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May 26, 2009

Alm on Quantization as a Kan Extension

Posted by Urs Schreiber

Recently I was contacted by Johan Alm, a beginning PhD student at Stockholm University, Sweden, with Prof. Merkulov.

He wrote that he had thought about formalizing and proving aspects of the idea that appeared as the The nn-Café Quantum Conjecture about the nature of [[path integral quantization ]].

After a bit of discussion of his work, we thought it would be nice to post some of his notes here:

Johan Alm, Quantization as a Kan extension (lab)

nnCafé-regulars may be pleased to meet some old friends in there, such as the [[Leinster measure]] starring in its role as a canonical path integral measure.

Posted at May 26, 2009 7:21 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1977

44 Comments & 1 Trackback

Re: Alm on Quantization as a Kan Extension

Hi Urs,

In the beggining of the notes, Johan Alm cites Witten’s achivements in understanding path integrals, which makes me remember that you once promised to talk and open a discussion about his article “Branes and Quantization”. Did you forget that?

Posted by: Daniel de Franša MTd2 on May 26, 2009 8:33 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

you once promised to talk and open a discussion about his article “Branes and Quantization”.

True.

Did you forget that?

No, I didn’t, but I didn’t quite find the time. What an awful feeling.

You are right to pester me about this, really. I promise to write an (nnLab)-entry on the article and discuss is here by, let me see, end of July, would that still be okay?

I mean, as long as Edward Witten doesn’t create an nnLab entry on his paper himself, before I do.

Posted by: Urs Schreiber on May 26, 2009 9:31 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

“would that still be okay?”
Okay! :) I feel really flattered. Would you mind talking more about the aspects of generlized complex geometry instead of only supersymmetries(if that makes sense)? I guess since you are going to bother to right that much, wouldn’t it cover a wider audience than those that specializes in strings and sugras?

Posted by: Daniel de Franša MTd2 on May 27, 2009 5:31 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

“to right that much” - to write that much

Excuse me for the typo.

Posted by: Daniel de Franša MTd2 on May 28, 2009 3:27 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

I’m sorry, the observation about witten is in the beginning of n-cat lab’s entry.

Posted by: Daniel de Franša MTd2 on May 26, 2009 8:35 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Beautiful! Thanks. I definitely want to work through this.

Posted by: Eric on May 26, 2009 9:13 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

It is great to hear that some details of the picture outlines in nlab months ago happen to be nailed down and proven.

On the other hand one should make clear (especially in nlab) that Feynman integral resists foundations only as far as situations involved are general enough, while toy examples and simple classes are not mathematically unclear. First of all, the situation in usual QM is much more settled than in QFT; especially simple case of quadratic action functionals in QM has been rigorously defined in several formalisms.

Now, the challenges are for example: wide class of action functionals, nontrivial cases of continuum QFT-s, QFT-s involving nontrivial geometric objects. Regarding that the integral involves some space of paths (where Lipshitz class paths dominate in practice, and smooth paths give negligible contribution) it is natural that in cases involving various connections, gerbes and stacks, one needs to make good geometrical picture of that space and the mechanics of Feynman integral. It is great to confirm that the Kan extension and Leinster measure are so useful concepts in doing this! On the other hand, all the summations involved in the considerations suffer the same lack of analytical detail which usual Feynman integral does for infinite spaces, infinite groups and so on. Thus I find this research COMPLEMENTARY rather
than a substitute to other attempts which focus on more trivial geometry but more nontrivial analysis (including e.g. stochastics, Wick rotation, nonstandard analysis, discretization and other strategies). Finally, I believe that there should be a picture combining combinatorics of category theory with nontrivial analysis (similar to usage of D-modules in some quantization programs) for real-life QFT-s (hence not TQFT-s, or finite models). It would be great to come as soon as possible to elementary cases of such synthesis. For example, the results similar to index theorems relating topology and analysis could play the role. Recall that the first TQFTs have been constructed in

A.Schwarz, Partition function of degenerate quadratic functional and Ray-Singer invariant. Letters in Math. Phys. 2 (1978), 247-252

(and his paper in Comm. Math. Phys. soon after that)

and his approach started from the analysis of certain operators. We should get back to the synthesis reminding this interplay between operator theory and geometry.

Posted by: Zoran Skoda on May 26, 2009 9:31 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Dear Zoran,

very good points, indeed. Yes, I agree.

Of course the aim is in this direction. Eventually the goal is to enhance the finitized setup that one experiments with to get some basic concepts right to a more refined setup. Finite groupoids should be generalized to smooth \infty-stacks, eventually, categories of vector spaces by suitable (,2)(\infty,2)-categories for instance, and in that context then one wants to see analysis, operator algebra, etc appear.

I think right this moment the genral idea is to explore which kind of abstract nonsense works most smoothly in the finitized context, so as to be sure about which structure then to internalize in a more sophisticated setup.

