Quantum mechanics smooths spacetime

There was a spirit of optimism among the scientists who met in the summer of 1995 at the Center for Physics in Aspen (US state Colorado) for a workshop on string theory. Several researchers were of the opinion that the final, all-encompassing theory (theory of everything or TOE for short) was finally within reach.

The physicists' dream would be a world formula of the kind that Werner Heisenberg (1901 to 1976; Nobel Prize 1932) tried in vain to set up in his later years: a single equation, the solution of which is able to describe our universe completely - with a three-dimensional space and a time dimension , in which quarks, electrons and other particles organize themselves into atoms, butterflies and stars, held together by nuclear forces, electromagnetic interaction and gravity, and with a big bang at the beginning of everything. Then it would become evident that the two previously unconnected great theoretical structures of physics - quantum mechanics and the theory of gravity - form a close unit. "Our physical terms will change fundamentally along the way," predicts Edward Witten of the Institute for Advanced Study in Princeton (New Jersey).

The string theory

The so-called string theory raised great hopes for such a standardization ten years ago; According to her, the basic building blocks of the universe are not punctiform particles, but inconceivably tiny strings, the vibrations of which are supposed to produce all observable particles and forces (Spektrum der Wissenschaft, November 1986, page 54). These strings - open or closed in loops - are only about 10-35 meters long and, like a violin string, can carry out many different natural vibrations. Every oscillation state has a certain energy and can be regarded as a quantum mechanical particle (Fig. 2).

But the string model soon encountered mathematical obstacles and broke up into five competing theories. Having multiple unified theories isn't just "unaesthetic," as Andrew Strominger of the University of California at Santa Barbara finds; on top of that, they have thousands of solutions, most of which do not see our universe at all alike. When Sheldon L. Glashow of Harvard University in Cambridge, Massachusetts was asked in 1986 to summarize the unified theory in no more than seven words, the stubborn skeptic replied in mock desperation: "My God, why did you leave me? "

A new symmetry

As if the call for help had been heard, a recently brought into play symmetry - duality - causes all the different strings to twist into one another, as it were. Fundamental particles - or strings - are redefined through duality. Now it seems as if the basic building blocks consist of the very particles that they themselves produce (Fig. 1). Witten believes that this principle not only points the way to an all-encompassing theory, but could also explain why our universe is just this way and not different. "We are approaching an explanation of quantum mechanics," he says. For the time being, such claims hardly meet with loud criticism, because string theory is mathematically too complicated for the vast majority of physicists.

On the other hand, under the aspect of the duality principle, the world becomes even more bizarre than it already is. Strings easily turn into black holes and vice versa; additional dimensions appear in different areas, and the universe is filled not only with strings, but also with bubbles and other interfaces. But for some researchers, the diverse connections point to a deeper layer of explanation - presumably the unified theory of everything.

For physicists, "dual" has many meanings. Roughly speaking, two theories are called dual if they differ outwardly but make the same physical predictions. For example, if you swap all electrical and magnetic quantities in Maxwell's equations, you get a different theory. But if one also hypothesizes that there are not only separate electrical charges, but also magnetic monopoles (something like the isolated north pole of a bar magnet), the two theories would be completely identical - that is, dual.

In a narrower sense, duality means that elementary and composite objects become interchangeable: whether an object is fundamental or is composed of even more fundamental beings is only a question of point of view. Both perspectives give the same physical results.

The first signs of duality were found in the quantum field theories; they describe particles as quantum mechanical waves that propagate in space-time. For example, in quantum chromodynamics (QCD), quarks are elementary particles that have a property similar to electrical charge called color. Due to the color charge, the quarks attract each other extremely strongly and form pairs or triplets, which correspond to directly observable particles such as pions or protons.

