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.