1 00:00:00,070 --> 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 00:00:21,177 --> 00:00:21,760 PROFESSOR: OK. 9 00:00:24,380 --> 00:00:28,530 Let's start with our standard starting 10 00:00:28,530 --> 00:00:30,940 point of the last few lectures. 11 00:00:30,940 --> 00:00:34,170 That is, we are looking at some system. 12 00:00:34,170 --> 00:00:36,850 We tried to describe it by some kind 13 00:00:36,850 --> 00:00:43,520 of a field, statistical field after some averaging. 14 00:00:43,520 --> 00:00:47,070 That is, a function of position. 15 00:00:47,070 --> 00:00:51,440 And we are interested in calculating something 16 00:00:51,440 --> 00:00:54,710 like a partition function by integrating over 17 00:00:54,710 --> 00:01:00,420 all configurations of the statistical field. 18 00:01:00,420 --> 00:01:06,670 And these configurations have some kind of a weight. 19 00:01:06,670 --> 00:01:09,930 This weight we choose to write as exponential 20 00:01:09,930 --> 00:01:18,020 of some kind of a-- something like an effective Hamiltonian 21 00:01:18,020 --> 00:01:19,900 that depends on the configuration 22 00:01:19,900 --> 00:01:20,870 that you're looking at. 23 00:01:25,540 --> 00:01:33,080 And of course, the main thing faced with a problem 24 00:01:33,080 --> 00:01:38,060 that you haven't seen before is to decide on what the field is 25 00:01:38,060 --> 00:01:40,740 that you are looking at to average. 26 00:01:40,740 --> 00:01:44,950 And what kind of symmetries and constraints 27 00:01:44,950 --> 00:01:49,320 you want to construct in this form of the weight. 28 00:01:49,320 --> 00:01:51,690 In particular, we are sort of focusing 29 00:01:51,690 --> 00:01:53,800 on this Landau-Ginzburg model that 30 00:01:53,800 --> 00:01:56,320 describes phase transitions. 31 00:01:56,320 --> 00:01:58,830 And let's say in the absence of magnetic field, 32 00:01:58,830 --> 00:02:04,166 we are interested in a system that is rotationally symmetric. 33 00:02:04,166 --> 00:02:07,610 But the procedure is reasonably standard. 34 00:02:07,610 --> 00:02:10,190 Maybe in some cases you can solve 35 00:02:10,190 --> 00:02:13,600 a particular part of this. 36 00:02:13,600 --> 00:02:15,080 Let's call that part beta H0. 37 00:02:17,690 --> 00:02:21,980 In the context that we are working with, 38 00:02:21,980 --> 00:02:25,250 it's the Gaussian part that we have looked at before. 39 00:02:25,250 --> 00:02:28,830 So that's the integral over space. 40 00:02:28,830 --> 00:02:31,380 We used this idea of locality. 41 00:02:31,380 --> 00:02:34,870 We had an expansion in all things 42 00:02:34,870 --> 00:02:37,610 that are consistent with this. 43 00:02:37,610 --> 00:02:42,270 But for the purposes of the exactly solvable part, 44 00:02:42,270 --> 00:02:44,800 we focus on the Gaussian. 45 00:02:44,800 --> 00:02:48,460 So there is a term that is proportional to m squared, 46 00:02:48,460 --> 00:02:50,833 gradient of m squared and higher-order terms. 47 00:03:00,710 --> 00:03:04,010 So clearly for the time being, I'm 48 00:03:04,010 --> 00:03:06,330 ignoring the magnetic field. 49 00:03:06,330 --> 00:03:11,180 So let's say in this formulation the problem that we 50 00:03:11,180 --> 00:03:14,460 are interested is how our partition function depends 51 00:03:14,460 --> 00:03:17,420 on this coefficient, which where it goes to 0, 52 00:03:17,420 --> 00:03:20,740 the Gaussian weight becomes kind of unsustainable. 53 00:03:23,710 --> 00:03:30,550 Now, of course, we said that the full problem has, in addition 54 00:03:30,550 --> 00:03:38,190 to this beta H0, a part that involves the interaction. 55 00:03:38,190 --> 00:03:41,690 So what I have done is I have written the weight 56 00:03:41,690 --> 00:03:47,090 as beta H0 and a part that is an interaction. 57 00:03:47,090 --> 00:03:49,080 By interaction, I really mean something 58 00:03:49,080 --> 00:03:55,620 that is not solvable within the framework of Gaussian. 59 00:03:55,620 --> 00:04:01,900 In our case, what was non-solvable is essentially 60 00:04:01,900 --> 00:04:05,690 anything-- and there is infinity of terms-- 61 00:04:05,690 --> 00:04:09,950 that don't have second-order powers of m. 62 00:04:09,950 --> 00:04:14,655 So we wrote terms like m to the fourth, m to the sixth, 63 00:04:14,655 --> 00:04:15,976 and so forth. 64 00:04:22,450 --> 00:04:27,040 Now, the key to being able to solve this problem 65 00:04:27,040 --> 00:04:31,130 was to make a transformation to Fourier modes. 66 00:04:31,130 --> 00:04:34,100 So essentially, what we did was to write 67 00:04:34,100 --> 00:04:40,140 our m of x as a sum over Fourier modes. 68 00:04:40,140 --> 00:04:43,610 You could write it, let's say, in the discrete form as e 69 00:04:43,610 --> 00:04:45,460 to the i q dot x. 70 00:04:45,460 --> 00:04:48,550 And whether I write e to the i q dot x or minus i q 71 00:04:48,550 --> 00:04:51,150 dot x is not as important as long as I'm 72 00:04:51,150 --> 00:04:56,260 consistent within one session at least. 73 00:04:56,260 --> 00:05:01,188 And the normalization that I used was 1 over V. 74 00:05:01,188 --> 00:05:04,500 And the reason I used this normalization was 75 00:05:04,500 --> 00:05:08,150 that if I went to the continuum, I could write it 76 00:05:08,150 --> 00:05:14,390 nicely as an integral over q divided 77 00:05:14,390 --> 00:05:15,820 by the density of states. 78 00:05:15,820 --> 00:05:18,110 The V would disappear. 79 00:05:18,110 --> 00:05:22,730 e to the i q x m tilde of q. 80 00:05:30,510 --> 00:05:35,760 Now in particular if I do that transformation, 81 00:05:35,760 --> 00:05:46,730 the Gaussian part simply becomes 1 over V sum over q. 82 00:05:46,730 --> 00:05:50,800 Then the Fourier transform of this kernel t 83 00:05:50,800 --> 00:06:00,091 plus k q squared and so forth divided by 2 m of q discrete 84 00:06:00,091 --> 00:06:00,590 squared. 85 00:06:03,220 --> 00:06:06,330 Which if I go to the continuum limit simply 86 00:06:06,330 --> 00:06:14,340 becomes an integral over q t plus k q 87 00:06:14,340 --> 00:06:22,020 squared and so forth over 2 m of q squared. 88 00:06:31,430 --> 00:06:36,290 Now, once I have the Gaussian weight, from the Gaussian 89 00:06:36,290 --> 00:06:40,780 weight I can calculate various averages. 90 00:06:40,780 --> 00:06:46,030 And the averages are best described 91 00:06:46,030 --> 00:06:49,630 by noting that essentially after this transformation 92 00:06:49,630 --> 00:06:56,300 I can also write my weight as a product over the contributions 93 00:06:56,300 --> 00:07:04,000 of the different modes of something that is of this form, 94 00:07:04,000 --> 00:07:08,110 e to the minus beta H0. 95 00:07:08,110 --> 00:07:10,450 Now written in terms of these q modes, 96 00:07:10,450 --> 00:07:14,630 clearly it's a product of independent contributions. 97 00:07:14,630 --> 00:07:19,050 And then of course, there will be the u to be added later on. 98 00:07:19,050 --> 00:07:22,690 But when I have a product of independent contributions 99 00:07:22,690 --> 00:07:26,720 for each q, I can immediately see that if I look at, 100 00:07:26,720 --> 00:07:32,090 say, m evaluated for some q, m evaluated 101 00:07:32,090 --> 00:07:35,820 for some different q with the Gaussian weight. 102 00:07:35,820 --> 00:07:38,980 And when I calculate things with the Gaussian weight, 103 00:07:38,980 --> 00:07:41,610 I put this index 0. 104 00:07:41,610 --> 00:07:45,320 So that's my 0 to order or exactly solvable theory. 105 00:07:45,320 --> 00:07:48,580 And of course, we are dealing here with a vector. 106 00:07:48,580 --> 00:07:51,350 So these things have indices alpha and beta 107 00:07:51,350 --> 00:07:53,230 associated with them. 108 00:07:53,230 --> 00:07:58,720 And if I look at the discrete version, 109 00:07:58,720 --> 00:08:04,570 I have a product over Gaussians for each one of them. 110 00:08:04,570 --> 00:08:08,670 Clearly, I will get 0 unless I am 111 00:08:08,670 --> 00:08:11,540 looking at the same components. 112 00:08:11,540 --> 00:08:15,310 And I'm looking at the same q. 113 00:08:15,310 --> 00:08:17,080 And in particular, the constraint really 114 00:08:17,080 --> 00:08:20,385 is that q plus q prime should add up to 0. 115 00:08:23,490 --> 00:08:26,340 And if those constraints are satisfied, 116 00:08:26,340 --> 00:08:28,200 then I am looking at the particular term 117 00:08:28,200 --> 00:08:30,260 in this Gaussian. 118 00:08:30,260 --> 00:08:32,299 And the expectation value of m squared 119 00:08:32,299 --> 00:08:34,710 is simply the variance that we can see 120 00:08:34,710 --> 00:08:40,770 is V divided by t plus k q squared, q to the fourth, 121 00:08:40,770 --> 00:08:42,195 and so forth. 122 00:08:46,000 --> 00:08:49,780 And the thing is that most of the time we will actually 123 00:08:49,780 --> 00:08:56,000 be looking at things directly in the limit of the continuum 124 00:08:56,000 --> 00:09:01,230 where we replace sums q's with integrals over q. 125 00:09:01,230 --> 00:09:05,810 And then we have to replace this discrete delta function 126 00:09:05,810 --> 00:09:08,390 with a continuum delta function. 127 00:09:08,390 --> 00:09:10,750 And the procedure to do that is that this 128 00:09:10,750 --> 00:09:13,060 becomes delta alpha beta. 129 00:09:13,060 --> 00:09:16,776 This combination gets replaced by 2 pi 130 00:09:16,776 --> 00:09:21,160 to the d delta function q plus q prime, 131 00:09:21,160 --> 00:09:24,952 where this is now a direct delta function 132 00:09:24,952 --> 00:09:32,240 t plus k q squared plus l q to the fourth and so forth. 133 00:09:32,240 --> 00:09:38,510 And the justification for doing that is simply 134 00:09:38,510 --> 00:09:42,820 that the Kronecker delta is defined such 135 00:09:42,820 --> 00:09:51,100 that if I sum over, let's say, all q, the delta that 136 00:09:51,100 --> 00:09:58,720 is Kronecker, the answer would be 0. 137 00:09:58,720 --> 00:10:01,770 Now, if I go to the continuum limit, 138 00:10:01,770 --> 00:10:06,530 the sum over q I have to replace with integral over q 139 00:10:06,530 --> 00:10:12,440 with a density of states, which is V divided by 2 pi to the d. 140 00:10:15,660 --> 00:10:20,500 So if I want to replace this with a continuum delta 141 00:10:20,500 --> 00:10:30,640 function of q, I have to get rid of this 2 pi to the d over V. 142 00:10:30,640 --> 00:10:32,200 And that's what I have done. 143 00:10:32,200 --> 00:10:40,280 So basically, you replace-- hopefully I 144 00:10:40,280 --> 00:10:42,305 didn't make a mistake. 