The membrane at the end of the Universe

In the quest for the ultimate theory of everything, theorists have given
superstrings another dimension. Supermembranes may wrap up the Universe in 11 dimensions

Michael Duff and Christine Sutton

In 1984, a revolution rocked the rarefied world of theoretical physics. "Superstrings" had arrived, bringing with them the promise of the ultimate theory of everything, the holy grail of the knights of the theoretical round table. Four years later, superstrings probably' still offer the best hope of discovering a theory to describe all physics. Recent work is beginning, however, to show that the theory of superstrings is not the only hope. A theory based on "supermembranes" might prove to be just as plausible. Before 1984, physicists seeking a theory of everything had been thwarted in one aspect of their work, in trying to combine two well-understood branches of theoretical physics. On one hand, they had a perfectly workable theory of gravity, in the form of Albert Einstein's theory of general relativity. On the other, they had the so-called "standard model", which incorporates quantum theories for the strong nuclear force and the electroweak force. The strong force binds quarks together within particles such as protons; the electroweak force is manifest in the world about us as two forces-the electromagnetic force and the weak nuclear force responsible for radioactivity. Between them, the standard model and general relativity describe all four known fundamental forces.

The difficulties arose in trying to combine general relativity with the standard model -- in other words, in attempting to form a quantum theory that described gravity as well as the electroweak and strong forces. Such "unified" theories always developed fatal flaws in the form of infinite quantities, and even worse, so-called anomalies. "Anomaly" is the word theorists use to describe what happens when a mathematical symmetry, displayed by' a theory at the "classical", non-quantum level, becomes spoiled once they take quantum effects into account. Such anomalies may lead to inconsistencies, particularly when calculating the probability of an event taking place-the approach that characterizes quantum theory. You can only assign a probability to each possible outcome of a specific process, and this probability must lie between 0 (it will not happen) and 1 (it will happen). Even in quantum theory, however, we still demand that the total probabilities for all possible processes add up to one. If the presence of an anomaly prevents this from being true, then we must reject the theory as mathematically inconsistent. Candidates for a theory of everything must, therefore, first prove that they are "anomaly free", before we can even think of testing them experimentally.

If the search for a theory of everything is proving so difficult, you might ask: "Why bother?" The answer is that the quest has to do with more than simple aesthetics. As it stands, the standard model leaves open many questions. For example, why do the known elementary particles have the masses we observe? Why are there apparently two kinds of basic "building brick"-the quarks, which constitute the matter of atomic nuclei, and the leptons, which include the electron among others? The hope among many theorists is that a theory describing all physical forces might automatically answer such questions. Theorists would also prefer such a theory to be so well constrained as to be "unique".

The excitement that arose in 1984 came from the work of Michael Green, at Queen Mary College, London, and John Schwarz at the California Institute of Technology. They showed that in certain special kinds of theory, anomalies arising from quantum gravity cancel those due to other forces. Specifically, they cancel only in a very limited kind of "superstring" theory. Here at last was a remarkable hint of a theory that was not only free from anomalies but also possessed the desired quality of uniqueness. What is a superstring theory? In theories such as those incorporated in the standard model, the fundamental particles or objects, -- are like points -- they have no dimensions in space. As one of these objects moves through spacetime, it draws out a one-dimensional "world line" through the point's successive positions. In string theories, the fundamental objects are one-dimensional, like very thin strings. In this case, the objects draw out two-dimensional "world sheets" as they move through space-time (see Figure below).

To the superstrings, these fundamental objects must obey a symmetry known as supersymmetry. Supersymmetry relates the two classes of fundamental object required by the standard model-bosons and fermions. Bosons (named after Satendra Bose, because they behave in a manner that he first described) have integer values of intrinsic spin. This means that they have an angular momentum of 0, 1, 2, ... times the basic unit. Fermions have half-integer spins of 1/2, 3/2, and so on. These particles are named after Enrico Fermi.

In the standard model, the building bricks of matter, the leptons and the quarks, are fermions. But the particles that carry the fundamental forces, such as the photon of the electromagnetic force, or the gluons that bind quarks together inside protons and neutrons, are bosons. Supersymmetry translates fermions into bosons and vice versa. In particular, it turns out that the net result of two consecutive supersymmetry operations is simply a translation in space-time.

The theory of general relativity involves similar operations on a large scale, leading to what we know as gravity. Thus, gravity is a natural consequence of supersymmetry, absolving us from the need to include it in some ad hoc fashion-which is why theorists are enthusiastic about supersymmetry, despite the lack of experimental evidence.

In this way, a superstring theory-describing string-like fundamental objects that behave according to the dictates of supersymmetry-automatically provides a quantum theory of gravity. But does such a theory work? The remarkable answer that Green and Schwarz found was: "No, not usually." Only in a very special case does a superstring theory work. The superstrings must move in a 10-dimentional space-time. Further, we can use only one of two special mathematical symmetries to describe the fundamental interactions. According to Green and Schwarz, such a theory did indeed appear to be free from anomalies; it also seemed to be close to being unique.

