1 00:00:03,870 --> 00:00:09,150 So thanks very much. It's a great honour to give this ninth Danny Schama lecture. 2 00:00:09,810 --> 00:00:21,870 So my thanks to Lidia, to the Sharma family, to the organisers, Joe and John, to All Souls College for sponsoring this lecture series. 3 00:00:22,260 --> 00:00:32,520 So as Joe says, my Denis was my dphil supervisor back in the 1970s something. 4 00:00:34,050 --> 00:00:47,820 And yeah, as, as he says for the last gosh, 35 years, I think I've got my maths right, my trajectory has taken a rather different uh, uh, direction. 5 00:00:49,350 --> 00:00:55,530 But what I want to do today is to try to look, uh, retrospectively, 6 00:00:55,530 --> 00:01:04,829 I guess at some of the issues which uh, in the seventies for sure excited Denis and excited me. 7 00:01:04,830 --> 00:01:13,320 Two issues to do with the marriage of general relativity and quantum mechanics, issues to do with the large scale structure of the universe. 8 00:01:13,830 --> 00:01:18,870 Uh, issues to do with the very sort of meaning and interpretation of quantum mechanics. 9 00:01:19,890 --> 00:01:30,660 And I say retrospectively, what I'm going to try to do here is, well, at least ask the question whether some of the mathematical concepts, 10 00:01:30,660 --> 00:01:36,569 which I guess have become pretty much second nature to me in the last, whatever it is, 11 00:01:36,570 --> 00:01:45,000 30 odd years, things to do with, uh, with nonlinear dynamics, with chaos, with fractal attractors and fractal geometry, 12 00:01:45,360 --> 00:01:51,000 things which are very central if one is looking at uh, uh, the climate system or the weather. 13 00:01:52,570 --> 00:01:58,209 But to ask whether these mathematical concepts might actually provide new insights 14 00:01:58,210 --> 00:02:03,400 into these types of problems which which certainly were problems that excited Dennis. 15 00:02:03,700 --> 00:02:06,550 Of course, these are problems still very much alive today. 16 00:02:07,120 --> 00:02:14,020 I think we still don't have any unanimity about how to bring general relativity and quantum mechanics together. 17 00:02:14,500 --> 00:02:18,640 And we still have fundamental problems trying to understand basic quantum mechanics. 18 00:02:19,030 --> 00:02:24,310 So this may be this may seem a tall order, 19 00:02:24,550 --> 00:02:29,140 but I'm going to try to persuade you at least that there are some new ways of 20 00:02:29,140 --> 00:02:35,080 thinking about these old problems that nonlinear dynamics might bring to the table. 21 00:02:35,980 --> 00:02:39,820 Well, at least you can make your own judgement about that by the end of the lecture. 22 00:02:41,810 --> 00:02:51,080 So let me start. The first of my protagonists is Ed Lorenz, protagonist in the title of My Talk. 23 00:02:51,650 --> 00:02:55,820 He's a meteorologist from MIT. Someday, actually, I got to know pretty well. 24 00:02:56,210 --> 00:02:59,300 He died, unfortunately, about three years ago now, I think. 25 00:03:01,580 --> 00:03:11,150 Very famous, of course, outside meteorology, famous around the world for his his 1963 paper on deterministic non periodic flow. 26 00:03:12,140 --> 00:03:20,090 If I had more time, I'd tell you a bit more about it. But I'm sure many of you are familiar with the equations which came out of that paper. 27 00:03:20,330 --> 00:03:29,480 Three coupled ordinary differential equations nonlinear equations which exhibited this concept we now call sensitive dependence on initial conditions. 28 00:03:30,200 --> 00:03:35,990 So here's two animations of two initial conditions, which are almost but not quite identical. 29 00:03:36,440 --> 00:03:41,240 They track each other for a while, and then they they diverge from each other. 30 00:03:42,380 --> 00:03:49,630 And this is often called the butterfly effect, although interestingly, this is actually a bit of a misnomer. 31 00:03:49,640 --> 00:03:53,360 What at Lorenz meant by the butterfly effect is actually somewhat different to this. 32 00:03:53,360 --> 00:03:56,960 But again, that's another title of another lecture, which I won't go into. 33 00:03:59,340 --> 00:04:06,270 Okay. Now, you might well ask if you know, your history was actually Lorenz, 34 00:04:06,270 --> 00:04:11,450 the first person to discover a sensitive dependence of evolution on initial conditions. 35 00:04:11,460 --> 00:04:20,280 And in fact, in some sense, he wasn't. At least 50 years earlier on Poincaré studying the gravitational three-body problem. 36 00:04:20,280 --> 00:04:25,050 It also discovered this phenomenon essentially of what we now call chaos. 37 00:04:25,800 --> 00:04:28,410 So the question you might ask is, well, what's special? 38 00:04:28,440 --> 00:04:34,710 What what's special that Lorenz brought to the table that Poincaré had not already brought to the table? 39 00:04:36,030 --> 00:04:36,749 And in a sense, 40 00:04:36,750 --> 00:04:46,860 this is very much the theme of my talk that what Lorenz brought to the table in the particular equations that he looked at which were not actually 41 00:04:46,860 --> 00:04:58,500 present in the gravitational three-body equations was this type of geometry in the state space of the equations which the system evolved towards. 42 00:04:58,740 --> 00:05:05,910 So take any initial state X, Y and z a time zero evolve it, 43 00:05:06,270 --> 00:05:14,520 and you see eventually it starts no matter where you start in the state space of X, Y and Z, it attracts towards this geometry. 44 00:05:16,220 --> 00:05:19,879 Laurence knew that this geometry had to have zero dimensions. 45 00:05:19,880 --> 00:05:25,010 By looking at the structure of the differential equations, he knew that this Aston toxic attracting set, 46 00:05:25,160 --> 00:05:30,020 as it's called, has to have a zero dimension, 000 volume. 47 00:05:30,380 --> 00:05:34,870 But he didn't. You know, he was struggling for a long time. 48 00:05:34,880 --> 00:05:40,220 It's very interesting reading his papers and his notes and things, how he struggled to try to understand what this was. 49 00:05:40,730 --> 00:05:45,140 He realised it couldn't be a point because the system didn't settle down to a steady state. 50 00:05:45,500 --> 00:05:49,860 It wasn't a circle. The system didn't wasn't periodic. 51 00:05:49,880 --> 00:05:53,540 The title of his paper was non periodic deterministic motion. 52 00:05:54,530 --> 00:05:59,149 It wasn't a surface. So what was it? And he agonised about this. 53 00:05:59,150 --> 00:06:06,049 And in fact, I think this is one of the real pieces of genius of Lawrence to realise that this geometry he was looking at, 54 00:06:06,050 --> 00:06:11,030 which came out of these differential equations, was actually a fractal a fractal structure. 55 00:06:12,680 --> 00:06:17,180 And what I want to talk about is that fractal structures. 56 00:06:17,510 --> 00:06:23,960 If we focus on the fractal structure associated with these equations rather than the differential equations per say, 57 00:06:24,650 --> 00:06:30,680 we can discover some remarkable connections into deep parts of 20th century mathematics. 58 00:06:30,710 --> 00:06:35,480 So I'm going to just highlight very briefly a couple of those things that you would never, 59 00:06:35,480 --> 00:06:39,650 I think, have gleaned just by looking at the differential equations as such. 60 00:06:40,760 --> 00:06:44,780 The reason I'm telling you this is I want to make the point that these fractal geometries, you know, 61 00:06:44,780 --> 00:06:52,670 often one sees fractals on the front covers of books or conference flyers and things just as a kind of a sexy piece of of geometry. 62 00:06:53,330 --> 00:06:59,660 But no, they're much more than this. They have very profound links into into deep areas of mathematics. 63 00:07:00,980 --> 00:07:02,450 So just a couple of examples. 64 00:07:03,050 --> 00:07:11,390 The question is how how might one characterise this fractal attractor if one didn't have the differential equations to use? 65 00:07:12,180 --> 00:07:17,780 And one technique that people mathematicians use to try to characterise the 66 00:07:18,110 --> 00:07:23,839 attractor are looking at periodic orbits that lie in some sense close to almost, 67 00:07:23,840 --> 00:07:28,070 you could say embedded in the attractor. They lie close to within the body of the attractor. 68 00:07:28,370 --> 00:07:31,849 So there are many, in fact periodic but unstable periodic orbits. 69 00:07:31,850 --> 00:07:33,260 These actually repeat each other. 70 00:07:34,460 --> 00:07:42,380 You can characterise these or describe these periodic orbits by a technique called symbolic dynamics or symbolic representation. 71 00:07:43,510 --> 00:07:50,440 So what I'm just there is you just partition the attractor into two lobes called the left hand lobe out on the right hand low bar. 72 00:07:51,400 --> 00:08:01,200 And then these particular periodic or the period in general periodic orbits that one looks at, one can specify a sum symbolic bit string. 73 00:08:01,960 --> 00:08:08,470 So for example, this corresponds to a periodic orbit that starts on the left hand lobe, moves to the right, then goes to the left, 74 00:08:08,470 --> 00:08:14,380 then goes to the right, then goes to the left, then goes stays on the left once more and then keeps repeating itself. 