This is a brief guide to other books on quantum field theory. The standard modern textbook is An Introduction to Quantum Field Theory, by Peskin and Schroeder [33]. I recommend especially their wonderful Chapter 5, and all of the calculational sections between 16.5 and 18.5, as well as Chapters 20 and 21. Every serious student of QFT should work out the final project on the Coleman–Weinberg potential, which can be found on page 469. Another standard is Weinberg's three-volume opus [131]. Here I recommend the marvelous sections on symmetries and anomalies in Volume II. The technical discussions of perturbative effective field theory are invaluable. The section on the Batalin–Vilkovisky treatment of general gauge equivalences is also useful. Volume I should probably be read after completing a first course on the subject. It presents an interesting but idiosyncratic approach to the logical structure of the field. Volume III on supersymmetry is full of gems. In my opinion, it is flawed by an idiosyncratic notation and a tendency to obscure relatively simple ideas in an attempt to give absolutely general discussions. Finally, let me mention a relatively new book by M. Srednicki [168]. I have not gone through it thoroughly, and I do not agree with the author's ordering of topics, but the pedagogical style of the sections I have read is wonderful. It is clear that everyone in the field will turn to this book for all those nasty little details about minus signs and spinor conventions.

Preface and conventions

This book is meant as a quick and dirty introduction to the techniques of quantum field theory. It was inspired by a little book (long out of print) by F. Mandl, which my advisor gave me to read in my first year of graduate school in 1969. Mandl's book enabled the smart student to master the elements of field theory, as it was known in the early 1960s, in about two intense weeks of self-study. The body of field-theory knowledge has grown way beyond what was known then, and a book with similar intent has to be larger and will take longer to absorb. I hope that what I have written here will fill that Mandl niche: enough coverage to at least touch on most important topics, but short enough to be mastered in a semester or less. The most important omissions will be supersymmetry (which deserves a book of its own) and finite-temperature field theory. Pedagogically, this book can be used in three ways. Chapters 1–6 can be used as a text for a one-semester introductory course, the whole book for a one-year course. In either case, the instructor will want to turn some of the starred exercises into lecture material. Finally, the book was designed for self-study, and can be assigned as a supplementary text. My own opinion is that a complete course in modern quantum field theory needs 3–4 semesters, and should cover supersymmetric and finite-temperature field theory.

We are finally ready to face up to the fact that most of the expressions we write in the Feynman-diagram solution of quantum field theory are nonsensical, which is to say infinite. The purpose of this introductory section is to demonstrate the simple and general counting rules which prove that this is so, and to give the reader a conceptual orientation to the general theory of renormalization, which we will discuss in the rest of this chapter.

We will begin with the conceptual. The formalism of quantum field theory makes statements about arbitrarily short distances in space-time and arbitrarily high energies. At a given moment in history experiment will never probe arbitrarily short distances or high energies. We should expect that the formalism will have to change to fit the data as we uncover more empirical facts about short distances. This has happened before. Hydrodynamics is a field theory, but we know that at short enough distance scales it is not a good description of water. A better one is given (in principle) by the Schrödinger equation for hydrogen and oxygen nuclei interacting with electrons, and that in turn is superseded by the standard model of particle physics as we go to even shorter distances and higher energies. Yet only a fool would imagine that one should try to understand the properties of waves in the ocean in terms of Feynman-diagram calculations in the standard model, even if the latter understanding is possible “in principle.”

Symmetries

The rules that limit the possibility of an initial state transforming into another state in a quantum process (collision or decay) are called conservation laws and are expressed in terms of the quantum numbers of those states. We shall not deal with the invariance under continuum transformations in space-time and the corresponding conservation of energy-momentum and of angular momentum, which are known to the reader. We shall consider the following types of quantum numbers.

Discrete additive If a quantum number is additive, the total quantum number of a system is the sum of the quantum numbers of its components. The ‘charges’ of all fundamental interactions fall into this category, the electric charge, the colour charges and the weak charges. They are conserved absolutely, as far as we know. The conservation of each of them corresponds to the invariance of the Lagrangian of that interaction under the transformations of a unitary group. The group is called the ‘gauge group’ and the invariance of the Lagrangian is called ‘gauge invariance’. The gauge group of the electromagnetic interaction is U(1), that of the strong interaction is SU(3) and that of the electroweak interaction is SU(2) ⊗ U(1). Other quantum numbers in this category are the quark flavours, the baryon number, the lepton flavours and the lepton numbers. They do not correspond to a gauge symmetry and are not necessarily conserved (actually, quark and lepton flavours are not).