I have a question about ADS/CFT Correspondence. There is a quote from Wikipedia that confuses me.

"Every entity in one theory has a counterpart in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory."

How can two theories that describe the same thing sound so different? For example, one theory may have one particle and the other would have multiple particles. Why wouldn't they have the same amount? Why don't these two theories that describe the same thing sound more similar?
Also, shouldn't only one of the theories be the correct one?

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Chalnoth
I do find it rather fascinating that two different theories that describe the exact same system can appear so wildly different. But yes, the two theories are mathematically identical: anything you can describe in one theory you can describe in the other theory, and produce the same predictions for how the system evolves over time. The two theories appear radically different on the surface, but are genuinely describing the same thing. Yes, it's weird. But the mathematics is pretty unambiguous.

I do find it rather fascinating that two different theories that describe the exact same system can appear so wildly different. But yes, the two theories are mathematically identical: anything you can describe in one theory you can describe in the other theory, and produce the same predictions for how the system evolves over time. The two theories appear radically different on the surface, but are genuinely describing the same thing. Yes, it's weird. But the mathematics is pretty unambiguous.

But shouldn't only one of the theories be the correct one? Even if they are both making the same prediction?

MathematicalPhysicist
Gold Member
How would you differenitate between the two theories, assuming both pass occam's razor criterion (even though it's not a scientific criterion).

Staff Emeritus
2021 Award
But shouldn't only one of the theories be the correct one?

Why do you think that? You can solve a roller-coaster problem using kinematic equations or with conservation of energy. Which theory is the correct one?

Demystifier
Chalnoth
But shouldn't only one of the theories be the correct one? Even if they are both making the same prediction?
No. They're the exact same theory, just a different perspective.

Why do you think that? You can solve a roller-coaster problem using kinematic equations or with conservation of energy. Which theory is the correct one?

They just sound like they contradict each other. Even if they do make the same prediction. One theory may have one particle while the other would say there are multiple particles. Wouldn't that be like if I had a box of 3 cookies and I predict that they will be eaten by tomorrow, then my friend says that there is only one cookie but it will be eaten by tomorrow, our predictions have the same end result, but only one of us should be correct.

Or am I just missing something?

Chalnoth
The number of particles is an observer-dependent quantity, though. If you accelerate rapidly, you see a bunch of photons (Unruh radiation) that do not exist for an observer that isn't accelerating.

Demystifier
Gold Member
Every entity in one theory has a counterpart in the other theory.
I would like to make a comment from a mathematical (rather than physical) point of view.

The quoted statement does not make much sense without precisely defining what one means by "every entity". In the literature on AdS/CFT, there is in fact no precise definition of the notion of "every entity". But quite generally, "every entity" certainly refers to all elements of a set. Moreover, it is almost certain that the corresponding set is infinite, with cardinality either ##\aleph_0## (countable infinity) or ##2^{\aleph_0}## (cardinality of the set of real numbers). But for any two sets of the same cardinality there exists a 1-1 correspondence, which makes the AdS/CFT correspondence uninteresting.

The only way to make AdS/CFT interesting is to show that there is some correspondence with a certain elegant mathematical structure. This, indeed, is what the physical papers on AdS/CFT claim to be the case. But from a mathematical point of view it is still far from being clear what exactly this elegant mathematical structure is supposed to be. So when physicists say that "every entity in one theory has a counterpart in the other theory", they do not really know what exactly do they mean by that.

Staff Emeritus
2021 Award
They just sound like they contradict each other

Well, they don't. I don't think it's possible to explain why to someone who understands neither CFT nor AdS and is just going by "how things sound". At some point you have to simply accept that when using a popularization of a simplification of another simplification of a theory you don't understand, things might not sound right.

Ok. Thanks everyone.

Demystifier
Gold Member
Why do you think that? You can solve a roller-coaster problem using kinematic equations or with conservation of energy. Which theory is the correct one?

No. They're the exact same theory, just a different perspective.
As the example by Vanadium 50 shows, just because two theories lead to some identical results doesn't mean they are the exact same theory. For example, the Newton equation in 3 dimensions can say something about the direction of motion which the energy-conservation equation cannot. In this sense, Newton equation is more fundamental and more complete than energy-conservation equation. Similarly it is conceivable that, for instance, AdS is more fundamental or more complete than CFT. The fact is that the correspondence between AdS and CFT is only partially understood, so it is not really justified to claim categorically that they are the exact same theory. They are certainly very much related theories, but it is still not completely clear whether they are really the same.

Chalnoth
As near as I can tell, for certain specific cases, the correspondence is exact, which means that indeed the two theories are just different descriptions of the same physical system. The conjecture is that this correspondence holds across a broader range of AdS/CFT theory pairs, but this hasn't yet been shown.

Demystifier
Gold Member
As near as I can tell, for certain specific cases, the correspondence is exact, which means that indeed the two theories are just different descriptions of the same physical system.
It looks as if you missed my last point entirely (as probably many others have), so let me explain that point again, in a different way. Even if the correspondence is exact, and even if it is proved absolutely rigorously, it still doesn't mean that these are different descriptions of the same physical system.

