1 00:00:00,090 --> 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 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 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 --> 00:00:16,610 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,610 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:21,940 --> 00:00:25,100 HONG LIU: OK, let us start. 9 00:00:25,100 --> 00:00:27,730 So let me first remind you what we 10 00:00:27,730 --> 00:00:29,190 did at the end of last lecture. 11 00:00:34,640 --> 00:00:36,640 So we start talking about how to calculate 12 00:00:36,640 --> 00:00:45,420 the Euclidean correlation functions 13 00:00:45,420 --> 00:00:47,370 or how to relate the Euclidean correlation 14 00:00:47,370 --> 00:00:53,895 functions in the boundary field series to the bulk quantity. 15 00:00:57,670 --> 00:01:02,710 So the question we are always interested in-- so the question 16 00:01:02,710 --> 00:01:06,407 we are always interested in in the field theory 17 00:01:06,407 --> 00:01:07,615 is the generating functional. 18 00:01:12,090 --> 00:01:18,030 And so this is defined, say, in the Euclidean series 19 00:01:18,030 --> 00:01:24,355 to be-- we insert this. 20 00:01:27,490 --> 00:01:34,750 You insert this information into your Lagrangian. 21 00:01:34,750 --> 00:01:37,720 And then you just calculate the correlation function 22 00:01:37,720 --> 00:01:39,360 of this exponential, OK? 23 00:01:43,740 --> 00:01:48,950 So even if I write down a single field phi, single source phi, 24 00:01:48,950 --> 00:01:52,800 and a single operator-- so you should really understand this 25 00:01:52,800 --> 00:01:55,580 as the collapsing of all possible operators-- 26 00:01:55,580 --> 00:01:58,770 all possible operators and all possible corresponding sources 27 00:01:58,770 --> 00:02:01,062 you can put in. 28 00:02:01,062 --> 00:02:02,770 And those are a lot of important things-- 29 00:02:02,770 --> 00:02:07,790 is that you should imagine this exponential as a power series, 30 00:02:07,790 --> 00:02:08,650 OK? 31 00:02:08,650 --> 00:02:11,750 So this expression should-- for the purpose of the generating 32 00:02:11,750 --> 00:02:12,740 functional. 33 00:02:12,740 --> 00:02:17,160 And this exponential should be understood 34 00:02:17,160 --> 00:02:20,010 as a power series in phi. 35 00:02:20,010 --> 00:02:22,380 So we expand it in phi. 36 00:02:22,380 --> 00:02:25,080 And then the term proportion to phi, 37 00:02:25,080 --> 00:02:29,290 then gives you the one-point function proportion to phi 38 00:02:29,290 --> 00:02:31,770 squared, give you the two-point function of O. 39 00:02:31,770 --> 00:02:34,565 And the cubic term would be the three-point function 40 00:02:34,565 --> 00:02:35,940 of O, et cetera. 41 00:02:35,940 --> 00:02:42,120 And so this should-- yeah, this exponential 42 00:02:42,120 --> 00:02:45,930 does not need to make sense non-perturbatively. 43 00:02:45,930 --> 00:02:49,690 You only need to make sense of it as a power series of phi, 44 00:02:49,690 --> 00:02:50,190 OK? 45 00:02:50,190 --> 00:02:52,440 You only need to make sense of it perturbatively 46 00:02:52,440 --> 00:02:57,340 and then just given by the correlation functions of O. 47 00:02:57,340 --> 00:03:00,340 OK, so this is a power series of phi, 48 00:03:00,340 --> 00:03:02,490 which is the coefficient given by the correlation 49 00:03:02,490 --> 00:03:03,997 functions of O, OK? 50 00:03:03,997 --> 00:03:04,580 So this clear? 51 00:03:11,434 --> 00:03:17,460 And then we argued last time that given the relation phi 52 00:03:17,460 --> 00:03:24,230 x is related to some operator to some field phi, 53 00:03:24,230 --> 00:03:25,520 to the boundary value of phi. 54 00:03:31,488 --> 00:03:36,730 Yeah, given that, say, the operator O 55 00:03:36,730 --> 00:03:42,140 is related to some bulk field phi 56 00:03:42,140 --> 00:03:43,880 and the nominal [INAUDIBLE] modes, 57 00:03:43,880 --> 00:03:48,820 the boundary value of phi should be related to the source 58 00:03:48,820 --> 00:03:52,710 corresponding to this o, then you can in some sense 59 00:03:52,710 --> 00:03:56,860 immediately write down what this generating function should 60 00:03:56,860 --> 00:03:59,220 correspond to on the gravity side. 61 00:03:59,220 --> 00:04:01,950 This can be conceivably just a partition function 62 00:04:01,950 --> 00:04:05,850 with this deformation-- with this deformation. 63 00:04:05,850 --> 00:04:11,490 And then this should be the same as the partition 64 00:04:11,490 --> 00:04:13,480 function on the gravity side. 65 00:04:13,480 --> 00:04:19,250 Then we should expect that CFT phi should 66 00:04:19,250 --> 00:04:25,970 be the same as the gravity partition function reaches 67 00:04:25,970 --> 00:04:33,880 the boundary condition, OK? 68 00:04:33,880 --> 00:04:39,890 So we expect this to be true, OK? 69 00:04:39,890 --> 00:04:42,890 And so this is just a heuristic way 70 00:04:42,890 --> 00:04:46,590 of writing the more precise relation. 71 00:04:46,590 --> 00:04:51,160 For example, for scalar, the way to understand 72 00:04:51,160 --> 00:04:58,580 this is that phi xz at z goes to 0 73 00:04:58,580 --> 00:05:03,640 should goes to phi x z d minus delta OK? 74 00:05:06,570 --> 00:05:09,785 So this is a precise mathematical meaning 75 00:05:09,785 --> 00:05:11,710 of this relation. 76 00:05:11,710 --> 00:05:14,100 So the boundary value of capital phi 77 00:05:14,100 --> 00:05:16,420 should be given by the small phi. 78 00:05:16,420 --> 00:05:18,720 And of course, depending on different fields, 79 00:05:18,720 --> 00:05:21,550 then this power may be different. 80 00:05:21,550 --> 00:05:24,800 So last time, we talked about the, say, 81 00:05:24,800 --> 00:05:28,500 for conserved current, we just do it to vector fields. 82 00:05:28,500 --> 00:05:31,500 And then this power will be 0. 83 00:05:31,500 --> 00:05:35,460 And then also for the [INAUDIBLE] perturbation 84 00:05:35,460 --> 00:05:37,410 will be a different power, et cetera, OK? 85 00:05:41,580 --> 00:05:42,960 So any questions regarding this? 86 00:05:52,660 --> 00:05:54,210 Good. 87 00:05:54,210 --> 00:05:59,960 So now, the thing is that in general, 88 00:05:59,960 --> 00:06:04,590 we cannot really verify this relation because the right-hand 89 00:06:04,590 --> 00:06:07,660 side is very hard to compute. 90 00:06:07,660 --> 00:06:11,330 Actually, we don't even know how to define the right-hand side 91 00:06:11,330 --> 00:06:14,390 in the full quantum gravitational regime. 92 00:06:14,390 --> 00:06:16,440 And it's just a formal expression. 93 00:06:16,440 --> 00:06:20,330 Say for quantum gravity exist, then maybe there should 94 00:06:20,330 --> 00:06:24,140 exist a partition function, OK? 95 00:06:24,140 --> 00:06:26,290 But there is a regime-- we do know 96 00:06:26,290 --> 00:06:28,570 how to calculate this side. 97 00:06:28,570 --> 00:06:32,710 So this is so-called classical gravity regime 98 00:06:32,710 --> 00:06:34,900 or also called semi classical regime. 99 00:06:39,510 --> 00:06:43,790 In the semi classical regime, which 100 00:06:43,790 --> 00:06:49,060 is the limit which alpha prime goes to 0 101 00:06:49,060 --> 00:06:53,590 and the G newton goes to 0 or similarly gs 102 00:06:53,590 --> 00:06:57,400 go to-- string coupling goes to zero. 103 00:06:57,400 --> 00:07:04,810 And in this limit, one can formally write this z bulk just 104 00:07:04,810 --> 00:07:08,490 as a Euclidean path integral of all the bulk fields. 105 00:07:11,680 --> 00:07:19,680 OK, so now I use the convention for the Euclidean path integral 106 00:07:19,680 --> 00:07:23,140 without the minus sign. 107 00:07:23,140 --> 00:07:26,830 So we just integrate, say-- so this is your action. 108 00:07:26,830 --> 00:07:30,730 Then you integrate over all possible gravity fields. 109 00:07:30,730 --> 00:07:33,370 Again, this should be integrated as a collection 110 00:07:33,370 --> 00:07:35,970 of all the fields on the gravity side, 111 00:07:35,970 --> 00:07:39,550 including whatever scalar field, gravity, vector field-- you 112 00:07:39,550 --> 00:07:42,220 catch everything, OK? 113 00:07:42,220 --> 00:07:45,529 And there'll be some appropriate boundary condition-- OK, so 114 00:07:45,529 --> 00:07:46,820 appropriate boundary condition. 115 00:07:49,750 --> 00:07:59,080 And so this SE-- so this SE, just the Euclidean action-- 116 00:07:59,080 --> 00:08:00,300 standard Euclidean action. 117 00:08:08,200 --> 00:08:11,120 Say the Einstein term and the plus-- 118 00:08:11,120 --> 00:08:13,270 say the matter terms, OK. 119 00:08:16,680 --> 00:08:24,010 And then the leading order in this limit-- and the meeting 120 00:08:24,010 --> 00:08:26,100 order in that limit, you can just 121 00:08:26,100 --> 00:08:33,299 perform this integral by saddle point of approximation, 122 00:08:33,299 --> 00:08:35,429 which is say, alpha prime goes to 0 limit, 123 00:08:35,429 --> 00:08:37,440 kappa goes to 0 limit. 124 00:08:37,440 --> 00:08:40,361 OK, kappa is essentially the G newton. 125 00:08:40,361 --> 00:08:41,986 So you can calculate this path integral 126 00:08:41,986 --> 00:08:47,760 by saddle point approximation. 127 00:08:47,760 --> 00:08:51,800 Just solve the equation motion following from this action, OK? 128 00:08:51,800 --> 00:08:56,049 I assume you all know what's saddle point approximation. 129 00:08:56,049 --> 00:08:58,090 So the saddle point approximation you just solve. 130 00:09:01,700 --> 00:09:08,740 So z bulk-- that's given by SE. 131 00:09:08,740 --> 00:09:10,460 Now you evaluate it at the solution 132 00:09:10,460 --> 00:09:11,970 of the classical equation motion. 133 00:09:16,390 --> 00:09:18,540 So phi c is just a classical solution. 134 00:09:25,400 --> 00:09:31,301 It's a classic solution which satisfies the right boundary 135 00:09:31,301 --> 00:09:31,800 condition. 136 00:09:35,320 --> 00:09:41,830 Correct boundary condition, OK? 137 00:09:53,170 --> 00:09:58,270 So in this limit, we can really calculate this quantity. 138 00:09:58,270 --> 00:10:01,040 And this is very good news. 139 00:10:01,040 --> 00:10:03,950 This is very good news because this 140 00:10:03,950 --> 00:10:16,490 limit-- because this limit is the-- what did we say before? 141 00:10:16,490 --> 00:10:17,920 N equal to infinity. 