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 00:00:21,540 --> 00:00:22,800 PROFESSOR: No, no minus sign. 9 00:00:22,800 --> 00:00:25,280 I think maybe-- yeah, it should be a plus sign. 10 00:00:25,280 --> 00:00:25,780 Yeah. 11 00:00:25,780 --> 00:00:28,500 Should be a plus sign. 12 00:00:28,500 --> 00:00:29,440 Yeah. 13 00:00:29,440 --> 00:00:32,090 Yeah, you should correct that minus sign to a plus sign. 14 00:00:32,090 --> 00:00:33,510 Yeah, both should be plus sign. 15 00:00:36,550 --> 00:00:40,760 Yeah-- I wrote on my notes was correct, 16 00:00:40,760 --> 00:00:45,510 but then on this board, somehow-- [INAUDIBLE] maybe 17 00:00:45,510 --> 00:00:48,270 I should have a minus sign and put it there. 18 00:00:48,270 --> 00:00:49,800 But didn't realize. 19 00:00:49,800 --> 00:00:50,300 Yeah. 20 00:00:50,300 --> 00:00:56,835 So let me first remind you what we did in last lecture. 21 00:00:59,570 --> 00:01:03,720 So first, we have some gauge symmetries 22 00:01:03,720 --> 00:01:06,645 on the [? worksheet. ?] And then we 23 00:01:06,645 --> 00:01:11,820 can use that, say, [INAUDIBLE] traditional symmetry. 24 00:01:11,820 --> 00:01:15,320 You can re-parametrize the [? worksheet ?] coordinates, 25 00:01:15,320 --> 00:01:19,090 and also to a [? while ?] scaling for the metric itself. 26 00:01:19,090 --> 00:01:22,980 Use those freedom, we can set the metric to, just 27 00:01:22,980 --> 00:01:25,510 to the flat Minkowski metric. 28 00:01:25,510 --> 00:01:27,260 [? On worksheet. ?] 29 00:01:27,260 --> 00:01:31,090 Now after you do that, then the [? worksheet ?] action 30 00:01:31,090 --> 00:01:34,650 then becomes just like a free scalar field. 31 00:01:34,650 --> 00:01:36,360 Then of course, the equation of motion 32 00:01:36,360 --> 00:01:39,730 is just given by the standard [? wave ?] equation. 33 00:01:39,730 --> 00:01:41,900 I then can solve it immediately. 34 00:01:41,900 --> 00:01:46,520 And essentially, you just have some left-moving wave, 35 00:01:46,520 --> 00:01:51,130 some right-moving wave, and plus, some zero modes. 36 00:01:51,130 --> 00:01:53,310 And then for a closed string, these 37 00:01:53,310 --> 00:01:55,620 are independent functions. 38 00:01:55,620 --> 00:01:58,640 So then for the closed string, closed, 39 00:01:58,640 --> 00:02:02,930 so you can have independent left-moving modes, 40 00:02:02,930 --> 00:02:09,489 and then also independent right-moving modes. 41 00:02:09,489 --> 00:02:11,155 And but for the open stream, because you 42 00:02:11,155 --> 00:02:13,530 have boundary conditions, and then 43 00:02:13,530 --> 00:02:18,840 you only have one set of modes, because you are-- yeah, 44 00:02:18,840 --> 00:02:22,210 because boundary conditions give you a constraint. 45 00:02:22,210 --> 00:02:24,500 So it's just like you have a standing wave. 46 00:02:24,500 --> 00:02:30,040 And then X l should be equal to L r. 47 00:02:30,040 --> 00:02:31,580 Yeah, X l should be equal to X r. 48 00:02:38,680 --> 00:02:41,740 But this is not-- but [? quantized ?] string theory, 49 00:02:41,740 --> 00:02:49,010 even in this gauge, is not just quantized scalar field theory, 50 00:02:49,010 --> 00:02:52,150 because we still need to solve the equation 51 00:02:52,150 --> 00:02:54,530 motion come from doing the variation of gamma 52 00:02:54,530 --> 00:02:56,560 a b themselves. 53 00:02:56,560 --> 00:02:59,560 And the way you do the variation of gamma a b themselves, 54 00:02:59,560 --> 00:03:03,040 they see [INAUDIBLE] to set the stress tensor of this scalar 55 00:03:03,040 --> 00:03:05,630 field theory zero. 56 00:03:05,630 --> 00:03:07,960 Yeah, to set the zero, the stress tensor. 57 00:03:07,960 --> 00:03:11,320 And so, already, just-- yeah, so they're 58 00:03:11,320 --> 00:03:12,890 two independent equations. 59 00:03:12,890 --> 00:03:14,550 Y is the diagonal component. 60 00:03:14,550 --> 00:03:15,580 It should be zero. 61 00:03:15,580 --> 00:03:18,140 And then the other is the off-diagonal component, 62 00:03:18,140 --> 00:03:19,120 should be zero. 63 00:03:19,120 --> 00:03:21,250 So you have two sets of equations. 64 00:03:21,250 --> 00:03:25,110 And those equations are in general hard to solve. 65 00:03:25,110 --> 00:03:28,020 So they are non-linear quadratic equations. 66 00:03:28,020 --> 00:03:29,670 Which I did not write them here. 67 00:03:29,670 --> 00:03:37,260 So they're non-linear constrained equations, which 68 00:03:37,260 --> 00:03:39,340 are, in general, hard to solve. 69 00:03:39,340 --> 00:03:44,370 Then the important trick we discussed 70 00:03:44,370 --> 00:03:48,510 is that you can go to the light-cone gauge. 71 00:03:48,510 --> 00:03:50,670 We said even after this fixing, there's still 72 00:03:50,670 --> 00:03:53,330 some remaining gauge freedom. 73 00:03:53,330 --> 00:03:59,720 That you can actually make one more choice to set the X plus, 74 00:03:59,720 --> 00:04:02,940 set it to be the same as tau. 75 00:04:02,940 --> 00:04:06,290 And the X plus, we always use this definition. 76 00:04:06,290 --> 00:04:09,760 So you can see that the all directions, X zero, X one, two, 77 00:04:09,760 --> 00:04:10,860 et cetera. 78 00:04:10,860 --> 00:04:13,460 And then we combine X zero and X one 79 00:04:13,460 --> 00:04:16,980 together to form X plus X minus. 80 00:04:16,980 --> 00:04:19,779 And then the rest we call them X i. 81 00:04:19,779 --> 00:04:23,220 So these are also sometimes called transverse directions. 82 00:04:23,220 --> 00:04:26,390 So also called transverse directions. 83 00:04:26,390 --> 00:04:30,670 And so now, then we can use this remaining gauge freedom 84 00:04:30,670 --> 00:04:33,080 to go to so-called light-cone gauge. 85 00:04:33,080 --> 00:04:36,980 Then the X plus becomes V plus times tau, some constant times 86 00:04:36,980 --> 00:04:38,220 tau. 87 00:04:38,220 --> 00:04:41,320 And then this light-cone gauge, then those equations 88 00:04:41,320 --> 00:04:44,010 can be written in a very simple way. 89 00:04:44,010 --> 00:04:48,300 In particular, the dependence on X minus becomes linear. 90 00:04:48,300 --> 00:04:50,600 Then you can actually use these two equations-- 91 00:04:50,600 --> 00:04:53,170 you can solve X minus exactly. 92 00:04:53,170 --> 00:04:58,450 And so this tells you that actually, 93 00:04:58,450 --> 00:05:00,850 the independent [? degree of ?] freedom just X i's. 94 00:05:00,850 --> 00:05:02,130 OK? 95 00:05:02,130 --> 00:05:05,040 The independent [? degree of ?] freedom just X i's. 96 00:05:05,040 --> 00:05:07,490 Any questions about this? 97 00:05:07,490 --> 00:05:09,670 So let me just put the equation number, 98 00:05:09,670 --> 00:05:12,800 because I'm going to use this equation number later. 99 00:05:15,610 --> 00:05:18,657 Which inherited from the last lecture. 100 00:05:18,657 --> 00:05:20,115 AUDIENCE: So the-- [? confidence ?] 101 00:05:20,115 --> 00:05:23,296 on the [? energy ?] [INAUDIBLE] because they're 102 00:05:23,296 --> 00:05:26,630 [? close together ?] [INAUDIBLE] 103 00:05:26,630 --> 00:05:27,270 PROFESSOR: No. 104 00:05:27,270 --> 00:05:27,770 No. 105 00:05:27,770 --> 00:05:31,900 This come from the equation motion for gamma a b. 106 00:05:31,900 --> 00:05:34,320 So [? remember, ?] previously, the gamma a b 107 00:05:34,320 --> 00:05:36,157 is also a dynamic variable. 108 00:05:36,157 --> 00:05:37,740 Even when you fix the gauge, you still 109 00:05:37,740 --> 00:05:39,330 impose the equation motion. 110 00:05:39,330 --> 00:05:42,460 And so that's essentially this. 111 00:05:42,460 --> 00:05:42,960 Yes? 112 00:05:45,329 --> 00:05:47,370 AUDIENCE: Aren't the remaining degrees of freedom 113 00:05:47,370 --> 00:05:50,610 for X minus just because of the boundary condition, 114 00:05:50,610 --> 00:05:52,835 or something like that? 115 00:05:52,835 --> 00:05:55,670 Is there a freedom in specifying that? 116 00:05:55,670 --> 00:05:58,950 PROFESSOR: What-- There's some constant, 117 00:05:58,950 --> 00:06:02,935 a [? possibility ?] constant and we can-- of all constants. 118 00:06:02,935 --> 00:06:06,590 Yeah, because this determines all the derivative dependents. 119 00:06:06,590 --> 00:06:09,660 And they might be, overall, constant. 120 00:06:09,660 --> 00:06:13,190 And that you can always-- yeah, it's not important. 121 00:06:13,190 --> 00:06:13,790 Yeah, yeah. 122 00:06:17,170 --> 00:06:18,834 Any other questions? 123 00:06:18,834 --> 00:06:19,334 Yes? 124 00:06:19,334 --> 00:06:21,375 AUDIENCE: So going back to what you said earlier. 125 00:06:21,375 --> 00:06:23,790 So for the boundary conditions for the open string, 126 00:06:23,790 --> 00:06:26,810 the derivative's also positive? 127 00:06:26,810 --> 00:06:28,130 So there's not a minus sign? 128 00:06:28,130 --> 00:06:30,040 PROFESSOR: Yeah. 129 00:06:30,040 --> 00:06:32,350 Let me also add here-- so for the open string, 130 00:06:32,350 --> 00:06:35,070 for the moment, we can see the so-called Neumann boundary 131 00:06:35,070 --> 00:06:36,500 condition. 132 00:06:36,500 --> 00:06:40,200 So this is equal to zero. 133 00:06:40,200 --> 00:06:42,090 So that, we'll keep that. 134 00:06:42,090 --> 00:06:43,180 Yeah, so one correction. 135 00:06:43,180 --> 00:06:47,990 In the last lecture, I wrote a minus sign there. 136 00:06:47,990 --> 00:06:50,230 That should be corrected by plus sign. 137 00:06:50,230 --> 00:06:54,307 The both equations should be corrected by plus sign. 138 00:06:54,307 --> 00:06:54,807 Yeah. 139 00:07:02,360 --> 00:07:02,860 Good? 140 00:07:06,580 --> 00:07:10,970 So with this set up, then we are ready for quantization. 141 00:07:10,970 --> 00:07:13,580 Because now, we only need to quantize the X i's, 142 00:07:13,580 --> 00:07:15,980 because X i's are independent-- only X i's 143 00:07:15,980 --> 00:07:18,430 are independent variables. 144 00:07:18,430 --> 00:07:21,770 And so we only need to quantize X i's. 145 00:07:21,770 --> 00:07:25,340 And X i's, they are just a free field. 146 00:07:25,340 --> 00:07:27,700 And that's very simple. 147 00:07:27,700 --> 00:07:29,380 OK? 148 00:07:29,380 --> 00:07:32,715 But before do that, let me just emphasize one point. 149 00:07:35,220 --> 00:07:36,360 Emphasize one point. 150 00:07:40,830 --> 00:07:47,940 So the light-cone gauge, by making this choice, 151 00:07:47,940 --> 00:07:51,460 you make those equations very simple, 152 00:07:51,460 --> 00:07:54,300 but you also sacrifice something. 153 00:07:54,300 --> 00:07:57,940 Because of the light-cone gauge, because the [? shaky ?] 154 00:07:57,940 --> 00:08:00,890 choice itself, it breaks Lorentz symmetry. 155 00:08:06,540 --> 00:08:09,400 OK? 156 00:08:09,400 --> 00:08:11,060 Because [INAUDIBLE] [? it lacks ?] 157 00:08:11,060 --> 00:08:13,170 two special directions. 158 00:08:13,170 --> 00:08:16,920 So X zero X one, and you form X plus from them, 159 00:08:16,920 --> 00:08:19,412 and you impose some conditions on X plus. 160 00:08:19,412 --> 00:08:20,745 So this breaks Lorentz symmetry. 161 00:08:28,950 --> 00:08:29,480 OK? 162 00:08:29,480 --> 00:08:33,960 So when we say "break," this does not really 163 00:08:33,960 --> 00:08:35,600 break the Lorentz symmetry. 164 00:08:35,600 --> 00:08:38,200 Just say this gauge choice itself does not 165 00:08:38,200 --> 00:08:39,630 respect Lorentz symmetry. 166 00:08:39,630 --> 00:08:42,230 OK? 167 00:08:42,230 --> 00:08:44,720 So the theory is still Lorentz symmetric. 