145 00:10:44,901 --> 00:10:45,400 Yes. 146 00:10:45,400 --> 00:10:49,020 So the discrete delta I replace with 1 over V. The V 147 00:10:49,020 --> 00:10:52,975 disappears and the 2 pi to the d appears. 148 00:10:52,975 --> 00:10:53,475 OK? 149 00:10:58,410 --> 00:11:06,010 Now, the thing that makes some difficulty 150 00:11:06,010 --> 00:11:12,040 is that whereas the rest of these things that we have not 151 00:11:12,040 --> 00:11:14,710 included as part of the Gaussian, 152 00:11:14,710 --> 00:11:20,900 because of the locality I could write reasonably simply 153 00:11:20,900 --> 00:11:23,660 in the space x. 154 00:11:23,660 --> 00:11:29,610 When I go to the space q, it becomes complicated. 155 00:11:29,610 --> 00:11:33,130 Because each m of x here I have to replace 156 00:11:33,130 --> 00:11:36,170 with a sum or an integral. 157 00:11:36,170 --> 00:11:39,460 And I have four of those m's. 158 00:11:39,460 --> 00:11:45,050 So here, let's say for the first term that involves u, 159 00:11:45,050 --> 00:11:49,043 I have in principle to go over an integral associated 160 00:11:49,043 --> 00:11:55,445 with conversion of the first m, conversion of the second m, 161 00:11:55,445 --> 00:11:57,640 third m. 162 00:11:57,640 --> 00:12:00,790 Each one of them will carry a factor of 2 pi to the d. 163 00:12:00,790 --> 00:12:04,300 So there will be three of them. 164 00:12:04,300 --> 00:12:07,540 And the reason I didn't write the fourth one 165 00:12:07,540 --> 00:12:11,240 is because after I do all of that transformation, 166 00:12:11,240 --> 00:12:17,178 I will have an integral over x of e to the i q1 167 00:12:17,178 --> 00:12:22,670 dot x plus q2 dot x plus q3 dot x plus q4 dot x. 168 00:12:22,670 --> 00:12:26,240 So I have an integral of e to the i q 169 00:12:26,240 --> 00:12:30,860 dot x where q is the sum of the four of them over x. 170 00:12:30,860 --> 00:12:32,570 And that gives me a delta function 171 00:12:32,570 --> 00:12:36,630 that ensures the sum of the four q's have to be 0. 172 00:12:36,630 --> 00:12:43,030 So basically, one of the m's will carry now index q1. 173 00:12:43,030 --> 00:12:46,610 The other will carry index q2. 174 00:12:46,610 --> 00:12:49,510 The third will carry index q3. 175 00:12:49,510 --> 00:12:58,190 The fourth will carry index that is minus q1 minus q2 minus q3. 176 00:12:58,190 --> 00:12:58,856 Yes? 177 00:12:58,856 --> 00:13:02,610 AUDIENCE: Does it matter which indices for squaring it, 178 00:13:02,610 --> 00:13:03,846 or whatever? 179 00:13:03,846 --> 00:13:04,670 Sorry. 180 00:13:04,670 --> 00:13:06,927 Do you need a subscript for alpha and beta? 181 00:13:06,927 --> 00:13:07,510 PROFESSOR: OK. 182 00:13:07,510 --> 00:13:08,760 That's what I was [INAUDIBLE]. 183 00:13:08,760 --> 00:13:13,990 So this m to the fourth is really m squared m squared, 184 00:13:13,990 --> 00:13:17,740 where m squared is a vector that is squared. 185 00:13:17,740 --> 00:13:22,290 So I have to put the indices-- let's say alpha alpha-- 186 00:13:22,290 --> 00:13:24,560 that are summed over all possibility 187 00:13:24,560 --> 00:13:26,680 to get the dot product here. 188 00:13:26,680 --> 00:13:28,600 And I have to put the indices beta 189 00:13:28,600 --> 00:13:33,380 beta to have the dot products here. 190 00:13:33,380 --> 00:13:34,770 OK? 191 00:13:34,770 --> 00:13:37,660 Now when I go to the next term, clearly I 192 00:13:37,660 --> 00:13:44,430 will have a whole bunch more integrals and things like that. 193 00:13:44,430 --> 00:13:51,970 So the e u terms do not look as nice and clean. 194 00:13:51,970 --> 00:13:55,650 They were local in real space. 195 00:13:55,650 --> 00:13:59,240 But when I go to this Fourier space, they become non-local. 196 00:13:59,240 --> 00:14:02,190 q's that are all over this [INAUDIBLE] 197 00:14:02,190 --> 00:14:07,400 are going to be coupled to each other through this expression. 198 00:14:07,400 --> 00:14:10,500 And that's also why it is called interaction. 199 00:14:10,500 --> 00:14:15,230 Because in some sense, previously each q was a mode 200 00:14:15,230 --> 00:14:21,350 by itself and these terms give interactions between modes that 201 00:14:21,350 --> 00:14:23,920 have different q's. 202 00:14:23,920 --> 00:14:24,420 Yes? 203 00:14:24,420 --> 00:14:26,003 AUDIENCE: Is there a way to understand 204 00:14:26,003 --> 00:14:33,628 that physically of why you get coupling in Fourier space? 205 00:14:33,628 --> 00:14:36,870 [INAUDIBLE] higher than [INAUDIBLE]. 206 00:14:39,680 --> 00:14:40,460 PROFESSOR: OK. 207 00:14:40,460 --> 00:14:46,650 So essentially, we have a system that 208 00:14:46,650 --> 00:14:49,230 has translational symmetry. 209 00:14:49,230 --> 00:14:51,500 So when you have translational symmetry, 210 00:14:51,500 --> 00:14:54,810 this Fourier vector q is a good conserved quantity. 211 00:14:54,810 --> 00:14:58,080 It's like a momentum. 212 00:14:58,080 --> 00:15:02,270 So one thing that we have is, in some sense, a particle 213 00:15:02,270 --> 00:15:05,140 or an excitation that is going by itself 214 00:15:05,140 --> 00:15:08,890 with some particular momentum. 215 00:15:08,890 --> 00:15:13,920 But what these terms represent is the possibility that you 216 00:15:13,920 --> 00:15:16,480 have, let's say, two of these momenta 217 00:15:16,480 --> 00:15:19,040 coming and interacting with each other 218 00:15:19,040 --> 00:15:22,370 and getting two that are going out. 219 00:15:22,370 --> 00:15:24,050 Why is it [INAUDIBLE]? 220 00:15:24,050 --> 00:15:25,940 It's partly because of the symmetries 221 00:15:25,940 --> 00:15:27,700 that we built into the problem. 222 00:15:27,700 --> 00:15:30,230 If I had written something that was m cubed, 223 00:15:30,230 --> 00:15:33,230 I had the possibility of 2 going to, 1 or 1 224 00:15:33,230 --> 00:15:34,275 going to 2, et cetera. 225 00:15:43,040 --> 00:15:45,580 All right. 226 00:15:45,580 --> 00:15:49,060 I forgot to say one more thing, which 227 00:15:49,060 --> 00:15:54,890 is that for the Gaussian theory, I can calculate essentially 228 00:15:54,890 --> 00:15:58,255 all expectation values most [INAUDIBLE] 229 00:15:58,255 --> 00:16:01,640 in this context of the Fourier representation. 230 00:16:01,640 --> 00:16:03,650 So this was an example of something 231 00:16:03,650 --> 00:16:06,330 that had two factors of m. 232 00:16:06,330 --> 00:16:09,450 But very soon, we will see that we would need terms 233 00:16:09,450 --> 00:16:13,380 that, let's say, involve m factors of n that 234 00:16:13,380 --> 00:16:14,810 are multiplied each other. 235 00:16:14,810 --> 00:16:24,260 So I have m alpha i of qi-- something like that. 236 00:16:24,260 --> 00:16:29,190 And again, 0 for the Gaussian expectation value. 237 00:16:29,190 --> 00:16:31,670 And if I have written things the way that I have-- that is, 238 00:16:31,670 --> 00:16:33,380 I have no magnetic field. 239 00:16:33,380 --> 00:16:37,260 So I have m2 minus m symmetry, clearly 240 00:16:37,260 --> 00:16:41,360 the answer is going to be 0 if l is odd. 241 00:16:44,060 --> 00:16:51,200 If l is even, we have this nice property of Gaussians 242 00:16:51,200 --> 00:16:54,190 that we described in 8.333, which 243 00:16:54,190 --> 00:17:05,890 is that this will be the sum over all pairs of averages. 244 00:17:05,890 --> 00:17:10,550 So something like m1, m2, m3, m4. 245 00:17:10,550 --> 00:17:15,060 You can have m1 m2 multiplying by m3 m4 average. 246 00:17:15,060 --> 00:17:18,500 m1 m3 average multiply m2 m4 average. 247 00:17:18,500 --> 00:17:22,990 m1 m4 average multiplied by m2 m3 average. 248 00:17:22,990 --> 00:17:28,770 And this is what's called a Wick's theorem, which 249 00:17:28,770 --> 00:17:32,730 is an important property of the Gaussian that we will use. 250 00:17:46,500 --> 00:17:53,250 So we know how to calculate averages of things of interest, 251 00:17:53,250 --> 00:17:57,070 which are essentially product of these factors of m, 252 00:17:57,070 --> 00:17:59,310 in the Gaussian theory. 253 00:17:59,310 --> 00:18:03,960 Now let's calculate these averages 254 00:18:03,960 --> 00:18:05,090 in perturbation theory. 255 00:18:15,260 --> 00:18:23,180 So quite generally, if I want to calculate the average of some O 256 00:18:23,180 --> 00:18:29,840 in a theory that involves averaging over some function 257 00:18:29,840 --> 00:18:33,520 l-- so this could be some trace, some completely 258 00:18:33,520 --> 00:18:38,050 unspecified things-- of a weight that is like, say, 259 00:18:38,050 --> 00:18:39,690 e to the minus beta H0. 260 00:18:39,690 --> 00:18:43,010 A part that I can do and a part that I 261 00:18:43,010 --> 00:18:48,530 want to treat as a small change to what I can do. 262 00:18:48,530 --> 00:18:51,080 The procedure of calculating the average 263 00:18:51,080 --> 00:18:55,210 is to multiply the probability by the quantity 264 00:18:55,210 --> 00:18:56,620 that I want to average. 265 00:18:56,620 --> 00:19:00,446 And of course, the whole thing has to be properly normalized. 266 00:19:00,446 --> 00:19:05,882 And this is the normalization, which is the partition function 267 00:19:05,882 --> 00:19:06,840 that we had previously. 268 00:19:10,780 --> 00:19:14,000 Now, the whole idea of perturbation 269 00:19:14,000 --> 00:19:17,870 is to assume that this quantity u is small. 270 00:19:17,870 --> 00:19:21,456 So we start to expand e to the minus u. 271 00:19:21,456 --> 00:19:23,825 I can certainly do that very easily, 272 00:19:23,825 --> 00:19:26,560 let's say, in the denominator. 273 00:19:26,560 --> 00:19:30,840 I have e to the minus beta H0, and then I 274 00:19:30,840 --> 00:19:38,560 have 1 minus u plus u squared over 2 minus u cubed over 6. 275 00:19:38,560 --> 00:19:43,700 Basically, the usual expansion of the exponential. 