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 --> 00:00:06,060 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 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:21,715 --> 00:00:22,340 HONG LIU: Good. 9 00:00:22,340 --> 00:00:24,510 So before we start, do you have any questions 10 00:00:24,510 --> 00:00:26,490 on what we did before? 11 00:00:32,930 --> 00:00:33,430 OK. 12 00:00:33,430 --> 00:00:39,600 So we have discussed that a black hole can be 13 00:00:39,600 --> 00:00:43,250 considered as a similar object. 14 00:00:43,250 --> 00:00:47,911 And that, actually, black holes obey the standard law 15 00:00:47,911 --> 00:00:48,660 of thermodynamics. 16 00:00:51,330 --> 00:00:59,260 And now we also mentioned that it's a natural question 17 00:00:59,260 --> 00:01:02,530 to ask whether the black hole should we 18 00:01:02,530 --> 00:01:06,200 be considered as an ordinary quantum mechanical object. 19 00:01:06,200 --> 00:01:09,020 And so this is related to the issue of information paradox. 20 00:01:11,680 --> 00:01:15,790 So by this regard, this information paradox, 21 00:01:15,790 --> 00:01:21,560 which I think now has a very good-- in principle 22 00:01:21,560 --> 00:01:25,950 has a reasonable resolution, which we will talk about later. 23 00:01:25,950 --> 00:01:29,770 But if we do treat the black hole as an ordinary quantum 24 00:01:29,770 --> 00:01:36,080 mechanical object, then they actually 25 00:01:36,080 --> 00:01:38,055 are very important implications. 26 00:01:42,920 --> 00:01:47,115 So one implication is called the holographic principle. 27 00:01:58,000 --> 00:02:04,290 So, the holographic principle by itself is, in some sense, 28 00:02:04,290 --> 00:02:11,280 a rather heuristic concept at the moment. 29 00:02:11,280 --> 00:02:19,450 And I will also only give you a very heuristic discussion. 30 00:02:19,450 --> 00:02:21,830 And it's possible to make it a little bit more rigorous. 31 00:02:21,830 --> 00:02:27,370 But we will do it a rather heuristic level, 32 00:02:27,370 --> 00:02:31,340 just to give you a feeling that there 33 00:02:31,340 --> 00:02:38,410 is something profound going on in quantum gravity because 34 00:02:38,410 --> 00:02:39,490 of black holes. 35 00:02:44,720 --> 00:03:03,960 So let's consider an isolated system of mass. 36 00:03:03,960 --> 00:03:17,900 Say E and, say, entropy as zero, say 37 00:03:17,900 --> 00:03:20,340 in asymptotic flat spacetime. 38 00:03:35,280 --> 00:03:47,600 And then we also denote A as the area of the smallest 39 00:03:47,600 --> 00:03:59,045 sphere that closes the system. 40 00:04:08,550 --> 00:04:13,745 And also let me introduce-- let MA 41 00:04:13,745 --> 00:04:27,350 be the mass of the black hole, of the same horizon area. 42 00:04:31,140 --> 00:04:33,330 Yes. 43 00:04:33,330 --> 00:04:42,415 Black hole with horizon area A. OK? 44 00:04:47,820 --> 00:04:53,490 So in GR, anything which is sufficiently massive 45 00:04:53,490 --> 00:04:56,980 will eventually collapse to form a black hole. 46 00:04:56,980 --> 00:05:10,720 Because of that-- so we must have E is smaller than MA, 47 00:05:10,720 --> 00:05:13,230 So we must have E-- so whatever energy 48 00:05:13,230 --> 00:05:16,790 you can put in this region surrounded by area 49 00:05:16,790 --> 00:05:22,850 A must be smaller than the mass of a black hole 50 00:05:22,850 --> 00:05:24,840 which you can put in that region. 51 00:05:24,840 --> 00:05:25,340 OK? 52 00:05:28,730 --> 00:05:30,750 Because otherwise, if we have more mass, 53 00:05:30,750 --> 00:05:34,215 then it will already be a black hole. 54 00:05:34,215 --> 00:05:35,549 AUDIENCE: Should that be equal? 55 00:05:35,549 --> 00:05:36,090 HONG LIU: Hm? 56 00:05:36,090 --> 00:05:36,946 AUDIENCE: Should it be equal? 57 00:05:36,946 --> 00:05:37,760 HONG LIU: Yeah. 58 00:05:37,760 --> 00:05:38,380 Say equal. 59 00:05:48,400 --> 00:05:50,670 So now let's suppose E is smaller than MA. 60 00:05:55,880 --> 00:06:04,100 So we can now-- so suppose E is smaller than MA 61 00:06:04,100 --> 00:06:09,720 and now we can add this amount of energy, 62 00:06:09,720 --> 00:06:21,840 say MA minus E amount of energy, to the system. 63 00:06:30,200 --> 00:06:30,700 OK. 64 00:06:30,700 --> 00:06:32,395 Suppose we can do that and keep A fixed. 65 00:06:38,950 --> 00:06:42,490 Suppose we can do that and keep A fixed. 66 00:06:42,490 --> 00:06:45,140 And then this will reach a black hole mass. 67 00:06:48,400 --> 00:06:59,410 Then the final system will have a black hole, 68 00:06:59,410 --> 00:07:01,790 will have the energy given by MA. 69 00:07:04,680 --> 00:07:13,380 And that object must be a black hole 70 00:07:13,380 --> 00:07:15,755 because that's the black hole with the same horizon area. 71 00:07:24,760 --> 00:07:26,740 Because this process also tells you 72 00:07:26,740 --> 00:07:32,050 that the black hole entropy must be greater than the S0, 73 00:07:32,050 --> 00:07:37,650 the initial entropy of this system, plus the S prime-- S 74 00:07:37,650 --> 00:07:39,635 prime is the entropy of added energy. 75 00:07:48,280 --> 00:07:49,690 OK. 76 00:07:49,690 --> 00:07:52,290 So just because of the second law of thermodynamics, 77 00:07:52,290 --> 00:07:55,450 your entropy should not decrease. 78 00:07:55,450 --> 00:08:02,376 So the final-- so you add this amount of energy so to MA, 79 00:08:02,376 --> 00:08:07,630 so then you will precisely form a black hole over the mass MA, 80 00:08:07,630 --> 00:08:11,450 and with the horizon area A. But this black hole entropy 81 00:08:11,450 --> 00:08:15,550 should be greater than its zero plus the entropy you 82 00:08:15,550 --> 00:08:17,550 added from the second law of thermodynamics. 83 00:08:20,530 --> 00:08:29,090 So then that means your initial entropy must be smaller 84 00:08:29,090 --> 00:08:31,650 than the black hole entropy. 85 00:08:35,530 --> 00:08:41,020 Must be smaller than A divided by 4 h bar G newton. 86 00:08:45,980 --> 00:08:59,630 So this argument tells you that the maximal entropy 87 00:08:59,630 --> 00:09:16,360 inside a region bounded by area A is given by-- 88 00:09:28,615 --> 00:09:29,115 OK. 89 00:09:33,280 --> 00:09:33,780 Yes? 90 00:09:33,780 --> 00:09:37,210 AUDIENCE: So will keeping A fixed add entropy 91 00:09:37,210 --> 00:09:38,680 to the system? 92 00:09:38,680 --> 00:09:40,038 HONG LIU: Yeah. 93 00:09:40,038 --> 00:09:44,299 AUDIENCE: So in that case, why don't we have such a term? 94 00:09:44,299 --> 00:09:44,965 HONG LIU: Sorry? 95 00:09:44,965 --> 00:09:47,760 AUDIENCE: So you said we want to keep A fixed. 96 00:09:47,760 --> 00:09:48,343 HONG LIU: Yes. 97 00:09:48,343 --> 00:09:50,760 AUDIENCE: Will that add extra entropy to the system? 98 00:09:50,760 --> 00:09:52,450 HONG LIU: Yes. 99 00:09:52,450 --> 00:09:56,370 AUDIENCE: So according to the second law of thermodynamics, 100 00:09:56,370 --> 00:10:00,630 why will we claim that the entropy of the hole 101 00:10:00,630 --> 00:10:05,000 is greater than or equal than S0 plus S-- 102 00:10:05,000 --> 00:10:06,144 HONG LIU: Yeah. 103 00:10:06,144 --> 00:10:06,685 AUDIENCE: Oh. 104 00:10:06,685 --> 00:10:07,185 OK. 105 00:10:07,185 --> 00:10:08,370 It's add the energy. 106 00:10:08,370 --> 00:10:08,600 HONG LIU: Yeah. 107 00:10:08,600 --> 00:10:09,350 Because it-- Yeah. 108 00:10:09,350 --> 00:10:11,356 This is the energy of added energy. 109 00:10:11,356 --> 00:10:14,670 AUDIENCE: How about the extra energy 110 00:10:14,670 --> 00:10:17,840 that might be introduced by keeping A fixed? 111 00:10:17,840 --> 00:10:18,915 Because it's not-- 112 00:10:18,915 --> 00:10:19,540 HONG LIU: Yeah. 113 00:10:19,540 --> 00:10:20,720 Whatever it is. 114 00:10:20,720 --> 00:10:21,670 Whatever it is. 115 00:10:21,670 --> 00:10:24,300 Suppose you can keep A fixed. 116 00:10:24,300 --> 00:10:25,930 Suppose you can keep A fixed. 117 00:10:25,930 --> 00:10:29,680 Then whatever things you added to that-- 118 00:10:29,680 --> 00:10:33,070 that total entropy-- must be smaller than the [INAUDIBLE] 119 00:10:33,070 --> 00:10:34,780 because black hole is the final state. 120 00:10:34,780 --> 00:10:40,036 AUDIENCE: You said it's an isolated system, so with that, 121 00:10:40,036 --> 00:10:42,210 how can you make sure that A is fixed? 122 00:10:42,210 --> 00:10:43,190 HONG LIU: Hm? 123 00:10:43,190 --> 00:10:44,760 AUDIENCE: Why is an isolated system-- 124 00:10:44,760 --> 00:10:45,255 HONG LIU: Oh. 125 00:10:45,255 --> 00:10:45,755 Yeah. 126 00:10:45,755 --> 00:10:48,010 I'm just saying, we are doing a thought experiment. 127 00:10:48,010 --> 00:10:52,130 Imagine you can do this by keeping A fixed. 128 00:10:52,130 --> 00:10:58,600 Yeah, just say you just put stuff in into this region. 129 00:10:58,600 --> 00:11:04,320 So that stuff does not come out. 130 00:11:04,320 --> 00:11:08,790 Yeah, you just have some region A. Let me do it here. 131 00:11:08,790 --> 00:11:16,570 You have some region A. And there is a mass inside here. 132 00:11:16,570 --> 00:11:18,330 There's some energy stuff inside here. 133 00:11:18,330 --> 00:11:23,980 So this is region A. And then you just keep putting stuff in. 134 00:11:23,980 --> 00:11:26,690 Just keep putting stuff in so that those things 135 00:11:26,690 --> 00:11:29,990 don't come out A-- don't come out of this region. 136 00:11:29,990 --> 00:11:32,830 That's what this means. 137 00:11:32,830 --> 00:11:35,670 AUDIENCE: Then A is not the smallest sphere? 138 00:11:35,670 --> 00:11:36,590 HONG LIU: Hm? 139 00:11:36,590 --> 00:11:40,730 AUDIENCE: Then A is not the smallest sphere? 140 00:11:40,730 --> 00:11:41,640 HONG LIU: A is what? 141 00:11:41,640 --> 00:11:43,690 AUDIENCE: Not the smallest sphere 142 00:11:43,690 --> 00:11:45,485 that can enclose the system. 143 00:11:45,485 --> 00:11:47,010 So it's not-- 144 00:11:47,010 --> 00:11:47,640 HONG LIU: No. 145 00:11:47,640 --> 00:11:50,370 A is not the smallest sphere in the system. 146 00:11:50,370 --> 00:11:56,750 A is just-- maybe I should call this-- A is 147 00:11:56,750 --> 00:12:01,164 the area of a sphere. 148 00:12:01,164 --> 00:12:02,640 Yeah. 149 00:12:02,640 --> 00:12:03,880 Sorry. 150 00:12:03,880 --> 00:12:04,800 Yeah. 151 00:12:04,800 --> 00:12:06,520 Thanks. 152 00:12:06,520 --> 00:12:11,820 So just enough A to be a sphere that encloses the system. 153 00:12:11,820 --> 00:12:12,410 Yeah. 154 00:12:12,410 --> 00:12:15,340 Should not put the smallest sphere for some reason. 155 00:12:18,270 --> 00:12:18,770 Yeah. 156 00:12:18,770 --> 00:12:19,290 No, no, no. 157 00:12:19,290 --> 00:12:20,970 A should not be the smallest. 158 00:12:20,970 --> 00:12:25,200 Just-- A is any sphere. 159 00:12:25,200 --> 00:12:29,680 Just some sphere which encloses the region. 160 00:12:29,680 --> 00:12:30,180 Yes? 161 00:12:30,180 --> 00:12:32,305 AUDIENCE: So it seems that this argument implicitly 162 00:12:32,305 --> 00:12:34,110 assumed that in order to add entropy, 163 00:12:34,110 --> 00:12:36,770 that sort of corresponds to adding some amount of energy. 164 00:12:36,770 --> 00:12:38,840 But is it possible to add entropy to the system 165 00:12:38,840 --> 00:12:40,840 without adding energy to the system? 166 00:12:40,840 --> 00:12:42,041 HONG LIU: Oh, you can. 167 00:12:42,041 --> 00:12:42,540 You can. 168 00:12:42,540 --> 00:12:44,420 They just make it even more. 169 00:12:44,420 --> 00:12:47,760 Whatever it is, this is the final state. 