168 00:08:44,720 --> 00:08:47,220 The theory should still have Lorentz symmetry. 169 00:08:47,220 --> 00:08:50,720 It's just Lorenz symmetry not manifest. 170 00:08:50,720 --> 00:08:55,145 Say, yeah, say just not manifest. 171 00:09:06,090 --> 00:09:08,100 Yeah, maybe I should-- using proper English, 172 00:09:08,100 --> 00:09:14,780 I should say, in the light-cone gauge, 173 00:09:14,780 --> 00:09:18,060 Lorentz symmetry is not manifest. 174 00:09:18,060 --> 00:09:20,954 I think this is a better way to say it. 175 00:09:20,954 --> 00:09:21,830 OK? 176 00:09:21,830 --> 00:09:23,730 It's not manifest. 177 00:09:23,730 --> 00:09:26,990 So this equation breaks Lorentz symmetry, 178 00:09:26,990 --> 00:09:30,370 and then the Lorentz symmetry is not manifest. 179 00:09:30,370 --> 00:09:37,000 So but there's a third group. 180 00:09:37,000 --> 00:09:38,960 Lorentz symmetry is still manifest. 181 00:09:38,960 --> 00:09:42,200 So that's the group which rotates the X i. 182 00:09:42,200 --> 00:10:01,435 So this remaining s o d minus two, which rotating X i's. 183 00:10:04,397 --> 00:10:05,105 This is manifest. 184 00:10:11,280 --> 00:10:11,850 OK. 185 00:10:11,850 --> 00:10:15,830 The remaining rotating X i is manifest. 186 00:10:18,570 --> 00:10:19,070 OK? 187 00:10:19,070 --> 00:10:19,950 So keep this in mind. 188 00:10:19,950 --> 00:10:20,200 OK? 189 00:10:20,200 --> 00:10:21,450 This is very important point. 190 00:10:29,030 --> 00:10:30,530 OK. 191 00:10:30,530 --> 00:10:34,100 So now we can proceed to quantize it. 192 00:10:34,100 --> 00:10:38,670 So before doing the quantization, 193 00:10:38,670 --> 00:10:43,830 let me develop a little bit further 194 00:10:43,830 --> 00:10:46,490 to make the task of the quantization easy. 195 00:10:55,140 --> 00:11:00,750 So I ask, what we do in free field theory is [? continue ?] 196 00:11:00,750 --> 00:11:03,260 to expand to those arbitrary functions in terms 197 00:11:03,260 --> 00:11:04,310 of Fourier modes. 198 00:11:04,310 --> 00:11:05,430 OK? 199 00:11:05,430 --> 00:11:08,010 And they are periodic functions of 2 pi, 200 00:11:08,010 --> 00:11:11,040 so we can easily expand them in Fourier modes. 201 00:11:11,040 --> 00:11:17,610 For example, X r X l mu sigma plus tau 202 00:11:17,610 --> 00:11:22,200 can be written as i over prime 2. 203 00:11:39,180 --> 00:11:39,680 OK? 204 00:11:42,330 --> 00:11:45,900 So this alpha prime, so this [? pre-factor ?] is purely 205 00:11:45,900 --> 00:11:46,710 convention. 206 00:11:46,710 --> 00:11:48,540 OK? 207 00:11:48,540 --> 00:11:51,240 Just for the later convenience. 208 00:11:51,240 --> 00:11:54,160 And this one over n is also purely convention. 209 00:11:54,160 --> 00:11:56,210 And you can choose what you want. 210 00:11:56,210 --> 00:11:58,870 So the key is that you are expanding 211 00:11:58,870 --> 00:12:01,800 this periodic function. 212 00:12:01,800 --> 00:12:03,390 Inferior modes. 213 00:12:03,390 --> 00:12:05,870 Because you have to be periodic in sigma, 214 00:12:05,870 --> 00:12:08,790 so the coefficient here must be integer n. 215 00:12:08,790 --> 00:12:10,510 And we exclude unequal to zero mode, 216 00:12:10,510 --> 00:12:13,100 because the unequal to zero, this is just a constant. 217 00:12:13,100 --> 00:12:15,590 And that is already included here. 218 00:12:15,590 --> 00:12:17,320 And that is already included here. 219 00:12:17,320 --> 00:12:21,750 So we say exclude unequal to zero. 220 00:12:21,750 --> 00:12:24,870 And also this is a real function. 221 00:12:24,870 --> 00:12:29,530 Then this means that the alpha n mu should 222 00:12:29,530 --> 00:12:32,930 be equal to alpha minus n. 223 00:12:32,930 --> 00:12:36,500 So with this convention over n here, and with this i 224 00:12:36,500 --> 00:12:40,580 here, would be alpha minus n mu [? ba. ?] OK? 225 00:12:46,900 --> 00:12:49,993 And similarly, you can write down the expansion for X r. 226 00:12:56,144 --> 00:12:58,310 Similarly, you can write down the expansion for X r. 227 00:13:04,390 --> 00:13:09,390 The same [? pre-factor. ?] And this n sum for minus infinity 228 00:13:09,390 --> 00:13:13,040 to plus infinity but not equal to-- unequal to zero, 229 00:13:13,040 --> 00:13:17,850 so we call the modes f n tilde divided by n. 230 00:13:17,850 --> 00:13:21,510 [INAUDIBLE] tau plus sigma, tau minus sigma. 231 00:13:25,842 --> 00:13:26,730 OK? 232 00:13:26,730 --> 00:13:31,060 And here, again, alpha tilde mu should 233 00:13:31,060 --> 00:13:36,500 be equal to [? ba ?] alpha tilde mu minus n, 234 00:13:36,500 --> 00:13:37,750 in order this to be real. 235 00:13:54,593 --> 00:13:55,093 OK. 236 00:13:55,093 --> 00:13:58,500 For the open string, you only have one set of modes. 237 00:13:58,500 --> 00:13:59,322 For open strings. 238 00:13:59,322 --> 00:14:00,530 So this is for closed string. 239 00:14:00,530 --> 00:14:09,150 For open string, you can-- you just have alpha. 240 00:14:09,150 --> 00:14:14,560 Yeah, just alpha and mu, you go to alpha and mu tilde. 241 00:14:14,560 --> 00:14:16,326 So you only have one set of modes, 242 00:14:16,326 --> 00:14:17,450 OK, because are they equal. 243 00:14:21,400 --> 00:14:22,290 OK. 244 00:14:22,290 --> 00:14:27,590 So you can plug this, those expressions, into here. 245 00:14:27,590 --> 00:14:29,930 Then you have most general form of X mu in terms 246 00:14:29,930 --> 00:14:33,600 of these Fourier modes. 247 00:14:33,600 --> 00:14:35,100 OK? 248 00:14:35,100 --> 00:14:39,220 So to save time, let me not do that step. 249 00:14:39,220 --> 00:14:42,570 So you should try to do-- yeah, just so 250 00:14:42,570 --> 00:14:46,170 a trivial step of yourself, write those things 251 00:14:46,170 --> 00:14:47,539 into a single equation. 252 00:14:47,539 --> 00:14:49,080 And then that's the most general form 253 00:14:49,080 --> 00:14:52,480 of X mu in terms of these specific modes. 254 00:14:55,160 --> 00:14:57,420 So now let me make some remarks. 255 00:15:10,464 --> 00:15:12,130 You might wonder, what's the implication 256 00:15:12,130 --> 00:15:15,820 of those zero modes here? 257 00:15:15,820 --> 00:15:16,570 OK? 258 00:15:16,570 --> 00:15:18,120 So those just describe the center 259 00:15:18,120 --> 00:15:19,630 of mass motion of the string. 260 00:15:22,546 --> 00:15:24,045 Center of mass motion of the string. 261 00:15:30,950 --> 00:15:40,040 It's given by-- say, you just average the motion of-- you 262 00:15:40,040 --> 00:15:43,190 average the motion of, say, of the string. 263 00:15:43,190 --> 00:15:47,240 So that gives you the center of mass motion of the string. 264 00:15:47,240 --> 00:15:48,910 And all those are periodic functions. 265 00:15:52,100 --> 00:15:55,230 So all of those are periodic functions. 266 00:15:55,230 --> 00:15:58,280 So all X a or X r periodic functions. 267 00:15:58,280 --> 00:16:00,560 So when they integrate, they give you zero. 268 00:16:00,560 --> 00:16:04,630 So the only thing remaining are those zero modes. 269 00:16:04,630 --> 00:16:12,320 And so you just get the X mu plus V mu tau. 270 00:16:12,320 --> 00:16:14,960 OK? 271 00:16:14,960 --> 00:16:17,340 So if you think of the center of mass of the string, 272 00:16:17,340 --> 00:16:19,210 that's a particle. 273 00:16:19,210 --> 00:16:21,507 It is a point, it's a point particle. 274 00:16:21,507 --> 00:16:23,340 So this essentially gives you the trajectory 275 00:16:23,340 --> 00:16:26,880 of the particle in terms of the proper time. 276 00:16:26,880 --> 00:16:29,925 So you can think of this tau as the proper time for the center 277 00:16:29,925 --> 00:16:32,610 of mass of the string. 278 00:16:32,610 --> 00:16:35,640 And then this just-- yeah. 279 00:16:35,640 --> 00:16:41,120 V mu is the velocity for the center 280 00:16:41,120 --> 00:16:45,740 of mass in terms of the proper time along the string. 281 00:16:45,740 --> 00:16:46,240 OK? 282 00:16:54,770 --> 00:16:59,460 And so all these different alpha n, alpha n 283 00:16:59,460 --> 00:17:03,050 tilde-- so those just parametrize 284 00:17:03,050 --> 00:17:05,730 the different oscillation modes. 285 00:17:05,730 --> 00:17:07,910 So they just parametrize the different oscillations 286 00:17:07,910 --> 00:17:08,609 of the string. 287 00:17:21,280 --> 00:17:22,260 Yeah, of a string. 288 00:17:26,020 --> 00:17:27,390 Oscillation of a string. 289 00:17:27,390 --> 00:17:29,590 So any particular alpha n is non-zero, 290 00:17:29,590 --> 00:17:32,100 means that particular harmonic is 291 00:17:32,100 --> 00:17:34,120 present in the string motion. 292 00:17:34,120 --> 00:17:34,980 Et cetera. 293 00:17:34,980 --> 00:17:36,340 OK? 294 00:17:36,340 --> 00:17:40,270 And for the closed string, as I said, 295 00:17:40,270 --> 00:17:42,140 before we have both left- and right-moving. 296 00:17:47,150 --> 00:17:55,687 Because you have left- plus right-moving waves. 297 00:17:55,687 --> 00:17:57,770 But for the open string, you essentially only have 298 00:17:57,770 --> 00:18:04,630 a standing wave, because of the boundary condition. 299 00:18:04,630 --> 00:18:05,600 OK? 300 00:18:05,600 --> 00:18:08,410 So you only have one set of oscillation modes, 301 00:18:08,410 --> 00:18:09,310 rather than two sets. 302 00:18:14,689 --> 00:18:16,130 So any questions on this? 303 00:18:25,310 --> 00:18:26,660 Good? 304 00:18:26,660 --> 00:18:29,120 So in the light-cone gauge, we can do further. 305 00:18:29,120 --> 00:18:32,740 Between the light-cone gauge, we can further solve this X minus. 306 00:18:32,740 --> 00:18:33,959 OK? 307 00:18:33,959 --> 00:18:35,375 So let's try to do it classically. 308 00:18:38,250 --> 00:18:57,530 In the light-cone gauge-- OK. 309 00:18:57,530 --> 00:19:09,950 So X minus can be solved by inserting the mode 310 00:19:09,950 --> 00:19:27,331 expansion into this equation 14 on the 15. 311 00:19:27,331 --> 00:19:27,830 OK? 312 00:19:27,830 --> 00:19:35,530 Then you're just equating different Fourier modes, 313 00:19:35,530 --> 00:19:48,215 the coefficients of different Fourier modes. 314 00:19:57,070 --> 00:19:58,880 OK? 315 00:19:58,880 --> 00:20:06,850 And then you can solve V minus, alpha n minus, alpha n minus, 316 00:20:06,850 --> 00:20:09,820 alpha n tilde minus, et cetera. 317 00:20:09,820 --> 00:20:10,320 OK? 318 00:20:22,270 --> 00:20:24,440 So now let's look at the zero modes. 319 00:20:24,440 --> 00:20:24,940 OK? 320 00:20:24,940 --> 00:20:27,405 So let's look at the equation satisfied by V minus. 321 00:20:30,177 --> 00:20:31,510 So let's look at the zero modes. 322 00:20:43,140 --> 00:20:48,350 So let's look at-- first look at equation 14. 323 00:20:48,350 --> 00:20:51,920 So if you say zero mode, means we look at the unequal 324 00:20:51,920 --> 00:20:54,440 to zero modes. 325 00:20:54,440 --> 00:20:57,220 And equal to zero modes here, partial tau X minus, 326 00:20:57,220 --> 00:20:59,670 you just have a V minus. 327 00:20:59,670 --> 00:21:05,310 So left-hand side-- so from 14-- so let's first 328 00:21:05,310 --> 00:21:06,565 do for the open string. 329 00:21:10,570 --> 00:21:12,595 So left-hand side you have 2 V [? 2 ?] minus. 330 00:21:15,160 --> 00:21:16,275 So the right-hand side. 331 00:21:19,040 --> 00:21:21,670 So if you look-- again, you look at the mode, 332 00:21:21,670 --> 00:21:23,430 which is unequal to zero. 333 00:21:23,430 --> 00:21:25,330 So there's one obvious term come from just 334 00:21:25,330 --> 00:21:27,570 taking derivative of this term. 335 00:21:27,570 --> 00:21:29,800 And so we have a V i squared term. 336 00:21:33,780 --> 00:21:34,930 OK? 337 00:21:34,930 --> 00:21:42,160 And then you will have a contribution from oscillators. 