1 00:00:00,040 --> 00:00:02,410 The following content is provided under a Creative 2 00:00:02,410 --> 00:00:03,790 Commons license. 3 00:00:03,790 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,100 continue to offer high quality educational resources for free. 5 00:00:10,100 --> 00:00:12,680 To make a donation or to view additional materials 6 00:00:12,680 --> 00:00:16,426 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,426 --> 00:00:17,050 at ocw.mit.edu. 8 00:00:22,060 --> 00:00:24,550 HONG LIU: OK, let us start. 9 00:00:24,550 --> 00:00:29,510 So first, let me remind you what we did 10 00:00:29,510 --> 00:00:33,330 at the end of the last lecture. 11 00:00:33,330 --> 00:00:38,480 So we can see there's a large N matrix field theory. 12 00:00:38,480 --> 00:00:48,020 And then we saw that when you write down your Feynman 13 00:00:48,020 --> 00:00:52,040 diagrams to calculate things, then 14 00:00:52,040 --> 00:00:56,850 there's a difference now between the order of contractions. 15 00:00:56,850 --> 00:00:58,970 Which is now refracted whether you 16 00:00:58,970 --> 00:01:02,950 have a Planar diagram or non-planar diagram, et cetera. 17 00:01:02,950 --> 00:01:07,190 And that in turn, also affects your end counting. 18 00:01:07,190 --> 00:01:13,920 So at the end, we discussed two observations. 19 00:01:13,920 --> 00:01:18,770 One observation we said, is for the example to be considered is 20 00:01:18,770 --> 00:01:23,840 the non-planar diagram, even though it cannot be drawn 21 00:01:23,840 --> 00:01:27,060 on the plane without crossing lines, 22 00:01:27,060 --> 00:01:32,560 can actually be drawn on the torus without crossing lines. 23 00:01:32,560 --> 00:01:37,900 So the diagram can actually be straightened out on the torus. 24 00:01:37,900 --> 00:01:44,200 And another observation, is that the power of N 25 00:01:44,200 --> 00:01:50,490 is related to the number of faces you have in your diagram 26 00:01:50,490 --> 00:01:51,960 after you have straightened it out. 27 00:01:55,070 --> 00:02:02,535 So now, I'm going to generalize these two observations. 28 00:02:08,560 --> 00:02:11,605 So first, I will tell you a fact. 29 00:02:14,230 --> 00:02:15,847 Many of you may already know this. 30 00:02:19,010 --> 00:02:28,270 So I'll first tell you a fact that any orientable two 31 00:02:28,270 --> 00:02:58,986 dimensional surface is classified topologically 32 00:02:58,986 --> 00:03:05,960 by integer, which I will call h. 33 00:03:05,960 --> 00:03:08,723 So this h is called a genus. 34 00:03:16,090 --> 00:03:35,010 So heuristically, this genus is equal to the number of holes 35 00:03:35,010 --> 00:03:36,020 a surface has. 36 00:03:40,890 --> 00:03:46,850 So for example, if you have a plane, 37 00:03:46,850 --> 00:03:49,860 the plane topologically is actually 38 00:03:49,860 --> 00:03:52,470 equivalent to a sphere, so when I 39 00:03:52,470 --> 00:03:55,380 draw a sphere, that means a plane, 40 00:03:55,380 --> 00:03:57,180 because if you identify the point 41 00:03:57,180 --> 00:04:01,770 on the plane of the infinity, then it becomes a sphere. 42 00:04:01,770 --> 00:04:04,590 So topologically, a plane is no different from a sphere. 43 00:04:04,590 --> 00:04:07,900 So this genus 0. 44 00:04:07,900 --> 00:04:08,723 So this is plane. 45 00:04:12,340 --> 00:04:13,330 So h is equal to 0. 46 00:04:16,290 --> 00:04:26,920 And then for torus, then there's one hole, 47 00:04:26,920 --> 00:04:27,980 and this is just genus 1. 48 00:04:30,720 --> 00:04:33,560 So this is torus. 49 00:04:33,560 --> 00:04:40,995 You can also draw surfaces with as many holes as you want. 50 00:04:40,995 --> 00:04:42,952 So this is surface with two holes. 51 00:04:46,950 --> 00:04:50,800 A surface with two holes, and so this is genius 2. 52 00:04:55,310 --> 00:04:59,050 So the remarkable thing is that this actually 53 00:04:59,050 --> 00:05:03,296 classifies the topology of all two-dimensional surfaces. 54 00:05:06,600 --> 00:05:12,090 And this so-called topological invariant, 55 00:05:12,090 --> 00:05:18,930 there's a topological number a so-called Euler number, 56 00:05:18,930 --> 00:05:25,710 which is defined to be Chi is equal to 2 minus 2h. 57 00:05:25,710 --> 00:05:32,720 So if h labels the topology, and so the Chi 58 00:05:32,720 --> 00:05:35,010 is related to h in this way. 59 00:05:35,010 --> 00:05:39,865 So any two surfaces with the same topology 60 00:05:39,865 --> 00:05:43,330 will have the same Chi, because if they have the same h, 61 00:05:43,330 --> 00:05:44,520 they have the same Chi. 62 00:05:44,520 --> 00:05:48,385 So Chi is what we call the topological invariants. 63 00:05:51,550 --> 00:05:54,790 So this is just a mathematical fact. 64 00:05:54,790 --> 00:05:58,700 So now I'm going to make two claims. 65 00:05:58,700 --> 00:06:02,430 So these two claims are in some sense, 66 00:06:02,430 --> 00:06:07,360 still evident after you have studied a little bit. 67 00:06:07,360 --> 00:06:10,390 I'm not going to prove it here, will just make the claim, 68 00:06:10,390 --> 00:06:16,980 and then I will leave the task to familiarize these two 69 00:06:16,980 --> 00:06:20,490 claims to yourself. 70 00:06:20,490 --> 00:06:24,150 But as I said, these two claims are actually self-evident, 71 00:06:24,150 --> 00:06:26,370 if you just do a little bit of studying. 72 00:06:29,270 --> 00:06:36,830 So claim one-- so this is regarding 73 00:06:36,830 --> 00:06:39,900 the structural Feynman diagrams. 74 00:06:39,900 --> 00:06:48,350 So remember, the example theory we conceded last time 75 00:06:48,350 --> 00:07:04,360 was something like this, which phi is [INAUDIBLE] matrix. 76 00:07:08,770 --> 00:07:10,690 So if you can look at the Feynman diagrams 77 00:07:10,690 --> 00:07:16,180 of the theories, the claim is the following. 78 00:07:16,180 --> 00:07:29,360 for any non-planar diagram, there 79 00:07:29,360 --> 00:07:49,430 exists integer h so that the diagram 80 00:07:49,430 --> 00:07:53,342 can be straightened out. 81 00:08:06,040 --> 00:08:11,950 Straightened out just means non-crossing on a genus h 82 00:08:11,950 --> 00:08:33,080 surface, but not on a surface with a smaller genus. 83 00:08:43,669 --> 00:08:52,190 So actually, you can classify all the non-planar diagrams 84 00:08:52,190 --> 00:09:00,630 by drawing it on a genus h surface 85 00:09:00,630 --> 00:09:04,900 on two-dimensional surfaces with non-trivial topology. 86 00:09:04,900 --> 00:09:07,410 And then you should be able to find the number h, which 87 00:09:07,410 --> 00:09:13,000 is the lowest genus you need to make this diagram 88 00:09:13,000 --> 00:09:17,670 to be non-crossing, instead of crossing. 89 00:09:17,670 --> 00:09:20,820 Last time, we saw an example which you can stretch it out 90 00:09:20,820 --> 00:09:24,160 on the torus, but it was a more complicated diagram 91 00:09:24,160 --> 00:09:26,230 and torus would not be enough. 92 00:09:26,230 --> 00:09:29,020 You would need, say, a genus 2 surface or higher genus 93 00:09:29,020 --> 00:09:29,520 surface. 94 00:09:34,640 --> 00:09:38,290 So you can easily convince yourself 95 00:09:38,290 --> 00:09:44,890 that this is doable just by some practice. 96 00:09:44,890 --> 00:09:46,146 Also, this is very reasonable. 97 00:09:46,146 --> 00:09:48,104 Do you have any questions regarding this claim? 98 00:09:52,330 --> 00:09:52,830 Good. 99 00:09:55,690 --> 00:09:56,670 So this was claim one. 100 00:09:56,670 --> 00:10:04,450 The claim two-- and this is a generalization 101 00:10:04,450 --> 00:10:06,434 of the second observation. 102 00:10:06,434 --> 00:10:07,850 It's a claim of the generalization 103 00:10:07,850 --> 00:10:10,300 of the first observation of last time, 104 00:10:10,300 --> 00:10:12,190 and claim two will be the generalization 105 00:10:12,190 --> 00:10:13,273 of the second observation. 106 00:10:23,730 --> 00:10:43,040 For any diagram, the power of n coming from contracting 107 00:10:43,040 --> 00:11:20,706 propagators is given by the number of faces 108 00:11:20,706 --> 00:11:24,390 on such a genus h surface. 109 00:11:37,640 --> 00:11:41,820 As we explained last time, the number of faces 110 00:11:41,820 --> 00:11:47,260 just means the number of these connected regions 111 00:11:47,260 --> 00:11:47,975 in the diagram. 112 00:11:51,730 --> 00:11:53,830 But if you are stretching it out, 113 00:11:53,830 --> 00:11:55,710 then there's an ambiguous way you 114 00:11:55,710 --> 00:12:01,010 can count the number of disconnected regions 115 00:12:01,010 --> 00:12:02,570 In the diagram. 116 00:12:02,570 --> 00:12:10,660 And that number is the power of n coming from the contracting 117 00:12:10,660 --> 00:12:12,180 propagators. 118 00:12:12,180 --> 00:12:16,290 So this claim is also self-evident 119 00:12:16,290 --> 00:12:20,790 because each power of n comes from a single connected line. 120 00:12:20,790 --> 00:12:22,840 And essentially, a single connected line 121 00:12:22,840 --> 00:12:26,830 will be circles some face and so the number of independents 122 00:12:26,830 --> 00:12:29,350 inside the face of course, are just the number of power n. 123 00:12:33,920 --> 00:12:35,695 Any questions regarding the second claim? 124 00:12:41,367 --> 00:12:41,866 Good. 125 00:12:44,560 --> 00:12:49,257 So based on these two claims, now immediately write down 126 00:12:49,257 --> 00:12:49,965 the independents. 127 00:12:53,610 --> 00:12:58,550 So from this, we can find a vacuum diagram. 128 00:12:58,550 --> 00:13:02,496 Again, we have been only talking about the vacuum diagram. 129 00:13:02,496 --> 00:13:07,810 Vacuum diagram means the diagram has no external lags, 130 00:13:07,810 --> 00:13:09,570 has the full independence. 