| I |
Following the media reports about string theory there was an immediate backlash. People naturally asked what this Theory Of Everything had to tell us. The answer was that it could not yet tell us anything, even about physics, yet. On closer examination it was revealed that the theory is not even complete. It exists only as a perturbation series with an infinite number of terms. Although each term is well defined and finite, the sum of the series will diverge.
To understand string theory properly it is necessary to define the action principle for a non-perturbative quantum field theory. In the physics of point particles it is possible to do this at least formally, but in string theory success has evaded all attempts. To get any useful predictions out of string theory it will be necessary to find non-perturbative results. The perturbation theory simply breaks down at the Planck scale where stringy effects should be interesting.
More bad news was to come. Systematic analysis showed that there were really several different ten-dimensional superstring theories which are well defined in perturbation theory. If you count the various open and closed string theories with all possible chirality modes and gauge groups which have no anomalies, there are five in all. This is not bad when compared to the infinite number of renormalisable theories of point particles, but one of the original selling points of string theory was its uniqueness. Worse still, to produce a four-dimensional string theory it is necessary to compactify six dimensions into a small curled up space. There are estimated to be many thousands of ways to do this. Each one predicts different particle physics. With the Heterotic string it is possible to get tantalisingly closed to the right number of particles and gauge groups. At the moment there are just too many possibilities and the problem is made more difficult because we do not know how the supersymmetry is broken.
All this makes string theory look less promising. Some critics called it a theory of nothing and advocated a more conservative approach to particle physics tied more closely to experimental results. Yet a large number of physicists have persisted. There is something about superstring theory which is very persuasive.
In 1950 it was known that physics holds fast to solid principles including the principle of relativity, causality and the quantum version of the principle of least action. These impose very tight mathematical constraints on the kind of theories you can build. One day those principles may be superseded but it is not easy to modify them without destroying the successes of the past. You cannot just replace linear quantum mechanics with some non-linear version and expect it to make sense, nor can you break the symmetries of relativity without invalidating the whole thing. There is more sense in thinking about how physical theories can be generalised within these principles and that is what Dirac was doing.
At the time particle physics was understood in terms of quantum field theory derived from quantised interaction of point particles. There is very limited scope for relativistic theories of this type which are renormalisable. We now know that Yang-Mills theory with spin half and spin zero particles with a few possible interaction terms is all that is permitted. Dirac considered the possibility that more general theories might start from string-like and membrane-like objects rather than point particles. It may seem like a wild idea but actually there is not much else you can do without revising our concepts of space-time or quantum mechanics. As a mathematical problem in its own right you can study the full class of possible theories of p-dimensional surfaces, known as p-branes moving in D-dimensional space. 0-branes are just particles, 1-branes are strings and 2-branes are membranes. You can work out all the ways these objects might interact which are consistent with relativity and then try to work out which of those can be consistently quantised and which are consistent with causality. The final step would be to see which of the remaining possibilities matches the real world. It is an ambitious program which is far from easy to complete.
As it turned out Dirac's ideas about strings and membranes were forgotten and history delivered string theory by a less direct route. In 1968 physicists were trying to understand the nature of the strong nuclear interactions which held the quarks together in nucleons. It was by no means clear that quantum field theory was adequate to solve the problem. Even the quark hypothesis was not universally accepted although experiments were just beginning to see signs of their effects. One way to tackle the problem was to work directly with the matrix of scattering amplitudes, the S-matrix, which describes how hadronic particles interact. Instead of trying to derive it from some underlying field theory it could be considered fundamental. The rules of quantum mechanics and relativity restrict the S-matrix to satisfy a set of equations. It was hoped that a few more additional principles might pin it down to some unique form.
An extra principle which would help was a form of duality. When two particles come together, interact and scatter off each other they could have done one of two things. It could be that they exchanged an intermediate particle, like an electron and positron exchanging a photon. Or, it could be that they join to form a new particle which then reverts back to the original two, like an electron and positron which annihilate briefly and are then recreated from a photon. These two scattering modes are known as the t-channel and s-channel respectively. For strong interactions it was found experimentally that these two amplitudes were approximately the same. There might be a principle which meant that the two channels were somehow really the same thing. Could there be an underlying interaction which possessed such duality exactly?
