Why is Quantum Gravity so hard? の全文書き起こし

we have a sum over all possible parts. 25 00:03:09,620 --> 00:03:20,960 And these these build these this song builds up with all of these face factors to give you the quantum mechanical aptitude for the process. 26 00:03:20,960 --> 00:03:28,400 Now, out of this, you can actually recover classical physics because if you take a bar to zero, 27 00:03:28,400 --> 00:03:38,090 then the integral has to be dominated by the stationary point of of the integral and in the exponential. 28 00:03:38,090 --> 00:03:44,120 So if you have eh eh eh eh classical. 29 00:03:44,120 --> 00:03:53,450 If you have a classical path and you add a small amount to it, you can compute the variation around the classical part of path of the action. 30 00:03:53,450 --> 00:03:58,520 And this is something that actually you did when you calculated equations of motion using 31 00:03:58,520 --> 00:04:07,710 the variation of principle in all the principle of least action in classical mechanics. 32 00:04:07,710 --> 00:04:11,270 And if you go through this calculation, it's just a very short number of steps. 33 00:04:11,270 --> 00:04:15,530 We won't do it in detail here, but you can pick it up from the slides afterwards. 34 00:04:15,530 --> 00:04:20,270 You end up with a classical equation of motion, which is the AirMax double dot. 35 00:04:20,270 --> 00:04:23,810 Is minder's degrading of the potential. Which is. Which is. 36 00:04:23,810 --> 00:04:25,190 Which is Newton's law. 37 00:04:25,190 --> 00:04:35,660 So this Pieman path integral formulation is completely consistent with what you know about classical physics and the ball goes to zero limit. 38 00:04:35,660 --> 00:04:43,200 Now we can use this formulation now to move on and try to understand something about field theory. 39 00:04:43,200 --> 00:04:49,790 So now it's the archetypal quantum field theory is quantum electro. 40 00:04:49,790 --> 00:04:57,170 That's the theory of electrons and positrons interacting with the electromagnetic field. 41 00:04:57,170 --> 00:05:06,410 And here the situation is slightly different. Now, space and time are our framework within within which the the action takes place. 42 00:05:06,410 --> 00:05:19,340 And that's the time that the time c0 that there is a configuration of the of the field Soliah electric field takes take some form that sa e0, 43 00:05:19,340 --> 00:05:28,070 which is a function of X. There are also some fields which describe the matter, the electrons and positrons. 44 00:05:28,070 --> 00:05:34,880 And then you let this system evolve and some time t one later the field has evolved, 45 00:05:34,880 --> 00:05:42,310 the electric field is evolved to E one and the matter fields have evolved to something which I'd call Cybele. 46 00:05:42,310 --> 00:05:49,370 So, so now. And of course the interschool electromagnetism or classical we can mechanics. 47 00:05:49,370 --> 00:05:53,720 What you would do is you get some bunch of differential equations which you would solve,

48 00:05:53,720 --> 00:05:57,990 which would tell you how how you how the field's evolved over time. 49 00:05:57,990 --> 00:06:04,610 But instead in quantum mechanics, what we have to do is calculate this amplitude for starting with a field 50 00:06:04,610 --> 00:06:12,350 configuration at T zero and ending up with a different field configuration at at T1. 51 00:06:12,350 --> 00:06:18,970 So that's the basic quantum mechanical object in a can in a quantum field theory now. 52 00:06:18,970 --> 00:06:22,150 So. So I'm not just. So remind you about it. About. 53 00:06:22,150 --> 00:06:28,780 About something to do with electromagnets, which is that you'll recall that you write in the magnetic field, 54 00:06:28,780 --> 00:06:37,380 which is a measurable quantity as the kernel of the facts, a potential which is Vector A. 55 00:06:37,380 --> 00:06:42,040 And you can write the electric field as mine as the gradient of the potential now. 56 00:06:42,040 --> 00:06:49,270 Actually, there's another term there, which is the time derivative or of the fact of potential. 