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 Quote by naima Could you tell me why with addition of these generators the group is "vectorial" and "axial" with subtraction ? Thanks
In the essence it's simple: The two chiral components $\psi_L$ and $\psi_R$
have some similarities with left and right circular polarized photons.
Independently they propagate at the speed of light, however if you
couple them together then the combined momentum is the sum of
the two individual momenta and the resulting propagation speed of
the particle can be anywhere between +c and -c.

For the combined momentum one first calculates the individual momenta.

$\psi_L\tilde{\sigma}^\mu\psi_L$ and $\psi_R\sigma^\mu\psi_R$

The sum of these two is the combined momentum, a vector.
However $\psi_L\tilde{\sigma}^\mu\psi_L$ is also a measure for left handedness while
$\psi_R\sigma^\mu\psi_R$ is a measure for right handedness.

This means that when you subtract the two you get their combined
"handedness", which is an axial vector.

Regards, Hans