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Does the universe obey the holographic principle due to Stokes' theorem?

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

Can this theorem be enough proof of our Universe being a hologram – the choice of $\omega$ and $\Omega$ is completely arbitrary!

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up vote 12 down vote accepted

No, it cannot be enough. Stokes' theorem says that the volume ($\Omega$) integral of $d\omega$, a form that is the exterior derivative of another one (of $\omega$), may be written as a surface integral. But it doesn't allow us to rewrite the volume integral of a general integrand (which isn't the exterior derivative of anything) such as the Lagrangian density ${\mathcal L}$ as a surface integral. So the Stokes' theorem is useless for dealing e.g. with the action $S$ that defines the dynamics of a general theory in the volume.

One should mention that when the action is topologically invariant, ${\mathcal L}$ may indeed be locally written as a "total derivative", and in that case, the theory has indeed a provable relationship with lower-dimensional theories (a major example is Chern-Simons theory in 3 dimensions and the related WZNW theories in 2D). But the general theories we know – the Standard Model coupled to gravity – aren't of this special type, at least not manifestly so. What's happening in the volume is general – we surely do care about values of some fields such as the electric field in particular places of the volume – and there apparently isn't any "counterpart degree of freedom" on the surface that we could associate it with.

Some people including Leonard Susskind and Steve Shenker etc. do suspect that there exists some "conceptually simple" proof of the holography in which almost all the degrees of freedom in the volume would be unphysical or topological – some huge gauge symmetry that allows one to eliminate all the bulk degrees of freedom except for some leftovers on the surface. But such a proof of holography remains a wishful thinking. Meanwhile, we have several frameworks – especially the AdS/CFT – that seem to unmask the actual logic behind holography. The surface theory is inevitably "strongly coupled" (i.e. strongly dependent on quantum corrections) if the volume description appears at all so things can't be as simple as you suggest, it seems.

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