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May 28, 2010

The Quantum Whisky Club

Posted by John Baez

I’m live blogging from the Quantum Whisky Club. We’re in a dimly-lit office in the Computing Lab at Oxford University, listening to a rap song about Pythagoras composed by Richard Garner, who is here along with a bunch of other folks who’ll be attending the Quantum Physics and Logic workshop.

Like who?

Well, from left to right you can see Eugenia Cheng’s foot — if you look incredibly hard, that is — and then Alissa Crans leaning on the windowpane, and then the mysterious black silhouette of Andrei Akhvlediani, who is facing away, in front of the glare of a desk lamp. Then comes Richard Garner basking in the golden glow of light at the edge of a curtain, and then, in front of him — much easier to see! — Aleks Kissinger cheerily holding his cup high, and Philip Atzemoglou way in back, and me happily blogging away on my laptop…

…and then Bruce Bartlett, and Ray Lal behind him way in back, and Chris Heunen, Simon Willerton right up front, then Andreas Döring grinning and holding a glass, Ross Duncan, and just a tiny bit of Bertfried Fauser on the right-hand edge of the photo. Present but invisible are Nadja Kutz… and the fellow taking this picture: the inimitable Jamie Vicary.

And look who just walked in! It’s Bob Coecke!

… and Jonathan Barrett!

I’m giving my talk at 9 a.m. tomorrow, so I shouldn’t have too much fun. Good night!

Posted at May 28, 2010 10:11 PM UTC

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87 Comments & 1 Trackback

Re: The Quantum Whisky Club

Hi folks, here’s a pic of John liveblogging for the above post…!

pic

Posted by: Anotherpic on May 29, 2010 12:16 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

Ah yes, the Fox room. I used to sleep there for the month before I moved into my nice room on Norham Rd last year. Been snowing here.

Posted by: Kea on May 29, 2010 4:34 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

Mathematicians + Alcohol + Live Blogging. Could get interesting :)

Posted by: Eric Forgy on May 29, 2010 6:17 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

I tried hard to keep it from getting too interesting.

Posted by: John Baez on May 29, 2010 11:35 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

What happens at the Quantum Whiskey Club stays at the Quantum Whiskey Club :)

Posted by: Eric Forgy on May 29, 2010 5:17 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

Indeed: quantum information cannot be duplicated.

But it’s called the ‘Quantum Whisky Club’. They drink whisky there, not whiskey.

Posted by: John Baez on May 30, 2010 11:53 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

Fans of Oz and the Wizard may like this photo taken at an Indian restaurant in Oxford last Wednesday:

From left to right: Oz’s wife Alison, Oz, and the Wizard in front, then Bruce Bartlett and Jamie Vicary in back.

Posted by: John Baez on May 29, 2010 11:40 AM | Permalink | Reply to this

“the h-bar”; Re: The Quantum Whisky Club

“Quantum Whisky Club” not to be confused with Caltech’s Hayman Lounge at the Athenaeum, known to all Physicists as “the h-bar.”

My wife and I explained that to Tommy Smothers when we bought him a drink there (it is Members only, and I was) but he could not see this as funny, and so did not use it in his monologue later to a Caltech audience. But you find it funny. Right?

By the way, Scotch whisky is whisky made in Scotland. In Britain, the term whisky is usually taken to mean Scotch unless otherwise specified.

Posted by: Jonathan Vos Post on May 29, 2010 6:15 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

Well, I get it, but I don’t find it all that funny either. I might smile indulgently if someone told me, but I can’t imagine Tommy Smothers getting more than a polite chuckle, no matter how he told it and no matter how geeky the crowd. He was right, I’m afraid.

Posted by: Todd Trimble on May 29, 2010 7:08 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

The members of the Quantum Whisky Club are sufficiently geeky that they might approve of calling it the “h-bar” — after enough whiskies, at least. But I forget if the term “bar”, as opposed to “pub”, is idiomatic in British English.

By the way, Scotch whisky is whisky made in Scotland. In Britain, the term whisky is usually taken to mean Scotch unless otherwise specified.

Yes, that’s what they drink at the Quantum Whisky Club. On Friday it was mainly Talisker… but last night, in a smaller ceremony, a few of us tried something else, I believe a Caol Illa.

Posted by: John Baez on May 30, 2010 11:47 AM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

I like the Progress Bar :)

Posted by: Eric Forgy on May 30, 2010 1:03 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

Reposted from Facebook conversation on this n-Category Cafe page:

Patricia Williams-King:
A quark, a proton and an electron walked into the Quantum Whisky Club…..

Jonathan Vos Post:
The proton says: “I’m positive that I’ve been here before.”

The electron asks the bartender: “What’s the charge?”

The quark says: “Can we split the bill 3 ways?”

Posted by: Jonathan Vos Post on May 30, 2010 3:34 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

“Bar” and “pub” are both used in British English, but a bar is different from a pub. It’s better.

A club is even better than a bar. It’s like a bar, but with dancing…

Posted by: Eugenia Cheng on June 1, 2010 11:09 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

My ordering is the opposite! I think mine goes something like:

The Turf > pub > bar > club

As for whisky, I spell it whiskey and it is most definitely Irish. The Norwegian equivalent appears to be aquavit, which has the same meaning both linguistically and inebriationally.

Posted by: Andrew Stacey on June 2, 2010 9:12 AM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

As for whisky, I spell it whiskey …

I love a good language debate, but this blog post certainly convinced me to stay away from this particular usage issue.

Posted by: Mark Meckes on June 2, 2010 2:55 PM | Permalink | Reply to this

inn-teammates

If I may offer an opinion: You can use the German “Schnaps” for everything that is made of of some sort of grain and has more than 30% ethanol.

The Scandinavian may recognize this from the tradition of “drinking snaps”.

As for whisky: An aunt of mine once had a dog named Whisky (or Whiskey, I don’t remember), so, when I took him for a walk, I had to call “Whisky!” out loud on the street once, which prompted some curious reactions from other pedestrians - I demanded to know why they did not name him “Bobby”, but never got an answer…

Posted by: Tim van Beek on June 2, 2010 3:23 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

I don’t think there’s much of a debate here Mark. At the Quantum Whisky Club they drink a spirit distilled in Scotland. Andrew was expressing, in his own style, his preference for the analogous Irish spirit which is differently tasting and differently spelt,.

Posted by: Simon Willerton on June 2, 2010 5:57 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

I see, I misinterpreted Andrew’s comment. Too bad, I find watching Andrew in a language debate even more entertaining than having one myself.

As for Tim’s suggestion above, unfortunately the German word Schnaps is easily mistaken for the derivative English word schnapps, which doesn’t mean the same thing.

Posted by: Mark Meckes on June 2, 2010 6:25 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

The Turf < pub < bar < club < whisky club < Quantum Whisky Club

Posted by: Eugenia Cheng on June 2, 2010 10:45 PM | Permalink | Reply to this

Re: “the h-bar”; Re: The Quantum Whisky Club

a bar is different from a pub. It’s better

how so? not the image I have over here

Posted by: jim stasheff on June 2, 2010 12:28 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

Eugenia wrote:

A club is even better than a bar. It’s like a bar, but with dancing…

Okay, then the Quantum Whisky Club is definitely a club.

I told someone at the workshop about this place. His jaw dropped and he said “What’s quantum whisky?

Posted by: John Baez on June 2, 2010 7:35 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

It’s clearly a superposition of both “whiskey” and “whisky”, which will satisfy everyone.

Posted by: John Armstrong on June 2, 2010 9:05 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

Now I am really jealous that I didn’t get to tag along. I am going to have to demand a whiskey night when you get back.

Posted by: Alex Hoffnung on May 30, 2010 4:36 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

Okay, but you need to finish your thesis, pass your thesis defense and watch Casablanca first.

Posted by: John Baez on May 30, 2010 11:50 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

Movies are today what Greek mythology was 100 years ago, I’d say that, if Freud lived today, it would not be “Oedipus complex” but rather “Skywalker complex”.

So you definitely need to watch Casablanca, it’s referenced everywhere.

Posted by: Tim van Beek on May 30, 2010 3:06 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

I can’t argue with that Tim. Plus watching Casablanca does sound way better than the first two items on the list. :)

Posted by: Alex Hoffnung on May 30, 2010 6:00 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

As for most influential/best books, I’m just now reading reviews of new books on Ayn Rand and a piece about the anniversary of To Kill a Mockingbird

Posted by: jim stasheff on May 31, 2010 1:24 PM | Permalink | Reply to this

What happened?

Àpart from the whisky-session, anything noteworthy?

(This question is meant to be addressing anyone who attended. I am aware that John might be offline travelling across the continents.)

I am looking at the abstracts.

I have some idea of what John was talking about in “Duality in Logic and Physics”.

John Barrett’s abstract is

John Barrett: State sum models, induced gravity and the spectral action

I will give a new proposal for the spectral action for gravity coupled to matter, taking into account the phenomenon of induced gravity. This is then used to give some perspectives for the construction of a quantum theory of gravity coupled to matter using state sum models based on a tricategory.

Is “spectral action” here meant in the sense of Connes, Chamseddine et al? A functional on a space of spectral triples that evaluates to something close to the Einstein-Hilbert action coupled to this and that in the case that the Dirac operator of the triple is one for an ordinary Riemannian manifold?

What tricategory does he consider?

Louis Crane’s abstract begins with

Louis Crane, The category of spacetime regions

The purpose of this talk is to propose a connection between the spin foam approach to quantum gravity and a branch of abstract homotopy theory called model category theory.

Ah! See (,1)(\infty,1)-categories enter the field of state sum models for quantum gravity.

It continues:

The spin foam approach models regions in spacetime by four dimensional simplicial complexes. The quantum theory of the metric on them is constructed by putting representations of the Lorentz group on the triangulation and combining them in a specific way, called a categorical state sum. Recently a technical advance has removed the problems that plagued the theory, so that a finite computational theory of quantum general relativity seems to be available. The deeper interpretational problems remain: what do the specific complexes mean? Are they triangulations of an underlying point set, or Feynman diagrams of a fundamental theory? If they are just approximations, how does the exact theory emerge? How does the Bekenstein bound relate to this picture? How does the classical limit of the models approach general relativity?

