1 00:00:00,125 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,030 under a Creative Commons license. 3 00:00:04,030 --> 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 00:00:21,000 --> 00:00:21,625 PROFESSOR: Hey. 9 00:00:21,625 --> 00:00:22,125 Let's start. 10 00:00:24,950 --> 00:00:31,620 So a few weeks ago we started with writing a partition 11 00:00:31,620 --> 00:00:34,510 function for a statistical field that 12 00:00:34,510 --> 00:00:41,170 was going to capture behavior of a variety of systems undergoing 13 00:00:41,170 --> 00:00:44,380 critical phase transitions. 14 00:00:44,380 --> 00:00:49,930 And this was obtained by integrating over configurations 15 00:00:49,930 --> 00:01:00,580 of this statistical field a rate that we wrote 16 00:01:00,580 --> 00:01:03,760 on the basis of a form of locality. 17 00:01:06,520 --> 00:01:10,330 And terms that were consistent with that 18 00:01:10,330 --> 00:01:15,596 were of the form m squared, m to the fourth. 19 00:01:15,596 --> 00:01:17,192 Let's say m to the sixth. 20 00:01:19,790 --> 00:01:22,280 Various types of gradient types of terms. 21 00:01:32,680 --> 00:01:36,927 And in principle, allowing for a symmetry-breaking field that 22 00:01:36,927 --> 00:01:39,659 was more in the form of h dot 1. 23 00:01:46,281 --> 00:01:50,020 And again, we always emphasized that in writing 24 00:01:50,020 --> 00:01:53,470 these statistical fields, we have to do averaging. 25 00:01:53,470 --> 00:01:58,590 We have to get rid of a lot of short wavelength fluctuations. 26 00:01:58,590 --> 00:02:01,880 And essentially, the future m of x, although I write it 27 00:02:01,880 --> 00:02:06,870 as a continuum, has an implicit short scale 28 00:02:06,870 --> 00:02:10,870 below which it does not fluctuate. 29 00:02:10,870 --> 00:02:15,290 OK, so we tried to evaluate this by certain point, 30 00:02:15,290 --> 00:02:17,260 and we didn't succeed. 31 00:02:17,260 --> 00:02:22,430 So we went phenomenologically and tried to describe things 32 00:02:22,430 --> 00:02:24,490 on the basis of scaling theory. 33 00:02:24,490 --> 00:02:27,430 Ultimately, this renormalization group 34 00:02:27,430 --> 00:02:33,770 procedure that we would like to apply to something like this. 35 00:02:33,770 --> 00:02:36,520 Now, there is a part of this that 36 00:02:36,520 --> 00:02:39,840 is actually pretty easy to solve. 37 00:02:39,840 --> 00:02:43,710 And that's when we ignore anything 38 00:02:43,710 --> 00:02:47,670 that is higher than second order in m. 39 00:02:47,670 --> 00:02:50,840 Because once we ignore them, we have essentially 40 00:02:50,840 --> 00:02:53,580 a generalized Gaussian integral. 41 00:02:53,580 --> 00:02:56,120 We can do Gaussian integrals. 42 00:02:56,120 --> 00:03:00,690 So what we are going to do is, in this lecture, 43 00:03:00,690 --> 00:03:05,150 focusing on understanding a lot about the behavior 44 00:03:05,150 --> 00:03:07,310 of the Gaussian version of the theory. 45 00:03:07,310 --> 00:03:09,400 Which is certainly a diminished version, 46 00:03:09,400 --> 00:03:13,740 because it doesn't have lots of essential things. 47 00:03:13,740 --> 00:03:16,820 And then gradually putting back all of those things 48 00:03:16,820 --> 00:03:20,390 that we have not considered at the Gaussian level. 49 00:03:20,390 --> 00:03:23,320 In particular, we'll try to do with them 50 00:03:23,320 --> 00:03:26,980 with a version of a perturbation theory. 51 00:03:26,980 --> 00:03:30,120 We'll see that standard perturbation theory has 52 00:03:30,120 --> 00:03:32,640 some limitations that we will eventually 53 00:03:32,640 --> 00:03:38,280 resolve by using this renormalization procedure. 54 00:03:38,280 --> 00:03:39,410 OK. 55 00:03:39,410 --> 00:03:41,750 So what happens if I do that? 56 00:03:41,750 --> 00:03:45,690 Why do I say that that theory is now solve-able? 57 00:03:45,690 --> 00:03:48,380 And the key to that is, of course, 58 00:03:48,380 --> 00:03:49,960 to go into Fourier representation. 59 00:03:54,420 --> 00:03:57,890 Which, because the theory that I wrote down 60 00:03:57,890 --> 00:04:01,450 has this inherent translational of symmetry, 61 00:04:01,450 --> 00:04:06,420 Fourier representation decouples the various m's 62 00:04:06,420 --> 00:04:09,530 that are currently connected to their neighborhood 63 00:04:09,530 --> 00:04:12,430 by these gradients and high orders. 64 00:04:12,430 --> 00:04:18,480 So let's introduce a m of q, which 65 00:04:18,480 --> 00:04:21,660 is the Fourier transform of m of x. 66 00:04:29,388 --> 00:04:32,770 Let's see. m of x. 67 00:04:32,770 --> 00:04:36,720 And these are all vectors. 68 00:04:36,720 --> 00:04:42,470 And I should really use a different symbol, such as m 69 00:04:42,470 --> 00:04:46,660 [INAUDIBLE], to indicate the Fourier 70 00:04:46,660 --> 00:04:50,120 components of this field m of x. 71 00:04:50,120 --> 00:04:53,480 But since in the context of renormalization group 72 00:04:53,480 --> 00:04:57,930 we had defined a coarse grained field that was in tilde, 73 00:04:57,930 --> 00:04:59,690 I don't want to do that. 74 00:04:59,690 --> 00:05:02,740 I hope that the argument of the function 75 00:05:02,740 --> 00:05:05,940 is sufficient indicator of whether we are in real space 76 00:05:05,940 --> 00:05:07,660 or in momentum space. 77 00:05:07,660 --> 00:05:10,870 Initially, I'll try to put a tail on the m 78 00:05:10,870 --> 00:05:13,490 to indicate that I'm doing Fourier space, 79 00:05:13,490 --> 00:05:16,450 but I suspect that very soon I'll forget about the tail. 80 00:05:16,450 --> 00:05:20,320 So keep that in mind. 81 00:05:20,320 --> 00:05:25,440 So if I-- oops. 82 00:05:25,440 --> 00:05:26,970 OK. 83 00:05:26,970 --> 00:05:28,580 m of q. 84 00:05:28,580 --> 00:05:34,980 So if I go back and write what this m of x 85 00:05:34,980 --> 00:05:49,230 is, it is an integral over 2, 2 pi to the, d to the minus iq 86 00:05:49,230 --> 00:05:54,930 dot x with m of q. 87 00:05:59,610 --> 00:06:03,190 Now, I also want to at some stage, 88 00:06:03,190 --> 00:06:09,650 since it would be cleaner to have this rate in terms 89 00:06:09,650 --> 00:06:15,860 of a product of q's, remind you that this could have obtained, 90 00:06:15,860 --> 00:06:19,690 if I hadn't gone to the continuum version-- 91 00:06:19,690 --> 00:06:26,390 if I had a finite system-- to a sum over q. 92 00:06:26,390 --> 00:06:28,720 And the sum over q would be basically 93 00:06:28,720 --> 00:06:32,070 things that are separated q values by multiples of 1 94 00:06:32,070 --> 00:06:33,960 over the size of the system. 95 00:06:33,960 --> 00:06:39,170 And e to the minus iq dot x. 96 00:06:39,170 --> 00:06:42,380 This m with the cues that are now discretized. 97 00:06:45,400 --> 00:06:48,180 But let's remember that the density of state 98 00:06:48,180 --> 00:06:52,230 has a factor of 1 over v. So if I use this definition, 99 00:06:52,230 --> 00:06:55,825 I really should put the 1 over v here when 100 00:06:55,825 --> 00:06:58,170 I go to the discrete version. 101 00:06:58,170 --> 00:07:00,890 And I emphasize this because previously, we 102 00:07:00,890 --> 00:07:04,950 had done Fourier decomposition where 103 00:07:04,950 --> 00:07:08,990 I had used the square root of v as a normalization. 104 00:07:08,990 --> 00:07:15,585 It really doesn't matter which normalization 105 00:07:15,585 --> 00:07:19,200 you use at the end as long as you are consistent. 106 00:07:19,200 --> 00:07:23,330 We'll see the advantages of this normalization shortly. 107 00:07:23,330 --> 00:07:26,735 AUDIENCE: Is there any particular reason 108 00:07:26,735 --> 00:07:28,735 for using the different sign in the exponential? 109 00:07:28,735 --> 00:07:30,410 PROFESSOR: Actually, no. 110 00:07:30,410 --> 00:07:35,140 I'm not sure even whether I used iqx here or minus iqx here. 111 00:07:35,140 --> 00:07:37,870 It's just a matter of which one you 112 00:07:37,870 --> 00:07:40,860 want to stick with consistently. 113 00:07:40,860 --> 00:07:45,460 At the end of the day, the phase will not be that important. 114 00:07:45,460 --> 00:07:48,570 So even if we mistake one form or the other, 115 00:07:48,570 --> 00:07:49,861 it doesn't make any difference. 116 00:07:55,160 --> 00:08:00,580 So if I do that, then again, to sort of be more precise, 117 00:08:00,580 --> 00:08:06,300 I have to think about what to do with gradients. 118 00:08:06,300 --> 00:08:14,300 Gradients, I can imagine, are the limit of something like n 119 00:08:14,300 --> 00:08:21,110 at x plus A minus n at x divided by A. 120 00:08:21,110 --> 00:08:24,660 If this is a gradient in the x direction. 121 00:08:24,660 --> 00:08:28,840 And I have to take the limit as A goes to 0. 122 00:08:28,840 --> 00:08:34,320 So when I'm thinking about this kind of functional integral, 123 00:08:34,320 --> 00:08:39,940 keeping in mind that I have a shortest landscape, maybe 124 00:08:39,940 --> 00:08:42,840 one way to do it is to imagine that I discretize 125 00:08:42,840 --> 00:08:47,360 my system over here into spacing of size A. 126 00:08:47,360 --> 00:08:52,162 And then I have a variable on each size, 127 00:08:52,162 --> 00:08:57,200 and then I integrate every place, 128 00:08:57,200 --> 00:09:01,530 subject this replacement for the gradient. 129 00:09:01,530 --> 00:09:05,630 Again, what you do precisely does not matter here. 130 00:09:05,630 --> 00:09:08,420 If you remember in the first lecture 131 00:09:08,420 --> 00:09:12,760 when we were thinking about the dl lattice system 132 00:09:12,760 --> 00:09:16,350 and then using these kinds of coupling between springs 133 00:09:16,350 --> 00:09:19,510 that they're connecting nearest neighbors, what ended up 134 00:09:19,510 --> 00:09:22,920 by using this was that when I Fourier transformed, 135 00:09:22,920 --> 00:09:25,080 I had things like cosine. 136 00:09:25,080 --> 00:09:29,650 And then when I expanded the cosine close to q, close to 0, 137 00:09:29,650 --> 00:09:32,080 I generated a series that had q squared, 138 00:09:32,080 --> 00:09:34,210 q to the fourth, et cetera. 