Baratin and Freidel: a spin foam model of ordinary particle physics

[Kea]

CarlB, can we tempt you to read a little more? Maybe some stuff about
quantum mechanical logic in diagrams?

Good! I hope you're healthy by now...

Yes, thank you John. I have started taking the big puppy for long walks in the bush.

I really like thinking of ordinary vector spaces this way. And soon perhaps we can teach the kids about them this way. One of my little nephews would probably refuse to do it any other way, because pictures make a lot more sense to him than lists of random looking rules.

The thing I really like (sorry to be so repetitive) about the 2-dimensional picture is that one day we can try and do Gray compositions of pieces of surfaces to make 3-dimensional pictures...and we can do this
sort of thing for ordinary vector spaces...which are like categorified numbers!

Well, it might be better to think of numbers (or polynomials) as tangles in a Riemann surface...but this is off topic...except that then by turning them into vector spaces we get things like sheaves! So it doesn't really matter how we try and do things...maybe we like doing String theory with Hecke eigensheaves...everything seems to end up at the same place at the end. It's always exciting to see that happening.

 

[CarlB]

Well the one thing that this sort of reminds me of is David Hestenes' comments on the Cambridge Geometric Algebra Group's gauge theory of gravity (GTG).

The comment was that it was significant that the theory could be put onto flat space, and Hestenes' reason for why this was something needed in the context of his "geometric algebra" seems to resonate with these ideas about vectors, particularly section IX, pages 21-23 of this link (which pages may be read without reading the rest of the paper):

Spacetime Geometry with Geometric Calculus
David Hestenes, To be published in the Preceedings of the Seventh International Conference on Clifford Algebra
http://modelingnts.la.asu.edu/pdf/SpacetimeGeometry.w.GC.proc.pdf

I'd give a brief description of the argument, but I don't think I can do it justice. Hetsenes does it so well and so clearly that I wouldn't want to butcher it by reducing its length and two pages is too long. Okay, but basically, the idea has to do with how one connects up an algebra to a manifold in such a way that one can do calculus on it.

For more on the geometric gauge theory of gravity, see the Cambridge geometric algebra group:
http://www.mrao.cam.ac.uk/~clifford/index.html

Carl

 

[Hurkyl]

Namely, we can conjugate an automorphism f by a group element g and get another automorphism f':

That, actually, is where I was getting stuck!

The problem is that something like R^n comes equipped with a mental image -- it's a space of points! But, my mental image of 2-automorphisms is very much like a homotopy of maps. While that generally works fine for natural transformations, I can't get it to mesh with my picture of R^n.

And to make things even more confusing... the 2-automorphisms of the vector space R^n look like the 1-automorphisms of the affine space R^n.

I think I'm okay if I pretend I don't know what R^n is, and just picture it as a dot with a bunch of loops hooked up to it... but I really don't think that's the right way to approach this problem.

 

[marcus]

JB began the thread with mention of a paper in the works by Baratin and Freidel-----extending to 4D what they have already done in 3D.

I think this other paper may be relevant. It just posted today and is also by Freidel, but with Starodubtsev and Kowalski-Glikman


http://arxiv.org/abs/gr-qc/0607014
Particles as Wilson lines of gravitational field
L. Freidel, J. Kowalski--Glikman, A. Starodubtsev
19 pages

"Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom."

 

[garrett]

I just read that new paper.

As far as I can tell, it works out exactly as you would expect point particles to behave in MacDowell-Mansouri BeeF gravity. Judging from their introduction, they seem oddly excited about it though, so maybe I'm missing something. It could be the excitement stems from their description of these particles as field monopoles, but I'm not sure why that's so different than putting the point particle actions in by hand. Anyway, it's a decent treatment and I like the approach.

 

[marcus]

I just read that new paper...

I am glad you had a look at it.
In a recent post, Baez mentioned that Freidel has 3 papers in the works with Starodubtsev and one in the work with Baratin. even if all don't come to fruition I'm inclined to expect at least a couple more in this same line of investigation.

It seems to me that you are especially well prepared to understand and comment, not only on this one but on the others when they come out.

The conclusions section speaks of a "forthcoming paper" in which they do a perturbation expansion in alpha

and thru that, they say, address the question of the flat limit of gravity and particles. I will get the quote

==quote==
First, since the alpha parameter is small, we can consider a perturbation theory of gravity coupled to particle(s) being the perturbation theory in alpha. The distinguished feature of this theory would be that it is, contrary to earlier approaches, manifestly diffeomorphism-invariant, so its framework it is possible to talk about weak gravitational field in the conceptual framework of full general relativity. These investigations, both in the case of beta = 0 and beta not = 0 will be presented in the forthcoming paper. The fuller control over the small alpha sector will presumably make it possible to address the outstanding question of what is the flat space limit of the theory of gravity, coupled to point particles. It has been claimed that such a theory will be not the special relativity, but some form of doubly special relativity
==endquote==

If they can show that the flatspace limit is not usual Lorentz but is, instead, some DSR, this would probably open up some possibilities to TEST. It would seem to me like considerable progress just to get a good flatspace limit of one sort or another.

 

[marcus]

I am mulling over this parameter alpha that they want to do the perturbation expansion in.
I think it came up in the earlier (Jan 2005?) Freidel Staro paper.
You see it on page 3 of this paper, equation 2.3

if alpha and beta were both zero then S would be a usual BF action, but alpha perturbs it and makes it deviate from the usual BF action. Am I wrong?

the nice thing is that we are now looking at a perturbation theory where we DO NOT HAVE A FIXED BACKGROUND GEOMETRY around which we perturb. I don't claim to have much grasp of this, but we seem to be contemplating the opportunity to "perturb around pure BeeF itself"

so they hold out the attractive notion of a background independent perturbation theory or I guess what they said was a "manifestly diffeomorphism invariant" perturbation theory. that was what they said in conclusions on page 15.

right now it looks to me as if they are proceeding with exactly what they promised in http://arxiv.org/hep-th/0501191 that they would do. rather than us getting new signals this time we are getting confirmation of progress along lines they said in january last year. Am I missing something?