142 00:10:17,920 --> 00:10:24,019 And the lambda goes to infinity limit of the boundary theory 143 00:10:24,019 --> 00:10:25,310 of the super Yang-Mills theory. 144 00:10:30,690 --> 00:10:43,630 OK, so we can actually-- so given this, 145 00:10:43,630 --> 00:10:45,900 then essentially we know the generating functional. 146 00:10:45,900 --> 00:10:49,360 Then we can actually calculate the correlation functions 147 00:10:49,360 --> 00:10:53,350 in the strongly coupled field series, OK? 148 00:10:56,850 --> 00:11:00,670 And it's just by solving the classical gravity equation, 149 00:11:00,670 --> 00:11:04,735 just by solving the classical gravity equations. 150 00:11:11,120 --> 00:11:14,600 So we conclude-- this is an important equation. 151 00:11:14,600 --> 00:11:16,560 So let me write it here. 152 00:11:16,560 --> 00:11:22,280 Yes, so let me just make a small remark-- make a small remark. 153 00:11:26,250 --> 00:11:29,790 So to [? read in ?] order, this correlation function 154 00:11:29,790 --> 00:11:33,150 essentially is controlled by this 1 over 2 kappa square. 155 00:11:33,150 --> 00:11:34,860 So this is 1 over 2 newton. 156 00:11:34,860 --> 00:11:45,610 So remember that previously, we-- 157 00:11:45,610 --> 00:11:48,560 when we talk about relation between the G newton 158 00:11:48,560 --> 00:11:51,570 and the field theory relation, this is proportional to N 159 00:11:51,570 --> 00:11:57,050 square, N square of the [INAUDIBLE] theory. 160 00:11:57,050 --> 00:12:00,230 And this is indeed consistent. 161 00:12:00,230 --> 00:12:02,756 So in the saddle point approximation, the SE, 162 00:12:02,756 --> 00:12:06,160 this is just controlled by 1 over 2 kappa square when we 163 00:12:06,160 --> 00:12:08,170 just [? put it ?] to N square. 164 00:12:08,170 --> 00:12:10,370 And this is indeed consistent with our expectation 165 00:12:10,370 --> 00:12:14,630 from [INAUDIBLE] theory that the leading order, the generating 166 00:12:14,630 --> 00:12:16,880 functional, should be proportional to N square, OK? 167 00:12:16,880 --> 00:12:18,088 We support for normalization. 168 00:12:22,960 --> 00:12:24,960 OK, so now we have obtained something powerful. 169 00:12:27,550 --> 00:12:30,870 So now we have obtained something very powerful. 170 00:12:30,870 --> 00:12:34,750 It said in the N equal to infinity 171 00:12:34,750 --> 00:12:44,180 and the lambda goes to infinity limit, the logarithm of CFT, 172 00:12:44,180 --> 00:12:58,280 the generating functional, is related to-- we 173 00:12:58,280 --> 00:13:00,639 said the boundary condition, the boundary value. 174 00:13:00,639 --> 00:13:02,180 Yeah, this is the boundary condition. 175 00:13:02,180 --> 00:13:04,080 Yeah, maybe. 176 00:13:04,080 --> 00:13:06,640 Just, we have this, which is the boundary 177 00:13:06,640 --> 00:13:19,800 value which phi c goes to z to some power of phi x. 178 00:13:19,800 --> 00:13:22,390 And thus, z goes to 0, OK? 179 00:13:29,730 --> 00:13:32,560 OK? 180 00:13:32,560 --> 00:13:36,710 And this is the part of the [? non-normalizable ?] mode 181 00:13:36,710 --> 00:13:41,280 for this bulk field phi, OK? 182 00:13:41,280 --> 00:13:46,730 And this a-- so this precise number a 183 00:13:46,730 --> 00:13:53,350 depends on-- so this exponent a depends on, say, 184 00:13:53,350 --> 00:13:55,120 dimension of the operator. 185 00:13:58,630 --> 00:14:03,430 Dimension, spin-- say, possible other quantum numbers 186 00:14:03,430 --> 00:14:03,985 of operator. 187 00:14:09,370 --> 00:14:15,210 So for scalar, we just have this guy, which we just wrote, OK? 188 00:14:15,210 --> 00:14:21,060 And so let me remind you for the vector, 189 00:14:21,060 --> 00:14:30,170 for the conserved current, which is due to a vector field, 190 00:14:30,170 --> 00:14:35,380 to a gauge field, to a massless vector field-- 191 00:14:35,380 --> 00:14:40,230 and then this gives you a equal to 0. 192 00:14:40,230 --> 00:14:44,260 Yeah, maybe this just gives you a equal to 0 in this case. 193 00:14:44,260 --> 00:14:46,420 So the leading, non-normalizable term-- 194 00:14:46,420 --> 00:14:50,360 so just the boundary value of the vector field 195 00:14:50,360 --> 00:14:55,780 is just given by the source to the current. 196 00:14:55,780 --> 00:14:59,990 And for the metric, we discussed last time, 197 00:14:59,990 --> 00:15:03,087 is that for the metric perturbation, 198 00:15:03,087 --> 00:15:04,545 we just do it to the stress tensor. 199 00:15:09,620 --> 00:15:14,760 So this is corresponding to a equal to 2, OK? 200 00:15:14,760 --> 00:15:19,260 So where this comes from is that, remember-- and last time, 201 00:15:19,260 --> 00:15:22,780 we say that A mu-- say if you have a gauge field, 202 00:15:22,780 --> 00:15:27,500 say a [? mass ?] term, when you go to the boundary, 203 00:15:27,500 --> 00:15:33,530 then you get a mu then plus some b mu Z d minus 2. 204 00:15:33,530 --> 00:15:37,960 So you see, there's no power here, Z equal to 0 here, OK? 205 00:15:37,960 --> 00:15:43,100 Similarly for the metric, component of the metric, 206 00:15:43,100 --> 00:15:47,590 go to 0, goes to 1 over Z square, say, 207 00:15:47,590 --> 00:15:48,520 the boundary value. 208 00:15:53,590 --> 00:15:58,170 And the boundaries-- yeah, delta g mu 209 00:15:58,170 --> 00:16:07,490 is the source to the-- for [? T mu. ?] So here, the a is 210 00:16:07,490 --> 00:16:10,700 actually equal to minus 2. 211 00:16:15,100 --> 00:16:18,104 Or a equal to 2, yeah-- a equal to minus 2. 212 00:16:18,104 --> 00:16:18,645 That's right. 213 00:16:21,381 --> 00:16:21,880 OK? 214 00:16:27,894 --> 00:16:28,810 Any questions on this? 215 00:16:40,740 --> 00:16:42,050 Now let me make some remarks. 216 00:16:53,230 --> 00:16:58,100 So the first remark is what I said a little bit earlier. 217 00:16:58,100 --> 00:17:07,589 It's that the phi x in this generating functional 218 00:17:07,589 --> 00:17:09,459 should be considered as infinitesimal. 219 00:17:23,810 --> 00:17:25,430 OK? 220 00:17:25,430 --> 00:17:27,614 So the reason we want to consider same term 221 00:17:27,614 --> 00:17:28,780 is that what we said before. 222 00:17:28,780 --> 00:17:31,410 We said we should consider the generating functional 223 00:17:31,410 --> 00:17:33,580 simply just as a power series. 224 00:17:33,580 --> 00:17:36,220 Because we don't want to take on perturbation 225 00:17:36,220 --> 00:17:39,826 to deform your regional series, to actually change 226 00:17:39,826 --> 00:17:40,700 your regional series. 227 00:17:40,700 --> 00:17:44,259 We just use it as a generating function. 228 00:17:44,259 --> 00:17:46,300 So similarly, the same thing should be considered 229 00:17:46,300 --> 00:17:49,860 the gravity side, OK? 230 00:17:49,860 --> 00:18:02,160 So similarly on the gravity side-- so this phi 231 00:18:02,160 --> 00:18:05,050 is due to the non-normalizable mode. 232 00:18:05,050 --> 00:18:08,653 So this non-normalizable mode should also be infinitesimal. 233 00:18:27,770 --> 00:18:34,330 OK, so don't disturb your asymptotic ADS geometry. 234 00:18:41,530 --> 00:18:42,280 OK, any questions? 235 00:18:49,130 --> 00:18:50,130 So this is first remark. 236 00:19:04,370 --> 00:19:11,560 The second remark-- now let me call this equation. 237 00:19:11,560 --> 00:19:12,810 Let me give the equation name. 238 00:19:19,090 --> 00:19:32,267 It said that both sides of star are actually divergent, OK? 239 00:19:32,267 --> 00:19:33,225 Are actually divergent. 240 00:19:38,430 --> 00:19:44,220 So the left-hand side, for a very simple reason, 241 00:19:44,220 --> 00:19:47,800 we are considering a quantum field theory. 242 00:19:47,800 --> 00:19:52,380 We are considering a generating function of composite operators 243 00:19:52,380 --> 00:19:53,750 in a quantum field theory. 244 00:19:53,750 --> 00:19:56,530 They're always short distance divergences when the two 245 00:19:56,530 --> 00:20:00,290 operators come together, OK? 246 00:20:00,290 --> 00:20:10,680 So this is just the usual uv divergences of a QFT. 247 00:20:18,150 --> 00:20:20,500 On the right-hand side, we will see soon 248 00:20:20,500 --> 00:20:24,775 in the example this is also divergent. 249 00:20:27,390 --> 00:20:32,976 And the reason it's divergent is also we have seen it before. 250 00:20:32,976 --> 00:20:36,115 So remember the ADS metric is like this. 251 00:20:38,920 --> 00:20:43,850 OK, so I write all boundary directions together 252 00:20:43,850 --> 00:20:46,870 in terms of dx squared. 253 00:20:46,870 --> 00:20:50,240 So because when Z-- when you go to the boundary, Z goes to 0. 254 00:20:50,240 --> 00:20:52,360 And then the whole metric blows up. 255 00:20:52,360 --> 00:20:54,500 So essentially, you have infinite volume. 256 00:20:54,500 --> 00:20:59,710 And yeah, essentially-- so roughly you 257 00:20:59,710 --> 00:21:08,440 can say this is a volume divergence near boundary 258 00:21:08,440 --> 00:21:18,390 of ADS, OK? 259 00:21:18,390 --> 00:21:22,230 So the relation between them, of course, 260 00:21:22,230 --> 00:21:27,770 is precisely what we called IR uv connection. 261 00:21:27,770 --> 00:21:31,650 So the IR-- so the volume divergences associated, say, 262 00:21:31,650 --> 00:21:33,830 with the space time have infinite volume 263 00:21:33,830 --> 00:21:38,178 is related to the uv divergences in the field series sites, OK? 264 00:21:44,050 --> 00:21:47,930 So in order to actually obtain the finite answer, 265 00:21:47,930 --> 00:21:49,370 we need to renormalize them. 266 00:21:58,890 --> 00:22:00,197 We need to renormalize them. 267 00:22:15,260 --> 00:22:18,600 So the idea to role match them from the field theory 268 00:22:18,600 --> 00:22:21,700 point of view, essentially, you need 269 00:22:21,700 --> 00:22:27,350 to add some counterterms to cancel the divergences, OK? 270 00:22:27,350 --> 00:22:31,320 You need to add the counterterms to cancel the divergences. 271 00:22:31,320 --> 00:22:35,740 So the similar logic we do on the gravity side. 272 00:22:35,740 --> 00:22:39,230 So that means that in order to make sense of this expression, 273 00:22:39,230 --> 00:22:46,820 we introduce a renormalized action 274 00:22:46,820 --> 00:22:52,195 which is defined by your original action-- say, 275 00:22:52,195 --> 00:22:53,950 the [INAUDIBLE] action. 