I’ve recently noticed that two very good new books on quantum field theory have become available, one aimed more at mathematicians, one purely for physicists.
What Is a Quantum Field Theory?
Available online now from Cambridge University Press (actual printed books to come soon) is mathematician Michel Talagrand’s What Is a Quantum Field Theory?. While it’s subtitled “A First Introduction for Mathematicians” and definitely aimed more at mathematicians than physicists, it’s a wonderful resource for anyone who wants to understand exactly what a quantum field theory is.
Like many mathematicians, Talagrand tried to learn about quantum field theory first from physics textbooks, which tend to avoid any precise definition of even the basics of the subject. He soon found what was the best source for someone looking for more precision, Gerald Folland’s 2008 Quantum Field Theory: A Tourist Guide for Mathematicians. Folland’s book is extremely good, but also extremely terse. In 325 pages it covers more carefully the material of an old-style QFT book such as Schweber’s 900 page or so An Introduction to Relativistic Quantum Field Theory from 1961. Talagrand is covering much the same material, but with 742 pages to work with he is able (unlike Folland) to work out many topics in full detail, providing something previously unavailable anywhere else.
Both Folland and Talagrand have written books with much the same goal: to as precisely as possible explain the details of the renormalized perturbative expansion of QED. There is little overlap with the work of mathematical physicists who have aimed at rigorous non-perturbative constructions of quantum field theories. They are using canonical quantization methods and don’t overlap much with many of the more recent physics QFT textbooks, which are based on path integral quantization and aimed at getting to non-abelian gauge theories and non-perturbative techniques as quickly as possible.
When I was learning QFT not that long after the advent of the Standard Model, I had little patience for fat QFT books about perturbative QED and canonical methods. Why not just write down the path integral and start calculating? Over the years I’ve realized that things are not so simple, with canonical quantization and operator fields giving a perspective complementary to that of the path integral. Among the more modern books, volume 1 of Weinberg’s three-volume series is the one that best gives this different perspective, and is most closely related to what Talagrand is covering.
For mathematicians, Talagrand’s book is a great place to start. For physicists, Weinberg’s is an important perspective to get to know. If you’re reading Weinberg and want more detail about precisely what is going on, Talagrand’s new book would be a very good place to turn for help.
Quantum Field Theory: An Integrated Approach
Over the years I’ve often consulted various parts of Eduardo Fradkin’s notes on quantum field theory on his web pages. On some basic topics I found these to give very clear explanations of things that were done in a confusing way elsewhere. After recently hearing that the notes are now a book from Princeton University Press, I ordered a copy, which recently arrived.
Fradkin’s book has not much overlap with the material in the Talagrand book described above, and is somewhat different than traditional high energy physics-oriented QFT books. It tries as much as possible to integrate the high energy physics point of view with that of condensed matter and statistical mechanics. Path integral methods are then fundamental. Unlike many other modern QFT textbooks that aim at getting to the details of perturbative Standard Model calculations, Fradkin is more oriented towards getting as quickly as possible to non-perturbative techniques and models of interest in statistical mechanics. He gives a good introduction to various of the modern non-perturbative QFT techniques that have been developed in recent decades, often motivated by the so far only partially successful attempt to come to terms with a strongly-interacting gauge theory like QCD.
While most of the book is quite good, the first few pages aren’t, and will immediately drive away mathematicians who might pick it up. The material in these pages about group theory uses bad terminology (for Fradkin, the “rank” of a Lie group is its dimension and the fundamental representation of SU(n) is the “spinor” representation) and sometimes is just completely wrong. On the second page of the first chapter after the introduction, he wants to explain why the Lorentz group is non-compact, in contrast to SO(3). To explain why SO(3) is compact he starts by mistakenly arguing that since it leaves the unit two-sphere invariant the points of SO(3) and of the unit two-sphere are in one-to-one correspondence, showing the volume of SO(3) is $4\pi^2$. This paragraph should be deleted in future editions of the book.
That this kind of thing can make it into a book like this is remarkable, but unfortunately relativistic QFT books and other sources (e.g. here) don’t always get right basic facts about the Lorentz and rotation groups. I once tried to do my part to remedy this, see here.
Update: John Collins has here an article that provides a careful discussion of scattering in QFT, starting with the basics, which could be thought of as part of a QFT book. This may be of interest to both physicists and mathematicians who want to see something less superficial than many text book discussions.