131 00:13:09,570 --> 00:13:13,140 So from here, we immediately conclude 132 00:13:13,140 --> 00:13:25,965 that the vacuum diagram has the following, g squared 133 00:13:25,965 --> 00:13:26,590 and dependence. 134 00:13:32,090 --> 00:13:34,810 So g squared is just the carpeting constant. 135 00:13:34,810 --> 00:13:39,990 So remember, each propagator is proportional to g squared 136 00:13:39,990 --> 00:13:47,233 and that each vertex is one over g squared. 137 00:13:49,950 --> 00:13:53,210 Just from there. 138 00:13:53,210 --> 00:13:55,940 So let's just take an arbitrary vacuum diagram. 139 00:13:55,940 --> 00:14:02,470 Let me call this amplitude to be A. And A should 140 00:14:02,470 --> 00:14:05,760 be proportional to g squared to the power 141 00:14:05,760 --> 00:14:21,620 E. E should be the number of propagators, 142 00:14:21,620 --> 00:14:24,450 because each propagator gives you a factor of g squared. 143 00:14:26,970 --> 00:14:33,630 And also minus V. And V is the number 144 00:14:33,630 --> 00:14:38,130 of vertices below to the number of vertices. 145 00:14:38,130 --> 00:14:40,970 And each vertex gives you one over g squared, 146 00:14:40,970 --> 00:14:45,890 so that should give you E minus V. 147 00:14:45,890 --> 00:14:50,530 And then we just multiply N to the power of F, 148 00:14:50,530 --> 00:14:52,010 and F is the number of faces. 149 00:15:03,810 --> 00:15:07,690 So without doing any calculation, 150 00:15:07,690 --> 00:15:15,222 so this has essentially characterized the N and the g 151 00:15:15,222 --> 00:15:17,222 and the accompanying dependents of any diagrams. 152 00:15:21,150 --> 00:15:25,470 Now, if you look at this expression, 153 00:15:25,470 --> 00:15:31,550 you say we are doomed, because this kind of expression 154 00:15:31,550 --> 00:15:36,730 does not have a sensible [INAUDIBLE] 155 00:15:36,730 --> 00:15:41,460 So remember our goal, the original goal of the 't Hooft 156 00:15:41,460 --> 00:15:44,690 is that you treat this N as a parameter 157 00:15:44,690 --> 00:15:48,990 and then you want to take it N to be large 158 00:15:48,990 --> 00:15:54,160 and then to it's ranking, one over N. Doing expansion in one 159 00:15:54,160 --> 00:15:59,170 over N means you are expanding around the including infinity. 160 00:15:59,170 --> 00:16:03,480 You want to expand around the including infinity. 161 00:16:03,480 --> 00:16:06,260 But if you look at this expression 162 00:16:06,260 --> 00:16:10,920 and then go to infinity, it's not a well-defined limit, 163 00:16:10,920 --> 00:16:15,850 because I can draw a [INAUDIBLE] Feynman diagram, which 164 00:16:15,850 --> 00:16:18,000 F can be as large as possible. 165 00:16:18,000 --> 00:16:22,720 So you just have sufficiently many vertices, 166 00:16:22,720 --> 00:16:28,290 and sufficient to be mainly propogators, F is unbounded. 167 00:16:31,050 --> 00:16:33,130 So then, there's another well-defined angle 168 00:16:33,130 --> 00:16:34,115 to infinity limits. 169 00:16:37,250 --> 00:16:45,340 So this expression is not a sensible N of infinity limit. 170 00:16:48,540 --> 00:16:52,880 If you don't have a sensible N infinity limit, 171 00:16:52,880 --> 00:16:56,140 then you cannot talk about doing a longer expansion, 172 00:16:56,140 --> 00:17:01,545 and you cannot even define the limit we just finished. 173 00:17:05,150 --> 00:17:06,224 Yes? 174 00:17:06,224 --> 00:17:10,705 AUDIENCE: I remember people do partial sum to-- can 175 00:17:10,705 --> 00:17:12,490 they do the same thing here? 176 00:17:12,490 --> 00:17:16,900 Put the large in the denominator pattern from series 177 00:17:16,900 --> 00:17:20,222 of singular bubbles and-- 178 00:17:20,222 --> 00:17:21,210 HONG LIU: Yeah. 179 00:17:21,210 --> 00:17:23,486 AUDIENCE: Like a random face approximation-- 180 00:17:23,486 --> 00:17:24,069 HONG LIU: Yes. 181 00:17:24,069 --> 00:17:27,109 AUDIENCE: Somehow they can infinite a sum 182 00:17:27,109 --> 00:17:31,053 over an infinite number of A, and put the margin 183 00:17:31,053 --> 00:17:36,360 into like 1 minus N something for the denominator-- 184 00:17:36,360 --> 00:17:37,770 HONG LIU: Yeah. 185 00:17:37,770 --> 00:17:41,450 So people certainly have been talking about expansion. 186 00:17:41,450 --> 00:17:46,540 And certainly, if this just failed here, 187 00:17:46,540 --> 00:17:48,100 we will not be talking about this. 188 00:17:54,102 --> 00:17:55,560 I'm just setting up a target, which 189 00:17:55,560 --> 00:17:57,510 I'm going to shoot it down. 190 00:18:02,900 --> 00:18:04,500 Right. 191 00:18:04,500 --> 00:18:11,470 Yeah, I would just say if you do the margin expansion, 192 00:18:11,470 --> 00:18:14,530 this probably would be the place which would turn you back. 193 00:18:14,530 --> 00:18:18,844 You say, ah, there's no margin limits, 194 00:18:18,844 --> 00:18:20,710 and that's from another problem. 195 00:18:23,400 --> 00:18:27,590 But hopefully, that's no ordinary person. 196 00:18:27,590 --> 00:18:31,900 And so remember, there are several nice mathematical 197 00:18:31,900 --> 00:18:33,670 tricks you have to go through. 198 00:18:33,670 --> 00:18:36,180 First, you have to invent the stop line notation so that you 199 00:18:36,180 --> 00:18:37,730 can count the end very easily. 200 00:18:37,730 --> 00:18:39,980 Yeah, first you have to come up with this margin idea. 201 00:18:39,980 --> 00:18:42,800 I think maybe not due to him, maybe other people 202 00:18:42,800 --> 00:18:45,090 already considered similar things like right here. 203 00:18:45,090 --> 00:18:47,610 But first you have to think about the double line notation 204 00:18:47,610 --> 00:18:51,420 to make your computation easier, and then even after you reach 205 00:18:51,420 --> 00:18:54,040 the staff and then you need to know this kind of topology, 206 00:18:54,040 --> 00:18:55,684 et cetera. 207 00:18:55,684 --> 00:18:57,350 After you reach this staff, still, there 208 00:18:57,350 --> 00:18:59,740 is a roadblock here. 209 00:18:59,740 --> 00:19:02,173 But 't Hooft found a very simple way to go around it. 210 00:19:05,810 --> 00:19:12,850 Because when I say this limit it is not well-defined, 211 00:19:12,850 --> 00:19:17,420 I made the assumption that when I go to infinity limit, 212 00:19:17,420 --> 00:19:19,735 the g is kept fixed. 213 00:19:26,570 --> 00:19:31,470 So that's the reason that this will blow up. 214 00:19:31,470 --> 00:19:37,070 But then 't Hooft came up with a different image. 215 00:19:37,070 --> 00:19:40,026 He said when you can see that N goes infinity, 216 00:19:40,026 --> 00:19:41,400 but at the same time, you can see 217 00:19:41,400 --> 00:19:42,720 that g squared goes to zero. 218 00:19:46,330 --> 00:19:49,800 At the same time, you can see that g squared goes to zero. 219 00:19:49,800 --> 00:19:52,930 Because the problem with this, is 220 00:19:52,930 --> 00:19:57,450 that if you have a diagram with lots of faces 221 00:19:57,450 --> 00:20:01,670 and if you multiply by a finite number, 222 00:20:01,670 --> 00:20:06,590 if you take g equal to 0, then you have infinity 223 00:20:06,590 --> 00:20:08,650 multiply something potentially goes to 0, 224 00:20:08,650 --> 00:20:12,020 maybe you will get something finite. 225 00:20:12,020 --> 00:20:13,030 One second. 226 00:20:13,030 --> 00:20:16,220 So you said let's consider this limit, 227 00:20:16,220 --> 00:20:20,520 so that the product of them is kept finite. 228 00:20:23,890 --> 00:20:27,030 Let's consider this limit. 229 00:20:27,030 --> 00:20:31,140 So in this limit, the end counting will be different. 230 00:20:31,140 --> 00:20:39,750 So this A will be g squared N, E minus l, because we 231 00:20:39,750 --> 00:20:42,040 want to keep this fixed. 232 00:20:42,040 --> 00:20:45,600 So we put an N factor here, then we take the N factor out. 233 00:20:45,600 --> 00:20:52,900 N plus F plus V minus E. And then 234 00:20:52,900 --> 00:20:55,230 let me just write this slightly differently. 235 00:20:55,230 --> 00:20:56,120 So this is Lambda. 236 00:20:56,120 --> 00:20:59,190 If I use this notation, Lambda now is finite. 237 00:20:59,190 --> 00:21:03,490 So this is E minus V. So let me just 238 00:21:03,490 --> 00:21:08,930 write it as Lambda L minus 1 N to the power of Chi. 239 00:21:08,930 --> 00:21:12,240 Now, let me explain [INAUDIBLE]. 240 00:21:12,240 --> 00:21:25,610 First, L is equal to E minus V plus 1, 241 00:21:25,610 --> 00:21:27,660 is the number of loops in the diagram. 242 00:21:34,330 --> 00:21:37,820 So do you guys remember this formula, why this is true? 243 00:21:40,780 --> 00:21:44,590 The reason this is true is very easy to understand. 244 00:21:44,590 --> 00:21:49,810 So if you look at the Feynman diagram, the number of loops 245 00:21:49,810 --> 00:21:51,790 is exactly the same as the number 246 00:21:51,790 --> 00:21:53,010 of undetermined momentum. 247 00:21:55,730 --> 00:22:01,510 And each propagator will carry momentum, and at each vertex, 248 00:22:01,510 --> 00:22:03,670 you have momentum conservation. 249 00:22:03,670 --> 00:22:05,630 But then this overall momentum conservation, 250 00:22:05,630 --> 00:22:09,370 which is guaranteed, so the number of independent momentum 251 00:22:09,370 --> 00:22:12,360 is E minus V and then plus 1. 252 00:22:12,360 --> 00:22:18,260 Yeah, it's just E minus V minus 1, 253 00:22:18,260 --> 00:22:21,970 because there's only V minus 1 independent momentum 254 00:22:21,970 --> 00:22:22,960 constraints. 255 00:22:22,960 --> 00:22:26,150 So this is the number of independent momentums, 256 00:22:26,150 --> 00:22:29,280 and so this is the number of loops. 