the relation between entropy and probability at quantum levelu [Archive]

mitchell porter

May23-11, 12:46 AM

You shouldn't imagine that what follows came to me all in an instant. What I did was to condense what you were saying into a slogan, and then tried to make that slogan meaningful.

Let's start with the standard AdS/CFT example of holography, the duality between d=4 N=4 Yang-Mills field theory and Type IIB string theory on the d=10 space "AdS5 x S5". Working from a different direction, Nima Arkani-Hamed and collaborators have come up with a "Grassmannian" representation of that same field theory which doesn't involve space or time.

If you examine Erik Verlinde's plan, you'll see that his fundamental entropic force is supposed to be pre-holographic. "The starting point is a microscopic theory that knows about time, energy and number of states. That is all, nothing more." The conventional holographic principle, in which a field theory in a space is equivalent to a gravitational theory on a space with at least one more dimension, somehow comes later.

So the first idea is that you could try to understand the progression from the Grassmannian to the field theory to the string theory, according to Verlinde's program.

Now for quantum randomness. The conventional attitude in physics is that it's fundamental, doesn't have a deeper explanation, and can't be understood as an extra "noise" term added to deterministic classical dynamics. Edward Nelson's stochastic mechanics tried to reproduce quantum mechanics in that way, but demonstrated that the noise has to be nonlocally correlated. You get this with the quantum potential of Bohmian mechanics, but Bohmian mechanics is artificial because it's only defined with respect to a particular reference frame. That is, if you pick a coordinate system, then you can construct a nonlocal force which will give you the same predictions as the standard quantum wavefunctions. But your formula will only be valid in that coordinate system, whereas fundamental physics is supposed to be independent of coordinate system.

The holographic principle, as realized in AdS/CFT, is an equivalence between two quantum theories: a field theory on the boundary of a space, and a string theory in the bulk of that space. The boundary quantum states already include information about what happens in the extra dimensions, away from the boundary. The simplest way to think about this is to think of the shadows in Plato's cave. If you have an object and hold it close to the cave wall, the shadow will be small, but as you move it further away, the shadow gets larger. In the same way, the further into the bulk an object is from the boundary, the larger its image in the boundary field theory. Since the boundary field theory is "conformal", which includes scale invariance, it allows structures of all sizes, so it can represent objects at an arbitrary distance from the boundary.

You wouldn't normally try to explain quantum mechanics using the holographic principle. The quantum framework is presupposed both by the field theory and the string theory. But what if you tried expressing both theories using Bohm's equations? Yes, it's artificial, but it's also mathematically well-defined, and it's something that no-one has done. What would the relationship be, between the quantum potential on the boundary, and the quantum potential in the bulk? It is an orthodox fact about the holographic principle that locality on the boundary and locality in the bulk are not directly related. Could a purely local interaction in a classical boundary field theory turn into a nonlocal interaction in its holographic image? In that case, could you understand quantum randomness in the bulk theory as arising from the holographic transformation of a local interaction on the boundary? It's an amazing idea and I think it's a new one.

I'm sure this isn't what you were thinking when you posted :-) but it was inspired by the unusual connections you were making.

dhillonv10

May25-11, 12:09 AM

no that was not what i was thinking but the last few lines are exactly what i tried to say, I need a bit more time to look up the stuff you've mentioned before I can reply back with how this idea is evolving :)

- V

atyy

May25-11, 12:35 AM

Without holography, there's conventional speculation about quantum mechanics and thermalization eg. http://arxiv.org/abs/1007.3957 , which also gives a nice overview of the literature in its introduction.

dhillonv10

May25-11, 09:44 PM

thanks for the link atyy.

mitchell: if its okay with you, could we possibly move to a private conversation? perhaps PM? I wouldn't PM anyone without their permission, that's simply rude, if not than that's cool too, i'll just post here.

mitchell porter

May26-11, 05:50 AM

It can't be rude to PM someone; they can always ignore you! Anyway, PM me if you wish.

I'll just add that I found some papers relating AdS/CFT to "stochastic quantization" (this paper (http://arxiv.org/abs/0912.2105), also its ref #1). That's the closest thing to a holographic explanation of quantum theory that I've seen.

dhillonv10

May26-11, 02:45 PM

Also I found another paper describing a lot of what I'll need http://arxiv.org/abs/1103.3427

dhillonv10

May27-11, 07:57 PM

mitchell, I send you a message in PM if you get time please reply back to that, also with the description of local interactions turning into non local ones, i think there's another way to approach the idea through the use of the paper i posted a link to, that paper describes the whole process in terms of PEPS and here's the abstract:

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models, like a deformed AKLT [1], an Ising-type [2], and Kitaev's toric code [3], both in finite ladders and infinite square lattices. In the latter case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield non-local Hamiltonians. As our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.

Seems like someone else has already done a lot of the work required, now from what I understand of this, the paper proposes a method to relate interactions (at least entaglement) on the bulk to boundary. Although we need to generalize the following:

- This method is proposed in lattice theory, does it really matter what framework is being used, our concern is holography.

- The holographic framework being proposed, see how quantum randomness arises from that, going back to your original idea

- Is entanglement enough as a type of interaction? Explore how to generalize to all types of interactions, for that the Hamiltonian would have to be modified.

- Explore other implications.

atyy

May28-11, 12:04 AM

I think the holographic principle mentioned in the OP was AdS/CFT.

However, there are other examples of holography, such as the above mentioned paper by Cirac et al. Some discussions of how the various types of holography may be related are found in Gukov et al's http://arxiv.org/abs/hep-th/0403225 and Swingle's http://arxiv.org/abs/0905.1317 .

dhillonv10

May28-11, 12:53 AM

thank you very much for the link atty, after the comments by mitchell the idea that i put up in the OP has evolved, I am trying to focus on what was brought up:

Could a purely local interaction in a classical boundary field theory turn into a nonlocal interaction in its holographic image? In that case, could you understand quantum randomness in the bulk theory as arising from the holographic transformation of a local interaction on the boundary?

mitchell porter

May28-11, 10:02 PM

1) Despite the authors' terminology, the paper on PEPS is not talking about the holographic principle. The holographic principle is about an equivalence between a higher-dimensional theory and a lower-dimensional theory, but their mapping is not an equivalence. You cannot reconstruct the whole of the bulk theory from their boundary theory (see the remark on page 3 about their mapping not being "injective" (http://en.wikipedia.org/wiki/Injective_function)). At best, they are describing some more general phenomenon which might correspond to holography in those cases where the mapping is invertible.

2) The holographic principle relates a quantum theory in the bulk space to another quantum theory on the boundary of that space. That is, in holography as conventionally conceived, you will have quantum uncertainty on both sides of the relationship. The only difference is that in the boundary theory, the quantum uncertainty is defined on top of a fixed space, but the bulk theory contains quantum gravity, so the metric and topology fluctuate as well. It's this extension of quantum uncertainty to space and time themselves which Craig Hogan hopes to detect, as "holographic noise".

So be aware that trying to derive quantum mechanics itself from the holographic relationship is a highly unusual idea, and possibly a wrong idea. Just to say it again: in holographic dualities as they are actually studied, there is already quantum randomness on the boundary, which extends to include space-time itself in the bulk theory on the other side of the equivalence. The innovative concept under discussion in this thread, as I understand it, is that we could start without quantum randomness on the boundary, and still end up with the appearance of quantum randomness in the bulk, because of the "local-to-nonlocal" aspect of the holographic mapping.

3) The origins of the holographic principle lie in the study of black holes. Without gravity, the number of possible states in a field theory will increase with energy as a function of spatial volume. But if one of your fields is gravity, then black holes will form at high densities, and the entropy (and therefore the number of possible states) of a black hole is a function of area (the area of the event horizon), not of volume. So a field theory containing gravity has to behave like an ordinary field theory in a space of one less dimension. That is the qualitative statement of the holographic principle, as proposed by Gerard 't Hooft (http://arxiv.org/abs/gr-qc/9310026) and Leonard Susskind (http://arxiv.org/abs/hep-th/9409089).

It is a very interesting fact that 't Hooft is trying to explain quantum mechanics itself, and not just quantum gravity, using cellular automaton models in one less dimension. So I was wrong to say that the idea of a holographic explanation of quantum mechanics is entirely new; Gerard 't Hooft, a Nobel Prize winner and one of the many parents of the standard model of particle physics, is an exponent of this idea! But it should be understood that this is 't Hooft in the "later Einstein" phase of his career. Einstein spent the last decades of his life working on unified field theories and away from the mainstream of theoretical and experimental physics. With respect to the holographic principle, the 1997 discovery of AdS/CFT by Juan Maldacena (http://arxiv.org/abs/hep-th/9711200) was a new stage in the development of the idea, because for the first time there was a quantitative example. Maldacena wasn't just proposing that a quantum gravity theory is equivalent to some unknown field theory; he was saying, string theory on a particular space is equivalent to a particular, already known field theory.

The years since then have involved the intensive study of these two theories, and the identification of many other dual pairs. But 't Hooft's recent work does not involve any of this. He is proceeding alone in a different direction.

So what about the people who are working on the concrete examples of holographic duality unearthed by Maldacena and others? They aren't trying to explain quantum mechanics. They take it as a given, and instead they use the duality between two quantum theories to learn about both. However, in some of the early papers by Arkani-Hamed et al (who I mentioned earlier in the thread), you will occasionally see the idea that maybe their new framework will even explain quantum mechanics. The usual headline for their approach is that they have abandoned manifest locality, and so perhaps they have found a level of description beyond space-time. But another feature of their Grassmannian formalism is that unitarity is not an input either. Unitarity is the feature which, in quantum mechanics, ensures that the probabilities add to one. So to have discovered a formalism in which unitarity doesn't have to be introduced may mean they have found something more fundamental than quantum mechanics. (Let me mention that their formalism involves twistors, and it was precisely Penrose's ambition to explain quantum mechanics as well as to go beyond space-time.)

