The relation between entropy and probability at quantum levelu

[mitchell porter]

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 https://www.physicsforums.com/showthread.php?p=3353934".

 

[dhillonv10]

That was a very interesting post, now from http://docs.google.com/viewer?a=v&q...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]

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 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]

By the way, 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]

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

 

[marcus]

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

 

[dhillonv10]

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]

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]

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_Bohm


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]

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, 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 existence
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]

I haven't decoded all of that yet. But it's interesting to see that 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]

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]

I finally got to see one of the http://pirsa.org/11060046/".

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 (http://arxiv.org/abs/1104.2605" , 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.

 

[dhillonv10]

Thanks for the link to the talk, I was looking around for the talk on Simulating the universe as a quantum computer when i found the talks from the Holographic Cosmology 2.0. There was some talk on the Denef paper as well. Anyways the idea of using a screen is very interesting, it reminds me of the Grassimian representation that we talked about before, the fact that you would use a third theory that is more fundamental. You make the bulk dual to the screen and then the screen to the boundary, so the paper I mentioned before: dS/CFT Correspondence in Two Dimensions: Scott Ness, George Siopsis (2002) might actually work. I'll comment again with questions and such as I watch the talk.

update: There is also a talk on uplifting, titled: Uplifting AdS/CFT to Cosmology

 

[Fyzix]

I ressurrect this thread.

Could you outline your main thoughts in layman terms?
What is it you are thinking is going on in the horizon that gives us the illusion of randomness and nonlocality?

 

[dhillonv10]

Fyzix: this thread isn't dead yet, we are simply waiting, at Strings 2011, Herman et. al announced that they had worked out a complete example of dS/CFT and the paper will be out later this month. Once that's in, then we can do a lot more instead of making guesses as to what really happens because of the lack of an example.