rbwang1225
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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!
Global Symmetries: Understanding ##T^a##s and (1.10)
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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!
Discussion
jfy4
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The finite transformation is the exponential
[tex] \exp(-i \theta_a T^a)[/tex]
here [itex]\theta[/itex] is the generator's parameter and [itex]T[/itex] is the generator of the transformation. This is to linear order
[tex] = 1 - i\theta_{a}T^{a}[/tex]
This is like in quantum mechanics where for a finite translation you would write
[tex] \exp(-ia\hat{p}) \rightarrow 1 - i a \hat{p}[/tex]
for an infinitesimal translation. But you need to know the action of the generator on the coordinates, i.e. how does [itex]x[/itex] look after it gets acted on by this transformation.
[tex] \exp(-i \theta_a T^a)[/tex]
here [itex]\theta[/itex] is the generator's parameter and [itex]T[/itex] is the generator of the transformation. This is to linear order
[tex] = 1 - i\theta_{a}T^{a}[/tex]
This is like in quantum mechanics where for a finite translation you would write
[tex] \exp(-ia\hat{p}) \rightarrow 1 - i a \hat{p}[/tex]
for an infinitesimal translation. But you need to know the action of the generator on the coordinates, i.e. how does [itex]x[/itex] look after it gets acted on by this transformation.