75 00:08:15,570 --> 00:08:22,800 And this turns out to be topologically equivalent to the, say, communal garden if that means something to you. 76 00:08:22,830 --> 00:08:29,610 Trefoil Not. It's one of the basic knots in knot theory and using this kind of jones polynomial type of language, 77 00:08:29,610 --> 00:08:36,870 one can classify these periodic orbits or alternatively these symbolic strings as knots. 78 00:08:37,170 --> 00:08:41,040 The knots of the of the periodic orbits. 79 00:08:43,640 --> 00:08:51,380 One, I think, quite remarkable result which a French mathematician and geese showed in 2000. 80 00:08:52,070 --> 00:08:57,410 And if you have time to write down, if you're interested at all in this, there's a fantastic online, uh, 81 00:08:57,680 --> 00:09:06,050 kind of, uh, discussion of this result with some fantastic movies showing how these knots evolved and so on. 82 00:09:06,350 --> 00:09:11,780 The result is that these Lorenz knots are entirely equivalent to what are called modular knots. 83 00:09:12,770 --> 00:09:15,440 Now, I'm not going to tell you what a modular nut is because don't have time. 84 00:09:15,710 --> 00:09:24,230 But suffice to say that it's related to properties of the modular group and one can think of the modular 85 00:09:24,230 --> 00:09:31,730 group as essentially the group of two by two matrices with integer elements and with unit determinant. 86 00:09:32,360 --> 00:09:34,520 So here's an element of the modular group, 87 00:09:35,480 --> 00:09:45,530 and it turns out it also can be written in as sort of a string of symbols where these symbols now define elemental two by two matrices. 88 00:09:46,760 --> 00:09:52,940 And I guess the work of DS is basically to show an equivalence between these symbolic strings for the modular elements of 89 00:09:52,940 --> 00:10:00,770 the modular group and the symbolic strings of the periodic orbits of the of the Lorenz associated with the Lorenz attractor. 90 00:10:03,040 --> 00:10:07,629 Now, if you go into the mathematics, if you go into this this weblink, 91 00:10:07,630 --> 00:10:13,300 you'll see very quickly that the sort of mathematics guess uses to prove this theorem 92 00:10:13,630 --> 00:10:17,230 is the sort of mathematics that number theorists would feel very at home with. 93 00:10:17,710 --> 00:10:20,770 He talks about lattices on the org and plane. 94 00:10:21,040 --> 00:10:25,450 He talks about via Strauss elliptic functions associated with those lattices. 95 00:10:26,170 --> 00:10:30,820 He talks about the Eisenstein series again associated with those lattices. 96 00:10:31,000 --> 00:10:35,250 And then one sort of moves into the field of elliptic curves and modular forms. 97 00:10:36,070 --> 00:10:41,320 And this is the area that that Andrew Wiles unified in his proof of Fermat's Last Theorem. 98 00:10:42,410 --> 00:10:50,120 So from a set of differential equations, one suddenly finds via the structure of the geometry which these equations produce. 99 00:10:50,420 --> 00:10:55,940 One is moving into areas of quite deep number theory, which you would never have guessed, 100 00:10:55,940 --> 00:10:59,330 I think to just look at the differential equations themselves. 101 00:11:01,640 --> 00:11:04,360 Another problem you might want to think about is this. 102 00:11:04,370 --> 00:11:12,770 If I, uh, if you were to give me, let's say, a point in this three dimensional state space of the Lawrence equations, 103 00:11:13,370 --> 00:11:21,650 and you asked me, is there an algorithm for determining whether that point lies on this, on this Lawrence fractal attractor? 104 00:11:22,640 --> 00:11:27,620 The answer is there is no algorithm. It's actually it's actually a non if you like. 105 00:11:27,620 --> 00:11:30,140 It's a problem that can't be solved by finite algorithms. 106 00:11:30,530 --> 00:11:35,359 And you can imagine there's not sort of a totally unreasonable to imagine why that should be, 107 00:11:35,360 --> 00:11:39,790 because even if one knew one of the points on the attractor, it might take you. 108 00:11:40,070 --> 00:11:45,350 No matter how far you integrated the equations, you might still not have reached the point that you are given. 109 00:11:45,800 --> 00:11:55,670 And so you would never have a procedure for deciding in finite time whether your given point was, was, was belonging to the attractor. 110 00:11:56,000 --> 00:12:03,200 Now, this was proven rigorously in in this book by Blue Mittal, which included the famous Steve Smail, incidentally, 111 00:12:03,650 --> 00:12:08,390 result being basically that's what are called halting sets, must have integral house stored dimension. 112 00:12:08,930 --> 00:12:14,150 And we know that these fractal attractors are characterised by fractional dimension. 113 00:12:15,490 --> 00:12:23,740 And indeed, one can say one can take many of the classic problems in computing science that are known not to be solvable 114 00:12:23,740 --> 00:12:30,670 by algorithm and show that they have an equivalence in terms of of of a of a fractal geometric problem. 115 00:12:31,660 --> 00:12:37,750 If people have heard of the post correspondence problem, this is one of the classic problems that can't be solved by algorithms. 116 00:12:38,200 --> 00:12:43,809 One can show it's equivalent to the question of asking, what does a point lie on the attractor? 117 00:12:43,810 --> 00:12:49,300 But does a line intersect the attractor of a of a of a chaotic dynamical system? 118 00:12:50,860 --> 00:12:54,700 So you can see we're going into the territory here of the girdle incompleteness 119 00:12:54,700 --> 00:13:02,290 theorem and the corresponding sharing non compute ability type of issue. 120 00:13:04,360 --> 00:13:12,370 So just to summarise then, so far, what I'm trying to tell you is that here are these Lorentz differential equations. 121 00:13:12,370 --> 00:13:16,750 I think Newton probably could have understood in principle what these equations were saying. 122 00:13:17,440 --> 00:13:22,329 After all, he did discover the calculus, but he would never have guessed. 123 00:13:22,330 --> 00:13:29,740 I am sure that he would never have guessed that these equations could generate this amazingly rich and deep type of geometry. 124 00:13:30,190 --> 00:13:38,319 And it's through that geometry that we see links into quite into some of the classic problems of 20th century mathematics, 125 00:13:38,320 --> 00:13:44,050 whether it's Wiles, proof of Fermat's Last Theorem or the Girdle Theorem or the Turing Non Compute ability theorems. 126 00:13:45,190 --> 00:13:53,650 So the reason I'm saying this is that it's by focusing on this geometry that one gets these links into these deep areas of maths. 127 00:13:55,410 --> 00:13:56,670 So what I want to do then, 128 00:13:56,880 --> 00:14:04,830 having convinced you that this is these these types of fractal geometries are really serious topics and worthy of serious discussion. 129 00:14:05,250 --> 00:14:10,170 I want to now switch to these three people at the bottom of the of the figure for three more. 130 00:14:11,790 --> 00:14:18,300 These are three, of course, two famous 20th century physicists, Schrodinger, Heisenberg and Dirac. 131 00:14:18,930 --> 00:14:29,460 And ask the question, do this does this type of geometry provide us with any new insights into these deep problems of 20th century physics, 132 00:14:29,700 --> 00:14:34,830 which are characterised by these three people, Schrödinger, Heisenberg and Dirac? 133 00:14:36,030 --> 00:14:38,510 Now, at the level of differential equations, 134 00:14:38,520 --> 00:14:46,259 you would have to say this is a barking mad concept that there might ever be any connection because Schrodinger's equation or 135 00:14:46,260 --> 00:14:52,740 Heisenberg's form of the Schrödinger equation or Dirac's relativistic form of the shooting equation are all linear equations, 136 00:14:53,160 --> 00:14:58,620 and this equation is manifestly nonlinear. It's got these terms X and Y and X, and so it's a nonlinear equation. 137 00:14:58,620 --> 00:15:07,380 So how could there ever be any connection at all? So that's the kind of a many I'm sure you've read books who make this point, you know, 138 00:15:07,800 --> 00:15:11,459 as the unpredictability of quantum theory have anything to do with unpredictability of chaos. 139 00:15:11,460 --> 00:15:16,440 And usually people say, no, they're not, because one is a linear century, a linear problem, the other is nonlinear. 140 00:15:16,440 --> 00:15:21,180 They don't have anything to do with each other. I want to make the point that I don't think this is correct. 141 00:15:21,300 --> 00:15:27,600 I think there are some profound links, but they need to be approached by looking at this intermediate geometry. 142 00:15:27,600 --> 00:15:33,330 This is the key. So this is sort of the basis of the talk to some extent. 143 00:15:35,970 --> 00:15:41,610 I thought I would start with a few quotes before we move on any further. 144 00:15:42,000 --> 00:15:54,600 This is from one of Hawking's popular books where he sets out, I guess, the standard model, if you like, of of quantum interpretation. 145 00:15:54,860 --> 00:16:01,229 The standard interpretation which says there really is no connection between quantum the 146 00:16:01,230 --> 00:16:07,740 unpredictability of quantum physics and the unpredictability of nonlinear dynamical systems. 147 00:16:08,160 --> 00:16:15,570 So according to quantum physics, says Hawking, no matter how much information we obtain or how powerful our computing abilities, 148 00:16:15,810 --> 00:16:22,260 the outcomes of physical processes cannot be predicted with certainty because they're not determined with certainty. 