








Do we really need halfadozen 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? A40yearold 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
20thcentury 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.
In the case of general relativity, the problem comes with "spacetime
singularities". These are the regions of time and space where everything
seems to go infinite and the laws of physics simply break down. Spacetime
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 spacetime 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
bigbang origin of the universe: this is again, of necessity, singular
in the classical theory, and curvatures of spacetime 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 stringlike loops, rather than pointlike 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 spacetime 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 spacetime 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 presentday 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 onedimensional as
viewed from a distance but can be seen to be twodimensional on close
examination. The six extra spatial dimensions in the GreenSchwarz 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 10^{15 }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 fourdimensional
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 spacetime 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 spacetime geometry, and this could ultimately
lead to a resolution of the basic paradoxes of presentday 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
spacetime, 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 spacetime. 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 spacetime, 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
19thcentury 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
twodimensional 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 spacetime, so it takes "lightray space"  the space whose individual
"points" represent entire light rays in spacetime  to be more fundamental
than spacetime itself. This turns out to have widereaching consequences.
In the normal view of spacetime, two points can sometimes be linked by a
light ray, which is usually represented as a line passing through the spacetime.
But in twistor theory, a light ray is defined as a single point in lightray
space, and a spacetime 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 lightray space. In fact, "twistor space", the full
space of twistor theory, is slightly bigger than lightray space, but the
surprise that emerges is that it is a space of three complex dimensions.
This translation of spacetime 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 complexspace 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 CalabiYau space. The strings themselves
are usually taken as lying inside the CalabiYau 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 spacetime of special relativity means that
twistor space now does double duty: it simultaneously supplies the needs
of both the CalabiYau spaces and spacetime 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 clearcut 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
It came from another dimension 









