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

The Twistor Approach to Space-Time Structures 483ticle in one place immediately forbids its detection at some distant place).Accordingly, twistor theory’s essentially non-local description of wavefunctionsis actually something rather closer to Nature than the conventionalpicture in terms of a space-time “field”. Indeed, it is frequently pointedout that this holistic character of a wavefunction distinguishes it fundamentallyfrom the kind of local behaviour that is exhibited by ordinaryphysical fields or wavelike disturbances, this distinction contributing to thecommon viewpoint that a wavefunction is not to be attributed any actual“physical reality”. However, we see that the cohomological character of thetwistor formulation of a wavefunction gives it precisely the kind of holistic(non-local) nature that wavefunctions actually posses, and I would contendthat the twistor formulation of a wavefunction assigns just the right kindof mathematical “reality” to a physical wavefunction.To emphasize the essential non-locality of the concept of cohomology, wemay take note of the fact that the 1st (and higher) holomorphic cohomologyof any Stein manifold always vanishes. (I have not defined cohomologyhigher than the 1st here; the basic difference is that for n-cochains, we needfunctions defined on (n + 1)-ple intersections, with a consistency conditionon (n+2)-ple intersections, the coboundaries being defined in terms of “h”son n-ple intersections. t ) It is important, therefore, that PT + is not Stein (itis, indeed, not pseudo-convex at its boundary PN). It is this that allows nontrivialholomorphic 1st cohomology elements to exist. However, from whathas just been said, we see that 1st cohomology of an open region alwaysvanishes locally, in the sense that it vanishes if we restrict it down to a smallStein set containing any chosen point. First (and higher) cohomology, foran open complex manifold, is indeed an essentially non-local notion.To end this section, we take note of the remarkable fact that the positivefrequencycondition for a wavefunction is now neatly taken care of by thefact that we are referring simply to the holomorphic (1st) cohomology ofPT + . Correspondingly, negative-frequency complex massless fields wouldbe those which are described by the holomorphic cohomology of PT − . Thisprovides a very close analogy to the way in which the positive/negativefrequency splitting of a (complex) function defined on the real line — compactifiedinto a circle S 1 — can be described in terms of holomorphic extendibilityinto the northern or southern hemispheres S + , S − of a Riemannsphere (a CP 1 ) whose equator represents this S 1 (and where I am arrangingthings so that the “north pole” is the point −i, with the “south pole” at+i). A complex function defined on S 1 can be split into a component whichextends holomorphically into S + , namely the positive-frequency part, and a