1 00:00:00,040 --> 00:00:02,410 The following content is provided under a Creative 2 00:00:02,410 --> 00:00:03,790 Commons license. 3 00:00:03,790 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,100 continue to offer high quality educational resources for free. 5 00:00:10,100 --> 00:00:12,680 To make a donation, or to view additional materials 6 00:00:12,680 --> 00:00:16,426 from 100 of MIT courses, visit MIT OpenCourseWare 7 00:00:16,426 --> 00:00:17,050 at ocw.mit.edu. 8 00:00:21,890 --> 00:00:22,950 HONG LIU: OK, good. 9 00:00:22,950 --> 00:00:26,590 So let me first remind you what we did last time. 10 00:00:26,590 --> 00:00:31,860 So last time we discussed that, for any operator, 11 00:00:31,860 --> 00:00:35,610 say some operator O in the field theory, 12 00:00:35,610 --> 00:00:38,810 we expect this one to one correspondence to some field, 13 00:00:38,810 --> 00:00:42,480 say, phi in the gravity side. 14 00:00:42,480 --> 00:00:46,770 And of course, the quantum numbers of those, 15 00:00:46,770 --> 00:00:52,050 they should match because of the four [? interpretations ?] 16 00:00:52,050 --> 00:00:55,564 of some symmetry groups and four [? interpretations ?] 17 00:00:55,564 --> 00:00:56,730 of the same symmetry groups. 18 00:00:56,730 --> 00:01:00,350 Of course, their quantum numbers should also match. 19 00:01:00,350 --> 00:01:05,880 And, in particular, on the field theory side, 20 00:01:05,880 --> 00:01:10,730 say if you have such an operator, 21 00:01:10,730 --> 00:01:13,070 it's natural to add the source term 22 00:01:13,070 --> 00:01:19,650 and then to deform the Lagrangian in such a form. 23 00:01:19,650 --> 00:01:22,890 And then we explained that, this phi 24 00:01:22,890 --> 00:01:29,340 may be considered as the boundary value of phi 25 00:01:29,340 --> 00:01:30,486 in the boundary of ADS. 26 00:01:33,590 --> 00:01:40,000 So the boundary value of phi, and it's mapped to the-- if phi 27 00:01:40,000 --> 00:01:41,570 have some boundary value, then that 28 00:01:41,570 --> 00:01:43,260 is corresponding to your boundary series 29 00:01:43,260 --> 00:01:45,250 Lagrangian have such a term. 30 00:01:45,250 --> 00:01:48,120 You know, Lagrangian. 31 00:01:48,120 --> 00:01:56,700 And then we also discussed this for the vector field, 32 00:01:56,700 --> 00:01:59,840 and for conserved current, say, the case 33 00:01:59,840 --> 00:02:03,120 which is a conserved current, then 34 00:02:03,120 --> 00:02:06,250 that implies that, in a particular example of this, 35 00:02:06,250 --> 00:02:09,699 if you have a conserved current that that's corresponding to, 36 00:02:09,699 --> 00:02:14,250 you must be dual to a gate field. 37 00:02:14,250 --> 00:02:17,300 And in particular, the stress tensor 38 00:02:17,300 --> 00:02:22,871 must be to dual to the magic deformation in the field series 39 00:02:22,871 --> 00:02:23,370 site. 40 00:02:29,990 --> 00:02:31,570 So any questions regarding this? 41 00:02:35,000 --> 00:02:35,500 Good. 42 00:02:35,500 --> 00:02:36,200 OK. 43 00:02:36,200 --> 00:02:43,560 So now let's try to develop more this relation. 44 00:02:43,560 --> 00:02:49,440 So this is normally called the operator field mapping. 45 00:02:49,440 --> 00:02:55,810 And so let's look at this relation a little bit more. 46 00:02:55,810 --> 00:02:57,570 So my immediate question, one can 47 00:02:57,570 --> 00:03:00,480 ask-- so the obvious question is regarding, 48 00:03:00,480 --> 00:03:02,980 say, the spin of the operator that 49 00:03:02,980 --> 00:03:06,270 should map to the spin of this operator, et cetera. 50 00:03:06,270 --> 00:03:09,460 But if you can see the conformal theory, which is the scaling 51 00:03:09,460 --> 00:03:14,510 variant, a very important quantum number 52 00:03:14,510 --> 00:03:18,095 for scaling operator is the scaling dimension. 53 00:03:18,095 --> 00:03:20,470 As we mentioned before, we said, in the conformal theory, 54 00:03:20,470 --> 00:03:22,170 everything is scaling variant. 55 00:03:22,170 --> 00:03:25,730 And then you can assign any operator-- 56 00:03:25,730 --> 00:03:30,200 any operator can be transform under 57 00:03:30,200 --> 00:03:32,930 appropriate representation of the conformal symmetry 58 00:03:32,930 --> 00:03:35,310 of the scaling dimension. 59 00:03:35,310 --> 00:03:37,900 That's one of the most important quantum numbers. 60 00:03:37,900 --> 00:03:40,790 So now let's look at how the scaling 61 00:03:40,790 --> 00:03:43,620 dimension of this operator are related 62 00:03:43,620 --> 00:03:46,230 to the physical properties here. 63 00:03:46,230 --> 00:03:49,780 Because here, there is no low scaling 64 00:03:49,780 --> 00:03:54,410 dimension in the sense this is just some gravity field 65 00:03:54,410 --> 00:03:59,610 and here is isometric it's not the scaling symmetry. 66 00:03:59,610 --> 00:04:02,330 And then we also can try to understand 67 00:04:02,330 --> 00:04:04,600 the worries of properties of this phi, how 68 00:04:04,600 --> 00:04:08,110 they are reflected in O. So that's 69 00:04:08,110 --> 00:04:12,770 the purpose of today's lecture. 70 00:04:12,770 --> 00:04:18,670 So for this purpose, now, let's look at the gravity side. 71 00:04:18,670 --> 00:04:20,490 So as we said last time, we always 72 00:04:20,490 --> 00:04:25,880 work in the regime of the semiclassical gravity. 73 00:04:25,880 --> 00:04:44,955 So on the gravity side-- OK. 74 00:04:47,895 --> 00:04:50,410 So what we will show is that, actually, the dimension 75 00:04:50,410 --> 00:04:53,500 of this operator can be directly related 76 00:04:53,500 --> 00:04:57,560 to the mass of the corresponding field in the gravity site. 77 00:04:57,560 --> 00:05:01,610 And so this is normally called a mass dimension relation. 78 00:05:01,610 --> 00:05:11,970 So now let's just consider, on the gravity side, 79 00:05:11,970 --> 00:05:15,640 so the action can generically be written as the following. 80 00:05:15,640 --> 00:05:18,910 So everything is controlled by the gravitational interaction. 81 00:05:18,910 --> 00:05:21,947 So now I have introduced a new rotation, which we explain. 82 00:05:29,260 --> 00:05:31,370 So you have Einstein gravity. 83 00:05:31,370 --> 00:05:33,080 And then you have some matter fields. 84 00:05:33,080 --> 00:05:35,882 OK? 85 00:05:35,882 --> 00:05:37,130 You have some matter fields. 86 00:05:37,130 --> 00:05:38,320 Matter Lagrangian. 87 00:05:38,320 --> 00:05:40,890 So this is the generic form, say, 88 00:05:40,890 --> 00:05:43,340 of action in the gravity side. 89 00:05:43,340 --> 00:05:48,170 So I've introduced this 2 kappa squared, which is essentially 90 00:05:48,170 --> 00:05:54,180 whatever is the Newton constant in this d plus one dimension. 91 00:05:54,180 --> 00:06:00,770 So normally we put the 6 pi G Newton constant here. 92 00:06:00,770 --> 00:06:02,970 So d plus 1 just means the Newton 93 00:06:02,970 --> 00:06:06,230 constant in d plus 1 dimension. 94 00:06:06,230 --> 00:06:11,130 And I just now called this 2 kappa squared for convenience. 95 00:06:11,130 --> 00:06:15,390 And typically, the matter fields, say, 96 00:06:15,390 --> 00:06:18,220 suppose you have a scalar field, then 97 00:06:18,220 --> 00:06:29,807 you will have-- say for inform, plus, say, nonlinear terms. 98 00:06:37,590 --> 00:06:42,640 And if you have, say, if you have vector fields, 99 00:06:42,640 --> 00:06:46,650 then you will have a Maxwell actions, et cetera. 100 00:06:46,650 --> 00:06:50,920 So just whatever matter field you have in the gravity side, 101 00:06:50,920 --> 00:06:57,130 and they just have this-- so now let's 102 00:06:57,130 --> 00:07:04,200 consider small perturbations around the pure AdS. 103 00:07:04,200 --> 00:07:07,270 So pure AdS, you only have a metric, 104 00:07:07,270 --> 00:07:10,500 and all the matter field is essentially zero. 105 00:07:10,500 --> 00:07:30,329 So now let's consider the small perturbations around the AdS. 106 00:07:30,329 --> 00:07:31,870 Or maybe I should say around the AdS. 107 00:07:40,310 --> 00:07:45,590 So first thing you notice is that these 2 kappa squared 108 00:07:45,590 --> 00:07:49,050 is multiplied over all action. 109 00:07:49,050 --> 00:07:52,990 So that means that all the interactions here, essentially, 110 00:07:52,990 --> 00:07:56,460 are controlled by this gravitational interaction. 111 00:07:56,460 --> 00:08:00,220 Yeah, it's all controlled by kappa. 112 00:08:00,220 --> 00:08:02,950 And when you can see the small perturbations, say, 113 00:08:02,950 --> 00:08:06,750 when you can quantize the small perturbations, et cetera, 114 00:08:06,750 --> 00:08:09,882 it's convenient to use a canonically normalized action. 115 00:08:12,480 --> 00:08:15,100 But if you look at the matter field here, 116 00:08:15,100 --> 00:08:19,560 the [INAUDIBLE] is this 1 over 2 kappa squared, 117 00:08:19,560 --> 00:08:24,180 which is a lot convenient. 118 00:08:24,180 --> 00:08:29,060 Actually, with that 2, then I don't need 2 here. 119 00:08:29,060 --> 00:08:31,080 So let me just get rid of the 2. 120 00:08:35,799 --> 00:08:43,010 So it's convenient to use a canonically normalized action. 121 00:08:43,010 --> 00:08:48,610 So it's considered a small [INAUDIBLE] around AdS, 122 00:08:48,610 --> 00:08:56,896 say, for convenient to canonically. 123 00:09:03,337 --> 00:09:03,836 normalize. 124 00:09:06,697 --> 00:09:07,780 AUDIENCE: Excuse me, why-- 125 00:09:07,780 --> 00:09:08,904 HONG LIU: Yeah, one second. 126 00:09:08,904 --> 00:09:19,290 Normalize the kinetic term of small perturbations. 127 00:09:19,290 --> 00:09:19,790 OK? 128 00:09:24,580 --> 00:09:27,150 Which means it's convenient when we can see, say, 129 00:09:27,150 --> 00:09:31,760 for example the series for phi, when we consider scaling, 130 00:09:31,760 --> 00:09:38,880 they take phi-- say k, phi equal to kappa phi prime, 131 00:09:38,880 --> 00:09:41,080 and then you can see that in the phi prime, 132 00:09:41,080 --> 00:09:45,570 we'll have a canonical action. 133 00:09:45,570 --> 00:09:49,030 And say, for example, if you can see the magic perturbations, 134 00:09:49,030 --> 00:09:56,780 and then you can write it as hmn. 135 00:09:56,780 --> 00:09:58,110 So this is the AdS matrix. 136 00:10:02,730 --> 00:10:03,740 OK? 137 00:10:03,740 --> 00:10:08,590 Then the phi prime and hmn will be canonically normalized. 138 00:10:14,580 --> 00:10:17,660 So later, we'll just drop this prime. 139 00:10:17,660 --> 00:10:19,410 I will just call this phi. 140 00:10:19,410 --> 00:10:20,903 So we just do a scaling operation. 141 00:10:23,500 --> 00:10:24,850 Yes? 142 00:10:24,850 --> 00:10:27,710 AUDIENCE: Why did the 1 over 2? 143 00:10:27,710 --> 00:10:30,780 HONG LIU: Oh no, it's just because there is 1 over 2 here. 144 00:10:30,780 --> 00:10:34,850 AUDIENCE: But what path, you have r-- 145 00:10:34,850 --> 00:10:39,460 you say the r is in the action and r should-- 146 00:10:39,460 --> 00:10:41,870 HONG LIU: No, I'm talking about the matter field. 147 00:10:41,870 --> 00:10:44,570 AUDIENCE: But in the action, you write the r minus 2 148 00:10:44,570 --> 00:10:46,931 and the plus [INAUDIBLE]. 149 00:10:46,931 --> 00:10:47,556 HONG LIU: Yeah. 150 00:10:47,556 --> 00:10:51,770 AUDIENCE: So ordinarily, should it be r plus 1 over 2 151 00:10:51,770 --> 00:10:54,060 [INAUDIBLE] something else? 152 00:10:54,060 --> 00:10:55,870 HONG LIU: It doesn't matter. 153 00:10:55,870 --> 00:10:57,950 It's just how I normalized things. 154 00:10:57,950 --> 00:11:00,680 You can normalize-- if you worry about the 1/2, 155 00:11:00,680 --> 00:11:02,300 it doesn't matter. 156 00:11:02,300 --> 00:11:04,120 Here we're not worried about 1/2. 157 00:11:04,120 --> 00:11:04,661 AUDIENCE: OK. 158 00:11:08,390 --> 00:11:10,490 HONG LIU: Good? 159 00:11:10,490 --> 00:11:13,089 So now here's the important point. 160 00:11:13,089 --> 00:11:14,380 So here is the important point. 161 00:11:14,380 --> 00:11:22,590 And remember, when we have the relation between the Newton 162 00:11:22,590 --> 00:11:27,130 constants, so the Kappa squared is a Newton constant. 163 00:11:27,130 --> 00:11:29,920 And remember, this is proportional to our n squared. 164 00:11:32,940 --> 00:11:35,790 Remember what we discussed last time. 165 00:11:35,790 --> 00:11:44,230 So that means, under this scaling, 166 00:11:44,230 --> 00:11:50,260 so kappa is actually of all the 1 over n, 167 00:11:50,260 --> 00:11:52,420 kappa is the Newton constant it's 168 00:11:52,420 --> 00:11:53,780 a square of Newton constant. 169 00:11:53,780 --> 00:11:55,830 It's very small. 170 00:11:55,830 --> 00:11:57,885 So this is in units of literature. 171 00:12:03,982 --> 00:12:05,940 And we always can see that in the large N image 172 00:12:05,940 --> 00:12:09,050 so that the Newton constant is small. 173 00:12:09,050 --> 00:12:10,565 And so this is small. 174 00:12:14,020 --> 00:12:17,820 So in this regime, if we keep phi-- 175 00:12:17,820 --> 00:12:22,460 so for canonically normalized field, say, 176 00:12:22,460 --> 00:12:27,250 if [INAUDIBLE] hmn are of order 1, 177 00:12:27,250 --> 00:12:29,991 then their perturbation is naturally small. 178 00:12:29,991 --> 00:12:31,740 Then their perturbation is naturally small 179 00:12:31,740 --> 00:12:37,420 because they are essentially controlled by the kappa. 180 00:12:37,420 --> 00:12:44,150 And it also means, if you plug this into the action, 181 00:12:44,150 --> 00:12:49,560 so it expands also long linear terms. 182 00:12:49,560 --> 00:13:03,146 So it also means that nonlinear terms 183 00:13:03,146 --> 00:13:06,198 are order kappa or higher. 184 00:13:13,700 --> 00:13:15,720 So this is very easy to understand. 185 00:13:15,720 --> 00:13:22,570 So let's consider, say, phi cubed term come from here. 186 00:13:22,570 --> 00:13:27,080 So previously, we have 1 over kappa squared, 187 00:13:27,080 --> 00:13:33,350 phi cubed term, so after this scaling, 188 00:13:33,350 --> 00:13:36,520 then I just have kappa phi cubed. 189 00:13:36,520 --> 00:13:38,750 So for phi of all the 1, then the cubic term 190 00:13:38,750 --> 00:13:41,830 will be suppressed compared to the quadratic term 191 00:13:41,830 --> 00:13:45,710 because the quadratic term, after the scaling, 192 00:13:45,710 --> 00:13:49,300 will be of all the 1, will be all the 1. 193 00:13:53,150 --> 00:14:03,350 So the lambda near term will be suppressed or higher 194 00:14:03,350 --> 00:14:04,196 and suppressed. 195 00:14:14,030 --> 00:14:17,145 So except if we are interested in considering 196 00:14:17,145 --> 00:14:19,742 the contribution of those higher order terms, 197 00:14:19,742 --> 00:14:21,950 then to leading order, we can actually just consider, 198 00:14:21,950 --> 00:14:23,910 for small fluctuations, we can just 199 00:14:23,910 --> 00:14:27,480 consider the quadratic action. 200 00:14:27,480 --> 00:14:28,961 This means that to leading order? 201 00:14:36,047 --> 00:14:37,713 So we can consider the quadratic action. 202 00:14:42,360 --> 00:14:50,120 So this means we can just-- so this corresponding to quantize 203 00:14:50,120 --> 00:14:52,637 the free series, OK? 204 00:14:52,637 --> 00:14:54,720 So this corresponding to quantize the free series. 205 00:14:58,100 --> 00:15:01,200 Remember, we always use h bar equal to 1. 206 00:15:01,200 --> 00:15:05,400 So for those matter fields, h bar is always equal to one. 207 00:15:05,400 --> 00:15:12,340 And after you do the scaling to get rid of the kappa, 208 00:15:12,340 --> 00:15:15,080 we just treat them as the free field theory quantization. 209 00:15:18,532 --> 00:15:19,370 Good? 210 00:15:19,370 --> 00:15:22,640 Any questions about this? 211 00:15:22,640 --> 00:15:26,770 So now let's look at scalar fields, just for illustration. 212 00:15:29,450 --> 00:15:35,758 Let's look at what's the behavior of a massive scalar 213 00:15:35,758 --> 00:15:36,257 field. 214 00:15:41,130 --> 00:15:47,570 So now let's consider a massive scalar field. 215 00:15:51,232 --> 00:15:51,940 for illustration. 216 00:15:56,620 --> 00:16:01,700 And we consider some scalar field phi in the [? back, ?] 217 00:16:01,700 --> 00:16:06,030 which is due to some scalar operator O in the boundary. 218 00:16:06,030 --> 00:16:06,690 Just imagine. 219 00:16:09,980 --> 00:16:10,670 OK? 220 00:16:10,670 --> 00:16:13,584 So now let's consider the quadratic action for phi. 221 00:16:27,190 --> 00:16:30,400 So we can write down the quadratic action. 222 00:16:30,400 --> 00:16:41,030 So after you have scaled out this kappa, 223 00:16:41,030 --> 00:16:43,660 and the quadratic action can be written explicitly 224 00:16:43,660 --> 00:16:53,574 as the following across all the kappa terms. 225 00:16:57,790 --> 00:16:59,650 So this is just a free action. 226 00:16:59,650 --> 00:17:04,839 And of the GMA, just the AdS. 227 00:17:04,839 --> 00:17:09,099 So the quadratic order, this just in the background AdS. 228 00:17:13,119 --> 00:17:17,240 So we'll use this notation to denote the [INAUDIBLE] 229 00:17:17,240 --> 00:17:18,339 component of AdS. 230 00:17:21,970 --> 00:17:27,446 So we're most of the time using this metric. 231 00:17:32,360 --> 00:17:34,210 This is my background AdS metric. 232 00:17:38,320 --> 00:17:41,850 And this is my action. 233 00:17:41,850 --> 00:17:43,590 So we'll also use of the notation 234 00:17:43,590 --> 00:17:49,063 that the x m actually, I should write this matrix here, 235 00:17:49,063 --> 00:17:50,480 but it doesn't matter. 236 00:17:50,480 --> 00:17:54,155 The x m should be considered as z x mu. 237 00:17:54,155 --> 00:17:57,590 And x mu is the coordinates parallel to the boundary. 238 00:18:06,707 --> 00:18:09,040 So do you remember how to quantize the free field series 239 00:18:09,040 --> 00:18:09,540 action? 240 00:18:12,620 --> 00:18:17,830 So this is the time we need to remember that. 241 00:18:17,830 --> 00:18:20,163 So the first thing is how you solve the equation motion. 242 00:18:35,460 --> 00:18:37,470 So we actually don't need to go to details. 243 00:18:37,470 --> 00:18:43,840 I will just mention a few important features. 244 00:18:43,840 --> 00:18:47,600 OK, so this is just the standard [INAUDIBLE] equation 245 00:18:47,600 --> 00:18:52,270 for massive scalar fields in the curved spacetime. 246 00:18:52,270 --> 00:18:55,425 And this gMN is just given by this metric. 247 00:18:58,190 --> 00:19:00,830 So this is a seemingly complicated 248 00:19:00,830 --> 00:19:03,400 partial differential equation. 249 00:19:03,400 --> 00:19:05,910 But here we have lots of symmetry. 250 00:19:05,910 --> 00:19:10,680 So there is a translational symmetry along the directions 251 00:19:10,680 --> 00:19:12,750 parallel to the boundary. 252 00:19:12,750 --> 00:19:16,150 Then we can just write down a Fourier transform. 253 00:19:16,150 --> 00:19:29,210 So because of the translation symmetries in x mu directions. 254 00:19:29,210 --> 00:19:31,620 So we can just do a Fourier transform. 255 00:19:31,620 --> 00:19:38,596 We can write phi z x mu in terms of, say, DDK. 256 00:19:53,040 --> 00:19:57,250 So this k.x is just the standard Minkowski contraction 257 00:19:57,250 --> 00:20:00,290 because of the t and x part. 258 00:20:00,290 --> 00:20:05,440 And essentially it's slightly a Minkowski metric. 259 00:20:05,440 --> 00:20:10,375 So [INAUDIBLE] here, k.x is just the standard Minkowski 260 00:20:10,375 --> 00:20:10,875 contraction. 261 00:20:23,440 --> 00:20:24,980 So now it's the first transform. 262 00:20:24,980 --> 00:20:30,470 Then you just plug this in. 263 00:20:30,470 --> 00:20:32,220 Then this is very easy to write down. 264 00:20:35,315 --> 00:20:39,800 The only derivative now would be just in the z direction. 265 00:20:39,800 --> 00:20:42,750 Because then all the derivatives in the t and x 266 00:20:42,750 --> 00:20:46,190 can be replaced by k. 267 00:20:46,190 --> 00:20:49,530 And then the equation becomes very simple. 268 00:20:49,530 --> 00:20:58,300 So it takes you a couple of seconds to write it down. 269 00:20:58,300 --> 00:21:01,249 So let me just write it down. 270 00:21:01,249 --> 00:21:03,040 So it can be written in the following form. 271 00:21:31,940 --> 00:21:35,000 And then the k squared just is the standard one. 272 00:21:35,000 --> 00:21:37,542 Again, the k squared is just as the standard Lorentz 273 00:21:37,542 --> 00:21:38,042 contraction. 274 00:21:41,416 --> 00:21:44,164 So omega would be the k zero in the time direction. 275 00:21:44,164 --> 00:21:46,122 We're just do the standard Lorentz contraction. 276 00:21:52,130 --> 00:21:53,420 So k mu. 277 00:22:00,630 --> 00:22:04,860 So now, this equation can actually be solved exactly. 278 00:22:04,860 --> 00:22:07,470 So normally, when you quantize it, 279 00:22:07,470 --> 00:22:10,550 you solve the equation motion, then 280 00:22:10,550 --> 00:22:13,440 you find the complete basis of solutions. 281 00:22:13,440 --> 00:22:17,620 Then you expand in terms of that complete the basis, 282 00:22:17,620 --> 00:22:21,350 the co-option of the expansion would be your creation 283 00:22:21,350 --> 00:22:23,240 and annihilation operators. 284 00:22:23,240 --> 00:22:25,890 And then that's how you quantize this series. 285 00:22:25,890 --> 00:22:28,040 So this equation can actually be solved exactly. 286 00:22:28,040 --> 00:22:31,860 So you can actually easily find out the complete set of modes. 287 00:22:31,860 --> 00:22:34,080 But it would be actually not needed 288 00:22:34,080 --> 00:22:35,830 for the current purpose right now. 289 00:22:35,830 --> 00:22:40,770 So maybe not solve it. 290 00:22:40,770 --> 00:22:42,760 So we'll leave it here. 291 00:22:42,760 --> 00:22:46,200 But what will be important for us 292 00:22:46,200 --> 00:22:48,810 is consider the symptotic behavior of the phi, 293 00:22:48,810 --> 00:22:51,742 the phi near the boundary. 294 00:22:51,742 --> 00:22:53,450 Now, let's consider the behavior of a phi 295 00:22:53,450 --> 00:23:07,040 near the boundary, which will be very important for us. 296 00:23:07,040 --> 00:23:10,526 Now let's just consider behavior of phi near the boundary. 297 00:23:13,850 --> 00:23:18,280 So near the boundary, z goes to 0. 298 00:23:18,280 --> 00:23:21,380 So this term compared to this term 299 00:23:21,380 --> 00:23:24,920 is always much smaller because z goes to 0. 300 00:23:24,920 --> 00:23:28,910 m squared R squared is finite. 301 00:23:28,910 --> 00:23:30,540 So we can drop this term. 302 00:23:30,540 --> 00:23:33,130 But kinetic term, typically, cannot drop because kinetic 303 00:23:33,130 --> 00:23:38,630 term depends on 1. 304 00:23:38,630 --> 00:23:44,580 So we just throw away this term. 305 00:23:44,580 --> 00:23:48,590 So as z goes to 0, and maybe also expand 306 00:23:48,590 --> 00:23:50,350 this term more explicitly. 307 00:23:50,350 --> 00:23:53,880 Then what you find, just drop this term 308 00:23:53,880 --> 00:23:55,870 and then write in this term more explicitly. 309 00:23:55,870 --> 00:23:58,036 Then you find the equation can written as following. 310 00:24:16,640 --> 00:24:22,200 So this is only valid when z equals to 0, leading order. 311 00:24:22,200 --> 00:24:24,120 So now this is equation has a very nice form 312 00:24:24,120 --> 00:24:27,320 because this equation is homogeneous. 313 00:24:27,320 --> 00:24:31,550 So z squared, then the partial z squared, 314 00:24:31,550 --> 00:24:35,270 z partial z, and then you have this thing 315 00:24:35,270 --> 00:24:37,660 which there's no derivative. 316 00:24:37,660 --> 00:24:39,290 And so this is not in some sense, 317 00:24:39,290 --> 00:24:41,041 homogeneous in the derivative. 318 00:24:41,041 --> 00:24:42,540 Such kind of equation you can always 319 00:24:42,540 --> 00:24:47,620 solve it by some power of z. 320 00:24:47,620 --> 00:24:49,640 So this just write down answers. 321 00:24:49,640 --> 00:24:53,280 So assuming phi is proportional z to some power. 322 00:24:53,280 --> 00:24:56,140 And this power, let me call it delta. 323 00:24:56,140 --> 00:24:59,690 So you plug these answers into this equation, 324 00:24:59,690 --> 00:25:02,830 then you get [INAUDIBLE] equation. 325 00:25:02,830 --> 00:25:04,970 So you can easily see, from first term, you 326 00:25:04,970 --> 00:25:07,560 get delta, delta minus 1. 327 00:25:07,560 --> 00:25:12,600 From the second equation, you get 1 minus d times delta, 328 00:25:12,600 --> 00:25:14,430 m squared r squared. 329 00:25:19,760 --> 00:25:21,660 And this is a simple quadratic equation now 330 00:25:21,660 --> 00:25:22,798 you can immediately solve. 331 00:25:29,426 --> 00:25:30,550 So you found two solutions. 332 00:25:38,860 --> 00:25:41,590 Find two solutions. 333 00:25:41,590 --> 00:25:47,270 So we are writing this in the form 1/2 d plus minus nu. 334 00:25:47,270 --> 00:25:50,370 I will define this thing to be my nu, so this is definition. 335 00:25:54,090 --> 00:25:56,320 So I will introduce the following notation. 336 00:25:56,320 --> 00:25:57,880 I will take the plus sign. 337 00:25:57,880 --> 00:25:59,930 I call the plus sign delta. 338 00:25:59,930 --> 00:26:05,040 So from now on, delta refers to plus sign. 339 00:26:05,040 --> 00:26:08,360 And sometimes I use the notation called 340 00:26:08,360 --> 00:26:13,380 delta minus, which is the minus sign, which 341 00:26:13,380 --> 00:26:14,920 is also equal to d minus delta. 342 00:26:19,910 --> 00:26:21,410 So this is the definition. 343 00:26:21,410 --> 00:26:23,010 So now I use this as the definition. 344 00:26:23,010 --> 00:26:25,010 So from now on, my delta is always 345 00:26:25,010 --> 00:26:27,568 d divided by 2 plus this number. 346 00:26:35,960 --> 00:26:43,570 So now what we have found is that as z 347 00:26:43,570 --> 00:26:57,490 go to 0, what we have found is that as z goes to 0, 348 00:26:57,490 --> 00:26:59,370 that destabilizes. 349 00:26:59,370 --> 00:27:08,090 So as z goes to 0, the phi kz has the following behavior. 350 00:27:08,090 --> 00:27:11,260 So you have two independent solutions. 351 00:27:11,260 --> 00:27:14,920 So then I can put arbitrary [INAUDIBLE], 352 00:27:14,920 --> 00:27:21,390 z d minus delta to exponents, and another one 353 00:27:21,390 --> 00:27:24,830 is Bk z to the delta. 354 00:27:24,830 --> 00:27:26,826 Then plus have all the corrections. 355 00:27:26,826 --> 00:27:27,326 OK? 356 00:27:31,590 --> 00:27:36,460 So this is two leading behaviors in z goes to the zero limit. 357 00:27:36,460 --> 00:27:39,290 And of course, I can also go to the coordinate space. 358 00:27:39,290 --> 00:27:55,758 Then this is just the x, z, which is Ax x z d minus OK? 359 00:28:02,760 --> 00:28:10,460 So first notice that this exponent, this d minus delta 360 00:28:10,460 --> 00:28:17,760 exponent is always smaller than delta. 361 00:28:17,760 --> 00:28:19,010 So this term always dominates. 362 00:28:22,130 --> 00:28:23,880 So this is the leading behavior. 363 00:28:23,880 --> 00:28:27,820 And this is the sub-leading behavior. 364 00:28:27,820 --> 00:28:30,480 So keep this in mind. 365 00:28:30,480 --> 00:28:32,815 So now we have found the fourth symptotic behavior 366 00:28:32,815 --> 00:28:33,523 of the solutions. 367 00:28:37,840 --> 00:28:41,190 And now we can discuss physics. 368 00:28:41,190 --> 00:28:43,860 Now we can discuss some physics. 369 00:28:43,860 --> 00:28:44,986 Now let me make remarks. 370 00:28:48,460 --> 00:28:54,110 For the first remark-- any questions so far? 371 00:28:57,420 --> 00:28:58,700 Good. 372 00:28:58,700 --> 00:29:10,540 So the first remark, you notice that this exponent 373 00:29:10,540 --> 00:29:20,370 are real if m squared R squared greater than d minus 4. 374 00:29:27,960 --> 00:29:32,470 Then you say, ah, for sure, these exponents are all real. 375 00:29:32,470 --> 00:29:34,120 Because normally, we only consider 376 00:29:34,120 --> 00:29:35,320 the positive mass squared. 377 00:29:38,460 --> 00:29:41,036 But [INAUDIBLE] idea is something very interesting 378 00:29:41,036 --> 00:29:41,536 happening. 379 00:29:44,440 --> 00:29:47,150 It's actually, in AdS, the [INAUDIBLE] mass 380 00:29:47,150 --> 00:29:50,490 squared also makes sense. 