A Modern Introduction to Quantum Field Theory , MicheleMaggiore , Oxford U. Press, New York, 2005. $114.50, $54.50 paper (291 pp.). ISBN 0-19-852073-5, ISBN 0-19-852074-3 paper

Quantum field theory, which marries the principles of quantum mechanics and special relativity, is one of the great intellectual edifices of the latter half of the 20th century. It is the language of modern particle physics. It has also become an essential part of the theoretical tool kit of condensed matter theorists and has found fruitful applications in diverse other fields.

But few undergraduates ever get exposed to quantum field theory, despite its importance to modern theoretical physics. More or less universally, it is offered only as a graduate course. That is a pity. No one expects in an undergraduate-level course to be able to treat such an admittedly difficult subject in the same depth as one would in a graduate course. Yet that limitation has not stopped instructors from teaching undergraduate quantum mechanics alongside the more thorough graduate course. We would consider an undergraduate education in physics incomplete without at least one course in quantum mechanics.

Other difficult subjects, such as general relativity, are routinely offered as undergraduate courses in many institutions. Even string theory now has an undergraduate text, A First Course in String Theory (Cambridge U. Press, 2004), by Barton Zwiebach, based on his MIT course for seniors (see the review by Marcelo Gleiser, Physics Today, September 2005, page 57). But no undergraduate text on quantum field theory had existed—until now. So I welcome Michele Maggiore’s A Modern Introduction to Quantum Field Theory. In 291 pages he introduces the basics of perturbative quantum field theory, the renormalization group, gauge theories, and the standard model.

Graduate texts, such as Michael Peskin and Daniel Schroeder’s An Introduction to Quantum Field Theory (Addison-Wesley, 1995) or Steven Weinberg’s The Quantum Theory of Fields (Cambridge U. Press, 1995–2000), are too advanced for an undergraduate course. Many physicists think that Weinberg’s two-volume opus on quantum field theory is too expansive, even for a standard full-year graduate course. Anthony Zee’s Quantum Field Theory in a Nutshell (Princeton U. Press, 2003) is written at the right level for undergraduates but is not focused enough to serve as a good undergraduate textbook (see the review by Zvi Bern, Physics Today, April 2004, page 88).

Obviously, many topics essential to the working field theorist are omitted in Maggiore’s book. He develops the rudiments of scattering theory, the LSZ reduction formula, and tree-level cross sections and decay rates. Loop amplitudes are discussed qualitatively, but none of the technical machinery—for instance, the Feynman-parameter trick for combining denominators—necessary for actual computations is developed. The author gives a nice conceptual discussion of divergences in loop amplitudes, the need to renormalize, and how both lead to the modern picture of the renormalization group, but most of the nitty-gritty of renormalization theory is omitted. When he finally arrives at non-abelian gauge theories, he does not discuss the necessity of gauge fixing and the introduction of ghosts.

For the most part, the simplifications that Maggiore makes are innocuous; he manages to convey the main ideas without getting lost in technical details. But occasionally the simplifications get in the way of understanding. For instance, in discussing Goldstone’s theorem, Maggiore breezily asserts that the generator of the spontaneously broken symmetry does not annihilate the vacuum and hence generates another state of the same energy. I think his assertion may leave the reader with a serious misapprehension that there is some big Hilbert space with a continuous degeneracy of states. In fact, although the charge density does exist as an operator, the global charge—the generator of the symmetry—does not. The vacua, which would have been related by the action of the generator, are in fact states in different Hilbert spaces.

All in all, Maggiore’s approach is precisely the one that should be taken in an undergraduate course: Introduce the “big” ideas and leave the computational and thornier technical details for subsequent courses. Toward the end, the book contains a brief discussion of critical phenomena and concludes with an introduction to spontaneous symmetry breaking, the Higgs mechanism, and the standard model—all the bases one would hope to cover in a one-year course in quantum field theory.

Will Maggiore’s text find a place in the undergraduate physics curriculum? I don’t know. For the most part, we physicists are terribly conservative about our undergraduate curricula. The most ambitious undergraduates at the best institutions take refuge in graduate courses, which is not altogether a bad thing. But the act is not the same as our delivering challenging undergraduate courses worthy of student’s attention.

Whatever its role in the undergraduate curriculum, Maggiore’s text would benefit another audience: graduate students who are working to become high-energy experimentalists. They really do need to learn a smattering of quantum field theory, if only to be able to communicate effectively with their theorist colleagues. For most experimentalists, a course on the level of Peskin and Schroeder’s book would be too heavy-duty. Consequently, most seem to shy away from tackling a course in quantum field theory. A course based on Maggiore’s text would be much more suitable than the standard graduate course geared toward theorists. Throw a little more particle physics into its content, and the book would make for an excellent course for high-energy experimentalists. With any luck, such a course will become the norm.