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

+ 10 like - 0 dislike
281 views

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 ? 

asked Aug 27, 2013 in Theoretical Physics by dimension10 (1,950 points) [ revision history ]
edited Apr 25, 2014 by dimension10

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

Thanks @Trimok !  

2 Answers

+ 6 like - 0 dislike

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

answered Aug 27, 2013 by joshphysics (810 points) [ revision history ]
edited Apr 25, 2014 by dimension10
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
@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
Ok, thanks for the clarification.

This post imported from StackExchange Physics at 2014-03-07 13:40 (UCT), posted by SE-user Dimensio1n0
+ 3 like - 0 dislike

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.

answered Apr 26, 2014 by Ryan Thorngren (1,405 points) [ revision history ]
edited Apr 26, 2014 by Ryan Thorngren

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