Prerequisites: The standard model -- Sypersymmetry
See also: M-theory -- Quantum gravity -- Higher dimensions of spacetime -- Mathematics and physics
It wasn't entirely necessary to make this assumption. Physicists characteristically make assumptions that "simplify" a physical system in many respects in order to make it easier to work with theoretically. As a result of this, we have such abstractions as "frictionless surfaces", "perfect gasses", "massless springs", and so on. Physicists realize that these do not correspond perfectly to reality in nature.
But physicists are ultimately interested in writing equations and solving them in order to make realistic predictions about the behavior of physical systems. If using simplifying assumptions and abstractions facilitates this activity -- and the resulting computations agree with experiment -- physicists are usually delighted, and tend not to further question the assumptions.
In light of this, assuming that genuinely elementary particles are mathematical points is not an unreasonable thing to do. We stress the "genuinely" qualifier, since from the earliest days of particle physics it was realized that "elementary" particles need not actually be elementary at all, but could be composed of smaller entities. After all, atoms had been found to consist of electrons and a nucleus. The nucleus had been found to consist of protons and neutrons. Protons and neutrons quickly showed evidence of having smaller constituents, which were eventually called quarks.
But at that point, two things happened. Despite many experiments, no evidence turned up that either quarks or leptons (such as electrons) were composed of smaller particles -- even though various theoretical ideas could be based on that assumption. Secondly, good theories of the electromagnetic, weak, and strong forces were developed in which the elementary particles really were elementary.
Even if a particle is elementary, it need not be a mathematical point without spatial extent -- it could be a solid 3-dimensional object of some sort or other, for instance. Like baseballs or pyramids, perhaps. However, the theories that were developed to describe the elementary particles were easiest to work out and apply at very small distances if it was assumed that elementary particles were points. Or, at least, so small that any spatial extent was irrelevant to the calculations, so that an excellent approximation could be obtained under the assumption that particles are points.
This is all well and good as long as the theories continue to agree with experiment. And that is exactly what they did, as far as any experiments that could be performed with any existing or reasonably feasible accelerator technology were concerned.
But that wasn't enough to satisfy theorists. It began to appear as though theories such as "grand unified theories" might explain how at least three of the fundamental forces were really just aspects of a simgle force. And not only that, but such theories might explain features at the cosmic scale as well -- why the universe has the shape and dynamics that it appears to have. However, in order to do that, it would be necessary to extrapolate the theories to vastly larger energy scales, and equally drastically smaller distance scales. In that case, the assumption that elementary particles are mathematical points might no longer be tenable. Assumptions about their precise size (however small) and shape could become important.
And in fact, that seems to have happened, in a rather spectacular way. Attempts to work with these vastly extrapolated theories didn't merely give wrong answers. The theories seemed to fail ignominously. This was particularly the case whenever theorists attempted to add the fourth known fundamental force -- gravity -- to their theories.
In all such attempts, the theories produced not merely wrong answers. They gave infinite answers, and had other sorts of mathematical inconsistencies as well. This was quite unacceptable. Of course, infinities had often appeared in earlier calculations as well, but it had always been possible to eliminate them with various tricks such as renormalization or supersymmetry.
But by the early 1980s, it was becoming apparent that such magic was just not working any more whenever gravity had to be dealt with. The magic seemed fundamentally broken. Not only that, but the theories themselves were not as aesthetically pleasing as they might be. In fact, they were pretty ugly, being filled with many unexplained arbitrary constants, free parameters, and gratuitous assumptions.
Existing assumptions needed re-examination. What assumptions might not be right? How about the assumption of point particles? We'll take a closer look at the problems with this assumption, but first let's consider the alternatives.
If elementary particles are not 0-dimensional points, then mathematically the next simplest assumption to try would be 1-dimensional objects, such as lines and curves. Since physicists like concrete, intuitive names, it was natural to call such things strings. However, unlike real, physical strings, a true 1-dimensional object has no thickness at all, only a length. Yes, an extremely thin string might be a good approximation of a 1-dimensional object. But it might not be good enough. There might be some very small distance scale where the thickness of a "string" would be of comparable size and could not be ignored. So if we are going to be honest and try to build a theory based on 1-dimensional objects, we have to keep this caution in mind.
What about theories in which the elementary objects are 2-dimensional (surfaces), 3-dimensional (solids), or perhaps are of even higher dimension? They are mathematically much harder to work with than string theories (which is plenty hard enough). In particular, it is very difficult in such theories to satisfy the usual requirements of a quantum theory, such as unitarity and causality.
So it's natural to try the simplest assumption first: 1-dimensional string theory. There are generalizations of string theory (called M-theory) in which higher dimensional elementary objects are considered, even though the successes of string theory seem to depend on features that don't generalize readily to higher dimensional objects. M-theory, in fact, is at the cutting edge of elementary particle theory, and we will deal with it elsewhere.
In string theory, elementary particles are assumed to be true 1-dimensional objects. They might be like strings with free ends, but in most forms of the theory they are assumed to be closed loops with no free ends. The length of such a string, or the diameter of a loop, is assumed to be exceedingly small, on the order of the "Planck length" (about 1.62 × 10-33 cm). Strings must be nearly this small so that gravity -- which is an integral part of string theory -- will be roughly as strong as the other forces at the string scale.
Since this size is 20 orders of magnitude smaller than the size of an atomic nucleus, treating such strings as 0-dimensional points is certainly a good approximation -- when dealing with phenomena we are actually capable of measuring. It is not, however, a good approximation when trying to understand what is "really" going on at a fundamental level, and in particular when trying to integrate a theory of gravity into the larger theory.
Strings are not expected to be quite as small as the Planck length. In fact, the ratio of their size to this length is an important parameter of the theory. The ratio of the Planck length to string length is a number somewhat like a charge. When it is small (so strings are relatively large), then strings will interact with each other only weakly. However, if it were large (perhaps even more than 1, so strings would be smaller than the Planck length), strings would interact strongly. Other charges like the electric charge are proportional to this ratio. The fact that the fundamental unit of electric charge is a small number is the reason for thinking the ratio of Planck length to string lengh is also small -- rather less than 1, in any case. The best current guess estimates the ratio is about 1/10, implying a string length scale of about 10-32 cm.
A most important property of strings is that they vibrate, like the strings of a violin or a piano. In fact, every string is assumed to be just like every other string, except for its mode of vibration. In a closed string (a loop), vibrations occur as "standing waves" in which there are an exactly integral number of waves, which must be the case if the displacement of the string is continuous along its length. Every different vibrational mode corresponds to a different type of elementary particle, and all known particles -- as well as many not yet known -- must correspond to a particular mode of vibration.
In any vibrating 1-dimensional object, there is a simplest vibrational mode -- the fundamental mode. This is the situation where the length of the standing wave is equal to the object length. In higher modes, there are an integral number (greater than one) of complete waves. It turns out that the energy in a vibrating string for all but the fundamental mode is very large -- in mass units it is on the order of the Planck mass. (Although that is something like 20 micrograms, about the mass of a dust speck, that's a huge amount for such a small object.)
The mass-energy of the fundamental mode, however, is quite small. That's fortunate, because this is what must correspond to the masses of known elementary particles. What's different about the fundamental mode? Why is its energy not around the Planck energy, as one would expect, but so much less? This is a good question, and the answer is "quantum effects". What happens is analogous to the way matter in bulk can't be cooled to exactly absolute zero. Due to the uncertainty principle, nothing can be precisely "at rest". There is always a little jittery motion of some kind. The energy from this effect turns out to cancel the energy in the fundamental mode almost exactly, resulting in masses for the elementary particles which are as small as we know them to be. In fact, for some particles like the graviton, the cancellation is exact, resulting in zero mass. Unfortunately, we do not know how to do the appropriate computations with sufficient precision to be able to derive masses for the elementary particles directly. This is what, at present, frustrates the most obvious way to set up an observational test of string theory, by deriving particles masses.
