Prof. Richard Kerner, Sorbonne, Paris

Lorentz covariance from discrete symmetries Z2 and Z3

Our aim is to derive the symmetries of the space-time, i.e. the Lorentz transformations, from symmetries of the interactions between the most fundamental constituents of matter, in particular quarks and leptons. We show how the discrete symmetries Z_2 and Z_3 combined with the superposition principle result in the SL(2, C) and SU(3) symmetries. 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 Pauli's principle is generalized for the case of the Z_3 grading replacing the usual Z_2 grading, leading to ternary commutation relations. We present the cubic and ternary algebras which are a direct generalization of fermionic algebras with Z_3-grading replacing the usual Z_2-grading. Elementary properties and structures of such algebras are discussed, with special interest in the low-dimensional ones, with two generators only. Invariant cubic forms on Z_3-graded algebra with two generators are introduced, a possible description of the isospin. It is shown how a Z_3-graded generalization of the SL(2,C) group arises naturally as the symmetry group preserving the Z_3-graded ternary isospin algebra. Vectorial and spinorial representations of the generalized Z_3-graded. Lorentz algebra are briefly discussed.