1 00:00:00,080 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high-quality educational resources for free. 5 00:00:10,150 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,600 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,600 --> 00:00:17,260 at ocw.mit.edu. 8 00:00:21,780 --> 00:00:23,280 HONG LIU: So let me first remind you 9 00:00:23,280 --> 00:00:26,430 what we did at the end of last lecture. 10 00:00:29,050 --> 00:00:31,380 Say, suppose you can see the string moving 11 00:00:31,380 --> 00:00:36,390 in some spacetime M. So it's the following metric, given 12 00:00:36,390 --> 00:00:41,420 by g mu nu and the coordinate, which I've denoted by X nu. 13 00:00:41,420 --> 00:00:48,600 So nu typically will go from, say, mu nu, go from 0, 1 to D 14 00:00:48,600 --> 00:00:49,640 minus 1. 15 00:00:49,640 --> 00:00:51,660 And capital D will be the [? number of ?] 16 00:00:51,660 --> 00:00:54,470 spacetime dimension. 17 00:00:54,470 --> 00:01:00,550 And then the worldsheets will be characterized by this X mu 18 00:01:00,550 --> 00:01:03,415 as a function of worldsheet coordinate sigma tau. 19 00:01:03,415 --> 00:01:04,965 Then this describe the [? mapping ?] 20 00:01:04,965 --> 00:01:08,430 of a two-dimensional surface embedded in the full spacetime. 21 00:01:11,510 --> 00:01:12,920 And then we also use the notation 22 00:01:12,920 --> 00:01:15,380 of X mu equal to sigma a, and sigma a 23 00:01:15,380 --> 00:01:20,390 will be sigma 0, sigma 1, and equal to tau and sigma, OK? 24 00:01:20,390 --> 00:01:23,760 And then the simplest action, which 25 00:01:23,760 --> 00:01:26,490 have a clear geometric meaning, is essentially 26 00:01:26,490 --> 00:01:31,540 just the action that is given by the area of the surface. 27 00:01:31,540 --> 00:01:34,640 Given by the area of the surface. 28 00:01:34,640 --> 00:01:38,300 And this delta h is the induced metric. 29 00:01:38,300 --> 00:01:42,480 So if you have embedding in this spacetime, 30 00:01:42,480 --> 00:01:46,020 then this embedding will induce a longitudinal metric 31 00:01:46,020 --> 00:01:47,060 on the worldsheets. 32 00:01:47,060 --> 00:01:51,149 So this h is the induced metric on the worldsheet. 33 00:01:51,149 --> 00:01:52,940 And then if you take the determinant of it, 34 00:01:52,940 --> 00:01:55,430 then it essentially gives you the area. 35 00:01:55,430 --> 00:01:57,780 [? Adamant ?] on the worldsheets. 36 00:01:57,780 --> 00:02:00,410 But we also showed that classically, this 37 00:02:00,410 --> 00:02:02,310 is action is equivalent to this action. 38 00:02:07,860 --> 00:02:11,770 You can rewrite it by introducing something 39 00:02:11,770 --> 00:02:14,706 like a Lagrangian multiplier like gamma ab. 40 00:02:14,706 --> 00:02:18,270 And then by eliminate gamma ab, then you cover this. 41 00:02:18,270 --> 00:02:19,100 OK? 42 00:02:19,100 --> 00:02:23,569 And the equation of motion for the gamma ab-- yeah, 43 00:02:23,569 --> 00:02:25,610 if you solve the equation for gamma ab, the thing 44 00:02:25,610 --> 00:02:28,600 you will get something like this, lambda hab. 45 00:02:28,600 --> 00:02:31,889 And lambda is arbitrary constant. 46 00:02:31,889 --> 00:02:32,680 Arbitrary function. 47 00:02:40,070 --> 00:02:42,210 Yeah, say if you solve the equation of motion 48 00:02:42,210 --> 00:02:44,300 for gamma ab, and then that's what you get. 49 00:02:49,250 --> 00:02:52,140 So for this final form, so this is 50 00:02:52,140 --> 00:02:56,010 the action we will use to quantize it, 51 00:02:56,010 --> 00:03:01,280 is because this action have a nice polynomial form. 52 00:03:01,280 --> 00:03:02,570 Have a nice polynomial form. 53 00:03:02,570 --> 00:03:06,140 Not like this action, which is a square root. 54 00:03:06,140 --> 00:03:07,860 It's always awkward. 55 00:03:07,860 --> 00:03:11,080 And so this have a nice polynomial form. 56 00:03:11,080 --> 00:03:15,080 And then when you quantize the action, 57 00:03:15,080 --> 00:03:18,930 say if you do path integral quantization, 58 00:03:18,930 --> 00:03:37,640 then you need to integrate both dynamic variables. 59 00:03:40,580 --> 00:03:47,700 So exponential i Sp gamma on X. OK. 60 00:03:47,700 --> 00:03:49,782 So both are dynamic variables. 61 00:03:49,782 --> 00:03:51,740 When you quantize, you have to do path integral 62 00:03:51,740 --> 00:03:54,170 over all of them. 63 00:03:54,170 --> 00:03:57,750 And so let me just say a little bit more 64 00:03:57,750 --> 00:04:00,050 about the geometric, which we mentioned at the end 65 00:04:00,050 --> 00:04:03,820 in last lecture, the geometric meaning of this action. 66 00:04:03,820 --> 00:04:07,340 So if you just look at this action itself, 67 00:04:07,340 --> 00:04:12,000 this just like a scalar field in a two-dimensional curved 68 00:04:12,000 --> 00:04:14,311 spacetime. 69 00:04:14,311 --> 00:04:16,019 And this two-dimensional curved spacetime 70 00:04:16,019 --> 00:04:19,490 have a longitudinal metric, which is given by gamma. 71 00:04:19,490 --> 00:04:22,210 And the scalar field is just this X, OK? 72 00:04:22,210 --> 00:04:25,510 So this gamma can be considered as some kind of intrinsic. 73 00:04:25,510 --> 00:04:30,170 And we can see that as an intrinsic metric 74 00:04:30,170 --> 00:04:31,950 on the worldsheet. 75 00:04:31,950 --> 00:04:39,530 And then this X still describes this embedding 76 00:04:39,530 --> 00:04:42,270 of this worldsheet in spacetime. 77 00:04:42,270 --> 00:04:45,040 But X, from the point of view of this two-dimensional 78 00:04:45,040 --> 00:04:47,750 worldsheet, it just behave like a scalar field. 79 00:04:47,750 --> 00:04:50,230 It just behave like a scalar field. 80 00:04:50,230 --> 00:04:52,690 So when we quantize this action, so 81 00:04:52,690 --> 00:04:58,340 when we do the path integral, if you just 82 00:04:58,340 --> 00:05:01,080 do the path integral over DX, then 83 00:05:01,080 --> 00:05:03,670 this is just like a standard two-dimensional quantum field 84 00:05:03,670 --> 00:05:05,497 theory, OK? 85 00:05:05,497 --> 00:05:07,580 Just like a standard two-dimensional quantum field 86 00:05:07,580 --> 00:05:07,990 theory. 87 00:05:07,990 --> 00:05:09,740 But what's unusual here, when you quantize 88 00:05:09,740 --> 00:05:13,370 a string, because the gamma is also a dynamic variable, 89 00:05:13,370 --> 00:05:15,860 you also have to integrate over gamma. 90 00:05:15,860 --> 00:05:18,010 And this make it more nontrivial, 91 00:05:18,010 --> 00:05:21,720 and then this is no longer just like quantizing the scalar 92 00:05:21,720 --> 00:05:22,640 fields. 93 00:05:22,640 --> 00:05:26,080 And because you are integrating over all possible intrinsic 94 00:05:26,080 --> 00:05:27,740 metrics on the worldsheet. 95 00:05:27,740 --> 00:05:36,080 So this is like quantizing 2D gravity. 96 00:05:38,870 --> 00:05:41,939 When you integrate over all possible metric, 97 00:05:41,939 --> 00:05:43,480 then this is like quantizing gravity. 98 00:05:43,480 --> 00:05:54,410 But this is only the gravity on the worldsheet, 99 00:05:54,410 --> 00:05:59,480 coupled to D scalar fields. 100 00:06:03,800 --> 00:06:04,300 OK? 101 00:06:08,020 --> 00:06:08,610 Yes. 102 00:06:08,610 --> 00:06:10,520 AUDIENCE: So in what sense is it 2D gravity? 103 00:06:10,520 --> 00:06:14,212 Because the metric is like a D by D object. 104 00:06:14,212 --> 00:06:14,920 HONG LIU: No, no. 105 00:06:17,980 --> 00:06:19,780 Gamma is a two-dimensional metric. 106 00:06:19,780 --> 00:06:24,080 It's intrinsic metric on the worldsheet. 107 00:06:24,080 --> 00:06:27,720 Yeah, so it's two by two. 108 00:06:27,720 --> 00:06:30,370 [? ab's ?] only goes two by two. 109 00:06:30,370 --> 00:06:34,060 ab is only going from 0 to 1. 110 00:06:34,060 --> 00:06:39,360 Yeah, so ab only runs over the worldsheet at 0 to 1. 111 00:06:39,360 --> 00:06:40,990 Yeah. 112 00:06:40,990 --> 00:06:42,790 So once you write that form, then 113 00:06:42,790 --> 00:06:45,930 you introduce an intrinsic metric on the worldsheet. 114 00:06:45,930 --> 00:06:48,370 They introduce an intrinsic curved spacetime 115 00:06:48,370 --> 00:06:50,770 on the worldsheet. 116 00:06:50,770 --> 00:06:53,974 And when you integrate over all possible metrics, 117 00:06:53,974 --> 00:06:55,890 then that's corresponding to take into account 118 00:06:55,890 --> 00:06:59,150 the fluctuations in the metric, from this two-dimensional point 119 00:06:59,150 --> 00:06:59,990 of view. 120 00:06:59,990 --> 00:07:03,410 So that's why we say this is like quantizing 2D gravity. 121 00:07:03,410 --> 00:07:05,850 By gravity we just mean the spacetime. 122 00:07:05,850 --> 00:07:07,950 OK, is this clear? 123 00:07:07,950 --> 00:07:11,560 So now we see the question of quantizing 124 00:07:11,560 --> 00:07:15,270 the string theory, essentially [INAUDIBLE] 125 00:07:15,270 --> 00:07:18,850 to quantize this two-dimensional gravity on worldsheets 126 00:07:18,850 --> 00:07:21,039 coupled to these scalar fields. 127 00:07:21,039 --> 00:07:22,955 And these scalar fields describe the embedding 128 00:07:22,955 --> 00:07:26,000 of your worldsheet. 129 00:07:26,000 --> 00:07:27,680 OK? 130 00:07:27,680 --> 00:07:31,960 So we will consider a simple example. 131 00:07:31,960 --> 00:07:33,445 We will consider the example which 132 00:07:33,445 --> 00:07:37,140 M is just Minkowski spacetime. 133 00:07:37,140 --> 00:07:39,970 Three-dimensional Minkowski spacetime, OK? 134 00:07:39,970 --> 00:07:45,085 So in this case, g mu nu [? come ?] [? equal ?] to eta 135 00:07:45,085 --> 00:07:45,710 [INAUDIBLE] nu. 136 00:07:50,360 --> 00:07:54,270 And then this Polyakov action can also be simplified. 137 00:07:57,090 --> 00:08:01,730 So this SP now just become 1 over [? 4 ?] 138 00:08:01,730 --> 00:08:11,300 alpha prime, d square sigma, gamma ab, partial a X mu, 139 00:08:11,300 --> 00:08:13,430 partial b X mu. 140 00:08:13,430 --> 00:08:16,850 So now when I write this X mu, X mu 141 00:08:16,850 --> 00:08:18,790 should be understood that they are 142 00:08:18,790 --> 00:08:22,440 contracted by the standard Minkowski metric. 143 00:08:25,790 --> 00:08:32,880 So now, this is really, if you just look at the X, 144 00:08:32,880 --> 00:08:34,760 this is like free scalar field series, 145 00:08:34,760 --> 00:08:38,289 because of the dependence on x, purely quadratic. 146 00:08:38,289 --> 00:08:47,800 So this is like a free scalar field in curved spacetime, 147 00:08:47,800 --> 00:08:51,070 in curved worldsheet, 2D worldsheet. 148 00:08:59,560 --> 00:09:03,660 So this still slightly unusual thing about this scalar field 149 00:09:03,660 --> 00:09:06,720 theory, from our standard one. 150 00:09:10,720 --> 00:09:18,230 Actually, I might be missing a minus sign somewhere. 151 00:09:18,230 --> 00:09:20,400 I might be missing overall minus sign. 152 00:09:20,400 --> 00:09:22,740 I think I'm missing overall minus sign. 153 00:09:22,740 --> 00:09:23,830 Just let me put it here. 154 00:09:23,830 --> 00:09:26,470 I think I'm missing overall minus signs. 155 00:09:26,470 --> 00:09:28,780 For our current purpose, it does not matter. 156 00:09:28,780 --> 00:09:31,720 But I think for the later purpose we need it. 157 00:09:31,720 --> 00:09:34,060 Because if you can see the standard field theory 158 00:09:34,060 --> 00:09:37,260 using this minus [? 1 ?] metric, then I need the minus sign. 159 00:09:37,260 --> 00:09:40,050 OK? 160 00:09:40,050 --> 00:09:44,190 So this is just like a free scalar field theory 161 00:09:44,190 --> 00:09:48,080 in this 2D curved spacetime. 162 00:09:48,080 --> 00:09:51,717 But the slightly [? thing ?] unusual about this one, 163 00:09:51,717 --> 00:09:53,050 so there are two unusual things. 164 00:09:53,050 --> 00:09:56,330 One is that we also need to integrate gamma. 165 00:09:56,330 --> 00:09:57,830 But there's a lot of unusual things, 166 00:09:57,830 --> 00:10:01,460 even from the perspective of scalar field theory. 167 00:10:01,460 --> 00:10:07,440 And so if you look at the 0-th component of X, 168 00:10:07,440 --> 00:10:15,970 because eta 0,0-- we are using the eta equal to minus 1, 1. 169 00:10:15,970 --> 00:10:19,310 Because eta 0,0 is minus 1. 170 00:10:19,310 --> 00:10:27,280 So the 0-th component of X, we have a opposite kinetic term 171 00:10:27,280 --> 00:10:30,780 to the other component, OK? 172 00:10:30,780 --> 00:10:35,810 So you might have learned in your quantum field theory class 173 00:10:35,810 --> 00:10:40,730 that actually you cannot change the sign of your kinetic term. 174 00:10:40,730 --> 00:10:43,830 Otherwise you get instability, because then your Hamiltonian 175 00:10:43,830 --> 00:10:46,130 will be unbounded from below, et cetera. 176 00:10:46,130 --> 00:10:52,670 So this action, so our story here have two unusual features. 177 00:10:52,670 --> 00:10:55,440 One, you have to integrate over gamma. 178 00:10:55,440 --> 00:10:57,610 And the second is that [? the 1 over ?] 179 00:10:57,610 --> 00:11:01,560 the scalar field have the wrong kinetic term. 180 00:11:01,560 --> 00:11:02,370 OK. 181 00:11:02,370 --> 00:11:04,260 The sign with kinetic term is wrong. 182 00:11:04,260 --> 00:11:06,620 And we will see that these two features actually 183 00:11:06,620 --> 00:11:07,960 solve for each other. 184 00:11:07,960 --> 00:11:11,147 And this is actually a consistent theory. 185 00:11:11,147 --> 00:11:12,480 So this is actually consistency. 186 00:11:15,960 --> 00:11:18,640 Any questions at this point? 187 00:11:18,640 --> 00:11:23,410 OK, so before you do any theory, what you should do? 188 00:11:23,410 --> 00:11:26,630 So before you work on any theory, 189 00:11:26,630 --> 00:11:29,154 what should you do first? 190 00:11:29,154 --> 00:11:31,400 AUDIENCE: [INAUDIBLE]. 191 00:11:31,400 --> 00:11:33,830 HONG LIU: No, no, that's too far away. 192 00:11:33,830 --> 00:11:36,320 Just the first thing you should do. 193 00:11:36,320 --> 00:11:38,967 After you write down the action [? Lagrange, ?] what should do? 194 00:11:38,967 --> 00:11:40,850 AUDIENCE: Solve the equation of motion. 195 00:11:40,850 --> 00:11:44,700 HONG LIU: That's still too far away. 196 00:11:44,700 --> 00:11:47,700 AUDIENCE: Check units. 197 00:11:47,700 --> 00:11:49,860 HONG LIU: That's something what you should do. 198 00:11:49,860 --> 00:11:51,783 Yeah, but that is done in kindergarten, 199 00:11:51,783 --> 00:11:53,635 so it's not included here. 200 00:11:56,880 --> 00:11:58,430 AUDIENCE: Constants of motion. 