Peter Woit is a senior lecturer in the department of mathematics at Columbia University. Educated at Harvard and at Princeton, Woit is known for his book Not Even Wrong, and for his blog of the same name.1 Quantum Theory, Groups and Representations is based on a series of lectures that he gave at Columbia University.
And it is excellent.
An introduction to quantum mechanics very often follows a well-worn path. Wave functions are defined on a Hilbert space. Observables are represented as operators acting on quantum states. The evolution of a system is specified by Schrödinger’s equation. Analytic functions, Fourier transforms, eigenvalues, and matrices all play their accustomed roles. Matrices are particularly useful in quantum mechanics because, unlike the numbers, they do not necessarily commute under multiplication.
Is there a deeper structure beneath the quantum formalism? Yes, of course there is. It is a structure that appears when the symmetries of a quantum system are under analysis. In 1915, Emmy Noether demonstrated that differentiable symmetries give rise to conservation laws. Her work is a foundational document in quantum theory because it verifies the ancient insight that what is most important in any physical system is what remains the same in the system as the system is changing.
Differentiable symmetries belong to the much larger class of symmetrical transformations. The basic mathematical object is a group. Mathematicians of the early nineteenth century knew that there was no algorithm determining the roots of a fifth-order polynomial equation. They did not know why. In 1832, Évariste Galois saw gold while studying the permutations among the roots of a polynomial equation. He realized that there was an abstract mathematical object that corresponded to those permutations. This was the gold. The impossibility of solving fifth-order polynomial equations by means of a formula followed at once. It took a number of years for the idea of a group to catch on among mathematicians, and even longer for it to be appreciated among physicists. Galois himself never saw his ideas flower: he lost his life at the age of twenty—the victim of an ignominious duel.
A set G closed under an associative binary operation G × G → G is a group if G includes an identity and an inverse. An identity I returns every element to itself: a o I = a. An inverse a–1 returns any element to the identity: a–1 o a = I. The ordinary integers form a group under the operation of addition.
An abstract group admits of many different representations. A group representation maps the elements of the group to some vector space. A representation π of the group G is defined as a homomorphism—a map of groups preserving the group operation
π: g ∈ G → π(g) ∈ GL(V)
where V is a vector space, and GL, the group of invertible linear maps.
As an example, consider the continuous rotation group S0(2). The generator J of this group, it is easy to demonstrate, is R(ø) = e–iøJ, where R(ø) is, of course, a continuous measure of rotation through a given angle. S0(2) is a Lie group; its structure is determined by group operations on J near the identity—so, too, its representations. Thus consider a representation of S0(2) defined in a finite dimensional vector space V. R(ø) and J both have associated operators R(ø)* and J* in V. Under certain circumstances J* may be understood as an angular momentum operator in the sense of quantum mechanics.
The discussion is rapidly getting technical, and if unimpeded, it is apt to become much more technical. The pedagogical challenge in all this is to communicate the elements of group theory to physicists in a way that persuades them of its use, and to mathematicians in a way that assures them of its honesty.
There is not really a good text that does this in a systematic way.2 What Woit has done is to introduce the basic idea of groups, and then, via a series of examples and simple cases, work his way up through geometry, complex numbers, basic quantum mechanics, and quantum field theory. By keeping his focus on quantum physics, he incorporates enough of the old standards to keep physics students interested when they are not lost, and to provide them guidance when they are.
At 585 pages, this book is not short. To profit from Woit’s text, physics students should already know quantum mechanics fairly well, and they should have at least some mathematical inclinations. This may seem daunting, but students wanting a deeper knowledge of quantum theory must know something about its underlying group theoretic structure. Woit stresses particle physics, his own area of study, ending the book with a survey of the Standard Model as an exercise in group theory—the study, most obviously, of SU(3) × SU(2) × U(1).
Woit is a fine teacher. He starts with the simplest examples, such as the group of two-dimensional rotations, and lays out all of the important elements of the group: their representation(s), their group action, their generators. By seeing how abstract objects work in more familiar contexts, the student is then prepared when more advanced algebras appear. Woit is at his best in explaining how the group-theoretic approach simplifies certain concepts. Solid body mechanics typically makes use of the unwieldy Euler angle formulas; they are gone in Woit’s text, replaced by matrix generators of a Lie algebra. If abstract concepts do not usually proceed for many pages without a clarifying example, some credit must be given to Woit’s artist and figure designer, Ben Dribus, because the figures are both simple and clear. Woit does not, for the most part, follow the death march of proposition, lemma, proof. He writes in the style of a theoretical physicist.
