Ternary generalization of Pauli's principle and the Z6-graded algebras

TL;DR

This paper explores a ternary extension of Pauli's principle using Z3-graded algebras, leading to new symmetries, a generalized wave equation, and insights into the origin of color SU(3) symmetry.

Contribution

It introduces a Z3-graded generalization of Grassmann algebra and constructs a corresponding wave equation, expanding the mathematical framework of quantum symmetries.

Findings

Derived a Z3-graded generalization of Grassmann algebra.

Constructed a sixth-order wave equation with non-propagating solutions.

Proposed a connection between ternary symmetries and color SU(3).

Abstract

We show how the discrete symmetries $Z_2$ and $Z_3$ combined with the superposition principle result in the $SL(2, {\bf C})$-symmetry of quantum states. The role of Pauli's exclusion principle in the derivation of the SL(2, C) symmetry is put forward as the source of the macroscopically observed Lorentz symmetry, then it is generalized for the case of the Z3 grading replacing the usual Z2 grading, leading to ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann algebra. Invariant cubic forms are introduced, and their symmetry group is shown to be the $SL(2,C)$ group The wave equation generalizing the Dirac operator to the Z3-graded case is constructed. Its diagonalization leads to a sixth-order equation. The solutions cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.