201 00:11:58,430 --> 00:12:01,550 HONG LIU: That's close. 202 00:12:01,550 --> 00:12:03,470 What is responsible for constant of motion? 203 00:12:03,470 --> 00:12:05,249 [INTERPOSING VOICES] 204 00:12:05,249 --> 00:12:06,040 AUDIENCE: Symmetry. 205 00:12:06,040 --> 00:12:07,664 HONG LIU: Yeah, that's right, symmetry. 206 00:12:07,664 --> 00:12:09,400 You first have to analyze the symmetries. 207 00:12:09,400 --> 00:12:13,000 OK, so that's what we will do. 208 00:12:13,000 --> 00:12:16,130 OK, let me call this equation three, 209 00:12:16,130 --> 00:12:18,910 just continue from my notation of last week. 210 00:12:21,840 --> 00:12:24,210 Yeah. 211 00:12:24,210 --> 00:12:28,640 So first we should analyze the symmetry of this action. 212 00:12:40,460 --> 00:12:49,550 So first, so this is a mapping to a Minkowski spacetime. 213 00:12:49,550 --> 00:12:51,340 The Minkowski [? spacetime ?] itself 214 00:12:51,340 --> 00:12:53,140 have a Poincare symmetry. 215 00:12:53,140 --> 00:12:56,520 And that Poincare symmetry will be reflected here. 216 00:12:56,520 --> 00:13:07,580 First we have the Poincare symmetry of X mu. 217 00:13:07,580 --> 00:13:11,320 In other words-- and this action is invariant-- 218 00:13:11,320 --> 00:13:18,660 if you take X mu to go to X mu plus a constant. 219 00:13:18,660 --> 00:13:20,250 OK, so a mu is a constant. 220 00:13:24,660 --> 00:13:29,190 So it's obvious this is invariant because everything 221 00:13:29,190 --> 00:13:31,600 on [? X ?] we said derivative. 222 00:13:31,600 --> 00:13:33,395 So if we shift by X mu with a constant, 223 00:13:33,395 --> 00:13:36,550 then I'll call that a symmetry. 224 00:13:36,550 --> 00:13:41,205 And also, this is invariant under Lorentz symmetry. 225 00:13:45,760 --> 00:13:46,340 OK. 226 00:13:46,340 --> 00:13:53,840 So lambda is the constant Lorentz transformation. 227 00:13:58,020 --> 00:13:59,230 OK? 228 00:13:59,230 --> 00:14:06,510 So this is also self-evident, because of this contraction, 229 00:14:06,510 --> 00:14:09,700 if you make a constant Lorentzian transformation, 230 00:14:09,700 --> 00:14:11,450 you don't see the derivatives. 231 00:14:11,450 --> 00:14:14,400 And then this is contracted as Lorentz scalar, 232 00:14:14,400 --> 00:14:15,530 then this is invariant. 233 00:14:15,530 --> 00:14:16,030 OK? 234 00:14:22,970 --> 00:14:25,335 So this is the first set of symmetry. 235 00:14:27,970 --> 00:14:31,470 The second set of symmetry essentially by construction, 236 00:14:31,470 --> 00:14:36,780 because we are writing this as a scalar 237 00:14:36,780 --> 00:14:39,730 field in the curved spacetime. 238 00:14:39,730 --> 00:14:46,325 Then this is invariant in the two-dimensional coordinate 239 00:14:46,325 --> 00:14:46,950 transformation. 240 00:14:54,530 --> 00:14:57,250 And in other words, this is invariant 241 00:14:57,250 --> 00:15:03,920 on the reparameterization of the worldsheet. 242 00:15:07,840 --> 00:15:09,815 OK. 243 00:15:09,815 --> 00:15:11,690 So this is essentially a less [? desirable ?] 244 00:15:11,690 --> 00:15:16,960 condition, which should satisfy by any string action, 245 00:15:16,960 --> 00:15:22,110 because as we said before, sigma tau, 246 00:15:22,110 --> 00:15:24,179 you just parameterize your worldsheet. 247 00:15:24,179 --> 00:15:25,720 And you actually should not depend on 248 00:15:25,720 --> 00:15:27,810 how you parameterize it, and that parameterization 249 00:15:27,810 --> 00:15:30,180 should be arbitrary. 250 00:15:30,180 --> 00:15:32,430 You should be able to change your parameterization. 251 00:15:32,430 --> 00:15:39,630 Because if you think about the surface in spacetime, 252 00:15:39,630 --> 00:15:43,620 no matter, you can say parameterization cause 253 00:15:43,620 --> 00:15:46,290 [INAUDIBLE] agreed on the surface. 254 00:15:46,290 --> 00:15:48,811 Whether you do it this way, or you do it in some other way, 255 00:15:48,811 --> 00:15:51,102 if you're not changing the surface, the property of the 256 00:15:51,102 --> 00:15:52,340 surface itself. 257 00:15:52,340 --> 00:15:54,510 So that means the action itself should not 258 00:15:54,510 --> 00:15:57,710 be invariant on the reparameterization 259 00:15:57,710 --> 00:15:58,580 of your worldsheet. 260 00:15:58,580 --> 00:16:00,330 And the parameterization of the worldsheet 261 00:16:00,330 --> 00:16:04,540 translates into just arbitrary coordinate transformations. 262 00:16:04,540 --> 00:16:07,480 So sigma and tau should be able to, say, 263 00:16:07,480 --> 00:16:12,590 if you make arbitrary coordinate transformation, 264 00:16:12,590 --> 00:16:15,190 and the action should be invariant. 265 00:16:15,190 --> 00:16:16,320 OK. 266 00:16:16,320 --> 00:16:19,750 And this is by construction. 267 00:16:19,750 --> 00:16:23,030 And this is action by construction in [? Lorentz. ?] 268 00:16:23,030 --> 00:16:24,440 And this number goes to action. 269 00:16:24,440 --> 00:16:27,217 And Polyakov action we obtained by rewriting 270 00:16:27,217 --> 00:16:28,300 this number called action. 271 00:16:28,300 --> 00:16:30,475 It's also about construction. 272 00:16:30,475 --> 00:16:32,225 Because its [? manner fits ?] because this 273 00:16:32,225 --> 00:16:35,140 is in [INAUDIBLE] in the curved space actions for scalar 274 00:16:35,140 --> 00:16:35,850 fields. 275 00:16:35,850 --> 00:16:37,940 So this automatically invariant under 276 00:16:37,940 --> 00:16:40,952 the coordinate transformation. 277 00:16:40,952 --> 00:16:42,660 And under this coordinate transformation, 278 00:16:42,660 --> 00:16:46,440 this X mu just transformed [? my ?] scalar field. 279 00:16:46,440 --> 00:16:48,720 So the scalar field transforms. 280 00:16:48,720 --> 00:16:53,410 Say as the following is going to X mu prime. 281 00:16:53,410 --> 00:16:56,560 And X mu prime, we've evaluated here 282 00:16:56,560 --> 00:16:59,540 that the new coordinate tau prime and sigma prime 283 00:16:59,540 --> 00:17:04,859 should be the same as your original X evaluated 284 00:17:04,859 --> 00:17:08,760 at the original location. 285 00:17:08,760 --> 00:17:12,560 So this is essentially the definition of a scalar field. 286 00:17:12,560 --> 00:17:16,400 You see that when you make a coordinate transformation, 287 00:17:16,400 --> 00:17:21,354 if you follow the point, the value of X does not change. 288 00:17:21,354 --> 00:17:23,020 Because when you go to the new location, 289 00:17:23,020 --> 00:17:26,011 the X prime evaluated at new location 290 00:17:26,011 --> 00:17:27,760 should be the same as your original value. 291 00:17:27,760 --> 00:17:28,396 OK. 292 00:17:28,396 --> 00:17:29,210 Yes. 293 00:17:29,210 --> 00:17:31,418 AUDIENCE: So basically we have a few more [INAUDIBLE] 294 00:17:31,418 --> 00:17:33,780 variants in real space, and the worldsheet. 295 00:17:33,780 --> 00:17:35,495 That's like-- 296 00:17:35,495 --> 00:17:38,120 HONG LIU: No, no, this is not if your [? morphism ?] invariant. 297 00:17:38,120 --> 00:17:39,640 No, this is not [? different ?] [? morphism ?] invariant. 298 00:17:39,640 --> 00:17:41,250 AUDIENCE: What about in a special case, though? 299 00:17:41,250 --> 00:17:42,680 HONG LIU: No, this is not [? different ?] [? morphism. ?] 300 00:17:42,680 --> 00:17:45,013 No, this is not different [? morphism. ?] This is global 301 00:17:45,013 --> 00:17:45,520 symmetry. 302 00:17:45,520 --> 00:17:47,852 AUDIENCE: Oh, yeah, OK. 303 00:17:47,852 --> 00:17:51,970 AUDIENCE: And also, did we ever consider curved real space? 304 00:17:51,970 --> 00:17:54,285 HONG LIU: Yeah, you can consider-- well, curved space. 305 00:17:57,930 --> 00:18:00,780 But right now we are at the high school level of string theory 306 00:18:00,780 --> 00:18:03,920 class, and if you go to curved space, 307 00:18:03,920 --> 00:18:06,380 there should be college in that level of string theory. 308 00:18:06,380 --> 00:18:08,287 It's beyond what we are doing right now. 309 00:18:08,287 --> 00:18:10,120 AUDIENCE: That's [INAUDIBLE] curve of space, 310 00:18:10,120 --> 00:18:12,770 we don't have Poincare. 311 00:18:12,770 --> 00:18:14,110 Will we have Poincare? 312 00:18:14,110 --> 00:18:16,360 HONG LIU: Yeah, then you don't have Poincare symmetry. 313 00:18:16,360 --> 00:18:19,210 So this is specific to this action. 314 00:18:19,210 --> 00:18:22,930 But this is [INAUDIBLE] general. 315 00:18:22,930 --> 00:18:25,660 So, of course, under this coordinate transformation, 316 00:18:25,660 --> 00:18:30,000 this gamma ab should transform as a tensor, 317 00:18:30,000 --> 00:18:36,910 as in standard [? in GR. ?] This evaluated gamma prime you 318 00:18:36,910 --> 00:18:41,770 evaluate at a new point, should be [? related ?] as a tensor. 319 00:18:54,710 --> 00:18:56,230 OK. 320 00:18:56,230 --> 00:18:59,265 So you can check yourself. 321 00:18:59,265 --> 00:19:00,640 This essentially by constructing, 322 00:19:00,640 --> 00:19:03,140 but you can check yourself that the action is indeed 323 00:19:03,140 --> 00:19:04,980 invariant under those transformations. 324 00:19:04,980 --> 00:19:06,262 OK? 325 00:19:06,262 --> 00:19:10,570 AUDIENCE: Is there also gauge symmetry for this, [INAUDIBLE] 326 00:19:10,570 --> 00:19:15,674 add a [? phase ?] to X and derivative with a it cancels. 327 00:19:15,674 --> 00:19:16,340 HONG LIU: Sorry? 328 00:19:16,340 --> 00:19:18,870 AUDIENCE: Is there [? any gauge's ?] theory in-- 329 00:19:18,870 --> 00:19:19,870 HONG LIU: Gauge in what? 330 00:19:19,870 --> 00:19:22,061 Gauge in which one? 331 00:19:22,061 --> 00:19:23,560 AUDIENCE: X, like in equation three, 332 00:19:23,560 --> 00:19:27,410 [INAUDIBLE] either global [? phase ?] to X and derivative 333 00:19:27,410 --> 00:19:28,105 with a and-- 334 00:19:28,105 --> 00:19:28,980 HONG LIU: No, no, no. 335 00:19:28,980 --> 00:19:30,350 No, X is real here. 336 00:19:30,350 --> 00:19:32,804 This is our full action. 337 00:19:32,804 --> 00:19:33,470 AUDIENCE: I see. 338 00:19:33,470 --> 00:19:34,030 OK. 339 00:19:34,030 --> 00:19:35,080 HONG LIU: Yeah, these are full action. 340 00:19:35,080 --> 00:19:35,810 X are real. 341 00:19:40,770 --> 00:19:42,840 So this is invariant on that. 342 00:19:42,840 --> 00:19:45,050 We will not show this is [? invariant. ?] 343 00:19:45,050 --> 00:19:47,320 This just essentially by construction. 344 00:19:47,320 --> 00:19:51,330 You should check it yourself, if you are not convinced. 345 00:19:51,330 --> 00:19:58,540 So the third symmetry is called a Weyl scaling of gamma ab. 346 00:19:58,540 --> 00:20:03,200 So the [INAUDIBLE] are slightly surprising. 347 00:20:07,290 --> 00:20:14,560 So under this symmetry, X mu does not change. 348 00:20:14,560 --> 00:20:20,100 But the metric we multiply it by overall prefactor. 349 00:20:23,680 --> 00:20:26,590 And this omega is arbitrary for any function. 350 00:20:34,680 --> 00:20:37,240 OK. 351 00:20:37,240 --> 00:20:41,520 So this is a very important symmetry, so let's check that. 352 00:20:41,520 --> 00:20:46,350 So when you multiply the gamma by our overall prefactor, 353 00:20:46,350 --> 00:20:49,930 when you go to gamma ab, then that's 354 00:20:49,930 --> 00:20:55,550 giving you the same prefactor, but with a minus sign here. 355 00:20:55,550 --> 00:20:58,040 And now, in the determinant for gamma, 356 00:20:58,040 --> 00:21:00,880 then that give you the square of these. 357 00:21:00,880 --> 00:21:03,770 And then take the square root, then give you a positive power. 358 00:21:03,770 --> 00:21:05,620 Then they cancel. 359 00:21:05,620 --> 00:21:08,030 Is it clear? 360 00:21:08,030 --> 00:21:12,870 So the [INAUDIBLE] power here cancels with the positive power 361 00:21:12,870 --> 00:21:14,290 here. 362 00:21:14,290 --> 00:21:14,870 OK? 363 00:21:14,870 --> 00:21:15,995 So this thing is invariant. 364 00:21:18,460 --> 00:21:20,310 On [INAUDIBLE] symmetry. 365 00:21:20,310 --> 00:21:26,970 So the way to see that-- another way to understand 366 00:21:26,970 --> 00:21:31,672 that somehow this action-- so the same thing 367 00:21:31,672 --> 00:21:32,380 with this action. 368 00:21:32,380 --> 00:21:35,290 This is not specific to we are working 369 00:21:35,290 --> 00:21:37,340 with Minkowski spacetime. 370 00:21:37,340 --> 00:21:40,539 The same feature here only depend on these two, OK? 371 00:21:40,539 --> 00:21:41,830 So the same thing happens here. 372 00:21:45,910 --> 00:21:49,720 So you can also see a feature why-- 373 00:21:49,720 --> 00:21:54,130 and you can also get a feeling why this is such symmetry, 374 00:21:54,130 --> 00:21:57,260 just by looking at the relation between the Nambu-Goto action 375 00:21:57,260 --> 00:21:59,290 with this action. 376 00:21:59,290 --> 00:22:03,160 So in this Nambu-Goto action, there's no gamma. 377 00:22:03,160 --> 00:22:06,240 OK, so in some sense you can see. 378 00:22:06,240 --> 00:22:07,650 You can say this is automatically 379 00:22:07,650 --> 00:22:09,620 invariant under this kind of symmetry, 380 00:22:09,620 --> 00:22:11,400 because there's no gamma. 381 00:22:11,400 --> 00:22:16,200 And now we see that the equivalence-- 382 00:22:16,200 --> 00:22:18,290 we write down equation of motion to go 383 00:22:18,290 --> 00:22:20,876 from here to Nambu-Goto action. 384 00:22:20,876 --> 00:22:23,330 The equivalence between them, they 385 00:22:23,330 --> 00:22:28,210 can be related by arbitrary function lambda. 386 00:22:28,210 --> 00:22:33,010 And really by aperture function lambda, 387 00:22:33,010 --> 00:22:36,260 yeah, this is another way to see this symmetry. 388 00:22:36,260 --> 00:22:37,420 OK? 389 00:22:37,420 --> 00:22:39,003 It's another way to see this symmetry. 390 00:22:41,350 --> 00:22:46,920 So in some sense, you can say that this symmetry is also 391 00:22:46,920 --> 00:22:52,390 required for the equivalence to the Nambu-Goto action, OK? 392 00:22:52,390 --> 00:23:04,480 So we can actually use-- we can also turn things around. 393 00:23:04,480 --> 00:23:13,145 We can also use as a symmetry principle. 394 00:23:22,601 --> 00:23:24,040 OK. 395 00:23:24,040 --> 00:23:26,300 I see major points that we can ask. 396 00:23:26,300 --> 00:23:29,280 Suppose we have action. 