1
00:00:11,700 --> 00:00:25,440
Great pleasure, it's a great honour. I'm particularly happy and feel grateful of having the chance to celebrate.
2
00:00:25,440 --> 00:00:37,800
Roger, here Roger has been an intellectual hero for me all through my life because he's not a normal, great scientist.
3
00:00:37,800 --> 00:00:42,900
He's something else. I've been always struck by two things about Roger.
4
00:00:42,900 --> 00:00:47,940
On the one hand, he's wide, extremely wide,
5
00:00:47,940 --> 00:01:00,410
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.
6
00:01:00,410 --> 00:01:05,610
I mean, his popular book that many of you know, show that if you think about Road to Reality,
7
00:01:05,610 --> 00:01:11,970
one of his books, which is probably one of the hardest of the popular book,
8
00:01:11,970 --> 00:01:26,040
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.
9
00:01:26,040 --> 00:01:37,200
But next to that, Roger has been always an extraordinarily creative and courageous thing thinker that has always
10
00:01:37,200 --> 00:01:44,610
gone in directions which were quite far away and independent from the common way of thinking.
11
00:01:44,610 --> 00:01:50,610
Is he my view of what's going on in physics today is that there there's a lot of wild
12
00:01:50,610 --> 00:01:58,470
speculations about the reality of the universe that everything many dimensions,
13
00:01:58,470 --> 00:02:10,560
many fields, many particles, many symmetries, but is always all of the same dimensions, political symmetries.
14
00:02:10,560 --> 00:02:16,620
Most of Roger's idea going in other directions and has been that for four years,
15
00:02:16,620 --> 00:02:26,040
and it's still going on this Roger's creativity, sometimes leaving the community perplexed.
16
00:02:26,040 --> 00:02:36,390
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.
17
00:02:36,390 --> 00:02:44,130
And that's why I'm so happy today to be able to participate to celebrate him.
18
00:02:44,130 --> 00:02:48,990
And I want to do that by telling your story when when I was asked to give this lecture,
19
00:02:48,990 --> 00:02:54,060
I said yes immediately because I have a story to tell and there's a story of networks.
20
00:02:54,060 --> 00:03:05,490
And I think it's an interesting story because it shows this aspect of Roger's creativity, his capacity of thinking in new directions,
21
00:03:05,490 --> 00:03:15,270
completely transversal to where people are going, and to anticipate also where the field is going.
22
00:03:15,270 --> 00:03:21,630
So the story I'm giving you is is a is a story of one idea, one object spin networks,
23
00:03:21,630 --> 00:03:29,160
which today is playing a big role in quantum gravity today in in the the part
24
00:03:29,160 --> 00:03:34,350
of the researching quantum gravity that goes around look quantum gravity.
25
00:03:34,350 --> 00:03:37,590
There is a maybe hundreds of scientists in the world,
26
00:03:37,590 --> 00:03:50,040
in the world that use B networks as a daily tool for describing quantum properties of space, time and space networks came into.
27
00:03:50,040 --> 00:03:52,800
I'll tell you what they are at some point, precisely.
28
00:03:52,800 --> 00:04:04,620
But Phoenix was came into the theory in the middle of the 90s from a long and complicated mathematical work within
29
00:04:04,620 --> 00:04:14,340
the complicated structure of loop quantum gravity and emerged as a proper language for describing quantum spacetime.
30
00:04:14,340 --> 00:04:23,190
This is I was deeply involved in in in in that work and eat.
31
00:04:23,190 --> 00:04:30,930
The appearance of this tool for describing quantum spacetime changed completely the way we were dealing,
32
00:04:30,930 --> 00:04:35,250
and also we were thinking about the micro spacetime.
33
00:04:35,250 --> 00:04:47,370
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,
34
00:04:47,370 --> 00:04:56,280
where there was no general relativity, no quantum theory of gravity, no big and complicated mathematical story.
35
00:04:56,280 --> 00:05:05,130
There were just some key ideas of how geometry and quantum could stay together.
36
00:05:05,130 --> 00:05:10,540
And out of that, there was the beautiful construction of this mathematical object that.
37
00:05:10,540 --> 00:05:22,360
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.
38
00:05:22,360 --> 00:05:26,920
So the story I'm going to tell you, it's a first of all, that paper.
39
00:05:26,920 --> 00:05:32,380
So I'll start from the 71. And what is in that paper? Bye bye, bye.
40
00:05:32,380 --> 00:05:35,680
Bye, Roger. A copy here.
41
00:05:35,680 --> 00:05:44,530
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.