The “path integral”-kind of quantization description at our σ\sigma-models is set up for \infty-stacks. I have had a bit of discussion with Johan Alm about how the Kan extension prescription relates to the pull-push prescription used there.

I had an idea for how they are related. Johan said he has proved that this idea works. But I didn’t undertstand his proof! :-) Now I think we are still trying to sort this out behind the scenes.

Maybe it would be best if we just presented the problem here to collect the esteemed expert advice from our readers. But Johan will have to decide this when he comes back online.

Posted by: Urs Schreiber on May 26, 2009 9:43 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

This note is very well written. Very clean. Very clear.

I hope to see more from Johan around here. Welcome to the nCafe!

In the conclusion you state:

There are two immediate directions in which the result we have proven begs to be generalized. Firstly, the result should extend to higher categories (the categorified versions of n-dimensional quantum field theories). Secondly, it ought to be possible to find at least traces of the universality described by the Kan extension formula also in the continuum (i.e. non-discretized) theory of particle quantum mechanics.

If I could make a suggestion, it would be to continue with this discretized setup as if it were a perfectly suitable model of nature and see where it takes you. I’ve found it helpful to take these finite models seriously. I have no doubt that the continuum version can be found, but I am doubtful that it will lead to any deeper understanding not already present in the finite version.

Posted by: Eric on May 27, 2009 12:20 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Basic question: what does it “mean” in general when the left and right Kan extensions agree? Is it like or related to the product and coproduct agreeing, for finite sets of vector spaces? (But not for finite sets of finite sets.)

Posted by: Allen Knutson on May 27, 2009 1:09 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

I’m no expert on Kan extensions, but I think you’ve got the right intuition: left and right Kan extensions are close relatives of coproducts and products, and these tend to agree in contexts with a ‘linear algebra’ flavor. Since Johan Alm is forming Kan extensions that take values in Vect, we shouldn’t be shocked that the left and right Kan extensions agree.

Indeed I often say (to myself) that the special thing about quantum mechanics is that it’s a context where left and right adjoints agree. In category theory we have to very carefully distinguish between the left and right adjoints:

hom(Lx,y)hom(x,Ry)hom(L x,y) \cong hom(x,R y)

because the homset hom(,)hom(-,-) is not symmetric in its two arguments. But in quantum mechanics we talk about adjoint operators without worrying about the difference between ‘left’ and ‘right’ adjoints:

T *x,y=x,Ty\langle T^* x, y \rangle = \langle x, T y \rangle

but also

Tx,y=T *x,y\langle T x, y \rangle = \langle T^* x, y \rangle

The reason is that here the inner product ,\langle -, - \rangle is symmetric in its two arguments. Well, almost — that’s the mystery of complex conjugation!

Anyway, the ‘almost-symmetry’ of the inner product in a Hilbert space is, I believe, a close relative of the almost-symmetry of the homset in a 2-Hilbert space, like HilbHilb, or even a 2-vector space, like VectVect.

There is, in short, something about quantum mechanics or linear algebra that doesn’t care much about who is the ‘input’ and who is the ‘output’. Matrices have transposes!

And this is why I’m not surprised that the left and right Kan extensions agree in Johan’s paper.

Posted by: John Baez on May 27, 2009 6:45 AM | Permalink | Reply to this

Kan extension, limit, product

Is it like or related to the product and coproduct agreeing, for finite sets of vector spaces?

Yes, since this is a special case of a Kan extension.

Limits and colimits are precisely Kan extensions down to a point.

Product and coproduct are precisely limits and colimits over categories without nontrivial morphisms.

Indeed, notice that in Johan’s description, the Kan extension assigns to each object nothing but the direct sum of the vector spaces “sitting above it”.

Well, or almost. If the configuration space groupoid has no nontrivial morphisms, then it’s just this direct sum. In general it is a the image of a projector inside this direct sum.

This gives me a chance to highlight the following very pleasing aspect of Johan’s result:

take his first example, quantization of particles that trace out paths of a certain parameter time.

There may be paths that “take no time to traverse” (their image under the projection along the functor along which we Kan extent is an identity morphism).

The formalism spits out the prescription: project onto the kernel of the operations induced by these 0-time paths. And do this by group averaging over them.

This means: the formalism regards paths of time length 0 automatically as internal gauge transformations and automatically projects onto the “physical states”: those that are invariant under translation along any 0-time path, i.e. those that are gauge invariant.

And it does so by group averaging over the gauge group. This is a famous technique used in gauge theory. Here it all drops out all by itself by just turning the Kan extension crank.

Posted by: Urs Schreiber on May 27, 2009 9:58 AM | Permalink | Reply to this

Re: Kan extension, limit, product

take his first example, quantization of particles that trace out paths of a certain parameter time.

