THE TWISTOR THEORY

INTRODUCTION.

The search for the Grand Unified Theory (GUT) of physics may be thought of in terms of the two great theories of the twentieth century: the general theory of relativity and quantum mechanics. The first, the general theory of relativity, is a theory of gravity that relates gravity to the curvature of spacetime. It is essentially a theory of structure of space and time. The curvature of spacetime is responsible for the force of gravity. The second, quantum mechanics, describes the behavior of the atomic and subatomic world; there are quantum theories which describe each of the other three forces of nature, apart from gravity; the quantum electrodynamic theory (QED) describes electromagnetism at quantum level, the quantum chromodynamics theory (QCD) describes the strong force, and the electroweak theory of weak interactions. A fully unified description of the Universe and all that it contains (called a "theory of everything", TOE) would also have to describe gravity and spacetime at the quantum level. This implies that spacetime itself must be, on a very short-range scale, quantized into discrete lumps, not smoothly continuous.

SuperString Theory

The extension of the string theory known as the SuperString Theory (SST), naturally produced a quantum description of gravity. But it took several years for gravity to fall out of the superstring theory. The string theory took off in the middle of the 1980s, after a new variation on the theme was developed by John Schwarz and Michael Green. They started working together at the end of the 1970s, after they met at a conference of CERN, and discovered that, unlike everybody else studying particle physics at the time, they were both interested in strings. They began almost immediately to produce results. The first step that they took was to realize that what was needed was a theory of everything - not just of hadrons but of all particles and fields. Naturally, from the outset the new version of string theory had SUSY built in. In such a theory, the strings would have to be very small - much smaller than Nambu's strings, which were only designed to describe hadrons. Even without knowing how the theory would develop, Schwarz and Green could predict what scale the strings would operate on, because they wanted to include gravity in the theory. Gravity becomes seriously affected by quantum effects at a scale about 10^-33 cm (about 10^-35 m), the distance scale at which the very structure of spacetime is affected by quantum uncertainty. The scale at which quantum effects become significant for a particular force depends upon the strength of the force. Since gravity is the weakest of the four forces of nature, the quantum effects only become dominant for gravity on such a tiny scale. It is by measuring the strength of gravity (the value of the gravitational constant, G) that the physicists can work out what scale quantum gravity operates on.

The first string model that was developed by Schwarz and Green, in 1980, dealt with open-ended strings vibrating in ten dimensions, able to link up with one another and break apart. Superficially, this model looked like a shrunken version of the Nambu's string theory. But in fact it went far further, including in principle string states corresponding to almost all known particles and fields, and all known symmetries affecting fermions and bosons, plus supersymmetry. There was one exception - gravity. In spite of their intentions, gravity still could not be explained by the new string theory.

But in spite of this deficiency, this early version of the superstring theory set the scene for what was to follow. The central idea of all subsequent superstring theories is that the conventional picture of fundamental particles (leptons and quarks) as points with no extension in any direction is replaced by the concept of particles as objects which have extension in one dimension, like a line drawn on a piece of paper, or the thinnest of strings. The extension is very small, about 10^-35 meters. It would take 10²⁰ such strings, laid end to end, to stretch across the diameter of a proton.

The next step in development of a superstring theory of everything came in 1981, when Schwarz and Green introduced a new twist (literally) to the theory. The open string theory became known as the Type I theory, and the new Type II theory introduced a key variation on the theme - closed loops of string. The Type I theory only had open-ended strings; the Type II theory had only closed loops of string. In a particularly neat piece of packaging, in closed loops, fermionic states correspond to ripples running around the loop one way, while the bosonic states correspond to ripples running around the loop the other way, demonstrating the power and influence of supersymmetry. The closed loop version had some advantages over the open model; it proved easier to deal with those infinities that plagued particle physicists in the open string model. But the Type II theory also had its difficulties, and did not seem at the time capable of predicting, or encompassing, all varieties of the known particle world.

The next step forward was actually a step backward. Dissatisfied with Type II theory, Schwarz and Green went back to the Type I theory, and tried to remove the infinities which plagued it. The problem was that there were many possible variations on their theme, and that all of them seemed to be beset by not just infinities, but by what were called anomalies - behavior that did not match the behavior of the everyday world, especially its conservation laws. For example, in more than one version of the theory, electric charge is not conserved, so charge can appear out of nothing at all, and disappear, as well.

But in 1984 Schwarz and Green found that there is one, and only one, form of symmetry [SO(32)] which, when applied to the Type I string theory, removed all the anomalities and all the infinities. They had a unique theory, free from all anomalities and infinities, that was a real candidate for the theory of everything. It was at this point that other physicists started to set up and take notice of strings once again.

The excitement was largely to do with the way gravity appeared naturally out of the superstring theory. Gravity may be thought of in two ways. Starting from Einstein's description of curved spacetime, we are led to the image of gravity waves, ripples in the fabric of spacetime, with, of course, an associated particle, the spin-2 graviton. This is the way the concept appeared historically. But we could, in principle, start out with a quantum field theory based on a zero mass, spin-2 particle, the graviton, and see what the equation describe. Carry the calculations through, and you end up with Einstein's general theory of relativity. The problem with all theories prior to superstrings (except, just possibly, N = 8 supergravity) is that when you add in a massless, spin-2 particle, they are beset with infinities that are impossible to remove even by renormalization. The dramatic discovery that emerged in the mid-1980s is that whenever theorists set up a mathematical description of superstring loops, tailored to describe the behavior of the known particles, a description of a massless, spin-2 particle always falls out of the equation and the quarks and leptons and the rest. What's more, it does so without causing those uncomfortable infinities to rear their ugly heads. One of the founders of superstring theory, John Schwarz, refers to this as a "Deep Truth", that must be telling us something about the way the Universe works.