170 00:12:47,760 --> 00:12:51,290 So the base assumption of this is 171 00:12:51,290 --> 00:12:57,310 that if you can put stuff inside this region, whatever 172 00:12:57,310 --> 00:13:02,146 is inside, the sum is, the maximal stuff you 173 00:13:02,146 --> 00:13:05,020 can put inside the region is a black hole with the horizon 174 00:13:05,020 --> 00:13:06,992 area. 175 00:13:06,992 --> 00:13:08,700 AUDIENCE: Well, I guess what I'm confused 176 00:13:08,700 --> 00:13:11,270 about is that it seems, that-- I certainly believe when 177 00:13:11,270 --> 00:13:13,170 you said, you can't have a certain amount 178 00:13:13,170 --> 00:13:14,050 of energy in an area. 179 00:13:14,050 --> 00:13:15,620 But I'm not sure why you can't have 180 00:13:15,620 --> 00:13:17,220 a certain amount of entropy in an area 181 00:13:17,220 --> 00:13:19,130 if energy and entropy are not-- 182 00:13:19,130 --> 00:13:19,950 HONG LIU: No. 183 00:13:19,950 --> 00:13:22,070 We're not making that assumption. 184 00:13:22,070 --> 00:13:26,720 The entropy is just a consequence of that. 185 00:13:26,720 --> 00:13:29,070 This is the final state entropy. 186 00:13:29,070 --> 00:13:31,080 And the final state entropy is always 187 00:13:31,080 --> 00:13:34,566 larger than the entropy of your initial state. 188 00:13:34,566 --> 00:13:35,107 AUDIENCE: OK. 189 00:13:35,107 --> 00:13:35,607 Sure. 190 00:13:35,607 --> 00:13:36,360 Fine. 191 00:13:36,360 --> 00:13:36,680 HONG LIU: Yeah. 192 00:13:36,680 --> 00:13:37,180 Yeah. 193 00:13:37,180 --> 00:13:38,157 Entropy is derived. 194 00:13:38,157 --> 00:13:39,490 This equation is not assumption. 195 00:13:39,490 --> 00:13:40,065 It's derived. 196 00:13:40,065 --> 00:13:40,690 AUDIENCE: Fine. 197 00:13:40,690 --> 00:13:41,680 OK. 198 00:13:41,680 --> 00:13:45,790 HONG LIU: The assumption is that the black hole 199 00:13:45,790 --> 00:13:49,460 is the maximal-- is the most massive object you 200 00:13:49,460 --> 00:13:51,860 can put inside this region. 201 00:13:51,860 --> 00:13:52,484 AUDIENCE: Sure. 202 00:13:52,484 --> 00:13:52,983 I see. 203 00:13:52,983 --> 00:13:54,790 HONG LIU: Yeah. 204 00:13:54,790 --> 00:13:57,550 AUDIENCE: But you can't have a less massive object 205 00:13:57,550 --> 00:13:59,929 with more than that entropy. 206 00:13:59,929 --> 00:14:00,470 HONG LIU: Hm? 207 00:14:00,470 --> 00:14:03,000 AUDIENCE: You can't have a less massive object with entropy 208 00:14:03,000 --> 00:14:04,670 that is more than that value. 209 00:14:04,670 --> 00:14:06,640 Inside the same region. 210 00:14:06,640 --> 00:14:10,750 HONG LIU: This is ruled out by this-- 211 00:14:10,750 --> 00:14:14,790 once you get this equation, then you don't ask about the mass. 212 00:14:14,790 --> 00:14:17,490 It just-- any entropy must be smaller than this value. 213 00:14:22,750 --> 00:14:24,610 Any other questions? 214 00:14:24,610 --> 00:14:28,656 AUDIENCE: How do we keep track of what the area of [INAUDIBLE] 215 00:14:31,554 --> 00:14:32,220 HONG LIU: Right. 216 00:14:32,220 --> 00:14:37,277 So that's why I say this is a little bit heuristic argument. 217 00:14:37,277 --> 00:14:39,360 And that's why I say this is a heuristic argument. 218 00:14:39,360 --> 00:14:40,514 Yeah. 219 00:14:40,514 --> 00:14:42,430 Yeah, indeed, you have to be a little bit more 220 00:14:42,430 --> 00:14:43,450 careful about that. 221 00:14:46,880 --> 00:14:58,060 Yeah but say if you-- the key thing is about the area. 222 00:14:58,060 --> 00:15:00,975 And then this other one cautions one maybe 223 00:15:00,975 --> 00:15:02,300 to have to think about it. 224 00:15:02,300 --> 00:15:04,015 When you include the quantum corrections 225 00:15:04,015 --> 00:15:07,550 et cetera-- the back reaction, et cetera. 226 00:15:07,550 --> 00:15:09,650 But the key thing is the area of the black hole. 227 00:15:09,650 --> 00:15:12,650 It must be proportionate to the area. 228 00:15:12,650 --> 00:15:14,010 AUDIENCE: Well. 229 00:15:14,010 --> 00:15:19,320 [INAUDIBLE] for some function that the area 230 00:15:19,320 --> 00:15:25,355 of the horizon with the black hole is only based on the mass. 231 00:15:25,355 --> 00:15:25,980 HONG LIU: Yeah. 232 00:15:25,980 --> 00:15:26,240 Yeah. 233 00:15:26,240 --> 00:15:26,740 Yeah. 234 00:15:26,740 --> 00:15:27,950 Yeah. 235 00:15:27,950 --> 00:15:31,440 So let me say again, there's two natures of this argument. 236 00:15:31,440 --> 00:15:35,750 The first nature of the argument is, 237 00:15:35,750 --> 00:15:40,440 the maximal energy you can put inside the region 238 00:15:40,440 --> 00:15:43,630 is a black hole whose horizon area is given 239 00:15:43,630 --> 00:15:45,387 by the area of this region. 240 00:15:45,387 --> 00:15:46,720 AUDIENCE: This is an assumption. 241 00:15:46,720 --> 00:15:46,860 HONG LIU: Yeah. 242 00:15:46,860 --> 00:15:47,920 This is assumption. 243 00:15:47,920 --> 00:15:49,840 Yeah, because we know, in the GR, 244 00:15:49,840 --> 00:15:51,430 any sufficiently massive thing will 245 00:15:51,430 --> 00:15:53,144 collapse to form a black hole. 246 00:15:53,144 --> 00:15:53,810 AUDIENCE: Mmhmm. 247 00:15:53,810 --> 00:15:54,435 HONG LIU: Yeah. 248 00:15:54,435 --> 00:15:57,030 And if you have a more mass, then the horizon area 249 00:15:57,030 --> 00:15:58,260 would be outside. 250 00:15:58,260 --> 00:16:00,330 And then you will be outside this area. 251 00:16:00,330 --> 00:16:04,420 You can no longer keep this A fixed. 252 00:16:04,420 --> 00:16:05,900 So that's the first assumption. 253 00:16:05,900 --> 00:16:11,000 So we said that assumption-- that assumption then 254 00:16:11,000 --> 00:16:13,950 just follows from the second law of thermodynamics. 255 00:16:13,950 --> 00:16:16,170 Because this is whatever-- the entropy 256 00:16:16,170 --> 00:16:20,650 of the matter inside-- then this is whatever the initial entropy 257 00:16:20,650 --> 00:16:22,935 of the energy-- you need to put it in 258 00:16:22,935 --> 00:16:24,630 to make it into a black hole. 259 00:16:24,630 --> 00:16:26,640 And then that must be smaller than the entropy 260 00:16:26,640 --> 00:16:29,680 of the final state, which is the entropy of the black hole. 261 00:16:29,680 --> 00:16:33,433 AUDIENCE: But my question is that by keeping A fixed, 262 00:16:33,433 --> 00:16:37,297 you might introduce extra entropy into the system. 263 00:16:37,297 --> 00:16:38,270 HONG LIU: Hm? 264 00:16:38,270 --> 00:16:38,820 Yeah. 265 00:16:38,820 --> 00:16:39,680 It doesn't matter. 266 00:16:39,680 --> 00:16:44,097 As far as the black hole is the final entropy state. 267 00:16:44,097 --> 00:16:45,680 What you're saying is that there might 268 00:16:45,680 --> 00:16:46,763 be some other things here. 269 00:16:46,763 --> 00:16:47,710 It's fine. 270 00:16:47,710 --> 00:16:51,630 That does not change the direction of this-- 271 00:16:51,630 --> 00:16:53,774 AUDIENCE: The entropy might be negative. 272 00:16:53,774 --> 00:16:57,089 I'm not sure-- just-- 273 00:16:57,089 --> 00:16:57,630 HONG LIU: No. 274 00:16:57,630 --> 00:16:58,130 No. 275 00:16:58,130 --> 00:17:01,710 Whatever it is-- this is my initial. 276 00:17:01,710 --> 00:17:03,760 So that's why this is an inequality here. 277 00:17:03,760 --> 00:17:05,270 This is not the equality. 278 00:17:05,270 --> 00:17:06,660 Inequality. 279 00:17:06,660 --> 00:17:10,260 And we're just using the second law of thermodynamics. 280 00:17:10,260 --> 00:17:11,750 This is your initial state entropy. 281 00:17:11,750 --> 00:17:13,739 This is my final state entropy. 282 00:17:13,739 --> 00:17:15,030 This must be greater than that. 283 00:17:15,030 --> 00:17:19,200 We're not asking the specific process. 284 00:17:19,200 --> 00:17:22,551 We're not asking the specific process. 285 00:17:22,551 --> 00:17:23,050 Yeah. 286 00:17:23,050 --> 00:17:25,591 Just, the final state must be greater than the initial state. 287 00:17:28,810 --> 00:17:29,310 Good? 288 00:17:32,320 --> 00:17:33,480 OK. 289 00:17:33,480 --> 00:17:38,950 So now let's see what we can try to go a little bit further 290 00:17:38,950 --> 00:17:41,030 from here. 291 00:17:41,030 --> 00:17:41,970 So now let's recall. 292 00:17:45,364 --> 00:17:46,905 So now we really treat the black hole 293 00:17:46,905 --> 00:17:49,840 as a quantum statistical object. 294 00:17:49,840 --> 00:18:00,950 So now recall the definition of entropy 295 00:18:00,950 --> 00:18:02,365 in quantum statistical physics. 296 00:18:41,741 --> 00:18:42,240 OK. 297 00:18:45,140 --> 00:18:48,230 So if you say I have some system, 298 00:18:48,230 --> 00:18:52,620 and the state of that system is described by some density 299 00:18:52,620 --> 00:18:55,162 operator, or density matrix, and then 300 00:18:55,162 --> 00:18:57,745 that the entropy of that system is just given by this formula. 301 00:19:03,670 --> 00:19:23,250 In particular, for a system with n-dimensional Hilbert space. 302 00:19:30,550 --> 00:19:32,350 OK. 303 00:19:32,350 --> 00:19:34,470 So this is the n-dimensional Hilbert space. 304 00:19:34,470 --> 00:19:38,220 This is a dimension of the Hilbert space. 305 00:19:38,220 --> 00:19:43,220 Let S max is log N. So the maximum 306 00:19:43,220 --> 00:19:48,154 possible entropy you can have is log N. 307 00:19:53,460 --> 00:19:53,960 OK. 308 00:19:53,960 --> 00:19:55,775 Is this fact familiar to you? 309 00:19:59,030 --> 00:20:02,492 So you can convince yourself the maximal entropy 310 00:20:02,492 --> 00:20:08,900 is realized when role is just given by 1/n times the identity 311 00:20:08,900 --> 00:20:10,700 matrix. 312 00:20:10,700 --> 00:20:17,400 And so is the density matrix which has low information. 313 00:20:17,400 --> 00:20:20,120 Because every state plays the same role. 314 00:20:20,120 --> 00:20:23,640 And this state-- then you can plug in here 315 00:20:23,640 --> 00:20:26,430 and give your log N. And the maximum entropy 316 00:20:26,430 --> 00:20:28,810 means you have minimal information. 317 00:20:28,810 --> 00:20:32,360 And that is the state with minimal information. 318 00:20:32,360 --> 00:20:38,350 And you can also just prove yourself using some algebra. 319 00:20:38,350 --> 00:20:42,490 But this is the physical argument for this. 320 00:20:45,730 --> 00:20:51,380 So now, if you compare with these two formula, 321 00:20:51,380 --> 00:20:55,160 then we can draw a conclusion, is 322 00:20:55,160 --> 00:21:00,080 that-- so let me add a word called 323 00:21:00,080 --> 00:21:08,775 effective-- effective dimension-- of the Hilbert 324 00:21:08,775 --> 00:21:36,540 space for a system inside a region of area A 325 00:21:36,540 --> 00:21:47,940 is bounded by log N must be smaller than A h bar GN. 326 00:21:56,390 --> 00:21:59,700 So that tells you-- so whatever you can put inside 327 00:21:59,700 --> 00:22:03,310 of this region of area A, that's the maximal entropy you 328 00:22:03,310 --> 00:22:05,180 can have. 329 00:22:05,180 --> 00:22:07,619 So if you have N degrees of freedom, 330 00:22:07,619 --> 00:22:09,160 then the maximal entropy you can have 331 00:22:09,160 --> 00:22:11,935 for this N degrees of freedom is log N. 332 00:22:11,935 --> 00:22:14,310 Then that tells you that log N must be smaller than this. 333 00:22:14,310 --> 00:22:14,900 OK? 334 00:22:14,900 --> 00:22:18,320 Smaller or equal to that. 335 00:22:18,320 --> 00:22:20,070 And so let me write this in terms 336 00:22:20,070 --> 00:22:24,660 of the-- so this can also, 4 h bar N 337 00:22:24,660 --> 00:22:28,950 is also Planck length square. 338 00:22:28,950 --> 00:22:30,930 So it's also four times a Planck length square. 339 00:22:34,318 --> 00:22:37,720 AUDIENCE: So for this part, really Hilbert space 340 00:22:37,720 --> 00:22:39,780 really can be even infinite. 341 00:22:39,780 --> 00:22:40,847 HONG LIU: Yeah. 342 00:22:40,847 --> 00:22:41,680 And let me say that. 343 00:22:41,680 --> 00:22:44,300 So that's why we need "effective." 344 00:22:44,300 --> 00:22:45,860 So now let me make some remarks. 345 00:22:48,570 --> 00:22:49,975 Let me make some remarks. 346 00:23:00,500 --> 00:23:02,550 So zero remarks. 