 

[garrett]

Yep, that all sounds right.

The \alpha term is what makes BF into gravity. With \alpha itself proportional to the gravitational constant. Rovelli wrote about this as well, in his propagator paper. And, urr, I do BF too -- although I came to it rather circuitously.

"BF, it's what's for dinner."

 

[marcus]

... And, urr, I do BF too -- although I came to it rather circuitously.

Yes! and I am looking for you to surf this BF wave!

there is much truth in the saying
"BF, it's what's for dinner." I may adopt it as a signature.

 

[selfAdjoint]

there is much truth in the saying
"BF, it's what's for dinner." I may adopt it as a signature.

Although Prof. Baez once remarked that it should be "EF" which spoils the pun. He said the so-called B part of BF theory was not in fact like magnetism (which B traditionally expresses) but like electricity, E.

 

[marcus]

Although Prof. Baez once remarked that it should be "EF" which spoils the pun. He said the so-called B part of BF theory was not in fact like magnetism (which B traditionally expresses) but like electricity, E.

the equations look prettier with E and F instead of B and F
and the analogy is more correct, true, but still everybody says BF.
maybe we have to go with it.

 

[selfAdjoint]

the equations look prettier with E and F instead of B and F
and the analogy is more correct, true, but still everybody says BF.
maybe we have to go with it.

Yeah, I agee. And who would want to give up that great Sig line? Even Baez seems to have bit the bullet.

 

[marcus]

Yeah, I agee. And who would want to give up that great Sig line? Even Baez seems to have bit the bullet.

Garrett can have it back anytime he wants

On Thursday, two days hence, John Baez student Derek Wise will give a talk at Perimeter.

It is along the general lines Baez has been talking about but expecially about the papers of Baez, Wise, Crans and of Baez Perez.

I hope they put a video at the streamer site. here is the abstract:

Derek Wise
Exotic statistics and particle types in 3- and 4d BF theory
Thursday July 13, 2006, 1:30 PM
"Gravity in 2+1 dimensions has the remarkable property that momenta live most naturally not in Minkowski vector space but in the 3d Lorentz group SO(2,1) itself. Having group-valued momentum has interesting consequences for particles, including exotic statistics and a modified classification of elementary particle types. These results generalize immediately to 3d BF theory with arbitrary gauge group. Better yet, they generalize to 4d BF theory, where matter shows up as string-like defects. These 'strings' exhibit exotic statistics governed not by the usual braid group, but by its higher dimensional cousin: the 'loop braid group'. We discuss these statistics as well as the classification of elementary 'string types' in 4d BF theory."

http://perimeterinstitute.com/activities/scientific/seminarseries/alltalks.cfm?CurrentPage=1&SeminarID=759

 

[marcus]

I had the wrong post here earlier. Here is a question. if anyone wants to comment.

In the first Freidel Starodubtsev paper they cited two "in preparation" papers
one of them was something we know for sure has NOT appeared
[13]Freidel Starodubtsev "perturbation gravity via spin foams"
that would be the SPIN FOAM QUANTIZATION OF THE CLASSICAL WORK WE JUST SAW
so if and when that paper comes out it will be kind of major.

the other was
[6] Freidel Kowalski-Glikman Starodubtsev "Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis"

Now my feeling is that Freidel has gotten cagey about saying "background independent" because that term is defined differently by string theorists and others and tends to provoke controversy. people feel threatened and start protesting that maybe string theory really IS "background independent" even though it might not be "manifestly" background independent, and then they go on to say "LQG" is not really background independent, and so on. The term irritates people---and has become associated with semantic conflict
So my suspicion is that the paper that JUST CAME OUT REALLY IS THIS PAPER but RETITLED in a kind of inconspicuous ivy-league coat-and-tie way.

the paper that just came out is titled
"PARTICLES AS WILSON LINES OF GRAVITATIONAL FIELD"
which is shocking if you think of it, but innocuous enough on the surface.
the number is
http://arxiv.org/gr-qc/0607014 (remember by Quatorze Juillet Bastille day)

So I guess the question is, what do you think? Do you also think that the promised paper
"Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis"

is actually the new one we have in hand called "Particles as Wilson Lines of Gravitational Field" but renamed?

Notice if you look at "Particles as Wilson Lines" actually wilson lines is only a part of what they are doing and
very much of what they are doing could be accurately described as a classical analysis of background independent (in the LQG sense) gravity-and-matter perturbation theory.

and if so, any idea why they decided on the new name?

================
to repeat another point, that I think JB made, or various people have: to say "background independent perturbation theory" is a real kicker of a headline. Because perturbation theory is the customary predominant way to do fields and UP TILL THIS MOMENT all the perturbation field theory ever done has used a fixed BACKGROUND SPACETIME geometry. so when you hear that phrase you hear a slight breaking noise.
(which among other things could motive people to deny that the paper could possibly be on the right track, causing the author a lot of bother answering them). I can understand how one might want the breaking noise to be inaudible.

 

[john baez]

So I guess the question is, what do you think? Do you also think that the promised paper
"Background Independent Perturbation Theory for Gravity Coupled to Particles: Classical Analysis" is actually the new one we have in hand called "Particles as Wilson Lines of Gravitational Field" but renamed?

I don't know. I remember Laurent saying they weren't even sure how many papers they were writing on this subject: two or three. They've done a lot of work, obviously, and for a big project like this one needs to keep rethinking the best way to slice the work into papers.

The paper they wrote doesn't actually do any "perturbation theory", apart from writing the MacDowell-Mansouri Lagrangian as the BF Lagrangian plus two extra terms, and analysing what this means... which they'd already done in a previous paper. The big new thing is to introduce particle worldlines as "defects" - curves removed from spacetime - much as had already been done in 3d gravity. So, it makes sense for their title to emphasize this.