149 00:16:23,280 --> 00:16:30,450 So is this fundamental issue that there is there is no determinism in quantum physics. 150 00:16:30,690 --> 00:16:40,979 That's the standard view. However, I want to contrast that with two other comments from also eminent, very eminent scientist. 151 00:16:40,980 --> 00:16:47,610 So here's Dirac himself, one of the the grandfathers father figures of quantum theory. 152 00:16:48,720 --> 00:16:50,580 One can always hope, he says, 153 00:16:50,580 --> 00:16:55,800 that there will be future developments which will lead to a drastically different theory from the present quantum mechanical theory, 154 00:16:56,130 --> 00:17:00,090 and for which there may be a partial return of determinism. 155 00:17:01,800 --> 00:17:11,370 So this, uh, this from the one of the guys that was the Founding Fathers, uh, at least suggesting that the, the door is not yet closed. 156 00:17:12,810 --> 00:17:17,580 And this is from my third protagonist of my title, Roger Penrose, 157 00:17:17,580 --> 00:17:25,460 whose work will feature more and more as we go through the talk with an even, I think, more positive statement. 158 00:17:25,470 --> 00:17:34,500 Still, he says, it seems quite plausible that the correct theory of quantum gravity might be a deterministic but non computable theory. 159 00:17:35,620 --> 00:17:44,530 Now this non computable action is interesting because we've already seen a maybe is this is this just a spurious link or is there something deep here? 160 00:17:44,770 --> 00:17:49,030 We've already seen these fractal geometries are non computable. 161 00:17:49,240 --> 00:17:54,340 They membership of that set cannot be determined by algorithm. 162 00:17:55,590 --> 00:17:58,680 So is that is that just a coincidence or is there something deep link here? 163 00:18:00,660 --> 00:18:07,800 Those three quotes were sort of deliberately taken from three people that link very closely to Dennis. 164 00:18:09,300 --> 00:18:14,080 Stephen was was one of Dennis's students. 165 00:18:15,180 --> 00:18:19,680 Dirac was Dennis's, uh, supervisor. 166 00:18:20,630 --> 00:18:32,180 And Roger was somebody that Denis inspired to move into mathematical physics from an earlier career career in pure mathematics. 167 00:18:32,930 --> 00:18:40,160 So Denis was very instrumental, pivotal in bringing Roger into the relativity community. 168 00:18:41,780 --> 00:18:48,110 Dennis has had an enormous number of very eminent students, and a few perhaps not quite so eminent students. 169 00:18:49,040 --> 00:19:00,620 Um, but, uh, I find that it's almost impossible not to be sort of proud of my linkage to Dennis because of this, 170 00:19:00,800 --> 00:19:04,100 an ability to talk about my academic brother, 171 00:19:04,100 --> 00:19:13,310 Stephen Hawking, my academic father, uh, Roger Penrose, my academic grandfather, Paul Dirac, you know, my academic uncle Roger Penrose. 172 00:19:13,430 --> 00:19:17,600 So that's I couldn't resist putting that slide up. 173 00:19:19,890 --> 00:19:24,090 Okay. Let's just move back then to the Lorenz system. 174 00:19:25,300 --> 00:19:30,060 And this is a slide, actually, I often show in my climate talks or weather type talks, 175 00:19:30,060 --> 00:19:38,430 because it shows how the predictability of the system is a function of initial conditions. 176 00:19:38,430 --> 00:19:43,560 So some parts of the attractor are extremely stable and predictable. 177 00:19:44,070 --> 00:19:48,600 So if one imagines this as some little error bowl, perhaps a cup through an error ball, 178 00:19:49,260 --> 00:19:54,870 the error actually doesn't grow at all as you propagate as you as you evolve the system forward in time. 179 00:19:55,710 --> 00:20:02,790 Other initial states are more unpredictable and others yet more unpredictable. 180 00:20:02,790 --> 00:20:11,850 Still, the fact that the predictability varies around the attractor is a direct result of the nonlinear arity of the equations. 181 00:20:12,390 --> 00:20:18,180 So if these these are Lorenz equations, we can linear ize them. 182 00:20:19,140 --> 00:20:25,590 So look at the growth of small perturbations and the operator is now the derivative of F with respect to x. 183 00:20:26,070 --> 00:20:34,470 So if F is at least quadratic in x, then df by the x is at least linear in x i.e. this operator depends on the underlying state, 184 00:20:34,500 --> 00:20:45,850 and that's what we're seeing here. So somewhere, you know, some weather forecasts as Michael Fish found to his costs can be very unpredictable. 185 00:20:46,230 --> 00:20:51,690 And in fact, in other cases, we can make very potentially quite long range forecasts with confidence. 186 00:20:51,870 --> 00:20:55,980 That's the sort of basis behind much of weather prediction. 187 00:20:57,030 --> 00:21:05,940 But I'm not going to go down that route. What I want to talk about is what equation is actually describing this evolution of probability. 188 00:21:07,050 --> 00:21:10,710 I mean, for example, one could take this point here and say this supply, 189 00:21:10,730 --> 00:21:16,860 this little ring defines a 90% probability that the true state lies within that ring. 190 00:21:17,670 --> 00:21:25,470 Then as you map this forward in time, you get to some future forecast time and there's a 90% probability that the true state lies in that ring. 191 00:21:25,560 --> 00:21:30,480 The forecast time or this would be a sort of a squashed banana, if you like, if you looked at it properly. 192 00:21:30,810 --> 00:21:38,160 There's a 90% probability it lies in that squashed banana. What's the equation that's describing that evolution of probability? 193 00:21:38,790 --> 00:21:44,640 It's this thing. It's called the Louisville equation. This is written in its general, most general form. 194 00:21:45,270 --> 00:21:49,360 By the way, some people think the Louisville equation only works for Hamiltonian systems. 195 00:21:49,380 --> 00:21:55,800 This is not true. Louisville Equation. Conservation of probability works for Hamiltonian and non Hamiltonian systems. 196 00:21:56,220 --> 00:22:01,860 This is a bit like the mass conservation equation for a compressed, compressible fluid. 197 00:22:02,640 --> 00:22:07,560 If it was incompressible, you could move the Volt. But if it's compressible, the V has to be in here. 198 00:22:07,980 --> 00:22:13,710 Now, this is not this is not mass density and volume and velocity in physical space. 199 00:22:14,010 --> 00:22:21,630 This is probability and velocity and state space, but it's still a conservation equation, nevertheless, conservation of probability. 200 00:22:22,440 --> 00:22:28,470 Now, one thing you can note if you look at this equation is it's linear and in probability it's linear in row. 201 00:22:30,130 --> 00:22:34,180 And this is an important point I want to make. If I gave you that equation. 202 00:22:35,350 --> 00:22:38,770 You would not be able to deduce because it is linear. 203 00:22:39,100 --> 00:22:43,120 It cannot be underpinned by some underlying nonlinear dynamic. 204 00:22:44,050 --> 00:22:48,550 The linearity of this equation says nothing about the weather. 205 00:22:48,550 --> 00:22:52,240 The dynamics that generates that probability is linear or nonlinear. 206 00:22:52,690 --> 00:22:56,830 And in fact, in this system, as I've just pointed out, the system is nonlinear. 207 00:22:58,010 --> 00:23:07,190 So the linearity of the alluvial equation is really just a statement of the fact that the probabilities are conserved as the flow evolves. 208 00:23:09,300 --> 00:23:19,590 Now, if we have a Hamiltonian system, we can write the Louisville equation in this form using Poisson brackets I'm sure familiar to people, 209 00:23:21,300 --> 00:23:27,780 or if indeed if we had a dynamical system which was approximated, approximated a Hamiltonian system, this would be a good approximation. 210 00:23:29,760 --> 00:23:36,180 So I want to contrast this with the direct form, at least of the of the Schrodinger equation. 211 00:23:39,010 --> 00:23:45,970 Which I've written here. And the first point I want to make is how remarkably similar it is formally. 212 00:23:48,560 --> 00:23:56,540 It has a row, a d row by d t, it has a sort of an anti symmetric bracket, it has hamiltonians and so on. 213 00:23:57,260 --> 00:24:02,150 It seems to be some formal close similarity with the classical legal equation. 214 00:24:04,160 --> 00:24:08,720 And I'm sure it's because of that that Dirac perhaps was inspired to make that comment that, 215 00:24:09,530 --> 00:24:15,020 you know, just just as this equation can be underpinned by deterministic dynamics. 216 00:24:15,320 --> 00:24:21,530 So maybe also this equation might be underpinned by deterministic dynamics. 217 00:24:24,090 --> 00:24:30,660 And of course, the point that this this equation is linear says nothing at all about whether 218 00:24:30,930 --> 00:24:34,890 the underpinning dynamics is linear or nonlinear might just be non-linear. 219 00:24:36,500 --> 00:24:44,930 Incidentally, just as an aside, I often find people who who look for nonlinear things that might nonlinear processes that might 220 00:24:44,930 --> 00:24:50,060 come in to quantum mechanics sometimes add non-linear terms to the Schrodinger equation itself. 