No sooner had the idea been thought of when Gabriele Veneziano came up with a simple formula for the scattering amplitude which did indeed possess this duality. He gave no model of what it was going on during the scattering process, just a formula which satisfied the constraints on the S-matrix. It was not long before the answer emerged suddenly from three different people. Lenny Susskind, then at Yeshiva University published his "Dual-symmetric theory of hadrons". Holger Nielsen of the Niels Bohr institute in Copenhagen called his paper "An almost physical interpretation of the dual N point function" while Yoichiro Nambu in Chicago produced "Quark model and the factorisation of the Veneziano amplitude". It was 1970 and string theory had been reborn.
By that time the evidence in favour of quarks as constituents of the proton and neutron was becoming more convincing, but nobody could understand why they were never seen on their own. They seemed to be bound together inside the hadrons. According to string theory "bound" was just the right word. The quarks were always attached to the end of strings which resisted them being pulled apart. When stretched too far it would break but a new quark anti-quark pair formed from the energy released would take hold of the lose ends. The process could also reverse when strings join together. In space-time the strings sweep out a surface or world sheet. The scattering of two mesons would now be described by a process in which two strings joined momentarily and then broke. When the world sheet is drawn the explanation for duality suddenly becomes clear. The same picture can be interpreted as either a t-channel or s-channel scattering mode.
String theory was considered as a theory of strong interactions for some time but it had problems. It only worked correctly in 26-dimensional space-time, not a very physical feature. Eventually this theory gave way to another theory called Quantum Chromo Dynamics which explained the strong nuclear interaction in terms of colour charge on gluons. In any case, string theory may have sounded good for mesons made of two quarks but protons have three. A string cannot have three ends. It looked like string theory was about to be lost for a second time.
String theory suffered from certain inconsistencies apart from its dependence on 26 dimensions of space-time. It also had tachyons, particles with imaginary mass which must travel faster than light. Tachyons could reek havoc with causality and would destabilise the vacuum, but string theory had already cast its spell on a small group of physicists who felt there must be something more to it. Pierre Ramond, Andre Neveu and John Schwarz looked for other forms of string theory and found one with fermions in place of bosons. The new theory in 10 dimensions was supersymmetric and, magically, the tachyon modes vanished.
What then was the interpretation of this new model? Schwarz teamed up with Joel Scherk and found that at low energies the strings would appear as particles. Only at very high energies would these particles be revealed as bits of string. The strings could vibrate in an infinite tower of quantised modes in an ever increasing range of mass, spin and charge. The lowest modes could correspond to all the known particles. Better still, the spin two modes would behave like gravitons. The theory was necessarily a unified theory of all interactions including quantum gravity. In 1978 the leading candidate for a super unified theory was eleven-dimensional supergravity and superstrings were largely ignored. Despite early hopes, supergravity was not quite renormalisable and it just failed to have the right properties to explain the left-right asymmetry of particle physics. Then came the historic 1984 paper of Green and Schwarz and their discovery of almost miraculous anomaly cancellations in one particular theory. Almost instantly superstrings took over as the hottest topic of research.
To come back to the original question, why string theory? The answer is simply that it has the right mathematical properties to be able to reduce to theories of point particles at low energies, while being a perturbatively finite theory which includes gravity. The simple fact is that there are no other known theories which accomplish so much. Of course physicists have now studied the mathematics of vibrating membranes in any number of dimensions. The fact is that there are only a certain number of possibilities to try and only the known string theories work out right in perturbation theory.
Of course it is possible that there are other completely different self-consistent theories but they would lack the important perturbative form of string theories. The fact is that string theorists are now turning to other p-brane theories. Harvey, Duff and others have found equations for certain p-branes which suggest that self-consistent field theories of this type might exist, even if they do not have a perturbative form.