347 00:23:02,550 --> 00:23:11,950 It says, for system of n spins say, n spins half particle-- 348 00:23:11,950 --> 00:23:15,850 so when I say spins, I always means spins half-- then 349 00:23:15,850 --> 00:23:18,660 the dimension of the Hilbert space is 2 to the power n. 350 00:23:21,370 --> 00:23:22,260 Just to remind you. 351 00:23:25,720 --> 00:23:37,200 But the dimension of h-- the Hilbert space-- so normally I 352 00:23:37,200 --> 00:23:41,090 just group h to denote the Hilbert space-- 353 00:23:41,090 --> 00:23:43,570 for a single harmonic oscillator. 354 00:23:49,890 --> 00:23:52,240 So the simplest system is spin. 355 00:23:52,240 --> 00:23:54,730 Then the next simplest system is essentially 356 00:23:54,730 --> 00:23:57,260 a harmonic oscillator. 357 00:23:57,260 --> 00:24:01,520 And even for a single harmonic oscillator, then 358 00:24:01,520 --> 00:24:03,070 the dimension of h is infinite. 359 00:24:06,010 --> 00:24:07,510 OK. 360 00:24:07,510 --> 00:24:10,280 And then you ask what I'm talking about here. 361 00:24:10,280 --> 00:24:12,110 Because here I say the Hilbert space 362 00:24:12,110 --> 00:24:13,300 should be smaller than that. 363 00:24:16,570 --> 00:24:19,252 But. 364 00:24:19,252 --> 00:24:21,460 So even though, even for a single harmonic oscillator 365 00:24:21,460 --> 00:24:28,325 the Hilbert space is infinite, but for a quantum mechanical 366 00:24:28,325 --> 00:24:38,780 system, for a quantum system with a finite number of degrees 367 00:24:38,780 --> 00:24:46,680 of freedom-- by finite number of degrees of freedom, 368 00:24:46,680 --> 00:24:49,040 for example, you can interpret this, for example, 369 00:24:49,040 --> 00:24:53,560 as a finite number of harmonic oscillators, OK-- 370 00:24:53,560 --> 00:25:02,130 the dimension of the Hilbert space 371 00:25:02,130 --> 00:25:12,160 below some finite energy scale. 372 00:25:12,160 --> 00:25:15,995 Energy is always finite. 373 00:25:21,740 --> 00:25:23,700 It's always finite. 374 00:25:23,700 --> 00:25:28,020 And so that's why I use the word "effective" there. 375 00:25:28,020 --> 00:25:32,280 And because normally we all consider the finite energy, 376 00:25:32,280 --> 00:25:34,460 because the system will have finite energy, 377 00:25:34,460 --> 00:25:36,790 and then you have, effectively, a finite dimension 378 00:25:36,790 --> 00:25:38,010 of Hilbert space. 379 00:25:38,010 --> 00:25:40,400 OK? 380 00:25:40,400 --> 00:25:43,080 So this equation is not that crazy. 381 00:25:47,000 --> 00:25:53,750 So I mean typical-- so actually, in typical physical systems, 382 00:25:53,750 --> 00:25:56,980 even for harmonic oscillators, both for spin or harmonic 383 00:25:56,980 --> 00:26:12,380 oscillators, essentially, number of degrees of freedom 384 00:26:12,380 --> 00:26:20,420 is of all the log N. Yeah, N is the effective dimension 385 00:26:20,420 --> 00:26:23,780 of your Hilbert space. 386 00:26:23,780 --> 00:26:27,350 Yeah, this is obvious from the spin. 387 00:26:27,350 --> 00:26:31,370 This is also obvious if you have a harmonic oscillator. 388 00:26:31,370 --> 00:26:34,980 So if each harmonic oscillator is finitely excited. 389 00:26:34,980 --> 00:26:39,110 And again, roughly the log N is the number 390 00:26:39,110 --> 00:26:40,110 of harmonic oscillators. 391 00:26:43,140 --> 00:26:47,900 So because of this we can also write this equation 392 00:26:47,900 --> 00:26:58,320 as the number of degrees of freedom of any quantum gravity 393 00:26:58,320 --> 00:27:12,260 system inside the region A can be-- 394 00:27:12,260 --> 00:27:15,410 should be smaller than-- should be bounded 395 00:27:15,410 --> 00:27:16,700 by the area of that region. 396 00:27:34,870 --> 00:27:49,710 So this bond is certainly violated 397 00:27:49,710 --> 00:28:03,200 in non-gravitational systems because 398 00:28:03,200 --> 00:28:06,460 in non-gravitational systems whose number 399 00:28:06,460 --> 00:28:15,340 of degrees of freedom and thus, the log 400 00:28:15,340 --> 00:28:19,420 N, the log of your dimension of your Hilbert space, 401 00:28:19,420 --> 00:28:31,620 is proportional to the volume rather than area. 402 00:28:37,900 --> 00:28:40,790 So when the reading is big enough, 403 00:28:40,790 --> 00:28:44,710 then you are guaranteed to violate this bound. 404 00:28:44,710 --> 00:28:49,670 Because, of course, the volume grows faster than the area 405 00:28:49,670 --> 00:28:53,320 if you have a sufficiently larger region. 406 00:28:53,320 --> 00:29:05,210 So for example, consider just a lattice of spins-- 407 00:29:05,210 --> 00:29:09,610 so a three dimensional lattice of spins-- of lattice spacing 408 00:29:09,610 --> 00:29:28,730 a-- say of lattice spacing a-- then the total number of spins 409 00:29:28,730 --> 00:29:35,240 that is given by volume divided by A cubed. 410 00:29:35,240 --> 00:29:38,180 So this is essentially the number of degrees of freedom. 411 00:29:38,180 --> 00:29:43,760 And then this is given by A, the area, divided by a squared, 412 00:29:43,760 --> 00:29:45,250 times L divided by a. 413 00:29:48,160 --> 00:29:49,710 And this can be much, much greater 414 00:29:49,710 --> 00:29:57,380 than a divided by Planck square, Planck length square, 415 00:29:57,380 --> 00:29:58,730 for L sufficiently large. 416 00:30:06,400 --> 00:30:09,750 They are large enough. 417 00:30:09,750 --> 00:30:12,930 So if you have a long gravitational system, 418 00:30:12,930 --> 00:30:14,680 then this boundary can be easily violated. 419 00:30:19,400 --> 00:30:23,190 In particular, this equation can also 420 00:30:23,190 --> 00:30:32,330 be violated because the dimension of the Hilbert space 421 00:30:32,330 --> 00:30:37,510 that in this case would 2 to the power a cubed, 422 00:30:37,510 --> 00:30:44,050 and similarly-- so a-- so the maximal entropy would be, 423 00:30:44,050 --> 00:30:50,410 say, v divided by a cubed log 2, because it's a spin 2 system, 424 00:30:50,410 --> 00:30:51,610 spin half system. 425 00:30:51,610 --> 00:30:55,050 So this can be larger than the entropy of a black hole. 426 00:30:58,090 --> 00:31:06,320 So it's key that we can see the gravitational systems. 427 00:31:06,320 --> 00:31:08,370 Only in gravitational systems you 428 00:31:08,370 --> 00:31:11,470 have this remarkable feature that if you-- things 429 00:31:11,470 --> 00:31:13,355 will collapse. 430 00:31:13,355 --> 00:31:14,230 Things will collapse. 431 00:31:17,110 --> 00:31:19,210 And then you will have this upper limit 432 00:31:19,210 --> 00:31:20,833 provided by the black hole. 433 00:31:20,833 --> 00:31:21,333 Yes? 434 00:31:21,333 --> 00:31:23,249 AUDIENCE: So in the real world, every system's 435 00:31:23,249 --> 00:31:24,216 a gravitational system. 436 00:31:24,216 --> 00:31:24,840 HONG LIU: Yeah. 437 00:31:24,840 --> 00:31:26,655 AUDIENCE: So this is just-- in some sense, 438 00:31:26,655 --> 00:31:28,850 this is not actually correct in the real world. 439 00:31:28,850 --> 00:31:32,216 HONG LIU: Yeah, but you need this L to be not enough. 440 00:31:32,216 --> 00:31:33,245 AUDIENCE: I see. 441 00:31:33,245 --> 00:31:33,870 HONG LIU: Yeah. 442 00:31:37,420 --> 00:31:38,330 Yes? 443 00:31:38,330 --> 00:31:39,380 AUDIENCE: So, right. 444 00:31:39,380 --> 00:31:42,370 In the real world, this bound has to be obeyed by systems. 445 00:31:42,370 --> 00:31:46,630 So does that put some bounds on QFT UV completions? 446 00:31:46,630 --> 00:31:47,820 HONG LIU: Hm? 447 00:31:47,820 --> 00:31:49,800 AUDIENCE: So, in the real world this bound 448 00:31:49,800 --> 00:31:51,316 has to be obeyed by everything. 449 00:31:51,316 --> 00:31:51,940 HONG LIU: Yeah. 450 00:31:51,940 --> 00:31:54,106 AUDIENCE: So does that put some kind of restrictions 451 00:31:54,106 --> 00:31:55,900 on QFT UV completion? 452 00:31:55,900 --> 00:31:57,830 Because we usually say it's infinite density 453 00:31:57,830 --> 00:32:00,010 of degrees of freedom. 454 00:32:00,010 --> 00:32:02,480 HONG LIU: No, because UV completion of QFT 455 00:32:02,480 --> 00:32:04,812 is provided by gravity. 456 00:32:04,812 --> 00:32:05,970 AUDIENCE: For any QFT. 457 00:32:05,970 --> 00:32:09,470 I just mean, like, a scalar fuel. 458 00:32:09,470 --> 00:32:12,042 HONG LIU: Then that's not the theory of gravity. 459 00:32:12,042 --> 00:32:13,750 AUDIENCE: Right, but if you have a scalar 460 00:32:13,750 --> 00:32:18,575 fuel in space time, which has GR in it, is otherwise like, flat? 461 00:32:18,575 --> 00:32:19,200 HONG LIU: Yeah. 462 00:32:19,200 --> 00:32:22,330 Then the GR-- then the gravity will provide the UV completion 463 00:32:22,330 --> 00:32:24,465 on that scalar field. 464 00:32:24,465 --> 00:32:25,090 AUDIENCE: Sure. 465 00:32:25,090 --> 00:32:25,715 HONG LIU: Yeah. 466 00:32:28,620 --> 00:32:30,360 Right. 467 00:32:30,360 --> 00:32:32,410 I was going to say something else, but I forgot. 468 00:32:32,410 --> 00:32:34,096 AUDIENCE: [LAUGHTER] 469 00:32:34,096 --> 00:32:34,720 HONG LIU: Haha. 470 00:32:34,720 --> 00:32:36,568 AUDIENCE: What about the ones that are to be complete? 471 00:32:36,568 --> 00:32:37,030 HONG LIU: Hm? 472 00:32:37,030 --> 00:32:38,460 AUDIENCE: What about the theories 473 00:32:38,460 --> 00:32:41,295 that are to be complete at least, formally? 474 00:32:41,295 --> 00:32:43,220 HONG LIU: Uh. 475 00:32:43,220 --> 00:32:44,270 Sorry, say it again? 476 00:32:44,270 --> 00:32:46,915 AUDIENCE: The QFTs that are to be complete. 477 00:32:46,915 --> 00:32:47,540 HONG LIU: Yeah. 478 00:32:47,540 --> 00:32:48,570 They are limited by gravity. 479 00:32:48,570 --> 00:32:48,840 AUDIENCE: OK. 480 00:32:48,840 --> 00:32:49,465 HONG LIU: Yeah. 481 00:32:49,465 --> 00:32:51,360 The gravity will still provide something. 482 00:32:51,360 --> 00:32:53,735 Because a QFT have infinite number of degrees of freedom. 483 00:32:58,210 --> 00:33:00,690 Yeah. 484 00:33:00,690 --> 00:33:03,790 So as an example of this, for example you 485 00:33:03,790 --> 00:33:09,330 can think of a classroom, say, full of photons, 486 00:33:09,330 --> 00:33:11,480 and then you can calculate entropy. 487 00:33:11,480 --> 00:33:14,090 Again, that entropy will be proportional to the volume. 488 00:33:14,090 --> 00:33:17,200 So if you make the classroom big enough, 489 00:33:17,200 --> 00:33:20,010 then that will violate this bound. 490 00:33:20,010 --> 00:33:22,612 But it's a simple exercise for you to do yourself, 491 00:33:22,612 --> 00:33:24,820 that if you put the classroom at a finite temperature 492 00:33:24,820 --> 00:33:28,780 of photons, then when the room is big enough, 493 00:33:28,780 --> 00:33:32,010 there's always a radius that will 494 00:33:32,010 --> 00:33:35,157 collapse to form a black hole before this bound is violated. 495 00:33:35,157 --> 00:33:35,990 AUDIENCE: [LAUGHTER] 496 00:33:35,990 --> 00:33:36,400 HONG LIU: OK. 497 00:33:36,400 --> 00:33:38,400 I think maybe I have this formula in the p-set. 498 00:33:38,400 --> 00:33:39,816 I forgot whether I have it or not. 499 00:33:39,816 --> 00:33:43,420 Yeah, you can do this simple exercise yourself. 500 00:33:43,420 --> 00:33:47,590 Yeah just imagine you have a box of photons 501 00:33:47,590 --> 00:33:49,970 at any non-zero temperature t, and then 502 00:33:49,970 --> 00:33:54,110 when you put the box big enough, then at a certain point then 503 00:33:54,110 --> 00:33:58,485 this whole box of photons will collapse to form a black hole. 504 00:34:01,870 --> 00:34:07,560 And that's within to make this bound to be satisfied. 505 00:34:10,489 --> 00:34:17,790 So what this means-- that if you have gravity, 506 00:34:17,790 --> 00:34:31,794 the quantum gravity leads to a huge reduction of number 507 00:34:31,794 --> 00:34:32,710 of degrees of freedom. 