WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.810 Commons license. 00:00:03.810 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.150 continue to offer high-quality educational resources for free. 00:00:10.150 --> 00:00:12.700 To make a donation, or to view additional materials 00:00:12.700 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.260 at ocw.mit.edu. 00:00:21.260 --> 00:00:22.430 HONG LIU: OK, let us start. 00:00:22.430 --> 00:00:24.590 Welcome, everybody. 00:00:24.590 --> 00:00:31.680 So first, let me just say a few general things regarding 00:00:31.680 --> 00:00:35.590 the logistics of this class. 00:00:35.590 --> 00:00:39.180 So you should have these three handouts. 00:00:39.180 --> 00:00:40.580 One, organization. 00:00:40.580 --> 00:00:42.070 One, outline. 00:00:42.070 --> 00:00:44.920 And then, one is for PSET 1. 00:00:47.960 --> 00:00:52.870 So let me just go over briefly this organization 00:00:52.870 --> 00:00:53.580 and the outline. 00:00:58.650 --> 00:01:03.245 So firstly in this class, there's no textbooks. 00:01:06.640 --> 00:01:13.180 However, there are many reviews available. 00:01:13.180 --> 00:01:16.240 But unfortunately, a lot of those reviews, 00:01:16.240 --> 00:01:19.145 none of the single review is suitable for the whole course. 00:01:23.430 --> 00:01:23.930 Yeah. 00:01:23.930 --> 00:01:26.180 So I have listed some of the reviews 00:01:26.180 --> 00:01:28.565 at the end of this organization. 00:01:31.381 --> 00:01:31.880 Yeah. 00:01:31.880 --> 00:01:35.380 So along the way, I will point it out, 00:01:35.380 --> 00:01:38.720 some specific references or some specific parts 00:01:38.720 --> 00:01:45.390 of certain reviews if they are directly relevant. 00:01:45.390 --> 00:01:49.160 And also for this class, there's no recitation. 00:01:49.160 --> 00:01:53.580 Yeah, for advanced courses like this, 00:01:53.580 --> 00:01:56.720 department does not assign TA. 00:01:56.720 --> 00:01:58.350 So we don't have recitation. 00:01:58.350 --> 00:02:06.220 So I have my own office hour, which is Monday 3:30 to 4:30. 00:02:06.220 --> 00:02:10.449 Just one hour before the lecture on Monday. 00:02:10.449 --> 00:02:10.949 Yeah. 00:02:10.949 --> 00:02:15.710 So if you have questions or if you have-- yeah, 00:02:15.710 --> 00:02:20.220 just feel free to come. 00:02:20.220 --> 00:02:23.625 So if you have some questions other time, 00:02:23.625 --> 00:02:28.070 say, if this office hour is not good for you, 00:02:28.070 --> 00:02:31.430 I'm also happy to set up some other time to meet you. 00:02:31.430 --> 00:02:37.420 Or, you can stay after the class to ask questions, et cetera. 00:02:37.420 --> 00:02:37.970 Yeah. 00:02:37.970 --> 00:02:41.280 So whichever way is more convenient to you. 00:02:46.250 --> 00:02:48.940 So today, we have this-- yeah, so form now on, 00:02:48.940 --> 00:02:51.870 everything will be just on the web. 00:02:51.870 --> 00:02:54.800 You should download the future PSETs. 00:02:54.800 --> 00:02:58.600 Or if I put any lecture notes or materials, 00:02:58.600 --> 00:03:00.940 you should find them on the web. 00:03:00.940 --> 00:03:08.700 And I assume you can all find out the website of the course. 00:03:08.700 --> 00:03:14.680 And it's also listed here in the organization page. 00:03:14.680 --> 00:03:18.060 In the organization page. 00:03:18.060 --> 00:03:19.530 So any questions so far? 00:03:24.790 --> 00:03:27.260 Good. 00:03:27.260 --> 00:03:31.550 So this class there's no exam. 00:03:31.550 --> 00:03:36.660 And the grades are solely based on PSETs and the final project. 00:03:36.660 --> 00:03:40.530 And the PSETs will be due every two weeks. 00:03:40.530 --> 00:03:45.940 So if you look at the calendar, there are about like 5 PSETs. 00:03:45.940 --> 00:03:49.090 Then, for the last three weeks, instead 00:03:49.090 --> 00:03:53.880 of, say, handing a PSET, you're handing in the small paper. 00:03:53.880 --> 00:03:58.810 Just whatever topics you would like to choose 00:03:58.810 --> 00:04:01.170 about the holographic duality. 00:04:01.170 --> 00:04:02.350 Yeah, just choose one topic. 00:04:02.350 --> 00:04:05.990 Write, say, 8 to 10-page paper. 00:04:05.990 --> 00:04:08.260 Either review or some calculation 00:04:08.260 --> 00:04:11.680 you would like to do, et cetera. 00:04:11.680 --> 00:04:12.820 Yeah. 00:04:12.820 --> 00:04:16.360 So 5 PSET is 75%. 00:04:16.360 --> 00:04:19.160 And this final project would be 25%. 00:04:23.460 --> 00:04:32.842 That's pretty much the logistics regarding this class. 00:04:32.842 --> 00:04:34.300 Do you have any questions regarding 00:04:34.300 --> 00:04:39.650 the grading, regarding the project or PSET, et cetera? 00:04:45.180 --> 00:04:47.750 Good. 00:04:47.750 --> 00:04:51.910 So let me say a few words about this outline. 00:04:51.910 --> 00:04:57.290 So this outline is way too ambitious. 00:04:57.290 --> 00:05:00.850 It gives you-- yeah, it just serves 00:05:00.850 --> 00:05:04.380 to give you a rough contour what we will cover. 00:05:04.380 --> 00:05:10.140 But really, don't take it too literally because-- first, we 00:05:10.140 --> 00:05:13.990 are not going to cover all seven chapters. 00:05:13.990 --> 00:05:15.320 That would be too much. 00:05:15.320 --> 00:05:18.325 So last time when I taught it, I covered only three chapters. 00:05:21.310 --> 00:05:25.610 And so hopefully, this time I will do four. 00:05:25.610 --> 00:05:28.130 Hopefully, time I will do four. 00:05:28.130 --> 00:05:31.370 And I may also just deviate from some of the things 00:05:31.370 --> 00:05:35.630 here, depending on the pace or depend on people's interests, 00:05:35.630 --> 00:05:37.410 et cetera. 00:05:37.410 --> 00:05:39.615 So this is going to be flexible. 00:05:39.615 --> 00:05:44.160 If there are certain things you would really very much like 00:05:44.160 --> 00:05:46.130 to hear, you can also let me know. 00:05:46.130 --> 00:05:47.760 I can think about it. 00:05:47.760 --> 00:05:50.250 In particular, among the five, six, seven. 00:05:50.250 --> 00:05:54.880 And I can think about it, maybe I'll discuss some of them. 00:05:54.880 --> 00:05:58.122 Discuss some of them. 00:05:58.122 --> 00:05:59.270 Yeah. 00:05:59.270 --> 00:06:05.880 Also, another thing if you keep in mind is that in this class, 00:06:05.880 --> 00:06:13.850 this is a class which will touch on many different subjects, 00:06:13.850 --> 00:06:16.380 say gravity, a little bit of string theory. 00:06:16.380 --> 00:06:20.600 And also, quantum field theories, et cetera. 00:06:20.600 --> 00:06:24.800 So in such a class, it's almost impossible-- I 00:06:24.800 --> 00:06:30.120 think it's just impossible to really derive everything 00:06:30.120 --> 00:06:32.000 in the self-contained way. 00:06:32.000 --> 00:06:37.710 And also, some of-- yeah, I think some of your background 00:06:37.710 --> 00:06:39.450 are also very different. 00:06:39.450 --> 00:06:44.380 So the certain part may not appear 00:06:44.380 --> 00:06:47.500 self-contained to some of you. 00:06:47.500 --> 00:06:51.670 So if there is anything which is not familiar to you, 00:06:51.670 --> 00:06:53.650 if I just mention it, if I just code it, 00:06:53.650 --> 00:06:56.380 if it's not familiar with you, then 00:06:56.380 --> 00:07:00.490 you make sure-- don't hesitate to ask me, 00:07:00.490 --> 00:07:03.010 either during the class or after the class, 00:07:03.010 --> 00:07:06.960 so that I can help you to make up on that. 00:07:06.960 --> 00:07:08.880 To make up on that. 00:07:08.880 --> 00:07:13.460 And also, to keep in mind, like other advanced graduate 00:07:13.460 --> 00:07:20.200 classes, you actually mostly learn outside the class. 00:07:20.200 --> 00:07:23.560 Inside the class, it's mostly try 00:07:23.560 --> 00:07:27.430 to provide you some motivations, guidance, et cetera, 00:07:27.430 --> 00:07:29.950 and give you a rough contour. 00:07:29.950 --> 00:07:33.340 And your own reading outside the class 00:07:33.340 --> 00:07:40.772 actually should be the main routes you learn things. 00:07:40.772 --> 00:07:42.105 The main route you learn things. 00:07:47.950 --> 00:07:48.715 So any questions? 00:07:52.380 --> 00:07:57.970 So during the class, please feel free to ask questions. 00:07:57.970 --> 00:07:58.470 Yeah. 00:08:01.154 --> 00:08:02.570 Asking questions are a great thing 00:08:02.570 --> 00:08:04.470 because if something you don't understand, 00:08:04.470 --> 00:08:09.460 there's a very good chance quite a few of your fellow students 00:08:09.460 --> 00:08:10.930 also don't understand. 00:08:10.930 --> 00:08:12.466 So if you ask the question, you're 00:08:12.466 --> 00:08:13.840 not only helping yourself, you're 00:08:13.840 --> 00:08:15.680 also helping other people. 00:08:15.680 --> 00:08:17.210 And you also help me because then I 00:08:17.210 --> 00:08:20.090 will know certain aspect is not clear to you. 00:08:20.090 --> 00:08:24.360 Then, I will try to emphasize it, or repeat it, et cetera. 00:08:24.360 --> 00:08:27.730 So please, do ask questions. 00:08:27.730 --> 00:08:31.320 And also, if you have any feedback on the class, 00:08:31.320 --> 00:08:36.090 whether it's too fast, too slow, or problems are too hard or too 00:08:36.090 --> 00:08:40.270 easy, the PSET is too short for you, 00:08:40.270 --> 00:08:44.360 just not challenging enough, just let me know. 00:08:44.360 --> 00:08:47.540 Just let me know. 00:08:47.540 --> 00:08:48.040 Yeah. 00:08:48.040 --> 00:08:50.420 Then, I will try to adjust. 