57 00:06:49,270 --> 00:06:53,860 And this can all be encapsulated in one object, which is called field strength tensor, 58 00:06:53,860 --> 00:07:01,900 which is if you knew and SMU New has an expression in terms of the four vector potential. 59 00:07:01,900 --> 00:07:10,000 Now, the important thing to realise is Acme New contains E fields and B fields and it something which is directly measurable. 60 00:07:10,000 --> 00:07:16,390 Whereas actually the full vector potential, you can't measure it, you can't measure it directly. 61 00:07:16,390 --> 00:07:21,220 All of these things have four components because there's a time and three space components. 62 00:07:21,220 --> 00:07:25,720 If we're in regular four dimensional, four dimensional space time. 63 00:07:25,720 --> 00:07:32,920 So that there's a desert and they these things behave well under under Lorentz transformation. 64 00:07:32,920 --> 00:07:38,590 So they would transforming and they're not what's known as covariance only underlines transformations. 65 00:07:38,590 --> 00:07:44,050 So it's it that the easy objects to. Right. Lorenz in various theories from. 66 00:07:44,050 --> 00:07:50,770 They have this property, that unengaging variance, so you can shift the vector potential by a total derivative. 67 00:07:50,770 --> 00:07:56,710 So that means different Teemu means deep differentiating with respect to each X. 68 00:07:56,710 --> 00:08:03,190 And what happens is F Munir doesn't change. So the physics doesn't change. 69 00:08:03,190 --> 00:08:14,410 So so now now we're in a situation where we can we can write down by mimicking the classical case, fine, final path integral. 70 00:08:14,410 --> 00:08:20,320 And so now we integrate over intermediate field configurations. 71 00:08:20,320 --> 00:08:25,480 And again, the some face factor, which is now a quite complicated looking action. 72 00:08:25,480 --> 00:08:29,020 But it's got bits in it which contain the electric and magnetic fields. 73 00:08:29,020 --> 00:08:37,620 That's the FBI new any new term. And it has other bits in it which contain the matter fields, which is the rest of this. 74 00:08:37,620 --> 00:08:43,600 This expression, if you if you compute what happens when each ball goes to zero, 75 00:08:43,600 --> 00:08:50,800 you do this stationary point calculation again is rather more complicated than in in the particle case. 76 00:08:50,800 --> 00:08:52,890 But actually, it gives you Maxwell's equations. So. 77 00:08:52,890 --> 00:09:02,620 So the in homogeneous Maxwell's equation is the team you deem you have, if you knew, is equal to the current. 78 00:09:02,620 --> 00:09:11,910 So the the charge and the and the current are sources of electromagnetic field because that's what happens if H. 79 00:09:11,910 --> 00:09:21,760 Baatar goes to zero. If they HBO is not zero, then this gives us Quantum at the famous quantum electrodynamics. 80 00:09:21,760 --> 00:09:31,630 And in principle, you can start from this. This formula here in the further the final path integral, 81 00:09:31,630 --> 00:09:40,610 you can compute any amplitude you like in terms of the electric charge, the mass or of the of the matter fields. 82 00:09:40,610 --> 00:09:47,200 The photon, of course, is massless and Planck's constant in practise. 83 00:09:47,200 --> 00:09:55,540 Of course, it's very hard work and there's a whole industry which has done this so over many years. 84 00:09:55,540 --> 00:10:05,860 But but the outcome the basic outcome is very interesting. So what you find is that you can if you take a couple of measurable quantities. 85 00:10:05,860 --> 00:10:12,100 I've just called them and be here and you do a quantum electrodynamics calculation. 86 00:10:12,100 --> 00:10:19,060 Then there's one extra element that you have to feed into the calculation, which is some ultra, which is some scale capital lambda. 87 00:10:19,060 --> 00:10:28,450 And the reason for that is that you find that there are ultraviolet divergences in the calculations that you do at your feet. 88 00:10:28,450 --> 00:10:37,990 Once you've fed this scale in, you can you can extract from your measurements A and B values for the mass or of the electron. 