381 00:29:50,490 --> 00:29:55,460 In fact, you can show, as far as those exponents are real, 382 00:29:55,460 --> 00:29:59,360 means that, as far as, so one can 383 00:29:59,360 --> 00:30:21,376 show, as far as the star is satisfied, 384 00:30:21,376 --> 00:30:22,876 the theory is actually well defined. 385 00:30:29,740 --> 00:30:32,240 So in AdS, you are now-- you have 386 00:30:32,240 --> 00:30:34,840 an [INAUDIBLE] mass squared. 387 00:30:34,840 --> 00:30:37,970 But if you violated this, then you find instability. 388 00:30:41,240 --> 00:30:43,470 This is a very nice story, but I don't have time 389 00:30:43,470 --> 00:30:46,260 to discuss in detail here. 390 00:30:46,260 --> 00:30:49,720 But my former student [? Nabil Ackbar ?], 391 00:30:49,720 --> 00:30:52,540 when he was doing TA for me last time, 392 00:30:52,540 --> 00:30:57,190 he wrote a very nice note explaining how this works. 393 00:30:57,190 --> 00:31:01,960 And I put that note on the web. 394 00:31:01,960 --> 00:31:05,530 So this equation is called the BF bound. 395 00:31:05,530 --> 00:31:08,660 F is of our colleague Freedman. 396 00:31:08,660 --> 00:31:14,030 And the B is a very complicated name I always pronounced wrong. 397 00:31:14,030 --> 00:31:16,450 So maybe not try here. 398 00:31:16,450 --> 00:31:22,456 Anyway, so [? Nabil ?] gave a very nice explanation 399 00:31:22,456 --> 00:31:25,640 of where this comes from. 400 00:31:25,640 --> 00:31:28,830 And he shows that when you violate this, 401 00:31:28,830 --> 00:31:30,800 you actually get instability. 402 00:31:30,800 --> 00:31:32,930 And this maps into a very nice quantum mechanics 403 00:31:32,930 --> 00:31:36,086 problem, the quantum mechanics problem of minus 1 404 00:31:36,086 --> 00:31:36,710 over x squared. 405 00:31:39,520 --> 00:31:43,060 Some of you may have experience with that problem. 406 00:31:43,060 --> 00:31:47,670 Anyway, so please take a look at those. 407 00:31:47,670 --> 00:31:49,090 I put it on the web. 408 00:31:49,090 --> 00:31:52,600 And it's a very instructive discussion. 409 00:31:52,600 --> 00:31:56,090 But we'll take a lecture to do that. 410 00:31:56,090 --> 00:31:59,140 Anyway, so now I just make the claim. 411 00:31:59,140 --> 00:32:02,327 It's that, as far as those exponents are real, 412 00:32:02,327 --> 00:32:03,410 the theory's well defined. 413 00:32:03,410 --> 00:32:03,910 OK? 414 00:32:06,730 --> 00:32:10,100 But if this exponent is violated, 415 00:32:10,100 --> 00:32:13,070 then there exist modes which are exponentially 416 00:32:13,070 --> 00:32:17,570 growing in time indicating the system in unstable. 417 00:32:20,190 --> 00:32:22,673 So this star is called the BF bound. 418 00:32:26,819 --> 00:32:27,360 The BF bound. 419 00:32:30,380 --> 00:32:33,270 So it's actually using instructive to compare 420 00:32:33,270 --> 00:32:39,800 with a standard-- actually, I forgot to use this blackboard. 421 00:32:39,800 --> 00:32:40,691 Anyway. 422 00:32:40,691 --> 00:32:41,190 Yeah. 423 00:32:52,540 --> 00:33:00,050 So [INAUDIBLE] to compare with Minkowski spacetime, 424 00:33:00,050 --> 00:33:04,640 so this compare with Minkowski case. 425 00:33:04,640 --> 00:33:07,830 So, in the Minkowski case, if you have a free particle, 426 00:33:07,830 --> 00:33:11,380 say, if you have a free massive scalar field-- so of course, 427 00:33:11,380 --> 00:33:16,330 we know this is our equation motion, OK? 428 00:33:16,330 --> 00:33:19,840 And from here, if you go to the Minkowski space, 429 00:33:19,840 --> 00:33:23,393 every direction is translation variant, 430 00:33:23,393 --> 00:33:25,184 then you just get this dispersion relation. 431 00:33:29,047 --> 00:33:30,630 You just get this dispersion relation. 432 00:33:30,630 --> 00:33:36,846 You can write phi equals to mass at omega t. 433 00:33:36,846 --> 00:33:40,080 Let me just write it more carefully here. 434 00:33:40,080 --> 00:33:46,140 You can write phi [INAUDIBLE] omega t plus ikx. 435 00:33:49,150 --> 00:33:50,480 Then plug it in. 436 00:33:50,480 --> 00:33:52,870 Then you find for omega squared equal to m 437 00:33:52,870 --> 00:33:54,645 squared plus k squared. 438 00:33:58,650 --> 00:34:03,320 So in Minkowski spacetime we don't 439 00:34:03,320 --> 00:34:10,320 allow m squared to be smaller than 0 because if m squared is 440 00:34:10,320 --> 00:34:18,110 smaller than 0, then that means, for certain range of k, 441 00:34:18,110 --> 00:34:24,770 say for k, for those values of k which are smaller 442 00:34:24,770 --> 00:34:28,981 than the absolute value of m squared, then 443 00:34:28,981 --> 00:34:29,897 this will be negative. 444 00:34:32,400 --> 00:34:35,712 So for those things, the omega will be purely imaginary. 445 00:34:47,020 --> 00:34:49,860 You can be plus minus purely imaginary. 446 00:34:49,860 --> 00:34:51,350 And then you plug it in here. 447 00:34:51,350 --> 00:34:54,510 Then you find it in phi and typically 448 00:34:54,510 --> 00:35:01,840 have exponential omega t plus exponential minus omega t. 449 00:35:01,840 --> 00:35:04,390 So the phi can have these two behaviors. 450 00:35:04,390 --> 00:35:08,410 And then, of course, this will exponentially grow. 451 00:35:08,410 --> 00:35:11,350 So this tells you that the magnitude of phi 452 00:35:11,350 --> 00:35:13,500 will exponentially grow with time. 453 00:35:13,500 --> 00:35:16,790 And so that indicated this instability. 454 00:35:16,790 --> 00:35:20,370 So this is the same as the analog I said earlier before. 455 00:35:20,370 --> 00:35:27,030 In the Minkowski spacetime, if you have a mass squared, 456 00:35:27,030 --> 00:35:31,680 and then your scalar field one wants to slide down, 457 00:35:31,680 --> 00:35:36,325 and this just tells you the behavior wants to increase. 458 00:35:41,490 --> 00:35:44,040 So this the phi. 459 00:35:44,040 --> 00:35:46,510 So this is V phi. 460 00:35:46,510 --> 00:35:48,110 Then the phi wants to increase. 461 00:35:48,110 --> 00:35:49,015 So this is phi to 0. 462 00:35:52,110 --> 00:35:56,980 But in AdS, something nicer happens. 463 00:35:56,980 --> 00:36:05,830 So in AdS, the instability only sets in if the mass squared 464 00:36:05,830 --> 00:36:07,579 becomes sufficiently [INAUDIBLE]. 465 00:36:07,579 --> 00:36:09,620 So if you just have the [INAUDIBLE] mass squared, 466 00:36:09,620 --> 00:36:13,160 if it's greater than this value, it's still OK. 467 00:36:13,160 --> 00:36:14,890 It's still OK. 468 00:36:14,890 --> 00:36:18,680 So the reason is actually very simple. 469 00:36:18,680 --> 00:36:21,010 So the reason is very simple. 470 00:36:21,010 --> 00:36:24,610 You can try to convince yourself by looking at the equations. 471 00:36:24,610 --> 00:36:27,100 And you can see the physics from the equations. 472 00:36:27,100 --> 00:36:29,570 But the basic physical reason is following. 473 00:36:29,570 --> 00:36:39,930 In AdS, so you can just look at that equation. 474 00:36:39,930 --> 00:36:53,910 Because of the spacetime curvature-- 475 00:36:53,910 --> 00:36:56,620 so the constant modes are not allowed. 476 00:36:56,620 --> 00:36:58,600 So you see here, the most dangerous modes 477 00:36:58,600 --> 00:37:00,260 are the constant modes. 478 00:37:00,260 --> 00:37:02,070 And when you have constant modes, 479 00:37:02,070 --> 00:37:05,740 and then m squared becomes-- no matter how small it is, 480 00:37:05,740 --> 00:37:11,446 no matter how small m squared is, if you have constant modes, 481 00:37:11,446 --> 00:37:14,070 and constant mode means k equal to 0, then 482 00:37:14,070 --> 00:37:18,650 that means that omega squared will always be negative. 483 00:37:18,650 --> 00:37:21,880 As far as m squared is nonzero in the negative, 484 00:37:21,880 --> 00:37:24,860 there's always constant mode which is unstable. 485 00:37:24,860 --> 00:37:26,975 But in AdS, because of the spacetime curvature, 486 00:37:26,975 --> 00:37:28,350 the constant mode is not allowed. 487 00:37:33,260 --> 00:37:36,230 So you always have some kind of kinetic term in the sense 488 00:37:36,230 --> 00:37:40,440 that you always have some kind of kinetic energy. 489 00:37:40,440 --> 00:37:55,550 So a field is forced to have some kinetic energy, which 490 00:37:55,550 --> 00:38:06,670 can compensate this negative m squared. 491 00:38:20,180 --> 00:38:24,390 But this won't happen when the mass becomes too negative. 492 00:38:24,390 --> 00:38:27,061 And then you have instability. 493 00:38:27,061 --> 00:38:28,144 Then you have instability. 494 00:38:33,490 --> 00:38:36,950 I suggest to you-- so this is just words, 495 00:38:36,950 --> 00:38:39,340 but those words can be reflected by looking 496 00:38:39,340 --> 00:38:41,300 at the structure of those equations 497 00:38:41,300 --> 00:38:47,860 and try to get some feeling about these 498 00:38:47,860 --> 00:38:52,190 by looking at those equations a little bit more carefully. 499 00:38:52,190 --> 00:38:55,120 OK, so this is the first remark. 500 00:38:55,120 --> 00:38:58,520 This is the first remark. 501 00:38:58,520 --> 00:39:02,120 So the second remark, is that we know 502 00:39:02,120 --> 00:39:03,435 that the AdS have a boundary. 503 00:39:11,610 --> 00:39:22,070 And the light array which of the boundaries 504 00:39:22,070 --> 00:39:24,900 in the front of time. 505 00:39:24,900 --> 00:39:26,510 So this is the one important feature 506 00:39:26,510 --> 00:39:28,710 we discussed at the beginning, which 507 00:39:28,710 --> 00:39:32,120 is the boundary in the finite time. 508 00:39:35,750 --> 00:39:40,320 In fact, in the AdS, it's just pi over 2 in the finite time. 509 00:39:42,940 --> 00:39:44,470 So that means, actually, energy can 510 00:39:44,470 --> 00:39:49,953 be exchanged on the boundary. 511 00:39:59,170 --> 00:40:03,270 Essentially, you can send stuff in or send things out. 512 00:40:07,210 --> 00:40:14,560 And so we actually need to impose the-- so in order 513 00:40:14,560 --> 00:40:17,615 to have a sensible physical system, 514 00:40:17,615 --> 00:40:19,240 you need to impose appropriate boundary 515 00:40:19,240 --> 00:40:29,707 conditions between appropriate boundary conditions. 516 00:40:33,120 --> 00:40:42,020 So the most important basic criterion-- again we 517 00:40:42,020 --> 00:40:44,130 are not going to the detail. 518 00:40:44,130 --> 00:40:47,310 And again, [? Nabil's ?] notes discussed this 519 00:40:47,310 --> 00:40:50,670 because he has to-- in order to discuss instability, 520 00:40:50,670 --> 00:40:54,330 he has also to discuss the boundary condition, et cetera. 521 00:40:54,330 --> 00:40:57,190 And you can look at [? Nabil's ?] notes. 522 00:40:57,190 --> 00:41:04,120 So basic criterion to impose a right boundary condition is 523 00:41:04,120 --> 00:41:06,940 to make sure that your energy is conserved 524 00:41:06,940 --> 00:41:09,650 and that your energy don't dissipate into nothing, 525 00:41:09,650 --> 00:41:14,708 don't dissipate at the boundary so that you 526 00:41:14,708 --> 00:41:15,833 have a well-defined system. 527 00:41:23,520 --> 00:41:24,720 Good. 528 00:41:24,720 --> 00:41:27,730 So now. 529 00:41:27,730 --> 00:41:39,140 So when you do canonical quantization, 530 00:41:39,140 --> 00:41:44,010 then what do you do is that you expand phi 531 00:41:44,010 --> 00:41:57,091 in terms of a complete set of normalizable modes. 532 00:42:01,315 --> 00:42:05,724 Of course they also have to be no normalizable as, normally, 533 00:42:05,724 --> 00:42:13,840 in the quantum mechanics, normalizable modes which 534 00:42:13,840 --> 00:42:21,195 satisfies the appropriate boundary condition. 535 00:42:28,490 --> 00:42:36,240 So before talking about the explicit boundary condition, 536 00:42:36,240 --> 00:42:40,160 let me first mention that issue of normalizability. 537 00:42:40,160 --> 00:42:41,968 So to talk about normalizability, 538 00:42:41,968 --> 00:42:43,426 you have to devalue in the product. 539 00:42:46,920 --> 00:42:51,809 So here we have the standard inner product for quantum field 540 00:42:51,809 --> 00:42:53,100 series in the curved spacetime. 541 00:42:57,700 --> 00:43:00,970 So here we have the standard inner product. 542 00:43:00,970 --> 00:43:02,220 And let me just write it down. 543 00:43:06,300 --> 00:43:08,180 So this is not specific to AdS. 544 00:43:08,180 --> 00:43:12,270 This just works for any quantum field theory, 545 00:43:12,270 --> 00:43:15,440 for any free scalar field series in the curved spacetime. 546 00:43:15,440 --> 00:43:19,820 So, for that series, you can define 547 00:43:19,820 --> 00:43:22,100 an inner product that follows. 548 00:43:22,100 --> 00:43:23,130 So consider. 549 00:43:23,130 --> 00:43:24,580 You have two modes, phi1, phi2. 550 00:43:27,170 --> 00:43:30,270 And then the inner product can be defined as the following. 551 00:43:30,270 --> 00:43:34,180 You choose a constant the time slice. 552 00:43:34,180 --> 00:43:35,840 You have to choose a constant time 553 00:43:35,840 --> 00:43:38,540 slice, which I called sigma t. 554 00:43:38,540 --> 00:43:43,043 So this is a constant slice. 555 00:43:46,320 --> 00:43:48,710 By time, I really mean this time. 556 00:43:48,710 --> 00:43:51,310 You choose a constant time slice. 557 00:43:51,310 --> 00:43:53,240 And then, of course, you integrate 558 00:43:53,240 --> 00:43:58,110 over spatial coordinates on these constant time slice. 559 00:44:20,570 --> 00:44:22,167 So this is the inner product. 560 00:44:22,167 --> 00:44:23,750 So this is the standard inner product. 561 00:44:23,750 --> 00:44:30,440 And you just plug in the AdS metric that gives you 562 00:44:30,440 --> 00:44:32,401 the explicit form for the AdS. 563 00:44:36,670 --> 00:44:41,590 So if you're not familiar with this, 564 00:44:41,590 --> 00:44:44,930 and you can check that this makes sense, 565 00:44:44,930 --> 00:44:54,690 that by showing that, actually, this thing is actually 566 00:44:54,690 --> 00:45:02,520 independent of t, so it does not depend, 567 00:45:02,520 --> 00:45:04,340 actually, your time slice. 