Abstract

Twistor theory was originated in late 1963 as a geometric approach to the quantum field theoretic requirement of splitting the positive-frequency modes from the negative frequency ones. This was addressed through translating the geometry of Minkowski space into the projective geometry CP3 , referred to as the projective twistor space PT, whose division into two halves PT+ and PT– geometrically described the required positive/negative frequency splitting.   However, as the geometry of twistor theory progressed, in relation to its physical interpretation in terms of the momentum/angular momentum of photon states, it became clear that the splitting of PT into PT+ and PT– had more directly to do with positive/negative helicity than with positive/negative frequency.   This confusion of interpretation became more manifest with the non-linear graviton construction, whereby twistor theory broadened its scope to describe curved complex space-times, where the helicity/frequency tension manifested itself into the “anti-self-dual” requirement for the curved 4-manifolds that could be directly described by curved twistor-space theory, and twistor theory itself bifurcated into a “positive-definite” version of more interest to pure mathematicians and th “Lorentzian” (or even “split-signature”) version of more direct interest to physicists.   The concept of a bi-twistor is introduced here to circumvent this asymmetry and (anti-)self-dual requirement, by involving both twistors and dual twistors together, subject to their quantum commutation laws, this providing a bi-twistor triple product (and a split-octonion algebra), enabling general space-times to be described by bi-twistors.