257 00:22:29,280 --> 00:22:37,950 And the Chi is just defined to be the number of faces 258 00:22:37,950 --> 00:22:41,300 plus the number of vertices minus the number of edge. 259 00:22:46,060 --> 00:22:46,640 Yes? 260 00:22:46,640 --> 00:22:48,810 AUDIENCE: Is there any-- in general, is there 261 00:22:48,810 --> 00:22:51,370 a relation between E, V, and F, like are they completely 262 00:22:51,370 --> 00:22:52,740 independent to each other? 263 00:22:52,740 --> 00:22:53,700 HONG LIU: E, V, and F? 264 00:22:53,700 --> 00:22:56,110 AUDIENCE: Yeah. 265 00:22:56,110 --> 00:22:59,630 HONG LIU: Yeah, that I'm going to explain. 266 00:22:59,630 --> 00:23:01,640 Any other questions? 267 00:23:01,640 --> 00:23:04,620 AUDIENCE: So the reason that you can take the F 268 00:23:04,620 --> 00:23:06,120 to infinity limits without taking it 269 00:23:06,120 --> 00:23:10,700 to 0, is that why we can't apply this to QCD to get a-- 270 00:23:10,700 --> 00:23:13,800 HONG LIU: No, because we apply this. 271 00:23:13,800 --> 00:23:16,044 AUDIENCE: But if g is not small? 272 00:23:16,044 --> 00:23:17,210 HONG LIU: It doesn't matter. 273 00:23:22,550 --> 00:23:28,130 So this is an expansion scheme. 274 00:23:28,130 --> 00:23:31,340 And you can apply it to QCD, then 275 00:23:31,340 --> 00:23:33,940 you can ask whether this particular expansion scheme is 276 00:23:33,940 --> 00:23:35,970 a good approximation or not. 277 00:23:35,970 --> 00:23:37,380 That's a separate question. 278 00:23:37,380 --> 00:23:39,960 And the focus is pretty good, it's not that bad. 279 00:23:42,660 --> 00:23:45,840 Yeah, focus of each squaring is finite. 280 00:23:45,840 --> 00:23:47,205 AUDIENCE: But it's [INAUDIBLE]. 281 00:23:47,205 --> 00:23:48,830 HONG LIU: No, here it does not tell you 282 00:23:48,830 --> 00:23:51,500 if lambda has to be small. 283 00:23:51,500 --> 00:23:54,200 Lambda can be very big here. 284 00:23:54,200 --> 00:23:58,000 It just tells you the lambda in this limit has to be finite. 285 00:23:58,000 --> 00:24:01,700 Lambda can be convening and it can be very big. 286 00:24:01,700 --> 00:24:03,700 And actually, in the future, we will take lambda 287 00:24:03,700 --> 00:24:04,450 to go to infinity. 288 00:24:12,030 --> 00:24:17,030 So now, in order to have a well-defined limit 289 00:24:17,030 --> 00:24:20,210 still depends on this chi. 290 00:24:20,210 --> 00:24:29,040 So now, you ask why this works because we still have this chi, 291 00:24:29,040 --> 00:24:31,358 but now again, we need another piece of mathematics. 292 00:24:37,340 --> 00:24:43,300 First, when you draw, let me remind 293 00:24:43,300 --> 00:24:51,590 you some diagram we had last time, 294 00:24:51,590 --> 00:24:53,994 so this is the simplest diagram. 295 00:24:53,994 --> 00:24:55,660 And there's a lot of non-planar diagrams 296 00:24:55,660 --> 00:24:59,610 which you can draw on the torus, which 297 00:24:59,610 --> 00:25:02,710 is the non-planar version of this, et cetera. 298 00:25:02,710 --> 00:25:05,224 And you can also have more complicated diagrams. 299 00:25:05,224 --> 00:25:06,640 So suppose you are on the torus, I 300 00:25:06,640 --> 00:25:10,250 can consider more complicated diagrams, like that. 301 00:25:10,250 --> 00:25:11,845 For example, such diagrams. 302 00:25:14,760 --> 00:25:22,440 So if you think about such diagrams, then in a sense, 303 00:25:22,440 --> 00:25:34,530 each Feynman diagram can be considered 304 00:25:34,530 --> 00:25:45,220 as a partition of the surface. 305 00:25:45,220 --> 00:25:55,640 So if you draw a diagram of the surface into polygons. 306 00:26:00,680 --> 00:26:04,370 Yeah, it's very clear here. 307 00:26:04,370 --> 00:26:08,660 I drew this now with wavy lines, just make them straight. 308 00:26:08,660 --> 00:26:13,270 And this one topologically, I can just draw like this, 309 00:26:13,270 --> 00:26:21,270 and then I have one, one, and the other part, similarly here. 310 00:26:21,270 --> 00:26:23,600 So each Feynman diagram can be concealed 311 00:26:23,600 --> 00:26:28,750 if you just partition whatever surface you draw the diagram. 312 00:26:28,750 --> 00:26:30,280 Is this clear to you? 313 00:26:30,280 --> 00:26:32,570 So this is a very important point. 314 00:26:32,570 --> 00:26:33,970 Yes? 315 00:26:33,970 --> 00:26:35,220 AUDIENCE: Can you repeat that? 316 00:26:35,220 --> 00:26:37,010 HONG LIU: Yes. 317 00:26:37,010 --> 00:26:38,500 Look at this diagram. 318 00:26:38,500 --> 00:26:41,890 So this is a Feynman diagram drawn on the torus. 319 00:26:41,890 --> 00:26:46,460 Does this look like a partition of torus into polygons? 320 00:26:46,460 --> 00:26:48,870 Yes, so that's what I'm talking about. 321 00:26:48,870 --> 00:26:51,260 And the statement that this would apply for any Feynman 322 00:26:51,260 --> 00:26:52,750 diagrams. 323 00:26:52,750 --> 00:26:56,250 Yes, so this is like a partition of a sphere. 324 00:26:56,250 --> 00:26:59,520 You separate a sphere into three regions, one, two, three. 325 00:27:06,340 --> 00:27:08,260 Any questions about this thing? 326 00:27:08,260 --> 00:27:10,780 Yes, do you have a question? 327 00:27:10,780 --> 00:27:13,798 AUDIENCE: Yeah, in that case the third one isn't abandoned, 328 00:27:13,798 --> 00:27:15,280 is it? 329 00:27:15,280 --> 00:27:15,820 The third-- 330 00:27:15,820 --> 00:27:16,720 HONG LIU: So this one I am going to show you, 331 00:27:16,720 --> 00:27:18,670 you should view the plane as a sphere. 332 00:27:18,670 --> 00:27:20,780 AUDIENCE: Oh, OK. 333 00:27:20,780 --> 00:27:23,960 HONG LIU: Topologically, it's the same as a sphere. 334 00:27:23,960 --> 00:27:30,460 So now, once you recognize this, now you 335 00:27:30,460 --> 00:27:37,200 can use a famous theorem due to Euler. 336 00:27:37,200 --> 00:27:42,110 Some of you could have learned it in junior in high school, 337 00:27:42,110 --> 00:28:00,490 because the-- so this theorem, given a surface composed 338 00:28:00,490 --> 00:28:08,770 of polygons-- so if you're not familiar with this, 339 00:28:08,770 --> 00:28:22,420 you can go to Wikipedia-- with F faces, E edges, and V vertices. 340 00:28:32,210 --> 00:28:36,690 Suppose you have a surface like this, and then 341 00:28:36,690 --> 00:28:39,730 this particular combination chi, which 342 00:28:39,730 --> 00:28:44,580 is defined to be F plus V minus E is precisely equal 343 00:28:44,580 --> 00:28:47,820 to 2 minus h. 344 00:28:47,820 --> 00:28:53,330 So this is the-- this is why this is called Euler's number. 345 00:28:53,330 --> 00:28:56,840 And they only depend on the topology. 346 00:28:56,840 --> 00:29:00,500 So this combination only depends on the topology of the surface 347 00:29:00,500 --> 00:29:01,355 and nothing else. 348 00:29:04,890 --> 00:29:08,450 So we can imagine 't Hooft a very good high school student. 349 00:29:08,450 --> 00:29:11,250 So he already knew this. 350 00:29:11,250 --> 00:29:15,320 So now, we can rewrite this thing 351 00:29:15,320 --> 00:29:23,540 as A, reading script A lambda to the power L minus 1 N 352 00:29:23,540 --> 00:29:24,890 to the power 2 minus 2h. 353 00:29:27,460 --> 00:29:29,680 So remember h is greater than zero. 354 00:29:33,030 --> 00:29:35,110 So now this expression has a well-defined limit. 355 00:29:38,939 --> 00:29:40,980 So now this has a well-defined analogy and limit. 356 00:29:44,010 --> 00:29:48,240 So to the leading order, and in particular, 357 00:29:48,240 --> 00:29:52,206 this N dependence only depends on the topology of the diagram. 358 00:29:52,206 --> 00:29:53,997 Only depend on the topology of the diagram. 359 00:29:58,970 --> 00:30:18,190 For example, to leading order in large N, 360 00:30:18,190 --> 00:30:20,926 then the leading order will be given by the planar diagrams. 361 00:30:24,580 --> 00:30:27,620 Because that should be the diagrams with h 362 00:30:27,620 --> 00:30:30,440 equal zero because h is not negative. 363 00:30:30,440 --> 00:30:33,920 So the leading term is given by h zero. 364 00:30:33,920 --> 00:30:39,190 So all of the planar diagrams in this limit, 365 00:30:39,190 --> 00:30:45,190 in this so-called 't Hooft limit, 366 00:30:45,190 --> 00:30:47,326 will have N dependence which is in N squared. 367 00:30:50,570 --> 00:30:51,890 N squared. 368 00:30:51,890 --> 00:30:54,700 So now if you go back to your loads, 369 00:30:54,700 --> 00:30:57,800 above the fourth diagram we started last time, 370 00:30:57,800 --> 00:31:00,510 above the two planar diagrams we started last time, 371 00:31:00,510 --> 00:31:02,415 you can immediately tell that indeed it 372 00:31:02,415 --> 00:31:04,010 is N squared in this limit. 373 00:31:09,720 --> 00:31:14,220 Then you still haven't explained in terms of lambda. 374 00:31:14,220 --> 00:31:16,830 So you can do Feynman diagrams, et cetera. 375 00:31:16,830 --> 00:31:19,560 Say lambda just depends on the number of loops. 376 00:31:19,560 --> 00:31:23,580 So if you have one loop, then start it 377 00:31:23,580 --> 00:31:27,940 with a lambda equal to zero, start it with some constant. 378 00:31:27,940 --> 00:31:33,680 If it's two loops, then lambda, three loops, lambda squared, 379 00:31:33,680 --> 00:31:34,180 et cetera. 380 00:31:37,650 --> 00:31:41,847 So the sum of four-- sum of all planar 381 00:31:41,847 --> 00:31:43,222 diagrams, we have this structure. 382 00:31:46,410 --> 00:31:49,280 So you can imagine you can sum all this, 383 00:31:49,280 --> 00:31:50,990 if you're powerful enough. 384 00:31:50,990 --> 00:31:53,030 And then you can write it as N squared, then 385 00:31:53,030 --> 00:31:56,580 some function of lambda. 386 00:31:56,580 --> 00:32:00,620 So the planar diagram would be just N squared time 387 00:32:00,620 --> 00:32:02,390 some function of lambda. 388 00:32:02,390 --> 00:32:08,870 So if you are powerful enough to compute this f0 lambda exactly, 389 00:32:08,870 --> 00:32:12,080 then you can say you have solved the planar. 