1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,050 Your support will help MIT OpenCourseWare 4 00:00:06,050 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,305 at ocw.mit.edu. 8 00:00:22,780 --> 00:00:26,230 PROFESSOR: Start by reminding you 9 00:00:26,230 --> 00:00:29,710 of what we did in last lecture. 10 00:00:29,710 --> 00:00:37,260 And, in particular, there was a little bit of confusion. 11 00:00:37,260 --> 00:00:41,330 It seems like, maybe some part, I went too fast. 12 00:00:41,330 --> 00:00:48,125 So we'll try to-- yeah. 13 00:00:48,125 --> 00:00:56,540 [AUDIO OUT] 14 00:00:56,540 --> 00:01:03,452 So again, we start, say, by choosing the worldsheet metric 15 00:01:03,452 --> 00:01:04,285 to be the Minkowski. 16 00:01:09,700 --> 00:01:13,040 Then the worldsheet action just reduce 17 00:01:13,040 --> 00:01:16,490 to that of a bunch of free scalar fields. 18 00:01:22,740 --> 00:01:26,780 Just reduced to a bunch of free scalar fields. 19 00:01:26,780 --> 00:01:29,250 And those free scalar field is a bit unusual, 20 00:01:29,250 --> 00:01:33,310 because, in particular, there's a 0 component, which we have 21 00:01:33,310 --> 00:01:37,090 a long sine kinetic term, OK? 22 00:01:37,090 --> 00:01:40,210 But, of course, this is not our full story. 23 00:01:40,210 --> 00:01:46,892 You still have to impose the so-called Virosoro constraint, 24 00:01:46,892 --> 00:01:49,100 because [INAUDIBLE], the stress tensor of this series 25 00:01:49,100 --> 00:01:53,645 should be 0, which is the consequence of the equation 26 00:01:53,645 --> 00:01:55,096 of motion for gamma. 27 00:01:58,597 --> 00:02:01,180 But nevertheless, you can write down the most general solution 28 00:02:01,180 --> 00:02:05,560 to this problem, so let me write it here. 29 00:02:05,560 --> 00:02:10,115 So the most general of a closed string, 30 00:02:10,115 --> 00:02:12,080 or the most general conclusion can be 31 00:02:12,080 --> 00:02:16,925 written as x mu plus p mu tau. 32 00:02:45,340 --> 00:02:49,830 So I've now written, what I called previous 33 00:02:49,830 --> 00:02:52,610 by Xl and the Xr, I have written them explicitly 34 00:02:52,610 --> 00:02:55,130 in terms of the Fourier modes, OK? 35 00:02:55,130 --> 00:02:56,630 And I have written them, explicitly, 36 00:02:56,630 --> 00:02:58,480 in terms of Fourier modes. 37 00:02:58,480 --> 00:03:03,120 And in this form, so you can compare with what we discovered 38 00:03:03,120 --> 00:03:05,970 before, previous, here, because v mu, 39 00:03:05,970 --> 00:03:09,470 then we discussed last time, that this v mu 40 00:03:09,470 --> 00:03:13,120 should be considered as related to the center of mass, 41 00:03:13,120 --> 00:03:15,294 momentum, over the whole string. 42 00:03:15,294 --> 00:03:17,835 And, for example, for the closed string, that's the relation. 43 00:03:22,760 --> 00:03:32,050 OK, so similarly, for the open string, 44 00:03:32,050 --> 00:03:36,070 if you use the Neumann boundary condition, here, 45 00:03:36,070 --> 00:03:37,940 use the Neumann boundary condition, 46 00:03:37,940 --> 00:03:42,575 then you find, OK, now we'll substitute 47 00:03:42,575 --> 00:03:49,980 the explicit expression for the Xl and the Xr. 48 00:03:49,980 --> 00:03:53,130 So now, it become 2 alpha prime p mu tau. 49 00:03:53,130 --> 00:03:56,200 So, again, this is our previous v mu. 50 00:03:56,200 --> 00:03:59,930 So this 2 is related to the open string. 51 00:03:59,930 --> 00:04:04,640 Sigma only go from 0 to pi, rather than 2 pi. 52 00:04:04,640 --> 00:04:09,400 And then, you can write the oscillator, 53 00:04:09,400 --> 00:04:16,399 this Xl Xr in terms of the explicit Fourier transform. 54 00:04:16,399 --> 00:04:18,150 So here, you only have one set of modes. 55 00:04:25,090 --> 00:04:26,800 And of course, n sigma just come from you 56 00:04:26,800 --> 00:04:31,120 have Xl plus Xr, and Xl, which is equal to Xr. 57 00:04:31,120 --> 00:04:34,170 And yeah, when these two become the same, 58 00:04:34,170 --> 00:04:37,650 when these two become the same, then these two combine. 59 00:04:37,650 --> 00:04:39,300 Because the sigma have opposite sign 60 00:04:39,300 --> 00:04:40,810 that combine to cause n sigma. 61 00:04:40,810 --> 00:04:41,789 OK? 62 00:04:41,789 --> 00:04:42,830 Yeah, it cause a m sigma. 63 00:04:50,050 --> 00:04:53,755 So this is the most general classical solution, 64 00:04:53,755 --> 00:05:09,610 and in the light-cone gauge, we say x plus can be setting to 0. 65 00:05:09,610 --> 00:05:16,240 And also everything related to alpha plus, and for tilde plus, 66 00:05:16,240 --> 00:05:18,965 all the oscillation modes also said to be 0. 67 00:05:18,965 --> 00:05:21,149 We raise it to the plus, set it to 0, 68 00:05:21,149 --> 00:05:22,690 so there's only [? tau ?] [INAUDIBLE] 69 00:05:22,690 --> 00:05:24,370 left in the light-cone gauge. 70 00:05:27,080 --> 00:05:36,980 And Virasoro constraint, they become-- I have tau x minus. 71 00:05:43,650 --> 00:05:47,460 So this v plus is the same as that, 72 00:05:47,460 --> 00:05:49,000 related to the p plus that way. 73 00:06:05,250 --> 00:06:06,820 only single v plus. 74 00:06:12,170 --> 00:06:15,120 OK, so you know the equation, so the i 75 00:06:15,120 --> 00:06:16,960 should be considered as a sum, OK? 76 00:06:16,960 --> 00:06:19,760 A sum of all directions, or transpose directions. 77 00:06:22,556 --> 00:06:23,930 So, from the Virasoro constraint, 78 00:06:23,930 --> 00:06:25,400 you can deduce X minus. 79 00:06:29,080 --> 00:06:31,600 You can deduce X minus. 80 00:06:31,600 --> 00:06:34,470 Yeah, again, let me just write here, our convention's 81 00:06:34,470 --> 00:06:44,530 always X mu is equal to X plus, X minus, then Xi. 82 00:06:44,530 --> 00:06:51,310 And i goes from 2 to D minus 1, and X plus minus 83 00:06:51,310 --> 00:06:58,540 is, say, 1 over square root 2, X0 plus minus X1, OK? 84 00:07:05,190 --> 00:07:09,390 So, from here, you can deduce the X minus. 85 00:07:09,390 --> 00:07:15,250 And also, it means that independent variables-- 86 00:07:15,250 --> 00:07:18,770 so this, we have determined the X minus up to a constant, 87 00:07:18,770 --> 00:07:21,540 because this determines the tau and the sigma derivative 88 00:07:21,540 --> 00:07:23,210 up to a constant. 89 00:07:23,210 --> 00:07:24,460 So, the independent variables. 90 00:07:29,200 --> 00:07:35,790 Xi, and then, also, this p plus, or v 91 00:07:35,790 --> 00:07:41,710 plus which appears in the plus, and then the X minus. 92 00:07:41,710 --> 00:07:44,510 The 0 modes for the X minus, which is not 93 00:07:44,510 --> 00:07:46,370 determined by those things. 94 00:07:46,370 --> 00:07:49,972 And these two, this is just a constant. 95 00:07:49,972 --> 00:07:51,430 And, again, these two are constant. 96 00:07:51,430 --> 00:07:54,520 These are two-dimensional fields, OK? 97 00:08:07,030 --> 00:08:08,140 Any questions so far? 98 00:08:12,070 --> 00:08:12,570 Good. 99 00:08:15,390 --> 00:08:18,530 So actually, [INAUDIBLE] that the zero mode part of this 100 00:08:18,530 --> 00:08:19,925 equation particularly important. 101 00:08:22,520 --> 00:08:28,090 Yeah, let me call this equation 1, this equation 2. 102 00:08:30,890 --> 00:08:33,394 So the 0 part of those equations are particularly important. 103 00:08:36,700 --> 00:08:42,419 And, for example, say, from equation 1, 104 00:08:42,419 --> 00:08:46,000 so the zero modes part, for the first equation-- 105 00:08:46,000 --> 00:08:53,440 so let's do it for closed string-- then zero mode part, 106 00:08:53,440 --> 00:08:57,990 you just-- alpha prime p minus, OK, 107 00:08:57,990 --> 00:08:59,620 you should take the root of alpha tau. 108 00:08:59,620 --> 00:09:02,390 So you just get the alpha prime p minus, 109 00:09:02,390 --> 00:09:05,720 and the right-hand side. 110 00:09:05,720 --> 00:09:08,460 So, right-hand side, let me also rewrite the v 111 00:09:08,460 --> 00:09:14,610 plus in terms of a prime, so this is 2 alpha prime p plus. 112 00:09:14,610 --> 00:09:22,940 And then, the zero mode means that you integrate over, 113 00:09:22,940 --> 00:09:25,075 you integrate over the string. 114 00:09:34,690 --> 00:09:35,190 OK. 115 00:09:38,650 --> 00:09:42,230 So this is the first equation. 116 00:09:42,230 --> 00:09:44,190 So the zero mode equation become like that. 117 00:09:47,850 --> 00:09:50,720 So, now, let me just make a brief comment, which 118 00:09:50,720 --> 00:09:57,230 I, at the beginning, I forgot to mention, 119 00:09:57,230 --> 00:10:01,515 last time, which apparently causes some confusion later. 120 00:10:01,515 --> 00:10:08,530 It's that, if you look at this expression, 121 00:10:08,530 --> 00:10:11,650 this is actually precisely-- so if you 122 00:10:11,650 --> 00:10:14,040 look at that two-dimensional field theory, 123 00:10:14,040 --> 00:10:17,380 this is actually precisely the Hamiltonian, 124 00:10:17,380 --> 00:10:20,990 the classical Hamiltonian for that field theory, OK? 125 00:10:20,990 --> 00:10:23,180 For the Xi part. 126 00:10:23,180 --> 00:10:26,770 This is just a free scalar field theory. 127 00:10:26,770 --> 00:10:29,960 Yeah, so, in particular-- so let me write that more precise. 128 00:10:29,960 --> 00:10:33,200 Let's take out this p plus, and then this 129 00:10:33,200 --> 00:10:35,890 become 4 pi alpha prime, and then 130 00:10:35,890 --> 00:10:39,030 this just become exactly the Hamiltonian of that theory, 131 00:10:39,030 --> 00:10:42,980 because 1 and 2 for the Xi, OK? 132 00:10:42,980 --> 00:11:03,630 So H Xi is the Hamiltonian for two-dimensional quantum fields. 133 00:11:03,630 --> 00:11:12,530 Quantum field theory of Xi, for the transverse directions, OK? 134 00:11:16,630 --> 00:11:18,890 So you can also write these explicitly 135 00:11:18,890 --> 00:11:21,420 in terms of those modes. 136 00:11:21,420 --> 00:11:28,305 In terms of those modes, then, for example, you write p minus. 137 00:11:28,305 --> 00:11:32,610 Yeah, if you can also write modes, then become p minus. 138 00:11:32,610 --> 00:11:40,320 You could, too, 2p plus, pi square, 139 00:11:40,320 --> 00:11:43,420 plus 1 over alpha prime. 140 00:11:43,420 --> 00:11:47,530 So if you just substitute those expansion into here, 141 00:11:47,530 --> 00:11:49,910 then you can also write it explicitly in terms of modes. 142 00:12:05,820 --> 00:12:06,320 OK. 143 00:12:13,810 --> 00:12:17,885 And then you can combine-- so this is p minus, 144 00:12:17,885 --> 00:12:19,760 then you can multiply this to the other side, 145 00:12:19,760 --> 00:12:22,580 combine all the p together. 146 00:12:22,580 --> 00:12:28,910 You can write it as M squared, which is defined to be p mu. 147 00:12:28,910 --> 00:12:33,070 P mu minus P mu P mu, which is then 2P 148 00:12:33,070 --> 00:12:36,000 plus P minus minus Pi squared. 149 00:12:38,670 --> 00:12:49,570 And then, this then become equal to 1 over alpha prime sum m 150 00:12:49,570 --> 00:12:56,720 mu equal to 0 alpha minus m i alpha m i plus alpha 151 00:12:56,720 --> 00:13:02,086 tilde minus m i alpha tilde m i, OK? 152 00:13:07,510 --> 00:13:11,140 Then, you see that this constraint 153 00:13:11,140 --> 00:13:15,820 for p prime, for p minus, now can be written, 154 00:13:15,820 --> 00:13:21,350 can be rewritten in terms of the relation 155 00:13:21,350 --> 00:13:25,160 of the mass of the whole string in terms of its oscillation 156 00:13:25,160 --> 00:13:27,640 modes, OK? 157 00:13:27,640 --> 00:13:30,830 In terms of its oscillation modes. 158 00:13:30,830 --> 00:13:33,200 And, similarity, for the open string-- so, 159 00:13:33,200 --> 00:13:36,790 this is for the closed string-- for the open, 160 00:13:36,790 --> 00:13:39,250 you could act the same thing applies, just you only have, 161 00:13:39,250 --> 00:13:41,285 now, one side of modes. 162 00:13:51,070 --> 00:13:55,876 So remember, in those expressions, sum over i 163 00:13:55,876 --> 00:13:56,665 is always assumed. 164 00:14:03,190 --> 00:14:06,160 And whenever I wrote m not equal to 0, 165 00:14:06,160 --> 00:14:09,070 it means you always sum for minus infinity plus infinity, 166 00:14:09,070 --> 00:14:11,120 and except where m equal to 0, OK? 167 00:14:18,170 --> 00:14:19,427 Yes? 168 00:14:19,427 --> 00:14:20,010 Any questions? 169 00:14:22,990 --> 00:14:27,320 So these are the consequence of the zero modes for 1. 170 00:14:27,320 --> 00:14:31,800 And the consequence for the zero modes for 2, 171 00:14:31,800 --> 00:14:34,830 again, you can just integrate over 172 00:14:34,830 --> 00:14:38,070 all direction of the string. 173 00:14:38,070 --> 00:14:46,030 Then the left-hand side just get 0, 174 00:14:46,030 --> 00:14:49,740 because the x minus is a periodic function, 175 00:14:49,740 --> 00:14:51,650 so this is a total derivative, and so 176 00:14:51,650 --> 00:14:55,500 this is give you exactly 0. 177 00:14:55,500 --> 00:15:09,865 And then, for the constraint, on the right-hand side, 178 00:15:09,865 --> 00:15:14,230 on this expression, which can be written explicitly. 179 00:15:14,230 --> 00:15:17,730 So you can also, then, plus into the explicit mode, 180 00:15:17,730 --> 00:15:24,876 and then they become sum m not equal to 0, alpha minus mi 181 00:15:24,876 --> 00:15:32,845 alpha mi equal to [INAUDIBLE]. 182 00:15:35,820 --> 00:15:41,440 OK, so this is sometimes called a level matching condition. 183 00:15:41,440 --> 00:15:43,070 So this tells you that the oscillation 184 00:15:43,070 --> 00:15:47,020 from the left-moving part-- this is for the closed string. 185 00:15:47,020 --> 00:15:48,870 For the open string, this equation just 186 00:15:48,870 --> 00:15:50,010 does not give you anything. 187 00:15:50,010 --> 00:15:51,390 For the closed string, this gives you 188 00:15:51,390 --> 00:15:52,431 a non-trivial constraint. 189 00:15:54,902 --> 00:15:55,610 Oh, I got a ring. 190 00:16:00,880 --> 00:16:02,575 So, for a closed string, this just 191 00:16:02,575 --> 00:16:04,950 tells you that the left-moving part and right-moving part 192 00:16:04,950 --> 00:16:06,680 have to be balanced. 193 00:16:06,680 --> 00:16:10,800 And this is related to that string. 