Free fermion fields are canonically quantized, proceeding from Weyl to Dirac and Majorana fermions, and from the massless to the massive case. We discuss properties like chirality, helicity, and the fermion number, as well as the behavior under parity and charge conjugation transformation. Fermionic statistics is applied to the cosmic neutrino background.

In modern physics, symmetries are a powerful tool to constrain the form of equations, namely the Lagrangian that describes the system. Equations are assumed to be invariant under the transformation of a given group, which may be discrete or a continuous Lie group. Classification of the various types of symmetry. The concept of spontaneous symmetry breaking. It will evolve into the Higgs mechanism, which gives origin to the masses of the vector bosons that mediate the weak interactions, of the quarks and of the charged leptons.

The discrete symmetries, in particular the parity and the particle–antiparticle conjugation operations and the corresponding quantum numbers.

An important dynamical symmetry of the hadrons, the invariance of the Lagrangian under rigid rotations in an ‘internal’ space, the isospin space. The unitary group is SU(2).

There are many representations of time reversal symmetry, including PT, CT, and CPT, but only the standard time reversal operator T is associated with an arrow of time itself.