1 00:00:00,090 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:21,300 --> 00:00:25,010 PROFESSOR: OK, let's start. 9 00:00:25,010 --> 00:00:29,350 So today hopefully we will finally 10 00:00:29,350 --> 00:00:31,810 calculate some exponents. 11 00:00:31,810 --> 00:00:34,600 We've been writing, again and again, 12 00:00:34,600 --> 00:00:38,220 how to calculate partition functions for systems, 13 00:00:38,220 --> 00:00:42,560 such as a magnet, by integrating over configurations 14 00:00:42,560 --> 00:00:50,080 of all shapes of a statistical field. 15 00:00:50,080 --> 00:00:58,090 And we have given weights to these configurations that 16 00:00:58,090 --> 00:01:03,720 are constructed as some kind of a function [? l ?] 17 00:01:03,720 --> 00:01:05,563 of these configurations. 18 00:01:08,550 --> 00:01:16,170 And the idea is that presumably, if I could do this, 19 00:01:16,170 --> 00:01:18,680 then I could figure out the singularities that 20 00:01:18,680 --> 00:01:22,140 are possible at a place where, for example, I 21 00:01:22,140 --> 00:01:26,700 go from an unmagnetized to a magnetized case. 22 00:01:26,700 --> 00:01:29,370 Now, one of the first things that we noted 23 00:01:29,370 --> 00:01:35,980 was that in general, I can't solve the types of Hamiltonians 24 00:01:35,980 --> 00:01:37,860 that I would like. 25 00:01:37,860 --> 00:01:41,290 And maybe what I should do is to break it into two 26 00:01:41,290 --> 00:01:45,210 parts, a part that I will treat perturbatively, 27 00:01:45,210 --> 00:01:47,690 and a part-- sorry, a [INAUDIBLE] 28 00:01:47,690 --> 00:01:50,620 part that I can calculate exactly, 29 00:01:50,620 --> 00:01:54,655 and a contribution that I can then treat as a perturbation. 30 00:02:00,660 --> 00:02:03,840 Now, we saw that there were difficulties 31 00:02:03,840 --> 00:02:06,890 if I attempted straightforward perturbation 32 00:02:06,890 --> 00:02:09,039 type of calculations. 33 00:02:09,039 --> 00:02:13,870 And what we did was to replace this 34 00:02:13,870 --> 00:02:18,420 with some kind of a renormalization group approach. 35 00:02:18,420 --> 00:02:20,645 The idea was something like this, 36 00:02:20,645 --> 00:02:24,450 that these statistical field theories that we write down 37 00:02:24,450 --> 00:02:27,950 have been obtained by averaging true microscopic degrees 38 00:02:27,950 --> 00:02:31,780 of freedom over some characteristic landscape. 39 00:02:31,780 --> 00:02:33,940 So this field m certainly does not 40 00:02:33,940 --> 00:02:37,790 have fluctuations that are very short wavelength. 41 00:02:37,790 --> 00:02:41,420 And, for example, if we were to describe things 42 00:02:41,420 --> 00:02:46,980 in the perspective of Fourier components, 43 00:02:46,980 --> 00:02:50,510 presumably the variables that I would have 44 00:02:50,510 --> 00:02:56,470 would have some maximum, q, that is related 45 00:02:56,470 --> 00:02:58,650 to the inverse of the wavelength. 46 00:02:58,650 --> 00:03:03,210 So there is some lambda. 47 00:03:03,210 --> 00:03:08,310 And if I were to in fact Fourier transform my modes in terms 48 00:03:08,310 --> 00:03:15,440 of q, then these modes will be defined [INAUDIBLE] this space. 49 00:03:15,440 --> 00:03:20,520 And for example, my beta is zero. 50 00:03:20,520 --> 00:03:23,940 In the language of Fourier modes would be the part 51 00:03:23,940 --> 00:03:29,610 that I can do exactly, which is the part that is quadratic 52 00:03:29,610 --> 00:03:31,320 and Gaussian. 53 00:03:31,320 --> 00:03:37,250 And the q vectors would be between the interval 0 54 00:03:37,250 --> 00:03:40,060 to whatever this lambda is. 55 00:03:40,060 --> 00:03:42,650 And the kind of thing that I can do exactly 56 00:03:42,650 --> 00:03:44,270 are things that are quadratic. 57 00:03:44,270 --> 00:03:47,490 So I would have m of q squared. 58 00:03:47,490 --> 00:03:51,290 And then some expansion, [INAUDIBLE] 59 00:03:51,290 --> 00:03:55,680 of q, that has a constant plus tq squared 60 00:03:55,680 --> 00:03:59,920 and potentially higher order [INAUDIBLE]. 61 00:03:59,920 --> 00:04:04,730 So this is the Gaussian theory that I can calculate. 62 00:04:04,730 --> 00:04:06,420 Problem with this Gaussian theory 63 00:04:06,420 --> 00:04:10,870 is that it only is meaningful for t positive. 64 00:04:10,870 --> 00:04:14,260 And in order to go to the space where t is negative, 65 00:04:14,260 --> 00:04:18,390 I have to include higher order terms in the magnetization, 66 00:04:18,390 --> 00:04:21,089 and those are non-perturbative. 67 00:04:21,089 --> 00:04:25,250 And for example, if I go back to the description in real space, 68 00:04:25,250 --> 00:04:29,370 I was writing something like um to the fourth 69 00:04:29,370 --> 00:04:32,780 plus higher order terms for the expansion of this u. 70 00:04:35,680 --> 00:04:41,480 When we attempted to do straightforward perturbative 71 00:04:41,480 --> 00:04:45,060 calculations, we encountered some singularities. 72 00:04:45,060 --> 00:04:48,110 And the perturbation didn't quite make sense. 73 00:04:48,110 --> 00:04:51,020 So we decided to combine that with the idea 74 00:04:51,020 --> 00:04:53,360 of renormalization group. 75 00:04:53,360 --> 00:04:58,980 The idea there was to basically, rather than integrate over 76 00:04:58,980 --> 00:05:02,590 all modes, to subdivide the modes into two 77 00:05:02,590 --> 00:05:08,090 classes, the modes that are long wavelength 78 00:05:08,090 --> 00:05:09,876 and I would like to keep, and I'll 79 00:05:09,876 --> 00:05:15,026 call that m tilde, and the modes that are sitting out here 80 00:05:15,026 --> 00:05:19,150 that I'm not interested in because they give rise 81 00:05:19,150 --> 00:05:23,800 to no singularities that I would like to get rid of. 82 00:05:23,800 --> 00:05:29,820 So my integration over all set of configurations 83 00:05:29,820 --> 00:05:33,840 is really an integration over both this m tilde 84 00:05:33,840 --> 00:05:36,670 and the sigma. 85 00:05:36,670 --> 00:05:42,590 And if I regard m tilde as a span over wave numbers 86 00:05:42,590 --> 00:05:47,030 to either be m tilde or sigma, I can basically 87 00:05:47,030 --> 00:05:49,960 write this as m tilde plus sigma, 88 00:05:49,960 --> 00:05:52,180 and this is m tilde plus sigma also. 89 00:05:56,280 --> 00:06:01,500 So this is just a rewriting of the partition function 90 00:06:01,500 --> 00:06:06,990 where I have just changed the names of the modes. 91 00:06:06,990 --> 00:06:10,860 Now, the first step in the renormalization group 92 00:06:10,860 --> 00:06:15,740 is the coarse graining, which is to average out fluctuations 93 00:06:15,740 --> 00:06:22,310 that have scale between a, and let's say this in Fourier space 94 00:06:22,310 --> 00:06:26,030 is lambda over b, in real space would be b times whatever 95 00:06:26,030 --> 00:06:32,440 your original base scale was for average. 96 00:06:32,440 --> 00:06:35,100 So getting rid of those modes would 97 00:06:35,100 --> 00:06:38,395 amount to basically changing the scale over which 98 00:06:38,395 --> 00:06:41,630 you're averaging by a factor of b. 99 00:06:41,630 --> 00:06:45,920 Once I do that, if I can do the integration, 100 00:06:45,920 --> 00:06:48,880 what I will be left if I integrate over sigma 101 00:06:48,880 --> 00:06:53,260 is just an integral over m tilde. 102 00:06:53,260 --> 00:06:55,570 OK? 103 00:06:55,570 --> 00:06:59,780 Now, what would be the form of this integration? 104 00:06:59,780 --> 00:07:01,980 The result. 105 00:07:01,980 --> 00:07:05,640 Well, first of all, if I take the Gaussian 106 00:07:05,640 --> 00:07:10,760 and separate it out between zero to lambda over b and lambda 107 00:07:10,760 --> 00:07:15,330 over b2 lambda and integrate over the modes between lambda 108 00:07:15,330 --> 00:07:19,380 over b and lambda, just as if I had the Gaussian, then 109 00:07:19,380 --> 00:07:21,600 I would get essentially the contribution 110 00:07:21,600 --> 00:07:24,700 of the logarithm of the determinants of all 111 00:07:24,700 --> 00:07:28,420 of these Gaussian types of variances. 112 00:07:28,420 --> 00:07:32,120 So there will be a contribution to the free energy that 113 00:07:32,120 --> 00:07:37,500 is effectively independent of m tilde will depend 114 00:07:37,500 --> 00:07:41,627 on the rescaling factor that you are looking at. 115 00:07:41,627 --> 00:07:42,460 But it's a constant. 116 00:07:42,460 --> 00:07:45,920 It doesn't depend on the different configurations 117 00:07:45,920 --> 00:07:48,720 of the field m tilde. 118 00:07:48,720 --> 00:07:51,060 The other part of the Gaussian-- so essentially, 119 00:07:51,060 --> 00:07:53,270 I wrote the Gaussian as 0 to lambda 120 00:07:53,270 --> 00:07:55,970 over b and lambda over b2 lambda. 121 00:07:55,970 --> 00:07:59,800 The part that is 0 over lambda b will simply remain, 122 00:07:59,800 --> 00:08:02,110 so I will have [? beta 8 ?] 0. 123 00:08:02,110 --> 00:08:05,760 That now depends only on these m tildes. 124 00:08:08,720 --> 00:08:14,130 Well, what do I have to do with this term? 125 00:08:14,130 --> 00:08:19,350 So it's an integration over sigma that has to be performed. 126 00:08:19,350 --> 00:08:22,340 I did the integration by taking out this them 127 00:08:22,340 --> 00:08:25,080 as if it was a Gaussian. 128 00:08:25,080 --> 00:08:28,820 So effectively, the result of the remaining integration 129 00:08:28,820 --> 00:08:33,110 is the average of e to the minus u. 130 00:08:33,110 --> 00:08:38,390 And when I take it to the log, I will get plus log of e 131 00:08:38,390 --> 00:08:44,690 to the minus u, which is a function of m tilde and sigma, 132 00:08:44,690 --> 00:08:50,890 where I have integrated out the modes that 133 00:08:50,890 --> 00:08:53,960 are out here, the sigmas. 134 00:08:53,960 --> 00:08:56,860 So it's only a function of m tilde. 135 00:08:56,860 --> 00:08:59,980 They have been integrated out using a Gaussian weight, 136 00:08:59,980 --> 00:09:01,580 such as the one that I have over here. 137 00:09:06,700 --> 00:09:10,180 So that's formally exact. 138 00:09:10,180 --> 00:09:12,270 But it hasn't given me any insights 139 00:09:12,270 --> 00:09:15,590 because I don't know what that entity is. 140 00:09:15,590 --> 00:09:18,160 What I can do with that entity is 141 00:09:18,160 --> 00:09:22,040 to make an expansion powers of u. 142 00:09:22,040 --> 00:09:26,160 So I will have a minus the average of u. 143 00:09:28,760 --> 00:09:32,450 And then the next term would be the variance of u. 144 00:09:32,450 --> 00:09:39,870 So I will have the average of u squared, average of u squared, 145 00:09:39,870 --> 00:09:42,530 and then higher order terms. 146 00:09:42,530 --> 00:09:49,090 So basically, this term can be expanded as a power series 147 00:09:49,090 --> 00:09:51,250 as I have indicated. 148 00:09:51,250 --> 00:09:56,640 And again, just to make sure, these averages 149 00:09:56,640 --> 00:09:59,750 are performed with this Gaussian weight. 150 00:09:59,750 --> 00:10:03,090 And in particular, we've seen that when we have a Gaussian 151 00:10:03,090 --> 00:10:11,300 weight, the different components and the different q values 152 00:10:11,300 --> 00:10:13,690 are independent of each other. 153 00:10:13,690 --> 00:10:19,130 So I get here a delta alpha beta, I get a delta of q plus q 154 00:10:19,130 --> 00:10:25,730 prime, and I will get t plus k q squared 155 00:10:25,730 --> 00:10:28,190 and potentially higher order powers of q 156 00:10:28,190 --> 00:10:30,570 that will appear in this series. 