We give a brief and basic introduction to perturbation theory. The main idea is to attempt to consider the situation of interest as a small perturbation of a simpler situation (which can be understood completely), and in particular to consider a system with a weak interaction as a perturbation of a non-interacting system. We develop the interaction picture, which allows approximating the time-evolution of an interacting system by the partial sums of the Dyson series, a fundamental tool for the sequel. We illustrate these ideas on a rudimentary model of the interaction of electrons and “photons”.

We give a very concise introduction to some concepts of Special Relativity.

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.

We try to take some of the mystery out of the physicist’s manipulations of Classical Fields.

We give some complements to Chapter 9.

We prove in complete generality that the BPHZ scheme assigns a finite value to each Feynman diagram. The proof uses only elementary mathematics, but is magnificently clever.

This chapter enters “theoretical physics”, giving significant examples of the arguments by which physicists try to justify the success of their methods. Formal series in diagrams take a life of their own. A major discovery is that the parameter which is called m in the model, and which is supposed to represent the mass of the particle cannot possibly be the mass of the particle as it is measured in the laboratory. In other words, self-interaction changes the mass of the particle. This is the phenomenon of mass renormalization. We also approach an even deeper mystery, the phenomenon of field renormalization, which forces us to revisit the method we used in Chapter 13 to compute the S-matrix. We also explain why all these efforts barely provide any convincing support of the physicist’s methods, because these use a huge leap of faith which is mostly kept implicit in their work.

We review contour integrals and the Feynman propagator as a fundamental solution of the Klein-Gordon equation.

We clarify the mathematical meaning of one aspect of scattering in Quantum Mechanics.

We spell out a useful lemma in the theory of distributions.

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.

We study the canonical commutation relation, and give a complete proof of a fundamental result of Stone and von Neumann: A finite set of operators satisfying (the proper form of) these relations is essentially unique. We also detail why this result miserably fails for infinite sets of operators.

The orthogonal group admits projective unitary representations which do not derive from true representations, and we describe a fundamental family of such representations. As a consequence there exist quantum systems that change state under a full turn rotation along a given axis (although a second full turn rotation brings them back to the original state). Amazingly, Nature has made essential use of this structure. In order to study the projective representations of the orthogonal and Lorentz groups, it is convenient to replace them by “better versions“; the groups SU(2) and SL(2,C), which are groups of 2 by 2 matrices, and for which projective representations are simply related to true representations. The orthogonal and Lorentz groups are then images of these groups under two-to-one group homomorphisms, and it is these isomorphisms that concentrate the behavior of their projective representations. Finally we describe how the introduction of parity in our theory leads to the discovery of the Dirac matrices.

We introduce the massive scalar field, the simplest Lorentz invariant Quantum Field which respects causality, a natural and canonical object, and we explain the formulas used to describe it in physics books.

This chapter provides an introduction to the notion of physical dimension, to the specific notations which are used in physics, as well a brief review of some basic mathematics: an introduction to informal distribution theory, to the delta function and the Fourier transform.

We explain why the experimentally established fact of conservation of electrical charges more or less forces the existence of anti-particles. Armed with this essential information we then turn to the study of Lorentz covariant families of quantum fields, of which the massive scalar field of Chapter 5 is the simplest example. These are the building blocks of the standard model, which describes the whole zoo of existing particles. We follow the steps of S. Weinberg to discover that simple linear algebra, combined with a few natural assumptions is all that is required to discover the main fields which are used by Nature (which we list and study), without having to resort to the contortions often seen in the physics literature. We give an example of these contortions by describing the attempts made to relate the Dirac field to classical mechanics.