Any specific answers to these questions?

And then:

What we are going to propose is that the state sum models on different simplicial complexes can be modified to fit together to form what is called a model category.

What are the weak equivalences?

We can then incorporate the Bekenstein bound into the model by using a technique from model category theory called localization. The flow of information from a system to its exterior would be a map in the category,

Hm, above it sounded as if the objects of the model category are supposed to be state sum models on different simplicial complexes. How would a morphism of such encode “the flow of information from a system to its exterior”??

and the localization of a model of the system with respect to this map would give a simplicial complex on which the exact effective theory would be computed.

Hm, does this refer to taking a state sum model on some simplicial complex and then passing to a fibrant-cofibrant replacement?

Does anyone have more details?

Posted by: Urs Schreiber on June 1, 2010 12:21 PM | Permalink | Reply to this

Re: What happened?

I just got home… gotta prepare for my classes. I’ll write a detailed report in This Week’s Finds when I have time. A lot of interesting stuff!

Is “spectral action” here meant in the sense of Connes, Chamseddine et al?

Yes; as you probably recall, Barrett came up with a nice formulation of the Standard Model using this approach. In this talk he merely suggested that if you get the right matter fields and the right matter-geometry couplings, you may not need to put an Einstein-Hilbert term in the action, since it could arise thanks to Sakharov’s ‘induced gravity’ idea. Most of his talk was concerned with describing matter in 3d gravity using a tricategorical approach, which is a warmup for describing matter in 4d gravity using a tetracategorical approach.

What tricategory does he consider?

Well, in fact, most of his talk was about some very nice general abstract nonsense involving the tricategory BicatBicat. I’ll either explain it or point people to the video when that comes online.

As for Louis Crane’s talk:

Hm, does this refer to taking a state sum model on some simplicial complex and then passing to a fibrant-cofibrant replacement?

Yes — all your guesses sound correct to me. He wants to use the state sum model to create some Bousfeld localization of the category of simplicial sets, which ‘trims it down’ enough to get the Bekenstein bound… or something like that.

Here’s a picture taken during his talk. The impressive blackboard also contains leftover pictures from John Barrett’s talk:

Does anyone have more details?

I have some, since I spent much of my spare time over the weekend talking to John and Louis — old friends whom I haven’t seen for a long time.

Sometime I will try to explain what I learned. Luckily Louis will soon come out with a paper on his ideas. But he said it’s still quite preliminary.

I also learned a lot of other great things during this conference. For example, I learned a lot about using symmetric monoidal categories with duals to describe ‘complementary observables’, the ‘collapse of the wavefunction’, and other phenomena in quantum theory. I want to explain this stuff too. We’ll see how much time I have to do this!

Posted by: John Baez on June 1, 2010 6:11 PM | Permalink | Reply to this

Re: What happened?

We’ll see how much time I have to do this!

Surely someone else in the audience could help out, or are we learning that as Chief Expositor you are inimitable?

Posted by: David Corfield on June 2, 2010 9:43 AM | Permalink | Reply to this

Re: What happened?

Not inimitable, merely unimitated. At least for this workshop.

Posted by: John Baez on June 2, 2010 4:13 PM | Permalink | Reply to this

Re: What happened?

Hmmm, quantum theory is the most accurate and thus valid scientific theory we have. Well, this new paper in Nature Physics demonstrates a quantum experimental violation of macroscopic reality:

I wonder if the “quantum feedback” described in this paper might have some future use in quantum computation? Also, I am very skeptical of any attempts to describe how the wavefunction “collapses”.

Note that the last author of the above paper is from UC Riverside (small world).

Posted by: Charlie Stromeyer on June 2, 2010 4:31 PM | Permalink | Reply to this

Re: What happened?

Charlie Stromeyer wrote:

Also, I am very skeptical of any attempts to describe how the wavefunction “collapses”.

Me too! The good thing about the new work I’m describing is that it avoids this issue and simply formalizes, using dagger-compact categories, what the effects of this ‘collapse’ are typically taken to be:

And the authors are too intelligent to use the term ‘collapse’. I mainly did that to get people interested. Instead, they use the language of environmentally induced decoherence. But what really matters — the real meat — is their diagrammatic formalism, introduced starting here after some preliminaries on dagger-compact categories and ‘classical structures’ as commutative dagger-Frobenius algebras in such categories.

Note that the last author of the above paper is from UC Riverside (small world).

Yeah, I know him — we’re on the library committee together. But we never get time to talk about physics! So thanks for pointing out that paper.

By the way, I added a link to the arXiv version of the paper. Nature sucks. The Nature Publishing Group is currently trying to boost the prices of 77 journals they sell to the University of California by a whopping 400%. When they bought Scientific American they raised the price of an institutional subscription seven-fold. I recently wrote an article for Scientific American (coming out soon), and now I feel like a real rat. At least I got the rights to keep the text on my website. But it galls me to have done work for these bastards.

Posted by: John Baez on June 2, 2010 7:18 PM | Permalink | Reply to this

Re: What happened?

Thanks, John!

I figured you’d be too busy with other things to live-blog on less alcoholic matters ;-)

Is “spectral action” here meant in the sense of Connes, Chamseddine et al?

Yes; as you probably recall, Barrett came up with a nice formulation of the Standard Model using this approach.

Ah, right. That insight that the Connes-Chamseddine model predicts the same K-theoretic dimension of the Kaluza-Klein compactification as the heterotic superstring: 6.

I have to admit to my shame that I did forget that this was that John Barrett. Thanks for reminding me.

In this talk he merely suggested that if you get the right matter fields and the right matter-geometry couplings, you may not need to put an Einstein-Hilbert term in the action, since it could arise thanks to Sakharov’s ‘induced gravity’ idea.

Hm, that sounds a bit weird, there must be some important fine-print here that I am missing: a central point of what makes the “spectral action” principle interesting is that none of the terms of the action is put in by hand. They all drop out automatically by forming a certain exponentiated trace of the Dirac operator.

Posted by: Urs Schreiber on June 2, 2010 9:17 AM | Permalink | Reply to this

Re: What happened?

Urs wrote:

Hm, that sounds a bit weird, there must be some important fine-print here that I am missing: a central point of what makes the “spectral action” principle interesting is that none of the terms of the action is put in by hand. They all drop out automatically by forming a certain exponentiated trace of the Dirac operator.

Maybe this is all I was trying to say. I’m a bit confused, since I only vaguely remember a slide in John Barrett’s talk where he wrote down the usual spectral action and then some other action. I’ll say something more coherent after the videos come out.

But it may only matter a little bit, at least for now, because John really didn’t do anything with the spectral action principle except use it to motivate the work he’s attempting now.

Namely: instead of trying to dream up spin foam models that approximate a quantum theory based on the Einstein–Hilbert action (like the Barrett–Crane model or the closely related but much better-behaved EPRL model), he wants to dream up spin foam models that approximate a quantum theory based on some sort of spectral action. For this, a key step will be to get fermions into the game. And he wants to do this in an elegant manner, using higher categories. And that’s what his talk was really about.

Posted by: John Baez on June 2, 2010 4:28 PM | Permalink | Reply to this

Re: What happened?

Okay, during a break in a conference here let me quickly say something about John Barrett’s talk, adding to what John said above. He came up with the following intriguing idea which struck me completely from left field:

If we use a higher category like 2Cat to model quantum gravity, then matter corresponds to the stuff inside a particular 2-category, while geometry corresponds to the external stuff in 2Cat.

Weird eh? But pretty nifty. Here’s roughly how it goes. Take a diagram in the 3-category 2Cat (objects are 2-categories, 1-morphisms are functors, 2-morphisms are transformations, 3-morphisms are modifications, everything weak of course!). Make sure you draw it in string diagram form instead of globular form. It will be a 3d diagram consisting of planes, lines and circles. Now, project that diagram onto a surface, so as to get a 2d diagram.

Now, draw a string diagram inside the source 2-category referred to above. So now we have two diagrams: one in 2Cat that has been projected to a plane (this will correspond to “geometry”), and one in CC, where CC is a 2-category (this internal diagram will correspond to “matter”).

Here’s the cool step: draw the two diagrams on top of each other (you might need to move the one a bit to the side so as to make the lines not lie directly on top of each other).

From this, you can do some hocus pocus and get out a quantity which will be the action in your theory of quantum gravity! So, hey presto, we’ve combined geometry and matter in one foul swoop. Something like that anyway! The videos will be online pretty soon anyhow…

Posted by: Bruce Bartlett on June 4, 2010 8:33 AM | Permalink | Reply to this

Re: What happened?

Actually, this idea has been around for some years now, but it does sound like a cool talk.

Posted by: Kea on June 4, 2010 8:36 AM | Permalink | Reply to this

Re: What happened?

By the way, the diagram I’m referring to (where you draw the two diagrams “on top” of each other) is available in John’s photo above (it’s the one with the white and the pink lines in the center roughly).

Posted by: Bruce Bartlett on June 4, 2010 8:38 AM | Permalink | Reply to this

Re: What happened?

Bruce wrote:

If we use a higher category like 2Cat to model quantum gravity, then matter corresponds to the stuff inside a particular 2-category, while geometry corresponds to the external stuff in 2Cat.

Thanks for taking the time to summarize a bit more of John Barrett’s talk!

By the way, I think you accidentally switched the words geometry and matter in this sentence.

Example: in the Ponzano–Regge model of 3d quantum gravity where our particular monoidal category is Rep(SU(2))Rep(SU(2)), irreducible objects in our monoidal category are spins and these describe lengths of edges of the tetrahedra in a triangulation of spacetime. So, here we are seeing geometry.