276 00:22:53,950 --> 00:22:55,670 But this is divergence. 277 00:22:55,670 --> 00:22:57,440 So we first need to regularize it. 278 00:22:57,440 --> 00:23:00,270 So we normally do regularization. 279 00:23:00,270 --> 00:23:03,300 So instead of evaluating all the way 280 00:23:03,300 --> 00:23:05,830 to z equal to 0, which will be divergent, 281 00:23:05,830 --> 00:23:10,360 we cut it off at some z equal to epsilon. 282 00:23:10,360 --> 00:23:13,430 So this is your full space time. 283 00:23:13,430 --> 00:23:15,580 So z equal to 0 is, again, your boundary. 284 00:23:19,176 --> 00:23:21,050 So you integrate all the way to z equal to 0. 285 00:23:21,050 --> 00:23:22,870 You will get a divergence. 286 00:23:22,870 --> 00:23:27,800 So we just cut it off at a z equal to epsilon, OK? 287 00:23:30,620 --> 00:23:37,150 And then we added the counterterm, 288 00:23:37,150 --> 00:23:39,180 which can be expressed in terms of the phi 289 00:23:39,180 --> 00:23:43,350 c evaluated at this epsilon, OK? 290 00:23:46,780 --> 00:23:49,590 And this [? displaced ?] row should cancel the divergence 291 00:23:49,590 --> 00:23:52,640 here, OK? 292 00:23:52,640 --> 00:23:54,050 Should cancel divergence here. 293 00:23:54,050 --> 00:23:56,900 But this St is much-- so St should 294 00:23:56,900 --> 00:23:58,240 satisfy some simple criterion. 295 00:24:02,990 --> 00:24:08,430 In principle, this St can be arbitrary, 296 00:24:08,430 --> 00:24:11,470 but should satisfy some criterion. 297 00:24:11,470 --> 00:24:28,320 First, this must be local functionals of phi c, OK? 298 00:24:33,970 --> 00:24:38,478 The reason it must be local functionals of phi c 299 00:24:38,478 --> 00:24:41,685 is because this should come from short distance physics. 300 00:24:44,500 --> 00:24:45,900 And short distance physics is not 301 00:24:45,900 --> 00:24:47,610 something we can really control even 302 00:24:47,610 --> 00:24:49,091 in the renormalizable theories. 303 00:24:49,091 --> 00:24:50,590 Even in the renormalizable theories, 304 00:24:50,590 --> 00:24:55,240 we just look at the quantity which 305 00:24:55,240 --> 00:24:57,345 are insensitive to those short distance physics. 306 00:25:00,810 --> 00:25:06,510 So the short distance physics, not something we can control. 307 00:25:06,510 --> 00:25:08,890 But one important thing about short distance physics 308 00:25:08,890 --> 00:25:14,900 is that they all must come from a single point, OK? 309 00:25:14,900 --> 00:25:19,350 So it means the counterterm you act must be local, OK? 310 00:25:19,350 --> 00:25:21,330 And of course must be covariant, et cetera. 311 00:25:21,330 --> 00:25:25,700 So local, covariant-- satisfy those basic conditions 312 00:25:25,700 --> 00:25:29,755 and the covariant, et cetera, with all the right symmetry. 313 00:25:32,260 --> 00:25:34,890 Essentially, this is the only condition 314 00:25:34,890 --> 00:25:38,194 those counterterms should satisfy. 315 00:25:38,194 --> 00:25:40,110 And then you can just choose your counterterms 316 00:25:40,110 --> 00:25:44,650 to cancel the divergent terms here, OK? 317 00:25:44,650 --> 00:25:48,240 But now the key thing is that this 318 00:25:48,240 --> 00:25:52,850 is different from just throwing away the divergent term here. 319 00:25:52,850 --> 00:25:58,630 Because this counterterm also contain some finite terms. 320 00:25:58,630 --> 00:26:01,720 So that's normally, in the [INAUDIBLE] field 321 00:26:01,720 --> 00:26:04,110 series, that may change the finite terms here. 322 00:26:04,110 --> 00:26:06,370 And those finite terms are constrained 323 00:26:06,370 --> 00:26:08,200 by the locality and the covariant 324 00:26:08,200 --> 00:26:09,620 and [? those ?] symmetries. 325 00:26:09,620 --> 00:26:13,200 And actually, there's no ambiguity in them, OK? 326 00:26:13,200 --> 00:26:15,780 Even though the divergent term, in some sense, 327 00:26:15,780 --> 00:26:16,730 there's ambiguity. 328 00:26:16,730 --> 00:26:18,880 But in the finite term, there's no ambiguity. 329 00:26:18,880 --> 00:26:22,630 And so we will see an example at the end of this class. 330 00:26:25,340 --> 00:26:28,870 OK, so we need to renormalize them. 331 00:26:28,870 --> 00:26:32,670 And this is the criterion to normalize it, OK? 332 00:26:32,670 --> 00:26:35,500 So once we have renormalized it, then we 333 00:26:35,500 --> 00:26:38,980 can just obtain the n-point function. 334 00:26:38,980 --> 00:26:50,460 So the general n-point function-- 335 00:26:50,460 --> 00:26:58,890 say, oy x1, on xn-- if you look at this n-point function, 336 00:26:58,890 --> 00:27:01,340 so let's say connected, then we'll 337 00:27:01,340 --> 00:27:04,610 be just corresponding to the standard story. 338 00:27:04,610 --> 00:27:11,515 You just take n derivatives on the partition function, not 339 00:27:11,515 --> 00:27:14,430 the normalized one. 340 00:27:14,430 --> 00:27:19,870 OK, so you take derivative over phi 1 corresponding to o1 341 00:27:19,870 --> 00:27:22,030 and then phi 2 corresponding to o2, 342 00:27:22,030 --> 00:27:27,410 et cetera, and the phi n corresponding to xn. 343 00:27:27,410 --> 00:27:29,060 And then at the end of the calculation, 344 00:27:29,060 --> 00:27:32,480 after you have done the derivative, 345 00:27:32,480 --> 00:27:36,760 you set all the phi to 0, OK? 346 00:27:36,760 --> 00:27:43,700 So then you have extracted n-point function, OK? 347 00:27:43,700 --> 00:27:45,550 So this is the same that you expand 348 00:27:45,550 --> 00:27:49,750 that exponential in power series and then only extract terms 349 00:27:49,750 --> 00:27:51,800 power n in phi, OK? 350 00:27:51,800 --> 00:27:54,110 So same thing. 351 00:27:54,110 --> 00:27:57,180 And now because this is now equal to that-- 352 00:27:57,180 --> 00:28:01,040 so we can write this in terms of gravity quantities. 353 00:28:01,040 --> 00:28:04,120 So writing as n-point function of SE 354 00:28:04,120 --> 00:28:17,790 will normalize SE as a function phi c, now phi 1 x1, phi n xn. 355 00:28:17,790 --> 00:28:21,470 Then at the end, you take phi equal to 0, OK? 356 00:28:21,470 --> 00:28:25,580 All the phi to 0, OK? 357 00:28:25,580 --> 00:28:31,900 So now you in principle, by calculating the gravity action 358 00:28:31,900 --> 00:28:34,520 with the appropriate boundary condition, 359 00:28:34,520 --> 00:28:39,104 then you would have obtained any n-point functions 360 00:28:39,104 --> 00:28:40,020 from the field theory. 361 00:28:47,960 --> 00:28:51,470 Any questions about this? 362 00:28:51,470 --> 00:28:52,220 Yes. 363 00:28:52,220 --> 00:28:54,845 AUDIENCE: Can we check that relations start on the lattice? 364 00:28:57,344 --> 00:28:58,510 HONG LIU: Can we check that? 365 00:28:58,510 --> 00:29:01,170 Yeah, in principle, you can. 366 00:29:01,170 --> 00:29:02,975 But it's just not easy to do. 367 00:29:02,975 --> 00:29:06,300 AUDIENCE: Have people tried? 368 00:29:06,300 --> 00:29:09,620 HONG LIU: So people have tried, not for super Yang-Mills 369 00:29:09,620 --> 00:29:11,200 series. 370 00:29:11,200 --> 00:29:15,720 So there's some version of these for some matrix quantum 371 00:29:15,720 --> 00:29:16,750 mechanics. 372 00:29:16,750 --> 00:29:17,597 People have tried. 373 00:29:17,597 --> 00:29:19,680 In matrix quantum mechanics on the left-hand side, 374 00:29:19,680 --> 00:29:22,520 you can actually calculate using the lattice 375 00:29:22,520 --> 00:29:23,450 or using Monte Carlo. 376 00:29:23,450 --> 00:29:24,800 So maybe not lattice. 377 00:29:24,800 --> 00:29:26,800 It's just a matrix integral. 378 00:29:26,800 --> 00:29:30,580 You can calculate it using a Monte Carlo 379 00:29:30,580 --> 00:29:32,840 and then can compare with the gravity. 380 00:29:32,840 --> 00:29:35,850 Then they turned out to agree with very high accuracy. 381 00:29:38,446 --> 00:29:40,570 But for super Yang-Mills series, the left-hand side 382 00:29:40,570 --> 00:29:46,770 is not doable yet using lattice calculation. 383 00:29:46,770 --> 00:29:49,910 So-- and the one particularly important thing 384 00:29:49,910 --> 00:29:51,830 is the one-point function. 385 00:29:51,830 --> 00:29:55,790 So maybe just separate it to mention elliptics. 386 00:29:55,790 --> 00:29:57,220 So one-point function is important 387 00:29:57,220 --> 00:30:00,500 because one-point function just expectation 388 00:30:00,500 --> 00:30:06,362 value, which of course we often are interested in. 389 00:30:06,362 --> 00:30:07,820 So if you have a symmetry breaking, 390 00:30:07,820 --> 00:30:09,710 whether you develop expectation value 391 00:30:09,710 --> 00:30:13,040 or if you want to extract the expectation value of a stress 392 00:30:13,040 --> 00:30:14,360 tensor, et cetera. 393 00:30:14,360 --> 00:30:18,720 So the one-point function is of very special importance 394 00:30:18,720 --> 00:30:20,290 in many cases. 395 00:30:20,290 --> 00:30:22,290 And also, if you know the one-point function 396 00:30:22,290 --> 00:30:25,650 in the presence of the source, then essentially, 397 00:30:25,650 --> 00:30:28,840 you actually know the full n-point function. 398 00:30:28,840 --> 00:30:31,050 Because you can just take the derivatives here. 399 00:30:31,050 --> 00:30:33,680 Then you can get to the higher point functions because 400 00:30:33,680 --> 00:30:36,536 of the-- yeah, if you know the source here, 401 00:30:36,536 --> 00:30:38,160 if you take one derivative, now you get 402 00:30:38,160 --> 00:30:39,760 two-point function, et cetera. 403 00:30:39,760 --> 00:30:41,980 OK? 404 00:30:41,980 --> 00:30:44,230 So one-point function-- so one-point 405 00:30:44,230 --> 00:30:46,540 function we can write as the following. 406 00:30:46,540 --> 00:30:53,806 So you just take one derivative, delta phi 407 00:30:53,806 --> 00:30:58,900 x, due to this operator, OK? 408 00:30:58,900 --> 00:31:01,490 So now let's do the scalar case. 409 00:31:01,490 --> 00:31:06,300 But similar can be applied to other case, OK? 410 00:31:06,300 --> 00:31:09,890 So you can just replace this phi x 411 00:31:09,890 --> 00:31:15,035 by the boundary value of capital phi. 412 00:31:17,660 --> 00:31:19,970 So this is for the scalar case. 413 00:31:19,970 --> 00:31:27,340 And you could write it as delta SE of phi c. 414 00:31:30,870 --> 00:31:35,300 On the right I said, derivative over phi c z, x. 415 00:31:35,300 --> 00:31:38,230 And as z goes to 0. 