In fact, their title is a bit more dramatic than what I might have chosen, because they don't really study these particle worldlines in the context of MacDowell-Mansouri gravity, except for one equation right near the end. Mostly they study these particles in the context of plain old 4d BF theory.

This nicely complements my own study, with http://arxiv.org/abs/gr-qc/0603085" .

Unfortunately, Crans, Wise, Perez and I studied strings coupled to 4d BF theory for a general gauge group but didn't work out the details for the gauge group Freidel uses, namely SO(4,1). We focused on SO(3,1). It should be easy to do the SO(4,1) case now, though since Freidel & Company have worked out a lot of the necessary stuff.

After a talk I gave, Freidel guessed that the strings may be related to gravitons... or replace them, somehow. It's a big mystery: a nice structure is emerging, but it's not clear what it means! This is what makes physics fun.

 

[selfAdjoint]

...This is what makes physics fun.

Are you now finding it as much fun as math again? I know you were fed up for a while.

It is surely a wonderful gift you have to be able to work back and forth in the two areas; not only are you able to spot unnobvious connections, but there always seems to be something in one field or the other that really floats your boat.

 

[marcus]

I think it's all one thing, basically.
just the formalities of which department and which journal
but if you see a glint in the eye of the universe or a little
smile on the face of nature it probably doesn't matter much
whether it is one or the other

 

[marcus]

... I explain this a bit more in the latest issue of This Week's Finds, http://math.ucr.edu/home/baez/week235.html" .

Great news! Glad you found time!

Unfortunately,...

UNFORTUNATELY? That's the way it's SUPPOSED to happen
general group case first, then specialize to SO(4,1)
couldnt be sweeter
certainly maximizes the pleasure and excitement for the sidelines observers like us anyway.

...Crans, Wise, Perez and I studied strings coupled to 4d BF theory for a general gauge group but didn't work out the details for the gauge group Freidel uses, namely SO(4,1). We focused on SO(3,1). It should be easy to do the SO(4,1) case now, though since Freidel & Company have worked out a lot of the necessary stuff.
...

hotdog

"After a talk I gave, Freidel guessed that the strings may be related to gravitons... or replace them, somehow. It's a big mystery: a nice structure is emerging, but it's not clear what it means! This is what makes physics fun."

Freidel: let's invent how spacetimematter works. My stuff can be the geometry and your stuff can be the gravitons that connect changes in the geometry, OK?"but it's not clear what it means! This is what makes physics fun" at some point, this begins to sound like a memorable understatement

thanks for posting here, enjoy Shanghai, and don't forget to figure spacetime out for us

 

[selfAdjoint]

I think it's all one thing, basically.
just the formalities of which department and which journal
but if you see a glint in the eye of the universe or a little
smile on the face of nature it probably doesn't matter much
whether it is one or the other

Sub specie aeternitatis, of course, you're right. But how few people are capable of working creatively in both! And you see so much (mostly tacit) dismissal of the concerns and interests of each field in so many practitioners of the other.

 

[Careful]

Sub specie aeternitatis, of course, you're right. But how few people are capable of working creatively in both! And you see so much (mostly tacit) dismissal of the concerns and interests of each field in so many practitioners of the other.

Many smart people are creatively working both in mathematics and physics (see Ed Witten, Roger Penrose, Stephen Hawking, George Ellis, Yau, Paul Dirac, ... and many lesser Gods as well). When I read some threads here, there seem to be very strange opinions wandering around about ``the way physics is done''. Good physics inventions always *started* with a coherent intuitive picture of (a part of) nature suggested by experiment; a physicist has fun when he/she can find out a mathematical model incoorporating these intuitions and delivering the correct numbers. In some rare cases, he can get excited when some unexpected solutions come out which require *new* experiments to be done (or the inventor might even dismiss these as unphysical). History confirms this thesis over and over again - Einstein for example had the physical picture of GR already in his mind (at least) six years prior to writing down his field equations. His theory got experimental support by Eddington in 1919 and he was so surprised by the Schwarzschild solutions that he did not hesitate to refute them. Mathematicians on the other hand have fun exploring structures per se and in these days are not shy at all to sell some weak (possibly accidental) correspondences with some established theories as ``physics''. The fact that some physicists are interested in these merely expresses the lack of good ideas from their side. Once I heard from a mathematician that GR was the most beautiful theory one could imagine until mathematicians started formalising it - what distinguishes supreme physicists is their powerful intuition to recogize what to do (and what not), mathematical ability only serves as a very useful tool. A beautiful example of this is given by the mathematical genius Dirac (one of the very few to have done so much useful mathematical physics), who kept on insisting that QED was not a good physical theory and that its miracles could very well be accidental.

Now, when some camp does not appreciate the worries of the other very well; it is usually so that the latter is not presenting a somewhat clear coherent picture of nature at all.

Careful

 

[selfAdjoint]

Many smart people are creatively working both in mathematics and physics (see Ed Witten, Roger Penrose, Stephen Hawking, George Ellis, Yau, Paul Dirac,

Umm, yes. Witten, Penrose, Hawking, are a world famous trio among hundreds of creative mathematicians and physicists. Yau is a mathematician whose work turned out to be significant to the physicists which has led him over; he certainly didn't prove Calabi's conjecture with string theory in mind. Elis I'm not familiar with. And Dirac, I believe, is Still Dead.

 

[marcus]

Ellis I'm not familiar with...

when he was quite a bit younger George Ellis co-authored a book with Stephen Hawking (The Large Scale Structure of Space-Time). He is a distinguished cosmologist among other things.

I am afraid it is off topic of me to say so, but what excites me about Ellis recent contribution is his essay on Philosophical Issues in Cosmology. It is comprehensive and lays the issues out very clearly. The philosophical issues in cosmology are substantial and interesting, not just abstract hot air (as might be the case in some other areas of philosophy.)

he is not as well known as the others. Careful could have tossed the name in knowing that he would please a few Ellis-fans like me.

http://arxiv.org/abs/astro-ph/0602280
Issues in the Philosophy of Cosmology
George F. R. Ellis
To appear in the Handbook in Philosophy of Physics, Ed J Butterfield and J Earman (Elsevier, 2006).