276 00:19:43,700 --> 00:19:49,270 In the numerator, I have the same thing, 277 00:19:49,270 --> 00:19:52,610 except that there is an I that is multiplying 278 00:19:52,610 --> 00:19:58,010 this expansion for the operator, or the object that I 279 00:19:58,010 --> 00:19:59,542 want to calculate the average. 280 00:20:03,190 --> 00:20:08,680 Now, the first term that I have in the denominator here, 281 00:20:08,680 --> 00:20:13,550 1 multiplied by all integrals of e to the minus H0 282 00:20:13,550 --> 00:20:18,680 is clearly what I would call the partition function, 283 00:20:18,680 --> 00:20:20,600 or the normalization that I would 284 00:20:20,600 --> 00:20:23,920 have for the Gaussian weight. 285 00:20:23,920 --> 00:20:25,840 So that's the first term. 286 00:20:25,840 --> 00:20:30,490 If I factor that out, then the next term 287 00:20:30,490 --> 00:20:33,950 is u integrated against the Gaussian weight 288 00:20:33,950 --> 00:20:36,780 and then properly normalized. 289 00:20:36,780 --> 00:20:41,610 So the next term will be the average of u with the Gaussian 290 00:20:41,610 --> 00:20:44,150 or whatever other 0 to order weight 291 00:20:44,150 --> 00:20:49,560 is that I can calculate things and I have indicated that by 0. 292 00:20:49,560 --> 00:20:53,180 And then I would have 1/2 average 293 00:20:53,180 --> 00:20:56,400 of u squared 0 and so forth. 294 00:20:56,400 --> 00:21:01,720 And the series in the numerator once I factor out the Z0 295 00:21:01,720 --> 00:21:05,265 is pretty much the same except that every term will have an o. 296 00:21:20,930 --> 00:21:26,420 The Z 0's naturally I can cancel out. 297 00:21:26,420 --> 00:21:30,640 So what I have is from the numerator o 298 00:21:30,640 --> 00:21:38,565 minus ou plus 1/2 ou squared. 299 00:21:41,550 --> 00:21:45,160 What I will do with the denominator 300 00:21:45,160 --> 00:21:51,090 is to bring it in the numerator regarding 301 00:21:51,090 --> 00:21:55,110 all of these as a small quantity. 302 00:21:55,110 --> 00:21:59,270 So if I were to essentially write this expression raised 303 00:21:59,270 --> 00:22:03,190 to the minus 1 power, I can make a series expansion 304 00:22:03,190 --> 00:22:04,820 of all of these terms. 305 00:22:04,820 --> 00:22:09,820 So the first thing, if I just had one over 1 minus u 306 00:22:09,820 --> 00:22:12,240 in the denominator, it would come 307 00:22:12,240 --> 00:22:21,570 from 1 plus u plus u squared and u cubed, et cetera. 308 00:22:21,570 --> 00:22:24,590 But I've only kept thing to order of u squared. 309 00:22:24,590 --> 00:22:26,900 So when I then correct it because 310 00:22:26,900 --> 00:22:28,820 of this thing in the denominator, 311 00:22:28,820 --> 00:22:35,570 the 1/2 becomes minus 1/2 u squared 0. 312 00:22:35,570 --> 00:22:40,120 And then, there will be order of u cubed terms. 313 00:22:40,120 --> 00:22:44,500 So the answer is the product of two brackets. 314 00:22:44,500 --> 00:22:50,022 And I can reorganize that product, again, in powers of u. 315 00:22:50,022 --> 00:22:54,460 The lowest-order term is the 0 to order, 316 00:22:54,460 --> 00:22:57,340 or unperturbed average. 317 00:22:57,340 --> 00:23:03,095 And the first correction comes from ou average. 318 00:23:06,250 --> 00:23:08,800 And then I have the average of u average of u. 319 00:23:12,710 --> 00:23:19,020 You can see that something like a variance or connected 320 00:23:19,020 --> 00:23:21,760 correlation or cumulant appears because I 321 00:23:21,760 --> 00:23:25,160 have to subtract out the averages. 322 00:23:25,160 --> 00:23:28,185 And then the next order term, what will I have? 323 00:23:28,185 --> 00:23:31,210 I will write it as 1/2. 324 00:23:31,210 --> 00:23:35,490 I start with ou squared 0. 325 00:23:35,490 --> 00:23:38,240 Then I can multiply this with this, 326 00:23:38,240 --> 00:23:44,700 so I will get minus 2 o u0 u0. 327 00:23:44,700 --> 00:23:49,570 And then I can multiply o0 with those two terms. 328 00:23:49,570 --> 00:24:03,058 So I will have minus o0 u squared 0 plus 2 o0-- 329 00:24:03,058 --> 00:24:05,530 this is over here. 330 00:24:05,530 --> 00:24:08,851 u0 squared and higher-order terms. 331 00:24:16,860 --> 00:24:22,370 So basically, we can see that the coefficients, 332 00:24:22,370 --> 00:24:25,530 as I have written, are going to be essentially the coefficients 333 00:24:25,530 --> 00:24:32,530 that I would have if I were to expand the exponential. 334 00:24:32,530 --> 00:24:36,540 So things like minus 1 to the n over n factorial. 335 00:24:39,380 --> 00:24:42,680 And the leading term in all cases 336 00:24:42,680 --> 00:24:48,578 is o u raised to the n-th power 0, 337 00:24:48,578 --> 00:24:53,390 out of which are subtracted various things. 338 00:24:53,390 --> 00:24:56,640 And the effect of those subtractions, 339 00:24:56,640 --> 00:24:59,740 let's say we define a quantity which 340 00:24:59,740 --> 00:25:06,060 is likely cumulants that we were using in 8.333, which describe 341 00:25:06,060 --> 00:25:09,778 the subtractions that you would have to define such an average. 342 00:25:16,620 --> 00:25:19,410 So that's the general structure. 343 00:25:19,410 --> 00:25:23,370 What this really means-- and I sometimes 344 00:25:23,370 --> 00:25:27,600 call it cumulant or connected-- will become apparent 345 00:25:27,600 --> 00:25:28,300 very shortly. 346 00:25:38,880 --> 00:25:40,740 This is the general result. 347 00:25:40,740 --> 00:25:43,980 We have a particular case above, which 348 00:25:43,980 --> 00:25:48,550 is this Landau-Ginzburg theory perturbed around the Gaussian. 349 00:25:48,550 --> 00:25:53,840 So let's calculate the simplest one of our averages, this m 350 00:25:53,840 --> 00:26:03,724 alpha of q m beta of q prime, not at the Gaussian level, 351 00:26:03,724 --> 00:26:04,640 but as a perturbation. 352 00:26:07,630 --> 00:26:13,390 And actually, for practical reasons, 353 00:26:13,390 --> 00:26:16,290 I will just calculate the effect of the first term, which 354 00:26:16,290 --> 00:26:18,000 is u m to the fourth. 355 00:26:18,000 --> 00:26:21,280 So I will expand in powers of u. 356 00:26:21,280 --> 00:26:23,070 But once you see that, you would know 357 00:26:23,070 --> 00:26:27,780 how to do it for m to the sixth and all the higher powers. 358 00:26:27,780 --> 00:26:30,870 So according to what we have here, 359 00:26:30,870 --> 00:26:35,860 the first term is m alpha of q m beta 360 00:26:35,860 --> 00:26:42,550 of q prime evaluated with the Gaussian theory. 361 00:26:42,550 --> 00:26:48,920 The next term, this one, involves the average 362 00:26:48,920 --> 00:26:54,540 of this entity and the u. 363 00:26:54,540 --> 00:26:58,780 So our u I have written up there. 364 00:26:58,780 --> 00:27:04,340 So I have minus from the first term. 365 00:27:04,340 --> 00:27:07,720 The terms that are proportional to u, 366 00:27:07,720 --> 00:27:13,840 I will group together coming from here. 367 00:27:13,840 --> 00:27:21,180 u itself involved this integration over q1 q2 q3. 368 00:27:28,500 --> 00:27:36,670 And u involves this m i of q1 m i 369 00:27:36,670 --> 00:27:46,210 of q2 mj of q3 mj of minus q1 minus q2 minus q3. 370 00:27:50,990 --> 00:27:53,420 And I have to multiply it. 371 00:27:53,420 --> 00:27:54,240 So this is u. 372 00:27:54,240 --> 00:27:56,720 I have to multiply it by o. 373 00:27:56,720 --> 00:28:03,030 So I have m alpha of q m beta of q prime. 374 00:28:03,030 --> 00:28:04,240 So this is my o. 375 00:28:04,240 --> 00:28:05,160 This is my u. 376 00:28:07,670 --> 00:28:11,140 And I have to take the average of this term. 377 00:28:11,140 --> 00:28:14,760 But really, the average operates on the m's, 378 00:28:14,760 --> 00:28:16,030 so it will go over here. 379 00:28:19,850 --> 00:28:23,610 So that's the average of ou. 380 00:28:23,610 --> 00:28:28,620 I have to subtract from that the average of o average of u. 381 00:28:28,620 --> 00:28:32,060 So let me, again, write the next term. 382 00:28:32,060 --> 00:28:36,420 u will be the same bunch of integrations. 383 00:28:36,420 --> 00:28:47,720 I have to do average of o and then average of u. 384 00:29:06,900 --> 00:29:14,080 This completes my first correction coming from u, 385 00:29:14,080 --> 00:29:17,460 and then there will be corrections to first order 386 00:29:17,460 --> 00:29:20,940 coming from V. There will be second-order corrections coming 387 00:29:20,940 --> 00:29:22,809 from u squared, all kinds of things 388 00:29:22,809 --> 00:29:23,850 that will come into play. 389 00:29:26,460 --> 00:29:29,870 But the important thing, again, to realize is the structure. 390 00:29:29,870 --> 00:29:34,400 u is this thing that involves four factors of m. 391 00:29:34,400 --> 00:29:36,590 The averages are over the m, so I 392 00:29:36,590 --> 00:29:39,560 can take them within the integral. 393 00:29:39,560 --> 00:29:44,390 And so I have one case which is an expectation value of six 394 00:29:44,390 --> 00:29:45,350 m's. 395 00:29:45,350 --> 00:29:48,020 Another case, a product of two and a product of four. 396 00:29:50,680 --> 00:29:53,580 So that's why I said I would need 397 00:29:53,580 --> 00:29:58,190 to know how to calculate in the Gaussian theory 398 00:29:58,190 --> 00:30:03,600 product of various factors of m because my interaction 399 00:30:03,600 --> 00:30:07,360 term involves various powers of m that will be added 400 00:30:07,360 --> 00:30:10,010 to whatever expectation value I'm calculating perturbation 401 00:30:10,010 --> 00:30:10,510 theory. 402 00:30:13,840 --> 00:30:17,740 So how do I calculate an expectation 403 00:30:17,740 --> 00:30:22,540 that involves six factors-- certainly, it's even-- of m? 404 00:30:22,540 --> 00:30:28,410 I have to group-- make all possible pairings. 405 00:30:28,410 --> 00:30:31,620 So this, for example, can be paired to this. 406 00:30:31,620 --> 00:30:32,962 This can be paired to this. 407 00:30:32,962 --> 00:30:34,140 This can be paired to this. 408 00:30:34,140 --> 00:30:38,090 That's a perfectly well-defined average. 