221 00:24:50,420 --> 00:24:57,680 And I always find this an extremely misguided idea personally, because if this is just reflecting a conservation of probability, 222 00:24:58,550 --> 00:25:01,580 you don't really want to tinker with it by adding non-linear terms. 223 00:25:01,850 --> 00:25:08,780 If you want to look for nonlinearity, try and find the equation that underpins this type of conservation of probability. 224 00:25:10,470 --> 00:25:17,760 Okay. Now of course I recognise that there are differences as well between the classical Louisville equation. 225 00:25:18,030 --> 00:25:21,960 There's a square root of minus one. There's a Planck's constant. 226 00:25:22,410 --> 00:25:29,040 And in fact, these things aren't just well, these aren't functions in state space operators in some Hilbert space. 227 00:25:29,580 --> 00:25:32,790 And this is an operator commentator bracket, not a person bracket. 228 00:25:34,730 --> 00:25:39,800 So there are similarities and there are differences. Do these differences matter? 229 00:25:40,220 --> 00:25:45,260 Does the fact it is operating on a Hilbert space rather than a classical state space, does it matter? 230 00:25:45,680 --> 00:25:48,140 Well, the answer, of course, is yes, it does matter. 231 00:25:48,770 --> 00:25:56,480 And there are a number of no go theorems in quantum mechanics, which, uh, which tell us very much that it does matter. 232 00:25:56,850 --> 00:26:06,830 And the most famous of these, I guess, is, is Bell's theorem, which in a nutshell says no physical theory based on local cores or hidden variables. 233 00:26:07,460 --> 00:26:14,330 So you posit some deterministic system which operates deterministically. 234 00:26:14,330 --> 00:26:19,130 The variables of the of the deterministic system may be hidden to you or hidden certainly hidden to quantum mechanics. 235 00:26:19,610 --> 00:26:25,330 But Bell's Theorem says that one can never replicate the predictions of quantum mechanics. 236 00:26:25,340 --> 00:26:29,120 It's based on this inequality, which I don't proceed. 237 00:26:29,120 --> 00:26:33,320 Don't propose to describe in detail. I don't propose to describe at all. 238 00:26:34,550 --> 00:26:40,310 So you either you either say from this that there is no such thing as deterministic reality, 239 00:26:40,880 --> 00:26:47,670 or you say, well, if there is deterministic reality, it somehow violates principles of relativity. 240 00:26:47,690 --> 00:26:49,790 There's something, to use Einstein's phrase, 241 00:26:49,790 --> 00:26:59,060 some spooky action at a distance going on in if we want to try to underpin quantum mechanics with some underlying deterministic model. 242 00:26:59,900 --> 00:27:09,460 That's the standard interpretation. Now I want to at this stage just talk about an idea which. 243 00:27:11,420 --> 00:27:20,090 Bell himself recognised was a potential way out of the constrictions of his theorem. 244 00:27:21,200 --> 00:27:25,069 I'll just say right from the beginning, it's it's an idea which he himself did not believe in. 245 00:27:25,070 --> 00:27:29,990 And I don't think anybody in the in the community or almost nobody believes in. 246 00:27:30,440 --> 00:27:39,050 So I'm not actually going to advocate this. You'll be pleased to know. But it's an important concept to get over in what I will talk about later on. 247 00:27:41,060 --> 00:27:42,139 And actually at first sight, 248 00:27:42,140 --> 00:27:52,040 it's a rather strange loophole or a escape from the constrictions of bells whom you might think to escape from the constrictions of battle serum. 249 00:27:52,490 --> 00:27:58,219 You might want to somehow relax this notion of determinism and imagine things being more stochastic. 250 00:27:58,220 --> 00:28:01,590 Maybe. But what Bell says actually is the opposite. 251 00:28:01,610 --> 00:28:10,610 There is a way to escape the inference of super luminal speeds and spooky action at a distance, but it involves absolute determinism in the universe. 252 00:28:10,940 --> 00:28:15,380 So going somehow even more deterministic. So what does that mean? 253 00:28:15,410 --> 00:28:20,780 What does this word phrase? Absolute determinism. People now generally use the word super determinism. 254 00:28:21,350 --> 00:28:25,430 So what is super determinism? What's the difference between determinism and super determinism? 255 00:28:27,120 --> 00:28:34,950 Well, this at least, is my understanding of the difference between these words and the normal deterministic system. 256 00:28:35,370 --> 00:28:39,990 You have maybe a set of differential equations describing the evolution of the system, 257 00:28:40,560 --> 00:28:47,160 and you start off with some initial condition and then the equation matches into the future. 258 00:28:47,970 --> 00:28:51,660 Now here the equations are deterministic, the dynamics is fixed, 259 00:28:52,170 --> 00:28:57,030 but you can imagine perhaps you've got a free choice to make in what the initial conditions are, 260 00:28:57,060 --> 00:29:02,330 so you don't have to start from a specific initial condition. In the Lawrence model. 261 00:29:02,330 --> 00:29:06,740 In principle, you can start the equations off from any point in space you like. 262 00:29:08,530 --> 00:29:16,060 So you have a free choice. And to some extent in this idea, the the the initial conditions are independent of the dynamics. 263 00:29:16,690 --> 00:29:17,990 So go anywhere you like. 264 00:29:18,010 --> 00:29:24,820 But when you made that choice, then you run the dynamics and the dynamics are fixed and that provides you with a future state. 265 00:29:26,280 --> 00:29:33,630 The super deterministic idea is where somehow not only is the dynamics fixed, but your initial state is fixed. 266 00:29:34,110 --> 00:29:37,290 Now, how this should be, how your initial state should be fixed? 267 00:29:37,290 --> 00:29:40,790 What determines that your initial state is fixed is another matter. 268 00:29:40,860 --> 00:29:43,560 We'll come on to try to discuss some of these issues. 269 00:29:43,980 --> 00:29:50,250 But just say for the moment, there is some reason, some principle that your initial state, there's no choice at all. 270 00:29:50,370 --> 00:29:56,310 Then absolutely everything is fixed. Your initial state's fixture, deterministic dynamics are fixed. 271 00:29:57,840 --> 00:30:05,760 The reason this is relevant is that in the Bell Theorem, if you posit a so-called hidden variable model, 272 00:30:06,690 --> 00:30:14,040 what that does in general is predict outcomes of of of physical experiments. 273 00:30:14,040 --> 00:30:24,990 You might do so you might measure the spin of a particle and your hidden variable model will tell you for a given particle what the result is, 274 00:30:25,290 --> 00:30:29,010 but only for experiments that you might actually do on that particle. 275 00:30:29,520 --> 00:30:33,240 But for experiments that you might have done but didn't do. 276 00:30:34,700 --> 00:30:39,210 So this is what philosophers would call counterfactual experiments. 277 00:30:39,220 --> 00:30:47,270 So I actually measured the particle in the X direction, but I might have measured it in the x direction. 278 00:30:49,340 --> 00:30:55,370 A standard hidden variable model will give you not only the prediction in the in the Z direction that you actually did. 279 00:30:55,610 --> 00:30:58,460 It would also tell you give you a prediction of that. 280 00:30:59,570 --> 00:31:05,030 Counterfactual experiment that you didn't actually do, but you somehow might have done in the extraction. 281 00:31:06,650 --> 00:31:09,740 Now, if you posit then the world is super deterministic, 282 00:31:10,280 --> 00:31:17,749 then obviously these so called counterfactual measurements become by definition they're not things that correspond to. 283 00:31:17,750 --> 00:31:23,200 There's only that one trajectories. Everything's fixed. So there is no there are no counterfactual worlds to consider. 284 00:31:23,210 --> 00:31:31,070 So it's a kind of almost a trivial way in which the Bell Theorem can be in some sense, 285 00:31:31,220 --> 00:31:36,320 negated by disallowing these types of counterfactual experiments. 286 00:31:39,000 --> 00:31:44,790 But the problem with it is, and this is the reason nobody actually believes this is a viable way in physics, 287 00:31:45,840 --> 00:31:53,850 is that it seems to say something about the world that's just uncomfortably fine tuned, uncomfortably special. 288 00:31:55,290 --> 00:32:04,590 So, for example, I might roll a dice and use that outcome to determine how I'm going to orientate my measuring apparatus. 289 00:32:05,190 --> 00:32:10,650 So if I roll the dice and it's a five, that will determine some angle for the measuring apparatus. 290 00:32:11,850 --> 00:32:23,670 Now, in the super deterministic world, the hidden variable will have to somehow be in harmony with the outcome of that dice in order that the in order 291 00:32:23,850 --> 00:32:29,280 in order to get the correct quantum mechanical probabilities or in order to violate the bell inequalities. 292 00:32:30,600 --> 00:32:35,610 But you might imagine well, just as I was about to roll that dice, I sneezed. 293 00:32:35,820 --> 00:32:40,070 And then the dice actually ended up as a six rather than the five or tiny gust of air. 294 00:32:40,750 --> 00:32:44,580 You know, slightly affected the dice as it was rolling and it ended up as a six. 295 00:32:45,360 --> 00:32:50,069 Then that tiny perturbation that's the sneeze or the or the gust of air would 296 00:32:50,070 --> 00:32:57,510 completely upset that harmony that is required to to to violate the bell inequalities. 