Great pleasure, it's a great honour. I'm particularly happy and feel grateful of having the chance to celebrate.
Roger, here Roger has been an intellectual hero for me all through my life because he's not a normal, great scientist.
He's something else. I've been always struck by two things about Roger.
On the one hand, he's wide, extremely wide,
the wide scope of his interests and the the wide area of the path of our current knowledge that he master in in in death.
I mean, his popular book that many of you know, show that if you think about Road to Reality,
one of his books, which is probably one of the hardest of the popular book,
the photography he gets of what we know and we don't know about the physical world today, I think is unique in death and a wide angle.
But next to that, Roger has been always an extraordinarily creative and courageous thing thinker that has always
gone in directions which were quite far away and independent from the common way of thinking.
Is he my view of what's going on in physics today is that there there's a lot of wild
speculations about the reality of the universe that everything many dimensions,
many fields, many particles, many symmetries, but is always all of the same dimensions, political symmetries.
Most of Roger's idea going in other directions and has been that for four years,
and it's still going on this Roger's creativity, sometimes leaving the community perplexed.
What is he doing? Because what he's proposing is often a truly new way of thinking about about reality, and that's what I've admired of him.
And that's why I'm so happy today to be able to participate to celebrate him.
And I want to do that by telling your story when when I was asked to give this lecture,
I said yes immediately because I have a story to tell and there's a story of networks.
And I think it's an interesting story because it shows this aspect of Roger's creativity, his capacity of thinking in new directions,
completely transversal to where people are going, and to anticipate also where the field is going.
So the story I'm giving you is is a is a story of one idea, one object spin networks,
which today is playing a big role in quantum gravity today in in the the part
of the researching quantum gravity that goes around look quantum gravity.
There is a maybe hundreds of scientists in the world,
in the world that use B networks as a daily tool for describing quantum properties of space, time and space networks came into.
I'll tell you what they are at some point, precisely.
But Phoenix was came into the theory in the middle of the 90s from a long and complicated mathematical work within
the complicated structure of loop quantum gravity and emerged as a proper language for describing quantum spacetime.
This is I was deeply involved in in in in that work and eat.
The appearance of this tool for describing quantum spacetime changed completely the way we were dealing,
and also we were thinking about the micro spacetime.
This was the middle of the 90s, except that the networks were actually introduced by Roger Penrose 20 years earlier in a paper in 71,
where there was no general relativity, no quantum theory of gravity, no big and complicated mathematical story.
There were just some key ideas of how geometry and quantum could stay together.
And out of that, there was the beautiful construction of this mathematical object that.
That later on turned out to be so key and central in the construction of what is today one of the main attempts to do a quantum theory of gravity.
So the story I'm going to tell you, it's a first of all, that paper.
So I'll start from the 71. And what is in that paper? Bye bye, bye.
Bye, Roger. A copy here.
It's a I don't think many people have read this paper because in the community, everybody knows about that, but people don't go back to that.
That paper is a marvellous paper because it just focus on the key physical ideas,
which is needed to bring two apparently contradictory things together.
I'll I'll I'll try to show you how and what, so I will explain what is that paper?
And then in the end, I will tell you how we how this will have evolved in today in a form which is essentially the same,
also a little bit richer and and with some additional mathematical structure, but not much.
But the idea is the same and how they are today a tool for thinking about quantum a quantum spacetime.
So that's, uh, that's the plan. So first of all, let me tell you what the spectrum is because it's a very simple object that.
Spin that I hope I'm writing large enough. So you can see, but I'm not going to write much, it's just it's just a few pictures.
And also I'm I'm going to talk to most of the audience are not mathematicians as far as I see enlisting, but I see seen the audience,
some friends and people I recognise which are doing more mathematics than me or more philosophy than me or I'm not talking to these people.
I will add here in there some a few technical comments just to connect with a mathematical which is behind that, but they will stay very basic.
So what is this network expecting is a graph with some numbers, some integer numbers.
So a graph. It's a set of points connected by lines.
So these are points. Connected by lines, point point, point, point, point point lines, lines, lines where it doesn't matter how you draw it.
It only matters how many points you have and who is connected to who. So whether a draw is B networker, say two points like that or like that,
it's the same object as a Nexus two point connected by three lines and all the lines sits integers numbers.
So say two. And the reason I'm looking here because I want to draw the same happy network that
Roger had in his first paper just to see his own initial picture of a spin for.
One one zero. I guess you need a one here, one.
So that's it. This is Alpha. So he's a graph, an abstract graph with some integers on it.
That's the object introduced in the 70s and attached to it.
There's a little bit of let him out. You can compute just out of this data, the connectivity of the graph, the graph and the numbers you can.