One of the strengths of string theory is that it also included massless spin two bosons in its repertoire. These were identified as gravitons; quantum particles of gravity. Physicists had thought before then that they could see how to fit together the electromagnetic and nuclear forces but the gravitational force had been a big problem. Now they were replacing quantum field theory, which could not include gravity, with string theory which must include it.
By 1981 Green and Schwarz had identified two separate types of superstring theory. Type I is the theory of open strings but it must include closed strings as well to be complete. The other known as Type II has only closed strings. In the Type II theories the bosons and fermions appear as wave modes which circle round the strings in opposite directions. There is a version of either type for each gauge group, but the breakthrough of 1984 was the discovery that the quantisation of Type I is only free of infinities when the gauge group is SO(32) .
They also found that Type II theory worked with the same group and that it had two versions Type IIa and Type IIb. In 1985 the family of string theories was enlarged by the arrival of the heterotic string. This version discovered at Princeton by David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm, also had two versions which were finite. One with gauge group SO(32) again, and the other with E8%E8. The total number of possibilities was therefore five, sometimes denoted I, IIa, IIb, HO and HE. No other theories with the same good behaviour can be found. String theorists would like to have a unique theory so five is an embarrassment of choice. On the other hand it is much better than the situation regarding quantum field theory which works with any gauge group and a whole variety of possible matter fields, yet cannot unify all the forces.
All five superstring theories only work in 10 dimensions, 9 space dimensions plus 1 time dimension. If they have anything to do with real physics then six of the space dimensions must be rolled up or compactified just as a two-dimensional sheet of paper can be rolled into a narrow tube which becomes a one-dimensional line. If the distance around the compact dimension is very small, perhaps the Planck length, then we would not be aware of it. While there is only one way to roll up one dimension giving a tubular cross-section which is a circle, more dimensions can be rolled up in many different ways. With two dimensions there is already the choice of a sphere, torus or other surfaces with more than one hole.
These are topologically distinct and for any given choice of compactification for each string theory a different theory of the universe with different particles is found. The number of ways you can go about reducing string theory to four dimensions in this fashion is just mind boggling. It is too difficult to find the one which should correspond to our universe.
String theory is a superb example of unification. Through supersymmetry, matter is united with force. There is only one type of object; the string. If it vibrates one way it can be a quark, another way it is an electron, change its mode again and it becomes a force carrying photon or even a graviton.
But by 1988 string theory was in trouble. Past history shows that breakthroughs in physics are at first largely ignored until experiment forces the community of physicists to accept them. Such had been the case with atoms, relativity, parity violation, quark theory and electroweak unification. By contrast string theory was immediately taken up by a huge proportion of physicists and then it failed to make any experimental predictions which could be tested. Richard Feynman was one of those who spoke against his mostly younger colleagues who supported string theory. He did not like the fact that string theorists were not calculating anything which would allow them to check their ideas empirically.
Yet they carried on. String theory was still young and rather than letting its critics stop them they would rise to the challenge. The acknowledged leader in the fight to understand string theory is Ed Witten. He speaks in a very different tone, explaining that the critics do not seem to have fully grasped the scope and richness of the structure involved in string theory. They are too impatient for quick answers.
A fine example of the rich and beautiful structure of string theory is T-duality, short for target space duality. The target space of a string theory is just the space-time in which it is placed. The five principal superstring theories are most at home in flat ten-dimensional space-time infinite in all directions, but they can also be placed in space-times where some of the dimensions have been compactified.
The simplest case is where one of the space dimensions is rolled up round a circle of radius R. A string theory in such a space-time appears like a nine-dimensional theory of strings. The rolled up dimension becomes invisible and the compactification radius R becomes just one of many arbitrary parameters.