42
00:05:44,530 --> 00:05:52,210
That paper is a marvellous paper because it just focus on the key physical ideas,
43
00:05:52,210 --> 00:05:56,700
which is needed to bring two apparently contradictory things together.
44
00:05:56,700 --> 00:06:03,610
I'll I'll I'll try to show you how and what, so I will explain what is that paper?
45
00:06:03,610 --> 00:06:18,340
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,
46
00:06:18,340 --> 00:06:23,620
also a little bit richer and and with some additional mathematical structure, but not much.
47
00:06:23,620 --> 00:06:30,610
But the idea is the same and how they are today a tool for thinking about quantum a quantum spacetime.
48
00:06:30,610 --> 00:06:44,550
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.
49
00:06:44,550 --> 00:06:59,230
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.
50
00:06:59,230 --> 00:07:09,220
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,
51
00:07:09,220 --> 00:07:18,490
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.
52
00:07:18,490 --> 00:07:29,500
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.
53
00:07:29,500 --> 00:07:34,570
So what is this network expecting is a graph with some numbers, some integer numbers.
54
00:07:34,570 --> 00:07:41,770
So a graph. It's a set of points connected by lines.
55
00:07:41,770 --> 00:07:57,120
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.
56
00:07:57,120 --> 00:08:12,390
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,
57
00:08:12,390 --> 00:08:22,620
it's the same object as a Nexus two point connected by three lines and all the lines sits integers numbers.
58
00:08:22,620 --> 00:08:29,940
So say two. And the reason I'm looking here because I want to draw the same happy network that
59
00:08:29,940 --> 00:08:38,860
Roger had in his first paper just to see his own initial picture of a spin for.
60
00:08:38,860 --> 00:08:43,480
One one zero. I guess you need a one here, one.
61
00:08:43,480 --> 00:08:55,630
So that's it. This is Alpha. So he's a graph, an abstract graph with some integers on it.
62
00:08:55,630 --> 00:09:00,400
That's the object introduced in the 70s and attached to it.
63
00:09:00,400 --> 00:09:10,660
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.
64
00:09:10,660 --> 00:09:18,660
Computer number called the volume. Of this network, which a certain number, which you can do just by doing something with its numbers,
65
00:09:18,660 --> 00:09:26,550
I'll tell you later how exactly you compute this number.
66
00:09:26,550 --> 00:09:40,890
And. This is the object I'm going to talk about for for one, 30, 40 minutes, and I'm going to talk now.
67
00:09:40,890 --> 00:09:54,090
Where does it come from and why? Roger introduced it and how it then became a key object for doing something about it.
68
00:09:54,090 --> 00:10:03,990
The problem, the physical problem, which is the problem of quantum gravity is doing, is the quantum aspect of space.
69
00:10:03,990 --> 00:10:13,320
So space is whatever is between my hands. What's in between in these two feet or whatever?
70
00:10:13,320 --> 00:10:20,490
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.
71
00:10:20,490 --> 00:10:26,700
Imagine you zero all the light, all the fields. Something remains in that space.
72
00:10:26,700 --> 00:10:36,030
And that space has a structure which is described by elementary geometry to some approximation or to a better approximation,
73
00:10:36,030 --> 00:10:41,340
describe ice and general activity because actually there's some curvature, something like that.
74
00:10:41,340 --> 00:10:45,710
So we know a lot about the space and how to describe it.
75
00:10:45,710 --> 00:10:57,560
But we also know that precisely because of general relativity, the space has dynamical properties, can bend, can curve.
76
00:10:57,560 --> 00:11:03,830
So it's a field is a physical field, and as such, it must have quantum properties.
77
00:11:03,830 --> 00:11:10,310
So we expect that it has quantum properties that show up somehow.
78
00:11:10,310 --> 00:11:13,550
And the problem of quantum gravity is understand what are these quantum properties?
79
00:11:13,550 --> 00:11:18,500
The quantum properties show up in a small scale at the very small scale.
80
00:11:18,500 --> 00:11:31,520
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,
81
00:11:31,520 --> 00:11:43,250
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.
82
00:11:43,250 --> 00:11:50,240
So what are the key aspect of space on which Roger did vocalise?
83
00:11:50,240 --> 00:12:01,250
What are the key aspects of quantum space? It's a set of points, given two points.
84
00:12:01,250 --> 00:12:08,660
You can measure the distance so you can measure distances. Distances, of course, satisfy all the relation, you know?
85
00:12:08,660 --> 00:12:12,650
Well. And if this, you know, this is a five. This is five.
86
00:12:12,650 --> 00:12:17,360
This is four. This is three reasons. This is the right angle and so on.
