# Bilinear covariants

## Main Question or Discussion Point

Can anyone explain to me how these fit into the bigger picture of the Dirac equation, or suggest a reference?

The only thing I've been able to absorb from reading about these is that they explain the choice of normalization for plane waves $\psi$ (since $\psi^\dag\psi$ is the fourth component of a 4-vector and hence must transform as the 4th component of the momentum-energy vector).

Incidentally, I've been reading about how solution to the charge conjugated Dirac equation is a negative energy state, thus giving support to the positron ~ negative energy solution to Dirac equation" theory.

Is there any physical reason why the bilinear covariants should be invariant under charge conjugation?

I am not sure if all bilinear covariants are invariant under charge conjugation, but only because I haven't explicitly checked this. However, I think that perhaps the Coleman-Mandula theorem would guarantee this somehow. It basically states that Lorentz and "internal" symmetries do not mix. Charge follows from the $$U(1)$$ symmetry of the fields.