Since there are five superstring theories in 10 dimensions and only one way to compactify to 9 dimensions, you would expect there to be five superstring theories in 9 dimensions too. In actual fact there are only three. The two different heterotic theories in 10 dimensions, HE and HO, reduce to the same nine-dimensional theory. The compactification radii RE for HE and RO for HO heterotic string appear as a parameter in this theory but they are related inversely RE = 
/RO . HE is recovered as the limit of the nine-dimensional string theory as RE is made large and HO is the limit as RO is made large. So the two heterotic string theories are really two aspects of the same theory. They are said to be T-dual. The same magic can be applied to the two TypeII theories. IIa is T-dual to IIb. This leaves us with just three separate superstring theories Type I, Type II and Heterotic.
That is how the situation stood in 1993 but then another kind of duality was found. It concerns a relation between electric charges and magnetic monopoles.
Maxwell's equations for electromagnetic waves in free space are symmetric between electric and magnetic fields. A changing magnetic field generates an electric field and a changing magnetic field generates an electric one. The equations are the same in each case, apart from a sign change. If you take the equations and switch the electric and magnetic fields, while changing the sign of one of them, you arrive back at the same form. The free fields without charges are invariant but if electric charges are included there must also be magnetic charges to complete the symmetry. However, it is an experimental observation that there are no magnetic monopole charges in nature which mirror the electric charge of electrons and other particles. Despite some quite careful experiments only dipole magnetic fields which are generated by circulating electric charges have ever been seen.
In classical electrodynamics there is no inconsistency in a theory which places both magnetic and electric monopoles together. In quantum electrodynamics this is not so easy. To quantise Maxwell's equations it is necessary to introduce a vector potential field from which the electric and magnetic fields are derived by differentiation. This procedure cannot be done in a way which is symmetric between the electric and magnetic fields.
Forty years ago Paul Dirac was not convinced that this ruled out the existence of magnetic monopoles. Again motivated by mathematical beauty in physics, he tried to formulate a theory in which the gauge potential could be singular along a string joining two magnetic charges in such a way that the singularity could be displaced through gauge transformations and must therefore be considered physically inconsequential. The theory was not quite complete but it did have one saving grace. It provided a tidy explanation for why electric charges must be quantised as multiples of a unit of electric charge.
In the 1970s it was realised by 't Hooft and Polyakov that grand unified theories which might unify the strong and electro-weak forces would get around the problem of the singular gauge potential because they had a more general gauge structure. In fact these theories would predict the existence of magnetic monopoles. Even their classical formulation could contain these particles which would form out of the matter fields as topological solitons.
There is a simple model which gives an intuitive idea of what a topological soliton is. Imagine first a straight wire pulled tight like a washing line with many clothes pegs strung along it. Imagine that the clothes pegs are free to rotate about the axis of the line but that each one is attached to its neighbours by elastic bands on the free ends. If you turn up one peg it will pull those nearby up with it. When it is let go it will swing back like a pendulum but the energy will be carried away by waves which travel down the line. The angles of the pegs approximate a field along the one-dimensional line.
The equation for the dynamics of this field is known as the sine-Gordon equation. It is a pun on the Klein-Gordon equation which is the correct linear equation for a scalar field and which is the first order approximation to the sine-Gordon equation for small amplitude waves. If the sine-Gordon equation is quantised it will be found to be a description of interacting scalar fields in one dimension.
The interesting behaviour of this system appears when some of the pegs are swung through a large angle of 360 degrees over the top of the line. If you grab one peg and swing it over in this way you would create two twists in the opposite sense around the line. These twists are quite stable and can be made to travel up and down the line. A twist can only be made to disappear in a collision with a twist in the opposite direction.
These twists are examples of topological solitons. They can be regarded as being like particles and antiparticles but they exist in the classical physics system and are apparently quite different from the scalar particles of the quantum theory. In fact the solitons also exist in the quantum theory but they can only be understood non-perturbatively. So the quantised sine-Gordon equation has two types of particle which are quite different.
What makes this equation so remarkable is that there is a non-local transformation of the field which turns it into another one-dimensional equation known as the Thirring model. The transformation maps the soliton particles of the sine-Gordon equation onto the ordinary quantum excitations of the Thirring model, so the two types of particle are not so different after all. We say that there is a duality between the two models, the sine-Gordon and the Thirring. They have different equations but they are really the same because there is a transformation which takes one to the other.
