The n-Category Café

remember their reaction to the gruppenpest

That’s precisely the example I mentioned in the discussion session.

I said, once upon a time, some mathematicians tried to point out that physics depends all over the place on one-object categories with invertible morphisms.

There was lots of opposition to this idea, among physicists, back then. Things have changed since then. If today you have a question about $E_8$, chances are that you find more answers in your physics theory group than in your math department.

What is it that made the difference?

That mathematicians published papers which conjectured that one-object categories with invertible morphsims (aka groups) might solve some far-out fancy almost unthinkable problems one might imagine on the horizon of physical research?

No. The difference came when it became clear that ignoring one-object categories with invertible morphisms in your physics research meant that you would be left behind soon, because for your those of your colleagues who already did accept it, the dust suddenly settled and they could see the light and progress.

The same will be true for categories with more than one object and potentitally non-invertible morphisms, eventually.

But it takes time to read a book on group theory. While physicist A was locked away reading books on group theory, physicist B kept cranking out papers on physics. At the end of the year physicist A was beginning to have a vision of the theory of the next century (eventually called quantum mechanics) while phyicist B had a vision of tenure.

While that period lasts, it’s hard. But towards the end of that period, suddenly things change.

Jamie Vicarie mentioned

quantum gravity

Ah, now I see where you are coming from.

I am slightly taken aback by how you want to jump from the harmonic oscillator right to quantum gravity, with a certain disdain for “ordinary” quantum field theory, but I have seen that happen before… ;-)

But okay, let’s discuss this, this will be getting interesting.

Right this moment I don’t at all have time for a detailed response, but two quick remarks, with more details later:

a) yes, the “quantum propagation $n$-transport” for $n$-dimensional gravity will be a functor not on $1Cob_{Riem}$ but on plain $n\mathrm{Cob}$. Or rather on an $n$-categorical refinement of that.

b) second quantization will be a procedure that reads in quantum $n$-transport of $n$-particles ($n=1$: points, $n=2$: string, $n=3$: membrane, etc.) propagating on $m$-dimensional target space, and spits out the quantom, $m$-transport of the corresponding second quantized field theory on target space ($n=1$: ordinary field theory, $n=2$: string field theory, etc.)

c) $1Cob_{Riem}$ is just a toy model to get started (well, and to make contact with standard non-relativistic textbook quantum mechanics).

More later.

We can even say what the “coherent states” are and everything. Cool.

I think the coolest thing is being able to define exponentials fully abstractly: starting with some endomorphism $f:A \rightarrow A$, and constructing an endomorphism $e ^f: A \rightarrow A$. This all satisfies $e ^0 = \mathrm{id} _{A}$ and $e ^f e^g = e ^{f+g}$ if $f\circ g = g \circ f$, just like you’d expect.

When I worked this out it was really a “who ordered this?” moment. I still don’t really know why they’re there…

Hi Jamie,

thanks for your reply! I agree with what you say. Here are some comments.

everybody else (i.e. Jeffrey Morton, John Baez) starts saying “quantum harmonic oscillator” whenever they get a whiff of the CCRs

To be precise, one should not really talk about the “oscillator” unless and until one has something satisfying the CCRs and then “equips” that something with the corresponding number operator (the “Hamiltonian”).

The point is that the space of states of the particle on the line in any potential is obtained by acting with raising operators on the state that would have been the vacuum state for the particle in the square potential. All these inf. dimensional Hilbert spaces are isomorphic, after all!

So it’s the Hamiltonian that makes a difference. And this relates to that other point I made above, by the way: most of what was said at the London conference about QM was purely kinematical. The dynamics was missing.

The dynamics gets into the game once we realize that the monoidal categories we are talking about (as well as their $C^*$-like algebras of (bounded) endomorphisms of their objects) are just the codomain of a propagation functor. On objects, that functor encodes the kinematics. On morphisms, it encodes the dynamics.

So, if you ask me: what is “the harmonic oscillator”, abstractly? When can I rightly say that my abstract categorical stuff is about generalized harmonic oscillators?

Then my answer would be:

precisely if you are looking at a functor $$QM : 1dCob_{Riem} \to C \,,$$ where $C$ is your monoidal category of choice such that it supports CCR algebra on its objects in the sense you nicely describe in your work, and where the morphism in $C$ associated by the functor $QM$ to the 1-dimensional Riemannian manifold of length $t$ is the endomorphism $$\exp( - t N ) \,,$$ where $N$ is the number operator on the object in $C$ which $QM$ assigns to the siungle 0-dimensional Riemannian manifold (the point) in your context.

$$QM_{\mathrm{oscillator}} : (\bullet \stackrel{t}{\to} \bullet) \mapsto H \stackrel{\exp(-t N)}{\to} H$$

(Strictly speaking, that would be the “euclidean” harmonic oscillator, unless you can somehow manage to make sense in your categorical framework of what it would mean to put a factor that behaves like the square root of -1 into the above exponent. But euclidean is fine for many purposes. Even though one might express the hope that a deep abstract understanding of QM will in the end also help explain what that business with complex numbers in QM really is about, and maybe make it arise automatically from some deeper reasoning. We have talked about that a couple of times here on the blog.)

a beautiful fact about constrained single-particle quantum mechanics

I think you mean to say: bound instead of constrained. Right? Systems on the line whose potential goes to $+ \infty$ to both sides?