157 00:10:34,480 --> 00:10:37,360 Now, we kind of started developing 158 00:10:37,360 --> 00:10:40,760 a diagrammatic perspective on all of this. 159 00:10:40,760 --> 00:10:45,140 Something is m to the fourth, since it 160 00:10:45,140 --> 00:10:49,980 was the dot product of two factors of m squared, 161 00:10:49,980 --> 00:10:53,380 we demonstrate it as a graph such as this. 162 00:10:53,380 --> 00:10:56,430 And we also introduced a convention 163 00:10:56,430 --> 00:11:03,280 where there solid lines would correspond to m tilde. 164 00:11:03,280 --> 00:11:09,230 Let's say wavy lines would correspond to sigma. 165 00:11:09,230 --> 00:11:12,560 And essentially, what I have to do 166 00:11:12,560 --> 00:11:17,630 is to write this object according 167 00:11:17,630 --> 00:11:22,780 to this, where each factor of m is replaced 168 00:11:22,780 --> 00:11:27,410 by two factors, which is the sum of this entity and that entity 169 00:11:27,410 --> 00:11:29,690 diagrammatically. 170 00:11:29,690 --> 00:11:34,660 So that's two to the four, or 16 different possibilities 171 00:11:34,660 --> 00:11:38,740 that I could have once I expand this. 172 00:11:38,740 --> 00:11:42,430 And what was the answer that we got for the first term 173 00:11:42,430 --> 00:11:44,170 in the series? 174 00:11:44,170 --> 00:11:51,450 So if I take u0 average, the kind of diagrams that I can get 175 00:11:51,450 --> 00:11:55,180 is essentially keeping this entity as it is. 176 00:11:55,180 --> 00:12:02,010 So essentially, I will get the original potential that I have. 177 00:12:02,010 --> 00:12:04,380 Rather than m to the fourth, I will simply 178 00:12:04,380 --> 00:12:07,010 have the equivalent m tilde to the fourth. 179 00:12:07,010 --> 00:12:09,780 So basically, diagrammatically this 180 00:12:09,780 --> 00:12:13,050 would correspond to this entity. 181 00:12:13,050 --> 00:12:15,890 There was a whole bunch of things that cancel out 182 00:12:15,890 --> 00:12:21,260 to zero in the diagram that I had with only one leg 183 00:12:21,260 --> 00:12:25,620 when I took the average because I had a leg by itself, 184 00:12:25,620 --> 00:12:29,370 which would make it an odd average, would give me zero. 185 00:12:29,370 --> 00:12:32,850 So I didn't have to put any of these. 186 00:12:32,850 --> 00:12:38,050 And then I had diagrams where I had 187 00:12:38,050 --> 00:12:42,030 two of the lines replaced by wavy lines. 188 00:12:42,030 --> 00:12:47,610 And so then I would get a contribution to u. 189 00:12:47,610 --> 00:12:53,545 There was a factor of 4n plus-- sorry, 2n plus 4. 190 00:12:57,690 --> 00:13:01,624 That came from diagrams in which I 191 00:13:01,624 --> 00:13:06,130 took two of the legs that were together, 192 00:13:06,130 --> 00:13:11,560 and the other two I made wavy, and I joined them together. 193 00:13:11,560 --> 00:13:13,810 And essentially, I had the choice 194 00:13:13,810 --> 00:13:17,080 of picking this pair of legs or that pair of legs, 195 00:13:17,080 --> 00:13:20,330 so that gave me a factor of two. 196 00:13:20,330 --> 00:13:24,020 And something that we will see again and again, whenever 197 00:13:24,020 --> 00:13:27,430 we have a loop that goes around by itself, 198 00:13:27,430 --> 00:13:30,470 it corresponds to something like a delta alpha alpha, 199 00:13:30,470 --> 00:13:35,280 which, when you sum over alpha, will give you a factor of n. 200 00:13:35,280 --> 00:13:38,320 The other contribution, the four, 201 00:13:38,320 --> 00:13:47,260 came from diagrams in which I had 202 00:13:47,260 --> 00:13:50,680 two wavy lines on different branches. 203 00:13:50,680 --> 00:13:54,060 And since they came originally from different branches, 204 00:13:54,060 --> 00:13:58,490 there wasn't a repeated helix to sum and give me a factor of n. 205 00:13:58,490 --> 00:14:00,560 So I just have as a factor of two 206 00:14:00,560 --> 00:14:04,380 from choice of one branch or the other branch. 207 00:14:04,380 --> 00:14:06,780 So that was a factor of four. 208 00:14:06,780 --> 00:14:10,820 And then associated with each one of these diagrams, 209 00:14:10,820 --> 00:14:16,680 there was then an integration over the index, k, 210 00:14:16,680 --> 00:14:20,050 that characterized these m tildes, 211 00:14:20,050 --> 00:14:23,460 in fact, the sigmas that had been integrated over. 212 00:14:23,460 --> 00:14:27,780 So I would have an integral from lambda over b2 lambda. 213 00:14:27,780 --> 00:14:32,210 Let's call that dbk 2 pi to the d 1 214 00:14:32,210 --> 00:14:35,233 over the variance, which is what I have here, 215 00:14:35,233 --> 00:14:37,275 t plus k, k squared, [INAUDIBLE]. 216 00:14:42,970 --> 00:14:46,940 There are diagrams then with three wavy lines, which again 217 00:14:46,940 --> 00:14:49,770 gave me zero because the average of three-- 218 00:14:49,770 --> 00:14:53,799 average of an odd number with a Gaussian weight is zero. 219 00:14:53,799 --> 00:14:55,340 And then there were a bunch of things 220 00:14:55,340 --> 00:15:00,210 that would correspond to all legs being wavy. 221 00:15:00,210 --> 00:15:02,590 There was something like this, and there 222 00:15:02,590 --> 00:15:05,090 was something like this. 223 00:15:05,090 --> 00:15:09,980 And basically, I didn't really have to calculate them. 224 00:15:09,980 --> 00:15:12,900 So I just wrote the answer to those things 225 00:15:12,900 --> 00:15:17,530 as being a contribution to the free energy 226 00:15:17,530 --> 00:15:20,380 and overall constant, such as the constant 227 00:15:20,380 --> 00:15:24,720 that I have over here, but not at the next order in u, 228 00:15:24,720 --> 00:15:26,343 but independent of the configurations. 229 00:15:29,420 --> 00:15:33,830 So this was straightforward perturbation. 230 00:15:33,830 --> 00:15:35,730 I forgot something very important 231 00:15:35,730 --> 00:15:49,070 here, which is that this entire coefficient was also 232 00:15:49,070 --> 00:15:53,770 coupled to these solid lines, whose meaning is that it 233 00:15:53,770 --> 00:16:01,540 is an integral over q 2 pi to the d m tilde of q squared, 234 00:16:01,540 --> 00:16:04,880 where the waves and numbers that are sitting 235 00:16:04,880 --> 00:16:09,133 on these solid lines naturally run from 0 to lambda. 236 00:16:14,340 --> 00:16:26,370 So we can see that if I were to add this to what I have above, 237 00:16:26,370 --> 00:16:30,080 I see that my z has now been written 238 00:16:30,080 --> 00:16:32,940 as an integral over these modes that I'm 239 00:16:32,940 --> 00:16:41,080 keeping of a new weight that I will call beta h tilde, 240 00:16:41,080 --> 00:16:42,290 depending on these m tildes. 241 00:16:46,350 --> 00:16:53,240 Where this beta h tilde is, first of all, these terms 242 00:16:53,240 --> 00:16:55,180 that are proportional. 243 00:16:55,180 --> 00:16:57,940 That's with a v here also. 244 00:16:57,940 --> 00:17:01,670 To contributions of the free energy coming 245 00:17:01,670 --> 00:17:04,420 from the modes that I have integrated out, 246 00:17:04,420 --> 00:17:07,869 either at the zero order or at the first order so far. 247 00:17:11,300 --> 00:17:22,329 I have the u, exactly the same u as I had before, 248 00:17:22,329 --> 00:17:24,089 but now acting on m tilde. 249 00:17:24,089 --> 00:17:27,230 So four factors of m tilde. 250 00:17:27,230 --> 00:17:34,330 The only thing that happened is that the Gaussian contribution 251 00:17:34,330 --> 00:17:42,230 now running from 0 to lambda over b, 252 00:17:42,230 --> 00:17:49,650 that is proportional to m tilde of q squared, 253 00:17:49,650 --> 00:17:54,490 is now still a series, such as the one that I had before, 254 00:17:54,490 --> 00:17:57,310 where the coefficient that was a constant has changed. 255 00:18:00,080 --> 00:18:02,020 All the other terms in the series, 256 00:18:02,020 --> 00:18:04,600 the term that is proportional to q squared, 257 00:18:04,600 --> 00:18:09,060 q to the fourth, et cetera, are left exactly as before. 258 00:18:11,670 --> 00:18:19,600 So what happened is that this beta h tilde pretty much looks 259 00:18:19,600 --> 00:18:23,010 like the beta h that I started with, 260 00:18:23,010 --> 00:18:26,030 with the only difference being that t tilde is 261 00:18:26,030 --> 00:18:30,110 t plus essentially what I have over there, 262 00:18:30,110 --> 00:18:38,190 4u n plus 2 integral lambda over b2 lambda 263 00:18:38,190 --> 00:18:46,120 ddk 2 pi to the d, 1 over t plus k, k squared, and so forth. 264 00:18:49,530 --> 00:18:53,000 But quite importantly, the parameter 265 00:18:53,000 --> 00:18:57,490 that I would associate with coefficient of q squared 266 00:18:57,490 --> 00:18:59,690 is left unchanged. 267 00:18:59,690 --> 00:19:02,580 If I had a coefficient of q to the fourth, 268 00:19:02,580 --> 00:19:05,060 its coefficient would be unchanged. 269 00:19:05,060 --> 00:19:07,790 And I have a coefficient for u. 270 00:19:07,790 --> 00:19:11,810 Its coefficient is unchanged also. 271 00:19:11,810 --> 00:19:14,370 So the only thing that happened is 272 00:19:14,370 --> 00:19:20,800 that the parameter that corresponded to t got modified. 273 00:19:20,800 --> 00:19:24,160 And you actually should recognize this 274 00:19:24,160 --> 00:19:28,250 as the inverse susceptibility, if I 275 00:19:28,250 --> 00:19:33,290 were to integrate all the way from 0 to lambda. 276 00:19:33,290 --> 00:19:37,630 And when we did that, this contribution was singular. 277 00:19:37,630 --> 00:19:40,265 And that's why straightforward perturbation theory 278 00:19:40,265 --> 00:19:41,770 didn't make sense. 279 00:19:41,770 --> 00:19:44,040 But now we are not integrating to 0, 280 00:19:44,040 --> 00:19:46,160 which would have given the singularity. 281 00:19:46,160 --> 00:19:48,560 We are just integrating over the shell 282 00:19:48,560 --> 00:19:52,680 that I have indicated outside. 283 00:19:52,680 --> 00:20:02,920 So this step was the first step of renormalization group 284 00:20:02,920 --> 00:20:04,450 that we call coarse graining. 285 00:20:11,500 --> 00:20:16,080 But rg had two other steps. 286 00:20:16,080 --> 00:20:19,270 That was rescale. 287 00:20:19,270 --> 00:20:23,180 Basically, the theory that I have 288 00:20:23,180 --> 00:20:27,090 has a cut-off that is lambda over b. 289 00:20:27,090 --> 00:20:31,290 So it looks grainier in real space. 290 00:20:31,290 --> 00:20:35,250 So what I can do in real space is to shrink it. 291 00:20:35,250 --> 00:20:40,170 In Fourier space, I have to blow up my momenta. 292 00:20:40,170 --> 00:20:43,770 So essentially, whenever I see q, 293 00:20:43,770 --> 00:20:47,730 I replace it with the inverse q prime so 294 00:20:47,730 --> 00:20:53,640 that q prime, that is bq, runs from zero to lambda, restoring 295 00:20:53,640 --> 00:20:57,180 the cut-off that I had originally. 296 00:20:57,180 --> 00:21:05,740 And the next step was to renormalize, 297 00:21:05,740 --> 00:21:15,520 which amounted to replacing the field m tilde with a new field 298 00:21:15,520 --> 00:21:20,362 m prime after multiplying or rescaling by a factor of z 299 00:21:20,362 --> 00:21:21,070 to be determined. 300 00:21:24,680 --> 00:21:28,720 Now, this amounts to simple dimensional analysis. 301 00:21:28,720 --> 00:21:33,360 So I go back into my equation, and whenever I see q, 302 00:21:33,360 --> 00:21:36,510 I replace it with b inverse q prime. 303 00:21:36,510 --> 00:21:40,740 So from the integration, I get a factor of b to the minus d, 304 00:21:40,740 --> 00:21:45,760 multiplying t tilde, replace m tilde by z times m prime. 