Computer number called the volume. Of this network, which a certain number, which you can do just by doing something with its numbers,
I'll tell you later how exactly you compute this number.
And. This is the object I'm going to talk about for for one, 30, 40 minutes, and I'm going to talk now.
Where does it come from and why? Roger introduced it and how it then became a key object for doing something about it.
The problem, the physical problem, which is the problem of quantum gravity is doing, is the quantum aspect of space.
So space is whatever is between my hands. What's in between in these two feet or whatever?
So if you imagine what is there a bit in my hand, if you imagine these air, you take away the air there is a light goes through.
Imagine you zero all the light, all the fields. Something remains in that space.
And that space has a structure which is described by elementary geometry to some approximation or to a better approximation,
describe ice and general activity because actually there's some curvature, something like that.
So we know a lot about the space and how to describe it.
But we also know that precisely because of general relativity, the space has dynamical properties, can bend, can curve.
So it's a field is a physical field, and as such, it must have quantum properties.
So we expect that it has quantum properties that show up somehow.
And the problem of quantum gravity is understand what are these quantum properties?
The quantum properties show up in a small scale at the very small scale.
So the question is what is the quantum properties of space here between my hands and what Roger did is to come up with a hint,
an idea of how to describe this quantum properties by taking some key aspect of space and key aspect of quantum and bringing them together.
So what are the key aspect of space on which Roger did vocalise?
What are the key aspects of quantum space? It's a set of points, given two points.
You can measure the distance so you can measure distances. Distances, of course, satisfy all the relation, you know?
Well. And if this, you know, this is a five. This is five.
This is four. This is three reasons. This is the right angle and so on.
A key aspect of space is continuity. It's a continuous space.
As we describe it, in standard geometry, it's continuous space.
In general, relativity is continuous. You can look at the properties based on another perspective,
just by picking one point and looking around and you have all the possible directions, you just take in all the possible directions.
From one point in these directions also are continuous.
So there is a continuity of possible directions and you measure their reaction with angles,
so you have to measure the angle with respect to something. Also, distance, you have to measure the distance with respect to something.
So distance is always a relational property between two points and angles are it's a relational point between two directions,
but the angles continuous right can go from zero continuously to by half to pay and so on.
So a continuity is a key aspect of of of of space.
And one way relies is a continuity of possible directions in space.
That's space. Now let's look a quantum one to this something which is sort of goes the other way around.
What is quantum mechanics? Quantum theory?
Well, you'll know that nobody knows what's quantum theory. But one way of summarising quantum theory is to capture it in three steps.
One way I like to think about quantum theory is to capture it in three three main sort of discoveries about the world,
which are the basis of quantum theory in Oxford,
where somebody who doesn't like to think quantum theory this way,
but I'm definitely not going into the discussion of what quantum how to think about quantum theory.
This is taking a standard textbook quantum theory, as it is. So one way is probability.
Quantum theory is discovery that generically we cannot predict.
We cannot know sufficiently about the state of the world to predict what we're going to see with the certainty.
You always some have some irreducible probabilities that you might interpret in different ways, but is always there,
which means that if we know something about directions,
we are going to have enough quantum theory to be able only to predict probabilities for adults.
This is going to come back. The second is a relation ality.
In quantum theory, in the standard formulation of quantum theory,
you always have a quantum system and you measure its property and they some to some observer something else that interact with the object.
And you measure these properties are seen by the apparatus that's in perhaps the most mysterious aspect aspect of quantum theory.
The properties always relate to something else that's not going to affect much other way of thinking about space,
because I just said that we can focus on the future of space, which are relational, right?
The distance between two points is a relation with these two in two points.
The angle and how you think about directions is always by measuring the actual mean, meaning measuring angles between two directions.
One direction by itself doesn't mean anything unless you compare it to something else.
So it's OK and just put this on the back. But let's focus on the third and key point, which is this greatness.
This creepiness as opposed to content continuity,
so a lot of quantum mechanics is the discovery that these discrete this in the world, I think, are discrete.
They're not continuous, they only can take certain values, right?
The light that comes to me is well described.
If I if I neglect quantum mechanics as a continuous wave whose amplitude can be take any continuous possible value,
we know from quantum mechanics that if we look in detail,
the light that arrives somewhere arise actually like a raining of droplets of photons, quantum of space.
So there is a granular aspect in a in light,
which is the photons and then the orbits of the electrons of around an atom cannot have any energy, but only certain energies.
So most observable quantities in quantum mechanics quantum mechanics take only certain values and not others.
And to some extent, one could even say that the ones who don't are extreme case of limiting case.
