# Can holography be applied to every model?

Can all models have a holographic representation?

I was talking with a physicist and he told me:

"Holography is not a property you can just apply to any model of the universe willy-nilly. It either emerges or it does not. There is no guarantee that an arbitrary universe has a valid holographic representation. This is an incredibly complicated subject, and it's actually still debated whether it is valid under any realistic conditions. But the holographic principle doesn't even arise for the same reason in different contexts - it's just a property that some models have. It's like expecting there to be a single reason that Anyway, no one tries modifying a model so that it will have a holographic representation - they simply look for holographic representations of models they are already interested in."

But is this right? Can holographic principle be modified or is/are it there any version(s) of it that can be applied to every model? Can there be holography without string theory?

Disclaimer: I'm not an expert on this subject, and I don't know the yes/no answers to your questions, but I'll offer some perspectives and quotes from real experts.

General arguments strongly suggest that a black hole has a finite entropy proportional to the area of its horizon, the Bekenstein-Hawking entropy. Since a black hole represents the largest amount of "stuff" that can be crammed into a given region of space (trying to fit more "stuff" inside just makes the black hole bigger), this suggests the holographic principle. This argument doesn't seem to depend on the large-scale structure of the universe, but it also doesn't tell us how to construct models that satisfy the holographic principle explicitly.

The AdS/CFT correspondence (aka gauge/gravity duality) gives a large family of models that appear to realize the holographic principle explicitly, but these models all involve a specific asymptotic structure of spacetime, namely that spacetime is asymptotically anti-de Sitter (AdS). This is the structure we'd expect in a universe with a negative cosmological constant. With this restriction, the correspondence otherwise appears to be very general. One paper ("Gauge/gravity duality", https://arxiv.org/abs/gr-qc/0602037) even starts with this bold assertion:

Hidden within every non-Abelian gauge theory, even within the weak and strong nuclear interactions, is a theory of quantum gravity.

This assertion is referring to the idea that if we have an ordinary quantum field theory (QFT) defined on a $$D$$-dimensional spacetime with the same topology as the "boundary" of a $$D+1$$-dimensional AdS spacetime, then the assertion says that the former is equivalent to a model of quantum gravity in the latter — specifically string theory. In fact, according to https://arxiv.org/abs/gr-qc/0602037 again (page 16),

...this duality is itself our most precise definition of string theory...

The paper concludes with this statement:

In conclusion, the embedding of [$$D+1$$-dimensional] quantum gravity in ordinary [$$D$$-dimensional] gauge theory is a remarkable and unexpected property of the mathematical structures underlying theoretical physics. We find it difficult to believe that nature does not make use of it, but the precise way in which it does so remains to be discovered.

(By the way, to get a big $$D+1$$-dimensional spacetime with familiar properties, the $$D$$-dimensional QFT needs to have certain special properties, so the first assertion quoted above should be interpreted with caution.)

However, at least in its current form, the AdS/CFT correspondence only works when the $$D+1$$-dimensional spacetime is asymptotically AdS — that is, for a negative cosmological constant. Since we now have good evidence that the cosmological constant is positive in the real world, the AdS/CFT correspondence does not directly address "realistic conditions". According to another paper ("The Holographic Bound in Anti-de Sitter Space", https://arxiv.org/abs/hep-th/9805114, page 9),

It remains to ask whether one can build a similarly sharpened holographic hypothesis for theories with zero (or even positive) cosmological constant. The answer will require some new ideas, since Minkowski space (or de Sitter space) has no obvious close analog of the `boundary at spatial infinity' by which holography is realized when the cosmological constant is negative.

That was written in 1998, and progress has been made since then. Still, as far as I know, our understanding of how to implement the holographic principle in non-AdS spacetimes is not nearly as well-developed as it is for AdS spacetimes. This is still a very active area of research, and Leonard Susskind (who is one of the pioneers of the holographic principle and a co-author on the 1998 paper cited above) is now suggesting an even broader correspondence (page 1 in http://arxiv.org/abs/1708.03040):

To summarize this viewpoint in a short slogan: Where there is quantum mechanics, there is also gravity. I suggest that this is true in a very strong sense; even for systems that are deep into the non-relativistic range of parameters – the range in which the Newton constant is negligibly small, and and the speed of light is much larger than any laboratory velocity.

That was written in 2017. Whether it's true or not (and whatever it means exactly), it illustrates the fact that we still have a lot to learn.