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THE TWISTOR THEORY
by Ray Shelton
Working with complex numbers enables theoreticians to call upon another important tool of cohomology. A combination of the very powerful mathematics that stems from complex numbers and cohomology has enabled Penrose to create new maps for the world where quantum theory and relativity meet.
The essential idea of cohomology can be understood by returning to the Penrose triangle, that “impossible figure” that was to inspire some of Escher’s etchings. Suppose that you were to observe the triangle, section by section, through a telescope. You would see each of the three segments that, taken together, make up the entire figure. Segment (a) looks perfectly reasonable. There is not ambiguity in the figure, and it would be perfectly possible to construct it out of two pieces of wood. The same applies to segments (b) and (c). Each of the three segments is a perfectly proper figure. Since these figures overlap slightly, it is possible to identify a point x that is on both the segment (a) and the segment (b). Likewise the point y and z can also be identified as common to othe pair of segments. Finally, we try to build the complete Penrose triangle by moving the segments toward or away from us until they all join up. This means that x, y and z on the segments must match exact. Of course, in the case of an impossible figure like the Penrose triangle, this can never be done. While each segment makes sense by itself, when taken together they cannot join smoothly to form a sensible figure.
Another way of seeing this is to realize that the point x must be different distance from the viewer depending on whether it is defined in the segment (a) or (b). Yet x is supposed to be a single point common to both segments. Clearly the figure must be paradoxical.
The properties of the Penrose triangle illustrate the general approach of cohomology. This topic deals with the properties of spaces that can be divided into a seies of small regions, each of which is well behaved. In general, however, it may not be possible to join all these regions together smoothly and unambiguously. Cohomology therefore enable physicists to work with complicated, distorted spaces by dividing them into overlapping regions, each of which is, on its own, well behaved. We shall return to cohomology again and again in the following sections.
Penrose’s twistors are part of what the physicist John Wheller has called “the great dream,” a vision that all of physics can be reduced to geometry. In the nineteenth century, the great dreamer was an English mathematician and philosopher, William Kingdom Clifford. Clifford had been investigating the new geometries of Bernhard Riemann and Nikolay Ivanovich Lobachevsky that went beyond the more familiar schoolbook geometry of Euclid. Speaking to the Cambridge Philosophical Society on February 21, 1870, Clifford said:
“I hold in fact
(1) That small portions of space are in fact of nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them
(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter.”
(By a stroke of irony, it was Clifford’s other major contribution in mathematics, the Clifford parallel, that first led Penrose to his fundamental twistor picture. While puzzling over the meaning of Ivor Robinson”s solution, Penrose was led to a geometrical construction in terms of what known as Clifford parallels. It was thinking about this diagram that Penrose was first struck by the idea of twistors.)
For Clifford, there was nothing in the physical world but this variation of geometry, these little hills and changes of curvature. Clifford’s dream was later taken up by Einstein, who extended the notion of space to space-time and demonstrated that the geometry of space-time and demonstrated that the geometry of space-time detortions the motion of matter. But Einstein wanted to go further, to complete the dream of Clifford and demonstrate that matter itself is nothing more than the knots and hills of space-time, and that the fields of force are also regions of curvature. If the force of gravity can be reduced to geometry, he speculated, then why not the magnetic and electrical forces?
This quest preoccupied Einstein in vain during the last decades of his life as he attempted to modify his basic equations of general relativity in attempt to make additional room within the geometry of space-time for matter and force. Herman Weyl, his contemporary, tried a slightly different approach. In Einstein’s theory, lengths and the rate of clocks can change, but if an object is taken on a circuitous route and is finally brought back to its initial point, the length should be unchanged. Weyl, however, allowed this length scale to vary from place to place but, at the same time, introduced a sort of standardizing field that enabled this varying scale to be fixed at each point – by a fundamental length gauge as it were. This gauge field, he argued, would look exactly like an electromagnetic field but would inhabit a space-time in which scale can vary.