508 00:34:38,449 --> 00:34:42,446 It's really bounded by your area. 509 00:34:42,446 --> 00:34:45,694 AUDIENCE: Now, in previous argument [INAUDIBLE], 510 00:34:45,694 --> 00:34:48,369 you have a conformal field theory. 511 00:34:48,369 --> 00:34:55,060 And so this other [INAUDIBLE] we don't need to have [INAUDIBLE]. 512 00:34:55,060 --> 00:34:55,749 HONG LIU: No. 513 00:34:55,749 --> 00:34:56,429 Yeah. 514 00:34:56,429 --> 00:35:01,520 As I was answering Pavel before, if you have a conformal field 515 00:35:01,520 --> 00:35:04,660 theory just by itself, then there's no bound. 516 00:35:04,660 --> 00:35:05,370 AUDIENCE: Yes. 517 00:35:05,370 --> 00:35:06,250 HONG LIU: Then there's no bound. 518 00:35:06,250 --> 00:35:06,900 AUDIENCE: Oh. 519 00:35:06,900 --> 00:35:09,191 HONG LIU: But if you couple this conformal field theory 520 00:35:09,191 --> 00:35:12,117 to gravity, then at sufficiently high energy, 521 00:35:12,117 --> 00:35:14,700 and then the degrees of freedom of this conformal field theory 522 00:35:14,700 --> 00:35:15,366 must be reduced. 523 00:35:15,366 --> 00:35:18,920 AUDIENCE: No, but, if you just had a conformal field theory, 524 00:35:18,920 --> 00:35:19,920 then there's no bound. 525 00:35:19,920 --> 00:35:20,711 HONG LIU: No bound. 526 00:35:20,711 --> 00:35:21,720 Yeah. 527 00:35:21,720 --> 00:35:23,410 Yeah, but you can ask what happens 528 00:35:23,410 --> 00:35:28,670 if we couple a conformal field theory to gravity. 529 00:35:28,670 --> 00:35:31,420 Not even conformal field theory, just any quantum field theory, 530 00:35:31,420 --> 00:35:33,047 because any quantum field theory has an infinite number 531 00:35:33,047 --> 00:35:35,505 of degrees of freedom, and the number of degrees of freedom 532 00:35:35,505 --> 00:35:37,690 is also proportional to the volume. 533 00:35:37,690 --> 00:35:42,420 And if you couple it to gravity, then this reduction 534 00:35:42,420 --> 00:35:44,180 must happen. 535 00:35:44,180 --> 00:35:45,640 This reduction must happen. 536 00:35:45,640 --> 00:35:47,430 OK. 537 00:35:47,430 --> 00:35:53,970 So now if we take a small leap of faith-- 538 00:35:53,970 --> 00:35:56,870 so we already have taken some leaps of faith, 539 00:35:56,870 --> 00:35:59,640 but let's take one more stab, or leap of faith-- then 540 00:35:59,640 --> 00:36:01,330 you can formulate something called 541 00:36:01,330 --> 00:36:02,415 the holographic principle. 542 00:36:09,302 --> 00:36:10,760 So given that the number of degrees 543 00:36:10,760 --> 00:36:16,720 of freedom inside the region is always bounded by the area, 544 00:36:16,720 --> 00:36:19,170 and this area actually has a very nice way, 545 00:36:19,170 --> 00:36:22,720 it's area in the unit of a Planck lens-- 546 00:36:22,720 --> 00:36:26,980 it looks like we have a minimal lens-- minimal area which 547 00:36:26,980 --> 00:36:29,140 is a Planck's square lens-- and just 548 00:36:29,140 --> 00:36:32,020 the area divided by that gives you the total number of degrees 549 00:36:32,020 --> 00:36:32,520 of freedom. 550 00:36:32,520 --> 00:36:33,820 OK. 551 00:36:33,820 --> 00:36:38,090 So if you try to extrapolate a little bit further, 552 00:36:38,090 --> 00:36:40,840 then we have the so-called holographic principle. 553 00:36:40,840 --> 00:37:02,860 It says, in quantum gravity, a region of boundary area A 554 00:37:02,860 --> 00:37:18,700 can be fully described-- can be fully described 555 00:37:18,700 --> 00:37:26,120 by no more than-- so essentially it's 556 00:37:26,120 --> 00:37:37,874 just A h bar hN equals A [INAUDIBLE] squared 557 00:37:37,874 --> 00:37:38,540 degrees freedom. 558 00:37:41,370 --> 00:37:49,380 In other words, one degree of freedom per Planck area. 559 00:38:02,030 --> 00:38:09,670 So this is a rough formulation of the holographic principle. 560 00:38:14,760 --> 00:38:22,560 So what we just said-- it just highlights 561 00:38:22,560 --> 00:38:27,750 a point we made earlier-- is that even though naively, 562 00:38:27,750 --> 00:38:32,270 from dimensional analysis, that the quantum gravity 563 00:38:32,270 --> 00:38:36,230 is only operating, say, at the Planck scale. 564 00:38:36,230 --> 00:38:39,136 Say, at the Planck mass or [INAUDIBLE], et cetera. 565 00:38:39,136 --> 00:38:44,450 And we see that the black hole actually 566 00:38:44,450 --> 00:38:47,300 brings the quantum gravity really 567 00:38:47,300 --> 00:38:49,350 need to macroscopic level. 568 00:38:49,350 --> 00:38:50,070 OK. 569 00:38:50,070 --> 00:38:53,740 Because this is, really macroscopic statement 570 00:38:53,740 --> 00:38:57,230 can apply to regions of arbitrary sides. 571 00:38:57,230 --> 00:39:01,420 But somehow, they are limited by constraints of quantum gravity. 572 00:39:01,420 --> 00:39:01,950 OK. 573 00:39:01,950 --> 00:39:07,950 So this is a remarkable manifestation 574 00:39:07,950 --> 00:39:11,470 of the quantum effect. 575 00:39:11,470 --> 00:39:12,490 Any questions on this? 576 00:39:12,490 --> 00:39:13,240 Yes? 577 00:39:13,240 --> 00:39:14,615 AUDIENCE: What's so special about 578 00:39:14,615 --> 00:39:17,089 the dimensional differences? 579 00:39:17,089 --> 00:39:17,630 HONG LIU: Oh. 580 00:39:17,630 --> 00:39:17,950 Yeah. 581 00:39:17,950 --> 00:39:18,110 Yeah. 582 00:39:18,110 --> 00:39:20,060 This we can generalize to any dimension. 583 00:39:20,060 --> 00:39:21,700 There's always core dimension two. 584 00:39:24,280 --> 00:39:27,410 So if you live in five dimensions, 585 00:39:27,410 --> 00:39:29,595 then becomes three dimensions-- yeah, 586 00:39:29,595 --> 00:39:34,750 it's always co-dimension y in the spatial region. 587 00:39:34,750 --> 00:39:39,070 Because the black hole-- just area of the black hole 588 00:39:39,070 --> 00:39:41,940 is always co-dimension y in terms of spatial. 589 00:39:41,940 --> 00:39:42,934 Yeah. 590 00:39:42,934 --> 00:39:48,401 AUDIENCE: Well, can't it be just [INAUDIBLE] sphere, 591 00:39:48,401 --> 00:39:49,760 and then it's [INAUDIBLE] 592 00:39:49,760 --> 00:39:50,450 HONG LIU: Yeah. 593 00:39:50,450 --> 00:39:52,660 10-dimensional space, then naively, 594 00:39:52,660 --> 00:39:58,072 this would be constrained by eight-dimensional surface. 595 00:39:58,072 --> 00:39:59,840 Yeah. 596 00:39:59,840 --> 00:40:01,579 Yeah, just, no matter what dimension, 597 00:40:01,579 --> 00:40:03,870 it's always constrained by the surface, not constrained 598 00:40:03,870 --> 00:40:05,409 by the volume. 599 00:40:05,409 --> 00:40:06,450 No matter what dimension. 600 00:40:10,589 --> 00:40:11,255 Other questions? 601 00:40:11,255 --> 00:40:13,705 AUDIENCE: I just have a question about [INAUDIBLE]. 602 00:40:13,705 --> 00:40:14,330 HONG LIU: Yeah? 603 00:40:14,330 --> 00:40:18,885 AUDIENCE: So it seems like we are constraining a system 604 00:40:18,885 --> 00:40:24,258 [INAUDIBLE] theory that its number of states for a given 605 00:40:24,258 --> 00:40:31,174 energy must grow slower somehow because we must be lower 606 00:40:31,174 --> 00:40:32,310 than some finite energy. 607 00:40:32,310 --> 00:40:32,935 HONG LIU: Yeah. 608 00:40:32,935 --> 00:40:34,991 AUDIENCE: The number of [INAUDIBLE] must be below 609 00:40:34,991 --> 00:40:35,490 certain-- 610 00:40:35,490 --> 00:40:36,370 HONG LIU: Yeah. 611 00:40:36,370 --> 00:40:44,760 Just say-- essentially, what we said earlier, 612 00:40:44,760 --> 00:40:47,390 is if you are in the theory of quantum gravity, 613 00:40:47,390 --> 00:40:50,860 there's a cutoff on the maximal energy you can have. 614 00:40:50,860 --> 00:40:54,270 And there's a cutoff also on the maximal energy you can have, 615 00:40:54,270 --> 00:40:56,570 because otherwise you form a black hole, 616 00:40:56,570 --> 00:40:59,780 and then that cutoff will translate 617 00:40:59,780 --> 00:41:02,090 into the state of the [INAUDIBLE] of entropy 618 00:41:02,090 --> 00:41:04,460 and into the number of degrees of freedom. 619 00:41:04,460 --> 00:41:05,500 AUDIENCE: Yes, I mean-- 620 00:41:05,500 --> 00:41:07,291 HONG LIU: Because much easier-- yeah, yeah. 621 00:41:07,291 --> 00:41:08,768 AUDIENCE: I mean, for example, we 622 00:41:08,768 --> 00:41:11,469 have-- first we start with a simple series, 623 00:41:11,469 --> 00:41:14,906 and maybe, for example, we add, for example, standard model, 624 00:41:14,906 --> 00:41:19,325 then we add super symmetry, then-- energy spectrum-- the 625 00:41:19,325 --> 00:41:23,660 limiting energy that is there, but the number of Hilbert space 626 00:41:23,660 --> 00:41:27,696 state is more, and we can keep doing things like that. 627 00:41:27,696 --> 00:41:28,320 HONG LIU: Yeah. 628 00:41:28,320 --> 00:41:29,650 AUDIENCE: But we can't. 629 00:41:29,650 --> 00:41:32,695 Because it must be below these values. 630 00:41:32,695 --> 00:41:33,320 HONG LIU: Yeah. 631 00:41:33,320 --> 00:41:37,580 But you see, there is something about the volume here. 632 00:41:37,580 --> 00:41:42,450 You have to talk about within some region. 633 00:41:42,450 --> 00:41:46,410 Of course, abstractly, you can put some degree-- yeah, 634 00:41:46,410 --> 00:41:49,590 maybe I'm not getting your question. 635 00:41:49,590 --> 00:41:52,100 AUDIENCE: How do you say-- naively, 636 00:41:52,100 --> 00:41:56,970 we think, we can just, for example, 637 00:41:56,970 --> 00:41:59,930 say if the energy boundary is here, and below this, 638 00:41:59,930 --> 00:42:01,665 how many states we could have? 639 00:42:01,665 --> 00:42:02,540 HONG LIU: No, no, no. 640 00:42:02,540 --> 00:42:03,956 This is not the direction in terms 641 00:42:03,956 --> 00:42:05,810 of the entropy-- energy bound. 642 00:42:05,810 --> 00:42:13,220 This is, in the end, is the bound on the [INAUDIBLE] 643 00:42:13,220 --> 00:42:17,780 degrees of freedom you can put inside a region. 644 00:42:17,780 --> 00:42:20,720 If you put too many degrees of freedom inside the region, 645 00:42:20,720 --> 00:42:22,840 then you need to enlarge the region. 646 00:42:22,840 --> 00:42:23,340 Yeah. 647 00:42:23,340 --> 00:42:25,800 Just say, look at the number of degrees of freedom 648 00:42:25,800 --> 00:42:26,960 is incompatible. 649 00:42:26,960 --> 00:42:29,684 Something like that. 650 00:42:29,684 --> 00:42:30,225 AUDIENCE: OK. 651 00:42:30,225 --> 00:42:33,111 I have a question regarding [INAUDIBLE]. 652 00:42:33,111 --> 00:42:34,710 HONG LIU: Yeah. 653 00:42:34,710 --> 00:42:36,174 AUDIENCE: I still don't understand. 654 00:42:36,174 --> 00:42:40,175 Shouldn't an isolated system mean also included 655 00:42:40,175 --> 00:42:42,719 the machinery to do these things? 656 00:42:42,719 --> 00:42:43,260 HONG LIU: No. 657 00:42:43,260 --> 00:42:45,760 I'm just saying you start with the isolated system. 658 00:42:45,760 --> 00:42:49,440 Then you can add something to that isolated system. 659 00:42:49,440 --> 00:42:51,403 AUDIENCE: What about if we take matters out 660 00:42:51,403 --> 00:42:55,670 of the black hole, these [INAUDIBLE]? 661 00:42:55,670 --> 00:42:57,550 HONG LIU: That thing, that isolated system, 662 00:42:57,550 --> 00:42:59,690 does not have to be a black hole. 663 00:42:59,690 --> 00:43:01,340 It can be anything. 664 00:43:01,340 --> 00:43:02,460 It just can be anything. 665 00:43:02,460 --> 00:43:05,310 AUDIENCE: So if we instead of putting energy inside, 666 00:43:05,310 --> 00:43:08,115 we take energy out, entropy will decrease? 667 00:43:08,115 --> 00:43:08,740 HONG LIU: Yeah. 668 00:43:08,740 --> 00:43:10,010 Just take entropy out. 669 00:43:10,010 --> 00:43:11,320 Yeah. 670 00:43:11,320 --> 00:43:14,011 Then this bound will be even more satisfied. 671 00:43:14,011 --> 00:43:15,935 [LAUGHTER] 672 00:43:15,935 --> 00:43:17,859 AUDIENCE: I have a question there, 673 00:43:17,859 --> 00:43:22,791 that the degree of freedom-- I mean log of the degree 674 00:43:22,791 --> 00:43:23,380 of freedom-- 675 00:43:23,380 --> 00:43:23,570 HONG LIU: No. 676 00:43:23,570 --> 00:43:24,070 No. 677 00:43:24,070 --> 00:43:27,364 N is the dimension of the Hilbert space. 