WEBVTT 00:00:00.080 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.810 under a Creative Commons license. 00:00:03.810 --> 00:00:06.550 Your support will help MIT OpenCourseWare continue 00:00:06.550 --> 00:00:10.160 to offer high quality educational resources for free. 00:00:10.160 --> 00:00:12.700 To make a donation or to view additional materials 00:00:12.700 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:21.112 at ocw.mit.edu 00:00:21.112 --> 00:00:23.070 PROFESSOR: As you know, Professor Guth is away. 00:00:23.070 --> 00:00:27.250 I'm substituting for today, he didn't leave me 00:00:27.250 --> 00:00:29.230 with a particularly coherent game plan, 00:00:29.230 --> 00:00:33.340 so I'm going to begin with where he thinks we should start. 00:00:33.340 --> 00:00:36.350 Please jump in if I am just repeating something 00:00:36.350 --> 00:00:39.259 that he has already described to you guys, 00:00:39.259 --> 00:00:41.550 or if there's anything you like me go over a little bit 00:00:41.550 --> 00:00:43.310 more detail, I will do my best here. 00:00:43.310 --> 00:00:48.470 So, I'm working off of a fairly rough plan. 00:00:48.470 --> 00:00:50.450 But let me just quickly describe what-- 00:00:50.450 --> 00:00:55.100 based on what Alan has explained to me --what we're planning 00:00:55.100 --> 00:00:57.686 to talk about today, and if there's 00:00:57.686 --> 00:00:59.102 any adjustments you think I should 00:00:59.102 --> 00:01:00.351 be making that would be great. 00:01:00.351 --> 00:01:02.800 So, the game plan for today. 00:01:05.780 --> 00:01:07.616 What I want to do very quickly is 00:01:07.616 --> 00:01:09.240 hit on a couple of the key points which 00:01:09.240 --> 00:01:11.330 I believe you talked about last week, 00:01:11.330 --> 00:01:16.040 which is a quick review of the essential features 00:01:16.040 --> 00:01:21.669 of symmetries of the gauge fields 00:01:21.669 --> 00:01:22.960 the make up the standard model. 00:01:33.690 --> 00:01:35.490 Now, I believe you guys did in fact talk 00:01:35.490 --> 00:01:37.400 about this last week, at least briefly. 00:01:37.400 --> 00:01:39.483 And you talked about how you can take these things 00:01:39.483 --> 00:01:43.141 and embed them in a larger gauge group, the group SU(5). 00:01:43.141 --> 00:01:44.890 I'm not going to talk about that too much, 00:01:44.890 --> 00:01:47.560 but I want to just quickly hit on a few elements related 00:01:47.560 --> 00:01:49.240 to this before we get into that. 00:01:49.240 --> 00:01:54.034 From this we'll then talk about the Higgs mechanism-- really 00:01:54.034 --> 00:01:55.700 I'm going to talk about the Higgs field, 00:01:55.700 --> 00:01:57.658 I'm not going to talk about the Higgs mechanism 00:01:57.658 --> 00:02:02.870 quite so much as motivate why it is necessary-- 00:02:02.870 --> 00:02:05.550 and then talk about how the Higgs field behaves 00:02:05.550 --> 00:02:09.870 and why it's important for the next problem, which 00:02:09.870 --> 00:02:15.230 is what is called the cosmological monopole problem. 00:02:19.166 --> 00:02:21.685 To be more specific magnetic monopole problem. 00:02:33.350 --> 00:02:35.560 I confess I feel a little bit awkward talking 00:02:35.560 --> 00:02:39.650 about this problem on behalf of Alan. 00:02:39.650 --> 00:02:42.370 This would be kind of like if you were planning 00:02:42.370 --> 00:02:45.840 on studying Hamlet and there was this guy W. Shakespeare who 00:02:45.840 --> 00:02:49.200 was listed as the instructor and you walk in 00:02:49.200 --> 00:02:51.890 and discover there's this guy Warren Shackspeare, who's 00:02:51.890 --> 00:02:54.098 actually going to be teaching or something like that. 00:02:54.098 --> 00:02:56.850 I kind of feel like Warren here. 00:02:56.850 --> 00:03:02.290 This stuff really is Allen's thing, so it's sort of, 00:03:02.290 --> 00:03:05.057 I'm probably going to leave this at the denouement of all this 00:03:05.057 --> 00:03:06.890 when you actually get into inflation to him. 00:03:06.890 --> 00:03:09.500 I may have a little bit of time at the end to just motivate 00:03:09.500 --> 00:03:14.030 it a little bit, but the grand summary will come from him. 00:03:14.030 --> 00:03:26.230 OK, so, as discussed by Alan the standard model 00:03:26.230 --> 00:03:28.480 describes all the fundamental interactions 00:03:28.480 --> 00:03:31.510 between particles via gauge theories. 00:03:53.460 --> 00:03:57.400 OK, and these gauge theories all have a combined symmetry 00:03:57.400 --> 00:04:06.620 group that is traditionally written in a somewhat 00:04:06.620 --> 00:04:17.990 awkward form, SU(3) cross SU(2) cross U(1). 00:04:17.990 --> 00:04:20.529 U(1) could be an SU(1) for reasons 00:04:20.529 --> 00:04:23.070 which I'll elucidate a little bit more 00:04:23.070 --> 00:04:24.450 clearly in just a moment. 00:04:24.450 --> 00:04:27.210 There's really no point in putting the S on that one. 00:04:27.210 --> 00:04:30.130 So each of these things essentially 00:04:30.130 --> 00:04:34.170 labels the particular symmetry group. 00:04:34.170 --> 00:04:45.910 So, the "S" an element of SU(n) is 00:04:45.910 --> 00:05:04.436 a matrix that is n x n, that is unitary-- that's the U. 00:05:04.436 --> 00:05:07.380 Unitary just means that the inverse and the transpose 00:05:07.380 --> 00:05:09.610 of the matrix at the same, actually the Hermitian 00:05:09.610 --> 00:05:13.169 conjugate because they can be complex, in fact, 00:05:13.169 --> 00:05:13.960 they generally are. 00:05:16.840 --> 00:05:18.955 And it has determinant of 1. 00:05:24.580 --> 00:05:28.460 That's what the special refers to, special, the S in SU(n) 00:05:28.460 --> 00:05:30.880 stands for special unitary n. 00:05:30.880 --> 00:05:32.990 So, the S means that the determinant 00:05:32.990 --> 00:05:35.640 is one-- that's what's special about it-- unitary 00:05:35.640 --> 00:05:38.730 is this idea that the inverse Hermitian conjugate are 00:05:38.730 --> 00:05:41.430 the same, and then n refers to all these things. 00:05:41.430 --> 00:05:45.656 So, that tells us that the gauge degrees of freedom 00:05:45.656 --> 00:05:47.530 are related by a symmetry that looks like a 3 00:05:47.530 --> 00:05:50.620 by 3 matrix with these properties, 00:05:50.620 --> 00:05:54.480 as listed there for the SU(3) piece of the symmetry. 00:05:54.480 --> 00:05:57.120 SU(2) means it's a 2 by 2 matrix. 00:05:57.120 --> 00:06:01.290 And U(1) means it's a one by one matrix, what's a 1 by 1 matrix? 00:06:01.290 --> 00:06:03.392 It's a number, its a complex number. 00:06:03.392 --> 00:06:04.850 And that's why we don't really need 00:06:04.850 --> 00:06:06.230 to put an "s" in front of it. 00:06:06.230 --> 00:06:08.740 If it's a complex number its determinant 00:06:08.740 --> 00:06:11.480 is 1 if it's just a complex number whose modules is one. 00:06:11.480 --> 00:06:13.860 That's why we don't bother with the S on the U(1). 00:06:16.720 --> 00:06:19.229 So, I think you've already hit on some of this 00:06:19.229 --> 00:06:21.270 but this is sort of useful to review because it's 00:06:21.270 --> 00:06:23.910 going to set up why we need to introduce a Higgs 00:06:23.910 --> 00:06:26.050 mechanism in a little bit. 00:06:26.050 --> 00:06:29.820 Let me just quickly hit on what the details 00:06:29.820 --> 00:06:33.110 structure of this looks like for you want to think 00:06:33.110 --> 00:06:46.150 is the easiest one understand, So, as I just said, 00:06:46.150 --> 00:06:49.540 a one by one matrix is just a complex number. 00:06:49.540 --> 00:06:54.820 So that means that any element of this group 00:06:54.820 --> 00:07:11.390 is a complex number, which we can write in the form z 00:07:11.390 --> 00:07:14.990 equals ei theta, where theta is a real number. 00:07:18.620 --> 00:07:20.460 Now, the thing which is I want to hit on 00:07:20.460 --> 00:07:22.960 in this, the reason I want to describe this a little bit is, 00:07:22.960 --> 00:07:26.140 this may not smell like the gauge symmetry 00:07:26.140 --> 00:07:28.620 that you're used to if you study classical E&M. Some of you 00:07:28.620 --> 00:07:30.190 here are in 807 with me right now, 00:07:30.190 --> 00:07:32.300 and we've gone over this quite a bit recently. 00:07:32.300 --> 00:07:36.160 How is this akin to the gauge group that we are normally 00:07:36.160 --> 00:07:38.799 used to when we talk about the gauge 00:07:38.799 --> 00:07:40.340 freedom of electricity and magnetism? 00:07:56.092 --> 00:07:57.550 Well, it turns out there's actually 00:07:57.550 --> 00:08:01.520 a very simple relationship between one and the other, 00:08:01.520 --> 00:08:04.880 rather between this view of it and the way we learn 00:08:04.880 --> 00:08:11.095 about it when we study classical E&M. It's simply 00:08:11.095 --> 00:08:13.720 that we use a somewhat different language, because when we talk 00:08:13.720 --> 00:08:16.340 about it in this group theoretic picture we're doing it 00:08:16.340 --> 00:08:20.310 in the way that is sort of tuned to a quantum field theory. 00:08:20.310 --> 00:08:22.800 So, the way we have learned about electromagnetic gauge 00:08:22.800 --> 00:08:43.739 symmetry in terms of the fields sort of goes as follows. 00:08:43.739 --> 00:08:46.030 We actually work with the potentials, and so what we do 00:08:46.030 --> 00:08:54.860 is we note that the potentials Amu, which you can write 00:08:54.860 --> 00:08:59.760 as a four vector, whose time-like component is 00:08:59.760 --> 00:09:02.710 the negative of the scalar potential, 00:09:02.710 --> 00:09:04.320 and whose spatial components are just 00:09:04.320 --> 00:09:08.160 the three components of the vector potential. 