SPEAKER 1: --despite having the
same last name as me is
not actually a relative
of mine.
but he is a friend who I have
known for very long time.
In fact, he and I wrote our very
first computer program
together, about almost
30 years ago now.
I was an adventure game
written in basic.
After that, Waddie went
on to college.
even when he was in high school,
he worked for Thinking
Machines Corporation, produces
the connection machine.
And he worked for them
again after college.
But then, unfortunately, the
intellectual rigor and complex
math of computer programming
became too much for him.
And he moved over into
theoretical physics.
He's now a professor at
MIT at the Center
for Theoretical Physics.
And we've lost our screen.
Well, he's here to talk about
how big the universe is, which
for Google means how many data
centers we're going to need.
so please welcome Professor
Washington Taylor.
WASHINGTON TAYLOR: Great.
Thank you very much, Ian.
And thanks to Ian and the
seminar organizers for
inviting me here.
It's a pleasure to be here.
I'm very pleased that so many
of you can come to a talk
which is probably going to be
about as far from practical
application as any talk you
might have heard here.
So what I'm going to talk about
today is a story which
has emerged in recent physics
having to do with how big the
universe is, and how we're
going to make predictions
about future experiments, things
that will be seen at
accelerators that are not yet
built, things that'll be seen
in cosmology, in the context of
a rather challenging world
view, which has come out a
string theory and cosmology in
the last few years.
So I'm going to start off--
It's a little shift
to the left.
Is there any way to fix that?
SPEAKER 1: The image on the
screen is a little off.
It is shifted a little off.
WASHINGTON TAYLOR: Well,
I'll carry on.
So let me summarize the big
picture in a few words.
And then I'll get into a little
more of the details of
this, and how this turns into
what can be a very difficult
computational problem.
So recent evidence coming from
two different directions.
On the one, hand from
cosmological experiments,
which measure things that
happened in very early history
of the universe that
we see at very far
distances away from us.
13, 14 billion light
years away.
These observations have combined
with some recent
theoretical developments in
string theory to give us a
picture of the universe which
is radically different from
that which we had in the past.
It's the next step in a
Copernican revolution where,
in our little patch 13 of 14
billion lightyears wide of the
universe, seems to be
potentially part of a much,
much, much larger system.
And in other regions of this
much larger universe, going
beyond the horizon to which we
can see with our experimental
apparatus, there may be regions
in which the physics
is governed by somewhat
different laws.
And in some places, there may
be regions where the laws of
physics look very different.
All governed, in some sense, by
an underlying theory, which
perhaps is string theory.
But from the point of view of
traditional particle physics,
these different regions in the
universe may have very
different kinds of physics.
So one of the challenges in
this picture, where the
universe is so bit and physics
is different in different
patches of the universe,
is making predictions.
So in some places in the
universe, physics may be very,
very different from ours.
And there may be some places
in the universe where
everything is almost exactly
like it is here, but if you
look at the electron mass, the
ninth digit of the electron
mass may be off by one.
And if that is, in fact, the
case, it means that that ninth
digit of the electron mass is
not something we can predict
based on fundamental
physical theory.
So the title is "When is a
Googol Not Enough?" And I'm
sure each of you at some point,
since coming here or,
coming here has gotten into some
discussion about how big
the number of googol
really is.
And as you all, I'm sure, know a
googol is pretty much bigger
than anything you could run into
in almost any physical
problem, or any problem of
relevance to nature.
Because a googol is
approximately the number of
particles in the observable
universe.
So you might think that you
will not need to deal with
numbers or computation problems.
Well, obviously,
some computational
problems search
exponentially big spaces.
So you might need to search
combinatorial
spaces of that size.
But you certainly wouldn't
expect to see this number
appearing in a characterization
of a physical system.
The point of this talk is that,
in this new picture of
the universe, it seems like
there may be far more than a
googol different regions of the
universe, each with their
own laws of physics.
String theory, in particular,
seems to have more than 10 to
the 1,000 different solutions,
each of which corresponds to
some kind of local physics.
And if the current paradigm that
is emerging is correct,
each of those may be realized
somewhere in the universe.
This poses a huge computational
problem.
Even if we can, in principle,
completely
understand string theory.
Even if we could define it,
write down the equations, and
even solve those equations, how
would we sort through this
10 to the 1,000 or more possible
solutions to find the
one the matches our world?
And so, what I want to talk
about today is filling in a
bit more as to why we ended up
at this picture, telling you
about how to think about the
problem of making some
prediction based on this model,
and then talking a
little bit about some of the
computational problems. I and
some other people have been
working on actually starting
to systematically look at this
space of solutions, and trying
to understand if there are
correlations, if there are
structure, if there are things
we can do, which make this
into a computationally
intractable problems. So I'll
talk about some of that.
I encourage questions from the
audience during the talk.
Yes.
AUDIENCE: So So [INAUDIBLE]
WASHINGTON TAYLOR: So the
question is could each of
these solutions be realized
just once?
No.
As will see as we get through
the story, probably each of
these solutions is realized
many, many times.
So the size of the universe is
probably much bigger than
this, if this picture
is correct.
You could take the currently
observed size of 13.5 billion
lightyears, you could tack on
pretty much as many zeros as
you want after that, and you
might still not be quite
approaching the size of the
spacetime that you would need
to capture all the physics
that might be realized.
And I'll talk a little bit about
how things got so big.
Why the universe might be so
big as we go through this.
But I encourage questions
as we go through.
I'm going to try to keep this
to a fairly simple and
straightforward core of what
I'm talking about.
But there's lots of things
I'm going to gloss over.
And if people are interested,
I'm happy to say a little more
about anything.
OK.
So here's a quick outline of
what I'm going to talk about.
I'll start by discussing the
idea of cosmological
expansion, how the universe
got to be so big, and some
issues which have to do with
recent cosmological
observation, telling us
about the rate of
expansion of the universe.
Then, I will talk about string
theory, which is a formal
development, a theoretical
framework in which people have
been trying to describe
quantum gravity.
And then, I'll bring them
together to talk about the
vacuum problem.
The problem of looking at these
many solutions of string
theory and figuring out
which one might
correspond to our world.
