Penrose is famous particularly for his proof that black holes are a generic feature of space-time: by now one detects black holes in many parts of the universe, for example at the center of our own galaxy. However my work lies more in another area founded by Penrose and the school of general relativity around Sir Hermann Bondi and Peter Bergmann, called twistor theory. This theory has been under development for around forty years, but is still considered to be radical. The overall aim has always been more or less clear: to construct a theory of space and time from which quantum mechanics and geometry naturally merge or emerge.

One of the key problems when one studies esoteric areas of knowledge is that of communication: to one’s immediate colleagues, to other experts in related areas and to the public at large. It is here that graphical representation of ideas can lay an important role. I will illustrate this with several examples taken from my recent work which goes beyond the more-established twistor theory.

If we go back to the beginning of this century, two physicists, Shou-Cheng Zhang and Jai-ping Hu, then both at Stanford University, published a theory in the journal Science, called by them ”A Four-Dimensional Generalization of the Quantum Hall Effect”. I came across their work by accident and soon realized that it could be reformulated into the language of twistor theory. Their theory produces a quantum fermionic fluid and relates it to the properties of massless quantum particles (such as the photon or graviton) that occur in nature.

Graphically I was able to summarize the formalism in a simple picture:

Without the wavy lines, this is a standard picture of the six-dimensional projective twistor space, due to Penrose. The space-time arises as a space of lines, lying entirely in the central zone, called the space of null twistors, which forms a five- dimensional boundary for the other two parts of the twistor space. Originally this was just a mathematical picture. However in my work I place the Zhang-Hu fluid in the picture (represented by the wavy lines), so, if the fluid is ”real”, then so is the twistor picture. Oscillations in the fluid are represented on the boundary by massless particles in space-time.

Although, at the time, this work seemed like a big breakthrough, it raised many questions. For example, the given picture represents the situation for flat space-time. However, as the space-time becomes curved, for technical reasons, the right and left parts of the picture tend to be squeezed or even disappear entirely, leaving no-where for the fluid to be.

The search for a better formalism dragged on for more than five years. By the beginning of 2006, I was reduced to writing a book summarizing my work to date, implicitly conceding to myself that I was not going to make further progress. After writing around two hundred pages of preamble I arrived at the point where I had to present my definition of a twistor. Since I had spent the last thirty years developing the theory, one would think that that would be the easy part. However I wanted my definition to be as wide as possible, whilst retaining the basic twistor philosophy. In particular I started playing with using twistor-type methods to generate solutions of wave equations in higher dimensions. Such tricks were basic to ordinary twistor theory and had been studied sporadically over the years by various people, including myself, Penrose and an old twistor hand, Lane Hughston, now a Professor of Economics at London University.

Whilst doing this, I stumbled upon a key new idea, which I now call the Ξ- transform. In the process, I had to overcome a conceptual blockage: in ordinary twistor theory, one uses functions of three twistor variables to codify solutions of the wave equation in space-time, which depend on four variables (one time variable and three space variables). Much of the elegance of twistor theory revolves around the gains in economy from using just the three variables. However, in the simplest generalization to higher dimensions, the wave equation is in six dimensions and the twistor variables also range over six variables, so there is apparently no gain. I had to realize that it is the relation between the various six-dimensional spaces which is at issue and which contains the important information.

Once I understood this the results began to flow. The key underlying concept turns out to be the number three, used in the sense of trinity. The philosophy of the trinity then becomes paramount.

The first picture illustrating this is the triality picture:

This picture illustrates an application of a famous idea due to the mathematician Elie Cartan. Each blob represents an eight-dimensional vector space. Two of the blobs represent twistor spaces, the third represents an enhanced version of space- time. The three spaces are connected by the triality tensor. In the past, I would have drawn this more like a Mercedes or peace symbol, with straight lines to the central vertex, rather than the present curved lines. The use of curved lines instead is intended to suggest dynamics (a rotating wheel, or a fireworks Saint Catherine’s
Wheel) and to suggest spin, which is the foundation of the present theory.

The new theory envisages a primordial theory which would transcend ordinary space and time, out of which the present world emerges by a process of symmetry breaking. Possible outlines of such a primordial theory were developed in the past five years in collaboration with my former students Philip Tillman and Dana Mihai.

The basic theory would be triality invariant, so would not distinguish between the three triality spaces. One needs to illustrate the underlying unity of these three spaces. I believe that this is achieved by the third graphic: