Timeline for Infinite-dimensional Hilbert spaces in QM vs. finite-dimensional Hilbert spaces in quantum gravity?
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16 events
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| Oct 17, 2022 at 16:34 | comment | added | Cuhrazatee | Possibly related? Stone-von Neumann Theorem | |
| Jul 27, 2018 at 14:39 | comment | added | Alex Nelson | @skids It is curious to note that Witten and the paper you cited both are working in de Sitter spacetime; I speculate it is folklore that the "Hilbert spaces" for quantum gravity in de Sitter spacetime is "finite dimensional", in the sense of "finite number of degrees of freedom". Actually this is discussed on page numbered 9 (page 10 of the pdf) et seq. in the Witten article linked. | |
| Jul 26, 2018 at 17:12 | comment | added | skids | @AlexNelson: fair enough, I may be confused about this; but then it seems to me that the confusion is widespread, see e.g. arxiv.org/pdf/1806.10134.pdf, in which the authors claim that it `becomes natural to consider theories where Hilbert space, or at least the factor of Hilbert space describing our observable region of the cosmos, is finite-dimensional' (and here they are unambiguously talking about the dimension of the space). | |
| Jul 25, 2018 at 22:10 | comment | added | Alex Nelson | Be careful, Witten seems to be using "dimension" in the sense of "number of degrees of freedom", whereas "Infinite-Dimensional Hilbert spaces" means two different things depending on context... | |
| Jul 25, 2018 at 21:00 | history | tweeted | twitter.com/StackPhysics/status/1022225196621725698 | ||
| Jul 25, 2018 at 18:36 | history | edited | skids | CC BY-SA 4.0 |
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| Jul 25, 2018 at 16:38 | comment | added | skids | You're obviously right about this; but still, from Robertson uncertainty relation in finitely many dimensions I won't be able to infer, say, the standard uncertainty principle for position and momentum. | |
| Jul 25, 2018 at 15:52 | comment | added | probably_someone | You definitely get uncertainty principles in finite-dimensional Hilbert spaces. Anytime you have two non-commuting Hermitian operators, the product of the standard deviations of their expectation values is bounded below by the Robertson uncertainty relation (which is just a generalization of the Heisenberg uncertainty principle). For example, in the two-dimensional spin space of a spin-1/2 particle, the spin operators along the $z$ direction and the $x$ direction do not commute with each other, so there is an uncertainty principle relating the spins measured along the $x$ and $z$ directions. | |
| Jul 25, 2018 at 15:41 | comment | added | skids | But things do seem to be different in finite-dimensional spaces, no matter how large they are. So for instance, you do not get uncertainty principles, as there are no unbounded operators in finitely many dimensions; in what sense exactly could one approximate uncertainty principles by a single large finite-dimensional Hilbert space? | |
| Jul 25, 2018 at 15:26 | comment | added | probably_someone | I think the point is that there shouldn't be much physical difference between an infinite-dimensional Hilbert space and a finite (but very, very, very large) dimensional Hilbert space. For a position-space analogy, treating space as a continuum would be a good approximation if space were discrete on an extremely, extremely fine-grained scale. This is what we do in fluid mechanics all the time - we make continuum approximations of materials that are made of (a huge number of) discrete particles. | |
| Jul 25, 2018 at 15:20 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
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| Jul 25, 2018 at 15:01 | comment | added | skids | The idea, I think, is that the dimension of the Hilbert space corresponds to the entropy of de Sitter space (cf. cds.cern.ch/record/447660/files/0007146.pdf). I don't quite see how this answers my initial questions, though... | |
| Jul 25, 2018 at 14:56 | history | edited | skids | CC BY-SA 4.0 |
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| Jul 25, 2018 at 14:42 | comment | added | Slereah | This may just be linked to the discreteness of space - if you have a discrete spacetime with de Sitter metric (which is closed), there will only be a finite numbers of points in space, which may only require a finite number of dimensions for the Hilbert space. | |
| Jul 25, 2018 at 14:20 | review | First posts | |||
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| Jul 25, 2018 at 14:15 | history | asked | skids | CC BY-SA 4.0 |