Finally, I'll talk about the
computational challenge that
appears when we have this
huge space of solutions.
OK.
So let's start with
cosmic expansion.
So the basic feature of gravity,
which makes it
different from all the other
forces, is that all objects
exert a gravitational
attraction
on all other objects.
That's not true of
electromagnetism.
If you have two positively
charged particles, they will
repel each other.
If you have oppositely charged
particles, they will attract.
In gravity, everything
attracts.
So although gravity is, by far,
the weakest of all the
forces in nature, this fact
that it is a universally
attractive force is really what
makes it such a relevant
force in our everyday lives.
Because we're interacting
with big objects,
the sun, the earth.
These things all have lots of
mass and they attract us, and
they attract each other, and
cause most of Newtonian
gravitational physics
to happen.
So in the absence of any other
effects, if you have a bunch
of objects floating around in
the universe, they will
gravitationally attract
each other.
And they will begin
to come together.
So one consequence of that
observation is that we can't
have a static universe if the
only force between these
objects is gravity.
If you just try to populate the
universe with a bunch of
objects and keep them at fixed
points, they'll all attract,
and things will come together.
There'll be some kind
of collapse.
So about a century ago, Einstein
was worried about--
Yes, please.
Question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: That's
a good idea.
Great.
Yeah, it's a little smaller.
But that's probably--
Yeah.
Let's do that.
Thank you for that suggestion.
OK.
So about a century ago, Einstein
worried about this.
And Einstein wrote down this
beautiful theory of general
relativity.
The basic equation of general
relativity is there's a thing
called Gmu, nu, which
characterizes the
curvature of spacetime.
And there's a thing on the right
called Tmu, nu, which
characterizes the amount of
matter and energy that's
moving around in
that spacetime.
And he had this beautiful and
simple equation, Gmu, nu
equals 8pi Tmu, nu.
I promise I won't have
very many equations.
But this is one of the very
few I will include.
He had this beautiful equation
saying that gravity
is caused by matter.
You have matter.
It causes spacetime to curve.
And in turn, the curvature of
spacetime causes things to
move on curved trajectories.
And that explains
all of gravity.
But he couldn't explain
why things weren't
all collapsing together.
So he stuck in a fudge factor,
a little extra term in this
equation, which is that thing
in blue over on the right.
He added a little gmu, nu--
That's called a metric.
It describes distance scales
in spacetime--
multiplied by a constant
called capital Lambda.
So capital Lambda is
something called a
cosmological constant.
And he stuck it in, basically,
to allow his theory of general
relativity to have a static
solution, a solution where all
the stars were fixed
in the heavens.
And they weren't coming towards
each other, and they
weren't going apart.
Well, Hubble pointed out, after
some observations, that,
in fact, it seems that the
universe is expanding.
All objects are moving apart
from all other objects at a
fairly rapid rate.
And what that means
is that we're
not in a static universe.
And Einstein said--
Oops.
And he characterized the
addition of this term into
this equation as his greatest
mistake in his career.
Because he said, we
don't need it.
We can get rid of it.
Basically, we have the
universe expanding.
And gravity will just cause the
rate of expansion of the
universe to gradually decrease
as the objects
attract each other.
So ever sense then, people have
basically believed that
this thing, the cosmological
constant, is zero.
You set it equal to zero, the
universe expands, there was
some big bang, the rate of
expansion gradually decreases
as gravity pulls everything
together.
And that, people thought, was
the end of the story.
Now, in the last few years,
people have done very careful
observations of distant
supernovas, of the large-scale
structure of microwave radiation
left over from just
after the Big Bang.
A lot of detailed cosmological
observations.
And what those things tell us is
that the rate at which the
universe is expanding is
no longer the race.
It's actually starting
to accelerate.
So distant objects, you would
have thought gravity would
just slow down their rate of
motion away from each other.
Distant objects are now starting
to accelerate away
from us at a faster
and faster rate.
So this implies that the
cosmological constant is not,
in fact, zero.
What the cosmological constant
does it exerts a constant
pressure on everything else,
which causes things to expand.
It can be characterized as
kind of a dark energy, an
energy which cannot be measured
in any particular
way, but which seems to be
causing things to move apart.
So it now seems, and there's
pretty convincing evidence for
this, that for about seven
billion years-- about half the
lifetime of our universe,
of our local
patch of the universe--
the rate of expansion
has been increasing.
Things have been moving
apart from each
other faster and faster.
And this is now a big
puzzle in cosmology.
You might have seen discussions
in some popular
science or more technical
things on dark energy.
This is the issue that we now
think that much of the energy
in the universe is made up of
this dark energy, which is
forcing everything to
accelerate apart.
So if you characterized the
amount of this dark energy,
the stuff which is causing the
rate of expansion to increase,
the thing which Einstein thought
was a blunder, but, in
fact, seems to be there, if you
characterize the amount of
that stuff in natural units--
which you can make everything
dimensionless so you end up with
a dimensionless number--
the number describing the
cosmological constant seems to
be about 0.00 and there's 119
0's, and then there's a 1.
So it's 10 to the minus 120.
It's an incredibly
small number.
So somehow, the universe is very
finely tuned, so that,
for a long time it was expanding
under the influence
of the initial energy causing
the Big Bang, and things were
slowing down.
And then, only very late in the
history of the universe--
after galaxies had formed and
lots of structure was here--
has this cosmological constant
taken over and forced things
to start moving apart
faster and faster.
OK.
So that's the cosmological
observation.
It seems that the universe
is expanding.
But the thing causing it to
expand is an incredibly
finely-tuned number.
And from basic physics, it's
not obvious why this number
should be so finally tuned.
Any questions about that?
OK.
So let me come to a slightly
different aspect of this
story, another set of
cosmological observations,
which also tell us about some
expansion, but in a slightly
different context.
So there was a thing called
the Wilkinson Microwave
Anisotropy Probe, which went
up in space and was very
carefully measuring the
microwave radiation coming
from the very early universe,
coming from different
directions.
You might have seen pictures
of that, also, if you read
some popular science things.