One final characteristic of strings that is of greatest importance is the way they move through space over time. Just as a moving point particle traces out a 1-dimensional curve -- called its "world line" -- as it moves through spacetime, a 1-dimensional string traces out a 2-dimensional surface -- called its "world sheet" -- as it moves through spacetime. A point particle moves in such a way that its world line is a geodesic in spacetime. That is, it moves in such a way that certain quantities are minimized. Intuitively, a particle moves so that it minimizes the amount of energy used. Likewise, the world sheet of a 1-dimensional string must also be minimal in certain respects, like the area of the surface of a soap bubble. There's a lot of sophisticated mathematics in describing such minimal surfaces, and it plays an important role in the theory of strings. Indeed, if we knew precisely how to calculate such minimal surfaces, we could predict all we want about the motion of strings.
That's it. That's all we need to say about the basics in order to proceed. We'll elaborate later, of course. Just take note that if we assumed elementary particles were higher dimensional objects -- 2-dimensional surfaces, for example -- the theory would become rapidly more complex. A 1-dimensional loop has a very simple, essentially unique geometry, whereas surfaces can have a lot more complexity. They can have an arbitrary number of holes, for example, and their boundaries (if any) are curves rather than (at most) points. The possible vibrational modes of a string are very simple compared to those of a surface. And when we consider the motion of strings, we have to add one more dimension, and deal with a surface. If we started working with 2-dimensional objects, the analog of the world-line would become a solid 3-dimensional object which has certain minimal properties when embedded in 4 spacetime dimensions.
So. We shall find that string theory itself is mathematically very complex -- so complex we have few adequate tools for dealing with it. The mathematical problems only become harder if we start with surfaces or higher dimensional elementary objects.
The most obvious problem was that gravity obeys a law of the following form. The gravitational force between two particles can be expressed as
F = G × M1 × M2 / r2where G is a quantity known as Newton's constant, M1 and M2 are the masses of the particles, and r is the distance between them. The same formula applies whether the two objects in question are very large, like planets, stars, and galaxies or very small, like elementary particles.
As far as Newton (who discovered the formula) was concerned, it was completely satisfactory, since G is a very small number, so the force is negligible unless (at least one of) the masses are very large, as in the case of astronomical objects. Further, astronomical objects are separated by large distances, so r is also large. But even if r weren't large, since astronomical objects are not 0-dimensional points, r is bounded away from 0 by the size of the objects.
This is important, of course, because if one were to consider what happens with the gravitational force when r is very small, it would appear that there is trouble. If r could be arbitrarily small, the force could be arbitrarily large. This probably didn't bother Newton, since he was concerned mainly with astronomical situations, and he certainly had nothing like our modern concept of elementary particles. Even the idea of atoms was a matter of pure speculation until long after Newton's time.
But for modern physicists, this is a serious problem. If elementary particles really are points, and if they can (therefore) approach each other arbitrarily closely, the gravitational force between them would become arbitrarily large, no matter how small the mass of the particles (if it weren't actually zero).
At what value of r does this become a serious concern? Well, it so happens that the physicist Max Planck noticed in 1899 that certain simple expressions involving Newton's gravitational constant (G), the speed of light (c), and a third constant -- now called Planck's constant -- yielded quantities of length, mass, time, and energy. (And also temperature if a constant known as Boltzmann's constant is thrown in.) Planck introduced his constant (usually denoted by the letter ℎ) in connection with his theory of blackbody radiation. (The constant is often used in the form of ℎ divided by 2π and denoted by ℏ.)
Planck's constant turned out to be fundamental in quantum mechanics. The Heisenberg uncertainty principle, for example, states that the product of the uncertainties in certain "conjugate" quantities (such as position and momentum or time and energy) must be larger than Planck's constant. Curiously, too -- given that the constant was introduced to deal with a problem in thermodynamics -- the constant has units of angular momentum. And in fact, elementary particle spins are actually integral or half-integral multiples of Planck's constant.
Anyhow, the fundamental units which can be defined in terms of Planck's constant, G, and c are also named after Planck and are as follows:
|Planck length||1.62 × 10-33 cm|
|Planck time||5.39 × 10-44 sec|
|Planck mass||2.17 × 10-5 gm|
|Planck energy||1.22 × 1019 GeV|
|Planck temperature||1.42 × 1032 degrees K|
The Planck length and Planck time are very small in comparison with anything we have any experience with. The length, for instance, is about 20 orders of magnitude (i. e. a factor of 1020) smaller than the diameter of an atomic nucleus. Yet the Planck mass, though it is only about 20 micrograms, is enormous in comparison to the masses of subatomic particles. And the Planck energy and temperature are just plain enormous, period.
Why does all this matter? It matters because, since the units are derived from fundamental constants important in relativity, gravitation, and quantum mechanics, they represent a scale at which quantum mechanics and gravity are of roughly equal importance. In particular, at a distance represented by the Planck length, the force of gravity is as strong as the other elementary forces -- even though gravity is exceedingly weak in comparison to the other forces at the atomic distance scale or any larger scale. One speaks of any scale with values near the Planck units as the "Planck scale".
The upshot is that gravity can't be neglected at the Planck scale. And at distances smaller than the Planck length, gravity only becomes even stronger, because of the r2 in the denominator of Newton's forumula.
If elementary particles really were mathematical points, there would be nothing obvious to stop them from approaching each other at distances smaller than the Planck length. It isn't hard to see that unbounded quantities -- "infinities" -- could plausibly arise in computations involving point particles. And this is precisely what has happened in all attempts to work with a quantum theory of gravity, at least until very recently.
Physicists have done much speculating on how to get around this sort of problem. There have, for instance, been many suggestions that space (and time too) itself is quantized and discrete rather than continuous. Perhaps distances much smaller than the Planck length are physically impossible, and spacetime at the Planck scale begins to resemble a chaotic "quantum foam" instead of a smooth continuum.
That might well be. However, a less drastic possibility is simply to suppose elementary particles are not mathematical points, but instead some other shape (such as a 1-dimensional string) with a non-zero size.
This problem of infinities arising when particles are allowed to come arbitrarily close to each other is not limited to gravity. Coulomb's law for the strength of electromagnetic force between charged objects also has r2 in its denominator. Of course, we now know that Coulomb's law is just a large-distance approximation for the laws governing electromagnetism -- which in quantum mechanical form are known as quantum electrodynamics.
The theory of quantum electrodynamics itself was plagued with infinities in its early days. The problem was with "virtual particles", which can materialize out of the vacuum for very small periods of time without violating conservation of energy, because of the Heisenberg uncertainty relations. Without going into all the gory details, we'll just note that this problem was ultimately solved by what is called a "renormalization" procedure, and this did not require abandoning the notion of a point particle.
Point particles, nevertheless, made the whole theory much more difficult and less rigorous. Richard Feynman was one of the key contributors to the formulation of a satisfactory theory of quantum electrodynamics. You're probably already familiar with one of Feynman's inventions -- the "Feynman diagram". This is a pictorial representation of particle interactions which helps organize the necessary calculations of results in the theory by means of "perturbation theory".
|External links on Feynman diagrams|
One of the things you'll note about such diagrams is that they may contain "loops", representing virtual particles, in between the vertices which represent "real" observable particles such as electrons, quarks, and neutrinos. Now, if those virtual particles are mathematical points, there can be an unlimited number of such particle loops in possible Feynman diagrams, because the loops themselves can be arbitrarily small.
The calculation rules of quantum electrodynamics say that there must be terms corresponding to every possible Feynman diagram, and that all the terms must be added up in order to compute a result. Of course, one cannot actually add up an infinite number of terms. But, mathematically speaking, if the terms become small "fast enough", the infinite sum is said to "converge" to a finite value. In practice, this means that computing with only a finite number of terms should yield results to any desired degree of accuracy. And indeed, computations in quantum electrodynamics have been made and compared successfully with experimental measurements with more accuracy than any other computation in physics.