139 00:09:34,210 --> 00:09:40,720 So essentially, any discretized version corresponds 140 00:09:40,720 --> 00:09:44,845 to an expansion like this with sufficient [INAUDIBLE] powers 141 00:09:44,845 --> 00:09:47,410 of q in both. 142 00:09:47,410 --> 00:09:53,310 So at the end of the day, when you go through this process, 143 00:09:53,310 --> 00:09:59,740 you find that you can write the partition function after 144 00:09:59,740 --> 00:10:05,490 the change of variables to m of x to m of q to doing a whole 145 00:10:05,490 --> 00:10:10,250 bunch of integrals over different q's. 146 00:10:10,250 --> 00:10:15,670 So, essentially you would have-- actually, maybe I 147 00:10:15,670 --> 00:10:23,460 will explicitly put the product over q outside 148 00:10:23,460 --> 00:10:27,060 to emphasize that essentially, for each q I would 149 00:10:27,060 --> 00:10:30,370 have to do independent integrals. 150 00:10:30,370 --> 00:10:33,990 Of course, for each q mode I have, 151 00:10:33,990 --> 00:10:39,840 since I've gone to this representation of a vector that 152 00:10:39,840 --> 00:10:46,160 is n-dimensional, I have to do n integrals on n tilde of q. 153 00:10:48,750 --> 00:10:51,100 On-- m will be the tail of q. 154 00:10:53,720 --> 00:11:03,600 And if I had chosen the square root 155 00:11:03,600 --> 00:11:08,840 of V type of normalization, the Jacobian 156 00:11:08,840 --> 00:11:11,910 of the transformation from here to here would have been 1. 157 00:11:11,910 --> 00:11:15,460 Because it's kind of a symmetric way of writing things. 158 00:11:15,460 --> 00:11:19,120 Because I chose this way of doing things, 159 00:11:19,120 --> 00:11:24,740 I will have a factor of V to the n over 2 in the denominator 160 00:11:24,740 --> 00:11:26,440 here. 161 00:11:26,440 --> 00:11:28,940 But again, it's just being pedantic, 162 00:11:28,940 --> 00:11:30,680 because at the end of the day, we 163 00:11:30,680 --> 00:11:32,690 don't care about these factors. 164 00:11:32,690 --> 00:11:34,850 We are interested in things like this singular 165 00:11:34,850 --> 00:11:39,120 part of the partition function as it 166 00:11:39,120 --> 00:11:41,470 depends on these coordinates. 167 00:11:41,470 --> 00:11:45,230 This really just gives you an overall constant. 168 00:11:45,230 --> 00:11:47,810 Of course, how many of these constants you have 169 00:11:47,810 --> 00:11:55,050 would depend basically how you have discretized the problem. 170 00:11:55,050 --> 00:11:59,800 But it is a constant independent of tnh, not something 171 00:11:59,800 --> 00:12:01,110 that we have to worry about. 172 00:12:04,630 --> 00:12:10,140 Now what happens to these Gaussian factors? 173 00:12:10,140 --> 00:12:13,930 Essentially, I have put the product over q outside. 174 00:12:13,930 --> 00:12:17,650 So when I transform this integral over xm squared 175 00:12:17,650 --> 00:12:22,790 goes over to an integral over q, m of q 176 00:12:22,790 --> 00:12:25,920 squared, which then I can write as a product 177 00:12:25,920 --> 00:12:28,500 over those contributions. 178 00:12:28,500 --> 00:12:33,770 And what you will get is t plus, from here, 179 00:12:33,770 --> 00:12:38,020 you will get a Kq squared, put in Lq 180 00:12:38,020 --> 00:12:41,800 to the fourth and all kinds of order terms 181 00:12:41,800 --> 00:12:44,840 that I have included. 182 00:12:44,840 --> 00:12:52,160 Multiplying this m component vector m of q squared. 183 00:12:52,160 --> 00:12:55,560 Again, reminding you this means m of q, 184 00:12:55,560 --> 00:12:59,840 m of minus q, which is the same thing as m star of q, 185 00:12:59,840 --> 00:13:03,700 if you go through these procedures over here. 186 00:13:03,700 --> 00:13:05,670 There is 2. 187 00:13:05,670 --> 00:13:07,838 And this factor of the v actually 188 00:13:07,838 --> 00:13:10,780 will come up over here. 189 00:13:10,780 --> 00:13:16,210 So previously, I had used the normalization square root of V, 190 00:13:16,210 --> 00:13:19,435 and I didn't have this factor of 1 over V. 191 00:13:19,435 --> 00:13:23,560 Now I have put if there, I will have that factor. 192 00:13:23,560 --> 00:13:24,060 Yes? 193 00:13:24,060 --> 00:13:29,097 AUDIENCE: m minus q is star q only if it is the real field, 194 00:13:29,097 --> 00:13:29,597 right? 195 00:13:29,597 --> 00:13:30,832 If m is real. 196 00:13:30,832 --> 00:13:31,860 PROFESSOR: Yes. 197 00:13:31,860 --> 00:13:34,874 And we are dealing with the field m of q of this. 198 00:13:34,874 --> 00:13:36,826 AUDIENCE: And in the case of superfluidity? 199 00:13:36,826 --> 00:13:38,492 PROFESSOR: In the case of superfluidity? 200 00:13:41,670 --> 00:13:43,868 So let's see. 201 00:13:43,868 --> 00:13:51,080 So we would have a psi of q integral d dx into the i q dot 202 00:13:51,080 --> 00:13:56,540 x psi of x. 203 00:13:56,540 --> 00:13:59,626 If I Fourier transform this, I will 204 00:13:59,626 --> 00:14:04,919 get a psi star of q integral into the x 205 00:14:04,919 --> 00:14:10,540 into the minus [INAUDIBLE] x psi star of x. 206 00:14:10,540 --> 00:14:15,170 So what you are saying is that in the case where psi of x 207 00:14:15,170 --> 00:14:25,420 is a complex number-- I have psi1 plus ipsi2-- here 208 00:14:25,420 --> 00:14:26,700 I would have psi1 minus ipsi2. 209 00:14:29,310 --> 00:14:35,230 So here I would have to make it a statement 210 00:14:35,230 --> 00:14:39,040 that the real part and the imaginary part 211 00:14:39,040 --> 00:14:43,760 come when you Fourier transform with an additional minus. 212 00:14:43,760 --> 00:14:45,530 But let's remember that something 213 00:14:45,530 --> 00:14:47,786 like this that we are interested is 214 00:14:47,786 --> 00:14:52,510 psi1 squared plus psi2 squared. 215 00:14:52,510 --> 00:14:55,664 So ultimately that minus sign did not make any difference. 216 00:15:00,922 --> 00:15:05,660 But it's good to sort of think of all of these issues. 217 00:15:05,660 --> 00:15:14,406 And in particular, we are used to thinking of Gaussians, 218 00:15:14,406 --> 00:15:19,850 where I would have a scalar and then I would have x squared. 219 00:15:19,850 --> 00:15:23,130 When I have this complex number and I have psi of q, 220 00:15:23,130 --> 00:15:26,687 psi of minus q, then I have a real part 221 00:15:26,687 --> 00:15:28,410 squared plus an imaginary part squared. 222 00:15:31,380 --> 00:15:35,520 And you have to think about whether or not 223 00:15:35,520 --> 00:15:39,762 you have changed the number of degrees of freedom. 224 00:15:39,762 --> 00:15:46,100 If you basically integrate over all q's, you may have problems. 225 00:15:46,100 --> 00:15:51,090 You may have at some point to think about seeing psi of q 226 00:15:51,090 --> 00:15:54,970 and psi of minus q star are the same thing. 227 00:15:54,970 --> 00:15:58,780 Maybe you have to integrate over just the positive values. 228 00:15:58,780 --> 00:16:02,800 But then at each q you will have two different variables, 229 00:16:02,800 --> 00:16:05,770 which is the real part and the imaginary part. 230 00:16:05,770 --> 00:16:09,805 So you have to think about all of those doublings and halvings 231 00:16:09,805 --> 00:16:13,100 that are involved in this statement. 232 00:16:13,100 --> 00:16:16,280 And in the notes, I have the writeup 233 00:16:16,280 --> 00:16:19,860 about that that you go and precisely check 234 00:16:19,860 --> 00:16:22,530 where the factors of one half and two go. 235 00:16:22,530 --> 00:16:25,063 But ultimately, it looks as if you're 236 00:16:25,063 --> 00:16:28,170 dealing with a simple scalar quantity. 237 00:16:28,170 --> 00:16:31,840 So I did not give you that detail explicitly, 238 00:16:31,840 --> 00:16:35,250 but you can go and check it in the important issue. 239 00:16:41,140 --> 00:16:44,410 The other term that we have. 240 00:16:44,410 --> 00:16:47,750 One advantage of this normalization 241 00:16:47,750 --> 00:16:51,345 is that h multiplies the integral of m 242 00:16:51,345 --> 00:16:57,920 of x, which is clearly this m with a tail for q equals to 0. 243 00:16:57,920 --> 00:17:05,069 So that's [INAUDIBLE] mh dotted by this m [INAUDIBLE]. 244 00:17:10,040 --> 00:17:10,600 Yes? 245 00:17:10,600 --> 00:17:14,659 AUDIENCE: This is assuming a uniform field? 246 00:17:14,659 --> 00:17:17,440 PROFESSOR: Yes, that's right. 247 00:17:17,440 --> 00:17:20,500 So we are thinking about the physics problem, 248 00:17:20,500 --> 00:17:23,258 but we added the uniform field. 249 00:17:23,258 --> 00:17:25,862 So if you are for some physical reason 250 00:17:25,862 --> 00:17:28,792 interested in a position where you 251 00:17:28,792 --> 00:17:32,102 feel you can modify that, then this would be h of q, 252 00:17:32,102 --> 00:17:33,131 m of minus q. 253 00:17:40,140 --> 00:17:47,670 Actually, one reason ultimately to choose this normalization is 254 00:17:47,670 --> 00:17:54,020 that clearly what appears here is a sum of q. 255 00:17:54,020 --> 00:18:00,200 If I go over to my integral over q, then the factor of 1 over V 256 00:18:00,200 --> 00:18:01,110 disappears. 257 00:18:01,110 --> 00:18:05,560 So that's one reason-- since mostly after this, going 258 00:18:05,560 --> 00:18:08,670 through the details we'll be dealing with the continuum 259 00:18:08,670 --> 00:18:13,130 version-- I prefer this normalization. 260 00:18:13,130 --> 00:18:20,520 And we can now do the Gaussian integrals. 261 00:18:20,520 --> 00:18:24,880 Basically, there's an overall factor of 1 262 00:18:24,880 --> 00:18:30,070 over V to the n over 2 for each q mode. 263 00:18:30,070 --> 00:18:36,980 Then each one of these Gaussian integrals 264 00:18:36,980 --> 00:18:44,370 will leave me a factor of root 2 pi times the variant. 265 00:18:44,370 --> 00:18:48,710 So I will get 2 pi. 266 00:18:48,710 --> 00:18:53,802 The variance is V divided by t plus k 267 00:18:53,802 --> 00:18:57,246 q squared plus lq to the fourth, and so forth. 268 00:19:00,200 --> 00:19:04,750 Square root, but there are n components, 269 00:19:04,750 --> 00:19:08,400 so I will get something like this. 270 00:19:08,400 --> 00:19:16,080 And then the term that corresponds to q equals to 0 271 00:19:16,080 --> 00:19:19,725 does not have any of this part. 272 00:19:19,725 --> 00:19:21,840 So it will give a contribution even 273 00:19:21,840 --> 00:19:25,370 for q equals to 0 that is like this. 274 00:19:25,370 --> 00:19:30,880 But you have a term that shifts the center of integration 275 00:19:30,880 --> 00:19:35,890 from m equals to 0 because of the presence of the field. 276 00:19:35,890 --> 00:19:40,670 So you will get a term that is exponential of essentially-- 277 00:19:40,670 --> 00:19:48,980 completing the square-- will give you V divided by 2t times 278 00:19:48,980 --> 00:19:49,580 h squared. 