I’m confused by this. Consider the case of a free particle on a Lorentzian manifold. Then the action is simply given by the proper time evolved, right? In this case, his propagator K(x,y;T)K(x,y;T) is independent of xx and yy.

The formalism spits out the prescription: project onto the kernel of the operations induced by these 0-time paths. And do this by group averaging over them.

How should one think of this in the free scalar particle case? There are a lot of paths with vanishing proper time…

Posted by: Tobias Fritz on May 27, 2009 11:39 AM | Permalink | Reply to this

concrete comparison to particle quantization

take his first example, quantization of particles that trace out paths of a certain parameter time.

I’m confused by this. Consider the case of a free particle on a Lorentzian manifold. Then the action is simply given by the proper time evolved, right?

One must beware that in its present state, configuration space is a groupoid and the “action” is a functor on that, it doesn’t naturally encode the kinetic part of the actions, actually, but just the gauge-coupling part in the case of a charged particle.

The formalism spits out the prescription: project onto the kernel of the operations induced by these 0-time paths. And do this by group averaging over them.

How should one think of this in the free scalar particle case? There are a lot of paths with vanishing proper time…

Apart from the above remark, here it should be noticed I think that the parameter ss that appears is not to be read as proper time, but as “parameter time” the way it appears when quantizing for instance the Klein-Gordon particle as a QFT on the line with fields being maps X:RM 4X : R \to M^4 and action involving X=XsX' = \frac{\partial X}{\partial s} for instance in the gauge coupling term. So ss is a formal parameter. It is related to proper time by

dτ=|X|ds d \tau = |X'| d s

with |||\cdot| the Minkowski norm.

That said, in it’s present state I think Johan’s example 1 could just as well be read as a model for non-relativistic QM, where ss would be the external time parameter. It’s a bit hard to tell without the kinetic part of the action! :-)

Johan and I have talked about this quite a bit. At the moment this is all really more like “topological quantum mechanics”.

Before you think that’s nonsense, notice that an example of just the kind that fits into Johan’s setup appears on p. 4 of, sigh, Freed-Hopkins-Lurie-Teleman.

To appreciate these phenomena better it will be necessary to raise the formalism to higher categorical dimension, where this is actually more familiar:

as indicated towards the end of Johan’s notes, once this is formulated not for 1-functors but for higher functors, it’s pretty clear how TFTs like Dijkgraaf-Witten theory fit in, and, by extension, Chern-Simons theory. Which of course is again topological: no kinetic part of the action, just gauge coupling to the 2-gerbe over BG\mathbf{B} G.

It’s maybe interesting in this context how the kinetic part of the string propagating on GG arises from the point of view of its holographically associated 3d TFT with target BG\mathbf{B} G : one transgresses the 2-gerbe to the mapping space [Σ 2,BG][\Sigma_2, \mathbf{B} G] and then there chooses a holomorphic structure of the resulting line bundle. It is this choice of holomorphic structure on [Σ 2,BG][\Sigma_2,\mathbf{B} G] that induces the conformal structure on Σ\Sigma (as reviewed for instance in Atiyah’s little red book on Chern-Simons theory).

So I am thinking that maybe to see the kinetic terms we want to see in abstract-nonsense quantization, we may have to better understand holographic boundary theories of TFTs. These somehow pick up an extra choice, which manifests itself then as a kinetic term. I think.

Posted by: Urs Schreiber on May 27, 2009 12:02 PM | Permalink | Reply to this

topological QM: (0+1)-dimensional DW-theory

Before you think that’s nonsense, notice that an example of just the kind that fits into Johan’s setup appears on p. 4 of, sigh, Freed-Hopkins-Lurie-Teleman.

To make this more concrete, notice that this 1-dimensional Dijkgraaf-Witten theory described there is, unless I am mixed up, a special case of Johan Alm’s examples:

The group character λ:GU(1)\lambda : G \to U(1) appearing in FHLT is nothing but a 1-functor

λ:BGBU(1). \lambda : \mathbf{B} G \to \mathbf{B} U(1) \,.

The standard representation of U(1)U(1) on \mathbb{C} yields the corresponding associated background field

exp(iS):BGλBU(1)Vect. \exp(i S) : \mathbf{B} G \stackrel{\lambda}{\to} \mathbf{B} U(1) \to Vect \,.

The (0+1)-dimensional σ\sigma-model quantum field theory described by this has as target space BG\mathbf{B} G, a field configuration is a functor ΣBG\Sigma \to \mathbf{B} G .

In Johan Alm’s setup, the collection of all field configurations is to be encoded in a bundle AΣA \to \Sigma over Σ\Sigma. So let A=(BG) *×Σ=BG×ΣA = (\mathbf{B} G)^{*} \times \Sigma = \mathbf{B} G \times \Sigma with AΣA \to \Sigma the canonical projection. exp(iS)\exp(i S ) extends to a functor exp(iS):AVect\exp(i S ) : A \to Vect in the obvious.