390 00:32:12,080 --> 00:32:15,610 You have solved the large N limit of this series. 391 00:32:15,610 --> 00:32:18,460 Unfortunately for lambda being in gauge theory we cannot do 392 00:32:18,460 --> 00:32:19,134 that. 393 00:32:19,134 --> 00:32:20,550 We don't know how to compute this. 394 00:32:20,550 --> 00:32:22,970 We can on only compute perturbatively which 395 00:32:22,970 --> 00:32:26,600 actually does not work for QCD. 396 00:32:26,600 --> 00:32:27,980 Yes? 397 00:32:27,980 --> 00:32:30,450 AUDIENCE: But N going to be infinite. 398 00:32:30,450 --> 00:32:34,140 It just seems like-- why does this overcome the same problem 399 00:32:34,140 --> 00:32:35,610 that we had before? 400 00:32:35,610 --> 00:32:37,940 HONG LIU: Because this is a specific power. 401 00:32:37,940 --> 00:32:39,430 Here there's no specific power. 402 00:32:41,940 --> 00:32:45,469 That F can be as large as you want. 403 00:32:45,469 --> 00:32:47,260 AUDIENCE: But wasn't the other problem also 404 00:32:47,260 --> 00:32:48,990 that N was going to infinity? 405 00:32:48,990 --> 00:32:50,550 HONG LIU: No that's another problem. 406 00:32:50,550 --> 00:32:53,400 Because here, there is a specification limit, when 407 00:32:53,400 --> 00:32:55,490 you take N going to infinity. 408 00:32:55,490 --> 00:32:57,270 In there, there's no specific limit. 409 00:32:57,270 --> 00:33:00,340 There's no limit when you take N go to infinity. 410 00:33:00,340 --> 00:33:03,350 No, the limit is N squared. 411 00:33:03,350 --> 00:33:08,750 I can say-- this is your essentially, 412 00:33:08,750 --> 00:33:12,410 your vacuum energy, right? 413 00:33:12,410 --> 00:33:15,709 I can say E0, divided by N squared 414 00:33:15,709 --> 00:33:16,750 has a well defined limit. 415 00:33:21,230 --> 00:33:24,850 So the key is that this is a specific dependence on N. 416 00:33:24,850 --> 00:33:26,544 But there it's unbounded. 417 00:33:26,544 --> 00:33:27,960 AUDIENCE: I guess another question 418 00:33:27,960 --> 00:33:31,067 is, we didn't assume that lambda is a small prohibiter. 419 00:33:31,067 --> 00:33:32,650 HONG LIU: No we did not assume lambda. 420 00:33:32,650 --> 00:33:35,060 AUDIENCE: So isn't this kind of perturbative analysis, 421 00:33:35,060 --> 00:33:38,340 saying that that series converges to some function, 422 00:33:38,340 --> 00:33:41,148 is that OK? 423 00:33:41,148 --> 00:33:46,350 HONG LIU: It's a very good question, and it is OK. 424 00:33:46,350 --> 00:33:49,080 But the reasoning is more complicated. 425 00:33:49,080 --> 00:33:51,285 The reasoning is more complicated. 426 00:33:51,285 --> 00:33:52,160 The keys are falling. 427 00:33:55,730 --> 00:33:59,870 What I'm writing here you can understand 428 00:33:59,870 --> 00:34:01,400 using two perspectives. 429 00:34:01,400 --> 00:34:03,220 Just one is heuristic. 430 00:34:03,220 --> 00:34:07,020 Just one say, suppose you can only do perturbation series, 431 00:34:07,020 --> 00:34:09,510 and that would be the thing you compute. 432 00:34:09,510 --> 00:34:11,710 And then say, if you are powerful enough 433 00:34:11,710 --> 00:34:13,620 to solve the theory with perturbity, 434 00:34:13,620 --> 00:34:16,409 then you're guaranteed just to find some general functions. 435 00:34:16,409 --> 00:34:18,969 Yeah but whether this series actually 436 00:34:18,969 --> 00:34:22,989 converges is an important mathematical question, 437 00:34:22,989 --> 00:34:24,760 of course, one has to answer. 438 00:34:24,760 --> 00:34:27,190 It turns out, actually this is convergent. 439 00:34:27,190 --> 00:34:29,730 You can mathematically prove this is convergent, 440 00:34:29,730 --> 00:34:30,789 for very simple reasons. 441 00:34:35,560 --> 00:34:37,780 Yeah this is a side remark, but let me mention it 442 00:34:37,780 --> 00:34:41,110 because this is a cool fact. 443 00:34:41,110 --> 00:34:47,050 So if you do just lambda of g 5, 4 series, 444 00:34:47,050 --> 00:34:50,770 just do the standard Feynman diagram 445 00:34:50,770 --> 00:34:53,409 calculation, input the basic series, 446 00:34:53,409 --> 00:34:56,489 the series is divergent. 447 00:34:56,489 --> 00:34:59,950 The series is divergent, no matter how 448 00:34:59,950 --> 00:35:03,790 g is small, just as the radius of convergence 449 00:35:03,790 --> 00:35:07,200 is 0 for any non-zero value. 450 00:35:07,200 --> 00:35:10,520 Yeah the radius of convergence is 0 in terms of g, 451 00:35:10,520 --> 00:35:12,370 no matter how small g is. 452 00:35:12,370 --> 00:35:14,440 The reason that it's divergent is 453 00:35:14,440 --> 00:35:16,940 because the number of diagrams. 454 00:35:16,940 --> 00:35:22,210 So when you go to Nth order, it increases exponentially in N. 455 00:35:22,210 --> 00:35:25,370 So the coefficient of g-- say to the power 456 00:35:25,370 --> 00:35:27,756 N-- the coefficient can become huge. 457 00:35:27,756 --> 00:35:29,130 Because of the number of diagrams 458 00:35:29,130 --> 00:35:31,680 to increase exponentially in N. So you think, 459 00:35:31,680 --> 00:35:35,630 I have g to the power N times N factorial, something like that. 460 00:35:35,630 --> 00:35:39,690 And that series is not convergent. 461 00:35:39,690 --> 00:35:42,640 But the thing that is convergent here, 462 00:35:42,640 --> 00:35:46,300 because can see the planar diagram, the number of planar 463 00:35:46,300 --> 00:35:48,620 diagram is very, very small. 464 00:35:48,620 --> 00:35:52,720 It only increases with N as a power of N, 465 00:35:52,720 --> 00:35:55,990 rather than as a factorial of N. And so you 466 00:35:55,990 --> 00:35:59,000 can make lambda small enough, then this can be convergent. 467 00:35:59,000 --> 00:35:59,500 Yeah. 468 00:36:01,973 --> 00:36:04,556 AUDIENCE: Didn't we already see at the end of the last lecture 469 00:36:04,556 --> 00:36:07,770 that only planar diagrams keep the biggest contribution, so 470 00:36:07,770 --> 00:36:10,770 why is that equal to-- 471 00:36:10,770 --> 00:36:12,220 HONG LIU: Sorry? 472 00:36:12,220 --> 00:36:14,790 AUDIENCE: So at the end of the last lecture, we saw that-- 473 00:36:14,790 --> 00:36:15,998 HONG LIU: No not necessarily. 474 00:36:18,870 --> 00:36:20,330 That is only for the two diagrams. 475 00:36:20,330 --> 00:36:23,680 If you look at this thing, I can draw arbitrary complicated 476 00:36:23,680 --> 00:36:27,400 non-planar diagrams, with a very large F. They only 477 00:36:27,400 --> 00:36:30,820 depend on F. It does not depend on anything else, 478 00:36:30,820 --> 00:36:31,616 if you do this. 479 00:36:41,570 --> 00:36:47,130 OK so in general, then of course if you 480 00:36:47,130 --> 00:36:50,720 include the higher non-planar diagram, et cetera-- 481 00:36:50,720 --> 00:36:56,820 so in general the vacuum energy, which you would normally 482 00:36:56,820 --> 00:36:58,260 call log z. 483 00:36:58,260 --> 00:37:03,260 OK log z is, essentially, the sum of all vacuum diagrams. 484 00:37:09,220 --> 00:37:11,390 So this is the partition functions, 485 00:37:11,390 --> 00:37:13,050 so it's a path integral. 486 00:37:13,050 --> 00:37:16,555 So log z, then we'll have the expansion from h 487 00:37:16,555 --> 00:37:21,560 equal to 0 to infinity, N two the power 2 488 00:37:21,560 --> 00:37:26,230 minus 2h, F h lambda. 489 00:37:26,230 --> 00:37:31,980 OK so at each genus level, you will 490 00:37:31,980 --> 00:37:33,330 have some function of lambda. 491 00:37:40,620 --> 00:37:47,300 Yeah the leading, order we just showed is f0. 492 00:37:47,300 --> 00:37:49,760 And then if you add the Taurus now-- 493 00:37:49,760 --> 00:37:52,490 we'll add to the torus now our on non-planar diagram. 494 00:37:52,490 --> 00:37:54,680 It's order one. 495 00:37:54,680 --> 00:37:56,780 And then, for the two genus, it's 496 00:37:56,780 --> 00:38:00,650 f over n squared, et cetera. 497 00:38:09,900 --> 00:38:12,900 So let me just write down z explicitly. 498 00:38:12,900 --> 00:38:22,430 Z is the partition function, i S phi. 499 00:38:22,430 --> 00:38:27,060 So if you compute this, with the right boundary condition-- 500 00:38:27,060 --> 00:38:29,000 if you compute this path integral, 501 00:38:29,000 --> 00:38:30,730 with the proper boundary conditions, 502 00:38:30,730 --> 00:38:35,290 then that gives you the vacuum diagrams. 503 00:38:35,290 --> 00:38:38,310 The log z is the sum of all connected vacuum diagrams. 504 00:38:38,310 --> 00:38:43,510 I should say, the sum of all connected vacuum diagrams. 505 00:39:02,530 --> 00:39:03,785 Any questions regarding this? 506 00:39:12,050 --> 00:39:13,280 Good. 507 00:39:13,280 --> 00:39:24,927 Now there is a heuristic way we can understand this expansion. 508 00:39:27,670 --> 00:39:38,260 So it's actually a heuristic way to understand 509 00:39:38,260 --> 00:39:41,150 this one-way expansion. 510 00:39:41,150 --> 00:39:46,690 So let's just look at this path integral. 511 00:39:46,690 --> 00:39:49,270 So let's look at the Lagrangian. 512 00:39:49,270 --> 00:39:52,070 So the Lagrangian I wrote there. 513 00:39:52,070 --> 00:39:53,830 In this 't Hooft limit, I can write it 514 00:39:53,830 --> 00:39:57,460 as minus N divided by lambda. 515 00:39:57,460 --> 00:40:00,280 So I want to write things in terms of lambda. 516 00:40:00,280 --> 00:40:04,630 So I multiply the pre-factor N downstairs, and then upstairs. 517 00:40:04,630 --> 00:40:06,240 So g squared N give me lambda. 518 00:40:09,750 --> 00:40:13,920 And then I have a trace, et cetera. 519 00:40:19,490 --> 00:40:24,170 So now it's easy to see that you're leading order, 520 00:40:24,170 --> 00:40:27,170 this things should give you order N squared. 