338 00:21:42,160 --> 00:21:43,930 Which I will not go into details here. 339 00:21:46,580 --> 00:21:48,840 So you can easily calculate it yourself. 340 00:21:48,840 --> 00:21:51,540 So you find for the open string. 341 00:21:51,540 --> 00:21:52,550 That's the answer. 342 00:21:52,550 --> 00:21:54,250 And so there's only one set of modes. 343 00:21:57,187 --> 00:21:59,145 And similarly, you can do it for the co-string. 344 00:22:03,970 --> 00:22:07,750 Again, on the left-hand side, just 2 V plus, 2 V minus, 345 00:22:07,750 --> 00:22:10,430 and the right-hand side you have V i squared. 346 00:22:10,430 --> 00:22:12,251 But now you have two sets of modes. 347 00:22:12,251 --> 00:22:14,125 And also there's some coefficient difference. 348 00:22:36,230 --> 00:22:38,870 So other than this [? pre-factor ?] 2 on the one, 349 00:22:38,870 --> 00:22:42,600 here, you can actually wrote down those expression just 350 00:22:42,600 --> 00:22:44,280 by closing your eyes. 351 00:22:44,280 --> 00:22:47,396 Just write them down. 352 00:22:47,396 --> 00:22:49,020 Because you have a zero mode, these two 353 00:22:49,020 --> 00:22:51,444 have to add up to zero. 354 00:22:51,444 --> 00:22:52,735 So that's the only possibility. 355 00:22:55,610 --> 00:22:57,520 So other than this factor of 2 in the one 356 00:22:57,520 --> 00:22:59,930 you need to check yourself, you can essentially 357 00:22:59,930 --> 00:23:03,200 write down this expression immediately, 358 00:23:03,200 --> 00:23:05,060 just from the structure of that equation. 359 00:23:05,060 --> 00:23:05,559 OK? 360 00:23:10,000 --> 00:23:13,710 AUDIENCE: What's the definition of V i? 361 00:23:13,710 --> 00:23:16,180 PROFESSOR: V i is there. 362 00:23:16,180 --> 00:23:17,550 V mu in the i direction. 363 00:23:20,850 --> 00:23:23,410 And V minus, V plus, it's all that. 364 00:23:26,440 --> 00:23:30,010 So those equations are of great importance. 365 00:23:30,010 --> 00:23:31,500 And let me box them. 366 00:23:35,700 --> 00:23:38,860 Which is also the reason I write them down. 367 00:23:38,860 --> 00:23:41,795 So but now, I did not leave space to give them numbers. 368 00:23:48,190 --> 00:23:50,960 Yeah, anyway, these are the two equations [? of ?] add 369 00:23:50,960 --> 00:23:52,310 the numbers [? nature. ?] 370 00:23:52,310 --> 00:23:59,220 From the 15, from the equation 15-- so 371 00:23:59,220 --> 00:24:04,840 this is a consequence from the 14, the zero modes from the 14. 372 00:24:04,840 --> 00:24:09,940 From the 15, you get nothing. 373 00:24:09,940 --> 00:24:12,900 You get zero equal to zero for open string. 374 00:24:16,580 --> 00:24:19,160 So this is obvious, because on the left-hand side, zero, 375 00:24:19,160 --> 00:24:27,002 because for the open string-- for the-- yeah. 376 00:24:27,002 --> 00:24:28,710 Yeah, because for the-- because left-hand 377 00:24:28,710 --> 00:24:31,800 is zero, because the zero mode does not have an [INAUDIBLE] 378 00:24:31,800 --> 00:24:33,980 independent sigma. 379 00:24:33,980 --> 00:24:35,610 So this is zero. 380 00:24:35,610 --> 00:24:39,020 And then you [? credited ?] a little bit checking yourself, 381 00:24:39,020 --> 00:24:41,450 that the right-hand side is also zero, 382 00:24:41,450 --> 00:24:44,647 essentially because of this structure X 0 equal 383 00:24:44,647 --> 00:24:48,066 to [? X y. ?] Yeah, X 0 equal to X r. 384 00:24:48,066 --> 00:24:49,690 And but for closed string, you actually 385 00:24:49,690 --> 00:24:52,870 find non-trivial equations. 386 00:24:52,870 --> 00:24:56,870 You find it-- the left-hand again, is zero. 387 00:24:56,870 --> 00:25:00,880 And the right-hand side-- the right-hand side, 388 00:25:00,880 --> 00:25:03,030 if you want to find the zero mode, essentially you 389 00:25:03,030 --> 00:25:04,529 just integrate over the whole thing. 390 00:25:07,422 --> 00:25:08,880 So let me just write it explicitly. 391 00:25:15,550 --> 00:25:17,655 So you can write it also in terms of modes. 392 00:25:55,410 --> 00:25:57,430 OK. 393 00:25:57,430 --> 00:26:01,220 So this equation is very interesting. 394 00:26:01,220 --> 00:26:08,040 So this tells you that the overall amplitude 395 00:26:08,040 --> 00:26:12,134 of the left-moving string, yeah, left-moving modes, 396 00:26:12,134 --> 00:26:13,550 when you add them together, should 397 00:26:13,550 --> 00:26:18,880 be the same as the overall moving, overall amplitude. 398 00:26:18,880 --> 00:26:27,800 Say-- yeah, in this combination of the right-moving string. 399 00:26:27,800 --> 00:26:30,110 OK? 400 00:26:30,110 --> 00:26:32,800 Yeah, they have to be balanced. 401 00:26:32,800 --> 00:26:37,020 So if we look at the source of this equation, 402 00:26:37,020 --> 00:26:40,570 and the reason the left-hand side is equal to zero 403 00:26:40,570 --> 00:26:44,417 is essentially due to the periodic boundary conditions. 404 00:26:44,417 --> 00:26:46,500 Because the periodic boundary-- for closed system, 405 00:26:46,500 --> 00:26:48,000 because periodic boundary condition, 406 00:26:48,000 --> 00:26:51,154 you cannot have any term linear in sigma. 407 00:26:51,154 --> 00:26:53,320 And the periodic boundary condition, in other words, 408 00:26:53,320 --> 00:26:56,460 it means that there is no special point on the string. 409 00:26:56,460 --> 00:26:56,960 OK? 410 00:26:56,960 --> 00:26:58,840 Any point is the same as the other. 411 00:26:58,840 --> 00:27:03,220 So your choice of origin is actually arbitrary. 412 00:27:03,220 --> 00:27:07,160 And so this equation essentially reflects that. 413 00:27:07,160 --> 00:27:10,740 So the fact along the string, there's no special point, 414 00:27:10,740 --> 00:27:16,297 and they give you a global constraint on the oscillation 415 00:27:16,297 --> 00:27:16,880 of the string. 416 00:27:16,880 --> 00:27:17,409 OK? 417 00:27:17,409 --> 00:27:19,700 Between the left-moving part and the right-moving part. 418 00:27:27,070 --> 00:27:31,005 And then, if you look at the-- so 419 00:27:31,005 --> 00:27:34,985 if you look at the nth modes, then 420 00:27:34,985 --> 00:27:39,380 you can find out alpha n minus, alpha n tilde minus, et cetera. 421 00:27:39,380 --> 00:27:42,105 OK? 422 00:27:42,105 --> 00:27:44,480 In the piece that you have a [? problem, ?] you will have 423 00:27:44,480 --> 00:27:45,438 [? found ?] doing this. 424 00:27:50,096 --> 00:27:50,970 Any questions so far? 425 00:27:54,990 --> 00:27:57,490 So now let's look at the physical meaning 426 00:27:57,490 --> 00:27:58,811 of those equations. 427 00:27:58,811 --> 00:27:59,310 OK? 428 00:27:59,310 --> 00:28:00,840 I boxed them. 429 00:28:00,840 --> 00:28:02,940 I said they are very important. 430 00:28:02,940 --> 00:28:06,800 And now let's look at the physical meaning 431 00:28:06,800 --> 00:28:09,610 of those equations. 432 00:28:09,610 --> 00:28:27,437 So this can be considered as consequence of no special point 433 00:28:27,437 --> 00:28:28,431 on the string. 434 00:28:35,444 --> 00:28:37,360 And now let's look at the physical [INAUDIBLE] 435 00:28:37,360 --> 00:28:39,370 of those modes, of those equations. 436 00:28:45,160 --> 00:28:47,310 So this is the fourth comment. 437 00:28:47,310 --> 00:28:50,630 The third remark-- fourth remark. 438 00:28:50,630 --> 00:28:55,380 So let me remind you that previously, the action 439 00:28:55,380 --> 00:28:56,505 have the global symmetries. 440 00:29:02,922 --> 00:29:04,380 So this comes from the translation. 441 00:29:09,260 --> 00:29:12,512 Sorry, by some constant [? m a ?] mu, 442 00:29:12,512 --> 00:29:13,845 all are Lorentz transformations. 443 00:29:16,920 --> 00:29:18,990 OK? 444 00:29:18,990 --> 00:29:20,902 So those global symmetries, as we know, 445 00:29:20,902 --> 00:29:23,027 they are [INAUDIBLE] to [? conserve ?] the current. 446 00:29:27,460 --> 00:29:34,567 Conserved current on the [? worksheet. ?] OK? 447 00:29:34,567 --> 00:29:35,525 On the [? worksheet. ?] 448 00:29:38,650 --> 00:29:43,175 So for the moment, let us consider the translation. 449 00:29:47,070 --> 00:29:50,350 So it's actually-- so it's a couple of lines, 450 00:29:50,350 --> 00:29:52,080 but I will leave you to do it yourself. 451 00:29:52,080 --> 00:29:55,420 I think you will also do it in your [? p ?] set. 452 00:29:55,420 --> 00:29:57,500 To derive the conserved current for this one. 453 00:29:57,500 --> 00:29:59,820 So let me just write down the answer. 454 00:29:59,820 --> 00:30:14,140 So the conserved current for the translation-- for translation, 455 00:30:14,140 --> 00:30:19,847 you can derive the conserved current because this [? t ?] 456 00:30:19,847 --> 00:30:20,430 such symmetry. 457 00:30:20,430 --> 00:30:23,460 For each mu, there's a symmetry. 458 00:30:23,460 --> 00:30:25,350 So the current is labeled by mu. 459 00:30:25,350 --> 00:30:27,690 But there's also [? worksheet-- ?] index 460 00:30:27,690 --> 00:30:31,550 to index, this is a current on the [? worksheet. ?] OK? 461 00:30:31,550 --> 00:30:35,310 So this can be written as 2 pi alpha prime. 462 00:30:39,770 --> 00:30:42,760 So I'm just writing down the answer, but you should-- 463 00:30:42,760 --> 00:30:44,510 you will check yourself in the [? piece ?] 464 00:30:44,510 --> 00:30:47,840 that this is the right answer. 465 00:30:47,840 --> 00:30:49,950 OK? 466 00:30:49,950 --> 00:30:53,320 So you can immediately see that this current is conserved, 467 00:30:53,320 --> 00:30:57,060 because the-- if you act partial a on here, 468 00:30:57,060 --> 00:31:00,010 you just get the equation motion when it's zero. 469 00:31:00,010 --> 00:31:01,690 Equation motion just partial a squared 470 00:31:01,690 --> 00:31:05,021 equal to partial a squared acting on x mu equal to zero. 471 00:31:05,021 --> 00:31:05,520 OK? 472 00:31:05,520 --> 00:31:08,845 So you can immediately see this is indeed conserved. 473 00:31:08,845 --> 00:31:09,595 This is conserved. 474 00:31:13,170 --> 00:31:16,270 And then this will lead us to a conserved charge. 475 00:31:16,270 --> 00:31:16,770 OK? 476 00:31:16,770 --> 00:31:20,820 Say, if I integrate it all along the string-- 477 00:31:20,820 --> 00:31:23,030 so let's do the-- for example, for the closed string, 478 00:31:23,030 --> 00:31:27,980 if I integrate it for the string, and this partial zero 479 00:31:27,980 --> 00:31:30,240 component. 480 00:31:30,240 --> 00:31:35,490 So the zero component, so the time component of this current, 481 00:31:35,490 --> 00:31:39,120 and then this is conserved charge on the worksheet. 482 00:31:39,120 --> 00:31:40,760 OK? 483 00:31:40,760 --> 00:31:43,330 So do you have any guess what should be this conserved charge 484 00:31:43,330 --> 00:31:49,585 equals 1 [? to? ?] Yes? 485 00:31:52,832 --> 00:31:54,194 AUDIENCE: [INAUDIBLE]. 486 00:31:54,194 --> 00:31:54,860 PROFESSOR: Yeah. 487 00:31:54,860 --> 00:31:58,551 Should be, must be, the space-time momentum. 488 00:31:58,551 --> 00:31:59,050 OK? 489 00:32:03,840 --> 00:32:08,230 In your-- maybe [? level ?] one, you 490 00:32:08,230 --> 00:32:11,340 should have learned that-- say, if you 491 00:32:11,340 --> 00:32:17,940 look in the classical mechanics, for particle move 492 00:32:17,940 --> 00:32:20,160 with the translation symmetry, then 493 00:32:20,160 --> 00:32:22,070 the momentum should be conserved. 