QG five principles: superpos. locality diff-inv. cross-sym. Lorentz-inv.

[Finbar]

That was my understanding also of the hypothesis of Asymptotic Safety, but then what does Litim mean by "A Wilsonian effective action for gravity Gk ... and, possibly, non-local operators in the metric field." http://arxiv.org/abs/0810.3675

Non-local operators appear in the effective action of gauge theories also.

http://arxiv.org/abs/hep-th/9808044

But the fields are still local functions on space-time.

My interpretation is that although gauge invariant observables can be non-local the
fundamental fields that are quantised may still be local quantum fields.

 

[marcus]

To follow up what I said here:

...
2. people wonder "If there is no manifold, what are the other fields defined on?"

To be consistent they must be defined on a graph---the labeled spin-network. And what Rovelli has to say about that is on page 14: the problem 17. It's a key question. Atyy has identified some other papers and other lines of development (spinfoam/GroupFieldTheory/Freidel Livine Rivasseau and others) where the inclusion of matter differs somewhat from what one gets from this brief problem statement. But at least for starters this is simple and concise:"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?"

Anyone who wants to read the paper we are discussing should keep in mind that R. is consistent about this terminology:
Spin-networks (3D boundary geometry) consist of nodes and links. Nodes can contribute to volume measurement, links to area measurement.

Spinfoams (possible 4D bulk histories) consist of vertices edges faces. So in the bulk history, edges "carry" volume and faces "carry" area.

On the other hand a vertex is where something happens. The dynamics is primarily defined by vertex amplitudes. Edges are like the worldlines of nodes. A vertex is where one or more nodes can be created or annihilated.

To deliver the goods on matter-coupling IMHO the researchers may have to expand the spinfoam vertex amplitude formula to include matter.

It is a serious question that I quoted: "Can we use the simple group theoretical argument that has selected the gravitational vertex also for coupling these matter fields?"

The spinfoam vertex for gravity alone turned out to be the very simple equation (45) that you see explained on page 8. It comes out of a mapping of an SU(2) manifold into an SL(2,C) manifold---or rather a map of functions defined on one into those defined on the other. Spin-network states are complex valued functions on a group manifold.
So suppose we enlarge the groups? Can the simple transparent method used for gravity be extended? One may feel some suspense.

Atyy has been following another possible way that matter might be included in the spin-network/group manifold picture. Spin-networks seem to be Feynman diagrams of a sort.
But I don't see the connection between the papers cited in that connection and Rovelli's brief statement of problem #17, so for simplicity I only explicate the latter here.

 

[atyy]

Non-local operators appear in the effective action of gauge theories also.

http://arxiv.org/abs/hep-th/9808044

But the fields are still local functions on space-time.

My interpretation is that although gauge invariant observables can be non-local the
fundamental fields that are quantised may still be local quantum fields.

I see. Thanks.

 

[Fra]

To follow up what I said here:

Anyone who wants to read the paper we are discussing should keep in mind that R. is consistent about this terminology:
Spin-networks (3D boundary geometry) consist of nodes and links. Nodes can contribute to volume measurement, links to area measurement.

Spinfoams (possible 4D bulk histories) consist of vertices edges faces. So in the bulk history, edges "carry" volume and faces "carry" area.

On the other hand a vertex is where something happens. The dynamics is primarily defined by vertex amplitudes. Edges are like the worldlines of nodes. A vertex is where one or more nodes can be created or annihilated.

To deliver the goods on matter-coupling IMHO the researchers may have to expand the spinfoam vertex amplitude formula to include matter.

It is a serious question that I quoted: "Can we use the simple group theoretical argument that has selected the gravitational vertex also for coupling these matter fields?"

The spinfoam vertex for gravity alone turned out to be the very simple equation (45) that you see explained on page 8. It comes out of a mapping of an SU(2) manifold into an SL(2,C) manifold---or rather a map of functions defined on one into those defined on the other. Spin-network states are complex valued functions on a group manifold.
So suppose we enlarge the groups? Can the simple transparent method used for gravity be extended? One may feel some suspense.

Atyy has been following another possible way that matter might be included in the spin-network/group manifold picture. Spin-networks seem to be Feynman diagrams of a sort.
But I don't see the connection between the papers cited in that connection and Rovelli's brief statement of problem #17, so for simplicity I only explicate the latter here.

These last possible expansion points Marcus made, and in particular the more abstract interpretation not unrelated to feynmann diagrains is the part of LQG I find most interesting, and it was what caused my first interest in it some years ago.