397 00:23:29,280 --> 00:23:33,755 Suppose we require an action to satisfy the a, b, c, 398 00:23:33,755 --> 00:23:35,830 to be invariant under a, b, c. 399 00:23:35,830 --> 00:23:38,090 What is the most general action? 400 00:23:38,090 --> 00:23:39,070 OK? 401 00:23:39,070 --> 00:23:42,960 And then, you can essentially uniquely determine the three. 402 00:23:42,960 --> 00:23:44,970 OK. 403 00:23:44,970 --> 00:23:46,785 We use actually symmetry principles 404 00:23:46,785 --> 00:24:00,784 to uniquely-- actually, it's almost uniquely determined. 405 00:24:04,580 --> 00:24:08,660 OK, so I will explain this "almost." 406 00:24:08,660 --> 00:24:10,530 So now I use this clear. 407 00:24:10,530 --> 00:24:12,890 Is the logic clear? 408 00:24:12,890 --> 00:24:14,550 So we do two ways. 409 00:24:14,550 --> 00:24:18,050 We first start with Nambu-Goto and then we go to this action. 410 00:24:18,050 --> 00:24:21,380 But then we observe this action having the symmetries. 411 00:24:21,380 --> 00:24:25,310 And now we ask ourselves, what is the most general action 412 00:24:25,310 --> 00:24:26,390 besides symmetries? 413 00:24:26,390 --> 00:24:28,130 OK. 414 00:24:28,130 --> 00:24:33,370 And that then leads us back to this term, to this action. 415 00:24:36,700 --> 00:24:39,491 Except there's one more term you can add. 416 00:24:39,491 --> 00:24:39,990 OK. 417 00:24:39,990 --> 00:24:41,448 There's one other term you can add. 418 00:24:44,680 --> 00:24:45,895 So that's why this is almost. 419 00:25:06,240 --> 00:25:08,750 So I will just immediately write it down. 420 00:25:08,750 --> 00:25:11,960 This is called an Euler action. 421 00:25:11,960 --> 00:25:15,190 So this action actually have a long history. 422 00:25:15,190 --> 00:25:17,070 So this is called Euler action. 423 00:25:17,070 --> 00:25:18,570 So this can be written as following. 424 00:25:24,250 --> 00:25:27,000 And this lambda is just a constant, 425 00:25:27,000 --> 00:25:29,990 some arbitrary constant. 426 00:25:29,990 --> 00:25:39,630 And R is the Ricci scalar for gamma ab. 427 00:25:39,630 --> 00:25:41,460 OK, gamma ab is symmetric. 428 00:25:41,460 --> 00:25:42,900 It's a two-dimensional metric. 429 00:25:42,900 --> 00:25:45,295 We will construct the corresponding Ricci scalar. 430 00:25:51,910 --> 00:25:54,730 So you can clearly check this action-- so 431 00:25:54,730 --> 00:25:57,340 let me call this equation four. 432 00:25:57,340 --> 00:26:01,080 So you can check easily that a four is invariant under a plus 433 00:26:01,080 --> 00:26:02,370 b under a. 434 00:26:02,370 --> 00:26:05,650 Because this does not depend on X. OK, of course, 435 00:26:05,650 --> 00:26:06,994 this is invariant under a. 436 00:26:06,994 --> 00:26:08,785 And this is invariant under b, because this 437 00:26:08,785 --> 00:26:12,520 is a covariant action, so this invariant on b. 438 00:26:12,520 --> 00:26:16,470 To see this is invariant under c requires 439 00:26:16,470 --> 00:26:18,810 a little bit of effort. 440 00:26:18,810 --> 00:26:20,790 Okay, requires a little bit of effort. 441 00:26:20,790 --> 00:26:25,270 But if I write down the formula, then it will become clear. 442 00:26:25,270 --> 00:26:30,830 So under c, if you make such a transformation, 443 00:26:30,830 --> 00:26:37,150 say under gamma prime equal to [? exponent ?] 2 omega gamma. 444 00:26:37,150 --> 00:26:38,460 [? 2w ?] gamma. 445 00:26:38,460 --> 00:26:43,640 And then you find that minus gamma prime and R prime-- R 446 00:26:43,640 --> 00:26:46,210 prime means the Ricci scalar for the gamma prime-- 447 00:26:46,210 --> 00:26:53,469 is equal to square root minus gamma, R minus 2. 448 00:26:53,469 --> 00:26:56,010 The 2-- yeah, this is a formula which you can check yourself. 449 00:26:56,010 --> 00:26:57,410 Maybe just directly write down. 450 00:27:02,920 --> 00:27:07,000 And then you see you get one more term, 451 00:27:07,000 --> 00:27:09,720 but this term is a total derivative. 452 00:27:09,720 --> 00:27:12,150 OK, so this term is a total derivative. 453 00:27:12,150 --> 00:27:14,660 And if we impose the right boundary conditions, 454 00:27:14,660 --> 00:27:16,790 or if you can see the compact worldsheets, 455 00:27:16,790 --> 00:27:19,060 then this term will automatically vanishes, 456 00:27:19,060 --> 00:27:21,100 and then this is invariant. 457 00:27:21,100 --> 00:27:23,210 OK? 458 00:27:23,210 --> 00:27:27,460 So we have shown this is invariant under everything, a 459 00:27:27,460 --> 00:27:28,700 to c. 460 00:27:28,700 --> 00:27:30,550 So essentially, that's it. 461 00:27:30,550 --> 00:27:34,050 So these two terms are only-- three and four 462 00:27:34,050 --> 00:27:37,680 are the only terms are invariant under a to c, OK? 463 00:27:37,680 --> 00:27:38,650 There's no other terms. 464 00:27:42,460 --> 00:27:43,420 So total action. 465 00:27:51,615 --> 00:27:52,115 Then. 466 00:28:17,970 --> 00:28:19,670 And then this is our total action. 467 00:28:31,150 --> 00:28:41,966 So actually some of you who knows two-dimensional gravity. 468 00:28:41,966 --> 00:28:44,900 AUDIENCE: Is there a minus sign? 469 00:28:44,900 --> 00:28:45,700 HONG LIU: Yeah. 470 00:28:45,700 --> 00:28:48,235 Maybe let me put the minus sign here. 471 00:28:48,235 --> 00:28:48,734 Good. 472 00:28:51,470 --> 00:28:54,640 So some of you who know the two-dimensional gravity 473 00:28:54,640 --> 00:28:57,834 will immediately point out, you actually 474 00:28:57,834 --> 00:28:59,000 don't need to add this term. 475 00:29:02,200 --> 00:29:06,090 Because this is actually a total derivative. 476 00:29:06,090 --> 00:29:11,230 So this is a property of-- I only make a claim here. 477 00:29:11,230 --> 00:29:15,840 So this thing is a total derivative. 478 00:29:15,840 --> 00:29:17,110 So you can check yourself. 479 00:29:17,110 --> 00:29:23,160 Go back, go to your favorite GR book, 480 00:29:23,160 --> 00:29:27,200 go to your favorite GR book, and we'll call it 481 00:29:27,200 --> 00:29:30,820 the scalar, Ricci scalar in 2D. 482 00:29:30,820 --> 00:29:33,670 And then you find actually the square root of gamma times 483 00:29:33,670 --> 00:29:39,260 i is actually a total derivative. 484 00:29:39,260 --> 00:29:42,290 So it will not affect the equation motion, et cetera. 485 00:29:45,240 --> 00:29:46,480 But it is important. 486 00:29:46,480 --> 00:29:49,900 Otherwise I would not have talked about it. 487 00:29:49,900 --> 00:29:52,220 Because when you go to the Euclidean signature, that's 488 00:29:52,220 --> 00:29:56,950 the standard way we evaluate the path integral. 489 00:29:56,950 --> 00:30:09,090 So when you go to Euclidean signature, 490 00:30:09,090 --> 00:30:13,510 this is only-- saying that this is a total derivative is only 491 00:30:13,510 --> 00:30:16,820 a local description. 492 00:30:16,820 --> 00:30:19,340 It's actually a local description. 493 00:30:19,340 --> 00:30:23,920 In fact, this is actually a topological term. 494 00:30:23,920 --> 00:30:26,867 So locally, you can write it as a total derivative. 495 00:30:26,867 --> 00:30:28,950 But if you put on the topological [? nontrivial ?] 496 00:30:28,950 --> 00:30:33,080 manifold, then you cannot do it globally. 497 00:30:33,080 --> 00:30:41,280 And so, in fact, in the Euclidean signature, 498 00:30:41,280 --> 00:30:48,460 it's a-- actually I don't know whether this is due to Euler. 499 00:30:48,460 --> 00:30:56,000 This Euler action is actually equal to lambda times chi. 500 00:30:56,000 --> 00:31:02,450 And the chi is equal to [? pi-- ?] 501 00:31:02,450 --> 00:31:07,920 yeah, which is defined to be-- so now 502 00:31:07,920 --> 00:31:12,237 it's R. It's actually 2 minus 2h, OK? 503 00:31:12,237 --> 00:31:13,820 And the h is the genus of the surface. 504 00:31:18,376 --> 00:31:20,176 AUDIENCE: [INAUDIBLE]. 505 00:31:20,176 --> 00:31:21,550 HONG LIU: No, this is only in 2D. 506 00:31:21,550 --> 00:31:23,010 AUDIENCE: This is only true in 2D. 507 00:31:23,010 --> 00:31:24,510 HONG LIU: Yeah, in 2D. 508 00:31:24,510 --> 00:31:28,770 So in 2D, this-- even though this is locally a total 509 00:31:28,770 --> 00:31:31,120 derivative, but it's not the global total-- 510 00:31:31,120 --> 00:31:34,500 But you cannot write a total derivative globally. 511 00:31:34,500 --> 00:31:37,906 And so we integrated over a nontrivial manifold. 512 00:31:37,906 --> 00:31:40,030 But manifold with a nontrivial topology-- actually, 513 00:31:40,030 --> 00:31:40,863 this is [INAUDIBLE]. 514 00:31:44,720 --> 00:31:47,460 It actually gives you this Euler number. 515 00:31:47,460 --> 00:31:50,210 So example, if you integrated on a sphere, 516 00:31:50,210 --> 00:31:52,600 then you find that this number is 2, 517 00:31:52,600 --> 00:31:54,251 because [? when ?] h equal to 0. 518 00:31:54,251 --> 00:31:55,750 And if you evaluate it on the torus, 519 00:31:55,750 --> 00:31:59,190 then you find that this is 0, et cetera. 520 00:31:59,190 --> 00:32:00,282 OK. 521 00:32:00,282 --> 00:32:00,865 So one second. 522 00:32:03,690 --> 00:32:19,110 So in the Euclidean path integral, 523 00:32:19,110 --> 00:32:23,790 so Euclidean path integral, this S minus-- 524 00:32:23,790 --> 00:32:29,290 so this exponential S Euler term-- 525 00:32:29,290 --> 00:32:36,560 just give you exponential [? s ?] lambda chi, OK? 526 00:32:36,560 --> 00:32:40,680 So now you may ring a bell, because this is precisely 527 00:32:40,680 --> 00:32:43,420 the [INAUDIBLE] term I added before when I say, 528 00:32:43,420 --> 00:32:46,740 when we sum over different genus. 529 00:32:46,740 --> 00:32:50,550 You have the freedom to add [? mechanical ?] potential, 530 00:32:50,550 --> 00:32:53,840 lambda times the Euler number. 531 00:32:53,840 --> 00:32:57,310 And actually this is dictated by the symmetry. 532 00:32:57,310 --> 00:33:00,770 And if you include the most general terms are 533 00:33:00,770 --> 00:33:02,720 consistent with those symmetries, 534 00:33:02,720 --> 00:33:06,840 then you are required to include this Euler term. 535 00:33:06,840 --> 00:33:09,440 And then you'll have automatically weight, 536 00:33:09,440 --> 00:33:13,740 that you would automatically have a weight 537 00:33:13,740 --> 00:33:16,490 for different genus, OK? 538 00:33:16,490 --> 00:33:23,350 But for this theory, the lambda is actually constant. 539 00:33:23,350 --> 00:33:28,600 So the choice of lambda is still arbitrary, 540 00:33:28,600 --> 00:33:32,030 but this does give you a nontrivial weight. 541 00:33:32,030 --> 00:33:36,100 So this is weight-- different genus. 542 00:33:41,553 --> 00:33:42,053 Yeah. 543 00:33:49,470 --> 00:33:51,650 Any questions about this, yes? 544 00:33:51,650 --> 00:33:55,140 AUDIENCE: So if you have [? a nontrivial ?] topology, 545 00:33:55,140 --> 00:33:57,330 wouldn't the boundary term from the symmetry also 546 00:33:57,330 --> 00:33:58,202 be nontrivial? 547 00:33:58,202 --> 00:33:59,510 So wouldn't the symmetry break? 548 00:33:59,510 --> 00:34:00,176 HONG LIU: Sorry? 549 00:34:00,176 --> 00:34:02,350 AUDIENCE: If you have a nontrivial topology, what 550 00:34:02,350 --> 00:34:05,170 is the total derivative boundary term, total derivative 551 00:34:05,170 --> 00:34:06,670 of the symmetry transformation also? 552 00:34:06,670 --> 00:34:07,330 HONG LIU: No. 553 00:34:07,330 --> 00:34:14,139 No, they are different kind of-- no, 554 00:34:14,139 --> 00:34:16,080 there's no global issue here. 555 00:34:16,080 --> 00:34:18,740 This is just a ordinary function. 556 00:34:18,740 --> 00:34:20,739 You have to write down this explicit to know it. 557 00:34:24,139 --> 00:34:24,639 Yes. 558 00:34:24,639 --> 00:34:27,792 AUDIENCE: So the claim was that [? efficient ?] work, no matter 559 00:34:27,792 --> 00:34:29,170 what value [INAUDIBLE] input. 560 00:34:29,170 --> 00:34:32,954 So this is why you were able to put in that factor at the sum. 561 00:34:32,954 --> 00:34:36,065 We could make lambda equal 0, in which case, we don't have it. 562 00:34:36,065 --> 00:34:36,690 HONG LIU: Yeah. 563 00:34:36,690 --> 00:34:41,639 Lambda is not dictated by symmetry, 564 00:34:41,639 --> 00:34:43,440 so lambda is just a free parameter here. 565 00:34:46,230 --> 00:34:51,030 But in principle, if I determine it exists, yeah. 566 00:34:51,030 --> 00:34:54,210 Just lambda equal to 0 is a special case. 567 00:34:57,010 --> 00:34:57,704 Yes. 568 00:34:57,704 --> 00:34:58,620 AUDIENCE: [INAUDIBLE]. 569 00:35:01,619 --> 00:35:03,160 HONG LIU: Yeah, in the Minkowski case 570 00:35:03,160 --> 00:35:05,160 it's a little bit tricky to talk about topology, 571 00:35:05,160 --> 00:35:08,635 [? et cetera. ?] So that's why we always, 572 00:35:08,635 --> 00:35:10,260 when we do the path integral, we always 573 00:35:10,260 --> 00:35:11,410 go to Euclidean signature. 574 00:35:14,360 --> 00:35:17,920 And for example, later when we quantized 575 00:35:17,920 --> 00:35:20,990 the action, which we work with at Lorentzian signature. 576 00:35:20,990 --> 00:35:23,500 And we only care, say when we solve the equation of motion, 577 00:35:23,500 --> 00:35:25,315 but we only care about the local structure. 578 00:35:25,315 --> 00:35:26,690 Then we can just ignore this term 579 00:35:26,690 --> 00:35:29,106 because this does not contribute [? to ?] local structure. 580 00:35:32,400 --> 00:35:34,240 Good. 581 00:35:34,240 --> 00:35:38,356 So to summarize, from the two-dimensional-- so again, 582 00:35:38,356 --> 00:35:40,230 we can signal this as a two-dimensional field 583 00:35:40,230 --> 00:35:44,110 theory coupled to in the curved spacetime. 584 00:35:44,110 --> 00:35:46,160 From that two-dimensional perspective, 585 00:35:46,160 --> 00:35:48,510 this is a global symmetry. 586 00:35:48,510 --> 00:35:52,225 OK, so a is a global symmetry. 587 00:35:59,820 --> 00:36:01,320 Because the symmetry does not depend 588 00:36:01,320 --> 00:36:05,630 on the sigma and the gamma, and the sigma and tau. 589 00:36:05,630 --> 00:36:08,920 So this should leave to conserve charge, conserve current, 590 00:36:08,920 --> 00:36:12,520 conserve charge and conserve current. 591 00:36:12,520 --> 00:36:13,045 OK? 592 00:36:13,045 --> 00:36:14,920 So we're going to see later this indeed plays 593 00:36:14,920 --> 00:36:18,210 a very important role in physics. 