Ternary generalization of Pauli’s principle and the Z ₆-graded algebras

Abstract

We show how the discrete symmetries Z ₂ and Z ₃ combined with the superposition principle result in the SL(2,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 Z ₃ grading replacing the usual Z ₂ grading, leading to ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann algebra. Invariant cubic forms on such algebras are introduced, and it is shown how the SL(2,C) group arises naturally in the case of two generators only, as the symmetry group preserving these forms. The wave equation generalizing the Dirac operator to the Z ₃-graded case is introduced, whose diagonalization leads to a sixthorder equation. The solutions of this equation 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.