If we forget about the spin networks beeing one way or the other embedded in manifolds (something I do not like) and instead look for a more abstract view, in particular a view where the spin-network more represents the system microstructures encoding the observers information, then I was always tempted to interpret nodes as distinguishable memory states, and then the edges as defining the flow of a internal remapping. In this interpretation the "spin" is more of "action" quanta, referring to a general space of distinguishable events and information states. And in this picture it seems also a priori not clear how to separate space indexes from other internal indexes (matter).

Maybe there is a way for these views to meet.

The historical LQG is starting from considering just the space and spacetime INDEX (which is really just a sub-index of the full event-index), and apply consistency conditions in order to yield the 3D/4D result and GR.

The othre view, which I would be extremely interested to see developmne int, is a more abstract starting point where one considers the general event-index (discretly and combinatorically), and then asks why sub-spaces (such as 3d space) will emerge as these observer interact, and further how the remaining event-index and subsequently hte other forces connect to that sub-space.

This could in fact be closely relate to the reconstruction of an intrinsic theory of measurement, where more general action-netoworks might be the choice of _representation_ that generalises the single microstructure framework that we have in the simple thermodynamics. It could also integrate well with the entropic ideas, as each network would then define a natural flow that is observer dependent, that should replace the hamiltonian framework.

/Fredrk

 

[marcus]

These last possible expansion points Marcus made, and in particular the more abstract interpretation not unrelated to feynmann diagrains is the part of LQG I find most interesting, and it was what caused my first interest in it some years ago.

If we forget about the spin networks beeing one way or the other embedded in manifolds (something I do not like) and instead look for a more abstract view, in particular a view where the spin-network more represents the system microstructures encoding the observers information, then I was always tempted to interpret nodes as distinguishable memory states, and then the edges as defining the flow of a internal remapping. In this interpretation the "spin" is more of "action" quanta, referring to a general space of distinguishable events and information states. And in this picture it seems also a priori not clear how to separate space indexes from other internal indexes (matter).

Maybe there is a way for these views to meet...

Fra, I'm glad you drew my attention back to this thread. First I should correct something I said in the passage you quoted. In the papers (for example by Laurent Freidel and by Vincent Rivasseau) which Atyy has pointed to, what is like a Feynman diagram is the spinfoam . I must have been in a hurry and not watching what I said, and I said it was the spin-network which was like a kind of Feynman diagram.

It sounds trivial as if it should make no confusion. "What's the difference?" But I find these ideas and this way of understanding are new enough that it is worth while to emphasize simple distinctions. To repeat, and to try out different ways of saying the basic message.

Your discussion helps to do that. You say you don't like embedding spin-networks into a manifold. I agree in general. Sometimes people will use embedded spin-networks temporarily--to prove a theorem or show an equivalence. But it is a yoking together of new and old.

===============

I think of a spin-network as an (idealized) web of preparation&measurment that one might imagine making. These could be analogous to the initial and final particle states in an experiment Say there are N nodes and L links in the graph. The N nodes correspond to measurements of volume. The L links correspond to measurements of area.

One can imagine that there are also some fermions which ride on the Nodes, and some YangMills fields expressed in the Links. Since as an experimenter one has only finite resources, one is dealing with a finite graph.

Now just to underline the distinction, the spin-network has no interaction vertices where something happens. It is the spinfoam that has the vertices where something happens.

The spin-network describes the boundary conditions that we control, the boundary surrounding the 4D bulk which we do not control.

It is the spinfoam that describes all the millions of possible processes that may or may not occur in the bounded 4D bulk.

I feel a little ridiculous highlighting repeating these simple obvious things. But this was exactly where I mis-spoke when I was trying to mention some papers that Atyy was talking about! You may not need me to repeat these things, but I myself need to repeat them for my own good!

So they make it easy for us to remember by using particular chosen terminology: a spinfoam consists of vertices, edges, faces.
And a spin-network consists of nodes and links.

 

[marcus]

I also want to emphasize that this approach uses group manifolds, for example to do its gauge theory. It merely does not use a manifold to represent space or spacetime continua. In a quantum theory we can no more assume that a spacetime continuum geometry exists than we can assume that the trajectory of a particle exists.

My personal view is that this is good. Historically the LQG approach used manifolds to represent the continua, and embedded the graphs and 2-complexes in those continua. Now it is more abstract.

But we still can use Lie groups, differential geometry, and manifolds. Suppose we are working with a graph that has N nodes and L links. Then we can take N-fold and L-fold cartesian products of the group G---and have for example the group manifold G^N consisting of all possible N-tuples of elements of G.