305 00:21:45,760 --> 00:21:48,140 So that's two factors of z. 306 00:21:48,140 --> 00:21:54,150 So what I get is that t prime is z squared b to the minus d, 307 00:21:54,150 --> 00:21:56,830 this t tilde that I have indicated above. 308 00:22:00,290 --> 00:22:05,960 Now, k prime is also something that 309 00:22:05,960 --> 00:22:07,570 appears in the Gaussian term. 310 00:22:07,570 --> 00:22:08,970 So it has a z squared. 311 00:22:08,970 --> 00:22:11,310 It came from two factors of m. 312 00:22:11,310 --> 00:22:15,870 But because it had an additional factor of q squared 313 00:22:15,870 --> 00:22:23,720 rather than b to the minus d, it is b to the minus d minus 2. 314 00:22:23,720 --> 00:22:28,740 And I can do the same analysis for higher order terms going 315 00:22:28,740 --> 00:22:32,150 with higher powers of q in the expansion that 316 00:22:32,150 --> 00:22:33,240 appears in the Gaussian. 317 00:22:35,830 --> 00:22:38,020 But then we get to the non-linear terms, 318 00:22:38,020 --> 00:22:42,550 and the first linearity that we have kept is this u prime. 319 00:22:42,550 --> 00:22:47,170 And what we see is it goes with four factors of m. 320 00:22:47,170 --> 00:22:49,960 So there will be z to the fourth. 321 00:22:49,960 --> 00:22:53,400 If I write things in Fourier space, 322 00:22:53,400 --> 00:22:56,210 m to the fourth in real space in Fourier space 323 00:22:56,210 --> 00:23:00,410 would involve m of g1, m of q2, m of q3. 324 00:23:00,410 --> 00:23:04,660 And the fourth m, that is minus q1, minus q2, minus q3. 325 00:23:04,660 --> 00:23:06,740 But there will be three integrations 326 00:23:06,740 --> 00:23:10,730 over q, which gives me three factors of b to the minus 3. 327 00:23:14,280 --> 00:23:17,500 So these are pretty much exactly what we had already 328 00:23:17,500 --> 00:23:23,850 seen for the Gaussian model-- forgot the k-- except that we 329 00:23:23,850 --> 00:23:27,840 replaced this t that was appearing for the Gaussian 330 00:23:27,840 --> 00:23:30,480 model with t tilde which is what I have up here. 331 00:23:34,440 --> 00:23:38,860 Now, you have to choose z such that the theory 332 00:23:38,860 --> 00:23:45,300 looks as much as possible as the original way that I had. 333 00:23:45,300 --> 00:23:48,590 And as I mentioned, our anchoring point 334 00:23:48,590 --> 00:23:50,730 would be the Gaussian. 335 00:23:50,730 --> 00:23:52,680 So for the Gaussian model, we saw 336 00:23:52,680 --> 00:23:56,240 that the appropriate choice, so that ultimately we 337 00:23:56,240 --> 00:24:01,240 were left with the right number of relevant directions 338 00:24:01,240 --> 00:24:05,820 was to set this combination to 1, which 339 00:24:05,820 --> 00:24:10,250 means that I have to choose z to be 340 00:24:10,250 --> 00:24:12,270 v to the power of 1 plus d over 2. 341 00:24:17,522 --> 00:24:22,300 Now, once I choose that factor for z, 342 00:24:22,300 --> 00:24:24,430 everything else becomes determined. 343 00:24:24,430 --> 00:24:30,930 This clearly has two factors of b with respect to the original. 344 00:24:30,930 --> 00:24:32,575 So this becomes b squared. 345 00:24:35,890 --> 00:24:39,570 This you have to do a little bit of work. 346 00:24:39,570 --> 00:24:42,230 Z to the fourth would be b to the 4 347 00:24:42,230 --> 00:24:47,150 plus 2d, then minus 3d gives me b to the 4 minus d. 348 00:24:50,480 --> 00:24:55,460 And I can similarly determine what the dimensions would 349 00:24:55,460 --> 00:25:01,220 be for additional terms that appear in the Gaussian, 350 00:25:01,220 --> 00:25:03,855 as well as additional nonlinearities that 351 00:25:03,855 --> 00:25:04,910 could appear here. 352 00:25:04,910 --> 00:25:09,600 All of them, by this analysis, I can assign some power of b. 353 00:25:14,220 --> 00:25:19,680 So this completes the rg in the sense 354 00:25:19,680 --> 00:25:24,340 that at least at this order in perturbation theory, 355 00:25:24,340 --> 00:25:27,420 I started with my original theory, 356 00:25:27,420 --> 00:25:30,660 and I see how the parameters of the new theory 357 00:25:30,660 --> 00:25:35,490 are obtained if I were to rescale and renormalize 358 00:25:35,490 --> 00:25:36,691 by this factor of b. 359 00:25:39,280 --> 00:25:43,600 Now, we did one thing else, which is quite common, 360 00:25:43,600 --> 00:25:46,410 which is rather than choosing factors 361 00:25:46,410 --> 00:25:53,780 like b equals to 2 or 3, making b to be infinitesimally small, 362 00:25:53,780 --> 00:25:58,740 at least on the picture that I have over there. 363 00:25:58,740 --> 00:26:05,300 What I'm doing is I'm making this b very close 364 00:26:05,300 --> 00:26:08,800 to 1, which means that effectively I'm 365 00:26:08,800 --> 00:26:10,470 putting the modes that I'm getting 366 00:26:10,470 --> 00:26:15,030 rid of in a tiny shell around lambda. 367 00:26:17,710 --> 00:26:24,350 So I have chosen b to be slightly larger than 1 368 00:26:24,350 --> 00:26:26,720 by an amount delta l. 369 00:26:26,720 --> 00:26:29,460 And I expect that all of the parameters 370 00:26:29,460 --> 00:26:33,180 will also change very slightly, such 371 00:26:33,180 --> 00:26:38,430 that this t prime evaluated at scale v 372 00:26:38,430 --> 00:26:42,660 would be what I had originally, plus something 373 00:26:42,660 --> 00:26:45,640 that vanishes as delta l goes to zero 374 00:26:45,640 --> 00:26:50,510 and presumably is linear in delta l dt by dl. 375 00:26:50,510 --> 00:26:56,741 And similarly, I can do the same thing for u 376 00:26:56,741 --> 00:27:01,540 and all the other parameters of the theory. 377 00:27:01,540 --> 00:27:08,300 Once I do that, these jumps from one parameter 378 00:27:08,300 --> 00:27:14,230 to another parameter can be translated into flows. 379 00:27:14,230 --> 00:27:21,270 And, for example, dt by dl gets a contribution 380 00:27:21,270 --> 00:27:25,350 from writing b squared as 1 plus 2 delta l. 381 00:27:25,350 --> 00:27:30,310 That is proportional to 2 times t. 382 00:27:30,310 --> 00:27:32,590 And then there's another contribution 383 00:27:32,590 --> 00:27:35,020 that is order of delta l. 384 00:27:35,020 --> 00:27:39,750 Clearly, if b equals to 1, this integral would vanish. 385 00:27:39,750 --> 00:27:43,580 So if b is very close to 1, this integral 386 00:27:43,580 --> 00:27:45,920 is off the order of delta l. 387 00:27:45,920 --> 00:27:50,560 And what it is is just evaluating the integrand when 388 00:27:50,560 --> 00:27:55,910 k equals to lambda at the shell, and then multiplying 389 00:27:55,910 --> 00:27:58,760 by the volume of that shell, which 390 00:27:58,760 --> 00:28:01,450 is the surface area times the thickness. 391 00:28:01,450 --> 00:28:05,430 So I will get from here a contribution order of delta l. 392 00:28:05,430 --> 00:28:12,270 I have divided through by delta l, which is 4u m plus 2 1 393 00:28:12,270 --> 00:28:17,650 over t plus k lambda squared [INAUDIBLE] integrand. 394 00:28:17,650 --> 00:28:22,460 And then I have the surface area divided by 2 pi to the d 395 00:28:22,460 --> 00:28:25,390 that we have always called kd. 396 00:28:25,390 --> 00:28:28,970 And then lambda to the d is the product 397 00:28:28,970 --> 00:28:31,710 of lambda to the d minus 1 and lambda 398 00:28:31,710 --> 00:28:34,390 delta l, which comes from the thickness. 399 00:28:34,390 --> 00:28:36,860 The delta l I have taken out. 400 00:28:36,860 --> 00:28:40,960 And this whole thing is the order of u contribution. 401 00:28:40,960 --> 00:28:43,100 And then they had a term that is du 402 00:28:43,100 --> 00:28:48,920 by dl, which is 4 minus d times u. 403 00:28:48,920 --> 00:28:57,070 So this is the result of doing this perturbative rg 404 00:28:57,070 --> 00:28:59,660 to the lowest order in this parameter u. 405 00:29:02,970 --> 00:29:08,790 Now, these things are really the important parameters. 406 00:29:08,790 --> 00:29:11,970 There will be other parameters that I have not specifically 407 00:29:11,970 --> 00:29:13,120 written down. 408 00:29:13,120 --> 00:29:16,390 And next lecture, we will deal with all of them. 409 00:29:16,390 --> 00:29:18,900 But let's focus on these two. 410 00:29:18,900 --> 00:29:21,180 So I have one parameter, which is 411 00:29:21,180 --> 00:29:26,270 t, the other parameter, which is u. 412 00:29:26,270 --> 00:29:34,570 But u can only be positive for the theory to make sense. 413 00:29:34,570 --> 00:29:37,930 I said that originally the Gaussian theory only makes 414 00:29:37,930 --> 00:29:42,860 sense if t is positive because once t becomes negative, 415 00:29:42,860 --> 00:29:47,470 then the weight gets shifted to large values of m. 416 00:29:47,470 --> 00:29:48,870 It is unphysical. 417 00:29:48,870 --> 00:29:52,220 So for physicalness of the Gaussian theory, 418 00:29:52,220 --> 00:29:57,960 I need to confine myself to the t positive plane. 419 00:29:57,960 --> 00:30:01,920 Now that I have u, I can have t that is negative 420 00:30:01,920 --> 00:30:04,635 and um to the fourth, as long as u positive, 421 00:30:04,635 --> 00:30:08,910 will make the weight well behaved. 422 00:30:08,910 --> 00:30:12,900 So this entire plane is now accessible. 423 00:30:12,900 --> 00:30:15,570 Within this plane, there is a point 424 00:30:15,570 --> 00:30:18,480 which corresponds to a fixed point, a point 425 00:30:18,480 --> 00:30:22,550 that if I'm at that location, then the parameters no longer 426 00:30:22,550 --> 00:30:23,410 change. 427 00:30:23,410 --> 00:30:28,230 Clearly, if u does not change, u at the fixed point should be 0. 428 00:30:28,230 --> 00:30:31,130 If u at the fixed point is 0 and t does not change, 429 00:30:31,130 --> 00:30:33,000 t at the fixed point is 0. 430 00:30:33,000 --> 00:30:36,760 So this is the fixed point. 431 00:30:36,760 --> 00:30:40,620 Since I'm looking at a two-dimensional projection, 432 00:30:40,620 --> 00:30:44,830 there will be two eigendirections associated 433 00:30:44,830 --> 00:30:47,920 with moving away from this fixed point. 434 00:30:47,920 --> 00:30:52,180 If I stick with the axis where u is 0, 435 00:30:52,180 --> 00:30:54,650 you can see that u will stay 0. 436 00:30:54,650 --> 00:30:57,980 But then dt by dl is 2t. 437 00:30:57,980 --> 00:31:01,430 So if I'm on the axis that u equals to 0, 438 00:31:01,430 --> 00:31:03,590 I will stay on this axis. 439 00:31:03,590 --> 00:31:06,540 So that's one of my eigendirections. 440 00:31:06,540 --> 00:31:09,850 And along this eigendirection, I will 441 00:31:09,850 --> 00:31:15,080 be flowing out with an eigenvalue of 2. 442 00:31:18,110 --> 00:31:24,380 Now, in general however, let's say if I go to t equals to 0, 443 00:31:24,380 --> 00:31:28,860 you can see that if t is 0, but u positive dt by dl 444 00:31:28,860 --> 00:31:30,330 is positive. 445 00:31:30,330 --> 00:31:33,640 So basically, the u direction you 446 00:31:33,640 --> 00:31:39,660 will be going if you start on the t equals to 0 axis, 447 00:31:39,660 --> 00:31:42,000 you will generate a positive t. 448 00:31:42,000 --> 00:31:44,850 And the typical flows that you would 449 00:31:44,850 --> 00:31:47,480 have would be in this direction. 450 00:31:47,480 --> 00:31:49,960 Actually, I should draw it with a different color. 451 00:31:49,960 --> 00:31:54,070 So quite generically, the flows are like this. 452 00:31:57,740 --> 00:32:08,960 But there is a direction along which the flow is preserved. 453 00:32:08,960 --> 00:32:11,330 So there is a straight line. 454 00:32:11,330 --> 00:32:14,530 This straight line you can calculate by setting dt 455 00:32:14,530 --> 00:32:20,136 by dl divided by du by dl to be the ratio of t over u. 456 00:32:20,136 --> 00:32:23,650 You can very easily find that it corresponds 457 00:32:23,650 --> 00:32:26,790 to a line of t being proportional 458 00:32:26,790 --> 00:32:28,940 to u with a negative slope. 