Global Symmetries: Understanding ##T^a##s and (1.10)

[rbwang1225]

I don't know how (1.10) pops up and why the $T^a$s satisfy the Lie algebra.
Is there any physical intuition?

Any comment would be very appreciated!

 

[jfy4]

The finite transformation is the exponential
$$\exp(-i \theta_a T^a)$$
here $\theta$ is the generator's parameter and $T$ is the generator of the transformation. This is to linear order
$$= 1 - i\theta_{a}T^{a}$$
This is like in quantum mechanics where for a finite translation you would write
$$\exp(-ia\hat{p}) \rightarrow 1 - i a \hat{p}$$
for an infinitesimal translation. But you need to know the action of the generator on the coordinates, i.e. how does $x$ look after it gets acted on by this transformation.