Right away in Rovelli's equation (3) on page 2, you can see how one uses any given N-tuple of group elements to twirl the gauge. At every given node one has chosen a group element to screw around with the links that are in-coming and out-going at that node.

Now I think about assignments of G-labels to the links of the graph. The group manifold G^L. And I think about "wave-functions" defined on G^L. Functions with values in the complex numbers. You can screw around with these functions simply by messing with the domain they are defined on, as described above and in equation (3).

We can define an equivalence between "wave-functions" defined on the group manifold. Two functions are equivalent if you can turn one into the other by screwing around with the domain it's defined on---that being G^L---as described in equation (3).

That's what I meant by "twirling the gauge" simultaneously at each node of the graph.
Two wave functions might actually describe the same physical conditions. So they might have a certain percentage of "gauge" in them: spurious non-physically-significant content like the air whipped into cheap icecream. Screwing around with the domain they're defined on---using all possible G^N assignments of group elements to the nodes---to see if you can make one equal the other, is a way to squeeze out the unphysical "air".

One nice thing is that ordinarily people might think of gauge theory only in the context of a differential geometry package like a bundle on a manifold. Here there is no manifold, there is only a graph of measurements you might imagine making in order to nail down the boundary conditions of your experiment---the geometric inputs outputs and such.

 

[Fra]

You say you don't like embedding spin-networks into a manifold. I agree in general. Sometimes people will use embedded spin-networks temporarily--to prove a theorem or show an equivalence. But it is a yoking together of new and old.

Yes, I also appreciate the value in connecting new abstractions to old ones. In this sense, embedding a discrete abstraction into a continuum one certainly shows the way how the old abstraction can be "emergent" from the new one.

But I think it's important when one has the ambition to explain something, to note the line of reasoning and not use what we want to show as motivation for the construction. That happens at times.

This is why own interest in LQG all along has been at it's outer edge. I've had clear objections to some of rovelli's reasoning. In particular the way he avoids analysing the foundations of QM and measurment theory itself - he just use it. I find that somehow incoherent and his initial reasoning somehow (to me at least) holds an higher ambition. But perhaps after all, there is some common touching points in the new development.

I think of a spin-network as an (idealized) web of preparation&measurment that one might imagine making.
...
Since as an experimenter one has only finite resources, one is dealing with a finite graph.
...
Now just to underline the distinction, the spin-network has no interaction vertices where something happens. It is the spinfoam that has the vertices where something happens.
...
The spin-network describes the boundary conditions that we control, the boundary surrounding the 4D bulk which we do not control.
...
Here there is no manifold, there is only a graph of measurements

This is the direction of abstracton I like!

But I think we could be more radical than rovelli is, and then we can not just assume like rovelli does, that all communication is perfectly described by the QM formalism.

The problem is that QM is an external description of communication, not an intrinsic one, this alone makes it unphysical - except for as Smolin points out, for studying subsystems.

I'm more interpreting the spin-network as a sub-set of the observers total "information state" - ie an inside view. This is always bounded, because as you say any experimenter or observer has finite resources to STORE information. It can be of arbitrary size, but each observer has a number associated to it which is it's complexity (in my view that is). But this would encode all events (all forces and "fields"), not just 4D spacetime events.

Then, a third observer, can imagine two other observers interaction. Where each of the observers has a certain microstructure. Then I expect the spin network to be emergent as they have equilibrated. Their common communication channels is subjectively indexed by spin networks, in a way that they are related by means of your equivalence transformations.

But, I see this as an equilibrium condition. The assumption that we must always have perfect equilibirum and perfect consistency I don't understand. In fact it does not match real life observations of any learning agent. Inconsistencies is what drives development, and it's the drive of evolution of time.

But then, the "residual" of the total event index structure, once we "substracted" the equilibrium spin network (or spacetimepart) should hopefully be further classified into the other forces (matter); since the internal structure of any real observer is supposedly made of matter.

Two pieces of matter, will _establish_ a space relation, ontop of which the residuals correspond to other fields. But I do not understand how there is a route to this, unless we admitt that space itself, (even discrete space) is emergent as a separation from the more general space encoded in a observer memory (matter).

/Fredrik

 

[Fra]

Now just to underline the distinction, the spin-network has no interaction vertices where something happens. It is the spinfoam that has the vertices where something happens.

I envision that even the information state = memory records IS in fact a subjective, re-encoded HISTORY of actual events. So in a sense one can still talke about frozen events as existing in a memory record, and would classify events as externa and internal. External events is the real observations (ie. "collapses"), the internal events require no external interaction, they are internal reequilibrations, or internal recoding of history.

