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A Combinatorial Perspective on Quantum Field Theory

Karen Yeats
Publication Date: 
Number of Pages: 
Springer Briefs in Mathematical Physics 15
[Reviewed by
Michael Berg
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QFT, universally short for quantum field theory, is one of those weird things from physics that, somewhat perversely, have become irresistible to mathematicians. I guess there is actually no other example of something where our cousins next door (or in my university’s case, from across campus: a geographical perversity) concoct something for their own parochial use which then evolves to the point of seducing sober, rigor-minded mathematicians. They then have to go out and learn a weird mathematical dialect: the physicists use our words, but they don’t quite mean what they should. Or they take standard things from our neck of the woods and tailor them for their own use. I first encountered this experience when reading Dirac’s Principles of Quantum Mechanics and his extremely practical but (to me, at least) unsettling way of handing dual bases for linear spaces — yes, the business of bras and kets. And then there is his admittedly brilliant and critical presentation of the principle of superposition. Dirac is my favorite physicist, together with Einstein, but his works need a caveat for us mathematicians: it’s a master physicist teaching physicist acolytes, not mathematicians. Even Einstein is not immune to doing weird things — witness his (in)famous summation notation, adopted by almost all differential geometers, for better or for worse. But there is no doubt that the physicists use these notational devices to huge advantage, particularly given that they need to get numbers out. After all, it’s still an experimental science.

Well, it is and it isn’t. First there’s quantum electrodynamics and yes, there are experiments there, some of which succeed in awarding QED the title for the best physical theory ever, given that some of the computations done by the theorists agree with experimental values to something like twenty-three places (if I remember correctly). But QED is just the start: it has evolved into an autonomous overarching formalism, namely QFT, and a lot more has to be said — much of it pretty weird. QED won the 1965 Nobel Prize in Physics for Schwinger, Feynman, and Tomonaga, with Freeman Dyson unfairly out of the running (in most people’s opinion). And the point here is that Feynman (and Dyson, as his interpreter to the world) provided an approach to QED which was fabulous physics, but anything but rigorous mathematics. The trouble is that his yoga of integrals over all possible paths in a certain patch of space-time, or (equivalently) his sum over all histories approach to quantum mechanical (or electrodynamic) processes is mathematically flawed: Feynman’s integrals lack a well-defined measure. But they work. The physicists win. So here is a notion that’s morally worse than the earlier examples of stuff done by Einstein and Dirac: Feynman’s integrals actually fail the well-definition test. So they’re not really proper fare for mathematics. Well, think again…

I guess the credit goes to Edward Witten, although he was preceded by, e.g., Simon Donaldson in this regard. It’s all about topology and using physics to get at invariants. Witten’s principal work in this area is his very famous paper, “Quantum field theory and the Jones polynomial,” now almost thirty years old, in which he uses a Feynman integral formalism with a Chern-Simons action to get at, yes, nothing short of knot theory. And that’s just the tip of the iceberg, really. A great deal has happened since, and there is a great deal of literature to be had, even at the expository level. And as regards knot theory, I can do no better than recommend the beautiful, deep, and readable tract, The Geometry and Physics of Knots, by Sir Michael Atiyah. Truly fabulous, and, as the song goes, don’t worry, be happy: it’s Atiyah, so the mathematics is safe (and the physics comes packaged with appropriate warnings and such).

So now we get to the book under review: not geometry or topology, but combinatorics. I guess this just adds weight to my point about the unreasonable effectiveness of QFT, to turn a saying of Wigner’s on its side (by the way, Wigner was Dirac’s brother-in-law).

What’s going on here? Well, it’s not quite as spectacular a case of cross-pollination as the geometry-and-physics business, and the author, Karen Yeats, presents her book as “a tour, shaped by the author’s biases, that a combinatorial perspective can be brought to bear on quantum field theory.” Accordingly she splits the presentation into three parts: preliminaries, Dyson-Schwinger equations, and Feynman graph periods. She elaborates: “Dyson-Schwinger equations are the quantum analogues of the equations of motions and so are physically important,” and as regards Feynman graphs (a.k.a. his diagrams, I guess) and integrals, she trains her focus on “a particular renormalization-scheme-independent residue known as the Feynman period.” Among other things, she looks at “graph symmetries that preserve the Feynman period” and then at “a graph invariant known to have these same symmetries, but for which it is not known how it relates to the period itself” — certainly a very tantalizing mathematical problem in and of itself. And this gives some of the flavor of this book. It’s really a demonstration of how the Feynman integral and all it has wrought is a fabulous playground for very interesting combinatorics, as well as an elegant presentation of the indicated physics through the agency of combinatorics.

Yeats does a beautiful job in presenting the wealth of material she has chosen to feature, with each chapter supplemented by a list of references. The book is well-written from a mathematical perspective, i.e. it reads like mathematics not physics, with definitions, propositions, and proofs. But there are plenty of remarks about physics qua physics. It’s a thin book, at 120 pages, but the print is small, so we have some serious density here. Be sure to bear that in mind when going at this material: it’s very nice stuff, but, yes, also pretty dense. 

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

See the table of contents in the publisher's webpage.