The book under review charts a path from classical field theory to the sexiest thing around, at least for the physicists, quantum field theory. In fact, it’s even more than that, namely, perturbative quantum field theory, the spectacularly successful yoga of getting numbers from, e.g., Feynman path integrals and Feynman diagrams by massaging certain series expansions in just the right way.

But physicists don’t work the way mathematicians do, and when mathematicians encounter physics, it’s really a clash of cultures, and it is exceedingly difficult to bridge the gap, from either direction. As a case in point, consider the most fundamental theme of modern physics, quantization. How is it defined? Perusing the papers and books by the pioneers (dating to the late 1920s, in many cases) doesn’t really repay the effort: an explicit definition, *klipp und klar*, is not really available. To be sure, the great papers by Heisenberg, Bohr, Pauli, Schrödinger, and of course Dirac, do not hide what they’re up to: change the classical notions of position and momentum to operators on a Hilbert space of states subject to either algebraic rules or to a dictionary that sets up an ersatz (non-classical) Hamiltonian fitted into a differential equation, the vitally important Schrödinger wave equation. All right, then --- first of all, what is a “state”? Well, it is a ray in a Hilbert space attached to a quantum mechanical system. How is it to be interpreted? Here we get non-classical statistics and the Born rules. This is certainly where the rubber hits the road, seeing that we get to the spectral analysis of this quantum Hamiltonian, obtain eigenstates as stationary states for the system, and accordingly acquire the wherewithal to compute probabilities regarding particles occupying certain states. Magically, this process produces numbers that jive with what one can observe in a laboratory --- well, sort of: the observer disturbs the observed, and somewhere Bohr and Heisenberg are smiling. There’s an awful lot hiding in these shadows, but all this still begs the question of defining quantization as such, i.e. as more than a procedure.

Fine, then let’s bring in a mathematician who was in on what the physicists were up to in those *Knabenphysik* days, Hermann Weyl. On p. 47 of his acknowledged 1928 classic, *The Theory of Groups and Quantum Mechanics*, we read that “the linear oscillator … and the fundamental notions of the theory of oscillations suggest the following … guiding principle (P): the frequencies derived from the energy levels by means of Bohr’s frequency rule shall correspond to the frequencies of the simple vibrations into which the actual motion of the atomic constituents can be resolved in accordance with the laws of dynamics. Such a resolution … is attainable … only if the system is ‘multiply’ or ‘conditionally periodic,’ and for this case it was actually found possible to sharpen the general principle (P) into a definite rule for quantization.” So it certainly again looks like it’s all about special frequencies, i.e. eigen-vibrations. But this characterization still is more of a description of a procedure than a self-contained definition.

All right, maybe that’s where things stand and need to stay. Here is Varadarajan on the subject (cf. p. 315 of his book, __Geometry of Quantum Theory__): “Roughly speaking, Quantization refers to a process by which one establishes a ‘correspondence’ between certain classical and quantum systems” and then he goes on to note that “[t]he mathematical requirements of this correspondence are, however, capable of being formulated in many ways and so there are several approaches to quantization …”

And this brings us to Feynman who famously quipped that “if you think you understand quantum mechanics, you don’t understand quantum mechanics,” and, perhaps in that spirit, he wrote his

PhD thesis, , on an alternative formulation of quantum mechanics than what we sketched above. Feynman worked from a Lagrangian perspective with the Principle of Least Action taking the point and before long the redoubtable path integrals and Feynman diagrams entered the game, which was now played in a new way.

So, if we switch to a Lagrangian perspective, can we characterize quantization more compactly? In a sense, the answer is yes, although Feynman doesn’t really get that explicit either in his thesis. More explicitly, going this Lagrangian route, the main step is to stipulate an action functional as the principal building block in a formalism of Feynman integration adequate to the task of producing probabilities, i.e. expectation values for measurements of observables, associated bijectively to a probability distribution. Observables are presented as the real elements in a complex associative algebra that can in turn be realized as an algebra of linear operators on some complex Hilbert space, and then the states are positive trace class operators with trace 1. The probability distributions are obtained as traces of density matrices defined relative to special projector functions of the given operator, and the desired numbers appear by Feynman integration, analysis of Feynman diagrams, and massaging perturbative expansions.

In the wake of Feynman’s innovations, quantum electrodynamics received its definitive treatment by Feynman, sharing the 1965 Nobel Prize with Julian Schwinger and Scin’ichiro Tomonaga, whose approaches to this subject were quite different. It was Freeman Dyson who established that these approaches were equivalent. Quantum electrodynamics is an example of a quantum field theory (as is quantum mechanics), and its progenitor was none other than Dirac.

With the Feynman formalism on the scene and the cat out of the bag, meaning that we are now talking about fields, we finally arrive at the book under review. Here is what the author, Michael Dütsch, has to say about quantization: “Quantization is done by deformation of classical field algebras” (p. xviii). All right, what does this mean? In his Chapter 1, “Classical Field Theory,” Dütsch presents us with a configuration space consisting of smooth paths in a d-1 dimensional real space (as the space-like part of a d-dimensional Minkowski space, the additional dimension given by time: particles travel along these paths as time passes); then the fundamental observable is “position at time t,” an evaluation functional on this configuration space. This is really the set-up for Newtonian physics, and we may conflate observables with fields to get the notion of a classical field. Says Dütsch: “… the space of fields F … will be used throughout the whole book --- also in Q[uantum] F[ield] T[heory]. In particular in view of deformation quantization, it is a main advantage of our approach that the fields of both the classical and the quantum theory are defined in terms of the *same* space of functionals. *Classical and quantum fields differ only by the algebraic structures that we will introduce on this space: Main building blocks are the Poisson bracket for the classical algebra and the (non-commutative) star product for the quantum algebra* [italics: mine].”

Thus, we have at least a vague idea of what we are dealing with in Dütsch’s book, and in particular a sense of what he means by quantization and how he looks at quantum field theory. That said, this is a very impressive piece of work. The big (500+ pages) book starts off with coverage of classical field theory, deformation quantization, and perturbative quantum field theory --- the latter covered to the tune of over 200 pages. Very happily, Dütsch adopts an axiomatic approach to this subject; cf. his §3.1 on the so-called retarded product, central to the developing interacting fields as formal power series. In this third chapter we encounter such standbys as time-ordered products and renormalization, which can be very roughly described as a yoga to sweep infinities and divergences away. The renormalization schemes dealt with here come in several flavors, e.g. that of Stückelberg-Petermann and that of Wilson, fitted into the general framework of Epstein-Glaser renormalization. The chapter closes with a “Comparison with the functional integral approach” [to renormalization] which is manifestly of particular interest given the ubiquity of the latter approach.

The final two chapters deal with, respectively, symmetries and quantum electrodynamics which got the ball rolling in the first place. Regarding the matter of symmetries, the focus falls on the so-called Master Ward Identity, “a universal formulation of symmetries --- it is the straightforward generalization to QFT of the most general classical identity for local fields [in the sense of the physicists, not the number theorists] which can be obtained from the off-shell field equation and the fact that classical fields can be multiplied pointwise.” This is obviously both highly non-trivial and deep: one can in fact say the same about this entire book, in the most positive sense of this characterization.

This book is a wonderful piece of scholarship, aimed at an audience of serious and correspondingly well-prepared mathematical physicists, or even mathematicians interested in this marvelous material and willing to play the long game. To paraphrase none other than Feynman himself, this book is chock-full of “most of the good” stuff in this area.

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