So I envision that the internal structure of the observer/matter (of which the spin-network would merely be a subset of) is to be thought of as a compressed (in the datacompression sense) history, where the compression algorithm has evolved for self preservation.

Thus, I expect the structure of this (be it spin networks, or some other structure, and it's vertex group rules) to be a result of a selection process to optimum representation.

Something along these lines has been the way I tried to understand LQG, but unfortunately it's been too different so far. As I see it, this is also related to unifying matter with it, so maybe if some new clever ideas come out of this my hope is that.

/Fredrik

 

[Fra]

Something along these lines has been the way I tried to understand LQG, but unfortunately it's been too different so far.

The idea I had so far was to picture the spinnetwork edges, as defining "dataflow" between two or more different microstructures (representing different encoding algorithms) where the IN and OUT nodes then belong to two different sets. And the observers microstructures is then really sets of sets, where each set has a different compression algortihm. Some rules of the network would then simply be determined by the complexity constraint (assume the networks doesn't grow and acquire more complexity, which could also happen which would make it more difficult). Some other rules would also follow from the compression algorithms chosen. And this would be a result of evolution.

However, this is the general case, and it has still remained how to separate out of the main communication channel where local observers can agree (within some connection transformation) of it's state.

At any stage, there is defined a flow in the entropic sense. But since it's not a single microstrucure anymore, the dynamics is far more complex that simple dissipation.

My idea was always to exploit the complexity constraint, and start at zero complexity, because there things are finite and computable, and then make conclusions, and then find how these conclusions scale. Zero complexity meaning a very small network, which also constrains it's possible interactions just by constraining permutations.

/Fredrik

 

[marcus]

The idea I had so far was to picture the spinnetwork edges, as defining "dataflow" ...

Do you mean spinfoam edges?

Or did you actually mean spinnetwork links? I find it is a big help to use the prevailing terms in the literature---not mixing up terminology helps me think straight. I was confused by your statement and could not tell which you meant---foams or networks?
============

BTW have you noticed in the standard LQG (1004.1780) treatment there is a kind of reciprocal interplay between boundary and bulk? It is interesting how the treatment of transition amplitudes goes back and forth between network (boundary state) and foam (bulk history). Like each scratching the other's back---or like for some jobs you need two hands.

You start out with (network) boundary states. Kinematics is defined, but still no dynamics---no amplitudes.

Then comes equation (43): you see that the amplitude of a network boundary state is going to be a sum over foam histories in the surrounded bulk.

Now each foam can be broken down to its constituent vertices. We need to define an amplitude for each foam vertex.* The amplitude for the whole history will be the product of all the amplitudes for the constituent vertices. Equation (44).

The most efficient way to define a single vertex's amplitude turns out to be to surround the individual vertex by a small private boundary, defining again a network! But this network is especially simple and turns out to have a natural and concise amplitude formula! Equation (45).

That then defines the individual vertex amplitude, and makes it computable.

So one has "walked" down a reductive path, stepping both with the "bulk foot" and the "boundary foot". From a large complex (network) boundary, to a sum over (foam) histories, each becoming a product over individual (foam) vertices, which were surrounded then by calculable individual (network) boundaries.

This is condensed into one equation, (52) on page 9. <W|\psi> = \sum_\sigma \prod_f d(j_f)\prod_v W_v(\sigma)

Here d(j_f) just stands for the vectorspace dimension of the representation j_f. In other words, d(j_f)=2j_f + 1.

And W_v(\sigma) is shorthand for the local vertex amplitude I was talking about. Equation (53) explains:

W_v(\sigma) = <W_v|\psi_v>

You will recognize ψ_v as the small private boundary one can always construct around an individual vertex, and evaluate to get the vertex amplitude.

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ...∏∑∫∂√ ...± ÷...←↓→↑↔~≈≠≡≤≥...½...∞...(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)
========================
*In case someone new is joining us, in standard Lqg terminology, spinnetworks do not have vertices (they are made of nodes and links). If someone says "vertex" in the Lqg context you know they are talking about a spinfoam (made of vertices, edges, faces). It makes communication more economical and convenient to remember these simple distinctions.

 

[Fra]

Do you mean spinfoam edges?

Or did you actually mean spinnetwork links? I find it is a big help to use the prevailing terms in the literature---not mixing up terminology helps me think straight. I was confused by your
...
*In case someone new is joining us, in standard Lqg terminology, spinnetworks do not have vertices (they are made of nodes and links). If someone says "vertex" in the Lqg context you know they are talking about a spinfoam (made of vertices, edges, faces). It makes communication more economical and convenient to remember these simple distinctions.