In our discussion here we're saying, what if quantum mechanics only applies in the bulk, but the boundary is classical? However, in Arkani-Hamed's recent talks, he interprets their work as the discovery of a third framework, neither string theory (bulk) nor field theory (boundary), but something else outside space-time entirely (twistor space - perhaps it is as simple as that - twistors are the answer). And this is the framework where neither space-time locality nor quantum unitarity is "manifest", i.e. visible - you have to switch to the other perspectives to see them.

So if we take a hint from the stories of Einstein, 't Hooft, and Arkani-Hamed, and say that the best guide to the truth is to focus on the concrete quantitative example of holography which we are lucky enough to have (AdS/CFT), rather than just guessing - then that would imply that the genesis of quantum mechanics is to be found, not in the boundary-to-bulk transformation, but in the transformation from the "third theory" into either of the space-time descriptions, boundary or bulk.

atyy

May28-11, 10:09 PM

atyy

May28-11, 10:26 PM

However, in Arkani-Hamed's recent talks, he interprets their work as the discovery of a third framework, neither string theory (bulk) nor field theory (boundary), but something else outside space-time entirely (twistor space - perhaps it is as simple as that - twistors are the answer). And this is the framework where neither space-time locality nor quantum unitarity is "manifest", i.e. visible - you have to switch to the other perspectives to see them.

Couldn't one say also say that unitarity isn't manifest in the Lagrangian description, compared to the Hamiltonian one?

dhillonv10

May28-11, 10:32 PM

So then could it be that the way Arkani-Hamed described it, the boundary-bulk holographic transformaiton that you (mitchell) postulated before are part of a third framework? That is also very interesting, I understand that the paper I posted before perhaps doesn't have what i am looking for, actually introducing a third choice give one more freedom, now using the Grassmannian formalism, one can describe interactions that occur in both the bulk and boundary, this third choice gives you a framework can independently describe both of them or at least that is the general idea. Thanks for your help atty and mitchell.

dhillonv10

May28-11, 11:30 PM

i just found this paper: http://arxiv.org/abs/physics/0611104 that talks about some very similar concepts that we have explored here:

Holographic Principle and Quantum Physics
Zoltan Batiz, Bhag C. Chauhan
(Submitted on 10 Nov 2006)

The concept of holography has lured philosophers of science for decades, and is becoming more and more popular on several fronts of science, e. g. in the physics of black holes. In this paper we try to understand things as if the visible universe were a reading of a low-dimensional hologram generated in hyperspace. We performed the whole process of creating and reading the hologram of a point in virtual space by using computer simulations. We claim that the fuzzieness in quantum mechanics, in statistical physics and thermodynamics is due to the fact that we do not see the real image of the object, but a holographic projection of it. We found that the projection of a point particle is a de Broglie-type wave. This indicates that holography could be the origin of the wave nature of a particle. We have also noted that one cannot stabilize the noise (or fuzzieness) in terms of the integration grid-points of the hologram, it means that one needs to give the grid-points a physical significance. So we further claim that the space is quantized, which supports the basic assumption of quantum gravity. Our study in the paper, although it is more qualitative, yet gives a smoking gun hint of a holographic basis of physical reality.

What do you guys think??

mitchell porter

May31-11, 03:26 AM

Couldn't one say also say that unitarity isn't manifest in the Lagrangian description, compared to the Hamiltonian one?Maybe it's a little less manifest. But in the Lagrangian formalism for QFT, you usually want to prove that the S-matrix, not the time evolution operator, is unitary.

You can see a statement about emergence of unitarity in the Grassmannian formalism here (http://arxiv.org/abs/0907.5418), in the final section. That's from two years ago, there has been a lot of progress, and I don't know how the authors think about it now. I have heard it stated, in recent lectures, that these amplitudes are the volumes of polytopes (in twistor space, not AdS, I think), and that different integrations correspond to different triangulations in which different properties (such as locality) are manifest. So it would be of interest to know whether there are triangulations in which unitarity is non-manifest.

One might also want to compare these discoveries with Penrose's original hopes for twistor theory. The original spin networks were like Feynman diagrams but they didn't involve complex numbers, a property which facilitated an interpretation in terms of traditional probability theory, but they only gave you back angular momentum. I think Penrose hoped that twistor diagrams would be a calculus in which quantum amplitudes in space-time would emerge from a combinatorics governed by classical probability. Isn't there a Euclidean-Euclidean version of AdS/CFT? In which case the boundary should have a Bohmian interpretation, shouldn't it?

But of course this wouldn't be a derivation of QM, since it the Bohmian interpretation is QM.I've seen Euclidean continuations of one or the other side of AdS/CFT (not sure about both), and this may even pertain to the construction of dS/CFT from AdS/CFT. But you can make a Bohmian version of Lorentzian quantum field theory anyway, it's just a matter of gauge fixing: pick a preferred reference frame, use the Schrodinger picture, and then construct the usual Bohmian trajectories from the resulting wavefunctional. You can do the same thing for quantum gravity too - e.g. by reifying the "lapse" and "shift" functions of the ADM formalism. I don't know how you apply Bohmian constructions to Euclidean QFT, however.

mitchell porter

Jun1-11, 04:06 AM

A simple fact about how the boundary-to-bulk mapping works (or indeed the mapping in the other direction) is that it relates points which are spacelike separated. You can see an example for AdS3 in figure 1 here (http://arxiv.org/abs/hep-th/0606141): the set of points on the boundary, relevant to a point in the bulk, is bounded by the past and future light-cones. Apply that principle in the other direction, so that a point on the boundary potentially maps to any bulk point lying between its past and future light cones... if you go here (http://relativity.livingreviews.org/Articles/lrr-2004-1/index.html), look under "figures", and look at figure 1, then it's the set of points in the interior of the cylinder lying between the two tilted lens shapes. The same principle should also apply in de Sitter space, where the holographic surface is either the cosmological horizon or the past/future conformal boundary, depending on which version of dS/CFT you use.

The paper in that first link tries to narrow the set of points on the boundary which contribute to what happens at a point in the bulk. They manage to reduce that set to a boundary region whose size depends on the radial distance from the boundary to the bulk point. This is more like the "shadow on the wall of Plato's cave". If you think of the light source as located at the center of the AdS space, so that a shadow is being cast on the boundary, then the size of the shadow of an object increases, the closer the object moves towards the light. But, a very interesting twist: to achieve this in AdS/CFT, that first paper had to use complexified boundary variables. Reminiscent of twistors! And of course twistor space is used in the neither-bulk-nor-boundary "third theory" of Arkani-Hamed et al.

Returning to the idea that boundary locality might be transformed into bulk nonlocality by a holographic mapping: what needs to be constructed is the inverse of the mappings mentioned above, which start with a point in the bulk and relate it to points on the boundary. For the approach using real variables, one can already see what the result will look like - the second link. But the approach using complex variables might be more challenging - to talk about the region of the bulk affected by a point in the complexified boundary, one may need to complexify the bulk as well.

edit: See figure 1 in this paper (http://arxiv.org/abs/0710.4334) for the de Sitter case. And they're saying that the de Sitter case, for real variables ... which ought to be the version of holography relevant for the real world ... is related to the anti de Sitter case, for complex variables! - the "complexified boundary" mentioned above. I'll write more when I understand it.

edit #2: Also see page 7 here (http://arxiv.org/abs/hep-th/0312282) for a boundary-to-bulk map for de Sitter space.

dhillonv10

Jun7-11, 01:48 PM

thanks for the information mitchell, I will read those papers thoroughly, I also found another very interesting paper: http://arxiv.org/abs/gr-qc/0609128 which describes the following: (pg 3)

Quantum state changes caused by local interactions between mass quanta have non-local consequences throughout the universe. Any local change in the quantum state of the mass distribution within the universe is instantly reﬂected in changes in the eigensolutions of the Helmholtz wave equation within the universe, as well as in the lattice gas representation of the universe on an observer’s de Sitter horizon. This instantly changes both the probability distribution of the bits of information on the horizon and the corresponding probability distribution of mass quanta throughout the universe. In this way, the holographic principle indicates a mechanism for non-locality in quantum processes throughout the universe.

Here we see a similar process being defined in new terms. So perhaps this approach can be extended to include bulk to boundary transformations through the use of complex variables as mentioned.

dhillonv10

Jun7-11, 02:08 PM

Perhaps another idea that came to me, and this is a little unrelated to our discussion of holographic mapping, but it is a consequence of it, let's assume what we are talking about is indeed correct. Now we know that entanglement can occur over very large areas, what i mean by that is, two particles can (theoretically) be hundreds of miles away and as long as they are isolated a spin measurement on one of them causes the other to take the opposite measure. Now from our mapping we can deduce that the idea of the particles being spatially separated isn't necessarily true, they appear to be far apart in the holographic image and the non-local interaction that occurs here is actually a local interaction occurring on the boundary.