Gravity must be included in superstring theory, and must arise naturally in a way that can be portrayed in simple physical terms. The simplest form of closed string that emerges automatically from the theory has the properties of a spin-2 vector boson, the quantum particle of gravity. Indeed, they are gravitons, the carriers of the gravitational force. Gravity, including Einstein's equations of the general theory of relativity, then emerges naturally from string theory as a quantum phenomenon.

But what of space-time itself? A proper superstring theory should generate its own space-time, since space and superstrings are irreducibly linked. But the best that physicists have been able to do is to put the strings in a flat, inert background space. Moreover, superstring theory, in its present formulation, begins with the quantum theory in its conventional formulation. It assumes an three dimensional Eucleadean space. Should not a theory of everything be expected to change quantum theory in some equally fundamental way? At such incredibly short distances, will not space-time and quantum theory both become transformed?

Twistor Theory

The present superstring theory leaves unanswered these and number of other questions. But before we look at these other questions, let us look at an alternative theory that starts from relativity and the mathematics of complex spaces, rather than from the mainstream of particle physics and quantum theory. This is the theory of twistors that was the invention of the mathematician Roger Penrose. This theory is very different from that of superstrings. It uses an extended object called a twistor that is different from the superstrings of the superstring theory. But at deepest levels there may be a subtle connnection between the two approaches, on the surface they reflect very different philosophies. Superstring theory, as we have seen, was built by many hands and did not evolve in any linear or straightforward way. In fact, while the origin of string theory lay in speculations of the elementary particle physicists, superstrings themselves ended up talking about gravity and space-time in addition to the elementary particles. In contrast, twistors are essentially the work of one man, Roger Penrose, and have their origins not in the the exciting mainstream of particle physicis but in the more sedate fields of relativity and the mathematics of complex spaces. Yet, like superstrings, twistors also end up having something to say about gravity, space-time, and the possible nature of elementary particles.

The history of the development of twistor theory has been relative more straight forward. The basic approach of Roger Penrose can be traced to his interest in the structure of space-time and its meaning in a universe that must include quantum theory. Assisted by a small and slowly-growing band of students and colleagues, Penrose's approach finally flowered to produce powerful insights, not only into the nature of quantum space-time, but also into a number of areas of mathematics and theoretical physics.

The Problem - Spin

Twistor theory begins with a series of questions that preoccupied Penrose when he was was a research student. In particular, he wondered what it means for an electron to spin in an up and down direction. It was known that electron spin in one of two possible directions. But what, Penrose asked, would this mean anything if there was nothing in the universe but one electron? What meaning would this have to talk about two alternative directions in an otherwise empty space? Can spatial directions have meaning in the absence of matter, or do they somehow arise through the actual relationships between material bodies? Penrose was eventually to answer this question by constructing spin networks, a precursor of the twistors.

As a mathematican, Roger Penrose was also interested in the rich and elegant forms of mathematics that are based on complex numbers. But complex numbers themselves are not simply the product of abstract mathematical operations (the square root of negative numbers), for they also play a significant role in quantum theory. Penrose speculated that complex numbers should possibly enter in some equally fundamental way, for nature must exhibit a unity of description. Complex numbers and their mathematics were to become an essential feature of twistor theory.

The Problem - Relativity and Particle Physics

Roger Penrose's days as a student of mathematics, in the early 1950's, happened in a period during which there was a marked sociological division between relativists and elementary particle physicists. Although Penrose did not become seriously interested in relativity theory until 1958, this division did exist during his student days. Elementary particle physics was the mainstream, and most university physics departments had groups of theoretical and experimental physicists engaged in investigating some aspect of the elementary particles. By contrast, relativity was a quieter and more relaxed pursuit in which there was little competition; there were few serious relativists in physics and only a handful of major centers in which large groups worked. Relativity was essentially a theoretical study, and many of its practitioners were trained in mathematics rather than in expermental physics.

In these early days, relativity and particle physics had little to say to each other. Of course, the relativitsts know that there were serious problems involving the unification of general relativity and quantum theory. But the elementary particle physicists, for their part they had little motivation for struggling with the ideas and the mathematics of general relativity. Admittedly there were some physicists, like Bruno Zumino at CERN in Switzerland, who realized that the phenomenon of gravity, an essential part of relativity, could not be ignored in the context of a unified model of the elementary particles. But he speculated that such unified approaches would have wait until the twentieth-first century. There were also theoretians who attempted to keep a foot in each field, for they were concerned to quantizing relativity and producing a quantum account of the phenomenon of gravity. But the problems involved in this approach were formidable.

In the face of this general lack of connection between general relativity and quantum theory, Penrose's early research was concerned with ways in which these two worlds could be brought together. General relativity is a theory about the geometry of space-time, but, Penrose asked, what does this geometry look like in a world that also takes into account the processes of quantum theory.

Quantum Theory and Space-Time

Some physicists had speculated upon the possible nature of geometry at the quantum level. They had pointed out that the energy within a small enough region of space-time will induce tremendous curvature. John Wheeler had even speculated that, at small enough distances, this space-time breaks apart into a foamlike structure. Other physicists had tried to hold onto a coherent geometry by proposing a shortest possible distance in nature, or that space-time has an underlying lattice structure.

But if quantum fluctuations have a profound effect on space-time structure, then do things also work in the opposite directions? Does the curvature of space-time change even the quantum theory itself? So far, most formulations of quantum theory have been written in a flat space-time. How would the dynamics of a curving space affect the quantum world?

