Total Operators and Inhomogeneous Proper Values Equations

TL;DR

This paper explores inhomogeneous proper-value equations involving the total angular momentum operator in Clifford algebra, developing solutions that resemble algebraic quark structures without asserting physical identification.

Contribution

It introduces and analyzes inhomogeneous proper-value equations with total angular momentum operators, providing solutions with algebraic structures akin to quarks.

Findings

Solutions with {} = 0 and {} = {} = 0 have three parts each.

A set of 36 distinct solutions forms an algebraic structure similar to quarks.

Develops a mathematical framework for inhomogeneous eigenvalue problems in Clifford algebra.

Abstract

Kaehler's two-sided angular momentum operator, K + 1, is neither vector-valued nor bivector-valued. It is total in the sense that it involves terms for all three dimensions. Constant idempotents that are "proper functions" of K+1's components are not proper functions of K+1. They rather satisfy "inhomogeneous proper-value equations", i.e. of the form (K + 1)U = {\mu}U + {\pi}, where {\pi} is a scalar. We consider an equation of that type with K+1 replaced with operators T that comprise K + 1 as a factor, but also containing factors for both space and spacetime translations. We study the action of those T's on linear combinations of constant idempotents, so that only the algebraic (spin) part of K +1 has to be considered. {\pi} is now, in general, a non-scalar member of a Kaehler algebra. We develop the system of equations to be satisfied by the combinations of those idempotents for which {\pi} becomes a scalar. We solve for its solutions with {\mu} = 0, which actually also makes {\pi} = 0: The solutions with {\mu} = {\pi} = 0 all have three constituent parts, 36 of them being different in the ensemble of all such solutions. That set of different constituents is structured in such a way that we might as well be speaking of an algebraic representation of quarks. In this paper, however, we refrain from pursuing this identification in order to emphasize the purely mathematical nature of the argument.