So we'll continue this greatness. It's a it's a core aspect of quantum mechanics.
It gives it the name quantum of quantum chunks.
A quantum is a it's a it's a it's a discrete bit of something. So I'm going to focus on this one.
Because it's clearly something that first societies seem to clash a bit with the
continuity of the property for space if space has to be quantum mechanical.
And in fact, those of you who know a little bit of quantum mechanics taking a course on
quantum mechanics have taken a course in what is called the angular momentum
theory and know that there is a precise science in quantum mechanics in which the angles between two things are not continue without discrete.
This happens when you are thinking about the direction, the direction.
Individuated picked up by something that rotates if something rotates,
it rotates around an axis and the axis picks up a rotation, then you put it visually.
This is something that rotates, right?
And it has an axis, and it can have this axis or effect on this way.
It has this axis, so any rotating thing picks up a direction and commensurate with they rotate this way the axis this way,
if rotate this way, the actions this way.
So in any classical rotating object can rotate with an arbitrary angle, the velocity and with an angle or direction of its axis of rotation.
Well, in quantum mechanics because of that, this greatness you show, in fact,
you learn in elementary courses, in quantum mechanics that that's not true. Everything that turns has discrete, angular momentum,
which is that the measure of the axis and the amount of its rotation is created to sense the velocity is discrete.
So give you the mass of the you have a certain possible velocity possible. And more importantly, for us, you don't have the possibility of all.
Possible directions, but only to some of them. How many of them depend of the of the will of the total velocity?
So for all of you, for those of you who have studied quantum mechanics, then it's not allowing women to is quantised.
So the length of this is quite nice. But the component of the angular momentum in One Direction L.Z. is also quantised.
So if you want to know whether the direction is up or down or in the middle or continuously between the two in quantum mechanics,
you cannot get all continuity,
but you can get only some possible values for an electron just to up or down for an atom was being one, three possibility or four and so on.
And this is the quantisation final moment. This is well known, studied in mathematical classes, Cisco physical classes.
It was discovered back in the thirties and is there's a lot of experimental support to that angle, experimental sort of things.
You measure the angle of name to something come up with test. All right.
So far is just standard thing. Now back to space.
Roger's point in the 70s was wait a minute.
He found the momentum is quantised. Directions happen twice that's likely to be a property, not just of a rotating object,
but of space itself, space itself must be such that direction so quantised.
Because space itself, this is a this is not this is a general properties of the geometrical variables that describe space.
If this geometrical variables describe space are quantum, they have to have this property and I should be able to find this greatness.
So Roger was trying to build up a way of thinking about quantised direction,
discrete directions out of which to derive continuous space only in some limit instead of seeing a continuous space.
And then some quantum stuff on it. Try to have continue space, emerge from purely quantum.
Underlying thing, and in fact, in the paper, he was just going to the most basic, basic thing at all and says, well,
the secret this is the basics just counting one two three fourths, just saying how elementary bits can stay together.
So it's combinatorics. And the question he put a bit at the beginning of the paper is Can I get space from combinatorics?
And the first reaction? You have it. No, I mean, space.
Continuous three dimensional space is most things, most distance us all from combinatorics, you can imagine.
And at the end of the paper, you say, no, you can get key aspect of space.
Space of all possible directions just from combinatorics. And that's it all.
And let me show how he did it because it's it's it's beautiful.
There's a few lines and it's actually quite simple.
So the idea is that imagine that all this,
each one of these lines of direction and somehow given one line splitting in two lines of two lines going to one line.
This is sort of combination of direction and whatever it means the quantum mechanics, a combination of high and low momentum.
And imagine these numbers. Here are the possible discrete value of total agreement.
So how much something is rotating? Nothing here seems to talk about directions.
So there's no directions here. Can I get directions out of that?
Can I compute angles out of that? And then remember, directions are always is.
Something is related to something else. But if I have one spinnaker here and another it there,
it makes no sense to talk about relative angle between the two because there's no connexion between the two.
What could make sense is they think about that a little bit in this direction, this direction.
So the question that Roger asked is that you have speed networking are some of some complicated graph have two two particular lines coming out,
one with number eight with a number of B, how I define the angle between these two.
And he writes a formula for the angle between the two only based on the spin across it inside here, more precisely probability.
He writes a formula for having the probability of one of the other values of possible angles between the two.
Given the spectrum here and how does it?
Oh, let me be technical for one minute and actually give the the core of the of the story?
You take this. Finished when you.
You are two point and you do another one with an ex.
And you think that this is the reaction a this is a direction b so this is a direction, a reference in the other direction B.
