To be continued…
It's a great pleasure for me tonight to introduce the Gibbs Lecture tonight.
Sir Michael Attia, I have to correct his address, he's Master of Trinity College, Cambridge,
not England.
Well maybe also England, but Trinity College, Cambridge.
Well I was planning to give an exhaustive review of Sir Michael's career and all the
honours that he's amassed, but I thought I would leave him a little bit of time for
his lecture.
So I'll only mention that, of course, besides being a professor at Oxford at times, various
times, he's also been a professor of the Institute for Advanced Studies, and at that time he
was also active in the affairs of the American Mathematical Society, in fact, serving on
the council.
He's now, as well as the Master of Trinity College, he is the new director of the forthcoming
Institute for Advanced Studies in Cambridge.
He's received almost all possible honours, I think, including the Fields Medal, and rumour
has it he may run for President of the American Mathematical Society.
It's one of the few gaps in his career, but perhaps this is an exaggeration.
Anyway, it's a great pleasure to introduce Sir Michael Attia, who will speak on the mysteries
of space.
Thank you.
Thank you, Bill.
First, I should say I'm very sorry to be speaking on such a sombre occasion, and perhaps it
wouldn't be inappropriate if I just paused for a few moments in respect to those who
are dying somewhere out in the Persian Gulf.
When I was asked to give this lecture, I'm, of course, very pleased and honoured to do
so.
Gibbs was, of course, a great mathematical physicist, and in recent years I have taken
an interest in physics and its relationship to mathematics, so I'm very pleased to be
asked to give this lecture.
The Secretary of the AMS very kindly sent me a list of all the previous Gibbs lecturers.
It's a very impressive list, going back to 1924, I think.
And I noticed going through that list, there were a few names I recognised.
In particular, there were two British mathematicians, only two on that list.
One was G.H. Hardy in 1928, and the other one is someone you might not recognise as
a British mathematician.
You might think of him as an American physicist, but Freeman Dyson in 1972.
I have something in common with both of them, since both of them came from my college in
Cambridge.
I also had other things in common.
Hardy held the Chair of Geometry in Oxford, which I subsequently held, and as you heard,
I was a colleague of Freeman Dyson's at Princeton for some years.
Now, Hardy's lecture was a very attractive introduction to elementary number theory,
which I did consult in the library while I was here.
And Dyson's, which most of you probably didn't hear, I don't suppose many of you were here
in 1928, but Dyson's lecture is more recent, and so I imagine quite a few of you either
heard it in person, or at least read it in the Bulletin of the AMS.
And that talk, which is entitled Missed Opportunities, obviously made quite an impression on the
AMS, because when they asked me to speak, to give me some advice on what kind of lecture
to give, they said, why don't you look at Dyson's lecture?
Now, actually, Dyson's talk was about the missed opportunities between mathematics
and physics, the fact that mathematicians and physicists hadn't been talking together
much and all sorts of possible exciting things had been missed out because of this.
Well, as it so happens, shortly after 1972, and possibly as a result of Dyson's talk,
this gap between mathematicians and physicists was rapidly closed.
And over the intervening 10 or 15 years, there has been a great coming together of mathematicians
and physicists.
So I could have given a talk perhaps entitled Opportunities of Grass, or something like
that.
But I thought that perhaps a little presumptuous.
But essentially, that's what I want to talk about.
While I was going through the list, I also came across another talk by somebody whom
I hadn't heard of before, although some of you may have done, a peer point who I think
was a professor at Yale, 1925.
And his talk had the intriguing title, Some Modern Views of Space.
Unfortunately, the MSRI library didn't go quite that far, so I couldn't get out the
copy to read it.
I would have liked to have done so.
I think what it shows certainly is that you should never use the word modern in your title.
At least, not if you want to be read by posterity.
Well, the exciting developments that I want, what I want to do is try and tell you about
some of the things that have been happening in the relationship between geometry and physics.
Of course, in a very, what you might call impressionistic way, because this is a large
hall, you're a lot of people, the common intersection of the knowledge of everybody in this audience
is probably small.
And so it isn't appropriate to go into any kind of technicalities.
Nevertheless, I think it's possible to convey something of what's been going on.
And the particular topics I want to talk about, and many of you will have heard about them
by reading things like the New York Times, on the one hand there is the spectacular work
of Vaughan Jones of this university, well, the university here in Berkeley, about knots
in three-dimensional space.
And a short while before, there was the equally spectacular work of Simon Donaldson in Oxford
about four-dimensional manifolds.
And trying to understand these results and make progress with them has been a great stimulus
and source of great inspiration, and perhaps the most insightful way of looking at these
things has been introduced by Ed Witten, who's a theoretical physicist, who has explained,
at least in terms that are satisfactory from physicists and also to sympathetic mathematicians,
that quantum field theory provides the right framework in which to understand these great
developments.
And you probably know all these gentlemen got Fields Medals for their work.
So we're talking about something which has been recognized.
Now what I want to do is to really take a very long historical view of this thing, take
you back to the beginning, and go through the development of geometry and its relationship
to physics so we get some kind of perspective.
So let us start at the beginning.
In the beginning, there was Euclid.
And what Euclid, of course, studied was the geometry of static objects in space, triangles,
squares, and so on.
