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.
Interesting review, and maybe Livio avoids to write about specific representations because he intuitively knows that the truly interesting symmetries of nature do not admit any non-trivial representation?
Weinberg said 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�. (Peter said: I don’t think I really understand what Weinberg has in mind here)
Well, I understand. On first thought you would think that mathematical consistency is a necessary condition. If the math is inconsistent then the theory is useless regardless of anything else.
Wenberg probably belong to the group of people who believes that the ultimate theory can explain everything from pure mathematical principles, like an exact formular to calculate alpha.
If the ultimate theory is pure math, then, probably, being mathematically self-consistent is all that is required to be correct in mathematics. That must be what Wenberg had in mind.
I do not believe the ultimate theory will be pure math. There will be at least one thing unexplainable in pure math, and that remains physics.
Quantoken
The paper mentioned misses the point (as usual) about Weyl’s 1918 work – namely that Weyl does NOT find a suitable action at all, because the equations for g are 4th order. In fact it is not possible in 4 dimensions to find a suitable action in this theory. Only first in 6 dimesions does it become possible to find an action leading to 2nd order equations for the g’s and have the g and A fields essentially coupled, without arbitrary constants.
http://cdsweb.cern.ch/search.py?recid=688763&ln=en
See
-drl
Peter Woit writes:
It’s forgivable not to explain what a representation of a group is. It’s unforgivable not to explain what an action of a group is.
Groups naturally arise as symmetries of sets with extra structure, and in this situation we say the group acts on the set. The first and most important way people discovered groups was by finding them acting on sets. We call these “concrete groups”. This is the sort of group Galois ran into: the group of symmetries of a field fixing some subfield. I don’t see how Livio could discuss Galois theory without at least implicitly talking about group actions, or at least concrete groups.
Only later was the concept of “abstract group” achieved. This is an incredibly powerful concept. But now, horribly, there are some abstract algebra texts that discuss group theory without a good discussion of group actions – of which representations are a special case.
Of course, anyone working on quantum theory needs the concept of group representation.
Yes, and sadly if group actions are discussed at all within the context of abstract algebra, it’s action on an abstract set with no additional structure. Much interesting theory arises when you attach a topology to an infinite set and consider notions such as orbital almost-periodicity and proximal and distal relations under action by the group. Everyone should know how concepts of periodicity can be supported within the context of co-compact (syndetic) subsets of the acting group, even when the spaces involved are not even necessarily metric spaces.
Concepts of group actions really should be woven more completely into the math curricula, and much theory is readily accessible even to undergraduates.
Given your critics, what book does give a good introduction to group theory? Let’s assume for example the reader to be an engineer with only trivial knowledge of quantum mechanics.
Unfortunately I don’t know of a good book that explains group theory and how to use it in quantum mechanics at an elementary level. Quite possibly such a thing is out there, but I just haven’t run across it. If anyone has any suggestions, I’d like to hear about them.
Not to butt in, but I love recommending books 🙂
There are any number of good books but the all-time classic for physicists is “Theory of Groups and Quantum Mechanics” by Weyl. This reprint suffers from horrible typesetting but is otherwise a masterpiece.
For a great introduction to the entire worldview of group theory, read “Elementary Mathematics from an Advanced Standpoint” by Felix Klein, both volumes (short but very intense).
For a standard textbook, try “Group Theory and Quantum Mechanics” by Tinkham.
A little formal algebra couldn’t hurt, for which I recommend “Abstract Algebra” by N. Herstein.
There must be modern texts that are readable but I don’t know of any.
-drl