00:09:08.160 --> 00:09:34.030 So, this potential and this potential-- --okay 00:09:34.030 --> 00:09:40.179 this is possibly module of factor of c somewhere in here 00:09:40.179 --> 00:09:41.970 but I'm going to imagine the speed of light 00:09:41.970 --> 00:09:44.490 has been set equal to 1. 00:09:44.490 --> 00:09:47.760 Both of those potentials generate the same E&B fields. 00:10:09.970 --> 00:10:12.589 OK, again you still should be looking at this 00:10:12.589 --> 00:10:14.380 and thinking to yourself what the hell does 00:10:14.380 --> 00:10:18.200 this have to do with the U(1) as we presented it here. 00:10:18.200 --> 00:10:19.960 I've given you a bunch of operations 00:10:19.960 --> 00:10:23.060 that involve some kind of a scale or function of time 00:10:23.060 --> 00:10:23.680 and space. 00:10:23.680 --> 00:10:26.490 And I've added particular components of this four vector 00:10:26.490 --> 00:10:29.790 in this way, what does that to do with this multiplication 00:10:29.790 --> 00:10:32.887 by a complex number? 00:10:32.887 --> 00:10:34.470 Well, where it comes from is that when 00:10:34.470 --> 00:10:37.930 we study E&M, not as a classical field theory 00:10:37.930 --> 00:10:40.250 but as a quantum field theory, we 00:10:40.250 --> 00:10:43.800 have a field that describes the electron. 00:10:43.800 --> 00:10:47.540 So, where it comes from is that when 00:10:47.540 --> 00:10:50.360 you examine the Dirac field, which is the quantum field 00:10:50.360 --> 00:10:54.270 theory that governs the electron, when you change gauge 00:10:54.270 --> 00:11:00.300 the electron field acquires a local phase change. 00:11:35.060 --> 00:11:36.520 So in particular, what we find is 00:11:36.520 --> 00:11:46.292 that if we have a field 5x, which those of you who 00:11:46.292 --> 00:11:48.250 have taken a little bit of quantum field theory 00:11:48.250 --> 00:11:50.240 should know this is actually a spinner field, 00:11:50.240 --> 00:11:53.140 but for now, just think of it as some kind of a field that 00:11:53.140 --> 00:11:57.970 under the field equations of quantum 00:11:57.970 --> 00:12:01.544 electrodynamics-- the Dirac equation or high order 00:12:01.544 --> 00:12:03.710 ones that have been developed by Feynman, Schwinger, 00:12:03.710 --> 00:12:06.860 and others-- under a change of gauge 00:12:06.860 --> 00:12:14.290 this goes over to si prime of x, which 00:12:14.290 --> 00:12:17.800 equals e to the-- terrible notation 00:12:17.800 --> 00:12:24.030 I realized-- 1e is obviously the root of natural logs, 00:12:24.030 --> 00:12:27.240 e sub 0 is the fundamental electric charge. 00:12:38.510 --> 00:12:40.200 OK, can everyone read that? 00:12:40.200 --> 00:12:42.870 I didn't block it too badly here I'm not used to this classroom. 00:12:42.870 --> 00:12:48.240 So, here's the thing to note, is that this field lambda, which 00:12:48.240 --> 00:12:52.425 we learned about in classical E&M 00:12:52.425 --> 00:12:58.165 directly connects to the phase function of the Dirac field 00:12:58.165 --> 00:12:59.290 in quantum electrodynamics. 00:13:04.030 --> 00:13:07.730 So, our gauge symmetry is simply expressed 00:13:07.730 --> 00:13:11.700 in the quantum version of electrodynamics 00:13:11.700 --> 00:13:18.660 by a function of the form e to the i real number, where 00:13:18.660 --> 00:13:21.840 that real number is the fundamental electric charge 00:13:21.840 --> 00:13:23.630 times the classical gauge generator. 00:14:03.930 --> 00:14:07.780 So, this is what is meant when people say that electrodynamics 00:14:07.780 --> 00:14:09.325 is a U(1) gauge theory. 00:14:26.080 --> 00:14:29.190 Now, I'm not going to go into this level of detail 00:14:29.190 --> 00:14:31.760 for the other two gauge symmetry that 00:14:31.760 --> 00:14:33.680 are built into the standard model. 00:14:33.680 --> 00:14:36.380 But, what I want you to understand 00:14:36.380 --> 00:14:39.690 is that the root idea is very, very similar. 00:14:39.690 --> 00:14:47.770 It's just now, instead of my gauge functions looking like e 00:14:47.770 --> 00:14:53.430 to the i, some kind of a local gauge phase of x multiplying 00:14:53.430 --> 00:14:57.820 my functions, my quantities which 00:14:57.820 --> 00:15:00.020 generate the gauge transformation are going 00:15:00.020 --> 00:15:02.340 to become complex value matrices. 00:15:02.340 --> 00:15:04.390 So that makes them a lot more complicated, 00:15:04.390 --> 00:15:05.830 and it's responsible for the fact 00:15:05.830 --> 00:15:07.860 that the weak and the strong interactions 00:15:07.860 --> 00:15:10.420 are non-abelian paid which order you perform the gauge 00:15:10.420 --> 00:15:11.620 transformation in matters. 00:15:11.620 --> 00:15:12.690 Question. 00:15:12.690 --> 00:15:14.440 AUDIENCE: What's the physical significance 00:15:14.440 --> 00:15:16.625 of them being non-abelian? 00:15:16.625 --> 00:15:17.250 PROFESSOR: Yes. 00:15:17.250 --> 00:15:18.470 So, what is a physical significance 00:15:18.470 --> 00:15:19.553 of them being non-abelian? 00:15:23.877 --> 00:15:29.790 I'm trying think of a really simple way to put this, 00:15:29.790 --> 00:15:36.510 it's-- Alan would have an answer to this right off the top 00:15:36.510 --> 00:15:38.120 of his head, so I apologize for this-- 00:15:38.120 --> 00:15:40.920 this isn't the kind of thing that I work on every day so I 00:15:40.920 --> 00:15:43.790 don't have an answer right at the very top of my head, 00:15:43.790 --> 00:15:44.920 unfortunately. 00:15:44.920 --> 00:15:46.604 Let me get back to you on that one, OK, 00:15:46.604 --> 00:15:48.770 that's something I can't give you a quick answer to. 00:15:48.770 --> 00:15:55.180 It's an excellent question and it's an important question. 00:15:55.180 --> 00:15:58.500 Any other questions? 00:15:58.500 --> 00:16:04.210 OK, so, here's a basic picture that we have. 00:16:10.160 --> 00:16:21.230 So, we find is that the strong interactions have 00:16:21.230 --> 00:16:29.680 a similar structure where my need to e to the i factor 00:16:29.680 --> 00:16:47.380 goes over to a 3 by 3 matrix, and the weak interactions 00:16:47.380 --> 00:16:57.070 in a similar structure with my e to the i factor going over 00:16:57.070 --> 00:16:58.760 to a 2 by 2 complex matrix. 00:17:07.865 --> 00:17:09.740 OK, what does this have to do with cosmology? 00:17:14.234 --> 00:17:16.400 In fact, as an enormous amount to do with cosmology, 00:17:16.400 --> 00:17:19.079 as we'll see over the course of the rest of course. 00:17:19.079 --> 00:17:24.240 Part of the thing which is interesting about all this 00:17:24.240 --> 00:17:28.600 is that we have strong experimental reasons, 00:17:28.600 --> 00:17:31.110 and theoretical reasons to believe, 00:17:31.110 --> 00:17:35.230 that the different symmetries that these interactions 00:17:35.230 --> 00:17:37.380 participate in, the different symmetries that we 00:17:37.380 --> 00:17:42.880 see them having, that isn't the way things have always been. 00:17:42.880 --> 00:17:45.580 So, in particular when the universe was a lot hotter 00:17:45.580 --> 00:17:49.550 and denser these different symmetries actually 00:17:49.550 --> 00:17:50.880 all began to look the same. 00:17:50.880 --> 00:17:53.395 In particular the one which is particularly important, 00:17:53.395 --> 00:18:04.410 and you guys have surely heard of this, is that the SU(2)-- 00:18:04.410 --> 00:18:15.360 if we just focus on electric and the weak piece of this-- 00:18:15.360 --> 00:18:18.530 SU(2) cross U(1). 00:18:18.530 --> 00:18:23.100 So, this is associated with the gauge boson that 00:18:23.100 --> 00:18:26.620 carry the weak force, OK, the z boson, the w plus, and the w 00:18:26.620 --> 00:18:27.460 minus. 00:18:27.460 --> 00:18:31.181 And your U(1) ends up being associated with the photon. 00:18:31.181 --> 00:18:32.680 In many ways, when you actually look 00:18:32.680 --> 00:18:34.730 at the equations that govern these things, 00:18:34.730 --> 00:18:36.830 they seem very, very similar to one 00:18:36.830 --> 00:18:42.025 another except that the-- here's partly 00:18:42.025 --> 00:18:43.400 an answer to your question I just 00:18:43.400 --> 00:18:49.280 realized-- the gauge generators of these things 00:18:49.280 --> 00:18:51.185 have a mass associated with them. 00:18:51.185 --> 00:18:52.560 That mass ends up being connected 00:18:52.560 --> 00:18:56.380 to the non-abelian nature of these things. 00:18:56.380 --> 00:18:59.400 That's not the whole answer, but it has a connection to that. 00:18:59.400 --> 00:19:01.940 That's one thing which I do remember, 00:19:01.940 --> 00:19:03.450 like I said I feel this is really 00:19:03.450 --> 00:19:07.140 Alan's perfect framework here and I'm just 00:19:07.140 --> 00:19:11.640 a posture in bad shoes. 00:19:11.640 --> 00:19:14.050 So if we look at this thing, what we see 00:19:14.050 --> 00:19:16.690 is that these symmetry groups, what's particularly interesting 00:19:16.690 --> 00:19:23.740 is that U(1) can be regarded as a piece of SU(2). 00:19:23.740 --> 00:19:29.530 And we would expect that in a perfect world 00:19:29.530 --> 00:19:31.900 they would actually be SU(2) governing 00:19:31.900 --> 00:19:34.430 both the electric and the weak interactions. 