Spinfoams are the Feynman diagrams of GFT (V. Rivasseau)

[marcus]

Vincent Rivasseau and four others just posted "July3150" (I mean 1007.3150, just easier to remember that way for some reason) where they made the observation in effect that
spinfoams are the Feynman diagrams of GFT.

It's not a new thing to point out, but it is very helpful to mention.
Spinfoams are a method to calculate QG amplitudes. One does not imagine that spacetime is "made" of spinfoams any more than electrodynamic particle interactions are "made" out of conventional Feynman diagrams. It is a way to catalog various possible histories and to organize one's calculations.

I think many people have now taken a look at the key April paper of Rovelli, and if you did then you know that he gives a GFT formulation of LQG. I will take another post to same more specifically what I mean.

Rivasseau's July3150 paper is called "Quantum Corrections to the Group Field Theory Formulation of the EPRL/FK Models."
To relate that to Rovelli's April paper one just has to know that Rovelli uses the term "new LQG" to refer to what Rivasseau calls EPRL (Engle, Pereira, Rovelli, Livine) model. We have been saying things like EPRL for a couple of years now and I guess by now it is really clear that there is a new LQG, and this is it, and it is much easier to say "new LQG".

I should start a new post, but first will mention for newcomers that a spinfoam is not a graph. A graph is made of nodes and links. Spinnetworks are labeled graphs.

A spinfoam is just the "one-more-dimension" extended idea of a graph. It is technically a "2-complex", made of vertices, edges, and faces (a face is just a 2d surface, an irregular polygon.)

A 2-complex can describe schematically how a graph might evolve in time. The track of a graph node gives an edge. The track of a graph link describes a face. As the graph evolves, new nodes can appear in it, old nodes can disappear. Old links can be broken and the nodes reconnect in changed ways by new links. The record of these "moves" is contained in the 2-complex. I think it was John Baez who called 2-complexes by the name "foam" when they are serving to diagram the history of a graph. Especially a graph labeled by spins (in which case the elements of the 2-complex also get labeled: the vertices edges faces of the foam.)

 

[marcus]

I think that anyone who looks at this thread most likely knows that the key idea of LQG is that quantum gravity is about making geometrical measurements.
Rather than what one imagines space to be "made of". So we are talking about geometry---the business of measuring distances angles areas volumes, the geometric relationships between events. (I cravenly avoided mentioning time.)

That is why the idea of a GRAPH is so important. We can think of geometry as consisting of chunks of volume which meet each other at flakes of surface area. The chunks of volume and flakes of area are the nodes and links of a graph.

The volume of any physical region will be the total volume of all the nodes it encloses. The area of any physical surface (like a desktop) will be the total area of all the links which the surface cuts---or which, to put it differently, pass through the surface.

So the Hilbert space of geometric states will be the sum of the Hilbert spaces of each possible graph. For each graph Gamma, we need to know the states of geometry which can live on that graph . That will be the GRAPH HILBERTSPACE H_Γ.

It is what turns out to be simple and elegant to describe using GFT.

And then the whole Hilbertspace of states of geometry is just the direct sum of all the graph Hilbertspaces.

One of the beauties of this approach is that one does not have to put in by hand the idea of labeling the graphs with spins. The labeling jumps out at you when you ask for a basis for the graph Hilbertspace. It is a consequence of the trusty and venerable Peter Weyl theorem.

The basic way we get H_Γ is as the space of square-integrable complex-valued functions on a group manifold---typically a direct sum of copies of SU(2). The graph Γ determines the detailed structure of the group manifold (how many copies of the group etc...). Every math or physics major knows the space of square-integrable functions L²(X) for some domain X. It is just all the functions you can define on that domain that aren't too wild for you to integrate their absolute value squared.

I'm following the development in the April paper http://arxv.org/abs/1004.1780 .