Also what about the time considerations? Could time be affected by such a transformation? The equations of physics are time invariant and we "feel" time through increase in entropy, a concept that Eddington calls the arrow of time. Another speculation, it may be the case that the instantaneous signalling of time is not instantaneous after all, the time is takes for an interaction to occur in the bulk seems instantaneous (this time we are going the other way) but through the holographic mapping, at the boundary it is actually stretched out or more precisely said, the time is dilated as we would experience in Einstein's GR.

qsa

Jun7-11, 02:55 PM

thanks for the information mitchell, I will read those papers thoroughly, I also found another very interesting paper: http://arxiv.org/abs/gr-qc/0609128 which describes the following: (pg 3)

Here we see a similar process being defined in new terms. So perhaps this approach can be extended to include bulk to boundary transformations through the use of complex variables as mentioned.

good find, this is very similar to my idea

http://www.physicsforums.com/showthread.php?t=399053&highlight=susskind post #4,

also,

post #50 http://www.physicsforums.com/showthread.php?p=2615567&highlight=susskind#post2615567

I do that in the most direct way which amounts to an extended Buffon's needle(radon) which is the non-complex Penrose transform.

qsa

Jun7-11, 03:02 PM

dhillonv10, if you want I can PM you some related info with very interesting results.

atyy

Jun7-11, 03:07 PM

Also what about the time considerations? Could time be affected by such a transformation? The equations of physics are time invariant and we "feel" time through increase in entropy, a concept that Eddington calls the arrow of time. Another speculation, it may be the case that the instantaneous signalling of time is not instantaneous after all, the time is takes for an interaction to occur in the bulk seems instantaneous (this time we are going the other way) but through the holographic mapping, at the boundary it is actually stretched out or more precisely said, the time is dilated as we would experience in Einstein's GR.

There is some discussion in this article by Hubeny and Rangamani about non-locality when in non-equilibrium situations http://arxiv.org/abs/1006.3675 (bottom of p6).

dhillonv10

Jun7-11, 03:58 PM

qsa: please do, I would very much appreciate that. Also thanks for the links =)

atyy: that was a good find, they are discussing some ideas very similar to what we were speculating. Although they seem to be approaching this through the use of guage/gravity duality. Nonetheless thanks for the link.

atyy

Jun7-11, 04:54 PM

atyy: that was a good find, they are discussing some ideas very similar to what we were speculating. Although they seem to be approaching this through the use of guage/gravity duality. Nonetheless thanks for the link.

I thought you were talking about holography in AdS/CFT?

dhillonv10

Jun7-11, 07:47 PM

Yes indeed, the original intent was to see the ADS/CFT type holography however i recently found an impressive paper: http://arxiv.org/abs/gr-qc/0609128 that finishes half the equation so to speak, i have to work out if the inverse of what is mentioned in that paper holds and I would be able to generalize the statement. I would probably working on this approach for a couple of days however if it meets a dead-end, i would have to take the older twistor-theory approach and the paper you mentioned would compliment that line of thought.

dhillonv10

Jun8-11, 08:30 PM

I found another paper that goes along the lines of what mitchell mentioned before:

QCD/String holographic mapping and glueball mass spectrum: http://arxiv.org/abs/hep-th/0209080

Abstract
We ﬁnd a one-to-one mapping between low-energy string dilaton states in AdS bulk and high-energy glueball states on the corresponding boundary. This holographic mapping leads to a relation between bulk and boundary scattering amplitudes. From this relation and the dilaton action we ﬁnd the appropriate momentum scaling for high-energy QCD amplitudes at ﬁxed angles.

Even though this may have started off as a crackpot idea, I think we are moving in a good direction.

dhillonv10

Jun8-11, 09:16 PM

Another update, in the paper mentioned in #23, could this be the mapping we are looking for: (pg 2, 1st column, near the bottom)

"So, a quantum description of the total amount of information available about the universe can be obtained by identifying each area (pixel) of size 4δ 2 ln 2 on the de Sitter horizon with one bit of information, associated with the wavefunction for a quantum of mass \frac{h}{c} \sqrt{\frac{\Lambda}{3}} with Compton wavelength 2πRF"

mitchell porter

Jun8-11, 10:24 PM

That glueball paper is using AdS/CFT in a completely standard way - the same way that 99% of the other thousands of AdS/CFT papers use the correspondence - as a way to relate a quantum field theory on the boundary to a quantum string theory in the bulk. There's no attempt to explain quantum mechanics itself in this paper.

If you look further into the mainstream use of AdS/CFT, you will often see talk of a correspondence between quantum field theory on the boundary and classical supergravity (or a classical string, or a classical string field) in the bulk. But that is just using the classical limit of the bulk quantum theory, it's still conceived as ultimately being a quantum-quantum correspondence.

I should say something about the role of anti de Sitter space in AdS/CFT and gauge/gravity duality. AdS space is hyperbolic (negatively curved). As such, it does not resemble the space we actually see. Its role in AdS/CFT might be approached on three levels.

First, AdS space has the property that it has ordinary "flat" space (Minkowski space) as its boundary at infinity, so it's mathematically suited as a space in which gravity duals of ordinary field theories in flat space can be examined.

Second, it approximates the geometry occurring in some beyond-standard-model theories which are meant to be realistic, such as braneworlds, the Randall-Sundrum model, and warped compactifications. In string theory, AdS space shows up when you have a stack of branes with an energy density high enough to be a black hole (or really, a "black brane"). The flat space outside the event horizon becomes completely separated from the hyperbolic space at the bottom of the gravitational well, which is then occupied by the open string fields living on the black brane stack. In the more realistic models, you have a partial decoupling between what happens on the brane and what happens in the bulk, but it's not a total decoupling as in AdS/CFT; the braneworld interacts with the bulk and vice versa.

Third, AdS space turns out to be a natural way to describe the dependence of a field theory's behavior on energy scale. This is still somewhat new and mysterious, but PF user Physics Monkey works in this area, so maybe he can say something. But there's a way to get an intuitive picture here. Suppose we focus on AdS3, the three-dimensional form of AdS space. It's like a cylinder: AdS is the interior of the cylinder, the boundary is the surface of the cylinder, time is the direction along the cylinder, space is the direction around the cylinder, and the radial direction (from the surface into the interior) is the extra holographic direction. So the boundary theory has one time dimension and one space dimension - it lives on a circle - and the bulk theory has one time dimension and two space dimensions - it lives on a disk, the interior of the circle, the cross-section of the cylinder.

I mentioned earlier the idea of placing a light at the center of AdS space (here, the center of the disk) and considering the shadow of an object that moves towards the center. The shadow on the boundary gets bigger. The inverse of this perspective is to think about events happening on the boundary (on the circle, the perimeter of the disk). There might be solitonic waves of different sizes, traveling around the rim. The size of a wave is like the size of a shadow; the longer the wave, the deeper its holographic information reaches into the bulk. A wave that is really really small corresponds to events in the bulk which are only a short distance away from the boundary, but a wave which wraps most of the way around the circle will map to points which are very close to the center.

The same thing applies to higher-dimensional situations. So consider our three space dimensions. Processes occurring in this space usually have a typical length scale - some physical processes occur at 10^-18 meters, others at 1 meter, others at 10^20 meters. If we were mapping events into a fourth, "AdS" spatial dimension, processes that are small would only be a short distance away from the boundary, processes that are really spread out would be much further away from the boundary, and processes that are completely spread out would be at the AdS "center". Note that I mean quantum processes, where there's quantum coherence or quantum entanglement existing on those scales.

So much for AdS/CFT. Then there's the attempt to apply holography to the space that we find ourselves living in cosmologically. This would be "dS/CFT", holography in de Sitter space. Compared to AdS/CFT, dS/CFT is in a very primitive state. AdS/CFT has hundreds of examples, in which a particular field theory is dual to a particular string theory or theory with gravity, which people study in great detail and use for calculations. dS/CFT doesn't have a single working example at the same level, just various approximations and guesses. Physicists can't even agree on whether the holographic boundary should be the observer-dependent "cosmological horizon" or the unobservable "boundary at infinity".

The paper by T.R. Mongan (gr-qc/0609128) is one of these guesswork papers. Without having an equation for the boundary theory or the bulk theory, the author is nonetheless trying to deduce consequences from a few principles and hypotheses, such as "there's 1 bit of information for every 4 ln 2 planck units of surface area on the cosmological horizon". Even the best physicists reason in this risky way from time to time, when they have no alternative (e.g. when they are thinking about something for which no established theory exists), and sometimes they succeed and sometimes they fail. But I think Mongan is approaching the subject in a somewhat dubious way. I say this without having worked my way through his reasoning, and without being a top physicist myself.

However, I can usually get a sense of the level of sophistication or depth of understanding at work in such arguments, and Mongan's argument doesn't go very "deep". He's basically just combining one relationship that he's been told - the holographic relationship between surface area and number of bits - with another relationship that he's been told - the quantum relationship between wavelength and mass - and deducing that for each holographic bit, there's a "mass quantum" of a certain size. It's too simplistic. In the case of AdS/CFT, the work of Vidal, Swingle, and others indicates that distance in the holographic direction (towards the center of the AdS space) is associated with entanglement length scales on the boundary, but Mongan's boundary pixels aren't entangled, or at least, he doesn't address this issue. I suppose that if we are trying to derive quantum nonlocality in the bulk from a holographic transformation of classical locality on the boundary, we shouldn't have entanglement on the boundary, only in the bulk.

Anyway, I simply can't see what Mongan's mapping is. He says the pixels of area provide boundary conditions for the wavefunctions of his "mass quanta", but how, exactly?

dhillonv10

Jun8-11, 11:52 PM

I just finished reading the paper by T.R. Mongan and i came to similar conclusions, that paper doesn't define any sort of mapping, although I would like to state an observation I made from his work to another paper:

The wavefunctions on the horizon are the boundary condition on the form of the wavefunctions specifying the probability distributions of the ﬁnite number of mass quanta distributed throughout the featureless background space of a closed universe with a constant vacuum energy density (cosmological constant). The wavefunction for the probability of ﬁnding a mass quantum anywhere in the universe is a solution to the Helmholtz wave equation in the closed universe.