416 00:31:38,230 --> 00:31:43,200 Because phi c has to approach phi with this power. 417 00:31:43,200 --> 00:31:45,200 And essentially, you just replace it, OK? 418 00:31:47,819 --> 00:31:50,360 But the reason we want to write this form-- because this form 419 00:31:50,360 --> 00:31:52,109 have a very nice geometric interpretation. 420 00:31:54,580 --> 00:32:00,290 This has a very nice geometric interpretation 421 00:32:00,290 --> 00:32:17,766 because delta SE, delta phi c evaluated at z, x 422 00:32:17,766 --> 00:32:23,320 should be equal to phi c z, x. 423 00:32:23,320 --> 00:32:35,490 And this pi, which is the canonical momentum 424 00:32:35,490 --> 00:32:52,460 conjugate to phi treating the z direction as time. 425 00:32:58,690 --> 00:33:02,090 This z direction treating as time. 426 00:33:02,090 --> 00:33:06,810 And so this pi c just means it's the classical-- means 427 00:33:06,810 --> 00:33:12,960 that this is evaluated to the classical solution-- 428 00:33:12,960 --> 00:33:14,600 classical solution phi c. 429 00:33:18,460 --> 00:33:25,180 So this expression-- is this expression obvious to you? 430 00:33:29,419 --> 00:33:31,774 AUDIENCE: No. 431 00:33:31,774 --> 00:33:33,190 Just from the other graphs. 432 00:33:33,190 --> 00:33:33,793 HONG LIU: Hm? 433 00:33:33,793 --> 00:33:36,168 AUDIENCE: It's just [INAUDIBLE] conjugation, essentially, 434 00:33:36,168 --> 00:33:37,110 right? 435 00:33:37,110 --> 00:33:43,930 HONG LIU: Yeah, this is the-- yeah, actually, 436 00:33:43,930 --> 00:33:46,450 I forgot the name of this equation. 437 00:33:46,450 --> 00:33:49,610 Maybe it's-- normally this equation is taught 438 00:33:49,610 --> 00:33:53,000 in the so-called Hamilton-Jacobi theory. 439 00:33:53,000 --> 00:33:57,325 It's normally called in the Hamilton-Jacobi theory. 440 00:33:57,325 --> 00:34:00,120 But this is a well-known fact in the classical mechanics. 441 00:34:00,120 --> 00:34:03,380 Let me just tell you one thing. 442 00:34:03,380 --> 00:34:07,960 So consider you have a, say, classical mechanical system. 443 00:34:07,960 --> 00:34:12,969 So you move from t0 to t1, OK? 444 00:34:12,969 --> 00:34:14,550 And then suppose you have already 445 00:34:14,550 --> 00:34:16,370 found your classical solution. 446 00:34:16,370 --> 00:34:20,730 So this is a classical trajectory. 447 00:34:20,730 --> 00:34:24,780 And then suppose you evaluate your classical action. 448 00:34:24,780 --> 00:34:33,060 Say so this is t0 to t1 dt, all evaluated at Xc, Xc dot, et 449 00:34:33,060 --> 00:34:34,650 cetera, OK? 450 00:34:34,650 --> 00:34:38,040 Xc is already classical solution. 451 00:34:38,040 --> 00:34:44,350 And so this function-- and let's suppose that Xc at t0 452 00:34:44,350 --> 00:34:49,530 is equal to X0, Xc at t1 equal to X1. 453 00:34:49,530 --> 00:34:55,260 So this is the initial value and the final value 454 00:34:55,260 --> 00:34:56,940 of your trajectory, OK? 455 00:34:59,570 --> 00:35:03,970 And then-- so clearly, because of this boundary condition, 456 00:35:03,970 --> 00:35:09,600 the S depends on X0 and X1. 457 00:35:09,600 --> 00:35:12,360 Then this is an important statement 458 00:35:12,360 --> 00:35:16,440 in classical mechanics-- is that delta S, 459 00:35:16,440 --> 00:35:24,710 you do the valuation over the final value of your location. 460 00:35:24,710 --> 00:35:29,320 This gives you the momentum at that point at P1. 461 00:35:29,320 --> 00:35:36,860 And if you do the valuation over delta X0, 462 00:35:36,860 --> 00:35:40,500 then that gives you the initial momentum, the [INAUDIBLE] 463 00:35:40,500 --> 00:35:43,240 or the initial momentum, at this point, OK? 464 00:35:46,150 --> 00:35:48,755 So if you're not familiar with this fact, 465 00:35:48,755 --> 00:35:53,770 try to derive it yourself when you go home. 466 00:35:53,770 --> 00:35:57,220 Take two minutes to derive it. 467 00:35:57,220 --> 00:35:59,980 If you find the right the way to do it, 468 00:35:59,980 --> 00:36:02,310 it's actually very simple. 469 00:36:02,310 --> 00:36:05,400 Yeah, but it's also in the chapter 1 of Landau, Lifshitz. 470 00:36:05,400 --> 00:36:08,760 And it's very-- essentially the first few pages 471 00:36:08,760 --> 00:36:10,310 of Landau, Lifshitz. 472 00:36:10,310 --> 00:36:12,430 You can also take a look at that. 473 00:36:12,430 --> 00:36:16,730 And now this situation we have here is very similar to that. 474 00:36:16,730 --> 00:36:20,390 So we are evaluating the classical action all the way 475 00:36:20,390 --> 00:36:22,430 to some value of z. 476 00:36:22,430 --> 00:36:26,750 And z essentially is the boundary-- is the time here. 477 00:36:26,750 --> 00:36:31,170 And the phi c is the value of phi at this value of z, 478 00:36:31,170 --> 00:36:34,892 which is the analog of our X1 here, OK? 479 00:36:34,892 --> 00:36:37,100 Then you take the derivative-- so this is a function, 480 00:36:37,100 --> 00:36:37,933 a version with that. 481 00:36:37,933 --> 00:36:40,750 Then you take the derivative over phi c, 482 00:36:40,750 --> 00:36:45,690 which is the value-- because [? one you ?] take derivative 483 00:36:45,690 --> 00:36:46,620 over X1. 484 00:36:46,620 --> 00:36:47,840 Then here, you get the P1. 485 00:36:47,840 --> 00:36:51,013 There, you get equals 1 in momentum conjugate to phi, OK? 486 00:36:51,013 --> 00:36:52,971 So this is just the functional version of that. 487 00:36:56,130 --> 00:36:59,237 Any questions on this? 488 00:36:59,237 --> 00:37:00,945 So now we have a very elegant expression. 489 00:37:06,834 --> 00:37:08,125 We said the one-point function. 490 00:37:11,340 --> 00:37:13,750 We said now, if you look at this-- now, 491 00:37:13,750 --> 00:37:19,530 the one-point function-- so even in the presence of the source, 492 00:37:19,530 --> 00:37:24,190 you just z equal to 0 limit-- z to the power 493 00:37:24,190 --> 00:37:30,450 d minus delta, pi c z, x. 494 00:37:30,450 --> 00:37:32,110 And of course, this should also be 495 00:37:32,110 --> 00:37:39,550 renormalized because of here, we renormalized R. 496 00:37:39,550 --> 00:37:41,590 So here, it should also be renormalized. 497 00:37:41,590 --> 00:37:43,790 And so here-- yeah, OK? 498 00:37:51,698 --> 00:37:54,190 Yeah, so this is a nice expression. 499 00:37:54,190 --> 00:37:57,946 So it tells you that the one-point function 500 00:37:57,946 --> 00:37:59,320 you see essentially just goes one 501 00:37:59,320 --> 00:38:07,370 into a limit of the canonical momentum to that field 502 00:38:07,370 --> 00:38:10,168 evaluated at the boundary, OK? 503 00:38:17,470 --> 00:38:20,180 So now, with a little bit of effort, 504 00:38:20,180 --> 00:38:28,630 we will show-- so you remember phi x, z, 505 00:38:28,630 --> 00:38:34,020 when you go to z equal to 0, as we said last time equal to Ax, 506 00:38:34,020 --> 00:38:35,455 you have two asymptotic behavior. 507 00:38:41,520 --> 00:38:45,570 So this A is related to the source. 508 00:38:45,570 --> 00:38:49,320 So A is related to the phi. 509 00:38:49,320 --> 00:38:55,750 So we discussed last time A is related to the phi, OK? 510 00:38:55,750 --> 00:39:00,470 So now you can show, using this expression, 511 00:39:00,470 --> 00:39:04,690 that the one-point function is actually 512 00:39:04,690 --> 00:39:11,550 exactly equal to 2 nu times B. 513 00:39:11,550 --> 00:39:14,790 And the nu-- yeah, let me just introduce my notation, 514 00:39:14,790 --> 00:39:17,120 remind you of which we had been used last time. 515 00:39:17,120 --> 00:39:18,990 You said the delta. 516 00:39:18,990 --> 00:39:21,350 So last time, we derived this delta 517 00:39:21,350 --> 00:39:24,736 equal to d divided by 2 plus nu. 518 00:39:24,736 --> 00:39:29,610 And the nu is equal to d squared over 4 519 00:39:29,610 --> 00:39:33,200 plus m squared R squared for scalar fields, OK? 520 00:39:36,750 --> 00:39:40,860 So you can show-- so we will do that. 521 00:39:40,860 --> 00:39:45,890 We will reach this point at the end of this class, OK? 522 00:39:45,890 --> 00:39:48,510 So you show that actually, the one-point function precisely 523 00:39:48,510 --> 00:39:50,200 is given by B. 524 00:39:50,200 --> 00:39:52,260 So this is very nice. 525 00:39:52,260 --> 00:39:53,630 This is very nice. 526 00:39:53,630 --> 00:39:56,070 So you have two asymptotic behavior. 527 00:39:56,070 --> 00:39:58,860 And one of them is related to the source. 528 00:39:58,860 --> 00:40:01,650 And the other is related to the expectation value, OK? 529 00:40:07,660 --> 00:40:17,500 And so I urge you to do a self-consistency check yourself 530 00:40:17,500 --> 00:40:22,210 as we did last time that the [INAUDIBLE] you can determine 531 00:40:22,210 --> 00:40:27,221 the dimension of this operator, which is precisely delta. 532 00:40:27,221 --> 00:40:28,720 And from this relation, you can also 533 00:40:28,720 --> 00:40:31,470 check that this relation is compatible. 534 00:40:31,470 --> 00:40:34,840 We said O have dimension delta, OK? 535 00:40:34,840 --> 00:40:38,210 And delta are given by that delta. 536 00:40:38,210 --> 00:40:43,020 Again, this you can check just by the scaling the two side, 537 00:40:43,020 --> 00:40:47,680 the how two sides scale under the scaling, OK? 538 00:40:47,680 --> 00:40:49,316 AUDIENCE: So usually in QFT there's 539 00:40:49,316 --> 00:40:52,360 no relation between the conjugate momentum 540 00:40:52,360 --> 00:40:55,720 and one-point function? 541 00:40:55,720 --> 00:40:57,622 There's no kind of a-- 542 00:40:57,622 --> 00:40:58,330 HONG LIU: No, no. 543 00:40:58,330 --> 00:41:01,840 No, this is conjugate momentum in the gravity side. 544 00:41:01,840 --> 00:41:02,420 This is not-- 545 00:41:02,420 --> 00:41:05,490 AUDIENCE: So within QFT, there isn't any-- 546 00:41:05,490 --> 00:41:08,240 HONG LIU: No, this is the gravity canonical momentum 547 00:41:08,240 --> 00:41:11,552 related to the field theory one-point function. 