409 00:30:38,090 --> 00:30:39,850 But you can see that if I do this, 410 00:30:39,850 --> 00:30:43,370 if I pair this one, this one, this one, 411 00:30:43,370 --> 00:30:46,430 I will get something that will cancel out against this one. 412 00:30:51,420 --> 00:30:56,060 So basically, you can see that any way that I 413 00:30:56,060 --> 00:31:03,150 do averaging that involves only things that are coming from o 414 00:31:03,150 --> 00:31:07,310 and separately the things that come from u 415 00:31:07,310 --> 00:31:10,840 will cancel out with the corresponding-- oops. 416 00:31:14,636 --> 00:31:19,710 With the corresponding averages that I do over here. 417 00:31:22,470 --> 00:31:25,630 That c stands for connected. 418 00:31:25,630 --> 00:31:31,650 So the only things that survive are pairings or contractions 419 00:31:31,650 --> 00:31:33,950 that pick something that is from o 420 00:31:33,950 --> 00:31:39,250 and connect it to something that is from u. 421 00:31:39,250 --> 00:31:44,170 And the purpose of all of these other terms at all higher 422 00:31:44,170 --> 00:31:48,480 orders is precisely to remove pieces 423 00:31:48,480 --> 00:31:51,700 where you don't have full connections among all 424 00:31:51,700 --> 00:31:53,764 of the o's and the u's that you are dealing with. 425 00:31:57,640 --> 00:32:02,810 So let's see what this is. 426 00:32:02,810 --> 00:32:12,975 So I will show you that using connections that involve both o 427 00:32:12,975 --> 00:32:27,890 and u, I will have two types of contractions joining o and u. 428 00:32:34,700 --> 00:32:36,550 The first type is something like this. 429 00:32:36,550 --> 00:32:40,870 I will, again, draw all-- or write down all of the fours. 430 00:32:40,870 --> 00:32:53,380 So I have m alpha of q m beta of q prime m i of q1 m i of q2 mj 431 00:32:53,380 --> 00:33:01,080 of q3 mj of minus q1 minus q2 minus q3. 432 00:33:01,080 --> 00:33:03,480 I have to take that average. 433 00:33:03,480 --> 00:33:05,120 And I do that the average according 434 00:33:05,120 --> 00:33:09,680 to Wick's theorem as a product of contractions. 435 00:33:09,680 --> 00:33:11,440 So let's pick this m alpha. 436 00:33:11,440 --> 00:33:15,010 It has to ultimately be paired with somebody. 437 00:33:15,010 --> 00:33:18,630 I can't pair it with m beta because that's 438 00:33:18,630 --> 00:33:22,500 a self-contraction and will get subtracted out. 439 00:33:22,500 --> 00:33:25,900 So I can pick any one of these fours. 440 00:33:25,900 --> 00:33:29,930 As far as I'm concerned, all four are the same, 441 00:33:29,930 --> 00:33:36,310 so I have a choice of four as to which one of these four 442 00:33:36,310 --> 00:33:38,300 operators from u I connect to. 443 00:33:38,300 --> 00:33:40,810 So that 4 is one of the numerical factors 444 00:33:40,810 --> 00:33:44,440 that ultimately we have to take care of. 445 00:33:44,440 --> 00:33:48,340 Then, the two types comes because the next m 446 00:33:48,340 --> 00:33:53,980 that I pick from o I have two possibilities. 447 00:33:53,980 --> 00:33:57,840 I can connect it either with the partner of the first one that 448 00:33:57,840 --> 00:34:01,950 also carries index i or I can connect 449 00:34:01,950 --> 00:34:06,310 to one of the things that carries the opposite index j. 450 00:34:06,310 --> 00:34:10,469 So let's call type 1 where I make the choice that I 451 00:34:10,469 --> 00:34:14,370 connect to the partner. 452 00:34:14,370 --> 00:34:17,190 And once I do that, then I am forced 453 00:34:17,190 --> 00:34:19,157 to connect these two together. 454 00:34:22,190 --> 00:34:28,800 Now, each one of these pairings connects one of these averages. 455 00:34:28,800 --> 00:34:31,790 So I can write down what that is. 456 00:34:31,790 --> 00:34:37,500 So the first one connected an alpha to an i as far as indices 457 00:34:37,500 --> 00:34:38,685 were concerned. 458 00:34:38,685 --> 00:34:46,510 It connected q to q1, so I have 2 pi to the d a delta function 459 00:34:46,510 --> 00:34:47,870 q plus q1. 460 00:34:50,630 --> 00:34:53,550 And the variance associated with that, 461 00:34:53,550 --> 00:34:57,809 which is t plus k q squared, et cetera. 462 00:35:04,850 --> 00:35:09,520 The second pairing connects a beta to an i. 463 00:35:09,520 --> 00:35:12,890 So that's a delta beta i. 464 00:35:12,890 --> 00:35:20,224 And it connects q prime to q2. 465 00:35:20,224 --> 00:35:23,220 And so the variance associated with that 466 00:35:23,220 --> 00:35:29,740 is t plus k q prime squared and so forth. 467 00:35:29,740 --> 00:35:34,690 And finally, the third pairing connects j to itself j. 468 00:35:34,690 --> 00:35:38,780 So I will get a delta jj. 469 00:35:38,780 --> 00:35:41,076 And then I have 2 pi to the d. 470 00:35:45,740 --> 00:35:50,240 q3 to minus q1 minus q2 minus q3. 471 00:35:50,240 --> 00:35:54,100 So I will get minus q1 minus q2, and then 472 00:35:54,100 --> 00:35:58,628 I have t plus, say, k q3 squared and so forth. 473 00:36:07,040 --> 00:36:15,700 Now, what I am supposed to do is at the next stage, 474 00:36:15,700 --> 00:36:33,050 I have to sum over indices i and j and integrate over q1 q2 q3. 475 00:36:37,500 --> 00:36:40,760 So when I do that, what do I get? 476 00:36:40,760 --> 00:36:43,410 There is an overall factor of minus u. 477 00:36:46,500 --> 00:36:48,900 Let's do the indices. 478 00:36:48,900 --> 00:36:54,830 When I sum over i, delta alpha i delta beta i 479 00:36:54,830 --> 00:36:57,370 becomes-- actually, let me put the factor of 4 480 00:36:57,370 --> 00:36:58,510 before I forget it. 481 00:36:58,510 --> 00:37:01,970 There is a factor of 4 numerically. 482 00:37:01,970 --> 00:37:06,330 Delta alpha i delta beta i will give me a delta alpha beta. 483 00:37:11,160 --> 00:37:17,200 When I integrate over q1, q1 is set to minus q. 484 00:37:17,200 --> 00:37:22,050 So this after the integration becomes q. 485 00:37:22,050 --> 00:37:25,120 When I integrate over q2, the delta 486 00:37:25,120 --> 00:37:31,710 function q2 forces minus q2 to be q prime. 487 00:37:31,710 --> 00:37:38,780 And through the process, two of these factors of 2 pi to the d 488 00:37:38,780 --> 00:37:40,390 disappear. 489 00:37:40,390 --> 00:37:43,480 So what I'm left with is 2 pi to the d. 490 00:37:43,480 --> 00:37:49,240 This delta function now involves q plus q prime. 491 00:37:49,240 --> 00:37:53,310 And then in the denominator, I have this factor of t 492 00:37:53,310 --> 00:37:56,030 plus k q squared. 493 00:37:56,030 --> 00:38:00,890 I have t plus k q prime squared. 494 00:38:00,890 --> 00:38:03,540 Although, q prime squared and q squared are the same. 495 00:38:03,540 --> 00:38:07,920 I could have collapsed these things together. 496 00:38:07,920 --> 00:38:10,855 I have one integration left over q3. 497 00:38:15,700 --> 00:38:18,660 And these two factors went outside 498 00:38:18,660 --> 00:38:21,120 the integral [INAUDIBLE] independent q3. 499 00:38:21,120 --> 00:38:23,115 The only thing that depends on q3 500 00:38:23,115 --> 00:38:26,852 is t plus k q3 squared and so forth. 501 00:38:31,770 --> 00:38:33,894 So that was easy. 502 00:38:33,894 --> 00:38:35,060 AUDIENCE: I have a question. 503 00:38:35,060 --> 00:38:36,710 PROFESSOR: Yes. 504 00:38:36,710 --> 00:38:40,780 AUDIENCE: If you're summing over j, won't you get an n? 505 00:38:40,780 --> 00:38:42,620 PROFESSOR: Thank you very much. 506 00:38:42,620 --> 00:38:44,790 I forgot the delta jj. 507 00:38:44,790 --> 00:38:48,960 Summing over j, I will get a factor of n. 508 00:38:48,960 --> 00:38:52,230 So what I had written here as 4 should be 4n. 509 00:38:55,770 --> 00:38:56,688 Yes. 510 00:38:56,688 --> 00:38:59,540 AUDIENCE: This may be a question too far back. 511 00:38:59,540 --> 00:39:05,090 But when you write a correlation between two different m's, 512 00:39:05,090 --> 00:39:07,410 why do you write delta function of q 513 00:39:07,410 --> 00:39:13,010 plus q prime instead of q minus q prime? 514 00:39:13,010 --> 00:39:14,470 PROFESSOR: OK. 515 00:39:14,470 --> 00:39:17,950 Again, go back all the way to here 516 00:39:17,950 --> 00:39:21,330 when we were doing the Gaussian integral. 517 00:39:21,330 --> 00:39:27,040 I will have for the first one, q1. 518 00:39:27,040 --> 00:39:31,230 For the second m, I will write q2. 519 00:39:31,230 --> 00:39:33,940 So when I Fourier transform this term, 520 00:39:33,940 --> 00:39:40,030 I will have e to the i q1 plus q2 dot x. 521 00:39:40,030 --> 00:39:43,600 And then when I integrate over x, 522 00:39:43,600 --> 00:39:48,040 I will get a delta function q1 plus q2. 523 00:39:48,040 --> 00:39:52,560 So that's why I write all of these as absolute value 524 00:39:52,560 --> 00:39:55,960 squared because I could have written this 525 00:39:55,960 --> 00:40:02,290 as m of q m of minus q, but I realized that m of minus q 526 00:40:02,290 --> 00:40:05,040 is the complex conjugate of m of q. 527 00:40:05,040 --> 00:40:07,010 So all of these are absolute values squared. 528 00:40:21,370 --> 00:40:27,480 Now, the second class of contraction is-- again, 529 00:40:27,480 --> 00:40:37,250 write the same thing, m alpha of q m beta of q prime m i of q1 m 530 00:40:37,250 --> 00:40:47,560 i of q2, mj of q3 mj of minus q1 minus q2 minus q3. 531 00:40:50,260 --> 00:40:52,750 The first step is the same. 532 00:40:52,750 --> 00:40:57,370 I pick m alpha of q and I have no choice but to pick 533 00:40:57,370 --> 00:41:01,380 one of the four possibilities that I have 534 00:41:01,380 --> 00:41:06,400 for the operators that appear in u. 535 00:41:06,400 --> 00:41:09,210 But for the second one, previously I 536 00:41:09,210 --> 00:41:12,100 chose to connect it to something that 537 00:41:12,100 --> 00:41:14,240 was carrying the same index. 538 00:41:14,240 --> 00:41:16,410 Now I choose to carry it to something 539 00:41:16,410 --> 00:41:20,430 that carries the other index, j in this case. 540 00:41:20,430 --> 00:41:23,290 And there are two things that carry index j, 541 00:41:23,290 --> 00:41:26,090 so I have two choices there. 542 00:41:26,090 --> 00:41:30,150 And then I have the remaining two have to be connected. 543 00:41:30,150 --> 00:41:31,771 Yes? 544 00:41:31,771 --> 00:41:34,910 AUDIENCE: Just going back a little bit. 545 00:41:34,910 --> 00:41:38,060 Are you assuming that your integral over q3 546 00:41:38,060 --> 00:41:40,110 converges because you're only integrating over 547 00:41:40,110 --> 00:41:42,780 the [INAUDIBLE] zone? 548 00:41:42,780 --> 00:41:44,066 PROFESSOR: Yes. 549 00:41:44,066 --> 00:41:45,470 AUDIENCE: OK. 550 00:41:45,470 --> 00:41:47,820 PROFESSOR: That's right. 