This is an outstandingly good book. Although aimed at physicists, it is a godsend to philosophers of physics who want to understand quantum field theory. Every such philosopher of physics should have it on their bookshelf. To know all that is here would be wisdom indeed.

There are, of course, many excellent textbooks of quantum field theory, but none of them, so far as I know, engage with the conceptual issues in the way that and to the extent that Duncan does. Besides, I believe that Duncan’s ideas about what are the conceptual questions, as well as what are the correct answers to them, will ring true with the philosophical reader, just as they no doubt will with the physics student. Agreed: physics is controversial, and there are undoubtedly several different ways to think about the deeper or subtler aspects of quantum field theory. So it would be more precise to say that Duncan’s ideas about the questions and their correct answers are undoubtedly very reasonable, and his elaboration of them is beautifully detailed. Thus, the philosopher setting out to understand quantum field theory could surely have no better guide than his discussions.

In his preface, Duncan explains his book’s overall goal. After opening the preface with praise for some of the classic books, such as Schweber’s An Introduction to Relativistic Quantum Field Theory (New York: Harper & Row, 1961) and Weinberg’s three-volume The Quantum Theory of Fields (Cambridge: Cambridge University Press, 1995–98), Duncan says that in teaching quantum field theory over 25 years, he has been struck by the number of “important conceptual issues raised by questions in the classroom which require a careful explanation not to be found in any of the readily available textbooks” (iii). He gives a list of such issues. I shall pick out two that he discusses in the first half of the book, since I will advertise the book by reporting his discussions:

• 1. “Why do so few quantum fields (basically only electromagnetism and gravity) have classical macroscopic correlates?” (iii).

• 2. “If the interaction-picture does not exist, as implied by Haag’s theorem, why (or in what sense) are the formulas derived in this picture for the S-matrix still valid?” (iv).

Duncan then remarks that “none of these questions require an answer if one’s attitude in learning quantum field theory amounts to a purely pragmatic desire to ‘start with a Lagrangian and compute a process to two loops’. However, if the aim is to arrive at a truly deep and satisfying comprehension of the most powerful, beautiful and effective theoretical edifice ever constructed in the physical sciences, the pedagogical approach taken by the instructor has to be quite a bit different from that adopted in the ‘classics’ enumerated above” (iv). (I indulge in the quotation so as to give a sense of Duncan’s prose: detailed, learned, but never ponderous; and conveying a passion that is all the more attractive for being understated.)

Before giving more details of Duncan’s book, it will help to relate it to the two classics by Schweber and Weinberg. Philosophers of physics will find it a joy and an education to take as an accompaniment to Duncan’s discussions the corresponding chapters of these books. For some topics, there is concordance despite the intervening years: for example, about the Wightman axioms and the Lehmann–Symanzik–Zimmermann (LSZ) formalism (chap. 9 of Duncan, and chap. 18 of Schweber). For other topics, there is complementarity (in the clear everyday sense, not Bohr’s!). An example is the relation between the various relativistic equations describing particles (Klein–Gordon, Dirac, etc.) and the representation theory of the Poincare group (chaps. 5–7 of Duncan, chaps. 2–5 of Schweber, and chaps. 2 and 4 of Weinberg). In particular, Duncan stresses (following Weinberg’s lead) the clustering principle (also known as the cluster decomposition principle), which requires, roughly speaking, that the amplitude for a process composed of two spatially far separated subprocesses should be the product of the subprocesses’ amplitudes. For yet other topics, there is more of a contrast. The most striking example is renormalization. For Schweber’s exposition of this (his chaps. 16 and 17), though masterly, is separated from both Duncan’s exposition (chaps. 16–19) and Weinberg’s (chaps. 12 and 18) by the ‘Wilsonian revolution’, that is, by the major developments in understanding renormalization that were launched by the work of Wilson, Fisher, Kadanoff, Kogut, and others in the years 1965–1975.

Duncan divides the book into four sections: “Origins,” “Dynamics,” “Symmetries,” and “Scales.” Though the overall avowed aim of the book is a modern and systematic exposition of quantum field theory, the first section, “Origins,” is historical. I will concentrate on reporting the longest section, “Dynamics,” since it constitutes about half of the book and includes Duncan’s answers to the two questions that I have listed. But before that, I will give some details about the “Origins” section, since it also is of great conceptual interest; it builds on articles coauthored with the historian and philosopher of physics Michel Janssen. And after my sketch of the “Dynamics” section, I will say a little about the “Scales” section, which covers the topic of renormalization. (I regretfully set aside the “Symmetries” section.)

The “Origins” section comprises two chapters. The first is on the origins of quantum field theory: first implicitly, in Planck’s and Einstein’s work on blackbody radiation in 1900; and second explicitly, in Jordan’s 1925 derivation of a formula that describes the energy fluctuations in electromagnetic radiation in a manner that combines both the wave and particle aspects of the radiation. The second chapter continues the story, starting with Dirac’s 1927 paper and ending with the vindication of quantum electrodynamics, in the late 1940s, by the techniques of renormalization, in the hands of Dyson, Feynman, Schwinger, and Tomonaga.

Thereafter, the systematic exposition of quantum field theory follows, in the successive sections: “Dynamics” (eight chapters), “Symmetries” (four chapters), and “Scales” (four chapters). Here follows a lightning sketch of “Dynamics” (chaps. 4–11), a sketch that is tailored to answering our questions 1 and 2 above.