But this is no thanks to the possible existence of diagrams with an arbitrarily large number of loops. From a theoretical standpoint, they make any theory analogous to quantum electrodynamics that uses such "perturbation" calculations mathematically hazardous. In particular, all attempts to formulate a quantum theory of this kind for gravity have been stopped dead by this sort of problem.
There is one more -- a third -- major sort of theoretical problem that comes from assuming elementary particles are 0-dimensional points. This one, too, can be understood by looking at Feynman diagrams. You will notice that all such diagrams have vertices -- points at which two lines in the diagram join or diverge. Now, a Feynman diagram is more than just a schematic way of representing particle interactions. The lines that make up a diagram are actually the world-lines of point particles, as discussed above. That is, they represent the paths through space and time taken by several interacting particles.
It is not a good thing to have paths like this which have sharp angles, corners, and kinks in them. Mathematically, it means that the paths are not "smooth". Although the paths are not actually broken -- i. e. discontinuous -- there are mathematical difficulties arising from these kinks in a path. Recall that the paths are supposed to be minimal in some sense. Mathematically, one uses differential equations to describe minimal conditions. That is, one has to compute certain kinds of derivatives, and these derivatives are not continuous at the corners -- they jump discontinuously from one value to another. In terms of elementary calculus, one sees that the slope of the lines in the diagram changes abruptly at the corners.
While there are ways of dealing with such "singularities" in the diagrams, it's a tricky business. One would much prefer not to have such things present in the geometrical objects one has to work with.
|A string theory Feynman diagram|
|Source: The Official String Therory Web Site|
Notice that there are no kinks, creases, or folds in the diagram. It is, mathematically speaking, a smooth (2-dimensional) "manifold". (The edges of the diagram at the top and bottom are just artifacts of the need to cut the diagram to fit the box -- in reality the surface would extend indefinitely in both directions.) Smooth manifolds are much preferred for any sort of calculus.
The diagram illustrates other things as well. If you look at the diagram as representing a 2-dimensional surface embedded in 3-space, then the view of it at any point in time can be represented as a plane that is approximately a horizontal cross section. A cross section near the bottom would show two strings, while one near the top shows a single string. Different planes correspond to the viewpoints of different observers at different points in time and travelling with differnt velocities. Such planes will not be parallel to each other if the observers aren't moving with the same velocity. Thus different observers may disagree about the exact instant when the two strings come into contact. This is just what we expect from special relativity theory -- yet it cannot be represented in a traditional Feynman diagram of point particles, where there is necessarily a unique place that world-lines meet.
One last remark, though it can't be seen directly from the diagram. It turns out that in order to do perturbation calculations with diagrams like this, it is necessary to consider only topological equivalence classes of diagrams. Further, the topological equivalence class of the diagram depends only on the number of "holes" in the surface. (Holes correspond to loops in a conventional Feynman diagram. There aren't any in the illustration here.) Although calculations in string theory remain enormously challenging, this little simplification is welcome.
All of these considerations of the difficulties of working with point particles that can be alleviated with strings are encouraging. They suggest that we may be on the right track. At least, we have something worth pursuing further.
Interestingly enough, all of this wasn't so obvious initially. The way in which string theory actually came about, historically, was a bit less direct. Looking at the history will help make a number of theoretical points clearer.
Veneziano took the discovery somewhat farther. He was able to express many phenomena of strongly interacting particles in terms of the beta function. This stimulated a lot of work on the part of others, using the beta function and generalizations of it, to summarize many experimental results involving particle "resonances". Veneziano called his approach a "dual resonance" model, because it had a kind of "duality" property. That is, there were two "dual" ways to describe the same physics and compute the same results. This is similar to, but not the same as, simultaneously thinking in terms of waves and particles. Take note, because dualities of other kinds will play a big role in much more recent theories.
The dual resonance model created a lot of excitement, but posed a mystery, in that initially no one knew why the formulas worked.
In 1969 several physicists (Keiji Kikkawa, Bungi Sakita, and Miguel Virasoro) found that adding extra terms to Veneziano's forumlas could restore unitarity. Although not recognized at the time, what they did amounted to adding extra terms to perturbation expansions, where the extra terms corresponded to Feynman string diagrams with multiple holes.
It should be understood that the "strings" in this sort of theory are not the purely 1-dimensional Planck-scale objects of current string theory. Instead, they were more like very thin tubes consisting of force lines of the strong force. But the important point is that mathematically they behaved a lot like modern string-theory strings. For example, the properties of such strings depended on their vibrational modes.
These force flux tubes joined together (for example) the quark and antiquark pair inside a meson. They also seemed to have the property that the force between the quarks increased with the distance separating them. If proven rigorously, this would explain the apparent property of "quark confinement".
(As an aside, we note that these strong force flux tubes have a lot in common with magnetic lines of force inside a superconductor, which exhibit what is known as the "Meissner effect". This means that the lines of force are confined in a very narrow space and cannot spread outward. The reason for this is that 2-dimensional cross sections of such tubes are mathematical objects known as "solitons", which is a kind of wave that does not spread or disperse like "normal" solutions of wave equations.)
Among the properties that could be derived from the vibrational mode of Nambu's strings was that of spin. However, only integral spins fell out of the initial theory, so the theory was called a bosonic string theory. This circumstance isn't too surprising, given that the strings arose out of lines of force, because force carrying particles are bosons.
An interesting and very important consequence discovered in these early string theories was that a certain condition on the world sheets of moving strings was required for quantum mechanical consistency. This condition was that the sheets had to be symmetric under relabeling of their points. That is, they should be symmetric under a change of coordinate system. This condition turns out to determine the number of spacetime dimensions in which a string can move, as we'll explain later. In other words, if a string theory is correct, quantum mechanics places strong contraints on the geometry of spacetime. This is a good property to have -- as long as the implications are correct.
With the addition of fermions, string theory seemed to be gathering momentum as a theory of elementary particles. But a number of theoretical problems still remained:
Interestingly enough, this approach to quantum gravity via string theory actually predated another theory of quantum gravity -- supergravity -- which grew out of the theory of supersymmetry. Supersymmetry itself as an important theory dates only from around 1973, and in fact was partly suggested by ideas from string theory, yet supergravity was by far the favored quantum gravity theory among physicists during the later 1970s. Supergravity, however, flamed out in the 1980s due to a variety of problems, including intractable renormalizability difficulties. These particular problems did not affect string theory, which at the time had serious but different problems of its own.
The discoveries of 1974 were significant in that they showed string theory was far more than a theory of the strong force (for which the gauge theory of quantum chromodynamics was proving to be quite adequate) -- it might actually be a viable quantum theory of gravity. Provided it could solve more of its own problems.
This was no mere cosmetic improvement, because it turns out that tachyons cannot have supersymmetric partners. They are, therefore, automatically excluded from any theory which incorporates supersymmetry.
Thus, by 1977, two of the serious theoretical problems in string theory listed above seemed to have been eliminated. Tachyons were banished, and two of the predicted bosons could be identified with gravitons and photons. What remained? Well, there were the "ghost particles". These are entities which Feynman first discovered in the 1960s in his own work on gauge theories of gravity. Although they do not really exist, they seem to emerge out of calculations of multiple graviton exchanges. Ghosts can be eliminated by various tricks, especially when the theory is forumuated in dimensions higher than 4. Hence they came to be viewed as more of a curiosity than a serious problem.
Of the remaining problems, there was still the little matter of the apparent need that strings had to live in at least 10 spacetime dimensions. This is not a mathematical inconsistency, but it certainly presents difficulties in being reconciled with our pervasive experience that there are only 4 spacetime dimensions. The precise way in which to effect this reconciliation remains an open question even today.
Then there was the puzzle of the other forces -- the weak and strong forces. Which, if any, of the bosons of the theory could account for those? And was any form of the theory provably free of all infinities and divergences? In addition to these concerns, there remmained problems that hadn't even been thought of yet.
