week185

August 30, 2002

This Week's Finds in Mathematical Physics (Week 185)

John Baez

I'd like to continue the story of "q-mathematics" which I was telling you in "week183" and "week184". Sorry for the enormous pause - I was travelling around a bunch.

Let's see... where were we? We were talking about "q-deformation" - a method of systematically modifying vast tracts of math and physics by introducing a new parameter "q", in such a way that everything reduces to stuff you already knew when q = 1.

First we talked about the q-derivative:

              f(qx) - f(x)
              ------------
                 qx - x

and how we can reinvent mathematics by replacing the ordinary derivative with this gadget: modifying the commutation relations in quantum mechanics, replacing groups by quantum groups, and so on. I didn't say too much about this, but there's a lot to say. Here's a good place to get started:

1) Yu. I. Manin, Quantum Groups and Noncommutative Geometry, Les Publ. du Centre de Recherches Math., Universite de Montreal, Montreal, 1988.

Next we took an idiosyncratic detour into "q-arithmetic". We started with the q-integers:

[n] = 1 + q + ... + q^n-1

which show up first from the fact that the q-derivative of xⁿ is

               (qx)ⁿ - xⁿ
              ------------ = [n] x^n-1
                 qx - x

From these we built q-factorials and q-binomial coefficients, and saw that these functions arise naturally from "q-deforming" combinatorics. In ordinary combinatorics, you count structures on sets. In q-deformed combinatorics, you instead count structures on projective spaces over the field with q elements, F_q. All the formulas look the same, except that wherever you had integers, you need to carefully replace them by q-integers!

We also saw that by taking these formulas and setting q = -1 and q = 1, we can calculate the Euler characteristics of some projective varieties defined over the real and complex numbers.

So: certain bits of combinatorics, projective geometry over finite fields, and real or complex projective geometry are all somehow part of a unified theory. We can prove theorems simultaneously for all these subjects and then specialize to the case we want just by setting q to the right value. It's sort of like tuning to whatever radio station you want by turning the dial. It's even good to tune q to be complex, when we're studying quantum groups... but I don't feel like listening to those stations right now! Right now I want to ponder the basics a bit more. Like: what does all this have to do with quantum mechanics?

In "week183" I described how to q-deform the "Schroedinger representation" of a quantum particle on a line, in which its state is described by a wavefunction. The basic idea was to leave the position operator alone, but replace the derivative in the momentum operator by a q-derivative.

However, there's another way to describe a quantum particle on the line, called the "Fock representation". Here we have an abstract basis of states |0>, |1>, |2>, ... where secretly we think of |n> as the nth eigenstate of the harmonic oscillator Hamiltonian. There are annihilation and creation operators a and a* which push us up and down this ladder of states. We can describe these very efficiently if we think of states as being polynomials, with

|n> = xⁿ / n!

In these terms, the creation operator a* is just multiplication by x, while the annihilation operator a is differentiation. These satisfy

 a|n>  =        |n-1>
a*|n>  =  (n+1) |n+1>

so we get the commutation relations

a a* - a* a = 1

In case you're wondering, my conventions differ slightly from the usual ones, because my states |n> aren't normalized - but there's a good reason for this, which will become clear in due course.

We can define other operators starting from the annihilation and creation operators. First, there's the harmonic oscillator Hamiltonian:

H = a* a

As you can easily check for yourself, it has our nice basis of states as eigenvectors:

H|n> = n |n>

It's also called the "number operator", because its eigenvalue in the nth state is just n.

Next, we can define position and momentum operators Q and P by:

        (a + a*) 
Q =    ----------
         sqrt(2)


        (a - a*) 
P =    ----------
        sqrt(2) i

It's easy to check that they satisfy the same commutation relations

PQ - QP = -i

as in the Schrodinger representation. To get a full-blown isomorphism between the Fock and Schrodinger representations, we just need to map the state |0> to a wisely chosen Gaussian function on the line, and the rest falls into place....

But anyway, having already q-deformed the Schroedinger representation, let's q-deform the Fock representation. It's pretty simple: we leave the creation operator alone, but use the q-derivative as our annihilation operator! This gives the q-deformed commutation relations:

a a* - q a* a = 1

If we now define a basis of states by

|n> = xⁿ / [n]!

we get

 a|n>  =        |n-1>
a*|n>  =  [n+1] |n+1>

We can also define a q-deformed Hamiltonian by

H = a* a

and we get

H|n> = [n] |n>

so we could call this operator the "q-number operator".