Then, I think, Johan’s Kan-tization for this setup gives precisely what FHLT mention on p. 4. Johan’s projector pp from the bottom of p.3 of his notes performs the group averaging of the character λ\lambda over GG that appears in the displayed formula on th bottom of p. 4 in FHLT.

Posted by: Urs Schreiber on May 27, 2009 12:23 PM | Permalink | Reply to this

Re: topological QM: (0+1)-dimensional DW-theory

Urs wrote:

Then, I think, Johan’s Kan-tization…

Lol, that’s a good one! Anyhow, welcome Johan to the n-category blogosphere!

Johan’s picture suggests – what is very natural in hindsight – that even for a σ\sigma-model with target XX, the space of fields on Σ\Sigma should not be just all maps [Σ,X][\Sigma, X], but instead there should be a bundle PΣP \rightarrow \Sigma such that fields are sections of this bundle.

I don’t get this. Maybe I am missing the main point? I don’t see how the standard geometric picture of quantization fits into this. For instance, in Chern-Simons theory (like in Atiyah’s little red book, as you mention above) the space of fields on a 2-manifold Σ\Sigma is the space of flat connections on Σ\Sigma. There is a line bundle sitting over this space of fields. What is the bundle PΣP \rightarrow \Sigma whose sections give rise to flat connections?

Before you think that’s nonsense, notice that an example of just the kind that fits into Johan’s setup appears on p. 4 of, sigh, Freed-Hopkins-Lurie-Teleman.

I’m thinking: this (0+1) dimensional example in Freed-Hopkins-Lurie-Teleman is something very simple, it’s almost not worth “abstracting”… unless the abstraction process has something interesting to say about the (1+1) theory or the (2+1) theory. That is, I’m reluctant to rack my brain wondering if it’s a left Kan extension or a push-pull or a new bundle-over-Σ\Sigma idea when there isn’t enough nontrivial geometric “footholds” to hold on to.

By the way, that quick calculation they give, namely that

(1)1volG Gλ(g)dg={0 λ1 1 λ=1 \frac{1}{vol G} \int_G \lambda(g) dg = \begin{cases} 0 & \lambda \neq 1 \\ 1 & \lambda = 1 \end{cases}

confused me for a bit… until I went back to Simon’s classic cheat sheet (thm 6).

There may be paths that “take no time to traverse” (their image under the projection along the functor along which we Kan extent is an identity morphism).

The formalism spits out the prescription: project onto the kernel of the operations induced by these 0-time paths. And do this by group averaging over them.

This is a new ingredient and I’m trying to figure it out. I don’t understand it; I was confused by the same thing that was confusing Tobias Fritz above. I think talking about Lorentzian manifolds may be a red herring, introducing confusion? For instance, the Hilbert space of the quantum theory shouldn’t be “functions on spacetime”, but rather “functions on space”. Doing that in a geometric canonical way has interested me for ages (see here and here).

On the whole, I would be happy if quantization can be thought of as a Kan extension. Not so much for myself, but more so that I could explain quantum stuff to abstract category theorists in language they might prefer :-) Myself, I’m quite happy with the “sum over all paths” explicit prescription. If someone were to tell me (speaking personally) that quantization is a Kan extension, I would find that more opaque then saying “you should perform a path integral”. This has more to do with my shoddy grasp of category theory… because I forget what things like a “Kan extension” are, I tend to prefer explicit formulas to things like “the universal arrow out of such and such”.

So I see the significance of the statement “Quantization is a Kan extension” more along the lines of “In certain cases, the abstract concept of Kan extension has a beautiful explicit geometric description: the path integral”. Perhaps even Johan thinks this way? Witness the paragraph at the top of page 3:

Usually, one has a classical ‘action’ of some kind defined for manifolds with some extra structure, e.g. a riemannian metric, a symplectic form, a principal bundle, or etc. Quantization is what happens when one tries to assign that same action to a manifold that does not have that structure! Hence one has to mathematically compensate for this by summing over all possible structures of the specified type.

Posted by: Bruce Bartlett on May 28, 2009 4:57 PM | Permalink | Reply to this

Re: topological QM: (0+1)-dimensional DW-theory

Hi Bruce!

“welcome Johan to the n-category blogosphere!”

Thanks for welcoming me into the fold! I’ve been following the blog for almost a year, but I haven’t felt I’ve had anything interesting to contribute with, until now.

I think Urs gave good answers to most of the questions you raised, so I will just make some brief comments.

First, I do not think the `group averaging’ is strange. It may be odd in Example 1 (the relativistic particle), but then, that model does not have the flavor of a gauge theory. We do want to see a projector onto gauge-inequivalent states appear somewhere in the abstract nonsense, no? The oddity is more in the model than in the presence of the projector.

