Do we really need half-a-dozen hidden dimensions? Not any more, says physicist Roger Penrose

MOST people have at least heard of string theory; even if their ideas about what it actually is may be a bit vague. One of the most widely publicised aspects of string theory is its need for extra "tiny" dimensions of space. But in the last few months there have been developments that are quietly beginning to do away with the need for these extra dimensions. The fantastical but perhaps appealingly exotic idea that our universe has myriad extra dimensions on top of the four we experience may soon be considered rather dated. The reason for this possible sea change? A40-year-old idea called twistor theory.
Both string theory and twistor theory are attempts to understand the fundamental structure of the universe. It has long been recognised that the two greatest 20th-century revolutions in physical understanding, general relativity and quantum theory (or, more accurately, quantum field theory), must somehow be combined into one comprehensive view of the world, a "quantum gravity" theory. There are serious inconsistencies in each of the great theories separately, and there are reasons for believing that each could benefit from its union with the other.The main reason is that in certain situations, the theories just don't make any physical sense. Click for more on Twists
In the case of general relativity, the problem comes with "space-time singularities". These are the regions of time and space where everything seems to go infinite and the laws of physics simply break down. Space-time singularities are an almost unavoidable feature of this classical theory. When matter collapses in on itself to form a black hole, for example, there will be a region in the deep interior of the hole where space-time singularities arise, and where matter as we know it gets crushed out of existence. And if we reverse the direction of time, we find the big-bang origin of the universe: this is again, of necessity, singular in the classical theory, and curvatures of space-time evidently become infinite. It is anticipated that the appropriate quantum gravity theory will serve to replace this singular classical behaviour by some mathematically comprehensible and physically consistent scheme.
Quantum field theory also has its problems with infinity. A strict adherence to the mathematical requirements of the theory almost inevitably leads to the nonsensical answer ''infinity''. If that were the whole story, quantum field theory would be essentially useless. Fortunately, physicists have come up with mathematical procedures such as "renormalisation" which, in many cases, will allow the physically observed values of various physical quantities (such as mass or electric charge) to be inserted into the expression in a way that sidesteps the problem of the infinite expressions. The guiding principle in theories of particle physics is that renormalisation has to work, and this has been used to great effect in arriving at the standard model for particle physics, which now enjoys a great deal of experimental support. However, the standard model of particle physics is still not a "finite theory"- it doesn't always give finite answer to reasonable questions -and it has something like 17 undetermined parameters, such as the masses of sorne fundamental particles. Theoretical physicists have sought deeper theories which might be finite, as the first incaritmation of string theory initially appeared to be.

"I have always had difficulties with these extra dimensions. The tiny balls that they curl up into are likely to be highly unstable and collapse"

Around 1970, on the basis of a study of certain properties of strong nuclear interactions, Yoichiro Nambu of the University of Chicago proposed that the fundamental constituents of matter might best be thought of as little string-like loops, rather than point-like particles. There seemed to be a distinct possibility that the resulting quantum theory would actually be finite. It was subsequently noticed, however, that a serious anomaly arose in the quantum version of this string theory, which could apparently only be resolved in a space-time of 26 dimensions. The situation was improved somewhat when, in 1984, John Schwarz of the California Institute of Technology in Pasadena and Michael Green, now at the University of Cambridge (with input also from Joel Scherk of the Ecole Normale Supefleure in Paris) showed that the anomaly problem could be resolved somewhat less drastically if ideas of "supersymmetry" were incorporated. Supersymmetry is a proposal, popular with many particle physicists, which takes advantage of the fact that the particles of nature are divided into the "bosons" (such as photons) and "fermions" (such as electrons). According to supersymmetry, each fundamental boson has a supersymmetry partner which is a fermion, and vice versa. This pairing off of bosons and fermions would result in the infinities coming from bosons cancelling out those coming from the fermions, giving hope for a finite quantum field theory. As yet, no such supersymmetry partner has been observed, giving essentially zero observational evidence, so far, for supersymmetry in nature. Nevertheless, supersymmetry enables the number of dimensions of string theory to be reduced to 10 (one time and nine space dimensions), so only six extra spatial dimensions are needed.
But that's still a problem. How are we to make sense of the extra spatial dimensions in these theories? To have a space-time in which there are six (or 22) extra spatial dimensions certainly seems, at first, to be very removed from experience.

"Twistor theory could ultimately lead to a resolution of the basic paradoxes of present-day quantum theory, such as the problem of Schrödinger's cat"

