WEBVTT 00:00:00.125 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.030 under a Creative Commons license. 00:00:04.030 --> 00:00:06.870 Your support will help MIT OpenCourseWare continue 00:00:06.870 --> 00:00:10.730 to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:21.000 --> 00:00:21.625 PROFESSOR: Hey. 00:00:21.625 --> 00:00:22.125 Let's start. 00:00:24.950 --> 00:00:31.620 So a few weeks ago we started with writing a partition 00:00:31.620 --> 00:00:34.510 function for a statistical field that 00:00:34.510 --> 00:00:41.170 was going to capture behavior of a variety of systems undergoing 00:00:41.170 --> 00:00:44.380 critical phase transitions. 00:00:44.380 --> 00:00:49.930 And this was obtained by integrating over configurations 00:00:49.930 --> 00:01:00.580 of this statistical field a rate that we wrote 00:01:00.580 --> 00:01:03.760 on the basis of a form of locality. 00:01:06.520 --> 00:01:10.330 And terms that were consistent with that 00:01:10.330 --> 00:01:15.596 were of the form m squared, m to the fourth. 00:01:15.596 --> 00:01:17.192 Let's say m to the sixth. 00:01:19.790 --> 00:01:22.280 Various types of gradient types of terms. 00:01:32.680 --> 00:01:36.927 And in principle, allowing for a symmetry-breaking field that 00:01:36.927 --> 00:01:39.659 was more in the form of h dot 1. 00:01:46.281 --> 00:01:50.020 And again, we always emphasized that in writing 00:01:50.020 --> 00:01:53.470 these statistical fields, we have to do averaging. 00:01:53.470 --> 00:01:58.590 We have to get rid of a lot of short wavelength fluctuations. 00:01:58.590 --> 00:02:01.880 And essentially, the future m of x, although I write it 00:02:01.880 --> 00:02:06.870 as a continuum, has an implicit short scale 00:02:06.870 --> 00:02:10.870 below which it does not fluctuate. 00:02:10.870 --> 00:02:15.290 OK, so we tried to evaluate this by certain point, 00:02:15.290 --> 00:02:17.260 and we didn't succeed. 00:02:17.260 --> 00:02:22.430 So we went phenomenologically and tried to describe things 00:02:22.430 --> 00:02:24.490 on the basis of scaling theory. 00:02:24.490 --> 00:02:27.430 Ultimately, this renormalization group 00:02:27.430 --> 00:02:33.770 procedure that we would like to apply to something like this. 00:02:33.770 --> 00:02:36.520 Now, there is a part of this that 00:02:36.520 --> 00:02:39.840 is actually pretty easy to solve. 00:02:39.840 --> 00:02:43.710 And that's when we ignore anything 00:02:43.710 --> 00:02:47.670 that is higher than second order in m. 00:02:47.670 --> 00:02:50.840 Because once we ignore them, we have essentially 00:02:50.840 --> 00:02:53.580 a generalized Gaussian integral. 00:02:53.580 --> 00:02:56.120 We can do Gaussian integrals. 00:02:56.120 --> 00:03:00.690 So what we are going to do is, in this lecture, 00:03:00.690 --> 00:03:05.150 focusing on understanding a lot about the behavior 00:03:05.150 --> 00:03:07.310 of the Gaussian version of the theory. 00:03:07.310 --> 00:03:09.400 Which is certainly a diminished version, 00:03:09.400 --> 00:03:13.740 because it doesn't have lots of essential things. 00:03:13.740 --> 00:03:16.820 And then gradually putting back all of those things 00:03:16.820 --> 00:03:20.390 that we have not considered at the Gaussian level. 00:03:20.390 --> 00:03:23.320 In particular, we'll try to do with them 00:03:23.320 --> 00:03:26.980 with a version of a perturbation theory. 00:03:26.980 --> 00:03:30.120 We'll see that standard perturbation theory has 00:03:30.120 --> 00:03:32.640 some limitations that we will eventually 00:03:32.640 --> 00:03:38.280 resolve by using this renormalization procedure. 00:03:38.280 --> 00:03:39.410 OK. 00:03:39.410 --> 00:03:41.750 So what happens if I do that? 00:03:41.750 --> 00:03:45.690 Why do I say that that theory is now solve-able? 00:03:45.690 --> 00:03:48.380 And the key to that is, of course, 00:03:48.380 --> 00:03:49.960 to go into Fourier representation. 00:03:54.420 --> 00:03:57.890 Which, because the theory that I wrote down 00:03:57.890 --> 00:04:01.450 has this inherent translational of symmetry, 00:04:01.450 --> 00:04:06.420 Fourier representation decouples the various m's 00:04:06.420 --> 00:04:09.530 that are currently connected to their neighborhood 00:04:09.530 --> 00:04:12.430 by these gradients and high orders. 00:04:12.430 --> 00:04:18.480 So let's introduce a m of q, which 00:04:18.480 --> 00:04:21.660 is the Fourier transform of m of x. 00:04:29.388 --> 00:04:32.770 Let's see. m of x. 00:04:32.770 --> 00:04:36.720 And these are all vectors. 00:04:36.720 --> 00:04:42.470 And I should really use a different symbol, such as m 00:04:42.470 --> 00:04:46.660 [INAUDIBLE], to indicate the Fourier 00:04:46.660 --> 00:04:50.120 components of this field m of x. 00:04:50.120 --> 00:04:53.480 But since in the context of renormalization group 00:04:53.480 --> 00:04:57.930 we had defined a coarse grained field that was in tilde, 00:04:57.930 --> 00:04:59.690 I don't want to do that. 00:04:59.690 --> 00:05:02.740 I hope that the argument of the function 00:05:02.740 --> 00:05:05.940 is sufficient indicator of whether we are in real space 00:05:05.940 --> 00:05:07.660 or in momentum space. 00:05:07.660 --> 00:05:10.870 Initially, I'll try to put a tail on the m 00:05:10.870 --> 00:05:13.490 to indicate that I'm doing Fourier space, 00:05:13.490 --> 00:05:16.450 but I suspect that very soon I'll forget about the tail. 00:05:16.450 --> 00:05:20.320 So keep that in mind. 00:05:20.320 --> 00:05:25.440 So if I-- oops. 00:05:25.440 --> 00:05:26.970 OK. 00:05:26.970 --> 00:05:28.580 m of q. 00:05:28.580 --> 00:05:34.980 So if I go back and write what this m of x 00:05:34.980 --> 00:05:49.230 is, it is an integral over 2, 2 pi to the, d to the minus iq 00:05:49.230 --> 00:05:54.930 dot x with m of q. 00:05:59.610 --> 00:06:03.190 Now, I also want to at some stage, 00:06:03.190 --> 00:06:09.650 since it would be cleaner to have this rate in terms 00:06:09.650 --> 00:06:15.860 of a product of q's, remind you that this could have obtained, 00:06:15.860 --> 00:06:19.690 if I hadn't gone to the continuum version-- 00:06:19.690 --> 00:06:26.390 if I had a finite system-- to a sum over q. 00:06:26.390 --> 00:06:28.720 And the sum over q would be basically 00:06:28.720 --> 00:06:32.070 things that are separated q values by multiples of 1 00:06:32.070 --> 00:06:33.960 over the size of the system. 00:06:33.960 --> 00:06:39.170 And e to the minus iq dot x. 00:06:39.170 --> 00:06:42.380 This m with the cues that are now discretized. 00:06:45.400 --> 00:06:48.180 But let's remember that the density of state 00:06:48.180 --> 00:06:52.230 has a factor of 1 over v. So if I use this definition, 00:06:52.230 --> 00:06:55.825 I really should put the 1 over v here when 00:06:55.825 --> 00:06:58.170 I go to the discrete version. 00:06:58.170 --> 00:07:00.890 And I emphasize this because previously, we 00:07:00.890 --> 00:07:04.950 had done Fourier decomposition where 00:07:04.950 --> 00:07:08.990 I had used the square root of v as a normalization. 00:07:08.990 --> 00:07:15.585 It really doesn't matter which normalization 00:07:15.585 --> 00:07:19.200 you use at the end as long as you are consistent. 00:07:19.200 --> 00:07:23.330 We'll see the advantages of this normalization shortly. 00:07:23.330 --> 00:07:26.735 AUDIENCE: Is there any particular reason 00:07:26.735 --> 00:07:28.735 for using the different sign in the exponential? 00:07:28.735 --> 00:07:30.410 PROFESSOR: Actually, no. 00:07:30.410 --> 00:07:35.140 I'm not sure even whether I used iqx here or minus iqx here. 00:07:35.140 --> 00:07:37.870 It's just a matter of which one you 00:07:37.870 --> 00:07:40.860 want to stick with consistently. 00:07:40.860 --> 00:07:45.460 At the end of the day, the phase will not be that important. 00:07:45.460 --> 00:07:48.570 So even if we mistake one form or the other, 00:07:48.570 --> 00:07:49.861 it doesn't make any difference. 00:07:55.160 --> 00:08:00.580 So if I do that, then again, to sort of be more precise, 00:08:00.580 --> 00:08:06.300 I have to think about what to do with gradients. 00:08:06.300 --> 00:08:14.300 Gradients, I can imagine, are the limit of something like n 00:08:14.300 --> 00:08:21.110 at x plus A minus n at x divided by A. 00:08:21.110 --> 00:08:24.660 If this is a gradient in the x direction. 00:08:24.660 --> 00:08:28.840 And I have to take the limit as A goes to 0. 00:08:28.840 --> 00:08:34.320 So when I'm thinking about this kind of functional integral, 00:08:34.320 --> 00:08:39.940 keeping in mind that I have a shortest landscape, maybe 00:08:39.940 --> 00:08:42.840 one way to do it is to imagine that I discretize 00:08:42.840 --> 00:08:47.360 my system over here into spacing of size A. 00:08:47.360 --> 00:08:52.162 And then I have a variable on each size, 00:08:52.162 --> 00:08:57.200 and then I integrate every place, 00:08:57.200 --> 00:09:01.530 subject this replacement for the gradient. 00:09:01.530 --> 00:09:05.630 Again, what you do precisely does not matter here. 00:09:05.630 --> 00:09:08.420 If you remember in the first lecture 00:09:08.420 --> 00:09:12.760 when we were thinking about the dl lattice system 00:09:12.760 --> 00:09:16.350 and then using these kinds of coupling between springs 00:09:16.350 --> 00:09:19.510 that they're connecting nearest neighbors, what ended up 00:09:19.510 --> 00:09:22.920 by using this was that when I Fourier transformed, 00:09:22.920 --> 00:09:25.080 I had things like cosine. 00:09:25.080 --> 00:09:29.650 And then when I expanded the cosine close to q, close to 0, 00:09:29.650 --> 00:09:32.080 I generated a series that had q squared, 00:09:32.080 --> 00:09:34.210 q to the fourth, et cetera. 