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ...∏∑∫∂√ ...± ÷...←↓→↑↔~≈≠≡≤≥...½...∞...(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[marcus]

Something f-h said, just over 4 years ago, in a thread which George Jones reminded us of.

... both seem extremely unphysical to me, but LQG with the better potential to get somewhere without experiment because its proponents are more conceptually minded...

I think he pointed to a key realization (sometimes called "taking GR seriously".) Recognizing that we must find a way to do field theory without any background spacetime.
So there must be a mathematical handle on the geometry of the universe providing a context within which (other) fields can be defined. The giant web of relationships which is the geometry of the universe must somehow be present as a mathematical object in the theory.
Or if not the whole geometric web, at least any arbitrarily large spacetime region.

 

[atyy]

Does GR make sense without matter? Remember that test particles are matter - and the use of test particles is not background independent.

 

[marcus]

Does GR make sense without matter?

I'm not sure what your point is, Atyy. QG for sure has to include matter fields. I don't see why that should compromise background independence.

Basically we are talking about work in progress. LQC models include some simple matter, and people are currently working on treating cosmology using spinfoams and loosening the restrictions, so that LQC looks more like the full theory. Francesca might be able to say something about that. I don't know the latest.

 

[MTd2]

What is the relation between GFT and spin networks?

 

[marcus]

What is the relation between GFT and spin networks?

I don't have anything to say, at the moment, about GFT in abstract generality. If you look in Oriti's papers about GFT he explains how it in kind of a unifying formalism that can be used in various applications.

Rivasseau simply says he is dealing with the "GFT formulation" of new LQG models.
I can say very easily what "GFT formulation" means to me. It means that you formulate a theory or model by writing down functions defined on group manifolds.

An example of a group manifold is the direct sum of N copies of a Lie group such as SU(2).

If you want to see how spin networks arise in a GFT formulation, you can look in http://arxiv.org/abs/1004.1780.

I will also try to go through the steps. I already started both in another thread and here.

Do you know the notation L²(X) for the square-integrable function defined on X?
I think you do but anybody to whom it is not familiar already should stop us and ask.

If you have an unlabeled graph with L links, where L is some number like 5 or 8 or 5 million, then you can think of the square integrable functions on the group manifold SU(2)^L.

The conventional notation is L²(SU(2)^L)

You can think of that as made up of all the possible ways of labeling the L links of the graph with spins. And all possible linear combinations of those ways of labeling. All the mixtures of possible ways of labeling the links with spins.

There is still some work to do but you can see that the L² functions on the manifold are like "wave functions" defined on the set of possible labelings. And then you can pick a set of basis vectors---"pure states". These turn out to be like spin networks---definite labelings of all L links of the graph.

I just want to give the idea in a handwave way. There is more work. One actually deals with directed graphs, and one has to remember about the nodes.
Details are in the 1004.1780 paper. I've also written some more in other posts.

The basic idea is that the GFT formulation let's you start simply with a group manifold and the familiar L² function space defined on it. And the desired structure of spin networks arises naturally from the group manifold.

It seems less arbitrary then.
http://en.wikipedia.org/wiki/Peter–Weyl_theorem

 

[atyy]

In GR, how do you get geometry without test particles?

 

[marcus]

In GR, how do you get geometry without test particles?

I regret to say I don't understand your question, Atyy. Why would one suppose that GR has to make do without the concept of a test particle?

 

[MTd2]

So, GFT is more primitive then spin networks?

 

[marcus]

So, GFT is more primitive then spin networks?

Oriti gave some general introductory talks about GFT at Perimeter a few years back, and also I've read some of his overview stuff about GFT. My impression is that GFT is not a theory of anything in particular.

It is an extremely broad versatile framework. You can use it to formulate several different theories including LQG.

My impression could be wrong and, as I said, I don't have anything to say about GFT in general. I am focusing on the GFT-style formulation of spinfoam LQG.