194 00:16:10,800 --> 00:16:12,680 It's periodic. 195 00:16:12,680 --> 00:16:14,590 Along the string, it's periodic, so there's 196 00:16:14,590 --> 00:16:16,410 no special point on the string. 197 00:16:16,410 --> 00:16:19,252 And then, there's no special, and then you can actually not 198 00:16:19,252 --> 00:16:20,710 distinguish between the left-moving 199 00:16:20,710 --> 00:16:22,160 and the right-moving part, OK? 200 00:16:27,320 --> 00:16:28,080 Good. 201 00:16:28,080 --> 00:16:31,205 So this is a the whole classical story. 202 00:16:33,810 --> 00:16:39,500 This is, in some sense, the complete classical story, OK? 203 00:16:39,500 --> 00:16:41,580 And then, quantum mechanically, we just 204 00:16:41,580 --> 00:16:44,745 need to quantize those guys. 205 00:16:44,745 --> 00:16:46,119 Quantize those guys. 206 00:16:46,119 --> 00:16:48,285 And those are just ordinary quantum mechanic degrees 207 00:16:48,285 --> 00:16:50,380 of freedom we don't need to worry about. 208 00:16:50,380 --> 00:16:54,340 And this just become a quantization 209 00:16:54,340 --> 00:16:57,120 of a free scalar field theory. 210 00:16:57,120 --> 00:16:57,620 OK. 211 00:17:00,310 --> 00:17:03,930 STUDENT: [INAUDIBLE] all these modes are massive? 212 00:17:03,930 --> 00:17:06,602 Or is there any conditions it can be massless? 213 00:17:06,602 --> 00:17:08,560 PROFESSOR: No, this is a massless field theory. 214 00:17:11,284 --> 00:17:13,830 We are quantizing this theory, right? 215 00:17:13,830 --> 00:17:18,035 Yeah, so this is massless scalar field theory in two-dimension. 216 00:17:18,035 --> 00:17:19,780 STUDENT: But so that mass is-- 217 00:17:19,780 --> 00:17:22,710 PROFESSOR: No, this mass is the mass of the string. 218 00:17:22,710 --> 00:17:24,450 No, this is the mass of the center 219 00:17:24,450 --> 00:17:27,660 of-- this is the total mass of the string. 220 00:17:31,240 --> 00:17:34,490 When I say the massless, this is a massless scalar field theory 221 00:17:34,490 --> 00:17:36,060 in the worldsheet. 222 00:17:36,060 --> 00:17:38,290 This is a spacetime description. 223 00:17:38,290 --> 00:17:42,480 So it's important to separate two things. 224 00:17:42,480 --> 00:17:45,420 Things happening on the worldsheet and things 225 00:17:45,420 --> 00:17:47,820 happening in spacetime. 226 00:17:47,820 --> 00:17:51,830 So this is the mass of the string viewed as object 227 00:17:51,830 --> 00:17:53,850 moving in the spacetime. 228 00:17:53,850 --> 00:17:57,700 And when I say quantizing a massless scalar field theory, 229 00:17:57,700 --> 00:18:00,640 it's to think of those, the mode. 230 00:18:00,640 --> 00:18:03,440 So Xi is saying here, describe the motion of the string. 231 00:18:03,440 --> 00:18:07,290 Think of them as a field theory on the worldsheet. 232 00:18:07,290 --> 00:18:09,121 And that's a massless field theory. 233 00:18:09,121 --> 00:18:09,620 STUDENT: OK. 234 00:18:09,620 --> 00:18:12,380 PROFESSOR: Right. 235 00:18:12,380 --> 00:18:13,030 Yes? 236 00:18:13,030 --> 00:18:15,874 STUDENT: [INAUDIBLE] definition of [INAUDIBLE]? 237 00:18:21,530 --> 00:18:24,440 PROFESSOR: It's the conserve the loss of charge for the string. 238 00:18:24,440 --> 00:18:26,781 Cause 1 into the translation. 239 00:18:26,781 --> 00:18:28,280 It's to conserve the loss of charge, 240 00:18:28,280 --> 00:18:30,980 cause 1 into the translation. 241 00:18:30,980 --> 00:18:33,470 Yeah, so that's what we derived last time. 242 00:18:33,470 --> 00:18:36,630 So be sure that this v-- but to be original, write v, 243 00:18:36,630 --> 00:18:40,180 here-- are related to the center of mass momentum 244 00:18:40,180 --> 00:18:43,500 in this particular way. 245 00:18:43,500 --> 00:18:44,374 Yes. 246 00:18:44,374 --> 00:18:48,246 STUDENT: Could you use the fact that the right [INAUDIBLE] 247 00:18:48,246 --> 00:18:49,214 looks like [INAUDIBLE]. 248 00:18:52,610 --> 00:18:53,814 PROFESSOR: Sorry, which one? 249 00:18:53,814 --> 00:18:55,688 STUDENT: The fact that this alpha [INAUDIBLE] 250 00:18:55,688 --> 00:18:57,446 P minus is actually the Hamiltonian. 251 00:18:57,446 --> 00:18:58,762 Can you use it anyway? 252 00:18:58,762 --> 00:19:00,450 Or it's just the-- 253 00:19:00,450 --> 00:19:02,470 PROFESSOR: No, this is a remark. 254 00:19:02,470 --> 00:19:04,100 And this will make them more lateral. 255 00:19:08,310 --> 00:19:11,450 For us, this is not the essential remark, 256 00:19:11,450 --> 00:19:18,730 but it will be more lateral, when we work out 257 00:19:18,730 --> 00:19:20,990 the zero-point energy. 258 00:19:20,990 --> 00:19:23,320 And when we work out zero-point energy, 259 00:19:23,320 --> 00:19:26,410 then that's a more lateral thing to consider, 260 00:19:26,410 --> 00:19:30,280 because that will actually give you a way of the computing 261 00:19:30,280 --> 00:19:31,662 zero-point energy. 262 00:19:31,662 --> 00:19:33,110 Yeah. 263 00:19:33,110 --> 00:19:35,282 Yes. 264 00:19:35,282 --> 00:19:38,060 STUDENT: So how do you-- [INAUDIBLE]. 265 00:19:38,060 --> 00:19:41,186 If you plug in this extension, does it really 266 00:19:41,186 --> 00:19:43,430 direct me [INAUDIBLE]? 267 00:19:43,430 --> 00:19:44,178 PROFESSOR: Yeah. 268 00:19:44,178 --> 00:19:44,678 [LAUGHTER] 269 00:19:44,678 --> 00:19:46,262 STUDENT: [INAUDIBLE] n times-- 270 00:19:46,262 --> 00:19:46,970 PROFESSOR: Sorry? 271 00:19:46,970 --> 00:19:50,050 STUDENT: Are you supposed to have n time [INAUDIBLE] 272 00:19:50,050 --> 00:19:53,740 times the other number of alpha? 273 00:19:53,740 --> 00:19:55,520 PROFESSOR: Sorry, what are you saying? 274 00:19:55,520 --> 00:19:57,520 STUDENT: Here, if you plug in this [INAUDIBLE]. 275 00:19:57,520 --> 00:20:00,333 And in the end, you've got m times alpha m 276 00:20:00,333 --> 00:20:00,999 PROFESSOR: Yeah. 277 00:20:00,999 --> 00:20:03,231 STUDENT: But you directly use this equation. 278 00:20:03,231 --> 00:20:03,897 PROFESSOR: Yeah. 279 00:20:03,897 --> 00:20:05,840 STUDENT: And you-- well, [INAUDIBLE]-- 280 00:20:05,840 --> 00:20:07,950 PROFESSOR: Oh. 281 00:20:07,950 --> 00:20:08,940 This is trivial to see. 282 00:20:08,940 --> 00:20:12,090 We can see it immediately here. 283 00:20:12,090 --> 00:20:14,920 First, if you take the derivative, we just get Pi, 284 00:20:14,920 --> 00:20:17,750 and then you just get that term. 285 00:20:17,750 --> 00:20:21,540 And when you take derivative here, then you cancel the m. 286 00:20:21,540 --> 00:20:23,630 And then, in order to get the zero modes, 287 00:20:23,630 --> 00:20:25,890 then the m and the minus m have to cancel. 288 00:20:25,890 --> 00:20:28,933 So the only structure that can happen is this one. 289 00:20:28,933 --> 00:20:30,660 Then, the only thing left remaining 290 00:20:30,660 --> 00:20:33,900 is to check the quotient, here, is 1. 291 00:20:33,900 --> 00:20:36,070 And that, we have to do the calculation. 292 00:20:36,070 --> 00:20:38,997 And the rest, you don't need to do the calculation. 293 00:20:38,997 --> 00:20:41,455 Yeah, the only thing to do the calculation is for this one. 294 00:20:45,770 --> 00:20:46,680 Yeah. 295 00:20:46,680 --> 00:20:49,390 STUDENT: So just kind of to bore you, 296 00:20:49,390 --> 00:20:52,120 so all we're doing is free fields 297 00:20:52,120 --> 00:20:55,542 in a two-dimensional worldsheet with the diffeomorphism 298 00:20:55,542 --> 00:20:57,495 and variance as a gauge symmetry, 299 00:20:57,495 --> 00:21:00,280 and that gauge symmetry gives us some complicated constraints, 300 00:21:00,280 --> 00:21:02,360 which we then get rid of. 301 00:21:02,360 --> 00:21:04,240 PROFESSOR: Solve the constraints, yeah. 302 00:21:04,240 --> 00:21:05,420 STUDENT: That's all there is to it. 303 00:21:05,420 --> 00:21:06,670 PROFESSOR: Yeah, that's right. 304 00:21:06,670 --> 00:21:08,690 So, as I said before, solving the constraint 305 00:21:08,690 --> 00:21:10,800 help you two things. 306 00:21:10,800 --> 00:21:13,540 First, you solve this constraint. 307 00:21:13,540 --> 00:21:16,210 And second, you get rid of this X plus and X minus. 308 00:21:18,822 --> 00:21:21,030 Because those are the dangerous field theory, and now 309 00:21:21,030 --> 00:21:22,130 we get rid of. 310 00:21:22,130 --> 00:21:26,107 And what's remaining, Xi, are good. 311 00:21:26,107 --> 00:21:27,440 Will be, hey, they're good boys. 312 00:21:27,440 --> 00:21:30,555 They're good, well-behaved field theories. 313 00:21:34,540 --> 00:21:36,336 Any other questions? 314 00:21:36,336 --> 00:21:38,510 STUDENT: Yeah, one question. 315 00:21:38,510 --> 00:21:41,430 So isn't knowing how to do, basically, string theory-- 316 00:21:41,430 --> 00:21:45,290 well, is it knowing how to deal with the Polyakov action, 317 00:21:45,290 --> 00:21:46,660 and not in the light-cone gauge? 318 00:21:46,660 --> 00:21:47,840 PROFESSOR: Oh, sure. 319 00:21:47,840 --> 00:21:49,170 But that take much longer time. 320 00:21:51,960 --> 00:21:54,830 Even in the light-cone gauge, I'm 321 00:21:54,830 --> 00:22:01,120 still trying to telling a long story short. 322 00:22:01,120 --> 00:22:01,900 STUDENT: OK. 323 00:22:01,900 --> 00:22:05,590 And so I'm trying to give you the essence. 324 00:22:05,590 --> 00:22:08,730 But make sure-- I'm trying to tell you the essence, 325 00:22:08,730 --> 00:22:11,500 but at the same time, I want you to understand. 326 00:22:11,500 --> 00:22:17,130 So make sure you ask, whatever, it's not clear. 327 00:22:17,130 --> 00:22:21,430 But, for example, in the next semesters of undergraduate, 328 00:22:21,430 --> 00:22:24,700 of string theory for undergraduate, 329 00:22:24,700 --> 00:22:28,750 you will reach this point, essentially, 330 00:22:28,750 --> 00:22:32,690 after maybe 15 lectures. 331 00:22:32,690 --> 00:22:35,740 And we've reached here maybe only two or three lectures. 332 00:22:35,740 --> 00:22:40,100 And so, I'm trying to just give you the essence, 333 00:22:40,100 --> 00:22:44,930 and to try not to give you too many technical details. 334 00:22:44,930 --> 00:22:46,600 Say, cross checks, et cetera. 335 00:22:46,600 --> 00:22:48,855 There are many cross checks you can do, et cetera. 336 00:22:48,855 --> 00:22:49,980 I'm not going those things. 337 00:22:49,980 --> 00:22:53,242 Just give you the essence. 338 00:22:53,242 --> 00:22:54,075 Any other questions? 339 00:22:57,030 --> 00:22:59,000 Good. 340 00:22:59,000 --> 00:23:01,100 OK. 341 00:23:01,100 --> 00:23:04,300 So this is the review of the classical story. 342 00:23:04,300 --> 00:23:05,820 So now, let's do the quantum story. 343 00:23:08,900 --> 00:23:13,280 Again, this is a quick review of what we did last time. 344 00:23:17,530 --> 00:23:26,060 So, quantum mechanically, all those become operators. 345 00:23:26,060 --> 00:23:28,220 All those become operators. 346 00:23:28,220 --> 00:23:29,780 And the alpha also become operators. 347 00:23:51,760 --> 00:23:58,030 OK, so this commutator, as we said last time, 348 00:23:58,030 --> 00:24:01,370 implies that 1 over square root of alpha m i 349 00:24:01,370 --> 00:24:06,090 should be considered as standard [INAUDIBLE] operators, 350 00:24:06,090 --> 00:24:08,840 for m greater than zero. 351 00:24:08,840 --> 00:24:13,420 And the mode will be [INAUDIBLE], 352 00:24:13,420 --> 00:24:19,220 should be considered as the creation operators, OK? 353 00:24:19,220 --> 00:24:20,880 Creation operators. 354 00:24:20,880 --> 00:24:26,750 And then, this is the standard of quantization, or field 355 00:24:26,750 --> 00:24:27,300 theory. 356 00:24:27,300 --> 00:24:31,640 You reduce to an infinite number of harmonic oscillators. 357 00:24:31,640 --> 00:24:37,590 And in particular, this product just 358 00:24:37,590 --> 00:24:49,830 reduced to m times this Ami dagger, Ami just become m Nmi. 359 00:24:49,830 --> 00:24:50,670 OK? 360 00:24:50,670 --> 00:24:55,732 So Nmi is the oscillator number, occupation number, say, 361 00:24:55,732 --> 00:24:56,940 for each harmonic oscillator. 362 00:25:04,550 --> 00:25:11,550 Then, the typical state of the system 363 00:25:11,550 --> 00:25:16,270 just obtained by acting those things on the vacuum. 364 00:25:16,270 --> 00:25:17,920 For example, for the open string, 365 00:25:17,920 --> 00:25:20,320 we only have one set of modes. 366 00:25:20,320 --> 00:25:23,202 So then, we would have this form. 367 00:25:23,202 --> 00:25:32,610 m1 m1 alpha minus mq, iq, mq on the vacuum. 368 00:25:32,610 --> 00:25:38,200 And the vacuum also have a quantum number, p mu, 369 00:25:38,200 --> 00:25:43,095 because those p, those p plus and the pi, 370 00:25:43,095 --> 00:25:46,140 here, are quantum operators. 371 00:25:46,140 --> 00:25:49,540 And so, we take the vacuum to be eigenstate of them. 372 00:25:49,540 --> 00:25:51,610 And they're independent of those alphas. 373 00:25:51,610 --> 00:25:56,120 So this vacuum is the only vacuum for the oscillators, 374 00:25:56,120 --> 00:26:00,210 but they are still labeled by a spacetime moment, OK? 375 00:26:00,210 --> 00:26:02,860 So labeled by the spacetime momentum of the string. 376 00:26:05,610 --> 00:26:06,740 So for the open string. 377 00:26:12,370 --> 00:26:22,570 And for the closed string, similarly, you just 378 00:26:22,570 --> 00:26:24,420 have two sets of modes. 379 00:26:24,420 --> 00:26:31,760 Minus i1 n2 alpha minus m2 i2 and 2. 380 00:26:34,950 --> 00:26:37,290 And then, also, you have the other. 381 00:26:37,290 --> 00:26:48,210 let me call it k1 j1 l1 alpha minus k2 j2 l2, et cetera. 382 00:26:48,210 --> 00:26:50,352 In the vacuum. 383 00:26:50,352 --> 00:26:52,435 So this is a typical state, for the closed string. 384 00:26:58,600 --> 00:27:04,310 But the allowed state should still satisfy this constraint. 385 00:27:04,310 --> 00:27:06,050 OK, should still satisfy this constraint. 386 00:27:08,560 --> 00:27:12,240 So this constrain can be written as, 387 00:27:12,240 --> 00:27:15,300 reach the following constraint. 388 00:27:15,300 --> 00:27:18,200 It said, the oscillator number on the left-hand side, 389 00:27:18,200 --> 00:27:22,650 for the left-moving mode, or tilde. 