At first sight, it might seem that a theory operating in 10-dimensional space-time should be nothing more than a mathematical curiosity. After all, we inhabit a Universe of four-dimensional space-time-three dimensions of space and one of time. However, as long ago as the 1920, theorists had found that a theory based in more than four dimensions could describe a four-dimensional Universe.

In 1919, Theodor Kaluza, at the University of Konigsberg in East Prussia decided to solve the equations of general relativity in five dimensions rather than four. In doing so, he ended up with the usual solutions of Einstein's four-dimensional relativity together with the equations of James Clerk Maxwell's theory of electromagnetism. It was as if electromagnetism was somehow the result of the "unseen" fifth dimension in space-time. (Why Kaluza took this step is unclear, but it was certainly one of the great moments of 20th-centuryscience.)

A few years later, in 1926, Oskar Klein at the University of Copenhagen showed how this fifth dimension could be real and yet unseen. Four of the dimensions spread out in opposite directions towards infinity, along the axes of a graph in Cartesian geometry; the fifth dimension, however, is periodic, curling round and round on itself as if following a circle. If the radius of this circle were very small, then we would not see this extra dimension. To see this idea, think of a hose pipe. From a far distance, the hosepipe appears one-dimensional, but if you look at it closer, you can then see the circular cross-section (see Figure below right).

So by 1984, theorists were already accustomed to the idea of theories that began life in more than four dimensions but which had extra dimensions that were "compactified" in order to explain the familiar four-dimensional Universe. Physicists had studied various "supergravity" theories based on supersymmetry together with the ideas of Kaluza and Klein in up to 11 dimensions. Theories in 11 dimensions were particularly interesting because 11 is the maximum number of dimensions that supersymmetry allows. It seems that such a limit might have some special significance - which in fact we do live in an 11-dimensional Universe, in which seven dimensions have compactified to provide "internal" particle properties, such as electric charge.

The trouble with 10

Why does superstring theory work only in 10dimensions, when the upper limit that supersymmetry imposes is II? There are also other questions that the superstring aficionados do not always answer. Is the 10000mensional superstring theory really finite; are there no infinite quantities in it? Is it really unique? Is there really no alternative?

First, consider the question of whether superstring theory in 10 dimensions is finite. As yet, there is no rigorous proof, but most experts are optimistic that a proof will be forthcoming. This optimism has to do with the fact that strings are "extended" objects, unlike the one-dimensional "points" of earlier theories.

More disturbing is the question of uniqueness. In 1985, it became clear that there are as many as five theories of superstrings in 10dimensions that are self-consistent. Each of these theories involves only one of the two allowed mathematical symmetries, but the theories differ slightly in detail. In one theory for example, the strings are open-ended, while in the others they are closed.

Already, then, the claim to uniqueness seemed to be weakening; in 1986, it became still shakier. There may be only five useful superstring theories in 10 dimensions, but what happens when the chosen theory is compactified-that is when it is brought down to the realities of the four-dimensional Universe, with its six extra dimensions reduced to unobservable proportions?

Kuman Narain, while at the Rutherford Appleton Laboratory, near Oxford showed that this compactification can lead to thousands of possible mathematical symmetries to describe the four-dimensional Universe. The "only two" of the original superstring theory were lost. Moreover, the resulting four-dimensional theory contains as many as 132 free parameters -- in other words, values such as the masses of particles, about which the theory is silent.

Other theorists have now developed many variations of Narain's original construction, leading to yet more four-dimensional superstring theories. Indeed, no one yet knows how many there are, but it surely runs into millions. One intriguing idea is that these are not actually different theories, but are the descriptions of millions of different phases of one and the same theory. Just as water can exist in three phases, as ice, liquid and steam, so superstring theory may exist in millions of different phases, only one of which we happen to inhabit.

The idea is theoretically tidy, but it does not help at all in extracting experimental predictions from superstrings. After all, if there are millions of different possible universes, why do we happen to be living in this one rather than any of the others? Some cosmologists working with superstrings have even suggeste dthat all the phases were created shortly after the big bang, and that they continue to coexist in different domains of our Universe. This view is difficult, if not impossible, to refute, but it sets a pessimistic note for the future of scientific enquiry. As Murray Gell-Mann, best known for suggesting the theory of quarks, has commented, physics would then be reduced to an environmental science such as botany.

It was against this background that a team working at the International Centre for Theoretical Physics in Trieste published their work on supermembranes early last year. Eric Bergshoeff, Ergin Sezgin and Paul Townsend considered objects that are, like strings, extended in space, but which differ from strings in being extended in more than one dimension. One of these was the supermembrane -- an object with two spatial dimensions (hence a membrane) which moves in the dimension of time against a background of Il-dimensional supersymmetric space-time. Now this turns out to be particularly interesting because 11 dimensions are the maximum allowed by supersymmetry.

