Spacetime Symmetries and Z3-graded Quark Algebra

Abstract

We investigate certain Z3-graded associative algebras with cubic Z3-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,C) group emerges naturally without any notion of Minkowskian metric, just as the invariance group of the Z3-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.