noncommutative balls in boxes
July 21, 2012 9:42 AM Subscribe
Morton and Vicary on the Categorified Heisenberg Algebra - "In quantum mechanics, position times momentum does not equal momentum times position! This sounds weird, but it's connected to a very simple fact. Suppose you have a box with some balls in it, and you have the magical ability to create and annihilate balls. Then there's one more way to create a ball and then annihilate one, than to annihilate one and then create one. Huh? Yes: if there are, say, 3 balls in the box to start with, there are 4 balls you can choose to annihilate after you've created one but only 3 before you create one..."
This insight, that funny facts about quantum mechanics are related to simple facts about balls in boxes, allows us to 'categorify' a chunk of quantum mechanics—including the Heisenberg algebra, which is the algebra generated by the position and momentum operators. Now is not the time to explain what that means. The important thing is that Mikhail Khovanov figured out a seemingly quite different way to categorify the same chunk of quantum mechanics, so there was a big puzzle about how his approach relates to the one I just described. Jeffrey Morton and Jamie Vicary have solved that puzzle.btw John Baez sez: "Don't even look at it unless you're an expert in mathematical physics, but it's important - and I think someday it could change our understanding of quantum mechanics." :P (I didn't; I know next to nothing about this stuff, but I found it interesting so I thought I'd share ;)
The big spinoff is this. Khovanov's approach showed that in categorified quantum mechanics, a bunch of new equations are true! These equations are 'higher analogues' of Heisenberg's famous formula
pq − qp = −i
So, these new equations should be important! ... Now Jeffrey and Jamie have shown how to get these equations just by thinking about balls in boxes... The important thing is this: those equations are not something we get to choose, or make up. They are what they are, and they're just sitting there waiting for us to discover them.
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