In 1980 they had synthesized a theory of open supersymmetric strings which had both fermionic and bosonic vibrational modes. These strings had a length near the Planck length, instead of the much larger strings of Nambu's theory. "Open" here means that the strings were assumed to be usually just 1-dimensional objects with two free ends. Strings in this theory vibrated in a 10-dimensional spacetime.
There is no restriction which says the ends of one string can't join together. So a theory of this type may include strings which (at times) are closed loops. Indeed, the theory has to include some closed strings in order that it can include among the vibrational states one that corresponds to a massless spin-2 graviton. Type I string are "unoriented", which means there is no difference in the threory between the two possible directions on the string, whether or not it is closed.
A year later, Green and Schwarz came up with a second theory in which the strings were always closed loops, with the ends joined. They were still of the same size and still required 10 dimensions. In this theory, however, the strings are oriented, and the fermionic vibrations move in one direction around the loop, while the bosonic vibrations move in the opposite direction. Their previous theory was now called a Type I theory, while the newer version was Type II.
A closed string generates a (roughly) cylindrical world sheet as it moves through spacetime. Two interacting strings produce nice, topologically well-behaved Feynman diagrams, as shown above. It was, therefore, possible to show that perturbation theory calculations with closed strings did not give rise to any of the infinities that plagued point particle theory. This was a definite plus of the Type II theory.
However, as we suggested, new problems were recognized that any reasonable theory had to solve, in addition to the list given above. For instance, the theory must provide a correct "phenomenological" description of the behavior of known particles and forces. That is, obviously, it has to make correct predictions about behavior observed in experiments. Now, one of the key discoveries of particle physics, though one of the oddest, was that the weak force does not preserve left-right symmetry (parity). Particles with a left-handed spin behave differently from particles with a right-handed spin when the weak force is involved. This circumstance is called "chirality".
It turns out that this concept of handedness (known as "chirality") can be defined only in an odd number of spatial dimensions, hence an even number of spacetime dimensions. And so a chiral (i. e., parity-violating) theory is possible only in an even number of spacetime dimensions. This was, in fact, one of the fatal flaws in supergravity, which required 11 dimensions. Although string theories were normally formulated in 10 or 26 dimensions, the chirality property could be easily lost for other reasons. This problem especially occurs if the "extra" dimensions (beyond 4) are not properly rolled up ("compactified"). So, preserving chirality in their theory was one problem that Schwarz and Green faced. They were eventually successful at proving chirality for both the Type I and Type II theories in 10 dimensions.
The first regarded chirality. Witten had earlier shown that spacetime needed to have an even number of dimensions to allow chirality, but he now showed that even in that case, chirality could be lost if some of the "extra" dimensions were compactified (as they had to be, since we can detect only four dimensions), instead of "flat" (like macroscopic dimensions).
The second problem involved the use of Kaluza-Klein theory. Ever since its origins, physicists hoped that this kind of theory could explain how gauge forces arise from the behavior of gravity in the compact "extra" dimensions. (For instance, Kaluza and Klein had shown that Einstein's equations implied Maxwell's equations when extended to a 5-dimensional spacetime.) The new results of Witten and Alvarez-Gaumé showed that this worked only if spacetime had an odd number of dimensions. Thus, it was impossible to have both chirality and a Kaluza-Klein type of theory. What this meant for string theory is that the gauge forces had to be somehow intrinsic to the theory instead of coming about by a Kaluza-Klein mechanism.
Finally, the new work raised an unexpected problem involving quantum "anomalies". These are violations, due to quantum effects, of a gauge symmetry in a quantized theory and (hence) violations of conservation of quantities that ought to be conserved, such as electric charge. Such anomalies had gone unnoticed, since in fact they cannot occur if spacetime has four dimensions. But they can occur if the dimension is two, six, or ten. Superstring theory had become thoroughly 10-dimensional. While it wasn't proven that anomalies were inevitable, the burden of proof was on string theory to show that they did not occur.
It turned out that under very special conditions, it is possible to forumlate a chiral string theory in 10 spacetime dimensions that is free of anomalies, and in fact this actually helped to weed out most otherwise possible theories as unrealistic.
Michael Green and John Schwarz succeeded in solving the anomaly problem for a certain form of the string theories they were working on. But the importance of their result, the reason it attracted so much attention, was that they were able to avoid giving up anything in the process. They simultaneously managed to satisfy many other properties that a reasonable thoery of particles and forces -- including gravity -- needed to have.
Here are some of the desiderata, the features that this ideal theory needed:
The last point on the list -- a proliferation of plausible theories -- turned out to be a key problem to focus on. Point particle grand unified models could be constructed using almost an infinite number of gauge symmetry groups. Each implied the existence of different types of elementary particles -- though they did not differ as far as any particles light enough to actually be observed were concerned.
It did not appear, initially, that string theories placed any new constraints on the gauge symmetry groups they might entail, so there seemed to be many possible string theories that differed largely in the gauge symmetries they allowed. Indeed, when string theories were considered in dimensions smaller than 10, there were even more gauge symmetries possible. But even 10-dimensional theories did not seem to be unique. Most of these theories, and perhaps all of them, seemed to have problems with anomalies. And beyond all that, there seemed to be no way to determine which of the theories correctly reflected nature.
Then a "miracle" happened. What Green and Schwarz found in 1984 was that for their "Type I" string theory with open strings in 10-dimensions, in order for the theory to lack anomalies, it must have SO(32) as its associated gauge symmetry group. So having SO(32) as its internal symmetry group was a necessary condition in a Type I string theory. This immediately eliminated a large number of candidate theories from contention.
But Green and Schwarz showed further that the condition was also sufficient -- a Type I theory with SO(32) definitely lacked anomalies. So the anomaly problem was solved as well. With the "right" symmetry group, anomalies automatically cancelled themselves out. This killed two birds with one stone -- requiring a solution to the anomaly problem was not only possible, but it also eliminated many other theories as unrealistic.
However, this was true only for Type I theories, using open strings. It said nothing about what happened with Type II theories, which had only closed strings. What about them?
Well, as it happened, the Type II theories were looking less attractive at the time. The reason for this is interesting, because it relates to another point on our list of desiderata. In a Type I theory, the strings are just short (very short) 1-dimensional curves, with free endpoints. Each string endpoint can carry a charge of one or more of the fundamental forces, and this is in fact how the forces enter the phenomenology of the theory. But a closed (Type II) string doesn't have endpoints, so it was less clear how any charge could be carried.
The net result was that Green and Schwarz did not extend their results to Type II theories. Surprisingly, however, this did not mean the demise of string theories with closed strings. Instead, as we noted, the success of Green and Schwarz in showing that one kind of string theory was both free of anomalies and also had a definite gauge symmetry group stimulated many other physicists to take an interest in strings.
It helps to recall one detail about the Type II strings. For these, the strings are oriented, and the vibrations corresponding to fermions move in one direction around the loop, while those that correspond to bosons move in the other direction. Both types of vibration, however, are confined to 10 dimensions.
Gross and his collaborators added one new twist to this picture. They allowed the bosonic vibrations to vibrate in 26 dimensions instead of 10. In other words, they reverted to the original Nambu theory for the bosonic part of their theory. Their resulting string was a hybrid that managed to live in both 10 and 26 dimensions at the same time, depending on which direction around the string one looked. They called this creation a "heterotic" string. (The term is from the Greek heterosis, which means "hybrid vigor" rather than something sexual.)
Gross et al then showed (in 1985) that their heterotic strings had just the same nice properties as the Type I strings of Green and Schwarz, with one addition. Specifically, a necessary and sufficient condition for the heterotic theory to be free of anomalies was that the associated gauge group be of a certain type. But in this case, there were two possibilities instead of one: the group could be either SO(32) or else a more exotic Lie group, E8×E8. These two theories are referred to as Heterotic-O and Heterotic-E, respectively. (For a time, it was thought that a third group was possible, namely SO(16)×SO(16). This did not, however, work out in the end.)
