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
 
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
I started reading a National Geographic article related to the Big Bang. It starts these statements: Gazing up at the stars at night, it’s easy to imagine that space goes on forever. But cosmologists know that the universe actually has limits. First, their best models indicate that space and time had a beginning, a subatomic point called a singularity. This point of intense heat and density rapidly ballooned outward. My first reaction was that this is a layman's approximation to...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
Back
Top