Bilinear Covariants & the Dirac Equation: Exploring the Bigger Picture

[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.

 

[denian]

The bilinear covariants play a crucial role in the Dirac equation and understanding their significance is important in exploring the bigger picture of this equation. These covariants are mathematical objects that are constructed from the Dirac spinors, and they have important properties such as being Lorentz invariant and being related to the conserved quantities in the Dirac equation. They also provide a way to understand the choice of normalization for plane waves in the Dirac equation.

Furthermore, the bilinear covariants are intimately connected to the concept of charge conjugation, which is a fundamental symmetry in particle physics. This symmetry relates particles to their anti-particles and plays a crucial role in the study of particle interactions. The fact that the bilinear covariants are invariant under charge conjugation is not a coincidence, but rather a consequence of the underlying mathematical structure of the Dirac equation.

In terms of the bigger picture, understanding the role of bilinear covariants in the Dirac equation helps us to gain a deeper understanding of the fundamental principles of quantum mechanics and the behavior of particles at the subatomic level. It also has implications for the study of particle interactions and the search for new physics beyond the Standard Model.

I would recommend further reading on the topic, such as the book "Quantum Field Theory" by Lewis H. Ryder or the paper "Bilinear Covariants and the Dirac Equation" by David G. C. McKeon for a deeper understanding of the connection between bilinear covariants and the Dirac equation.