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English Theodore Mitchell

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Quantum theory, the world of the smallest particles, is actually influencing mathematics, and

quantum theory the world of the smallest particles is actually influencing mathematics and

I would even say that the future is for quantum mathematicians.

i would even say that the future is for quantum mathematicians

Mathematics is also what mathematicians do, and we do this with our brain, and so how is our brain created?

mathematics is also what mathematicians do and we do this with our brain and so how is our brain created well uh many

Well, many many million years of evolution. And so I think our concepts

many million years of evolution

are very much influenced by the things to see around us.

and so i think our concepts are very much influenced by the things that you

So we talk about space because we kind of move in space,

see around us so we talk about space because we

We move in time. We count because we see objects. We throw stones,

kind of move in space we move in time we count because we see objects

we want to describe the path of a stone and its velocity.

we throw stones so we want to describe the path of a stone and its velocity

And so calculus, for instance, is based on the idea of speed or slope. So I think actually our mind is very much kind of set by everyday experience.

and so calculus for instance is based on the idea of speed uh or slope so i think actually our mind is very much kind of set by

So you might think that mathematics kind of stops being effective if we move into a new territory.

everyday experience so you might think that you know mathematics kind of stops being effective

if you move into a new territory for

For instance, the very large, the universe, or the very small, the world of elementary particles.

instance the very large the universe or the very small the world of

But a remarkable thing: this hasn't happened. So mathematics has been

elementary particles but a remarkable thing this hasn't happened so mathematics has been

extremely successful in conquering, certainly I think, the physical experience.

extremely successful in conquering certainly i think the physical experience [Music]

General relativity,

general relativity einstein's theory is

Einstein's theory is a good example.

He needed a concept of curved space. And it turns out that Riemann and others,

he needed a concept of curved space and it turns out that

in the 19th century, had already thought about it. They thought, you have two dimensions, three dimensions,

riemann and others in the 19th century had already thought about it they thought well you have two dimensions

why not have an arbitrary number of dimensions?

three dimensions why not have an arbitrary number of dimensions

Abstract thinking of mathematicians was kind of ahead. In quantum theory,

abstract thinking of mathematicians was kind of ahead

you know, even if you sit and meditate for a million years you wouldn't come up with the laws of quantum

in quantum theory you know even if you sit and meditate for a million years you

mechanics because it's so bizarre. So in that sense nature had to force that upon us,

wouldn't come up with the laws of quantum mechanics because it's so bizarre so in that sense nature had to force

and it was like a very painful process. We had to let go of many very cherished concepts.

that upon us and it was like a very painful process we had to let go of many very cherished concepts for

For instance, the fact that the world is a predictable system. Quantum mechanics is just the way nature works.

instance the fact that the world is a predictable system quantum mechanics is just the way nature

You see it very clearly at the level of atoms and elementary particles,

works you see it very clearly at the level of atoms and elementary particles

and if you take many many atoms together, then in some sense many of the features of quantum mechanics

and if you take many many atoms together then in some sense many of the features of quantum mechanics are

are kind of washed away. We don't observe it because we are made out of like 10 to the 24 molecules typically.

kind of washed away we don't observe it because we are made out of like 10 to 24 molecules typically

There are several things that you have to give up in quantum mechanics. The first is that you cannot know the answer to all questions.

there are several things that you have to give up on quantum mechanics the first is that you cannot know the answer

So, for instance, typically, I would say, well, this is pen, I know where it is, and if I drop it,

to all questions so for instance typically i would say well this is pen i know where it is and if i drop it i

I know what the velocity is; that it's moving. Quantum mechanics says that's all fine,

know what the velocity is that it's moving quantum mechanics that's all fine you

you can know either the position or the velocity, so roughly half the questions are unanswerable.

can know either the position or the velocity so roughly half the questions are

That's really bizarre, because how would I then describe a pen?

unanswerable that's really bizarre because how would i then describe a pen

But there's a second thing, which is that even if you want to describe

but there's a second thing which is that even if you want to

how things move, they never have a specific answer. So, like if you are a

describe how things move they never have a specific answer so like if

classical object like a pen, now you would say "How do I go from A to B?"

you are a classical object like a pen now you would say how do i go from a to b

Well there will be a shortest path. If I throw a stone it will go like this from A to B.

well there will be a shortest path if i throw a stone it will go like this from

And I will know exactly how it went,

a to b and i know exactly how it went and this

and this is like the optimal path, and I can calculate this, and mathematics is about this. Now, if you're a

is like the optimal path and i can calculate this and mathematics is about this

quantum mechanical particle, you are basically free to go on any possible

now if you're a quantum mechanical particle you are basically free to go on any

path, so you know, and you could even have really very bizarre paths that goes all the way like that.

possible path so you know and you could even have really very bizarre path that goes all the way like that

