1 00:00:00,060 --> 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 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:23,560 --> 00:00:26,470 PROFESSOR: OK, let's start. 9 00:00:26,470 --> 00:00:30,856 So last lecture we started with the strategy of using 10 00:00:30,856 --> 00:00:40,040 perturbation theory to study our statistical themes. 11 00:00:40,040 --> 00:00:45,030 For example, we need to evaluate a partition function 12 00:00:45,030 --> 00:00:50,220 by integrating over all configurations of a field. 13 00:00:50,220 --> 00:00:54,120 Let's say n components in d dimensions 14 00:00:54,120 --> 00:01:00,100 with some kind of a weight that we can write as e to the minus 15 00:01:00,100 --> 00:01:05,740 beta H. And the strategy of perturbation 16 00:01:05,740 --> 00:01:10,270 was to find part of this Hamiltonian 17 00:01:10,270 --> 00:01:13,480 that we can calculate exactly. 18 00:01:13,480 --> 00:01:18,080 And the rest of it, hopefully treating as a small quantity 19 00:01:18,080 --> 00:01:21,980 and doing perturbative calculations. 20 00:01:21,980 --> 00:01:25,540 Now, in the context of the Landau-Ginzburg theory 21 00:01:25,540 --> 00:01:32,050 that we wrote down, this beta H0 was 22 00:01:32,050 --> 00:01:34,890 bets described in terms of Fourier modes. 23 00:01:34,890 --> 00:01:38,398 So basically, we could make a change of variables 24 00:01:38,398 --> 00:01:41,672 to integrate over all configurations of Fourier 25 00:01:41,672 --> 00:01:43,380 modes. 26 00:01:43,380 --> 00:01:48,701 And the same breakdown of the weight 27 00:01:48,701 --> 00:01:52,090 in the language of Fourier modes. 28 00:01:52,090 --> 00:01:55,360 Since the underlying theories that we were writing 29 00:01:55,360 --> 00:01:59,740 had translational symmetry, every point in space 30 00:01:59,740 --> 00:02:04,240 was the same as any other, the composition in to modes 31 00:02:04,240 --> 00:02:06,262 was immediately accomplished by going 32 00:02:06,262 --> 00:02:09,199 to Fourier representation. 33 00:02:09,199 --> 00:02:12,030 And each component of each q-value 34 00:02:12,030 --> 00:02:15,550 would correspond to essentially an independent weight 35 00:02:15,550 --> 00:02:19,450 that we could expand in some power series 36 00:02:19,450 --> 00:02:23,590 in this parameter q, which is an inverse in the wave length. 37 00:02:23,590 --> 00:02:26,260 And the lowest order terms are what 38 00:02:26,260 --> 00:02:30,440 determines the longer and longer wavelengths. 39 00:02:30,440 --> 00:02:38,230 So there is some m of q squared characterizing 40 00:02:38,230 --> 00:02:41,180 this part of the Hamiltonian. 41 00:02:41,180 --> 00:02:45,800 And since is real space we had emphasized 42 00:02:45,800 --> 00:02:53,020 some form of locality, the interaction part in real space 43 00:02:53,020 --> 00:02:58,660 could be written simply in terms of a power series, let's say, 44 00:02:58,660 --> 00:02:59,650 in m. 45 00:03:03,120 --> 00:03:08,580 Which means that if we were to then go to Fourier space, 46 00:03:08,580 --> 00:03:11,160 things that are local in real space 47 00:03:11,160 --> 00:03:13,580 become non-local in Fourier space. 48 00:03:13,580 --> 00:03:17,346 And the first of those terms that we treated 49 00:03:17,346 --> 00:03:19,980 as a perturbation would involve an integral 50 00:03:19,980 --> 00:03:25,985 over four factors of m tilde. 51 00:03:28,800 --> 00:03:33,800 Again, translational invariance forces the four q's 52 00:03:33,800 --> 00:03:40,700 that appear in the multiplication to add up to 0. 53 00:03:40,700 --> 00:03:48,410 So I would have m of q1 dot for dot with m of q2, 54 00:03:48,410 --> 00:03:54,460 m of q3 dot for dot with m of q4, 55 00:03:54,460 --> 00:03:59,210 which is minus q1 minus q2 minus q3. 56 00:03:59,210 --> 00:04:03,280 And I can go on and do higher order. 57 00:04:07,620 --> 00:04:10,260 So once we did this, we could calculate 58 00:04:10,260 --> 00:04:13,670 various-- let's say, two-point correlation functions, 59 00:04:13,670 --> 00:04:16,440 et cetera, in perturbation theory. 60 00:04:16,440 --> 00:04:20,000 And in particular, the two-point correlation function 61 00:04:20,000 --> 00:04:23,260 was related to the susceptibility. 62 00:04:23,260 --> 00:04:27,020 And setting q to 0, we found an expression 63 00:04:27,020 --> 00:04:33,010 for the inverse susceptibility where the 0 order just 64 00:04:33,010 --> 00:04:36,270 comes from the t that we have over here. 65 00:04:36,270 --> 00:04:42,580 And because of this perturbation calculated to order of u, 66 00:04:42,580 --> 00:04:51,240 we had 4u n plus 2 and integral over modes 67 00:04:51,240 --> 00:04:54,770 of just the variance of the modes if y like. 68 00:05:03,100 --> 00:05:06,000 Now, first thing that we noted was 69 00:05:06,000 --> 00:05:09,992 that the location of the point at which the susceptibility 70 00:05:09,992 --> 00:05:13,110 vanishes, or susceptibility diverges 71 00:05:13,110 --> 00:05:18,810 or its inverse vanishes is no longer at t equals to 0. 72 00:05:18,810 --> 00:05:22,675 But we can see just setting this expression to 0 that we have 73 00:05:22,675 --> 00:05:35,250 a tc, which is minus 4u n plus 2 integral d dk 2 pi to the d 1 74 00:05:35,250 --> 00:05:40,210 over-- let's put the k here, k squared, potentially 75 00:05:40,210 --> 00:05:41,366 higher-order terms. 76 00:05:45,350 --> 00:05:51,760 Now, this is an integral that in dimensions above 2-- 77 00:05:51,760 --> 00:05:56,210 let's for time being focus on dimensions above 2-- there 78 00:05:56,210 --> 00:06:00,090 is no singularity as k goes to 0. 79 00:06:00,090 --> 00:06:03,850 k goes to 0, which is long wavelength, is well-behaved. 80 00:06:03,850 --> 00:06:06,830 The integral could potentially be singular 81 00:06:06,830 --> 00:06:10,060 if I were allowed to go all the way to infinity, 82 00:06:10,060 --> 00:06:12,910 but I don't go all the infinity. 83 00:06:12,910 --> 00:06:18,520 All of my theories have an underlying short wavelength. 84 00:06:18,520 --> 00:06:22,240 And hence, there is a maximum in the Fourier modes 85 00:06:22,240 --> 00:06:26,504 which would render this completely well-behaved 86 00:06:26,504 --> 00:06:28,746 integral. 87 00:06:28,746 --> 00:06:31,870 In fact, if I forget higher-order term, 88 00:06:31,870 --> 00:06:33,150 I could put them. 89 00:06:33,150 --> 00:06:35,375 But if I forget them, I can evaluate 90 00:06:35,375 --> 00:06:38,080 what this correction tc is. 91 00:06:38,080 --> 00:06:43,190 It is minus 4u n plus 2 over k. 92 00:06:43,190 --> 00:06:49,800 This-- I've been writing-- symmetry at surface area 93 00:06:49,800 --> 00:06:57,660 of a d dimensional unit sphere divided by 2 pi to the d. 94 00:06:57,660 --> 00:07:03,240 And then I have the integral of k to the d minus 3, 95 00:07:03,240 --> 00:07:07,660 which integrates to lambda to the d minus 2 96 00:07:07,660 --> 00:07:11,780 divided by d minus 2. 97 00:07:11,780 --> 00:07:15,270 I wrote that explicitly because we 98 00:07:15,270 --> 00:07:20,880 are going to encounter this combination a lot of times. 99 00:07:20,880 --> 00:07:24,470 And so we will give it a name k sub d. 100 00:07:24,470 --> 00:07:29,710 So it's just the solid angle in d dimensions divided by 2 pi 101 00:07:29,710 --> 00:07:31,030 to the d. 102 00:07:31,030 --> 00:07:36,030 OK, so essentially, in dimensions greater than 2, 103 00:07:36,030 --> 00:07:37,410 nothing much happens. 104 00:07:37,410 --> 00:07:42,120 There is a shift in the location of the singularity compared 105 00:07:42,120 --> 00:07:43,080 to the Gaussian. 106 00:07:43,080 --> 00:07:45,800 Because you are no longer at the Gaussian, 107 00:07:45,800 --> 00:07:49,530 you are at a theory that has additional stabilizing 108 00:07:49,530 --> 00:07:51,810 terms such as m to the fourth, et cetera. 109 00:07:51,810 --> 00:07:57,190 So there is no problem now for p going to negative values. 110 00:07:57,190 --> 00:08:01,170 The thing that was more interesting 111 00:08:01,170 --> 00:08:04,700 was that when we looked at what happens 112 00:08:04,700 --> 00:08:11,170 in the vicinity of this new tc, and to lowest order 113 00:08:11,170 --> 00:08:14,910 we got this form of a divergence. 114 00:08:14,910 --> 00:08:21,550 And then at the next order, I had a correction. 115 00:08:21,550 --> 00:08:27,470 Again, coming from this form, 4u n plus 2, an integral. 116 00:08:30,560 --> 00:08:34,650 And actually, this was obtained by taking the difference of two 117 00:08:34,650 --> 00:08:38,080 of these factors evaluated at p and tc. 118 00:08:38,080 --> 00:08:41,410 That's what gave me the t minus tc outside. 119 00:08:41,410 --> 00:08:51,450 And then I had an integral that involved two of these factors. 120 00:08:51,450 --> 00:08:54,690 Presumably to be consistent to lowest order, 121 00:08:54,690 --> 00:08:59,220 I have to evaluate them as small as I can. 122 00:08:59,220 --> 00:09:02,660 And so I would have two factors of k squared or k squared 123 00:09:02,660 --> 00:09:03,570 plus something. 124 00:09:08,500 --> 00:09:09,918 Presumably, higher-order terms. 125 00:09:14,310 --> 00:09:19,350 The thing about these integrals as opposed to the previous one 126 00:09:19,350 --> 00:09:21,800 is that, again, I can try to look 127 00:09:21,800 --> 00:09:25,330 at the behavior at large k and small k. 128 00:09:25,330 --> 00:09:28,330 At large k, no matter how many terms 129 00:09:28,330 --> 00:09:31,070 I add to the series, ultimately, I 130 00:09:31,070 --> 00:09:36,180 will be concerned by cutting it off by lambda. 131 00:09:36,180 --> 00:09:38,390 Whereas, if I have something that I 132 00:09:38,390 --> 00:09:43,030 have set t equals to 0 in both of these denominator factors, 133 00:09:43,030 --> 00:09:45,920 I now have a singularity at k goes 134 00:09:45,920 --> 00:09:49,200 to 0 in dimensions less than 4. 135 00:09:49,200 --> 00:09:52,600 The integral would blow up in dimensions less than 4 136 00:09:52,600 --> 00:09:56,830 if I am allowed to go all the way to 0, which is arbitrarily 137 00:09:56,830 --> 00:09:59,480 long wavelengths. 