular momentum. This table looks as if it were a representation of real life quarks. We did not consider all options for solutions of the system of equations that gave rise to it. We shall not, therefore, claim that the present distribution of those components as a well ordered table has strict physical relevance. It, however, is of great interest for the purpose of developing methodology, which may then be used for other solutions. We insert into our present table concepts that parallel those of the phenomenology of HEP: generations, color, flavor, isospin, etc. Breaking then loose from that distribution, we consider simpler alternatives for algebraic "quarks" of primary color (The mathematics speaks of each generation having its own primary color). We use them to show how stoichiometric argument allows one to reach what appear to be esthetically appealing idempotent representation of particles for other than electrons and positrons (K\"ahler already provided these half a century ago with idempotents similar to our hypothetical quarks). We then use neutron decay to obtain formulas for also hypothetical algebraic neutrinos and the $Z_{0}$, and use pair annihilation to obtain formulas for gamma particles. We go back to the system of equations and develop an alternative option. We solve the system but stop short of studying it along the present line. This will thus be an easy entry point to this theory by HEP physicists, who will be able to go aster and further.