"After a survey of the present state of cosmological theory and observations, this article discusses a series of major themes underlying the relation of philosophy to cosmology. These are: A: The uniqueness of the universe; B: The large scale of the universe in space and time; C: The unbound energies in the early universe; D: Explaining the universe -- the question of origins; E: The universe as the background for existence; F: The explicit philosophical basis; G: The Anthropic question: fine tuning for life; H: The possible existence of multiverses; I: The natures of existence. Each of these themes is explored and related to a series of Theses that set out the major issues confronting cosmology in relation to philosophy."

(probably far more than anyone wants to know about Ellis )

 

[Careful]

Umm, yes. Witten, Penrose, Hawking, are a world famous trio among hundreds of creative mathematicians and physicists. Yau is a mathematician whose work turned out to be significant to the physicists which has led him over; he certainly didn't prove Calabi's conjecture with string theory in mind. Elis I'm not familiar with. And Dirac, I believe, is Still Dead.

Dirac will be very much alive again (and Yau's first significant result I know of (in physics) is the positive energy theorem in GR - and that dates back from the seventies). You know, there are very many scientists around who can do good mathematics and physics (and know how to pick interesting subjects). To name a few (this is just a random list !) : Eric Poisson, Fred Cooperstock, Luca Lusanna, Brian Greene, AP Balachandran, Alain Connes ...

Careful

 

[Careful]

**Careful could have tossed the name in knowing that he would please a few Ellis-fans like me.**

Oh dear ... No Marcus, the reason why I mention Ellis is because of his down to Earth rock solid work which has clearly something to do with physics.

 

[john baez]

Are you now finding it as much fun as math again? I know you were fed up for a while.

Math is still more fun for me, because I see lots of beautiful stuff just waiting to be done that will surely be useful, while everything I do on fundamental physics has a high chance of being on the wrong track. And, I feel I can go much deeper in math, and reach much more mind-blowing realms.

But, I like keeping my finger in physics, in part because people seem to appreciate physics more than math... I used to, too.

 

[Careful]

Math is still more fun for me, because I see lots of beautiful stuff just waiting to be done that will surely be useful, while everything I do on fundamental physics has a high chance of being on the wrong track. And, I feel I can go much deeper in math, and reach much more mind-blowing realms.

But, I like keeping my finger in physics, in part because people seem to appreciate physics more than math... I used to, too.

Again, could you give us an *example* relevant to physics where CT can offer more - as a mathematician you surely know the importance of the latter (all good inventions in topology -say- started with that). We have a common situation here (in modern times) : person 1 says that stuff X is important and offers little or no evidence, person 2 thinks ``hmm perhaps in 150 years and even then ... ''. The only way to resolve this is naked evidence (and I have asked Kea for that at least 5 times already).

Careful

 

[john baez]

category theory and physics

Again, could you give us an *example* relevant to physics where CT can offer more...

A lot of my work is on this subject. I tried to pull together all the strands in my http://math.ucr.edu/home/baez/quantum_spacetime/" , so that's probably a good place to start. I tried to sketch how "categories with duals" connect general relativity and quantum mechanics, and how "2-categories with duals" will eventually unify string theory, spin foam models and higher gauge theory. It's a big story, so you'll probably need to read some of the references to get what's going on.

Another decent place to start is my paper with Aaron Lauda, http://math.ucr.edu/home/baez/history.pdf" . It's only in draft form, but it tells the story pretty much from the beginning.

For some applications of higher categories to strings, try the physics introduction to my paper with Urs Schreiber on http://math.ucr.edu/home/baez/2conn.pdf" develop these applications in more detail.

But, the applications of higher gauge theory are not just to strings! BF theory has http://arxiv.org/abs/gr-qc/9905087" .

If this isn't enough for you, Urs has also made a http://golem.ph.utexas.edu/string/archives/000775.html" . Dig in!

 

[selfAdjoint]

Professor Baez, your link for "A History of n-Categorical Physics" was not current. I have taken the liberty of correcting it. I hope this is all right with you.

 

[marcus]

in case anyone wants to steer back closer to topic, the original Baratin Freidel paper didnt have category theory in it, but still had a lot of non-trivial mathematics.
http://arxiv.org/abs/gr-qc/0604016
Hidden Quantum Gravity in 3d Feynman diagrams
Aristide Baratin, Laurent Freidel
35 pages, 4 figures

"In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results."

this is the Baratin Freidel 3D paper (april 2006)
the Baratin Freidel 4D paper is what JB started this thread about

when the Baratin Freidel 4D paper comes out, we will probably all want to have understood the 3D paper, because they explicitly say they are using similar methods and get similar results
================

discussing Category Theory is also helpful though, so whatever happens is all to the good

 

[marcus]

even though the Baratin Freidel 3D paper does not have HGT or two-groups----it is very down to earth---in the original post JB suggested how the other stuff could come in----might sort of INSIST on getting into the picture actually.

With any luck, sometime soon you can read this paper on the arXiv:

Aristide Baratin and Laurent Freidel
Hidden quantum gravity in 4d Feynman diagrams: emergence of spin foams

The idea is that any ordinary quantum field theory in 4d Minkowski spacetime can be reformulated as a spin foam model. This spin foam model is thus a candidate for the G -> 0 limit of any spin foam model of quantum gravity and matter!

In other words, we now have a precise target to shoot at. We don't know a spin foam model that gives gravity in 4 dimensions, but now we know one that gives the G -> 0 limit of gravity: i.e., ordinary quantum field theory. So, we should make up a spin foam model that reduces to Baratin and Freidel's when G -> 0.