678 00:43:27,364 --> 00:43:29,030 N is the dimension of the Hilbert space. 679 00:43:29,030 --> 00:43:30,817 That's why I called it N, here. 680 00:43:30,817 --> 00:43:31,400 AUDIENCE: Yes. 681 00:43:31,400 --> 00:43:33,812 HONG LIU: N, the dimension of the Hilbert space. 682 00:43:33,812 --> 00:43:35,720 AUDIENCE: But-- the degree of freedom is-- 683 00:43:35,720 --> 00:43:38,420 HONG LIU: Typically, the number of-- the dimension 684 00:43:38,420 --> 00:43:41,960 of the Hilbert space is exponential of the lambda 685 00:43:41,960 --> 00:43:42,940 of degrees of freedom. 686 00:43:42,940 --> 00:43:43,481 AUDIENCE: Oh. 687 00:43:43,481 --> 00:43:44,130 OK. 688 00:43:44,130 --> 00:43:45,000 HONG LIU: Just look at here. 689 00:43:45,000 --> 00:43:45,260 AUDIENCE: Oh. 690 00:43:45,260 --> 00:43:45,890 I see. 691 00:43:45,890 --> 00:43:47,056 HONG LIU: Just look at here. 692 00:43:47,056 --> 00:43:48,780 Yeah. 693 00:43:48,780 --> 00:43:50,791 Any other questions? 694 00:43:50,791 --> 00:43:51,290 OK. 695 00:43:55,162 --> 00:43:56,995 AUDIENCE: If we have a quantum field theory, 696 00:43:56,995 --> 00:43:58,680 it has a certain vacuum energy, right? 697 00:43:58,680 --> 00:44:02,460 And then if you now consider that to be the vacuum energy 698 00:44:02,460 --> 00:44:05,020 and you expand it at some point, it will actually 699 00:44:05,020 --> 00:44:06,820 collapse into a black hole? 700 00:44:06,820 --> 00:44:09,660 So if you take the vacuum energy to be 701 00:44:09,660 --> 00:44:13,185 that predicted by QFT, then for the universe, 702 00:44:13,185 --> 00:44:15,430 it would not work. 703 00:44:15,430 --> 00:44:17,758 Even theoretically-- I know the [INAUDIBLE] constant 704 00:44:17,758 --> 00:44:20,450 is like 120 orders less, but. 705 00:44:20,450 --> 00:44:24,870 HONG LIU: Yeah, that's a little bit tricky because-- yeah, 706 00:44:24,870 --> 00:44:27,737 so the vacuum energy will make it into, say, 707 00:44:27,737 --> 00:44:30,320 depending on whether the vacuum energy is positive or negative 708 00:44:30,320 --> 00:44:34,230 will make it into de Sitter, anti-de Sitter, then, yeah, 709 00:44:34,230 --> 00:44:40,237 then you try to see whether-- it depends on details. 710 00:44:40,237 --> 00:44:41,695 AUDIENCE: But if you go big enough, 711 00:44:41,695 --> 00:44:44,289 it's going to collapse, or-- 712 00:44:44,289 --> 00:44:44,830 HONG LIU: No. 713 00:44:44,830 --> 00:44:45,430 Not necessarily. 714 00:44:45,430 --> 00:44:47,180 I'm just saying that you have to formulate 715 00:44:47,180 --> 00:44:48,525 a little bit differently. 716 00:44:48,525 --> 00:44:50,150 You're talking about a particular state 717 00:44:50,150 --> 00:44:53,080 of certain energy to couple to gravity, 718 00:44:53,080 --> 00:44:56,110 then you just take that state coupled to gravity 719 00:44:56,110 --> 00:44:57,950 to see what you get. 720 00:44:57,950 --> 00:44:59,758 Yeah, then ask questions from there. 721 00:44:59,758 --> 00:45:01,900 Yeah. 722 00:45:01,900 --> 00:45:03,440 Good. 723 00:45:03,440 --> 00:45:06,780 So now let's move to a new topic. 724 00:45:06,780 --> 00:45:15,720 Still, it's in the preparation to go to the duality 725 00:45:15,720 --> 00:45:23,200 but we need a lot of pieces of important intuition 726 00:45:23,200 --> 00:45:26,045 in order to fully appreciate the duality. 727 00:45:29,320 --> 00:45:34,100 And in this piece of intuition is Large N theories. 728 00:45:37,910 --> 00:45:39,710 And let me call it Large N matrix theories. 729 00:45:49,480 --> 00:45:57,220 So now we try to look at the hint for holographic duality 730 00:45:57,220 --> 00:46:01,060 from the field theory side. 731 00:46:01,060 --> 00:46:05,110 So so far, we have been talking about the hint for holographic 732 00:46:05,110 --> 00:46:08,910 from the gravity side, and now we 733 00:46:08,910 --> 00:46:10,800 look at it from the field theory side, 734 00:46:10,800 --> 00:46:13,430 to see, actually from field theory side, 735 00:46:13,430 --> 00:46:17,470 there also some hints to relate a long gravity 736 00:46:17,470 --> 00:46:19,720 system to a gravity system. 737 00:46:19,720 --> 00:46:21,100 OK. 738 00:46:21,100 --> 00:46:22,200 So let's start with QCD. 739 00:46:28,580 --> 00:46:37,890 So QCD is a SU(3) gauge theory plus quarks. 740 00:46:43,720 --> 00:46:45,470 So let me just write down it's Lagrangian. 741 00:47:07,240 --> 00:47:13,670 So this F describes the gluon fields and this side 742 00:47:13,670 --> 00:47:14,990 describes the quarks. 743 00:47:14,990 --> 00:47:16,510 OK. 744 00:47:16,510 --> 00:47:24,010 So this F is built out of the standard partial mu 745 00:47:24,010 --> 00:47:33,110 A mu minus partial mu A mu minus A mu A mu, 746 00:47:33,110 --> 00:47:37,760 and A mu-- each gauge field is a 3 by 3 Hermitian matrix. 747 00:47:43,220 --> 00:47:46,030 So these are 3 times 3 Hermitian matrix. 748 00:47:49,840 --> 00:47:51,540 So this is using the matrix notation, 749 00:47:51,540 --> 00:47:53,010 so F is also a matrix. 750 00:47:53,010 --> 00:47:55,150 So there's a trace here. 751 00:47:55,150 --> 00:47:57,100 OK. 752 00:47:57,100 --> 00:47:57,845 Hermitian matrix. 753 00:48:03,820 --> 00:48:06,688 And then the T to a is just generators of SU(3). 754 00:48:12,400 --> 00:48:14,740 So it doesn't matter if you don't know QCD. 755 00:48:14,740 --> 00:48:19,590 It doesn't matter if you have not seen QCD before. 756 00:48:19,590 --> 00:48:23,150 The only thing that matters is just 757 00:48:23,150 --> 00:48:28,140 to remember that QCD is a series of matrices. 758 00:48:28,140 --> 00:48:30,130 AUDIENCE: [LAUGHTER] 759 00:48:30,130 --> 00:48:33,221 HONG LIU: QCD is a series of 3 by 3 matrices. 760 00:48:33,221 --> 00:48:33,720 OK. 761 00:48:33,720 --> 00:48:38,270 So the gluon field is 3 by 3 matrices. 762 00:48:38,270 --> 00:48:42,750 It's described by matrices. 763 00:48:42,750 --> 00:48:44,550 So of course, one of the remarkable things 764 00:48:44,550 --> 00:48:48,710 about the QCD-- what's the most remarkable thing about QCD, 765 00:48:48,710 --> 00:48:49,838 do you know? 766 00:48:49,838 --> 00:48:52,086 AUDIENCE: [INAUDIBLE]. 767 00:48:52,086 --> 00:48:52,710 HONG LIU: Yeah. 768 00:48:52,710 --> 00:48:53,766 That's one. 769 00:48:53,766 --> 00:48:54,700 AUDIENCE: [LAUGHTER] 770 00:48:54,700 --> 00:48:57,110 HONG LIU: And what other remarkable thing about the QCD? 771 00:48:57,110 --> 00:48:58,852 AUDIENCE: Soft coupling? 772 00:48:58,852 --> 00:48:59,393 HONG LIU: Hm? 773 00:48:59,393 --> 00:49:00,340 AUDIENCE: Soft coupling. 774 00:49:00,340 --> 00:49:00,630 HONG LIU: Yeah. 775 00:49:00,630 --> 00:49:01,700 It's related to that. 776 00:49:01,700 --> 00:49:02,392 AUDIENCE: Confinement, then. 777 00:49:02,392 --> 00:49:04,683 HONG LIU: Yeah, confinement, it's also related to that. 778 00:49:04,683 --> 00:49:09,840 It's just-- yeah, one of the most remarkable things 779 00:49:09,840 --> 00:49:12,040 about QCD is it's still not solvable. 780 00:49:12,040 --> 00:49:16,540 AUDIENCE: [LAUGHTER] 781 00:49:16,540 --> 00:49:18,414 HONG LIU: And the reason it's not solvable 782 00:49:18,414 --> 00:49:20,080 is because when you've got low energies, 783 00:49:20,080 --> 00:49:22,160 the coupling becomes strong, then we 784 00:49:22,160 --> 00:49:23,700 don't know how to deal with it. 785 00:49:23,700 --> 00:49:25,500 OK. 786 00:49:25,500 --> 00:49:27,250 So we still don't know how to calculate it 787 00:49:27,250 --> 00:49:28,250 from first principle. 788 00:49:28,250 --> 00:49:31,290 Of course, we can put it on the computer, 789 00:49:31,290 --> 00:49:36,810 but just from the-- in terms of theoretical understanding, 790 00:49:36,810 --> 00:49:39,420 it's hard. 791 00:49:39,420 --> 00:49:49,750 So in 1974-- so this is an old problem-- so starting in 1974-- 792 00:49:49,750 --> 00:49:54,960 starting in 1971, people already tried to solve it. 793 00:49:54,960 --> 00:49:57,850 So in 1974-- so the big problem with of this thing 794 00:49:57,850 --> 00:50:01,530 is that in quantum field theory, normally we do expansions. 795 00:50:01,530 --> 00:50:02,680 OK. 796 00:50:02,680 --> 00:50:06,670 Say we find some starting point, so in physics, that's 797 00:50:06,670 --> 00:50:07,730 what we always do. 798 00:50:07,730 --> 00:50:10,700 We find the starting point, which can be solved. 799 00:50:10,700 --> 00:50:13,600 Then we expand around that starting point. 800 00:50:13,600 --> 00:50:17,490 And that starting point is often a free theory. 801 00:50:17,490 --> 00:50:22,430 And then it may expand upon the free theory. 802 00:50:22,430 --> 00:50:26,570 So the difficult thing about this theory is at first, 803 00:50:26,570 --> 00:50:30,310 it's hard, when you go to low energies-- first 804 00:50:30,310 --> 00:50:32,630 it's hard to find the starting point. 805 00:50:32,630 --> 00:50:35,330 And the second is that's there's nothing to expand. 806 00:50:35,330 --> 00:50:37,250 Because there's no small parameters. 807 00:50:37,250 --> 00:50:37,750 OK. 808 00:50:37,750 --> 00:50:40,034 There's no small parameters. 809 00:50:40,034 --> 00:50:41,700 When you need to expand, you will always 810 00:50:41,700 --> 00:50:43,710 need a small parameter. 811 00:50:43,710 --> 00:50:48,722 So in 1974, t' Hooft came up with a very nice idea. 812 00:50:48,722 --> 00:50:49,680 He said, this is SU(3). 813 00:50:52,812 --> 00:50:54,270 He said, let's just think of taking 814 00:50:54,270 --> 00:51:04,075 the number of colors, which is N equal to 3, as a parameter. 815 00:51:08,830 --> 00:51:14,770 So even though you realize, he said, there's 816 00:51:14,770 --> 00:51:16,700 only three types of colors. 817 00:51:16,700 --> 00:51:20,030 But let's imagine we treat it as a free parameter. 818 00:51:20,030 --> 00:51:21,300 OK. 819 00:51:21,300 --> 00:51:31,480 So this just implies, say you promote the gauge field 820 00:51:31,480 --> 00:51:33,140 by M matrices. 821 00:51:33,140 --> 00:51:34,481 And by N Hermitian matrices. 822 00:51:34,481 --> 00:51:34,980 OK. 823 00:51:39,480 --> 00:51:44,252 He said, let's just treat this in terms of N by N matrices. 824 00:51:44,252 --> 00:51:46,710 And then this let's consider that N goes to infinity limit. 825 00:52:04,271 --> 00:52:06,270 So let's consider that N goes to infinity limit. 826 00:52:13,110 --> 00:52:17,180 So he reasoned, if N goes to infinity limit, 827 00:52:17,180 --> 00:52:21,880 it's sufficiently simple, then maybe we can solve that limit. 828 00:52:21,880 --> 00:52:26,340 And then do 1/N expansion. 829 00:52:26,340 --> 00:52:28,090 1/N is a small parameter. 830 00:52:28,090 --> 00:52:31,620 Then we can treat 1/N as a small parameter 831 00:52:31,620 --> 00:52:34,540 and then do one of the expansions. 832 00:52:34,540 --> 00:52:36,760 OK. 833 00:52:36,760 --> 00:52:38,540 So that's the rough idea. 834 00:52:38,540 --> 00:52:41,150 And then you get a small parameter of 1/N. 835 00:52:41,150 --> 00:52:43,270 So in real life, N will be three. 836 00:52:43,270 --> 00:52:45,630 So 1/3. 837 00:52:45,630 --> 00:52:49,060 It's not very small, but it's at least smaller than 1. 838 00:52:51,600 --> 00:52:57,230 OK, so, this turns out to be a genius idea. 839 00:52:57,230 --> 00:52:59,590 Turns out to be a genius idea. 840 00:52:59,590 --> 00:53:02,845 But unfortunately, QCD still cannot be solved this way. 841 00:53:02,845 --> 00:53:05,475 AUDIENCE: [LAUGHTER] 842 00:53:05,475 --> 00:53:06,850 HONG LIU: But there's a surprise. 843 00:53:06,850 --> 00:53:09,430 But there's a surprise. 