00:09:34.210 --> 00:09:40.720 So essentially, any discretized version corresponds 00:09:40.720 --> 00:09:44.845 to an expansion like this with sufficient [INAUDIBLE] powers 00:09:44.845 --> 00:09:47.410 of q in both. 00:09:47.410 --> 00:09:53.310 So at the end of the day, when you go through this process, 00:09:53.310 --> 00:09:59.740 you find that you can write the partition function after 00:09:59.740 --> 00:10:05.490 the change of variables to m of x to m of q to doing a whole 00:10:05.490 --> 00:10:10.250 bunch of integrals over different q's. 00:10:10.250 --> 00:10:15.670 So, essentially you would have-- actually, maybe I 00:10:15.670 --> 00:10:23.460 will explicitly put the product over q outside 00:10:23.460 --> 00:10:27.060 to emphasize that essentially, for each q I would 00:10:27.060 --> 00:10:30.370 have to do independent integrals. 00:10:30.370 --> 00:10:33.990 Of course, for each q mode I have, 00:10:33.990 --> 00:10:39.840 since I've gone to this representation of a vector that 00:10:39.840 --> 00:10:46.160 is n-dimensional, I have to do n integrals on n tilde of q. 00:10:48.750 --> 00:10:51.100 On-- m will be the tail of q. 00:10:53.720 --> 00:11:03.600 And if I had chosen the square root 00:11:03.600 --> 00:11:08.840 of V type of normalization, the Jacobian 00:11:08.840 --> 00:11:11.910 of the transformation from here to here would have been 1. 00:11:11.910 --> 00:11:15.460 Because it's kind of a symmetric way of writing things. 00:11:15.460 --> 00:11:19.120 Because I chose this way of doing things, 00:11:19.120 --> 00:11:24.740 I will have a factor of V to the n over 2 in the denominator 00:11:24.740 --> 00:11:26.440 here. 00:11:26.440 --> 00:11:28.940 But again, it's just being pedantic, 00:11:28.940 --> 00:11:30.680 because at the end of the day, we 00:11:30.680 --> 00:11:32.690 don't care about these factors. 00:11:32.690 --> 00:11:34.850 We are interested in things like this singular 00:11:34.850 --> 00:11:39.120 part of the partition function as it 00:11:39.120 --> 00:11:41.470 depends on these coordinates. 00:11:41.470 --> 00:11:45.230 This really just gives you an overall constant. 00:11:45.230 --> 00:11:47.810 Of course, how many of these constants you have 00:11:47.810 --> 00:11:55.050 would depend basically how you have discretized the problem. 00:11:55.050 --> 00:11:59.800 But it is a constant independent of tnh, not something 00:11:59.800 --> 00:12:01.110 that we have to worry about. 00:12:04.630 --> 00:12:10.140 Now what happens to these Gaussian factors? 00:12:10.140 --> 00:12:13.930 Essentially, I have put the product over q outside. 00:12:13.930 --> 00:12:17.650 So when I transform this integral over xm squared 00:12:17.650 --> 00:12:22.790 goes over to an integral over q, m of q 00:12:22.790 --> 00:12:25.920 squared, which then I can write as a product 00:12:25.920 --> 00:12:28.500 over those contributions. 00:12:28.500 --> 00:12:33.770 And what you will get is t plus, from here, 00:12:33.770 --> 00:12:38.020 you will get a Kq squared, put in Lq 00:12:38.020 --> 00:12:41.800 to the fourth and all kinds of order terms 00:12:41.800 --> 00:12:44.840 that I have included. 00:12:44.840 --> 00:12:52.160 Multiplying this m component vector m of q squared. 00:12:52.160 --> 00:12:55.560 Again, reminding you this means m of q, 00:12:55.560 --> 00:12:59.840 m of minus q, which is the same thing as m star of q, 00:12:59.840 --> 00:13:03.700 if you go through these procedures over here. 00:13:03.700 --> 00:13:05.670 There is 2. 00:13:05.670 --> 00:13:07.838 And this factor of the v actually 00:13:07.838 --> 00:13:10.780 will come up over here. 00:13:10.780 --> 00:13:16.210 So previously, I had used the normalization square root of V, 00:13:16.210 --> 00:13:19.435 and I didn't have this factor of 1 over V. 00:13:19.435 --> 00:13:23.560 Now I have put if there, I will have that factor. 00:13:23.560 --> 00:13:24.060 Yes? 00:13:24.060 --> 00:13:29.097 AUDIENCE: m minus q is star q only if it is the real field, 00:13:29.097 --> 00:13:29.597 right? 00:13:29.597 --> 00:13:30.832 If m is real. 00:13:30.832 --> 00:13:31.860 PROFESSOR: Yes. 00:13:31.860 --> 00:13:34.874 And we are dealing with the field m of q of this. 00:13:34.874 --> 00:13:36.826 AUDIENCE: And in the case of superfluidity? 00:13:36.826 --> 00:13:38.492 PROFESSOR: In the case of superfluidity? 00:13:41.670 --> 00:13:43.868 So let's see. 00:13:43.868 --> 00:13:51.080 So we would have a psi of q integral d dx into the i q dot 00:13:51.080 --> 00:13:56.540 x psi of x. 00:13:56.540 --> 00:13:59.626 If I Fourier transform this, I will 00:13:59.626 --> 00:14:04.919 get a psi star of q integral into the x 00:14:04.919 --> 00:14:10.540 into the minus [INAUDIBLE] x psi star of x. 00:14:10.540 --> 00:14:15.170 So what you are saying is that in the case where psi of x 00:14:15.170 --> 00:14:25.420 is a complex number-- I have psi1 plus ipsi2-- here 00:14:25.420 --> 00:14:26.700 I would have psi1 minus ipsi2. 00:14:29.310 --> 00:14:35.230 So here I would have to make it a statement 00:14:35.230 --> 00:14:39.040 that the real part and the imaginary part 00:14:39.040 --> 00:14:43.760 come when you Fourier transform with an additional minus. 00:14:43.760 --> 00:14:45.530 But let's remember that something 00:14:45.530 --> 00:14:47.786 like this that we are interested is 00:14:47.786 --> 00:14:52.510 psi1 squared plus psi2 squared. 00:14:52.510 --> 00:14:55.664 So ultimately that minus sign did not make any difference. 00:15:00.922 --> 00:15:05.660 But it's good to sort of think of all of these issues. 00:15:05.660 --> 00:15:14.406 And in particular, we are used to thinking of Gaussians, 00:15:14.406 --> 00:15:19.850 where I would have a scalar and then I would have x squared. 00:15:19.850 --> 00:15:23.130 When I have this complex number and I have psi of q, 00:15:23.130 --> 00:15:26.687 psi of minus q, then I have a real part 00:15:26.687 --> 00:15:28.410 squared plus an imaginary part squared. 00:15:31.380 --> 00:15:35.520 And you have to think about whether or not 00:15:35.520 --> 00:15:39.762 you have changed the number of degrees of freedom. 00:15:39.762 --> 00:15:46.100 If you basically integrate over all q's, you may have problems. 00:15:46.100 --> 00:15:51.090 You may have at some point to think about seeing psi of q 00:15:51.090 --> 00:15:54.970 and psi of minus q star are the same thing. 00:15:54.970 --> 00:15:58.780 Maybe you have to integrate over just the positive values. 00:15:58.780 --> 00:16:02.800 But then at each q you will have two different variables, 00:16:02.800 --> 00:16:05.770 which is the real part and the imaginary part. 00:16:05.770 --> 00:16:09.805 So you have to think about all of those doublings and halvings 00:16:09.805 --> 00:16:13.100 that are involved in this statement. 00:16:13.100 --> 00:16:16.280 And in the notes, I have the writeup 00:16:16.280 --> 00:16:19.860 about that that you go and precisely check 00:16:19.860 --> 00:16:22.530 where the factors of one half and two go. 00:16:22.530 --> 00:16:25.063 But ultimately, it looks as if you're 00:16:25.063 --> 00:16:28.170 dealing with a simple scalar quantity. 00:16:28.170 --> 00:16:31.840 So I did not give you that detail explicitly, 00:16:31.840 --> 00:16:35.250 but you can go and check it in the important issue. 00:16:41.140 --> 00:16:44.410 The other term that we have. 00:16:44.410 --> 00:16:47.750 One advantage of this normalization 00:16:47.750 --> 00:16:51.345 is that h multiplies the integral of m 00:16:51.345 --> 00:16:57.920 of x, which is clearly this m with a tail for q equals to 0. 00:16:57.920 --> 00:17:05.069 So that's [INAUDIBLE] mh dotted by this m [INAUDIBLE]. 00:17:10.040 --> 00:17:10.600 Yes? 00:17:10.600 --> 00:17:14.659 AUDIENCE: This is assuming a uniform field? 00:17:14.659 --> 00:17:17.440 PROFESSOR: Yes, that's right. 00:17:17.440 --> 00:17:20.500 So we are thinking about the physics problem, 00:17:20.500 --> 00:17:23.258 but we added the uniform field. 00:17:23.258 --> 00:17:25.862 So if you are for some physical reason 00:17:25.862 --> 00:17:28.792 interested in a position where you 00:17:28.792 --> 00:17:32.102 feel you can modify that, then this would be h of q, 00:17:32.102 --> 00:17:33.131 m of minus q. 00:17:40.140 --> 00:17:47.670 Actually, one reason ultimately to choose this normalization is 00:17:47.670 --> 00:17:54.020 that clearly what appears here is a sum of q. 00:17:54.020 --> 00:18:00.200 If I go over to my integral over q, then the factor of 1 over V 00:18:00.200 --> 00:18:01.110 disappears. 00:18:01.110 --> 00:18:05.560 So that's one reason-- since mostly after this, going 00:18:05.560 --> 00:18:08.670 through the details we'll be dealing with the continuum 00:18:08.670 --> 00:18:13.130 version-- I prefer this normalization. 00:18:13.130 --> 00:18:20.520 And we can now do the Gaussian integrals. 00:18:20.520 --> 00:18:24.880 Basically, there's an overall factor of 1 00:18:24.880 --> 00:18:30.070 over V to the n over 2 for each q mode. 00:18:30.070 --> 00:18:36.980 Then each one of these Gaussian integrals 00:18:36.980 --> 00:18:44.370 will leave me a factor of root 2 pi times the variant. 00:18:44.370 --> 00:18:48.710 So I will get 2 pi. 00:18:48.710 --> 00:18:53.802 The variance is V divided by t plus k 00:18:53.802 --> 00:18:57.246 q squared plus lq to the fourth, and so forth. 00:19:00.200 --> 00:19:04.750 Square root, but there are n components, 00:19:04.750 --> 00:19:08.400 so I will get something like this. 00:19:08.400 --> 00:19:16.080 And then the term that corresponds to q equals to 0 00:19:16.080 --> 00:19:19.725 does not have any of this part. 00:19:19.725 --> 00:19:21.840 So it will give a contribution even 00:19:21.840 --> 00:19:25.370 for q equals to 0 that is like this. 00:19:25.370 --> 00:19:30.880 But you have a term that shifts the center of integration 00:19:30.880 --> 00:19:35.890 from m equals to 0 because of the presence of the field. 