We could march on like this, but now I want to take a quantum leap. If we "categorify" the ordinary Fock representation, we get the combinatorics of structures on finite sets. And if we categorify the q-deformed Fock representation, we get the combinatorics of structures on projective spaces over the field with q elements!

Let me explain....

Ordinary combinatorics counts structures on finite sets. It's fun to do this using "generating functions". To do this, suppose we have some type of structure that we can put on a finite set - like an ordering, or a partition, or a way of coloring the set, or making it into a graph of some sort, or whatever: anything we might want to count! Let's call this type of structure F, and let F_k stand for the set of all ways we can put this structure on a k-element set, and let |F_k| be the number of all ways we can put this structure on a k-element set. Then we can define a function |F| by

               |F_k| 
|F|(x) =   ∑  -----  x^k
                k!

This is called the "generating function" of F. Of course, the sum might not converge; it's really just a formal power series.

For example, suppose F is "2-colorings" - to put a structure like this on a finite set, we color each element either red or blue. There are 2^k ways to do this to a k-element set, so

|F_k| = 2^k

and thus

              2^k 
|F|(x) =  ∑  ----  x^k
              k!

       = exp(2x)

More generally, if F is "n-colorings", its generating function is

|F|(x) = exp(nx)

Here's another example that's even simpler. Suppose G is "being an n-element set". This is such a boring structure that you might never have thought about it. There's exactly one way to put this structure on an k-element set if k = n, and none if k is different from n, so

|G|(x) =  xⁿ / n!

You should recognize this function: a while back, I called it the nth eigenstate of the harmonic oscillator Hamiltonian! This is cool, because in physics we often think of this state as one in which there are n identical bosons present - for example, n photons. That's why the harmonic oscillator is also called the "number operator". Now we're seeing that this "n-particle state" is also the generating function of "being an n-element set". So the quantum mechanics of identical bosons may not be so weird after all.

The generating function

|F|(x) = exp(nx)

also corresponds to a famous state in Fock space, called a "coherent state". For example, a laser beam is a coherent state of photons. If you're curious about the details, see:

2) John Baez and Michael Weiss, Photons, schmotons, available at http://math.ucr.edu/home/baez/photon

But don't worry about it too much: my main point is just that it's fun to take types of structure, work out their generating functions, and think of these as states in Fock space.

To take this a step further, let's see how the creation and annihilation operators fit into the picture.

First, since these are linear operators, we should think about how addition fits into the picture! In quantum mechanics, adding states is called "superposition". But what about in combinatorics? What corresponds to adding generating functions?

It's very nice. Given two types of structure, say F and G, we can define a type of structure F+G by saying an F+G-structure on the set S consists of either an F-structure on S or a G-structure on S. This gives us

|F+G| = |F| + |G|

which justifies the notation F+G. It means we can think of F+G as a "superposition" of structure types. Of course you might complain that in quantum mechanics we can do more than add states: we can also multiply them by complex numbers! We can't do this with structure types; we can only multiply those by natural numbers, via repeated addition.

So the combinatorics of structures on finite sets is like a bare-bones version of quantum mechanics, without the complex numbers or even subtraction. You might think we're doing quantum mechanics over the natural numbers, and that's close - but we're actually doing quantum mechanics over the category of finite sets!

To make the idea of "categorified quantum mechanics" really precise, I'll need to jack up the math level a fair amount. This may be a bit scary, so I'll do it later in this article, after everyone has already stopped reading.

But now, what about the creation operator? Since this involves multiplication, I'd better tell you how to multiply structure types.

We can define a type of structure FG by saying an FG-structure on S consists of a way of chopping S into two disjoint subsets and putting an F-structure on the first subset and a G-structure on the second. If we make this definition, we get

|FG| = |F| |G|

I'll let you check this!

Now let's invent a creation operation A* on structure types that reduces to the usual creation operator a* when we take their generating functions. In other words, we want an operation A* with

|A*F| = a*|F|

The operator a* is multiplication by x, and we've seen that x is the generating function of the structure type "being a 1-element set". So if we call that structure type X, the operation

A*F = XF

does what we want.

But what is A*F really like?

Well, to put a structure of this type on a set S, we chop it into two parts, put an X-structure on one part, and put an F-structure on the other. So putting an A*F-structure on a set really just means picking a point from that set, removing it, and putting an F-structure on what's left!

This business about "removing a point" may sound more like annihilation than creation. But you can check that if you have an F-structure on a set with n elements, you get an A*F-structure on a set with n+1 elements. It's just like how you translate the function f(x) to the left one notch by forming the new function f(x+1). You might have thought that would translate the function to the right - but pushing points to the right pushes functions to the left.