390 00:27:22,650 --> 00:27:25,840 And the oscillator number for the right-moving mode, 391 00:27:25,840 --> 00:27:29,090 they have to be the same. 392 00:27:29,090 --> 00:27:32,560 So let me rewrite this equation, in terms of oscillator numbers. 393 00:27:32,560 --> 00:27:35,665 So this means i sum. 394 00:27:35,665 --> 00:27:38,220 So let me, now, write them from m to infinity. 395 00:27:38,220 --> 00:27:40,176 So m Nm i. 396 00:27:55,840 --> 00:27:59,320 OK, so this just form that equation, OK? 397 00:28:01,362 --> 00:28:03,570 Yeah, so this is called the level matching condition. 398 00:28:11,570 --> 00:28:16,390 So, for closed string, you cannot just act arbitrarily 399 00:28:16,390 --> 00:28:17,595 alpha and alpha tilde. 400 00:28:20,660 --> 00:28:24,740 The number of modes, the total thing 401 00:28:24,740 --> 00:28:27,730 you act from the left-hand side, from the alpha 402 00:28:27,730 --> 00:28:29,650 and those from the tilde, they have 403 00:28:29,650 --> 00:28:33,000 to be balanced by this equation, OK? 404 00:28:33,000 --> 00:28:34,900 Which is a consequence that there's 405 00:28:34,900 --> 00:28:36,460 no spatial point on the string. 406 00:29:00,270 --> 00:29:04,750 So now, we can also rewrite those equations 407 00:29:04,750 --> 00:29:07,130 at the quantum level. 408 00:29:07,130 --> 00:29:09,230 OK, so those equations, the quantum level, 409 00:29:09,230 --> 00:29:16,970 those equation just tells you that, for those states, 410 00:29:16,970 --> 00:29:19,780 the P mu are not arbitrary. 411 00:29:19,780 --> 00:29:21,150 P mu are not arbitrary. 412 00:29:21,150 --> 00:29:24,680 P mu must satisfy this kind of constraint. 413 00:29:24,680 --> 00:29:27,340 P mu must satisfy this kind of constraint. 414 00:29:27,340 --> 00:29:29,810 And that's can, in turn, be integrated 415 00:29:29,810 --> 00:29:34,210 as the determine the mass of the string, OK? 416 00:29:34,210 --> 00:29:35,640 Determine the mass of the string. 417 00:29:35,640 --> 00:29:41,320 So the mass of the string, so then for the open, 418 00:29:41,320 --> 00:29:46,205 mass of string can be written 1 over alpha prime sum 419 00:29:46,205 --> 00:29:53,440 over i sum over m infinity m Nm i. 420 00:29:53,440 --> 00:29:58,140 Then plus some zero-point energy. 421 00:29:58,140 --> 00:30:00,020 So this is just the frequency of each mode. 422 00:30:07,620 --> 00:30:11,935 So this is just the frequency of each mode, 423 00:30:11,935 --> 00:30:14,290 a frequency of each mode. 424 00:30:14,290 --> 00:30:16,890 And then, this is the occupation number. 425 00:30:16,890 --> 00:30:22,590 So this is a standard harmonic oscillator result. 426 00:30:22,590 --> 00:30:26,480 And then, at the quantum level, of course, 427 00:30:26,480 --> 00:30:29,780 there's ordinary issue, because the alpha m 428 00:30:29,780 --> 00:30:31,800 and m don't commute. 429 00:30:31,800 --> 00:30:34,200 Because [INAUDIBLE] creation, [INAUDIBLE]. 430 00:30:34,200 --> 00:30:36,130 So the ordering here matters. 431 00:30:36,130 --> 00:30:37,650 So the ordering here matters. 432 00:30:40,330 --> 00:30:47,340 So, in principle, yeah. 433 00:30:47,340 --> 00:30:47,840 It matters. 434 00:30:47,840 --> 00:30:50,810 Immediately, you can write it. 435 00:30:50,810 --> 00:30:56,430 So that will give rise to this zero-point energy, 436 00:30:56,430 --> 00:30:58,510 or ordering number. 437 00:30:58,510 --> 00:31:01,950 So, similarity, for the closed string, 438 00:31:01,950 --> 00:31:07,186 then you just have two sets of modes. 439 00:31:07,186 --> 00:31:07,685 Similarly. 440 00:31:16,515 --> 00:31:17,015 that's a0. 441 00:31:21,580 --> 00:31:33,420 So a0 can be-- so this is the place this remark is useful, 442 00:31:33,420 --> 00:31:38,970 because this just come from-- essentially, 443 00:31:38,970 --> 00:31:41,250 this part, just come from, essentially, it's 444 00:31:41,250 --> 00:31:44,960 just the Hamiltonian of the Xi. 445 00:31:44,960 --> 00:31:48,880 Just the Xi viewed as a field theory on the worldsheet. 446 00:31:48,880 --> 00:31:51,130 And the field theory of Xi, essentially, 447 00:31:51,130 --> 00:31:53,620 is a bunch of harmonic oscillators. 448 00:31:53,620 --> 00:31:55,710 And for each harmonic oscillator, 449 00:31:55,710 --> 00:31:58,510 we do know what is the ordinary number. 450 00:31:58,510 --> 00:31:59,420 It's just 1/2. 451 00:31:59,420 --> 00:32:01,370 Zero-point energy, just 1/2. 452 00:32:01,370 --> 00:32:03,970 And then you just add all of them together, OK? 453 00:32:03,970 --> 00:32:05,880 Just add all of them together. 454 00:32:05,880 --> 00:32:08,810 So, for example, for the open string, 455 00:32:08,810 --> 00:32:11,520 this just become alpha D minus 2. 456 00:32:11,520 --> 00:32:14,790 D minus 2 because that D minus 2 derive directions. 457 00:32:14,790 --> 00:32:23,040 And then, your sum m equal to infinity 1/2 omega. 458 00:32:23,040 --> 00:32:27,210 OK, 1/2 omega, and omega is m. 459 00:32:27,210 --> 00:32:30,060 Then, I told you this beautiful trick last time, 460 00:32:30,060 --> 00:32:36,160 that this should be equal to D minus 2 divided by 24 1 461 00:32:36,160 --> 00:32:38,320 over alpha prime, because this guy, 462 00:32:38,320 --> 00:32:44,800 the sum over m 1 to infinity m, give you minus 1/12. 463 00:32:44,800 --> 00:32:45,990 OK, so this is for the open. 464 00:32:48,750 --> 00:32:50,890 So, similarly, for the closed string, 465 00:32:50,890 --> 00:32:52,860 you can do the similar thing. 466 00:32:52,860 --> 00:32:55,560 Just differ by sum 2, et cetera. 467 00:32:55,560 --> 00:33:02,670 So this give you D minus 2 24 4 divided by alpha prime. 468 00:33:02,670 --> 00:33:04,581 Closed. 469 00:33:04,581 --> 00:33:05,080 OK? 470 00:33:09,910 --> 00:33:16,230 So this vacuum energy-- So this can also 471 00:33:16,230 --> 00:33:19,180 be integrated as a vacuum energy on the circle. 472 00:33:19,180 --> 00:33:21,774 Vacuum energy of this quantum field theory on the circle. 473 00:33:21,774 --> 00:33:23,190 And, in quantum field theory, this 474 00:33:23,190 --> 00:33:25,890 is sometimes called the Casmir energy. 475 00:33:25,890 --> 00:33:30,070 And you can check yourself, that those answers agree 476 00:33:30,070 --> 00:33:34,140 with the standard expression for the Casmir energy 477 00:33:34,140 --> 00:33:35,572 on the circle. 478 00:33:35,572 --> 00:33:37,530 Yeah, if you choose here, [INAUDIBLE] property, 479 00:33:37,530 --> 00:33:40,750 because we have chosen the sides to be 2 pi. 480 00:33:46,540 --> 00:33:47,300 Good. 481 00:33:47,300 --> 00:33:55,640 So this summarized what we did, summarized what we did so far. 482 00:33:58,480 --> 00:33:59,560 Any questions on this? 483 00:34:09,699 --> 00:34:12,150 Good, no more questions? 484 00:34:12,150 --> 00:34:15,670 Everything is crystal clear? 485 00:34:15,670 --> 00:34:17,160 I should immediately have a quiz. 486 00:34:19,699 --> 00:34:20,489 Yes? 487 00:34:20,489 --> 00:34:26,210 STUDENT: So the 26 dimensions comes from making this 0? 488 00:34:26,210 --> 00:34:27,850 PROFESSOR: No. 489 00:34:27,850 --> 00:34:31,480 No, if you put the 26 here, this is not 0. 490 00:34:31,480 --> 00:34:32,505 STUDENT: I'm sorry. 491 00:34:32,505 --> 00:34:35,516 I'm thinking 1 over-- I apologize. 492 00:34:35,516 --> 00:34:36,389 Not 0. 493 00:34:36,389 --> 00:34:39,630 So, you said, the 26 dimensions comes from the fact 494 00:34:39,630 --> 00:34:42,260 that, somehow, 26 minus 2 over 24 is 1. 495 00:34:42,260 --> 00:34:44,010 It's like it's numerology, but it's like-- 496 00:34:44,010 --> 00:34:45,520 PROFESSOR: Yeah. 497 00:34:45,520 --> 00:34:48,870 Yeah, that's a very good observation. 498 00:34:48,870 --> 00:34:52,520 That's exactly the reason I write in this way. 499 00:34:52,520 --> 00:34:53,159 Right. 500 00:34:53,159 --> 00:34:55,699 So let me just make a comment. 501 00:34:55,699 --> 00:34:56,969 Maybe I make a comment later. 502 00:35:00,120 --> 00:35:08,060 Anyway, so from here, from these two expression, 503 00:35:08,060 --> 00:35:12,570 from this expression and this expression, 504 00:35:12,570 --> 00:35:14,760 you see this picture, which I said at the beginning. 505 00:35:18,190 --> 00:35:25,660 So each of these describe a state of a string. 506 00:35:25,660 --> 00:35:29,930 And the state of a string, the state of such a string, they 507 00:35:29,930 --> 00:35:31,900 oscillate in this particular way. 508 00:35:35,500 --> 00:35:39,000 Say they have those oscillation modes, 509 00:35:39,000 --> 00:35:42,250 and then it moves in spacetime, besides your center 510 00:35:42,250 --> 00:35:46,390 of mass momentum, besides center of mass momentum. 511 00:35:46,390 --> 00:35:52,600 So such object, you look at it from afar, 512 00:35:52,600 --> 00:35:55,070 it's just like a particle, OK? 513 00:35:55,070 --> 00:35:57,380 It's just like a particle. 514 00:35:57,380 --> 00:35:59,110 So we have established it. 515 00:35:59,110 --> 00:36:01,540 So now, let's look at the spectrum. 516 00:36:09,960 --> 00:36:20,250 So each state of a string can be considered a map 517 00:36:20,250 --> 00:36:21,570 to a spacetime particle. 518 00:36:30,520 --> 00:36:33,580 So let's now work out. 519 00:36:33,580 --> 00:36:37,370 And the mass of the particle can be worked out 520 00:36:37,370 --> 00:36:39,400 by those formulas. 521 00:36:39,400 --> 00:36:42,020 OK, so let's now just work out what 522 00:36:42,020 --> 00:36:44,840 are the mightiest particles, because we 523 00:36:44,840 --> 00:36:50,130 are interested in the mightiest particles, typically, OK? 524 00:36:50,130 --> 00:36:53,760 So now, let's start from the beginning. 525 00:36:53,760 --> 00:36:55,830 So now, let's start with open string. 526 00:36:59,250 --> 00:37:02,610 So the lowest mode, of course, is just the vacuum. 527 00:37:05,410 --> 00:37:07,410 P mu. 528 00:37:07,410 --> 00:37:08,565 OK, there's no oscillators. 529 00:37:11,120 --> 00:37:19,285 So, for such a mode, Nm i just equal to 0 for all m and i, OK? 530 00:37:24,050 --> 00:37:26,439 So this should be just a spacetime scalar. 531 00:37:26,439 --> 00:37:28,230 So this should describe a spacetime scalar, 532 00:37:28,230 --> 00:37:32,680 because there's no other quantum number, other 533 00:37:32,680 --> 00:37:33,980 than the momentum. 534 00:37:33,980 --> 00:37:36,650 So it should be a spacetime scalar. 535 00:37:36,650 --> 00:37:40,230 It should describe a scalar particle. 536 00:37:44,490 --> 00:37:46,925 And the mass of the particle, we can just read from here. 537 00:37:50,800 --> 00:37:55,680 So the M square equal to minus-- so this is for the open string, 538 00:37:55,680 --> 00:37:57,584 so we use this formula. 539 00:37:57,584 --> 00:37:58,500 That's the only thing. 540 00:37:58,500 --> 00:38:00,390 Because this here is 0. 541 00:38:00,390 --> 00:38:03,510 So the only thing come from a 0 term. 542 00:38:03,510 --> 00:38:11,650 So you're just given by D minus 2 divided by 24 1 543 00:38:11,650 --> 00:38:12,688 over alpha prime. 544 00:38:17,680 --> 00:38:20,350 So why you need anything in lower case 545 00:38:20,350 --> 00:38:26,030 is that this guy is smaller than 0 for D greater than 2. 546 00:38:26,030 --> 00:38:29,390 Say, for any spacetime dimension greater than 2, 547 00:38:29,390 --> 00:38:31,210 you actually find the mass square, 548 00:38:31,210 --> 00:38:32,210 [INAUDIBLE] mass square. 549 00:38:34,890 --> 00:38:38,660 So people actually gave a fancy name for such kind of particle. 550 00:38:38,660 --> 00:38:40,830 They call it tachyon. 551 00:38:40,830 --> 00:38:43,310 And, in the '60s, actually, people 552 00:38:43,310 --> 00:38:46,940 designed an experiment to look for such particles, particles 553 00:38:46,940 --> 00:38:51,150 of negative mass, negative mass square. 554 00:38:51,150 --> 00:39:00,515 Anyway, we are not going to here. 555 00:39:05,210 --> 00:39:10,810 Let me just say, for the following, 556 00:39:10,810 --> 00:39:15,800 in the theory, if you see excitations, 557 00:39:15,800 --> 00:39:18,665 if it's a negative mass square, typically, it 558 00:39:18,665 --> 00:39:22,310 tells you that the system have instability, that you're not 559 00:39:22,310 --> 00:39:24,580 in the lowest energy state. 560 00:39:24,580 --> 00:39:27,180 That you are not in a low-energy state. 561 00:39:27,180 --> 00:39:32,570 So what this tells you is that this open string propagate 562 00:39:32,570 --> 00:39:36,240 in the flat Minkowski spacetime may not 563 00:39:36,240 --> 00:39:40,720 be the lowest configuration of the string, but that's is OK. 564 00:39:40,720 --> 00:39:44,120 If you're not in the lowest configuration, 565 00:39:44,120 --> 00:39:46,899 it's not a big deal, and it just means you have not 566 00:39:46,899 --> 00:39:48,190 found the correct ground state. 567 00:39:48,190 --> 00:39:52,240 It does not mean the theory is inconsistent, OK? 568 00:39:52,240 --> 00:39:56,130 And so, even though this is unpleasant, 569 00:39:56,130 --> 00:39:58,361 this thing is tolerable, OK? 570 00:39:58,361 --> 00:39:59,360 This thing is tolerable. 571 00:40:02,610 --> 00:40:05,430 Any problem with this? 572 00:40:05,430 --> 00:40:06,350 STUDENT: [INAUDIBLE]. 573 00:40:06,350 --> 00:40:09,470 You already set [INAUDIBLE] m equal to 0, 574 00:40:09,470 --> 00:40:11,790 so that is the ground state. 575 00:40:11,790 --> 00:40:14,145 PROFESSOR: No, this is a ground state on the worldsheet. 576 00:40:16,830 --> 00:40:20,630 But in the spacetime, this goes one into a particle. 577 00:40:20,630 --> 00:40:24,450 This goes one into excitation in the spacetime. 578 00:40:24,450 --> 00:40:26,795 And so this goes one into excitation in the spacetime, 579 00:40:26,795 --> 00:40:28,987 with an actual mass square. 580 00:40:28,987 --> 00:40:31,320 And, typically, if you have something with a actual mass 581 00:40:31,320 --> 00:40:32,986 square-- let me just say one more words, 582 00:40:32,986 --> 00:40:37,050 here-- then that means you are sitting on the top of a hill, 583 00:40:37,050 --> 00:40:40,150 and that's where you have a actual mass square. 584 00:40:40,150 --> 00:40:44,420 And so, it means you are in some kind of unstable state. 