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

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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A) Put in a ball, take a ball out.
B) Take a ball out, put a ball in.

There are four ways to do operation A and six to do operation B, but only if you treat the balls as non-identical.
Thanks for the warning - I didn't look
+John Brockman - yes, I know it's confusing.  The reason they're called 'creation' and 'annihilation' operators is that should imagine not so much 'putting a ball in a box' as 'creating a ball out of thin air', and not so much 'taking a ball out of a box' as 'making a ball disappear'.

If we have 3 balls in a box, we can annihilate one (there are 3 choices) and then create a ball (just one choice: it doesn't make sense to ask 'which' ball is created, it's just appearing out of thin air).  That's 3 choices.

Or, we can create a ball (just one choice) and then annihilate one (now there are 4 choices).  That's 4 choices.

So, there's one more way to create and then annihilate, than to annihilate and create.

The math is sound, I just had trouble explaining it clearly in a single short paragraph.  I could try to fix that paragraph, but it's probably more helpful to let people wonder about it, read your comment, then read this, and then ask or say stuff.
do you deeply believe in this? is that something like believing in virtual time and assuming sqar root of (-1) real? and excuse me. my mathematics is not advanced
Yes, I deeply believe in this.  It's like deeply believing that 16 x 16 = 256  It's not at all obvious at first... but then you check it and find it's true, and it becomes very hard to doubt.

This is a math paper.   Math doesn't really concern itself with whether virtual time exists (maybe you mean imaginary time) or whether the square root of -1 is real.   Math is more concerned with taking precisely stated assumptions and deriving conclusions from them using precisely stated rules.  Of course the assumptions and rules need to be interesting for this process to be worth bothering with.  But I think that relating formulas in quantum mechanics to things that are easier to understand, like balls in boxes, is very much worthwhile.
This non-commutative multiplication of related operations is an expression of path dependence and shows itself in many forms;

- Between differentiation and integration
- The flip-flop memory of the digital electronics
- The hysteresis of magnetism

and seems to be a worthwhile topic to pursue.
Oh, I understand the math, John; I've a few degrees in chemistry and spent a number of years working with exactly that math.

But that's what you get for using a metaphor instead of math. You didn't say "create and annihilate a ball", you said "put in and take out".  :P
That's how I often think about it, since I don't care about the balls that aren't in the box - they might as well be gone.  It might be better to say "create and annihilate a ball", but I suspect a bunch of people would go "Huh?  Create a ball?   What is, this black magic?" and not read any further.
The creation and annihilation operations and the lifetimes of quanta of quantum mechanics seem to invite an entropy definition similar to the one you have been talking about in the life context. Maybe there is such a thing already.
+John Baez: I'm curious how momentum and position are expressible via creation/annihilation -- is it via the idea that moving particle can be seen as disappearing in one place and appearing in another? 
what i understand is that there are many compelicated not easy to explain and understand rules that have let the existance of the world we see. you use the example of the ball and box and they used the example of a boomerang to explain the forces between particles
+Dmitri Manin To answer your question in the most superficial way possible, in the quantum harmonic oscillator (for example) you can write the position and momentum operators as x = a + a† and p = i(a† - a) (to within some constant factors).  Algebraically, it has to do with factoring the Hamiltonian.  There's probably a deeper explanation of this, though.  The fact that the Hamiltonian can be expressed in terms of creation and annihilation operators is the really mysterious thing to me.  (For the QHO it's H = a†a + 1/2, the 1/2 being the zero-point energy.)
For me it's less mysterious that the Hamiltonian of the quantum harmonic oscillator (QHO) can be expressed in terms of creation and annihilation operators than it is for position and momentum.   From one point of view, the quantum harmonic oscillator describes a system consisting of a finite number of 'quanta' of energy.  These quanta are like the balls in our box.

