Quantization and Cohomology (Week 17)

March 7, 2007

Quantization and Cohomology (Week 17)

Posted by John Baez

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This week in our course on Quantization and Cohomology we sketched an avenue of attack for getting a Hilbert space and an operator algebra from a category $C$ equipped with a functor $S: C \to \mathbb{R}.$ We think of the objects of $C$ as ‘configurations’ and the morphisms as ‘processes’ or ‘paths’; the functor $S$ assigns a real number to each morphism called its ‘action’. There are a lot of technical issues involved in getting a Hilbert space from this data, but the basic idea is simple: use path integrals!

Week 17 (Mar. 6) - Hilbert spaces and operator algebras from categories. Under what conditions can we obtain a Hilbert space from a category $C$ equipped with an "action" functor $S: C \to \mathbb{R}$ ? The importance of time reversal: groupoids versus ∗-categories (also known as †-categories). The ‘category algebra’ of $C$ .

Last week’s notes are here; next week’s notes are here.

Posted at March 7, 2007 9:52 PM UTC

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Re: Quantization and Cohomology (Week 17)

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It is interesting how you set up the inner product in this week’s notes.

After declaring your Hilbert spaces to be spaces of functions on configuration space $X$ , you don’t set $\langle \phi | \psi \rangle = \int_{X} \bar \phi(x) \psi(x)\; dx$ as the naive reader might have expected you would. Instead, you do something more like a correlator and set $\langle \phi | \psi \rangle = \int_{x \stackrel{\gamma}{\to} y} \bar \phi(x) \psi(y) \; e^{i S(\gamma)}d\mu \,.$

What precisely this will mean will in particular depend on the precise nature of paths that are integrated over.

I think I know where you are heading: constrained dynamics. Your way to look at the inner product will be more immediately applicable to the relativistic than to the non-relativistic particle, for instance.

And that’s also why you base category $C$ looks like paths in $Q \times \mathbb{R}$ , instead of like paths in $Q$ , for a particle propagating on space $Q$ .

I need to think more about this point of view.

Differing approaches to this issue might, for instance, affect discussions like I had with Jeffrey, over on the canonical measure thread (his comment, my reply).

Posted by: urs on March 12, 2007 8:17 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 17)

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Urs wrote:

It is interesting how you set up the inner product in this week’s notes.

After declaring your Hilbert spaces to be spaces of functions on configuration space X…

You understand exactly what I’m doing! But just for those who don’t:

I didn’t declare my Hilbert spaces to consist of functions on configuration space $X$ ; I declared them to consist of functions on the set of objects of some category $C$ .

I do indeed sometimes call these objects ‘configurations’, but in the example I keep coming back to, they are points in spacetime ( $\mathbb{R} \times Q$ ) rather than points in space ( $Q$ ). So, perhaps the word ‘configuration’ is misleading.

Sometimes people use the phrase ‘extended configuration space’ to stand for the product of time ( $\mathbb{R}$ ) and the usual configuration space ( $Q$ ).

Your way to look at the inner product will be more immediately applicable to the relativistic than to the non-relativistic particle, for instance.

It’s certainly designed to work well for the relativistic particle: I don’t want a quantization procedure that requires me to split spacetime into space and time!

But, it also works perfectly well for the nonrelativistic particle. My student Alex Hoffnung and I are working on that example as a test case.

It’s fun: seeing quantization of a point particle as a special case of getting a Hilbert space from a category. But the real goal is to categorify all this stuff, and get a new way of quantizing strings, as a special case of getting a 2-Hilbert space from a 2-category.

Please don’t work out all the details before my student does. You could do it much faster… but he needs the practice!

Posted by: John Baez on March 13, 2007 5:28 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 17)

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I don’t want a quantization procedure that requires me to split spacetime into space and time!

Sure!

Except maybe for the non-relativistic particle…

I think the issue here is about dealing with the two relevant spaces that enter the game of quantization:

the target space $Q$ and the extended parameter space (= the “worldvolume”).

For the particle, we usually identify the worldvolume with $\mathbb{R}$ .

This makes it a little hard to distinguish, whenever we encounter the object $Q \times \mathbb{R} \,,$ whether we should think of it as spacetime, or as space times extended parameter space (and to decide whether we should care about the difference! :-).

(This is closely related to what I discussed with Bruce recently here and here).

As far as I understood what Rovelli talks about (which I guess is what you have in mind, too), the “extended configuration space” is in fact the product of target space with the worldvolume (manifestly so for instance in equation (18)).

So, let’s look at the relativistic particle: what would be the space of objects of the category $C$ in your notes?

If you follow Rovelli, it should be spacetime times $\mathbb{R}$ - right?

By the way, there is possibly something very deep about this product space

extended config space = target space times worldvolume.

I realized this when thinking about the things that then lead me to write the entry Sheaves of Observables (and in particular when thinking about the stuff that is indicated at the very end of the entry, but did not make it into it this time):

As Witten explains in in hep-th/0504078 (ignore the first half of the paper – only the second half is of relevance here), the sheaves of operator algebras that are so crucial in the Heisenberg picture of quantum theory (well, that’s not the way he puts it ;-) are best thought of indeed as sheaves on target space times the worldvolume.

(This is section 3.5, in particular top of p. 30. The example in section 5.3 is also relevant.)

Let me use his notation. Call target space $X$ (and never mind whether this is Riemannian or pseudo-Riemannian!) and call the worldvolume $\Sigma$ .