00:19:35.890 --> 00:19:40.670 So you will get a term that is exponential of essentially-- 00:19:40.670 --> 00:19:48.980 completing the square-- will give you V divided by 2t times 00:19:48.980 --> 00:19:49.580 h squared. 00:19:57.665 --> 00:20:01.120 Now, clearly the thing that I'm interested 00:20:01.120 --> 00:20:09.050 is log of Z as a function of t and h. 00:20:09.050 --> 00:20:12.700 I'm interested in t and h dependents. 00:20:12.700 --> 00:20:14.672 So there is a bunch of things that 00:20:14.672 --> 00:20:18.608 are constants that I don't really care. 00:20:18.608 --> 00:20:24.890 And then there is a, from here, minus 1/2, actually 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. 00:20:33.190 --> 00:20:37.717 And plus here, I have V a squared over 2t. 00:20:42.690 --> 00:20:45.100 So I can define something that's like 00:20:45.100 --> 00:20:51.425 if the energy from log of Z divided by the volume. 00:20:54.360 --> 00:20:56.870 And you can see that once I replace 00:20:56.870 --> 00:21:00.445 this sum of a q with an integral, 00:21:00.445 --> 00:21:04.370 I will get a factor of volume that I can disregard then. 00:21:04.370 --> 00:21:06.360 So there's some other constant. 00:21:06.360 --> 00:21:15.150 And then I have plus n over 2 integral over q divided by q pi 00:21:15.150 --> 00:21:22.190 to the d log of q plus k q squared, and so forth. 00:21:22.190 --> 00:21:25.212 Minus V k squared divided by 2t. 00:21:33.970 --> 00:21:36.770 Now, again, the question is what's 00:21:36.770 --> 00:21:41.740 the range of q's that I have to integrate, 00:21:41.740 --> 00:21:48.178 given that I'm making things that are coarse grained. 00:21:48.178 --> 00:21:51.890 Now, if I were to really discretize my system 00:21:51.890 --> 00:21:55.020 and, say, put it on q and you plot this, 00:21:55.020 --> 00:21:59.380 then the allowed values of q would leave on 00:21:59.380 --> 00:22:01.660 the [INAUDIBLE] zone. 00:22:01.660 --> 00:22:05.370 [INAUDIBLE] zone, say, in the different directions in q 00:22:05.370 --> 00:22:08.830 would be something like the q that 00:22:08.830 --> 00:22:14.050 would be centered around pi over a. 00:22:14.050 --> 00:22:17.590 But it would be centered at 0, but then you 00:22:17.590 --> 00:22:19.808 would have pi plus pi over a. 00:22:19.808 --> 00:22:21.272 Yes? 00:22:21.272 --> 00:22:23.712 AUDIENCE: The d would disappear, right? 00:22:23.712 --> 00:22:26.923 PROFESSOR: The d would disappear because I divided by it. 00:22:31.640 --> 00:22:37.970 So in principle, if I had done the discretization 00:22:37.970 --> 00:22:45.030 to a cube and plot this, I would have been integrating over q 00:22:45.030 --> 00:22:49.840 that this would find to a cube like this. 00:22:49.840 --> 00:22:51.670 But maybe I chose some other lattice 00:22:51.670 --> 00:22:55.900 like a diamond lattice, et cetera. 00:22:55.900 --> 00:22:58.862 Then the shape of this thing would change. 00:22:58.862 --> 00:23:01.495 But what's the meaning of doing the whole thing on a lattice 00:23:01.495 --> 00:23:03.590 anyway? 00:23:03.590 --> 00:23:06.200 The thing that I want to do is to make sure 00:23:06.200 --> 00:23:10.400 that I have done some averaging in order to remove 00:23:10.400 --> 00:23:13.260 short wavelength fluctuations. 00:23:13.260 --> 00:23:19.460 So a much more natural way to do that averaging and removing 00:23:19.460 --> 00:23:22.550 short wavelength operations is to say 00:23:22.550 --> 00:23:31.770 that my field has only Fourier components that 00:23:31.770 --> 00:23:36.706 are from 0 to some maximum value of lambda, which 00:23:36.706 --> 00:23:40.080 is the inverse of some radiant. 00:23:40.080 --> 00:23:43.400 And if you are worried about the difference in integration 00:23:43.400 --> 00:23:46.470 between doing things on this nice mirror that 00:23:46.470 --> 00:23:50.930 has nice symmetry and maybe doing it on a cube, 00:23:50.930 --> 00:23:53.292 then the difference is essentially 00:23:53.292 --> 00:23:59.370 the bit of integration that you would have to do over here. 00:23:59.370 --> 00:24:02.620 But the function that you are integrating 00:24:02.620 --> 00:24:07.930 his no singularities for large values of q. 00:24:07.930 --> 00:24:10.020 You are interested in the singularities 00:24:10.020 --> 00:24:13.480 of the function when t goes to 0. 00:24:13.480 --> 00:24:15.520 And then the log has singularities 00:24:15.520 --> 00:24:18.470 when its argument goes to 0. 00:24:18.470 --> 00:24:20.740 So I should be interested, as far as singularities 00:24:20.740 --> 00:24:27.120 are concerned, only in the vicinity of this point anyway. 00:24:27.120 --> 00:24:30.870 What I do out there, whether I replace the sphere 00:24:30.870 --> 00:24:33.942 with the cube or et cetera, will add 00:24:33.942 --> 00:24:36.616 some other non-singular term over here, 00:24:36.616 --> 00:24:37.699 which I don't really care. 00:24:41.590 --> 00:24:43.950 Actually, if I do that, this non-singular term 00:24:43.950 --> 00:24:46.840 here could be actually functions of t. 00:24:46.840 --> 00:24:50.110 But they would be very perfect and regular functions of t. 00:24:50.110 --> 00:24:52.070 Like constant plus alpha t, plus pheta q 00:24:52.070 --> 00:24:57.190 squared, et cetera, that have no singularities. 00:24:57.190 --> 00:25:01.730 So if I'm interested in singularities, 00:25:01.730 --> 00:25:03.342 I am going to be focused on that. 00:25:06.000 --> 00:25:11.900 Now actually, we encountered this integral 00:25:11.900 --> 00:25:16.745 before when we were looking at corrections 00:25:16.745 --> 00:25:19.756 to the saddle-point approximation. 00:25:19.756 --> 00:25:23.830 And if you remember what we did then was to take, 00:25:23.830 --> 00:25:27.100 let's say, C of d of h across 0 while 00:25:27.100 --> 00:25:33.720 taking two derivatives of this free energy with respect to t. 00:25:33.720 --> 00:25:38.056 And then we ended up with an integral. 00:25:38.056 --> 00:25:40.900 There's a minus sign here over d. 00:25:40.900 --> 00:25:45.980 n over 2 integral dt 2 pi squared. 00:25:45.980 --> 00:25:48.380 2 pi to the d. 00:25:48.380 --> 00:25:51.040 Taking two derivatives of the log. 00:25:51.040 --> 00:25:53.610 The first derivative will give me 1 over the argument. 00:25:53.610 --> 00:25:56.910 The second derivative will give me 1 over the argument squared. 00:25:56.910 --> 00:25:58.878 One side take care of the minus sign. 00:26:18.558 --> 00:26:24.020 Now, I think this is a kind of integral, 00:26:24.020 --> 00:26:30.335 after I have focused on the singular part, 00:26:30.335 --> 00:26:34.736 that I can replace when integrating over a sphere. 00:26:39.640 --> 00:26:43.230 Now, when I integrate over a sphere, 00:26:43.230 --> 00:26:47.740 I may be concerned about what's going on at small values. 00:26:47.740 --> 00:26:51.350 At q, at small values of q, as long as t is around, 00:26:51.350 --> 00:26:53.120 I have no problem. 00:26:53.120 --> 00:26:55.380 When t goes to 0, I will have to worry 00:26:55.380 --> 00:26:59.080 about the singularity that comes from 1 over k, 00:26:59.080 --> 00:27:00.690 2 squared, et cetera. 00:27:00.690 --> 00:27:04.510 So that's really the singularity that I'm interested in. 00:27:04.510 --> 00:27:07.150 Exactly what happens at large q, I'm 00:27:07.150 --> 00:27:11.360 not really all that interested in. 00:27:11.360 --> 00:27:20.050 And in particular, what I can do is I can rescale things. 00:27:20.050 --> 00:27:27.730 I can call q squared over t to the x squared. 00:27:27.730 --> 00:27:35.060 So I can essentially make that change over there. 00:27:35.060 --> 00:27:40.890 So that whenever I see a factor of q, 00:27:40.890 --> 00:27:44.330 I replace it with t over k to the 1/2 x. 00:27:49.940 --> 00:27:51.417 What happens here? 00:27:51.417 --> 00:27:54.970 I have, first of all, n over 2. 00:27:54.970 --> 00:27:57.768 I have 1 over 2 pi to the d. 00:28:01.680 --> 00:28:09.340 Writing this in terms of spherical symmetry, 00:28:09.340 --> 00:28:13.670 I will have the solid angle d dimensions. 00:28:13.670 --> 00:28:18.510 And then I would have q to d minus 1 q. 00:28:18.510 --> 00:28:23.530 Every time I put a factor of q, I can replace it with this. 00:28:23.530 --> 00:28:30.130 So I would have a t over k with a power of q/2. 00:28:30.130 --> 00:28:34.960 And then I have my integral that becomes the x, 00:28:34.960 --> 00:28:41.230 x to the d minus 1, 1 plus x squared plus potentially 00:28:41.230 --> 00:28:43.130 higher order things like this. 00:28:49.790 --> 00:28:56.680 Now, the upper cut-off for x is in fact 00:28:56.680 --> 00:29:03.430 square root k over t times lambda. 00:29:03.430 --> 00:29:10.910 And we are interested in the limit of when t goes to 0. 00:29:10.910 --> 00:29:17.100 So that upper limit is essentially going to infinity. 00:29:17.100 --> 00:29:20.630 Now, whether or not this integral, 00:29:20.630 --> 00:29:24.140 if I learn to ignore higher order terms 00:29:24.140 --> 00:29:28.890 and focus on the first term, exists really 00:29:28.890 --> 00:29:36.950 depends on whether d is larger, d minus 1 plus 1 d minus 4 00:29:36.950 --> 00:29:40.050 is positive or negative. 00:29:40.050 --> 00:29:44.810 And in particular, if I learn to get rid 00:29:44.810 --> 00:29:47.830 of all those higher order terms. 00:29:47.830 --> 00:29:51.170 And basically, the argument for that 00:29:51.170 --> 00:29:54.370 is the things that would go with x to the fourth, et cetera, 00:29:54.370 --> 00:29:57.090 if we carry additional factors of t-- 00:29:57.090 --> 00:30:01.598 and hopefully getting rid of them as to go to 0-- 00:30:01.598 --> 00:30:03.970 will give me an integral like this. 00:30:03.970 --> 00:30:08.562 This will exist only if I am in dimensions 00:30:08.562 --> 00:30:11.540 d that is less than 4. 