A Proof of covariant derivative of spinor

baba26
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Looking for a proof that the covariant derivative defined using spin connection transforms as expected.
I have read that we can define covariant derivative for spinors using the spin connection. But I can't see its proof in any textbook. Can anyone point to a reference where it is proved that such a definition indeed transforms covariantly ?
 
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baba26 said:
TL;DR Summary: Looking for a proof that the covariant derivative defined using spin connection transforms as expected.

I have read that we can define covariant derivative for spinors using the spin connection. But I can't see its proof in any textbook. Can anyone point to a reference where it is proved that such a definition indeed transforms covariantly ?
There are many textbook references. One example: Weinberg Gravitation and Cosmology (1972), section 12.5.
 
Does this answer your question, baba26?

Covariant derivative using spin connection 1 of 2.jpg

Covariant derivative using spin connection 2 of 2.jpg
 
@pellis , in the (second)last line of the proof, why did you drop the partial mu of S(Λ) term ? Is it zero for some reason ?
I am talking about the line before "Thus".
 
@baba26 Yes, good that you noticed this, and it does cancel out, as follows:
Covariant derivative using spin connection Reply to query.jpg
 
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