This post is based on a conversation I had with Allan Adams at Mathcamp a few summers ago, and I was reminded of it by an aside in Mike Freedman’s talk in Scott’s backyard on Friday. As usual with blog posts based on other people’s talks, all good ideas in this post should be attributed to Allan and Mike and all mistakes to me. Furthermore I think everything I say here is obvious to people who actually know physics.
My basic confusion was how physical intuition (in particular in quantum field theory) could be applied to so many mathematical settings when there’s only one physical world so there’s no reason to think any intuition built up within that single example would apply any more generally than that one example. What Allan pointed out to me is that it’s not true that physicists are only studying one example. Although there may only be one fundamental theory of physics, by looking at various particular physical systems the limiting behavior becomes its own theory. The physics at the surface of a black hole can be thought of as its own example; the physics of superconductors is its own example; etc. Because all of these examples are physical (they involve minimizing actions, they’re quantum, etc.) they have a lot of attributes in common, so intuition and general techniques can be developed by understanding their commonalities.
Mike made two comments in his talk (on K-theory and superconductors) that flesh out this idea further. He was discussing the BCS superconductor and explained that when physicists refer to a theory by initials they’re not just being polite, what they mean is that you’re studying the mathematical model rather than any particularly instantation of it. In particular, the model doesn’t care if there are exactly 10^9 electron pairs or the exact composition of the material, it is studying the abstract setting that appears in the limit. By calling it the “BCS superconductor” they mean that in some sense they’re studying the physics of a different world. In particular, in the BCS setting since you’re assuming that there’s a huge sea of electron pairs the “vacuum” consists of this huge sea. This explains how physicists can develop intuition for more general notions of vacuum: they’re not always studying the absolute vacuum, they’re also studying other systems with states that have the properties of being a “vacuum.” This particular vacuum has a delightfully strange property. Since a new electron pair doesn’t change the underlying vacuum, in this “world” electric charge isn’t preserved!