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It's not a "conceptual breakthrough" if it's unproven. Making the conjecture showed fine insight, and it is clear that valuable results will flow from verifying its status. If it's true then the QCD particle spectrum can be analysed using perturbation theory; if it's false then the explanation, juxtaposed with the fact that it often works, will tell us something interesting. But to build on conjectures is risky, and effort is better spent on checking whether it's true. - AG, Stockport, UK.

I have two questions:

• Have any instances of the correspondence been proven?
• In the correspondence between AdS5×S5 and N=4 Yang-Mills theory on the conformal boundary, is our real universe supposed to be the former or the latter? In other words, are we living in the higher dimensional space whose image is the hologram, or in the hologram itself?

Thanks, AxelBoldt (talk) 15:52, 9 April 2009 (UTC)

• No, nothing proven.
• If there is a correspondence, it means they are identical and it doesn't matter which one the universe is. In this particular case our universe is neither as there is no $\mathcal N=4$ supersymmetry or conformality realized in reality.
MuDavid (talk) 15:10, 17 March 2010 (UTC)
Comment: it's still a breakthrough. Also, the mathematical correspondence is being applied in condensed matter physics a lot right now. Someone should include that in the article. Also, given the importance of this topic, I would like to see an introduction for lay-readers, (if that is even possible) or something for those who are not gods of theoretical physics (such as myself). Danski14(talk) 22:24, 8 September 2010 (UTC)

The article need to expand with a section about Holographic Superconductors. —Preceding unsigned comment added by 79.145.145.119 (talk) 04:22, 30 April 2011 (UTC)?

wtf? totaly unreadable, even for somebody who understands the basics of concepts such as quantum stats, quantum entanglement, many world theory and such... this was totaly unreadable. — Preceding unsigned comment added by 85.226.1.123 (talk) 08:04, 5 January 2012 (UTC)

## Can Someone Explain Why This Is Important?

I agree with the above. If you know what this is telling you, you might be able to understand it. But the purpose of an encyclopedia is to explain the topic to people who do not already know about it.

I was directed to this article from the philosophy of time entry. I have a good layman's understanding of quantum theory and string theory, but what this means, why it's important, and what its implications are do not appear to be available in this article. Can we have it interpreted for the rest of us?

Important because it shows how far in the weeds modern physics is with its substitution of mathematics for real things in nature. The crowning glory is nothing with properties (general relativity) and a real existent without extent (strings). Only the overall backwardness of human culture could bring such a thing about, an excellent example of how the 98% steer, govern, and culturally determine the 2%. So important in a nude emperor kind of way (besides the fun maths). 198.255.198.157 (talk) 20:58, 8 December 2013 (UTC)

## Picture?

Could someone help me find a free picture for the section on the quark-gluon plasma? I'm thinking something like this would be good:

Thanks. Polytope24 (talk) 03:16, 22 August 2013 (UTC)

## GA Review

Reviewer: SPat (talk · contribs) 00:11, 14 September 2013 (UTC)

## Initial remarks

I'm starting a review. On first read, I'm highly impressed by your ability to convey such a technical topic to a fairly wide audience while still not omitting too many details. I have some specific issues with style and presentation that I'll get to over the next couple of days; and I haven't had much time to look at the references; but overall, good job! SPat talk 00:11, 14 September 2013 (UTC)

### Layout

The basic layout seems to be Overview/Background->Applications->History. Is there a reason you've put history at the end? I believe the norm is to put history up front (right after the lead).

### Overview

I think your goal here is to explain the main ideas behind the correspondence. However, to make things accessible you spend a fair amount of effort on introducing the basic terminology which end up looking like digressions. I think one way to solve this would be to split this section into two; a "Background" section describing to set up the terminology (holography/quantum gravity/general relativity/...) and an "Overview" section, where you actually describe the AdS/CFT correspondence.