00:30:11.540 --> 00:30:12.175 Yes? 00:30:12.175 --> 00:30:13.925 AUDIENCE: Are you missing the factors of t 00:30:13.925 --> 00:30:15.575 over t that comes with the denominator? 00:30:18.796 --> 00:30:20.834 PROFESSOR: Yes. 00:30:20.834 --> 00:30:23.738 There is a factor of 1 over t here. 00:30:29.550 --> 00:30:33.500 So I have to put out the factor of t. 00:30:33.500 --> 00:30:37.756 Write this as 1 plus k over t plus the element 00:30:37.756 --> 00:30:39.492 of t, et cetera. 00:30:39.492 --> 00:30:41.972 So there is a factor of 1 over t. 00:30:41.972 --> 00:30:43.460 AUDIENCE: t squared. 00:30:43.460 --> 00:30:45.196 PROFESSOR: And that's a factor of t 00:30:45.196 --> 00:30:46.932 squared, because that's two powers. 00:30:56.870 --> 00:31:02.385 So if I'm in dimensions d less than 4, what I can write 00:31:02.385 --> 00:31:09.530 is that this c singular, this as t goes to 0. 00:31:09.530 --> 00:31:13.550 The leading behavior, this goes to the constant. 00:31:13.550 --> 00:31:17.347 So as we discussed, after all of the mistakes that I made, 00:31:17.347 --> 00:31:22.204 there will be some overall coefficient A. The power of t 00:31:22.204 --> 00:31:25.400 will be d over 2 minus 2. 00:31:25.400 --> 00:31:27.700 d over 2 came from the integrations. 00:31:27.700 --> 00:31:32.310 1 over t squared came from the denominator. 00:31:32.310 --> 00:31:36.405 And then if I were to expand all of these other terms 00:31:36.405 --> 00:31:39.590 that we've ignored, higher powers of-- here 00:31:39.590 --> 00:31:43.660 I will get various series that will correct this. 00:31:43.660 --> 00:31:48.840 But the leading key dependents in dimensions less than 4 00:31:48.840 --> 00:31:51.032 is this thing that we had seen previously. 00:31:54.476 --> 00:31:58.340 Now I can take this, and you see that in dimensions 00:31:58.340 --> 00:32:02.294 d less than 4, this is a singular 00:32:02.294 --> 00:32:04.130 term that is diversion. 00:32:04.130 --> 00:32:08.100 If I were to say what kind of thing 00:32:08.100 --> 00:32:12.330 was of the energy that gave result to this? 00:32:12.330 --> 00:32:16.480 Then it would say that if the energy must 00:32:16.480 --> 00:32:19.737 have had some other constant that was proportionate of the t 00:32:19.737 --> 00:32:25.500 to the d over 2, that when I put two derivatives, 00:32:25.500 --> 00:32:27.920 I got something like this. 00:32:27.920 --> 00:32:29.790 Of course, if the energy could also 00:32:29.790 --> 00:32:33.940 have had a term that was linear in t, I wouldn't have seen it. 00:32:36.770 --> 00:32:39.170 So there is a singular part. 00:32:39.170 --> 00:32:43.095 Essentially, if I were to do that integral in dimensions 00:32:43.095 --> 00:32:47.690 less than fourth, I will get a leading singularity 00:32:47.690 --> 00:32:49.779 that is applied. 00:32:49.779 --> 00:32:51.570 I will get a singularity that is like this. 00:32:51.570 --> 00:32:54.520 I will get additional terms per constant-- t, t squared, 00:32:54.520 --> 00:32:58.482 et cetera-- and singular terms that 00:32:58.482 --> 00:32:59.880 are subbing in to this one. 00:33:03.394 --> 00:33:04.810 And then, of course, I have a term 00:33:04.810 --> 00:33:12.856 that is minus h squared over t if I were to include this here. 00:33:12.856 --> 00:33:23.783 So why don't I write the answer as B minus h divided by t 00:33:23.783 --> 00:33:29.039 to the 1/2 plus d/4, the whole thing squared. 00:33:33.830 --> 00:33:37.470 So what I did was essentially I divided and multiplied 00:33:37.470 --> 00:33:42.600 by inputting d and put the whole thing in the form of h divided 00:33:42.600 --> 00:33:46.410 by t to something squared? 00:33:46.410 --> 00:33:48.030 Why did I do that? 00:33:48.030 --> 00:33:53.425 It's because we had first related a singular form 00:33:53.425 --> 00:33:59.120 for the energies in the scaling picture that had the E to the 2 00:33:59.120 --> 00:34:03.818 minus alpha in front of them and the function of h t 00:34:03.818 --> 00:34:05.590 to the delta. 00:34:05.590 --> 00:34:11.960 And all I wanted to emphasize is that this picture, 2 minus 00:34:11.960 --> 00:34:16.860 alpha is d over 2. 00:34:16.860 --> 00:34:20.000 And the thing that we call the gap exponent 00:34:20.000 --> 00:34:22.034 is 1/2 plus d over 4. 00:34:30.630 --> 00:34:32.840 Of course, I can't use this theory 00:34:32.840 --> 00:34:36.060 as a description of the case. 00:34:36.060 --> 00:34:41.659 And the reason for that is that the Gaussian theory 00:34:41.659 --> 00:34:45.109 exists and is well-defined only as long as t is positive. 00:34:49.989 --> 00:34:57.310 Because once t becomes negative, then the rate 00:34:57.310 --> 00:35:01.250 essentially becomes ill-defined. 00:35:01.250 --> 00:35:05.080 Because if I look at the various rates that I have here, 00:35:05.080 --> 00:35:08.550 we certainly-- the rate for q equals to 0. 00:35:08.550 --> 00:35:13.700 It is proportional to minus t over 2v. 00:35:13.700 --> 00:35:16.670 If the t changes sign, rather than having a Gaussian, 00:35:16.670 --> 00:35:21.900 I have essentially a rate that is maximized as [INAUDIBLE]. 00:35:21.900 --> 00:35:26.420 So clearly, again, by issue of stability, 00:35:26.420 --> 00:35:30.940 the theory for t negative does not describe a stable theory. 00:35:30.940 --> 00:35:33.740 And that's why n to the fourth and all of those terms 00:35:33.740 --> 00:35:36.735 will be necessary to describe that side of the phase 00:35:36.735 --> 00:35:37.780 transition. 00:35:37.780 --> 00:35:42.920 So if you like, this is a kind of a description 00:35:42.920 --> 00:35:51.700 of a singularity that exists only in this half of the space. 00:35:51.700 --> 00:35:55.250 Kind of reminiscent of coming from the disordered side, 00:35:55.250 --> 00:36:00.730 but I don't want to give it more reality than that. 00:36:00.730 --> 00:36:02.010 It's a mathematical construct. 00:36:02.010 --> 00:36:05.260 If we want to venture to make the connection 00:36:05.260 --> 00:36:07.646 to the actual phase transition, we 00:36:07.646 --> 00:36:09.329 have to prove the n to the fourth. 00:36:19.600 --> 00:36:24.410 Now, the only reason to go and recap this Gaussian theory 00:36:24.410 --> 00:36:30.100 is because since it is solve-able, 00:36:30.100 --> 00:36:32.710 we can try to use it as a toy model 00:36:32.710 --> 00:36:37.350 to apply the various steps of renormalization group 00:36:37.350 --> 00:36:39.980 that we had outlined last lecture. 00:36:39.980 --> 00:36:45.580 And once we understand the steps of renormalization group 00:36:45.580 --> 00:36:50.590 for this theory, then it gives us an anchoring point 00:36:50.590 --> 00:36:52.650 when we describe the full theory that 00:36:52.650 --> 00:36:55.930 has n to the fourth, et cetera-- how 00:36:55.930 --> 00:36:58.687 to sort of start with the renormalization approach 00:36:58.687 --> 00:37:04.316 to the theory as we understand and do the more complicated. 00:37:04.316 --> 00:37:08.060 So essentially, as I said, it's not really a phase transition 00:37:08.060 --> 00:37:10.980 that can be described by this theory. 00:37:10.980 --> 00:37:14.310 It's a singularity. 00:37:14.310 --> 00:37:18.910 But its value is that it is this fully-modelled anchoring 00:37:18.910 --> 00:37:22.407 point for the full theory that we are describing. 00:37:27.397 --> 00:37:32.886 So what we want to do is to do an RG for the Gaussian model. 00:37:48.890 --> 00:37:51.260 So what is the procedure. 00:37:55.145 --> 00:38:00.940 We have a theory best described in the space 00:38:00.940 --> 00:38:04.330 of variables q, the Fourier variables. 00:38:04.330 --> 00:38:11.967 Where I have modes that exist between 0-- very long 00:38:11.967 --> 00:38:16.075 wavelength-- to lambda, which is the inverse of some shortest 00:38:16.075 --> 00:38:19.456 wavelength that I'm allowing. 00:38:19.456 --> 00:38:27.400 And so basically, I have a bunch of modes m of q 00:38:27.400 --> 00:38:29.910 that are defined in this range of qx. 00:38:32.790 --> 00:38:39.005 The first step of RG was to coarse grain. 00:38:43.616 --> 00:38:50.225 The idea of coarse graining was to change the scale over which 00:38:50.225 --> 00:38:56.330 you were doing the averaging from some a to ba. 00:38:56.330 --> 00:39:04.510 So average from a to ba of fluctuations. 00:39:07.090 --> 00:39:11.940 So once I do that, at the end of the day 00:39:11.940 --> 00:39:16.370 I have fluctuations whose minimum wavelength has 00:39:16.370 --> 00:39:18.175 gone from a to ba. 00:39:21.320 --> 00:39:29.750 So that means that q max, after I go and do this procedure, 00:39:29.750 --> 00:39:37.956 is the previous q max that I had divided by a factor of b. 00:39:37.956 --> 00:39:45.480 So basically, at the end of the day I want to have, 00:39:45.480 --> 00:39:51.530 after coarse graining, variables that only exist up 00:39:51.530 --> 00:39:55.468 to lambda over b. 00:39:55.468 --> 00:39:59.896 Whereas previously, they existed after that. 00:40:02.850 --> 00:40:07.330 So this is very easy at this level. 00:40:07.330 --> 00:40:12.800 All I can do is to replace this m tilde of q 00:40:12.800 --> 00:40:15.520 in terms of two sets. 00:40:15.520 --> 00:40:22.400 I will call it to be sigma if q is 00:40:22.400 --> 00:40:27.840 greater than this lambda over b. 00:40:27.840 --> 00:40:33.350 That is, everybody that is out here, their q-- 00:40:33.350 --> 00:40:36.100 I will call it q larger. 00:40:36.100 --> 00:40:41.400 Everybody that is here, their q I will call q lesser. 00:40:41.400 --> 00:40:44.620 And all the modes that were here, 00:40:44.620 --> 00:40:46.810 I will give them a different name. 00:40:46.810 --> 00:40:50.320 The ones here I will call sigma. 00:40:50.320 --> 00:41:03.145 The ones here, if q less than lambda over b, 00:41:03.145 --> 00:41:06.620 will get called m tilde. 