1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,122 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,122 --> 00:00:17,084 at ocw.mit.edu. 8 00:00:21,625 --> 00:00:22,750 HONG LIU: And let us start. 9 00:00:25,940 --> 00:00:31,040 So let me again start by reminding you 10 00:00:31,040 --> 00:00:31,995 what we did before. 11 00:00:35,320 --> 00:00:42,170 So consider, say, we have a scalar field in AdS 12 00:00:42,170 --> 00:00:45,170 So let me just summarize what we did before. 13 00:00:49,900 --> 00:00:59,360 So consider, say, a scalar field, 14 00:00:59,360 --> 00:01:09,350 phi, in AdS and with mass times square, so with mass 15 00:01:09,350 --> 00:01:13,130 squared given by m squared, then we know, 16 00:01:13,130 --> 00:01:15,725 when you go to the boundary of AdS-- so let's 17 00:01:15,725 --> 00:01:22,010 take z goes to 0-- then phi will have 18 00:01:22,010 --> 00:01:26,080 the following asymptotic behavior. 19 00:01:26,080 --> 00:01:33,120 So these are the two independent modes of phi [? to ?] infinity. 20 00:01:46,060 --> 00:01:48,760 Of course, there are higher order corrections in z. 21 00:01:51,470 --> 00:01:59,170 And this delta is given by 1/2 d plus nu. 22 00:01:59,170 --> 00:02:02,310 And nu is given by this square divided 23 00:02:02,310 --> 00:02:05,326 by 4 plus m squared r squared. 24 00:02:10,315 --> 00:02:11,314 So this is the behavior. 25 00:02:16,210 --> 00:02:24,625 And so this phi is assumed to be dual to some boundary, scalar 26 00:02:24,625 --> 00:02:25,125 operator. 27 00:02:28,290 --> 00:02:29,960 And then all of those quantities, 28 00:02:29,960 --> 00:02:32,010 on the gravity side, they all have counterparts 29 00:02:32,010 --> 00:02:34,050 on field theory side. 30 00:02:34,050 --> 00:02:39,340 For example, the delta, which, essentially, 31 00:02:39,340 --> 00:02:42,470 is a function of mass, then, essentially, it's 32 00:02:42,470 --> 00:02:51,410 the scaling dimension of O. It's a scaling dimension of O. 33 00:02:51,410 --> 00:02:54,730 And this Ax essentially translates 34 00:02:54,730 --> 00:02:58,000 into the source for phi. 35 00:02:58,000 --> 00:03:02,330 Or you can consider the momentum space version, say, you 36 00:03:02,330 --> 00:03:06,080 can do Fourier transform, Ak, then go to the phi k. 37 00:03:09,630 --> 00:03:16,410 And then we also showed, last time, that the 2 nu times Bx 38 00:03:16,410 --> 00:03:23,420 is actually related to the expectation value of O. 39 00:03:23,420 --> 00:03:25,815 And again, there's a momentum space version of it. 40 00:03:25,815 --> 00:03:27,440 So you can just Fourier transform, then 41 00:03:27,440 --> 00:03:28,910 become O k and Bk. 42 00:03:34,300 --> 00:03:36,160 And in particular, in the example, 43 00:03:36,160 --> 00:03:37,760 we can see the scalar field example. 44 00:03:37,760 --> 00:03:39,759 In particular, in the example, you can see that. 45 00:03:52,510 --> 00:03:57,850 So we derived this relation from a free scalar field. 46 00:04:00,959 --> 00:04:03,500 but you can actually show this relation is actually generally 47 00:04:03,500 --> 00:04:05,560 true at non-linear level. 48 00:04:05,560 --> 00:04:09,330 So if you also include the high orders, et cetera, so including 49 00:04:09,330 --> 00:04:12,260 non-linear dependence on phi, actually this relation 50 00:04:12,260 --> 00:04:15,305 remains true. 51 00:04:15,305 --> 00:04:19,670 But due to the time, we will not go into that. 52 00:04:19,670 --> 00:04:24,410 The proof is actually not very difficult. 53 00:04:24,410 --> 00:04:26,440 So you can see that the free scalar examples-- 54 00:04:26,440 --> 00:04:28,273 so, in the example, we can see that actually 55 00:04:28,273 --> 00:04:35,360 the B-- so in momentum space, the B is actually 56 00:04:35,360 --> 00:04:44,485 proportional to A. If you remember what we did last time. 57 00:04:44,485 --> 00:04:46,210 So in other words, this one-point 58 00:04:46,210 --> 00:04:49,890 function-- so B is related to the one-point function. 59 00:04:49,890 --> 00:04:52,080 So in other words, the one-point function 60 00:04:52,080 --> 00:04:55,090 it's proportional to the source. 61 00:04:55,090 --> 00:04:56,540 So this is the standard story. 62 00:05:00,630 --> 00:05:04,040 Say all x equal to 0 when phi equal to 0. 63 00:05:06,820 --> 00:05:09,540 So if you don't have the source, then this one-point function 64 00:05:09,540 --> 00:05:10,940 is 0. 65 00:05:10,940 --> 00:05:17,830 So you don't have a spontaneous expectation value for the O. 66 00:05:17,830 --> 00:05:23,640 And now the fact that B is proportional to A, 67 00:05:23,640 --> 00:05:36,640 so this is just reflects that, in the presence of the source, 68 00:05:36,640 --> 00:05:40,430 then, of course, the expectation value-- so 69 00:05:40,430 --> 00:05:44,020 let me write it in momentum space for simplicity. 70 00:05:44,020 --> 00:05:45,400 Then we're no longer [INAUDIBLE]. 71 00:05:50,530 --> 00:05:54,470 Assume the source is small, then you can expand the expectation 72 00:05:54,470 --> 00:05:57,880 value in power series of phi. 73 00:05:57,880 --> 00:06:00,590 So to the 0, so that would be phi, and then 74 00:06:00,590 --> 00:06:01,610 phi squared, et cetera. 75 00:06:06,440 --> 00:06:10,420 So at linearized level, say we just find 76 00:06:10,420 --> 00:06:12,226 O k is proportional to phi k. 77 00:06:20,040 --> 00:06:24,870 We just find that O k is proportional to phi k. 78 00:06:24,870 --> 00:06:27,740 And the proportional constant is, in fact, 79 00:06:27,740 --> 00:06:28,740 the two-point function. 80 00:06:36,090 --> 00:06:38,979 So at linear level, so O k would be just proportional to phi k, 81 00:06:38,979 --> 00:06:41,520 but the proportional constant is just the two-point function. 82 00:06:45,782 --> 00:06:46,990 So the reason is very simple. 83 00:06:52,576 --> 00:06:54,200 First, let me write down the definition 84 00:06:54,200 --> 00:06:57,120 of this two-point function. 85 00:06:57,120 --> 00:06:59,560 So GEx in coordinate space is defined 86 00:06:59,560 --> 00:07:06,115 to be O x of 0, because of translational symmetry, 87 00:07:06,115 --> 00:07:08,050 so I can put the other point to be 0. 88 00:07:08,050 --> 00:07:12,007 So this is only the function of one variable, x. 89 00:07:12,007 --> 00:07:13,465 And then you can Fourier transform, 90 00:07:13,465 --> 00:07:15,196 and then you get GEk. 91 00:07:15,196 --> 00:07:16,570 So this is how we define the GEk. 92 00:07:20,050 --> 00:07:24,920 So to see that this is true, you need 93 00:07:24,920 --> 00:07:28,800 to remember that the GEk is given 94 00:07:28,800 --> 00:07:39,290 from taking the generating functional, delta phi k 95 00:07:39,290 --> 00:07:44,570 and delta phi minus k, then you set phi equal to 0. 96 00:07:44,570 --> 00:07:50,002 So this is the definition of the two-point function. 97 00:07:50,002 --> 00:07:52,376 And then next, you can see that the [? one derivative ?] 98 00:07:52,376 --> 00:07:59,110 of phi minus k-- so you can see that this takes 1 derivative 99 00:07:59,110 --> 00:07:59,860 on phi minus k. 100 00:07:59,860 --> 00:08:04,610 That just gives you the one-point function 101 00:08:04,610 --> 00:08:07,650 in the presence of phi and then delta phi k. 102 00:08:10,440 --> 00:08:13,060 And take phi equal to 0. 103 00:08:13,060 --> 00:08:15,140 So I just evaluated this derivative. 104 00:08:15,140 --> 00:08:17,679 So if you take this generating functional, 105 00:08:17,679 --> 00:08:19,845 take one derivative, we just get one-point function. 106 00:08:19,845 --> 00:08:23,620 And then this is the other point function. 107 00:08:23,620 --> 00:08:27,650 Then at the linear level-- so if O k 108 00:08:27,650 --> 00:08:29,980 is linear in phi k, of course, taking the derivative 109 00:08:29,980 --> 00:08:32,960 is just the same as dividing it. 110 00:08:32,960 --> 00:08:33,970 So this is the same. 111 00:08:39,409 --> 00:08:42,100 So continue over. 112 00:08:42,100 --> 00:08:49,290 This is the same just as to the phi k divided by phi k 113 00:08:49,290 --> 00:08:50,610 at the linearized level. 114 00:08:50,610 --> 00:08:54,062 And then so this tells you that this pre-factor is just 115 00:08:54,062 --> 00:08:55,020 the two-point function. 116 00:09:02,340 --> 00:09:06,160 And so we can now write a two-point function explicitly 117 00:09:06,160 --> 00:09:08,760 in terms of B and A, so this is just given, 118 00:09:08,760 --> 00:09:12,690 because this is 2 nu Bk. 119 00:09:12,690 --> 00:09:17,080 So this is just given by that. 120 00:09:22,080 --> 00:09:23,830 So this is just a quick summary of what we 121 00:09:23,830 --> 00:09:25,670 did at the end of last lecture. 122 00:09:30,000 --> 00:09:33,870 So that's how you can find the two-point function. 123 00:09:33,870 --> 00:09:37,480 So you work out what B is, what the A is, 124 00:09:37,480 --> 00:09:39,770 and then the rest of them is the two-point function. 125 00:09:42,350 --> 00:09:47,527 And the way we work out the B and A is the following. 