Srednicki's normalization choice for lie algebra generators

[hideelo]

IN Srednicki's QFT he seems to make two different choices for normalizing the generators of lie algebras. In chapter 24 (eqn 24.5) he chooses Tr (T^aT^b) = 2 δ^ab and in chapter 69 (eqn 69.8) he chooses Tr (T^aT^b) = (1/2) δ^ab

Is there a reason for this? Is there any particular reason to make one choice over the other?

Thanks

 

[fresh_42]

IN Srednicki's QFT he seems to make two different choices for normalizing the generators of lie algebras. In chapter 24 (eqn 24.5) he chooses Tr (T^aT^b) = 2 δ^ab and in chapter 69 (eqn 69.8) he chooses Tr (T^aT^b) = (1/2) δ^ab

Is there a reason for this? Is there any particular reason to make one choice over the other?

Thanks

I don't think that it is an essential difference, as a basis ##T^a## is as good as a basis ##c_aT^a\; , \;c_a\in \mathbb{R}##. However, one will get different eigenvalues: ##T^a.X= \lambda X \Longrightarrow (c_aT^a).X= (c_a \lambda)X## so it is indeed a normalization. I assume we want to end up with half integers as possible values for spins. So the task in chapter ##69## (gauge theory) is different than in chapter ##24## (group theory).

(24.5) is due to a special choice of basis vectors (infinitesimal generators) which makes the following computations easier. It's a kind of natural choice here. E.g. (24.9) is easy to remember. You may chose any other basis, but it will probably lead to more calculations.

It is the same with (69.8). It is a special choice of basis vectors. Other would do as well with different equations then. Basically it is even another Lie algebra, ##\mathfrak{su}(n)## in (69.8) instead of ##\mathfrak{so}(n)## in (24.5). But this isn't the reason for his choice. It is simply a matter of convenience in regard to the following calculations. And in this case for the role of the eigenvalues. The spin of Fermions have been set to half integers first, I think, and QED as their description came second.