In the Fall of 2008 Matilde Marcolli taught a class at Caltech, titled “Geometry and Quantum Fields,” in which she brought the burgeoning methods of Alain Connes’ non-commutative geometry to bear on certain things Feynman: “Though it inevitably feels somewhat strange to be teaching Feynman diagrams at Caltech, I hope that having made the main focus of the lectures the as yet largely unexplored relationship between quantum field theory and Grothendieck’s theory of motives in algebraic geometry may provide a sufficiently different viewpoint of the quantum field theoretic notions to make the resulting combination of topics appealing to mathematicians and physicists alike.”
And how does non-commutative geometry fit into this scheme (if you’ll pardon the pun)? Well, Marcolli goes on to say that she is using a “top-down approach” to the interplay between Feynman graphs and motives, “based on the formal properties that the category of mixed Tate motives satisfies, which are sufficiently rigid to identify it… with a category of representations of an affine group scheme. One then approaches the question of the relation to Feynman integrals by showing that the data of Feynman integrals for all graphs and arbitrary scalar field theories also fit together to form a category with the same properties. This… approach was the focus of my joint work with Connes on renormalization and the Riemann-Hilbert correspondence…” Marcolli refers to their recent book, Non-commutative Geometry, Quantum Fields, and Motives, which appeared in 2008 as volume 55 in the AMS Colloquium Publications series.
To be sure, the top-down approach, as sketched above, would benefit from a additional architectural bolstering, and, no surprise to any one, that comes from a complementary (dare I say, “dual”?) approach, the bottom-up approach: “[One] looks at individual Feynman integrals for given Feynman graphs and, using the parametric representation in terms of Schwinger and Feynman parameters, identifies… the Feynman integral (modulo … divergences [Oh, those physicists!]) with an integral of an algebraic differential form on a cycle in an algebraic variety.” This, by itself, is extremely evocative: it’s a rapid transit from QFT, looked at like a mathematician looks at it (after all, physics is just “a special case,” no?), to algebraic geometry. And this wins us a huge advantage, courtesy of Grothendieck’s legacy: “[O]ne then tries to understand the motivic nature of the piece of the relative cohomology of the algebraic variety involved in the computation of the period, trying to identify conditions under which it will be a realization of … a mixed Tate motive.”
Coincidentally, and by way of a breather from all this hard-core avant garde algebraic geometry dancing a tango with QFT, here’s a story. Some years ago I had occasion to attend a lecture by Serre, appropriately at Caltech, at which he wrote on the black-board, almost as a closing afterthought following upon an audience question that took him in the direction of the treacherous theory of motives (much of which is still speculative), “All my motives are pure.” Well, what can one say to that? The modern French school of algebraic geometry has given us an argot meant for punning. Just think of the opportunities afforded by such gems as “perverse sheaves” and “pointless spaces.” And that’s only in English!
All right, back to Marcolli’s book. It’s clear as vodka that the material in its 220 pages is state of the art, and then some: Connes and the author only launched it a couple of years ago, and, heavens to Murgatroyd, they’re playing with motives as though there’s no tomorrow. In his wonderful review (Bull. AMS, Vol. 39, 2002, No. 1), of V. Voevodsky’s Cycles, Transfers, and Motivic Homology Theories (Annals of Maths. Studies, No. 143, 2000), Chalres Weibel states flatly that “an actual category of mixed motives is not yet fully known to exist, even with coefficients in Q.” Zowie! What’s all this then? What are Marcolli and Connes doing? Well, it’s not as bad as all that: “It turns out that in order to construct the motivic cohomology of a variety we do not need to know whether or not (mixed) motives exist. [Can you hear constructivist logicians gasping all across the universe?] Using homological algebra, all we need is a candidate for the derived category of motives.” Weibel then concludes that “[Voevodsky’s] book ... constructs a candidate for the derived category of mixed motives, and hence motivic cohomology…”
But it’s not necessary to delve into motivic cohomology separately, even given the appeal of Voevodsky’s book. Says Marcolli: “I am not an expert in the theory of motives and this fact is clearly reflected in the way this text is organized,” and so it is that of her nine chapters in Feynman Motives only the twenty-five pages of the second chapter concern themselves with the requisite algebraic geometry per se: it is all quite user-friendly. It is necessary to note, however, that the reader should be disposed to take a number of things on faith and agree to treat certain things as “black boxes” (to steal a phrase from Nicholas Katz’s “Travaux de Laumon” (Sém. Bourbaki, 1987-1988, No. 691). For example, given that Marcolli’s section 2.3, “Mixed motives and triangulated categories,” spans just pp. 34–36, after which she races on to motivic sheaves, it’s clear that no proofs can be expected as regards some rather intricate stuff. Happily she provides good references: the business of triangulated categories is capped off by a nod in the direction of the original source for this material, Beilinson–Bernstein–Deligne’s famous Astérisque 100 (1982), Faisceaux Pervers (= perverse sheaves), and my absolute favorite book in this genre, Sheaves on Manifolds, by M. Kashiwara and P. Schapira (in Springer Grundlehren series, no. 292, 1990).
The point is that, without doubt, Marcolli’s book is more in the way of a high level survey and exposé of a subject that is not only brand new, but is growing very fast. The reader is accordingly in a position to approach the material in different ways, as a tour of the landscape, or as a flow-chart of sorts for further exploration. This is of course altogether proper for a graduate course of the indicated type, what with Feynman Motives evolving from Marcolli’s lecture notes.
So it is also the case that when proofs can reasonably be given Marcolli does so, and does so in a clear and concise style. And the latter phrase indeed characterizes the entire book. Furthermore, the book’s aforementioned genesis as lecture notes for a graduate course also means that, modulo the obvious limitations already described, the mathematics itself is appropriately serious. To plagiarize Feynman, it’s full of “the good stuff,” for us mathematicians, even though the chapter headings tend to mislead one into thinking he’s on the wrong floor of Sloan (Caltech perversely houses mathematics in the Sloan Laboratory of Mathematics and Physics)
And, speaking of Caltech physics, let’s take another breath and recall parenthetically that Feynman graphs, or diagrams, are popularly the best known aspect of Feynman’s way of doing QFT — they provide a big bag of magic tricks used to compute Feynman integrals, whose integrands are Gaussian kernels multiplied by polynomials and which are taken over sets of paths on a chunk of a space-time manifold (Aargh! — There are (notorious) measure theoretic gyrations required here…). It’s something like graph theory and combinatorics meet integration by parts on anabolic steroids.
So it’s only fitting that the back cover of the book under review gives us the propaganda that “[t]he text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry… [I]t can also be used by graduate students interested in working in this area.” I agree: Feynman Motives is well-written, and presents a vast amount material in a very attractive form — to me, irresistible; in fact:, if I may end on a personal note, my own work in number theory has various points of contact with themes Marcolli discusses in her book and I intend to use it heavily in conjunction with the texts mentioned earlier.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.