138 00:09:59,480 --> 00:10:03,430 Now in principle, if I am not exactly at tc 139 00:10:03,430 --> 00:10:07,390 and I'm looking at singularity as being away from tc, 140 00:10:07,390 --> 00:10:11,476 I expect on physical grounds that fluctuations 141 00:10:11,476 --> 00:10:16,530 will persist up to some correlation length. 142 00:10:16,530 --> 00:10:21,320 So the shortest value of k that I should really physically 143 00:10:21,320 --> 00:10:27,570 be able to go to, irrespective of how careful or careless 144 00:10:27,570 --> 00:10:31,640 I am with the factors of t and t minus tc that I put here, 145 00:10:31,640 --> 00:10:34,830 is of the order of the physical correlation length. 146 00:10:34,830 --> 00:10:37,870 And as we saw, this means that there 147 00:10:37,870 --> 00:10:42,080 is a correction that is of the form of u k 148 00:10:42,080 --> 00:10:47,560 to the 4-- k squared psi to the power of 4 minus d. 149 00:10:47,560 --> 00:10:53,620 And I emphasized that the dimensionless combination 150 00:10:53,620 --> 00:10:56,320 of the parameter u that potentially 151 00:10:56,320 --> 00:10:59,880 can be added as a correction to a number of order of 1 152 00:10:59,880 --> 00:11:04,650 is u k squared divided by some-- multiplied by some length scale 153 00:11:04,650 --> 00:11:06,775 to the power of 4 minus d. 154 00:11:06,775 --> 00:11:09,210 Above four dimensions, the integral 155 00:11:09,210 --> 00:11:12,720 is convergent at small values. 156 00:11:12,720 --> 00:11:14,630 And the integral will be dominated 157 00:11:14,630 --> 00:11:17,440 and the length scale that would appear here 158 00:11:17,440 --> 00:11:21,440 would be some kind of a short distance cutoff, 159 00:11:21,440 --> 00:11:23,170 like the averaging length. 160 00:11:23,170 --> 00:11:29,140 Whereas in four dimensions with divergence of the correlation 161 00:11:29,140 --> 00:11:33,085 length is the thing that will lead this perturbation 162 00:11:33,085 --> 00:11:35,771 theory to be kind of difficult and [INAUDIBLE]. 163 00:11:40,490 --> 00:11:48,690 So this is an example of a divergent perturbation theory. 164 00:11:48,690 --> 00:11:50,930 So what we are going to do in order 165 00:11:50,930 --> 00:11:53,930 to be able to make sense out of it, 166 00:11:53,930 --> 00:11:58,600 and see how this divergence here can be translated 167 00:11:58,600 --> 00:12:01,940 to a change in exponent, which is what we are physically 168 00:12:01,940 --> 00:12:06,330 expecting to occur, we reorganize this perturbation 169 00:12:06,330 --> 00:12:10,190 theory in a conceptual way that is 170 00:12:10,190 --> 00:12:17,258 helped by this perturbative renormalization group approach. 171 00:12:17,258 --> 00:12:21,490 So we keep the perturbation theory, 172 00:12:21,490 --> 00:12:25,200 but change the way that we look at perturbation theory 173 00:12:25,200 --> 00:12:26,910 by appealing to renormalization. 174 00:12:31,900 --> 00:12:36,626 So you can see that throughout doing this perturbation theory, 175 00:12:36,626 --> 00:12:44,890 I end up having to do integrals over modes that are defined 176 00:12:44,890 --> 00:12:49,016 in the space of the Fourier parameter q. 177 00:12:49,016 --> 00:12:54,280 And a nice way to implement this coarse graining that 178 00:12:54,280 --> 00:12:58,310 has led to this field theory is to imagine 179 00:12:58,310 --> 00:13:03,700 that this integration is over some sphere where 180 00:13:03,700 --> 00:13:09,410 the maximum inverse wavelength that is allowed, 181 00:13:09,410 --> 00:13:14,090 or q number that is allowed is some lambda. 182 00:13:14,090 --> 00:13:18,050 And so the task that I have on the first line 183 00:13:18,050 --> 00:13:24,430 is to integrate over all modes that live in this. 184 00:13:24,430 --> 00:13:26,950 And just on physical grounds, we don't 185 00:13:26,950 --> 00:13:31,235 expect to get any singularities from the modes that 186 00:13:31,235 --> 00:13:33,410 are at the edge. 187 00:13:33,410 --> 00:13:37,890 We expect to get singularities by considering what's 188 00:13:37,890 --> 00:13:43,200 going on at long wavelengths or 0q. 189 00:13:43,200 --> 00:13:47,980 So the idea of renormalization group 190 00:13:47,980 --> 00:13:50,650 was to follow three steps. 191 00:13:50,650 --> 00:13:58,960 The first step was to do coarse graining, which 192 00:13:58,960 --> 00:14:05,140 was to take whatever your shortest wavelength was, 193 00:14:05,140 --> 00:14:10,350 make it b times larger and average. 194 00:14:10,350 --> 00:14:11,700 That's in real space. 195 00:14:11,700 --> 00:14:13,570 In Fourier space, what that amounts 196 00:14:13,570 --> 00:14:23,640 to is to get rid of all of the variations and frequencies that 197 00:14:23,640 --> 00:14:26,753 are up to lambda over b. 198 00:14:29,980 --> 00:14:37,680 So what I can do is to say that I had a whole bunch of modes 199 00:14:37,680 --> 00:14:38,962 m of q. 200 00:14:41,620 --> 00:14:47,120 I am going to subdivide them into two classes. 201 00:14:47,120 --> 00:14:51,500 I will have the modes sigma of q. 202 00:14:51,500 --> 00:14:54,950 Maybe I will write it in different color, sigma of q. 203 00:14:58,180 --> 00:15:03,610 That are the ones that sitting here. 204 00:15:03,610 --> 00:15:06,390 So these are the sigmas. 205 00:15:06,390 --> 00:15:09,520 And these correspond to wave numbers 206 00:15:09,520 --> 00:15:15,240 that are between lambda over b and lambda. 207 00:15:15,240 --> 00:15:20,340 And I will have a bunch of other variables 208 00:15:20,340 --> 00:15:22,387 that I will call m tilde. 209 00:15:27,750 --> 00:15:35,080 And this wave close to the singularity, 210 00:15:35,080 --> 00:15:40,930 but now removed by an amount lambda over b. 211 00:15:40,930 --> 00:15:45,260 So essentially, getting rid of the picture 212 00:15:45,260 --> 00:15:49,150 where it had fluctuations at short length scale 213 00:15:49,150 --> 00:15:54,370 amounts to integrating over Fourier modes that 214 00:15:54,370 --> 00:15:58,352 represent your field that lie in this integral. 215 00:16:02,220 --> 00:16:06,585 So I want to do that as an operation that is performed, 216 00:16:06,585 --> 00:16:10,580 let's say, at the level of the partition function. 217 00:16:10,580 --> 00:16:16,190 So I can say that my original integration can be broken up 218 00:16:16,190 --> 00:16:22,840 into integration over this m tilde 219 00:16:22,840 --> 00:16:25,822 and the integration over sigma. 220 00:16:25,822 --> 00:16:33,860 So that's just rewriting that rightmost integral up there. 221 00:16:33,860 --> 00:16:40,760 And then I have a weight exponential. 222 00:16:40,760 --> 00:16:44,420 OK, let's write it out explicitly. 223 00:16:44,420 --> 00:16:52,220 So the weight is composed of beta H0 and the u. 224 00:16:52,220 --> 00:16:58,160 Now, we note that the beta H0 part, just as we did already 225 00:16:58,160 --> 00:17:00,670 for the case of the Gaussian, does not 226 00:17:00,670 --> 00:17:04,869 mix up these two classes of modes. 227 00:17:04,869 --> 00:17:08,460 So I can write that part as an integral from 0 228 00:17:08,460 --> 00:17:16,269 to lambda over b dd 2 pi over d. 229 00:17:16,269 --> 00:17:19,800 And these are things that are really inside, 230 00:17:19,800 --> 00:17:23,289 so I could also label them by q lesser. 231 00:17:23,289 --> 00:17:27,955 So I have m tilde of q lesser squared. 232 00:17:30,530 --> 00:17:35,940 And this multiplies t plus k q lesser 233 00:17:35,940 --> 00:17:38,869 squared and so forth over 2. 234 00:17:43,560 --> 00:17:49,650 I have a similar term, which is the modes that 235 00:17:49,650 --> 00:17:54,330 are between lambda over b and lambda. 236 00:17:54,330 --> 00:18:01,652 So I just simply changed or broke my overall integration 237 00:18:01,652 --> 00:18:06,380 in beta H0 into two parts. 238 00:18:06,380 --> 00:18:11,340 Now I have the higher q numbers. 239 00:18:11,340 --> 00:18:18,570 And these are sigma of q larger squared. 240 00:18:18,570 --> 00:18:23,820 Again, same weight, t plus k q larger 241 00:18:23,820 --> 00:18:26,447 squared and so forth over 2. 242 00:18:30,279 --> 00:18:34,460 Make sure this minus is in line with this. 243 00:18:34,460 --> 00:18:36,700 And then I have, of course, the u. 244 00:18:36,700 --> 00:18:42,060 So then I have a minus u. 245 00:18:42,060 --> 00:18:43,870 Now, I won't write this explicitly. 246 00:18:43,870 --> 00:18:47,510 I will write it explicitly on the next board. 247 00:18:47,510 --> 00:18:57,510 But clearly, implicitly it involves both m tilde and sigma 248 00:18:57,510 --> 00:18:58,780 mixed up into each other. 249 00:19:07,930 --> 00:19:11,580 So I have just rewritten my partition function 250 00:19:11,580 --> 00:19:16,170 after subdividing it into these two classes of modes 251 00:19:16,170 --> 00:19:19,150 and just hiding all of the complexity 252 00:19:19,150 --> 00:19:21,454 in this function that mixes the two. 253 00:19:25,570 --> 00:19:29,690 So let's rewrite this. 254 00:19:29,690 --> 00:19:34,830 I have integral over the modes that I would like to keep. 255 00:19:34,830 --> 00:19:37,320 The m tilde I would like to keep. 256 00:19:40,010 --> 00:19:43,370 And there is a weight associated with them 257 00:19:43,370 --> 00:19:46,095 that will, therefore, not be integrated. 258 00:19:46,095 --> 00:19:55,870 This is the integral from 0 to lambda over b dd k lesser 2 259 00:19:55,870 --> 00:20:02,720 pi to the d t plus k, k lesser q lesser squared, et cetera. 260 00:20:02,720 --> 00:20:10,642 And then I have tilde of q lesser squared. 261 00:20:15,720 --> 00:20:22,260 Now, if I didn't have this there, the u, 262 00:20:22,260 --> 00:20:27,680 I could immediately perform the Gaussian integrals over sigmas. 263 00:20:27,680 --> 00:20:30,030 Indeed, we already did this. 264 00:20:30,030 --> 00:20:32,840 And the answer would be e to the minus-- 265 00:20:32,840 --> 00:20:35,860 there are n-components to this vector. 