Main Question or Discussion Point
Hello! I understand that the free Dirac equations has spinors as solutions, of dimension 4, and one can't discard the negative energy solutions (as one needs a complete basis to span the Hilbert space of solutions), and these negative energy particles are interpreted as positive energy antiparticles. Now, if one adds an EM field, the Dirac equation becomes: $$\gamma^\mu(\partial_\mu-ieA_\mu)\psi+im\psi = 0$$ This equation still requires 4 linearly independent solutions, but I am not sure how to think of them in this case. One can't pretend that 2 of them are particles and 2 are anti-particles, because the charge sign appears explicitly here and at the same time, one can't discard 2 of the solutions. So how should I think of the 4 linearly independent solutions to this equations? Thank you!