Group Theory: Application to the Physics of Condensed Matter , Mildred S. Dresselhaus , Gene Dresselhaus , and Ado Jorio , Springer, New York, 2008. $89.95 (582 pp.). ISBN 978-3-540-32897-1
In my experience, there is no ideal way to teach the application of group theory to the quantum mechanics of atoms, molecules, and solids; the reason is the inherent nature of the subject. Students have to invest a large amount of intellectual capital before they reap the benefits in applications that often fall out almost trivially. Moreover, the mathematics of group representations is quite sophisticated and unfamiliar to most physics and electrical engineering graduate students, and that is rather off-putting for many if they do not already have some idea of the physical applications.
Group Theory: Application to the Physics of Condensed Matter, by Mildred S. Dresselhaus, Gene Dresselhaus, and Ado Jorio, has had enormous consumer testing in the way it steers around those obstacles: It was developed for a graduate course taught mostly by Millie Dresselhaus at MIT for more than 30 years, with many revisions of lecture notes. Very much a graduate text or specialist monograph, the book covers a wealth of applications across solid-state physics. In fact, the only two topics I could not find covered in its pages were selection rules for Raman transitions in solids and lines of accidental degeneracy when electron energy bands cross, both somewhat esoteric.
Following a four-chapter introduction to the mathematics—with more math promised for later—and another two chapters on its applications to quantum systems, two chapters on electronic states and vibrations in molecules (including Raman activity) introduce the basics of group theory in molecular systems. The authors use crystal-field effects to show how to treat the perturbative splitting of degenerate energy levels. Nearly half the book is taken up with the space-group symmetry of solids, its application to phonons and electron energy bands, and the use of double groups to account for spin. The authors give detailed discussions of effective-mass tensors, g-factors in a magnetic field, and other topics. Three more chapters cover time-reversal symmetry, tensors in elasticity and nonlinear optics, and the permutation group. The point of including the last subject is not clear. The authors use the permutation group to derive the spectroscopic terms (which denote the total angular momentum L and total spin S) in various orbital configurations such as p 5 or d 5, but that can be done much more simply with John Slater’s scheme, which originally introduced Slater determinants. That approach is not mentioned, perhaps because it is assumed known from atomic theory.
The book takes a how-to approach throughout. For example, the authors introduce irreducible representations of the full rotation group from spherical harmonics, assumed to be familiar, and they write the extension to spin representations by analogy. Even so, the background mathematics covers the first 70 pages, though the authors provide examples from physics to illustrate concepts. A novice would be advised to have a guide through that early material, because, of course, not all the mathematics is needed for many simple applications.
In one or two places, where the ideas are a matter of physics and not of mathematics, the authors’ how-to approach left me a bit dissatisfied. The most basic points of the subject are that quantum energy levels have associated group representations and that those representations are “normally” irreducible, as stated on page 58. But why should they be irreducible? The fact is that they are irreducible only if the group includes all the symmetries of the system and if neither quirky mathematical symmetries such as those inherent in the hydrogen atom’s simple 1/r potential nor “accidental” degeneracies such as the band crossings mentioned above are present, while keeping in mind that time-reversal symmetry treated afterward as an add-on can also produce degeneracy. The authors include those ideas, of course, but in a conversational way without a crisp formulation of where the irreducibility comes from. I noticed only one small typo, and found the index satisfyingly complete.
The book can be warmly recommended to students and researchers in solid-state physics, either to serve as a text for an advanced lecture course or for individual study, preferably with an instructor to help select mathematical background and applications of interest. But even if one is never going to use the detailed machinery of group theory, the concepts give a precise way of thinking about symmetry and degeneracy in physics.