Quantum Field Theory, of QFT for short, is one of those marvels of modern physics. It provides mathematical techniques of such power and potential that it becomes necessary for mathematicians, even pure ones, to cross the Rubicon and learn some physics. The metaphor is flawed on several counts, of course, since we mathematicians certainly do not intend our invasion of physics as a bellicose act (unlike Julius Caesar’s) and our act is not irreversible. But it is unquestionably true that even number theorists (or maybe especially number theorists) are learning physics in larger and larger numbers and often, like me, later in life.
A major problem is, however, that all of our goodwill and zeal notwithstanding, studying physics as done by physicists is ultimately a deeply unnatural act: they don’t really prove things the way we do and they have different ideas about what definitions mean. (Can anyone really define what an elementary particle is? Well, I guess that’s not a fair one … Or is it? See below.) Reading physics is not at all like reading mathematics, and studying it, even — or especially — for mathematical purposes, is a difficult proposition.
Accordingly, the market has begun to support greater and greater numbers of books dealing with the presentation of some very good stuff (the phrase is borrowed from none other than Richard Feynman, of course) in mathematical translation. The book under review, Mathematical Aspects of Quantum Field Theory, by de Faria and de Melo (of São Paulo and Rio de Janeiro, respectively), is a player in this game. And a promising player it is, coming as it does with an imprimatur by Dennis Sullivan including the following appraisal:
The method of the text is to explain the meaning of a large number of ideas in theoretical physics via the splendid medium of mathematical communication … [T]here are descriptions of objects in terms of the precise definitions of mathematics. There are clearly defined statements about these objects expressed as mathematical theorems. Finally there are clearly defined statements about these objects, expressed as mathematical theorems.
Sullivan closes his Foreword with a caveat which delineates the role this book is projected to play:
After closing the book, one has not arrived at the kind of understanding of physics [physicists have] … but then, maybe, armed with the information so elegantly provided by the authors, the process of infusion, assimilation, and deeper insight based on further rumination and study can begin.
And what do de Faria and de Melo have to say along these lines? Here it is:
Despite the great importance of … physical ideas … mostly coming from quantum theory, they remain utterly unfamiliar to most mathematicians. This we find quite sad. As mathematicians, while researching for this book, we found it quite difficult to absorb physical ideas, not only because of … lack of rigor — this is rarely a priority for physicists — but primarily because of the absence of clear definitions and statements of the concepts involved. This book aims at patching some of these gaps in communication.
What do they address? Well, after a fist chapter dealing with classical mechanics (and closing with a treatment of Emmy Nöther’s theorem on symmetry and conservation laws, worked out by her for David Hilbert — apparently by request!), we get to quantum mechanics, culminating with Feynman’s integral formalism; after that it’s time for (special) relativity and Dirac’s equation.
After this, with the fourth chapter, we get a respite from physics: it’s about geometry now: fibre bundles, connections, some Hodge theory, Clifford algebras, spinor bundles, and, finally, representations.
Back to physics: field theory is next, with classical field theory setting the stage, in chapter five, for quantization in chapter six, perturbation methods in chapter seven, and renormalization in chapter eight.
Then Mathematical Aspects of Quantum Field Theory closes (modulo appendices on Hilbert space, and C* algebras and spectral theory) with a discussion of the standard model. What a wonderful trajectory.
Indeed, the list of topics covers reads like a road map into QFT as well as QED (quantum electrodynamics), at least to some extent (gratia Feynman). It should serve its intended role as a springboard for mathematicians into these physical theories beautifully: the exposition is marvelous and the mathematics is solid and elegant. A fine example is found in de Faria and de Melo’s treatment of path integral quantization (p. 40 ff.), which includes a truly wonderful discussion of Trotter’s formula as well as a properly detailed exposition of “the path integral in Hamiltonian form” in the formalism of Dirac.
I was particularly struck by the authors’ treatment (in their fourth chapter, on representations) of Eugene Wigner’s classification of particles, starting off with the following gem: “Definition 4.25. A quantum mechanical particle is a projective, irreducible unitary representation of the Poincaré group.” Wow! And they go on to note that
Wigner’s classification theorem provides the (correct) mathematical framework for the study of elementary particles, and has stimulated a great deal of research in the theory of group representations. The classification amounts to finding all unitary representations of the [universal double covering group of the Poincaré group].
Turn the page and get the Peter-Weyl theorem … Again, good stuff.
There are exercises at the end of each chapter; they look to be perfectly pitched. Here is a particularly nice one: “ … show that Schrödinger’s equation (Chapter 2) can also be derived from a variational principle .” This is the first exercise for chapter five; it comes equipped with a number of hints, and leads to something rather deep.
Clearly, then, Mathematical Aspects of Quantum Field Theory is obviously a hugely valuable contribution to the literature, bound to be of immense use to any of us who want to cross the aforementioned Rubicon: it makes the makes the crossing smooth.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.