126 00:09:47,527 --> 00:09:50,110 So at the linearized level, when you solve the equation motion 127 00:09:50,110 --> 00:09:54,580 for phi, then you impose the boundary conditions 128 00:09:54,580 --> 00:09:59,604 at infinity, A should be equal to phi. 129 00:09:59,604 --> 00:10:01,020 But then there's another condition 130 00:10:01,020 --> 00:10:03,290 being proposed to make sure the phi is actually 131 00:10:03,290 --> 00:10:05,320 regular in the interior. 132 00:10:05,320 --> 00:10:08,700 So the regularity condition, there's still 133 00:10:08,700 --> 00:10:12,900 a relation between A and B. And then 134 00:10:12,900 --> 00:10:17,950 that leads to this two-point function. 135 00:10:17,950 --> 00:10:18,870 Any questions on this? 136 00:10:21,630 --> 00:10:24,060 Is everything clear? 137 00:10:24,060 --> 00:10:25,364 Good. 138 00:10:25,364 --> 00:10:27,405 So now let's consider the higher-point functions. 139 00:10:50,260 --> 00:10:54,610 So this is, in principle, straightforward to do. 140 00:10:54,610 --> 00:11:05,110 Because, recall, that the log CFT phi, this generating 141 00:11:05,110 --> 00:11:09,700 functional, in the field theory, is just 142 00:11:09,700 --> 00:11:11,640 given by the classical action on the gravity 143 00:11:11,640 --> 00:11:16,770 side, which is the boundary condition given by phi. 144 00:11:19,719 --> 00:11:21,260 You're solving the classical equation 145 00:11:21,260 --> 00:11:23,170 motion, the initial action. 146 00:11:26,730 --> 00:11:29,640 Then you just, as we did earlier, 147 00:11:29,640 --> 00:11:32,590 you just solve the equation motion for phi 148 00:11:32,590 --> 00:11:35,110 to the [? linear ?] order. 149 00:11:35,110 --> 00:11:37,940 And then you validated the initial action. 150 00:11:41,890 --> 00:11:44,890 Then you can express, for example, this SE 151 00:11:44,890 --> 00:11:46,700 in power series of phi. 152 00:11:46,700 --> 00:11:52,950 And then the quotient for the power series of this phi 153 00:11:52,950 --> 00:11:56,540 then just gives you correlation functions. 154 00:11:56,540 --> 00:11:59,620 So the principle of this is straightforward to do. 155 00:11:59,620 --> 00:12:02,620 We know how to solve the classical equation motion. 156 00:12:02,620 --> 00:12:06,100 We know how to plug back into the action, 157 00:12:06,100 --> 00:12:08,950 and then you are done. 158 00:12:08,950 --> 00:12:11,522 So just to be able to be more specific, 159 00:12:11,522 --> 00:12:12,605 let's consider an example. 160 00:12:21,660 --> 00:12:25,671 Let's just consider two examples. 161 00:12:25,671 --> 00:12:27,545 Again, let we just consider the scalar field. 162 00:12:35,450 --> 00:12:36,590 Now, I add the linear term. 163 00:12:43,670 --> 00:12:47,780 For illustration, let me give you two examples. 164 00:12:47,780 --> 00:12:52,870 And so a very important thing is that in our set-up, the lambda, 165 00:12:52,870 --> 00:12:57,440 this cubic quotient is proportional to kappa. 166 00:12:57,440 --> 00:12:59,830 And then if you translate into the field series side, 167 00:12:59,830 --> 00:13:02,180 this is 1/n. 168 00:13:02,180 --> 00:13:06,300 So remember, for anything in the gravity action, 169 00:13:06,300 --> 00:13:08,990 you always have the 1 over kappa squared, 1 170 00:13:08,990 --> 00:13:12,370 over G Newton before [INAUDIBLE] action. 171 00:13:12,370 --> 00:13:16,870 And then when we try to change phi into canonical 172 00:13:16,870 --> 00:13:20,370 normalized, and then you rescale phi. 173 00:13:20,370 --> 00:13:24,560 And the consequence is that this cubic term or quartic term, 174 00:13:24,560 --> 00:13:28,820 then they will have quotients proportionate to kappa. 175 00:13:28,820 --> 00:13:32,080 And according to this rule, if you have a quartic term, then 176 00:13:32,080 --> 00:13:35,070 that will be proportional to kappa squared, et cetera. 177 00:13:35,070 --> 00:13:39,774 And so the key is that as far as [? phi is ?] order 1, 178 00:13:39,774 --> 00:13:41,940 then you can actually treat those higher-order terms 179 00:13:41,940 --> 00:13:44,760 by perturbation because the quotient is small. 180 00:13:51,667 --> 00:13:53,750 And of course, you can also consider the situation 181 00:13:53,750 --> 00:13:55,208 in which phi is large, and then you 182 00:13:55,208 --> 00:13:57,072 have to solve the full nonlinear problem. 183 00:13:57,072 --> 00:13:59,930 Then you cannot do a perturbative expansion. 184 00:13:59,930 --> 00:14:04,080 You have to do the problem. 185 00:14:04,080 --> 00:14:05,740 So for our purpose, if you want to find 186 00:14:05,740 --> 00:14:08,080 the higher-point function for phi, 187 00:14:08,080 --> 00:14:09,790 then you need to solve this equation. 188 00:14:14,940 --> 00:14:16,700 So box phi is just a Laplace operator. 189 00:14:19,228 --> 00:14:20,860 I just save time. 190 00:14:25,650 --> 00:14:28,660 So you need to solve this equation. 191 00:14:28,660 --> 00:14:44,090 And with the boundary condition still the boundary condition, 192 00:14:44,090 --> 00:14:46,810 the boundary value of capital phi to equal to this phi x. 193 00:14:50,110 --> 00:14:53,900 And let me call this equation star. 194 00:14:53,900 --> 00:14:56,690 So as I said, because this lambda is small, 195 00:14:56,690 --> 00:15:00,744 you can actually try to solve this equation just 196 00:15:00,744 --> 00:15:01,785 perturbatively in lambda. 197 00:15:04,320 --> 00:15:08,720 And essentially, you're solving it perturbatively in this phi, 198 00:15:08,720 --> 00:15:11,440 because everything, in the end, will be expressed in this phi. 199 00:15:23,360 --> 00:15:34,850 So one can solve star perturbatively 200 00:15:34,850 --> 00:15:40,760 in phi, which is the boundary value of the field. 201 00:15:40,760 --> 00:15:47,822 And say phi c, you can expand it as phi 1 plus phi 2, et cetera, 202 00:15:47,822 --> 00:15:49,280 so the classical solution expanded. 203 00:15:49,280 --> 00:15:50,350 And this linear in phi. 204 00:15:53,780 --> 00:16:01,010 And this is quadratic in phi-- and et cetera. 205 00:16:03,990 --> 00:16:10,130 And then when you plug back this into the action, 206 00:16:10,130 --> 00:16:13,240 then you find on the initial action 207 00:16:13,240 --> 00:16:16,310 can also be expanded in power series of this phi. 208 00:16:16,310 --> 00:16:22,365 So you will have S2 phi plus-- started with quadratic order-- 209 00:16:22,365 --> 00:16:27,030 and 3 phi, et cetera. 210 00:16:27,030 --> 00:16:31,270 And then the quotient here gives you the two-point function. 211 00:16:31,270 --> 00:16:33,999 And then quotient here gives you the three-point function. 212 00:16:33,999 --> 00:16:36,540 And the quotient here gives a four-point function, et cetera. 213 00:16:46,432 --> 00:16:48,045 I hope the procedure is clear. 214 00:16:56,207 --> 00:16:57,415 Any questions regarding this? 215 00:17:06,068 --> 00:17:07,859 So even though this is very straightforward 216 00:17:07,859 --> 00:17:09,800 to do, conceptually, but in practice 217 00:17:09,800 --> 00:17:16,720 that it not what we would do, because this is a very tedious. 218 00:17:16,720 --> 00:17:18,859 It's very tedious. 219 00:17:18,859 --> 00:17:21,770 And our knowledge of quantum field theory 220 00:17:21,770 --> 00:17:26,290 gives us something much simpler to deal with. 221 00:17:26,290 --> 00:17:31,410 Because this, essentially, if you think about it, 222 00:17:31,410 --> 00:17:35,790 is not very different from doing a correlation function 223 00:17:35,790 --> 00:17:39,130 calculation, just in the flat-space quantum field 224 00:17:39,130 --> 00:17:40,320 theory. 225 00:17:40,320 --> 00:17:42,430 And there, whey you calculate the Green function 226 00:17:42,430 --> 00:17:46,380 in flat-space-- in flat-space quantum field 227 00:17:46,380 --> 00:17:48,810 theory, what do you do? 228 00:17:48,810 --> 00:17:51,509 Do we try to solve the classical equation motion and you iterate 229 00:17:51,509 --> 00:17:52,050 the equation? 230 00:17:52,050 --> 00:17:52,907 What do you do? 231 00:17:52,907 --> 00:17:54,100 AUDIENCE: Draw a diagram. 232 00:17:54,100 --> 00:17:56,270 HONG LIU: Yeah, you use Feynman diagrams. 233 00:17:56,270 --> 00:17:58,680 So here, it's exactly the same thing. 234 00:17:58,680 --> 00:18:04,720 So here it's much easier if you just do the Feynman diagrams. 235 00:18:04,720 --> 00:18:08,810 So let me first remind you what we do in the flat-space. 236 00:18:13,440 --> 00:18:15,740 So let's now go to the field theory 237 00:18:15,740 --> 00:18:24,040 one, standard, ordinary flat-space QFT. 238 00:18:24,040 --> 00:18:29,570 So now let's consider the lambda phi cubed theory in flat-space 239 00:18:29,570 --> 00:18:33,040 Just the same theory now using the flat-space. 240 00:18:36,710 --> 00:18:42,131 Let me call this star-star. 241 00:18:42,131 --> 00:18:44,005 So essentially, let's just consider star-star 242 00:18:44,005 --> 00:18:51,486 in flat-space, now in flat Euclidean space. 243 00:19:03,080 --> 00:19:05,680 So if we are given a field theory like that, 244 00:19:05,680 --> 00:19:08,790 so how would you calculate the following correlation function? 245 00:19:08,790 --> 00:19:15,600 Say, suppose I want to calculate phi x1, phi xn. 246 00:19:15,600 --> 00:19:19,495 So now this x1, xn, they're just flat-space coordinates. 