String Field Theory and Background Independence?

[Haelfix]

Reread the section again, you missed the point or don't understand what physicists mean.. UV finiteness, by original classical definition, is one possible outcome of a specific set of procedures physicists do in a quantum theory.

If you do not follow the procedure, it makes no sense to say xyz is UV finite unless you redefine it that way. It is NOT a statement that says divergences are absent in some intermediate step in a calculation. It says roughly that all counterterm coefficients possible in the most general effective lagrangian (preserving all the original symmetries) after an *arbitrary* regularization and under a massless renormalization scheme like msbar are zero and the regularizer drops out *always*. In short no renormalization is necessary or even possible.

Note how this is absolutely NOT what LQG are doing.. The symmetries of the original system are traded off early on, and a very specific set of regularization rules are imposed (ones that by necessity are compatible with diffeomorphism invariant states), the original desired symmetries must then for consistency reemerge much later in a limit (this is the famous task of trying to get poincare invariance to flow out of the theory).

Its very much analogous to picking a particular path along a renormalization group orbit in a conventional QFT and looking at the divergence structure thereof. Eg you might see something very singularity free along that particularly trajectory, but you are very far from proving finiteness for the original thing. You essentially have to hope (or guess right) that the particular trajectory you picked flows to an attractor. In LQG's case this ambiguity or choice is traded into uncertainties in the hamiltonian constraint, but does NOT mean the original EH action that is being loop quantized is UV finite.

Anyway, different definitions.

 

[marcus]

...
If you do not follow the procedure, it makes no sense to say xyz is UV finite unless you redefine it that way. It is NOT a statement that says divergences are absent in some intermediate step in a calculation. It says roughly that all counterterm coefficients possible in the most general effective lagrangian (preserving all the original symmetries) after an *arbitrary* regularization and under a massless renormalization scheme like msbar are zero and the regularizer drops out *always*. In short no renormalization is necessary or even possible.
...

You are limiting "UV finite" to a particular context where you presumably encountered it.
I don't think the 400-some string and HEP people at the String 2008 conference where Rovelli spoke were unable to understand that in his case UV finite means exactly that, no UV divergences. Hermann Nicolai obviously understands, and in his second paper about LQG he drops the scare quotes. He no longer pretends that LQG people and HEP-sters mean something different by the term!

Anyone who is curious can watch the video of the talk.
Here is the video:
http://cdsweb.cern.ch/record/1121957?ln=en
The string audience don't look confused by unaccustomed terminology, they look interested. So in this case I will go with them and with Hermann Nicolai---and with Demystifier

...as far as I know, UV finiteness of LQG is shown rigorously (LQG is defined non-perturbatively) to be a consequence of compactness of SU(2) or SO(3).

I won't go with you on this Haelfix. I don't think your objections are credible: in this case I think you've a trumped up language barrier which isn't really there.

In case anyone still hasn't checked out the slides to the talk, here they are:
http://indico.cern.ch/getFile.py/access?contribId=30&resId=0&materialId=slides&confId=21917

 

[Haelfix]

Marcus, I am getting bored of this, email Nicolai and have him explain it to you if you want b/c you don't understand what you are talking about.

Posting completely irrelevant conference talks and the attitudes of its participants is not going to unfog this nonsense, nor is it going to redefine completely standard terminology that's existed since the late 70s.

I'll repeat Nicolai's comments

"First of all it should be evident from the foregoing discussion that the notions of ‘finiteness’ and ‘regulator independence’ as currently used in LQG are not the same as in conventional quantum field theory."

 

[marcus]

...email Nicolai and have him explain it to you if you want b/c you don't understand what you are talking about.
...
I'll repeat Nicolai's comments
...

You choose to repeat from Nicolai's 2005 paper! He got a fair amount of flak for it and came out with an improved updated version the next year. To wit:

=quote Nicolai 2006==
At least in its present incarnation, the canonical formulation of LQG does not encounter any UV divergences,.. Just like canonical LQG, spin foams thus show no signs of ultraviolet divergences...
==endquote==
http://arxiv.org/abs/hep-th/0601129

You introduced Nicolai into discussion as an authority on HEP terminology. Now you are trying to suggest that Nicolai is not talking standard HEP English. You insist that your vocabulary is the standard, and that Nicolai and Rovelli both speak in a pecuiliar nonstandard way. With respect, I have to point out Nicolai is probably the most influential String theorist in Germany if not Europe as a whole.

Since you are going against his use of terminology, I'd say it is up to you to write him and get clarification, Haelfix.

Demystifier, another knowledgeable person here, does not agree with you on this point either.

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

To continue our discussion, I wanted to ask you about something you said several posts back. You quoted Rovelli and offered what you said was an easy counterexample. In what sense is N=8 Supergravity a counterexample to what he said?

"In gravity, (unrenormalizable) UV divergences are consequences of a perturbation expansion around a wrong vacuum"

Trivial and modern counterexample: N=8 Supergravity.

Note that he did not claim that a perturbative treatment of gravity would always, in every case, cause unrenormalizable UV divergences. His point is that if you see UV divergences they may be caused by a misplaced perturbative expansion.

 

[Haelfix]

The finiteness of N=8 Supergravity has nothing to do with perturbing around a wrong vacuum. Instead the maximal measure of supersymmetry combined with a process called the 'no triangle conjecture' (KLT relations + the unitary method) contribute to unexpected cancellations between bubble and triangle diagrams, leaving each loop amplitude free of counterterms.

Anyway, I didn't get my information from Nicolai in the first place, he just spells it out clearly.

