Twistors and plane waves

Recent joint work of the speaker and George Sparling show how to construct spacetime fields spectrally, provided the target geometry is compactified Minkowski space. Given a quantization of the twistor commutation relations, the spacetime points are maximal abelian von Neumann algebras generated by linear twistor observables. Spinors are then the eigenvalues of these algebras, and fields are reconstructed spectrally from the eigenstates. The purpose of this talk is to investigate how much of this picture can be generalized to non-flat spacetimes, specifically the class of plane waves. We start with the same twistor space and the same quantization, and pick a single twistor ray representing a null geodesic at infinity. A moduli space of possible points is then the class of maximal abelian von Neumann algebras that are fixed by some dilation compatible with the twistor at infinity. Within this large family, we identify a natural first order compatibility condition that encodes when two point algebras are adjacent. We then show that the maximal regular families of point algebras satisfying this adjacency condition, together with twistor incidence giving the conformal structure, is exactly the family of conformal plane wave metrics in four dimensions. We conclude with the spectral description of spinors and fields on a plane wave.