594 00:36:18,210 --> 00:36:25,270 And the b and the c are what we call the gauge symmetries, 595 00:36:25,270 --> 00:36:32,670 or local symmetries, because they depend on sigma and tau. 596 00:36:32,670 --> 00:36:37,080 They have [? arbitrary ?] dependence on sigma and tau. 597 00:36:37,080 --> 00:36:39,860 There and here can also depend on sigma and tau. 598 00:36:39,860 --> 00:36:41,875 So this is what we call the gauge symmetries. 599 00:36:44,840 --> 00:36:50,020 So for those of you who have started Quantum Field Theory II 600 00:36:50,020 --> 00:36:52,720 or even Quantum Field Theory I, for those of you who have 601 00:36:52,720 --> 00:36:57,040 quantized [? QED, ?] you know that the gauge symmetries are 602 00:36:57,040 --> 00:36:58,920 not [? general ?] symmetries. 603 00:36:58,920 --> 00:37:04,550 They just tells you they are redundant degrees of freedom. 604 00:37:04,550 --> 00:37:05,159 OK? 605 00:37:05,159 --> 00:37:07,200 And they just redundantly [? with the ?] freedom. 606 00:37:07,200 --> 00:37:09,800 And then when you quantize the theory, 607 00:37:09,800 --> 00:37:12,470 then you need to rid of those redundant degrees of freedom 608 00:37:12,470 --> 00:37:13,760 first. 609 00:37:13,760 --> 00:37:17,120 Because otherwise you will have a consistency problem. 610 00:37:17,120 --> 00:37:22,055 And the process of getting rid of redundancy problems-- 611 00:37:22,055 --> 00:37:27,300 those redundancies are called gauge fixing. 612 00:37:27,300 --> 00:37:29,460 OK, so that's what we'll get to a little bit later. 613 00:37:35,350 --> 00:37:37,520 So in principle, you can just proceed 614 00:37:37,520 --> 00:37:41,120 to do the path integral quantization as we did before. 615 00:37:41,120 --> 00:37:43,640 And the path integral quantization 616 00:37:43,640 --> 00:37:48,450 is conceptually simple and geometrically elegant. 617 00:37:48,450 --> 00:37:52,645 But it does not give you, say, string theory spectrum. 618 00:37:52,645 --> 00:37:54,770 And because it's a path integral, it's [INAUDIBLE]. 619 00:37:54,770 --> 00:37:57,800 If you really want to find what are the spectrums of the string 620 00:37:57,800 --> 00:38:01,210 theory, you actually have to do the standard canonical 621 00:38:01,210 --> 00:38:02,280 quantization. 622 00:38:02,280 --> 00:38:06,370 So that's what we are going to do next, OK? 623 00:38:06,370 --> 00:38:10,950 Now we are going to quantize the string in the canonical way. 624 00:38:13,770 --> 00:38:16,580 There are many ways of doing it. 625 00:38:16,580 --> 00:38:19,590 I will use the simplest way, so [INAUDIBLE] 626 00:38:19,590 --> 00:38:22,220 the simplest way, also conceptually simplest way. 627 00:38:22,220 --> 00:38:23,886 It's called the light-cone quantization. 628 00:38:30,080 --> 00:38:33,400 And the meaning of this word will be clear. 629 00:38:33,400 --> 00:38:45,260 So the basic picture will emerge from this quantization 630 00:38:45,260 --> 00:38:50,350 is that each oscillation mode-- so if you have a string, 631 00:38:50,350 --> 00:38:51,870 then you can oscillate, right? 632 00:38:51,870 --> 00:38:53,530 Because that's what the string does. 633 00:38:53,530 --> 00:38:55,640 It can oscillate. 634 00:38:55,640 --> 00:38:59,900 So each oscillation mode, so the bottom line 635 00:38:59,900 --> 00:39:07,290 is that we will show that each oscillation mode of the string 636 00:39:07,290 --> 00:39:09,165 gives rise to a spacetime particle. 637 00:39:15,440 --> 00:39:15,940 OK. 638 00:39:15,940 --> 00:39:17,150 From the string point of view, it 639 00:39:17,150 --> 00:39:19,066 looks like some kind of oscillation of string. 640 00:39:19,066 --> 00:39:20,890 But from the spacetime point of view, 641 00:39:20,890 --> 00:39:23,410 it's like a spacetime particle. 642 00:39:23,410 --> 00:39:26,500 In particular, if you have a closed string, 643 00:39:26,500 --> 00:39:29,350 then that will give rise, say, graviton. 644 00:39:29,350 --> 00:39:33,810 Say one particular mode will give you graviton and then 645 00:39:33,810 --> 00:39:34,980 many other particles. 646 00:39:34,980 --> 00:39:37,090 Actually, we have an infinite number of particles from string 647 00:39:37,090 --> 00:39:38,923 because there's infinite number of vibration 648 00:39:38,923 --> 00:39:40,350 modes with a string. 649 00:39:40,350 --> 00:39:44,370 And if you have an open string, say the string which does not 650 00:39:44,370 --> 00:39:46,650 close, and then actually then you 651 00:39:46,650 --> 00:39:52,450 find the gauge field-- a rise in one of the vibration modes. 652 00:39:52,450 --> 00:39:59,960 Say this can either be photon or gluon, et cetera. 653 00:39:59,960 --> 00:40:01,739 OK. 654 00:40:01,739 --> 00:40:03,530 Yeah, so this is the thing we like to show. 655 00:40:11,310 --> 00:40:11,810 Yes. 656 00:40:11,810 --> 00:40:14,297 AUDIENCE: [INAUDIBLE]. 657 00:40:14,297 --> 00:40:16,630 HONG LIU: Oh, there are an infinite number of particles. 658 00:40:16,630 --> 00:40:19,470 So because the string can oscillate, 659 00:40:19,470 --> 00:40:21,410 just any string oscillate in principle 660 00:40:21,410 --> 00:40:22,674 in an infinite number of ways. 661 00:40:22,674 --> 00:40:24,340 And if each oscillation mode [? cause ?] 662 00:40:24,340 --> 00:40:25,910 one into a particle, then you can have an infinite number 663 00:40:25,910 --> 00:40:26,714 of particles. 664 00:40:26,714 --> 00:40:27,630 AUDIENCE: [INAUDIBLE]. 665 00:40:30,875 --> 00:40:32,000 HONG LIU: Oh yeah, oh yeah. 666 00:40:32,000 --> 00:40:34,290 They're completely different from gravitons and protons. 667 00:40:34,290 --> 00:40:36,123 They're just completely different particles. 668 00:40:36,123 --> 00:40:39,065 AUDIENCE: So [INAUDIBLE] particles [INAUDIBLE]. 669 00:40:39,065 --> 00:40:43,280 What other particles in the same class with gravitons? 670 00:40:43,280 --> 00:40:45,620 HONG LIU: Depend on what you mean by class. 671 00:40:45,620 --> 00:40:48,830 AUDIENCE: Well, can you give a second [INAUDIBLE]? 672 00:40:48,830 --> 00:40:51,680 In the same line as graviton, [? what else can you do? ?] 673 00:40:51,680 --> 00:40:54,900 HONG LIU: Oh, for example, mass will spin two particles. 674 00:40:54,900 --> 00:40:56,978 Mass will spin three particles. 675 00:40:56,978 --> 00:40:58,952 AUDIENCE: Everything will spin more than one? 676 00:40:58,952 --> 00:41:00,410 HONG LIU: It can also have a scalar 677 00:41:00,410 --> 00:41:02,118 particle, [? massive ?] scalar particles, 678 00:41:02,118 --> 00:41:03,945 but they're all massive. 679 00:41:03,945 --> 00:41:06,839 AUDIENCE: That'd be boson? 680 00:41:06,839 --> 00:41:08,130 HONG LIU: They can be fermions. 681 00:41:08,130 --> 00:41:10,100 In this theory they can only be bosons. 682 00:41:10,100 --> 00:41:12,150 But we can play tricks to get fermions. 683 00:41:15,740 --> 00:41:17,700 AUDIENCE: So does this predict that-- so if you 684 00:41:17,700 --> 00:41:19,635 were to have a string theory model which would reproduce 685 00:41:19,635 --> 00:41:20,980 the standard model, this basically 686 00:41:20,980 --> 00:41:22,604 says that at higher and higher energies 687 00:41:22,604 --> 00:41:25,269 that we expect to always see new types of particles 688 00:41:25,269 --> 00:41:25,810 or something? 689 00:41:25,810 --> 00:41:27,018 HONG LIU: Yeah, that's right. 690 00:41:27,018 --> 00:41:28,520 AUDIENCE: Interesting. 691 00:41:28,520 --> 00:41:31,240 HONG LIU: But we will see. 692 00:41:31,240 --> 00:41:33,960 You need to go to a specific time-- we already 693 00:41:33,960 --> 00:41:36,849 go to a specific energy scale to see. 694 00:41:36,849 --> 00:41:37,390 AUDIENCE: OK. 695 00:41:37,390 --> 00:41:38,640 We can see how they're spaced. 696 00:41:38,640 --> 00:41:39,514 HONG LIU: Yeah, yeah. 697 00:41:39,514 --> 00:41:40,070 That's right. 698 00:41:40,070 --> 00:41:41,486 Yeah, but that space is determined 699 00:41:41,486 --> 00:41:44,570 by some energy scale, and that scale can be very high. 700 00:41:47,710 --> 00:41:51,080 OK, so now we will quantize this theory using 701 00:41:51,080 --> 00:41:54,060 canonical quantization, OK? 702 00:41:54,060 --> 00:41:56,669 So by canonical quantization-- so let me remind you, 703 00:41:56,669 --> 00:41:57,585 this is the procedure. 704 00:42:01,900 --> 00:42:04,220 So this is a procedure we typically 705 00:42:04,220 --> 00:42:09,830 explain, say, in the second day of free quantum field theory. 706 00:42:09,830 --> 00:42:13,200 And so we are doing the same thing here. 707 00:42:13,200 --> 00:42:16,620 So the quantization procedure is the following. 708 00:42:16,620 --> 00:42:19,270 The first step is you write down equation of motion. 709 00:42:23,690 --> 00:42:25,190 OK. 710 00:42:25,190 --> 00:42:30,134 And in the case which you have gauge symmetries, 711 00:42:30,134 --> 00:42:31,425 then we fix the gauge symmetry. 712 00:42:40,200 --> 00:42:44,967 And the third is find the complete set 713 00:42:44,967 --> 00:42:45,925 of classical solutions. 714 00:42:56,240 --> 00:42:58,640 OK. 715 00:42:58,640 --> 00:43:05,800 And the four is you promote the classical field 716 00:43:05,800 --> 00:43:11,470 on the worldsheet, worldsheet field, 717 00:43:11,470 --> 00:43:13,835 say the X or gamma to quantum operators. 718 00:43:24,914 --> 00:43:25,910 OK. 719 00:43:25,910 --> 00:43:28,220 So the quantization procedure is that you 720 00:43:28,220 --> 00:43:30,030 promote them to quantum operators, 721 00:43:30,030 --> 00:43:31,720 which satisfy the so-called canonical 722 00:43:31,720 --> 00:43:33,530 [? computation ?] relation. 723 00:43:33,530 --> 00:43:36,930 And the previous equation of motion 724 00:43:36,930 --> 00:43:39,910 then becomes operator equations for those operators. 725 00:43:39,910 --> 00:43:41,280 OK? 726 00:43:41,280 --> 00:43:44,610 In particular, then the classical solution 727 00:43:44,610 --> 00:43:53,930 you find in three becomes the solutions 728 00:43:53,930 --> 00:43:54,855 to operator equations. 729 00:44:03,200 --> 00:44:08,620 And in particular, the parameters 730 00:44:08,620 --> 00:44:18,930 we use to parameterize your classical solutions-- 731 00:44:18,930 --> 00:44:22,240 so those become, say, the creation and annihilation 732 00:44:22,240 --> 00:44:24,990 operators. 733 00:44:24,990 --> 00:44:27,950 Just as what we do in the free field theory quantization. 734 00:44:34,468 --> 00:44:37,510 And then the last step, once you have the creation 735 00:44:37,510 --> 00:44:40,595 and annihilation operators, then you can just find the spectrum. 736 00:44:44,300 --> 00:44:46,392 By acting the creation and annihilation operators 737 00:44:46,392 --> 00:44:48,350 on your vacuum, then you can find the spectrum. 738 00:44:52,470 --> 00:44:55,030 So find the spectrum. 739 00:44:55,030 --> 00:44:57,610 So this is the standard procedure. 740 00:44:57,610 --> 00:45:00,680 Yeah, so this is almost exactly the same as how we quantize 741 00:45:00,680 --> 00:45:02,320 our free scalar field theory. 742 00:45:02,320 --> 00:45:08,550 In Quantum Field Theory I, the only tricky thing is now, 743 00:45:08,550 --> 00:45:11,804 this is a system with gauge symmetries. 744 00:45:11,804 --> 00:45:13,220 That's the only tricky thing here. 745 00:45:18,820 --> 00:45:19,952 Any questions here? 746 00:45:25,610 --> 00:45:28,649 So now let me just again say some boundary 747 00:45:28,649 --> 00:45:29,940 conditions we are going to use. 748 00:45:33,617 --> 00:45:34,533 AUDIENCE: [INAUDIBLE]. 749 00:45:38,840 --> 00:45:39,660 HONG LIU: Sorry? 750 00:45:39,660 --> 00:45:43,186 AUDIENCE: What are parameters in classical solutions 751 00:45:43,186 --> 00:45:45,470 permutations. [INAUDIBLE]. 752 00:45:45,470 --> 00:45:47,240 HONG LIU: What other-- 753 00:45:47,240 --> 00:45:49,120 AUDIENCE: Parameters in classical solutions 754 00:45:49,120 --> 00:45:52,340 in case of [INAUDIBLE]. 755 00:45:52,340 --> 00:45:57,410 HONG LIU: It's different modes for your classical waves. 756 00:45:57,410 --> 00:46:00,780 The amplitude for your different classical waves. 757 00:46:04,021 --> 00:46:06,660 AUDIENCE: [INAUDIBLE]? 758 00:46:06,660 --> 00:46:10,090 HONG LIU: Yeah, it is like that, yeah. 759 00:46:10,090 --> 00:46:14,362 For scalar field theory, yeah, just for harmonic oscillator, 760 00:46:14,362 --> 00:46:15,320 you can write a cosine. 761 00:46:20,550 --> 00:46:25,090 a explains your i omega X plus a star, [INAUDIBLE] 762 00:46:25,090 --> 00:46:30,090 omega X. So a is the parameter, and a and a star 763 00:46:30,090 --> 00:46:33,239 become creation and annihilation operator. 764 00:46:33,239 --> 00:46:34,280 It's the same thing here. 765 00:46:37,340 --> 00:46:39,675 So we will consider both closed and open strings. 766 00:46:47,180 --> 00:46:49,880 So suppose closed and open strings. 767 00:46:49,880 --> 00:46:58,610 So if you have a closed string, then, just by convention 768 00:46:58,610 --> 00:47:01,470 we always take sigma to go from 0 to 2 pi. 769 00:47:05,390 --> 00:47:07,015 So the closed string means it's closed. 770 00:47:11,550 --> 00:47:13,840 Then we take this from to 0 to 2 pi. 771 00:47:13,840 --> 00:47:16,820 That means whatever function should 772 00:47:16,820 --> 00:47:18,680 be a periodic function of 2 pi. 773 00:47:23,530 --> 00:47:25,280 OK, should be a periodic function of 2 pi, 774 00:47:25,280 --> 00:47:27,720 because it's closed, OK? 775 00:47:27,720 --> 00:47:30,620 So the same thing with gamma. 776 00:47:30,620 --> 00:47:34,430 Same thing with gamma, OK? 777 00:47:34,430 --> 00:47:44,050 But if you have open string, by convention 778 00:47:44,050 --> 00:47:48,680 we take sigma goes to 0 to pi. 779 00:47:48,680 --> 00:47:51,510 So one end of the string is at sigma equal to 0. 780 00:47:51,510 --> 00:47:53,660 The other end is at sigma equal to pi. 781 00:47:53,660 --> 00:47:57,800 So for open strings, so here is sigma equal to 0. 782 00:47:57,800 --> 00:48:00,490 Here is sigma equal to pi. 783 00:48:00,490 --> 00:48:03,720 So the sigma is directly on the string. 784 00:48:03,720 --> 00:48:06,090 So we need to-- for open strings, then, 785 00:48:06,090 --> 00:48:07,740 because you have boundaries, then 786 00:48:07,740 --> 00:48:16,440 we need to supply boundary conditions at the end points. 787 00:48:26,760 --> 00:48:27,720 Good. 788 00:48:27,720 --> 00:48:30,470 So now with this set-up, we can now just 789 00:48:30,470 --> 00:48:32,785 follow that five steps, OK? 790 00:48:32,785 --> 00:48:36,740 We can just follow that five steps and quantize the string. 