So the creation operator really does push the particle number up by one. In particular, if we stretch our notation and let |n> stand for the structure type "being an n-element set", we get

A*|n> = (n+1) |n+1>

just like we should.

The annihilation operator for structure types is similar. Let's call it A. To put an AF-structure on the set S, we pick an extra point, say *, and put an F-structure on the disjoint union S U {*}. I'll let you check that with this definition,

|AF| = a|F|

and

A|n> = |n-1>

as desired.

The creation and annihilation operators are linear:

 A(F+G) =  AF + AG
A*(F+G) = A*F + A*G

where the equals sign is secretly an isomorphism... you see, we're categorifying! We also have an isomorphism

A A* = A* A + 1

which is just a categorified version of

a a* = a* a + 1,

cleverly rewritten to avoid subtraction. You should prove this yourself! If you get stuck, the answer is here:

3) John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Bjorn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. Also available as math.QA/0004133.

... along with lots of other stuff, like the inner product on our categorified Fock representation - and indeed, a categorification of the whole theory of Feynman diagrams. However, to describe these we need to go a bit beyond the concept of "structure type" and talk about "stuff types", which would be too much of a digression here.

At this point I should mention that the idea of categorifying the Fock representation was worked out by Jim Dolan and myself in a lengthy series of coffee-shop conversations. On the other hand, people have used generating functions in combinatorics for a long time. There are a lot of really fun things you can do with them! For a nice easy introduction, try this:

4) Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics: a Foundation for Computer Science, 2nd edition, Addison-Wesley, Reading, Massachusetts, 1994.

To dig deeper, try these:

5) Herbert Wilf, Generatingfunctionology, Academic Press, Boston, 1994. Also available for free at http://www.cis.upenn.edu/~wilf/

6) Richard P. Stanley, Enumerative Combinatorics, two volumes, Cambridge U. Press, Cambridge, 1999.

However, it was only in the 1980s that Andre Joyal gave a precise definition of a "structure type" - he called them "especes de structures", so English speakers often call them "species":

7) Andre Joyal, Une theorie combinatoire des series formelles, Adv. Math. 42 (1981), 1-82.

8) Andre Joyal, Foncteurs analytiques et especes de structures, in Combinatoire Enumerative, Springer Lecture Notes in Mathematics 1234, Springer, Berlin (1986), 126-159.

I also urge you to read this excellent book:

9) F. Bergeron, G. Labelle, and P. Leroux, Combinatorial species and tree-like structures, Cambridge, Cambridge U. Press, 1998.

But now let me get to the punchline. We can talk about structures not just on finite sets, but on projective spaces over the field with q elements, where q is any prime power. In "week184" I started trying to convince you that there is a very fruitful analogy between these. If V is a n-dimensional vector space over this field, and P(V) is the projective space consisting of all lines through the origin in V, we should think of P(V) as a q-deformed version of an n-element set. For example, the number of points in P(V) is the q-integer

[n] = 1 + q + ... + q^n-1

So, let F be any type of structure we can put on a projective space like this. Let F_k stand for the set of all ways we can put this structure on P(V) when V is our favorite k-dimensional vector space. Let |F_k| be the number of all ways we can do this. Then we can define the generating function |F| by

              |F_k| 
|F|(x) =  ∑  -----  x^k
              [k]!

Now there's a q-factorial in the denominator!

We can add structure types just as before, and get

|F+G| = |F| + |G|

However, we have to multiply them differently. To put an FG-structure on P(V), we pick a subspace U of V and put an F-structure on P(U) and a G-structure on P(V/U). With this sneaky definition we get

|FG| = |F| |G|

This only works because there's q-factorial in our definition of generating function!

Now for the new creation and annihilation operators. To put an A*F-structure on P(V), we pick a 1-dimensional subspace L in V and put an F-structure on P(V/L). To put an AF-structure on P(V) we take a 1-dimensional vector space L and put an F-structure on P(V+L). Note that these definitions are almost like the old ones! But now we get the q-deformed commutation relation:

A A* = q A* A + 1

The equation here is really an isomorphism.

If we let |n> be the structure of "being the projectivization of an n-dimensional vector space", we have

 A|n>  =        |n-1>
A*|n>  =  [n+1] |n+1>

We can also define a Hamiltonian by

H = A* A

and we get

H|n> = [n] |n>

where now the eigenvalues are q-integers.

In short, we've categorified the q-deformed Fock representation!

To wrap up, I'd like to make the underlying category theory in this story a bit more precise. I'm afraid I'll have to turn up the math level a notch now.