297 00:32:58,920 --> 00:33:02,220 So you can take all those little perturbations back to the beginning of time. 298 00:33:02,580 --> 00:33:09,690 And the conundrum you're left with is that any almost infinitesimal perturbation, let's say, to the Big Bang, 299 00:33:10,710 --> 00:33:16,290 would produce a world where this perfect harmony that you'd set up had been somehow destroyed. 300 00:33:17,830 --> 00:33:20,559 So the problem for a super determinist is to say, well, 301 00:33:20,560 --> 00:33:30,370 why should the initial conditions be precisely big bang initial conditions and not some just marginal sort of quasi random perturbation of Big Bang. 302 00:33:31,150 --> 00:33:36,380 And I think all a super determinist could do would say, well, maybe God decided this. 303 00:33:36,850 --> 00:33:40,930 This is on the big bang initial condition and not this other one. 304 00:33:42,030 --> 00:33:46,530 Well, as theorists, we don't like you know, we like our theories to be self-contained. 305 00:33:46,530 --> 00:33:51,420 We don't want external agents to somehow determine important parts of the theory. 306 00:33:51,430 --> 00:34:01,590 So this is this is considered then a sort of an implausible reconciliation of of the Bell Theorem. 307 00:34:01,830 --> 00:34:05,790 And so as things stand, people do indeed believe either. 308 00:34:06,890 --> 00:34:11,990 We live in a world where there isn't any deterministic reality or if there is something, 309 00:34:11,990 --> 00:34:16,520 something spooky action at a distance, something super luminal is happening. 310 00:34:19,090 --> 00:34:24,970 Okay. I'm not satisfied with either of these explanations, so I. 311 00:34:25,420 --> 00:34:33,309 This is why I've been going back and thinking and, as I say, trying to think about the things which are pretty much second nature to me over the years 312 00:34:33,310 --> 00:34:39,970 and ask whether these provide some new ways of thinking about this rather old problem. 313 00:34:41,420 --> 00:34:49,160 So we talked about the Lorenz attractor. At the heart of any fractal attractor is a cantor set. 314 00:34:49,220 --> 00:34:52,890 This is perhaps the simplest of all fractals. 315 00:34:52,910 --> 00:34:58,280 I'm sure most of you are familiar with this idea. You start with the real line between nought and one. 316 00:34:58,310 --> 00:35:01,940 Think of that as your zeroth iterate of the cantor set. 317 00:35:02,270 --> 00:35:05,300 Your first iterate will be throwing away the middle third. 318 00:35:06,190 --> 00:35:10,990 Your second district will be throwing away the middle thirds of the two remaining pieces. 319 00:35:11,320 --> 00:35:14,980 Your next district will be throwing the middle third of all the remaining pieces. 320 00:35:15,400 --> 00:35:19,870 You carry on with those iterates and then you take the intersection of the iterates. 321 00:35:20,830 --> 00:35:24,790 Overall, the all the iterates. And this is the cancel set. 322 00:35:28,140 --> 00:35:33,959 Now you can define from that. A dynamical system which operates on that cancel set. 323 00:35:33,960 --> 00:35:38,130 So this is a very simple dynamical system called an iterated function system. 324 00:35:38,550 --> 00:35:44,520 In this example, there are just two functions which two different things to the state, the current state, 325 00:35:45,000 --> 00:35:51,299 but basically the action of either F one or F two, which is to be considered as a kind of time evolution. 326 00:35:51,300 --> 00:35:55,740 Operator keeps the state on this canter set. 327 00:35:59,670 --> 00:36:07,649 By the way, the council set is another nice and very simple example about how number theory kind of comes into these geometries in this in this case, 328 00:36:07,650 --> 00:36:18,390 in a very simple way, because one can represent a point on the cancel set by just writing a number, a fraction say between zero and one in base three. 329 00:36:19,690 --> 00:36:27,760 And if the base three expansion of the number has no digit one in it but has digits zero and two that it's a point on the cancel set. 330 00:36:28,700 --> 00:36:31,820 By definition, because you've thrown away all the bits where you'd have the digit one. 331 00:36:33,900 --> 00:36:40,770 So now I'm going to consider a perturbation of a point on the on the on the cancel set. 332 00:36:41,190 --> 00:36:47,819 And let's suppose for the sake of argument, my I have no knowledge of this cancel set when I do this perturbation. 333 00:36:47,820 --> 00:36:54,770 So I'm going to take a very small perturbation, but in some sense it'll be random with respect to the real line itself. 334 00:36:55,110 --> 00:37:05,730 So imagine I have no knowledge of the geometry or I choose to ignore the geometry of the cancel set in defining my, my, my small perturbation. 335 00:37:08,660 --> 00:37:14,050 So here we are. I've chosen a what I'm calling a geometrically unconstrained perturbation at some. 336 00:37:14,060 --> 00:37:19,280 It's a number that's very small, but it's been somehow chosen from that, from the real line itself. 337 00:37:19,550 --> 00:37:25,100 And therefore, somewhere there may be lots of zeros to start with, because it's a small perturbation, but somewhere there'll be ones. 338 00:37:26,330 --> 00:37:34,160 And because there are ones, as soon as I in general added on to my point on the capital set, I'll I'll move it off. 339 00:37:35,470 --> 00:37:40,120 So here x prime is being perturbed off the or set. 340 00:37:42,720 --> 00:37:48,240 Now what this is saying is that if you view the cantor set from the outside. 341 00:37:49,960 --> 00:37:56,560 It looks incredibly special. Any tiny perturbation will take you off the capital set, 342 00:37:57,190 --> 00:38:05,260 and indeed that is consistent with a cantor set having zero volume or zero measure relative to the real line. 343 00:38:05,620 --> 00:38:10,300 It looks incredibly special. It looks very high, very finely tuned. 344 00:38:10,330 --> 00:38:14,050 Any tiny perturbation will take you off the cantor set. 345 00:38:15,670 --> 00:38:21,010 So this initial state x nought looks incredibly fine tuned when viewed from the outside. 346 00:38:22,830 --> 00:38:27,510 I'm calling this, by the way, fractal determinism because it it's a type of determinism. 347 00:38:28,680 --> 00:38:32,790 Which either looks deterministic or super deterministic, depending on how you look at it. 348 00:38:33,330 --> 00:38:35,250 And when you look at it from the outside. 349 00:38:36,280 --> 00:38:47,710 It looks a bit super deterministic because any tiny perturbation, uh, uh, destroys this balance associated with the system being on the cancel set. 350 00:38:49,390 --> 00:38:57,340 On the other hand, if I am aware of the of this geometry when I construct these perturbations, 351 00:38:58,570 --> 00:39:05,650 so I'm going to construct now perturbation that what I would call geometrically constrained perturbation, 352 00:39:05,950 --> 00:39:11,230 which ensures that I stay on the cantor set when I add this perturbation. 353 00:39:11,530 --> 00:39:13,050 It stays on the cantor set. 354 00:39:15,730 --> 00:39:24,640 Now at first sight, you might think that these this these perturbations are much less numerous than the more random ones relative to the real line. 355 00:39:25,600 --> 00:39:33,270 But the remarkable property of the Cantor set is that there are as many points in the Cantor set as there are actually on the real line itself. 356 00:39:33,280 --> 00:39:36,700 There's an undeniable, infinite number of points on both. 357 00:39:38,170 --> 00:39:48,310 And a way to see this is if you take a point on the on the rail line between zero and one and express it as a binary number just with zeros and ones. 358 00:39:49,550 --> 00:39:58,550 And then right instead of the one replace each one with a digit two, then that becomes a point in base three on the cancel set. 359 00:39:58,790 --> 00:40:05,120 So as a 1 to 1 correspondence to points on the real line and points on the on the cancel set. 360 00:40:05,540 --> 00:40:09,100 So when you're inside the kantor set, it looks absolutely enormous. 361 00:40:09,110 --> 00:40:12,200 It's it's as big as the real line ever was. 362 00:40:13,890 --> 00:40:19,230 The aficionados of Doctor Who may liken the cancer set to the TARDIS. 363 00:40:21,030 --> 00:40:24,390 If you look at it from the outside, it looks incredibly small. 364 00:40:25,020 --> 00:40:28,350 But open the door and go in. And it's fantastically big and spacious. 365 00:40:30,670 --> 00:40:38,740 This is now, of course, us. Colleagues at the time thought he was completely crazy and I think that drove him a little bit to despair. 366 00:40:40,150 --> 00:40:41,650 But he was absolutely dead right. 367 00:40:42,070 --> 00:40:50,740 And also, these things underpin all of these wonderful geometric attractors, which I talked about at the beginning, such as the Lorenz attractor. 368 00:40:53,770 --> 00:40:57,160 We can do a similar thing for Laurent 63. 369 00:40:59,890 --> 00:41:03,570 Take. Uh, take a point. 370 00:41:04,160 --> 00:41:11,970 Uh, if we start on the attractor and evolve along trajectories of the Lorentz attractor, then we'll stay on the attractor. 