Timeline for Is there a decomposition exists for $e^{c(K_++K_-)^2}$
Current License: CC BY-SA 3.0
13 events
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| Feb 11, 2018 at 3:03 | review | First posts | |||
| Feb 11, 2018 at 7:01 | |||||
| Feb 10, 2018 at 14:29 | comment | added | Paul Levy | No, I don't think that's the way to put it. The spectrum is a subset of ${\mathbb C}$ (roughly akin to the set of eigenvalues), not a vector space. | |
| Feb 10, 2018 at 13:01 | comment | added | user120129 | Thanks for the answer. Sorry for my ignorance. The operators have infinite dimensional spectrum. I guess that is the way to put it, right? | |
| Feb 10, 2018 at 12:52 | comment | added | Paul Levy | You are taking the square of an operator, which does depend on the representation. If you take the standard two-dimensional representation then $(K_++K_-)^2$ is a multiple of the identity matrix so the question is almost trivial (as any traceless $2\times 2$ matrix squares to a scalar multiple of the identity - to find the scalar you would have to write down the matrices explicitly). I am fairly certain that this is not true for a unitary representation of $\mathfrak{su}(1,1)$. | |
| Feb 10, 2018 at 12:39 | comment | added | user120129 | Does it really depend on any particular representation?? I am working on quantum optics and the operators are bosonic mode operators. But as far as I understood what makes Lie algebraic methods useful is that they doesn't depend on representations but only to their commutation relations. Also even if it was a homework question it is still an interesting question I think, your reaction is strange. | |
| Feb 10, 2018 at 12:23 | comment | added | Paul Levy | Now I'm satisfied that you aren't a student asking for help with homework, but you haven't posed the question very well. When you say "the square of the operator", you haven't specified what operator is associated to each basis element. The standard way of thinking of ${\mathfrak su}(1,1)$ is via $2\times 2$ matrices, but I guess you have a particular unitary representation in mind. | |
| Feb 10, 2018 at 9:53 | comment | added | user120129 | Ok then my next question for mathflow community is: is it possible that such a question can be a homework question? Is it that simple. If you can decompose such operator you can decompose whole range of physically important operators. Up to now we use Trotter type approximations. I have emailed to you. If you want I can give you explicit operators. I am even ready to pay for who can do it. | |
| Feb 10, 2018 at 9:26 | comment | added | Paul Levy | That doesn't really explain why you are interested in this particular question. It looks suspiciously like a homework question. In any case, I don't think it is about research-level mathematics. | |
| Feb 10, 2018 at 0:10 | comment | added | user120129 | Ok there has to be $\pm$ for the second commutation relation so I have edited the question now. Thank you for that. Such operators appears in physics so I would like to now if there exist decompositions for these type of operators. $K_+^2$ and the other operators are just the square of the operator that obeys the defined group relation. | |
| Feb 10, 2018 at 0:07 | history | edited | user120129 | CC BY-SA 3.0 |
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| Feb 9, 2018 at 7:36 | comment | added | Paul Levy | I think your commutation relations are wrong. What do you mean by e.g. $K_+^2$? Also, why are you interested in this question? | |
| Feb 7, 2018 at 20:05 | history | edited | user120129 | CC BY-SA 3.0 |
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| Feb 7, 2018 at 19:24 | history | asked | user120129 | CC BY-SA 3.0 |