Spacetime Symmetries and Z_3-graded Quark Algebra

Details Spacetime Symmetries and Z_3-graded Quark Algebra Spacetime Symmetries and Z_3-graded Quark Algebra

2011-11-02

arxiv.org/abs/1111.0518v1

Physics

High Energy Physics

High Energy Physics - Theory

15 pages; To appear in the Proceedings of the conference "Quantum Theory and Symmetries" held in August 2011 in Prague

Scientific paper

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.

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