But the ‘center’ of Rep(SU(2))Rep(SU(2)) can be found in 2Cat by looking inside hom(Rep(SU(2)),Rep(SU(2))hom(Rep(SU(2)), Rep(SU(2)). And irreducible objects of the center are spins and masses, which describe particles. So, here we are seeing matter.

For more details try this paper:

and the many references therein!

John’s real goal is to tackle the 4d version of this idea. Then instead of particles with braid group statistics you can get particles with ordinary Bose or Fermi statistics, and also string-like excitations with very interesting statistics.

Posted by: John Baez on June 4, 2010 5:56 PM | Permalink | Reply to this

Re: What happened?

Ok, thanks for setting me straight here. In some sense it’s a bit of a shame really that it’s the matter that’s the “external” stuff, and the geometry that’s the “internal” stuff. Because now suddenly quantum gravity and category theory are at odds: in quantum gravity, it would be nice to “express everything, including the matter, as secretly coming from the geometry”, while in category theory, one often wants to express everything in terms of “external structure”, i.e. “ask not what concrete objects make up your category… ask how your category relates to other categories”! Anyhow, that’s just a vague philosophical ramble.

Posted by: Bruce Bartlett on June 6, 2010 11:59 AM | Permalink | Reply to this

Re: What happened?

The above discussion brings Bruce to the conclusion:

now suddenly quantum gravity and category theory are at odds

Sorry, but I don’t see anything in the above discussion that is even remotely precise/non-vague/done-deal enough enough to justify making any statements along these lines. Remember that there may be readers here who do not have a good feeling for the process that involves playing around with mathematical structures in order to see if and how they may match aspects of physical reality. I am afraid they can get away with a wrong impression of that process.

I’d rather somebody tried again explaining what Barrett was actually suggesting. From the above summaries, I still don’t get a clue, to be frank.

Posted by: Urs Schreiber on June 6, 2010 4:36 PM | Permalink | Reply to this

Re: What happened?

All John Barrett actually did in his talk is describe a diagrammatic calculus that simultaneously handles objects, morphisms and 2-morphisms inside a fixed 2-category CC and also objects, morphisms, 2-morphisms and 3-morphisms in 2Cat2Cat. The former is what Bruce is calling ‘internal’, the latter ‘external’.

To understand what this might have to do with physics one needs to look at examples. The most familiar example is matter in the Ponzano-Regge model of 3d quantum gravity, where geometry is described using the one-object 2-category CC coming from the monoidal category Rep(SU(2))Rep(SU(2)), while matter is described using a small portion of 2Cat, namely the ‘center’ of CC,

Z(C)=End(1 1 C)Z(C) = End(1_{1_{C}})

John Barrett didn’t actually recall this example, but I know this is what is motivating his endeavor.

I don’t share Bruce’s feeling that “suddenly quantum gravity and category theory are at odds” in this story. On the contrary, I think it’s a tremendously wonderful thing how starting from C=Rep(SU(2))C = Rep(SU(2)) we can get, not only 3d Riemannian quantum gravity, but also matter — just by using higher categories.

I’ve been excited about this for a long time. I wrote about it back in week222, saying:

The second most exciting thing at Loops ‘05, in my biased opinion, was the work of John Barrett, Laurent Freidel, Karim Noui and others on "matter without matter" in 3d quantum gravity. Simply by carving a Feynman-diagram-shaped hole in 3d spacetime and doing quantum gravity on the spacetime that’s left over, you get a good theory of quantum gravity coupled to matter! You can even take the limit as Newton’s gravitational constant goes to zero and get ordinary quantum field theory on flat spacetime!

Check these out:

23) John Barrett, Feynman diagams coupled to three-dimensional quantum gravity, available as gr-qc/0502048.

John Barrett, Feynman loops and three-dimensional quantum gravity, Mod. Phys. Lett. A20 (2005) 1271. Also available as gr-qc/0412107.

24) Laurent Freidel and David Louapre, Ponzano-Regge model revisited I: gauge fixing, observables and interacting spinning particles, Class. Quant. Grav. 21 (2004) 5685-5726. Also available as hep-th/0401076.

Laurent Freidel and David Louapre, Ponzano-Regge model revisited II: equivalence with Chern-Simons, available as gr-qc/0410141

Laurent Freidel and Etera R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and effective field theory, available as hep-th/0502106.

25) Laurent Freidel, Daniele Oriti, and James Ryan, A group field theory for 3d quantum gravity coupled to a scalar field, available as gr-qc/0506067.

26) Karin Noui and Alejandro Perez, Three dimensional loop quantum gravity: coupling to point particles, available as gr-qc/0402111.

This is mindblowingly beautiful, especially because lots of it is already mathematically rigorous, and we can easily make more so. It’s even related to n-categories: my student Jeffrey Morton presented a poster on this aspect.

Together with my student Derek Wise, Jeffrey Morton and I plan to have a lot of fun studying this stuff. So, I won’t talk about it more now - I’ll probably get around to saying more someday, especially about how the whole story generalizes to 4 dimensions.

And then Derek Wise and Jeff Morton wrote papers on this stuff, which I explained in week232 and week242.

The basic idea is simple: extended TQFTs give Hilbert spaces not just for closed manifolds describing ‘space’, but also for manifolds with labelled boundaries, which describe ‘space with matter’. And it connects beautifully to the cobordism hypothesis. The big question is just whether we can use this idea to get physically realistic theories, not just toy models.

Posted by: John Baez on June 6, 2010 7:09 PM | Permalink | Reply to this

Re: What happened?

Simply by carving a Feynman-diagram-shaped hole in 3d spacetime and doing quantum gravity on the spacetime that’s left over, you get a good theory of quantum gravity coupled to matter!

There must be some fine print to this that is a little subtle, I imagine.

A priori if we talk about encoding quantum gravity in terms of cobordism representations we imagine that we start with an action functional on a space of metrics and matter fields on a given cobordism, imagine we are able to make sens of the path integral over that in one way or another, and end up with a cobordism representation that to each codimension 1 space assigns a space of states of configurations of the gravity and matter fields, and to each cobordism a propagator between these.

So gravity coupled to matter, as usually imagined, already is supposed to give a cobordism representation on unmarked cobordisms.

What you are talking about is a curious step where we consider cobordisms with marked singularities on them that you want to think of as Feynman diagrams of matter fields. Curiously, on a single cobordisms with singularities, this would assign the fully quantized dynamics of the gravitational field, and at the same time, from what you say, encodes now a single Feynamn diagram for the matter fields.

So to get from there to the actual quantum field theory of the matter fields, one would now have to do a sum of our representation of marked cobordisms over all possible ways to draw Feynman diagrams into them, renormalize that sum, Borel resum it. What is the result of that? While we do all these operations in order to get the perturbation series for the matter field, at the same time this would amount doing them to the field of gravity, which already as assumed to be quantized, and non-perturbatively so.

I find it a bit difficult to see what is supposed to be going on here.

I am wondering: the Feynman-diagram-like singularities that you are looking at: are they really to be thought of as encoding matter, or not rather something like branes? If I think of them as branes, i seem to get a more natural interpretation that matches with what is being thought about in other corners.

Posted by: Urs Schreiber on June 8, 2010 11:43 AM | Permalink | Reply to this

Re: What happened?

Urs wrote:

So gravity coupled to matter, as usually imagined, already is supposed to give a cobordism representation on unmarked cobordisms.

That’s what one might usually imagine, but it’s definitely not what people are studying in the papers I listed. Instead they are looking at pure 3d quantum gravity, without matter, and treating that as an extended TQFT. Then matter shows up automatically when one considers labellings.

(Technical subtlety: if we work with 3d Riemannian gravity with cosmological constant equal to zero, we get the Ponzano–Regge theory. Most of the papers I cited study this theory. This theory has some divergences that would need to be tamed to get a full-fledged extended TQFT. If we work with a nonzero cosmological constant with the right sign, we get the Turaev–Viro theory. This is an honest extended TQFT without divergences.)

What you are talking about is a curious step where we consider cobordisms with marked singularities on them that you want to think of as Feynman diagrams of matter fields. Curiously, on a single cobordisms with singularities, this would assign the fully quantized dynamics of the gravitational field, and at the same time, from what you say, encodes now a single Feynman diagram for the matter fields.

Right.

By the way, I only expect this to work in 3d quantum gravity, not 4d. In 3d, gravity has the special property that spacetime is flat away from matter. In 4d, gravity has propagating degrees of freedom, and things get a lot more complicated.

More later — gotta teach a calculus review session!

Posted by: John Baez on June 8, 2010 11:04 PM | Permalink | Reply to this

Re: What happened?

Okay, I’m back….

Urs wrote:

So to get from there to the actual quantum field theory of the matter fields, one would now have to do a sum of our representation of marked cobordisms over all possible ways to draw Feynman diagrams into them, renormalize that sum, Borel resum it.

Right.

What is the result of that?

Well, I don’t think anyone has actually carried out all these steps yet. I believe they have only obtained the Feynman rules — i.e. the rules for writing down a possibly divergent integral giving the amplitude of any fixed Feynman diagram. These rules depend on Newton’s gravitational constant G NG_N. At G N=0G_N =0 they’ve been shown to reduce to the usual rules for quantum field theory in Minkowski spacetime. For nonzero G NG_N they give something new and interesting, which may describe 3d quantum gravity coupled to matter, assuming the further steps you describe can be carried out.

I urge you to look at this paper:

  • Laurent Freidel, Etera R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory.

    Abstract: We study the no gravity limit G N0G_{N}\to 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G NG_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators.

I am wondering: the Feynman-diagram-like singularities that you are looking at: are they really to be thought of as encoding matter, or not rather something like branes?

I can see why you’d want to call them branes, and that’s fine. Indeed, once upon a time I gave a talk at the Perimeter Institute on the 4d analogue of these ideas, which involves 1d extended objects — which I called ‘strings’ — coupled to 4d BF theory, instead of 0d extended objects (particles) coupled to 3d BF theory (= for example, 3d gravity). And afterwards, Malcolm Perry came up and said the word ‘1-branes’ would be better than ‘strings’ for these 1d extended objects. And he explained why.