521 00:40:27,170 --> 00:40:30,880 Because there is already a factor of N here, and the trace 522 00:40:30,880 --> 00:40:33,960 is the sum of N things. 523 00:40:33,960 --> 00:40:37,290 So generically, this should be of the order N squared. 524 00:40:43,070 --> 00:40:47,200 So now we-- a little bit leap of faith-- 525 00:40:47,200 --> 00:40:53,880 say supposing the large N limit, since the leading order 526 00:40:53,880 --> 00:40:56,540 is order N squared, you can argue that actually all the N 527 00:40:56,540 --> 00:40:58,250 squared is the expansion parameter, 528 00:40:58,250 --> 00:41:00,240 if you do scatter point approximation. 529 00:41:00,240 --> 00:41:03,750 And then naturally, you will see the power will 530 00:41:03,750 --> 00:41:05,760 be given by 1 over N squared. 531 00:41:18,070 --> 00:41:20,100 any questions regarding this? 532 00:41:20,100 --> 00:41:21,322 Yes? 533 00:41:21,322 --> 00:41:24,770 AUDIENCE: What does N over lambda factor to? 534 00:41:24,770 --> 00:41:26,459 HONG LIU: It's-- 535 00:41:26,459 --> 00:41:27,000 AUDIENCE: OK. 536 00:41:33,530 --> 00:41:35,060 HONG LIU: So clearly this discussion 537 00:41:35,060 --> 00:41:39,580 actually does not-- so when I say clearly here, 538 00:41:39,580 --> 00:41:42,670 it requires a little bit of practice. 539 00:41:42,670 --> 00:41:48,660 But clearly, our discussion only depend 540 00:41:48,660 --> 00:41:53,173 on the matrix nature of the Lagrangian and the fields. 541 00:41:55,810 --> 00:41:58,610 So I'm going to make a claim. 542 00:41:58,610 --> 00:42:19,520 It says for any Lagrangian, of matrix valued fields 543 00:42:19,520 --> 00:42:29,790 of the form L, which is N divided by some coupling 544 00:42:29,790 --> 00:42:32,967 constant, times a trace of something. 545 00:42:32,967 --> 00:42:34,800 Doesn't matter what you put inside the trace 546 00:42:34,800 --> 00:42:37,720 here, as far as you have a single trace. 547 00:42:40,350 --> 00:42:45,983 Such a series will always have the expansion like this. 548 00:43:00,450 --> 00:43:06,040 It will always have an expansion like this. 549 00:43:21,297 --> 00:43:22,338 So let me just summarize. 550 00:43:39,080 --> 00:43:57,460 In the 't Hooft limit-- by 't Hooft limit, 551 00:43:57,460 --> 00:44:05,770 I always mean this form-- the coupling constant is defined 552 00:44:05,770 --> 00:44:10,830 such that you have some coupling concept here, 553 00:44:10,830 --> 00:44:13,220 and then you have over N factor. 554 00:44:13,220 --> 00:44:16,660 And then of course, you can also have some coupling constant 555 00:44:16,660 --> 00:44:17,160 inside here. 556 00:44:17,160 --> 00:44:18,410 It doesn't matter. 557 00:44:18,410 --> 00:44:21,100 As far as those coupling constants are independent of N. 558 00:44:21,100 --> 00:44:26,040 And as far as things inside the trace are independent of N. 559 00:44:26,040 --> 00:44:36,050 So for such a kind of series, the 1 over N expansion 560 00:44:36,050 --> 00:44:43,100 is equal to the topological expansion. 561 00:44:49,050 --> 00:45:06,905 It's the expansion in terms of topology of Feynman diagrams. 562 00:45:18,880 --> 00:45:23,090 So this is a very, very beautiful-- 563 00:45:23,090 --> 00:45:30,420 and as I said, it's [INAUDIBLE], because in principle, it 564 00:45:30,420 --> 00:45:36,429 puts a very simple structure into something that's, 565 00:45:36,429 --> 00:45:37,720 in principle, very complicated. 566 00:45:41,000 --> 00:45:44,520 Yeah, any questions regarding this? 567 00:45:44,520 --> 00:45:45,290 Yes? 568 00:45:45,290 --> 00:45:48,090 AUDIENCE: Maybe it's not can I just understand it 569 00:45:48,090 --> 00:45:50,270 in such a way that he kind of treats the Feynman 570 00:45:50,270 --> 00:45:56,730 diagram as a triangulation of different spaces. 571 00:45:56,730 --> 00:45:59,300 HONG LIU: Yeah for example, you can think of from that point. 572 00:45:59,300 --> 00:46:02,660 Yeah, so we use that to derive, to use this formula. 573 00:46:09,180 --> 00:46:14,020 Yeah and that picture will be very useful in a little bit 574 00:46:14,020 --> 00:46:15,460 from now on. 575 00:46:15,460 --> 00:46:16,880 Keep that picture in mind. 576 00:46:16,880 --> 00:46:23,690 The Feynman diagram is like the partition of some surfaces. 577 00:46:23,690 --> 00:46:25,570 And that will be very useful later. 578 00:46:25,570 --> 00:46:27,530 AUDIENCE: Does there ever arise a situation 579 00:46:27,530 --> 00:46:29,830 in which you care not about two surfaces, 580 00:46:29,830 --> 00:46:32,740 but Feynman diagrams on three surfaces or something 581 00:46:32,740 --> 00:46:33,750 like that? 582 00:46:33,750 --> 00:46:35,904 Because this asks the question, you 583 00:46:35,904 --> 00:46:37,320 don't necessarily have to consider 584 00:46:37,320 --> 00:46:38,870 the topology of two surfaces. 585 00:46:38,870 --> 00:46:41,290 Are there any situations in which it's more complicated? 586 00:46:41,290 --> 00:46:43,373 HONG LIU: Yeah but we always draw Feynman diagrams 587 00:46:43,373 --> 00:46:47,260 on the paper, which is two dimensional. 588 00:46:47,260 --> 00:46:48,200 Yeah it's enough. 589 00:46:48,200 --> 00:46:49,560 Two dimensions is enough. 590 00:46:49,560 --> 00:46:51,694 You don't need to go to three dimensions. 591 00:46:51,694 --> 00:46:53,110 Yeah and this structure only comes 592 00:46:53,110 --> 00:46:54,610 when you go to two dimensions, because if you 593 00:46:54,610 --> 00:46:56,735 go to three dimension, of course, they don't cross. 594 00:46:58,619 --> 00:47:00,160 In three dimension, you can no longer 595 00:47:00,160 --> 00:47:03,489 distinguish planar or non-planar diagram. 596 00:47:03,489 --> 00:47:04,780 AUDIENCE: Well if you did-- OK. 597 00:47:10,224 --> 00:47:11,682 HONG LIU: Good any other questions? 598 00:47:14,300 --> 00:47:16,300 AUDIENCE: Why is it always orientable surfaces? 599 00:47:19,440 --> 00:47:25,340 HONG LIU: That's a good question. 600 00:47:25,340 --> 00:47:31,480 It's because those lines are orientable because when 601 00:47:31,480 --> 00:47:34,590 we draw the double roation, so you 602 00:47:34,590 --> 00:47:36,140 have this two, double rotation. 603 00:47:36,140 --> 00:47:38,390 So essentially those lines are orientable. 604 00:47:38,390 --> 00:47:39,810 They have a direction. 605 00:47:39,810 --> 00:47:42,450 And essentially, this is, of course, one and two. 606 00:47:42,450 --> 00:47:45,960 Yeah so those Feynman diagrams actually have a direction, 607 00:47:45,960 --> 00:47:48,670 have a sense of orientation. 608 00:47:48,670 --> 00:47:50,750 So I'm going to mention, by passing in nature, 609 00:47:50,750 --> 00:47:52,570 but let me just mention also now. 610 00:47:52,570 --> 00:47:57,890 So if it's not [INAUDIBLE] matrix, 611 00:47:57,890 --> 00:48:01,500 say if it's a real symmetric matrix, then the then 612 00:48:01,500 --> 00:48:03,530 there's no difference between two index. 613 00:48:03,530 --> 00:48:05,270 And then there's no orientation. 614 00:48:05,270 --> 00:48:08,690 And that would be related to m orientable surfaces. 615 00:48:08,690 --> 00:48:11,115 And then you need to slightly generalize this. 616 00:48:16,070 --> 00:48:18,662 Any other questions? 617 00:48:18,662 --> 00:48:20,120 AUDIENCE: Now you mentioned that it 618 00:48:20,120 --> 00:48:22,325 has something to do with string theory, 619 00:48:22,325 --> 00:48:25,607 but does that it has anything to do with scatter particles-- 620 00:48:25,607 --> 00:48:27,690 HONG LIU: Yeah, so we're going to talk about that. 621 00:48:30,122 --> 00:48:31,580 No, we're going to talk about that. 622 00:48:35,590 --> 00:48:36,090 Good? 623 00:48:36,090 --> 00:48:37,965 So now let me talk about general observables. 624 00:48:41,182 --> 00:48:42,890 I think we're a little bit short on time, 625 00:48:42,890 --> 00:48:44,650 if we want to reach the punchline today. 626 00:48:48,210 --> 00:48:54,260 So right now we only have looked at the vacuum diagrams. 627 00:48:56,900 --> 00:49:01,972 So now let's look at the general observables. 628 00:49:01,972 --> 00:49:03,680 Before talking about general observables, 629 00:49:03,680 --> 00:49:06,640 let me just again make a side remark, 630 00:49:06,640 --> 00:49:09,800 Which is the gauge versus global symmetries. 631 00:49:17,047 --> 00:49:18,630 So in the example we talk about here-- 632 00:49:18,630 --> 00:49:22,780 let me call this a, equation a. 633 00:49:22,780 --> 00:49:24,450 Then let me write down another equation 634 00:49:24,450 --> 00:49:27,078 b, which is a Yang-Mills theory. 635 00:49:42,260 --> 00:49:51,240 So the difference between a and b-- so a is this guy. 636 00:49:51,240 --> 00:49:53,630 So the difference between a and b, 637 00:49:53,630 --> 00:49:55,340 is that a, as we discussed last time, 638 00:49:55,340 --> 00:49:57,200 is invariant on the global symmetry, 639 00:49:57,200 --> 00:49:58,967 is a U(N) global symmetry. 640 00:50:06,650 --> 00:50:16,810 It's that phi is invariant under the acting of a unitary matrix. 641 00:50:16,810 --> 00:50:18,195 But this U must be constant. 642 00:50:22,080 --> 00:50:25,480 Only for a constant U is this a symmetry. 643 00:50:25,480 --> 00:50:28,754 But if you have studied a little bit of gauge theory, 644 00:50:28,754 --> 00:50:30,420 or if you have not studied gauge theory, 645 00:50:30,420 --> 00:50:45,320 just take my word for it, the b is 646 00:50:45,320 --> 00:50:53,852 invariant under a local symmetry, a local U(N) 647 00:50:53,852 --> 00:50:54,351 symmetry. 648 00:50:57,060 --> 00:51:00,460 He said A miu-- so A mu is what make up 649 00:51:00,460 --> 00:51:12,370 the F-- U x A mu x, U dagger x minus i partial mu U x. 650 00:51:12,370 --> 00:51:14,820 It doesn't matter. 651 00:51:14,820 --> 00:51:21,755 The only thing I want to say is that this U x is arbitrary. 