• A holographic description of physics
Do you think a picture to illustrate the idea of a hologram would help? I can't think of anything in particular, but something that would illustrate the notion of depth being captured in a two-dimensional image. I'm sure commons will have something helpful. Also, you may not want to do it here itself, but you should provide some details about what exactly you mean by "equivalent formulation"
• Quantum theory and gravity
Again, would an illustration for compactification help?
• Gravity in anti-de Sitter space
This section, I think, has an issue in that you spend too much time making the reader familiar with the idea of non-Euclidean spaces at the expense of actually explaining AdS itself. I'll take a leap and say that majority of readers who come to this article will have at least heard of the idea of non-Euclidean geometry, and it might be enough to mention the non-flatness of space in a couple of sentences at most. Instead, I think it is much more important to explain what exactly an AdS space is, what (a sufficiently simplified version of) the metric tensor in this geometry look like, what is the importance of the negative Ricci scalar is, etc.
"In anti-de Sitter space, it is possible to define a notion of "boundary" of spacetime at infinity." This sentence needs a lot more explanation and details, as it pretty much seems to be the crux of the matter. Maybe a mathematical example would help. Also I don't know if the fact that this boundary is conformal is trivial or not - might be worth explaining.
It's probably going to be helpful to use atleast a few images describing the AdS geometry (some suggestions) The Escher picture, great as it is, is more of a distraction at the moment, I think. (Sorry if I'm appearing too picky about images!)
I'd consider this subsection to be the heart of the article, and I think it needs a major expansion - maybe make it an entire section instead of a subsection.
"...can be regarded as the "spacetime" for a special type of quantum field theory called a conformal field theory." If you haven't done so earlier, this is a good place to explain what exactly a CFT is. For the "dictionary", is it possible to give a specific example of a correspondence? Maybe you could put that in the subsequent "Examples" subsection, but if possible you could take a specific (simplified?) example like AdS5/N=4SYM and write out some correspondences ("X-field in AdS goes to Y-interaction in CFT"). I don't know if it's actually possible to do something like that, I just think it would be helpful to add if you can.

(I've to leave now, will continue later...) SPat talk 19:47, 14 September 2013 (UTC)

Thank you very much for these comments! I don't know if you want me to start making changes now or wait until you finish reviewing. In any case, when you're ready, I will make the following changes to the article:
1. I will make a separate "Background" section combining material from the subsections entitled "A holographic description of physics" and "Quantum theory and gravity". This will include a more detailed discussion of the notion of duality in physics and (hopefully) some more pictures.
2. This will be followed by an "Overview" section explaining the geometry of AdS space and including a longer discussion of the idea of AdS/CFT. It is unfortunately not very easy to give a precise definition of AdS space without taking a lengthy detour with lots of mathematics. As explained in the anti-de Sitter space article, it is usually defined by taking a hyperboloid in an ambient space with two timelike directions and then passing to the universal cover. Instead of trying to explain all this, I think I'll try to make my own version of the picture in this article of a stack of Escher prints. This would be much more accessible, and it would help to address your concerns about pictures. I will also say more about conformal field theory and the notion of conformal boundary.
I was thinking of an explanation similar to the one given in the AdS article, in particular, emphasizing the fact that it is always a negative curvature. SPat talk 22:26, 15 September 2013 (UTC)
When I was writing the article, I decided to put the "History" section near the end because I felt that the reader would need to first have some understanding of the topic. The thinking was that it would make more sense to the reader after he or she understands what QCD is about and how N=4 SYM is related to string theory via AdS/CFT. Please let me know what you think about this. I'll move it if you think it's necessary, though I'm not sure whether it belongs before or after the "Background" section…
Thanks again for the feedback. Polytope24 (talk) 22:46, 14 September 2013 (UTC)

(Thanks for the quick response! I'll continue with the rest of my review and get back to your reply later) SPat talk 22:18, 15 September 2013 (UTC)

• Examples of the correspondence
Mostly ok. Maybe you could spend a couple of lines each explaining the newly introduced CFTs - the N=4 super Yang Mills, (2,0)-theory, ABJM superconformal field theory? It's very abstract as it stands.