266 00:20:35,860 --> 00:20:40,520 So the answer is going to be multiplied by n. 267 00:20:40,520 --> 00:20:44,440 1/2 is because the square root that I get from each mode. 268 00:20:44,440 --> 00:20:47,916 I get a factor of volume integral dd 269 00:20:47,916 --> 00:20:53,560 q larger 2 pi to the d integrated from lambda over v 270 00:20:53,560 --> 00:21:01,090 to lambda log of t plus k q greater squared and so forth. 271 00:21:01,090 --> 00:21:04,190 So if I didn't have the u, this would 272 00:21:04,190 --> 00:21:08,840 be the answer for doing the Gaussian integration. 273 00:21:08,840 --> 00:21:12,030 But I have the u, so what should I do? 274 00:21:12,030 --> 00:21:14,150 The answer is very simple. 275 00:21:14,150 --> 00:21:22,900 I write it as e to the minus u m tilde m sigma average. 276 00:21:27,840 --> 00:21:37,270 So what I have done is to say that with this weight, that 277 00:21:37,270 --> 00:21:41,490 is a Gaussian weight for sigma, I 278 00:21:41,490 --> 00:21:45,760 average the function e to the minus u. 279 00:21:45,760 --> 00:21:53,020 If you like, this is a Gaussian sigma. 280 00:21:53,020 --> 00:21:57,960 So to sort of write it explicitly, what I have stated 281 00:21:57,960 --> 00:22:03,280 is an average where I integrate out 282 00:22:03,280 --> 00:22:07,920 the high-frequency short wavelength modes 283 00:22:07,920 --> 00:22:14,740 is by definition integrate over all configurations of sigma 284 00:22:14,740 --> 00:22:24,800 with the Gaussian weight whatever object you have, 285 00:22:24,800 --> 00:22:29,490 and then normalize by the Gaussian. 286 00:22:35,310 --> 00:22:41,284 Of course, in our case, our O depends both on sigma and m 287 00:22:41,284 --> 00:22:44,370 tilde, so the result of these averaging 288 00:22:44,370 --> 00:22:48,810 will be a function of in tilde. 289 00:22:48,810 --> 00:23:00,590 And indeed, I can write this as an integral over m tilde of q 290 00:23:00,590 --> 00:23:05,530 with a new weight, e to the minus beta H tilde, which 291 00:23:05,530 --> 00:23:08,760 only depends on m tilde because I got rid of 292 00:23:08,760 --> 00:23:10,860 and I integrated over the sigmas. 293 00:23:13,520 --> 00:23:22,740 And by definition, my beta H tilde 294 00:23:22,740 --> 00:23:28,810 that depends only on m tilde has a part 295 00:23:28,810 --> 00:23:40,380 that is the integral from 0 to lambda over b dd q lesser 296 00:23:40,380 --> 00:23:56,340 2 pi to the d the Gaussian weight over the range of modes 297 00:23:56,340 --> 00:23:59,530 that are allowed. 298 00:23:59,530 --> 00:24:03,250 There is a part that is just this constant term when 299 00:24:03,250 --> 00:24:06,580 I take care-- if I write it in this fashion, 300 00:24:06,580 --> 00:24:09,040 there is an overall constant. 301 00:24:09,040 --> 00:24:15,360 Clearly, what this constant is, is the free energy of the modes 302 00:24:15,360 --> 00:24:19,380 that I have integrated, assuming that they are Gaussian, 303 00:24:19,380 --> 00:24:21,700 in this interval. 304 00:24:21,700 --> 00:24:24,670 The answer is proportional to volume. 305 00:24:24,670 --> 00:24:26,610 But as usual, when we are thinking 306 00:24:26,610 --> 00:24:30,270 about weights and probabilities, overall constants don't matter. 307 00:24:30,270 --> 00:24:34,160 But I can certainly continue to write that over here. 308 00:24:34,160 --> 00:24:39,750 So that part went to here, this part went to here. 309 00:24:39,750 --> 00:24:43,130 And so the only part that is left 310 00:24:43,130 --> 00:24:51,510 is minus log of d to the minus u of m tilde and sigma 311 00:24:51,510 --> 00:24:53,982 after I get rid of the sigmas. 312 00:25:02,340 --> 00:25:06,955 So far, I have done things that are extremely general. 313 00:25:09,490 --> 00:25:16,180 But now I note that I am interested in doing 314 00:25:16,180 --> 00:25:19,100 perturbation. 315 00:25:19,100 --> 00:25:23,900 So the only place that I haven't really evaluated things 316 00:25:23,900 --> 00:25:27,860 is where this u is appearing inside the exponential log 317 00:25:27,860 --> 00:25:30,500 average, et cetera. 318 00:25:30,500 --> 00:25:34,670 So what I can do is I can perturbatively 319 00:25:34,670 --> 00:25:38,376 expand this exponential over here. 320 00:25:38,376 --> 00:25:43,830 So I will get log of 1 minus u, which is approximately minus u. 321 00:25:43,830 --> 00:25:50,890 So the first term here would be u averaged Gaussian. 322 00:25:50,890 --> 00:25:57,440 The next term will be minus 1/2 u squared 323 00:25:57,440 --> 00:26:06,440 average minus u average squared and so forth. 324 00:26:06,440 --> 00:26:11,850 So you can see that the variance appeared in this stage. 325 00:26:11,850 --> 00:26:17,120 And generally, the l-th term in the series would be minus 1 326 00:26:17,120 --> 00:26:22,470 to the l divided by l factorial. 327 00:26:22,470 --> 00:26:26,660 And we saw this already. 328 00:26:26,660 --> 00:26:32,490 The log of e to the something is the generator of cumulants. 329 00:26:32,490 --> 00:26:37,890 So this would be the l-th power of u, 330 00:26:37,890 --> 00:26:41,500 the cumulant here would appear. 331 00:26:41,500 --> 00:26:45,205 And again, the cumulant would serve the function 332 00:26:45,205 --> 00:26:48,770 of cutting off connected pieces as we shall see shortly. 333 00:26:55,580 --> 00:26:59,270 So that's what we are going to do. 334 00:26:59,270 --> 00:27:05,290 We are going to insert the u's, all things that 335 00:27:05,290 --> 00:27:07,920 go beyond the Gaussian-- but initially, 336 00:27:07,920 --> 00:27:13,005 just the m to the fourth part-- inside this series 337 00:27:13,005 --> 00:27:18,060 and term by term calculate the corrections to this weight 338 00:27:18,060 --> 00:27:24,950 that we get after we integrate out the long lambda. 339 00:27:24,950 --> 00:27:30,291 Or sorry, the short wavelength modes or the long q modes. 340 00:27:30,291 --> 00:27:30,790 OK? 341 00:27:33,870 --> 00:27:38,430 So let's focus on this first term. 342 00:27:38,430 --> 00:27:45,050 So what is this u that depends on both m tilde and sigma? 343 00:27:48,830 --> 00:27:52,577 And I have the expression for u up there. 344 00:27:55,560 --> 00:28:07,030 So I can write it as u integral dd q1 dd q2 dd q3 to be 345 00:28:07,030 --> 00:28:10,686 symmetric in all of the four q's. 346 00:28:10,686 --> 00:28:15,740 I write an integration over the fourth q, 347 00:28:15,740 --> 00:28:19,060 but then enforce it by a delta function 348 00:28:19,060 --> 00:28:23,620 that the sum of the q's should be 0. 349 00:28:29,260 --> 00:28:34,460 And then I have four factors of m, 350 00:28:34,460 --> 00:28:40,540 but an m depending on which part of the q space 351 00:28:40,540 --> 00:28:45,990 I am encountering is either a sigma or an m tilde. 352 00:28:45,990 --> 00:28:48,320 So without doing anything wrong, I 353 00:28:48,320 --> 00:28:55,070 can replace each m with an m tilde plus sigma. 354 00:29:02,790 --> 00:29:07,200 So depending on where my q1 is in the integrations from 0 355 00:29:07,200 --> 00:29:11,630 to lambda, I will be encountering either this 356 00:29:11,630 --> 00:29:13,660 or this. 357 00:29:13,660 --> 00:29:20,230 And then I have the dot product of that with m tilde q2 358 00:29:20,230 --> 00:29:23,015 plus sigma of q2. 359 00:29:23,015 --> 00:29:25,230 And then I have the dot product that 360 00:29:25,230 --> 00:29:33,914 would correspond to m tilde of q3 plus sigma of q3 361 00:29:33,914 --> 00:29:40,096 with m tilde of q4 plus sigma of q4. 362 00:29:43,670 --> 00:29:45,550 So that's the structure of my [INAUDIBLE]. 363 00:29:49,850 --> 00:29:52,900 And again, what I have to do in principle 364 00:29:52,900 --> 00:29:59,540 is to integrate out the sigmas keeping the m tildes when 365 00:29:59,540 --> 00:30:02,910 I perform this averaging over here. 366 00:30:09,470 --> 00:30:12,530 So let's write down, if I were to expand 367 00:30:12,530 --> 00:30:16,670 this thing before the integration, what 368 00:30:16,670 --> 00:30:20,580 are the types of terms that I would get? 369 00:30:20,580 --> 00:30:23,264 And I'll give them names. 370 00:30:26,050 --> 00:30:29,800 One type of term that is very easy 371 00:30:29,800 --> 00:30:36,260 is when I have m tilde of q1 dotted with m tilde of Q2, 372 00:30:36,260 --> 00:30:41,880 m tilde of q3 dotted with m tilde of q4. 373 00:30:47,460 --> 00:30:50,670 If I expand this so there's 2 terms per bracket 374 00:30:50,670 --> 00:30:54,540 and there are 4 brackets, so there are 16 terms, only 1 375 00:30:54,540 --> 00:31:01,210 of these terms is of this variety out of the 16. 376 00:31:01,210 --> 00:31:04,520 What I will do is also now introduce 377 00:31:04,520 --> 00:31:08,030 a diagrammatic representation. 378 00:31:08,030 --> 00:31:15,740 Whenever I see an m tilde, I will include a straight line. 379 00:31:15,740 --> 00:31:22,140 Whenever I see a sigma, I will include a wavy line. 380 00:31:22,140 --> 00:31:24,790 So this entity that I have over here 381 00:31:24,790 --> 00:31:28,980 is composed of four of these straight lines. 382 00:31:28,980 --> 00:31:40,000 And I will indicate that by this diagram, q1, q2, q3, q4. 383 00:31:40,000 --> 00:31:43,010 And the reason is, of course-- first of all, 384 00:31:43,010 --> 00:31:45,136 there are four of these. 385 00:31:45,136 --> 00:31:49,780 So this is a so-called vertex in a diagrammatic representation 386 00:31:49,780 --> 00:31:51,940 that has four lines. 387 00:31:51,940 --> 00:31:57,830 And secondly, the lines are not all totally equivalent because 388 00:31:57,830 --> 00:32:01,030 of the way that the dot products are arranged. 389 00:32:01,030 --> 00:32:06,250 Say, q1 and q2 that are dot product together are distinct, 390 00:32:06,250 --> 00:32:11,020 let's say, from q1 and q3 that are not dot product together. 391 00:32:11,020 --> 00:32:14,240 And to indicate that, I make sure that there 392 00:32:14,240 --> 00:32:19,040 is this dotted line in the vertex that separates 393 00:32:19,040 --> 00:32:22,445 and indicates which two are dot product to each other. 394 00:32:27,470 --> 00:32:32,200 Now, the second class of diagram comes 395 00:32:32,200 --> 00:32:36,410 when I replace one of the m tildes with a sigma. 