Quantization, Geometry and Noncommutative Structures in Mathematics and Physics

Non-commutative geometry is magical stuff. Alain Connes is the driving force in this area, which has connections to everything from algebraic geometry to quantum physics, and over the years he has written substantially on the subject.

The natural place to start on this austere material is probably Connes’ own writings on the subject, with the caveat of a great deal of serious preparation on the part of the reader. But it is apparently not necessary to prepare for the book currently under review with a painstaking preliminary study of all things Connes: non-commutativity per se (as defined by the book’s index) only occurs on a couple of pages in C. Kassel’s article, “Principal fibre bundles in non-commutative geometry,” and there the learning curve is by no means too steep. Indeed, taking Kassel’s article, Chapter 3 of the book, as a bellwether for this compendium of articles, it’s really all quite accessible modulo (and it’s a serious modulo, I admit) the reader’s interest in the interphase between quantum theory in the modern (very broad) sense (replete with quantum fields, strings, etc.) and geometry. Regarding this interplay, the first thing that likely comes to mind is the work of Graeme Segal, Simon Donaldson, and Michael Atiyah, i.e. low-dimensional topology and Morse theory making contact with Yang-Mills theory and knot theory, and the beautiful results of Edward Witten (e.g. getting at the Jones polynomial by means of quantum field theory). And this is certainly an area in modern mathematics, or even mathematical physics — or even physical mathematics! — which is capable of tempting any one. I, for one, as a dyed-in-the-wool number theorist, have been at it for years now, seeing that low-dimensional topology (knot theory) and TQFT (topological quantum field theory) have come to relate rather dramatically to the analytic number theory I do. But TQFT is very pretty and seductive in its own right, just by itself, and, accordingly, so is this book.

For me, the first things that catch the eye are the two articles, “Quantum Field Theory in Curved Space-Time,” by Reyes Lega, and “Split Chern-Simons Theory in the BV-BFV Formalism,” by Cattaneo, Mnev, and Wernli. The reason for my antennae quivering like this is that Chern-Simons theory is the physics alchemy employed by Witten in the late 1980s to get at the Jones polynomial of knot theory by Langrangian quantum field theory, i.e. the yoga of Feynman integrals: here the action is the Chern-Simons action, and then you’re airborne. And then, doing QFT in curved space-time is, well, how can it get any better than that, if you recall what Dirac said, viz. his remarks in a May 1963 Scientific American article: “A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.”

Speaking of marvelous aphorisms, the editors of this book pepper their Preface with an inversion of an observation on the part of Dirac’s brother-in-law, the physicist Eugene Wigner. In 1960 the latter famously noted the “unreasonable effectiveness” of mathematics in physics — which is in and of itself accounted for, of course, by Dirac’s preceding observation — but the present book’s editors go for the inversion maneuver pulled in 2010 by Atiyah (yes!), Dijkgraaf, and Hitchin, who argued that it’s equally surprising that physics should turn out to be so effective in mathematics. And, given how QFT has come to influence low-dimensional topology, there’s no denying that, is there? In that spirit, the present book’s intent is “for curious readers to get acquainted with one of the … topics, … Quantization, Geometry, and Non-commutative structures, and the relations between them.” I must say that I am a bit puzzled by this limitation: why not get acquainted with all three themes, and then move around between and among them? But I guess one has to favor one over the others: for us mathematicians, can it be other than Geometry? I don’t see how…

So, there is lot of Feynman’s “good stuff” in this book. The Prelude (Chapter 1), by Cardona, Paycha, and Reyes Lega, includes the following passage: “Let us name a few transversal concepts to various lectures that can serve as guiding threads for the reader: Group actions … Hopf algebras … Fibrations … Supersymmetry … Quantization … Non-commutativity and deformation …” They elaborate (e.g.): “Group actions … arise where there are symmetries” (and somewhere Emmy Noether is smiling). “Hopf algebras [are] the dual counterparts to groups.” “Fibrations … arise whenever quantization meets geometry.” We indeed get a very tantalizing prelude to a symphonic presentation of physics dancing with geometry (raising the question, who leads?).

I think this is a very important book, not least because it’s pitched at the level of serious beginners, i.e. folks embarking on research in these areas, keen to get the lay of the land. Indeed the lectures stem from the Summer School Geometric, Algebraic, and Topological Methods for Quantum Field Theory, run biennially in Villa de Leyva in Colombia since 1999. The book’s editors note, too, that “[t]he present volume … will reflect the effort of the authors … to adapt the lectures to the needs of the participants.” Good stuff, indeed.

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