Timeline for Conserved topological charge for d=3 Yang-Mills. G=U(2)
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 28, 2013 at 18:06 | vote | accept | Federico Carta | ||
| Oct 28, 2013 at 18:03 | answer | added | Trimok | timeline score: 3 | |
| Oct 27, 2013 at 13:55 | comment | added | Federico Carta | $\epsilon^{\mu\nu\rho}D_{\mu} F_{\nu\rho}=\epsilon^{\mu\nu\rho}\partial_{\mu}F_{\nu\rho}+\epsilon^{\mu\nu\rho}[A^a_{\mu} t_a,F^b_{\nu\rho}t_b]=\epsilon^{\mu\nu\rho}\partial_{\mu}F_{\nu\rho}+\epsilon^{\mu\nu\rho}f^{abc}A^a_{\mu}F^b_{\nu\rho}$ And therefore for $b=0$ (which is the current associated to $t_0$) I have that it is simply conserved. While for $b=i \quad i=1,2.3$ it is not simply but covariantly conserved. Therefore can I answer question 4) saying that the conserved topological charge arises from the factor $U(1)$ of the gauge group and not the $U(1)$ generated by $t_3$ ?! | |
| Oct 27, 2013 at 13:24 | comment | added | Trimok | @FedericoCarta : Hints : For each matrix $X= X^aT_a$, the definition of $"D_\mu X"$ or $"[D_\mu,X]"$ (a notation for instance used in the Bianchi identities) is $D_\mu X = [D_\mu,X] = \partial_\mu X -ig [A_\mu, X]$ (see for instance formulae $4.29, 4.30$ in this paper). You don't need to have a detailed expression for the $j^\mu$, just use the definition of the $j^\mu$ in function of the $F_{\mu\nu}$ | |
| Oct 26, 2013 at 22:18 | history | edited | Federico Carta | CC BY-SA 3.0 |
added 4 characters in body
|
| Oct 26, 2013 at 21:38 | history | edited | Federico Carta | CC BY-SA 3.0 |
added 925 characters in body
|
| Oct 26, 2013 at 20:02 | comment | added | Federico Carta | $f^{0ij}=0 \ \forall \ i,j$ while $f^{ijk}=\epsilon^{ijk}$. I got this, and I have that the Bianchi Identity for $F^0_{\mu\nu}$ is off course different than the one for $F^i_{\mu\nu}$,but still dont understand if they are simply or covariantly conserved. In few minutes I'll post all the computations I did | |
| Oct 26, 2013 at 19:53 | comment | added | Trimok | @FedericoCarta : Hints : From Wiki and Bianchi identities (you may replace explicitely the indices $\mu,\nu,\rho$, etc.. by $1,2,3$ if it is clearer for you), you have the equation for "conservation" of your dual current. Look at the difference between $f^{oij}$ and $f^{ijk}$, and you will see the difference between "conservation" of $j^\mu_0$ and "conservation" of $j^\mu_i$ | |
| Oct 26, 2013 at 19:51 | comment | added | Federico Carta | Correct. The only difference is in the structure costants. I will post it as soon as possible the computation I did. I have computed the four currents, and they are obviously different, but none of them is simply conserved. Is it possible? And how can one define a conserved charge without an ordinary continuity equation? Is it possible that a linear combination on the current associated to $t_0$ and the current associated to $t_3$ is simply conserved, thus answering to 4) that "the conserved charge" arises from both the U(1) factors? | |
| Oct 26, 2013 at 19:46 | comment | added | Vibert | Well, the only difference between $SU(3)$ and your case really lies in the structure constants, correct? In turn, they show up in the expression for $F_{\mu \nu}.$ Try to write down (if you haven't already done so) the different $j_\mu^a$ in terms of the structure constants. Also in general, if you have done some work, you should always post it - this helps people to see what's going wrong and where you need some tips. | |
| Oct 26, 2013 at 19:41 | comment | added | Federico Carta | I have made an effort. I have done part of the computation (as far as I can get) but can't arrive at the correct answer. I can post it if you want to, or do not believe I first tried and then asked. I have also looked at Peskin-Schroeder but there is nothing similar to this in the whole book. They only treat ordinary Yang Mills in 4 dimension, and not in d=3. If it really is textbook material, could you suggest a book in which they treat Yang-Mills in d=3? I believe that there are 4 different currents, but then I don't understand point 4). | |
| Oct 26, 2013 at 19:21 | comment | added | Vibert | You're asking us to do all the work without making an effort yourself. For 1), you write that the current is conserved because of the Bianchi identity. But in the next sentence you ask us to prove the conservation of the current. 2) is really textbook material. It's the same as asking whether there's only one "gluon" $A_\mu$ or several. 3.) is also a textbook question. Many of us know the answer but it's a waste of time to write down all these computations if they're done in every QFT textbook, e.g. Peskin-Schroeder (the chapter on non-Abelian gauge theories). | |
| Oct 26, 2013 at 18:39 | history | tweeted | twitter.com/#!/StackPhysics/status/394171448660852737 | ||
| Oct 26, 2013 at 17:35 | comment | added | Federico Carta | This article explains in the first pages why the current is conserved in the case of d=3 QED (which is U(1) Yang-Mills) but still is quite different... | |
| Oct 26, 2013 at 17:11 | history | edited | Federico Carta |
edited tags
|
|
| Oct 26, 2013 at 17:04 | history | asked | Federico Carta | CC BY-SA 3.0 |