*Space-Time Topology Change in String Theory*, Aspinwall, Greene, Morrison*Unity of Superstring Dualities*, C. Hull*Classical Quantum Duality*, Michael Duff*What is String Theory?*, Joseph Polchinski*String Theory Dynamics in Various Dimensions*, Edward Witten*The Power of M Theory*, John Schwarz*Black Hole Condensation and the Duality in String Theory*, Andrew Strominger*World Volumes an String Target Spaces*, Michael Green

Part 6 of 9

In 1982 Michael Green and John Schwarz made a discovery which might turn out be the greatest scientific advance of all time, if it is right. What they found was that a particular quantum field theory of supersymmetric strings in 10 dimensions gives finite answers at all orders in perturbation theory.

This was a breakthrough because the superstring theory had the
potential to include all the particles and forces in nature. It
could be a completely unified theory of physics. By 1985 the
press had got hold of it. Articles appeared in Science and
New Scientist. They called strings a *Theory Of
Everything*.

The term *Theory of Everything* is a desperately
misleading one. Physicists usually try to avoid it but the media
can't help themselves. If physicists find a complete unified
set of equations for the laws of physics, then that would be
a fantastic discovery. The implications would be enormous,
but to call it a theory of everything would be nonsense.

For one thing, it would be necessary to solve the equations
to understand anything. No doubt many problems in particle
physics could be solved from first principles, perhaps it
would be possible to derive the complete spectrum of
elementary particles. However, there would certainly be
limits to the solvability of the equations. We already find
that it is almost impossible to derive the spectrum of
hadrons composed of quarks, even though we believe we have
an accurate theory of strong interactions. *In principle*
any set of well defined equations can be solved numerically
given enough computer power. The whole of nuclear physics and
chemistry ought to be possible to calculate from the laws we
now have. *In practice* computers are limited and
experiments will always be needed.

Furthermore, it is not even possible to derive everything
*in principle* from the basic laws of physics. Many
things in science are determined by historical accident. The
foundations of biology fall into this category. The final
theory of physics will not help us to understand how life
on Earth originated. The most ardent reductionist would
retort that, in principle, it would be possible to derive a
list of all possible forms of life from the basic laws of
physics.

Finally it must be said that even given a convincing unified
theory of physics, it is likely that it would still have the
indeterminacy of quantum mechanics. This would mean that no
argument could finally lay to rest questions about paranormal,
religion, destiny or other such things, and beyond that there
are many matters of philosophy and metaphysics which might not
be resolved, not to mention an infinite number of mathematical
problems. Clearly the term *Theory of Everything* is
misleading.

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. 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 10 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 four 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 physicists called it a *theory of nothing* and
advocated a more conservative approach to particle physics
tied more closely to experimental results. But a large
number of physicists have persisted. There is something about
superstring theory which is very persuasive.

The most common question from lay-people about
string theory is *Why?*. To understand why
physicists study string theory rather than theories of
surfaces or other objects we have to go back to its
origins. In 1968 physicists were trying to understand
the nature of the strong nuclear interactions which
held the quarks together in nucleons. There was an
idea about duality between scattering interactions which
led Veneziano and Virasoro to suggest exact forms for the
dual resonance amplitude. These amplitudes turned out
to have interesting properties in 26 dimensions and
various independent lines of research by Nambu, Nielson
and Susskind led to the revelation that
the amplitudes were derivable from a theory of strings.

String theory was considered as a theory of strong interactions for some time. Physicists thought that the explanation for confinement of quarks was that they were somehow bound together by strings. 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.

String theory suffered from certain inconsistencies apart
from its dependence on 26 dimensions of space-time. It also
had *Tachyonic modes* which destabilised 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.
Ramond, Neveu and 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 were
removed.

But what was the interpretation of this new model? Scherk and Schwarz found that at low energies the strings would appear as particles. Only at very high energies would these particles be revealed as loops 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. Still only a small group pursued this idea until the historic paper of Green and Schwarz with the discovery of almost miraculous anomaly cancellations in one particular theory.

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 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 membrane theories, or
*p-brane theories* as they are known, where *p*
is the number of dimensions of the membrane. 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.

In the past couple of years there have been some new developments which have inspired a revival of interest in string theory. The first of these concerns duality between electric 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 which is irrelevant here. However, it is an experimental fact 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 observed.

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 can not be done in a way which is symmetric between the electric and magnetic fields.

40 years ago Paul Dirac was not convinced that this ruled out the existence of magnetic monopoles. He always professed that he was 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 1970's 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 angle 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 Klien-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.

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 dimensional of the clothes line. Wouldn't it be nice if there was a similar duality between electric and magnetic charges as the one discovered for the sine-Gordon equation? If there was then a duality between electric and magnetic fields would be demonstrated. It would not quite be 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 Seiberg and Witten made a fantastic breakthrough. 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.

Now at last we turn to string theory with the realisation that duality in string theory is very natural. In the last year physicists have discovered how to apply tests of duality to different string and p-brane theories in various dimensions. A series of 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. 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. The latest result in this effort is the discovery that all four string theories which are known to be perturbatively finite are now thought to be derivable from a single theory in 11 dimensions known as M-theory. M-theory is a hypothetical quantum field theory which describes 2-branes and 5-branes related through a duality.

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.

As if one major conceptual breakthrough was not enough, string theorists have been coming to terms with another which turned up last year. Just as physicists have been quietly speculating about electro-magnetic duality for decades, a few have also speculated that somehow elementary particles could be the same things as black holes so that matter could be regarded as a feature of the geometry of space-time. The idea actually goes back at least as far as Riemann.

The theory started to look a little less ridiculous when Hawking postulated that black holes actually emit 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. 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 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.

The recent breakthrough due to Strominger, Greene and Morrison is the discovery 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.

Superstring theory is full of symmetries. There are gauge symmetries, supersymmetries, covariance, dualities, conformal symmetries and many more. But superstring theory is supposed to be a unified theory which should mean that its symmetries are unified. In the perturbative formulation of string theory that we have, the symmetries are not unified.

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 1980's that physicists such as Witten and Gross explored the significance of this for string theory.

The Hagedorn temperature of superstring theory is almost
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 figure 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 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 when he 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 a 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.

This page was last updated 15 May 1996 by
Phil Gibbs.

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