Guide to this website

Overview

My work is concerned mostly with the quantum mechanics side of this programme. Twistor diagrams, which Roger Penrose first wrote down in about 1970, give a description of how fundamental particles and forces act on each other. Twistor diagrams involved difficult mathematics in many-dimensional spaces. But from the start it became clear that they manifested a strange and beautiful structure, bringing out some amazing properties of fundamental interactions.

Work on twistor diagrams made slow progress for thirty years, plagued by mathematical difficulties, and seemed rather remote from mainstream developments in physics. But in 2003 the leading theoretical physicist Edward Witten came up with a new paper which related string theory and twistor geometry. People working in more orthodox research in physics suddenly started taking an interest in twistors.

Quantum field theorists began talking about 'the twistor revolution'. And twistor theorists like myself started catching up with the wonderful discoveries that leading quantum field theorists had been making in gauge theory, which is essentially the study of fundamental forces in Nature.

My main contribution was showing in 2005 that one of these discoveries (the 'Britto-Cachazo-Feng-Witten recursion relation' for gauge theory), was in fact best represented in terms of twistor diagrams. This came as a rather unexpected and surprising idea, which took until 2008 to be absorbed.

In March 2009, Nima Arkani-Hamed, Freddy Cachazo, Clifford Cheung, and Jared Kaplan published a paper ( The S-matrix in Twistor Space ), which greatly extended this idea (by identifying its Grassmannian structure) and made it much more powerful and better known. My Oxford colleagues David Skinner and Lionel Mason did parallel work.

In May 2009 I noticed that another new field-theoretic development ('dual conformal invariance') could also naturally be described using twistor structure. This has also proved to be a very helpful idea, but it is only one component of the many further advances made in the summer of 2009 by Nima Arkani-Hamed's group, by my Oxford colleagues, and by other leading field theorists.

By September 2009, the fundamental importance of Roger Penrose's twistor-geometric approach had been firmly established. The next stage required the extension of these ideas to encompass the 'loop amplitudes' of quantum field theory. Much of this was achieved in 2010. A new generation of young mathematicians and physicists became involved, and the fast pace of progress was sustained.

Nima Arkani-Hamed and Roger Penrose meet at the White Hart, Wytham, Oxford, 18 June 2009.

(Photo by AH).

For a taste of the 'twistor revolution' in 2011 see this video of Nima Arkani-Hamed's January 2011 lecture at the Perimeter Institute, and these blog comments.

In 2012, twistor diagrams came right back into the centre of the theory, when Nima Arkani-Hamed and five collaborators published a completely new account of their properties, making a connection with the advanced geometrical ideas of the 'Positive Grassmannian'. The entire theory of amplitudes in gauge theory has now been transformed.

Sources for the classical background of twistor theory

Roger Penrose adds extra detail about the origin of his ideas in an on-line video entitled Extra Time: Professor Sir Roger Penrose in conversation with Andrew Hodges, from the Mathematical Institute, University of Oxford.

S. A. Huggett and K. P. Tod, An introduction to twistor theory, London Mathematical Society Student Texts 4 (New edition, Cambridge University Press, 1994).

There is much further material in

R. Penrose and W. Rindler, Spinors and Space-time Vol. 2, Cambridge University Press (1986)

For an introduction to research in twistor theory in its early period, see the collection of papers in

London Mathematical Society Lecture Notes Series 156, Twistors in Mathematics and Physics, eds. T. N. Bailey and R. J. Baston (1990)

Twistor theory since 2003

Go to a summary of recent developments, giving links to ArXiv versions of papers.