Regarding

“…Perhaps even Johan thinks this way?”

I have to say that no, not entirely. At the moment our Kantization project is about verifying the intuition that, indeed, “the abstract concept of Kan extension has a beautiful explicit geometric description: the path integral”, as you put it. But my personal hope (still very distant) is to find a way to redefine the path integral (at least for TQFT) as particular example of Kan extension, so that path integrals become examples of Kan extensions instead of vice versa. That would be very pleasing esthetically, but even more importantly, I think, very practical. Once settled, it would be a magic black box: turn the crank (no thinking needed) and out comes quantization …no worries about choice of measure or anything.

THANKS to everyone for the supportive comments!

Posted by: Johan Alm on May 28, 2009 7:39 PM | Permalink | Reply to this

Re: topological QM: (0+1)-dimensional DW-theory

until I went back to Simon’s classic cheat sheet (thm 6).

Thanks for pointing to that and reminding us. Right, this is good to keep in mind.

BTW, do you realize that this means insights are being established here in our discussion? Whenever this happens, we should move the corresponding material to the nnLab, for here it will just be buried by time and noise.

We should create

[[Dijkgraaf-Witten theory]].

If you don’t beat me to it (hopefully you do!), I will do this after I post now another reply to another comment of yours in another thread, which will itself invove creating an nnLab entry…

Posted by: Urs Schreiber on May 28, 2009 8:45 PM | Permalink | Reply to this

Re: topological QM: (0+1)-dimensional DW-theory

Then, I think, Johan’s Kan-tization…

Lol, that’s a good one!

Due to Johan, by the way. That was the working title of the document.

What is the bundle PΣP \to \Sigma whose sections give rise to flat connections?

See example 2. It’s

(BG Σ 2×P 1())p 2P 1() (\mathbf{B}G^{\Sigma_2} \times P_1(\mathbb{R})) \stackrel{p_2}{\to} P_1(\mathbb{R})

In the full extended 3-functorial 3d theory it will look a bit simpler

(BG×Σ 3)p 2Σ 3 (\mathbf{B}G \times \Sigma_3) \stackrel{p_2}{\to} \Sigma_3

this (0+1) dimensional example in Freed-Hopkins-Lurie-Teleman is something very simple,

Sure. It’s (0+1)(0+1)-dimensional DW theory. This is what you want to check a formalism for quantization of 1-dimensional QFT against.

I hope it’s clear that this here is supposed to sort out the right starting point: once you know what quantization is by abstract nonsense in a 1-categorical context, you can say the same thing then in the \infty-categorical context.

But let’s go step-by-step. I don’t get the impression that the (0+1)(0+1)-dimensional DW example is so trivial (even though it is very trivial) that it’s structural meaning is immediately clear.

when there isn’t enough nontrivial geometric “footholds” to hold on to.

Right, the insinuation is of course that (n+1)(n+1)-d DW theory drops out turning the same crank in the corresponding (n+1)(n+1)-catgeorical machine. But first things first.

the Hilbert space of the quantum theory shouldn’t be “functions on spacetime”, but rather “functions on space”.

Generally, the space of states assigned to ss is those functions on the slice sitting over ss that are invariant under the “0-time path” evolution.

(Ah, now I see, example 1 in the current version maybe deserves some tuning, let me check…)

Myself, I’m quite happy with the “sum over all paths” explicit prescription.

Which is, precisely? Where does the DW measure come from? The Yetter measure? How does it work for extended QFT in codimension higher than 1?

The point here is that we know what

the “sum over all paths” explicit prescription

means in a handful explicit cases where one tunes it to make sense. Notice that the DW and the Yetter path integral measure are chosen by fiddling around until it so happens that the path integral becomes independent of the triangulation used to compute it. That’s clearly asking for a more systematic and fundamental description. The question is: What’s really going on?

Posted by: Urs Schreiber on May 28, 2009 5:41 PM | Permalink | Reply to this

Re: topological QM: (0+1)-dimensional DW-theory

In the full extended 3-functorial 3d theory it will look a bit simpler: (BG×Σ 3)p 2Σ 3(BG \times \Sigma_3) \stackrel{p_2}{\rightarrow} \Sigma_3.

Okay… but what’s confusing me is when you said “the collection of all field configurations is to be encoded in a bundle AΣA \rightarrow \Sigma over Σ\Sigma”. So I thought to myself: “Okay, in Chern-Simons theory a field configuration is a connection (dope I was wrong before, it doesn’t need to be flat does it) on a principal GG-bundle on the manifold”. Now I don’t see how the concept “a connection on a principal GG-bundle over Σ\Sigma” can be encoded geometrically as “a section of some bundle sitting over Σ\Sigma”. Can it?