I emphasize that they are static objects.
He wasn't studying how they moved.
And also, they were in space.
And although, of course, the platonic idea was that there was some sort of abstract platonic
world where these perfect figures lived, nevertheless, there's no doubt that when they thought about
triangles and measurements and lengths, they had in mind, of course, that you took a ruler
and measured the length between one vertex and another one.
So space was the physical space.
You actually measured it with some device, rather, and then you built the mathematical
model that described that.
And that was what Euclidean geometry was.
So it was the mathematical development of the things in physical space in this elementary
form.
Now, of course, we move on to later times.
And this is a rapid historical survey, so you won't mind if I jump a few centuries here
and there.
And so we'll now move on to another man from Trinity College, Cambridge, Isaac Newton.
I'm sorry for the plug, but you know, when you're the head of an organization, you have
to take every possible advantage.
Contribution to the door would be great if you received.
So Newton, of course, went on beyond studying statics and studied dynamics.
The whole question of how things move in space was the fundamental object of study.
And in that, of course, the notion of force, what it is that makes things move or what
it is that makes things deviate from straight line motion into curved motion, these were
the fundamental objects of study in Newtonian mechanics.
And of course, alongside of that, there was this very difficult notion which people had
great difficulty with, but eventually, they accepted it.
Nowadays, we think of it as obvious, but in many ways, it's still mysterious, namely
the idea of action at the distance, the fact that things which are far apart can still
nevertheless exert some kind of force on one another.
People didn't accept that originally.
They thought it had to be done by some kind of intermediate medium.
But nowadays, we accept it and we understand or think we understand that there are fields
in space.
Space is full of these things called fields, which are the manifestations of force and
a field is produced by something, a particle, a gravitational field is produced by a particle
which attracts other particles, and there are, of course, other kinds of fields which
are produced by charged particles which produce electric fields.
So the development of a notion of field and force, these were the beginnings of modern
physics.
And I emphasize that they had this geometrical content in terms of studying the curved motion
of particles going beyond the straight line study of Euclidean geometry.
Now one of the major subsequent developments, as you no doubt know, in geometry was the
development of non-Euclidean geometry, which is, of course, attributed in various forms
to Boliay, Lomachevsky, Gauss, Riemann.
And this was certainly a great point in the development of geometry.
And what the conventional view is that it really represented the emancipation of geometry
from physics, whereas in the old days, geometry was tied to physics.
It was the description of the physical space.
When people took a more tolerant view, they said, there are many geometries.
We can write down axioms for Euclidean geometry, non-Euclidean geometry, curved geometries,
and one of them may be the right geometry for space, but nothing to stop us studying
all sorts of geometries.
Geometry is not restricted to the study of the actual physical space.
It's freer.
Mathematics is not earthbound.
And this, more generally, led on, had a big impact in mathematics in what one would call
the axiomatic era.
I mean, of course, Euclid based his geometry on axioms, but I think the axiomatic era,
in the modern sense, begins much later with people like Hilbert.
And Euclidean geometry certainly prepared the ground for the subsequent development
of axiomatic systems, the idea that mathematicians could lay down rules for themselves independent,
perhaps, of physical reality and develop the mathematical consequences.
And, of course, this has been, in many ways, a remarkable success.
And, of course, the fundamental notion in particular Riemannian geometry is the notion
of the curvature, not just the curvature of lines, but now the curvature of space.
And as you all know, this was subsequently fully used and justified back in a physical
context by Einstein's theory of general relativity.
So this is usually told as a great success story for mathematics, namely, mathematics
was originally tied to the physical space.
We were constrained by thinking in terms of straight lines and so on.
But when we got our freedom and were allowed to develop things in much greater generality,
we sawed into the stratosphere.
We invented these marvelous ideas of Riemannian geometry and curvature, all for its own sake
in a way.
And then subsequently, this was triumphantly justified when Einstein's general relativity
required precisely this kind of mathematics for its explanation.
And in many ways, that is, of course, true.
So the emancipation of physics was a preliminary necessary requisite to being able to come
back and help physics at a subsequent stage.
And in addition to Einstein's theory of general relativity, we have, of course, Maxwell's
theory of electromagnetism, which nowadays is viewed in a similar sort of light and understood
in terms of the curvature in some way.
The ideas of force and curvature are married together.
So this is, if you like, the culmination of the classical theory in which force is no
longer.
You see, originally, you had space, then you had things moving in space.
There were forces inside space.
But now subsequently, the force has become kind of ingrained in the space.
It is part of the structure of the space.
It represents the curvature of space.
So again, space has become much more complicated.
It is something which incorporates the physics into it.
And this is really the theme.
I'm sorry for my cold, which makes me hoarse.
But it does have the effect of slowing me down.
My theme, really, you see, is this interplay between physics and geometry over the centuries
in different ways.
And it's a very fascinating story if you really try to think about it a little bit not-so-superficially.
Now let me leave the development of classical physics, and let me go on to talk about knots,
the subject which Vaughan Jones has made such a spectacular progress with, knots and links.
And a knot is something that looks like that.
I can't draw complicated knots.
I try and draw a trefoil knot, and only about the third attempt do I get it right.
So I won't try and draw anything much more complicated.
And then a link is, of course, something which has several components which may link together
in that sort of way.
