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{\bf Conformal Field Theory: A Bridge Over Troubled Waters} \\[2ex]
W. Nahm\\
University of Bonn
Physics Institute\\
Nussallee 12, D-53115 Bonn, Germany
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{\bf Abstract}\\
Since 70 years, quantum field theory has been one of the most
important research areas of physics. For further progress, it must become
a standard domain of mathematical research, too. The article reviews the
historical obstacles and shows how they can be overcome. Already now,
conformally invariant quantum field theory in two dimensions has become
a well defined and beautiful structure. Its emergence is considered in the
context of high energy physics, statistical physics, and string theory.
\\[3ex]

In his 1972 address to the American Mathematical Society, Dyson deplored
the
'divorce' between mathematics and physics over the issue of quantum
field
theory. The present book on the impact of field theory on modern
physics, timed
in accordance with the International Mathematics Year of 2000 AD, gives
hope
that the rift will soon be bridged. In conformal quantum field theory in
two
dimensional spacetime, conditions are particularly favorable for gaining
common
ground. This area can attract both mathematicians and physicists by its
beauty,
its transparent mathematical structure and its many applications. Thus
its
investigation has moved to a rather central position in quantum field
theory as
a whole.  Perhaps one can say that a bridge exists since many years, but
has been used much below its capacity.


Analogues in more than two dimensions will be mentioned in the present
article, but they have not been developed very far nor found
mathematical applications yet. For convenience, the expression
'conformal field
theory' will refer to conformally invariant quantum field theories in
two
dimensions, if nothing else is said.  

Its applications are suprisingly diverse. In mathematics, one particular
theory
has become well known due to its automorphism group, the Fischer-Griess
monster, and by the Fields medal for Borcherds. Currently research on
the many
mirror symmetric cases makes rapid progress, whereas the explicit
construction 
of K\"ahler-Einstein metrics is just at a planning stage. In physics,
conformal
field theory became essential for the study of continuous phase
transitions in
condensed matter physics, but the most important applications concern
string
theory. Even independently of its status as the best candidate for a
theory of
quantum gravity, string theory has become an important tool in quantum
field
theory. In particular, it yields a map from two-dimensional conformal
theories
to quantum field theories in higher dimensions, even to rather realistic
examples in four-dimensional spacetime.    

It will be argued that conformal field theory can satisfy all
mathematicians
who want to understand quantum field theory without giving up their
standards of
clarity and rigour. In return, many of its important aspects need
advanced
mathematical techniques. Some still remain out of the reach of
physicists,
others are handled in a manner which produces lots of undiscovered
errors in the
literature. A serious involvement of mathematicians will yield much
firmer
foundations for the things to come. 

Apart from conformal field theory, this article will discuss some string
theory.
The perturbative aspects of the latter are well understood by now and
should be
easily accessible. Indeed, their bulk just constitutes a direct
application of
conformal field theory. By themselves, these aspects do not yet
transcend the
old principles of 20th century physics, but they are incomplete and seem
to point in one unique direction of progress. There, non-perturbative
string
theory (or M-theory, or however one chooses to call it) has started to
emerge
and should lead to a new insight. Its birth might be a lot easier if
mathematicians again get into the habit of acting as good midwives.

There is no lack of good will. Many conferences have been attended by
mixed
groups of mathematicians and physicists. Most importantly, Princeton has
brought together many of the best people of both communities in a
dedicated
program. Nevertheless, one sometimes gets the feeling that a new
generation will
be needed to overcome the difficulties. 

It may help to recognize what the main obstacles
have been, since a discussion
of past successes and mistakes can prepare the way for future
achievements.
In any case, the present article is supposed to cover the early stages
of the
discoveries. Of course, history is complex, full of turbulence
and countercurrents, such that precise statements would need more
hedging or
much more research. The historical content of the present article should
therefore be taken as a signpost, not as a map. Its indications will be
approximate, but may be helpful. Moreover, a low density of formulas
should make
the article accessible to a wider readership. Methodological qualms of
the
historians will be brazenly ignored. If they deny that it makes sense to
ask
what might have happened, what about an advanced quantum computer which
reruns
history, starting from a reasonable subspace of initial conditions.

An invitation for mathematicians to cross the bridge also needs some
technical
parts, however. Mathematicians still think of quantum field theory as a
useful source of ideas (cf. the Seiberg-Witten equations), but otherwise
as
impenetrable, though some related structures are regarded as good
mathematics
(topological quantum field theory, probably conformal field theory).
>From the
point of view of a physicist, this is a strange attitude. By nature,
conformal
field theory  was presented in the context of quantum field theory as a
whole,
so this is the way it should be discussed. Nature has a habit of posing
such problems, recall infinitesimals and differential equations. Again,
we
should have confidence in her guidance. 

The article starts with a short introduction to the history of quantum
field theory. For readers who want to get a fuller picture and the
necessary
references, there are good reviews and reprint volumes at different
technical
levels [Schwinger 1958, Pais 1986, Crease and Mann 1987]. In those
reviews the
aim is to show how nature was explained. Here, mathematically well-built
structures will be regarded as equally important for further progress. 
We shall see that the tools to build the bridge were available
at the end of the 60's, maybe even ten years earlier. 

Because of the focus on conformally invariant theories, other issues of
mathematically rigorous quantum field theory, like the work of Glimm and
Jaffe
on superrenormalizable theories will not be discussed, however.  
Even for the main themes of the article, the selection is partial.
Current
algebras are one of the major themes of conformal field theory, but here
the
collaboration between mathematicians and physicists has been very
fruitful
since twenty years, and there are already nice reviews, e.g. [Goddard
and
Olive 1988], so the topic will receive less emphasis than would be
necessary in a complete survey. Renormalization will be discussed in
some
detail, since most mathematicians regard it as the major stumbling block
which prevents an understanding of quantum field theory. Thus it may
help
to see that this procedure is rather easy from a mathematical
point of view and follows a well known idea of the 19th century. Path
integrals
will not be mentioned. In quantum field theory as a whole the
corresponding
ideas have not yet found a satisfactory form, and even in conformal
field theory
one needs a discussion in terms of categories and operads, which only
seems
simple to the very good or the very young. The discussion of string
theory is
limited to the perspective of conformal field theory. A serious
consideration of its non-perturbative aspects would have to start with a
description of instantons and solitons. Many connections with conformal
invariance could be explained, but not within the scope of the present
article.
Altogether, it is unavoidable that many readers will have reasons to
complain,
but at least they should feel encouraged to do better.\\[2ex]
{\bf After a golden age}

A theoretical physicists who looks back to the beginning of the past
century
has reasons to feel rather humble. A few decades witnessed three
revolutionary insights in structures of nature, namely special
relativity,
quantum mechanics and Einstein's theory of gravity. We still have to
embed these
in a unified theory. Meanwhile, we covered much territory in the study
of the complexities of nature, but further understanding of her basic
features
proceeds at a snail's pace.

Evidently, the initial rapid progress relied on a close
interaction between physicists and mathematicians. We cannot even talk
about
those three structures without alluding to this fact. Just think of the
Poincar\'e group, Hilbert space and Riemannian geometry. 

In the first years of the century, many physicists felt doubtful or
uneasy about the importance of contemporary mathematical methods. 
For example, Mittag-Leffler struggled in vain to get a Nobel prize for
Poincar\'e out of the old establishment. Around 1920, the movement had
become
irresistible, however. Einstein's Nobel prize document avoids to mention
special or general relativity, but this was hardly more than a funny
detail.
Everyone went to G\"ottingen,  Weyl solved the Schr\"odinger equation
for the
hydrogen atom, and Hilbert was an eager competitor when Einstein
approached the
final form of his equation for gravity. Of longer lasting importance was
the
clarification of the mathematical structure of quantum mechanics by v.
Neumann
and Weyl. Von Neumann had to immerse himself in physics because of the
bomb and the computer, but Weyl's publication list in physics journals
is
impressive, too, and his exchange of ideas with Einstein and Pauli was 
particularly fruitful.  
 
Mathematicians made contributions of three kinds. Sometimes
they solved concrete problems which the physicists found too difficult,
but this
was rare. More importantly, the internal logic of mathematics had led to
the
discovery of deep structures which found unexpected applications.
Finally, the
analysis of discoveries made in physics uncovered new mathematical
worlds and
allowed the physicists to think more clearly and efficiently about their
own
results.

It is no surprise that the latter task attracts some of the best
mathematical
minds. Already in 1900, Hilbert saw that a renewed interest in physics
would be
productive, and he put the development of good axioms for mechanics as
the sixth
problem on his famous list. Twenty-five years later, classical mechanics
had the
necessary clear conceptual basis to allow the wonderful emergence of
quantum
mechanics. In contrast, the infinite dimensional spaces of classical
field
theory remained less well understood. This contributed a bit to the
confusion about quantum field theory, as we shall see.

When a science is restructured during and after a major advance, it is
particularly important to put what has been known before in a new
context and to
provide it with deeper foundations. Conformal invariance in physics
emerged in
this way, though at first its place seemed to be marginal. Its discovery
was triggered by special relativity. This theory had underlined the
importance
of symmetry groups and stimulated a new mathematical look at Maxwell's
equations. Cunningham [1910] and Bateman [1910] determined the maximal
group of symmetries of the latter and discovered their conformal
invariance. In
other words, Maxwell's system of equations is invariant under the
maximal group
of spacetime transformations which preserve angles, but not necessarily
distances. Addressing mathematics and physics audiences, F. Klein
repeatedly
asked for an explanation, but in vain. 

Some progress became possible with the work of E. Noether [1918]. She
was a
creator of modern abstract algebra and only had a marginal interest in
physics,
but was motivated by Einstein's gravity theory to explain the general
connection between symmetries of Lagrangians and conservation laws. 
Prompted by Klein to analyse the problem with Noether's method,
Bessel-Hagen 
determined the conserved quantities corresponding to the conformal
invariance of
Maxwell's equations [1921]. In the following years, several other
differential
equations were investigated. Most importantly, Pauli proved that the
Dirac
equation is conformally invariant when the mass vanishes [1940].

By that time, Hodge had found the tools for a deeper analysis [1941],
which
needed much longer to make headway into the physics community. He
had investigated topological questions like Poincar\'e duality from
the point of view of differential forms. In this language, the
electromagnetic field can be described by a 2-form $F$. In the absence
of
charges,  Maxwell's equations take the form 
$$dF=0, \hskip1cm d\ast F=0\ .$$  
This simple form even applies to the curved
spacetime of Einstein's theory. Indeed, the differential operator $d$
does not
depend on any kind of metric structure. The Hodge duality operator
$\ast$,
which acts linearly on differential forms, depends on the metric. When
applied
to $k$-forms in a spacetime of $2k$ dimensions, however, the dependence
on the
distance scale cancels out. In particular, in the most important case of
four-dimensional spacetime, $\ast dF$ only depends on the angles, in
other words
on the conformal structure. Moreover, the Lagrangian density of the
electromagnetic field in empty space is given by the integral over
$F\wedge\ast
F$. Since the metric again appears only through the Hodge star
operation, the 
Poisson brackets derived from this Lagrangian have the same conformal
invariance. This remains true for the corresponding quantum system, but
in
general conformal invariance is broken when one introduces charges.

Eventually, this approach turned out to be very productive for physics,
see e.g.
[Atiyah, Hitchin, Singer 1978], but mainstream physicists learned about
it only
in the 70's, largely through the efforts of Atiyah, who had heard
Hodge's
lectures as a student. Indeed, in the 30's a fault-line between the
communities
of the physicists and the mathematicians had started to develop. It must
have
been hard to spot at the time, since there were greater and more
immediate
concerns. Within physics, the split between theoreticians and
experimentalists
now became complete. Einstein still had done moderately respectable
experimental
work, but Heisenberg's PhD exam was a near disaster, since he had
concentrated
all his efforts on theory. Some misgivings of the experimentalists were
quite
natural. Though there was no reason to expect that mathematical physics
would
regain the prestigeous position it had had in the times of Newton, Euler
and
Lagrange, the incredible popularity of Einstein and the difficulties of
his
theories must have suggested to many that something had gone wrong.

Finally, from the 30's to the 50's physics was hopelessly entangled in
far more
important events (fascism, the war, the bomb, Stalin, McCarthy, ...).
This was a
worldwide phenomenon, which left little refuge. It had one important
positive
aspect, however. Due to the international contacts, progress in physics
was no
longer the prerogative of Europe or North America. Most visibly, Japan
and
India had started to take part.

All of this left little room for concern about the spreading rift
between
theoretical physics and mathematics. Around 1970, however, it had become
too big
to overlook. In his Gibbs Lecture, Dyson put it bluntly: "the marriage
between
mathematics and physics ... has recently ended in divorce" [Dyson 1972].

The main culprit seemed to be quantum field theory. Here is a fairly
typical
quotation from a highly regarded textbook:
"The mathematically inclined reader undoubtedly by now will have had
serious
misgivings about the validity and meaningfulness of the renormalization
program,
since this program has at its point of departure a set of meaningless
equations
which it then proceeds to manipulate according to rules which are
outside the
bounds of conventional mathematics to obtain (presumably) finite results
(not to
mention the fact these prescriptions, as outlined in the present
chapter, are
applicable only to the power series expansion of the 'meaningless
equations,'
which power series expansion in all probability does not converge!)"
[Schweber 1964, p. 645].

It is clear that something had gone wrong. In a sense, one may put the
blame on
nature, since she gave ambiguous directions. We considered the
discoveries of
special relativity, quantum mechanics and Einstein's theory of gravity,
but it
is somewhat misleading to talk about them on a par, since the three
theories do
not occupy the same logical level. Einstein's name for his theory of
gravity was
general relativity, because compatibility with the principles of special
relativity was incorporated from its inception. Thus the task of
unification
would be finished, if one could join gravity and quantum mechanics in
one move. 
>From today's point of view, this problem was too difficult and led into
a thick fog.

