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!

In my opinion, “physical” and “geometric” are at times rather pretentious descriptions of arguments that might not add much to the explanation. In particular, so-called “physical” arguments and models include many quantum field theories, such as Chern-Simons on a closed 3-manifold, that are not realistic. Yes, Chern-Simons for braids is thought to arise as a sector of certain condensed-matter system, but it is a struggle to make these systems to witness anything.

What is really going on is that quantum field theory is a semi-separate island of reasoning that is knit together by consistency checks. The consistency checks actually are rigorous, but quantum field theory as a whole is missing a rigorous grounding in the usual definitions and axioms of mathematics. There is by now hundreds of tons of evidence that quantum field theory could be assimilated into rigorous mathematics in principle, but it is a sprawling program to do so in practice.

In fact, the problem is more the fields than the quantum. Stochastic field theory, based on classical probability rather than quantum probability, already shows some of the problems (with path integrals) as quantum field theory. On the other hand quantum probability itself is entirely rigorous, even thought it can be hard to believe that quantum probability is realistic.

Anyway, when “physical intuition” for quantum field theory means anything tangible that mathematicians don’t have, it mainly means long experience with the consistency checks of quantum field theory, in lieu of rigorous definitions. For realistic quantum field theories, the experimental checks are great too, but they are an optional side of the story for us mathematicians. The consistency checks are the mandatory side.

A slight digression into philosophy…

If you are some sort of neo-Kantian as I am, then you think of physics not as the study of a real world, but rather as the study of ideas we(*) have which are useful in describing and predicting some of our perceptions. Mathematics is then something like the study of what systems of ideas are coherent, with particular attention to some systems of ideas which have been historically studied. It’s not surprising then that looking carefully at how we think about certain particular perceptions helps us look at how we can think more generally, especially when our notion of ‘coherence’ has many historical links to physics.

(*) I’m being deliberately vague about who is included in ‘we’.

I wonder to what extent you really do need multiple instances in order to at least start developing intuition for the general case. I certainly agree that it’s very helpful, and probably in most cases essential, but it’s not obvious that it always is. After all, even if you’ve just got one example of a type of system, you might still be able to get some sense of which features of it are essential, and which are special features due to the nature of this particular example. Sure you’ll be wrong about some cases, but that’s normally the way things work with intuition.

Of course, there are some other ways of thinking about intuition. In the tradition going back to Kant (I’m not sure if Alex is adopting all of this picture or only part of it), the idea of intuition is not that it’s something you develop for some external system, but rather than it’s a sort of pre-condition for thought about that sort of system at all. I believe Kant’s idea is that geometry (and in particular, Euclidean geometry) is a pre-condition for any sort of thought involving vision and probably other senses, and that arithmetic is a pre-condition for any sort of thought about time, and that the notions of cause and effect are pre-conditions for any sort of thought about a physical world. Thus, all these things are actually knowable

a priori– we don’t even need to have encountered one example in order to have these intuitions.(He further went on to argue that these truths are

synthetic, meaning that they can’t be derived just by analyzing the concepts involved. I think most people sympathetic to any version of this picture these days think of mathematics at least as a set ofanalytictruths, because the notion of the synthetic a priori is a bit mysterious.)But in general, the notion of intuition is a bit mysterious, and it’s definitely hard to see how intuition not shaped by experience (or by some sort of evolutionary or other teleological process) could possibly be reliable. But it still does seem plausible that some intuition at least can reliably be built from considering even just a single example, at least as long as one is paying attention to what features of this example seem like they might be special cases and which features might not be.