Just as there are no magnetic monopoles in the known world, there are also no particles with a color magnetic charge. But in 1974 Gerard 't Hooft from the University of Utrecht (Netherlands) and Alexander Polyakow (at that time at the Landau Institute near Moscow) described how the fields assigned to the quarks can aggregate into small structures that carry a color magnetic charge. Such lumps - imagined as spheres from which vector arrows protrude like hedgehog spikes - are actually solitons (that is, compact wave packets that do not dissolve over time) and behave like particles. Thus, from a theory for quarks with a color charge, the existence of solitons with a color magnetic charge could follow. These color magnetic monopoles would be composite particles that emerged from the fields of the more fundamental quarks.

In 1977 David Olive and Claus Montonen at CERN (the European Laboratory for Particle Physics near Geneva) pursued the idea that field theories with color are dual - so instead of viewing quarks as elementary and monopoles as composite, one might as well consider monopoles as elementary particles look at. One would then start from a field theory of interacting monopoles and arrive at solitons that correspond to the quarks. Both the theory based on quarks and the theory based on monopoles should physically amount to the same thing (Fig. 3).

Most theorists were skeptical. Even if the duality were to exist, it was thought to be hardly provable: the QCD is extremely complicated mathematically, and from it one would have to derive two comparable sets of exact predictions. "In physics, something can only be calculated very precisely in exceptional cases," explained Nathan Seiberg of Rutgers University in New Brunswick, New Jersey. But in February 1994 Ashoke Sen from the Tata Institute in Bombay (India) showed that duality can sometimes be exactly proven. This calculation converted the string theorists. "Witten used to tell everyone it was a waste of time. Now he thinks it's the most important thing," joked Jeffrey A. Harvey of the University of Chicago, Illinois at the meeting in Aspen. After all, Witten - often called Pope of string theory by scoffers - has established new trends in particle physics several times over the past twenty years.

Supersymmetry and duality

In the meantime, Seiberg has developed a computational method that is extremely useful for QCD research and is based on so-called supersymmetry; According to this theory, each particle of matter is assigned a force-transmitting partner particle and vice versa (Spektrum der Wissenschaft, August 1986, page 68). Supersymmetry cannot be proven with today's particle accelerators, but theorists like to invoke it.

By using supersymmetry to limit the theoretical variety of interactions between particles, Seiberg was able to perform some previously impossible QCD calculations. He and Witten also showed that supersymmetric versions of QCD are dual.

This has amazing advantages: quantum chromodynamic calculations are extremely difficult because of the strong coupling between the quarks. On the other hand, the interaction between monopoles is weak and easy to calculate. Because of the duality, theorists can deal with monopolies - and thus solve all QCD problems. "It's like a magic trick," says Harvey. "However, we still don't understand why it works." In this way, Seiberg and Witten calculated in great detail why one never observes free quarks in nature; in doing so, they proved a claim made by 't Hooft and Stanley Mandelstam of the University of California at Berkeley back in the 1970s.

Of course, all these results are based on the assumption that supersymmetry does indeed exist. Seiberg hopes, however, that the principle of duality will remain valid even without supersymmetry, so that "the results are qualitatively correct even if they depend quantitatively on supersymmetry".

But duality is much more than a mere calculation tool, it is actually a new worldview. "Something that was previously thought to be composed is becoming fundamental," stated Harvey - and vice versa. Even the otherwise rather conservative physicist Seiberg is now speculating about whether the quarks might be solitons, that is, dual partner particles of even more fundamental particles.

Intertwined strings

Although the concept of duality grew out of field theories, Sen said it "fits into string theory much more casually". The duality is able to unite different strings that exist in different dimensions and in spacetime of different shapes. This could cause string theory to lose its flaws and become an all-encompassing explanation.

This used to fail because it required too many string types and provided an undesirable variety of solutions. The reason is that the theory requires a ten-dimensional spacetime. The real world only has four dimensions: three for space and one for time. The other six should curl up so tightly that they do not even influence the behavior of the quarks - let alone that of everyday objects (Fig. 4). "It's like a garden hose," said Brian R. Greene from Cornell University in Ithaca (New York State): "From a distance it looks like a one-dimensional line. Only up close does it actually appear has a two-dimensional surface with one dimension curled tightly. "

To the chagrin of string theorists, the six additional dimensions can curl up in a number of ways. "Official estimates run into the tens of thousands," Strominger scoffed on the sidelines of the Aspen conference. Each alternative gives a different solution for string theory and its own picture of the four-dimensional world - not exactly what one expects from a unified theory.

