Jeffrey Morton - 2006-01
Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called “categorification”. Examples include the interpretation of N as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of N[x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s “species”, a new generalization called “stuff types”, and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these “stuff operators” can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic oscillator in which the group of phases, U(1), plays a special role. We describe a general notion of “M-Stuff Types” for any monoid M, and see that the case M = U(1) provides an interpretation of time evolution in the combinatorial setting, as well as other quantum mechanical features of the harmonic oscillator.
Who wants to talk or write about mathematical physics with me? I am primarily interested in the following ideas and in particular their potential interplay:
It became clear to me that there was no communication between what I was trying to say and [what] they were thinking. Bohr thought that I didn’t know the uncertainty principle, and was actually not doing quantum mechanics right either. He didn’t understand at all what I was saying. I got a terrible feeling of resignation. I said to myself, ‘I’ll just have to write it all down and publish it, so that they can read it and study it, because I know it’s right!’ That’s all there is to it.
Richard Feynman, reminiscing about the 1948 Pocono conference
I am not going to tell you that quantum mechanics is weird, bizarre, confusing, or alien. QM is counterintuitive, but that is a problem with your intuitions, not a problem with quantum mechanics. Quantum mechanics has been around for billions of years before the Sun coalesced from interstellar hydrogen. Quantum mechanics was here before you were, and if you have a problem with that, you are the one who needs to change. QM sure won’t. There are no surprising facts, only models that are surprised by facts; and if a model is surprised by the facts, it is no credit to that model.