The relevance of this is that the magnetic monopoles predicted in GUT's are also topological solitons, though the configuration in three-dimensional space is more difficult to visualise than for the one dimension of the clothesline. It would be nice if there was a similar duality between electric and magnetic charges as the one discovered for the sine-Gordon and Thirring equations. If there was then a duality between electric and magnetic fields would be demonstrated. It would not be quite a perfect symmetry because we know that magnetic monopoles must be very heavy if they exist.
In 1977 Olive and Montenen conjectured that this kind of duality could exists, but the mathematics of field theories in 3 space dimensions is much more difficult than that of one dimension and it seems beyond hope that such a duality transformation can be constructed. But they made one step further forward. They showed that the duality could only exist in a supersymmetric version of a GUT. This is quite tantalising given the increasing interest in supersymmetric GUT's which are now considered more promising than the ordinary variety of GUT's for a whole host of reasons.
Until 1994 most physicists thought that there was no good reason to believe that there was anything to the Olive-Montenen conjecture. Then Nathan Seiberg and Ed Witten made a breakthrough which rocked the worlds of physics and mathematics. By means of a special set of equations they demonstrated that a certain supersymmetric field theory did indeed exhibit electro-magnetic duality. As a bonus their method can be used to solve many unsolved problems in topology and physics. The duality exchanges strong coupling with weak coupling. This is very significant for theories like QCD where the strong coupling limit is not understood.
This kind of duality is now known as S-duality to distinguish it from T-duality. In string theory S-duality is very natural. There is a general rule about the dimensions of dual objects. An "electric" p1-brane which is a fundamental construct of a theory in D dimensions can have a p2-brane "magnetic" soliton when p1 + p2 = D - 4. In the familiar case the electric and magnetic charges in D=4 are particles, i.e. 0-branes. In D=10 string theory the strings are 1-branes so their duals must be (10-4-1)-dimensional 5-branes. In the last year physicists have discovered how to apply tests of duality to different string and p-brane theories in various dimensions. Conjectures have been made and tested. This does not prove that the duality is correct but each time a test has had the potential to show an inconsistency it has failed to destroy the conjectures.
It now seems that any string theory with sufficient supersymmetry must have an S-dual waiting to be found. What makes this discovery so useful is that the dualities are a non-perturbative feature of string theory. Now many physicists see that p-brane theories can be as interesting as string theories in a non-perturbative setting. Using T-duality we made reduced the five superstring theories to three. Now with S-duality we can make further links which leave them all connected. Type I is S-dual to HO while HE is S-dual to IIa (but only when compactified to six dimensions). The last of the five IIb is self dual.
That was not quite the end of the story. If these five theories are all part of the same thing then what is that thing? The answer, it seems, is that they are all derived from something called M-theory in 11 dimensions. M-theory is like string theory except that it is a theory of membranes (2-branes) rather than strings (1-branes). It also has an S-duality between its 2-branes and solitonic 5-branes. All five string theories are special points in the parameter space of this one theory, but so is eleven-dimensional supergravity theory, the same theory that string theory ousted as the most popular super-unified theory in 1984.
This may be too simple a picture of M-theory which really includes open and closed strings, membranes, p-branes etc. Each of the string theories appears in some corner of M-theory where particular states become weakly coupled and can be described using perturbation theory.
It would be wrong to say that very much of this is understood yet. There is still nothing like a correct formulation of M-theory or p-brane theories in their full quantum form, but there is new hope because now it is seen that all the different theories can be seen as part of one unique theory. The best way to formulate that theory is not yet known.
It is curious that various stellar objects under the influence of strong gravity parallel various entities from particle physics. A white dwarf star is like an atom in that it resists collapse due to the Pauli exclusion principle. A more massive star will collapse further to a neutron star which is like a stable nucleus. A stronger gravitational force can reduce it to a quark star which is like a neutron. The final stage of gravitational collapse reduces the star to a black hole. If the analogy continues to hold, the black hole should be like a quark or other elementary particle.