It is not really necessary to regard the bosonic vibrations as occurring in 26 spacetime dimensions. Strictly speaking, what is required is 16 additional "degrees of freedom" beyond what is required for the fermionic vibrations. These degrees of freedom do not need to be part of spacetime as such. They could exist in an "internal space", something like the internal spaces associated with the symmetries of gauge groups (where different particle types transform into each other under an operation of the group). The mathematical distinction between a "dimension" and a "degree of freedom" is subtle, but important in order to avoid problems in formulating superstring theory in more than 10 "real" dimensions.
The extra 16 degrees of freedom of the bosonic vibrations allow for the internal gauge symmetries that give rise to the Yang-Mills forces. Heterotic strings are able to carry charges associated with the different forces even though they do not have free ends on which charge can reside. In the heterotic model, the charges and their associated quantum numbers propagate along the whole length of the string. The model provides what is necessary for such strings to respond to fundamental forces in a way that was lacking in the Type II closed string model.
The first symmetry to break would be that between the two E8 factors in the product group. This symmetry breaking is the point at which gravity differentiates from the other fundamental forces. Each of these factors is still very large, and each represents a very large number of particles. Most of these particles aren't observable, because of their large masses, but among them are all the particles we do observe. The symmetry group which encompasses all of these latter particles is some subgroup we still can't identify, though it certainly contains SU(3)×SU(2)×U(1). Call this subgroup G.
Both factors of E8 will contain a copy of G, hence there will exist duplicate, but distinct, versions of all the elementary particles we know or could know. The key point is that in the universe after the full E8×E8 symmetry breaks, corresponding "mirror" particles will not interact with each other, though they will continue to interact among themselves according to group representations associated with each factor group. That is, these two sets of particles will still experience strong, weak, and electromagnetic forces in interactions with others of their own "kind", but not with particles of the "other kind". This "other kind" of matter different from that we know is called "shadow matter".
All of this is still completely hypothetical, because it is based on the assumption that a specific string theory with E8×E8 symmetry gives the correct description of our universe. However, if this turned out to be the case, then corresponding to all the matter we know of existing in planets, stars, galaxies, and so forth, there could exist exactly the same "stuff", composed entirely of shadow matter, and obeying the same physical laws we already know. We simply could not see it or observe it in any way, except through gravitational interaction. (Such shadow matter is completely different from anti-matter, by the way. Anti-matter is known for certain to exist, though probably in at most very small quantities, because it does interact, quite violently, with ordinary matter.)
Shadow matter, therefore, could in principle consititure all or part of the "dark matter" which we know, for many reasons, must exist. However, there isn't any observational evidence for shadow matter as such. Certainly there isn't any near our solar system, since we can fully account for planetary motions on the basis of matter we can actually observe. We've never seen any other possible evidence either, such as stars or galaxies orbiting around completely undetectable companions of a similar size.
So, shadow matter is an interesting possibility based on the hypothetical E8×E8 heterotic string theory. But hypothetical is all it is at the present time.
You may recall from our discussion of added dimensions in connection with supersymmetry and supergravity that theories, called Kaluza-Klein theories, allow the equations governing other forces to arise naturally out of Einstein's equations of general relativity when formulated in spacetime with more than four dimensions. This was a hint at a deep relationship between the forces of nature.
Further investigations over many years have shown a general principle. This is that extra dimensions of spacetime which are compactified (i. e., "curled up" into a very small size) lead to local gauge symmetries. When several extra dimensions are involved, one obtains non-Abelian gauge symmetries, as required for the SU(2) electroweak theory and the SU(3) theory of the strong force. The exact symmetries depend on the topology of the compact dimensions.
This is extremely suggestive, and looked very promising when incorporated in 11-dimensional supergravity theory of point particles. Unfortunately, supergravity was ultimately a failure for various reasons. One of these, as noted, is that chirality is not possible in a theory involving an odd number of dimensions, and yet Kaluza-Klein theories seem to work only in an odd number of dimensions.
On the other hand, all this applies to point particle theories rather than string theories. Of course, at accessible distance scales (around 10-13 cm) a much smaller string is quite well approximated by a point. So the Kaluza-Klein theories may be approximately true if we assume that the difficulties can be ignored as a result of the necessary approximations. And in fact, 11-dimensional supergravity seems to be one facet of the generalization of string theory known as M-theory. The one additional dimension (over the 10 of string theory) may be something like the "thickness" of a string.
Now, all of this Kaluza-Klein type of theory is an example of how higher dimensions can be incorporated deliberately in a theory in order to achieve certain results. Something else, however, is also going on with string theory. Namely, the presence of added dimensions seems to be required to ensure the mathematical consistency of the theory.
By consistency, we mean that the equations which describe the motion of strings (as opposed to point particles) seem to place very restrictive conditions on the spacetime in which the strings move. This is because the equations must satisfy both the axioms of relativity (covariance) and quantum mechanics. Consistency means that all of the various mathematical problems such as quantum anomalies, tachyons, ghost particles, infinities, and negative probabilities must be avoided.
Some of these problems can be avoided by means that are not specific to the dimension of spacetime. For instance, incorporating supersymmetry in string theory eliminated tachyons. However, when looking at the equations in detail, the dimension of spacetime occasionally occurs explicitly. As an example, in Nambu's original theory, some equations involved a factor like this:
1 - (D - 2) / 24Here, D is the spacetime dimension. The term which contained that factor needed to be 0 for relativistic correctness. In this case, the alternative was having "ghost" particles and loss of covariance. Assuming D = 26 was necessary and sufficient to make things work right. There wasn't any way around it. Although it may seem like cheating, it worked to achieve the needed consistency and -- what is rather amazing -- did not cause other problems. That is, there were no other situations where a different value of D would be required. The result was a theory that worked (within its limited scope). That is reason enough to, at least, take the theory seriously.
Similar though slightly different considerations apply to the more highly refined superstring theories of Green and Schwarz, Gross, and others. Again, one finds factors like (10 - D) occurring in formulas, where D is dimension. One generally wants terms that contain such factors to drop out in order to achieve mathematical consistency. The obvious solution is to take D = 10. And again, while this may seem like cheating, it also seems like it must be significant that there are no terms containing, say, (11 - D) or (9 - D) that one wants to get rid of. Somehow, mysteriously, D = 10 seems like the "right" choice.
What's "really" going on here? Some theorists believe that properties of special mathematical functions are somehow involved. One of these is the so-called "modular" function of Srinivasa Ramanujan. This function, as well as the whole class of modular functions, plays a very important -- though incompletely understood -- role in advanced number theory and the mathematical analysis of Riemann surfaces. Riemann surface theory, in particular, is relevant to describing the behavior of the world sheets of strings (the path they trace out in spacetime as they move). These world sheets must represent minimal surfaces in order to minimize a quantum mechanical quantity called "action". The surfaces must also move in such a way that they maintain a kind of symmetry called "conformal invariance".
The reason for requiring conformal invariance is that there's nothing spcial about how points on the surface are labeled. If all points on the surface are relabeled in a consistent way, the physics should not change. In Riemann surface theory, such a relabeling is called a "conformal map". The equations of string theory must be invariant under such maps.
When one writes down the equations of how moving strings must behave, expressions naturally arise that involve the Ramanujan function. It is these particular expressions which contain the factors such as (10 - D) that include the spacetime dimension D. Is there some deeper mathematical significance in this, perhaps some connection with number theory? We don't know.
Actually, equations containing such factors are associated with particular solutions of the more general equations related to each type of string theory. There are solutions in which D may be less than 10, provided that 10 - D is interpreted as the dimension of a "compact" space, just like the extra dimensions of a Kaluza-Klein theory. So the total dimension of the space in which strings vibrate -- spacetime plus the compact part -- is still 10. This is quite convenient, since it allows for the 4-dimensional spacetime we experience, with the remaining (six) dimensions too small to be observed.
One consequence of any interpretation like this is that D could never be more than 10, since a space with a negative dimension doesn't make sense. Hence, string theory just doesn't work in more than 10 total dimensions.