According to those measurements,
there's an
incredible similarity between
patches of the universe in
different directions
in the sky.
And what that means physically
is that it seems like those
regions must have somehow
been in causal contact.
Things that happened in one of
those places that we see at
the very far distance must
have influenced others.
But according to a simple model
of the expansion of the
universe and the history of the
universe, there's no time
in history when those
things would have
been in causal contact.
Because the light from those two
things, the two different
directions, is just
now meeting us.
So if you have light rays coming
from two points coming
together and just now meeting
us, it's very hard for a light
ray from one of those to have
gotten to the other in the far
distant past of the universe.
So people found this
very puzzling.
But there's a nice simple and
beautiful explanation for
this, which is the notion of
inflation, which a fellow
named Alan Guth, who's now
at MIT, came up with.
The idea of inflation is that,
in the very early universe,
probably even before the Big
Bang, there may have been a
significantly large cosmological
constant.
And that large cosmological
constant, as we've discussed
now, causes the universe
to expand.
And if that number is not 10 to
the minus 120, but 1/2 in
natural units, then it
causes the universe
to expand very quickly.
In every Planck unit of time,
the size of the universe will
multiply by a factor of e.
So you rapidly get exponential
growth.
And like stock markets until
they undergo corrections, it
would just get bigger and bigger
and bigger, faster and
faster until some correction
would kick in and move us into
the phase of the universe we're
now in at this time.
So the scale of the universe
in that period of expansion
would go as an exponential
in time.
Now, we do not now live in a
universe which is incredibly
rapidly inflating.
So there was an initial picture
of inflation which
Guth came up with, which I will
describe here 'cause it's
relevant to the later
part of the story.
And we can combine the two
things we've talked about.
In the distant history,
the cosmological
constant was bigger--
and now, it's incredibly
small--
in this little picture, where we
have some kind of potential
function, sort of a landscape,
where there's one minimum of
this potential at
a high value.
So it might be that the universe
used to be living in
this red dot, Vacuum 1 where the
cosmological constant was
fairly high and things were
inflating very rapidly.
In classical mechanics, if
you're in a local hill valley
like that, you'll never
get out of it.
But in quantum mechanics, you
have a wave function which can
creep out over the hill, and
suddenly, you can sneak out--
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: It's true I
didn't want to move too far
away from the microphone
because then
nobody will hear me.
So if we moved out int the
green minimum by quantum
tunneling, we get a wave
function which moves out of
the first vacuum into
the second vacuum.
At a certain point, we can
suddenly get a region, a
bubble of the second
vacuum popping out.
And we would move into, by
quantum tunneling, the vacuum
that we now live in, which
was a much smaller
cosmological constant.
So let's just keep that
picture in mind.
This is one scenario for how
we could get a very high
cosmological constant
in the past and a
smaller one in the present.
This scenario, as described,
can't be exactly right.
This is the old version
of inflation.
And there's some newer ideas
which are taken into account.
But this is actually more
relevant for the story I'm
telling today.
Yes.
Question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: Yeah.
I'm actually talking about the
whole universe, or a region in
the universe.
So here's the way to
think about it.
If we live in a situation
where everywhere in the
universe we're in Vacuum 1,
what can happen through
quantum tunneling is a patch
of a certain size can creep
into Vacuum 2.
And if that patch is
big enough, that
patch will take over.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: Well,
what it'll do is
it'll form a bubble.
And that bubble will expand
under it's own
cosmological constant.
It'll expand at the
speed of light.
The region outside it will
also be expanding.
And I'll come to you a picture
in a little bit where you'll
see what the universe
starts to look like.
You get a background where the
universe has a large domain in
region one with a bubble
inside it in domain
two, in Vacuum 2.
And both of them
are expanding.
So each one of them is getting
bigger and bigger over time,
due to its own cosmological
constant.
Yes.
Another question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: Good.
So the question is, is there
anything which rules out
negative matter, something which
would cause a repulsive
force in gravity?
So there have been a lot of
experiments testing gravity in
great detail, and looking for
so-called fifth forces, which
would be additional forces which
might be interpreted as
negative gravitational forces.
So far, none of those
experiments have come up with
anything definitive.
There are alternatives to
Einstein's theory of general
relativity.
None of them are, in my mind
at this time, equally
attractive as explanations
of what we see in nature.
But there are some puzzles
having to do with both dark
energy and a more prosaic kind
of dark matter, which is extra
material that seems to
be in the universe.
These detailed observations by
WMAP tell us that only a small
fraction, less than 10%, of the
energy in the universe is
the stuff we can see.
Barions and the stuff
that makes us up.
Some fraction of it is made out
of dark matter, which is
some kind of matter we
can't yet observe.
And some part of it is made
out of the dark energy.
And there are alternatives to
Einstein's general relativity.
One, in particular, is called
MOND, which modified Newtonian
dynamics, which purport
to give an alternative
explanation.
I would say, at this point in
time, these are on the fringe
of what's widely accepted.
It's pretty uncontroversial
that most experiments very
strongly support Einstein's
theory of general relativity,
which would give us a picture
where the mass that appears in
the term sourcing the
gravitational field is the
same as the thing which
is affected by the
gravitational field.
And that just always gives
you an attractive force.
Good question, though.
Other questions?
OK.
So we've talked a little
about cosmology.
Next, I want to move
to string theory.
So what is string theory?
And why do I want to
talk about it?
So there are four forces in
nature that we have observed
and verified with clear
experiments.
There's electromagnetism,
which we're all
very familiar with.
It underlies all our
electronics.
There's the strong and the weak
nuclear forces, which are
equally important.
The strong nuclear force holds
together the charged particles
in the nuclei of atoms. And the
weak nuclear forces, also
relevant in nuclear
interactions.
And then finally,
there's gravity.
As we've said, gravity is the
weakest of these forces, but
the only one which is
universally attractive.
Now, we have this really snazzy
mathematical framework
for describing theories
of forces.
It combines quantum mechanics
with field theories like
electromagnetism.
It's called quantum
field theory.
And it's a very elegant
theory.
It accurately describes
three of these forces.
It describes electromagnetism,
the strong and the weak
nuclear forces.
