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Noncommutative Geometry, Quantum Fields and Motives

Alain Connes and Matilde Marcolli
American Mathematical Society
Publication Date: 
Number of Pages: 
Colloquium Publications 55
[Reviewed by
Michael Berg
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The scope of Non-commutative Geometry, Quantum Fields and Motives is titanic, as the title already indicates. Non-commutative geometry is of course closely associated with Alain Connes, and this has been the case for many years now, certainly ever since his deep and definitive work hit the presses over a quarter of a century ago, at that time focused primarily on the (vast) area where algebraic and differential topology, algebraic geometry, von Neumann and C* algebras, and K-theory meet. Over the intervening years non-commutative geometry has taken shape more and more and rumors surfaced from time to time that Connes’ focus was ultimately nothing less than the Riemann Hypothesis. Well, there’s a lot more to it than rumor, but it’s only fair to say that there’s even more to it than that.

It’s really all about a philosophy of how deep themes in mathematics, specifically number theory and theoretical physics (particularly quantum field theory) are interrelated, and how different themes along these lines influence one another. For instance, there’s Chapter 1, weighing in at well over 300 pages: a great deal of quantum electrodynamics-spawned material featuring Feynman’s formalism of “path integrals” and diagrams as well as the notorious business of renormalization, non-commutative spaces, and Grothendieck’s motives in a non-commutative setting.

Next, there’s truly spellbinding material on Riemann’s zeta function and non-commutative geometry (Chapter 2, at only about 100 pages), followed by quantum statistical mechanics connected to a good deal of arithmetic and algebraic geometry (everything from class field theory and Kronecker-Weber to elliptic curves and Shimura varieties) and Galois theoretic themes in the broader sense (Chapter 3: almost 150 pages). Finally we get almost 200 pages (Chapter 4) on “endomotives, thermodynamics, and the Weil explicit formula.”

It is revealing to quote from Connes-Marcolli’s § 5.7 (p. 671) to get an idea of what forces are conspiring to reach a common goal:

The fact that we do not work here with Hilbert spaces means … that we do not have the restriction of unitarity … [and] will be able to obtain a trace formula [realizing Weil’s explicit formula] which is not only [i.e. merely] semilocal … The RH [Yes! It’s the Riemann Hypothesis] will then be equivalent to a positivity statement … [W]e formulate the trace formula in a cohomological version, which is closer to what happens in the classical setting with the action of Frobenius on étale cohomology.

And one discerns shades of RH for function fields, of course, and to be sure, on p. 674 ff. we read, as a prelude to §6, “We now proceed to compare the setting we have developed in terms of the non-commutative geometry of adèle class space with the classical algebra-geometric setting of the Weil proof of RH for function fields.”

So the cat is out of the bag, really: what greater mathematical objective can there be than to realize RH for number fields and RH for function fields as two sides of the same coin, the discriminators being, as it were, algebraic geometry and (or versus) non-commutative geometry? This is manifestly one aspect of the rationale for what the entire sweeping (and evolving) program is about, with a complementary aspect being quantum physics in its post-Feynman form. What a wild, wild ride!

So, what is required of the reader, then? Well, a lot. It would be folly to try to read this book without a preliminary knowledge of QED + QFT (well, I guess QFT will do), done for mathematicians. Here there is the (likewise titanic, but actually very readable) two-volume source, Quantum Fields and Strings: A Course for Mathematicians, now available in paperback. There are other sources, of course, but it’s useful to stick with things written for us mathematicians. The critical thing is to learn the yoga of Feynman diagrams, and to get a good idea of what renormalization is all about. The latter topic is certainly covered at length in the aforementioned Quantum Fields and Strings.

Next, you had better be comfortable with the French approach to algebraic geometry (Leray à H. Cartan à Serre à Grothendieck), as opposed to, say, the Zariski approach. Then there’s Chevalley and Weil, of course: adèles and idèles and (at least) Tate’s thesis. Furthermore, given the focus of RH for function fields, it would be a good idea if some familiarity were present there, too. Beyond this, well, it’s important to have a good “graduate school and beyond” grounding in commutative algebra, homological algebra, functional analysis, and so on. And, oh yes, be sure to know a load of number theory, e.g. elliptic curves, modular forms — well, you get the idea. And it would also be good if you know a bit about Grothendieck’s motifs.

So there it is. There are no two ways about it: this is a very dense book, and has to be worked through very slowly, with lots of margin work, and outside reading. But to go that route is both virtuous and psychologically instructive for a scholar, and the material at hand is obviously of huge importance, depth, and elegance.

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

  • Quantum fields, noncommutative spaces, and motives
  • The Riemann zeta function and noncommutative geometry
  • Quantum statistical mechanics and Galois symmetries
  • Endomotives, thermodynamics, and the Weil explicit formula
  • Appendix
  • Bibliography
  • Index