In this chapter, we will quantize the Dirac field using canonical quantization, i.e. with a similiar method as was used for the Klein–Gordon field (cf. Chapter 6). However, we will observe that we need to replace the canonical commutation relations with canonical anticommutation relations in order to obtain positive energy for the quantized Dirac field and to obey the Pauli exclusion principle. In addition, we will study the transformations of parity, time reversal, and charge conjugation as well as the CPT symmetry for this field. Especially, we will investigate the Majorana field, which is a special case of the Dirac field. We will also derive Green's functions and propagators and briefly investigate interactions.

The Dirac equation (cf. Chapter 3) establishes a single-particle theory (usually known as the Dirac theory), since it cannot take into account creation and annihilation of particles. Therefore, the Dirac equation for wave functions has to be replaced by the Dirac equation for quantum fields, which means that the problems of Dirac theory are circumvented by introducing a quantum field theory reformulation of this theory; thus abandoning wave functions in favour of quantum fields. In fact, adding to the theory of the Dirac equation for quantum fields the quantized version of the electromagnetic field (see Chapter 8), we will end up with the theory of QED, which will be investigated in detail in Chapters 11–13.