371 00:41:12,510 --> 00:41:15,659 There are many perturbations as a non de numeral, 372 00:41:15,660 --> 00:41:23,490 infinite number of perturbations to the initial conditions which keep the initial conditions on the on the attractor. 373 00:41:24,700 --> 00:41:31,870 So there's there's a there's a lot of freedom. So it looks from inside on the attractor, it looks deterministic. 374 00:41:31,870 --> 00:41:35,500 You have a big choice of of degrees of freedom. 375 00:41:36,070 --> 00:41:43,300 But on the other hand, if you do something dynamically unconstrained, you can take yourself off the attractor. 376 00:41:43,960 --> 00:41:49,420 And a possible way of doing of taking yourself off is to is to take a perturbation which keeps, 377 00:41:50,230 --> 00:41:55,000 say, one component unchanged and just perturbs the other two components. 378 00:41:55,600 --> 00:42:02,810 This will take you off the attractor. Now, when I was thinking about this, this struck me as a very. 379 00:42:03,500 --> 00:42:12,500 In my mind, it was kind of reminiscent of those counterfactual experiments where you you think, here's a particle. 380 00:42:12,950 --> 00:42:16,490 I actually measure it in this way. I could have measured it this way. 381 00:42:16,520 --> 00:42:23,890 So when you say that you're perturbing, the sort of conceptual experiment you're thinking about is one where the particle stays fixed. 382 00:42:23,900 --> 00:42:30,770 You do nothing to the particle, but you somehow perturb your measuring system so that instead of the actual direction, 383 00:42:31,760 --> 00:42:35,930 you perturb some degrees of freedom to make the direction slightly different. 384 00:42:36,170 --> 00:42:45,180 And you say, well, what would have happened there? So let's just imagine the toy, the Lorenz model now is a toy universe. 385 00:42:45,850 --> 00:42:48,540 And just think about what this type of perturbation might mean. 386 00:42:48,720 --> 00:42:54,960 It might mean perturbing two degrees of freedom, which correspond to the orientation of your measuring apparatus, 387 00:42:55,380 --> 00:43:02,250 keeping fixed the degree of freedom associated with the hidden variable associated with the particle you're about to measure. 388 00:43:02,790 --> 00:43:05,909 So this would then be a non reasonable, I would say, 389 00:43:05,910 --> 00:43:12,720 physically unreasonable type of perturbation if you believe the universe was constrained to this underlying geometry. 390 00:43:14,360 --> 00:43:18,740 And similarly, the experiment where you say, well, I threw my I threw the dice. 391 00:43:19,040 --> 00:43:23,120 And that determined, again, how I would measure this particle. 392 00:43:23,390 --> 00:43:30,050 I will now consider a hypothetical universe where the particle again was the same particle, but now I sneezed, 393 00:43:30,320 --> 00:43:36,290 which caused the dice to land and on different number and that to to lead to a different measurement orientation. 394 00:43:37,550 --> 00:43:44,180 Again, that corresponds to keeping fixed one of the variables, one of the components of the state vector and varying the other two. 395 00:43:45,380 --> 00:43:49,400 And this would be a perturbation that would be unconstrained. 396 00:43:49,400 --> 00:43:53,510 It would be something that you were just thinking up out of your head as a possible type of perturbation. 397 00:43:53,780 --> 00:43:59,800 But if you had the view that there was some underlying geometric constraint associated with the, uh, 398 00:44:00,260 --> 00:44:08,630 the universe somehow lying on a on or belonging to an invariant set, then this would be a physically unreasonable perturbation. 399 00:44:09,470 --> 00:44:13,760 A physically reasonable perturbation would be one where all three components were perturbed. 400 00:44:14,210 --> 00:44:18,050 So not only that, so there are plenty of universes, perhaps where you did, in fact, 401 00:44:18,050 --> 00:44:24,110 had the orientation of the measuring device was as it was in the counterfactual world. 402 00:44:24,230 --> 00:44:27,590 But in that universe, the hidden variable was also different. 403 00:44:27,830 --> 00:44:30,200 To keep the whole system on the invariant set. 404 00:44:31,070 --> 00:44:41,570 So this is something which I have been, uh, as I say, rather interested in as a kind of this is not my day job, if you like, but it's something that. 405 00:44:43,520 --> 00:44:50,420 I find a compelling idea myself whether I can convince my colleagues of that. 406 00:44:50,450 --> 00:44:58,759 I don't know. But this may actually what I call fractal determinism may actually provide a conspiracy 407 00:44:58,760 --> 00:45:05,800 free loophole for the Bell Theorem and a paper in crisis a few years ago about it. 408 00:45:07,550 --> 00:45:12,860 But there is a fundamental assumption here to make this postulate make any sort of sense, 409 00:45:13,130 --> 00:45:18,800 and that is that one can really think of the universe as a dynamical system in its own right. 410 00:45:19,870 --> 00:45:23,410 But it's a dynamical system to beat all dynamical systems, I suppose. 411 00:45:25,210 --> 00:45:33,010 But more than that, it's a dynamical system which evolves on one of these special types of fractal geometries. 412 00:45:34,720 --> 00:45:41,470 Now. I'm I will. I actually there's I mean one could talk about what is the evidence, 413 00:45:41,470 --> 00:45:48,430 the cosmological evidence that the universe is a dynamical system that evolves on a fractal invariant set. 414 00:45:49,990 --> 00:45:55,000 That's a great topic for discussion, but I'm not going to. I'm going to leave that because we'll we'll run out of time. 415 00:45:56,460 --> 00:46:00,780 So let's just assume that is a viable model for the universe that it actually 416 00:46:00,780 --> 00:46:07,940 evolves on on one of these zero volume invariant sets in its state space. 417 00:46:10,350 --> 00:46:20,340 I want to come back now to then, uh, these three key differences between the Louisville equation and the Schrodinger equation. 418 00:46:21,240 --> 00:46:25,860 Planck's constant. Square root of minus one. And the fact it's a Hilbert space. 419 00:46:27,670 --> 00:46:34,360 And if you do, you want a bit more vivid? Hilbert Space means that cats are both alive and dead. 420 00:46:35,590 --> 00:46:40,810 So the question I want to ask is if this does provide a potential loophole, 421 00:46:41,650 --> 00:46:50,530 can we actually go from just postulating this as a possible way out of the dilemma of underpinning quantum mechanics with something deterministic? 422 00:46:50,920 --> 00:47:01,299 Can we go from that to an actual theory? Can we actually put some meat on the bones, as it were, and construct a real theory based on this this idea? 423 00:47:01,300 --> 00:47:07,270 So can we in particular, can we construct a fractal set bit like I did for the Cantor set, 424 00:47:07,270 --> 00:47:13,390 but something which will obviously be more complex than that from which quantum statistics would emerge naturally? 425 00:47:15,800 --> 00:47:25,310 So I just want to spend the last 10 minutes of the talk going through some slightly more technical stuff, 426 00:47:25,310 --> 00:47:34,580 which is in a paper on the archive which I wrote last year, trying to at least give some outline for how one might do this. 427 00:47:36,750 --> 00:47:40,290 So I want to start by talking about Planck's constant. How might that emerge? 428 00:47:40,980 --> 00:47:44,430 Let's start by thinking about a stern Gerlach experiment. 429 00:47:46,700 --> 00:47:56,720 Now I'm going to be very much motivated by people that have been and Roger is is is certainly well known in the field. 430 00:47:56,990 --> 00:48:03,620 People have been motivated by thinking about gravity as a possible mechanism for collapse, state vector collapse. 431 00:48:03,950 --> 00:48:14,510 Now, I'm not talking about collapse in this model because I'm trying not to view superpositions as something of of fundamental significance. 432 00:48:15,020 --> 00:48:22,130 So but nevertheless, I'm going to use the order of magnitude calculations to motivate this work. 433 00:48:22,940 --> 00:48:26,749 So the idea is we have now a sort of bundle of space times, 434 00:48:26,750 --> 00:48:34,100 a bundle of trajectories on this supposed invariant set, this fractal geometry of the universe as a whole. 435 00:48:35,510 --> 00:48:46,430 And I'm going to use this language of symbolic dynamics. So I want to label each space time with a with a colour which represents its symbolic label. 436 00:48:48,420 --> 00:48:57,660 So these are trajectories corresponding to space times where a particle has gone through the magnet and is on its way up towards the spin up detector. 437 00:48:58,080 --> 00:49:03,480 And here are some trajectories of space times where there's a particle on its way to the spin down. 438 00:49:04,110 --> 00:49:10,230 So this this time t one is after it's left the magnet, but long before it's got to the detector. 439 00:49:11,740 --> 00:49:17,260 I want to ask the question, are these space times gravitationally distinct from one another? 440 00:49:17,980 --> 00:49:23,560 And to do that, I'm going to take pairs of space times and ask whether their interaction, 441 00:49:23,830 --> 00:49:31,030 the gravitational interaction, energy integrated over the trajectory exceeds Planck's constant. 