The n-Category Café

The very basics.

While I want to do it the Kindergarten way, we need at least some basic idea about what a phase space in classical mechanics is, and which role the symplectic 2-form on that space plays in physics.

But really all the information we need is recalled in John’s lecture

Quantization and Cohomology (Week 6).

If you feel you need a little more details, just go back to

Quantization and Cohomology (Week 5).

By ignoring lots of more general cases which we could just as well take into account, but which would just distract us from a nice simple main point, we will be content with considering the following basic setup:

The configurations of some physical system form a space $$\mathrm{X} \,.$$ The configuration space. Clearly.

But a physical system weren’t a physical system if it would just sit there, pointlike as it is, and do nothing. Instead, physical systems want to undergo processes and move from here to there in the space of configurations.

A first simple approximation to the notion of a space of configurations $X$ equipped with a way to move from here to there is $$T X$$ the tangent bundle to $X$. This provides us for each configuration $x$ of the system with an inkling of what it would be like to evolve that system in time in various way: a tangent vector at $x$.

But some processes may be more likely than others. The dynamics of the system is encoded in a function which assigns a measure of likeliness of each trajectory in $X$ that a system traces out over time. Therefore, as soon as the system indeed does start processing in time, we assign a weight to that. This means we are really looking at the cotangent space $$P := T^* X \,.$$ That space is called phase space. For no good reason. But that’s what it’s called. (Okay, everybody feel free to teach me about the deep reason to call phase space phase space.)

By abstract magic, the very definition of the cotangent bundle indeed provides us with a canonical way to weight each process. This is given by the canonical 1-form $$\alpha$$ on $P$, which John describes in week 5, and which is best known under its local name, which reads $$\alpha = p_i \; d q^i \,.$$

So given any path $$\gamma : [0,1] \to P$$ in phase space, we can immediately assign a weight to it simply by integrating $\alpha$ over it. $$\gamma \mapsto \int_\gamma \alpha \,.$$

Around here, in the $n$-Café I suspect it doesn’t come as a surprise to anyone that assigning weights to paths in a manner which nicely respects composition of paths makes us want to think of parallel transport of a connection along that path.

(Okay, if you’re really just a Kindergarten graduate, then you should learn about how to color handrails before reading on. Or else, get used to reading stuff which you don’t understand. It’s never too early to develop that skill.)

In that context, we are tempted to turn the above weight into an exponential, $$\gamma \mapsto \mathrm{exp}(i \int_\gamma \alpha) \,.$$

Well, first I am trying to be so extraordinarily elementary here, and now I introduce an imaginary unit and an exponential for apparently no good reason!

That’s what it might look like. While I don’t feel awake enough to try to indicate this all the way down to the very roots of it all (that would involve mentioning $n$-transformations and their relation to holography, underlying all this), I feel that it will actually prove so very useful for understanding the gist of the Kontsevich-Cattaneo-Felder insight to think of those “weights” which we are assigning to paths as parallel transport of some connection, that I do want to adopt that point of view now. For a gentle introduction to this way of thinking go to

Quantization and Cohomology (Week 22).

So those who know how it works are kindly invited to think of our 1-form $\alpha$ (locally) as containing the contribution of a connection of a line bundle on configuration space $X$. All others simply ignore this.

What we shall not ignore is the Stokes theorem, otherwise known as the Fundamental theorem of calculus.

Suppose our path $\gamma$ happens to be a loop. Then we are entitled to compute the parallel transport of $\alpha$ along $\gamma$ $$\mathrm{tra}_\alpha(\gamma) = \exp(i \int_\gamma \alpha)$$ using – guess what –

a) a curious shift in dimension

b) together with a certain “topological” invariance in that higher dimension.

