Classical field theory plays a key role in fundamental physics. Of the four fundamental forces, three of them, the strong nuclear, weak nuclear, and electromagnetic forces, are described by quantum field theories that are formulated by quantizing a corresponding classical field theory. The remaining fundamental force, gravity, does not have a quantum description yet and is currently best described by the classical field theory of general relativity. Franklin's book provides an introduction to classical field theory, which will be accessible to advanced undergraduates and beginning graduate students. The book consists of four chapters, covering special relativity, the electrodynamics of point particles, Lagrangian methods in field theory, and gravity. There are also three appendices that discuss analytical mathematical methods, numerical methods, and the derivation of classical electrodynamics from an action.

The focus of the first three chapters is largely on classical electrodynamics. Any textbook covering this material will invite comparison with the two standards: Griffiths at the undergraduate level and Jackson at the graduate level. In terms of difficulty, Franklin's book is intermediate between the two: it is more advanced than Griffiths but is written in a similar informal, conversational style. It differs from both books, however, in that the intent here is not to systematically develop the theory of classical electrodynamics, but rather to use classical electrodynamics as a stepping stone towards graduate-level study in quantum field theory and general relativity. To this end, three topics are particularly emphasized: the use of tensor notation, Green's functions, and field Lagrangians. Tensor notation is widely used in both quantum field theory and general relativity. Green's functions serve as the foundation of Feynman diagram methods and the perturbative formulation of quantum field theories. Field Lagrangians provide a compact way of describing classical and quantum field theories and, via Noether's theorem, of understanding the important role that symmetries and conservation laws play in these theories. Franklin introduces each of these topics by showing how it applies to the familiar example of classical electrodynamics, but the ultimate goal here is to develop the student's facility with these techniques so they can be applied to general field theories.

The fourth chapter, on gravity, is the most novel. Here, the author considers how Newtonian gravity might be modified to incorporate some of the features of special relativity. Such modifications are shown to qualitatively account for several classic predictions of general relativity, such as the deflection of starlight by gravity and the perihelion precession of Mercury, and the discussion offers some physical insight into the origin of these effects. This approach puts the reader in the position of Einstein faced with the challenge of formulating a relativistic theory of gravity: it is shown how a new theory might be developed, based on physical reasoning and extrapolation from the known theories of classical electrodynamics and Newtonian gravity. The full theory of general relativity is not discussed, but by the end of the book the student will be well prepared to take up its study. Indeed, the author has already paved the way by outlining many of the problems encountered in developing a relativistic theory of gravity and sketching some of the features that one might expect of such a theory. For example, the author uses the field theory techniques developed in previous chapters to show that scalar and vector field theories yield unphysical predictions, but that a theory involving a symmetric rank-two tensor field might be viable.

One feature of the book that I particularly liked is that often before solving a problem, the author first uses physical reasoning to deduce as much about the qualitative behavior of the solution as possible. Only then does he proceed to a detailed calculation. This approach offers greater physical insight than would be achieved by a direct calculation and provides a good model of how a working physicist would actually approach a real problem.

Another nice feature is that the author often incorporates numerical results into the discussion. Computers are powerful tools for developing physical insight and intuition, but their full potential has not yet been realized in physics education. There are two reasons why students would benefit from a greater emphasis on computers: first, computer simulation provides an excellent way to interact and tinker with physical systems that would be impractical to study in the lab, and second, numerical methods are an increasingly important part of the toolkit of most physicists. One appendix of Franklin's book is entirely devoted to numerical methods that are relevant to classical field theory, and at several places in the main text he indicates how such methods could be applied to the topic at hand. For example, in the chapter on special relativity, the equations of motion for a relativistic harmonic oscillator are numerically integrated, and in the chapter on point particles, there are diagrams of radiation fields that are produced numerically. Also, many radiation problems of the sort described in Chapter 3 involve retarded time calculations that cannot be performed analytically, so, numerical methods significantly expand the repertoire of radiation problems that can be treated.

Although on the whole the book is carefully written, I did notice several mistakes. In particular, in two places (sections B.2.3 and C.3.1) the author discusses a particle moving in the electromagnetic field generated by a charged wire sliding parallel to itself at constant velocity. In both instances, the described particle trajectory is incorrect: the particle is shown as “bouncing” along the wire, periodically moving away and then cycling back to a location further along the wire, whereas in fact this behavior does not occur, as can be seen by working in the rest frame of the wire. In the first instance, the incorrect trajectory is obtained by integrating a Newtonian equation of motion in a regime where the Newtonian approximation breaks down; in the second instance, it appears that the full relativistic equation of motion has been incorrectly integrated. There is also a minor pet peeve: the book uses SI units throughout, whereas Gaussian units with c = 1 would be more appropriate. For an introductory textbook, SI is preferable because students are already familiar with units like Amps and Ohms, but here, where the focus is on relativistic invariance, the profusion of c's, ϵ0's, and μ0's only serves to obscure the symmetry and physical meaning of the equations.

Despite these slight flaws, Franklin's book is a worthwhile introduction to classical field theory. There are three settings in which I think the book would be particularly useful:

  1. The book could be used to supplement the treatment of special relativity or radiation in an undergraduate course on classical electrodynamics. Although much of this material is standard, there are still features here that are new. I particularly liked the discussion of the relativistic harmonic oscillator in Chapter 1 and the derivation of the fields of a charged particle moving at the speed of light in Chapter 2. There is also a good selection of problems, some of which involve numerical computation.

  2. The book could be used as a preliminary to graduate-level quantum field theory and general relativity. The topics of tensor methods, Green's functions, and field Lagrangians covered here are of fundamental importance in these subjects, and students will greatly benefit from having a solid grounding in these techniques.

  3. The book could be easily mined for undergraduate thesis projects, especially those involving the numerical exploration of classical field theories. The appendices in particular provide just what one would want for this purpose: a succinct discussion of the relevant topic, an illustrative example, and references to a more extensive discussion.

Addendum: The author has contacted the reviewer to inform him that conceptual errors, including those noted in the review, have been deliberately introduced into the text without warning in order to test readers.

David Boozer has a Ph.D. in physics from the California Institute of Technology; his interests include classical and quantum field theory, atomic physics, and statistical mechanics.