494 00:32:22,070 --> 00:32:22,770 OK? 495 00:32:22,770 --> 00:32:26,600 So here, we have a string with the translation symmetry-- 496 00:32:26,600 --> 00:32:29,220 moving the space time is a translation symmetry-- 497 00:32:29,220 --> 00:32:33,900 and then the corresponding momentum must be conserved. 498 00:32:33,900 --> 00:32:36,681 And this conserved charge must be that momentum. 499 00:32:36,681 --> 00:32:37,180 OK? 500 00:32:47,380 --> 00:32:50,789 So these scenes-- so if I can see the p mu, 501 00:32:50,789 --> 00:32:52,330 so let me write in the p [INAUDIBLE], 502 00:32:52,330 --> 00:32:57,795 if you can see the p mu, which is this integration of 2 pi. 503 00:32:57,795 --> 00:32:59,170 So this is for the closed string. 504 00:32:59,170 --> 00:33:01,680 Similarly, for the open string, you integrate over pi. 505 00:33:14,260 --> 00:33:18,510 So this must be-- so this is the space time. 506 00:33:37,130 --> 00:33:39,020 So this-- oh, I'm sorry. 507 00:33:39,020 --> 00:33:41,000 This is a space-time momentum over the string. 508 00:33:41,000 --> 00:33:41,500 OK? 509 00:33:54,610 --> 00:33:55,110 Yes? 510 00:33:55,110 --> 00:33:57,026 AUDIENCE: What's the definition of d partial a 511 00:33:57,026 --> 00:34:00,880 in the definition of the current? 512 00:34:00,880 --> 00:34:03,760 PROFESSOR: So this is a conserved current, 513 00:34:03,760 --> 00:34:06,520 because [? one ?] into this global translation. 514 00:34:06,520 --> 00:34:08,737 AUDIENCE: But what's d sub a? 515 00:34:08,737 --> 00:34:09,320 PROFESSOR: Oh. 516 00:34:09,320 --> 00:34:11,350 Just the derivative of a. 517 00:34:11,350 --> 00:34:12,679 Derivative of sigma a. 518 00:34:15,224 --> 00:34:17,320 So I have always used a notation here. 519 00:34:20,370 --> 00:34:24,230 That the partial a is partial sigma a. 520 00:34:32,429 --> 00:34:35,219 So I should-- this goes to one into the space time momentum 521 00:34:35,219 --> 00:34:35,889 over the string. 522 00:34:38,449 --> 00:34:39,230 OK? 523 00:34:39,230 --> 00:34:42,110 So for the closed string, you integrate from zero to 2 pi. 524 00:34:42,110 --> 00:34:45,245 From open string, you're not going from only from zero 525 00:34:45,245 --> 00:34:46,690 to pi. 526 00:34:46,690 --> 00:34:49,925 So now if you plug in that mode expansion to here, 527 00:34:49,925 --> 00:34:51,300 again, all the oscillatory modes, 528 00:34:51,300 --> 00:34:54,080 they should not contribute. 529 00:34:54,080 --> 00:34:55,090 OK? 530 00:34:55,090 --> 00:34:58,880 And because this is a-- you can check they do not contribute. 531 00:34:58,880 --> 00:35:02,260 So the only thing contribute is zero modes. 532 00:35:02,260 --> 00:35:09,200 So you find that this V mu, p mu, 533 00:35:09,200 --> 00:35:14,086 is equal to v mu divided by alpha prime. 534 00:35:14,086 --> 00:35:15,210 So this is a closed string. 535 00:35:18,469 --> 00:35:20,010 So this is [? overviews ?] from here, 536 00:35:20,010 --> 00:35:22,700 because this gives you v mu times 2 pi, 537 00:35:22,700 --> 00:35:26,270 and then 2 pi can still give you alpha prime. 538 00:35:26,270 --> 00:35:30,050 But for the open string, you only integrate over pi. 539 00:35:30,050 --> 00:35:32,120 So you get 2 alpha prime. 540 00:35:32,120 --> 00:35:34,670 So this is for the open. 541 00:35:34,670 --> 00:35:36,670 OK? 542 00:35:36,670 --> 00:35:40,560 Or in other words-- or in other words, 543 00:35:40,560 --> 00:35:42,600 in terms of the space time momentum-- so this 544 00:35:42,600 --> 00:35:45,890 is a more physical quantity then this v mu-- so 545 00:35:45,890 --> 00:35:48,330 in terms of the space time momentum, 546 00:35:48,330 --> 00:35:51,975 the v mu can be written as alpha prime p 547 00:35:51,975 --> 00:35:56,420 mu, which is for the closed. 548 00:35:56,420 --> 00:36:01,860 And 2 alpha prime p mu, this is for the open. 549 00:36:07,670 --> 00:36:14,420 And then this itself, of course, should be in the [INAUDIBLE] 550 00:36:14,420 --> 00:36:28,160 that space-time momentum density along the string. 551 00:36:33,640 --> 00:36:34,140 OK? 552 00:36:52,450 --> 00:37:00,610 So now we have found that the v mu secretly is just 553 00:37:00,610 --> 00:37:02,071 the space-time momentum. 554 00:37:02,071 --> 00:37:02,570 OK? 555 00:37:02,570 --> 00:37:04,111 Essentially just space-time momentum, 556 00:37:04,111 --> 00:37:06,670 up to some overall constant. 557 00:37:06,670 --> 00:37:10,510 So now we can integrate those equations. 558 00:37:10,510 --> 00:37:13,730 Now we can integrate those equations. 559 00:37:13,730 --> 00:37:16,294 Then we can write them. 560 00:37:16,294 --> 00:37:17,960 So for example, for the first equation-- 561 00:37:17,960 --> 00:37:19,335 so now, let me give you a number. 562 00:37:22,530 --> 00:37:26,680 Yeah, actually, on my notes here, it's equation 18. 563 00:37:26,680 --> 00:37:29,856 Let me just write the equation 18 here. 564 00:37:29,856 --> 00:37:31,685 Only this is equation 19 for the closed. 565 00:37:37,800 --> 00:37:41,430 So I have more equation number in my notes than here, 566 00:37:41,430 --> 00:37:47,550 because I did not copy all the equations. 567 00:37:47,550 --> 00:37:50,830 Anyway, so that's-- for example, equation 18. 568 00:37:55,920 --> 00:38:02,590 Now I can rewrite it in terms of the momentum. 569 00:38:02,590 --> 00:38:03,780 OK? 570 00:38:03,780 --> 00:38:08,660 So let me put all v to a single side. 571 00:38:08,660 --> 00:38:13,790 So I can write as 2 p plus p minus, 572 00:38:13,790 --> 00:38:18,720 minus p i squared equal to. 573 00:38:18,720 --> 00:38:22,380 So each p [? rated ?] 2 alpha. 574 00:38:22,380 --> 00:38:26,450 Under that 2 alpha here, so that's 1 over 2 alpha sum 575 00:38:26,450 --> 00:38:34,040 over m equal to zero alpha m i, alpha minus m i. 576 00:38:34,040 --> 00:38:34,540 OK? 577 00:38:37,330 --> 00:38:38,930 OK? 578 00:38:38,930 --> 00:38:41,670 So now what is this? 579 00:38:41,670 --> 00:38:44,228 Do you recognize what's this? 580 00:38:44,228 --> 00:38:45,100 AUDIENCE: p squared. 581 00:38:45,100 --> 00:38:46,475 PROFESSOR: This is just p square. 582 00:38:50,765 --> 00:38:54,920 Or depend on your rotation, is minus p square. 583 00:38:54,920 --> 00:38:55,760 OK? 584 00:38:55,760 --> 00:38:57,090 Minus p mu, p mu. 585 00:38:57,090 --> 00:39:01,100 Maybe-- let me write [INAUDIBLE]. 586 00:39:01,100 --> 00:39:04,540 Minus p mu p mu. 587 00:39:04,540 --> 00:39:09,802 So what is minus p mu p mu? 588 00:39:09,802 --> 00:39:10,760 AUDIENCE: Mass squared. 589 00:39:10,760 --> 00:39:12,460 PROFESSOR: Yeah it's the mass squared. 590 00:39:12,460 --> 00:39:17,510 So this equation should be interpreted as a mass equation. 591 00:39:17,510 --> 00:39:22,780 So this tells you-- so it tells you 592 00:39:22,780 --> 00:39:39,800 the mass of the string should be written as the form-- OK? 593 00:39:42,450 --> 00:39:46,160 So now we have obtained the relation, 594 00:39:46,160 --> 00:39:54,140 we see that this relation-- so this is for the open string-- 595 00:39:54,140 --> 00:39:57,732 so we have obtained the relation between the mass of the string. 596 00:39:57,732 --> 00:39:59,690 So if you think about the center-of-mass motion 597 00:39:59,690 --> 00:40:01,820 of a string-- think, it's like a particle, 598 00:40:01,820 --> 00:40:03,959 then this particle can have a mass-- 599 00:40:03,959 --> 00:40:06,500 than this mass over the string, mass squared over the string, 600 00:40:06,500 --> 00:40:12,080 is related, can be expressed in terms of oscillation 601 00:40:12,080 --> 00:40:14,420 modes in this way. 602 00:40:14,420 --> 00:40:15,700 OK? 603 00:40:15,700 --> 00:40:23,290 And similarly, this equation 19 for the closed string, 604 00:40:23,290 --> 00:40:28,370 you find that m squared equal to-- again, 605 00:40:28,370 --> 00:40:38,240 just differ by pre-factor-- then alpha minus m i alpha 606 00:40:38,240 --> 00:40:44,016 m i, plus alpha tilde minus m i, alpha m i. 607 00:40:46,860 --> 00:40:54,790 So this is the mass equation for the closed string. 608 00:40:54,790 --> 00:40:59,510 So these are typically called the mass-shell conditions. 609 00:40:59,510 --> 00:41:00,589 OK? 610 00:41:00,589 --> 00:41:01,755 So these are for the closed. 611 00:41:11,260 --> 00:41:12,310 Any questions on this? 612 00:41:12,310 --> 00:41:14,615 AUDIENCE: [INAUDIBLE]. 613 00:41:14,615 --> 00:41:15,537 PROFESSOR: Hm? 614 00:41:15,537 --> 00:41:17,032 AUDIENCE: [INAUDIBLE] tilde? 615 00:41:17,032 --> 00:41:17,740 PROFESSOR: Sorry? 616 00:41:17,740 --> 00:41:19,290 AUDIENCE: Would there be another tilde? 617 00:41:19,290 --> 00:41:19,940 PROFESSOR: Oh that's right. 618 00:41:19,940 --> 00:41:20,290 Yeah. 619 00:41:20,290 --> 00:41:20,789 Thanks. 620 00:41:34,466 --> 00:41:35,400 Good. 621 00:41:35,400 --> 00:41:38,570 AUDIENCE: [INAUDIBLE] have a question that here, 622 00:41:38,570 --> 00:41:40,974 before, it's a expression that we'll 623 00:41:40,974 --> 00:41:46,008 have to enter to find the mass of the string. 624 00:41:46,008 --> 00:41:47,815 So this is a kind of a definition 625 00:41:47,815 --> 00:41:48,815 of the mass of the str-- 626 00:41:48,815 --> 00:41:49,565 PROFESSOR: No, no. 627 00:41:49,565 --> 00:41:53,940 The definition of the mass of the string 628 00:41:53,940 --> 00:42:00,561 is determined by how the center-of-mass motion, right? 629 00:42:00,561 --> 00:42:02,060 How would you tell the [? power ?]-- 630 00:42:02,060 --> 00:42:05,730 the particle moves is from the center-of-mass motion. 631 00:42:05,730 --> 00:42:08,680 And that determines its mass. 632 00:42:08,680 --> 00:42:13,540 This is a conserved quantity, p is a conserved quantity, 633 00:42:13,540 --> 00:42:18,110 and the dispersion relation determines your mass, right? 634 00:42:18,110 --> 00:42:20,860 Yes, so this is the mass in the [? right. ?] Yeah. 635 00:42:25,407 --> 00:42:26,240 Any other questions? 636 00:42:30,610 --> 00:42:31,637 Good. 637 00:42:31,637 --> 00:42:33,470 So now, finally, we can do the quantization, 638 00:42:33,470 --> 00:42:34,636 with all those preparations. 639 00:42:37,140 --> 00:42:38,020 OK? 640 00:42:38,020 --> 00:42:40,566 So now we can finally do the quantization. 641 00:42:54,060 --> 00:42:57,506 So similarly, I will not go into here-- similarly, yeah, 642 00:42:57,506 --> 00:43:02,675 let me just also mention by looking at this transformation, 643 00:43:02,675 --> 00:43:05,480 you can write down the conserved [? chart, ?] conserved current 644 00:43:05,480 --> 00:43:08,170 [? running ?] the Lorentz transformation. 645 00:43:08,170 --> 00:43:11,520 Then that will give you rise to the angular momentum. 646 00:43:11,520 --> 00:43:15,310 And also give rise to the corresponding charge associated 647 00:43:15,310 --> 00:43:17,111 with the boost, et cetera. 648 00:43:17,111 --> 00:43:17,610 OK? 649 00:43:21,020 --> 00:43:23,660 And we will not go into there, they actually 650 00:43:23,660 --> 00:43:27,860 play a very important role understand the various aspects 651 00:43:27,860 --> 00:43:31,460 of the string also. 652 00:43:31,460 --> 00:43:31,960 OK. 653 00:43:31,960 --> 00:43:34,927 So now let's-- now we can quantize it. 654 00:43:34,927 --> 00:43:37,385 And as we said before, we only need to quantize independent 655 00:43:37,385 --> 00:43:40,770 [? degrees of ?] freedom, so those X i's. 656 00:43:40,770 --> 00:43:51,530 So only need to quantize x i. 657 00:43:54,442 --> 00:43:56,650 Because these are the independent degrees of freedom. 