Well, of course, there are much more complicated ones.
Well, I can draw something a bit more complicated.
Here's one.
That's what you get by forming a braid and then joining the ends together to make a complicated
knot.
So knots are simply closed circles which are embedded in three-dimensional space, and links
are collections of circles which are embedded in three-dimensional space.
And of course, each component of a link may itself be knotted.
The question, of course, is how do you classify or distinguish between different knots in
space?
And anybody who's tried to undo his shoelaces when they get tied up realizes this is a non-trivial
problem.
Of course, it is a problem not in geometry, but it is a problem in topology, at least
in principle, although the size of a shoelace doesn't have much to do with the knot, nor
does the color of the shoelace, and so on.
It's a question in topology to classify knots or classify links and try to put them in some
kind of organized structure.
And as you see, writing down pictures isn't adequate because the same knot can be predicted
in many ways onto a piece of paper, and you have to know when two of these diagrams really
represent the same knot.
So knot theory is, by the way, a marvelous subject to explain to a lay audience, since
everybody knows what a knot is.
And I must admit, in my early years, when we were young, we always know better, and
I thought of knot theory as a very dull subject, which only specialists worked on and beavered
away.
So we, knowledgeable people, worked with much more abstract, highfalutin ideas, and knot
theory was not for us.
Well, we were wrong.
So knot theory is, in fact, a fascinating subject, at least in the present light.
A subject doesn't become fascinating until the right ideas emerge, by the way.
Now the history of knot theory is itself a very remarkable story because it does in fact
tie up with ideas in physics.
And the history of knot theory is the following, that in the middle of the last century, Kelvin,
the famous mathematical physicist, a contemporary of Gibbs, amongst, together with other, of
course, physicists at the time, they understood about forces, electromagnetic forces, and
so on.
They knew about fluids, but the structure of matter was still a mystery.
The modern atomic theory was still some decades into the future.
And Kelvin had the remarkable imaginative idea that an atom might in fact be a knot,
and he developed what he called a theory of vortex atoms.
Now the idea is based on the analogy with hydrodynamics.
A vortex, you know, is what goes down the plug when the water goes round, or it's one
of these cyclones that runs across the Midwest, or spiraling round.
And vortices have a habit of not liking to get, crossing each other.
If you have two vortices, for example, which are linked, and they'll move around, then
normally they won't cross each other.
And in particular, if you had a vortex, which happens for some reason to be in the shape
of a knot, it would remain in the shape of that knot, although it would change its exact
dimensions, but it would remain the same topological knot because it wouldn't want to cross itself.
So this idea that vortices had this kind of stability for knots suggested to Kelvin that
perhaps that's what atoms were.
So he put forward this theory that an atom is a knotted vortex tube, and if you ask him
what is a tube of, the answer was probably that it was something like ether.
If you ask what the ether is, well, you don't know.
But actually, one shouldn't be too down on people like that.
We use glibly many words nowadays that if we were actually pushed into a corner and
asked what they mean, we don't really know either.
Now, the theory that Kelvin put forward had many arguments in his favor.
I mean, Kelvin was a serious mathematical physicist, he wouldn't have put, and this
idea was not put forward just as a kind of jurisprudence, it was put forward seriously.
And the arguments, roughly speaking, are the following.
First of all, a knot is by its very nature stable.
As I explained, the knot, the particular kind of knot, if you move it around a bit, doesn't
change its character.
So you have stability, and of course, stability of matter is an experimental fact of life,
so you need to explain where stability comes from, so the idea that topology provides stability
is really a marvelous idea.
Secondly, of course, there are a large number of different knots, which is just as well
if you want to explain the large number of different atoms.
So there are plenty of them, and that's very satisfactory.
And then finally, it was the idea that, well, if you had a knot, it could vibrate, this
might produce the spectral lines, which you would emit by atoms.
So these were the kind of three basic arguments, and I can add a fourth one now, which is,
perhaps after the event, we don't know that in certain situations, in fact, you can convert
one kind of atom into another one.
We can transmute atoms at high energies.
And of course, that's not unrealistic from the point of view of knots, where there's
knots normally wouldn't change their character, if you forced the knot to cross itself, you
could convert it into another knot.
So under certain circumstances, you could change them.
This theory was regarded very seriously.
And for example, Maxwell said one time that this was the best theory of atoms yet put
forward at the time.
And for about 10 or 20 years, it was regarded as a serious contender.
And Tate, who was a mathematician who worked with Kelvin and also worked with Hamilton,
spent many years of his life trying to classify knots, because of course, if knots were going
to be the basis of the atomic table, you better understand how to classify them, put them
in some kind of order.
And so he spent many years producing marvelous tables of knots, which are still of great
interest.
Now, of course, that was the beginning of the history of knot theory as a branch of
physics.
But of course, I don't need to tell you that subsequently atomic theory took a different
term.
This was one of the brilliant ideas, which was just wrong.
So physicists dropped knots, they were left as a plaything for the mathematicians.
Once the mathematicians' appetites had been wetted and they saw all these knots around,
they began to study them for their own sake.
So the history of knot theory for the next 50 or more years was a straightforward part
of ordinary topology, in which basic ideas of topology of a general kind were used systematically
and certain amount of machinery was developed using basic ideas of homology theory, which
is closely related to ideas of linking numbers, the number of times the two links intertwine.