In a certain sense, its a little absurd to even call a supposedly nonperturbative theory UV finite, as that sort of definition is essentially vacuous. The difficulty lies in showing the known behaviour of quantum gravity (the infinite amount of ambiguity in making predictions, starting with a bad 2 loop term) goes away somehow.

 

[marcus]

"In gravity, (unrenormalizable) UV divergences are consequences of a perturbation expansion around a wrong vacuum"

Trivial and modern counterexample: N=8 Supergravity.

The finiteness of N=8 Supergravity has nothing to do with perturbing around a wrong vacuum. Instead the maximal measure of supersymmetry combined with a process called the 'no triangle conjecture' (KLT relations + the unitary method) contribute to unexpected cancellations between bubble and triangle diagrams, leaving each loop amplitude free of counterterms.
...

So N=8 SUGRA, as I suspected, is not a counterexample to Rovelli's statement (as you claimed earlier). Thanks, that is what I was asking about.

 

[marcus]

In a certain sense, its a little absurd to even call a supposedly nonperturbative theory UV finite, as that sort of definition is essentially vacuous...

Take it up with Hermann Nicolai, or any number of others, who think it is a substantive point. Maybe you could provide some sources, besides self opinion?

 

[Haelfix]

Despite my better judgement, one last time for interested others.. Marcus is beyond hope.

Consider a general nonrenormalizable field theory. Let us follow Wilson and use an exact renormalization group method and impose a regularization scheme with a hard cutoff at some *specified* scale lambda. After Wick rotating the theory and Fourier transforming in Momentum space, we insist on calculating the integral appearing in the partition function only at values P^2 < lambda^2.

Now, generically you will encounter radiative terms that are badly divergent (say worse than log or quadratic divergences). Now the general prescription (say MSbar) is to write down a counterterm that absorbs the divergence (+ a constant) and rewrite it into the definition of the action, hoping that the following dissappears at the next order (it won't, in general for nonrenormalizable theories).

Now assume instead that we simply insisted upon cancelling the entire term by picking the counterterms coefficient by hand so that it matches the bad piece and that we continue doing this order by order until such time as we pass the cutoff scale. Ladies and gentlemen a miracle occurs, we are left with a theory with no divergences present! We've bruteforced the entire thing free of singularities. I could take the resultant expression, and call it "UV finite" and it wouldn't be that far from what this discussion has been about.

Of course, what's happened is simply effective field theory and is of course not what's traditionally call UV finiteness (you could do this in principle for any theory, not to mention by definition each coefficient must be identically zero if you really want to be exact). We've essentially shifted the entire infinite set of ambiguities past the cutoff scale by judicious choices (picked by experiment say). You might ask yourself, why pick the scale lambda over say lambda /10, it seems completely arbitrary? And you would be right, the whole point is to make sure the theory is free of such an ambiguity.

This is almost, but not quite what LQG people are doing with the spinfoam method. Instead what they do is morally similar to something like lattice regularization (a nonperturbative regularization). Here, what happens is the regulator manifests itself into values like the lattice spacing and volume. Again, divergences can be made to dissappear using a similar trick (ambiguities are flushed into irrelevant and marginal operators), but the problem must reemerge or be solved when you take the continuum limit or upon each refinement of the lattice spacing (eg you want all traces and choices made in the regularization process to dissappear).

In LQG we aren't really dealing with a lattice though. Instead its a spin network, where we have to sum over various spin labels, but the situation is analogous for the purposes of this discussion.

Thus what's being *defined* here as UV finiteness is really similar to what I've explained above, it is NOT what HEP people usually mean. This isn't a bad thing or something nefarious going on, its just the language that's used. The hard part is showing the resultant theories (LQG on one hand and gravity on the other) are isomorphic and well that's the whole point of the program and what all the continued research and funding is aimed at.

 

[marcus]

The main topic of the thread is background independence (the related issue of diffeo-invariance comes in as well.) UV finiteness is connected to the main topics. I'd like to steer us back towards the main subject matter by recalling some earlier posts.

To write the string action, you have to pick up some background spacetime metric (usually taken to be flat). To write the string-field action (e.g., Witten's cubic bosonic open-string-field action), you don't have to pick up some background spacetime metric. Doesn't it mean that string field theory is more background independent than regular string theory?

Demystifier makes the essential point. BI means you can build the theory either without specifying a differential manifold (continuum) at all, or, if you specify a manifold, you don't specify a metric. So the continuum you start with is limp, formless. Has not geometry.

That's the way GR is constructed.

BI doesn't mean that you have a wide range of choice of what geometry you can specify. It means you don't specify any geometry at all.

BI is a term introduced by GR-people to describe one of the outstanding distinguishing features of GR, which is taken as a guiding principle in constructing extensions or quantized versions of GR.

On the issue of the BI of string theory, appealing to authority:

In the words of Ed Witten:

“Finding the right framework for an intrinsic, background independent formulation
of string theory is one of the main problems in string theory, and so far has remained out
of reach.” ... “This problem is fundamental because it is here that one really has to address the question of what kind of geometrical object the string represents.”

I'm sure he wouldn't have made this statement if ST was only non-manifestly BI but BI all the same...

Julian addresses a question that one naturally asks---why is BI important? Obviously Ed Witten thought it was important in the 1990s when he wrote that paper Julian quotes from. There are several reasons. It would probably help for us to discuss not only what BI means (in the original quantum gravity sense) which is after all pretty straightforward but why it matters.

I'll offer some suggested reasons why it matters.

...In BI theories ... small and large distances are gauge equivalent... Hence the reason why BI theories like lqg are manifestly UV finite...