844 00:53:09,430 --> 00:53:12,290 So, still, QCD, you cannot solve this way. 845 00:53:12,290 --> 00:53:16,200 Even N going to infinity limit turned out to be too hard. 846 00:53:16,200 --> 00:53:18,120 But there's a surprise. 847 00:53:18,120 --> 00:53:32,530 It turns out this 1/N expansion gives you a string theory. 848 00:53:37,580 --> 00:53:40,160 OK. 849 00:53:40,160 --> 00:53:44,460 So I will now explain this, OK. 850 00:53:44,460 --> 00:53:46,830 I will now explain this. 851 00:53:46,830 --> 00:53:50,350 So there's a byproduct of his large N expansion. 852 00:53:50,350 --> 00:53:54,710 You said, you find that this actually-- such a gauge 853 00:53:54,710 --> 00:53:58,920 theory actually sticks with real string theory. 854 00:53:58,920 --> 00:54:04,880 So the key is that these fields are matrices. 855 00:54:04,880 --> 00:54:05,380 OK. 856 00:54:11,920 --> 00:54:13,537 So the key reason this emerges is 857 00:54:13,537 --> 00:54:15,870 because we are working with a field, which are matrices. 858 00:54:19,820 --> 00:54:26,621 So to illustrate this, I will not do the QCD. 859 00:54:26,621 --> 00:54:27,870 It's a little bit complicated. 860 00:54:27,870 --> 00:54:29,790 I will just do a simple scalar field theory. 861 00:54:29,790 --> 00:54:31,700 OK. 862 00:54:31,700 --> 00:54:33,680 So I will just do a simple scalar field theory 863 00:54:33,680 --> 00:54:35,311 as an illustration. 864 00:54:35,311 --> 00:54:36,936 So let's consider the following theory. 865 00:55:02,240 --> 00:55:05,080 OK. 866 00:55:05,080 --> 00:55:09,040 So this g is still coupling constants. 867 00:55:09,040 --> 00:55:11,045 But what we do is actually convenient to put it 868 00:55:11,045 --> 00:55:12,780 in the front. 869 00:55:12,780 --> 00:55:15,810 Of course you can rescale it-- you can rescale it, 870 00:55:15,810 --> 00:55:18,300 you put it here, which is the normal place. 871 00:55:18,300 --> 00:55:20,470 But I put it in the front. 872 00:55:20,470 --> 00:55:23,620 Just like what we normally do also for QCD. 873 00:55:23,620 --> 00:55:26,940 So this g is coupling constant which we put in the front. 874 00:55:26,940 --> 00:55:29,010 And then each phi, again, I take it to be 875 00:55:29,010 --> 00:55:30,790 N by N in the Hermitian matrix. 876 00:55:33,450 --> 00:55:37,550 Phi x, each one is a matrix a, b. 877 00:55:37,550 --> 00:55:40,190 So I have two index. 878 00:55:40,190 --> 00:55:41,860 So this N by N Hermitian matrix. 879 00:55:44,270 --> 00:55:44,770 Yeah. 880 00:55:44,770 --> 00:55:45,890 N by N, Hermitian matrix. 881 00:55:49,840 --> 00:56:04,610 So which means that phi a b, equal to star equal to phi b a. 882 00:56:04,610 --> 00:56:05,110 OK. 883 00:56:10,190 --> 00:56:12,690 So there's a difference between the upper and lower indices. 884 00:56:12,690 --> 00:56:13,189 OK. 885 00:56:16,575 --> 00:56:18,866 So I can write it more explicit in terms of components. 886 00:56:21,580 --> 00:56:24,610 So just to give you, in case some of you 887 00:56:24,610 --> 00:56:29,802 are not familiar with this kind of matrix kind of field theory, 888 00:56:29,802 --> 00:56:32,225 so let me just also write it in terms of components. 889 00:56:36,810 --> 00:56:43,658 So writing in terms of components, 890 00:56:43,658 --> 00:56:50,710 it's 1/2 partial mu, phi a b, then partial mu, 891 00:56:50,710 --> 00:57:09,380 phi b a, then plus 1/4 phi a b, phi b c, phi c a, no-- phi c d, 892 00:57:09,380 --> 00:57:11,210 phi d a. 893 00:57:11,210 --> 00:57:14,000 OK. 894 00:57:14,000 --> 00:57:17,320 So this is the just having writing down explicitly 895 00:57:17,320 --> 00:57:18,040 using components. 896 00:57:22,330 --> 00:57:35,670 So this L, by definition, is invariant under, 897 00:57:35,670 --> 00:57:38,300 in addition to the usual translation 898 00:57:38,300 --> 00:57:46,140 rotation symmetries, et cetera, has also internal symmetry-- 899 00:57:46,140 --> 00:57:48,860 under UN, global symmetry. 900 00:57:54,840 --> 00:57:55,620 OK. 901 00:57:55,620 --> 00:58:04,500 So say phi x goes to U, phi x u dagger, 902 00:58:04,500 --> 00:58:13,522 for any u, any constant, is u N matrix. 903 00:58:19,060 --> 00:58:21,400 So this series is invariant under-- so this 904 00:58:21,400 --> 00:58:25,770 is obvious from here, because of this trace, 905 00:58:25,770 --> 00:58:27,751 so this is invariant under this. 906 00:58:27,751 --> 00:58:28,250 Hm? 907 00:58:31,180 --> 00:58:33,240 So let me just make some remarks. 908 00:58:33,240 --> 00:58:35,770 Before we proceed to talk about the features of this series, 909 00:58:35,770 --> 00:58:37,695 let me just make some quick remarks. 910 00:58:44,540 --> 00:58:46,170 Zero. 911 00:58:46,170 --> 00:58:48,005 This is a theory of N squared scalar field. 912 00:58:52,600 --> 00:58:53,510 OK. 913 00:58:53,510 --> 00:58:57,300 So if this is intuitive to you, then this just 914 00:58:57,300 --> 00:58:59,530 means you have many, many fields coupled together 915 00:58:59,530 --> 00:59:00,540 in some specific way. 916 00:59:04,440 --> 00:59:09,130 And also we don't have to consider Hermitian matrices. 917 00:59:09,130 --> 00:59:11,740 To consider Hermitian matrices just for simplicity, 918 00:59:11,740 --> 00:59:20,230 one can also consider other types of matrices. 919 00:59:24,880 --> 00:59:31,330 For example, real symmetric matrices. 920 00:59:33,970 --> 00:59:38,485 So it's real symmetrical, anti-symmetric matrices. 921 00:59:42,040 --> 00:59:46,840 So in those cases, if you consider, say, real matrices, 922 00:59:46,840 --> 00:59:49,620 then symmetric or anti-symmetric matrices, in that case, 923 00:59:49,620 --> 00:59:56,175 this global symmetry becomes s o N, rather than u n. 924 01:00:01,400 --> 01:00:04,580 So in here, this u is a global symmetry. 925 01:00:04,580 --> 01:00:07,380 This u has become and cannot depend on x. 926 01:00:07,380 --> 01:00:10,390 You see from here, it cannot depend on x. 927 01:00:10,390 --> 01:00:26,350 But you can also-- can also introduce gauge fields 928 01:00:26,350 --> 01:00:30,088 to make this u n symmetry local. 929 01:00:40,500 --> 01:00:42,320 So I will not do it now, but this will be 930 01:00:42,320 --> 01:00:44,730 important a little bit later. 931 01:00:44,730 --> 01:00:48,410 So later, we will indeed make it into a local gauge symmetry. 932 01:00:48,410 --> 01:00:53,190 But for now, let we just consider this simple example. 933 01:00:53,190 --> 01:00:55,648 Because I don't want to introduce too many fields, 934 01:00:55,648 --> 01:00:56,148 et cetera. 935 01:00:59,640 --> 01:01:02,539 So any questions regarding the setup of this theory? 936 01:01:02,539 --> 01:01:05,080 AUDIENCE: Yes, can you explain that last point one more time? 937 01:01:05,080 --> 01:01:06,853 What do you mean the-- if you-- how 938 01:01:06,853 --> 01:01:09,102 does introducing gauge fields make the symmetry local? 939 01:01:12,615 --> 01:01:13,240 HONG LIU: Yeah. 940 01:01:13,240 --> 01:01:19,167 You have to take 8324 to know that. 941 01:01:19,167 --> 01:01:20,000 AUDIENCE: [LAUGHTER] 942 01:01:20,000 --> 01:01:25,801 HONG LIU: But the simple way to explain it-- 943 01:01:25,801 --> 01:01:28,130 yeah, there's no simple way to explain it. 944 01:01:28,130 --> 01:01:29,364 [LAUGHTER] 945 01:01:29,364 --> 01:01:31,030 AUDIENCE: I'll draw the fields, I guess. 946 01:01:31,030 --> 01:01:31,654 HONG LIU: Yeah. 947 01:01:31,654 --> 01:01:38,130 Yeah, just say-- let me just explain the statement. 948 01:01:38,130 --> 01:01:40,560 If you introduce some gauge fields, 949 01:01:40,560 --> 01:01:45,160 then you can make this u to be some arbitrary function of x, 950 01:01:45,160 --> 01:01:47,032 rather than a constant. 951 01:01:47,032 --> 01:01:48,490 So that's the statement of a local. 952 01:01:48,490 --> 01:01:49,210 AUDIENCE: That's fine. 953 01:01:49,210 --> 01:01:50,030 HONG LIU: Yeah. 954 01:01:50,030 --> 01:01:52,690 Just so I can generalize this theory, 955 01:01:52,690 --> 01:01:55,880 so that this symmetry becomes local-- 956 01:01:55,880 --> 01:01:56,421 AUDIENCE: Ah. 957 01:01:56,421 --> 01:01:57,260 I see. 958 01:01:57,260 --> 01:01:59,000 HONG LIU: Becomes a symmetry so that u 959 01:01:59,000 --> 01:02:01,350 can have arbitrary x dependence-- can have 960 01:02:01,350 --> 01:02:03,510 arbitrary spacetime dependence. 961 01:02:03,510 --> 01:02:06,577 So right now, u must be a constant. 962 01:02:06,577 --> 01:02:07,410 Any other questions? 963 01:02:12,405 --> 01:02:12,905 Good. 964 01:02:19,120 --> 01:02:19,660 OK. 965 01:02:19,660 --> 01:02:25,670 So now, let's just try to do perturbation theory. 966 01:02:25,670 --> 01:02:28,150 Think g squared small. 967 01:02:28,150 --> 01:02:30,240 Try to do perturbation theory. 968 01:02:30,240 --> 01:02:31,640 OK. 969 01:02:31,640 --> 01:02:34,164 So the first thing you do-- of course, 970 01:02:34,164 --> 01:02:35,830 you should write down the Feynman rules. 971 01:02:42,322 --> 01:02:44,530 Say, first thing, you have to look at the propagator. 972 01:02:50,560 --> 01:02:52,590 Look at the propagator. 973 01:02:52,590 --> 01:02:57,880 So you can, essentially-- you are all field theory experts, 974 01:02:57,880 --> 01:03:00,610 so just by looking at that Lagrangian, 975 01:03:00,610 --> 01:03:03,320 then you can immediately write down this propagator. 976 01:03:06,190 --> 01:03:09,470 We can just write down x minus y. 977 01:03:09,470 --> 01:03:12,660 So whatever propagator-- whatever spacetime dependence, 978 01:03:12,660 --> 01:03:15,320 let me just call it x minus y. 979 01:03:15,320 --> 01:03:18,120 But also there's the nontrivial index structure. 980 01:03:18,120 --> 01:03:21,050 First there's also g squared. 981 01:03:21,050 --> 01:03:21,770 OK. 982 01:03:21,770 --> 01:03:24,640 Because there's a prefactor 1/g squared. 983 01:03:24,640 --> 01:03:26,140 So when you invert the connection, 984 01:03:26,140 --> 01:03:28,960 there's a g squared. 985 01:03:28,960 --> 01:03:33,390 But the index structure is at a b, b a. 986 01:03:33,390 --> 01:03:39,080 So that a must contract with d and b must be the same as c. 987 01:03:39,080 --> 01:03:45,900 So that means we have delta a d and delta b c. 988 01:03:45,900 --> 01:03:48,740 OK. 989 01:03:48,740 --> 01:03:51,610 Because in the Lagrangian, only those fields are paired. 990 01:03:51,610 --> 01:03:55,450 So the propagator must have this structure. 991 01:03:55,450 --> 01:03:58,742 So similarly, we can write down the-- 992 01:03:58,742 --> 01:04:00,700 so there is a four particle interaction vertex. 993 01:04:04,280 --> 01:04:08,818 So each particle has two index, so this is a b, 994 01:04:08,818 --> 01:04:15,795 c d, e f, oh yeah, right, sorry. 995 01:04:15,795 --> 01:04:17,150 Let me add one more. 996 01:04:17,150 --> 01:04:19,240 And then I can denote this diagramatically just 997 01:04:19,240 --> 01:04:24,710 as a b c d, OK. 998 01:04:24,710 --> 01:04:27,090 Just denote this diagramatically as the propagator, 999 01:04:27,090 --> 01:04:29,650 which is wavy line a b c d. 1000 01:04:29,650 --> 01:04:31,770 OK. 1001 01:04:31,770 --> 01:04:35,820 And now, also, we have this quartic interaction vertices, 1002 01:04:35,820 --> 01:04:37,850 which is involving four fields. 1003 01:04:37,850 --> 01:04:42,245 So each field has two index, because a b, so a b c d, f g h. 1004 01:04:46,002 --> 01:04:47,710 But this is another arbitrary interaction 1005 01:04:47,710 --> 01:04:48,585 they have contracted. 1006 01:04:48,585 --> 01:04:51,770 In the interaction, they contract in a specific way. 1007 01:04:51,770 --> 01:04:55,730 So you get 1/g squared because of [INAUDIBLE] factor 1008 01:04:55,730 --> 01:04:56,660 g squared. 1009 01:04:56,660 --> 01:05:01,560 And then you have delta b c, delta 1010 01:05:01,560 --> 01:05:07,009 d e, delta f g, delta h a. 1011 01:05:07,009 --> 01:05:08,633 OK, because they have to be contracted. 