568 00:45:04,340 --> 00:45:08,892 Because you don't want to your definition of inner product 569 00:45:08,892 --> 00:45:10,350 depend on specific time you choose. 570 00:45:19,280 --> 00:45:22,380 Then the normalizable would be the standard condition, 571 00:45:22,380 --> 00:45:25,830 whether this thing is finite, or infinity, et cetera. 572 00:45:25,830 --> 00:45:29,020 And you plug-in some modes with itself. 573 00:45:29,020 --> 00:45:31,879 And if it is finite, then it's normalizable. 574 00:45:31,879 --> 00:45:33,920 And if it's infinite, then it's not normalizable. 575 00:45:38,320 --> 00:45:40,156 So now this is our symptotic behavior. 576 00:45:42,890 --> 00:45:51,090 And you can check yourself, you can also convince yourself 577 00:45:51,090 --> 00:45:56,520 that the most dangerous behavior because the metric diverges 578 00:45:56,520 --> 00:45:58,700 near the boundary. 579 00:45:58,700 --> 00:46:02,370 So the most dangerous part of this integral 580 00:46:02,370 --> 00:46:07,960 is near the boundary because your metric diverges there. 581 00:46:07,960 --> 00:46:11,490 And you can just plug this behavior 582 00:46:11,490 --> 00:46:13,060 and this metric behavior. 583 00:46:13,060 --> 00:46:16,742 And then just pay attention to the behavior near the boundary. 584 00:46:16,742 --> 00:46:18,700 Then you can decide which mode is normalizable, 585 00:46:18,700 --> 00:46:21,750 which mode is non-normalizable. 586 00:46:21,750 --> 00:46:25,550 Then you find, again, this takes you a few minutes to do. 587 00:46:25,550 --> 00:46:26,670 So you'll not do it here. 588 00:46:29,810 --> 00:46:33,160 So I will only talk about the real nu. 589 00:46:33,160 --> 00:46:35,790 From now on. 590 00:46:35,790 --> 00:46:37,610 Just real nu. 591 00:46:37,610 --> 00:46:40,550 It means that we always satisfy that bound. 592 00:46:40,550 --> 00:46:51,800 So we find that, for that mode, so this mode always goes to 0. 593 00:46:51,800 --> 00:46:53,147 So you have these two modes. 594 00:46:53,147 --> 00:46:55,230 This mode always goes to 0 because delta is always 595 00:46:55,230 --> 00:47:00,110 positive because nu is positive, and nu is non-active, 596 00:47:00,110 --> 00:47:02,970 and d minus t over 2 is positive. 597 00:47:02,970 --> 00:47:06,277 So this is always goes to 0. 598 00:47:06,277 --> 00:47:08,360 And you can show that this is always normalizable. 599 00:47:16,630 --> 00:47:17,130 OK. 600 00:47:17,130 --> 00:47:23,510 So there's no divergence at the boundary. 601 00:47:23,510 --> 00:47:33,690 But for these modes, you're going 602 00:47:33,690 --> 00:47:37,080 to see these modes become dangerous when 603 00:47:37,080 --> 00:47:39,380 the mass become very large. 604 00:47:39,380 --> 00:47:43,850 When the mass becomes very large, then the nu is big. 605 00:47:43,850 --> 00:47:49,480 When nu is big, eventually the delta can be pretty big. 606 00:47:49,480 --> 00:47:52,080 And then can be greater than d. 607 00:47:52,080 --> 00:47:53,950 And then this will be an active exponent. 608 00:47:53,950 --> 00:47:55,230 And then this will blow up. 609 00:47:59,110 --> 00:48:06,526 So it turns out this is actually non-normalizable indeed. 610 00:48:10,250 --> 00:48:14,540 So this is non-normalizable for nu to be greater than 1. 611 00:48:14,540 --> 00:48:16,510 You actually don't need to for, mass 612 00:48:16,510 --> 00:48:18,149 to be very large, because for nu to be 613 00:48:18,149 --> 00:48:20,690 greater than or equal to 1, this is already non-normalizable. 614 00:48:23,600 --> 00:48:26,860 But it's an interesting thing. 615 00:48:26,860 --> 00:48:31,180 You said this mode is also normalizable 616 00:48:31,180 --> 00:48:32,580 when nu is in this range. 617 00:48:51,971 --> 00:48:52,470 Good. 618 00:48:52,470 --> 00:48:53,117 Any questions? 619 00:48:56,100 --> 00:49:00,600 Yes 620 00:49:00,600 --> 00:49:04,540 AUDIENCE: Can we define inner product some other way? 621 00:49:04,540 --> 00:49:06,610 HONG LIU: No, this is the only way. 622 00:49:06,610 --> 00:49:09,240 This is the canonical way, just generic for quantum field 623 00:49:09,240 --> 00:49:10,830 theory in the curved spacetime. 624 00:49:10,830 --> 00:49:12,180 This is not specific AdS. 625 00:49:15,804 --> 00:49:18,220 Yeah, essentially, this is a unique one, yeah I would say. 626 00:49:20,435 --> 00:49:22,810 AUDIENCE: So I'm just trying to understand inner product. 627 00:49:22,810 --> 00:49:26,672 In the case that I take, make it a norm in the sense 628 00:49:26,672 --> 00:49:28,630 that I take inner product of a field by itself. 629 00:49:28,630 --> 00:49:33,170 So why, for example, should this be positive? 630 00:49:33,170 --> 00:49:37,100 HONG LIU: No, no this is in general not positive. 631 00:49:37,100 --> 00:49:43,400 In general, this is a what we know in the Klein-Gordon field, 632 00:49:43,400 --> 00:49:46,270 say, if you do in the Minkowski spacetime, this in general, not 633 00:49:46,270 --> 00:49:48,200 positive definite. 634 00:49:48,200 --> 00:49:50,600 AUDIENCE: Right, sure. 635 00:49:50,600 --> 00:49:54,630 HONG LIU: Yeah, same thing in the Klein-Gordon fields. 636 00:49:54,630 --> 00:49:58,950 Yeah, but the [INAUDIBLE] is, so that 637 00:49:58,950 --> 00:50:00,520 means this thing does not make sense 638 00:50:00,520 --> 00:50:02,440 to interpolate as a probability. 639 00:50:02,440 --> 00:50:06,415 But you can still use this as a definition of normalizability. 640 00:50:06,415 --> 00:50:08,040 AUDIENCE: So it still does the property 641 00:50:08,040 --> 00:50:12,604 that something will only have so it's not 642 00:50:12,604 --> 00:50:14,520 a semi-norm in the sense-- like it is actually 643 00:50:14,520 --> 00:50:15,395 a good inner product? 644 00:50:15,395 --> 00:50:17,257 HONG LIU: Yeah. 645 00:50:17,257 --> 00:50:18,715 AUDIENCE: Isn't it this it just the 646 00:50:18,715 --> 00:50:22,680 generalization of probability current of [INAUDIBLE]? 647 00:50:22,680 --> 00:50:24,180 HONG LIU: Yeah, no, this is the same 648 00:50:24,180 --> 00:50:25,388 as the Klein-Gordon equation. 649 00:50:29,320 --> 00:50:32,310 This is the same inner product as you 650 00:50:32,310 --> 00:50:38,610 would define for the Klein-Gordon field, 651 00:50:38,610 --> 00:50:41,600 a free massive scalar field in Minkowski spacetime. 652 00:50:41,600 --> 00:50:43,310 And that this is just the generalization 653 00:50:43,310 --> 00:50:44,560 to a general curved spacetime. 654 00:50:48,520 --> 00:50:51,320 Good. 655 00:50:51,320 --> 00:50:56,170 So now, yeah, so we see that this is always normalizable. 656 00:50:56,170 --> 00:50:58,780 And this sometimes is normalizable and sometimes not 657 00:50:58,780 --> 00:50:59,542 normalizable. 658 00:50:59,542 --> 00:51:01,625 So now, let me talk about the boundary conditions. 659 00:51:08,360 --> 00:51:09,890 So for the time reason, I will not 660 00:51:09,890 --> 00:51:21,090 show that the boundary condition I'm going to talk about 661 00:51:21,090 --> 00:51:24,370 will give you a well-defined energy in the sense 662 00:51:24,370 --> 00:51:27,310 that there's no energy exchange at the boundary. 663 00:51:27,310 --> 00:51:32,010 And for that, take a look at [? Nabil's ?] notes. 664 00:51:32,010 --> 00:51:34,390 I will not go into detail on that. 665 00:51:34,390 --> 00:51:38,026 But the boundary condition I say will satisfy that property. 666 00:51:40,620 --> 00:51:45,340 So let's first look at when nu is greater than or equal to n. 667 00:51:45,340 --> 00:51:48,360 In this case, the answer is obvious. 668 00:51:48,360 --> 00:51:52,530 You don't even need to check the energy conservation. 669 00:51:52,530 --> 00:51:55,900 Because you only have one possibility because this mode 670 00:51:55,900 --> 00:51:58,200 is non-normalizable. 671 00:51:58,200 --> 00:52:01,302 And a non-normalizable mode, you cannot keep them when you do 672 00:52:01,302 --> 00:52:02,010 the quantization. 673 00:52:02,010 --> 00:52:04,064 Quantization, you have to use normalizable modes. 674 00:52:04,064 --> 00:52:05,480 So the boundary condition you have 675 00:52:05,480 --> 00:52:11,260 to choose in order that conical quantization is equal to 0. 676 00:52:13,850 --> 00:52:18,535 So you should only keep this mode of a nu equal to 1. 677 00:52:18,535 --> 00:52:19,910 Because this is non-normalizable. 678 00:52:22,810 --> 00:52:32,694 So now, when nu is in this range, 679 00:52:32,694 --> 00:52:35,110 so both modes are normalizable, but you cannot choose both 680 00:52:35,110 --> 00:52:35,610 of them. 681 00:52:35,610 --> 00:52:38,580 You cannot allow both of them because then it leads 682 00:52:38,580 --> 00:52:41,210 to the non-conversation. 683 00:52:41,210 --> 00:52:44,400 You need to put some boundary conditions there. 684 00:52:44,400 --> 00:52:48,889 So then, naturally, the equal to 0 also works in this range. 685 00:52:48,889 --> 00:52:50,722 So this is called the standard quantization. 686 00:52:59,030 --> 00:53:02,760 You can still impose, even though the A modes become 687 00:53:02,760 --> 00:53:06,630 normalizable, you can still impose A equal is 0. 688 00:53:06,630 --> 00:53:11,384 So tentatively you can also impose B equal to 0. 689 00:53:11,384 --> 00:53:12,300 Just take one of them. 690 00:53:12,300 --> 00:53:16,170 Because A now is normalizable. 691 00:53:16,170 --> 00:53:18,755 So this is called the alternative quantization. 692 00:53:30,730 --> 00:53:33,680 So sometimes, it's also possible to impose the mixed boundary 693 00:53:33,680 --> 00:53:37,290 condition between them-- to impose a mixed boundary 694 00:53:37,290 --> 00:53:38,260 condition. 695 00:53:38,260 --> 00:53:41,120 I will not go into there. 696 00:53:41,120 --> 00:53:45,790 Anyway, so for example, if you look at [? Nabil's ?] notes, 697 00:53:45,790 --> 00:53:50,350 you can easily check for those boundary conditions 698 00:53:50,350 --> 00:53:52,940 that there's no energy flux, through the boundary. 699 00:53:55,580 --> 00:54:00,034 So, from now on, I will use the following terminology. 700 00:54:02,700 --> 00:54:05,697 From now on, we'll use the following terminology. 701 00:54:14,870 --> 00:54:26,820 So from now on, when I say normalizable, 702 00:54:26,820 --> 00:54:31,510 I always say the mode which are picked by quantization, OK? 703 00:54:31,510 --> 00:54:33,760 So when I say normalizable, in this case, of course, 704 00:54:33,760 --> 00:54:36,550 this is normalizable. 705 00:54:36,550 --> 00:54:41,810 When I say normalizable, for nu greater than 1, 706 00:54:41,810 --> 00:54:43,460 it's no ambiguity, just the B mode. 707 00:54:46,440 --> 00:54:50,770 And when nu is between 0 and 1, the normalizable 708 00:54:50,770 --> 00:54:54,330 refers to the mode which you choose to quantize. 709 00:54:54,330 --> 00:54:57,790 And then the other you consider to be non-normalizable. 710 00:54:57,790 --> 00:55:00,780 Is it clear, the terminology? 711 00:55:00,780 --> 00:55:05,420 So in the standard quantization, the B-mode is normalizable, 712 00:55:05,420 --> 00:55:08,570 and the A is non-normalizable, even though, mathematically, A 713 00:55:08,570 --> 00:55:10,070 is normalizable. 714 00:55:10,070 --> 00:55:12,450 And in the alternative quantization, 715 00:55:12,450 --> 00:55:15,450 the B is non-normalizable, even though, mathematically, it 716 00:55:15,450 --> 00:55:16,340 is normalizable. 717 00:55:19,770 --> 00:55:30,970 So normalizable would mean, the behavior specified. 718 00:55:37,740 --> 00:55:41,240 I'll call the other modes non-normalizable modes. 719 00:55:41,240 --> 00:55:43,470 So with this terminology, I always 720 00:55:43,470 --> 00:55:45,980 have 1 normalizable mode, one non-normalizable modes. 721 00:56:01,520 --> 00:56:05,620 OK, let's move on a little bit more. 722 00:56:05,620 --> 00:56:08,670 So this is the second remark. 723 00:56:08,670 --> 00:56:12,300 And the third remark-- let me just erase this. 724 00:56:17,150 --> 00:56:22,350 So the third remark, so normalizable modes, 725 00:56:22,350 --> 00:56:30,760 by definition, is the one we use to do a canonical quantization. 726 00:56:30,760 --> 00:56:33,578 And it's the one we use to build up of the Hilbert space. 727 00:56:52,852 --> 00:56:55,060 But now, since we have an equivalence between the two 728 00:56:55,060 --> 00:56:58,900 series, the Hilbert space of the two series 729 00:56:58,900 --> 00:57:02,377 will be the same because, otherwise, the state of the two 730 00:57:02,377 --> 00:57:03,460 series should be the same. 731 00:57:03,460 --> 00:57:05,960 Otherwise, we cannot talk about the equivalence. 732 00:57:05,960 --> 00:57:11,430 So that means that any normalizable mode, so anything 733 00:57:11,430 --> 00:57:16,640 which fall off, say, as, for example, 734 00:57:16,640 --> 00:57:20,960 in the standard quantization, like z to the powers delta 735 00:57:20,960 --> 00:57:24,740 at infinity near the boundary, you 736 00:57:24,740 --> 00:57:30,420 should consider as mapped to some states of the boundary 737 00:57:30,420 --> 00:57:30,920 theory. 738 00:57:46,520 --> 00:57:49,630 So this is, actually, a very general statement. 739 00:57:49,630 --> 00:57:52,490 So this will also work if you include those higher order 740 00:57:52,490 --> 00:57:54,360 corrections which are suppressed by kappa. 741 00:57:58,010 --> 00:58:01,300 So, in general, you can apply it to a complicated 742 00:58:01,300 --> 00:58:04,770 interacting nonlinear series in the gravity side. 743 00:58:04,770 --> 00:58:07,080 And the normalizable modes are always 744 00:58:07,080 --> 00:58:10,150 the modes which are in your Hilbert space in the gravity 745 00:58:10,150 --> 00:58:11,480 side. 746 00:58:11,480 --> 00:58:13,530 And so that means the mass corresponding 747 00:58:13,530 --> 00:58:16,630 to some states in the boundary field series. 748 00:58:20,820 --> 00:58:21,724 Good. 749 00:58:21,724 --> 00:58:23,390 So now, let's look at the interpretation 750 00:58:23,390 --> 00:58:24,837 of the non-normalizable modes. 751 00:58:44,869 --> 00:58:46,410 Non-normalizable modes, suddenly they 752 00:58:46,410 --> 00:58:47,925 are not part of the Hilbert space. 