 

[atyy]

Thus what's being *defined* here as UV finiteness is really similar to what I've explained above, it is NOT what HEP people usually mean. This isn't a bad thing or something nefarious going on, its just the language that's used. The hard part is showing the resultant theories (LQG on one hand and gravity on the other) are isomorphic and well that's the whole point of the program and what all the continued research and funding is aimed at.

So the main problem of LQG is that we don't know it produces gravity at low energies. Does that mean that say lattice QCD was also not know to be UV finite until it was recently (at least partially) shown to have correct low energy behavior?

Also, do effective theories have to be UV finite? I thought Einstein gravity was itself an effective theory that isn't UV finite.

 

[marcus]

So the main problem of LQG is that we don't know it produces gravity at low energies. Does that mean that say lattice QCD was also not know to be UV finite until it was recently (at least partially) shown to have correct low energy behavior?
...

Heh heh
Haelfix has painted himself into a strange corner. A restrictive, overly parochial definition of UV finite, seemingly invented to deny LQG, will end up excluding other things he probably wants to call UV finite.

What I hope is that we can get back to some discussion of Background Independence (and diffeo-invariance). Haelfix strained semantic contortions seem kind of sterile. So let's try for some BI.

In informal conversation it's convenient not to distinguish between BI and general covariance---diffeomorphism invariance. General Relativity has both features and they are so closely interconnected as to be more or less inseparable.

BI means you don't need to begin by specifying a metric on the manifold (a geometry) in fact GR makes metrics or geometries emerge as solutions to the equation. If your theory is such that you HAVE to provide a metric on a manifold in order to get started, then your theory is not BI. It depends on haveing a background geometry to start with.

DI (diffeo-invariance) means you can moosh a solution around by any smooth mapping and it will give you another solution.
You take any smooth mapping and use it to re-map the metric (i.e. the gravitational field) onto the manifold and also at the same time re-map whatever matter there is, and presto what you get is still a solution. Two solutions, if you can morph one into becoming the other, are physically equivalent. There is no reason to distinguish.

Because points of the manifold have no absolute meaning in and of themselves, only relationships involving material events have meaning.

The diffeomorphism group of smooth morphings is a GR gauge group. If one thing can be morphed into another they are gauge equivalent. There is a certain amount of jargon here, verbal fads that caught on with physicists by some historical accident and won't go away.

One reason BI and DI are important is they are central features of GR which is our theory of what space time and gravity are.

1. No theory can really be a quantum version of GR unless it is both BI and DI.

2. No theory can be fundamental if you have to give it an ad hoc spacetime to build on. A fundamental theory will say where space time comes from and why it looks and acts the way it does.

This seems to have been what was on Witten's mind when he said what Julian quoted. If ST is supposed to include gravity, well gravity is the geometry of spacetime. Geometry is dynamic and it's not a given. It has to emerge from any fundamental theory. So you don't really know what superstring theory is until you have a version you can define without specifying some ad hoc spacetime geometry. You don't know what a string is, he is saying.
If you have any doubts, go back and read the quote again. He is saying it is a major, almost existential, challenge.

3. That isn't all. There are other ways BI (and DI) are important. One is that the Landscape of Umpteen different versions of physics that plagues ST arises from the fact that the theory is given pre-packaged ad hoc geometries to build on. With extra dimensions rolled up invarious ways. Each ad hoc geometry is called a vacuum, and it has different physics. There are Umpteen different ways to construct and stabilize vacua. This has caused turmoil and decline in recent years. One obvious solution: don't coddle the theory by choosing an initial geometric setup for it to build on. Force it to evolve its own geometry, the way General Relativity does, and the extensions that derive from GR also do.

4. And then there is the plus that Julian mentioned. BI and DI are hard to implement, but they come with some bonuses. You get UV finiteness free. If you are wondering about that, you can watch Rovelli's talk at Strings 2008. Essentially Julian's post was a condensed version of what Rovelli said over the course of several slides.

That is not the only reason that BI is desirable, but it belongs on the list. Thanks to Demystifier and Julian for those earlier contributions to this thread!

 

[friend]

1. No theory can really be a quantum version of GR unless it is both BI and DI.

Thanks for the post. This was a very informative summary. Tell me, is number 1 above a guess? Or is there a relationship between BI, DI, and the quantization procedure? Thanks.

 

[marcus]

Thanks for the post. This was a very informative summary. Tell me, is number 1 above a guess? Or is there a relationship between BI, DI, and the quantization procedure? Thanks.

friend, I'm really glad of the feedback. You are most welcome! About #1, you could call it a guess. The non-string QG people, LQG people especially, treat it as an axiom. They see BI and DI as essential properties of GR and they make it their goal to find a quantum GR which carries over those properties.

But the correctness of the axiom, or of this way to define the goal, will not be demonstrated until their program succeeds.

So I cannot, should not, state #1 as a fact.
==============

All the same, to me it looks as if the real world has dynamic geometry which is BI and DI, because GR works. I have a hard time imagining how #1 could fail to be correct. There would have to be some deeper structure which is NOT BI or DI, from which the appearance of GR emerges. To me those two features seem like minimum requirement for a deeper math that underlies GR. Just my opinion.

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

BI and DI are stated mathematically but they carry in them an idea of the nature of space (include time if you want). The idea is space has no reality in and of itself. Points of space have no physical existence. There is no space, only relationships. Geometric relationships between events. The gravitational field, the metric, is the essential. It is what exists. Once you have defined it you can throw the manifold away.
I think this is the message of those two principles. The manifold, the Minkowski space, the continuum---it is just gauge: you can moosh it around, you can throw it away. All there is is this web of geometric relationships.