While Weyl did not ultimately succeed with this idea that the electromagnetic field is a gauge field, the notion did, however, surface in a very different form many decades later, when force fields were treated as gauge fields in elementary particle theories. The 1920s also saw the attempt of Theodor Kaluza and Oskar Klein to increase the dimensions of space-time from four to five and in this way try to portray the electromagnetic field in terms of the effects that would appear as this extra dimension is curled up.
But, in the last analysis, this dream of Einstein and his contemporaries could never work, for it took no account of the great advances that were being made in quantum theory. Despite his significant contribution to the birth of quantum theory, Einstein felt uneasy with its final form and could not accept the theory as a complete account of nature. Yet quantum theory claims to offer a full description of the ultimate structure of matter. So how can a theory like the one Einstein was searching for, which purports to explain matter in geometric terms, ignore this complementary quantum mechanical description? Some physicists proposed to take account of this by trying to give space-time a quantum structure. They attempted, for example, to add in rules of quantum theory to a curved space-time. But such research projects always ran into serious difficulties.
In addition, the experiments of elementary particle physicists had added to physics two new forces, the weak and strong nuclear force. Einstein’s unified theory would have to take account of both of these, in addition to electromagnetism and gravity. But these new forces are essentially quantum mechanical in nature, and the sorts of theories that Einstein and his contemporaries were working on where purely classical.
Clearly a new unified account of nature demanded something much deeper. Roger Penrose, for his part, believed that a true geometrical account of nature would have to involve a unification of quantum theory and space-time. But this could never be done using some simple extension of the classical space-time of Einsterin and Minkowski with its four real dimensions and its foundation in the dimensionless point.
For Penrose, the starting point had to involve a space of complex dimensions, since complexity also occurs in a fundamental way in quantum theory and, in addition, complex geometry is far richer than real geometry. A complex space, he believed, would be able to account not only for the space of everyday, macroscopic objects but the space of quantum process as well, and offer a geometrical account for photons and a single quantum of curvature. It would be the primordial space that eventually give birth to our own space-time.
The basic objects of this space are not points but one-dimensional objects that are extended in space. More general than points, they are at the same time quantum mechanical in nature and elements of a complex geometry. By the late 1960s, Penrose had found his space, this space of the great dream. It is a space of complex dimensions, and its basic objects are twistors. These twistors can be thought of as generalizations of light rays or null lines and are defined by complex numbers. And, although twistors can be pictured in a space-time of real dimensions, their real home is twistor space.
Just as matter has its foundations in the elementary particles, so space-time will ultimately have its origins in twistor space. Twistor space becomes the new arena of physics, the ground on which quantum processes are played out. Clifford’s great dream was of a physics reduced to the geometry of space-time. Now Penrose had transformed that dream: the proper arena of physics is not space-time but the complex dimensions of twistor space.
Is it possible to give a picture of the twistor and an image of twistor space? That rather old-fashioned topic, projective geometry, which Penrose had studied as a student, now comes to our aid, for with its help it becomes possible to go back and forth between two pictures, one in twistor space and the other in space-time. First, let us examine the underlying twistor picture. (The twistor space that we shall be exploring in this paper is built with three complex dimensions and is called projective twistor space. But there is also a full twistor space having four complex dimensions. But the important properties and arguemnts of this and the following chapter can be derived from the three complex dimensions of projective twistor space alone. We shall only need to refer to full twistor space agin in the next chapter – when we set up a very curous space-time in which the wave function for a single graviton can be written down. For the moment, however, let us forget about the space of four complex dimensions and concentrate on the projective twistor space of three complex dimensions.)
Since the twistor, which we shall call Z, is built to have both angular and linear momentum, it is more general than a spinor. It can be defined by complex numbers, which are in fact its coordinates in twistor space. A twistor Z therefore becomes a point in twistor space. Being complex, the twistor Z also has a partner, its complex conjugate called Z*. As we would expect from complex numbers, the result of multiplying the twistor Z by its complex conjugate Z* is a real number; in fact such a product can be used to define s, the helicity or degree of twist of the twistor. (The symbol ⋅ between the two twistors indicates multiplication.)
