Timeline for Factoring a the exponential form of a group element of a Lie group, using subgroups
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 16, 2021 at 8:53 | comment | added | Martin Johnsrud | As my edit above says, it seems that the original paper only require the weaker claim of "in a neighbourhood of the identity of $G$". I guess the strong claim might not be true, nor needed, but i have to look closer at it. | |
| Nov 15, 2021 at 22:19 | comment | added | Valter Moretti | However it should be a known fact if it is true. For instance it is known that if the group is compact then the exponential map is surjective (not necessarily injective outside a neighborhood of the identity). | |
| Nov 15, 2021 at 22:13 | comment | added | Martin Johnsrud | Thank you for considering the question, it has been bugging me. It is always good to hear that your issues are not trivial. | |
| Nov 15, 2021 at 22:07 | comment | added | Valter Moretti | Have a look at Barut Raczak's book on representation theory. | |
| Nov 15, 2021 at 22:05 | comment | added | Valter Moretti | I am not saying that it is not true. Maybe it is. But the proof is not trivial. | |
| Nov 15, 2021 at 22:01 | comment | added | Martin Johnsrud | Well, that is not strong enough. Weinberg claims "Because $t_a$ and $x_i$ span the Lie algebra of $G$, any finite element of $G$ may be expressed in the form $g = \exp(i \xi_i x_i)\exp(i \theta_a t_a)$". Are you saying this is not true? As far as i can tell, this is necessary for the construction of the realization of the Goldstone bosons. | |
| Nov 15, 2021 at 21:45 | comment | added | Valter Moretti | No, not every element. However every element is a finite product of thes products of two factors only. This is consequence of the fact that the group is connected. | |
| Nov 15, 2021 at 21:39 | comment | added | Martin Johnsrud | Is this enough to conclude that any element of $G$ can be written in this way? | |
| Nov 15, 2021 at 21:35 | history | answered | Valter Moretti | CC BY-SA 4.0 |