339. Twistors: getting more formal
ASTRO, HEP-TH/PH — By Dmitry Podolsky on April 6, 2009 at 10:55 pm
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After discussing (or rather musing about) generalities related to twistor formalism, let me now get a bit more formal – I hope it will finally help you to understand what I was talking about in the previous posts 
As was mentioned before, the twistor space corresponding to 4-dimensional real Minkowski spacetime is a complex projective space 
– that is, by definition we introduce complex coordinates 
such that 
and the points 
and 
are identified for arbitrary 
.
It is possible to describe lines in by a pair 
, 
for any . The set of these straight lines therefore depends on four complex parameters. Conformal structure in twistor space is defined by the condition that the distance between any straight line 
crossing a given line 
and the line itself is zero (therefore the line belongs to the light cone with origin somewhere in ).
Let us consider a real hypersurface (called Hermitian quadric) given by the equation
.
It divides the twistor space into two parts: the form 
is positive in one part and negative in the other. The set 
of all straight lines belonging to this Hermitian quadric depends only on 4 real parameters and therefore represents conformal compactification of Minkowski space. Cones of all lines from the set that cross the given line are just light cones. If we want to get the usual Minkowski space, we need to choose a particular line 
(for example, 
, 
) and remove all the lines belonging to the set 
from the twistor space.
That’s how 4-dimensional flat space with Minkowski signature is embedded into the twistor space … Euclidean space (
) can be embedded into as follows. Consider a set of straight lines connecting points and 
. All these lines either do not intersect or coincide. This way we can divide into classes of non-intersecting lines or introduce fibration in other words.
Finally, let us talk a bit about various symmetries associated with twistor space and various embeddings discussed above. First of all, the group of projective transformations of space is 
. It has a subgroup 
which conserves the quadric above. It therefore induces the group of conformal transformations of Minkowski space. If we want to keep the line defined above fixed, the corresponding subgroup of the group is nothing but Poincare group of Minkowski space. Finally, if we fix one more line not crossing , we will get Lorentz group.
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