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
<br /> \exp(-i \theta_a T^a)<br />
here \theta is the generator's parameter and T is the generator of the transformation. This is to linear order
<br /> = 1 - i\theta_{a}T^{a}<br />
This is like in quantum mechanics where for a finite translation you would write
<br /> \exp(-ia\hat{p}) \rightarrow 1 - i a \hat{p}<br />
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.