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Group theory. A physicist’s survey. (English)
Cambridge: Cambridge University Press (ISBN 978-0-521-89603-0/hbk). ix, 310~p. \sterling~40.00; \$~70.00; \$~56.00/e-book (2010).
The nature, the world surrounding us, is a wealthy applied area for the concept of symmetry that manifests itself via a manifold of diverse symmetries. The word ‘symmetry’ originates from the Greek ‘$σνμμϵτρϵιν$’ that means ‘to measure together’ [see e.g. {\it G. Hon} and {\it B. R. Goldstein}, From summetria to symmetry: the making of a revolutionary scientific concept. Dordrecht: Springer (2008; Zbl 1355.01003)]. It is true, as the author of the present book notes, that “Symmetric objects are so singular in the natural world that our ancestors must have noticed them very early.” According to Vitruvius who lived c. 10 AD [{\it Vitruvius}, The ten books on architecture. New York: Dover Publications(1960), p. 14], symmetry expresses a proper harmony of the parts to each other and to the whole, defining a kind of beauty. Interestingly, in his famous 1939-year James Scott lecture, Dirac advocated the principle of mathematical beauty in seeking physical theories. In mathematics, precisely in geometry, the concept of symmetry was introduced by A.-M. Legendre who defined an equality of solid angles by symmetry [{\it A.-M. Legendre}, Eléments de Géometrie. Paris: F. Didot (1794)]: “Two equal solid angles which are formed (by the same plane angles) but in reverse order will be called angles equal by symmetry, or simply symmetrical angles.” Mathematically speaking, symmetry is a rather abstract concept, independent of the nature of the object under study: whether it is a geometric figure, a crystal, a hydrogen atom, or a molecule of benzene or fullerene, quark, Maxwell, Robert Dautray, Jacques-Louis Lions’ equation, and et cetera et cetera, and on observer as well. Symmetry is such an operation or transformation that, as being applied to a given object, preserves it unchanged, invariant [see {\it H. Weyl}, Symmetry. Princeton: Princeton University Press (1952; Zbl 0046.00406)]. The major part of objects reveal either no symmetry at all, or little symmetry, or are almost symmetric. A perfect or, equivalently, in some sense, high symmetry ‒ recall a sphere, platonic solids, and at last the buckyball fullerene C$_{60}$, e.g. ‒ is indeed rare and not an exception, though. A set of symmetries of a given object generates a group, a so called group of symmetry or group of transformations. The pioneers in the theory of abstract groups were N. H. Abel and E. Galois. Studying the problem of finding the roots of polynomials of degree of five and higher, E. Galois reduced it to that of the set of substitutions that form a mathematical structure that was defined as a group. The concept of group also goes back to S. Lie’s famous paper “Über Gruppen von Transformationen”, that was written in 1874 [Gött. Nachr. 1874, 529‒542 (1874; JFM 06.0093.01)]. A group can be either finite or infinite, either discrete or continuous, the latter is exactly that group that S. Lie referred to in the above work. A group is an abstract concept, too. However, in our daily life groups are percepted via their representations. Traditionally, representations are the mappings of the actions of a group on vector spaces. Let us recall Hilbert spaces where nearly all quantum mechanics acts on. It is natural then to characterize a group representation by its trace, simply imagine a finite-dimensional linear vector space and finite-matrix mapping ‒ this is the character of the representation, the concept that was proposed by F. G. Frobenius for finite groups in 1896, as the solution of the problem posed to him by P. G. L. Dirichlet. In 1983, E. Wigner recalled [{\it M. Michel, V. L. Fitch, F. Gursey, A. Pais, R. U. Sexl, V. L. Telegdi} and {\it E. P. Wigner}, “Round table on the evolution of symmetries”, in: Proceedings of Symmetries in physics (1600‒1980). 