Now we see here that he poses some vague relationship between the wavefunction on the horizon (the boundary) to the probability distribution of mass quanta distributed throughout the universe, in Lee Smolin's paper on the real ensemble framework (arXiv:1104.2822) the members of the ensemble are spread out throughout the universe and there is a nonlocal interaction between the members through which they can copy another system's beables. Now Lee claims (in his talk to the quantum foundation) that there is a deeper theory that already knows how the members of the ensemble are spread out and so on, however from this quote above, I sense some relation between the wavefunction on the boundary and the probability distribution of the members of an ensemble, perhaps the wavefunction like mitchell stated:

The inverse of this perspective is to think about events happening on the boundary (on the circle, the perimeter of the disk). There might be solitonic waves of different sizes, traveling around the rim. The size of a wave is like the size of a shadow; the longer the wave, the deeper its holographic information reaches into the bulk. A wave that is really really small corresponds to events in the bulk which are only a short distance away from the boundary, but a wave which wraps most of the way around the circle will map to points which are very close to the center.

Let's not think of that for a second as applicable to the AdS spacetime, but instead as something that happens on the dS boundary, then we can speculate that the elements of the ensemble that are far away are there as a product of their "shadow" and on the boundary, as the bigger wave interacts with the smaller wave, we see the result as copy-dynamics. I will admit this last part doesn't make much sense since I am attributing a phenomenon that would otherwise occur on AdS to occur on our boundary, but there maybe a link there.

Currently however, its time to fall back to mitchell's approach and study complex variables.

dhillonv10

Jun9-11, 12:17 AM

Just a clarification mitchell, in #22 you mention the following:

edit: See figure 1 in this paper for the de Sitter case. And they're saying that the de Sitter case, for real variables ... which ought to be the version of holography relevant for the real world ... is related to the anti de Sitter case, for complex variables! - the "complexified boundary" mentioned above. I'll write more when I understand it.

Referring to this paper: http://arxiv.org/abs/0710.4334

Now if I understand correctly, is it the case that if the inverse mapping that you postulated can be done using complex variables, then the same can be applied to the de Sitter space? Thanks.

edit: I found another paper talking about what mitchell mentioned before: Bulk versus boundary quantum states, arXiv:hep-th/0106108

This paper although not in the way we need it, states the following relationship:

This reduces the dimensionality of the bulk space of states and makes it possible to ﬁnd a one to one mapping into the boundary states.

mitchell porter

Jun9-11, 02:08 AM

Well, let's just look more closely at the mapping from that paper, a mapping from a region on the boundary to a point in the bulk. Mathematically, it's defined on page 3, equation 2.4. Also see slide 10 from this talk (http://www.phys.vt.edu/~sowers/talks/kabat.pdf).

In both references, the equation describes the field at a point in anti de Sitter space, but the picture is of a point in de Sitter space. In the picture of de Sitter space, time is in the vertical direction, so it's saying that the boundary is in the past. The "smearing", which defines the region on the boundary which maps to the point, is in two space directions on the boundary (I'm referring to the circle at the base of the light cone in the picture).

In the equation, which is for anti de Sitter space, we are defining "phi" in the bulk in terms of "phi0" on the boundary. You'll see that phi depends on three coordinates, T, bold X, and Z; but phi0 just depends on T and X. T is time (on the boundary or in the bulk), bold X is a vector (which is why it's printed in bold) and represents the space coordinates on the boundary, and Z is the extra space dimension in the bulk. So we're constructing something at a point in the bulk (with coordinates T, X, and Z) as an integral over a region on the boundary (ranging over values of T and X).

Now look at the phi0 term that we are integrating over. You'll see that time ranges over T+T', while space ranges over X+iY'. Both T' and Y' are real numbers that range over positive and negative values, and represent points on the boundary away from (T,X,0) - that is, away from the (T,X) point on the boundary where the bulk coordinate Z is just 0. So when T' is negative, it's back in time, when T' is positive, it's forward in time. But we're adding iY' to X; the boundary coordinates are supposed to become complex numbers. What does that mean?

In calculus, it's very common, when solving an equation for real numbers, to switch to complex numbers first, where it is often easier to solve, and then to later return just to real numbers. But here there is actually a physical meaning too.

In relativity, there's a formula for the length of the "space-time interval" (http://en.wikipedia.org/wiki/Spacetime#Basic_concepts) between two points. The square of the length is negative for timelike separation, positive for spacelike separation, and zero for lightlike separation. "Timelike separation" means that one point is definitely (causally) in the future of the other point. Because of the way that space and time change in relativity, when viewed from different reference frames, sometimes A can be in the future of B, in one frame, but in its past in another frame. So you can't just use the time coordinate to determine the ordering of events. If A is, not just in the future of B according to the coordinate system, but also close enough to B in space that it is in the "future light cone" of B, then it is definitely in the future of B, there's no physically valid coordinate change which will put it into the past. Timelike separation refers to this relationship of definitely being in the future. Something that is spacelike separated, you could think of as "quasi-simultaneous". There will be coordinate systems where it's in the future, others where it's in the past, but it's always too far away for a causal connection at the speed of light.

These are the basics of special relativity. Now notice, just as a fact of algebra, that if you could somehow have something which was an imaginary-number amount of time into the future, the square of the spacetime interval would now be positive, even though timelike is supposed to be negative, because the i factor in the interval length would produce an extra factor of -1 in the square of the length.

It is also a fact that de Sitter space and anti de Sitter space are closely connected geometrically. I wrote a little about it here (http://www.physicsforums.com/showthread.php?p=3222927). You can get both spaces from the same geometric object, but choosing a different direction for time.

So, returning to the paper: their region-on-the-boundary-to-point-in-the-bulk map for anti de Sitter space is an analytic continuation of the region-on-the-past-boundary-to-point-in-the-present-bulk map for de Sitter space. More precisely, they use a formula which makes sense in de Sitter space, because all the space and time distances in the formula are specified in real numbers, and then they transform that into a formula which looks like the anti de Sitter formula by making some quantities imaginary. So the resulting formula is a peculiar intermediate thing: it's motivated by how de Sitter space works, but it's applied in anti de Sitter space, but it relies on treating coordinates in anti de Sitter space as if they were complex numbers rather than real numbers. (Among other things, that would double the number of real dimensions, because now you have x+yi wherever you previously just had an x.)

Most physicists are rather unconcerned about formal manipulations like this, because they are just intermediate steps in a larger calculation. For example, you might be computing the probabilities of various outcomes of a particle collision in which the motion of the incoming particles are specified by momentum vectors. Those probabilities will be complicated functions of the momentum vectors. It is an utterly routine thing for such functions to be computed by treating the momentum vectors as vectors of complex numbers, and then later restricting back to real numbers in some way. The same thing happens here with the "complexified boundary" coordinates. We are actually talking about fields whose value varies across space and time in a way that depends on space-time coordinates, so using complex-valued space-time coordinates really means, using complex numbers as an input to the function which defines how the field varies with space and time. In this case, we are then figuring out something about the value of the field at a point in the bulk - a point whose coordinates are definitely just real numbers - by an integration over the behavior of the field on the complexified boundary.

According to the usual pragmatic philosophy, we don't care too much about the implicit doubling of dimensions on the boundary that this involves, because that's just an intermediate step. What we start out with is a specification of how the field behaves on the boundary, we mysteriously extend that specification to "the way the field would behave if the boundary coordinates were complex numbers rather than real numbers", we perform a big integral, and since we get an answer which once again involves just real-valued coordinates, we don't have to worry about whether the complex-valued space-time coordinates correspond to something real.

But if we want to invert this mapping, we're trying to go from a region in the bulk to a point on the boundary. Therefore, we either have to go back to the uncomplexified boundary, or we have to start taking the complexified boundary literally.

Earlier in this thread I mentioned Roger Penrose's twistors. They also derive from a complexification of space-time coordinates. But Penrose, at least, wanted to consider them as a fundamental theory (most of the people now using twistors regard them just as a mathematical tool). So maybe, if we want to map bulk nonlocality to boundary locality, we have to use twistors somehow. And there's the fact that the "third theory" of Arkani-Hamed et al, which adds a third description to the bulk/boundary duality, is expressed in terms of twistors. The only problem is, I don't think twistors look local in terms of the boundary either (at least, not in terms of the usual boundary, with real-only coordinates). So if we're chasing after the origins of quantum mechanics itself, this may be an indication that seeking classical locality on the boundary is not the answer - that the boundary will remain quantum. And after all, that's how it is in the orthodox use of AdS/CFT.

mitchell porter

Jun9-11, 02:20 AM

However, having said all that, let me mention one thought arising from your comments on Mongan's paper. I complained that I didn't see how Mongan wanted his boundary to provide boundary conditions to quantum wavefunctions in the bulk. But the stipulation that the boundary should be local in the classical sense - that you can describe it reductionistically, in terms of states confined to the "pixels of surface area" - could perhaps be expressed in quantum terms, as a requirement that the wavefunction on the boundary can be factorized into local states. In other words, no entanglement, it's just a tensor product of wavefunctions on the "pixels". If you could define a Schrodinger equation for evolution of a bulk wavefunction, whose restriction to the boundary remained factorized in this way, maybe you'd be getting somewhere; but I don't think this would resemble the concrete examples of gauge/gravity holography that have been discovered so far, because the dynamics of the boundary theory should produce entanglement on the boundary in all such cases.