And you ask how they combine and they can combine depending on the angle,
by sort of summing up or by subtracting one from the other with all possibility between the two.
So this x here can be any number between a plus b.
And a minus B thinking of two vectors with an angle between.
And each number in between. Of course, is clearly related to an angle theta between between the two, which is given by sort of elementary geometry.
And then you want a probability of having that angle theta just in terms of these two.
And he and Roger writes a formula which is a some simple formula,
which is something related to the value of this pinnacle divided by the value of this big network.
So let's call this beta, and this alpha is sort of the ratio of something which is the normal alpha and all of beta with some coefficients here,
which I'm not writing just depend on AB and X with the normal network is just the value.
Of the to square there are some technicalities is not important, will this minute to spare is by taking to SpiNET?
Will you join all the free, the free ends and you get the computer the number just to know what is its value because it's going to play?
So the value?
Office network, and I'll see how pretty this and how really combinatorial it is, I don't even want you to feel the detail, so I want you to get the.
The gift, the sense of it. Imagine you have a simple spin area here.
Two one one. I'll tell you how to compute the value.
The value is given by a sum divided by a product, and you do like that every time you have a line,
a link with a number end, you copy it with any little line.
So here you have to. You rewrite it with two lines.
And this is a single one and this a single one. And then when you have two or more, you copy this many times and you.
Cemeteries are. You can you can consider all possible permutation of the way this to attach to this to this case, you have only two possibilities.
So you open up your spine network in some expanded spinnaker with individual
little lines and then you count the number of closed loops here you call it.
See, that case is to that case is one, and you count the number of intersection.
You see, there's one intersection here, zero here is one, and the value is the sum over this open up network of minus one.
To the eye. Minus two to the seat divided the product on all the links of the number of the links to Factorial.
That's it. So I've given you the full formula with up some details of the angle.
You get a formula for the angle purely combinatorial out of just minus one plus one's minus two counting nothing else.
And then you show that this angle has all the property of the angle of the angle joined together with all the property of of geometry.
So you get a reconstruction of all the possible angles and the relations from combinatorics.
Now it seems magic is, of course not magic, as Roger says in his article, and some of you immediately applies,
you just reverse engineering representation theory of the rotation group.
So that's what it is. It's angular momentum theory unpacked behind this is the this is the representational spin one.
This is a particular spin one half. This expansion is the fact that higher for those of you who know the magmatic,
higher representation can be written as tensor product of a smaller representation of fundamental content and so on.
So it's sort of a working backward we know from the theory of representation of rotational group.
But if you forget all that, that's spectacular. Yes, just.
Counting and out of counting, you get the mathematics finals.
That was it for the 71 paper and the core ideas.
Again, our space is continuous. Quantum mechanics is discrete sweetness.
At the core, we expect space to be quantum mechanical.
Can we get continuous peace out of this greatness?
It also is, yes, at least for what regards the elections.
We can all by using the sweetness, by using the fact that the property I'm looking for are relational.
It's one direction. We expect another one. And by not computing angles, preferably two different angles the core ingredients of quantum mechanics.
Nineteen seventy one, Roger Penrose, twenty years later, twenty three years later, right here.
You see here. James Colour Nineteen ninety five spy networks enter loop quantum gravity.
Just to tell you where we were. Nineteen eighty five, roughly.
It's loops. The beginning of loop quantum gravity.
Hear the story. It's not going to cover the full story, but just to orient you where you are.
It's a long and complicated story of the quantisation general activity. It starts with the rock.
It starts with an Einstein goes to the rock. The Re writing generative Hamiltonian form.
I went with David Misner and so on. It's a long and complicated and very technically heavy story with the idea of doing the Hamiltonian formulation
of relativity for doing quantum version and therefore going to the quantum properties for space time.
And then these are in the 60s. There is a very celebrated work by two American physicist Wheeler John Wheeler and race the V2 Ray,
the will of the white equation with a sort of screening equation for general activity.
It's a very, very complicated the fact set of infinite set of functional differential equations coupled which nobody knows how to deal with.
And then magically Ted Jacobson and Lee smiling in the air in the 80s.
Find solutions to this equation, and these solutions are readable by loops, closed lines in space, and nobody knows what how to encode.
And then that's when I entered the game, and I remember the confusion at the time.
Somehow, they present an exhibition of space where you have geometry along this loop and nowhere else, whatever.
And during this 10 years,
there was a lot of discussion about the interpretation of this complicated loop solutions and two key steps were taken by a Polish mathematician,
Yurek, Levandowski and a Argentinian physicist haaga puling who realised the first of all
that the theory is not complete unless you consider loops that can cross themselves.