00:19:34.430 --> 00:19:36.350 Whereby perfect I mean everything 00:19:36.350 --> 00:19:40.090 is a nice balmy 10 to the 16th GeV throughout all of space 00:19:40.090 --> 00:19:43.470 time, and all the different vector bosons happily 00:19:43.470 --> 00:19:47.569 exchange with one another, not caring with who is who. 00:19:47.569 --> 00:19:49.110 It's actually not very perfect if you 00:19:49.110 --> 00:19:51.568 want to teach a physics class and have a nice conversation, 00:19:51.568 --> 00:19:54.050 but if you are interested in perfect symmetry among gauge 00:19:54.050 --> 00:19:58.520 interactions it's very, very nice. 00:19:58.520 --> 00:20:12.130 So, the fact that these are separate 00:20:12.130 --> 00:20:15.870 is now-- I was about to use the word believed but it's stronger 00:20:15.870 --> 00:20:17.780 and that, we now know this for sure thanks 00:20:17.780 --> 00:20:21.590 to all the exciting work that happened at the LHC 00:20:21.590 --> 00:20:26.440 over the past year or two-- the fact that these symmetries are 00:20:26.440 --> 00:20:31.650 separate is due to what is called spontaneous symmetry 00:20:31.650 --> 00:20:32.150 breaking. 00:20:45.610 --> 00:20:51.100 So, let's talk very briefly about what 00:20:51.100 --> 00:20:53.590 goes into this spontaneous symmetry breaking. 00:21:09.220 --> 00:21:11.560 So SU(2) turns out to actually be 00:21:11.560 --> 00:21:15.204 isomorphic to the group of rotations on a sphere. 00:21:15.204 --> 00:21:16.620 So, when you think about something 00:21:16.620 --> 00:21:19.120 that has perfect SU(2) symmetry it's 00:21:19.120 --> 00:21:22.010 as though you have perfect symmetry when you move around 00:21:22.010 --> 00:21:24.624 through a whole host of different angles. 00:21:24.624 --> 00:21:26.790 OK, so you move through all of your different angles 00:21:26.790 --> 00:21:31.050 and everyone looks exactly identical to all the others. 00:21:31.050 --> 00:21:32.820 If you break that symmetry it may 00:21:32.820 --> 00:21:35.530 mean you're picking out one angle as being special, 00:21:35.530 --> 00:21:39.090 and then you only retain a symmetry with respect 00:21:39.090 --> 00:21:42.250 to the other angle. 00:21:42.250 --> 00:21:45.920 And essentially, that is what happens when SU(2) breaks off 00:21:45.920 --> 00:21:47.600 in a U(1) piece of it. 00:21:47.600 --> 00:21:50.280 Something has occurred that picked out 00:21:50.280 --> 00:21:52.220 one of these directions. 00:21:52.220 --> 00:21:55.142 And by the way, you have to think very abstractly here. 00:21:55.142 --> 00:21:57.350 This is not necessarily a direction in physical space 00:21:57.350 --> 00:21:58.725 we're talking about here but it's 00:21:58.725 --> 00:22:01.450 a direction in the space of gauge fields. 00:22:01.450 --> 00:22:04.714 So, if we imagine that all of these, my gauge fields in some 00:22:04.714 --> 00:22:06.630 sense the different components of them defined 00:22:06.630 --> 00:22:08.840 in some abstract space direction, 00:22:08.840 --> 00:22:11.120 initially these things are completely 00:22:11.120 --> 00:22:12.970 symmetric with respect to rotations 00:22:12.970 --> 00:22:15.400 in some kind of an abstract notion of a sphere. 00:22:15.400 --> 00:22:17.570 And then something happens to freeze 00:22:17.570 --> 00:22:19.825 one of the directions and only symmetries 00:22:19.825 --> 00:22:21.950 with respect to one of the angles remains the same. 00:22:27.180 --> 00:22:29.390 Let's just write that out, when SU(2)'s symmetry 00:22:29.390 --> 00:22:52.830 is broken so one of the directions in the space 00:22:52.830 --> 00:23:02.090 of gauge fields is picked out as special. 00:23:07.790 --> 00:23:09.980 That direction then ends up being 00:23:09.980 --> 00:23:12.740 associated with your U(1) symmetry. 00:23:39.400 --> 00:23:44.570 So, what is the mechanism that actually breaks the symmetry 00:23:44.570 --> 00:23:46.720 and causes this to happen. 00:23:46.720 --> 00:23:50.490 Well, this is what the Higgs field is all about. 00:23:50.490 --> 00:23:57.120 The idea is there is some field that fills all of space time. 00:24:00.720 --> 00:24:05.490 It has the property that at very high energies 00:24:05.490 --> 00:24:10.310 it is extremely symmetric, with to respect all these gauge 00:24:10.310 --> 00:24:13.330 fields, all directions and sort of gauge field space 00:24:13.330 --> 00:24:14.980 look exactly the same. 00:24:14.980 --> 00:24:19.370 And then as things cool, as the energy density goes down 00:24:19.370 --> 00:24:21.580 by the temperature of the expanding universe, cooling 00:24:21.580 --> 00:24:26.190 everything off, the Higgs field moves to a particular place 00:24:26.190 --> 00:24:30.460 that picks out some direction in the space of gauge fields 00:24:30.460 --> 00:24:31.470 as being special. 00:24:34.514 --> 00:24:36.430 So let's make this a little bit more concrete. 00:25:13.070 --> 00:25:13.570 OK. 00:25:13.570 --> 00:25:15.903 You guys have probably heard quite a lot about the Higgs 00:25:15.903 --> 00:25:17.710 field over the past couple years, 00:25:17.710 --> 00:25:21.510 months-- what actually is it? 00:25:21.510 --> 00:25:38.410 Well, the field itself is described by a complex doublet. 00:25:45.340 --> 00:25:49.000 So, if you actually see someone write down a Higgs field what 00:25:49.000 --> 00:25:53.080 they will actually write down is h, 00:25:53.080 --> 00:26:00.900 being a two components spinner, whose components are 00:26:00.900 --> 00:26:06.450 h1 of x, h2 of x-- where x really 00:26:06.450 --> 00:26:08.840 stands for space time coordinates, so that's time 00:26:08.840 --> 00:26:10.690 and all of your spatial coordinates-- 00:26:10.690 --> 00:26:15.940 and both h1 and h2 are complex fields. 00:26:44.850 --> 00:26:47.760 The thing which is particularly key to understanding 00:26:47.760 --> 00:26:53.040 the importance of this thing is that h transforms, 00:26:53.040 --> 00:26:59.300 under gauge transformations, with elements of SU(2). 00:27:06.495 --> 00:27:08.120 So, if you want to change gauge the way 00:27:08.120 --> 00:27:13.060 you're going to do it is you're going to have some new Higgs 00:27:13.060 --> 00:27:13.560 field. 00:27:16.610 --> 00:27:30.930 So remember, if U(2) is an element of SU(2) 00:27:30.930 --> 00:27:34.250 we call it the two by two matrix. 00:27:34.250 --> 00:27:43.150 This is what they look like in a new gauge OK-- pardon me 00:27:43.150 --> 00:27:45.380 a second I don't see a clock in this room, 00:27:45.380 --> 00:27:47.920 I just want to make sure I know the time, thank you. 00:27:50.750 --> 00:27:54.266 OK, so, what are we going to do with this? 00:27:54.266 --> 00:27:56.390 Well, there's a couple features which it must have, 00:27:56.390 --> 00:28:01.330 so the Higgs field fills all of space time 00:28:01.330 --> 00:28:04.780 and it has an energy density associated 00:28:04.780 --> 00:28:07.680 with it, which we will call just the potential energy. 00:28:07.680 --> 00:28:13.612 It's really an energy density, but, whatever. 00:28:13.612 --> 00:28:15.820 The energy density that is associated with this thing 00:28:15.820 --> 00:28:16.915 must be gauge invariant. 00:28:39.660 --> 00:28:41.870 OK, even when you're working with strong fields 00:28:41.870 --> 00:28:46.390 and weak fields, the lesson of gauge invariance from E&M 00:28:46.390 --> 00:28:47.190 still holds. 00:28:47.190 --> 00:28:50.450 OK, one of the key points was that the gauge fields affect 00:28:50.450 --> 00:28:54.760 potentials, they allow us to manipulate our equations 00:28:54.760 --> 00:28:57.800 to put things into a form where the calculation may be easier. 00:28:57.800 --> 00:29:00.425 But at the end of the day, there are certain things it actually 00:29:00.425 --> 00:29:03.080 exert forces that cause things to happen, 00:29:03.080 --> 00:29:05.770 those must be invariant to the gauge transformation. 00:29:05.770 --> 00:29:07.410 Energy density is of those things. 00:29:07.410 --> 00:29:10.016 If you were to get into your spaceship 00:29:10.016 --> 00:29:11.390 and go back to the early universe 00:29:11.390 --> 00:29:13.640 and actually take a little scoop of early universe out 00:29:13.640 --> 00:29:17.590 and measure the energy density, A, that would be cool, but B 00:29:17.590 --> 00:29:20.350 it would be something that couldn't actually 00:29:20.350 --> 00:29:22.417 depend on what gauge you were using 00:29:22.417 --> 00:29:23.500 to make your measurements. 00:29:23.500 --> 00:29:26.024 That is something that is a complete artifice of how 00:29:26.024 --> 00:29:27.440 you want to set up the convenience 00:29:27.440 --> 00:29:28.770 of your calculation. 00:29:28.770 --> 00:29:36.300 So, in order for the energy density to be gauge invariant 00:29:36.300 --> 00:29:38.570 we have to find a gauge invariant quantity that 00:29:38.570 --> 00:29:42.160 is constructed from this, which is the only thing the energy 00:29:42.160 --> 00:29:44.060 density can depend on. 00:29:44.060 --> 00:29:47.360 This means, let's call our energy density V, 00:29:47.360 --> 00:29:49.150 it's the potential energy density. 00:30:04.560 --> 00:30:11.380 So, it can only depend on the following combination 00:30:11.380 --> 00:30:23.050 of the fundamental fields Pretty much just what you'd expect. 