Timeline for Hermitian (?) representations of $su(2)$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 9, 2020 at 7:19 | history | edited | MAS | CC BY-SA 4.0 |
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| Sep 10, 2019 at 20:09 | vote | accept | Laurent Claessens | ||
| Aug 19, 2019 at 16:57 | answer | added | Torsten Schoeneberg | timeline score: 5 | |
| Aug 11, 2019 at 10:44 | history | edited | Laurent Claessens | CC BY-SA 4.0 |
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| Aug 11, 2019 at 10:34 | comment | added | Laurent Claessens | For me, the definition of an hermitian operator $A$ is that $<Av,w>=<v,Aw>$ for every vectors $v,w$. Since a representation is a purely algebraic thing, when a representation $(\rho,V)$ is given, I can craft a scalar product on $V$ for which the equality $<\rho(J)v,w>=<v,\rho(J)w>$ does not hold. The point is that the conditions about "being a representation" do not mention the scalar product on the representation space. I edit my question to make it more clear. | |
| Aug 11, 2019 at 10:03 | comment | added | Cosmas Zachos | In the algebra of the Js you wrote, Hermitean transposition has them Hermitean. The rep satisfies the same algebra. | |
| Aug 11, 2019 at 6:24 | comment | added | Laurent Claessens | About my previous comment, the answer is "no". The represented operators do not have to be hermitian. If $(\rho, V)$ is a representation, I can craft a scalar product on V such that the matrices $\rho(J_i)$ are not (anti)hermitian. | |
| Aug 11, 2019 at 6:21 | comment | added | Laurent Claessens | The matrices of su(2) are (anti)hermitian, ok. But my question is : why the REPRESENTED matrices have to be (anti)hermitian ? | |
| Aug 10, 2019 at 22:53 | comment | added | Cosmas Zachos | Most physics texts address those. The J s are hermitean, and not antihermitean, as you started with, so the group elements are ~$\exp (i\theta J)$. The eigenvalues of all eigenvectors are thus real, no? | |
| Aug 10, 2019 at 8:30 | history | asked | Laurent Claessens | CC BY-SA 4.0 |