791 00:48:36,740 --> 00:48:38,680 So do you want to have a break? 792 00:48:38,680 --> 00:48:41,660 OK, let us start. 793 00:48:41,660 --> 00:48:45,511 So we just follow these steps. 794 00:48:45,511 --> 00:48:47,260 First let's write down equation of motion. 795 00:49:04,550 --> 00:49:07,860 OK, first we just do a variation of gamma ab. 796 00:49:11,430 --> 00:49:13,200 So gamma is the worldsheet metric 797 00:49:13,200 --> 00:49:17,430 from the point of view of two-dimensional field theory. 798 00:49:17,430 --> 00:49:19,375 The gamma is the worldsheet metric. 799 00:49:19,375 --> 00:49:21,740 When you do evaluation over gamma, essentially what 800 00:49:21,740 --> 00:49:31,510 you get is the stress tensor for this worldsheet theory, OK? 801 00:49:31,510 --> 00:49:34,180 So when you do the [? evaluation over ?] gamma, 802 00:49:34,180 --> 00:49:38,430 essentially, it's just the statement 803 00:49:38,430 --> 00:49:41,760 that the stress tensor had to be 0. 804 00:49:41,760 --> 00:49:42,260 OK. 805 00:49:46,570 --> 00:49:50,316 And then you can work this out explicitly. 806 00:49:50,316 --> 00:49:52,700 And the stress tensor, which I call Tab. 807 00:50:19,950 --> 00:50:21,054 OK. 808 00:50:21,054 --> 00:50:22,470 So I just [? did the evaluation ?] 809 00:50:22,470 --> 00:50:25,740 of that action with respect to gamma ab. 810 00:50:25,740 --> 00:50:29,660 This is what you get You find this thing has 811 00:50:29,660 --> 00:50:31,320 to be equal to 0. 812 00:50:31,320 --> 00:50:32,970 And I just call this thing Tab. 813 00:50:32,970 --> 00:50:37,200 And this Tab is essentially the stress tensor of this. 814 00:50:37,200 --> 00:50:40,090 If you [INAUDIBLE] the scalar field theory, 815 00:50:40,090 --> 00:50:43,280 so this Tab is essentially the stress tensor 816 00:50:43,280 --> 00:50:44,630 of this scalar field theory. 817 00:50:44,630 --> 00:50:45,130 OK? 818 00:50:49,660 --> 00:50:51,070 Yeah. 819 00:50:51,070 --> 00:50:54,280 AUDIENCE: If we add the topological term and said, 820 00:50:54,280 --> 00:50:58,090 there's a Ricci scalar there, there 821 00:50:58,090 --> 00:50:59,800 must be Einstein's tensor equal to Tab. 822 00:50:59,800 --> 00:51:01,550 HONG LIU: No, no, no. 823 00:51:01,550 --> 00:51:03,820 Because there's a total derivative term 824 00:51:03,820 --> 00:51:05,820 in two dimensions-- total derivative term 825 00:51:05,820 --> 00:51:08,110 never contribute to equation of motion. 826 00:51:08,110 --> 00:51:08,840 Yeah. 827 00:51:08,840 --> 00:51:14,990 AUDIENCE: So but if we calculated the Einstein tensor 828 00:51:14,990 --> 00:51:17,580 for that R [INAUDIBLE] get 0. 829 00:51:17,580 --> 00:51:18,996 HONG LIU: You will get 0, yeah. 830 00:51:23,712 --> 00:51:25,295 AUDIENCE: Sorry, so you just said that 831 00:51:25,295 --> 00:51:27,620 from the equation of motion, that gamma ab can get 832 00:51:27,620 --> 00:51:30,514 gamma ab equals 1/2 lambda hab. 833 00:51:30,514 --> 00:51:31,014 Is that it? 834 00:51:31,014 --> 00:51:31,860 HONG LIU: Yeah. 835 00:51:31,860 --> 00:51:34,277 It's the same thing. 836 00:51:34,277 --> 00:51:35,110 It's the same thing. 837 00:51:38,625 --> 00:51:40,030 But this is a very good question. 838 00:51:40,030 --> 00:51:43,230 Let me do that here. 839 00:51:43,230 --> 00:51:52,660 Say if I write gamma ab equal to 1/2 lambda hab, 840 00:51:52,660 --> 00:51:55,890 I can write this lambda explicitly. 841 00:51:55,890 --> 00:51:58,390 Then I can modify this by hab on both sides 842 00:51:58,390 --> 00:52:04,420 and this means hab gamma ab equal to lambda. 843 00:52:04,420 --> 00:52:08,812 OK, that means lambda is equal to gamma ab times hab. 844 00:52:08,812 --> 00:52:10,520 And so this equation, I can also write it 845 00:52:10,520 --> 00:52:20,960 as gamma ab is equal to 1/2 hab gamma ab hab. 846 00:52:24,830 --> 00:52:25,785 Oh, gamma cd. 847 00:52:31,200 --> 00:52:36,910 Yeah, which you can show is equivalent to that equation. 848 00:52:36,910 --> 00:52:37,660 OK. 849 00:52:37,660 --> 00:52:42,770 Or maybe I should do gamma ab. 850 00:52:42,770 --> 00:52:45,560 No, if you do that one, it doesn't matter. 851 00:52:45,560 --> 00:52:53,290 I can do-- to be exactly the same as that equation, 852 00:52:53,290 --> 00:52:56,100 let me just do both sides. 853 00:52:56,100 --> 00:52:57,620 I can track with gamma ab. 854 00:53:02,160 --> 00:53:05,950 Let me see, maybe I can [? track ?] both sides. 855 00:53:05,950 --> 00:53:07,500 Let me think. 856 00:53:07,500 --> 00:53:18,460 So that is [? HCD ?] so that is h gamma ab, times that thing, 857 00:53:18,460 --> 00:53:19,580 is equal to-- 858 00:53:24,120 --> 00:53:30,200 Yeah, you can-- just take this equation. 859 00:53:30,200 --> 00:53:31,616 You get this equation. 860 00:53:31,616 --> 00:53:34,240 Then you can show that equation is equivalent to this equation, 861 00:53:34,240 --> 00:53:35,990 OK? 862 00:53:35,990 --> 00:53:38,890 [INAUDIBLE] right now, they're not reading the identical way, 863 00:53:38,890 --> 00:53:40,306 but they can show the equivalence. 864 00:53:48,310 --> 00:53:51,100 So yeah, actually that's your homework. 865 00:53:53,720 --> 00:54:03,645 So now if you look at equation of motion for X mu, 866 00:54:03,645 --> 00:54:05,145 look at the equation of motion for X 867 00:54:05,145 --> 00:54:08,220 mu, then this is just [INAUDIBLE] 868 00:54:08,220 --> 00:54:10,220 your standard scalar field. 869 00:54:10,220 --> 00:54:20,730 So this just give you-- OK, and we call this equation six. 870 00:54:28,420 --> 00:54:31,452 So these are all of your equation of motion. 871 00:54:31,452 --> 00:54:35,070 If you have open string, then when 872 00:54:35,070 --> 00:54:41,290 you do the evaluation of X, you have 873 00:54:41,290 --> 00:54:43,765 to do integration by parts. 874 00:54:43,765 --> 00:54:45,140 Then in the standard [? story, ?] 875 00:54:45,140 --> 00:54:48,750 you have to have a boundary term. 876 00:54:48,750 --> 00:54:49,529 OK? 877 00:54:49,529 --> 00:54:51,820 But you actually have to be careful about that boundary 878 00:54:51,820 --> 00:54:52,320 term. 879 00:54:56,330 --> 00:55:05,780 For the open string, you will get a boundary term, 880 00:55:05,780 --> 00:55:13,796 which is given by delta mu gamma sigma b partial b X 881 00:55:13,796 --> 00:55:21,291 mu, evaluated at sigma equal to 0 and pi, should be equal to 0. 882 00:55:21,291 --> 00:55:21,790 OK? 883 00:55:25,650 --> 00:55:31,230 So this just come from where you get this second-order equation, 884 00:55:31,230 --> 00:55:33,260 you always have to integration by parts 885 00:55:33,260 --> 00:55:34,800 when you do the evaluation. 886 00:55:34,800 --> 00:55:36,550 That just comes from the boundary term 887 00:55:36,550 --> 00:55:37,980 of that [? variation, ?] evaluated 888 00:55:37,980 --> 00:55:40,650 at the end point of the string you 889 00:55:40,650 --> 00:55:42,611 have in the sigma direction. 890 00:55:42,611 --> 00:55:43,110 OK. 891 00:55:43,110 --> 00:55:43,693 Is this clear? 892 00:55:46,040 --> 00:55:46,920 Good. 893 00:55:46,920 --> 00:55:49,874 Again, I will not go through all the algebraic details. 894 00:55:49,874 --> 00:55:51,540 You should check those details yourself. 895 00:55:54,490 --> 00:56:01,410 You should check them, and if you find mistakes, 896 00:56:01,410 --> 00:56:02,410 it's greatly encouraged. 897 00:56:06,217 --> 00:56:07,550 Yeah, I will add to your points. 898 00:56:10,430 --> 00:56:14,000 Yeah, if you find mistakes, then I will add your points. 899 00:56:14,000 --> 00:56:16,240 If you have found enough mistakes, 900 00:56:16,240 --> 00:56:18,456 then you may not need to do the [? PSAT ?] anymore. 901 00:56:25,760 --> 00:56:30,155 So this tells us we can actually impose two kinds of boundary 902 00:56:30,155 --> 00:56:30,655 conditions. 903 00:56:35,470 --> 00:56:39,330 Say delta X mu have to be 0. 904 00:56:39,330 --> 00:56:47,720 So evaluated at sigma equal to 0, pi, equal to 0. 905 00:56:47,720 --> 00:56:50,100 Sigma 0 or pi-- it does not have to be the same. 906 00:56:53,440 --> 00:56:57,470 Or this thing is 0, gamma sigma b. 907 00:56:57,470 --> 00:56:59,740 The reason the sigma here, because we are interested 908 00:56:59,740 --> 00:57:02,130 only in the boundary term in the sigma direction. 909 00:57:02,130 --> 00:57:04,380 The boundary term in the time direction we [INAUDIBLE] 910 00:57:04,380 --> 00:57:05,291 care about. 911 00:57:05,291 --> 00:57:07,290 Because the boundary term in the time direction, 912 00:57:07,290 --> 00:57:10,890 we always assume in the far past and the far future, 913 00:57:10,890 --> 00:57:12,350 nothing happens. 914 00:57:12,350 --> 00:57:14,610 We always assume that. 915 00:57:14,610 --> 00:57:17,211 But you cannot assume a boundary condition at the spatial 916 00:57:17,211 --> 00:57:17,710 condition. 917 00:57:17,710 --> 00:57:19,390 You have-- yeah, in the spatial boundary, 918 00:57:19,390 --> 00:57:20,806 you have to be careful, because we 919 00:57:20,806 --> 00:57:23,530 have a finite-- [? a special ?] extension. 920 00:57:23,530 --> 00:57:26,050 OK, so that's why this sigma. 921 00:57:26,050 --> 00:57:28,720 So a lot of the possible boundary conditions, 922 00:57:28,720 --> 00:57:34,720 gamma sigma b, times partial b X mu, so evaluated at sigma 923 00:57:34,720 --> 00:57:37,240 is equal to 0 or pi, equal to 0. 924 00:57:41,010 --> 00:57:43,700 So this is normally called the Dirichlet boundary condition. 925 00:57:43,700 --> 00:57:47,550 This is normally call the Neumann boundary condition. 926 00:57:47,550 --> 00:57:50,410 The reason this is called Dirichlet boundary condition 927 00:57:50,410 --> 00:57:56,260 because if that X is fixed at the end point, 928 00:57:56,260 --> 00:58:00,820 it means that the value of X are not allowed to vary. 929 00:58:00,820 --> 00:58:03,600 So this is normally called the Dirichlet, OK? 930 00:58:03,600 --> 00:58:05,660 And for the Neumann boundary condition, 931 00:58:05,660 --> 00:58:08,110 you only constrain the derivative. 932 00:58:08,110 --> 00:58:10,467 And the value of X can be anything. 933 00:58:10,467 --> 00:58:12,050 And you only constrain the derivative. 934 00:58:12,050 --> 00:58:14,091 So this is called the Neumann boundary condition. 935 00:58:20,710 --> 00:58:27,210 So for simplicity, later we will treat both of them. 936 00:58:27,210 --> 00:58:29,430 But for now, let me just look at this one, 937 00:58:29,430 --> 00:58:33,140 because this one is slightly simpler. 938 00:58:33,140 --> 00:58:36,000 So for the moment, let me just consider the Neumann boundary 939 00:58:36,000 --> 00:58:36,880 condition, OK. 940 00:58:39,800 --> 00:58:42,065 So now the second step is fixing the gauge. 941 00:58:45,440 --> 00:58:47,330 Second step is fixing the gauge. 942 00:58:47,330 --> 00:58:50,700 So we have two gauge freedom. 943 00:58:50,700 --> 00:58:54,430 One is to change the coordinates. 944 00:58:54,430 --> 00:58:57,870 And the one is to do a scaling. 945 00:58:57,870 --> 00:58:59,210 OK. 946 00:58:59,210 --> 00:59:00,670 So before we fix the gauge, let's 947 00:59:00,670 --> 00:59:04,170 do a little bit of counting. 948 00:59:04,170 --> 00:59:08,850 So here I can change coordinates by arbitrary two functions, OK? 949 00:59:08,850 --> 00:59:14,280 So here I essentially have two arbitrary functions 950 00:59:14,280 --> 00:59:16,900 with freedom of the gauge freedom. 951 00:59:16,900 --> 00:59:20,010 And here, I have one arbitrary functions 952 00:59:20,010 --> 00:59:25,520 for both gauge freedom, because I can choose omega [INAUDIBLE] 953 00:59:25,520 --> 00:59:27,170 arbitrarily. 954 00:59:27,170 --> 00:59:32,280 And if you look at gamma ab, the worldsheet metric-- so ab 955 00:59:32,280 --> 00:59:33,590 is from 0 to 1. 956 00:59:33,590 --> 00:59:38,020 So this also has three freedoms, three degrees of freedom. 957 00:59:43,220 --> 00:59:47,840 So that means actually, in [INAUDIBLE] theory, 958 00:59:47,840 --> 00:59:55,820 at least you can expect by taking account 959 00:59:55,820 --> 00:59:59,010 those degrees of freedom, taking account of these three gauge 960 00:59:59,010 --> 01:00:05,040 degrees freedom, you can set this gamma ab 961 01:00:05,040 --> 01:00:07,330 to some fixed metric. 962 01:00:07,330 --> 01:00:10,230 Because there's only three freedom here, and we have three 963 01:00:10,230 --> 01:00:11,140 gauge freedom. 964 01:00:11,140 --> 01:00:14,600 And you can completely fix them. 965 01:00:14,600 --> 01:00:18,545 So [? not ?] let me erase it now. 966 01:00:23,470 --> 01:00:25,017 Now let me do the second step. 967 01:00:29,400 --> 01:00:37,470 Gauge fixing using this different morphism of b, 968 01:00:37,470 --> 01:00:40,020 so these have two parameters. 969 01:00:40,020 --> 01:00:46,390 You can show that you can just make 970 01:00:46,390 --> 01:00:47,930 a coordinate transformation. 971 01:00:47,930 --> 01:00:52,790 Take any gamma ab, you can show that by using 972 01:00:52,790 --> 01:00:55,220 a coordinate transformation, you can transform the gamma 973 01:00:55,220 --> 01:00:56,740 ab to the following form. 974 01:01:05,700 --> 01:01:07,500 OK. 975 01:01:07,500 --> 01:01:12,210 So I will not prove it here, but I'll 976 01:01:12,210 --> 01:01:16,530 let you go back to try to convince yourself of this fact, 977 01:01:16,530 --> 01:01:18,270 OK? 978 01:01:18,270 --> 01:01:20,090 Just by using the coordinate transformation 979 01:01:20,090 --> 01:01:24,070 starting with generic gamma ab, you 980 01:01:24,070 --> 01:01:25,570 can make a coordinate transformation 981 01:01:25,570 --> 01:01:28,400 so that in the new coordinates, the gamma ab 982 01:01:28,400 --> 01:01:34,180 is given by a prefactor times the Minkowski metric. 983 01:01:34,180 --> 01:01:40,586 And now we can use the c, that guy, 984 01:01:40,586 --> 01:01:43,680 because we have the freedom to change the gamma 985 01:01:43,680 --> 01:01:46,460 ab by arbitrary function. 986 01:01:46,460 --> 01:01:48,920 And then we can just choose that function 987 01:01:48,920 --> 01:01:50,600 to be [INAUDIBLE] to this guy. 988 01:01:50,600 --> 01:01:58,840 Then we can set gamma ab just equal to eta ab. 989 01:01:58,840 --> 01:01:59,340 OK. 990 01:02:03,150 --> 01:02:05,560 So we can use these two freedom to set the gamma 991 01:02:05,560 --> 01:02:06,490 ab with eta ab. 992 01:02:06,490 --> 01:02:07,824 Yes? 993 01:02:07,824 --> 01:02:10,234 AUDIENCE: So you're saying that we can [INAUDIBLE] gauge 994 01:02:10,234 --> 01:02:12,162 [? toward ?] locally on the worldsheet. 