LQC and string theory combined in one theory

[marcus]

But there is no indication of this yet.

Actually there are many indications, Atyy, and more continue to accumulate.

Pages 11-17 of the February review paper have some discussion relevant to this.
BTW did you see that version 2 of 1102.3660 has been posted just in the last few days.
Looks like typos corrected and some additional explanation inserted.

 

[atyy]

Actually there are many indications, Atyy, and more continue to accumulate.

Pages 11-17 of the February review paper have some discussion relevant to this.
BTW did you see that version 2 of 1102.3660 has been posted just in the last few days.
Looks like typos corrected and some additional explanation inserted.

I think there are none, the updated version of 1102.3660 notwithstanding.

http://arxiv.org/abs/1101.5061 , p54: "On the negative side, the actual setting for practical calculations of the spinfoam graviton correlations has been much too simple up to now. These basic calculations were done mainly for a single 4-simplex, which is indeed the simplest space-time triangulation. They typically do not involve summing over bulk internal associated to internal spinfoam vertices. Thus, these calculations don’t allow to truly test the quantum gravity dynamics defined by the spinfoam models and the gluing of 4-simplices (“space-time atoms”) used to construct the amplitudes. They should be considered as kinematical checks."

 

[marcus]

Thanks for the page reference! I see you chose a negative paragraph out of Livine's tutorial/survey of spinfoam LQG. Here is the rest of the page 54 section surrounding it, to give context and provide balance:

==quote Etera Livine http://arxiv.org/abs/1101.5061 ==
3.1.2 The Practical Calculations of the Graviton Propagator

This framework for the spinfoam graviton propagator is based on a very simple setting. There has been a lot of research work done on this subject. Results are, up to now, both full of promise and very restricted.
On the positive side, we are able to compute systematically at leading order the spinfoam graviton propagator at large scale (for large values of the boundary areas j∂) for all the spinfoam models which have been defined. We have even developed techniques to extract (in principle) all quantum corrections of arbitrarily higher order (interpreted as “loop corrections”). This leads to recover the proper scaling of Newton’s law for gravity, with the gravitational potential going as the inverse distance, and even the correct spin-2 tensorial structure of the graviton (correlations) for specific spinfoam models. We even understand the relation between the spinfoam path integral and Regge calculus at large scale. The short scale behavior has also been investigated. It appears that the graviton propagator is regularized (as expected) by quantum gravity effects and that we have the emergence of a dynamical minimal length scale close to the Planck scale. All this has been tested analytically and numerically.

On the negative side, the actual setting for practical calculations of the spinfoam graviton correlations has been much too simple up to now. These basic calculations were done mainly for a single 4-simplex, which is indeed the simplest space-time triangulation. They typically do not involve summing over bulk internal associated to internal spinfoam vertices. Thus, these calculations don’t allow to truly test the quantum gravity dynamics defined by the spinfoam models and the gluing of 4-simplices (“space-time atoms”) used to construct the amplitudes. They should be considered as kinematical checks. It thus remains a challenge to go beyond the single 4-simplex and work with refined space-time triangulations, which would allow local fluctuations of the curvature in the bulk.

Here is nevertheless a (almost-exhaustive) list of the works done on the programme of the spinfoam graviton propagator:

• Definition of the framework [81, 82].

• Analytical study of the asymptotic ansatz for the spinfoam vertex amplitude in order to recover at leading the correct tensorial structure for the graviton propagator [83, 84, 85].

• Group integrals techniques to compute explicitly analytically the graviton propa- gator for the Barrett-Crane model (generalizable to arbitrary spinfoam models ex- pressed in the connection representation) [88].

• Numerical investigations of the behavior of the graviton propagator for the Barrett- Crane model, both for the large scale and short scale, both at leading order and at next-to-leading order (first order quantum gravity corrections) [89, 90]

• Calculations of the asymptotics of the graviton propagator for the EPRL-FK spin- foam model [91].

• Definition of a 3d toy model using the Ponzano-Regge model [92], numerical investi- gations and development of the tools to compute the full expansion of the correlations and solve the model analytically [93, 94].

• Study of the propagation of coherent wave-packets of geometry within a 4-simplex [95].

