Ternary generalization of Dirac's equation

par Richard Kerner (LPTMC, Univ. Pierre et Marie Curie, Paris)

jeudi 24 nov. 2016, 11:00 → 12:30 Europe/Paris

Petit Amphi (Annecy-le-Vieux)

We start by a discussion of similarities between the $Z_2$ and $Z_3$ symmetry groups and their role in physics. 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 $SL(2, {\bf C})$ 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 and Clifford algebras. We show that the product group $Z_3 \times Z_2 \times Z_2$ is the most appropriate symmetry for an adequate description of quarks endowed with color and with a dychotomic spin-like observable, and the anti-quarks, leading to twelve-valued state vectors. The wave equation generalizing the Dirac operator to the $Z_3$-graded case is introduced, whose diagonalization leads to a sixth-order 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.