I just finished reading an interesting new book by astrophysicist Mario Livio. It’s called The Equation That Couldn’t Be Solved, and the subtitle is “How Mathematical Genius Discovered the Language of Symmetry”. Livio’s topic is the idea of a symmetry group, concentrating on its origins in Galois theory.
The first part of the book contains a wonderful detailed history of the discovery of the formulas for the roots of third and fourth order polynomials, and the much later proofs that no such formulas existed for general fifth order polynomials. The romantic stories of the short and tragic lives of Abel and Galois are well-told, in much more detail than in other popular books that I’ve seen. Galois was the one responsible for first really understanding the significance of the concept of a group, and using it to get deep insights into the structure of the solutions of polynomial equations.
The latter part of the book deals with the important role of symmetry in modern theoretical physics, and this is a topic treated in many other places in more detail. Livio gives the standard party-line about string theory, but he does do one very interesting thing. He notices that while string theory implies various sorts of symmetries, e.g. supersymmetry, it lacks a fundamental symmetry principle itself, and this leaves open a very important question. Does physics at its most fundamental level involve a symmetry principle, or are symmetry principles an artifact of our throwing out complexity and only focussing on simple situations that we can understand? Perhaps symmetry is not fundamental, but only an artifact of our limited abilities to understand things. Livio asks several people this question, and gets the following answers:
Weinberg: symmetry might not be the most fundamental concept in the ultimate theory, and “I suspect that at the end the only firm principle will be that of mathematical consistency”. (I don’t think I really understand what Weinberg has in mind here)
Witten: “there are still missing, or unknown ingredients in string theory” and “some concepts, such as Riemannian geometry in general relativity, may prove to be more fundamental than symmetry.”
Atiyah: “We come to describe nature with certain spectacles… Our mathematical description is accurate, but there may be better ways. The use of exceptional Lie groups may be an artifact of how we think of it.”
Dyson: “I feel that we are not even at the beginning of understanding why the universe is the way it is.”
There is one interesting thing that Livio gets wrong. He explains Klein’s Erlangen program of identifying the notion of symmetry with the notion of a geometry, but then says that this is precisely what Riemannian geometry is. This isn’t really right, since the non-Euclidean geometries Klein was using are basically homogeneous spaces of Lie groups, whereas Riemann’s notion was more general, just insisting that the geometry be locally Euclidean. To unify these two points of view, you need the later ideas of Elie Cartan about Cartan geometries and connections. A related distinction is that Klein was considering finite dimensional symmetry groups, whereas in Riemannian geometry you don’t have a global symmetry group. You do have infinite dimensional groups of local symmetries, e.g. the diffeomorphism group, and the gauge group of frame rotations. By the way, a nice article about the early history of gauge theory has just appeared on the arXiv.
My main problem with Livio’s book is that he only discusses the groups themselves, and doesn’t even try to explain what a representation of a group is. For the applications to quantum mechanical systems and to particle physics, it is this notion of a representation of a group that is absolutely crucial.