½Z ⋅ Z* = s
This helicity turns out to be an important factor. It can be positive, negative, or zero. As we shall see, twistors with zero helicity have a special role to play in space-time, for they look exactly like rays of light. Since twistors are built to have a positive, zero, or negative twist, this means that they also have a natural right- or left-handedness — which is also called chirality. Chirality, therefore, is totally natural within the twistor picture and does not become a major problem for physicists as it did the early string approach.
All the twistors with zero helicity, Z ⋅ Z* = 0, lie in a special region of twistor space which we shall label PN. This region PN has the effect of dividing twistor space in half, into regions, PT+ and PT–. This division is the geometrical analogue of the way in which solutions in quantum theory are divided into positive- and negative-frequency parts.
Above PN can be found the region PT+, which contains only twistors of positive twist. Below PN lies PT–, whose points correspond only to twistors with a negative twist or helicity, while all the points on PN itself are the coordinates of twistors with zero helicity. This division of twistor space in half by the region PN turns out to have a profound physical meaning. Remember how Penrose had been interested in the way physics must pick out the real, or physical, solutions from a quantum field. This process of selection involves picking out positive-frequency solutions from negative-frequency ones, and it had always struck Penrose that this should have a purely geometrical interpretation. But now, working in twistor space, this becomes possible. The selection of solutions in quantum field theory is related to whether points lie in PT+ or in PT–. We shall also find, when the twistor picture is extended to include curvature, that the space-time of individual graviton wave functions is also broken down into curvature with a right- or a left-handed sense.
Having established twistor space, whose points are the twistors and which divided by PN into two parts, it is now possible to create the complementary picture in terms of the more familiar space-time first created by Minkowski. It turns out that twistor space has a far richer structure than the flat Minkowski space-time of Einstein’s theory of special relativity. Not only is it possible to recover or reproduce this space-time out of twistor space, but a whole structure of geometric relationships can be recovered as well. While the traditional approaches have emphasized points and other local features, the essential strength of twistor space will lie in its ability to describe large-scale – also called global – structures in space-time.
Take, for example, a point in that special slice of twistor space that we have labeled PN. Points in PN represent twistors with zero twist, and it turns out that they correspond to light rays or null lines in space-time. In fact, by taking all the points in this section (PN) of twistor space, we can fill space-time with null lines or light rays.
Since the twistor is the most fundamental aspect of the geometry of twistor space, we now see that the corresponding basic object in space-time is not a point but a null line. Space-time is to be recovered from the more fundamental twistor space, and its foundation becomes lines rather than points. Its geometry is based upon twistors, which look very like light rays or the tracks made by massless particles. In fact, there is a fundamental duality between the null line in space-time and a point in projective twistor space PN. One way of thinking about this is that the local structure, the points, of twistor space encode global or large-scale information about space-time.
If these extended null lines are the essential geometric objects of space-time, then what about points? These are now secondary or derived objects that are defined by the intersection of twistors. By the way of an example, think of an interchange on a highway. This is represented by a point on a map, but it could also be thought of more globally as something held in common by several different routes. In other words, these routes have their meeting at this intersection. From the perspective of a traveler in a car, who is concerned with journeying from state to state, the various routes are key importance, and the intersection is a secondary thing used for getting from one route to the next. Intersections are therefore perceived from a global perspective.
It is also possible to go in the other direction and look at a complementary picture. Points in that special region PN of twistor space correspond to twistors of zero helicity – that is, light ray – in space-time. On the other hand, points in space-time will now correspond to extended structures in twistor space. In fact, points in space-time define lines in the special region PN of twistor space. The duality is complete: Points in PN correspond to null lines in space time. Points in space-time correspond to line in PN, a special region of twistor space. Yet the twistor picture is more fundamental, for we take the points in twistor space as the start of geometry, while the points in space-time are secondary objects. Space-time appears to be fundamentally non-local.
Twistors are essentially nonlocal objects concerned with the global structure of space-time; points in space-time are of secondary importance. This aspect of the twistor picture has some immediate implications. To begin with, it opens the door on a totally new way of understanding space-time and the elementary particles: relationships are now to be viewed at a global rather than a local level.