459 00:32:28,940 --> 00:32:31,350 And the eigenvalue along that direction 460 00:32:31,350 --> 00:32:34,540 is determined by 4 minus d. 461 00:32:34,540 --> 00:32:37,640 So that the picture that I have actually drawn for you 462 00:32:37,640 --> 00:32:42,140 here corresponds to dimensions greater than four. 463 00:32:42,140 --> 00:32:47,000 In dimensions greater than four along this other direction, 464 00:32:47,000 --> 00:32:49,940 you will be flowing towards the fixed point. 465 00:32:49,940 --> 00:32:52,790 And in general, the flows look something like this. 466 00:33:00,980 --> 00:33:02,530 So what does that mean? 467 00:33:02,530 --> 00:33:05,490 Again, the whole thing that we wrote down 468 00:33:05,490 --> 00:33:07,400 was supposed to describe something 469 00:33:07,400 --> 00:33:10,770 like a magnet at some temperature. 470 00:33:10,770 --> 00:33:13,910 So when I fix my temperature of the magnet, 471 00:33:13,910 --> 00:33:18,680 I presumably reside at some particular point 472 00:33:18,680 --> 00:33:20,800 on this diagram. 473 00:33:20,800 --> 00:33:24,330 Let's say in the phase that is up here, 474 00:33:24,330 --> 00:33:28,860 eventually I can see that I go to large t and u goes to 0. 475 00:33:28,860 --> 00:33:31,845 So the eventual weight is very much like a Gaussian, 476 00:33:31,845 --> 00:33:34,760 e to the tm squared over 2. 477 00:33:34,760 --> 00:33:38,030 So this is essentially independent patches 478 00:33:38,030 --> 00:33:42,360 of the system randomly pointing to different directions. 479 00:33:42,360 --> 00:33:46,560 If I change my system to have a lower temperature, 480 00:33:46,560 --> 00:33:49,720 I will be looking at a point such as this. 481 00:33:49,720 --> 00:33:51,200 As I lower the temperature, I will 482 00:33:51,200 --> 00:33:54,600 be looking at some other point presumably. 483 00:33:54,600 --> 00:33:56,880 But all of these points that correspond 484 00:33:56,880 --> 00:33:59,630 to lowering temperatures, if I also 485 00:33:59,630 --> 00:34:05,200 now look at increasing land scale, will flow up here. 486 00:34:05,200 --> 00:34:09,340 Presumably, if I go below tc, I will 487 00:34:09,340 --> 00:34:13,800 be flowing in the other direction, where t is negative, 488 00:34:13,800 --> 00:34:16,400 and then the u is needed for stability, 489 00:34:16,400 --> 00:34:19,139 which means that I have to spontaneously choose 490 00:34:19,139 --> 00:34:21,880 a direction in which I order things. 491 00:34:21,880 --> 00:34:27,730 So the benefit of doing this renormalization and this study 492 00:34:27,730 --> 00:34:32,530 was that in the absence of u, I could not 493 00:34:32,530 --> 00:34:36,350 achieve the low temperature part of the system. 494 00:34:36,350 --> 00:34:40,000 With the addition of u, I can describe both sides, 495 00:34:40,000 --> 00:34:44,150 and I can see on the rescaling which set of points 496 00:34:44,150 --> 00:34:46,880 go to what is the analog of the high temperature, which 497 00:34:46,880 --> 00:34:50,650 set of points go to what is the analog of low temperature. 498 00:34:50,650 --> 00:34:53,750 And the point that corresponds to the transition 499 00:34:53,750 --> 00:34:58,390 between the two is on the basing of attraction of the Gaussian 500 00:34:58,390 --> 00:35:01,360 fixed point that is asymptotically, 501 00:35:01,360 --> 00:35:04,730 the theory would be described by just gradient of m squared. 502 00:35:07,260 --> 00:35:11,130 But this picture does not work if I go too 503 00:35:11,130 --> 00:35:12,840 d that is less than four. 504 00:35:15,670 --> 00:35:20,320 And d less than four, I can again draw u. 505 00:35:20,320 --> 00:35:24,040 I can draw t. 506 00:35:24,040 --> 00:35:30,110 And I will again find the fixed point at 0, 0. 507 00:35:30,110 --> 00:35:34,350 I will again find an eigendirection, 508 00:35:34,350 --> 00:35:40,640 at u equals to 0, which pushes things out along the u equals 509 00:35:40,640 --> 00:35:42,490 to 0 axis. 510 00:35:42,490 --> 00:35:46,380 Going from d of above four to d of below four 511 00:35:46,380 --> 00:35:48,630 does not really materially change 512 00:35:48,630 --> 00:35:52,660 the location of this other eigendirection by much. 513 00:35:52,660 --> 00:35:55,480 It pretty much stays where it was. 514 00:35:55,480 --> 00:36:00,910 The thing that it does change is the eigenvalue. 515 00:36:00,910 --> 00:36:05,590 So basically, here I will find that the flow 516 00:36:05,590 --> 00:36:07,070 is in this direction. 517 00:36:07,070 --> 00:36:11,550 And if I were to generalize the picture that I have, 518 00:36:11,550 --> 00:36:17,740 I would get things that would be going like this 519 00:36:17,740 --> 00:36:19,455 or going like this. 520 00:36:23,770 --> 00:36:26,820 Once again, there are a set of trajectories 521 00:36:26,820 --> 00:36:29,300 that go on one side, a set of trajectories 522 00:36:29,300 --> 00:36:31,150 that go on the other side. 523 00:36:31,150 --> 00:36:35,390 And presumably, by changing temperature, 524 00:36:35,390 --> 00:36:37,540 I will cross from one set of trajectories 525 00:36:37,540 --> 00:36:40,520 to the other set of trajectories. 526 00:36:40,520 --> 00:36:43,250 But the thing is that the point that 527 00:36:43,250 --> 00:36:47,530 corresponds to hitting the basin that separates the two 528 00:36:47,530 --> 00:36:53,050 sets of trajectories, I don't know what it corresponds to. 529 00:36:53,050 --> 00:36:56,100 Here, for d greater than 4, it went 530 00:36:56,100 --> 00:36:57,930 to the Gaussian fixed point. 531 00:36:57,930 --> 00:37:00,350 Here currently, I don't know where it is going. 532 00:37:02,970 --> 00:37:09,590 So I have no understanding at this level of what the scale 533 00:37:09,590 --> 00:37:13,960 invariant properties are that describe magnets 534 00:37:13,960 --> 00:37:17,720 in three dimensions at their critical temperature. 535 00:37:23,030 --> 00:37:27,960 Now, the thing is that the resolution and everything 536 00:37:27,960 --> 00:37:36,230 that we need comes from staring more at this expansion 537 00:37:36,230 --> 00:37:37,860 that we had. 538 00:37:37,860 --> 00:37:42,060 We can see that this is an alternating theory because I 539 00:37:42,060 --> 00:37:44,630 started with e to the minus u. 540 00:37:44,630 --> 00:37:50,140 And so the next term is likely to have the opposite sign 541 00:37:50,140 --> 00:37:52,450 to the first term. 542 00:37:52,450 --> 00:37:58,380 So I anticipate that at the end of doing the calculation, if I 543 00:37:58,380 --> 00:38:02,340 go to the next order, there will be a term here 544 00:38:02,340 --> 00:38:06,270 that is minus vu squared. 545 00:38:06,270 --> 00:38:09,940 Actually, there will be a contribution to dt by dl also 546 00:38:09,940 --> 00:38:14,590 that is minus, let's say, au squared. 547 00:38:14,590 --> 00:38:18,140 So I expect that if I were to do things at the next order, 548 00:38:18,140 --> 00:38:22,040 and we will do that in about 15 minutes, 549 00:38:22,040 --> 00:38:25,140 I will get these kinds of terms. 550 00:38:25,140 --> 00:38:27,660 Once I have that kind of term, you 551 00:38:27,660 --> 00:38:33,160 can see that I anticipate then a fixed point occurring 552 00:38:33,160 --> 00:38:38,699 at the location u star, which is 4 minus d divided by b. 553 00:38:42,830 --> 00:38:47,020 And then, by looking in the vicinity of this fixed point, 554 00:38:47,020 --> 00:38:49,460 I should be able to determine everything 555 00:38:49,460 --> 00:38:53,400 that I need about the phase transition. 556 00:38:53,400 --> 00:39:00,140 But then you can ask, is this a legitimate thing to do? 557 00:39:00,140 --> 00:39:04,810 I have to make sure I do things self consistently. 558 00:39:04,810 --> 00:39:07,580 I did a perturbation theory assuming 559 00:39:07,580 --> 00:39:11,990 that u is a small quantity, so that I can organize things 560 00:39:11,990 --> 00:39:16,960 in power of u, u squared, u cubed. 561 00:39:16,960 --> 00:39:23,740 But what does it mean that I have control over powers of u? 562 00:39:23,740 --> 00:39:28,550 Once I have landed at this fixed point, where at the fixed 563 00:39:28,550 --> 00:39:33,600 point, u has a value that is fixed and determined. 564 00:39:33,600 --> 00:39:37,120 It is this 4 minus d over b. 565 00:39:37,120 --> 00:39:41,340 So in order for the series to make sense and be 566 00:39:41,340 --> 00:39:45,080 under control, I need this u star 567 00:39:45,080 --> 00:39:49,800 to be under control as a small parameter. 568 00:39:49,800 --> 00:39:53,560 So what knob do I have to ensure that this u 569 00:39:53,560 --> 00:39:56,180 star is a small parameter? 570 00:39:56,180 --> 00:39:59,800 Turns out that practically the only knob that I have 571 00:39:59,800 --> 00:40:02,540 is that this 4 minus d should be small. 572 00:40:05,070 --> 00:40:09,830 So I can only make this into a systematic theory 573 00:40:09,830 --> 00:40:15,300 by making it into an expansion in a small quantity, which 574 00:40:15,300 --> 00:40:18,130 is 4 minus d. 575 00:40:18,130 --> 00:40:19,920 Let's call that epsilon. 576 00:40:19,920 --> 00:40:23,490 And now we can hopefully, at the end of the day, 577 00:40:23,490 --> 00:40:28,940 keep track of appropriate powers of epsilon. 578 00:40:28,940 --> 00:40:31,940 So the Gaussian theory describes properly 579 00:40:31,940 --> 00:40:34,900 the behavior at four dimensions. 580 00:40:34,900 --> 00:40:38,340 At 4 minus epsilon dimensions, I can 581 00:40:38,340 --> 00:40:40,820 figure out where this fixed point is 582 00:40:40,820 --> 00:40:44,630 and calculate things correctly. 583 00:40:44,630 --> 00:40:46,500 All right? 584 00:40:46,500 --> 00:40:50,230 So that means that I need to do this calculation 585 00:40:50,230 --> 00:40:51,500 of the variance of u. 586 00:40:54,720 --> 00:41:00,600 So what I will do here is to draw a diagram 587 00:41:00,600 --> 00:41:03,250 to help me do that. 588 00:41:03,250 --> 00:41:04,990 So let's do something like this. 589 00:41:29,510 --> 00:41:32,510 OK, let's do something like this. 590 00:42:06,760 --> 00:42:10,830 Six, seven rows and seven columns. 591 00:42:10,830 --> 00:42:15,060 The first row is to just tell you what we are going to plot. 592 00:42:15,060 --> 00:42:19,420 So basically, I need a u squared average, 593 00:42:19,420 --> 00:42:23,750 which means that I need to have two factors of u. 594 00:42:23,750 --> 00:42:28,838 Each one of them depends on m tilde and sigma. 595 00:42:28,838 --> 00:42:35,260 And so I will indicate the two sets. 596 00:42:35,260 --> 00:42:38,920 Actually already, we saw when we were 597 00:42:38,920 --> 00:42:43,210 doing the case of the first order calculation, 598 00:42:43,210 --> 00:42:48,600 how to decompose this object that has four lines. 599 00:42:48,600 --> 00:42:53,920 And we said, well, the first thing that I can do is to just 600 00:42:53,920 --> 00:42:56,840 use the m's. 601 00:42:56,840 --> 00:42:58,840 The next thing that I can do is I 602 00:42:58,840 --> 00:43:02,810 could replace one of the m's with a sigma. 603 00:43:02,810 --> 00:43:07,030 And there was a choice of four ways to do so. 604 00:43:07,030 --> 00:43:15,040 Or I could choose to replace two of the m's with wavy lines. 605 00:43:15,040 --> 00:43:17,900 And question was, the right branch or the left branch? 606 00:43:17,900 --> 00:43:20,590 So there's two of these. 607 00:43:20,590 --> 00:43:28,190 I could put the wavy lines on two different branches. 