Proof of covariant derivative of spinor

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SUMMARY

The covariant derivative for spinors can be defined using the spin connection, and its transformation properties are confirmed in the textbook "Gravitation and Cosmology" by Steven Weinberg, specifically in section 12.5. The discussion highlights the need for a clear proof that this definition transforms covariantly, which is addressed through references to established literature. Participants in the forum confirm that certain terms, such as the partial derivative of S(Λ), cancel out during the proof process, ensuring the validity of the transformation.

PREREQUISITES
  • Understanding of covariant derivatives in differential geometry
  • Familiarity with spinors and their mathematical properties
  • Knowledge of the spin connection and its role in general relativity
  • Basic grasp of tensor calculus and transformation laws
NEXT STEPS
  • Study the covariant derivative of spinors in detail using "Gravitation and Cosmology" by Steven Weinberg
  • Research the properties of the spin connection in the context of general relativity
  • Explore advanced topics in tensor calculus related to spinor transformations
  • Examine additional proofs and examples of covariant derivatives in other textbooks or papers
USEFUL FOR

This discussion is beneficial for theoretical physicists, mathematicians specializing in differential geometry, and students studying general relativity who seek a deeper understanding of spinor calculus and covariant derivatives.

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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