658 00:43:59,970 --> 00:44:05,170 And from here, from here, you can immediately write down 659 00:44:05,170 --> 00:44:07,470 the canonical momentum. 660 00:44:07,470 --> 00:44:09,900 Say, the canonical [? worksheet ?] momentum 661 00:44:09,900 --> 00:44:13,670 for the string-- for the x i. 662 00:44:13,670 --> 00:44:20,370 So the canonical [? worksheet ?] momentum. 663 00:44:20,370 --> 00:44:26,430 So I should distinguish this with this guy. 664 00:44:26,430 --> 00:44:29,920 So that is the-- this goes one into the current, 665 00:44:29,920 --> 00:44:32,190 goes one into the space-time momentum. 666 00:44:32,190 --> 00:44:35,400 And this is just treating so this is just treating the x 667 00:44:35,400 --> 00:44:36,652 i as a two-dimensional field. 668 00:44:36,652 --> 00:44:38,610 We can write down its [? worksheet ?] canonical 669 00:44:38,610 --> 00:44:40,420 momentum. 670 00:44:40,420 --> 00:44:42,340 And we just take the time derivative 671 00:44:42,340 --> 00:44:46,170 of that action for the i direction, 672 00:44:46,170 --> 00:44:56,560 and then you find it's given by 2 pi alpha prime partial tau x 673 00:44:56,560 --> 00:44:57,790 i. 674 00:44:57,790 --> 00:45:00,540 OK? 675 00:45:00,540 --> 00:45:04,030 So this is the standard to the canonical momentum 676 00:45:04,030 --> 00:45:06,330 for the two-dimensional field series. 677 00:45:06,330 --> 00:45:08,030 So this happens to agree. 678 00:45:08,030 --> 00:45:10,030 So this happens to be the same as that momentum. 679 00:45:13,470 --> 00:45:16,240 But you should keep in mind that the physical interpretations 680 00:45:16,240 --> 00:45:17,700 were a bit different. 681 00:45:17,700 --> 00:45:19,710 There, it's corresponding to the density 682 00:45:19,710 --> 00:45:22,145 for the space-time momentum, and here, this just 683 00:45:22,145 --> 00:45:25,550 goes one into the canonical momentum, 684 00:45:25,550 --> 00:45:26,710 into the quantization. 685 00:45:26,710 --> 00:45:27,210 OK? 686 00:45:32,370 --> 00:45:34,195 So now you've found the canonical momentum, 687 00:45:34,195 --> 00:45:36,320 then we can just impose the quantization condition. 688 00:45:47,029 --> 00:45:48,820 You can just impose quantization condition. 689 00:45:58,910 --> 00:46:02,930 So now, we will-- you promote all this as operators, 690 00:46:02,930 --> 00:46:07,150 all the classical field is promoted as operators, 691 00:46:07,150 --> 00:46:12,750 and then you imposes a canonical quantization condition. 692 00:46:12,750 --> 00:46:18,580 So x i x j, for different x sigma prime, but evaluated 693 00:46:18,580 --> 00:46:20,565 same tau, should commute. 694 00:46:23,390 --> 00:46:25,430 You commute. 695 00:46:25,430 --> 00:46:36,375 Same thing if you do the pi at a different sigma prime, but tau 696 00:46:36,375 --> 00:46:37,000 should commute. 697 00:46:40,640 --> 00:46:50,430 But x i sigma tau and the pi j sigma prime tau 698 00:46:50,430 --> 00:46:55,310 then should give you i delta i j, then delta function 699 00:46:55,310 --> 00:46:57,030 sigma minus sigma prime. 700 00:46:57,030 --> 00:46:57,530 OK? 701 00:47:05,790 --> 00:47:09,750 So this is just the-- you just impose the-- this a free field, 702 00:47:09,750 --> 00:47:11,960 you just impose a standard canonical quantization 703 00:47:11,960 --> 00:47:16,420 condition as a two-dimensional field. 704 00:47:16,420 --> 00:47:20,900 And now all of these modes, all these different modes-- 705 00:47:20,900 --> 00:47:26,770 oh, which I just erased half of it-- and all 706 00:47:26,770 --> 00:47:35,040 these different modes-- so x i p i-- so p i-- v 707 00:47:35,040 --> 00:47:41,640 i is the same as v i-- and then alpha n i, alpha n tilde i, 708 00:47:41,640 --> 00:47:42,655 they are all operators. 709 00:47:51,300 --> 00:47:53,790 So classically, because one into integration 710 00:47:53,790 --> 00:47:59,130 constant for the equation of motion, 711 00:47:59,130 --> 00:48:00,740 and then quantum mechanically, they 712 00:48:00,740 --> 00:48:04,830 become integration constant for your operator equation. 713 00:48:04,830 --> 00:48:10,630 So they're just constant operators quantum mechanically. 714 00:48:10,630 --> 00:48:11,920 OK? 715 00:48:11,920 --> 00:48:17,170 And then when you plug in those mode expansions into here, just 716 00:48:17,170 --> 00:48:24,810 as usual, in your free field theory quantization, 717 00:48:24,810 --> 00:48:27,420 then you just find the accommodation relation. 718 00:48:27,420 --> 00:48:35,010 For example, you just find x i p j equal to i delta i j. 719 00:48:35,010 --> 00:48:43,170 Then you also find alpha m i alpha n j equal to alpha 720 00:48:43,170 --> 00:48:55,007 m tilde i alpha n tilde j equal to m delta i j delta. 721 00:48:55,007 --> 00:48:56,840 So I'm just writing down the answer for you. 722 00:49:07,839 --> 00:49:08,830 OK? 723 00:49:08,830 --> 00:49:14,377 So this is all straightforward, other than unconventional 724 00:49:14,377 --> 00:49:14,960 organizations. 725 00:49:26,640 --> 00:49:29,560 AUDIENCE: So that's things like delta m minus n, or something? 726 00:49:29,560 --> 00:49:30,810 PROFESSOR: Yeah, that's right. 727 00:49:30,810 --> 00:49:31,310 Yeah. 728 00:49:34,010 --> 00:49:37,880 So now if you look at this, OK, if you 729 00:49:37,880 --> 00:49:40,340 look at this, so this is m. 730 00:49:40,340 --> 00:49:45,840 So this m is related to this normalization 731 00:49:45,840 --> 00:49:47,570 we were choosing here. 732 00:49:47,570 --> 00:49:50,110 Here we are choosing n. 733 00:49:50,110 --> 00:49:53,000 [INAUDIBLE] here, there's appearing some m here. 734 00:49:53,000 --> 00:49:54,640 OK? 735 00:49:54,640 --> 00:50:00,510 And from this scene, you can see-- 736 00:50:00,510 --> 00:50:06,024 so these all others all other commutation relation, all 737 00:50:06,024 --> 00:50:07,065 other commutators vanish. 738 00:50:09,420 --> 00:50:09,920 OK. 739 00:50:09,920 --> 00:50:13,860 So these are the only non-vanishing ones. 740 00:50:13,860 --> 00:50:17,940 So if you look at this equation. 741 00:50:17,940 --> 00:50:22,620 So let's see for m greater than zero, 742 00:50:22,620 --> 00:50:27,567 so for m greater than zero, then this is m, 743 00:50:27,567 --> 00:50:29,400 then this is only [? non-manageable ?] for m 744 00:50:29,400 --> 00:50:30,441 equal to m equal to zero. 745 00:50:30,441 --> 00:50:35,400 So this alpha m alpha minus m, then you go to m something. 746 00:50:35,400 --> 00:50:37,540 It's a positive number. 747 00:50:37,540 --> 00:50:41,170 So that means, tells you, that in terms 748 00:50:41,170 --> 00:50:44,872 of a standard location, square root m alpha 749 00:50:44,872 --> 00:50:49,206 m i should be interpreted as the, say, the standard 750 00:50:49,206 --> 00:50:50,080 [INAUDIBLE] operator. 751 00:50:53,740 --> 00:50:58,470 And the square root m alpha minus m i 752 00:50:58,470 --> 00:51:01,840 should be interpreted as a i dagger. 753 00:51:01,840 --> 00:51:05,890 So this is all for m greater than zero. 754 00:51:05,890 --> 00:51:07,350 OK? 755 00:51:07,350 --> 00:51:10,119 Similarly for the tildes. 756 00:51:10,119 --> 00:51:11,160 Similarly for the tildes. 757 00:51:11,160 --> 00:51:13,820 So that's what the equation means. 758 00:51:13,820 --> 00:51:16,145 OK? 759 00:51:16,145 --> 00:51:17,020 So it's clear to you? 760 00:51:23,750 --> 00:51:28,940 Then we essentially-- then we have essentially finished 761 00:51:28,940 --> 00:51:30,330 the quantization of the string. 762 00:51:30,330 --> 00:51:30,830 OK? 763 00:51:34,390 --> 00:51:39,090 We have solved the Heisenberg equations. 764 00:51:39,090 --> 00:51:48,260 And so, this is now-- we saw those modes [? expression ?] 765 00:51:48,260 --> 00:51:50,350 plug in. 766 00:51:50,350 --> 00:51:55,830 This is now interpreted as to the solution to the Heisenberg 767 00:51:55,830 --> 00:51:57,710 equations. 768 00:51:57,710 --> 00:52:00,620 And then the integration constant in these equations, 769 00:52:00,620 --> 00:52:04,470 they satisfy those commutation relations. 770 00:52:04,470 --> 00:52:07,120 They satisfy those commutation relations. 771 00:52:07,120 --> 00:52:10,535 And also, the commutation relation between x i 772 00:52:10,535 --> 00:52:13,810 and the p, indeed is it what you would 773 00:52:13,810 --> 00:52:16,980 expect for point particle. 774 00:52:16,980 --> 00:52:20,040 So this goes one into the center-of-mass location 775 00:52:20,040 --> 00:52:21,920 and the center-of-mass momentum. 776 00:52:21,920 --> 00:52:26,140 And indeed, you would expect between the position 777 00:52:26,140 --> 00:52:27,316 and the momentum. 778 00:52:27,316 --> 00:52:27,816 OK? 779 00:52:32,232 --> 00:52:32,815 Any questions? 780 00:52:35,361 --> 00:52:35,860 Yes? 781 00:52:35,860 --> 00:52:39,208 AUDIENCE: Are we dropping the index m on the [? eight? ?] 782 00:52:39,208 --> 00:52:41,864 Or is that a subscript? 783 00:52:41,864 --> 00:52:42,530 PROFESSOR: Yeah. 784 00:52:47,040 --> 00:52:48,950 But then we always use the notation of alpha. 785 00:52:48,950 --> 00:52:50,210 I will not use a. 786 00:52:50,210 --> 00:52:53,180 Yeah, but just keep in mind that their relation is like that. 787 00:52:53,180 --> 00:52:56,160 And then a and a dagger will have the standard commutation 788 00:52:56,160 --> 00:52:58,110 relation equal to 1. 789 00:52:58,110 --> 00:52:58,610 OK? 790 00:53:01,740 --> 00:53:06,110 So now, we can now work out the spectrum. 791 00:53:12,121 --> 00:53:13,620 So now we can work out the spectrum. 792 00:53:26,270 --> 00:53:29,150 Let me see. 793 00:53:29,150 --> 00:53:30,785 So we can work out the spectrum. 794 00:53:33,740 --> 00:53:39,660 So the lowest state is so-called oscillator vacuum. 795 00:53:39,660 --> 00:53:42,020 So when we quantize it, we define our vacuum. 796 00:53:45,254 --> 00:53:46,670 So the vacuum [? does ?] run into, 797 00:53:46,670 --> 00:53:50,870 of course, the states, which are [INAUDIBLE] by all [INAUDIBLE] 798 00:53:50,870 --> 00:53:53,022 operators. 799 00:53:53,022 --> 00:53:55,105 [INAUDIBLE] by all our [? relational ?] operators. 800 00:54:03,075 --> 00:54:03,575 OK? 801 00:54:13,310 --> 00:54:15,560 Greater than zero and i. 802 00:54:15,560 --> 00:54:16,060 OK. 803 00:54:16,060 --> 00:54:19,880 So we first define-- so in order to-- so we are quantizing-- 804 00:54:19,880 --> 00:54:22,055 so this is like a free field theory, 805 00:54:22,055 --> 00:54:25,250 and so we define our vacuum, which is a [INAUDIBLE] 806 00:54:25,250 --> 00:54:27,600 by all [INAUDIBLE] operator. 807 00:54:27,600 --> 00:54:33,890 But the difference from the standard quantum fields theory, 808 00:54:33,890 --> 00:54:36,850 two-dimensional quantum fields theory, is that now, 809 00:54:36,850 --> 00:54:39,100 the vacuum here is labeled [? by ?] p. 810 00:54:39,100 --> 00:54:39,600 OK? 811 00:54:39,600 --> 00:54:42,550 This is the space-time, center-of-mass momentum 812 00:54:42,550 --> 00:54:43,970 of the string. 813 00:54:43,970 --> 00:54:46,446 So we are taking the vacuum to be, 814 00:54:46,446 --> 00:54:47,820 say a momentum [INAUDIBLE] state. 815 00:54:47,820 --> 00:54:48,540 OK? 816 00:54:48,540 --> 00:54:50,920 In terms of space-time. 817 00:54:50,920 --> 00:54:55,550 And so this zero p, which because one and two 818 00:54:55,550 --> 00:54:59,900 have low oscillator exactly on the string, 819 00:54:59,900 --> 00:55:03,790 but can still have a momentum, a space-time momentum, 820 00:55:03,790 --> 00:55:08,000 so this is labeled by center-of-mass momentum. 