These are ideas related to homology, numbers of holes and so on.
And all of this took a very elegant form in the polynomial introduced by J.W.
Alexander in 1928, which is a very convenient and effective invariant of knots, which you
can write down.
So this was a very useful invariant of knots, which was discovered that far back.
And on the other hand, there is, of course, the more general idea of studying the complement
of the knot, the region outside the knot, and studying the fundamental group you get
by forming closed paths.
So that black line is meant to be a closed path outside the trefoil knot.
These form a group under composition, if you identify two paths which can be deformed into
one another, called the fundamental group of the space, in this case the complement
of the knot.
Of course, this fundamental group is quite a complicated thing, it's a non-abelian group.
And so to say that the knot may be determined by the fundamental group of the complement
is not in itself a very effective way of computing the knot.
Nevertheless, that's part of the story.
Now that was the situation in 1928, and although much work was done on knot theory afterwards
by topologists, the really big breakthrough which quite surprised all the experts in the
field came when Jones discovered his polynomial in about 1985.
This polynomial was one which superficially looked very similar to the Alexander polynomial
in the sense that both of them are polynomials in one variable with integer coefficients
which you could associate intrinsically to a knot.
In other words, if I give you the diagram of a knot, you can write down the polynomial
and it won't depend on the particular way in which that knot is represented on the plane.
So it had the same properties in that sense as the Alexander polynomial, but it was different.
Not only was it different, it had one very important differentiating feature, and that
was it could distinguish between mirror images of knots.
So, for example, the trefoil comes in two versions, the right-handed trefoil and the
left-handed trefoil, and don't ask me which is which.
But they're clearly – well, they look different.
I mean, of course, it doesn't follow because they look different.
They are different.
You might be able to take this one up, turn it around, and show it's the same as the
other one, but in fact they are different.
However, the Alexander polynomial can't tell the difference, whereas the Jones polynomial
can tell the difference, and that's true much more generally.
The Jones polynomial can be sensitive to orientation and distinguish knots from their mirror images,
whereas the Alexander polynomial never has that property.
This was a very deep difference, and that wasn't the only difference.
Another difference was that whereas the Alexander polynomial is just one, the Jones polynomial
turned out to have a large number of generalizations, so it was somehow part of a much more general
theory.
Moreover, these generalizations seemed to have some relationship to physics, in fact
more than one relationship to physics, although the exact nature of that relationship wasn't
really very clear.
Moreover, these Jones polynomials turned out to be extremely useful in practice, and
in particular they led to the solution of many conjectures which Tait had made in the
last century.
Well, more precisely, Tait had been producing tables of knots, and had more or less discovered
experimentally certain facts, and these facts, although he didn't give general proofs, were
then christened the Tait conjectures, and mathematicians tried to produce general proofs
over the next hundred years, but without success.
So it was really a very signal achievement of the Jones theory that he was able to deal
with these conjectures very satisfactorily.
There are a number of conjectures of Tait, and I believe at the present time that they've
all been established.
And just to give you a feeling for what these conjectures are, let me mention one of them.
And to describe this conjecture, I have to give you a little bit of terminology.
This is about the only bit of serious mathematics I'm going to do.
First of all, you have to know what an alternating knot is.
An alternating knot is when you draw it on the plane, and you go around the knot, then
the crossings are alternately over and under.
So the trefoil knot I drew before was, of course, an example with three crossings, and
they were alternately over and under.
Then of course, if you have some crossings which aren't over and under, normally you
would expect to be able to pull them apart.
So alternating knots, you expect, are fairly widespread.
Secondly, there are what are called reduced knots.
For example, if you get a picture like this, where the little shaded box is meant to indicate
some horribly complicated knot, which I don't want to examine in detail, the proverbial
black box, the green box in this case.
And if I have two boxes like that, which are tied together with two lines which cross over,
I can, of course, give them a half twist and undo and remove that central crossing without
increasing the number of crossings elsewhere.
So I've reduced the number of crossings by one, and you go on doing that until you can't
do it anymore, and then you have what's called the reduced knot.
So now if you have a knot which is both reduced and alternating, that looks like the kind
of rock bottom you might get down to.
And now the conjecture of Tate is that if you've done this, then at least the following
should be true.
That the number of crossings that you've got now is actually an invariant of the knot,
independent of the diagram.
In general, of course, if you write down a diagram, you may have a lot of redundant
crossings.
But if we've got rid of the obvious redundancies in this way, the conjecture is that the number
you got is now the smallest you can get, and any two which are reduced in this way should
have the same number of crossings.
This is a very, very plausible conjecture, and experimentally we've discovered to be true
for all the knots that were classified up to about 10 or 9 or 10 crossings.
Well this conjecture has now been proved, but not by elementary methods, but by using
these sophisticated ideas of Jones.
Well now, that was digression into knot theory.
Let us return to physics.
Now we ended up, I left you with physics in the classical era of Einstein, Maxwell, ideas
of curvature of space, forces, and so on.
But as you know, subsequent history of physics has been dominated by quantum theory.