Instead one could follow the geometrical path indicated by Einstein's
gravity
theory. This was natural for mathematicians, but not immediately
productive for physics. Nevertheless, the search in these directions
provided a favorable environment for the development of gauge theories,
as we
shall see later. 

For physicists, a different path was indicated by nature. After quantum
mechanics had matured around 1926 (the year of the Schr\"odinger
equation), the
next fundamental problem was to put together quantum theory and special
relativity. The essential guidance came from the experiments, whereas
the
mathematical structures remained rather obscure. In favorable
circumstances, it
still might have been possible to advance together, but many of the
links
between mathematics and physics were broken by the war. When a deeper
study of
the weak and strong interactions led to gauge theories, a convergence of
the two
paths was indicated, but this came too late for a reestablishment of the
old
contacts. In this sense the lack of mathematical accessability of
relativistic
quantum field theory is rather a consequence of the separation of
mathematics 
and physics than its cause.  

Let us come back to the perspective of 1926. Classical physics deals
with rigid
bodies and with fields. Now the former were to be regarded as a low
velocity
approximation, since extended rigid bodies are incompatible with special
relativity. When an object is touched, it cannot be affected all at
once, since
this would surpass the speed of light. 

Rigid bodies often had been approximated by point particles. Now they
had to be
considered in terms of pointlike constituents. In one space dimension,
the
latter can interact by collisions, but in more dimensions this makes
little
sense. Thus the only available candidates for the description of
interactions
in the real world were field theories. Conversely, the discovery of
special
relativity depended on an analysis of Maxwell's equations for the 
electromagnetic field. In quantum physics, matter in the form of point
particles was easily incorporated in this frame, since the Schr\"odinger
wave
function could be regarded as the avatar of a relativistic field. Thus
the
unification of special relativity and quantum theory demanded the
formulation 
of quantum field theory. 

These ideas were well understood in the 1920's. They came very
naturally, since
already the first steps of quantum mechanics were guided by quantum
field
theory: The first formula for a quantal phenomenon was Planck's
radiation law
for the electromagnetic fields emitted by a heated black body. Thus
quantum
electrodynamics took shape immediately after the birth of modern quantum
mechanics, in a 1927 paper by Dirac and, in more appropriate guise, in a
paper by Heisenberg and Pauli in 1929. In its initial form, it was
sufficient 
for a calculation of the semiclassical electromagnetic processes
observable at
that time.

Soon, however, quantum field theory was put in doubt by experimental
results
and problems of consistency. For experimentalists, further work on the
unification of special relativity and quantum theory posed a single
basic
challenge - study particle interactions at velocities close to the speed
of
light. Here early researchers were confronted with a bewildering wealth
of data,
from nuclear physics to cosmic rays. It took a long time until things
were
sorted out to reveal the underlying structures. In particular, it was
far from
obvious that the experimental results could be described by any kind of
quantum
field theory. For a long time, electromagnetism was the only interaction
for
which it made real sense. 

For mathematicians and physicists alike, this greatly diminished the
attractiveness of quantum field theory. Most importantly, it contributed
to the
persistent but unproductive expectation of another revolution in the
foundations of physics. In view of the previous decades, this
expectation was
quite understandable. Quantum mechanics had been developed for atomic
physics,
particle physics might need something equally revolutionary and
exciting.
Oppenheimer even gave a number: don't believe the old ideas beyond 100
MeV. For
a while, there was a concrete reason for this attitude. Yukawa had
predicted a
particle of 100 MeV to explain the strong interaction, but when it
seemed to be
discovered, most of its other properties were wrong. Eventually, muons
and pions
were distinguished and the paradoxes dissolved away, but the basic
attitude
surfaced again on many occasions. 

Quantum electrodynamics itself set other obstacles against the joint
development
of quantum field theory in a common effort of physicists and
mathematicians. In
particular, one immediately had to face  one of the old problems of
classical
physics, namely the infinite energy in the electric field of point
charges. In
classical physics, spreading out the charge yielded a temporary excuse,
but
special relativity and quantum mechanics demanded the consideration of
point
charges, such that a clash was inevitable. There soon came a reason for
hope,
however. Against his expectations, Weisskopf (with a little help from
Furry)
showed in 1934 that in quantum electrodynamics the pole divergence of
the
classical theory is replaced by a mild logarithmic one. In the following
years,
Kramers explained the basic principles of regularization and
renormalization.
Weisskopf apparently was slowed down by discontent about his small
mistake, but
in 1939 he published a clear argument which indicated that any intrinsic
inconsistencies of quantum electrodynamics were many orders of magnitude
away from the experimentally observable domain.

Altogether, in the 1930's the stage was set for the further development
of the
theory, and a concerted effort of physicists and mathematicians was not
completely out of the question. There was no single overwhelming
obstacle.
Still, the effort would have demanded an unlikely amount of patience and
persistence against many stumbling blocks. Quantum electrodynamics has
the
typical difficulties of a gauge theory, and mathematics was not quite
ready to
provide elegant tools for their resolution. For the physicists,
experiments had
not yet provided a compelling reason to put much effort in the study of
the
small quantum electrodynamical effects of higher order, and to some
extent the
bewildering features of the other interactions undermined the faith in
quantum
field theory as a whole. The mathematicians had no reason to invest much
work in
something which might not last. They still were busy to consolidate the
advances
of quantum mechanics and  gravity theory. Above all, they saw no
compelling
internal mathematical reason to develop quantum field theory. In
hindsight,   
v. Neumann's operator algebras came close, but they hardly became part
of the
mathematical mainstream, and v. Neumann soon had more important things
to do.

Since the time for quantum electrodynamics was not yet ripe, joint
mathematical
and physical progress would have needed another stroke of genius. 
One possibility would have been the creation of a rigorously solvable
but
non-free toy model, and from today's point of view conformally invariant
theories in two dimensions were by far the best thing to be tried.
Indeed,
there was a little chance. Einstein had thought about conformal
invariance,
and Dirac got interested in 1936. At that time, Heisenberg just had
started to
work on quantum field theories with four-fermion interactions. These can
be made
conformally invariant in two dimensions, such that a joint effort might
have led
directly to the Thirring model and perhaps to its solution. After the
war,
G\"ursey searched for a way to make Heisenberg's four-fermion theory
conformally
invariant, in the line of thought which went back to Cunningham and
Bateman. He
didn't think about two dimensions, however, and wrote down a
four-dimensional
version with a cube root, which is impossible to quantize [G\"ursey
1956]. For
Dirac and Heisenberg, it is unlikely, too, that they considered playing
around
in two dimensions. Moreover, Dirac became increasingly discontent with
quantum
field theory as a whole.  

Many of Heisenberg's efforts were still creative and successful,
but his flirt with mathematics was over. Still, his 1932 concept of a
nucleon
with two states, put four years later in the language of SU(2)
invariance by
Cassen and Condon, had initiated the group theoretic studies which from
the 50's
onward became one of the major themes of particle physics. Here was
perhaps a
better chance for joint work with mathematicians, for which such
considerations
soon became very natural. Some physicists also looked at related
structures,
even before the experimentalists found convincing reasons to study
internal symmetry groups or gauge symmetries. 

Einstein had long decided to concentrate on classical gravity and
electromagnetism and kept away from quantum theory and the nuclear
interactions. He continued to work in the context of classical
differential
geometry and played around with five dimensions and connections with
torsion.
These efforts are not highly regarded nowadays, but they familiarized
the
physics community with the work of E. Cartan and gave much support to
the
Kaluza-Klein ideas of five-dimensional spacetime. Yang argued that
Einstein somehow was looking for the gauge theory found in 1954 by him
and
Mills [Yang 1982]. Indeed, Einstein repeatedly contemplated parallel
transport
without the metric constraint of the Levi-Civita connection.

In the five-dimensional line of research,
O. Klein himself performed an amazing miracle by writing down the
Lagrangian of
SU(2) gauge theory during a 1938 conference in Warsaw. He only
recognized
the U(1) part of the symmetry and saw no clear physical applications,
since
charged vector mesons had not been found yet [Klein 1939, p. 93].

In 1953, Pauli rediscovered the same SU(2) gauge theory in a
conceptually
clearer way, when he pushed the Kaluza-Klein ideas one dimension higher
and
compactified two dimensions on $S^2$, such that the SO(3) symmetry
became
manifest. Pauli liked the result and described it in a letter to Pais.
He did
not publish, however, because he saw no mechanism to give mass to the
gauge
bosons. Together with Heisenberg he started to work on a fermionic
Lagrangian
with a four-fermion interaction, but he quickly saw that it made not
much sense.
Cut down from four to two dimensions it would have been transformed from
a
wrong unified theory to a fascinating mathematical toy. Altogether,
Pauli and
Weyl were probably the only ones of the pioneering giants who where both
close
to the mainstream and imaginative enough to push quantum field theory by
inventing, e.g., a mathematically nice conformal field theory. In more
fortunate times, Z\"urich might have witnessed such a step ahead, but it
is hard
to play in the shadow of war and persecution. 

With roots in a dominating wave of mood, the Nazi aversion against
Jewish
mathematics and physics had pervaded the German universities and the
G\"ottingen
environment was destroyed. The focus of research shifted from Europe to
the USA. \\[2ex]
{\bf Progress in the face of mathematics}

After the war, the seminal event in the further development of quantum
field theory was the Shelter Island conference. The decisive input came
from the
experimentalists, who made good use of the technology created in the
years
before. Their results implied that quantum electrodynamics had to be
taken very
seriously. Mathematicians were absent. Apparently, it had occurred to
nobody
that they might be of help.
 
It seems that progress needed a new generation: Very attentive to the
experiments, much less to new mathematical structures, conservative in
its
attachment to the old principles found in the golden age, careful and
innovative
in calculations. In Princeton, the lonely walks of Einstein and G\"odel
were
parts of a different world, faint reverberations of revolutions in a
distant
past. The young mathematicians had happy times. They developed fibre
bundles,
connections, characteristic classes, the deformation theory of complex
structures and many other nice things. They laid the groundwork for
modern
physics, and couldn't care less.

In 1947, work on interacting quantum fields started in earnest. 
Bethe explained the Lamb shift, and soon after Schwinger calculated the
anomalous magnetic moment of the electron. The calculations still were
done
with the theoretical tools of the prewar period. They started from the
quantum theory of free fields and introduced perturbations according to
the
standard rules of quantum mechanics. Since quantum mechanical
perturbation
theory makes no use of Lorentz invariance, this procedure compounded the
intrinsic difficulties. Soon after, Schwinger and Feynman developed
relativistically invariant formalisms, and comparison of quantum
electrodynamics with the experiments became very successful.
Stueckelberg
and Tomonaga had done earlier work in this direction, unfortunately with
less impact.

These calculational procedures were correct, but were derived from
wrong standard assumptions by dubious mathematical methods. Soon, the
standard
assumptions were proven to be wrong by a small group of mathematical
physicists,
whose work was based on an uncontested set of axioms. This caused some
uneasiness among the calculators, and kept the mathematical community at
a
distance. To explain what happened, we have to consider some details of
the
Heisenberg and Pauli paper of 1929. They were the first to derive the
equal time
commutators of free fields.

With respect to differentiation by space and time
coordinates, quantum fields can satisfy differential equations. In the
simplest
case, the latter are linear, as for Maxwell's equation. Such fields are
called
free, since a linear combination of two solutions describes two waves
which pass
through each other without mutual influence. In the language of today,
one
starts with the space of classical solutions of the linear
differential equation. On this space one needs a symplectic structure,
given
by a Poisson bracket, or equivalently a Heisenberg Lie algebra. In
most cases one has an invariance with respect to a time translation
group, 
the generator of which is the energy. Polarization with respect to the
sign of the energy yields the appropriate Hilbert space representation
of the
Heisenberg Lie algebra (Fock space). 

Apparently, in the 20's the symplectic structure of the space of
classical
solutions had not yet been grasped in depth. Thus the historical
procedure was
slightly more complicated and involved a special and somewhat formal
choice of
the classical observables. For free fields, it yielded canonical
commutation
relations in perfect analogy with Heisenberg's commutation relations
$$[x_i,p_j]=i\delta_{ij}\ ,$$
and the vanishing equal time commutators $[x_i,x_j]$ and $[p_i,p_j]$.
Analogously, for a free real scalar field $\phi$ satisfying the Laplace
differential equation, the equal time commutators  $[\phi(x),\phi(y)]$
and
$[\partial_t\phi(x),\partial_t\phi(y)]$ both are zero. The remaining
equal time
commutator takes the form 
$$[\phi(x),\partial_t\phi(y)]=i\delta(x-y)\ .$$ 

Note that distribution theory was not yet developed. This caused no
physical problems at all, but meant that the use of Dirac's $\delta$
had no firm mathematical base yet. One may wonder, if this
encouraged the physics community to ignore mathematical niceties.
Of course, the mathematical justification was provided in the late 40's
by
L. Schwartz, in a splendid case of interaction between the two
communities.
When this was done, one had a good description for free quantum fields
as distributions over three-dimensional space. Once a fixed field $\phi$ 
is paired 
with a test function $f$ (physicists write $\int \phi(x)f(x)d^3x$), the
result
is an element of the Heisenberg Lie algebra and acts on the Hilbert
space of
the system. For real $f$ the operator is hermitean and describes an
observable,
as usual in quantum mechanics. 