The annihilation operator a annhilates one quantum, while the creation operator a† creates one.  Since there's one more way to create one and then annihilate one than the other way around, we have

a a† = a† a + 1

If we let |0> be the vacuum, meaning the state with no quanta, we have

a|0> = 0

meaning that there's no way to remove something that isn't there.  We can define a state with n quanta by starting with the vacuum and creating n quanta:

|n> = (a†)^n |0>

A little calculation using the three formulas I just wrote down gives

a† a |n> = n |n>

So, the operator a†a is called the number operator: it counts the number of quanta in a state.  If each quantum has the same energy and the state with no quanta has no energy, the operator for energy (called the 'Hamiltonian') will be proportional to this number operator.

All this is perfectly sensible and self-contained. 

The "+1/2" at the end of +Nathan Reed's comment is more tricky, but in many situations we can ignore it, because we can't really measure energies, only energy differences - so we can often get away with setting the energy of the vacuum to 0.

The fact that we can reinterpret the quantum harmonic oscillator as a little quantum rock hanging on a spring, and reinterpret the annihilation and creation operators in terms of the position q and momentum p of this rock using the formulas

q = (a + a†)/2


p = (a - a†)/2i

(up to some constant factors) is, I think, deeper.  Of course it's taught in every quantum mechanics course, but this is where the square root of minus one first shows up, and this is where the concept of 'space' (implicit in the concept of position) first shows up.... at least, starting from the end of the story where I started. 

So something rather interesting is happening here, which will continue to repay further study... though course many geniuses have already thought hard about this for about 75 years, so there's a lot that's already known.  One nice observation is that the formulas above say that q and p are like the 'real and imaginary parts' of the operator a.  And this ties in nicely with the fact that classically, we can unify the position and momentum of a harmonic oscillator into a single complex number, q+ip, which goes round and round in circles. 
+Dmitri Manin wrote: "I'm curious how momentum and position are expressible via creation/annihilation -- is it via the idea that moving particle can be seen as disappearing in one place and appearing in another?"

It's definitely not that, since I'm talking about momentum and position operators in quantum mechanics, not quantum field theory - so the annihilation and creation operators are not destroying and creating particles at a specific location in space, but rather, destroying 'quanta of energy' for the harmonic oscillator.    Why should destroying a quantum of energy plus creating a quantum of energy be the same as measuring your harmonic oscillator's position:

q = (a + a†)/2

(again, up to a constant factor)?  It works just fine, but I can't explain it in plain English, so I don't deeply understand it.
This looks like a bit of quite impressive mathematics. But I do not understand what is actually derived. I mean: what do eqs. 25-29 actually represent? Can they be related to some extension of the algebra of the simple harmonic oscillator? What is their possible use?
+Francesco Virotta - I wrote a more technical blog article about their paper here:

I think if you read that, you'll be more prepared to understand Jeff and Jamie's discussion of what those equations 25-29 actually mean.  In their paper, they explain the meaning of these equations in something like plain English.  They are fairly simple facts about annihilating and creating balls in boxes.

The applications of this will start out being mathematical - stuff about the representation theory of symmetric groups.  But if we understand it well enough, it should be saying something about the physical world.
This looks like extra hard core group algebra.... Now I recall why I gave up on theoretical physics and became an experimentalist :)
Respect for those who stuck with it! 
Im just beginning to look at this, and Im definitely planning on taking a closer look, but right off the bat Im struck that Ive seen exactly this kind of visual representation of quantum operators before. I wonder to what degree they overlap and differ:

I noticed he has one of your papers in the references, John, based on a quick glance.

Once again I find myself wishing I could upload an encyclopedia of category theory into my brain...
+Cliff Harvey - that paper by +Bob Coecke is a great way to start learning the category theory that underlies the use of Feynman diagrams in quantum physics.  I have a paper ( that might be the encyclopedia you want: it takes a historical approach to the development of category theory and physics, starting with the birth of special relativity and quantum mechanics, going through the work of Feynman, Eilenberg and MacLane in the 1940s, and on up to modern developments in string theory and loop quantum gravity. 