608 00:43:28,190 --> 00:43:32,820 And there was four ways to do this one. 609 00:43:32,820 --> 00:43:40,120 I could have three wavy lines, and the one solid line 610 00:43:40,120 --> 00:43:44,260 could then be in one of four positions. 611 00:43:44,260 --> 00:43:52,250 Or I had all wavy lines, so there is this. 612 00:43:52,250 --> 00:43:57,320 So that's one of my factors of u on the vertical for this table. 613 00:43:57,320 --> 00:44:00,290 On the horizontal, I will have the same thing. 614 00:44:00,290 --> 00:44:03,400 I will have one of these. 615 00:44:03,400 --> 00:44:08,080 I will have four of these. 616 00:44:08,080 --> 00:44:13,260 I will have two of these. 617 00:44:13,260 --> 00:44:16,490 I will have four of these. 618 00:44:19,150 --> 00:44:28,747 I will have four of these, and one which is all wavy lines. 619 00:44:36,380 --> 00:44:40,670 Now I have to put two of these together 620 00:44:40,670 --> 00:44:43,900 and then do the average. 621 00:44:43,900 --> 00:44:47,590 Now clearly, if I put two of these together, 622 00:44:47,590 --> 00:44:49,290 there's no average to be done. 623 00:44:49,290 --> 00:44:52,770 I will get something that is order of m to the fourth. 624 00:44:52,770 --> 00:44:56,410 But remember that I'm calculating the variance. 625 00:44:56,410 --> 00:45:00,110 So that would subtract from the average squared 626 00:45:00,110 --> 00:45:01,540 of the same quantity. 627 00:45:01,540 --> 00:45:03,790 It's a disconnected piece. 628 00:45:03,790 --> 00:45:07,640 And I have stated that anything that is disconnected 629 00:45:07,640 --> 00:45:09,820 will not contribute. 630 00:45:09,820 --> 00:45:13,460 And in particular, there is no way to join this to anything. 631 00:45:13,460 --> 00:45:17,660 So everything that we log here in this row 632 00:45:17,660 --> 00:45:21,040 would correspond to no contribution 633 00:45:21,040 --> 00:45:25,190 once I have subtracted out the average of u squared. 634 00:45:25,190 --> 00:45:27,610 And there is symmetry in this table. 635 00:45:27,610 --> 00:45:30,470 So the corresponding column is also 636 00:45:30,470 --> 00:45:33,496 all things that are disconnected entities. 637 00:45:39,260 --> 00:45:42,050 All right. 638 00:45:42,050 --> 00:45:45,330 Now let's see the next one. 639 00:45:45,330 --> 00:45:49,880 I have a wavy line here, a sigma here, and a sigma here. 640 00:45:49,880 --> 00:45:53,020 I can potentially join them together 641 00:45:53,020 --> 00:45:57,870 into a diagram that looks something like this. 642 00:46:01,850 --> 00:46:06,820 So I will have this, this. 643 00:46:06,820 --> 00:46:08,490 I have a leg here. 644 00:46:08,490 --> 00:46:12,820 I will have this line gets joined to that line. 645 00:46:12,820 --> 00:46:15,580 And then I have this, this, this. 646 00:46:19,401 --> 00:46:21,280 Now, what is that beast? 647 00:46:21,280 --> 00:46:27,820 It is something that has six factors of m tilde. 648 00:46:27,820 --> 00:46:29,600 So this is something that is order 649 00:46:29,600 --> 00:46:35,750 of m tilde to the sixth power. 650 00:46:35,750 --> 00:46:37,900 So the point is that we started here 651 00:46:37,900 --> 00:46:40,710 saying that I should put every term that 652 00:46:40,710 --> 00:46:42,075 is consistent with symmetry. 653 00:46:44,650 --> 00:46:48,230 I just focused on the first fourth order term, 654 00:46:48,230 --> 00:46:49,940 but I see this is one of the things that 655 00:46:49,940 --> 00:46:53,180 happens under renormalization group. 656 00:46:53,180 --> 00:46:55,530 Everything that is consistent with symmetry, 657 00:46:55,530 --> 00:46:57,840 even if you didn't put it there at the beginning, 658 00:46:57,840 --> 00:46:59,970 is likely to appear. 659 00:46:59,970 --> 00:47:02,580 So this term appeared at this order. 660 00:47:02,580 --> 00:47:04,880 You have to think of ultimately whether that's 661 00:47:04,880 --> 00:47:07,210 something to worry about or not. 662 00:47:07,210 --> 00:47:09,110 I will deal with that next time. 663 00:47:09,110 --> 00:47:10,640 It is not something to worry about. 664 00:47:10,640 --> 00:47:13,390 But let's forget about that for the time being. 665 00:47:13,390 --> 00:47:18,750 Next term, I have one wavy line here and two wavy lines there. 666 00:47:18,750 --> 00:47:21,260 So it's something that is sigma cubed. 667 00:47:21,260 --> 00:47:24,490 Against the Gaussian weight, it gives me 0. 668 00:47:24,490 --> 00:47:29,740 So because of it being an odd term, I will get a 0 here. 669 00:47:29,740 --> 00:47:32,355 What color [INAUDIBLE] a 0 here. 670 00:47:37,770 --> 00:47:45,670 Somehow I need this row to be larger in connection 671 00:47:45,670 --> 00:47:46,825 with future needs. 672 00:47:54,900 --> 00:47:57,370 Next one is also something that involves 673 00:47:57,370 --> 00:48:02,140 three factors of sigma, so it is 0 by symmetry. 674 00:48:02,140 --> 00:48:06,510 And again, since this is a diagram that 675 00:48:06,510 --> 00:48:13,010 has symmetry along the diagonal, there will be 0's over here. 676 00:48:13,010 --> 00:48:15,030 Next diagram. 677 00:48:15,030 --> 00:48:17,730 I can somehow join things together 678 00:48:17,730 --> 00:48:21,630 and create something that has four legs. 679 00:48:21,630 --> 00:48:23,970 It will look something like this. 680 00:48:23,970 --> 00:48:31,530 I will have this leg. 681 00:48:31,530 --> 00:48:41,310 This leg can be joined, let's say, with this leg, 682 00:48:41,310 --> 00:48:44,940 giving me something out here. 683 00:48:44,940 --> 00:48:48,905 And these two wavy lines can be joined together. 684 00:48:48,905 --> 00:48:49,780 That's a possibility. 685 00:48:52,360 --> 00:48:53,020 You say, OK. 686 00:48:53,020 --> 00:48:55,210 This is a diagram that corresponds 687 00:48:55,210 --> 00:49:00,130 to four factors of m tilde. 688 00:49:00,130 --> 00:49:05,990 So that should contribute over here. 689 00:49:05,990 --> 00:49:09,790 Actually, the answer is that diagram is 0. 690 00:49:09,790 --> 00:49:12,840 The reason for that is the following. 691 00:49:12,840 --> 00:49:16,680 Let's look at this vortex over here. 692 00:49:16,680 --> 00:49:19,850 It describes four momenta that have come together. 693 00:49:23,260 --> 00:49:27,400 And the sum of the four has to be 0. 694 00:49:27,400 --> 00:49:28,710 Same thing holds here. 695 00:49:31,360 --> 00:49:33,530 The sum of these four has to be 0. 696 00:49:37,440 --> 00:49:50,990 Now, if we look at this diagram, once I 697 00:49:50,990 --> 00:49:53,780 have joined these two together, I 698 00:49:53,780 --> 00:49:58,000 have ensured that the sum of these two is 0. 699 00:49:58,000 --> 00:49:59,660 The sum of all of four is 0. 700 00:49:59,660 --> 00:50:01,820 The sum of these two is 0. 701 00:50:01,820 --> 00:50:04,380 So the sum of these two should be 0 too. 702 00:50:04,380 --> 00:50:06,190 But that's not allowed. 703 00:50:06,190 --> 00:50:10,385 Because one of them is outside this shell, and the other 704 00:50:10,385 --> 00:50:12,650 is inside the shell. 705 00:50:12,650 --> 00:50:17,310 So just kinematically, there's no choice of momenta 706 00:50:17,310 --> 00:50:21,190 that I could make that would give a contribution to this. 707 00:50:21,190 --> 00:50:24,710 So this is 0 because of what I will 708 00:50:24,710 --> 00:50:26,910 write as momentum type of conservation. 709 00:50:30,120 --> 00:50:34,710 Again, because of that, I will have here as 0 momentum 710 00:50:34,710 --> 00:50:37,630 down here. 711 00:50:37,630 --> 00:50:42,640 The next diagram has one sigma from here and four sigmas. 712 00:50:42,640 --> 00:50:44,970 So that's an odd number of sigmas. 713 00:50:44,970 --> 00:50:47,410 So this will be 0 too, just because 714 00:50:47,410 --> 00:50:49,939 of up-down symmetry in m tilde. 715 00:50:54,530 --> 00:51:00,450 So we are gradually getting rid of places in this table. 716 00:51:00,450 --> 00:51:03,260 But the next one is actually important. 717 00:51:03,260 --> 00:51:07,150 I can take these two and join them to those two 718 00:51:07,150 --> 00:51:10,700 and generate a diagram that looks like this. 719 00:51:10,700 --> 00:51:12,700 So I have these two hands. 720 00:51:12,700 --> 00:51:17,730 These two hands get joined to the corresponding two hands. 721 00:51:17,730 --> 00:51:21,440 And I have a diagram such as this. 722 00:51:21,440 --> 00:51:22,300 Yes. 723 00:51:22,300 --> 00:51:27,934 AUDIENCE: [INAUDIBLE] Is there another way for them 724 00:51:27,934 --> 00:51:28,961 to join also? 725 00:51:28,961 --> 00:51:30,460 PROFESSOR: Yes, there is another way 726 00:51:30,460 --> 00:51:32,780 which suffers exactly the same problem. 727 00:51:32,780 --> 00:51:38,050 Ultimately, because you see the problem is here. 728 00:51:38,050 --> 00:51:40,540 I will have to join two of them together, 729 00:51:40,540 --> 00:51:42,810 and the other two will be incompatible. 730 00:51:49,850 --> 00:51:55,110 Now, just to sort of give you ultimately an idea, associated 731 00:51:55,110 --> 00:52:00,510 with this diagram there will be a numerical factor of 2 times 2 732 00:52:00,510 --> 00:52:05,400 from the horizontal times the vertical choices. 733 00:52:05,400 --> 00:52:08,150 But then there's another factor of 2 734 00:52:08,150 --> 00:52:10,320 because this diagram has two hands. 735 00:52:10,320 --> 00:52:12,370 The other diagram has two hands. 736 00:52:12,370 --> 00:52:15,350 They can either join like this, or they can join like this. 737 00:52:15,350 --> 00:52:19,650 So there's two possibilities for the crossing. 738 00:52:19,650 --> 00:52:25,240 If you kind of look ahead to the indices that carry around, 739 00:52:25,240 --> 00:52:27,770 these two are part of the same branch. 740 00:52:27,770 --> 00:52:29,630 They carry the same index. 741 00:52:29,630 --> 00:52:33,470 These two would be carrying the same index, let's say j. 742 00:52:33,470 --> 00:52:36,570 These two would be carrying the same index, j prime. 743 00:52:36,570 --> 00:52:39,290 So when I do the sum, I will have a sum over j 744 00:52:39,290 --> 00:52:41,790 and j prime of delta j, j prime. 745 00:52:41,790 --> 00:52:44,620 I will have a sum over j delta jj, which 746 00:52:44,620 --> 00:52:47,130 will give me a factor of n. 747 00:52:47,130 --> 00:52:51,110 Any time you see a closed loop, you generate a factor of n, 748 00:52:51,110 --> 00:52:53,250 just like we did over here. 749 00:52:53,250 --> 00:52:55,346 It generated a factor of n. 750 00:52:55,346 --> 00:52:57,550 OK, so there's that. 751 00:52:57,550 --> 00:53:01,040 The next diagram looks similar, but does not 752 00:53:01,040 --> 00:53:02,870 have the factor of n. 753 00:53:02,870 --> 00:53:09,650 I have from over there the two hands that I have to join here. 754 00:53:09,650 --> 00:53:14,385 I have to put my hands across, and I 755 00:53:14,385 --> 00:53:18,160 will get something like this. 756 00:53:18,160 --> 00:53:24,230 So it's a slightly different-looking diagram. 757 00:53:24,230 --> 00:53:31,700 The numerical factor that goes with that is 2 times 4 times 2. 758 00:53:31,700 --> 00:53:33,054 There is no factor of n. 759 00:53:35,720 --> 00:53:38,230 Now, again, because of symmetry, there's 760 00:53:38,230 --> 00:53:41,330 a corresponding entity that we have over here. 761 00:53:41,330 --> 00:53:43,800 If I just rotate that, I will essentially 762 00:53:43,800 --> 00:53:46,470 have the same diagram. 763 00:53:46,470 --> 00:53:49,995 Opposite way, I have essentially that. 764 00:53:55,280 --> 00:54:01,350 The two hands reach across to these 765 00:54:01,350 --> 00:54:03,870 and give me something that is like this. 766 00:54:08,680 --> 00:54:10,478 To that, sorry. 