Recent content by Othin

Oh, that's right. Thank you very much, I now have the right result.

I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...

I didn't find a way to edit the first post (which is strange, I could swear to have edited questions that were, at the time, older than this one is), so I'll write it as a new reply rather than creating a new thread. Looking up other sources, I realized I might be making a confusion about the...

Thank you for answering. I should have added that the boundary conditions are strong enough to compactify the plane, which if I understand correctly happens because they restrict the scalar field to be of the form ##\phi=e^{in\theta}## at infinity. That makes sense. I think I've seen a similar...

Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...

wow, thanks guys! This is very comprehensive and I think it really solved my problem! :smile:

Oh, that's right. It's my bad here. It should read ##D_{\mu}\Phi=\partial_{\mu}\Phi + [A_{\mu},\Phi] ##. That's how the book states it. I used the right one on my calculations though, so it shouldn't be the source of the problem.

The expressions were taken from Topological Solitons from Niicholas Manton and Paul Sutcliffe .##\Phi## is stated to be an element of the Lie Algebra, as is ##A_{\mu}##. Wouldn't ##F_{\mu\nu}## be a 2x2 matrix (in the case of SU(2)) since ##[A_{\mu},A_{\nu}]## is?

Well, ##\Phi## is here a scalar field valued in the underlying Lie Algebra. The notation is that of the book Topological Solitons from Niicholas Manton and Paul Sutcliffe (for example, equation (8.40) and (8.41)). I too used the distributive law, but I don't see why...

Must it be zero, tough? I thought I'd get rid of it by using the product rule to write ## \partial_{\mu}\operatorname{ad}A_{\nu}\Phi=\operatorname{ad}A_\mu (\partial_\nu\Phi) + (\partial_{\mu}\operatorname{ad}A_\nu )\Phi ##.

I do see the problem with the notation, I'll try using the one you suggested instead. But still, if equation (6) is the same expression as that in (3), why isn't it ##F_{\mu\nu}=\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} +\rm{ad} ([A_{\mu},A_{\nu}])##? Isn't ##\rm{ad}...

But doesn't ##[X,\Phi]=X\Phi - \Phi X?##

I'm trying yo verify the relation \begin{equation} [D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi, \end{equation} where the scalar field is valued in the lie algebra of a Yang-Mills theory. Here, \begin{equation} D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi], \end{equation} and \begin{equation}...

From (6) to (7) must I do another expansion (around ##\mu_o##)? There is no equation relating ## \mu_o \text{and } \mu## that I remember, but neither my expansions around ##\mu_o## nor T=0 lead to the desired expression :(.

Wow, thanks a lot! I was carrying the expansion around the ## \mu_o##. So, let me see if I got it right: I make a Taylor expansion around x=0, but that implies T=0, and, for that reason and the definition of the fermi energy, I wrtite ##\mu_o ## insead of ## \mu##?