The latest New York Review of Books has an article by Steven Weinberg entitled Symmetry: A ‘Key to Nature’s Secrets’. It’s a bit unusual for the NYRB, since it is both scientifically more technical than usual for them (coming from a write-up of Weinberg’s talk at this conference), and doesn’t review any books. The printed version tells readers to go to the web version for footnotes, but some of these just note that things are being over-simplified. One of the footnotes is worse than useless: the editors have replaced x^3=x as an example of an equation with solutions that break a symmetry (x goes to -x) by “x 3 equals x”, an equation with the same symmetry but only a symmetric solution (x=0). The idea seems to have been to remove or replace any symbols in the equation that might upset people.
Weinberg tells the conventional story of how the Standard Model emerged during the 60s and early 70s out of the realization that non-abelian gauge symmetries were important and an understanding of what happens when symmetries are spontaneously broken. He tries to do some much more ambitious things, explaining the idea of “accidental symmetries” that are due to the limited number of possible renormalizable terms you can build out of a specified list of fields, but I’m not sure the typical reader of the NYRB is going to get much out of this.
The question of how to explain the notion of “symmetry” is an interesting one, and I thought a lot about it when writing Not Even Wrong, the book. To my mind, most such explanations mix up two conceptually distinct things: the group of symmetries (a group), and the action of the group on some other mathematical object (the representation: mathematically a homomorphism from the group to the group of automorphisms of something). It’s both the group and the representation that are important in the use of symmetries in physics, although often what is important is the trivial representation. From a mathematician’s point of view, the simplest representations to look at are unitary representations on a complex vector space, so the mathematical structure of quantum mechanics is very natural. To each symmetry generator you get a conserved quantity, and it appears in quantum mechanics as the thing you exponentiate (a self-adjoint operator) to get a unitary representation. In Weinberg’s piece, which aims at sophisticated issues in particle theory, the question of the basic relation of symmetries and conservation laws is relegated to a footnote which says only “For reasons that are difficult to explain without mathematics…”.
Weinberg ends with a landscape sort of picture, involving symmetries emerging only when a specific ground state emerges out of an initial chaotic inflation state. Philosophically this is a popular view of the future of the subject these days, but one that has so far led nowhere, and one that I think even in principle can never lead anywhere. Much more interesting would be to try and draw lessons from what has worked well in the past: exactly the gauge symmetries and spontaneous symmetry breaking phenomena that led to the standard model. We may very well soon find out there is no Higgs particle, turning this whole subject into a wide-open one. Future progress may come from exactly the same place as in the past: new ideas about how to exploit the mathematical structures inherent in quantum mechanical symmetries.
Update: The missing exponentiation in the on-line footnote has been fixed.