652 00:51:21,755 --> 00:51:27,230 It can have arbitrary space time dependence. 653 00:51:42,820 --> 00:51:46,040 Just like of the generalization of the QED, 654 00:51:46,040 --> 00:51:49,340 it's the gauge symmetry. 655 00:51:49,340 --> 00:51:51,050 So the key difference between the two. 656 00:51:55,900 --> 00:51:59,170 the key difference between this local and the global symmetry, 657 00:51:59,170 --> 00:52:05,360 are manifested in what kind of observables we can see there. 658 00:52:05,360 --> 00:52:07,280 For example, for a-- 659 00:52:07,280 --> 00:52:10,180 AUDIENCE: I think that should be U dagger, partial mu and then 660 00:52:10,180 --> 00:52:11,122 U dagger. 661 00:52:16,110 --> 00:52:18,210 HONG LIU: HONG LIU: So this difference, 662 00:52:18,210 --> 00:52:19,610 you can say what's the big deal? 663 00:52:19,610 --> 00:52:22,100 in one case, this is constant, and this 664 00:52:22,100 --> 00:52:24,130 is dependent on space time. 665 00:52:24,130 --> 00:52:27,100 So the key difference between them 666 00:52:27,100 --> 00:52:38,480 is that in the case for a, operator like this phi squared, 667 00:52:38,480 --> 00:52:39,330 phi is a matrix. 668 00:52:39,330 --> 00:52:41,600 So phi squared is a matrix. 669 00:52:41,600 --> 00:52:44,530 Phi squared x is an allowed operator. 670 00:52:54,640 --> 00:52:56,590 So this operator is not invariant under 671 00:52:56,590 --> 00:52:59,420 this global U(N) symmetry. 672 00:52:59,420 --> 00:53:01,930 But it doesn't matter because this is a global symmetry. 673 00:53:01,930 --> 00:53:03,450 So this is an allowed operator. 674 00:53:06,080 --> 00:53:11,181 But if you have gauge symmetry, all operators 675 00:53:11,181 --> 00:53:12,180 must be gauge invariant. 676 00:53:21,898 --> 00:53:24,530 That means that all operator must 677 00:53:24,530 --> 00:53:27,310 be invariant under this kind of transformations. 678 00:53:27,310 --> 00:53:32,854 So the analog of this is not allows operators. 679 00:53:35,740 --> 00:53:38,600 So observables in the gauge theory are much more limited. 680 00:53:45,360 --> 00:53:49,220 So we will be interested gauge theories. 681 00:53:49,220 --> 00:53:51,440 We will be interested in gauge theories. 682 00:53:51,440 --> 00:53:55,812 So that means we are always interested in observables, 683 00:53:55,812 --> 00:53:57,436 which are invariant under the symmetry. 684 00:54:21,737 --> 00:54:23,320 So we're interested in gauge theories. 685 00:54:36,170 --> 00:54:38,830 So that means we're always interested in gauge invariant 686 00:54:38,830 --> 00:54:41,720 operators. 687 00:54:41,720 --> 00:54:44,820 So the kind of Lagrangian does not matter. 688 00:54:44,820 --> 00:54:46,732 So you can have the gauge fields. 689 00:54:46,732 --> 00:54:48,190 You can also have some other field, 690 00:54:48,190 --> 00:54:51,600 say some matrix phi, et cetera. 691 00:54:51,600 --> 00:54:55,720 As phi is the Lagrangian of this form, it's OK. 692 00:54:55,720 --> 00:54:59,100 We always only consider the Lagrangian of that form. 693 00:54:59,100 --> 00:55:02,280 But it doesn't have an arbitrary number of fields, 694 00:55:02,280 --> 00:55:04,330 and with arbitrary kind of interactions. 695 00:55:14,260 --> 00:55:15,305 So you start your theory. 696 00:55:19,440 --> 00:55:22,247 So let's for simplicity, this can see the local operators. 697 00:55:25,010 --> 00:55:34,750 In this kind of theory, then allowed, say local operators, 698 00:55:34,750 --> 00:55:37,820 must always have some kind of trace in it. 699 00:55:37,820 --> 00:55:44,390 Say you must have some form trace, F mu U, F mu U, 700 00:55:44,390 --> 00:55:50,790 a trace phi squared, et cetera. 701 00:55:50,790 --> 00:55:57,370 A trace phi to some power F N, phi to some power k, et cetera. 702 00:56:01,790 --> 00:56:04,650 You can also have operators with more than one 703 00:56:04,650 --> 00:56:18,090 trace, say trace phi squared, trace F mu U, F mu U. 704 00:56:18,090 --> 00:56:22,430 So we are going to make a distinction 705 00:56:22,430 --> 00:56:25,380 because the operator with only a single trace, 706 00:56:25,380 --> 00:56:26,892 we call them single trace operator. 707 00:56:31,210 --> 00:56:33,810 And the operator with more than one trace, 708 00:56:33,810 --> 00:56:35,512 we call them multiple trace operators. 709 00:56:42,260 --> 00:56:45,400 OK so the reason for this distinction will be clear soon. 710 00:56:52,040 --> 00:56:57,884 So multiple trace operators-- it's self-evident again-- 711 00:56:57,884 --> 00:56:59,550 that multiple trace operators can always 712 00:56:59,550 --> 00:57:01,760 be written as products of single trace operators. 713 00:57:04,834 --> 00:57:06,000 AUDIENCE: I have a question. 714 00:57:06,000 --> 00:57:10,890 Is it a possible case to have a local gauge invariant operator, 715 00:57:10,890 --> 00:57:12,846 the F mu U times F mu U? 716 00:57:12,846 --> 00:57:15,220 HONG LIU: Yeah I always can see the local gauge invariant 717 00:57:15,220 --> 00:57:16,910 operators. 718 00:57:16,910 --> 00:57:20,500 We can see the gauge theories. 719 00:57:20,500 --> 00:57:24,309 AUDIENCE: So this combination, F mu U times F mu U, it's only-- 720 00:57:24,309 --> 00:57:25,850 HONG LIU: No this is gauge invariant. 721 00:57:25,850 --> 00:57:29,944 AUDIENCE: Is that the only gauge invariant component? 722 00:57:29,944 --> 00:57:30,610 HONG LIU: Sorry? 723 00:57:30,610 --> 00:57:33,120 AUDIENCE: Is this is the only gauge invariant 724 00:57:33,120 --> 00:57:36,160 component that we can use to construct the gauge 725 00:57:36,160 --> 00:57:38,087 invariant local operators? 726 00:57:38,087 --> 00:57:40,420 HONG LIU: Sorry, I don't quite understand what you mean. 727 00:57:40,420 --> 00:57:44,311 No this is just one specific operator. 728 00:57:44,311 --> 00:57:50,420 No, you can take F to the power N, an arbitrary number of-- as 729 00:57:50,420 --> 00:57:53,820 far as they're inside the trace, it's always gauge invariant. 730 00:57:53,820 --> 00:57:56,410 AUDIENCE: I see. 731 00:57:56,410 --> 00:57:59,380 HONG LIU: I'm just writing down a particular example. 732 00:57:59,380 --> 00:58:02,220 So just for notational simplicity, 733 00:58:02,220 --> 00:58:05,610 I will just write-- so from now on I will just 734 00:58:05,610 --> 00:58:15,990 denote the single trace operators collectively just 735 00:58:15,990 --> 00:58:19,540 as O with some script n, which denotes 736 00:58:19,540 --> 00:58:22,080 the different operators. 737 00:58:22,080 --> 00:58:24,820 so n denotes different operators. 738 00:58:24,820 --> 00:58:28,810 And then for the multiple trace operator, then you have O1, O2. 739 00:58:28,810 --> 00:58:30,830 That would be a double trace operator. 740 00:58:30,830 --> 00:58:34,730 And O1, O2, O3, say times O3 would 741 00:58:34,730 --> 00:58:38,530 be a triple trace operator. 742 00:58:43,600 --> 00:58:46,400 n just labels different operators. 743 00:58:46,400 --> 00:58:48,245 I'm just using abstract notation. 744 00:58:53,500 --> 00:59:01,880 And the reason for this-- a distinction will be clear soon. 745 00:59:01,880 --> 00:59:15,540 Then for such gauge theories, the general observables, 746 00:59:15,540 --> 00:59:26,362 in the quantum field theory is just correlation functions 747 00:59:26,362 --> 00:59:27,570 of gauge invariant operators. 748 00:59:40,591 --> 00:59:42,090 So by gauge invariant operators, you 749 00:59:42,090 --> 00:59:44,465 can have local operators, non-local operators, et cetera. 750 00:59:48,100 --> 00:59:52,580 So for simplicity, I will restrict my discussion only 751 00:59:52,580 --> 00:59:54,010 on the local operators. 752 00:59:54,010 --> 00:59:56,400 But local operators means that the fields 753 00:59:56,400 --> 00:59:57,812 evaluated at a single point. 754 01:00:05,100 --> 01:00:09,870 So the typical correlation functions, 755 01:00:09,870 --> 01:00:12,660 then the typical observables will have this form, 756 01:00:12,660 --> 01:00:15,070 will be just a product of some correlation functions, 757 01:00:15,070 --> 01:00:19,000 a product of some operators, and I say their correlation 758 01:00:19,000 --> 01:00:20,260 functions. 759 01:00:20,260 --> 01:00:23,993 By c I mean the connected correlation functions. 760 01:00:29,600 --> 01:00:32,330 So you can see immediately, these multi 761 01:00:32,330 --> 01:00:34,440 trace operators are just the product 762 01:00:34,440 --> 01:00:36,440 of a single trace operator. 763 01:00:36,440 --> 01:00:38,900 And the correlation function of a multiple trace operator 764 01:00:38,900 --> 01:00:41,840 can be obtained from those of a single trace one. 765 01:00:41,840 --> 01:00:44,230 You just identify some of the acts. 766 01:00:44,230 --> 01:00:45,685 Then that will be enough. 767 01:00:48,320 --> 01:00:50,750 So we only need to talk about the correlation function 768 01:00:50,750 --> 01:00:51,850 of single trace operators. 769 01:00:56,730 --> 01:00:59,950 So now the question-- let me call this equation one. 770 01:00:59,950 --> 01:01:03,890 So now the question follows what we discussed before, 771 01:01:03,890 --> 01:01:07,932 is how do we decide the N dependence of the guy? 772 01:01:07,932 --> 01:01:09,640 Previously we determined the N dependence 773 01:01:09,640 --> 01:01:13,496 of this vacuum diagrams of this petition function. 774 01:01:13,496 --> 01:01:15,120 But now want should be the N dependence 775 01:01:15,120 --> 01:01:16,494 of general correlation functions? 776 01:01:23,090 --> 01:01:26,460 One way to do it, you just start immediately calculating. 777 01:01:26,460 --> 01:01:29,590 And then you can find some root, et cetera. 778 01:01:29,590 --> 01:01:32,680 But actually there's a very simple trick. 