00:30:23.050 --> 00:30:27.330 This is sort of the equivalent to saying 00:30:27.330 --> 00:30:31.220 that if you're working in spherical symmetry 00:30:31.220 --> 00:30:33.220 the electrostatic potential can only 00:30:33.220 --> 00:30:35.200 depend on the distance from a point charge. 00:30:35.200 --> 00:30:37.130 This is a very similar kind of construct 00:30:37.130 --> 00:30:40.160 here, where I'm taking the only quantity that 00:30:40.160 --> 00:30:43.150 follows in a fully symmetric way, of calling 00:30:43.150 --> 00:30:47.447 the fact that this is a special unitary matrix that I 00:30:47.447 --> 00:30:48.780 can construct from these things. 00:30:52.480 --> 00:30:55.670 So then, where all the magic comes 00:30:55.670 --> 00:31:01.890 in is in how the Higgs field potential energy density 00:31:01.890 --> 00:31:06.980 varies as a function of this h, this magnitude of h. 00:31:40.240 --> 00:31:42.857 So, as I plot v as a function of h, 00:31:42.857 --> 00:31:44.940 in order to get your spontaneous symmetry breaking 00:31:44.940 --> 00:31:49.070 to happen what you want is for the minimum of V, 00:31:49.070 --> 00:31:53.540 the minimum potential energy, to occur somewhere 00:31:53.540 --> 00:32:21.310 out at a non-zero value of the Higgs field H. 00:32:21.310 --> 00:32:24.135 Now, why is that so special? 00:32:38.850 --> 00:32:41.030 The thing that is so special about that 00:32:41.030 --> 00:32:46.320 is that when I constructed this magnitude of h, 00:32:46.320 --> 00:32:48.350 I actually lost a lot of information 00:32:48.350 --> 00:32:51.000 about the Higgs field. 00:32:51.000 --> 00:32:52.860 OK, let's just say for the sake of argument 00:32:52.860 --> 00:32:56.120 that this minimum occurs at a place where 00:32:56.120 --> 00:33:01.580 the Higgs field in some system of units has a value of 1. 00:33:01.580 --> 00:33:05.990 So, all I need to do is as my universe cools 00:33:05.990 --> 00:33:08.100 what I'm going to want is energetically, 00:33:08.100 --> 00:33:11.160 my potential is going to want to go down to its minimum. 00:33:11.160 --> 00:33:14.620 So, that just means that as the universe is cooling, maybe 00:33:14.620 --> 00:33:17.080 at very, very early times when everything is extremely 00:33:17.080 --> 00:33:21.020 hot and dense, I'm up here where the potential energy is 00:33:21.020 --> 00:33:23.020 very high. 00:33:23.020 --> 00:33:24.960 As the universe expands, as everything cools, 00:33:24.960 --> 00:33:26.459 it moves over to here, it just moves 00:33:26.459 --> 00:33:29.780 to someplace where the Higgs field takes on a value of 1. 00:33:29.780 --> 00:33:33.200 And that's exactly correct, that is what ends up happening. 00:33:33.200 --> 00:33:48.340 But remember, the minimum occurs at some value 00:33:48.340 --> 00:33:52.360 in which the magnitude of this field 00:33:52.360 --> 00:33:56.140 does not equal zero, but given that value-- where again let's 00:33:56.140 --> 00:33:58.110 just say for this for sake of specificity 00:33:58.110 --> 00:34:00.340 that we set it equal to the magnitude of this thing 00:34:00.340 --> 00:34:03.370 equal to 1 in some units-- there's actually 00:34:03.370 --> 00:34:07.090 an infinite number of configurations that correspond 00:34:07.090 --> 00:34:10.380 to that because this is a complex number, 00:34:10.380 --> 00:34:12.159 this is a complex number. 00:34:12.159 --> 00:34:15.539 I could put it all into little h1, 00:34:15.539 --> 00:34:18.080 and I could set into the value where that thing is completely 00:34:18.080 --> 00:34:22.540 real, or I could put it all into little h2 being completely 00:34:22.540 --> 00:34:27.310 imaginary or all on to h1 being all imaginary, halfway into h1, 00:34:27.310 --> 00:34:28.800 halfway into h2. 00:34:28.800 --> 00:34:31.460 There are literally an infinite number of combinations 00:34:31.460 --> 00:34:34.429 that I can choose which are consistent 00:34:34.429 --> 00:35:20.958 with this value of the magnitude of H. So, yeah-- 00:35:20.958 --> 00:35:22.704 AUDIENCE: So, I don't know if I'm 00:35:22.704 --> 00:35:24.870 putting too much physical significance on the gauge, 00:35:24.870 --> 00:35:27.480 but with the other cases of spontaneous symmetry, 00:35:27.480 --> 00:35:29.640 briefly, that we discussed you can always measure. 00:35:29.640 --> 00:35:32.140 OK, I've broken my symmetry, and now it's lined up this way, 00:35:32.140 --> 00:35:34.280 or there's something measurable. 00:35:34.280 --> 00:35:35.896 Now, the field has to be physical 00:35:35.896 --> 00:35:38.736 because the fact that you have gauge symmetry 00:35:38.736 --> 00:35:41.350 gives you some concerned quantity, right? 00:35:41.350 --> 00:35:44.710 But, how can I measure what direction in gauge space 00:35:44.710 --> 00:35:46.300 that I picked out? 00:35:46.300 --> 00:35:47.960 PROFESSOR: So, that is, let me talk 00:35:47.960 --> 00:35:51.090 about this just a little bit more. 00:35:51.090 --> 00:35:53.390 I think answering your question completely 00:35:53.390 --> 00:35:55.450 is not really possible, but there 00:35:55.450 --> 00:35:58.406 is a residue of that is in fact very interesting, 00:35:58.406 --> 00:36:01.030 and let me just lay out a couple more facts about what actually 00:36:01.030 --> 00:36:02.550 happens with this gauge symmetry, 00:36:02.550 --> 00:36:04.500 and it's not going to answer your question 00:36:04.500 --> 00:36:06.666 but it's going to give you something to think about. 00:36:06.666 --> 00:36:10.130 OK, so that's an excellent and very deep question, 00:36:10.130 --> 00:36:12.120 and there are really interesting consequences. 00:36:12.120 --> 00:36:15.645 And this is a case where my failure 00:36:15.645 --> 00:36:17.520 to answer the previous one is because there's 00:36:17.520 --> 00:36:19.160 details I can't remember, in this case, 00:36:19.160 --> 00:36:21.160 I think it's because there's details we actually 00:36:21.160 --> 00:36:22.260 don't understand fully. 00:36:24.709 --> 00:36:27.250 Research into the mechanism of electroweak symmetry breaking, 00:36:27.250 --> 00:36:29.194 which is what this is all about, is 00:36:29.194 --> 00:36:31.360 one of the hot topics in particle physics right now. 00:36:39.492 --> 00:36:41.200 AUDIENCE: I was just wondering if gravity 00:36:41.200 --> 00:36:43.660 has any gauge symmetry associated with it. 00:36:43.660 --> 00:36:47.390 PROFESSOR: It does, but it fits in a very, very different way, 00:36:47.390 --> 00:36:54.210 and with the exception of the fairly speculative framework 00:36:54.210 --> 00:36:56.810 of string theory-- which I think is very, very promising, 00:36:56.810 --> 00:36:58.280 but it's just sufficiently removed 00:36:58.280 --> 00:37:00.290 from experimental verification that I'm 00:37:00.290 --> 00:37:03.050 going to have to label it speculative-- it doesn't quite 00:37:03.050 --> 00:37:04.700 tie in in the same way. 00:37:04.700 --> 00:37:07.870 And that's the best I can say right now. 00:37:07.870 --> 00:37:10.030 The gauge symmetries of general relativity 00:37:10.030 --> 00:37:13.110 are, at the classical level, they correspond 00:37:13.110 --> 00:37:16.790 to coordinate transformations, at a quantum level, 00:37:16.790 --> 00:37:22.450 there's not such a simple way to put it. 00:37:22.450 --> 00:37:24.530 All right, where was I, OK, sorry 00:37:24.530 --> 00:37:25.780 I didn't get to your question. 00:37:25.780 --> 00:37:29.540 So, the point we made here is that we have spontaneously, 00:37:29.540 --> 00:37:32.400 when we actually choose which one of these infinite number 00:37:32.400 --> 00:37:35.040 of values we're going to have, we just 00:37:35.040 --> 00:37:36.500 randomly break the symmetry. 00:38:08.514 --> 00:38:10.180 OK, and you guys apparently have already 00:38:10.180 --> 00:38:12.900 talked a little bit about spontaneous symmetry breaking. 00:38:12.900 --> 00:38:14.730 The analogy that people often make 00:38:14.730 --> 00:38:17.430 is to the freezing of water, OK, prior 00:38:17.430 --> 00:38:21.620 to the water entering its solid phase its completely 00:38:21.620 --> 00:38:26.480 rotationally symmetric, then at a certain point 00:38:26.480 --> 00:38:28.310 crystalline planes start to form, 00:38:28.310 --> 00:38:30.850 the water forms, all the molecules 00:38:30.850 --> 00:38:32.990 get set into a particular orientation, 00:38:32.990 --> 00:38:35.940 you lose that rotational symmetry. 00:38:35.940 --> 00:38:41.130 In this case, we started out with a theory, 00:38:41.130 --> 00:38:44.740 with a set of interactions that were completely symmetric 00:38:44.740 --> 00:38:46.590 in sort of gauge field space. 00:38:46.590 --> 00:38:49.290 And now by settling down and picking 00:38:49.290 --> 00:38:52.730 a particular special value of h1 and h2 00:38:52.730 --> 00:38:55.435 we have at least nailed down one direction. 00:38:55.435 --> 00:38:58.110 It's like we've defined a crystalline plane, 00:38:58.110 --> 00:39:02.330 and so now things, suddenly, aren't as symmetric. 00:39:02.330 --> 00:39:06.770 And we start to pick out preferred directions 00:39:06.770 --> 00:39:12.150 in our gauge fields. 00:39:16.620 --> 00:39:20.610 What we can do with this is really 00:39:20.610 --> 00:39:23.200 a topic for a whole other course, 00:39:23.200 --> 00:39:25.550 and that course is called quantum field theory, 00:39:25.550 --> 00:39:27.539 but I will sketch a couple of the consequences 00:39:27.539 --> 00:39:29.705 and this gets directly to the answer your questions. 