279 00:19:57,665 --> 00:20:01,120 Now, clearly the thing that I'm interested 280 00:20:01,120 --> 00:20:09,050 is log of Z as a function of t and h. 281 00:20:09,050 --> 00:20:12,700 I'm interested in t and h dependents. 282 00:20:12,700 --> 00:20:14,672 So there is a bunch of things that 283 00:20:14,672 --> 00:20:18,608 are constants that I don't really care. 284 00:20:18,608 --> 00:20:24,890 And then there is a, from here, minus 1/2, actually 285 00:20:24,890 --> 00:20:33,190 minus n 1/2 sum over q log of t plus k q squared and so forth. 286 00:20:33,190 --> 00:20:37,717 And plus here, I have V a squared over 2t. 287 00:20:42,690 --> 00:20:45,100 So I can define something that's like 288 00:20:45,100 --> 00:20:51,425 if the energy from log of Z divided by the volume. 289 00:20:54,360 --> 00:20:56,870 And you can see that once I replace 290 00:20:56,870 --> 00:21:00,445 this sum of a q with an integral, 291 00:21:00,445 --> 00:21:04,370 I will get a factor of volume that I can disregard then. 292 00:21:04,370 --> 00:21:06,360 So there's some other constant. 293 00:21:06,360 --> 00:21:15,150 And then I have plus n over 2 integral over q divided by q pi 294 00:21:15,150 --> 00:21:22,190 to the d log of q plus k q squared, and so forth. 295 00:21:22,190 --> 00:21:25,212 Minus V k squared divided by 2t. 296 00:21:33,970 --> 00:21:36,770 Now, again, the question is what's 297 00:21:36,770 --> 00:21:41,740 the range of q's that I have to integrate, 298 00:21:41,740 --> 00:21:48,178 given that I'm making things that are coarse grained. 299 00:21:48,178 --> 00:21:51,890 Now, if I were to really discretize my system 300 00:21:51,890 --> 00:21:55,020 and, say, put it on q and you plot this, 301 00:21:55,020 --> 00:21:59,380 then the allowed values of q would leave on 302 00:21:59,380 --> 00:22:01,660 the [INAUDIBLE] zone. 303 00:22:01,660 --> 00:22:05,370 [INAUDIBLE] zone, say, in the different directions in q 304 00:22:05,370 --> 00:22:08,830 would be something like the q that 305 00:22:08,830 --> 00:22:14,050 would be centered around pi over a. 306 00:22:14,050 --> 00:22:17,590 But it would be centered at 0, but then you 307 00:22:17,590 --> 00:22:19,808 would have pi plus pi over a. 308 00:22:19,808 --> 00:22:21,272 Yes? 309 00:22:21,272 --> 00:22:23,712 AUDIENCE: The d would disappear, right? 310 00:22:23,712 --> 00:22:26,923 PROFESSOR: The d would disappear because I divided by it. 311 00:22:31,640 --> 00:22:37,970 So in principle, if I had done the discretization 312 00:22:37,970 --> 00:22:45,030 to a cube and plot this, I would have been integrating over q 313 00:22:45,030 --> 00:22:49,840 that this would find to a cube like this. 314 00:22:49,840 --> 00:22:51,670 But maybe I chose some other lattice 315 00:22:51,670 --> 00:22:55,900 like a diamond lattice, et cetera. 316 00:22:55,900 --> 00:22:58,862 Then the shape of this thing would change. 317 00:22:58,862 --> 00:23:01,495 But what's the meaning of doing the whole thing on a lattice 318 00:23:01,495 --> 00:23:03,590 anyway? 319 00:23:03,590 --> 00:23:06,200 The thing that I want to do is to make sure 320 00:23:06,200 --> 00:23:10,400 that I have done some averaging in order to remove 321 00:23:10,400 --> 00:23:13,260 short wavelength fluctuations. 322 00:23:13,260 --> 00:23:19,460 So a much more natural way to do that averaging and removing 323 00:23:19,460 --> 00:23:22,550 short wavelength operations is to say 324 00:23:22,550 --> 00:23:31,770 that my field has only Fourier components that 325 00:23:31,770 --> 00:23:36,706 are from 0 to some maximum value of lambda, which 326 00:23:36,706 --> 00:23:40,080 is the inverse of some radiant. 327 00:23:40,080 --> 00:23:43,400 And if you are worried about the difference in integration 328 00:23:43,400 --> 00:23:46,470 between doing things on this nice mirror that 329 00:23:46,470 --> 00:23:50,930 has nice symmetry and maybe doing it on a cube, 330 00:23:50,930 --> 00:23:53,292 then the difference is essentially 331 00:23:53,292 --> 00:23:59,370 the bit of integration that you would have to do over here. 332 00:23:59,370 --> 00:24:02,620 But the function that you are integrating 333 00:24:02,620 --> 00:24:07,930 his no singularities for large values of q. 334 00:24:07,930 --> 00:24:10,020 You are interested in the singularities 335 00:24:10,020 --> 00:24:13,480 of the function when t goes to 0. 336 00:24:13,480 --> 00:24:15,520 And then the log has singularities 337 00:24:15,520 --> 00:24:18,470 when its argument goes to 0. 338 00:24:18,470 --> 00:24:20,740 So I should be interested, as far as singularities 339 00:24:20,740 --> 00:24:27,120 are concerned, only in the vicinity of this point anyway. 340 00:24:27,120 --> 00:24:30,870 What I do out there, whether I replace the sphere 341 00:24:30,870 --> 00:24:33,942 with the cube or et cetera, will add 342 00:24:33,942 --> 00:24:36,616 some other non-singular term over here, 343 00:24:36,616 --> 00:24:37,699 which I don't really care. 344 00:24:41,590 --> 00:24:43,950 Actually, if I do that, this non-singular term 345 00:24:43,950 --> 00:24:46,840 here could be actually functions of t. 346 00:24:46,840 --> 00:24:50,110 But they would be very perfect and regular functions of t. 347 00:24:50,110 --> 00:24:52,070 Like constant plus alpha t, plus pheta q 348 00:24:52,070 --> 00:24:57,190 squared, et cetera, that have no singularities. 349 00:24:57,190 --> 00:25:01,730 So if I'm interested in singularities, 350 00:25:01,730 --> 00:25:03,342 I am going to be focused on that. 351 00:25:06,000 --> 00:25:11,900 Now actually, we encountered this integral 352 00:25:11,900 --> 00:25:16,745 before when we were looking at corrections 353 00:25:16,745 --> 00:25:19,756 to the saddle-point approximation. 354 00:25:19,756 --> 00:25:23,830 And if you remember what we did then was to take, 355 00:25:23,830 --> 00:25:27,100 let's say, C of d of h across 0 while 356 00:25:27,100 --> 00:25:33,720 taking two derivatives of this free energy with respect to t. 357 00:25:33,720 --> 00:25:38,056 And then we ended up with an integral. 358 00:25:38,056 --> 00:25:40,900 There's a minus sign here over d. 359 00:25:40,900 --> 00:25:45,980 n over 2 integral dt 2 pi squared. 360 00:25:45,980 --> 00:25:48,380 2 pi to the d. 361 00:25:48,380 --> 00:25:51,040 Taking two derivatives of the log. 362 00:25:51,040 --> 00:25:53,610 The first derivative will give me 1 over the argument. 363 00:25:53,610 --> 00:25:56,910 The second derivative will give me 1 over the argument squared. 364 00:25:56,910 --> 00:25:58,878 One side take care of the minus sign. 365 00:26:18,558 --> 00:26:24,020 Now, I think this is a kind of integral, 366 00:26:24,020 --> 00:26:30,335 after I have focused on the singular part, 367 00:26:30,335 --> 00:26:34,736 that I can replace when integrating over a sphere. 368 00:26:39,640 --> 00:26:43,230 Now, when I integrate over a sphere, 369 00:26:43,230 --> 00:26:47,740 I may be concerned about what's going on at small values. 370 00:26:47,740 --> 00:26:51,350 At q, at small values of q, as long as t is around, 371 00:26:51,350 --> 00:26:53,120 I have no problem. 372 00:26:53,120 --> 00:26:55,380 When t goes to 0, I will have to worry 373 00:26:55,380 --> 00:26:59,080 about the singularity that comes from 1 over k, 374 00:26:59,080 --> 00:27:00,690 2 squared, et cetera. 375 00:27:00,690 --> 00:27:04,510 So that's really the singularity that I'm interested in. 376 00:27:04,510 --> 00:27:07,150 Exactly what happens at large q, I'm 377 00:27:07,150 --> 00:27:11,360 not really all that interested in. 378 00:27:11,360 --> 00:27:20,050 And in particular, what I can do is I can rescale things. 379 00:27:20,050 --> 00:27:27,730 I can call q squared over t to the x squared. 380 00:27:27,730 --> 00:27:35,060 So I can essentially make that change over there. 381 00:27:35,060 --> 00:27:40,890 So that whenever I see a factor of q, 382 00:27:40,890 --> 00:27:44,330 I replace it with t over k to the 1/2 x. 383 00:27:49,940 --> 00:27:51,417 What happens here? 384 00:27:51,417 --> 00:27:54,970 I have, first of all, n over 2. 385 00:27:54,970 --> 00:27:57,768 I have 1 over 2 pi to the d. 386 00:28:01,680 --> 00:28:09,340 Writing this in terms of spherical symmetry, 387 00:28:09,340 --> 00:28:13,670 I will have the solid angle d dimensions. 388 00:28:13,670 --> 00:28:18,510 And then I would have q to d minus 1 q. 389 00:28:18,510 --> 00:28:23,530 Every time I put a factor of q, I can replace it with this. 390 00:28:23,530 --> 00:28:30,130 So I would have a t over k with a power of q/2. 391 00:28:30,130 --> 00:28:34,960 And then I have my integral that becomes the x, 392 00:28:34,960 --> 00:28:41,230 x to the d minus 1, 1 plus x squared plus potentially 393 00:28:41,230 --> 00:28:43,130 higher order things like this. 394 00:28:49,790 --> 00:28:56,680 Now, the upper cut-off for x is in fact 395 00:28:56,680 --> 00:29:03,430 square root k over t times lambda. 396 00:29:03,430 --> 00:29:10,910 And we are interested in the limit of when t goes to 0. 397 00:29:10,910 --> 00:29:17,100 So that upper limit is essentially going to infinity. 398 00:29:17,100 --> 00:29:20,630 Now, whether or not this integral, 399 00:29:20,630 --> 00:29:24,140 if I learn to ignore higher order terms 400 00:29:24,140 --> 00:29:28,890 and focus on the first term, exists really 401 00:29:28,890 --> 00:29:36,950 depends on whether d is larger, d minus 1 plus 1 d minus 4 402 00:29:36,950 --> 00:29:40,050 is positive or negative. 403 00:29:40,050 --> 00:29:44,810 And in particular, if I learn to get rid 404 00:29:44,810 --> 00:29:47,830 of all those higher order terms. 405 00:29:47,830 --> 00:29:51,170 And basically, the argument for that 406 00:29:51,170 --> 00:29:54,370 is the things that would go with x to the fourth, et cetera, 407 00:29:54,370 --> 00:29:57,090 if we carry additional factors of t-- 408 00:29:57,090 --> 00:30:01,598 and hopefully getting rid of them as to go to 0-- 409 00:30:01,598 --> 00:30:03,970 will give me an integral like this. 410 00:30:03,970 --> 00:30:08,562 This will exist only if I am in dimensions 411 00:30:08,562 --> 00:30:11,540 d that is less than 4. 412 00:30:11,540 --> 00:30:12,175 Yes? 413 00:30:12,175 --> 00:30:13,925 AUDIENCE: Are you missing the factors of t 414 00:30:13,925 --> 00:30:15,575 over t that comes with the denominator? 