00:08:50.420 --> 00:08:52.670 And again, this is a very important. 00:08:52.670 --> 00:08:57.270 Again, this is very important because the-- yeah. 00:08:57.270 --> 00:09:00.580 This will help us to improve the class. 00:09:00.580 --> 00:09:02.670 To improve the class. 00:09:02.670 --> 00:09:04.447 Any other questions? 00:09:04.447 --> 00:09:05.155 Or any questions? 00:09:11.030 --> 00:09:12.590 OK. 00:09:12.590 --> 00:09:17.790 Then, let us start our lecture. 00:09:23.020 --> 00:09:35.175 So first, we look at the hints for the holographic duality. 00:10:07.380 --> 00:10:07.880 OK. 00:10:07.880 --> 00:10:10.750 So we start by doing a multiple choice problem. 00:11:10.300 --> 00:11:10.800 OK. 00:11:10.800 --> 00:11:11.880 So what is your answer? 00:11:15.235 --> 00:11:16.110 AUDIENCE: [INAUDIBLE] 00:11:18.745 --> 00:11:19.370 HONG LIU: Good. 00:11:22.370 --> 00:11:23.270 So why is gravity? 00:11:28.080 --> 00:11:30.995 AUDIENCE: We don't have a good quantum theory of gravity. 00:11:30.995 --> 00:11:32.050 HONG LIU: Good. 00:11:32.050 --> 00:11:34.360 So we don't have a good quantum theory of gravity. 00:11:34.360 --> 00:11:35.068 Any other reason? 00:11:35.068 --> 00:11:37.200 AUDIENCE: Except string theory. 00:11:37.200 --> 00:11:38.890 HONG LIU: Yeah, string theory is still 00:11:38.890 --> 00:11:43.930 not the-- I would say not yet at the [INAUDIBLE] quantum 00:11:43.930 --> 00:11:44.680 theory of gravity. 00:11:47.065 --> 00:11:47.940 AUDIENCE: [INAUDIBLE] 00:11:50.880 --> 00:11:51.690 HONG LIU: Yeah. 00:11:51.690 --> 00:11:53.590 Yeah, this is the similar to that. 00:11:53.590 --> 00:11:58.476 We don't have a full theory of gravity nonrealizable. 00:11:58.476 --> 00:12:00.690 AUDIENCE: [INAUDIBLE] 00:12:00.690 --> 00:12:02.870 HONG LIU: Good. 00:12:02.870 --> 00:12:04.950 Anything else? 00:12:04.950 --> 00:12:06.080 Or any other opinions? 00:12:06.080 --> 00:12:11.998 Say you should choose any other interaction other than gravity? 00:12:11.998 --> 00:12:13.741 AUDIENCE: Strong is the strongest one. 00:12:13.741 --> 00:12:16.772 Strong is the strongest one. 00:12:16.772 --> 00:12:17.730 HONG LIU: That's right. 00:12:17.730 --> 00:12:20.188 You can also choose the gravity because this is the weakest 00:12:20.188 --> 00:12:22.450 one. 00:12:22.450 --> 00:12:24.170 Yeah, so the gravity is the weakest one. 00:12:24.170 --> 00:12:25.040 OK. 00:12:25.040 --> 00:12:26.360 So yeah. 00:12:26.360 --> 00:12:29.500 So I think you have said most of the things. 00:12:29.500 --> 00:12:32.670 And the gravity is very different from the others. 00:12:32.670 --> 00:12:35.070 And in some sense, all the others are understood. 00:12:38.000 --> 00:12:44.450 So a to c are now understood to be 00:12:44.450 --> 00:12:57.150 described by gauge theories in the fixed spacetime. 00:12:57.150 --> 00:13:00.410 So the fixed spacetime. 00:13:00.410 --> 00:13:06.250 In fact, mostly we use Minkowski. 00:13:10.096 --> 00:13:10.595 Minkowski. 00:13:13.740 --> 00:13:22.120 And for example, electromagnetism, [INAUDIBLE] 00:13:22.120 --> 00:13:23.710 by QED. 00:13:23.710 --> 00:13:26.710 And the QED [INAUDIBLE] gauge field. 00:13:26.710 --> 00:13:29.810 You want gauge theory plus a Dirac fermion. 00:13:29.810 --> 00:13:30.590 Dirac theory. 00:13:33.770 --> 00:13:35.220 A Dirac theory. 00:13:35.220 --> 00:13:44.750 And similarly, if you include the weak interaction-- 00:13:44.750 --> 00:13:49.540 so there's something called [INAUDIBLE] weak-- 00:13:49.540 --> 00:13:55.460 it's described by su2 times u1 gauge theory. 00:13:55.460 --> 00:13:58.910 Described by su2 times u1 gauge theory. 00:13:58.910 --> 00:14:04.440 And also, the strong interaction you know now 00:14:04.440 --> 00:14:06.690 is described by su3 gauge theory. 00:14:10.390 --> 00:14:16.000 So for all those cases, the basic theoretical structure 00:14:16.000 --> 00:14:17.214 is understood. 00:14:17.214 --> 00:14:17.755 So the basic. 00:14:34.100 --> 00:14:34.640 OK. 00:14:34.640 --> 00:14:37.470 In principle, we can formulate those theories 00:14:37.470 --> 00:14:39.345 from first principle, just use path integral. 00:14:44.450 --> 00:14:47.890 Say, [INAUDIBLE] where [INAUDIBLE] 00:14:47.890 --> 00:14:53.770 group just based on principle, relation group. 00:14:53.770 --> 00:14:56.210 And then you can actually formulate all the theory 00:14:56.210 --> 00:14:59.220 in the [INAUDIBLE] way. 00:14:59.220 --> 00:15:09.750 So in the sense, then any calculation of theories 00:15:09.750 --> 00:15:19.140 can be reduced to certain algorithm. 00:15:27.020 --> 00:15:29.280 OK. 00:15:29.280 --> 00:15:32.670 Of course, this does not imply we actually 00:15:32.670 --> 00:15:35.720 know how to calculate many quantities here. 00:15:35.720 --> 00:15:38.990 It just say in principle, they are calculable. 00:15:38.990 --> 00:15:41.430 We often don't have the technical tool 00:15:41.430 --> 00:15:42.250 to calculate them. 00:15:42.250 --> 00:15:44.864 For example, even in QCD. 00:15:44.864 --> 00:15:47.280 [INAUDIBLE] joint action, we don't have the technical tool 00:15:47.280 --> 00:15:49.240 to calculate certain things. 00:15:49.240 --> 00:15:50.970 But they are, in principle, calculable. 00:15:50.970 --> 00:15:55.800 And we can write down the algorithm to calculate it. 00:15:55.800 --> 00:15:58.435 But for gravity, this is different. 00:16:01.340 --> 00:16:02.840 So for gravity, it's very different. 00:16:07.060 --> 00:16:07.755 So for gravity. 00:16:14.580 --> 00:16:23.802 So now we understand that classical gravity, 00:16:23.802 --> 00:16:25.010 it's just equal to spacetime. 00:16:27.820 --> 00:16:31.000 It's just a theory of spacetime. 00:16:31.000 --> 00:16:33.125 So this is the content over the general relativity. 00:16:35.252 --> 00:16:36.460 The general quantum activity. 00:16:41.260 --> 00:16:47.890 But at a quantum level, we really 00:16:47.890 --> 00:16:53.980 don't have a precise idea how to formulate the theory. 00:16:53.980 --> 00:16:57.740 So there are many questions still not understood. 00:16:57.740 --> 00:17:00.800 Many, say, conceptual questions still not understood. 00:17:00.800 --> 00:17:10.494 For example, a spacetime should fluctuate. 00:17:19.760 --> 00:17:22.359 OK. 00:17:22.359 --> 00:17:24.040 The spacetime should fluctuate because 00:17:24.040 --> 00:17:26.420 of the quantum fluctuation [INAUDIBLE] 00:17:26.420 --> 00:17:29.220 in quantum anything fluctuates. 00:17:29.220 --> 00:17:38.510 And then, the natural question is whether this is still-- 00:17:38.510 --> 00:17:41.315 so natural question is whether spacetime is still continuous. 00:17:47.560 --> 00:17:49.870 OK. 00:17:49.870 --> 00:17:52.280 So this is still [INAUDIBLE]. 00:17:52.280 --> 00:17:59.820 You need to replace it by something discrete. 00:17:59.820 --> 00:18:04.040 And it's not even clear right now 00:18:04.040 --> 00:18:09.917 whether spacetime will remain a fundamental notion when 00:18:09.917 --> 00:18:11.000 you go to quantum gravity. 00:18:13.650 --> 00:18:15.300 It could well be that spacetime would 00:18:15.300 --> 00:18:16.830 be replaced by something else. 00:18:24.640 --> 00:18:29.220 And also, there are many other questions. 00:18:29.220 --> 00:18:35.645 So for example, quantum nature of black holes. 00:18:45.760 --> 00:18:59.990 Or what should be the beginning of the universe, et cetera. 00:18:59.990 --> 00:19:03.060 So all these questions are not understood. 00:19:03.060 --> 00:19:11.630 Also, gravity is the weakest interaction. 00:19:17.390 --> 00:19:20.340 It is actually much, much weaker. 00:19:20.340 --> 00:19:23.320 We will see, it's much, much weaker 00:19:23.320 --> 00:19:26.060 than all the other interactions. 00:19:26.060 --> 00:19:30.040 And also, people have been speculate-- maybe actually 00:19:30.040 --> 00:19:34.570 this feature can also be actually a fundamental feature 00:19:34.570 --> 00:19:36.750 underlying in the special role of the gravity. 00:19:47.382 --> 00:19:48.840 For many years, it seemed to people 00:19:48.840 --> 00:19:53.490 that these questions and the gravity questions, 00:19:53.490 --> 00:19:55.882 they're completely unrelated. 00:19:55.882 --> 00:19:57.090 They're completely unrelated. 00:19:57.090 --> 00:19:59.612 They're a completely different subject. 00:19:59.612 --> 00:20:02.120 And those are well understood, but somehow those we 00:20:02.120 --> 00:20:05.390 need completely new ideas. 00:20:05.390 --> 00:20:06.800 So it came out a great surprise. 00:20:18.710 --> 00:20:22.020 So it came out that great surprise in 1997 00:20:22.020 --> 00:20:33.930 by [INAUDIBLE] that actually they actually are equivalent. 00:20:33.930 --> 00:20:38.190 So the proposal of the [INAUDIBLE] 00:20:38.190 --> 00:20:41.970 is that the-- chalk. 00:20:45.710 --> 00:20:55.270 Is that quantum gravity is actually 00:20:55.270 --> 00:21:07.720 equal to field theories on a fixed spacetime. 00:21:16.990 --> 00:21:18.750 So on a fixed spacetime just means 00:21:18.750 --> 00:21:23.510 that there's no gravity here on the right-hand side. 00:21:23.510 --> 00:21:26.310 Because if there's gravity, then you cannot have a fixed 00:21:26.310 --> 00:21:27.502 spacetime. 00:21:27.502 --> 00:21:29.065 And spacetime should be dynamical. 00:21:32.150 --> 00:21:35.180 So this is the equivalence between quantum gravity 00:21:35.180 --> 00:21:39.010 and just our ordinary quantum field theories. 00:21:39.010 --> 00:21:42.300 And this is ordinary quantum mechanics. 00:21:42.300 --> 00:21:44.490 And this is a very nontrivial quantum mechanics, 00:21:44.490 --> 00:21:47.350 which we don't yet understand. 