First, here's how Joyal's theory works. A "structure type" is really a functor

F: FinSet₀ → Set

where FinSet₀ is the groupoid of finite sets and bijections, and Set is the category of sets and functions.

So: if you feed F a finite set X it spits out F(X), the set of all structures on X of the given type. For example, if F is the structure type of "orderings", F(X) would be the set of all orderings of X.

But also: if you feed your structure type a bijection f: X → Y, it spits out a function F(f): F(X) → F(Y). This describes how we can transfer any structure on X to a structure on Y using the bijection f. For example, we can use our bijection to turn any ordering of X into an ordering of Y.

There is actually a category of structure types, where the objects are functors

F: FinSet₀ → Set

and the morphisms are natural transformations between these. I'll call this category Set[[x]], because it's really a categorification of the set of formal power series with natural number coefficients, N[[x]]. But I want to explain exactly what this means!

In Platonic heaven, there's an enormous chart showing how you can categorify all sorts of concepts. It starts out something like this:


   MATHEMATICS BASED ON SETS          MATHEMATICS BASED ON CATEGORIES

            sets                              categories
    functions between sets              functors between categories
 equations between functions       natural isomorphisms between functors

      elements of sets                   objects of categories
 equations between elements           isomorphisms between objects

... and it goes on forever. In particular, if you look further down this chart, you'll see that N appears in the left-hand column as the free commutative rig on no generators, Set appears in the right-hand column as the free symmetric 2-rig on no generators.

Huh?

A "rig" is a "ring but without negatives" - hence the missing letter n. More precisely, it's a set with two monoid structures, + and x, where + is commutative and x distributes over +. We call a rig "commutative" if the multiplication is also commutative. The most important rig of all is the natural numbers, since this is the free rig on no generators. It's also the free commutative rig on no generators.

There are actually different ways to categorify the concept of rig and get a notion of "2-rig", but one nice way is to define it as a category with colimits equipped with a monoidal structure that distributes over the colimits. Having colimits is like having addition; the monoidal structure is like multiplication. We call a 2-rig "symmetric" if the monoidal structure is symmetric; this is like being commutative. The most important 2-rig of all is the category Set, since this is the free 2-rig on no generators. It's also the free symmetric 2-rig on no generators.

The free commutative rig on one generator is N[x], the rig of polynomials in x with natural number coefficients. We need to do a kind of "completion" process, throwing in certain infinite sums, to get N[[x]], the rig of formal power series in x with natural number coefficients. The theory of 2-rigs allows infinite sums automatically, so the free symmetric 2-rig on one generator is called Set[[x]] - and this is the category of structure types! Addition and multiplication in this 2-rig turn out to work exactly as I've already described.

There's a lot more to say about this, but the interesting thing to me now is that when we q-deform Set[[x]], we get the category of structures on projective spaces over the field with q elements. And the really interesting part is that while this is a monoidal category, it's no longer symmetric. However, it's almost braided. Actually, Joyal and Street showed this in a related situation, namely where one considers not a set of structures on a projective space, but a complex vector space of structures:

10) Andre Joyal and Ross Street, The category of representations of the general linear groups over a finite field, Jour. Alg. 176 (1995), 908-945.

They even show that the braiding satisfies the Hecke relations, familiar from the theory of the quantum group SL_q(n)! This shows there's a really deep relationship between the q-deformation in the theory of quantum groups and the strange q-deformation I'm talking about here, where q is a power of a prime number. There are indeed other clues pointing to a relation of this sort, but this seems like the most fundamental one I've seen so far... and I'm trying to get to the bottom of things!

I hope the general picture is clear:


  q = 1                         q = a power of a prime number    

  finite sets                   projective spaces over F_q
  permutation groups S_n         projective special linear groups PSL(n,F_q)
  structure types               q-deformed structure types
  Fock representation           q-deformed Fock representation

We're thinking of the groupoid formed by the projective spaces and their symmetry groups PSL(n,F_q) as a q-deformed version of the groupoid formed by the finite sets and their symmetry groups S_n. The functors from these groupoids to Set are "structure types", and taking generating functions of these we get the Fock representation.

In a sense, all this relies on the analogy between the permutation groups S_n and the groups PSL(n). The groups PSL(n) have Dynkin diagrams like this:


                    o----o----o----o----o----o

and we call this series of Dynkin diagrams the "A" series. So, you should wonder if there is a grand generalization of everything I've said so far to other Dynkin diagrams. And the answer appears to be: yes!

I'll talk a bit about this next week.