551 00:41:47,820 --> 00:41:51,480 Any time I see a divergent integral, 552 00:41:51,480 --> 00:41:54,290 I have a reason to go back to my physics 553 00:41:54,290 --> 00:41:59,030 and see why physics will avoid infinities. 554 00:41:59,030 --> 00:42:04,360 And in this case, because all of my theories 555 00:42:04,360 --> 00:42:08,600 have an underlying length scale associated with them 556 00:42:08,600 --> 00:42:12,900 and there is an associated maximum value 557 00:42:12,900 --> 00:42:14,535 that I can go in Fourier space. 558 00:42:19,450 --> 00:42:23,850 The only possible singularities that I 559 00:42:23,850 --> 00:42:27,500 want to get are coming from q goes to 0. 560 00:42:27,500 --> 00:42:31,540 And again, if I really want to physically cut that off, 561 00:42:31,540 --> 00:42:33,550 I would put the size of the system. 562 00:42:33,550 --> 00:42:37,056 But I'm interested in systems that become infinite in size. 563 00:42:42,040 --> 00:42:53,080 So the first term for this way of contracting things 564 00:42:53,080 --> 00:42:54,380 is as follows. 565 00:42:54,380 --> 00:42:56,420 There are eight such terms. 566 00:42:56,420 --> 00:42:58,130 I should have really put the four here. 567 00:42:58,130 --> 00:43:03,390 There are eight such types of contractions. 568 00:43:03,390 --> 00:43:08,940 Then I have a delta alpha i 2 pi to the d delta 569 00:43:08,940 --> 00:43:16,440 function q plus q1 divided by t plus k q squared and so forth. 570 00:43:16,440 --> 00:43:19,740 The first contraction is exactly the same as before. 571 00:43:19,740 --> 00:43:27,080 The next contraction I connect i to j and q prime to q3. 572 00:43:27,080 --> 00:43:33,700 So I have a delta beta j 2 pi to the d delta function q 573 00:43:33,700 --> 00:43:39,240 prime going to q3 divided by t plus k q prime squared 574 00:43:39,240 --> 00:43:39,800 and so forth. 575 00:43:42,640 --> 00:43:47,680 And the last contraction connects an i to a j. 576 00:43:47,680 --> 00:43:49,710 Delta ij. 577 00:43:49,710 --> 00:43:54,270 I have 2 pi to the d. 578 00:43:54,270 --> 00:43:59,140 Connecting q2 to minus q1 minus q2 minus q3 579 00:43:59,140 --> 00:44:04,200 will give me a delta function which is minus q1 minus q3. 580 00:44:04,200 --> 00:44:08,950 And then I have t plus k-- I guess in this case-- q2 squared 581 00:44:08,950 --> 00:44:10,342 and so forth. 582 00:44:17,850 --> 00:44:23,340 So once more sum ij. 583 00:44:23,340 --> 00:44:30,190 Integrate q1 q2 q3 and let's see what happens. 584 00:44:30,190 --> 00:44:35,270 So again, it's a term that is proportional to minus u. 585 00:44:35,270 --> 00:44:40,250 The numerical coefficient that it carries is 8. 586 00:44:40,250 --> 00:44:47,090 And there is no n here because when I sum over i, 587 00:44:47,090 --> 00:44:51,270 you can see that j is set to be the same as alpha. 588 00:44:51,270 --> 00:44:56,390 Then when I sum over j, I set alpha to be the same as beta. 589 00:44:56,390 --> 00:44:58,250 So there is just a delta alpha beta. 590 00:45:02,400 --> 00:45:07,650 When I integrate over q1, q1 is set to minus q. 591 00:45:07,650 --> 00:45:12,120 q3 is set to minus q prime. 592 00:45:12,120 --> 00:45:16,450 So this factor becomes the same as q plus q prime. 593 00:45:20,790 --> 00:45:26,250 And the two variances, which are in fact the same, 594 00:45:26,250 --> 00:45:30,610 I can continue to write as separate entities 595 00:45:30,610 --> 00:45:34,200 but they're really the same thing. 596 00:45:34,200 --> 00:45:36,310 And then the one integral that is 597 00:45:36,310 --> 00:45:42,270 left-- I did q1 and Q3-- it's the integral over q2, 598 00:45:42,270 --> 00:45:51,390 2 pi to the d 1 over t plus K q2 squared and so forth. 599 00:45:51,390 --> 00:45:53,790 It is, in fact, exactly the same integral 600 00:45:53,790 --> 00:45:57,090 as before, except that the name of the dummy integration 601 00:45:57,090 --> 00:46:01,837 variable has changed from q2 to q3, or q3 to q2. 602 00:46:06,070 --> 00:46:18,890 So we have calculated m alpha of q m beta of q prime 603 00:46:18,890 --> 00:46:21,730 to the lowest order in perturbation theory. 604 00:46:24,560 --> 00:46:31,040 To the first order, what I had was a delta alpha beta 2 605 00:46:31,040 --> 00:46:38,830 pi to the d delta function q plus q prime divided by t 606 00:46:38,830 --> 00:46:40,460 plus k q squared. 607 00:46:44,970 --> 00:46:48,400 Now, note that all of these factors 608 00:46:48,400 --> 00:46:51,030 are present in the two terms that I 609 00:46:51,030 --> 00:46:52,720 had calculated as corrections. 610 00:46:55,400 --> 00:47:00,920 So I can factor this out and write it 611 00:47:00,920 --> 00:47:06,090 as the correction as 1 plus or minus something. 612 00:47:06,090 --> 00:47:08,020 It is proportional to u. 613 00:47:10,910 --> 00:47:15,180 The coefficient is 4n plus 8, which 614 00:47:15,180 --> 00:47:18,560 I will write as 4 n plus 2. 615 00:47:21,530 --> 00:47:25,435 I took out one factor of t plus k q squared. 616 00:47:25,435 --> 00:47:28,120 There is one factor that's will be remaining. 617 00:47:28,120 --> 00:47:33,750 Therefore, t plus k q squared. 618 00:47:33,750 --> 00:47:39,280 And then I have one integration over some variable. 619 00:47:39,280 --> 00:47:40,520 Let's call it k. 620 00:47:40,520 --> 00:47:44,180 It doesn't matter what I call the integration variable. 621 00:47:44,180 --> 00:47:47,882 1 over t plus k, k squared, and so forth. 622 00:47:51,990 --> 00:47:56,177 And presumably, there will be higher-order terms. 623 00:48:09,100 --> 00:48:12,790 Now again, I did the calculation specifically 624 00:48:12,790 --> 00:48:15,970 for the Landau-Ginzburg, but the procedure you 625 00:48:15,970 --> 00:48:18,960 would have been able to do for any field theory. 626 00:48:18,960 --> 00:48:20,790 You could have started with a part 627 00:48:20,790 --> 00:48:22,620 that you can solve exactly and then 628 00:48:22,620 --> 00:48:25,112 look at perturbations and corrections. 629 00:48:28,420 --> 00:48:33,960 Now, there is, in fact, a reason why 630 00:48:33,960 --> 00:48:36,570 this correction that we calculated 631 00:48:36,570 --> 00:48:40,320 had exactly the same structure of delta functions 632 00:48:40,320 --> 00:48:42,560 as the original one. 633 00:48:42,560 --> 00:48:47,070 And why I anticipate that higher-order terms, if I were 634 00:48:47,070 --> 00:48:49,680 to calculate, will preserve that structure. 635 00:48:52,500 --> 00:48:57,070 And the reason has to do with symmetries. 636 00:48:57,070 --> 00:49:02,580 Because quite generally, I can write for anything-- m 637 00:49:02,580 --> 00:49:08,930 alpha m beta of q prime without doing perturbation theory. 638 00:49:08,930 --> 00:49:13,300 Again, let's remember m alphas of q 639 00:49:13,300 --> 00:49:17,790 are going to be related to m of x by inverse Fourier 640 00:49:17,790 --> 00:49:19,380 transformation. 641 00:49:19,380 --> 00:49:26,340 So m alpha of q I can write an integral d dx e to the-- I 642 00:49:26,340 --> 00:49:32,810 guess by that convention, it has to be minus i q dot x. 643 00:49:32,810 --> 00:49:35,670 m alpha of x. 644 00:49:35,670 --> 00:49:38,920 And again, m beta of q prime I can 645 00:49:38,920 --> 00:49:43,700 write as minus i q prime dot x prime. 646 00:49:43,700 --> 00:49:52,060 And I integrate also over an x prime 647 00:49:52,060 --> 00:49:56,420 of m alpha of x m beta of x prime. 648 00:49:56,420 --> 00:50:00,930 Now, these are evaluated in real space as opposed to Fourier 649 00:50:00,930 --> 00:50:01,980 space. 650 00:50:01,980 --> 00:50:04,910 And the average goes over here. 651 00:50:08,172 --> 00:50:10,230 At this stage, I don't say anything 652 00:50:10,230 --> 00:50:14,460 about perturbation theory, Gaussian, et cetera. 653 00:50:14,460 --> 00:50:22,280 What I expect is that this is a function, 654 00:50:22,280 --> 00:50:26,600 that in a system that has translational symmetry, only 655 00:50:26,600 --> 00:50:28,810 depends on x minus x prime. 656 00:50:39,040 --> 00:50:39,540 m beta. 657 00:50:42,130 --> 00:50:46,180 Furthermore, in a system that has rotational symmetry that 658 00:50:46,180 --> 00:50:48,020 is not spontaneously broken, that 659 00:50:48,020 --> 00:50:51,580 is approaching from the high temperature side, 660 00:50:51,580 --> 00:50:55,800 then just rotational symmetry forces you 661 00:50:55,800 --> 00:50:58,340 that-- the only tensor that you have 662 00:50:58,340 --> 00:51:01,940 has to be proportional to delta alpha beta. 663 00:51:01,940 --> 00:51:06,150 So I can pick some particular component-- let's say m1-- 664 00:51:06,150 --> 00:51:07,700 and I can write it in this fashion. 665 00:51:11,240 --> 00:51:17,020 So the rotational symmetry explains the delta alpha beta. 666 00:51:17,020 --> 00:51:22,170 Now, knowing that the function that I'm integrating over two 667 00:51:22,170 --> 00:51:27,620 variables actually only depends on the relative position 668 00:51:27,620 --> 00:51:34,880 means that I can re-express this in terms of the relative 669 00:51:34,880 --> 00:51:36,550 and center of mass coordinates. 670 00:51:36,550 --> 00:51:38,900 So I can express that expression as e 671 00:51:38,900 --> 00:51:47,670 to the minus i q minus q prime x minus x prime over 2. 672 00:51:47,670 --> 00:51:50,750 And then I will write it as minus i 673 00:51:50,750 --> 00:51:56,260 q plus q prime x plus x prime over 2. 674 00:51:56,260 --> 00:52:00,020 If you expand those, you will see that all of the cross terms 675 00:52:00,020 --> 00:52:04,160 will vanish and I will get q dot x and q prime dot x prime. 676 00:52:04,160 --> 00:52:04,990 Yes. 677 00:52:04,990 --> 00:52:05,906 AUDIENCE: [INAUDIBLE]? 678 00:52:09,065 --> 00:52:09,690 PROFESSOR: Yes. 679 00:52:09,690 --> 00:52:10,260 Thank you. 680 00:52:18,420 --> 00:52:24,156 So now I can change integration variables 681 00:52:24,156 --> 00:52:33,110 to the relative coordinate and the center of mass coordinate 682 00:52:33,110 --> 00:52:37,330 rather than integrating over x and x prime. 