By Josephine Evans
By Ornella Mattei
By Graeme Milton
By Maryna Kachanovska
Appears in collection : Combinatorics and Arithmetic for Physics: special days
We show how the fundamental statistical properties of quantum fields combined with the superposition principle lead to continuous symmetries including the $SL(2,\mathbb C)$ group and the internal symmetry groups $SU(2)$ and $SU(3)$. The exact colour symmetry is related to ternary $\mathbb Z_3$-graded generalization of the fermionic commutation relations for quarks. A $\mathbb Z_3$-graded generalization of the Dirac equation is presented, and its invariance properties are analyzed. They lead to an enlarged $\mathbb Z_3$-graded Lorentz group, operating in the Hilbert space of quark states including flavors and generations.