The relation between entropy and probability at quantum levelu

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

atyy
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).

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
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.

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.

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.

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"

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?

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.

Last edited:
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.

Last edited:
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 http://www.phys.vt.edu/~sowers/talks/kabat.pdf" [Broken].

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 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 https://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.

Last edited by a moderator:
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 said
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.
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.

Last edited:
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, http://arxiv.org/abs/hep-th/0312282" [Broken] 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:

Code:
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.

Last edited by a moderator:
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.

Here we are talking about a free (non-interacting) http://en.wikipedia.org/wiki/Scalar_field" [Broken], 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 http://www.flickr.com/photos/ethanhein/2680541012/" [Broken]. 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.

Last edited by a moderator:
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.

Last edited:
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.

edit: i just finished reading your explanation of that email, and wow that's all i can say, in the beginning this idea was a mere speculation, but now i think this maybe taking a serious direction.

Last edited:
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

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 :)

Last edited:
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.

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.

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 http://arxiv.org/abs/hep-th/0201253" [Broken] 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 http://en.wikipedia.org/wiki/Trace_%28linear_algebra%29" [Broken] 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.

Last edited by a moderator:
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.

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.

Last edited:
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 http://en.wikipedia.org/wiki/Wick_rotation" [Broken].

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.

Attachments

11.3 KB · Views: 349
Last edited by a moderator:
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:

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