683 00:52:37,330 --> 00:52:41,410 The integration over the center of mass, the x plus x 684 00:52:41,410 --> 00:52:47,060 prime variable, couples to q plus q prime. 685 00:52:47,060 --> 00:52:49,800 So it will immediately tell me that the answer has 686 00:52:49,800 --> 00:52:56,350 to be proportional to q plus q prime. 687 00:52:56,350 --> 00:53:00,820 I had already established that there is a delta alpha beta. 688 00:53:00,820 --> 00:53:05,500 So the only thing that is left is the integration 689 00:53:05,500 --> 00:53:10,980 over the relative coordinate of e to the minus i 690 00:53:10,980 --> 00:53:17,900 some q-- either one of them. 691 00:53:17,900 --> 00:53:19,030 q dot r. 692 00:53:19,030 --> 00:53:22,685 Since q prime is minus q, I can replace it with e 693 00:53:22,685 --> 00:53:27,060 to the i q dot the relative coordinate. 694 00:53:27,060 --> 00:53:29,800 m1 of r m1 of 0. 695 00:53:36,590 --> 00:53:42,310 So that's why for a system that has translational symmetry 696 00:53:42,310 --> 00:53:45,280 and rotational symmetry, this structure 697 00:53:45,280 --> 00:53:48,880 of the delta functions is really imposed 698 00:53:48,880 --> 00:53:51,460 for this type of expectation value. 699 00:53:51,460 --> 00:53:57,450 Naturally, perturbation theory has to obey that. 700 00:53:57,450 --> 00:54:04,160 But then, this is a quantity that we had encountered before. 701 00:54:06,810 --> 00:54:09,060 If you recall when we were scattering 702 00:54:09,060 --> 00:54:13,600 light out of the system, the amplitude 703 00:54:13,600 --> 00:54:18,460 of something that was scattered was proportional 704 00:54:18,460 --> 00:54:23,940 to the Fourier transform of the correlation function. 705 00:54:23,940 --> 00:54:27,860 And furthermore, in the limit where 706 00:54:27,860 --> 00:54:34,540 S is evaluated for q equal to 0, what we are doing 707 00:54:34,540 --> 00:54:38,440 is we're essentially integrating the correlation function. 708 00:54:38,440 --> 00:54:41,155 We've seen that the integrals of correlation function 709 00:54:41,155 --> 00:54:42,613 correspond to the susceptibilities. 710 00:54:47,770 --> 00:54:52,200 So you may have thought that what I was calculating 711 00:54:52,200 --> 00:54:55,780 was a two-point correlation function in perturbation 712 00:54:55,780 --> 00:54:56,340 theory. 713 00:54:56,340 --> 00:54:58,992 But what I was actually leading up to 714 00:54:58,992 --> 00:55:01,690 is to know what the result is for scattering 715 00:55:01,690 --> 00:55:02,980 from this theory. 716 00:55:02,980 --> 00:55:04,800 And in some limit of it, I've also 717 00:55:04,800 --> 00:55:07,510 calculated what the susceptibility is, 718 00:55:07,510 --> 00:55:09,065 how the susceptibility is corrected. 719 00:55:11,930 --> 00:55:16,440 And again, if you recall the typical structure 720 00:55:16,440 --> 00:55:20,750 that people see for S of q is that S of q 721 00:55:20,750 --> 00:55:25,996 is something like 1 over something like this. 722 00:55:25,996 --> 00:55:28,770 This is the Lorentzian line shapes 723 00:55:28,770 --> 00:55:31,000 that we had in scattering. 724 00:55:31,000 --> 00:55:34,680 And clearly, the Lorentzian line shape 725 00:55:34,680 --> 00:55:40,600 is obtained by Fourier transformation and expectation 726 00:55:40,600 --> 00:55:44,900 values of these expansions that we make. 727 00:55:44,900 --> 00:55:49,780 So it kind of makes sense that rather 728 00:55:49,780 --> 00:55:55,940 than looking at this quantity, I should look at its inverse. 729 00:55:55,940 --> 00:55:59,770 So I have calculated s of q, which 730 00:55:59,770 --> 00:56:03,290 is the formula that I have up there. 731 00:56:03,290 --> 00:56:15,520 So this whole thing here is S of q. 732 00:56:18,490 --> 00:56:23,310 If I calculate its inverse, what do I get? 733 00:56:23,310 --> 00:56:25,370 First of all, I have to invert this. 734 00:56:25,370 --> 00:56:33,600 I have t plus k q squared, which is what would have given me 735 00:56:33,600 --> 00:56:36,680 the Lorentzian if I were to invert it. 736 00:56:36,680 --> 00:56:39,740 And now we have found the correction to the Lorentzian 737 00:56:39,740 --> 00:56:44,610 if you like, which is this object raised 738 00:56:44,610 --> 00:56:47,210 to the power of minus 1. 739 00:56:47,210 --> 00:56:49,750 But recall that I've only calculated things 740 00:56:49,750 --> 00:56:52,520 to lowest order in u. 741 00:56:52,520 --> 00:56:57,110 So whenever I see something and I'm inverting it 742 00:56:57,110 --> 00:57:00,020 just like I did over here, I better 743 00:57:00,020 --> 00:57:02,900 be consistent to order of u. 744 00:57:02,900 --> 00:57:07,850 So to order of u, I can take this thing from the numerator, 745 00:57:07,850 --> 00:57:10,780 bring it to the num-- from denominator 746 00:57:10,780 --> 00:57:13,490 to numerator at the expense of just changing the sign. 747 00:57:32,580 --> 00:57:34,740 Order of u squared that we haven't really 748 00:57:34,740 --> 00:57:35,656 bothered to calculate. 749 00:57:41,110 --> 00:57:44,420 So now it's nice because you can see that when I expand this, 750 00:57:44,420 --> 00:57:48,890 this factor will cancel that factor. 751 00:57:48,890 --> 00:57:52,900 So the inverse has the structure that we would like. 752 00:57:52,900 --> 00:57:58,750 It is t plus something that is a constant, 753 00:57:58,750 --> 00:58:04,260 doesn't depend on q, 4 n plus 2 u. 754 00:58:04,260 --> 00:58:05,600 Well, actually, no. 755 00:58:05,600 --> 00:58:06,210 Yeah. 756 00:58:06,210 --> 00:58:12,250 Because this denominator gets canceled. 757 00:58:12,250 --> 00:58:20,150 I will get 4 n plus 2 u integral over k 2 pi 758 00:58:20,150 --> 00:58:26,690 to the d 1 over t plus k k squared and so forth. 759 00:58:26,690 --> 00:58:31,100 And then I have my k q squared. 760 00:58:31,100 --> 00:58:33,580 And presumably, I will have higher-order terms 761 00:58:33,580 --> 00:58:36,042 both in u and higher powers of q, et cetera. 762 00:58:43,920 --> 00:58:51,960 And in particular, the inverse of the susceptibility 763 00:58:51,960 --> 00:58:53,790 is simply the first part. 764 00:58:53,790 --> 00:58:56,150 Forget about the k q squared. 765 00:58:56,150 --> 00:58:58,420 So the inverse of susceptibility is 766 00:58:58,420 --> 00:59:11,190 t plus 4 n plus 2 u integral d dk 2 pi to the d 1 over t 767 00:59:11,190 --> 00:59:14,690 plus k k squared and so forth. 768 00:59:14,690 --> 00:59:17,045 Plus order of things that we haven't computed. 769 00:59:35,900 --> 00:59:40,000 So why is it interesting to look at susceptibility? 770 00:59:40,000 --> 00:59:46,590 Because susceptibility is one of the quantities-- 771 00:59:46,590 --> 00:59:52,270 it always has to be positive-- that we were associating 772 00:59:52,270 --> 00:59:56,700 previously with singular behavior. 773 00:59:56,700 --> 01:00:01,340 And in the absence of the perturbative correction 774 01:00:01,340 --> 01:00:03,980 from the Gaussian, the susceptibility 775 01:00:03,980 --> 01:00:05,460 we calculated many times. 776 01:00:05,460 --> 01:00:08,040 It was simply 1 over t. 777 01:00:08,040 --> 01:00:11,290 If I had added a field, the field h 778 01:00:11,290 --> 01:00:12,770 would have changed the free energy 779 01:00:12,770 --> 01:00:16,370 by an amount that would be h squared over 2t as we saw. 780 01:00:16,370 --> 01:00:18,270 Take two derivatives, I will get 1 781 01:00:18,270 --> 01:00:20,440 over t for the susceptibility. 782 01:00:20,440 --> 01:00:25,760 So the 0 order sustainability that I will indicate by chi sub 783 01:00:25,760 --> 01:00:31,340 0 was something that was diverging at t equals to 0. 784 01:00:31,340 --> 01:00:34,670 And we were identifying the critical exponent 785 01:00:34,670 --> 01:00:38,830 of the divergence as gamma equals to 1. 786 01:00:38,830 --> 01:00:44,330 So here, I would have added gamma 0 equals to 1. 787 01:00:44,330 --> 01:00:46,830 Because of the linear divergence-- 788 01:00:46,830 --> 01:00:49,410 and the linear divergence can be traced back 789 01:00:49,410 --> 01:00:52,930 to the linearity of the vanishing 790 01:00:52,930 --> 01:00:55,000 of the inverse susceptibility at temperature. 791 01:00:59,020 --> 01:01:01,540 Now, let's see whether we have calculated a correction 792 01:01:01,540 --> 01:01:04,139 to gamma. 793 01:01:04,139 --> 01:01:05,680 Well, the first thing that you notice 794 01:01:05,680 --> 01:01:15,370 that if I evaluated the new chi inverse at 0, all I need to do 795 01:01:15,370 --> 01:01:18,970 is to put 0 in this formula. 796 01:01:18,970 --> 01:01:23,890 I will get 4 n plus 2 u. 797 01:01:23,890 --> 01:01:31,740 This integral d dk 2 pi to the d 1 over k k squared. 798 01:01:31,740 --> 01:01:33,722 I set t equals to 0 here. 799 01:01:36,814 --> 01:01:42,580 Now this is, indeed, an integral that if I integrate all the ay 800 01:01:42,580 --> 01:01:45,150 to infinity would diverge on me. 801 01:01:45,150 --> 01:01:48,690 I have to put an upper cutoff. 802 01:01:48,690 --> 01:01:50,070 It's a simple integral. 803 01:01:50,070 --> 01:01:57,106 I can write it as integral 0 to lambda dk 804 01:01:57,106 --> 01:02:04,650 k to the d minus 1 with some surface of a d dimensional 805 01:02:04,650 --> 01:02:06,230 sphere. 806 01:02:06,230 --> 01:02:09,650 I have this 2 pi to the d out front 807 01:02:09,650 --> 01:02:13,000 and I have a k squared here. 808 01:02:13,000 --> 01:02:14,960 I can put the k out here. 809 01:02:14,960 --> 01:02:18,730 So you can see that this is an integral that's just a power. 810 01:02:18,730 --> 01:02:20,710 I can simply do that. 811 01:02:20,710 --> 01:02:32,180 The answer ultimately will be 4 n plus 2 u Sd 2 pi to the d. 812 01:02:32,180 --> 01:02:35,830 There is a factor of 1 over k that comes into play. 813 01:02:35,830 --> 01:02:37,880 The integral of this will give me 814 01:02:37,880 --> 01:02:43,938 the upper cutoff to the d minus 2 divided by d minus 2. 815 01:02:47,270 --> 01:02:54,480 So what we find is that the corrected susceptibility 816 01:02:54,480 --> 01:03:00,650 to lowest order does not diverge at t equals to 0. 817 01:03:00,650 --> 01:03:03,980 Its inverse is a finite value. 