Quantum theory was a shattering blow for mathematicians in a way, it altered and revolutionized
people's points of view, and in particular it seemed to be rather disastrous from the
point of view of the relationship to geometry.
So I think for a long time, geometers kept away from quantum theory, believing that it
simply didn't fit into their conventional framework.
But in fact, that was a mistake, and we've just learnt to discover that mistake, and
that's where the progress has come from.
So let me try and explain how that comes about.
First of all, there is a very famous experiment, or effect, which can be carried out and was
actually present in the early days, but wasn't realized until some time later, which is basically
the following.
Imagine you have here a cross-section of some magnetic flux tube, some solenoid, so inside
of that green thing there is a magnetic field, but there is no magnetic field in the region
outside, out there.
Then you take a beam of electrons, which you shine past this cylinder, and you see what
happens to the electrons when they come out the other side, and you make them hit some
plate.
And apparently what happens, if you carry out the experiment, is that there is some
kind of interference pattern which is produced there.
And the interpretation of an interference pattern is, of course, that quantum mechanically
that the electrons have, of course, a wave function, or they have a phase, and the phase
of the electrons that go around one side is different from the phase that goes around
the other side.
So if you prefer, an electron that went right around the whole thing would alter its phase.
Now phase is something which is not normally immediately visible, but when you have two
different phases you get interference.
So this is an effect, you see, where particles that move in a region without a magnetic field,
without forces, nevertheless undergo a physical change.
Now this is really very fundamental, you see, because it's going beyond the idea of action
at a distance, because now we have an action at a distance without forces.
Action at a distance was fine when we talked about forces reaching out into space with
fields which made things move, but now we have something happening in a region where
there are no forces.
So this is one step more mysterious, one step deeper, and this is a quantum mechanical phenomenon.
So this is the thing which we have to get to grips with and understand from the geometrical
mathematical point of view.
Now the point in this example is that if you go around a, if you have an electron that
goes around such a cylinder, then you have an effect, but if you have electrons that
keep away from that cylinder and don't go around it, there is no effect.
So that the physical effect is due to the fact that paths that go around something,
link it, have an effect, and paths that don't link it have no effect.
In other words, it's sensitive to the topology of the situation.
And so we can summarize that in the following way, say that classical physics is concerned
with forces which we understand or represent in terms of curvature, but quantum physics
goes beyond that and is concerned with things which have to do with topology.
And in fact, this fact here was present and known in a way much earlier in the work of
Dirac, which led to the arguments why electric charge occurs in multiple units of the charge
of the electron, an argument which is now fully recognized to be a topological argument.
So quantum theory, when you examine it closely, shows that even in regions where there are
no forces, nevertheless things can happen.
And the things that happen depend there on the topology of the situation.
Well, that's a lesson which ought to have been learned.
But mathematicians are slow to learn these lessons.
But still, we could ask now, since Jones discovered this Jones polynomial, it has something to
do with physics.
And I just told you that quantum physics has something to do with topology.
We could ask, is it possible that the Jones polynomial has really some kind of quantum
origin?
In other words, unlike the Alexander polynomial, which has to do with classical ideas of homology,
is it perhaps possible the Jones polynomial has something to do with quantum theory?
Well, the answer to that is yes.
Witten showed, essentially, that is a correct interpretation of the Jones polynomial, that
you can interpret it as arising from a quantum background.
And let me try to spend a little time trying to explain this to you.
Of course, it will be oversimplified.
Everything I say will be actually wrong.
But nevertheless, there'll be a germ of truth in what I say.
So first of all, I have to tell you that in present-day physical theories, we have other
forces in the universe besides electromagnetic forces and gravitational forces.
And these other forces are nowadays treated in a similar way to electromagnetic theory.
And one can understand those by saying that the Maxwell theory can be generalized to deal
with matrices instead of scalars.
And because matrices don't commute under multiplication, these are what physicists call non-obligatistic.
These are now the kind of present-day bread and butter of elementary particle physicists.
So in these more general situations, when you replace complex scalars by matrices, then
the phases, which normally would be an angle, something in a complex plane, these phases
can become like matrices or, if you like, like rotations in higher-dimensional space.
So these phases are non-abelian in the sense that they don't just add.
So in these more general theories, the phases are non-abelian.
Suppose we had a situation which is something like this.
This is meant to be some, again, a cross-section of the sort of thing we had before with some
kind of solenoids.
But this time, these solenoids are meant to be somehow non-abelian solenoids.
But again, the idea is that in the outside region, there is a vacuum, there are no forces.
And if you took a suitable kind of particle around this one, it will pick up some kind
of internal rotation, a non-abelian phase.
And if you went around that one, you'd get another non-abelian phase and so on.
And therefore, if you want to describe the nature of that background vacuum, it will
be specified by saying what all these different rotations were.
So there will be some parameters.
The vacuum won't be unique, but there will be some choices.
The choices will be these rotations and phases.
And that, of course, mathematically, you could put together and say, what you have is something
with some representation of the fundamental group, which in this case just has one generated
for each hole, with values in the appropriate set of matrices.
So a vacuum is, in this sense, the complicated thing.
Now that was trying to describe for you a static situation.
Suppose we now imagine things moving.
Now I'm going to draw for you a braid.
I drew you one example of a braid before, a couple of strands which overlap.