The fascinating thing I noticed is that their spin foam model seems to be based on the Poincare 2-group. I invented this 2-group in my http://www.arxiv.org/abs/hep-th/0206130" . The physical meaning of their spin foam model was unclear, and some details were not worked out, but it was very tantalizing. What did it mean?

I now conjecture - and so do Baratin and Freidel - that when everything is properly worked out, Crane and Sheppeard's spin foam model is the same as Baratin and Freidel's. So, it gives ordinary particle physics in Minkowski spacetime, at least after matter is included (which Baratin and Freidel explain how to do).

If this is true, one can't help but dream...

... that deforming the Poincare 2-group into some sort of "quantum 2-group" could give a more interesting spin foam model: ideally, something that describes 4d quantum gravity coupled to matter! This more interesting spin foam model should reduce to Baratin and Freidel's in the limit G -> 0.

Of course this dream sounds "too good to be true", but there are some hints that it might work, to be found in http://arxiv.org/abs/hep-th/0501191" . In particular, they describe gravity in way (equation 26) which reduces to BF theory as G -> 0.

Optimistic hopes in quantum gravity are usually dashed, but stay tuned.

I will quote the Crane Sheppeard abstract again for completeness

http://www.arxiv.org/abs/math.QA/0306440
2-categorical Poincare Representations and State Sum Applications
L. Crane, M.D. Sheppeard
16 pages, 1 figure

"This is intended as a self-contained introduction to the representation theory developed in order to create a Poincare 2-category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2-category appropriate to Lie 2-group symmetries and discuss its application to the problem of finding a state sum model for Quantum Gravity. There is a remarkable richness in its details, reflecting some desirable characteristics of physical 4-dimensionality. We begin with a review of the method of orbits in Geometric Quantization, as an aid to the intuition that the geometric picture unfolded here may be seen as a categorification of this process."

 

[john baez]

Thanks!

Professor Baez, your link for "A History of n-Categorical Physics" was not current. I have taken the liberty of correcting it. I hope this is all right with you.

Sure, thanks a million! I guess I just made a typo or something.

I'm quite oblivious to the details of how people are keeping things on track behind the scenes here - and I love it! :!) I used to moderate sci.physics.research, so I know some of the work involved in running a flame-free forum with a low http://math.ucr.edu/home/baez/crackpot.html" . It's great not having to think about this for a change.

It's also fun being able to use really childish emoticons.

 

[marcus]

It's also fun being able to use really childish emoticons.

Our emoticons are unquestionably excellent

I don't want to interrupt the discussion (which might have some bearing on HGT, twogroups etc.), so no need to respond to this---anybody can just hop over and continue. But this occurred to me today about Baratin Freidel 3D paper which I was trying to assimilate.

Freidel is offering us a new thing. the "FEYNMAN DIAGRAM SPIN FOAM". He uses that name somewhere, for the object he is constructing. Sometimes simply naming a new thing helps to make it palpable.

You give him any old Feynman diagram and he constructs a spin foam out of it. Actually the Feynman diagram has to be of a restricted sort (IIRC scalar and closed) because so far they didnt bother to extend it to certain cases but IMO they can.

So modulo some uninteresting restrictions he can make a FLAT GRAVITYLESS SPIN FOAM that imitates any Feynman diagram you want, with amplitudes as observables whose expectation values are defined using the spinfoam's own partition function.

So "Freidel transcription service" is open for business---you can take in all your old Feynman diagrams and get them transcribed into the new spinfoam format.

... I used to moderate sci.physics.research, so I know some of the work involved in running a flame-free forum with a low http://math.ucr.edu/home/baez/crackpot.html" .

Despite much dedicated hard work by the mods it's my impression that we aren't always super low on the crackpot index---or entirely flame-free either. Sometimes one just has to ignore stuff.

 

[Careful]

***A lot of my work is on this subject. I tried to pull together all the strands in my http://math.ucr.edu/home/baez/quantum_spacetime/" , so that's probably a good place to start. I tried to sketch how "categories with duals" connect general relativity and quantum mechanics, and how "2-categories with duals" will eventually unify string theory, spin foam models and higher gauge theory. It's a big story, so you'll probably need to read some of the references to get what's going on. ***

First, thanks for the references. Second, a good deal of your talk deals with the paper I read and commented upon on another thread where I consider this kinematical analogy to be merely an artifact of the particular model (which can be easily ``undone'') - and therefore not deep at all. So, I would appreciate it if you could respond to those and/or tell us whether you have any relationship between QM/GR which goes further than those due to some abstractions of specifications in the spinfoam models floating around. Sorry, but I have met enough people giving references which turned out to be entirely useless (and actually without any meat/relevance at all) in the end; therefore I ask (this is nothing personal !) in order to calibrate my reading filter since we all have limited amount of time. Likewise, I have seen enough big stories without any specific applications and those are classified under theology in my mind.

Careful

 

[john baez]

So, I would appreciate it if you could respond to those and/or tell us whether you have any relationship between QM/GR which goes further than those due to some abstractions of specifications in the spinfoam models floating around.

I do, and I explained this in the first half of my http://math.ucr.edu/home/baez/quantum_spacetime/" . Quantum mechanics and general relativity are similar in ways which suggest the outlines of a theory that combines them. If you understand the similarities, you'll see that quantum teleportation is no stranger than stretching out a wiggle in a piece of rope! Check it out!

 

[Careful]

I do, and I explained this in the first half of my http://math.ucr.edu/home/baez/quantum_spacetime/" . Quantum mechanics and general relativity are similar in ways which suggest the outlines of a theory that combines them. If you understand the similarities, you'll see that quantum teleportation is no stranger than stretching out a wiggle in a piece of rope! Check it out!