548 00:41:11,552 --> 00:41:14,010 This has nothing to do with canonical momentum on the field 549 00:41:14,010 --> 00:41:14,970 theory side. 550 00:41:14,970 --> 00:41:17,750 So this is the canonical momentum on the gravity side. 551 00:41:17,750 --> 00:41:21,590 The boundary value of the canonical momentum 552 00:41:21,590 --> 00:41:24,190 on the gravity side with the rate of direction 553 00:41:24,190 --> 00:41:27,490 treated as the time, which gives you the one-point function 554 00:41:27,490 --> 00:41:28,645 in the field theory. 555 00:41:34,560 --> 00:41:35,060 Good. 556 00:41:42,654 --> 00:41:43,410 Good, OK. 557 00:41:43,410 --> 00:41:46,880 So now let's look at them more explicitly. 558 00:41:46,880 --> 00:41:48,920 So let's now give you an example to compute 559 00:41:48,920 --> 00:41:55,680 the two-point function for scalar operator. 560 00:41:55,680 --> 00:41:58,930 And then we'll also derive this relation. 561 00:41:58,930 --> 00:42:02,520 We'll also derive this relation, OK? 562 00:42:02,520 --> 00:42:05,015 So before we do that, do you have any questions? 563 00:42:08,360 --> 00:42:09,940 Yeah. 564 00:42:09,940 --> 00:42:13,262 AUDIENCE: I guess usually in QFT, 565 00:42:13,262 --> 00:42:16,664 when you have Gaussian integrals and one-point function is 566 00:42:16,664 --> 00:42:21,660 0, in what case does this happen here? 567 00:42:21,660 --> 00:42:24,990 HONG LIU: Say-- in many cases, in QFT-- 568 00:42:24,990 --> 00:42:30,790 so if you have any state with a non-zero [? energy, ?] 569 00:42:30,790 --> 00:42:33,070 any state above the ground state where you 570 00:42:33,070 --> 00:42:35,170 have a non-zero stress tensor. 571 00:42:35,170 --> 00:42:38,490 And then the stress tensor will have a one-point function. 572 00:42:38,490 --> 00:42:41,540 And so for example, if you have a symmetry breaking 573 00:42:41,540 --> 00:42:44,645 in the scalar fields, we also have expectation value, yeah. 574 00:42:47,320 --> 00:42:50,280 And also, if you have charge density, than 575 00:42:50,280 --> 00:42:52,969 the current will have expectation value. 576 00:42:52,969 --> 00:42:53,469 Yeah. 577 00:42:58,050 --> 00:43:00,310 Any other questions? 578 00:43:00,310 --> 00:43:04,390 So yeah, actually, before we do the example, 579 00:43:04,390 --> 00:43:07,097 let me also make a brief remark why we actually first 580 00:43:07,097 --> 00:43:09,055 considered the Euclidean correlation functions. 581 00:43:11,680 --> 00:43:17,420 So now first, can I ask the reason 582 00:43:17,420 --> 00:43:20,598 why we want to do the Euclidean correlation function first? 583 00:43:24,530 --> 00:43:25,030 Hm? 584 00:43:25,030 --> 00:43:25,946 AUDIENCE: It's easier? 585 00:43:31,558 --> 00:43:33,430 HONG LIU: It's true. 586 00:43:33,430 --> 00:43:33,930 It's true. 587 00:43:38,930 --> 00:43:40,470 And why it's easier? 588 00:43:43,541 --> 00:43:44,040 Hm? 589 00:43:44,040 --> 00:43:45,270 AUDIENCE: You can-- 590 00:43:45,270 --> 00:43:46,120 HONG LIU: Sorry? 591 00:43:46,120 --> 00:43:48,700 AUDIENCE: Usually Euclidean takes less time. 592 00:43:48,700 --> 00:43:49,560 HONG LIU: It's true. 593 00:43:49,560 --> 00:43:54,390 Yeah, so there's several reasons maybe to save time. 594 00:43:54,390 --> 00:43:57,190 Actually, we don't have much time. 595 00:43:57,190 --> 00:44:03,186 Several reasons-- first Euclidean interval, 596 00:44:03,186 --> 00:44:05,435 you can more easily define it using the Euclidean path 597 00:44:05,435 --> 00:44:07,580 integral. 598 00:44:07,580 --> 00:44:10,330 And so there's no ordering issues, et cetera. 599 00:44:10,330 --> 00:44:14,402 And we just have Euclidean correlation functions. 600 00:44:14,402 --> 00:44:16,110 But the Lorentzian correlation functions, 601 00:44:16,110 --> 00:44:17,500 there are many of them. 602 00:44:17,500 --> 00:44:19,820 So it depends on the orderings, et cetera. 603 00:44:19,820 --> 00:44:23,870 And the typical Lorentzian function cannot be defined 604 00:44:23,870 --> 00:44:26,190 straightforwardly using path integral. 605 00:44:26,190 --> 00:44:28,140 So except the Feynman correlation functions, 606 00:44:28,140 --> 00:44:29,417 say time-ordered. 607 00:44:29,417 --> 00:44:32,000 So if you want to look at other ordered correlation functions, 608 00:44:32,000 --> 00:44:33,770 they cannot be defined straightforward. 609 00:44:33,770 --> 00:44:35,945 They cannot be defined straightforward using path 610 00:44:35,945 --> 00:44:37,020 integral. 611 00:44:37,020 --> 00:44:39,180 And so it's much more complicated. 612 00:44:39,180 --> 00:44:42,570 Actually, for many applications-- in particular, 613 00:44:42,570 --> 00:44:46,990 say, for the linear response for the many body system-- 614 00:44:46,990 --> 00:44:48,760 say for condensed matter or for QCD, 615 00:44:48,760 --> 00:44:51,160 et cetera, mostly we're interested in the retarded 616 00:44:51,160 --> 00:44:52,350 Green function. 617 00:44:52,350 --> 00:44:55,130 And that is not easy to define using path integral. 618 00:44:55,130 --> 00:44:58,250 And so you need to develop some other tricks. 619 00:44:58,250 --> 00:45:00,574 So that's why we are doing the Euclidean first. 620 00:45:00,574 --> 00:45:01,073 OK? 621 00:45:05,240 --> 00:45:16,547 So now let's look at example two-point function of a scalar, 622 00:45:16,547 --> 00:45:17,046 OK? 623 00:45:23,047 --> 00:45:24,880 So this is essentially the simplest example. 624 00:45:29,230 --> 00:45:34,940 And for a two-point function, they only 625 00:45:34,940 --> 00:45:39,124 need to have the quadratic term in phi, OK? 626 00:45:39,124 --> 00:45:40,540 And that means here, you only need 627 00:45:40,540 --> 00:45:43,520 to include the quadratic term in capital phi. 628 00:45:43,520 --> 00:45:46,640 So that means it's enough just to consider the action 629 00:45:46,640 --> 00:45:48,955 to the quadratic reliable, OK? 630 00:45:48,955 --> 00:45:50,580 So that means that if we are interested 631 00:45:50,580 --> 00:45:53,800 in the two-point function to leading order, then 632 00:45:53,800 --> 00:45:57,280 you can just look at the quadratic action. 633 00:45:57,280 --> 00:46:00,780 And for scalar, then, for scalar operator O, 634 00:46:00,780 --> 00:46:03,522 then you are just [INAUDIBLE] massive scalar fields. 635 00:46:03,522 --> 00:46:05,105 So we can write down Euclidean action. 636 00:46:31,230 --> 00:46:33,180 OK? 637 00:46:33,180 --> 00:46:36,740 You don't have to worry about the higher order corrections. 638 00:46:36,740 --> 00:46:39,230 And then the metric is the Euclidean version of this. 639 00:46:39,230 --> 00:46:42,250 It's just the Euclidean metric. 640 00:46:42,250 --> 00:46:46,270 And this is dx squared-- so dx squared, 641 00:46:46,270 --> 00:46:48,640 Euclidean metric, dx squared will 642 00:46:48,640 --> 00:46:52,000 be just delta [? mu ?] mu, dx mu, dx mu. 643 00:46:52,000 --> 00:46:56,140 It's really just the standard Euclidean metric. 644 00:46:59,360 --> 00:47:01,690 So essentially, we take the-- so here, 645 00:47:01,690 --> 00:47:04,020 where previously was minus dt squared 646 00:47:04,020 --> 00:47:05,790 plus the vector x square. 647 00:47:05,790 --> 00:47:08,680 Then we take that dt square into Euclidean signature. 648 00:47:08,680 --> 00:47:10,710 Then it just becomes a pure Euclidean. 649 00:47:10,710 --> 00:47:12,110 All right, I hope this is clear. 650 00:47:16,030 --> 00:47:28,920 And then our goal is just find renormalized phi c using 651 00:47:28,920 --> 00:47:34,078 this action, which satisfies the right boundary condition, OK? 652 00:47:34,078 --> 00:47:35,830 OK, so this is our goal. 653 00:47:35,830 --> 00:47:38,310 And if we find this guy, then you take the derivative. 654 00:47:38,310 --> 00:47:39,684 Then you get two-point functions. 655 00:47:44,860 --> 00:47:47,500 OK, first, let me-- first, you can write down 656 00:47:47,500 --> 00:47:53,760 the canonical momentum with the z as the time direction. 657 00:47:53,760 --> 00:48:01,235 So this is just-- OK, so this is canonical momentum. 658 00:48:03,770 --> 00:48:18,490 Under the equation motion-- OK, equation motion. 659 00:48:21,620 --> 00:48:23,660 And again, because of a translation symmetry 660 00:48:23,660 --> 00:48:26,890 in those X in the boundary directions, 661 00:48:26,890 --> 00:48:30,421 we can write it in the Fourier transform. 662 00:48:30,421 --> 00:48:38,620 You can write x, z in the form of k, z-- 663 00:48:38,620 --> 00:48:45,370 so [? expression ?] ikx, just as we did last time when we solved 664 00:48:45,370 --> 00:48:47,890 the [? free ?] theory. 665 00:48:47,890 --> 00:48:49,795 Then this gives you-- so you plug 666 00:48:49,795 --> 00:48:55,340 this in when you put in the explicit metric here. 667 00:48:55,340 --> 00:49:01,310 Then you get z d plus 1 partial z z 1 minus 668 00:49:01,310 --> 00:49:11,800 d partial z phi minus k squared z squared phi 669 00:49:11,800 --> 00:49:19,544 and then minus m square R square phi equal to 0, OK? 670 00:49:33,230 --> 00:49:35,147 So this is very similar to what we did before, 671 00:49:35,147 --> 00:49:37,688 what we did last time when we talked about normalizable modes 672 00:49:37,688 --> 00:49:39,650 and non-normalizable modes. 673 00:49:39,650 --> 00:49:41,189 This is the same equation. 674 00:49:41,189 --> 00:49:43,730 The only difference is now we are in the Euclidean signature. 675 00:49:53,160 --> 00:49:56,630 So this equation can be solved exactly-- so 676 00:49:56,630 --> 00:49:58,990 in terms of Bessel functions. 677 00:49:58,990 --> 00:50:01,405 But actually, we will not need the explicit solution here 678 00:50:01,405 --> 00:50:02,680 at the end. 679 00:50:02,680 --> 00:50:06,210 So for now, we will not try to solve it explicitly, OK? 680 00:50:06,210 --> 00:50:09,440 So we only do that at the end. 681 00:50:09,440 --> 00:50:13,637 So now let's first look at the on-shell action, OK? 682 00:50:13,637 --> 00:50:14,720 So there's a simple trick. 683 00:50:14,720 --> 00:50:18,350 Because this is a quadratic, you can actually do an integration 684 00:50:18,350 --> 00:50:20,080 by part in the action. 685 00:50:20,080 --> 00:50:28,900 So let's say plug in the-- let's plug in the classical solution 686 00:50:28,900 --> 00:50:35,340 to this equation, to the classic equation, into this action. 687 00:50:35,340 --> 00:50:37,465 So let's do an integration by parts in this term. 688 00:50:45,720 --> 00:50:51,200 Then when you do integration by parts, then you get two terms. 