(Constrained systems are those where we have a Hilbert space of states such that not every vector in that Hilbert space has an interpretation as a physical state of the system. They, and their abstract categorical formilation, are extremely important, but I don’t think that’s what you are looking at.)

difficult to motivate the definition of the Hamiltonian

[…]

(Does anybody have any ideas?)

This, I think, cannot (and has no right to) be solved by just looking at the internalized CCR structure.

You are asking the question: “Why does that particular quantum propagation functor come to us?”

Pretty much generally the answer will be: “Because it arises as the quantization of some classical transport functor.”

(Remove the term “transport functor” from these two statements if you don’t like them, and read “dynamics” in both cases.)

So then the question is: what’s the thing about classical transport/dynamics that makes the harmonic oscillator appear so prominently?

I guess I could give something like an answer to that. But it won’t be really short. It comes down to understanding why free field theory Lagrangians like to be of the form $$\phi \mapsto (\nabla \phi)^2 - m^2 \phi^2$$ for “fields” $\phi : X \to \mathbb{R}$ of “mass” $m$.

It’s that $\phi^2$ term here which makes the oscillator appear.

Why is that square term singled out, though?

The best answer we have to that (which I think is actually the final answer already) is

a) fermions

b) Higgs mechanism.

(I really do think we need to go all that way, but I can sense readers thinking: “now he is overdoing it”.)

Together this says: we see square potentials all around us in field theory, because they come from bilinear fermion pairings $$\langle \psi , \Phi \psi\rangle \,,$$ where the endomorphism-valued field $\Phi$ (the Higgs field) is assumed to be constant.

If that sounds far-fetched from the internal category-theoretical point from which we started out looking at the world around us here, it should be emphasized that there is, in turn, a beautiful abstract way to explain this: that’s Alain Connes’ spectral action way of understanding things.

With that in hand, we get full circle back to the transport functor point of view:

we learn: quantum propagation is not just a functor on 1-dimensional Riemannian cobordisms, it is really thought of as a functor on 1-dimensional super-Riemannian cobordisms:

$$QM : 1dCob_{superRiem} \to C$$

If that functor is smooth (as it should be), then such functors are no longer characterized just by a Hamiltonian, but by a Dirac operator, a spectral triple in fact.

(Have a look at the first part of Stolz-Teichner’s What is an elliptic object? the here relevant aspects of which I review towards the end of Seminar on 2-Vector Bundles and Elliptic Cohomology, V).

Once we accept that 1-particles should be taken to be superparticles (which for 1-particles just means that they are spinning particles, which indeed our particles are) then we see that Connes’ spectral action is the natural action for the second quantized version, and there we see that it’s the fact that fermions want to pair in bilinears which gives rise to the Higgs effect and the appearance of quadratic potentials all over the place.

This sketches, I think, one big aspect of the full story which eventually needs to be told in detail.

There’s also the fact that I want mathematicians and computer scientists to be interested in my paper, and ‘second quantisation’ perhaps sounds a bit more esoteric and technical than ‘the quantum harmonic oscillator’. But this is a small point.

I don’t know about computer scientists, but I imagine that mathematicians will be rather pleased to learn (if they hadn’t heard about it before) that “second quantization is a functor”, and that this functor even arises as naturally and abstractly as you show it does.

Jamie wrote:

…everybody else (i.e. Jeffrey Morton, John Baez) starts saying “quantum harmonic oscillator” whenever they get a whiff of the CCRs

Just to be clear: Jeffrey Morton has given a specific prescription for categorifying not only the canonical commutation relations and the harmonic oscillator Hamiltonian, but also a large class of perturbed harmonic oscillator Hamiltonians and the time evolution operators these give rise to. He expresses these time evolution operators as a sum of Feynman diagrams in the standard way, but categorified.

So, he’s gotten quite a bit more when a ‘whiff’ of the quantum harmonic oscillator. Indeed, he’s categorified enough of the usual techniques of perturbative quantum field theory that I think the way is clear to march down this road and do much more. And, there’s a nice relation to the ‘generating functions’ familiar in combinatorics, since the categorified version of the Fock space for an oscillator with $n$ degrees of freedom is the groupoid of $n$-tuples of finite sets.

Right now in the seminar we’re starting to:

1. review this material,
2. integrate it with our new improved understanding of groupoidification,
3. $q$-deform a bunch of this stuff, replacing the groupoid of finite sets by the groupoid of finite-dimensional vector spaces over $F_q$.
4. show how the resulting math is related to the Hecke and Hall algebras associated to the $A_n$ Dynkin diagrams.

(I know you were trying to defend your use of the term ‘harmonic oscillator’, not attack ours. But, you did it in a self-and-other-deprecatory British style that Americans don’t understand. Also, you should know that a good thesis advisor defends its students as a mother bear defends her cubs!)