Of course the notation $\Sigma$ comes from thinking about the 2-particle now, but that’s quite irrelevant for the general point, I think. For the $n$ -particle $\Sigma$ will be some $n$ -dimensional worldvolume. For our particle we take $\Sigma = \mathbb{R}$ .

Then, as Witten discusses, the pre-sheaves of algebras of observables that we are dealing with are best thought of as sheaves on the product $X \times \Sigma \,,$ which is precisely what Rovelli would call the extended configuration space!

There is a subtle gluing condition involved here, which I think I am beginning to understand “from first principles” (indicating this was the point of the entry on “Sheaves of Operator Algebras” linked to above).

Gluing may fail on the $X$ factor and on the $Y$ factor and on both factors together. These failures (if present) directly translate into various more or less known “anomalies”.

In principle one could discuss all this just for the point particle. Only problem is that everything is so simple here!

Sometimes simplicity is a bigger hazard to understanding than complexity - it hides so much!

Anyway, last time I thought about it I did not find a way to set up the point particle from this perspective such that I get a nontrivial anomaly along these lines.

Probably no surprise, since it would be known.

Ah, of course what I should look at is spinning point particles… yes…

Maybe that makes a good exercise: reformulate the old very work by Witten on anomalies of spinning (and super-)particles into the languge of these sheaves of operator algebras on extended configuration space.

The other thing I still need to think more about is how the way you set up the inner product in your notes does what we want it to do. I am slightly puzzled about this. Sorry. Looking forward to the next sessions of “quantization and cohomology”! :-)

Posted by: urs on March 13, 2007 7:24 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 17)

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Alas, I’m really exhausted, so this will be very short:

You’re right — Rovelli’s ‘extended configuration space’ is

$spacetime \times worldvolume$

hence for particles

$spacetime \times \mathbb{R} .$

But, there’s another approach where you take as your ‘configuration space’ for a particle simply

$spacetime$

and this is the one I’m using now. In the context of classical mechanics, I discussed it back in week 6 of the Fall lectures, on page 2.

There are so many different approaches to classical mechanics! I would like someday to write a nice book about them all. So far I just have rough draft. I’m going to teach classical mechanics again next year, and hopefully this draft will be refined.

Posted by: John Baez on March 14, 2007 6:48 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 17)

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Here is a general observation which occured to me this morning. It’s a half-baked thought, but maybe somebody can see something worthwhile in it.

First, I have to apologize for writing “”configuration space” in my above comment # a couple of times when I really should have said “target space”. This is the worst kind of mistake one can make in this kind of discussion, where everything is about the subtle differences between all these concepts, and I am feeling really bad about it. I have now corrected that in the above comment.

So what really happened this morning was that I noticed this recurring misprint. But as it goes with mistakes, they sometimes lead us to deeper insights.

So let me amplify the correct statement once more:

configuration space is the space of maps from parameter space into target space $\mathrm{conf} = [\mathrm{par},\mathrm{tar}]$ (or a subthing thereof, but let’s not worry about such details at the moment).

history space is the space of maps from the worldvolume into target space $\mathrm{hist} = [\mathrm{worldvol},\mathrm{tar}]$

Now, what people like Rovelli call the “extended configuration space” is the product of these two spaces $\mathrm{extconf} = \mathrm{worldvol}\times\mathrm{tar} \,.$ Moreover, it turns out that sheaves (of algebras of observables) on the extended configuration space play an important role.

Staring at these facts for a moment seems to suggest that this is telling us that we should pass from functors $\mathrm{hist} \ni \gamma : \mathrm{worldvol} \to \mathrm{tar}$ to the corresponding profunctors $\tilde \gamma : \mathrm{worldvol}^{\mathrm{op}}\times \mathrm{tar} \to \mathrm{Set} \,.$

These now can loosely be thought of as set-valued functions on the extended configuration space. That might be remarkable, given that we noticed that some things in quantum theory might become much clearer would we be dealing with set-valued functions instead of with ordinary functions that take values in numbers.

Maybe even better, this might be a way to see why we should in fact consider not just sets, but $\mathbb{C}$ -sets:

because what are these sets that $\tilde \gamma$ associates to a point in the extended configuration space? Well, these are the collection of morphisms to that point from a given point, of course, but here this means that these are the set of paths between these two points.

So given some functor $\mathrm{tar} \to \mathrm{phas}$ that associates phases to paths, we can naturally compose it with our $\tilde \gamma$ in some way – and this will label all elements of these sets with their associated phase.

So, given such an action functor, we naturally get a profunctor $(\tilde \gamma,\mathrm{tra}) : \mathrm{extconf} \to \mathbb{C}\mathrm{Set} \,.$

The process of path integration would then be the postcomposition of this with the cardinality operation.

Hm…

Posted by: urs on March 14, 2007 11:34 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 17)

what is the glueing problem? of what with respect to what?

jim

Posted by: jim stasheff on March 16, 2007 6:18 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 17)

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what is the glueing problem? of what with respect to what?

I will try to answer this in detail. But not right now, I am short of time.

I have mentioned several aspects of this before, though. Maybe have a look at what I wrote in Sheaves of CDOs.

I have started giving a derivation of this issue from first and elementary principles in QFT of Charged n-Particle: Sheaves of Observables.

More later…

Posted by: urs on March 16, 2007 7:12 PM | Permalink | Reply to this

Read the post QFT of Charged n-Particle: The Canonical 1-Particle
Weblog: The n-Category Café
Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:09 PM

Read the post A Groupoid Approach to Quantization
Weblog: The n-Category Café
Excerpt: On Eli Hawkins' groupoid version of geometric quantization.
Tracked: June 12, 2008 5:55 PM

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March 7, 2007