247 00:19:23,180 --> 00:19:25,150 To distinguish them, let me just call y1, 248 00:19:25,150 --> 00:19:30,470 yn, just not to confuse about the AdS coordinate. 249 00:19:30,470 --> 00:19:37,120 So in flat-space with y, y mu, OK? 250 00:19:37,120 --> 00:19:40,854 In flat-space with coordinate y mu. 251 00:19:40,854 --> 00:19:43,020 So now suppose you want to calculate the correlation 252 00:19:43,020 --> 00:19:43,770 function for this. 253 00:19:48,260 --> 00:19:49,110 What do you do? 254 00:19:53,980 --> 00:19:55,330 So what do you do? 255 00:19:55,330 --> 00:19:55,990 It's easy. 256 00:19:55,990 --> 00:20:05,980 What you do is that this theory has a propagator, G, 257 00:20:05,980 --> 00:20:07,530 [? with ?] [? arrow ?] propagator. 258 00:20:07,530 --> 00:20:11,270 And also, this theory have a interaction vertex, 259 00:20:11,270 --> 00:20:14,320 which is controlled by lambda. 260 00:20:14,320 --> 00:20:21,280 And then, when you calculate such endpoint functions, 261 00:20:21,280 --> 00:20:24,310 you just fix your endpoint. 262 00:20:24,310 --> 00:20:26,680 So now remember, this is slightly different than what 263 00:20:26,680 --> 00:20:30,549 you would normally do in field theory one, 264 00:20:30,549 --> 00:20:32,340 because, here, we're doing coordinate space 265 00:20:32,340 --> 00:20:34,360 rather than momentum space. 266 00:20:34,360 --> 00:20:37,800 In field theory one, you're more used to doing momentum space. 267 00:20:37,800 --> 00:20:46,360 So here what you do is that you fix y1, y2 y3, extends to yn. 268 00:20:46,360 --> 00:20:51,930 So this is essentially the location of your insertion, 269 00:20:51,930 --> 00:20:53,820 of your field insertion. 270 00:20:53,820 --> 00:20:56,240 So essentially, imagine there's a source there. 271 00:20:56,240 --> 00:21:01,230 And then you just connect all the extended points 272 00:21:01,230 --> 00:21:09,250 by propagators and with these kind of interaction vertices. 273 00:21:09,250 --> 00:21:19,940 So for example, you can have something like this, et cetera. 274 00:21:19,940 --> 00:21:23,830 And this connects to some boundary point, 275 00:21:23,830 --> 00:21:26,870 some other point, et cetera-- say y's. 276 00:21:29,590 --> 00:21:31,940 And this also have lines. 277 00:21:31,940 --> 00:21:34,810 We just draw all possible diagrams, 278 00:21:34,810 --> 00:21:37,840 and you calculate that diagram. 279 00:21:37,840 --> 00:21:42,320 You calculate those diagrams, and they give you this thing. 280 00:21:42,320 --> 00:21:44,846 And essentially, the diagrams automatically 281 00:21:44,846 --> 00:21:46,220 give you the iteration procedure. 282 00:21:49,490 --> 00:21:51,400 So now, back to AdS, essentially, we 283 00:21:51,400 --> 00:21:53,000 can just do exactly the same thing. 284 00:22:00,989 --> 00:22:02,530 Just remind yourself of the procedure 285 00:22:02,530 --> 00:22:06,510 of calculating this and what we are going to do here, 286 00:22:06,510 --> 00:22:10,790 and you easily convince yourself. 287 00:22:10,790 --> 00:22:13,620 So it requires a couple of minutes' thinking, 288 00:22:13,620 --> 00:22:16,820 but we'll leave it to yourself. 289 00:22:16,820 --> 00:22:20,250 Actually, that procedure is no different from just doing 290 00:22:20,250 --> 00:22:21,380 the calculation like this. 291 00:22:29,150 --> 00:22:30,975 Of course, the difference is that now we 292 00:22:30,975 --> 00:22:33,804 are in the curved spacetime rather than flat spacetime. 293 00:22:33,804 --> 00:22:35,345 But there's another major difference. 294 00:22:45,910 --> 00:22:49,800 So here, the source all lies in the interior of your spacetime, 295 00:22:49,800 --> 00:22:52,560 where you inserted the operator. 296 00:22:52,560 --> 00:22:56,470 But here, all the source, phi x, they lie at the boundary. 297 00:23:07,157 --> 00:23:08,865 That's, essentially, the only difference. 298 00:23:12,520 --> 00:23:16,630 So if I just schematically-- so this is AdS. 299 00:23:16,630 --> 00:23:19,184 This is the boundary. 300 00:23:19,184 --> 00:23:21,600 And essentially, all the points are lying on the boundary, 301 00:23:21,600 --> 00:23:25,020 like we have x1, x2. 302 00:23:25,020 --> 00:23:29,720 So let's consider four-point function. 303 00:23:29,720 --> 00:23:34,770 Because each one of them is labeled by the boundary point. 304 00:23:34,770 --> 00:23:39,720 And then for the four-point function, you just connect. 305 00:23:39,720 --> 00:23:43,810 So you can connect the diagram like this. 306 00:23:43,810 --> 00:23:45,960 Again, you have the vertices, which 307 00:23:45,960 --> 00:23:48,180 is the [? cube ?] of vertices. 308 00:23:48,180 --> 00:23:51,690 But your endpoint is all lying on the boundary. 309 00:23:57,160 --> 00:23:59,260 So this structure also tells you, actually, 310 00:23:59,260 --> 00:24:02,070 there's two types of propagators. 311 00:24:02,070 --> 00:24:06,120 Because one type of propagator connects the two bulk points. 312 00:24:06,120 --> 00:24:08,150 And then there's this kind of propagator 313 00:24:08,150 --> 00:24:12,150 which connects the bulk point into the boundary points. 314 00:24:12,150 --> 00:24:14,870 So this is the difference, one major difference, 315 00:24:14,870 --> 00:24:16,360 from the flat-space case. 316 00:24:16,360 --> 00:24:20,330 So here, all the propagators are the same. 317 00:24:20,330 --> 00:24:23,491 So here, the difference is now, you also 318 00:24:23,491 --> 00:24:25,740 have propagators which come from boundary to the bulk. 319 00:24:28,376 --> 00:24:30,500 So in this case, there are two kind of propagators. 320 00:24:44,770 --> 00:24:50,990 Here, you have two type of propagators. 321 00:24:58,710 --> 00:25:02,420 So one type of propagator is what we call 322 00:25:02,420 --> 00:25:03,770 the bulk-to-bulk propagator. 323 00:25:11,345 --> 00:25:13,350 Somehow this is going up and up. 324 00:25:21,330 --> 00:25:23,435 OK, I lost some blackboard space. 325 00:25:33,510 --> 00:25:38,090 So this the complete analog, this 326 00:25:38,090 --> 00:25:45,750 is the precise counterpart of the flat-space propagator. 327 00:25:45,750 --> 00:25:51,875 So this connects two points, z, x, z prime, x prime. 328 00:25:51,875 --> 00:25:55,684 So I still use the notation that z is the bulk 329 00:25:55,684 --> 00:25:56,600 [INAUDIBLE] direction. 330 00:25:56,600 --> 00:25:58,349 And the x is along the boundary direction. 331 00:26:04,030 --> 00:26:07,180 And this bulk-to-bulk will satisfy the standard Laplace 332 00:26:07,180 --> 00:26:14,260 equation just as in flat-space. 333 00:26:14,260 --> 00:26:15,970 So this gives you the delta function. 334 00:26:22,140 --> 00:26:35,550 Let me just express it, z minus z prime delta x minus x prime. 335 00:26:35,550 --> 00:26:39,780 So this is a complete analog of the standard flat-space 336 00:26:39,780 --> 00:26:40,790 propagator. 337 00:26:40,790 --> 00:26:43,160 It's just now in Anti-de Sitter space. 338 00:26:51,210 --> 00:27:07,278 So this is a counterpart in AdS standard flat-space propagator. 339 00:27:12,530 --> 00:27:27,420 So in particular, so as a propagator, 340 00:27:27,420 --> 00:27:56,207 this should be normalizable in either z or z prime, 341 00:27:56,207 --> 00:27:57,790 so when you take them to the boundary. 342 00:28:04,060 --> 00:28:10,970 So more explicitly, for example, the z at z prime, 343 00:28:10,970 --> 00:28:15,775 x prime, should scale as z prime to the power delta, when 344 00:28:15,775 --> 00:28:17,190 you take z prime goes to 0. 345 00:28:23,639 --> 00:28:25,430 And of course, this should also be regular. 346 00:28:28,220 --> 00:28:31,940 We should not have singularities as z goes to infinity. 347 00:28:35,100 --> 00:28:38,184 They should also be regular. 348 00:28:38,184 --> 00:28:40,100 So essentially, these condition will precisely 349 00:28:40,100 --> 00:28:41,950 define those propagators. 350 00:28:51,590 --> 00:28:55,940 But the idea is we also, because the source 351 00:28:55,940 --> 00:29:00,010 lies on the boundary, because of this boundary condition, 352 00:29:00,010 --> 00:29:02,225 we also have so-called boundary-to-bulk propagators. 353 00:29:46,440 --> 00:29:48,730 Actually, the standard in flat-space one, 354 00:29:48,730 --> 00:29:51,720 it propagates the field from one point to the other point. 355 00:29:51,720 --> 00:29:54,220 So that's why there's a delta function here. 356 00:29:54,220 --> 00:29:56,520 So the standard story, there's a delta function here. 357 00:29:56,520 --> 00:30:02,020 And you propagate the field starting 358 00:30:02,020 --> 00:30:05,040 from this point to that point. 359 00:30:05,040 --> 00:30:12,590 So this boundary-to-bulk propagator normally is written 360 00:30:12,590 --> 00:30:18,470 as k z,x, but the second index only has x prime, 361 00:30:18,470 --> 00:30:20,970 because this is when you take the boundary point to a bulk 362 00:30:20,970 --> 00:30:21,470 point. 363 00:30:21,470 --> 00:30:24,180 A boundary point, of course, there's no z anymore. 