89 00:10:37,990 --> 00:10:42,550 And for the electromagnetic fine structure, constant output alpha. 90 00:10:42,550 --> 00:10:50,800 Even an H ball, by the way, is kind of. You defy a modern metrology age bar is defined to be a certain value. 91 00:10:50,800 --> 00:10:56,890 And then you can take these numbers m an alpha and you feed them into another QED calculation. 92 00:10:56,890 --> 00:11:02,950 Put the scale in again what you may not. Where did you predict the outcome for another process. 93 00:11:02,950 --> 00:11:11,130 C. And the interesting thing is that actually, when when you when you've done this, 94 00:11:11,130 --> 00:11:23,490 you realise that you can relate what your

prediction for the measurable speed is entirely back to the measurable for and be using or QED calculations. 95 00:11:23,490 --> 00:11:30,120 And this scale parameter lambda that you put in and the intermediate steps disappears. 96 00:11:30,120 --> 00:11:33,870 And that's because QED is a renormalise zable theory. 97 00:11:33,870 --> 00:11:45,000 So it has this remarkable property that it predicts unambiguous relationships between measure rouble's and an expansion in Alpha Electromagnetic, 98 00:11:45,000 --> 00:11:49,350 which is roughly 100, well over 137. So it's a fairly small number. 99 00:11:49,350 --> 00:12:01,230 This works spectacularly well. And probably the best demonstration of this is the anomalous magnetic dipole moment of the electron. 100 00:12:01,230 --> 00:12:12,630 So according to Dirac's relativistic wave equation, the magnetic dipole ratio of the electron is exactly two. 101 00:12:12,630 --> 00:12:17,280 So the magnetic dipole moment, the electron is likely to be amongst photons, 102 00:12:17,280 --> 00:12:23,990 but there's a small deviation from the this which is generated by by quantum mechanical effects. 103 00:12:23,990 --> 00:12:28,500 And you can measure this small deviation. And this is the result that you get. 104 00:12:28,500 --> 00:12:34,560 You can see it's spectacularly accurate. The number 28 is the error in the last two digits of the result. 105 00:12:34,560 --> 00:12:41,810 Seventy three. And you can also calculate you have to work very hard to calculate to this degree of accuracy. 106 00:12:41,810 --> 00:12:46,200 And these are calculations which have only been developed over many years. 107 00:12:46,200 --> 00:12:51,510 But you can see that the agreement is spectacularly good. 108 00:12:51,510 --> 00:12:59,200 So so quantum electrodynamics is a very powerful theory and it's a very accurate theory. 109 00:12:59,200 --> 00:13:08,340 And it works. It works extremely well. It's one of these are some of the most accurate measurements and calculations of the main. 110 00:13:08,340 --> 00:13:17,130 So that's field theory in a nutshell. Never can tell about gravity, we better remind ourselves a little bit about general relativity. 111 00:13:17,130 --> 00:13:21,810 Already heard some of this from from John and in his talk. 112 00:13:21,810 --> 00:13:30,030 So the basic point is he is here that what one of them was the really important ingredients. 113 00:13:30,030 --> 00:13:37,110 Well, we have a space M which has some fixed topology for a minute. And on that space, there is a metric. 114 00:13:37,110 --> 00:13:45,780 And so so the the the degree of freedom which really matters is the distance between points. 115 00:13:45,780 --> 00:13:52,310 So you you take a point X mbewe and a point X bew plus dxp, which is quite close to it. 116 00:13:52,310 --> 00:13:58,110 And, and what matters is what's the real distance between these two points or the proper distance. 117 00:13:58,110 --> 00:14:04,570 If we're talking rigourously and that's determined by the metric which is this this object. 118 00:14:04,570 --> 00:14:11,880 Jemmy, you and Jamie knew is in general dependent on where you are in the manifold. 119 00:14:11,880 --> 00:14:17,190 Now, some examples are the Euclidian plane is the easiest example, and that's flat. 120 00:14:17,190 --> 00:14:24,350 And you know that the DSC ad is just the X squared plus dy Y squared. 121 00:14:24,350 --> 00:14:32,210 Another example is the surface of the sphere, which is most easily written down in polar coordinates. 