A fundamental fact about quantum theory is that these operators fail to commute: aa* – a*a = 1 in units where Planck’s constant is 1. Remarkably, this fact has a simple combinatorial interpretation: there is one more way to add an element to a finite set and then remove one, than to remove one and then add one.

— Categorifying Fundamental Physics, John Baez

2010.01.06 Wednesday ACHK