Mathematics of Quantization and Quantum Fields

Quantization is without doubt one of the greatest modern ideas in science, coming first into play in physics as a hypothesis Planck reluctantly introduced into his analysis of black-body radiation, thereby launching the quantum theory revolution. The effects on and reactions from mathematics were equally profound, although perhaps not as visibly dramatic. As Constance Reid put it in her biography Hilbert (cf. p.182),

The Courant-Hilbert book [Methoden der mathematischen Physik] … which had appeared … before both Heisenberg’s and Schrödinger’s work … seemed to have been written expressly for the physicists … “Indirectly Hilbert exerted the strongest influence on quantum mechanics in Göttingen,” Heisenberg was later to write.

It was none other than Hilbert’s dazzling assistant John von Neumann who, in his seminal work, Mathematical Foundations of Quantum Mechanics, presented what he had wrought, namely, a definitive formulation of quantum mechanics in terms of unitary representation theory and functional analysis. Much of modern functional analysis is beholden to von Neumann in this manner. Similarly, an earlier assistant to Hilbert, the redoubtable Hermann Weyl, was responsible for a version of quantization that is in many ways the most adaptable to sundry mathematical purposes (including, e.g., various facets of symplectic geometry, and the non-archimedean versions of quantum mechanics proposed in the last few decades).

In the context of quantum mechanics, the attendant commutation rules came in three flavors, historically speaking: in the respective contexts of Heisenberg’s matrix mechanics, Schrödinger’s wave mechanics, and then Weyl’s approach via unitary representation theory proper. The subtext of all this is of course Lie theory, at least in the broadest sense. Somewhat more precisely, the equivalence of what Dirac called the Heisenberg and Schrödinger pictures is expressed by the fact that in one case time is held fixed and space varies, so to speak, while in the other it’s the other way around.. Going over to Weyl’s version of quantum mechanics then involves exponentiation of operators in the Lie theoretic sense. In this menagerie of exciting mathematical maneuvers one begins to discern connections to mathematical structures with a certain individual autonomy, e.g., symplectic spaces, Clifford algebras, von Neumann algebras, and so on.

All of this is found in the book under review, but there’s a lot more to be had there, as the very title suggests: it’s also about quantum fields, which suggests some hypermodern connections at least inasmuch as quantum field theory is very, very hot these days — even for mathematicians. But the focus is on free fields which are, so to speak, closer down to earth, and the mathematics takes center stage more naturally.

Mathematics of Quantization and Quantum Fields is a very useful book for mathematicians, if only because of the absence of the physicists’ black magic of inferring mathematical things from experiments and then “proving” their mathematical assertions and prognostications by means of more experimental evidence — and then there is the business of thought experiments, of course. (This just underscores the intrepidity of men like Hilbert, Weyl, and von Neumann.)

Happily, even less intrepid mathematicians need fear no such specters in the pages of the book under review. Despite the authors’ solid physical credentials, they take great care to do the mathematics as mathematics (with theorems and proofs everywhere!), and we encounter sophisticated material in a most palatable and accessible format, modulo sufficient mathematical preparation. But this isn’t too bad at all: the book starts out with linear algebra, Hilbert space methods, measure theory, and material on algebras, all before even getting to commutation relations. But then it’s a quick ascent: canonical commutation relations, representation theoretic aspects (and then some) of symplectic (and metaplectic!) groups, Fock spaces (which are constructs of Hilber spaces — check your local Wikipedia stub (or entry, I guess)), canonical anticommutation relations, bosonic and fermionic states, and then a large chunk of material on quantum field theory (with chapter 20, at p. 555, hitting “diagrammatics,” including of course Feynman’s).

I think this is a very good book, doing justice to some wonderful mathematics in an context that is physically exceedingly meaningful. The topics covered are woven together very cogently and elegantly, as it should be when it comes to this kind of mathematical physics about which Dirac famously insisted that it should be beautiful “because [the mathematics] was chosen by God,” and, as I already noted, it is very mathematician-friendly. I am happy to recommend the book very strongly.

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