585 00:40:44,420 --> 00:40:48,030 But, of course, you are allowed to sit on the top of a hill. 586 00:40:48,030 --> 00:40:49,080 It's not a big deal. 587 00:40:49,080 --> 00:40:51,802 STUDENT: Why is this unstable, like a [INAUDIBLE]. 588 00:40:54,530 --> 00:40:56,590 PROFESSOR: On the top of the hill, is it stable? 589 00:40:56,590 --> 00:41:00,730 STUDENT: No, why [INAUDIBLE] means [? uncomplicated. ?] 590 00:41:00,730 --> 00:41:04,770 PROFESSOR: Oh, if you write a scalar field theory. 591 00:41:04,770 --> 00:41:08,360 Yeah, so this goes one into a scalar field in spacetime, 592 00:41:08,360 --> 00:41:09,580 now, OK? 593 00:41:09,580 --> 00:41:14,680 So now, if I write a scalar field theory in spacetime, 594 00:41:14,680 --> 00:41:17,510 say, let me call phi, with a actual mass squared. 595 00:41:20,200 --> 00:41:25,140 So that means, the potential for this mass square is like this. 596 00:41:25,140 --> 00:41:27,670 Then, that means that the phi wants to increase. 597 00:41:30,978 --> 00:41:33,220 STUDENT: Is that the same thing as example 4, 598 00:41:33,220 --> 00:41:33,500 with spontaneous-- 599 00:41:33,500 --> 00:41:34,750 PROFESSOR: Yeah, that's right. 600 00:41:34,750 --> 00:41:36,540 Similar to 4, [INAUDIBLE]. 601 00:41:36,540 --> 00:41:39,600 Except, here, we don't know what is the bottom. 602 00:41:39,600 --> 00:41:40,930 We are just sitting on the top. 603 00:41:44,640 --> 00:41:47,480 Anyway, so later, we will be able to find the way 604 00:41:47,480 --> 00:41:48,730 to get rid of this. 605 00:41:48,730 --> 00:41:49,370 So it's OK. 606 00:41:49,370 --> 00:41:52,870 So you don't need to worry about this, here. 607 00:41:52,870 --> 00:41:56,060 So the second mode, the second lowest mode, 608 00:41:56,060 --> 00:42:05,390 you just add alpha minus 1 on this worldsheet ground state, 609 00:42:05,390 --> 00:42:06,430 OK? 610 00:42:06,430 --> 00:42:09,750 So now, this thing is interesting. 611 00:42:09,750 --> 00:42:19,350 First, this index i, this index i is a spacetime index. 612 00:42:19,350 --> 00:42:21,670 It's a spacetime index. 613 00:42:21,670 --> 00:42:27,750 And so, this actually means, this transform 614 00:42:27,750 --> 00:42:33,720 means this state transform as a vector 615 00:42:33,720 --> 00:42:41,060 under So D minus 2 the rotation of the Xi directions. 616 00:42:41,060 --> 00:42:44,540 And, remember, in the light-cone gauge, 617 00:42:44,540 --> 00:42:46,480 the [INAUDIBLE] symmetry's broken, 618 00:42:46,480 --> 00:42:49,340 and this is the only symmetry which is manifest. 619 00:42:49,340 --> 00:42:53,550 And so, that suggests this state should be a spacetime vector, 620 00:42:53,550 --> 00:42:54,050 OK? 621 00:42:54,050 --> 00:42:55,280 Should be a spacetime vector. 622 00:43:01,812 --> 00:43:03,145 So now, let's work out its mass. 623 00:43:06,220 --> 00:43:09,730 So now, m equal to 1. 624 00:43:09,730 --> 00:43:15,640 So now, m equal to 1, and so, you just have 1, here. 625 00:43:15,640 --> 00:43:21,560 And so, then you just put the 1 here. 626 00:43:21,560 --> 00:43:32,910 So this is the alpha prime 1 minus D minus 2 divided by 24. 627 00:43:32,910 --> 00:43:38,000 So this is 26 minus D divided by alpha prime. 628 00:43:38,000 --> 00:43:45,330 OK, so now you see the 26 divided by 24 alpha prime. 629 00:43:48,600 --> 00:43:50,330 So now, you see this magic number, 26. 630 00:44:05,470 --> 00:44:15,040 So now, we emphasized before, in the light-cone gauge, 631 00:44:15,040 --> 00:44:18,040 even though only this So D minus 2 632 00:44:18,040 --> 00:44:20,070 is manifest, because you break Lorentz symmetry. 633 00:44:20,070 --> 00:44:21,445 The gauge break Lorentz symmetry. 634 00:44:24,120 --> 00:44:28,305 But your theory is still, secretly, Lorentz symmetric. 635 00:44:28,305 --> 00:44:30,710 It should still be Lorentz [INAUDIBLE], 636 00:44:30,710 --> 00:44:35,410 because the string is propagate in the flat Minkowski 637 00:44:35,410 --> 00:44:37,100 spacetime. 638 00:44:37,100 --> 00:44:40,220 That means, all your particle spectrum, 639 00:44:40,220 --> 00:44:43,840 they must fall into [? representations ?] 640 00:44:43,840 --> 00:44:47,150 of the four Lorentz group, OK? 641 00:44:47,150 --> 00:44:49,200 They must fall into the [? representations ?] 642 00:44:49,200 --> 00:44:52,080 of the four Lorentz group, even though the Lorentz symmetry 643 00:44:52,080 --> 00:44:53,200 is not manifest, here. 644 00:44:56,060 --> 00:45:00,013 And now, let us recall an important fact. 645 00:45:09,190 --> 00:45:14,550 A Lorentz vector, a vector field. 646 00:45:17,090 --> 00:45:18,340 Yeah, just a vector particle. 647 00:45:22,560 --> 00:45:28,610 In D Minkowski spacetime, D dimension 648 00:45:28,610 --> 00:45:35,650 of Minkowski spacetime, if this particle is massive, 649 00:45:35,650 --> 00:45:43,420 then have D minus 1 independent components, 650 00:45:43,420 --> 00:45:46,490 so independent modes. 651 00:45:46,490 --> 00:45:52,640 And if it's massless, then I have D 652 00:45:52,640 --> 00:45:56,030 minus 2 independent modes, OK? 653 00:45:56,030 --> 00:45:58,020 So the situation we have for many of these 654 00:45:58,020 --> 00:46:00,600 is the D equal for four, four-dimensional spacetime. 655 00:46:00,600 --> 00:46:04,090 So in four-dimensional spacetime, 656 00:46:04,090 --> 00:46:07,210 a massless vector is a photon. 657 00:46:07,210 --> 00:46:10,350 Photons have two polarizations, have two independent modes. 658 00:46:13,780 --> 00:46:15,200 But if you have a massive vector, 659 00:46:15,200 --> 00:46:19,710 then you actually have three polarizations, rather than two, 660 00:46:19,710 --> 00:46:21,450 OK? 661 00:46:21,450 --> 00:46:24,690 But now, we see a problem. 662 00:46:24,690 --> 00:46:29,050 Here, we see a vector, but this factor only 663 00:46:29,050 --> 00:46:31,190 have D minus 2 components. 664 00:46:34,750 --> 00:46:39,560 Because i-- because these are the only independent modes. 665 00:46:39,560 --> 00:46:55,765 Here, we have a vector which has only D minus 2 components. 666 00:46:59,600 --> 00:47:00,780 Independent components. 667 00:47:00,780 --> 00:47:08,200 OK, so if you compare with this list-- 668 00:47:08,200 --> 00:47:09,540 because there's nothing else. 669 00:47:09,540 --> 00:47:13,640 Because these are the only independent modes, here. 670 00:47:13,640 --> 00:47:17,050 In the last [INAUDIBLE], there's nothing else. 671 00:47:17,050 --> 00:47:22,010 So by compare with our knowledge of the Lorentz symmetry, 672 00:47:22,010 --> 00:47:25,820 we conclude the only way this particle can be mathematically 673 00:47:25,820 --> 00:47:30,190 consistent, we said, it has to be a massless particle. 674 00:47:30,190 --> 00:47:34,750 So that means M square have to be 0, OK? 675 00:47:34,750 --> 00:47:40,520 So M square has to be 0, means that D must be equal to 26. 676 00:47:45,770 --> 00:47:50,020 And we actually find massless particles. 677 00:47:50,020 --> 00:47:52,295 We actually find the photon. 678 00:47:52,295 --> 00:47:55,740 So we actually find the photon in the string excitations. 679 00:47:59,300 --> 00:48:00,090 Yeah, one second. 680 00:48:00,090 --> 00:48:01,590 Let me just finish this. 681 00:48:01,590 --> 00:48:08,225 For D not equal to 26, Lorentz symmetry is lost. 682 00:48:17,280 --> 00:48:20,030 Lorentz symmetry is lost. 683 00:48:20,030 --> 00:48:23,410 It's because that means this particle, 684 00:48:23,410 --> 00:48:27,590 where M square is not 0, no matter what, 685 00:48:27,590 --> 00:48:32,150 these states cannot fall into a [? representation ?] 686 00:48:32,150 --> 00:48:34,900 of a Lorentz group. 687 00:48:34,900 --> 00:48:37,200 And so, being said, Lorentz symmetry 688 00:48:37,200 --> 00:48:41,610 is not maintained, even the Lorentz symmetry is 689 00:48:41,610 --> 00:48:44,230 a symmetry of the classical action, 690 00:48:44,230 --> 00:48:47,490 but it's not maintained at quantum level. 691 00:48:47,490 --> 00:48:51,880 Somehow, in the quantization procedure, 692 00:48:51,880 --> 00:48:56,860 a symmetry which is in your classical theory, 693 00:48:56,860 --> 00:48:58,645 it's lost, OK? 694 00:49:02,040 --> 00:49:06,270 And this tells you that the quantization is not consistent. 695 00:49:17,720 --> 00:49:19,940 It's inconsistent. 696 00:49:19,940 --> 00:49:23,820 Because it means, whatever it is, 697 00:49:23,820 --> 00:49:28,900 if you have something propagate in Minkowski spacetime, 698 00:49:28,900 --> 00:49:31,000 is has two [INAUDIBLE] [? representations ?] 699 00:49:31,000 --> 00:49:33,910 of the Lorentz group. 700 00:49:33,910 --> 00:49:37,900 That means that, yeah, this just cannot be the right-- 701 00:49:37,900 --> 00:49:40,146 you have to go back, to redo your thing. 702 00:49:40,146 --> 00:49:42,270 This is not propagating in the Minkowski spacetime. 703 00:49:45,990 --> 00:49:51,620 So, alternatively-- so this is a conclusion 704 00:49:51,620 --> 00:49:55,430 that the D must be equal to 26, OK? 705 00:49:55,430 --> 00:49:58,560 So you can reach the same conclusion the following way. 706 00:50:01,080 --> 00:50:08,190 So, right now, so the way we did this, we said we fudge this 0. 707 00:50:08,190 --> 00:50:11,240 Yeah, we did not fudge it, but we 708 00:50:11,240 --> 00:50:16,360 did something to an infinite sum, and find a valid answer. 709 00:50:16,360 --> 00:50:19,500 Yeah, we have to do an infinite sum of positive numbers, 710 00:50:19,500 --> 00:50:22,810 and then find the active number. 711 00:50:22,810 --> 00:50:26,140 And then we find, somehow, there's something. 712 00:50:26,140 --> 00:50:29,505 D minus 2 and D minus 2 somehow missing 1. 713 00:50:29,505 --> 00:50:30,005 Anyway. 714 00:50:33,880 --> 00:50:36,690 Anyway, but this is actually a deep story. 715 00:50:36,690 --> 00:50:40,710 It's not, say, just missing 1, or something like that. 716 00:50:40,710 --> 00:50:43,680 So you can reach the same conclusion 717 00:50:43,680 --> 00:50:45,540 by doing the following. 718 00:50:45,540 --> 00:50:52,480 Say, you put here a 0 as undetermined quotient. 719 00:50:52,480 --> 00:50:57,830 And it turns out, the same 0-- yeah, so you can check. 720 00:50:57,830 --> 00:51:01,400 So here, it tells you that Lorentz symmetry is lost. 721 00:51:01,400 --> 00:51:05,120 So you can double check this conclusion as follows. 722 00:51:05,120 --> 00:51:08,340 So, remember, I said that that classical action 723 00:51:08,340 --> 00:51:10,650 is [INAUDIBLE] on the Lorentz symmetry, 724 00:51:10,650 --> 00:51:12,430 and then this conserve the loss of charge, 725 00:51:12,430 --> 00:51:15,660 because one to the Lorentz transformation. 726 00:51:15,660 --> 00:51:19,035 And those charge, they become generators 727 00:51:19,035 --> 00:51:21,010 of Lorentz symmetry at the quantum level, 728 00:51:21,010 --> 00:51:23,660 just in quantum field theory. 729 00:51:23,660 --> 00:51:29,320 And then, by consistency, those Lorentz-- 730 00:51:29,320 --> 00:51:33,320 conserve the Lorentz charges, as a quantum operator, 731 00:51:33,320 --> 00:51:37,930 they must satisfy Lorentz algebra, OK? 732 00:51:37,930 --> 00:51:41,820 Then, you can check with a general D, 733 00:51:41,820 --> 00:51:43,970 and with a general a0. 734 00:51:46,850 --> 00:51:51,350 And that Lorentz algebra is only satisfied 735 00:51:51,350 --> 00:51:54,880 in the D equal to 26 dimension, and this a0 given 736 00:51:54,880 --> 00:51:56,330 by these formulas. 737 00:51:56,330 --> 00:52:00,360 OK, so that will be a rigorous way to derive it. 738 00:52:00,360 --> 00:52:02,790 Rigorous way to derive it, but we are not doing it here, 739 00:52:02,790 --> 00:52:05,040 because that will take a little bit of time. 740 00:52:07,902 --> 00:52:11,718 STUDENT: Does that also involve the [INAUDIBLE] of the small-- 741 00:52:11,718 --> 00:52:13,625 PROFESSOR: Hm? 742 00:52:13,625 --> 00:52:19,440 STUDENT: That [INAUDIBLE] of the [INAUDIBLE], alpha mode? 743 00:52:19,440 --> 00:52:20,106 PROFESSOR: Yeah. 744 00:52:20,106 --> 00:52:20,606 Yes. 745 00:52:20,606 --> 00:52:22,137 Yeah. 746 00:52:22,137 --> 00:52:23,012 STUDENT: [INAUDIBLE]. 747 00:52:27,154 --> 00:52:28,100 PROFESSOR: Sorry? 748 00:52:28,100 --> 00:52:28,975 STUDENT: [INAUDIBLE]. 749 00:52:31,294 --> 00:52:32,210 PROFESSOR: No, no, no. 750 00:52:32,210 --> 00:52:35,835 You're just assuming some general a0, here. 751 00:52:35,835 --> 00:52:39,970 You determine this by requiring that the Lorentz 752 00:52:39,970 --> 00:52:41,100 algebra is satisfied. 753 00:52:41,100 --> 00:52:46,672 STUDENT: [INAUDIBLE], in terms of this [INAUDIBLE]. 754 00:52:46,672 --> 00:52:48,130 PROFESSOR: Yeah, in terms of alpha. 755 00:52:48,130 --> 00:52:49,332 That's right. 756 00:52:49,332 --> 00:52:51,490 STUDENT: Then, [INAUDIBLE]. 757 00:52:51,490 --> 00:52:54,850 PROFESSOR: Oh, those commutators are fine. 758 00:52:54,850 --> 00:52:57,637 Those commutators are just from standard quantization. 759 00:52:57,637 --> 00:53:02,210 STUDENT: Oh, there's [INAUDIBLE]. 760 00:53:02,210 --> 00:53:05,660 PROFESSOR: Sorry, which one over 24? 761 00:53:05,660 --> 00:53:07,580 No, forget about 1/24. 762 00:53:07,580 --> 00:53:09,430 There's no 1/24. 763 00:53:09,430 --> 00:53:12,020 There's no 1/24. 764 00:53:12,020 --> 00:53:15,850 You just write a 0, here. 765 00:53:15,850 --> 00:53:17,620 It's undetermined constant. 766 00:53:17,620 --> 00:53:19,180 And check it by consistency. 767 00:53:19,180 --> 00:53:21,420 Determined by consistency. 768 00:53:21,420 --> 00:53:23,240 STUDENT: [INAUDIBLE]. 769 00:53:23,240 --> 00:53:24,380 PROFESSOR: Yeah, no. 770 00:53:24,380 --> 00:53:25,654 Yes? 771 00:53:25,654 --> 00:53:31,920 STUDENT: [INAUDIBLE] we'll have to work [INAUDIBLE] anyway. 772 00:53:31,920 --> 00:53:36,652 So why do we worry so much about [INAUDIBLE]? 773 00:53:36,652 --> 00:53:37,360 PROFESSOR: Sorry? 774 00:53:37,360 --> 00:53:40,236 STUDENT: I mean, we'll have to work D minus 4 dimensions 775 00:53:40,236 --> 00:53:40,820 anyway. 776 00:53:40,820 --> 00:53:45,401 So then, should we just begin with, like, 777 00:53:45,401 --> 00:53:47,750 Minkowski space D is just not correct? 