The n-Category Café

Ah, you mean that maybe this puzzle about how to think of classical mechanics properly as a deformed-rig version of quantum mechanics becomes maybe clearer once we do take care of the “categorical refinement” also in the classical case?

Could be. Should be, in fact. If the conjecture is true. But I don’t know.

“classical mechanics should be cast at the same category theoretic level as quantum mechanics” –

this seems as plausible, without detailed construction, as –

“Newtonian mechanics should be cast at the same category theoretic level as (Special/General) Relativity.”

That is, the normative “should be” as a goal for n-categorification rather than a claim about the Physics of the cosmos we happen to inhabit?

Narn i Groupoidification

HAH!

Do you think I have the grammar about right?

I bought the Narn recently when I missed a train, at the railway station. I made the interesting observation that it had the greatest impact on me when I read it after thinking about technical things all day. Then I found the language alone really moving.

This effect was less pronounced after a leisurely spent day of a weekend.

Maybe that’s a reason why it’s a geeky thing. It acts as a counterbalance of something, somehow. :-)

Why do objects internal to nCat require mathematically ill defined renormalization for their proper definition?

While renormalization need not be ill defined (see Connes-Kreimer) it is true that quantization tends to be a subtle and difficult step.

Hence one might wonder:

How is that non-straightforwardness reflected in the category-theoretic formulation we are discussing here?

This has a nice answer:

the action functor which we are talking about can naturally be pulled back. But what we need for quantization is its push forward.

This is an operation defined only indirectly, by means of a property.

For one, the push-forward here need not exist. If it exists, it may be hard to find.

There will, in general, not be a strightforward prescription to compute such a push-forward. The best one can in general hope for is a direct prescription to check whether a given guess satisfies all the properties.

So when I say that “quantization is a push-forward” (slightly simplifying, by the way, since it is really a combination of pullbacks and push forwards) this doesn’t mean that, if true, all problems in quantum field theory become trivial instantly.

Perhaps it is a bit like the group $String_G$ - if we try to construct it as an object in a category, it is defined only up to homotopy. But if we define it as an object in a 2-category, it is (Frechet-) smooth.

Would this change in viewpoint be more or less natural formulated in a “discrete” spacetime, e.g. an n-diamond?

Yes. In fact, that’s currently part of the best evidence I have that the above conjecture is actually true:

In

QFT of the Qharged $n$-Particle: The canonical 1-particle

I try to work out the push forward which we are talking about for the case of an ordinary charged particle 1-particle which propagates on a 1-dimensional space, which is modeled by a discrete lattice of points $\simeq \mathbb{Z}$.

The main result there is that, indeed, by going throught the pull-push quantization prescription, we do indeed obtain the (discretized version of) the (euclidean) quantum propagator $$t \mapsto \exp(- t \Delta) \,,$$ where $\Delta$ is the covariant Laplace operator in one dimension.

This may have sensible continuum analogs, but in the discretized setup it is an elementary computation.

Hi Urs,

I know my gushing is probably getting old to some people, but I think you are doing some fantastic stuff. My gut reaction after reading this is that you have essentially found a nice graphical way to teach quantum mechanics purely with pictures and no need at all to resort to formulas. In this regard, you could almost get away with teaching this to high school students (or earlier) who can draw directed graphs.

That was one of the very cool things that comes out of elementary electromagnetic theory via differential forms on a “discretization” of spacetime, i.e. it becomes trivial to explain Maxwell’s equations without ever writing down a formula simply be looking at stuff flowing out of the boundary of a 4-dimensional domain (maybe projected down to 2+1-dimensions for drawing purposes :)).

I am so insanely busy with work (and can’t forget family) that I haven’t been able to sit down and learn the arrow theoretic stuff yet, but I am looking forward to calmer days when I finally can.

One unfortunate thing about all this is that you are just too smart. Your brain operates on a level so far beyond mere mortals that you’ve, to a certain extent, lost touch with mere mortal’s (like mine) ability to understand stuff.

What you need is a Rosetta stone for arrow theory. I’ve grown to trust my gut to a certain extent and my gut tells me that what you are doing is not so insanely unreachable for mere mortals and, at its heart, it is probably quite simple once you break through the linguistic barrier.

Simple yet powerful. If I understand, by turning a fairly simple crank on some, almost laughably trivial, structures you end up with an arrow theoretic (or picture theoretic) description of simple quantum mechanics (0-quantization?). Turning the crank once again gives an elegant/simple desctiption of quantum field theory (1-quantization?). Iterating the process once more gives second quantization.

I’m sure I’ve got the details muddled, but it is probably not completely wrong.

Beautiful!

Then, moving along a different axis of the cube you end up with quantum strings, quantum field of strings, and second quantized string field theory.

Sounds like what you were after from the beginning to me :)

So when is the book coming out?

By the way, when the history books are written can you give my grandkids (who haven’t been born yet) a gift and mention my name ;)

Eric

Another way to see what I just said (or at least another aspect of what should be the same phenomenon) is to remember that $U(1)$-bundles are the same as $\Sigma \mathbb{Z}$-2-bundles.

(We once talked about that here).

$U(1)$-1-torsors play the same roles as $\Sigma \mathbb{Z}$-2-torsors.

Here I am saying: $U(1)$-representations should play the same roles as $\Sigma \mathbb{Z}$-2-representations.

Let me get this straight. Are you reconstructing QM without the voodoo? :)

Here is yet another aspect of what I was saying about getting $U(1)$-phases from $\mathbb{Z}$.

The following is essentially the triviality $$U(1) \simeq \mathbb{R}/\mathbb{Z}$$ but interpreted in “the right way”, if you wish.

Namely, a fancy way to restate this triviality is to say that there is an equivalence of strict 2-groups

$$U(1) \simeq (\mathbb{Z} \to \mathbb{R}) \,,$$ where on the left the group $U(1)$ is regraded as a 2-group in the obvious way and where on the right we have the 2-group coming from the obvious crossed module of groups, coming from the inclusion $\mathbb{Z} \hookrightarrow \mathbb{R}$.

So, this is a triviality. But luckily we are not afraid of the trivial here.

A slightly less trivial sounding statement I can make now is the following:

First, as a roughly correct slogan:

Principal $(\mathbb{Z} \to \mathbb{R})$-2-bundles with connection are the same as principal $U(1)$-bundles with connection.

Here is the more precise statement:

let $P^t_2(X)$ be a version of the 2-path 2-groupoid of $X$ defined as follows:

- objects are the points of $X$

- morphisms are parameterized paths in $X$ cobounding two points

- 2-morphisms are thin homotopopy classes of thin maps $[0,1]^2 \to X$ cobounding two paths.

Then:

locally trivializable smooth 2-functors $$\P_2^t(X) \to (\mathbb{Z} \to \mathbb{R})\mathrm{Tor}$$ are equivalent to locally trivializable smooth 1-functors $$\P_1(X) \to U(1)\mathrm{Tor} \,,$$ which in turn we know are equivalent to smooth principal $U(1)$-bundles with connection.