By means of a form of duality, mirror symmetry, it was possible at the end of the 1980s to merge at least some alternative solutions by showing that the strings in two differently rolled up spaces sometimes produce the same particles. For example, if a dimension becomes very small, a loop of string wrapped around it (similar to a rubber band around a hose) can create the same particles as a loop that moves around a less curled dimension.

In string theory, the residual size of a shrunk dimension is related to the strength of the force between particles. In 1990, Anamaria Font, Luis E. Ibáñez, Dieter Lüst and Fernando Quevedo at CERN therefore hypothesized that there is also a kind of mirror symmetry for binding forces. Just as large spaces can be physically equivalent to small ones, a string theory with strong interaction might give the same results as one with weak coupling.

This conjecture linked different string theories, just as duality had done in the case of field theory. In addition, strings look like particles from a distance; therefore a duality in field theory follows from the duality in string theory and vice versa. In both cases, the duality passed all tests with flying colors and brought the two areas closer together.

Meanwhile, duality appeared in an entirely different realm - supergravity (Spectrum of Science, May 1985, p. 78). Albert Einstein's theory of gravity (1879 to 1955; Nobel Prize 1921) seeks to expand this standardization to include the aspect of supersymmetry. (On the other hand, one wants to modify the particle theory with string theory so that gravity fits into it.) In 1986 Michael J. Duff at Imperial College London developed a model of supergravity based on the vibrations of a bubble and thus a completely new fundamental entity introduced; while strings meander through ten dimensions, this bubble swims in an eleven-dimensional space-time.

"Most string theorists weren't interested in that at all," recalls Duff, who is now a researcher at Texas A&M University (Headquarters College Station) - probably because nobody knew how to do calculations with such bubbles. Nevertheless, Duff continued to work on different models with closed interfaces.

He found out that a five-dimensional membrane that moves in a ten-dimensional space could be an alternative description of string theory: If the membrane wraps around a space rolled up in it like a skin around the sausage and the enclosed space shrinks down Nothing together, the bubble at the end resembles a string. According to Duff, this coiled string was actually the same object as in string theory; he practically postulated a string-string duality.

At the same time in England, Christopher M. Hull of Queen Mary and Westfield College and Paul K. Townsend of Cambridge University were making many string theoretic generalizations of the duality principle. "But no group," says Duff, "cared a lot about the work of the others back then."

Duality and solitons

That changed suddenly in March 1995 at a meeting at the University of Southern California in Los Angeles. Witten gave the opening lecture and gathered evidence of duality from various areas. He noticed that Hull, Townsend, and Duff actually had the same idea, and claimed that Duff's bubbles in eleven dimensions were solitons of a particular ten-dimensional string.

Seiberg spoke to Witten. "He was so impressed by Witten's lecture," said John H. Schwarz of the California Institute of Technology later, "that he said: I should be a truck driver." But Seiberg also presented so many new results that Schwarz - a founder of string theory - began his lecture with the words: "I'll get myself a tricycle."

Since then there has been hectic activity. Every day the scientists find around ten new treatises on the subject in the electronic preprint library of the National Laboratory in Los Alamos. "This is how every morning begins for us," remarked Anna Ceresole from the Technical University of Turin during a conversation at the Aspener workshop a few months later, "as with others with the newspaper." In the process, incoherent and strange references to duality appear; they connect strings and bubbles with solitons of every type and shape.

One of these solitons resembles a line that stares like a hairy caterpillar of vector arrows; it turned out to be dual to a fundamental string. On top of that, it resembles a so-called cosmic string - a kind of hypothetical fault line in space-time that is said to have originated during the Big Bang (Spektrum der Wissenschaft, February 1988, p. 94). In addition, it has been found that different types of strings are dual when they are squeezed into our four-dimensional world by rolling in the excess dimensions. "The reasons are different, but in the end it all fits together," says Seiberg. "It feels like magic."

The hectic search for dualities has a method. "Many string theories are certainly not realistic," stresses Sen, "but we have to understand them all in order to find the right one." The variants can be linked and thus reduced by means of duality. According to Witten, the five ten-dimensional string models that are currently preferred will eventually prove to be aspects of one and the same basic string theory.

Duff even speaks of a "duality of dualities": the duality between different spaces could be related to that between elementary particles and composite objects. From this idea it would follow that the size of a rolled-up space influences the strength of the particle interactions and vice versa: If an internal dimension is large, the particle coupling should therefore be strong.

In addition, the size of the internal dimension can vary from place to place, explains Leonard Susskind from Harvard University: When a rolled-up dimension bursts open in a distant corner of the universe, space-time there also grows a fifth dimension. If, on the other hand, it remains tightly compressed - as in our immediate environment - quantum effects occur.

In fact, the fundamental measure of quantum theory - Planck's constant - is closely related to duality. This quantity relates, for example, the mass of a particle or a string to that of the dual partner. "For me, this is the strongest indication that we can learn about quantum mechanics from string theory," said Stephen H. Shenker of Rutgers University in Aspen.

The Brit Townsend also proposed a kind of fundamental democracy there: the membranes, which appear as solitons in string theory, are perhaps as fundamental as the strings themselves. However, our American colleagues are not yet convinced of this idea; they emphasized that calculations with membranes still made no sense. Cumrun Vafa from Harvard University commented skeptically: "It's a weird start - but you never know."

Small black holes

In April 1995 a connection between strings and black holes became apparent - and thus a way out of the second main problem of string theory. As Strominger, Greene, and David R. Morrison of Duke University in Durham, North Carolina found, black holes can be used to combine thousands of tens of thousands of string theory solutions into a complex tissue. This makes it much easier to find the solution that fits our universe.

In a sense, the black holes have always lurked on the edge of string theory. If enough mass accumulates in a place, it collapses under its own gravity to form a black hole (the size of which is determined by the so-called event horizon, depending on its mass). According to Stephen W. Hawking from the University of Cambridge, although a black hole, viewed classically, swallows everything - even light - it can still emit particles for quantum mechanical reasons; it gradually loses mass and continues to shrink. If the original mass consisted of strings, the decay would ultimately create an object without any expansion: a so-called extreme black hole, which actually looked more like a single particle.

Susskind, of course, objects that these tiny black holes have nothing to do with the collapsed stars astrophysicists are looking for: "Strominger's work is great; but I find it a bit far-fetched to talk about black holes here."

Indeed, extremal black holes - or black bubbles or black areas - are simply clumps of string fields called solitons. Strominger investigated how extremal black holes behave when one dimension of space-time curls up very closely. Take an infinitely long tube, bend it round, and tuck its ends together so that it looks like a swimming ring or torus. In this way the two dimensions of its surface can shrink to such an extent that a much smaller (but still infinite) space is created. Assume that this torus now constricts very tightly at one point. As Strominger found, some black holes, which consist of membranes wrapped around the compressed dimension, become completely massless. He interpreted these objects as quantum mechanical waves and included them in his calculations.

There were two amazing results. On the one hand, in string theory, the calculations used to fail whenever the hose contracted into a line; but with the quantum mechanical black holes one arrives at a satisfactory result even in this extreme case. This is thanks to quantum physics, explains Gary T. Horowitz of the University of California in Santa Barbara: "In classical physics, an electron that crashes into the point charge of a proton creates a singularity. It is only with quantum mechanics that one realizes that Electron goes into orbit. "

Second, a large number of massless black holes suddenly appear: The system has undergone a phase transition - similar to the condensation of steam to water.

The phase transition reflects a change in the torus itself. The structure tears at the thinnest point - physicists and mathematicians have always shied away from this violent act - and reshapes itself into a sphere; thus it continues to form a closed two-dimensional surface. In this way, two topologically very differently curled spaces are connected to one another in string theory (see box on page 47). "Mathematicians don't like that because something tears in the process," admits Strominger, "but the quantum effects smooth the transition."

Different types of cracks could thus connect thousands of solutions to string theory. With the internal spaces linked in this way, the strings can wander around and find the right one. Similar to the way water freezes in the Arctic and evaporates in the Sahara, strings would take on a shape appropriate to their respective environment. Finding the right solution becomes a dynamic problem.

According to Strominger, somewhere in the universe there could be a droplet in which Strings have found a different internal space. Upon entering this exotic droplet, black holes would turn into strings and vice versa. Such droplets may even appear in our immediate surroundings as virtual universes for tiny periods of time and then disappear again before we are able to perceive them.

The theory at all

Despite such speculations, physicists soberly concede that the final theory is still a long way off. "If we have found a nice formulation, it may no longer be called string theory," says Schwarz. "Maybe we'll just call it the theory." As early as the 1980s, claims that one had the all-embracing explanation of the world at hand met with such biting ridicule that theorists today shy away from the label TOE.

But many also doubt that the great theory can be formulated in the foreseeable future. "The string theorists clan tend to exaggerate," says' t Hooft snappy. An immense problem is that you may never be able to examine strings experimentally; Even with the most modern devices, the maximum achievable resolution is only 10-18 meters. The theorists hope that the large hadron collider (LHC), which is due to go into operation at CERN in 2005, will at least be able to demonstrate supersymmetry (see Spektrum der Wissenschaft, April 1994, page 54).

But even then, a persistent problem will remain: the familiar four-dimensional space-time is flat; but the broken supersymmetry favored by theorists forces space and time to curl up unimaginably close in most dimensions.

Witten dreams of a way out based on the duality between theories with different dimensions. Perhaps one can assume a universe in which only three of the four known dimensions are initially flat - one is still rolled up. Such spacetime has strange properties, but it eliminates the difficulties of supersymmetry. Eventually the fourth dimension would also unfold until the world we are familiar with emerged. "Witten's idea is pretty daring," said Schwarz in Aspen, "but maybe he's right."

The nature of gravity also raises difficult questions. According to Einstein, gravity arises from the curvature of space-time. Thus, the quantization of gravity is synonymous with a quantization of space and time. In this case, Horowitz explained, "space and time may lose their usual meaning and only emerge again as something like a structure when one considers great distances."

However, string theory does not yet meet such demands. In addition, the great theory will also have to work under the most extreme conditions, for example when describing the formation of the universe or the interior of a black hole. "String theorists have blind faith in their theory and are convinced that it can handle anything," says' t Hooft. "They don't understand a gravitational collapse any more than anyone else."

But at least the string theorists gathered in Aspen - fascinated by the mathematical treasures whose sparkle they think they can already see - were apparently hardly put off by skeptics: The genius loci of the small town between high peaks of the Rocky Mountains, which was discovered by adventurers in 1878 had been founded by silver deposits, blew again. After dinner, they debated in the open air about the wave function of the universe, which is supposed to describe the entire universe as a single quantum mechanical object. "It's like walking through a valley, knocking over a stone and finding an enchanted staircase," said Pierre M. Ramond of the University of Florida in Gainesville, caught up in the mood. "We're currently clearing the steps." Not knowing where they're going - that's what makes things exciting.


- Particles, fields and symmetries. Second edition. Spectrum of Science: Understandable Research. Spectrum Academic Publishing House, Heidelberg 1995.

- Theories for everything. The philosophical approaches of modern physics. By John D. Barrow. Spectrum Academic Publishing House, Heidelberg 1992.

- Cosmology and Particle Physics. With an introduction by Immo Appenzeller. Spectrum of Science: Understandable Research. Spectrum Academic Publishing House, Heidelberg 1990.

From: Spectrum of Science 3/1996, page 42
© Spektrum der Wissenschaft Verlagsgesellschaft mbH

This article is included in Spectrum of Science 3/1996