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Some time ago, I heard about a simple analogy for the AdS/CFT correspondence to something in everyday life. Consider a room filled with furniture, with the walls of the room covered in mirrors. The 2D mirrors we can think of as an N-dimensional CFT--the furniture is the corresponding entities in the N+1-dimensional AdS.

I feel like this is an over-simplification. What is lost in the actual AdS/CFT correspondence in this analogy? I feel like information in the 3D world is lost in the 2D mirror--is this true in the AdS/CFT correspondence? That is, is there information in the AdS that can't be expressed in the CFT?

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The analogy is not perfect. In principle the AdS/CFT correspondence is exact, however the mirror analogy is not. It is well known in psychophysics that recovering the structure of a scene from a 2d image is an ill posed problem. There are may heuristics you can use to recover the shape and depth of objects based only on the projected image, and these usually give accurate enough results, but there is an infinite number of solutions, all compatible with the same image.

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I feel like this is an over-simplification.

I agree. IMHO it's a popscience analogy that's presented as a genuine explanation, but it isn't.

What is lost in the actual AdS/CFT correspondence in this analogy?

A dimension. You can't just discard this. If you do, you lay yourself wide open to dropping another dimension, and saying the flat 2D image is equivalent to a 1D line. It just isn't true. Nor is it true that an electron is a point-particle or some kind of plane or string. Its field is what it is.

I feel like information in the 3D world is lost in the 2D mirror--is this true in the AdS/CFT correspondence?

For that, let's take a look at the Wikipedia AdS/CFT correspondence article:

"In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles. On the other side are anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory".

I don't have much to say about quantum field theory, because light has its E=hf quantum nature, and we can diffract photons and electrons and protons and neutrons. At the fundamental level, everything is fields and waves. There's hard scientific evidence for that. However I have plenty to say about quantum gravity, because gravity has no discernible quantum nature, a photon has a non-zero "active gravitational mass", it doesn't approach you in steps, and gravitating bodies don't exchange field like the electron and proton such that the hydrogen atom has very little field left. In addition, we have no evidence whatsoever that strings or branes exist. So what do we really have? Let's look at Wikipedia again:

"...because it is the most successful realization of the holographic principle, an idea in quantum gravity originally proposed by Gerard 't Hooft and promoted by Leonard Susskind".

What holographic principle? It's "a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a boundary to the region". Only we have no evidence whatsoever for this speculation. None. There is nothing whatsoever that supports the notion that the world is a hologram. But what we have instead is AdS/CFT correspondence, where proponents of a speculation attempt to justify it. Not with hard scientific evidence, as per Clifford M Will's Confrontation between General Relativity and Experiment. But with mathematical assertions. I wouldn't be the first person to say there's more than a whiff of Emperor's New Clothes to all this.

That is, is there information in the AdS that can't be expressed in the CFT?

It would seem there's no information in the Anti-de Sitter space, because WMAP and Planck have provided good scientific evidence that the universe is flat rather than negatively curved. It's a hypothesis that was found to be incorrect. So IMHO what we have here is a 20-year-old mathematical claim that was trying to keep alive a 50-year-old speculation riding on top of a 90-year-old hypothesis, for which there's no evidence at all. Ho hum. And people why physics has funding problems.

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