The analogy between particles and free fields given by passing from the
Kronecker function $\delta_{ij}$ to Dirac's $\delta(x-y)$ was
compelling, but
proved to be very misleading. Heisenberg's commutation relations for
$x_i,p_i$
remain valid when interactions are present. In contrast, Haag showed
that an interacting field theory cannot have canonical commutation
relations.
Indeed, interacting fields cannot even be understood as distributions
over
three-dimensional space at fixed time. Time averaging is necessary, too
[Haag 1955]. 

In a special relativistic context, this might not have come as a big
surprise.
There even was a paper by Bohr and Rosenfeld which argued that a careful
analysis of measurements implies a spacetime average [1933]. The
arguments were
clear enough and the paper was never forgotten, but its somewhat obscure
style
missed its mark on most of the new generation. 

Instead, adherence to the canonical commutation relations for quantum
fields
remained pervasive in the physics literature till recent times, inspite
of the
fact that everyone knew it was wrong. Most probably, it was much more
this
attitude than the difficulties of renormalization which made it
impossible for
mathematicians to digest the intricate and important structures of
quantum field
theory.

Historians will have to weigh this issue when the dust has settled.
Despite of
what has been said, they hardly can find a better starting point than
the
following classic quotation: "In the thirties, under the demoralizing
influence
of quantum-theoretic perturbation theory, the mathematics required of a
theoretical physicist was reduced to a rudimentary knowledge of the
Latin and
Greek alphabets." (Jost) [Streater and Wightman 1964, p. 31].  

The insistence of the physics
community on using a wrong basis for successful calculations would be
easy to
understand, if no alternative formalism had been available. Due to
Schwinger
and Dyson, this was not the case. Dyson had read much mathematics and
brought 
clarity of thinking to the muddled field. By 1949, Schwinger and Dyson
had
started to analyse quantum fields in terms of the n-point functions (or
rather
distributions) $T\langle \phi(x_1,t_1)\ldots\phi(x_n,t_n)\rangle$. 
Here for an operator $A$ the real or complex number $\langle A\rangle$
is
its expectation value in the vacuum state of the Hilbert space, and the
analogous notation applies to distributions. The time ordering imposes
the
condition $t_1>\ldots>t_n$ on the support of the test functions.
Moreover, in
1951 Schwinger published his action principle, which describes how an
n-point
function varies when one changes the parameters of the interaction.

Thus most of the theoretical tools were ready. On reading the tributes
to Schwinger published after his death [Ng 1996], it seems that some
obstacles to
progress were personal. Schwinger had been a prodigy and the centre of
attention. Apparently, he didn't mind that his calculations remained
almost
incomprehensible. All that changed after 1948. In Schwinger's
own
words: "Like the silicon chip of more recent years, the Feynman diagram
was
bringing computation to the masses" [Schwinger 1983, p. 343]. Dyson had
a
particularly clear understanding of the issues: "The advantages of the
Feynman
theory are simplicity and ease of application, while those of
Tomonaga-Schwinger
are generality and theoretical completeness" [Dyson 1949, p. 486].
Schwinger forbade his students to mention Feynman or Dyson, or to use
Feynman
graphs. From a European perspective it seems that Einstein and Weyl
would have
had more reasons for grudges against Hilbert and Schr\"odinger, but one
has to
respect a difference of culture. 

In 1953, the Wightman axioms [Streater, Wightman 1964] were presented in
lectures at Princeton. They were something of a mixed blessing. On one
hand,
they allowed clear proofs of structural statements, in particular of
Haag's 
insight that the canonical commutation relations are wrong for
interacting
theories [Haag 1955]. On the other hand, the axioms sacrificed the
connection
to the concrete quantum field theories which were under development.

One technical detail needs comment. The Wightman axioms concern n-point
distributions $\langle \phi(x_1,t_1)\ldots\phi(x_n,t_n)\rangle$, but
without
time ordering. This seems mathematically convenient, for example when
one wants
to take Fourier transforms. Nevertheless, for contact with the
experiments, the
time ordering is natural. This became particularly clear with the LSZ
formalism of Lehmann, Symanzik and Zimmermann, which provided a direct
calculation of the results of scattering experiments in terms of the
time
ordered distributions. Different time orderings correspond to different
experiments. 

The three authors were members of Heisenberg's group, which attracted
most of
the young people who wanted to work on elementary particles in postwar
Germany.
Unfortunately, Heisenberg was hardly interested in mathematics and too
occupied
by his world formula to have much regard for the LSZ achievements. When
Lehmann
returned from the States, Heisenberg greeted him: "Na, Herr Lehmann, wie
geht's
der Mathematik?" (how is mathematics?), an episode which Lehmann never
forgot.
So much for the superiority of European culture. 

As an aside, any Third World country which wants to strengthen her
scientific basis would be well advised to do a few case studies.
The decline of physics in Germany is particularly interesting. One
cannot put
all of the blame on fascism, since mathematics did not suffer the same
fate
after the war, largely due to the achievements of Hirzebruch.  

The n-point distributions made mathematical sense, but were difficult to
deal
with. The next big advance was the introduction of the euclidean
formalism, as
discussed in [Osterwalder 1973]. Early on, Dyson had recognized that
some
calculations become much easier when one performs an analytic
continuation to
imaginary values of time (Wick rotation). The gestation of the idea took
most of
the 1950's, with contributions from Wick, Nakano and, in condensed
matter
physics, Matsubara [1955]. It first appears in complete form in papers
of
Schwinger. 
 
In his 1993 lecture in Nottingham [Ng 1996], Schwinger states that it
could have
been published any time after 1951, but in fact "The Euclidean Structure
of
Relativistic Field Theory" appeared in 1958. Schwinger made an analytic
continuation of the time-ordered n-point distributions to purely
imaginary values
of time. As Wightman had seen already, the analytic continuation
allows to
consider the distributions as boundary values of ordinary analytic
functions.
Thus Schwinger's idea allows to describe physics by functions of some
$D$-dimensional euclidean space instead of distributions with
testfunctions over
$D$-dimensional spacetime.  By that time, mathematical physicists had
mastered
the difficulties of distribution theory, such that the due expression of
relief
was rather muted.  Often, the euclidean n-point functions are regarded
as
distributions, too, but the present article will not follow this
habit.

As usual nowadays, Schwinger's euclidean n-point functions will just be
written
in the form $\langle \phi(x_1)\ldots\phi(x_n)\rangle$, where the $x_i$
now
denote points in $D$-dimensional euclidean space. These functions are
real
analytic and defined everywhere except on the partial diagonals
$x_i=x_j$.
Since there is no causal structure in euclidean space, the necessity of
time
ordering disappears. Accordingly, the functions are symmetric under
permutation
of the $x_i$. If one considers several fields $\phi_1,\phi_2,\ldots$,
one has
instead 
$$\langle A_1 \phi_i(x_i)\phi_{i+1}(x_{i+1}) A_2\rangle=
\langle A_1 \phi_{i+1}(x_{i+1})\phi_i(x_i) A_2\rangle\ ,$$
where the $A_k$ stand for products of fields at points different from
$x_i,x_{i+1}$. In spacetime, all possible time orderings can be reached
by
analytic continuations starting from the same euclidean  n-point
function, a
fact called crossing symmetry. 

Since the choice of quantum field theories is quite limited, their
n-point
functions should be special functions with very interesting properties.
Not much is known about them, however. For free theories, they vanish
unless
$n$ is even, in which case they reduce to sums of products over 2-point
functions. The latter are variants of Bessel functions. For conformal
field
theories, one obtains functions of hypergeometric type. In some other
cases in
two dimensions, at least the 2-point functions are under good numerical
control,
but little is known about their analytic properties. It is quite
possible that
some examples will yield functions of Painlev\'e type. Unfortunately,
interest
in special functions was at a low ebb in the past century, but this
certainly
will change again. 

Most quantum field theories have free parameters. The latter take values
in some
differentiable manifold which is called moduli space. Accordingly, the
n-point
functions can be differentiated with respect to these parameters. Let
$\partial_\lambda$ be a tangent vector in moduli space.
According to Schwinger's action principle, each tangent vector
corresponds to
some field $t(x)$, such that formally
$$\partial_\lambda \langle \phi(x_1)\ldots\phi(x_n)\rangle=
\int \langle t(x)\phi(x_1)\ldots\phi(x_n)\rangle d^Dx\ .$$
The expression is formal, since the integral diverges when $x$
approaches one
of the $x_i$ and needs to be regularized.

In general, there is no easy way to normalize the field $\phi$. Of
course, the
canonical commutation relations would have provided a natural
normalization,
but they are wrong. When one changes the normalization by some factor
$f(\lambda)$, the derivative of the n-point function changes by a term
proportional to $n\langle \phi(x_1)\ldots\phi(x_n)\rangle$. If the
divergence
of Schwinger's integral is of exactly this type, the freedom of
normalization
can be used to cancel it. This is the renormalization procedure, which
will be
discussed in more generality below.

In principle, vector fields can be integrated, such that Schwinger's
action
principle should allow to recover the moduli space from any of its
regular
points by higher order derivatives and the summation of the Taylor
expansion.
In many practical cases, however, the only explicitly known points of
the moduli
space lie at the boundary, where the space is no longer regular. As a
consequence, the Taylor expansion is only asymptotic. This problem can
be
avoided for conformal field theories, but it will be mentioned
again in the context of string theory. 

Many moduli spaces do not have a natural metric, such that the
integration of a
vector field has to follow an arbitrary smooth curve. Equivalently, one
can
choose local coordinates, also known as renormalization scheme.
Indeed,
without a metric on moduli space, the perturbing field $t(x)$ does not
have a
natural normalization. Typically it lives in some finite dimensional
vector
bundle over moduli space which includes mass perturbations and coupling
constant
perturbations. When one takes higher order derivatives of the n-point
functions,
all of these parameters have to be considered together, which requires
mass and
coupling constant renormalizations of $t(x)$. The finite ambiguities of
the
latter are fixed by the choice of a renormalization scheme. Changing
them leads
to a different curve for the integration. 

If one wants, one can include the constant field 1 in the vector
bundle, but since one wants $\langle 1\rangle=1$ it is usually more
convenient
to require $\langle t(x)\rangle=0$. This is called the renormalization
of the
vacuum energy density.

In the 50's, renormalization was well understood on a computational
level, but before Wilson's work in the late 60's the concepts were not
particularly clear. Nevertheless, the time was ripe for the first
quantum field
theory which was not free and made complete mathematical sense.
\\[2ex]
{\bf The Thirring model: Conformally invariant quantum field theory is
born}

In 1958, W. Thirring published a paper with the title 'A Soluble
Relativistic
Field Theory' (in Mathematical Reviews, it was described by
Raychaudhury,
Calcutta). The paper kept the promise of its title. Let me quote a few
sentences: 'In spite of the great efforts of many people the
mathematical
structure of relativistic quantum fields is still in the dark. ... In
order to
study those (features) we propose in the present paper a model of a
relativistic
field theory... Since the reduction of the number of fields does not
simplify
the problem sufficiently ... one has to take recourse to a reduction of
the
dimensionality of the problem... Thus the simplest nontrivial case seems
to be a
one-dimensional Fermi-field with an interaction
$\lambda\bar\psi\psi\bar\psi\psi$. Although the problem is of
considerable
complexity it turns out to be soluble. ... (The model) shows explicitly
what a
relativistic theory can look like. Furthermore it can serve as a testing
ground
for field theorists."  

All of this is true. Perhaps the most remarkable part is the courage to
do
something simple in two dimensions. Here Thirring was inspired by the
investigation of many-body systems in terms of the Bethe ansatz. In two
dimensions, one can get interactions by collisions only, without fields.
This
knowledge led to the correct conjecture that the model would be
solvable.
Thirring also made some entirely correct remarks about Heisenberg's
unified
four-fermion interaction theory in four-dimensional spacetime, which may
have
contributed to some tension between Munich and Vienna. Indeed, despite
of the
fact that part of  Thirring's work had been done at MIT and at the IAS,
Princeton, one almost gets the impression that the creation of the model
was a
provincial non-event. The leading soluble model of the time was due to
Lee
(1954) and not  relativistic. Thirring's remark about the Lee model in
his 1958
paper is not particularly deferential, but in his textbook with Henley
[1962] 
he gives it two chapters, whereas his own model does not even seem to be
hinted
at. Schweber's 1964 textbook doesn't cite it either.  

Nevertheless, some of Schwinger's former students had paid attention,
and
Johnson from MIT devoted a paper to the model [1961]. I quote from the
introduction: "Thirring has proposed a two dimensional ... model which
is of
some interest because its exact solubility enables one to study some of
the
general conjectures which have been proposed in regard to the behaviour
of local
relativistic fields. In spite of the model, no general solutions have
been
proposed which are free from possible criticism because of the rather
formal
manner in which they have been obtained." In the conclusion, Johnson
states: 
"We have shown how it is possible to solve the two dimensional model of
Thirring
by making use of the existence of the two vector density conservation
laws. ...
It was shown how it is possible to define the products of the singular
operators
$\psi(x)$, in order to determine other covariant operators but that
these
singular field products do not satisfy the equal time commutation
relations with
the field operators $\psi(x)$, that one would obtain by means of the
canonical
commutation relations ...". Again, all of this is correct. Still, some
mathematical problems were left, but they were settled in the subsequent
years.

Let us describe the model in more detail. It is obtained by perturbing
the theory
of a massless complex fermion in two dimensions. In the euclidean
formulation, the Dirac equation reduces to the Cauchy-Riemann equation
and its
complex conjugate. Real and imaginary parts of the fermion yield two
holomorphic
field $\psi_i(z)$ and two anti-holomorphic fields $\bar\psi_i(\bar z)$,
$i=1,2$.
At this point in moduli space, the two conserved vector densities
mentioned by
Johnson are $j(z)=\psi_1(z)\psi_2(z)$ and 
$\bar j(\bar z)= \bar\psi_1(\bar z)\bar\psi_2(\bar z)$. 
The conservation equations are the Cauchy-Riemann equation for $j$ and
its
conjugate for $\bar j$. The 2-point functions have the form
$$\langle j(z_1)j(z_2)\rangle =(z_1-z_2)^{-2}$$
and analogously for $\langle \bar j\bar j\rangle$, whereas
$\langle j\bar j\rangle=0$.