821 00:55:15,260 --> 00:55:15,760 OK? 822 00:55:20,690 --> 00:55:24,880 And then you can also build up other state. 823 00:55:24,880 --> 00:55:28,220 Then you can-- so this is the lowest state, on the-- so this 824 00:55:28,220 --> 00:55:31,449 is, in some sense, the vacuum state on the string. 825 00:55:31,449 --> 00:55:33,490 And then you can also build up the excited state. 826 00:55:37,000 --> 00:55:45,630 And so you just elect arbitrary number 827 00:55:45,630 --> 00:55:47,130 of alpha and alpha tildes. 828 00:55:56,925 --> 00:55:59,721 On this vacuum state. 829 00:55:59,721 --> 00:56:00,220 OK? 830 00:56:11,030 --> 00:56:14,010 So yeah, let me also label the equations. 831 00:56:14,010 --> 00:56:16,471 So let me call this equation 20. 832 00:56:16,471 --> 00:56:16,970 OK? 833 00:56:21,930 --> 00:56:32,610 So for the open string-- yeah, so this 834 00:56:32,610 --> 00:56:34,480 is for the closed string. 835 00:56:34,480 --> 00:56:39,810 For the open string, you only have one set. 836 00:56:39,810 --> 00:56:43,910 You just have one set of, say, alpha n i. 837 00:56:43,910 --> 00:56:45,080 And you still have two sets. 838 00:56:45,080 --> 00:56:45,580 OK? 839 00:56:52,120 --> 00:56:55,350 It's also convenient to define oscillator number. 840 00:57:02,510 --> 00:57:03,440 OK? 841 00:57:03,440 --> 00:57:06,130 So which we call-- define in terms 842 00:57:06,130 --> 00:57:12,490 of the standard [INAUDIBLE] And then the alpha minus m i, 843 00:57:12,490 --> 00:57:16,791 alpha m i, so it would be equal to m N m i. 844 00:57:16,791 --> 00:57:17,290 OK? 845 00:57:17,290 --> 00:57:21,360 So this is oscillator number for the m's mode 846 00:57:21,360 --> 00:57:22,980 and in i directions. 847 00:57:22,980 --> 00:57:24,440 OK? 848 00:57:24,440 --> 00:57:27,200 So in this equation there's no summation of i. 849 00:57:27,200 --> 00:57:33,420 Also there's no summation of m, just a-- OK? 850 00:57:33,420 --> 00:57:35,060 So I hope this is clear. 851 00:57:45,900 --> 00:57:46,400 Good. 852 00:57:46,400 --> 00:57:46,983 Any questions? 853 00:57:55,960 --> 00:58:00,010 So now let's look at those equations. 854 00:58:00,010 --> 00:58:03,620 Now let's look at the-- those mass-shell conditions. 855 00:58:03,620 --> 00:58:05,873 So the quantum [? variable. ?] OK? 856 00:58:05,873 --> 00:58:07,160 AUDIENCE: Excuse me. 857 00:58:07,160 --> 00:58:10,329 What does operators x and p do in those vacuums? 858 00:58:10,329 --> 00:58:10,954 PROFESSOR: Hmm? 859 00:58:10,954 --> 00:58:12,930 AUDIENCE: Operators x and p. 860 00:58:12,930 --> 00:58:15,757 What do they do with those right here? 861 00:58:15,757 --> 00:58:16,590 PROFESSOR: No no no. 862 00:58:16,590 --> 00:58:19,780 So we take-- so back here, so we take the vacuum 863 00:58:19,780 --> 00:58:22,940 to be a [INAUDIBLE] state of p. 864 00:58:22,940 --> 00:58:24,542 And then yeah, then that's it. 865 00:58:24,542 --> 00:58:25,041 Yeah. 866 00:58:25,041 --> 00:58:27,807 AUDIENCE: [INAUDIBLE] p i [INAUDIBLE] right here, 867 00:58:27,807 --> 00:58:29,190 what do they do? 868 00:58:29,190 --> 00:58:31,730 PROFESSOR: So this is [INAUDIBLE] state of p. 869 00:58:31,730 --> 00:58:34,010 And the action of x on p, it just 870 00:58:34,010 --> 00:58:36,613 asks what do you do in quantum mechanics. 871 00:58:36,613 --> 00:58:37,112 Yeah. 872 00:58:40,721 --> 00:58:41,220 Yes? 873 00:58:41,220 --> 00:58:43,910 AUDIENCE: [INAUDIBLE] kind of a dumb thing, but. 874 00:58:43,910 --> 00:58:48,647 So the p mu, that only includes the i [INAUDIBLE], or all? 875 00:58:48,647 --> 00:58:49,730 PROFESSOR: All components. 876 00:58:49,730 --> 00:58:52,150 All components. 877 00:58:52,150 --> 00:58:53,512 But then they must be satisfied. 878 00:58:53,512 --> 00:58:54,470 AUDIENCE: Right, right. 879 00:58:54,470 --> 00:58:55,440 So they have to satisfy-- 880 00:58:55,440 --> 00:58:56,280 PROFESSOR: Then they have to satisfy 881 00:58:56,280 --> 00:58:57,135 this kind of constraint. 882 00:58:57,135 --> 00:58:57,801 AUDIENCE: I see. 883 00:59:08,910 --> 00:59:11,240 PROFESSOR: Other questions? 884 00:59:11,240 --> 00:59:12,371 Good. 885 00:59:12,371 --> 00:59:13,400 Good. 886 00:59:13,400 --> 00:59:17,610 So now, we can write down-- so now 887 00:59:17,610 --> 00:59:20,510 let's write down the mass-shell condition in terms of-- so this 888 00:59:20,510 --> 00:59:23,900 is a classical equation. 889 00:59:23,900 --> 00:59:24,400 OK? 890 00:59:24,400 --> 00:59:25,816 These are the classical equations. 891 00:59:30,755 --> 00:59:32,880 So now we can write down the quantum version of it. 892 00:59:44,642 --> 00:59:46,600 So we can write down the quantum version of it. 893 00:59:49,340 --> 00:59:58,400 So each state, so the typical state, here, will carry some p, 894 00:59:58,400 --> 01:00:00,720 will carry some p. 895 01:00:00,720 --> 01:00:06,610 So p i are independent, but then p minus p i and p plus, 896 01:00:06,610 --> 01:00:09,530 for example, yeah, for x plus, this 897 01:00:09,530 --> 01:00:11,420 will also give you a p plus. 898 01:00:11,420 --> 01:00:14,340 For p i and p plus are independent. 899 01:00:14,340 --> 01:00:17,560 But then the p minus, then it's determined 900 01:00:17,560 --> 01:00:20,780 from those equations, determined from 901 01:00:20,780 --> 01:00:22,450 these mass-shell conditions. 902 01:00:22,450 --> 01:00:23,115 OK? 903 01:00:23,115 --> 01:00:25,240 So now let's write down those mass-shell conditions 904 01:00:25,240 --> 01:00:26,330 in terms of quantum form. 905 01:00:29,029 --> 01:00:31,320 So now let's, again, look at the mass-shell conditions. 906 01:00:40,630 --> 01:00:44,910 So let's first, again, do the open string. 907 01:00:44,910 --> 01:00:46,515 So we just take these. 908 01:00:53,042 --> 01:00:54,500 So when you go to quantum, then you 909 01:00:54,500 --> 01:00:57,470 have to worry about the ordering. 910 01:00:57,470 --> 01:00:57,970 OK? 911 01:00:57,970 --> 01:01:00,840 Then you have to worry about the ordering, because now they 912 01:01:00,840 --> 01:01:02,540 don't [? commute. ?] OK? 913 01:01:02,540 --> 01:01:05,380 So now you have-- about the ordering. 914 01:01:05,380 --> 01:01:08,570 So let me-- for the moment, don't-- then 915 01:01:08,570 --> 01:01:11,990 there's an ordering constant, et cetera. 916 01:01:11,990 --> 01:01:13,710 And it's easy to understand what should 917 01:01:13,710 --> 01:01:19,040 be the ordering constant, because this is a string, 918 01:01:19,040 --> 01:01:21,740 and each oscillation mode behaves like an-- it's just 919 01:01:21,740 --> 01:01:23,690 a harmonic oscillator. 920 01:01:23,690 --> 01:01:29,850 And then you just add up all the zero-point energy 921 01:01:29,850 --> 01:01:32,100 of the harmonic oscillator, and that will give you 922 01:01:32,100 --> 01:01:33,350 the ordering constant. 923 01:01:33,350 --> 01:01:33,850 OK? 924 01:01:33,850 --> 01:01:37,715 It's just the same as a harmonic oscillator problem. 925 01:01:37,715 --> 01:01:41,720 So to translate this into a quantum operator expression, 926 01:01:41,720 --> 01:01:44,830 so I just do it-- [? isolate ?] the harmonic oscillator. 927 01:01:44,830 --> 01:01:46,470 OK? 928 01:01:46,470 --> 01:01:47,990 Because I just-- have essentially 929 01:01:47,990 --> 01:01:50,350 have m harmonic oscillators. 930 01:01:50,350 --> 01:01:52,490 Have all these different harmonic oscillators. 931 01:01:52,490 --> 01:01:54,000 Then this is easy. 932 01:01:54,000 --> 01:01:56,660 So I just copy this one of alpha prime. 933 01:01:56,660 --> 01:02:02,010 So this is m from minus infinity to plus infinity. 934 01:02:02,010 --> 01:02:04,910 So let me rewrite it from m equal to 1, to plus infinity. 935 01:02:04,910 --> 01:02:07,060 Then I have a factor of 2. 936 01:02:07,060 --> 01:02:11,540 Then the overall factor becomes 1 over i, 1 over alpha prime. 937 01:02:11,540 --> 01:02:13,410 OK? 938 01:02:13,410 --> 01:02:15,890 And now, let me-- also making the sum 939 01:02:15,890 --> 01:02:19,640 over i explicit, so from 2 to D minus 1, 940 01:02:19,640 --> 01:02:23,380 all the transverse directions, and then I sum 941 01:02:23,380 --> 01:02:27,060 over m equal to 1 to infinity. 942 01:02:27,060 --> 01:02:29,590 And this, I have written down before. 943 01:02:29,590 --> 01:02:34,480 This is essentially just this m times the oscillation number. 944 01:02:34,480 --> 01:02:37,390 m N m i. 945 01:02:37,390 --> 01:02:40,050 OK? 946 01:02:40,050 --> 01:02:42,610 So now I have to write down the zero-point energy. 947 01:02:42,610 --> 01:02:45,820 So let me call it a zero. 948 01:02:45,820 --> 01:02:47,930 So the zero-point energy, so you just 949 01:02:47,930 --> 01:02:50,040 do it as a harmonic oscillator. 950 01:02:50,040 --> 01:02:52,170 So this is for the oscillator number. 951 01:02:52,170 --> 01:02:54,530 So for harmonic oscillator, we have m plus 1/2, OK? 952 01:02:54,530 --> 01:02:57,590 So for each oscillator, you have 1/2. 953 01:02:57,590 --> 01:03:04,200 So a zero must be equal to 1 over alpha prime sum over i sum 954 01:03:04,200 --> 01:03:09,626 over m then 1/2 times m. 955 01:03:09,626 --> 01:03:10,125 OK? 956 01:03:13,370 --> 01:03:14,840 So is this clear to you? 957 01:03:14,840 --> 01:03:15,440 Yes? 958 01:03:15,440 --> 01:03:17,830 AUDIENCE: So in here, we're not actually summing over i, 959 01:03:17,830 --> 01:03:18,330 are we? 960 01:03:18,330 --> 01:03:20,390 Is there an implicit sum over i? 961 01:03:20,390 --> 01:03:21,910 PROFESSOR: Oh yeah, oh, sorry. 962 01:03:21,910 --> 01:03:23,120 Here. 963 01:03:23,120 --> 01:03:25,584 Here, repeated index always means we summed. 964 01:03:25,584 --> 01:03:27,500 AUDIENCE: Even though they're both up indices? 965 01:03:27,500 --> 01:03:29,090 PROFESSOR: Yeah, yeah. 966 01:03:29,090 --> 01:03:33,318 Yeah, because we are working with Minkowski metric. 967 01:03:33,318 --> 01:03:33,984 AUDIENCE: I see. 968 01:03:33,984 --> 01:03:34,484 OK. 969 01:03:41,320 --> 01:03:42,220 PROFESSOR: OK? 970 01:03:42,220 --> 01:03:47,050 So essentially, you have the harmonic oscillator 971 01:03:47,050 --> 01:03:48,450 with a frequency m. 972 01:03:48,450 --> 01:03:48,950 OK? 973 01:03:48,950 --> 01:03:52,210 Because that's what each the-- yeah, 974 01:03:52,210 --> 01:03:55,185 I should have emphasized one step. 975 01:03:55,185 --> 01:03:56,540 Let me see. 976 01:03:59,484 --> 01:04:02,350 Yeah, I did not write it down explicitly here. 977 01:04:02,350 --> 01:04:05,320 But if you remind yourself, when you quantize a free quantum 978 01:04:05,320 --> 01:04:09,380 fields theory, the harmon-- now I 979 01:04:09,380 --> 01:04:11,900 have erased my harmonic expansion-- 980 01:04:11,900 --> 01:04:15,100 and in the harmonic expansion, under the nth mode, 981 01:04:15,100 --> 01:04:16,920 we have frequency n. 982 01:04:16,920 --> 01:04:17,420 OK? 983 01:04:17,420 --> 01:04:20,380 Yeah, say-- let me just add it here. 984 01:04:20,380 --> 01:04:33,690 So when we write down [? pi-- ?] say tau minus sigma-- yeah, 985 01:04:33,690 --> 01:04:35,500 it'll be some pre-factor. 986 01:04:35,500 --> 01:04:40,590 So this n, this just the frequency of each mode, 987 01:04:40,590 --> 01:04:43,690 OK, as a harmonic oscillator. 