00:39:40.180 --> 00:39:42.740 So, one of the consequences of this 00:39:42.740 --> 00:39:48.950 is that once we have picked out a particular direction, 00:39:48.950 --> 00:39:55.210 electrons and neutrinos are different. 00:40:00.630 --> 00:40:02.780 When the Higgs field is equal to zero 00:40:02.780 --> 00:40:06.350 there is no difference between an electron and a neutrino. 00:40:06.350 --> 00:40:08.810 They obey exactly the same equation, 00:40:08.810 --> 00:40:11.450 there's literally no difference between them. 00:40:11.450 --> 00:40:14.940 Once we have actually settled on an h1 and an h2 00:40:14.940 --> 00:40:17.680 some combination of the fundamental underlying fields 00:40:17.680 --> 00:40:21.100 comes together, acquires a mass, acquires an electric charge, 00:40:21.100 --> 00:40:25.780 and we say A-HA thou beist an electron. 00:40:25.780 --> 00:40:31.093 It wasn't like that in the original unbroken symmetry. 00:40:31.093 --> 00:40:32.420 AUDIENCE: Also, [INAUDIBLE]? 00:40:32.420 --> 00:40:33.919 PROFESSOR: Presumably, but I'm going 00:40:33.919 --> 00:40:35.680 to stick with just these for now, 00:40:35.680 --> 00:40:40.490 but I've I'm pretty sure that's the case, yeah. 00:40:40.490 --> 00:40:42.800 That gets into even more complications of course 00:40:42.800 --> 00:40:44.770 because the additional generations are actually 00:40:44.770 --> 00:40:47.680 consequence presumably of some broken higher level symmetry, 00:40:47.680 --> 00:40:51.540 which is even poorly, more poorly understood. 00:40:51.540 --> 00:40:54.280 But you raise a good point. 00:40:54.280 --> 00:40:56.870 So, that's one partial answer your question. 00:40:56.870 --> 00:40:59.360 How one can actually walk that backwards 00:40:59.360 --> 00:41:01.730 to understand this thing about the initial state? 00:41:01.730 --> 00:41:05.410 That's hard to say. 00:41:16.760 --> 00:41:18.260 I actually think this particular one 00:41:18.260 --> 00:41:20.745 is one of the profound and interesting aspects 00:41:20.745 --> 00:41:27.440 of this, in part because we now know the neutrino has a mass. 00:41:27.440 --> 00:41:31.270 We have no idea what that is, and in fact we only really 00:41:31.270 --> 00:41:35.460 have bounds on the mass, such that we know it is non-zero, 00:41:35.460 --> 00:41:37.150 and we have upper limits that are 00:41:37.150 --> 00:41:39.450 set by very indirect measurements. 00:41:39.450 --> 00:41:41.190 But the actual values of the mass 00:41:41.190 --> 00:41:42.810 are very, very poorly constrained. 00:41:45.790 --> 00:41:48.590 Within the standard model you just 00:41:48.590 --> 00:41:50.610 take the electroweak interaction, 00:41:50.610 --> 00:41:52.780 introduce a Higgs coupling and allow the symmetry 00:41:52.780 --> 00:41:55.880 to be spontaneously broken, the neutrino mass is zero. 00:41:55.880 --> 00:41:57.494 Full stop zero. 00:41:57.494 --> 00:41:59.160 So something's not right, we're actually 00:41:59.160 --> 00:42:00.950 missing something here. 00:42:00.950 --> 00:42:03.120 People have kind of jury rigged the standard model 00:42:03.120 --> 00:42:06.920 to put in the masses by hand, and it works OK, 00:42:06.920 --> 00:42:09.770 but it's not completely satisfying. 00:42:09.770 --> 00:42:12.000 And a lot of experiments going on right now 00:42:12.000 --> 00:42:14.060 to explore the neutrino sector are hopefully 00:42:14.060 --> 00:42:17.000 going to open us up to a deeper understanding of this 00:42:17.000 --> 00:42:21.330 and may say a lot about all this physics, which is at present, 00:42:21.330 --> 00:42:24.140 pretty poorly understood. 00:42:24.140 --> 00:42:26.540 The consequence, which has received the most popular 00:42:26.540 --> 00:42:28.920 press, and what you guys have certainly 00:42:28.920 --> 00:42:31.860 seen about in newspapers, given the results that came out 00:42:31.860 --> 00:42:40.330 from the LHC over the past year is that quarks and leptons have 00:42:40.330 --> 00:42:45.000 mass, or put more specifically, rest mass. 00:42:50.930 --> 00:42:52.810 To understand what this actually means 00:42:52.810 --> 00:42:55.440 I think you really need to ask yourself what is mass 00:42:55.440 --> 00:42:56.000 meant to be. 00:43:01.560 --> 00:43:06.470 Well, the idea is you calculate the spectrum of oscillations 00:43:06.470 --> 00:43:36.560 associated with the fields of your theory, 00:43:36.560 --> 00:43:39.660 and then if your theory predicts a discrete spectrum 00:43:39.660 --> 00:43:42.180 of oscillations, it doesn't even have to be discrete 00:43:42.180 --> 00:43:44.560 but predict some spectrum of oscillations, 00:43:44.560 --> 00:43:47.790 then for every oscillation frequency omega 00:43:47.790 --> 00:43:53.340 there's an associated mass that is just 00:43:53.340 --> 00:43:58.190 H bar omega over c squared. 00:43:58.190 --> 00:44:03.630 If your omega has some lower bound that 00:44:03.630 --> 00:44:10.030 is greater than zero, then your theory 00:44:10.030 --> 00:44:14.310 has particles with nonzero rest mass. 00:44:56.082 --> 00:44:58.040 Without going into the details-- and this again 00:44:58.040 --> 00:44:59.970 is something which those of you who 00:44:59.970 --> 00:45:02.930 are going to go on to study this in more detail in a higher 00:45:02.930 --> 00:45:05.730 level course, which is fairly standard stuff is done 00:45:05.730 --> 00:45:09.897 in probably the first or maybe late in the first or early 00:45:09.897 --> 00:45:12.230 in the second semester of a typical quantum field theory 00:45:12.230 --> 00:45:16.250 course-- what you'll find is that when the Higgs field is 00:45:16.250 --> 00:45:19.470 zero then quarks and leptons have, 00:45:19.470 --> 00:45:22.890 the field that describes quarks and leptons-- and yes including 00:45:22.890 --> 00:45:25.210 mu and tau, so including all the leptons, this one 00:45:25.210 --> 00:45:27.530 I'm very confident on-- the spectrum 00:45:27.530 --> 00:45:30.550 goes all the way to zero if the Higgs field is zero. 00:45:58.150 --> 00:46:01.260 But when the Higgs field becomes non-zero, roughly speaking, 00:46:01.260 --> 00:46:04.500 it shifts the spectrum over for these particles. 00:46:04.500 --> 00:46:08.740 There's an interaction between the things like the electron 00:46:08.740 --> 00:46:12.880 field in the Higgs field or the up quark field and the Higgs 00:46:12.880 --> 00:46:16.400 field, which shifts the spectrum over just enough 00:46:16.400 --> 00:46:18.470 so that the frequency is never allowed 00:46:18.470 --> 00:46:20.598 to go below some minimum. 00:46:20.598 --> 00:46:22.590 AUDIENCE: Going back a bit, I'm confused 00:46:22.590 --> 00:46:25.950 about how picking a specific value to the Higgs field 00:46:25.950 --> 00:46:29.562 is breaking SU(2) symmetry and not U(1), 00:46:29.562 --> 00:46:32.060 because it seems like we're fixed on a circle, right? 00:46:32.060 --> 00:46:34.559 PROFESSOR: That's right what U(1) is a symmetry on a circle, 00:46:34.559 --> 00:46:37.354 SU(2) is kind of like symmetry on a sphere, essentially. 00:46:37.354 --> 00:46:40.763 AUDIENCE: Right, so how are we not picking a specific value 00:46:40.763 --> 00:46:42.932 [INAUDIBLE] circle [INAUDIBLE]? 00:46:42.932 --> 00:46:45.390 PROFESSOR: Well, what we're doing is, think of it this way, 00:46:45.390 --> 00:46:48.590 imagine SU(2) is a symmetry on a sphere, 00:46:48.590 --> 00:46:51.670 and then when we break the SU(2) symmetry 00:46:51.670 --> 00:46:54.460 it's like we're picking some circle on that sphere. 00:46:54.460 --> 00:46:57.370 So, we've broken one circle, we've picked one circle, 00:46:57.370 --> 00:46:58.870 but now we're allowed to go anywhere 00:46:58.870 --> 00:47:02.770 on that remaining circle, which is a U(1) symmetry. 00:47:02.770 --> 00:47:04.160 Does that help? 00:47:04.160 --> 00:47:05.145 Yeah, OK good. 00:47:09.110 --> 00:47:11.890 And it comes down to the fact if you sort of count up 00:47:11.890 --> 00:47:15.110 your degrees of freedom, it has to do with the fact 00:47:15.110 --> 00:47:20.930 you you've got four, you have two complex numbers, 00:47:20.930 --> 00:47:23.650 so there's four real parameters associated with this thing, 00:47:23.650 --> 00:47:28.060 and they are isomorphic to sort of rotations in a three space 00:47:28.060 --> 00:47:29.435 and you're adding one constraint. 00:47:33.220 --> 00:47:35.960 OK, so let me just finish making this point here again. 00:47:35.960 --> 00:47:41.240 So, when h does not equal zero, spectrum 00:47:41.240 --> 00:48:00.340 get shifted for the quarks and leptons, 00:48:00.340 --> 00:48:03.280 so everything picks up a little bit of a mass. 00:48:03.280 --> 00:48:06.025 And the final one, final consequence 00:48:06.025 --> 00:48:11.400 which we're going to talk about today, 00:48:11.400 --> 00:48:15.765 is that the universe is filled with magnetic monopoles. 00:48:25.550 --> 00:48:28.040 We all remember studying Maxwell's equations 00:48:28.040 --> 00:48:31.140 learning that del dot b is equal to 4 pi 00:48:31.140 --> 00:48:32.750 times the density of magnetic charge-- 00:48:32.750 --> 00:48:35.520 this all makes perfect sense, right? 00:48:35.520 --> 00:48:42.000 Well, this is actually something that when it first sort of came 00:48:42.000 --> 00:48:45.570 out and people begin to appreciate this thing with sort 00:48:45.570 --> 00:48:50.510 of a "Um, well everything else works so well, 00:48:50.510 --> 00:48:55.340 maybe we're just not looking hard enough. " So, 00:48:55.340 --> 00:48:57.085 it was a bit of a surprise. 