778 00:53:47,750 --> 00:53:49,870 PROFESSOR: Right, yeah. 779 00:53:49,870 --> 00:53:53,810 So will find, actually, this conclusion 780 00:53:53,810 --> 00:53:56,730 does not depend on the details. 781 00:53:56,730 --> 00:54:00,890 It does not depend on details. 782 00:54:00,890 --> 00:54:06,580 Yeah, so you can generalize this to more general case. 783 00:54:06,580 --> 00:54:08,920 Say, curve spacetime, et cetera. 784 00:54:08,920 --> 00:54:13,460 And the [INAUDIBLE] spacetime [INAUDIBLE]-- then, 785 00:54:13,460 --> 00:54:16,780 you'll find the same conclusion will happen. 786 00:54:16,780 --> 00:54:18,320 The same conclusion will happen. 787 00:54:18,320 --> 00:54:20,656 And then, you reduce to four dimensions, 788 00:54:20,656 --> 00:54:22,280 then you will find the fourth dimension 789 00:54:22,280 --> 00:54:24,710 a massive vector which only have two polarizations. 790 00:54:27,230 --> 00:54:28,055 Yes? 791 00:54:28,055 --> 00:54:29,955 STUDENT: Maybe related to this question. 792 00:54:29,955 --> 00:54:32,674 We have no evidence that Lorentz symmetry holds 793 00:54:32,674 --> 00:54:33,965 in the compactified dimensions. 794 00:54:33,965 --> 00:54:36,140 So why do we want to keep it there? 795 00:54:36,140 --> 00:54:38,500 PROFESSOR: We just say, doesn't matter. 796 00:54:38,500 --> 00:54:41,030 It's only a question as far as you 797 00:54:41,030 --> 00:54:43,880 have some uncompact directions. 798 00:54:43,880 --> 00:54:46,600 As far as you have some Lorentz directions, then 799 00:54:46,600 --> 00:54:49,400 this will apply. 800 00:54:49,400 --> 00:54:50,230 Yes. 801 00:54:50,230 --> 00:54:52,206 STUDENT: Kind of going back a little bit to the [INAUDIBLE] 802 00:54:52,206 --> 00:54:52,706 thing. 803 00:54:52,706 --> 00:54:56,010 How do we know for a fact that the string tension causes-- 804 00:54:56,010 --> 00:54:58,453 because, for example, I think, in QCD strings, 805 00:54:58,453 --> 00:55:01,374 isn't string tension negative? 806 00:55:01,374 --> 00:55:02,290 PROFESSOR: Not really. 807 00:55:02,290 --> 00:55:05,000 How do you define a negative string tension? 808 00:55:05,000 --> 00:55:06,170 STUDENT: A negative tension? 809 00:55:06,170 --> 00:55:07,544 I don't know. 810 00:55:07,544 --> 00:55:09,492 I just read that, the QCD string, 811 00:55:09,492 --> 00:55:11,930 they have negative string tension. 812 00:55:11,930 --> 00:55:17,559 PROFESSOR: No, I think, here-- so alpha prime is a scale. 813 00:55:17,559 --> 00:55:18,475 It's a physical scale. 814 00:55:21,830 --> 00:55:25,400 Tension is-- it's defined to be positive. 815 00:55:25,400 --> 00:55:27,050 Just by definition, it's positive. 816 00:55:27,050 --> 00:55:27,550 Yeah. 817 00:55:31,540 --> 00:55:33,600 Other questions? 818 00:55:33,600 --> 00:55:40,850 OK, so let me just say a little bit more 819 00:55:40,850 --> 00:55:52,600 regarding-- so now we have found a tachyon 820 00:55:52,600 --> 00:55:54,530 and a massless vector, and, also, we 821 00:55:54,530 --> 00:55:57,960 have fixed the spacetime dimension to be 26. 822 00:55:57,960 --> 00:56:00,140 We have fixed spacetime dimension to 26. 823 00:56:02,650 --> 00:56:08,657 So now, if you now fix 0 equal to 26, 824 00:56:08,657 --> 00:56:10,490 then the higher excitations are all massive. 825 00:56:23,530 --> 00:56:28,420 OK, so for the photon, essentially, it's this guy. 826 00:56:28,420 --> 00:56:30,300 So this guy's inactive. 827 00:56:30,300 --> 00:56:32,190 This guy cancel this guy. 828 00:56:32,190 --> 00:56:35,570 So, when you go to higher excitation, 829 00:56:35,570 --> 00:56:38,910 then this guy will dominate, and the m square 830 00:56:38,910 --> 00:56:40,470 will be all positive. 831 00:56:40,470 --> 00:56:44,530 And the scale is controlled by this 1 over alpha prime, OK? 832 00:56:44,530 --> 00:56:57,110 So will be all massive with spacing, given by, say, 1 over 833 00:56:57,110 --> 00:56:58,280 alpha prime. 834 00:56:58,280 --> 00:57:02,930 So, we said, spacing m squared given by 1 over alpha prime. 835 00:57:02,930 --> 00:57:08,120 For example, the next level would be-- so alpha 836 00:57:08,120 --> 00:57:15,090 minus 1 alpha minus 1 some i some j, or alpha minus 2 i 837 00:57:15,090 --> 00:57:18,370 acting on 0, P, OK? 838 00:57:18,370 --> 00:57:21,370 So those things acting on-- to avoid confusion, 839 00:57:21,370 --> 00:57:23,470 let me write it more clearly. 840 00:57:23,470 --> 00:57:29,290 So this acting on 0, P, and alpha minus 2 [INAUDIBLE] 841 00:57:29,290 --> 00:57:33,250 on 0, P. OK? 842 00:57:33,250 --> 00:57:37,390 So this would be like a tensor, because these have two index. 843 00:57:37,390 --> 00:57:39,860 And this, again, like a vector. 844 00:57:39,860 --> 00:57:41,769 Again, like a vector. 845 00:57:41,769 --> 00:57:44,060 So those would be, obviously, the mass square of 1 over 846 00:57:44,060 --> 00:57:44,940 alpha prime. 847 00:57:48,730 --> 00:57:52,010 And you can check that, actually, they actually fall 848 00:57:52,010 --> 00:57:54,340 into the four [INAUDIBLE] [? representations ?] 849 00:57:54,340 --> 00:57:57,570 of the Lorentz group, OK? 850 00:57:57,570 --> 00:57:59,490 For [INAUDIBLE] [? representation ?] 851 00:57:59,490 --> 00:58:02,230 of the Lorentz group. 852 00:58:02,230 --> 00:58:03,040 Yes? 853 00:58:03,040 --> 00:58:08,679 STUDENT: Where is the one with only one index [INAUDIBLE]? 854 00:58:08,679 --> 00:58:09,970 PROFESSOR: Sorry. say it again? 855 00:58:09,970 --> 00:58:12,650 STUDENT: So the one with one index, since it's massive, 856 00:58:12,650 --> 00:58:15,420 it's supposed to have P minus 1 degrees of freedom, right? 857 00:58:15,420 --> 00:58:21,330 PROFESSOR: They have-- oh. 858 00:58:21,330 --> 00:58:26,590 What's happening-- that's a very good question-- so, what's 859 00:58:26,590 --> 00:58:30,370 happening is that this should give 860 00:58:30,370 --> 00:58:33,190 a tensor [? representation, ?] but this not enough. 861 00:58:33,190 --> 00:58:37,620 And this acts together to form a tensor [? representation. ?] 862 00:58:37,620 --> 00:58:42,266 Yeah, because i only going from to 2 to D minus 1, 863 00:58:42,266 --> 00:58:43,641 so you need to add them together. 864 00:58:47,460 --> 00:58:47,960 Good. 865 00:58:50,950 --> 00:58:53,310 So just to summarize story for the open string, 866 00:58:53,310 --> 00:58:54,170 we find the tachyon. 867 00:58:54,170 --> 00:58:57,990 We find the massless vector, which can be integrated, maybe, 868 00:58:57,990 --> 00:58:58,866 as a photon. 869 00:58:58,866 --> 00:59:00,740 And then, you find lots of massive particles, 870 00:59:00,740 --> 00:59:07,020 infinite number of massive particles, OK? 871 00:59:07,020 --> 00:59:10,698 So any other questions, or do you want to have a break? 872 00:59:13,770 --> 00:59:16,031 We are a little bit out of time. 873 00:59:16,031 --> 00:59:18,030 Yeah, maybe let me give you three minutes break. 874 00:59:21,920 --> 00:59:24,790 Yeah, let's have a break, now. 875 00:59:24,790 --> 00:59:29,730 So, again, the lowest state is just the 0, P. 876 00:59:29,730 --> 00:59:31,570 And then, again, all the N is 0. 877 00:59:37,390 --> 00:59:39,210 All the N are 0. 878 00:59:39,210 --> 00:59:41,445 So we just read the answer from here. 879 00:59:44,015 --> 00:59:47,090 Then M square, for the closed string, 880 00:59:47,090 --> 00:59:50,520 you just equal to a0 for the closed string. 881 00:59:50,520 --> 01:00:00,420 So, here, is now m squared minus 4 divided by alpha prime. 882 01:00:00,420 --> 01:00:04,510 D minus 2 divide 24. 883 01:00:04,510 --> 01:00:10,630 So, again, this is tachyonic for D greater than 2, OK? 884 01:00:10,630 --> 01:00:15,570 Tachyonic for D greater than 2, and this is a scalar. 885 01:00:15,570 --> 01:00:18,785 And this is a scalar, because there's no other quantum 886 01:00:18,785 --> 01:00:19,285 number. 887 01:00:26,289 --> 01:00:28,330 Yeah, so now we are familiar with these tachyons, 888 01:00:28,330 --> 01:00:31,732 so we don't need to worry about it. 889 01:00:31,732 --> 01:00:32,815 So let's look at the next. 890 01:00:35,880 --> 01:00:39,750 So, next, naively, you may say, let's do 891 01:00:39,750 --> 01:00:48,790 this one as open string case, but this is not allowed. 892 01:00:52,380 --> 01:00:54,191 This is not allowed. 893 01:00:54,191 --> 01:00:54,690 Why? 894 01:00:54,690 --> 01:00:55,731 STUDENT: [INAUDIBLE]. 895 01:00:55,731 --> 01:00:56,730 PROFESSOR: That's right. 896 01:00:56,730 --> 01:00:59,510 It does not satisfy this condition. 897 01:00:59,510 --> 01:01:02,280 Because, this one, you only have the left-moving excitations, 898 01:01:02,280 --> 01:01:04,960 but does not have the right-moving excitations. 899 01:01:04,960 --> 01:01:07,270 You are not balanced. 900 01:01:07,270 --> 01:01:16,460 So you also need to add-- so the next one will be this guy, OK? 901 01:01:19,970 --> 01:01:22,370 So, now-- oh, here, have a j. 902 01:01:26,370 --> 01:01:29,510 Now have a j. 903 01:01:29,510 --> 01:01:37,500 So this, we'll have m squared 26 minus D divided 904 01:01:37,500 --> 01:01:40,750 by 20 alpha 4 alpha prime. 905 01:01:40,750 --> 01:01:41,940 6 alpha prime. 906 01:01:50,970 --> 01:01:58,430 Again, now, look for what representations of the Lorentz 907 01:01:58,430 --> 01:02:00,585 group will give you this. 908 01:02:03,230 --> 01:02:08,910 OK, you'll find none, unless you're in the D equal to 26, 909 01:02:08,910 --> 01:02:09,410 OK? 910 01:02:14,060 --> 01:02:21,680 The same story happens, again, only for D equal to 26, 911 01:02:21,680 --> 01:02:29,070 fall into the representations of Lorentz group. 912 01:02:35,180 --> 01:02:43,710 And reach the m squared, again, it's massless. 913 01:02:43,710 --> 01:02:44,820 m square is massless. 914 01:02:53,550 --> 01:02:56,970 So, it turns out, actually, this does not 915 01:02:56,970 --> 01:02:59,759 transform under any reducible [? representation ?] 916 01:02:59,759 --> 01:03:00,550 of a Lorentz group. 917 01:03:03,930 --> 01:03:06,040 It's actually a reducible, so it can be 918 01:03:06,040 --> 01:03:10,060 separated into several subsets. 919 01:03:10,060 --> 01:03:13,190 So this can be further decomposed to-- 920 01:03:30,760 --> 01:03:37,000 So you can take all the i and j together, 921 01:03:37,000 --> 01:03:47,820 take the same i, some i. 922 01:03:47,820 --> 01:03:52,060 So take this guy, take these two index 923 01:03:52,060 --> 01:03:56,460 to be the same, and the sum of all the directions. 924 01:03:56,460 --> 01:04:00,285 So this does not transform under the rotation 925 01:04:00,285 --> 01:04:02,290 of i's, so this is a scalar. 926 01:04:05,850 --> 01:04:07,280 But it's a massless scalar. 927 01:04:11,160 --> 01:04:17,660 And you can also have the situation-- 928 01:04:17,660 --> 01:04:20,265 so all the state are in this [? representation ?] 929 01:04:20,265 --> 01:04:24,100 of both states, so they are D minus 2. 930 01:04:24,100 --> 01:04:28,840 So there are D minus 2 times D minus 2 of them. 931 01:04:28,840 --> 01:04:31,450 D minus 2 times D minus 2 of them. 932 01:04:31,450 --> 01:04:34,250 So this, D minus 2, say, one of them 933 01:04:34,250 --> 01:04:37,960 can be decomposing to a scalar. 934 01:04:37,960 --> 01:04:43,680 And I can also take the linear slope [INAUDIBLE] of them 935 01:04:43,680 --> 01:04:45,145 with a symmetric traceless. 936 01:04:51,320 --> 01:04:54,840 So the trace part is, essentially, this scalar. 937 01:04:54,840 --> 01:04:58,460 And I can also take a traceless part. 938 01:04:58,460 --> 01:05:02,849 Traceless e i j, OK, because the trace part 939 01:05:02,849 --> 01:05:04,140 is already covered by this one. 940 01:05:04,140 --> 01:05:07,110 I don't want to repeat. 941 01:05:07,110 --> 01:05:09,810 And this is precisely the generalization, 942 01:05:09,810 --> 01:05:14,390 what we normally call the spin-2 representation, 943 01:05:14,390 --> 01:05:17,100 to general dimension. 944 01:05:17,100 --> 01:05:23,230 OK, so we have found a massless spin-2 particle. 945 01:05:23,230 --> 01:05:25,330 So this is a massless spin-2 particle. 946 01:05:31,644 --> 01:05:33,060 And then, you can also, of course, 947 01:05:33,060 --> 01:05:36,220 take it to be antisymmetric. 948 01:05:39,100 --> 01:05:40,675 So that's the only possibility, now. 949 01:05:46,390 --> 01:05:47,550 bij, antisymmetric bij. 950 01:05:57,450 --> 01:05:59,200 So these are called antisymmetric. 951 01:05:59,200 --> 01:06:03,169 So these will give rise to an antisymmetric tensor 952 01:06:03,169 --> 01:06:03,710 in spacetime. 953 01:06:08,430 --> 01:06:11,600 So this is an object with a 2 index, 954 01:06:11,600 --> 01:06:15,014 and the 2 index are antisymmetric, OK? 955 01:06:26,340 --> 01:06:35,120 So, similarly, the higher modes are all massive. 956 01:06:41,730 --> 01:06:48,090 For example, at the next level, next mass level, 957 01:06:48,090 --> 01:06:51,500 m square is equal to 4 divided by alpha prime. 958 01:07:04,610 --> 01:07:07,237 Any questions on this? 959 01:07:07,237 --> 01:07:10,231 STUDENT: This antisymmetric tensor, what is that? 960 01:07:10,231 --> 01:07:13,730 STUDENT: It's like a [INAUDIBLE]. 961 01:07:13,730 --> 01:07:15,760 PROFESSOR: It's a antisymmetric tensor. 962 01:07:15,760 --> 01:07:17,826 STUDENT: But what's the spin? 963 01:07:20,627 --> 01:07:22,710 PROFESSOR: Yeah, normally, in the fourth dimension 964 01:07:22,710 --> 01:07:25,490 will be something what we normally call 1 comma 965 01:07:25,490 --> 01:07:27,620 1, a representation of the Lorentz group. 966 01:07:30,952 --> 01:07:34,200 Yeah, it's not 0, 2. 967 01:07:34,200 --> 01:07:37,261 It's what we normally call 1 comma 1, yeah. 968 01:07:37,261 --> 01:07:37,760 Yes. 969 01:07:37,760 --> 01:07:41,220 STUDENT: It's a form of bij [INAUDIBLE]? 970 01:07:41,220 --> 01:07:44,280 PROFESSOR: No, this is just arbitrary. 971 01:07:44,280 --> 01:07:49,740 Yeah, so this is your state space, at this level, right? 972 01:07:49,740 --> 01:07:53,460 And so, the general state would be [INAUDIBLE] of them. 973 01:07:53,460 --> 01:07:57,620 And those states, they transform separately, 974 01:07:57,620 --> 01:07:58,720 on the Lorentz symmetry. 