In terms of Schwinger's action principle, the perturbation corresponds
to 
the field $t=j\bar j$. It turns out that the n-point functions of
$j$ and $\bar j$ are unaffected by the perturbation. In particular, the
two
fields and their product $t$ have a natural continuation over Thirring's
moduli space and need no renormalization. Moreover, the conservation
equations
do not change, which accounts for the solvability of the model.

The special properties of $j,\bar j$ arise because they are currents,
i.e.
quantum analogues of the conserved densities which arise by Noether's
theorem
from continuous symmetries. Because of their close relation to
observable
quantities, they behave similarly to free fields. This led to the
concept of
current algebra. In two dimensional theories, the currents of simple Lie
groups
generate the corresponding affine Kac-Moody algebra, at least when
space is
compactified to a circle. Unfortunately, the mathematical potential of
current
algebras was not realized for many years. The work of Kac and Moody in
1967 was
independent of physics. In the context of string theory, it was
introduced in
the physics literature by the mathematicians Lepowsky and Wilson [1978]
and
again by G. Segal [1981], and became a rare success story of physics and
mathematics in cooperation. 

The Thirring model fields $\psi_i$ do not remain holomorphic under the
perturbation by $j\bar j$. Instead, one obtains
$$\langle \psi_i(z_1)\psi_j(z_2)\rangle =
(z_1-z_2)^{-1}|z_1-z_2|^{-s} \delta_{ij}\ ,$$
where the real number $s$ changes under perturbation. Under the
conformal
transformation $z\mapsto z'=(az+b)/(cz+d)$ with $ad-bc=1$,
$\psi(z)\mapsto
(cz+d)^{-1}|cz+d|^{-s}\psi(z')$, the two-point functions remain
invariant. This
remains true for all the n-point functions, such that the Thirring model
is a
conformally invariant theory. Initially, this seems to have been
overlooked,
and only the special case of invariance under scale transformations
$z\mapsto \lambda z$ was commented upon. This is a bit surprising, since
in
these years Thirring was very much concerned with conformal invariance.
In the
important 1962 paper where Gell-Mann introduced current algebra to the
theory of
the strong interactions, he acknowledges that Thirring introduced him to
the
conformal group. Moreover, conformal invariance had become an issue
between
Munich and Vienna. There was little internal logic in this local
turbulence, but
it turned out to be important and may be of interest to historically
inclined
people. 

Heisenberg had developed an interacting spinor theory for all of
particle
physics and pushed it for many years, though it made no sense. At the
time, the
new quantum number of strangeness demanded an explanation. Due to
Noether,
an invariance of the theory had to be found. Heisenberg tried scale
invariance,
though the theory has a length scale and a non-compact group has a hard
time to
yield discrete quantum numbers. The contemporary fashion for negative
norm
states, also present in the Lee model, gave some hope for a cure [D\"urr
1959].

In Vienna, Cunningham and Bateman were remembered and Wess used
Heisenberg's
attempts as justification for the resurrection of the conformal group.
In a
brief remark, he hinted at a possible use of the conformal group at high
energies. Otherwise, he showed in a few pages that Heisenberg had missed
the
mark [Wess 1960]. Since several of the few good young German
theoreticians had
flocked around Heisenberg, the paper triggered new interest in the
conformal
group, and Kastrup started to work on it, though Heisenberg did not pay
much
attention. Kastrup published papers on the possible importance of
conformal
invariance at high energies. During a visit to Russia, he explained it
to
Polyakov, as acknowledged in the first paper of the latter on conformal
symmetry
[1970]. This paper showed that scale invariance implies full conformal
invariance. On the other side of the Atlantic, in his historic paper on
the
short distance expansion, Wilson ascribes the idea of scale invariance
at short
distance to Kastrup and his student Mack [Wilson 1969]. The fact that
scale
invariance implies full conformal invariance was recognized by Callan,
Coleman
and Jackiw, slightly before Polyakov's work and in a different context
[1969].
On the physical relevance of scale and conformal invariance, they cite
1969
papers by Mack and Salam and by Gross and Wess. 

Wilson's short distance expansion was the main concept which still was
lacking
for a rigorous and calculationally efficient description of quantum
field
theory.  It concerns the behaviour of the n-point functions along there
singularities. Wilson considered them in Minkowskian spacetime, but the
euclidean case is much easier.

It has been mentioned that the euclidean n-point functions  $\langle
\phi(x_1)\ldots\phi(x_n)\rangle$ are not well defined on the partial
diagonal
$x_i=x_j$. In general, the functions diverge on these diagonals. For a
free
field $\phi$ of dimension $h$, the leading term at $x_1=x_2$ is
proportional to 
$|x_1-x_2|^{-2h}\langle \phi(x_3)\ldots\phi(x_n)\rangle$. The case of
several
different fields needs a bit more discussion, but is not complicated
either. 
 
There had been some speculation on the corresponding behaviour for
interacting
fields. One idea was that the singularity might be the same as for free
fields.
In 1964 Wilson conjectured that perturbations just introduce some
logarithmic
corrections. This was wrong, but one of Wilson's talents was to talk to
the
right people for correcting mistakes. In particular, he had crucial
discussions
with Johnson, who familiarized him with the Thirring model. Wilson
learned that
the latter indeed is scale invariant, but that the dimension $h$ changes
with
the strength of the interaction. Independently, the same modification to
Wilson's original ideas was made by Lowenstein. 

Wilson was a mainstream theorist on the way to a Nobel prize, but he did
not
fear to go against the tide: "The assumption that integrating
an operator over space only gives an observable is a basic tenet of
canonical
field theory... The assumption has been rejected by axiomatic field
theory from
the beginning" [Wilson 1970, p. 1484]. In the same paper, he discusses a
related
issue and concludes: "The axiomatic view must in the end replace the
popular
view" [p. 1483]. It seems that the time was ripe to discuss all of
quantum field
theory in terms of statements which are at least potentially true.

Before we discuss other aspects of Wilson's work, let us
continue the history of the Thirring model. At the end of the 60's,
string
theory was invented and soon it was recognized that conformal field
theory
is an essential ingredient [Galli 1970]. Halpern recognized the
importance of
the Thirring model in this context and informed Virasoro, who gave it
publicity
[1971]. A comparative investigation of the Thirring model and string
physics in
the context of conformal field theory was made by Ferrara, Grillo and
Gatto
[1972]. 

By 1974, it had become popular to elucidate the properties of quantum
field
theory by a study of two-dimensional examples. A particularly
interesting one
was the sine-Gordon model, which describes a bosonic scalar field with
trigonometric interaction term. Coleman wrote an elegant and deep paper
where he showed that the perturbation by a fermion mass term makes the
Thirring
model isomorphic to the sine-Gordon model [1975]. This took everyone by
surprise, since superficially the two models look entirely different and
equally
impenetrable in a strict mathematical sense.

On hindsight, people remembered that the equivalence between fermions
and
bosons in two dimensions had been prefigured by Skyrme [1958,1961], but
Skyrme
had been too far ahead to have an immediate impact. 

After Coleman's paper, at last, one leading mathematician was shocked
enough to
take things seriously. G. Segal regarded the mass term as an unessential
complication and concentrated on the boson-fermion equivalence. This was
Coleman's starting point and concerns an isomorphism between two
conformally
invariant theories. Initially, Segal felt quite sure that boson-fermion
equivalence made no sense. When it turned out in the late 70's that the
equivalence leads to a combinatorial identity known already to Euler, a
dam had
been broken. Segal developed a beautiful system of axioms for
conformally
invariant quantum field theories in two dimensions and transformed the
latter
into a legitimate field of study for mathematicians [Segal 1988]. But
even in
their book on loop groups [Pressley, Segal 1986, p. 215] the authors
state 
that a mathematically clear formulation of the isomorphism between the
massive
Thirring model and the sine-Gordon model still seems not to have been
found.
\\[2ex]
{\bf Nature's helping hand}

The long delay in the gestation of a correct theory of quantum fields
would
have been even longer without some direct help from nature. 
One reason is that the investigation of two-dimensional toy models was
not
taken very seriously by the particle physicists. Here is a quotation
from
a paper which reports the discovery of a fundamental
property of the Thirring model: "The results are of interest ... because
they
allow one to see very readily (a) why the Thirring model is solvable
and (b)
why it has trivial physical consequences. As will be clear from the
following,
the solvability of this model depends critically on the fact that it is
a 2-dimensional model. It is not likely that any of the specific 
features of this
model can be generalized to more realistic cases, or that they will
provide a
useful guide to the state of affairs in the real world"
[Callan, Dashen and Sharp 1968, p. 1883].
 
Indeed, the highly non-trivial physical consequences of such conformal
field
theories in the context of string theory could not have been guessed in
1967.
No wonder that the authors permitted themselves some sloppiness in the
analysis:
"At this point, one could introduce the Fock representation for the
scalar
field, annihilation and creation operators, etc., and verify in detail
that the
energy and momentum operators have the expected properties, but there is
little
to be gained by going over these well-known details" [p. 1885]. This was
a
missed opportunity. For example, the commutation relations for the
energy-momentum tensor given in the paper miss the central extension of
what is
now called the Virasoro algebra. What would have happened, if some
interested
mathematics student had tried to digest the paper?  

Since it seems that no mathematicians were interested, it was very kind
of
nature to provide her own motivation for the study of such models. In
the 50's,
physicists were confronted unexpectedly with a rich class of quantum
field
theory in condensed matter laboratories, which turned out to be
conformal field
theories in the real world of two-dimensional surface coatings or three
dimensional liquids. 

After Feynman's breakthrough in 1948, his graph methods soon were
transferred to other fields of physics. Their application in condensed
matter
physics was pioneered by Salam [1953] and Matsubara [1955]. In
particular,
Matsubara recognized the perfect analogy of imaginary time and
temperature,
due to the relation between the time translation $\exp(iHt)$ in quantum
mechanics and the Boltzmann factor $\exp(-H/T)$ in statistical
mechanics.

When continuous phase transitions were studied, it turned out
that the analogies went much deeper. At the critical
temperatures, the behaviour of the materials is dominated by long range
fluctuations of arbitrary scales, and the details of the molecular
structure
become unimportant. The theory approaches a continuum limit. The
correlation
functions of the limiting theory behave exactly like the euclidean
n-point
functions of quantum field theory. In this way, many statistical systems
at
continuous phase transitions are related to quantum field theories in
spacetime by analytic continuation. 

Thus nature herself had declared that the Wick rotation introduced by
Schwinger
makes good sense. Of course, the dimensions of the observed examples are
different, since the phase transitions happen in two or three
dimensional
systems, whereas spacetime has four dimensions. Moreover, the natural
constraints on the field theories are not the same.
Quantum field theories need a probability interpretation, which is
realized by
positive scalar products. Under Wick rotation, this becomes
Osterwalder-Schrader positivity, which is not a necessary property of
phase
transitions. On the other hand, statistical observables are given by
real
numbers. This real structure yields a time reversal invariance of the
corresponding quantum field theory, a property not shared by all
examples and
only approximately true in nature. On a purely mathematical level, these
difficulties are not particularly serious, however. 

Lab experiments on phase transitions were much cheaper than particle
physics
with high energy accelerators. Moreover, there were no worries that a
breakthrough in the domain of the fundamental laws was necessary. Thus
progress
was rather steady, both on the experimental and the theoretical side.
Soon
it became clear that the physics at the critical phase transition
point is scale invariant [Kadanoff 1966]. Much of the relevant work on
these
euclidean quantum field theories was done in the Soviet Union, and
Polyakov was
one of the most important contributors. He found convincing arguments
that scale invariance implies full conformal invariance at the critical
point
and recognized that this invariance allowed a calculation of the 3-point
functions up to a constant factor [Polyakov 1970].  

Further developments depended on the analysis of a soluble example in
the 
context of statistical mechanics. This might have been provided by the
Thirring
model, which had occurred in its bosonic description, and was
called the gaussian model. Because of the somewhat misleading simplicity
of the
bosonic formulation, the subtle features of its fermionic fields
were not
recognized in this context. Instead, the Ising model played a r\^ole for
the
study of continuous phase transitions which was parallel to the one of
the
Thirring model for particle physicists.   

The states of the Ising model put a number 1 or -1 to each site of a
rectangular 
lattice. The latter are called values of the Ising spin. Pairs of
nearest
neighbours have an interaction energy which depends on the product of
their
Ising spins. The total energy $E$ is given by a sum over the interaction
energies of such pairs. The thermodynamic partition functions at
temperature
$T$ is given by the average of $\exp(-E/T)$ over all states. 

Rectangular lattices can be considered in various dimensions. The
thermodynamic
functions for the linear or one dimensional model are very easy to
calculate.
The problem was given by Lenz as part of a PhD thesis to a rather weak
student,
who did not do any later scientific work. One hardly can imagine an
easier way
to lasting fame. The two-dimensional model, where the Ising spins sit on
a square lattice, was solvable but very hard. The breakthrough 
calculation was due
to Onsager [1944]. There is a critical temperature where the model turns
into a
rather simple euclidean quantum field theory. In particular, at this
point the
spin waves of the model satisfy the two-dimensional Dirac equation
for free
massless fermions, as first noted by Kadanoff [1969]. This equation is
conformally invariant, as in the more complicated four-dimensional
situation. In
contrast to the complex fermion of the Thirring model, the fermion field
of the
Ising model is real. In this sense, the Ising model at its critical
temperature
has half as many degrees of freedom as the Thirring model. 