00:49:06.940 --> 00:49:09.460 So, where do these magnetic monopoles come from? 00:49:31.580 --> 00:49:34.120 And essentially, the magnetic monopoles 00:49:34.120 --> 00:49:38.010 are going to turn out to be a consequence of the fact 00:49:38.010 --> 00:49:41.770 that when spontaneous symmetry breaking happens 00:49:41.770 --> 00:49:44.715 it doesn't happen everywhere simultaneously. 00:50:26.210 --> 00:50:28.319 So, think again about-- yeah? 00:50:28.319 --> 00:50:30.110 AUDIENCE: Doesn't that bring up possibility 00:50:30.110 --> 00:50:32.068 that the symmetry could break in different ways 00:50:32.068 --> 00:50:32.940 in different places? 00:50:32.940 --> 00:50:34.780 PROFESSOR: That is in fact exactly what this 00:50:34.780 --> 00:50:35.750 is going to be. 00:50:35.750 --> 00:50:38.430 Magnetic monopoles are in fact exactly a consequence 00:50:38.430 --> 00:50:42.052 of this, yes. 00:50:42.052 --> 00:50:43.510 Give me a few moments to step ahead 00:50:43.510 --> 00:50:46.517 to fill in a couple of the gaps, but you're basically 00:50:46.517 --> 00:50:47.100 already there. 00:50:50.160 --> 00:50:56.480 So, think about crystalline crystal formation again. 00:50:56.480 --> 00:51:03.030 Imagine you have, we could do ice 00:51:03.030 --> 00:51:05.604 if you like or choose something that's got a little bit more 00:51:05.604 --> 00:51:07.270 of an interesting crystalline structure. 00:51:07.270 --> 00:51:12.420 Imagine you have a big bucket full of molten quarts, OK. 00:51:12.420 --> 00:51:14.550 So, if you have a big thing of quartz 00:51:14.550 --> 00:51:18.740 that you want to sort of freeze into a single gigantic crystal, 00:51:18.740 --> 00:51:21.380 what you typically do if you'd like to do this is 00:51:21.380 --> 00:51:23.710 you actually seed it with a little bit 00:51:23.710 --> 00:51:26.340 of a starter crystal. 00:51:26.340 --> 00:51:29.060 So, you put a little bit of crystal into this thing, 00:51:29.060 --> 00:51:33.390 and what that does is it sort of defines a preferred orientation 00:51:33.390 --> 00:51:37.570 of the crystal axes, so that as things 00:51:37.570 --> 00:51:39.920 start to cool in the vicinity of that they have 00:51:39.920 --> 00:51:42.420 a preferred orientation to grab on to. 00:51:42.420 --> 00:51:44.330 And that seed then gradually gets 00:51:44.330 --> 00:51:47.560 bigger and bigger and bigger, and all the little crystals 00:51:47.560 --> 00:51:50.120 as they form near it tend to latch 00:51:50.120 --> 00:51:52.350 onto the preexisting crystalline structure, 00:51:52.350 --> 00:51:54.010 and that allows you to grow actually 00:51:54.010 --> 00:51:55.819 extremely large crystals. 00:51:55.819 --> 00:51:58.360 I don't know if anyone here is doing a year off with the LIGO 00:51:58.360 --> 00:52:02.800 project but these guys have to make these sort of 100 kilogram 00:52:02.800 --> 00:52:07.990 mirrors of very pure either Sapphire or silicon dioxide, 00:52:07.990 --> 00:52:10.574 and when you make 100 kilograms of crystal 00:52:10.574 --> 00:52:12.490 you need to build it really, really carefully. 00:52:12.490 --> 00:52:14.531 It's extremely important for the optical purposes 00:52:14.531 --> 00:52:17.150 that all the axes associated with the crystal 00:52:17.150 --> 00:52:18.970 will be pointing in the right direction. 00:52:18.970 --> 00:52:20.780 Otherwise you spend $100,000 on this thing 00:52:20.780 --> 00:52:25.070 and it ends up being the world's prettiest paperweight. 00:52:25.070 --> 00:52:28.880 So, similar things happen when the Higgs field cools. 00:52:28.880 --> 00:52:34.150 Let's imagine that we've got our universe, 00:52:34.150 --> 00:52:39.730 time going forward like this, and at some point over here 00:52:39.730 --> 00:52:45.390 the universe cools enough that's the Higgs field condenses 00:52:45.390 --> 00:52:46.900 into some particular direction. 00:52:46.900 --> 00:52:52.370 And symmetry is spontaneously broken right at this one point 00:52:52.370 --> 00:52:54.199 over here. 00:52:54.199 --> 00:52:55.990 So, I'm going to draw my diagram over there 00:52:55.990 --> 00:52:57.198 and put some words over here. 00:53:18.622 --> 00:53:20.330 I shouldn't say Higgs field cools enough, 00:53:20.330 --> 00:53:22.590 the universe cools enough so that the Higgs 00:53:22.590 --> 00:53:23.905 field breaks the symmetry. 00:53:43.230 --> 00:53:49.370 So, just to be concrete, let's imagine that at 0.1 over here 00:53:49.370 --> 00:54:00.476 it takes on a field of the value one for h1 and I for h2. 00:54:00.476 --> 00:54:02.600 So just for concreteness imagine it looks something 00:54:02.600 --> 00:54:05.297 like this at this point. 00:54:05.297 --> 00:54:07.380 And so what happens is as the university continues 00:54:07.380 --> 00:54:11.020 to expand other areas are going to cool off. 00:54:11.020 --> 00:54:12.620 The bits that are closest to it are 00:54:12.620 --> 00:54:15.010 going to see that there is already 00:54:15.010 --> 00:54:18.650 a preferred orientation defined by the Higgs field. 00:54:18.650 --> 00:54:21.280 And so it's energetically favorable for those regions 00:54:21.280 --> 00:54:23.750 of the universe to fall into the same alignment 00:54:23.750 --> 00:54:26.390 and so there'll be a region in space times 00:54:26.390 --> 00:54:31.080 that grows here as the universe cools, in which the Higgs 00:54:31.080 --> 00:54:34.640 field all falls into this configuration, which 00:54:34.640 --> 00:54:35.660 I will call h1. 00:54:44.920 --> 00:54:49.000 But suppose somewhere over here at 0.2, 00:54:49.000 --> 00:54:51.920 and the key thing is that initially 0.2 00:54:51.920 --> 00:54:55.580 is going to be so far away from 0.1 that these points are 00:54:55.580 --> 00:54:57.210 out of causal contact with one another. 00:54:57.210 --> 00:55:01.410 I can not send a message from event one to event two. 00:55:01.410 --> 00:55:03.450 The Higgs field also reaches a point 00:55:03.450 --> 00:55:07.220 that the universe cools enough that at 0.2, just you know, 00:55:07.220 --> 00:55:10.717 it's a system that's not in thermal equilibrium. 00:55:10.717 --> 00:55:12.550 So, some places are going to be a little bit 00:55:12.550 --> 00:55:13.604 hotter than others, some are going 00:55:13.604 --> 00:55:14.687 to be a little bit cooler. 00:55:14.687 --> 00:55:18.040 And so, at these two points it just so happened 00:55:18.040 --> 00:55:19.850 that the Higgs field got to the point 00:55:19.850 --> 00:55:22.250 where it could spontaneously break the symmetry. 00:55:22.250 --> 00:55:31.215 So at 0.2 the Higgs field also got to the point 00:55:31.215 --> 00:55:33.215 where it could spontaneously break its symmetry. 00:56:00.295 --> 00:56:01.920 And the only thing that's got to happen 00:56:01.920 --> 00:56:04.140 is, remember the only constraint we have is 00:56:04.140 --> 00:56:05.762 that the magnitude of the Higgs field 00:56:05.762 --> 00:56:08.220 be equals to some value-- I should normalize that to root 2 00:56:08.220 --> 00:56:09.720 in units I want to use but whatever. 00:56:09.720 --> 00:56:15.510 Let's say on this one my h1 is equal to y, 00:56:15.510 --> 00:56:17.890 and h2 is equal to minus 1. 00:56:21.265 --> 00:56:22.640 So, it's basically the same thing 00:56:22.640 --> 00:56:25.727 but all the fields are multiplied by i. 00:56:25.727 --> 00:56:27.310 It's the same magnitude, so it's going 00:56:27.310 --> 00:56:29.370 to have the same potential energy. 00:56:29.370 --> 00:56:31.150 So that's cool. 00:56:31.150 --> 00:56:36.370 Clearly this is allowed, and now all the regions 00:56:36.370 --> 00:56:38.311 in the universe that are close to the this 00:56:38.311 --> 00:56:40.560 are going to sort of smell this particular arrangement 00:56:40.560 --> 00:56:42.101 of the Higgs field and say OK, that's 00:56:42.101 --> 00:56:45.410 preferred arrangement I want to go into. 00:56:45.410 --> 00:56:48.740 So, we have two separate values of the Higgs field 00:56:48.740 --> 00:56:53.320 that are happily swooping out space time here. 00:56:53.320 --> 00:56:55.180 This gets to the excellent question 00:56:55.180 --> 00:57:00.670 I was just asked a moment ago-- what happens when they collide? 00:57:00.670 --> 00:57:04.190 As the universe expands and gets cooler, all of it 00:57:04.190 --> 00:57:07.060 is going to end up getting swooped 00:57:07.060 --> 00:57:09.810 into either the field that was seeded at event one, 00:57:09.810 --> 00:57:11.754 or the field that was seeded in to event two, 00:57:11.754 --> 00:57:13.170 but at a certain point we're going 00:57:13.170 --> 00:57:15.890 to get the bits where they're smashing into one another. 00:57:55.810 --> 00:57:58.630 So what happens when these different domains come 00:57:58.630 --> 00:58:00.530 into contact with one another? 00:58:05.640 --> 00:58:08.210 The absolutely full and probably correct 00:58:08.210 --> 00:58:11.390 answer is we don't know. 00:58:11.390 --> 00:58:14.470 The reason is that we don't really, to be perfectly blunt, 00:58:14.470 --> 00:58:20.150 fully understand every little detail about the symmetry 00:58:20.150 --> 00:58:24.580 breaking, or about the structure of whatever 00:58:24.580 --> 00:58:26.880 grand unified theory brings all these things together 00:58:26.880 --> 00:58:28.310 at the temperatures at which this is happening. 