Are Stokes' theorem and Gauss's theorem examples of the Holographic Principle?

Question

Before I write this question, I'd want to say that I've read this question , and Lubos Motl's answer to it (I found it through the "Questions that may already have your answer").

My question isn't exactly that. I'm asking whether Stokes' theorem and Gauss's theorem are Examples of the Holographic principle . My impression is that it is, since Stokes' theorem, for example, in it's all-intiuitive most general sense, tells us that:

\begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation}

In other words, it relates something (the RHS) on the region to something (the LHS) on its boundary.

So, I had written a blog post about that to summarise my thoughts on Holography and AdS/CFT. However, Mitchell Porter corrected me saying that it really isn't.

So, I just need to confirm whether it is at least an example (of course not the basis) for Holography ? 

Comments

You might be interested by a previous answer

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user Trimok

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Thanks @Trimok !  

Answer 1

The following assumes that the holography to which the OP refers is that which is studied in high energy theory. Holography is not just a framework that relates

something (the RHS) on the region to something (the LHS) on its boundary

It is a framework for studying the equivalence of certain theories, one of which is defined in the bulk of some spacetime manifold with boundary, and the other of which is defined on its boundary. On one side of the equivalence, one has a theory of gravity. On the other side of the equivalence, one has a quantum field theory. In particular, in order to produce an example of holography, one needs to find two such theories, and one needs to show that the quantities that characterize the boundary theory (e.g. correlation functions in a quantum field theory) can be computed in terms of the quantities that characterize the bulk gravity theory, and vice versa.

Stoke's theorem is a mathematical fact about integrating differential forms on manifolds with boundary; it is not an equivalence between a theory of gravity and a quantum field theory. Therefore it would, in my opinion, be quite a terminological stretch to say that it is an example of holography.

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user joshphysics

Comments

Thanks. But the "holography" you mention is simply AdS/CFT (and it's generalisations), isn't it? Can't there be other sorts of holography? .

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user Dimensio1n0

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@DImension10AbhimanyuPS If you go to any high energy physics group at a university, and you say you're giving a talk about holographic so-and-so, they will most likely think you'll be talking about AdS/CFT or one of its generalizations in the way I describe. Of course the English word "holography" can be used in other contexts, but that would just be a different word. Of course, in some loose sense where holography just refers to relating something in the bulk to something on the boundary, Stokes' theorem is a "holographic" statement, but such broad usage in high energy physics is uncommon.

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user joshphysics

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Ok, thanks for the clarification.

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user Dimensio1n0

Answer 2

Often in topological gauge theories, especially Dijkgraaf-Witten theory, the action is gauge invariant on closed manifolds iff the Lagrangian is a closed form. If the Lagrangian isn't closed, then one can put the theory on the boundary of some spacetime one dimension higher and try to use Stokes' theorem to cure the anomaly on the larger composite system. Recently Anton Kapustin and I discussed how and when this works in http://arxiv.org/abs/1404.3230 ;. One can probably greatly generalize this example by proving a Stokes' theorem for the BRST differential.

This means that Stokes' theorem has something to do with anomaly in-flow. Does anomaly in-flow have anything to do with holography? I'm no expert by any means, but I think so. In AdS/CFT the boundary CFT develops a dynamical scale that becomes the AdS curvature radius. This seems to me to mean that the AdS space is there in some sense to cure the conformal anomaly. I'd appreciate any thoughts on this.

Comments

@RyanThorngren Excuse me, but what do you mean by "In AdS/CFT the boundary CFT develops a dynamical scale that becomes the AdS curvature radius."? AdS radius $R$ in units of $\sqrt{\alpha^{\prime}}$ is dual to $\lambda^{1/4}$ in the boundary CFT in the framework of AdS/CFT correspondence. In the canonical example of $\cal{N}=4$ SYM theory, there is no dynamical scale at all on the gauge theory side.

Answer 3

The key point to be aware of is that

• The holographic principle is not about plain restrictions to the boundary.

The principle does not speak about relating A) fields in the bulk to B) their restriction to the boundary.

Instead it speaks about a more subtle and more interesting relation, namely between

A) fields in the bulk

and

B) sources on the boundary.

The asymptotic boundary value \(\phi|_{\partial }\) of a field \(\phi\) in the bulk of an AdS spacetime is not to be identified with a field on the boundary CFT. Instead, it is to be identified with a source of the boundary CFT.

• AdS/CFT :   bulk fields \(\leftrightarrow\) boundary sources

Therefore none of the vast land of discussion of boundary value problems (which includes Stokes' and Gauss' law) is about holography.

After quantization, this relation becomes

• quantum AdS/CFT :   bulk wave function \(\leftrightarrow\) boundary generating partition functions

Just like a wave function in the bulk is a function of the fields, so a generating function for a partition function is a function of the sources: you differentiate it with respect to the sources to get the correlation functions.

Here is an example of a setup much simpler than full AdS/CFT that nevertheless does capture correctly the basic mechanism of the holographic principle (which may be what you are after):

The relation by which

1) states of 3d Chern-Simons theory

are identified with

2) pre-correlators of the 2d WZW model

    ("conformal blocks" namely with functions satisfying the conformal Ward identities, among which the actual acoorelators are to be found)

is an example of the holographic principle, and one that is understood at a mathematically rigorous level. Hence if you want to see a toy example in which to understand what's really going on with the holographic principle, then check out the 3dCS/2dWZW correspondence .