753 00:59:08,760 --> 00:59:20,445 So if they are present, in particular-- 754 00:59:20,445 --> 00:59:22,100 let me just finish this. 755 00:59:22,100 --> 00:59:22,850 then we'll say it. 756 00:59:22,850 --> 00:59:30,460 So, if present, it should be considered-- so sometimes, 757 00:59:30,460 --> 00:59:33,620 they're just not present. 758 00:59:33,620 --> 00:59:35,590 But, if in some situations, they are present, 759 00:59:35,590 --> 00:59:37,760 then you should view them, view such kind 760 00:59:37,760 --> 00:59:51,746 of non-normalizable modes-- if I'm in the background. 761 01:00:07,770 --> 01:00:14,021 So by background, for example, the AdS, pure AdS, 762 01:00:14,021 --> 01:00:16,020 which we are expanding around is our background. 763 01:00:19,780 --> 01:00:22,310 Yeah. 764 01:00:22,310 --> 01:00:25,140 So you can imagine, say, [INAUDIBLE] 765 01:00:25,140 --> 01:00:27,755 non-normalizable modes deform the AdS, 766 01:00:27,755 --> 01:00:32,170 then you have a new background, then you have a new background. 767 01:00:32,170 --> 01:00:34,010 And in that background, you're not 768 01:00:34,010 --> 01:00:36,990 supposed to quantize the cell, because these 769 01:00:36,990 --> 01:00:39,790 are non-normalizable modes. 770 01:00:39,790 --> 01:00:41,368 These are non-normalizable modes. 771 01:00:41,368 --> 01:00:43,320 AUDIENCE: Yes, sorry, I guess I didn't quite 772 01:00:43,320 --> 01:00:48,700 get the terminology normalizable for 2, I guess? 773 01:00:48,700 --> 01:00:51,222 You mentioned it, but could you repeat it? 774 01:00:51,222 --> 01:00:51,888 HONG LIU: Sorry? 775 01:00:51,888 --> 01:00:53,880 AUDIENCE: Normalizable, not normalizable. 776 01:00:53,880 --> 01:00:56,420 HONG LIU: Yeah. 777 01:00:56,420 --> 01:00:59,830 Normalizable means the behavior specified by the quantization. 778 01:00:59,830 --> 01:01:05,075 AUDIENCE: So I don't think-- I don't quite understand-- 779 01:01:05,075 --> 01:01:07,240 HONG LIU: Yeah, let me just repeat. 780 01:01:07,240 --> 01:01:10,560 For nu greater than 1, normalizable, it's 781 01:01:10,560 --> 01:01:13,730 just normalizable, because there's only one normalizable 782 01:01:13,730 --> 01:01:15,300 mode. 783 01:01:15,300 --> 01:01:21,030 For nu between 0 and 1, mathematically, both modes 784 01:01:21,030 --> 01:01:22,780 become normalizable. 785 01:01:22,780 --> 01:01:27,290 But when we quantize them, and we [INAUDIBLE] one of them. 786 01:01:27,290 --> 01:01:29,510 OK? 787 01:01:29,510 --> 01:01:32,950 As I said, in order to preserve the energy conservation-- so 788 01:01:32,950 --> 01:01:35,160 we have to select one of them. 789 01:01:35,160 --> 01:01:36,730 So there are two possibilities. 790 01:01:36,730 --> 01:01:39,056 You select the standard quantization, 791 01:01:39,056 --> 01:01:40,430 which you said A equals to 0, you 792 01:01:40,430 --> 01:01:42,150 said, the alternative quantization 793 01:01:42,150 --> 01:01:44,500 you set B equal to 0. 794 01:01:44,500 --> 01:01:46,960 So the 1 you set 0 is considered to be 795 01:01:46,960 --> 01:01:50,660 non-normalizable in this terminology. 796 01:01:50,660 --> 01:01:55,990 And the 1 you allowed, which means 797 01:01:55,990 --> 01:02:00,240 the one you are used to expand as a canonical connotation 798 01:02:00,240 --> 01:02:01,698 are called normalizable. 799 01:02:15,550 --> 01:02:17,550 I will elaborate this a little bit more. 800 01:02:23,660 --> 01:02:25,050 Yeah. 801 01:02:25,050 --> 01:02:28,330 Yeah, after five minutes, then you will see. 802 01:02:28,330 --> 01:02:29,167 Yes? 803 01:02:29,167 --> 01:02:31,625 AUDIENCE: What do you mean by the other ones being present? 804 01:02:31,625 --> 01:02:36,200 If we don't quantize them, have they appeared? 805 01:02:36,200 --> 01:02:39,580 HONG LIU: Yeah, they can appear. 806 01:02:39,580 --> 01:02:43,810 I will, again, explain in one minute. 807 01:02:43,810 --> 01:02:50,370 So now, I'm going to make a connection for this statement 808 01:02:50,370 --> 01:02:52,220 to the statement we are making last time. 809 01:02:54,800 --> 01:02:56,630 We're making last time. 810 01:02:56,630 --> 01:03:06,620 So last time we said, if a gravity field has a boundary 811 01:03:06,620 --> 01:03:10,680 value, then this boundary value can 812 01:03:10,680 --> 01:03:15,770 be considered as a source multiply the operator which 813 01:03:15,770 --> 01:03:18,830 you can add to a Lagrangian. 814 01:03:18,830 --> 01:03:20,960 You can add to the action. 815 01:03:20,960 --> 01:03:23,880 And so this is precisely like that. 816 01:03:23,880 --> 01:03:26,110 It's precisely like that. 817 01:03:26,110 --> 01:03:29,030 So let me just repeat. 818 01:03:33,440 --> 01:03:38,670 Say, let's consider the standard quantization just 819 01:03:38,670 --> 01:03:41,705 to be specific and say standard quantization. 820 01:03:45,800 --> 01:03:47,960 In this case, the non-normalizable mode 821 01:03:47,960 --> 01:03:54,180 corresponding to A not equal to 0, 822 01:03:54,180 --> 01:03:58,250 so if the non-normalizable modes are present, 823 01:03:58,250 --> 01:04:03,356 and this is corresponding to A not equal to 0. 824 01:04:08,900 --> 01:04:13,290 And we see from here, this term, A term, 825 01:04:13,290 --> 01:04:17,250 always dominated the other term. 826 01:04:17,250 --> 01:04:19,500 So, in other words, when you go to the boundary limit, 827 01:04:19,500 --> 01:04:23,813 it's the A which determines the boundary value of the field. 828 01:04:30,590 --> 01:04:31,500 Let me just say. 829 01:04:31,500 --> 01:04:34,486 The A, which is determined boundary value of the field. 830 01:04:40,260 --> 01:04:41,385 I put the code. 831 01:04:44,117 --> 01:04:45,033 AUDIENCE: [INAUDIBLE]. 832 01:04:51,147 --> 01:04:53,730 HONG LIU: A equal to 0 when you do the canonical quantization. 833 01:04:53,730 --> 01:04:55,370 AUDIENCE: Oh, OK. [INAUDIBLE] 834 01:04:55,370 --> 01:04:57,160 HONG LIU: But A equal to 0, you mean 835 01:04:57,160 --> 01:05:00,566 this is the condition in the normalizable modes. 836 01:05:00,566 --> 01:05:02,440 So here, we are discussing the interpretation 837 01:05:02,440 --> 01:05:04,160 of the non-normalizable mode. 838 01:05:04,160 --> 01:05:08,220 So, now, suppose A is nonzero. 839 01:05:08,220 --> 01:05:10,230 Suppose a non-normalizable mode is non-zero. 840 01:05:12,322 --> 01:05:13,947 Suppose, now, the non-normalizable mode 841 01:05:13,947 --> 01:05:15,630 is non-zero. 842 01:05:15,630 --> 01:05:17,350 We are down with normalizable modes. 843 01:05:17,350 --> 01:05:19,480 This is a condition for the normalizable modes. 844 01:05:19,480 --> 01:05:19,980 OK? 845 01:05:26,060 --> 01:05:30,930 So since this term dominates, the A 846 01:05:30,930 --> 01:05:35,190 should be interpolated as a boundary value of the fields. 847 01:05:35,190 --> 01:06:04,060 And if you have a non-zero A, so if you have a non-zero A, 848 01:06:04,060 --> 01:06:10,630 say, if A x is equal to some phi x, 849 01:06:10,630 --> 01:06:16,000 according to what we discussed last time, 850 01:06:16,000 --> 01:06:21,110 now up to this kinematic factor, this thing, 851 01:06:21,110 --> 01:06:24,790 which you need to strip away because this just determines 852 01:06:24,790 --> 01:06:27,300 your equation motion. 853 01:06:27,300 --> 01:06:29,420 So if A equal to 5 equal to 0, that 854 01:06:29,420 --> 01:06:38,260 implies that the boundary series action should contain the term. 855 01:06:59,190 --> 01:07:01,810 So what this means is that the non-normalizable, 856 01:07:01,810 --> 01:07:04,880 there are certainly not part of the Hilbert space. 857 01:07:04,880 --> 01:07:07,260 If they are present, they actually 858 01:07:07,260 --> 01:07:14,890 determine the boundary theory itself. 859 01:07:14,890 --> 01:07:34,882 So the non-normalizable modes determine the boundary theory 860 01:07:34,882 --> 01:07:35,382 itself. 861 01:07:45,174 --> 01:07:53,540 So we will later see examples of this. 862 01:07:56,960 --> 01:08:04,080 But let me just repeat, so if you look at the gravity 863 01:08:04,080 --> 01:08:09,610 solution, then you look at its symptotic behavior 864 01:08:09,610 --> 01:08:12,190 near the boundary. 865 01:08:12,190 --> 01:08:17,149 And if you see-- let's, say, consider two solutions, 866 01:08:17,149 --> 01:08:19,370 two different solutions. 867 01:08:19,370 --> 01:08:21,359 If the two different solutions have 868 01:08:21,359 --> 01:08:26,279 the same non-normalizable behavior-- 869 01:08:26,279 --> 01:08:31,990 say the non-normalizable modes are the same, but only 870 01:08:31,990 --> 01:08:34,700 the difference in the normalizable modes, 871 01:08:34,700 --> 01:08:38,029 then you will say these two solutions describe 872 01:08:38,029 --> 01:08:42,300 two states of the same theory. 873 01:08:42,300 --> 01:08:44,410 Because they only differ by normalizable modes. 874 01:08:44,410 --> 01:08:48,700 Then it describes the different states in the same theory. 875 01:08:48,700 --> 01:08:53,279 But now, if I look at two such solutions, 876 01:08:53,279 --> 01:08:56,670 and these two solutions differ by the behavior 877 01:08:56,670 --> 01:09:00,870 of the non-normalizable modes, then we would say, 878 01:09:00,870 --> 01:09:03,569 these two solutions corresponding 879 01:09:03,569 --> 01:09:07,189 to two different series because one 880 01:09:07,189 --> 01:09:09,090 is due to two different series. 881 01:09:09,090 --> 01:09:13,429 Because non-normalizable modes needs to add additional terms 882 01:09:13,429 --> 01:09:14,720 to your boundary theory action. 883 01:09:18,229 --> 01:09:21,147 Have I answered your questions? 884 01:09:21,147 --> 01:09:25,464 AUDIENCE: So basically, you just assume any field which is 885 01:09:25,464 --> 01:09:29,160 a type of-- maybe a combination of those two modes, 886 01:09:29,160 --> 01:09:30,550 but if the-- 887 01:09:30,550 --> 01:09:33,451 HONG LIU: Forget about linear combination of the modes. 888 01:09:33,451 --> 01:09:36,979 Just say, if you consider two solutions, which if there only 889 01:09:36,979 --> 01:09:39,180 differ by normalizable modes, then they 890 01:09:39,180 --> 01:09:42,160 correspond to different states of the same theory. 891 01:09:42,160 --> 01:09:44,239 And if they differ by non-normalizable modes, 892 01:09:44,239 --> 01:09:46,944 that means they describe different theories. 893 01:09:46,944 --> 01:09:48,860 Even the theories they describe are different. 894 01:09:54,140 --> 01:09:58,440 And so that's why we call this background. 895 01:09:58,440 --> 01:10:01,300 We call this background because the symptotic conditions 896 01:10:01,300 --> 01:10:04,610 at infinity has changed. 897 01:10:04,610 --> 01:10:06,450 AUDIENCE: I have a question. 898 01:10:06,450 --> 01:10:09,828 So, in the alternative standard upon the quantization 899 01:10:09,828 --> 01:10:13,666 case that-- which would be equal to 0. 900 01:10:13,666 --> 01:10:14,960 HONG LIU: Right. 901 01:10:14,960 --> 01:10:18,120 AUDIENCE: But we know that the d minus delta is always 902 01:10:18,120 --> 01:10:19,660 larger than delta. 903 01:10:19,660 --> 01:10:23,590 But in that case, it seems near the boundary. 904 01:10:23,590 --> 01:10:25,482 The boundary determines, not the number. 905 01:10:25,482 --> 01:10:26,440 HONG LIU: Right, right. 906 01:10:26,440 --> 01:10:26,610 Yeah. 907 01:10:26,610 --> 01:10:28,030 That's a very good question. 908 01:10:28,030 --> 01:10:30,490 So that, you have to stretch a little bit. 909 01:10:30,490 --> 01:10:31,210 Yeah. 910 01:10:31,210 --> 01:10:32,160 Yeah. 911 01:10:32,160 --> 01:10:33,370 Yeah. 912 01:10:33,370 --> 01:10:33,870 Good? 913 01:10:36,720 --> 01:10:41,620 So let me finish my last equation before we break. 914 01:10:41,620 --> 01:10:47,380 So more precisely, because of this z term, 915 01:10:47,380 --> 01:10:52,060 so in the standard quantization, yeah 916 01:10:52,060 --> 01:10:55,715 let me just write it there, so write it here. 917 01:10:58,827 --> 01:11:00,035 Let me just replace this one. 918 01:11:03,290 --> 01:11:06,350 In the standard quantization, if you 919 01:11:06,350 --> 01:11:12,630 have such a term in your boundary series Lagrangian, 920 01:11:12,630 --> 01:11:20,510 that's corresponding to this phi x is equal to limit z 921 01:11:20,510 --> 01:11:28,560 goes to 0, z to the power delta minus d phi zx. 922 01:11:28,560 --> 01:11:31,270 So I have to multiply a power so that it will just 923 01:11:31,270 --> 01:11:36,785 extract this A factor from the behavior of the phi. 924 01:11:40,910 --> 01:11:42,844 And the so this is the relation. 925 01:11:45,700 --> 01:11:48,500 So this is the important relation. 926 01:11:48,500 --> 01:11:51,254 So, last time, we only said that phi 927 01:11:51,254 --> 01:11:53,670 should corresponding to the boundary value of this capital 928 01:11:53,670 --> 01:11:54,300 phi. 929 01:11:54,300 --> 01:11:57,300 But now, by working out the symptotic behavior, 930 01:11:57,300 --> 01:11:59,490 we have refined that statement. 931 01:11:59,490 --> 01:12:01,600 Now it's a really precise mathematical statement. 932 01:12:06,980 --> 01:12:11,185 Any questions regarding this? 933 01:12:11,185 --> 01:12:13,672 AUDIENCE: Is it a guess or-- 934 01:12:13,672 --> 01:12:15,380 HONG LIU: No, this is just a [INAUDIBLE]. 935 01:12:20,000 --> 01:12:27,180 This limit precisely extracts this A term. 936 01:12:27,180 --> 01:12:30,970 This term, because that opposite power to this one, and this one 937 01:12:30,970 --> 01:12:33,010 will be smaller. 938 01:12:33,010 --> 01:12:37,160 This one will be smaller because that multiplier, this one, when 939 01:12:37,160 --> 01:12:39,070 z goes to 0 will go to 0. 940 01:12:39,070 --> 01:12:41,585 So you automatically extract this term. 941 01:12:41,585 --> 01:12:42,460 AUDIENCE: [INAUDIBLE] 942 01:12:46,732 --> 01:12:48,190 HONG LIU: It's just this statement. 