87
00:12:17,360 --> 00:12:22,070
A key aspect of space is continuity. It's a continuous space.
88
00:12:22,070 --> 00:12:27,890
As we describe it, in standard geometry, it's continuous space.
89
00:12:27,890 --> 00:12:35,430
In general, relativity is continuous. You can look at the properties based on another perspective,
90
00:12:35,430 --> 00:12:46,920
just by picking one point and looking around and you have all the possible directions, you just take in all the possible directions.
91
00:12:46,920 --> 00:12:52,080
From one point in these directions also are continuous.
92
00:12:52,080 --> 00:12:58,140
So there is a continuity of possible directions and you measure their reaction with angles,
93
00:12:58,140 --> 00:13:06,180
so you have to measure the angle with respect to something. Also, distance, you have to measure the distance with respect to something.
94
00:13:06,180 --> 00:13:16,770
So distance is always a relational property between two points and angles are it's a relational point between two directions,
95
00:13:16,770 --> 00:13:22,920
but the angles continuous right can go from zero continuously to by half to pay and so on.
96
00:13:22,920 --> 00:13:29,400
So a continuity is a key aspect of of of of space.
97
00:13:29,400 --> 00:13:34,470
And one way relies is a continuity of possible directions in space.
98
00:13:34,470 --> 00:13:46,440
That's space. Now let's look a quantum one to this something which is sort of goes the other way around.
99
00:13:46,440 --> 00:13:57,480
What is quantum mechanics? Quantum theory?
100
00:13:57,480 --> 00:14:07,020
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.
101
00:14:07,020 --> 00:14:16,140
One way I like to think about quantum theory is to capture it in three three main sort of discoveries about the world,
102
00:14:16,140 --> 00:14:21,540
which are the basis of quantum theory in Oxford,
103
00:14:21,540 --> 00:14:25,020
where somebody who doesn't like to think quantum theory this way,
104
00:14:25,020 --> 00:14:32,010
but I'm definitely not going into the discussion of what quantum how to think about quantum theory.
105
00:14:32,010 --> 00:14:42,850
This is taking a standard textbook quantum theory, as it is. So one way is probability.
106
00:14:42,850 --> 00:14:49,810
Quantum theory is discovery that generically we cannot predict.
107
00:14:49,810 --> 00:14:56,620
We cannot know sufficiently about the state of the world to predict what we're going to see with the certainty.
108
00:14:56,620 --> 00:15:04,120
You always some have some irreducible probabilities that you might interpret in different ways, but is always there,
109
00:15:04,120 --> 00:15:08,680
which means that if we know something about directions,
110
00:15:08,680 --> 00:15:16,960
we are going to have enough quantum theory to be able only to predict probabilities for adults.
111
00:15:16,960 --> 00:15:31,630
This is going to come back. The second is a relation ality.
112
00:15:31,630 --> 00:15:34,540
In quantum theory, in the standard formulation of quantum theory,
113
00:15:34,540 --> 00:15:44,110
you always have a quantum system and you measure its property and they some to some observer something else that interact with the object.
114
00:15:44,110 --> 00:15:52,540
And you measure these properties are seen by the apparatus that's in perhaps the most mysterious aspect aspect of quantum theory.
115
00:15:52,540 --> 00:16:00,700
The properties always relate to something else that's not going to affect much other way of thinking about space,
116
00:16:00,700 --> 00:16:08,770
because I just said that we can focus on the future of space, which are relational, right?
117
00:16:08,770 --> 00:16:13,720
The distance between two points is a relation with these two in two points.
118
00:16:13,720 --> 00:16:25,330
The angle and how you think about directions is always by measuring the actual mean, meaning measuring angles between two directions.
119
00:16:25,330 --> 00:16:29,920
One direction by itself doesn't mean anything unless you compare it to something else.
120
00:16:29,920 --> 00:16:41,940
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.
121
00:16:41,940 --> 00:16:46,410
This creepiness as opposed to content continuity,
122
00:16:46,410 --> 00:16:53,280
so a lot of quantum mechanics is the discovery that these discrete this in the world, I think, are discrete.
123
00:16:53,280 --> 00:16:59,490
They're not continuous, they only can take certain values, right?
124
00:16:59,490 --> 00:17:02,280
The light that comes to me is well described.
125
00:17:02,280 --> 00:17:11,070
If I if I neglect quantum mechanics as a continuous wave whose amplitude can be take any continuous possible value,
126
00:17:11,070 --> 00:17:15,690
we know from quantum mechanics that if we look in detail,
127
00:17:15,690 --> 00:17:22,380
the light that arrives somewhere arise actually like a raining of droplets of photons, quantum of space.