It seems "reasonable" to expect that a solution should have a spacetime with 4 dimensions, plus 6 compact dimensions. For better or worse, it turns out, there are a vast number of solutions, even with this restriction. Each solution corresponds to a different topology of the 6 extra dimensions, and also to a different "phenomenology" -- i. e., concrete physics that we can observe.
The trick, then, is to try to find the "right" topology for the extra dimensions. And in the best case, it would be very nice if there is just one choice that yields the right physics. Suffice it to say, this trick hasn't yet been pulled off.
When the situation is looked at even more closely it is found that there are additional constraints. These arise from the way a string may interact with the background space. Later we will say a bit more about how calculations are performed on the world sheet of a string in order to describe string motions and interactions by a "sum over histories". For now, the point is that the extra compact dimensions must be curled up in a special way in order to form a certain kind of topological space, called an orbifold.
In order to go into more detail, we need to explain a little terminology. First, "topological space" is just a fancy name mathematicians use for a geometrical object such as a curve, surface, solid, or something of even higher dimension. Although mathematicians consider very general types of topological spaces, in physics only a type of a somewhat more special sort is commonly encountered. This is the type of space known as a "manifold" (and some slight generalizations, including orbifolds).
In Euclidean geometry, which is about 2500 years old, the types of geometric object most often dealt with are made up of (straight) lines and (flat) planes. Much later, thanks to René Descartes, it was found that such objects could be handled analytically in terms of coordinate systems -- hence "analytic" geometry. A coordinate system assigns a single number to a point on a line, a pair of two numbers to a point on a plane, and so on. Ordinarily the numbers involved are "real" numbers, which are the numbers of ordinary discourse, the ones which can be represented as a familiar (but perhaps infinite) decimal expansion.
Mathematicians use the symbol R for the set of all real numbers. An (infinite) straight line is identified with R by means of a 1-to-1 correspondence, which is nothing more nor less than a coordinate system. Similarly, a plane can be given a coordinate system consisting of pairs of numbers ("Cartesian coordinates"), which is designated mathematically as R×R or R2. Likewise, the 3-dimensional space of ordinary experience can be given a coordinate system with triples of numbers, so it is denoted by R3.
These geometric objects are collectively known as "Euclidean spaces" of a certain dimension, which is just how many numbers are needed to specify the coordinates of each point. In general, an n-dimensional Euclidean space is denoted by Rn, where the dimension appears as the superscript. Such spaces are singled out for special attention because their coordinate systems make them the easiest sort of geometric objects to deal with.
A manifold, then, is nothing but a generalization of Euclidean space. It is a geometric object in which the immediate vicinity of every point "looks like" a Euclidean space of a certain dimension that is the same for all points of the object. So a manifold has a specific dimension, which is that of the associated Euclidean space. The only difference between a manifold and a Euclidean space is that the latter has a single "global" coordinate system, whereas the manifold can have a different "local" coordinate system in the vicinity of each of its points.
There are many technicalities associated with defining precisely what is the "vicinity" of a point (called a "neighborhood"), how different local coordinate systems must be consistent where they overlap, etc. But the essence of being a manifold is the existence of a Euclidean coordinate system around each point.
A curve such as a circle is a 1-dimensional manifold. The reason it is not itself a Euclidean space is that it is finite and closed in an obvious way, which is not true of R. But the neighborhood of any point on a circle is "just like" R, and that's what counts. Similarly, the surface of a sphere is a 2-dimensional manifold, which is also closed and finite.
Now, in fact, there are a number of different types of manifolds, which are distinguished from each other by the technical details of how the local coordinate systems are required to be compatible when they overlap. In physics, the most common type of manifold encountered is a "smooth" manifold. What this means is that the maps between local coordinate systems are "smooth", that is, "differentiable" in the sense of calculus. This is important, because being able to compute derivatives is necessary in order to express physical laws, which are usually expressed in terms of derivatives and differential equations.
In the case of 1-dimensional manifolds -- curves -- the smoothness condition means the curve has no kinks. (It also has no self-crossings, because the neighborhood of a point of crossing is not even 1-dimensional.) For 2-dimensional manifolds -- surfaces -- smoothness means there are no creases.
Given all that, we can say that the extra six dimensions required for string theory must take the form of a special type of manifold, known as a Calabi-Yau manifold (after the mathematicians Eugenio Calabi and Shing-Tung Yau). Actually, it may also be an "orbifold", which has the same technical definition as a Calabi-Yau manifold, except the condition on smoothness is slightly relaxed. (An orbifold is allowed to have sharp corners, like the point of a cone in 2 dimensions.)
One of the defining properties of a Calabi-Yau manifold (or orbifold) is that it be "compact". Basically, this means that it doesn't extend to inifinity, as a plane does, but must be curled up, like a sphere. In fact, it must be very tightly curled up, because it must be very small in comparison to the normal four macroscopic dimensions of spacetime. (Otherwise, we would actually be able to observe more than four dimensions.)
The total 10 dimensional-space in which strings live is also a manifold, with a special structure. Part of the structure is the normal 4-dimensional spacetime. Then, associated with every point of spacetime, there is a separate, distinct 6-dimensional Calabi-Yau manifold (or orbifold). Although the Calabi-Yau manifolds associated with each point are distinct, all have the exactly the same topology. Every point in this large 10-dimensional space can be uniquely identified by specifying a point in ordinary spacetime (with 4 coordinates) and a point on the associated Calabi-Yau space (with 6 additional coordinates).
A structure like this is called, by mathematicians, a "fiber bundle". In this case, to each point of the "base space" (ordinary spacetime) there is a "fiber" sticking out of it -- the associated Calabi-Yau space. There is an elaborate and subtle mathematical theory of fiber bundles. One of the features of this theory is that there is a certain kind of consistency condition that relates the fibers of points in the base space that are close together. Physicists have adapted fiber bundle theory for a variety of purposes -- including gauge field theories of the fundamental forces. In fact, in mathemtical language, a Yang-Mills gauge field turns out to be an example of a consistency condition between fibers, which is known as a "connection".
Now, given that gauge fields can be realized as connections in a fiber bundle where the base space is ordinary 4-dimensional spacetime and the fibers are some sort of Calabi-Yau space, it should not be surprising to find that the specific physical properties of such gauge fields depend on the precise details of the topology of the Calabi-Yau space.
This is wonderful, because it helps us eliminate unrealistic solutions. The situation is that each different Calabi-Yau space is associated with a different particular solution of the string theory. Although the theory is nearly unique (or rather, has apparently only a handful of alternate forms), a large multitude of different solutions are possible. The question is: which of these different solutions (Calabi-Yau spaces) corresponds to physical reality? And the answer is that a great many (though not all) solutions can be rejected simply by looking at the topology of the fiber. If the topology is wrong, the physics will be wrong, and one potential solution can be rejected.
How does the topology actually become involved with the physics? To answer this, we need a few remarks on how the topologies of spaces can differ. Consider closed (finite) 2-dimensional manifolds as an example. The simplest is the surface of a sphere. The next simplest 2-manifold is (the surface of) a doughnut shape, called a torus. This is equivalent to a sphere with one handle. By "equivalent" we mean in the topological sense -- there are maps of an appropriate sort that transform one manifold into another in an allowable manner (i. e., without disturbing the manifold structure). In looser terms, this means without tearing or cutting the surface. It turns out that the only essential difference between 2-manifolds in this sense of equivalence is in the number of holes, i. e. in the number of handles attached to a sphere. Because of the importance of the number of holes in a 2-manifold, this number is called the "genus" of the space.
There are similar sorts of numbers and algebraic properties for higher dimensional manifolds which help distinguish between toplogically inequivalent classes. Various aspects of the physics associated with each different Calabi-Yau space that can arise as a solution of some string theory may be computed directly from such numbers.