396 00:32:36,410 --> 00:32:41,240 So I have sigma of q1 dotted with m tilde 397 00:32:41,240 --> 00:32:51,700 of q2, m tilde of q3 dotted with m tilde of q4. 398 00:32:51,700 --> 00:32:55,100 Now clearly, in this case, I had a choice 399 00:32:55,100 --> 00:33:00,420 of four factors of m tilde to replace with this. 400 00:33:00,420 --> 00:33:04,780 So of the 16 terms in this expansion, 4 of them 401 00:33:04,780 --> 00:33:06,880 belong to this class. 402 00:33:06,880 --> 00:33:10,040 Which if I were to represent diagrammatically, 403 00:33:10,040 --> 00:33:12,480 I would have one of the legs replaced 404 00:33:12,480 --> 00:33:16,430 with a wavy line and all the other legs 405 00:33:16,430 --> 00:33:18,820 staying as solid lines. 406 00:33:25,530 --> 00:33:30,200 The third class of terms correspond 407 00:33:30,200 --> 00:33:34,630 to replacing two of the m tildes with sigmas. 408 00:33:34,630 --> 00:33:38,250 Now here again, I have a choice whether the second one 409 00:33:38,250 --> 00:33:40,170 is a partner of the first one that 410 00:33:40,170 --> 00:33:44,260 became sigma, such as this one, sigma of q1 411 00:33:44,260 --> 00:33:51,240 dotted with sigma of q2, m tilde of q3 dotted 412 00:33:51,240 --> 00:33:54,140 with m tilde of q4. 413 00:33:54,140 --> 00:33:56,360 And then clearly, I could have chosen 414 00:33:56,360 --> 00:34:01,040 one pair or the other pair to change into sigmas. 415 00:34:01,040 --> 00:34:04,830 So there are two terms that are like this. 416 00:34:04,830 --> 00:34:09,050 And diagrammatically, the wavy lines 417 00:34:09,050 --> 00:34:13,400 belong to the same branch of this object. 418 00:34:20,409 --> 00:34:21,239 OK, next. 419 00:34:21,239 --> 00:34:23,270 Keep going. 420 00:34:23,270 --> 00:34:25,429 Actually, I have another thing when 421 00:34:25,429 --> 00:34:28,840 I replace two of the m tildes with sigma, 422 00:34:28,840 --> 00:34:32,389 but now belonging to two different elements of this dot 423 00:34:32,389 --> 00:34:33,173 product. 424 00:34:33,173 --> 00:34:39,659 So I could have sigma of q1 dotted with m tilde of q2. 425 00:34:39,659 --> 00:34:44,699 And then I have sigma of q3 dotted with m tilde of q4. 426 00:34:48,830 --> 00:34:54,179 In which case, in each of the pairs I had a choice of two 427 00:34:54,179 --> 00:34:56,590 for replacing m tilde with sigma. 428 00:34:56,590 --> 00:34:58,390 So that's 2 times 2. 429 00:34:58,390 --> 00:35:01,990 There are four terms that have this character. 430 00:35:01,990 --> 00:35:05,560 And if I were to represent them diagrammatically, 431 00:35:05,560 --> 00:35:08,070 I would need to put two wavy lines 432 00:35:08,070 --> 00:35:09,189 on two different branches. 433 00:35:14,460 --> 00:35:20,410 And then I have the possibility of three things replaced. 434 00:35:20,410 --> 00:35:32,450 So I have sigma of q1 sigma of q2 sigma of q3 m tilde of q4. 435 00:35:32,450 --> 00:35:35,800 And again, now it's the other way around. 436 00:35:35,800 --> 00:35:39,120 One term is left out of 4 to be m tilde. 437 00:35:39,120 --> 00:35:43,990 So this is, again, a degeneracy of 4. 438 00:35:43,990 --> 00:35:50,790 And diagrammatically, I have three lines that are wavy 439 00:35:50,790 --> 00:35:54,060 and one line that is solid. 440 00:35:54,060 --> 00:35:58,040 And at the end of story, 6, I will 441 00:35:58,040 --> 00:36:10,840 have one diagram which is all sigmas, which 442 00:36:10,840 --> 00:36:14,113 can be represented essentially by all wavy lines. 443 00:36:21,160 --> 00:36:25,190 And to check that I didn't make any mistake in my calculation, 444 00:36:25,190 --> 00:36:28,330 the sum of these numbers better be 16. 445 00:36:28,330 --> 00:36:31,728 So that's 5, 7, 11, 15, 16. 446 00:36:34,440 --> 00:36:34,940 All right? 447 00:36:38,220 --> 00:36:44,320 Now, the next step of the story is to do these averages. 448 00:36:44,320 --> 00:36:46,144 So I have to do the average. 449 00:37:02,060 --> 00:37:07,230 Now, the first term doesn't involve any sigmas. 450 00:37:07,230 --> 00:37:09,610 All of my averages here are obtained 451 00:37:09,610 --> 00:37:12,250 by integrating over sigmas. 452 00:37:12,250 --> 00:37:14,985 If there is no sigmas to integrate, 453 00:37:14,985 --> 00:37:18,120 after I do the averaging here I essentially 454 00:37:18,120 --> 00:37:19,680 get the same thing back. 455 00:37:19,680 --> 00:37:27,670 So I will get this same expression. 456 00:37:27,670 --> 00:37:30,640 And clearly, that would be a term 457 00:37:30,640 --> 00:37:34,710 that would contribute to my beta H tilde, which 458 00:37:34,710 --> 00:37:37,460 is identical to what I had originally. 459 00:37:37,460 --> 00:37:40,300 It is, again, m to the fourth. 460 00:37:40,300 --> 00:37:43,630 So that we understand. 461 00:37:43,630 --> 00:37:48,920 Now, the second term here, what is 462 00:37:48,920 --> 00:37:50,620 the average that I have to do here? 463 00:37:54,140 --> 00:37:58,270 I have one factor of sigma with which I can average. 464 00:37:58,270 --> 00:38:03,290 But the weight that I have is even in sigma. 465 00:38:03,290 --> 00:38:08,640 So the average of sigma, which is Gaussian-distributed, is 0. 466 00:38:08,640 --> 00:38:10,413 So this will give me 0. 467 00:38:13,180 --> 00:38:17,420 And clearly, here also there is a term 468 00:38:17,420 --> 00:38:19,885 that involves three factors of sigma. 469 00:38:19,885 --> 00:38:22,022 Again, by symmetry this will average out to 0. 470 00:38:31,850 --> 00:38:40,060 Now, there is a way of indicating what happens here. 471 00:38:40,060 --> 00:38:43,370 See, what happens here is that I will 472 00:38:43,370 --> 00:38:47,145 have to do an average of this thing. 473 00:38:47,145 --> 00:38:50,080 The m tildes are not part of the averaging. 474 00:38:50,080 --> 00:38:51,980 They just go out. 475 00:38:51,980 --> 00:38:54,550 The average moves all the way over here. 476 00:38:57,540 --> 00:39:02,460 And the average of sigma of q1, sigma of q2, I know what it is. 477 00:39:02,460 --> 00:39:06,620 It is going to be-- I could have just written it over here. 478 00:39:06,620 --> 00:39:13,740 It's 2 pi to the d delta function q1 plus q2 479 00:39:13,740 --> 00:39:16,560 divided by k q squared. 480 00:39:16,560 --> 00:39:18,590 Maybe I'll explicitly write it over here. 481 00:39:18,590 --> 00:39:21,500 So what we have here is that the average of sigma 482 00:39:21,500 --> 00:39:31,086 of q1 with some index sigma of q2 with some other index 483 00:39:31,086 --> 00:39:33,377 is-- first of all, the two indices have to be the same. 484 00:39:36,000 --> 00:39:39,332 I have a delta function q1 plus q2. 485 00:39:39,332 --> 00:39:46,084 And then I have t plus k q1 squared and so forth. 486 00:39:46,084 --> 00:39:48,850 So it's my usual Gaussian. 487 00:39:48,850 --> 00:39:53,370 So essentially, you can see that one immediate consequence 488 00:39:53,370 --> 00:39:57,520 of this averaging is that previously these things had 489 00:39:57,520 --> 00:39:59,770 two different momenta and potentially 490 00:39:59,770 --> 00:40:01,750 two different indices. 491 00:40:01,750 --> 00:40:05,670 They get to be the same thing. 492 00:40:05,670 --> 00:40:10,690 And the fact that the labels that 493 00:40:10,690 --> 00:40:14,690 were assigned to this, the q and the index alpha, 494 00:40:14,690 --> 00:40:21,890 are forced to be the same, we can diagrammatically 495 00:40:21,890 --> 00:40:25,570 indicate by making this a closed line. 496 00:40:25,570 --> 00:40:30,230 So we are going to represent the result of that averaging 497 00:40:30,230 --> 00:40:36,250 with essentially taking-- these two lines are unchanged. 498 00:40:36,250 --> 00:40:37,880 They can be whatever they were. 499 00:40:37,880 --> 00:40:42,640 These two lines really are joined together 500 00:40:42,640 --> 00:40:44,840 through this process. 501 00:40:44,840 --> 00:40:48,850 So we indicate them that way. 502 00:40:48,850 --> 00:40:53,510 And similarly, when I do the same thing over here, 503 00:40:53,510 --> 00:40:59,120 I do the averaging of this and the answer 504 00:40:59,120 --> 00:41:05,470 I can indicate by leaving these two lines by themselves 505 00:41:05,470 --> 00:41:08,845 and joining these two wavy lines together in this fashion. 506 00:41:16,770 --> 00:41:19,680 Now, when you do-- this one we said is 0. 507 00:41:19,680 --> 00:41:23,480 So there's essentially one that is left, which is number 6. 508 00:41:23,480 --> 00:41:25,395 For number 6, we do our averaging. 509 00:41:27,920 --> 00:41:33,090 And for that we have to use- for average or a product of four 510 00:41:33,090 --> 00:41:36,050 sigmas that are Gaussian-distributed Wick's 511 00:41:36,050 --> 00:41:37,550 theorem. 512 00:41:37,550 --> 00:41:41,840 So one possibility is that sigma 1 and sigma 2 are joined, 513 00:41:41,840 --> 00:41:45,520 and then sigma 4 and sigma 3 have to be joined. 514 00:41:45,520 --> 00:41:48,760 So basically, I took sigma 1 and sigma 2 515 00:41:48,760 --> 00:41:53,670 and joined them, sigma 3 and sigma 4 that I joined them. 516 00:41:53,670 --> 00:41:56,210 But another possibility is I can take 517 00:41:56,210 --> 00:41:58,980 sigma 1 with sigma 3 or sigma 4. 518 00:41:58,980 --> 00:42:01,650 So there are really two choices. 519 00:42:01,650 --> 00:42:05,098 And then I will have a diagram that is like this. 520 00:42:19,050 --> 00:42:24,010 Now, each one of these operations and diagrams 521 00:42:24,010 --> 00:42:28,430 really stands for some integration and result. 522 00:42:28,430 --> 00:42:32,730 And let's for example, pick our number 3. 523 00:42:38,670 --> 00:42:42,890 For number 3, what we are supposed to do 524 00:42:42,890 --> 00:42:46,090 is to do the integration. 525 00:42:46,090 --> 00:42:47,030 Sorry. 526 00:42:47,030 --> 00:42:52,750 First of all, number 3 has a numerical factor of 2. 527 00:42:52,750 --> 00:42:55,180 This is something that is proportional to u 528 00:42:55,180 --> 00:42:56,730 when we take the average. 529 00:42:59,590 --> 00:43:11,710 I have in principle to do integration over q1 q2 q3 q4. 