975 01:07:58,720 --> 01:08:00,390 So we separate them. 976 01:08:00,390 --> 01:08:02,180 And so, for example, symmetric strings, 977 01:08:02,180 --> 01:08:04,950 they transform separately on the Lorentz transformation 978 01:08:04,950 --> 01:08:07,310 under these guys. 979 01:08:07,310 --> 01:08:10,500 So this should cause 1 into different spacetime fields. 980 01:08:10,500 --> 01:08:15,590 So each of those things should correspond 981 01:08:15,590 --> 01:08:17,760 to a spacetime field, OK? 982 01:08:23,590 --> 01:08:25,955 So, now, let me just summarize what we have found. 983 01:08:36,620 --> 01:08:40,810 So we have found-- so let's collect 984 01:08:40,810 --> 01:08:42,810 the massless excitations we've found. 985 01:08:42,810 --> 01:08:44,851 Because, as we will see, the massless excitations 986 01:08:44,851 --> 01:08:46,100 are the most important one. 987 01:08:46,100 --> 01:08:50,069 Let me also mention, let me just emphasize. 988 01:08:53,260 --> 01:08:55,850 In physics, it's always massless particle give you 989 01:08:55,850 --> 01:08:59,370 something interesting, OK? 990 01:08:59,370 --> 01:09:02,270 For example, here, even for D not 991 01:09:02,270 --> 01:09:07,430 equal to 26, those massive particles-- as I said, 992 01:09:07,430 --> 01:09:10,390 maybe I did not emphasize-- even for D not equal to 26, 993 01:09:10,390 --> 01:09:13,319 these massive particles that do form into representations 994 01:09:13,319 --> 01:09:15,722 or Lorentz symmetry, for any dimension, 995 01:09:15,722 --> 01:09:18,762 only for those massless particles, OK? 996 01:09:18,762 --> 01:09:21,399 It almost seems funny. 997 01:09:21,399 --> 01:09:24,810 And so, of course, we also know the massless [INAUDIBLE] 998 01:09:24,810 --> 01:09:27,200 that give rise to long-range [INAUDIBLE], et cetera. 999 01:09:27,200 --> 01:09:31,070 So now, let's say, let's collect the massless particles. 1000 01:09:35,450 --> 01:09:36,324 Massless excitations. 1001 01:09:41,550 --> 01:09:45,689 So we will write them in terms of the spacetime fields. 1002 01:09:45,689 --> 01:09:51,479 So, for the open string, essentially, we 1003 01:09:51,479 --> 01:09:53,640 find the massless vector field. 1004 01:09:56,320 --> 01:09:58,280 So this is our photon. 1005 01:09:58,280 --> 01:10:00,830 So, at the moment, I put it as a, 1006 01:10:00,830 --> 01:10:04,550 quote, photon, because we only find the massless particle. 1007 01:10:04,550 --> 01:10:11,180 We don't know whether this is our own beloved photon, yet. 1008 01:10:11,180 --> 01:10:16,640 And, for the closed string, then we find a symmetric tensor. 1009 01:10:18,704 --> 01:10:20,620 So this is what we normally call the graviton. 1010 01:10:24,580 --> 01:10:26,750 So again, quoted. 1011 01:10:26,750 --> 01:10:28,980 We find the massless spin-2 particle, 1012 01:10:28,980 --> 01:10:30,610 which, if you write in terms of field, 1013 01:10:30,610 --> 01:10:35,180 would be like a symmetric tensor, 1014 01:10:35,180 --> 01:10:39,740 Or B mu mu, which is antisymmetric tensor. 1015 01:10:39,740 --> 01:10:42,220 And now, this mu mu, this all wrong in all spacetime 1016 01:10:42,220 --> 01:10:43,590 dimensions, OK? 1017 01:10:43,590 --> 01:10:45,280 And then we have a phi. 1018 01:10:45,280 --> 01:10:46,910 Then you have a scalar field. 1019 01:10:46,910 --> 01:10:49,440 So this is just called antisymmetric tensor, 1020 01:10:49,440 --> 01:10:52,930 and this phi is called a [INAUDIBLE]. 1021 01:10:52,930 --> 01:10:56,630 Phi, which is that scalar field, is often called a [INAUDIBLE]. 1022 01:11:08,470 --> 01:11:12,380 So, so far, even we call them photon, call 1023 01:11:12,380 --> 01:11:16,340 this one photon, and the h mu, they 1024 01:11:16,340 --> 01:11:20,586 are not-- to call them photon and the graviton is, actually, 1025 01:11:20,586 --> 01:11:21,960 a little bit cheating, because we 1026 01:11:21,960 --> 01:11:24,380 don't know whether they really behave like a photon 1027 01:11:24,380 --> 01:11:26,860 or like a graviton, OK? 1028 01:11:26,860 --> 01:11:30,020 We just find a massless spin-1 particle 1029 01:11:30,020 --> 01:11:32,735 and a massless spin-2 particle. 1030 01:11:32,735 --> 01:11:36,430 But, actually, there is something very general 1031 01:11:36,430 --> 01:11:39,345 one can say, just from general principle. 1032 01:11:43,990 --> 01:11:46,280 Just from general principle, one can 1033 01:11:46,280 --> 01:11:58,050 show, based on Lorentz covariance, 1034 01:11:58,050 --> 01:12:02,790 and, say, an absence of [INAUDIBLE] physical states, 1035 01:12:02,790 --> 01:12:13,870 et cetera, say [INAUDIBLE], et cetera, 1036 01:12:13,870 --> 01:12:16,430 just based on those general principle, 1037 01:12:16,430 --> 01:12:33,620 one can argue that, at the low energies, 1038 01:12:33,620 --> 01:12:50,090 that the dynamics of any massless vector field 1039 01:12:50,090 --> 01:12:55,820 should be Maxwell. 1040 01:12:59,270 --> 01:13:05,180 And, for the massless spin-2 particle, 1041 01:13:05,180 --> 01:13:06,990 must be Einstein gravity, OK? 1042 01:13:15,910 --> 01:13:20,060 So that's why, say, tomorrow, supposing 1043 01:13:20,060 --> 01:13:24,150 if you mend this theory yourself, and suppose 1044 01:13:24,150 --> 01:13:28,780 that's a quantum theory, and that theory just 1045 01:13:28,780 --> 01:13:33,220 happens to have a massive spin-2 particle, 1046 01:13:33,220 --> 01:13:36,060 then you don't have to do calculations. 1047 01:13:36,060 --> 01:13:38,830 Then you say, if my theory is consistent, 1048 01:13:38,830 --> 01:13:43,804 this spin-2 particle must behave like a graviton, OK? 1049 01:13:43,804 --> 01:13:46,012 STUDENT: Are you saying that there's no other Lorentz 1050 01:13:46,012 --> 01:13:46,928 invariant [INAUDIBLE]? 1051 01:13:49,079 --> 01:13:51,620 PROFESSOR: Yeah, essentially, you can show, the low energies, 1052 01:13:51,620 --> 01:13:54,800 it's always just based on gauge symmetry, et cetera. 1053 01:13:54,800 --> 01:13:57,460 The only thing you can have is [INAUDIBLE], 1054 01:13:57,460 --> 01:14:00,840 yeah, is Einstein gravity. 1055 01:14:00,840 --> 01:14:03,810 Good. 1056 01:14:03,810 --> 01:14:08,790 And this can be, actually, checked explicitly. 1057 01:14:08,790 --> 01:14:11,820 So now, let me erase those things, now. 1058 01:14:11,820 --> 01:14:13,305 Don't need them. 1059 01:14:18,760 --> 01:14:22,410 So this can actually be checked explicitly. 1060 01:14:22,410 --> 01:14:28,052 So, in string theory, not only can you find the spectrum, 1061 01:14:28,052 --> 01:14:30,640 you can also compute the scattering amplitude 1062 01:14:30,640 --> 01:14:33,320 among those particles. 1063 01:14:33,320 --> 01:14:37,290 OK, so I said [INAUDIBLE] before, essentially, perform 1064 01:14:37,290 --> 01:14:39,420 path integrals with some initial string 1065 01:14:39,420 --> 01:14:43,550 states going to some final string states, et cetera, OK? 1066 01:14:43,550 --> 01:14:47,530 So, of course, this will be too far for us, 1067 01:14:47,530 --> 01:14:48,710 so we will not go into that. 1068 01:14:48,710 --> 01:14:50,120 Let me just tell you the answer. 1069 01:14:53,400 --> 01:14:56,690 So you can confirm this. 1070 01:14:56,690 --> 01:15:00,040 So this expectation, this confirmed. 1071 01:15:07,130 --> 01:15:26,815 This confirmed by explicit string theory calculation 1072 01:15:26,815 --> 01:15:32,865 of scattering of these particles, OK? 1073 01:15:41,040 --> 01:15:44,240 For example, let's consider-- so we 1074 01:15:44,240 --> 01:15:48,730 have this massless spin-2 particle, which I called h. 1075 01:15:48,730 --> 01:15:52,600 So let's consider, you start with initial state with two h, 1076 01:15:52,600 --> 01:15:56,220 then scatter to get its own two final stage, which, again, h. 1077 01:15:56,220 --> 01:15:58,625 So this is, say, graviton graviton scattering. 1078 01:15:58,625 --> 01:16:03,180 Start with two graviton, scatter them. 1079 01:16:03,180 --> 01:16:06,520 So, as in string theory, we'll be, say, at the lowest order, 1080 01:16:06,520 --> 01:16:09,375 we will have a diagram like this. 1081 01:16:09,375 --> 01:16:11,490 So a nice [INAUDIBLE] order. 1082 01:16:19,600 --> 01:16:20,900 I'm not drawing very well. 1083 01:16:31,940 --> 01:16:35,080 Anyway, I hope this is clear. 1084 01:16:51,080 --> 01:16:54,320 So you start with two initial string states. 1085 01:16:54,320 --> 01:16:57,040 You scatter into some final states, OK? 1086 01:16:57,040 --> 01:16:59,660 So this is an obvious string theory scattering diagram. 1087 01:16:59,660 --> 01:17:02,400 You can, in principle, compute this using path integral, which 1088 01:17:02,400 --> 01:17:04,590 I outlined earlier. 1089 01:17:04,590 --> 01:17:06,950 Of course, we will not compute this path integral. 1090 01:17:06,950 --> 01:17:09,009 And so, you see, there are two vertex here. 1091 01:17:09,009 --> 01:17:10,300 One is proportionate to string. 1092 01:17:10,300 --> 01:17:14,680 You have two string merging to one string, 1093 01:17:14,680 --> 01:17:18,060 and then you have a string that's split into two. 1094 01:17:18,060 --> 01:17:20,430 So they're two. 1095 01:17:20,430 --> 01:17:22,605 Remember, each of [INAUDIBLE] equals 1096 01:17:22,605 --> 01:17:24,360 1 and 2 [INAUDIBLE] string, OK? 1097 01:17:27,000 --> 01:17:30,050 So this will be the process, in string theory. 1098 01:17:30,050 --> 01:17:32,810 So this will be the process in string theory. 1099 01:17:32,810 --> 01:17:39,820 So now, If you go to low energies, 1100 01:17:39,820 --> 01:17:41,520 low energies means that, if you can 1101 01:17:41,520 --> 01:17:44,170 see that the energy of the initial and the final particles 1102 01:17:44,170 --> 01:17:46,695 to be much, much smaller than 1 over alpha prime. 1103 01:17:46,695 --> 01:17:49,430 So, remember, 1 over alpha prime, 1104 01:17:49,430 --> 01:17:53,342 it's the scale which go from massless to massive particles. 1105 01:17:53,342 --> 01:17:55,550 So, if you can see the very low-energy process, which 1106 01:17:55,550 --> 01:17:59,060 E is much, much smaller than 1 over alpha prime, then 1107 01:17:59,060 --> 01:18:01,720 the contribution of the mass-- so, in some sense, 1108 01:18:01,720 --> 01:18:05,210 in the string, in this intermediate channel, when you 1109 01:18:05,210 --> 01:18:07,140 go from 2 initial state to 2 final state, 1110 01:18:07,140 --> 01:18:09,760 this intermediate channel, the infinite number 1111 01:18:09,760 --> 01:18:16,560 of string states can participate in this intermediate process. 1112 01:18:16,560 --> 01:18:19,935 But, in the process, if your energy's sufficiently low, 1113 01:18:19,935 --> 01:18:24,790 then, from your common sense, we can do a calculation. 1114 01:18:24,790 --> 01:18:27,020 And then, the contribution of the massive state 1115 01:18:27,020 --> 01:18:29,410 becomes, actually, not important. 1116 01:18:29,410 --> 01:18:32,490 So, essentially, what is important is 1117 01:18:32,490 --> 01:18:37,030 those massless particles propagating between them. 1118 01:18:37,030 --> 01:18:41,510 And then, you can show that it's actually just precisely reduced 1119 01:18:41,510 --> 01:18:44,400 to the Einstein gravity. 1120 01:18:44,400 --> 01:18:46,660 Precisely reduced to Einstein gravity. 1121 01:18:46,660 --> 01:18:52,650 So more explicitly-- so yeah, so when you go to low energies, 1122 01:18:52,650 --> 01:19:02,780 then only massless modes exchange, dominate. 1123 01:19:08,350 --> 01:19:09,410 Terminates. 1124 01:19:09,410 --> 01:19:11,660 And then, you find that the answer is precisely agreed 1125 01:19:11,660 --> 01:19:15,140 in the [INAUDIBLE] limit. 1126 01:19:15,140 --> 01:19:22,850 Agree with that from Einstein gravity. 1127 01:19:29,300 --> 01:19:32,025 Because you not only have graviton, 1128 01:19:32,025 --> 01:19:36,040 you also have this B and phi, they are also massless modes. 1129 01:19:36,040 --> 01:19:39,420 So this is a slight generalization 1130 01:19:39,420 --> 01:19:42,140 of Einstein gravity. 1131 01:19:42,140 --> 01:19:48,230 It's Einstein gravity coupled to such B and the phi, OK? 1132 01:19:52,390 --> 01:19:57,210 So, in fact, you can write down the so-called low-energy 1133 01:19:57,210 --> 01:19:58,090 effective action. 1134 01:20:01,180 --> 01:20:04,870 So-called low-energy effective action, and we call LEE, here. 1135 01:20:09,924 --> 01:20:10,840 let me see, like this. 1136 01:20:19,850 --> 01:20:25,190 So phi is this phi, here, and R is the standard reach scalar 1137 01:20:25,190 --> 01:20:28,810 for the Einstein gravity. 1138 01:20:34,000 --> 01:20:50,720 H is this reindexed tensor formed out of B. 1139 01:20:50,720 --> 01:20:57,620 So H square just a kinetic term for B. So, more precisely, 1140 01:20:57,620 --> 01:21:01,330 you can show that the scattering amplitude 1141 01:21:01,330 --> 01:21:04,205 you obtained from string theory, then you 1142 01:21:04,205 --> 01:21:11,710 take a low-energy limit, that answer precisely the same 1143 01:21:11,710 --> 01:21:15,910 as the scattering amplitude you calculated from this theory, 1144 01:21:15,910 --> 01:21:20,020 say, expanded around Minkowski spacetime, OK? 1145 01:21:20,020 --> 01:21:22,060 So this is Einstein gravity coupled 1146 01:21:22,060 --> 01:21:24,660 to [? home ?] scalar field, OK? 1147 01:21:31,451 --> 01:21:31,950 Yes? 1148 01:21:31,950 --> 01:21:33,838 STUDENT: So what about higher loops 1149 01:21:33,838 --> 01:21:37,620 or higher energies [INAUDIBLE]? 1150 01:21:37,620 --> 01:21:40,440 PROFESSOR: Of course, then, it will not be the same. 1151 01:21:40,440 --> 01:21:44,510 This is low-energies, OK? 1152 01:21:44,510 --> 01:21:46,080 So let me make one more remark. 1153 01:21:51,220 --> 01:21:53,110 Make one more remark. 1154 01:21:53,110 --> 01:22:01,580 In Einstein gravity, say, like this. 1155 01:22:01,580 --> 01:22:05,150 So Einstein gravity coupled to the matter field. 