An especially simple example involves the number of "generations" of quarks and leptons. We know, from various lines of evidence, that there are in fact exactly three generations each of quarks and leptons. (For instance, within the leptons the three generations are represented by electons, muons, and tau particles.) It turns out that the number of generations must be exactly half of a quantity called the "Euler number" of the Calabi-Yau space. (This is the same Euler whose beta function caused string theory to be stumbled upon in the first place.) How the Euler number is computed in general is not easy to explain, but for a 2-manifold it is simply 2 minus twice the genus of the surface.
So how do such topological properties of these compact manifolds come to affect the physics? The general picture is pretty easy to understand. Recall that these manifolds occur as fibers in the 10-dimensional fiber bundle of spacetime. Every string moves and vibrates in this 10-dimensional total space. When a part of the space like the Calabi-Yau fibers has a complex topology with higher dimensional holes, it is an easy matter for strings to become entangled in these holes. That is, the strings may actually loop through the holes multiple times. This obviously affects how they can move and vibrate. In the case of an orbifold, which has "conical singularities" in addition to holes, strings may also get snagged on the singularties. When a string gets trapped in a hole or singularity, the result is a massive particle with peculiarities like magnetic charge (monopoles) or fractional electric charge.
The net result is that the number of holes in a Calabi-Yau manifold in fact determines the number of low mass vibrational string states -- exactly the states that correspond to observable elementary particles. So the particle physics we observe actually depends on the topology of these compact manifolds.
The gauge fields may be thought of in terms of "lines of force", and these lines too will be affected by the topology of the compact space. When such lines get trapped, the result is a breaking of the related gauge symmetry. If the 10-dimensional space were totally flat, i. e. equivalent to R10, we would have the full symmetry -- SO(32) or E8×E8. But when this is broken because some of the dimensions curl up, we get a smaller symmetry group like E6 -- which is one of the groups that has been investigated as the symmetry of a grand unified theory. Whatever symmetry group survives after breaking determines the particles and forces we actually observe -- as well as others which are not yet observable. The symmetry group that results depends on precise details of the compact space.
Witten derived yet another effect of the topology of the compact dimensions on physics. In a flat 10-dimensional spacetime with Yang-Mills forces resulting from SO(32) or E8×E8 symmetry, the average values of the fields are zero. But if there are compactified dimensions, the large curvature of the Calabi-Yau spaces contributes to a nonzero average value of the gauge fields within the spaces. A nonzero average value of a gauge field is typical of a transition to a phase with less symmetry.
In summary, an important consequence of symmetry breaking is that the details of how it occurs affects the masses of elementary particles and the relative strenghts of fundamental forces. So indirectly, particle masses and force strengths are influenced by the topology of the compact dimensions. Unfortunately, we don't know how to calculate these influences precisely enough to test anything.
It's clear that topological issues are of key importance in understanding the ultimate implications of string theory for physics. There is, however, a fundamental reason why this is going to be very difficult to work out. That is, the size of the compact dimensions -- as well as strings themselves -- is expected to be close to the Planck length, 10-33 cm. At that scale, the interactions between gravity, spacetime, and quantum effects are all-important. Indeed, it's rather likely that spacetime itself has a grainy or foamy structure at this scale. The idea of strings as objects moving independently in a fixed background spacetime is probably not accurate, but at best an approximation. At the very least, some sort of "quantum geometry" needs to be much further developed to understand what is going on with the topology at this scale.
This is a large part of what the topics of M-theory and quantum gravity are all about. We'll deal with those elsewhere.
There's a basic fact here, which often isn't well-emphasized. Namely, physicists don't actually know what the basic equations of string theory really are. And even if they did, those equations would be like Einstein's equation of general relativity or Shrödinger's equation of quantum mechanics. Such equations are fine things, but in order to derive much useful physical information, one needs actual solutions of the equations.
Ideally, a solution consists of functions that describe physical quantities such as particle properties, black hole characteristics, the outcomes of interactions among particles and fields, etc. Or at least, more specialized (differential) equations whose solutions are functions like that. But in practice, one sometimes doesn't even get that much.
Quantum field theory began with Dirac's Lorentz-invariant equation for relativistic electrons. But that did not become the more useful theory of quantum electrodynamics until Feynman, Schwinger, Tomonaga, and others figured out how to perform useful computations. In their approach, one does not actually obtain functions that describe physical quantities. Instead, one expresses a solution as an infinite series of terms by which one can obtain a result to some desired degree of accuracy by summing up enough initial terms of the series. (Mathematicians had done something like this for a long time, when they found "solutions" of differential equations in the form of a power series or similar series expansion.)
Surprisingly, this in fact produces answers to a high degree of accuracy -- at least 11 decimal places, some of the most precise computations in physics. The technique is called "perturbation theory", because one starts with an approximate solution and then "perturbs" it a little by adding corrections that (one hopes) are increasingly small.
Feynman diagrams as mentioned above are the key to this process. Each diagram corresponds to one term of the perturbation series. The value associated with a diagram is a "probability amplitude" which can be computed by specific rules.
The idea behind this method is what Feynman called a "sum over histories". What one tries to do is to enumerate all possible ways that something can go from an initial state to a final state. Each different possibility is one particular history. For instance, when a photon travels from the Sun to the Earth, one possible path is a straight line (in the geodesic sense). But another possible path might have the photon go all the way to the Andromeda galaxy and back before it reaches Earth. Although only one of the paths (the geodesic) is minimal, every such possibility has to be included in the sum in order to get a final answer. Such sums are called "Feynman path integrals" because they add up contributions from an infinite number of separate paths. Mathematically, this operation is still not rigorously defined. But, as implemented in perturbation theory, it happens to work very well.
As we noted, when this procedure is applied to classical Feynman diagrams for point particles, troublesome infinities arise, which are essentially due to the assumption that particles are mathematical points. Such infinities can often be overcome by the method of "renormalization", as long as gravity isn't involved.
The lines that compose a conventional Feynman diagram are the world-lines of particles. In string theory, using 1-dimensional strings, the equivalent notion is a world-sheet corresponding to each string. Computations are done in string theory using perturbation theory with Feynman diagrams that involve such world-sheets. Although this looks as though it must be a far more complicated kind of calculation, in fact things become much simpler.
Again, the computation requries one to sum over an infinite number of possible paths (world-sheets). However, it turns out that the computation simplifies enormously, since it is not necessary to add up terms for every possible path, but only for paths which are distinct, in a topological sense.
We've encountered the notion of topological equivalence before. The idea is that two geometrical objects can be essentially "the same" if there is a 1-to-1 map between them which doesn't change their topological nature. Loosely, the map just needs to avoid cutting or tearing the object.
Now, the world-sheet of a string is obviously a 2-dimensional surface. In fact, it is a 2-dimensional manifold. Such objects have been thoroughly studied in mathematics as part of "differential geometry" for over 150 years. Some of the earliest examples were investigated by G. F. B. Riemann, who also originated the "Riemannian geometry" that is fundamental to Einstein's general theory of relativity. The objects Riemann studied came to be known as "Riemann surfaces", which are a very special type of 2-dimensional manifold.
Riemann surfaces are special because they can be viewed as 1-dimensional manifolds over the complex numbers (the totality of which is denoted by C) as well as 2-dimensional manifolds over the real numbers R). What this means is that any point on a Riemann surface can be specified, at least locally, by a single complex number as well as by a pair of real numbers. (This is intuitively obvious, since a complex number is composed of a pair of real numbers with specific rules of addition and multiplication of pairs.) Riemann surfaces also have a specialized notion of equivalence, where the maps between them are called "conformal maps".
Although not obvious, it turns out that Riemann surfaces and conformal maps have very elegant and almost "magical" properties. The axioms which define them impart a very rich structure that makes them easy to work with (given the necessary mathematics). The world-sheets of strings can be regarded as objects of this kind, so in effect a great deal is already understood about them.
One of the key facts is that, for the surfaces which occur as stringy world-sheets, a complete classification up to equivalence type can be given just by knowing the number of holes in the surface. As noted earlier, this is the "genus" of the surface, which is the same as the number of handles that would be added to a (2-dimensional) sphere in order to get a topologically equivalent surface.