988 01:04:43,690 --> 01:04:46,750 So that's why, here, for the zero point [INAUDIBLE], 989 01:04:46,750 --> 01:04:51,700 so each one, each mode, contribute 1/2 m. 990 01:04:51,700 --> 01:04:53,320 OK? 991 01:04:53,320 --> 01:04:54,590 So is this clear to you? 992 01:05:00,080 --> 01:05:02,670 AUDIENCE: And you can also get that from the computation 993 01:05:02,670 --> 01:05:04,010 relationship? 994 01:05:04,010 --> 01:05:04,745 PROFESSOR: Yeah. 995 01:05:04,745 --> 01:05:05,806 AUDIENCE: [INAUDIBLE]. 996 01:05:05,806 --> 01:05:07,180 PROFESSOR: Yeah, let me just say, 997 01:05:07,180 --> 01:05:09,960 I want to write down the quantum version of this equation. 998 01:05:09,960 --> 01:05:12,910 The way I do it is that I write down the quantum version 999 01:05:12,910 --> 01:05:15,300 of the harmonic oscillator. 1000 01:05:15,300 --> 01:05:20,212 Treat each mode as a harmonic oscillator, and then the zero-- 1001 01:05:20,212 --> 01:05:22,295 then this will be just a sum of an infinite number 1002 01:05:22,295 --> 01:05:23,990 of harmonic oscillators. 1003 01:05:23,990 --> 01:05:27,920 And each harmonic oscillator have a frequency n. 1004 01:05:27,920 --> 01:05:32,230 So each harmonic oscillator have frequency n. 1005 01:05:32,230 --> 01:05:33,070 OK? 1006 01:05:33,070 --> 01:05:36,270 So each harmonic oscillator has a frequency n. 1007 01:05:36,270 --> 01:05:41,660 Then this is the oscillator number times the frequency. 1008 01:05:41,660 --> 01:05:46,600 And the zero mode is just 1/2 times the frequency. 1009 01:05:46,600 --> 01:05:47,970 Yeah. 1010 01:05:47,970 --> 01:05:49,630 Clear? 1011 01:05:49,630 --> 01:05:51,480 Good. 1012 01:05:51,480 --> 01:05:53,584 And similarly, we can write down the equation 1013 01:05:53,584 --> 01:05:54,500 for the closed string. 1014 01:06:05,920 --> 01:06:14,000 Again, sum over i sum over m not equal to zero, 1015 01:06:14,000 --> 01:06:19,600 or sum m equal to one to infinity, then m, 1016 01:06:19,600 --> 01:06:24,630 m N i plus m N m i tilde. 1017 01:06:24,630 --> 01:06:25,590 OK? 1018 01:06:25,590 --> 01:06:28,640 And then, again, plus a zero. 1019 01:06:28,640 --> 01:06:33,070 So this here, a zero, it's essentially the same. 1020 01:06:33,070 --> 01:06:40,510 Is the-- so now you have this equation. 1021 01:06:40,510 --> 01:06:46,260 So this is alpha 2 sum over i sum over m from minus 1 1022 01:06:46,260 --> 01:06:53,280 to infinity, from 1 to infinity, 1/2 m plus m. 1023 01:06:56,300 --> 01:07:00,963 So 1/2 m plus m because there's two modes here. 1024 01:07:00,963 --> 01:07:01,770 OK? 1025 01:07:01,770 --> 01:07:07,950 Yeah, maybe I should write m 1/2 plus 1/2. 1026 01:07:13,890 --> 01:07:14,390 Yeah? 1027 01:07:14,390 --> 01:07:16,265 AUDIENCE: I'm still confused about something. 1028 01:07:16,265 --> 01:07:18,345 So where exactly did the a zero come from? 1029 01:07:18,345 --> 01:07:20,547 Where is it defined before? 1030 01:07:20,547 --> 01:07:21,380 PROFESSOR: No no no. 1031 01:07:21,380 --> 01:07:24,160 No, a zero come from here, come from I'm 1032 01:07:24,160 --> 01:07:27,920 rewriting this equation as a quantum equation. 1033 01:07:27,920 --> 01:07:34,430 So this is-- so if you look at it here, here, 1034 01:07:34,430 --> 01:07:36,890 each one is a dagger and a. 1035 01:07:36,890 --> 01:07:39,450 So there's an ordering issue here. 1036 01:07:39,450 --> 01:07:42,640 And there's an ordering issue here. 1037 01:07:42,640 --> 01:07:44,490 So I'm just giving you a simple trick 1038 01:07:44,490 --> 01:07:48,130 to find out what is the-- how to resolve the ordering issue, 1039 01:07:48,130 --> 01:07:51,190 because each term is like a harmonic oscillator. 1040 01:07:51,190 --> 01:07:53,780 And then this will be just like, exactly 1041 01:07:53,780 --> 01:07:55,101 like a harmonic oscillator. 1042 01:07:55,101 --> 01:07:55,642 AUDIENCE: OK. 1043 01:07:55,642 --> 01:07:58,317 So you're just taking all those contributions and packaging it? 1044 01:07:58,317 --> 01:08:00,150 PROFESSOR: Yeah, that's right, that's right. 1045 01:08:00,150 --> 01:08:01,930 Yeah, just like you're writing down 1046 01:08:01,930 --> 01:08:03,610 the energy of a harmonic oscillator. 1047 01:08:03,610 --> 01:08:04,100 AUDIENCE: Gotcha. 1048 01:08:04,100 --> 01:08:04,600 OK. 1049 01:08:08,510 --> 01:08:09,300 PROFESSOR: Yes? 1050 01:08:09,300 --> 01:08:12,516 AUDIENCE: I'm a bit surprised that these terms contribute 1051 01:08:12,516 --> 01:08:14,748 to m squared, not to m. 1052 01:08:14,748 --> 01:08:19,020 They look like [INAUDIBLE] of a-- 1053 01:08:19,020 --> 01:08:19,939 PROFESSOR: Which one? 1054 01:08:19,939 --> 01:08:21,750 AUDIENCE: I mean [INAUDIBLE] so the a 1055 01:08:21,750 --> 01:08:24,760 zero, all zero-point energies? 1056 01:08:24,760 --> 01:08:27,920 Intuitively I would have thought that they would add up 1057 01:08:27,920 --> 01:08:30,776 to-- they would contribute to the mass of the string, 1058 01:08:30,776 --> 01:08:32,250 not to the mass squared. 1059 01:08:35,790 --> 01:08:40,333 PROFESSOR: It depends on where is your intuition come from. 1060 01:08:40,333 --> 01:08:43,750 AUDIENCE: I don't know, it-- is it-- 1061 01:08:43,750 --> 01:08:46,810 PROFESSOR: Yeah, this is a-- this is in the-- yeah. 1062 01:08:46,810 --> 01:08:53,790 I think, again, so, so here, I'm assuming the-- yeah, 1063 01:08:53,790 --> 01:08:57,140 let me just-- I understand where you've-- yeah, 1064 01:08:57,140 --> 01:08:59,430 let me just explain one more thing. 1065 01:08:59,430 --> 01:09:02,340 Let me just explain one more thing. 1066 01:09:02,340 --> 01:09:06,450 So that equation, which I write in that form, 1067 01:09:06,450 --> 01:09:07,934 m square equal to that thing. 1068 01:09:07,934 --> 01:09:09,170 OK? 1069 01:09:09,170 --> 01:09:14,700 And this should be understood as the 2 p plus p 1070 01:09:14,700 --> 01:09:21,380 minus-- minus p i square, say minus-- 1071 01:09:21,380 --> 01:09:23,800 yeah, minus p i squared equal to that thing. 1072 01:09:23,800 --> 01:09:24,859 OK? 1073 01:09:24,859 --> 01:09:28,830 And then this can also-- so this essentially gives you p minus. 1074 01:09:28,830 --> 01:09:32,010 You could do this, the rest. 1075 01:09:32,010 --> 01:09:35,880 And the p minus is the image in the light-cone frame. 1076 01:09:35,880 --> 01:09:38,310 And so, you are just [? conceding ?] the image, 1077 01:09:38,310 --> 01:09:39,599 but in the light-cone frame. 1078 01:09:42,449 --> 01:09:44,286 AUDIENCE: Why does that get p minus, sorry? 1079 01:09:44,286 --> 01:09:44,952 PROFESSOR: Yeah. 1080 01:09:44,952 --> 01:09:49,160 I'm just saying this equation itself-- think about how 1081 01:09:49,160 --> 01:09:52,270 we derived this equation. 1082 01:09:52,270 --> 01:09:55,150 This come from here. 1083 01:09:55,150 --> 01:09:55,960 Come from here. 1084 01:09:55,960 --> 01:09:59,960 Yeah, I'm doing it a little bit fast today. 1085 01:09:59,960 --> 01:10:01,770 So come from here. 1086 01:10:01,770 --> 01:10:05,050 So this equation should be considered 1087 01:10:05,050 --> 01:10:08,990 as a constrained equation which you solve for p minus. 1088 01:10:08,990 --> 01:10:13,756 And p minus is precisely the image in the light-cone frame. 1089 01:10:13,756 --> 01:10:18,660 Yeah, this is a precise image in the light-cone frame. 1090 01:10:18,660 --> 01:10:20,360 Yeah. 1091 01:10:20,360 --> 01:10:22,404 That's why. 1092 01:10:22,404 --> 01:10:22,904 Yeah. 1093 01:10:26,240 --> 01:10:27,781 So is it clear? 1094 01:10:27,781 --> 01:10:29,780 AUDIENCE: I need to think about it a little bit. 1095 01:10:29,780 --> 01:10:30,446 PROFESSOR: Yeah. 1096 01:10:30,446 --> 01:10:33,101 And so, that's, yeah. 1097 01:10:33,101 --> 01:10:33,600 Yeah. 1098 01:10:33,600 --> 01:10:34,750 This is a good question. 1099 01:10:34,750 --> 01:10:36,749 I should have emphasized this point a little bit 1100 01:10:36,749 --> 01:10:40,340 earlier, in jump from here to there. 1101 01:10:40,340 --> 01:10:40,840 Yeah. 1102 01:10:40,840 --> 01:10:46,650 You should think of this-- so this 1103 01:10:46,650 --> 01:10:49,410 is an equation which you can imagine it 1104 01:10:49,410 --> 01:10:51,310 as solve for p minus. 1105 01:10:51,310 --> 01:10:53,990 And p minus is the image in the light-cone frame. 1106 01:10:53,990 --> 01:10:57,140 And then all these contributions, 1107 01:10:57,140 --> 01:10:59,630 the right-hand side, can be considered as the contribution 1108 01:10:59,630 --> 01:11:01,560 to the light-cone image. 1109 01:11:01,560 --> 01:11:06,116 And then we're just repackaging it in terms of the mass term. 1110 01:11:06,116 --> 01:11:06,615 Yeah. 1111 01:11:09,819 --> 01:11:10,860 So now we have a problem. 1112 01:11:13,440 --> 01:11:15,390 Because if you look at the zero-point energy-- 1113 01:11:15,390 --> 01:11:18,510 so any other questions? 1114 01:11:18,510 --> 01:11:20,808 AUDIENCE: So the mass square, the mass 1115 01:11:20,808 --> 01:11:24,360 of the string [INAUDIBLE]. 1116 01:11:24,360 --> 01:11:26,640 PROFESSOR: The mass over the-- no no no no. 1117 01:11:26,640 --> 01:11:29,330 That's not the way you should think about-- sorry, say it 1118 01:11:29,330 --> 01:11:29,830 again? 1119 01:11:29,830 --> 01:11:32,070 AUDIENCE: Because you said p minus the energy, and then. 1120 01:11:32,070 --> 01:11:33,870 PROFESSOR: p minus is the light-cone energy. 1121 01:11:33,870 --> 01:11:34,650 AUDIENCE: Light-cone, yeah. 1122 01:11:34,650 --> 01:11:35,316 PROFESSOR: Yeah. 1123 01:11:35,316 --> 01:11:37,840 So the light-cone image-- yeah, the light-cone energy. 1124 01:11:37,840 --> 01:11:41,050 Yeah, this is a light-cone [? image. ?] Yes? 1125 01:11:41,050 --> 01:11:42,512 AUDIENCE: But then is-- what he was 1126 01:11:42,512 --> 01:11:47,120 asking is why the contribution from the zero point energy 1127 01:11:47,120 --> 01:11:50,451 is-- you are contributing to the m squared, right? 1128 01:11:50,451 --> 01:11:50,950 So. 1129 01:11:50,950 --> 01:11:52,190 PROFESSOR: Yeah. 1130 01:11:52,190 --> 01:11:56,700 My answer is that the zero-point energy is contributing 1131 01:11:56,700 --> 01:11:59,680 to the light-cone image. 1132 01:11:59,680 --> 01:12:03,870 It's contributing to the light-cone image. 1133 01:12:03,870 --> 01:12:06,680 Yeah, just repackage it in terms of the mass-squared term. 1134 01:12:12,006 --> 01:12:13,130 AUDIENCE: One more comment. 1135 01:12:13,130 --> 01:12:16,980 That [INAUDIBLE] in the [INAUDIBLE] case 1136 01:12:16,980 --> 01:12:19,890 and that you will find the [INAUDIBLE] p minus. 1137 01:12:19,890 --> 01:12:21,754 That is [INAUDIBLE] interpretation. 1138 01:12:21,754 --> 01:12:22,420 PROFESSOR: Yeah. 1139 01:12:22,420 --> 01:12:26,190 Just say because we are going to the light cone, 1140 01:12:26,190 --> 01:12:30,874 and in the light cone, the p minus is the image. 1141 01:12:30,874 --> 01:12:33,040 And then we repackage it into the mass-squared term. 1142 01:12:38,632 --> 01:12:41,770 Is it clear? 1143 01:12:41,770 --> 01:12:47,470 Yeah, so what he was asking is-- so his intuition is right, 1144 01:12:47,470 --> 01:12:51,660 because this-- at least in the light-cone gauge, 1145 01:12:51,660 --> 01:12:55,020 you can interpret it-- so this zero-point image 1146 01:12:55,020 --> 01:12:57,340 is contributing to something like image, 1147 01:12:57,340 --> 01:13:00,300 rather than something like mass square, 1148 01:13:00,300 --> 01:13:03,240 even though this equation is acting like a mass square. 1149 01:13:03,240 --> 01:13:04,620 Yeah. 1150 01:13:04,620 --> 01:13:07,550 That was my explanation to him. 1151 01:13:07,550 --> 01:13:08,050 Yeah? 1152 01:13:08,050 --> 01:13:09,383 AUDIENCE: So one other question. 