The two-dimensional Ising model is not just a theory of free fermions,
however.
The average values of the Ising spins turn into a field with scaling
dimension
1/8. This result proved to be a highly non-trivial check which uncovered
the
failures of many calculational methods.

Now two different conformally invariant quantum field theories were
available,
the Ising model in statistical mechanics and the Thirring model in
conventional
relativistic quantum field theory. They were used for very much the same
theoretical tools, in particular the short distance expansion. Wilson
discovered it in 1964 in in the Minkowskian context, Polyakov and
Kadanoff in
1969 in the euclidean. Polyakov called it correlation coalescence,
Kadanoff
reduction hypothesis. Wilson called it operator product expansion, and
this
terminology has survived, because it clearly has the priority. In the
context 
of statistical mechanics it is not appropriate, however, since there are
no
operators around. Since it has advantages to have a unique name in both
contexts, we use the common synonym short distance expansion.

A scale invariant n-point function of type
 $\langle\phi(x)\phi(y)A\rangle$ has a leading
singularity at $x=y$ proportional to $|x-y|^{-2h}$, where $h$ is the
scaling dimension of $\phi$. When this leading singularity is
subtracted, the
next term behaves  like $|x-y|^{-2h+h_1}\langle\chi_1(y)A\rangle$. Here
$\chi_1$
is some other field of scaling dimension $h_1>0$, which can be measured
in the
way just described. Subtracting this subleading term one finds 
$|x-y|^{-2h+h_2}\langle\chi_2(y)A\rangle$, where $\chi_2$ now has a
larger scaling dimension $h_2>h_1$. The procedure can be repeated as far
as one
wants to go. One will find an infinity of fields of ever higher scaling
dimension. Note that the $\chi_i$ are independent of the fields included
in $A$. 

One can apply the same procedure to other n-point functions like
$\langle \phi(x_1) \linebreak
\chi_1(x_2) \ldots\rangle$ and so on and produce as many new
fields as
possible. The short distance expansion now states that for any real
number $h_0$ there is only a finite number of linearly independent
fields of
scaling dimensions $\leq h_0$. This property can be verified in many
concrete
examples and may very well be taken as part of the mathematical
definition of a
quantum field theory. 

Lattice systems are scale invariant at the exact temperature of 
a continuous phase transition.
When the temperature is changed a bit, the correlations will show 
an exponential decay at large distances. When one is sufficiently close
to the
critical temperature, the corresponding correlation length is still very
large
compared to the distance between neighbours. With a suitable limiting
procedure,
one obtains the n-point functions of a quantum field theory which is no
longer
conformally invariant. In this case, more complicated expressions than
$|x-y|^{-2h}$ will occur in the n-point functions. At the very least,
one
expects logarithmic correction factors. Nevertheless, the basic idea of
the
short distance expansion applies as before. 

Let us consider a euclidean n-point function $\langle
\phi(x)\chi(y)A\rangle$,
where $A$ is a product of local fields at positions different from
$x,y$. An
experimentalist may study the behaviour of this function when $x$
approaches
$y$. Each such measurement can be interpreted as the measurement of some
field
at $y$. This is the physical content of the short distance expansion.
We can axiomatize it in the following way. Let $\Gamma(y)$ be the vector
space
of germs of functions which are defined near $y$, but not at the point
$y$
itself. We give a topology to this space by using $o(|x-y|^s)$, $s\in
{\bf R}$
as a basis of neighborhoods of 0 in $\Gamma(y)$. Let $\gamma$ be an
element of
the dual of $\Gamma(y)$. Then for each pair of fields $\phi,\chi$ and
each $h$
there must be a field $\psi$ such that 
$\gamma\langle \phi(x)\chi(y)A\rangle=\langle \psi(y)A\rangle$
for arbitrary $A$. One just can write $\gamma(\phi(x),\chi(y))=\psi(y)$.

Consider the vector space $F$ of all fields of a quantum field theory.
This
vector space is filtered by the scaling dimension. Let $F(h)$ be the
subspace of
all fields of scaling dimension less or equal to $h$. We assume that
these
subspaces are finite dimensional. We also assume that the theory has
some degree
of asymptotic scale invariance. More precisely, $\psi\in F(h_1+h_2+h_0)$
when
$\phi\in F(h_1)$, $\chi\in F(h_2)$ and $\gamma$ vanishes on $o(|x-y|^h)$
for
$h>h_0$.
This condition will be important for renormalizability.
Finally, $\dim F(h)$ should not increase faster than for free
theories. In two dimensions, this yields $\log(\dim F(h))=O(\sqrt h)$.

In this way one obtains a nice algebraic structure which is well adapted
to
calculational purposes. It does not contradict the Wightman axioms, but
emphasizes quite different aspects. Whereas those axioms concentrate
on one field, or maybe a few, the short distance expansion considers all
possible fields at once. For mathematicians, this is certainly the more
natural
procedure. To some extent, it eliminates the surprise one first feels
about
the equivalence of the sine-Gordon and the massive Thirring model, since
in the
latter one immediately has to include its bosonic fields, too.
\\[2ex]
{\bf Regularization and renormalization}


With the help of the short distance expansion, it is rather easy to put
renormalization in a standard mathematical frame. First we have to
generalize
the change of normalization of the fields which we considered above.
Instead, we will use all the linear transformations of $F$ which
conserve the subspaces $F(h)$. The group of these linear
transformations will be called $L(F)$.

We want to regard a perturbation of some theory. In accordance with
Schwinger's
action principle, the deformation is described by a field $t(x)$. 
We shall see that in a spacetime of $D$ dimensions, the scaling
dimension of
$t$ must be $D$ or less.

The corresponding derivative of an n-point function $\langle
\phi_1(x_1)\ldots\phi_n(x_n)\rangle$ is given by
$\int d^k x \langle t(x)\phi_1(x_1)\ldots\phi_n(x_n)\rangle$.
The integral behaves well at infinity, but diverges when $x$ approaches
one
of the $x_i$. Thus we regularize it by excluding a small neighborhood of
size
$\epsilon$ around each $x_i$ from the integration domain. Let us denote
the
resulting integral by $\int_\epsilon$.

The idea of renormalization means that the divergence can be absorbed by
a
redefinition of the fields. Such a redefinition is given by a linear
transformation in $L(F)$ of the fields which maps every
subspace $F(h)$ into itself. Using Wilson's short distance expansion,
one sees
easily that there are transformations $f(\epsilon)\in L(F)$ such that
$$\int_\epsilon d^D x \langle t(x)\phi_1(x_1)\ldots\phi_n(x_n)\rangle
-\sum_{i=0}^n\langle \phi_1(x_1)\ldots
(f(\epsilon)\phi_i)(x_i)\ldots\rangle
$$
has a well defined limit when $\epsilon$ goes to zero. Indeed, any
divergent
contribution $\gamma$ to the integral near $x_i$ vanishes when the
n-point
function behaves as $o(|x-x_i|^{-D})$, such that $\gamma(t(x)\phi)\in
F(h_i)$,
when $h_i$ is the scaling dimension of $\phi_i$.

The transformation
$f(\epsilon)$ is only defined up to addition of a finite linear
transformation in $L(F)$. Any choice defines a connection on the
filtered
vector bundle $F$ over the moduli space. Altogether, we now have well
defined
first derivatives in the moduli space of a quantum field theory. The
calculation
gets harder when one looks at higher derivatives, since the perturbing
field
$t(x)$ will have to be renormalized, too, but this is just a technical
difficulty. 

As one sees, renormalization is nothing particularly problematic. On the
contrary, regularization of divergencies has a long history in
mathematics. For
example, the Weierstrass product formula for entire function needs the
regularization of an infinite product. 
Let us consider it in more detail.
One wants a product formula for an entire holomorphic function $P(z)$
with 
zeros exactly at given positions $z_i$,  $i=1,2,\ldots$,
more precisely a function with $\sum (z_i)$ as zero divisor. 
The sequence $z_i$ must have no accumulation point in the Gauss plane.
When the number of zero positions is finite, the product $\prod (z-z_i)$
will
do. The most general function with this divisor is $\exp(f(z))\prod
(z-z_i)$,
where $f(z)$ is an arbitrary entire function. 

Now let us consider the case of an infinite number of positions.
Factoring out a power of $z$ if necessary, we may assume that none of
the $z_i$
is zero. Let us formulate Weierstrass' solution in terms of the language
of
quantum field theory. We regularize the problem by restricting the set
of zeros
to $z_i$,  $i=1,\ldots,N$. Then we order the $z_i$ in accordance with
their
absolute value and renormalize the function $\prod_{i=1}^N (z-z_i)$ in
the form 
$$P_N(z)=\exp(f_N(z))\prod_{i=1}^N (z-z_i)\ ,$$
such that the limit
$\lim_{N\rightarrow\infty}P_N$ is finite.

The situation in quantum field theory is quite analogous. The cut-off
by $\epsilon$ is analogous to the cut-off by $N$, the achievement of
convergence by the renormalization transformation $f(\epsilon)$ is
analogous
to the multiplication by $\exp(f_N)$. In renormalizable quantum field
theories,
fixing a finite number of parameters is sufficient to determine the
n-point
functions of a given finite set of fields. In the case of the
Weierstrass
products, this is analogous to the situation where it is sufficient to
take for
the $f_N$ polynomials of fixed order $r$. In this case, one can
normalize
$P$ by demanding that $P(0)$ and the first $r$ derivatives of $P$ at
$z=0$ have
prescribed values. This means that the solution $P$ has $r+1$ free
parameters.
For $r=0$, the solution is
$$P(z)=P(0) \lim_{N\rightarrow\infty}\prod_{i=1}^N (1-z/z_i)\ .$$ 
For $r=1$ one obtains
$$P(z)=P(0) \exp(zP'(0)/P(0)) \lim_{N\rightarrow\infty}\prod_{i=1}^N
(1-z/z_i)\exp(z/z_i)$$
and so on.

When polynomials do not suffice, the number of free parameters becomes
infinite. Quantum field theory is simpler, since the latter case does 
not seem to
have an analogue. Moreover, quantum field theories are far more
constrained than
entire functions, since they only have a finite number of parameters, in
contrast to the infinite set of the $z_i$. 

For conformal field theories, the Weierstrass
product formula is more than a far-fetched analogue, since many
correlation
functions involve Jacobi's theta-functions or Dedekind's
$\eta$-function.
Examples will be given below. Many important properties of these
functions are
best understood by their product formulas.

As one sees, regularization and renormalization are perfectly standard
mathematical procedures. Their unfamiliar context was bound to cause
some delay
in understanding, but it is hard to comprehend how a delay of many
decades could come about.
\\[2ex]
{\bf The structure of conformally invariant theories}

One important way to deform a quantum field theory has not been
introduced so
far. One can change all n-point functions by a simple rescaling of the
distances. When this change can be compensated by a transformation in
$L(F)$,
the theory is called scale invariant. More generally, the change is
equivalent
to such a transformation in addition to a change of the parameters of
the
theory. Infinitesimally, this equivalence is expressed by the
Callan-Symanzik
equation. 

When a deformation should respect some symmetry, the corresponding field
$t(x)$
must be invariant under the symmetry group. In particular, this is true
for
Lorentz invariance. Indeed, our formalism does not require Lorentz
invariance
and can easily be adapted to quantum field theories on general
spacetimes.
One just has to replace the vector space $F$ of fields by a bundle over
spacetime. Let us conserve translational invariance, however, such that
fields
can be transported in canonical ways between arbitrary points of
spacetime.
When some component $g_{\mu\nu}$ of the Riemannian metric is changed
in a translationally invariant way, the corresponding field $t$ is the
component $T^{\mu\nu}$ of the energy momentum tensor. For a rescaling of
the
distances, this yields $t=T^\mu_\mu$. For a scale invariant theory
this means that the trace of the energy momentum tensor vanishes.
Moreover, the integral $\int T^{\mu\nu} d^Dx$ must not depend on the
distance
scale, which means that the scaling dimension of the energy momentum
tensor is
equal to $D$.
 
Scale invariant quantum field theories are conformally invariant, too.
This
implies that the three point functions are known explicitly. The
four-point
functions reduce to functions of a single variable. Such theories have a
good
chance to be solvable in a rather explicit form, but for theories in
more than
two dimensions, the situation is still rather unclear. Nevertheless,
recent
developments indicate that these theories are important, too [Maldacena
1998,
Witten 1998]. Suppose that you have a quantum field theory in $k$
dimensional
Minkowski space which admits a deformation to the corresponding
Anti-de-Sitter
space. Recall that this is a homogeneous space of negative spatial
curvature,
with symmetry group $SO(k-1,2)$. Anti-de-Sitter space has a
$(k-1)$-dimensional
boundary at infinity with a conformal structure, on which $SO(k-1,2)$
acts as
the group of conformal transformations. When one takes suitable limits
of the
n-point functions, the theory in Anti-de-Sitter space reduces to a
conformally
invariant theory in a space of one lower dimension. In principle, the
higher
dimensional theory can be recovered from the boundary theory by
techniques of algebraic quantum field theory [Rehren 1999]. 

Perhaps this procedure can be iterated. In this way, the properties of 
theories in higher dimension would be encoded in conformal field
theories in
two dimensions. This possibility is due to the typically quantum field
theoretical fact that there is more freedom to construct conformal
theories than higher dimensional quantum field theories in homogeneous
spaces.
In other words, the moduli spaces in higher spacetime dimensions have
lower
dimensions as manifolds, and can be embedded in the moduli spaces of
quantum
field theories in lower spacetime dimensions. As we shall see, string
theory
also performs such an encoding. It would be interesting to see if the
two
encodings are related. 

