We give a short introduction to some rigorous aspects of distribution theory.

We prove that the singularities occurring from the form of the propagator are a simple technical nuisance, and can be removed in the limit, provided we accept to deal with distributions rather than with functions.

In the first few sections we examine how we might define Hamiltonians which make physical sense, and we observe that the interaction picture, on which the entire approach is built, is fraught with mathematical inconsistencies. Nonetheless we proceed using it to compute the S-matrix in some of the simplest possible models. This is the heart of the theory. In a very progressive fashion we introduce the main tools, Wick’s theorem and the Feynman propagator, a very special tempered distribution. We then introduce Feynman’s diagrams. Each diagram encodes a term of a complicated calculation, and we give an algorithm to compute the value of such a diagram by a complicated integral. We pay great attention to clarify the nature and the role of the so-called symmetry factors. We then receive the bad news. As soon as the diagrams contain loops the integral giving its value has an irresistible tendency to diverge, a consequence of having attempted an ill-defined multiplication of distributions. We then show how to get a sensible physical prediction out of these infinite integrals, first in the relatively easy case of diagrams with one loop, and then in the much deeper case of diagrams with two loops, which involves a remarkable “cancellation of infinities”. We also introduce the physicist’s counter-term method to produce such cancellations.

Two well-polished metallic plates very close to each other in a vacuum feel a (slight) mutual attraction. This is the experimentally verified Casimir effect, deeply linked to the fact that the ground state energy of the harmonic oscillator is positive, and to the far more intriguing fact that in a sense and infinity of such oscillators live in the vacuum. Modeling the Casimir effect we have to first confront the great plague of the theory, the occurrence of infinite quantities in the calculations. In this case, though, a simple procedure allows canceling the infinites and getting a meaningful result.

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.

We give a brief introduction to the rigorous approach to Quantum Field Theory through the Wightman axioms, in the direction of Haag’s theorem on the inconsistencies of the interaction picture.

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.