1156 01:22:05,150 --> 01:22:07,900 So, when I say Einstein gravity, I always 1157 01:22:07,900 --> 01:22:10,620 imply Einstein gravity plus some other matter 1158 01:22:10,620 --> 01:22:13,040 field which you can add. 1159 01:22:13,040 --> 01:22:16,400 So, in Einstein gravity, start your scattering process. 1160 01:22:16,400 --> 01:22:19,930 It's that lowest order, we all know, 1161 01:22:19,930 --> 01:22:23,990 is proportional to G Newton . 1162 01:22:23,990 --> 01:22:28,490 OK, so this is the same G Newton, the G newton observed. 1163 01:22:31,180 --> 01:22:37,150 Yeah, let me call this, say, scattering for this is A4. 1164 01:22:37,150 --> 01:22:41,380 And then, the scattering for the Einstein gravity 1165 01:22:41,380 --> 01:22:43,970 is proportionate to G Newton. 1166 01:22:43,970 --> 01:22:46,750 So one graviton is changed, and it's 1167 01:22:46,750 --> 01:22:48,530 proportionate to Newton constant. 1168 01:22:48,530 --> 01:22:52,610 This is an attractive force between two objects. 1169 01:22:52,610 --> 01:22:57,330 And, if you look at this string diagram, 1170 01:22:57,330 --> 01:23:00,670 then this string diagram is proportional to gs square. 1171 01:23:00,670 --> 01:23:04,910 So we conclude that the relation between the Newton constant 1172 01:23:04,910 --> 01:23:08,370 and the string coupling must be G Newton proportional to gs 1173 01:23:08,370 --> 01:23:11,722 square, up to some, say, dimensional numbers 1174 01:23:11,722 --> 01:23:12,805 or some numerical factors. 1175 01:23:16,130 --> 01:23:17,550 So this is very important relation 1176 01:23:17,550 --> 01:23:18,910 you should always keep in mind. 1177 01:23:24,540 --> 01:23:29,240 So now, here's the important point. 1178 01:23:29,240 --> 01:23:35,096 [INAUDIBLE] just asked, what happens at the loop levels? 1179 01:23:35,096 --> 01:23:36,970 So you can compare the three-level processes, 1180 01:23:36,970 --> 01:23:38,844 because, actually, find they agree very well. 1181 01:23:38,844 --> 01:23:43,000 We say, what happens at loop levels? 1182 01:23:43,000 --> 01:23:44,660 So now, let me call this equation 1. 1183 01:23:48,420 --> 01:24:01,309 So, at loop level, this 1 is notoriously divergent. 1184 01:24:01,309 --> 01:24:03,350 So if you can calculate some scattering amplitude 1185 01:24:03,350 --> 01:24:09,810 to the loop level, then find the results are divergent. 1186 01:24:09,810 --> 01:24:11,440 In particular, more and more divergent 1187 01:24:11,440 --> 01:24:17,280 when you go to more and more higher loops, OK? 1188 01:24:17,280 --> 01:24:23,092 More divergent at higher orders in [INAUDIBLE] series, say, 1189 01:24:23,092 --> 01:24:23,925 at the higher loops. 1190 01:24:28,430 --> 01:24:31,670 So that's what we normally mean, say a theory is non-realizable. 1191 01:24:41,470 --> 01:24:46,020 So this tells you-- so, if you take this gravity theory-- 1192 01:24:46,020 --> 01:24:49,290 so this is just, essentially, our Einstein gravity 1193 01:24:49,290 --> 01:24:54,200 coupled to some fields-- if you take the Einstein gravity, 1194 01:24:54,200 --> 01:24:58,090 expand the long flat space, quantize 1195 01:24:58,090 --> 01:25:03,640 that spin-2 excitations, then that will fail. 1196 01:25:03,640 --> 01:25:06,520 Because, at certain point, you don't know what you are doing, 1197 01:25:06,520 --> 01:25:10,350 because you get all divergences, which you cannot renormalize. 1198 01:25:10,350 --> 01:25:13,800 OK, you can normalize. 1199 01:25:13,800 --> 01:25:18,800 So, of course, what this tells you 1200 01:25:18,800 --> 01:25:22,090 is that this equation itself likely 1201 01:25:22,090 --> 01:25:25,710 does not describe the right UV physics. 1202 01:25:25,710 --> 01:25:27,840 So that's why you see all these divergences, 1203 01:25:27,840 --> 01:25:31,602 because you maybe lost some more important physics, 1204 01:25:31,602 --> 01:25:32,810 which you cannot renormalize. 1205 01:25:38,860 --> 01:25:43,150 But now, so this is supposed to only agree with the string 1206 01:25:43,150 --> 01:25:47,310 theory at low energies, which the maximum modes are not 1207 01:25:47,310 --> 01:25:48,260 important. 1208 01:25:48,260 --> 01:25:51,820 But, in string theory, there are this infinite number 1209 01:25:51,820 --> 01:25:54,480 if massive modes, et cetera. 1210 01:25:54,480 --> 01:25:59,940 So you find, in string theory, if you do similar loop 1211 01:25:59,940 --> 01:26:07,920 calculating string theory, the string loop diagrams, 1212 01:26:07,920 --> 01:26:10,020 magically, are all UV finite. 1213 01:26:16,000 --> 01:26:19,810 Or UV finite, so there's no such divergences. 1214 01:26:19,810 --> 01:26:22,990 There's no such divergences. 1215 01:26:22,990 --> 01:26:25,309 So this is the first hint. 1216 01:26:25,309 --> 01:26:27,100 So this was the first hint of string theory 1217 01:26:27,100 --> 01:26:30,820 as a consistent theory of gravity. 1218 01:26:30,820 --> 01:26:40,660 And because, at least, at the pertubative level, 1219 01:26:40,660 --> 01:26:45,930 you can really quantize massive spin-2 particles, 1220 01:26:45,930 --> 01:26:51,460 and to calculate their physics in the self-consistant way, OK? 1221 01:26:54,400 --> 01:26:57,450 Any questions on this? 1222 01:26:57,450 --> 01:26:58,269 Yes. 1223 01:26:58,269 --> 01:26:59,727 STUDENT: In that pertubation, you'd 1224 01:26:59,727 --> 01:27:02,760 have to use all those upper-- 1225 01:27:02,760 --> 01:27:04,900 PROFESSOR: Yeah, that's crucial. 1226 01:27:04,900 --> 01:27:07,677 So that's why this kind of thing is not good enough, 1227 01:27:07,677 --> 01:27:09,760 because that does not have enough degrees freedom. 1228 01:27:09,760 --> 01:27:11,850 So, in string theory, you have all these additional degrees 1229 01:27:11,850 --> 01:27:14,030 of freedom that make your UV structure completely 1230 01:27:14,030 --> 01:27:14,530 difference. 1231 01:27:14,530 --> 01:27:15,027 Yes. 1232 01:27:15,027 --> 01:27:16,777 STUDENT: Then if you take that [INAUDIBLE] 1233 01:27:16,777 --> 01:27:19,500 and you just [INAUDIBLE] from above on [INAUDIBLE], 1234 01:27:19,500 --> 01:27:20,991 will it make [INAUDIBLE]? 1235 01:27:24,040 --> 01:27:28,000 PROFESSOR: Yeah, so this will work, 1236 01:27:28,000 --> 01:27:32,200 as what normally works as a low-energy effective theory. 1237 01:27:32,200 --> 01:27:34,710 A low-energy [INAUDIBLE]. 1238 01:27:34,710 --> 01:27:41,310 When you consistently integrate out the massive modes. 1239 01:27:41,310 --> 01:27:45,390 And yeah, so, even at loop level, 1240 01:27:45,390 --> 01:27:48,550 this can capture, indeed, at low energies, in my loop level, 1241 01:27:48,550 --> 01:27:54,270 this can capture some of the string theory with that. 1242 01:27:54,270 --> 01:27:59,120 But you have to normalize properly, et cetera, yeah. 1243 01:27:59,120 --> 01:27:59,620 Yes. 1244 01:27:59,620 --> 01:28:02,540 STUDENT: Does this imply that, for large objects, 1245 01:28:02,540 --> 01:28:05,320 not things on these very small scales, that we 1246 01:28:05,320 --> 01:28:07,430 should reproduce the Einstein field questions? 1247 01:28:07,430 --> 01:28:08,412 PROFESSOR: Yeah. 1248 01:28:08,412 --> 01:28:10,120 STUDENT: This is sufficient to show that? 1249 01:28:10,120 --> 01:28:13,420 PROFESSOR: Yeah, that's what it tells you. 1250 01:28:13,420 --> 01:28:18,140 Yeah, for example, if you measure 1251 01:28:18,140 --> 01:28:23,670 the gravity between you and me, you won't see the difference. 1252 01:28:23,670 --> 01:28:27,022 Yeah, actually, you will see a difference. 1253 01:28:27,022 --> 01:28:28,730 So this theory is a little bit different, 1254 01:28:28,730 --> 01:28:30,430 because of this massive scalar field. 1255 01:28:34,600 --> 01:28:38,420 So, in ordinary gravity, the attractive force 1256 01:28:38,420 --> 01:28:41,570 between you and me just come from graviton. 1257 01:28:41,570 --> 01:28:43,710 But in this theory, because this scalar is 1258 01:28:43,710 --> 01:28:48,210 massless and, actually, have additional attractive force. 1259 01:28:48,210 --> 01:28:50,220 And so, this theory is, actually, not the same. 1260 01:28:53,080 --> 01:28:56,740 So this theory, even though it's very similar generalization 1261 01:28:56,740 --> 01:28:59,070 of Einstein gravity, but actually 1262 01:28:59,070 --> 01:29:02,190 give you a different gravity force. 1263 01:29:02,190 --> 01:29:04,300 So that's why, with the string theory, 1264 01:29:04,300 --> 01:29:07,530 each going to describe the real life, somehow the scalar field 1265 01:29:07,530 --> 01:29:09,480 has to become massive. 1266 01:29:09,480 --> 01:29:12,720 Some other mechanism has to make the scalar field massive. 1267 01:29:12,720 --> 01:29:14,337 STUDENT: I see. 1268 01:29:14,337 --> 01:29:16,670 PROFESSOR: Yeah, of course, we also don't observe B mu m 1269 01:29:16,670 --> 01:29:18,378 and this also, obviously, become massive. 1270 01:29:22,670 --> 01:29:26,234 STUDENT: And will we find out that's the mechanism 1271 01:29:26,234 --> 01:29:27,970 to make [INAUDIBLE]? 1272 01:29:27,970 --> 01:29:29,550 PROFESSOR: Yeah. 1273 01:29:29,550 --> 01:29:33,924 So this is one of the very important questions, 1274 01:29:33,924 --> 01:29:35,340 since early days of string theory. 1275 01:29:35,340 --> 01:29:38,500 People have been trying to look for all kinds of mechanisms 1276 01:29:38,500 --> 01:29:40,794 to do it, et cetera, yeah. 1277 01:29:40,794 --> 01:29:43,650 STUDENT: So there is an agreed-upon way of doing this? 1278 01:29:43,650 --> 01:29:45,140 Or is it still sort of [INAUDIBLE]? 1279 01:29:45,140 --> 01:29:47,960 PROFESSOR: It depend, yeah. 1280 01:29:47,960 --> 01:29:50,620 This goes to a Lex point. 1281 01:29:50,620 --> 01:29:55,040 Yeah, wait for my Lex point. 1282 01:29:55,040 --> 01:29:56,029 Yes. 1283 01:29:56,029 --> 01:29:59,736 STUDENT: Can I ask, how do we know that this effective theory 1284 01:29:59,736 --> 01:30:01,777 couples all the fields as Einstein's theory does, 1285 01:30:01,777 --> 01:30:05,610 though quandrant derivative? 1286 01:30:05,610 --> 01:30:07,230 PROFESSOR: So what do you mean? 1287 01:30:07,230 --> 01:30:13,370 You can add-- the coupling between them, 1288 01:30:13,370 --> 01:30:16,940 this and gravity is through the-- Yeah, here, of course, 1289 01:30:16,940 --> 01:30:19,240 you should use quadrant derivative. 1290 01:30:19,240 --> 01:30:20,760 Yeah. 1291 01:30:20,760 --> 01:30:24,760 Yeah, here, I did not-- I'm not very careful in defining this, 1292 01:30:24,760 --> 01:30:29,010 but here, you use quadrant derivative, et cetera. 1293 01:30:29,010 --> 01:30:32,286 STUDENT: What about coupling to the open strings, 1294 01:30:32,286 --> 01:30:35,110 or to photons, or to matter? 1295 01:30:35,110 --> 01:30:37,032 PROFESSOR: Would be the same thing. 1296 01:30:37,032 --> 01:30:38,431 STUDENT: As? 1297 01:30:38,431 --> 01:30:40,680 PROFESSOR: As what you would expect, that [INAUDIBLE]. 1298 01:30:40,680 --> 01:30:48,820 Just saying, it just governed by general covariance. 1299 01:30:48,820 --> 01:30:51,440 General covariance have to arise at low energies. 1300 01:30:51,440 --> 01:30:55,130 STUDENT: So that general principle is effective 1301 01:30:55,130 --> 01:30:57,080 including matter, as well? 1302 01:30:57,080 --> 01:30:58,720 PROFESSOR: Yeah, then you can check. 1303 01:30:58,720 --> 01:30:59,500 They can check. 1304 01:30:59,500 --> 01:31:01,860 It it's a string theory, anything you can check 1305 01:31:01,860 --> 01:31:03,571 is consistent with that principle. 1306 01:31:07,470 --> 01:31:08,030 Good. 1307 01:31:08,030 --> 01:31:09,320 So now, it's another point. 1308 01:31:09,320 --> 01:31:12,490 So let me just make a side remark 1309 01:31:12,490 --> 01:31:16,170 on the physical consequence of this scales field. 1310 01:31:16,170 --> 01:31:18,350 So we see that this scalar field is important, 1311 01:31:18,350 --> 01:31:25,670 because it, actually, can mediate, say, attractive force. 1312 01:31:25,670 --> 01:31:28,330 But, actually, there's another very important role 1313 01:31:28,330 --> 01:31:30,890 of this scalar field task. 1314 01:31:36,400 --> 01:31:40,630 It's that, if you look at this low-energy effective action-- 1315 01:31:40,630 --> 01:31:46,360 so let me now write this G Newton 1316 01:31:46,360 --> 01:31:47,770 as 1 over g string square. 1317 01:31:50,480 --> 01:31:52,770 Then this have the structure proportional to 1 1318 01:31:52,770 --> 01:32:00,800 over g string square times exponential minus 2 phi, OK. 1319 01:32:00,800 --> 01:32:04,570 So now, there's a very important thing. 1320 01:32:04,570 --> 01:32:14,280 So now you see, if phi behave non-trivially-- 1321 01:32:14,280 --> 01:32:18,900 this is, actually, modify this guy-- it's actually factively-- 1322 01:32:18,900 --> 01:32:25,220 yeah, because this is multiplied by the Einstein scalar-- 1323 01:32:25,220 --> 01:32:30,290 so this, effectively, modifies your Newton constant, OK? 1324 01:32:30,290 --> 01:32:32,086 In fact, they modify the Newton constant. 1325 01:32:35,780 --> 01:32:40,610 In fact, you can actually integrate this gs 1326 01:32:40,610 --> 01:32:46,710 as the expectation value of these phi, OK? 1327 01:32:46,710 --> 01:32:50,700 That's the expectation value of this phi. 1328 01:32:50,700 --> 01:32:52,580 Yeah, because if you can change your phi, 1329 01:32:52,580 --> 01:32:54,660 and then change effective gs, then the gs 1330 01:32:54,660 --> 01:32:58,224 can reintegrate as expectation value of phi. 1331 01:33:01,540 --> 01:33:06,710 So this is something very important and very deep, 1332 01:33:06,710 --> 01:33:12,150 because, remember, gs is, essentially, The only parameter 1333 01:33:12,150 --> 01:33:13,250 in string theory. 1334 01:33:13,250 --> 01:33:15,780 The only dimension is parameter in string theory, 1335 01:33:15,780 --> 01:33:18,400 which characterize the strings of the string. 1336 01:33:21,760 --> 01:33:25,950 And now, we see this constant is not arbitrary. 1337 01:33:25,950 --> 01:33:30,309 It's actually determined by some dynamic field, OK? 1338 01:33:30,309 --> 01:33:32,100 So that actually means that, string theory, 1339 01:33:32,100 --> 01:33:33,710 there's no free parameter. 1340 01:33:33,710 --> 01:33:36,310 There's no free dimensionless parameter. 