767 00:54:13,620 --> 00:54:17,250 They join to that one. 768 00:54:17,250 --> 00:54:20,330 And the corresponding thing here looks like this. 769 00:54:25,440 --> 00:54:30,000 Numerical factors, this would be 2 times 4 times 2. 770 00:54:30,000 --> 00:54:31,790 It is exactly the same as this. 771 00:54:31,790 --> 00:54:34,622 This would be 4 times 4 times 2. 772 00:54:38,730 --> 00:54:41,090 At the end of the day, I will convince you 773 00:54:41,090 --> 00:54:48,700 that this block of four diagrams is really the only thing 774 00:54:48,700 --> 00:54:51,390 that we need to compute. 775 00:54:51,390 --> 00:54:54,580 But let's go ahead and see what else we have. 776 00:54:54,580 --> 00:54:58,000 If I take this thing that has two hands, 777 00:54:58,000 --> 00:55:00,470 try to join this thing that has three hands, 778 00:55:00,470 --> 00:55:05,160 I will get, of course, 0, based on symmetry. 779 00:55:05,160 --> 00:55:09,950 If I take this term with two hands, 780 00:55:09,950 --> 00:55:11,760 join this thing with four hands, I 781 00:55:11,760 --> 00:55:14,610 will generate a bunch of diagrams, 782 00:55:14,610 --> 00:55:18,170 including, for example, this one. 783 00:55:18,170 --> 00:55:23,520 I can do this. 784 00:55:23,520 --> 00:55:27,280 There are other diagrams also. 785 00:55:27,280 --> 00:55:33,040 So these are ultimately diagrams with two hands left over. 786 00:55:33,040 --> 00:55:38,390 So they will be contributions to m tilde squared. 787 00:55:38,390 --> 00:55:43,230 And they will indeed give me modifications 788 00:55:43,230 --> 00:55:44,710 of this term over here. 789 00:55:47,920 --> 00:55:51,230 But we don't need to calculate them. 790 00:55:51,230 --> 00:55:52,500 Why? 791 00:55:52,500 --> 00:55:55,190 Because we want to do things consistently 792 00:55:55,190 --> 00:55:58,220 to order of epsilon. 793 00:55:58,220 --> 00:56:04,100 In the second equation, we start already with epsilon u. 794 00:56:04,100 --> 00:56:07,760 So this term was order of epsilon squared. 795 00:56:07,760 --> 00:56:10,140 Since u star will be order of epsilon, 796 00:56:10,140 --> 00:56:12,050 this term will be epsilon squared. 797 00:56:12,050 --> 00:56:14,120 The two terms I have to evaluate, 798 00:56:14,120 --> 00:56:16,470 they are both of the same order. 799 00:56:16,470 --> 00:56:18,590 But in the first equation, I already 800 00:56:18,590 --> 00:56:21,740 have a contribution that is order of epsilon. 801 00:56:21,740 --> 00:56:25,970 If I'm calculating things consistently to lowest order, 802 00:56:25,970 --> 00:56:29,170 I don't need to calculate this explicitly. 803 00:56:29,170 --> 00:56:31,520 I would need to calculate it explicitly 804 00:56:31,520 --> 00:56:33,470 if I wanted to calculate things to order 805 00:56:33,470 --> 00:56:37,090 of epsilon squared, which I'm not about to do. 806 00:56:37,090 --> 00:56:40,460 But to our order, this diagram exists, 807 00:56:40,460 --> 00:56:42,660 but we don't need to evaluate. 808 00:56:42,660 --> 00:56:47,040 Again, going because of the symmetry along the diagonal 809 00:56:47,040 --> 00:56:50,170 of the diagram, we have something here 810 00:56:50,170 --> 00:56:54,770 that is order of m tilde squared that we don't evaluate. 811 00:56:54,770 --> 00:56:57,570 OK, let's go further. 812 00:56:57,570 --> 00:57:00,280 Over here, we have two hands, three hands. 813 00:57:00,280 --> 00:57:03,220 By symmetry, it will be zero. 814 00:57:03,220 --> 00:57:06,820 Over here, we have two hands, four hands. 815 00:57:06,820 --> 00:57:13,710 I will get a whole bunch of other things 816 00:57:13,710 --> 00:57:16,740 that are order of m tilde squared. 817 00:57:16,740 --> 00:57:24,500 So there are other terms that are of this same form that 818 00:57:24,500 --> 00:57:26,730 would modify the factor of a, which 819 00:57:26,730 --> 00:57:30,340 I don't need to explicitly evaluate. 820 00:57:30,340 --> 00:57:31,910 All right. 821 00:57:31,910 --> 00:57:34,770 What do we have left? 822 00:57:34,770 --> 00:57:37,610 There is a diagram here that is interesting 823 00:57:37,610 --> 00:57:41,720 because it also gives me a contribution that 824 00:57:41,720 --> 00:57:51,930 is order of m tilde squared, which we may come back to 825 00:57:51,930 --> 00:57:53,120 at some point. 826 00:57:53,120 --> 00:57:56,840 But for the time being, it's another thing 827 00:57:56,840 --> 00:58:00,400 that gives us a contribution to a. 828 00:58:00,400 --> 00:58:01,700 Here, what do we have? 829 00:58:01,700 --> 00:58:04,400 We have three hands, four hands, zero by symmetry, 830 00:58:04,400 --> 00:58:07,150 zero by symmetry. 831 00:58:07,150 --> 00:58:10,410 Down here, we have no solid hands. 832 00:58:10,410 --> 00:58:13,850 So we will get a whole bunch of diagrams, 833 00:58:13,850 --> 00:58:18,920 such as this one, for example, other things, which 834 00:58:18,920 --> 00:58:21,960 collectively will give a second order 835 00:58:21,960 --> 00:58:24,860 correction to the free energy. 836 00:58:24,860 --> 00:58:28,942 It's another constant that we don't need to evaluate. 837 00:58:35,540 --> 00:58:41,345 So let's pick one of these diagrams, this one 838 00:58:41,345 --> 00:58:44,400 in particular, and explicitly see what that is. 839 00:58:52,750 --> 00:58:56,530 It came out of putting two factors of u together. 840 00:58:56,530 --> 00:58:57,620 Let's be explicit. 841 00:58:57,620 --> 00:59:04,540 Let's call the momenta here q1, q2, and k1, k2. 842 00:59:04,540 --> 00:59:10,090 And the other u here came from before I joined them, 843 00:59:10,090 --> 00:59:13,460 there was a q3, q4. 844 00:59:13,460 --> 00:59:16,397 There was a k1 prime, k2 prime. 845 00:59:19,320 --> 00:59:22,850 So let's say the first u-- this is a diagram that 846 00:59:22,850 --> 00:59:26,280 will contribute at order of u squared. 847 00:59:26,280 --> 00:59:28,745 Second order terms in the series all 848 00:59:28,745 --> 00:59:30,510 come with a factor of one half. 849 00:59:30,510 --> 00:59:34,100 It is u to the n divided by n factorial. 850 00:59:34,100 --> 00:59:38,600 So this would be explicitly u squared over 2. 851 00:59:38,600 --> 00:59:41,230 For the choice of the left diagram, 852 00:59:41,230 --> 00:59:43,990 we said there were two possibilities. 853 00:59:43,990 --> 00:59:46,120 For the choice of right diagram, there 854 00:59:46,120 --> 00:59:48,900 were two branches, one of which I could have taken. 855 00:59:48,900 --> 00:59:51,480 In joining the two hands together, 856 00:59:51,480 --> 00:59:53,940 I had a degeneracy of two, so I have all of that. 857 00:59:57,570 --> 01:00:04,500 A particular one of these is an integral over q1, q2. 858 01:00:04,500 --> 01:00:08,590 And from here, I would have integrations over q3, q4. 859 01:00:12,170 --> 01:00:16,250 These are all integrations that are 860 01:00:16,250 --> 01:00:19,570 for variables that are in the inner shell. 861 01:00:19,570 --> 01:00:22,970 So this is lambda to lambda over b. 862 01:00:22,970 --> 01:00:26,830 I have integrations from lambda over v2 lambda 863 01:00:26,830 --> 01:00:33,420 for the variables k1, k2, k1 prime, k2 prime. 864 01:00:38,610 --> 01:00:43,720 And if I explicitly decided to write all four 865 01:00:43,720 --> 01:00:46,910 momenta associated with a particular index, 866 01:00:46,910 --> 01:00:50,310 I have to explicitly include the delta functions that 867 01:00:50,310 --> 01:00:54,664 say the sum of the momenta has to add up to 0. 868 01:01:08,840 --> 01:01:14,530 Now, what I did was to drawing these two sigmas together. 869 01:01:14,530 --> 01:01:17,350 So I calculated one of those Gaussian averages 870 01:01:17,350 --> 01:01:20,410 that I have over there. 871 01:01:20,410 --> 01:01:23,080 Actually, before I do that, I note 872 01:01:23,080 --> 01:01:26,010 that these pairs are dotted together. 873 01:01:26,010 --> 01:01:29,825 So I have m tilde of q1 dotted with m tilde 874 01:01:29,825 --> 01:01:42,580 of q3, m tilde of q4 dotted with m tilde of q1, q2, q3, q4. 875 01:01:42,580 --> 01:01:43,880 These two are dot products. 876 01:01:43,880 --> 01:01:46,620 These two are dot products. 877 01:01:46,620 --> 01:01:51,260 Here, I joined the two sigmas together. 878 01:01:51,260 --> 01:01:53,560 The expectation value gives me 2 pi 879 01:01:53,560 --> 01:01:57,770 to the d at delta function k1 plus k1 880 01:01:57,770 --> 01:02:04,240 prime divided by t plus k, k1 squared, and so forth. 881 01:02:04,240 --> 01:02:06,110 And the delta function, if I call 882 01:02:06,110 --> 01:02:11,540 these indices j, j, j prime, j prime, 883 01:02:11,540 --> 01:02:13,590 I will have a delta j j prime. 884 01:02:17,220 --> 01:02:20,440 And from the lower two that I have connected together, 885 01:02:20,440 --> 01:02:27,460 I have 2 pi to the d delta function k2 plus k2 prime. 886 01:02:27,460 --> 01:02:33,550 Another delta j j prime, t plus k, k2 squared, and so forth. 887 01:02:38,050 --> 01:02:40,190 Now I can do the integrations. 888 01:02:40,190 --> 01:02:47,350 But first of all, numerical factors, I will get 4u squared. 889 01:02:47,350 --> 01:02:50,560 As I told you, delta j j prime, delta j j prime 890 01:02:50,560 --> 01:02:51,910 will give me delta jj. 891 01:02:51,910 --> 01:02:54,780 Sum over j, I will get a factor of n. 892 01:02:54,780 --> 01:02:59,070 That's the n that I anticipated and put over there. 893 01:02:59,070 --> 01:03:07,250 I have the integrations 0 to lambda over v, ddq1, ddq2, 894 01:03:07,250 --> 01:03:13,370 ddq3, ddq4, 2 pi to the 4d. 895 01:03:16,980 --> 01:03:25,780 And then this m tilde q1 m tilde q2 m tilde q3 896 01:03:25,780 --> 01:03:27,220 dotted with m tilde q4. 897 01:03:30,570 --> 01:03:32,500 Now, note the following. 898 01:03:32,500 --> 01:03:36,940 If I do the integration over k1 prime, 899 01:03:36,940 --> 01:03:40,570 k1 prime is set to minus k1. 900 01:03:40,570 --> 01:03:43,220 If I do the integration over k2 prime, 901 01:03:43,220 --> 01:03:47,510 k2 prime is set to minus k2. 902 01:03:47,510 --> 01:03:51,410 If I now do the integration over k2, 903 01:03:51,410 --> 01:03:56,560 k2 is set to minus q1 minus q2 minus k1, which 904 01:03:56,560 --> 01:04:02,340 if I insert over here, will give me a delta function that simply 905 01:04:02,340 --> 01:04:06,988 says that the four external q's have to add up to 0. 906 01:04:13,320 --> 01:04:17,490 So there is one integration that is left, which is over k1. 907 01:04:17,490 --> 01:04:19,750 So I have to do the integral lambda 908 01:04:19,750 --> 01:04:25,650 over v2 lambda, dd of k1 2 pi to the d. 909 01:04:25,650 --> 01:04:31,120 So basically, there's k1 running across the upper line gives me 910 01:04:31,120 --> 01:04:37,500 a factor of 1 over t plus k, k1 squared, and so forth. 911 01:04:37,500 --> 01:04:39,350 And then there is what is running 912 01:04:39,350 --> 01:04:42,720 along the bottom line, which is k2 squared. 913 01:04:42,720 --> 01:04:48,790 And k2 squared is the same thing as q1 plus q2 914 01:04:48,790 --> 01:04:51,115 plus k1, the whole thing squared. 915 01:04:59,230 --> 01:05:05,580 So the outcome of doing the averages that 916 01:05:05,580 --> 01:05:09,360 appear in this integral is to generate 917 01:05:09,360 --> 01:05:14,710 a term that is proportional to m to the fourth, which 918 01:05:14,710 --> 01:05:20,390 is exactly what we had, with one twist. 919 01:05:20,390 --> 01:05:22,380 The twist is that the coefficient that 920 01:05:22,380 --> 01:05:28,710 is appearing here actually depends on q1 and q2. 