The fields of matter are ultimately to be defined on this web, not on some Euclidean 3-space This is my own subjective interpretation, just for whatever it's worth. I am trying to get a glimpse of the significance of requiring that a theory have background independence and diffeo-invariance, the significance beyond the mathematical expression of it. Not to worry too much about this, just interpretation.

 

[ensabah6]

friend, I'm really glad of the feedback. You are most welcome! About #1, you could call it a guess. The non-string QG people, LQG people especially, treat it as an axiom. They see BI and DI as essential properties of GR and they make it their goal to find a quantum GR which carries over those properties.

But the correctness of the axiom, or of this way to define the goal, will not be demonstrated until their program succeeds.

So I cannot, should not, state #1 as a fact.
==============

All the same, to me it looks as if the real world has dynamic geometry which is BI and DI, because GR works. I have a hard time imagining how #1 could fail to be correct. There would have to be some deeper structure which is NOT BI or DI, from which the appearance of GR emerges. To me those two features seem like minimum requirement for a deeper math that underlies GR. Just my opinion.

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

BI and DI are stated mathematically but they carry in them an idea of the nature of space (include time if you want). The idea is space has no reality in and of itself. Points of space have no physical existence. There is no space, only relationships. Geometric relationships between events. The gravitational field, the metric, is the essential. It is what exists. Once you have defined it you can throw the manifold away.
I think this is the message of those two principles. The manifold, the Minkowski space, the continuum---it is just gauge: you can moosh it around, you can throw it away. All there is is this web of geometric relationships.

The fields of matter are ultimately to be defined on this web, not on some Euclidean 3-space This is my own subjective interpretation, just for whatever it's worth. I am trying to get a glimpse of the significance of requiring that a theory have background independence and diffeo-invariance, the significance beyond the mathematical expression of it. Not to worry too much about this, just interpretation.

Marcus,
Do you think it is possible that GR's BI and DI might be "emergent" from a more fundamental microscopic physics that is not BI nor DI? There are emergent gravity and composite gravity scenarios that suggest GR might not be fundamental.

thakns

 

[marcus]

Marcus,
Do you think it is possible that GR's BI and DI might be "emergent" from a more fundamental microscopic physics that is not BI nor DI? There are emergent gravity and composite gravity scenarios that suggest GR might not be fundamental.

thakns

You are asking my opinion---our opinion is the least valuable type of information we can offer each other. And putting bets on winners and future outcomes of research is something none of us are notably successful at doing. But I don't think I'm any worse than anybody else at it, and you ask me, so I'll speculate.

It is actually not too hard. I think I know that GR is not fundamental because it predicts singularities. So it must be wrong. It clearly breaks down at very high density and curvature.

So it must arise from some more fundamental math. The appearance at large scale of a smooth continuum obeying GR dynamic geometry must be an illusion.

Something much more chaotic and fractally is probably going on with the geometry at very small scale.

Nowadays we sideliners are watching the development of a new idea of the continuum, that will not be smooth at small scale (like Riemann's 1850 smooth manifold everybody still uses) and which may have dimensionality declining gradually with scale (e.g. Loll spacetime 4d at large, 2d at small). Look at Loll's SciAm article. The link is in my sig. It will give you an idea of what a new math model of the continuum might look like. Here, I brought it up from my sig. If you haven't studied this you really should.
http://www.signallake.com/innovation/SelfOrganizingQuantumJul08.pdf This new continuum will have maps analogous to diffeomorphisms. But they won't be diffeomorphisms because the continuum won't be smooth at small scale, or even have the same dimensionality, or even be certain---it may have geometric Heisenberg jitters at small scale.
So there will be a deeper principle which is not the same as diffeomorphism invariance but is analogous to it. It can't be the same because a new continuum will have new morphisms. Ones that preserve the fractal small scale structure, if there is such, and the scale-dependent properties like variable dimensionality.

So whatever the new model continuum looks like I expect the principle of DI (and BI) to persist. Because those principles are the most essential things that General Relativity tells us. But the principles will have to be reformulated in a somewhat different mathematical language, because differential manifolds and differential geometry as we know them today will not apply.

So one part of your question gets the answer YES
because General Relativity is certainly not fundamental and it certainly emerges from something more fundamental.

The other part of the question is could the basic principles of GR be missing from that deeper layer of reality? The answer to to that is NO!
The basic features of GR will have to be reformulated to suit that layer, because we won't have differential geometry language to formulate them in. But there will be some basic principles corresponding to BI and DI.

Even if the frigging mess of noodles is represented by networks of finite-state automata, which I hope not.

BTW here's some interesting work:
https://www.physicsforums.com/showthread.php?t=270975

 

[Fra]

So whatever the new model continuum looks like I expect the principle of DI (and BI) to persist. Because those principles are the most essential things that General Relativity tells us. But the principles will have to be reformulated in a somewhat different mathematical language, because differential manifolds and differential geometry as we know them today will not apply.

Thanks for your elaborations Marcus! This discussion partly replies to the question I raised on DI as a new thread.

I have often wondered, what Einstein would have come up with, if he was born after QM was matured so that he had gotten over his realist issues, so he would have in Bohrs measurement spirit, added general covariance. I always speculated that he would haved broadened the meaning of "that the laws of physics looks the same to all observers" to broaden the class of observers, and maybe also noted that the inside deduction of wether the laws of physics ARE in fact the same, is rather an induction, because there isn't enough information on hand for the inside observer to make a deduction. So one possible solution could be thta the spirit of GR is more to be seen as a deep "induction principle". This step from deduction to induction, might be requied to not clash with bohr spirit to stay away from realist visions. Realist visisions aren't wrong per see, I think they are rather speculative. So the choice is between a "speculative" deduction, or a confident "induction".