And the thing is not complete unless you consider also the possibility that troops run over themselves,
so they go through the same path, so to say whatever that means.
That was what Urich understood and what pulling understood is that these crossing points is not a crucial.
Because the states are represented by this funny solution that will admit that question have in a very precise technical sense,
zero volume unless there are these notes. So, Nord's.
Are essential, the name Lupe Fluke, antigravity, slipping, misleading,
because if you only have lupus, you can get anything interesting and the volume sit here.
Which gives a sort of picture like this one, if you have a spy network roughly.
So imagine you must be network. So imagine you have a graph.
Then you have the we have to think about is that the volume sort of sits on these.
Nods. And what this really represent is a triumph of space here, a chunk of space here, a chunk of space here attached to one another.
And then in 95,
Lee Smalling was visiting Verona and we were working on a technical problem which is trying to understand the actual geometry in these states.
And in particular, we were thinking of this problem here. Suppose this is a chunk of space.
Let me just draw to trade to for simplicity. And this is another one.
And they're attached to this common face, common triangle.
The question we're asking is how big is this area and we attack we we study this question
with standard kind of calculation that you do in quantum mechanics and quantum mechanics.
You compute this this crudeness, which is common through a technical calculation in quantum mechanics,
which is called the spectral analysis of a sulphur joint operator.
And in the theory, the theory was about gravity and gravity geometry.
So there was a lot of geometry. So the geometry, the area of this, it's.
A observable of the geometry and the geometry is a gravitational field,
so the gravitational field is a quantum field, so there was a quantum appraisal associated with it.
And we say the metrics of this going from a plate of all this one to repudiate is for all this.
And what we find, we find that the spectrum is discrete and so we can look technically the island states of this operator.
So the quantum state of space that have these particular values of the area.
And this particular value of the area are labelled by integers so we can write the states with integers here one two one zero three,
etcetera, where these numbers represent the area of this triangle, which of course, defines the direction space.
And what if we want to know the angle between the direction of this and the direction of this?
So the direction of the face? Of this chunk of space which goes out and this one.
Well, not surprising is given by pendulous formula.
And the possible angle between the two are discreet and the possible areas are discrete.
So here we are, taking out from the theory a description of Michael structural space where there are a small chunks.
A volume. That defines their actions.
But this election not only defied relative to one another.
And probabilistically and can only take discrete values, and at the core of a loop, quantum gravity is a.
In mathematics, which is essentially just a bunch of representational theory of the rotation group,
which is exactly the object that Roger has introduced to take more than 20 years earlier without any consideration of general activity of willow,
the weight equation or apparatus or anything like that.
And nowadays, this pinnacle which are used are essentially the same thing, just built with the same mathematics.
A little bit some technical complication, but are the same thing.
So I found I found this story marvellous. Marvellous,
because what Roger had done in the seventies was a sidestepping all the construction of the theory on the
basis of the details of a current knowledge about the generative structural space and quantum mechanics,
and just zoomed down to the core idea. Continuity, discreet, necessary rationality, and found the natural way of putting them together today.
Let me to move toward the end today. This provides spiritual provides a picture of a of the small structure of mental space time and the.
I would like you to appreciate what I think is the beauty of this picture, which can be said in the following way this cuanto space,
this elementary, these things which describe the elementary quantum of space.
This is a little chunk of space, a love triangle space. This is a is the connexion between the two.
So this number here says how big is the connexion? How large is the area between the two and the relative direction?
These are not like particles of quantum field theory like photons.
These are not quanta that move around in a given space.
These are quantum of space itself. So these don't do not fit into a space.
You cannot ask what is in between two of this point. These are the elementary thing that built up.
The structure in which we live, which we at the very large scale perceive as a continuous space.
This is a way of doing physics, which takes us suddenly completely away from a field theory as it's done by particle physics,
by people do electromagnetism, where the main ingredients are space a priori continuous time space, time for correct group energy momentum.
All these quantities disappear from the theory because there's no space.
There's no time. I'm not talking about time here, but there's also no time the same sense.
There is no energy. There is no momentum.
The quantum numbers that describe something don't have a p and momentum like in particle physics or an X a position.
Because position momentum are only in the small things, relatively to one another, but not relative to a background structure, which is space.
The core physical intuition behind loop quantum gravity is this.
And it was largely anticipated by by Roger.
So that's a story I wanted to tell you. And I think to me has been a great story because a lot of my scientific career was
developing around the construction of this theory around the set of key ideas,
which I've tried to to explain. And it has been a privilege to sort of walk in the step of Roger.
And I want to take this opportunity to say to Roger, Thank you very much for this and for the many other wonderful ideas has given us.
Thank you.