But if you try to use quantum
field theory to describe
gravity, you run into
significant problems. All
kinds of hell breaks loose.
And you really can't make sense
of what you've got.
The issue is essentially that,
in quantum mechanics, you are
looking at all possible ways
that a system might fluctuate
or make a trajectory from one
point to another in its space
of possible configurations.
And in the case of quantum field
theories, that means you
imagine that a particle could
take all possible routes to
get from A to B,
and the you sum
something over all of them.
In the case of gravity, that
means that you have to sum
over all possible ways
that spacetime
itself could fluctuate.
And that would include things
like little wormholes
appearing and disappearing, and
very complicated things
happening with the geometry and
topology, which we don't
have any mathematics
to really handle.
So there's some foundational
level at which we don't have
the right mathematics
to do it.
And there's a practical level
where you just try to naively
apply the rules of quantum
field theory.
And you get infinities
you can't deal with.
There are infinities in all
quantum field theories, but
some of them you can sweep under
the rug very easily.
Those are called renormalizable
theories.
Gravity is not a renormalizable
theory.
It has further problems.
So over the last two or three
decades, people have started
to work on an alternative to
quantum field theory which is
premised on a very simple
starting assumption, which is
just that, rather than
point-like particles, you
assume that the objects
moving around are
little loops of string.
They're little one-dimensional
objects.
And if you make that assumption,
you change the
picture of spacetime where a
point-like particle will move
along and emit a photon.
That's an electron emitting a
photon which then connects to
another electron.
In place of that, you get a
picture where two loops of
string will interact
by sending a
loop of string across.
This gives you a smooth picture
of how strings interact.
And that smoothness of the
picture on the right, compared
to the picture on the left, is
what makes it a much better
behaved theory in some sense.
So if we just do this simple
technical thing, we replace
point-like particles with little
loops of string, we
discover that we automatically
get a quantum field theory
which has forces like
electromagnetism.
But it also has gravity in it.
It just pops out
automatically.
So that's great.
I think we're definitely
on the right track.
The biggest catch here is that
this theory lives most
naturally in 10 spacetime
dimensions.
So if you think that this is
the right theory describing
quantum gravity-- and right
now, it is really the only
consistent and robust framework
we have to describe
quantum gravity and a higher
number of spacetime dimensions
in its variance--
you're stuck dealing with
10 spacetime dimensions.
So how do we deal with that?
Well, there's a notion which
people have pursued for many
years, which is called
compactification.
And basically, the idea is maybe
not all the dimensions,
of those 10 dimensions,
are big.
Maybe you can get rid of six
of them by curling them up
into some very small
dimensions.
Like, imagine a very
long straw.
If you look at a very long straw
from a ways away, it
just looks like a line.
Actually, the surface of the
straw is two-dimensional.
It's curled up in a
little cylinder.
But if you're far enough away,
you don't notice that.
So we may be living in a
spacetime which is really 10
dimensional, but we're only
seeing four of those
dimensions.
The other six are curled
up very small.
And the structure of those
extra six dimensions,
depending on exactly how you
curl them up, can you need
different kinds of quantum
field theories.
That is, it can give you
electromagnetism, and the
strong and weak nuclear
forces.
Or it can give you things which
are slightly different.
And that's where we're going
to get to this issue that
there's lots of different
solutions.
Yes.
Question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: Good.
So the question is whether
there's a natural reason for
the dimensions to curl
up like that.
There are a number
of suggestions.
And it's obviously very
attractive to try to find a
reason why we would live in four
dimensions rather than
different number
of dimensions.
I would say, at this time,
none of them are really
convincing.
There are some notions that
there are extended objects
that might wrap around some of
these dimensions, and they
would keep some of them
from expanding and
allow others to expand.
And there's some numerology
that might make that a
plausible story.
But I would not say that it's
yet a really robust story.
So in particular, you can
already see that the number of
dimensions of space time that
we observe is not obviously
predicted by string theory.
So in some of those other
regions I mentioned, the
apparent dimension of spacetime
may be different.
If you go far enough away, our
next door neighbors in the
next patch may have a
six-dimensional apparent universe.
That's one of the things that
might be different.
One of the other things that
might be different is they
might have a different
compactification.
They might be compactified down
to four dimensions on a
different manifold, which would
give some other quantum
field theories.
So you can start to see how
you're going to get a lot of
different solutions.
OK.
So an important question
is, is string
theory a unique theory?
So there's a famous parable of
the Indian parable of the
blind man and the elephant which
people like to trot out
at this time.
So I will trot it out.
Here's the elephant.
You have a bunch of different
blind men.
And there's an elephant.
And each one of them approaches
the elephant and
touches a different part
of the elephant.
And one of them describes it
as being sort of hard and
smooth, because they were
touching the toenails.
And one of them describes it as
being kind of like a snake
because they were touching
the trunk.
Anyway, string theory is
a little bit like that.
For a long time, we thought
there were five different
string theories.
And they had these names
you can see up here.
Type IIA, type IIB, heterotic
SO(32) et cetera.
I've got six, and I'll mention
that in a minute.
What was realized about 10 years
ago is that these are
all different pieces of the
same underlying structure.
There are things called duality
symmetries which
relate all these and show that
they're really all part of one
underlying mathematical
framework.
They're all limits of
the same theory.
So string theory is,
in fact, unique.
Not only that, but there's
another thing over there
called m-theory, which is not
even quite a string theory.
It's 11-dimensional
rather than 10.
And it doesn't have
strings at all.
It's got two-dimensional
membranes.
It's also tied into
this thing.
So there is some underlying
mathematical thing.
We can't quite define it.
But we know how to define
certain limits of it.
And the limits of it that we
can define are the little
pieces of that elephant
that the blind
men have been touching.
So string theory is
a unique theory.
It's got some unique underlying
mathematical
structure, eve though can't
articulate it yet.
But it has many different
realizations in terms of the
kind of physics you can
see coming out of it.
That comes from the fact that
there are many different
compactifications, many ways
of curling up those extra
dimensions.
And each one of those can give
you different kinds of physics
and spacetime.