442 00:49:32,220 --> 00:49:39,600 The gravitational interaction. Energy is if you have two, let's say two lumps of matter, the gravitational interaction, 443 00:49:39,600 --> 00:49:46,139 energy and Newtonian gravity anyway is easily defined by merging moving one lump of matter to 444 00:49:46,140 --> 00:49:51,150 the position of the second lump of matter against the gravitational field of the second lump. 445 00:49:51,570 --> 00:49:55,830 So how much energy do you need to move it against the gravitational field of, say, this one? 446 00:49:56,400 --> 00:50:03,960 That's the gravitational interaction energy. Now, at this time, it's this small time here, then for sure. 447 00:50:04,590 --> 00:50:08,110 Although Planck's constant is a small number, this thing will be minuscule. Is small. 448 00:50:08,490 --> 00:50:12,270 And so this criterion will not be satisfied. 449 00:50:12,570 --> 00:50:18,420 And I view that as a statement that each of these will be given the same symbolic label. 450 00:50:18,420 --> 00:50:21,960 They're not distinct enough to be given distinct symbolic labels. 451 00:50:23,310 --> 00:50:28,680 But again, using the work DFC, Penrose, Kebble, Percival, many others, 452 00:50:29,130 --> 00:50:35,160 there are good indications that when the particle is started to interact with atoms 453 00:50:35,160 --> 00:50:41,700 in the detectors and you're starting to see macroscopic differences between the bit, 454 00:50:41,700 --> 00:50:46,740 between whether the particle's excited, the spin up detector or the spin down detector, 455 00:50:47,400 --> 00:50:52,260 then this interaction, gravitational energy can start to exceed Planck's constant. 456 00:50:53,250 --> 00:50:59,520 So this might be a small term, but Planck's concerns are also small and the order of magnitude estimates. 457 00:50:59,610 --> 00:51:06,540 I'm just going to take this on faith now. When when the when you start to get macroscopic effects in the detectors. 458 00:51:06,840 --> 00:51:10,800 So at this point, then you can give these distinct labels. 459 00:51:11,640 --> 00:51:17,400 So Planck's constant in this picture is entering through essentially through gravitational effects. 460 00:51:19,400 --> 00:51:23,500 One can analyse multiple stern Gerlach experiments this way. 461 00:51:23,510 --> 00:51:29,510 So here's here's a particle. Here's a series of trajectories that go through one stern Gerlach apparatus. 462 00:51:29,840 --> 00:51:34,460 So this one is and this one is the top one is blocked or measured somehow. 463 00:51:34,760 --> 00:51:38,480 So these three have the same symbolic label, which is different to that one. 464 00:51:39,440 --> 00:51:46,700 You can take this through a second sequential Stern Gerlach apparatus, and now these three trajectories start to this. 465 00:51:46,730 --> 00:51:50,810 This trajectory starts to become distinguished from these two gravitationally. 466 00:51:51,740 --> 00:52:00,650 And then finally in the third one, these two and one can derive the whole of Stern Gerlach statistics in this in this way, quite straightforwardly. 467 00:52:00,920 --> 00:52:08,120 This, by the way, is how Schwinger introduces his students to quantum mechanics through these sequential Stern Gerlach experiments. 468 00:52:11,650 --> 00:52:15,370 You might. Oh, yeah. So I couldn't resist this, you might say. 469 00:52:15,580 --> 00:52:20,200 Gravitational interaction, energy. It is very Newtonian. How do you define this relativistic li? 470 00:52:20,470 --> 00:52:25,780 Uh, this is actually what I did for my thesis under Dennis defining gravitational energy momentum. 471 00:52:26,260 --> 00:52:31,989 And, uh, we came up with, I think, a rather neat solution to this rather old problem. 472 00:52:31,990 --> 00:52:37,210 How do you define gravitational energy momentum in space times which don't have killing vectors? 473 00:52:38,080 --> 00:52:42,820 And the proposal was a not a tensor field on space time, 474 00:52:43,120 --> 00:52:48,880 but a tensor field on the tangent bundle two space time, which is a bit like a state space for space time. 475 00:52:50,710 --> 00:52:59,820 So this is, this is my, this is in the days when you had typewriters and ink and stuff, uh, it appeared, it appeared eventually in Fitzrovia. 476 00:53:00,400 --> 00:53:03,700 So actually there is a way to do this. I just. 477 00:53:04,770 --> 00:53:12,149 To change this because it amuses me. It links back a bit to my thesis work that there is a way to develop these ideas for 478 00:53:12,150 --> 00:53:16,740 how Planck's constant might be based on relativistic gravitational energy momentum. 479 00:53:18,150 --> 00:53:25,260 So conscious of the time I'm moving on, I want to talk about the second aspect planks, the square root of minus one. 480 00:53:26,700 --> 00:53:34,830 One of the interesting properties of a fractal is itself similarity is an essential part of a fractal, in fact. 481 00:53:35,220 --> 00:53:40,860 So if one was to zoom into a cancel set, you'd see a precise copy of the whole thing. 482 00:53:40,860 --> 00:53:46,259 And you could zoom in again and you'd see another copy. If you need to. 483 00:53:46,260 --> 00:53:55,840 Really, if your theory requires you to keep that, you have that notion of self similarity at the fore. 484 00:53:56,100 --> 00:54:08,200 It turns out that it's very useful to describe the dimension, the house of dimension of the fractal using complex numbers rather than real numbers. 485 00:54:08,700 --> 00:54:15,220 And the, the, the second, if you like, the complex number characterises not only if you like the fractal too, 486 00:54:15,630 --> 00:54:18,850 but also this notion of the self similarity. 487 00:54:18,870 --> 00:54:24,750 As you zoom in, how does it how quickly does it amplify to the back to the original scale? 488 00:54:26,040 --> 00:54:29,280 I guess the mother of all I'm so similar. 489 00:54:30,180 --> 00:54:32,990 I mean, this, this, this, this. I'll just show this. 490 00:54:33,000 --> 00:54:38,069 We people have seen this before, but I feel I haven't done sort of justice to the wonders of fractals. 491 00:54:38,070 --> 00:54:41,160 So I'll just show this movie, which just illustrates this notion of. 492 00:54:42,030 --> 00:54:51,180 If self similarity if you haven't seen it before for the Mandelbrot set, we're focusing on a bit now which where this kind of content, 493 00:54:51,190 --> 00:54:57,930 this scaling symmetry part, which is pretty, pretty manifest, it just keeps repeating and repeating ad infinitum. 494 00:55:01,510 --> 00:55:04,570 Now, if one is I'll just leave this point very briefly. 495 00:55:04,840 --> 00:55:15,250 If one is trying to develop a relativistic theory, a theory that's invariant under Lorentz, Lorentz with a T now transformations. 496 00:55:18,440 --> 00:55:25,790 So you might say suppose you had to spatially extend a fractal set which describes a spatially extended dynamical system, 497 00:55:26,330 --> 00:55:31,790 and you wanted that scaling invariance, that scale invariance to be Lorentz invariant. 498 00:55:33,030 --> 00:55:41,130 Okay. Now, in Lorenz. Lorenz transformation, one man's, one man's time is another man's space and time. 499 00:55:41,640 --> 00:55:51,540 So if you want your scaling symmetry to be Lorenz invariant in a spatially extended dynamical system, 500 00:55:51,840 --> 00:56:02,190 you better have some oscillatory type of structure in the configuration space and the spatial degrees of freedom of spatially extended system. 501 00:56:03,480 --> 00:56:11,520 So in some sense, my I think the picture which I I'm sort of coming to for, you know, why is the wave function wavy? 502 00:56:12,180 --> 00:56:21,240 Is it is actually I mean it's it's needed to make this type of scale invariant a symmetry. 503 00:56:21,360 --> 00:56:27,180 Lorentz invariant under a boost for a system with a spatially extended system. 504 00:56:29,560 --> 00:56:34,720 All right. I'm going to move on. This is the last sort of this is the last bit. 505 00:56:35,140 --> 00:56:40,120 A little bit. Well, no, I didn't start till 25, so I'm okay. But I might have run by a couple of minutes. 506 00:56:40,510 --> 00:56:44,110 I just want a very focussed come back to this last the last bit now the Hilbert space. 507 00:56:44,110 --> 00:56:49,880 How would that fit into this picture of fractal invariant sets? 508 00:56:50,260 --> 00:56:57,309 So I just want to remind a very standard picture about Hilbert space for a simple qubit there. 509 00:56:57,310 --> 00:57:10,120 There's the Hilbert space state for this state, state here on the equator relative to a basis where the north poles up, the south poles down. 510 00:57:10,570 --> 00:57:15,370 But now I want to do a unitary transformation where I rotate the basis. 511 00:57:15,370 --> 00:57:24,400 So now if you like the North Pole points here. And ask, how does this state transform under that rotation of the basis? 512 00:57:24,760 --> 00:57:27,880 And quantum mechanics gives us a very clear answer. 513 00:57:28,770 --> 00:57:31,530 Now one can think of this as this counterfactual experiment. 514 00:57:31,530 --> 00:57:38,490 So the state is well defined that say this is the actual experiment I did, but I ask what would have happened had I done this experiment? 515 00:57:38,790 --> 00:57:46,800 Quantum mechanics gives us a well-defined state, at least in which for giving us probabilities relative to this rotated basis. 516 00:57:50,200 --> 00:57:53,830 In this fractal theory, and I'm jumping over quite a bit now. 517 00:57:55,180 --> 00:57:59,380 The equivalent of the Hilbert space is what I'm calling a symbolic skeleton. 