# In twistor theory, what's the relation between points with dual Plucker coordinates? Also about a special null line

In twistor theory, each point $Z=[Z0,Z1,Z2,Z3]$ in the complexified Minkowski space $CM$ has a correspondent Plucker coordinate $P(Z)$ embedded in $CP^5$ and we can also find its dual $P(Z)^{*}$. My questions are:

(1) If we have two points $Z_{a},Z_{b}\in CM$ so that their Plucker coordinates are dual to each other so that $P(Z_{a})=P(Z_{b})^{*}$, then what's the relation between $Z_a,Z_b$? Is there a physical picture/interpretation for it?

(2) Another question is: if there is a point $Z \in CM$, $P(Z)$ and $P(Z)^{*}$ are null-separated so that $P(Z)\rightarrow P(Z)^{*}$ is a null line, i.e., $Z$ and $Z',(P(Z')=P(Z)^{*})$ fall on a light ray. It seems for me that this is a very special null line since it connects two dual points. But how special? What's the difference between this 'special' null line and other normal null lines? Does it have a correspondent physical picture? This is a little contradictory since all the light rays should be equivalent,but this special null line seems to hint the existence 'abnormal' light rays. Is there an error or misunderstanding here? Is it possible that in fact every null line do connect a pair of dual points, that's to say there is always a pair of points $Z,Z'$ on any light ray in $CM$ that leads to dual points $P(Z)$ and $P(Z')=P(Z)^{*}$ in $CP^5$? So the 'special' null line is in fact just a normal null line? (Contradiction again! then $P$ and $P'$ become special points in $CM$, but all points should be equivalent!)

(3) Assuming: For every light ray in $CM$, we can always find two points on it, which are dual to each other in $CP^5$; at the same time, for any point $Z \in CM$, we can find a light ray passing $Z$ which corresponds to a null line in $CP^5$ that connects $P(Z)$ and $P(Z)*$. Then nobody is 'special'. The first half of the assumption may be possible, but the second half seems wrong. So the point $Z$ that leads to the null line $P(Z)\rightarrow P(Z)^*$ is really special in the sense that the set of such points is a subset (with 0-measure I guess) of the Minkowski space. A special point in $CM$? It sounds really strange for me.

(4) Assuming that we find the points $Z$,$Z'$ and the correspondent null line or light ray connecting them, they are 'special', then which one is more fundamental? the point $Z$ or the light ray $ZZ'$?

Anyone has an idea? Thanks.

• I saw this yesterday and I must say: I wish I knew enough twistor theory to answer this. It's a bit of a specialized question but I'm sure someone has an answer. – CR Drost Sep 15 '15 at 4:48
• Crossposted to math.stackexchange.com/q/1437447/11127 – Qmechanic Sep 16 '15 at 21:39