818 01:03:03,980 --> 01:03:07,270 So actually, you can see that I've 819 01:03:07,270 --> 01:03:11,060 added something positive to the denominator 820 01:03:11,060 --> 01:03:15,860 so the value of susceptibility is always reduced. 821 01:03:15,860 --> 01:03:17,250 So what does that mean? 822 01:03:17,250 --> 01:03:20,970 Does that mean that the susceptibility does not 823 01:03:20,970 --> 01:03:23,350 have a singularity anymore? 824 01:03:23,350 --> 01:03:24,760 The answer is no. 825 01:03:24,760 --> 01:03:29,340 It's just that the location of the singularity has changed. 826 01:03:29,340 --> 01:03:31,520 The presence of u m to the fourth 827 01:03:31,520 --> 01:03:35,900 gives some additional stiffness that you have to overcome. 828 01:03:35,900 --> 01:03:38,930 t equals to 0 is not sufficient for you. 829 01:03:38,930 --> 01:03:43,470 You have to go to some other point tc. 830 01:03:43,470 --> 01:03:45,720 So I expect that this thing will actually 831 01:03:45,720 --> 01:03:52,090 diverge at a new point tc that is negative. 832 01:03:52,090 --> 01:03:58,730 And if it diverges, then its inverse will be 0. 833 01:03:58,730 --> 01:04:05,120 So I have to solve the equation tc plus 4 n 834 01:04:05,120 --> 01:04:17,250 plus 2 u integral d dk 2 pi to the d of 1 over tc plus k 835 01:04:17,250 --> 01:04:19,406 k squared and so forth. 836 01:04:23,350 --> 01:04:26,620 So this seems like an implicit [INAUDIBLE] equation in tc 837 01:04:26,620 --> 01:04:30,430 because I have to evaluate the integral that depends on tc, 838 01:04:30,430 --> 01:04:33,930 and then have to set that function to 0. 839 01:04:33,930 --> 01:04:37,140 But again, we have calculated things only correctly 840 01:04:37,140 --> 01:04:38,680 to order of u. 841 01:04:38,680 --> 01:04:43,160 And you can see already that tc is proportional to u. 842 01:04:43,160 --> 01:04:48,000 So this answer here is something that is order of u presumably. 843 01:04:48,000 --> 01:04:51,410 And a u compared to the u out front 844 01:04:51,410 --> 01:04:54,500 will give me a correction that is order of u squared. 845 01:04:54,500 --> 01:04:57,532 So I can ignore this thing that is over here. 846 01:05:00,770 --> 01:05:05,430 So to order of u, I know that tc is actually minus 847 01:05:05,430 --> 01:05:07,450 what I had calculated before. 848 01:05:07,450 --> 01:05:14,400 So I get that tc is minus this 4 n plus 2 u k 849 01:05:14,400 --> 01:05:22,380 Sd lambda to the d minus 2 2 pi to the d d minus 2. 850 01:05:22,380 --> 01:05:26,900 It doesn't matter what it is, it's some non-universal value. 851 01:05:26,900 --> 01:05:29,470 Point is that, again, the location of the phase 852 01:05:29,470 --> 01:05:33,250 transition certainly will depend on the parameters 853 01:05:33,250 --> 01:05:34,790 that you put in your theory. 854 01:05:34,790 --> 01:05:39,210 We readjusted our theory by putting m to the fourth. 855 01:05:39,210 --> 01:05:43,505 And certainly, that will change the location of the transition. 856 01:05:43,505 --> 01:05:46,050 So this is what we found. 857 01:05:46,050 --> 01:05:50,390 The location of the transition is not universal. 858 01:05:50,390 --> 01:05:53,680 However, we kind of hope and expect 859 01:05:53,680 --> 01:05:58,700 that the singularity, the divergence of susceptibility 860 01:05:58,700 --> 01:06:00,180 has a form that is universal. 861 01:06:00,180 --> 01:06:04,770 There is an exponent that is characteristic of that. 862 01:06:04,770 --> 01:06:08,720 So asking the question of how these corrected chi 863 01:06:08,720 --> 01:06:13,420 divergence diverges at this tc is the same 864 01:06:13,420 --> 01:06:18,440 as asking the question of how its inverse vanishes at tc. 865 01:06:18,440 --> 01:06:21,530 So what I am interested is to find out 866 01:06:21,530 --> 01:06:27,660 what the behavior of chi inverse is in the vicinity of the point 867 01:06:27,660 --> 01:06:28,810 that it goes to 0. 868 01:06:32,177 --> 01:06:35,650 So basically, what this singularity is. 869 01:06:35,650 --> 01:06:39,080 This is, of course, 0. 870 01:06:39,080 --> 01:06:42,830 By definition, chi inverse of tc is 0. 871 01:06:42,830 --> 01:06:51,640 So I am asking how chi vanishes its inverse when I approach tc. 872 01:06:51,640 --> 01:06:57,395 So I have the formula for chi once I substitute t, 873 01:06:57,395 --> 01:07:00,170 once I substitute tc and I subtract them. 874 01:07:00,170 --> 01:07:03,656 To lowest order I have t minus tc. 875 01:07:03,656 --> 01:07:11,770 To next order, I have this 4 u n plus 2 integral over k 2 pi 876 01:07:11,770 --> 01:07:13,710 to the d. 877 01:07:13,710 --> 01:07:23,430 I have for chi inverse of t 1 over t plus k k squared minus 1 878 01:07:23,430 --> 01:07:27,490 over tc plus k k squared. 879 01:07:27,490 --> 01:07:30,040 And terms that I have not calculated 880 01:07:30,040 --> 01:07:32,003 are certainly order of u squared. 881 01:07:36,000 --> 01:07:39,430 Now, you can see that if I combine these two 882 01:07:39,430 --> 01:07:42,820 into the same denominator that is the product, 883 01:07:42,820 --> 01:07:46,954 in the numerator I will get a factor of t minus tc. 884 01:07:46,954 --> 01:07:49,280 The k q squareds cancel. 885 01:07:49,280 --> 01:07:55,910 So I can factor out this t minus tc between the two terms. 886 01:07:55,910 --> 01:08:00,170 Both terms vanish at t equals to tc. 887 01:08:00,170 --> 01:08:06,050 And then I can look at what the correction is 888 01:08:06,050 --> 01:08:09,595 to one, just like I did before. 889 01:08:09,595 --> 01:08:12,670 The correction is going to-- actually, this 890 01:08:12,670 --> 01:08:17,153 will give me tc minus t, so I will have a minus 4 u 891 01:08:17,153 --> 01:08:25,240 n plus 2 integral d dk 2 pi to the d. 892 01:08:25,240 --> 01:08:34,600 The product of two of these factors, t plus k-- tc plus k k 893 01:08:34,600 --> 01:08:44,585 squared t plus k k squared, and then order of u squared. 894 01:08:49,535 --> 01:08:50,525 OK? 895 01:08:50,525 --> 01:08:51,515 Happy with that? 896 01:08:55,990 --> 01:09:01,410 Now again, this tc we've calculated is order of u. 897 01:09:01,410 --> 01:09:05,260 And consistently, to calculate things to order of u, 898 01:09:05,260 --> 01:09:06,340 I can drop that. 899 01:09:09,130 --> 01:09:13,960 And again, consistently to doing things to order of u, 900 01:09:13,960 --> 01:09:16,729 I can add a tc here. 901 01:09:16,729 --> 01:09:19,790 And that's also a correction that is order of u. 902 01:09:19,790 --> 01:09:22,770 And this answer would not change. 903 01:09:22,770 --> 01:09:26,000 The justification of why I choose to do that 904 01:09:26,000 --> 01:09:30,069 will become apparent shortly, but it's consistent 905 01:09:30,069 --> 01:09:32,460 that this is left. 906 01:09:32,460 --> 01:09:37,220 So what I find at this stage is that I 907 01:09:37,220 --> 01:09:42,109 need to evaluate an integral of this form. 908 01:09:58,750 --> 01:10:01,800 And again, with all of these integrals, 909 01:10:01,800 --> 01:10:06,850 we better take a look as to what the most significant 910 01:10:06,850 --> 01:10:09,950 contribution to the integral is. 911 01:10:09,950 --> 01:10:16,730 And clearly, if I look at k goes to 0, 912 01:10:16,730 --> 01:10:20,580 there are various factors out there 913 01:10:20,580 --> 01:10:24,920 that as long as t minus tc is positive, 914 01:10:24,920 --> 01:10:28,870 I will have no worries because this k squared 915 01:10:28,870 --> 01:10:31,930 will be killed off by factors of k 916 01:10:31,930 --> 01:10:36,170 to the d minus 1 in dimensions above 2. 917 01:10:36,170 --> 01:10:39,620 But if I go to large k-values, I find 918 01:10:39,620 --> 01:10:43,880 that large k-values, the singularity 919 01:10:43,880 --> 01:10:48,790 is governed by k to the power of d minus 4. 920 01:10:48,790 --> 01:10:57,670 So as long as I'm dealing with things 921 01:10:57,670 --> 01:11:01,130 that have some upper cutoff, I don't 922 01:11:01,130 --> 01:11:05,455 have to worry about it even in dimensions greater than 4. 923 01:11:05,455 --> 01:11:12,380 In dimensions greater than 4, what 924 01:11:12,380 --> 01:11:15,470 happens is that the integral is going 925 01:11:15,470 --> 01:11:21,700 to be dominated by the largest values of k. 926 01:11:21,700 --> 01:11:26,360 But those largest values of k will be cutoff by lambda. 927 01:11:26,360 --> 01:11:29,410 The answer ultimately will be proportional to 1 928 01:11:29,410 --> 01:11:35,980 over k squared, and then k to the power of d minus 4 929 01:11:35,980 --> 01:11:38,680 replaced by lambda to the d minus 4-- 930 01:11:38,680 --> 01:11:42,210 various overall coefficient of d minus 4 or whatever. 931 01:11:42,210 --> 01:11:43,190 It doesn't matter. 932 01:11:46,130 --> 01:11:49,430 On the other hand, if you go to dimensions less than 4-- 933 01:11:49,430 --> 01:11:50,850 again, larger than 2, but I won't 934 01:11:50,850 --> 01:11:58,500 write that for the time being-- then the behavior at large k 935 01:11:58,500 --> 01:12:00,820 is perfectly convergent. 936 01:12:00,820 --> 01:12:05,340 So you are integrating a function that goes up, 937 01:12:05,340 --> 01:12:06,720 comes down. 938 01:12:06,720 --> 01:12:11,440 You can extend the integration all the way to infinity, 939 01:12:11,440 --> 01:12:14,480 end up with a definite integral. 940 01:12:14,480 --> 01:12:18,290 We can rescale all of our factors of k 941 01:12:18,290 --> 01:12:23,150 to find out what that definite integral is dependent on. 942 01:12:23,150 --> 01:12:30,000 And essentially, what it does is it replaces this lambda 943 01:12:30,000 --> 01:12:34,170 with the characteristic value of k that corresponds roughly 944 01:12:34,170 --> 01:12:35,550 to the maximum. 945 01:12:35,550 --> 01:12:37,910 And that's going to occur at something 946 01:12:37,910 --> 01:12:45,200 like t minus tc over k to the power of 1/2. 947 01:12:45,200 --> 01:12:50,430 So I will get d minus 4 over 2. 948 01:12:50,430 --> 01:12:54,990 There is some overall definite integral that I have to do, 949 01:12:54,990 --> 01:12:58,440 which will give me some numerical coefficient. 