Timeline for $\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Feb 7, 2020 at 22:48 | answer | added | MannyC | timeline score: 1 | |
| Jun 4, 2015 at 16:45 | answer | added | user82794 | timeline score: 11 | |
| Dec 6, 2013 at 16:40 | vote | accept | Hunter | ||
| Dec 6, 2013 at 14:39 | comment | added | Hunter | @AstoundingJB that makes complete sense. I think I understand where I went wrong with your help. Thanks! If you'd like to, you can put your message in an answer below and I can formally accept your answer. | |
| Dec 6, 2013 at 14:33 | comment | added | AstoundingJB | Let me clarify the answer to your last question. Spacing in transformation matrices like $U_i^{\ j}$ makes sense because it make sense to find their transpose $(U^T)_i^{\ j}\equiv U^i_{\ j}$, which also belongs to $SU(N)$ and describe transformations. Spacing in base tensors do not. | |
| Dec 6, 2013 at 14:06 | comment | added | AstoundingJB | This notation was confusing me as well..! :S Well, without adopting the pompous notation of Giorgi's book, simply denote transformations with upper-case letters, like $U,V,...$, and base states for this tonsorial rep. with lowercase letters, as Zuber does, like $v^i$ or Greek letters, as you did, $\psi^i$. Once you've done this distinction, its straightforward not confusing states and transformations and you won't ask how a transformation $U_i^{\ j}$ transforms or what is the transpose of a state, say $\varphi^i_j$.. $(U^T)_i^{\ j}=U_{\ j}^i$ makes sense but $(\varphi^T)_i^j$.. no! | |
| Dec 6, 2013 at 13:57 | comment | added | Hunter | @AstoundingJB what you wrote actually makes sense and that was indeed my confusion. Now, I also understand why it doesn't make sense to make a distinction on the spacing of the indices for $S^i_j$. But it does still make sense for to space apart the indices on $U^i{}_j$, right? If this is true and you write your answer below, then I will accept it as an answer. Thank you | |
| Dec 6, 2013 at 13:53 | comment | added | Hunter | I think I am actually confusing those two. I will read Georgi again when I come home tonight and pay more attention to this. | |
| Dec 6, 2013 at 13:49 | comment | added | AstoundingJB | Uhm.. wait! I'm wandering if you're confusing the meaning of a transformation matrix, say those you named $U_i^{\phantom{i}j}$, and a base state for this representation, say $S^i_j$ or $T^i_j$... They are very different objects, indeed Georgi write these bases like $\big| ^i_j\big\rangle$, while the $U$s are matrices... The point is that the transpose of a base tensor like $S^i_j=\big| ^i_j\big\rangle$ doesn't actually have much sense... It has no sense mixing the two indices, one fro $\boldsymbol{3}$ and one from $\bar{\boldsymbol{3}}$... | |
| Dec 6, 2013 at 13:27 | history | edited | Hunter | CC BY-SA 3.0 |
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| Dec 6, 2013 at 13:19 | comment | added | Hunter | Thank you both (@Olof and @AstoundingJB) for your answers. The reason why I think it is natural to space apart the indices is because then we could write something like: $S^i{}_j = (S^T)_j{}^i$ where $S^T$ denotes the transpose of $S$. If we don't space the indices apart, then this would become $S^i_j = (S^T)^i_j$ and this would make much sense, right? | |
| Dec 6, 2013 at 11:33 | answer | added | Trimok | timeline score: 7 | |
| Dec 6, 2013 at 11:08 | comment | added | AstoundingJB | Good one @Olof ! +1 I didn't get your suggestion at first! :) Probably, this is the key point for one can't decompose further $\boldsymbol{8}$ into... something! | |
| Dec 6, 2013 at 10:57 | comment | added | Olof | @AstoundingJB: That's my point. It looks like Hunter tries to raise and lower the indices in order to construct the "symmetric" and "anti-symmetric" tensors, but there is no way to do this in $SU(3)$. So the only sensible interpretation is $S_j{}^i - S^i{}_j = 0$, which of course gives an irreducible representation, but not a very interesting one :) | |
| Dec 6, 2013 at 10:49 | comment | added | AstoundingJB | @Olof: there's no need in $SU(N)$ to space apart the indices of the base states of the tensor representation; that is, the state $S_i^{\phantom{i}j}=S^j_{\phantom{j}i}=S^j_i$. Anyway, Hunter , I think that the equation you listed for the symmetric and the antisymmetric part of your $S_i^j$ is meaningless since you are symmetrizing and antisymmetrizing about different objects... | |
| Dec 6, 2013 at 10:01 | answer | added | AstoundingJB | timeline score: 8 | |
| Dec 6, 2013 at 9:28 | comment | added | AstoundingJB | I think there is a problem in the transformations of the symmetric and antisymmetric parts... I don't agree with these equations, let me check it! | |
| Dec 6, 2013 at 8:56 | comment | added | Olof | How would the tensor $S_j{}^i$ be related to $S^i{}_j$? | |
| Dec 6, 2013 at 2:09 | history | edited | Hunter | CC BY-SA 3.0 |
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| Dec 6, 2013 at 1:41 | history | asked | Hunter | CC BY-SA 3.0 |