00:21:47.350 --> 00:21:50.210 OK. 00:21:50.210 --> 00:21:55.590 This equality should be really understood 00:21:55.590 --> 00:21:58.810 as the different descriptions of the same quantum systems. 00:22:06.710 --> 00:22:25.460 So two sides are different descriptions 00:22:25.460 --> 00:22:27.459 of the same quantum system. 00:22:35.360 --> 00:22:38.400 So it just depends on how you want to see it. 00:22:38.400 --> 00:22:41.860 So one way you see it, you see a quantum gravity. 00:22:41.860 --> 00:22:43.360 Then from some other way you see it, 00:22:43.360 --> 00:22:44.568 you don't see gravity at all. 00:22:44.568 --> 00:22:46.710 You just see ordinary quantum mechanics. 00:22:46.710 --> 00:22:47.210 OK. 00:22:51.280 --> 00:22:57.280 Yeah, so this is what this equality is supposed to mean. 00:23:00.520 --> 00:23:04.710 So this equality essentially-- let me call this equation 1, 00:23:04.710 --> 00:23:08.870 which I will often use later. 00:23:08.870 --> 00:23:14.700 So this equation 1 is really a unification. 00:23:21.720 --> 00:23:24.540 Unification, which unifies quantum gravity 00:23:24.540 --> 00:23:28.320 with other field theory in a rather unconventional way. 00:23:28.320 --> 00:23:31.470 Not in a standard sense of unification. 00:23:31.470 --> 00:23:34.220 But that's bringing them together. 00:23:34.220 --> 00:23:34.720 OK. 00:23:37.230 --> 00:23:39.870 So now, let me make some philosophical remark. 00:23:42.600 --> 00:23:44.980 Now, let me make some philosophical remark. 00:23:44.980 --> 00:23:48.540 So last time, I have checked when 00:23:48.540 --> 00:23:51.160 preparing this class this [INAUDIBLE] original paper 00:23:51.160 --> 00:23:55.700 has been cited more than 10,000 times 00:23:55.700 --> 00:23:59.780 in the [INAUDIBLE] Database. 00:23:59.780 --> 00:24:02.180 And so there's a huge amount of work 00:24:02.180 --> 00:24:07.430 has been devoted to understanding this relation. 00:24:07.430 --> 00:24:12.040 But in my opinion, this subject's still in its infancy. 00:24:12.040 --> 00:24:13.800 And with many elementary [INAUDIBLE] 00:24:13.800 --> 00:24:16.550 we will see in the course as course goes on, 00:24:16.550 --> 00:24:19.150 there's still many elementary issues 00:24:19.150 --> 00:24:21.800 which are not understood. 00:24:21.800 --> 00:24:26.500 And to us, this relation is still like a magic. 00:24:26.500 --> 00:24:30.830 And we don't really have very good idea 00:24:30.830 --> 00:24:32.960 where this relation comes home. 00:24:32.960 --> 00:24:37.270 We have some rough idea, but not very precise idea. 00:24:37.270 --> 00:24:40.000 So the purpose of physics, of course, 00:24:40.000 --> 00:24:46.690 is to turn what looks like a magic or art into some rules 00:24:46.690 --> 00:24:51.240 so that it becomes something trivial. 00:24:51.240 --> 00:24:54.410 So at the beginning of course of the gravity, 00:24:54.410 --> 00:24:55.930 Newtonian gravity was magic. 00:24:55.930 --> 00:24:58.290 But in the end-- and what Newton did 00:24:58.290 --> 00:25:03.280 is to make the gravity to be not different from the force we 00:25:03.280 --> 00:25:05.730 see in ordinary life. 00:25:05.730 --> 00:25:11.550 And similarly, for other-- yeah, for other interactions 00:25:11.550 --> 00:25:14.880 we have seen before. 00:25:14.880 --> 00:25:16.930 Anyway. 00:25:16.930 --> 00:25:20.340 So personally, I believe when we really fully understand 00:25:20.340 --> 00:25:23.710 this relation, this will really be 00:25:23.710 --> 00:25:27.710 a huge landmark in the physics. 00:25:27.710 --> 00:25:32.585 And comparable, say, maybe to Newton's understanding 00:25:32.585 --> 00:25:36.480 of gravity or Maxwell-Boltzmann, et cetera. 00:25:40.530 --> 00:25:44.450 So the goal of this course is to help 00:25:44.450 --> 00:25:52.680 you understand how we-- our current understanding 00:25:52.680 --> 00:25:54.950 of this relation. 00:25:54.950 --> 00:25:59.670 And also, to help you to derive the related-- yeah, 00:25:59.670 --> 00:26:03.490 so these are two very different objects. 00:26:03.490 --> 00:26:05.170 And in order to set up an inequality, 00:26:05.170 --> 00:26:07.794 you also need to define a dictionary. 00:26:07.794 --> 00:26:09.210 For example, this is like Chinese. 00:26:09.210 --> 00:26:10.580 This is like English. 00:26:10.580 --> 00:26:13.940 So you need to define a dictionary between the two 00:26:13.940 --> 00:26:16.250 and what one said to the other. 00:26:16.250 --> 00:26:18.590 So we will also work out the dictionary. 00:26:18.590 --> 00:26:22.190 And also, we will develop with the tools 00:26:22.190 --> 00:26:24.180 and how to use this relation. 00:26:24.180 --> 00:26:25.920 And how to use the relation. 00:26:25.920 --> 00:26:29.940 And also, discuss many features of this relation 00:26:29.940 --> 00:26:32.630 and the implications. 00:26:32.630 --> 00:26:35.650 So the purpose is to help you to-- really, 00:26:35.650 --> 00:26:42.257 to bring to the forefront of this very exciting subject. 00:26:42.257 --> 00:26:42.965 So any questions? 00:26:53.360 --> 00:26:54.060 Good. 00:26:54.060 --> 00:26:55.980 No questions? 00:26:55.980 --> 00:27:05.240 OK, so now let's move to-- so after this introduction. 00:27:05.240 --> 00:27:07.310 So let's talk about this. 00:27:17.240 --> 00:27:18.352 OK. 00:27:18.352 --> 00:27:19.685 So if you look at this relation. 00:27:22.790 --> 00:27:26.040 So looking from the right-hand side. 00:27:26.040 --> 00:27:29.150 So you just have some kind of ordinary field theory-- 00:27:29.150 --> 00:27:31.850 ordinary quantum mechanical system, which 00:27:31.850 --> 00:27:36.070 we, in principle, know very well conceptually, 00:27:36.070 --> 00:27:39.140 without any gravity at all. 00:27:39.140 --> 00:27:46.090 Somehow, if you view it in some different way than the gravity 00:27:46.090 --> 00:27:48.460 and the spacetime, and the dynamical spacetime 00:27:48.460 --> 00:27:51.760 should emerge out of this. 00:27:51.760 --> 00:28:02.250 So in some sense, this 1 implies that the quantum gravity-- so 00:28:02.250 --> 00:28:06.140 let me just use the quantum gravity 00:28:06.140 --> 00:28:07.580 plus the dynamical spacetime. 00:28:15.160 --> 00:28:16.930 OK, dynamical spacetime. 00:28:16.930 --> 00:28:22.788 So it's important it's a dynamical spacetime-- 00:28:22.788 --> 00:28:33.330 can really merge from non-gravitational system. 00:28:45.420 --> 00:28:47.500 It can emerge from a non-gravitational system. 00:28:51.120 --> 00:28:56.980 So the idea of emergence of gravity from some other degrees 00:28:56.980 --> 00:29:01.820 freedom is not a really new idea. 00:29:01.820 --> 00:29:13.150 So [INAUDIBLE] in 1967, Sakharov, 00:29:13.150 --> 00:29:18.080 who was a Russian physicist. 00:29:18.080 --> 00:29:22.290 Sakharov is a very common name, a very common Russian name. 00:29:22.290 --> 00:29:24.080 But this is the most famous one. 00:29:26.930 --> 00:29:30.670 This is the most famous Sakharov who 00:29:30.670 --> 00:29:36.390 has invented the hydrogen bomb for the former Soviet Union. 00:29:36.390 --> 00:29:41.370 And he was also the guy who also later, I think, 00:29:41.370 --> 00:29:45.320 got Nobel Peace Prize for some human rights stuff. 00:29:49.230 --> 00:29:54.410 But he was also a very excellent physicist, including this idea 00:29:54.410 --> 00:29:57.040 of emergence of the gravity. 00:29:57.040 --> 00:30:00.340 So what he observed is that-- he found 00:30:00.340 --> 00:30:01.900 actually certain [INAUDIBLE]. 00:30:01.900 --> 00:30:05.530 He would study, say, some materials, et cetera. 00:30:05.530 --> 00:30:09.030 Then, he finds that certain [INAUDIBLE] systems 00:30:09.030 --> 00:30:13.630 can actually have mathematical description which 00:30:13.630 --> 00:30:16.920 looks like a magic connection, et cetera. 00:30:21.240 --> 00:30:23.150 Yeah, just mathematical, it looks 00:30:23.150 --> 00:30:25.150 like the equation for the magic and the equation 00:30:25.150 --> 00:30:27.290 for the connection, et cetera. 00:30:27.290 --> 00:30:30.450 And then he was speculating, maybe the general relativity 00:30:30.450 --> 00:30:34.560 can actually arise just from some kind of electron systems, 00:30:34.560 --> 00:30:37.170 just from the ordinary [INAUDIBLE] systems, 00:30:37.170 --> 00:30:40.055 and as effective description. 00:30:46.730 --> 00:30:51.140 And in fact, in the 1950s, even before the Sakharov 00:30:51.140 --> 00:30:56.330 have [INAUDIBLE] idea, people working in GR, 00:30:56.330 --> 00:30:59.170 in General Relativity, they already 00:30:59.170 --> 00:31:08.690 found many striking parallel between Einstein's equations 00:31:08.690 --> 00:31:14.600 and also associated many phenomena we hydrodynamics. 00:31:14.600 --> 00:31:17.860 And so we know that hydrodynamics is really 00:31:17.860 --> 00:31:20.200 just an effective description. 00:31:20.200 --> 00:31:23.895 And if you just look at the river-- so in fact, 00:31:23.895 --> 00:31:26.890 it's the discrete water molecules. 00:31:26.890 --> 00:31:30.210 But if you describe the motion of the water molecules 00:31:30.210 --> 00:31:33.340 at a microscopic level, then you find the hydrodynamics. 00:31:33.340 --> 00:31:35.256 But the hydrodynamics does not apply 00:31:35.256 --> 00:31:36.380 at more fundamental levels. 00:31:36.380 --> 00:31:40.610 So it's more like some kind of effective theory. 00:31:40.610 --> 00:31:43.750 So in the 1950s, people already found 00:31:43.750 --> 00:31:48.570 that there are many features of Einstein equation-- 00:31:48.570 --> 00:31:49.984 it's reminiscent. 