A Proof of covariant derivative of spinor

baba26
Messages
4
Reaction score
1
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 ?
 
Physics news on Phys.org
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
 
I asked a question here, probably over 15 years ago on entanglement and I appreciated the thoughtful answers I received back then. The intervening years haven't made me any more knowledgeable in physics, so forgive my naïveté ! If a have a piece of paper in an area of high gravity, lets say near a black hole, and I draw a triangle on this paper and 'measure' the angles of the triangle, will they add to 180 degrees? How about if I'm looking at this paper outside of the (reasonable)...
Thread 'Relativity of simultaneity in actuality'
I’m attaching two figures from the book, Basic concepts in relativity and QT, by Resnick and Halliday. They are describing the relativity of simultaneity from a theoretical pov, which I understand. Basically, the lightning strikes at AA’ and BB’ can be deemed simultaneous either in frame S, in which case they will not be simultaneous in frame S’, and vice versa. Only in one of the frames are the two events simultaneous, but not in both, and this claim of simultaneity can be done by either of...
Back
Top