I meant spinnetwork links.

I'm sorry for the confusion, I guess I was confused what the standard terminology was in LQG, I don't follow the LQG development regularly. It was a couple of years ago when I looked into Rovelli's book and papers.

Not that wikipedia is a sensible reference but I see that also used the word edge and vertex even for spin network in the abstract sense.

BTW have you noticed in the standard LQG (1004.1780) treatment there is a kind of reciprocal interplay between boundary and bulk? It is interesting how the treatment of transition amplitudes goes back and forth between network (boundary state) and foam (bulk history). Like each scratching the other's back---or like for some jobs you need two hands.

Yes, just like we have transitions between quantum states in normal QM, we have transitions between spin networks, or equivalence classes of spin networks.

But what I was after is to suggest that the "structure" of ANY quantum state, may be seen as a state of a system of memory records. And that the information processing taking place in the observer MAY be represented abstractly like a system of distinguishable indexes, in between there is a possible directional communication obeying certain rules.

So take a regular time-history of events and picture this data physically stored, then you get a historical combinatorical probability. But then picture that to this record, one can increase the capacity by recoding the actual history, maybe doing an FFT to split the memory into both a historical probability and transformations of the same.

In this sense a HISTORY of events, should be related to inertia.

I'm struggling howto represent these things. As of right now my best idea is sets of sets of distinguishable events (microstructures), where each set in the set comes with a transformation which is interpreted as a lossy data compression. The overall complexity of the set of all sets are constrained by the observers recourses (memory capacity). Now if we could somehow count the set of intrinsically COMPUTABLE transformations, the number of possibilities for each such a construction would be finite, or even in the large complexity limit, countable. Then the laws of physics, coded as symmetries would correspond to the most proable one in the entropic sense. Thus all information processing rules would have an entropic origin.

Symbolically, one could represent internal recoding of the history, as directed links between different sets or more specifically between each element in the sets.

I was seeing to what extend the LQG spin network might fit in some remote connection there. I think the fit is more likely if matter is introduced. It doesn't seem out of the question.

You start out with (network) boundary states. Kinematics is defined, but still no dynamics---no amplitudes.

In my picture the boundary states and state space is defined by the lossy compression of history of interactions - in this sense there is no timeless state spaces. We need a history as I see it. And the only accesible history is the one implicit in the observer.

/Fredrik

 

[Fra]

Symbolically, one could represent internal recoding of the history, as directed links between different sets or more specifically between each element in the sets.

As for the spinfoam, or the evolution of the spinnetwork, my take on that in the context of my proposed analogy here would be that the instability of the spinnetwork itself, defines a flow - a direction of change, which is the expected evolution. a generalization of 2nd law.
Ie. a static spinnetwork is simply not a likely solution, no more than a static universe is. This instability when quantified, defines a flow (not unlike GR of course - but constructed from more first princples).

The "quantum part" I expect to follow naturally from the generalized statistics that follow when you do probability not on a probability space, but a set of such sets, that has certain relations and are subject to constraints. In this sense even QM would be emergent. I guess that's a point where rovelli is at right angle. But maybe things can change.

/Fredrik

 

[marcus]

...Yes, just like we have transitions between quantum states in normal QM, we have transitions between spin networks, or equivalence classes of spin networks.

But what I was after is to suggest that the "structure" of ANY quantum state, may be seen as a state of a system of memory records...

It is interesting that you are thinking in terms of what, in Computer Science, are called "data structures" used for storage and retrieval. Just for explicitness, I will mention that some examples of data structures are graphs, trees, linked lists, stacks, heaps. Not something I know much about. It is also intriguing that you mention a type of Fourier transform (the FFT).
====================

I think that primarily for pragmatic reasons the game (in QG) is now to find SOMETHING that works. Not necessarily the most perfect or complete, but simply some solution to the problem of a manifold-less quantum theory of geometry and matter.

If one could just get one manifold-less quantum theory that reduced to General Relativity in the large limit, that would provide pointers in the right direction---could be improved-on gradually, and so forth.

Actually I suspect it is likely that the LQG we now see will turn out to be such a theory--a "first" manifoldless QG+M. It is the projective limit of finite graph-based group field theories, so it gets the benefit of being operationally finite but incorporating the possibility of arbitrarily large and, in effect, infinitely complicated graphs.

It may be that in the present situation the graph-based path is the only profitable way to go for a QG+M theory. The graph represents our finite information about volume and adjacency---the essence of geometry is information about bits of volume and the areas through which neighbors communicate or across which they meet. So we see the labeled graph (the spin-labeled network) proving to be an increasingly fertile idea.