00:31:49.984 --> 00:31:51.900 Actually, just reminiscent of certain features 00:31:51.900 --> 00:31:53.055 of hydrodynamics. 00:31:53.055 --> 00:31:54.800 So there's already this suspicion 00:31:54.800 --> 00:32:01.170 that maybe the gravity or spacetime actually can 00:32:01.170 --> 00:32:03.250 emerge out of something else. 00:32:03.250 --> 00:32:06.600 Just like the hydrodynamics is not a fundamental theory. 00:32:06.600 --> 00:32:12.130 It actually emerges from molecules, 00:32:12.130 --> 00:32:13.580 dynamics of the molecules. 00:32:16.225 --> 00:32:16.725 Anyway. 00:32:20.600 --> 00:32:27.650 So from the field theory perspective, 00:32:27.650 --> 00:33:12.770 it's also natural to ask whether massless spin-2 particles can 00:33:12.770 --> 00:33:29.535 arise as bound state, say, of lower spin particles. 00:33:38.880 --> 00:33:41.280 Say, like spin-1. 00:33:41.280 --> 00:33:42.620 Like photons, which are spin-1. 00:33:45.230 --> 00:33:46.916 So photons. 00:33:46.916 --> 00:33:48.430 Or gluons in QCD. 00:33:51.670 --> 00:33:55.600 Electrons, or quarks, et cetera. 00:33:55.600 --> 00:34:00.710 So these are the spin [INAUDIBLE] objects. 00:34:00.710 --> 00:34:12.170 And if yes-- so massless spin-2 particle in some sense-- 00:34:12.170 --> 00:34:16.520 so when you learn QR, you might have 00:34:16.520 --> 00:34:21.580 learned that the massless spin-2 particle may be considered 00:34:21.580 --> 00:34:25.537 as a hallmark form of gravity because of the-- so 00:34:25.537 --> 00:34:29.830 the basic propogation. 00:34:29.830 --> 00:34:32.230 The gravitational wave. 00:34:32.230 --> 00:34:35.179 So the basic propogation of the gravity 00:34:35.179 --> 00:34:36.800 is the gravitational wave. 00:34:36.800 --> 00:34:40.030 And that's propogated as spin-2 particles. 00:34:40.030 --> 00:34:46.159 And if you quantize the theory-- a propogation as spin-2 object. 00:34:46.159 --> 00:34:50.060 And when you quantize the gravitational waves, 00:34:50.060 --> 00:34:53.350 then they become massless spin-2 particles. 00:34:53.350 --> 00:34:55.659 They become spin-2 particles. 00:34:55.659 --> 00:34:58.810 So in some sense, if your theory can 00:34:58.810 --> 00:35:03.970 allow the massless spin-2 particles 00:35:03.970 --> 00:35:07.160 as-- so if your theory contains the massless spin-2 particles, 00:35:07.160 --> 00:35:08.841 then they must contain gravity. 00:35:08.841 --> 00:35:09.340 OK. 00:35:26.020 --> 00:35:46.490 So for example, even in strong interaction, which are, say, 00:35:46.490 --> 00:35:49.810 the theory describes gluons and the quarks. 00:35:49.810 --> 00:35:52.770 And the gluons and the quarks, they can indeed form. 00:35:52.770 --> 00:35:56.660 So these are the gluons and the quarks. 00:36:03.642 --> 00:36:12.435 In [INAUDIBLE], they're indeed spin-2 bound states. 00:36:16.360 --> 00:36:18.920 The gluons are spin-1, quarks are spin [INAUDIBLE]. 00:36:18.920 --> 00:36:20.345 But you combine them together, you 00:36:20.345 --> 00:36:22.870 can make [INAUDIBLE] the spin-2 bound state, even just 00:36:22.870 --> 00:36:23.828 in strong interactions. 00:36:26.930 --> 00:36:30.647 But of course in nature, the spin-2 bound state, 00:36:30.647 --> 00:36:31.480 they're all massive. 00:36:36.340 --> 00:36:38.660 They're all massive. 00:36:38.660 --> 00:36:42.750 And there are some very unstable, massive particles. 00:36:42.750 --> 00:36:45.510 When you create them, immediate decay. 00:36:45.510 --> 00:36:47.265 So they cannot be gravity. 00:36:47.265 --> 00:36:49.622 For gravity, you need massless spin-2 particles. 00:36:55.430 --> 00:36:57.890 But if you look at this fact, you cannot have the feeling. 00:36:57.890 --> 00:37:01.950 Say, maybe I take the QCD. 00:37:01.950 --> 00:37:04.240 Maybe I take a little bit. 00:37:04.240 --> 00:37:07.040 Maybe I can make that spin-2 massive particles 00:37:07.040 --> 00:37:10.680 into some massless spin-2 particles. 00:37:10.680 --> 00:37:15.180 Then I would have generated the gravity from a QCD like series. 00:37:15.180 --> 00:37:17.400 And then, that would be a revolution. 00:37:17.400 --> 00:37:20.270 And then you will be immortal. 00:37:23.740 --> 00:37:26.920 But unfortunately, even though this 00:37:26.920 --> 00:37:30.360 was a very promising idea for many years, 00:37:30.360 --> 00:37:37.376 and this hope was actually completely dashed 00:37:37.376 --> 00:37:43.070 by a powerful theorem of Weinberg and Witten. 00:37:43.070 --> 00:37:45.470 So there's a powerful theorem from Weinberg-Witten 00:37:45.470 --> 00:37:48.380 that say this can never happen. 00:37:48.380 --> 00:37:53.665 So this is the-- they're not possible. 00:38:10.550 --> 00:38:13.380 So this is 1980. 00:38:13.380 --> 00:38:15.864 So the paper. 00:38:15.864 --> 00:38:18.510 If you want, you can take a look at the paper. 00:38:26.780 --> 00:38:30.290 So this is volume 96 and the page 59. 00:38:30.290 --> 00:38:36.150 So they proved two theorems in that paper. 00:38:36.150 --> 00:38:38.180 So let me just copy the theorem down. 00:38:42.070 --> 00:38:44.780 Copy the theorem down. 00:38:44.780 --> 00:39:04.240 So they say a theory that allows the construction 00:39:04.240 --> 00:39:23.410 of a Lorentz-covariant and the conserved current 00:39:23.410 --> 00:39:26.100 for a vector-- current. 00:39:30.370 --> 00:39:56.470 Say, j mu, cannot contain massless particles. 00:40:02.610 --> 00:40:10.180 Cannot contain massless particles of spin greater than 00:40:10.180 --> 00:40:29.454 1/2 with non-vanishing value of-- OK, 00:40:29.454 --> 00:40:30.495 so this is first theorem. 00:40:34.110 --> 00:40:46.100 So the second theorem-- I think I should have enough space. 00:40:46.100 --> 00:40:47.120 So the second theorem. 00:40:54.490 --> 00:41:05.360 Second theorem says that a theory that 00:41:05.360 --> 00:41:19.980 allows a covariant, Lorentz-covariant, 00:41:19.980 --> 00:41:24.430 and the conserved stress tensor. 00:41:27.570 --> 00:41:29.080 So let me just call it T mu mu. 00:41:29.080 --> 00:41:53.460 A stress tensor cannot contain massless particles of spin j 00:41:53.460 --> 00:41:54.210 greater than 1. 00:42:05.930 --> 00:42:08.980 So the key-- so let me just repeat 00:42:08.980 --> 00:42:11.640 the theorem a little bit. 00:42:11.640 --> 00:42:16.140 So the first theorem said if you have a Lorentz-covariant 00:42:16.140 --> 00:42:21.510 and the conserved current, and in such a theory 00:42:21.510 --> 00:42:29.960 there's no charged particles can have spin more than 1/2. 00:42:29.960 --> 00:42:34.210 So there's no charged particle can have spin more than 1/2. 00:42:34.210 --> 00:42:37.990 Of course, this cannot contain graviton. 00:42:37.990 --> 00:42:42.590 And the second theorem says if you have a conserved stress 00:42:42.590 --> 00:42:47.550 tensor, Lorentz-covariant and a conserved stress tensor, 00:42:47.550 --> 00:42:50.210 then this theory cannot contain any particles with spin greater 00:42:50.210 --> 00:42:53.020 than 1. 00:42:53.020 --> 00:42:55.410 So of course, this cannot also contain a graviton. 00:42:55.410 --> 00:42:57.100 Graviton would be a spin-2. 00:43:00.820 --> 00:43:03.570 So this theorem-- yes? 00:43:03.570 --> 00:43:06.001 AUDIENCE: Is there anything like RG flow of mass? 00:43:06.001 --> 00:43:06.500 [INAUDIBLE] 00:43:09.170 --> 00:43:10.940 HONG LIU: This is final mass. 00:43:10.940 --> 00:43:12.250 Yeah. 00:43:12.250 --> 00:43:14.875 Just the [INAUDIBLE] mass. 00:43:14.875 --> 00:43:16.750 You don't talk about even normalization here. 00:43:22.086 --> 00:43:25.090 Yeah, this is just [INAUDIBLE] statement. 00:43:29.080 --> 00:43:33.240 So this theorem turned out to be actually very easy to prove. 00:43:33.240 --> 00:43:36.080 And in some sense, it's rather instructive. 00:43:36.080 --> 00:43:41.250 And just to give you the sense of the power of this theorem, 00:43:41.250 --> 00:43:43.340 we will prove it in a little bit. 00:43:46.750 --> 00:43:48.850 So before proving it, I will first 00:43:48.850 --> 00:43:52.970 make some remarks to make you appreciate 00:43:52.970 --> 00:43:54.840 what these two theorems means. 00:43:54.840 --> 00:43:58.222 Do you have any questions before I do that? 00:44:07.570 --> 00:44:08.700 OK, good. 00:44:08.700 --> 00:44:11.160 So let me just make some remarks on this theorem. 00:44:41.780 --> 00:44:51.810 So the first remark is that the theorems really apply 00:44:51.810 --> 00:44:57.860 to any particles, to both. 00:44:57.860 --> 00:44:59.700 So normally we say the fundamental particle 00:44:59.700 --> 00:45:04.640 is the particle which you put in your theory is 00:45:04.640 --> 00:45:09.040 the particle which say-- yeah, let me say [INAUDIBLE]. 00:45:09.040 --> 00:45:11.300 Rather fundamental, let me call it elementary. 00:45:11.300 --> 00:45:13.780 So the particle which already appear in your Lagrangian. 00:45:18.100 --> 00:45:21.520 And composite particles. 00:45:21.520 --> 00:45:24.955 Composite particles, say, would be some kind of bound state 00:45:24.955 --> 00:45:27.020 or some particles which don't appear 00:45:27.020 --> 00:45:30.870 in your original Lagrangian. 00:45:30.870 --> 00:45:35.050 So this theorem does not distinguish them. 00:45:35.050 --> 00:45:39.690 As far as you have some particles, [INAUDIBLE] region. 