995 01:02:12,162 --> 01:02:14,630 [INAUDIBLE] gauge transform the metric [? globally ?] 996 01:02:14,630 --> 01:02:15,962 [INAUDIBLE]. 997 01:02:15,962 --> 01:02:16,920 HONG LIU: That's right. 998 01:02:16,920 --> 01:02:20,320 Yeah, we can transform it locally into [? flat ?] metric. 999 01:02:20,320 --> 01:02:24,020 And then, indeed, there is a global issue of which 1000 01:02:24,020 --> 01:02:25,740 I will not worry about now. 1001 01:02:25,740 --> 01:02:27,900 Yeah, because the question which I'm interested in, 1002 01:02:27,900 --> 01:02:29,830 say, the spectrum only [? consists ?] 1003 01:02:29,830 --> 01:02:32,550 of the local questions. 1004 01:02:32,550 --> 01:02:34,322 For example, if you do the path integral, 1005 01:02:34,322 --> 01:02:36,530 then you don't have to worry about the global issues. 1006 01:02:39,440 --> 01:02:40,410 AUDIENCE: Why? 1007 01:02:40,410 --> 01:02:44,130 Why is it that when you [INAUDIBLE]? 1008 01:02:44,130 --> 01:02:47,650 HONG LIU: Yeah, because you need to integrate 1009 01:02:47,650 --> 01:02:49,670 over all possible configurations. 1010 01:02:49,670 --> 01:02:54,670 And you don't want to do the overcounting, or count mass. 1011 01:02:54,670 --> 01:02:57,410 Yeah. 1012 01:02:57,410 --> 01:02:59,270 AUDIENCE: So we don't have to work this out. 1013 01:02:59,270 --> 01:03:00,895 So why does that mean we don't have to? 1014 01:03:00,895 --> 01:03:01,950 HONG LIU: Yeah. 1015 01:03:01,950 --> 01:03:03,500 So let me give you an example. 1016 01:03:03,500 --> 01:03:10,580 For example, if you do the genus g surfaces, and then say 1017 01:03:10,580 --> 01:03:11,701 let's do Euclidean again. 1018 01:03:11,701 --> 01:03:12,200 It's easier. 1019 01:03:12,200 --> 01:03:14,040 Do Euclidean. 1020 01:03:14,040 --> 01:03:16,370 Indeed, you can locally transform to eta ab 1021 01:03:16,370 --> 01:03:19,064 the actual set of discrete parameters. 1022 01:03:19,064 --> 01:03:21,230 And then you have to be careful about those discrete 1023 01:03:21,230 --> 01:03:22,910 parameters, et cetera. 1024 01:03:22,910 --> 01:03:25,400 Yeah, just those discrete parameters 1025 01:03:25,400 --> 01:03:27,372 have to do with global issues, et cetera. 1026 01:03:27,372 --> 01:03:28,830 AUDIENCE: I see what you're saying. 1027 01:03:28,830 --> 01:03:29,653 HONG LIU: Yes? 1028 01:03:29,653 --> 01:03:30,569 AUDIENCE: [INAUDIBLE]. 1029 01:03:34,749 --> 01:03:36,290 HONG LIU: Yeah, just imagine you have 1030 01:03:36,290 --> 01:03:37,498 some two-dimensional surface. 1031 01:03:37,498 --> 01:03:40,620 Right now let's not consider about the topology. 1032 01:03:40,620 --> 01:03:42,400 Right now just consider. 1033 01:03:42,400 --> 01:03:48,240 So in this case, for the closed string, the simplest thing-- 1034 01:03:48,240 --> 01:03:50,096 you can see this [? sitting ?] there. 1035 01:03:50,096 --> 01:03:51,220 So this is sigma direction. 1036 01:03:51,220 --> 01:03:53,300 This is the tau direction. 1037 01:03:53,300 --> 01:03:54,425 Yeah. 1038 01:03:54,425 --> 01:03:56,300 Yeah, for the closed string it's [INAUDIBLE]. 1039 01:03:56,300 --> 01:03:58,730 But the open string, roughly just like this. 1040 01:04:04,130 --> 01:04:07,480 And right now we don't consider complicated topology. 1041 01:04:07,480 --> 01:04:09,020 We just consider local question. 1042 01:04:09,020 --> 01:04:12,349 Then we can consider the simplest topology. 1043 01:04:12,349 --> 01:04:13,890 AUDIENCE: Yeah, sorry, I didn't quite 1044 01:04:13,890 --> 01:04:15,010 get the path integral being-- 1045 01:04:15,010 --> 01:04:16,465 HONG LIU: Yeah, don't worry about it. 1046 01:04:16,465 --> 01:04:18,330 AUDIENCE: Don't you have to integrate around 1047 01:04:18,330 --> 01:04:20,344 both sides of every [? pole that ?] you 1048 01:04:20,344 --> 01:04:21,200 have in the-- 1049 01:04:21,200 --> 01:04:23,785 HONG LIU: Yeah, you have to integrate over everything. 1050 01:04:23,785 --> 01:04:25,760 You have to be careful about everything. 1051 01:04:25,760 --> 01:04:27,805 AUDIENCE: So global [? effect is a factor. ?] 1052 01:04:27,805 --> 01:04:28,430 HONG LIU: Yeah. 1053 01:04:28,430 --> 01:04:29,350 AUDIENCE: It does matter. 1054 01:04:29,350 --> 01:04:29,570 OK. 1055 01:04:29,570 --> 01:04:29,890 HONG LIU: Of course. 1056 01:04:29,890 --> 01:04:31,806 AUDIENCE: I thought you were saying the other. 1057 01:04:31,806 --> 01:04:33,070 HONG LIU: No, no, no. 1058 01:04:33,070 --> 01:04:35,340 I'm saying the global effects are very important when 1059 01:04:35,340 --> 01:04:38,030 you do the path integral, and if you want to do the path 1060 01:04:38,030 --> 01:04:39,210 integral correctly. 1061 01:04:39,210 --> 01:04:42,802 But here we are interested in the spectrum. 1062 01:04:42,802 --> 01:04:44,010 And this is a local question. 1063 01:04:48,400 --> 01:04:54,460 OK, so now with this gauge fixing, 1064 01:04:54,460 --> 01:04:57,800 we can see how the equation of motion simplifies. 1065 01:05:00,530 --> 01:05:09,530 First we have fixed all gauge freedom, OK? 1066 01:05:09,530 --> 01:05:13,460 So let's look at the six first. 1067 01:05:13,460 --> 01:05:17,470 So six just become a free scalar equation. 1068 01:05:17,470 --> 01:05:19,820 Because when gamma ab become a Minkowski metric, 1069 01:05:19,820 --> 01:05:23,010 this is just like a scalar field, OK? 1070 01:05:23,010 --> 01:05:24,662 This is like a scalar field. 1071 01:05:24,662 --> 01:05:25,995 So this we can write explicitly. 1072 01:05:32,160 --> 01:05:34,510 So this is a free scalar field. 1073 01:05:34,510 --> 01:05:42,344 And let me call this somehow-- oh, yeah, 1074 01:05:42,344 --> 01:05:45,950 let me call this seven. 1075 01:05:45,950 --> 01:05:48,340 This one seven, this one eight, just 1076 01:05:48,340 --> 01:05:51,690 to be consistent with my notes. 1077 01:05:51,690 --> 01:05:52,640 Then I call this nine. 1078 01:05:56,850 --> 01:06:00,080 But if we still have to worry about the five, 1079 01:06:00,080 --> 01:06:06,190 OK, so the five can now also be written explicitly. 1080 01:06:09,140 --> 01:06:13,530 So first thing you can check is that T0,0 is equal to T1,1, 1081 01:06:13,530 --> 01:06:16,330 so the diagonal elements are the same. 1082 01:06:16,330 --> 01:06:21,480 It's given by 1/2 partial tau X mu partial tau 1083 01:06:21,480 --> 01:06:32,310 X mu, plus partial sigma X mu partial sigma X mu equal to 0. 1084 01:06:32,310 --> 01:06:47,570 And T0,1 component will give you partial tau X mu [? times ?] 1085 01:06:47,570 --> 01:06:51,590 partial sigma X mu equal to 0. 1086 01:06:51,590 --> 01:06:54,525 So I'm just writing everything explicitly. 1087 01:06:54,525 --> 01:06:55,900 So these are the three components 1088 01:06:55,900 --> 01:06:59,910 of that equation, when I set gamma ab equal to eta ab. 1089 01:07:07,540 --> 01:07:08,790 You can easily check yourself. 1090 01:07:08,790 --> 01:07:10,380 Just plug that in. 1091 01:07:10,380 --> 01:07:11,237 OK? 1092 01:07:11,237 --> 01:07:20,290 So the reason that T0,0 is equal to T1,1 is because we actually 1093 01:07:20,290 --> 01:07:22,550 have a scaling symmetry. 1094 01:07:22,550 --> 01:07:27,050 So in the case when this action-- yeah, 1095 01:07:27,050 --> 01:07:29,260 this is a free massless scalar field. 1096 01:07:29,260 --> 01:07:31,030 It's a scaling symmetry. 1097 01:07:31,030 --> 01:07:33,580 And you know that when you have a scaling symmetry under 1098 01:07:33,580 --> 01:07:35,579 the stress tensor, the [? trace ?] of the stress 1099 01:07:35,579 --> 01:07:41,060 tensor always vanishes, so that's why T0,0 and T1,1. 1100 01:07:41,060 --> 01:07:42,986 So for the open string, we also have 1101 01:07:42,986 --> 01:07:44,611 to worry about the boundary conditions. 1102 01:07:51,350 --> 01:07:55,740 And so this just become the Minkowski metric. 1103 01:07:55,740 --> 01:07:58,900 And the sigma sigma, only the sigma sigma is non-zero. 1104 01:07:58,900 --> 01:08:05,620 So you just get the partial sigma X mu equal to 0. 1105 01:08:05,620 --> 01:08:08,747 OK, so this should be evaluated at sigma equal to 0 and pi. 1106 01:08:11,634 --> 01:08:14,050 So right now, we only consider Neumann boundary condition. 1107 01:08:18,470 --> 01:08:20,380 And let me call it 12. 1108 01:08:20,380 --> 01:08:21,540 So this is for Neumann. 1109 01:08:25,750 --> 01:08:27,914 Good? 1110 01:08:27,914 --> 01:08:28,830 Any questions on this? 1111 01:08:53,279 --> 01:08:54,170 Yeah. 1112 01:08:54,170 --> 01:08:58,409 AUDIENCE: For this Dirichlet boundary condition, 1113 01:08:58,409 --> 01:09:00,910 it's a [? valuation. ?] So you already may say 1114 01:09:00,910 --> 01:09:06,920 the [? valuation ?] is arbitrary by [INAUDIBLE] can be 1115 01:09:06,920 --> 01:09:07,420 arbitrary. 1116 01:09:07,420 --> 01:09:12,300 So it seems like only Neumann boundary condition is valid. 1117 01:09:12,300 --> 01:09:14,540 HONG LIU: Yeah, this is your choice. 1118 01:09:14,540 --> 01:09:17,300 This is your choice. 1119 01:09:17,300 --> 01:09:26,300 So normally in the field theory, what we do-- indeed, 1120 01:09:26,300 --> 01:09:29,080 normally in the field theory, yeah, this is your choice. 1121 01:09:32,310 --> 01:09:35,569 In the normal field theory, where you consider, say, 1122 01:09:35,569 --> 01:09:38,170 an infinite space, we just assume everything 1123 01:09:38,170 --> 01:09:39,590 goes to 0 at infinity. 1124 01:09:39,590 --> 01:09:42,700 And we don't even need to worry about the boundary condition. 1125 01:09:42,700 --> 01:09:45,300 Then the boundary term automatically vanishes. 1126 01:09:45,300 --> 01:09:47,009 And so that's why, normally, in the field 1127 01:09:47,009 --> 01:09:48,591 theory in the infinite space, we never 1128 01:09:48,591 --> 01:09:50,080 worry about the boundary conditions 1129 01:09:50,080 --> 01:09:52,520 because we made that assumption. 1130 01:09:52,520 --> 01:09:57,070 So here, because the string has a finite [? lens, ?] 1131 01:09:57,070 --> 01:09:58,850 it's a finite boundary. 1132 01:09:58,850 --> 01:10:02,780 And then you actually have a freedom to introduce both. 1133 01:10:02,780 --> 01:10:04,030 Yeah. 1134 01:10:04,030 --> 01:10:06,284 You do have the freedom to introduce both. 1135 01:10:21,120 --> 01:10:22,090 Good? 1136 01:10:22,090 --> 01:10:25,770 So let me emphasize again. 1137 01:10:25,770 --> 01:10:30,650 So now in this gauge, we just have a free scalar field. 1138 01:10:30,650 --> 01:10:33,129 So this is a standard wave equation in one 1139 01:10:33,129 --> 01:10:33,920 plus one dimension. 1140 01:10:36,560 --> 01:10:39,155 But we are not quantizing a free scalar field theory, 1141 01:10:39,155 --> 01:10:42,082 because we still have to solve those things. 1142 01:10:42,082 --> 01:10:44,370 We have to solve these two equations. 1143 01:10:44,370 --> 01:10:46,470 And these are constraints. 1144 01:10:46,470 --> 01:10:48,354 Because in this equation, in principle, 1145 01:10:48,354 --> 01:10:49,770 you completely solve the equation. 1146 01:10:49,770 --> 01:10:52,270 We know that this is a wave equation. 1147 01:10:52,270 --> 01:10:54,550 You already solved X completely. 1148 01:10:54,550 --> 01:10:57,409 But these two equation, coming from the gamma equation 1149 01:10:57,409 --> 01:10:59,200 of motion, they impose a [? longitudinal ?] 1150 01:10:59,200 --> 01:11:02,090 constraint we have to satisfy. 1151 01:11:02,090 --> 01:11:04,920 And those constraints actually last [INAUDIBLE] 1152 01:11:04,920 --> 01:11:06,320 because they are non-linear. 1153 01:11:06,320 --> 01:11:07,820 This is a linear equation, but these 1154 01:11:07,820 --> 01:11:10,450 are non-linear constraints. 1155 01:11:10,450 --> 01:11:14,830 And this is the 10, 11-- I'll normally 1156 01:11:14,830 --> 01:11:17,130 call them Virasoro constraints. 1157 01:11:20,450 --> 01:11:23,170 Virasoro was a person. 1158 01:11:23,170 --> 01:11:25,130 So this is the Virasoro constraints. 1159 01:11:25,130 --> 01:11:27,320 OK? 1160 01:11:27,320 --> 01:11:28,755 And so this allows me to constrain 1161 01:11:28,755 --> 01:11:29,796 the equation [INAUDIBLE]. 1162 01:11:34,640 --> 01:11:38,390 So now let's first solve the easy equation. 1163 01:11:38,390 --> 01:11:41,210 So let's go to the third step, trying to solve 1164 01:11:41,210 --> 01:11:42,750 the equation of motion. 1165 01:11:42,750 --> 01:11:44,430 And now we have fixed the gauge. 1166 01:11:44,430 --> 01:11:46,560 And now we try to solve the equation of motion, OK? 1167 01:11:50,390 --> 01:11:56,469 So for people at your level, I can write down 1168 01:11:56,469 --> 01:11:57,760 a solution of line immediately. 1169 01:12:06,480 --> 01:12:11,870 So I can have a constant, of course, satisfy this equation. 1170 01:12:11,870 --> 01:12:14,670 I can have a linear term in tau, which of course will 1171 01:12:14,670 --> 01:12:18,370 satisfy this equation. 1172 01:12:18,370 --> 01:12:20,095 For closed string, in principle, I 1173 01:12:20,095 --> 01:12:24,420 can also have a linear term in sigma, 1174 01:12:24,420 --> 01:12:27,745 but that won't satisfy the periodic boundary condition. 1175 01:12:32,270 --> 01:12:33,920 So that's not allowed. 1176 01:12:33,920 --> 01:12:37,025 And then I can add the traveling wave solutions. 1177 01:12:39,810 --> 01:12:43,030 So this is a left-moving wave, a right-moving wave, 1178 01:12:43,030 --> 01:12:46,560 and this is a left-moving wave. 1179 01:12:46,560 --> 01:12:48,210 OK? 1180 01:12:48,210 --> 01:12:53,540 So this is a full set of solutions to-- so 1181 01:12:53,540 --> 01:12:54,875 x mu are arbitrary constants. 1182 01:12:58,320 --> 01:13:00,295 x mu and v mu are arbitrary constants. 1183 01:13:03,010 --> 01:13:09,550 And XL and XR are arbitrary for closed strings. 1184 01:13:14,050 --> 01:13:23,560 So XL and XR just are independent, 1185 01:13:23,560 --> 01:13:34,900 should be independent periodic functions of period 2 pi. 1186 01:13:38,620 --> 01:13:41,270 OK? 1187 01:13:41,270 --> 01:13:43,250 Because they have to be. 1188 01:13:43,250 --> 01:13:47,530 So XL and XR are function of single variable. 1189 01:13:47,530 --> 01:13:49,460 And they have to be period in sigma means 1190 01:13:49,460 --> 01:13:50,501 they have to be periodic. 