I am currently thinking about the Dirac equation in curved (1+1)-dimensional spacetime. First I have tried to understand how vectors can be defined in curved space and how the covariant derivative comes off. In the case of the 2D Dirac equation the partial derivative indeed acts on a spinor; that’s why we need the spin connection here.
I always thought about spacetime and spinor space as two different spaces. When we discussed the spin of the electron, we learned that it is described by what are called spinors, which live in a 2-dimensional complex vector space.
My question is the following:
If the spacetime the electron "lives" in is curved, why does the space the spinor "lives" in also get curved? What is the connection between this spaces? Do both of them get curved in some sense in the same way?