Penrose realized that something radical was called for if quantum theory and space-time be united. In fact, he believed that this had to be done, not in our usual space-time, but in a complex space in which the fundamental objects are not points but twisting lines.

Our familiar space-time may not in fact be the background in which the elementary particles play out their lives, rather quantum systems may define their lines, rather quantum systems may define their own space-times! Penrose's speculation was a particularly bold one, for it suggested that somehow quantum particles are not born into a background space-time, but rather this space-time is created out of quantum processes themselves at the aubstomic level. Space-time should not therefore exist before quantum theory but must somehow emerge out of a deeper level.

These idea had its seeds to the speculations of the eightenth century mathematician and philosopher Gottfried Wilhelm Leibnitz (1646-1716), who objected to the assumption of absolute space and time that lay at the base of Newton's new theory of motion. But for Leibniz, space arises in the relationship between material bodies. Penrose was now taking this idea one step further by suggesting that quantum systems define their own geometries. If this were true, then it would mean that quantum geometry is improtant not just at the incredibly small distances of Wheeler's spece-time foam, but whenever quantum effects are manifest. It therefore becomes even more important to solve the problem quantum geometry.

The Quantum Paradox

The significance of Penrose's proposals on a quantum geometry can be illustrated by the famous double slit experiment. Photons are sent toward a barrier that contained two slits. After passing through these slits, they fall on the screen behind, where they are recorded. First let's look at the classical case, in which a wave of light is used in place of quantum particles. In this pre-quantum account of the experiment, physicists would say that part of the light wave goes through one slit and part through the other. Behind the slits, these two wave fronts meet up and create what is called interference, a pattern of light and dark lines on the screen.

Interfrence is a perfectly general phenomenon where waves of all kinds are concerned. It seems, for example, when water waves meet up after passing through gaps in a dock. Wherever the waves have to travel different distances in order to meet, in those places they are out of phase. This means that peaks will meet troughs and tend to cancel out; however, a short distance further on, troughs will meet troughs, and peaks meet peaks and reinforce each other. The result is a complex pattern of disturbances.

Such interference patterns occur when light passes through two slits in a screen. However, if one of these slits is blocked off, then light can pass only through the other slit, and since no interference is possible, it simply illuminates the corresponding part of screen.

In the quantum case, something unexpected occurs, for it is technically possible to cut down the beam of light to the point where only one photon goes through the slit at a time. Common sense tells us that single photon can only go through one slit of the other. In other words, interference just does not make sense when a single photon is observed. After all, how could a single photon go through two slits at once and then interfere with itself? Photons are indivisible. Nevertheless the experimental results are quite different. Interference fringes are produced even when the photons pass through the slits one at a time. (In an actual experiment, the results of a large number of these single-event processes are recorded on a photographic plate.) But how is this possible? How can a single particle go through two slits and be in two places at the same time? Since this effect was first observed, physicists have attempted a number of different explanations for this quantum paradox.

Penrose's solution to the double-slit experiment is particularly radical. What, he asked, happens if the single photon defines its own geometry? What if, with respect to the photon's space-time, the apparatus appears to have such a curious geometry that it seems to have only a single slit? The photon will pass through this single slit and fall onto the screen. But the screen's geometry will also be so distorted that a complex pattern is built up by successive photons, each with its own geometry. In our own geometry, which includes the laboratory apparatus, the single photon appears to split itself and go through both slits, but with respect to the geometry generated by the photon, the laboratory appears to be distorted so that the two slits become one. The geometry of the world, it seems, depends on one's viewpoint.

At this stage, the idea of quantum particles creating their own space was an exciting speculation with no formal mathematical demonstration to back it up. Would it indeed be possible to create a geometry of space-time out of quantum objects alone? And would these individual quantum space-times then weave together and recreate the geometry of our own large-scale world?

Spin Networks

Penrose was spurred on to create such a new geometry using what he called spin networks. Such an approach, he hoped, would help to answer another problem: What is the role of the continuum in physics? This problem had bothered Penrose for some time. Mathematicans claim that there are as many points in a finite line segment as there are in the entire universe. This is an infinite number that is called aleph null. And no matter how finely this line is subdivied into segments, there will still remain an infinite number of points between the ends of each segment. This concept which is called continuity, and occurs both in numbers and in space, made Penrose uneasy. He was also concerned about the way in which continuity crops up in quantum theory when different solutions called wave functions are superimposed.

Penrose asked, why should it be possible to divide space without limit into infinitely small parts? After all, the quantum theory has set a limit on such division when it comes to energy. Light, for example, is not continuous but must ultimately be represented in terms of descrete quantum particles called photons; matter similarly reaches its limits in the elementary particles. Why should space be the exception and capable of infinite subdivision?

Penrose also believed that the universe should somehow be created out of integers alone, using combinatorial processes, that is, simple arithmetic operations such as addition, substraction, ratio, and permutation. If God is a mathematician, then He creates the universe by counting. In this way space, while appearing continuous at one scale of things, would not be infinitely divisible or continuous without limit but would have its origin in finite processes of counting. Some decades later Penrose was to soften this position and admit the significance of the complex mathematics. Possibly the power of complex numbers will eventulally be found to provide a complementary picture to counting and combinatorial processes.

But to return to the line segment. From our perspective, it appears continuous. But would it be possible to create another form of space though simple arithmetic processes like counting in which infinities and continuities do not crop up? What looks like continuity to us could be the result of the particular energy range in which we live and carry out our elementary particle experiments. If space-time were to have a grainy nature, it could be that our experiments have not yet detected this limit, so that everything still looks continuous. But at such smaller distances, space-time could have a totally different structure.