00:58:28.310 --> 00:58:30.268 Because this is happening when the universe has 00:58:30.268 --> 00:58:32.960 a temperature of like 10 to the 16th GeV. 00:58:32.960 --> 00:58:34.730 And so it's way beyond the domain 00:58:34.730 --> 00:58:36.830 of where we can push things. 00:58:36.830 --> 00:58:41.000 But we can, as physicists are fond of doing, 00:58:41.000 --> 00:58:43.570 we can paramaterize our ignorance, 00:58:43.570 --> 00:58:45.960 and we can ask ourselves, well what 00:58:45.960 --> 00:58:49.480 happens if these various parameters that characterize 00:58:49.480 --> 00:58:52.770 my grand unified theory take on the following plausible kinds 00:58:52.770 --> 00:58:54.000 of parameters. 00:58:54.000 --> 00:58:56.216 And what we find is that generically, 00:58:56.216 --> 00:58:58.340 when you have two different domains where the Higgs 00:58:58.340 --> 00:59:01.480 field takes on different values like this, when 00:59:01.480 --> 00:59:03.560 these domains come into contact you 00:59:03.560 --> 00:59:05.345 get what are called topological defects. 00:59:35.880 --> 00:59:39.299 The topological defects come in three different flavors. 00:59:39.299 --> 00:59:41.090 To understand something about those flavors 00:59:41.090 --> 00:59:42.756 you have to know a little bit about what 00:59:42.756 --> 00:59:47.430 happens in general when you have phase transitions, 00:59:47.430 --> 00:59:51.410 and different regions of your medium 00:59:51.410 --> 00:59:53.690 go through a phase transition with different values 00:59:53.690 --> 00:59:55.070 of the parameters. 00:59:55.070 --> 00:59:57.100 So, it's a general case that whenever 00:59:57.100 --> 01:00:02.250 you have some kind of a phase transition 01:00:02.250 --> 01:00:04.560 and you have domains of different phase 01:00:04.560 --> 01:00:07.930 that come into contact with one another, 01:00:07.930 --> 01:00:29.300 your field will attempt to smoothly match itself 01:00:29.300 --> 01:00:30.235 across the boundary. 01:00:35.790 --> 01:00:37.110 But that can be very difficult. 01:00:37.110 --> 01:00:38.980 So if you imagine these particular two cases 01:00:38.980 --> 01:00:41.150 that I have here, that's essentially 01:00:41.150 --> 01:00:42.690 saying that when these two domains 01:00:42.690 --> 01:00:44.530 coming to contact with one another 01:00:44.530 --> 01:00:46.830 there's going to be sort of a transition zone 01:00:46.830 --> 01:00:49.590 where the field is attempting to rotate 01:00:49.590 --> 01:00:52.130 from one value of the Higgs to the other. 01:00:52.130 --> 01:00:54.630 And it's going to pick some value that is in some sense 01:00:54.630 --> 01:00:56.240 intermediate to those two things. 01:00:56.240 --> 01:00:58.390 So that, let's say we continue these up here, 01:00:58.390 --> 01:01:03.890 so that the collision is occurring right in this place 01:01:03.890 --> 01:01:06.920 here, in this little locus of events in space time. 01:01:06.920 --> 01:01:10.830 I have Higgs field 2 over here, Higgs field 1 over here, 01:01:10.830 --> 01:01:13.232 and I've got some crazy intermediate field that 01:01:13.232 --> 01:01:14.690 goes between the two of them, which 01:01:14.690 --> 01:01:19.760 is trying to sort of force itself to smoothly transition 01:01:19.760 --> 01:01:21.570 from one to the other. 01:01:21.570 --> 01:01:25.010 In so doing, I might end up pushing my field away 01:01:25.010 --> 01:01:27.120 from the minimum, in which case there will then 01:01:27.120 --> 01:01:31.320 be some energy trapped in that layer. 01:01:31.320 --> 01:01:34.920 And there's a reason we do this level the class 01:01:34.920 --> 01:01:36.540 in a bit of a hand wavy way, I mean 01:01:36.540 --> 01:01:39.490 it's very, very complicated to get the details right. 01:01:39.490 --> 01:01:43.770 But the key thing we see is that in doing this match, 01:01:43.770 --> 01:01:46.782 the field has to do some pretty silly shenanigans order 01:01:46.782 --> 01:01:48.240 to make everything kind of match up 01:01:48.240 --> 01:01:52.430 and we can be left with odd observable consequences 01:01:52.430 --> 01:01:55.110 from the energy associated with the Higgs field getting 01:01:55.110 --> 01:01:57.210 pinned down at that boundary here. 01:01:57.210 --> 01:02:02.480 Now, the details of the forms of this boundary 01:02:02.480 --> 01:02:05.730 vary a lot depending upon to the specific assumptions 01:02:05.730 --> 01:02:08.330 you make about your underlying grand unified theory. 01:02:36.230 --> 01:02:38.720 OK, so I should back up for a bit. 01:02:38.720 --> 01:02:41.270 I'm sort of assuming here when I discuss all this that there 01:02:41.270 --> 01:02:45.750 is some underlying SU(5) theory which describes 01:02:45.750 --> 01:02:48.590 the strong weak and electromagnetic interactions 01:02:48.590 --> 01:02:51.114 are very, very high temperatures as one gigantic thing. 01:02:51.114 --> 01:02:52.530 And we're getting to the point now 01:02:52.530 --> 01:02:54.060 where all the different interactions are beginning 01:02:54.060 --> 01:02:55.643 to just sort of crystallize out of it. 01:02:59.620 --> 01:03:02.920 There's a lot of different ways you can pack your underlying, 01:03:02.920 --> 01:03:05.470 fundamental, what we now think of as our standard model, 01:03:05.470 --> 01:03:10.010 into SU(5) grand unified theories. 01:03:10.010 --> 01:03:11.560 And so the ways in which we can get 01:03:11.560 --> 01:03:13.640 different topological defects depend 01:03:13.640 --> 01:03:15.580 upon how we choose to do that. 01:03:15.580 --> 01:03:26.580 So defect flavor one is you get something called a domain wall. 01:03:34.870 --> 01:03:37.600 When we do this the fields attempts 01:03:37.600 --> 01:03:42.170 to make itself smoothly match from one region of Higgs field, 01:03:42.170 --> 01:03:44.850 say from Higgs 1 to Higgs 2. 01:03:44.850 --> 01:03:46.510 It succeeds, but you end up with kind 01:03:46.510 --> 01:03:52.070 of a two dimensional structure-- a wall-- in which there's 01:03:52.070 --> 01:03:55.570 some kind of anomalous field that is just pinned down there. 01:03:59.290 --> 01:04:01.145 And so we end up with a big sheet. 01:04:36.430 --> 01:04:38.240 So in a theory like this, it would 01:04:38.240 --> 01:04:40.600 predict that somewhere out in the universe if there were 01:04:40.600 --> 01:04:42.350 regions in which the Higgs field had taken 01:04:42.350 --> 01:04:43.725 on a different value than the one 01:04:43.725 --> 01:04:45.969 that we encounter around us right now, 01:04:45.969 --> 01:04:48.260 it could be somewhere out gigaparsecs away, essentially 01:04:48.260 --> 01:04:50.477 a giant sheet of some kind. 01:04:50.477 --> 01:04:52.310 And there would be weird, anomalous behavior 01:04:52.310 --> 01:04:53.620 associated with it. 01:04:53.620 --> 01:04:55.940 People have really looked long and hard 01:04:55.940 --> 01:04:58.500 to try to find things like this and in fact it 01:04:58.500 --> 01:05:01.380 would be expected to leave interesting residuals 01:05:01.380 --> 01:05:03.630 in the cause of microwave background. 01:05:03.630 --> 01:05:05.230 My understanding of the literature 01:05:05.230 --> 01:05:07.290 is that there are actually now very 01:05:07.290 --> 01:05:10.187 strong bounds on the possibility of having 01:05:10.187 --> 01:05:12.270 a grand unified theory that leads to domain walls. 01:05:12.270 --> 01:05:15.110 And so this kind of a topological defect 01:05:15.110 --> 01:05:17.055 is observationally disfavored. 01:05:19.730 --> 01:05:23.820 So this, I should mention, only occurs 01:05:23.820 --> 01:05:27.210 in some grand unified theories. 01:05:38.360 --> 01:05:48.920 Basically, As we move on to the other flavors of defects 01:05:48.920 --> 01:05:53.950 we end up just going down a step in dimensionality associated 01:05:53.950 --> 01:05:55.845 with the little kinks that are left over when 01:05:55.845 --> 01:05:58.220 the different domains come into contact with one another. 01:06:02.550 --> 01:06:08.070 Flavor two, we would get what's called a cosmic string. 01:06:11.150 --> 01:06:12.850 Some of you may have heard of this. 01:06:12.850 --> 01:06:18.920 This is essentially, at its core, just a one dimensional, 01:06:18.920 --> 01:06:22.684 it could be gigparsecs long, but one dimensional, truly 01:06:22.684 --> 01:06:24.100 one dimensional-- essentially just 01:06:24.100 --> 01:06:28.590 a point in the other two dimensions-- string of mismatch 01:06:28.590 --> 01:06:31.150 Higgs field with some kind of an energy density associated 01:06:31.150 --> 01:06:32.995 with it when the different domains get in contact. 01:06:32.995 --> 01:06:34.390 AUDIENCE: Do we have any estimate 01:06:34.390 --> 01:06:38.110 of how close in actual space these different regions would 01:06:38.110 --> 01:06:39.050 have started? 01:06:39.050 --> 01:06:42.430 PROFESSOR: We do and I'm actually going to get to that. 01:06:42.430 --> 01:06:45.390 So, let me give you two answers to that. 01:06:45.390 --> 01:06:48.870 One of them is you are going to estimate that apparently 01:06:48.870 --> 01:06:52.760 on PSET 10, according to the notes that Alan left for me. 01:06:52.760 --> 01:06:55.440 But I'm going to spell out for you the arguments that 01:06:55.440 --> 01:06:58.320 go into it in the last 10 minutes of a class. 