Alright, time to discuss some physics again. A while back I outlined the basic dichotomy of quantum gravity. In brief: we have classical general relativity as an excellent description of all observed gravitational phenomena, but when we go to extremely short distances we need to have a good description of quantum gravity. There are two possibilities: either the description of gravitational phenomena by the machinery of general relativity (metric, the principle of equivalence, etc.) holds all the way down to those extremely short distances, or it doesn’t. If it doesn’t, which is my own prejudice, classical general relativity and its variables are never to be quantized, there is no range of energies (or distance scales) which is described by quantum metrics obeying a quantum version of Einstein’s equations.
This situation is similar to the hydrodynamical description of fluid motion: it is a classical effective field theory, which breaks down long before quantum mechanics is needed. We can observe granularity in the fluid on distance scales much larger than those relevant to quantum mechanics. From that perspective, quantizing Einstein’s equations makes as much sense as quantizing the Navier-Stokes equations, which is to say not too much sense.
But, since lots of smart people do not share my prejudice, and continue to look for just the right trick to quantize the metric field (be it dynamical triangulations, loop quantization, or anything else), let us just agree that there is this basic dichotomy, and one has to choose their own path. The path I find plausible is that where the gravitational field is a manifestation of some deeper structure at short distances, distances shorter than the ones currently probed by experiment, but not as short as the Planck length. This path is sometime dubbed emergent gravity, a name which I find somewhat misleading, for reasons I’ll describe below.
It is important to keep in mind that many of the problems of what is labeled “quantum gravity” are just problems of one of the approaches to the problem, that of quantizing the metric field. When dropping this assumption, life seems to simplify quite a bit, so let’s do that – imagine we are quantizing some dynamical system, and in a certain classical limit that system starts to look like a gravitational system. By that I mean very simply that it has objects moving in some large spacetime, and attract each other with universal Newtonian potential. Or in other words, it has apples and they fall from trees.
But which dynamical system? fundamentally, it doesn’t have to look at all like Einstein’s gravity, it could be pretty much anything. This looks like hopeless guesswork, but we have a few things working for us. First, it is not that easy to obtain a theory that look like classical general relativity at low energies: Lorentz invariance and the principle of equivalence are very restrictive. We understand the structure well enough to rule out many attempts a priori, without further examination. This is very valuable for busy people who can attend only so many conferences…
Secondly, we do have some examples of the idea working perfectly, within string theory, so we know of possible loopholes. There is much more to say, but for now let me borrow the slogan from the beautiful paper of Elvang and Polchinski: The emergence of general relativity requires the emergence of spacetime itself.
What does that mean? Recall the idea of duality: some notions are not a property of the physics, just of a certain description thereof. Surprising as it sounds, this includes the question of how many dimensions spacetime has. The attempt to derive general relativity from something else, living in the same spacetime, is likely to fail. The examples we have working all have the property of holography: the microscopic system from which general relativity emerges lives in a different spacetime, usually a lower dimensional one. In that sense the subject is different from what is usually titled as emergence, for example in quantum condensed matter physics, so we probably need to use a different word. Besides, emergence was all the rage in the 1970s, maybe it’s time for it to join old ABBA albums and John Travolta posters and all those things…
And that is it for today, hope everyone has enough to chew on for Turkey day. If everyone excuse me now, I have an exam to give.
Update: for a thoughtful summary and elaboration on the discussion here, you can turn to Lubos.
Shores of the Dirac Sea 
As a true know-nothing in these matters I must say that the whole concept of emergence of gravity has not made much sense in the typical use of the word ’emergence’. Does not 4d gravity just become more apparent instead or more in line with the concept of limit. You just cannot say that gravity is more of a gravity at this end of things or the other end. Maybe 4d is just a growth or peeling from the 10d structure leaving the other 6d in the extreme ultraviolet space.
Hi Moshe– I completely agree with your philosophy concerning quantum gravity, but I don’t think the hydrodynamical analogy is a very good one. There can certainly be regimes in which a quantum description of the metric is useful, even if you are nowhere near the Planck scale; gravitational waves from inflation, which lead to testable predictions for the CMB, are the most obvious example. More generally, it can often be useful to quantize collective excitations of some underlying degrees of freedom; otherwise the word “phonon” would not exist.
Yes, you are basically right, though there is a sense in which the analogy is not too inaccurate. Basically it comes down to what you mean by quantum gravity. As you say you can always linearize and quantize, pretty much anything, including Einstein’s gravity. The question of whether or not you need to use quantum mechanics is normally not that of distance scales, it depends more on the actions involved. So you can have an EFT which summarizes the physics of the long wavelength modes, which depending on the regime of actions you are interested in can be taken to be either quantum of classical.