I too think a deeper formalism is necessary and as for the specific action of gravity, say E-H action. I think that will be emergent. The breaking of DI invariance that I picture in the emergent picture, doesn't necessarily mean that there is global breaking, I rather think of the breaking a result of indeterminsm, where you consider the symmetry itself as observable.

I think a basic conceptual problem, that relates also to the foundations of QM, is that whatever theory we come up with, this theory is constrained to a subsystem of the universe, to an observer. In this sense, I don't think there is enough information in a subsystem to deduce the certainty of a symmetry. I think this also applies generally to the physical basis of symmetries.

/Fredrik

 

[Fra]

Such a vision, of a hypothetical "Einsteinian reasoning" if he had some of bohr's spirit, might even provide unification of the GR "induction principle" and QM itself. The Heinseinberg uncerainty principle might emerge in the domain of small complexity, as a manifestation of induction based on incomplete information. The emergence of non-commuting observables might be a result of this SAME induction principle. A self-organizing observer, evolves to maintain answers to non-commuting questions, because it's a form of clever "data compression". I find this also very much sniffing Wilczek's reflections of "profound simplicity" and his comparasion with data compression. This is exactly what I associated when I read his book. But they way he talked about it in the rest of the book, I don't think Wilczek's himself fully spelled out the consquences of hte idea of the fact that the problem of optimum data compression is an intrinsically relative concept. It it context dependent.

/Fredrik

 

[marcus]

Thanks for your elaborations Marcus! This discussion partly replies to the question I raised on DI as a new thread.

I checked out your thread and added some response.
https://www.physicsforums.com/showthread.php?t=275482
It's encouraging to hear that some of what I'm saying makes sense to you!

 

[friend]

So it must arise from some more fundamental math. The appearance at large scale of a smooth continuum obeying GR dynamic geometry must be an illusion.

Something much more chaotic and fractally is probably going on with the geometry at very small scale.

Nowadays we sideliners are watching the development of a new idea of the continuum, that will not be smooth at small scale (like Riemann's 1850 smooth manifold everybody still uses) and which may have dimensionality declining gradually with scale (e.g. Loll spacetime 4d at large, 2d at small).

These kinds of statements confuse me. Afterall, when we look at Schrodinger's equation or Feynman path integrals, it would seem that all quantum observables are derived by using an underlying continuous spacetime in the equations. How then can spacetime itself be quantized? Have we developed a new quantization procedure that does not use underlying continuous parameters (a.k.a. spacetime)? Or is it the case that the underlying spacetime is continuous, but any observable that involves the metric is quantized?

 

[marcus]

...Or is it the case that the underlying spacetime is continuous, but any observable that involves the metric is quantized?

You hit the nail on the head! That is how LQG is constructed. It is based on a continuous manifold without any metric specified. So it is initially limp, shapeless, without geometry. Then, instead of metrics, there are defined quantum states of geometry, a hilbertspace of these. Observables are operators on that hilbertspace and some of the geometric observables turn out to have discrete spectrum.

===================
But keep in mind that it is not essential in quantization to divide something up into little bits.

The leading approaches to quantum gravity are LQG and CDT (Loll Triangulation approach) and neither of them divides space up into little bits. In LQG some discreteness comes in at the level of measurement, so you can say that there is a minimal nonzero length that you can get as read-out from a measurement--or at least a minimal nonzero area. This means that space as an observer measures it appears to have a certain graininess.

But in CDT, the other approach we hear most about, there is not even that kind of apparent graininess! The mathematical construction is based on a limp continuum, like for example topologically it could be the 3-sphere cross the real line: S³ x R. The continuum is without geometry, it is formless, because no metric distance function is defined on it. Then one specifies a quantum rule by which it can triangulate itself in millions of different ways.

Analogous to how a particle can get from point A to point B in millions of different ways in a Feynman path integral.

Each path will take the universe from an initial to a final state and it will have an amplitude. A path is a spacetime geometry. The amplitude-weighted average can be taken.
So they can produce millions of sample universes in the computer and study each one's properties (dimensionality, how radii and volumes are related, correlations over time etc) and they can also sum up analogous to a path integral.

Now the CDT theory says let the size of the triangles go to zero. So you see there is finally no discreteness! There is no minimal length.
And the entire construction is still based on a topological continuum----like the three-sphere cross R that I mentioned earlier.

The triangulations are simply a regularization which allows the quantum path integral to be computed. The method is discrete only in the same sense as the Feynman path integral is discrete because it uses polygonal paths, piecewise linear paths, to approximate curved paths, at a certain stage in the calculation. No one pretends that Feynman's particle travels along a path made of straightline segments. And no one should pretend that the Loll quantum continuum is made of little triangles . The triangles could as well be squares or any other tile shape, their size is taken to zero and what shape they are doesn't matter.

Have a look at the Loll SciAm article. It's excellent. There is also a growing technical literature for CDT available on arxiv, but I recommend the SciAm article. It gives a good idea of what is likely to come out of the current multipronged research into quantum gravity. There are a number of approaches and signs that they may have begun to converge. Space doesn't necessarily get broken up into little chunks, but it may reveal a more chaotic, less smooth structure at very small scale. At the micro level it may have the geometric Heisenberg jitters.