415 00:30:18,796 --> 00:30:20,834 PROFESSOR: Yes. 416 00:30:20,834 --> 00:30:23,738 There is a factor of 1 over t here. 417 00:30:29,550 --> 00:30:33,500 So I have to put out the factor of t. 418 00:30:33,500 --> 00:30:37,756 Write this as 1 plus k over t plus the element 419 00:30:37,756 --> 00:30:39,492 of t, et cetera. 420 00:30:39,492 --> 00:30:41,972 So there is a factor of 1 over t. 421 00:30:41,972 --> 00:30:43,460 AUDIENCE: t squared. 422 00:30:43,460 --> 00:30:45,196 PROFESSOR: And that's a factor of t 423 00:30:45,196 --> 00:30:46,932 squared, because that's two powers. 424 00:30:56,870 --> 00:31:02,385 So if I'm in dimensions d less than 4, what I can write 425 00:31:02,385 --> 00:31:09,530 is that this c singular, this as t goes to 0. 426 00:31:09,530 --> 00:31:13,550 The leading behavior, this goes to the constant. 427 00:31:13,550 --> 00:31:17,347 So as we discussed, after all of the mistakes that I made, 428 00:31:17,347 --> 00:31:22,204 there will be some overall coefficient A. The power of t 429 00:31:22,204 --> 00:31:25,400 will be d over 2 minus 2. 430 00:31:25,400 --> 00:31:27,700 d over 2 came from the integrations. 431 00:31:27,700 --> 00:31:32,310 1 over t squared came from the denominator. 432 00:31:32,310 --> 00:31:36,405 And then if I were to expand all of these other terms 433 00:31:36,405 --> 00:31:39,590 that we've ignored, higher powers of-- here 434 00:31:39,590 --> 00:31:43,660 I will get various series that will correct this. 435 00:31:43,660 --> 00:31:48,840 But the leading key dependents in dimensions less than 4 436 00:31:48,840 --> 00:31:51,032 is this thing that we had seen previously. 437 00:31:54,476 --> 00:31:58,340 Now I can take this, and you see that in dimensions 438 00:31:58,340 --> 00:32:02,294 d less than 4, this is a singular 439 00:32:02,294 --> 00:32:04,130 term that is diversion. 440 00:32:04,130 --> 00:32:08,100 If I were to say what kind of thing 441 00:32:08,100 --> 00:32:12,330 was of the energy that gave result to this? 442 00:32:12,330 --> 00:32:16,480 Then it would say that if the energy must 443 00:32:16,480 --> 00:32:19,737 have had some other constant that was proportionate of the t 444 00:32:19,737 --> 00:32:25,500 to the d over 2, that when I put two derivatives, 445 00:32:25,500 --> 00:32:27,920 I got something like this. 446 00:32:27,920 --> 00:32:29,790 Of course, if the energy could also 447 00:32:29,790 --> 00:32:33,940 have had a term that was linear in t, I wouldn't have seen it. 448 00:32:36,770 --> 00:32:39,170 So there is a singular part. 449 00:32:39,170 --> 00:32:43,095 Essentially, if I were to do that integral in dimensions 450 00:32:43,095 --> 00:32:47,690 less than fourth, I will get a leading singularity 451 00:32:47,690 --> 00:32:49,779 that is applied. 452 00:32:49,779 --> 00:32:51,570 I will get a singularity that is like this. 453 00:32:51,570 --> 00:32:54,520 I will get additional terms per constant-- t, t squared, 454 00:32:54,520 --> 00:32:58,482 et cetera-- and singular terms that 455 00:32:58,482 --> 00:32:59,880 are subbing in to this one. 456 00:33:03,394 --> 00:33:04,810 And then, of course, I have a term 457 00:33:04,810 --> 00:33:12,856 that is minus h squared over t if I were to include this here. 458 00:33:12,856 --> 00:33:23,783 So why don't I write the answer as B minus h divided by t 459 00:33:23,783 --> 00:33:29,039 to the 1/2 plus d/4, the whole thing squared. 460 00:33:33,830 --> 00:33:37,470 So what I did was essentially I divided and multiplied 461 00:33:37,470 --> 00:33:42,600 by inputting d and put the whole thing in the form of h divided 462 00:33:42,600 --> 00:33:46,410 by t to something squared? 463 00:33:46,410 --> 00:33:48,030 Why did I do that? 464 00:33:48,030 --> 00:33:53,425 It's because we had first related a singular form 465 00:33:53,425 --> 00:33:59,120 for the energies in the scaling picture that had the E to the 2 466 00:33:59,120 --> 00:34:03,818 minus alpha in front of them and the function of h t 467 00:34:03,818 --> 00:34:05,590 to the delta. 468 00:34:05,590 --> 00:34:11,960 And all I wanted to emphasize is that this picture, 2 minus 469 00:34:11,960 --> 00:34:16,860 alpha is d over 2. 470 00:34:16,860 --> 00:34:20,000 And the thing that we call the gap exponent 471 00:34:20,000 --> 00:34:22,034 is 1/2 plus d over 4. 472 00:34:30,630 --> 00:34:32,840 Of course, I can't use this theory 473 00:34:32,840 --> 00:34:36,060 as a description of the case. 474 00:34:36,060 --> 00:34:41,659 And the reason for that is that the Gaussian theory 475 00:34:41,659 --> 00:34:45,109 exists and is well-defined only as long as t is positive. 476 00:34:49,989 --> 00:34:57,310 Because once t becomes negative, then the rate 477 00:34:57,310 --> 00:35:01,250 essentially becomes ill-defined. 478 00:35:01,250 --> 00:35:05,080 Because if I look at the various rates that I have here, 479 00:35:05,080 --> 00:35:08,550 we certainly-- the rate for q equals to 0. 480 00:35:08,550 --> 00:35:13,700 It is proportional to minus t over 2v. 481 00:35:13,700 --> 00:35:16,670 If the t changes sign, rather than having a Gaussian, 482 00:35:16,670 --> 00:35:21,900 I have essentially a rate that is maximized as [INAUDIBLE]. 483 00:35:21,900 --> 00:35:26,420 So clearly, again, by issue of stability, 484 00:35:26,420 --> 00:35:30,940 the theory for t negative does not describe a stable theory. 485 00:35:30,940 --> 00:35:33,740 And that's why n to the fourth and all of those terms 486 00:35:33,740 --> 00:35:36,735 will be necessary to describe that side of the phase 487 00:35:36,735 --> 00:35:37,780 transition. 488 00:35:37,780 --> 00:35:42,920 So if you like, this is a kind of a description 489 00:35:42,920 --> 00:35:51,700 of a singularity that exists only in this half of the space. 490 00:35:51,700 --> 00:35:55,250 Kind of reminiscent of coming from the disordered side, 491 00:35:55,250 --> 00:36:00,730 but I don't want to give it more reality than that. 492 00:36:00,730 --> 00:36:02,010 It's a mathematical construct. 493 00:36:02,010 --> 00:36:05,260 If we want to venture to make the connection 494 00:36:05,260 --> 00:36:07,646 to the actual phase transition, we 495 00:36:07,646 --> 00:36:09,329 have to prove the n to the fourth. 496 00:36:19,600 --> 00:36:24,410 Now, the only reason to go and recap this Gaussian theory 497 00:36:24,410 --> 00:36:30,100 is because since it is solve-able, 498 00:36:30,100 --> 00:36:32,710 we can try to use it as a toy model 499 00:36:32,710 --> 00:36:37,350 to apply the various steps of renormalization group 500 00:36:37,350 --> 00:36:39,980 that we had outlined last lecture. 501 00:36:39,980 --> 00:36:45,580 And once we understand the steps of renormalization group 502 00:36:45,580 --> 00:36:50,590 for this theory, then it gives us an anchoring point 503 00:36:50,590 --> 00:36:52,650 when we describe the full theory that 504 00:36:52,650 --> 00:36:55,930 has n to the fourth, et cetera-- how 505 00:36:55,930 --> 00:36:58,687 to sort of start with the renormalization approach 506 00:36:58,687 --> 00:37:04,316 to the theory as we understand and do the more complicated. 507 00:37:04,316 --> 00:37:08,060 So essentially, as I said, it's not really a phase transition 508 00:37:08,060 --> 00:37:10,980 that can be described by this theory. 509 00:37:10,980 --> 00:37:14,310 It's a singularity. 510 00:37:14,310 --> 00:37:18,910 But its value is that it is this fully-modelled anchoring 511 00:37:18,910 --> 00:37:22,407 point for the full theory that we are describing. 512 00:37:27,397 --> 00:37:32,886 So what we want to do is to do an RG for the Gaussian model. 513 00:37:48,890 --> 00:37:51,260 So what is the procedure. 514 00:37:55,145 --> 00:38:00,940 We have a theory best described in the space 515 00:38:00,940 --> 00:38:04,330 of variables q, the Fourier variables. 516 00:38:04,330 --> 00:38:11,967 Where I have modes that exist between 0-- very long 517 00:38:11,967 --> 00:38:16,075 wavelength-- to lambda, which is the inverse of some shortest 518 00:38:16,075 --> 00:38:19,456 wavelength that I'm allowing. 519 00:38:19,456 --> 00:38:27,400 And so basically, I have a bunch of modes m of q 520 00:38:27,400 --> 00:38:29,910 that are defined in this range of qx. 521 00:38:32,790 --> 00:38:39,005 The first step of RG was to coarse grain. 522 00:38:43,616 --> 00:38:50,225 The idea of coarse graining was to change the scale over which 523 00:38:50,225 --> 00:38:56,330 you were doing the averaging from some a to ba. 524 00:38:56,330 --> 00:39:04,510 So average from a to ba of fluctuations. 525 00:39:07,090 --> 00:39:11,940 So once I do that, at the end of the day 526 00:39:11,940 --> 00:39:16,370 I have fluctuations whose minimum wavelength has 527 00:39:16,370 --> 00:39:18,175 gone from a to ba. 528 00:39:21,320 --> 00:39:29,750 So that means that q max, after I go and do this procedure, 529 00:39:29,750 --> 00:39:37,956 is the previous q max that I had divided by a factor of b. 530 00:39:37,956 --> 00:39:45,480 So basically, at the end of the day I want to have, 531 00:39:45,480 --> 00:39:51,530 after coarse graining, variables that only exist up 532 00:39:51,530 --> 00:39:55,468 to lambda over b. 533 00:39:55,468 --> 00:39:59,896 Whereas previously, they existed after that. 534 00:40:02,850 --> 00:40:07,330 So this is very easy at this level. 535 00:40:07,330 --> 00:40:12,800 All I can do is to replace this m tilde of q 536 00:40:12,800 --> 00:40:15,520 in terms of two sets. 537 00:40:15,520 --> 00:40:22,400 I will call it to be sigma if q is 538 00:40:22,400 --> 00:40:27,840 greater than this lambda over b. 539 00:40:27,840 --> 00:40:33,350 That is, everybody that is out here, their q-- 540 00:40:33,350 --> 00:40:36,100 I will call it q larger. 541 00:40:36,100 --> 00:40:41,400 Everybody that is here, their q I will call q lesser. 542 00:40:41,400 --> 00:40:44,620 And all the modes that were here, 543 00:40:44,620 --> 00:40:46,810 I will give them a different name. 544 00:40:46,810 --> 00:40:50,320 The ones here I will call sigma. 545 00:40:50,320 --> 00:41:03,145 The ones here, if q less than lambda over b, 546 00:41:03,145 --> 00:41:06,620 will get called m tilde. 547 00:41:06,620 --> 00:41:09,620 So I just renamed my variables. 548 00:41:09,620 --> 00:41:17,740 So essentially, right here I had integration 549 00:41:17,740 --> 00:41:22,120 over all of the modes. 