1191 01:13:53,000 --> 01:13:57,010 So these are the periodic function. 1192 01:13:57,010 --> 01:13:59,040 OK, is this clear to you? 1193 01:13:59,040 --> 01:14:02,170 Do I need to explain this equation? 1194 01:14:02,170 --> 01:14:02,890 Good. 1195 01:14:02,890 --> 01:14:03,390 Yes? 1196 01:14:03,390 --> 01:14:07,579 AUDIENCE: Why a function of tau minus the sigma? 1197 01:14:07,579 --> 01:14:09,620 HONG LIU: It's because you have a traveling wave. 1198 01:14:09,620 --> 01:14:13,520 Because first we can immediately see this 1199 01:14:13,520 --> 01:14:16,477 satisfies this equation. 1200 01:14:16,477 --> 01:14:18,060 And then you need to convince yourself 1201 01:14:18,060 --> 01:14:27,820 these are the only solutions, in the sense that-- yeah, 1202 01:14:27,820 --> 01:14:29,890 because these are the two arbitrary functions. 1203 01:14:29,890 --> 01:14:31,464 There's no other choices anymore. 1204 01:14:35,340 --> 01:14:36,510 Any more questions on this? 1205 01:14:36,510 --> 01:14:37,426 AUDIENCE: [INAUDIBLE]. 1206 01:14:42,160 --> 01:14:47,600 HONG LIU: Also not, for other reasons. 1207 01:14:47,600 --> 01:14:52,385 Not for the Neumann boundary condition. 1208 01:14:52,385 --> 01:14:53,760 Not for this boundary connection. 1209 01:14:53,760 --> 01:14:55,870 And now we are come to open string. 1210 01:14:59,270 --> 01:15:09,677 So for open string, you can then be-- again, we have these. 1211 01:15:09,677 --> 01:15:11,510 We can check that if you have a linear sigma 1212 01:15:11,510 --> 01:15:15,490 term [? it will be ?] incompatible with this boundary 1213 01:15:15,490 --> 01:15:18,460 condition, so we don't worry about that. 1214 01:15:18,460 --> 01:15:20,180 So we don't worry about that. 1215 01:15:20,180 --> 01:15:25,650 And then, from this equation, then plug that 1216 01:15:25,650 --> 01:15:33,420 into this equation, we conclude that the XL prime tau 1217 01:15:33,420 --> 01:15:39,630 should be equal to XR prime tau at sigma equal to 0. 1218 01:15:42,310 --> 01:15:47,365 And XL prime, tau minus pi, should 1219 01:15:47,365 --> 01:15:54,750 be equal to XR prime, tau plus pi, at sigma equal to pi. 1220 01:15:54,750 --> 01:15:59,160 OK, so you just plug this into here. 1221 01:15:59,160 --> 01:16:01,390 So the first two terms does not matter, 1222 01:16:01,390 --> 01:16:03,920 and then you just have the last two terms. 1223 01:16:03,920 --> 01:16:09,980 So the last two terms, then if you evaluate it 1224 01:16:09,980 --> 01:16:16,510 at sigma equal to 0, then you just get this equal to that. 1225 01:16:16,510 --> 01:16:19,970 And if you evaluate it at sigma equal to pi, 1226 01:16:19,970 --> 01:16:21,200 then find this, OK? 1227 01:16:24,550 --> 01:16:25,110 Yeah. 1228 01:16:25,110 --> 01:16:26,526 AUDIENCE: Why is it that you don't 1229 01:16:26,526 --> 01:16:29,650 package those original two terms into the right 1230 01:16:29,650 --> 01:16:31,500 and left functions? 1231 01:16:31,500 --> 01:16:35,710 So why is that you write x mu plus v mu tau? 1232 01:16:35,710 --> 01:16:39,141 HONG LIU: Because those equation don't have this form. 1233 01:16:39,141 --> 01:16:41,682 They're not in the form of tau plus sigma or tau minus sigma. 1234 01:16:46,600 --> 01:16:48,050 No, this is just tau. 1235 01:16:48,050 --> 01:16:49,712 This is not tau plus sigma. 1236 01:16:49,712 --> 01:16:51,295 Now, I can [? pack it ?] in that form. 1237 01:16:51,295 --> 01:16:51,770 It doesn't matter. 1238 01:16:51,770 --> 01:16:53,520 I'm just saying here I'm not doing this. 1239 01:17:03,720 --> 01:17:04,280 Yeah. 1240 01:17:04,280 --> 01:17:07,610 What I'm saying-- these are trivially periodic functions. 1241 01:17:07,610 --> 01:17:09,865 And yeah, I [? just separate ?] that. 1242 01:17:12,570 --> 01:17:13,070 OK? 1243 01:17:16,450 --> 01:17:29,700 So now, from here, you can find that-- so this equation 1244 01:17:29,700 --> 01:17:36,030 tells you that XL is essentially equal to XR, OK? 1245 01:17:36,030 --> 01:17:40,810 So up to a constant we can absorb into X mu, for example. 1246 01:17:40,810 --> 01:17:44,420 Then these two functions could be the same. 1247 01:17:44,420 --> 01:17:47,870 And if we take these two function to be the same, 1248 01:17:47,870 --> 01:17:51,490 then this function tells us this is 1249 01:17:51,490 --> 01:17:56,410 a function-- the second line tells 1250 01:17:56,410 --> 01:18:01,790 us this is periodic in 2 pi. 1251 01:18:01,790 --> 01:18:02,290 OK? 1252 01:18:13,470 --> 01:18:17,290 Yeah, I actually think I missed a minus sign here. 1253 01:18:17,290 --> 01:18:20,220 So if I take the sigma derivative, 1254 01:18:20,220 --> 01:18:22,970 yeah, I can put a minus sign here. 1255 01:18:22,970 --> 01:18:24,730 But these two conclusions still are right. 1256 01:18:39,130 --> 01:18:41,170 So now for both open and closed string, 1257 01:18:41,170 --> 01:18:44,820 you have solved nine completely. 1258 01:18:44,820 --> 01:18:46,358 Solved nine completely. 1259 01:18:46,358 --> 01:18:47,399 Any questions about this? 1260 01:18:57,950 --> 01:19:02,150 OK, but we still have to solve 10 and 11. 1261 01:19:05,480 --> 01:19:07,860 So what 10 and 11 does is to impose 1262 01:19:07,860 --> 01:19:12,020 some nontrivial constraints between XL and XR 1263 01:19:12,020 --> 01:19:15,130 and x mu and v, et cetera. 1264 01:19:15,130 --> 01:19:18,210 I'd just say plug in this most general solution 1265 01:19:18,210 --> 01:19:19,950 into 10 and 11. 1266 01:19:19,950 --> 01:19:23,300 And we'll impose some constraints on those functions. 1267 01:19:23,300 --> 01:19:26,100 So I urge you to try it a little bit yourself. 1268 01:19:26,100 --> 01:19:31,840 And those equations are not nice equations. 1269 01:19:35,150 --> 01:19:38,590 There are lots of nice people. 1270 01:19:38,590 --> 01:19:42,000 Yeah, there are lots of nice equations. 1271 01:19:42,000 --> 01:19:54,010 And so at this stage, we face a decision. 1272 01:19:54,010 --> 01:19:57,836 So when you see this 10 and 11 are hard to solve, 1273 01:19:57,836 --> 01:19:59,210 then you have a decision to make. 1274 01:19:59,210 --> 01:20:00,320 Yeah, whenever you have something 1275 01:20:00,320 --> 01:20:02,070 that you don't know how to do immediately, 1276 01:20:02,070 --> 01:20:03,650 you have a decision to make. 1277 01:20:03,650 --> 01:20:05,020 We have some decision to make. 1278 01:20:07,860 --> 01:20:09,730 Or you just give up. 1279 01:20:09,730 --> 01:20:15,580 And first-- so one option is that we could just 1280 01:20:15,580 --> 01:20:17,500 quantize this theory. 1281 01:20:17,500 --> 01:20:21,680 We know how to quantize this free scalar field theory. 1282 01:20:21,680 --> 01:20:25,160 We just quantized this free scalar field theory first. 1283 01:20:25,160 --> 01:20:28,420 And then we can construct the Hilbert space, et cetera. 1284 01:20:28,420 --> 01:20:30,440 And then we can impose those constraints 1285 01:20:30,440 --> 01:20:32,750 at the quantum level. 1286 01:20:32,750 --> 01:20:37,790 We can impose those constraints at the quantum level. 1287 01:20:37,790 --> 01:20:39,000 Yeah, so this is one option. 1288 01:20:41,770 --> 01:20:45,070 The second option is that you find some ingenious way 1289 01:20:45,070 --> 01:20:49,010 to solve those equations so that you can actually 1290 01:20:49,010 --> 01:20:52,550 find independent variables. 1291 01:20:52,550 --> 01:20:54,620 OK, because those are constraint equations, 1292 01:20:54,620 --> 01:20:57,550 means some degrees of freedom allow the independence 1293 01:20:57,550 --> 01:20:59,080 from others. 1294 01:20:59,080 --> 01:21:04,220 So the alternative is, I guess, you just solve it. 1295 01:21:04,220 --> 01:21:07,730 And finally, independently, with freedom, and then quantize only 1296 01:21:07,730 --> 01:21:09,960 [? those ?] independently with freedom. 1297 01:21:09,960 --> 01:21:11,190 OK. 1298 01:21:11,190 --> 01:21:15,600 So the first option, this stage is easy, 1299 01:21:15,600 --> 01:21:18,810 because you can quantize the first [? gauge ?] field theory. 1300 01:21:18,810 --> 01:21:21,690 But the process will impose the constraint at a quantum level. 1301 01:21:21,690 --> 01:21:23,270 It's actually rather tricky. 1302 01:21:23,270 --> 01:21:24,850 So I will not go there. 1303 01:21:24,850 --> 01:21:27,030 So I will do the second method, [? rather ?] 1304 01:21:27,030 --> 01:21:30,130 we try to solve this constraint, find some trick 1305 01:21:30,130 --> 01:21:33,964 to solve this constraint, and then quantize 1306 01:21:33,964 --> 01:21:35,130 independent degrees freedom. 1307 01:21:40,110 --> 01:21:41,760 Is it clear? 1308 01:21:41,760 --> 01:21:44,400 So I will take the second approach. 1309 01:21:44,400 --> 01:21:50,430 So this is the idea of the light-cone quantization, OK? 1310 01:21:50,430 --> 01:21:53,230 So this is the idea of going to the light-cone gauge. 1311 01:22:01,570 --> 01:22:06,220 So this is an ingenious way to solve those constraints. 1312 01:22:06,220 --> 01:22:07,730 OK? 1313 01:22:07,730 --> 01:22:12,480 And it was first developed by four people, 1314 01:22:12,480 --> 01:22:16,240 including our colleague here, Goldstone, Jeffrey Goldstone, 1315 01:22:16,240 --> 01:22:22,380 who was among the first people to quantize this theory, 1316 01:22:22,380 --> 01:22:24,900 using the light-cone gauge. 1317 01:22:24,900 --> 01:22:26,830 Actually they quantized the Nambu-Goto action. 1318 01:22:26,830 --> 01:22:28,260 They did not quantize this theory. 1319 01:22:28,260 --> 01:22:30,146 They quantized the Nambu-Goto action 1320 01:22:30,146 --> 01:22:31,270 using the light-cone gauge. 1321 01:22:38,750 --> 01:22:41,800 So the key idea is the following. 1322 01:22:48,180 --> 01:22:57,800 He said after fixing this gauge, after fixing gamma ab, 1323 01:22:57,800 --> 01:23:00,940 you go to eta ab. 1324 01:23:00,940 --> 01:23:04,887 There are actually still some residue gauge freedom. 1325 01:23:04,887 --> 01:23:06,470 We actually have not fixed everything. 1326 01:23:19,240 --> 01:23:20,640 OK. 1327 01:23:20,640 --> 01:23:24,630 By residue gauge degrees freedom-- 1328 01:23:24,630 --> 01:23:27,740 so if you fix the gauge completely, 1329 01:23:27,740 --> 01:23:30,930 then that means that any operation which previously-- 1330 01:23:30,930 --> 01:23:36,680 what we did b and c-- when you act on gamma, 1331 01:23:36,680 --> 01:23:42,460 we will take this away from this configuration, OK? 1332 01:23:42,460 --> 01:23:44,020 So by residue degrees freedom, it 1333 01:23:44,020 --> 01:23:46,546 means I can still find the combination 1334 01:23:46,546 --> 01:23:50,590 of coordinate transformation and the Weyl scaling 1335 01:23:50,590 --> 01:23:53,780 so that after those operations, I'm still 1336 01:23:53,780 --> 01:23:56,510 going back to this metric. 1337 01:23:56,510 --> 01:24:00,560 OK, so is it clear what we mean by the residue gauge freedom? 1338 01:24:00,560 --> 01:24:02,980 Good. 1339 01:24:02,980 --> 01:24:08,880 So to see this I have to introduce 1340 01:24:08,880 --> 01:24:12,680 so-called the light-cone coordinate on the worldsheet, 1341 01:24:12,680 --> 01:24:17,261 introduce sigma plus/minus equal to square root 2 tau plus/minus 1342 01:24:17,261 --> 01:24:17,760 sigma. 1343 01:24:24,520 --> 01:24:27,890 And then the worldsheet metric can 1344 01:24:27,890 --> 01:24:31,255 be written as minus d tau square plus d sigma 1345 01:24:31,255 --> 01:24:35,820 square-- because we take the Minkowski metric. 1346 01:24:35,820 --> 01:24:39,670 So this can be written as minus 2 d sigma plus, d sigma minus. 1347 01:24:44,820 --> 01:24:48,630 So now here is the key observation 1348 01:24:48,630 --> 01:25:01,520 is that this metric is preserved by the following coordinate 1349 01:25:01,520 --> 01:25:03,290 transformation. 1350 01:25:03,290 --> 01:25:08,680 I take sigma plus goes to sigma tilde plus, which is only 1351 01:25:08,680 --> 01:25:11,880 a function of sigma plus. 1352 01:25:11,880 --> 01:25:16,280 And sigma minus goes to some function sigma tilde minus, 1353 01:25:16,280 --> 01:25:18,570 some new coordinate sigma tilde minus, which is only 1354 01:25:18,570 --> 01:25:19,927 a function of sigma minus. 1355 01:25:23,910 --> 01:25:27,490 So if you've plugged this in here, 1356 01:25:27,490 --> 01:25:31,680 then you'll find under this coordinate transformation, 1357 01:25:31,680 --> 01:25:35,320 we get this ds tilde square. 1358 01:25:35,320 --> 01:25:39,450 If you just plug this in here, this is, 1359 01:25:39,450 --> 01:25:43,220 say, 2d sigma tilde plus, 2 sigma tilde minus. 1360 01:25:43,220 --> 01:25:44,947 You go to new coordinates. 1361 01:25:44,947 --> 01:25:46,655 And [? these ?] new coordinates expressed 1362 01:25:46,655 --> 01:25:48,510 in terms of original coordinates, 1363 01:25:48,510 --> 01:25:56,010 you get 2f prime sigma plus, g prime sigma minus, 1364 01:25:56,010 --> 01:25:58,120 and d sigma d minus. 1365 01:26:03,220 --> 01:26:07,870 So now the key is that under this coordinate transformation, 1366 01:26:07,870 --> 01:26:12,710 you only transform your metric by an overall prefactor. 1367 01:26:12,710 --> 01:26:15,800 And then you can get rid [? of it ?] by Weyl scaling. 1368 01:26:15,800 --> 01:26:21,404 It is preserved by this, followed by Weyl scaling. 1369 01:26:26,179 --> 01:26:27,970 Because this coordinate transformation only 1370 01:26:27,970 --> 01:26:31,560 changed your metric by an overall prefactor. 1371 01:26:31,560 --> 01:26:35,510 And any overall prefactor you can get rid of by Weyl scaling. 1372 01:26:35,510 --> 01:26:37,460 OK, is this clear? 1373 01:26:37,460 --> 01:26:43,561 So we actually still have some residue of gauge freedom left. 1374 01:26:43,561 --> 01:26:44,061 OK? 1375 01:26:49,320 --> 01:26:52,908 So now we can use this freedom to do something good for us. 1376 01:27:06,560 --> 01:27:10,960 So these new coordinates, say, in this new tilde coordinate, 1377 01:27:10,960 --> 01:27:18,220 tau tilde, then have the form of square root sigma plus tilde 1378 01:27:18,220 --> 01:27:20,660 plus sigma tilde minus. 1379 01:27:20,660 --> 01:27:21,980 OK. 1380 01:27:21,980 --> 01:27:31,580 So this have the form 1 over 2 f sigma plus tau, 1381 01:27:31,580 --> 01:27:35,790 and g sigma minus tau, say tau minus sigma. 