The discussion revolves around questions related to string theory, specifically focusing on indices and quantum field theory (QFT). The original poster presents multiple inquiries regarding the properties of spinors, transposition of expressions involving Dirac spinors, and the implications of certain equations of motion in string theory.
The discussion is ongoing, with participants providing insights and seeking clarification on various points. Some have offered explanations regarding the properties of Grassmann variables and the nature of spinor indices, while others express confusion and seek further elaboration on specific topics.
Participants note the complexity of the questions posed, indicating that they may not be directly related to one another. There is also mention of the need for foundational knowledge in group theory to fully grasp the discussions on spinor indices.
fzero said: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.
latentcorpse said: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 said: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 said: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 said:I expect that it's because you're using Latin indices for spacetime indices, whether spacetime is Minkowski or curved.
latentcorpse said: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?