Khovanov's diagrams above certainly rely on the same basic idea as ordinary Feynman diagrams: morphisms in monoidal, braided and symmetric categories can be drawn as pictures in 2, 3, and 4 dimensions, respectively.  But he's using this idea in a higher-level way!  In a Feynman diagram, a "cap"

denotes a particle and an antiparticle colliding and turning into nothing.  In the diagrams above, a cap denotes a map that goes from the process of first creating and then annihilating a particle to the process of doing nothing! 

This map is part of a 'meta-level' that you don't see in ordinary Feynman diagrams.   I describe what's going on in a bit more detail here:
I wish we'd get more talkative mathematicians and physicists onto Google+.
+John Baez I have been reading your blog entry/looking again at the paper, and I think I'd like to understand what is going on in categorification / decategorification of QM. I  see that to really grasp it I need to refresh my algebra skills....  What I would like to understand is how the standard picture of QM fits in. How does the Hilbert space structure survive categorification, for instance?
In Khovanov's setup, the categorified vector spaces are called 2-vector spaces.  Just as a vector space is a set like C^n where C is the set of complex numbers, a 2-vector space is a category like Vect^n where Vect is the category of vector spaces.   So, his categorified Heisenberg algebra is, for starters, a 2-vector space (with n being infinite).  But it also has a 'multiplication', making it a kind of categorified algebra.

Just as some vector spaces are Hilbert spaces, some 2-vector spaces are 2-Hilbert spaces.  The paper introducing that concept is here:

but another good paper, which clarifies the idea further, is here:

Khovanov's categorified Heisenberg algebra should act on a categorified Fock space, which should be a 2-Hilbert space.  I think I see how this works, but I don't know if anyone has written it up! 
Thanks for the explanation. I will take a look at the papers and eventually come back with questions :) 
+Edward O'Callaghan - I think it's a wonderfully written paper.  I just didn't want crowds of people who don't know any math or quantum theory to try to read it and become disappointed.
+reza zademohamadi : the square roots of -1 (both of them) are of course not real - they're imaginary numbers :) But that doesn't stop them from being highly useful.
More seriously, are they able to use this to enumerate free and fixed polyminoes for any n?
+Douglas Summers-Stay - Fig. 54 in which paper?  And which "they" are you asking about in your last question?  If you mean Jeffrey Morton and Jamie Vicary, the math they're studying is a new twist on a very well-studied tradition (the study of so-called "symmetric groups", consisting of all permutations of a finite set).  This tradition is already widely used to study enumeration problems, and Jeffrey knows a lot about this, but if people haven't been able to use it to enumerate polyminoes yet (and I guess they haven't), it's unlikely that Morton and Vicary are going to be able to do so. 
+John Baez Yes, I meant Morton and Vicary's new paper, the topic of your post. The illustration shows how the (n+1)-ominoes can be derived from the n-ominoes by different paths. It was interesting to me because I had read about the problem in high school (probably in a Martin Gardner column) and thought about it a bit then. 
Oh, okay.  I hadn't even realized their paper had 54 figures it.  Looking at it, I see they're not discussing polynomioes in general, but just certain very special ones called 'Young diagrams':

A Young diagram is a finite collection of square boxes, arranged in left-justified rows, with each row having the same or shorter length than all the rows above.

These diagrams are hugely important throughout mathematics - I wrote an nLab article where I listed about 10 different ways they show up.   I could talk to you about them for weeks, literally.  But I won't. 

Morton and Vicary aren't doing anything really new with Young diagrams, instead, they're giving a new explanation of a known fact, namely that Young diagrams form a basis of Khovanov's 'categorified Heisenberg algebra'.
+John Baez By the way, A bit off topic however. What did you think of my category theory notes I wrote? If you have a few minutes to check them out, I would love some feedback from someone who is much more qualified such as your good self.
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.
+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, :)
Incidentally, this research, and in particular your n-category blog post about it, +John Baez, are being discussed over at MetaFilter ( 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.) 
+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.
Hmm, sorry, there are too many people there who sound grumpy, complaining about me and 'these sciencey types'.
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