550 00:41:22,120 --> 00:41:26,403 I just renamed some of the modes that were inside 551 00:41:26,403 --> 00:41:32,040 q lesser and sigma-- and tilde, the ones that 552 00:41:32,040 --> 00:41:35,600 are outside q greater. 553 00:41:35,600 --> 00:41:40,570 So what I have to do for my Gaussian theory. 554 00:41:40,570 --> 00:41:44,350 Let's write it rather than in this form 555 00:41:44,350 --> 00:41:48,574 that was discrete in terms of the continuum. 556 00:41:51,360 --> 00:41:55,370 I have to iterate over all configurations of these Fourier 557 00:41:55,370 --> 00:41:56,720 modes. 558 00:41:56,720 --> 00:41:58,793 So I have these m tilde of q's. 559 00:42:02,950 --> 00:42:05,921 And the wave that I have to assign to them when 560 00:42:05,921 --> 00:42:11,580 I look at the continuum is exponential, integral in dq 561 00:42:11,580 --> 00:42:12,760 q to pi to the d. 562 00:42:12,760 --> 00:42:16,736 T plus kq squared, and so forth. 563 00:42:19,718 --> 00:42:22,203 And tilde of q squared. 564 00:42:26,180 --> 00:42:33,554 And then I had the one term that was hm of 0. 565 00:42:38,790 --> 00:42:43,030 What I have done is to simply rewrite 566 00:42:43,030 --> 00:42:50,704 this as two sets of integrations over the-- whoops. 567 00:42:50,704 --> 00:42:52,135 This was m. 568 00:42:55,951 --> 00:43:01,450 m, let's call is sigma first-- sigma of q 569 00:43:01,450 --> 00:43:06,692 larger integrate over m tilde of q lesser. 570 00:43:13,664 --> 00:43:18,670 And actually, you can see that the modes here 571 00:43:18,670 --> 00:43:21,950 and the modes here don't talk to each other. 572 00:43:21,950 --> 00:43:27,000 And that's really the advantage of doing the Gaussian theory. 573 00:43:27,000 --> 00:43:29,930 And the thing that allowed me to solve the problem here 574 00:43:29,930 --> 00:43:32,860 and also to do the coarse graining there. 575 00:43:32,860 --> 00:43:36,560 Once we do things like n to the fourth, 576 00:43:36,560 --> 00:43:38,856 then I will have couplings between modes 577 00:43:38,856 --> 00:43:42,410 that go across between the three sets. 578 00:43:42,410 --> 00:43:44,846 And then the problem becomes difficult. 579 00:43:44,846 --> 00:43:48,100 But now that I don't have that, I 580 00:43:48,100 --> 00:43:52,085 can actually separately write the integral as two parts. 581 00:44:03,700 --> 00:44:07,000 And this is for q lesser. 582 00:44:07,000 --> 00:44:11,741 And for each one of them, I essentially have the same rate. 583 00:44:11,741 --> 00:44:19,240 The integral over q greater goes between lambda 584 00:44:19,240 --> 00:44:20,818 over d and lambda. 585 00:44:36,626 --> 00:44:40,578 The integral over m tilde of q lesser 586 00:44:40,578 --> 00:44:43,048 is essentially the same thing. 587 00:44:51,446 --> 00:44:58,160 Exponential minus integral 0 to lambda over d. dv 588 00:44:58,160 --> 00:45:05,650 q lesser to five to the d, t plus kq lesser squared, 589 00:45:05,650 --> 00:45:08,720 and so forth. 590 00:45:08,720 --> 00:45:14,406 And q lesser squared. 591 00:45:14,406 --> 00:45:22,448 And then I have the additional term which sits at 0. 592 00:45:22,448 --> 00:45:31,760 It is part of the modes that are assigned with q lesser. 593 00:45:36,093 --> 00:45:36,593 OK? 594 00:45:36,593 --> 00:45:37,620 Fine. 595 00:45:37,620 --> 00:45:40,590 Nothing particularly profound here. 596 00:45:40,590 --> 00:45:42,090 In fact, it's very simple. 597 00:45:42,090 --> 00:45:46,040 It's just renaming two sets of modes. 598 00:45:46,040 --> 00:45:51,420 And the averaging that I have to do, 599 00:45:51,420 --> 00:45:53,764 and getting rid of the fluctuations 600 00:45:53,764 --> 00:45:59,250 at short wavelength, here is very trickier. 601 00:45:59,250 --> 00:46:03,938 Because this is just a bunch of integrations 602 00:46:03,938 --> 00:46:09,380 that I had to do over here, but it is only over things 603 00:46:09,380 --> 00:46:12,826 that are sitting close to the edge of this [INAUDIBLE]. 604 00:46:16,810 --> 00:46:21,292 So essentially, the integrations over these modes 605 00:46:21,292 --> 00:46:24,280 is doing this integral over here, 606 00:46:24,280 --> 00:46:30,260 from lambda over d to lambda, and none of the singularities 607 00:46:30,260 --> 00:46:34,020 has anything to do with the range of integration 608 00:46:34,020 --> 00:46:37,020 from lambda over d to lambda. 609 00:46:37,020 --> 00:46:40,020 So the result of doing all of that 610 00:46:40,020 --> 00:46:45,206 is simply just a constant-- but not a constant. 611 00:46:45,206 --> 00:46:51,220 It's a function of t that is completely non-singular 612 00:46:51,220 --> 00:46:55,690 and have a nice state of expansion powers of t. 613 00:46:55,690 --> 00:47:01,080 A kind of [INAUDIBLE] I call non-singular functions 614 00:47:01,080 --> 00:47:01,660 sometimes. 615 00:47:01,660 --> 00:47:03,539 Constant thing is that eventually 616 00:47:03,539 --> 00:47:05,288 if you take sufficiently high derivatives, 617 00:47:05,288 --> 00:47:10,900 I guess, of this value, the t dependents [INAUDIBLE]. 618 00:47:16,250 --> 00:47:24,550 So all of the interesting thing is really in this m tilde of k 619 00:47:24,550 --> 00:47:25,800 lesser. 620 00:47:25,800 --> 00:47:31,220 And really, the eventual process of renormalization 621 00:47:31,220 --> 00:47:35,400 in this picture is something like this. 622 00:47:35,400 --> 00:47:37,330 That all of the singularities are 623 00:47:37,330 --> 00:47:43,310 sitting at the center of this kind of orange-shaped entity. 624 00:47:43,310 --> 00:47:46,115 And rather than biting the whole thing, 625 00:47:46,115 --> 00:47:51,390 you kind of cut it slowly and slowly from the edge, 626 00:47:51,390 --> 00:47:54,610 approaching to where all of the exciting things 627 00:47:54,610 --> 00:47:55,680 are at the center. 628 00:47:55,680 --> 00:47:57,550 For this problem of the Gaussian, 629 00:47:57,550 --> 00:48:00,440 it turns out to be trivial to do so. 630 00:48:00,440 --> 00:48:02,980 But for the more general problem, 631 00:48:02,980 --> 00:48:06,570 it can be interesting because procedure is the same. 632 00:48:06,570 --> 00:48:09,030 We are interested in what's happening here, 633 00:48:09,030 --> 00:48:13,450 but we gradually peel of things that we 634 00:48:13,450 --> 00:48:17,540 know don't cause anything difficult for the problems. 635 00:48:21,540 --> 00:48:24,600 So then I have to multiply with this, 636 00:48:24,600 --> 00:48:31,440 and I have found in some sense a probability for configurations 637 00:48:31,440 --> 00:48:35,490 of the coarse grain system, which is simply given by this. 638 00:48:38,100 --> 00:48:42,842 But then renormalization group has two other steps. 639 00:48:45,690 --> 00:48:50,520 The second step was to say, well, in real space, 640 00:48:50,520 --> 00:48:53,870 as we said, the picture that is represented 641 00:48:53,870 --> 00:48:57,730 by these coarse grain variables is grainy. 642 00:48:57,730 --> 00:49:01,410 If my pixels were previously one by one by one, 643 00:49:01,410 --> 00:49:04,746 now my pixels are d by d by d. 644 00:49:04,746 --> 00:49:10,380 So I can make my picture look to have the same resolution 645 00:49:10,380 --> 00:49:14,630 as my initial picture if I rescale 646 00:49:14,630 --> 00:49:16,590 all of the events for a factor of t. 647 00:49:19,180 --> 00:49:23,310 In momentum representation, or intuitive presentation, 648 00:49:23,310 --> 00:49:33,160 it corresponds to rescaling all of the q's by a factor of B. 649 00:49:33,160 --> 00:49:36,640 And clearly, what that serves to achieve 650 00:49:36,640 --> 00:49:44,020 is that if I replace q lesser with B times q prime, 651 00:49:44,020 --> 00:49:50,820 then the maximum value will go back to 0 to lambda. 652 00:49:50,820 --> 00:49:53,350 So by doing this one in formation, 653 00:49:53,350 --> 00:50:00,514 I can ensure that the upper cut-off is, in fact, lambda 654 00:50:00,514 --> 00:50:03,240 again. 655 00:50:03,240 --> 00:50:07,160 Now, there was another thing, which in real space 656 00:50:07,160 --> 00:50:15,330 we said that we defined m prime to be m tilde rescaled 657 00:50:15,330 --> 00:50:16,819 by some factor zeta. 658 00:50:21,130 --> 00:50:24,990 I had to do a change of the contrast. 659 00:50:24,990 --> 00:50:27,590 I did have to do the same change of contrast 660 00:50:27,590 --> 00:50:32,200 here, except that the variables that I am dealing with here, 661 00:50:32,200 --> 00:50:36,470 it was in x coordinates. 662 00:50:36,470 --> 00:50:40,570 What I want to do it is in the q coordinate. 663 00:50:40,570 --> 00:50:49,884 So I will call m with a tail prime of q prime 664 00:50:49,884 --> 00:50:59,510 to be m tilde of q prime by a factor of z. 665 00:50:59,510 --> 00:51:03,200 The difference between the z and the zeta, which 666 00:51:03,200 --> 00:51:06,280 is real space and Fourier space is just the fact 667 00:51:06,280 --> 00:51:11,210 that in going from one to the other, 668 00:51:11,210 --> 00:51:14,280 you have to do integrations over space. 669 00:51:14,280 --> 00:51:17,070 So dimensionally, there is a factor of b 670 00:51:17,070 --> 00:51:20,670 to the d difference between the rescaling of this quantity 671 00:51:20,670 --> 00:51:26,665 and that quantity, and if you want to use 672 00:51:26,665 --> 00:51:32,180 or the other zeta against b to the minus d and z. 673 00:51:32,180 --> 00:51:36,250 But since we would be doing everything in Fourier space, 674 00:51:36,250 --> 00:51:38,500 we would just use this factor traditionally. 675 00:51:42,950 --> 00:51:46,440 So if I do that, what do I find? 676 00:51:46,440 --> 00:51:57,430 I find that Z of t of h is exponential of some singular, 677 00:51:57,430 --> 00:52:00,700 non-singular dependents. 678 00:52:00,700 --> 00:52:08,550 And then I have to integrate over these new variables, 679 00:52:08,550 --> 00:52:11,940 m prime of q prime. 680 00:52:11,940 --> 00:52:13,740 Yes? 681 00:52:13,740 --> 00:52:17,270 AUDIENCE: In your real space renormalization 682 00:52:17,270 --> 00:52:21,769 your m tilde is a function of an x. 683 00:52:21,769 --> 00:52:23,435 But in your Fourier space representation 684 00:52:23,435 --> 00:52:28,885 your m tilde is a function of q prime? 685 00:52:28,885 --> 00:52:32,053 PROFESSOR: I guess I could have written here x prime, also. 686 00:52:32,053 --> 00:52:33,094 It doesn't really matter. 