As I recall, Laurent Freidel introduced the GFT idea over 10 years ago. He and others have used the idea without a making a big deal of it. The person who put the spotlight on it AFAICS is Dan Oriti. He gave it name-recognition. Up to then, other people had just been quietly utilizing the method from time to time. I think.

Again I could be wrong, but my impression is that the basic GFT idea is to define functions (or "fields") on a GROUP MANIFOLD.

That way you don't have to construct a naive explicit manifold-representation of spacetime!
You can express the GEOMETRY of the universe abstractly, without having an explicit spacetime continuum.

Here is a simple example of a group manifold: the cartesian product of a million copies of SU(2). Call that M. It is a manifold. It is in fact a Lie group (so very nice.)

A point in M is a choice of a million SU(2) group elements----a million 2x2 matrices---a million possible Lorentzian symmetries or rotations or whatever. These could represent a web of 2x2 matrices that are "happening" all over the universe, each in a different place, in a million different places.
With a little extra work, they could be labels of a spin network that describes the geometry of the the universe.

This would need more detail to give substance but it suggests the idea. Basically the versatility of working on a group manifold. You can do lots of different things with a group manifold. Not just spin foam! You can also do Regge. Probably other things I have not bothered to learn about. GFT is a multipurpose framework, not a particular theory of any thing AFAICS.

There is a category theory angle angle which I dimly perceive. Group manifolds do not have to be simple cartesian products, they can also be quotients of products where you "mod out". It is possible that there is a kind of "functorial" relation of directed GRAPHS with group manifolds. Given a group G and a directed graph Γ there may be a natural way to construct a group manifold out of copies of G which expresses the structure of Γ.
I am only mildly curious about this now. I have too much else to think about. Some people might be interested in that aspect. Or perhaps John Baez wrote about it 10 years ago and all the questions are already addressed. I haven't looked enough to know anything about the "category" aspect of graphs and group manifolds. Powerful mathematical tools in any case.

In Rovelli's April paper you can see him making intensive use of the GFT-style approach. It is how the new LQG formulation is built. A countable infinity of group manifolds, plus the Peter Weyl theorem.

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ...∏∑∫∂√ ...± ÷...←↓→↑↔~≈≠≡≤≥...½...∞...(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[atyy]

I regret to say I don't understand your question, Atyy. Why would one suppose that GR has to make do without the concept of a test particle?

In my understanding, background independence means that you do not know the gravitational field at all (except for topology, signature) until you know the matter configuration, or at least its stress-energy tensor. A test particle is a particle whose stress-energy does not enter into the determining the solution of the Einstein field equation, so the use of test particles to give the gravitational field a geometrical meaning is not background independent.

 

[marcus]

... background independence means that you do not know the gravitational field at all...until you know the matter configuration...

Seems inconsequential, Atyy. You have the same thing in Newton gravity. The test particle is something of such small mass that its effect on the field can be neglected. Same thing with GR which is background indep.

You seem to be saying there is some philosophical problem with 1915 General Relativity which Einstein did not notice. I don't think it is anything to worry about, either in the classical or quantum context. But you could start a thread about it in the Relativity forum, I guess.

 

[atyy]

Seems inconsequential, Atyy. You have the same thing in Newton gravity. The test particle is something of such small mass that its effect on the field can be neglected. Same thing with GR which is background indep.

You seem to be saying there is some philosophical problem with 1915 General Relativity which Einstein did not notice. I don't think it is anything to worry about, either in the classical or quantum context. But you could start a thread about it in the Relativity forum, I guess.

I am saying there is a conceptual problem with LQG (in the light of your quote of f-h above). I like GFT, because if does produce quantum gravity, I believe it will do so in an emergent way - with matter automatically coming along - just like in string theory.

Incidentally, I don't think Freidel introduced GFTs. I think it was Boulatov and then Ooguri who introduced the first GFTs, then Reisenberger and Rovelli who showed that spin foams come from GFTs.