122 00:14:32,210 --> 00:14:37,190 And and then there's a piece which is Defeater squared in a piece, which is Defi Square. 123 00:14:37,190 --> 00:14:41,810 But you see the component of D5 squared is not constant. 124 00:14:41,810 --> 00:14:48,860 OK. And K is basically the radius of a sphere. And then a final example is Binkowski Space Time. 125 00:14:48,860 --> 00:14:56,180 OK. So now we're really talking that now we're really talking about a proper, proper interval. 126 00:14:56,180 --> 00:14:59,840 And that again, is duty's square by the X squared. 127 00:14:59,840 --> 00:15:07,580 And the metric for Mankowski space time is often given the name item you knew, which we will come back to in a minute. 128 00:15:07,580 --> 00:15:12,710 Now, the point about this is that we can we can change coordinate systems. 129 00:15:12,710 --> 00:15:13,580 Right. For example, 130 00:15:13,580 --> 00:15:23,510 you can change from Euclidean coordinates to plain polar coordinates on the on the Euclidean plane and the coordinates of the two points and B, 131 00:15:23,510 --> 00:15:31,130 change. But the proper distance, which is actually just a Judi's existence in this case between them, doesn't. 132 00:15:31,130 --> 00:15:39,080 So so there is actually a hidden in all of this, a reprivatization of variance, which is that you can change the coordinates. 133 00:15:39,080 --> 00:15:51,320 And when you change the code, that's a metric function changes. But the distances between points are independent of that reprivatization. 134 00:15:51,320 --> 00:16:03,120 Now, proper distance is actually not the only independent frame, character frame, independent characteristic of of the space and the match. 135 00:16:03,120 --> 00:16:12,590 Right. There's another frame, independent characteristic, which is the intrinsic curvature. 136 00:16:12,590 --> 00:16:17,450 So that's illustrated in the end, the picture on on the right hand side. 137 00:16:17,450 --> 00:16:23,050 You can you can see that if a space is curved so that there are two. 138 00:16:23,050 --> 00:16:27,180 Right. Independent principal wages of radii of curvature. 139 00:16:27,180 --> 00:16:39,590 So just think of a surface. And this is the curvature of the space is it is again, independent of the coldness. 140 00:16:39,590 --> 00:16:49,330 So, for example, if you are if you are looking on the surface of the O of the sphere. 141

00:16:49,330 --> 00:16:56,330 Right. It doesn't matter what sorts of coordinates you use to describe that surface, it is still curved. 142 00:16:56,330 --> 00:17:06,440 So the curvature you're clearly implying is co zero because it flat the curvature of this spherical service is just one of the rageous square. 143 00:17:06,440 --> 00:17:11,980 It's a product of the inverse principle radii. 144 00:17:11,980 --> 00:17:20,110 Curvature of Binkowski space time is again zero. Binkowski space time is is flat now. 145 00:17:20,110 --> 00:17:24,710 So. So in this, G.M. is the dynamical degree of freedom. 146 00:17:24,710 --> 00:17:33,950 It's sort of like the vector potential intellectual dynamics. But so the whole posture of general relativity is that Jemmy New is a dynamical 147 00:17:33,950 --> 00:17:39,710 degree of freedom and the curvature of space time is generated by mass and energy. 148 00:17:39,710 --> 00:17:43,940 And that leads to the Einsteins equations, which which we have written down here. 149 00:17:43,940 --> 00:17:48,080 And the the important bit is the on the left hand side. 150 00:17:48,080 --> 00:17:51,920 We have all we have the curvature on the right hand side. 151 00:17:51,920 --> 00:17:57,980 We have the stuff that generates the curvature, which is the stress energy tense. 152 00:17:57,980 --> 00:18:05,340 And then we've also written down this other term. This is the cosmological term which Einstein wasn't very happy about. 153 00:18:05,340 --> 00:18:10,960 But but which there is no no reason in general not to have that. 154 00:18:10,960 --> 00:18:16,760 And one of one of the puzzles of nature is why the cosmological term is rather small. 