Special Issue on Progress in Twistor Theory
The Guest Editors for this special issue are
Preface The motivational origins of twistor theory were initially aimed at providing a possible framework for physical theory (as formulated in December 1963 — though there were many mathematical precursors, dating back to work of Sophus Lie and Felix Klein in around 1870, or to earlier related ideas of Grassmann, Cayley, and Plücker). The intention was for a formalism that would be very specific to the 4dimensionality of physical spacetime, with its relativistic Lorentzian signature, as is directly relevant to the geometry of the physical world we perceive about us. This spacetime geometry was taken to be that of Minkowskian 4space, whereby special relativity but not gravitation is accommodated. In the “twistor correspondence” that emerged, entire light rays (null geodesics) in spacetime were to be regarded as more primary elements than would actual spacetime points (“events”). Various underlying physical motivations provided reasons for taking this specific correspondence seriously as a plausibly fruitful viewpoint for the description of basic physics, and some justification for this optimism was subsequently found in the surprising way in which the field equations for free massless particles of arbitrary helicity …, −2, −3/2, −1, −1/2, 0, 1/2, 1, 3/2, 2, … could all be directly encoded as holomorphic sheaf cohomology classes in twistor space. At the time, this specific twistor correspondence presented no evident generalization that would allow it to be extended to the curved Lorentzian spacetimes of Einstein's general relativity. Twelve years later, however, by a roundabout route arising from attempts to address some unresolved issues of general relativity, where certain quantum considerations also contributed significantly, an unexpected connection between twistor theory and Einstein's curvedspace picture of gravity actually emerged, this being what was referred to as the “nonlinear graviton construction”. This brought some basic features of quantum mechanics and general relativity together in a surprising way. About a year later, Richard Ward then saw how to provide a corresponding construction for general gauge fields whereby not only the interactions of Maxwell's electromagnetism could now be expressed in twistor terms, but so also could the selfinteracting Yang–Mills equations of the strong and weak nuclear forces. Yet, these nonlinear constructions were explicitly limited to the lefthanded helicity (in standard conventions), which meant that the “spacetimes” (and “fields”) described according to these constructions were fundamentally restricted, and had to have a conformal curvature (and gauge fields) that are necessarily antiselfdual (or selfdual, if opposite conventions are adopted). With the Lorentzian signature of the ordinary spacetimes of Einstein's classical theory, this restriction is huge, and the constructions directly apply, only to spacetimes that are conformally flat. Despite the seemingly devastating limitation that this restriction imposed on the direct physical applicability of this aspect of twistor theory, such (anti)selfduality was found to be intriguing from a puremathematical perspective, because for metrics that are of positivedefinite (or split) signature, much nontrivial geometry arises. Indeed, this restriction is revealed to be a fruitful one to study for is own sake, leading to many deep connections between twistor methods and integrable systems, and with such topics as Kähler and hyperKähler geometry. All these areas are elegantly illustrated in various articles in this volume. We see from the fascinating variety of articles provided here that there is much to be gained in investigating the roles played by twistortype correspondences in numerous different geometrical contexts. Most of these articles explore such geometrical relationships and roles for twistorrelated methods of proof, finding remarkable new contexts and ingenious novelty of puremathematical application. Yet, the physical applications of twistor theory are themselves not left behind, and in this collection we also find a fine illustration of what is currently by far the most active area of application of twistor theory in physics, this involving powerful new procedures for the calculation of scattering amplitudes in highenergy processes. Impressive as these modern developments indeed are, both on the puremathematical side and in the calculation of physical processes, the (anti)selfdual restriction for the full (nonperturbative) applicability of twistor methods in nonlinear theories has represented a fundamental barrier to twistor theory being regarded as providing a fully plausible formalism for the description of basic physics. This is most blatant with regard to gravitation, but it applies also to the full description of the other forces of Nature. As it turns out, a broader outlook has recently been under development (palatial twistor theory) that appears to provide a way of surmounting this barrier in a very natural way. It will be interesting to see how such developments may affect both the physical and the mathematical applicability of twistor theory in future years.
Roger Penrose, September 2014
Papers in this Issue:
