This Week's Finds 18: categorifying the quantum harmonic oscillator

Classically, light in a mirrored box can be described as a collection of harmonic oscillators, one for each vibrational mode of the light. Planck ‘quantized’ the electromagnetic field by assuming that energy of each oscillator could only take on discrete, evenly spaced values. Later Einstein took this seriously, and realized that light comes in discrete energy packets called 'quanta'. Surprisingly, when we categorify the mathematics describing this situation we are led to the theory of 'species' - one of the basic tools of combinatorics. A species is any type of structure we can put on finite sets. The commutation relations between annihilation and creation operators, and the inner product on the Hilbert space of a quantum harmonic oscillator, then receive a natural interpretation in terms of species.
This is my last lecture on combinatorics and categorification, loosely following this paper:
John Baez and James Dolan, From finite sets to Feynman diagrams, arxiv.org/abs/m...
And here's some more reading material - free books:
François Bergeron, Gilbert Labelle, and Pierre Leroux, Introduction to the Theory of Species of Structures, bergeron.math.u...
Phillipe Flajolet and Robert Sedgewick, Analytic Combinatorics,
algo.inria.fr/f...
Herbert S. Wilf, Generatingfunctionology, www.math.upenn...
This is one of a series of lectures at the University of Edinburgh on topics drawn from my column This Week's Finds. This is the 6th lecture of 2023. For other lectures go here:
math.ucr.edu/ho...
The cover image was created by AllenMcC. and placed on Wikicommons here:
upload.wikimed...