These is a question about the Hilbert space structure of quantum mechanics in the context of holography. See page 175 of Thomas Hartman's notes on Quantum Gravity and Black Holes.

In quantum mechanics in the context of holography, why is the Hilbert space finite-dimensional only for a finite region?

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    $\begingroup$ Your questions are pretty unclear, you should clarify exactly what you mean. $\endgroup$ – Javier May 24 '17 at 23:50
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    $\begingroup$ In quantum mechanics, why is the Hilbert space finite-dimensional only for a finite region? – as a counterexample, consider a 1-dimensional particle in the finite cavity. The Hilbert space is given by a superposition of (an infinite number of) Fourier modes, which is what mathematicians indeed call the Hilbert space. $\endgroup$ – Prof. Legolasov May 25 '17 at 0:02
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    $\begingroup$ Yep, that's what I mean. It is almost always infinite-dimensional. $\endgroup$ – Prof. Legolasov May 26 '17 at 4:57
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    $\begingroup$ I took a quick look in the notes, it seems like we are in the context of holographic principle. This is highly non-obvious, quantum gravity-related, etc. You should've emphasized it in the title, the body and the tags of your question. $\endgroup$ – Prof. Legolasov May 26 '17 at 5:02
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    $\begingroup$ In the context of holography, it is conjectured that Hilbert spaces of compact regions are finite and that their dimensionality depends exponentially on the area of the region's boundary. This is just a conjecture, it needs both a more rigorous context (i.e. AdS/CFT or spin networks in LQG) in which it is strictly realized, and lots of experimental evidence. $\endgroup$ – Prof. Legolasov May 26 '17 at 5:04

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