1012 01:05:11,710 --> 01:05:14,910 Better to write it as a b, this way. a b. 1013 01:05:14,910 --> 01:05:15,750 OK. 1014 01:05:15,750 --> 01:05:19,715 So a b, b contracted with c, d contracted with e, e 1015 01:05:19,715 --> 01:05:22,090 contracted with g, h contracted with a, so this gives you 1016 01:05:22,090 --> 01:05:23,840 a matrix product. 1017 01:05:23,840 --> 01:05:25,624 So just like that, OK. 1018 01:05:25,624 --> 01:05:27,540 AUDIENCE: So it doesn't propagate or anything. 1019 01:05:27,540 --> 01:05:28,659 It just changes a and b. 1020 01:05:28,659 --> 01:05:29,200 HONG LIU: Hm? 1021 01:05:31,960 --> 01:05:32,610 a b 1022 01:05:32,610 --> 01:05:34,035 AUDIENCE: --contracted with d. 1023 01:05:34,035 --> 01:05:34,660 HONG LIU: Yeah. 1024 01:05:34,660 --> 01:05:35,826 Let me just [INAUDIBLE] d c. 1025 01:05:35,826 --> 01:05:36,619 That's right. 1026 01:05:36,619 --> 01:05:37,119 OK. 1027 01:05:41,960 --> 01:05:43,081 So this is a Feynman rule. 1028 01:05:43,081 --> 01:05:44,455 So now you can start calculating. 1029 01:05:47,790 --> 01:05:50,354 So if you have calculated this for a while, 1030 01:05:50,354 --> 01:05:52,145 then you find this is a little bit tedious. 1031 01:05:52,145 --> 01:05:54,440 AUDIENCE: [LAUGHTER] 1032 01:05:54,440 --> 01:05:59,540 HONG LIU: With all these indices, it's quite annoying. 1033 01:05:59,540 --> 01:06:04,720 But t' Hooft was a smart fellow. 1034 01:06:04,720 --> 01:06:07,540 He said, let me find a way so that I don't have to keep 1035 01:06:07,540 --> 01:06:10,760 track of all these indices. 1036 01:06:10,760 --> 01:06:12,846 Then he invented a very nice way. 1037 01:06:12,846 --> 01:06:14,470 So this is called double line notation. 1038 01:06:24,450 --> 01:06:33,470 So first, this propagator a b c d, 1039 01:06:33,470 --> 01:06:36,610 since that a is contracted with c, c 1040 01:06:36,610 --> 01:06:39,040 is contracted with b separately, he said, 1041 01:06:39,040 --> 01:06:40,680 let's just write it as a double line. 1042 01:06:43,460 --> 01:06:46,700 But since there's a difference between the upper the lower 1043 01:06:46,700 --> 01:06:51,440 indices, so let's define a direction for this double line. 1044 01:06:51,440 --> 01:06:54,060 So let's define the direction to go from the upper index 1045 01:06:54,060 --> 01:06:55,660 to the lower index. 1046 01:06:55,660 --> 01:07:00,690 Then you will have a b, then you have d, then c b. 1047 01:07:00,690 --> 01:07:04,810 Because for c, is upper to lower index, to go from c d. 1048 01:07:11,680 --> 01:07:16,180 So now you don't have to worry about this thing anymore, 1049 01:07:16,180 --> 01:07:19,820 because it's automatically taken care of by this notation. 1050 01:07:19,820 --> 01:07:26,800 Similarly, this one is also automatically taken 1051 01:07:26,800 --> 01:07:35,050 care of because each one, if you replace it, by the double line 1052 01:07:35,050 --> 01:07:43,190 notation, then you just make it a, 1053 01:07:43,190 --> 01:07:47,980 b, then b is equal to c, b equal to it here, c equal to c, d 1054 01:07:47,980 --> 01:07:52,660 equal to d, and a to a. 1055 01:07:52,660 --> 01:07:57,240 And then we have g squared, here you have g squared. 1056 01:08:00,940 --> 01:08:03,690 So once you do all the double line notation-- so 1057 01:08:03,690 --> 01:08:15,040 let me also draw the arrow-- once you have double line 1058 01:08:15,040 --> 01:08:21,410 notation, then all of those annoying deltas, 1059 01:08:21,410 --> 01:08:22,921 they're automatically taken care of. 1060 01:08:26,080 --> 01:08:30,068 AUDIENCE: On the left, that side, [INAUDIBLE] 1061 01:08:30,068 --> 01:08:30,609 HONG LIU: Hm? 1062 01:08:30,609 --> 01:08:33,386 AUDIENCE: c is on the top of d. 1063 01:08:33,386 --> 01:08:34,510 HONG LIU: Sorry, which one? 1064 01:08:34,510 --> 01:08:36,995 AUDIENCE: That [INAUDIBLE] 1065 01:08:36,995 --> 01:08:37,620 HONG LIU: Here? 1066 01:08:37,620 --> 01:08:39,840 AUDIENCE: Yes, c is on the top. 1067 01:08:39,840 --> 01:08:41,359 HONG LIU: That-- 1068 01:08:41,359 --> 01:08:43,472 AUDIENCE: Yeah, if we write [INAUDIBLE] 1069 01:08:43,472 --> 01:08:44,013 HONG LIU: Oh. 1070 01:08:44,013 --> 01:08:45,605 OK. 1071 01:08:45,605 --> 01:08:46,520 delta b c. 1072 01:08:46,520 --> 01:08:48,520 That's right. 1073 01:08:48,520 --> 01:08:49,520 b c. 1074 01:08:51,920 --> 01:08:52,420 Good. 1075 01:08:52,420 --> 01:08:52,919 Good. 1076 01:08:52,919 --> 01:08:55,550 Good. 1077 01:08:55,550 --> 01:08:58,609 So now let us give you some examples of using this diagram 1078 01:08:58,609 --> 01:09:01,399 to calculate things. 1079 01:09:01,399 --> 01:09:03,479 And also try to calculate the independents. 1080 01:09:03,479 --> 01:09:06,439 Because eventually we want to take a large N limit. 1081 01:09:06,439 --> 01:09:06,941 OK. 1082 01:09:06,941 --> 01:09:08,815 So we want to keep track of the independents. 1083 01:09:17,370 --> 01:09:18,340 HONG LIU: Yes? 1084 01:09:18,340 --> 01:09:20,034 AUDIENCE: The four point coupling? 1085 01:09:20,034 --> 01:09:23,090 Does that come from the commutator in the f [INAUDIBLE] 1086 01:09:23,090 --> 01:09:24,495 in the gauge theory? 1087 01:09:24,495 --> 01:09:26,370 HONG LIU: No, I just wrote it down by myself. 1088 01:09:26,370 --> 01:09:28,200 AUDIENCE: OK, so it's an additional thing. 1089 01:09:28,200 --> 01:09:28,529 HONG LIU: Yeah. 1090 01:09:28,529 --> 01:09:28,880 Yeah. 1091 01:09:28,880 --> 01:09:29,880 AUDIENCE: So [INAUDIBLE] 1092 01:09:29,880 --> 01:09:31,600 HONG LIU: It's actually-- yeah. 1093 01:09:31,600 --> 01:09:33,968 Just arbitrary field theory. 1094 01:09:33,968 --> 01:09:34,634 Just an example. 1095 01:09:37,529 --> 01:09:38,029 OK. 1096 01:09:38,029 --> 01:09:42,029 So now let's start calculating. 1097 01:09:42,029 --> 01:09:44,709 You already have the Feynman rule. 1098 01:09:44,709 --> 01:09:45,750 What are you waiting for? 1099 01:09:45,750 --> 01:09:46,224 AUDIENCE: [LAUGHTER] 1100 01:09:46,224 --> 01:09:47,172 HONG LIU: Haha. 1101 01:09:47,172 --> 01:09:50,970 So let's just calculate. 1102 01:09:50,970 --> 01:09:55,750 So the first thing, let's look at the vacuum energy. 1103 01:09:55,750 --> 01:09:59,440 So that's the first thing to calculate. 1104 01:09:59,440 --> 01:10:02,140 So what's the vacuum energy? 1105 01:10:02,140 --> 01:10:05,577 The vacuum energy is all the diagrams. 1106 01:10:05,577 --> 01:10:07,910 It's a summation of all of the diagrams without external 1107 01:10:07,910 --> 01:10:08,450 [INAUDIBLE]. 1108 01:10:08,450 --> 01:10:08,950 OK. 1109 01:10:15,850 --> 01:10:22,017 So the simplest diagram is just with one vertex. 1110 01:10:27,760 --> 01:10:28,260 OK. 1111 01:10:28,260 --> 01:10:30,220 So this is the simplest diagram. 1112 01:10:33,650 --> 01:10:37,830 But now, so if you do lambda 5 4 theory, 1113 01:10:37,830 --> 01:10:40,430 then that will be the end of the story. 1114 01:10:40,430 --> 01:10:42,410 OK. 1115 01:10:42,410 --> 01:10:44,440 You just contract them. 1116 01:10:44,440 --> 01:10:51,920 But now, the matrix, because we have contracting matrices, 1117 01:10:51,920 --> 01:10:55,000 so this is like contracting to four matrices. 1118 01:10:55,000 --> 01:10:57,060 When you're contracting matrices-- but matrices 1119 01:10:57,060 --> 01:10:58,770 don't commute. 1120 01:10:58,770 --> 01:11:01,750 So actually there's a difference. 1121 01:11:01,750 --> 01:11:12,740 So you should convince yourself there's 1122 01:11:12,740 --> 01:11:14,690 a difference which I actually connect 1123 01:11:14,690 --> 01:11:23,070 these two from which I connect these two and these two. 1124 01:11:23,070 --> 01:11:24,777 OK. 1125 01:11:24,777 --> 01:11:27,360 Because this corresponding to I connect with neighboring ones. 1126 01:11:27,360 --> 01:11:30,100 And this one I connect with diagonal ones. 1127 01:11:30,100 --> 01:11:32,740 And the order actually matters. 1128 01:11:32,740 --> 01:11:34,380 The order actually matters. 1129 01:11:34,380 --> 01:11:37,356 Because matrices don't commute. 1130 01:11:37,356 --> 01:11:46,807 So if I call this a and the b, and a not equal to b. 1131 01:11:46,807 --> 01:11:48,265 Because the matrices don't commute. 1132 01:11:51,720 --> 01:11:55,880 So now let's do the N counting. 1133 01:11:55,880 --> 01:11:57,880 So to do the N counting is actually much simpler 1134 01:11:57,880 --> 01:12:00,400 if we use the double line notation. 1135 01:12:00,400 --> 01:12:02,810 So let's now use the double line notation. 1136 01:12:02,810 --> 01:12:06,990 Because all these N, so when you do the contractions, 1137 01:12:06,990 --> 01:12:11,270 you have all these indices, you sum all those indices-- when 1138 01:12:11,270 --> 01:12:13,710 you sum all those indices, you will get some factors of N. 1139 01:12:13,710 --> 01:12:18,560 So that's how N will come into this theory. 1140 01:12:18,560 --> 01:12:23,849 But double line notation makes the constraint all manifest, 1141 01:12:23,849 --> 01:12:25,390 so the contraction will be automatic. 1142 01:12:25,390 --> 01:12:28,940 You don't have to worry about all those deltas. 1143 01:12:28,940 --> 01:12:31,954 So let's just do that. 1144 01:12:31,954 --> 01:12:33,370 Let's draw this four point vertex. 1145 01:12:36,590 --> 01:12:39,430 For this one, we just contract the neighboring ones. 1146 01:12:42,420 --> 01:12:44,310 For this one, we do the same thing. 1147 01:12:51,810 --> 01:12:55,384 We contract the neighboring ones. 1148 01:13:01,390 --> 01:13:01,890 Yeah. 1149 01:13:01,890 --> 01:13:03,180 I did not draw very well. 1150 01:13:03,180 --> 01:13:06,496 Let me-- 1151 01:13:16,120 --> 01:13:18,280 OK. 1152 01:13:18,280 --> 01:13:22,910 So this one should be contracted with this one. 1153 01:13:22,910 --> 01:13:26,870 This one should be contracted with that one. 1154 01:13:26,870 --> 01:13:31,620 And this one contract with this one, and this one 1155 01:13:31,620 --> 01:13:32,730 is contracted there. 1156 01:13:32,730 --> 01:13:34,220 OK. 1157 01:13:34,220 --> 01:13:37,720 There's some intersection. 1158 01:13:37,720 --> 01:13:39,250 So now let's come to the end. 1159 01:13:43,870 --> 01:13:45,055 So now here is the key. 1160 01:13:48,160 --> 01:13:54,370 Each loop corresponding to a contracted indices. 1161 01:13:54,370 --> 01:13:59,460 So for example, in here, this is already a and a. 1162 01:13:59,460 --> 01:14:02,560 And then if you contract these two, then you sum over this a 1163 01:14:02,560 --> 01:14:04,320 when you calculate the propagator. 1164 01:14:04,320 --> 01:14:06,760 And then that gives you a factor of n. 1165 01:14:06,760 --> 01:14:09,630 So each contract, each closed loop, essentially 1166 01:14:09,630 --> 01:14:12,550 gives you a factor of n. 1167 01:14:12,550 --> 01:14:15,660 Just because you contract-- yeah. 1168 01:14:15,660 --> 01:14:17,570 So there are three loops here. 1169 01:14:17,570 --> 01:14:19,190 Three contracted loops. 1170 01:14:19,190 --> 01:14:20,493 So this gives you n cubed. 1171 01:14:23,660 --> 01:14:26,940 And if you look at the g squared dependence, 1172 01:14:26,940 --> 01:14:30,050 so you have two propagators here. 1173 01:14:30,050 --> 01:14:31,570 You have two propagators. 1174 01:14:31,570 --> 01:14:33,370 Each propagator gives you a g squared. 1175 01:14:33,370 --> 01:14:35,750 Then you have one vertex. 1176 01:14:35,750 --> 01:14:40,710 Each vertex gives you 1/g squared, so you have g squared. 1177 01:14:40,710 --> 01:14:43,140 OK. 1178 01:14:43,140 --> 01:14:46,090 So now let's look at this one. 1179 01:14:46,090 --> 01:14:48,030 This one, this says follow this line-- oh, you 1180 01:14:48,030 --> 01:14:49,904 should also draw the arrow, because the arrow 1181 01:14:49,904 --> 01:14:52,370 should follow. 