But what would be the starting point, the fundamental building block of such a space? Penrose decided to start with what is called a spinor. The spinor is a mathematical object that is used in the quantum theory to describe the spin properties of the elementary particles. In fact, it is the simplest mathematical object in the quantum theory. By a remarkable coincidence, mathematical spinors also play a singnificent role in relativity theory. Penrose's intuition led him to the spinor as the possible building block for a space-time.

In fact, Penrose had been puzzling about the meaning of electron spin for some time. His question goes right back to one asked by Ernst Mach, the great nineteenth century physicist and philosopher whose writing had deeply influenced the young Albert Einstein. Mach not only made significant contributions to the foundations of physics, but his writings also deeply influenced a school of philosophy known as logical positivism; he was the type of philosopher who was always asking troublesome questions.

Mach had been particularly puzzled by the meaning of certain propeties, like linear and angular momentum, in an otherwise empty space. In fact, Einstein specifically referred to this question of Mach in the first pages of his great paper "The Foundations of the General Theory of Relativity." Mach asked, what would it mean to say that a planet is rotating in space if there were nothing else in the universe against which this rotation can be measured? For example, if the sky were totally empty, how would we ever know that the earth spins on its axis?

It is possible to answer that the earth bulges at the equator because of the effect of centrifugal force, and since we can observe this bulge, it implies that the earth must indeed be spinning. But what exactly is this centrifugal force, and how does it arise? If we choose a set of axes that rotate at exactly the same speed as the planet, then everything would apppear stationary in this otherwise empty space. How can a centrifugal force arise in such a case? Where does this force have its origin? What does it mean for there to be a direction for spin when nothing else is present?

Penrose asked this same question in a new way. Quantum theory gives a meaning to the spin of the electron. It claims that an electron can spin in one of two alternative directions. But what meaning would those alternatives have when the universe is totally empty? What difference is there between a spin up and a spin down? We should expect such differences to manifest themselves only when a number of other reference points are present.

So if the distinction between having a spin up and spin down is to have meaning within a quantum theory set in empty space, it seems to imply that, rather than living in some sort of general background space, electrons actually create their own spaces; a sort of quantum version of our own space-time. Each electron would therefore have associated with it a sort of primitive space, possibly at this stage nothing like our own space-time. However, as large numbers of electrons come together, Penrose conjectured, it is possible that their individual protospaces will give rise to a collective space, a sort of shared spatial relationship which may then begin to look something like our own space-time.

To Pentrose, spin seemed an excellent starting point, and the mathematical building block for spin is a spinor. As was pointed out earlier, this spinor is the most primitive element in quantum theory. Moreover, the quantum rules for putting spinors together involve pure addition and subtraction and have nothing to do with the ideas of continuity. Spinors were tailor-made for Penrose's combinatorial approach.

It is difficult to give a picture of a quantum mechanical spinor, because the spin of an electron is something very different from our classical, Newtonian motion of spin. A ball or a planet can spin on its axis and have a whole continuous range of spins, depending on how fast or slow it rotates. Another way of thinking about the spin of this ball would be to refer to its angular momentum. In the quantum case, however, something very curious happens, for the spin of an electron can have only one of two possible values: up or down. A lone spinor is therefore the simplest possible quantum object, and it can have one of two values. It is a binary object, a single bit of quantum space.

But just as a coherent message is built up in a computer out of many bits, can space be built out of many spinors? The next step, therefore, was to add a second electron, which would join quantum mechanically with the first. At the moment, no idea or property of space is present in the theory; the spinors simply join according to a basic combinatorial rule of quantum theory. Now add a third, fourth, fifth, and so on, building up a whole network of spinors. If Penrose is right, then when it became large enough, this spinor network should begin to develop some of the properties we normally associate with space.

Penrose tested his idea by, theoretically, bringing an additional spinor up to this giant spin network. Will there now, he asked, be a meaning to the direction of its spin? To his delight, the spin network answered: direction has meaning, direction emerges out of the relationships between spinors; a relationship, moreover, that is based purely on addition and subtraction. Moreover, the network gave its answer in the language of three-dimensional angles, the same spatial properties one meets in a school textbook of geometry. The spin network had generated the properties of angular directions in a three-dimensional space!

This is a really exciting result, for spinors are two-valued objects yet have created the properties of a space that is three-dimensional. Moreover, this space is generated at the quantum level out of very simple rules of addition and subtraction. Part of Penrose's dream of a combinatorial origin of space had been achieved; this spin network approach involves only the operations of arthmetic and avoids the ideas of infinity and continuity that had bothered Penrose.

The next question is: what happens when two very large networks are brought together? It turns out that, in general, their respective spaces do not cohere. The result is like a patchwork quilt in which adjacent parts do not match. Is this a failure of the theory? No, it is a marvelous result, for it is exactly how space should look from the perspective of general relativity. General relativity deals with space-time that is curved and distorted by the presence of matter.

It is possible to think of a curved space as one that is covered by a patchwork of small, flat spaces. When the overall space is curved, these individual flat segments will not join smoothly at their edges. The effect is similar to trying to cover a beach ball with a series of flat cards; there will always be discontiniuies at their edges. But this is exactly what the spin network seems to be suggesting.

This idea can be thought of in another way. What we experience as gravity, the pulling of our bodies to the floor, is the result of living in a curved space-time. But suppose we lived in a freely falling elevator. We would no longer experience any pull to the floor, and all our observations would convince us that gravity is absent and that we live in a flat space-time. In other words, with respect to the small region inside the falling elevator, space-time can be treated as flat. Now take the inhabitants of another falling elevator. They also believe that their space-time is flat and gravity is absent. Each set of observers in each elevator believe that they are in a flat space. However, it is not possible to join all these flat spaces together in a smooth way.

