It is an extremely well-established experimental fact that the speed of light is the same for all “inertial observers” (those who do not undergo accelerations). The analysis of the consequences of this remarkable fact has forced a complete revision of Newton’s ideas: Space and time are not different entities but are different aspects of one single entity, space-time. Different inertial observers may use different coordinates to describe the points of space-time, but these coordinates must be related in a way that preserves the speed of light. The changes of coordinates between observers form a group, the Lorentz group. To a large extent the mathematics of Special Relativity reduce to the study of this group. Physics appears to respect causality, a strong constraint in the presence of a finite speed of light. We introduce the Poincaré group, related to the Lorentz group. We develop Wigner’s idea that to each elementary particle is associated an irreducible unitary representation of the Poincaré group and we describe the representation corresponding to a spinless massive particle, explaining also how the physicists view these matters.

This chapter provides a self-contained introduction to the basic aspects of Quantum Mechanics, focusing on what is must for Quantum Field Theory. The notions of state space, unitary operators, self-adjoint operators, and projective representation are covered as well as Heisenberg’s uncertainty principle. A complete proof of Stone’s theorem is given, but the spectral theory is covered only at the heuristic level. We provide an introduction to Dirac’s formalism, which is almost universally used in physics literature. The time-evolution is described in both the Schrödinger and the Heisenberg picture. A complete treatment of the harmonic oscillator, providing an introduction to the fundamental idea of creation and annihilation operators concludes the chapter.