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In this paper, we will find a way to apply the Gell-Mann transformations made by the λi matrices using Geometric Algebra Cl3,0. And without the need of adding the time as an ad-hoc dimension, but just considering that:
The transformations are as follows. Considering the original ψ:
The new ψ’ obtained when applying each of the Gell-Mann matrices λi is:
Considering that Gell-Mann matrices do not consider at all the existence I if and , it is possible that we should consider them zero form the beginning. Anyhow, above relations would correspond with the most general case.
We have also worked in the bra-ket product using geometric algebra. For the Euclidean case we have the equation (where the cross sign means reverse and the asterisk means conjugate, both mean the same in Cl3,0:
Being ρ the probability density:
And the fermionic current:
We have made the same in the case of orthogonal buy not orthonormal metric, leading to:
But in this case:
It has been also shown that the g-2 issue of the muon could be related to gravitational (non-Euclidean metric) issues without needing another natural force.
The difference of the values of g-2 of the muon are:
And the effect of the non-Euclidean metric on the surface of Earth is: