1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,810 Commons license. 3 00:00:03,810 --> 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 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 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 00:00:21,260 --> 00:00:22,430 HONG LIU: OK, let us start. 9 00:00:22,430 --> 00:00:24,590 Welcome, everybody. 10 00:00:24,590 --> 00:00:31,680 So first, let me just say a few general things regarding 11 00:00:31,680 --> 00:00:35,590 the logistics of this class. 12 00:00:35,590 --> 00:00:39,180 So you should have these three handouts. 13 00:00:39,180 --> 00:00:40,580 One, organization. 14 00:00:40,580 --> 00:00:42,070 One, outline. 15 00:00:42,070 --> 00:00:44,920 And then, one is for PSET 1. 16 00:00:47,960 --> 00:00:52,870 So let me just go over briefly this organization 17 00:00:52,870 --> 00:00:53,580 and the outline. 18 00:00:58,650 --> 00:01:03,245 So firstly in this class, there's no textbooks. 19 00:01:06,640 --> 00:01:13,180 However, there are many reviews available. 20 00:01:13,180 --> 00:01:16,240 But unfortunately, a lot of those reviews, 21 00:01:16,240 --> 00:01:19,145 none of the single review is suitable for the whole course. 22 00:01:23,430 --> 00:01:23,930 Yeah. 23 00:01:23,930 --> 00:01:26,180 So I have listed some of the reviews 24 00:01:26,180 --> 00:01:28,565 at the end of this organization. 25 00:01:31,381 --> 00:01:31,880 Yeah. 26 00:01:31,880 --> 00:01:35,380 So along the way, I will point it out, 27 00:01:35,380 --> 00:01:38,720 some specific references or some specific parts 28 00:01:38,720 --> 00:01:45,390 of certain reviews if they are directly relevant. 29 00:01:45,390 --> 00:01:49,160 And also for this class, there's no recitation. 30 00:01:49,160 --> 00:01:53,580 Yeah, for advanced courses like this, 31 00:01:53,580 --> 00:01:56,720 department does not assign TA. 32 00:01:56,720 --> 00:01:58,350 So we don't have recitation. 33 00:01:58,350 --> 00:02:06,220 So I have my own office hour, which is Monday 3:30 to 4:30. 34 00:02:06,220 --> 00:02:10,449 Just one hour before the lecture on Monday. 35 00:02:10,449 --> 00:02:10,949 Yeah. 36 00:02:10,949 --> 00:02:15,710 So if you have questions or if you have-- yeah, 37 00:02:15,710 --> 00:02:20,220 just feel free to come. 38 00:02:20,220 --> 00:02:23,625 So if you have some questions other time, 39 00:02:23,625 --> 00:02:28,070 say, if this office hour is not good for you, 40 00:02:28,070 --> 00:02:31,430 I'm also happy to set up some other time to meet you. 41 00:02:31,430 --> 00:02:37,420 Or, you can stay after the class to ask questions, et cetera. 42 00:02:37,420 --> 00:02:37,970 Yeah. 43 00:02:37,970 --> 00:02:41,280 So whichever way is more convenient to you. 44 00:02:46,250 --> 00:02:48,940 So today, we have this-- yeah, so form now on, 45 00:02:48,940 --> 00:02:51,870 everything will be just on the web. 46 00:02:51,870 --> 00:02:54,800 You should download the future PSETs. 47 00:02:54,800 --> 00:02:58,600 Or if I put any lecture notes or materials, 48 00:02:58,600 --> 00:03:00,940 you should find them on the web. 49 00:03:00,940 --> 00:03:08,700 And I assume you can all find out the website of the course. 50 00:03:08,700 --> 00:03:14,680 And it's also listed here in the organization page. 51 00:03:14,680 --> 00:03:18,060 In the organization page. 52 00:03:18,060 --> 00:03:19,530 So any questions so far? 53 00:03:24,790 --> 00:03:27,260 Good. 54 00:03:27,260 --> 00:03:31,550 So this class there's no exam. 55 00:03:31,550 --> 00:03:36,660 And the grades are solely based on PSETs and the final project. 56 00:03:36,660 --> 00:03:40,530 And the PSETs will be due every two weeks. 57 00:03:40,530 --> 00:03:45,940 So if you look at the calendar, there are about like 5 PSETs. 58 00:03:45,940 --> 00:03:49,090 Then, for the last three weeks, instead 59 00:03:49,090 --> 00:03:53,880 of, say, handing a PSET, you're handing in the small paper. 60 00:03:53,880 --> 00:03:58,810 Just whatever topics you would like to choose 61 00:03:58,810 --> 00:04:01,170 about the holographic duality. 62 00:04:01,170 --> 00:04:02,350 Yeah, just choose one topic. 63 00:04:02,350 --> 00:04:05,990 Write, say, 8 to 10-page paper. 64 00:04:05,990 --> 00:04:08,260 Either review or some calculation 65 00:04:08,260 --> 00:04:11,680 you would like to do, et cetera. 66 00:04:11,680 --> 00:04:12,820 Yeah. 67 00:04:12,820 --> 00:04:16,360 So 5 PSET is 75%. 68 00:04:16,360 --> 00:04:19,160 And this final project would be 25%. 69 00:04:23,460 --> 00:04:32,842 That's pretty much the logistics regarding this class. 70 00:04:32,842 --> 00:04:34,300 Do you have any questions regarding 71 00:04:34,300 --> 00:04:39,650 the grading, regarding the project or PSET, et cetera? 72 00:04:45,180 --> 00:04:47,750 Good. 73 00:04:47,750 --> 00:04:51,910 So let me say a few words about this outline. 74 00:04:51,910 --> 00:04:57,290 So this outline is way too ambitious. 75 00:04:57,290 --> 00:05:00,850 It gives you-- yeah, it just serves 76 00:05:00,850 --> 00:05:04,380 to give you a rough contour what we will cover. 77 00:05:04,380 --> 00:05:10,140 But really, don't take it too literally because-- first, we 78 00:05:10,140 --> 00:05:13,990 are not going to cover all seven chapters. 79 00:05:13,990 --> 00:05:15,320 That would be too much. 80 00:05:15,320 --> 00:05:18,325 So last time when I taught it, I covered only three chapters. 81 00:05:21,310 --> 00:05:25,610 And so hopefully, this time I will do four. 82 00:05:25,610 --> 00:05:28,130 Hopefully, time I will do four. 83 00:05:28,130 --> 00:05:31,370 And I may also just deviate from some of the things 84 00:05:31,370 --> 00:05:35,630 here, depending on the pace or depend on people's interests, 85 00:05:35,630 --> 00:05:37,410 et cetera. 86 00:05:37,410 --> 00:05:39,615 So this is going to be flexible. 87 00:05:39,615 --> 00:05:44,160 If there are certain things you would really very much like 88 00:05:44,160 --> 00:05:46,130 to hear, you can also let me know. 89 00:05:46,130 --> 00:05:47,760 I can think about it. 90 00:05:47,760 --> 00:05:50,250 In particular, among the five, six, seven. 91 00:05:50,250 --> 00:05:54,880 And I can think about it, maybe I'll discuss some of them. 92 00:05:54,880 --> 00:05:58,122 Discuss some of them. 93 00:05:58,122 --> 00:05:59,270 Yeah. 94 00:05:59,270 --> 00:06:05,880 Also, another thing if you keep in mind is that in this class, 95 00:06:05,880 --> 00:06:13,850 this is a class which will touch on many different subjects, 96 00:06:13,850 --> 00:06:16,380 say gravity, a little bit of string theory. 97 00:06:16,380 --> 00:06:20,600 And also, quantum field theories, et cetera. 98 00:06:20,600 --> 00:06:24,800 So in such a class, it's almost impossible-- I 99 00:06:24,800 --> 00:06:30,120 think it's just impossible to really derive everything 100 00:06:30,120 --> 00:06:32,000 in the self-contained way. 101 00:06:32,000 --> 00:06:37,710 And also, some of-- yeah, I think some of your background 102 00:06:37,710 --> 00:06:39,450 are also very different. 103 00:06:39,450 --> 00:06:44,380 So the certain part may not appear 104 00:06:44,380 --> 00:06:47,500 self-contained to some of you. 105 00:06:47,500 --> 00:06:51,670 So if there is anything which is not familiar to you, 106 00:06:51,670 --> 00:06:53,650 if I just mention it, if I just code it, 107 00:06:53,650 --> 00:06:56,380 if it's not familiar with you, then 108 00:06:56,380 --> 00:07:00,490 you make sure-- don't hesitate to ask me, 109 00:07:00,490 --> 00:07:03,010 either during the class or after the class, 110 00:07:03,010 --> 00:07:06,960 so that I can help you to make up on that. 111 00:07:06,960 --> 00:07:08,880 To make up on that. 112 00:07:08,880 --> 00:07:13,460 And also, to keep in mind, like other advanced graduate 113 00:07:13,460 --> 00:07:20,200 classes, you actually mostly learn outside the class. 114 00:07:20,200 --> 00:07:23,560 Inside the class, it's mostly try 115 00:07:23,560 --> 00:07:27,430 to provide you some motivations, guidance, et cetera, 116 00:07:27,430 --> 00:07:29,950 and give you a rough contour. 117 00:07:29,950 --> 00:07:33,340 And your own reading outside the class 118 00:07:33,340 --> 00:07:40,772 actually should be the main routes you learn things. 119 00:07:40,772 --> 00:07:42,105 The main route you learn things. 120 00:07:47,950 --> 00:07:48,715 So any questions? 121 00:07:52,380 --> 00:07:57,970 So during the class, please feel free to ask questions. 122 00:07:57,970 --> 00:07:58,470 Yeah. 123 00:08:01,154 --> 00:08:02,570 Asking questions are a great thing 124 00:08:02,570 --> 00:08:04,470 because if something you don't understand, 125 00:08:04,470 --> 00:08:09,460 there's a very good chance quite a few of your fellow students 126 00:08:09,460 --> 00:08:10,930 also don't understand. 127 00:08:10,930 --> 00:08:12,466 So if you ask the question, you're 128 00:08:12,466 --> 00:08:13,840 not only helping yourself, you're 129 00:08:13,840 --> 00:08:15,680 also helping other people. 130 00:08:15,680 --> 00:08:17,210 And you also help me because then I 131 00:08:17,210 --> 00:08:20,090 will know certain aspect is not clear to you. 132 00:08:20,090 --> 00:08:24,360 Then, I will try to emphasize it, or repeat it, et cetera. 133 00:08:24,360 --> 00:08:27,730 So please, do ask questions. 134 00:08:27,730 --> 00:08:31,320 And also, if you have any feedback on the class, 135 00:08:31,320 --> 00:08:36,090 whether it's too fast, too slow, or problems are too hard or too 136 00:08:36,090 --> 00:08:40,270 easy, the PSET is too short for you, 137 00:08:40,270 --> 00:08:44,360 just not challenging enough, just let me know. 138 00:08:44,360 --> 00:08:47,540 Just let me know. 139 00:08:47,540 --> 00:08:48,040 Yeah. 140 00:08:48,040 --> 00:08:50,420 Then, I will try to adjust. 141 00:08:50,420 --> 00:08:52,670 And again, this is a very important. 142 00:08:52,670 --> 00:08:57,270 Again, this is very important because the-- yeah. 143 00:08:57,270 --> 00:09:00,580 This will help us to improve the class. 144 00:09:00,580 --> 00:09:02,670 To improve the class. 