364 00:30:24,180 --> 00:30:26,100 So z prime is 0 here. 365 00:30:26,100 --> 00:30:29,270 You don't need to write it. 366 00:30:29,270 --> 00:30:33,560 And this boundary-to-bulk propagator, 367 00:30:33,560 --> 00:30:35,695 again, should satisfy the Laplace equation. 368 00:30:40,800 --> 00:30:44,750 But now the key point is that, on the right-hand side, 369 00:30:44,750 --> 00:30:47,320 because there's no bulk source, so the right-hand side 370 00:30:47,320 --> 00:30:49,790 should be 0 rather than the delta function. 371 00:30:49,790 --> 00:30:54,070 So that delta function is due to the bulk source. 372 00:30:54,070 --> 00:30:57,560 But we have to make sure this k have 373 00:30:57,560 --> 00:30:59,360 the right non-normalizable boundary 374 00:30:59,360 --> 00:31:01,950 conditions, so you have the right boundary 375 00:31:01,950 --> 00:31:04,050 condition to the boundary. 376 00:31:04,050 --> 00:31:16,260 So this when it approach to the boundary, as z goes to 0, 377 00:31:16,260 --> 00:31:22,680 should be like z d delta, delta x minus x prime. 378 00:31:29,010 --> 00:31:35,340 So the k satisfies the Laplace equation without a source. 379 00:31:35,340 --> 00:31:43,520 But when you take this propagator to the boundary, 380 00:31:43,520 --> 00:31:46,660 then it should give you the non-normalizable boundary 381 00:31:46,660 --> 00:31:49,540 conditions, because you should approach a source. 382 00:31:49,540 --> 00:31:55,690 And the delta function is put here. 383 00:31:55,690 --> 00:31:59,800 So if you write, say, a bulk field, 384 00:31:59,800 --> 00:32:07,570 which is sourced by some boundary source, say k z, x, 385 00:32:07,570 --> 00:32:15,450 x prime times phi x prime, and then this 386 00:32:15,450 --> 00:32:23,330 will have right boundary condition 387 00:32:23,330 --> 00:32:32,155 that the phi x, z goes to z d minus delta phi x. 388 00:32:42,650 --> 00:32:46,110 So this boundary-to-bulk propagator 389 00:32:46,110 --> 00:32:49,460 does not have a source in the bulk 390 00:32:49,460 --> 00:32:52,690 but does have a source in the boundary, 391 00:32:52,690 --> 00:32:55,825 non-normalizable off in the boundary. 392 00:32:59,260 --> 00:33:01,780 So now with this two propagator, then the story 393 00:33:01,780 --> 00:33:04,750 is just actually as in the standard way 394 00:33:04,750 --> 00:33:08,420 you obtain the Green functions. 395 00:33:08,420 --> 00:33:09,750 It's just really no different. 396 00:33:15,850 --> 00:33:20,565 So let me just summarize what we have discussed. 397 00:33:39,640 --> 00:33:47,830 The boundary endpoint function can 398 00:33:47,830 --> 00:33:54,260 be obtained by, essentially, endpoint function of the phi 399 00:33:54,260 --> 00:34:04,340 field in the gravity side-- let me write it here-- just 400 00:34:04,340 --> 00:34:08,389 related to the endpoint function of the gravity field 401 00:34:08,389 --> 00:34:18,310 on the gravity side, with all these points, x1, xn, 402 00:34:18,310 --> 00:34:19,850 they're lying on the boundary. 403 00:34:28,780 --> 00:34:31,800 This side is treated as ordinary field theory. 404 00:34:31,800 --> 00:34:35,170 Then compute this endpoint function of your field 405 00:34:35,170 --> 00:34:37,320 but just put the field on the boundary. 406 00:34:45,150 --> 00:34:48,110 And then that would just give you the AdS correlation 407 00:34:48,110 --> 00:34:49,830 functions. 408 00:34:49,830 --> 00:34:53,050 Yeah, and then you just need to distinguish 409 00:34:53,050 --> 00:34:57,000 two types of propagators, a bulk-to-bulk propagator 410 00:34:57,000 --> 00:35:00,000 and bulk-to-boundary propagator. 411 00:35:00,000 --> 00:35:01,763 So any questions regarding this? 412 00:35:01,763 --> 00:35:06,230 AUDIENCE: Now, here, [INAUDIBLE] d minus delta. 413 00:35:06,230 --> 00:35:11,035 Right there, the G goes to [? d ?] phi in the delta. 414 00:35:11,035 --> 00:35:11,660 HONG LIU: Yeah. 415 00:35:14,558 --> 00:35:18,262 AUDIENCE: But why aren't they different? 416 00:35:18,262 --> 00:35:20,220 HONG LIU: It's because this must have a source. 417 00:35:20,220 --> 00:35:24,290 So the boundary-to-bulk propagator-- 418 00:35:24,290 --> 00:35:27,120 you propagate the source, from the boundary to the bulk, 419 00:35:27,120 --> 00:35:29,730 so that means that field must have the right boundary 420 00:35:29,730 --> 00:35:30,230 condition. 421 00:35:33,310 --> 00:35:34,990 So that's the reason for this. 422 00:35:34,990 --> 00:35:38,530 We must have non-normalizable boundary conditions. 423 00:35:38,530 --> 00:35:42,300 And that propagator is just the standard flat-space propagator. 424 00:35:42,300 --> 00:35:44,620 Of course, it's always normalizable. 425 00:35:44,620 --> 00:35:47,480 And in the flat-space, when you construct 426 00:35:47,480 --> 00:35:49,084 a propagator, of course, it's always 427 00:35:49,084 --> 00:35:50,625 used in a normalizable wave function. 428 00:35:53,355 --> 00:35:55,655 AUDIENCE: This is that's non-normalizable? 429 00:35:55,655 --> 00:35:57,280 HONG LIU: No, this is non-normalizable. 430 00:35:57,280 --> 00:36:01,190 This is designed so that you have the right boundary 431 00:36:01,190 --> 00:36:01,690 condition. 432 00:36:04,470 --> 00:36:08,390 So this is designed so that you can propagate 433 00:36:08,390 --> 00:36:11,280 just as in flat-space. 434 00:36:11,280 --> 00:36:14,070 If you have phi at this point, and then 435 00:36:14,070 --> 00:36:16,480 you can propagate to the other point by convolution. 436 00:36:19,310 --> 00:36:23,000 And of course, when you write down the bulk field phi, 437 00:36:23,000 --> 00:36:28,050 in general, you only construct them out 438 00:36:28,050 --> 00:36:29,340 of the normalizable modes. 439 00:36:33,542 --> 00:36:34,125 Any questions? 440 00:36:40,771 --> 00:36:41,270 Good. 441 00:36:44,727 --> 00:36:46,060 So now let me make some remarks. 442 00:36:55,970 --> 00:37:00,000 So here, this procedure of iterating 443 00:37:00,000 --> 00:37:02,000 the classical equation of motion, 444 00:37:02,000 --> 00:37:03,780 solving the classical equation of motion, 445 00:37:03,780 --> 00:37:07,910 then iterating the action, of course, 446 00:37:07,910 --> 00:37:11,391 again, from our experience with quantum field theory, that 447 00:37:11,391 --> 00:37:13,015 only gives you the tree-level diagrams. 448 00:37:16,110 --> 00:37:20,440 So if I really write the bulk path integral, 449 00:37:20,440 --> 00:37:25,450 so Z CFT phi, then, in the same classical limit, 450 00:37:25,450 --> 00:37:30,590 it can be written as Z gravity, which 451 00:37:30,590 --> 00:37:33,720 is phi with the right boundary condition. 452 00:37:33,720 --> 00:37:37,775 And then on the gravity side, at semi-classical level, 453 00:37:37,775 --> 00:37:42,139 we can actually write down the path integral for this gravity 454 00:37:42,139 --> 00:37:42,930 partition function. 455 00:37:48,430 --> 00:37:50,560 So that's what we discussed before. 456 00:37:50,560 --> 00:37:57,230 And then at the leading order, in performing this path 457 00:37:57,230 --> 00:38:14,310 integral, of course, you just get 458 00:38:14,310 --> 00:38:17,170 what we have been doing so far. 459 00:38:17,170 --> 00:38:19,470 You just evaluate it at the classical solutions. 460 00:38:22,219 --> 00:38:24,385 But in principle, you can also see the fluctuations. 461 00:38:27,850 --> 00:38:36,330 So this is the fluctuations around, say, 462 00:38:36,330 --> 00:38:39,510 the classical solution, phi c. 463 00:38:39,510 --> 00:38:45,040 And then the action for the fluctuations 464 00:38:45,040 --> 00:38:48,050 is just given by phi c plus phi. 465 00:38:48,050 --> 00:38:51,550 The minus, of course, is the classical action. 466 00:38:51,550 --> 00:38:53,900 So we just expand around the phi c. 467 00:38:59,040 --> 00:39:02,140 So if we expand this, the leading order 468 00:39:02,140 --> 00:39:04,680 would be quadratic in small phi. 469 00:39:04,680 --> 00:39:06,960 I should now quote small phi. 470 00:39:06,960 --> 00:39:12,540 Maybe let me quote chi, because that is the same as that phi. 471 00:39:12,540 --> 00:39:16,320 So this chi is the small fluctuations around the phi c. 472 00:39:18,930 --> 00:39:22,670 So if you expanded this in chi, the linear order 473 00:39:22,670 --> 00:39:26,180 is 0, because phi c satisfies the equation of motion. 474 00:39:26,180 --> 00:39:29,610 So this will start in quadratic order and then cubic order, et 475 00:39:29,610 --> 00:39:30,150 cetera. 476 00:39:30,150 --> 00:39:33,500 So you validated the quadratic order as a Gaussian. 477 00:39:33,500 --> 00:39:35,610 And then that's the one-loop diagram. 478 00:39:35,610 --> 00:39:37,490 And then if you go to higher orders, 479 00:39:37,490 --> 00:39:39,310 then it'll give you higher-order diagrams. 480 00:39:39,310 --> 00:39:45,960 So if you now include the fluctuations, 481 00:39:45,960 --> 00:39:51,900 so the SE phi c, which is a classical action. 482 00:39:54,630 --> 00:39:56,520 This encodes only tree-level diagrams. 483 00:40:04,870 --> 00:40:11,560 And when you include the fluctuations, phi, 484 00:40:11,560 --> 00:40:15,780 the fluctuations, chi, and then this corresponding 485 00:40:15,780 --> 00:40:16,640 to include loops. 