Timeline for Canonical quantization of time-dependent lagrangians
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14, 2018 at 13:07 | vote | accept | Prof. Legolasov | ||
| Aug 11, 2018 at 15:31 | answer | added | Qmechanic♦ | timeline score: 1 | |
| Aug 11, 2018 at 0:01 | history | tweeted | twitter.com/StackPhysics/status/1028068802607624194 | ||
| Aug 10, 2018 at 22:55 | comment | added | Prof. Legolasov | @AccidentalFourierTransform still, if you could sketch the general picture, I would appreciate it a lot (upvote included :) ). But no pressure | |
| Aug 10, 2018 at 22:53 | comment | added | AccidentalFourierTransform | Ah, I wish I could. The general definition of the P-DW bracket is much more elaborate than that of the P bracket, and while I am familiar with the general picture, I don't remember the details. Also, the concept is so rich and interesting that it deserves a discussion more thorough that what one may write in an answer here. Just so that you know, if you are willing to download the pdf, it is on Library Genesis (it is not like DeWitt is going to get any money if you buy it anyway...) | |
| Aug 10, 2018 at 22:51 | comment | added | Prof. Legolasov | @AccidentalFourierTransform if you are satisfied with the algebraic approach to quantum theory, then yes, you can promote $x$ and $p$ to a $C^{*}$ algebra straight away and then deduce its representation. However, in some cases (not related to the question at hand) that last part is very hard: there could be multiple representations, and it is always hard to find the one which is physically relevant. Using $a$ and $a^{*}$ ensures that the Hilbert space is built along the quantization correctly, and all operators are represented on it explicitly. I find it nice. | |
| Aug 10, 2018 at 22:49 | comment | added | Prof. Legolasov | @AccidentalFourierTransform thanks for the reference, but the book is behind a paywall. I would appreciate if you could sketch how to use it in the answer to this question. I would be particularly happy if you could first explain how to introduce the bracket in the most general case (arbitrary spacetime manifold, any kind of lagrangian, any parity, etc.), and then briefly explain the aspects relevant to my question here. | |
| Aug 10, 2018 at 22:48 | comment | added | AccidentalFourierTransform | Out of curiosity, why do you promote $a,a^*$ to operators at step 3? I noticed that this is what everyone does, but to me it makes much more sense to promote $x,p$ to operators right at step 1. Why wait? Formulate the theory as a quantum theory from the onset! | |
| Aug 10, 2018 at 22:42 | comment | added | AccidentalFourierTransform | If you want a general recipe for canonical quantisation (one where the variables may have any Grassmann parity; be defined on an arbitrary manifold; where the Lagrangian is completely arbitrary; the system may have any sort of constraint and/or gauge symmetries; etc.), you need to introduce the Peierls-DeWitt bracket, which replaces the Poisson bracket and is much more powerful. The procedure is essentially equivalent, but promoting the P-DW bracket to a super-commutator instead of the Poisson one. You will find a great discussion on DeWitt's The global approach to Quantum Field Theory. | |
| Aug 10, 2018 at 22:39 | history | edited | Prof. Legolasov | CC BY-SA 4.0 |
added 136 characters in body
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| Aug 10, 2018 at 22:34 | history | asked | Prof. Legolasov | CC BY-SA 4.0 |