Ohw are you going to knitpick now on the mere fact that strictly speaking this entanglement aspect does not belong to the spin foam formalism. I did not miss that kinematical analogy which is quite simple to imagine and does not require nCob at all. But again you are not answering my questions, neither do I understand why you suggest we should take these things (which were long known already) seriously. First, I explained previously why your conclusions concerning the duplication map and cartesian product are ``unstable'' (while you put quite some stress on that analogy), second your ``solution'' to quantum entanglement has been studied in one form or another for many decades: for example it was well known how to do this using backwards causation (hence playing around with two arrows of time) in Minkowski - Aharonov has toyed with this in the eighties. Models where entangled particles are connected by some invisible rope and where a twist is somehow communicated over a spacelike distance are old. However, very few claim these have something to do with general relativity unless you take for example (a generalization of) the idea of ``invisible'' Einstein-Rosen bridges connecting the two particles seriously (or a small fifth dimension which serves as a superluminal hosepipe) ; but in this way you can virtually explain everything and (almost) classical field theory is sufficient for constructing QG. In that context, I have already mentioned that topology change is not something which belongs to GR in my view. So you poored some known ideas into particular categories nCob and Hilb (some stable, some not) so where is the new physics ? We do not seem to get to the same conclusions, hmm must be because I did not understand the similarities.

Why should this help me in finding and/or implementing a suitable dynamics ? What about the relation between the problem of time and the measurement problem (yes, that is an issue in quantum gravity) ? After eighty years of research, one could at least expect any modern proposal to adress these core issues, so I am curious how you see these.

Careful

 

[marcus]

Many of us are so eager to discuss categorics---a sign of how important it has become as a way of transforming how we view and understand mathematics these days. But actually part of the theme here is not explicitly involved with categories, as far as I know so far-----namely the BARATIN FREIDEL work on
Feynman graph spinfoams

which might turn out to involve categories explicitly but so far didnt AFAIK.

so as not to lose sight of the non-categoric side, I have highlighted parts of Francesca's post earlier in this thread :

We are waiting for...
meanwhile I call back the former paper:

http://arxiv.org/abs/gr-qc/0604016
Hidden Quantum Gravity in 3d Feynman diagrams
Aristide Baratin, Laurent Freidel
35 pages, 4 figures

"... The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories[/color] and can also be applied in higher dimensions. We also show that this Feynman graph spin foam[/color] model, which encodes the geometry of flat space-time, can be ...
... the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. ..."

Anyone is welcome to correct me if I am wrong, but to me it seems
that so far THERE NEVER HAS BEEN a background independent perspective on usual field theories[/color]

So far Feynman graphs have always been defined on a flat space or some setup curved space which is morally flat, but never defined in a background independent way (without reference to any background metric).

and the appliance gadget by which this is accomplished is a new thing, namely a Feynman graph spin foam[/color]
====================

I have two main questions that I am waiting for the answers----only time will tell about this:

will Baratin and Freidel be able to extend the results to 4D and if so, when will the paper appear? (the 3D paper was back in April)

supposing they are successful, and the 4D paper appears, then will it explicitly have a categories angle?

 

[marcus]

*... correct me if I am wrong, but to me it seems
that so far THERE NEVER HAS BEEN a background independent perspective on usual field theories[/color]*

So what prior field theories does this make obsolete? Or what prior field theories does this approach NOT make obsolete?

The gadget accomplishing this is a new thing, namely a Feynman graph spin foam.
It seems that Freidel et al have TAUGHT A FEYNMAN GRAPH TO LIE FLAT, in the dimension space they want, as one can teach a dog to sit when one says "SIT!"

the Freidel-trained Feynman graph can still adopt all possible postures and proportions, with legs all different lengths, except without being embedded it nevertheless knows how to assume only those positions that are appropriate to living in a particular space.
so it INTRINSICALLY has the right geometry. it is INATELY at home in the proper dimension.

that is what a "Feynman graph spinfoam" is, AFAICS. It is a Feynman graph that does not have to be embedded in some rented Minkowski apartment because it is has an instinctive sense of geometry.

Psssst! this means we don't need space any more.
========
Please let me know if you disagree with either the interpretation or the attributed significance.
disagreement would be welcome!

 

[Kea]

...supposing they are successful, and the 4D paper appears, then will it explicitly have a categories angle?

Oh, yes! Even if it looks a lot like the last one, secretly it is really categories through and through. One is really taking geometric realisations of n-functors into the representation (Poincare) category, philosophically like in TFTs. Recursive triangulations are secretly related to Street's oriental diagrams. Balls with marked points and their dual trees are just the sort of thing the operad people deal with...

...and so on.

 

[marcus]

secretly it is really categories through and through.

...and so on.

but I will still be curious to see if it is explicitly categories or not.

we could bet.

I will bet that Baratin Freidel DO bring out their 4D paper sometime this year
and that it does indeed show how to rewrite Feynman graphs as spinfoams
(thus giving a background independent version of usual field theory)
BUT that there are no explicit categorics.

I hope you bet to the contrary, that there will be explicit categorics.

then I will win: W :!) T !
===============

I think that they will still have to turn on gravity. Maybe that will require some twogroup construction.
If they succeed in the next paper that is still only describing the flat
G -> 0 limit of what the real theory has to be. It will have usual field theory, and thus matter, in a zero-gravity spinfoam world. And I bet not a word about categories. Shall we bet?

 

[Kea]

And I bet not a word about categories. Shall we bet?

All right. I don't think there'll be much category theory, but I do think there will be a word or two. And I don't mind losing a bet to you, Marcus!

 

[selfAdjoint]

All right. I don't think there'll be much category theory, but I do think there will be a word or two. And I don't mind losing a bet to you, Marcus!

Thing about categories, which even their partisans grant makes skeptics smile, is that with sufficient skill and ingenuity you can do ANY math in categories, especially now that n-categories (recursive categorization) has/have been added to the tool kit. Categories can be to math as macros are to programming.

The perennial question about categories is not "Can we do this theory in categories?" but "Can categories give us answers to these questions that we couldn't get without categories?" (Never mind "easier"; that's in the eye of the beholder. Some people get off on down and dirty hard analysis; look at Hardy; his hobby was simplifying awful complicated integrals).