The first three chapters develop the ideas that (a) although the principles of quantum mechanics and special relativity allow a plethora of theories, most of these imply unphysical behavior at long distances; and (b) this behavior can be avoided by imposing the clustering principle. Then, in chapters 7–11, Duncan gives a detailed treatment of free and interacting fields. Chapter 7 describes how the familiar equations (Klein–Gordon, Dirac, etc.) are best seen as consequences of the representation theory of the Poincare group. Chapter 8 discusses the classical limit of quantum fields. Its answer to question 1 above is in section 8.2, especially page 228. In short, classical behavior requires a state with (a) every mode having a high occupation number (and so a bosonic field) and (b) stable particles, so that a classical configuration can be long-lived.

Chapters 9–11 expound interacting quantum fields. Chapter 9 deals with general aspects, especially the connection between particles and fields. This includes an exposition of the Wightman axioms and two associated important formalisms for treating asymptotic states: the scattering theory of Haag and Ruelle, and the LSZ formalism. Duncan stresses (253 and 308) that these formalisms stand up, independently of both perturbation theory and the use of the interaction-picture, which, as mentioned in question 2 above, unfortunately does not exist—as spelled out by Haag’s theorem. Of course, to calculate most of the physically important quantities treated by these formalisms, for example, vacuum expectation values of time-ordered products of field operators, one needs approximation schemes: either perturbative (and interaction-picture) schemes, which are treated in chapter 10, or nonperturbative schemes, which are treated in chapter 11.

I shall concentrate on chapter 10. For after expounding Feynman graphs and rules—first using operator methods, and then with path integrals—this chapter closes with Duncan’s answer to question 2 above (sec. 10.5, with the enticing title “How to Stop Worrying about Haag’s Theorem,” especially 361, 369–70).

The answer is that the interaction-picture formalism can be justified in two steps, as follows:

• 1. First, one makes it mathematically rigorous by regularizing it with a finite lattice, with the lattice spacing introducing an ultraviolet cutoff, and the finiteness of the lattice (‘box’) introducing an infrared cutoff. So the system now has only a finite number of degrees of freedom, that is, it is quantum mechanical, not quantum field theoretic! Thus, perturbative contributions to quantities of interest, such as expectation values of products of operators, are well defined at each order of perturbation theory.

• 2. Second, one argues that one recovers sensible, in particular Poincare-invariant, results for these contributions in the limit where the cutoffs are removed. For the infrared cutoff, this chapter’s technical discussion suffices for the argument, while for the ultraviolet cutoff, the argument needs to appeal to the theory of renormalization, which is anyway needed to make sense of interacting field theories in 3 + 1 space-time dimensions. The details of this are in the book’s last section, “Scales,” but the idea is this: one shows that amplitudes computed with a cutoff are insensitive, order by order in perturbation theory, to the details of the physics at the scale of the lattice spacing (where ‘insensitive’ means that they differ from the continuum amplitudes by only a small amount). Thus, Duncan summarizes, “In the opinion of this author, therefore, the proper response to Haag’s theorem is simply a frank admission that the same regularizations needed to make proper mathematical sense of the dynamics of an interacting field theory at each stage of a perturbative calculation will do double duty in restoring the applicability of the interaction-picture at intermediate stages of the calculation” (370).

I now turn briefly to the last section, “Scales” (chaps. 16–19). Its topic—renormalization and effective field theories—is an important part of the education of any philosopher of physics. For in the past fifteen years, this has become a hot topic in philosophical discussion of reductionism and emergence—that is, in philosophers’ attempts to characterize precisely the evident autonomy, conceptually and methodologically, of the special (i.e., higher-level) sciences. Various philosophical authors such as Cao, Schweber, Batterman, and Morrison have stressed the importance of the Wilsonian revolution in physics. It is also often emphasized by physicists; thus, Cardy ends his recent eulogy for Wilson by saying that his “understanding of the renormalisation group revolutionised our view of the whole world” (“The Legacy of Ken Wilson,” Journal of Statistical Mechanics, October 2013, 4).

Thus, it is a boon to find Duncan in these chapters expounding the details with the sort of emphasis on concepts that we philosophers want and need. To convey the flavor, I shall end this short advertisement for the book by indulging again in quotation from the preambles to chapters 16 and 17. He writes, “A remarkable property of local quantum field theory … is that it is structurally amenable to a precise mathematical description of the way in which the form of dynamical laws changes as the phenomena are examined at varying distance scales. … The representation of the theory at a given distance (or energy) scale will turn out to be specified by an ‘effective Lagrangian’ fixed in terms of an infinite vector of dimensionless couplings, and the variation of these couplings with the distance scale determined by a ‘renormalization group equation’ which [corresponds to] finite changes of scale at which we examine the theory” (569). And similarly later: “The inescapable presence of new physics (quantum gravity, string theory, or what have you) at short distances implies … that we must necessarily imagine a cut off of some kind at high energies: the question of infinities in the (unphysical!) limit where this cutoff is taken to infinity is then replaced by the issue of the sensitivity of the measurable low-energy amplitudes of the theory to the value (and type) of this cutoff” (611).

So to sum up, this book is an invaluable aid to truly understanding quantum field theory. With Duncan’s help, we philosophers need not be defeated by this “most powerful, beautiful and effective theoretical edifice ever constructed in the physical sciences.”