This book gives a solid introduction to the simplest of gauge theories, that of the Abelian gauge field governing the interactions between photons and charged particles. The emphasis is on doing calculations, and so readers who need a more in-depth mathematical or "foundational" overview of quantum electrodynamics may be disappointed. Quantum field theory of course was not founded on the need for mathematical rigor in physics, but instead has its origins in reconciling quantum mechanics with the theory of special relativity. This reconciliation has sometimes been a rough road, and in many places employs some sophisticated but eccentric "trickery" on the part of the researchers. It is these tricks that are the most difficult to generalize, to the annoyance of mathematicians who want to put quantum field theory on a more rigorous mathematical foundation. But in spite of the use of these oddities quantum field theory is not magical, and has proven to be one of the most precise physical theories ever constructed.
Some of the highlights of the book:
1. The chapter on exact propogators and vertex parts is particularly illuminating, especially the discussions on Dyson's equation, Ward's identity, and the physical conditions needed for renormalization. Dyson's equation relates the vertex part to the exact propagator, and the authors derive it using two different approaches in the book: one using the concepts of reducible and irreducible diagrams, the other using direct calculation and taking the Fourier transform. Readers who go on in quantum field theory will find that this equation is usually called the Dyson-Schwinger equation and can be derived using "functional methods." Ward's identity is a relation that connects the momentum derivative of the electron propagator to the vertex part, but can derived solely by using gauge invariance. Applying a gauge transformation to the electron propagator will result in an expression involving an external (photon) field. This expression though has a contribution coming from photons with longitudinal components in their momentum, but the expression is shown to vanish. Hence, as expected, gauge invariance results in an electron propagator that does not involve massive photon fields, and its momentum derivatives are equal to the vertex part. The authors point out that this identity generalizes the expression for the case of the free-particle propagator.
2. The discussion on the radiative corrections to Coulomb's law, resulting from the "polarization of the vacuum" around a point charge. The corrections are done via the use of an "effective field", thus introducing the reader to a very common approach these days. After taking Fourier transforms the authors show that the polarization of the vacuum alters the Coulomb field in a region inversely proportional to the electron mass. Beyond this region the change drops off exponentially. The authors point out though that they have ignored the contributions of pions and muons in their calculation of the correction. At distances less than one over the muon (or pion) mass, the strong interaction must be taken into account and quantum electrodynamics breaks down.
3. The discussion on photon-photon scattering, which is a strictly quantum effect since it cannot occur in classical electrodynamics, due to the linearity of Maxwell's equations. It is the electron-positron annihilation which is responsible for this effect, and this is one example of the matter-antimatter duality that seems to always occur in quantum theories that must respect the principle of relativity (although, strictly speaking, another assumption, called "cluster decomposition" is needed to show this in a convincing way).
4. The (short) chapter on hadron electrodynamics, with "electromagnetic form factors" used to finesse the problem of the strong interaction. One thus gets a purely phenomonological theory, but one that still allows the calculation of electron-hadron and photon-hadron scattering.