Maggiore Book misunderstanding

[kroni]

Well, Look at the image.
If T is a generator so VTV* (with V unitary) is another basis of the representation too, i am totally agree because it satisfy the structure equation. Now, he say that we can find V that set Gij = tr(TiTj) diagonal BUT when i try, i have :
Gij = Tr(VTiV*VTjV*)
= Tr(VTiTjV*) because V is unitary
= Tr(TiTj) because Tr(AB) = Tr(BA)
So V as no effect and it can't diagonize it. I don't understand why it don't work ?

Thanks for all

 

[Demystifier]

I think you are right. Indeed, the Killing form (Cartan metric) can be expressed in terms of structure constants, which clearly don't depend on V.
https://en.wikipedia.org/wiki/Killing_form
https://www.encyclopediaofmath.org/index.php/Killing_form

To diagonalize $G_{ij}$ (for the case it is not already diagonal), the diagonalization matrix should act in the vector space in which $G_{ij}$ are components of a tensor, i.e. the diagonalization matrix should itself have the $ij$ components. It seems that the author of the book failed to distinguish different vector spaces, which is a mistake similar to that in
https://www.physicsforums.com/threads/do-we-really-mean-hermitian-conjugate-here.858987/

 

[kroni]

To conclude, i send an email to Maggiore himself, he said that the matrix V act directly on Gij, that seems logic but the sentence in the book is confusing because he speak of VTiV* implying that V act on the générators.

Thanks for the answer.

 

[Demystifier]

To conclude, i send an email to Maggiore himself, he said that the matrix V act directly on Gij, that seems logic but the sentence in the book is confusing because he speak of VTiV* implying that V act on the générators.

Thanks for the answer.

At the very least, I think he would need to rewrite this (small and inessential) part of the book.

 

[vanhees71]

I'd say that's one of the most essential parts of any book on QT, because Lie algebras are at the heart of all QT :-).

 

[Demystifier]

I'd say that's one of the most essential parts of any book on QT, because Lie algebras are at the heart of all QT :-).

Then why books on non-relativistic QM (which is also a part of quantum theory) rarely mention Lie algebras?
I'm sure every branch of theoretical physics can be expressed in terms of Lie algebras, but I think they are really essential only in Yang-Mills gauge theories.

 

[vanhees71]

That speaks against the books. Already angular-momentum algebra is a (non-)abelian Lie algebra. Also, how do you motivate the commutation relations of the observables if not via the Lie algebra of the Galilei group? I think, you can not overstate the importance of Lie algebras and Lie groups in QT!

 

[Demystifier]

Also, how do you motivate the commutation relations of the observables if not via the Lie algebra of the Galilei group?

Ask Heisenberg!

 

[Demystifier]

I think, you can not overstate the importance of Lie algebras and Lie groups in QT!

I certainly can't, but you can.