Yes, I am indeed just overselling the Kindergarten fact that for any disk $$\Sigma : D \to X$$ in $X$ which co-bounds our loop $\gamma$, we may compute the transport of $\alpha$ along $\gamma$ equivalently by computing the surface transport of the 2-form $$\omega = d \alpha$$ over $\Sigma$: $$\exp( i \int_\gamma \alpha) = \exp( i \int_\Sigma \omega) \,.$$

For those readers (anyone?) who learned how to do all this by painting doorknobs, handrails and walls: think of solving the task of painting a handrail that is tightly attached to a wall by painting the entire wall above the handrail, being sufficiently careless while doing so such that the handrail gets painted, too, in the process.

With all that out of the way, let’s finally pass on to the genuine point of what this here is supposed to be about.

A jump in dimension

With the ingredients mentioned above

- the configuration space $X$

- the corresponding phase space $T^* X$

- our canonical “weight” $\alpha$ on that

we can finally cook up the true dynamics of our system.

Let’s do some reverse engineering of something fancy to something obvious.

Suppose our system evolves for a while. That means we have a path $$\gamma : [0,1] \to P \,.$$ Suppose further that after time $1/3$ we want to check how the system is doing, using a probe we call $\hat F$.

Whatever that means (and in fact I will explain in a moment what it means), it makes us want to write $$\hat F(t= 1/3) \,.$$ When that is done, we let the system evolve a little more. But, nervous as we are, we check again how it is doing a little later. To keep track of that, we write $$\hat F(t= 1/3) \hat G(t = 2/3) \,.$$ After that is done, we wait still a little further. Then, let’s say, we check if the system happens to be in confuguration $x$. Somebody in the last millenium thought that a good way to note this down is to write $$\langle x | \hat F(t= 1/3) \hat G(t = 2/3) | x \rangle \,.$$ I could go on at this point about how this funny notation is in fact so very close to the right underlying $n$-categorical arrow theory, which I think is really at the very heart of the phenomenon to be unravelled here, but I did so elsewhere and don’t have the nerve to do so again here. Instead I just tell you what this funny notation is actually really suppposed to mean: When we write $$\langle x | \hat F(t= 1/3) \hat G(t = 2/3) | x \rangle$$ we are imagining we have a measure $d\mu$ which in this context people like to write $$[D\gamma]$$ on the space of maps $$\{[0,1] \to P\}$$ of sorts, and that we have functions $$F, G \in C^\infty(P)$$ which we integrate using this measure, weighted by our weight. So: $$\langle x | \hat F(t= 1/3) \hat G(t = 2/3) | x \rangle := \int_{\gamma(0) = \gamma(1) = x} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\gamma (\alpha - H d t) \,.$$

This is the infamous path integral.

I have sneaked in here a previously undefined symbol: $H$. The cognoscenti immediately recognize this old friend as the Hamiltonian. But all others may immediately forget that I even introduced that symbol – for the following reason:

I do want to consider the case where actually no time passes between our probe $$\hat F$$ and our probe $$\hat G$$ of our system. We simply probe it twice in a row, quickly, all at $x$.

Those who know what this means may imagine that I write a little limit sign in front of everything, which sends the parameter length of all my paths to zero. All others will hopefully gladly trust me that whatever that limiting procedure means, its effect will simply be to suppress precisely that term which I did not mention in the first place.

So then we are left with

$$\langle x | \hat F \hat G | x \rangle := \int_{\gamma(0) = \gamma(1) = x} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\gamma \alpha ) \,.$$

If you want to compare this expression with what Cattaneo and Felder have in their footnote on p. 5, you need only note that what I am writing using angular brackets (that’s Dirac’s notation) is what they write using a star (that’s traditional in deformation quantization), so $$\langle x | \hat F \hat G | x \rangle := f \star g (x) \,.$$

In any case, we should note three things:

- using our weight on physical processes, we can take the averaged value of the product of two functions on phase space, each evaluated at one point of this process

- while at the same time we assume that this “process” requires no time at all

- but still the order in which we evaluate $F$ and $G$ crucially matters, as the formula clearly indicates

(This does make sense and is the way nature works. If you find that part troubling, I cannot help much more right now except for offering the canonical Wikipedia entry: Quantum Mechanics)

Also notice that, since we are integrating over a space of maps whose domain is a 1-dimensional space, we say we are doing the “1-dimensional quantum field theory of the 1-particle”.

