If you haven’t seen the paper yet, I highly recommend reading it:
A categorical framework for the quantum harmonic oscillator Authors: Jamie Vicary
Abstract: This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role. Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Surprisingly, generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.
I haven’t read a whole lot of it, and what I have read I don’t pretend to understand, but it is impressive to say the least.
6 June 2007 at 10:26 pm
It looks nice, but Jeff Morton (student of John Baez) categorified the QHO already.
7 June 2007 at 9:49 pm
No one actually dies when they do something beautiful, but impractical, in physics.
The human tendency to prize beauty over practicality is worse for us now than it ever was for the warriors of 400 years ago, and yet Miyamoto Musashi, in his classic text “The Book of Five Rings” (1645), wrote: “When we look at the world, we see the commercialization of arts. People use objects in order to sell their talents. As with the nut and the flower, the nut has become less important than the flower. In this manner, the Way of Strategy, both among teachers and among students, has become a show of technique, out of their desire to rush the blooming of the flower.”