Nevertheless, it is often said that quantum gravity becomes relevant at the Planck scale. That is the regime, the distance scale, where both quantum mechanics and the non-linearities of gravity are simultaneously important. If we reserve the name quantum gravity for that theory, then there is a somewhat strained analogy to hydrodynamics: both are non-linear effective theories which only apply classically.
Hi Moshe
I like the idea of ADS/CFT as an emergent theory of gravity and space-time; in the sense that a 10 dimensional gravity theory emerges from a 4 dimensional gauge theory where gravity is absent. Although it would be nice if this string (gravity)/gauge duality would be valid independently of the background geometry.
Someone could also say that the seed of emergent gravity and space-time is already present in the perturbative string theory in the sense that in the sigma model the background metric field is derived from (or it is related to) the oscillations of the string (i.e. the graviton field of the spectrum).
BR
Giotis, in AdS/CFT all dynamical aspects of the 10 dimensional geometry emerge from the gauge theory, whereas non-dynamical aspects (like the boundary conditions of the geometry) remain at the level of background. To my taste that is the maximal background independence one can achieve in a physical theory. Also, though perturbative ST is not BI, there is indeed a sense of emergent geometry there as well. More in the paper I referred to in my post on background independence.
Metric gravitation postulates the Equivalence Principle (spacetime curvature, pseudo-Riemannian spacetime). Teleparallel gravitation ignores the EP (spacetime torsion, Weitzenböck spacetime). The disjoint non-overlap is empirical EP violation. PSR J1903+0327 (arxiv/0805.2396) strong-field eliminates all candidates but one:
Is EP=true for chemically identical, opposite geometric parity atomic mass configurations? Do left and right shoes fall identically?
Atoms self-similarly, gaplessly, 3-D pack in 230 unique ways – the crystallographic space groups. Eliminating improper rotation symmetries and racemic screw axes leaves three enantiomorphic pairs: P3(1)21/P3(2)21 (quartz group), P3(1)12/P3(2)12 (e.g., CrCl3), and P3(1)/P3(2).
Load an Eötvös balance with solid spherical single crystal quartz test masses, space group P3(1)21 opposing P3(2)21. Does a parity Eötvös experiment obtain a net non-zero output? If so, there’s your problem.
Parity is not a Noetherian external symmetry (coupled to rotation and translation). Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. The only strongly allowed EP violation has never been examined. Somebody should look.
perturbative string theory
Theory arises from symmetries, observables arise from symmetry breakings. Perturbative treatments exclude symmetry breakings. Empirically falsify BRST invariance uniting the effects of a massive body and an accelerated geometry. Demonstrate an Equivalence Principle violation. String theory’s 10^(10^5) acceptable vacua, containing not a single testable prediction, are then empirically wrong.
Test spacetime geometry with opposite parity atomic mass distribution geometries – quartz, as above. The experiment feeds SOP publication mills grinding out annual null results. No less than a null result can obtain. The worst that can happen is… success.
(Controls are each parity of quartz against amorphous fused silica. Density difference requires silica to be 0.99123 radius brought to size with additional gold plating.)
The first argument for why GR shouldn’t be quantized could also be made for electrodynamics, where short distances give trouble already classically. But maybe the devil doesn’t need any more advocates.
Hi Lionel, I didn’t give any detailed argument for my prejudice (and pretty much that of everyone else) that (nonlinear) GR is just a classical EFT. When I do, the difference from QED will hopefully be clear. In any event, it is not the appearance of singularities in the classical theory that is the issue.
I agree It’s a very important point whether a classical theory should be quantized or not, and *how* (whatever that means…). But in this case I’m not sure whether quantization is more ‘fundamental’ than GR actually. The quantization process itself may be ‘effective’. It is true we do not quantize thermodynamics, but we *do* quantize field excitations in a flat background, and seems to work pretty well, e.g. photons, etc. After all relativistic ED itself is an ‘effective’ description too, and we put hats on perturbations around the ‘vacuum’. Honestly I don’t see anything wrong in quantizing gravity, as we quantize any other field, specially when it comes to gravitons, but also because we would run into all sort of problems if we don’t, although we ignore them since it’s so damn small, at least at our energy scales. I’m tempted to believe actually that QG has no quantum nor GR in it, as we know it. Nevertheless, for our ‘effective’ description of the world, it works to do what we do, however we do it, although maybe only as an S matrix approximation as Heisenberg taught us… 🙂
I agree, the description will break down at some point, as we talk about the hydrogen atom in classical physics, and then Schroedinger, and then vacuum polarization, and then gravitons (damn Lambda 😉 ), and then…
The whole point here is where is it that GR breaks down, is at the planck scale, or we hit some other scale before… after?
