A Proof of covariant derivative of spinor

[baba26]

TL;DR

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 ?

 

[renormalize]

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.

 

[pellis]

Does this answer your question, baba26?

 

[baba26]

@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".

 

[pellis]

@baba26 Yes, good that you noticed this, and it does cancel out, as follows: