In any case, in 1977 string models received a big boost when Ferdinando Gliozzi, Scherk, and David Olive discovered a way to remove the tachyon from string theories in ten dimensions. Their solution appeared to involve supersymmetry as a symmetry not just on the string itself, as it had originally in fact been discovered, but throughout the full tendimensional space-time in which the strings moved. All of the particle states on the string involved equal numbers of fermions and bosons, a hallmark of space-time supersymmetry. Interestingly, this finding appeared well before four-dimensional supersymmetric GUT models were explored, four years later.
In 1981 John Schwarz and his collaborator Michael Green, another well-established string theorist, actually proved that Gliozzi and colleagues’ construction indeed involved supersymmetry as a symmetry on the full ten-dimensional space, and not just on the string itself. String theory had officially become superstring theory.
The significance of this proof cannot be overemphasized, because with the unphysical tachyon state done away with, and with full supersymmetry in ten dimensions, a host of new and elegant mathematical techniques could then be applied to the problem of determining if the theory was fully consistent as a possible quantum theory of gravity. Within two years Green and Schwarz had their answer, and it rocked the physics world. In 1984 they submitted a paper to the European journal Physics Letters in which they demonstrated that superstrings in ten dimensions could yield fermions, bosons, Yang-Mills fields, and gravitons in a way in which all nasty infinities appeared to be completely absent. It was a fully finite quantum theory that in principle had the potential to be, as it quickly became known, a Theory of Everything—the holy grail of physics ever since Einstein had first set out to unify gravity with the other forces in nature. Suddenly all the diverse pieces that had occupied theorists over the past decade seemed to come together in a most remarkable way. Perhaps the most unexpected result was that this theory appeared to not only produce finite results instead of infinite ones when dealing with what seemed otherwise intractable physical processes, but if the ten-dimensional superstring had attached to it a sufficiently large set of Yang-Mills fields, then it turned out that it would be possible to break left–right symmetry. As you will recall, this is required if the theory is ultimately to incorporate in four dimensions the measured weak interaction—which has no such symmetry—without producing a mathematical horror called an anomaly. The response to these dramatic results from the particle physics community was thunderous. The first result—the lack of infinities—was perhaps not so surprising. After all, strings had tamed infinities when they were proposed as models of the strong interaction. Recall that the mechanism of producing finite results was apparently based on a mathematical trick: An infinite sum of terms can add up to a finite number even if the individual terms appear to increase indefinitely. In the case of strings, because an infinite set of states exist with every higher energy, as vibrations of a string become more pronounced, the possibility of infinite sums contributing to any physical process is immediate. What was far less obvious was that the physical conditions associated with the quantum mechanics of strings would allow the infinite sums to, in fact, converge to a finite value. In retrospect, there is a more concrete way of understanding this particular string miracle. Remember that quantum mechanics and relativity tell us that forces between particles occur via the exchange of virtual particles—those objects that can appear momentarily and then disappear so quickly that they cannot be directly observed. In this case, a virtual particle can be emitted by one object and absorbed by the other on an exceedingly small timescale.
Now, the troublesome mathematical infinities arise when virtual particles of arbitrarily high energy are exchanged. Because the uncertainty principle tells us that if virtual particles carry a great deal of energy, they can exist for only a very short time, so the particles that can emit and absorb them, respectively, must be very, very close together. High-energy processes such as this are therefore really probing the nature of very short distance scales. Strings solve this problem because on very short distance scales what we would otherwise view as elementary particles could instead be seen as excitations of strings. Below some distance scale, then, elementary particles must be treated as spread-out vibrations of a string. Thus, by changing the rules at short distances, strings provide a new limit (or “cutoff,” as it is referred to by physicists), thus taming the otherwise potentially nasty short-distance, or high-energy, behavior of virtual processes involving point particles. This kind of smoothing mechanism actually has another precedent—in this case arising not from earlier considerations of the strong interaction, but rather from the weak interaction. Before the weak and electromagnetic interactions were unified in a Yang-Mills–type theory, Enrico Fermi developed an approximate theory that could be used for calculating weak processes. While this theory was very good at low energies, it was well known that it would eventually produce nonsensical results if the energies involved got too high. In the Fermi theory, weak interactions resulted from four different particles interacting at a single point (for example, when a neutron might decay into a proton, an electron, and an antineutrino). In the refined electroweak theory, however, it was seen that, in fact, what appeared at large distances to be four particles interacting at a single point was really two particles emitting a virtual particle that traveled a very short distance before either being absorbed by or producing via its decay, the other two particles. The short distance scale—at which this new picture becomes manifest—provided a short distance cutoff in calculations. Namely, the calculations of the old theory were only valid if one considered processes on scales larger than this short distance-limiting scale. On smaller scales new rules would apply, which, in fact, turned the previously nonsensical results into finite, sensible predictions that could be compared with experiments. This new short-distance scale where the rules change, called the “weak scale,” turns out to be precisely the scale below which the particles that convey the weak force behave differently than photons, which convey the electromagnetic force. On smaller scales, the two forces would appear to behave quite similarly.
String theory had the potential to solve similar nonsensical predictions of the naive quantum version of general relativity. In this theory, recall, the gravitational force occurs because of the exchange of virtual particles, called gravitons. Because of the complicated structure of general relativity, it turns out that there are an infinite tower of possible interactions of gravitons with each other, so that one can find interactions of three, four, five, or more gravitons at a single point.
It turns out that, in a way similar to that in which the interactions of four particles at a single point in the weak interaction produced nonsensical results, these many-particle interactions in general relativity ultimately produce a host of infinities if one allows the energies involved to become arbitrarily large.
But string theory offered a new opportunity to once again change the rules at small distances. If the particle we call the graviton is, at sufficiently small scales, resolved instead to be a vibrating string, then what is allowed at small scales will change. It turns out that, for technical reasons, a graviton is required to be made up of a closed string loop rather than a string segment whose two ends are not connected. In this case, one can redraw what would otherwise appear at large distances to be an interaction involving four gravitons at a single point. The picture becomes more complicated than simply having two graviton particles exchange some other particle with two other gravitons located some distance away, as in the weak case. Rather, one imagines a more complex process in which the vibrating string loops that masquerade as gravitons at large distances bifurcate and exchange other vibrating loops as shown in the second diagram below, which looks like two pairs of trousers sewed together. But while this is more complicated to draw, the effect is the same: The seemingly pointlike interaction of gravitons is instead spread out over some region of space, providing a new lower-scale cutoff that yields
results that are finite for such physical processes, even as the energies of the particles involved become very large.
In superstring language there is another way of viewing this effect, and that is that the string has a fundamental symmetry, called “conformal invariance.” This symmetry would imply that the physical nature of string interactions is independent of how one might stretch the string. Thus, for example, two strings that might seem to be otherwise close together during an interaction can in fact be stretched farther apart, and one would still get the same answer for the contribution of this process to physically measurable quantities. But, as we have seen, if the interaction points are spread out in space, then the dangerous infinities tend to be removed. This conformal, or stretching symmetry turns out to have unexpected implications when strings interact in certain exotic spaces. For example, in the particular case where one has closed string loops moving on a space that looks like a donut, called a torus, then it turns out that a string loop having a very small size around one circle of the donut behaves identically to a large loop stretched around the other circle (the circle spanning the horizontal direction around the donut).
This was a remarkable result and its implication was very important in the attempt to understand why strings might universally tame quantum infinities. For if there is a symmetry that says that string loops of radius smaller than some quantity—say, R0 —produce identical physical effects to those of strings of a size much bigger than R0 , the implication is that R0 represents some fundamental physical scale below which distances have no physical meaning. If you do try to probe smaller scales using strings that appear to be smaller in size, you end up producing phenomena that could instead be equally well pictured as involving strings of a much larger size. This “duality” between large and small strings, as it is called, can therefore be seen as providing a clear physical cutoff on how small a region can be over which virtual processes can occur. Once again, this small-scale cutoff has the effect of rendering otherwise potentially infinite virtual processes finite. While spreading out the interactions of gravitons is one way to turn gravity from a quantum theory beset with infinities to a quantum theory that is apparently finite, having a finite theory does not imply that one has the finite theory. A host of other issues, both physical and mathematical, must be addressed before we might gain confidence that this is the case. This brings us to the truly unexpected string miracle. It was also discovered in 1984 that the quantum theory of supersymmetric strings in ten dimensions can, in certain circumstances, naturally avoid another type of more subtle and dangerous mathematical inconsistency I mentioned earlier, which physicists call an “anomaly.” An anomaly occurs when quantum mechanical virtual processes destroy the mathematical symmetries that one would otherwise expect a theory to possess. It is as if one produced a theory that predicted the earth should be a perfect sphere without any imperfections, so that any place on the planet would be identical to any other place, but when one considered quantum mechanical effects one would instead find that on small scales the sphere would contain mountains and valleys, so that some of its points would be very different than other points. Thus, the beautiful spherical symmetry of the theory would be destroyed.
Such nasty quantum mechanical anomalies have been found to generically occur in one particular type of quantum theory: that which distinguishes left from right. Unfortunately, as we have seen, the weak interaction is precisely such a theory, in which “left-handed” electrons behave differently than “right-handed” electrons. To step back a bit, it was somewhat of a surprise that strings in higher dimensions even allow for such a possibility of “handedness” in the first place. Careful studies of Kaluza-Klein theories in higher dimensions by Ed Witten, in particular, had earlier demonstrated “no-go” theorems implying that there was no straightforward way to distinguish left-and right-handed fermions in higher-dimensional theories.
It turned out, however, that one can avoid these no-go theorems if one changes the rules a bit. Namely, if instead of pure Kaluza-Klein gravity in the higher dimensions, one supplements the theory by having extra YangMills fields living in these higher dimensions—precisely the situation that, I remind you, arises in supersymmetric string theories in ten dimensions—then these fields can impact upon the fermions living on strings in complicated new ways in order to produce right-handed and left-handed objects that behave differently.
But with this realization came the concern about anomalies. In general, once left-and right-handed fermions behave differently, then the quantum mechanical contributions of virtual left-and right-handed particles to various processes can destroy the very symmetries that are required in order to keep the theory mathematically consistent. These anomalies essentially undo the very careful cancellations of various otherwise infinite quantities that are ensured by the Yang-Mills symmetries, as well as resulting in a host of other nonsensical predictions. Actually, things are even worse in ten dimensions than in four, because not only can the Yang-Mills symmetries get destroyed by anomalies, so can the symmetries that underlie general relativity. Thus, there is actually a greater chance that any given theory of gravity will prove to be nonsensical as a quantum theory in ten dimensions than it will in four dimensions. What Green and Schwarz showed in 1984 was that for two specific kinds of supersymmetric string theories in ten dimensions, the theory was not only finite, but even with left-and right-handed fermions acting differently, all anomalies disappear. What might result therefore could be a completely finite and consistent quantum theory of gravity. Within months of the Green and Schwarz discovery, feverish activity by two different groups produced two more dramatic developments that ultimately generated enough excitement to induce much of the rest of the particle physics community to drop what they were working on and begin to explore this new possible Theory of Everything.
The first development involved a group led by David Gross, who, you may recall, helped to kill the first incarnation of dual string models when he discovered the phenomenon of asymptotic freedom, which demonstrated that QCD, and not a dual string model, was the proper theory of the strong interaction. As I indicated earlier, David’s graduate career had begun at Berkeley, and continued at Princeton, where he did important work on dual string models with Neveu, Scherk, and Schwarz. His return to this subject, after having abandoned it a decade earlier, was nothing short of triumphant, and he has taken it up again with all the fervor, as he himself suggested, of a converted atheist.
Gross, along with his colleague Jeff Harvey and students Emil Martinec and Ryan Rohm, developed, in a tour de force, something with the memorable name of “heterotic string.” The name does not derive from the word erotic, but rather from the root heterosis, although there is also no doubt that the model is kinky, both metaphorically and literally. Indeed, it is so imaginative as to be considered sexy by many theorists, which says something either about the model or about theorists.
When Green and Schwarz discovered that superstring theories in ten dimensions could be consistent, finite, and anomaly-free, they identified two possible symmetries of strings that would allow this. They explicitly demonstrated three different sorts of superstring solutions that exhibited one type of symmetry, but none that exhibited the other type, which for a number of technical reasons seemed like it might produce more interesting grand unified scenarios. The heterotic string, on the other hand, could work with either symmetry and thus was of special interest. What made this particular string theory so exciting, however, was not merely that it could produce potentially more interesting Yang-Mills symmetries, but that the existence of this Yang-Mills symmetry was forced upon it, not by the seemingly ad hoc need for anomaly cancellation, but by the requirements of formulating the string theory itself. This suggested some potentially deep connection between the possible existence of strings in ten dimensions and the observed Yang-Mills symmetries of nature in four dimensions.
The heterotic string model involves closed string loops, which on first glance is unusua
l, because closed strings, while they incorporate gravity, do not generally incorporate Yang-Mills symmetries. Gross and his collaborators, however, realized that if one is bold enough then this limitation can be circumvented. In particular, on a closed string, the vibrations that travel in one direction around the string are completely decoupled—that is, they do not interact with the vibrations that travel in the other direction around the string. There is a classical analogy for this: If you take a regular string, and jiggle it from one end to send a wave down it, while at the same time jiggling it from the other end to send a wave in the opposite direction, you will be able to see the two waves pass directly through each other at the center of the string. The two wave modes do not interact. Now for an amazing feat of mathematical sleight-of-hand: It is possible to imagine a sort of “hybrid” string in which the left-moving and rightmoving vibrations on a string are quite different. In fact, Gross and his coworkers argued that these different modes could actually be pictured as living in different sets of dimensions!
For the ten right-moving sets of vibrations on strings, Gross and colleagues treated them precisely as Green and Schwarz did for their tendimensional superstring: with ten normal coordinates, and with sixteen of those strange Grassmann anticommuting coordinates. Recall that the effect of this construction is to produce equal numbers of fermion and boson excitations on the string.
Gross and coworkers then imagined that the poor left-moving vibrational excitations were bereft of supersymmetry. You may recall, however, that the quantum mechanics of vibrating strings without supersymmetry can only be formulated consistently in twenty-six dimensions. In a leap of creative chutzpa that is hard to beat, Gross and his colleagues then simply imagined that the left-moving vibrations on strings act as if they live in a twenty-six-dimensional space!
Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone) Page 20