While the effects of gravity can be abolished locally by freely falling, this can never be done over a large region of space-time. Therefore, while curved spaces can be approximated by a patchwork of small flat regions, these regions will always have discontinuities where we try to join them at their edges. The mathematics involved in fitting regions of space together to cover some larger curved space is called cohomology. The idea of cohomology will crop up again in Penrose's work.

In a similar way, Penrose's giant spin networks do not join smoothly. This could be taken to mean that the overall space is curved, or to put it another way, the very fact of this failure to join is the curvature of space.

Twistor Origins

Penrose's spin network were a provcative concept, but in the end they did not go far enough. They could not in fact be used as a base for the unification of quantum theory with geometry. To begin with, the space they create is incomplete, for it contains no sense of distance or of separation, only Euclidean angles. Moreover, spin networks create space alone, and not space-time. These spinors are static and nonrelativistic. As a starting point, they are simply not rich enough to create a full space-time. What Penrose needed was a new building block, something that had both a quantum and a relativistic nature. He was eventually to create such a novel object and name it the twistor. The twistor project therefore had its origins partly in this idea of extending spin networks (quantum spaces) to a more general notion of space-time.

During the early 1960s, Penrose began to think about a new quantum object that, like the spinor, could be used to build up the concept of a quantum space. But this time the starting point had to be rich enough so that it would lead to a full four-dimensional space-time. At least this was Penrose's program back in the 1960s, but as it turned out the twistor, when it was finally invented, was to take Penrose in unexpected directions and to create powerful new insights in the world of theoretical physics.

The spinor is not rich enough. All it does is sit there and spin. In combinatorial form, the networks can only point out Euclidean angles. What was needed was a quantum object that emodies some notion of linear movement. Spin is associated with what is called angular momentum. The new quantum object must combine angular momentum with linear momentum, and on an equal footing. It must be an object that is both spinning and moving along. In addition, it must be both quantum mechanical and relativistic. Penrose's twistor was to fulfill all these requirements. It was also to bring together a number of other key ideas that Penrose had been working on: the significance of complex numbers and their geometry; the rules played by light rays in relativity (also called null lines), and the special ways in which physical solutions are singled out in quantum field theory.

All these ideas were to meet in twistors, and from these twistors would flow new intuitions on the nature of quantum fields, the internal structures of the elementary particles, a theory of quantum spaces, and new mathematical results in several important branches of mathematics. By the end of the 1960s, twistors had been created, and in the next two decades, their implications were unfolded in a variety of directions. Today one of the major researchers into the topic of superstrings, Edward Witten, is suggesting that these same twistors may be the true starting point for superstrings.

Complex Numbers

To understand the full significance of what Penrose had created, it is necessary to know something abut the power of complex algebra and why complex geometry may be most natural way to make the connection between quantum theory and the general theory of relativity. To enter the world twistors, it is first necessary to open the door of complex numbers.

The algebra of complex numbers and the geometry of the spaces they generate, is a powerful and elegant branch of modern mathematics. Not only is it a subject of study for mathematicians, but it also plays a significant role in physics, particularly in quantum theory. It was for these reasons that Penrose became convinced that twistor theory must be based within that field which mathematicans call complex analysis. Today Witten and other superstring scientists are also working in complex analysis, the mathematics that unfolds from the properties of complex numbers.

What exactly are complex numbers? You may remember that they have something to do with the square root of -1, that mysterious number that is always called i, the imaginary number. In fact, perhaps without realizing it, you have already been exposed to the power of complex numbers and complex analysis. The dual resonance model was created to explain how elementary particles scatter off each other during high-energy experiments. Since the effects of the strong nuclear force were too strong to solve directly, physicists resorted to a fundamental property of the S-matrix, which is itself a complex function. Without having to solve a number of very difficult equations, they were able to learn something about the S-matrix by relating what called its real and immaginary parts. Significant deductions could then be made about the elementary particles based on the mathematical good behavior of complex functions.

One way of thinking about the complex numbers is thay they allow algebra to continue in the face of a certain class of equations. Consider the following equations.
x² + x - 6 = 0
x² - 9 = 0
x² + 1 = 0
At first sight they look more or less the same, but there is something very strange about the third one.

The first, x² + x - 6 = 0, can be written as
(x + 3)(x - 2) = 0. For the whole expression to vanish either
x + 3 = 0, or x - 2 = 0. In other words, the solution to this equation is either x = -3 or x = 2.

Likewise, x² - 9 = 0 has two solutions. It can be written as
(x + 3)(x - 3) = 0, indicating the solutions are x = 3 and x = -3.

The third equations, x² + 1 = 0, runs into problems when we try to write it in the form
(x + ?)(x - ?) = 0. This can only work if ? is replaced by √(-1).
Yet there seems to be no way to define the square root of a negative number; no such number exists in the early history of mathematics.

But what if we assume that √(-1) is a perfectly legitimate number in its own right? In fact, why not define a whole continuum of imaginary numbers? If these imaginary numbers are allowed into mathematics, then it become possible for algebra to go on to solve equations like
x² + 4 = 0, x² + 2x - 4 = 0, etc.