530 00:43:22,895 --> 00:43:23,395 OK. 531 00:43:26,260 --> 00:43:34,470 The m tilde of q3 and m tilde of q4 in this diagram 532 00:43:34,470 --> 00:43:37,850 were not averaged over. 533 00:43:37,850 --> 00:43:39,240 So that term remains. 534 00:43:42,140 --> 00:43:48,840 I did the averaging over q1 and q2. 535 00:43:48,840 --> 00:43:51,760 When I did that averaging, I, first of all, 536 00:43:51,760 --> 00:43:56,720 got a delta alpha alpha because those were two things that 537 00:43:56,720 --> 00:43:58,780 were dot product to each other, so they 538 00:43:58,780 --> 00:44:01,690 were carrying the same index to start with. 539 00:44:01,690 --> 00:44:06,220 I have a 2 pi to the d, a delta function q1 plus q2. 540 00:44:06,220 --> 00:44:11,485 And I have t plus k q1 squared. 541 00:44:17,780 --> 00:44:19,790 Now, delta alpha alpha. 542 00:44:19,790 --> 00:44:25,060 Summing over alpha gives a factor of n. 543 00:44:25,060 --> 00:44:28,420 And when you look at these diagrams, 544 00:44:28,420 --> 00:44:32,836 quite generally whenever you see a loop, 545 00:44:32,836 --> 00:44:37,080 with a loop you would associate the factor of n because 546 00:44:37,080 --> 00:44:42,080 of the index that runs and gets summed over. 547 00:44:42,080 --> 00:44:47,840 So this answer is going to be proportional to 2 u n. 548 00:44:51,660 --> 00:44:57,960 Now, q1 and q2 are said to be 0, the sum. 549 00:44:57,960 --> 00:44:59,810 So this is 0. 550 00:44:59,810 --> 00:45:05,140 So q3 and q4 have to add up to 0. 551 00:45:05,140 --> 00:45:09,290 So the part that involves q3 and q4, the m tilde, 552 00:45:09,290 --> 00:45:13,720 essentially I will get an integral dd-- 553 00:45:13,720 --> 00:45:15,980 let's say whatever q3. 554 00:45:15,980 --> 00:45:19,810 It doesn't matter because it's an index of integration. 555 00:45:19,810 --> 00:45:26,732 I have m tilde of q3 squared. 556 00:45:26,732 --> 00:45:31,180 And again, q3, it is something that 557 00:45:31,180 --> 00:45:33,590 goes with one of these m tildes. 558 00:45:33,590 --> 00:45:38,952 So this is an integration that I have to do between 0 and lambda 559 00:45:38,952 --> 00:45:39,944 over b. 560 00:45:42,920 --> 00:45:45,130 So there is essentially one integration 561 00:45:45,130 --> 00:45:48,702 left because q1 and q2 are left to be the same. 562 00:45:48,702 --> 00:45:49,660 So this is an integral. 563 00:45:49,660 --> 00:45:52,450 Let's call the integration variable 564 00:45:52,450 --> 00:45:56,240 that was q1-- I could k, it doesn't matter-- 2 pi 565 00:45:56,240 --> 00:46:02,140 to the d the same integral that we've seen before. 566 00:46:02,140 --> 00:46:07,640 Except that since this originated from the sigmas, 567 00:46:07,640 --> 00:46:11,870 the integration here is from lambda over b lambda. 568 00:46:19,160 --> 00:46:24,340 So this is basically a number that I can take, say, out here 569 00:46:24,340 --> 00:46:27,640 and regard as a coefficient that multiplies 570 00:46:27,640 --> 00:46:32,890 a term that is m tilde squared. 571 00:46:32,890 --> 00:46:37,990 And similarly, 4. 572 00:46:37,990 --> 00:46:38,490 4. 573 00:46:38,490 --> 00:46:41,410 We said we have four diagrams of his variety, 574 00:46:41,410 --> 00:46:46,820 so this would be a contribution that is 4u. 575 00:46:46,820 --> 00:46:49,620 I can read out the whole-- write out the whole thing. 576 00:46:49,620 --> 00:46:52,130 Certainly, I have all of this. 577 00:46:52,130 --> 00:46:54,290 I have all of this. 578 00:46:54,290 --> 00:47:03,980 In that case, I have m tilde q3 m tilde-- well, let's see. 579 00:47:03,980 --> 00:47:08,860 I have m tilde of q2 m tilde of q4. 580 00:47:08,860 --> 00:47:13,730 And they carry different indices because they 581 00:47:13,730 --> 00:47:18,320 came from two different dot products. 582 00:47:18,320 --> 00:47:23,725 And then I have to do an average over sigma 1 583 00:47:23,725 --> 00:47:27,580 and sigma 3 which carry different indices that 584 00:47:27,580 --> 00:47:29,410 are for beta. 585 00:47:29,410 --> 00:47:38,654 2 pi to the d delta function q1 plus q3 e plus k q1 squared. 586 00:47:44,170 --> 00:47:49,490 Again, since q1 plus q3 is 0 and the sum of the four q's is 0, 587 00:47:49,490 --> 00:47:52,675 these two have to add up to 0. 588 00:47:52,675 --> 00:47:57,270 So the answer, again, will be written as 4u integral 0 589 00:47:57,270 --> 00:48:02,955 to lambda over b dd of some q divided by 2 pi 590 00:48:02,955 --> 00:48:07,890 to the b m tilde of q squared. 591 00:48:07,890 --> 00:48:09,620 And then actually, the same integration, 592 00:48:09,620 --> 00:48:18,812 lambda over d lambda dd k 2 pi to the d 1 over 2 593 00:48:18,812 --> 00:48:20,255 plus k squared. 594 00:48:28,440 --> 00:48:31,990 So out of the six terms, two are 0. 595 00:48:31,990 --> 00:48:35,590 Two are explicitly calculated over here. 596 00:48:35,590 --> 00:48:39,200 One is trivially just m to the fourth. 597 00:48:39,200 --> 00:48:43,940 The last one is basically summing up all of these things. 598 00:48:46,750 --> 00:48:52,805 But these explicitly do not depend on m tilde. 599 00:48:52,805 --> 00:48:58,290 So I'll just call the result of doing all of this sum V delta f 600 00:48:58,290 --> 00:49:01,260 v at level 1. 601 00:49:01,260 --> 00:49:08,230 In the same way that integrating the modes sigma that I'm not 602 00:49:08,230 --> 00:49:10,870 interested and averaging over them 603 00:49:10,870 --> 00:49:15,380 gave a constant of integration, that constant of integration 604 00:49:15,380 --> 00:49:20,280 gets corrected to order of u over here. 605 00:49:20,280 --> 00:49:23,373 I don't need to explicitly take care of it. 606 00:49:28,650 --> 00:49:32,010 So given all of this information, 607 00:49:32,010 --> 00:49:36,760 let's write down what our last line from the previous board 608 00:49:36,760 --> 00:49:37,670 is. 609 00:49:37,670 --> 00:49:43,000 So our intent was to calculate a weight that 610 00:49:43,000 --> 00:49:48,050 governed these coarse-grained modes. 611 00:49:48,050 --> 00:49:50,730 And our answer is that, first of all, 612 00:49:50,730 --> 00:49:58,977 we will get a bunch of constants delta f v 0 plus delta f v 1 613 00:49:58,977 --> 00:50:00,060 that we don't really care. 614 00:50:00,060 --> 00:50:04,040 They're just an overall change that 615 00:50:04,040 --> 00:50:05,940 doesn't matter for the probabilities. 616 00:50:05,940 --> 00:50:10,650 It's just contribution to the free energy. 617 00:50:10,650 --> 00:50:14,180 And then we start to get things. 618 00:50:14,180 --> 00:50:17,980 And to the lowest order what we had 619 00:50:17,980 --> 00:50:24,740 was replacing the Gaussian weight, 620 00:50:24,740 --> 00:50:31,070 but only over this permitted set of wavelengths. 621 00:50:31,070 --> 00:50:35,460 So I have dd q lesser, let's say, 622 00:50:35,460 --> 00:50:43,550 2 pi to the d t plus k q lesser squared and so forth. 623 00:50:43,550 --> 00:50:50,150 Divided by 2 m tilde of q squared. 624 00:50:54,370 --> 00:50:59,150 Then, term number 1 in the series gave me what? 625 00:50:59,150 --> 00:51:06,290 It gave me something that was equivalent to my u 626 00:51:06,290 --> 00:51:09,160 if I were to Fourier transform back to real space, 627 00:51:09,160 --> 00:51:10,920 m to the fourth. 628 00:51:10,920 --> 00:51:19,050 Except that my cutoff has been shifted by lambda over b. 629 00:51:19,050 --> 00:51:22,210 So I don't want to bother to write down 630 00:51:22,210 --> 00:51:25,660 that full form in terms of Fourier modes. 631 00:51:25,660 --> 00:51:28,796 Essentially, if I want to write this explicitly, 632 00:51:28,796 --> 00:51:31,390 it is just like that line that I have, 633 00:51:31,390 --> 00:51:33,575 except that for the integrations I'll 634 00:51:33,575 --> 00:51:37,190 have to explicitly indicate 0 to lambda over b. 635 00:51:39,950 --> 00:51:46,900 So the only terms that we haven't included 636 00:51:46,900 --> 00:51:48,400 are the ones that are over here. 637 00:51:51,120 --> 00:51:54,740 Now, you look at those terms and you 638 00:51:54,740 --> 00:51:59,290 find that the structure of these terms 639 00:51:59,290 --> 00:52:01,720 is precisely what we have over here. 640 00:52:07,080 --> 00:52:10,870 Except that there is a modification. 641 00:52:10,870 --> 00:52:15,500 There is a constant term that is added from this one 642 00:52:15,500 --> 00:52:18,666 and there's a constant term that is added from that one. 643 00:52:18,666 --> 00:52:23,000 So the effect of those things I can capture 644 00:52:23,000 --> 00:52:31,090 by changing the parameters t to something else t tilde. 645 00:52:31,090 --> 00:52:36,230 So you can see that to order of u squared that I haven't 646 00:52:36,230 --> 00:52:39,720 calculated, to order of u, the only effect 647 00:52:39,720 --> 00:52:45,110 of this coarse graining is to modify this one parameter so 648 00:52:45,110 --> 00:52:48,140 that t goes to t tilde. 649 00:52:48,140 --> 00:52:52,590 It certainly depends on how much I coarse grain things. 650 00:52:52,590 --> 00:52:59,330 And this is the original t plus the sum of these things. 651 00:52:59,330 --> 00:53:12,670 So I will have 2-- n plus 2 u integral dd k 2 652 00:53:12,670 --> 00:53:17,364 pi to the d 1 over t plus k k squared. 653 00:53:17,364 --> 00:53:18,780 And presumably, higher-order terms 654 00:53:18,780 --> 00:53:24,980 are allowed, going from 0 to lambda over here. 655 00:53:24,980 --> 00:53:26,390 AUDIENCE: Question. 656 00:53:26,390 --> 00:53:29,080 PROFESSOR: Yes. 657 00:53:29,080 --> 00:53:29,886 Question? 658 00:53:29,886 --> 00:53:30,850 AUDIENCE: Yeah. 659 00:53:30,850 --> 00:53:35,090 When you have an integration over x, if you have previously 660 00:53:35,090 --> 00:53:41,480 defined lambda to be a cutoff in k space, 661 00:53:41,480 --> 00:53:45,180 might it be-- is it 1 over b lambda then? 662 00:53:45,180 --> 00:53:48,135 Or, is it b over lambda? 663 00:53:48,135 --> 00:53:48,637 Because-- 664 00:53:48,637 --> 00:53:49,220 PROFESSOR: OK. 665 00:53:52,870 --> 00:53:53,460 You're right. 