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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

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

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.

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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- Okay, +Edward O'Callaghan - you may regret asking! :-)

Personally I find that very few people can stomach long lists of definitions without lots of motivating examples and intuitive hand-waving mixed in. You list definitions of "epimorphism", "monomorphism", etcetera, but - so far at least - you don't say why these are interesting. Maybe you do later in the text... but few readers will wait for 'later'.

Of course, there will be readers who already know why they want to learn these definitions. But such people can look them up on Wikipedia, which actually explains them reasonably well, though one could do much better. So it seems to me the point of a textbook is to present definitions and theorems*as part of a story*: a kind of adventure story where each step is motivated by the previous ones, and seems natural and exciting.

I find Goldblatt's*Topoi: The Categorial Analysis of Logic*to be - despite its off-putting title - a gentle, friendly introdution to many of the concepts you list in your notes. (Not monads, though.) Experts despise it, but students love it because it provides nice examples that explain each new concept that's introduced.

If I wrote a book on category theory it would be a bit like that, but it would include a lot more example, including lots that*aren't*categories of mathematical structures and structure-preserving maps, but categories of physical systems and processes going between these systems. My papers "Physics, Topology, Logic and Computation: a Rosetta Stone" and "A Prehistory of n-Categorical Physics" can be seen as fragments of this book I'll never actually get around to writing.Jul 13, 2012 - +John Baez Thanks very much.

I don't regret asking, you are very right.

I do plan to give many motivating examples.

This was stage one, just to get the overall content down pat. It took me a little over a week to get it all down in LaTex and work out what I wanted to include.

This is not a book, more lecture notes. Perhaps I shall break them up with boomer or whatever that macro thing is called later. At least once I worked out good examples, which I think is actually the tricky part.

The actual motivation behind this was for people wanting to learn a functional programming language called Haskell.

Perhaps I should have made all that a bit more clear at the onset. However, it is all still a work in progress.

Many thanks for the feedback! Kind Regards, :)Jul 13, 2012 - Incidentally, this research, and in particular your n-category blog post about it, +John Baez, are being discussed over at MetaFilter (http://www.metafilter.com/118119/noncommutative-balls-in-boxes). There are some people over there who'd be interested in a more lay explanation if anyone feels like putting the work into giving one! (A small tangent: MetaFilter is where that link to the old ambient albums on Ubuweb came from.)Jul 21, 2012
- +Louigi Addario-Berry - thanks, I'll check that out. I'm not sure I'll have the energy to explain this stuff more simply, but it could be done very nicely with a video camera, me, a box, and a bunch of golf balls.Jul 21, 2012
- Hmm, sorry, there are too many people there who sound grumpy, complaining about me and 'these sciencey types'.Jul 21, 2012
- where you stand depends on where you sit in the domain 😊Feb 8, 2015