Consider the horizontal real number line in which real number are placed so that number zero is placed at its middle and the negative numbers -1, -2, -3, etc. are placed on it increasingly to the left of zero, and the positive numbers 1, 2, 3, etc. are place onit increasingly to the right of zero.
Now let us consider a vertical number line for the imaginary numbers in which imaginary number 0 is placed at its middle and the positive imaginary numbers i, 2i, 3i, etc. are placed on it increasingly above zero, and the negative imaginary numbers -i, -2i, -3i, etc. are placed on it increasingly below zero. In such a line, imaginary numbers add and subtract the way real numbers do:
2i + 3i = 5i, 6i - 3i = 3i, and so on.
Addition and subtraction keep us on the number line. But what about multiplication?
3i × 2i = -6, since i × i = -1.
Multiplication takes us off from the imaginary number line to the real number line. Now the obvious next step is to make the whole thing consistent and combine the two lines, placing them perpendicular to each other with number zero as their intersection, thus making them axes of a number space. The space of these two numbers lines will be a complex number space. Numbers that lie on the axes are either pure imaginary numbers or pure real numbers. But off these axes, each number is a mixture of real and imaginary parts. Such a number as 3 + 3i is called a complex number and is therefore represented by a point in the complex number space.

One advantage of using complex numbers is that they enable us to locate any point in a plane using a single complex number and therefore provide an alternative scheme to the more familiar Cartesian coordinates. You may have seen photographs of a modern mathematical application of this idea, called the Mandelbraot set. This is related to the general topic of fractals, and, in this case, rich and infinitely complicated images can be generated in a simple way starting from a single complex number.

Complex numbers are also used when physicists and mathematicians work with vectors. With their help, the rules for combining vectors become straightforward; indeed, there is a whole algebra of vectors based on complex numbers.

But defining points and vectors in terms of complex numbers is relatively simple. The real power of complex numbers comes in the way they can generalize algebra and geometry. Being able to describe a system in terms of functions of a complex variable rather than a real variable has tremendous advantages. To begin with, there are formal relationships between the real and imaginary parts of such functions which stem from their mathematical good behavior (called "complex analyticity"). Just as a complex number always has a real and an imaginary part, a complex function can be similarly divided. For example, the relationship between the real and imaginary parts of a function determines how the absorption of light, as it passes through a piece of glass, is related to the way this light is bent and broken down into its component colors.

What is truly amazing about this relationship is that it follows directly from the mathematical behavior of complex functions and not from the physical details of a particular piece of glass. There are a host of other important physical relationships between the real and imaginary parts of a complex funtion. The dispersion relations of the S-matrix are a case in point. They tell us about the relationship between elementary particle resonances and scattering experiments simply by relating the real to the imaginary parts of the scattering matrix. In many instances, mathematicians and physicists find that by pushing a function into complex regions - "analytic continuity" as they would put it - they can bring more powerful techniques to bear. Clearly if Penrose's God is a mathematician, then while He may have created the universe by counting, He certainly had the beauty of complex geometry in mind when He did so.

But if this complexity is so important, why do we never see it directly? The length of something is 2 feet, never 2i feet or 2 + 3i feet. Likewise, the gasoline in your car is measured in gallons, not complex gallons. It appears that while complexity underlies the physical world, each time we try to see it by making a measurement, this complexity hides its face.

Mathematicians have a formal recipe for getting real real results from complex numbers. Every complex number N has its mirror image reflected in the real axis. This is called its complex conjugate N*. For example, the complex conjugate of 2 + 3i is 2 - 3i; it is obtained by reflecting the complex number in the real axis. Therefore, whenever a complex number is multiplied by its complex conjugate, the result will be real number, for
N × N* = (2 + 3i)(2 - 3i) = 4 - 9i².
And since i² = -1, the result is 4 - 9(-1) = 4 + 9 = 13, a real result obtained from two complex numbers.

Whenever a complex number and its conjugate pair up, the result is always real. Or to put it another way, it could be that underlying our real universe there are products of complex and their conjugates. This, as we shall see, is exactly what happens in quantum theory. So mathematical comlexity is always hidden from us by the pairing up of complex things with their conjugates to give real answers.

That complexity must underlie our real world can be most powerfully seen in the case of the quantum theory. In 1925, a matter of weeks after Werner Hdisenberg had discovered quantum mechanics, Erwin Schrodinger came along with an alternative approach. At first sight, Schrodinges's equation had a direct intuitive appeal, for it looked like the sort of equation that describes waves in water and air. This differential equation of Schrodinger was called the wave equation, and its solutions were called wave functions. All observed quantities can be calculated using the wave equation and turn out to be real.

At first it seemed as if electrons in an atom could thought of a real standing waves - like the vibratins of a string. But closer examination of the mathematics showed that things were not that simple, for the wave function itself was a complex function rather than a real function. Being complex, the wave function can never be directly observed or measured. However, the physical and observable quantities that are predicted by the theory are always obtained by a mathematical operation that involves taking the wave function Ψ and multiplying by its complex conjugate Ψ*. Simply taking the product of a complex wave function and its complex conjugate, for example, gives the probability of locating a quantum particle in given region of space. But the product of a complex function and its complex conjugate must alway give a real result!

So although the underlying formalism of the quantum theory is complex, its predictions always involve real numbers. Again complexity hides itself under the cloak of reality. Penrose believed that such complexity is so fundamental to the quantum world that it must enter explicitly into the describing of quantum space-time. In generalizing his spin network to twistors, it would therefore be necessary to work in complex spaces, and the whole power of complex mathematics would have to be used.

Null Lines

Another key idea in Penrose's approach is the light ray, also called a null line. In fact, the null line is also connected with the idea of complex numbers. You will recall from earlier discussion that, because the transformations of Einstein's theory of special relativity act to mix together space and time coordinates, Hermann Minkowski (1864-1909) decided to introduct time on an equal footing as the fourth dimension in physics.