String theorists would hold that these extra dimensions are "small", just as a length of garden hose is seemingly one-dimensional as viewed from a distance but can be seen to be two-dimensional on close examination. The six extra spatial dimensions in the Green-Schwarz scheme would be curled up into a tiny ball of overall dimension not much greater than what is referred to as the "Planck scale" of 1015 metres, which is some 20 orders of magnitude smaller than the ordinary scale of nuclear particles. I have always had difficulties with these extra dimensions. The tiny balls that the extra dimensions curl up into are likely to be highly unstable and to collapse into singularities in the same kind of way that occurs with the black holes and the big bang in standard four-dimensional general relativity, but now in a characteristic timescale of the absurdly tiny "Planck time" 10-43 seconds. So with extra dimensions we are, essentially, back where we started: facing infinities -but now the infinities of space-time singularities, not just those of quantum field theory. However, new ideas suggest that "twistor theory", a proposal that I put forward in the 1960s, may offer a way out. Twistor theory operates within the viewpoint that the very rules of quantum theory may need to be modified at the macroscopic level. Moreover, it provides some possibility that these rules may become subtly modified when applied to space-time geometry, and this could ultimately lead to a resolution of the basic paradoxes of present-day quantum theory, such as Schrödinger's cat (New Scientist, 9 March 2002, p27).
Its underlying methodology is to seek basic features of the geometry of space-time, and other manifestations of macroscopic physics, that provide links with fundamental aspects of quantum theory. The essential key is the profound role of complex numbers in quantum theory - numbers of the form a + ib, where a and bare ordinary real numbers and i is the square root of -1. These numbers, their remarkable algebra and their even more remarkable calculus, have turned out to be essential ingredients of the equations of quantum mechanics. Twistor theory is rooted in the observation that they also have a deep role to play in the geometry of space-time. What is this role? Well imagine you are out in space, observing the celestial sphere of stars all around. At the moment you see a distant star, you are, in effect involved in the entire history of some photon, whose track through space-time, or "light ray", contains both the point on the star where it was emitted (and the time when that event happened) and your eye (and the time of absorption). Your celestial sphere represents the totality of such light rays. Another observer nearby, looking out at the same sky, also sees a celestial sphere.If there is no significant relative velocity between the two of you, the two celestial spheres are geometrically identical and could be put into coincidence by a rotation at most. But suppose that the two observers are in high relative velocity, of some sizeable fraction of the speed of light. Stellar aberration, which arises because of the finite nature of the speed of the light travelling from each star, now creates some distortion between one celestial sphere and the other.
A simple rotation is no longer enough to map one to the other; there is a geometrical transformation involved that is, remarkably, precisely the one that comes about naturally from the algebra and calculus of complex numbers.

Complex curves
Another way of putting this is that the celestial sphere can be regarded as a "complex curve" -sometimes called a Riemann surface, after the great 19th-century mathematician Bernhard Riemann, whose geometry turned out to be basic to Einstein's general relativity and whose complex curves underlie string theory. How can a two-dimensional sphere be acurve? Well, a complex space of dimension n, when regarded in the conventional, "real" way, always has 2n dimensions - for the simple reason that the single complex number a + ib contains the information of the pair of real numbers a and b. Riemann surfaces can have various topologies; the celestial sphere is the particular case of spherical topology known as a Riemann sphere. At this point, this looks like just a mathematical curiosity. But twistor theory takes this apparent curiosity as fundamental. The twistor viewpoint is to regard light rays as more primitive than points in space-time, so it takes "light-ray space" - the space whose individual "points" represent entire light rays in space-time - to be more fundamental than space-time itself. This turns out to have wide-reaching consequences. In the normal view of space-time, two points can sometimes be linked by a light ray, which is usually represented as a line passing through the space-time. But in twistor theory, a light ray is defined as a single point in light-ray space, and a space-time point P is represented by the celestial sphere's worth of light rays through P - in other words by a complex curve (a Riemann sphere) running through light-ray space. In fact, "twistor space", the full space of twistor theory, is slightly bigger than light-ray space, but the surprise that emerges is that it is a space of three complex dimensions. This translation of space-time to twistor space turns out to be a surprisingly fruitful endeavour. In particular, the equations of certain important physical fields, such as the Maxwell field of electromagnetism, can be solved and very simply expressed in terms of twistors. Moreover, there are extensive partial solutions in terms of simple twistor geometry of the Einstein equations for describing gravity, and of the corresponding equations describing the basic forces of particle physics. The emerging link between twistors and string theory arises from the complex-space nature of twistor space. In modern string theory, the space representing the six hidden dimensions is a space of three complex dimensions known as a Calabi-Yau space. The strings themselves are usually taken as lying inside the Calabi-Yau spaces, but these spaces are artificially imposed. In twistor theory, we can put the strings into twistor space instead. The geometrical correspondence between twistor space and the four-'real' dimensional space-time of special relativity means that twistor space now does double duty: it simultaneously supplies the needs of both the Calabi-Yau spaces and space-time itself. Accordingly, the strings now become complex curves - Riemann surfaces, to be specific - in twistor space! The development of these ideas is due mostly to Edward Witten of the Institute for Advanced Study in Princeton, New Jersey, who has been the main driving force behind new directions in string theory since the late 1950s. It seems to me that there are some striking developments coming from this new approach, and it will be exciting to see how far these ideas can be extended towards a comprehensive physical theory with something serious to say about quantum gravity. Twistor theory has been around for a little over four decades now. Like string theory, it has had more impact on pure mathematics than on clear-cut physical results, but as the string theorists begin to take it up it may just becoming into its own as a physical theory. And, since fully fledged twistor theory calls for just three space dimensions and one time dimension, the first result of this emerging union may well be that those extra dimensions of string theory slip quietly away.

Further Reading

Roger Penrose is Emeritus Rouse Ball Professor of mathematics at the University of Oxford. His new book, The Road To Reality: a complete guide to the laws of the universe which gives accounts of both string theory and twistor theory, is published this week by Jonathan Cape

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