00:45:45.260 --> 00:45:50.670 So now, let's try to see whether this theorem is compatible 00:45:50.670 --> 00:45:55.280 to [INAUDIBLE] these things we already know. 00:45:55.280 --> 00:45:59.045 So let's first try to apply to, say, QED. 00:46:01.970 --> 00:46:15.400 So the theorem is compatible with QED. 00:46:23.140 --> 00:46:37.860 So we know that QED is a Maxwell theory plus Dirac's theory. 00:46:42.070 --> 00:46:47.150 And then the Dirac-- and the fermions in the Dirac theory 00:46:47.150 --> 00:46:50.760 interact with the photons in the Maxwell theory. 00:46:50.760 --> 00:46:55.540 So this theory have a massless photon of spin-1. 00:46:55.540 --> 00:47:00.300 And this will have a conserved charge because of, say, 00:47:00.300 --> 00:47:02.656 electrons that do have a charge. 00:47:02.656 --> 00:47:04.155 This theory has a conserved current. 00:47:10.770 --> 00:47:12.490 Has a conserved current. 00:47:12.490 --> 00:47:16.770 But this theory does allow a spin-1 particle. 00:47:16.770 --> 00:47:19.340 It does allow the spin-1 particle. 00:47:19.340 --> 00:47:24.580 And this is not contradictory with the theorem 1. 00:47:24.580 --> 00:47:27.170 So theorem 1 said if you have a Lorentz [INAUDIBLE] 00:47:27.170 --> 00:47:30.890 and the conserved current, then you cannot have massless 00:47:30.890 --> 00:47:33.505 particle of spin 1/2 with non-vanishing charge. 00:47:39.270 --> 00:47:41.036 In this case, photon is massless. 00:47:41.036 --> 00:47:42.410 But photon does not carry charge. 00:47:51.720 --> 00:47:56.910 Photon is neutral under the electric charge. 00:48:00.330 --> 00:48:02.182 So the existence of the photon is 00:48:02.182 --> 00:48:03.390 compatible with that theorem. 00:48:09.840 --> 00:48:12.870 And now, let's look at the [INAUDIBLE] theory 00:48:12.870 --> 00:48:16.591 which also contain massless spin-1 particle. 00:48:16.591 --> 00:48:19.090 And the [INAUDIBLE] theory which contain the spin-1 particle 00:48:19.090 --> 00:48:20.285 is the Yang-Mill theory. 00:48:27.930 --> 00:48:29.430 So Yang-Mill theory. 00:48:29.430 --> 00:48:31.783 For example, let's consider su2 Yang-Mill theory. 00:48:34.860 --> 00:48:40.840 So the Yang-Mill theory have three gauge field, A mu a. 00:48:40.840 --> 00:48:43.792 And a is equal to 1, 2, 3. 00:48:43.792 --> 00:48:49.175 Because 1 and 2 [INAUDIBLE] of su2 gauge symmetry. 00:49:00.981 --> 00:49:02.480 And from this three gauge field, you 00:49:02.480 --> 00:49:07.010 can combine them into the following form. 00:49:07.010 --> 00:49:09.670 I have A mu 3. 00:49:09.670 --> 00:49:12.130 Then, I have A mu plus minus. 00:49:12.130 --> 00:49:20.360 You could do 1 over square root 2 a mu 1 plus minus A mu 2. 00:49:20.360 --> 00:49:20.860 OK. 00:49:25.850 --> 00:49:28.510 So [INAUDIBLE]. 00:49:28.510 --> 00:49:34.670 So you can easily check yourself that this a mu plus minus 00:49:34.670 --> 00:49:39.625 is massless spin-1 particle. 00:49:45.785 --> 00:49:46.285 Particles. 00:49:49.280 --> 00:49:51.430 Can create massless spin-1 particles 00:49:51.430 --> 00:50:13.290 charged under the u1 subgroup generated by sigma 3. 00:50:16.430 --> 00:50:20.770 So sigma 3 is the generator associated with A mu 3. 00:50:20.770 --> 00:50:23.780 And so those two are charged under this generator. 00:50:26.330 --> 00:50:30.430 So now, [INAUDIBLE] we have a contradiction 00:50:30.430 --> 00:50:33.570 because those are massless spin-1 particles charged 00:50:33.570 --> 00:50:34.100 under this. 00:50:38.440 --> 00:50:43.230 And the theorem 1 said you have a conserved current, 00:50:43.230 --> 00:50:47.200 then you cannot have any particle recharged-- 00:50:47.200 --> 00:50:50.542 any charged particle with spin greater than 1/2. 00:50:50.542 --> 00:50:52.000 But those are the spin-1 particles. 00:50:55.050 --> 00:50:58.150 But actually, we're consistent. 00:51:00.750 --> 00:51:07.030 Because as you may remember, in Yang-Mills theory 00:51:07.030 --> 00:51:13.440 there does not exist-- that actually does not exist. 00:51:17.288 --> 00:51:19.140 The way out [INAUDIBLE] there does not 00:51:19.140 --> 00:51:43.350 exist a conserved Lorentz-covariant current 00:51:43.350 --> 00:51:47.504 for actually-- for this u1. 00:51:47.504 --> 00:51:53.390 For the u1 generated by this sigma 3 divided by 2. 00:51:53.390 --> 00:51:56.692 So this is actually compatible. 00:51:56.692 --> 00:51:59.150 This is actually compatible because such a current does not 00:51:59.150 --> 00:52:00.351 exist. 00:52:00.351 --> 00:52:00.850 OK. 00:52:05.010 --> 00:52:09.210 So you will show this fact yourself in your PSET. 00:52:09.210 --> 00:52:10.750 But you may remember, if you have 00:52:10.750 --> 00:52:14.500 studied quantum field theory 2, then 00:52:14.500 --> 00:52:18.240 you may remember this fact from the discussion 00:52:18.240 --> 00:52:20.750 of [INAUDIBLE] gauge theories. 00:52:20.750 --> 00:52:26.570 So in this course, we will from time to time 00:52:26.570 --> 00:52:30.300 use various facts of, say, [INAUDIBLE] theories. 00:52:33.420 --> 00:52:36.310 But they're not essential. 00:52:36.310 --> 00:52:39.570 If you don't allow [INAUDIBLE] theories, 00:52:39.570 --> 00:52:42.090 you can still understand this duality. 00:52:42.090 --> 00:52:44.600 You can still understand the relation one. 00:52:44.600 --> 00:52:47.910 But of course, if you know gauge field, it will help you a lot. 00:52:53.790 --> 00:52:55.030 Any questions about this? 00:53:01.630 --> 00:53:03.760 Good. 00:53:03.760 --> 00:53:04.490 Let's continue. 00:53:15.770 --> 00:53:38.950 So this theorem does not forbid graviton from GR. 00:53:43.900 --> 00:53:48.780 So if you try to quantize Einstein's general relativity 00:53:48.780 --> 00:53:51.540 around the flash space for example, 00:53:51.540 --> 00:53:54.776 then you will find the massless spin-2 particle. 00:53:54.776 --> 00:53:56.150 So this is what we call graviton. 00:54:01.700 --> 00:54:08.260 And this is not contradictory with that theorem 2. 00:54:08.260 --> 00:54:17.270 It's because you may still remember 00:54:17.270 --> 00:54:21.080 when you learn GR is that in GR, General Relativity, actually 00:54:21.080 --> 00:54:22.995 there is not. 00:54:22.995 --> 00:54:23.870 There's no conserved. 00:54:27.080 --> 00:54:30.050 So here, it's conserved. 00:54:30.050 --> 00:54:33.260 Not covariantly converged, just conserved. 00:54:33.260 --> 00:54:38.860 Lorentz-covariant stress tensor. 00:54:41.805 --> 00:54:42.305 OK. 00:54:45.670 --> 00:54:48.554 So in GR, actually there's no conserved 00:54:48.554 --> 00:54:52.000 in the Lorentz-covariant stress tensor. 00:54:52.000 --> 00:54:55.810 So actually, the GR [INAUDIBLE] graviton. 00:55:01.230 --> 00:55:07.517 But this theorem 2, it's nevertheless, very powerful. 00:55:07.517 --> 00:55:08.850 It's nevertheless very powerful. 00:55:13.470 --> 00:55:22.700 It said very powerful. 00:55:22.700 --> 00:55:26.496 It said all [INAUDIBLE]. 00:55:36.938 --> 00:55:50.696 None of [INAUDIBLE] QFTs in Minkowski spacetime, 00:55:50.696 --> 00:55:59.660 which is the one we normally work with, 00:55:59.660 --> 00:56:03.585 can have emergent gravity. 00:56:21.850 --> 00:56:24.210 OK. 00:56:24.210 --> 00:56:28.100 So QCD like theories, no matter how 00:56:28.100 --> 00:56:35.100 you try to trick the theory, the spin-2 massive bound state 00:56:35.100 --> 00:56:37.330 can never become massless. 00:56:37.330 --> 00:56:40.630 They can never become massless. 00:56:40.630 --> 00:56:44.275 Because those series, they do-- for example, 00:56:44.275 --> 00:56:46.160 our QCD, our strong interaction, they 00:56:46.160 --> 00:56:51.070 do have a covariantly conserved stress tensor. 00:56:51.070 --> 00:56:53.670 A Lorentz-covariant conserved stress tensor. 00:56:58.690 --> 00:57:06.380 But this theorem does not validate 00:57:06.380 --> 00:57:10.130 this equation 1 I just erased. 00:57:10.130 --> 00:57:14.740 Because this theorem, even though it's very powerful, 00:57:14.740 --> 00:57:17.330 it does have a hidden assumption. 00:57:20.056 --> 00:57:21.430 It does have a hidden assumption. 00:57:25.940 --> 00:57:35.772 This assumption is so obvious and so self-evident, 00:57:35.772 --> 00:57:39.620 that even though Weinberg and Witten, 00:57:39.620 --> 00:57:41.820 they're extremely careful people, 00:57:41.820 --> 00:57:43.870 they did not bother to mention it. 00:57:48.810 --> 00:57:52.130 In those times, nobody with their right mind 00:57:52.130 --> 00:57:55.050 would have mentioned such a thing. 00:57:55.050 --> 00:58:02.190 So the hidden assumption is that the kind of particles 00:58:02.190 --> 00:58:08.400 they are talking about-- is that the particles are they talking 00:58:08.400 --> 00:58:18.170 about, the particles, which they tried to rule out, 00:58:18.170 --> 00:58:29.261 live in the same spacetime as original theory. 00:58:36.411 --> 00:58:36.910 OK. 00:58:39.510 --> 00:58:44.900 So if you write down, say, QED in Minkowski 00:58:44.900 --> 00:58:47.640 spacetime, and then you ask about 00:58:47.640 --> 00:58:53.586 whether you can have some spin-2 particles coming out from QED. 00:58:53.586 --> 00:58:54.960 And then of course, you ask about 00:58:54.960 --> 00:58:56.890 whether you have particles come out 00:58:56.890 --> 00:58:59.470 QED living in the same Minkowski spacetime. 00:59:04.271 --> 00:59:04.770 Yeah. 00:59:04.770 --> 00:59:09.070 So in some sense, this assumption is self-evident. 00:59:09.070 --> 00:59:14.710 But precisely, this is assumption, 00:59:14.710 --> 00:59:21.120 which taking advantage by this relation, which first 00:59:21.120 --> 00:59:25.000 envisioned by [INAUDIBLE], is that precisely 00:59:25.000 --> 00:59:30.840 in this relation, in this equation 1 which I raised, 00:59:30.840 --> 00:59:33.330 that the gravity actually does not 00:59:33.330 --> 00:59:37.710 live in the same spacetime as the original theory. 