128
00:17:22,380 --> 00:17:27,330
So there is a granular aspect in a in light,
129
00:17:27,330 --> 00:17:38,100
which is the photons and then the orbits of the electrons of around an atom cannot have any energy, but only certain energies.
130
00:17:38,100 --> 00:17:46,500
So most observable quantities in quantum mechanics quantum mechanics take only certain values and not others.
131
00:17:46,500 --> 00:17:56,370
And to some extent, one could even say that the ones who don't are extreme case of limiting case.
132
00:17:56,370 --> 00:18:05,220
So we'll continue this greatness. It's a it's a core aspect of quantum mechanics.
133
00:18:05,220 --> 00:18:09,060
It gives it the name quantum of quantum chunks.
134
00:18:09,060 --> 00:18:17,650
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.
135
00:18:17,650 --> 00:18:26,590
Because it's clearly something that first societies seem to clash a bit with the
136
00:18:26,590 --> 00:18:34,480
continuity of the property for space if space has to be quantum mechanical.
137
00:18:34,480 --> 00:18:39,790
And in fact, those of you who know a little bit of quantum mechanics taking a course on
138
00:18:39,790 --> 00:18:44,590
quantum mechanics have taken a course in what is called the angular momentum
139
00:18:44,590 --> 00:18:55,340
theory and know that there is a precise science in quantum mechanics in which the angles between two things are not continue without discrete.
140
00:18:55,340 --> 00:19:03,590
This happens when you are thinking about the direction, the direction.
141
00:19:03,590 --> 00:19:09,290
Individuated picked up by something that rotates if something rotates,
142
00:19:09,290 --> 00:19:18,390
it rotates around an axis and the axis picks up a rotation, then you put it visually.
143
00:19:18,390 --> 00:19:25,410
This is something that rotates, right?
144
00:19:25,410 --> 00:19:30,690
And it has an axis, and it can have this axis or effect on this way.
145
00:19:30,690 --> 00:19:41,680
It has this axis, so any rotating thing picks up a direction and commensurate with they rotate this way the axis this way,
146
00:19:41,680 --> 00:19:44,610
if rotate this way, the actions this way.
147
00:19:44,610 --> 00:19:57,130
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.
148
00:19:57,130 --> 00:20:03,220
Well, in quantum mechanics because of that, this greatness you show, in fact,
149
00:20:03,220 --> 00:20:15,220
you learn in elementary courses, in quantum mechanics that that's not true. Everything that turns has discrete, angular momentum,
150
00:20:15,220 --> 00:20:24,580
which is that the measure of the axis and the amount of its rotation is created to sense the velocity is discrete.
151
00:20:24,580 --> 00:20:38,080
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.
152
00:20:38,080 --> 00:20:45,370
Possible directions, but only to some of them. How many of them depend of the of the will of the total velocity?
153
00:20:45,370 --> 00:20:53,200
So for all of you, for those of you who have studied quantum mechanics, then it's not allowing women to is quantised.
154
00:20:53,200 --> 00:20:59,980
So the length of this is quite nice. But the component of the angular momentum in One Direction L.Z. is also quantised.
155
00:20:59,980 --> 00:21:09,610
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,
156
00:21:09,610 --> 00:21:11,290
you cannot get all continuity,
157
00:21:11,290 --> 00:21:20,410
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.
158
00:21:20,410 --> 00:21:33,860
And this is the quantisation final moment. This is well known, studied in mathematical classes, Cisco physical classes.
159
00:21:33,860 --> 00:21:44,670
It was discovered back in the thirties and is there's a lot of experimental support to that angle, experimental sort of things.
160
00:21:44,670 --> 00:21:49,290
You measure the angle of name to something come up with test. All right.
161
00:21:49,290 --> 00:21:55,810
So far is just standard thing. Now back to space.
162
00:21:55,810 --> 00:21:59,880
Roger's point in the 70s was wait a minute.
163
00:21:59,880 --> 00:22:07,720
He found the momentum is quantised. Directions happen twice that's likely to be a property, not just of a rotating object,
164
00:22:07,720 --> 00:22:16,420
but of space itself, space itself must be such that direction so quantised.
165
00:22:16,420 --> 00:22:28,690
Because space itself, this is a this is not this is a general properties of the geometrical variables that describe space.
166
00:22:28,690 --> 00:22:39,220
If this geometrical variables describe space are quantum, they have to have this property and I should be able to find this greatness.
167
00:22:39,220 --> 00:22:45,520
So Roger was trying to build up a way of thinking about quantised direction,
168
00:22:45,520 --> 00:22:56,470
discrete directions out of which to derive continuous space only in some limit instead of seeing a continuous space.
169
00:22:56,470 --> 00:23:06,570
And then some quantum stuff on it. Try to have continue space, emerge from purely quantum.
170
00:23:06,570 --> 00:23:15,780
Underlying thing, and in fact, in the paper, he was just going to the most basic, basic thing at all and says, well,
171
00:23:15,780 --> 00:23:24,390
the secret this is the basics just counting one two three fourths, just saying how elementary bits can stay together.
172
00:23:24,390 --> 00:23:32,640
So it's combinatorics. And the question he put a bit at the beginning of the paper is Can I get space from combinatorics?
173
00:23:32,640 --> 00:23:36,210
And the first reaction? You have it. No, I mean, space.
174
00:23:36,210 --> 00:23:42,270
Continuous three dimensional space is most things, most distance us all from combinatorics, you can imagine.
175
00:23:42,270 --> 00:23:48,060
And at the end of the paper, you say, no, you can get key aspect of space.
176
00:23:48,060 --> 00:23:53,290
Space of all possible directions just from combinatorics. And that's it all.
177
00:23:53,290 --> 00:23:56,860
And let me show how he did it because it's it's it's beautiful.
178
00:23:56,860 --> 00:24:05,610
There's a few lines and it's actually quite simple.
179
00:24:05,610 --> 00:24:12,540
So the idea is that imagine that all this,
180
00:24:12,540 --> 00:24:21,270
each one of these lines of direction and somehow given one line splitting in two lines of two lines going to one line.
181
00:24:21,270 --> 00:24:27,750
This is sort of combination of direction and whatever it means the quantum mechanics, a combination of high and low momentum.
182
00:24:27,750 --> 00:24:32,880
And imagine these numbers. Here are the possible discrete value of total agreement.
183
00:24:32,880 --> 00:24:38,010
So how much something is rotating? Nothing here seems to talk about directions.
184
00:24:38,010 --> 00:24:42,300
So there's no directions here. Can I get directions out of that?
185
00:24:42,300 --> 00:24:48,690
Can I compute angles out of that? And then remember, directions are always is.
186
00:24:48,690 --> 00:24:56,850
Something is related to something else. But if I have one spinnaker here and another it there,
187
00:24:56,850 --> 00:25:05,220
it makes no sense to talk about relative angle between the two because there's no connexion between the two.
188
00:25:05,220 --> 00:25:09,930
What could make sense is they think about that a little bit in this direction, this direction.
189
00:25:09,930 --> 00:25:19,140
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,
190
00:25:19,140 --> 00:25:26,040
one with number eight with a number of B, how I define the angle between these two.
191
00:25:26,040 --> 00:25:37,140
And he writes a formula for the angle between the two only based on the spin across it inside here, more precisely probability.
192
00:25:37,140 --> 00:25:47,440
He writes a formula for having the probability of one of the other values of possible angles between the two.
193
00:25:47,440 --> 00:25:51,490
Given the spectrum here and how does it?
194
00:25:51,490 --> 00:26:02,050
Oh, let me be technical for one minute and actually give the the core of the of the story?
195
00:26:02,050 --> 00:26:08,190
You take this. Finished when you.
196
00:26:08,190 --> 00:26:13,060
You are two point and you do another one with an ex.
197
00:26:13,060 --> 00:26:25,840
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.
198
00:26:25,840 --> 00:26:30,610
And you ask how they combine and they can combine depending on the angle,
199
00:26:30,610 --> 00:26:37,810
by sort of summing up or by subtracting one from the other with all possibility between the two.
200
00:26:37,810 --> 00:26:45,350
So this x here can be any number between a plus b.
201
00:26:45,350 --> 00:26:51,440
And a minus B thinking of two vectors with an angle between.
202
00:26:51,440 --> 00:27:03,860
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.
203
00:27:03,860 --> 00:27:13,540
And then you want a probability of having that angle theta just in terms of these two.
204
00:27:13,540 --> 00:27:17,270
And he and Roger writes a formula which is a some simple formula,
205
00:27:17,270 --> 00:27:25,490
which is something related to the value of this pinnacle divided by the value of this big network.
206
00:27:25,490 --> 00:27:36,470
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,
207
00:27:36,470 --> 00:27:47,070
which I'm not writing just depend on AB and X with the normal network is just the value.
208
00:27:47,070 --> 00:27:52,870
Of the to square there are some technicalities is not important, will this minute to spare is by taking to SpiNET?
209
00:27:52,870 --> 00:28:07,680
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?
