Mathematics – Logic
Scientific paper
Apr 2011
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2011cqgra..28h9003p&link_type=abstract
Classical and Quantum Gravity, Volume 28, Issue 8, pp. 089003 (2011).
Mathematics
Logic
Scientific paper
Anthony Zee is not only a leading theoretical physicist but also an author of popular books on both physics and non-physics topics. I recommend especially `Swallowing Clouds', on Chinese cooking and its folklore. Thus, it is not surprising that his textbook has a unique flavor. Derivations end, not with `QED' but with exclamation points. At the end of one argument, we read `Vive Cauchy!', in another `the theorem practically exudes generality'. This is quantum field theory taught at the knee of an eccentric uncle; one who loves the grandeur of his subject, has a keen eye for a slick argument, and is eager to share his repertoire of anecdotes about Feynman, Fermi, and all of his heroes.
A one-page section entitled `Electric Charge' illustrates the depth and tone of the book. In the previous section, Zee has computed the Feynman diagram responsible for vacuum polarization, in which a photon converts briefly to a virtual electron-positron pair. In the first paragraph, he evaluates this expression, giving a concrete formula for the momentum-dependence of the electric charge, an important effect of quantum field theory. Next, he dismisses other possible diagrams that could affect the value of the electric charge. Most authors would give an explicit argument that these diagrams cancel, but for Zee it is more important to make the point that this result is expected and, from the right point of view, obvious. Finally, he discusses the implications for the relative size of the charges of the electron and the proton. If the magnitudes of charges are affected by interactions, and the proton has strong interactions but the electron does not, can it make sense that the charges of the proton and the electron are exactly equal and opposite? The answer is yes, and also that this was the real point of the whole derivation.
The book takes on the full range of topics covered in typical graduate course in quantum field theory, and many additional topics: magnetic monopoles, solitons and topology, and applications to condensed matter systems including the Peierls instability and the quantum Hall fluid. It is a large amount of territory to cover in a single volume. Few derivations are more than one page long. Those that fit in that space are very smooth, but others are too abbreviated to be fully comprehensible. The prose that accompanies the derivations, though, is always enticing. Zee misses no opportunity to point out that an argument he gives opens the door to some deeper subject that he encourages the reader to explore. I do warn students that it is easy to learn from this book how to talk quantum field theory without understanding it. To avoid this pitfall, it is important (as Zee emphasizes) to fill in the steps of his arguments with hard calculation.
One topic from which Zee does not restrain himself is the quantum theory of gravity. In the first hundred pages we find a `concise introduction to curved spacetime' that includes a very pretty derivation of the Christoffel symbol from the geodesic equation. Toward the end of the book, there is a set of chapters devoted to the quantization of the gravitational field. The structure of the graviton propagator is worked out carefully. The van Dam-Veltman discontinuity between massless and massive spin 2 exchange is explained clearly. But after this Zee runs out of steam in presenting fully worked arguments. Still, there is room for more prose on connections to the great mysteries of the subject: the ultraviolet behavior, the cosmological constant, and the unification of forces. A new chapter added to the second edition discusses `Is Einstein Gravity The Square Of Yang-Mills Theory?' and suggests an affirmative answer, based on brand-new developments in perturbative quantum field theory.
Quantum field theory is a large subject that still has not reached its definitive form. As such, there is room for many textbooks of complementary character. Zee states frankly, `It is not the purpose of this book to teach you to calculate cross sections for a living.' Students can use other books to dot the i's. This one can help them love the subject and race to its frontier.
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