921 01:05:28,710 --> 01:05:32,430 Of course, q1 and q2 being inner momenta 922 01:05:32,430 --> 01:05:36,840 are much smaller than k1, which is one of the shared momenta. 923 01:05:36,840 --> 01:05:39,010 So in principle, I can expand this. 924 01:05:39,010 --> 01:05:49,560 I can expand this as ddk 2 pi to the d lambda 925 01:05:49,560 --> 01:05:56,310 over b2 lambda, 1 over t plus k, k squared. 926 01:05:56,310 --> 01:06:02,680 I've renamed k1 to k squared to lowest order in q, this is 0. 927 01:06:02,680 --> 01:06:13,140 And then I can expand this thing as 1 plus k q1 plus q2 squared 928 01:06:13,140 --> 01:06:17,300 and so forth, divided by t plus k k 929 01:06:17,300 --> 01:06:23,210 squared raised to the minus 1 power. 930 01:06:23,210 --> 01:06:27,320 Point is that if I set the q's to 0, 931 01:06:27,320 --> 01:06:31,190 I have obtained a constant addition 932 01:06:31,190 --> 01:06:35,150 to the coefficient of my m to the fourth. 933 01:06:35,150 --> 01:06:39,650 But I see that further down, I have generated also 934 01:06:39,650 --> 01:06:42,620 terms that depend on q. 935 01:06:42,620 --> 01:06:45,630 What kind of terms could these be? 936 01:06:45,630 --> 01:06:47,930 If I go back to real space, these 937 01:06:47,930 --> 01:06:50,420 are terms that are after order of m 938 01:06:50,420 --> 01:06:55,900 to the fourth, which was, if you remember, m squared m squared. 939 01:06:55,900 --> 01:06:59,300 But carry additional gradients with them. 940 01:06:59,300 --> 01:07:04,100 So, for example, it could be something like this. 941 01:07:04,100 --> 01:07:07,760 It has two factors of q, various factors of n. 942 01:07:07,760 --> 01:07:12,870 Or it could be something like m squared gradient of m squared. 943 01:07:12,870 --> 01:07:18,440 The point is that we have again the possibility, when 944 01:07:18,440 --> 01:07:22,030 we write our most general term, to introduce 945 01:07:22,030 --> 01:07:25,650 lots and lots of non-linearities that I didn't explicitly 946 01:07:25,650 --> 01:07:27,130 include. 947 01:07:27,130 --> 01:07:31,040 But again, I see, if I forget them at the beginning, 948 01:07:31,040 --> 01:07:34,730 the process will generate them for you. 949 01:07:34,730 --> 01:07:37,720 So I should have really included these types of terms 950 01:07:37,720 --> 01:07:41,870 in the beginning because they will be generated under the RG, 951 01:07:41,870 --> 01:07:44,700 and then I can track their evolution of all 952 01:07:44,700 --> 01:07:46,190 of the parameters. 953 01:07:46,190 --> 01:07:49,180 I started with 0 for this type of parameter. 954 01:07:49,180 --> 01:07:50,830 I generated it out of nothing. 955 01:07:50,830 --> 01:07:53,640 So I should really go back and put it there. 956 01:07:53,640 --> 01:07:56,780 But for the time being, let's again ignore that. 957 01:07:56,780 --> 01:07:59,970 And next time, I'll see what happens. 958 01:07:59,970 --> 01:08:05,500 So what we find at the end of evaluating 959 01:08:05,500 --> 01:08:15,800 all of these diagrams is that this beta h tilde evaluated 960 01:08:15,800 --> 01:08:19,100 at the second order, first of all, 961 01:08:19,100 --> 01:08:26,109 has a bunch of constants which in principle, now 962 01:08:26,109 --> 01:08:28,220 we can calculate to the next order. 963 01:08:33,600 --> 01:08:42,029 Then we find that we get terms that 964 01:08:42,029 --> 01:08:47,979 are proportional to m tilde squared, the Gaussian. 965 01:08:51,240 --> 01:08:58,740 And I can get the terms that I got out of second order 966 01:08:58,740 --> 01:09:00,170 and put it here. 967 01:09:00,170 --> 01:09:04,220 So I have my original t tilde now 968 01:09:04,220 --> 01:09:11,520 evaluated at order of u squared. 969 01:09:11,520 --> 01:09:16,590 Because of all those diagrams that I said I have to do. 970 01:09:16,590 --> 01:09:22,410 I will get a k tilde q squared and so forth, 971 01:09:22,410 --> 01:09:27,680 all of them multiplying m tilde squared. 972 01:09:27,680 --> 01:09:33,560 And I see that I generated terms that 973 01:09:33,560 --> 01:09:37,270 are of the order of m tilde to the fourth. 974 01:09:37,270 --> 01:09:44,569 So I have ddq1, ddq4, 2 pi to the 4d, 975 01:09:44,569 --> 01:09:58,880 2 pi to the d delta function, q1, q4, m tilde q1, m tilde q2, 976 01:09:58,880 --> 01:10:05,410 m tilde q3, m tilde q4. 977 01:10:05,410 --> 01:10:11,590 And what I have to lowest order is u. 978 01:10:11,590 --> 01:10:13,780 And then I have a bunch of terms that 979 01:10:13,780 --> 01:10:17,600 are of the form something like this. 980 01:10:17,600 --> 01:10:22,300 So they are corrections that are proportional 981 01:10:22,300 --> 01:10:28,150 to the integral lambda over b lambda ddk 2 pi 982 01:10:28,150 --> 01:10:35,110 to the d, 1 over t plus k, k squared squared. 983 01:10:37,740 --> 01:10:44,300 So essentially, I took this part of that diagram. 984 01:10:44,300 --> 01:10:47,590 That diagram has a contribution at order 985 01:10:47,590 --> 01:10:51,120 of u squared, which is 4n. 986 01:10:51,120 --> 01:10:54,950 So if I had written it as u squared over 2, 987 01:10:54,950 --> 01:10:59,160 I would have put 8n, the 8 coming 988 01:10:59,160 --> 01:11:02,470 from just the multiplication that I have there, 2 times 989 01:11:02,470 --> 01:11:05,578 2 times 2 times n. 990 01:11:05,578 --> 01:11:09,140 Now, if you calculate the other three diagrams 991 01:11:09,140 --> 01:11:13,600 that I have boxed, you'll find that they 992 01:11:13,600 --> 01:11:17,690 give exactly the same form of the contribution, 993 01:11:17,690 --> 01:11:21,580 except that the numerical factor for them is different. 994 01:11:21,580 --> 01:11:32,890 I will get 16, 1632, adding up together to a factor of 64 995 01:11:32,890 --> 01:11:35,120 here. 996 01:11:35,120 --> 01:11:40,230 And then the point is that I will generate additional terms 997 01:11:40,230 --> 01:11:43,380 that are, let's say, order of q squared 998 01:11:43,380 --> 01:11:46,360 and so forth, Which are the kinds 999 01:11:46,360 --> 01:11:48,192 of terms that I had not included. 1000 01:11:52,350 --> 01:11:58,430 So what we find is that-- question? 1001 01:12:02,390 --> 01:12:04,370 AUDIENCE: Where did you add the t total? 1002 01:12:07,340 --> 01:12:08,930 PROFESSOR: OK. 1003 01:12:08,930 --> 01:12:12,170 So let's maybe write this explicitly. 1004 01:12:12,170 --> 01:12:15,560 So what would be the coefficient that I have to put over here? 1005 01:12:19,890 --> 01:12:24,610 I have t at the 0-th order. 1006 01:12:24,610 --> 01:12:33,640 At order of u, I calculated 4u n plus 2 integral. 1007 01:12:42,700 --> 01:12:47,540 The point is that when I add up all of those diagrams 1008 01:12:47,540 --> 01:12:51,170 that I haven't explicitly calculated, 1009 01:12:51,170 --> 01:12:54,550 I will get a correction here that is order 1010 01:12:54,550 --> 01:12:59,105 of u squared whose coefficient I will call a. 1011 01:13:03,180 --> 01:13:07,140 But then this is the 0-th order in the momenta, 1012 01:13:07,140 --> 01:13:11,340 and then I have to go and add terms 1013 01:13:11,340 --> 01:13:14,940 that are at the order of q squared and higher order terms. 1014 01:13:23,910 --> 01:13:26,370 So this was, again, the course graining step 1015 01:13:26,370 --> 01:13:28,020 of RG, which is the hard part. 1016 01:13:28,020 --> 01:13:33,110 The rescaling and renormalization are simple. 1017 01:13:33,110 --> 01:13:35,600 And what they give me at the end of the day 1018 01:13:35,600 --> 01:13:40,130 are the modifications to dt by dl and du 1019 01:13:40,130 --> 01:13:42,810 by dl that we expected. 1020 01:13:42,810 --> 01:13:46,210 dt by dl we already wrote. 1021 01:13:46,210 --> 01:13:47,860 0-th order is 2t. 1022 01:13:47,860 --> 01:13:50,040 First order is a correction. 1023 01:13:50,040 --> 01:13:55,260 4u n plus 2 integral, which, when evaluated on the shell, 1024 01:13:55,260 --> 01:14:01,010 gives me kt lambda to the d t plus k lambda 1025 01:14:01,010 --> 01:14:02,170 squared and so forth. 1026 01:14:05,120 --> 01:14:10,790 Now, this a here would involve an integration. 1027 01:14:10,790 --> 01:14:14,830 Again, this integration I evaluated on the shell. 1028 01:14:14,830 --> 01:14:18,130 So the answer will be some a that 1029 01:14:18,130 --> 01:14:23,330 will depend on t, k, and other things, 1030 01:14:23,330 --> 01:14:26,180 would be a contribution that is order of u squared. 1031 01:14:29,020 --> 01:14:32,350 I haven't explicitly calculated what this a is. 1032 01:14:32,350 --> 01:14:36,100 It will depend on all of these other parameters. 1033 01:14:36,100 --> 01:14:40,010 Now, when I calculate the u by dl, I will get this 4 minus 1034 01:14:40,010 --> 01:14:44,120 d times u to the lowest order. 1035 01:14:44,120 --> 01:14:49,540 To the next order, I essentially get this integral. 1036 01:14:49,540 --> 01:15:06,830 So I have minus 4n minus 4n plus 8 u squared. 1037 01:15:06,830 --> 01:15:09,760 Evaluate that integral on the shell. 1038 01:15:09,760 --> 01:15:14,850 kd lambda to the d t plus k lambda 1039 01:15:14,850 --> 01:15:16,640 squared and so forth squared. 1040 01:15:19,440 --> 01:15:21,330 And presumably, both of these will 1041 01:15:21,330 --> 01:15:24,260 have corrections at higher orders, order 1042 01:15:24,260 --> 01:15:25,627 of u squared, et cetera. 1043 01:15:29,740 --> 01:15:34,700 So this generalizes the picture that we had over here. 1044 01:15:37,260 --> 01:15:40,300 Now we can ask, what is the fixed point? 1045 01:15:46,180 --> 01:15:49,400 In fact, there will be two of them. 1046 01:15:49,400 --> 01:15:56,700 There is the old Gaussian fixed point at t star 1047 01:15:56,700 --> 01:15:59,620 u star equals to 0. 1048 01:15:59,620 --> 01:16:03,460 Clearly, if I said t and u equals to 0, I will stay at 0. 1049 01:16:03,460 --> 01:16:06,190 So the old fixed point is still there. 1050 01:16:06,190 --> 01:16:09,010 But now I have a new fixed point, 1051 01:16:09,010 --> 01:16:12,860 which is called the ON fixed point because it explicitly 1052 01:16:12,860 --> 01:16:16,040 depends on the symmetry of the order 1053 01:16:16,040 --> 01:16:19,750 parameter, the number of components, n, 1054 01:16:19,750 --> 01:16:21,240 as well as dimensionality. 1055 01:16:21,240 --> 01:16:23,520 It's called the ON fixed point. 1056 01:16:23,520 --> 01:16:31,430 So setting this to 0, I will find that u star is essentially 1057 01:16:31,430 --> 01:16:35,780 epsilon divided by whatever I have here. 1058 01:16:35,780 --> 01:16:46,530 I have 4n plus 8 kd lambda to the d. 1059 01:16:46,530 --> 01:16:51,810 In the numerator, I would have t star 1060 01:16:51,810 --> 01:16:55,850 plus k lambda squared squared. 1061 01:16:59,500 --> 01:17:03,920 And then I can substitute that over here 1062 01:17:03,920 --> 01:17:06,820 to find what t star is. 1063 01:17:06,820 --> 01:17:16,250 So t star would be minus 2 n plus 2 kd lambda to the d 1064 01:17:16,250 --> 01:17:21,230 divided by t star plus k lambda squared, 1065 01:17:21,230 --> 01:17:26,870 et cetera, times u star, which is what I have the line above, 1066 01:17:26,870 --> 01:17:30,790 which is t star plus k lambda squared, 1067 01:17:30,790 --> 01:17:37,960 et cetera, squared, divided by 4n plus 8 kd lambda to the t. 1068 01:17:46,290 --> 01:17:49,610 Now, over here, this is in principle 1069 01:17:49,610 --> 01:17:54,130 an implicit equation for t star. 1070 01:17:54,130 --> 01:17:57,700 But I forgot the epsilon that I have here. 1071 01:17:57,700 --> 01:18:02,150 But it is epsilon multiplying some function of t star. 1072 01:18:02,150 --> 01:18:05,430 So clearly, t star is order of epsilon. 