Remember too, that whatever continuum we come to define and use will always be merely a mathematical model. Nobody should confuse it with reality. At present almost all physics is done on some sort of differential manifold---a thing invented around 1850 by Riemann. A thing which generalized classic Euclidean space by allowing internally measureable curvature, among other things. Just because that model of space works well and is typically what is used does not mean it corresponds to reality. Most likely it doesn't! Most likely Riemann gives a very bad picture of space at microscopic scale. (And this could be at the heart of physicists' unrenormalizable divergence pains---they use a continuum which is vintage 1850 and totally unrealistic at small scale.)

How then can spacetime itself be quantized? Have we developed a new quantization procedure that does not use underlying continuous parameters (a.k.a. spacetime)?

Well I've tried to suggest how geometry is quantized in the two leading approaches LQG and CDT. They don't actually quantize spacetime itself. They quantize the geometry. Gravity = geometry so quantizing geometry is the name of the game.

And there is an underlying continuum in both cases. Neither space nor spacetime is broken up into little chunks. So, in your sense, we continue to use continuous parameters. I think the answer to your second question "Have we developed...", if I understand it right, is no.
Because we don't need any revolutionary new proceedure---the geometry being defined on a continuum. (in those two cases)

I would urge you to read the Loll SciAm article on CDT. Here is the link:
http://www.signallake.com/innovation/SelfOrganizingQuantumJul08.pdf
The link is also in my sig.
CDT is easier to grasp than LQG, at intro level, and in certain respects it is currently more complete.

 

[atyy]

I would urge you to read the Loll SciAm article on CDT. Here is the link:
http://www.signallake.com/innovation/SelfOrganizingQuantumJul08.pdf
The link is also in my sig.
CDT is easier to grasp than LQG, at intro level, and in certain respects it is currently more complete.

Thanks for the link - that's a nice article!

 

[friend]

You hit the nail on the head! That is how LQG is constructed. It is based on a continuous manifold without any metric specified. So it is initially limp, shapeless, without geometry. Then, instead of metrics, there are defined quantum states of geometry, a hilbertspace of these. Observables are operators on that hilbertspace and some of the geometric observables turn out to have discrete spectrum.

So the only questions now are why and how to quantize gravity/geometry. It seems to me that this is something we should try simply because Einstein's Field Equations of GR can be derived from the Hilbert-Einstein Action. And Action integrals in general appear in Feynman Path Integrals to produce quantum theories. But if we could find a more fundamental reason for the existence of the Hilbert-Einstein action in the Feynman path integral, then that would prescribe the necessity of quantizing gravity.

 

[marcus]

... GR can be derived from the Hilbert-Einstein Action. And Action integrals in general appear in Feynman Path Integrals to produce quantum theories...

Indeed that is just how the Loll CDT approaches quantizes gravity/geometry!
They use the Einstein-Hilbert action, with a positive cosmological constant, to build the path integral.

It turns out that the E-H action has a nice combinatorial form based on counting identical triangles. Because in piecewise linear (identically triangulated) manifolds the curvature is expressed by how many triangles are gathered around a point, or more generally how many D-simplexes are joined to a D-2 simplex. After a big adding up and cancelation one sees that the E-H action is given simply by taking a census of the different simplex types---the different types of triangle. It is remarkably elegant.

Again, I urge you to have a look at the Loll CDT article in my sig. But you may also be wanting a more technical CDT article, if it interests you at all. So I'll see what would be a good arxiv link to suggest. Or you can just ask, if you want more technical detail.

Thanks for the link - that's a nice article!

atyy, I'm so glad you had a look at it! Loll is a good communicator. It makes a difference. I wish she'd write a book.

 

[friend]

Again, I urge you to have a look at the Loll CDT article in my sig. But you may also be wanting a more technical CDT article, if it interests you at all. So I'll see what would be a good arxiv link to suggest. Or you can just ask, if you want more technical detail.

Yes, I looked at the article. It is interesting. I wonder if the method they use to come up with the dimensionality at various scales has a general procedure in a continuous form. It would be nice if the 4D world could be made to pop out of any continuous, closed, non-perturbative form of Quantum Gravity.

 

[marcus]

...I wonder if the method they use to come up with the dimensionality at various scales has a general procedure in a continuous form. ...

It does! They used two methods to measure dimensionality, both of which work in the continuous case. I think in fact they were both invented to use in the continuous case and Loll has adapted them to her situation of a triangulated manifold.

The two methods which Loll's group has applied to investigate the small quantum universes they generate in the computer are:

A. Hausdorff dimension
B. Spectral dimension

Both can give fractional non-integer results, like 1.72 and 2.36.

A. Hausdorff dimension works in any space where you can define radius and volume. You just look at volume of balls of radius R and if the volume grows as R^d then d is the Hausdorff dimension.

B. Spectral dimension measuring works in any space where you can set up a diffusion process or Brownian motion. It doesn't have to be a space which is in any sense discrete.

Loll's group has a 2005 paper about measuring the spectral dimension. I'll get the link.
http://arxiv.org/abs/hep-th/0505113

 

[Fra]

But if we could find a more fundamental reason for the existence of the Hilbert-Einstein action in the Feynman path integral, then that would prescribe the necessity of quantizing gravity.

This is what I want to se as well. To just pull a particular action form, is not satisfactory. I agree that it's good that some try it, but I fail to see that such strategy addresses the full question. So even if there is partial success, I can't see how it can possibly be the full answer. I think the emergence of actions in general, unavoidably also deals with the foundational issues of QM. But of course to start with another fixed non-trivial action, like string action, doesn't solve the problem.