We clearly have some funky non-local behavior going on we haven’t fully understood yet, at least not from the geometric point of view which I believe is a valid one as a good effective field theorist 🙂
This brings us back to whether ADS/CFT *solves* the BH info paradox. I believe it does not, I don’t buy this “new paradigm” answer where I’m supposed to believe that the geometric picture doesn’t make any sense anymore, well, yes it does, and breaks down somehow and I’d like to know why/where/when/how!
Of course we have hints from all over the place, no local observables, time dependent backgrounds, do horizons form at all?
I agree it is a *global* question, and perhaps in som sense *locally* (as much as we can define local observations) nothing is really happening, but there are non-local correlations which are damn small but accumulate over time. Anyhow, I think we could learn something from the answer, after all we live in the world where the BHs (or whatever the end point is) are, not in the boundary 😉
R
Gravity is a manifestation of the curvature of spacetime while the other three forces are a manifestation of the curvature of internal spaces (I mean fibre bundles).
In this sense, gravity is quite different from the other three forces. So, if a unification of forces really makes sense, in which the four forces are different manifestations of a single force, it should be desirable to reconcile gravity with the others. A usual wisdom to resolve the difference of the geometrization of forces is the idea of Kaluza and Klein.
The basic idea of the Kaluza-Klein theory (including string theory) is to construct spin-1 gauge fields plus gravity in lower dimensions from spin-2 gravitons in higher dimensions. An underlying view in this program is
that a “fundamental” theory exists as a theory of gravity in higher dimensions and a lower dimensional
theory of spin-1 gauge fields is derived from the higher dimensional gravitational theory. Though it
is mathematically beautiful and elegant, it seems to be physically unnatural if the higher spin theory
should be regarded as a more fundamental theory.
Anyway the Kaluza-Klein theory is based on the following core relation; schematically,
(1 x 1)_s <– 2 + 0 or
(open string x open string) “. Salient examples are AdS/CFT correspondence and matrix models, e.g., IKKT, BFSS, etc. Of course, in order to realize the scheme “–>”, we have to overcome the Weinberg-Witten theorem.
Here is the point where we need the slogan of Elvang and Polchinski: “The emergence of general relativity requires the emergence of spacetime itself”.
Then the question is how to realize the slogan of Elvang and Polchinski. To be specific, what is the first principle to realize spacetime geometry from gauge fields ? I think we need a paradigm shift about the spacetime geometry which has been exclusively based on Riemannian geometry so far. But we have to remember that there are two kinds of geometries in sharp contrast each other: Riemannian geometry and Symplectic geometry.
What is a use of symplectic geometry for spacetime geometry ? Surprisingly the symplectic geometry provides
a noble form of equivalence principle for the geometrization of electromagnetic force. It is nothing but the Darboux theorem in symplectic geometry. Note that noncommutative spacetime intrinsically carries its own symplectic structure. (Well time is a bit differently realized.)
As a consequence, the electromagnetism in NC spacetime can be realized as a geometrical property of spacetime like gravity. Indeed the geometrization of the electromagnetic force in the context of symplectic geometry seems to be the origin of gravity. Anyway the noncommutative spacetime precisely provides us everything to realize the scheme “–>” and the slogan of Elvang and Polchinski. You may find more details in the paper arXiv:0809.4728. Sorry for my too long comment and rude citation of my paper.
Hi Moshe,
I read you paper. I’ve heard about this argument before (i.e. that ADS/CFT is fully background independent) but your paper presents it in a systematic way. Basically what you are saying (correct me if I’m wrong) is that the ADS metric field could be fully dynamical in the bulk as long as it remains fixed at the boundary of ADS space (boundary condition). Moreover you are arguing that this way the boundary conditions of the various fields living in the bulk cannot be changed by the fluctuating dynamical metric. Thus, nothing will change for the gauge theory at the boundary of ADS (since its physics is determined by the boundary conditions), even if the bulk metric is fully dynamical (not at the boundary though) without a fixed background (i.e. without a perturbation around a fixed metric).