A few obligatory criticisms now follow. I am a firm believer in problem solving as a way of learning, and do not think you really learn any subject in either mathematics or physics without struggling over the exercises. Woit does provide problems, but they are all at the back of the book. It would have been better to see them between chapters. That provides a natural break in the material and gives the student a quick check on his understanding. Woit’s review of linear algebra is oddly placed in chapter four, after the introduction and his treatment of U(1) and SU(2). It would be better served as a mathematical preliminary or as an appendix. In the introductory chapters, some symbols are introduced that are not defined until chapter four, such as the notation for a general linear group. Woit uses the overbar for complex conjugation; we physicists prefer the raised asterisk. I would also encourage Woit in the next edition to expand the index of the book: there is no entry for homeomorphism. A text of this sort must inevitably contain a few terminological oddities. Canonical transformations among physicists are called symplectomorphisms among mathematicians. Who knew?
If nothing else, Woit’s book allows its readers a sense of vicarious sympathy for physicists of the early twentieth century forced to face the facts and learn some group theory. The term gruppenpest has entered the literature, the spontaneous outburst, one gathers, of John Slater, the chair of MIT’s Department of Physics. Given the great triumphs of group theory in particle physics, it would be hard to imagine a better way of going wrong.
In following Woit’s way, a plainspoken physicist—a Watson to Woit’s Holmes—may sometimes feel the hankering to put the mechanic back into quantum mechanics by means of an honest matrix or a differential operator. Fortunately, Woit does just that from time to time. After the metaplectic representations of chapters 20.1 and 20.2, he proceeds to the real calculations of 20.3, just in order to show how the thing actually works.
For that plainspoken physicist—there are many chapters that read more or less like other books on quantum mechanics or quantum field theory. Woit’s treatment of Green functions, angular momentum, the harmonic oscillator, coherent states, and Feynman path integrals is more or less standard. Topics in which group theory peeps out of graduate textbooks—the generators of continuous symmetries, the fact that SO(3) is homomorphic to SU(2) (but with a double cover), or the SO(4) treatment of the hydrogen atom—are treated in great detail. And Woit is sensitive to the insights provided by group theory when it comes to the quantization of a classical theory. Following Paul Dirac, he observes that quantizing classical mechanics is, “completely equivalent to asking for a unitary representation of the infinite dimensional functional Lie algebra defined on the phase space of a classical system.”3
First-order quantization is followed in Woit’s text by multiparticle physics, second quantization, and field theory. There is much to admire in Woit’s treatment of the Clifford algebras as the fundamental fermionic algebras. This sets the stage for his discussion of supersymmetry in a nonrelativistic context. It is a discussion that I was delighted to see. Woit introduces a supersymmetric oscillator, with both bosonic and fermionic degrees of freedom, and assesses its supersymmetry in a simple example, even pointing to its physical realization as a spin-1/2 particle in a magnetic field with a gyromagnetic ratio of 2. Later chapters contain a more-or-less standard discussion of second quantization. Relativity makes an even later appearance, with its own symmetric and group structure intact, and these, Woit shows, can be used to construct relativistically invariant field theories and their quantization. Still later chapters make a daring entry into non-abelian gauge theories and discuss Yang–Mills theory in a way that I have never seen before. It is in these chapters that Woit draws on the analogy between Yang–Mills fields and electromagnetic fields with an anti-commutative vector potential.
Somewhere along a later line, Woit puts a toe into the quantized electromagnetic field, and discusses the photon, but there he toes the line. There is no discussion of BRST (Becchi, Rouet, Stora, and Tyutin) quantization. The last two chapters of the book return to group theory, discussing Majorana fermions, which are not usually treated in field theory books, as well as Weyl spinors, which are.
Quantum Theory, Groups, and Representations, is, in addition to being a tour d’horizon, a tour de force. It is well written; it is quite readable; and it fills a gap in the literature between texts in quantum mechanics that brush up against group theory, and texts on the mathematics of group theory that are mainly concerned with the formal aspects of mathematical structure. Woit’s book helps to abstract quantum physics away from the representations of wave functions and Hilbert space, and teach us something deeper about the structure of the theory that cannot be easily seen within the representations.
It is a fine text; I recommend it.