Smolin's copy dynamics has the problem, which he mentions in his section VI, that there's no rule governing the dynamics of a hierarchy of composite systems. A molecule contains an atom contains a proton: does the proton copy its state from another proton, or does the whole atom copy its state from another whole atom, overriding the proton's copy dynamics? To understand how the holographic dynamics of nested systems works, it might be better to obtain guidance from a worked example in AdS/CFT, if we can find one.

edit: I saidI don't think this would resemble the concrete examples of gauge/gravity holography that have been discovered so far, because the dynamics of the boundary theory should produce entanglement on the boundary in all such cases.But if we suppose that the version of dS/CFT that is relevant for the real world involves past and future boundaries, maybe this doesn't matter, because the boundary theory has no dynamics! In this version of dS/CFT, the time direction is the bulk - space-time holographically emerges from a boundary which is purely spatial.

However, we may end up finding out that, even though the boundary here has no dynamics, it still requires an entangled quantum state on the boundary to give rise holographically to quantum dynamics over time in the bulk.

dhillonv10

Jun9-11, 04:06 PM

mitchell, thanks for that post, I haven't studied the local nature of twistors for the boundary before so I'll look around, study and then come back to address the concerns you mentioned, and it may be as you said, the boundary remains quantum.

Another observation, this one however may not be that relevant, I just saw another paper: Constructing local bulk observables in interacting AdS/CFT (I think this was mentioned here before, the problem again is that it is in AdS)

6.2 Bulk Feynman diagrams

In this section we show how the Feynman diagrams associated with a local theory in the bulk can be mapped over to CFT calculations. This will provide yet another way of deriving the CFT operators which are dual to local bulk observables.

Although in the mean time, regarding this paper: A new twist on dS/CFT, (arXiv) (http://arxiv.org/abs/hep-th/0312282) I emailed one of the authors to ask about the point brought up at pg. 7

edit #2: Also see page 7 here for a boundary-to-bulk map for de Sitter space.

Here's the whole email:

Vikram Dhillon, <Thu, Jun 9, 2011 at 1:30 AM>
To: lowe@brown.edu

Hi Prof. Lowe,

I recently came across your paper on A new twist on dS/CFT and on page 7
you mention a mapping from boundary to bulk by promoting the modes on
the circle to the modes on the de Sitter. I have a question about that,
is it possible to formulate an inverse of this mapping? Can the inverse
of this mapping be written down where we have a function mapping
bulk-to-boundary in dS/CFT? Thanks for your time.

- Vikram

David Lowe <david_lowe@brown.edu> Thu, Jun 9, 2011 at 4:36 PM
To: Vikram Dhillon

Yes, the inverse map is easier -- you just look at the asymptotics of
the bulk mode near infinity (I think we had in mind past infinity),
and extract the appropriate coefficient of the time dependent piece.
If the bulk mode is a positive frequency mode with respect to the
Euclidean vacuum, this time dependent factor should be uniquely
defined.

I don't fully understand what he is stating in the email, mostly because I haven't read that paper but this email shows that it is possible to construct an inverse mapping. This case isn't similar to what we were discussing before mostly because in the other paper they focused on an AdS space and then mentioned the local buk operators, this new approach, if it works, is more direct now that we are focusing on the dS space.

Now my question with this post is that, say for instance, we are able to construct a mapping from bulk-to-boundary, then how would that show the holographic transformation of boundary locality to bulk nonlocality? and that also raises the concern if the boundary is local at all.

dhillonv10

Jun9-11, 10:45 PM

Actually I think I can somewhat answer my question, please correct me where needed, ds/CFT correspondence implies the duality between the bulk and boundary, the mapping will show, when two points, that are far away interact, that interaction (through the use of the constructed mapping) is actually occurring between those points locally.

mitchell porter

Jun10-11, 03:49 AM

Here we are talking about a free (non-interacting) scalar (http://en.wikipedia.org/wiki/Scalar_field) field on two-dimensional de Sitter space-time. From this picture (page 5) (http://arxiv.org/abs/astro-ph/0307311), you can see that space in dS2 is just a circle (time runs vertically, up the hyperboloid surface in the diagram).

In Fourier analysis, you can express an arbitrary oscillating curve as a weighted sum of periodic curves, the Fourier modes. You can do the same thing on a circle (http://www.flickr.com/photos/ethanhein/2680541012/). So classically, the behavior of this free field just consists of the waves in each of its component modes, moving around the circle. On the diagram, if you followed just one peak in one mode, you would see it trying to trace out a spiral up the diagram (movement around the circle in space translates to movement in an upward spiral on the space-time diagram); but the accelerating expansion of dS2 (represented by the vertical spreading out of the hyperboloid) would outrun it, so that it never got any further than halfway around the circle. Since we are talking about a quantum field, we also have to talk in terms of probabilities, but since it's a free field, the probabilities for each mode are independent, so it's not too complicated (compared to interacting fields).

If you look at the bottom of the picture of dS2, you'll see space is a circle (horizontal cross-section of the hyperboloid), just as it is at every other time in this coordinate system, infinitely far into the past or the future. So the dS2 space-time theoretically extends infinitely far into the past, and this allows us to define a "circle at time = -infinity". This is the "past infinity" to which David Lowe refers. We can also extrapolate the behavior of the field modes endlessly back in time - this is their asymptotic behavior at past infinity. For example, if the activity in one field mode just subsides to zero, it asymptotes to zero. But if the mode just oscillates endlessly as you extrapolate it back, it doesn't converge to anything. However, you may still be able to say something about its asymptotic behavior - for example, that the size of the oscillations approaches a constant. This is the time-dependent piece of the asymptotic behavior.

For the final detail, we have to remember we are talking about quantum field modes, so we are talking about probabilities (which may be expressed in terms of "correlation functions"). So really we're extrapolating the quantum correlation functions for the scalar field modes endlessly back in time, and this gives us correlation functions for the scalar field "at past infinity". Here is where the holographic magic happens: we re-express the correlation functions at past infinity in terms of correlation functions for "conformal primary operators" on the circle at infinity, and then we discover that these conformal operators also gives us a language for talking about dS2 correlation functions at any point in the history of dS2, not just at past infinity. These operators are built from "conformal fields" which are defined to exist only on the circle at past infinity, but which allows us to extrapolate the behavior of the scalar field at any time and place in space-time (in the diagram, that's any time and place on the hyperboloid surface).

To extend this construction to higher dimensions, the boundary at past infinity would be a sphere or a hypersphere (e.g. the past boundary of dS3 would be the surface of an ordinary sphere, the past boundary of dS4 would be "S^3", a hypersphere), and we would start with Fourier modes in multidimensional space, not just on a circle. Also, it needs to be done for other types of field (spinor, vector, tensor) and for interactions between fields.

mitchell porter

Jun10-11, 04:15 AM

This is about as simple a prototype of dS/CFT as we are likely to find, so we should try to understand it in detail. One thing to understand is that the mapping between modes on de Sitter and "modes on the circle" is only halfway to the full holographic correspondence; it's just the extrapolation of the bulk field's behavior back to past infinity. The real heart of the correspondence is the re-expression of the bulk correlation functions at past infinity, in terms of CFT operators. The CFT is the boundary theory, a completely different set of fields which nonetheless implicitly contain all the information about how the fields in the future "bulk" will behave.

With respect to locality and nonlocality, a description in terms of Fourier coefficients is about as nonlocal as you can get: instead of stating the value of the scalar field at a particular point on the circle, instead you state the strength of all the different modes stretching around the circle - and if you do the resulting Fourier sum at that point, you get back the field strength at that point. But the correlation functions on the boundary can easily be re-expressed in terms of position rather than mode strength - see the sentence under equation 13 in "A new twist on dS/CFT", which refers to "delta(theta'-theta) in coordinate space". Delta functions like that equal 1 if the two variables are the same, and equal 0 otherwise, so what that seems to be saying is that nothing at past infinity can move (probability for propagation equals zero, if the particle has to move from one location on the circle, theta, to a different location, theta') - which makes some sense if you think about the nature of de Sitter space; space itself expands so quickly that every particle eventually gets turned into an island, unable to reach its neighbors. Asymptotically (at infinite time), any surviving matter is stuck in its own patch of space, which will shrink to a point on the circle at infinity.

Since there's no time in the CFT here, it seems like we will just start with entanglement of the conformal fields around the circle (or across the (hyper)sphere, for higher-dimensional dS), and then, with the holographic emergence of a time dimension, that entanglement at past infinity will be turned into temporary correlations, and thus temporary opportunities for interaction, during the bulk lifetime of the universe.

dhillonv10

Jun10-11, 04:26 PM

qsa

Jun10-11, 08:07 PM

So then studying the ds/CFT paper is probably a good idea, I think I will be doing that for the next few days. Originally I had in mind to go study twister theory and its implication that you provided but now in the light of these new developments, mitchell is it a good idea to spend time on twisters? The boundary-to-bulk mapping that is provided in that paper is derived from another paper so I'll probably start there and then come back to this one.