155 00:18:16,760 --> 00:18:23,870 Why is this cosmological constant lambda? Sorry, this is a different quantity from our previous Cut-Off previous scale. 156 00:18:23,870 --> 00:18:27,750 Why is the cosmological constant so small? 157 00:18:27,750 --> 00:18:32,270 Is one of one of the mysteries which modern physics still wrestles with. 158 00:18:32,270 --> 00:18:36,200 A capital G is just Newton's constant. 159 00:18:36,200 --> 00:18:43,130 So this is very nice, neat way of writing things which are actually non-linear, partial differential equations for the metric. 160 00:18:43,130 --> 00:18:46,360 Basically, all you have to do is specify the initial conditions and solve. 161 00:18:46,360 --> 00:18:51,110 Right now, of course, that's easier said than done in practise for complicated situations. 162 00:18:51,110 --> 00:18:58,850 But nonetheless, you get excellent agreement with observation and with experiment. 163 00:18:58,850 --> 00:19:02,970 So how we get so then how are we going to quantised gravity? 164 00:19:02,970 --> 00:19:07,010 What what what must be the basic ingredients? Well, 165 00:19:07,010 --> 00:19:12,410 the first basic ingredient has to be the physical amplitudes that you talk about must be reprivatization 166 00:19:12,410 --> 00:19:18,140 invariant if they're not read traumatisation invariant and they just turn out to be zero. 167 00:19:18,140 --> 00:19:24,440 And so they don't actually tell you anything. Okay, so this is a very easy, reprivatization, 168 00:19:24,440 --> 00:19:32,870 invariant physical amplitude is basically an amplitude that the exact universe exists for proper time as. 169 00:19:32,870 --> 00:19:40,670 It starts at some event p0 and ends up at some event P1 after proper time. 170 00:19:40,670 --> 00:19:43,100 And then the forum appal thing to grow. 171 00:19:43,100 --> 00:19:52,790 You might speculate, which gives us these Ankit amplitude is basically a sum overall, such overall, such metrics. 172 00:19:52,790 --> 00:20:00,770 So here I've written down a find important to grow where I sum overall the old metrics which have this characteristic I saw. 173 00:20:00,770 --> 00:20:14,990 I've got some matter fields in there as well. And the hypothesis is this final part integral would enable you to calculate this amplitude. 174 00:20:14,990 --> 00:20:20,330 Now, this part this part that we've written down here, this is called the Einstein Hilbert action. 175 00:20:20,330 --> 00:20:25,340 If you take a bar to zero, the stationary point just this gives you about Eisenstein's equations, 176 00:20:25,340 --> 00:20:30,740 which we wrote down or on on on the previous previous slide. 177 00:20:30,740 --> 00:20:39,020 So, again, the pathological gives us as each bar goes to zero, limit the classical field equations, these bets here. 178 00:20:39,020 --> 00:20:46,280 If I if I just integrate. If I just some over matter configurations and I just use the maths apart in the phase. 179 00:20:46,280 --> 00:20:50,060 This basically gives me quantum field theory and a fixed background spacetime, 180 00:20:50,060 --> 00:20:58,550 which which is part of what John was talking about in his lecture, where we have quantum electrons in quantum photons. 181 00:20:58,550 --> 00:21:07,070 But we have a classical background, spacetime. 182 00:21:07,070 --> 00:21:11,670 But that's a very easy way of writing down something that could easily be very complicated. 183 00:21:11,670 --> 00:21:14,880 So what does this integral over the metrics mean? 184 00:21:14,880 --> 00:21:25,230 What what what do we mean when we are going to sum over all these interpolating metrics between two space time events? 185 00:21:25,230 --> 00:21:29,400 So we're going to look at the two to two attempts at doing this. 186 00:21:29,400 --> 00:21:42,120 So the first attempt is to basically copy QED. And you do that with a combined with a perturbation expansion in Newton's constant. 187 00:21:42,120 --> 00:21:44,970 And you might think that, well, gravity is weak. 188 00:21:44,970 --> 00:21:53,670 So so perhaps the way to think about this is that basically there is a sort of small wrinkles in in space time. 189 00:21:53,670 --> 00:21:59,820 And although the metric is a function of position, actually what it is, is