486 00:40:23,300 --> 00:40:27,856 So again, the whole procedure can be captured 487 00:40:27,856 --> 00:40:28,855 by the Feynman diagrams. 488 00:40:32,990 --> 00:40:34,657 So the advantage of the Feynman diagram 489 00:40:34,657 --> 00:40:36,990 is actually there's a natural generalization to the loop 490 00:40:36,990 --> 00:40:38,190 diagrams. 491 00:40:38,190 --> 00:40:39,930 And then just include the loops here. 492 00:40:47,700 --> 00:40:52,655 For example, here, you can do something like this. 493 00:40:52,655 --> 00:40:54,405 Now you can also have something like this. 494 00:40:59,287 --> 00:41:01,120 So that's, of course, one and two, including 495 00:41:01,120 --> 00:41:05,810 the fluctuations, are under the saddle point, 496 00:41:05,810 --> 00:41:08,538 are including the fluctuation around the saddle point. 497 00:41:13,722 --> 00:41:15,180 Any questions regarding this point? 498 00:41:19,990 --> 00:41:20,490 Good. 499 00:41:30,340 --> 00:41:31,130 Yes? 500 00:41:31,130 --> 00:41:33,890 AUDIENCE: So when people draw these like so-called Witten 501 00:41:33,890 --> 00:41:35,825 diagrams, where they draw up like this circle. 502 00:41:35,825 --> 00:41:36,630 HONG LIU: Right. 503 00:41:36,630 --> 00:41:38,030 AUDIENCE: Is that basically what we're doing? 504 00:41:38,030 --> 00:41:38,390 HONG LIU: Yeah. 505 00:41:38,390 --> 00:41:39,000 Yeah, right. 506 00:41:39,000 --> 00:41:41,300 Yeah, so let me just explain one thing. 507 00:41:41,300 --> 00:41:46,130 In the Euclidean, I have been drawing the ideas like this. 508 00:41:46,130 --> 00:41:50,040 So this is z equal to 0, et cetera. 509 00:41:50,040 --> 00:41:54,800 So in the Euclidean signature, as you will do in your PSet, 510 00:41:54,800 --> 00:41:59,240 then this z equal to infinity, you 511 00:41:59,240 --> 00:42:02,780 can argue is actually a single point. 512 00:42:02,780 --> 00:42:05,580 So let me just write down the metric. 513 00:42:05,580 --> 00:42:12,002 So the metric is R square, z square. 514 00:42:12,002 --> 00:42:13,460 When you go to Euclidean signature, 515 00:42:13,460 --> 00:42:15,000 you just have dx squared. 516 00:42:15,000 --> 00:42:19,900 So this is a full Euclidean metric. 517 00:42:19,900 --> 00:42:22,289 So there's something interesting about the difference 518 00:42:22,289 --> 00:42:23,830 between the Euclidean and Lorentzian, 519 00:42:23,830 --> 00:42:27,700 even though, from this picture, it's roughly the same. 520 00:42:27,700 --> 00:42:31,600 So if it's equal to 0, you go to z notch. 521 00:42:31,600 --> 00:42:35,580 And then in the Euclidean case, you can actually 522 00:42:35,580 --> 00:42:38,920 show that z equal infinity, even though, 523 00:42:38,920 --> 00:42:43,224 [? naively, ?] you still have a full Euclidean plane, 524 00:42:43,224 --> 00:42:45,390 but you can argue, actually, that the whole thing is 525 00:42:45,390 --> 00:42:46,940 actually a point. 526 00:42:46,940 --> 00:42:50,570 And you can roughly see, when you go to z equals infinity, 527 00:42:50,570 --> 00:42:54,560 the overall factor goes to 0. 528 00:42:54,560 --> 00:43:00,160 So essentially, the whole space shrinks to a point. 529 00:43:00,160 --> 00:43:02,280 The whole space shrinks to a point. 530 00:43:02,280 --> 00:43:05,500 So in Euclidean, if it's equal to infinity, 531 00:43:05,500 --> 00:43:07,220 it's essentially a point. 532 00:43:07,220 --> 00:43:12,100 And then topologically, this is equivalent to having a disk. 533 00:43:16,780 --> 00:43:20,360 Topologically, the whole space is just disk. 534 00:43:20,360 --> 00:43:24,670 And the z equal to infinity is one point on the boundary. 535 00:43:24,670 --> 00:43:27,125 And then the other is z equal to 0. 536 00:43:27,125 --> 00:43:28,890 And so this is the interior of the space. 537 00:43:33,140 --> 00:43:33,640 Yes? 538 00:43:33,640 --> 00:43:36,265 AUDIENCE: So in other words, the entire perimeter of the circle 539 00:43:36,265 --> 00:43:37,054 is identi-- 540 00:43:37,054 --> 00:43:38,220 HONG LIU: It's the boundary. 541 00:43:38,220 --> 00:43:38,770 AUDIENCE: [INAUDIBLE]. 542 00:43:38,770 --> 00:43:40,311 HONG LIU: Yeah, this is the boundary. 543 00:43:43,290 --> 00:43:48,727 Essentially, you can imagine, each point here is R to d. 544 00:43:48,727 --> 00:43:50,185 Then you add the point at infinity, 545 00:43:50,185 --> 00:43:51,514 and then it becomes a sphere. 546 00:43:51,514 --> 00:43:52,430 AUDIENCE: [INAUDIBLE]. 547 00:43:52,430 --> 00:43:54,610 HONG LIU: Yeah, it becomes a sphere. 548 00:43:54,610 --> 00:43:57,240 And then the whole thing becomes like a disk. 549 00:44:03,700 --> 00:44:07,046 So normally, when people will draw-- so topologically, 550 00:44:07,046 --> 00:44:08,670 you're creating a disk, just like this. 551 00:44:08,670 --> 00:44:10,680 Then people will just draw diagrams like this. 552 00:44:15,400 --> 00:44:18,434 Yeah, I draw diagrams like this. 553 00:44:18,434 --> 00:44:21,200 AUDIENCE: If you add a single point to the boundary, 554 00:44:21,200 --> 00:44:25,110 that single point is equal to infinity to the boundary, which 555 00:44:25,110 --> 00:44:27,010 is z equals 0? 556 00:44:27,010 --> 00:44:28,220 HONG LIU: Sorry? 557 00:44:28,220 --> 00:44:31,440 AUDIENCE: You mean that you have a point that [INAUDIBLE]. 558 00:44:31,440 --> 00:44:32,210 HONG LIU: Yeah. 559 00:44:32,210 --> 00:44:35,310 Yeah, this point turns out, it can actually 560 00:44:35,310 --> 00:44:36,780 lie on the boundary. 561 00:44:36,780 --> 00:44:40,215 AUDIENCE: But then, on the boundary, the value of t 562 00:44:40,215 --> 00:44:42,470 is not continuous. 563 00:44:42,470 --> 00:44:43,730 HONG LIU: Hm? 564 00:44:43,730 --> 00:44:44,230 Yeah. 565 00:44:52,450 --> 00:44:54,430 So this is due to the coordinate choice. 566 00:44:54,430 --> 00:44:59,200 It's just this coordinate becomes singular at that point. 567 00:44:59,200 --> 00:45:01,437 You can rewrite it. 568 00:45:01,437 --> 00:45:04,020 Yeah, the simplest thing is that you rewrite these [INAUDIBLE] 569 00:45:04,020 --> 00:45:06,900 coordinates to the coordinate which has a parameter that has 570 00:45:06,900 --> 00:45:08,610 a boundary really as a sphere. 571 00:45:08,610 --> 00:45:09,680 And then you will see it. 572 00:45:09,680 --> 00:45:12,860 Because when you write the sphere as a plane, 573 00:45:12,860 --> 00:45:15,220 then this one point becomes singular. 574 00:45:15,220 --> 00:45:17,634 This is just the standard story. 575 00:45:17,634 --> 00:45:19,300 Just that coordinate choice is singular. 576 00:45:23,047 --> 00:45:23,880 Any other questions? 577 00:45:32,830 --> 00:45:34,450 Good. 578 00:45:34,450 --> 00:45:37,410 So this is the first remark. 579 00:45:37,410 --> 00:45:44,960 The secondary remark is that this 580 00:45:44,960 --> 00:45:47,260 is a little bit of an unusual correlation function. 581 00:45:47,260 --> 00:45:50,010 So this is the bulk correlation functions, 582 00:45:50,010 --> 00:45:52,830 but with all points lying on the boundary. 583 00:46:01,680 --> 00:46:10,650 So the counterpart or just the exact analog 584 00:46:10,650 --> 00:46:33,110 of standard flat-space correlation functions in AdS-- 585 00:46:33,110 --> 00:46:46,882 you can see the end bulk point and look 586 00:46:46,882 --> 00:46:48,090 at its correlation functions. 587 00:46:52,050 --> 00:46:57,345 So this then will be really just bring you to the standard QFT 588 00:46:57,345 --> 00:46:59,530 one correlation functions. 589 00:46:59,530 --> 00:47:07,620 Then you have endpoints, z1, x1, or in the bulk, z2, x2, et 590 00:47:07,620 --> 00:47:08,320 cetera. 591 00:47:08,320 --> 00:47:14,380 And then you just draw a diagram between them, et cetera. 592 00:47:14,380 --> 00:47:16,000 The area propagator here would be 593 00:47:16,000 --> 00:47:17,250 just bulk-to-bulk propagator. 594 00:47:17,250 --> 00:47:19,819 There's no boundary-to-bulk propagator. 595 00:47:19,819 --> 00:47:21,610 There's no boundary-to-bulk propagator just 596 00:47:21,610 --> 00:47:24,767 all bulk-to-bulk propagator. 597 00:47:32,560 --> 00:47:36,876 So this is a complete analog of your ordinary flat-space Green 598 00:47:36,876 --> 00:47:37,375 functions. 599 00:47:39,910 --> 00:47:46,420 But we can easily imagine that this correlation function must 600 00:47:46,420 --> 00:47:49,634 be related to those correlations functions 601 00:47:49,634 --> 00:47:51,550 if you just take those points to the boundary. 602 00:48:20,260 --> 00:48:33,670 So it's natural to expect, say, those correlations functions, 603 00:48:33,670 --> 00:48:38,150 since they're at the boundary, must 604 00:48:38,150 --> 00:48:43,340 be related to those kind of standard correlation 605 00:48:43,340 --> 00:48:48,550 functions, in which you would take the point to the boundary. 606 00:49:03,540 --> 00:49:07,091 So those, somehow, in the end, must be the same. 607 00:49:07,091 --> 00:49:08,840 So if I take those points to the boundary, 608 00:49:08,840 --> 00:49:10,230 I should recover that guy. 609 00:49:13,870 --> 00:49:20,040 So indeed, this relation is true. 610 00:49:20,040 --> 00:49:22,860 And you will work it out yourself in the PSet. 611 00:49:27,210 --> 00:49:29,510 You see, the only difference is the following. 