And the thing that physics will do, it will give a certain probability for all these

and the thing that physics will do it will give a certain probability for all these

possible scenarios. If I for instance want to go from my work to home I know pretty good what the path is I'm taking,

possible scenarios if i for instance want to go from my work to home i know pretty good

but if I'm an electron that's moving, say, from one atom to another atom,

what the path is i'm taking but if i'm an electron that's moving say from one atom to

it's more like these, these curves, so everything is in terms of chance.

another atom it's more like these these curves so everything is in terms of chance

The quantum probability is something, it's like an, it's a complex number, and if you take the absolute value squared,

the quantum probability is something is like an it's it's a complex number and if you take

it becomes actual probabilities.

the absolute value squared it becomes actual probabilities so for

So for instance, one thing that, because of these kind of funny things, what can happen is that

instance one thing that a part because of these kind of funny things what can happen

instead, you know, even if you think about where a particle is, now it could be either here, or it could be here, and you might say, well there's a certain probability

is that instead you know even if you think about where a particle is now it could be either here or it could be here and you might

say well there's a certain probability

it's an A, and a probability it's in B.

it's an a and a probability it's in b

But a quantum mechanical particle can be for half be here and half be there, at the same time.

but a quantum mechanical particle can be for half b here and half be there at the

So this is very strange.

It's like you take a point, and you divide it in two fractions of a point, put one here and put one there.

so this is very strange it's like you take a point and you divide it in two fractions of a point

put one here and put one there and so

And so the rules of quantum mechanics are partly captured probability,

the rules of quantum mechanics are partly captured probability

but they have some kind of very strange phenomena built in them, and it's ... you have to kind of dig deep and basically things

but they they have some kind of very strange phenomena built in them and it's you have to kind of dig deep and

happening, you know, all at the same time

basically things happening you know all at the same time

makes quantum mechanics very difficult to absorb.

makes quantum mechanics very difficult

It's been said, you know, if you say you understand quantum mechanics that you're just lying, because this is

to absorb being said you know if you say you understand quantum mechanics that you're

something that in our intuition we simply cannot understand.

just lying because this is something that in our intuition we simply cannot understand

Physicists were thinking about, well, we need some objects, for instance, to describe the transitions in the atom,

physicists were thinking about well we need some object for instance to describe the transitions in the atom

how we go from one state to another state,

how we go from one state to another

so well, we need something, and then they discovered it's called a matrix, and it's actually known, of course, in the mathematical literature.

state say well you need something and then they discovered it's called a matrix it is actually known of course in the

But to understand the

mathematical literature but to understand

concepts that came in, like Hilbert spaces and wave functions, that was mathematics that partly was being developed at that time,

the the concepts that came in like hilbert spaces and wave functions that was mathematics that partly was being developed at that time

but I would say then the physicists really ran off with it,

but i would say then the physicists really ran off with it

and at this point, I would say mathematicians are

and at this point i would say mathematicians are

trying to catch up and understand, you know,

trying to catch up and understand

not only the language, but also the power of the concept. Because there's something incredibly

not only the language but also the power of the concept because there's something

natural in quantum theory which is perfect for the mathematical mind.

incredibly natural in quantum theory which is perfect for the mathematical mind

Because mathematicians, they're not studying one object.

because mathematicians they're not studying one object

They actually really always think about the whole family; the category, all possible objects. So, all numbers at the same time, or all possible

they actually really always think about the whole family the category all possible objects so all numbers at the same time or all

geometrical shapes. And what quantum mechanics says is, that's like perfect, because I'm not considering one path,

possible geometrical shapes and what quantum mechanics says is that's like perfect because

I'm considering the space of all paths. And in fact, I'm giving you something else. I'm giving you a way,

i'm not considering one path i'm considering the space of all paths and in fact i'm giving you something else i'm giving you a way

indeed, a probability of each possible path,

and indeed a probability of each

so in some way, you can sum all over these different paths. We call this the sum over histories

possible path so in some way you can sum all over these different paths we call this the sum over

which is a ... again, very funny concept, because we think of history as something, well,

histories which is uh uh again very funny concept because we think of history

certain things, certain facts that happens. For the quantum mechanical mathematician,

as something well certain things certain facts that happened for the quantum mechanical mathematician

history is all possible scenarios at the same time. And so there's a certain way to

history is all possible scenarios at the same time and so there's a certain way to consider

consider many different things at the same time. And so one

many different things at the same time and so

wonderful application on all of this was in, in knot theory. So if you have a knot, like this,

one wonderful application of all of this was in in in not theory so if you have a knot

connected like this, so it's a ... it's a... I'm just

like this connect it like this so it's a

drawing here the projection, but there's a certain trajectory in three dimensions, and, you know,

it's a i'm just drawing here the projection which is a certain trajectory in three dimensions and

mathematicians want to distinguish. This one is knotted, this one is not. And so, if you're a knot

you know mathematicians want to distinguish this one is knotted this one is not and so if you are not theorist you want

theorist, you want to study knots in generality; the space of all possible knots. Actually

to study nuts in generality the space of all possible

quantum theory comes to the rescue.