it is a sum of two bits. 190 00:21:59,820 --> 00:22:03,720 It's the sum of something which is completely flat, which is just I mean, calfskin metric. 191 00:22:03,720 --> 00:22:11,580 And then there's H meunière who which gives this these small wrinkles, the oscillations. 192 00:22:11,580 --> 00:22:18,900 If you if you if you make that substitution for Jim, you and I start as equations and we just drop the cosmological constant term. 193 00:22:18,900 --> 00:22:23,160 It goes in practise. It's very small. In this case. 194 00:22:23,160 --> 00:22:24,330 OK. So you drop. 195 00:22:24,330 --> 00:22:37,320 So then then what you find is, is that the these small oscillations h they satisfy this equation in the middle, which is basically a wave equation. 196 00:22:37,320 --> 00:22:43,890 This does tells you that the matter is the source of these small fluctuations. 197 00:22:43,890 --> 00:22:56,630 And this is essentially the equation which leads to to gravitational waves, for example, being being emitted by by match or in some kind of motion. 198 00:22:56,630 --> 00:23:03,890 So that's some thought that that's one. That's what that's what happens classically if you try. 199 00:23:03,890 --> 00:23:06,740 If you do this quantum mechanically. 200 00:23:06,740 --> 00:23:12,950 And so you're now viewing this as a quantum field theory, then you need an analogue of the fine structure constant. 201 00:23:12,950 --> 00:23:17,600 So in this little quantum general relativity theory we're talking about. 202 00:23:17,600 --> 00:23:26,420 We need to have dimensionless fine structure constant. And unfortunately, Newton's constant has dimension. 203 00:23:26,420 --> 00:23:30,800 So the dimensionless object is actually Newton's constant times ours, 204 00:23:30,800 --> 00:23:39,320 a scale squared and then divided by by age bonder all the time where you actually can use it in units where the speed of light, 205 00:23:39,320 --> 00:23:47,130 you said equal of one and all these, these calculations. So it's that scale is the mass of the electron. 206 00:23:47,130 --> 00:23:52,430 Then this alpha for faja general relativity is about ten to the minus forty six. 207 00:23:52,430 --> 00:23:57,440 So he's really. The small and the certainly weak. 208 00:23:57,440 --> 00:24:08,000 But what you should worry because if you now set the scale to be about 10 to the 22 times the mass of the electron or so, 209 00:24:08,000 --> 00:24:15,240 then Alpha Alpha will will be about one. 210 00:24:15,240 --> 00:24:20,350 Now, that seems like a very large energy scale, but it's not really a very light Jannuzi scale, 211 00:24:20,350 --> 00:24:24,970 10 to the 22 times the mass of the electron is only about 20 at 10. 212 00:24:24,970 --> 00:24:30,640 The mine is eight kilograms. So this looks like there might be a problem. 213 00:24:30,640 --> 00:24:36,280 And in fact, there is a problem. The problem is that if you start taking measurable. 214 00:24:36,280 --> 00:24:45,940 And you start doing field theory calculations and you feed in your scale and you try to to deduce what Alpha is and all the other parameters, 215 00:24:45,940 --> 00:24:50,260 what you find is that the parameters cascade. 216 00:24:50,260 --> 00:24:58,960 And you can never take the outcomes of a number of measurements and feed them into a calculation, unpredicted other measurement. 217 00:24:58,960 --> 00:25:05,470 And this is this is basically the statement that this theory we're talking about is not renormalise. 218 00:25:05,470 --> 00:25:08,920 Now, actually proving it's not renormalise is a tour de force. 219 00:25:08,920 --> 00:25:21,010 And the famous theoretical physicist Envelopment who died this week was an important figure in that endeavour. 220 00:25:21,010 --> 00:25:28,870 But she is not totally dead. You can keep your scale and just work in a regime very small. 221 00:25:28,870 --> 00:25:31,250 And that's called defective field theory. 