So that leads us to the vacuum
problem, which is what I
started discussing
at the beginning.
So until about five years ago,
if you asked most people that
thought about these things,
they would have told you,
well, the cosmological
constant is
almost certainly zero.
And it seems like
there's lots of
compactifications of string theory.
But there's probably some
mechanism which just forces it
to choose one.
So when we really understand
the theory, we'll probably
have some dynamics, which
will force us into
one particular vacuum.
It will have Lambda
equals zero.
That will be the world, and
we'll be able to predict the
mass of the electron.
There was a quote, I don't know,
20 years ago, 15 years
ago, by an outstanding physicist
who commented that,
well, it seems like it's now
only a matter technical
details before we can predict
the mass of the
electron to 20 digits.
Something to that effect.
Hasn't been done yet.
And it's now not quite so clear
that we will be able to
do that for the following
reason.
As we've discussed,
experimentally, it's been
observed that the cosmological
constant is not zero.
It's 10 to the minus 120.
And in a parallel
development--
not necessarily related to that,
but certainly the idea
of looking for solutions with
positive Lambda has been
promoted quite a bit by
this observation--
it's been realized that, in
string theory, we really don't
have a natural mechanism
to pick up on vacuum.
There's many, many solutions
with Lambda equals zero.
And there's many, many solutions
it seems with Lambda
not equal to zero.
It's harder to mathematically
characterize the ones with
Lambda not equal to zero.
But in fact, we can show that
there an infinite number of
solutions to string theory,
each of which gives you a
different local spacetime
physics.
Now, if you put in some really
simple cut offs, like that the
size of the compactification
manifold is not too, too big,
so we would have seen it
already, then you can cut it
down to a finite number that
we can actually compete.
It's not to say that there has
to be a finite number.
But the ones we know about are
on the order of 10 to the
1,000, it seems. From basically
putting together a
bunch of pieces people
understand and making the most
natural conclusion.
So there are an enormous number
of different ways of
using string theory,
compactifying it and coming up
with some kind of
four-dimensional physics like
ours but different.
And that number starts
to be on the order
of 10 to the 1,000.
Yes, please.
Question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: OK.
So the question was, is there
any notion that some of these
solutions might be more
likely or more stable?
And those are precisely the
questions that a lot of string
theorists are trying very hard
to answer right now.
Let me address those two
things separately.
One is the likelihood.
The likelihood would involve
choosing some kind of a
measure, a distribution on
the space of these vacua.
And to do that, we would really
have to understand the
dynamics of the theory and the
dynamics of quantum gravity
better than we do.
This picture of inflation with
this tunneling process that I
talked about is something people
do not mathematically
understand very well.
And, in fact, it's very hard
to assign probabilities.
There's a sense in which we
really have no idea how to
define probabilities
for these things.
Because you get an infinite
number of
things at some point.
And it's like if I give you all
the integers, and I ask
you, what fraction of the
integers are even?
Your natural response is half.
But then, I can list them
in a very different way.
I can list uneven integers,
every third.
One, three, two, five, seven,
four, et cetera.
And I would say, well, only
a third of them are even.
It really depends on
what you put in
as a cut-off, basically.
So we don't know
how to do that.
There may be a sense in
which that's true.
But we don't know
the mathematics
to describe it yet.
Right now, all we can do is
talk about the solutions.
The second question is whether
some of them can be stable and
others can be unstable.
That is probably the most
promising way in which to get
rid of this rather disturbing
picture I'm painting where
there's so many solutions.
It is possible that many of the
solutions are unstable.
All the solutions with positive
cosmological constant
are probably unstable but on
a very long time scale.
Time scale is much, much
longer than the
history of our universe.
In particular, if you have a
small cosmological constant,
you can always decay into
an even smaller
cosmological constant.
But it takes a long time.
On the other hand, if your
cosmological constant is 1/2
in natural units, and there's
a nearby vacuum with a
cosmological constant of 1/4,
you're going to be unstable to
that vacuum in a very short
period of time.
So there is a sense in which
that instability will
naturally take you down further
and further to smaller
cosmological constants
eventually.
I hope that answer
your question.
OK.
So is this picture
at all sensible?
Should we take this
at all seriously?
This is sounding pretty wacky.
We're talking about enormous
universe with different
regions that we can't
really observe.
Why would we really take
this seriously?
So there's of prescient argument
by a Nobel Prize
winning physicist named Steven
Weinberg who made an argument,
which, I think, is one of the
more compelling arguments for
taking the story seriously.
And his argument
was as follows.
Before anybody had observed
that there was a non-zero
cosmological constant, he did
a little calculation.
He said, if the cosmological
constant was bigger than a
certain value--
remember, the cosmological
constant causes things to move
apart from each other--
if it was bigger than a certain
value, then before
galaxies had a time to form,
everything would have just
drifted apart, or been
pushed apart by the
cosmological constant.
Galaxies would never
have formed.
We would not have the nice
structure we see in our patch
of the universe.
And we would probably
not be here.
And he computed the number.
And the number was 10
to the minus 118.
So if the cosmological constant
was bigger than 10 to
the minus 118, then we
wouldn't be here.
Because galaxies would never
have formed, structure of the
kind we see would not have
formed, and we would just have
a diffuse gas of particles
moving apart from one another.
Or at least, we might have some
smaller structures, but
nothing as big as what we need
to get our kind of structure
in the universe.
So he made this argument before
anybody measured that
the cosmological constant
was 10 to the minus 120.
And then, people measured
that yes, it's
just below his bound.
It's 10 to the minus 120.
To me, that's actually a
compelling observation.
Because it says that, if this
story is correct, that there's
a lot of different patches with
different cosmological
constants, then the fact that
we live in the patch with 10
to the minus 120 is just an
environmental feature of the
fact that stuff like what we see
can only occur in certain
places in the universe.
You don't ask the following
question.
You don't look around and say,
look, most of the universe is
empty space.
Why are we not sitting in
empty space right now?
Statistically, we should be
in empty space, right?
You don't even have
that discussion.
Because it's clear that for us
to be here, we would have to
be living in a patch where
there's a bunch of mass and a
planet and all that stuff.
