A discrete model for Gell-Mann matrices

Abstract

I propose a discrete model for the Gell-Mann matrices, which allows them to participate in discrete symmetries of three generations of four types of elementary fermions, in addition to their usual role in describing a continuous group $SU(3)$ of colour symmetries. This model sheds new light on the mathematical (rather than physical) necessity for `mixing' between the various gauge groups $SU(3)$, $SU(2)$ and $U(1)$ of the Standard Model. In particular it shows how the anti-Hermitian version of Pauli matrices can act non-trivially on a unitary version of the Gell-Mann matrices, which leads to a non-trivial mixing between the weak and strong nuclear forces. The unitary version of the Gell-Mann matrices can in turn act non-trivially on a quaternionic version of Dirac matrices, which leads to a non-trivial mixing between the strong force and the shape of spacetime defined by the Dirac matrices. Hence this model implies a mixing between the electro-weak-strong forces on the one hand and gravity, as described by General Relativity, on the other. This mixing in turn implies the necessity for both general relativistic corrections to the Standard Model of Particle Physics, and quantum corrections to General Relativity. Contrary to general expectation, both types of corrections seem to be large enough to be tested experimentally.