210
00:28:07,680 --> 00:28:10,290
So the value?
211
00:28:10,290 --> 00:28:21,380
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.
212
00:28:21,380 --> 00:28:26,690
The gift, the sense of it. Imagine you have a simple spin area here.
213
00:28:26,690 --> 00:28:31,460
Two one one. I'll tell you how to compute the value.
214
00:28:31,460 --> 00:28:38,900
The value is given by a sum divided by a product, and you do like that every time you have a line,
215
00:28:38,900 --> 00:28:45,260
a link with a number end, you copy it with any little line.
216
00:28:45,260 --> 00:28:52,640
So here you have to. You rewrite it with two lines.
217
00:28:52,640 --> 00:29:05,610
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.
218
00:29:05,610 --> 00:29:18,520
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.
219
00:29:18,520 --> 00:29:23,650
So you open up your spine network in some expanded spinnaker with individual
220
00:29:23,650 --> 00:29:29,110
little lines and then you count the number of closed loops here you call it.
221
00:29:29,110 --> 00:29:35,950
See, that case is to that case is one, and you count the number of intersection.
222
00:29:35,950 --> 00:29:50,500
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.
223
00:29:50,500 --> 00:30:00,280
To the eye. Minus two to the seat divided the product on all the links of the number of the links to Factorial.
224
00:30:00,280 --> 00:30:07,720
That's it. So I've given you the full formula with up some details of the angle.
225
00:30:07,720 --> 00:30:17,650
You get a formula for the angle purely combinatorial out of just minus one plus one's minus two counting nothing else.
226
00:30:17,650 --> 00:30:25,570
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.
227
00:30:25,570 --> 00:30:35,260
So you get a reconstruction of all the possible angles and the relations from combinatorics.
228
00:30:35,260 --> 00:30:42,130
Now it seems magic is, of course not magic, as Roger says in his article, and some of you immediately applies,
229
00:30:42,130 --> 00:30:46,810
you just reverse engineering representation theory of the rotation group.
230
00:30:46,810 --> 00:30:56,230
So that's what it is. It's angular momentum theory unpacked behind this is the this is the representational spin one.
231
00:30:56,230 --> 00:31:03,460
This is a particular spin one half. This expansion is the fact that higher for those of you who know the magmatic,
232
00:31:03,460 --> 00:31:10,330
higher representation can be written as tensor product of a smaller representation of fundamental content and so on.
233
00:31:10,330 --> 00:31:16,120
So it's sort of a working backward we know from the theory of representation of rotational group.
234
00:31:16,120 --> 00:31:21,430
But if you forget all that, that's spectacular. Yes, just.
235
00:31:21,430 --> 00:31:33,170
Counting and out of counting, you get the mathematics finals.
236
00:31:33,170 --> 00:31:37,590
That was it for the 71 paper and the core ideas.
237
00:31:37,590 --> 00:31:43,410
Again, our space is continuous. Quantum mechanics is discrete sweetness.
238
00:31:43,410 --> 00:31:47,790
At the core, we expect space to be quantum mechanical.
239
00:31:47,790 --> 00:31:51,670
Can we get continuous peace out of this greatness?
240
00:31:51,670 --> 00:31:57,000
It also is, yes, at least for what regards the elections.
241
00:31:57,000 --> 00:32:07,470
We can all by using the sweetness, by using the fact that the property I'm looking for are relational.
242
00:32:07,470 --> 00:32:17,320
It's one direction. We expect another one. And by not computing angles, preferably two different angles the core ingredients of quantum mechanics.
243
00:32:17,320 --> 00:32:32,470
Nineteen seventy one, Roger Penrose, twenty years later, twenty three years later, right here.
244
00:32:32,470 --> 00:32:53,760
You see here. James Colour Nineteen ninety five spy networks enter loop quantum gravity.
245
00:32:53,760 --> 00:33:00,150
Just to tell you where we were. Nineteen eighty five, roughly.
246
00:33:00,150 --> 00:33:06,060
It's loops. The beginning of loop quantum gravity.
247
00:33:06,060 --> 00:33:13,140
Hear the story. It's not going to cover the full story, but just to orient you where you are.
248
00:33:13,140 --> 00:33:19,710
It's a long and complicated story of the quantisation general activity. It starts with the rock.
249
00:33:19,710 --> 00:33:25,650
It starts with an Einstein goes to the rock. The Re writing generative Hamiltonian form.
250
00:33:25,650 --> 00:33:35,160
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
251
00:33:35,160 --> 00:33:44,190
of relativity for doing quantum version and therefore going to the quantum properties for space time.