Oh I see, maybe I interpreted it wrong. The collection of all field configurations isn’t to be thought of as arising from taking the space of sections of some bundle… rather, the space of field configurations is the bundle! Is that it?

Posted by: Bruce Bartlett on May 28, 2009 6:02 PM | Permalink | Reply to this

Re: topological QM: (0+1)-dimensional DW-theory

Now I don’t see how the concept “a connection on a principal G-bundle over Σ” can be encoded geometrically as “a section of some bundle sitting over Σ”. Can it?

DW/CS theory is a σ\sigma-model, so its fields are all maps ΣBG\Sigma \to \mathbf{B} G for suitable realization of parameter space Σ\Sigma and target space BG\mathbf{B} G. (For instanc in DW theory Σ\Sigma a skeleton of the fundamental groupoid of a manifold and BG\mathbf{B} G a one-object groupoid).

In a more general non-σ\sigma-model QFT on Σ\Sigma the fields are not to be expected to be just plain maps out of Σ\Sigma, but to be sections of a bundle over Σ\Sigma.

But the former situation is trivially a special case of the latter, when we take that bundle to be BG×ΣΣ\mathbf{B}G \times \Sigma \to \Sigma. A section of that is just a functor ΣBG\Sigma \to \mathbf{B}G.

Posted by: Urs Schreiber on May 28, 2009 6:12 PM | Permalink | Reply to this

Re: concrete comparison to particle quantization

One must beware that in its present state, configuration space is a groupoid and the “action” is a functor on that, it doesn’t naturally encode the kinetic part of the actions, actually, but just the gauge-coupling part in the case of a charged particle.

Uh? Is this not a rather serious limitation?

Apart from the above remark, here it should be noticed I think that the parameter s that appears is not to be read as proper time, but as parameter time;

But can’t you just fix a gauge by setting t = q0(t), So the Lagrangian is L = sqrt(1 - dqi/dt dqi/dt)?

Posted by: Thomas Larsson on May 28, 2009 4:27 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

what does it “mean” in general when the left and right Kan extensions agree?

When we restrict Kan extensions to the case of products and coproducts, then this means that we have a category with [[biproducts]], which in turn means that we are at least pretty close to an additive or abelian category (namely that we are enriched over abelian monoids, at least).

Posted by: Urs Schreiber on May 27, 2009 11:04 AM | Permalink | Reply to this

Laplace Transform and quantified Z-transform

It appears that Alm is doing a lot of Laplace Transform like stuff. The ‘discrete’ version of the Laplace Transform is the Z-transform that operates on digitally quantified signals.

While digital ‘quantification’ appears to be a different beast from the ‘quantum’ part of QM, see the discussion of z-transforms in the nLab discussion of Day Convolution.

Maybe QM and signal quantification can be unified.

Posted by: RodMcGuire on May 27, 2009 2:44 AM | Permalink | Reply to this

Re: Laplace Transform and quantified Z-transform

It appears that Alm is doing a lot of Laplace Transform like stuff.

Could you expand on that? I am not sure I see which part of Alm’s considerations you are thinking of.

Posted by: Urs Schreiber on May 27, 2009 10:02 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

A very nice paper! It should be polished up and published! It makes me very happy that new people are joining the battle to understand quantum theory using categories. Congratulations, Johan!

There are lots of typos, but I’m too busy fixing my own typos.

Posted by: John Baez on May 27, 2009 6:55 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

This summing over all possible choices behaves mathematically as if it ZZ is trying to compensate for the loss of information in the passage from 𝒫 1(M)\mathcal{P}_1(M) to 𝒫 1(R)\mathcal{P}_1(R).

Remember how once we were considering classical mechanics as a matrix mechanics with ‘superposition principle’, just with a different rig?

Is it perhaps that classical mechanics is unusual in that it uses a rig which allows taking as codomain 𝒫 1(M)\mathcal{P}_1(M)?

Posted by: David Corfield on May 27, 2009 9:06 AM | Permalink | Reply to this

Kan extension and pull-push

I have a question to our abstract nonsense expert readers.

With Johan, I am trying to find the relation, if any, between quantization by Kan extension and by pull-push.

I have the following simple observation:

Johan’s picture suggests – what is very natural in hindsight – that even for a σ\sigma-model with target XX, the space of fields on Σ\Sigma should not be just all maps [Σ,X][\Sigma,X], but instead there should be a bundle PΣP \to \Sigma such that fields are sections of this bundle.

So, in this spirit, there is an obvious slight variation of the quantization presciption from section 7 of Cocycles and σ\sigma-models, where [Σ,X][\Sigma,X] is replaced by some category of weak sections, which is very close to this, but slightly larger.

You need none of the QFT motivation background to follow this, especially if you are the abstract nonsense wizard that I am hoping to address here (Todd? :-). The simple idea/observation I have typed into this pdf

Relation Kan extension/pull-push ?