In the same way, it may be that
the universe is very big,
and we're not in one of these
other patches for the same
reason we're not
in empty space.
Because nothing interesting
happens there.
So to me, this is actually a
somewhat compelling argument.
It certainly is one which
indicates that we should
perhaps take this point of view
seriously and at least
try to debunk it.
So the picture we've assembled
by this time is often referred
to as the string landscape.
One of your neighbors at
Stanford University here, who
I've have been talking to a lot
this semester and the last
few years, is a guy named
Lenny Susskind.
He recently wrote a nice book,
which I recommend to anybody
interested in learning a little
bit more about this,
called The Cosmic Landscape.
He, I think, was at least partly
instrumental in coining
the term the landscape to
describe this scenario.
Basically, the picture we end
up with is a landscape where
there are lots of hills and
valleys in this cosmological
constant, scattered over the
space of possible string
configurations.
And there's a bunch of minimum,
local minima, denoted
by the green points.
And the different minima
correspond to
different kinds of vacua.
The lake there is a continuous
many-parameter family of vacua
with Lambda equals zero.
And the various local hills are
things with Lambda may be
greater than zero.
But some of them will have
Lambda less than zero.
But there could be
lots of these.
If there are enough of these,
in particular, if there are
greater than 10 to the 120 of
these, and if Lambda is
relatively uniformly distributed
between zero and
one or minus one and one, then
generically, just by
probability, some of these
vacua will have 10 to the
minus 120 or smaller for their
cosmological constant.
So in fact, the best way to
realize a vacuum which has
such a small cosmological
constant is to just have a
whole bunch of them, and have
some of them just by random
statistics have to be small.
Otherwise, you have to somehow
fine-tune nature
So in some sense, it's the
simplest explanation for how
to get a small cosmological
constant.
It's just that there's so many
different possibilities, we
just happen to have enough
possibilities
that we get some naturally.
So you're actually naturally
led, in the absence of some
kind of really detailed
fine-tuning of how the
universe works, to a picture
where there are lots of
different vacua.
And most of them don't
have such a small
cosmological constant.
OK.
So what this leads to, if we
put this in the context of
this inflation and bubbling
story we talked about, is now
we have this potential.
And if you're sitting in one
vacuum, say Vacuum 1, which is
the red vacuum, There's always a
possibility that some little
patch of the universe will
bubble up a patch by quantum
tunneling that's in Phase 2 in
the second of these vacua.
Phase 1 will expand.
Because when you have a
cosmological constant, space
expands at an exponential
rate.
Phase 2 will also expand.
So the amount of space that is
in each of these two phases
will expand exponentially.
The amount of space in Phase 1
will expand faster than the
amount of space in Phase 2.
But they'll both expand
exponentially.
And then, if there's a region
three, we can either tunnel
from one to three or from two
to three, and we'll get
patches of three appearing
inside the
patches of one and two.
And if you go on from three to
10 to the 1,000, you get some
idea what the picture of the
universe might look like.
There might be regions which are
in one phase with bubbles
that are in another phase
containing bubbles in another
phase up to 10 to the 1,000
different kinds of phases, all
scattered around the universe.
OK.
So that's the picture
we end up with.
Now, the question I want to
addresses is if that's the
universe, then what
do we do about it?
And what can we actually do in
terms of carrying out the goal
of physics, which is to actually
make predictions, or
figure out how to explain the
physics where we are?
So the vacuum problem is where
are we in this picture?
Yes.
Question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: There would
actually be boundaries
separating these regions.
It might be hard to get
to those boundaries.
Because if you're in a given
patch, because of the fact
that the universe is expanding,
things that are
distant are moving away
faster and faster.
So for instance, if we live in
a patch with our cosmological
constant that we've observed,
because of this accelerated
expansion of the universe, if
you started traveling at the
speed of light out towards the
boundary of the universe, by
the time you got there, it would
have moved so far away
that you would just have to
keep going and going.
And you would, in fact, never
be able to reach the things
which are past a certain point,
called the Hubble radius.
So we can't actually
get there.
But there may be a boundary
out there.
If you had a global picture of
the spacetime, then you would
have a function where you would
be living in one vacuum,
and then there would be some
kind of a kink where the thing
would move into another
vacuum.
And then, you'd be living
in the other vacuum.
So there would be boundaries,
domain walls separating these
different kinds of physics.
But unless you were lucky, you
wouldn't be living in a region
where you could see one.
Yeah, there was another
question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: So the
question is, is it possible
that there's a boundary heading
towards us at the
speed of light?
And yes, it is possible.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: It is not
required because, in our patch
of the universe--
Our patch of the universe
has this Hubble size.
Anything that happens outside
the Hubble size will
never get in to us.
Now, it's possible that, out
at Alpha Centauri, it might
happen that a little patch of
another vacuum forms, which is
just big enough in some next--
Maybe we're in Vacuum 2.
Maybe a patch of Vacuum
3 could form
somewhere out there.
It's close enough to us, and
it's expanding fast enough,
that we would not get
away from that.
It would just come and hit us.
So it is possible.
It's very unlikely.
Because we have such a small
cosmological constant that
probably the time for the decay
is at least billions of
years or much longer.
But it is not logically
impossible in this picture.
Yes.
Another question.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: OK.
So I should add here that a
lot physicists really hate
this picture.
And there's a lot of controversy
about whether it
makes sense to talk about this,
what you do with this
picture of it's right,
or really
whether this is physics.
Because when we talk about one
of these other patches, we
are, in principle, unable to
move to one of those other
patches and measure things.
In our patch of the universe,
even if 30 billion lightyears
away there's a domain wall,
we're never see it.
Because it's outside
our Hubble radius.
So parts of this are very
hard to verify directly.
Now, there are other things in
physics that are hard to
verify directly.
Nobody's ever seen
a lone quark.
And yet, we believe that
there are quarks.
Quarks come in sets of two or
three that are tied up in
little bundles which have no
charge under the strong
nuclear force.
The fact that nobody's ever
seen a quark doesn't make
people not believe it.
It just means you have to have
more sophisticated indirect
evidence that this is true.