The idea for the supermembranes did not come straight out of the blue. James Hughes, Jun Liu and Joseph Polchinski at the University of Texas at Austin had written an earlier paper on the subject. Moreover, the original proposal that membranes might describe elementary particles dates back to 1962 to work by one of 20th century's greatest theoretical physicists, Paul Dirac at Cambridge University. After the paper by Bergshoeff and his colleagues appeared, the supermembrane quickly became still more interesting.

At the department of applied mathematics and theoretical physics in Cambridge, another group of theorists, again including Townsend, made a different discovery about the supermembrane in 11 dimensions. Together with Anna Achucarro, Jonathan Evans, and David Wiltshire, Townsend found that only 12 extended objects are consistent with supersymmetry. One of these is the superstring in 10 dimensions, of the kind discovered first by Green and Schwartz. Closely related to it is the supermembrane in 11dimensions.

Indeed, around the same time, a group of theorists visiting CERN, the European centre for research in particle physics at Geneva, showed that they could derive one of the five consistent 10-dimensional superstring theories starting from the 11-dimensional super- membrane theory. Michael Duff, Paul Howe, Takeo Inami and Kelly Stelle used the technique of "simultaneous dimensional reduction". In other words, they could decrease the dimensions of the basic object from two (a membrane) to one (a string) while the background of space-time changed from having 11to 10dimensions. This development suggests that it is the 11-dimensional theory that is the more fundamental.

This was all very well. But could a supermembrane theory even ~n to make the same claims that had launched the superstring theories to fame? At first it seemed unlikely. There appeared to be good reasons why a supermembrane theory would fail, violating several of the basic conditions required of a theory of everything. Some of the difficulties were easy to overcome, but others seemed daunting.

In particular, the theory would need to accommodate massless particles, especially with those with a spin of 2. A quantum theory of gravity requires such particles to carry the gravitational force-the so-called gravitons. But early work suggested that a theory based on membranes could not allow these particles. In addition, the theory would, of course, need to be free from infinities and anomalies.

Progress came in August 1987, when Itzak Bars, Christopher Pope and Sezgin showed that massless particles could exist in a supermembrane theory. Moreover, they found that only in the supermembrane in II dimensions did the massless particles include the necessary graviton, with spin 2. Here again was evidence pointing to the importance of II dimensions.

Around the same time, Bars overcame another problem. He showed that there are definitely unavoidable anomalies in all "super" theories, except those based on the superstring in 10 dimensions or the supermembrane in 11 dimensions. Green and Schwarz had shown that these anomalies cancel out in the case of the superstring in 10 dimensions. Bars and Pope went on to prove that, at least in some circumstances, the anomalies would cancel out also for the supermembrane in 11 dimensions. The case for the supermembrane was looking stronger.

Meanwhile, Bergshoeff, Duff, Pope and Sezgin had shown that they could find stable solutions to the theory in 11-dimensional space-time that corresponded to a four-dimensional space-time plus seven compactified dimensions, which manifest themselves as a seven-sphere (S^7) (New Scientist, 2 June 1988, p 44). It appeared that the three-dimensional supermembrane (with two spatial dimensions and one time dimension) would form the boundary of the four-dimensional space-time of the real world, rather as the two-dimensional surface of a soap bubble encloses a three-dimensional volume-the membrane at the end of the Universe. Here at last was a tantalizing "explanation" forwhy we live in a four-dimensional Universe.

There remains, however, the thorny question of infinities. The hope is that because, like strings, supermembranes are extended objects in space, then they will avoid the infinities that normally arise in theories based on points. The rigorous proof is turning out to be exceedingly difficult to find for strings, however, so it is likely to be even more of a problem for membranes. Indeed, some string theorists have attacked membranes on the $founds that they might introduce yet more infinities. This 1Sbecause, in addition to the infinities in 11-dimensional space-time of the membrane itself, you have to worry about infinities in the three-dimensional space-time. (These kinds of infinities turn out to be trivial in the case of strings, because the world sheet is only two-dimensional.)

Some indications that the problem of infinities in supermembrane theories might be overcome have recently)' appeared. Towards the end of 1987, Miles Blencowe and Duff found that the symmetry of the four-dimensional space-time, produced in the compactification of the supermembrane theory from II dimensions, led to a striking conclusion. The theory must be free from infinities in three-dimensional space-time; this much is guaranteed by the special kind of symmetry involved. Hermann Nicolai, Sezgin and Yoshiaki Tanii have independently reached the same conclusion; thus supermembrane theory has overcome another major obstacle.

What next for supermembranes? Whatever the future holds. Theoretical physics will be the richer. Even if supermembrane theory proves to be wrong, theorists will be able to use that fact to understand better the uniqueness of superstrings. If supermembranes prove to be right, the outcome will be still more interesting. The supermembrane may cast light on why we live in a four-dimensional space-time. Whatever happens, however, supermembranes, superstrings, or both will have to confront experiment with testable predictions before the accolade of "theory of everything~" can be finally awarded. And that may prove the most difficult hurdle of all.

Michael Duff is a reader in theoretical physics at Imperial College and a staff member of the theory division at CERN.

Paul Dirac was the first to suggest that particles may be more like membranes