However, the ‘0-branes coupled to 3d gravity’ really do act exactly like ordinary matter, at least in the limit where the gravitational constant G NG_N goes to zero. So, people want to use them describe matter.

Posted by: John Baez on June 9, 2010 3:01 AM | Permalink | Reply to this

Re: What happened?

Something which might be related to this stuff is the following which has been confusing me lately: a 1-2-3 extended 3d TQFT is a 2-functor

(1)Z:3Cob 22Hilb Z \colon 3Cob_2 \rightarrow 2Hilb

where 3Cob_2 is the 2-category whose objects are closed oriented 1-manifolds, whose 1-morphisms are cobordisms between these, and whose 2-morphisms are diffeomorphism classes of cobordisms between those, and the 2-category 2Hilb can be replaced if necessary by your favourite “2-vector space”.

It’s pretty clear that the category C:=Z(S 1)C := Z(S^1) assigned to the circle is a ribbon category. But suddenly this is somewhat of a paradox to me: we get out invariants of knots and links from a formalism which only apriori had to do with invariants of manifolds and cobordisms.

I know this sounds silly, after all this is what all that 3d TQFT / knots stuff was all about in the 1990’s… but if you look at the books (eg. Turaev, Bakalov and Kirillov) you’ll see that they don’t want to (or didn’t know how to) work with 2-categories of cobordisms, so they put in knots and links explicitly by hand into their formalism (they will talk about a “CC“-extended 3d TQFT” by which they mean CC is some fixed semisimple category and “extended” means the cobordisms between the 2d surfaces come with knots and links sitting inside them). So it wasn’t surprising there: they put in knots and get out invariants of knots. But in our modern formalism of higher-categorical cobordisms it seems more interesting… we are getting out more than we explicitly put in (analogous to “putting in geometry gives you matter as well”).

Posted by: Bruce Bartlett on June 9, 2010 4:56 PM | Permalink | Reply to this

Re: What happened?

I said:

So gravity coupled to matter, as usually imagined, already is supposed to give a cobordism representation on unmarked cobordisms.

John replied:

That’s what one might usually imagine, but it’s definitely not what people are studying in the papers I listed.

I understand that this is not what these articles are studying. What I do not understand is how what these articles are studying is “quantum gravity coupled to matter”.

To me it looks like what they are studying is a curious fact that some TFTs on cobordisms with embedded markings compute amplitudes of single Feynman graphs (and that for some QFT on Minkowski space, right!?).

Instead of giving me the feeling that suddenly this gives a deep explanation for how matter appears in pure functorial QFT, it confuses me: what on earth does it mean that a non-perturbative QFT that is allegedly a quantum theory of gravity computes single summands of some perturbation series of matter?

Not that I don’t find the fact interesting or remarkable, I just don’t follow how this fact can be read as giving “quantum gravity coupled to matter”. Can you tell my why you think that is the right way to describe this fact?

Posted by: Urs Schreiber on June 9, 2010 9:21 PM | Permalink | Reply to this

Re: What happened?

Urs wrote:

To me it looks like what they are studying is a curious fact that some TFTs on cobordisms with embedded markings compute amplitudes of single Feynman graphs (and that for some QFT on Minkowski space, right!?).

No, not for a quantum field theory on Minkowski space! For a quantum field theory of a novel sort, where the motion of the particles affects the geometry of spacetime.

Only in the the limit where the gravitational constant goes to zero do the Feynman amplitudes reduce to the usual thing you’d get for a quantum field theory on Minkowski space. That’s just a consistency check, though a remarkable result in itself.

The interesting part is that when the gravitational constant is nonzero, the particles affect the geometry of spacetime exactly as you would hope for in a theory of 3d gravity. You can compute gravitational holonomies around the particle’s worldlines and they work out exactly right.

Posted by: John Baez on June 9, 2010 9:46 PM | Permalink | Reply to this

Re: What happened?

You can compute gravitational holonomies around the particle’s worldlines and they work out exactly right.

Sorry for being a pain on this point, but I still don’t understand why in quantum-\langlegravity coupled to matter\rangle it should make sense to speak of any worldlines of particles at all.

If there is a fixed worldline of a single particle, then the matter field content of which that particle might be a quantum has not yet been quantized.

And my confusion goes further: even if I assume that I did understand what it would mean to have a classical particle trajectory in a quantum gravitational background, then I would be hard-pressed to see what gravitational holonomies around it were.

I would know what gravitational holonomies of a classical gravitational field background around a line-like singularity would be. But somehow the claim is that we are speaking about quantum gravity coupled to quantum/classical(?) matter here, instead. Which confuses me.

Posted by: Urs Schreiber on June 9, 2010 10:16 PM | Permalink | Reply to this

Re: What happened?

Concerning particle trajectories in 3d quantum gravity I want to point out that the situation is completely analogous to the one found in other more standard situations.

The notion of the trajectory makes sense only at the classical level. One writes an action principle that describes the coupling of the 3d gravitational degrees of freedom to those of a particle moving along a one dimensional world line. The world line geometry (namely the dynamics of the particle in 3d spacetime) as well as the spacetime geometry are determined by Einstein’s equations.

Important remark: the above statement means that the particle trajectory is not given a priori so interpreting where the particle is and what it does as it evolves in spacetime is a subtle issue related to what is observable and what is not in a diff invariant theory as general relativity.

Now, one takes this classical system and quantizes it.
At the quantum level is no longer true that the notion of particle trajectory remains. It is gone just as it is gone in usual quantum mechanics due to the uncertainty principle. Even when in spin foams there seem to be a “trajectory” along which the conical singularity is placed one should keep in mind that a spin foam is a term in a sum over histories producing physical transition amplitudes. It is just the analog of the sum over histories introduced by
Feynman to describe the unitary evolution operator in QM. In that case one can also loosely interpret a single term in the sum as defining a particle trajectory but we know that this interpretation has its limitations. In 3d quantum gravity it is just the same. When one has summed over spin foams to compute some transition amplitude the notion of where the particle went through is simply lost. There are no trajectories in 3d quantum gravity.

The above analogy with Feynman’s path integral is precise in the non perturbative sense. (No Feynman diagrams in this version of the argument, although I also agree with John’s remarks.)

Posted by: Alejandro Perez on June 10, 2010 1:03 PM | Permalink | Reply to this

Matter?

Alejandro Perez writes:

One writes an action principle that describes the coupling of the 3d gravitational degrees of freedom to those of a particle moving along a one dimensional world line.

Ah, so the Lagrangian is roughly of the form

(X 3,g,γ) X 3L EH(g)+ ΓL par(g,γ), (X_3, g,\gamma) \mapsto \int_{X_3} L_{EH}(g) + \int_{\Gamma} L_par(g,\gamma) \,,

where X 3X_3 is some 3-manifold, gg a metric on it, Γ\Gamma some graph, γ:ΓX 3\gamma : \Gamma \to X_3 some embedding of this graph into X 3X_3, L EHL_{EH} the Einstein-Hilbert Lagrangian or similar and L par(g,γ)L_ {par}(g,\gamma) is some action functional for a particle propagating in the gravitational background gg

?

And it is not of the form

(X 3,g,ϕ) X 3L EH(g)+L mat(ϕ,g), (X_3, g,\phi) \mapsto \int_{X_3} L_{EH}(g) + L_{mat}(\phi, g) \,,

where ϕ:X 3V\phi : X_3 \to V is some kind of function on X 3X_3 representing the configurations of a matter field and L mat(ϕ,g)L_{mat}(\phi,g) is a typical matter Lagrangian, say as for a free bosonic particle ϕ|ϕ| 2\phi \mapsto |\nabla \phi|^2

?

The latter is what is usually meant by “gravity coupled to matter”. The former is a setup that is usually known as the action functional for gravity in the presence of a brane.

Posted by: Urs Schreiber on June 10, 2010 2:08 PM | Permalink | Reply to this

Re: Matter?

Since nobody else is answering, and I’m in a slightly better mood today:

The Lagrangian is indeed of the ‘brane’ type you describe, Urs. But people in quantum gravity typically call this the Lagrangian for a ‘particle’ coupled to gravity.

Regardless of what this theory is called, the work we’re discussing shows that by quantizing a theory of this sort, we can derive a new set of ‘Feynman rules’ which reduce to the usual Feynman rules for perturbative quantum field theory in the limit where the gravitational constant G NG_N goes to zero. And so, this opens up a project of studying a new sort of ‘G NG_N-deformed quantum field theory’.

Jacques has raised a few of the questions one might worry about, here. My hope is that such questions can be addressed and one can use this theory – in the case of 3d spacetime! — to describe quantum matter in the presence of quantum gravity just as ordinary quantum field theory can be used to describe quantum matter in the absence of gravity. And part of why I have this hope is that the G NG_N-deformed Feynman rules are almost exactly like the ordinary Feynman rules. The only difference (unless I’m confused) is that the usual 3\mathbb{R}^3 momentum space is replaced by an S 3S^3 momentum space whose radius is inversely proportional to G NG_N.

Posted by: John Baez on June 14, 2010 9:43 PM | Permalink | Reply to this

Re: Matter?

The Lagrangian is indeed of the ‘brane’ type you describe, Urs.

Thanks for confirming this.

But people in quantum gravity typically call this the Lagrangian for a ‘particle’ coupled to gravity.

At least we have identified a source of the misunderstanding. A bit further down you write instead:

[…] to describe quantum matter in the presence of quantum gravity […]

I think I see now what is going on, but I find this use of language still confusing. I think one has to be careful here, with the different levels of what’s going on, as far as I can see:

  1. there is a worldvolume TFT that we think of as being nonperturbatively quantized gravity with “defects” (or “particles”, if you wish), and it

  2. gives rise to an effective perturbative background theory that looks like matter fields (this time in the usual sense!) in some deformed geometric background.