779 01:01:32,680 --> 01:01:35,580 There's a very simple trick to determine 780 01:01:35,580 --> 01:01:37,220 the N dependence of this. 781 01:01:37,220 --> 01:01:39,010 So now I will explain. 782 01:01:39,010 --> 01:01:41,460 Now I will tell you. 783 01:01:41,460 --> 01:01:42,970 So I will not give you any examples 784 01:01:42,970 --> 01:01:46,820 because this trick is so nice, and it just works very easy. 785 01:01:55,720 --> 01:02:06,270 And so the question, what is N dependence of one? 786 01:02:06,270 --> 01:02:08,363 So this is the question we want to address. 787 01:02:11,030 --> 01:02:14,110 so here is a very simple, beautiful, trick. 788 01:02:14,110 --> 01:02:18,232 So let's consider the following generating functional. 789 01:02:18,232 --> 01:02:19,690 So in quantum field theory, when we 790 01:02:19,690 --> 01:02:21,590 talk about correlation functions, 791 01:02:21,590 --> 01:02:24,040 it's always convenient to talk about the generating 792 01:02:24,040 --> 01:02:24,540 functional. 793 01:02:32,340 --> 01:02:37,830 So whatever is your field, you do the path integral 794 01:02:37,830 --> 01:02:41,140 of all fields. 795 01:02:41,140 --> 01:02:45,470 And then you look at the action. 796 01:02:45,470 --> 01:02:50,260 So you have your regional action, which I call S0. 797 01:02:50,260 --> 01:02:54,322 And then that's add those operators. 798 01:02:58,010 --> 01:03:06,285 Ji x, O i x. 799 01:03:19,670 --> 01:03:21,990 Yes so this is a standard story. 800 01:03:21,990 --> 01:03:24,910 When you take the derivative over Ji, 801 01:03:24,910 --> 01:03:28,130 then you will bring down a factor of O i. 802 01:03:28,130 --> 01:03:30,880 Then that essentially give you the correlation function. 803 01:03:30,880 --> 01:03:33,280 You have, for example, a correlation function, 804 01:03:33,280 --> 01:03:34,738 the connected correlation function. 805 01:03:37,880 --> 01:03:41,680 O n, the connected correlation function 806 01:03:41,680 --> 01:03:49,290 would be just you take the derivative of log z. 807 01:03:49,290 --> 01:03:58,220 And then delta J1 x delta Jn xn. 808 01:03:58,220 --> 01:04:00,310 And then you set J equal to zero. 809 01:04:00,310 --> 01:04:02,560 You set all the J equal to zero. 810 01:04:02,560 --> 01:04:06,660 And that gives you the end point function. 811 01:04:06,660 --> 01:04:10,096 I should write i here, so i to the power n. 812 01:04:17,540 --> 01:04:21,890 So now here is the beautiful trick. 813 01:04:21,890 --> 01:04:26,062 You can determine this in a single shot, the N 814 01:04:26,062 --> 01:04:27,295 dependence of this guy. 815 01:04:30,640 --> 01:04:33,090 And this simple trick is just to add N here. 816 01:04:37,580 --> 01:04:39,640 You add N here. 817 01:04:39,640 --> 01:04:41,530 In order to get the correlation function, 818 01:04:41,530 --> 01:04:45,200 we need to divide it by N to the power N. Now you 819 01:04:45,200 --> 01:04:48,460 wanted to get O i, you need to divide-- take the root of N 820 01:04:48,460 --> 01:04:55,190 times Ji, so you also need to divide it by 1 over N. 821 01:04:55,190 --> 01:04:57,880 But why does this help? 822 01:04:57,880 --> 01:05:00,220 Why does this help? 823 01:05:00,220 --> 01:05:01,650 It helps for the following reason. 824 01:05:01,650 --> 01:05:06,020 Let me call this whole thing iS effective. 825 01:05:10,430 --> 01:05:15,070 iS effective, so the key thing is 826 01:05:15,070 --> 01:05:26,360 that this O i, single trace operators, 827 01:05:26,360 --> 01:05:37,670 then this S effective then has the form 828 01:05:37,670 --> 01:05:39,807 N times the trace something. 829 01:05:43,690 --> 01:05:48,060 Because you already know S0, which is our starting point, 830 01:05:48,060 --> 01:05:49,720 has this form. 831 01:05:49,720 --> 01:05:54,920 And now the term you added in, precisely, also has this form, 832 01:05:54,920 --> 01:05:57,635 is the N factor times something single trace. 833 01:06:04,630 --> 01:06:09,370 So that means the whole thing still has this form. 834 01:06:09,370 --> 01:06:13,010 Then now we can immediately conclude 835 01:06:13,010 --> 01:06:22,200 this log z J1, Jn, must have this expansion. 836 01:06:22,200 --> 01:06:28,960 h from 0 to infinity, and to the 2 minus 2h, f 837 01:06:28,960 --> 01:06:31,771 h lambda J1, et cetera. 838 01:06:41,330 --> 01:06:47,130 So adding that N is a powerful, powerful trick. 839 01:06:47,130 --> 01:06:52,790 So now you can immediately, just from here, 840 01:06:52,790 --> 01:07:13,200 we can immediately find out that for endpoint function, 841 01:07:13,200 --> 01:07:18,860 connected endpoint function, the leading order is 2 minus n, 842 01:07:18,860 --> 01:07:20,910 because the leading order is n squared. 843 01:07:20,910 --> 01:07:23,790 Yeah it's 2 minus n. 844 01:07:23,790 --> 01:07:28,408 And then suppressed by 1 over N squared, et cetera. 845 01:07:39,400 --> 01:07:40,800 Good? 846 01:07:40,800 --> 01:07:44,830 So for example, if you look at a 0 point function, which 847 01:07:44,830 --> 01:07:48,440 is essentially the partition function, so this is order 848 01:07:48,440 --> 01:07:49,030 N squared. 849 01:07:49,030 --> 01:07:51,113 So this is what we found before, to leading order. 850 01:07:53,510 --> 01:07:55,370 And if you look at the one point function, 851 01:07:55,370 --> 01:07:57,110 some operator, then, will be order 852 01:07:57,110 --> 01:08:00,750 N. If you look at the two point function, connected two point 853 01:08:00,750 --> 01:08:03,910 function of some operator, so it would be order one. 854 01:08:03,910 --> 01:08:07,560 And the three point function of some operator 855 01:08:07,560 --> 01:08:10,860 will be order 1 over N, as the leading order. 856 01:08:14,010 --> 01:08:16,680 And then all higher order just down by 1 over N 857 01:08:16,680 --> 01:08:19,010 squared, compared to the leading order. 858 01:08:19,010 --> 01:08:23,149 And again, the leading order is given by the planar diagrams. 859 01:08:23,149 --> 01:08:25,910 Because of the leading order contribution to here, 860 01:08:25,910 --> 01:08:29,210 in terms of this S effective is planar diagram. 861 01:08:29,210 --> 01:08:32,180 And then they must be, under those things, 862 01:08:32,180 --> 01:08:35,080 just obtained by through [INAUDIBLE], so they 863 01:08:35,080 --> 01:08:37,569 must be planar diagrams. 864 01:08:37,569 --> 01:08:39,435 so again this comes from planar diagrams. 865 01:08:44,170 --> 01:08:46,930 Good? 866 01:08:46,930 --> 01:08:51,189 So now let's talk about the physical implications of this. 867 01:08:51,189 --> 01:08:54,439 So what does this mean? 868 01:08:54,439 --> 01:09:03,710 So we have found out, this is our N dependence for our gauge 869 01:09:03,710 --> 01:09:06,229 invariant operators. 870 01:09:06,229 --> 01:09:07,716 So what does this mean? 871 01:09:07,716 --> 01:09:09,549 Now let me talk about physical implications. 872 01:09:21,994 --> 01:09:25,403 AUDIENCE: What did one of those [INAUDIBLE] have with-- 873 01:09:28,840 --> 01:09:31,529 HONG LIU: Yeah, I defined them without chi. 874 01:09:31,529 --> 01:09:34,180 AUDIENCE: Then something like-- 875 01:09:34,180 --> 01:09:38,140 HONG LIU: You define something. 876 01:09:38,140 --> 01:09:40,740 So when you write down your theory, 877 01:09:40,740 --> 01:09:42,935 you define this 't Hooft limit. 878 01:09:42,935 --> 01:09:44,810 Then everything is already in terms of lambda 879 01:09:44,810 --> 01:09:46,560 or some other order one number. 880 01:09:46,560 --> 01:09:48,900 So that can depend on those numbers in an arbitrary way. 881 01:09:48,900 --> 01:09:50,983 It doesn't matter can depend on coupling constants 882 01:09:50,983 --> 01:09:53,710 in an arbitrary, but it cannot have N dependence defined 883 01:09:53,710 --> 01:09:56,606 inside the operator. 884 01:10:00,550 --> 01:10:04,750 Once we introduce this 't Hooft limit, 885 01:10:04,750 --> 01:10:12,670 then the operator can depend on the coupling in the 't Hooft 886 01:10:12,670 --> 01:10:15,300 limit in an arbitrary way. 887 01:10:15,300 --> 01:10:17,515 Because they're all just all the one constant. 888 01:10:21,270 --> 01:10:24,320 Let's talk about physical implications of this. 889 01:10:24,320 --> 01:10:27,210 It turns out, these simple and [INAUDIBLE] behavior 890 01:10:27,210 --> 01:10:35,760 actually has a very simple physical picture behind it. 891 01:10:35,760 --> 01:10:41,180 So first he said, in the large N limit-- so let's just 892 01:10:41,180 --> 01:10:44,630 look at leading order behavior. 893 01:10:44,630 --> 01:10:51,445 So in a large N limit, if we consider this state of O 894 01:10:51,445 --> 01:10:55,250 i x acting on the vacuum. 895 01:10:55,250 --> 01:10:58,990 So some single trace operator acting on the vacuum. 896 01:10:58,990 --> 01:11:00,860 So i, again, just labels different state, 897 01:11:00,860 --> 01:11:02,040 different operators. 898 01:11:02,040 --> 01:11:18,560 These can be interpreted as creating a "single particle" 899 01:11:18,560 --> 01:11:19,060 state. 900 01:11:34,000 --> 01:11:35,500 I'm first describing the conclusion. 901 01:11:35,500 --> 01:11:37,510 Then I will explain why. 902 01:11:37,510 --> 01:11:39,210 So you can see that the state obtained 903 01:11:39,210 --> 01:11:43,030 by adding a single trace operator on the vacuum, 904 01:11:43,030 --> 01:11:46,860 then this can be interpreted as a single particle state. 905 01:11:46,860 --> 01:11:51,560 So if you add on the double trace operator on the vacuum, 906 01:11:51,560 --> 01:11:53,762 then this can be interpreted as two particle state. 907 01:11:57,090 --> 01:12:03,110 Similarly, say O n acting on the vacuum 908 01:12:03,110 --> 01:12:04,950 would be N particle states. 