00:59:37.710 --> 00:59:40.410 And does not live in the same spacetime as original theory. 00:59:44.780 --> 00:59:50.810 So even in the early days, when people 00:59:50.810 --> 00:59:56.850 were trying to dream having such kind of spin-2 particles 00:59:56.850 --> 00:59:59.800 that come from QCD like theories, 00:59:59.800 --> 01:00:07.040 there's already an immediate puzzle. 01:00:13.250 --> 01:00:18.250 Suppose the QCD like theories can have emergent graviton 01:00:18.250 --> 01:00:21.280 in the same spacetime. 01:00:21.280 --> 01:00:25.500 Then firstly, your QCD theory is defined in a fix spacetime. 01:00:25.500 --> 01:00:28.240 But now if it contains spin-2 particle in the spacetime 01:00:28.240 --> 01:00:31.500 itself, it will be dynamical. 01:00:31.500 --> 01:00:34.570 Then, where does the original QCD theory [INAUDIBLE]? 01:00:34.570 --> 01:00:35.660 OK. 01:00:35.660 --> 01:00:38.810 And then you will just similarly go 01:00:38.810 --> 01:00:40.890 into this kind of self-contradiction, 01:00:40.890 --> 01:00:44.410 like somehow-- how can that-- yeah. 01:00:44.410 --> 01:00:45.270 Anyway. 01:00:45.270 --> 01:00:59.370 So in this relation, in the holographic duality, 01:00:59.370 --> 01:01:12.550 gravity does not live in the same spacetime. 01:01:17.960 --> 01:01:19.200 OK. 01:01:19.200 --> 01:01:25.370 So this, in fact, live in one dimension or higher, 01:01:25.370 --> 01:01:27.490 or they can be several dimension higher depending 01:01:27.490 --> 01:01:29.560 on the examples. 01:01:29.560 --> 01:01:32.130 So this void this theorem in obvious way. 01:01:35.020 --> 01:01:39.670 And also, void this conundrum I said earlier. 01:01:39.670 --> 01:01:41.590 Because in a different spacetime. 01:01:41.590 --> 01:01:44.090 So the field theory is still in the fixed spacetime, 01:01:44.090 --> 01:01:48.166 but then the gravity then can be dynamical. 01:01:48.166 --> 01:01:50.540 Then, the spacetime which gravity lives can be dynamical. 01:01:54.610 --> 01:01:56.610 So any questions regarding this so far? 01:01:56.610 --> 01:01:57.850 Yes. 01:01:57.850 --> 01:01:59.530 AUDIENCE: [INAUDIBLE] in GR, there 01:01:59.530 --> 01:02:01.670 was no conserved Lorentz-covariant stress 01:02:01.670 --> 01:02:02.170 tensor. 01:02:02.170 --> 01:02:04.850 Because the covariant derivative of the stress 01:02:04.850 --> 01:02:07.160 tensor [INAUDIBLE]. 01:02:07.160 --> 01:02:10.190 HONG LIU: No covariant derivative is not conserved. 01:02:10.190 --> 01:02:13.700 This covariant is not conserved. 01:02:13.700 --> 01:02:15.740 Conserved means the ordinary derivatives, not 01:02:15.740 --> 01:02:19.550 the covariant derivatives. 01:02:19.550 --> 01:02:22.816 A covariant conserved stress tensor is not conserved. 01:02:26.270 --> 01:02:30.360 If you write down a conserved stress tensor, 01:02:30.360 --> 01:02:34.520 then it's not covariant. 01:02:34.520 --> 01:02:36.610 Then it's not covariant. 01:02:36.610 --> 01:02:38.450 Yeah. 01:02:38.450 --> 01:02:42.580 Similar thing with this case for the gauge fields. 01:02:42.580 --> 01:02:51.690 And in this case, gauge invariant currents-- 01:02:51.690 --> 01:02:54.735 the gauge invariant and the conserved currents 01:02:54.735 --> 01:02:57.000 is not Lorentz-covariants. 01:02:57.000 --> 01:03:01.480 And the Lorentz-covariants, gauge invariant current 01:03:01.480 --> 01:03:02.738 then is not conserved. 01:03:10.073 --> 01:03:10.874 Any other question? 01:03:10.874 --> 01:03:12.040 AUDIENCE: I have a question. 01:03:12.040 --> 01:03:13.505 HONG LIU: Yes. 01:03:13.505 --> 01:03:14.380 AUDIENCE: [INAUDIBLE] 01:03:24.884 --> 01:03:25.550 HONG LIU: Sorry. 01:03:25.550 --> 01:03:26.856 Say it again? 01:03:26.856 --> 01:03:29.230 AUDIENCE: [INAUDIBLE] theory, every particle is massless. 01:03:29.230 --> 01:03:29.813 HONG LIU: Yes. 01:03:29.813 --> 01:03:33.900 AUDIENCE: And the [INAUDIBLE] spin-2 particle, which carries 01:03:33.900 --> 01:03:34.400 [INAUDIBLE] 01:03:39.350 --> 01:03:40.730 HONG LIU: Yeah. 01:03:40.730 --> 01:03:44.240 For [INAUDIBLE] field theory, indeed everything's massless. 01:03:44.240 --> 01:03:47.200 You can construct Lorentz-covariants 01:03:47.200 --> 01:03:48.980 conserved stress tensor. 01:03:48.980 --> 01:03:56.590 So that theory [INAUDIBLE] does not allow spin-2 excitations 01:03:56.590 --> 01:03:57.955 in the same spacetime. 01:03:57.955 --> 01:03:58.830 AUDIENCE: [INAUDIBLE] 01:04:03.284 --> 01:04:05.200 HONG LIU: There's no massless spin-2 particle. 01:04:08.345 --> 01:04:09.220 AUDIENCE: [INAUDIBLE] 01:04:16.865 --> 01:04:17.490 HONG LIU: Yeah. 01:04:17.490 --> 01:04:20.840 Yeah, but you don't have massless spin-2. 01:04:20.840 --> 01:04:25.040 You don't have massless spin-2 particles. 01:04:25.040 --> 01:04:27.000 AUDIENCE: [INAUDIBLE] 01:04:27.000 --> 01:04:28.350 HONG LIU: No. 01:04:28.350 --> 01:04:31.250 It depends. 01:04:31.250 --> 01:04:34.550 Actually, talking about the massless in the [INAUDIBLE] 01:04:34.550 --> 01:04:36.500 field theory is a little bit tricky. 01:04:36.500 --> 01:04:39.150 It's a little bit tricky. 01:04:39.150 --> 01:04:41.700 They come in all kind of spectrum. 01:04:41.700 --> 01:04:44.120 They come in the continuous spectrum. 01:04:44.120 --> 01:04:45.160 Yeah. 01:04:45.160 --> 01:04:50.990 Yeah, but this theorem does apply to ordinary [INAUDIBLE] 01:04:50.990 --> 01:04:52.820 field theories. 01:04:52.820 --> 01:04:57.862 So in those theories, you won't have massless. 01:04:57.862 --> 01:04:59.570 Yeah, when we prove the theorem, then you 01:04:59.570 --> 01:05:02.270 can see precisely what do we mean 01:05:02.270 --> 01:05:04.190 by massless spin-2 particles. 01:05:13.170 --> 01:05:16.922 If you stay in the course, then you will know. 01:05:16.922 --> 01:05:18.860 Yeah. 01:05:18.860 --> 01:05:22.270 Yeah, we will see it later on. 01:05:22.270 --> 01:05:25.040 But not for a while. 01:05:25.040 --> 01:05:27.050 Not for a couple of weeks. 01:05:27.050 --> 01:05:29.360 And maybe in a couple of weeks, we will see it. 01:05:32.930 --> 01:05:33.685 In two weeks. 01:05:33.685 --> 01:05:35.217 One or two weeks. 01:05:35.217 --> 01:05:36.050 Any other questions? 01:05:39.960 --> 01:05:40.700 OK. 01:05:40.700 --> 01:05:45.970 So now, let me give you the proof, which is pretty simple. 01:05:45.970 --> 01:05:51.610 And use some elementary facts, very elementary facts. 01:05:51.610 --> 01:05:56.820 Because you see, this theorem contains very little input. 01:05:56.820 --> 01:06:00.316 That theorem really contained very, very little input. 01:06:00.316 --> 01:06:01.690 So if you can prove such a thing, 01:06:01.690 --> 01:06:03.470 it cannot be complicated. 01:06:03.470 --> 01:06:06.480 If you can prove it, it cannot be complicated. 01:06:06.480 --> 01:06:07.920 Because the input is so little. 01:06:07.920 --> 01:06:10.520 It does not require any details. 01:06:10.520 --> 01:06:13.540 It does not require any details of a theory. 01:06:13.540 --> 01:06:15.250 So if you can prove such a theorem, 01:06:15.250 --> 01:06:18.480 it must be just from kinematics, not come from any dynamics. 01:06:21.590 --> 01:06:25.720 So indeed, that's how the theorem goes. 01:06:25.720 --> 01:06:26.805 So the proof. 01:06:32.500 --> 01:06:34.480 So we will do proof by contradiction 01:06:34.480 --> 01:06:38.140 as you would naturally expect. 01:06:38.140 --> 01:06:53.420 So let us suppose there exists a massless spin-j particle. 01:06:53.420 --> 01:06:55.590 Massless particles. 01:06:55.590 --> 01:06:57.660 So let's assume this theory contains 01:06:57.660 --> 01:07:02.700 some theories [INAUDIBLE] exist massless spin-2 particles. 01:07:02.700 --> 01:07:04.340 Massless particles of spin-j. 01:07:11.510 --> 01:07:13.110 And just from the Lorentz symmetry, 01:07:13.110 --> 01:07:15.080 you can immediately write down-- so 01:07:15.080 --> 01:07:20.630 from what we learned in QFT 1 and 2. 01:07:20.630 --> 01:07:23.600 So just by Lorentz symmetry-- it doesn't matter what 01:07:23.600 --> 01:07:24.850 is the nature of those states. 01:07:28.220 --> 01:07:30.240 Because the mass formal representations 01:07:30.240 --> 01:07:34.760 over the, say, Lorentz group and [INAUDIBLE] group. 01:07:34.760 --> 01:07:39.560 So [INAUDIBLE] immediately write down the one particle state 01:07:39.560 --> 01:07:41.330 of such particles. 01:07:41.330 --> 01:07:45.220 And you normally can write them as k and sigma. 01:07:45.220 --> 01:07:47.685 So k just is the ordinary spacetime momentum. 01:07:55.650 --> 01:07:57.760 So for massless particle, of course 01:07:57.760 --> 01:08:00.640 the k0 squared should be equal to k squared. 01:08:03.210 --> 01:08:06.960 And the sigma is the helicity. 01:08:06.960 --> 01:08:09.210 Helicity is a projection of the angular momentum 01:08:09.210 --> 01:08:12.010 in the direction of your momentum. 01:08:12.010 --> 01:08:16.689 And for a massless particle of spin-j, 01:08:16.689 --> 01:08:19.149 the sigma can only be plus minus j. 01:08:19.149 --> 01:08:20.315 So this is helicity. 01:08:24.680 --> 01:08:28.300 So if you don't remember this, you 01:08:28.300 --> 01:08:34.399 can go back to review Wemberg's book. 01:08:34.399 --> 01:08:39.420 Wemberg's volume 1 QFT. 01:08:39.420 --> 01:08:46.580 Volume 1, Section 2.5 discussed how you write down 01:08:46.580 --> 01:08:52.870 the general one particle states for, say, 01:08:52.870 --> 01:08:56.060 any quantum-- in any quantum theory 01:08:56.060 --> 01:08:57.609 as far as you have Lorentz symmetry. 