 

[marcus]

I think you are right about the history---I remember hearing that the mathematical approach was developed in the work of Boulatov, Ooguri, Reissenberger and Rovelli and that the NAME Group Field Theory was later introduced by Freidel.

I think Freidel contributed materially to the subject, as well as "christening" it GFT. And I could be wrong---don't have time to check.

I disagree with you, however, on there being a conceptual problem with a background independent theory (such as classical General Relativity and other B.I.) stemming from the need to imagine test particles.

Another thing that occurs to me is that the use of test particles is typically where you want to measure FORCES, but in many contexts the gravitational field is measured by geometrical measurements---areas, volumes, angles made by different geodesic paths, timechange, redshift etc. etc.
Since one is measuring geometry one does not need test particles in order to be operationally defined and on a solid philosphical footing. But I think you would probably agree that it is a quibble in any case?

 

[atyy]

I disagree with you, however, on there being a conceptual problem with a background independent theory (such as classical General Relativity and other B.I.) stemming from the need to imagine test particles.

Another thing that occurs to me is that the use of test particles is typically where you want to measure FORCES, but in many contexts the gravitational field is measured by geometrical measurements---areas, volumes, angles made by different geodesic paths, timechange, redshift etc. etc.
Since one is measuring geometry one does not need test particles in order to be operationally defined and on a solid philosphical footing. But I think you would probably agree that it is a quibble in any case?

There is no problem with a background independent theory because that is a theory in which one doesn't need test particles.

However, it is geometry that needs test particles - the geodesic paths you mention, for example, are traversed by test particles.

So I would say you can have background independence, but then you shouldn't have geometry - unless you have matter.

 

[marcus]

...So I would say you can have background independence, but then you shouldn't have geometry - unless you have matter.

Maybe one way to understand what you pose here is to ask "how soon will matter be added to the full LQG?"
We know that LQC paper involve matter and we know that matter can be added to LQG at the kinematic level, but how soon will this be working at the level of dynamics?

You probably read, and may recall, Rovelli's Open problem #17 on page 14 of the April paper. (You recall he gave a list of problems to work on, at the end.)
http://arxiv.org/abs/1004.1780
==quote==

17. How to couple fermions and YM fields to this formulation? The kinematics described above generalizes very easily to include fermions (at the nodes) and Yang Mills fields (on the links). Can we use the simple group theoretical argument that has selected the gravitational vertex also for coupling these matter fields?

In conclusion, the theory looks simple and beautiful to me, both in its kinematical and its dynamical parts. Some preliminary physical calculations have been performed and the results are encouraging. The theory is moving ahead fast. But we do not yet know if it really works, and there is still very much to do.
==endquote==

"Simple group theoretical argument" could be what we are calling GFT methods. I think GFT was a name that Laurent Freidel started calling group manifold methods later. As you point out, Reissenberger and Rovelli played a role in getting the methodology started, but they didn't call it GFT. I think that is what he means by "simple group theoretical argument".

So the question #17 is, can group manifold methods be used to couple matter to spinnetworks?

I'd give it a fair chance, Atyy. Loop is having a kind of "Wunderjahr" heyday this year. I just have to look at three major papers appearing inside of 3 months:

April1780 the "new LQG" paper that we are talking about---a GFT formulation if you like that term.
May0817 "a regularization of the hamiltonian" to be compatible with spinfoam
June1294 "physical boundary Hilbertspace and volume operator in the Lorentzian new spin-foam theory"

Note that the 3D boundary of a spacetime region is how Rovelli has learned to get a grip on matter---to compute 2-point functions, propagators.
Have to go. See what you think.

 

[atyy]

So the question is, can group manifold methods be used to couple matter to spinnetworks?

