Spacetime Symmetries and Z_3-graded Quark Algebra

TL;DR

This paper explores $Z_3$-graded associative algebras, revealing how Lorentz symmetry naturally arises without Minkowski metrics and relating the algebraic structures to quark states and higher-order wave equations.

Contribution

It introduces a novel $Z_3$-graded algebraic framework that naturally encodes Lorentz symmetry and connects to quark operator structures and third-order wave equations.

Findings

Lorentz symmetry emerges from $Z_3$-graded algebra without Minkowski metric

Representation of symmetry group via Pauli matrices

Third-order analogue of Klein-Gordon equation introduced

Abstract

We investigate certain $Z_3$-graded associative algebras with cubic $Z_3$-invariant constitutive relations. The invariant forms on finite algebras of this type are given in the low dimensional cases with two or three generators. We show how the Lorentz symmetry represented by the $SL(2, {\bf C})$ group emerges naturally without any notion of Minkowskian metric, just as the invariance group of the $Z_3$-graded cubic algebra and its constitutive relations. Its representation is found in terms of Pauli matrices. The relationship of this construction with the operators defining quark states is also considered, and a third-order analogue of the Klein-Gordon equation is introduced. Cubic products of its solutions may provide the basis for the familiar wave functions satisfying Dirac and Klein-Gordon equations.