689 00:50:51,200 --> 00:50:54,310 You get a boundary term come from integration by parts. 690 00:50:54,310 --> 00:50:55,762 Then you get the bulk term. 691 00:50:55,762 --> 00:50:57,970 So the bulk term can be written in the following way. 692 00:51:01,450 --> 00:51:07,780 So let me first copy the m square phi c term. 693 00:51:07,780 --> 00:51:10,510 And then it's the term which you obtained by integration 694 00:51:10,510 --> 00:51:12,040 by part of that term. 695 00:51:12,040 --> 00:51:23,410 So that term can be written as partial m 696 00:51:23,410 --> 00:51:38,380 and then plus the boundary term plus the total derivative term. 697 00:51:49,954 --> 00:51:51,245 Just did a simple manipulation. 698 00:51:51,245 --> 00:51:52,950 I did integration by parts here. 699 00:51:56,170 --> 00:51:57,880 So now you recognize this guy. 700 00:52:00,747 --> 00:52:03,330 It's precisely the combination which appeared in that equation 701 00:52:03,330 --> 00:52:05,900 motion. 702 00:52:05,900 --> 00:52:12,520 This is always the case when you have a quadratic action, OK? 703 00:52:12,520 --> 00:52:15,080 So this is just 0. 704 00:52:15,080 --> 00:52:16,680 So we don't need to worry about this. 705 00:52:16,680 --> 00:52:21,638 So this term just said [? N ?] equal to 0 on shell. 706 00:52:21,638 --> 00:52:25,030 So this is just equal to 0. 707 00:52:25,030 --> 00:52:28,090 And so we only have a boundary term. 708 00:52:28,090 --> 00:52:29,340 So this is a total derivative. 709 00:52:29,340 --> 00:52:32,530 But the boundary-- so we always see the boundary term 710 00:52:32,530 --> 00:52:35,317 in, say, the directions along the boundary. 711 00:52:35,317 --> 00:52:37,150 So you only need to worry about the boundary 712 00:52:37,150 --> 00:52:38,195 term in the z direction. 713 00:52:42,650 --> 00:52:46,180 So then this just translated into the z equal to 0 714 00:52:46,180 --> 00:52:49,238 and the z equal to infinity, OK? 715 00:52:54,790 --> 00:53:05,840 So this is translating to minus 1/2 d dx, then square root g, 716 00:53:05,840 --> 00:53:06,650 gzz. 717 00:53:06,650 --> 00:53:09,240 Only the z direction matters. 718 00:53:09,240 --> 00:53:13,442 Then partial z phi times pi c. 719 00:53:13,442 --> 00:53:14,900 Then z equal to 0 and the infinity. 720 00:53:20,950 --> 00:53:24,160 So now you recognize this guy is precisely 721 00:53:24,160 --> 00:53:31,640 our pi with some minus signs-- precisely our pi. 722 00:53:31,640 --> 00:53:38,515 So this can also be written as 1/2 d dx pi 723 00:53:38,515 --> 00:53:44,500 c and the phi c you write it as a 0, infinity-- is equal to 0, 724 00:53:44,500 --> 00:53:48,260 is equal to infinity. 725 00:53:48,260 --> 00:53:50,630 So sometimes, we also write it in momentum space, 726 00:53:50,630 --> 00:53:52,900 which is convenient because we can solve it 727 00:53:52,900 --> 00:53:54,490 in the momentum space. 728 00:53:54,490 --> 00:54:06,330 So it can also be written as d dk 2 pi d phi c kz and the pi 729 00:54:06,330 --> 00:54:11,775 c minus k, z, then 0, infinity. 730 00:54:31,520 --> 00:54:37,830 So we have now reduced it to evaluate this on-shell action 731 00:54:37,830 --> 00:54:41,860 to just two boundary terms at z equal to infinity and z equal 732 00:54:41,860 --> 00:54:45,750 to 0, OK? 733 00:54:45,750 --> 00:54:59,590 So now, we will later see that at infinity, with pi 734 00:54:59,590 --> 00:55:05,270 c times phi c at z equal to infinity just equal to 0, OK. 735 00:55:05,270 --> 00:55:13,106 We will see, OK? 736 00:55:16,610 --> 00:55:20,260 So we will see explicitly that in the right solution, that 737 00:55:20,260 --> 00:55:23,080 both pi c and the phi c, they go to 0. 738 00:55:23,080 --> 00:55:26,060 But actually, there's also a simple way 739 00:55:26,060 --> 00:55:32,570 to argue without looking at the [INAUDIBLE] solution 740 00:55:32,570 --> 00:55:34,170 that this is actually generically also 741 00:55:34,170 --> 00:55:36,265 0 for the following reason. 742 00:55:39,280 --> 00:55:44,850 It's that at infinity, of course, 743 00:55:44,850 --> 00:55:47,750 we want the full solution to be well-defined. 744 00:55:50,280 --> 00:55:54,550 So we should also require phi c or partial z 745 00:55:54,550 --> 00:56:04,300 phi c to be finite, OK? 746 00:56:04,300 --> 00:56:05,970 We don't want them to diverge. 747 00:56:05,970 --> 00:56:09,380 Then you have a singular solution, OK? 748 00:56:09,380 --> 00:56:12,970 So we want them to be finite. 749 00:56:12,970 --> 00:56:14,030 And now you see the pi. 750 00:56:16,620 --> 00:56:20,620 The pi-- you work out the fact of the pi when 751 00:56:20,620 --> 00:56:28,220 the z goes to infinity is proportional to 1 minus d 752 00:56:28,220 --> 00:56:31,301 partial z phi, OK? 753 00:56:34,530 --> 00:56:37,740 Because of the square root g-- square root 754 00:56:37,740 --> 00:56:40,370 g there take the determinant. 755 00:56:40,370 --> 00:56:45,340 It's 1 over z to the power d, d plus 1. 756 00:56:45,340 --> 00:56:48,310 And the gzz inverse is the z square. 757 00:56:48,310 --> 00:56:50,822 So you get this power, OK? 758 00:56:50,822 --> 00:56:52,140 You get this power. 759 00:56:52,140 --> 00:56:55,160 So this-- we generically go to 0 for z 760 00:56:55,160 --> 00:56:59,210 goes to infinity when d is greater than 1, OK? 761 00:56:59,210 --> 00:57:01,140 So for any dimension greater than 1, 762 00:57:01,140 --> 00:57:03,310 if we impose this kind of regularity condition, 763 00:57:03,310 --> 00:57:05,830 then this is always 0. 764 00:57:05,830 --> 00:57:08,380 But in fact, for this case, if you solve those equations 765 00:57:08,380 --> 00:57:11,020 explicitly, you find actually even for d equal to 1, 766 00:57:11,020 --> 00:57:14,334 this goes to 0, OK? 767 00:57:14,334 --> 00:57:15,750 So this is actually generically 0. 768 00:57:15,750 --> 00:57:17,291 So we don't need to worry about them. 769 00:57:22,160 --> 00:57:24,720 So now let's just focus on the z equal to 0 side. 770 00:57:28,370 --> 00:57:30,760 Now let's focus on z equal to 0 side. 771 00:57:40,000 --> 00:57:48,090 OK, so we have this condition for phi 772 00:57:48,090 --> 00:57:51,310 which we derived last time. 773 00:57:51,310 --> 00:57:56,840 It says phi c should satisfy some Ax z to the power 774 00:57:56,840 --> 00:58:01,779 d minus delta plus some Bx, the leading term. 775 00:58:01,779 --> 00:58:03,695 You know, because there are sub leading terms, 776 00:58:03,695 --> 00:58:04,653 this is a leading term. 777 00:58:08,030 --> 00:58:12,660 And so from that expression, you can also obtain the pi c 778 00:58:12,660 --> 00:58:17,410 from that expression-- pi c. 779 00:58:17,410 --> 00:58:19,800 And then you can easily work out. 780 00:58:19,800 --> 00:58:27,300 And that's proportional to, say, minus A d minus 781 00:58:27,300 --> 00:58:31,350 delta, z to the minus delta minus delta 782 00:58:31,350 --> 00:58:36,071 B z delta minus d, OK? 783 00:58:36,071 --> 00:58:40,780 So you just take the derivative, and you find that. 784 00:58:40,780 --> 00:58:43,950 So just intuitively, you have the same thing, 785 00:58:43,950 --> 00:58:48,630 this z minus d factor multiplying 786 00:58:48,630 --> 00:58:50,550 the derivative of phi. 787 00:58:50,550 --> 00:58:53,800 So intuitively, now we go to z equal to 0. 788 00:58:53,800 --> 00:58:55,330 Then this factor actually is bad. 789 00:58:55,330 --> 00:58:58,300 Because this factor goes to infinity, OK? 790 00:58:58,300 --> 00:58:59,884 This factor goes to infinity. 791 00:59:02,780 --> 00:59:09,010 So you see that the pi c indeed is generically 792 00:59:09,010 --> 00:59:12,100 divergent because of this term, OK? 793 00:59:12,100 --> 00:59:14,356 Generically divergent because of this term. 794 00:59:14,356 --> 00:59:16,230 So now let's look at the product between them 795 00:59:16,230 --> 00:59:19,400 because this involves the product between them. 796 00:59:19,400 --> 00:59:26,900 So the leading term product between them, phi c pi c, 797 00:59:26,900 --> 00:59:34,730 is proportional to z to the power of d minus 2 delta, OK? 798 00:59:37,410 --> 00:59:39,270 So this is all the expression for the delta. 799 00:59:42,850 --> 00:59:47,124 And nu is always greater than or equal to 0. 800 00:59:47,124 --> 00:59:49,040 So we see that this term was always divergent. 801 00:59:55,540 --> 01:00:02,680 It's always divergent for nu greater than 0. 802 01:00:08,270 --> 01:00:13,644 And for nu equal to 0, it actually 803 01:00:13,644 --> 01:00:15,310 requires a little bit special treatment. 804 01:00:15,310 --> 01:00:20,890 Because for nu equal to 0, then delta equal to d divided by 2. 805 01:00:20,890 --> 01:00:23,280 Then these two terms have the same power, then 806 01:00:23,280 --> 01:00:25,676 become degenerated in this logarithm term. 807 01:00:25,676 --> 01:00:27,050 And again, it will be divergence. 808 01:00:27,050 --> 01:00:30,620 But I will not go into that, OK? 809 01:00:30,620 --> 01:00:34,259 So the story is that this is always divergent. 810 01:00:34,259 --> 01:00:35,300 This is always divergent. 811 01:00:39,020 --> 01:00:42,050 So this will confirm what we said earlier. 812 01:00:42,050 --> 01:00:50,150 So this tells you that SE phi c is divergent. 813 01:01:02,630 --> 01:01:05,980 So we need to renormalize it. 814 01:01:05,980 --> 01:01:08,480 We need to renormalize it. 815 01:01:08,480 --> 01:01:12,650 So we need to add the counterterm to it, OK? 816 01:01:12,650 --> 01:01:16,350 We need to add the counterterm to it. 817 01:01:16,350 --> 01:01:21,440 So the choice of the counterterm, as we said before, 818 01:01:21,440 --> 01:01:23,050 should be local on the covariant. 819 01:01:30,790 --> 01:01:32,670 So here, it's very simple. 820 01:01:32,670 --> 01:01:34,730 We only work out the quadratic order. 821 01:01:34,730 --> 01:01:37,650 We only need to write down things quadratic in phi. 822 01:01:37,650 --> 01:01:40,180 So the counterterm should also be quadratic in phi. 823 01:01:40,180 --> 01:01:48,710 So we can write counterterm as the following. 824 01:01:48,710 --> 01:01:54,190 So it's easier to write in the momentum space, say, 825 01:01:54,190 --> 01:01:56,065 evaluated at this epsilon. 826 01:01:56,065 --> 01:02:00,530 So we need to introduce our cut-off equal to epsilon 827 01:02:00,530 --> 01:02:03,220 and then introduce a counterterm. 