00:41:06.620 --> 00:41:09.620 So I just renamed my variables. 00:41:09.620 --> 00:41:17.740 So essentially, right here I had integration 00:41:17.740 --> 00:41:22.120 over all of the modes. 00:41:22.120 --> 00:41:26.403 I just renamed some of the modes that were inside 00:41:26.403 --> 00:41:32.040 q lesser and sigma-- and tilde, the ones that 00:41:32.040 --> 00:41:35.600 are outside q greater. 00:41:35.600 --> 00:41:40.570 So what I have to do for my Gaussian theory. 00:41:40.570 --> 00:41:44.350 Let's write it rather than in this form 00:41:44.350 --> 00:41:48.574 that was discrete in terms of the continuum. 00:41:51.360 --> 00:41:55.370 I have to iterate over all configurations of these Fourier 00:41:55.370 --> 00:41:56.720 modes. 00:41:56.720 --> 00:41:58.793 So I have these m tilde of q's. 00:42:02.950 --> 00:42:05.921 And the wave that I have to assign to them when 00:42:05.921 --> 00:42:11.580 I look at the continuum is exponential, integral in dq 00:42:11.580 --> 00:42:12.760 q to pi to the d. 00:42:12.760 --> 00:42:16.736 T plus kq squared, and so forth. 00:42:19.718 --> 00:42:22.203 And tilde of q squared. 00:42:26.180 --> 00:42:33.554 And then I had the one term that was hm of 0. 00:42:38.790 --> 00:42:43.030 What I have done is to simply rewrite 00:42:43.030 --> 00:42:50.704 this as two sets of integrations over the-- whoops. 00:42:50.704 --> 00:42:52.135 This was m. 00:42:55.951 --> 00:43:01.450 m, let's call is sigma first-- sigma of q 00:43:01.450 --> 00:43:06.692 larger integrate over m tilde of q lesser. 00:43:13.664 --> 00:43:18.670 And actually, you can see that the modes here 00:43:18.670 --> 00:43:21.950 and the modes here don't talk to each other. 00:43:21.950 --> 00:43:27.000 And that's really the advantage of doing the Gaussian theory. 00:43:27.000 --> 00:43:29.930 And the thing that allowed me to solve the problem here 00:43:29.930 --> 00:43:32.860 and also to do the coarse graining there. 00:43:32.860 --> 00:43:36.560 Once we do things like n to the fourth, 00:43:36.560 --> 00:43:38.856 then I will have couplings between modes 00:43:38.856 --> 00:43:42.410 that go across between the three sets. 00:43:42.410 --> 00:43:44.846 And then the problem becomes difficult. 00:43:44.846 --> 00:43:48.100 But now that I don't have that, I 00:43:48.100 --> 00:43:52.085 can actually separately write the integral as two parts. 00:44:03.700 --> 00:44:07.000 And this is for q lesser. 00:44:07.000 --> 00:44:11.741 And for each one of them, I essentially have the same rate. 00:44:11.741 --> 00:44:19.240 The integral over q greater goes between lambda 00:44:19.240 --> 00:44:20.818 over d and lambda. 00:44:36.626 --> 00:44:40.578 The integral over m tilde of q lesser 00:44:40.578 --> 00:44:43.048 is essentially the same thing. 00:44:51.446 --> 00:44:58.160 Exponential minus integral 0 to lambda over d. dv 00:44:58.160 --> 00:45:05.650 q lesser to five to the d, t plus kq lesser squared, 00:45:05.650 --> 00:45:08.720 and so forth. 00:45:08.720 --> 00:45:14.406 And q lesser squared. 00:45:14.406 --> 00:45:22.448 And then I have the additional term which sits at 0. 00:45:22.448 --> 00:45:31.760 It is part of the modes that are assigned with q lesser. 00:45:36.093 --> 00:45:36.593 OK? 00:45:36.593 --> 00:45:37.620 Fine. 00:45:37.620 --> 00:45:40.590 Nothing particularly profound here. 00:45:40.590 --> 00:45:42.090 In fact, it's very simple. 00:45:42.090 --> 00:45:46.040 It's just renaming two sets of modes. 00:45:46.040 --> 00:45:51.420 And the averaging that I have to do, 00:45:51.420 --> 00:45:53.764 and getting rid of the fluctuations 00:45:53.764 --> 00:45:59.250 at short wavelength, here is very trickier. 00:45:59.250 --> 00:46:03.938 Because this is just a bunch of integrations 00:46:03.938 --> 00:46:09.380 that I had to do over here, but it is only over things 00:46:09.380 --> 00:46:12.826 that are sitting close to the edge of this [INAUDIBLE]. 00:46:16.810 --> 00:46:21.292 So essentially, the integrations over these modes 00:46:21.292 --> 00:46:24.280 is doing this integral over here, 00:46:24.280 --> 00:46:30.260 from lambda over d to lambda, and none of the singularities 00:46:30.260 --> 00:46:34.020 has anything to do with the range of integration 00:46:34.020 --> 00:46:37.020 from lambda over d to lambda. 00:46:37.020 --> 00:46:40.020 So the result of doing all of that 00:46:40.020 --> 00:46:45.206 is simply just a constant-- but not a constant. 00:46:45.206 --> 00:46:51.220 It's a function of t that is completely non-singular 00:46:51.220 --> 00:46:55.690 and have a nice state of expansion powers of t. 00:46:55.690 --> 00:47:01.080 A kind of [INAUDIBLE] I call non-singular functions 00:47:01.080 --> 00:47:01.660 sometimes. 00:47:01.660 --> 00:47:03.539 Constant thing is that eventually 00:47:03.539 --> 00:47:05.288 if you take sufficiently high derivatives, 00:47:05.288 --> 00:47:10.900 I guess, of this value, the t dependents [INAUDIBLE]. 00:47:16.250 --> 00:47:24.550 So all of the interesting thing is really in this m tilde of k 00:47:24.550 --> 00:47:25.800 lesser. 00:47:25.800 --> 00:47:31.220 And really, the eventual process of renormalization 00:47:31.220 --> 00:47:35.400 in this picture is something like this. 00:47:35.400 --> 00:47:37.330 That all of the singularities are 00:47:37.330 --> 00:47:43.310 sitting at the center of this kind of orange-shaped entity. 00:47:43.310 --> 00:47:46.115 And rather than biting the whole thing, 00:47:46.115 --> 00:47:51.390 you kind of cut it slowly and slowly from the edge, 00:47:51.390 --> 00:47:54.610 approaching to where all of the exciting things 00:47:54.610 --> 00:47:55.680 are at the center. 00:47:55.680 --> 00:47:57.550 For this problem of the Gaussian, 00:47:57.550 --> 00:48:00.440 it turns out to be trivial to do so. 00:48:00.440 --> 00:48:02.980 But for the more general problem, 00:48:02.980 --> 00:48:06.570 it can be interesting because procedure is the same. 00:48:06.570 --> 00:48:09.030 We are interested in what's happening here, 00:48:09.030 --> 00:48:13.450 but we gradually peel of things that we 00:48:13.450 --> 00:48:17.540 know don't cause anything difficult for the problems. 00:48:21.540 --> 00:48:24.600 So then I have to multiply with this, 00:48:24.600 --> 00:48:31.440 and I have found in some sense a probability for configurations 00:48:31.440 --> 00:48:35.490 of the coarse grain system, which is simply given by this. 00:48:38.100 --> 00:48:42.842 But then renormalization group has two other steps. 00:48:45.690 --> 00:48:50.520 The second step was to say, well, in real space, 00:48:50.520 --> 00:48:53.870 as we said, the picture that is represented 00:48:53.870 --> 00:48:57.730 by these coarse grain variables is grainy. 00:48:57.730 --> 00:49:01.410 If my pixels were previously one by one by one, 00:49:01.410 --> 00:49:04.746 now my pixels are d by d by d. 00:49:04.746 --> 00:49:10.380 So I can make my picture look to have the same resolution 00:49:10.380 --> 00:49:14.630 as my initial picture if I rescale 00:49:14.630 --> 00:49:16.590 all of the events for a factor of t. 00:49:19.180 --> 00:49:23.310 In momentum representation, or intuitive presentation, 00:49:23.310 --> 00:49:33.160 it corresponds to rescaling all of the q's by a factor of B. 00:49:33.160 --> 00:49:36.640 And clearly, what that serves to achieve 00:49:36.640 --> 00:49:44.020 is that if I replace q lesser with B times q prime, 00:49:44.020 --> 00:49:50.820 then the maximum value will go back to 0 to lambda. 00:49:50.820 --> 00:49:53.350 So by doing this one in formation, 00:49:53.350 --> 00:50:00.514 I can ensure that the upper cut-off is, in fact, lambda 00:50:00.514 --> 00:50:03.240 again. 00:50:03.240 --> 00:50:07.160 Now, there was another thing, which in real space 00:50:07.160 --> 00:50:15.330 we said that we defined m prime to be m tilde rescaled 00:50:15.330 --> 00:50:16.819 by some factor zeta. 00:50:21.130 --> 00:50:24.990 I had to do a change of the contrast. 00:50:24.990 --> 00:50:27.590 I did have to do the same change of contrast 00:50:27.590 --> 00:50:32.200 here, except that the variables that I am dealing with here, 00:50:32.200 --> 00:50:36.470 it was in x coordinates. 00:50:36.470 --> 00:50:40.570 What I want to do it is in the q coordinate. 00:50:40.570 --> 00:50:49.884 So I will call m with a tail prime of q prime 00:50:49.884 --> 00:50:59.510 to be m tilde of q prime by a factor of z. 00:50:59.510 --> 00:51:03.200 The difference between the z and the zeta, which 00:51:03.200 --> 00:51:06.280 is real space and Fourier space is just the fact 00:51:06.280 --> 00:51:11.210 that in going from one to the other, 00:51:11.210 --> 00:51:14.280 you have to do integrations over space. 00:51:14.280 --> 00:51:17.070 So dimensionally, there is a factor of b 00:51:17.070 --> 00:51:20.670 to the d difference between the rescaling of this quantity 00:51:20.670 --> 00:51:26.665 and that quantity, and if you want to use 00:51:26.665 --> 00:51:32.180 or the other zeta against b to the minus d and z. 00:51:32.180 --> 00:51:36.250 But since we would be doing everything in Fourier space, 00:51:36.250 --> 00:51:38.500 we would just use this factor traditionally. 00:51:42.950 --> 00:51:46.440 So if I do that, what do I find? 00:51:46.440 --> 00:51:57.430 I find that Z of t of h is exponential of some singular, 00:51:57.430 --> 00:52:00.700 non-singular dependents. 00:52:00.700 --> 00:52:08.550 And then I have to integrate over these new variables, 00:52:08.550 --> 00:52:11.940 m prime of q prime. 00:52:11.940 --> 00:52:13.740 Yes? 00:52:13.740 --> 00:52:17.270 AUDIENCE: In your real space renormalization 00:52:17.270 --> 00:52:21.769 your m tilde is a function of an x. 00:52:21.769 --> 00:52:23.435 But in your Fourier space representation 00:52:23.435 --> 00:52:28.885 your m tilde is a function of q prime? 