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

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:

$$\int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega.$$

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 courese not the basis) for Holography ? .

• You might be interested by a previous answer Commented Aug 27, 2013 at 18:41

The following assumes that the holography to which the OP refers is that which is studied in high energy thoery. 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.

• @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. Commented Aug 28, 2013 at 2:27
• To me, the holographic principle is much more specific than the asker of the original question implies, but definitely more general than just "generalizations of AdS/CFT". It's the statement that the number of degrees of freedom within a region of space is proportional to the surface area of its boundary (its rigorous form being essentially Bousso's covariant entropy bound). I don't think the physics on the boundary has to be describable by a quantum field theory. But it seems reasonable to say that the bulk theory must be a theory of quantum gravity. Commented Sep 11, 2022 at 4:22