In the following, we only will consider conformal field theories in two
dimensions. The amount of technical details will just about suffice to
put
string theory in context. For a history of the crucial years 1984-88 and
the relations to statistical mechanics, see [Itzykson, Saleur, Zuber
1988],
which contains many references. A recent textbook is [di Francesco,
Mathieu, Senechal 1997]. 

When one starts with a Minkowskian conformal field theory in flat
spacetime,
Wick rotation yields a euclidean theory on the Gauss plain. By conformal
invariance, it is possible to compactify it to a theory on the Riemann
sphere.
As symmetry group, one obtains the group of linear rational
transformations
$z\mapsto (az+b)/(cz+d)$ of the Riemann sphere. This will be the
symmetry group
of the n-point functions. 

In two dimensions, the energy momentum tensor is a symmetric $2\times 2$
matrix.
Because of scale invariance, its trace vanishes, such that it has only
two
independent components. By the Noether theorem, they are conserved
quantities.
More precisely, one linear combination is holomorphic, another
anti-holomorphic.
These are the famous Virasoro fields, which were first discovered in
string
theory [Virasoro 1970]. Their short-distance expansions are fixed by
conformal
invariance. 

The symmetry transformations $z\mapsto \lambda z$ introduce a change of
the
n-point functions which can be compensated by a linear transformation in
$L(F)$.
In most cases of interest, this transformation can be diagonalized.
When a field transforms as $\phi\mapsto \lambda^h\bar\lambda^{h'}\phi$,
we say that $\phi$ has conformal dimensions $(h,h')$. When $\lambda$ is
real,
we have a rescaling transformation. Thus $h+h'$ is the scaling dimension
of
$\phi$. When $|\lambda|=1$, we obtain a rotation, with an action
described by
the conformal spin $h-h'$. Since a rotation by $2\pi$ is trivial, the
conformal
spin must be integral for bosonic fields. For holomorphic fields,
$h'=0$.
Since the scaling dimension of the energy momentum tensor is 2, its
holomorphic
component has conformal dimensions (2,0) and its anti-holomorphic
component
has conformal dimensions (0,2).

One could proceed in a purely algebraic way, completely within the
framework for
quantum field theories which was described above. Instead, let us
shorten the
path by some geometric intuition. Let us look at some  holomorphic
transformation $z\mapsto f(z)$ of a neighborhood of $z=0$. Locally, this
is a
symmetry, since it does not change the angles. When $f(0)=0$, it induces
a
transformation in $L(F)$, since $F$ can be considered as the space of
fields at 
the point 0. The action of the transformations $z\mapsto \lambda z$ on a
field
$\phi$ of conformal dimensions $(h,h')$ can be described by stating that
the
form $\phi(z)(dz)^h(d\bar z)^{h'}$ is invariant. If this remains true
for all
$f$, the field $\phi$ is called primary. The primary fields span a
subspace of
$F$. If this subspace is finite dimensional, the corresponding conformal
field
theory is called minimal. The Ising model is minimal and has a three
dimensional subspace of primary fields, but the Thirring model is not
minimal.

The short distance expansion of a holomorphic field $\phi$ of conformal
dimensions $(h,0)$ on an arbitrary field $\chi$ is a Laurent expansion,
since it depends holomorphically on $z$. We write it in the form

$$\phi(z)\chi(w)=\sum_n (z-w)^{n-h}(\phi_n\chi)(w)\ .$$

For all integers $n$, this defines linear operators $\phi_n$ on $F$.
They are called the Fourier components of $\phi$.
When $\chi$ has conformal dimensions $(\tilde h,\tilde h')$, then
$\phi_n\chi$ has conformal dimensions $(n+\tilde h,\tilde h')$. We
regard
$F$ as graded by the conformal dimensions and see that $\phi_n$ is an
operator
of degree $(n,0)$. The action of local conformal transformations on $F$
is
given by the linear operators $L_n, \bar L_n$ obtained from the
holomorphic and
anti-holomorphic Virasoro fields.

For holomorphic fields $\chi$, the fields $\phi_n\chi$ are holomorphic,
too,
such that one obtains a new algebraic structure [Zamolodchikov 1985,
Borcherds 1986, Goddard 1989].
A standard name in the physics literature is W-algebra, but
mathematicians
prefer to talk about vertex operator algebras. The latter name has the
advantage
of a clear history in string theory, whereas the W seems to be due to
the
accidental naming of some field as $W(z)$ by Fateev and Zamolodchikov.
Proposed
allusions to Weyl, Wigner or Wilson are apocryphal, but may justify the
name,
which has the advantage of being short. 

The field $\phi_h\chi$ is called the normal ordered product of $\phi$
and
$\chi$. It is the first field which occurs in the regular part of the
short
distance expansion. In the Thirring model, the currents $j,\bar j$ have
conformal dimensions (1,0) and (0,1). The Virasoro fields are given by
the normal ordered products $j_1j/2$ and $\bar j_1\bar j/2$
[Callan, Dashen, Sharp 1967]. When $n<h$, the
field $\phi_n\chi$ occurs in the singular part. It turns out that it can
be
described in terms of commutators $[\phi_n,\chi_m]$. Thus one part of
the
operations of the W-algebra just describes a Lie algebra. For the
components
$L_n$ of the energy momentum tensor this is the Virasoro algebra. It was
discovered by Gelfand and Fuks [1968] and is a central extension of the
Lie
algebra of vector fields on a circle. The value of the central extension
is
universally called $c$ for the holomorphic Virasoro field and $\bar c$
for the
anti-holomorphic one. In many models, they are equal. The values of $c$
for the
minimal models lie in a countable set. All of them have $c<1$, whereas
the
Thirring model has $c=1$. When $\phi$ is holomorphic and $\chi$ is
anti-holomorphic, then $[\phi_n,\chi_m]=0$.
 
The action of the $\phi_n$ on the space $F$ of all fields yields a
representation of the W-algebra. With some effort, the representations
of a
fixed W-algebra can be given the structure of a tensor category, like
the
representations of a Lie algebra. The corresponding tensor product is
called
fusion product. The representation on the holomorphic fields themselves
is
called the basic representation and behaves as the neutral element under
fusion.

Some W-algebras only have finitely many irreducible representations.
These are
called rational. In conformally invariant theories with rational
W-algebras,
all scaling dimensions are rational numbers. Such theories themselves
are
called rational, too. The minimal theories are characterized by the
porperty
that already the Virasoro part of the W-algebra has only finitely many
irreducible representations. It is sufficient to consider the
holomorphic
Virasoro field, since for the anti-holomorphic one the situation is
analogous.
The properties of the Virasoro algebra only depend on the central
extension $c$.
The most interesting values occur for those minimal models where all 
the representations are unitary. This happens for $c=1-6/(p(p+1))$, $p$ 
an integer
greater 2. For $p=3$ one finds $c=1/2$ and the Ising model.
 
The first investigation of these questions was due to Mack and
L\"uscher.
They found that $c=1/2$ is the lowest possible value and that there is a
gap
above $1/2$. Here is one of the rare cases where progress depended on
difficult
calculations performed by a mathematician. V. Kac determined the
structure of the representations [1979], which later allowed Belavin,
Polyakov and Zamolodchikov to determine the values of $c$ for all
minimal
models [1984]. Soon afterwards, Friedan, Qiu and Shenker determined the
unitary
cases [1984]. 

The discovery of the minimal models and their explicit solution 
by Belavin, Polyakov, Zamolodchikov was the breakthrough event in the
history of conformal field theory. It quickly became clear that these
models
are beautiful and fundamental mathematical structures. For reasons which
are hard
to understand in depth, very different kinds of such structures, from
Platonic
solids to singularities, can be classified in terms of the ADE Dynkin
diagrams.
The same is true for the minimal models [Cappelli, Itzykson, Zuber
1987].

Part of the excitements about these early publications came from the
relationship to continuous phase transitions in statistical mechanics.
Besides the Ising model, many other well known continuous phase
transitions were
recognized as minimal models, like the ones for the 3-states Potts
model,
the tricritical Ising model, and the Lee-Yang edge singularity. Some of
them
have been realized in the lab, and measurements agree very
well with the theoretical calculations.

Many properties of continuous phase transitions now fell into place. For
example, some phase transitions are characterized by universal rational
numbers, 
others have free continuous parameters. The former now are described by
conformal field theories which have no conformally invariant
deformations.
In particular, this is true for the minimal models, like the continuum
limit of
the Ising model. When conformally invariant deformations exist, then
they do not
change $c$. The first example was Baxter's eight vertex model, which
at the critical point becomes isomorphic to the older but more
difficult Ashkin-Teller model. They yield $c=1$, as for the closely
related
Thirring model. For a study of all unitary $c=1$ models, see [Ginsparg
1988].


In some respects even simpler than the minimal models are those for
which the
Virasoro fields can be described in terms of normal ordered products of
fields of conformal dimensions (1,0) and (0,1). Such fields are called
currents, and the corresponding conserved integral quantities 
are called charges. The Thirring model is of this type, with currents 
$j,\bar j$  and single holomorphic and anti-holomorphic charges
$j_0,\bar j_0$.
In more complex models where the two types of charges form simple Lie
algebras,
the short distance expansion of the currents yields the corresponding
affine
Kac-Moody algebras [Goddard, Olive 1988]. 

The Thirring model yields the simplest continuous family of conformal
theories
and has $c=1$, too. In its bosonic description, it is given by a the
statistical mechanics of maps to a circle. The model is rational when
the
area of this circle is a rational number. This means that the set of
points
for which the model is rational is dense in the whole family.
At one particular rational point, another continuous deformation is
possible,
which generates the moduli space of the Ashkin-Teller phase transitions.
This is a first example of the rather intricate geometry of
such moduli spaces, with many number theoretic aspects. As a first step,
It would be important to know the rational points of more complex moduli
spaces, since rational theories have very explicit descriptions. So far,
there
are very few results.

Within a moduli space of conformal theories, consider a perturbations 
by a field $t(x)$. The integral
$\int t(x) dz\,d\bar z$ must be invariant under conformal
transformations,
such that $t$ should be a primary field of conformal dimensions (1,1).
In the Thirring model, the field $j\bar j$ has these properties.
The dimension of the vector space of such fields counts the number of
possible
infinitesimal deformations. Thus it is an upper bound on the dimension
of the
moduli space. For generic points of this space, one expects that the two
dimensions are equal. For the Thirring model, they are both equal to 1.

The short distance expansion is a local property of the theory. When one
wants
to calculate the n-point functions, one also has to specify a Riemann
surface
on which the fields live (in the language of algebraic geometry, an
algebraic
curve). The simplest case is the Riemann sphere. Here the n-point
functions of
holomorphic fields are just rational functions. For more general fields,
the
results are much more complicated. For example, the four-point functions
of minimal models already yield hypergeometric functions.  

The Riemann sphere is unique, but
more complicated Riemann surfaces (or equivalently algebraic curves)
have their own continuous parameters. For example, a torus is described
by the
ratio $\tau$ of two independent periods. When these are correctly
ordered and
varied continuously, $\tau$ varies over the upper complex half-plane.
The
latter is called the Teichm\"uller space of the curves with torus
topology.
Points of Teichm\"uller space describe the same torus when they are 
related by a
different choice of periods. Changes of the periods are described by
the
modular group. This is the group of linear rational transformations 
$\tau \rightarrow (a\tau+b)/(c\tau+d)$ with integral coefficients.
More complicated curves behave in an analogous, but of course more
complex way.

The 0-point function on the torus is essentially the partition function
of the
theory. Since energy and momentum are given by linear combinations of
the
Virasoro field components $L_0$ and $\bar L_0$, the latter can be
defined by
$$Z=\quad tr\; \exp(2\pi i(L_0\tau -\bar L_0\bar\tau))\ ,$$
where the trace goes over the vector space $F$ of all fields. The
0-point
function on a torus with parameter $\tau$ has the form
$$\tilde Z= \exp(-2\pi i(c\tau -\bar c\bar\tau)/24) Z\ ,$$
where $c,\bar c$ are the central extensions of the theory.
The prefactor is necessary to get invariance under the modular group.
For the Ising model one obtains

$$\tilde Z= {1\over 2}\sum_{i=2}^4 
\left\vert \theta_i(\tau)/\eta(\tau)\right\vert\ ,$$
where the $\theta_i$ are Jacobi's theta functions and $\eta$ is
Dedekind's
function. Note that the scaling dimension 1/8 of the Ising spin can be
read off
from $\theta_2$.

For a free complex fermion one obtains
$$\tilde Z= {1\over 2}\sum_{i=2}^4 
\left\vert \theta_i(\tau)/\eta(\tau)\right\vert^2\ .$$
This function arises at the parameter $R=\sqrt 2$ of the Thirring model
partition function

$$\tilde Z= \vert \eta(\tau)\vert^{-2}\quad
\sum_{m,n}\exp \left({\pi i\over 2}
\left(({m\over R}+nR)^2\tau - ({m\over R}-nR)^2\bar\tau \right)\right)
\ .$$
Here $m,n$ vary over the integers. The equality of the latter two
functions for $R=\sqrt 2$
is an example of the fermion-boson equivalence mentioned
above.
In the gaussian description, only the terms with $m=n=0$ were obvious,
which explains why the model was considered to be uninteresting.
\\[2ex]
{\bf String theory}

Contrary to the historical developments, we have considered conformal
field
theory before coming to string theory. The reason is that string theory
is more
complex. Conformal field theory is just one ingredient, albeit an
essential one.
For general introductions to string theory and more references, see
[Green, Schwarz, Witten 1987] and Polchinski [1998].