 

[marcus]

... Categories can be to math as macros are to programming.
... "Can categories give us answers to these questions that we couldn't get without categories?"...

Pragmatic. proof of the pudding. Maybe my attitude is similar to yours. I also try to look at results, particularly does it make people SMARTER? if they use categorics part of the time, do they see analogies quicker? is their inventiveness speeded up?

it is a kind of "smart pills" (as in the expression "now you're taking smart pills") and using categorics seems to make some people frazzled or even wacky and some more creative. the result is not always good, but sometimes is.

In this regard I am only interested in research say since 2002 because only lately did I see it impinge on physics (in ways that are explicit and make sense to my limited perception). Maybe all categorics was useless to physics before that---I don't know about that.

but now I am beginning to see a correlation. the hidebound rejectionist attitude may be correlated with mediocrity and lack of inspiration. and some sense of "higher algebra" (whether categoric or some other) seems correlated to promising new physics ideas.

I am waiting to see---my attitude is "by their fruits ye shall know them". We will just see if the people who come up with the necessary new ideas are the people who are taking smart pills, or the others.

Probably trigonometry was not necessary. Hipparchus invented it around 140 BC roughly, and it was convenient but you PROBABLY COULD DO EVERYTHING just using geometry. nevertheless he made trig tables.
Probably Cartesian graph paper was not necessary. You probably could do everything with elaborate geometric constructions and not using plotted formulas. Probably some hidebound rejectionists were scoffing this. But Descartes went ahead and promoted his coordinate methods.
YOU CAN ALWAYS DO EVERYTHING THE OLD WAY. the question is whether the people using the new way appear to be more clever and do they invent the necessary things. and the question is do the people who reject the new way, do they seem mediocre and uninventive. Or is it different? I can only learn by watching the outcomes.

 

[marcus]

All right. I don't think there'll be much category theory, but I do think there will be a word or two. And I don't mind losing a bet to you, Marcus!

We have a bet, Dea Kea!
If they include a word or two of categorics then you win.
If they have no explicit mention of categorics then I win.

this is only if the paper comes out this year. if the problem proves unexpectedly intractible and they get stalled, bets are off.

I wish someone would speculate what the Baratin Freidel 4D case will look like. I can see how they construct a flat Feynman spinfoam in 3D spacetime. It is just a PARTITION FUNCTION that somehow remembers that it is supposed to dwell in 3D even without a surrounding 3D spacetime to remind it. Like one of those shape-remembering pieces of metal, that go *boink* and flip back to their imprintment.

Formally it is all seemingly straightforward, the trick is to get the right partition function. but spinfoams in 3D are regarded as somewhat rudimentary. maybe in the 4D case the partition function will be similar but just a bit gruesome.

Is that all, do you think? Will everything look like the 3D case except messier? I think I could stand that, at least if I had a chocolate malted milkshake to steady my nerves.

 

[Careful]

***
but now I am beginning to see a correlation. the hidebound rejectionist attitude may be correlated with mediocrity and lack of inspiration. and some sense of "higher algebra" (whether categoric or some other) seems correlated to promising new physics ideas.

I am waiting to see---my attitude is "by their fruits ye shall know them". We will just see if the people who come up with the necessary new ideas are the people who are taking smart pills, or the others.
***

So you say :
(a) 99,8 percent of physicists is unimaginative and more mediocre than category theorists
(b) you have to know category theory in order to be smarter

Moreover, there is only a hidebound rejectionist attitude when a large community accepts the use of the subject under consideration.

****
Probably Cartesian graph paper was not necessary. You probably could do everything with elaborate geometric constructions and not using plotted formulas. Probably some hidebound rejectionists were scoffing this. But Descartes went ahead and promoted his coordinate methods.
YOU CAN ALWAYS DO EVERYTHING THE OLD WAY. the question is whether the people using the new way appear to be more clever and do they invent the necessary things. and the question is do the people who reject the new way, do they seem mediocre and uninventive. Or is it different? I can only learn by watching the outcomes ***

This shows that you do not understand history. The method of Descartes was immediately recognized, just as special relativity, Maxwell theory and so on... . Instead of making erroneous political manifests you could contribute by explaining the useful, new insights for physics, as I asked you before.

Careful

 

[Kea]

This shows that you do not understand history. The method of Descartes was immediately recognized, just as special relativity, Maxwell theory and so on...

Classic.

 

[marcus]

Classic.

I think you meant classic blooper. Hope you did anyway.
My point was that historically there were some holdouts to the method of Descartes. Indeed there were lots! Newton for example.

Descartes explained his coordinates in 1637 (Discourse on Method, Geometry) and Newton's Principia appeared in 1687. Fifty years later. You can see him strictly avoiding Cartesian method. The example of Newton suggests that Cartesian coordinates WERE NOT FASHIONABLE at least in some circles even 50 years after exposition.

Here
http://members.tripod.com/~gravitee/booki2.htm
you can see Newton using Euclidean method to discuss circular motion in a plane, where we would today normally use Cartesian coordinates.

To make my point (the analogy with category theory) I only need to know that there were SOME holdouts

...Cartesian graph paper was not necessary. You probably could do everything with elaborate geometric constructions and not using plotted formulas. Probably some hidebound rejectionists were scoffing this. But Descartes went ahead and promoted his coordinate methods.

Cartesian coordinates are a good analogy to categorics. Even though they were available and would have been convenient, Newton made do with a pre-Cartesian approach. At least here in Book I section 2 and IIRC more generally. And unquestionably so did many others. Indeed 300 years later there were still people who strenuously avoided coordinates and preferred Euclid's methods. I knew one of them personally.
Newton of the Principia Book I was hardly the sole holdout, Greek style plane geometry still has class (it is classic after all).

What I am trying to say with this example, about categorics, is that one should not look for something that you CAN'T DO without the new method. There will often be some way to kludge around and make do, and that doesn't prove anything. It can even be a matter of taste. What one should look for is cases where someone GETS DIFFERENT IDEAS by solving the same problem by way of a different conceptual framework.