I am not sure if this will help but this paper is FQXI contest paper that did not win but I like it because it is close to my idea.but my guess is that the use of time in modelling is the reason for all the problems, that is why it does not appear in mine naturally.

http://www.fqxi.org/community/forum/topic/950

I am somewhat disappointed in their winners, but I find Zenils paper is also good and he won third prize. also quantum graphity which I gather you like also won second prize.

http://www.fqxi.org/community/essay/winners/2011.1

dhillonv10

Jun10-11, 08:47 PM

thanks for the links qsa, currently i've given up on the quantum graphity approach to this problem and i'm studying that ds/CFT paper. I'll look at the first paper more closely, that one appears to have something nice, atleast it addresses some of the things i'm interested in.

edit: that first paper actually describes precisely something i was speculating earlier, the reason why entanglement occurs instantaneously when the universe is supposed to follow a speed limit for light, but for all i know following VSL i could be wrong :)

mitchell porter

Jun10-11, 11:32 PM

Herman Verlinde (Erik Verlinde's twin brother, also a physicist) gave a talk a few months ago called "Twistors and De Sitter Holography". No details are available, so it must be work in progress. (He coauthored another paper about twistors but it's not about dS/CFT.)

I can't say what order you should investigate these topics. They are all advanced, they are all connected, and they all depend on a lot of simpler ideas in mathematics and physics.

dhillonv10

Jun10-11, 11:43 PM

I see, well since the dS/CFT paper is readily available, I'm gonna start with that and the correlation functions. By the time I finish that, the twistors and de Sitter holography talk might be around.

mitchell porter

Jun11-11, 12:30 AM

I meant to add another indication of how the development of dS/CFT lags the development of AdS/CFT.

In examples of AdS/CFT, we have a precise definition of the field theory on the boundary, and a precise or semi-precise definition of the gravity theory in the bulk. For example, consider the original example, d=4 N=4 Yang-Mills (which is the CFT in this example) dual to Type IIB superstring on AdS5 x S^5. I'll use hep-th/0201253 (http://arxiv.org/abs/hep-th/0201253) as a reference. On the boundary side, we know exactly what the fields are and how they interact (page 16). In the bulk, we at least have approximate equations of motion for the Type IIB string (page 29) and we can specify the space it is moving through (pages 43-45). And then we have a mapping between combinations of the boundary fields and states of the string - outline on page 49, some details on page 50.

In the first column on page 50, you will see many expressions of the form "tr ABC". A, B, C are fields from the boundary theory, ABC is their product, tr ABC is the trace (http://en.wikipedia.org/wiki/Trace_%28linear_algebra%29) of the product. In the third column, you see fields from the bulk. (These are all actually vibrational or other modes of the superstrings in the bulk.) So the holographic correspondence is telling us that, for example, correlations between those bulk fields can actually be calculated from correlators of the corresponding boundary operators (the "tr ABC" products of boundary fields). None of this detail was visible from the beginning, by the way; Maldacena guessed the equivalence of the two theories, on the hypothesis that they are two ways of describing the same "black brane" in string theory, and then people painstakingly confirmed that the boundary operators in the first column have the right properties to match the bulk fields in the third column.

Now what do we have in dS/CFT? Basically, for all proposed examples of dS/CFT, we don't have the CFT - we can't list the boundary fields or say how they interact. (If anyone out there can prove me wrong, please do so.) It's as if, in the first column of the table on page 50, you just had "operator 1, operator 2,... operator 20", but you didn't have any of the "tr ABC" expressions providing the details. All people can do is specify the gross properties of the operators, especially the "conformal dimension", but they're just guessing that a CFT exists, in which there are field operator products with the necessary properties.

If this sounds like it might all be based on an illusion... Maldacena's original (1997) paper contained three examples of AdS/CFT duality. For the first case, he was able to say right away what the boundary theory was (it's N=4 YM, mentioned above). For the second case, it took ten years for the right theory to be found (in the "ABJM" paper - those are the initials of the authors). For the third case, he could specify the boundary theory but the theory in question lacks a tractable definition - people are working on this right now.

So while the lack of a fully realized concrete example of dS/CFT is a serious problem, it doesn't mean that it's an illusory idea, and in fact a lot of ideas and knowledge has been accumulated in the ten years since Andrew Strominger wrote the original dS/CFT paper. It's just that all of those ideas and all that knowledge is still preliminary; people are waiting for the breakthrough, and probably there has to be a conceptual breakthrough, some twist that no-one has thought of yet. For the second example of AdS/CFT in Maldacena's original paper, people were originally trying to employ a different Yang-Mills theory, but John Schwarz suggested that it might be a "Chern-Simons" theory, and eventually ABJM figured it out. For dS/CFT, David Lowe's technical idea for how to make it work, was to represent the geometric symmetries of the bulk differently in the CFT (using "principal series representations"), and also to modify ("deform") the CFT by a new parameter, q, and also to modify the bulk geometry in a way that he didn't quite specify... The last follow-up, to that "new twist" paper from 2003, seems to be 2006, so maybe the idea didn't work, or maybe it's in hibernation.

dhillonv10

Jun11-11, 03:16 PM

Currently reading through the new twist paper, from a preliminary analysis these guys are aiming to reformulate the ds/CFT correspondence by replacing the classical isometry group with this new q version, and introduce the principal series representation. Also one interesting line I found was this:

Such a reformulation of dS/CFT is natural from the bulk point of view, since the quantization of a scalar field on ...

So that in some sense means that this new formulation of the theory, called qdS/CFT, may be what is needed to finish up

The real heart of the correspondence is the re-expression of the bulk correlation functions at past infinity, in terms of CFT operators.

The problem however is that even thought they state some parts of their new theory, they never explicitly mention how is it natural from the bulk's point of view. I think that part will be accomplished by expressing the bulk correlation functions in terms of CFT operators. I'll be finishing up this paper probably before the end of the coming week and in the meantime i'll post any other observations I make.

edit: please clarify this mitchell:

So the holographic correspondence is telling us that, for example, correlations between those bulk fields can actually be calculated from correlators of the corresponding boundary operators (the "tr ABC" products of boundary fields).

In that paper, here's the section that describes what you stated:

5.6 Mapping Type IIB Fields and CFT Operators
Given that we have established that the global symmetry groups on both sides of the
AdS/CFT correspondence coincide, it remains to show that the actual representations of
the supergroup SU(2, 2|4) also coincide on both sides.

So does that imply that there is to some extent a mapping established from bulk to the boundary in terms of AdS space?? If so couldn't that be extended out to dS spacetime using what you said earlier: (using the complex numbers approach)

So, returning to the paper: their region-on-the-boundary-to-point-in-the-bulk map for anti de Sitter space is an analytic continuation of the region-on-the-past-boundary-to-point-in-the-present-bulk map for de Sitter space. More precisely, they use a formula which makes sense in de Sitter space, because all the space and time distances in the formula are specified in real numbers, and then they transform that into a formula which looks like the anti de Sitter formula by making some quantities imaginary.

mitchell porter

Jun11-11, 11:17 PM

So does that imply that there is to some extent a mapping established from bulk to the boundary in terms of AdS space?? If so couldn't that be extended out to dS spacetime using what you said earlier: (using the complex numbers approach)
I made a diagram for reference (see attachment)... I mentioned that you could analyse a holographic mapping, from bulk to boundary, into two stages. First, you go from the interior of the bulk to the edge of the bulk: for example, from a point in the interior to a region on the boundary. Then, you re-express everything in terms of the boundary theory, so that bulk fields become boundary operators.

The analytic continuation applies to the first stage, where you go from the interior to the edge. See my diagram. In AdS3/CFT2, you're going "sideways". The two-dimensional region on the boundary (black disk) has a space direction (around the cylinder) and a time direction (up the cylinder). But in dS3/CFT2, you just go back in time to past infinity, and the black disk is now entirely spacelike. Turning a space direction into a time direction, or the other way around, is where the complex numbers enter; it's called Wick rotation (http://en.wikipedia.org/wiki/Wick_rotation).

In both cases, what the diagram means is that you calculate something to do with the point at the tip of the cone, by summing over all the points in the black disk at the base of the cone. For example, you might be computing a two-point correlation function in the interior of the bulk. Each point would be the tip of a separate cone based on the boundary, and you would be re-expressing the bulk-to-bulk two-point correlation function as a double integral over correlation functions between every point in one black-disk region on the boundary and the other black-disk region on the boundary. And the analytic continuation means that you can express the integral for AdS space in terms of the integral for dS space with complex coordinate values, or vice versa.

OK, great. However, there are two problems. First, this is only the easy part of the true bulk-to-boundary mapping, we don't yet have the change of variables into the boundary CFT. Second, these cones are only localized structures; the global structure of AdS and dS space is different. It's similar to the difference between a Mobius strip and an ordinary untwisted rubber band. If you just look at one section, they look the same, but because of the twist, the Mobius strip can't fit into two dimensions in the way that an untwisted strip could. To globally transform the whole of a particular AdS/CFT correspondence into the whole of a particular dS/CFT correspondence, it would be as if the whole of the AdS boundary was covered in the black disks, and then you transformed the AdS boundary into the dS boundary.

In my diagram, the boundary of AdS3 is the outside of the cylinder, and the boundary of dS3 is supposed to be the surface of a sphere. It is actually possible to map a cylinder onto a sphere - if you make holes at two opposite points on a sphere and stretch them out into circles, and then straighten the sphere. So maybe some combination of this, with the analytic continuation into complex-valued coordinates, could be attempted, for a particular AdS/CFT pairing of theories. The question is, what do you end up with? Because for dS/CFT to work, you need to have much more than just a mapping between points in the bulk and points on the boundary. The field theory on the boundary has symmetries and they have to include the symmetries of the bulk theory. Or, to look at it another way, the boundary space has its own symmetries, and they have to be present in the bulk theory; in your quote, this is what "the supergroup SU(2,2|4)" refers to - the "superconformal" symmetries of the boundary theory.