518 00:57:59,920 --> 00:58:10,629 These symbols appear as as long symbolic sequences, a bit like for the lowest Lorenz orbits, the North Pole would have just zeros. 519 00:58:10,630 --> 00:58:20,080 The South Pole have just ones. On the equator you have mixtures of zeros and ones at a particular and antipodal points have opposite, 520 00:58:20,610 --> 00:58:24,819 you know, zero gets flipped to one and a particular co latitude feature. 521 00:58:24,820 --> 00:58:31,780 You'd have a symbolic sequence where the frequency of a zero would be given by cosine squared peter over to. 522 00:58:34,410 --> 00:58:38,360 And these are related to. Certain operators, 523 00:58:38,360 --> 00:58:43,459 which I don't have time to talk to talk about a key point about this symbolic 524 00:58:43,460 --> 00:58:51,200 skeleton is that these and these cosine of co latitude angles are rational. 525 00:58:51,200 --> 00:58:54,230 The cosine is facts are based to rational number. 526 00:58:55,570 --> 00:59:03,400 So these can be as dense as you like these. I've just drawn a rather coarse representation of what is a really dense set of points. 527 00:59:05,400 --> 00:59:16,860 So I want to consider this counterfactual and how, uh, how this, how this, this works in this alternative sort of fractal picture. 528 00:59:17,520 --> 00:59:23,940 So I'm taking this point here. So these dots in some sense of the set of all allowable points, and this is very dense. 529 00:59:25,380 --> 00:59:33,210 This is the set of all the blue points and the red points on the equator, of the set of all the allowable states relative to the rotated basis. 530 00:59:33,600 --> 00:59:38,219 But now I want to know this. If I keep this point, this is this is my original state. 531 00:59:38,220 --> 00:59:41,550 And I'll say, what's that state relative to this rotated basis? 532 00:59:43,090 --> 00:59:50,560 In other words, just this coincides with one of the allowable states. Then another very interesting kind of number theoretic result comes out. 533 00:59:52,820 --> 01:00:01,250 This angle between here and here is pi over eight by construction and cosine of pi over eight is not a rational number. 534 01:00:02,840 --> 01:00:07,010 So it doesn't lie on this set of points which have rational cosine. 535 01:00:07,670 --> 01:00:12,830 And in fact it doesn't matter. This can be any based to rational number between zero and one. 536 01:00:13,460 --> 01:00:18,200 Symbol number. Number. Theoretic calculation tells you this cosine is always irrational. 537 01:00:19,960 --> 01:00:25,630 So this construction captures perfectly this notion of of counterfactual indefinite. 538 01:00:25,630 --> 01:00:31,150 This. This point has no representation relative to this rotated basis. 539 01:00:36,920 --> 01:00:38,600 Uh, Heisenberg. 540 01:00:38,870 --> 01:00:48,590 Usually you will occasionally use this German phrase, thus under-estimate Princip to describe the uncertainty, what we call the uncertainty principle. 541 01:00:48,860 --> 01:00:55,310 But this actually translates better as the indeterminacy principle, rather the uncertainty principle. 542 01:00:55,610 --> 01:01:01,250 And I think this indeterminacy characterises this construction pretty well. 543 01:01:03,750 --> 01:01:12,090 So I view the Hilbert space as a sort of completion onto the continuum of this symbolic skeleton. 544 01:01:12,960 --> 01:01:18,980 A bit like the reals are the completion of the rationals. But you know, and the reels were fantastic. 545 01:01:18,980 --> 01:01:22,430 Of course, for for physics, we would nobody would be without the real numbers. 546 01:01:22,820 --> 01:01:25,790 But you don't want to take the real numbers too seriously, in my view. 547 01:01:25,820 --> 01:01:29,900 You can do some pretty strange things with the real numbers if you take them too seriously. 548 01:01:30,200 --> 01:01:34,129 People might be familiar with the Banach task construction. 549 01:01:34,130 --> 01:01:39,260 You can take a bar of gold, you can chop it up into a finite number of pieces, stick it back together again. 550 01:01:39,260 --> 01:01:48,330 And the bar of gold is twice as big as it was. So here's a way to either get rich very quickly or to devalue the value of gold very quickly. 551 01:01:50,010 --> 01:01:56,880 But of course, the reason it's got twice as big is that you've treated the real numbers too seriously in terms of physics. 552 01:01:58,280 --> 01:02:03,610 My own picture and this is, I suppose, a bit controversial, is that the Hilbert's face is a bit like this. 553 01:02:03,620 --> 01:02:06,739 It's a fantastic tool for doing calculations. 554 01:02:06,740 --> 01:02:11,620 And of course all students should be taught the Hilbert space because you have to use it to do calculations. 555 01:02:12,110 --> 01:02:19,220 But you shouldn't take it too seriously, because if you do take it too seriously, you end up with strange paradoxes like Schrödinger's cat. 556 01:02:20,570 --> 01:02:24,440 And if you. So. So that's my that's my picture of how this. 557 01:02:24,560 --> 01:02:28,280 The Hilbert space comes out. It's a completion of this skeleton. 558 01:02:28,880 --> 01:02:34,080 Uh. But it's and it's and in that sense, it's a useful tool for calculations. 559 01:02:34,080 --> 01:02:40,110 But don't take it too seriously. Right. 560 01:02:40,130 --> 01:02:48,350 I'm just going to finish now. So my claim is that if you if you if the universe evolves on a on a fractal set, 561 01:02:49,280 --> 01:02:55,370 one can overcome the Bell Theorem without any need for conspiracy, no need for God. 562 01:02:55,880 --> 01:02:59,660 And in fact, Bell's implausible conspiracy, like all good conspiracy theories, 563 01:03:00,080 --> 01:03:06,649 is really just all in the mind because the sort of conceptual issues that that lead you to think there 564 01:03:06,650 --> 01:03:15,590 might be some conspiracy are are what I would call geometrically unconstrained types of considerations. 565 01:03:17,850 --> 01:03:22,020 So I have a lot of sympathy with with Dirac and with with Roger. 566 01:03:22,440 --> 01:03:25,980 So I'm less wedded to Stephen's view about quantum physics. 567 01:03:28,000 --> 01:03:31,960 I do want to just finish with the word about quantum gravity, 568 01:03:32,380 --> 01:03:39,459 because it seems to me that if this picture if there's any sense in this picture what one is proposing, 569 01:03:39,460 --> 01:03:47,960 here is some extension of of general relativity theory taking, if you like, Einstein's insight about the importance of geometry. 570 01:03:49,060 --> 01:03:52,810 But now, not only for space time, but for state space itself. 571 01:03:53,680 --> 01:03:54,970 And the claim then is that. 572 01:03:58,190 --> 01:04:07,750 That from this extended type of idea about about geometry and in particular than about gravity, quantum physics, maybe emergent. 573 01:04:09,730 --> 01:04:16,460 Now if that's the case. Then the whole quantum gravity program seems to me to be put at B to be the wrong way round. 574 01:04:16,470 --> 01:04:22,110 It's putting the cart before the horse, the horse being gravity and the cart being quantum mechanics. 575 01:04:22,290 --> 01:04:24,300 So if I had to sort of finish with the prediction, 576 01:04:24,540 --> 01:04:29,520 I would I would say if we ever got to the stage of being able to detect gravitons, we won't ever find them. 577 01:04:29,850 --> 01:04:37,840 Because I think it's personally, I think it's a misguided concept. So my last statement is to say the three. 578 01:04:39,520 --> 01:04:45,250 Great theories of 20th century physics, quantum mechanics, relativity, chaos theory. 579 01:04:45,500 --> 01:04:49,810 They're a little bit just disparate theories as we standard have them currently. 580 01:04:49,870 --> 01:04:56,320 Quantum mechanics and relativity are not combined. Chaos and quantum mechanics are not considered different. 581 01:04:56,470 --> 01:05:03,040 Actually, even chaos. If we think of chaos as defined in terms of of of leaping of exponents, 582 01:05:03,040 --> 01:05:08,619 that actually is not a very relativistic idea because you could scale time logarithmic and then a leaping 583 01:05:08,620 --> 01:05:14,169 of the exponentially diverging trajectories would just look like linearly diverging trajectories, 584 01:05:14,170 --> 01:05:17,649 and then it wouldn't be chaotic. All these things. 585 01:05:17,650 --> 01:05:21,640 My view can come together by thinking about this underlying geometry. 586 01:05:22,450 --> 01:05:26,469 And I suppose if I had to summarise my talk in one sentence, 587 01:05:26,470 --> 01:05:34,600 is that we should think perhaps of the laws of physics in their most primitive expression in terms of state space geometry, 588 01:05:34,600 --> 01:05:40,990 move away from the old paradigm of differential equations and think like Einstein taught us about geometry. 589 01:05:42,980 --> 01:05:53,330 Well. So just to finish by saying Dennis, of course, was an enormously eminent and distinguished scientist. 590 01:05:54,310 --> 01:06:00,880 But he was also somebody who was phenomenally good at inspiring young scientists like myself. 591 01:06:01,770 --> 01:06:05,820 And I hope if you come away with nothing else, 592 01:06:06,360 --> 01:06:13,410 you will understand that it's an education under Dennis Sharma is really something one can never shake off. 593 01:06:14,910 --> 01:06:15,540 Thank you very much.