Now, I believe that if we can
get string theory to the point
where we can say, look, this
is a really robust theory.
This is what it predicts.
And it also predicts the
following things, which we can
verify by experiment.
Then, we will be in a situation
where we have to
believe that this is the
simplest explanation for our
universe, and, therefore, this
is probably correct.
But we may never be able to
measure those things directly.
Yes.
Another question.
AUDIENCE: Is there any
preference [INAUDIBLE]
WASHINGTON TAYLOR: Good.
So the question is, is there
a preference, when you're
traveling from one vacuum to
another, whether you're
increasing or decreasing
Lambda?
Now, the process of decreasing
Lambda is called a
[? Kalman ?]
[? dilutia ?]
[? instanton. ?]
And it's been fairly well
understood in physics.
It is pretty straightforward to
construct a solution, in a
certain Euclideanized version.
There's some technical
stuff you have to do.
But there is a relatively robust
sense in which people
think they can make sense of
the tunneling from a higher
cosmological constant
to a lower one.
There is some controversy,
I would say, as to
whether you can go up.
I don't think there's universal
agreement on that.
The general feeling, I think,
is you probably can't.
But I would not want to go out
on a limb on that one.
It's a harder thing
to understand.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: I'm sorry.
The dark energy is proportional
to lambda.
AUDIENCE: [INAUDIBLE]
WASHINGTON TAYLOR: Yeah, the
trick about quantum mechanics
is, for short periods of time,
you can tunnel into things
with higher energies.
You can disobey conservation
laws for very short periods of
time in quantum mechanics.
And we don't understand quantum
gravity well enough to
know whether, if you momentarily
disobey the law of
conservation of energy by
formulating a little bubble of
more energy, that thing is going
to start inflating under
its own cosmological constant.
It will just keep on going.
We don't understand whether
you're allowed
to do that or not.
Another question.
AUDIENCE: I just want to warn
you about [INAUDIBLE]
WASHINGTON TAYLOR: OK.
I have nine minutes.
I will, I think, conclude
in nine minutes.
Thank you very much.
So the problem is
where are we?
So this leads to the last part
of my talk which is actually
the computational challenge
here of is this something
where we might hope to make
some progress in terms of
figuring out how we can make a
prediction, even if this is
outlandish paradigm
is correct?
So there are two things that
we can try to do which, I
think, will help us to move
towards making physical
predictions.
The first thing, obviously, is
to try to identify our vacuum.
Which vacuum do we live in out
of these 10 to the 1,000?
That's a hard problem, because
you have to look a lot of
different vacua.
A simpler thing you might try
to do is to try to find
structure in this landscape.
You might try to find, for
instance, things which are
always correlated.
That is, if you find that all
vacua with a particular
feature also had some other
feature, and we have the first
feature in our world, then
you might hope to
find the second feature.
But there is this issue, which
is that, even if we know the
theory, and we can
mathematically write it down,
and we can compute precisely
each of the vacua one at a
time-- although we obviously
can't run through all of them,
because we don't have enough
particles in the
universe to do it--
if we could in some way compute
each of them, if all
the observables are uniformly
distributed and independent,
then we are in trouble.
Because we may not be able
to make predictions.
If there's another versions of
the universe right next door
which could have any possible
imaginable value of that
electron mass of the fine
structure constant governing
the strength of
electromagnetism.
If all these things can be tuned
to arbitrary precision
by just choosing some vacuum,
we have far less than 1,000
digits of experimental
information that goes into our
characterization of the standard
model of particle
physics right now.
We could, presumably, dial
each of those parameters
arbitrarily, if all these
things are uniformly and
independently distributed.
So there's a semi-organized
effort going on.
It's basically just a bunch
of people doing their own
individual things.
But there's some effort to
organize this little bit.
Mike Douglas from Rutgers,
Gordie Kane from Michigan,
myself from MIT, lots of other
good physicists from some of
the best institutions around,
both here and in Europe are
thinking about this.
No one mentions skewed towards
the younger end of the age
distribution.
A lot of the older physicists
are more skeptical of this
approach to understanding
physics.
And this controversy is an
interesting sociological
phenomenon.
But there's also an east
coast, west coast
split on this one.
I'm one of the renegades
on the east coast
who's more into this.
But anyway, lots of people,
both here in Europe, are
working to try to analyze
this vacua.
And what we need to do is
there's a number of steps that
have to be gone through.
We need a mathematical
formulation of the problem.
And we don't actually have
a complete mathematical
definition of string
theory yet.
So we need to work
towards that.
But along the way, we can
characterize these vacua
fairly well, in many cases,
without even knowing how to
define the theory, which
may sound difficult.
But it's true.
Once we can mathematically
characterize the problem, we
need to come up with efficient
algorithms. You're all
familiar with that issue.
You have some mathematical
framework that's
in some paper somewhere.
And you have to figure out how
to make an efficient algorithm
to actually solve the problem.
And then, once you know what the
algorithm is, you have to
actually perform a systematic
search involving implementing
the algorithm, putting it on
a bunch of computers, or a
computer, and running
through things.
And then you get a whole
bunch of numbers out.
And you have to interpret those
results, and say, does
this tell us anything
about physics?
So each of these four steps, I
think, is a major challenge.
And there's interesting things
to do on all of these.
I think of particular interest
perhaps to the people here,
there are a lot of challenging
questions having to do with
algorithms and systematic
searches, which really form a
major computational
challenge here.
We're obviously not going to
search through 10 to the 1,000
vacua by just having
a computer run
through them one time.
Because we'd never get there.
We need to be smart about
how to do that.
Yeah.
I've got about five minutes.
So I'll just whip
through this.
Let me give you a little bit
of a sense of how you
construct different vacuas.
So the first thing you do is
you have to pick a six or
seven dimensional manifold in
which to compactify things.
One class of manifolds which
people have compactified are
called Calabi-Yau manifolds.
They're certain mathematical
gadgets that we can
characterize by certain
combinatorial data in certain
cases in a very simple way.
One of the first computational
efforts in this direction was
done by Kreuzer and Skarke.
