The pion, the mediator of the nuclear force proposed in 1935 by Yukawa. The first particle discovered in the cosmic rays looked like the pion, but was later found to be a lepton, the muon. Experiments at high altitudes on cosmic rays led finally to the discovery of the pion. More experiments soon showed other surprises, the strange particles.

How the properties of the charged pion have been measured.

The discoveries of the charged leptons and of the neutrinos.

How, in 1928, A. M. Dirac found the fundamental relativistic wave equation and the Dirac Lagrangian. Dirac’s fundamental predictions of the existence for each fermion of an antiparticle with the same mass but opposite ‘charges’. How the positron and the antiproton were discovered. The important concepts of helicity and chirality.

The Majorana equation for completely neutral fermions.

Problems on properties of the Dirac equation and emphasis on nontrivial topology and specific materials (such as graphene) are included.

This chapter introduces the formal second quantization method for fermions in quantum field theory, and the connection to second quantization of bosons is shown. The picture of fermions as rotations between two states is presented, which helps the reader to see where the Pauli exclusion rule comes from. Finally, Dirac’s original derivation of his equation for relativistic motion of fermions is given.

This chapter addresses generalizations of the Schrödinger equation. It tries to convey that the Schrödinger equation is not the whole story when it comes to quantum physics. This is illustrated by expanding the framework in two rather orthogonal directions: relativistic quantum physics and open quantum systems. The former is introduced by taking the Klein–Gordon equation as the starting point, before shifting attention to the Dirac equation. Its time-independent version is solved numerically for a one-dimensional example, and its relation to the Schrödinger equation is derived. Also here, the Pauli matrices play crucial roles. The notion of open quantum systems is motivated by the fact that it is hard to keep a quantum system completely isolated from its surroundings – and that this necessitates a different approach than the one provided by wave functions. To this end, reduced density matrices and the notion of master equations are introduced. It is explained why master equations of the form of the generic Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation are desirable. Two particular phenomena following this equation are studied quantitatively: amplitude damping for a single quantum bit system and particle capture in a confining potential. Again, these examples draw directly on previous ones.

Wigner’s idea, that to each elementary particle is associated an irreducible representation of the Poincare group gives fundamental importance to these representations. They are non-trivial mathematical objects. We strive to give a mathematically sound and complete description of the physically relevant representations, and the multiple ways they can be presented, while avoiding the pitfall of relying on advanced representation theory. The representations corresponding to massive particles depend, besides the mass, on a single non-negative integer which corresponds to the spin of the particle. The representations which correspond to massless particles depend on an integer, the helicity, which is a property somewhat similar to the spin. We investigate that action of parity and the operation of taking a “mirror image” of a particle. Finally we provide a brief account of Dirac’s equation.