943 01:12:48,190 --> 01:12:51,440 The A is related to this. 944 01:12:51,440 --> 01:12:51,940 OK. 945 01:12:54,800 --> 01:12:57,920 AUDIENCE: No, this statement is a guess from the example 946 01:12:57,920 --> 01:12:59,850 we discussed last time. 947 01:12:59,850 --> 01:13:02,660 So last time, by looking at this line example, 948 01:13:02,660 --> 01:13:07,480 we deduce that the boundary value of a field 949 01:13:07,480 --> 01:13:10,740 should be related to the source. 950 01:13:10,740 --> 01:13:12,550 The boundary value of a [INAUDIBLE] field 951 01:13:12,550 --> 01:13:17,260 should be related to the source of a boundary operator. 952 01:13:17,260 --> 01:13:19,890 And at that time, we don't know what we mean, precisely, 953 01:13:19,890 --> 01:13:22,280 by boundary value. 954 01:13:22,280 --> 01:13:25,810 And now we have worked out a precise symptotic behavior. 955 01:13:25,810 --> 01:13:28,490 And now you can talk about what you mean by boundary behavior. 956 01:13:28,490 --> 01:13:31,060 And then that refines that statement 957 01:13:31,060 --> 01:13:34,220 to that precise mathematical statement. 958 01:13:34,220 --> 01:13:34,720 Good. 959 01:13:34,720 --> 01:13:35,355 So let's start. 960 01:13:45,610 --> 01:13:48,525 So let's see what we can deduce from this relation. 961 01:13:51,114 --> 01:13:53,530 So turns out that there's a very important information you 962 01:13:53,530 --> 01:13:57,880 can deduce from this equation. 963 01:13:57,880 --> 01:13:59,130 And this is my number five. 964 01:14:04,580 --> 01:14:06,085 So let me call this equation star. 965 01:14:09,270 --> 01:14:14,590 So this equation star, relation star, 966 01:14:14,590 --> 01:14:29,490 actually tells you that this delta, which defined before, 967 01:14:29,490 --> 01:14:31,970 in this way, and appears in your symptotic formula 968 01:14:31,970 --> 01:14:36,140 like this, this delta is actually, 969 01:14:36,140 --> 01:14:49,140 it's the scaling dimension of the operator dual to phi of O. 970 01:14:49,140 --> 01:14:53,230 So we're assuming this phi is due to some operator 971 01:14:53,230 --> 01:14:55,570 O in the boundary. 972 01:14:55,570 --> 01:14:57,910 And so this is actually this precise in the scaling 973 01:14:57,910 --> 01:15:01,654 dimension of O. So now, let me explain that. 974 01:15:10,477 --> 01:15:12,435 So before I do this, do you have any questions? 975 01:15:15,550 --> 01:15:16,440 Good. 976 01:15:16,440 --> 01:15:21,110 OK, so let me first remind you how 977 01:15:21,110 --> 01:15:23,150 we define the scaling dimension of an operator 978 01:15:23,150 --> 01:15:26,950 in conformal field theory. 979 01:15:26,950 --> 01:15:29,890 So if you have a scaling [INAUDIBLE] theory, 980 01:15:29,890 --> 01:15:35,940 then the overall scaling is a symmetry. 981 01:15:35,940 --> 01:15:38,630 So such a scaling is a symmetry. 982 01:15:41,920 --> 01:15:44,320 And so under such symmetry transformation, 983 01:15:44,320 --> 01:15:48,450 say, operator, we call it a scaling operator 984 01:15:48,450 --> 01:15:49,800 if you transform. 985 01:15:49,800 --> 01:15:51,860 And under this symmetry has the following. 986 01:15:51,860 --> 01:15:55,220 So transform into a new operator prime. 987 01:15:55,220 --> 01:15:58,630 And evaluated at the new x prime is 988 01:15:58,630 --> 01:16:02,060 equal to just differ by a scaling 989 01:16:02,060 --> 01:16:03,602 from the previous operator. 990 01:16:06,260 --> 01:16:07,690 OK, so we pause. 991 01:16:07,690 --> 01:16:13,900 So the operator, with such property, a scaling operator. 992 01:16:13,900 --> 01:16:23,680 So good operators, which are representations 993 01:16:23,680 --> 01:16:27,630 of the conformal symmetry always have this behavior. 994 01:16:27,630 --> 01:16:31,500 And this delta defines the scaling dimension. 995 01:16:31,500 --> 01:16:34,405 So this delta is the scaling dimension. 996 01:16:34,405 --> 01:16:36,530 So this is the definition of the scaling dimension. 997 01:16:42,075 --> 01:16:44,345 So this is almost similar to a scalar field. 998 01:16:44,345 --> 01:16:46,659 So a scalar field means that the new scalar field, 999 01:16:46,659 --> 01:16:47,950 you evaluate it at a new point. 1000 01:16:47,950 --> 01:16:51,500 It should be the same as your old scalar field. 1001 01:16:51,500 --> 01:16:53,060 You evaluate it at the old point. 1002 01:16:53,060 --> 01:16:57,100 And here the difference is just of a pre-factor. 1003 01:16:57,100 --> 01:17:02,620 And this number defines the scalar information. 1004 01:17:02,620 --> 01:17:04,900 Good? 1005 01:17:04,900 --> 01:17:12,690 So now I will derive that this relation implies 1006 01:17:12,690 --> 01:17:16,440 that this operator transforms precisely 1007 01:17:16,440 --> 01:17:17,440 according to this delta. 1008 01:17:24,010 --> 01:17:34,050 So let me first I remind you that the boundary scaling x mu 1009 01:17:34,050 --> 01:17:38,410 goes to x mu prime equal to lambda 1010 01:17:38,410 --> 01:17:44,940 x mu is related to the bark, also 1011 01:17:44,940 --> 01:17:51,270 to isometry, which is the x mu prime equal to lambda x mu. 1012 01:17:51,270 --> 01:17:53,860 But, at the same time, you also scale z. 1013 01:17:59,780 --> 01:18:03,180 So this scaling symmetry, the counterpart 1014 01:18:03,180 --> 01:18:05,086 of this scaling in the gravity side 1015 01:18:05,086 --> 01:18:06,752 is that you scale both of them together. 1016 01:18:09,920 --> 01:18:12,700 And the gravity metric is invariant under this. 1017 01:18:27,450 --> 01:18:30,600 So now let's consider transformation. 1018 01:18:30,600 --> 01:18:32,820 So now let's consider the relation between the phi x 1019 01:18:32,820 --> 01:18:35,052 z and the O x. 1020 01:18:38,610 --> 01:18:43,136 So let's consider such a transformation on both sides. 1021 01:18:43,136 --> 01:18:45,010 So on the gravity side, such a transformation 1022 01:18:45,010 --> 01:18:46,293 leads to a new field. 1023 01:18:50,370 --> 01:18:53,530 And this should be dual to transform the field 1024 01:18:53,530 --> 01:18:55,550 on the field theory side. 1025 01:19:06,485 --> 01:19:07,610 AUDIENCE: Which [INAUDIBLE] 1026 01:19:13,392 --> 01:19:14,600 HONG LIU: Yeah, that's right. 1027 01:19:14,600 --> 01:19:15,100 Yeah. 1028 01:19:15,100 --> 01:19:16,030 Yeah. 1029 01:19:16,030 --> 01:19:16,640 That's right. 1030 01:19:16,640 --> 01:19:17,560 Now, this is [INAUDIBLE]. 1031 01:19:17,560 --> 01:19:18,351 This is a boundary. 1032 01:19:21,480 --> 01:19:23,020 OK? 1033 01:19:23,020 --> 01:19:25,129 But I think, since this is a symmetry, 1034 01:19:25,129 --> 01:19:27,670 so this transformation of both symmetries in the field series 1035 01:19:27,670 --> 01:19:32,380 side on the gravity side, that should have low consequences. 1036 01:19:32,380 --> 01:19:34,680 That should have low consequence. 1037 01:19:34,680 --> 01:19:35,347 That means. 1038 01:19:39,060 --> 01:19:39,598 That means. 1039 01:19:57,130 --> 01:20:01,950 And the phi prime, this mu phi is the boundary value 1040 01:20:01,950 --> 01:20:06,680 of the corresponding capital phi, OK? 1041 01:20:06,680 --> 01:20:10,674 This should still be the same because your physics should not 1042 01:20:10,674 --> 01:20:11,590 be changed under this. 1043 01:20:16,770 --> 01:20:22,400 So now let me remind you that phi is a scalar field. 1044 01:20:22,400 --> 01:20:26,090 In the gravity side, this is a scalar fields. 1045 01:20:26,090 --> 01:20:27,810 So scalar fields transform, under such 1046 01:20:27,810 --> 01:20:33,900 a coordinate transformation, transforms as phi prime, 1047 01:20:33,900 --> 01:20:35,720 X prime, z prime. 1048 01:20:35,720 --> 01:20:37,484 You go to phi x z. 1049 01:20:45,507 --> 01:20:47,340 So this is just the definition of the scalar 1050 01:20:47,340 --> 01:20:49,460 field in the gravity site. 1051 01:20:49,460 --> 01:20:53,200 And now let's apply this to the boundary value. 1052 01:20:53,200 --> 01:20:58,780 Then we find that the phi prime x, which is a boundary value 1053 01:20:58,780 --> 01:21:14,560 capital phi prime, so this should be equal to that. 1054 01:21:14,560 --> 01:21:20,540 So now we use that this relation of z prime is lambda times Z. 1055 01:21:20,540 --> 01:21:23,280 And this is just equal to that. 1056 01:21:23,280 --> 01:21:27,670 So this just becomes lambda to the delta minus t phi x. 1057 01:21:37,690 --> 01:21:38,190 Clear? 1058 01:21:40,970 --> 01:21:45,990 So now you can just plug this back into here. 1059 01:21:45,990 --> 01:21:50,620 So there's a factor of lambda minus delta minus d. 1060 01:21:50,620 --> 01:21:56,210 And this comes with a factor of lambda to the power d and then 1061 01:21:56,210 --> 01:22:01,985 tells you that O prime x prime equal to lambda 1062 01:22:01,985 --> 01:22:03,266 minus delta O x. 1063 01:22:06,310 --> 01:22:11,530 And we show that this delta, which appears here, 1064 01:22:11,530 --> 01:22:15,150 or, whatever, here, is that delta. 1065 01:22:15,150 --> 01:22:17,742 It's the dimension of the operator. 1066 01:22:17,742 --> 01:22:20,718 AUDIENCE: But in standard quantization, 1067 01:22:20,718 --> 01:22:22,710 you mentioned that-- 1068 01:22:22,710 --> 01:22:25,030 HONG LIU: No, this is a standard quantization. 1069 01:22:25,030 --> 01:22:29,410 AUDIENCE: A equals 0 in standard quantization. 1070 01:22:29,410 --> 01:22:32,599 HONG LIU: No, no, we're talking about the source. 1071 01:22:32,599 --> 01:22:34,390 We're talking about non-normalizable modes. 1072 01:22:34,390 --> 01:22:36,680 We're not talking about canonical quantization. 1073 01:22:36,680 --> 01:22:38,638 We're not talking about canonical quantization. 1074 01:22:38,638 --> 01:22:42,020 We're talking about attending non-normalizable modes, 1075 01:22:42,020 --> 01:22:46,452 because when we're adding on such a term to action. 1076 01:22:46,452 --> 01:22:49,368 AUDIENCE: But isn't that a field operator correspondence? 1077 01:22:49,368 --> 01:22:50,340 HONG LIU: Sorry? 1078 01:22:50,340 --> 01:22:53,270 AUDIENCE: Isn't that a field operator correspondence? 1079 01:22:53,270 --> 01:22:54,650 HONG LIU: Yeah, this is. 1080 01:22:54,650 --> 01:22:56,358 AUDIENCE: But these are non-normalizable. 1081 01:23:00,305 --> 01:23:02,410 HONG LIU: Yeah, that's what we, yeah, yeah, yeah. 1082 01:23:02,410 --> 01:23:03,230 Yeah. 1083 01:23:03,230 --> 01:23:05,740 So we're talking about the non-normalizable modes. 1084 01:23:05,740 --> 01:23:08,490 And the non-normalizable modes corresponding 1085 01:23:08,490 --> 01:23:11,290 to [INAUDIBLE] operator, and [INAUDIBLE] source 1086 01:23:11,290 --> 01:23:14,090 into an operator in the boundary series. 1087 01:23:14,090 --> 01:23:19,292 And just this relation shows that delta is this dimension. 1088 01:23:23,760 --> 01:23:26,024 Any other questions? 1089 01:23:26,024 --> 01:23:28,464 AUDIENCE: So then, when we quantize everything, 1090 01:23:28,464 --> 01:23:31,757 this operator O is dual to normalizable [INAUDIBLE], 1091 01:23:31,757 --> 01:23:32,590 or non-normalizable? 1092 01:23:35,530 --> 01:23:36,320 HONG LIU: No. 1093 01:23:36,320 --> 01:23:40,830 Here we are talking about the question-- here 1094 01:23:40,830 --> 01:23:43,460 we're talking about this relation 1095 01:23:43,460 --> 01:23:47,580 that if there is a boundary value of capital phi, which 1096 01:23:47,580 --> 01:23:49,310 is this boundary value, then that's 1097 01:23:49,310 --> 01:23:53,180 corresponding to add such a term to your Lagrangian. 1098 01:23:53,180 --> 01:23:55,720 And under such a correspondence, then we 1099 01:23:55,720 --> 01:24:03,030 deduce that this delta is the dimension of this operators. 1100 01:24:03,030 --> 01:24:04,800 It's the dimension of this operator. 1101 01:24:04,800 --> 01:24:08,520 And this is deduced from the non-normalizable perspective. 1102 01:24:08,520 --> 01:24:13,910 And in indeed, you can ask, say, suppose 1103 01:24:13,910 --> 01:24:20,700 I create normalizable modes using this O, 1104 01:24:20,700 --> 01:24:23,550 whether that is consistent with that. 1105 01:24:23,550 --> 01:24:26,550 Indeed, that you can double check. 1106 01:24:26,550 --> 01:24:27,396 Is it clear? 1107 01:24:40,750 --> 01:24:45,477 Anyway, so this is what we have right now. 1108 01:24:45,477 --> 01:24:47,310 So later, we will discuss the correspondence 1109 01:24:47,310 --> 01:24:50,040 for the normalizable modes. 1110 01:24:50,040 --> 01:24:51,517 And we worry about later. 1111 01:24:51,517 --> 01:24:53,350 Later, we will talk about the correspondence 1112 01:24:53,350 --> 01:24:54,530 for normalizable modes. 1113 01:24:54,530 --> 01:24:58,056 And then you will see it's compatible with this relation. 1114 01:25:04,050 --> 01:25:16,090 So now we find, for scalar, for a standard quantization, 1115 01:25:16,090 --> 01:25:28,330 for scalar, in the standard quantization, 1116 01:25:28,330 --> 01:25:32,890 for scalar in the standard quantization, 1117 01:25:32,890 --> 01:25:35,080 that the dimension of the operator 1118 01:25:35,080 --> 01:25:38,327 related to the mass in the gravity side 1119 01:25:38,327 --> 01:25:39,410 by the following relation. 1120 01:25:46,929 --> 01:25:48,762 [INAUDIBLE] to [INAUDIBLE] by this relation. 1121 01:25:52,530 --> 01:25:56,170 So we have found that explicit relation 1122 01:25:56,170 --> 01:26:03,134 between the conformal dimension and the mass. 1123 01:26:08,010 --> 01:26:13,910 So now, let me elaborate a little bit on this relation. 1124 01:26:19,190 --> 01:26:21,358 So let's consider, suppose the m equal to 0. 1125 01:26:24,160 --> 01:26:30,777 Well, m equal to 0, then you find the delta equal to d. 1126 01:26:36,300 --> 01:26:39,797 So delta equal to d, on the field theory side, 1127 01:26:39,797 --> 01:26:48,140 this is a marginal operator, and so we 1128 01:26:48,140 --> 01:26:51,890 see that the massless field on the gravity side 1129 01:26:51,890 --> 01:26:56,430 is mapped to a marginal operator. 1130 01:27:00,922 --> 01:27:03,130 So are you familiar with the terminology of marginal, 1131 01:27:03,130 --> 01:27:04,890 operator, relevant operator? 1132 01:27:04,890 --> 01:27:05,990 Et cetera. 