This is a doubt about Polchinski's Volume 1, equation 8.3.21 and below. There is a Matrix $M_{ij}=j_i \otimes \bar j_j$, where $j_i$ is a current (right or left) transforming in the adjoint representation of $SU(2)$. Polchinski then uses a ''$SU(2) \times SU(2)$ rotation to diagonalize M''.
My question is how this is done (or if anyone can give me a reference).
I am confused because I know that under a rotation, $\sigma_x$ (which transforms in the adjoint, and can be identified with $j_x$) can be transformed to $\sigma_y$ by a rotation around $\hat z$ direction: $\sigma_x\rightarrow Exp[i \phi \sigma_3] \sigma_x Exp[-i \phi/2 \sigma_3]=\sigma_x cos\phi - \sigma_y sin\phi$. However, I don't see how to ''eliminate'' components from the matrix $M_{ij}$ defined above to take it to a diagonal form.
Is it possible to do the same thing with other gauge groups? For example $SU(3) \times SU(3)$.
Thanks for reading
If you live in a country that’s majority x, of course the majority of characters are going to be a part of that. Movies, tv shows, and games are all going to be majority x. Statistically it makes sense because in order to change that you would have to actively go out of your way to exclude that group. Even more importantly, movies and tv shows and games are designed to make the most amount of money possible. In order to do that they market towards the largest market that might buy their product. That means they will almost always market to the majority, not the minority. One exception being in the middle of a franchise. If you already have a bunch of successful movies out and you are releasing more, you can generally afford to throw in a movie not marketed towards the majority because they’ll watch it anyway to stick with the storyline.
Absolutely unforgivable historical inaccuracy
Dr Mark Spokane entered the private ward just as tea was served. Jill Hakath clapped her hands in delight, then motioned for the service staff to set another cup on the table. The soft clink of porcelain accompanied the wafting aroma of tea and scones.
“No, no,” said Mark, “I can’t stay long, I’ve just come in to make sure that the patient is-”
“I insist!” said Jill, with faux sternness. “My father’s recovering just fine, and I’ve still not had the chance to properly thank you or your clinic yet. Please! Join us!”
Mark bit his lip. Though the corners of her lip were turned up in amusement, there was no hiding the strength of command in her voice. He patted his pockets, just to make sure the package was safe. That set his mind at ease somewhat, and he eased into the armchair next to Jill. From his backpack he retrieved the folder on Bram Hakath, which he cracked open on his lap.
“How is his appetite? Has he managed to-”
Jill laughed, then patted Mark’s knee. “You worry too much. I have never seen him better than he is now, see for yourself.”
That much was true. Bram sat a distance away from them, his back to the window, painting easel and brush in his hands. Under the warm sunlight, Bram seemed like an entirely different person. There was colour in his cheeks, and the wispy hairs on his head were smoothened down, tucked neatly behind his ears. He was dressed in plaid trousers and a blue polo, which were a far cry from the rags Mark had first saw him in. Mark blinked just to make sure this was the same person he had rescued the day before.
“Shouldn’t we invite him to join us?”
“Oh, let him be,” said Jill. “That’s his first love, you know. Painting means the world to him, and for him to be able to enjoy it again, without all the… stress and pain… it just means so much to see him so happy…”
There were tears in Jill’s eyes, which Mark found interesting. He bit his lip, then focused on shuffling a scone onto his plate. Maybe he was wrong. Maybe he had gotten it all wrong. Maybe there really wasn’t anything to worry about after all.
“That reminds me,” said Jill, as she dipped into the purse next to her. She unfolded a check, then slid it across the table towards Mark. The number of zeroes printed neatly on the slip represented a multi-year income for Mark’s medical practice. “A small token of appreciation from the Hakaths. If you had not stepped in when you did, our father may have been seriously hurt.”
“I did what any doctor would have,” said Mark.
... keep reading on reddit ➡>According to friends' and colleagues' online posts, Grigoryeva was worried about her safety after she found her name and personal information listed on the snuff site.
>
>“I learned today that Lena asked a mutual friend to take care of her cat in the event of her death when she was threatened with murder,” friend and fellow activist Dinar Idrisov wrote on Facebook.
>
>“The state of Russia was obliged to guarantee her the right to life,” Idrisov wrote. “Lena and her lawyer appealed to law enforcement agencies both on the fact of violence and on the fact of threats, but there was no noticeable reaction.”
>
>On Tuesday, Idrisov updated his post to note that Grigoryeva's mother had identified her body and that investigators in St. Petersburg had taken over the case. Other activists said on social media that Grigoryeva's body was found near her home over the weekend with stab wounds and signs of strangulation, The Moscow Times reported.
>
>On July 18, just days before she was killed, Grigoryeva posted an alert on her Facebook page about the “Saw” website.
>
>Grioryeva wrote that the site, which organizes a “hunt for homosexual, bisexual and transgender people,” went online in spring 2018 and was shut down several times but always popped back up. The website posted the personal data of “presumably LGBT+” activists, “including photos and addresses,” and offered prizes to those who completed an attack.
>
>“Law enforcement agencies have still not done anything to find the creators of this ‘game’ and bring them to justice,” Grigoryeva wrote. She called on those opposed to the website to contact the Russian prosecutor’s office and the FSB, Russia’s intelligence agency.
>
>“IMPORTANT!” Grigoryeva wrote. “The Russian LGBT network has repeatedly tried to find people affected by the actions of this group, but failed. We did not find a single attack case directly related to this group.”
>
>Three days after her post, Grigoryeva was dead.
And finally,
>**Police in St. Petersburg confirmed she had reported being threatened repeatedly, but said the threats had no
... keep reading on reddit ➡Let G be a locally-compact group.
It is well-known that if G is abelian then any irreducible unitary representation is one-dimensional.
And if G is compact then any irreducible unitary representation is finite-dimensional.
What is weird about that is that for a group being abelian and being compact are two very different things. The first one is an algebraic property and the second one is a topological property. They don't seem to be related at all. Yet when it comes to the dimension of the irreducible unitary representations they imply very similar properties.
Is there any deeper reason for that or is that just a coincidence?
On a related note: Are there any implications the other way around? That is:
Are there non-abelian groups such that every irreducible unitary representation is one-dimensional?
Are there non-abelian non-compact groups such that every irreducible unitary representation is finite-dimensional?
This is all I have at the moment, I'm picturing it as a cross between The Lion in Winter and His Girl Friday.
Thoughts?