612 00:49:29,510 --> 00:49:33,920 When you take those points to the boundary, what you get 613 00:49:33,920 --> 00:49:34,920 is the boundary limit. 614 00:49:37,530 --> 00:49:44,160 So now let's imagine you take those points to the boundary. 615 00:49:44,160 --> 00:49:47,106 Because here, it's all bulk propagator. 616 00:49:47,106 --> 00:49:49,230 So those propagator, which connect to the boundary, 617 00:49:49,230 --> 00:49:53,170 would be the boundary limit of the bulk propagator. 618 00:49:53,170 --> 00:49:55,940 And then, it just boils down to, what 619 00:49:55,940 --> 00:49:57,440 is the relation between the boundary 620 00:49:57,440 --> 00:50:00,742 limit of the bulk propagator and this boundary 621 00:50:00,742 --> 00:50:01,700 to the bulk propagator? 622 00:50:09,430 --> 00:50:15,157 So the above relation-- let me just-- I'd rather just 623 00:50:15,157 --> 00:50:18,630 call it a number. 624 00:50:18,630 --> 00:50:21,350 So this star-star-star just boils 625 00:50:21,350 --> 00:50:36,183 down to the relation between the k 626 00:50:36,183 --> 00:50:42,950 z, x, x prime, which is our boundary-to-bulk propagator, 627 00:50:42,950 --> 00:50:55,860 and limit z prime goes to 0, G x, z, x prime, z prime. 628 00:50:58,970 --> 00:51:02,420 So you will find that these two guys 629 00:51:02,420 --> 00:51:04,660 are proportional to each other. 630 00:51:04,660 --> 00:51:11,190 And once you extract out that proportional factor, 631 00:51:11,190 --> 00:51:15,220 then you just relate them. 632 00:51:15,220 --> 00:51:21,690 So this is just another way to calculate. 633 00:51:21,690 --> 00:51:24,690 Because this is the bounty correlation function. 634 00:51:24,690 --> 00:51:27,110 So this is the boundary correlation function. 635 00:51:27,110 --> 00:51:30,960 So this is related to the CFT correlation functions. 636 00:51:30,960 --> 00:51:33,970 And then you can also calculate the CFT correlation function 637 00:51:33,970 --> 00:51:37,205 just by taking the limit of the bulk correlation functions. 638 00:51:39,810 --> 00:51:44,080 So this you will work out a little bit. 639 00:51:44,080 --> 00:51:48,200 Yeah, you will work out the precise relation in your PSet. 640 00:51:48,200 --> 00:51:51,790 Maybe some of you have looked at it already. 641 00:51:51,790 --> 00:51:53,904 So any questions on this? 642 00:51:53,904 --> 00:51:56,264 AUDIENCE: What is the advantage of maybe 643 00:51:56,264 --> 00:51:59,580 using another to express it? 644 00:51:59,580 --> 00:52:02,510 HONG LIU: Yeah, just different ways of calculating things. 645 00:52:02,510 --> 00:52:07,510 And of course, normally, I have many different ways 646 00:52:07,510 --> 00:52:09,310 to calculate things. 647 00:52:09,310 --> 00:52:11,301 Some of them may be convenient this way. 648 00:52:11,301 --> 00:52:13,675 So of them may be convenient for that purpose, et cetera. 649 00:52:13,675 --> 00:52:14,230 It depends. 650 00:52:16,460 --> 00:52:17,960 Normally, we just directly are using 651 00:52:17,960 --> 00:52:21,196 the bulk-to-boundary propagator. 652 00:52:21,196 --> 00:52:23,820 If you just do the calculation, the bulk-to boundary propagator 653 00:52:23,820 --> 00:52:25,980 actually is simpler. 654 00:52:25,980 --> 00:52:29,130 But sometimes, for certain conceptual questions, 655 00:52:29,130 --> 00:52:30,520 this actually becomes simpler. 656 00:52:34,630 --> 00:52:39,518 So now let's look at Wilson loops. 657 00:52:45,210 --> 00:52:49,140 So how do you calculate the Wilson loops using gravity? 658 00:52:53,640 --> 00:53:02,140 So in the gauge theory-- so I assume 659 00:53:02,140 --> 00:53:05,490 you have already done the QFT II and the gauge theory? 660 00:53:10,645 --> 00:53:12,789 In the gauge theory, we also know 661 00:53:12,789 --> 00:53:15,080 it's essentially one of the most important observables. 662 00:53:18,715 --> 00:53:21,090 The Wilson loop is normally defined in the following way. 663 00:53:34,890 --> 00:53:36,352 So now I will explain my notation. 664 00:53:41,100 --> 00:53:44,400 So let me also add in subscript r here. 665 00:53:47,320 --> 00:53:48,910 So c is a closed path. 666 00:53:52,400 --> 00:53:54,320 So this is defined for closed paths. 667 00:53:57,130 --> 00:53:58,720 And A mu is a matrix. 668 00:53:58,720 --> 00:54:02,960 So we can see the [INAUDIBLE] in gauge theory. 669 00:54:02,960 --> 00:54:08,750 A mu is a matrix, writing the standard away, 670 00:54:08,750 --> 00:54:11,940 in terms of the generator, in some representations r. 671 00:54:14,920 --> 00:54:19,140 So this is in some representations. 672 00:54:19,140 --> 00:54:22,140 So this is just the generators of the gauge 673 00:54:22,140 --> 00:54:26,750 group in some representation, which we'll call r. 674 00:54:52,620 --> 00:54:54,550 And P is the path ordering. 675 00:55:05,700 --> 00:55:09,330 So in general, you can choose any representations you want. 676 00:55:09,330 --> 00:55:17,665 But often, we choose r, say, in fundamental representations. 677 00:55:20,700 --> 00:55:23,250 For example, that's normally what we do, in QCD, 678 00:55:23,250 --> 00:55:25,170 in fundamental representation. 679 00:55:25,170 --> 00:55:28,695 But you can choose it to be any representation. 680 00:55:35,130 --> 00:55:39,644 So the physical meaning of this, so this operator, 681 00:55:39,644 --> 00:55:41,060 by definition, is gauge invariant, 682 00:55:41,060 --> 00:55:42,930 because of this trace. 683 00:55:42,930 --> 00:55:45,840 And need the path ordering, because you have a matrix here. 684 00:55:45,840 --> 00:55:48,420 And the matrix, at different points, don't commute. 685 00:55:48,420 --> 00:55:50,550 So you need to specify ordering. 686 00:55:50,550 --> 00:55:53,900 And so you just specify it by the order of the path. 687 00:55:56,830 --> 00:56:05,180 And so the physical meaning of the Wilson loop, 688 00:56:05,180 --> 00:56:19,240 so this is essentially the phase factor associated 689 00:56:19,240 --> 00:56:38,640 with transporting an external particle, in a given 690 00:56:38,640 --> 00:56:44,220 representation, say, in r representation, along c. 691 00:56:50,080 --> 00:56:53,480 So you transport the particle along some path. 692 00:56:53,480 --> 00:56:57,600 And you come back to the same point. 693 00:56:57,600 --> 00:56:59,840 Then you find that that phase does not necessarily 694 00:56:59,840 --> 00:57:02,890 go back to 0. 695 00:57:02,890 --> 00:57:04,680 And so this is a nontrivial phase. 696 00:57:04,680 --> 00:57:07,346 It essentially tells you there's a nontrivial gauge field there. 697 00:57:11,120 --> 00:57:15,480 Yeah, so this provides a probe of the gauge fields. 698 00:57:15,480 --> 00:57:15,980 Yes? 699 00:57:15,980 --> 00:57:17,354 AUDIENCE: I know that people talk 700 00:57:17,354 --> 00:57:19,519 about-- I know that it's somehow and observable. 701 00:57:19,519 --> 00:57:20,810 But is it actually observable ? 702 00:57:20,810 --> 00:57:22,428 Can you actually do an experiment 703 00:57:22,428 --> 00:57:23,518 to measure this phase? 704 00:57:27,496 --> 00:57:28,620 HONG LIU: I don't think so. 705 00:57:28,620 --> 00:57:29,119 Ha. 706 00:57:39,800 --> 00:57:40,385 It depends. 707 00:57:46,450 --> 00:57:48,460 Yeah, in some situations, you can. 708 00:57:53,180 --> 00:57:56,900 Let me mention something else, then the answer 709 00:57:56,900 --> 00:57:58,250 to this question will be seen. 710 00:58:03,230 --> 00:58:06,840 So the simplest case would be the W c, just 711 00:58:06,840 --> 00:58:11,800 the single-point function of W c, say, in a vacuum. 712 00:58:11,800 --> 00:58:14,060 So this is the simplest observable. 713 00:58:14,060 --> 00:58:19,310 But of course, you can also consider the generic, say, 714 00:58:19,310 --> 00:58:21,700 a large number, several [? routes ?] in some general 715 00:58:21,700 --> 00:58:23,950 state, say, for example, [? finite ?] [? telemetry ?], 716 00:58:23,950 --> 00:58:24,450 et cetera. 717 00:58:27,199 --> 00:58:28,740 So these are the typical observables. 718 00:58:32,370 --> 00:58:34,330 So now let me emphasize this external. 719 00:58:37,410 --> 00:58:40,470 So this external is very important. 720 00:58:40,470 --> 00:58:44,880 So normally, by calling something external, 721 00:58:44,880 --> 00:58:46,950 we mean that this particle has infinite mass. 722 00:58:46,950 --> 00:58:49,290 It's infinitely heavy. 723 00:58:49,290 --> 00:58:52,550 And the reason we want it to be infinitely heavy 724 00:58:52,550 --> 00:58:57,500 is because only for heavy particles, 725 00:58:57,500 --> 00:59:00,130 in principle, you can localize the lower path. 726 00:59:00,130 --> 00:59:05,040 Then you can make mathematical sense of the precise path. 727 00:59:05,040 --> 00:59:07,530 Otherwise, if it's a fluctuating particle, 728 00:59:07,530 --> 00:59:14,004 then you can not specify a path-- yeah, ambiguously. 729 00:59:14,004 --> 00:59:15,920 So when we say, an external particle, I always 730 00:59:15,920 --> 00:59:18,294 say a particle which is assumed to be infinitely massive. 