Categorified Algebra and Quantum Mechanics

The process some call ‘categorification’ consists of interpreting set-theoretic structures in mathematics as derived from category-theoretic structures. Examples include the interpretation of as the Burnside rig of the category of finite sets with product and coproduct, and of [x] in terms the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s ‘combinatorial species’, and a new generalization called ‘stuff types’ described by Baez and Dolan, which are a special case of Kelly’s “clubs”. Operators between stuff types 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 describe a categorification of the quantum harmonic oscillator in which the group of “phases” that is, U(1), the circle group 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 recovering the usual Feynman rules for the quantum harmonic oscillator.