625‒637 (1983); see also {\it A. Pais}, The genius of science. Oxford: Oxford University Press (2000; Zbl 1023.01010)] that “As to representation theory: I realized that there must be such a theory but I had no knowledge of it. Dr. von Neumann to whom I presented the problem (and I presented to him the representations of the permutation groups of three and four elements because those I could establish by long hand calculations) gave me a reprint of the article of Frobenius and Schur [1906]. And that was wonderful!” E. Wigner was talking about the paper by {\it F. G. Frobenius} and {\it I. Schur} entitled “Über die reellen Darstellungen der endlichen Gruppen” which was published in 1906 [Berl. Ber. 1906, 186‒208 (1906; JFM 37.0161.01)]. A key concern of physicists in the concept of group is how any group acts on a Hilbert space. This is precisely the basic theme of the present book “Group theory. A physicist’s survey" by Pierre Ramond. It is composed of eleven chapters and two appendices. The methodology of presenting the whole material is quite specific: mathematical portion is given in the form of schola and mathematical proofs are rather short or absent at all. Chapter 1 is an introduction to finite groups that presents the axioms of a group, Lagrange’s theorem, and the basic concepts of the group theory, and discusses the finite groups of order less than thirteen, among which are in particular abelian and non-abelian groups, the dihedral group, sometimes called $V$ (from Vierergruppe) or Klein’s four-group, quaternion groups, groups of permutations, simple groups, Sylow’s groups (see Appendix 1 for more details). Few subtle features of groups are sprinkled throughout the Chapter that brings it a sort of zest. The most attractive is definitely the discussion of the number of groups of order up to 200 given in Table 2.2 and the observation that there exists therein only one group of prime order. Well, does that be of any relevance to physics is likely another question. Chapter 2 continues the theme of finite groups and studies their representations mostly in a standard way, starting with Schur’s lemmas, irreducible representations, characters, induced representations. It also deals with some representations and their characters of particular groups. Since the scene of quantum mechanics is Hilbert space, Chapter 3 is briefly devoted to them, mostly focusing on finite Hilbert spaces and Fermi oscillators as a favorite way of building the former, and the infinite ones. Chapter 4 introduces us to continuous groups and this introduction embarks on the famous $SU(2)$ group, the group of rotations, and its Lie algebra. Some selected examples of applications of this group in quantum mechanics are considered: these are the isotropic harmonic oscillator, the Bohr atom, isotopic spin, including the Fermi-Yang model and the Wigner supermultiplet model. The next Chapter discusses the more complicated $SU(3)$ algebra and its quantum mechanical manifestations, among which are the harmonic oscillator, the Elliot model, the Sakata model, and the Eightfold Way. Mathematical studies of Lie algebras are presented in Chapters 7 and 8 and extended in Appendix 2. Chapter 9 returns to finite groups taking on “the road of simplicity", i.e. via treating finite groups through the composition series, in terms of simple finite groups. True, most simple groups can be considered as finite elements of Lie groups which parameters belong to finite Galois fields and which construction relies on the Chevalley basis of the Lie algebra and the topology of its Dynkin diagram. The rest of simple groups are the magnificent 26 sporadic groups. Altogether, these compose the content of Chapter 9. The next chapter, entitled “Beyond Lie algebras", focuses on the so called Serre presentation, the procedure to build the simple Lie algebras from the Cartan matrix, on infinite-dimensional affine Kac-Moody algebras, and on super-Lie algebras and their classification. In Chapter 10, the group-theoretical aspects of the standard model and grand unification are studied. Among these aspects are: space-time symmetries, the Lorentz and Poincaré groups, the conformal group, and finite subgroups of $SU(2)$, $SO(3)$, and $SU(3)$. As noticed in the beginning, objects that carry symmetries are quite singular or unusual. Among them, there exist those that occupy even a more special place. Such are e.g. the platonic solids which are associated with exceptional groups. The latter comprise the content of the last Chapter 11 that studies these groups and their representations by means of their unique underlying algebraic structures. Within the focus of Chapter 11 fall Hurwitz algebras and the magic square.
Reviewer: Eugene Kryachko (Liège)
Classification: H45 M55