nuts actually quantum theory comes to the

It's a very efficient way to describe these knots. And young physicists are not that surprised anymore as

rescue it's a very efficient way to describe these nuts and young physicists are not that surprised

for instance Niels Bohr or

anymore as for instance niels bohr or albert

Albert Einstein was, who were really fighting the concept. I often think, you know, at some point there will be a generation of mathematicians

einstein was who was really fighting the concept i often think you know at some point there will be a generation of

that kind of grow, grow up with quantum theory, and they will probably apply these kinds of rules in a very natural way.

mathematicians that kind of grow grow up with quantum theory and they will probably apply these kind of rules in a very natural way

Brady: "They are starting to steal your stuff, arent they?"

they are starting to steal your stuff

Exactly. So, the remarkable thing is that there are some very deep

are they exactly so the remarkable thing is that there are some very

mathematical problems; abstract mathematical problems, like this classification of all knots,

deep mathematical problems abstract mathematical problems like this classification of all nuts

or thinking of four- or six-dimensional spaces, really purely interested from mathematical arguments,

or thinking of four or six dimensional spaces really purely interested from

and now there is a bunch of these problems that actually have been solved using quantum theory. I think of it like a good mathematician

mathematical arguments and now there's a bunch of these problems that actually have been solved using quantum theory i think of it like

has a toolbox that's filled with everything: geometry,

a good mathematician has a toolbox that's filled with everything

calculus, number theory,

geometry calculus number theory

and, you know, the rules of modern physics should be there.

and the rules of modern physics should

It's just a very efficient way to deal with certain problems, and I am actually ... I think many mathematicians were surprised that the

be there it's just a very efficient way to deal with certain problems and i'm actually i think many mathematicians were surprised

information was flowing in the other direction. Physicists are surprised, because they see the quantum

that the information was flowing in the other direction physicists are surprised because they

mathematicians, so to say, take these ideas and

see the quantum mathematicians so to say take these ideas and generalize them

generalise them even further.

It's like you have a very brilliant student and you're teaching the student everything you know,

it's like you have a very brilliant student that you're teaching get the

and then the student takes that much further, because again,

student everything you know and then the student takes that much further

and that's the beauty of math. It is, in some sense, rooted in reality, but then it's not. It's hard to meet a

because again and that's the beauty of math it is in some sense rooted in reality but then it's not

mathematician that's not ...

it's hard to meet a mathematician that's

platonic; thinks that, you know, the prime numbers are just out there, and they have nothing to do with the fact that there are

doesn't think platonic it thinks that you know the prime numbers are just out there and they have nothing to do with the

three pens here. That the number three is there.

fact that there are three pens here the number three

And so, there's this kind of

is there and so uh there's this kind of

interesting kind of an

uh interesting kind of uh oscillation

oscillation between math

absorbing facts from reality,

absorbing facts from reality

and then kind of take a flight of the imagination, and go in directions that the physicist says, wait a moment that was not

and then kind of take a flight of the imagination and go in directions that the physicists say

actually our intent to go and do, to do this kind of crazy stuff.

wait a moment that was not actually our intent to go indeed to do these kind of crazy stuff

Brady: "Has this emergence of mathematics coming out of physics been a difficult birth? Has it been

has this emergence of mathematics coming out of physics been a difficult birth has it been has been

"something that has caused problems?"

something that has

Yes. In two directions, I think.

caused problems in two in two directions

You know, the physicists would say math is very attractive,

i think you know the the physicist would say math is very attractive but it's like a

but it's like a black hole in which you can disappear, because you will be divorced from reality.

black hole in which you can disappear because you will be divorced from

Odysseus, you have to tie yourself to the to the mast of the ship not to be seduced by the beauty of mathematics

reality with this size you have to tie yourself to the to the mast of the ship not to be seduced

by the beauty of mathematics because

many of the great insights in physics were looked very painful and inelegant and ugly

many of the great insights in physics were looked very painful and unelegant and ugly to begin with

to begin with. It took some time to appreciate

it took some time to appreciate quantum

quantum theory as something beautiful. So that's the rivalry from the physics part. From the math part,

theory as something beautiful so that's the rivalry from the physics part

I think it is that, you know, physicists are sometimes seen as kind of plumbers, they, you know, they make their hands dirty,

from the math part i think it is that uh you know physicists are sometimes seen as kind of plumbers they you know they

they deal with this messy thing called reality.