950 01:12:58,440 --> 01:13:00,680 But at this time, let's forget about 951 01:13:00,680 --> 01:13:02,410 the numerical coefficient. 952 01:13:02,410 --> 01:13:05,190 Let's see what the structure is. 953 01:13:05,190 --> 01:13:08,543 So the structure then is that chi inverse 954 01:13:08,543 --> 01:13:16,150 of t, the singularity that it has 955 01:13:16,150 --> 01:13:20,940 is t minus tc to the 0 order. 956 01:13:20,940 --> 01:13:24,370 Same thing as you would have predicted for the Gaussian. 957 01:13:24,370 --> 01:13:27,270 And then we have a correction, which 958 01:13:27,270 --> 01:13:32,820 is this minus something that goes after all of these things 959 01:13:32,820 --> 01:13:33,860 with some coefficient. 960 01:13:33,860 --> 01:13:36,140 I don't care what that coefficient is. 961 01:13:36,140 --> 01:13:39,205 u n plus 2 divided by k squared. 962 01:13:42,070 --> 01:13:48,090 And then multiplied by lambda to the power of d minus 4, 963 01:13:48,090 --> 01:13:55,420 or t minus tc over k to the power of d minus 4 over 2. 964 01:13:55,420 --> 01:13:57,535 And then presumably, higher-order terms. 965 01:14:00,830 --> 01:14:05,000 And whether you have the top or the bottom 966 01:14:05,000 --> 01:14:09,403 will depend on d greater than 4 or d less than 4. 967 01:14:24,430 --> 01:14:26,850 So you see the problem. 968 01:14:26,850 --> 01:14:32,110 If I'm above four dimensions, this term 969 01:14:32,110 --> 01:14:34,240 is governed by the upper cutoff. 970 01:14:34,240 --> 01:14:37,510 But the upper cutoff is just some constant. 971 01:14:37,510 --> 01:14:41,520 So all that happens is that the dependence 972 01:14:41,520 --> 01:14:44,590 remains as being proportional to t minus tc. 973 01:14:44,590 --> 01:14:47,630 The overall amplitude is corrected 974 01:14:47,630 --> 01:14:50,270 by something that depends on u. 975 01:14:50,270 --> 01:14:51,170 You are not worried. 976 01:14:51,170 --> 01:14:54,120 You say that the leading singularity is the same thing 977 01:14:54,120 --> 01:14:55,500 as I had before. 978 01:14:55,500 --> 01:14:58,770 Gamma will stay to be 1. 979 01:14:58,770 --> 01:15:04,380 I try to do that in less than four dimensions 980 01:15:04,380 --> 01:15:10,790 and I find that as I approach tc, the correction that I had 981 01:15:10,790 --> 01:15:12,858 actually itself becomes divergent. 982 01:15:16,210 --> 01:15:20,850 So now I have to throw out my entire perturbation theory 983 01:15:20,850 --> 01:15:26,270 because I thought I was making an expansion in quantity 984 01:15:26,270 --> 01:15:28,520 that I can make sufficiently small. 985 01:15:28,520 --> 01:15:31,680 So in usual perturbation theory, you say 986 01:15:31,680 --> 01:15:35,730 choose epsilon less than 10 to the minus 100, or whatever, 987 01:15:35,730 --> 01:15:38,560 and then things will be small correction 988 01:15:38,560 --> 01:15:40,790 to what you had at 0 order. 989 01:15:40,790 --> 01:15:44,740 Here, I can choose my u to be as small as I like. 990 01:15:44,740 --> 01:15:49,550 Once I approach tc, the correction will blow up. 991 01:15:49,550 --> 01:15:52,680 So this is called a divergent perturbation theory. 992 01:15:52,680 --> 01:15:53,180 Yes. 993 01:15:53,180 --> 01:15:55,596 AUDIENCE: So could we have known a priori that we couldn't 994 01:15:55,596 --> 01:15:58,290 get a correction to gamma from the perturbation theory 995 01:15:58,290 --> 01:15:59,995 because the only way for gamma to change 996 01:15:59,995 --> 01:16:01,786 is for the correction to have a divergence? 997 01:16:04,550 --> 01:16:07,910 PROFESSOR: You are presuming that that's what happens. 998 01:16:07,910 --> 01:16:11,560 So indeed, if you knew that there 999 01:16:11,560 --> 01:16:15,530 is a divergence with an exponent that is larger than gamma, 1000 01:16:15,530 --> 01:16:17,320 you probably could have guessed that you 1001 01:16:17,320 --> 01:16:20,410 wouldn't get it this way. 1002 01:16:20,410 --> 01:16:24,720 Let's say that we are choosing to proceed mathematically 1003 01:16:24,720 --> 01:16:29,090 without prior knowledge of what the experimentalists have told 1004 01:16:29,090 --> 01:16:33,414 us, then we can discover it this way. 1005 01:16:33,414 --> 01:16:34,830 AUDIENCE: I was thinking if you're 1006 01:16:34,830 --> 01:16:38,760 looking for how gamma changes due to the higher-order things, 1007 01:16:38,760 --> 01:16:43,210 if we found that our perturbation diverged 1008 01:16:43,210 --> 01:16:44,940 with a lower exponent than gamma, 1009 01:16:44,940 --> 01:16:46,810 then the leading one would still be there, 1010 01:16:46,810 --> 01:16:47,870 original gamma would be gone. 1011 01:16:47,870 --> 01:16:48,060 PROFESSOR: Yes. 1012 01:16:48,060 --> 01:16:48,560 AUDIENCE: And then if it's higher, 1013 01:16:48,560 --> 01:16:50,140 then we have the same problem. 1014 01:16:50,140 --> 01:16:51,480 PROFESSOR: That's right. 1015 01:16:51,480 --> 01:16:53,840 So the problem that we have is actually 1016 01:16:53,840 --> 01:16:59,840 to somehow make sense of this type of perturbation theory. 1017 01:16:59,840 --> 01:17:01,530 And as you say, it's correct. 1018 01:17:01,530 --> 01:17:02,910 We could have actually guessed. 1019 01:17:02,910 --> 01:17:07,350 And I'll give you another reason why the perturbation 1020 01:17:07,350 --> 01:17:10,100 theory would not have worked. 1021 01:17:10,100 --> 01:17:12,310 But the only thing that we can really do 1022 01:17:12,310 --> 01:17:14,370 is perturbation theory, so we have 1023 01:17:14,370 --> 01:17:18,260 to be clever and figure out a way of making sense 1024 01:17:18,260 --> 01:17:22,340 of this perturbation theory, which we will do by combining 1025 01:17:22,340 --> 01:17:25,400 it with the normalization group. 1026 01:17:25,400 --> 01:17:28,850 But a better way or another way to have seen maybe 1027 01:17:28,850 --> 01:17:33,630 why this does not work is good old-fashioned dimensional 1028 01:17:33,630 --> 01:17:35,360 analysis. 1029 01:17:35,360 --> 01:17:40,960 I have within the exponent of the weight that I wrote down 1030 01:17:40,960 --> 01:17:47,920 terms that are of this formula, t m squared k gradient of m 1031 01:17:47,920 --> 01:17:53,080 squared u m to the fourth and so forth. 1032 01:17:53,080 --> 01:17:57,170 Since whatever is in the exponent should 1033 01:17:57,170 --> 01:18:01,240 be dimensionless-- I usually write beta H for example-- 1034 01:18:01,240 --> 01:18:06,210 we know that this t has some dimension. 1035 01:18:06,210 --> 01:18:11,060 Square the dimension of m multiplied by length to the d. 1036 01:18:11,060 --> 01:18:13,906 This should be dimensionless. 1037 01:18:13,906 --> 01:18:20,930 Similarly, k m squared again. 1038 01:18:20,930 --> 01:18:24,830 Because of the gradient l to the d minus 2, 1039 01:18:24,830 --> 01:18:28,870 that combination should be dimensionless. 1040 01:18:28,870 --> 01:18:37,470 And my u m to the fourth l to the d should be dimensionless. 1041 01:18:41,400 --> 01:18:46,130 So we can get rid of the dimensions of m by dividing, 1042 01:18:46,130 --> 01:18:52,040 let's say, u m to the fourth with the square of k m squared. 1043 01:18:52,040 --> 01:18:57,570 So we can immediately see that u divided by k 1044 01:18:57,570 --> 01:19:02,822 squared, I get rid of the dimensions of m. 1045 01:19:02,822 --> 01:19:08,010 l to the power of d l to the 2d minus 4, 1046 01:19:08,010 --> 01:19:11,550 giving me l to the 4 minus d is dimensionless. 1047 01:19:19,030 --> 01:19:24,530 So any perturbation theory that I write down ultimately where 1048 01:19:24,530 --> 01:19:30,630 I have some quantity x, which is at 0 order 1, 1049 01:19:30,630 --> 01:19:34,770 and then I want to make a correction where u appears, 1050 01:19:34,770 --> 01:19:38,330 I should have something, u over k 1051 01:19:38,330 --> 01:19:44,090 squared, and then some power of length to make the dimensions 1052 01:19:44,090 --> 01:19:46,280 work out. 1053 01:19:46,280 --> 01:19:49,650 So what lengths do I have available to me? 1054 01:19:49,650 --> 01:19:53,910 One length that I have is my microscopic length a. 1055 01:19:53,910 --> 01:19:59,250 So I could have put here a to the power of 4 minus d. 1056 01:19:59,250 --> 01:20:01,490 But there is also an emergent length 1057 01:20:01,490 --> 01:20:03,760 in the problem, which is the correlation length. 1058 01:20:08,600 --> 01:20:15,480 And there is no reason why the dimensionless form that 1059 01:20:15,480 --> 01:20:19,380 involves the correlation length should not appear. 1060 01:20:19,380 --> 01:20:23,720 And indeed, what we have over here to 0 order, 1061 01:20:23,720 --> 01:20:28,580 our correlation length had the exponent 1/2 divergence. 1062 01:20:28,580 --> 01:20:32,560 So this is really the 0 order correlation length 1063 01:20:32,560 --> 01:20:37,400 that is raised to the power of 4 minus d. 1064 01:20:37,400 --> 01:20:43,140 So even before doing the calculation, 1065 01:20:43,140 --> 01:20:47,500 we could have guessed on dimensional ground 1066 01:20:47,500 --> 01:20:51,640 that it is quite possible that we are expanding in u, 1067 01:20:51,640 --> 01:20:52,910 we think. 1068 01:20:52,910 --> 01:20:56,020 But at the end of the day, we are expanding in u c 1069 01:20:56,020 --> 01:20:57,850 to the power of 4 minus d. 1070 01:20:57,850 --> 01:21:00,890 And there is no way that that's a small quantity on approaching 1071 01:21:00,890 --> 01:21:02,690 the phase transition. 1072 01:21:02,690 --> 01:21:05,730 And that hit us on the face and also is 1073 01:21:05,730 --> 01:21:10,060 the reason why I replaced this t over here with t minus tc 1074 01:21:10,060 --> 01:21:14,020 because the only place where I expect singularities to emerge 1075 01:21:14,020 --> 01:21:16,835 in any of these expansions is at tc. 1076 01:21:16,835 --> 01:21:22,010 I arranged things so they would appear at the right place. 1077 01:21:22,010 --> 01:21:27,870 So should we throw out perturbation theory completely 1078 01:21:27,870 --> 01:21:29,680 since the only thing that we can do 1079 01:21:29,680 --> 01:21:32,080 is really perturbation theory? 1080 01:21:32,080 --> 01:21:33,920 Well, we have to be clever about it. 1081 01:21:33,920 --> 01:21:37,051 And that's what we will do next lectures.