01:08:57.609 --> 01:08:58.109 OK. 01:09:03.550 --> 01:09:06.990 And the one properties of this sigma 01:09:06.990 --> 01:09:08.494 is that let's now consider rotation. 01:09:11.850 --> 01:09:14.490 Now, let's consider rotation. 01:09:14.490 --> 01:09:15.990 So let's consider rotation operator. 01:09:25.060 --> 01:09:35.370 This [INAUDIBLE] by angle theta around the direction of the k. 01:09:38.700 --> 01:09:41.359 Around the direction of the k, which is the direction of the k 01:09:41.359 --> 01:09:41.859 here. 01:09:49.140 --> 01:09:52.410 So let's consider the action of these 01:09:52.410 --> 01:09:59.870 under state, this one particle-- this massless particle state. 01:09:59.870 --> 01:10:07.930 Then by definition, this just give you 01:10:07.930 --> 01:10:13.230 i sigma theta k, sigma. 01:10:13.230 --> 01:10:17.050 So this is essentially come from the definition of the helicity. 01:10:17.050 --> 01:10:22.050 And the k does not change because this is the rotation 01:10:22.050 --> 01:10:25.430 along the direction of the k. 01:10:25.430 --> 01:10:26.960 The k itself does not change. 01:10:30.400 --> 01:10:35.890 But this eigenstate of this rotation [INAUDIBLE]. 01:10:35.890 --> 01:10:36.470 OK. 01:10:36.470 --> 01:10:42.910 Again, if you don't remember this, go back to Wemberg's book 01:10:42.910 --> 01:10:44.760 and you can easily find it. 01:10:47.940 --> 01:11:08.990 So now if you have a conserved current, 01:11:08.990 --> 01:11:16.800 if you have a covariant conserved current, 01:11:16.800 --> 01:11:19.460 J mu, then you can always construct 01:11:19.460 --> 01:11:23.280 the charge, the conserved charge, 01:11:23.280 --> 01:11:26.640 which is the integral over J0. 01:11:26.640 --> 01:11:31.250 And similarly, if you have T mu mu, then from T mu mu, 01:11:31.250 --> 01:11:44.610 you can construct a momentum operator, which is the T mu. 01:11:44.610 --> 01:11:47.225 So this is just the standard momentum operator 01:11:47.225 --> 01:11:50.527 in your field theory. 01:11:50.527 --> 01:11:51.860 So this is just still the setup. 01:11:57.262 --> 01:12:07.530 Of course just by definition, the p mu acting on k sigma, 01:12:07.530 --> 01:12:11.007 you just get back k mu. 01:12:11.007 --> 01:12:12.840 OK, because this is the momentum eigenstate. 01:12:18.180 --> 01:12:22.420 And if this particle is also charged 01:12:22.420 --> 01:12:40.230 under this conserved charge, then-- so if charged under k, 01:12:40.230 --> 01:12:47.150 under this conserved charge, then Q act on this state 01:12:47.150 --> 01:12:51.420 will also give you the charge q. 01:12:51.420 --> 01:12:53.262 So this is operator. 01:12:53.262 --> 01:12:53.970 This is operator. 01:12:56.762 --> 01:12:57.720 These are all operator. 01:13:04.860 --> 01:13:05.470 Operator. 01:13:05.470 --> 01:13:09.400 And then, this should give you charge q. 01:13:09.400 --> 01:13:14.460 So now we want to show this 2 theorem implies-- 01:13:14.460 --> 01:13:21.500 so with this set up, this theorem can-- 01:13:21.500 --> 01:13:25.080 so we want to show two things. 01:13:25.080 --> 01:13:34.070 1, if q is nonzero, that means if the particle is charged 01:13:34.070 --> 01:13:41.945 under a conserved charge, then j must be smaller than 1/2. 01:13:41.945 --> 01:13:43.730 It cannot be greater than 1/2. 01:13:48.200 --> 01:13:50.870 So we assume it's a conserved stress tensor. 01:13:50.870 --> 01:13:54.970 Otherwise, j should be always smaller or equal to 1 at most. 01:13:54.970 --> 01:13:55.470 OK. 01:13:58.040 --> 01:14:00.930 So that's what that theorem means 01:14:00.930 --> 01:14:03.760 translated in this language. 01:14:03.760 --> 01:14:05.430 OK. 01:14:05.430 --> 01:14:07.520 Any questions? 01:14:07.520 --> 01:14:10.920 So now, let's prove it. 01:14:10.920 --> 01:14:13.210 Let me just do it here. 01:14:13.210 --> 01:14:15.852 So now, let's prove it. 01:14:15.852 --> 01:14:18.470 I think we just have enough time, maybe to prove it. 01:14:25.730 --> 01:14:29.554 So to prove it, let's just first list some elementary facts. 01:14:34.880 --> 01:14:37.000 First, just do some elementary fact. 01:14:41.950 --> 01:14:51.690 First, from Lorentz symmetry-- we 01:14:51.690 --> 01:14:53.330 do have to assume Lorentz symmetry. 01:14:53.330 --> 01:14:54.980 OK. 01:14:54.980 --> 01:14:59.650 You can show-- so this I will just write down the answer 01:14:59.650 --> 01:15:01.160 and leave yourself to show it. 01:15:01.160 --> 01:15:02.106 OK? 01:15:02.106 --> 01:15:03.515 In your PSET. 01:15:06.430 --> 01:15:09.790 So if you look at the matrix [INAUDIBLE] 01:15:09.790 --> 01:15:12.640 of this conserved current between two such states 01:15:12.640 --> 01:15:19.530 of different k, in the limit which k goes to k prime, 01:15:19.530 --> 01:15:23.480 you have q k mu k0. 01:15:28.960 --> 01:15:50.516 And similarly, in this limit-- so this 01:15:50.516 --> 01:15:51.640 we will do it in your PSET. 01:15:55.480 --> 01:15:59.140 So let me call this equation 2, equation 3. 01:15:59.140 --> 01:16:03.010 So this is based on the following normalization. 01:16:05.650 --> 01:16:27.750 Based on the following normalization between the two 01:16:27.750 --> 01:16:28.250 states. 01:16:28.250 --> 01:16:28.750 OK. 01:16:35.420 --> 01:16:39.370 So this relation is actually very easy to see intuitively. 01:16:44.090 --> 01:16:46.750 Yeah, you can prove it by-- yeah, 01:16:46.750 --> 01:16:50.200 let me give me just a quick hint for [INAUDIBLE] regarding 01:16:50.200 --> 01:16:57.150 2, is that you can first show that k sigma J0 01:16:57.150 --> 01:17:03.290 sandwiched between the two, in the limit when 01:17:03.290 --> 01:17:07.130 this k become the same is just given 01:17:07.130 --> 01:17:08.820 by q divided by 2 pi cubed. 01:17:13.240 --> 01:17:16.665 So this is, again, somewhat intuitive 01:17:16.665 --> 01:17:21.860 because k is eigenstate of Q. Because the k is 01:17:21.860 --> 01:17:24.940 an eigenstate of this Q hat. 01:17:24.940 --> 01:17:27.990 And Q hat is just the integral of J0. 01:17:27.990 --> 01:17:30.154 It is integral of J0. 01:17:30.154 --> 01:17:32.070 And so you can imagine on the right-hand side, 01:17:32.070 --> 01:17:34.653 there must be a Q because these are essentially the eigenstate 01:17:34.653 --> 01:17:36.796 of the integration with J0. 01:17:36.796 --> 01:17:39.170 And then when you take into account all these [INAUDIBLE] 01:17:39.170 --> 01:17:42.910 functions here, and the integral-- and the relation 01:17:42.910 --> 01:17:45.470 between the integral, then you will naturally 01:17:45.470 --> 01:17:49.940 find the right-hand side is just q divided by this. 01:17:49.940 --> 01:17:52.250 So once you know this, then you can immediately 01:17:52.250 --> 01:17:55.550 see the first [INAUDIBLE] equation 2 there. 01:17:55.550 --> 01:17:57.520 Because at first, there's a j mu there. 01:17:57.520 --> 01:18:00.330 Then the right-hand side must have index mu. 01:18:00.330 --> 01:18:02.450 So the only thing can carry index mu is the k mu. 01:18:02.450 --> 01:18:04.350 There's nothing else. 01:18:04.350 --> 01:18:08.270 And also, you know when the mu equal to 0, 01:18:08.270 --> 01:18:09.800 you should just get the q. 01:18:09.800 --> 01:18:13.840 Then, downstairs we might have a k0 to cancel them. 01:18:13.840 --> 01:18:16.400 So this relation, in principle you 01:18:16.400 --> 01:18:19.640 can just-- just by intuition, you 01:18:19.640 --> 01:18:21.950 can just-- by dimensional analysis, [INAUDIBLE] just 01:18:21.950 --> 01:18:24.080 directly write it down. 01:18:24.080 --> 01:18:25.760 And similar in the second relation-- 01:18:25.760 --> 01:18:30.180 in the second relation, the analog of q. 01:18:30.180 --> 01:18:33.140 So q is the charge associated with Q hat. 01:18:33.140 --> 01:18:39.040 The analog of the q for the stress tensor is just the k mu. 01:18:39.040 --> 01:18:45.010 So you just replace this by k mu. 01:18:45.010 --> 01:18:47.485 It just relates to the q by k mu. 01:18:47.485 --> 01:18:50.810 Then you get the second line. 01:18:50.810 --> 01:18:53.740 Because k by k mu [INAUDIBLE]. 01:18:53.740 --> 01:18:54.470 OK. 01:18:54.470 --> 01:18:59.440 So I will leave yourself to prove this rigorously. 01:18:59.440 --> 01:19:02.670 So this is the second observation. 01:19:02.670 --> 01:19:11.230 It said for massless particles, which 01:19:11.230 --> 01:19:15.460 have k squared equal to 0, and k prime squared equal to 0, 01:19:15.460 --> 01:19:17.170 because [INAUDIBLE] particles. 01:19:17.170 --> 01:19:21.450 So k squared and k prime squared should be equal to 0. 01:19:21.450 --> 01:19:28.120 And that means for any two particles, a massless momentum 01:19:28.120 --> 01:19:34.230 like this, you see immediately just by writing this down 01:19:34.230 --> 01:19:37.280 this must be smaller than 0. 01:19:37.280 --> 01:19:42.180 So this should come from your high school physics. 01:19:42.180 --> 01:19:45.160 The k dot k prime should be smaller than 0. 01:19:45.160 --> 01:19:49.560 And that means that k plus k prime must be time-like. 01:19:54.490 --> 01:19:57.110 OK, must be time-like. 01:19:57.110 --> 01:20:04.280 Because this square is essentially given by this guy 01:20:04.280 --> 01:20:06.500 because k square and k prime equal to 0. 01:20:06.500 --> 01:20:08.610 So this guy must be time-like. 01:20:08.610 --> 01:20:10.660 OK. 01:20:10.660 --> 01:20:12.570 And so that means you can always choose 01:20:12.570 --> 01:20:17.440 a frame that the special part of this is 0. 