With this chapter starts the process of renormalization, how to assign in a consistent way a finite value to all these diverging integrals. The procedure itself is fully rigorous. It is the fact that the result of this procedure brings us information about the physical world which is largely an act of faith (which is however supported by experimental data to an astonishing degree of precision). We perform preliminary work, giving necessary conditions for rational functions to be integrable by comparing the degrees of the numerator and denominator, conditions which turn out to be sufficient for the rational functions of interest in computing the values of Feynman’s diagrams, a result known as Weinberg’s power counting theorem.

We make a short remark on the use of distribution theory to describe functions on regular surfaces.

We describe the BPHZ scheme of renormalization. The diverging integrals are tamed by replacing the integrand by a regularized version of it with better properties. Historically the scheme was recursive, removing in turn the divergences from the small diagrams to the larger ones, but now the regularization of the integrand is directly given as a sum of terms by the forest formula. In the end, the scheme can be seen as a very clever use of Taylor’s formula.

We answer some natural mathematical questions concerning representations. We develop the theory of induced representations for finite groups, which sheds considerable light on the structure of the induced representation of the Poincaré group studied in Chapter 8.

We provide the briefest introduction to Lagrangian and Hamiltonian Mechanics, and we explore several routes by which the physicists argue that the massive scalar field is a quantization of a natural classical field, the Klein-Gordon field.

The majority of our data concerning the particle world comes from scattering experiments, and the theoretical analysis of these is of fundamental importance. This analysis has two parts. First, we encode the properties of the scattering in an object called the S-matrix, whose computation is a main objective of the theory. Second, we relate the S-matrix to quantities that can actually be measured in our laboratory, the so-called cross-sections. We explain heuristically, through the analysis of situations of increasing complexity, what the S-matrix is, but we do not try yet to compute it. We then turn to the relation between the S-matrix and cross-sections, proving that indeed the S-matrix contains the information needed to predict the outcome of these experiments.

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.