145 00:09:02,670 --> 00:09:04,447 Any other questions? 146 00:09:04,447 --> 00:09:05,155 Or any questions? 147 00:09:11,030 --> 00:09:12,590 OK. 148 00:09:12,590 --> 00:09:17,790 Then, let us start our lecture. 149 00:09:23,020 --> 00:09:35,175 So first, we look at the hints for the holographic duality. 150 00:10:07,380 --> 00:10:07,880 OK. 151 00:10:07,880 --> 00:10:10,750 So we start by doing a multiple choice problem. 152 00:11:10,300 --> 00:11:10,800 OK. 153 00:11:10,800 --> 00:11:11,880 So what is your answer? 154 00:11:15,235 --> 00:11:16,110 AUDIENCE: [INAUDIBLE] 155 00:11:18,745 --> 00:11:19,370 HONG LIU: Good. 156 00:11:22,370 --> 00:11:23,270 So why is gravity? 157 00:11:28,080 --> 00:11:30,995 AUDIENCE: We don't have a good quantum theory of gravity. 158 00:11:30,995 --> 00:11:32,050 HONG LIU: Good. 159 00:11:32,050 --> 00:11:34,360 So we don't have a good quantum theory of gravity. 160 00:11:34,360 --> 00:11:35,068 Any other reason? 161 00:11:35,068 --> 00:11:37,200 AUDIENCE: Except string theory. 162 00:11:37,200 --> 00:11:38,890 HONG LIU: Yeah, string theory is still 163 00:11:38,890 --> 00:11:43,930 not the-- I would say not yet at the [INAUDIBLE] quantum 164 00:11:43,930 --> 00:11:44,680 theory of gravity. 165 00:11:47,065 --> 00:11:47,940 AUDIENCE: [INAUDIBLE] 166 00:11:50,880 --> 00:11:51,690 HONG LIU: Yeah. 167 00:11:51,690 --> 00:11:53,590 Yeah, this is the similar to that. 168 00:11:53,590 --> 00:11:58,476 We don't have a full theory of gravity nonrealizable. 169 00:11:58,476 --> 00:12:00,690 AUDIENCE: [INAUDIBLE] 170 00:12:00,690 --> 00:12:02,870 HONG LIU: Good. 171 00:12:02,870 --> 00:12:04,950 Anything else? 172 00:12:04,950 --> 00:12:06,080 Or any other opinions? 173 00:12:06,080 --> 00:12:11,998 Say you should choose any other interaction other than gravity? 174 00:12:11,998 --> 00:12:13,741 AUDIENCE: Strong is the strongest one. 175 00:12:13,741 --> 00:12:16,772 Strong is the strongest one. 176 00:12:16,772 --> 00:12:17,730 HONG LIU: That's right. 177 00:12:17,730 --> 00:12:20,188 You can also choose the gravity because this is the weakest 178 00:12:20,188 --> 00:12:22,450 one. 179 00:12:22,450 --> 00:12:24,170 Yeah, so the gravity is the weakest one. 180 00:12:24,170 --> 00:12:25,040 OK. 181 00:12:25,040 --> 00:12:26,360 So yeah. 182 00:12:26,360 --> 00:12:29,500 So I think you have said most of the things. 183 00:12:29,500 --> 00:12:32,670 And the gravity is very different from the others. 184 00:12:32,670 --> 00:12:35,070 And in some sense, all the others are understood. 185 00:12:38,000 --> 00:12:44,450 So a to c are now understood to be 186 00:12:44,450 --> 00:12:57,150 described by gauge theories in the fixed spacetime. 187 00:12:57,150 --> 00:13:00,410 So the fixed spacetime. 188 00:13:00,410 --> 00:13:06,250 In fact, mostly we use Minkowski. 189 00:13:10,096 --> 00:13:10,595 Minkowski. 190 00:13:13,740 --> 00:13:22,120 And for example, electromagnetism, [INAUDIBLE] 191 00:13:22,120 --> 00:13:23,710 by QED. 192 00:13:23,710 --> 00:13:26,710 And the QED [INAUDIBLE] gauge field. 193 00:13:26,710 --> 00:13:29,810 You want gauge theory plus a Dirac fermion. 194 00:13:29,810 --> 00:13:30,590 Dirac theory. 195 00:13:33,770 --> 00:13:35,220 A Dirac theory. 196 00:13:35,220 --> 00:13:44,750 And similarly, if you include the weak interaction-- 197 00:13:44,750 --> 00:13:49,540 so there's something called [INAUDIBLE] weak-- 198 00:13:49,540 --> 00:13:55,460 it's described by su2 times u1 gauge theory. 199 00:13:55,460 --> 00:13:58,910 Described by su2 times u1 gauge theory. 200 00:13:58,910 --> 00:14:04,440 And also, the strong interaction you know now 201 00:14:04,440 --> 00:14:06,690 is described by su3 gauge theory. 202 00:14:10,390 --> 00:14:16,000 So for all those cases, the basic theoretical structure 203 00:14:16,000 --> 00:14:17,214 is understood. 204 00:14:17,214 --> 00:14:17,755 So the basic. 205 00:14:34,100 --> 00:14:34,640 OK. 206 00:14:34,640 --> 00:14:37,470 In principle, we can formulate those theories 207 00:14:37,470 --> 00:14:39,345 from first principle, just use path integral. 208 00:14:44,450 --> 00:14:47,890 Say, [INAUDIBLE] where [INAUDIBLE] 209 00:14:47,890 --> 00:14:53,770 group just based on principle, relation group. 210 00:14:53,770 --> 00:14:56,210 And then you can actually formulate all the theory 211 00:14:56,210 --> 00:14:59,220 in the [INAUDIBLE] way. 212 00:14:59,220 --> 00:15:09,750 So in the sense, then any calculation of theories 213 00:15:09,750 --> 00:15:19,140 can be reduced to certain algorithm. 214 00:15:27,020 --> 00:15:29,280 OK. 215 00:15:29,280 --> 00:15:32,670 Of course, this does not imply we actually 216 00:15:32,670 --> 00:15:35,720 know how to calculate many quantities here. 217 00:15:35,720 --> 00:15:38,990 It just say in principle, they are calculable. 218 00:15:38,990 --> 00:15:41,430 We often don't have the technical tool 219 00:15:41,430 --> 00:15:42,250 to calculate them. 220 00:15:42,250 --> 00:15:44,864 For example, even in QCD. 221 00:15:44,864 --> 00:15:47,280 [INAUDIBLE] joint action, we don't have the technical tool 222 00:15:47,280 --> 00:15:49,240 to calculate certain things. 223 00:15:49,240 --> 00:15:50,970 But they are, in principle, calculable. 224 00:15:50,970 --> 00:15:55,800 And we can write down the algorithm to calculate it. 225 00:15:55,800 --> 00:15:58,435 But for gravity, this is different. 226 00:16:01,340 --> 00:16:02,840 So for gravity, it's very different. 227 00:16:07,060 --> 00:16:07,755 So for gravity. 228 00:16:14,580 --> 00:16:23,802 So now we understand that classical gravity, 229 00:16:23,802 --> 00:16:25,010 it's just equal to spacetime. 230 00:16:27,820 --> 00:16:31,000 It's just a theory of spacetime. 231 00:16:31,000 --> 00:16:33,125 So this is the content over the general relativity. 232 00:16:35,252 --> 00:16:36,460 The general quantum activity. 233 00:16:41,260 --> 00:16:47,890 But at a quantum level, we really 234 00:16:47,890 --> 00:16:53,980 don't have a precise idea how to formulate the theory. 235 00:16:53,980 --> 00:16:57,740 So there are many questions still not understood. 236 00:16:57,740 --> 00:17:00,800 Many, say, conceptual questions still not understood. 237 00:17:00,800 --> 00:17:10,494 For example, a spacetime should fluctuate. 238 00:17:19,760 --> 00:17:22,359 OK. 239 00:17:22,359 --> 00:17:24,040 The spacetime should fluctuate because 240 00:17:24,040 --> 00:17:26,420 of the quantum fluctuation [INAUDIBLE] 241 00:17:26,420 --> 00:17:29,220 in quantum anything fluctuates. 242 00:17:29,220 --> 00:17:38,510 And then, the natural question is whether this is still-- 243 00:17:38,510 --> 00:17:41,315 so natural question is whether spacetime is still continuous. 244 00:17:47,560 --> 00:17:49,870 OK. 245 00:17:49,870 --> 00:17:52,280 So this is still [INAUDIBLE]. 246 00:17:52,280 --> 00:17:59,820 You need to replace it by something discrete. 247 00:17:59,820 --> 00:18:04,040 And it's not even clear right now 248 00:18:04,040 --> 00:18:09,917 whether spacetime will remain a fundamental notion when 249 00:18:09,917 --> 00:18:11,000 you go to quantum gravity. 250 00:18:13,650 --> 00:18:15,300 It could well be that spacetime would 251 00:18:15,300 --> 00:18:16,830 be replaced by something else. 252 00:18:24,640 --> 00:18:29,220 And also, there are many other questions. 253 00:18:29,220 --> 00:18:35,645 So for example, quantum nature of black holes. 254 00:18:45,760 --> 00:18:59,990 Or what should be the beginning of the universe, et cetera. 255 00:18:59,990 --> 00:19:03,060 So all these questions are not understood. 256 00:19:03,060 --> 00:19:11,630 Also, gravity is the weakest interaction. 257 00:19:17,390 --> 00:19:20,340 It is actually much, much weaker. 258 00:19:20,340 --> 00:19:23,320 We will see, it's much, much weaker 259 00:19:23,320 --> 00:19:26,060 than all the other interactions. 260 00:19:26,060 --> 00:19:30,040 And also, people have been speculate-- maybe actually 261 00:19:30,040 --> 00:19:34,570 this feature can also be actually a fundamental feature 262 00:19:34,570 --> 00:19:36,750 underlying in the special role of the gravity. 263 00:19:47,382 --> 00:19:48,840 For many years, it seemed to people 264 00:19:48,840 --> 00:19:53,490 that these questions and the gravity questions, 265 00:19:53,490 --> 00:19:55,882 they're completely unrelated. 266 00:19:55,882 --> 00:19:57,090 They're completely unrelated. 267 00:19:57,090 --> 00:19:59,612 They're a completely different subject. 268 00:19:59,612 --> 00:20:02,120 And those are well understood, but somehow those we 269 00:20:02,120 --> 00:20:05,390 need completely new ideas. 270 00:20:05,390 --> 00:20:06,800 So it came out a great surprise. 271 00:20:18,710 --> 00:20:22,020 So it came out that great surprise in 1997 272 00:20:22,020 --> 00:20:33,930 by [INAUDIBLE] that actually they actually are equivalent. 273 00:20:33,930 --> 00:20:38,190 So the proposal of the [INAUDIBLE] 274 00:20:38,190 --> 00:20:41,970 is that the-- chalk. 275 00:20:45,710 --> 00:20:55,270 Is that quantum gravity is actually 276 00:20:55,270 --> 00:21:07,720 equal to field theories on a fixed spacetime. 277 00:21:16,990 --> 00:21:18,750 So on a fixed spacetime just means 278 00:21:18,750 --> 00:21:23,510 that there's no gravity here on the right-hand side. 279 00:21:23,510 --> 00:21:26,310 Because if there's gravity, then you cannot have a fixed 280 00:21:26,310 --> 00:21:27,502 spacetime. 281 00:21:27,502 --> 00:21:29,065 And spacetime should be dynamical. 282 00:21:32,150 --> 00:21:35,180 So this is the equivalence between quantum gravity 283 00:21:35,180 --> 00:21:39,010 and just our ordinary quantum field theories. 284 00:21:39,010 --> 00:21:42,300 And this is ordinary quantum mechanics. 285 00:21:42,300 --> 00:21:44,490 And this is a very nontrivial quantum mechanics, 286 00:21:44,490 --> 00:21:47,350 which we don't yet understand. 287 00:21:47,350 --> 00:21:50,210 OK. 288 00:21:50,210 --> 00:21:55,590 This equality should be really understood 289 00:21:55,590 --> 00:21:58,810 as the different descriptions of the same quantum systems. 290 00:22:06,710 --> 00:22:25,460 So two sides are different descriptions 291 00:22:25,460 --> 00:22:27,459 of the same quantum system. 292 00:22:35,360 --> 00:22:38,400 So it just depends on how you want to see it. 293 00:22:38,400 --> 00:22:41,860 So one way you see it, you see a quantum gravity. 294 00:22:41,860 --> 00:22:43,360 Then from some other way you see it, 295 00:22:43,360 --> 00:22:44,568 you don't see gravity at all. 296 00:22:44,568 --> 00:22:46,710 You just see ordinary quantum mechanics. 297 00:22:46,710 --> 00:22:47,210 OK. 298 00:22:51,280 --> 00:22:57,280 Yeah, so this is what this equality is supposed to mean. 299 00:23:00,520 --> 00:23:04,710 So this equality essentially-- let me call this equation 1, 300 00:23:04,710 --> 00:23:08,870 which I will often use later. 301 00:23:08,870 --> 00:23:14,700 So this equation 1 is really a unification. 302 00:23:21,720 --> 00:23:24,540 Unification, which unifies quantum gravity 303 00:23:24,540 --> 00:23:28,320 with other field theory in a rather unconventional way. 304 00:23:28,320 --> 00:23:31,470 Not in a standard sense of unification. 305 00:23:31,470 --> 00:23:34,220 But that's bringing them together. 306 00:23:34,220 --> 00:23:34,720 OK. 307 00:23:37,230 --> 00:23:39,870 So now, let me make some philosophical remark. 308 00:23:42,600 --> 00:23:44,980 Now, let me make some philosophical remark. 309 00:23:44,980 --> 00:23:48,540 So last time, I have checked when 310 00:23:48,540 --> 00:23:51,160 preparing this class this [INAUDIBLE] original paper 311 00:23:51,160 --> 00:23:55,700 has been cited more than 10,000 times 312 00:23:55,700 --> 00:23:59,780 in the [INAUDIBLE] Database. 313 00:23:59,780 --> 00:24:02,180 And so there's a huge amount of work 314 00:24:02,180 --> 00:24:07,430 has been devoted to understanding this relation. 315 00:24:07,430 --> 00:24:12,040 But in my opinion, this subject's still in its infancy. 316 00:24:12,040 --> 00:24:13,800 And with many elementary [INAUDIBLE] 317 00:24:13,800 --> 00:24:16,550 we will see in the course as course goes on, 318 00:24:16,550 --> 00:24:19,150 there's still many elementary issues 319 00:24:19,150 --> 00:24:21,800 which are not understood. 320 00:24:21,800 --> 00:24:26,500 And to us, this relation is still like a magic. 321 00:24:26,500 --> 00:24:30,830 And we don't really have very good idea 322 00:24:30,830 --> 00:24:32,960 where this relation comes home. 323 00:24:32,960 --> 00:24:37,270 We have some rough idea, but not very precise idea. 324 00:24:37,270 --> 00:24:40,000 So the purpose of physics, of course, 325 00:24:40,000 --> 00:24:46,690 is to turn what looks like a magic or art into some rules 326 00:24:46,690 --> 00:24:51,240 so that it becomes something trivial. 327 00:24:51,240 --> 00:24:54,410 So at the beginning of course of the gravity, 328 00:24:54,410 --> 00:24:55,930 Newtonian gravity was magic. 329 00:24:55,930 --> 00:24:58,290 But in the end-- and what Newton did 330 00:24:58,290 --> 00:25:03,280 is to make the gravity to be not different from the force we 331 00:25:03,280 --> 00:25:05,730 see in ordinary life. 332 00:25:05,730 --> 00:25:11,550 And similarly, for other-- yeah, for other interactions 333 00:25:11,550 --> 00:25:14,880 we have seen before. 334 00:25:14,880 --> 00:25:16,930 Anyway. 335 00:25:16,930 --> 00:25:20,340 So personally, I believe when we really fully understand 336 00:25:20,340 --> 00:25:23,710 this relation, this will really be 337 00:25:23,710 --> 00:25:27,710 a huge landmark in the physics. 338 00:25:27,710 --> 00:25:32,585 And comparable, say, maybe to Newton's understanding 339 00:25:32,585 --> 00:25:36,480 of gravity or Maxwell-Boltzmann, et cetera. 340 00:25:40,530 --> 00:25:44,450 So the goal of this course is to help 341 00:25:44,450 --> 00:25:52,680 you understand how we-- our current understanding 342 00:25:52,680 --> 00:25:54,950 of this relation. 343 00:25:54,950 --> 00:25:59,670 And also, to help you to derive the related-- yeah, 344 00:25:59,670 --> 00:26:03,490 so these are two very different objects. 345 00:26:03,490 --> 00:26:05,170 And in order to set up an inequality, 346 00:26:05,170 --> 00:26:07,794 you also need to define a dictionary. 347 00:26:07,794 --> 00:26:09,210 For example, this is like Chinese. 348 00:26:09,210 --> 00:26:10,580 This is like English. 349 00:26:10,580 --> 00:26:13,940 So you need to define a dictionary between the two 350 00:26:13,940 --> 00:26:16,250 and what one said to the other. 351 00:26:16,250 --> 00:26:18,590 So we will also work out the dictionary. 352 00:26:18,590 --> 00:26:22,190 And also, we will develop with the tools 353 00:26:22,190 --> 00:26:24,180 and how to use this relation. 354 00:26:24,180 --> 00:26:25,920 And how to use the relation. 355 00:26:25,920 --> 00:26:29,940 And also, discuss many features of this relation 356 00:26:29,940 --> 00:26:32,630 and the implications. 357 00:26:32,630 --> 00:26:35,650 So the purpose is to help you to-- really, 358 00:26:35,650 --> 00:26:42,257 to bring to the forefront of this very exciting subject. 359 00:26:42,257 --> 00:26:42,965 So any questions? 360 00:26:53,360 --> 00:26:54,060 Good. 361 00:26:54,060 --> 00:26:55,980 No questions? 362 00:26:55,980 --> 00:27:05,240 OK, so now let's move to-- so after this introduction. 363 00:27:05,240 --> 00:27:07,310 So let's talk about this. 364 00:27:17,240 --> 00:27:18,352 OK. 365 00:27:18,352 --> 00:27:19,685 So if you look at this relation. 366 00:27:22,790 --> 00:27:26,040 So looking from the right-hand side. 367 00:27:26,040 --> 00:27:29,150 So you just have some kind of ordinary field theory-- 368 00:27:29,150 --> 00:27:31,850 ordinary quantum mechanical system, which 369 00:27:31,850 --> 00:27:36,070 we, in principle, know very well conceptually, 370 00:27:36,070 --> 00:27:39,140 without any gravity at all. 371 00:27:39,140 --> 00:27:46,090 Somehow, if you view it in some different way than the gravity 372 00:27:46,090 --> 00:27:48,460 and the spacetime, and the dynamical spacetime 373 00:27:48,460 --> 00:27:51,760 should emerge out of this. 374 00:27:51,760 --> 00:28:02,250 So in some sense, this 1 implies that the quantum gravity-- so 375 00:28:02,250 --> 00:28:06,140 let me just use the quantum gravity 376 00:28:06,140 --> 00:28:07,580 plus the dynamical spacetime. 377 00:28:15,160 --> 00:28:16,930 OK, dynamical spacetime. 378 00:28:16,930 --> 00:28:22,788 So it's important it's a dynamical spacetime-- 379 00:28:22,788 --> 00:28:33,330 can really merge from non-gravitational system. 380 00:28:45,420 --> 00:28:47,500 It can emerge from a non-gravitational system. 381 00:28:51,120 --> 00:28:56,980 So the idea of emergence of gravity from some other degrees 382 00:28:56,980 --> 00:29:01,820 freedom is not a really new idea. 383 00:29:01,820 --> 00:29:13,150 So [INAUDIBLE] in 1967, Sakharov, 384 00:29:13,150 --> 00:29:18,080 who was a Russian physicist. 385 00:29:18,080 --> 00:29:22,290 Sakharov is a very common name, a very common Russian name. 386 00:29:22,290 --> 00:29:24,080 But this is the most famous one. 387 00:29:26,930 --> 00:29:30,670 This is the most famous Sakharov who 388 00:29:30,670 --> 00:29:36,390 has invented the hydrogen bomb for the former Soviet Union. 389 00:29:36,390 --> 00:29:41,370 And he was also the guy who also later, I think, 390 00:29:41,370 --> 00:29:45,320 got Nobel Peace Prize for some human rights stuff. 391 00:29:49,230 --> 00:29:54,410 But he was also a very excellent physicist, including this idea 392 00:29:54,410 --> 00:29:57,040 of emergence of the gravity. 393 00:29:57,040 --> 00:30:00,340 So what he observed is that-- he found 394 00:30:00,340 --> 00:30:01,900 actually certain [INAUDIBLE]. 395 00:30:01,900 --> 00:30:05,530 He would study, say, some materials, et cetera. 396 00:30:05,530 --> 00:30:09,030 Then, he finds that certain [INAUDIBLE] systems 397 00:30:09,030 --> 00:30:13,630 can actually have mathematical description which 398 00:30:13,630 --> 00:30:16,920 looks like a magic connection, et cetera. 399 00:30:21,240 --> 00:30:23,150 Yeah, just mathematical, it looks 400 00:30:23,150 --> 00:30:25,150 like the equation for the magic and the equation 401 00:30:25,150 --> 00:30:27,290 for the connection, et cetera. 402 00:30:27,290 --> 00:30:30,450 And then he was speculating, maybe the general relativity 403 00:30:30,450 --> 00:30:34,560 can actually arise just from some kind of electron systems, 404 00:30:34,560 --> 00:30:37,170 just from the ordinary [INAUDIBLE] systems, 405 00:30:37,170 --> 00:30:40,055 and as effective description. 406 00:30:46,730 --> 00:30:51,140 And in fact, in the 1950s, even before the Sakharov 407 00:30:51,140 --> 00:30:56,330 have [INAUDIBLE] idea, people working in GR, 408 00:30:56,330 --> 00:30:59,170 in General Relativity, they already 409 00:30:59,170 --> 00:31:08,690 found many striking parallel between Einstein's equations 410 00:31:08,690 --> 00:31:14,600 and also associated many phenomena we hydrodynamics. 411 00:31:14,600 --> 00:31:17,860 And so we know that hydrodynamics is really 412 00:31:17,860 --> 00:31:20,200 just an effective description. 413 00:31:20,200 --> 00:31:23,895 And if you just look at the river-- so in fact, 414 00:31:23,895 --> 00:31:26,890 it's the discrete water molecules. 415 00:31:26,890 --> 00:31:30,210 But if you describe the motion of the water molecules 416 00:31:30,210 --> 00:31:33,340 at a microscopic level, then you find the hydrodynamics. 417 00:31:33,340 --> 00:31:35,256 But the hydrodynamics does not apply 418 00:31:35,256 --> 00:31:36,380 at more fundamental levels. 419 00:31:36,380 --> 00:31:40,610 So it's more like some kind of effective theory. 420 00:31:40,610 --> 00:31:43,750 So in the 1950s, people already found 421 00:31:43,750 --> 00:31:48,570 that there are many features of Einstein equation-- 422 00:31:48,570 --> 00:31:49,984 it's reminiscent. 423 00:31:49,984 --> 00:31:51,900 Actually, just reminiscent of certain features 424 00:31:51,900 --> 00:31:53,055 of hydrodynamics. 425 00:31:53,055 --> 00:31:54,800 So there's already this suspicion 426 00:31:54,800 --> 00:32:01,170 that maybe the gravity or spacetime actually can 427 00:32:01,170 --> 00:32:03,250 emerge out of something else. 428 00:32:03,250 --> 00:32:06,600 Just like the hydrodynamics is not a fundamental theory. 429 00:32:06,600 --> 00:32:12,130 It actually emerges from molecules, 430 00:32:12,130 --> 00:32:13,580 dynamics of the molecules. 431 00:32:16,225 --> 00:32:16,725 Anyway. 432 00:32:20,600 --> 00:32:27,650 So from the field theory perspective, 433 00:32:27,650 --> 00:33:12,770 it's also natural to ask whether massless spin-2 particles can 434 00:33:12,770 --> 00:33:29,535 arise as bound state, say, of lower spin particles. 435 00:33:38,880 --> 00:33:41,280 Say, like spin-1. 436 00:33:41,280 --> 00:33:42,620 Like photons, which are spin-1. 437 00:33:45,230 --> 00:33:46,916 So photons. 438 00:33:46,916 --> 00:33:48,430 Or gluons in QCD. 439 00:33:51,670 --> 00:33:55,600 Electrons, or quarks, et cetera. 440 00:33:55,600 --> 00:34:00,710 So these are the spin [INAUDIBLE] objects. 441 00:34:00,710 --> 00:34:12,170 And if yes-- so massless spin-2 particle in some sense-- 442 00:34:12,170 --> 00:34:16,520 so when you learn QR, you might have 443 00:34:16,520 --> 00:34:21,580 learned that the massless spin-2 particle may be considered 444 00:34:21,580 --> 00:34:25,537 as a hallmark form of gravity because of the-- so 445 00:34:25,537 --> 00:34:29,830 the basic propogation. 446 00:34:29,830 --> 00:34:32,230 The gravitational wave. 447 00:34:32,230 --> 00:34:35,179 So the basic propogation of the gravity 448 00:34:35,179 --> 00:34:36,800 is the gravitational wave. 449 00:34:36,800 --> 00:34:40,030 And that's propogated as spin-2 particles. 450 00:34:40,030 --> 00:34:46,159 And if you quantize the theory-- a propogation as spin-2 object. 451 00:34:46,159 --> 00:34:50,060 And when you quantize the gravitational waves, 452 00:34:50,060 --> 00:34:53,350 then they become massless spin-2 particles. 453 00:34:53,350 --> 00:34:55,659 They become spin-2 particles. 454 00:34:55,659 --> 00:34:58,810 So in some sense, if your theory can 455 00:34:58,810 --> 00:35:03,970 allow the massless spin-2 particles 456 00:35:03,970 --> 00:35:07,160 as-- so if your theory contains the massless spin-2 particles, 457 00:35:07,160 --> 00:35:08,841 then they must contain gravity. 458 00:35:08,841 --> 00:35:09,340 OK. 459 00:35:26,020 --> 00:35:46,490 So for example, even in strong interaction, which are, say, 460 00:35:46,490 --> 00:35:49,810 the theory describes gluons and the quarks. 461 00:35:49,810 --> 00:35:52,770 And the gluons and the quarks, they can indeed form. 462 00:35:52,770 --> 00:35:56,660 So these are the gluons and the quarks. 463 00:36:03,642 --> 00:36:12,435 In [INAUDIBLE], they're indeed spin-2 bound states. 464 00:36:16,360 --> 00:36:18,920 The gluons are spin-1, quarks are spin [INAUDIBLE]. 465 00:36:18,920 --> 00:36:20,345 But you combine them together, you 466 00:36:20,345 --> 00:36:22,870 can make [INAUDIBLE] the spin-2 bound state, even just 467 00:36:22,870 --> 00:36:23,828 in strong interactions. 468 00:36:26,930 --> 00:36:30,647 But of course in nature, the spin-2 bound state, 469 00:36:30,647 --> 00:36:31,480 they're all massive. 470 00:36:36,340 --> 00:36:38,660 They're all massive. 471 00:36:38,660 --> 00:36:42,750 And there are some very unstable, massive particles. 472 00:36:42,750 --> 00:36:45,510 When you create them, immediate decay. 473 00:36:45,510 --> 00:36:47,265 So they cannot be gravity. 474 00:36:47,265 --> 00:36:49,622 For gravity, you need massless spin-2 particles. 475 00:36:55,430 --> 00:36:57,890 But if you look at this fact, you cannot have the feeling. 476 00:36:57,890 --> 00:37:01,950 Say, maybe I take the QCD. 477 00:37:01,950 --> 00:37:04,240 Maybe I take a little bit. 478 00:37:04,240 --> 00:37:07,040 Maybe I can make that spin-2 massive particles 479 00:37:07,040 --> 00:37:10,680 into some massless spin-2 particles. 480 00:37:10,680 --> 00:37:15,180 Then I would have generated the gravity from a QCD like series. 481 00:37:15,180 --> 00:37:17,400 And then, that would be a revolution. 482 00:37:17,400 --> 00:37:20,270 And then you will be immortal. 483 00:37:23,740 --> 00:37:26,920 But unfortunately, even though this 484 00:37:26,920 --> 00:37:30,360 was a very promising idea for many years, 485 00:37:30,360 --> 00:37:37,376 and this hope was actually completely dashed 486 00:37:37,376 --> 00:37:43,070 by a powerful theorem of Weinberg and Witten. 487 00:37:43,070 --> 00:37:45,470 So there's a powerful theorem from Weinberg-Witten 488 00:37:45,470 --> 00:37:48,380 that say this can never happen. 489 00:37:48,380 --> 00:37:53,665 So this is the-- they're not possible. 490 00:38:10,550 --> 00:38:13,380 So this is 1980. 491 00:38:13,380 --> 00:38:15,864 So the paper. 492 00:38:15,864 --> 00:38:18,510 If you want, you can take a look at the paper. 493 00:38:26,780 --> 00:38:30,290 So this is volume 96 and the page 59. 494 00:38:30,290 --> 00:38:36,150 So they proved two theorems in that paper. 495 00:38:36,150 --> 00:38:38,180 So let me just copy the theorem down. 496 00:38:42,070 --> 00:38:44,780 Copy the theorem down. 497 00:38:44,780 --> 00:39:04,240 So they say a theory that allows the construction 498 00:39:04,240 --> 00:39:23,410 of a Lorentz-covariant and the conserved current 499 00:39:23,410 --> 00:39:26,100 for a vector-- current. 500 00:39:30,370 --> 00:39:56,470 Say, j mu, cannot contain massless particles. 501 00:40:02,610 --> 00:40:10,180 Cannot contain massless particles of spin greater than 502 00:40:10,180 --> 00:40:29,454 1/2 with non-vanishing value of-- OK, 503 00:40:29,454 --> 00:40:30,495 so this is first theorem. 504 00:40:34,110 --> 00:40:46,100 So the second theorem-- I think I should have enough space. 505 00:40:46,100 --> 00:40:47,120 So the second theorem. 506 00:40:54,490 --> 00:41:05,360 Second theorem says that a theory that 507 00:41:05,360 --> 00:41:19,980 allows a covariant, Lorentz-covariant, 508 00:41:19,980 --> 00:41:24,430 and the conserved stress tensor. 509 00:41:27,570 --> 00:41:29,080 So let me just call it T mu mu. 510 00:41:29,080 --> 00:41:53,460 A stress tensor cannot contain massless particles of spin j 511 00:41:53,460 --> 00:41:54,210 greater than 1. 512 00:42:05,930 --> 00:42:08,980 So the key-- so let me just repeat 513 00:42:08,980 --> 00:42:11,640 the theorem a little bit. 514 00:42:11,640 --> 00:42:16,140 So the first theorem said if you have a Lorentz-covariant 515 00:42:16,140 --> 00:42:21,510 and the conserved current, and in such a theory 516 00:42:21,510 --> 00:42:29,960 there's no charged particles can have spin more than 1/2. 517 00:42:29,960 --> 00:42:34,210 So there's no charged particle can have spin more than 1/2. 518 00:42:34,210 --> 00:42:37,990 Of course, this cannot contain graviton. 519 00:42:37,990 --> 00:42:42,590 And the second theorem says if you have a conserved stress 520 00:42:42,590 --> 00:42:47,550 tensor, Lorentz-covariant and a conserved stress tensor, 521 00:42:47,550 --> 00:42:50,210 then this theory cannot contain any particles with spin greater 522 00:42:50,210 --> 00:42:53,020 than 1. 523 00:42:53,020 --> 00:42:55,410 So of course, this cannot also contain a graviton. 524 00:42:55,410 --> 00:42:57,100 Graviton would be a spin-2. 525 00:43:00,820 --> 00:43:03,570 So this theorem-- yes? 526 00:43:03,570 --> 00:43:06,001 AUDIENCE: Is there anything like RG flow of mass? 527 00:43:06,001 --> 00:43:06,500 [INAUDIBLE] 528 00:43:09,170 --> 00:43:10,940 HONG LIU: This is final mass. 529 00:43:10,940 --> 00:43:12,250 Yeah. 530 00:43:12,250 --> 00:43:14,875 Just the [INAUDIBLE] mass. 531 00:43:14,875 --> 00:43:16,750 You don't talk about even normalization here. 532 00:43:22,086 --> 00:43:25,090 Yeah, this is just [INAUDIBLE] statement. 533 00:43:29,080 --> 00:43:33,240 So this theorem turned out to be actually very easy to prove. 534 00:43:33,240 --> 00:43:36,080 And in some sense, it's rather instructive. 535 00:43:36,080 --> 00:43:41,250 And just to give you the sense of the power of this theorem, 536 00:43:41,250 --> 00:43:43,340 we will prove it in a little bit. 537 00:43:46,750 --> 00:43:48,850 So before proving it, I will first 538 00:43:48,850 --> 00:43:52,970 make some remarks to make you appreciate 539 00:43:52,970 --> 00:43:54,840 what these two theorems means. 540 00:43:54,840 --> 00:43:58,222 Do you have any questions before I do that? 541 00:44:07,570 --> 00:44:08,700 OK, good. 542 00:44:08,700 --> 00:44:11,160 So let me just make some remarks on this theorem. 543 00:44:41,780 --> 00:44:51,810 So the first remark is that the theorems really apply 544 00:44:51,810 --> 00:44:57,860 to any particles, to both. 545 00:44:57,860 --> 00:44:59,700 So normally we say the fundamental particle 546 00:44:59,700 --> 00:45:04,640 is the particle which you put in your theory is 547 00:45:04,640 --> 00:45:09,040 the particle which say-- yeah, let me say [INAUDIBLE]. 548 00:45:09,040 --> 00:45:11,300 Rather fundamental, let me call it elementary. 549 00:45:11,300 --> 00:45:13,780 So the particle which already appear in your Lagrangian. 550 00:45:18,100 --> 00:45:21,520 And composite particles. 551 00:45:21,520 --> 00:45:24,955 Composite particles, say, would be some kind of bound state 552 00:45:24,955 --> 00:45:27,020 or some particles which don't appear 553 00:45:27,020 --> 00:45:30,870 in your original Lagrangian. 554 00:45:30,870 --> 00:45:35,050 So this theorem does not distinguish them. 555 00:45:35,050 --> 00:45:39,690 As far as you have some particles, [INAUDIBLE] region. 556 00:45:45,260 --> 00:45:50,670 So now, let's try to see whether this theorem is compatible 557 00:45:50,670 --> 00:45:55,280 to [INAUDIBLE] these things we already know. 558 00:45:55,280 --> 00:45:59,045 So let's first try to apply to, say, QED. 559 00:46:01,970 --> 00:46:15,400 So the theorem is compatible with QED. 560 00:46:23,140 --> 00:46:37,860 So we know that QED is a Maxwell theory plus Dirac's theory. 561 00:46:42,070 --> 00:46:47,150 And then the Dirac-- and the fermions in the Dirac theory 562 00:46:47,150 --> 00:46:50,760 interact with the photons in the Maxwell theory. 563 00:46:50,760 --> 00:46:55,540 So this theory have a massless photon of spin-1. 564 00:46:55,540 --> 00:47:00,300 And this will have a conserved charge because of, say, 565 00:47:00,300 --> 00:47:02,656 electrons that do have a charge. 566 00:47:02,656 --> 00:47:04,155 This theory has a conserved current. 567 00:47:10,770 --> 00:47:12,490 Has a conserved current. 568 00:47:12,490 --> 00:47:16,770 But this theory does allow a spin-1 particle. 569 00:47:16,770 --> 00:47:19,340 It does allow the spin-1 particle. 570 00:47:19,340 --> 00:47:24,580 And this is not contradictory with the theorem 1. 571 00:47:24,580 --> 00:47:27,170 So theorem 1 said if you have a Lorentz [INAUDIBLE] 572 00:47:27,170 --> 00:47:30,890 and the conserved current, then you cannot have massless 573 00:47:30,890 --> 00:47:33,505 particle of spin 1/2 with non-vanishing charge. 574 00:47:39,270 --> 00:47:41,036 In this case, photon is massless. 575 00:47:41,036 --> 00:47:42,410 But photon does not carry charge. 576 00:47:51,720 --> 00:47:56,910 Photon is neutral under the electric charge. 577 00:48:00,330 --> 00:48:02,182 So the existence of the photon is 578 00:48:02,182 --> 00:48:03,390 compatible with that theorem. 579 00:48:09,840 --> 00:48:12,870 And now, let's look at the [INAUDIBLE] theory 580 00:48:12,870 --> 00:48:16,591 which also contain massless spin-1 particle. 581 00:48:16,591 --> 00:48:19,090 And the [INAUDIBLE] theory which contain the spin-1 particle 582 00:48:19,090 --> 00:48:20,285 is the Yang-Mill theory. 583 00:48:27,930 --> 00:48:29,430 So Yang-Mill theory. 584 00:48:29,430 --> 00:48:31,783 For example, let's consider su2 Yang-Mill theory. 585 00:48:34,860 --> 00:48:40,840 So the Yang-Mill theory have three gauge field, A mu a. 586 00:48:40,840 --> 00:48:43,792 And a is equal to 1, 2, 3. 587 00:48:43,792 --> 00:48:49,175 Because 1 and 2 [INAUDIBLE] of su2 gauge symmetry. 588 00:49:00,981 --> 00:49:02,480 And from this three gauge field, you 589 00:49:02,480 --> 00:49:07,010 can combine them into the following form. 590 00:49:07,010 --> 00:49:09,670 I have A mu 3. 591 00:49:09,670 --> 00:49:12,130 Then, I have A mu plus minus. 592 00:49:12,130 --> 00:49:20,360 You could do 1 over square root 2 a mu 1 plus minus A mu 2. 593 00:49:20,360 --> 00:49:20,860 OK. 594 00:49:25,850 --> 00:49:28,510 So [INAUDIBLE]. 595 00:49:28,510 --> 00:49:34,670 So you can easily check yourself that this a mu plus minus 596 00:49:34,670 --> 00:49:39,625 is massless spin-1 particle. 597 00:49:45,785 --> 00:49:46,285 Particles. 598 00:49:49,280 --> 00:49:51,430 Can create massless spin-1 particles 599 00:49:51,430 --> 00:50:13,290 charged under the u1 subgroup generated by sigma 3. 600 00:50:16,430 --> 00:50:20,770 So sigma 3 is the generator associated with A mu 3. 601 00:50:20,770 --> 00:50:23,780 And so those two are charged under this generator. 602 00:50:26,330 --> 00:50:30,430 So now, [INAUDIBLE] we have a contradiction 603 00:50:30,430 --> 00:50:33,570 because those are massless spin-1 particles charged 604 00:50:33,570 --> 00:50:34,100 under this. 605 00:50:38,440 --> 00:50:43,230 And the theorem 1 said you have a conserved current, 606 00:50:43,230 --> 00:50:47,200 then you cannot have any particle recharged-- 607 00:50:47,200 --> 00:50:50,542 any charged particle with spin greater than 1/2. 608 00:50:50,542 --> 00:50:52,000 But those are the spin-1 particles. 609 00:50:55,050 --> 00:50:58,150 But actually, we're consistent. 610 00:51:00,750 --> 00:51:07,030 Because as you may remember, in Yang-Mills theory 611 00:51:07,030 --> 00:51:13,440 there does not exist-- that actually does not exist. 612 00:51:17,288 --> 00:51:19,140 The way out [INAUDIBLE] there does not 613 00:51:19,140 --> 00:51:43,350 exist a conserved Lorentz-covariant current 614 00:51:43,350 --> 00:51:47,504 for actually-- for this u1. 615 00:51:47,504 --> 00:51:53,390 For the u1 generated by this sigma 3 divided by 2. 616 00:51:53,390 --> 00:51:56,692 So this is actually compatible. 617 00:51:56,692 --> 00:51:59,150 This is actually compatible because such a current does not 618 00:51:59,150 --> 00:52:00,351 exist. 619 00:52:00,351 --> 00:52:00,850 OK. 620 00:52:05,010 --> 00:52:09,210 So you will show this fact yourself in your PSET. 621 00:52:09,210 --> 00:52:10,750 But you may remember, if you have 622 00:52:10,750 --> 00:52:14,500 studied quantum field theory 2, then 623 00:52:14,500 --> 00:52:18,240 you may remember this fact from the discussion 624 00:52:18,240 --> 00:52:20,750 of [INAUDIBLE] gauge theories. 625 00:52:20,750 --> 00:52:26,570 So in this course, we will from time to time 626 00:52:26,570 --> 00:52:30,300 use various facts of, say, [INAUDIBLE] theories. 627 00:52:33,420 --> 00:52:36,310 But they're not essential. 628 00:52:36,310 --> 00:52:39,570 If you don't allow [INAUDIBLE] theories, 629 00:52:39,570 --> 00:52:42,090 you can still understand this duality. 630 00:52:42,090 --> 00:52:44,600 You can still understand the relation one. 631 00:52:44,600 --> 00:52:47,910 But of course, if you know gauge field, it will help you a lot. 632 00:52:53,790 --> 00:52:55,030 Any questions about this? 633 00:53:01,630 --> 00:53:03,760 Good. 634 00:53:03,760 --> 00:53:04,490 Let's continue. 635 00:53:15,770 --> 00:53:38,950 So this theorem does not forbid graviton from GR. 636 00:53:43,900 --> 00:53:48,780 So if you try to quantize Einstein's general relativity 637 00:53:48,780 --> 00:53:51,540 around the flash space for example, 638 00:53:51,540 --> 00:53:54,776 then you will find the massless spin-2 particle. 639 00:53:54,776 --> 00:53:56,150 So this is what we call graviton. 640 00:54:01,700 --> 00:54:08,260 And this is not contradictory with that theorem 2. 641 00:54:08,260 --> 00:54:17,270 It's because you may still remember 642 00:54:17,270 --> 00:54:21,080 when you learn GR is that in GR, General Relativity, actually 643 00:54:21,080 --> 00:54:22,995 there is not. 644 00:54:22,995 --> 00:54:23,870 There's no conserved. 645 00:54:27,080 --> 00:54:30,050 So here, it's conserved. 646 00:54:30,050 --> 00:54:33,260 Not covariantly converged, just conserved. 647 00:54:33,260 --> 00:54:38,860 Lorentz-covariant stress tensor. 648 00:54:41,805 --> 00:54:42,305 OK. 649 00:54:45,670 --> 00:54:48,554 So in GR, actually there's no conserved 650 00:54:48,554 --> 00:54:52,000 in the Lorentz-covariant stress tensor. 651 00:54:52,000 --> 00:54:55,810 So actually, the GR [INAUDIBLE] graviton. 652 00:55:01,230 --> 00:55:07,517 But this theorem 2, it's nevertheless, very powerful. 653 00:55:07,517 --> 00:55:08,850 It's nevertheless very powerful. 654 00:55:13,470 --> 00:55:22,700 It said very powerful. 655 00:55:22,700 --> 00:55:26,496 It said all [INAUDIBLE]. 656 00:55:36,938 --> 00:55:50,696 None of [INAUDIBLE] QFTs in Minkowski spacetime, 657 00:55:50,696 --> 00:55:59,660 which is the one we normally work with, 658 00:55:59,660 --> 00:56:03,585 can have emergent gravity. 659 00:56:21,850 --> 00:56:24,210 OK. 660 00:56:24,210 --> 00:56:28,100 So QCD like theories, no matter how 661 00:56:28,100 --> 00:56:35,100 you try to trick the theory, the spin-2 massive bound state 662 00:56:35,100 --> 00:56:37,330 can never become massless. 663 00:56:37,330 --> 00:56:40,630 They can never become massless. 664 00:56:40,630 --> 00:56:44,275 Because those series, they do-- for example, 665 00:56:44,275 --> 00:56:46,160 our QCD, our strong interaction, they 666 00:56:46,160 --> 00:56:51,070 do have a covariantly conserved stress tensor. 667 00:56:51,070 --> 00:56:53,670 A Lorentz-covariant conserved stress tensor. 668 00:56:58,690 --> 00:57:06,380 But this theorem does not validate 669 00:57:06,380 --> 00:57:10,130 this equation 1 I just erased. 670 00:57:10,130 --> 00:57:14,740 Because this theorem, even though it's very powerful, 671 00:57:14,740 --> 00:57:17,330 it does have a hidden assumption. 672 00:57:20,056 --> 00:57:21,430 It does have a hidden assumption. 673 00:57:25,940 --> 00:57:35,772 This assumption is so obvious and so self-evident, 674 00:57:35,772 --> 00:57:39,620 that even though Weinberg and Witten, 675 00:57:39,620 --> 00:57:41,820 they're extremely careful people, 676 00:57:41,820 --> 00:57:43,870 they did not bother to mention it. 677 00:57:48,810 --> 00:57:52,130 In those times, nobody with their right mind 678 00:57:52,130 --> 00:57:55,050 would have mentioned such a thing. 679 00:57:55,050 --> 00:58:02,190 So the hidden assumption is that the kind of particles 680 00:58:02,190 --> 00:58:08,400 they are talking about-- is that the particles are they talking 681 00:58:08,400 --> 00:58:18,170 about, the particles, which they tried to rule out, 682 00:58:18,170 --> 00:58:29,261 live in the same spacetime as original theory. 683 00:58:36,411 --> 00:58:36,910 OK. 684 00:58:39,510 --> 00:58:44,900 So if you write down, say, QED in Minkowski 685 00:58:44,900 --> 00:58:47,640 spacetime, and then you ask about 686 00:58:47,640 --> 00:58:53,586 whether you can have some spin-2 particles coming out from QED. 687 00:58:53,586 --> 00:58:54,960 And then of course, you ask about 688 00:58:54,960 --> 00:58:56,890 whether you have particles come out 689 00:58:56,890 --> 00:58:59,470 QED living in the same Minkowski spacetime. 690 00:59:04,271 --> 00:59:04,770 Yeah. 691 00:59:04,770 --> 00:59:09,070 So in some sense, this assumption is self-evident. 692 00:59:09,070 --> 00:59:14,710 But precisely, this is assumption, 693 00:59:14,710 --> 00:59:21,120 which taking advantage by this relation, which first 694 00:59:21,120 --> 00:59:25,000 envisioned by [INAUDIBLE], is that precisely 695 00:59:25,000 --> 00:59:30,840 in this relation, in this equation 1 which I raised, 696 00:59:30,840 --> 00:59:33,330 that the gravity actually does not 697 00:59:33,330 --> 00:59:37,710 live in the same spacetime as the original theory. 698 00:59:37,710 --> 00:59:40,410 And does not live in the same spacetime as original theory. 699 00:59:44,780 --> 00:59:50,810 So even in the early days, when people 700 00:59:50,810 --> 00:59:56,850 were trying to dream having such kind of spin-2 particles 701 00:59:56,850 --> 00:59:59,800 that come from QCD like theories, 702 00:59:59,800 --> 01:00:07,040 there's already an immediate puzzle. 703 01:00:13,250 --> 01:00:18,250 Suppose the QCD like theories can have emergent graviton 704 01:00:18,250 --> 01:00:21,280 in the same spacetime. 705 01:00:21,280 --> 01:00:25,500 Then firstly, your QCD theory is defined in a fix spacetime. 706 01:00:25,500 --> 01:00:28,240 But now if it contains spin-2 particle in the spacetime 707 01:00:28,240 --> 01:00:31,500 itself, it will be dynamical. 708 01:00:31,500 --> 01:00:34,570 Then, where does the original QCD theory [INAUDIBLE]? 709 01:00:34,570 --> 01:00:35,660 OK. 710 01:00:35,660 --> 01:00:38,810 And then you will just similarly go 711 01:00:38,810 --> 01:00:40,890 into this kind of self-contradiction, 712 01:00:40,890 --> 01:00:44,410 like somehow-- how can that-- yeah. 713 01:00:44,410 --> 01:00:45,270 Anyway. 714 01:00:45,270 --> 01:00:59,370 So in this relation, in the holographic duality, 715 01:00:59,370 --> 01:01:12,550 gravity does not live in the same spacetime. 716 01:01:17,960 --> 01:01:19,200 OK. 717 01:01:19,200 --> 01:01:25,370 So this, in fact, live in one dimension or higher, 718 01:01:25,370 --> 01:01:27,490 or they can be several dimension higher depending 719 01:01:27,490 --> 01:01:29,560 on the examples. 720 01:01:29,560 --> 01:01:32,130 So this void this theorem in obvious way. 721 01:01:35,020 --> 01:01:39,670 And also, void this conundrum I said earlier. 722 01:01:39,670 --> 01:01:41,590 Because in a different spacetime. 723 01:01:41,590 --> 01:01:44,090 So the field theory is still in the fixed spacetime, 724 01:01:44,090 --> 01:01:48,166 but then the gravity then can be dynamical. 725 01:01:48,166 --> 01:01:50,540 Then, the spacetime which gravity lives can be dynamical. 726 01:01:54,610 --> 01:01:56,610 So any questions regarding this so far? 727 01:01:56,610 --> 01:01:57,850 Yes. 728 01:01:57,850 --> 01:01:59,530 AUDIENCE: [INAUDIBLE] in GR, there 729 01:01:59,530 --> 01:02:01,670 was no conserved Lorentz-covariant stress 730 01:02:01,670 --> 01:02:02,170 tensor. 731 01:02:02,170 --> 01:02:04,850 Because the covariant derivative of the stress 732 01:02:04,850 --> 01:02:07,160 tensor [INAUDIBLE]. 733 01:02:07,160 --> 01:02:10,190 HONG LIU: No covariant derivative is not conserved. 734 01:02:10,190 --> 01:02:13,700 This covariant is not conserved. 735 01:02:13,700 --> 01:02:15,740 Conserved means the ordinary derivatives, not 736 01:02:15,740 --> 01:02:19,550 the covariant derivatives. 737 01:02:19,550 --> 01:02:22,816 A covariant conserved stress tensor is not conserved. 738 01:02:26,270 --> 01:02:30,360 If you write down a conserved stress tensor, 739 01:02:30,360 --> 01:02:34,520 then it's not covariant. 740 01:02:34,520 --> 01:02:36,610 Then it's not covariant. 741 01:02:36,610 --> 01:02:38,450 Yeah. 742 01:02:38,450 --> 01:02:42,580 Similar thing with this case for the gauge fields. 743 01:02:42,580 --> 01:02:51,690 And in this case, gauge invariant currents-- 744 01:02:51,690 --> 01:02:54,735 the gauge invariant and the conserved currents 745 01:02:54,735 --> 01:02:57,000 is not Lorentz-covariants. 746 01:02:57,000 --> 01:03:01,480 And the Lorentz-covariants, gauge invariant current 747 01:03:01,480 --> 01:03:02,738 then is not conserved. 748 01:03:10,073 --> 01:03:10,874 Any other question? 749 01:03:10,874 --> 01:03:12,040 AUDIENCE: I have a question. 750 01:03:12,040 --> 01:03:13,505 HONG LIU: Yes. 751 01:03:13,505 --> 01:03:14,380 AUDIENCE: [INAUDIBLE] 752 01:03:24,884 --> 01:03:25,550 HONG LIU: Sorry. 753 01:03:25,550 --> 01:03:26,856 Say it again? 754 01:03:26,856 --> 01:03:29,230 AUDIENCE: [INAUDIBLE] theory, every particle is massless. 755 01:03:29,230 --> 01:03:29,813 HONG LIU: Yes. 756 01:03:29,813 --> 01:03:33,900 AUDIENCE: And the [INAUDIBLE] spin-2 particle, which carries 757 01:03:33,900 --> 01:03:34,400 [INAUDIBLE] 758 01:03:39,350 --> 01:03:40,730 HONG LIU: Yeah. 759 01:03:40,730 --> 01:03:44,240 For [INAUDIBLE] field theory, indeed everything's massless. 760 01:03:44,240 --> 01:03:47,200 You can construct Lorentz-covariants 761 01:03:47,200 --> 01:03:48,980 conserved stress tensor. 762 01:03:48,980 --> 01:03:56,590 So that theory [INAUDIBLE] does not allow spin-2 excitations 763 01:03:56,590 --> 01:03:57,955 in the same spacetime. 764 01:03:57,955 --> 01:03:58,830 AUDIENCE: [INAUDIBLE] 765 01:04:03,284 --> 01:04:05,200 HONG LIU: There's no massless spin-2 particle. 766 01:04:08,345 --> 01:04:09,220 AUDIENCE: [INAUDIBLE] 767 01:04:16,865 --> 01:04:17,490 HONG LIU: Yeah. 768 01:04:17,490 --> 01:04:20,840 Yeah, but you don't have massless spin-2. 769 01:04:20,840 --> 01:04:25,040 You don't have massless spin-2 particles. 770 01:04:25,040 --> 01:04:27,000 AUDIENCE: [INAUDIBLE] 771 01:04:27,000 --> 01:04:28,350 HONG LIU: No. 772 01:04:28,350 --> 01:04:31,250 It depends. 773 01:04:31,250 --> 01:04:34,550 Actually, talking about the massless in the [INAUDIBLE] 774 01:04:34,550 --> 01:04:36,500 field theory is a little bit tricky. 775 01:04:36,500 --> 01:04:39,150 It's a little bit tricky. 776 01:04:39,150 --> 01:04:41,700 They come in all kind of spectrum. 777 01:04:41,700 --> 01:04:44,120 They come in the continuous spectrum. 778 01:04:44,120 --> 01:04:45,160 Yeah. 779 01:04:45,160 --> 01:04:50,990 Yeah, but this theorem does apply to ordinary [INAUDIBLE] 780 01:04:50,990 --> 01:04:52,820 field theories. 781 01:04:52,820 --> 01:04:57,862 So in those theories, you won't have massless. 782 01:04:57,862 --> 01:04:59,570 Yeah, when we prove the theorem, then you 783 01:04:59,570 --> 01:05:02,270 can see precisely what do we mean 784 01:05:02,270 --> 01:05:04,190 by massless spin-2 particles. 785 01:05:13,170 --> 01:05:16,922 If you stay in the course, then you will know. 786 01:05:16,922 --> 01:05:18,860 Yeah. 787 01:05:18,860 --> 01:05:22,270 Yeah, we will see it later on. 788 01:05:22,270 --> 01:05:25,040 But not for a while. 789 01:05:25,040 --> 01:05:27,050 Not for a couple of weeks. 790 01:05:27,050 --> 01:05:29,360 And maybe in a couple of weeks, we will see it. 791 01:05:32,930 --> 01:05:33,685 In two weeks. 792 01:05:33,685 --> 01:05:35,217 One or two weeks. 793 01:05:35,217 --> 01:05:36,050 Any other questions? 794 01:05:39,960 --> 01:05:40,700 OK. 795 01:05:40,700 --> 01:05:45,970 So now, let me give you the proof, which is pretty simple. 796 01:05:45,970 --> 01:05:51,610 And use some elementary facts, very elementary facts. 797 01:05:51,610 --> 01:05:56,820 Because you see, this theorem contains very little input. 798 01:05:56,820 --> 01:06:00,316 That theorem really contained very, very little input. 799 01:06:00,316 --> 01:06:01,690 So if you can prove such a thing, 800 01:06:01,690 --> 01:06:03,470 it cannot be complicated. 801 01:06:03,470 --> 01:06:06,480 If you can prove it, it cannot be complicated. 802 01:06:06,480 --> 01:06:07,920 Because the input is so little. 803 01:06:07,920 --> 01:06:10,520 It does not require any details. 804 01:06:10,520 --> 01:06:13,540 It does not require any details of a theory. 805 01:06:13,540 --> 01:06:15,250 So if you can prove such a theorem, 806 01:06:15,250 --> 01:06:18,480 it must be just from kinematics, not come from any dynamics. 807 01:06:21,590 --> 01:06:25,720 So indeed, that's how the theorem goes. 808 01:06:25,720 --> 01:06:26,805 So the proof. 809 01:06:32,500 --> 01:06:34,480 So we will do proof by contradiction 810 01:06:34,480 --> 01:06:38,140 as you would naturally expect. 811 01:06:38,140 --> 01:06:53,420 So let us suppose there exists a massless spin-j particle. 812 01:06:53,420 --> 01:06:55,590 Massless particles. 813 01:06:55,590 --> 01:06:57,660 So let's assume this theory contains 814 01:06:57,660 --> 01:07:02,700 some theories [INAUDIBLE] exist massless spin-2 particles. 815 01:07:02,700 --> 01:07:04,340 Massless particles of spin-j. 816 01:07:11,510 --> 01:07:13,110 And just from the Lorentz symmetry, 817 01:07:13,110 --> 01:07:15,080 you can immediately write down-- so 818 01:07:15,080 --> 01:07:20,630 from what we learned in QFT 1 and 2. 819 01:07:20,630 --> 01:07:23,600 So just by Lorentz symmetry-- it doesn't matter what 820 01:07:23,600 --> 01:07:24,850 is the nature of those states. 821 01:07:28,220 --> 01:07:30,240 Because the mass formal representations 822 01:07:30,240 --> 01:07:34,760 over the, say, Lorentz group and [INAUDIBLE] group. 823 01:07:34,760 --> 01:07:39,560 So [INAUDIBLE] immediately write down the one particle state 824 01:07:39,560 --> 01:07:41,330 of such particles. 825 01:07:41,330 --> 01:07:45,220 And you normally can write them as k and sigma. 826 01:07:45,220 --> 01:07:47,685 So k just is the ordinary spacetime momentum. 827 01:07:55,650 --> 01:07:57,760 So for massless particle, of course 828 01:07:57,760 --> 01:08:00,640 the k0 squared should be equal to k squared. 829 01:08:03,210 --> 01:08:06,960 And the sigma is the helicity. 830 01:08:06,960 --> 01:08:09,210 Helicity is a projection of the angular momentum 831 01:08:09,210 --> 01:08:12,010 in the direction of your momentum. 832 01:08:12,010 --> 01:08:16,689 And for a massless particle of spin-j, 833 01:08:16,689 --> 01:08:19,149 the sigma can only be plus minus j. 834 01:08:19,149 --> 01:08:20,315 So this is helicity. 835 01:08:24,680 --> 01:08:28,300 So if you don't remember this, you 836 01:08:28,300 --> 01:08:34,399 can go back to review Wemberg's book. 837 01:08:34,399 --> 01:08:39,420 Wemberg's volume 1 QFT. 838 01:08:39,420 --> 01:08:46,580 Volume 1, Section 2.5 discussed how you write down 839 01:08:46,580 --> 01:08:52,870 the general one particle states for, say, 840 01:08:52,870 --> 01:08:56,060 any quantum-- in any quantum theory 841 01:08:56,060 --> 01:08:57,609 as far as you have Lorentz symmetry. 