So I gave a little talk shortly before leaving London for Christmas. I had mostly written it up, but then I’ve been on the road for a while in Montreal, Ottawa, and Calgary, without consistent net access. However, now I have a moment to put this up.

The talk carried on from the previous one I described last post. It began to move in the direction of representation theory of 2-groups on 2-vector spaces and 2-Hilbert spaces, but didn’t get that far. This was partly because I had to finish describing what 2-linear maps and 2-maps look like for such spaces, and then because I had to explain about 2-groups and give some examples. I’ll say more about the representation-theory stuff in January. But here I’ll just summarize at least the rest of the description of the category $Meas$, and also $2Hilb$ by describing 2-linear maps and so forth. Then I’ll comment a little more philosophically about what these are about.

So I explain how there’s a 2-vector space (in some suitable sense, not the KV sense) of measurable fields of Hilbert spaces on a space $X$, analogous to the vector space of complex functions on a space. Also similarly, given a measure on $X$, we get an inner product. Then there’s a (2-)Hilbert space where this inner product is always well-defined (as a complex scalar, or a genuine Hilbert space – which is the equivalent of a scalar at the next level up).

Well, then Crane and Yetter’s paper describes constructively how to get 2-linear maps (additive, linear functors) between such 2-vector spaces. They don’t as far as I can see, show that all functors arise this way, but it seems likely. The way is to say you get a functor $T: Meas(X) \rightarrow Meas(Y)$ from:

1) A measurable field of Hilbert spaces $T \in Meas(X \times Y)$ (this is similar to the linear maps between KV 2-vector spaces, which are like matrices of vector spaces)

2) A $Y$-indexed family of measures $d \mu_y (x)$ on $X$ – these give you the measures you need to do the “inner product” involved in “matrix multiplication” at each $y \in Y$ (note that this stuff is only well-defined up to sets of measure 0, as usual). So we have, on objects:

$(T \mathcal{H})_y = \int^{\oplus}_X d \mu_y(x) T_{(x,y)} \otimes \mathcal{H}_x$

and a related expression for morphisms, using the identity on $T_{(x,y)}$.

It’s probably worth pointing out that the measures on $X$ are used in the direct integral here, and so their only real role is to define the inner product on $(T\mathcal{H})_y$ – the underlying vector space at each point in the new field would be the same no matter what these measures were (up to the fact that if the resulting inner product is degenerate, we need a quotient space where it’s not).

So this gives 2-linear maps, which are functors. Natural transformation between these functors come from the fact that $Meas(X \times Y)$ is itself a category, and in fact a 2-vector space in the sense we’re using here ($Meas$ is “enriched over itself”). So morphisms between these fields of Hilbert spaces basically amount to fields of bounded operators as usual. This is actually not quite right, because we need to account for the different measures: basically, you use a measure which is the geometric mean of those associated to source and target – check out Crane and Yetter’s paper if you want the details.

That finishes up a summary of how 2-Hilbert spaces work. The next thing I’ll be talking to our group about is how to use these for a categorified form of representation theory.

But first, what is the point of all this stuff? Not yet asking about representation theory in this setting – why is it interesting enough to bother? It’s worth thinking about what a categorification of a Hilbert space is supposed to be. In particular, let’s try locating them in the world of quantum mechanics.

A quantum system is usally portrayed as having states represented by vectors in a Hilbert space. The only things you can “do” to states involve applying operators to the whole space: project them into subspaces, “rotating” them by some unitary evolution operator, and so on. In a 2-Hilbert space, states, or “2-vectors” are objects in a category, which means there are not only these “macro” operations on the whole space, but also morphisms between any two states you pick. In fact, this is the source of the inner product on a 2-Hilbert space – there is a Hilbert space (in the usual sense) of morphisms between any two states, and in the world of 2-Hilbert spaces, this is the equivalent of a scalar.

In QM, the inner product $\langle x , y \rangle$ is telling you an amplitude to observe a system in state $y$ if it was set up in state $x$ – this is saying something about “how related” $x$ is to $y$. The categorifed picture saying this is just $hom(x,y)$ makes more explicit what kind of relationship this is.

Now, if you happen to pick the same vector to start and end with, considering $\langle x , x \rangle = hom(x,x)$, what this is saying is that there’s some bunch of “symmetry operations” on a state. (Taking just the invertible ones gives an actual symmetry group for a given state.) This is saying that “state 2-vectors” have some internal degrees of freedom. Their amplitudes give a measure of how many such degrees of freedom there are.

The fact that a 2-Hilbert space is described as an enriched category means that the usual picture of a quantum system returns when you look in individual components of a state 2-vector. In particular, the coefficients of a 2-state vector can be thought of as Hilbert spaces representing a system in that particular component. So, for instance, part of the big project I’m describing in these notes is to depict quantum gravity (at least in 3 dimensions) as an extended TQFT, which represents a physical system with these 2-Hilbert spaces. A 2-state vector here describes the situation on a boundary of space – matrix elements of a 2-linear map are Hilbert spaces of connections on a given manifold interpolating between chosen boundary states. Natural transformations between 2-linear maps are what give amplitudes for spacetimes joining such slices of space.

So what is a state 2-vector? All these properties should fit together into some nice scheme: classical configurations can exist in a “2-state” in some kind of superposition, where each configuration gets its own internal degrees of freedom. The inner product emerges naturally from this, considering morphisms between 2-states. Every morphism between 2-states has to respect the classical configurations, giving for each one a map between the internal spaces associated to it in the two 2-states. Is there a more elegant way to sum this up? Probably so, but at the moment I don’t quite see how to put it.

However: next time, I’ll carry on with some representation theory.