If anyone wants to see more of Newton Principia
http://members.tripod.com/~gravitee/toc.htm

 

[Kea]

If they include a word or two of categorics then you win.
If they have no explicit mention of categorics then I win.

What do I win, if I win?

Yes, classic blooper.

 

[marcus]

What do I win, if I win?

Well, I could write a (slightly disrespectful) rhyming poem about how wonderful you are.
Let me think about it. It probably wouldn't be a limerick. most likely a doggerel quatrain.
But I'm the one who is going to win! Can you write just-a-touch disrespectful light verse?
================
I wrote this next when out of sorts, before I saw your post:

I think we should just avoid or ignore complaining about category theory in this thread. People should use it if it gives them good ideas and inspires them to solve problems. And NOT use it if it DOESN'T.
People who don't get any good from it should simply not bother. After a point, more talking to them will not help them. In some way it seems silly to argue about the Goods and Bads of some (to an extent optional) mathematical method or framework, with someone with a mindset unsuited to it.
==============

From my viewpoint, Baez has already made abundantly clear to me as observer that it is a great source of new ways to look at things and that it is coming into physics. Also Urs Schreiber is a bellweather in this respect. So I will be sure to keep my eye out for things happening with categorics and physics. I am also glad to see new stuff come out that does NOT use category theory. Whatever floats the researcher's boat.

So I will do what I can to ignore arguing about the merit of categorics, or lack thereof, and hope I succeed.

 

[Careful]

**I think you meant classic blooper. Hope you did anyway.
My point was that historically there were some holdouts to the method of Descartes. Indeed there were lots! Newton for example.**

This example is not even a counterexample to what I said. The method of descartes was for sure accepted by more than 0,2 percent of scientists.

By the way Marcus, for someone with a nonexpert opinion, you often refer to the notion of wrong/right mindset and to what is hopeful/sufficient evidence for something.

Careful

 

[john baez]

Ohw are you going to knitpick now on the mere fact that strictly speaking this entanglement aspect does not belong to the spin foam formalism. I did not miss that kinematical analogy which is quite simple to imagine and does not require nCob at all. But again you are not answering my questions, neither do I understand why you suggest we should take these things (which were long known already) seriously.

You win; I give up. In fact I gave up online debates some time ago.

[...] your ``solution'' to quantum entanglement has been studied in one form or another for many decades: for example it was well known how to do this using backwards causation (hence playing around with two arrows of time) in Minkowski - Aharonov has toyed with this in the eighties. Models where entangled particles are connected by some invisible rope and where a twist is somehow communicated over a spacelike distance are old.

For those who are interested:

Such models aren't http://math.ucr.edu/home/baez/quantum_spacetime/" . I'm talking about how the category of Hilbert spaces (Hilb) and the category of n-dimensional cobordisms (nCob) are both monoidal categories with duals. The fact that Hilb has duals allows for quantum teleportation; the fact that nCob also has them is what allows you to straighten out a kink in a rope (in the case n = 1).

This is simply a fact, not a "model" - and certainly not a model where quantum entangled particles are connected in some way, e.g. by an "invisible rope". Quantum entanglement arises from the fact that Hilb is non-cartesian, unlike the category of sets. nCob is also non-cartesian.

(For a monoidal category to have duals, it must be non-cartesian, but not vice versa. Or, in physics speak: we need entangled states to carry out quantum teleportation, but we also need more. All this is nicely explained in Bob Coecke's paper on http://arxiv.org/abs/quant-ph/0510032" .)

While these are just mathematical facts, they point the way towards models of quantum gravity, by showing us which class of mathematical structures combine the physically important features of general relativity and quantum mechanics.

But, we need to take another step or two - and probably many more we haven't seen yet. For starters, nCob is better thought of as a monoidal n-category with duals. This describes all the ways we can stick together small pieces of n-dimensional spacetime; it captures the n-dimensionality of spacetime in a way that a mere category can't do.

This suggests trying to define "nHilb" - an n-category of "n-Hilbert spaces" - and showing it's a monoidal n-category with duals. I did this for http://arxiv.org/abs/q-alg/9609018" a while ago, and it turns out to be quite interesting. In particular: just as Hilb gives rise to Feynman diagrams, 2Hilb gives rise to "spin foams" - a 2-dimensional generalization of Feynman diagrams. If we went to nHilb for higher n, we'd get still higher-dimensiaonal diagrams.

I've never emphasized this aspect in my papers on spin foams, since I know most physicists don't like higher categories. But, I explain how it works in weeks 1-3 of the http://math.ucr.edu/home/baez/qg-winter2005/" from my quantum gravity seminar.

A lot of work has been done on spin foam models by now, but they're still mysterious. For example, we've all heard a lot about the Barrett-Crane model, but it's still unclear why Simone Speziale and Dan Christensen are getting really good agreement with the graviton propagator based on calculations involving a single big 4-simplex, refinements of http://arxiv.org/abs/gr-qc/0508124" . They made a lot of progress on this last week: Dan's supercomputer calculations match what Simone is getting analytically. But why should these calculations work at all - after all, if any model like this is right, you'd expect spacetime to be made of lots of small 4-simplexes. Viqar Husain has some ideas...

And then there's the http://arxiv.org/abs/math.QA/0306440" . This explicitly uses infinite-dimensional 2-Hilbert spaces, namely representations of the Poincare 2-group. But what does it mean, physically? Is it related to Baratin and Freidel's spin foam model for ordinary quantum field theory on Minkowski spacetime? I guessed it was... but my students Jeff Morton and Derek Wise have been doing a bunch of calculations with Baratin and Freidel, and they seem to be concluding that it's not.

However, they found the Crane-Sheppeard model includes the Barrett-Crane model in a certain sneaky way. And, perhaps the best part is: Freidel now understands 2-Hilbert spaces and 2-groups, and he wants to keep studying models based on them!