Superconformal symmetry includes supersymmetry, and supersymmetry is always broken in de Sitter space, so that's already a problem. And in fact problems like these are part of the reason why David Lowe suggested a "qdS/CFT" using a different sort of symmetry representation; he's trying to invent something which is tailored to de Sitter space. So there are at least two possibilities. One is that there are AdS/CFT dualities that can be Wick-rotated to dS/CFT with the whole structure intact. (Maybe they would need to be completely non-supersymmetric AdS/CFT dualities, given that dS/CFT won't allow unbroken supersymmetry.) Another possibility is that dS/CFT is a separate thing from AdS/CFT, and that the algebraic details of AdS/CFT never cross over to dS/CFT.

Those are deep questions, but maybe I can say something to clarify what's happening in the original "analytic continuation to de Sitter space". In effect, they are saying "let's pretend that locally we are in de Sitter space, because the calculation is easier". When they integrate over a boundary region (black disk in the diagram), and perform analytic continuations, they are slipping back and forth between my AdS picture and my dS picture, but all they care about is the cone, they don't care about the global structure. So the fact that they can do this doesn't necessarily imply that that all the details of the second part of the correspondence (re-expressing the bulk correlation functions in terms of CFT operators) can also be swapped back and forth between the two pictures - the analytic continuation here only pertains to an intermediate step.

dhillonv10

Jun12-11, 07:13 PM

thanks again for the clarification mitchell, so far you have presented a lot of information, and i want to make this post to collect the main ideas since its spread across 4 pages, then we can carry on the discussion. Please PM me if you want changes in this post, that way we can keep one post and I'll edit what changes are required.

Goal: Could a purely local interaction in a classical boundary field theory turn into a nonlocal interaction in its holographic image? Or more precisely stated: Re-expression of the bulk correlation functions at past infinity, in terms of CFT operators.

Approaches:

Explain quantum mechanics using the holographic principle by expressing both theories using Bohm's equations (Bohmian mechanics)

Grassmannian formalism: the discovery of a third framework, neither string theory (bulk) nor field theory (boundary), but something else outside space-time entirely, this is the framework where neither space-time locality nor quantum unitarity is "manifest", i.e. visible - you have to switch to the other perspectives to see them.

Map bulk nonlocality to boundary locality using twistors somehow. The "third theory" of Arkani-Hamed et al, which adds a third description to the bulk/boundary duality, is expressed in terms of twistors.

Papers:

Stochastic Quantization: http://arxiv.org/abs/0912.2105 in AdS/CFT

Entanglement spectrum and boundary theories with projected entangled-pair states: http://arxiv.org/abs/1103.3427

Chern-Simons Gauge Theory and the AdS(3)/CFT(2) Correspondence: http://arxiv.org/abs/hep-th/0403225

Entanglement Renormalization and Holography: http://arxiv.org/abs/0905.1317

A Duality For The S Matrix: http://arxiv.org/abs/0907.5418

Local bulk operators in AdS/CFT and the fate of the BTZ singularity: http://arxiv.org/abs/0710.4334

A new twist on dS/CFT: http://arxiv.org/abs/hep-th/0312282

A holographic view on physics out of equilibrium: http://arxiv.org/abs/1006.3675

mitchell porter

Jun13-11, 01:28 AM

That covers most of it. I want to return to the topics of holographic noise and entropic gravity, too, eventually.

Meanwhile, I also realized (something I already knew), that the way you go from globally AdS to globally dS is by adding ingredients (branes, field fluxes) which will add enough positive curvature to outweigh the negative curvature of the AdS geometry. I mention an example in this new post on the "dS/dS correspondence" (http://www.physicsforums.com/showthread.php?p=3353934).

dhillonv10

Jun14-11, 08:56 PM

That was a very interesting post, now from this (http://docs.google.com/viewer?a=v&q=cache:IU-7mTA7gfcJ:www.ugr.es/~prestrings2007/talks/lifschytz.pdf+bulk+operators&hl=en&gl=us&pid=bl&srcid=ADGEESgIff9ZPo_vQ1gztcwcBH0ksU2t3rU5THdlMUwJ 1sY1h5ekQ8vr6JobbBrIblalvXvVDcczhgfWlvZDpP2n7wBybB bLLX057xe6WVH-tS7QyabXRqi1Yw1LRfrNtRalDx-dbvBi&sig=AHIEtbTmz7irHqIdm1cAnD4vvaGI7uyuKQ&pli=1) presentation (slide 2) the bulk correlation functions have been written in terms of boundary operators in AdS (if I understand it correctly). Now we need to explore more on the AdS-to-dS uplift, I am going to look more into the papers mentioned in that post and see what I can find.

edit: This is basically the same thing you mentioned in #36, the idea of expressing bulk scalar field phi in terms of the boundary field phi_0. Now using the uplifting, we then maybe able to transform the AdS space to dS space and then we can have that scalar field in dS space.

edit 2: Just found another interesting paper: http://arxiv.org/abs/hep-th/0203208

They mention this on page 2:

Here we discuss a diﬀerent proposal of extrapolating from AdS to dS spaces. We establish a duality between the two spaces which interchanges the role of coordinates and momenta for a scalar ﬁeld. We thus show that a massive mode in dS space is dual to a tachyonic mode in AdS space.

and they are again using complex variables to accomplish a lot of this for instance the analytical continuation. The tacyonic nature however worries me in this case.

mitchell porter

Jun15-11, 03:41 AM

This is basically the same thing you mentioned in #36, the idea of expressing bulk scalar field phi in terms of the boundary field phi_0. Now using the uplifting, we then maybe able to transform the AdS space to dS space and then we can have that scalar field in dS space.
Almost certainly counterparts of these formulas for de Sitter space do exist. However, the other part of holography is specifying the conformal field theory on the boundary, the fields of which are combined to create the "O" operators that are equivalent to the bulk fields close to the boundary. We now have many examples of AdS/CFT where the CFT is known, but we have no examples of dS/CFT where the CFT is known; and most of the examples of AdS-to-dS uplift that were constructed in string theory since 2004 start with an AdS model where the boundary CFT also isn't known. The 2009 paper by Polchinski and Silverstein, which I mention in the "dS/dS" post, was a first step towards finding AdS/CFT dual pairs which were also suitable for uplifting. So in those cases, at least the CFT is known on one side of the AdS-to-dS uplift - but only on one side.
Just found another interesting paper ... The tachyonic nature however worries me in this case.A tachyonic mode of a field is now usually understood as an artefact of the field being in an unstable vacuum state. For example, see this explanation of the Higgs mechanism (http://www.ift.uni.wroc.pl/~rdurka/index/Higgs.pdf). If you start the field in a quantum state at the top of the Mexican hat potential, it will immediately move towards a lower-energy state in the valley below.

Particles in quantum field theory come about from quantum probability distributions over the Fourier modes we discussed earlier on. A quantum Fourier mode has an "occupation number" which is the number of particles with momentum corresponding to the wavelength of the mode (i.e. this is their de Broglie wavelength). A vacuum state is a quantum state for the field in which the occupation number is zero everywhere. Being at the top of the Mexican hat potential defines a vacuum state for the Higgs field in which excitations of the Fourier modes correspond to particles with negative mass squared. In terms of relativity, that would mean faster-than-light propagation, but here it means that the field is in an unstable state, and it decays to the stable lower energy state before any such tachyonic excitations could go anywhere. In the lower, more stable vacuum state, the Higgs particle now have positive mass squared.

So in contemporary physics, tachyons don't mean "faster than light", they mean "unstable vacuum". It's the same thing - particles with imaginary mass - but the second consequence turns out to be the relevant meaning. For example, you can see string theory papers about tachyon condensation between brane-antibrane configurations - all it means is that the brane configuration is unstable and will immediately annihilate into something else.

De Sitter vacua are usually and perhaps always unstable in string theory, so a fact about tachyonic modes in de Sitter space probably has to do with this instability, but the exact significance of this particular "duality" eludes me. The fact that it is an exchange between coordinate space and momentum space reminds me of the dual superconformal symmetry which exists in the most-studied examples of AdS/CFT. But there might be no connection; I'd have to study it properly to be sure.

mitchell porter

Jun15-11, 03:54 AM

By the way, here (http://arxiv.org/abs/1102.2910) is Kabat, Lowe et al's latest on locality in the bulk - just a few months old! So even they won't be much further along than what you can read there.

In a sense, they're looking at the opposite topic to what's being discussed in this thread. Here we want to understand how locality on the boundary gets turned into nonlocality in the bulk. But what they are trying to do, is to see how to represent bulk locality on the boundary. That means two things - just being able to talk about a point in the bulk, and then, being able to talk about causal locality in the bulk (see what they say about commutation of spacelike operators - if they commute, that means there's no causal interaction at a distance). Everyone agrees that the bulk theory has to be nonlocal in some sense, but also that it ought to be approximately local - it's not an unanalyzable mess of nonlocal connectedness. So they are working in the language of the boundary CFT to describe bulk physics in a way that is as local as possible. But if they make any progress on that question, it should be relevant for us, because any left-over nonlocality that they can't eliminate must be a big clue to the exact nature of holographically induced nonlocality.

sanoy19

Jun15-11, 09:15 AM

does quantum entaglement create all field that exist!!!! as everything were bound together before big bang??

marcus

Jun15-11, 10:14 AM

I started a separate thread for Sanoy's question:
http://physicsforums.com/showthread.php?t=507150

dhillonv10

Jun16-11, 09:39 PM

I'm currently reading that paper and it is an impressive result that they are able to show how a massless scalar field in EAdS space is dual to massive scalars in dS space. But see that also brings up another concern I have, the scalar field phi that we have been discussing in various papers so far, those seem to be massless, I think. Now how'd things change if that scalar was to gain mass? If it doesn't change anything that we are already most of the way there to the mapping. We already have a well defined mapping from bulk to boundary in AdS in terms of a scalar field and now if there exists an uplift mechanism to dS then the mapping in AdS should hold as well. That last part, I need a bit of time to properly define it.

mitchell porter

Jun18-11, 01:50 AM

I'm not sure which paper you're reading, and maybe I shouldn't distract you from this topic, but all of that is just about relating properties of a field in an AdS bulk to properties of a field in a dS bulk. The formulas for one field will be similar to formulas for the other field, except for a few alterations corresponding to the change from a negatively curved space to a positively curved space.