Now let me draw your braid and think of it this way.
This is meant to be a braid with three strands, not a very good one, I'm afraid, but there
you are.
So three points here, the twist in some way, and end up on the plane above.
Now you want to think of this as a spacetime graph.
So the extra dimension here is not the third dimension of space, which we disregarded when
we took a cross-section of the solenoid, we've thrown that one away, but it's time.
And therefore, you want to think of this picture as being a spacetime graph of the whole situation.
And so we can ask ourselves, how does the thing evolve with time?
Well, so we ask ourselves more precisely the following question.
We have a braid, this braid determines somehow what's going on, and that should lead some
kind of evolution of the vacuum from here to there.
And from what I've said, you might believe me that the effect of this depends only on
the topology of the braid and not on the exact position of the braid.
This vacuum is, of course, well, the first place you can think of the classical vacuum
I've described, namely the vacuum outside.
I took quantum mechanics into account only in so far that I took a single particle moving
in the background and thought of the wave function of that particle.
So I quantized the particle, but I didn't quantize the actual background field.
Now in modern day physics, you have to quantize everything, in particular you have to quantize
the background fields and quantum field theory.
So you have to treat the background quantum mechanically as well.
So when you ask how does the vacuum behave, evolve in time, you can't expect an answer
just in terms of dynamics in the conventional kind.
You have to explain the answer in terms of some kind of probability matrix that says
if you start in a certain quantum state here, what's the probability that you end up in
a quantum state there?
So if i and j are two different states, then there will be some matrix, Pij, that gives
you this probability.
And these matrices are in fact then invariants of the braid.
They depend only on the braid, some well-defined matrix, and in particular if you choose for
example to take the trace of the matrix, you will get a single numerical invariant of
the braid.
Now a braid is not quite the same thing as a knot or a link, because a braid has ends
at the bottom and the top.
However, as in the first example I gave you, you can join up the ends like that to close
the braid up and make at least a link, it may have several components.
So if you do that, with every braid you can get a knot or a link, which you might call
B with a hat on it.
And the marvelous thing is that, of course, when you think of this thing as a knot, you
no longer want to distinguish between the vertical direction and the horizontal direction,
because this is a knot in three-dimensional space and you're supposed to be able to move
it around as you like.
Whereas when it was thought of as a braid, it was thought of as preferentially moving
in a preferred direction.
However, if you're dealing with a physical theory, which has the fundamental properties
of relativistic invariance, which all good physical theories should have, then it should
be a consequence of relativistic invariance that what you get is independent of the way
you separated out space and time, and therefore what you thought of as an invariant of a braid
is in fact an invariant of the knot.
And that essentially, when I say essentially, left out all the mathematics, all the details,
but that gives you a feeling, I think, for how the Jones invariance can form quantum
theory in this way.
Now let me make a small comment here, which is the digression really, about three-dimensional
manifolds.
Talking about knots and links in three-dimensional space is very simple, but sometimes it's easier
or almost equivalent to talk about closed three-dimensional manifolds.
For example, the sphere is one, but you make much more complicated ones, and there's a
certain standard procedure by which if you start off with a knot in three-dimensional
space, I've just drawn a simple ordinary circle there, you can think of this lying inside
the three-dimensional sphere, then you can bore a tube out around the knot.
You take this tube out, you give it an appropriate twist, and then you put it back inside.
That alters the sphere and produces a different manifold, and in fact if you do this with
enough links and enough twists, you can generate all three-dimensional manifolds, and therefore
in fact the theory of three-dimensional manifolds and the theory of knots in three-dimensional
space are very closely related and of equal difficulty.
And of course the fundamental group of the component of the knot is in some way going
to be related to the fundamental group of the closed three-dimensional manifold.
I mention this partly in passing because I'm going to say something about four dimensions
in a moment.
And the other thing to point out is that I told you the Jones invariant was sensitive
to orientation of space, it can distinguish between a knot and its mirror image.
Now where does this come from in terms of the physics?
So far in physics what have we used?
We use quantum theory, we use non-abelian gauge theories, we use relativity theory,
you might think that's about everything.
Well what we haven't used is what's called parity violation.
It used to be thought that the laws of physics were invariant under mirror images, but it
was discovered by Li and Yang, or at least put forward by Li and Yang, and subsequently
verified experimentally that there are certain physical processes which do distinguish between
right-handed and left-handed systems.
And the mathematical theories that allow for that are the non-abelian gauge theories, and
those are the ingredients that go into the Jones polynomial, and therefore it's no surprise
that what we get is something that is sensitive to orientation.
So we've used that aspect of the physics as well.
So what's remarkable is that goes into this is not just a little bit of quantum theory
and so on, but every conceivable piece of modern sophisticated elementary particle theory.
Now let me move to the other topic I want to talk about which I will deal with more
briefly, partly because I'm short of time, partly because it's more difficult and I couldn't
do it so well in the short period I've got.
But in Donaldson theory, we started in about 1982 under which those of you who are interested
can consult the recent book by Donaldson and Cronheimer, Oxford University Press, 1990,
and very roughly what Donaldson showed was that the usual kind of topological invariance
which topologists know about were not sufficient to classify four-dimensional smooth manifolds.
So I just talked about closed three-dimensional manifolds to give you a kind of warm-up.