Isn’t that so?

It seems to me to be crucially important to distinguish these two levels. For instance it is not clear to me that there is indication that the effective background theory is quantum gravity. It seems more like a fixed, albeit deformed background, from what you say. But in any case, I understand now what you mean.

I’d be interested in some more details:

You say you get deformed Feynman rules. Feynman rules of what? Of a scalar particle? A free particle? Or a ϕ n\phi^n theory? Can one change this? Can one do this for fermionic particles? For different interaction terms?

Given a fixed shape of a Feynman diagram, how do you obtain the corresponding Feynman amplitude? Do you sum the 3d TFT over all possible cobordisms with all possible ways of drawing the given fixed Feynman diagram on them? How many such ways are there? Are there any extra rules or constraints for how to draw a Feynman diagram on a 3d cobordism, in this game? If there are no further rules, how do you see that the resulting sum converges (for a fixed single Feynman diagram)? If there are further rules, how do you determine them?

How do you non-perturbatively quantize the worldvolume theory, that on the 3d cobordisms, gravity coupled to defects/particles? Is this making use of non-perturbative Chern-Simons theory? I wasn’t aware that non-perturbative Chern-Simons with codimension 2-defects is understood to this extent. So maybe that’s not the way you do it? Where does your rule for how to assign amplitudes to marked 3d cobordisms come from?

Hm, too many questions. I suppose there is a chance that you’ll point me to the literature…

Posted by: Urs Schreiber on June 14, 2010 11:59 PM | Permalink | Reply to this

Re: Matter?

Urs wrote:

I think one has to be careful here, with the different levels of what’s going on, as far as I can see:

  1. there is a worldvolume TFT that we think of as being nonperturbatively quantized gravity with “defects” (or “particles”, if you wish), and it
  2. gives rise to an effective perturbative background theory that looks like matter fields (this time in the usual sense!) in some deformed geometric background.

Isn’t that so?

Yes, something like that. This subtlety is precisely the reason people find this stuff interesting. But they’re not string theorists, so they don’t use the same language you do.

… it is not clear to me that there is indication that the effective background theory is quantum gravity. It seems more like a fixed, albeit deformed background, from what you say.

It doesn’t start out as a fixed background: we are quantizing a classical theory of point particles that interact gravitationally. In that classical theory, each particle affects the spacetime geometry: it creates a curvature singularity along its worldline, following the rules of 3d general relativity.

But, when we quantize theory, the path integral — or at least a restricted portion of the path integral! (see below) — gives rise to deformed Feynman rules. They’re very much like the usual ones, but I believe the usual flat momentum space is replaced by a 3-sphere whose radius depends on the gravitational constant.

So perhaps for this aspect of the theory we could say we are working on a fixed ‘momentum space background’, or dually working on some sort of fixed noncommutative geometry as a ‘position space background’.

But in any case, I understand now what you mean.

Good!

You say you get deformed Feynman rules. Feynman rules of what? Of a scalar particle? A free particle? Or a ϕ n\phi^n theory? Can one change this? Can one do this for fermionic particles? For different interaction terms?

There’s a gap between what I believe can be done and what I’ve seen people do. The most advanced paper I know on the subject dates back to 2005. But this is suspiciously long ago — and suspiciously close to the time when I stopped working on quantum gravity! So, I bet more has been done by now. I can tell you what I know, but I hope someone else will know more.

You should be able to freely specify the masses and spins of your particles, and the interactions. The paper I just cited restricts attention to spin-0 particles, except at the end. A previous paper spends more time on spinning particles but apparently doesn’t include interactions (except for gravity).

Given a fixed shape of a Feynman diagram, how do you obtain the corresponding Feynman amplitude? Do you sum the 3d TFT over all possible cobordisms with all possible ways of drawing the given fixed Feynman diagram on them?

I’m not sure anybody has tried such an ambitious sum. Instead, people focus on fixing a cobordism with an embedded graph. A relevant buzzword is braided Feynman diagram.

Hm, too many questions. I suppose there is a chance that you’ll point me to the literature…

I did. But I really wish someone would point me to the more recent literature. I would hope there’s been progress since 2005, or at least a clear discussion of some obstacle that blocked further progress.

Posted by: John Baez on June 15, 2010 2:38 AM | Permalink | Reply to this

Re: Matter?

Thanks, John. I’ll make some further remarks.

I suggested that the situation has to be thought of as two-layered

[First] there is a worldvolume TFT that we think of as being nonperturbatively quantized gravity with “defects” (or “particles”, if you wish), and it

[Second] gives rise to an effective perturbative background theory that looks like matter fields (this time in the usual sense!) in some deformed geometric background.

You replied:

Yes, something like that.

Okay, so maybe we agree, then. But maybe not! Because next in response to my remark

[…] it is not clear to me that there is indication that the effective background theory is quantum gravity. It seems more like a fixed, albeit deformed background, from what you say.

you reply:

It doesn’t start out as a fixed background: we are quantizing a classical theory of point particles that interact gravitationally…

But that’s on the worldvolume, where you quantize a certain BF-theory (according to FL, p. 7)

…But, when we quantize theory, the path integral […] gives rise to deformed Feynman rules.

But that’s on an effective target space now. That’s something different! At least as I understand what’s going on. Maybe we haven’t agreed on that after all.

But actually it looks like Freidel-Livine would agree with the way I am phrasing it, at this point (I don’t think their Conclusions agree with me, though): on the bottom of p. 8 they write

[We] identify the Ponzano-Regge amplitudes [i.e. correlators of a 3d TFT] as providing a perturbative expansion […] which is interpreted as QFT amplitudes on a non-commutative geometry.

That “non-commutative geometry” is not the 3-sphere that the Ponzano-Regge/BF QFT is evaluated on, but a different effective background that depends on a choice of group GG which governs the TFT that is evaluated on the 3-sphere. That this non-commutative geometry still looks 3-dimensional is entirely a coincidence due to the fact that Freidel-Livine happen to choose first G= 3G = \mathbb{R}^3 (which, while being simple, serves to amplify the point that this is not the worldvolume 3-sphere that the 3dTFT is evaluated on ) and then G=SU(2)G = SU(2). If they had chosen more general groups, there would have been less risk of mixing up the worldvolume 3-sphere that we are restricting attention to with the effective (non-commutative) target space.

Then after computing the Feynman amplitudes of their graphs as correlators of their worldvolume theory, they identify in section 5 the effective action on target space that gives rise to these Feynman diagrams by perturbative quantization.

This is entirely as in worldline formalism for QFT and in string theory: you have a QFT on low dimensional cobordisms and interpret its correlators as Feynman amplitudes for a different theory on some effective background.

Then you look for an ordinary quantum field theory on the target space whose ordinary perturbative expansion produces these Feynman rules. This is the effective background theory that is defined by the given worldvolume theory. Both of these theories are closely connected but crucially very different: one is the “second quantization” of the other!

In particular, the “particles” (which I would call “defects”) of the worldvolume theory are very different from the matter on the effective target space that is described by the effective Feynman amplitudes: as we discussed, the “particles” of the worldvolume theory have an action given by a 1-particle action along a fixed graph (as on the worldline in the worldline formalism for QFT!) but quantizing such an action perturbatively does not give rise to Feynman diagrams at all. Feynman diagram describe matter whose action is of the usual matter form target|ϕ| 2+V(ϕ)\int_{target} |\nabla \phi|^2 + V(\phi), which is not the form of the action for your “particles” in the worldvolume.

I have to say that the effective background field theory which Freidel-Livine find on page 24 looks nothing like it contains “quantum gravity” on the target. It is precisely a ϕ 3\phi^3-theory on a single background NC-geometry.

They’re very much like the usual ones, but I believe the usual flat momentum space is replaced by a 3-sphere whose radius depends on the gravitational constant.

Yes, but that’s because Freidel-Livine choose the gauge group of their 3d TFT BF-theory to be SU(2)SU(2). It’s this choice that gives the deformed 3-dimensional background. A different choice here would yield a different effective background. In particular if they choose the group 3\mathbb{R}^3 instead, then the effective background theory is that of matter on ordinary flat space., despite the fact that the worldvolume theory is fully quantum and lives on the 3-sphere.

This subtlety is precisely the reason people find this stuff interesting.

Maybe just for emphasis, I would like to say: this “subtlety” is the whole starting point of worldline formalism of QFT and of perturbative string theory.

For the sake of it, let me recall how it goes:

  1. one observes that the Feynman amplitudes for the familiar theories of quantum field theory in 4-dimensional spacetime can be expressed as correlators of a 1-dimensional worldvolume QFT on 1-dimensional cobordisms with singularities.

    So perturbative quantum field theory can be understood as follows: we start with a cobordism representation of marked 1-d cobordisms, a non-perturbatively quantized field theory “on the worldline”, and find from it another quantum field theory, in a larger number of dimensions, whose perturbation series for correlators (the “S-matrix”) is obtained by summing the cobordism representation over all possible cobordisms with fixed in- and out markings.

  2. Then one observes that nothing in this prescription crucially depends on the worldvolume theory being 1-dimensional. In fact, the whole process becomes better behaved if the worldvolume theory is not 1-dimensional, since then interactions that used to be given by singular 1-manifolds becomes non-singular d-manifolds.

    So people tried this with d=2d = 2 and checked what happens if one postulates that the Feynman diagrams for a target space theory are secretly computed from cobordism representations of a 2-dimensionl theory. The result of this approach is called perturbative string theory.

  3. Now you seem to go a step further, even. You have now a 3-dimensional worldvolume theory, and want to interpret the correlators that it assigns to a given marked cobordism as the Feynman amplitude of another, effective theory. Some kind of effective target space theory of a membrane theory. In fact, you consider a mixture of this with the above, because your membranes are assumed to come with 1-d graphs drawn on them. (So one part of the point here seems to be the reproduce a worldline theory from embedding the worldline into a higher dimensional worldvolume and having a topological theory with defects there.)