666 00:53:53,460 --> 00:53:57,460 So previously, maybe the best way to write this 667 00:53:57,460 --> 00:54:05,120 would have been that there is a shortest length scale a. 668 00:54:05,120 --> 00:54:08,760 So I should really indicate what's 669 00:54:08,760 --> 00:54:13,070 happening here as shortest length scale having gone 670 00:54:13,070 --> 00:54:17,770 to v. And there is always a relationship 671 00:54:17,770 --> 00:54:23,330 between the a and lambda, which is inverse relation, 672 00:54:23,330 --> 00:54:26,835 but there are factors of 2 pi and things like that which 673 00:54:26,835 --> 00:54:28,676 I don't really want to bother. 674 00:54:28,676 --> 00:54:29,610 It doesn't matter. 675 00:54:33,980 --> 00:54:34,480 Yes. 676 00:54:34,480 --> 00:54:36,915 AUDIENCE: Shouldn't it be t tilde is 677 00:54:36,915 --> 00:54:40,319 equal to 2 plus 4 multiplied by? 678 00:54:40,319 --> 00:54:41,110 PROFESSOR: Exactly. 679 00:54:41,110 --> 00:54:41,680 Good. 680 00:54:41,680 --> 00:54:46,750 Because the coefficients here are divided by 2. 681 00:54:46,750 --> 00:54:47,975 So that 2 I forgot. 682 00:54:47,975 --> 00:54:49,175 And I should restore it. 683 00:54:52,520 --> 00:54:57,800 And if I had gone a little bit further, 684 00:54:57,800 --> 00:55:01,734 I would have then started comparing this formula 685 00:55:01,734 --> 00:55:05,320 with this formula. 686 00:55:05,320 --> 00:55:08,890 And I realized that I should have had the 4. 687 00:55:08,890 --> 00:55:14,130 Clearly, the two formula are telling me the same thing. 688 00:55:14,130 --> 00:55:16,330 You can see that they are almost exactly 689 00:55:16,330 --> 00:55:20,760 the same with the exception of how much integration. 690 00:55:20,760 --> 00:55:21,657 Yes? 691 00:55:21,657 --> 00:55:22,740 AUDIENCE: One other thing. 692 00:55:22,740 --> 00:55:25,710 Are bounds of those integrals for your tb, 693 00:55:25,710 --> 00:55:29,540 shouldn't they be lambda over d to lambda? 694 00:55:29,540 --> 00:55:30,165 PROFESSOR: Yes. 695 00:55:33,630 --> 00:55:36,170 Lambda over d to lambda. 696 00:55:36,170 --> 00:55:39,520 And it is because of that that I don't really 697 00:55:39,520 --> 00:55:43,210 have to worry because we saw that when we were doing 698 00:55:43,210 --> 00:55:46,830 straightforward perturbation theory, 699 00:55:46,830 --> 00:55:50,000 the reason that perturbation theory was blowing up 700 00:55:50,000 --> 00:55:54,000 in my face was integrating all the way to the origin. 701 00:55:54,000 --> 00:56:00,040 And the trick of renormalization group is by averaging of there, 702 00:56:00,040 --> 00:56:06,190 I don't really yet reach the singularity 703 00:56:06,190 --> 00:56:09,690 that I would have at k equals to 0. 704 00:56:09,690 --> 00:56:13,860 This integral by itself is not problematic at k equals to 0, 705 00:56:13,860 --> 00:56:15,410 but future integrals would. 706 00:56:19,640 --> 00:56:22,378 Any other mistakes? 707 00:56:22,378 --> 00:56:24,760 No. 708 00:56:24,760 --> 00:56:27,450 All right. 709 00:56:27,450 --> 00:56:29,430 But the other part of this story is 710 00:56:29,430 --> 00:56:34,400 that the effect of this coarse graining at lowest order mu 711 00:56:34,400 --> 00:56:38,600 was to modify this parameter t. 712 00:56:38,600 --> 00:56:41,260 But importantly, to do nothing else. 713 00:56:41,260 --> 00:56:45,432 That is, we can see that in this Hamiltonian that we 714 00:56:45,432 --> 00:56:49,340 had written, the k that is rescaled 715 00:56:49,340 --> 00:56:51,610 is the same as the old k. 716 00:56:51,610 --> 00:56:57,970 And the u is the same as the old u. 717 00:56:57,970 --> 00:57:06,330 That is, to the lowest order the only effect of coarse graining 718 00:57:06,330 --> 00:57:10,955 was to modify the parameter that covers coefficient 719 00:57:10,955 --> 00:57:11,736 of t squared. 720 00:57:18,880 --> 00:57:21,940 But we have not completed our task 721 00:57:21,940 --> 00:57:24,866 of constructing a renormalization group. 722 00:57:27,715 --> 00:57:31,080 Renormalization group had three steps. 723 00:57:31,080 --> 00:57:34,820 The most difficult step, which was the coarse graining 724 00:57:34,820 --> 00:57:40,220 we have completed, but now we have generated a grainy picture 725 00:57:40,220 --> 00:57:44,660 in which the shortest wavelengths are a factor of b 726 00:57:44,660 --> 00:57:47,160 larger than what we had before. 727 00:57:47,160 --> 00:57:50,850 In order to make our pictures look the same, 728 00:57:50,850 --> 00:57:59,326 we had to do steps 2 and 3 of rg. 729 00:57:59,326 --> 00:58:03,060 Step 2 was to shrink all of the lengths 730 00:58:03,060 --> 00:58:10,800 in real space, which amounts to q prime being b times q. 731 00:58:10,800 --> 00:58:15,650 And that will restore the upper part of the q integration 732 00:58:15,650 --> 00:58:17,510 to be lambda. 733 00:58:17,510 --> 00:58:22,120 And there was a rescaling that we 734 00:58:22,120 --> 00:58:26,540 had to perform for the magnitude of the fluctuations which 735 00:58:26,540 --> 00:58:34,690 amounted to replacing the m tilde of q with z N prime. 736 00:58:34,690 --> 00:58:38,290 So this would be, if you like, q lesser 737 00:58:38,290 --> 00:58:41,200 and this would be m prime. 738 00:58:51,410 --> 00:58:53,220 Now, the reason these steps are trivial 739 00:58:53,220 --> 00:58:56,140 is because just whenever I see a q, 740 00:58:56,140 --> 00:58:59,700 I replace it with b inverse q prime. 741 00:58:59,700 --> 00:59:05,190 Whenever I see an m tilde, I replace it with z m prime. 742 00:59:05,190 --> 00:59:14,140 So then I will find the Hamiltonian that characterizes 743 00:59:14,140 --> 00:59:22,030 the m prime variables, the rg implemented variables, 744 00:59:22,030 --> 00:59:26,764 which is-- OK, there is a bunch of constants out front. 745 00:59:26,764 --> 00:59:29,942 There is the fv. 746 00:59:29,942 --> 00:59:34,860 Actually, it's 0 delta fv1. 747 00:59:34,860 --> 00:59:40,030 In the sign, I had to put as plus. 748 00:59:40,030 --> 00:59:42,140 But really, it doesn't matter. 749 00:59:46,740 --> 00:59:51,660 Then, I go and write down what I have. 750 00:59:51,660 --> 00:59:58,310 The integration over q prime after the rescaling is 751 00:59:58,310 --> 01:00:04,225 performed goes back to the same cutoff or the same shortest 752 01:00:04,225 --> 01:00:05,500 wavelength as before. 753 01:00:08,060 --> 01:00:12,160 Except that when I do this replacement of q 754 01:00:12,160 --> 01:00:18,040 lesser with q prime, I will get a factor of b to the minus d 755 01:00:18,040 --> 01:00:19,716 down here. 756 01:00:19,716 --> 01:00:21,860 And there are b integrations. 757 01:00:21,860 --> 01:00:24,650 And then I have t tilde. 758 01:00:24,650 --> 01:00:32,720 The next term is k tilde, but k tilde is the same thing as k. 759 01:00:32,720 --> 01:00:34,230 Goes with q squared. 760 01:00:34,230 --> 01:00:37,440 So this becomes q prime squared, and then I 761 01:00:37,440 --> 01:00:40,040 will get a b to the minus 2. 762 01:00:40,040 --> 01:00:43,766 Higher orders will get more factors of b to the minus 2 763 01:00:43,766 --> 01:00:46,040 over 2. 764 01:00:46,040 --> 01:00:51,590 And then I have m tilde, which is replaced by z. 765 01:00:51,590 --> 01:00:53,578 And there are two of them. 766 01:00:53,578 --> 01:00:57,805 I will get m prime of q prime squared. 767 01:01:03,395 --> 01:01:05,770 If I were to explicitly now write 768 01:01:05,770 --> 01:01:12,340 the factors that go in construction of u, 769 01:01:12,340 --> 01:01:17,410 since u had three integrations over q-- 770 01:01:17,410 --> 01:01:21,980 left board over there-- I will get three integrations 771 01:01:21,980 --> 01:01:33,160 over q prime giving me a factor of b to the minus 3d. 772 01:01:36,780 --> 01:01:44,087 And then I have four factors of m tilde that become m prime. 773 01:01:56,820 --> 01:02:01,450 Along the way, I will pick up four factors of z. 774 01:02:01,450 --> 01:02:04,658 And then, of course, order of u squared. 775 01:02:07,860 --> 01:02:14,050 So under this three steps of rg, what 776 01:02:14,050 --> 01:02:21,430 happened was that I generated t prime, which was z 777 01:02:21,430 --> 01:02:28,650 squared b to the minus d t tilde. 778 01:02:28,650 --> 01:02:34,390 I generated u prime, which was z to the fourth 779 01:02:34,390 --> 01:02:40,702 b to the minus 3d u, the original u. 780 01:02:40,702 --> 01:02:44,510 And I generated the k prime, which 781 01:02:44,510 --> 01:02:50,570 was z squared b to the minus d minus 2 k. 782 01:02:50,570 --> 01:02:53,667 And I could do the same for various other parameters. 783 01:02:58,440 --> 01:03:01,655 Now again, we come up with this issue 784 01:03:01,655 --> 01:03:06,760 of what to choose for zeta. 785 01:03:06,760 --> 01:03:08,630 Sorry, for z. 786 01:03:08,630 --> 01:03:11,770 And what I had said previously was 787 01:03:11,770 --> 01:03:15,070 that we went through all of these exercise of doing 788 01:03:15,070 --> 01:03:18,530 the Gaussian model via rg in order 789 01:03:18,530 --> 01:03:21,150 to have an anchoring point. 790 01:03:21,150 --> 01:03:24,640 And there we saw that the thing that we were interested 791 01:03:24,640 --> 01:03:28,840 was to look at the point where k prime was the same as k. 792 01:03:28,840 --> 01:03:32,770 So let's stick with that choice and see 793 01:03:32,770 --> 01:03:34,720 what the consequences are. 794 01:03:34,720 --> 01:03:38,885 So choose z such that k prime is the same 795 01:03:38,885 --> 01:03:43,720 a k, which means the I choose my z to be b to the 1 796 01:03:43,720 --> 01:03:47,100 plus d over 2 exactly as I had done previously 797 01:03:47,100 --> 01:03:49,105 for the Gaussian model. 798 01:03:49,105 --> 01:03:52,900 So now I can substitute these values. 799 01:03:52,900 --> 01:03:55,660 And therefore, see that following 800 01:03:55,660 --> 01:04:02,080 a rescaling by a factor of b, the value of my t prime. 801 01:04:02,080 --> 01:04:03,970 z squared b to the minus d. 802 01:04:03,970 --> 01:04:08,830 I will get b to the power of 2 plus d minus d. 803 01:04:08,830 --> 01:04:11,960 So that would give me b squared. 804 01:04:11,960 --> 01:04:16,850 And I have t plus 4u n plus 2. 