00:52:28.885 --> 00:52:32.053 PROFESSOR: I guess I could have written here x prime, also. 00:52:32.053 --> 00:52:33.094 It doesn't really matter. 00:52:42.860 --> 00:52:44.150 So do you have here? 00:52:44.150 --> 00:52:49.880 You have exponential minus the integral. 00:52:49.880 --> 00:52:56.820 The integration for q prime now is going back to 0 to lambda. 00:52:56.820 --> 00:53:03.390 I have db of q prime divided by 2 pi to the d. 00:53:09.150 --> 00:53:19.490 Now, you see that every time I have a q-- V or q, q lesser, 00:53:19.490 --> 00:53:24.250 in fact-- I have to go to q prime 00:53:24.250 --> 00:53:28.592 by introducing a factor of the inverse. 00:53:28.592 --> 00:53:33.440 So there will be a total factor of V to the minus V 00:53:33.440 --> 00:53:37.246 that comes from this integration. 00:53:37.246 --> 00:53:40.000 And that will multiply t. 00:53:40.000 --> 00:53:45.590 That will multiply kb to the minus d. 00:53:45.590 --> 00:53:51.650 But then here I have to q's because of the q squared there. 00:53:51.650 --> 00:53:58.940 Again, doing the same thing, I will get V to the d minus two. 00:53:58.940 --> 00:54:02.790 I had q plus 2, if you like. 00:54:02.790 --> 00:54:09.195 And then the next l would be lb to the d minus 4. 00:54:09.195 --> 00:54:12.130 And you can see that as I have higher and higher derivatives 00:54:12.130 --> 00:54:15.850 of q, I get higher and higher powers with negative 00:54:15.850 --> 00:54:16.350 [INAUDIBLE]. 00:54:21.060 --> 00:54:28.760 But then I have m tilde that I want to replace with m prime. 00:54:28.760 --> 00:54:34.090 And that process will give me a factor of z squared. 00:54:34.090 --> 00:54:39.340 And then I have m prime of q prime squared. 00:54:43.225 --> 00:54:46.410 There is no integration for this terms. 00:54:46.410 --> 00:54:48.260 It's just one mode. 00:54:48.260 --> 00:54:53.460 But each mode I have rescaled by a factor of z. 00:54:53.460 --> 00:55:03.262 So I will have a term that is z h dot m prime of 0. 00:55:13.690 --> 00:55:17.470 So what we see is that what we have managed to do 00:55:17.470 --> 00:55:21.590 is to make the Gaussian integration over 00:55:21.590 --> 00:55:26.410 here precisely the same thing as the Gaussian integration 00:55:26.410 --> 00:55:28.650 that I started with. 00:55:28.650 --> 00:55:33.486 So I can conclude that this function tnh 00:55:33.486 --> 00:55:39.090 that I am interested in has a path that is non-singular. 00:55:42.780 --> 00:55:45.670 But its singular part is the same 00:55:45.670 --> 00:55:52.427 as the same z calculated for a bunch of new parameters. 00:55:55.840 --> 00:56:02.250 And in particular, the new t is v to the minus 00:56:02.250 --> 00:56:07.755 d z squared the old t. 00:56:07.755 --> 00:56:15.320 The new k is b to the minus d minus 2 z squared q. 00:56:15.320 --> 00:56:21.515 The new L would be to the minus d minus 4 z squared L, 00:56:21.515 --> 00:56:23.500 and so forth. 00:56:23.500 --> 00:56:27.858 And the new h is zh. 00:56:27.858 --> 00:56:28.792 Yes? 00:56:28.792 --> 00:56:33.465 AUDIENCE: There should be q prime squared and q prime 4? 00:56:33.465 --> 00:56:34.090 PROFESSOR: Yes. 00:56:49.342 --> 00:56:50.326 Yes. 00:56:50.326 --> 00:56:53.770 This is my day to do a lot of algebraic errors. 00:57:00.440 --> 00:57:00.940 OK. 00:57:05.180 --> 00:57:07.630 So what is the change in parameters? 00:57:07.630 --> 00:57:10.970 So I wrote it over there. 00:57:10.970 --> 00:57:17.610 So this kind of captures the very simplest type 00:57:17.610 --> 00:57:20.160 of renormalization. 00:57:20.160 --> 00:57:24.510 Actually, all I did was a scaling analysis. 00:57:24.510 --> 00:57:29.260 If I were to change positions by a factor of b 00:57:29.260 --> 00:57:34.460 and change the magnitude of my field m by a factor z or zeta, 00:57:34.460 --> 00:57:37.302 this is the kind of results that I will get. 00:57:40.470 --> 00:57:46.500 Now, how can we make this capture the kind of picture 00:57:46.500 --> 00:57:52.190 that we have over here in the language of renormalization? 00:57:52.190 --> 00:57:57.830 Want to be able to change two parameters 00:57:57.830 --> 00:57:59.918 and reach a fixed point. 00:58:03.590 --> 00:58:09.940 So we know that kind of [INAUDIBLE] that t and h 00:58:09.940 --> 00:58:11.020 have to go to 0. 00:58:11.020 --> 00:58:15.910 They are the variables that determine essentially 00:58:15.910 --> 00:58:19.260 whether you are at this said similar point. 00:58:19.260 --> 00:58:24.460 So if t and h I forget, the next most important term that 00:58:24.460 --> 00:58:29.850 comes into play is k prime, which is some function of k. 00:58:29.850 --> 00:58:32.680 And if I want to be at the fixed point, 00:58:32.680 --> 00:58:36.540 I may want to choose the factor z such 00:58:36.540 --> 00:58:40.960 that k prime is the same as k. 00:58:40.960 --> 00:58:50.390 So choose z such that k prime is k. 00:58:50.390 --> 00:58:54.346 And that tells me immediately that z 00:58:54.346 --> 00:58:57.552 would be b to the power of 1 plus d over 2. 00:59:02.760 --> 00:59:09.756 If I choose that particular form of z, then what do I get? 00:59:09.756 --> 00:59:15.132 I get t prime is z squared b to the minus b. 00:59:15.132 --> 00:59:20.600 So when I do that, I will get b squared t. 00:59:20.600 --> 00:59:27.780 I get that h prime is just z times h. 00:59:27.780 --> 00:59:33.280 So it is b to the 1 plus b over 2 times h. 00:59:33.280 --> 00:59:41.291 These are both directions that as b becomes larger than 1, 00:59:41.291 --> 00:59:45.970 b prime becomes larger than th prime, becomes larger than h. 00:59:45.970 --> 00:59:48.040 These are relevant directions. 00:59:48.040 --> 00:59:53.230 I would associate with them eigenvalues y dt minus 2. 00:59:53.230 --> 00:59:54.316 Divide h. 00:59:54.316 --> 00:59:56.696 That is 1 plus d over 2. 01:00:00.510 --> 01:00:07.300 So if I go according to the scaling construction that we 01:00:07.300 --> 01:00:14.310 had before, f singular of tnh is t to the power 01:00:14.310 --> 01:00:19.470 d over y dt, some scaling function of h, 01:00:19.470 --> 01:00:24.640 g to the power of divide h over y dt. 01:00:24.640 --> 01:00:27.480 This is what we have established before. 01:00:27.480 --> 01:00:32.430 With these values I will get t to the d over 2, 01:00:32.430 --> 01:00:41.888 some scaling function of h, t to the power of 1/2 plus d over 4. 01:00:44.660 --> 01:00:51.700 We can immediately compare this expression and this expression 01:00:51.700 --> 01:00:52.670 that we have over here. 01:00:52.670 --> 01:00:52.915 Yes? 01:00:52.915 --> 01:00:53.540 AUDIENCE: Wait. 01:00:53.540 --> 01:00:54.970 What's the reason to choose scale 01:00:54.970 --> 01:00:57.011 as the parameter that maps onto itself and not L? 01:00:57.011 --> 01:00:57.670 PROFESSOR: OK. 01:00:57.670 --> 01:00:59.310 I'll come to that. 01:00:59.310 --> 01:01:05.140 So having gone this far, let's see what l is doing. 01:01:05.140 --> 01:01:07.730 So if I put here-- you can see that clearly L 01:01:07.730 --> 01:01:12.540 has v to the minus 2 compared to k. 01:01:12.540 --> 01:01:15.620 So currently, the way that we established, 01:01:15.620 --> 01:01:19.860 L prime is b to the minus 2m. 01:01:19.860 --> 01:01:22.310 If I had a higher derivative, it would 01:01:22.310 --> 01:01:26.530 be b to a minus larger number, et cetera. 01:01:26.530 --> 01:01:32.390 So L, out of these other things, are irrelevant variables. 01:01:32.390 --> 01:01:36.950 So they are essentially under rescaling, 01:01:36.950 --> 01:01:40.832 under looking at the system in larger and larger scale, 01:01:40.832 --> 01:01:42.125 they will go to 0. 01:01:42.125 --> 01:01:48.410 And I did get a system that has the same topological structure 01:01:48.410 --> 01:01:51.133 as what I had established here. 01:01:51.133 --> 01:01:53.625 Because I have to tune two parameters 01:01:53.625 --> 01:01:56.880 in order to reach the critical point. 01:01:56.880 --> 01:02:00.510 Let's say I had chosen something else. 01:02:00.510 --> 01:02:11.530 If I had chosen z such that L prime equals to L. 01:02:11.530 --> 01:02:13.790 I could do that. 01:02:13.790 --> 01:02:19.780 Then all of the derivatives that are higher factors of q in this 01:02:19.780 --> 01:02:22.810 [INAUDIBLE], they would be all irrelevant. 01:02:22.810 --> 01:02:28.140 But then I would have k, t, and h 01:02:28.140 --> 01:02:30.219 all with irrelevant variables. 01:02:34.390 --> 01:02:39.000 So yeah, it could be that there is some physics. 01:02:39.000 --> 01:02:42.040 I mean, certainly mathematically I 01:02:42.040 --> 01:02:48.410 can ask the system what happens if k goes to 0. 01:02:48.410 --> 01:02:50.670 I kind of ignore the k dependencies 01:02:50.670 --> 01:02:54.340 that I have in all of these expressions, 01:02:54.340 --> 01:02:59.660 but there are going to be singular dependencies on k. 01:02:59.660 --> 01:03:04.070 So if there is indeed some experimental system 01:03:04.070 --> 01:03:10.170 in which you have to tune, in addition temperature, 01:03:10.170 --> 01:03:12.160 something that has to do with the way 01:03:12.160 --> 01:03:14.740 that the spins or degrees of freedom 01:03:14.740 --> 01:03:17.885 are coupled to each other, and that coupling changes sign 01:03:17.885 --> 01:03:20.056 from being positive to being negative, 01:03:20.056 --> 01:03:23.995 you go from one type of behavior to another type of behavior, 01:03:23.995 --> 01:03:26.350 maybe this would be a good thing for it. 01:03:26.350 --> 01:03:29.040 But you can see the kind of structure 01:03:29.040 --> 01:03:34.190 you would get if k has to go to 0, you go from a structure 01:03:34.190 --> 01:03:36.860 where things want to be in the same direction 01:03:36.860 --> 01:03:39.410 to things that want to be anti-parallel. 