Straightforward twistor theory/spinor calculus question

1. Jan 10, 2013

Bballer152

Not sure where to post this, but this question has been frustrating me for hours now because I'm pretty sure it has a very straightforward answer. I'll reproduce the problem in a moment, but for those with the Huggett and Tod's Intro to Twistor Theory, I don't understand how we get the expression immediately before eq. 8.5 from the given info.

We're given the twistor function $$f(Z^\alpha)=\frac{1}{(A_\alpha Z^\alpha)(B_\beta Z^\beta)}$$ and we restrict the function to the line $$L_x$$ so that

$$A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'}\equiv \alpha^{A'} \pi_{A'}$$ and
$$B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}\equiv \beta^{A'} \pi_{A'}$$.

Now comes the part I don't understand. Apparently, by supposing that the dual twistors $$A_\alpha$$ and $$B_\alpha$$ meet in the line $$L_y,$$ then we should get $$\alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2.$$ When I do this I get a similar answer but my contractions don't work out this way. In particular, my spinors indices contract in a different order and I have no idea how to "split them up", if that makes sense. Any help would be GREATLY appreciated!

Last edited by a moderator: Jan 10, 2013