805 01:04:16,850 --> 01:04:21,490 This integral from lambda over b to lambda 806 01:04:21,490 --> 01:04:28,560 to dk 2 pi to the d 1 over d plus k, 807 01:04:28,560 --> 01:04:30,909 k squared, and so forth. 808 01:04:30,909 --> 01:04:35,750 Presumably, order of u squared. 809 01:04:35,750 --> 01:04:44,255 And that my factor of u rescaled by b. 810 01:04:44,255 --> 01:04:49,670 I have to put four factors of z. 811 01:04:49,670 --> 01:04:52,900 So I will get b to the 4 plus 2d, 812 01:04:52,900 --> 01:04:57,740 and then 3d gets subtracted, so I will get b to the 4 minus 813 01:04:57,740 --> 01:05:00,330 d and u again. 814 01:05:00,330 --> 01:05:02,730 But presumably, order of u squared. 815 01:05:15,470 --> 01:05:20,130 So the first factors are precisely the things 816 01:05:20,130 --> 01:05:25,280 that we had done and obtained for the Gaussian model. 817 01:05:25,280 --> 01:05:29,440 So the only thing that we gained by this exercise so far 818 01:05:29,440 --> 01:05:31,045 is this correction to t. 819 01:05:33,720 --> 01:05:37,670 We will see that that correction is not that important. 820 01:05:37,670 --> 01:05:41,480 And in order to really gain an understanding, 821 01:05:41,480 --> 01:05:44,030 we have to go to the next order. 822 01:05:44,030 --> 01:05:48,700 But let's use this opportunity to also make 823 01:05:48,700 --> 01:05:51,120 some changes of terminology that is useful. 824 01:05:53,980 --> 01:05:58,450 So clearly, the way that we had constructed this-- 825 01:05:58,450 --> 01:06:00,220 and in particular, if you are thinking 826 01:06:00,220 --> 01:06:04,600 about averaging over spins, et cetera, in real space-- 827 01:06:04,600 --> 01:06:06,500 the natural thing to think about is 828 01:06:06,500 --> 01:06:10,350 that maybe your b is a factor of 2, or a factor of 3. 829 01:06:10,350 --> 01:06:14,010 You sort of group things by a factor of twice as much 830 01:06:14,010 --> 01:06:16,440 and then do the averaging. 831 01:06:16,440 --> 01:06:19,230 But when you look at things from the perspective 832 01:06:19,230 --> 01:06:25,620 of this momentum shell, this b can be anything. 833 01:06:25,620 --> 01:06:30,730 And it turns out to be useful just as a language tool, not as 834 01:06:30,730 --> 01:06:35,670 a conceptual tool, to make this b to be very close to 1 835 01:06:35,670 --> 01:06:39,900 so that effectively you are just removing a very, very 836 01:06:39,900 --> 01:06:45,380 tiny, thin shell around the boundary. 837 01:06:45,380 --> 01:06:54,620 So essentially what I am saying is choose b that is almost 1, 838 01:06:54,620 --> 01:06:58,550 but maybe a little bit shifted from 1. 839 01:07:01,340 --> 01:07:06,060 Then clearly, as b goes to 1, then t prime 840 01:07:06,060 --> 01:07:11,020 has to go to t-- prime has to go to u. 841 01:07:11,020 --> 01:07:15,580 So what I can do is I can define t prime scaled 842 01:07:15,580 --> 01:07:22,850 by a factor of 1 plus delta l to be t plus a small amount dt 843 01:07:22,850 --> 01:07:25,830 by dl. 844 01:07:25,830 --> 01:07:27,835 And in order of delta l squared. 845 01:07:31,930 --> 01:07:39,570 And similarly, I can write u prime to be u plus delta l du 846 01:07:39,570 --> 01:07:44,680 by dl and higher order. 847 01:07:44,680 --> 01:07:53,230 And the reason is that if we now look at the parameter space-- 848 01:07:53,230 --> 01:07:58,060 t, u, whatever-- the effect of this procedure 849 01:07:58,060 --> 01:08:01,610 rather than being some jump from some point to another point 850 01:08:01,610 --> 01:08:04,100 because we rescaled by a factor of b 851 01:08:04,100 --> 01:08:07,850 is to go to a nearby point. 852 01:08:07,850 --> 01:08:11,980 So these things, dt by dl, du by dl, 853 01:08:11,980 --> 01:08:15,690 essentially point to the direction 854 01:08:15,690 --> 01:08:19,279 in which the parameters would change. 855 01:08:19,279 --> 01:08:23,260 And so they allow you to replace these jumps 856 01:08:23,260 --> 01:08:28,464 that you have by things that are called flows in this parameter 857 01:08:28,464 --> 01:08:29,580 space. 858 01:08:29,580 --> 01:08:34,819 So basically, you have constructed flows and vectors 859 01:08:34,819 --> 01:08:39,390 that describe the flows in this parameter space. 860 01:08:39,390 --> 01:08:48,399 Now, if I do that over here, we can also 861 01:08:48,399 --> 01:08:51,310 see some other things emerging. 862 01:08:51,310 --> 01:08:57,279 So b squared I can write as 1 plus 2 delta l. 863 01:09:00,779 --> 01:09:02,920 And then here, I have t. 864 01:09:06,330 --> 01:09:12,000 Now clearly here, if d was 1, the integral would be 0. 865 01:09:12,000 --> 01:09:14,955 I am integrating over a tiny shell. 866 01:09:14,955 --> 01:09:18,430 So the answer here when b goes to 1 867 01:09:18,430 --> 01:09:24,760 is really just the area of that sphere multiplied 868 01:09:24,760 --> 01:09:26,720 by the thickness. 869 01:09:26,720 --> 01:09:28,790 And so what do I get? 870 01:09:28,790 --> 01:09:35,109 I get 4u n plus 2 is just the overall coefficient. 871 01:09:35,109 --> 01:09:39,649 Then, what is the surface area? 872 01:09:39,649 --> 01:09:43,609 I have the solid angle divided by 2 pi to the d. 873 01:09:43,609 --> 01:09:46,475 OK, you divide solid angle divided by 2 pi 874 01:09:46,475 --> 01:09:49,960 to the d to be kd. 875 01:09:49,960 --> 01:09:53,350 What's the value of the integrand on shell? 876 01:09:53,350 --> 01:09:56,530 On shell, I have to replace k with the lambda. 877 01:09:56,530 --> 01:10:01,680 So I will get t plus k lambda squared. 878 01:10:01,680 --> 01:10:09,000 Actually, the surface area is Sd lambda to the d minus 1. 879 01:10:09,000 --> 01:10:14,000 But then the thickness is lambda delta l. 880 01:10:21,164 --> 01:10:23,320 The second one is actually very easy. 881 01:10:23,320 --> 01:10:28,755 It is 1 plus 4 minus b delta l times u. 882 01:10:32,870 --> 01:10:42,460 So you can see that if I match things to order of delta l, 883 01:10:42,460 --> 01:10:45,010 I get the following rg flows. 884 01:10:47,550 --> 01:10:53,650 So the left-hand side is t plus delta l dt by dl. 885 01:10:53,650 --> 01:10:56,920 The right-hand side-- if I expand, there is a factor of t. 886 01:10:56,920 --> 01:10:59,020 So that it gets rid of. 887 01:10:59,020 --> 01:11:03,930 And I'm left with a term that is proportional to delta l, whose 888 01:11:03,930 --> 01:11:07,370 coefficient on the left-hand side is dt by dl. 889 01:11:07,370 --> 01:11:11,393 And the right-hand side, I get either a factor of 2t 890 01:11:11,393 --> 01:11:16,460 from multiplying the 2 delta l with t or from multiplying 891 01:11:16,460 --> 01:11:20,140 the 1 with the result of this integration. 892 01:11:20,140 --> 01:11:27,035 I will get a 4u n plus 2 kd lambda 893 01:11:27,035 --> 01:11:32,170 to the d divided by t plus k lambda squared. 894 01:11:32,170 --> 01:11:35,820 And I don't need to evaluate any integrals. 895 01:11:38,410 --> 01:11:44,390 And the second flow equation for the parameter u du by dl 896 01:11:44,390 --> 01:11:46,650 is 4 minus d u. 897 01:11:56,540 --> 01:12:00,680 Now, clearly in the language of flows, 898 01:12:00,680 --> 01:12:06,282 a fixed point is when there is no flow. 899 01:12:06,282 --> 01:12:12,560 So I could have dt by dl du by dl need to be 0. 900 01:12:15,210 --> 01:12:18,530 And du by dl being 0 immediately tells ms 901 01:12:18,530 --> 01:12:23,000 that u star has to be 0. 902 01:12:23,000 --> 01:12:25,630 And if I set u star equals to 0, I 903 01:12:25,630 --> 01:12:29,960 will see that t star has to be 0. 904 01:12:29,960 --> 01:12:33,370 So these equations have one and only 905 01:12:33,370 --> 01:12:36,480 one fixed point at this order. 906 01:12:36,480 --> 01:12:42,900 And then looking for relevance and irrelevance of going away 907 01:12:42,900 --> 01:12:46,855 from the fixed point can be captured by linearizing. 908 01:12:46,855 --> 01:12:51,660 That is, I write my t to be t star plus delta l. 909 01:12:51,660 --> 01:12:54,765 Of course, my t star and u star are both 0. 910 01:12:54,765 --> 01:12:59,560 But in general, if they were not at 0, 911 01:12:59,560 --> 01:13:07,650 I would linearize my in general, non-linear rg recursions 912 01:13:07,650 --> 01:13:11,620 by going slightly away from the fixed point. 913 01:13:11,620 --> 01:13:14,940 And then the linearized form of the equation 914 01:13:14,940 --> 01:13:21,030 would say that going away be a small amount delta t delta u-- 915 01:13:21,030 --> 01:13:25,350 and there could be more and more of these operators-- an 916 01:13:25,350 --> 01:13:32,800 be written in terms of a matrix multiply delta t and delta u. 917 01:13:32,800 --> 01:13:36,460 And clearly, the matrix for u is very simple. 918 01:13:36,460 --> 01:13:42,600 It is simply proportional to 4 minus d delta u. 919 01:13:42,600 --> 01:13:45,720 The matrix for t? 920 01:13:45,720 --> 01:13:47,170 Well, there's two terms. 921 01:13:47,170 --> 01:13:50,060 First of all, there is this 2. 922 01:13:50,060 --> 01:13:53,070 And then if I am taking a derivative, 923 01:13:53,070 --> 01:13:55,340 there will be a derivative of this expression 924 01:13:55,340 --> 01:13:57,700 because there is a t-dependence here. 925 01:13:57,700 --> 01:13:59,930 But ultimately, since I'm evaluating it 926 01:13:59,930 --> 01:14:06,230 at u star equals to 0, I don't need to include that term here. 927 01:14:06,230 --> 01:14:08,800 But if I now make a variation in u, 928 01:14:08,800 --> 01:14:15,870 I will get an off-diagonal term here, 929 01:14:15,870 --> 01:14:19,086 which is k lambda squared. 930 01:14:19,086 --> 01:14:22,394 So these two can combine with each other. 931 01:14:27,180 --> 01:14:32,730 Now, looking for relevance or irrelevance 932 01:14:32,730 --> 01:14:36,110 is then equivalent-- previously, we were talking about it 933 01:14:36,110 --> 01:14:42,520 in terms of the full equations with b that could have been 934 01:14:42,520 --> 01:14:45,190 anything, but now we have gone to this limit 935 01:14:45,190 --> 01:14:47,210 of infinitesimal b. 936 01:14:47,210 --> 01:14:52,190 Then, what I have to do is to find the eigenvalues 937 01:14:52,190 --> 01:14:54,973 of the matrix that I have for these flows. 