Spin $1/2$ motivation for twistor theory in general relativity?

Question

I was watching this Youtube video (I have linked it at the relevant timestamp) and to paraphrase Dr. Woit' s motivation for twistor theory:

Within the standard way of thinking about general relativity framework it is difficult and alien to describe the spin $1/2$ particle. You can't do it at all. You have develop a mathematically more sophisticated way of thinking of general relativity and it's very artificial.

Question

Can someone flesh out the mathematical details of this argument (more specifically how twistor solves it and why the standard way is artificial)?

Answer 1

Most standard GR textbooks approach the problem of curved space-time using the notion of tensor fields. Spin 1/2 particles, on the other hand, are defined using spinors which are semi-tensor like quantities. Although one can define these particles using the Dirac equation, it works in flat space-time... and there doesn't seem to be an intuitive and natural way to extend this equation to a general curved space-time. The issue here is that we are trying to define the transformation of spinor like quantities using tensor fields, which is quite artificial (at least how I interpreted Dr Woit's statement).

So a better way is to start with 2-component spinor fields. To ensure that a manifold M admits spin bundle, it should satisfy two conditions: (1) M must be space-time orientable (2)The topological invariant - second Stieffel-Whitney class must vanish

These Spinors are complex quantities and it can be shown that the real tensor fields are a special subclass of these spinorial quantities. So all of the standard tensor transformation that we use in GR can be derived from a general class of spinor transformation. In other words, given that M admits spin bundle, these 2 component spinor fields can be interpreted as more fundamental than tensor quantities. It follows that you can describe any arbitrary spinor field ( spin n/2) in a general curved space-time using a 2-component spinor treatment. (Point to note: this description is very particular to (3+1) dim space-time) Read: " Spinors and Spacetime " Volume I - R. Penrose, W. Rindler.

Twistors are defined using 2-components spinor fields only. A Twistor is essentially a pair of spinors that describes the momentum and angular momentum of a massless particle. In Twistor theory, you can describe massless spinor fields in a flat space-time using Twistor functions (via what is known as Penrose transform)...and this is a 1-1 correspondence. The motivation for Twistor theory is to give an alternate geometrization of conventional physics where your space-time plays a secondary role...and various fields and curvatures are essentially an emergent property. Either way, these 2-component spinor fields play a key role in all of this description. ( Suggested reading: "Twistor theory- An approach to quantisation of fields and space-time" by R. Penrose, MacCallum (1972) )