1133 01:27:05,990 --> 01:27:06,973 Good. 1134 01:27:06,973 --> 01:27:11,810 AUDIENCE: Wait, can you remind us what is it? 1135 01:27:11,810 --> 01:27:13,990 HONG LIU: This is defined in terms of the RG flow. 1136 01:27:13,990 --> 01:27:16,467 Are you taking the quantum field theory too? 1137 01:27:16,467 --> 01:27:17,050 AUDIENCE: Yes. 1138 01:27:17,050 --> 01:27:18,841 We haven't quite gotten there. [INAUDIBLE]. 1139 01:27:18,841 --> 01:27:22,132 HONG LIU: Yeah, then you will get there, maybe, 1140 01:27:22,132 --> 01:27:22,840 in a week or two. 1141 01:27:25,182 --> 01:27:27,015 Yeah, certainly by the end of this semester. 1142 01:27:31,350 --> 01:27:34,420 Yeah, so let me just say one-- yeah 1143 01:27:34,420 --> 01:27:36,990 I will say a few more words. 1144 01:27:36,990 --> 01:27:39,240 And then you will see what it means. 1145 01:27:39,240 --> 01:27:43,040 Yeah, just I will say a few more words. 1146 01:27:43,040 --> 01:27:48,820 Anyway, so will m squared smaller than 0, 1147 01:27:48,820 --> 01:27:49,750 then he is negative. 1148 01:27:52,980 --> 01:27:57,030 Then this is d squared over 4 matter something. 1149 01:27:57,030 --> 01:28:00,220 So that means the delta is smaller than d. 1150 01:28:02,850 --> 01:28:04,700 When the dimension of the operator 1151 01:28:04,700 --> 01:28:08,020 is smaller than the spacetime dimension, 1152 01:28:08,020 --> 01:28:10,296 so that, from field theory, according to you, 1153 01:28:10,296 --> 01:28:16,180 is called the relevant operator. 1154 01:28:16,180 --> 01:28:22,360 And now, when m squared is greater than 0, 1155 01:28:22,360 --> 01:28:24,800 it's mapped to delta greater than d, 1156 01:28:24,800 --> 01:28:27,950 so this is mapped to an irrelevant operator. 1157 01:28:37,630 --> 01:28:39,410 AUDIENCE: Why do they have [INAUDIBLE]? 1158 01:28:39,410 --> 01:28:41,640 HONG LIU: Why do they have these names? 1159 01:28:41,640 --> 01:28:43,580 Marginal, relevant, or irrelevant? 1160 01:28:43,580 --> 01:28:46,070 Yeah, I will say a little bit about that. 1161 01:28:46,070 --> 01:28:47,946 Just wait one second. 1162 01:28:53,470 --> 01:29:00,590 So now, let's say what this means from the perspective 1163 01:29:00,590 --> 01:29:01,356 of this relation. 1164 01:29:10,136 --> 01:29:11,510 So from the field series side, we 1165 01:29:11,510 --> 01:29:18,320 have phi x O x mapped into-- so this phi should be identified 1166 01:29:18,320 --> 01:29:23,400 we A. So this is identified with this term, d minus delta. 1167 01:29:23,400 --> 01:29:24,770 OK, leading term. 1168 01:29:31,770 --> 01:29:33,360 OK, so now let me remind you what 1169 01:29:33,360 --> 01:29:35,806 this marginal and relevant, irrelevant means. 1170 01:29:42,160 --> 01:29:48,880 So operator marginal means, when you add such a term 1171 01:29:48,880 --> 01:29:52,490 to your Lagrangian, the effect of such a term 1172 01:29:52,490 --> 01:29:54,380 does not change with scale. 1173 01:29:54,380 --> 01:29:56,390 So whether you're looking at a very high energy, 1174 01:29:56,390 --> 01:29:58,150 looking at very low energy, the effect 1175 01:29:58,150 --> 01:30:00,770 does not change very much. 1176 01:30:00,770 --> 01:30:02,940 OK, so that's why it's called a marginal operator. 1177 01:30:02,940 --> 01:30:07,540 So in the regulation group, there's a very simple scaling 1178 01:30:07,540 --> 01:30:10,310 argument to show that, when the delta is equal to d, 1179 01:30:10,310 --> 01:30:13,010 and then the effect of the operator 1180 01:30:13,010 --> 01:30:16,758 does not change with the scale. 1181 01:30:21,380 --> 01:30:27,330 In particular, when you go to the UV, this does not change. 1182 01:30:27,330 --> 01:30:28,330 For a marginal operator. 1183 01:30:33,510 --> 01:30:36,950 When you go UV, say, when you go to higher and higher energies, 1184 01:30:36,950 --> 01:30:41,980 the effect of these operators remains the same, roughly. 1185 01:30:41,980 --> 01:30:45,930 So now, we know from this UV connection, 1186 01:30:45,930 --> 01:30:48,242 UV mapped on the gravity side to z 1187 01:30:48,242 --> 01:30:52,490 goes to 0 means we are approaching the boundary. 1188 01:30:52,490 --> 01:30:56,390 And now we see, precisely for d equal to delta, 1189 01:30:56,390 --> 01:30:58,740 this term is actually constant. 1190 01:30:58,740 --> 01:31:02,470 So go to smaller and smaller z, and the effect of this term 1191 01:31:02,470 --> 01:31:03,420 does not change. 1192 01:31:03,420 --> 01:31:04,990 It's just a constant. 1193 01:31:04,990 --> 01:31:06,398 So again, does not change. 1194 01:31:12,440 --> 01:31:15,024 So now, let's look at the relevant operator. 1195 01:31:17,750 --> 01:31:22,380 So the relevant operator, so in the field theory, 1196 01:31:22,380 --> 01:31:25,180 when you do the RG, the relevant and the irrelevant 1197 01:31:25,180 --> 01:31:33,464 means, whether this term becomes relevant when you go to the IR. 1198 01:31:33,464 --> 01:31:36,130 By relevant, means that the term becomes more and more important 1199 01:31:36,130 --> 01:31:39,480 when you to the low energies. 1200 01:31:39,480 --> 01:31:41,700 So that means that, for the relevant operator, when 1201 01:31:41,700 --> 01:31:44,320 you go t higher energies, it becomes 1202 01:31:44,320 --> 01:31:46,580 less and less important. 1203 01:31:46,580 --> 01:31:48,380 By definition, the relevant operator means, 1204 01:31:48,380 --> 01:31:49,530 when you go to low energies, it becomes 1205 01:31:49,530 --> 01:31:50,550 more and more important. 1206 01:31:50,550 --> 01:31:52,425 And it means, when you go to higher energies, 1207 01:31:52,425 --> 01:31:54,220 it become less and less important. 1208 01:31:54,220 --> 01:31:56,390 So it means that, when you go to UV, 1209 01:31:56,390 --> 01:31:58,700 the relevant operator becomes less and less important. 1210 01:32:05,910 --> 01:32:09,032 So now, let's look at this side. 1211 01:32:09,032 --> 01:32:10,490 The relevant operator corresponding 1212 01:32:10,490 --> 01:32:14,070 to delta smaller than d, then you have a positive power. 1213 01:32:17,200 --> 01:32:20,510 And that means, when z goes to 0, when you go to UV, 1214 01:32:20,510 --> 01:32:23,260 actually, this term goes to 0. 1215 01:32:23,260 --> 01:32:25,520 So this is precisely compatible with the field theory 1216 01:32:25,520 --> 01:32:29,090 behavior we expect when you add such a term to a Lagrangian. 1217 01:32:29,090 --> 01:32:30,590 And the effect of such a term should 1218 01:32:30,590 --> 01:32:34,980 be less and less important when you go to higher energies. 1219 01:32:34,980 --> 01:32:38,160 And here, we see a precise counterpart here. 1220 01:32:42,280 --> 01:32:47,120 And similarly, now, if you go through the irrelevant 1221 01:32:47,120 --> 01:32:47,680 operator. 1222 01:32:47,680 --> 01:32:50,796 Irrelevant means irrelevant when you go to IR. 1223 01:32:50,796 --> 01:32:53,260 When you go to UV means more and more important. 1224 01:33:01,910 --> 01:33:06,180 So that means-- but here, when delta is greater than d, 1225 01:33:06,180 --> 01:33:09,080 then these here have elective power. 1226 01:33:09,080 --> 01:33:11,270 So here, goes to infinity, become 1227 01:33:11,270 --> 01:33:13,580 bigger and bigger and, indeed, this term 1228 01:33:13,580 --> 01:33:16,390 becomes more and more important. 1229 01:33:16,390 --> 01:33:18,200 So we see a very nice correspondence 1230 01:33:18,200 --> 01:33:20,750 between this behavior we expect just 1231 01:33:20,750 --> 01:33:26,450 from the ordinary RG and the behavior of this term 1232 01:33:26,450 --> 01:33:28,300 in the gravity side. 1233 01:33:28,300 --> 01:33:30,500 So this is a very important self consistency check 1234 01:33:30,500 --> 01:33:39,318 of our matching, both of this relation and of this relation. 1235 01:33:54,850 --> 01:33:56,760 Any questions on this? 1236 01:33:56,760 --> 01:33:57,902 Yes. 1237 01:33:57,902 --> 01:34:00,698 AUDIENCE: What's the topology of AdS boundary? 1238 01:34:00,698 --> 01:34:01,630 HONG LIU: Sorry? 1239 01:34:01,630 --> 01:34:03,680 AUDIENCE: What's the topology of AdS boundary? 1240 01:34:03,680 --> 01:34:05,580 HONG LIU: It's a Minkowski space. 1241 01:34:05,580 --> 01:34:11,704 AUDIENCE: But it's 5-- it's 5 times something. 1242 01:34:11,704 --> 01:34:13,120 HONG LIU: It is a Minkowski space. 1243 01:34:13,120 --> 01:34:16,880 Boundary of AdS is just Minkowski space. 1244 01:34:16,880 --> 01:34:19,380 AUDIENCE: I have a question. 1245 01:34:19,380 --> 01:34:27,630 [INAUDIBLE] in RG why [INAUDIBLE] 1246 01:34:27,630 --> 01:34:30,800 HONG LIU: That take some story to do. 1247 01:34:30,800 --> 01:34:33,860 Yeah, that you have to do in the quantum field theory too. 1248 01:34:40,820 --> 01:34:44,830 Yeah, but it's very easy to understand 1249 01:34:44,830 --> 01:34:50,810 if you just know a little bit of free field theory. 1250 01:34:50,810 --> 01:34:52,930 And in the field theory point of view, 1251 01:34:52,930 --> 01:34:55,610 when the dimension of this operator 1252 01:34:55,610 --> 01:34:57,950 is smaller than the dimension of this operator, 1253 01:34:57,950 --> 01:35:00,140 and then this is dimensional, then this 1254 01:35:00,140 --> 01:35:02,020 has a positive mass dimension. 1255 01:35:02,020 --> 01:35:03,969 And this is, like, mass term. 1256 01:35:03,969 --> 01:35:06,010 And the mass term becomes more and more important 1257 01:35:06,010 --> 01:35:09,374 when you go to lower energies. 1258 01:35:09,374 --> 01:35:11,415 And that's important because the higher energies. 1259 01:35:11,415 --> 01:35:13,712 Yeah, just from dimensional analysis. 1260 01:35:13,712 --> 01:35:17,000 Yeah, it takes maybe five minutes 1261 01:35:17,000 --> 01:35:19,390 to explain very clearly. 1262 01:35:19,390 --> 01:35:21,160 But we don't have these five minutes. 1263 01:35:26,170 --> 01:35:29,070 OK, so let me just summarize. 1264 01:35:34,700 --> 01:35:38,612 So we have this relation. 1265 01:35:42,100 --> 01:35:45,375 Let me maybe just summarize here. 1266 01:35:45,375 --> 01:35:46,625 So we have the other relation. 1267 01:35:49,360 --> 01:35:49,980 So summary. 1268 01:35:56,530 --> 01:36:01,560 So we have phi due to go to some O. 1269 01:36:01,560 --> 01:36:08,670 So here, normalizable modes should 1270 01:36:08,670 --> 01:36:11,879 be considered different normalizable modes 1271 01:36:11,879 --> 01:36:13,295 corresponding to different states. 1272 01:36:16,850 --> 01:36:31,532 And here, non-normalizable modes is mapped to different actions. 1273 01:36:31,532 --> 01:36:33,490 So you have a different non-normalizable modes, 1274 01:36:33,490 --> 01:36:34,906 then you map to different actions. 1275 01:36:34,906 --> 01:36:37,010 You map to different series. 1276 01:36:37,010 --> 01:36:40,204 So in other words, map to different series. 1277 01:36:45,362 --> 01:36:47,320 So when you turn on the non-normalizable modes, 1278 01:36:47,320 --> 01:36:48,530 you are deforming the theory. 1279 01:36:51,920 --> 01:36:56,680 So just to elaborate, just more precisely, 1280 01:36:56,680 --> 01:37:04,500 in the standard quantization, which 1281 01:37:04,500 --> 01:37:17,660 is A modes are non-normalizable, when the A-modes are 1282 01:37:17,660 --> 01:37:21,290 non-normalizable, then A x. 1283 01:37:21,290 --> 01:37:28,035 On the gravity side, is mapped to this last term on the field 1284 01:37:28,035 --> 01:37:29,900 theory side. 1285 01:37:29,900 --> 01:37:34,560 If you have a non-0 x, A x mapped to the following term 1286 01:37:34,560 --> 01:37:37,470 in your action. 1287 01:37:37,470 --> 01:37:41,450 And then your mass, we deduce that this side 1288 01:37:41,450 --> 01:37:45,440 is mapped to the dimension, give the delta dimension 1289 01:37:45,440 --> 01:37:49,280 of this operator O. 1290 01:37:49,280 --> 01:37:52,440 So later we will explain, as part of this normalizable mode 1291 01:37:52,440 --> 01:37:57,090 story, so this B-- so A is non-normalizable 1292 01:37:57,090 --> 01:38:00,000 but B is normalizable. 1293 01:38:00,000 --> 01:38:05,305 So later, as we will explain, in the normalizable mode story, 1294 01:38:05,305 --> 01:38:09,660 is the B x here, we are actually mapped 1295 01:38:09,660 --> 01:38:13,760 to the expectation value of O. 1296 01:38:13,760 --> 01:38:15,620 So B is a normalizable mode, then, actually, 1297 01:38:15,620 --> 01:38:19,251 that maps to a expectation value of O 1298 01:38:19,251 --> 01:38:20,417 in the corresponding states. 1299 01:38:24,600 --> 01:38:27,630 So A mapped to how you deform the theory, 1300 01:38:27,630 --> 01:38:30,030 and the B just mapped to the state which 1301 01:38:30,030 --> 01:38:32,190 specify your expectation value. 1302 01:38:32,190 --> 01:38:33,656 So this, we'll see later. 1303 01:38:36,712 --> 01:38:39,170 And in alternative quantization, you just do it oppositely. 1304 01:38:41,760 --> 01:38:50,220 In alternative quantization, you just switch A and B. 1305 01:38:50,220 --> 01:38:57,520 In alternative quantization, you just switch A and B. 1306 01:38:57,520 --> 01:39:10,770 So now B maps to B x O. And A x maps to the expectation value. 1307 01:39:10,770 --> 01:39:13,980 And m now maps to d minus delta. 1308 01:39:16,630 --> 01:39:18,990 So now, the dimension of the operator is d minus delta 1309 01:39:18,990 --> 01:39:21,090 rather than delta. 1310 01:39:21,090 --> 01:39:23,710 So now, because you just take the different exponents. 1311 01:39:23,710 --> 01:39:26,285 It's the same thing, just take different exponents. 1312 01:39:26,285 --> 01:39:30,505 So now, the dimension of the operator is d minus delta. 1313 01:39:34,089 --> 01:39:36,130 So now, let me also make you some other examples. 1314 01:39:39,500 --> 01:39:41,940 So you can consider gauge fields. 1315 01:39:41,940 --> 01:39:48,780 You can consider, say, a Maxwell field in the gravity side. 1316 01:39:48,780 --> 01:39:52,250 Then that's [INAUDIBLE] dual to some conserved currents, 1317 01:39:52,250 --> 01:39:55,520 so besides earlier. 