That is how I understand your paper and BI in ADS/CFT in general but I’m not sure if this is the maximal background independence that can be achieved or that it is equivalent to the background independence of GR at conceptual level at least. Moreover it seems to me that the background dependence of the perturbative string theory in the bulk has been transferred at the boundary. What i mean to say is that the information that the bulk string theory is highly background dependent cannot be just lost; it has to be transferred somehow at the boundary via the boundary conditions. Anyway I am not quite sure.
BTW what did they tell you at the conference? Did the loop guys accept that this notion of BI is equivalent to the notion of background independence as we encounter it in GR? Did they have any objections to your arguments?
BR
Giotis:”That is how I understand your paper and BI in ADS/CFT in general but I’m not sure if this is the maximal background independence that can be achieved or that it is equivalent to the background independence of GR at conceptual level at least. ”
Let me squeeze my two cents here.
Giotis, let me ask you a question. Can you find a solution for the metric in classical GR (which is BI by your standards) which can smoothly take you from AdS to Minkowski to dS. The answer is NO, of course there are no such solutions! Apart from topology change, such transitions would require an infinite amount of energy and are therefore not allowed in classical GR. In other words, GR is only BI as long as you stay in the same superselection sector (chosen by the boundary conditions). BI in the context of AdS/CFT is precisely in the same category. The big deal is that AdS/CFT is fully quantum and BI in the same sence as GR is.
Moreover, once you realize that there are non-perturbative objects (branes) contained in string theory, topology changing transitions become possible, implying that backgrounds with different topologies are connected. This is clearly a step above GR.
Hi Mark,
Maybe I didn’t phrase correctly what i wanted to say. My problem is with the string theory in the ADS space. As I understand it, in order ADS/CFT to be more than a conjecture, you must first define independently the string theory as a non-perturbative theory. Today you can’t even define it properly as a perturbative theory in ADS, you can only study its low energy approximation.
So you must first find a non-perturbative, BI language for the string theory in the bulk in order to prove the ADS/CFT conjecture and not the other way around i.e. to use the conjecture to prove that string theory is BI.
BR
Giotis, I have some catching up to do, replying to your comments, including a future post describing the restricted sense in which the field theory dual to asymptotically AdS quantum gravity lives on the boundary. I think that this viewpoint misses the point of the correspondence in many ways.
Let me start with your last comment. You can define string theory in asymptotically AdS space by the gauge theory. This is the only full non-perturbative definition of the theory, and I claim it is BI in precisely the same sense that classical GR is BI, no more but also no less.
You are right then that calling this a correspondence is somewhat misleading. This is not a statement of full equivalence between two independently defined objects, but a statement that this non-perturbative definition, even though it looks very different, is consistent with everything else we expect of string theory in asymptotically AdS space, including the fact that it is a quantum theory of gravity, has a specific perturbative expansion and non-perturbative effects, D-branes and black holes, etc. etc.
And, just anticipating a little bit, I hope we are not going to get into a discussion of what is proven or even rigorously stated. Evidence for the above statement is overwhelming, that’s good enough for me.
Also:
The people at the loops 2007 were very generous with their time and comments, some of which were incorporated in the text. I have no intention to convert anyone, I was (am) just trying to explain the alternative viewpoint and its internal logic, which I believe for many people is out of their field of view, or sometimes transmitted to them via lossy channels…Once they understand it, my work is done, it is up to them what to do with that information.
Giotis:”Today you can’t even define it properly as a perturbative theory in ADS, you can only study its low energy approximation.”
It’s possible to do it if you consider the pp-wave limit of AdS. If I recall correctly, Metsaev quantized strings on the pp-wave background in light-cone gauge and then Spradlin and Volovich gave a recipe how to compute string correlation functions using light-cone string field theory formalism. These computations are purely stringy, not just supergravity. It fact, as I recall, if you examine the spectrum, the stringy modes and supergravity modes generically do not decouple because in addition to the string scale one has an effective mass parameter related to the value of the background RR flux. So if you compute even 3-point tree level amplitudes, you already get alpha-prime corrections. Then you can do computations on the gauge theory side and again verify that there is indeed a match, going well beyond the supergravity regime.
“So you must first find a non-perturbative, BI language for the string theory in the bulk in order to prove the ADS/CFT conjecture and not the other way around i.e. to use the conjecture to prove that string theory is BI.”
As Moshe has pointed out already, the boundary CFT must be regarded as the full non-perturbative BI formulation of strings on all backgrounds with AdS asymptotics. In fact, a smooth manifold with AdS asymptotics only emerges as one takes a geometric limit. In a generic case the thing to compute is the partition function of the CFT which contains all the info about correlation functions (the observables). In this BI formulation, not only is there no particular background metric to talk about, there is actually no manifold either. So your point about having to quantize strings in the AdS bulk in a background-independent way is misplaced because the AdS bulk(geometry) is not fundamental. Loosely speaking, on the bulk side you just have a bunch of oscillating strings and D-branes, described by the boundary CFT, which only when you take a certain limit turn into some smooth AdS geometry.
