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Bballer152

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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 [tex] f(Z^\alpha)=\frac{1}{(A_\alpha Z^\alpha)(B_\beta Z^\beta)} [/tex] and we restrict the function to the line [tex] L_x [/tex] so that

[tex]A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'}\equiv \alpha^{A'} \pi_{A'} [/tex] and

[tex]B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}\equiv \beta^{A'} \pi_{A'} [/tex].

Now comes the part I don't understand. Apparently, by supposing that the dual twistors [tex]A_\alpha [/tex] and [tex]B_\alpha [/tex] meet in the line [tex] L_y, [/tex] then we should get [tex] \alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2. [/tex] 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!

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

[tex]A_\alpha Z^\alpha = (iA_A x^{AA'}+A^{A'})\pi_{A'}\equiv \alpha^{A'} \pi_{A'} [/tex] and

[tex]B_\alpha Z^\alpha = (iB_A x^{AA'}+B^{A'})\pi_{A'}\equiv \beta^{A'} \pi_{A'} [/tex].

Now comes the part I don't understand. Apparently, by supposing that the dual twistors [tex]A_\alpha [/tex] and [tex]B_\alpha [/tex] meet in the line [tex] L_y, [/tex] then we should get [tex] \alpha^{A'}\beta_{A'}=\frac{1}{2}A_AB^A(x-y)^2. [/tex] 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!

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