731 00:59:22,850 --> 00:59:29,480 And also, I've introduced terminology. 732 00:59:29,480 --> 00:59:33,290 When this r is in the fundamental representation-- 733 00:59:33,290 --> 00:59:34,740 yeah, maybe not right here. 734 00:59:34,740 --> 00:59:38,070 When this r is in the fundamental representation, 735 00:59:38,070 --> 00:59:43,060 we'll call the corresponding particle a quark. 736 00:59:43,060 --> 00:59:45,415 If it is in the fundamental representation, 737 00:59:45,415 --> 00:59:47,950 I will call it a quark. 738 00:59:47,950 --> 00:59:51,657 I will often just consider the fundamental representation. 739 00:59:51,657 --> 00:59:53,740 In the phase factor, where we're corresponding to, 740 00:59:53,740 --> 00:59:55,000 you transport a quark. 741 01:00:05,340 --> 01:00:18,930 So one of the very often used loops is a rectangular loop. 742 01:00:24,150 --> 01:00:25,920 So this is along some spatial direction, 743 01:00:25,920 --> 01:00:28,710 so this is the time direction. 744 01:00:28,710 --> 01:00:32,562 So let's say the spatial direction is L, 745 01:00:32,562 --> 01:00:34,270 and the time direction, using the length, 746 01:00:34,270 --> 01:00:39,020 is T. So imagine you have a loop like this. 747 01:01:11,970 --> 01:01:13,902 So normally, we can see that the T, say, 748 01:01:13,902 --> 01:01:15,777 the length of the loop in the time direction, 749 01:01:15,777 --> 01:01:19,200 is much, much greater than the L. 750 01:01:19,200 --> 01:01:26,620 So in such a limit, then, essentially, 751 01:01:26,620 --> 01:01:29,660 you can ignore the contribution from this side, 752 01:01:29,660 --> 01:01:30,970 from these two short sides. 753 01:01:30,970 --> 01:01:36,650 So in the limit, in which this T goes to infinity, 754 01:01:36,650 --> 01:01:40,085 to leading order, you can ignore the short side. 755 01:01:40,085 --> 01:01:43,930 And then you can think of this loop 756 01:01:43,930 --> 01:01:47,164 when it just has a particle-- so this is a particle moving 757 01:01:47,164 --> 01:01:47,830 forward in time. 758 01:01:47,830 --> 01:01:49,690 This is a particle going backward in time. 759 01:01:49,690 --> 01:01:53,520 Then you can consider this as a particle and an anti-particle 760 01:01:53,520 --> 01:01:54,670 moving forward in time. 761 01:01:58,160 --> 01:02:02,830 So in this case, we can argue, on general grounds, 762 01:02:02,830 --> 01:02:04,510 that this will just, to leading order, 763 01:02:04,510 --> 01:02:07,350 will give you something iET. 764 01:02:07,350 --> 01:02:10,320 And this e, it just can be interpreted 765 01:02:10,320 --> 01:02:31,680 as a potential energy between a single quark and an anti-quark. 766 01:02:41,260 --> 01:02:45,304 So in this example, this is the same. 767 01:02:45,304 --> 01:02:47,720 In this case, you may be able to measure this Wilson loop, 768 01:02:47,720 --> 01:02:51,095 because if you can measure-- experimentally, 769 01:02:51,095 --> 01:02:53,975 you may be able to measure this energy between the two 770 01:02:53,975 --> 01:02:54,475 particles. 771 01:02:59,389 --> 01:03:01,930 And then, essentially, you can say, I have measured the loop. 772 01:03:01,930 --> 01:03:04,305 And then you can compare with the prediction of the loop. 773 01:03:12,260 --> 01:03:16,574 So again, in order to make this interpretation make sense, 774 01:03:16,574 --> 01:03:18,865 this particle having infinite energy is very important. 775 01:03:22,200 --> 01:03:27,720 And again, only for the very heavy particle, 776 01:03:27,720 --> 01:03:29,420 you can really localize them, and then 777 01:03:29,420 --> 01:03:32,850 talk about the potential energy between these two particles. 778 01:03:32,850 --> 01:03:36,160 Otherwise, if they fluctuate a lot, 779 01:03:36,160 --> 01:03:39,815 then it's hard to make it precise. 780 01:03:39,815 --> 01:03:44,340 So now the question is how do we calculate this quantity in AdS? 781 01:04:11,420 --> 01:04:14,170 When I don't put an index here, I just mean that this 782 01:04:14,170 --> 01:04:15,573 the fundamental representation. 783 01:04:29,880 --> 01:04:33,000 The question is how we do that. 784 01:04:33,000 --> 01:04:35,850 So in order to answer this question, 785 01:04:35,850 --> 01:04:50,176 we first need to understand how to introduce 786 01:04:50,176 --> 01:05:07,685 a fundamental external particle, external quark in the N 787 01:05:07,685 --> 01:05:09,060 equals 4 super Yang-Mills theory. 788 01:05:18,980 --> 01:05:21,708 So how to introduce a fundamental external quark 789 01:05:21,708 --> 01:05:23,457 in the N equals 4 super Yang-Mills theory. 790 01:05:26,800 --> 01:05:30,600 So we know, as we discussed before, 791 01:05:30,600 --> 01:05:33,340 that everything in the N equals 4 super Yang-Mills theory, so 792 01:05:33,340 --> 01:05:36,295 all fundamental fields, all the fields in the N 793 01:05:36,295 --> 01:05:37,670 equals 4 super Yang-Mills theory, 794 01:05:37,670 --> 01:05:40,360 they are in a joint representation. 795 01:05:40,360 --> 01:05:43,090 There's nothing in the fundamental representation. 796 01:05:43,090 --> 01:05:46,530 Everything is in the joint representation. 797 01:05:46,530 --> 01:05:50,287 So first we need to introduce a fundamental object, 798 01:05:50,287 --> 01:05:52,620 an object we transform under fundamental representation. 799 01:05:56,780 --> 01:06:01,350 After we have done that, then we need to translate that quantity 800 01:06:01,350 --> 01:06:02,260 to the gravity side. 801 01:06:05,800 --> 01:06:19,082 And then we understand how the gravity description 802 01:06:19,082 --> 01:06:20,290 of such an external particle. 803 01:06:38,150 --> 01:06:41,870 So this is easy to do. 804 01:06:41,870 --> 01:06:43,355 This is easy to do. 805 01:06:43,355 --> 01:06:45,820 So let's first think about how to do this in the N 806 01:06:45,820 --> 01:06:48,050 equals 4 super Yang-Mills theory. 807 01:06:48,050 --> 01:06:51,640 And then this is a very intuitive by using 808 01:06:51,640 --> 01:06:54,860 this [INAUDIBLE] picture. 809 01:06:54,860 --> 01:07:01,180 So we know the N equals 4 super Yang-Mills 810 01:07:01,180 --> 01:07:04,860 theory comes from low-energy theory of N D3-branes. 811 01:07:07,834 --> 01:07:10,000 They come from the low-energy theory of N D3-branes. 812 01:07:24,100 --> 01:07:28,500 So how do we introduce an external particles? 813 01:07:28,500 --> 01:07:36,080 So now let's imagine, consider N plus 1 of them, 814 01:07:36,080 --> 01:07:38,780 and let me separate one from the rest. 815 01:07:43,440 --> 01:07:49,320 So this is still N, separate one from the rest. 816 01:07:49,320 --> 01:07:53,860 So then there can be open string connects between them. 817 01:07:53,860 --> 01:07:56,890 So suppose I separate them by some distance, r, 818 01:07:56,890 --> 01:07:59,080 in the transverse direction to the D3-brane. 819 01:08:04,580 --> 01:08:12,180 So as we described before, such as a separation breaks the SU N 820 01:08:12,180 --> 01:08:19,380 plus 1 symmetry of the D3-brane SU N 821 01:08:19,380 --> 01:08:25,439 and then a single U1 of this 1-brane. 822 01:08:32,930 --> 01:08:39,390 In particular, such an open string-- 823 01:08:39,390 --> 01:08:41,424 so let's look at the endpoint. 824 01:08:41,424 --> 01:08:43,715 Then the endpoint of this string can only have indices. 825 01:08:46,580 --> 01:08:49,102 Because this is a single stream, one 826 01:08:49,102 --> 01:08:50,310 end ending on the other side. 827 01:08:50,310 --> 01:08:52,180 There's only one index. 828 01:08:52,180 --> 01:08:54,760 And here, they can only have N possible index. 829 01:08:54,760 --> 01:09:09,279 So such a fundamental string transforms 830 01:09:09,279 --> 01:09:10,760 in the fundamental representation. 831 01:09:21,149 --> 01:09:23,569 So from the point of view of this SU N gauge theory, 832 01:09:23,569 --> 01:09:25,270 this is a quark. 833 01:09:25,270 --> 01:09:27,829 So this endpoint is a quark. 834 01:09:27,829 --> 01:09:28,620 So this is a quark. 835 01:09:36,109 --> 01:09:46,060 And the mass of this quark is equal to-- so let 836 01:09:46,060 --> 01:09:52,960 me call the mass capital M-- is equal to r, 837 01:09:52,960 --> 01:09:58,439 the length between them divided by tension of the string. 838 01:09:58,439 --> 01:09:59,730 So this we have derived before. 839 01:10:02,126 --> 01:10:04,000 So this just gives you the mass of the quark. 840 01:10:07,040 --> 01:10:10,390 So when you separate one D-brane from the other, 841 01:10:10,390 --> 01:10:13,840 so you have introduced this quark, with a mass. 842 01:10:21,750 --> 01:10:24,190 So now let's consider that there's no low-energy limit 843 01:10:24,190 --> 01:10:24,690 [INAUDIBLE]. 844 01:10:48,500 --> 01:10:56,960 So in the low-energy limit of [INAUDIBLE], 845 01:10:56,960 --> 01:11:03,944 so we want to keep this quark, so there, of course, 846 01:11:03,944 --> 01:11:05,485 you want to take alpha prime go to 0. 847 01:11:11,354 --> 01:11:13,770 But we want to keep this quark in the low-energy spectrum, 848 01:11:13,770 --> 01:11:15,350 because we want this to remain in N 849 01:11:15,350 --> 01:11:16,725 equals 4 super Yang-Mills theory, 850 01:11:16,725 --> 01:11:19,250 because we want to introduce this particle. 