make their hands dirty they they deal with this messy thing called reality

It's beautiful, as a mathematician, you can go this platonic world where there's a certain perfectness. So there's also

or it's beautiful as a mathematician you can go in this platonic world where there is a certain perfectness so there

I think quite a few mathematicians that weren't originally

was also i think quite a few mathematicians that weren't

impressed with the physical ideas. They might say, well,

originally impressed with the physical

you know, you might kind of be ... this might ... one or two areas where this is relevant, but not to my field,

ideas they might say wow you know you you might kind of be there's a might one or two areas where this is relevant but not to my field

which is pristine. And then certainly,

uh which is pristine and and then

out of left field,

there were some physical ideas that proved to be very very very powerful,

out of left field there were some physical ideas that proved to be very very very powerful and

and I feel it's like a wave, you know, you're, you think it won't hurt you until the wave hits you, and then suddenly you are underwater.

i feel it's like a wave you know you're you think it won't hurt you till the wave hits you and then certainly you are underwater

In physics we have really a crisis, and the crisis is that we have two

in physics we have really a crisis and the crisis is that we have two

wonderful theories that describe the world.

wonderful theories that describe the

world one is quantum theory it's perfect

theory: It's perfect for elementary particles. The other one is relativity, and it's used to describe large structures in the universe.

for elementary particles the other one is relativity and it's used to describe large structures in the

But the point is these two things clash, and they clash because these physical laws of the large. They include

universe but the point is these two things clash and they clash because these physical laws of the large

events like black holes, and perhaps more important, the big bang, where the rules of physics break down.

they include events like black holes and perhaps more important the big bang where the rules of physics

So this is quite remarkable that the theory that you have to describe reality has its own failure built into it; and

break down so this is quite remarkable that the theory that you have to describe reality has its own failure built into it

we think that, in some sense, the only solution is

and we think that um in some sense the only solution

to bring quantum mechanics into it. So we somehow have to marry these two pictures. It's not enough to do quantum mechanics.

is to bring quantum mechanics into two so we some somehow have to marry these two pictures it's not enough to do quantum mechanics

It's not enough to do geometry. And we know this is happening, because the universe is working,

it's not enough to do geometry and we know this is happening

you know, right now while we speaking, the laws of nature are perfectly working,

because the universe is working you know it's right now while we're speaking the laws of nature are perfectly working

and there's no breakdown anywhere. So that means that we have to come up with something

and there's no breakdown anywhere so that means that we have to come up with

deeper than geometry. Now what is that?

something deeper than geometry now what is that

That's, that's, I think, the big mystery. And we have some ideas.

um that's that's i think the big mystery and we have

So, the idea could be that geometry is something like

some ideas so the idea could be that geometry is something like

thermodynamics, which is the theory that describes, basically, materials. So if I think of this table, it feels pretty solid to me,

thermodynamics which is the theory that describes basically materials so if i think of this table it feels

but I know it's a large collection of molecules.

pretty solid to me but i know it's a large collection of molecules

I feel the air around me, and I feel a certain temperature.

i feel the air around me and i feel a

But there's no such thing as temperature, these are just molecules bouncing up and down.

certain temperature but there's no such thing as temperature these are just molecules bouncing

So could it be that just as this physical sheet of paper, if I zoom in, zoom in, zoom in,

up and down so could it be that just as this physical sheet of paper if i zoom in

I will certainly see molecules and atoms and elementary particles, and there's no paper anymore,

zoom in zoom in i will certainly see molecules and atoms and elementary particles and there's no paper anymore

could it be that if I zoom into space itself, at some point I see the pixels?

could it be that if i zoom into space itself at some point i see the pixels i see

I see something else? You know, we were thinking perhaps it's pure informations, like zeroes and ones. So, I find that absolutely

something else you know we were thinking perhaps it's pure information like zeros and ones so i find that absolutely breathtaking

breathtaking. Because if you are able to come up with a piece of mathematics

because if we are able to come up with a piece

that is more fundamental than space, than geometry, I think it would be

of mathematics that is more fundamental than space than geometry i think it would be

revolutionising all of math.

revolutionizing all of math

Just imagine trying to imagine these two huge objects what they're doing to the matter, to the space-time around it as

just imagine trying to imagine these two huge objects what they're doing to the matter to the space time around it as they and

they, and the space-time must be going "oh my god what's happening here", and it's flipping up and down up and down.

the space time must be going oh my god what's happening here and it's flipping up and down up and

And they're just generating, these waves are beginning to propagate out. 350 kilometers is about the distance,

down and they're just generating these waves are beginning to propagate out 350 kilometers means about the

what, to London? Then you've got two thirty solar mass black holes sort of orbiting one another in this region

distance what to london and you've got two thirty solar mass black holes sort of orbiting one another

And so they're going...

in this region and so they're going and at close to the speed

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