But all that is still just preliminary. The real AdS/CFT correspondence involves what I was talking about in #47: the re-expression of bulk fields near the boundary, in terms of a completely different set of fields on the boundary.

The fields of the boundary theory - call them A, B, C... - and combinations of them - dA/dx . B^2, or whatever - transform in a certain way under conformal transformations (re-scalings, mostly) of the boundary space. When you do this coordinate transformation on the boundary, the correlation functions, etc, have to be multipled by a quantity of the form z^n, where z expresses the magnitude of the re-scaling. For a given combination of boundary fields, the exponent n is called the "conformal dimension" of that combination. You can estimate n just from counting "tensor indices" (whether A, B are scalar, vector, how many space derivatives there are, etc), but then there's an extra "anomalous" contribution that comes from quantum mechanics. The full "anomalous conformal dimension" of a combination of field operators from the boundary theory then maps onto the mass of the corresponding field in the bulk. Also note, one field in the bulk corresponds to one combination of fields from the boundary. The capital-O operators which show up in these papers by David Lowe refer to unspecified combinations of fields from the unspecified boundary CFT (the "trace ABC" expressions I mentioned in comment #47).

This is the algebraic complexity which is at the heart of AdS/CFT (or at least, it was the first really difficult aspect of the correspondence to be investigated and confirmed, since it's quite hard to calculate these anomalous dimensions). It starts out as an algebraic relation between combinations of boundary field operators, and bulk fields near the boundary. Once you have that, it's a much simpler thing to extend the relation so it also applies to bulk fields away from the boundary - that just corresponds to higher energy scales on the boundary, or (same thing) to summing over increasingly large regions on the boundary, as in these papers by Lowe.

But so far as I can see, no-one has any real understanding of what happens to this algebraic relation when you go from AdS to dS. How could they, when they don't have any full examples of dS/CFT to work with, just guesses? We don't know if the AdS/CFT algebraic relation survives but gets changed, or if it is completely destroyed and a completely different one takes its place. So, this is a hard problem, even before you start trying to derive quantum mechanics itself from holography. :-)

Finbar

Jun18-11, 05:12 AM

Isn't there a Euclidean-Euclidean version of AdS/CFT? In which case the boundary should have a Bohmian interpretation, shouldn't it?

But of course this wouldn't be a derivation of QM, since it the Bohmian interpretation is QM.

atyy,

I wonder what your getting at here? can you elaborate a little. I find it interesting that Bohm had a "holographic" interpretation of QM.

http://en.wikipedia.org/wiki/Implicate_and_explicate_order_according_to_David_B ohm

These ideas come at least 14 years or so before the "holographic principle" for QG was first put forward!

Of coarse its not clear whether the two ideas of holographic are directly related. But it would seem likely to me.

dhillonv10

Jun18-11, 01:27 PM

thanks for the reply mitchell, in that last post I was referring to: An AdS/dS duality for a scalar particle. I do understand to some extent that reading that paper won't provide the full correspondence that we need, the ideas presented there seemed interesting though. But it is as you say, this problem is indeed very difficult and part of the reason is the lack of fully worked examples, in that case I think we may simply have to use indirect methods or guesses :) first of which is the wick rotation, this (http://arxiv.org/abs/0710.4334) paper explores some of that. Now I found another very interesting indirect method that is a little more than half-way developed:

Conformal anomaly from dS/CFT correspondence:

Abstract:
In frames of dS/CFT correspondence suggested by Strominger we cal-
culate holographic conformal anomaly for dual euclidean CFT. The holo-
graphic renormalization group method is used for this purpose. It is ex-
plicitly demonstrated that two-dimensional and four-dimensional conformal
anomalies (or corresponding central charges) have the same form as those
obtained in AdS/CFT duality.

And in the conclusion they mention this:

We should note that holographic conformal anomaly obtained from
dS/CFT duality seems to be identical with that from AdS/CFT one. This
shows that obtained central charge from dS/CFT duality itself is the same
with that from AdS/CFT. Nevertheless, it does not mean that boundary
CFTs should be necessarily the same because there may exist several diﬀer-
ent theories with the same central charge. Finally, let us note that the fact
that holographic conformal anomaly from AdS/CFT or dS/CFT duality is
the same suggests that both these dualities are the consequence of some un-
derlying fundamental principle. Even more, one can speculate on existance
of more dualities of such sort, also for other spaces.

Even though we are not looking for conformal anomalies, this gives some evidence that there is an indirect method that can be applied to uplift the AdS space to dS.

mitchell porter

Jun20-11, 03:25 AM

I haven't decoded all of that yet. But it's interesting to see that the original reference on this subject (http://arxiv.org/abs/hep-th/9912012) (computing the conformal anomaly using holographic RG flow) employs the Hamilton-Jacobi equations, because they also offer a path to the Bohmian approach to quantum mechanics. You may have seen news stories recently about the reconstruction of definite trajectories for photons in a double-slit experiment, using "weak-valued measurements"; those were "Bohmian trajectories". So, here we're close to something very basic about how quantum mechanics works.

dhillonv10

Jun26-11, 09:59 PM

Just an update, I've been working on a related problem in the meantime however, today two very interesting papers came up, I am not sure if anyone mentioned those already or not.

1. dS/CFT Duality on the Brane with a Topological Twist: A C Petkou, G Siopsis (2001)

Abstract:

We consider a brane universe in an asymptotically de Sitter background spacetime of arbitrary dimensionality. In particular, the bulk spacetime is described by a ``topological de Sitter'' solution, which has recently been investigated by Cai, Myung and Zhang. In the current study, we begin by showing that the brane evolution is described by Friedmann-like equations for radiative matter. Next, on the basis of the dS/CFT correspondence, we identify the thermodynamic properties of the brane universe. We then demonstrate that many (if not all) of the holographic aspects of analogous AdS-bulk scenarios persist. These include a (generalized) Cardy-Verlinde form for the CFT entropy and various coincidences when the brane crosses the cosmological horizon.

This in some sense goes back to the idea bf being able to uplift AdS to dS and that may preserve some of the holographic dualities.

2. dS/CFT Correspondence in Two Dimensions: Scott Ness, George Siopsis (2002)

Abstract:

We discuss the quantization of a scalar particle moving in two-dimensional de Sitter space. We construct the conformal quantum mechanical model on the asymptotic boundary of de Sitter space in the infinite past. We obtain explicit expressions for the generators of the conformal group and calculate the eigenvalues of the Hamiltonian. We also show that two-point correlators are in agreement with the Green function one obtains from the wave equation in the bulk de Sitter space.

Restricted dimensionality however, if I understand this correctly, it has something to do with the wave function from the quantum mechanical model constructed at the boundary to the correlators in bulk.

mitchell porter

Jun30-11, 03:48 AM

I finally got to see one of the talks on twistorial holography (http://pirsa.org/11060046/) by the other Verlinde brother, Herman, mentioned in comment #45. It was technically fascinating and connected to many other topics discussed on this forum in the past, like conformal gravity (http://www.physicsforums.com/showthread.php?t=502733&page=2#21) and how the twistor string works (http://www.physicsforums.com/showthread.php?t=415677).

But the sense in which it's a holographic construction eludes me. Holography is mentioned in the first ten minutes, and then again in the very last minute. There are mappings, q and q_T (q transpose, the inverse of q), which are not bulk-to-boundary mappings but bulk-to-"screen" mappings, where a screen is a surface in the bulk of one less dimension. There is a remark at 34 minutes that space-time points become the lowest Landau level of something in one extra dimension. At 45 minutes the matrices q and q_T show up again, as noncommutative space-time coordinates for strings stretching between a stack of N D4-branes and a cloud of k D0-branes. Then all this gets uplifted to a six-dimensional space of the form S^4 x S^2 - the D4-brane become space-filling D6-branes and the D0-branes become D2-branes wrapping the S^2 - and this six-dimensional space happens to be twistor space! - 4-dimensional space with an extra "sphere" at each point, corresponding to directions in 3-dimensional space. Again, the strings between these branes implement a version of twistor string theory, with one part being equivalent to the self-dual part of N=4 Yang-Mills, and another part giving you the rest of N=4 Yang-Mills coupled to conformal supergravity. Verlinde (along with Heckman (http://arxiv.org/abs/1104.2605)) has a "fuzzy twistor" construction which acts as a regulator for this theory (i.e. it eliminates divergences due to short-distance interactions (http://en.wikipedia.org/wiki/Ultraviolet_divergence), because the classical continuum picture no longer applies at very short distances), and he says it's holographic too - but that's the part I don't understand - at the end he says there's a projection onto the "twistor line", but I thought that was equivalent to one of the "S^2"s, so if he's talking about the reduction from 6d perspective to 4d perspective, it just seems like Kaluza-Klein - approximating in a way that neglects the compact extra dimensions - and not the dramatic holographic elimination of one large dimension.

So I don't get it, but it's extremely interesting, and will hopefully make more sense to me in the near future.