01:03:39.410 --> 01:03:43.370 And then clearly you need higher order terms to stabilize things 01:03:43.370 --> 01:03:45.340 so that your singularity does not 01:03:45.340 --> 01:03:48.880 go all the way to 0 wavelength, et cetera. 01:03:48.880 --> 01:03:54.090 So one can actually come up with physical systems 01:03:54.090 --> 01:03:56.226 that kind of resemble that, were there 01:03:56.226 --> 01:03:58.942 is some landscape that is also chosen. 01:03:58.942 --> 01:04:02.910 But for this very simplest thing that we are doing, 01:04:02.910 --> 01:04:05.440 this is what is going on. 01:04:05.440 --> 01:04:09.080 But you could have also asked the other question. 01:04:09.080 --> 01:04:12.010 So clearly we understand what happens 01:04:12.010 --> 01:04:15.790 if you choose z so that some term is fixed 01:04:15.790 --> 01:04:18.750 and everything above it is relevant, everything below it 01:04:18.750 --> 01:04:20.370 is irrelevant. 01:04:20.370 --> 01:04:31.998 But why not choose z such that t is fixed? 01:04:31.998 --> 01:04:35.225 So that's going to be b to the d over 2, 01:04:35.225 --> 01:04:37.120 then t prime equals to t. 01:04:40.048 --> 01:04:45.060 If I choose that, then clearly the coupling k 01:04:45.060 --> 01:04:46.295 will already be irrelevant. 01:04:49.270 --> 01:04:54.640 So this is actually a reasonable fixed point. 01:04:54.640 --> 01:04:57.360 It's a fixed one that corresponds to a system 01:04:57.360 --> 01:05:00.530 where k has gone to 0, which means 01:05:00.530 --> 01:05:04.150 that the different points don't talk to each other. 01:05:04.150 --> 01:05:06.180 Remember, when we were discussing 01:05:06.180 --> 01:05:09.410 the behavior of correlation lens at fixed points, 01:05:09.410 --> 01:05:11.995 there was two possibilities-- either the correlation 01:05:11.995 --> 01:05:14.570 lens was infinite or it was 0. 01:05:14.570 --> 01:05:20.730 So if I choose this, then k prime will go eventually to 0. 01:05:20.730 --> 01:05:24.410 I go towards a system in which the degrees of freedom 01:05:24.410 --> 01:05:28.630 are completely decoupled from each other. 01:05:28.630 --> 01:05:30.020 Perfectly well-behaved. 01:05:30.020 --> 01:05:34.380 Fixed behavior that corresponds to 0 correlation lens. 01:05:34.380 --> 01:05:38.200 And you can see that if I go through this formula 01:05:38.200 --> 01:05:44.640 that I told you over here, zeta in real space 01:05:44.640 --> 01:05:49.250 would be b to the minus d over 2. 01:05:49.250 --> 01:05:53.960 And what that means is that if you average 01:05:53.960 --> 01:05:59.130 independent variables over a size b, 01:05:59.130 --> 01:06:02.290 the scale of fluctuation is because of the central limit 01:06:02.290 --> 01:06:04.720 theeorem is the square root of the volume. 01:06:04.720 --> 01:06:05.720 So that's how it scales. 01:06:08.580 --> 01:06:12.310 So essentially, what's at the end of the story? 01:06:12.310 --> 01:06:16.820 That's a behavior in which there is only one coefficient event-- 01:06:16.820 --> 01:06:18.150 forget about h. 01:06:18.150 --> 01:06:23.480 The eventual rate is just t over 2m squared at different points. 01:06:23.480 --> 01:06:26.120 That's the central limit here. 01:06:26.120 --> 01:06:30.660 So through a different route, we have rediscovered, if you like, 01:06:30.660 --> 01:06:32.300 the central limit theorem. 01:06:32.300 --> 01:06:36.330 Because if you average lots of uncorrelated variables, 01:06:36.330 --> 01:06:39.700 you will generate Gaussian rates. 01:06:39.700 --> 01:06:45.070 So what we are really after in this language 01:06:45.070 --> 01:06:49.922 is how to generalize the central limit theorem, how to-- 01:06:49.922 --> 01:06:53.290 as we find the analog of a Gaussian, the degrees 01:06:53.290 --> 01:06:55.730 of freedom that are not correlated 01:06:55.730 --> 01:06:59.580 but talk to their neighborhood. 01:06:59.580 --> 01:07:04.110 So the kind of field theory that we are after 01:07:04.110 --> 01:07:06.950 are these generalizations of central limit theorem 01:07:06.950 --> 01:07:10.700 to the types of field theories that 01:07:10.700 --> 01:07:12.025 have some locality enablement. 01:07:15.985 --> 01:07:16.980 AUDIENCE: Question. 01:07:16.980 --> 01:07:17.680 PROFESSOR: Yes. 01:07:17.680 --> 01:07:22.660 AUDIENCE: So wherever you can define the renormalization 01:07:22.660 --> 01:07:25.720 you're finding different z's? 01:07:25.720 --> 01:07:26.380 PROFESSOR: Yes. 01:07:26.380 --> 01:07:29.660 AUDIENCE: We can tune how many parameters we 01:07:29.660 --> 01:07:30.770 want to be able to-- 01:07:30.770 --> 01:07:31.880 PROFESSOR: Exactly. 01:07:31.880 --> 01:07:32.900 Yes. 01:07:32.900 --> 01:07:36.300 And that's where the physics comes into play. 01:07:36.300 --> 01:07:40.030 Mathematically, there's a whole set of different fixed points 01:07:40.030 --> 01:07:43.390 that you can construct for choosing different z's. 01:07:43.390 --> 01:07:45.540 You have to decide which one of them 01:07:45.540 --> 01:07:47.260 corresponds to the physical problem 01:07:47.260 --> 01:07:49.036 that you are working on. 01:07:49.036 --> 01:07:49.970 AUDIENCE: Yes. 01:07:49.970 --> 01:07:53.052 So if the fixed point stops being just defined 01:07:53.052 --> 01:07:57.955 by the nature of the system, but it's also 01:07:57.955 --> 01:08:00.925 depends on how we define renormalization? 01:08:00.925 --> 01:08:04.390 On mathematical descriptions and-- 01:08:07.855 --> 01:08:13.176 PROFESSOR: If by how we define renormalization looking 01:08:13.176 --> 01:08:15.080 to choose z, yes, I agree with you. 01:08:15.080 --> 01:08:16.510 Yes. 01:08:16.510 --> 01:08:20.990 But again, you have this possibility 01:08:20.990 --> 01:08:25.005 of looking at the system at different scales. 01:08:25.005 --> 01:08:30.260 But we have been very agnostic about what that system is. 01:08:30.260 --> 01:08:33.300 And so you how many ways of doing things. 01:08:33.300 --> 01:08:36.474 Ultimately, you need some reality to come and choose 01:08:36.474 --> 01:08:38.609 among these different ways. 01:08:38.609 --> 01:08:41.444 Yes? 01:08:41.444 --> 01:08:42.860 AUDIENCE: So you do want to keep k 01:08:42.860 --> 01:08:46.930 a relevant variable in group problems, right? 01:08:46.930 --> 01:08:49.666 PROFESSOR: No. 01:08:49.666 --> 01:08:50.999 I make k to be a fixed variable. 01:08:50.999 --> 01:08:53.590 AUDIENCE: Oh, exactly. 01:08:53.590 --> 01:08:57.939 Why don't you add a small amount, like an absolute 01:08:57.939 --> 01:09:02.020 to the power of bf and [INAUDIBLE] point z. 01:09:02.020 --> 01:09:04.680 Plus or minus, doesn't matter. 01:09:04.680 --> 01:09:07.880 Why the equality assumption exactly? 01:09:07.880 --> 01:09:10.310 And the smaller one doesn't change anything? 01:09:10.310 --> 01:09:16.420 All the other variables like L become irrelevant? 01:09:16.420 --> 01:09:18.029 PROFESSOR: OK. 01:09:18.029 --> 01:09:25.090 So the point is that it is b raised to some power. 01:09:25.090 --> 01:09:30.890 So here I had, I don't know, Katie k prime was k. 01:09:30.890 --> 01:09:33.130 And you say, why not kb to the absolute? 01:09:33.130 --> 01:09:34.834 AUDIENCE: Yeah, exactly. 01:09:34.834 --> 01:09:37.420 PROFESSOR: Now, the thing that I'm 01:09:37.420 --> 01:09:41.649 interested in what happens at larger and larger scale. 01:09:41.649 --> 01:09:44.920 So in principle, I should be able to make v 01:09:44.920 --> 01:09:47.390 as large as I want. 01:09:47.390 --> 01:09:52.585 So I don't have the freedom that you mentioned. 01:09:52.585 --> 01:09:56.516 And you are right in the sense that, OK, 01:09:56.516 --> 01:09:58.470 what does it mean whether this ratio is 01:09:58.470 --> 01:10:01.355 larger than or smaller than what? 01:10:01.355 --> 01:10:04.970 But the point is that once you have selected 01:10:04.970 --> 01:10:07.914 some parameter in your system-- L or whatever you have, 01:10:07.914 --> 01:10:12.210 some value-- you can, by playing around with this, 01:10:12.210 --> 01:10:15.950 choose a value of V for any epsilon such 01:10:15.950 --> 01:10:18.730 that you reach that limit. 01:10:18.730 --> 01:10:26.090 So by doing this, you in a sense have defined a lens scale. 01:10:26.090 --> 01:10:28.760 The lens scale would depend on epsilon, 01:10:28.760 --> 01:10:30.637 and you would have different behaviors, 01:10:30.637 --> 01:10:33.160 whether you have shorter than that lens scale 01:10:33.160 --> 01:10:35.556 or larger than that lens scale. 01:10:35.556 --> 01:10:39.410 So this has to be done precisely because of this freedom 01:10:39.410 --> 01:10:41.318 of making b larger, and so on. 01:10:47.996 --> 01:10:50.080 Now, if you are dealing with a finite system 01:10:50.080 --> 01:10:53.032 and you can't make your b much larger than something 01:10:53.032 --> 01:10:54.740 or whatever, then you're perfectly right. 01:10:59.110 --> 01:10:59.640 Yes? 01:10:59.640 --> 01:11:02.210 AUDIENCE: Physically, z or zeta should 01:11:02.210 --> 01:11:08.445 be whatever type quantity is needed to actually make 01:11:08.445 --> 01:11:10.721 it look exactly the same-- where it keeps coming out. 01:11:10.721 --> 01:11:11.720 PROFESSOR: Exactly, yes. 01:11:11.720 --> 01:11:12.261 That's right. 01:11:12.261 --> 01:11:14.300 AUDIENCE: And then we know, because we already 01:11:14.300 --> 01:11:16.200 know that we have two relevant variables, 01:11:16.200 --> 01:11:19.010 that z has to look this way for a system that 01:11:19.010 --> 01:11:20.160 has two relevant variables. 01:11:20.160 --> 01:11:22.651 PROFESSOR: For the Gaussian one, right. 01:11:22.651 --> 01:11:23.276 AUDIENCE: Yeah. 01:11:23.276 --> 01:11:25.840 But then if we had a different kind of system, 01:11:25.840 --> 01:11:28.460 then actually, just going from the physical perspective, 01:11:28.460 --> 01:11:30.836 we would need a different z to make things look the same. 01:11:30.836 --> 01:11:32.543 And that would give us a different number 01:11:32.543 --> 01:11:33.390 of variables here. 01:11:33.390 --> 01:11:34.229 PROFESSOR: Yes. 01:11:34.229 --> 01:11:34.770 That's right. 