252
00:33:44,190 --> 00:33:54,810
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,
253
00:33:54,810 --> 00:33:59,670
the will of the white equation with a sort of screening equation for general activity.
254
00:33:59,670 --> 00:34:10,590
It's a very, very complicated the fact set of infinite set of functional differential equations coupled which nobody knows how to deal with.
255
00:34:10,590 --> 00:34:15,030
And then magically Ted Jacobson and Lee smiling in the air in the 80s.
256
00:34:15,030 --> 00:34:28,510
Find solutions to this equation, and these solutions are readable by loops, closed lines in space, and nobody knows what how to encode.
257
00:34:28,510 --> 00:34:33,330
And then that's when I entered the game, and I remember the confusion at the time.
258
00:34:33,330 --> 00:34:42,990
Somehow, they present an exhibition of space where you have geometry along this loop and nowhere else, whatever.
259
00:34:42,990 --> 00:34:45,840
And during this 10 years,
260
00:34:45,840 --> 00:34:57,990
there was a lot of discussion about the interpretation of this complicated loop solutions and two key steps were taken by a Polish mathematician,
261
00:34:57,990 --> 00:35:06,870
Yurek, Levandowski and a Argentinian physicist haaga puling who realised the first of all
262
00:35:06,870 --> 00:35:14,940
that the theory is not complete unless you consider loops that can cross themselves.
263
00:35:14,940 --> 00:35:21,960
And the thing is not complete unless you consider also the possibility that troops run over themselves,
264
00:35:21,960 --> 00:35:26,490
so they go through the same path, so to say whatever that means.
265
00:35:26,490 --> 00:35:40,010
That was what Urich understood and what pulling understood is that these crossing points is not a crucial.
266
00:35:40,010 --> 00:35:47,460
Because the states are represented by this funny solution that will admit that question have in a very precise technical sense,
267
00:35:47,460 --> 00:35:55,430
zero volume unless there are these notes. So, Nord's.
268
00:35:55,430 --> 00:36:03,920
Are essential, the name Lupe Fluke, antigravity, slipping, misleading,
269
00:36:03,920 --> 00:36:13,570
because if you only have lupus, you can get anything interesting and the volume sit here.
270
00:36:13,570 --> 00:36:23,770
Which gives a sort of picture like this one, if you have a spy network roughly.
271
00:36:23,770 --> 00:36:41,010
So imagine you must be network. So imagine you have a graph.
272
00:36:41,010 --> 00:36:48,830
Then you have the we have to think about is that the volume sort of sits on these.
273
00:36:48,830 --> 00:37:02,180
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.
274
00:37:02,180 --> 00:37:04,370
And then in 95,
275
00:37:04,370 --> 00:37:15,710
Lee Smalling was visiting Verona and we were working on a technical problem which is trying to understand the actual geometry in these states.
276
00:37:15,710 --> 00:37:21,050
And in particular, we were thinking of this problem here. Suppose this is a chunk of space.
277
00:37:21,050 --> 00:37:28,730
Let me just draw to trade to for simplicity. And this is another one.
278
00:37:28,730 --> 00:37:36,110
And they're attached to this common face, common triangle.
279
00:37:36,110 --> 00:37:44,120
The question we're asking is how big is this area and we attack we we study this question
280
00:37:44,120 --> 00:37:48,530
with standard kind of calculation that you do in quantum mechanics and quantum mechanics.
281
00:37:48,530 --> 00:37:54,680
You compute this this crudeness, which is common through a technical calculation in quantum mechanics,
282
00:37:54,680 --> 00:37:59,990
which is called the spectral analysis of a sulphur joint operator.
283
00:37:59,990 --> 00:38:04,490
And in the theory, the theory was about gravity and gravity geometry.
284
00:38:04,490 --> 00:38:11,130
So there was a lot of geometry. So the geometry, the area of this, it's.
285
00:38:11,130 --> 00:38:16,170
A observable of the geometry and the geometry is a gravitational field,
286
00:38:16,170 --> 00:38:21,090
so the gravitational field is a quantum field, so there was a quantum appraisal associated with it.
287
00:38:21,090 --> 00:38:26,460
And we say the metrics of this going from a plate of all this one to repudiate is for all this.
288
00:38:26,460 --> 00:38:37,890
And what we find, we find that the spectrum is discrete and so we can look technically the island states of this operator.
289
00:38:37,890 --> 00:38:44,700
So the quantum state of space that have these particular values of the area.
290
00:38:44,700 --> 00:38:55,110
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,
291
00:38:55,110 --> 00:39:10,680
etcetera, where these numbers represent the area of this triangle, which of course, defines the direction space.
