An Introduction to Quantum Field Theory

Ever since Edward Witten, Sir Michael Atiyah, and Graeme Segal opened this Pandora’s Box of magical things given to us by the physicists, things have been stirring in mathematical circles. The promise of quantum field theory (QFT), particularly topological quantum field theory (TQFT), as a new angle on deep geometrical themes has been irresistible, and certain parts of algebraic topology have acquired a new look. To be sure, Atiyah and Segal did brilliant work on axiomatizing QFT, as exemplified by Atiyah’s famous The Geometry and Physics of Knots and, for example, Segal’s online lectures “What is quantum field theory?”

We are now in a position to regard (T)QFT as something like another algebraic topology toolkit: the indicated axioms postulate a functor from manifolds and cobordisms to vector spaces and mappings between them. Ideally we’re in a Hilbert space setting: after all this is where classical quantum mechanics (QM) belongs — it can be regarded as a QFT thanks to, e.g., Feynman and his notorious integrals. But here is where much of the trouble begins: except in a few cases (for example, where the Feynman-Kac theorem applies), Feynman’s formalism of diagrams and path integrals (his “sum over histories” approach) involves an ersatz integration without a well-defined measure and we’re in a mathematical no-man’s land, even aside from the fact that physicists don’t play by axiomatic rules. To push the allusion to Greek sagas and myths even further: beware of physicists bearing gifts. Still, the dividends are stupendous, seeing that quantum electrodynamics, or QED, the playground par excellence of Feynman’s methods, sports the single most precise set of agreements between theoretically determined data and experimental results. This is all very serious stuff.

Of course, another mathematical fly in the ointment is the business of renormalization: to single out two famous examples, the physicists’ vaunted ploy of tweaking their QED or QCD (quantum chromodynamics) formalisms so as to make divergences disappear, is also mathematically unheimlich. Well, not just mathematically — here is what none other than the incomparable Paul Dirac had to say about renormalization: “It's just a stop-gap procedure. There must be some fundamental change in our ideas, probably a change just as fundamental as the passage from Bohr's orbit theory to quantum mechanics. When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number” (cf. a 1970 interview with Dirac conducted by David Peat and Paul Buckley for the CBC show, “Physics and Beyond”). This critique is particularly apt given that Dirac is considered to be the originator of QED and therefore QFT. So, to exploit an obvious play on words, there’s something rotten in Denmark, specifically in Bohr’s Copenhagen.

But nothing succeeds like success, and, as already mentioned, (T)QFT is very sexy geometrically and topologically, so even we mathematicians are now diving into these waters. On the other hand, diving into physics is, for many of us, and emphatically so for me, an unsettling affair: they’re not like us. Their proofs are, well, different; their intuition is very different; and they’re always driving toward calculating nasty numbers that, granted, have great physical significance, but certainly mystify the uninitiated, like me. To wit, what, exactly, is the magnetic moment of the electron, for example, and why should I want to calculate it? Well, I do not, of course, but the authors of the book under review do: see pp. 196–198. And, indeed, this section of the book (Peskin & Schroeder, from now on) sports some very beautiful facts: QED predicts a relation between this number and the fine structure constant, and sets the stage for exquisitely accurate approximations of the latter. (Roughly, the fine structure constant is 1/137, and 137 is of course a gorgeous number arithmetically: it’s a prime, a sum of two squares, 121 and 16, and so on: fun, fun, fun — it’s my favorite number.) On p. 198 we find a table of values for this constant which is nothing if not awe-inspiring, and Peskin & Schroeder finish the section with the emphatic phrase: “On the evidence presented in this table, QED is the most stringently tested — and the most dramatically successful — of all physical theories.”

All right, then, QED, QCD, and QFT: in Peskin & Schroeder’s Introduction to QFT we encounter QED in Part I, with Feynman diagrams properly featured: it all adds up to over 250 pages of physics, but we certainly get our money’s worth with a very thorough and accessible treatment of Feynman diagrams in Chapter 4. Next, Part II, weighing in at a little over 200 pages, is devoted to renormalization: how could it be otherwise, Dirac notwithstanding? Finally, Part III, at a beefy 300 pages, covers non-abelian gauge theory, and we find QCD in Chapter 17. Thus, the book comes to over 750 pages — in fact with the Epilogue and Appendix added we get to over 800 pages. But, again, it can’t be otherwise, not if the physics is to be done justice to, and the coverage is to be pedagogically on target.

The book is an introduction to this big, beautiful, difficult, and important subject, pitched at the level of second year physics graduate students, and is written in an appropriately accessible style. It’s not aimed at mathematicians, but we’ll just have to suck it up and be fellow travelers for a while, reserving the right to pick and choose topics a bit. But it’s really a coherent whole, and is written as a teaching text at the indicated level. Added to the mix are many problems for students to do, and a lot of discussions of experiments, interpretations, etc.: the stuff that physics is made of. And then there is the mathematics, too, of course: consider, e.g., pp. 486–491, where the Yang-Mills Lagrangian is discussed (and recall the seminal work of Atiyah-Bott in this connection, introducing equivariant Morse theory), and, quickly thereafter, on pp. 495–502, coverage of relevant “Basic Facts about Lie Algebras” — it all sets the stage for Chapter 16 and “Quantization of Non-Abelian Gauge Theories,” and we are approaching the frontier.

An Introduction to Quantum Field Theory is an impressive and important text, and it’s a marvelous thing that Westview Press has launched it at under $35.00 as a “Student Economy Edition.”

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