1073 01:18:05,430 --> 01:18:10,700 And I can set t star equal to 0 in all of the calculation, 1074 01:18:10,700 --> 01:18:14,340 if I'm calculating things consistently to epsilon. 1075 01:18:14,340 --> 01:18:19,060 You can see that this kd lambda to the d cancels that. 1076 01:18:19,060 --> 01:18:25,350 One of these factors cancels what I have over here. 1077 01:18:25,350 --> 01:18:32,290 At the end of the day, I will get minus n plus 2 divided 1078 01:18:32,290 --> 01:18:37,740 by n plus 8 k lambda squared epsilon. 1079 01:18:42,110 --> 01:18:46,190 And similarly, over here I can get rid of t star 1080 01:18:46,190 --> 01:18:48,070 because it's already order of epsilon, 1081 01:18:48,070 --> 01:18:50,740 and I have epsilon out here. 1082 01:18:50,740 --> 01:18:55,670 So the answer is going to be k squared lambda 1083 01:18:55,670 --> 01:19:02,420 to the power of 4 minus d divided by 4n plus 1084 01:19:02,420 --> 01:19:06,850 8 kd lambda to the d. 1085 01:19:06,850 --> 01:19:10,112 Presumably, both of these plus order of epsilon squared. 1086 01:19:14,540 --> 01:19:17,040 So you can see that, as anticipated, 1087 01:19:17,040 --> 01:19:19,750 there's a fixed point at negative 2 t 1088 01:19:19,750 --> 01:19:21,870 star and some particular u star. 1089 01:19:21,870 --> 01:19:22,990 There was a question. 1090 01:19:22,990 --> 01:19:23,864 [INAUDIBLE] 1091 01:19:23,864 --> 01:19:24,739 AUDIENCE: [INAUDIBLE] 1092 01:19:28,690 --> 01:19:29,999 PROFESSOR: What is unnecessary? 1093 01:19:29,999 --> 01:19:30,874 AUDIENCE: [INAUDIBLE] 1094 01:19:33,800 --> 01:19:36,250 AUDIENCE: You already did 4 minus [INAUDIBLE]. 1095 01:19:36,250 --> 01:19:38,202 AUDIENCE: The u started. 1096 01:19:38,202 --> 01:19:39,949 Yeah, that one. 1097 01:19:39,949 --> 01:19:40,615 PROFESSOR: Here. 1098 01:19:40,615 --> 01:19:41,980 AUDIENCE: Erase it. 1099 01:19:41,980 --> 01:19:44,061 PROFESSOR: Oh, lambda to the d is 0. 1100 01:19:44,061 --> 01:19:44,560 Right. 1101 01:19:44,560 --> 01:19:45,200 Thank you. 1102 01:19:45,200 --> 01:19:47,590 AUDIENCE: For t star, does factor of 2. 1103 01:19:47,590 --> 01:19:50,450 PROFESSOR: t star, does it have a factor of 2? 1104 01:19:50,450 --> 01:19:52,310 Yes, 2 divided by 4. 1105 01:19:52,310 --> 01:19:53,640 There is a factor of 2 here. 1106 01:19:57,690 --> 01:19:58,190 Thank you. 1107 01:20:02,960 --> 01:20:03,680 Look at this. 1108 01:20:03,680 --> 01:20:08,790 You don't really see much to recommend it. 1109 01:20:08,790 --> 01:20:13,830 The interesting thing is to find what 1110 01:20:13,830 --> 01:20:17,800 happens if you are not exactly at the fixed point, 1111 01:20:17,800 --> 01:20:20,130 but slightly shifted. 1112 01:20:20,130 --> 01:20:26,240 So we want to see what happens if t is t star plus delta t, 1113 01:20:26,240 --> 01:20:32,860 u is u star plus delta u, if I shift a little bit. 1114 01:20:32,860 --> 01:20:37,610 If I shift a little bit, linearizing the equation 1115 01:20:37,610 --> 01:20:42,540 means I want to know how the new shifts are 1116 01:20:42,540 --> 01:20:45,030 related to the old shift. 1117 01:20:45,030 --> 01:20:48,880 And essentially doing things at the linear level 1118 01:20:48,880 --> 01:20:53,600 means I want to construct a two-by-two matrix that relates 1119 01:20:53,600 --> 01:20:58,470 the changes delta t delta u to the shifts originally 1120 01:20:58,470 --> 01:21:00,690 of delta t and delta u. 1121 01:21:00,690 --> 01:21:02,970 What do I have to do to get this? 1122 01:21:02,970 --> 01:21:07,240 What I have to do is to take derivatives of the terms for dt 1123 01:21:07,240 --> 01:21:10,560 by dl with respect to t, with respect to u. 1124 01:21:10,560 --> 01:21:12,700 Take the derivative with respect to t. 1125 01:21:12,700 --> 01:21:13,380 What do I get? 1126 01:21:13,380 --> 01:21:15,120 I will get two. 1127 01:21:15,120 --> 01:21:24,900 I will get minus 4u n plus 2 kd lambda to the father of d 1128 01:21:24,900 --> 01:21:30,640 divided by t plus k lambda squared squared. 1129 01:21:30,640 --> 01:21:35,160 So the derivative of 1 over t became minus 1 over t squared. 1130 01:21:35,160 --> 01:21:36,700 There is a second order term. 1131 01:21:36,700 --> 01:21:39,810 So there will be a derivative of that with respect 1132 01:21:39,810 --> 01:21:43,730 to t multiplying u squared. 1133 01:21:43,730 --> 01:21:46,480 I want you to calculate it. 1134 01:21:46,480 --> 01:21:49,655 Delta u, if I make a change in u, 1135 01:21:49,655 --> 01:21:55,300 there will be a shift here, which is 4n 1136 01:21:55,300 --> 01:22:01,140 plus 2 kd lambda to the d divided 1137 01:22:01,140 --> 01:22:05,390 by t plus k lambda squared. 1138 01:22:05,390 --> 01:22:08,370 From the second order term, I will get minus 2au. 1139 01:22:11,950 --> 01:22:14,750 For the second equation, if I take the derivative 1140 01:22:14,750 --> 01:22:23,700 of this variation in t, I will get a plus 4n plus 8u squared 1141 01:22:23,700 --> 01:22:29,180 kd lambda to the d t plus k lambda squared 1142 01:22:29,180 --> 01:22:32,690 and so forth cubed. 1143 01:22:32,690 --> 01:22:43,150 And the fourth place, I will get epsilon minus 8 n plus 8 kd 1144 01:22:43,150 --> 01:22:50,190 lambda to the d u divided by t plus k lambda 1145 01:22:50,190 --> 01:22:51,820 squared and so forth squared. 1146 01:22:55,888 --> 01:23:00,950 Now, I want to evaluate this matrix at the fixed point. 1147 01:23:00,950 --> 01:23:04,460 So I have to linearize in the vicinity of fixed point. 1148 01:23:04,460 --> 01:23:10,300 Which means that I put the values of t star and u star 1149 01:23:10,300 --> 01:23:11,142 everywhere here. 1150 01:23:20,900 --> 01:23:26,064 And then I have to calculate the eigenvalues of this matrix. 1151 01:23:26,064 --> 01:23:29,870 Now, note that this element of the matrix 1152 01:23:29,870 --> 01:23:33,410 is proportional to u star squared. 1153 01:23:33,410 --> 01:23:36,453 So this is certainly evaluated at the fixed point 1154 01:23:36,453 --> 01:23:40,310 order of epsilon squared. 1155 01:23:40,310 --> 01:23:42,760 Order of epsilon squared to me is zero. 1156 01:23:42,760 --> 01:23:45,670 I don't see order of epsilon squared. 1157 01:23:45,670 --> 01:23:47,050 So I can get rid of this. 1158 01:23:47,050 --> 01:23:49,890 Think of a zero here at this order. 1159 01:23:49,890 --> 01:23:52,570 Which means that the matrix now has 1160 01:23:52,570 --> 01:23:55,480 zeroes on one side of the diagonal, which 1161 01:23:55,480 --> 01:23:57,645 means that what is appearing here 1162 01:23:57,645 --> 01:24:01,070 are exactly the eigenvalues. 1163 01:24:01,070 --> 01:24:04,220 Let's calculate the eigenvalue that 1164 01:24:04,220 --> 01:24:06,760 corresponds to this element. 1165 01:24:06,760 --> 01:24:09,230 I will call it yu. 1166 01:24:09,230 --> 01:24:18,700 It is epsilon minus 8 n plus 8 kd lambda to the d t star. 1167 01:24:18,700 --> 01:24:22,470 Well, since I'm calculating things to order of epsilon, 1168 01:24:22,470 --> 01:24:25,350 I can ignore that t star down there. 1169 01:24:25,350 --> 01:24:30,820 I have k squared lambda to the four or k squared, 1170 01:24:30,820 --> 01:24:33,220 lambda squared, and so forth squared. 1171 01:24:33,220 --> 01:24:35,730 Multiplied by u star. 1172 01:24:35,730 --> 01:24:38,830 Where is my u star? u star is here. 1173 01:24:38,830 --> 01:24:43,975 So it is multiplied by n plus 2. 1174 01:24:47,860 --> 01:24:51,110 Sorry, my u star is up here. 1175 01:24:51,110 --> 01:25:02,840 k squared lambda to the 4 minus d 4 n plus 8 kd epsilon. 1176 01:25:02,840 --> 01:25:04,170 Right. 1177 01:25:04,170 --> 01:25:06,500 Now the miracle happens. 1178 01:25:06,500 --> 01:25:10,960 So k squared cancels the k squared. 1179 01:25:10,960 --> 01:25:14,050 Lambda to the four and lambda to the d 1180 01:25:14,050 --> 01:25:16,940 cancel this lambda to the four minus d. 1181 01:25:16,940 --> 01:25:18,560 The kd cancels the kd. 1182 01:25:18,560 --> 01:25:22,340 The n plus 8 cancels the n plus a. 1183 01:25:22,340 --> 01:25:24,140 8 cancels the 2. 1184 01:25:24,140 --> 01:25:27,050 The answer is epsilon minus 2 epsilon, 1185 01:25:27,050 --> 01:25:28,080 which is minus epsilon. 1186 01:25:32,260 --> 01:25:34,138 OK? 1187 01:25:34,138 --> 01:25:39,010 [LAUGHTER] 1188 01:25:39,010 --> 01:25:43,235 So this direction has become irrelevant. 1189 01:25:43,235 --> 01:25:47,960 The epsilon here turn to a minus epsilon. 1190 01:25:47,960 --> 01:25:51,200 This irrelevant direction disappeared. 1191 01:25:51,200 --> 01:25:54,530 There is this relevant direction that is left, 1192 01:25:54,530 --> 01:25:57,350 which is a slightly shifted version of what 1193 01:25:57,350 --> 01:26:02,170 my original [INAUDIBLE] direction was. 1194 01:26:02,170 --> 01:26:04,730 And you can calculate yt. 1195 01:26:04,730 --> 01:26:08,180 So you go to that expression, do the same thing 1196 01:26:08,180 --> 01:26:09,970 that I did over here. 1197 01:26:09,970 --> 01:26:11,800 You'll find that at the end of the day, 1198 01:26:11,800 --> 01:26:19,630 you will find 2 minus n plus 2 over n plus 8 epsilon. 1199 01:26:19,630 --> 01:26:24,140 All these unwanted things, like kd's, these lambdas, et cetera, 1200 01:26:24,140 --> 01:26:25,610 disappear. 1201 01:26:25,610 --> 01:26:30,580 You expected at the end of the day to get pure numbers. 1202 01:26:30,580 --> 01:26:32,450 The exponents are pure numbers. 1203 01:26:32,450 --> 01:26:34,820 They don't depend on anything. 1204 01:26:34,820 --> 01:26:38,350 So we had to carry all of this baggage. 1205 01:26:38,350 --> 01:26:41,590 And at the end of the day, all of the baggage miraculously 1206 01:26:41,590 --> 01:26:43,580 disappears. 1207 01:26:43,580 --> 01:26:48,760 We get a fixed point that has only one relevant direction, 1208 01:26:48,760 --> 01:26:51,510 which is what we always wanted. 1209 01:26:51,510 --> 01:26:53,770 And once we have the exponent, we 1210 01:26:53,770 --> 01:26:55,960 can calculate everything that we want, 1211 01:26:55,960 --> 01:26:58,820 like the exponent for divergence of correlation length 1212 01:26:58,820 --> 01:27:00,950 is the inverse of that. 1213 01:27:00,950 --> 01:27:04,825 You can calculate how it has shifted from one half. 1214 01:27:04,825 --> 01:27:10,510 It is something like n plus 2 over n plus 8 epsilon. 1215 01:27:10,510 --> 01:27:14,820 And we see that the exponents now explicitly depend 1216 01:27:14,820 --> 01:27:18,320 on dimensionality of space because of this epsilon. 1217 01:27:18,320 --> 01:27:20,060 They explicitly depend on the number 1218 01:27:20,060 --> 01:27:23,370 of components of your order parameter n. 1219 01:27:23,370 --> 01:27:28,200 So we have managed, at least in some perturbative sense, 1220 01:27:28,200 --> 01:27:33,490 to demonstrate that there exists a kind of scale invariance 1221 01:27:33,490 --> 01:27:37,900 that characterizes this ON universality class. 1222 01:27:37,900 --> 01:27:40,950 And we can calculate exponents for that, at least 1223 01:27:40,950 --> 01:27:42,580 perturbatively. 1224 01:27:42,580 --> 01:27:44,720 In the process of getting that number, 1225 01:27:44,720 --> 01:27:47,070 I did things rapidly at the end, but I also 1226 01:27:47,070 --> 01:27:49,890 swept a lot of things under the rug. 1227 01:27:49,890 --> 01:27:54,050 So the task of next lecture is to go and look under the rug 1228 01:27:54,050 --> 01:27:57,240 and make sure that we haven't put anything that is important 1229 01:27:57,240 --> 01:27:58,742 away.