Anthony Zee is a talented and imaginative expositor who has written three
popular books on physics and cosmology. He is also a talented and imaginative
physicist who has an unusually broad range of interests. In this book Zee deploys
his imaginative style as a writer in the service of his incisive understanding of
physics to produce a unique book on quantum field theory. Quantum field theory
in a nutshell in more than 500 pages sounds like an oxymoron, but Zee’s book lives
up to the title, although we should think of the nutshell as a large pomegranate
with many piquant seeds inside.

Zee gets off to a fast start by introducing the Feynman path integral using the
limit of a double slit experiment with more and more holes and then more and
more screens. By page 15 he has shown how to calculate Gaussian integrals over n
variables with external sources, explained the gist of Wick’s theorem and of the
steepest descent approximation. He gives simple arguments to show why like
electric charges repel and masses attract. Still within Part I he calculates
the Planck mass in extra dimensions in terms of the four-dimensional Planck mass
and the compactification radius, outlines canonical quantization, gives Feynman
rules for scalar Phi4 theory, outlines a simplified calculation of the Casimir effect,
derives Noether’s theorem, and finishes by giving the rationalized
action for a scalar particle in a gravitational field.

Zee doesn’t slow down in the rest of the book. Rather than give a laundry list of
all the other topics he treats, I will first highlight a discussion I particulary like,
and then make some general comments. In discussing quantum electrodynamics
Zee absorbs the electric charge in the vector potential so that the charge enters the
photon propagator rather than the electron-positron-photon vertex. This
automatically leads to having the renormalized charges equal if their bare charges
are equal and makes the electron field strength renormalization constant equal the
vertex renormalization constant (sometimes called Z2 and Z1) equal, a fact that
usually requires a detailed argument.

The exposition is colloquial, in the style of somebody at the blackboard talking to
a small class of bright students, flavoring the discussion with relevant and amusing
anecdotes. The explanations, while succinct, are understandable. The exercises
lead to further development of topics in the text. Topics not usually presented in a
first text on quantum field theory find their rightful place; in particular
applications to collective phenomena and condensed matter, such as critical
phenomena (but only Landau-Ginzburg mean field theory), the integer and fractional
quantum Hall effects, and Anderson localization are included. The general
philosophy is effective field theory and the associated emphasis on symmetry and
the renormalization group. Some topics, such as Hawking radiation, will have to
be augmented by the instructor if included in a course, but by and large the book
is self contained. Appendices include a user’s guide to group theory and, most
welcome, a discussion of van der Waerden’s undotted and dotted indices for
spinors.

Who is the audience for this book? First of all, any physicist or mathematician
who wants an intuitive introduction to quantum field theory. Working theoretical
physicists will find it useful both as a quick introduction to topics they don’t know
and as a source of novel and pithy explanations of topics they do know. Working
experimental physicists will find it useful to bridge the gap between theory and
experiment. Students will find it an inspiring, if at times a bit intimidating, text
on quantum field theory in all its guises. For use as a text in a field theory course
oriented toward particle physics, a supplementary text such as the book by Peskin
and Schroeder would be helpful. The best way to capture the spirit of this book is
to quote Zee quoting his advisor Sidney Coleman that quantum field theory made
“the spectator gasp with awe and laugh with joy.” Zee has conveyed that awe and
joy in this book.

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