Perturbative Algebraic Quantum Field Theory

In this day and age, quantum field theory (QFT), famously the playground of physicists, is being frequented more and more by mathematicians, and mathematicians of different confessions, so to speak. It’s not just the mathematical physicists and the geometers (gratia Witten, Atiyah, Segal, &c.) who play here now, but even number theorists. It is in order to serve my own number theoretic pursuits, for example, that I turned to this part of physics many moons ago — well, maybe a little over six years back. For me, and I guess for most of the mathematicians who would enter this domain where “Physics” is spoken, this is really an unnatural act: these guys don’t define things the way we do, they have a different idea of what “proof” means, they use funny notations. (I still don’t like the raising and lowering indices business of their tensor analysis — although I guess that’s also part of the Riemannian geometers’ bag of tricks — and the Einstein summation convention. “Sigmas? Sigmas? We don’t need no stinking sigmas…” is still painful for my mathematical sensibilities.) They really care about little more than getting numbers out, hopefully to match experimental data.

Of course, the success of QED (quantum electrodynamics) in this latter regard is truly stunning, so the physicists are holding a very good hand. But is that why mathematicians might want to do this stuff? No, unless of course they want to be called physicists, and that is fine: one of my favorite modern day heroes, Freeman Dyson, went from the purest of mathematics (learned from G. H. Hardy) to such heights in theoretical physics that he really should have gotten part of the Nobel Prize that went to Feynman, Schwinger, and Tomonaga. But Dyson actually set out to be re-schooled as a physicist by apprenticing with Hans Bethe right after the Second World War. This is not the way of most mathematicians who enter the aforementioned playground: semper fidelis to mathematics. But this all means that we need to get at QFT in our own way, in our own language, formulated replete with real definitions, rigorous proofs, our own familiar notation (hopefully), and our own particular mathematical goals. There is, for example, Atiyah’s formulation of topological quantum field theory as a method in differential geometry (smooth manifolds and cobordisms, &c.) that has algebraic topological overtones (to an extent a TQFT is a functor to Hilbert spaces) and illuminates aspects of low-dimensional topology, specifically knot theory (cf. his beautiful book, The Geometry and Physics of Knots). To further these laudable if parochial objectives, there are a number of books in existence now that go at QFT in “Mathematicalese,” and Rejzner’s text, now under review, is a case in point.

I must say right off that I am sympathetic ab initio to what Rejzner is doing. She fits her discussion into what I think is the most satisfying and elegant framework for quantum theory, namely, functional analysis: this is part and parcel of her (critical) second chapter. To boot, in § 2.4.2 she provides nothing less that axioms for a perturbative Algebraic QFT (or, briefly, a pAQFT), following Haag and Kastler (cf. § 2.3, ff.). This is really excellent stuff: we get a full treatment of QFT in a form that is entirely free of the Unheimlichkeit that physics can bring about, starting off with functional analysis in Hilbert space — canonical quantization involves replacing classical observables by unbounded operators (densely defined) on a Hilbert space of quantum mechanical states —, followed by discussions of the Schrödinger picture, the Dirac-von Neumann axioms, and so forth.

That said, Rejzner doesn’t disappoint in what follows either: she hits much of what that QFT maven par excellence, Richard Feynman, would refer to as “the good stuff,” including, besides physics mainstays, a number of mathematical samplings of great interest, taken from differential geometry (including Kähler stuff), algebraic topology (dropping the names of Koszul-Tate and Chevalley-Eilenberg, all in the service of gauge theory), and of course (pervasively) functional analysis. I am particularly happy with her focus on Feynman graphs (cf. § 6.5, which includes a section titled, “Explicit construction and Feynman graphs,” after discussion time ordered products) and of course on gauge theory.

Thus, from my point of view as someone who is interested in QFT for purely mathematical reasons, and is accordingly keen on getting at it on truly mathematical terms, Rejzner’s book is irresistible, and I think it will make a very good impact in the right circles. Rejzner describes herself in the book’s introduction as “both a physicist and a mathematician,” and more power to her, but she shows good mathematical colors: “I will not use the excuse that ‘physicists often do something that is not well defined,’ so as mathematicians we don’t need to bother and just turn around for a while, until it’s over. Instead, I will jump straight into the lion’s den and will try to make sense of perturbative QFT all the way from the initial definition of the model to the interpretation of the results.” And I guess that really says it all.

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