1182 01:14:52,370 --> 01:14:55,200 I did not draw the arrow, but arrows should all follow. 1183 01:15:03,120 --> 01:15:04,880 The arrow should be consistent. 1184 01:15:04,880 --> 01:15:06,590 Similarly, here. 1185 01:15:06,590 --> 01:15:08,270 So here, if you go around, you find 1186 01:15:08,270 --> 01:15:12,150 there is only a single contracted line. 1187 01:15:12,150 --> 01:15:14,620 So this, only a factor of n. 1188 01:15:14,620 --> 01:15:16,860 Of course, g squared dependency is the same. 1189 01:15:16,860 --> 01:15:19,279 So you have g squared. 1190 01:15:19,279 --> 01:15:20,820 So now you already see the difference 1191 01:15:20,820 --> 01:15:22,285 between these two diagrams. 1192 01:15:24,800 --> 01:15:28,490 One is proportionate to n, one is proportionate to n cubed. 1193 01:15:33,190 --> 01:15:45,095 So a, diagram a, can be drawn on the plane-- we are drawing it 1194 01:15:45,095 --> 01:15:56,700 on the plane without crossing lines, 1195 01:15:56,700 --> 01:15:58,420 so we call it a planar diagram. 1196 01:16:05,270 --> 01:16:12,520 And b cannot be drawn on the plane without crossing lines, 1197 01:16:12,520 --> 01:16:15,432 so we call it a non-planar diagram. 1198 01:16:24,110 --> 01:16:26,807 So let's do one more step. 1199 01:16:26,807 --> 01:16:28,015 So let's do one more diagram. 1200 01:16:39,100 --> 01:16:40,320 So let's do one more diagram. 1201 01:16:46,750 --> 01:16:54,760 So next order, you just get two, next order [INAUDIBLE] 1202 01:16:54,760 --> 01:17:01,630 diagram without [INAUDIBLE] you have two vertices. 1203 01:17:01,630 --> 01:17:03,260 OK. 1204 01:17:03,260 --> 01:17:04,450 So you can connect this way. 1205 01:17:08,250 --> 01:17:10,230 So this is one diagram. 1206 01:17:10,230 --> 01:17:14,152 You can also have a non-planar diagram-- yeah, 1207 01:17:14,152 --> 01:17:16,110 let me just draw the straight line, rather than 1208 01:17:16,110 --> 01:17:20,480 the wavy line, just to save trouble. 1209 01:17:20,480 --> 01:17:23,000 You can, for example, you can draw like this, 1210 01:17:23,000 --> 01:17:24,560 so this is a non-planar diagram. 1211 01:17:29,190 --> 01:17:35,090 So now I'll give you an exercise today when you go home. 1212 01:17:35,090 --> 01:17:39,760 Turn those into double line diagrams. 1213 01:17:39,760 --> 01:17:42,800 Then you should find yourself that this 1214 01:17:42,800 --> 01:17:45,375 is n to the power of 4. 1215 01:17:45,375 --> 01:17:45,875 Yeah. 1216 01:17:45,875 --> 01:17:46,833 Maybe I should do this. 1217 01:17:49,630 --> 01:17:52,960 Yeah, let me do this one for you. 1218 01:17:52,960 --> 01:17:55,156 So let me just do this one quickly for you. 1219 01:18:03,870 --> 01:18:06,900 But I will not draw the arrows, just to save time. 1220 01:18:22,380 --> 01:18:27,390 So this one, you count how many contracted lines? 1221 01:18:27,390 --> 01:18:31,460 One, two, three, four, so this gives you n to the 4th 1222 01:18:31,460 --> 01:18:34,500 for the area. 1223 01:18:34,500 --> 01:18:36,410 And this I will not do it now. 1224 01:18:36,410 --> 01:18:38,100 You do it yourself. 1225 01:18:38,100 --> 01:18:40,220 So, also, if you count the g dependents, 1226 01:18:40,220 --> 01:18:42,820 this gives you g to the power of 4. 1227 01:18:42,820 --> 01:18:50,600 And this gives you n squared, g to the power of 4. 1228 01:18:50,600 --> 01:18:51,910 Is it g to the power of 4? 1229 01:18:51,910 --> 01:18:53,730 Maybe I'm counting wrong. 1230 01:18:53,730 --> 01:18:54,946 One, two, three, four. 1231 01:18:54,946 --> 01:18:57,490 Four propagators is g squared. 1232 01:18:57,490 --> 01:18:58,610 Yeah. g to the power of 4. 1233 01:18:58,610 --> 01:19:01,430 Yeah. 1234 01:19:01,430 --> 01:19:01,970 Yes? 1235 01:19:01,970 --> 01:19:05,060 AUDIENCE: So in general, the more-- the more crossings, 1236 01:19:05,060 --> 01:19:06,000 would that decrease n? 1237 01:19:06,000 --> 01:19:09,010 HONG LIU: Yeah, that's right. 1238 01:19:09,010 --> 01:19:10,420 OK. 1239 01:19:10,420 --> 01:19:12,950 So now you can just continue. 1240 01:19:12,950 --> 01:19:17,210 With three vertices, with four vertices, et cetera. 1241 01:19:17,210 --> 01:19:22,500 But of course, the smart fellows will not stop, and they ask, 1242 01:19:22,500 --> 01:19:25,950 can I find some rules in doing this? 1243 01:19:25,950 --> 01:19:30,180 So, how do you get the general n counting? 1244 01:19:30,180 --> 01:19:34,340 And whether that's the old planar diagram with the same, 1245 01:19:34,340 --> 01:19:37,120 or does one need to classify them further? 1246 01:19:37,120 --> 01:19:37,710 OK. 1247 01:19:37,710 --> 01:19:39,870 So these are the obvious questions. 1248 01:19:39,870 --> 01:19:42,740 So let me just tell you some observations. 1249 01:19:42,740 --> 01:19:46,656 Then we are closed for today. 1250 01:19:46,656 --> 01:19:48,370 Then we will close for today. 1251 01:19:52,550 --> 01:19:53,690 So the first observation. 1252 01:19:59,534 --> 01:20:02,200 So just from those examples, let me just draw some observations. 1253 01:20:06,770 --> 01:20:14,209 Observation one-- so this one requires a little bit 1254 01:20:14,209 --> 01:20:14,750 of ingenuity. 1255 01:20:18,655 --> 01:20:24,400 It says, even a and b-- so this a and b-- 1256 01:20:24,400 --> 01:20:28,130 they cannot be drawn on the plane. 1257 01:20:28,130 --> 01:20:33,700 They cannot be drawn on the plane without crossing lines. 1258 01:20:33,700 --> 01:20:47,080 But a and b can be drawn on a torus without crossing lines. 1259 01:20:57,180 --> 01:21:01,390 So now let me show it to you. 1260 01:21:01,390 --> 01:21:04,270 So what is a torus? 1261 01:21:04,270 --> 01:21:07,380 So we can draw a torus on the plane, like follows, 1262 01:21:07,380 --> 01:21:08,960 so we draw a square. 1263 01:21:08,960 --> 01:21:12,000 Then we identify this side with that side. 1264 01:21:12,000 --> 01:21:14,730 Identify this side with that side. 1265 01:21:14,730 --> 01:21:16,460 Then this gives us a torus. 1266 01:21:16,460 --> 01:21:18,060 OK. 1267 01:21:18,060 --> 01:21:22,722 Because when you do that, that give you cylinder. 1268 01:21:22,722 --> 01:21:24,930 And then you do that again, and it gives you a torus. 1269 01:21:27,780 --> 01:21:32,170 But on the plane, just simple identification. 1270 01:21:32,170 --> 01:21:34,720 So let's try to see this diagram. 1271 01:21:34,720 --> 01:21:38,160 Now let me show this diagram can be drawn on the torus 1272 01:21:38,160 --> 01:21:41,865 without crossing, because the vertex is like this. 1273 01:21:45,110 --> 01:21:47,990 We want to contract with this guy with that guy, and that guy 1274 01:21:47,990 --> 01:21:49,520 with that guy, right? 1275 01:21:49,520 --> 01:21:51,060 Not labeling lines. 1276 01:21:51,060 --> 01:21:54,550 But now I can just go up. 1277 01:21:54,550 --> 01:21:56,330 Now this is identified with here, 1278 01:21:56,330 --> 01:21:58,230 then I can come back here. 1279 01:21:58,230 --> 01:22:01,590 Can go here, come back here. 1280 01:22:01,590 --> 01:22:03,200 So this is contracted. 1281 01:22:03,200 --> 01:22:07,370 And I don't have to cross lines. 1282 01:22:07,370 --> 01:22:08,780 Yes? 1283 01:22:08,780 --> 01:22:09,280 Good. 1284 01:22:09,280 --> 01:22:12,880 Similarly, let me do that one. 1285 01:22:12,880 --> 01:22:16,090 Yeah, that one, maybe I will let you do yourself. 1286 01:22:16,090 --> 01:22:20,020 Yeah, do the-- so this a-- no, this is not a and b. 1287 01:22:20,020 --> 01:22:22,440 This is b and d. 1288 01:22:22,440 --> 01:22:29,380 Let me call this-- b, this is-- let me call it c and d. 1289 01:22:29,380 --> 01:22:32,870 So a and c are planar diagrams and b 1290 01:22:32,870 --> 01:22:35,020 and d are non-planar diagrams. 1291 01:22:35,020 --> 01:22:37,360 OK. 1292 01:22:37,360 --> 01:22:40,070 So this is b. 1293 01:22:40,070 --> 01:22:43,045 So try to do it yourself, the d. 1294 01:22:43,045 --> 01:22:44,670 So you essentially you have two vertex. 1295 01:22:47,720 --> 01:22:49,450 How to make that non-planar diagram 1296 01:22:49,450 --> 01:22:53,920 to be planar on this torus when you identify here 1297 01:22:53,920 --> 01:22:56,200 with there, and there with there? 1298 01:22:56,200 --> 01:22:56,700 OK. 1299 01:22:56,700 --> 01:22:57,930 So this observation one. 1300 01:23:00,791 --> 01:23:02,290 Actually, let me just do it for you. 1301 01:23:02,290 --> 01:23:03,686 AUDIENCE: [LAUGHTER] 1302 01:23:03,686 --> 01:23:05,060 HONG LIU: So we connect because I 1303 01:23:05,060 --> 01:23:09,680 need to do-- because it's better I have this example 1304 01:23:09,680 --> 01:23:13,550 to make my second point. 1305 01:23:13,550 --> 01:23:16,420 So now I need to connect this to that guy, 1306 01:23:16,420 --> 01:23:21,012 but I can do it by going up. 1307 01:23:21,012 --> 01:23:23,160 I can-- this guy can go here. 1308 01:23:23,160 --> 01:23:24,130 And then go there. 1309 01:23:24,130 --> 01:23:25,870 Similar thing, OK. 1310 01:23:25,870 --> 01:23:30,750 And then this is-- I can connect them without crossing lines. 1311 01:23:30,750 --> 01:23:34,100 So this is observation one. 1312 01:23:34,100 --> 01:23:40,850 So now, observation two, again, is a little bit ingenious, 1313 01:23:40,850 --> 01:23:46,936 but maybe slightly easier if you have done it long enough. 1314 01:23:51,900 --> 01:23:57,410 For all the a to d, the Fourier rule is true. 1315 01:23:57,410 --> 01:24:08,390 The power of n is equal to number of faces. 1316 01:24:08,390 --> 01:24:14,000 So this one, we call-- so even though they cannot be crossed, 1317 01:24:14,000 --> 01:24:17,510 they cross nice on the plane, but we can actually straighten 1318 01:24:17,510 --> 01:24:18,860 them out on the torus. 1319 01:24:18,860 --> 01:24:19,760 OK. 1320 01:24:19,760 --> 01:24:23,090 So here we can straighten them out on the torus. 1321 01:24:23,090 --> 01:24:24,940 So the second [INAUDIBLE] is a power of n, 1322 01:24:24,940 --> 01:24:28,525 it's the number of faces in each diagram. 1323 01:24:32,810 --> 01:24:36,990 After we straighten it out. 1324 01:24:46,911 --> 01:24:47,410 OK. 1325 01:24:47,410 --> 01:24:48,855 First, let's look at this diagram. 1326 01:24:51,460 --> 01:24:53,490 This is a planar diagram. 1327 01:24:53,490 --> 01:24:55,570 So there's one face. 1328 01:24:55,570 --> 01:24:57,096 There's two faces. 1329 01:24:57,096 --> 01:24:59,720 And then there's a face outside, because there's a whole plane. 1330 01:24:59,720 --> 01:25:01,240 So there's three faces. 1331 01:25:01,240 --> 01:25:04,380 So this is n cubed. 1332 01:25:04,380 --> 01:25:10,640 And the b, on the torus, there's only one face, 1333 01:25:10,640 --> 01:25:13,500 because each region is connected to each other. 1334 01:25:13,500 --> 01:25:14,359 OK. 1335 01:25:14,359 --> 01:25:15,400 So there's only one face. 1336 01:25:15,400 --> 01:25:17,900 You see the power n. 1337 01:25:17,900 --> 01:25:22,370 So now you see you had one face, you have two faces, 1338 01:25:22,370 --> 01:25:24,380 we have three faces, we have four faces, 1339 01:25:24,380 --> 01:25:27,730 so this is n to the power of 4. 1340 01:25:27,730 --> 01:25:32,060 And for this one, you have one face corresponding to here. 1341 01:25:34,690 --> 01:25:38,150 And then you easily convince yourself everything 1342 01:25:38,150 --> 01:25:40,460 outside this region, they're connected 1343 01:25:40,460 --> 01:25:44,541 through the identification, so you get n squared. 1344 01:25:44,541 --> 01:25:45,040 OK. 1345 01:25:45,040 --> 01:25:47,140 Then n squared. 1346 01:25:47,140 --> 01:25:49,010 So there's two observations. 1347 01:25:49,010 --> 01:25:51,110 So let me stop here. 1348 01:25:51,110 --> 01:25:54,380 And then, from here, you can figure out the general rules 1349 01:25:54,380 --> 01:25:55,650 for everything. 1350 01:26:03,660 --> 01:26:05,210 [APPLAUSE]