Now imagine going one higher and look at four-dimensional manifolds, and we'll come back to the relationship
with physics shortly.
Now to give you a flavor of the sort of results that Donaldson produced, why they were so
spectacular, it's helpful to think about the case of surfaces.
Now surfaces, theory of closed surfaces, orientable surfaces, reman surfaces is of course
a classical part of mathematics, and reman surfaces are also the same as algebraic curves
over the complex numbers.
And one knows algebraic curves or reman surfaces have a very simple topological structure,
namely they have a genus and every surface is essentially going to be built up in this
way.
For example, in this picture there are three tori, which you connect by bridges and that
gives you a surface of genus three.
And every surface is simply a connected sum in this way of g copies of the basic torus.
So there's a very simple structure for reman surfaces and also for algebraic curves.
Now for four-dimensional manifolds, well, of course, not every, unlike the case of surfaces,
not every four-dimensional manifold can actually be obtained by algebraic geometry, but nevertheless
many can.
If you have an algebraic surface now, something with two independent complex variables, then
that becomes four real variables, therefore algebraic surfaces, for example, given by
single polynomial equation in three-space, give examples of four-dimensional manifolds.
Now the sort of results that Donaldson proved here are very roughly two of his main statements.
The first is, unlike the case of curves, it's not true that every algebraic surface in some
sense breaks up into simple, irreducible surfaces.
On the contrary, an algebraic surface essentially can't be broken up at all, except in some
rather trivial way, which I won't go into.
So this is a very remarkable fact.
All these millions of algebraic surfaces, you can write down by equations.
Even if you regard them as differentiable manifolds, they cannot be analyzed into smaller
building blocks.
So this is a very spectacular fact.
And also at the same time, here's another fact.
There are lots and lots of algebraic surfaces, which you can show are topologically the same,
but differentially, from the point of view of their smooth structure, turn out to be
different.
So there is something very subtle going on, which topology alone cannot detect, but nevertheless
differential calculus, if you like, can distinguish, but in some very subtle way, because you can't
see it by conventional methods.
So the question is, what is it that enables one to detect these subtleties?
Well, as with the Jones theory, where Jones produced invariants of three-dimensional knots
in three-space or three-dimensional manifolds, so Donaldson produces invariants of four-dimensional
manifolds.
I'm not going to tell you exactly how he produced them, let's just say he had invariants
of four-dimensional manifolds, and with these invariants, he was able to prove the sort
of results I just mentioned.
Now, more than that, not only did he have invariants of four-dimensional manifolds,
but he had invariants in the following situation.
Suppose you have a picture like this.
Now, x0 and x1 are meant to represent closed three-dimensional manifolds, and y is meant
to represent some four-dimensional manifold which interpolates between them, and has x0
and x1 as its two ends.
And from the point of view of physics, so to speak, you want to think of, again, this
as being something like our picture of the braid evolving in time.
You want to think of this as time, and these as space.
Now we're dealing with three plus one dimensions, actual physical spacetime.
So whereas the Jones theory is related to quantum physics in two plus one dimensions,
in other words, one of the spatial dimensions is regarded as thrown away because things
are invariant in that direction, the Donaldson theory is really concerned with things in
fully four-dimensional spacetime, and therefore, in that sense, is deeper.
Now, the idea is that you imagine that at both these ends, you give yourself what you
might call a vacuum, namely a representation of the fundamental group of the three-dimensional
manifolds in question.
I told you that representations of fundamental groups in the two-dimensional case were something
like a physical vacuum, so you might accept this as some description of the vacuum states
here and here.
And now, approximately, and this is very approximate, what Donaldson does is to define invariance
of this intervening manifold y, which give you a matrix M alpha beta, or telling you
which given beta and alpha produces a number, and as you vary alpha and beta, you write
those out as a matrix, if you like, and these are then invariants of the manifold y with
the two boundaries and the two bits of data given by representations of the fundamental
group.
Now, these, the formal properties of these invariants of Donaldson, if you examine them,
turn out to be very similar or reminiscent of the formulas which quantum probabilities
would have.
And in fact, that's exactly what they are, Witten has shown that Donaldson's theory
can be interpreted in terms of a suitable quantum field theory, so these numbers are
in fact the quantum probabilities of going from this vacuum state to that vacuum state
through, so to speak, this intervening four-dimensional manifold y.
So, you see, the spirit of this is very similar to the spirit of the Jones invariance, everything
going up one dimension, and of course, a different quantum field theory, the particular choice
of quantum field theory, of course, if you like, depends on how you're interpreting
this in terms of particular forces and so on.
Now, both these examples, the Jones theory and the Donaldson theory, are examples of
what Witten has now christened the notion of a topological quantum field theory.
And let me just try to end up by explaining what this notion means to you.
A quantum field theory usually is defined somewhere in space, space-time, where you
give yourself a background metric, a Lorentz metric in space-time, of course, metrics
are essential in order that you can measure things, so all physical measurements, observables
depend on measurement, so there is a background metric in usual physics.
However, it may sometimes turn out that if you write down the formalism of a quantum
field theory by the usual rules that physicists do, you may discover something miraculous
happening occasionally, which is the outcome, the results, or the probability that you want
to calculate turn out to be independent of the background metric and depend only on the
underlying topology.