Here is a trivial example. One of the first equations one learns in physics is
s=vt
i.e. the path traveled is velocity multiplied by time. Another equally known equation in physics is the Ohm's law
U=IR
It can be proved absolutely rigorously that there is an exact mathematical correspondence between these two equations. (The proof, of course, is trivial. I am using such a trivial example such that even a half-idiot can understand my point. Indeed, half-idiots will probably understand my point before the sophisticated experts in AdS/CFT.) My point is, despite the exact mathematical correspondence, nobody thinks that these two equations describe the same physics.

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My point is, despite the exact mathematical correspondence, nobody thinks that these two equations describe the same physics.

I think some of the implications are lost in the simple examples although correct.
We can have a peak at regimes normally not available in the theories e.g. Weak coupling <-> Strong coupling between AdS <-> CFT.
This is why one success is found explaining/motivating experimental evidence from heavy ion collisions.

Don't get me wrong, your point is right. Its just some of the implications in the case of AdS/CFT are lost in the simple examples given.
Maybe we can compare it to popular accounts of physics on TV which makes people believe they get the idea while it is much more involved.

tl;dr Oversimplification calls for caution and often sidenotes

Demystifier
Chalnoth
It looks as if you missed my last point entirely (as probably many others have), so let me explain that point again, in a different way. Even if the correspondence is exact, and even if it is proved absolutely rigorously, it still doesn't mean that these are different descriptions of the same physical system.

Here is a trivial example. One of the first equations one learns in physics is
s=vt
i.e. the path traveled is velocity multiplied by time. Another equally known equation in physics is the Ohm's law
U=IR
It can be proved absolutely rigorously that there is an exact mathematical correspondence between these two equations. (The proof, of course, is trivial. I am using such a trivial example such that even a half-idiot can understand my point. Indeed, half-idiots will probably understand my point before the sophisticated experts in AdS/CFT.) My point is, despite the exact mathematical correspondence, nobody thinks that these two equations describe the same physics.
Right. But we're not talking about a relationship that that is nearly that simple.

From everything I've read on the subject, the correspondence is across the entire theory, not just an equation or two that takes the same form. For instance, it appears that the N=4 supersymmetric Yang-Mills theory and the 5-dimensional AdS Type IIB string theory have an exact one-to-one correspondence.

If it's true that this one-to-one correspondence is exact, then the two actually are two different descriptions of the same physics. In order for this not to be the case you'd need some hidden degrees of freedom that aren't described by either theory.

atyy
Marolf claims an even more amazing duality QG = QM!

http://arxiv.org/abs/1409.2509
Returning to the gravitational context, it is clear that the consequences of our theorem can be avoided by introducing a priori kinematic non-localities violating our assumptions. The gauge/gravity dualities of string theory are examples of this strategy. Indeed, any (Hamiltonian) quantum theory of gravity defined on a separable Hilbert space is completely equivalent to some local field theory - and in fact to a quantum mechanical theory describing a single particle in one dimension - via a sufficiently non-local map. One simply uses the fact that all separable Hilbert spaces are isomorphic to transcribe the Hamiltonian to the Hilbert space of a single non-relativistic particle. As a 0+1-dimensional field theory the result trivially satisfies definition II. The dynamics are also local in time, though when written (perhaps only formally) in terms of the usual position and momentum operators the Hamiltonian need not bear any resemblance to energy functions of Newtonian mechanics. Of course, the above construct requires one to first know the exact spectrum of the gravitational Hamiltonian. This is tantamount to solving the theory. And any construction which first [requires] the theory to be solved will be of very limited use. Allowing the map between theories to be arbitrary non-locality thus seems unproductive.

Demystifier
Gold Member
In order for this not to be the case you'd need some hidden degrees of freedom that aren't described by either theory.
That's a good point. But it's always possible that there are hidden degrees (not described by given theory) which makes the physics of two mathematically-equivalent theories different. For an example in string theory, see
http://lanl.arxiv.org/abs/hep-th/0605250

atyy
That's a good point. But it's always possible that there are hidden degrees (not described by given theory) which makes the physics of two mathematically-equivalent theories different. For an example in string theory, see
http://lanl.arxiv.org/abs/hep-th/0605250

Noooooooo...... There should be a Bohmian AdS/CFT - I see you are working on it from your question about the classical limit :p

Actually, I don't know whether the AdS side is really defined, because the string perturbation is probably divergent. So I tend to think of the CFT as the non-perturbative definition.

Demystifier
Gold Member
Noooooooo...... There should be a Bohmian AdS/CFT - I see you are working on it from your question about the classical limit :p
Maybe I do, maybe I don't. ;)

atyy
Is the Wikipedia article on it any good? AdS/CFT correspondence - Wikipedia

This correspondence may reflect something deeper, but if so, nobody has discovered it.

That aside, some similar correspondences have been discovered in the past.

• Quantum mechanics - Classical limit (equivalent to Newtonian / SR / GR equations of motion)
• Path integral - Lagrangian
• Heisenberg - Hamiltonian
• Schroedinger - Hamilton-Jacobi
The quantum ones are all equivalent, as are the classical ones.

Also, the inverse-square force law gives a conserved quantity in addition to the energy and the angular momentum: the Runge-Lenz vector or the eccentricity vector. It points in the pericenter direction. Using quantum-mechanical operator algebra in n space dimensions, angular momentum generates Lie algebra SO(n), the n-D rotation one, while adding the R-L vector generates Lie algebra SO(n+1) -- rotation in an additional dimension.