In 1968, Veneziano invented an amplitude for a scattering process with
two
incoming and two outgoing particles which shared several features with
strong
interaction processes. When a natural generalization to arbitrary
particle
numbers was found, Nambu, Nielsen and Susskind recognized that these
amplitudes
describe a one-dimensional object moving in space. The surface described
by its
motion is called a worldsheet. Its embedding into spacetime is described
by
functions $X^\mu (\sigma,\tau)$, where $\sigma,\tau$ are coordinates on
the
worldsheet and $X^\mu$ yields the corresponding spacetime positions.

Calculational problems arise, because there is no canonical
parametrization of
the worldsheet. Some natural choice can be made, however. The causal
structure
of the ambient spacetime induces a causal structure on the worldsheet,
with two lightlike tangent directions at each point. These directions
can be integrated to lightlike curves. One chooses the coordinates such
that
their equations are given by $d\tau=d\sigma$ and $d\tau=-d\sigma$. 
This introduces a Minkowskian conformal structure on the $\sigma,\tau$
parameter space. One chooses $d\tau$ to be timelike and $d\sigma$ to be
spacelike.

Strings have finite spatial extent, such that the range of $\sigma$ is
compact.
For open strings, the standard choice is an interval of length $\pi$,
for
closed strings a circle of circumference $\pi$. No further natural
choices can
be made, which means that the worldsheet dynamics is conformally
invariant. In other words, the possible states of a single string are
described
by a conformal field theory. When one continues to a euclidean conformal
field
theory, one must make a Wick rotation in $\tau$, not in the time
coordinate of
$X$. The euclidean coordinate is called $z$. 

By the analytic continuation, the worldsheet becomes a Riemann
surface. Let us consider the case of closed strings only. Both in
Minkowskian
and in euclidean space, the worldsheet has the topology of a cylinder.
By
conformal invariance, it can be compactified to a Riemann sphere with
two
special points, one for the incoming and one for the outgoing state.
Such special points are called punctures. String interactions are
introduced by
considering arbitrary Riemann surfaces with different numbers of
punctures.
Calculating the scattering of $n$ strings involves three steps. First,
the string states have to be identified with fields on the worldsheet.
Secondly,
one has to calculate the corresponding n-point functions for all 
Riemann surfaces with n punctures. The surfaces are not necessarily 
connected,
since some groups of strings can interact independently of the others.
Thirdly,
one has to integrate over all of these configurations, in particular
over the
position of the punctures. In addition, one has to integrate over the
finite
dimensional moduli space of complex structures on Riemann surfaces with
a given genus (number of handles).
This integral is not needed when one applies
conformal field theory to statistical systems, since there the Riemann
surface
is fixed. 

Finally, one has to sum over the genera. Each term is multiplied by a
power of
the coupling constant. The exponent is proportional to an
integral over the curvature, and can be normalized to $g-1$. The leading
contribution is given by $g=0$ and as many connected components as
possible. For
vanishing coupling, this leads to a free theory, exactly as for a
quantum field
theory. Indeed, a string theory can be regarded as a quantum field
theory which
includes graviton fields. In some limit, gravity decouples and one
obtains a
field theory of conventional type. For the latter, the perturbation
series is a
sum over Feynman diagrams. A tubular neighborhood of such a graph yields
a Riemann surface of some genus $g$. This allows to identify one of the
field
theory couplings with the string coupling. We shall see that the others
correspond to parameters of a conformal field theory on the string
worldsheet.

The sum over $g$ is certainly not convergent, which provides a
technical reason to develop non-perturbative string theory.  
A deeper reason is the following. As for quantum field theory, free
string
theory can be considered as a boundary stratum on some moduli space.
This
stratum is characterized by the vanishing of a coupling constant, but in
many
cases its codimension is larger than one. Thus an expansion in the
coupling
constant cannot recover the full theory. In particular, it has no reason
to be
convergent. One example is given by quantum electrodynamics, where the
following
picture can be conjectured. To get a well defined quantum field theory,
one has
to introduce magnetic monopoles. These become infinitely heavy when the
interaction goes to zero and their effects are not included in the
perturbation
expansion. Since monopoles can have an electric charge, one has an
additional
dimension of the moduli space which cannot be captured by perturbation
theory.

In string theory, the r\^ole of the magnetic monopoles is taken over by
branes
of various dimensions. One can approach the full picture by a
description of all
possible boundary strata, but this goes much beyond the scope of the
present
article. Nevertheless, the reader should keep in mind that the following
description of conformal worldsheet physics is perturbative and thus
incomplete.

When a string state is described by a field $\phi$ on the worldsheet,
the integration over the corresponding puncture position takes the form
$\int \phi(z)\,dz\,d\bar z$. This must make sense independently of the
choice
of the coordinate $z$. In other words, string states are described by
primary
fields of conformal dimensions (1,1). There is another way to get the
same
result. When the string is considered in the background of some particle
wave in
spacetime, this yields a conformally invariant deformation of the
theory, at
least infinitesimally. Since deformations are described by the primary
fields of
conformal dimensions (1,1), the same must be true for the particle
states
arising from the string.  With reference to spontaneous symmetry
breaking in
quantum field theory, the existence of particle states may be described
as a
Goldstone phenomenon. 

When one considers strings in flat spacetime, the coordinates of the
latter can
be regarded separately. For a space coordinate $X^i$ appropriate fields
are
given by $\exp(ip_iX^i(z))$, with arbitrary $p_i$. With a conventional
choice of the length scale, the scaling dimension of this field is
$p_i^2/4$.

A new situation appears for the time coordinate $X^0$. Due to Lorentz
invariance, the field $\exp(ip_0X^0(z))$ has scaling dimension
$-p_0^2/4$.
Fields with negative scaling dimensions of arbitrary size do not occur
in
statistical mechanics, but they can be made to fit in the framework of
conformal field theory. Indeed, without such negative contributions to
the
scaling dimension, one never would get an infinite number of particle
states.
Here we can take an arbitrary field with $h=h'$ and adjust the value of
$p_0^2$ such that the scaling dimension becomes 2. This produces at
least a
(1,1) field, though in general it will not be primary. 

When we disregard the latter problem, we can consider the fields
$\partial X^\mu\bar\partial X^\nu \\
\exp(ipX)$. They have conformal
dimensions
(1,1) when $p^2=0$, such that they describe massless particles. When one
considers their behaviour under spacetime rotations, one sees that they
include
spin 2 particles, i.e. states which behave like gravitons. For general
reasons,
the coupling of such states must be described by Einstein's theory. Thus
any
consistent string theory is a theory of quantum gravity.

Later, this fact was recognized as the best feature of string theory,
but it
was a nuisance as long as the theory was supposed to work for the strong
interaction. Other problems of the original string theory had to be
solved 
quite apart of this deeper issue, namely the existence of tachyons and
the wrong
dimension of spacetime. 

A tachyon appears when one considers the simple field $\exp(ipX)$. This
is a
primary (1,1) field, if $p^2=-8$. To get rid of this unwanted particle
with
negative squared mass, the conformal symmetry had to be extended to a
superconformal one.  The fields of such theories can have integral or
half-integral conformal spin. Those with a half-integral difference
$h-h'$ are
fermionic. In addition to the Virasoro fields one has fields $G$ and
$\bar G$ of
conformal dimensions (3/2,0) and (0,3/2). There are two different
fermion
numbers associated to the holomorphic and anti-holomorphic variables.
The short
distance expansion with $G$ changes the first one by one unit, that with
$\bar
G$ the second one. The Fourier components $L_n$ of the Virasoro field
and
those of $G$ together yield a superalgebra, in which the Virasoro
algebra is
embedded. The model has two sectors (discovered separately by Ramond and
by
Neveu and Schwarz), but we shall consider just the latter. In this
sector, the
fermionic fields have half-integral coefficients. Apart from such
modifications, superconformal field theory can be regarded as a special
case of conformal field theory, so most of the preceding description
remains valid. 

Fields related by the action of $G_{1/2},\bar G_{1/2}$ are
called superpartners. For superconformal deformations, the corresponding
(1,1)
fields must be superpartners of (1/2,1/2) fields. The physically
relevant
deformations are described by bosonic fields, such that the (1/2,1/2)
fields
must be fermionic with respect to both fermion numbers. The superstring
still
has fields $\exp(ipX)$, which have conformal dimensions (1/2,1/2) for
$p^2=-4$,
but these fields are of bosonic nature and do not correspond to physical
particles. This elimination of the tachyonic fields is due to Gliozzi,
Olive
and Scherk. 

The issue of the spacetime dimension arose in a different way. When one
calculates the norm of a field of type $\partial X^\mu$, Lorentz
invariance
yields a result proportional to $g_{\mu\mu}$. In particular, one can
find
negative norms which are incompatible with a probability interpretation.
In the 50's and 60's much ink had flown in unsuccessful attempts to make
sense
out of negative norms and no one was motivated to try again.
Fortunately,
Virasoro recognized that not all fields yield physical states [1970].
The concepts of primary fields and conformal dimensions did not exist
yet, but
he only found the correct constraints and described them by the Fourier
modes of the Virasoro fields. One year later, Galli obtained the
interpretation
in terms of conformal invariance [1970]. 

Numerical investigations showed up to a certain degree of complexity
that the
physically allowed fields all have positive norm, but a general proof
was
difficult to obtain. Then it turned out that allowed negative norm
fields do
exist when the spacetime dimension is greater than 26, or 10 for the
superstring. This made sense of an observation of Lovelace [1971], which
had
not been taken very seriously because it was too outlandish. Looking at
Riemann
surfaces of torus topology, Lovelace had shown that the bosonic string
theory
was found to require a spacetime of 26 dimensions. Now it became clear
that this
number was a deep structural property of the bosonic string theory and
would not
go away. Indeed, Brower [1972] and Goddard and Thorn [1972] used the 26
dimensions to prove that the norms make physical sense (the no-ghost
theorem). Later it turned out that the value of this critical dimension
has deep relations to the conformal invariance of the world sheet
physics
and the corresponding modular invariance [Brink, Nielsen 1973].
Moreover,
Beilinson and Manin found out that the strange 26 was closely related to
analytic torsion results of Mumford, which allowed them to write the
measure
for the integration over the moduli space of Riemann surfaces in a very
elegant
form [1986]. 

The critical dimension translates into the value $c=\bar c=26$ of the
central
extensions. For the superstring one needs 10 dimensions and $c=\bar
c=15$. The
latter value is due to the superpartners of the 10 coordinates $X^\mu$,
which
contribute half as much to the central extension. The simplest way to
obtain a
model in four dimensions is the old Kaluza-Klein idea. One just wraps up
all
superfluous dimensions in a small circle. For the bosonic string this
yields 22
copies of the Thirring model. The corresponding 44 currents of type $j$
and
$\bar j$ yield 44 photons, all with separate interactions of
electromagnetic
type. The values of $c,\bar c$ do not change. Obviously, this model is
not
particularly realistic. It exemplifies, however, that the spacetime
dimension of the model can be changed at will, as long as one keeps
conformal
invariance and the correct central extensions. For the superstring,
similar
remarks apply.

To write down a general bosonic string model in four dimensions, one
just needs
to replace the 22 copies of the Thirring model by an arbitrary conformal
field theory with $c=\bar c=22$. The latter is called the internal
conformal
theory. In analogy to the Kaluza-Klein case, one still says that it
describes 22
compactified dimensions, even if this is not always a geometrically
correct
interpretation. To compactify the superstring to fourdimensional
spacetime, one
needs six compactified dimensions and $c=\bar c=9$ instead. Every
possible
compactification corresponds to a theory in a space of less than 10
spacetime
dimensions. In this way one gets, e.g., an encoding of four-dimensional
quantum field theories by conformal or superconformal field theories in
two
dimensions. 

In particular, consider a field $\phi(z)\exp(iXp)$, where $\phi$ belongs
to the
internal theory. When one adjusts $p^2$ to get overall conformal
dimensions (1,1), one sees that for a particle state of mass $m$ the
corresponding field must have contributions $h=h'=1+m^2/8$ from the
internal conformal theory. Of particular interest are the internal (1,1) 
fields, which correspond to massless Higgs bosons. 

When the conformal theory includes an affine Kac-Moody algebra with
holomorphic
currents $j_a$ of conformal dimensions (1,0), the fields $j_a
\bar\partial X^\mu \\
\exp(iXp)$ with $p^2=0$ describe the quanta of a vector potential
$A_a^\mu$
belonging to the corresponding finite dimensional gauge group. Thus the
states
of the string now include non-abelian gauge fields, and the theory
starts to
look a bit more like the standard model. In the superstring theory, one
also
gets fermions. Their interactions with the Higgs bosons and the gauge
fields are of standard type, though one has not yet managed to obtain
precisely
the standard model. 

Of course, the bosonic model always will have the tachyonic $\exp(ipX)$
fields
and cannot be used by itself. Nevertheless, the bosonic string can be
used for
either the holomorphic or the anti-holomorphic coordinates. To get rid
of the
tachyon, it is indeed sufficient to use a field $G$ but no $\bar G$.
This
yields models with $c=26$ but $\bar c=15$, called heterotic string
models.
They were found by Gross, Harvey, Martinec and Rohm.
Heterotic strings do not have a pure spacetime version, since the
spacetime
contributions to $c$ and $\bar c$ have to match. The archetypal
heterotic string
lives in 10 dimensions, where the compactified part is purely
holomorphic, with
$c=16$. There are many constraints on purely holomorphic conformal
theories
which exist on arbitrary Riemann surfaces.  In particular, $c$ must be a
multiple of 8. For $c=8$, the only example is the affine Kac-Moody
algebra based
on $E_8$. For $c=16$, one either can take the tensor product of two
$c=8$ models
or use the affine Kac-Moody algebra based on $SO(32)$. For $c=24$, there
are 71
possibilities [Schellekens 1993]. One of them has a remarkable symmetry
group of
about $10^{54}$ elements, the Fischer-Griess monster [Borcherds 1986].
This closeness of string
theory to beautiful exotic structures is still a deep mystery. To get to
four
dimensions, one needs a less exotic internal conformal field theory with
$c=22$
and $\bar c=9$. 