I expect in the similar spirit that GR wants BI, I think the actions are also part of this. The ACTION is part of the "larger background" consisting of all possible observers IMO. The action does IMO corresponds to a kind of logic that governs behaviour. If you CHOOSE a background action, you can have anything you want emerge - just like by the choice of ergodic hypothesis and microstructure, you can prove that anything is "probable".

This I see as the same dog buried under the physical basis of probability in measurement theory.

I think this is a profound problem, that goes down to our use of logic. It also relates to the processes of deduction and induction.

This is IMHO at least, the reason what the concept of "background independece" is really not trivial.

/Fredrik

 

[friend]

This is what I want to se as well. To just pull a particular action form, is not satisfactory. I agree that it's good that some try it, but I fail to see that such strategy addresses the full question. So even if there is partial success, I can't see how it can possibly be the full answer. I think the emergence of actions in general, unavoidably also deals with the foundational issues of QM. But of course to start with another fixed non-trivial action, like string action, doesn't solve the problem.

Another question is how matter fits into the picture of Quantum Gravity. I hear language like "coupling matter" into the equations. I take this to mean that they just add lagrangians for particles in the Hilbert-Einstein action with some coupling constant in front of the matter lagrangian. Is this the case? If so it seems some justification is need to do this. Why not instead reinterpret the particle lagrangian as a lagrangian for gravity/geometry? Particle physics may just be a local form of gravity.

 

[marcus]

Another question is how matter fits into the picture of Quantum Gravity.

In the case of canonical LQG, matter is added to the picture by adding extra labels to the links (edges) of the spin network.
It looks to me as if the idea is to make the matter fields ride on top of the quantum state of the universe's geometry. The matter fields are riding on the back of the gravitational field, which describes the state of geometry.

At this point in the development, there is nothing but fields. fields defined on top of fields. The lattice that the matter fields are defined on, that lattice is itself the gravitational field.

The most recent introductory-level overview of LQG that we have is the video and slides of Rovelli's talk at Strings 2008.
Here are the links again.
Video:
http://cdsweb.cern.ch/record/1121957?ln=en
Slides:
http://indico.cern.ch/getFile.py/access?contribId=30&resId=0&materialId=slides&confId=21917[/QUOTE]

To get more grasp of the notation and ideas, one should watch the video, and also listen to the questions which the string theorist audience asked at the end of the talk. For most of them the talk probably was about new stuff, so it is instructive to see what they asked about.

Look at slide #28 where it says

|S> = |gamma, i_n, j_l, k_l>

the i's are socalled intertwiner labels at the nodes, the j's are spin labels at the links, the k's are quantum number labels for matter, also at the links

I can't shed any special light on this. It looks to me like it is analogous to a lattice treatment of matter fields, but with a highly random and highly irregular lattice.

The spin network, by itself without matter, is
|S> = |gamma, i_n, j_l>

representing a quantum state of geometry. (The continuum has been washed out of the picture by this point, there is no more space apart from this bare-bones skeleton of geometric relationships)

Now I should say more about gamma:

LQG has borrowed an idea from a branch of topology called knot theory. A knot is an equivalence class of embeddings. Two embedded knots are equivalent if you can deform one into the other by a diffeomorphism (by a smooth mooshing around of the space.). That makes sense. The two are really the same knot if you can make one into the other just by moving the rope around a little. One starts with a knot in space, and then one throws away the space and has only the pure knot itself. Can you picture this? I hope this idea is OK.

So we can do the same thing with networks. Networks are just complicated knots. We can start with a continuum and have networks embedded in it. Then a spin network is an equivalence class of embedded networks. It is abstracted from the continuum we started with, and the continuum is thrown away. It is this network that bears the labels.

What I just said is shown with pictures on slide #13

BTW there is reference to matter at slide #4, where he states the main result:
==quote==
Main result
Definition of diffeomorphism-invariant quantum field theory (for gauge fields plus fermions),
in canonical and in covariant form.
==endquote==

 

[Fra]

It's encouraging to hear that some of what I'm saying makes sense to you!

Just for the record - this wasn't the first time this happened

/Fredrik

 

[Fra]

Another question is how matter fits into the picture of Quantum Gravity. I hear language like "coupling matter" into the equations. I take this to mean that they just add lagrangians for particles in the Hilbert-Einstein action with some coupling constant in front of the matter lagrangian. Is this the case? If so it seems some justification is need to do this. Why not instead reinterpret the particle lagrangian as a lagrangian for gravity/geometry? Particle physics may just be a local form of gravity.

In addition to Marcus reference to LQG, which I think he knows best on here; since you responded to my comment and vision on emergent actions, I though I might just add one *conceptual possibility* I see consistent with that particular reasoning is that matter is really just the physical manifestation of observers. Thus I envision that the "choice of action" and "choice of observer" is simply nothing but the specificaiton of matter. And the evolution of actions, are synonymous to emergence of matter in my abstraction.

The objection of fixing the actions forms, can thus also be interpreted as the objection of fixing the microstructure of matter. Thus there seems to be a duality between emergence of matter, and emergence of spacetime. This is I think even plausible, since if we think of spacetime as relations between "stuff", and "stuff" usually means matter, it really makes sense.

The difficulty is to see that this is not circular reasoning, it really prescribed an evolution. And of course, one would hope that this would turn out to be time.

/Fredrik

 

[marcus]

Just for the record - this wasn't the first time this happened

/Fredrik

Awwww. I didn't mean it that way.
What I meant was about my post #50, which was not guaranteed to make sense to you or anybody else.