It’s really, brief, two pages nominally, but the idea itself is barely a page.

The observation there is that with this slight modification, the pull-push quantization happens to coincide with the Kan extension over points.

The big question is: what’s the relation over paths?

I am not sure yet about this. There are some obvious guesses here, but we haven’t been able to understand this to the end.

All comments are welcome.

Posted by: Urs Schreiber on May 27, 2009 1:19 PM | Permalink | Reply to this

Re: Kan extension and pull-push

I wrote:

The simple idea/observation I have typed into this pdf

Relation Kan extension/pull-push ?

It’s really, brief, two pages nominally, but the idea itself is barely a page.

The observation there is that with this slight modification, the pull-push quantization happens to coincide with the Kan extension over points.

The big question is: what’s the relation over paths?

I am not sure yet about this. There are some obvious guesses here,

One of these guesses is clearly: the Kan extension L gfL_g f

A f Vect g L gf B \array{ A &&\stackrel{f}{\to}& Vect \\ \downarrow^g && \nearrow_{L_g f} \\ B }

of a functor ff is computed as the colimit over its transgression to the category Γ(g)\Gamma(g) of weak sections

Γ(g)={ A ϕ g B Id B} \Gamma(g) = \left\{ \array{ && A \\ & {}^\phi \nearrow & \downarrow^g \\ B &\stackrel{Id}{\to}& B } \right\}

of gg

(all diagrams here with implicit 2-cells filled in)

as

Lan gfcolim(Γ(g)[B,A][B,f][B,Vect]) Lan_g f \simeq colim ( \Gamma(g) \to [B,A] \stackrel{[B,f]}{\to} [B, Vect])

not only over objects, but also over moprhisms.

Now Johan Alm suggests a proof for this, for the situation we are looking at here.

You can find it at the end of the new version here.

I am feeling a bit weird now: if this is right (looks right to me, but elementary as the steps are, it’s a bit intricate) then one would wonder if this is not known in basic Cat theory.

What’s going on?

Posted by: Urs Schreiber on May 27, 2009 9:48 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

There is a nice colloquium made Gukov, posted on PIRSA, about Branes and Quantization.

Posted by: Daniel de Franša MTd2 on June 10, 2009 1:49 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Over lunch at Strings, Fields, Topology I hear Peter Teichner talk about the idea that quantization of σ\sigma-model QFTs is pushforward of the classical action functor.

I don’t remember that this was accepted as so obvious when I talked about this idea in Notre Dame last spring #, so something has happened since then.

Posted by: Urs Schreiber on June 10, 2009 1:15 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Is there any fundamental new insight on this quantization scheme or is this just some fancy mathematics with only the hope for something new?

Posted by: Daniel de Franša MTd2 on June 10, 2009 5:49 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Is there any fundamental new insight on this quantization scheme or is this just some fancy mathematics with only the hope for something new?

Hope or not, the point is to notice that there is an open problem here that is waiting to be solved, this way or, if not this way, then some way:

given some differential nn-cocycle on some space, what is the extended nn-dimensional QFT nn-functor that quantizes it?

- for instance: what is the 3-functor corresponding to Chern-Simons theory? And more importantly: how does it arise from a systematic process from the Chern-Simons actions functional (instead of by guessing and plausibility checking as done these days).

- If that sounds as if it were too obvious a problem, maybe it is helpful to ask the next question: what is the 7-functor corresponding to 7-dimensional Chern-Simons theory? And more importantly: how does it arise from a systematic process from the 7-dimensional Chern-Simons action functional?

You see, recently lots of mathematicians talk about TQFT and QFT, because there has been much progress on formalizing the right axioms that a full QFT should satisfy, so mathematicians can now happily study the space of all solutions to these axioms. For instance for pure TFT it turns out that the “landscape” of TFTs with values in some VV is precisely the space of “fully dualizable objects in VV. Whatever that means.

So that’s fantastic. The landscape of all TQFTs has been fully identified. Why is the random physicist from the street only vaguely aware and even less excited about this fact:

simple: because we have at the moment only a highly insufficient understanding of how these axiomatically characterized TQFTs correspond to the physics they are supposed to describe: its not known which “action functional” or similar gives rise to them. It’s not known which problem they actually solve, which role they play in this world.

The missing link is the formalization of quantization in general and of σ\sigma-model quantization in particular. We don’t just want to say: an nn-d QFT is an nn-functor on Bord n SBord_n^S. We want to say: this physical setup has associated with it this quantum nn-functor.

Ideally and eventually we’d want to be able to say: this string background defined by such and such topology, metric, and flux and whatnot, gives rise to this 2d CFT, if at all. Currently this is done essentially by educated wild guessing.