At the moment I cannot imagine anything simpler or more obviously serviceable than a labeled graph, if one wants a manifold-less data structure capturing the essence of geometry. So I draw two conclusions, which I'll toss out as suggestions:

1. 4d dynamics is to be formulated with spinfoams since spinfoams are the possible "trajectories" of labeled graphs.

2. Matter has to ride on graphs and therefore its motion will also be described by spinfoams.

==============

Fra, the next thing we should talk about is Section IV Expansions, which starts on page 10 of the paper which this thread is about.
In case anyone has not read the paper yet, it is http://arxiv.org/abs/1004.1780.

Little if any physics is possible without the series expansions which provide for finite calculations giving arbitrarily precise numbers. So Rovelli, having described the theory in the first 9 pages, goes on to say (in Section IV) how it leads to various kinds of approximations.

==============
John Norton has a good account of diff-invariance and the hole argument, in case we need it here:
http://www.pitt.edu/~jdnorton/papers/decades.pdf

 

[marcus]

An important realworld process we can see and should try to understand is the explosive growth of LQG research in recent years, say since 2005.

Some of the growth has been in its application to cosmology ("LQC") but just in the past year papers by Ashtekar and Rovelli, with others, have merged the two effectively enough that we don't need to make the distinction. Some of the growth has been stimulated by the 2008 reformulation of core LQG.

In any case there has been a dramatic increase in job openings---including permanent hires---for LQG researchers, and also in the number of active research groups worldwide. The LQG research output has more than tripled since 2005 as well.
https://www.physicsforums.com/showthread.php?p=2839234#post2839234

I've suggested a reason that may partly explain this. LQG has come to be seen as a practical proposal for manifoldless QG+M.

A quantum geometric theory of gravity and matter that does not use a manifold to represent space or spacetime. It CAN use manifolds to represent space and indeed the new LQG developed from older versions which were continuum-based. But these are now just stepping-stones or scaffolding. At some point after the construction is finished one can throw away the spacetime continuum. The manifold is "gauge" in that sense.

This is the essential message of Rovelli's April 2010 survey of LQG which we are looking at in this thread.

 

[marcus]

A few posts back when I was discussing crossing symmetry, I used the image of a freeway interchange.

Here is a picture of an interchange:
http://en.wikipedia.org/wiki/Interchange_(road)
This one happens to be in Dallas, Texas, and is known as "high five" interchange.

Back in that post I forgot several times and said vertex when I meant node, so I will have to re-do it sometime.
The main thing is that boundary state is expressed as a graph of nodes and links. Nodes are volume, links are area.
A foam consists of vertices, edges, faces. So if someone is speaking LQG consistently "vertex" always means foam vertex.

A foam describes a complex geometric process. A foam can be imagined as the trajectory of a graph showing its evolution---as nodes (volume chunks) appear and disappear and dance/travel around so that they constantly need to be reconnected in various ways. They change their "adjacency" relations as they churn about and the foam is the kind of minimal picture that diagrams that kind of graph evolution.

When you pass from the graph to the foam picture of evolution, "nodes become roads". The graph elments which carry volume, i.e. the nodes, become linear in the foam. Several of these converge like roads going into a vertex and then several others diverge out from it.
(Officially we call these roads edges---a foam consists of vertices edges faces.)
The vertices of the foam are elementary geometric processes or events that we can think of as highway interchanges where some roads come in and some roads go out.

 

[atyy]

Taking a look at Rovelli's reference 46 I see it says "A spin foam model is a procedure to compute an amplitude from a triangulated manifold". So how is this manifoldless?

 

[marcus]

Taking a look at Rovelli's reference 46 I see it says "A spin foam model is a procedure to compute an amplitude from a triangulated manifold". So how is this manifoldless?

Well it's certainly more general than that. What you are quoting is what Barrett said in February 2009. That's merely the limited way Barrett et al were thinking of them at that time in that context.
Later that year the restriction to triangulations (even for this limited case) was broken by Lewandowski. So that invalidated the words Barrett et al used in their introduction. It doesn't invalidate Barrett et al excellent valuable mathematical result! (Only the parochial way they were thinking about what they were doing.)

This is a mathematical subject. You cannot think verbally about it. If you just cherrypick some nonessential words that somebody says in the introduction to give a general perspective on what they are doing---that is fairly meaningless. It gets dated quickly and you can't believe it or carry it over from paper to paper. The mathematical result is the essential message and value. That carries over.
Barrett et al had a key result in that paper about foam vertex asymptotics as I recall.
That carries over even though what they suggested about foams limited to a triangulated manifold is not true.

