Conformal compactification and the use of spinors (Twistor theory)

Question

I was reading the book from Huggett and Tod "An introduction to twistor theory" and as the book evolves they reach to the necessity to "found" a Lie derivative of a spinor respect to a conformal vector field. Then, they explain the problem that one has when one wants to work with conformal vector fields, because they don't have a complete representation in the Minkowski space (a type of conformal transformations, defined by this vector fields, sends the light cone to the "infinity"). In order for everything works well, they Compactify the Minkowski space and then all conformal transformation can be represented in this other space without worries.

Done this, they continue working with the spinors of the Minkowski space. They found the Lie derivative of a spinor, and proceed to found the way the zero-rest field equation and twistor equation changes under a conformal re-scaling, and also how their solutions change under conformal transformations (for this, they occupy the Lie derivative).

$\textbf{So here is my question:}$ Why they compactified the Minkowski space if they would still working with the geometry of Minkowski space (which is filed in the spinor representation)?. I mean, if they still working with the spinors of the Minkowski space then the problem that is in doing a conformal transformation still remains, right?

Answer 1

Twistor equations are invariant under conformal rescaling of metric and all geometric notions used in twistors are essentially conformally invariant. Twistors are reduced spinors of the pseudo-orthogonal group O(2,4) which happens to be locally isomorphic with conformal group of Minkowski space. So the description of space-time that you get using twistor coordinates enjoys the fifteen parameter conformal symmetry group, and not just the ten parameter Poincare group. This explains why we need a compactified Minkowski space.