The late discovery of the heterotic string was due to the fact that
string
research slowed down a lot after 1974. At that time it had become clear
that
QCD is a better theory for the strong interaction. Though Scherk and
Schwarz
had shown that one could reinterpret string theory as a theory of
quantum gravity [1974], there was not very much support for such an
arcane
research line. One of the ideas which appeared shortly before the theory
entered
a long hibernation period was the classification of the possible
rational
theories by modular forms. In particular, this yielded a candidate which
later
would be interpreted as the partition function of the compactified part
of the
heterotic string [Nahm 1977]. Unfortunately, present mathematical
techniques
only allow to apply this procedure to rational conformal fields
theories, for
which the partition function can be written as a finite sum $Z=\sum_i
Z_i\bar
Z_i$ with holomorphic functions $Z_i$ and anti-holomorphic functions
$\bar Z_i$.
Nevertheless, it was striking that the method indicated an incredibly
large
number of possible theories, in stark contrast to the initial hopes that
one
was heading for something unique. At present, the situation has not
changed
very much. There are vague hopes that non-perturbative string theory
will
select particular models, but it also is possible that one will end up
on a
moduli space with more parameters than in the standard model.
For every choice of parameters, one will have a quantum version of
Einstein's gravity theory, however. 

The bold switch from the interpretation of string theory as a theory of
the
strong interaction to a theory of quantum gravity by Scherk and
Schwarz must have been one of the strangest events in the history of
physics. In particular, the basic distance scale had to be changed
by twenty orders of magnitude, from the proton diameter to the Planck
length.
For mathematicians this should be easier to digest than for physicists,
since no
change in the mathematical structure is involved. For physicists,
however, the
emergence of string theory now appears as an accident. It would not have
happened if the discovery of the SU(3) gauge interaction of the quarks
had come
a few years earlier. Even in hindsight, one sees no way how a direct
study of
quantum gravity could have led to this theory. Indeed, a direct attack
has been
tried from several points of view, but with very limited success. It
seems that
one cannot unify quantum theory and gravitation without incorporating
much
knowledge about other interactions. 

In any case, present research on quantum gravity cannot follow the
traditional
pattern of physics. One hundred years ago, Planck himself
estimated its characteristic length scale by combining Newton's
constant,
the speed of light, and his new quantum of action. He found
$4,13\cdot 10^{-33}cm\,$
[1900]. At that time, physicists and chemists were getting the
first precise ideas about what happens at $10^{-8}cm$, so Planck must
have felt
like looking into an abyss. Of the 25 orders of magnitude to be covered
we now
have explored not quite ten, thus a naive extrapolation predicts
another 150 years before we really understand what is going on at the
basic
scale. Planck's report about his discovery is brief and sober.
Nevertheless, he
states that the units he found would keep their meaning for all times
and all
cultures, including extraterrestrian and non-human ones. 

Without the ability to do experiments in quantum gravity, it is hard to
know
if theoretical investigations are on the right track. Everyone who keeps
trying is inspired by Einstein's success with general relativity. He had
little
experimental input, but relied on his keen sense for structure
and mathematical beauty. His belief in the harmony of the spheres was 
as deep as
Kepler's, and when he had found an indication of congruence between
nature and
a mathematical structure he did everything to uncover it fully. Quantum
mechanics remained as a jarring note like the irrational numbers to the
early
Pythagoreans. Thus string theory would have disappointed Einstein as far
as
quantum physics is concerned. But if it is correct, it justifies some of
his attempts in the search for a unified theory. On one hand, he wanted
to generalize the metric tensor $g_{\mu\nu}$ to an object with an
antisymmetric
part. String theory has such an object, called the $B$ field.
Together with the metric tensor it is obtained from the fields
$\partial X^\mu\bar\partial X^\nu \exp(ipX)$ which have been considered
above.
Einstein also was right in his high regard for the Kaluza-Klein
approach.

Einstein's example can be used as an encouragement and as a warning. His
successful gravity theory was based on at least one elementary fact
which no one
else could explain - the equality of inertial and gravitational mass.
When he
let loose of such guidance, he still did important research, but went
astray.
String theory does not do too badly on this account. On one hand, one
can at
least come close to the standard model. Moreover, superstring theory at
least
suggests that the experimentalists will find supersymmetry in the near
future.
The theory develops in a search for deep and beautiful structures, but
it
has the advantage of holding on to the tenuous guide offered by low
energy
experiments. Currently, no other theory of quantum gravity can make such
claims.
Despite its unbelievable origin, string theory is by far the most
promising
approach to unify all of the known interactions. The one possible
exception of
the latter claim is the cosmological constant, since it is separated by
another abyss of many orders of magnitude from the rest of physics.
\\[2ex]
{\bf Missed and open opportunities}

Let us come back to
Dyson's 1972 address to the American Mathematical Society. It was titled
'Missed
Opportunities' and concerned problems in the communication between
mathematicians and physicists. In particular, he considered quantum
field
theory and the unification with gravity, but his first example concerned
a
communication problem between Dyson the physicist and Dyson the
mathematician.
As a mathematician, he had played around with
powers of Dedekind's $\eta$-function and obtained nice identities for
the
exponents 3,8,10,14,15,21,24,26,28,35,36,... In this combinatorial
context he
did not recognize the dimensions of the simple Lie groups, 
plus 26, which would
have been
evident to him in a physics context. From today's point of view, the
regret
about this little failure may have caused him to miss a greater
opportunity.
He must have heard about the incredible 26 dimensions of the bosonic
string, which Lovelace had found the year before, but apparently thought
little about this coincidence. Otherwise he would have
stumbled on the importance of Dedekind's $\eta$-function in string
theory.
Two years later Scherk and Schwarz established the importance of string
theory for the unification of quantum field theory and gravity.

Many of the present author's missed and taken opportunities also concern
the
interaction with mathematics. A very rapid course by D. Zagier
led to a classification of string theories by modular functions. On the
other
hand, searching the CERN library for books discussing infinite
dimensional Lie
algebras was a frustrating enterprise. Even worse was the inability to
find
the dimensions of the next representations of $E_8$, when the
classification
yielded $q^{-1/3}+248q^{2/3}+\ldots$. Unfortunately, the visit of Kac to
CERN came far too late, but to the author it proved the value of an
environment
where physicists and mathematicians could make the effort to learn about
their
respective discoveries. Princeton and some other places made a good
start, but
it would be nice to have a few more. Here are some problems which may be
tackled in such an environment.  

In the Kaluza-Klein formalism, a fifth dimension is hiding because it is
compactified to a circle. When one considers experiments at fixed energy
and
makes the period of the compactified dimension very large, the five
dimensional
geometry emerges again. This process can be generalized -- taking
suitable
limits of quantum field theories one can obtain classical geometries.
In this context, the latter are called target spaces. 

Let us look at the Kaluza-Klein situation in the context of string
theory.
We have considered fields $\exp(iXp)$, where for simplicity we consider
a
single position component $X$. When it is compactified with period $l$,
the
choice of $p$ is constrained by $\exp(ilp)=1$, or $p=2\pi n/l$ with
integral
$n$. The scaling dimension of such a field is proportional to $(n/l)^2$,
thus
small for large $l$ and small $n$. In particular, the integer $h-h'$ has
to
vanish. The short distance expansion of such fields involves weak
singularities
only. In the limit where $l$ becomes large, it reduces to the ordinary
product
$\exp(iXp_1)\exp(iXp_2)=\exp(iX(p_1+p_2))$. Because of scale invariance,
the large $l$ limit produces a unique commutative algebra of all fields
whose scaling dimension approaches zero. 

In the classical limit, every smooth function on the Kaluza-Klein circle
can be
Fourier expanded with basis $\exp(iXp)$. Thus one obtains the algebra of
all
smooth functions on the circle. Moreover, the space of these functions
is
graded by the scaling dimension $2h$, which is the eigenvalue of the
Laplace
operator. The geometry of the circle can easily be reconstructed from
this
information.

This example can be generalized to all kinds of manifolds. Particularly
attractive are Calabi-Yau manifolds, where one can work with the highly
constrained superconformal theories. Much less is known about these
manifolds
than about the circle, for example about their Einstein metrics. In
these
cases, the conformal theories may be easier to control than their
classical
limits. One certainly may hope to obtain the algebra of smooth functions
and
the corresponding eigenvalues of the Laplace operator from the conformal
data.

When the parameters of a conformal field theories are varied, one may
obtain
quite different classical limits. In particular, a connected moduli
space may
have several different boundary components. In this way, it is possible
to
relate different classical geometries by non-classical paths. As a
simple
example, consider again a string on the Kaluza-Klein circle. There are
more
fields than we have considered so far, since one can wind the string
around the
circle. When the circle is large, this yields particle states of large
mass. When the period $l$ becomes very small, winding costs hardly any
energy,
whereas the $\exp(iXp)$ fields have large scaling dimensions and
describe
particles of large mass. When $l$ goes to zero, the short distance
expansion
of the basic winding fields is just given by the additive group of the
winding
numbers. In this way, a Kaluza-Klein theory with period $l$ becomes
isomorphic
to one with period $l^{-1}$. This isomorphism is known as T-duality. It
is one
of many dualities which arise in string theory, so the name 'dual model'
used
around 1970 was quite prescient. Perhaps the most famous of the
dualities is mirror symmetry, which is a specific property of conformal
field theories with a high degree of supersymmetry. 

The winding fields are examples of solitonic objects, since the winding
number
is time-independent. In the euclidean conformal theory, one also has
instanton
contributions given by a map of the string Riemann surface to the target
space.
The most studied case is the one of embeddings of Riemann spheres in
algebraic
target manifolds, since mathematicians have been much interested in
counting
the number of different embeddings. As shown by Candelas' group, mirror
symmetry
yields the correct numbers for the quintic in four dimensional
projective space.
This started the huge interest of mathematicians in this topic. Usually,
mathematicians try to replace the quantum field theoretic approach by
more
classical methods, but in the end this may well turn out to be the more
arduous
approach, quite comparable to a proof of the prime number theorem
without using
analysis. 

The moduli space of superconformal theories of fixed central extensions
seems to
be connected. For $c=\bar c=9$ each Calabi-Yau manifold of three complex
dimensions is a possible target space and yields one boundary component
of the
moduli space. By following all ramifications of the moduli space it
should be
possible to classify all such Calabi-Yau manifolds, for a start.

In modern algebraic geometry, geometric and number theoretic problems
occur
side by side. The same is true of conformal field theory, though
physicists so
far are ill equipped to handle these issues. For example, what is
the meaning of Dysons formula for $\eta^{26}$? For the moment, this is a
mystery without much of a clue, but one can start with simpler problems.
Above, we have discussed rational points in the moduli space of string
theories.
At these points, coupling constants like the Yukawa couplings can be
calculated
explicitly. Usually, the rational models are among the few which such
calculations are possible at present. For example, let us
take models with $c=\bar c=6$ and sufficiently large supersymmetry. In
this case
one obtains K3 surfaces as target space. The moduli space turns out to
have 80
dimensions, but the largest submanifold under explicit control has just
16
dimensions. In addition, however, there are many rational points
sprinkeled
around which are perfectly well understood. Will the rational points
turn out to
be densely distributed? More specifically, the same moduli space occurs
for
torus compactifications of the heterotic string to six-dimensional flat
spacetime. In the latter case the rational points are well known, and
they
correspond precisely to the complex multiplication points of the K3
moduli
space. Are those the rational points of the latter moduli space?

In simple examples, the conformal dimensions of rational models are
obtained by
applying the dilogarithm function to algebraic numbers. The
corresponding sums
are described by torsion elements in the Bloch group [Nahm, Recknagel,
Terhoeven 1993]. Thus there is a link between conformal field theory and
one of
the most active areas of present mathematical research. In particular,
there
seem to be relations to Grothedieck's program for a description of the
Galois
group of all algebraic numbers and to the theory of motives. Kontsevich
recently
conjectured that the motivic Galois group acts on the moduli space of
conformal
field theories [1999].  

Obviously, mathematicians have much to gain from physics. In view of
the higher reliability of the answer (and regarding costs as
irrelevant) physicists were more inclined to ask nature than to ask a
mathematician. Quite typically, Schweber's textbook concludes with the
following
sentences: "In the final analysis, however, it will probably be the new
information that will be obtained from the high energy machines and
colliding
beam machines to go into operation in the next few years which will help
unravel
the puzzle of the elementary particles and their interactions. In
particular, we
may discover whether the notions of space and time upon which
present-day field
theories are based are in fact valid." But meanwhile we have learned
more
respect for the sixteen orders of magnitude which separate us from the
Planck
scale. If it gets too expensive to ask the direct questions, we just
have to
push the mathematical analysis of what little clues there are. There is
hope,
since sometimes it did work. Kepler managed to extract the secrets of
the
planetary motions from pretelescopic data, but it would have been much
harder
without some knowledge about ellipses.

As a final encouragement for those willing to use the bridge, let me
quote a
German poet: "Nur Beharrung f\"uhrt zum Ziele, nur die F\"ulle f\"uhrt
zur
Klarheit und im Abgrund wohnt die Wahrheit" (to reach the goal you must
be
persistent, to see clearly you have to understand a wealth of phenomena,
and
truth lives in the abyss).  Schiller's poem talks about causal time and
three-dimensional space, but two euclidean dimensions make a good start.
\\[2ex]
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\end{description}


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