Or maybe no (additional) coupling is needed:

"In this sense, the work presented here is in striking contrast with the GFT models containing extra data to model the particle content of the theory [18, 19]. In fact, our work shows that no extra data is needed: matter is a particular phase of the geometry and is already somehow contained in our quantum gravity models." http://arxiv.org/abs/gr-qc/0702125

"We show that N = 1 supersymmetric BF theory in 3d leads to a supersymmetric spin foam amplitude via a lattice discretisation. Furthermore, by analysing the supersymmetric quantum amplitudes, we show that they can be re-interpreted as 3d gravity coupled to embedded fermionic Feynman diagrams." http://arxiv.org/abs/1004.0672

 

[marcus]

You point to interesting papers. The first is a 3D LQG+matter one:
http://arxiv.org/abs/gr-qc/0702125
3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory
Winston Fairbairn, Etera R. Livine
17 pages, 1 figure
(Submitted on 23 Feb 2007)
"An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models in hep-th/0512113. In this paper, we show how this relates to group field theories and generalized matrix models. In the first part, we realize that the effective field theory can be recasted as a matrix model where couplings between matrices of different sizes can occur. In a second part, we provide a family of classical solutions to the three-dimensional group field theory. By studying perturbations around these solutions, we generate the dynamics of the effective field theory. We identify a particular case which leads to the action of hep-th/0512113 for a massive field living in a flat non-commutative space-time. The most general solutions lead to field theories with non-linear redefinitions of the momentum which we propose to interpret as living on curved space-times. We conclude by discussing the possible extension to four-dimensional spinfoam models."

It refers back to a Freidel Livine 3D LQG+matter paper from 2005.
The approach is background independent and involves both matter and geometry. I remember being pretty excited by several of Laurent Freidel's papers that year, partly because involving matter Feynman diagrams. The big question was could that 2005 3D work be extended to 4D?
Etera Livine gave a talk at Perimeter earlier this year which was kind of a status report on that effort, the various things they were trying. You may have watched the video.

The second paper you mention is also about 3D gravity+matter, with Livine as one of the authors:
http://arxiv.org/abs/1004.0672.
The particle interpretation of N = 1 supersymmetric spin foams
V. Baccetti, E. R. Livine, J. P. Ryan
(Submitted on 5 Apr 2010)
"We show that N = 1 supersymmetric BF theory in 3d leads to a supersymmetric spin foam amplitude via a lattice discretisation. Furthermore, by analysing the supersymmetric quantum amplitudes, we show that they can be re-interpreted as 3d gravity coupled to embedded fermionic Feynman diagrams."

 

[atyy]

It refers back to a Freidel Livine 3D LQG+matter paper from 2005.
The approach is background independent and involves both matter and geometry. I remember being pretty excited by several of Laurent Freidel's papers that year, partly because involving matter Feynman diagrams. The big question was could that 2005 3D work be extended to 4D?

There is some progress to 4D: http://arxiv.org/abs/0903.3475

If GFT renormalization works out, there will be a debt to string theory. I believe Rivasseau is interested in GFTs because non-commutative field theories can emerge from them. Historically, a lot of interest in non-commutative field theories came from string theory.

"In view of these difficulties some physicists have started to openly criticize what they consider a disproportionate amount of intellectual resources devoted to the study of string theory compared to other alternatives [32]. I do not share these critics. I think in particular that string theory has been very successful as a brain storming tool. It has lead already to many spectacular insights into pure mathematics and geometry. But my personal bet would be that if somewhere in the mountains near the Planck scale string theory might be useful, or even correct, we should also search for other complementary and more reliable principles to guide us in the maze of waterways at the entrance of terra incognita. ...

It is a rather natural remark that since gravity alters the very geometry of ordinary space, any quantum theory of gravity should quantize ordinary space, not just the phase space of mechanics, as quantum mechanics does. Hence at some point at or before the Planck scale we should expect the algebra of ordinary coordinates or observables to be generalized to a non commutative algebra. Alain Connes, Michel Dubois-Violette, Ali Chamseddine and others have forcefully advocated that the classical Lagrangian of the current standard model arises much more naturally on simple non-commutative geometries ...