    In your case it seems that by coincidence the worldvolume theory has the same dimension as the effective theory that it describes, and this leads to language that does not distinguish the two, even though they are very different objects…

Finally, we had this exchange:

Given a fixed shape of a Feynman diagram, how do you obtain the corresponding Feynman amplitude? Do you sum the 3d TFT over all possible cobordisms with all possible ways of drawing the given fixed Feynman diagram on them?

I’m not sure anybody has tried such an ambitious sum. Instead, people focus on fixing a cobordism with an embedded graph.

Oh. Now I am checking Freidel-Livine. On p. 8 they write:

Let us also point out that the Feynman evaluation corresponds to the simplest topology - the one of the 3-sphere. The spin foam framework allows to generalize these Feynman graph amplitudes to arbitrary topologies. It should be interesting to understand better what effect the non trivial topology of the ambient manifold has on Feynman graph evaluation.

Oh. So that means the pictures you have been posting here do not actually correspond to what is discussed in the literature??

Posted by: Urs Schreiber on June 15, 2010 8:59 AM | Permalink | Reply to this

Re: Matter?

A few comments:

Maybe we havenít agreed on that after all.

I think we agree, mostly, except that things you find important I find distracting and vice versa.

Part of the confusion may arise from the fact that there are two recipes for computing Feynman amplitudes in the Ponzano–Regge model:

1. There’s the original ‘position space’ recipe, which gives an operator for a labelled graph embedded in a 3d cobordism. The idea is to triangulate the cobordism, label edges with spins, and sum something over those labellings.

2. There’s the dual ‘momentum space’ recipe, which is equivalent to first in the special case of a closed graph embedded in a 3-sphere. Here we use a slight generalization of the Fourier transform to rewrite the amplitude as a multiple integral over the gauge group SU(2)SU(2). This is investigated in this series of papers:

  • Laurent Freidel and David Louapre, Ponzano-Regge model revisited I: gauge fixing, observables and interacting spinning particles, Class. Quant. Grav. 21 (2004) 5685-5726. Also available as hep-th/0401076.
  • Laurent Freidel and David Louapre, Ponzano-Regge model revisited II: equivalence with Chern-Simons, available as gr-qc/0410141.
  • Laurent Freidel and Etera R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and effective field theory, available as hep-th/0502106.

Since the 3-sphere is a special case of a 3d cobordism, it can show up as our ‘position space’. Since the ‘momentum space’ SU(2)SU(2) is also a 3-sphere, there’s room to get confused. But it’s not really bad: it’s really just a generalization of how Feynman diagrams can be evaluated in both position space (e.g. 3\mathbb{R}^3 in a 3d quantum field theory) or momentum space (also 3)\mathbb{R}^3).

Yes, but that’s because Freidel-Livine choose the gauge group of their 3d TFT BF-theory to be SU(2)SU(2). It’s this choice that gives the deformed 3-dimensional background.

Right. Of course this choice is not arbitrary — SU(2)SU(2) is the gauge group for 3d Riemannian quantum gravity, and the fact that the dimension of this group matches the dimension of spacetime is one of the key coincidences that makes 3d gravity work so beautifully! If this weren’t true, the triad field ee and the SU(2)SU(2) connection AA could not be canonically conjugate, and 3d gravity would not admit a BFB F description. This BFB F description of 3d gravity is what allows for a beautiful discretization of the theory, namely the Ponzano–Regge model.

(You probably know all this, but I think you’ve decided we’re also supposed to be setting a good example by educating the audience.)

A different choice here would yield a different effective background. In particular if they choose the group 3\mathbb{R}^3 instead, then the effective background theory is that of matter on ordinary flat space, despite the fact that the worldvolume theory is fully quantum and lives on the 3-sphere.

Right. As I’ve said a couple of time, the 3\mathbb{R}^3 theory is the limit of the SU(2)SU(2) theory as Newton’s gravitational constant goes to zero, since we should think of the ‘radius’ of the group SU(2)SU(2) as inversely proportional to Newton’s gravitational constant. We say that 3\mathbb{R}^3 is an ‘Inonu–Wigner contraction’ of SU(2)SU(2). And in this G N0G_N \to 0 limit where SU(2)SU(2) flattens out to the usual flat momentum space 3\mathbb{R}^3, the theory Freidel and coauthors consider reduces to the usual theory of Feynman diagrams in the absence of gravity.

Oh. So that means the pictures you have been posting here do not actually correspond to what is discussed in the literature??

It corresponds to what Jeffrey Morton discussed in his thesis on this subject. His thesis and Freidel’s work are treating different aspects of the same theory: the Ponzano–Regge model. Freidel is mainly focused on the case where our 3d cobordism is a 3-sphere: see equation (15) here. Morton is interested in general 3d cobordisms with corners, but he replaces SU(2)SU(2) by a finite group to make some things easier.

You might see a bit better how these stories are related by reading the work of Barrett, Perez and others. Freidel’s papers also have enormous bibliographies! The Ponzano–Regge model has been studied quite a bit. But quite possibly nobody has connected up all the viewpoints.

One reason may be that the loop quantum gravity crowd considers 3d quantum gravity ‘too easy’. In particular, Barrett, Freidel and others consider this 3d theory as just a warmup for a hoped-for 4d theory, and they’re putting most of their energy into that. In particular, while Barrett’s talk at Oxford was about 3d stuff, where it’s easy to draw pictures, he is really working on 4d stuff.

So, it’s too bad you aren’t having this conversation with them. They might actually get some ideas for research projects from the questions you and Jacques are raising. In particular, I don’t know how much anyone has studied what it’s like to sum over ‘G NG_N-deformed Feynman diagrams’.

In a previous life I considered writing a book on 3d quantum gravity. It’s fascinating, it’s like a dream world where you can actually calculate things and relate them beautifully to higher category theory. In that previous life, I might have spent a year trying to work out some of the answers here…

Posted by: John Baez on June 15, 2010 4:40 PM | Permalink | Reply to this

Re: What happened?

Urs wrote:

Sorry for being a pain on this point, but I still don’t understand why in quantum-⟨gravity coupled to matter⟩ it should make sense to speak of any worldlines of particles at all.

Of course you should already be worried about this in quantum field theory. And indeed, Bohr didn’t like it when Feynman first drew those famous diagrams, because the edges of those diagrams seem to describe particles having definite positions at each time, and that’s not how it works in quantum theory. But really, of course, these diagrams are symbols standing for the process of integrating over all possible worldlines. And that’s roughly what’s going on here, too. But now of course it’s even more tricky to interpret.

So perhaps I shouldn’t have said ‘worldline’: perhaps I should have said ‘Feynman diagram edge’. In fact it’s probably dangerous to use any pre-existing terminology for a theory of this new sort. So let me try to say some things of a more mathematical character:

For any value of the gravitational constant one can calculate an amplitude for a Feynman diagram embedded in a 3-manifold, with edges labelled by masses and spins and with vertices labelled by intertwining operators. You can interpret this amplitude as an integral over geometries that are flat away from the Feynman diagram, and have specified conjugacy classes of holonomies around the diagram’s edges — where these conjugacy classes are specified by the masses and spins. But in fact these amplitudes can be computed in a purely algebraic way starting from Rep(SU(2))Rep(SU(2)).

In the limit where the gravitational constant goes to zero, these Feynman amplitudes reduces to the usual Feynman amplitudes for a quantum field theories on 3d Euclidean space. Otherwise you get corrections depending on the gravitational constant.

You can also add extra observables corresponding to Wilson loops around the Feynman diagram edges. The expectation values are what you’d expect from a connection that was flat away from the Feynman diagram edges and singular along the edges, with the singularity you’d get from a classical particle with the specified mass and spin.

I should perhaps again emphasize that we only expect something so simple to happen in 3d quantum gravity, where spacetime is flat except where there’s matter. This is why, for example, the mass and spin of a classical particle determines the conjugacy class of the holonomy around its worldline! It’s also why we can smoothly deform the worldline of the classical particle without changing the holonomy around a loop that encircles that worldline. Neither of these things would be true in 4 dimensions… and these things make the path integral computation of the Wilson loop expectation values infinitely simpler than they would be otherwise. So if your intuition is based on the more interesting higher-dimensional theories, everything I’m saying should seem completely crazy.

Posted by: John Baez on June 9, 2010 11:33 PM | Permalink | Reply to this

Re: What happened?

I’m not sure I understand this.

Feynman diagrams represent terms in a perturbation expansion of a QFT. They’re not ‘the QFT itself’. Indeed, QFTs are invariant under field-redefinitions. But, under a field-redefinition, the expansion in terms of Feynman diagrams changes dramatically.

Is this theory invariant under field-redefinitions of the scalar field theory?

Posted by: Jacques Distler on June 10, 2010 12:04 AM | Permalink | PGP Sig | Reply to this

Re: What happened?

Jacques writes:

Is this theory invariant under field-redefinitions of the scalar field theory?

Good question. This may be a calculation that someone could just sit down and do… but I don’t think about this stuff anymore. Someone like Laurent Freidel might know.

Posted by: John Baez on June 10, 2010 12:37 AM | Permalink | Reply to this

Re: What happened?

Is this theory invariant under field-redefinitions of the scalar field theory?

Good question.

It seems to me that an affirmative answer is a necessary (but perhaps not sufficient) condition for answering Urs’s query as to why this construction has any right to be called “quantum gravity coupled to a QFT.”

This may be a calculation that someone could just sit down and do… but I don’t think about this stuff anymore. Someone like Laurent Freidel might know.

If he does, it would be nice if he wrote it up. It’s

  1. highly germane to whether one should take this stuff seriously and
  2. very, very non-obvious.
Posted by: Jacques Distler on June 10, 2010 1:53 AM | Permalink | PGP Sig | Reply to this

Re: What happened?

So if your intuition is based on the more interesting higher-dimensional theories, everything I’m saying should seem completely crazy.

I don’t think that’s my problem.

I should perhaps again emphasize that we only expect something so simple to happen in 3d quantum gravity, where spacetime is flat except where there’s matter.