### Applications to quantum gravity

• A non-perturbative formulation of string theory
Is it factually accurate to say perturbative QFT was developed by Feynman? If not, it might suffice to say something like "...Richard Feynman and others".
"...and there are many problems for which one requires a non-perturbative formulation." An example? Isn't gravity the canonical example here? (I might be wrong about that last bit)
"The problem of developing... ...the AdS/CFT correspondence." [Maldacena 1998] doesn't seem to directly imply this. Maybe you could include a secondary reference?
Mostly ok. I didn't quite understand how AdS/CFT solves the problem though - maybe you could use an explanation similar to the one given in [Maldacena 2005]?

### Applications to quantum field theory

• Nuclear physics
It seems the main arguments of this section rely on two primary sources, [Kovtun et al 2001, Luzum et al 2008]. You need to replace/supplement these with secondary sources. In other words, you need secondary sources that says something like "AdS/CFT calculations predicted a lower bound for a quantity which was experimentally confirmed at RHIC". Also, some details about the calculations would be nice.
• Condensed matter physics
• Criticism
Looks pretty good.

### History

The first two paragraphs seem to be a history of string theory in general and not AdS/CFT in particular. I'd suggest you condense these to a couple of sentences or so. But the rest of the section definitely needs major expansion - I'd say more than double. Is the large-N QCD stuff t'Hooft's only contribution to holography? I believe he also did some work on holographic cosmology and Hawking radiation. You should definitely say something about Susskind, and Stephen Hawking and Hawking radiation definitely deserve a mention. Are there any biographical accounts of Maldacena's first paper? That might be interesting to add. Also, Son's QCD stuff and Sachdev's CMT stuff should be discussed as important applications.

My theory about history sections is that they're the ones non-technical audiences will focus on, and so we should emphasize historical details as much as possible. That is things like the broad questions in the field, the dramatis personae, and locations/collaborations etc. This is also why I prefer to have them at the beginning. It's fine even if you end up repeating some stuff later in the article - as long as you re-word it properly.

• Most of the history section seems to be a repetition of things said earlier in the article. Maybe the history section could be collapsed to a simple list of sentence-long references further explained in the other sections? — Preceding unsigned comment added by 88.192.19.110 (talk) 16:04, 8 December 2013 (UTC)

## Summary

Whew - I've put up a huge list of suggestions for improvement. I'm not saying all of them need to be incorporated to achieve GA status. However, I think the couple of points you do need to address are A) providing sufficient details in the main "Overview" section so that things don't end up looking too hand-wavy. (My interpretation of WP:TECHNICAL is that mathematical details should not be included if they act as a digression, but should definitely be included if they help in illustrating arguments.) B) you need expand and re-structure the history section so that it can function as a stand-alone piece. As I mentioned earlier, many non-technical readers will only read that one section and we shouldn't leave them empty handed. C) most of the references are ok, but you need to be careful about using secondary and not primary sources for citing important statements.

Let me know if you need any clarifications, and congratulations on the good job so far! SPat talk 22:18, 15 September 2013 (UTC)

Thanks for the very thoughtful review. Should I get started on revising the article now? Polytope24 (talk) 04:09, 16 September 2013 (UTC)
Sure, go ahead. Tip: to keep a track of things, it might be helpful if you respond to my comments immediately below that comment. SPat talk 14:44, 16 September 2013 (UTC)

## Revised article

Alright. I just finished some major revisions to the article. In particular, I have made the following changes:

1. I've created a new "Background" section in which I review the basic ideas of quantum gravity, string theory, and quantum field theory.