Posted by: Urs Schreiber on June 10, 2009 7:45 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

The goal seems to be to reduce the amount of wild guessing at the current stage of development in order to help advance to the next stage so that we can make further wild guesses about even deeper questions. Repeat :)

Posted by: Eric on June 10, 2009 8:03 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

“It’s not known which problem they actually solve, which role they play in this world.”

So, how do you know that you are quantizing anything?

Posted by: Daniel de Franša MTd2 on June 11, 2009 6:45 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

So, how do you know that you are quantizing anything?

That’s exactly the point I was trying to make:

given just an abstract TQFT given by a representation of Bord nBord_n, you don’t a priori know what it is the quantization of, if anything.

And one central problem that currently prevents us from knowing so is that we have no systematic theory for what it even means precisely to quantize classical background field data to a (T)QFT.

The proposal that this thread is about is a proposal for formalizing what quantization of classical background field data to an extended (T)QFT should mean.

Namely: represent the classical background field data by a nn-functor on field configurations itself and then push that forward from field configurations to the bordisms that they are field configurations on.

Posted by: Urs Schreiber on June 11, 2009 9:42 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

given just an abstract TQFT given by a representation of Bord n, you don’t a priori know what it is the quantization of, if anything.

implying there is no way to pass to a classical `limit’?

Posted by: jim stasheff on June 11, 2009 3:58 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Urs, my question was not limited to that. How things like positive norm, unitarity, measurement, wave functions and operators are addressed and make consistent with Quantum Mechanics? All I see are extremely abstract constructions that makes me see no physical sense at all.

Posted by: Daniel de Franša MTd2 on June 11, 2009 6:07 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Hi Daniel,

There is a very pretty picture here. Unfortunately, Urs and others are too busy working it out to stop and bring it to earth for the rest of us. Urs made a valient attempt at getting me to understand it here:

An Exercise in Groupoidification: The Path Integral

We never did finish, but I started this to maybe help resurrect the low brow version:

An Exercise in Groupoidification

Alm’s paper itself is ALMOST understandable to me, but you can find some further discussion here:

Quantization as a Kan Extension

If you wanted to help me work through it in a way that we ultimately can explain it to others, that’d be great. The first step, of course, is to understand it ourselves :)

I’m at least convinced that it is pretty enough to make the effort to try to understand it.

Maybe we could start a more elementary “Journal Club” or something.

Posted by: Eric on June 11, 2009 7:15 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Eric, I will take a look, but as you can see in the video I provided above, Gukov and Witten works out simple examples the quantization in a circle… and well, the mathematics they use are very complicated, but the examples worked out are extremely simple and nice and they progress to more evolved ones.

If Witten can help us mere mortals with more trivial examples, perhaps it is not to much… well… to ask for simple examples, perhaps on behalf of the popularization of such ideas.

Posted by: Daniel de Franša MTd2 on June 11, 2009 8:31 PM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

How things like positive norm, unitarity, measurement, wave functions and operators are addressed and make consistent with Quantum Mechanics?

Yes, these are all questions eventually to be better understood. There is lack of that at the moment. But I take all this still as supporting my point that we need to think about the systematic relation of quantization to extended TQFTs.

By the way: concerning the relation to extended QFT to operator formalism in quantum theory that you mention: I suggested some observations on that here.

Posted by: Urs Schreiber on June 12, 2009 1:31 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

I will study that only if you try to explain me something. I don’t know is ok as well.

It seems that juat just an infinitesimaly small quantity of smooth manifolds seems to be subject of study of 4D TQFT, but not the infinitely many exotic ones.In other terms, if you are trying to avoid an essential singularity in space time, and you have to go quantum, that is, trying all possible allowed states in a theory, then you HAVE to consider these infinitely many smooth cases in a topological theory. This is, crudely comparing, like quantizing the hydrogen atom but just taking into consideration the spherical solutions…

That is not an intrinsically esoteric subject given that the culprit for the exotic smoothness can be related to topological sigma models, where topological strings try to avoid a singularity in 4D dimensions, and there are hundreds of, unfortunantely, disconnected articles about each of these subjects…

Posted by: Daniel de Franša MTd2 on June 12, 2009 5:17 AM | Permalink | Reply to this

Re: Alm on Quantization as a Kan Extension

Alright, you don’t have to explain anything…I will study without conditions.

Posted by: Daniel de Franša MTd2 on June 14, 2009 12:23 AM | Permalink | Reply to this

An Exercise in Kantization

I started an elementary version of the Journal Club to help work through Alm’s paper:

An Exercise in Kantization

For experts, we could use your help!

For lurkers, please feel free to join in with questions and/or contributions (even anonymously!).

Posted by: Eric on June 17, 2009 7:01 PM | Permalink | Reply to this
Read the post Question about ∞-Colimits
Weblog: The n-Category Café
Excerpt: What can one say about hocolimits over domains that are themselves hocolimits?
Tracked: August 10, 2009 9:56 PM

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