222 00:25:31,250 --> 00:25:38,410 And you can learn a lot about the relationship between quantum versions of gravity and classical gravity in that way. 223 00:25:38,410 --> 00:25:45,650 But it's not it's not a self-contained, fundamental theory of all of or all of everything, because you have to keep this role. 224 00:25:45,650 --> 00:25:50,680 Small scale in the other wave of other route. 225 00:25:50,680 --> 00:25:54,570 Is that string theory actually contain this field? H we do this. 226 00:25:54,570 --> 00:25:59,680 This is in fact as so kind of like a spin to field. 227 00:25:59,680 --> 00:26:08,230 And it also contains a consistent minimum distance scale. And it is a way of constructing a consistent quantum field theory of gravity. 228 00:26:08,230 --> 00:26:17,150 And Fernandinho will talk about that soon. But of course you pay a price and the price you pay is that string theory doesn't just contain gravity. 229 00:26:17,150 --> 00:26:26,740 It contains lots of other degrees of freedom as well. So so the the question is, can we do something else? 230 00:26:26,740 --> 00:26:36,460 Is there some other way of all of looking at this problem? So so this is the second possibility that I get talked about very briefly. 231 00:26:36,460 --> 00:26:44,140 So the first word looking at it, we just looked at fluctuations around them and kasky metric. 232 00:26:44,140 --> 00:26:49,390 But there isn't really any particularly good reason to think that that's the case. 233 00:26:49,390 --> 00:26:56,980 I mean, why aren't on metrics sort of equal in the eyes of the universe? 234 00:26:56,980 --> 00:27:00,640 So if we think about metric democracy. OK. 235 00:27:00,640 --> 00:27:10,340 Then I've hear I've drawn a metric which is very uneven, uneven and higgledy-piggledy, which nonetheless connects these two spacetime points. 236 00:27:10,340 --> 00:27:14,870 Those are manifold, which is higgledy piggledy worth bearing in mind. 237 00:27:14,870 --> 00:27:24,040 By the way, the path

integral, even in the particle case, includes particle trajectories which are not smooth. 238 00:27:24,040 --> 00:27:31,910 It doesn't just include smooth trajectories. It includes all possible very, very smart only beauty trajectories. 239 00:27:31,910 --> 00:27:38,750 So there's one. There's another. There's another. 240 00:27:38,750 --> 00:27:43,910 And the Parthenon Group will be the sum over all of these potentially very higgledy-piggledy, 241 00:27:43,910 --> 00:27:51,170 these bases which have the character had the sole common characteristic that 242 00:27:51,170 --> 00:27:58,800 they have these two events separated by by a fixed a fixed proper distance. 243 00:27:58,800 --> 00:28:02,680 Okay, but then if you have metric democracy, you should worry about. 244 00:28:02,680 --> 00:28:08,080 Well, shouldn't one. Why would you have? Why wouldn't you have topology democracy? 245 00:28:08,080 --> 00:28:13,000 Why should you just consider spaces of one particular topology? 246 00:28:13,000 --> 00:28:17,830 Surely you should allow the topology to change as well. 247 00:28:17,830 --> 00:28:24,100 So once you allow the topology to change, make all kinds of complicated things can appear. 248 00:28:24,100 --> 00:28:30,520 So you can get worm host, so little branches of the universe which split off and then join, read, join later. 249 00:28:30,520 --> 00:28:37,630 For example, you can get breeks, splits, big joints. 250 00:28:37,630 --> 00:28:42,100 And so there's a huge number of possibilities. 251 00:28:42,100 --> 00:28:48,940 And this is really very tough, because if you look at four dimensional manifolds, you're trying to do is Inforum attentions. 252 00:28:48,940 --> 00:28:54,290 They're only partially tortes charted territory. In fact, this is an extremely hard problem. 253 00:28:54,290 --> 00:28:58,690 So to finish with what we're going to do is just look at the right toy model 254 00:28:58,690 --> 00:29:04,120 toy model to give you a very simple flavour of how these things might work. 255 00:29:04,120 --> 00:29:10,120 So what you're supposed to do is you're supposed to consider a graph of equilateral triangles. 