687 00:52:42,860 --> 00:52:44,150 So do you have here? 688 00:52:44,150 --> 00:52:49,880 You have exponential minus the integral. 689 00:52:49,880 --> 00:52:56,820 The integration for q prime now is going back to 0 to lambda. 690 00:52:56,820 --> 00:53:03,390 I have db of q prime divided by 2 pi to the d. 691 00:53:09,150 --> 00:53:19,490 Now, you see that every time I have a q-- V or q, q lesser, 692 00:53:19,490 --> 00:53:24,250 in fact-- I have to go to q prime 693 00:53:24,250 --> 00:53:28,592 by introducing a factor of the inverse. 694 00:53:28,592 --> 00:53:33,440 So there will be a total factor of V to the minus V 695 00:53:33,440 --> 00:53:37,246 that comes from this integration. 696 00:53:37,246 --> 00:53:40,000 And that will multiply t. 697 00:53:40,000 --> 00:53:45,590 That will multiply kb to the minus d. 698 00:53:45,590 --> 00:53:51,650 But then here I have to q's because of the q squared there. 699 00:53:51,650 --> 00:53:58,940 Again, doing the same thing, I will get V to the d minus two. 700 00:53:58,940 --> 00:54:02,790 I had q plus 2, if you like. 701 00:54:02,790 --> 00:54:09,195 And then the next l would be lb to the d minus 4. 702 00:54:09,195 --> 00:54:12,130 And you can see that as I have higher and higher derivatives 703 00:54:12,130 --> 00:54:15,850 of q, I get higher and higher powers with negative 704 00:54:15,850 --> 00:54:16,350 [INAUDIBLE]. 705 00:54:21,060 --> 00:54:28,760 But then I have m tilde that I want to replace with m prime. 706 00:54:28,760 --> 00:54:34,090 And that process will give me a factor of z squared. 707 00:54:34,090 --> 00:54:39,340 And then I have m prime of q prime squared. 708 00:54:43,225 --> 00:54:46,410 There is no integration for this terms. 709 00:54:46,410 --> 00:54:48,260 It's just one mode. 710 00:54:48,260 --> 00:54:53,460 But each mode I have rescaled by a factor of z. 711 00:54:53,460 --> 00:55:03,262 So I will have a term that is z h dot m prime of 0. 712 00:55:13,690 --> 00:55:17,470 So what we see is that what we have managed to do 713 00:55:17,470 --> 00:55:21,590 is to make the Gaussian integration over 714 00:55:21,590 --> 00:55:26,410 here precisely the same thing as the Gaussian integration 715 00:55:26,410 --> 00:55:28,650 that I started with. 716 00:55:28,650 --> 00:55:33,486 So I can conclude that this function tnh 717 00:55:33,486 --> 00:55:39,090 that I am interested in has a path that is non-singular. 718 00:55:42,780 --> 00:55:45,670 But its singular part is the same 719 00:55:45,670 --> 00:55:52,427 as the same z calculated for a bunch of new parameters. 720 00:55:55,840 --> 00:56:02,250 And in particular, the new t is v to the minus 721 00:56:02,250 --> 00:56:07,755 d z squared the old t. 722 00:56:07,755 --> 00:56:15,320 The new k is b to the minus d minus 2 z squared q. 723 00:56:15,320 --> 00:56:21,515 The new L would be to the minus d minus 4 z squared L, 724 00:56:21,515 --> 00:56:23,500 and so forth. 725 00:56:23,500 --> 00:56:27,858 And the new h is zh. 726 00:56:27,858 --> 00:56:28,792 Yes? 727 00:56:28,792 --> 00:56:33,465 AUDIENCE: There should be q prime squared and q prime 4? 728 00:56:33,465 --> 00:56:34,090 PROFESSOR: Yes. 729 00:56:49,342 --> 00:56:50,326 Yes. 730 00:56:50,326 --> 00:56:53,770 This is my day to do a lot of algebraic errors. 731 00:57:00,440 --> 00:57:00,940 OK. 732 00:57:05,180 --> 00:57:07,630 So what is the change in parameters? 733 00:57:07,630 --> 00:57:10,970 So I wrote it over there. 734 00:57:10,970 --> 00:57:17,610 So this kind of captures the very simplest type 735 00:57:17,610 --> 00:57:20,160 of renormalization. 736 00:57:20,160 --> 00:57:24,510 Actually, all I did was a scaling analysis. 737 00:57:24,510 --> 00:57:29,260 If I were to change positions by a factor of b 738 00:57:29,260 --> 00:57:34,460 and change the magnitude of my field m by a factor z or zeta, 739 00:57:34,460 --> 00:57:37,302 this is the kind of results that I will get. 740 00:57:40,470 --> 00:57:46,500 Now, how can we make this capture the kind of picture 741 00:57:46,500 --> 00:57:52,190 that we have over here in the language of renormalization? 742 00:57:52,190 --> 00:57:57,830 Want to be able to change two parameters 743 00:57:57,830 --> 00:57:59,918 and reach a fixed point. 744 00:58:03,590 --> 00:58:09,940 So we know that kind of [INAUDIBLE] that t and h 745 00:58:09,940 --> 00:58:11,020 have to go to 0. 746 00:58:11,020 --> 00:58:15,910 They are the variables that determine essentially 747 00:58:15,910 --> 00:58:19,260 whether you are at this said similar point. 748 00:58:19,260 --> 00:58:24,460 So if t and h I forget, the next most important term that 749 00:58:24,460 --> 00:58:29,850 comes into play is k prime, which is some function of k. 750 00:58:29,850 --> 00:58:32,680 And if I want to be at the fixed point, 751 00:58:32,680 --> 00:58:36,540 I may want to choose the factor z such 752 00:58:36,540 --> 00:58:40,960 that k prime is the same as k. 753 00:58:40,960 --> 00:58:50,390 So choose z such that k prime is k. 754 00:58:50,390 --> 00:58:54,346 And that tells me immediately that z 755 00:58:54,346 --> 00:58:57,552 would be b to the power of 1 plus d over 2. 756 00:59:02,760 --> 00:59:09,756 If I choose that particular form of z, then what do I get? 757 00:59:09,756 --> 00:59:15,132 I get t prime is z squared b to the minus b. 758 00:59:15,132 --> 00:59:20,600 So when I do that, I will get b squared t. 759 00:59:20,600 --> 00:59:27,780 I get that h prime is just z times h. 760 00:59:27,780 --> 00:59:33,280 So it is b to the 1 plus b over 2 times h. 761 00:59:33,280 --> 00:59:41,291 These are both directions that as b becomes larger than 1, 762 00:59:41,291 --> 00:59:45,970 b prime becomes larger than th prime, becomes larger than h. 763 00:59:45,970 --> 00:59:48,040 These are relevant directions. 764 00:59:48,040 --> 00:59:53,230 I would associate with them eigenvalues y dt minus 2. 765 00:59:53,230 --> 00:59:54,316 Divide h. 766 00:59:54,316 --> 00:59:56,696 That is 1 plus d over 2. 767 01:00:00,510 --> 01:00:07,300 So if I go according to the scaling construction that we 768 01:00:07,300 --> 01:00:14,310 had before, f singular of tnh is t to the power 769 01:00:14,310 --> 01:00:19,470 d over y dt, some scaling function of h, 770 01:00:19,470 --> 01:00:24,640 g to the power of divide h over y dt. 771 01:00:24,640 --> 01:00:27,480 This is what we have established before. 772 01:00:27,480 --> 01:00:32,430 With these values I will get t to the d over 2, 773 01:00:32,430 --> 01:00:41,888 some scaling function of h, t to the power of 1/2 plus d over 4. 774 01:00:44,660 --> 01:00:51,700 We can immediately compare this expression and this expression 775 01:00:51,700 --> 01:00:52,670 that we have over here. 776 01:00:52,670 --> 01:00:52,915 Yes? 777 01:00:52,915 --> 01:00:53,540 AUDIENCE: Wait. 778 01:00:53,540 --> 01:00:54,970 What's the reason to choose scale 779 01:00:54,970 --> 01:00:57,011 as the parameter that maps onto itself and not L? 780 01:00:57,011 --> 01:00:57,670 PROFESSOR: OK. 781 01:00:57,670 --> 01:00:59,310 I'll come to that. 782 01:00:59,310 --> 01:01:05,140 So having gone this far, let's see what l is doing. 783 01:01:05,140 --> 01:01:07,730 So if I put here-- you can see that clearly L 784 01:01:07,730 --> 01:01:12,540 has v to the minus 2 compared to k. 785 01:01:12,540 --> 01:01:15,620 So currently, the way that we established, 786 01:01:15,620 --> 01:01:19,860 L prime is b to the minus 2m. 787 01:01:19,860 --> 01:01:22,310 If I had a higher derivative, it would 788 01:01:22,310 --> 01:01:26,530 be b to a minus larger number, et cetera. 789 01:01:26,530 --> 01:01:32,390 So L, out of these other things, are irrelevant variables. 790 01:01:32,390 --> 01:01:36,950 So they are essentially under rescaling, 791 01:01:36,950 --> 01:01:40,832 under looking at the system in larger and larger scale, 792 01:01:40,832 --> 01:01:42,125 they will go to 0. 793 01:01:42,125 --> 01:01:48,410 And I did get a system that has the same topological structure 794 01:01:48,410 --> 01:01:51,133 as what I had established here. 795 01:01:51,133 --> 01:01:53,625 Because I have to tune two parameters 796 01:01:53,625 --> 01:01:56,880 in order to reach the critical point. 797 01:01:56,880 --> 01:02:00,510 Let's say I had chosen something else. 798 01:02:00,510 --> 01:02:11,530 If I had chosen z such that L prime equals to L. 799 01:02:11,530 --> 01:02:13,790 I could do that. 800 01:02:13,790 --> 01:02:19,780 Then all of the derivatives that are higher factors of q in this 801 01:02:19,780 --> 01:02:22,810 [INAUDIBLE], they would be all irrelevant. 802 01:02:22,810 --> 01:02:28,140 But then I would have k, t, and h 803 01:02:28,140 --> 01:02:30,219 all with irrelevant variables. 804 01:02:34,390 --> 01:02:39,000 So yeah, it could be that there is some physics. 805 01:02:39,000 --> 01:02:42,040 I mean, certainly mathematically I 806 01:02:42,040 --> 01:02:48,410 can ask the system what happens if k goes to 0. 807 01:02:48,410 --> 01:02:50,670 I kind of ignore the k dependencies 808 01:02:50,670 --> 01:02:54,340 that I have in all of these expressions, 809 01:02:54,340 --> 01:02:59,660 but there are going to be singular dependencies on k. 810 01:02:59,660 --> 01:03:04,070 So if there is indeed some experimental system 811 01:03:04,070 --> 01:03:10,170 in which you have to tune, in addition temperature, 812 01:03:10,170 --> 01:03:12,160 something that has to do with the way 813 01:03:12,160 --> 01:03:14,740 that the spins or degrees of freedom 814 01:03:14,740 --> 01:03:17,885 are coupled to each other, and that coupling changes sign 815 01:03:17,885 --> 01:03:20,056 from being positive to being negative, 816 01:03:20,056 --> 01:03:23,995 you go from one type of behavior to another type of behavior, 817 01:03:23,995 --> 01:03:26,350 maybe this would be a good thing for it. 818 01:03:26,350 --> 01:03:29,040 But you can see the kind of structure 819 01:03:29,040 --> 01:03:34,190 you would get if k has to go to 0, you go from a structure 820 01:03:34,190 --> 01:03:36,860 where things want to be in the same direction 821 01:03:36,860 --> 01:03:39,410 to things that want to be anti-parallel. 822 01:03:39,410 --> 01:03:43,370 And then clearly you need higher order terms to stabilize things 823 01:03:43,370 --> 01:03:45,340 so that your singularity does not 824 01:03:45,340 --> 01:03:48,880 go all the way to 0 wavelength, et cetera. 