Nonlocality, for example, enters into the quantum picture of nature in an essential way. It was Niels Bohr who first stressed that quantum mechanics is essentially a nonlocal theory. Indeed, one way of avoiding paradoxical interpretations about quantum measurements is to assume that quantum theory refers always to nonlocal descriptions. A famous thought experiment suggested by Albert Einstein, Boris Podilsky, and Nathan Rosen can only be properly understood if nonlocal connections are assumed between different parts of the quantum system – that is, distant connection that do not involve the causal action of any force of a physical field. (This whole topic of nonlocality will be treated in greater detail in the author’s next book.) Since twistors are fundamentally nonlocal themselves, they are in accord with this very basic property of the quantum description of nature.
It may well turn out that the twistor approach is also able to shed new light on the description of black holes, which contain regions called space-time singularities. These singularities are points at which the very structure of space-time breaks down and the laws of physics no longer apply, and they are a major headache for relativist – what John Wheeler calls “the crisis in physics.” Since the description of space-time now begins in a global way, with points having their origin in the coincidence of nonlocal objects, it follows that it should also be possible to have a nonlocal description of a black hole.
The black hole sigularity occurs where matter and energy collapse down to a single point in space-time. At this same point, the basic structure of space-time is also supposed to break down. But now it becomes possible to be begin with a global description that refers to all space-time including the singularity itself. In this way, it may be possible to retain a description of a black hole in which the laws of nature need not break down. Even more importantly, this description of could include the first singularity of the universe — the big bang.
The fact that space-time points are secondary objects, derived from twistor intersections, also implies that we need not expect them to survive when quantum processes are admitted into the twistor picture. In effect, certain transformations or processes in twistor space turn out to be equivalent to quantum processes in space-time. And, as with any transformation, the basic geometrical units become interchanged. In this case, a quantum transform of twistor space mixes up the twistors. But since points in space-time are defined in terms of the conjuctions of these twistors this means that the space-time point will smear out. At the quantum level, the twistor space picture suggests that points in space-time lose their distinction and become “fuzzy.”
At this point, it is a good idea to make a direct comparison between superstrings and twistors. Superstrings are massless, one-dimensional objects having an incredibly short length. Twistors, as null lines or light rays, have no length, no sense of scale, and no mass.
Superstrings are defined in a ten-dimensional space, which, it is assumed, will compactify down to our own four-dimensional space-time. Twistors, by contrast, are defined in a space of complex dimensions. This complex twistor space is then used to generate our four-dimensional space-time, along with its rich structure of null lines.
Superstings carry a series of internal symmetries, which are broken as the ten-dimensional space compactifies. Such symmetry breaking does not necessarily occur in the twistor picture, rather certain symmetries such as that between right-handed and left-handed photons and gravitons are violated at the outset. At the same time the “twist” of the twistors means that its approach is basically chiral.
As closed loops, superstrings have a natural interaction with the vacuum of space, which can be pictured in terms of the creation and annihilation of gravitons. In the twistor picture, gravity and quantum processes are interconnected in a different way.
In the present formulation, superstring theory accepts the basic formulation of the quantum theory. Penrose, however, believes that this must change in the twistor approach so that quantum theory and space-time have to be described in a new way.
There are also a number of subtle connections between superstrings and twistors. Penrose’s colleagues, Lane Hughston, William Shaw, and Mike Singler have shown that a relativistic string corresponds to a general curve in twistor space. This means that relativistic strings can be derived from the twistor space picture. In addition, certain important twistor transformations can ge generalized to the ten-dimensional space of superstringss. More recently a connection has been made between the trouser diagrams of string theory and twistor diagrams. Researchers in the superstring and twistor fields are currently inverstigating the same rich fields of complex geometryand complex analysis, called cohomology theory, which may reveal yet deeper connections between the two approaches. Indeed, Edward Witten believes that the twistor formulation may be the proper starting point for superstring theory.