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Group Theory: A Physicist's Survey

Cambridge University Press
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The title of this ambitious book is somewhat disingenuous, at least as far as we mathematicians are concerned: it is really only our physicist cousins who all but identify group theory with group representation theory, and this would be closer to the mark as a description of what the book under review is all about. But I suppose that the subtitle, “A Physicist’s Survey,” goes a long way to justify the title in statu quo, so we’ll leave it at that.

Well, no. It is actually complementarily disingenuous to characterize the present book as exclusively about the representation theory physicists do: it’s about a lot more than that. Says the author, Pierre Ramond, “…with the advent of the Standard Model, [group theory] has become a powerful conceptual tool [as well as] an important computational tool…,” and he goes on to note that, accordingly, his book “introduces physicists to many of the fascinating mathematical aspects of group theory, and mathematicians to its physics applications.” (It is interesting — and irresistible — to note that only a physicist would use the construction, “the fascinating mathematical aspects of group theory,” betraying as it does his subtext that group theory has other than mathematical aspects, i.e. physical ones per se. Mathematicians and physicists are indeed separated by a common language.)

Group Theory: A Physicist’s Survey starts off with a discussion of finite groups, stressing their permutation presentation, touching on what Ramond calls “Sylow’s criteria,” semi-direct products, and Young tableaux. Then (yes) it’s on to representations of finite groups, including coverage of Schur’s lemma and induced representations, and, after this, to Hilbert space. It’s a bit surreal for a mathematician, of course, to see the sections “Finite Hilbert spaces” and “Infinite Hilbert spaces” flank “Fermi oscillators,” but one has to suck it up, I guess. After all, the next chapters, “SU(2)” and “SU(3),” not only introduce the crucial Lie algebra and Lie group angle, they also include sections titled, “The isotropic harmonic oscillator,” “The Bohr atom,” “Isotopic spin,” and, yes, “The Eightfold Way” (famously of Murray Gell-Mann and Yuval Ne’eman).

Against this backdrop (we’re only through about half the book at this point), Ramond downshifts and presents, in order, a discussion of the classification of compact simple Lie algebras, the representation theory of Lie algebras (including a discussion of oscillator representations and Verma modules), the classification of the finite simple groups, and then two chapters about hypermodern stuff: Chapter 10, “Beyond Lie algebras,” talks about affine Kac-Moody algebras and superalgebras, while Chapter 11, “The groups of the Standard Model,” includes discussion of quarks, non-abelian gauge theories, the Standard Model (of course), and grand unification. Then the book ends with a chapter on exceptional structures.

It is accordingly evident that whereas the mathematics Ramond focuses on is generally well within reach of many graduate students in our discipline, the flavor of the presentation is heavily oriented toward physics. For example, the discussion in Chapter 4 of infinite Hilbert spaces is rife with bras and kets à la Dirac, and, while Ramond is exceptionally clear, the discussion is bound to be a bit unheimlich to most of us pure mathematicians.

Another caveat that bears mentioning, perhaps, is that the simplicity of A5, surely a staple of all self respecting group theory books (my favorite, J. J. Rotman’s, presents it on p. 39, as a lemma in proving the simplicity of An for n>4), has to wait till p. 190 before it makes its appearance in Ramond’s book. (By the way, there’s a particularly slick presentation of the broader result on pp. 90–91 of Kurzweil-Stallmacher, The Theory of Groups, an Introduction, and I cannot resist mentioning that on p. 109 of Andreas Speiser’s Die Theorie der Gruppen von Endlicher Ordnung we find as Satz 94: “Die alternierende Gruppe von n Variablen ist einfach, sobald n größer 4 ist” — Doesn’t it simply sound better in German? And, by the way, Speiser credits the result to none other that Evariste Galois himself!)

The point is that Group Theory: A Physicist’s Survey will require a bit of effort, a good deal of openmindedness, and a pretty well-developed interest in particle physics (and/or high energy physics, and/or quantum chromodynamics) on the part of any mathematician–reader. But, to quote Richard Feynman (whose diagrams are discussed on p. 166 in a somewhat oblique way — interestingly, Ramond discusses not Feynman diagrams as such, but, rather, diagrammatic techniques in representation theory as developed in another book I had the pleasure to review, namely, P. Cvitanovic’s Group Theory: Birdtracks, Lie’s and Exceptional Groups), the book is full of (some of) “the good stuff.” You’ll enjoy it.



Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Monday, June 28, 2010
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Pierre Ramond
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Michael Berg

1. Preface: the pursuit of symmetries; 2. Finite groups: an introduction; 3. Finite groups: representations; 4. Hilbert Spaces; 5. SU(2); 6. SU(3); 7. Classification of compact simple Lie algebras; 8. Lie algebras: representation theory; 9. Finite groups: the road to simplicity; 10. Beyond Lie algebras; 11. The groups of the Standard Model; 12. Exceptional structures; Appendices; References; Bibliography; Index.

Publish Book: 
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Sunday, September 19, 2010