825 01:03:48,880 --> 01:03:54,090 So one can actually come up with physical systems 826 01:03:54,090 --> 01:03:56,226 that kind of resemble that, were there 827 01:03:56,226 --> 01:03:58,942 is some landscape that is also chosen. 828 01:03:58,942 --> 01:04:02,910 But for this very simplest thing that we are doing, 829 01:04:02,910 --> 01:04:05,440 this is what is going on. 830 01:04:05,440 --> 01:04:09,080 But you could have also asked the other question. 831 01:04:09,080 --> 01:04:12,010 So clearly we understand what happens 832 01:04:12,010 --> 01:04:15,790 if you choose z so that some term is fixed 833 01:04:15,790 --> 01:04:18,750 and everything above it is relevant, everything below it 834 01:04:18,750 --> 01:04:20,370 is irrelevant. 835 01:04:20,370 --> 01:04:31,998 But why not choose z such that t is fixed? 836 01:04:31,998 --> 01:04:35,225 So that's going to be b to the d over 2, 837 01:04:35,225 --> 01:04:37,120 then t prime equals to t. 838 01:04:40,048 --> 01:04:45,060 If I choose that, then clearly the coupling k 839 01:04:45,060 --> 01:04:46,295 will already be irrelevant. 840 01:04:49,270 --> 01:04:54,640 So this is actually a reasonable fixed point. 841 01:04:54,640 --> 01:04:57,360 It's a fixed one that corresponds to a system 842 01:04:57,360 --> 01:05:00,530 where k has gone to 0, which means 843 01:05:00,530 --> 01:05:04,150 that the different points don't talk to each other. 844 01:05:04,150 --> 01:05:06,180 Remember, when we were discussing 845 01:05:06,180 --> 01:05:09,410 the behavior of correlation lens at fixed points, 846 01:05:09,410 --> 01:05:11,995 there was two possibilities-- either the correlation 847 01:05:11,995 --> 01:05:14,570 lens was infinite or it was 0. 848 01:05:14,570 --> 01:05:20,730 So if I choose this, then k prime will go eventually to 0. 849 01:05:20,730 --> 01:05:24,410 I go towards a system in which the degrees of freedom 850 01:05:24,410 --> 01:05:28,630 are completely decoupled from each other. 851 01:05:28,630 --> 01:05:30,020 Perfectly well-behaved. 852 01:05:30,020 --> 01:05:34,380 Fixed behavior that corresponds to 0 correlation lens. 853 01:05:34,380 --> 01:05:38,200 And you can see that if I go through this formula 854 01:05:38,200 --> 01:05:44,640 that I told you over here, zeta in real space 855 01:05:44,640 --> 01:05:49,250 would be b to the minus d over 2. 856 01:05:49,250 --> 01:05:53,960 And what that means is that if you average 857 01:05:53,960 --> 01:05:59,130 independent variables over a size b, 858 01:05:59,130 --> 01:06:02,290 the scale of fluctuation is because of the central limit 859 01:06:02,290 --> 01:06:04,720 theeorem is the square root of the volume. 860 01:06:04,720 --> 01:06:05,720 So that's how it scales. 861 01:06:08,580 --> 01:06:12,310 So essentially, what's at the end of the story? 862 01:06:12,310 --> 01:06:16,820 That's a behavior in which there is only one coefficient event-- 863 01:06:16,820 --> 01:06:18,150 forget about h. 864 01:06:18,150 --> 01:06:23,480 The eventual rate is just t over 2m squared at different points. 865 01:06:23,480 --> 01:06:26,120 That's the central limit here. 866 01:06:26,120 --> 01:06:30,660 So through a different route, we have rediscovered, if you like, 867 01:06:30,660 --> 01:06:32,300 the central limit theorem. 868 01:06:32,300 --> 01:06:36,330 Because if you average lots of uncorrelated variables, 869 01:06:36,330 --> 01:06:39,700 you will generate Gaussian rates. 870 01:06:39,700 --> 01:06:45,070 So what we are really after in this language 871 01:06:45,070 --> 01:06:49,922 is how to generalize the central limit theorem, how to-- 872 01:06:49,922 --> 01:06:53,290 as we find the analog of a Gaussian, the degrees 873 01:06:53,290 --> 01:06:55,730 of freedom that are not correlated 874 01:06:55,730 --> 01:06:59,580 but talk to their neighborhood. 875 01:06:59,580 --> 01:07:04,110 So the kind of field theory that we are after 876 01:07:04,110 --> 01:07:06,950 are these generalizations of central limit theorem 877 01:07:06,950 --> 01:07:10,700 to the types of field theories that 878 01:07:10,700 --> 01:07:12,025 have some locality enablement. 879 01:07:15,985 --> 01:07:16,980 AUDIENCE: Question. 880 01:07:16,980 --> 01:07:17,680 PROFESSOR: Yes. 881 01:07:17,680 --> 01:07:22,660 AUDIENCE: So wherever you can define the renormalization 882 01:07:22,660 --> 01:07:25,720 you're finding different z's? 883 01:07:25,720 --> 01:07:26,380 PROFESSOR: Yes. 884 01:07:26,380 --> 01:07:29,660 AUDIENCE: We can tune how many parameters we 885 01:07:29,660 --> 01:07:30,770 want to be able to-- 886 01:07:30,770 --> 01:07:31,880 PROFESSOR: Exactly. 887 01:07:31,880 --> 01:07:32,900 Yes. 888 01:07:32,900 --> 01:07:36,300 And that's where the physics comes into play. 889 01:07:36,300 --> 01:07:40,030 Mathematically, there's a whole set of different fixed points 890 01:07:40,030 --> 01:07:43,390 that you can construct for choosing different z's. 891 01:07:43,390 --> 01:07:45,540 You have to decide which one of them 892 01:07:45,540 --> 01:07:47,260 corresponds to the physical problem 893 01:07:47,260 --> 01:07:49,036 that you are working on. 894 01:07:49,036 --> 01:07:49,970 AUDIENCE: Yes. 895 01:07:49,970 --> 01:07:53,052 So if the fixed point stops being just defined 896 01:07:53,052 --> 01:07:57,955 by the nature of the system, but it's also 897 01:07:57,955 --> 01:08:00,925 depends on how we define renormalization? 898 01:08:00,925 --> 01:08:04,390 On mathematical descriptions and-- 899 01:08:07,855 --> 01:08:13,176 PROFESSOR: If by how we define renormalization looking 900 01:08:13,176 --> 01:08:15,080 to choose z, yes, I agree with you. 901 01:08:15,080 --> 01:08:16,510 Yes. 902 01:08:16,510 --> 01:08:20,990 But again, you have this possibility 903 01:08:20,990 --> 01:08:25,005 of looking at the system at different scales. 904 01:08:25,005 --> 01:08:30,260 But we have been very agnostic about what that system is. 905 01:08:30,260 --> 01:08:33,300 And so you how many ways of doing things. 906 01:08:33,300 --> 01:08:36,474 Ultimately, you need some reality to come and choose 907 01:08:36,474 --> 01:08:38,609 among these different ways. 908 01:08:38,609 --> 01:08:41,444 Yes? 909 01:08:41,444 --> 01:08:42,860 AUDIENCE: So you do want to keep k 910 01:08:42,860 --> 01:08:46,930 a relevant variable in group problems, right? 911 01:08:46,930 --> 01:08:49,666 PROFESSOR: No. 912 01:08:49,666 --> 01:08:50,999 I make k to be a fixed variable. 913 01:08:50,999 --> 01:08:53,590 AUDIENCE: Oh, exactly. 914 01:08:53,590 --> 01:08:57,939 Why don't you add a small amount, like an absolute 915 01:08:57,939 --> 01:09:02,020 to the power of bf and [INAUDIBLE] point z. 916 01:09:02,020 --> 01:09:04,680 Plus or minus, doesn't matter. 917 01:09:04,680 --> 01:09:07,880 Why the equality assumption exactly? 918 01:09:07,880 --> 01:09:10,310 And the smaller one doesn't change anything? 919 01:09:10,310 --> 01:09:16,420 All the other variables like L become irrelevant? 920 01:09:16,420 --> 01:09:18,029 PROFESSOR: OK. 921 01:09:18,029 --> 01:09:25,090 So the point is that it is b raised to some power. 922 01:09:25,090 --> 01:09:30,890 So here I had, I don't know, Katie k prime was k. 923 01:09:30,890 --> 01:09:33,130 And you say, why not kb to the absolute? 924 01:09:33,130 --> 01:09:34,834 AUDIENCE: Yeah, exactly. 925 01:09:34,834 --> 01:09:37,420 PROFESSOR: Now, the thing that I'm 926 01:09:37,420 --> 01:09:41,649 interested in what happens at larger and larger scale. 927 01:09:41,649 --> 01:09:44,920 So in principle, I should be able to make v 928 01:09:44,920 --> 01:09:47,390 as large as I want. 929 01:09:47,390 --> 01:09:52,585 So I don't have the freedom that you mentioned. 930 01:09:52,585 --> 01:09:56,516 And you are right in the sense that, OK, 931 01:09:56,516 --> 01:09:58,470 what does it mean whether this ratio is 932 01:09:58,470 --> 01:10:01,355 larger than or smaller than what? 933 01:10:01,355 --> 01:10:04,970 But the point is that once you have selected 934 01:10:04,970 --> 01:10:07,914 some parameter in your system-- L or whatever you have, 935 01:10:07,914 --> 01:10:12,210 some value-- you can, by playing around with this, 936 01:10:12,210 --> 01:10:15,950 choose a value of V for any epsilon such 937 01:10:15,950 --> 01:10:18,730 that you reach that limit. 938 01:10:18,730 --> 01:10:26,090 So by doing this, you in a sense have defined a lens scale. 939 01:10:26,090 --> 01:10:28,760 The lens scale would depend on epsilon, 940 01:10:28,760 --> 01:10:30,637 and you would have different behaviors, 941 01:10:30,637 --> 01:10:33,160 whether you have shorter than that lens scale 942 01:10:33,160 --> 01:10:35,556 or larger than that lens scale. 943 01:10:35,556 --> 01:10:39,410 So this has to be done precisely because of this freedom 944 01:10:39,410 --> 01:10:41,318 of making b larger, and so on. 945 01:10:47,996 --> 01:10:50,080 Now, if you are dealing with a finite system 946 01:10:50,080 --> 01:10:53,032 and you can't make your b much larger than something 947 01:10:53,032 --> 01:10:54,740 or whatever, then you're perfectly right. 948 01:10:59,110 --> 01:10:59,640 Yes? 949 01:10:59,640 --> 01:11:02,210 AUDIENCE: Physically, z or zeta should 950 01:11:02,210 --> 01:11:08,445 be whatever type quantity is needed to actually make 951 01:11:08,445 --> 01:11:10,721 it look exactly the same-- where it keeps coming out. 952 01:11:10,721 --> 01:11:11,720 PROFESSOR: Exactly, yes. 953 01:11:11,720 --> 01:11:12,261 That's right. 954 01:11:12,261 --> 01:11:14,300 AUDIENCE: And then we know, because we already 955 01:11:14,300 --> 01:11:16,200 know that we have two relevant variables, 956 01:11:16,200 --> 01:11:19,010 that z has to look this way for a system that 957 01:11:19,010 --> 01:11:20,160 has two relevant variables. 958 01:11:20,160 --> 01:11:22,651 PROFESSOR: For the Gaussian one, right. 959 01:11:22,651 --> 01:11:23,276 AUDIENCE: Yeah. 960 01:11:23,276 --> 01:11:25,840 But then if we had a different kind of system, 961 01:11:25,840 --> 01:11:28,460 then actually, just going from the physical perspective, 962 01:11:28,460 --> 01:11:30,836 we would need a different z to make things look the same. 963 01:11:30,836 --> 01:11:32,543 And that would give us a different number 964 01:11:32,543 --> 01:11:33,390 of variables here. 965 01:11:33,390 --> 01:11:34,229 PROFESSOR: Yes. 966 01:11:34,229 --> 01:11:34,770 That's right. 