For a second slogan about quantum mechanics I’ve chosen:
Quantum mechanics is evidence of a grand unification of mathematics and physics.
I’m not sure whether this slogan is likely to annoy physicists or mathematicians more, but in any case Edward Frenkel deserves some of the blame for this, since he describes (see here) the Langlands program as a Grand Unified Theory of mathematics, which further is unified with gauge field theories similar to the Standard Model.
This week I’m in Berkeley and have been attending some talks at an MSRI workshop on New Geometric Methods in Number Theory and Automorphic forms. Number theory is normally thought of as a part of mathematics about as far away from physics as you can get, but I’m struck by the way the same mathematical structures appear in the representation theory point of view on quantum mechanics and in the modern point of view on number theory. For example, the lectures on Shimura varieties have taken as fundamental example the so-called Siegel upper-half space, which is the space Sp(2n,R)/U(n). Exactly the same space occurs in the quantization of the harmonic oscillator (see chapters 21 and 22 of my notes), where it parametrizes possible ground states. Different aspects of the structure play central roles in the math and the physics. In the simplest physics examples one works at a fixed point in this space, with Bogoliubov transformations taking one to other points, something which becomes significant in condensed matter applications. In number theory, one is interested not just in this space, but in the action of certain arithmetic groups on it, with the quotient by the arithmetic group giving the object of fundamental interest in the theory.
The workshop is the kick-off to a semester long program on this topic. It will run simultaneously with another program with deep connections to physics, on the topic of Geometric Representation Theory. This second program will deal with a range of topics relating quantum field theory and representation theory, with the geometric Langlands program a major part of the story, one that provides connections to the number theoretical Langlands program topics of this week’s workshop. I’ve got to be in New York teaching this semester, so I’m jealous of those who will get to participate in the two related MSRI programs here in Berkeley. Few physicists seem to be involved in the programs, but these are topics with deep relations to physics. I do think there is a grand unified theory of some kind going on here, although of course one needs to remember that grand unified theories in physics so far haven’t worked out very well. Maybe the problem is just that one hasn’t been been ambitious enough, that one needs to unify not just the interactions of the standard model, but number theory as well…