And that's such a theory, which is rather rare and wasn't thought of before, is what
Witten calls a topological quantum field theory, and the Donaldson theory and the Jones theory
are both examples of this notion, and there are many others.
Now, such a theory has, of course, no real dynamics, only statics, but now it's quantum
statics because things aren't considering any serious movement, everything is now in
the vacuum state.
There are no energy levels, everything is down at the ground state, but nevertheless
the vacuum is now a very complicated object, and the topological content of the theory
lies in the structure, the quantum structure of the vacuum.
Well, that's meant to give you some flavor of what these things mean.
Let me perhaps, for those of you who are familiar with Hodge theory, make some comparisons which
might be helpful.
Ordinary Hodge theory is concerned with the study of Riemannian manifolds, and on a Riemannian
manifold you can first of all define the generalized Laplace operator, the Laplace-Beltrami operator,
which acts on functions, and where the only zero eigenvalue is the constant functions,
but then you can consider the Laplacian acting on differential forms.
This is the Hodge organization of the Laplacian, and this has the property that the zero eigenvalues
are called the harmonic forms and have a topological interpretation as giving you the comology
of the manifold.
And of course the homology of the manifold is independent of the metric, but you use
the metric in defining the Laplacian, but you get back to the topology.
And of course here we are talking about the zero eigenvalue, which is the ground state.
So here we see an example where the ground state of these operators, which you can think
of as operators of quantum theory in some sense, are purely topological.
Because these topological quantum field theories are some much more sophisticated analogue
of the Hodge theory in this sense.
They're much more sophisticated for lots of reasons, they're not linear, they only work
in certain dimensions, and so on and so forth.
So we might just say that topological quantum field theory is the rock-bottom topological
content of physics.
Now you have to give a kind of quote for the newspaper, journalists, and that's what I'm
offering, headline for the San Francisco Chronicle tomorrow.
So let me sort of reflect on what I've been saying, what we learned.
So classical physics was related to classical geometry, measurements, distances, curvature,
forces, and so on.
And there was an interplay in which the forces of physics were first of all used to motivate
the geometry, and then there was this feedback, and so on.
But now what we're saying is that to probe the deeper aspects of geometry, these topological
aspects represented by the Jones theory and the Donaldson theory, we need to do quantum
physics.
And what I'm trying to suggest is really that here we want to understand space.
And the title I gave you was this mysteries of space.
And it really is mysterious.
And we now discover that the best way to understand some of the subtler things is to probe the
thing with quantum theory.
In the old days, we'd probe things just with classical methods, with rulers, and so on.
Now to really get to the bottom of it, we have to insert quantum ideas to probe into
the geometry.
And so you see, moral of this is that the emancipation of geometry from physics, or
if you like, the axiomatization, was too early.
We thought we knew what geometry and physics were.
We had these ideas of curvature and space and so on.
And we thought that was all there was to geometry.
And that should be enough to enable us to understand all the problems of geometry.
Well, we've learned now that that was a mistake.
And we've learned it by going back to the physics and realizing that there's more to
physics than classical physics, and the quantum physics has to be put into the geometry as
well.
And so, in a sense, we threw things out too early when we axiomatized our notions of geometry.
Well, mystery is a matter of wonder and inspiration.
I mean, I don't claim that we really can understand all these things.
They are, I find them quite miraculous, the way the quantum theory and geometry interact.
What their real meaning is, is still mysterious.
For example, what is the significance of these topological ideas for real physics?
I emphasize that these topological quantum field theories are sort of physics with no
dynamics, just statics.
They're not about what we usually think of as physics.
So the question is, what do these topological ideas have to do with real physics?
Well, we don't know yet, but it would be surprising in view of current developments, if they didn't
turn out to have a very important role in theories that can ultimately emerge.
And if that turns out to be the case, or even from what we know at the moment, I think you
might say this is some ultimate justification of Kelvin's ideas.
You see, Kelvin had this marvelous idea that topology should be useful for physics.
Well, like all ideas, it turned out not to be quite right in the form in which it was
applied.
But a good idea has the habit of coming back again and being used in a different way.
So now we see that topology is being again used, and again it's being used in the same
kind of fundamental way.
It's things like stability, which are really involved.
But now we deal with things at a much smaller scale.
We no longer satisfy with atoms, we think of those very large.
We go down to really minute particles.
And so this very, very small world that we physicists are now dealing with, these ideas
of topology are coming back into their own.
So I think it's a justification for saying that a good idea, even if it is discarded
one level, can come back at a different level.
Well, let me just finish with the following philosophical comments.
Some of my friends, some of my best friends, my best friends have the habit of saying these
things, tell me that when I talk about things like this, I am not actually doing mathematics.
I'm indulging in fiction.
This is some sort of marvelous scenario, but actually when you try to spell it all out
in precise mathematics, it tends to be a little bit elusive.
And I will concede that things aren't quite as well understood and as simple as I've tried
to indicate.
But mathematics, I think we all agree, is really a language for science.
And like all language, it can be used for many different purposes.
It can be used for fiction, it can be used for poetry, and it can be used for legal jargon.
And well, some of you may prefer the legal jargon, but I prefer the fiction and the poetry.
Thank you.