The theory started to look a little less ridiculous when Hawking postulated that black holes actually radiate particles. The process could be likened to a very massive particle decaying. If a black hole were to radiate long enough it would eventually lose so much energy that its mass would reduce to the Planck scale. This is still much heavier than any elementary particle we know but quantum effects would be so overwhelming on such a black hole that it would be difficult to see how it might be distinguished from an extremely unstable and massive particle in its final explosion.
To make such an idea concrete requires a full theory of quantum gravity and since string theory claims to be just that, it seems a natural step to compare string states and black holes. We know that strings can have an infinite number of states of ever increasing spin, mass and charge. Likewise a black hole, according to the no hair conjecture is also characterised only by its spin, mass and charge.
With magnetic duality we can add magnetic charge to the list. It is therefore quite plausible that there is a complementarity between string states and black hole states, and in fact this hypothesis is quite consistent with all mathematical tests which have been applied. It is not something which can be established with certainty simply because there is not a suitable definition of string theory to prove the identity. Nevertheless, many physicists now consider it reasonable to regard black holes as being single string states which are continually decaying to lower states through Hawking radiation.
It was discovered that if you consider Planck mass black holes in the context of string theory then it is possible for space-time to undergo a smooth transition from one topology to another. This means that many of the possible topologies of the curled up dimensions are connected and may pave a way to a solution of the selection of vacuum states in string theory.
One thing about string theory which was discovered very early on was that at high temperatures it would undergo a phase transition. The temperature at which this happens is known as the Hagedorn temperature after a paper written by Hagedorn back in 1968, but it was in the 1980s that physicists such as Witten and Gross explored the significance of this for string theory.
The Hagedorn temperature of superstring theory is very high, such temperatures would only have existed during the first 10-43 seconds of the universe existence, if indeed it is meaningful to talk about time in such situations at all. Calculations suggest that certain features of string theory simplify above this temperature. The implication seems to be that a huge symmetry is restored. This symmetry would be broken or hidden at lower temperatures, presumably leaving the known symmetries as residuals.
The problem then is to understand what this symmetry is. If it was known, then it might be possible to work out what string theory is really all about and answer all the puzzling questions it poses. This is the superstring mystery.
A favourite theory is that superstring theory is described by a topological quantum field theory above the Hagedorn temperature. TQFT is a special sort of quantum field theory which has the same number of degrees of gauge symmetry as it has fields, consequently it is possible to transform away all field variables except those which depend on the topology of space-time. Quantum gravity in (2+1)-dimensional space-time is a TQFT and is sufficiently simple to solve, but in the real world of (3+1)-dimensional Einstein Gravity this is not the case, or so it would seem.
But TQFT in itself is not enough to solve the superstring mystery. If space-time topology change is a reality then there must be more to it than that. Most physicists working in string theory believe that a radical change of viewpoint is needed to understand it. At the moment we seem to be faced with the same kind of strange contradictions that physicists faced exactly 100 years ago over electromagnetism. That mystery was finally resolved by Einstein and Poincaré when they dissolved the ether. To solve string theory it may be necessary to dissolve space-time altogether.
In string theory as we understand it now, space-time curls up and changes dimension. A fundamental minimum length scale is introduced, below which all measurement is possible. It will probably be necessary to revise our understanding of space-time to appreciate what this means. Even the relation between quantum mechanics and classical theory seems to need revision. String theory may explain why quantum mechanics works according to some string theorists.
All together there seem to be rather a lot of radical steps to be made and they may need to be put together into one leap in the dark. Those who work at quantum gravity coming from the side of relativity rather than particle physics see things differently. They believe that it is essential to stay faithful to the principles of diffeomorphism invariance from general relativity rather than working relative to a fixed background metric as string theorists do. They do not regard renormalisability as an essential feature of quantum gravity.
Working from this direction they have developed the canonical theory of quantum gravity which is also incomplete. It is a theory of loops, tantalisingly similar in certain ways to string theory, yet different. Relativists such as Lee Smolin hope that there is a way to bridge the gap and develop a unified method