• Analytical and numerical study of the asymptotics of the spinfoam vertex amplitude relevant to the calculations of the large scale behavior of the graviton propagator, in 3d [96, 97] and in 4d for both the Barrett-Crane model [98] and the EPRL vertex amplitude [99, 100, 101].

• Discussion of the potential use of the recursion relations satisfied by the spinfoam vertex amplitudes to the computation of the graviton correlations and to derivation of Ward-Takahashi identities for spinfoam amplitudes [97, 102].

• Tentative calculations of the 3-point correlation functions [103].

What is very nice about this framework is that it provides a physical interpretation to the correlations computed using spinfoam models and in particular shows how to recover the classical Newton’s law for gravity from our complicated and intricate model for a quantum gravity path integral. Moreover, we can actually compute analytically these correlations, plot them numerically, check that everything is consistent, and see explicit the first elements of the spinfoam dynamics with our own eyes.

However, progress in this direction is completely coupled with necessary progress that needs to be done on the coarse-graining and renormalization of spinfoam models. Indeed, we need to be able both to repeat the same graviton correlation computations for more refined and complex bulk triangulations and to say something about the non- perturbative sum over all 2-complexes. The main hope for this is put in exploiting the group field theory formalism and studying its renormalization as a quantum field theory.

3.2 From Spinfoam Amplitudes to Non-Commutative Field Theory
Besides looking at the quantum gravity corrections to the gravitational interaction, an- other way to probe the semi-classical regime of quantum gravity and extract potential...
==endquote==

 

[Haelfix]

A theory which can only recover GR in a perturbative setting (expanding around a fixed, e.g. flat, metric geometry) cannot properly be said to recover GR.

Because GR is a theory of fully dynamic geometry. It misses the essential thing about GR.

This statement is incorrect, and I believe its the root cause of most of the inordinate amount of confusion on this forum. You can always recover GR from perturbation series, it is done for instance on page 435 of MTW, or alternatively any of the classic texts by Deser et al, or by Weinberg in his Gravitation text. It has nothing to do with whether GR is dynamical either, it is simply consistency criteria. Said another way, the full nonlinearity of GR can be bootstrapped starting merely from a weak field expansion.

In other words, if historically Einstein had only known about perturbation theory and linearized gravity he would have been able to derive the full equations from a few mild observations.

"Quantum Gravity" may however be less favorable to perturbation series in that there could be regimes and objects that perturbation series cannot see. For instance, pertubation series apparently misses whatever it is that unitarizes black hole physics, even when done around a nice and 'under control' slice, far away from any violent curvature regimes (where you are not allowed to perturb around).

The important thing is that the theory is ABLE to be formulated non-pert..

The important thing is that you should be able to do both! What is also wrong, is to simply ignore whatever insights perturbation series tells you.

 

[atyy]

This statement is incorrect, and I believe its the root cause of most of the inordinate amount of confusion on this forum. You can always recover GR from perturbation series, it is done for instance on page 435 of MTW, or alternatively any of the classic texts by Deser et al, or by Weinberg in his Gravitation text. It has nothing to do with whether GR is dynamical either, it is simply consistency criteria. Said another way, the full nonlinearity of GR can be bootstrapped starting merely from a weak field expansion.

Can this really be said to be due to perturbation series? I think it is fair to say perturbation series alone does not recover GR, since one needs the consistency criteria, which are imposed in addition to the perturbation series. Eg. on perturbation series alone, can one get below the event horizon of a black hole?

 

[marcus]

Can this really be said to be due to perturbation series? I think it is fair to say perturbation series alone does not recover GR, since one needs the consistency criteria, which are imposed in addition to the perturbation series. Eg. on perturbation series alone, can one get below the event horizon of a black hole?

I certainly agree! We are talking about chapter 18 of MTW. "Weak gravitational fields" They do not say that one can recover GR solely from its weak field version. It is an interesting part of the book, if anyone is interested they can take a look at, say, pages 420-440 on line. Go here:
http://www.google.com/search?client...hl=en&tab=wp&bav=on.1,or.&fp=1d35b10d87834f0f

Or just go to google-books and search "weak gravitational fields". Click on the first hit.

