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What are Twistors? Why are they important?

This particular statement in Wikipedia is intriguing

According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers.

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Your question can be answered at many levels. I'll keep it simple; I'm not very well versed with the grand picture.

Twistors provide an efficient (and possibly natural) means to encode the kinematics of massless particles and the resulting conformal symmetry. For a nice and clear introduction, check Witten's lecture notes from PiTP 2004: http://www.sns.ias.edu/pitp2/2004/schedule.html

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