Proof of covariant derivative of spinor

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Discussion Overview

The discussion centers on the proof of the covariant derivative of spinors using the spin connection, specifically addressing the transformation properties of this definition. Participants seek references and clarifications regarding the proof's presence in textbooks.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant requests a reference for the proof that the covariant derivative defined using the spin connection transforms covariantly.
  • Another participant suggests a specific textbook, Weinberg's "Gravitation and Cosmology," as a potential reference for this proof.
  • A participant questions a specific step in a proof regarding the treatment of the partial derivative of a term, seeking clarification on why it was dropped.
  • A later reply confirms that the term does cancel out, implying a resolution to the previous question but without detailing the reasoning.

Areas of Agreement / Disagreement

Participants appear to have differing levels of understanding regarding the proof, with some seeking clarification and others providing references and partial confirmations. The discussion remains unresolved regarding the completeness of the proof and its presentation in available literature.

Contextual Notes

The discussion highlights potential gaps in textbook references and the need for clarity on specific mathematical steps in the proof. There is an assumption that the transformation properties are well-defined but not universally agreed upon in the context of the discussion.

baba26
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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 ?
 
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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".
 
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@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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