1153 01:13:09,383 --> 01:13:12,450 If indeed there is an infinite constant [INAUDIBLE], 1154 01:13:12,450 --> 01:13:15,910 like there is normal quantum physics, 1155 01:13:15,910 --> 01:13:17,500 how does string theory get around 1156 01:13:17,500 --> 01:13:19,374 this issue of-- so it seems that you're still 1157 01:13:19,374 --> 01:13:22,408 looking at this issue with infinite energy in a vacuum. 1158 01:13:25,390 --> 01:13:27,057 PROFESSOR: We're not going into that. 1159 01:13:27,057 --> 01:13:28,390 Yeah, we're not going into that. 1160 01:13:32,010 --> 01:13:35,440 So this is a standard issue. 1161 01:13:35,440 --> 01:13:43,470 So now we see that you have-- so now you 1162 01:13:43,470 --> 01:13:45,780 see you have a sum of an infinite number of modes, 1163 01:13:45,780 --> 01:13:49,580 and each mode contributing to 1/2 m. 1164 01:13:49,580 --> 01:13:51,360 So each mode contribute to 1/2 m. 1165 01:13:58,840 --> 01:14:01,250 So this is apparently an infinite answer. 1166 01:14:01,250 --> 01:14:03,490 OK? 1167 01:14:03,490 --> 01:14:06,345 But now we have a trick. 1168 01:14:06,345 --> 01:14:07,345 But now we have a trick. 1169 01:14:10,390 --> 01:14:17,700 The trick is this sum is actually equal to-- 1170 01:14:21,137 --> 01:14:22,610 AUDIENCE: Oh my gosh. 1171 01:14:22,610 --> 01:14:26,540 [GIGGLING] 1172 01:14:26,540 --> 01:14:27,910 PROFESSOR: So we have a trick. 1173 01:14:27,910 --> 01:14:30,940 And the trick is that this is actually-- we equate it 1174 01:14:30,940 --> 01:14:34,110 to minus 1/2-- minus 1/12. 1175 01:14:34,110 --> 01:14:36,400 OK? 1176 01:14:36,400 --> 01:14:41,900 So there are many ways to justify this thing. 1177 01:14:41,900 --> 01:14:43,690 But I will not do it here. 1178 01:14:43,690 --> 01:14:45,950 I will only do it in one way. 1179 01:14:45,950 --> 01:14:49,480 One way, which is the quickest way, but maybe 1180 01:14:49,480 --> 01:14:52,280 it's most unsatisfying way physically. 1181 01:14:52,280 --> 01:14:55,450 But it's the-- mathematically, the quickest way. 1182 01:14:55,450 --> 01:14:57,540 OK? 1183 01:14:57,540 --> 01:14:59,702 And this is so-called zeta function regularization. 1184 01:15:13,600 --> 01:15:20,590 So typically-- so a trick we often 1185 01:15:20,590 --> 01:15:25,690 used in physics is that when you go 1186 01:15:25,690 --> 01:15:34,850 to-- so when you get an infinite quantity, what you do 1187 01:15:34,850 --> 01:15:39,800 is you slightly change the form of your quantity 1188 01:15:39,800 --> 01:15:40,790 so that this is finite. 1189 01:15:43,450 --> 01:15:45,910 And then you manage to take the limit. 1190 01:15:49,570 --> 01:15:52,340 To go back to the original limit. 1191 01:15:52,340 --> 01:15:57,270 And yeah, let me first just write down the answer for you. 1192 01:15:57,270 --> 01:15:59,167 And then I talk about philosophy. 1193 01:15:59,167 --> 01:16:01,185 [LAUGHTER] 1194 01:16:01,185 --> 01:16:02,810 PROFESSOR: So the key is a [INAUDIBLE]. 1195 01:16:02,810 --> 01:16:06,020 And let's define a function called the zeta function. 1196 01:16:06,020 --> 01:16:09,580 So this is the so-called Riemann zeta function. 1197 01:16:09,580 --> 01:16:13,776 And so this is sum of n equal to 1 to infinity 1 over n 1198 01:16:13,776 --> 01:16:14,810 to the power s. 1199 01:16:14,810 --> 01:16:17,340 OK? 1200 01:16:17,340 --> 01:16:25,470 So as you all know, this sum is only convergent 1201 01:16:25,470 --> 01:16:28,980 only for s greater than 1. 1202 01:16:28,980 --> 01:16:31,450 And it's a famous sum that when s equal to 1, 1203 01:16:31,450 --> 01:16:33,220 this is a logarithm divergent. 1204 01:16:33,220 --> 01:16:35,350 OK? 1205 01:16:35,350 --> 01:16:40,670 So this function have [? a pole. ?] So [? this ?] 1206 01:16:40,670 --> 01:16:42,540 thing over these [? a ?] function. 1207 01:16:42,540 --> 01:16:44,780 So what you can do is you can do the sum. 1208 01:16:44,780 --> 01:16:47,920 For s greater than 1, then you can actually-- we'll 1209 01:16:47,920 --> 01:16:51,300 call the analytic function letter s. 1210 01:16:51,300 --> 01:16:53,800 Use this as a definition for this function. 1211 01:16:53,800 --> 01:16:56,602 And this function is well defined for s greater than 1. 1212 01:16:56,602 --> 01:16:58,640 And then you find this function has a pole. 1213 01:17:03,650 --> 01:17:06,820 And the zeta equal to, say, 1. 1214 01:17:06,820 --> 01:17:08,560 At s, you could do 1. 1215 01:17:08,560 --> 01:17:13,280 Because 1 into this-- as a logarithm divergence, 1216 01:17:13,280 --> 01:17:17,692 yeah, for 1 plus 1/2, et cetera. 1217 01:17:17,692 --> 01:17:27,580 But this function-- or, I can do it here-- 1218 01:17:27,580 --> 01:17:29,676 but this function have an analytical condition. 1219 01:17:39,331 --> 01:17:41,080 But this function, this analytic function, 1220 01:17:41,080 --> 01:17:44,960 can be analytically continued beyond s equal to 1 1221 01:17:44,960 --> 01:17:47,310 to smaller value of s. 1222 01:17:47,310 --> 01:17:50,700 In particular, you find can be analytically continued 1223 01:17:50,700 --> 01:17:52,150 to minus 1. 1224 01:17:52,150 --> 01:17:54,250 So minus 1 is a situation, what we 1225 01:17:54,250 --> 01:17:58,600 are seeing-- looking at here, with s equal to minus 1, 1226 01:17:58,600 --> 01:18:00,149 this equal to that. 1227 01:18:00,149 --> 01:18:02,440 And then this function when you [INAUDIBLE] to minus 1, 1228 01:18:02,440 --> 01:18:06,770 you find it's actually finite, and then given by minus 1/12. 1229 01:18:06,770 --> 01:18:07,970 OK? 1230 01:18:07,970 --> 01:18:12,050 So this is a rationale for the manipulation. 1231 01:18:14,590 --> 01:18:18,830 So this is very similar-- this is 1232 01:18:18,830 --> 01:18:21,620 very similar to dimensional regularization 1233 01:18:21,620 --> 01:18:24,720 you used in quantum field theory. 1234 01:18:24,720 --> 01:18:28,990 And you see, certain integrals are divergent. 1235 01:18:28,990 --> 01:18:32,680 Say, certain integrals are divergent in four dimensions. 1236 01:18:32,680 --> 01:18:36,750 And then, the trick we did, when you do your quantum field 1237 01:18:36,750 --> 01:18:43,140 theory, is that we promote the dimension as a variable. 1238 01:18:43,140 --> 01:18:46,890 Rather than equal to 4, we called it general d. 1239 01:18:46,890 --> 01:18:49,460 And then, for that general d, then there 1240 01:18:49,460 --> 01:18:52,275 exists some range of that d, which 1241 01:18:52,275 --> 01:18:54,350 is that integral is finite. 1242 01:18:54,350 --> 01:18:56,880 And then you can evaluate it as an analytic function of d. 1243 01:18:56,880 --> 01:19:00,270 And [? analytic ?] continue to d equal to 4. 1244 01:19:00,270 --> 01:19:04,990 And sometimes that just give you a finite answer. 1245 01:19:04,990 --> 01:19:07,240 And sometimes that still gives you a divergent answer, 1246 01:19:07,240 --> 01:19:09,149 and then you need to do [? regularization. ?] 1247 01:19:09,149 --> 01:19:10,690 But sometimes when you do that trick, 1248 01:19:10,690 --> 01:19:12,960 you just get a finite answer, and then 1249 01:19:12,960 --> 01:19:15,660 there's even no need to do a [? regularization. ?] 1250 01:19:15,660 --> 01:19:17,540 That trick just works. 1251 01:19:17,540 --> 01:19:21,370 So this is exactly the same as that trick. 1252 01:19:21,370 --> 01:19:25,920 So this trick is exactly in the same sense as that. 1253 01:19:25,920 --> 01:19:29,210 But of course, this is only a mathematical trick. 1254 01:19:29,210 --> 01:19:32,420 And to justify it physically, you 1255 01:19:32,420 --> 01:19:35,320 have to think about the-- to justify it physically, 1256 01:19:35,320 --> 01:19:41,220 it's the same as the philosophy for your [INAUDIBLE] 1257 01:19:41,220 --> 01:19:43,730 in general quantum field theories. 1258 01:19:43,730 --> 01:19:49,280 It's that this system have many, many high-energy modes. 1259 01:19:49,280 --> 01:19:52,140 So the divergence come from very large m, 1260 01:19:52,140 --> 01:19:55,170 have very, very high-energy modes. 1261 01:19:55,170 --> 01:19:58,360 And so the justification is that-- 1262 01:19:58,360 --> 01:20:01,942 so those high-energy modes, so the low-energy physics, 1263 01:20:01,942 --> 01:20:03,900 which is related to the zero point [? energy ?] 1264 01:20:03,900 --> 01:20:05,880 [? excitement, ?] should be insensitive to the physics 1265 01:20:05,880 --> 01:20:07,060 of those high-energy modes. 1266 01:20:07,060 --> 01:20:09,080 Then you can subtract the infinity, et cetera. 1267 01:20:11,800 --> 01:20:13,980 So that philosophy's the same as what you normally 1268 01:20:13,980 --> 01:20:17,170 do in the [? regularization ?] quantum fields theory. 1269 01:20:17,170 --> 01:20:18,290 OK. 1270 01:20:18,290 --> 01:20:21,198 Any other questions here? 1271 01:20:21,198 --> 01:20:24,315 AUDIENCE: So can we say that the true physical theory doesn't 1272 01:20:24,315 --> 01:20:27,150 have any divergences, but it's just this effective theory 1273 01:20:27,150 --> 01:20:31,980 that kind of works out, has divergences that we then, 1274 01:20:31,980 --> 01:20:35,520 kind of, get rid of by the strings? 1275 01:20:35,520 --> 01:20:40,280 But the real physical theory, that real wouldn't, shouldn't 1276 01:20:40,280 --> 01:20:42,120 wouldn't have any divergences? 1277 01:20:42,120 --> 01:20:47,480 PROFESSOR: So you just say the physics-- say 1278 01:20:47,480 --> 01:20:49,550 this particular quantity does not 1279 01:20:49,550 --> 01:20:53,810 depend on the details of your UV physics. 1280 01:20:53,810 --> 01:20:59,520 So this divergence itself comes from a particular assumption 1281 01:20:59,520 --> 01:21:02,710 of the UV physics. 1282 01:21:02,710 --> 01:21:06,090 But you can modify your UV physics in a certain way, 1283 01:21:06,090 --> 01:21:07,540 and this particular answer should 1284 01:21:07,540 --> 01:21:10,880 be independent of those details of that UV physics. 1285 01:21:10,880 --> 01:21:11,380 Yeah. 1286 01:21:11,380 --> 01:21:15,560 So this is a standard philosophy behind the regularization. 1287 01:21:15,560 --> 01:21:17,740 And so you can justify this answer 1288 01:21:17,740 --> 01:21:20,940 using many different ways. 1289 01:21:20,940 --> 01:21:24,160 You can also justify it-- so you can also 1290 01:21:24,160 --> 01:21:27,489 just put some constant in there, and then 1291 01:21:27,489 --> 01:21:29,405 impose some other self-consistency conditions, 1292 01:21:29,405 --> 01:21:30,821 and you can determine this answer. 1293 01:21:30,821 --> 01:21:33,710 Anyway, there are many ways you can derive this answer. 1294 01:21:33,710 --> 01:21:36,690 And this is one way, or the one quickest way of doing it, 1295 01:21:36,690 --> 01:21:39,022 so we are just do it here. 1296 01:21:39,022 --> 01:21:39,980 So any other questions? 1297 01:21:44,020 --> 01:21:44,520 Good. 1298 01:21:44,520 --> 01:21:45,500 OK. 1299 01:21:45,500 --> 01:21:50,620 So now, from here, we can find that for the open string, 1300 01:21:50,620 --> 01:21:54,780 the a zero is just equal to minus, d minus 2 divided 1301 01:21:54,780 --> 01:22:00,250 by 24, 1 over alpha prime, for the open. 1302 01:22:00,250 --> 01:22:05,980 And then for the closed string, is equal to minus d minus 2 1303 01:22:05,980 --> 01:22:11,440 divided by 24 alpha prime over 4. 1304 01:22:11,440 --> 01:22:12,940 Closed. 1305 01:22:12,940 --> 01:22:14,490 OK?