(That term makes strict intuitive sense here only if you imagine our configuration space $X$ to be the space of different ways a particle can sit in some spacetime. If otherwise you feel puzzled by the dimensions here, do make that assumption on $X$. If not, all the better: keep in mind that by the same kind of argument we should be able to say what I am going to say next for any $d$-dimensional field theory.)

But then – wait a second. Is that definition of dimension well defined? Is there any intrinsic way in which we could determine if any given expression is the integral over a space of paths?

No, there isn’t. Dimension of quantum field theory is not an intrinisic concept. In a way.

We already know this well from the classical differential geometry: there is no really intrinsic sense in which the parallel transport $$\exp(i \int_\gamma \alpha)$$ is “1-dimensional”. That’s because the value of this formula is just plain equal to one involving 2-dimensional integration $$\exp(i \int_\gamma \alpha) = \exp(i \int_\Sigma d \alpha) \,.$$ And that very tautological observation we now insert into our path integral – and turn it into a disk integral $$\begin{aligned} & \int_{\gamma(0) = \gamma(1) = x} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\gamma \alpha ) \\ &= \int_{\gamma(0) = \gamma(1) = x, \gamma = \partial\Sigma} [D\gamma] F(\gamma(1/3)) G(\gamma(2/3)) \exp(i \int_\Sigma \omega ) \end{aligned} \,.$$

Notice that I am not saying anything non-tautological here. In part, that’s the whole point. In other part it is of course due to the fact that I am evading some more subtle issues.

Like this one: we can easily imagine that instead of integrating over all paths and picking one cobounding disk for each path, as above, we integrate over the space of all disks in the first place – simply by dividing out by the overcounting introduced by the fact that there are many disks (quite a few, actually) with the same boundary.

While this is easily imagined, the main intellectual problem with quantum field theory is that of statements like that, involving integrals over huge spaces divided out by huge quotients.

Cattaneo and Felder give a very detailed discussion of how to make this as precise as the current level of understanding QFT allows. That means mainly: they use BV formalism.

And that’s of course where all the technicalities enter and where true glory can be earned.

So let’s pause for 30 seconds and commemorate all the trouble Cattaneo and Felder went through for figuring out how to deal with all this.

But then let’s allow ourselves to extract the basic mechanism, the underlying structure, all that which makes this here work and which is not a technicality. Imagine, if you like, that everything is taking place in the world of plain old finite cell complexes. Then everything technical evaporates. And we are left with a surprisingly simple and at the same time surprisingly powerful statement:

we can equivalently think of the product of two operators in quantum mechanics as being computed by a “path” integral over surfaces, hence as being computed by a 2-dimensional quantum field theory. $$f \star g (x) = \int [D \Sigma] f(\partial \Sigma(1/3)) g(\partial \Sigma(2/3)) \exp(i \int_\Sigma \omega) \,.$$

And the way this works is just

- apply Stokes’ theorem in the path integral.

How dare I bore you with such trivialities?

I did warn those not interested in Kindergarten quantum mechanics not to read this.

Anyone who did read up to this point but is feeling really annoyed but how trivial it all seems, I would be really interested in asking the following question:

Something to ponder. Characterize the relation between Stokes’ theorem in $n$-dimensions and transformations of $n$-functors. What does this imply for $n$-functorial extended QFT? Is there hence a rather high-brow generalization of the above Kindergarten formula?

It does pay to step back and think about what’s really going on here, I believe. The issue discussed at Making AdS/CFT precise is a case in point.

There needs to go more into this last paragraph here. But I am just way too tired.