Is it clear that this is true in 3d quantum gravity? And do you mean on average?

But my real problem is, as I tried to say before: if we are really talking about quantum gravity coupled to matter, then it does not make sense to say “where there’s matter” as if we had a classical trajectory of a particle.

So I am still confused. But I do gather that there are some interesting Feynman amplitudes here dropping out of some 3d TFT. And I gather that you think that these are the terms in a perturbation series of some matter quantum field in a gravitational background. But I find it quite opaque how that should relate to the original 3d TFT. If there is a relation, it would seem to me to require some nontrivial checks. But maybe I am still missing something.

[edit: composing this message I overlapped with Jacques’ latest]

Posted by: Urs Schreiber on June 10, 2010 2:08 AM | Permalink | Reply to this

Re: What happened?

Sorry for being a pain on this point, but I still don’t understand why in quantum-⟨gravity coupled to matter⟩ it should make sense to speak of any worldlines of particles at all.

Of course you should already be worried about this in quantum field theory.

I don’t actually think I should be worried about this. The Feynman perturbation series over Feynman graphs can be pretty well motivated as computing an approximation to the would-be path integral in a quantum field theory.

So in perturbation theory Feynman diagrams are all fine. Kind of by definition. But what does the appearance of a single Feynman diagram in the background of a supposedly non-perturbative quantum gravity theory mean? I can’t help but feel that this seems like mixing up concepts that do not live in the same context. Unless there is some deeper story at work here that I am missing.

Posted by: Urs Schreiber on June 10, 2010 2:22 AM | Permalink | Reply to this

Re: What happened?

Are the authors aware of this conversation? If not, they are missing a good opportunity to discuss their paper.

Posted by: Eric on June 10, 2010 2:28 AM | Permalink | Reply to this

Re: What happened?

I can’t help but feel that this seems like mixing up concepts that do not live in the same context. Unless there is some deeper story at work here that I am missing.

My understanding of what they were doing was the following:

In 3D, you can start with gravity coupled to matter, and integrate out the gravity, to obtain an effective theory of the matter alone.

(Of course, you can’t do that in 4-dimensions, because 4D gravity has massless degrees of freedom, so the resulting matter theory would be hopelessly nonlocal. But in 3D, since there are no local degrees of freedom – massive or massless – in the pure gravity theory, you can in principle do it.)

The claim was that the effect of integrating out gravity turned the ordinary (“commutative”) field theory into a noncommutative field theory.

What’s not at all clear to me (hence my question above) is that, if you start with equivalent field theories (related, say, by a field redefinition) this procedure yields equivalent noncommutative field theories.

If this were true, studying the matter theory, Feynman diagram-by-Feynman diagram (which is what they do), would not be the way you would go about understanding it.

Posted by: Jacques Distler on June 10, 2010 3:25 AM | Permalink | PGP Sig | Reply to this

Re: What happened?

I feel quite sure that this theory is a good thing and that it’s just our job to understand it. (Ever hear that before?) It’s got no room for tweaks or fudge factors — unlike loop quantum gravity in 4 dimensions. It all falls out starting from Rep(SU(2))Rep(SU(2)), or its qq-deformation, plus a bunch of higher category theory. It’s pure and beautiful!

So, none of the worries that Urs or Jacques are expressing bother me very much. These worries are definitely worth thinking about… for someone who works on this stuff. But the great thing about no longer working on quantum gravity is that I can relax, do something else for 10 or 20 years, and then take a peek at the arXiv and see what happened.

Posted by: John Baez on June 10, 2010 6:37 AM | Permalink | Reply to this

Chern-Simons theory with defects?

I wrote about having a Feynman diagram drawn on a cobordism on which one means to have a quantum theory of gravity:

I can’t help but feel that this seems like mixing up concepts that do not live in the same context. Unless there is some deeper story at work here that I am missing.

Jacques said:

My understanding of what they were doing was the following:

In 3D, you can start with gravity coupled to matter, and integrate out the gravity, to obtain an effective theory of the matter alone.

Thanks, that sounds plausibly like a proper way to think about what is going on.

If it is true though, I would still feel that the description of the situation that we discussed above is not quite right: because above we are talking about how that cobordism on which the Feynman diagram is drawn is supposed to be the target space of a theory of gravity, and the same target space as that of the theory of matter that we are talking about. This can’t quite make sense. Whereas what you just said would have us think of it as being the worldvolume of something.

So if I adopt the point of view that what is going on here is that we see Feynman rules for matter obtained after integrating out gravity, then it would seem to me that we are talking about this:

a theory of 2-branes with 1-dimensional defects on their worldvolume, such that performing the 2-brane worldvolume path integral for fixed defect labels produces an assignment of amplitudes to defect labels that looks like an assignment of Feynman amplitudes to Feynman graphs.

That description would make a lot of sense to me.

Posted by: Urs Schreiber on June 10, 2010 8:42 AM | Permalink | Reply to this

Trying to understand it

John wrote:

I feel quite sure that this theory is a good thing and that it’s just our job to understand it.

And I am trying to understand it, this is why I am pointing out where I can’t understand your description of it.

It’s pure and beautiful!

Clearly. But what is “it”?

So, none of the worries that Urs or Jacques are expressing bother me very much.

Maybe to clarify: I am not worrying that the formalism you are talking about is not of value. I am worried that the interpretation of it that you mention does not quite make sense to me.

But I am beginning to see a way to think of the situation that would make sense to me, for whatever that’s worth:

the formalism at hand describes the worldvolume theory of a 2-brane with line defects, which is quantum gravity on the worldvolume with defects, as usual, and whose effective target space theory is a QFT of some kind of matter field on some curved or noncommutative background.

Could that be? That would make a lot of sense to me.

Posted by: Urs Schreiber on June 10, 2010 8:57 AM | Permalink | Reply to this

Re: Trying to understand it

John wrote:

I feel quite sure that this theory is a good thing and that it’s just our job to understand it.

Urs wrote:

And I am trying to understand it, this is why I am pointing out where I can’t understand your description of it.

Right, that’s good. But when I said “it’s just our job to understand it”, I expressed myself poorly.

By “us” I really meant “the physicists who are working on quantum gravity”. And I don’t actually include myself in that group anymore.

Maybe to clarify: I am not worrying that the formalism you are talking about is not of value.

Okay, good. I was getting the feeling that you (and especially Jacques) wanted me to put up a “defense” of this formalism, as if it were “on trial”. And that’s the kind of thing I have no time for anymore.

But I am beginning to see a way to think of the situation that would make sense to me, for whatever that’s worth:

the formalism at hand describes the worldvolume theory of a 2-brane with line defects, which is quantum gravity on the worldvolume with defects, as usual, and whose effective target space theory is a QFT of some kind of matter field on some curved or noncommutative background.

Could that be?

Could be.

Posted by: John Baez on June 10, 2010 2:30 PM | Permalink | Reply to this

Re: Trying to understand it

John writes

I was getting the feeling that you (and especially Jacques) wanted me to put up a “defense” of this formalism, as if it were “on trial”. And that’s the kind of thing I have no time for anymore.

I see that you have no time for this anymore, but we are left then with a somewhat unsatisfactory situation: if bold claims such as this one…

Simply by carving a Feynman-diagram-shaped hole in 3d spacetime and doing quantum gravity on the spacetime that’s left over, you get a good theory of quantum gravity coupled to matter!

… are publicly voiced, it is the most natural thing in the world that the scientific community does not just nod happily and contuinues doing whatever they are doing, but that people try to understand it and try to poke holes into it. That’s the scientific process. Making bold claims and then not taking the time to defend these may give a bad impression.

For instance, if I am honest, I can tell you my current impression of this discussion here:

I asked what Barrett’s talk was about. I got a sketchy reply from Bruce who seemed to have gotten away with the impression and seemed to take for granted that there is a theory of quantum-\langlegravity coupled to matter \rangle in 3 dimensions encoded by marked 3-dimensional cobordisms which is all settled and of which Barrett was only looking for a more elegant or deeper understanding.

So I kept asking about how that would work, but so far none of what i have heard has given me much confidence, to be frank. For the time being, I am instead getting away with the impresssion that some possibly nice structure has been found but not truly understood for what it is. (Possibly because of lack of attempts to poke holes into it?)

Now, that may be a wrong impression. I invite everyone reading this to try to convince me, and maybe Jacques. I am open for being convinced, but not for accepting claims without understanding them myself.

Posted by: Urs Schreiber on June 10, 2010 3:02 PM | Permalink | Reply to this

Re: Trying to understand it

My understanding of what they were doing was the following:

In 3D, you can start with gravity coupled to matter, and integrate out the gravity, to obtain an effective theory of the matter alone.

Thanks, that sounds plausibly like a proper way to think about what is going on.

Please understand that that’s my best-faith attempt to make sense of Freidel-Livine. Anyone who actually understands their paper should feel free to chime in with a more accurate explanation.

Now, that may be a wrong impression. I invite everyone reading this to try to convince me, and maybe Jacques. I am open for being convinced, but not for accepting claims without understanding them myself.

We’ve discussed this topic before. I chimed in again, not because I expect answers this time. I just hope to save someone — who wasn’t party to the previous discussions — some time, by cutting to the chase.

Posted by: Jacques Distler on June 10, 2010 7:31 PM | Permalink | PGP Sig | Reply to this

Re: Trying to understand it

Urs wrote:

Making bold claims and then not taking the time to defend these may give a bad impression.

That’s true. I have thoughts on quantum gravity, but I avoid saying anything about them, because the subject excites certain people, and they want to talk about it — and I really don’t want to!

But this time I slipped, as follows:

First you wondered what John Barrett spoke about, and I didn’t think anyone else would answer, so I took pity on you and briefly replied. Then Bruce added more to my reply, mostly quite correctly, but mixing up the words “matter” and “geometry”… so I felt I had to correct him. Then he dramatically concluded that “now suddenly quantum gravity and category theory are at odds”, and I could not resist reacting… and so on. With each step I got pulled deeper into a conversation I didn’t want to be having at all!

