From Classical to Quantum Fields

Nine hundred and thirty-one pages on field theory — not the algebraic version, as in finite fields, Galois fields and algebraically closed fields, but the fields the physicists play with: a horse of a different color. This tome, by Baulieu, Iliopoulos, and Sénéor, covers the spectrum (I won’t apologize for these puns any more, since they’re just too easy to pass up) from classical fields to quantum fields. What this means is that we go from Hamiltonian and Lagrangian mechanics all the way to nothing less than “Beyond the Standard Model.”

Given how exciting current quantum field theory (henceforth just QFT) and its outgrowths are, and with so very much going on in physical circles these days, it is fitting and proper that symmetry is badly broken in this book: there’s comparatively little that’s classical, and the bulk of the book is about things quantum or even relatively quantum. Specifically, after doing a number on themes from relativity and electromagnetism, quantum mechanics is covered, in its relativistic form: the book’s seventh chapter is suggestively titled, “Towards a Relativistic Quantum Mechanics,” and features the equations of Klein-Gordon and Dirac. At this stage we’re only 161/931 of the way through, and it’s pretty fair to say that, so far, the authors have not taken us too far afield: the material thus far is, so to speak, at least quasi-classical.

But then a jump occurs: in chapter eight we meet Feynman. This is truly a watershed chapter:  8.4 sports the topics of the QM of a free particle, a particle in a potential, and then the sine qua non of the Schrödinger equation, and after that  8.5 introduces Feynman’s famous (if mathematically controversial) bossa nova version of QM based on the yoga of his “integral.” Well, the cat’s among the pigeons now, and the next three chapters deal with functional integrals and culminate in a section (  11.5) devoted to the critical theme of going from Feynman path integrals in QM to their counterparts in field theory. This is of huge importance since it addresses the business of how to finesse quantum fields as distinct from quantum mechanical systems.

But it’s still relatively early in the game: the authors are faced, next, with the matter of developing the notion of relativistic quantum fields, and it is fair to say that Chapter 12, kicking in on p.260, is at the very heart of the entire discussion. Starting with the issue of axiomatic formulation, we get to Feynman diagrams, the all-important Maxwell field (classically as well as considered as a quantum field — and in two flavors), reduction formulae for photons and fermions, and then the paradigm of a QFT, quantum electrodynamics (QED: and do read Feynman’s book by the same title, if you haven’t already). But we’re still not even half-way through. What’s next?

Well, it all begins with Chapter 14, “Geometry and Quantum Dynamics,” which is music to any mathematician’s ear, seeing that this is part of the wonderland where Graeme Segal, Michael Atiyah, and Ed Witten hold forth. Gauge theory enters the scene, Yang and Mills appear, and that’s only the start. Soon symmetry is broken, renormalization and symmetry are discussed, Yang-Mills theory is renormalized, and on and on it goes. But we are now, at last, in the second half of the tome and it’s perhaps fair to say that the focus begins to shift more and more in the direction of boots-on-the-ground modern physics. Consider the following topics, for example: infrared singularities, coherent states and the classical limit of QED, field theories beyond the perturbation expansion (who knew?), and then the very sexy business of fundamental interactions. The latter stuff is ultracool, as we encounter, in sequence, the topics, “What is an ‘elementary particle’?” (what indeed!), the four interactions, the Standard Model (yeah!: leptons, hadrons, and neutrinos — somewhere Pauli is smiling…), and then gauge theory for strong interactions, where we meet quantum chromodynamics. Inter alia, just to assuage the anxiety that grips mathematicians (well, me, anyway) when the physicists talk “gauge” (a notion due to Hermann Weyl), I first experienced something like a palliative when I read the following line in Atiyah’s The Geometry and Physics of Knots: “This [business of mathematicians and physicists interacting like two ships that pass each other in the night] has now been rectified with gauge theory (alias the theory of connections) providing the common ground.” We are saved, psychologically, by differential and Riemannian geometry.

At last we’re at, or at least near, the end of the voyage, with the last two chapters dealing with matters “beyond the standard model,” as already mentioned: “supersymmetry, or the defence [recall that the authors use the Queen’s English, not our colonial form] of scalars.” Supersymmetry (SUSY) is indeed very sexy, and is popular even among mathematicians. The even more tomistic (two huge volumes) Quantum Fields and Strings: A Course for Mathematicians, whose editors include both Pierre Deligne and Ed Witten, presents supersymmetric algebraic structures right from the start. More to the point, the book under review also ends with mathematically (or geometrically or topologically) seductive themes, namely, TQFT (topological QFT), including a Yang-Mills version, and also supergravity.

There are five beefy appendices including coverage of tensors (both algebraically and geometrically: excellent!), exterior differential calculus, Lie theory, and even an “Extract from Maxwell’s A treatise on Electricity and Magnetism.” What good taste.

After all this it’s important to stress that we’re still dealing with a textbook, its sweep notwithstanding: there are problem sets throughout the big book, everything is done in great detail, and the reader who stays with the discussion from beginning to end will obtain a fabulous education in modern physics, as well as a load of very solid ancillary mathematics. What a book!

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.