828 01:02:03,220 --> 01:02:06,320 So the counterterm introduced should be local. 829 01:02:09,810 --> 01:02:12,640 So the local term-- so at a quadratic level, 830 01:02:12,640 --> 01:02:16,420 all possible local terms can be written as follows. 831 01:02:16,420 --> 01:02:19,330 We can write it in momentum space as the following. 832 01:02:22,520 --> 01:02:24,740 Say, as something which is a function of k 833 01:02:24,740 --> 01:02:38,250 squared-- phi c, k, say, z phi c minus k, z, OK? 834 01:02:38,250 --> 01:02:40,390 So the criterion is said fk. 835 01:02:40,390 --> 01:02:52,315 So the locality means that fk must be analytic in k square. 836 01:02:59,870 --> 01:03:02,440 So let me just give you some intuition. 837 01:03:02,440 --> 01:03:05,760 So this is a compact expression using momentum space. 838 01:03:05,760 --> 01:03:09,050 But in coordinate space, it's easy to understand. 839 01:03:09,050 --> 01:03:10,503 So this is analytic in k squared. 840 01:03:10,503 --> 01:03:12,510 So you expand it in k square. 841 01:03:12,510 --> 01:03:16,340 So k square is this Lorentz covariant k square-- 842 01:03:16,340 --> 01:03:19,490 using this Lorenz covariant k squared. 843 01:03:23,570 --> 01:03:27,460 Yeah, the same k square is here, which is a Lorentz covariant k 844 01:03:27,460 --> 01:03:29,520 square. 845 01:03:29,520 --> 01:03:33,000 So you can expand this in k square. 846 01:03:33,000 --> 01:03:35,652 So the 0-th order term just a constant. 847 01:03:35,652 --> 01:03:38,110 And the constant, when you go back to the coordinate space, 848 01:03:38,110 --> 01:03:40,050 this is just phi square, OK? 849 01:03:40,050 --> 01:03:40,720 This is simple. 850 01:03:40,720 --> 01:03:41,470 It's a local term. 851 01:03:41,470 --> 01:03:44,270 You can write in terms of two phis. 852 01:03:44,270 --> 01:03:47,130 And the next order term will be k square. 853 01:03:47,130 --> 01:03:49,460 k square, when you go back to coordinate space, 854 01:03:49,460 --> 01:03:52,770 this will be just partial phi square and et cetera. 855 01:03:52,770 --> 01:03:55,890 And indeed, these are the only local terms you can write down. 856 01:03:55,890 --> 01:04:00,870 And so essentially, you can summarize in the momentum space 857 01:04:00,870 --> 01:04:05,550 just as something which is analytic in k square, OK? 858 01:04:08,390 --> 01:04:10,100 So now, we want to choose the k square 859 01:04:10,100 --> 01:04:12,810 to cancel the divergence, OK? 860 01:04:12,810 --> 01:04:15,020 So want to chose the case-- I chose this 861 01:04:15,020 --> 01:04:16,724 to cancel the divergence, OK? 862 01:04:39,480 --> 01:04:45,600 So now, in order to do a-- so now I 863 01:04:45,600 --> 01:04:49,310 need to introduce a little bit of notation. 864 01:04:49,310 --> 01:04:53,640 Because if we want to worry about the locality-- and it's 865 01:04:53,640 --> 01:04:58,570 not enough just to look at this leading term, OK? 866 01:04:58,570 --> 01:05:00,900 Look at leading term-- so remember, 867 01:05:00,900 --> 01:05:08,080 those things-- so those behavior obtained solving this equation 868 01:05:08,080 --> 01:05:09,960 by setting this to be 0. 869 01:05:09,960 --> 01:05:12,590 So when z goes to 0 limit to leading order, 870 01:05:12,590 --> 01:05:15,740 the kz just goes to 0 compared to this term. 871 01:05:15,740 --> 01:05:18,700 So that expression, which we did last time, 872 01:05:18,700 --> 01:05:22,290 are obtained by solving this equation 873 01:05:22,290 --> 01:05:24,610 without this kz squared term. 874 01:05:24,610 --> 01:05:27,460 So we include this kz squared term 875 01:05:27,460 --> 01:05:30,482 so that we include some higher order corrections, OK? 876 01:05:30,482 --> 01:05:32,190 We include some higher order corrections. 877 01:05:32,190 --> 01:05:35,712 But now since we worry about whether this fk squared 878 01:05:35,712 --> 01:05:38,170 is analytic-- so we have to worry about those k dependence. 879 01:05:43,270 --> 01:05:48,750 OK, so now let me introduce some notation, OK? 880 01:05:48,750 --> 01:05:57,710 So let's consider the basis of solution-- 881 01:05:57,710 --> 01:06:05,520 the basis of solutions to this equation, phi 1, 882 01:06:05,520 --> 01:06:10,260 which I call phi 1, and phi 2. 883 01:06:10,260 --> 01:06:12,690 So phi 1 and phi 2 precisely are defined by these two 884 01:06:12,690 --> 01:06:15,530 asymptotic behavior. 885 01:06:15,530 --> 01:06:20,580 So phi 1 has the behavior that goes to d minus delta. 886 01:06:20,580 --> 01:06:22,470 And the phi 2 then have the behavior 887 01:06:22,470 --> 01:06:25,690 when you go to z equal to 0, it goes to z to the power delta 888 01:06:25,690 --> 01:06:27,520 as z goes to 0, OK? 889 01:06:30,620 --> 01:06:33,580 So this is the leading term for phi. 890 01:06:33,580 --> 01:06:42,710 But for small z, this can be written as a power series. 891 01:06:42,710 --> 01:06:46,370 And then the leading term will be just like that. 892 01:06:46,370 --> 01:06:50,570 Then you will have, say, some a1 k 893 01:06:50,570 --> 01:06:55,980 squared z square term come from treating the next order list 894 01:06:55,980 --> 01:07:04,530 term and also higher order terms-- say, k4, z4, et cetera. 895 01:07:04,530 --> 01:07:11,680 So similarly, phi 2 z, k would be start with z 896 01:07:11,680 --> 01:07:13,660 to the power of delta. 897 01:07:13,660 --> 01:07:15,480 Then will be also have higher order terms. 898 01:07:20,440 --> 01:07:22,350 OK? 899 01:07:22,350 --> 01:07:26,040 So the key is that you can work out those coefficients, OK? 900 01:07:26,040 --> 01:07:28,820 Because this is local expansion of infinity, 901 01:07:28,820 --> 01:07:33,160 and you can work out those coefficients explicitly. 902 01:07:33,160 --> 01:07:36,910 And all the dependents on k are local in the sense 903 01:07:36,910 --> 01:07:40,220 that they all are dependant only on k square. 904 01:07:40,220 --> 01:07:42,800 It's because the k square is what k square appears 905 01:07:42,800 --> 01:07:44,900 in the original equations, OK? 906 01:07:48,220 --> 01:07:50,780 And similarly, you can work out the pi. 907 01:07:50,780 --> 01:07:56,200 So the corresponding pi 1 and the pi 2 908 01:07:56,200 --> 01:08:01,095 corresponding canonical momentum for them equal pi 1, pi 2, OK? 909 01:08:01,095 --> 01:08:03,500 Then the pi 1, pi 2, each of them 910 01:08:03,500 --> 01:08:05,680 will have these two behaviors, OK? 911 01:08:05,680 --> 01:08:07,560 Each of them will have these two behaviors 912 01:08:07,560 --> 01:08:09,984 and along with these higher order terms, OK? 913 01:08:19,161 --> 01:08:19,660 Good? 914 01:08:23,359 --> 01:08:44,420 So now-- so now let's look at this on-shell action. 915 01:08:44,420 --> 01:08:51,779 So this on-shell action, before we regularize it, evaluate it 916 01:08:51,779 --> 01:08:54,770 at z equal to epsilon. 917 01:08:54,770 --> 01:09:02,370 So this would be-- oh right, there's one more step. 918 01:09:02,370 --> 01:09:04,160 Sorry. 919 01:09:04,160 --> 01:09:05,920 So using this notation, we can just 920 01:09:05,920 --> 01:09:16,550 write our phi c equal to A phi 1 plus Bk same momentum case-- 921 01:09:16,550 --> 01:09:17,819 phi 2. 922 01:09:17,819 --> 01:09:22,729 So now these are exact expression because the phi 1, 923 01:09:22,729 --> 01:09:25,479 phi 2 are two bases and Ak and AB 924 01:09:25,479 --> 01:09:27,609 are just linear coefficients of that basis, OK? 925 01:09:30,950 --> 01:09:34,434 And the pi c would be similarly Ak, 926 01:09:34,434 --> 01:09:40,882 where pi 1 plus Bk pi 2, OK? 927 01:09:50,850 --> 01:10:09,880 So under the SE phi c would be, if you plug this in, 928 01:10:09,880 --> 01:10:17,770 just have A square pi 1 and phi 1 plus B square pi 929 01:10:17,770 --> 01:10:34,420 2 phi 2 plus AB pi 1 phi 2 plus phi 1 pi 2, OK? 930 01:10:41,400 --> 01:10:44,689 And this is the one that's divergent. 931 01:10:44,689 --> 01:10:46,230 So this is the one which is divergent 932 01:10:46,230 --> 01:10:50,890 coming from the expression we said before. 933 01:10:50,890 --> 01:10:52,610 OK, this is divergent. 934 01:10:52,610 --> 01:10:56,814 And so this is phi 1, this is pi 1, OK? 935 01:10:56,814 --> 01:10:57,750 So this is divergent. 936 01:11:03,860 --> 01:11:07,360 And the other term you can all check are finite, OK? 937 01:11:07,360 --> 01:11:08,670 So this is divergent. 938 01:11:08,670 --> 01:11:10,155 So we need to cancel this term. 939 01:11:13,310 --> 01:11:17,010 But you cannot just abstract this term. 940 01:11:17,010 --> 01:11:20,040 You cannot just abstract this term because this term is not 941 01:11:20,040 --> 01:11:21,860 covariant. 942 01:11:21,860 --> 01:11:25,220 Because A-- because this term is not covariant. 943 01:11:25,220 --> 01:11:28,970 This term does not have the form of the phi square et 944 01:11:28,970 --> 01:11:33,690 cetera, does not have this covariant form, OK? 945 01:11:33,690 --> 01:11:35,580 So this term by itself is not covariant. 946 01:11:35,580 --> 01:11:37,891 So you cannot just subtract this term. 947 01:11:44,020 --> 01:11:49,680 But now, you can easily-- so now, we have phi square here. 948 01:11:49,680 --> 01:11:53,460 So now you can easily pick a phi just to cancel this term, OK? 949 01:11:53,460 --> 01:11:56,340 Just by comparing these two, we can just 950 01:11:56,340 --> 01:11:59,740 pick an f to cancel that term. 951 01:11:59,740 --> 01:12:19,940 So it turns out that f can be written as pi 1 phi 1 times 952 01:12:19,940 --> 01:12:21,722 phi square, OK? 953 01:12:25,600 --> 01:12:29,300 So essentially, this is our f. 954 01:12:29,300 --> 01:12:31,284 It's the ratio between pi 1 and phi 1. 955 01:12:34,180 --> 01:12:38,710 So now if you expand it, since I'm out of time-- so 956 01:12:38,710 --> 01:12:40,400 let me not do it here. 957 01:12:40,400 --> 01:12:43,340 If you expand it, you can easily see this term 958 01:12:43,340 --> 01:12:47,790 cancels that term, OK? 959 01:12:47,790 --> 01:12:49,670 But there are other terms from the covariant. 960 01:12:49,670 --> 01:12:53,430 So there are also finite terms brought by this term. 961 01:12:53,430 --> 01:12:57,450 But there's also finite terms brought by this term. 962 01:12:57,450 --> 01:13:03,200 And this is local because this is a ratio of pi 1 and phi 1. 