Box 18.1 on page 437 is interesting: it compares the Einstein (geometric) derivation of GR with the "spin-2" derivation (weak field+bunch strong assumptions to make the bootstrapping work)

==quote page 436==
Just as one can "descend" from general relativity to linearized theory by linearizing about flat spacetime (see below), so one can "bootstrap" one's way back up from linearlized theory to general relativity by imposing consistency between the linearlized field equations and the equations of motion, or, equivalently, by asking about [a ton of stuff, five separate technical issues which they list, and then say "and so on..."]
...But because the outlook is far from geometric (see Box 18.1), the details of the derivation are not presented here...
==endquote==

 

[atyy]

In the context of perturbative GR being a limit of string theory, shouldn't the consistency conditions come from string theory itself, and lead to string theory, rather than GR?

Also, I can imagine that a non-perturbative formulation can consist of perturbative formulations, as long as the different formulations overlap uniquely, and cover the whole space. I'd imagine the various string dualities do this at least partially.

 

[Haelfix]

Can this really be said to be due to perturbation series? I think it is fair to say perturbation series alone does not recover GR, since one needs the consistency criteria, which are imposed in addition to the perturbation series.

This paper by Deser is one of the standard references for how you bootstrap up from the weak field equations to the full nonlinear form: arXiv:gr-qc/0411023 or alternatively Weinberg 1965 Phys. Rev. 138, B988 (1965).

There are of course a number of very simple assumptions (that turn out to be empirically correct) to mathematically go from linearized gravity to the full nonlinear form. However, that shouldn't be a surprise to anyone and indeed it is part of the power and beauty of General relativity that it admits multiple isomorphic mathematical formulations.

What is unclear (and likely untrue) is whether you can do this same thing in pure quantum gravity. The reason is that you most likely do need some sort of UV completion, but that is a story for another day.

 

[Haelfix]

In the context of perturbative GR being a limit of string theory, shouldn't the consistency conditions come from string theory itself, and lead to string theory, rather than GR?

To get GR from say the bosonic string, you need basically only invoke Weyl Invariance or the vanishing of the beta function. This is a consistency criteria of string theory proper.

 

[suprised]

To get GR from say the bosonic string, you need basically only invoke Weyl Invariance or the vanishing of the beta function. This is a consistency criteria of string theory proper.

That's exactly my point, that marcus apparently does not appreciate. I do not mean here just graviton scattering processes around a backround; I mean that the whole lagrangian of GR: L = Sqrt (g) R + corrections, pops out as consistency condition in string theory (by expanding around flat space, graviton scattering can be recovered if one wishes to do so).

Thus string theory acts as a black box:

input = 2d CFT plus ghosts, integrated over 2d fields, moduli
output = Einstein gravity (plus Planck scale suppressed corrections).

That LQG provides an analogous black box, once properly defined and understood, is not unlikely but not still proven after some 25 years of research.

Thus, as of today there is no point to dismiss string theory as "purely mathematical" construct, when comparing both approaches!

 

[tom.stoer]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

 

[suprised]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

It is not necessariy a flat backround; but the formalism is on-shell only and space-time is not dynamical, this is correct. Indeed no one claims that this is the final answer!

 

[atyy]

That's exactly my point, that marcus apparently does not appreciate. I do not mean here just graviton scattering processes around a backround; I mean that the whole lagrangian of GR: L = Sqrt (g) R + corrections, pops out as consistency condition in string theory (by expanding around flat space, graviton scattering can be recovered if one wishes to do so).

Thus string theory acts as a black box:

input = 2d CFT plus ghosts, integrated over 2d fields, moduli
output = Einstein gravity (plus Planck scale suppressed corrections).

That LQG provides an analogous black box, once properly defined and understood, is not unlikely but not still proven after some 25 years of research.

Thus, as of today there is no point to dismiss string theory as "purely mathematical" construct, when comparing both approaches!

The whole GR Lagrangian pops out as a a consistency requirement for the background on which perturbative string theory is done.

But isn't the claim for string theory to reproduce GR more than that, since that only reproduces the vacuum Einstein equations. Matter minimally coupled to gravity comes from the string excitations, which are perturbative, so is it still true that the whole Lagrangian of GR pops out if one discusses non-vacuum GR?