842 01:08:57,609 --> 01:08:58,109 OK. 843 01:09:03,550 --> 01:09:06,990 And the one properties of this sigma 844 01:09:06,990 --> 01:09:08,494 is that let's now consider rotation. 845 01:09:11,850 --> 01:09:14,490 Now, let's consider rotation. 846 01:09:14,490 --> 01:09:15,990 So let's consider rotation operator. 847 01:09:25,060 --> 01:09:35,370 This [INAUDIBLE] by angle theta around the direction of the k. 848 01:09:38,700 --> 01:09:41,359 Around the direction of the k, which is the direction of the k 849 01:09:41,359 --> 01:09:41,859 here. 850 01:09:49,140 --> 01:09:52,410 So let's consider the action of these 851 01:09:52,410 --> 01:09:59,870 under state, this one particle-- this massless particle state. 852 01:09:59,870 --> 01:10:07,930 Then by definition, this just give you 853 01:10:07,930 --> 01:10:13,230 i sigma theta k, sigma. 854 01:10:13,230 --> 01:10:17,050 So this is essentially come from the definition of the helicity. 855 01:10:17,050 --> 01:10:22,050 And the k does not change because this is the rotation 856 01:10:22,050 --> 01:10:25,430 along the direction of the k. 857 01:10:25,430 --> 01:10:26,960 The k itself does not change. 858 01:10:30,400 --> 01:10:35,890 But this eigenstate of this rotation [INAUDIBLE]. 859 01:10:35,890 --> 01:10:36,470 OK. 860 01:10:36,470 --> 01:10:42,910 Again, if you don't remember this, go back to Wemberg's book 861 01:10:42,910 --> 01:10:44,760 and you can easily find it. 862 01:10:47,940 --> 01:11:08,990 So now if you have a conserved current, 863 01:11:08,990 --> 01:11:16,800 if you have a covariant conserved current, 864 01:11:16,800 --> 01:11:19,460 J mu, then you can always construct 865 01:11:19,460 --> 01:11:23,280 the charge, the conserved charge, 866 01:11:23,280 --> 01:11:26,640 which is the integral over J0. 867 01:11:26,640 --> 01:11:31,250 And similarly, if you have T mu mu, then from T mu mu, 868 01:11:31,250 --> 01:11:44,610 you can construct a momentum operator, which is the T mu. 869 01:11:44,610 --> 01:11:47,225 So this is just the standard momentum operator 870 01:11:47,225 --> 01:11:50,527 in your field theory. 871 01:11:50,527 --> 01:11:51,860 So this is just still the setup. 872 01:11:57,262 --> 01:12:07,530 Of course just by definition, the p mu acting on k sigma, 873 01:12:07,530 --> 01:12:11,007 you just get back k mu. 874 01:12:11,007 --> 01:12:12,840 OK, because this is the momentum eigenstate. 875 01:12:18,180 --> 01:12:22,420 And if this particle is also charged 876 01:12:22,420 --> 01:12:40,230 under this conserved charge, then-- so if charged under k, 877 01:12:40,230 --> 01:12:47,150 under this conserved charge, then Q act on this state 878 01:12:47,150 --> 01:12:51,420 will also give you the charge q. 879 01:12:51,420 --> 01:12:53,262 So this is operator. 880 01:12:53,262 --> 01:12:53,970 This is operator. 881 01:12:56,762 --> 01:12:57,720 These are all operator. 882 01:13:04,860 --> 01:13:05,470 Operator. 883 01:13:05,470 --> 01:13:09,400 And then, this should give you charge q. 884 01:13:09,400 --> 01:13:14,460 So now we want to show this 2 theorem implies-- 885 01:13:14,460 --> 01:13:21,500 so with this set up, this theorem can-- 886 01:13:21,500 --> 01:13:25,080 so we want to show two things. 887 01:13:25,080 --> 01:13:34,070 1, if q is nonzero, that means if the particle is charged 888 01:13:34,070 --> 01:13:41,945 under a conserved charge, then j must be smaller than 1/2. 889 01:13:41,945 --> 01:13:43,730 It cannot be greater than 1/2. 890 01:13:48,200 --> 01:13:50,870 So we assume it's a conserved stress tensor. 891 01:13:50,870 --> 01:13:54,970 Otherwise, j should be always smaller or equal to 1 at most. 892 01:13:54,970 --> 01:13:55,470 OK. 893 01:13:58,040 --> 01:14:00,930 So that's what that theorem means 894 01:14:00,930 --> 01:14:03,760 translated in this language. 895 01:14:03,760 --> 01:14:05,430 OK. 896 01:14:05,430 --> 01:14:07,520 Any questions? 897 01:14:07,520 --> 01:14:10,920 So now, let's prove it. 898 01:14:10,920 --> 01:14:13,210 Let me just do it here. 899 01:14:13,210 --> 01:14:15,852 So now, let's prove it. 900 01:14:15,852 --> 01:14:18,470 I think we just have enough time, maybe to prove it. 901 01:14:25,730 --> 01:14:29,554 So to prove it, let's just first list some elementary facts. 902 01:14:34,880 --> 01:14:37,000 First, just do some elementary fact. 903 01:14:41,950 --> 01:14:51,690 First, from Lorentz symmetry-- we 904 01:14:51,690 --> 01:14:53,330 do have to assume Lorentz symmetry. 905 01:14:53,330 --> 01:14:54,980 OK. 906 01:14:54,980 --> 01:14:59,650 You can show-- so this I will just write down the answer 907 01:14:59,650 --> 01:15:01,160 and leave yourself to show it. 908 01:15:01,160 --> 01:15:02,106 OK? 909 01:15:02,106 --> 01:15:03,515 In your PSET. 910 01:15:06,430 --> 01:15:09,790 So if you look at the matrix [INAUDIBLE] 911 01:15:09,790 --> 01:15:12,640 of this conserved current between two such states 912 01:15:12,640 --> 01:15:19,530 of different k, in the limit which k goes to k prime, 913 01:15:19,530 --> 01:15:23,480 you have q k mu k0. 914 01:15:28,960 --> 01:15:50,516 And similarly, in this limit-- so this 915 01:15:50,516 --> 01:15:51,640 we will do it in your PSET. 916 01:15:55,480 --> 01:15:59,140 So let me call this equation 2, equation 3. 917 01:15:59,140 --> 01:16:03,010 So this is based on the following normalization. 918 01:16:05,650 --> 01:16:27,750 Based on the following normalization between the two 919 01:16:27,750 --> 01:16:28,250 states. 920 01:16:28,250 --> 01:16:28,750 OK. 921 01:16:35,420 --> 01:16:39,370 So this relation is actually very easy to see intuitively. 922 01:16:44,090 --> 01:16:46,750 Yeah, you can prove it by-- yeah, 923 01:16:46,750 --> 01:16:50,200 let me give me just a quick hint for [INAUDIBLE] regarding 924 01:16:50,200 --> 01:16:57,150 2, is that you can first show that k sigma J0 925 01:16:57,150 --> 01:17:03,290 sandwiched between the two, in the limit when 926 01:17:03,290 --> 01:17:07,130 this k become the same is just given 927 01:17:07,130 --> 01:17:08,820 by q divided by 2 pi cubed. 928 01:17:13,240 --> 01:17:16,665 So this is, again, somewhat intuitive 929 01:17:16,665 --> 01:17:21,860 because k is eigenstate of Q. Because the k is 930 01:17:21,860 --> 01:17:24,940 an eigenstate of this Q hat. 931 01:17:24,940 --> 01:17:27,990 And Q hat is just the integral of J0. 932 01:17:27,990 --> 01:17:30,154 It is integral of J0. 933 01:17:30,154 --> 01:17:32,070 And so you can imagine on the right-hand side, 934 01:17:32,070 --> 01:17:34,653 there must be a Q because these are essentially the eigenstate 935 01:17:34,653 --> 01:17:36,796 of the integration with J0. 936 01:17:36,796 --> 01:17:39,170 And then when you take into account all these [INAUDIBLE] 937 01:17:39,170 --> 01:17:42,910 functions here, and the integral-- and the relation 938 01:17:42,910 --> 01:17:45,470 between the integral, then you will naturally 939 01:17:45,470 --> 01:17:49,940 find the right-hand side is just q divided by this. 940 01:17:49,940 --> 01:17:52,250 So once you know this, then you can immediately 941 01:17:52,250 --> 01:17:55,550 see the first [INAUDIBLE] equation 2 there. 942 01:17:55,550 --> 01:17:57,520 Because at first, there's a j mu there. 943 01:17:57,520 --> 01:18:00,330 Then the right-hand side must have index mu. 944 01:18:00,330 --> 01:18:02,450 So the only thing can carry index mu is the k mu. 945 01:18:02,450 --> 01:18:04,350 There's nothing else. 946 01:18:04,350 --> 01:18:08,270 And also, you know when the mu equal to 0, 947 01:18:08,270 --> 01:18:09,800 you should just get the q. 948 01:18:09,800 --> 01:18:13,840 Then, downstairs we might have a k0 to cancel them. 949 01:18:13,840 --> 01:18:16,400 So this relation, in principle you 950 01:18:16,400 --> 01:18:19,640 can just-- just by intuition, you 951 01:18:19,640 --> 01:18:21,950 can just-- by dimensional analysis, [INAUDIBLE] just 952 01:18:21,950 --> 01:18:24,080 directly write it down. 953 01:18:24,080 --> 01:18:25,760 And similar in the second relation-- 954 01:18:25,760 --> 01:18:30,180 in the second relation, the analog of q. 955 01:18:30,180 --> 01:18:33,140 So q is the charge associated with Q hat. 956 01:18:33,140 --> 01:18:39,040 The analog of the q for the stress tensor is just the k mu. 957 01:18:39,040 --> 01:18:45,010 So you just replace this by k mu. 958 01:18:45,010 --> 01:18:47,485 It just relates to the q by k mu. 959 01:18:47,485 --> 01:18:50,810 Then you get the second line. 960 01:18:50,810 --> 01:18:53,740 Because k by k mu [INAUDIBLE]. 961 01:18:53,740 --> 01:18:54,470 OK. 962 01:18:54,470 --> 01:18:59,440 So I will leave yourself to prove this rigorously. 963 01:18:59,440 --> 01:19:02,670 So this is the second observation. 964 01:19:02,670 --> 01:19:11,230 It said for massless particles, which 965 01:19:11,230 --> 01:19:15,460 have k squared equal to 0, and k prime squared equal to 0, 966 01:19:15,460 --> 01:19:17,170 because [INAUDIBLE] particles. 967 01:19:17,170 --> 01:19:21,450 So k squared and k prime squared should be equal to 0. 968 01:19:21,450 --> 01:19:28,120 And that means for any two particles, a massless momentum 969 01:19:28,120 --> 01:19:34,230 like this, you see immediately just by writing this down 970 01:19:34,230 --> 01:19:37,280 this must be smaller than 0. 971 01:19:37,280 --> 01:19:42,180 So this should come from your high school physics. 972 01:19:42,180 --> 01:19:45,160 The k dot k prime should be smaller than 0. 973 01:19:45,160 --> 01:19:49,560 And that means that k plus k prime must be time-like. 974 01:19:54,490 --> 01:19:57,110 OK, must be time-like. 975 01:19:57,110 --> 01:20:04,280 Because this square is essentially given by this guy 976 01:20:04,280 --> 01:20:06,500 because k square and k prime equal to 0. 977 01:20:06,500 --> 01:20:08,610 So this guy must be time-like. 978 01:20:08,610 --> 01:20:10,660 OK. 979 01:20:10,660 --> 01:20:12,570 And so that means you can always choose 980 01:20:12,570 --> 01:20:17,440 a frame that the special part of this is 0. 981 01:20:17,440 --> 01:20:27,950 We can choose a frame k plus k prime equal to 0. 