01:11:37.530 --> 01:11:42.900 Now, in terms of that practically in all cases 01:11:42.900 --> 01:11:44.920 we either are dealing with a phase that 01:11:44.920 --> 01:11:48.005 has 0 correlation on that, and then 01:11:48.005 --> 01:11:51.190 this Gaussian behavior and central limit theorem 01:11:51.190 --> 01:11:55.345 is what we are dealing-- and the averaging is by 1 over volume. 01:11:55.345 --> 01:11:57.700 Or we have something that is very pretty 01:11:57.700 --> 01:12:01.420 close to this big [INAUDIBLE] that we have now discovered, 01:12:01.420 --> 01:12:04.460 which is just the gradient squared. 01:12:04.460 --> 01:12:07.490 And that has its own scaling according 01:12:07.490 --> 01:12:10.750 to these powers that I have found here, 01:12:10.750 --> 01:12:14.326 and I will explain that more deeply. 01:12:14.326 --> 01:12:16.910 It turns out that at the end of the day, 01:12:16.910 --> 01:12:20.910 that when we look at real phase transitions, 01:12:20.910 --> 01:12:26.680 all of these exponents will change, but not too much. 01:12:26.680 --> 01:12:30.590 So this Gaussian fixed point is actually in some sense 01:12:30.590 --> 01:12:34.890 rather close to where we want to end up. 01:12:34.890 --> 01:12:38.560 So that's why it's also an important anchoring point, 01:12:38.560 --> 01:12:40.440 as I just mentioned. 01:12:45.140 --> 01:12:48.510 Again, I said that essentially what we did 01:12:48.510 --> 01:12:52.620 was take the rate that we had originally, 01:12:52.620 --> 01:12:54.400 and we did a rescaling. 01:12:54.400 --> 01:13:02.125 So basically, we replace x by-- let 01:13:02.125 --> 01:13:06.682 me get the directions there. 01:13:06.682 --> 01:13:12.730 So we replace x by bx prime. 01:13:12.730 --> 01:13:16.323 If I had started being in real space, 01:13:16.323 --> 01:13:22.470 I would have replaced m with zeta m prime. 01:13:22.470 --> 01:13:28.765 m after getting rid of some degrees of freedom. 01:13:28.765 --> 01:13:32.160 Again, zeta m prime. 01:13:32.160 --> 01:13:36.946 Before I just do that to the rate that I had written before, 01:13:36.946 --> 01:13:39.550 there was a beta h. 01:13:39.550 --> 01:13:46.870 Which was we could derive d d x t over 2m squared, um 01:13:46.870 --> 01:13:50.314 to the fourth and higher order terms, 01:13:50.314 --> 01:13:59.170 k over 2 gradient m squared, L over 2 Laplacian of m squared 01:13:59.170 --> 01:14:01.870 and so forth. 01:14:01.870 --> 01:14:06.350 Just do this replacement of things. 01:14:06.350 --> 01:14:08.720 What do I get? 01:14:08.720 --> 01:14:14.230 I get that t prime is b to the d. 01:14:14.230 --> 01:14:18.430 Whenever I see x, I replace it with dx prime. 01:14:18.430 --> 01:14:22.470 Whenever I see m, I replace it with zeta m prime. 01:14:22.470 --> 01:14:23.840 So I get here the zeta squared. 01:14:26.850 --> 01:14:34.331 u prime would be b to the d zeta to the fourth. 01:14:38.259 --> 01:14:44.642 k prime would be b to the b minus 2 zeta squared. 01:14:44.642 --> 01:14:48.240 L prime would be b to the d minus 4 01:14:48.240 --> 01:14:49.629 zeta squared, and so forth. 01:14:52.410 --> 01:14:52.920 Essentially. 01:14:52.920 --> 01:15:01.700 All I did was replace x with b times x prime and m with zeta m 01:15:01.700 --> 01:15:02.990 prime. 01:15:02.990 --> 01:15:05.190 If I do that throughout, you can see 01:15:05.190 --> 01:15:08.210 how the various factors will change. 01:15:08.210 --> 01:15:11.020 So I didn't do all of these integrations, 01:15:11.020 --> 01:15:13.580 et cetera that I did over here. 01:15:13.580 --> 01:15:17.230 I just did the dimensional analysis, if you like. 01:15:17.230 --> 01:15:21.560 And within that dimensional analysis now in real space, 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 01:15:29.550 --> 01:15:31.015 minus d over 2. 01:15:37.460 --> 01:15:42.670 And again, you can see that once I have fixed k, 01:15:42.670 --> 01:15:46.150 all of the things that have the same power of m but two higher 01:15:46.150 --> 01:15:49.670 derivatives would get a factor of b to the minus 2, 01:15:49.670 --> 01:15:53.030 just as we had over here. 01:15:53.030 --> 01:15:55.410 Again, with this choice, you can check 01:15:55.410 --> 01:16:00.420 that if I put it back here, I would get b squared. 01:16:00.420 --> 01:16:07.320 But let's imagine that I have a generalization of m to the n. 01:16:07.320 --> 01:16:12.950 If I have a term that multiplies m to some power p-- 01:16:12.950 --> 01:16:18.420 with the coefficient up-- then under this kind of rescaling 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. 01:16:32.890 --> 01:16:36.030 And with this choice of zeta, what do I get? 01:16:36.030 --> 01:16:38.840 I will get b to the d. 01:16:38.840 --> 01:16:48.464 And then I will get plus p 1 minus d over 2 times up. 01:16:51.750 --> 01:16:56.914 Which I can define to be b to some power yp times up. 01:16:56.914 --> 01:16:57.872 Look here to make sure. 01:17:01.250 --> 01:17:07.230 So my yp, the dimension of something 01:17:07.230 --> 01:17:11.790 that multiplies m to some power p 01:17:11.790 --> 01:17:18.982 is simply p plus d 1 minus p or 2. 01:17:24.910 --> 01:17:28.280 And let's check some things. 01:17:28.280 --> 01:17:29.205 I have y1. 01:17:32.190 --> 01:17:35.290 y1 would correspond to a magnetic field, something 01:17:35.290 --> 01:17:38.760 that is proportional to the m itself. 01:17:38.760 --> 01:17:43.860 And if I push p close to 1, I will get 1 plus d over 2. 01:17:43.860 --> 01:17:49.374 And that is, indeed, the yh that we had over here. 01:17:49.374 --> 01:17:52.552 1 plus d over 2. 01:17:52.552 --> 01:17:55.400 So this is yh. 01:17:55.400 --> 01:18:00.095 If I ask what is multiplying m squared, 01:18:00.095 --> 01:18:02.060 I put p equals to 2 here. 01:18:02.060 --> 01:18:07.220 I will get 2, and then here I would get 1 minus 2 over 2. 01:18:07.220 --> 01:18:09.589 So that's the same thing. 01:18:09.589 --> 01:18:11.630 This is the thing that we were calling before yt. 01:18:14.640 --> 01:18:17.710 We didn't include any nq term in the theory, 01:18:17.710 --> 01:18:19.110 didn't make sense to us. 01:18:19.110 --> 01:18:22.050 But we certainly included the u that 01:18:22.050 --> 01:18:24.670 was multiplied in m to the fourth. 01:18:24.670 --> 01:18:27.442 AUDIENCE: So is the p [INAUDIBLE] in the yp? 01:18:32.876 --> 01:18:40.576 PROFESSOR: There is p times 1 plus d 1 minus p over 2. 01:18:40.576 --> 01:18:42.548 p over 2. 01:18:42.548 --> 01:18:44.027 Just rewrote it. 01:18:46.990 --> 01:18:51.900 If I look at 4, here would be 4. 01:18:51.900 --> 01:18:56.210 And then I would put 1 minus 4 over 2, which is 1 minus 2, 01:18:56.210 --> 01:18:57.750 which is minus 1. 01:18:57.750 --> 01:18:59.810 So I would get 4 minus z. 01:19:03.670 --> 01:19:09.302 If I look at y6, I would get 6 minus 2d. 01:19:09.302 --> 01:19:09.843 And so forth. 01:19:15.140 --> 01:19:18.780 So if I just do dimensional analysis 01:19:18.780 --> 01:19:22.080 and I say that I start with a fixed point that corresponds 01:19:22.080 --> 01:19:27.740 to gradient of m squared, and everybody else 0, 01:19:27.740 --> 01:19:30.880 and I ask, if in the vicinity of the fixed point 01:19:30.880 --> 01:19:33.660 where k is fixed and everybody else 01:19:33.660 --> 01:19:39.310 is 0 I put on a little bit up any of these other terms, 01:19:39.310 --> 01:19:41.200 what happens? 01:19:41.200 --> 01:19:45.970 And I find that what happens is that certainly the h 01:19:45.970 --> 01:19:48.285 term, the term that is linear, will be relevant. 01:19:48.285 --> 01:19:53.250 The term that is m squared is relevant. 01:19:53.250 --> 01:19:57.720 Whether or not all the other terms in the series-- 01:19:57.720 --> 01:20:00.320 like m to the fourth, m to the sixth, et cetera-- 01:20:00.320 --> 01:20:04.670 will be relevant depends on dimension. 01:20:04.670 --> 01:20:10.240 So once more we've hit this dimensional fork. 01:20:10.240 --> 01:20:13.720 So the term m to the fourth that we 01:20:13.720 --> 01:20:19.090 said is crucial to getting this theory to have some meaning-- 01:20:19.090 --> 01:20:23.080 and there's no reason for it to be absent-- is, in fact, 01:20:23.080 --> 01:20:24.580 relevant. 01:20:24.580 --> 01:20:27.030 In fact, close to three dimensions 01:20:27.030 --> 01:20:29.700 you would say that that's really the only other term that 01:20:29.700 --> 01:20:31.210 is relevant. 01:20:31.210 --> 01:20:34.890 And you'd say, well, it's almost good enough. 01:20:34.890 --> 01:20:37.950 But almost good enough is not sufficient. 01:20:37.950 --> 01:20:42.390 If we want to describe a physical theory that has only 01:20:42.390 --> 01:20:47.170 two relevant directions, we cannot use this fixed point, 01:20:47.170 --> 01:20:51.110 because this fixed point has three relevant directions 01:20:51.110 --> 01:20:54.370 in three dimensions. 01:20:54.370 --> 01:21:01.070 We have to deal with this somehow. 01:21:01.070 --> 01:21:03.080 So what will we do? 01:21:03.080 --> 01:21:08.110 Next is to explicitly include this m to the fourth. 01:21:08.110 --> 01:21:11.100 In fact, we will include all the other terms, also. 01:21:11.100 --> 01:21:13.810 But we will see that all the other terms, all the higher 01:21:13.810 --> 01:21:16.670 powers, are irrelevant in the same sense 01:21:16.670 --> 01:21:18.530 that all of these higher derivative terms 01:21:18.530 --> 01:21:19.740 are irrelevant. 01:21:19.740 --> 01:21:22.950 But that m to the fourth term is something 01:21:22.950 --> 01:21:25.760 that we really have to take care of. 01:21:25.760 --> 01:21:27.310 And we will do that.