If you Google Kreuzer and Skarke
and Calabi-Yau, you
will find their online
searchable database of 473
million Calabi-Yau manifolds.
So there, you can just a
little sample of a tiny
fraction of the number
of possible string
compactification.
These are 473 million different
things you could
compactify on.
And each one would give
you different physics.
There are probably many
more Calabi-Yau's.
Nobody even knows if the number
is finite or not.
But certainly this is probably
a small subset.
But it's a nice subset which
is easy to characterize.
There are also other spaces
possible, and even spaces
which are not really
geometric.
But I don't time to
get into that.
Generally, these Calabi-Yau's
will live in the
lakes in that landscape.
They will have Lambda
equals zero.
And they will have continuous
moduli.
If you take a circle, the
circle could have
an arbitrary radius.
The simplest string
compactification is you
compactify on a circle,
and then you're in
nine-dimensional spacetime.
That circle can have
arbitrary radius.
That will change the coupling
constants in the
nine-dimensional theory.
You want to get rid of these
things which are called
moduli, which are those
parameters.
So you put in some things called
fluxes, which are like
generalizations of
electromagnetic fluxes.
And when you put in those
fluxes, you generate the
landscape I described,
where you have
lots of isolated vacua.
So there's been some progress in
classifying these different
so-called flux vacua.
If you combine Calabi-Yau
manifold with fluxes, in
principle this gives you a
compactification with a
certain cosmological constant.
Now, there's a real trick,
which is that nobody can
calculate the cosmological
constant in a case where it's
non-zero for any
of these vacua.
And the reason is the
cosmological constant is the
energy density which comes from
summing up a huge number
of terms, which are like plus
1, minus 1, plus 7, plus 0.7
minus 0.3, et cetera.
A whole bunch of numbers with
opposite signs, and these
things have to add up and cancel
to zero to within 120
decimal places to get
10 to the minus 120.
It's really hard to compute
those things.
We don't even have
mathematically the
formalism to do it.
But even if we could, there was
a nice argument by Denef
and Douglas which shows that,
even in the simple models, the
problem of finding a vacuum
with a small lambda, even
given some simple models for
how to compute it, is A NP
hard problem.
It's a lot like the minimal
length vector problem.
This problem where you're given
a set of vectors in a
space, and question is what's
the integral linear
combination of those vectors
of the shortest length?
That's A NP hard problem.
The cosmological constant
problem, even in the very
simplest cartoon
characterization is at least
that heard.
So even if somebody could write
down all the models, and
then ask you to calculate the
cosmological constants, it
would be very hard to locate
our vacuum with 10 to the
minus 120 cosmological
constant.
OK.
So you might worry at this point
and say, we're totally
out of luck.
It is true that, because of
this mathematical and
computational complexity of the
cosmological constant, it
may be intractable to identify
our precise vacuum, or even to
compute those vacua which
have such a small
cosmological constant.
But that is not the
end of the story.
I would say that all
is not lost.
If, in fact, the cosmological
constant is relatively
independent of other features,
we may be able to predict
everything else about physics
and just not the
cosmological constant.
And then, we would just have
to say, look we found 10 to
the 400 vacua with all these
properties, the cosmological
constant is uniformly
distributed.
One of the must be ours
and have a small
cosmological constant.
So if we can show that all
models with particular
features-- let's say, features
x and y, which
are observed in nature--
also have some other feature,
z, then we can make a
prediction.
We can say, if you do a
particle accelerator
experiment, which should measure
z, we should see it.
Because every model we found
in string theory has that.
That would be a prediction out
of string theory, even in the
face of this compositional
complexity.
So to do that, what we have to
do is we have to look for
correlations in the landscape.
So then we have various
different ways of getting into
more detail.
Now, we have to not only have
a compactification, but we
have to pick the forces of
nature and the particles.
And there's one way to do this,
which is by putting
higher dimensional things
called branes--
I won't get too much
into this.
But we can put in things
wrapping the thing we're
compactifying on.
And they intersect in
certain places.
And the way that they intersect
will give rise to
some number of quarks and
some number of electrons
and things like that.
So skipping a lot of the
complexity, we can formulate
this problem, essentially,
as a partition problem.
Given a big vector, we're
given a bunch of little
vectors, and the question is,
how many ways are there of
adding up an integral linear
combination of a little
vectors to get the big vector?
So in this case, the integers
will correspond to
the forces of nature.
If n equals 1, it's EM, if it's
2, it's the weak force,
if it's 3, it's the strong
force, if it's 4, it's
something we don't have in
our part of the universe.
So there's a partition
problem.
If we want to evaluate all
partitions, you can think
about that as a computational
problem.
It's an exponentially
hard problem.
There's an exponentially large
number of solutions, if
there's lots of solutions.
Some a group in Germany,
Blumenhagen et al, spent a
year on computer time computing
through all these.
In some recent work
with Mike Douglas,
we've been doing this.
But we took a different
approach.
We basically are just looking
for the class of models which
include the standard model.
So we're saying, look, if you
fix the standard model, the
partition problem becomes
a cubic time problem.
So actually, in the
last couple of
months, we put on a computer.
We've got a bunch of algorithms.
And we can show
that, in this particular class
of models, there's only on the
order of 10 to seventh or 10 to
the eighth which have the
standard model gauge group.
And of those, only
about 10 have the
right set of particles.
Now, this is only a piece
of this landscape.
But it's an example of how,
by using computational
cleverness, we can actually
start to find data sets of
vacua, which we might compare
to our universe.
Summary is the universe may be
very big, really, really big,
as many zeros as you
can imagine.
Physics in other places may be
very, very different from the
physics we have here.
String theory seems to have a
lot of different solutions,
and can describe many
of these solutions.
But there's a real challenge
in figuring out how to make
predictions based
on this model.
And, I think, not only are they
mathematical, but also
many computational challenges.
And I think it's pretty
interesting.
It's a place where mathematics
and physics and computer
science all come together.
We want to do a bunch
of searches.
And so I hope you've
found this amusing.
And if any of you are really
interested, I can give you
more references.
Or if you want to spend some
of your 10% time trying to
find vacua, I'd be happy to
talk to you about that.