909 01:12:10,990 --> 01:12:13,710 So why is this so? 910 01:12:13,710 --> 01:12:14,726 Why this is so. 911 01:12:17,540 --> 01:12:24,790 So I will support this statement using three arguments. 912 01:12:29,820 --> 01:12:36,150 First remember O i O j, the connecting Green function 913 01:12:36,150 --> 01:12:38,350 of any two operators of order one. 914 01:12:42,050 --> 01:12:46,500 So we can actually just diagonalize them. 915 01:12:46,500 --> 01:13:00,450 If you can just diagonalize them, so that O i O j 916 01:13:00,450 --> 01:13:03,960 are proportional to delta i j. 917 01:13:08,350 --> 01:13:10,940 So in some sense, a two point function, 918 01:13:10,940 --> 01:13:13,277 those operators can be considered as independent. 919 01:13:18,400 --> 01:13:19,723 And now the second statement. 920 01:13:23,400 --> 01:13:26,420 So if you want to call this single particle, 921 01:13:26,420 --> 01:13:30,870 this two particle, multi particle, then they 922 01:13:30,870 --> 01:13:32,430 should be that they don't overlap. 923 01:13:32,430 --> 01:13:34,888 Because a single particle cannot overlap with two particle, 924 01:13:34,888 --> 01:13:37,260 et cetera. 925 01:13:37,260 --> 01:13:40,100 And then to see the overlap between a single particle 926 01:13:40,100 --> 01:13:44,340 with a multi particle, you look at these correlation functions. 927 01:13:44,340 --> 01:13:50,530 So if you look at the overlap of a single particle state, 928 01:13:50,530 --> 01:13:53,940 with a two particle state, say some double trace operator. 929 01:13:53,940 --> 01:13:55,980 Let me just, to avoid confusion, let 930 01:13:55,980 --> 01:14:08,030 me just use the inside-- use this notation 931 01:14:08,030 --> 01:14:09,610 to see this as a single operator. 932 01:14:15,240 --> 01:14:17,010 So you can see that the overlap we 933 01:14:17,010 --> 01:14:19,950 saw of the single trace operator with this double trace 934 01:14:19,950 --> 01:14:22,190 operator. 935 01:14:22,190 --> 01:14:24,150 And they start off-- this whole thing 936 01:14:24,150 --> 01:14:26,860 is like a three point function. 937 01:14:26,860 --> 01:14:29,616 Just put these two over the same point. 938 01:14:29,616 --> 01:14:31,990 So from our discussion here, is how you connect the Green 939 01:14:31,990 --> 01:14:36,710 functions of order 1 over N. 940 01:14:36,710 --> 01:14:45,580 So this goes to 0, compared to this overlap with itself. 941 01:14:45,580 --> 01:14:53,350 So that means in the N goes to infinity limit, 942 01:14:53,350 --> 01:15:03,350 there is no mixing between what we called single particle 943 01:15:03,350 --> 01:15:12,936 state, single trace, and the multiple trace states. 944 01:15:31,860 --> 01:15:41,190 So the third thing is that now let's 945 01:15:41,190 --> 01:15:56,160 look at the two point function of two double trace operator. 946 01:15:56,160 --> 01:16:08,740 So O1 O2 x, say O1 O2 and y. 947 01:16:08,740 --> 01:16:11,266 OK double trace operators. 948 01:16:11,266 --> 01:16:12,890 So let's look at the two point function 949 01:16:12,890 --> 01:16:14,056 with double trace operators. 950 01:16:31,530 --> 01:16:34,980 So there's an even contribution to this. 951 01:16:34,980 --> 01:16:36,230 So let's include all diagrams. 952 01:16:36,230 --> 01:16:40,240 Also these can all be connected So leading order 953 01:16:40,240 --> 01:16:42,200 is a disconnected diagram, which is essentially 954 01:16:42,200 --> 01:16:51,520 O1 x O1 y and O2 x and O2 y. 955 01:16:51,520 --> 01:16:56,440 Because I have a diagonal next to them. 956 01:16:56,440 --> 01:16:59,780 have And then plus the connected Green 957 01:16:59,780 --> 01:17:05,856 functions, which is order 1 over N squared. 958 01:17:10,390 --> 01:17:13,850 So if you see this leading order contribution, 959 01:17:13,850 --> 01:17:17,120 it's just like two independent particle propagating. 960 01:17:17,120 --> 01:17:20,650 Just like the product of two independent propagators. 961 01:17:20,650 --> 01:17:24,480 So it's sensible to interpret this 962 01:17:24,480 --> 01:17:26,350 as just the propagating of two particles. 963 01:17:26,350 --> 01:17:28,660 So it's sensible to interpret this two particle 964 01:17:28,660 --> 01:17:31,100 state, this double trace operator 965 01:17:31,100 --> 01:17:32,900 as just creating some two particle state. 966 01:17:39,160 --> 01:17:42,293 Yeah, again this goes to zero, in the large N limit. 967 01:17:47,680 --> 01:17:52,920 So I should emphasize when we call this a single particle 968 01:17:52,920 --> 01:17:56,550 state, it's not necessary they really 969 01:17:56,550 --> 01:18:01,180 exist on a shell particle corresponding to this state. 970 01:18:01,180 --> 01:18:04,370 We're just saying that the behavior of these states 971 01:18:04,370 --> 01:18:07,660 can be interpreted as some kind of single particles. 972 01:18:12,506 --> 01:18:16,170 The behavior you can just interpret as single particles. 973 01:18:16,170 --> 01:18:18,810 It's not really necessary they exist 974 01:18:18,810 --> 01:18:21,210 as stable on shell particles. 975 01:18:21,210 --> 01:18:23,900 In certain cases, there might be. 976 01:18:23,900 --> 01:18:25,980 There might be actual particles, actual stable, 977 01:18:25,980 --> 01:18:28,605 on shell particles associated with these kind of states. 978 01:18:28,605 --> 01:18:30,230 But for this interpretation to be true, 979 01:18:30,230 --> 01:18:32,320 it does not have to be. 980 01:18:32,320 --> 01:18:37,970 So in QCD actually sometimes-- so I just say they exist. 981 01:18:37,970 --> 01:18:39,471 So we call them "glueball" state. 982 01:18:42,400 --> 01:18:48,570 For example, in QCD, the analog of this kind of operator 983 01:18:48,570 --> 01:18:52,560 can create some state, which they are short-lived. 984 01:18:52,560 --> 01:18:53,920 They are not long-lived. 985 01:18:53,920 --> 01:18:56,030 They quickly decay. 986 01:18:56,030 --> 01:19:00,180 And so they're typically called the glueball state. 987 01:19:00,180 --> 01:19:04,077 So from now on we just call them glueballs. 988 01:19:04,077 --> 01:19:04,910 Call them glueballs. 989 01:19:09,410 --> 01:19:15,740 So this is one of the first implication-- 990 01:19:15,740 --> 01:19:17,850 the first indication is that they're just 991 01:19:17,850 --> 01:19:20,060 a single particles. 992 01:19:20,060 --> 01:19:21,900 A single trace operator can be interpreted 993 01:19:21,900 --> 01:19:24,150 as creating single particle states, 994 01:19:24,150 --> 01:19:25,990 and the multiple trace operator can 995 01:19:25,990 --> 01:19:28,688 be interpreted as creating multi particle states. 996 01:19:31,870 --> 01:19:38,900 Another second, he said if this glueball operators, so 997 01:19:38,900 --> 01:19:52,966 fluctuations of glueballs suppressed. 998 01:19:59,470 --> 01:20:02,940 So let me explain what this means. 999 01:20:02,940 --> 01:20:08,630 So let me suppose some single trace operator, O, 1000 01:20:08,630 --> 01:20:14,210 which has a non-zero expectation value, 1001 01:20:14,210 --> 01:20:17,360 suppose some state has a non-zero expectation value. 1002 01:20:22,650 --> 01:20:25,420 And then let's look at the variance of this operator, 1003 01:20:25,420 --> 01:20:30,440 the variance of the expectation value, which 1004 01:20:30,440 --> 01:20:36,452 is given by O squared, minus O squared for the fluctuation. 1005 01:20:39,450 --> 01:20:44,860 And this is, by definition, just a collected Green function of O 1006 01:20:44,860 --> 01:20:46,440 squared. 1007 01:20:46,440 --> 01:20:49,280 The is the full O squared. 1008 01:20:49,280 --> 01:20:51,860 This is the disconnected one. 1009 01:20:51,860 --> 01:20:55,560 So this is just a collected part of the O squared. 1010 01:20:55,560 --> 01:20:57,250 And this one, N dependence, we know 1011 01:20:57,250 --> 01:20:59,977 this is order one, order N to the power 0. 1012 01:21:05,080 --> 01:21:09,680 So that means that these to below this order 1013 01:21:09,680 --> 01:21:22,480 N. So the variance of this, compared to the expectation 1014 01:21:22,480 --> 01:21:26,890 value of this operator itself is surprising, the large N limit. 1015 01:21:31,720 --> 01:21:37,050 So essentially, in the large N limit, 1016 01:21:37,050 --> 01:21:41,380 so that means the variance provided by the operator 1017 01:21:41,380 --> 01:21:45,600 itself is 1 over N because of the 0 in the large N limit. 1018 01:21:49,310 --> 01:21:53,010 So assuming that if you have a two point function-- so 1019 01:21:53,010 --> 01:21:58,430 suppose each two point function, so each O2 1020 01:21:58,430 --> 01:22:01,340 have a non-zero expectation value, 1021 01:22:01,340 --> 01:22:05,530 then you can factorize this O1 O2, plus O1 O2. 1022 01:22:05,530 --> 01:22:09,510 So this is disconnected part. 1023 01:22:09,510 --> 01:22:13,290 And again this will be of order one. 1024 01:22:13,290 --> 01:22:17,328 But this part is of order N squared. 1025 01:22:20,930 --> 01:22:25,770 So essentially the disconnected the part is always factorized. 1026 01:22:25,770 --> 01:22:29,860 So the disconnected part is always factorized. 1027 01:22:29,860 --> 01:22:32,367 Yeah this connected part is always small, 1028 01:22:32,367 --> 01:22:33,783 compared to the disconnected part. 1029 01:22:37,820 --> 01:22:39,480 So this is like a classical theory. 1030 01:22:50,450 --> 01:22:52,440 AUDIENCE: We have just five minutes. 1031 01:22:52,440 --> 01:22:54,989 We can proceed before it's 5:00 1032 01:22:54,989 --> 01:22:56,780 HONG LIU: Yeah but it may go to 10 minutes. 1033 01:22:56,780 --> 01:22:59,840 It's just hard to say. 1034 01:22:59,840 --> 01:23:01,790 It's just hard to say. 1035 01:23:06,880 --> 01:23:09,630 Yeah, let me do it next time.