 

[marcus]

If you were to ask Barrett today about that I feel sure he would not make the same statement. Sure one thing about a spinfoam model is that it can be used with a triangulated manifold.

You just take the dual of the triangulation and that gives a 2 complex and that's your spin foam---so apply the model and calculate. That's one thing you can do.

But I hardly think Barrett would tell you that this is the only use of a spinfoam model . They also apply to manifolds that are not triangulated, but are divided up more generally. And they apply where you do not have a manifold at all!

 

[atyy]

So that invalidated the words Barrett et al used in their introduction. It doesn't invalidate Barrett et al excellent valuable mathematical result! (Only the parochial way they were thinking about what they were doing.)

Good, just wanted to make sure you were being consistent.

In my view Barrett et al are not necessarily parochial. The view for manifoldless spin foams goes all the way back to at least '97 http://arxiv.org/abs/gr-qc/9712067 . Barrett et al are surely not ignorant of this, I think there are at least 3 strands of interpretation of spin foams - algebraic (Markopoulou and Smolin), geometric (Barrett), GFT which agains splits into at least 2 strands - one geometric, very close to Barrett, the other going for unification with matter (Livine and Oriti). Rovelli (or at least your interpretation of Rovelli) is striving here for a Markopoulou and Smolin viewpoint, but the key result is via a Barrett interpretation, and I'm not sure you can easily keep the result and throw away the interpretation.

 

[marcus]

... and I'm not sure you can easily keep the result and throw away the interpretation.

That's mathematics for you. You keep a result and throw away the intuitive picture that led up to it. That's why a result is put out in the form of a theorem with explicit assumptions from which the theorem is proven.

It makes it portable so it can be taken into new contexts. If Barrett's result needed strengthening to apply in Rovelli's context we would have heard of it and some people would have already gotten to work on it. I haven't heard anything. Have you?

I do know that Barrett and a bunch of his co-authors took a trip to Marseille later that year (2009) to give seminar talks on their results to the Marseille team. (Team is what they call it équipe de gravité quantique.)

BTW you quoted ref. 46 and I believe that was actually superseded by Rovelli's ref. 45 ( http://arxiv.org/abs/0907.2440 ). That is probably the paper we should be looking at and quoting if we are interested in the details of how Barrett's 2009 work supports Rovelli's 2010 formulation. The July paper.

 

[atyy]

Strictly speaking yes, you are right, but there is no finished theory yet is there? So intuition is still important (and really, I mean, how can you throw away the intuition that led to the proof, even after the finished theory!)

On the other hand, I've never understood intuitively why large spin should be the semi-classical limit, so maybe that intuition will be a red herring.

 

[marcus]

Of course this is work in progress Atyy. I've been watching it and have a sense of the people and the momentum. You may have a different feel. Either way we both know parts definitely still have to be nailed down!

Just for nuance, I will quote Rovelli's section on page 12 where he cites the good work of Barrett group:

== http://arxiv.org/pdf/1004.1780 ==
The analysis of the vertex (49) as well as that of its euclidean analog (55) in this limit has been carried out in great detail for the 5-valent vertex, by the Nottingham group [26, 27, 45, 46]. The remarkable result of this analysis is that in this limit the vertex behaves as

W_v ∼ e ^iS_Regge

where S_Regge is a function of the boundary variables given by the Regge action, under the identifications of these with variables describing a Regge geometry. The Regge action codes the Einstein equations’ dynamics. Therefore this is an indication that the vertex can yield general relativity in the large distance limit. More correctly, this result supports the expectation that the boundary amplitude reduces to the exponential of the Hamilton function of the classical theory.
==endquote==

Supports, does not yet prove.

And we are still just looking at a 5-valence vertex. Which BTW is in line with your mention of the triangulated manifold picture, because a 4-simplex has 5 sides (the dual replaces it with a vertex and replaces each of its 5 sides by an edge). My hunch is that graduate students can extend the result to higher-valence vertices. It's how I'm used to seeing things go, but who knows? You think not?

 

[marcus]

In fact someone WAS working on a missing detail Rovelli describes right after what I quoted on page 12. The two terms in the vertex amplitude. A Marseille postdoc and a couple of grad students. They posted in April soon after the survey paper appeared.
http://arxiv.org/abs/1004.4550
"We show how the peakedness on the extrinsic geometry selects a single exponential of the Regge action in the semiclassical large-scale asymptotics of the spinfoam vertex."

