Timeline for Deriving the Poisson bracket relation of the Ashtekar variables
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2015 at 6:29 | comment | added | thenumbernine | This paper still only states fundamental Poisson brackets for each of the different actions it describes, without explaining how they are chosen, as the Romano, Giulini, and Pullin papers I reference under "my rationality for this assumption" in the question. Maybe I should've asked how fundamental Poisson brackets are chosen in the first place, because that seems to be done in these papers rather than deriving one set from another set. | |
| Oct 31, 2015 at 10:05 | comment | added | thenumbernine | The original question wasn't how to differentiate the square root of a matrix (as it has wildly deviated into). It was how to solve the Poisson brackets of Ashtekar variables. This Addendum is a much better candidate for an answer. Thanks for the link -- I will check it out and see if I can get something from this. | |
| Oct 27, 2015 at 21:54 | history | edited | Alex Nelson | CC BY-SA 3.0 |
added 706 characters in body
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| Oct 21, 2015 at 11:51 | comment | added | thenumbernine | This way works only if the tensor $e\delta e$ is symmetric -- I wrote out my work in the comments. Is there an easy proof that $e\delta e$ is symmetric? I'll start messing with that now... | |
| Oct 20, 2015 at 21:36 | comment | added | Alex Nelson | @thenumbernine It's just the product rule: $\delta_{jk}({e_{c}}^{j} (\partial {e_{d}}^{k}/\partial_{ab}) + {e_{d}}^{k} (\partial {e_{c}}^{j}/\partial_{ab}))$, then set it equal to $\delta^{(a}_{c}\delta^{b)}_{d}$, and you're done. | |
| Oct 20, 2015 at 17:09 | comment | added | thenumbernine | Thanks for the tip! That looks like the approach I've got listed. The $\approx$ is bugging me -- I'm looking for something with a $=$. | |
| Oct 20, 2015 at 15:32 | history | answered | Alex Nelson | CC BY-SA 3.0 |