963 01:13:03,200 --> 01:13:08,410 And both pi 1 and phi 1 are obtained 964 01:13:08,410 --> 01:13:13,600 by solving this equation locally at the infinity 965 01:13:13,600 --> 01:13:16,900 just by power series and dependence on k squared. 966 01:13:16,900 --> 01:13:19,380 It's always analytic, OK? 967 01:13:19,380 --> 01:13:21,270 So this is guaranteed. 968 01:13:21,270 --> 01:13:23,497 So this is analytic function of k squared. 969 01:13:29,101 --> 01:13:32,630 OK, so you take a little bit of exercise. 970 01:13:32,630 --> 01:13:38,087 You work out what is a here, a1 here, et cetera. 971 01:13:38,087 --> 01:13:39,920 Then you can write this term more explicitly 972 01:13:39,920 --> 01:13:43,130 in terms of standard term like phi square, partial phi 973 01:13:43,130 --> 01:13:44,570 square, et cetera, OK? 974 01:13:47,350 --> 01:13:50,869 So now you add these two together. 975 01:13:50,869 --> 01:13:52,285 So now you add these two together. 976 01:13:58,390 --> 01:14:07,010 You get a renormalized counterterm, OK? 977 01:14:07,010 --> 01:14:08,760 Then you find the renormalized expression. 978 01:14:19,790 --> 01:14:23,810 Then you find the renormalized expression 979 01:14:23,810 --> 01:14:30,940 is SR phi c is given by 1/2, again 980 01:14:30,940 --> 01:14:40,180 in momentum space, 2 nu A, minus k, Bk. 981 01:14:48,140 --> 01:14:51,770 So you find everything depends on phi 1, pi. 982 01:14:51,770 --> 01:14:53,300 They all disappear in the end. 983 01:14:53,300 --> 01:14:55,150 You just get something like this. 984 01:14:55,150 --> 01:14:58,910 You get a finite term like this. 985 01:14:58,910 --> 01:15:00,760 So now this is ambiguous. 986 01:15:00,760 --> 01:15:02,020 It's all finite and ambiguous. 987 01:15:07,080 --> 01:15:10,589 So this is your on-shell action. 988 01:15:10,589 --> 01:15:12,880 So normally, what we do is that we solve this equation. 989 01:15:16,250 --> 01:15:20,020 Then we can find the explicit A and B. 990 01:15:20,020 --> 01:15:22,850 So we solve this equation with the following boundary 991 01:15:22,850 --> 01:15:24,177 conditions. 992 01:15:24,177 --> 01:15:26,135 It said we first impose the boundary condition. 993 01:15:28,900 --> 01:15:35,630 You have Ak equal to phi k because A is the source. 994 01:15:35,630 --> 01:15:38,940 A is identified with the source. 995 01:15:38,940 --> 01:15:47,610 But we also need to impose regularity 996 01:15:47,610 --> 01:15:50,340 at z equal to infinity. 997 01:15:50,340 --> 01:15:53,650 So we wanted the solution to be well-defined 998 01:15:53,650 --> 01:15:56,210 at z equal to infinity. 999 01:15:56,210 --> 01:15:59,230 So it turns out these are the two conditions which uniquely 1000 01:15:59,230 --> 01:16:00,959 determine the solution. 1001 01:16:00,959 --> 01:16:02,750 So that means these two conditions together 1002 01:16:02,750 --> 01:16:09,680 which will determine B in terms of A. OK, 1003 01:16:09,680 --> 01:16:13,350 so this will determine B in terms of A. 1004 01:16:13,350 --> 01:16:19,540 So in particular, this is a linear solution. 1005 01:16:19,540 --> 01:16:24,939 So B will have the form which-- some number chi times A. So chi 1006 01:16:24,939 --> 01:16:26,730 is the thing which you determine by solving 1007 01:16:26,730 --> 01:16:29,950 the equation explicitly, OK? 1008 01:16:29,950 --> 01:16:36,492 This is a form-- chi you obtain from solving the equation. 1009 01:16:44,110 --> 01:16:45,595 Just short a couple minutes. 1010 01:16:45,595 --> 01:16:50,811 It's just-- so now you plug this into that. 1011 01:16:50,811 --> 01:16:51,810 You plug this into that. 1012 01:17:00,400 --> 01:17:01,610 Plug this into that. 1013 01:17:04,160 --> 01:17:06,470 Unfortunately, I erased my equation. 1014 01:17:10,650 --> 01:17:12,090 OK, so you plug this into that. 1015 01:17:12,090 --> 01:17:15,550 So chi you assume is some known function 1016 01:17:15,550 --> 01:17:17,770 which you obtain after you are solving 1017 01:17:17,770 --> 01:17:21,000 the equation explicitly. 1018 01:17:21,000 --> 01:17:25,620 Then what you get-- so without that page. 1019 01:17:25,620 --> 01:17:33,110 So what you get is you will get S renormalized action phi 1020 01:17:33,110 --> 01:17:36,610 c equal to 1/2 the integration. 1021 01:17:36,610 --> 01:17:42,485 So 2 nu Ak-- 2 nu lambda-- or we here 1022 01:17:42,485 --> 01:17:49,040 call chi-- Ak and A, minus k, OK? 1023 01:17:49,040 --> 01:17:50,860 So you remember this is just all phi k. 1024 01:17:53,920 --> 01:17:55,150 This is just all phi k. 1025 01:17:55,150 --> 01:17:58,710 So now you can obtain the one-point function. 1026 01:17:58,710 --> 01:18:04,610 Just take the derivative over phi-- say, minus k. 1027 01:18:04,610 --> 01:18:05,820 You get the Ok. 1028 01:18:08,610 --> 01:18:15,470 And this is precisely just 2 nu chi times Ak. 1029 01:18:15,470 --> 01:18:19,677 And this is precisely 2 nu B, OK? 1030 01:18:19,677 --> 01:18:21,510 And you can write it in the coordinate space 1031 01:18:21,510 --> 01:18:27,310 also, Ox equal to 2 nu B. 1032 01:18:27,310 --> 01:18:29,630 So this we derive in the linear order 1033 01:18:29,630 --> 01:18:32,120 by looking at the quadratic equation motions. 1034 01:18:32,120 --> 01:18:34,420 But actually, this turns out you can prove it actually 1035 01:18:34,420 --> 01:18:35,550 to all order. 1036 01:18:35,550 --> 01:18:37,965 You work with nonlinear equations, et cetera, 1037 01:18:37,965 --> 01:18:41,460 you can show this is always true that the one-point function is 1038 01:18:41,460 --> 01:18:43,510 always given by this B, OK? 1039 01:18:43,510 --> 01:18:46,844 One-point function is always given by this B. 1040 01:18:46,844 --> 01:18:49,196 So let me say one thing. 1041 01:18:49,196 --> 01:18:58,470 So now the two-point function-- so the two-point function, you 1042 01:18:58,470 --> 01:18:59,690 just take two derivatives. 1043 01:19:04,700 --> 01:19:07,670 You take two derivatives. 1044 01:19:07,670 --> 01:19:14,240 So that just gives you 2 nu chi, OK? 1045 01:19:14,240 --> 01:19:17,002 Just the 2 nu chi. 1046 01:19:17,002 --> 01:19:20,370 So this can also be written as 2 nu B divided 1047 01:19:20,370 --> 01:19:28,140 by A because the chi is equal to-- 1048 01:19:28,140 --> 01:19:29,940 and this makes perfect sense physically 1049 01:19:29,940 --> 01:19:35,910 because this is our Euclidean correlation function, OK? 1050 01:19:35,910 --> 01:19:38,110 And the 2 nu divided by A can also 1051 01:19:38,110 --> 01:19:40,640 be written using what we write there. 1052 01:19:40,640 --> 01:19:44,110 So this is the expectation value divided 1053 01:19:44,110 --> 01:19:47,700 by phi k divided by the source, so A because it's phi 1054 01:19:47,700 --> 01:19:50,190 and 2 nu B is expectation value. 1055 01:19:50,190 --> 01:19:52,620 And then we see this expression is precisely 1056 01:19:52,620 --> 01:19:55,170 a so-called linear response. 1057 01:19:55,170 --> 01:19:58,250 The expectation value in the presence of the source 1058 01:19:58,250 --> 01:20:05,186 is the correlation function times the source-- 1059 01:20:05,186 --> 01:20:07,440 the correlation times the source. 1060 01:20:07,440 --> 01:20:09,840 So again, this equation is completely general. 1061 01:20:09,840 --> 01:20:11,610 So for any two-point function case, 1062 01:20:11,610 --> 01:20:15,130 we'll always find this B divided by A times 2 nu. 1063 01:20:15,130 --> 01:20:19,270 And you can also alternatively written it 1064 01:20:19,270 --> 01:20:25,430 as the ratio O divided by phi, OK? 1065 01:20:25,430 --> 01:20:26,565 So just the final minute. 1066 01:20:29,136 --> 01:20:30,886 Now you can solve the equation explicitly. 1067 01:20:34,700 --> 01:20:38,660 So now you can solve the equation explicitly. 1068 01:20:38,660 --> 01:20:41,380 You can solve the equation explicitly. 1069 01:20:41,380 --> 01:20:46,598 And you can solve your [? eom ?] using the Bessel function. 1070 01:20:52,340 --> 01:20:54,810 And then you find that actually, the function which 1071 01:20:54,810 --> 01:20:58,860 is no [INAUDIBLE] the infinity at z equal to infinity 1072 01:20:58,860 --> 01:21:01,590 is given by this form. 1073 01:21:01,590 --> 01:21:07,330 k-- so this Bessel k function with index nu. 1074 01:21:07,330 --> 01:21:08,900 Nu is the same as this nu. 1075 01:21:12,540 --> 01:21:15,510 And then from here, you just expand in small z. 1076 01:21:15,510 --> 01:21:18,590 Then you can find the B divided by A. 1077 01:21:18,590 --> 01:21:22,940 Then you find B divided by A as gamma, minus nu, gamma nu. 1078 01:21:22,940 --> 01:21:25,650 So from here, you just expand in small z. 1079 01:21:25,650 --> 01:21:27,320 And then you extract different power. 1080 01:21:27,320 --> 01:21:30,740 You obtain the ratio of B divided by A, OK? 1081 01:21:30,740 --> 01:21:34,290 So as you get gamma nu-- minus nu, 1082 01:21:34,290 --> 01:21:39,670 gamma nu k divided by 2 to the power 2 nu. 1083 01:21:43,890 --> 01:21:45,680 So you find the two-point function. 1084 01:21:50,160 --> 01:21:53,330 So since the two-point function is just the 2 nu B, 1085 01:21:53,330 --> 01:21:56,400 A-- so you find the Euclidean two-point function 1086 01:21:56,400 --> 01:21:58,890 in momentum space. 1087 01:21:58,890 --> 01:22:04,600 It's just 2 nu gamma, minus nu, gamma nu, 1088 01:22:04,600 --> 01:22:08,860 k divided by 2 to the 2 nu. 1089 01:22:08,860 --> 01:22:11,330 So in some sense, this is precisely what 1090 01:22:11,330 --> 01:22:12,940 you expect because we are working 1091 01:22:12,940 --> 01:22:15,830 with the conformal field theory. 1092 01:22:15,830 --> 01:22:19,320 And the two-point function can only depend on k as a power 1093 01:22:19,320 --> 01:22:21,400 because there's no other scale. 1094 01:22:21,400 --> 01:22:22,560 There's no other scale. 1095 01:22:22,560 --> 01:22:25,800 And the power is determined by the dimension of the operator. 1096 01:22:25,800 --> 01:22:30,080 And for example, if you go to the coordinate space-- 1097 01:22:30,080 --> 01:22:32,370 so you Fourier transform, go to coordinate space. 1098 01:22:32,370 --> 01:22:35,842 So that just gives you the X to the power 2 delta. 1099 01:22:39,560 --> 01:22:43,240 Sorry-- I will stop here.