2. The "Overview" section has been significantly expanded. I've added new pictures and more explanation of the geometry of AdS space. As I was saying before, it's a little complicated to give a precise definition of AdS space (i.e., you can't just write down a metric on R^n or something). In order keep the article reasonably accessible, I therefore decided to explain (2+1)-dimensional AdS space as a stack of hyperbolic disks, following Maldacena 2005. This should be accessible to non-experts, and it gives a pretty good idea of what AdS space actually is.

3. I've clarified parts of the section entitled "Applications to quantum gravity", following your suggestions.

4. I've found secondary sources supporting the claims in the section entitled "Applications to quantum field theory", and I removed the reference to Bose-Einstein Condensates.

5. I have significantly expanded the "History" section with lots of details about the Thorne-Hawking-Preskill bet, the holographic principle, and Son's work. I condensed the discussion of the early history of string theory but also explained why this is relevant to AdS/CFT. I still think it makes sense to put the "History" section near the end of the article since AdS/CFT is such an abstract idea which requires so much background to understand. Nevertheless, the section is non-technical enough that it can be read independently of the rest of the article.

Please let me know if you have any questions about the choices I've made. Thanks again. Polytope24 (talk) 01:29, 18 September 2013 (UTC)

I am, unfortunately, very busy at the moment and haven't had time to look carefully. But you (and other editors) have put in a very impressive effort - especially with the "Overview" and "History" sections. It looks like you're mostly good to go - I'll look over the references etc. one last time before signing off sometime in the next couple of days. One minor style point: per MOS:IMAGES, staggering of images is good, but may be a problem if the text gets sandwiched between images, especially for readers using small screens. Maybe you could fix that? SPat talk 11:47, 19 September 2013 (UTC)
Okay, I made come changes that should help with this. Polytope24 (talk) 15:54, 19 September 2013 (UTC)
Done It may not be perfect, but it easily satisfies all GAC. Cheers! SPat talk 23:10, 19 September 2013 (UTC)

## A welcome effort... a few comments

I'm glad to see this kind of serious content being presented here! As a non-physicist trying to make sense of this I should, however, make some comments...

• The purpose of the anti-de Sitter space needs to be clarified a bit. I'm assuming that the point of the pretty picture is that it represents a space with 450 degrees in every circle (three 90-degree angles and three 60-degree angles). I assume this is just a conceptual diagram? We have an open universe but not a noticeable increased number of degrees in a circle, and the way the diagram looks that is true for every point in the space, so our diagram would look almost but not exactly quite like a checkerboard? Or is this a matter of scale???
• "hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher dimensional models of anti-de Sitter space." -- this is beside an image of a cylindrical prism of anti-de Sitter universes for the time dimension. My assumption is that for the three spatial dimensions this "stacking" is actually a higher-level sphere - that you have little pointy-edged cubes and tetrahedrons (or higher dimensional equivalents) coming together. Also, is time the one true-blue Euclidean straight line dimension, no extra degrees there? Then there are the compactified dimensions - do those have to be closed with < 180 degrees in a circle? (I'm out to sea by now)
• It seems like you give the N=4 super Yang–Mills theory as an example at least three separate times throughout the text, not going into it all that much each time - my feeling is you need to round up those stray sections and condense in one place.
• The absolute core of the article, "the idea of AdS/CFT", needs beefing up with more specific examples. I just don't get how the boundary of a cylinder looks like a four-dimensional spacetime! I don't get what sort of calculation on a single ?something? in the anti-de Sitter space looks like multiple particles in spacetime. This is the door of our perception - get out the cleanser.