1382 01:27:38,770 --> 01:27:40,140 So I have this form. 1383 01:27:40,140 --> 01:27:43,060 So in this new coordinate, yes, so you 1384 01:27:43,060 --> 01:27:44,290 can change the coordinates. 1385 01:27:44,290 --> 01:27:47,800 So you're allowed to make this kind of coordinate change. 1386 01:27:50,570 --> 01:27:53,780 So you are allowed to make this kind of coordinate change. 1387 01:27:53,780 --> 01:27:57,310 Yes, so the residue degrees freedom 1388 01:27:57,310 --> 01:27:59,810 means we are allowed to make this kind of coordinate change, 1389 01:27:59,810 --> 01:28:01,590 OK? 1390 01:28:01,590 --> 01:28:04,120 And that means-- and you see, this 1391 01:28:04,120 --> 01:28:12,080 is precisely the combination of this form, a left-moving wave 1392 01:28:12,080 --> 01:28:13,345 plus a right-moving wave. 1393 01:28:17,300 --> 01:28:22,245 So that means we can use this freedom. 1394 01:28:25,530 --> 01:28:28,630 We can choose. 1395 01:28:28,630 --> 01:28:39,420 So this means we can choose f and g 1396 01:28:39,420 --> 01:28:45,920 so that tau is given by any combinations of this X-- 1397 01:28:45,920 --> 01:28:47,930 the solutions of X, OK? 1398 01:28:47,930 --> 01:28:50,540 Because solutions of X precisely have this form. 1399 01:28:50,540 --> 01:28:56,356 So what we will do is we will-- maybe should I erase this? 1400 01:28:56,356 --> 01:29:01,640 Yeah, let me erase this because I think you all know this. 1401 01:29:16,780 --> 01:29:21,267 So I did a level of the equation of motion. 1402 01:29:21,267 --> 01:29:23,350 You have the freedom. [? You're going to ?] choose 1403 01:29:23,350 --> 01:29:30,380 tau tilde to be any combinations of those X, OK? 1404 01:29:30,380 --> 01:29:33,670 Because those X precisely have that freedom. 1405 01:29:33,670 --> 01:29:40,460 And the smart choice is what we call the light-cone gauge-- 1406 01:29:40,460 --> 01:29:47,060 we can choose so that tau tilde is proportional to so-called X 1407 01:29:47,060 --> 01:29:48,960 plus. 1408 01:29:48,960 --> 01:29:56,380 And X plus is defined by square root of 2, 1409 01:29:56,380 --> 01:30:00,380 say, the 0-th component of X plus, say, 1410 01:30:00,380 --> 01:30:02,190 one of the spatial directions. 1411 01:30:02,190 --> 01:30:03,410 OK, say, take one. 1412 01:30:09,130 --> 01:30:14,610 So to fix this gauge, to fix to fix this residue gauge freedom, 1413 01:30:14,610 --> 01:30:17,960 I have the freedom to choose tau tilde 1414 01:30:17,960 --> 01:30:21,360 to be some combination of X. And this is the combination 1415 01:30:21,360 --> 01:30:22,380 that we will choose. 1416 01:30:22,380 --> 01:30:24,790 So this is what we call the light-cone gauge. 1417 01:30:24,790 --> 01:30:27,580 So I will suppress this tilde. 1418 01:30:27,580 --> 01:30:33,430 So in the light-cone gauge means we go to a coordinate 1419 01:30:33,430 --> 01:30:38,630 so that tau become equal to X plus, and up to some constant 1420 01:30:38,630 --> 01:30:40,270 which I called V plus. 1421 01:30:40,270 --> 01:30:41,320 OK. 1422 01:30:41,320 --> 01:30:42,455 So V plus is some constant. 1423 01:30:48,510 --> 01:30:49,740 Is this clear to you? 1424 01:30:54,440 --> 01:30:59,320 And this is so-called light-cone coordinate in the target space. 1425 01:30:59,320 --> 01:31:03,550 So this [INAUDIBLE] identify the worldsheet's time 1426 01:31:03,550 --> 01:31:07,660 with the light-cone time in the target space. 1427 01:31:13,940 --> 01:31:18,325 Or in other words, X plus equal to V plus, tau. 1428 01:31:22,470 --> 01:31:24,310 This is what we call light-cone gauge. 1429 01:31:26,950 --> 01:31:30,790 So why do we want to do that? 1430 01:31:30,790 --> 01:31:34,580 And the ingenious thing is the following. 1431 01:31:34,580 --> 01:31:39,880 It's that if you write X mu, so we 1432 01:31:39,880 --> 01:31:46,060 can write X mu as X plus, X minus, and Xi. 1433 01:31:46,060 --> 01:31:49,570 And the i come from 2 to D minus 1. 1434 01:31:49,570 --> 01:31:52,850 So X plus and X minus is X0 and X1. 1435 01:31:52,850 --> 01:31:56,880 So that's right in this form. 1436 01:31:56,880 --> 01:32:01,270 OK, then you can easily check yourself, 1437 01:32:01,270 --> 01:32:03,800 or you maybe already know it, is that 1438 01:32:03,800 --> 01:32:10,690 this particular contraction is given by 2X plus, X minus, 1439 01:32:10,690 --> 01:32:12,020 plus Xi squared. 1440 01:32:23,250 --> 01:32:27,640 In particular [INAUDIBLE], just as here, which you have 1441 01:32:27,640 --> 01:32:29,640 this off-diagonal structure. 1442 01:32:29,640 --> 01:32:32,220 This kind of contraction has this off-diagonal structure 1443 01:32:32,220 --> 01:32:34,850 in X plus and X minus. 1444 01:32:34,850 --> 01:32:38,480 And this turns out to be key. 1445 01:32:38,480 --> 01:32:39,351 Yes. 1446 01:32:39,351 --> 01:32:41,225 AUDIENCE: Do you mean for that to be 0 and 1, 1447 01:32:41,225 --> 01:32:42,970 and then that to be plus 1? 1448 01:32:42,970 --> 01:32:44,000 HONG LIU: Sorry? 1449 01:32:44,000 --> 01:32:45,900 AUDIENCE: Do you mean for that to be 0 and 1? 1450 01:32:45,900 --> 01:32:50,990 HONG LIU: So these two are equivalent to X0 and X1. 1451 01:32:50,990 --> 01:32:53,170 Let me include the rest, starting 1452 01:32:53,170 --> 01:32:55,338 from i equal to 2 to D minus 1. 1453 01:32:59,330 --> 01:33:10,440 So X mu have X0, X1, then what I would call Xi, which 1454 01:33:10,440 --> 01:33:12,190 i start from 2 to D minus 1. 1455 01:33:12,190 --> 01:33:15,650 I just renamed these two to be X plus and X minus. 1456 01:33:15,650 --> 01:33:20,027 Just change this two to X plus and X minus. 1457 01:33:20,027 --> 01:33:20,860 AUDIENCE: Oh, I see. 1458 01:33:20,860 --> 01:33:22,608 OK. 1459 01:33:22,608 --> 01:33:24,096 HONG LIU: Yes? 1460 01:33:24,096 --> 01:33:25,088 Yes. 1461 01:33:25,088 --> 01:33:29,056 AUDIENCE: So when we do another one, 1462 01:33:29,056 --> 01:33:30,544 [? I thought ?] the Weyl scaling's 1463 01:33:30,544 --> 01:33:36,010 already fixed when you're trying to fix [INAUDIBLE]. 1464 01:33:36,010 --> 01:33:38,120 HONG LIU: No, no. 1465 01:33:38,120 --> 01:33:44,050 I'm just saying I do these two operation, but in sequence. 1466 01:33:44,050 --> 01:33:46,939 So I do that operation, and then I choose the Weyl scaling 1467 01:33:46,939 --> 01:33:48,230 so that precisely cancels this. 1468 01:33:53,450 --> 01:33:55,460 Of course, if you just do a single Weyl scaling, 1469 01:33:55,460 --> 01:33:57,800 this will be violated. 1470 01:33:57,800 --> 01:33:59,630 But if I do that two operation together, 1471 01:33:59,630 --> 01:34:02,190 then this will be preserved. 1472 01:34:02,190 --> 01:34:04,330 Yeah, I have to do that two together. 1473 01:34:04,330 --> 01:34:08,070 Yeah, if you do a single one, this will be invalidated. 1474 01:34:08,070 --> 01:34:10,830 OK, so this is the key point. 1475 01:34:10,830 --> 01:34:14,460 So we will see that this structure will play a key role. 1476 01:34:14,460 --> 01:34:18,012 So this is the smart thing which our friend Jeffrey did. 1477 01:34:24,050 --> 01:34:28,010 This is in fact the ingenious thing of our friend Jeffrey 1478 01:34:28,010 --> 01:34:29,450 and his friends [? did. ?] 1479 01:34:34,350 --> 01:34:39,900 So now let's look at these two equations, OK? 1480 01:34:39,900 --> 01:34:41,856 Now let's look at these two equations. 1481 01:34:46,020 --> 01:34:51,325 So the first thing-- so now let's look at 10 plus 11. 1482 01:34:56,140 --> 01:35:08,890 So first let's note, because of this, partial tau X 1483 01:35:08,890 --> 01:35:14,770 plus is equal to V plus, and partial sigma X 1484 01:35:14,770 --> 01:35:17,890 plus is equal to 0. 1485 01:35:17,890 --> 01:35:19,561 OK? 1486 01:35:19,561 --> 01:35:21,310 Because there's no sigma dependence there, 1487 01:35:21,310 --> 01:35:25,730 only tau dependence, which isn't in here. 1488 01:35:25,730 --> 01:35:28,600 And then let me just plug it in. 1489 01:35:28,600 --> 01:35:31,730 So let's look at here. 1490 01:35:31,730 --> 01:35:34,600 Let's forget about the 1/2. 1491 01:35:34,600 --> 01:35:38,350 You can also think about the 1/2, doesn't matter. 1492 01:35:38,350 --> 01:35:45,590 So look at plus and minus structure here. 1493 01:35:45,590 --> 01:35:46,285 Use this here. 1494 01:35:49,700 --> 01:36:05,590 Then you get minus 2 V plus, partial tau X minus, then 1495 01:36:05,590 --> 01:36:11,230 plus partial tau Xi squared. 1496 01:36:11,230 --> 01:36:14,570 This is for the first term. 1497 01:36:14,570 --> 01:36:17,810 In the second term, the X plus and the X minus part 1498 01:36:17,810 --> 01:36:22,310 is 0 because of these. 1499 01:36:22,310 --> 01:36:26,400 Because of this and because of the off-diagonal structure 1500 01:36:26,400 --> 01:36:32,114 here, OK? 1501 01:36:32,114 --> 01:36:34,140 Do you see it? 1502 01:36:34,140 --> 01:36:42,660 So the first term here would be partial tau minus 2, 1503 01:36:42,660 --> 01:36:45,730 partial sigma X plus, partial sigma X 1504 01:36:45,730 --> 01:36:51,870 minus plus partial sigma Xi squared. 1505 01:36:51,870 --> 01:36:54,840 So this would be 0. 1506 01:36:54,840 --> 01:37:00,400 But our friend here is 0, because of that is 0. 1507 01:37:00,400 --> 01:37:03,180 So now we can erase it. 1508 01:37:03,180 --> 01:37:05,610 So now let's change this to the other side. 1509 01:37:05,610 --> 01:37:06,930 OK. 1510 01:37:06,930 --> 01:37:08,430 Let's change this to the other side, 1511 01:37:08,430 --> 01:37:09,715 and then change this sign. 1512 01:37:12,660 --> 01:37:14,930 So the equation I get is the following. 1513 01:37:22,590 --> 01:37:24,600 OK, so this is equation 10. 1514 01:37:24,600 --> 01:37:28,100 So now let's look at equation 11. 1515 01:37:28,100 --> 01:37:33,510 In 11, so we have partial tau X plus, 1516 01:37:33,510 --> 01:37:37,410 then partial sigma X minus. 1517 01:37:37,410 --> 01:37:43,497 Then you go to partial tau Xi and partial sigma Xi. 1518 01:37:46,060 --> 01:37:47,790 And then this one just give us V plus. 1519 01:37:53,740 --> 01:37:57,113 So that two equation become these two equation. 1520 01:37:57,113 --> 01:37:58,112 And let me call this 14. 1521 01:38:07,280 --> 01:38:09,160 This is 15. 1522 01:38:09,160 --> 01:38:11,850 So is it clear to you, these equations? 1523 01:38:11,850 --> 01:38:15,280 So if it's not immediately clear to you, just check it yourself. 1524 01:38:15,280 --> 01:38:15,780 OK. 1525 01:38:18,850 --> 01:38:23,520 Urge you to do the check, because then you 1526 01:38:23,520 --> 01:38:25,400 will see the magic. 1527 01:38:25,400 --> 01:38:30,635 Then you will see the magic and the beauty of this trick. 1528 01:38:30,635 --> 01:38:31,551 AUDIENCE: [INAUDIBLE]. 1529 01:38:42,010 --> 01:38:43,168 HONG LIU: Sorry? 1530 01:38:43,168 --> 01:38:46,230 AUDIENCE: Can you tell X minus [INAUDIBLE] sigma X plus? 1531 01:38:46,230 --> 01:38:51,240 HONG LIU: Yeah, that's 0 because partial 0 X plus is 0. 1532 01:38:51,240 --> 01:38:53,780 So that's why there's only one here, not two. 1533 01:38:53,780 --> 01:38:56,440 Because the other term is 0. 1534 01:38:56,440 --> 01:39:01,650 OK, so now the beauty of these two equation 1535 01:39:01,650 --> 01:39:08,070 is that they express the X minus solely in terms of Xi. 1536 01:39:08,070 --> 01:39:27,006 So 14 and 15 tells you, is that X minus can be fully solved 1537 01:39:27,006 --> 01:39:32,180 in terms of Xi. 1538 01:39:37,390 --> 01:39:41,940 Because this equation give you the tau derivative of X minus, 1539 01:39:41,940 --> 01:39:46,100 this equation give you the sigma derivative of X minus, 1540 01:39:46,100 --> 01:39:50,810 and all expressed only in terms of Xi. 1541 01:39:50,810 --> 01:39:57,890 So the X minus is completely solved, completely expressed, 1542 01:39:57,890 --> 01:40:00,160 in terms of Xi. 1543 01:40:00,160 --> 01:40:03,470 And then we have solved those constraints [? similarly ?] 1544 01:40:03,470 --> 01:40:10,640 [INAUDIBLE] constraint explicitly, OK? 1545 01:40:10,640 --> 01:40:12,660 So right now we are doing it a little bit fast. 1546 01:40:12,660 --> 01:40:15,650 I urge you to go through the algebra yourself to appreciate 1547 01:40:15,650 --> 01:40:18,980 the beauty of this step. 1548 01:40:18,980 --> 01:40:30,310 So we conclude that independent degrees freedoms are only Xi. 1549 01:40:38,110 --> 01:40:43,090 Because X plus we already know by fixing the gauge. 1550 01:40:43,090 --> 01:40:47,530 And X minus now is completely solving in terms of Xi, 1551 01:40:47,530 --> 01:40:51,900 so the independent freedom are only Xi's. 1552 01:40:51,900 --> 01:40:56,470 And this will make our life very easy. 1553 01:40:56,470 --> 01:41:01,820 Because Xi are just free scalar fields. 1554 01:41:01,820 --> 01:41:06,980 Xi-- just free scalar field with the right kinetic term. 1555 01:41:06,980 --> 01:41:10,690 You have also get rid of X0, because we said that 1556 01:41:10,690 --> 01:41:13,935 [? before ?] X0, we have a [? long-sign ?] kinetic term. 1557 01:41:13,935 --> 01:41:15,810 But now we are only independent-- [INAUDIBLE] 1558 01:41:15,810 --> 01:41:20,140 only Xi's, and they have the right number of kinetic terms. 1559 01:41:20,140 --> 01:41:23,480 They have the right sign for the kinetic term. 1560 01:41:23,480 --> 01:41:27,260 And they are just free scalar fields. 1561 01:41:27,260 --> 01:41:29,560 So we can quantize them as what you 1562 01:41:29,560 --> 01:41:34,530 did in your first day of Quantum Field Theory class. 1563 01:41:34,530 --> 01:41:38,310 And then you have quantized the string theory. 1564 01:41:38,310 --> 01:41:40,690 And then you have quantized string theory. 1565 01:41:40,690 --> 01:41:43,640 OK, yeah. 1566 01:41:43,640 --> 01:41:47,180 I think I'm going very slow here. 1567 01:41:47,180 --> 01:41:48,770 Maybe I'm a little bit too emotional. 1568 01:41:52,980 --> 01:41:55,420 Anyway, so maybe let's stop here. 1569 01:42:00,120 --> 01:42:05,400 The next time, then we will just quantize those things. 1570 01:42:05,400 --> 01:42:07,390 And those equations still will give us 1571 01:42:07,390 --> 01:42:09,900 some very important information, and then we 1572 01:42:09,900 --> 01:42:11,280 will quantize those things. 1573 01:42:11,280 --> 01:42:16,550 And then we will be able to see that the string theory contains 1574 01:42:16,550 --> 01:42:17,470 gravity. 1575 01:42:17,470 --> 01:42:20,290 It contains gravitons, OK?