Holes in a world-sheet correspond to loops in a point particle Feynman diagram. An example of such a loop would be a very high-energy photon (gamma ray) which can branch into an electron and a positron, where the two antiparticles subsequently recombine to give a photon which is is indistinguishable from the original.
Loops like this are the bane of point particle perturbation theory because they can occur arbitrarily often and arbitrarily close together. Since strings are extended objects, this can not occur in their case. In any finite segment of a world-sheet, corresponding to a finite period of time, there can be only a finite number of holes -- with a specific upper bound on the number of such. This is the property which makes perturbation theory using strings so much more civilized.
Let's summarize. Ideally, the path of a string over time is represented by a world-sheet which is a minimal surface (in two dimensions), like a soap bubble. However, to apply Feynman's idea of a sum over histories in order to compute the actual probability of a given process, we have to sum over all possible path with the same initial and final states. The minimal path corresponds to a geodesic, and is the most probable. But other paths will have nonzero probabilities as well. Each of the other paths may be regarded as a quantum fluctuation of the minimal world-sheet.
The important thing is that each such fluctuation is a surface that is toplogically (i. e. "conformally") equivalent to the minimal surface. Nevertheless, among this collection of surfaces are some that have very narrow branches or fingers sticking out -- anything as long as the branch doesn't reconnect with the main surface to form a loop. When you look at a particular cross section near one of these branches, what it appears to be is a small string breaking off from the original one and floating off into the vacuum. Or else the reverse, in which a string appears in the vacuum and joins up with the main string.
Each surface like this represents a process in which the string has multiple interactions with the background spacetime (the vacuum). When doing the sum over histories, all such processes are automatically included. The result is that the interactions of a string with the background is included in the theory, and this is largely what puts tight constraints on the spacetime in which the string moves. The exact form of the constraints arises naturally out of the mathematics of Riemann surface theory.
Just as with point particle theory, in between the initial and final states, loops may occur. (The simplest case, perhaps, being a loop with an electron-positron pair in the middle of a propagating photon.) All these possibilities must be accounted for as well. Although world-sheets containing holes (loops) are not topologically equivalent to the minimal sheet (without holes), the great thing about string theory is that the only topologically inequivalent cases that need to be considered are surfaces with an integral number of holes. There simply isn't any other sort of surface with a materially different kind of topology.
All terms involving topologically equivalent surfaces can be collected together into a single term. The complete sum over histories, then, amounts to a sum of terms, one for each topologically distinct case. That is, one term for each class of surface having a specific number of holes.
In addition to the constraints on spacetime necessitated by the way that string world-sheets are embedded, additional requirements may be imposed as a result of other considerations. A very important one is that the sum over histories should be finite -- we want to avoid infinities. For instance, in heterotic string theory, this is the condition that forces the Yang-Mills gauge symmetry group to be either SO(32) or E8×E8. (In a type II closed string theory, finite results are always obtained. But therefore, we don't get any limitation on the type of gauge symmetry group allowed.)
It is true that even in the best of circumstances the actual perturbation calculations are impossible to perform numerically. We just don't know enough about the mathematics of strings to do that. So, unfortunately, we can't compute experimentally measurable results. But important theoretical conclusions can be reached anyhow. The two most important conclusions are these:
Most of the difficulties of string theory result from the fact that it is fundamentally a theory about physics at the Planck scale. That is, the size of a string is assumed to be something close to the Planck length, and the energy in most vibrational modes of a string is close to the Planck energy. The reason for this is strings are assumed to exist at a scale where full symmetry prevails and the four fundamental forces are unified -- the Planck scale by definition.
Even if we required only the three other forces besides gravity to be unified, that is still an energy scale around 2x1016 GeV -- nearly three orders of magnitude, a "mere" factor of 500, from the Planck scale. Compared to the highest energy scales presently accessible to experiment -- about 104 GeV -- even the lower unification scale is very large and far away.
The nature of the Planck scale and its vast disparity compared to any scale we can actually explore present several serious problems:
As for the second problem, it is merely a symptom of a larger problem. That is, what we really want in a satisfactory theory is to be able to explain exactly how spacetime behaves at the Planck scale -- and why. What is it structure of spacetime at this scale? If it's not a smooth manifold, what is a better mathematical description? How do objects which we conceive of as strings fit into this? No theory that can't answer such questions is really satisfactory.
Finally, for the last problem, we might reasonably expect that solving the other two will guide us to observations and experiments that can test the eventual theory.
Solving these problems will most probably require a respectable theory of quantum gravity. Although that is what string theory was hoped to be, at least in its early stages, most physicists now expect some more complete theory of quantum gravity to explain string theory, instead of the other way around.
That doesn't mean, however, that string theory is just another dead end. It still makes sense to continue extending it in a "bottom up" manner, while awaiting other answers from a "top down" theory of quantum gravity. Such extensions, in fact, have occurred, and the newer, extended theory is generally called "M-theory".
Such extension has come about from several different directions. One of these is to consider fundamental objects of two or more dimensions, rather than 1-dimensional strings. We've already noted this possibility, as well as its mathematical difficulty (preserving unitarity and causality). Nevertheless, some progress has been made. Higher dimensional fundamental objects are referred to as membranes, or simply "branes". Mathematically, these are just higher dimensional manifolds. M-theory is so-named (partly) because of the connection with membranes and manifolds.
Another direction is a consequence of the discovery of tantalizing new types of symmetries, called "dualities". Such dualities are distantly akin to the duality between waves and particles. In other words, a duality is a way of looking at the same thing from two different points of view, using two different mathematical formulations. Because the same reality underlies both points of view it is possible to work with whichever of the two happens to be easier in a given situation. We can also derive conclusions because the two points of view must entail the same results.
A (perhaps) surprising conclusion has emerged from these considerations. We've seen that there are several consistent "solutions" of string theory possible: the "Type I" theory of open and closed strings, the "Type II" theory of closed strings (which has two alternative forms, "Type IIa" and "Type IIb"), and the two forms of heterotic string theory. Adding to these the 11-dimensional theory of supergravity (which isn't even a string theory, of course) yields 6 theories. What's so surprising is that the dualities seem to relate each of these theories to exactly one other, pairwise. And each of the theories seems to be just a special case of a more comprehensive theory: M-theory.
We know almost nothing about what M-theory actually is, apart from these six special cases. Theorists have simply given a name to something of which little more is known than what has just been sketched. ("Mystery" and "magic" are other interpretations of the whimsical name.)
But the feeling is pretty strong that something is there, under the covers, to account for the suspicious duality relations between otherwise disparate theories.
More about M-theory
String theory solves these problems. In particular, a handful of quite specific versions of the theory are singled out by their avoidance of infinities and divergences, without needing to apply a renormalization procedure.
At first Hawking himself doubted this suggestion. But in 1974 he proved the now well-known result that black holes can radiate energy and therefore have a well-defined temperature. However, it's not much: the temperature of a black hole of several solar masses is only about 10-8 degrees. (Microscopic black holes radiate much more strongly and therefore have much higher temperatures.) At the same time, Hawking found that Bekenstein was right, and he could calculate the entropy. For a typical 3-solar-mass black hole, this is a rather large number, about 1078.
In a classical thermodynamic system, such as a volume of gas, entropy is a measure of disorder, and can be calculated from the microscopic details of the system using quantum statistical mechanics. Hawking's calculation used quantum mechanics but made no assumptions about microstates of the black hole. Consequently, it was hard to understand the origin of black hole entropy.
In 1996, Andrew Strominger and Cumrun Vafa were able to obtain Hawking's exact formula for the entropy by counting the number of microstates in a special class of black holes. (The results have since been generalized.) The techniques they used include supersymmetry and string theory, In fact, they used a nonperturbative form of string theory, based on M-theory and "D-branes".
This is considered to be an impressive solution of a 25-year-old problem, called the "black hole information problem", to account for the entropy of a black hole in terms of its microscopic properties.
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