938 01:14:54,973 --> 01:14:58,460 Now, a matrix that has this structure where there's 939 01:14:58,460 --> 01:15:01,790 0 on one side of the diagonal, I immediately 940 01:15:01,790 --> 01:15:07,260 know that the eigenvalues are 2 and 4 minus d. 941 01:15:07,260 --> 01:15:12,130 So basically, depending on whether I'm 942 01:15:12,130 --> 01:15:15,060 in dimensions greater than 4 or less than 4, 943 01:15:15,060 --> 01:15:20,410 I will have either two or one relevant direction. 944 01:15:20,410 --> 01:15:23,780 In particular, if I look at what is happening 945 01:15:23,780 --> 01:15:28,730 for d that is greater than 4-- for d greater than 4, 946 01:15:28,730 --> 01:15:31,970 I will just have one relevant direction. 947 01:15:31,970 --> 01:15:36,480 And if I look at the behavior and the flows 948 01:15:36,480 --> 01:15:44,000 that I am allowed to have in the two parameters t and u. 949 01:15:44,000 --> 01:15:47,530 Now, if my Hamiltonian only has t and u, 950 01:15:47,530 --> 01:15:49,620 I'm only allowed to look at the case 951 01:15:49,620 --> 01:15:52,530 where u is positive in order not to have 952 01:15:52,530 --> 01:15:55,140 weights that are unbounded. 953 01:15:55,140 --> 01:16:00,142 My fixed point occurs at 0, 0. 954 01:16:00,142 --> 01:16:10,480 And then one simple thing is that if u is 0, it will stay 0. 955 01:16:10,480 --> 01:16:13,950 And then dt by dl is 2t. 956 01:16:13,950 --> 01:16:18,320 So I know that I always have an eigen-direction that 957 01:16:18,320 --> 01:16:23,810 is along the t and is flowing away with a velocity of 2 958 01:16:23,810 --> 01:16:24,405 if you like. 959 01:16:27,660 --> 01:16:32,250 The other eigen-direction is not the axis t equals to 0. 960 01:16:32,250 --> 01:16:37,060 Because if t is 0 originally but 0 is nonzero, 961 01:16:37,060 --> 01:16:41,940 this term will generate some positive amount of t for me. 962 01:16:41,940 --> 01:16:45,085 So if I start somewhere on this axis, 963 01:16:45,085 --> 01:16:49,210 t will go in the direction of becoming positive. 964 01:16:49,210 --> 01:16:52,060 Above four dimensions, you will become less. 965 01:16:52,060 --> 01:16:54,280 So above four dimension, you can see 966 01:16:54,280 --> 01:16:57,930 that the general trend of flows is 967 01:16:57,930 --> 01:17:01,092 going to be something like this. 968 01:17:01,092 --> 01:17:04,700 Indeed, there has to be a second eigen-direction 969 01:17:04,700 --> 01:17:07,430 because I'm dealing with a 2 by 2 matrix. 970 01:17:07,430 --> 01:17:09,110 And if you look at it carefully, you'll 971 01:17:09,110 --> 01:17:14,360 find that the second eigen-direction is down here 972 01:17:14,360 --> 01:17:17,730 and corresponds to a negative eigenvalue. 973 01:17:17,730 --> 01:17:22,290 So I basically would be having flows that go towards that. 974 01:17:22,290 --> 01:17:25,180 And in general, the character of the flows, 975 01:17:25,180 --> 01:17:30,380 if I have parameters somewhere here, 976 01:17:30,380 --> 01:17:31,919 they would be flowing there. 977 01:17:31,919 --> 01:17:33,915 If I have parameters somewhere here, 978 01:17:33,915 --> 01:17:35,412 they would be flowing there. 979 01:17:38,410 --> 01:17:43,220 So physically, I have something like iron in five dimension, 980 01:17:43,220 --> 01:17:43,990 for example. 981 01:17:43,990 --> 01:17:47,880 And it corresponds to being somewhere here. 982 01:17:47,880 --> 01:17:51,213 When I change the temperature of iron in five dimension 983 01:17:51,213 --> 01:17:55,900 and I execute some trajectory such as this, 984 01:17:55,900 --> 01:17:59,680 all of the points that are on this side of the trajectory 985 01:17:59,680 --> 01:18:01,980 on their flow will go to a place where 986 01:18:01,980 --> 01:18:04,780 u becomes small and t becomes positive. 987 01:18:04,780 --> 01:18:09,130 So I will essentially go to this Gaussian-like fixed point 988 01:18:09,130 --> 01:18:12,520 that describes independent spins. 989 01:18:12,520 --> 01:18:14,780 All the things that are down here, 990 01:18:14,780 --> 01:18:16,950 which previously in the Gaussian model 991 01:18:16,950 --> 01:18:19,130 we could not describe, because the Gaussian model 992 01:18:19,130 --> 01:18:21,870 did not allow me to go to t negative. 993 01:18:21,870 --> 01:18:23,670 Because now I have a positive view, 994 01:18:23,670 --> 01:18:25,600 I have no problem with that. 995 01:18:25,600 --> 01:18:29,410 And I find that essentially I go to large negative t 996 01:18:29,410 --> 01:18:31,270 corresponding to some positive u. 997 01:18:31,270 --> 01:18:34,621 I can figure out what the amount of magnetization is. 998 01:18:34,621 --> 01:18:36,900 And so then in between them, there 999 01:18:36,900 --> 01:18:41,790 is a trajectory that separates paramagnetic and ferromagnetic 1000 01:18:41,790 --> 01:18:42,370 behavior. 1001 01:18:42,370 --> 01:18:45,985 And clearly, that trajectory at long scales 1002 01:18:45,985 --> 01:18:48,420 corresponds to the fixed point that 1003 01:18:48,420 --> 01:18:52,690 is simply the gradient squared. 1004 01:18:52,690 --> 01:18:55,640 Because all the other terms went to 0, so 1005 01:18:55,640 --> 01:18:57,950 we know all of the correlation functions, 1006 01:18:57,950 --> 01:19:02,400 et cetera, that we should see in this system. 1007 01:19:02,400 --> 01:19:12,690 Unfortunately, if I look for d less than 4, what happens 1008 01:19:12,690 --> 01:19:16,520 is that I still have the same fixed point. 1009 01:19:16,520 --> 01:19:19,410 And actually , the eigen-directions don't change 1010 01:19:19,410 --> 01:19:20,760 all that much. 1011 01:19:20,760 --> 01:19:23,500 u equals to 0 is still an eigen-direction 1012 01:19:23,500 --> 01:19:26,390 that has relevance, too. 1013 01:19:26,390 --> 01:19:30,334 But the other direction changes from being 1014 01:19:30,334 --> 01:19:32,190 irrelevant to being relevant. 1015 01:19:32,190 --> 01:19:35,470 So I have something like this. 1016 01:19:35,470 --> 01:19:38,600 And the natural types of flows that I 1017 01:19:38,600 --> 01:19:41,250 get kind of look like this. 1018 01:19:45,820 --> 01:19:49,890 And now if I take my iron in three dimensions 1019 01:19:49,890 --> 01:19:53,950 and change its temperature, I go from behavior 1020 01:19:53,950 --> 01:19:57,120 that is kind of paramagnetic-like 1021 01:19:57,120 --> 01:20:00,240 to ferromagnetic-like. 1022 01:20:00,240 --> 01:20:04,120 And there is a transition point. 1023 01:20:04,120 --> 01:20:06,180 But that transition point I don't 1024 01:20:06,180 --> 01:20:07,983 know what fixed point it goes to. 1025 01:20:07,983 --> 01:20:09,000 I have no idea. 1026 01:20:11,780 --> 01:20:14,550 So the only difference by doing this analysis 1027 01:20:14,550 --> 01:20:18,190 from what we had done just on the basis of scaling 1028 01:20:18,190 --> 01:20:20,420 and Gaussian theory, et cetera, is 1029 01:20:20,420 --> 01:20:23,080 that we have located the shift in t 1030 01:20:23,080 --> 01:20:27,640 with u, which is what we had done before. 1031 01:20:27,640 --> 01:20:30,780 So was it worth it? 1032 01:20:30,780 --> 01:20:34,000 The answer is up to here, no. 1033 01:20:34,000 --> 01:20:37,080 But let's see if you had gone one step further what 1034 01:20:37,080 --> 01:20:39,470 would have happened. 1035 01:20:39,470 --> 01:20:42,670 The series is an alternating series. 1036 01:20:42,670 --> 01:20:46,860 So the next order term that I get here I expect 1037 01:20:46,860 --> 01:20:49,300 will come with some negative u squared. 1038 01:20:49,300 --> 01:20:52,880 So let's say there will be some amount of work 1039 01:20:52,880 --> 01:20:53,750 that I have to do. 1040 01:20:53,750 --> 01:20:57,680 And I calculate something that is minus a u squared. 1041 01:20:57,680 --> 01:21:00,005 I do some amount of work and I calculate 1042 01:21:00,005 --> 01:21:04,410 something that is minus b u squared. 1043 01:21:04,410 --> 01:21:09,340 But then you can see that if I search for the fixed point, 1044 01:21:09,340 --> 01:21:13,570 I will find another one at u star, which 1045 01:21:13,570 --> 01:21:17,340 is 4 minus d over b. 1046 01:21:17,340 --> 01:21:21,890 So I will find another point. 1047 01:21:21,890 --> 01:21:26,310 And indeed, we'll find that things that left here 1048 01:21:26,310 --> 01:21:29,860 will go to that point. 1049 01:21:29,860 --> 01:21:34,650 And so that point has one direction here. 1050 01:21:34,650 --> 01:21:37,310 It was relevant that becomes irrelevant. 1051 01:21:37,310 --> 01:21:39,530 And the other direction is the analog 1052 01:21:39,530 --> 01:21:43,780 of this direction, which still remains relevant, 1053 01:21:43,780 --> 01:21:45,740 but with modified exponents. 1054 01:21:45,740 --> 01:21:49,870 We will figure out what that is. 1055 01:21:49,870 --> 01:21:52,300 So the next step is to find this b, 1056 01:21:52,300 --> 01:21:55,490 and then everything would be resolved. 1057 01:21:55,490 --> 01:22:01,920 The only thing to then realize is, what are we perturbing it? 1058 01:22:01,920 --> 01:22:04,210 Because the whole idea of perturbation 1059 01:22:04,210 --> 01:22:09,190 theory is that you should have a small parameter. 1060 01:22:09,190 --> 01:22:13,905 And if we are perturbing a u and then 1061 01:22:13,905 --> 01:22:17,720 basing our results on what is happening here, 1062 01:22:17,720 --> 01:22:20,330 the location of this fixed point better 1063 01:22:20,330 --> 01:22:23,080 be small-- close to the original one 1064 01:22:23,080 --> 01:22:26,010 around which I am perturbing. 1065 01:22:26,010 --> 01:22:28,590 So what do I have to make small? 1066 01:22:28,590 --> 01:22:30,860 4 minus d. 1067 01:22:30,860 --> 01:22:33,700 So we thought we were making a perturbation in u, 1068 01:22:33,700 --> 01:22:37,840 but in order to have a small quantity the only thing that we 1069 01:22:37,840 --> 01:22:40,760 can do is to stay very close to four dimension 1070 01:22:40,760 --> 01:22:43,630 and make a perturbation expansion around four 1071 01:22:43,630 --> 01:22:45,180 dimension.