1341 01:33:36,310 --> 01:33:39,285 Everything, in some sense, determined by dynamics. 1342 01:33:43,250 --> 01:33:45,420 So this is a very remarkable feature. 1343 01:33:45,420 --> 01:33:47,870 So this makes people think, in the early days of string 1344 01:33:47,870 --> 01:33:53,990 theory, that you actually may be able to derive 1345 01:33:53,990 --> 01:33:55,815 the mass of the electron. 1346 01:33:55,815 --> 01:33:57,940 Because there's no free parameter in string theory, 1347 01:33:57,940 --> 01:33:59,690 so you should be able to derive everything 1348 01:33:59,690 --> 01:34:01,920 from first principle. 1349 01:34:01,920 --> 01:34:10,585 Anyway, but this also create a problem for the issue 1350 01:34:10,585 --> 01:34:12,620 I just mentioned. 1351 01:34:12,620 --> 01:34:17,110 Again, this is a side remark, but it's a fun remark. 1352 01:34:25,426 --> 01:34:27,050 But string theory, we mentioned before, 1353 01:34:27,050 --> 01:34:29,830 is a summation of a topology. 1354 01:34:29,830 --> 01:34:32,660 And the topology's weighted by g string. 1355 01:34:32,660 --> 01:34:34,870 So, if g string is small, they you 1356 01:34:34,870 --> 01:34:36,730 only need to look at the lowest topology, 1357 01:34:36,730 --> 01:34:40,340 because the higher topology are suppressed by higher power of g 1358 01:34:40,340 --> 01:34:41,780 string, OK? 1359 01:34:41,780 --> 01:34:44,260 And, particularly, if g string become [INAUDIBLE] 1, 1360 01:34:44,260 --> 01:34:48,620 then, to calculate such a scattering, 1361 01:34:48,620 --> 01:34:52,000 then you need to sum if all plausible topologies, and then 1362 01:34:52,000 --> 01:34:53,720 that would be unmanageable problem, which 1363 01:34:53,720 --> 01:34:55,000 we don't know how to do. 1364 01:34:55,000 --> 01:35:00,260 And so, want g string to be very small, 1365 01:35:00,260 --> 01:35:02,340 so that, actually, we can actually 1366 01:35:02,340 --> 01:35:05,410 control this theory, OK? 1367 01:35:05,410 --> 01:35:08,760 But now, we have a difficulty. 1368 01:35:08,760 --> 01:35:11,310 But we also said we want the scalar 1369 01:35:11,310 --> 01:35:16,120 field to to develop a mass. 1370 01:35:16,120 --> 01:35:19,930 We want this scalar field to develop a mass. 1371 01:35:19,930 --> 01:35:22,020 And, turns out, this is actually not easy, 1372 01:35:22,020 --> 01:35:24,850 to arrange this scalar field to develop a mass, 1373 01:35:24,850 --> 01:35:27,570 and, at the same time, to make this expectation 1374 01:35:27,570 --> 01:35:29,150 to be very small. 1375 01:35:29,150 --> 01:35:34,036 K. And, actually, that turns out to be a non-trivial problem. 1376 01:35:34,036 --> 01:35:35,577 Yeah, so it's actually not that easy. 1377 01:35:40,250 --> 01:35:41,805 OK, so my last comment. 1378 01:35:47,120 --> 01:35:49,530 So, earlier, we said, the tachyon. 1379 01:35:52,850 --> 01:35:54,360 So what we described so far, these 1380 01:35:54,360 --> 01:35:58,500 are called the bosonic string, because we only have bosons. 1381 01:35:58,500 --> 01:35:59,745 We only have bosons. 1382 01:35:59,745 --> 01:36:01,570 It's called the bosonic string. 1383 01:36:01,570 --> 01:36:05,074 So this bosonic string can be generalized to what's 1384 01:36:05,074 --> 01:36:06,240 called a superstring theory. 1385 01:36:13,290 --> 01:36:15,410 Of course, superstring. 1386 01:36:15,410 --> 01:36:17,830 So what superstring does is the following. 1387 01:36:17,830 --> 01:36:25,130 After you fix this gauge, again, the superstring 1388 01:36:25,130 --> 01:36:28,760 can be written as some covariant worldsheet theory 1389 01:36:28,760 --> 01:36:30,890 with some intrinsic metric. 1390 01:36:30,890 --> 01:36:33,110 And, in a superstring, after you fix this gauge, 1391 01:36:33,110 --> 01:36:36,260 the worldsheet metric to be Minkowski, then 1392 01:36:36,260 --> 01:36:38,799 the superstring action can be written as the following. 1393 01:36:41,700 --> 01:36:50,520 You just, essentially, have the previously free scalar action. 1394 01:36:50,520 --> 01:36:53,300 But then, you add some fermions. 1395 01:36:59,404 --> 01:37:00,320 You add some fermions. 1396 01:37:05,302 --> 01:37:07,260 So these are just some two-dimensional fermions 1397 01:37:07,260 --> 01:37:09,089 living on the worldsheet. 1398 01:37:13,819 --> 01:37:15,860 Yeah, so these are two-dimensional spinner fields 1399 01:37:15,860 --> 01:37:17,720 living on the worldsheet. 1400 01:37:17,720 --> 01:37:19,360 So the reason you can add such a thing, 1401 01:37:19,360 --> 01:37:21,930 because those things don't have obvious geometric 1402 01:37:21,930 --> 01:37:23,140 interpretation. 1403 01:37:23,140 --> 01:37:26,230 So you can consider them as describe some internal degrees 1404 01:37:26,230 --> 01:37:28,800 of freedom of the string. 1405 01:37:28,800 --> 01:37:32,800 And so, these guys provide the spacetime in the interpretation 1406 01:37:32,800 --> 01:37:34,630 of moving the spacetime. 1407 01:37:34,630 --> 01:37:38,190 And those are just some additional internal degrees 1408 01:37:38,190 --> 01:37:40,670 of freedom on the worldsheet. 1409 01:37:40,670 --> 01:37:44,260 It turns out, by adding these fermions, actually, 1410 01:37:44,260 --> 01:37:46,540 things change a lot. 1411 01:37:46,540 --> 01:37:49,612 Actually, they do not change very much. 1412 01:37:49,612 --> 01:37:51,570 It turns out, things does not change very much, 1413 01:37:51,570 --> 01:37:53,410 because this is a free fermion series, also 1414 01:37:53,410 --> 01:37:55,380 very easy to quantize. 1415 01:37:55,380 --> 01:37:58,210 And everything we did before just 1416 01:37:58,210 --> 01:38:01,900 carry over, except you need to add those fermions 1417 01:38:01,900 --> 01:38:04,134 You need to also quantize those free fermions. 1418 01:38:09,081 --> 01:38:09,580 Turns 1419 01:38:09,580 --> 01:38:12,360 Out, when you do that, there are actually 1420 01:38:12,360 --> 01:38:22,200 two different quantization scheme, 1421 01:38:22,200 --> 01:38:25,880 quantization procedure, quantization process. 1422 01:38:25,880 --> 01:38:29,520 Two different quantizations exist. 1423 01:38:29,520 --> 01:38:34,965 So, when you add these fermions with no tachyon. 1424 01:38:34,965 --> 01:38:36,590 So, you actually can get rid of tachyon 1425 01:38:36,590 --> 01:38:40,940 by including these fermions, OK? 1426 01:38:40,940 --> 01:38:48,600 So then, the lowest mode is just your massless mode, OK? 1427 01:38:48,600 --> 01:38:53,000 And the reason that you can have more than one quantization 1428 01:38:53,000 --> 01:38:55,180 is that, when you have fermions-- 1429 01:38:55,180 --> 01:38:57,980 and this is fermion defined on the circle. 1430 01:38:57,980 --> 01:38:59,750 So fermion on the circle, you can define 1431 01:38:59,750 --> 01:39:01,910 to be periodic or antiperiodic. 1432 01:39:01,910 --> 01:39:03,329 So now, you have some choices. 1433 01:39:03,329 --> 01:39:05,120 And, depending on whether you chose fermion 1434 01:39:05,120 --> 01:39:07,220 to be periodic, antiperiodic, et cetera, 1435 01:39:07,220 --> 01:39:10,490 and then the story become different, OK? 1436 01:39:10,490 --> 01:39:14,110 And, sorry, we're not going into there. 1437 01:39:14,110 --> 01:39:18,515 But, in principle, just waste enough time, in principle, 1438 01:39:18,515 --> 01:39:20,330 now, you can do it yourself. 1439 01:39:20,330 --> 01:39:23,430 Because, just quantize free field theory. 1440 01:39:23,430 --> 01:39:26,225 Actually, you still cannot do it yourself. 1441 01:39:26,225 --> 01:39:28,350 Yeah, this there's still a little bit more subtlety 1442 01:39:28,350 --> 01:39:32,250 than that, but the principle is very similar. 1443 01:39:32,250 --> 01:39:36,640 So, in this case, you get rid of tachyon. 1444 01:39:36,640 --> 01:39:42,960 So these two procedures cause type IIA and type IIB string. 1445 01:39:42,960 --> 01:39:45,045 So they give you two type of string theory. 1446 01:39:45,045 --> 01:39:49,217 One is called type IIA, and one is called type IIB. 1447 01:39:49,217 --> 01:39:51,050 So, also, a lot of the important difference, 1448 01:39:51,050 --> 01:39:55,830 instead of D equal to 26, now only requires D equal to 10, 1449 01:39:55,830 --> 01:39:56,330 OK? 1450 01:40:04,420 --> 01:40:08,160 So now, let me just write down the massive spectrum. 1451 01:40:08,160 --> 01:40:11,030 So now, because you have fermions, 1452 01:40:11,030 --> 01:40:15,010 now, actually, this spacetime particle 1453 01:40:15,010 --> 01:40:15,970 can also have fermions. 1454 01:40:15,970 --> 01:40:18,552 Previously, it's all bosonic particles. 1455 01:40:18,552 --> 01:40:20,260 Now, by adding these worldsheet fermions, 1456 01:40:20,260 --> 01:40:21,968 it turns out that these can also generate 1457 01:40:21,968 --> 01:40:23,860 the spacetime fermions. 1458 01:40:23,860 --> 01:40:27,460 It can generate spacetime fermions. 1459 01:40:27,460 --> 01:40:29,623 So now, the massive spectrum. 1460 01:40:33,780 --> 01:40:35,900 So now, these are the lowest particles. 1461 01:40:35,900 --> 01:40:39,010 There's no tachyon. 1462 01:40:39,010 --> 01:40:43,060 So these, now, become all 10 dimensional fields. 1463 01:40:43,060 --> 01:40:51,370 So for type IIA, again, you have this graviton, this B mu mu, 1464 01:40:51,370 --> 01:40:53,580 then you have this theta [INAUDIBLE]. 1465 01:40:53,580 --> 01:40:55,904 And then you have a lot of active fields 1466 01:40:55,904 --> 01:40:57,070 come from the closed string. 1467 01:40:57,070 --> 01:40:58,665 And so, this is for the closed string. 1468 01:41:01,990 --> 01:41:04,700 Now, you actually have a gauge field. 1469 01:41:04,700 --> 01:41:07,990 Also, in the closed string, you want gauge field and the three 1470 01:41:07,990 --> 01:41:11,150 form tensor fields. 1471 01:41:11,150 --> 01:41:15,026 So you have three indexes, 40 antisymmetric. 1472 01:41:15,026 --> 01:41:16,790 STUDENT: Is the gauge field just coming 1473 01:41:16,790 --> 01:41:19,424 from the fermion bilinear? 1474 01:41:19,424 --> 01:41:20,590 PROFESSOR: Yeah, yeah, yeah. 1475 01:41:20,590 --> 01:41:20,930 Right. 1476 01:41:20,930 --> 01:41:21,550 That's right. 1477 01:41:21,550 --> 01:41:26,770 They are related, actually, to fermion bilinears, yeah. 1478 01:41:26,770 --> 01:41:29,950 So those things are exactly the same as before, 1479 01:41:29,950 --> 01:41:32,096 and then you get some additional fields. 1480 01:41:32,096 --> 01:41:34,345 So these are normally called the Ramond-Ramond fields. 1481 01:41:42,220 --> 01:41:43,564 And then, plus fermions. 1482 01:41:43,564 --> 01:41:44,980 So let me write down the fermions. 1483 01:41:47,550 --> 01:41:50,230 It turns out that, these theories, actually was magic. 1484 01:41:50,230 --> 01:41:53,410 So, each of those fields, they have some super fermionic part 1485 01:41:53,410 --> 01:41:56,650 So it's actually a supersymmetric theory. 1486 01:41:56,650 --> 01:41:59,930 So then, you also have type IIB. 1487 01:41:59,930 --> 01:42:06,510 And, again, the bosonic string, you have H mu mu, B mu mu, phi. 1488 01:42:06,510 --> 01:42:11,860 And then, you have the larger scalar field, chi, 1489 01:42:11,860 --> 01:42:16,180 then the larger antisymmetric tensor C mu mu 2, 1490 01:42:16,180 --> 01:42:17,800 and it's in the four form field. 1491 01:42:21,950 --> 01:42:25,870 OK, so this, again, is so-called the Ramond-Ramond. 1492 01:42:25,870 --> 01:42:26,735 Then plus fermions. 1493 01:42:33,490 --> 01:42:36,240 And then, the [INAUDIBLE] theory of them 1494 01:42:36,240 --> 01:42:40,280 becomes, so-called, type IIA and type IIB supergravity. 1495 01:42:40,280 --> 01:42:43,190 So these are supersymmetric theories, and then 1496 01:42:43,190 --> 01:42:47,370 the corresponding generation of gravity for the supergravity. 1497 01:42:47,370 --> 01:42:50,162 So we do not need to go in to there. 1498 01:42:54,780 --> 01:42:55,300 Yes. 1499 01:42:55,300 --> 01:42:56,480 STUDENT: So, if you were going to do 1500 01:42:56,480 --> 01:42:58,646 string theory, for example, for a the generalization 1501 01:42:58,646 --> 01:43:00,700 on membranes, as you had mentioned before, 1502 01:43:00,700 --> 01:43:02,470 if you were doing this on two manifolds, 1503 01:43:02,470 --> 01:43:05,630 would you also have to include anion contributions? 1504 01:43:05,630 --> 01:43:08,460 PROFESSOR: You may. 1505 01:43:08,460 --> 01:43:10,180 I don't know. 1506 01:43:10,180 --> 01:43:13,630 Nobody have succeeded in doing this. 1507 01:43:13,630 --> 01:43:14,590 Yeah, you may. 1508 01:43:18,430 --> 01:43:27,350 OK, actually, today I have to be particularly slow, 1509 01:43:27,350 --> 01:43:31,180 because I have a much more grand plan for today. 1510 01:43:33,990 --> 01:43:36,680 That also means that there's definitely-- 1511 01:43:36,680 --> 01:43:41,740 you can only do the first two problems in your pset. 1512 01:43:41,740 --> 01:43:43,620 There's only four problems in your pset, 1513 01:43:43,620 --> 01:43:46,310 but you can only do the first two. 1514 01:43:46,310 --> 01:43:49,650 So you want to start, either, the first two, 1515 01:43:49,650 --> 01:43:52,335 or you want to wait until next week to do it all together. 1516 01:43:56,040 --> 01:43:59,170 STUDENT: If we do it all together next week, 1517 01:43:59,170 --> 01:44:02,950 so how about the next homework? 1518 01:44:02,950 --> 01:44:04,364 [INAUDIBLE] one week later? 1519 01:44:04,364 --> 01:44:05,030 PROFESSOR: Yeah. 1520 01:44:09,400 --> 01:44:14,981 OK, yeah, then let's just defer to one week. 1521 01:44:14,981 --> 01:44:15,730 Yeah, next Friday. 1522 01:44:21,590 --> 01:44:26,270 Yeah, then the next time-- so these, in sum, 1523 01:44:26,270 --> 01:44:31,250 conclude our basic discussion of the string theory. 1524 01:44:31,250 --> 01:44:35,520 You have seen most of the magics of string theory, 1525 01:44:35,520 --> 01:44:36,785 even though it's very fast. 1526 01:44:40,810 --> 01:44:45,070 So, next time, we will talk about D-branes, 1527 01:44:45,070 --> 01:44:50,560 which is another piece of magic from string theory. 1528 01:44:50,560 --> 01:44:52,110 Yeah.