Dear Mark and Moshe,
You said classical GR is BI. But that is not true. Einstein equations say “Geometry = Matter”, that is, matter and energy distributions determine a spacetime geometry. In this sense Einstein gravity is almost BI, but it is not the case for “flat spacetime”. If spacetime metric is flat, the LHS of the Einstein quations vanishes, so the energy-momentum tensor of RHS should also vanish. Therefore Einstein gravity says that a flat spacetime is free gratis, i.e., costs no energy. Of course it is an inevitible structure of Einstein gravity since it is an extension of relativity to non-inertial frame and a flat spacetime is a geometry of special relativity rather than general relativity.
But more serious musing on the picture about the flat spacetime in Einstein gravity will reveal that it is not a sensible result since it causes some conflicts with the concept of vacuum in QFT and the spacetime picture in inflation theory. If the Einstein equation is really meaning ful, the vacuum of the matter sector should determine the one of geometry. But QFT, which explains the RHS of Einstein equations, say that QFT vacuum is not empty but full of quantum fluctuations whose weight is roughly Planck mass. So there is a conflict about the vacuum between geometry and matters. Another conflict is that inflation (if it is true) explains an almost infinite extension of spacetime geometry for a tiny moment, e.g., Planck time. But to get infinitely large spacetime through the inflation we need a huge vacuum energy. Note that after inflation our spacetime is very very large, so whose spacetime curvature will be small since the curvature is roughly 1/L^2 where L is a size of space. Anyway a “nearly” flat spacetime has been generated due to the inflation which required the huge amount of vacuum energy. Therefore a flat spacetime would not be free gratis, but a result of huge energy condensation in a vacuum.
Also I think in order to completely make sense the slogan of Elvang and Polchinski: “The emergence of general relativity requires the emergence of spacetime itself”, there should be a fundamental theory which naturally explains the dynamical origin of flat spacetime, which is certainly absent in Einstein gravity. I am to say that in this sense the classical GR is not BI. Furthermore this silence about the dynamical origin of flat spacetime seems to be a core reason of the cosmological constant problem in Einstein gravity.
there was a nice paper last week (arXiv:0811.2081) in which the expectation value of the polyakov loop (and hence, indirectly, the schwarzschild radius of the black hole in AdS) was obtained from a first principles (i.e. numerical) field theory computation.
if anyone needed convincing that AdS/CFT is correct I think this might just nail it…. there is no supersymmetry or integrability in sight and it shows a very precise sense in which the field theory defines the quantum gravity. you put the theory on a lattice and compute.
Hi Moshe and Mark
I understand your arguments and they are strong arguments indeed.
Nevertheless when I’m talking about the string theory in the bulk is because i want to address some conceptual issues regarding the “actual” string theory. Even after the ADS/CFT we still consider strings as the fundamental building blocks of nature. The conceptual image of a string moving in a certain background has not been changed with ADS/CFT. You still need the notion of a background where the string would be able to propagate in order your theory to have a physical meaning. That is exactly my problem. As I see it, if the strings and branes are the fundamental building blocks of nature, everything including space-time should be derived from them. To put it schematically, they must create their own space-time dynamically. So for me BI has a deep conceptual and physical importance and it is not just empty rhetoric.
In GR on the other hand the gravitational field is not located somewhere and does not propagate in a fixed space-time manifold. It is the space-time manifold and it is fully dynamical; it doesn’t need a background. Thus there is no fixed space-time in nature. So if you want to say that gravity is derived from strings, you must reproduce this dynamical gravitational field in a certain approximation but from scratch, without assuming that the strings are located somewhere or that they propagate in some space-time. This notion of BI introduced by GR is a fundamental property of nature and it must be followed all the way down to Planck scale.
So the important question is whether ADS/CFT can help me in that direction. After ADS/CFT can someone say that we don’t really need anymore the physical notion of a string located and moving in a pre-defined space-time manifold in order to describe nature? In other words, can ADS/CFT be formulated as a closed, self contained physical theory which does need that notion *at any stage* to describe nature? If the answer is yes then I would be pleased.
That is why i argued that you must first find an independent, non-perturbative, BI language for the string theory. As far as i know there are string theorists who are still trying to find that language.
BR