100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

496 R. Penrosecorresponding notion should be for a “power series” in ∂/∂Z α .It would appear to be probable that some insights into the appropriate“quantum twistor geometry” are to be obtained from the procedures ofnon-commutative geometry hh applied to the original twistor space T, sincehere we have basic “coordinates” Z α and ¯Z α which do not commute, thesebehaving formally like Z α and −∂/∂Z α I am not aware of any detailedwork in this direction, however. In the absence of this, I wish to makesome pertinent comments that seem to address this kind of issue from asomewhat different angle.There is at least one way in which the replacements¯Z α − ∂∂Z αand∂∂¯Z α Z αdo find a clear mathematical representation in important twistor expressions.This is in the (positive definite) Hermitian scalar product 〈f|g〉 betweenpositive-frequency twistor functions (1st cohomology elements) f andg, each of a given homogeneity degree r. Let us choose another such twistorfunction h, but now of homogeneity r − 1. Then we find the relations〈 ∂f∂Z α |h〉 = −〈f|Zα h〉 ,〈Z α h|f〉 = −〈h| ∂f∂Z α 〉 ,which is consistent with the above replacements, where we must bearin mind that the quantities in the “〈. . . |” actually appear in complexconjugateform and that in the first of these relations a minus sign comesabout when the action of “∂/∂¯Z α ” is transferred from leftward to rightward.To see how to ensure that these relations are satisfied, we need thegeneral form of the scalar product, but where (for the moment) I restrictattention to cases for which r > −4. We find that this scalar product takesthe form∮〈f|g〉 = c ¯f(W α )[W α Z α ] −r−4 g(Z α )d 4 W ∧ d 4 Zwhere c is some constant, independent of r, where d 4 W = 1 24 εαβρσ dW α ∧dW β ∧ dW β ∧ dW β and correspondingly for d 4 Z (which is the 4-form φabove), and where (with n > 0)so that[x] −1 = x −1[x] −n = −(−x) −n (n − 1)!andd[x] −ndx= [x] −n−1 .