Wnt (talk) 02:09, 8 December 2013 (UTC)

Thanks for your comments, Wnt. I'm not going to try to edit the content of the article while it's on the Main Page, but here are some answers to your questions:
1. You basically have the right idea about hyperbolic space. If you look at any triangle in the hyperbolic plane (see the picture), then its angles will sum to less than 180 degrees, and the value this sum depends on the size of the triangle. For very small triangles in the hyperbolic plane, the sum of the angle measures will be very close to 180 degrees. Thus the geometry of anti-de Sitter space is approximated by more familiar Euclidean geometry at very short distances.
2. Just as the boundary of the hyperbolic plane is a circle, the boundary of three-dimensional hyperbolic space will be an ordinary two-dimensional sphere, and the boundary of higher dimensional hyperbolic spaces will be higher dimensional spheres (which, obviously, are not so easy to visualize).
3. The example discussed in the article involves three-dimensional anti-de Sitter space. The boundary of this space is not four-dimensional spacetime, but an imaginary two-dimensional spacetime. If you want to model, say, nuclear physics using the AdS/CFT correspondence, then you have to consider an example of the duality where the boundary is four-dimensional.
Polytope24 (talk) 03:13, 8 December 2013 (UTC)
1. So to be clear, the "triangles" and "squares" really do have smaller angles, and at any given point there are only 360 degrees. But, say, if you shine a light down one of the lines of the triangle it will end up going "straight" to the next vertex? And you could tile our universe with these "tetrahedrons" and "cubes", three cubes around the vertex of a tetrahedron, three tetrahedrons around the vertex of a cube, for a total of I think three cubes and four tetrahedrons meeting at a point? Is there a way to calculate how big the side of each polytope is?
2. OK, so the anti-de Sitter space is the physical dimensions of our space (not time) ... plus the compactifed dimensions? And time remains a separate dimension orthogonal to everything else, and not subject to any transformations.
3. The boundary of the two dimensional space is a cylinder (counting the time dimension); in space it is a line that wraps around on itself, and the boundary of the four dimensional space would be a three dimensional space that wraps around on itself? Is the geometry/manifold specified by the theory? And if an infinite anti-de Sitter space has a boundary of finite size, what determines the size of the boundary?  :I still am confused at what actually "lives" in the anti-de Sitter space. Given the lack of precision of measurements in our space, what is this apparently "infinite resolution" needed for? (I'm waaay out to sea by now, distant ship smoke on the horizon...) Wnt (talk) 06:23, 8 December 2013 (UTC)
if you shine a light down one of the lines of the triangle it will end up going "straight" to the next vertex?
The edges of the triangles in the picture are what mathematicians call geodesics. This means that they are as straight as they can possibly be in the curved space.
you could tile our universe with [these shapes]?
On large distance scales, the real universe is not curved in the same way as anti-de Sitter space, so you can't have a similar hyperbolic tiling of real physical space.
the boundary of the four dimensional space would be a three dimensional space that wraps around on itself?
Correct.
And if an infinite anti-de Sitter space has a boundary of finite size, what determines the size of the boundary?
Technically, what we're calling the "boundary" should really be called a "conformal boundary". This is a mathematical notion that makes it possible to have a boundary that's infinitely far from any point in the interior. Also, the boundary theory appearing in the correspondence is a conformal field theory, which means in particular that it is scale invariant, and so one does not need to talk about the size of the boundary. Polytope24 (talk) 07:11, 8 December 2013 (UTC)
Great answers so far! And I shouldn't have confused the tiling of the space and the tiling of our universe. But...
• Is it necessary for the AdS/CFT correspondence that our universe actually be a closed universe? Of finite size? With positive curvature, with divergent lines intersecting?
• An event in the de Sitter space maps to one or more events in our universe. Every point in our universe, this conformal boundary, is infinitely far from the point in de Sitter space where the event occurs, and vice versa. So why do the events in our universe occur in one tiny region?
Wnt (talk) 17:50, 8 December 2013 (UTC)
The AdS/CFT correspondence describes gravity only in a certain approximation, so the large scale structure of our universe (and in particular the question of whether it is closed) is not really relevant.
I'm not sure if I understand your second bullet. Let me just say that the AdS/CFT dictionary is highly nontrivial. In AdS/CFT, you have physical objects in one theory mapping to a priori completely different physical objects in the dual theory. It's not as simple as events in one description mapping to events in the other. Polytope24 (talk) 19:30, 8 December 2013 (UTC)