Q&A on String Theory Course Indices/QFT

[latentcorpse]

The first expression he writes down is for flat space. The second one is correct for curved space because we're using the flat coordinates.

I still don't follow. I thought we were using the veirbein to put Latin indices in the curved case and basis indices for the flat case. Surely this would mean the 1st eqn would be for curved space and the second for flat.

 

[fzero]

I still don't follow. I thought we were using the veirbein to put Latin indices in the curved case and basis indices for the flat case. Surely this would mean the 1st eqn would be for curved space and the second for flat.

There's two problems with trying to define spinors on a curved manifold. The first is that a general curved manifold is not invariant under Lorentz transformations, so you can't even define the transformation of a spinor in curved indices. You can define tensors because you have a $$GL(n)$$ group acting on the tangent space, but $$GL(n)$$ does not have spinor representations. So you must introduce local frames to define spinors.

 

[latentcorpse]

There's two problems with trying to define spinors on a curved manifold. The first is that a general curved manifold is not invariant under Lorentz transformations, so you can't even define the transformation of a spinor in curved indices. You can define tensors because you have a $$GL(n)$$ group acting on the tangent space, but $$GL(n)$$ does not have spinor representations. So you must introduce local frames to define spinors.

OK. This makes sense to me. Can you explain the choice of indices in those two equations though? Like, why does he chose latin for one and greek for the other?

 

[fzero]

OK. This makes sense to me. Can you explain the choice of indices in those two equations though? Like, why does he chose latin for one and greek for the other?

I expect that it's because you're using Latin indices for spacetime indices, whether spacetime is Minkowski or curved.

 

[latentcorpse]

I expect that it's because you're using Latin indices for spacetime indices, whether spacetime is Minkowski or curved.

so to summarise, the transformation in Minkowski space (the first one i wrote down) had latin indices because in Minkowski space we can define spinor transformations and this will hold in any basis so we can move to abstract indices.

However, in a curved spacetime we cannot define spinor transformations so we need to move to flat spacetime (where we use greek indices) and the objects we use in the second transformation I wrote down will be related to their corresponding curved space quantities using veirbeins?

 

[fzero]

so to summarise, the transformation in Minkowski space (the first one i wrote down) had latin indices because in Minkowski space we can define spinor transformations and this will hold in any basis so we can move to abstract indices.

However, in a curved spacetime we cannot define spinor transformations so we need to move to flat spacetime (where we use greek indices) and the objects we use in the second transformation I wrote down will be related to their corresponding curved space quantities using veirbeins?

Roughly correct. As far as the curved-space quantities, I can't see any reason why you would learn anything from transforming the local Lorentz transformation matrix into curved indices.

 

[Afunk]

This is beyond me...