01:20:17.440 --> 01:20:27.950 We can choose a frame k plus k prime equal to 0. 01:20:27.950 --> 01:20:33.870 So such a frame is just k mu, say E, 0. 01:20:33.870 --> 01:20:38.330 Say we put all of these in the z-direction, 01:20:38.330 --> 01:20:41.470 in the 3-direction. 01:20:41.470 --> 01:20:49.700 Then you can choose k prime k mu equal to-- OK? 01:20:59.530 --> 01:21:00.940 And this just follows. 01:21:00.940 --> 01:21:05.650 These two are two [INAUDIBLE] particles. 01:21:05.650 --> 01:21:06.640 OK, good. 01:21:06.640 --> 01:21:10.610 So now, we can almost envision this. 01:21:10.610 --> 01:21:13.240 Then, there's still a third elementary observation 01:21:13.240 --> 01:21:16.210 we need to put here. 01:21:16.210 --> 01:21:23.880 So the third observation which uses this formula. 01:21:27.350 --> 01:21:29.430 So let's now take sigma just to be j. 01:21:37.230 --> 01:21:55.972 So now, under a rotation of theta around 3-direction-- 01:21:55.972 --> 01:21:58.555 around this direction, which we have the [INAUDIBLE] momentum. 01:21:58.555 --> 01:22:03.530 So we choose a momentum to be in this direction. 01:22:03.530 --> 01:22:08.140 Then you should have-- so let me call 01:22:08.140 --> 01:22:10.820 that operator to be R theta is the rotation 01:22:10.820 --> 01:22:13.270 in the z-direction, in the 3-direction. 01:22:13.270 --> 01:22:16.260 So this operate on k, j. 01:22:16.260 --> 01:22:20.016 So let's take sigma equal to j. 01:22:20.016 --> 01:22:23.040 Take sigma equal to j. 01:22:23.040 --> 01:22:32.550 So this should be just equal to exponential i, j theta k, j. 01:22:32.550 --> 01:22:39.270 And if this acting on k prime j. 01:22:42.170 --> 01:22:47.520 So k prime have opposite orientation as k 01:22:47.520 --> 01:22:48.635 because this is minus e. 01:22:51.821 --> 01:22:54.320 And the opposite orientation means your helicity should also 01:22:54.320 --> 01:22:56.290 change sign. 01:22:56.290 --> 01:23:02.860 So that means here should be minus i j theta k prime, j. 01:23:02.860 --> 01:23:05.380 OK. 01:23:05.380 --> 01:23:07.720 Yeah, because this is the rotation 01:23:07.720 --> 01:23:10.200 along the positive 3-direction. 01:23:12.981 --> 01:23:13.480 OK. 01:23:13.480 --> 01:23:16.250 So now we are in business. 01:23:16.250 --> 01:23:20.820 So now finally, we will try to-- from those facts together, 01:23:20.820 --> 01:23:24.335 we will now see a contradiction. 01:23:24.335 --> 01:23:27.566 Now see a contradiction. 01:23:27.566 --> 01:23:28.941 Now you can see that this object. 01:23:34.840 --> 01:23:40.410 Consider k, j-- k prime, j. 01:23:40.410 --> 01:23:52.107 Say this R minus 1 theta acting on J mu R theta k, j. 01:23:52.107 --> 01:23:53.565 So now, let's consider this object. 01:23:56.410 --> 01:24:00.090 So this object, I can evaluate it by two ways. 01:24:00.090 --> 01:24:04.160 First, I can just act R theta on both states. 01:24:04.160 --> 01:24:05.700 OK. 01:24:05.700 --> 01:24:10.530 So R theta on this one just gives me this. 01:24:10.530 --> 01:24:15.720 R minus 1 acting on that just give me opposite of this. 01:24:15.720 --> 01:24:19.670 So all together, if I do that way, 01:24:19.670 --> 01:24:29.914 then I get exponential 2 i j theta k prime, j J mu k, j. 01:24:33.680 --> 01:24:39.060 But I can also act this operator on this current. 01:24:39.060 --> 01:24:42.950 So this current should transform as a Lorentz vector. 01:24:42.950 --> 01:24:52.050 So if I do this way, then I should get lambda mu new k 01:24:52.050 --> 01:24:59.880 prime, j J new k, j. 01:24:59.880 --> 01:25:04.220 So these acting on that should just effect a Lorentzian 01:25:04.220 --> 01:25:05.232 summation on j. 01:25:05.232 --> 01:25:07.315 And so this is in the Lorentzian summation matrix. 01:25:10.230 --> 01:25:13.976 So lambda mu mu just is the standard rotation matrix 01:25:13.976 --> 01:25:14.850 acting on the vector. 01:25:18.147 --> 01:25:23.340 So lambda mu mu just the standard rotation matrix. 01:25:23.340 --> 01:25:28.170 For example, just given by 1, 0, 0, 0. 01:25:28.170 --> 01:25:32.150 Say, cosine theta minus sine theta. 01:25:32.150 --> 01:25:37.910 So just a rotation along the 3-direction. 01:25:43.460 --> 01:25:47.640 So this just the rotation acting on the j. 01:25:52.150 --> 01:26:02.320 So now similarly, if you examine this quantity. 01:26:10.150 --> 01:26:17.340 Yeah, similarly if I just replace here by T mu mu, OK? 01:26:17.340 --> 01:26:22.776 by T mu mu-- we are out of time, so let me 01:26:22.776 --> 01:26:25.490 be a little bit faster now. 01:26:25.490 --> 01:26:34.600 Similarly, if we replace that by T mu mu, 01:26:34.600 --> 01:26:42.230 then we will get the same thing 2 i j theta k prime, 01:26:42.230 --> 01:26:53.250 j T mu mu k, j, which obtained by acting directly on k, j. 01:26:53.250 --> 01:26:57.330 By the way, that R acting on T mu mu, then 01:26:57.330 --> 01:27:00.690 T mu mu transform as a 2 tensor. 01:27:00.690 --> 01:27:04.960 Then we should get lambda mu rho mu lambda. 01:27:04.960 --> 01:27:12.903 Lambda mu rho k prime, j T lambda rho k, j. 01:27:23.670 --> 01:27:24.710 So now, here's the key. 01:27:27.640 --> 01:27:29.270 Now, here's the key. 01:27:29.270 --> 01:27:31.708 So do you already see the contradiction? 01:27:37.440 --> 01:27:42.110 So let's look at this equation. 01:27:42.110 --> 01:27:43.880 This is just like a vector. 01:27:43.880 --> 01:27:47.460 The whole thing is like a vector. 01:27:47.460 --> 01:27:48.880 The same vector. 01:27:48.880 --> 01:27:52.320 So this is like an eigenvalue equation. 01:27:52.320 --> 01:27:55.830 So this lambda mu mu acting on this vector 01:27:55.830 --> 01:27:57.650 get eigenvalue 2 i j theta. 01:27:57.650 --> 01:28:01.360 So this is like an eigenvalue equation. 01:28:01.360 --> 01:28:10.040 But if you look at this matrix, this 01:28:10.040 --> 01:28:21.130 can only have eigenvalue exponential 01:28:21.130 --> 01:28:25.010 plus minus i theta and the 1. 01:28:25.010 --> 01:28:28.060 So this is thing we are familiar with. 01:28:28.060 --> 01:28:32.280 So [INAUDIBLE] minus i theta, 1. 01:28:32.280 --> 01:28:36.210 So similarly with this equation, just you have 2 lambda, OK? 01:28:38.760 --> 01:28:42.510 So now, that means if this quantity 01:28:42.510 --> 01:28:51.360 is nonzero, which we know it is nonzero from 2 and 3. 01:28:51.360 --> 01:28:53.440 It is nonzero. 01:28:53.440 --> 01:28:57.610 That these 2 i j theta can only take a value 01:28:57.610 --> 01:29:00.510 plus minus i theta and the 1. 01:29:00.510 --> 01:29:05.060 So that means that j-- from here, 01:29:05.060 --> 01:29:09.730 let's say equation 4 and 5. 01:29:09.730 --> 01:29:17.130 From equation 4, that j must be smaller than 1/2. 01:29:17.130 --> 01:29:21.160 Because there's a 2j there. 01:29:21.160 --> 01:29:27.080 Similarly, from equation 5, means j must be smaller than 1. 01:29:30.640 --> 01:29:34.050 So this is what we want to prove. 01:29:34.050 --> 01:29:35.540 So this is what we want to prove. 01:29:35.540 --> 01:29:36.040 Yes? 01:29:36.040 --> 01:29:37.831 AUDIENCE: Aren't k and k prime [INAUDIBLE]? 01:29:41.220 --> 01:29:46.484 I mean, 2 and 3, the limit k goes to k prime and k plus k 01:29:46.484 --> 01:29:47.360 prime is 0. 01:29:47.360 --> 01:29:48.040 HONG LIU: Yeah. 01:29:48.040 --> 01:29:52.930 So I can take them-- if j is greater than 1/2. 01:29:52.930 --> 01:29:56.025 For example in this relation, if j is greater than 1/2, 01:29:56.025 --> 01:29:59.380 then the only possibility is for this to be 0. 01:29:59.380 --> 01:30:01.750 But we know when you take k equal to k prime, at least 01:30:01.750 --> 01:30:03.820 in that limit this is nonzero. 01:30:03.820 --> 01:30:05.400 And so this cannot be 0. 01:30:05.400 --> 01:30:08.934 So this quantity cannot be identical to 0. 01:30:12.251 --> 01:30:12.750 OK. 01:30:15.864 --> 01:30:17.030 So this gives you the proof. 01:30:20.470 --> 01:30:23.100 So if you find the loophole in this proof, 01:30:23.100 --> 01:30:29.470 other than they have live in the same dimension-- yeah, 01:30:29.470 --> 01:30:32.840 that would be great. 01:30:32.840 --> 01:30:35.707 AUDIENCE: What is the whole matrix element is 0? 01:30:35.707 --> 01:30:37.290 What if the whole matrix element is 0? 01:30:37.290 --> 01:30:39.470 HONG LIU: No, the point is that the whole matrix element cannot 01:30:39.470 --> 01:30:39.990 be 0. 01:30:39.990 --> 01:30:40.573 AUDIENCE: Yes. 01:30:40.573 --> 01:30:41.820 HONG LIU: Because of this. 01:30:41.820 --> 01:30:43.682 AUDIENCE: Isn't the limit k goes to k prime? 01:30:43.682 --> 01:30:45.640 And there, you assume that k plus k prime is 0. 01:30:45.640 --> 01:30:48.447 So it means both k and k prime should be 0, no? 01:30:48.447 --> 01:30:49.280 HONG LIU: I'm sorry? 01:30:49.280 --> 01:30:49.779 No. 01:30:49.779 --> 01:30:50.800 k plus k prime is not-- 01:30:50.800 --> 01:30:51.674 AUDIENCE: [INAUDIBLE] 01:30:51.674 --> 01:30:53.510 HONG LIU: No, this is a spatial vector. 01:30:53.510 --> 01:30:54.385 AUDIENCE: [INAUDIBLE] 01:30:54.385 --> 01:30:55.970 HONG LIU: No, this is spatial vector. 01:30:55.970 --> 01:30:58.290 No, this is spatial vector. 01:30:58.290 --> 01:30:59.630 Yeah. 01:30:59.630 --> 01:31:00.400 Yeah. 01:31:00.400 --> 01:31:01.358 This is spatial vector. 01:31:03.581 --> 01:31:04.080 OK. 01:31:04.080 --> 01:31:07.400 Sorry, I'm a little bit late. 01:31:07.400 --> 01:31:08.000 Yeah. 01:31:08.000 --> 01:31:10.150 That's all for today.