967 01:11:37,530 --> 01:11:42,900 Now, in terms of that practically in all cases 968 01:11:42,900 --> 01:11:44,920 we either are dealing with a phase that 969 01:11:44,920 --> 01:11:48,005 has 0 correlation on that, and then 970 01:11:48,005 --> 01:11:51,190 this Gaussian behavior and central limit theorem 971 01:11:51,190 --> 01:11:55,345 is what we are dealing-- and the averaging is by 1 over volume. 972 01:11:55,345 --> 01:11:57,700 Or we have something that is very pretty 973 01:11:57,700 --> 01:12:01,420 close to this big [INAUDIBLE] that we have now discovered, 974 01:12:01,420 --> 01:12:04,460 which is just the gradient squared. 975 01:12:04,460 --> 01:12:07,490 And that has its own scaling according 976 01:12:07,490 --> 01:12:10,750 to these powers that I have found here, 977 01:12:10,750 --> 01:12:14,326 and I will explain that more deeply. 978 01:12:14,326 --> 01:12:16,910 It turns out that at the end of the day, 979 01:12:16,910 --> 01:12:20,910 that when we look at real phase transitions, 980 01:12:20,910 --> 01:12:26,680 all of these exponents will change, but not too much. 981 01:12:26,680 --> 01:12:30,590 So this Gaussian fixed point is actually in some sense 982 01:12:30,590 --> 01:12:34,890 rather close to where we want to end up. 983 01:12:34,890 --> 01:12:38,560 So that's why it's also an important anchoring point, 984 01:12:38,560 --> 01:12:40,440 as I just mentioned. 985 01:12:45,140 --> 01:12:48,510 Again, I said that essentially what we did 986 01:12:48,510 --> 01:12:52,620 was take the rate that we had originally, 987 01:12:52,620 --> 01:12:54,400 and we did a rescaling. 988 01:12:54,400 --> 01:13:02,125 So basically, we replace x by-- let 989 01:13:02,125 --> 01:13:06,682 me get the directions there. 990 01:13:06,682 --> 01:13:12,730 So we replace x by bx prime. 991 01:13:12,730 --> 01:13:16,323 If I had started being in real space, 992 01:13:16,323 --> 01:13:22,470 I would have replaced m with zeta m prime. 993 01:13:22,470 --> 01:13:28,765 m after getting rid of some degrees of freedom. 994 01:13:28,765 --> 01:13:32,160 Again, zeta m prime. 995 01:13:32,160 --> 01:13:36,946 Before I just do that to the rate that I had written before, 996 01:13:36,946 --> 01:13:39,550 there was a beta h. 997 01:13:39,550 --> 01:13:46,870 Which was we could derive d d x t over 2m squared, um 998 01:13:46,870 --> 01:13:50,314 to the fourth and higher order terms, 999 01:13:50,314 --> 01:13:59,170 k over 2 gradient m squared, L over 2 Laplacian of m squared 1000 01:13:59,170 --> 01:14:01,870 and so forth. 1001 01:14:01,870 --> 01:14:06,350 Just do this replacement of things. 1002 01:14:06,350 --> 01:14:08,720 What do I get? 1003 01:14:08,720 --> 01:14:14,230 I get that t prime is b to the d. 1004 01:14:14,230 --> 01:14:18,430 Whenever I see x, I replace it with dx prime. 1005 01:14:18,430 --> 01:14:22,470 Whenever I see m, I replace it with zeta m prime. 1006 01:14:22,470 --> 01:14:23,840 So I get here the zeta squared. 1007 01:14:26,850 --> 01:14:34,331 u prime would be b to the d zeta to the fourth. 1008 01:14:38,259 --> 01:14:44,642 k prime would be b to the b minus 2 zeta squared. 1009 01:14:44,642 --> 01:14:48,240 L prime would be b to the d minus 4 1010 01:14:48,240 --> 01:14:49,629 zeta squared, and so forth. 1011 01:14:52,410 --> 01:14:52,920 Essentially. 1012 01:14:52,920 --> 01:15:01,700 All I did was replace x with b times x prime and m with zeta m 1013 01:15:01,700 --> 01:15:02,990 prime. 1014 01:15:02,990 --> 01:15:05,190 If I do that throughout, you can see 1015 01:15:05,190 --> 01:15:08,210 how the various factors will change. 1016 01:15:08,210 --> 01:15:11,020 So I didn't do all of these integrations, 1017 01:15:11,020 --> 01:15:13,580 et cetera that I did over here. 1018 01:15:13,580 --> 01:15:17,230 I just did the dimensional analysis, if you like. 1019 01:15:17,230 --> 01:15:21,560 And within that dimensional analysis now in real space, 1020 01:15:21,560 --> 01:15:29,550 if I set k prime to be k, you can see that zeta is d to the 2 1021 01:15:29,550 --> 01:15:31,015 minus d over 2. 1022 01:15:37,460 --> 01:15:42,670 And again, you can see that once I have fixed k, 1023 01:15:42,670 --> 01:15:46,150 all of the things that have the same power of m but two higher 1024 01:15:46,150 --> 01:15:49,670 derivatives would get a factor of b to the minus 2, 1025 01:15:49,670 --> 01:15:53,030 just as we had over here. 1026 01:15:53,030 --> 01:15:55,410 Again, with this choice, you can check 1027 01:15:55,410 --> 01:16:00,420 that if I put it back here, I would get b squared. 1028 01:16:00,420 --> 01:16:07,320 But let's imagine that I have a generalization of m to the n. 1029 01:16:07,320 --> 01:16:12,950 If I have a term that multiplies m to some power p-- 1030 01:16:12,950 --> 01:16:18,420 with the coefficient up-- then under this kind of rescaling 1031 01:16:18,420 --> 01:16:28,190 I will get up prime is b to the d zeta to the power of p, up. 1032 01:16:32,890 --> 01:16:36,030 And with this choice of zeta, what do I get? 1033 01:16:36,030 --> 01:16:38,840 I will get b to the d. 1034 01:16:38,840 --> 01:16:48,464 And then I will get plus p 1 minus d over 2 times up. 1035 01:16:51,750 --> 01:16:56,914 Which I can define to be b to some power yp times up. 1036 01:16:56,914 --> 01:16:57,872 Look here to make sure. 1037 01:17:01,250 --> 01:17:07,230 So my yp, the dimension of something 1038 01:17:07,230 --> 01:17:11,790 that multiplies m to some power p 1039 01:17:11,790 --> 01:17:18,982 is simply p plus d 1 minus p or 2. 1040 01:17:24,910 --> 01:17:28,280 And let's check some things. 1041 01:17:28,280 --> 01:17:29,205 I have y1. 1042 01:17:32,190 --> 01:17:35,290 y1 would correspond to a magnetic field, something 1043 01:17:35,290 --> 01:17:38,760 that is proportional to the m itself. 1044 01:17:38,760 --> 01:17:43,860 And if I push p close to 1, I will get 1 plus d over 2. 1045 01:17:43,860 --> 01:17:49,374 And that is, indeed, the yh that we had over here. 1046 01:17:49,374 --> 01:17:52,552 1 plus d over 2. 1047 01:17:52,552 --> 01:17:55,400 So this is yh. 1048 01:17:55,400 --> 01:18:00,095 If I ask what is multiplying m squared, 1049 01:18:00,095 --> 01:18:02,060 I put p equals to 2 here. 1050 01:18:02,060 --> 01:18:07,220 I will get 2, and then here I would get 1 minus 2 over 2. 1051 01:18:07,220 --> 01:18:09,589 So that's the same thing. 1052 01:18:09,589 --> 01:18:11,630 This is the thing that we were calling before yt. 1053 01:18:14,640 --> 01:18:17,710 We didn't include any nq term in the theory, 1054 01:18:17,710 --> 01:18:19,110 didn't make sense to us. 1055 01:18:19,110 --> 01:18:22,050 But we certainly included the u that 1056 01:18:22,050 --> 01:18:24,670 was multiplied in m to the fourth. 1057 01:18:24,670 --> 01:18:27,442 AUDIENCE: So is the p [INAUDIBLE] in the yp? 1058 01:18:32,876 --> 01:18:40,576 PROFESSOR: There is p times 1 plus d 1 minus p over 2. 1059 01:18:40,576 --> 01:18:42,548 p over 2. 1060 01:18:42,548 --> 01:18:44,027 Just rewrote it. 1061 01:18:46,990 --> 01:18:51,900 If I look at 4, here would be 4. 1062 01:18:51,900 --> 01:18:56,210 And then I would put 1 minus 4 over 2, which is 1 minus 2, 1063 01:18:56,210 --> 01:18:57,750 which is minus 1. 1064 01:18:57,750 --> 01:18:59,810 So I would get 4 minus z. 1065 01:19:03,670 --> 01:19:09,302 If I look at y6, I would get 6 minus 2d. 1066 01:19:09,302 --> 01:19:09,843 And so forth. 1067 01:19:15,140 --> 01:19:18,780 So if I just do dimensional analysis 1068 01:19:18,780 --> 01:19:22,080 and I say that I start with a fixed point that corresponds 1069 01:19:22,080 --> 01:19:27,740 to gradient of m squared, and everybody else 0, 1070 01:19:27,740 --> 01:19:30,880 and I ask, if in the vicinity of the fixed point 1071 01:19:30,880 --> 01:19:33,660 where k is fixed and everybody else 1072 01:19:33,660 --> 01:19:39,310 is 0 I put on a little bit up any of these other terms, 1073 01:19:39,310 --> 01:19:41,200 what happens? 1074 01:19:41,200 --> 01:19:45,970 And I find that what happens is that certainly the h 1075 01:19:45,970 --> 01:19:48,285 term, the term that is linear, will be relevant. 1076 01:19:48,285 --> 01:19:53,250 The term that is m squared is relevant. 1077 01:19:53,250 --> 01:19:57,720 Whether or not all the other terms in the series-- 1078 01:19:57,720 --> 01:20:00,320 like m to the fourth, m to the sixth, et cetera-- 1079 01:20:00,320 --> 01:20:04,670 will be relevant depends on dimension. 1080 01:20:04,670 --> 01:20:10,240 So once more we've hit this dimensional fork. 1081 01:20:10,240 --> 01:20:13,720 So the term m to the fourth that we 1082 01:20:13,720 --> 01:20:19,090 said is crucial to getting this theory to have some meaning-- 1083 01:20:19,090 --> 01:20:23,080 and there's no reason for it to be absent-- is, in fact, 1084 01:20:23,080 --> 01:20:24,580 relevant. 1085 01:20:24,580 --> 01:20:27,030 In fact, close to three dimensions 1086 01:20:27,030 --> 01:20:29,700 you would say that that's really the only other term that 1087 01:20:29,700 --> 01:20:31,210 is relevant. 1088 01:20:31,210 --> 01:20:34,890 And you'd say, well, it's almost good enough. 1089 01:20:34,890 --> 01:20:37,950 But almost good enough is not sufficient. 1090 01:20:37,950 --> 01:20:42,390 If we want to describe a physical theory that has only 1091 01:20:42,390 --> 01:20:47,170 two relevant directions, we cannot use this fixed point, 1092 01:20:47,170 --> 01:20:51,110 because this fixed point has three relevant directions 1093 01:20:51,110 --> 01:20:54,370 in three dimensions. 1094 01:20:54,370 --> 01:21:01,070 We have to deal with this somehow. 1095 01:21:01,070 --> 01:21:03,080 So what will we do? 1096 01:21:03,080 --> 01:21:08,110 Next is to explicitly include this m to the fourth. 1097 01:21:08,110 --> 01:21:11,100 In fact, we will include all the other terms, also. 1098 01:21:11,100 --> 01:21:13,810 But we will see that all the other terms, all the higher 1099 01:21:13,810 --> 01:21:16,670 powers, are irrelevant in the same sense 1100 01:21:16,670 --> 01:21:18,530 that all of these higher derivative terms 1101 01:21:18,530 --> 01:21:19,740 are irrelevant. 1102 01:21:19,740 --> 01:21:22,950 But that m to the fourth term is something 1103 01:21:22,950 --> 01:21:25,760 that we really have to take care of. 1104 01:21:25,760 --> 01:21:27,310 And we will do that.