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.

We describe a most entertaining and imaginative solution to the difficult problem of quantization of the electromagnetic field.

We prove the converse of Stone’s theorem.

We explain why the existence of a true position operator is a dubious proposition.

We introduce tensor products, which are the appropriate tool to construct the state space for systems that can be decomposed into non-interacting simpler components. The boson Fock space, built from such tensor products, is the appropriate state space to describe collections of any number of identical bosons. The annihilation and creation operators act on this Fock space. We then introduce the idea of quantum fields, which are operator-valued distributions, not trying yet to incorporate ideas from special relativity, and we argue that these quantum fields can be viewed as a quantized version of certain spaces of functions.

The physicist’s counter-term method enriches the class of possible diagrams by adding new types of vertices. We explain in simple cases how this method can be used to re-parameterize a theory, and how the physicists use it to tame the diverging integrals by having “the counter terms cancel the divergences”. Assuming that the BPHZ method succeeds in producing finite results for the scattering amplitudes, we prove that the counter-term method succeeds too.

We take a short dip into a popular model for interactions.

We provide in a self-contained way the basics of the theory of Lie Algebras which are used in Quantum Mechanics.