My usual posts I think you certainly get the sense of, as much as you want, but in that case, ensabah had asked me to hazard a pure personal opinion speculation about the future course of research. I had to either ignore the question or gamble on my own intuitive hunches alone. Which I normally don't like to do. I'd rather report directions in current research, based on people's publications.

So in the case of #50 I was going out on a limb and not sure anyone would respond. So was very pleased by your understanding response.

 

[Fra]

Thanks for the clarification Marcus. I wasn't sure, and just didn't want to leave that possibility :)

The objection of fixing the actions forms, can thus also be interpreted as the objection of fixing the microstructure of matter. Thus there seems to be a duality between emergence of matter, and emergence of spacetime. This is I think even plausible, since if we think of spacetime as relations between "stuff", and "stuff" usually means matter, it really makes sense.

The difficulty is to see that this is not circular reasoning, it really prescribed an evolution. And of course, one would hope that this would turn out to be time.

As I've said before, I don't like string theory as it stands, but nevertheless there are elements of it that partly make sense to me. So while I don't advocate string theory at all, I think that similary with the outlined reasoning above, is the way string theory SHOULD emerge spacetime as well. As "relations between two interaction strings", and here the purpose of the string is a representation of the mictrostructure of matter.

This is not a bad idea IMO.

What string theory IMHO either misses, or has missed to convey to me at least, is

1) what the meaning of the background spacetime they use is. I think that this is not really a background space in the ordinary sense. It sure looks like that mathematically, but I think there might be another way.

2) the evolutionary character of the above reasoning is missing too as far as I can see.

I think this is related to string theories own quest for B/I. And it may also realte to the landscape, because if you really think that the background space relative to what the string action formulates is an ordinary space, then i am not surprised that you are lead to a landscape. Instead, it could be that the landscape really just represents different observers, and that the main problem is that since they have lost the evolutionary progression, they are LOST in the landscape.

This is what I would look into if I was a string theorist, but I am not. But I thought it might be worth noting the analogy, of a different reasoning.

/Fredrik

 

[friend]

In the case of canonical LQG, matter is added to the picture by adding extra labels to the links (edges) of the spin network.

The spin network, by itself without matter, is
|S> = |gamma, i_n, j_l>

representing a quantum state of geometry. (The continuum has been washed out of the picture by this point, there is no more space apart from this bare-bones skeleton of geometric relationships)

IIRC, LQG is derived using Hamiltonian in a Schrodinger type equation with commutation relations using cononically conjugate momentum, etc. What I'd like to know is if there is a Path Integral formulation and what these spin networks would look like in that formulation. Does such a thing exist? Thanks.

 

[marcus]

PF autolink for the word relation in case anyone wants it explained.
================

IIRC, LQG is derived using Hamiltonian in a Schrodinger type equation with commutation relations using cononically conjugate momentum, etc. What I'd like to know is if there is a Path Integral formulation and what these spin networks would look like in that formulation. Does such a thing exist? Thanks.

The development of the Path Integral formulation is an ongoing process, as is the knitting together of the two formulations. The latest paper on this was posted September 2008, and there was a June 2008 paper on it by the same authors

http://arxiv.org/abs/0806.4640
Path integral representation of spin foam models of 4d gravity
Florian Conrady, Laurent Freidel (Perimeter Inst. Theor. Phys.)
29 pages, 6 figures
(Submitted on 28 Jun 2008)

"We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 <= gamma < 1, the states can be restricted to SU(2) spin networks."

And here is the most recent work, continuing from the June paper:

http://arxiv.org/abs/0809.2280
On the semiclassical limit of 4d spin foam models
Florian Conrady, Laurent Freidel
32 pages, 5 figures
(Submitted on 15 Sep 2008)

The path integral version of LQG (more precisely "sum over histories" version) is called spinfoams. The earlier canonical version is called either canonical LQG, or simply LQG (although confusion can result from not specifying.)
A spinfoam is what you get if you make a spin network evolve in time.
Freidel and Conrady have established a connection between spinfoam sum over spacetime histories and the more usual kind of path integral with an action (as one gets e.g. in Loll CDT, the triangulations approach). A spinfoam is dual to a spacetime triangulation.

There are several competing versions of canonical LQG and several competing versions of spinfoam LQG. This paper by Conrady-Freidel is part of a shakedown and consolidation process where some of the alternatives on both sides are being eliminated. And one is seeing which path integral version best fits together with which canonical version. In the end I think there will be at most one combined version left standing.

Most of the recent results (like n-point functions, graviton propagator, classical limit in special cases...) have been achieved using the spinfoam, i.e. path integral, version.There's a spinfoam quantum amplitude formula is called the spinfoam vertex amplitude, or simply the spinfoam vertex. A spinfoam has a finite number of vertices and the dynamics depends critically on how the amplitudes for these are calculated. It's not clear which version of the vertex formula is going to win out. The closely related (dual in a sense?) path integral formulation does not need a vertex formula because it has an action and a more conventional setup. All this stuff is in the process of being hammered out.Sorry this is so fragmentary. I haven't taken the time to organize.

Anyway it is a very interesting and active area of current research, what you asked about.

 

[atyy]

Hmm, it seems string field theory may be background independent in the sense that a particular background specifies the field configuration (like in GR choosing the metric specifies the stress-energy tensor, in the geometric coordinate independent sense). So one is not free to choose the background, then put arbitrary fields on it as is done in SR or QFT in curved spacetime.

String Field Theory
Washington Taylor
http://arxiv.org/abs/hep-th/0605202