But this fourth coordinate does not quite enter as the other three coordinates. It is not written as t but as ict, where c is the speed of light. That is, time enters as an imaginary number. But we never see the imaginary or complex side of space-time, because all measurements turn out to be real. Take, for example, the distance traveled on a journey through space-time. The distance on any graph is calculated by first measuring the distance along each of the axes. For example, to find the distance AB on the graph, we measure three units on the x axis and four on the y axis. The total distance s is obtained using the Pythagorean theorem:
s² = x² + y² = 3² + 4² = 9 + 16 = 25 = 5².
Therefore s = 5.
The Pythagorean relationship is also used in calculating distances in space-time. In this case, the time coordinate appears as ict. Hence.
AB² = x² + (ict)²
AB² = x² - c²t².
When x = ct, the measure of the disnace AB becomes zero. This condition specifically holds for anthing that moves with the speed of light c.

But what happens when one of these axes is the time axis? Suppose we measure the distance taken by a rocket on its journey through space-time. This distance is
√(x² - c²t²).
Provided the rocket moves at speeds less than that of light, this distance has a positive value. The result is a real number, and the imaginary nature of the fourth coordinate does not show up.

But what happens when our rocket moves at the speed of light? In this case,
x = ct, and the distance is zero. We have what appears to be a finite line, yet it has a length of zero. This is one of the famous null lines of relativity. (Null lines are to take a key role in twistor theory.) They are the tracks taken by massless particles, which move at the speed of light.
(Note that the speed of an automobile or rocket is given by distance/time:
v = xt.
This means that when the rocket moves at a speed v, the spatial distance x that it will travel in t seconds is vt:
x = vt.
the spatial distance x that it will travel in t seconds is vt. It is possible to substitute this value of vt for x in the space-time distance equation. The equation therefore becomes
√(v²t² - c²t²) = √[t²(v² - c²)] = distance,
which also indicates that when v = c, the space-time length vanishes.)

But what does it mean to say that the length of a null line is zero? Traveling at constant speed on a highway, we can measure our journey in hours - it is five hours from the city to our vacation destination, two hours to our summer cottage. Now, in the theory of relativity, we must always refer times and distances to particular observers. Take, for example, what happens to the internal time experienced by an elementary particle that is traveling very close to the speed of light. When high-energy cosmic rays strike the earth's upper atmosphere, they collide with the nuclei of oxygen and nitrogen atoms and create a whole cascade of "secondary particles" - mesons and the like whose speeds are very close to the speed of light. But some of these mesons are very short-lived when measured in the laboratory. Knowing how long these particles have to live before they disintegrate and how fast they are going. it is a simple matter to calculate that they can travel only a short distance before decaying. Nevertheless, these particles are registered in laboratories on the earth's surface after traveling for several miles through the atmosphere. How is this possible? The answer is that while, with respect to a stationary observer, the particles who live only for a very short time, according to their own internal clocks, they live much longer and are therefore able to travel for many miles in their lifetime. Another way of looking at this is that, with respect to the speeding particles, the distance from the earth's upper atmosphere to its surface is only is only a few feet, and not many miles.

What happens if we go even faster - if a clock could be carried on board a null line? (Of course, a material body like a clock cannot move at the speed of light, but our argument is hypothetical.) In fact, no time would have elapsed between leaving A and arriving at B! All distances would have shrunk to zero. For light, not even one second ticks away from the time it leaves a distant galaxy until it reaches your eye. Look up at the night sky and realize that, along a light ray, the distance to the stars is zero. When you look along a null line, nothing separates you from all that you see in the universe around you!

Of course, we generally say that a star is so many light-years away. This means that, for us as stationary observers on the earth, the light takes several years to travel from the star to earth, or that the information that now reaches us from the star concerns events that happened on the star's surface several years ago. However, with respect to light itself, this time interval is zero, and the distance vanishes. This is a direct consequence of the fact that time, in relativity theory, enters as a imaginary quantity. In fact, the structure of a beam of light in space-time, what relativists like Roger Penrose would would call the light cone structure, is most naturally expressed in terms of complex spinors.

The light cone is a key geometrical feature in relativity theory. Light from O spreads out in space with increasing time. Note that the point P, which lies within or on the light cone, can be reached from O, but not the point P', which lies outside the line cone. P', being outside the light cone, cannot therefore be causally connected in any way to O. Events at O and P' can have no influence on each other.

Null lines and the speed of light occupy a very special place both in relativity theory and in twistor theory. That lines can have the special measure of zero length is a direct consequence of the fact that time enters space-time with an imaginary rather than a real factor. Complex numbers enter physics at many levels in relativity. For example, they appear in certain of the solutions to Einstein's field equations. Penrose therefore believed that they must form the foundations of twistor theory.

Null Lines and Conformal Geometry

Null lines, or the paths taken by light and massless particles, are of such importance in relativity that Penrose has suggested only null directions are really "there." Indeed, it is possible to create a geometry based on null lines alone. But in a universe having such a geometry, mass would not exit, for only massless particles can move at the speed of light. Since null lines always have zero length, scale and distancewould have no meaning in such a universe. Change the scale of this universe, and all physical quantities remain the same. The property of having unchanged under changes of scale is called conformal invariance.

Conformal geometry is the sort of geometry that exists on a ballon that is being inflated and let down again. A face drawn on a ballon remains a face as it is stretched or contracted, yet the distances between the various features is never fixed. Similarly there is a rich set of relationships, particularly as regards the causal connectins, that is unchanged by conformal transformations.

For Penrose, conformal geometry based on the properties of null lines, was another key to quantum geometry. Mass and scales of length, he conjectured, are not primary quantities but emerge in a secondary way. The universe may have begun with a conformal geometry, a universe of light and massless bodies moving along null lines. However, as these null lines began to interact, the basic conformal invariance was broken and mass was born into the universe.

