The Dirac equation

In this chapter the Dirac equation will be discussed, which is neccessary for the quantum field theoretical description of fermions, such as electrons.

To start with, the Schrödinger equation will be revisited and a relativistically invariant formulation, the Klein-Gordon equation will be introduced. This equation, however, has a number of intrinsic problems, tied to the fact that it is quadratic in the derivative of the field w.r.t. time. Historically, these problems led to the Dirac equation, which is linear in this and the other derivatives.

Independently of that, it will become obvious that the Klein-Gordon equation, being an equation for scalar fields, does not describe particles with internal degrees of freedom such as spin. Thus, in the next part, spin will be briefly discussed.

Only then, the Dirac equation in its full wealth will be presented and discussed, and finally, various ways to represent its solutions will be investigated.