256 00:29:10,120 --> 00:29:19,450 And you imagine that the minimum distance between two points on your triangulation, that's denoted in red on this on this picture, 257 00:29:19,450 --> 00:29:26,150 which is just three edges, you imagine you imagine that this is actually the Judi's existence. 258 00:29:26,150 --> 00:29:36,900 Okay. And then then you look at the configurations of triangles you've ever three-fold vertex and you make all of these triangles equilateral, right? 259 00:29:36,900 --> 00:29:43,020 Then you can't put this on a plane. It sticks up a tetrahedron. 260 00:29:43,020 --> 00:29:47,910 And that's that. So that says approximation of a place of positive curvature. 261 00:29:47,910 --> 00:29:52,110 If you have exactly six of them joined, on the other hand, then you can put them flat on a plane. 262

00:29:52,110 --> 00:29:58,590 That's a place of zero curvature. If you have seven, then you get a place of negative curvature. 263 00:29:58,590 --> 00:30:03,150 Because if you put light in the drawing and if you put five of them, you make a flap. 264 00:30:03,150 --> 00:30:07,450 Then the other two were the last two will stick up like this. 265 00:30:07,450 --> 00:30:14,830 OK, so the only idea is. Firstly, fixed its apology from an IT case. 266 00:30:14,830 --> 00:30:18,460 So, for example, you draw these graphs on a sphere, 267 00:30:18,460 --> 00:30:29,140 then you sum over all possible triangulations and because something over all possible triangulations sums overall possible Judi's distances. 268 00:30:29,140 --> 00:30:33,880 That's a bit like integrating overall possible metrics. OK. 269 00:30:33,880 --> 00:30:36,580 So then you sum. And then finally you sum over the Jenice. 270 00:30:36,580 --> 00:30:47,480 So the end result is that the amplitude in this case is a a sum over triangulations or something, which is a sum over Janice. 271 00:30:47,480 --> 00:30:53,080 So the simplest one is a sphere. The next most complicated one is a Taurus. 272 00:30:53,080 --> 00:30:54,390 Then you got a double Taurus. 273 00:30:54,390 --> 00:31:05,200 And so you just have these two points, which you keep a fixed graph distance fixed Gertie's existence as denoted by the by the red line. 274 00:31:05,200 --> 00:31:09,490 So this is actually a toy model which has which has caused. 275 00:31:09,490 --> 00:31:13,840 Which has given great entertainment to theoretical physicists so over a long time. 276 00:31:13,840 --> 00:31:22,240 And to this day, contained contains interesting stuff which is not possible, not properly understood, but it causes basically two dimensional. 277 00:31:22,240 --> 00:31:23,320 It's not four dimensional. 278 00:31:23,320 --> 00:31:31,760 And the reason why it works is because we understand geometry in two dimensions and manifolds in two dimensions and topology in two dimensions. 279 00:31:31,760 --> 00:31:37,210 Estrela. So what do our universities look like if we do that? 280 00:31:37,210 --> 00:31:46,930 Well, the most interesting characteristic is that if you if you ask the question, well, how much how much universe, 281 00:31:46,930 --> 00:32:00,770 how much universe is so it, how much universe is there in a volume which is inside a fixed distance as zero like. 282 00:32:00,770 --> 00:32:12,000 And the answer is it S02 the fourth. But if you're on a flat space, it would be a zero O squared. 283 00:32:12,000 --> 00:32:20,520 So actually, the spaces that you get in this model are a long, long way from being a flat, two dimensional space. 284 00:32:20,520 --> 00:32:27,280 And that's really the message of this. Once you let quantum gravity out of the bag, all right, 285 00:32:27,280 --> 00:32:36,450 then it is very naive to expect that you will get spaces which are nice and simple and flat and look very friendly. 286 00:32:36,450 --> 00:32:45,360 And in fact, these extraordinary, non-trivial that the quantum universe is so very four dimensional, both locally and globally. 287 00:32:45,360 --> 00:32:57,990 And this is absolutely something that we will have to have to understand when we understand properly understand quantum gravity and in a complete way. 288 00:32:57,990 --> 00:33:05,175 So we'll leave it there. Thank you very much.