Question: What's so extremely difficult about the mathematics used in superstring theory?
I think saying an object is dimensionless differs from saying it can be regarded as dimensionless.
Question: What's so philosophical about realizing that this physical extension must be...
I did some graduate student research on superstring theory, but I lost my enthousiasm for doing any more research in this topic totally.
I also do mathematics, but a lot of mathematics, used in many of the articles I had to study, isn't always used in a clear way. It took ages to realize it's...
Well, you didn't ask to show it.
I wrote: "The 3-sphere (S^3) can be considered as a 'manifold' isomorphic to SU(2) and the real projective space of dimension 3, which is a quotient of S^3 under 'antipodal identification' is isomorphic to SO(3). Both are orientable manifolds."
Working out...
Yes, SO(3) can be interpreted as a quotient group of SU(2).
{I,-I} is an abelian subgroup and it is a normal subgroup of SU(2). A quotient group G/H is always trivially related to a group G and a normal subgroup H of G.
Thus, indeed SO(3) equals (or is at least isomorphic to) SU(2) after...
Well, the Ricci curvature tensor R is symmetric and it is an object with 2 lower indices, but these indices are used to represent R with respect to a coordinate. As it has two lower indices it is cotensor of rank 2, so we need to feed it with 2 vectors.
We can represent R without coordinates...