A Space That Divides

By the early 1960s, Penrose was thinking seriously about a complex geometry that would be built of null lines and would employ all the powers of complex mathematics. This geometry would also connect with the basic ideas of spin networks. With the help of such a geometry, Penrose hoped to be able to make deep interconnections to quantum theory. The question was, how to proceed?

The answer lay in his creation of twistors, objects that lie partway between relativity and quantum theory and, being more general that spinors, combine linear and angular momentum and live in a world of complex dimensions. Indeed, this space of the first elementary particles. In the early years of twistor theory, Penrose had also hoped that this twistor space could be used to generalize the original notion of a spinor network and produce a full four dimensional space-time. But this project ran into serious difficulties. Nevertheless, the twistor space itself was to provide powerful new insights.

Penrose was also concerned abut a fundamental problem in quantum field theory. It concerned the way in which physicists have to pick out real, physical solutions in quantum field theory. Quantum field theory is a generalization of the quantum theory of Heisenberg and Schrodinger and deals with the quantization of nature's fields. Light, for example, is a property of the electromagnetic field, and in quantum field theory this electromagnetic field must be quantized. The electromagnetic field's various quantized vibrations are discrete quanta that can be identified with individual photons of light. In a similar way, one can write down for matter a quantum field whose excitations are elementary particles such as electrons.

The problem with this mathematical formulation is that it allows for a whole spectrum of solutions having both positive and negative frequencies. But only half of these solutions, the one with positive frequencies, show up in nature. An elegant theory should not leave its last step unfinished. If quantum field theory is to explain the origin of quantum particles, then it must also tell how to pick out the physical, positive frequencies. Penrose, for his part, believed that this principle of selection should be built in a geometrical fashion. In the field of complex numbers, for example, the real axis bisects the complex space and divides it into a positive and negative part. Would it be possible to create a space of quantum field solutions that is similarly divided, in a very natural way, into a positive and a negative-frequency part?

By the fall of 1963, these various ideas were running through Penrose's head and extension of the spinor that formed the basis of his spin network, a conformal geometry based on null lines, the power of complex functions and complex geometry, and a space that would have a fundamental division into a positive and a negative part. At that time he was staying with the famous relativity group at Austin. Following a weekend break in San Antonio, the group was driving home when Penrose began to play around in his imagination with a diagram he had recently constructed.

A friend of Penrose, the physicist Ivor Robinson, had been wrestling with a difficult problem involving the electromagnetic field. On his way toward creating a solution, Robinson had recently come up with an algebraic expression but did not know exactly what this result would look like. Penrose had, for his part, worked out a geometrical visualizatin, a complicated picture using a series of null lines (or light rays). Now, as he relaxed on the drive home from his weekend vacation, this configuration came back again into his mind. He was struck by the idea that such a diagram demands a space of complex dimensions. Moreover, this space would naturally divide itself into two regions. Such a division, Penrose realized, must be related to division of the quantum field into physical and unphysical solutions.

Suddenly all the pieces of the puzzle began to fall into place. Penrose realized that what he had been looking for was a complex space built out of complicated congrucences of null lines. It would underlie the space of quantum theory, the primodial space to which the first photons moved, a space in which a single quantum of curvature would have meaning. Just as the matter around us is built out of primitive quantum entities, so too space-time would be derived from this more elemental space. It would be a space built not out of points but of twisting congruences created out of straight null lines. Penrose would go even further, the lines themselves, these basic twistors, are not only the components of geometry but of the elementary particles as well. Twistors are the generalization of quantum mechanical spinors. They are dynamic and not static objects, for not only do they have angular momentum, but linear momentum as well. These twistors are extended objects, and the space that is built out of them must now be seen in an entirely new way. In twistor space, lines are fundamental; points are only secondary objects created through the intersection of lines. In a twistor theory, we may expect the whole idea of points to take second place and be replaced by nonlocal descriptions of space-time.

On the one hand, twistor space would contain the rich conformal geometry of null lines; on the other, it is related to the physical and unphysical solutions of quantum field theory. Suddenly a fundamental property of the quantum field had been directly connected with an equally fundamental geometrical and relativistic property of space-time. The idea was exhilarating. The twistor was a Janus-like object, unified, yet with one face pointing toward quantum theory and the other toward general relativity.

In that fundamental insight, twistors were born, and during the next two decades, Penrose and his group would have to unfold the meaning of their geometry. Although it was to take a long time before these ideas caught on with other physicists and mathematicisna, eventually a major research program was begun with such goals as creating a twistor structure for the elementary particles, generalizing spinor networks using twistors, understanding the meaning of a single graviton in space-time, and developing the mathematics of massless and massive fields. Some of these projects were to lead to new insights and advances; others became bogged down by formal difficulties or for want of new ideas. In some areas, twistor geometry developed in new and unexpected ways and began to find applications in formal mathematics and theoretical physics and in fields that were far from those Penrose had originally considered.

Today members of Penrose's twistor group can be found in several countries. They communicate via the Twistor Newsletter, a mimeographed series of papers, notes, and abstracts, some of the handwritten. The cover of the newsletter bears a drawing by Penrose of a group of twistors that represent the Robinson congress - the original image that had started the whole thing in the early 1960s.

The story of how twistor geometry and the mathematics of twistor space were unfolded will continue in the next section.
In the final section, we shall explore some possible connectins between twistors and superstrings and learn how both approaches are really taking us into some very deep questions that may one day lead physicists to a super theory in which space-time structure and quantum theory both take on new forms.