Barrett's group left it with both a +iRegge and a -iRegge. One wanted to get rid of or suppress the negative exponential, and just have a single exponential term. So Bianchi et al took care of that.

There's been a kind of stampede of results in the past 6 months or year, bringing us closer to what appears may be a satisfactory conclusion.

 

[atyy]

My hunch is that graduate students can extend the result to higher-valence vertices. It's how I'm used to seeing things go, but who knows? You think not?

I don't knoow - what I would like to see aesthetically is that it's a GFT, and that GFT renormalization is essential, and that matter must somehow come along automatically. But Barrett et al's, and also Conrady and Freidel's are the most intriguing results I have seen from the manifold point of view. But in which case, I think there must be Asymptotic Safety somehow, and a link via what Dittrich et al are saying.

 

[marcus]

As further motivation for the move towards manifoldless QG+M, I should quote (again) that passage from Marcolli's May 2010 paper. Marcolli mentions the view of Chamseddine and Connes. This is section 8.2 page 45.

==quote http://arxiv.org/abs/1005.1057==
8.2. Spectral triples and loop quantum gravity.

The Noncommutative Standard Model, despite its success, still produces an essentially classical conception of gravity, as seen by the Einstein–Hilbert action embedded in eq. (8.2). Indeed, the authors of [36] comment on this directly in the context of their discussion of the mass scale Λ, noting that they do not worry about the presence of a tachyon pole near the Planck mass since, in their view, “at the Planck energy the manifold structure of spacetime will break down and one must have a completely finite theory.”

Such a view is precisely that embodied by theories of quantum gravity, including of course loop quantum gravity—a setting in which spin networks and spin foams find their home. The hope would be to incorporate such existing work toward quantizing gravity into the spectral triple formalism by replacing the “commutative part” of our theory’s spectral triple with something representing discretized spacetime.

Seen from another point of view, if we can find a way of phrasing loop quantum gravity in the language of noncommutative geometry, then the spectral triple formalism provides a promising approach toward naturally integrating gravity and matter into one unified theory.
==endquote==

More discussion of the Marcolli May 2010 paper in this thread:
https://www.physicsforums.com/showthread.php?t=402234

 

[marcus]

I guess one way to put the point is to observe that LQG is amphibious.

The graph description of geometry (nodes of elemental volume, linked to neighbors by elemental area) can live embedded in a manifold and also out on its own---as combinatoric data structure.

In the April 2010 status report and survey of LQG, the main version presented is the manifoldless formulation which Rovelli calls "combinatorial". But he also, in small print, describes earlier manifoldy formulations using embedded graphs. In my view those are useful transitional formulations. They can be used to transfer concepts and to prove large limits and to relate to classical GR. Stepping stones, bridges, scaffolding.

It's not unusual to prove things in two or more stages--first prove the result for an intermediate or restricted case, then show you can remove the restriction. But as I see it, the manifoldless version is the real McCoy.

 

[atyy]

So what's the manifoldless take on renormalization?

In the manifoldy view it has to happen somewhere, since one started with a triangulation of the manifold.

 

[marcus]

...In the manifoldy view it has to happen somewhere, since one started with a triangulation of the manifold.

Historically, the LQG of the 1990s did not start with a triangulation of a manifold. It started with loops, which were superseded by slightly more complicated objects: spin networks. These have nothing to do with triangulations.

Spin networks can be embedded in a manifold. But the matter fields, if they enter the picture, are defined on the spin network---by labeling the nodes and links.

So what's the manifoldless take on renormalization?

Nodes carry fermions. Links carry Yang-Mills fields. Geometry is purely relational. The basic description is a labeled graph. The graph carries matter fields and there are no infinities.
See the statement of problem #17 on page 14 of the April paper. This points to what i think is now the main outstanding problem---going from QG to QG+M---including dynamics in what is (so far at best) a kinematic description of matter and geometry.

 

[marcus]

Thiemann uses an old-fashioned version of LQG here, but it gives the general idea:

http://arxiv.org/abs/gr-qc/9705019
QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories
Thomas Thiemann
(Submitted on 10 May 1997)
"It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum. Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized."

 

[atyy]

Historically, the LQG of the 1990s did not start with a triangulation of a manifold. It started with loops, which were superseded by slightly more complicated objects: spin networks. These have nothing to do with triangulations.

But aren't we talking about spin foams?

Also, if we take the Barrett result seriously, they only get to something like the Regge action. That needs a continuum limit to look like GR - that's why Loll et al - who started with the Regge! - try to link to Asymptotic Safety or some hopefully well defined theory in the continuum limit.