A second line of argument ends at the same conclusion. String theorists realized in the late 90's that NCQFT is an effective theory of strings [34, 35]. ...

These two lines of arguments, starting at both ends of terra incognita converge to the same conclusion: there should be an intermediate regime between QFT and string theory where NCQFT is the right formalism. ...

The simplest NCQFT on Moyal space, were found not to be renormalizable because of a surprising phenomenon called uv/ir mixing. ...

However three years ago the solution out of this riddle was found. H. Grosse and R. Wulkenhaar ..." http://arxiv.org/abs/0705.0705

 

[marcus]

Here is another QG+matter approach to consider. Can one show that it must necessarily confront an intractable landscape predicament with no clear selection principle?
One would have to investigate this as a special case, different from either NCG/SF or AsymSafe approaches.

Maybe one way to understand what you pose here is to ask "how soon will matter be added to the full LQG?"
We know that LQC papers involve matter and that matter can be added to LQG at the kinematic level, but how soon will this be working at the level of dynamics?

You probably read, and may recall, Rovelli's Open problem #17 on page 14 of the April paper. (You recall he gave a list of problems to work on, at the end.)
http://arxiv.org/abs/1004.1780
==quote==

17. How to couple fermions and YM fields to this formulation? The kinematics described above generalizes very easily to include fermions (at the nodes) and Yang Mills fields (on the links). Can we use the simple group theoretical argument that has selected the gravitational vertex also for coupling these matter fields?

In conclusion, the theory looks simple and beautiful to me, both in its kinematical and its dynamical parts. Some preliminary physical calculations have been performed and the results are encouraging. The theory is moving ahead fast. But we do not yet know if it really works, and there is still very much to do.
==endquote==

"Simple group theoretical argument" could be what we are calling GFT methods. I think GFT was a name that Laurent Freidel started calling group manifold methods later. As you point out, Reissenberger and Rovelli played a role in getting the methodology started, but they didn't call it GFT. I think that is what he means by "simple group theoretical argument".

So the question #17 is, can group manifold methods be used to couple matter to spinnetworks?...

Or maybe no (additional) coupling is needed:

"In this sense, the work presented here is in striking contrast with the GFT models containing extra data to model the particle content of the theory [18, 19]. In fact, our work shows that no extra data is needed: matter is a particular phase of the geometry and is already somehow contained in our quantum gravity models." http://arxiv.org/abs/gr-qc/0702125

"We show that N = 1 supersymmetric BF theory in 3d leads to a supersymmetric spin foam amplitude via a lattice discretisation. Furthermore, by analysing the supersymmetric quantum amplitudes, we show that they can be re-interpreted as 3d gravity coupled to embedded fermionic Feynman diagrams." http://arxiv.org/abs/1004.0672

The paper you cite here raises the prospect of coupling matter to spin foam QG in an appealingly natural way. However it is the 3D case, as you point out. For clarity I will copy the abstract:

http://arxiv.org/abs/gr-qc/0702125
3d Spinfoam Quantum Gravity: Matter as a Phase of the Group Field Theory
Winston Fairbairn, Etera R. Livine
17 pages, 1 figure
(Submitted on 23 Feb 2007)
"An effective field theory for matter coupled to three-dimensional quantum gravity was recently derived in the context of spinfoam models in hep-th/0512113. In this paper, we show how this relates to group field theories and generalized matrix models. In the first part, we realize that the effective field theory can be recasted as a matrix model where couplings between matrices of different sizes can occur. In a second part, we provide a family of classical solutions to the three-dimensional group field theory. By studying perturbations around these solutions, we generate the dynamics of the effective field theory. We identify a particular case which leads to the action of hep-th/0512113 for a massive field living in a flat non-commutative space-time. The most general solutions lead to field theories with non-linear redefinitions of the momentum which we propose to interpret as living on curved space-times. We conclude by discussing the possible extension to four-dimensional spinfoam models."