For the time being, I am instead getting away with the impression that some possibly nice structure has been found but not truly understood for what it is.

Okay, that’s fine with me. I hope you figure it out.

And — here I go again, damn it! — I hope your final understanding somehow accounts for the fact that even though we’re starting with a purely topological theory, namely 3d BFBF theory with gauge group SU(2)SU(2), treating it as an ‘extended’ theory gives a way of deforming the usual Feynman rules for a quantum field theory on a Euclidean 3d spacetime, with the deformation parameter being Newton’s gravitational constant. This is the really interesting part, to me at least.

I believe this deformation can be understood as follows (though it’s been a while since I’ve thought about it): instead of doing momentum space Feynman integrals in 3\mathbb{R}^3, we do them on SU(2)SU(2) instead, with the radius of the SU(2)SU(2) being inversely proportional to the gravitational constant. Perhaps this ‘curved momentum space’ corresponds to a ‘noncommutative position space’ — I’m not sure.

(Admittedly, it’s not really ‘usual’ to work with quantum field theory on Euclidean spacetime, with the Laplace equation taking the place of the wave equation, and so on — I’m not talking about a Wick-rotated version of a theory on Minkowski spacetime! It would be less peculiar, physically speaking, to write 2,1\mathbb{R}^{2,1} wherever I’d written 3\mathbb{R}^3 above, and SL(2,)SL(2,\mathbb{R}) where I’d written SU(2)SU(2). And for conceptual purposes this is probably what you should think about. But the noncompactness of SL(2,)SL(2,\mathbb{R}) makes integrals more likely to diverge, so there are more technical difficulties with this less peculiar theory.)

(Possibly because of lack of attempts to poke holes into it?)

On the contrary, ‘poking holes into it’ is exactly what people were doing. You can see the holes here:

Posted by: John Baez on June 10, 2010 8:38 PM | Permalink | Reply to this

Re: Trying to understand it

Hi! Where does that cool screenshot come from?

Posted by: Jamie Vicary on February 12, 2011 2:50 PM | Permalink | Reply to this

Re: Trying to understand it

Sorry, I should have said — it’s from here:

You and Jeffrey are in Lisbon right now, I think… working on categorified oscillators?

Posted by: John Baez on February 13, 2011 8:27 AM | Permalink | Reply to this

Re: What happened?

Bruce wrote:

It’s pretty clear that the category C:=Z(S 1)C:=Z(S^1) assigned to the circle is a ribbon category. But suddenly this is somewhat of a paradox to me: we get out invariants of knots and links from a formalism which only a priori had to do with invariants of manifolds and cobordisms.

It’s very cool, but it’s not a paradox.

Why not? Because a once-extended 3d TQFT is not just an invariant of 2d manifolds and 3d cobordisms. It’s an invariant of 1d manifolds, 2d manifolds, and 3d manifolds with corners. And we can get a 3d manifold with corners by taking a 3d cobordism and cutting out a thickened tangle… or a thickened graph!

And the way it works, we also get invariants where each edge of our thickened graph is labelled by an object of the category you’re calling C=Z(S 1)C = Z(S^1) — the category for the circle.

But such a thing is just a Feynman diagram.

But in our modern formalism of higher-categorical cobordisms it seems more interesting… we are getting out more than we explicitly put in (analogous to “putting in geometry gives you matter as well”).

It’s not just ‘analogous’ — it’s the exact same thing! The physics is just a way of talking about the math.

Jeff Morton did his thesis on this. Check out this page from a talk he gave at the Perimeter Institute in 2006:

Posted by: John Baez on June 9, 2010 6:36 PM | Permalink | Reply to this

Re: What happened?

Ok, thanks for this. This is an aspect of the story I haven’t appreciated properly, though it has indeed been stressed eg. in Jeffrey’s thesis.

Posted by: Bruce Bartlett on June 9, 2010 7:32 PM | Permalink | Reply to this

Re: What happened?

Ah, this is the sort of discussion I enjoy at this place … I’m sitting in the sitting room of a beautiful german Schloss, and am a little bit too schlosshed myself to make any useful contribution to this. But I do remember when Jeffrey explained this stuff to me a few years ago when we met in Nottingham once, and again when we met at the PI — it’s beautiful!

This stuff gives you a way to turn string diagrams into 2d cobordisms, and so gives a nice way to turn ‘internal’ things in your modular tensor category into ‘external’ things.

Posted by: Jamie Vicary on June 9, 2010 8:13 PM | Permalink | Reply to this

Re: What happened?

Don’t get so schlosshed that you fall out of the Schloss!

By the way, regarding that slide from Jeffrey’s talk… irreducible representations of SU(2)SU(2) are labelled by spins, so you might wonder why he’s talking about ‘mass and spin’.

The point is that in the Ponzano-Regge theory, the category for the circle is not Rep(SU(2))Rep(SU(2)) but the center Z(Rep(SU(2))Z(Rep(SU(2)), which consists of representations of the ‘quantum double’ of SU(2)SU(2). Irreducible representations of this gadget are labelled by masses and spins.

Again, it’s a special feature of 3 dimensions that the (double cover of the) Euclidean group is

SU(2)su(2)SU(2) \ltimes \mathbf{su}(2)

and thus has representations that are rather similar to those of the quantum double of SU(2)SU(2).

Indeed, Freidel and company show how the quantum double can be thought of as a ‘G NG_N-deformed’ version of the Euclidean group, where G NG_N is Newton’s gravitational constant. As I explained in week232, this deformation has the effect of putting an upper limit on the mass of particles — roughly the Planck mass!

In addition to this G NG_N-deformation we get a further qq-deformation when we turn on the cosmological constant.

Posted by: John Baez on June 9, 2010 8:28 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

Well, now I see from the Facebook photos who is about. That Woit character, for instance.

Posted by: Kea on June 2, 2010 5:16 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

What do you mean, “about”? Peter Woit was certainly not at the Quantum Physics and Logic workshop in Oxford, nor was he at the week-long school preceding that school. I’m sure he was “about”, somewhere, but not there.

Posted by: John Baez on June 2, 2010 6:35 AM | Permalink | Reply to this

Re: The Quantum Whisky Club

Here are Bruce Bartlett, Peter Selinger, Jamie Vicary and Chris Heunen talking things over at lunch at the QPL workshop:

I don’t recognize the person facing away from the camera. Who is he?

Posted by: John Baez on June 2, 2010 4:09 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

Posted by: bob on June 2, 2010 4:48 PM | Permalink | Reply to this
Read the post Categories, Logic and Physics 7 at Birmingham
Weblog: The n-Category Café
Excerpt: The 7th Categories in Logic and Physics workshop will be in Birmingham, England on September 21st, 2010.
Tracked: June 2, 2010 6:48 PM

Re: The Quantum Whisky Club

Actually, come to think of it, Brecht refers to it as a “whiskey bar” rather than a “whisky club”. Also: if we don’t find the next one, I tell you we must die.

Posted by: John Armstrong on June 2, 2010 11:56 PM | Permalink | Reply to this

Re: The Quantum Whisky Club

Because you would no doubt miss this, here is another interesting new paper about complementary observables.

Posted by: Kea on June 3, 2010 10:39 PM | Permalink | Reply to this

duals

My first attempt to find out what happened at the Quantum physics and logic workshop was not so successful. After a total of 76 messages in this thread we still mainly know what was not said ;-/

But there must have been some talk with digestible information. (I mean, of course John’s was, but of that I expect I already know the content. Or do I? )

I hear that Bruce Bartlett and Jamie Vicary announced a result that fills a decade-old gap in the description of 3dTFTs by modular tensor categories. Does anyone feel like telling us a bit about their talk?

Posted by: Urs Schreiber on June 11, 2010 9:54 PM | Permalink | Reply to this

Re: duals

The main points in my talk were these:

  • There are versions of matrix mechanics describing both quantum and classical physics. Both involve dagger-compact categories. There is a no-cloning theorem in classical mechanics.
  • The study of duality unifies real, complex and quaternionic quantum mechanics into a single theory which is already implicit in standard physics.
  • Dagger-compact categories are the n = 1, k = 3 example of k-monoidal n-categories with duals — the case most relevant to particles in 4d spacetime, but just one of many.
  • Treating profunctors as categorified linear operators relates propositional logic to categorified 2d topological quantum field theories in a somewhat mysterious way.

In ‘week299’ — not yet written, but coming soon — I’ll give a summary not just of the Quantum Physics and Logic workshop but also of the week-long school that preceded this workshop. This featured lots of nice applications of categorical ideas to quantum theory.

But, I would love it if other people described some of these talks! There is no way that I can summarize all of them. And surely Bruce and Jamie can do the best job of describing their own talk.

You can also click on the links above and see videos of talks. So, for example, you can see what John Barrett and Louis Crane actually said.

Posted by: John Baez on June 11, 2010 11:10 PM | Permalink | Reply to this

Re: duals

The rough recordings of both the QICS School and QPL are now online:
* QICS School videos
* QPL VII videos
This is our new talks archive, which before was at the Categories, logic and Foundations of Physics website. Thanks to Andrei Akhvlediani, Philip Atzemoglou, Bill Edwards, Aleks Kissinger, Ray Lal, Shane Mansfield, Alex Merry, Johan K. Paulsson, Jakub Zavodny, Jamie Vicary and Chris Heunen for the filming!

Posted by: bob on June 13, 2010 12:47 PM | Permalink | Reply to this

Re: duals

Jamie and I gave a talk on ‘Autonomous categories as local Frobenius algebras’, basically looking at how to characterize rigid/autonomous monoidal categories ‘externally’. It’s hopefully part of a project we’re working on with other authors, where we’re still trying to understand some of the details, which is why we haven’t spoken much about it.

Posted by: Bruce Bartlett on June 14, 2010 12:36 PM | Permalink | Reply to this

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