292
00:39:10,680 --> 00:39:17,070
And what if we want to know the angle between the direction of this and the direction of this?
293
00:39:17,070 --> 00:39:23,730
So the direction of the face? Of this chunk of space which goes out and this one.
294
00:39:23,730 --> 00:39:30,680
Well, not surprising is given by pendulous formula.
295
00:39:30,680 --> 00:39:39,980
And the possible angle between the two are discreet and the possible areas are discrete.
296
00:39:39,980 --> 00:39:54,040
So here we are, taking out from the theory a description of Michael structural space where there are a small chunks.
297
00:39:54,040 --> 00:39:59,820
A volume. That defines their actions.
298
00:39:59,820 --> 00:40:03,550
But this election not only defied relative to one another.
299
00:40:03,550 --> 00:40:14,680
And probabilistically and can only take discrete values, and at the core of a loop, quantum gravity is a.
300
00:40:14,680 --> 00:40:22,180
In mathematics, which is essentially just a bunch of representational theory of the rotation group,
301
00:40:22,180 --> 00:40:37,270
which is exactly the object that Roger has introduced to take more than 20 years earlier without any consideration of general activity of willow,
302
00:40:37,270 --> 00:40:41,500
the weight equation or apparatus or anything like that.
303
00:40:41,500 --> 00:40:50,000
And nowadays, this pinnacle which are used are essentially the same thing, just built with the same mathematics.
304
00:40:50,000 --> 00:40:55,300
A little bit some technical complication, but are the same thing.
305
00:40:55,300 --> 00:40:58,450
So I found I found this story marvellous. Marvellous,
306
00:40:58,450 --> 00:41:06,970
because what Roger had done in the seventies was a sidestepping all the construction of the theory on the
307
00:41:06,970 --> 00:41:16,510
basis of the details of a current knowledge about the generative structural space and quantum mechanics,
308
00:41:16,510 --> 00:41:28,530
and just zoomed down to the core idea. Continuity, discreet, necessary rationality, and found the natural way of putting them together today.
309
00:41:28,530 --> 00:41:41,560
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.
310
00:41:41,560 --> 00:41:50,200
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,
311
00:41:50,200 --> 00:41:56,980
this elementary, these things which describe the elementary quantum of space.
312
00:41:56,980 --> 00:42:03,580
This is a little chunk of space, a love triangle space. This is a is the connexion between the two.
313
00:42:03,580 --> 00:42:12,130
So this number here says how big is the connexion? How large is the area between the two and the relative direction?
314
00:42:12,130 --> 00:42:18,340
These are not like particles of quantum field theory like photons.
315
00:42:18,340 --> 00:42:22,420
These are not quanta that move around in a given space.
316
00:42:22,420 --> 00:42:27,490
These are quantum of space itself. So these don't do not fit into a space.
317
00:42:27,490 --> 00:42:38,210
You cannot ask what is in between two of this point. These are the elementary thing that built up.
318
00:42:38,210 --> 00:42:45,170
The structure in which we live, which we at the very large scale perceive as a continuous space.
319
00:42:45,170 --> 00:42:55,610
This is a way of doing physics, which takes us suddenly completely away from a field theory as it's done by particle physics,
320
00:42:55,610 --> 00:43:07,340
by people do electromagnetism, where the main ingredients are space a priori continuous time space, time for correct group energy momentum.
321
00:43:07,340 --> 00:43:10,880
All these quantities disappear from the theory because there's no space.
322
00:43:10,880 --> 00:43:17,030
There's no time. I'm not talking about time here, but there's also no time the same sense.
323
00:43:17,030 --> 00:43:20,000
There is no energy. There is no momentum.
324
00:43:20,000 --> 00:43:30,380
The quantum numbers that describe something don't have a p and momentum like in particle physics or an X a position.
325
00:43:30,380 --> 00:43:40,100
Because position momentum are only in the small things, relatively to one another, but not relative to a background structure, which is space.
326
00:43:40,100 --> 00:43:45,590
The core physical intuition behind loop quantum gravity is this.
327
00:43:45,590 --> 00:43:50,780
And it was largely anticipated by by Roger.
328
00:43:50,780 --> 00:44:02,840
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
329
00:44:02,840 --> 00:44:06,950
developing around the construction of this theory around the set of key ideas,
330
00:44:06,950 --> 00:44:16,460
which I've tried to to explain. And it has been a privilege to sort of walk in the step of Roger.
331
00:44:16,460 --> 00:44:24,530
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.
332
00:44:24,530 --> 00:44:49,683
Thank you.