851 01:11:19,250 --> 01:11:23,950 So in this low-energy limit, when we take off a prime, 852 01:11:23,950 --> 01:11:26,900 go to 0 limit, we also want to take, at the same time, 853 01:11:26,900 --> 01:11:33,740 r goes to 0 limit, so that this mass term, r divided by alpha 854 01:11:33,740 --> 01:11:35,605 prime, will remain finite. 855 01:11:40,616 --> 01:11:44,920 Then that means this will remain, 856 01:11:44,920 --> 01:11:48,450 this particle, such kind of fundamental representation, 857 01:11:48,450 --> 01:11:51,438 will remain in the N equals 4 super Yang-Mills theory. 858 01:11:58,430 --> 01:12:00,311 Because other modes are massive. 859 01:12:00,311 --> 01:12:02,060 And then when alpha prime goes to 0 limit, 860 01:12:02,060 --> 01:12:03,530 it goes to infinity. 861 01:12:03,530 --> 01:12:06,065 And under those modes, we will remain with the N 862 01:12:06,065 --> 01:12:07,440 equals 4 super Yang-Mills theory. 863 01:12:09,950 --> 01:12:12,345 So now let's see what happens on the gravity side. 864 01:12:15,120 --> 01:12:19,250 So here, you will need to fill in some details yourself. 865 01:12:22,310 --> 01:12:23,860 I'll only tell you the answer. 866 01:12:36,760 --> 01:12:40,720 On the gravity side, you have this N D3-brane. 867 01:12:40,720 --> 01:12:44,680 They have one brane which [? is ?] [? r ?] separated from 868 01:12:44,680 --> 01:12:46,580 this N D3-brane. 869 01:12:46,580 --> 01:12:48,420 And they're separation is such that, when 870 01:12:48,420 --> 01:12:50,230 you take off alpha prime and go to 0 limit, 871 01:12:50,230 --> 01:12:51,320 their ratio is finite. 872 01:12:54,894 --> 01:12:56,310 Now, in the gravity side, you have 873 01:12:56,310 --> 01:12:59,580 to take this so-called near-horizon limit, 874 01:12:59,580 --> 01:13:02,220 with taking into account. 875 01:13:05,110 --> 01:13:13,130 So what you find in the gravity side is that, 876 01:13:13,130 --> 01:13:34,110 as before, that N D3-brane now disappeared into r equal to 0. 877 01:13:34,110 --> 01:13:38,460 Remember, r equal to 0 is that infinite proper distance away. 878 01:13:38,460 --> 01:13:42,060 And now we are using this r coordinate. 879 01:13:42,060 --> 01:13:44,380 And when we take this low limit, we're 880 01:13:44,380 --> 01:13:47,040 using this r coordinate rather than this z coordinate. 881 01:13:47,040 --> 01:13:50,180 So in that case, you have i equal infinity. 882 01:13:50,180 --> 01:13:53,950 Then the brane, essentially, goes to r equal to 0. 883 01:13:57,964 --> 01:13:59,380 which has infinite proper distance 884 01:13:59,380 --> 01:14:04,510 away from any finite r. 885 01:14:04,510 --> 01:14:10,310 So now this is an exercise for yourself just 886 01:14:10,310 --> 01:14:14,700 to repeat our previous argument. 887 01:14:14,700 --> 01:14:18,050 Then you will find that in this regime, when 888 01:14:18,050 --> 01:14:21,300 you take alpha prime go to 0, this r finite, 889 01:14:21,300 --> 01:14:23,640 this D3-brane does not disappear. 890 01:14:23,640 --> 01:14:29,900 The single D3-brane will remain in the AdS. 891 01:14:29,900 --> 01:14:34,270 So the difference, from our previous story, 892 01:14:34,270 --> 01:14:42,450 is that, in addition to the AdS, now 893 01:14:42,450 --> 01:14:48,450 you have an additional D3-brane at some point in S5. 894 01:14:48,450 --> 01:14:51,170 Because D3-brane is in some transverse direction, 895 01:14:51,170 --> 01:14:52,910 which now become S5. 896 01:14:52,910 --> 01:14:55,760 So you have a D3-brane, which is parallel 897 01:14:55,760 --> 01:15:02,240 to the boundary coordinates, but sitting at one point in S5. 898 01:15:02,240 --> 01:15:05,370 And I think this picture is reasonable, 899 01:15:05,370 --> 01:15:07,420 but you should check yourself. 900 01:15:07,420 --> 01:15:10,960 Because in this regime, this D3-brane does not go away. 901 01:15:14,310 --> 01:15:19,800 And a remarkable thing you will check yourself 902 01:15:19,800 --> 01:15:22,571 is that when you take the [INAUDIBLE] limit, 903 01:15:22,571 --> 01:15:24,070 in this picture, this is flat-space. 904 01:15:28,330 --> 01:15:31,300 Then you have this formula. 905 01:15:31,300 --> 01:15:35,130 And now, when you take this limit, this now become AdS. 906 01:15:35,130 --> 01:15:44,210 Now, this AdS at some radius r, controlled by this r. 907 01:15:44,210 --> 01:15:45,890 And now you can check one nice thing. 908 01:15:55,670 --> 01:15:59,390 In AdS, if you consider a string, straight string, 909 01:15:59,390 --> 01:16:03,420 from this D3-brane all the way to r equal to 0, 910 01:16:03,420 --> 01:16:06,010 which is this string, this string 911 01:16:06,010 --> 01:16:07,510 has exactly the same mass. 912 01:16:10,450 --> 01:16:15,250 So the mass is equal to r divided by 2 pi alpha prime-- 913 01:16:15,250 --> 01:16:18,590 also in AdS5. 914 01:16:21,800 --> 01:16:23,290 So this is a consistency check. 915 01:16:25,940 --> 01:16:28,580 Because when you take the low-energy limits, of course, 916 01:16:28,580 --> 01:16:30,670 you should not change the mass. 917 01:16:30,670 --> 01:16:33,055 The mass of the string will not change. 918 01:16:33,055 --> 01:16:35,430 The only question is whether it will stay in the spectrum 919 01:16:35,430 --> 01:16:37,920 or will not stay in the low-energy spectrum. 920 01:16:37,920 --> 01:16:39,350 The mass will not change. 921 01:16:39,350 --> 01:16:41,540 And you will see that when you take that limit, 922 01:16:41,540 --> 01:16:45,130 you find this D3-brane remains here. 923 01:16:45,130 --> 01:16:50,410 And this string, which now will connect some r to r equal to 0, 924 01:16:50,410 --> 01:16:54,140 which is now just some AdS. 925 01:16:54,140 --> 01:16:56,240 This is just some interior or AdS, infinite proper 926 01:16:56,240 --> 01:16:57,460 distance away. 927 01:16:57,460 --> 01:16:59,250 And then you can check that, in the AdS 928 01:16:59,250 --> 01:17:11,660 metric, the image of this brane, viewed from the boundary time-- 929 01:17:11,660 --> 01:17:14,900 you have to this ratio factor, et cetera-- 930 01:17:14,900 --> 01:17:18,022 remains exactly equal to this. 931 01:17:18,022 --> 01:17:19,980 So this is an important self-consistency check, 932 01:17:19,980 --> 01:17:23,390 which I will not do here. 933 01:17:23,390 --> 01:17:24,860 So now, the story becomes simple. 934 01:17:29,620 --> 01:17:34,180 Now we generate such a string, such a fundamental particle 935 01:17:34,180 --> 01:17:35,880 with such a mass. 936 01:17:35,880 --> 01:17:50,660 But in order to introduce an external particle-- in order 937 01:17:50,660 --> 01:17:55,070 to have an external particle take a mass to go to infinity, 938 01:17:55,070 --> 01:17:58,070 so that means we need to take r goes to infinity. 939 01:17:58,070 --> 01:18:00,435 So that means we want to this D3-brane 940 01:18:00,435 --> 01:18:02,400 to lie on the boundary of AdS. 941 01:18:06,110 --> 01:18:13,765 So to summarize, now we want to move this D3-brane-- in order 942 01:18:13,765 --> 01:18:15,744 for the external particle, we want 943 01:18:15,744 --> 01:18:17,410 to take M goes to infinity, then we want 944 01:18:17,410 --> 01:18:18,600 to take r goes to infinity. 945 01:18:18,600 --> 01:18:20,610 So we want this D3-brane to actually lie 946 01:18:20,610 --> 01:18:23,590 on the boundary of AdS. 947 01:18:23,590 --> 01:18:25,423 So let us just conclude, summarize. 948 01:18:32,530 --> 01:18:45,250 An external quark with infinite mass, 949 01:18:45,250 --> 01:18:51,550 in N equals 4 super Yang-Mills theory, 950 01:18:51,550 --> 01:18:57,300 is described by a string ending on the boundary of AdS. 951 01:18:59,920 --> 01:19:01,590 So this is r equal to infinity. 952 01:19:04,360 --> 01:19:06,777 Because now, we have put in this D3-brane to the boundary, 953 01:19:06,777 --> 01:19:08,901 and then we'll, of course, run into a string ending 954 01:19:08,901 --> 01:19:09,882 on the boundary of AdS. 955 01:19:14,220 --> 01:19:16,950 A external quark with [INAUDIBLE] AdS, of course, 956 01:19:16,950 --> 01:19:31,516 run into a string ending at boundary of AdS. 957 01:19:31,516 --> 01:19:33,390 And in particular, the location of the string 958 01:19:33,390 --> 01:19:36,470 can be mapped to the location of the quark. 959 01:19:36,470 --> 01:19:45,931 So then endpoint of the string is the location of the quark. 960 01:19:54,120 --> 01:19:58,550 So now we have found a very nice picture that introduces 961 01:19:58,550 --> 01:20:01,530 a fundamental quark, in N equals 4 super Yang-Mills theory, 962 01:20:01,530 --> 01:20:02,490 because [? what you ?] [? need to ?] [? do with ?] 963 01:20:02,490 --> 01:20:03,073 such a string. 964 01:20:06,410 --> 01:20:08,740 And now, when you do a Wilson loop, 965 01:20:08,740 --> 01:20:17,440 then corresponding to transport this string around some path, 966 01:20:17,440 --> 01:20:19,700 then endpoint of the strings runs 967 01:20:19,700 --> 01:20:21,160 on a path in the field theory. 968 01:20:27,460 --> 01:20:29,910 I thought we would stop here.