1318 01:39:55,520 --> 01:39:58,079 And then you can check, then again, you 1319 01:39:58,079 --> 01:40:00,120 just solve the Maxwell equation, which I will not 1320 01:40:00,120 --> 01:40:01,300 have time to do it here. 1321 01:40:04,130 --> 01:40:07,020 But I may ask you to do it in your p set. 1322 01:40:07,020 --> 01:40:09,880 It said A mu as z goes to infinity 1323 01:40:09,880 --> 01:40:14,960 becomes-- so the path of A parallel to the boundary 1324 01:40:14,960 --> 01:40:16,970 goes to a constant. 1325 01:40:16,970 --> 01:40:17,780 There's no z. 1326 01:40:17,780 --> 01:40:23,010 It's really a constant plus b mu some number to the z 1327 01:40:23,010 --> 01:40:23,990 to the d minus 2. 1328 01:40:26,820 --> 01:40:32,050 So this is a leading behavior in the case of the vector field. 1329 01:40:32,050 --> 01:40:34,320 So using the very similar scaling argument 1330 01:40:34,320 --> 01:40:36,410 I was doing there, you can deduce 1331 01:40:36,410 --> 01:40:42,780 that the operator due to A must have dimension d minus 1. 1332 01:40:42,780 --> 01:40:45,370 This I will leave to you as an exercise. 1333 01:40:45,370 --> 01:40:49,100 Just exactly parallel statement for the scalar case. 1334 01:40:49,100 --> 01:40:53,000 And with this symptotic behavior, you can show, 1335 01:40:53,000 --> 01:40:54,660 just using the argument, you can show 1336 01:40:54,660 --> 01:40:58,790 that the operator dual to A must have dimension d minus 1. 1337 01:41:01,870 --> 01:41:04,335 And this is precisely the dimension 1338 01:41:04,335 --> 01:41:07,420 a conserved current must have. 1339 01:41:07,420 --> 01:41:10,520 So conserved current because it's conserved. 1340 01:41:10,520 --> 01:41:14,052 So its dimension, it's actually protected. 1341 01:41:14,052 --> 01:41:15,760 So no matter in what kind of interaction, 1342 01:41:15,760 --> 01:41:17,718 no matter in what kind of case, you always have 1343 01:41:17,718 --> 01:41:20,580 the same dimension d minus 1. 1344 01:41:20,580 --> 01:41:23,650 So this precisely agrees with that. 1345 01:41:23,650 --> 01:41:24,833 So do you understand why? 1346 01:41:29,890 --> 01:41:31,900 AUDIENCE: Why. 1347 01:41:31,900 --> 01:41:33,570 HONG LIU: Yeah, the reason is simple. 1348 01:41:33,570 --> 01:41:42,186 It's just because j0, the dimension of the j0. 1349 01:41:42,186 --> 01:41:43,560 You can understand the following. 1350 01:41:43,560 --> 01:41:46,115 The j0 is the charge density. 1351 01:41:46,115 --> 01:41:47,990 And the charge density charge, by definition, 1352 01:41:47,990 --> 01:41:49,780 does not have a dimension. 1353 01:41:49,780 --> 01:41:52,670 And by definition the density will always 1354 01:41:52,670 --> 01:41:55,670 have a dimension minus 1, d minus 1. 1355 01:41:55,670 --> 01:41:57,777 And then the other components, they just 1356 01:41:57,777 --> 01:41:58,902 follow by the conservation. 1357 01:42:05,170 --> 01:42:08,886 Yes, so, indeed, this is, again, this is a consistency check. 1358 01:42:08,886 --> 01:42:10,260 AUDIENCE: So for this case, there 1359 01:42:10,260 --> 01:42:12,084 is no choice of quantization? 1360 01:42:12,084 --> 01:42:14,500 HONG LIU: Yeah, there's no choice for quantization, right. 1361 01:42:14,500 --> 01:42:21,412 Actually, in AdS four, in some special dimensions, 1362 01:42:21,412 --> 01:42:23,120 you can also do a different quantization. 1363 01:42:23,120 --> 01:42:26,810 But the story is a little bit subtle. 1364 01:42:26,810 --> 01:42:30,740 In some special dimensions, sometimes you can do it. 1365 01:42:30,740 --> 01:42:36,080 Yeah, in AdS four, you can do a different boundary condition. 1366 01:42:36,080 --> 01:42:38,530 So now let me also mention the stress tensor, 1367 01:42:38,530 --> 01:42:41,764 which is the other important one. 1368 01:42:41,764 --> 01:42:43,430 And we mentioned that the stress tensor, 1369 01:42:43,430 --> 01:42:52,730 which is due to the perturbation of the metric component. 1370 01:42:52,730 --> 01:42:58,170 So if you write the AdS metric as some function times 1371 01:42:58,170 --> 01:43:04,330 dz squared plus g mu mu dx mu dx mu, 1372 01:43:04,330 --> 01:43:06,190 so this is the part of the metric parallel 1373 01:43:06,190 --> 01:43:12,460 to the boundary, then we discussed, 1374 01:43:12,460 --> 01:43:15,110 last time, if g mu mu goes to the boundary as something 1375 01:43:15,110 --> 01:43:22,980 like this, then eta mu mu plus some perturbation, 1376 01:43:22,980 --> 01:43:27,060 then that means that, in the field theory site, 1377 01:43:27,060 --> 01:43:32,540 you're turning on a stress tensor perturbation. 1378 01:43:32,540 --> 01:43:35,080 So that's what we argued last time. 1379 01:43:35,080 --> 01:43:42,590 If the boundary value of your [INAUDIBLE] metric, say, 1380 01:43:42,590 --> 01:43:48,874 is perturbed, away from the Minkowski metric, 1381 01:43:48,874 --> 01:43:51,165 then corresponding from the field theory point of view, 1382 01:43:51,165 --> 01:43:54,100 we turn on a source for the stress tensor. 1383 01:43:54,100 --> 01:43:57,840 So again, by using this relation, 1384 01:43:57,840 --> 01:44:00,620 you can deduce that the stress tensor must 1385 01:44:00,620 --> 01:44:03,890 have dimension, the corresponding operator must 1386 01:44:03,890 --> 01:44:05,530 have dimension d. 1387 01:44:05,530 --> 01:44:07,910 Again, this gives you exercise for yourself. 1388 01:44:07,910 --> 01:44:10,850 And again, that is precisely the stress tensor. 1389 01:44:10,850 --> 01:44:12,850 The dimension should satisfy that stress tensor. 1390 01:44:12,850 --> 01:44:14,120 And again, because the stress tensor 1391 01:44:14,120 --> 01:44:15,650 is a conserved current et cetera, 1392 01:44:15,650 --> 01:44:17,875 always, no matter in what kind of series always 1393 01:44:17,875 --> 01:44:18,583 have dimension d. 1394 01:44:24,450 --> 01:44:26,780 So now, after I have discussed this relation, 1395 01:44:26,780 --> 01:44:30,790 let's now discuss how to compute the correlation functions. 1396 01:44:42,100 --> 01:44:45,580 So let's discuss how to compute correlation functions. 1397 01:44:45,580 --> 01:44:49,540 So the basic observable in the field theory, 1398 01:44:49,540 --> 01:44:52,330 even more so in the conformal field theory, 1399 01:44:52,330 --> 01:44:55,010 is just correlation functions of local operators. 1400 01:44:55,010 --> 01:44:57,450 It's an important part of the [INAUDIBLE]. 1401 01:44:57,450 --> 01:44:59,180 And in the field theory, typically, 1402 01:44:59,180 --> 01:45:01,690 when we can see the correlation function of local operators, 1403 01:45:01,690 --> 01:45:04,305 we use something called a generating function, though. 1404 01:45:08,220 --> 01:45:10,410 I hope you're familiar with this. 1405 01:45:10,410 --> 01:45:16,530 Say, again, in the rating function, 1406 01:45:16,530 --> 01:45:18,957 though-- so now let's consider Euclidean, 1407 01:45:18,957 --> 01:45:20,206 which is a little bit simpler. 1408 01:45:23,857 --> 01:45:26,440 So essentially, it's a Euclidean creating correlation function 1409 01:45:26,440 --> 01:45:30,593 by adding such a term to your field theory. 1410 01:45:38,200 --> 01:45:41,350 And if you want to calculate correlation functions of O, 1411 01:45:41,350 --> 01:45:43,710 then just take derivative with phi 1412 01:45:43,710 --> 01:45:45,730 and then satisfy equal to 0. 1413 01:45:45,730 --> 01:45:47,620 Then you get the correlation function of O. 1414 01:45:47,620 --> 01:45:49,161 So I hope you are familiar with this. 1415 01:45:54,110 --> 01:45:59,710 so here, you should think of phi and O as just some, really, 1416 01:45:59,710 --> 01:46:02,140 I have a collection of all possible operators, 1417 01:46:02,140 --> 01:46:03,930 even though I just write the single one. 1418 01:46:03,930 --> 01:46:07,100 You should think of them as a collection of all operators. 1419 01:46:07,100 --> 01:46:09,470 You can put anything here, stress 1420 01:46:09,470 --> 01:46:11,900 tensor, conserved current, any scalar operator, 1421 01:46:11,900 --> 01:46:12,650 any spin operator. 1422 01:46:12,650 --> 01:46:14,024 Yeah, you can put anything there. 1423 01:46:16,900 --> 01:46:25,210 So in a simpler situation, you just 1424 01:46:25,210 --> 01:46:27,740 don't turn on any operator. 1425 01:46:27,740 --> 01:46:32,199 You just take phi equal to 0. 1426 01:46:32,199 --> 01:46:33,490 So this is a simpler situation. 1427 01:46:33,490 --> 01:46:36,870 Then you just have the partition function. 1428 01:46:36,870 --> 01:46:39,500 So this correlation function is defined by the Euclidean path 1429 01:46:39,500 --> 01:46:41,067 integral, for example. 1430 01:46:41,067 --> 01:46:43,400 And you just insert this in the Euclidean path integral. 1431 01:46:46,412 --> 01:46:48,870 And when phi is equal to 0, this is just the Euclidean path 1432 01:46:48,870 --> 01:46:51,242 integral. 1433 01:46:51,242 --> 01:46:53,200 And this is just gives you a partition function 1434 01:46:53,200 --> 01:46:56,580 of the whole system. 1435 01:46:56,580 --> 01:47:00,470 And since we believe the duality should be 2, 1436 01:47:00,470 --> 01:47:02,090 then this partition function should 1437 01:47:02,090 --> 01:47:03,360 be exactly the same as the partition 1438 01:47:03,360 --> 01:47:04,568 function on the gravity side. 1439 01:47:08,199 --> 01:47:09,740 Suppose you can compute the partition 1440 01:47:09,740 --> 01:47:11,079 function from the gravity side. 1441 01:47:11,079 --> 01:47:12,120 Then it must be the same. 1442 01:47:12,120 --> 01:47:15,820 Otherwise, there's no equivalence. 1443 01:47:15,820 --> 01:47:18,150 So this is the partition function of the gravity side. 1444 01:47:26,280 --> 01:47:28,830 So these have to be true. 1445 01:47:28,830 --> 01:47:42,830 So now, with this relation, again, we 1446 01:47:42,830 --> 01:47:46,340 are right in the heuristic form that some operator O 1447 01:47:46,340 --> 01:47:50,820 is due to some [INAUDIBLE] phi. 1448 01:47:50,820 --> 01:47:56,180 And the source for the O is due to the boundary value of phi-- 1449 01:47:56,180 --> 01:47:58,382 again, I write in the heuristic form. 1450 01:47:58,382 --> 01:47:59,840 So the boundary value of phi should 1451 01:47:59,840 --> 01:48:01,300 be understood in that limit. 1452 01:48:04,810 --> 01:48:08,760 Because of this, then that means we 1453 01:48:08,760 --> 01:48:19,730 must have-- if we turn on a [INAUDIBLE] or a phi, 1454 01:48:19,730 --> 01:48:22,480 deforming the series by this phi, 1455 01:48:22,480 --> 01:48:25,890 the mass corresponding to the gravity side of the partition 1456 01:48:25,890 --> 01:48:33,460 function, with this phi, have the boundary value 1457 01:48:33,460 --> 01:48:36,370 given by phi. 1458 01:48:36,370 --> 01:48:44,720 Then it must be the same because we have identified 1459 01:48:44,720 --> 01:48:48,770 turning on the boundary value to turning on the operator, 1460 01:48:48,770 --> 01:48:53,660 turning on this thing in the field series. 1461 01:48:53,660 --> 01:48:58,880 So we have now related the generating 1462 01:48:58,880 --> 01:49:04,450 functional to the gravity partition 1463 01:49:04,450 --> 01:49:06,430 function with some non-normalizable boundary 1464 01:49:06,430 --> 01:49:06,930 conditions. 1465 01:49:16,220 --> 01:49:25,370 So, in general, give me one more minute. 1466 01:49:25,370 --> 01:49:30,387 So in general, this equation is actually empty 1467 01:49:30,387 --> 01:49:32,595 because we don't know how to define the gravity site. 1468 01:49:36,190 --> 01:49:38,920 Full quantum gravity partition function. 1469 01:49:38,920 --> 01:49:40,870 What are you talking about? 1470 01:49:40,870 --> 01:49:43,340 If you don't know how to do that. 1471 01:49:43,340 --> 01:49:56,590 But we know how to compute this guy in the semiclassical limit, 1472 01:49:56,590 --> 01:49:58,981 which is the string coupling goes to 0, 1473 01:49:58,981 --> 01:50:00,480 and the other prime goes to 0 limit. 1474 01:50:04,260 --> 01:50:09,270 So in this limit, we can just write 1475 01:50:09,270 --> 01:50:14,060 the gravity path in function, essentially 1476 01:50:14,060 --> 01:50:16,812 as the Euclidean path integral of all gravity fields. 1477 01:50:19,860 --> 01:50:21,750 Again, I'm using this heuristic form 1478 01:50:21,750 --> 01:50:23,083 to write the boundary condition. 1479 01:50:23,083 --> 01:50:25,135 But it should be understood in that form. 1480 01:50:27,820 --> 01:50:31,520 And again, I'm by integrating over phi, 1481 01:50:31,520 --> 01:50:34,354 you should imagine integrating of all gravity fields. 1482 01:50:38,340 --> 01:50:44,030 So now, in the limit, so this SE is proportional to 1 1483 01:50:44,030 --> 01:50:47,610 over kappa squared, which is the Newton constant. 1484 01:50:47,610 --> 01:50:49,790 So in the semiclassical limit, in this limit, 1485 01:50:49,790 --> 01:50:52,542 the kappa squared goes to 0. 1486 01:50:52,542 --> 01:50:54,500 So you can actually evaluate this path integral 1487 01:50:54,500 --> 01:50:55,958 using the site point approximation. 1488 01:51:02,590 --> 01:51:11,200 So as the leading order, so this gravity 1489 01:51:11,200 --> 01:51:13,240 can actually be evaluated. 1490 01:51:13,240 --> 01:51:15,650 So leading order in the semiclassical limit, 1491 01:51:15,650 --> 01:51:18,130 you can just evaluate it by site point approximation, 1492 01:51:18,130 --> 01:51:20,950 then this is just given by SE. 1493 01:51:20,950 --> 01:51:35,130 Evaluate it as a classical solution 1494 01:51:35,130 --> 01:51:38,350 which satisfies the appropriate boundary conditions. 1495 01:51:38,350 --> 01:51:42,320 So that means, actually, in this limit, 1496 01:51:42,320 --> 01:51:46,170 we can actually evaluate this explicitly. 1497 01:51:46,170 --> 01:51:47,660 And the nice thing is, as we said 1498 01:51:47,660 --> 01:51:49,330 before, the semiclassical limit actually 1499 01:51:49,330 --> 01:51:54,200 corresponding to the [INAUDIBLE] limit in the field theory. 1500 01:51:54,200 --> 01:51:58,480 So we can actually compute explicitly 1501 01:51:58,480 --> 01:52:00,540 the generating functional in the [INAUDIBLE] 1502 01:52:00,540 --> 01:52:03,510 of the limit over the field theory. 1503 01:52:03,510 --> 01:52:05,060 So let's stop here.