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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?

nightmarish
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    Your questions are pretty unclear, you should clarify exactly what you mean. – Javier May 24 '17 at 23:50
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    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. – Prof. Legolasov May 25 '17 at 00:02
  • @ Solenodon Paradoxus: Do you mean that the Hilbert space of a quantum mechanical system in a finite region not necessarily finite-dimensional? – nightmarish May 26 '17 at 04:52
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    Yep, that's what I mean. It is almost always infinite-dimensional. – Prof. Legolasov May 26 '17 at 04:57
  • But the first line in page 175 of the linked notes say otherwise! – nightmarish May 26 '17 at 04:57
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    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. – Prof. Legolasov May 26 '17 at 05:02
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    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. – Prof. Legolasov May 26 '17 at 05:04
  • In the context of ordinary quantum mechanics, the rule of thumbs is that quantum theories on a finite-dimensional Hilbert space corresponds to classical theories on a compact phase space (roughly you expect the dimension of the Hilbert space to be $V/\hbar$ where $V$ is the volume of the phase space, this is related to eg this question). – Luzanne May 26 '17 at 05:16
  • Note that typically phase spaces arise as cotangent bundles on configuration spaces, and are thus non compact, even if the configuration space is. But the 2-sphere can for example be given the structure of a phase space, of which eg the finite-dimensional spin $1/2$ is an admissible quantization, see this answer. – Luzanne May 26 '17 at 05:23

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