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I have recently read about the information paradox of black holes and how it lead eventually to the formulation of the holographic principle. Quoting Wikipedia:

"the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — preferably a light-like boundary like a gravitational horizon."

There is a branch of classical physics, where a very similar sounding principle is nowadays almost trivially well-known, namely the field of linear wave propagation (in the language of mathematics, linear hyperbolic PDEs of second order). Huygens' principle, as it was later encoded in Kirchhoff's integral theorem, states roughly that in order to "understand" (i.e. predict, calculate) the wave propagation in a volume, you just need boundary values on the surface of the volume.

Is there any deeper mathematical connection between the holographic and Huygens' principle, or is this pure coincidence?

Qmechanic
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Bryson of Heraclea
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2 Answers2

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Kirchhoff's integral theorem is derived from Green's second identity and that identity is derived from the divergence theorem. Furthermore, the divergence theorem is of course just a special case of the Generalized Stokes theorem.

So we could conclude that Kirchhoff's integral theorem stems from the Generalized Stokes theorem.

Now, looking at this answer which says that there is no physical connection between the Generalized Stokes theorem and the Holographic principle, we can conclude that the answer to the OP's question is negative.

User198
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I know that undergrads usually have a way of approaching AdS/CFT (or more broadly holography, whatever) by saying it is similar to Stokes theorem, but while they look the same (superficially), there are huge differences between these. Here is a very simple way of seeing this. In a holographic theory, given access to some boundary subregion, you can reconstruct very precisely one corresponding dual bulk subregion. That is, given some $R$ subregion on the boundary, you can reconstruct fully, a corresponding entanglement wedge and this is non-local. There is a paper by Godet-Raju-Papadoulaki-someone-else where they remark that from a canonical QG perspective, it is important to note that holography is not the statement that "something" in the bulk can be reconstructed on the boundary, but that everything -- spacelike, null and timelike -- inside the entanglement wedge can be reconstructed from $R$. On the other hand, your wave thingy is described as being strictly timelike/null bound.

meowdib
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