982 01:20:27,950 --> 01:20:33,870 So such a frame is just k mu, say E, 0. 983 01:20:33,870 --> 01:20:38,330 Say we put all of these in the z-direction, 984 01:20:38,330 --> 01:20:41,470 in the 3-direction. 985 01:20:41,470 --> 01:20:49,700 Then you can choose k prime k mu equal to-- OK? 986 01:20:59,530 --> 01:21:00,940 And this just follows. 987 01:21:00,940 --> 01:21:05,650 These two are two [INAUDIBLE] particles. 988 01:21:05,650 --> 01:21:06,640 OK, good. 989 01:21:06,640 --> 01:21:10,610 So now, we can almost envision this. 990 01:21:10,610 --> 01:21:13,240 Then, there's still a third elementary observation 991 01:21:13,240 --> 01:21:16,210 we need to put here. 992 01:21:16,210 --> 01:21:23,880 So the third observation which uses this formula. 993 01:21:27,350 --> 01:21:29,430 So let's now take sigma just to be j. 994 01:21:37,230 --> 01:21:55,972 So now, under a rotation of theta around 3-direction-- 995 01:21:55,972 --> 01:21:58,555 around this direction, which we have the [INAUDIBLE] momentum. 996 01:21:58,555 --> 01:22:03,530 So we choose a momentum to be in this direction. 997 01:22:03,530 --> 01:22:08,140 Then you should have-- so let me call 998 01:22:08,140 --> 01:22:10,820 that operator to be R theta is the rotation 999 01:22:10,820 --> 01:22:13,270 in the z-direction, in the 3-direction. 1000 01:22:13,270 --> 01:22:16,260 So this operate on k, j. 1001 01:22:16,260 --> 01:22:20,016 So let's take sigma equal to j. 1002 01:22:20,016 --> 01:22:23,040 Take sigma equal to j. 1003 01:22:23,040 --> 01:22:32,550 So this should be just equal to exponential i, j theta k, j. 1004 01:22:32,550 --> 01:22:39,270 And if this acting on k prime j. 1005 01:22:42,170 --> 01:22:47,520 So k prime have opposite orientation as k 1006 01:22:47,520 --> 01:22:48,635 because this is minus e. 1007 01:22:51,821 --> 01:22:54,320 And the opposite orientation means your helicity should also 1008 01:22:54,320 --> 01:22:56,290 change sign. 1009 01:22:56,290 --> 01:23:02,860 So that means here should be minus i j theta k prime, j. 1010 01:23:02,860 --> 01:23:05,380 OK. 1011 01:23:05,380 --> 01:23:07,720 Yeah, because this is the rotation 1012 01:23:07,720 --> 01:23:10,200 along the positive 3-direction. 1013 01:23:12,981 --> 01:23:13,480 OK. 1014 01:23:13,480 --> 01:23:16,250 So now we are in business. 1015 01:23:16,250 --> 01:23:20,820 So now finally, we will try to-- from those facts together, 1016 01:23:20,820 --> 01:23:24,335 we will now see a contradiction. 1017 01:23:24,335 --> 01:23:27,566 Now see a contradiction. 1018 01:23:27,566 --> 01:23:28,941 Now you can see that this object. 1019 01:23:34,840 --> 01:23:40,410 Consider k, j-- k prime, j. 1020 01:23:40,410 --> 01:23:52,107 Say this R minus 1 theta acting on J mu R theta k, j. 1021 01:23:52,107 --> 01:23:53,565 So now, let's consider this object. 1022 01:23:56,410 --> 01:24:00,090 So this object, I can evaluate it by two ways. 1023 01:24:00,090 --> 01:24:04,160 First, I can just act R theta on both states. 1024 01:24:04,160 --> 01:24:05,700 OK. 1025 01:24:05,700 --> 01:24:10,530 So R theta on this one just gives me this. 1026 01:24:10,530 --> 01:24:15,720 R minus 1 acting on that just give me opposite of this. 1027 01:24:15,720 --> 01:24:19,670 So all together, if I do that way, 1028 01:24:19,670 --> 01:24:29,914 then I get exponential 2 i j theta k prime, j J mu k, j. 1029 01:24:33,680 --> 01:24:39,060 But I can also act this operator on this current. 1030 01:24:39,060 --> 01:24:42,950 So this current should transform as a Lorentz vector. 1031 01:24:42,950 --> 01:24:52,050 So if I do this way, then I should get lambda mu new k 1032 01:24:52,050 --> 01:24:59,880 prime, j J new k, j. 1033 01:24:59,880 --> 01:25:04,220 So these acting on that should just effect a Lorentzian 1034 01:25:04,220 --> 01:25:05,232 summation on j. 1035 01:25:05,232 --> 01:25:07,315 And so this is in the Lorentzian summation matrix. 1036 01:25:10,230 --> 01:25:13,976 So lambda mu mu just is the standard rotation matrix 1037 01:25:13,976 --> 01:25:14,850 acting on the vector. 1038 01:25:18,147 --> 01:25:23,340 So lambda mu mu just the standard rotation matrix. 1039 01:25:23,340 --> 01:25:28,170 For example, just given by 1, 0, 0, 0. 1040 01:25:28,170 --> 01:25:32,150 Say, cosine theta minus sine theta. 1041 01:25:32,150 --> 01:25:37,910 So just a rotation along the 3-direction. 1042 01:25:43,460 --> 01:25:47,640 So this just the rotation acting on the j. 1043 01:25:52,150 --> 01:26:02,320 So now similarly, if you examine this quantity. 1044 01:26:10,150 --> 01:26:17,340 Yeah, similarly if I just replace here by T mu mu, OK? 1045 01:26:17,340 --> 01:26:22,776 by T mu mu-- we are out of time, so let me 1046 01:26:22,776 --> 01:26:25,490 be a little bit faster now. 1047 01:26:25,490 --> 01:26:34,600 Similarly, if we replace that by T mu mu, 1048 01:26:34,600 --> 01:26:42,230 then we will get the same thing 2 i j theta k prime, 1049 01:26:42,230 --> 01:26:53,250 j T mu mu k, j, which obtained by acting directly on k, j. 1050 01:26:53,250 --> 01:26:57,330 By the way, that R acting on T mu mu, then 1051 01:26:57,330 --> 01:27:00,690 T mu mu transform as a 2 tensor. 1052 01:27:00,690 --> 01:27:04,960 Then we should get lambda mu rho mu lambda. 1053 01:27:04,960 --> 01:27:12,903 Lambda mu rho k prime, j T lambda rho k, j. 1054 01:27:23,670 --> 01:27:24,710 So now, here's the key. 1055 01:27:27,640 --> 01:27:29,270 Now, here's the key. 1056 01:27:29,270 --> 01:27:31,708 So do you already see the contradiction? 1057 01:27:37,440 --> 01:27:42,110 So let's look at this equation. 1058 01:27:42,110 --> 01:27:43,880 This is just like a vector. 1059 01:27:43,880 --> 01:27:47,460 The whole thing is like a vector. 1060 01:27:47,460 --> 01:27:48,880 The same vector. 1061 01:27:48,880 --> 01:27:52,320 So this is like an eigenvalue equation. 1062 01:27:52,320 --> 01:27:55,830 So this lambda mu mu acting on this vector 1063 01:27:55,830 --> 01:27:57,650 get eigenvalue 2 i j theta. 1064 01:27:57,650 --> 01:28:01,360 So this is like an eigenvalue equation. 1065 01:28:01,360 --> 01:28:10,040 But if you look at this matrix, this 1066 01:28:10,040 --> 01:28:21,130 can only have eigenvalue exponential 1067 01:28:21,130 --> 01:28:25,010 plus minus i theta and the 1. 1068 01:28:25,010 --> 01:28:28,060 So this is thing we are familiar with. 1069 01:28:28,060 --> 01:28:32,280 So [INAUDIBLE] minus i theta, 1. 1070 01:28:32,280 --> 01:28:36,210 So similarly with this equation, just you have 2 lambda, OK? 1071 01:28:38,760 --> 01:28:42,510 So now, that means if this quantity 1072 01:28:42,510 --> 01:28:51,360 is nonzero, which we know it is nonzero from 2 and 3. 1073 01:28:51,360 --> 01:28:53,440 It is nonzero. 1074 01:28:53,440 --> 01:28:57,610 That these 2 i j theta can only take a value 1075 01:28:57,610 --> 01:29:00,510 plus minus i theta and the 1. 1076 01:29:00,510 --> 01:29:05,060 So that means that j-- from here, 1077 01:29:05,060 --> 01:29:09,730 let's say equation 4 and 5. 1078 01:29:09,730 --> 01:29:17,130 From equation 4, that j must be smaller than 1/2. 1079 01:29:17,130 --> 01:29:21,160 Because there's a 2j there. 1080 01:29:21,160 --> 01:29:27,080 Similarly, from equation 5, means j must be smaller than 1. 1081 01:29:30,640 --> 01:29:34,050 So this is what we want to prove. 1082 01:29:34,050 --> 01:29:35,540 So this is what we want to prove. 1083 01:29:35,540 --> 01:29:36,040 Yes? 1084 01:29:36,040 --> 01:29:37,831 AUDIENCE: Aren't k and k prime [INAUDIBLE]? 1085 01:29:41,220 --> 01:29:46,484 I mean, 2 and 3, the limit k goes to k prime and k plus k 1086 01:29:46,484 --> 01:29:47,360 prime is 0. 1087 01:29:47,360 --> 01:29:48,040 HONG LIU: Yeah. 1088 01:29:48,040 --> 01:29:52,930 So I can take them-- if j is greater than 1/2. 1089 01:29:52,930 --> 01:29:56,025 For example in this relation, if j is greater than 1/2, 1090 01:29:56,025 --> 01:29:59,380 then the only possibility is for this to be 0. 1091 01:29:59,380 --> 01:30:01,750 But we know when you take k equal to k prime, at least 1092 01:30:01,750 --> 01:30:03,820 in that limit this is nonzero. 1093 01:30:03,820 --> 01:30:05,400 And so this cannot be 0. 1094 01:30:05,400 --> 01:30:08,934 So this quantity cannot be identical to 0. 1095 01:30:12,251 --> 01:30:12,750 OK. 1096 01:30:15,864 --> 01:30:17,030 So this gives you the proof. 1097 01:30:20,470 --> 01:30:23,100 So if you find the loophole in this proof, 1098 01:30:23,100 --> 01:30:29,470 other than they have live in the same dimension-- yeah, 1099 01:30:29,470 --> 01:30:32,840 that would be great. 1100 01:30:32,840 --> 01:30:35,707 AUDIENCE: What is the whole matrix element is 0? 1101 01:30:35,707 --> 01:30:37,290 What if the whole matrix element is 0? 1102 01:30:37,290 --> 01:30:39,470 HONG LIU: No, the point is that the whole matrix element cannot 1103 01:30:39,470 --> 01:30:39,990 be 0. 1104 01:30:39,990 --> 01:30:40,573 AUDIENCE: Yes. 1105 01:30:40,573 --> 01:30:41,820 HONG LIU: Because of this. 1106 01:30:41,820 --> 01:30:43,682 AUDIENCE: Isn't the limit k goes to k prime? 1107 01:30:43,682 --> 01:30:45,640 And there, you assume that k plus k prime is 0. 1108 01:30:45,640 --> 01:30:48,447 So it means both k and k prime should be 0, no? 1109 01:30:48,447 --> 01:30:49,280 HONG LIU: I'm sorry? 1110 01:30:49,280 --> 01:30:49,779 No. 1111 01:30:49,779 --> 01:30:50,800 k plus k prime is not-- 1112 01:30:50,800 --> 01:30:51,674 AUDIENCE: [INAUDIBLE] 1113 01:30:51,674 --> 01:30:53,510 HONG LIU: No, this is a spatial vector. 1114 01:30:53,510 --> 01:30:54,385 AUDIENCE: [INAUDIBLE] 1115 01:30:54,385 --> 01:30:55,970 HONG LIU: No, this is spatial vector. 1116 01:30:55,970 --> 01:30:58,290 No, this is spatial vector. 1117 01:30:58,290 --> 01:30:59,630 Yeah. 1118 01:30:59,630 --> 01:31:00,400 Yeah. 1119 01:31:00,400 --> 01:31:01,358 This is spatial vector. 1120 01:31:03,581 --> 01:31:04,080 OK. 1121 01:31:04,080 --> 01:31:07,400 Sorry, I'm a little bit late. 1122 01:31:07,400 --> 01:31:08,000 Yeah. 1123 01:31:08,000 --> 01:31:10,150 That's all for today.