Timeline for How to implement a Hilbert space on a manifold?
Current License: CC BY-SA 4.0
16 events
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| Jan 24, 2023 at 18:31 | comment | added | Níckolas Alves | @RyderRude A KG wavefunction doesn't uniquely determine the state of the system (you also need its derivative), so you are correct that it fails as a wavefunction. This is one of the failures of Relativistic QM that led to the development of field theory. However, as you said, we are making a correspondence between the classical solutions and the one-particle Hilbert space to get to a Fock space in which evolution is indeed unitary | |
| Jan 24, 2023 at 16:44 | comment | added | Ryder Rude | @NíckolasAlves Much thanks :) I initially thought that the classical solutions of Klein Gordon Eqn. did not follow unitary evolution, so we couldn't directly use them as wavefunctions, which is why my mind went into assigning one basis vector to each solution. But now I think I understood. We're merely getting a one-to-one correspondence between the classical solutions and a Hilbert space. | |
| Jan 24, 2023 at 15:59 | comment | added | Níckolas Alves | Do notice my knowledge of geometric quantization is quite limited, though | |
| Jan 24, 2023 at 15:59 | comment | added | Níckolas Alves | As for geometric quantization, I also can't say for sure because I know little about geometric quantization. I do believe (perhaps "I guess" would be a better word lol) they are related: if you check Wald's book, you'll notice his detailed approach involves writing the classical theory in a symplectic manifold, describing the dynamics in terms of a symplectic form and later use this symplectic form to define the inner product on the one-particle Hilbert space, for example. I am guessing that this might be geometric quantization because one is using geometry to quantize the theory | |
| Jan 24, 2023 at 15:57 | comment | added | Níckolas Alves | Perhaps an intuitive view of why this is a one-particle Hilbert space is to think about trying a similar construction with non-relativistic QM. In this case, the one-particle Hilbert space is given by the solutions to Schrödinger's equation. If we had two particles, the state would be on the tensor product of this Hilbert space with itself, and so on for more particles. Hence, each classical solution of the Schrödinger equation (i.e., each wavefunction) corresponds to a one-particle state | |
| Jan 24, 2023 at 15:55 | comment | added | Níckolas Alves | @RyderRude While I've heard the term "wavefunctional" before, I'm not familiar with it, so I can't comment on it properly. However, in the construction I outlined, there is a vector for each classical solution, not a basis. Perhaps the construction you suggested will lead to an equivalent theory, but I can't state it for sure. | |
| Jan 24, 2023 at 13:00 | comment | added | Ryder Rude | Also, this reminds me of geometric quantisation where we construct a pre-Hilbert space by complexifying the phase space. Since each phase space point corresponds to a classical solution, it seems similar to me. Is this related to geometric quantisation? | |
| Jan 24, 2023 at 12:51 | comment | added | Ryder Rude | I'm confused about the part where you build the Hilbert space (sixth paragraph). You say that the vector space of classical solutions corresponds to the Hilbert space. I am thinking that we label our Hilbert space basis vectors by the classical solutions :there's one basis vector for each solution. Then the Hilbert space spanned by this basis will be the space of wavefunctionals, which is equivalent to a Fock space after a change of basis. Then why do you say that this Hilbert space that we constructed is merely the single particle Hilbert space? | |
| Dec 31, 2021 at 0:37 | comment | added | Níckolas Alves | @tonetillo4 I've added a new section on it! =D | |
| Dec 31, 2021 at 0:35 | history | edited | Níckolas Alves | CC BY-SA 4.0 |
New section on Schwarzschild spacetime
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| Dec 30, 2021 at 11:42 | comment | added | tonetillo 4 | there aren't enough points to give you in the forum if you could write specifically about the schwarzschild case hahaha. I would appreciate that very much. | |
| Dec 30, 2021 at 9:08 | comment | added | Níckolas Alves | @tonetillo4 You are welcome! Yes, a Schwarzschild spacetime is stationary, and hence one can define a quantum field theory on it by using the generic prescription for stationary spacetimes. If you are particularly interested on the case for a Schwarzschild spacetime, I don't mind editing the answer to add an extra section mentioning an example of a consequence obtained in this case (namely, one can get a generalization of the Unruh effect) | |
| Dec 30, 2021 at 8:50 | comment | added | tonetillo 4 | Just what I was looking for, thank you. The discussion is valid for a schwarzschild ST right? | |
| Dec 29, 2021 at 20:56 | vote | accept | tonetillo 4 | ||
| Dec 29, 2021 at 10:35 | history | edited | DanielC | CC BY-SA 4.0 |
added 11 characters in body
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| Dec 29, 2021 at 6:26 | history | answered | Níckolas Alves | CC BY-SA 4.0 |