BTW, I do of course agree there is no reason to dismiss string theory as "purely mathematical". Even Smolin would agree string theory is much more than that: "it seems that any acceptable quantum theory of gravity, whatever its ultimate formulation, is likely to reduce to a perturbative string theory in the appropriate limit." http://arxiv.org/abs/gr-qc/9508064

 

[tom.stoer]

Short question: instead of repeating where theory X fails - wouldn't it be better to discuss where theory Y might help?

 

[Haelfix]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

There is a terminology point that I think you understand Tom, but I think that others on this board (Marcus in particular) have repeatedly struggled with over the years.

When we say that we recover GR as a consistency condition of String theory, we mean the full dynamical nonlinear classical theory. You bootstrap the theory in a completely analogous way as you do the classical linearized theory. In other words if you were a theorist in 1910 and someone handed you string theory in its current form, you would automatically read out not just a linearized version of gravity, but the whole shebang in the correct limit where you were justified in ignoring all the decidedly quantum effects. You really do limit to exactly the 1915 theory with all the associated geometry and so forth.

However, as you said and as Surprised has noted, there is still background dependence here at the level of the formalism and that may or may not be an impediment for defining the full offshell *QUANTUM* theory. It is in this sense that morally speaking one does not quite have the exact spirit of Einstein's theory, or its corresponding ability to calculate things in a simple and elegant manner. But, having said that, it is just completely wrong to suggest that they don't get the correct *classical* dynamics out (like Marcus has suggested a few posts up).

Incidentally, the word 'dynamic' is one of those loaded physics words that has evidently created some amount of confusion here for multiple years.
It is actually simpler to talk about what is fixed in string theory... What is 'fixed' is an object in an intermediate calculational stage, namely one's choice of an appropriate classical solution (like flat space or say the Schwarzschild solution) in a background field approximation.

This approximation method was initially invented to deal with quantum gravity, but it has been since utilitzed all across QFT in different situations. For instance we might do a background field split around a gluon field in QCD. In so far as this makes sense (and it is very limited), one doesn't really think of the gluon field as being 'non dynamic', it is merely approximated to ignore the backreaction effects by the small variation that is propagating. At the end of the calculation, you sum up all the contributions and you recover the full thing. I like to bring this up, b/c it demystifies the apparently peculiar position on this board whereby gravity is somehow special or mysterious in this regard.

 

[Haelfix]

But isn't the claim for string theory to reproduce GR more than that, since that only reproduces the vacuum Einstein equations. Matter minimally coupled to gravity comes from the string excitations, which are perturbative, so is it still true that the whole Lagrangian of GR pops out if one discusses non-vacuum GR?

"it seems that any acceptable quantum theory of gravity, whatever its ultimate formulation, is likely to reduce to a perturbative string theory in the appropriate limit."

Whether you get any matter at all is going to depend on the nature of the Stringy geometry and details of the moduli. Of course once you are far enough away from the Planck scale you can treat all the matter as an effective field theory and it will act like a stress energy term sourcing the Einstein equations.

As far as Lee's quote. Well, no one knows if String theory is correct or not, but of course whatever final theory of gravity is the truth, it will have to at the very least explain and subsume all the perturbative results from semiclassical gravity in the correct regime. That was my point earlier. We know a few things about perturbation series and gravity, and this does constrain our options.

 

[marcus]

But this is still a pure argument.

You quantize string theory on flat spacetime and get back (as a consistency condition) the Einstein equations. But you are still on flat spacetime; spacetime hasn't become fully dynamical in that setup. It's is still a consistency condition for the classical background.

I do not doubt that this is a hint towards some underlying truth but it is definately not this truth itself.

It is not necessariy a flat backround; but the formalism is on-shell only and space-time is not dynamical, this is correct. Indeed no one claims that this is the final answer!

Good. So in the end the "bootstrap" breaks and does not go the last way to recovering GR. But gets a strong suggestion of it.
And Suprised is right that the fixed background does not have to be flat. There is work using fixed but curved backgrounds.
We've had other discussions, over the years, that brought these same points out.
I appreciate your clarity and frankness, Suprised, in this instance.