Wow!  Jeff Morton and Jamie Vicary's paper is finally out! 

http://arxiv.org/abs/1207.2054

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.    

In quantum mechanics, position times momentum does not equal momentum times position!   This sounds weird, but it's connected to a very simple fact: there's one more way to put a ball into a box and then take one out, than to take one out and then put one in.  Huh?  Yes: if there are, say, 3 balls in the box to start with, there are 4 balls you can choose to take out after you've put one in, but only 3 before you put one in. 

In quantum mechanics, these operations of 'taking out' and 'putting in' are called annihilation and creation operators.  You can express position and momentum in terms of these.  So the fact that position and momentum 'don't commute' - position times momentum does not equal momentum times position - can be seen as coming from the fact that the annihilation and creation operators don't commute.

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.  Now is not the time to explain what that means - especially since I spent the last decade explaining it to anyone who would listen, and I'm sick of it now.  Suffice it to say that a famous mathematician named Mikhail Khovanov figured out a somewhat 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.  Jeff and Jamie have solved that puzzle.

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 saying precisely how position and momentum fail to commute:

pq - qp = -iℏ

where p is momentum, q is position, i is the square root of -1 and ℏ is Planck's constant.  So, these new equations should be important!  But they look very mysterious at first - you can see them below.  In fact, when I first saw them, it seemed as if Khovanov had just plucked them randomly from thin air (though of course he had not). 

Now Jeff and Jamie have shown how to get these equations just by thinking about balls in boxes.  They in fact make sense!  They are discussed in Section 2.8 of their paper.  Don't expect this to be readable without a lot of struggle if you don't have a PhD in math.  But at the bottom it's very simple.

I have no idea what the spinoffs will be, but understanding fundamental concepts more deeply can make things happen that seem utterly miraculous if you haven't been paying attention.
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