How do bilinear covariants relate to the Dirac equation's broader implications?

[jdstokes]

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?

 

[lbrits]

Mmm... I wouldn't normally do this, but I think Peskin & Schroeder do a good job of explaining these. Essentially, you show how various spinor and dirac matrices transform under Lorentz transformations. Then you build the most general Lagrangian you can that is Lorentz invariant. There isn't much room to work with when you're done.

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.