01:06:58.320 --> 01:07:01.350 But yeah, so let me just quickly finish up this one because this 01:07:01.350 --> 01:07:04.100 again-- so a cosmic string is sort of like a one dimensional 01:07:04.100 --> 01:07:05.570 analog of a domain wall. 01:07:14.596 --> 01:07:16.470 And because it would be this sort of long one 01:07:16.470 --> 01:07:20.140 dimensional structure, that has actually up a lot of energy 01:07:20.140 --> 01:07:22.560 sort of pinned down to it by the fact it has a Higgs 01:07:22.560 --> 01:07:25.974 anomaly associated with it, it would be strongly gravitating 01:07:25.974 --> 01:07:28.140 and so it would leave really interesting signatures. 01:07:28.140 --> 01:07:30.910 It was thought for a while that cosmic strings might 01:07:30.910 --> 01:07:33.490 have been the sort of original gravitational anomalies 01:07:33.490 --> 01:07:35.510 that seeded some of the structures we see 01:07:35.510 --> 01:07:36.700 in the universe today. 01:07:36.700 --> 01:07:38.960 Again, it's now pretty highly disfavored. 01:07:41.690 --> 01:07:43.910 If cosmic strings exist, they don't 01:07:43.910 --> 01:07:49.944 appear to contribute very much to the budget of mass 01:07:49.944 --> 01:07:50.610 in our universe. 01:07:53.266 --> 01:07:54.890 I should also mention that this is only 01:07:54.890 --> 01:07:57.120 predicted by some grand unifying theories. 01:08:03.244 --> 01:08:04.660 If you guys are curious about this 01:08:04.660 --> 01:08:06.590 I suggest when Alice back you ask him 01:08:06.590 --> 01:08:08.650 what the difference between these sums, 01:08:08.650 --> 01:08:10.150 why some predict a domain wall, some 01:08:10.150 --> 01:08:11.275 predict the cosmic strings. 01:08:14.820 --> 01:08:18.680 Flavor three is where you end up with the Higgs 01:08:18.680 --> 01:08:21.609 field essentially being able to smoothly transition 01:08:21.609 --> 01:08:25.000 without leaving any defect anywhere except at a zero 01:08:25.000 --> 01:08:27.840 dimensional point. 01:08:27.840 --> 01:08:42.029 So you end up with just a little knot in the Higgs field. 01:08:45.300 --> 01:08:51.500 And for reasons that I will outline very soon, 01:08:51.500 --> 01:08:55.604 it turns out that this little not must carry magnetic charge, 01:08:55.604 --> 01:08:57.479 and so it must be a magnetic monopole. 01:09:03.420 --> 01:09:06.290 The domain walls and the cosmic strings 01:09:06.290 --> 01:09:11.460 are, as I've emphasized, only predicted 01:09:11.460 --> 01:09:14.609 by certain specific grand unified theories. 01:09:14.609 --> 01:09:17.290 Magnetic monopoles are actually predicted by all of them. 01:09:17.290 --> 01:09:18.020 Question. 01:09:18.020 --> 01:09:21.490 AUDIENCE: What does it mean to have a one dimensional domain 01:09:21.490 --> 01:09:24.440 wall, because there's no different region separated 01:09:24.440 --> 01:09:25.451 by one [INAUDIBLE]. 01:09:25.451 --> 01:09:26.450 PROFESSOR: That's right. 01:09:26.450 --> 01:09:27.779 So what ends up happening, and this 01:09:27.779 --> 01:09:29.320 is where I think you're going to have 01:09:29.320 --> 01:09:31.850 to ask Alan to sort of follow up on this a little bit. 01:09:31.850 --> 01:09:35.330 So, as the domains come into contact with one another. 01:09:35.330 --> 01:09:38.970 The fields do their best to smoothly transition from one 01:09:38.970 --> 01:09:39.880 to the other. 01:09:39.880 --> 01:09:42.810 And grand unified theories that predict a cosmic string, 01:09:42.810 --> 01:09:44.852 they succeed pretty much everywhere. 01:09:44.852 --> 01:09:47.060 They're able to actually smoothly make it all go away 01:09:47.060 --> 01:09:49.018 so you don't end up with feel being pinned down 01:09:49.018 --> 01:09:53.290 anywhere, except in a little one dimensional singularity that 01:09:53.290 --> 01:09:57.020 is somewhere along where the two dimensional services originally 01:09:57.020 --> 01:09:58.050 met. 01:09:58.050 --> 01:10:00.224 And that is-- there's details there 01:10:00.224 --> 01:10:01.890 that I'm not even pretending to explain. 01:10:05.190 --> 01:10:07.120 And as I say, those are only predicted 01:10:07.120 --> 01:10:09.100 by certain kinds of grand unified theories. 01:10:09.100 --> 01:10:11.475 All of them will then predict that even if you don't have 01:10:11.475 --> 01:10:14.730 that, that cosmic string will then shrink itself down 01:10:14.730 --> 01:10:17.270 and it'll just be left with a little knot of Higgs field, 01:10:17.270 --> 01:10:20.380 where there's a little bit of residual mismatch 01:10:20.380 --> 01:10:23.041 between the two regions. 01:10:23.041 --> 01:10:24.915 AUDIENCE: Do all three types of defects carry 01:10:24.915 --> 01:10:26.560 a magnetic charge, or only the knots? 01:10:26.560 --> 01:10:28.230 PROFESSOR: I think only the knots. 01:10:28.230 --> 01:10:29.950 They do carry other kinds of fields, 01:10:29.950 --> 01:10:32.830 though, in particular the other ones gravitate, in fact 01:10:32.830 --> 01:10:34.240 all them gravitate, and so that's 01:10:34.240 --> 01:10:35.400 one of the ways in which people have 01:10:35.400 --> 01:10:37.546 tried to set observational limits on these things. 01:10:37.546 --> 01:10:38.920 In particular there have recently 01:10:38.920 --> 01:10:40.740 been a fair amount of work of people trying 01:10:40.740 --> 01:10:43.920 to set limits on cosmic strings from gravitational lensing, 01:10:43.920 --> 01:10:46.274 and there was really a lot of excitement 01:10:46.274 --> 01:10:48.440 because people thought they discovered want a couple 01:10:48.440 --> 01:10:50.436 years ago. 01:10:50.436 --> 01:10:54.342 And they saw basically two quasars 01:10:54.342 --> 01:10:56.550 that looked absolutely identical, that were separated 01:10:56.550 --> 01:11:00.904 or scale that was just right to be a cosmic string. 01:11:00.904 --> 01:11:03.320 And then people actually looked at with better telescopes, 01:11:03.320 --> 01:11:05.778 and saw they had absolutely nothing to do with one another. 01:11:05.778 --> 01:11:09.580 They were not cosmic, they were not lenses, it's just every now 01:11:09.580 --> 01:11:12.260 and then God is screwing with you. 01:11:12.260 --> 01:11:18.590 OK, so without going into some of the details what you have, 01:11:18.590 --> 01:11:24.500 these little point like defects-- and I'm short on time 01:11:24.500 --> 01:11:27.490 so I'm going to kind of go through this a little bit 01:11:27.490 --> 01:11:31.090 in a sketchy way enough so that I can pay for you how 01:11:31.090 --> 01:11:33.250 to do some calculations you're going to need to do. 01:11:36.230 --> 01:11:46.250 So the point like defects end up being regions, 01:11:46.250 --> 01:11:55.850 where at that point the Higgs field actually takes 01:11:55.850 --> 01:11:57.480 the value zero. 01:11:57.480 --> 01:12:00.150 So remember I was describing how when you have two regions where 01:12:00.150 --> 01:12:07.890 the Higgs fields are both taking on values such as there 01:12:07.890 --> 01:12:10.597 at the minimum of the Higgs potential energy, 01:12:10.597 --> 01:12:12.180 and they come in to match one another, 01:12:12.180 --> 01:12:13.721 and what we have a boundary condition 01:12:13.721 --> 01:12:16.770 that very far away the Higgs field has values 01:12:16.770 --> 01:12:19.190 such as the energy is minimized. 01:12:19.190 --> 01:12:25.140 And there is a theorem, which in his notes Alan-- the way 01:12:25.140 --> 01:12:26.930 he describes it is he gives you a figure 01:12:26.930 --> 01:12:30.090 and outlines the various things that 01:12:30.090 --> 01:12:32.310 are necessary for the theorem to be true, 01:12:32.310 --> 01:12:34.410 and invites you to think deeply for a moment 01:12:34.410 --> 01:12:40.330 and until insight comes to you, I guess. 01:12:40.330 --> 01:12:43.340 And when you put this ingredient that the Higgs field has 01:12:43.340 --> 01:12:46.020 this asymptotic, very far away value that drives you 01:12:46.020 --> 01:12:49.580 to the minimum of the field, and yet it must change value 01:12:49.580 --> 01:12:51.810 somewhere in the middle, the theorem 01:12:51.810 --> 01:12:56.089 requires that there be one point at which H equals 0. 01:12:56.089 --> 01:12:57.630 And apparently, this is a consequence 01:12:57.630 --> 01:13:00.520 in all grand unified theories. 01:13:00.520 --> 01:13:02.640 So, recall, H equals 0. 01:13:02.640 --> 01:13:14.990 This is a point at which the potential energy 01:13:14.990 --> 01:13:16.580 density can be huge. 01:13:29.760 --> 01:13:32.470 So, when you have a little point like defect like this, 01:13:32.470 --> 01:13:36.440 it looks like a massive nugget, little massive particle. 01:13:47.640 --> 01:13:51.170 You can in fact calculate the total amount 01:13:51.170 --> 01:13:53.620 of energy associated with this particle. 01:13:53.620 --> 01:14:16.897 If you do so just including the influence of the Higgs field, 01:14:16.897 --> 01:14:19.440 the calculation basically goes like this. 01:14:19.440 --> 01:14:22.110 It's very similar to the way we calculate the energy associated 01:14:22.110 --> 01:14:25.850 with electric and magnetic fields in electrodynamics. 01:14:25.850 --> 01:14:33.010 Ask yourself, how much energy is contained 01:14:33.010 --> 01:14:39.800 in a sphere of radius, capital R, 01:14:39.800 --> 01:14:44.520 centered on this little knot of Higgs field. 01:14:44.520 --> 01:14:48.380 Well, it's going to look like 4pi times 01:14:48.380 --> 01:14:53.460 an integral of the gradient of the Higgs field 01:14:53.460 --> 01:15:06.080 squared r squared dr It turns out, 01:15:06.080 --> 01:15:10.260 when you calculate the [INAUDIBLE] of the Higgs field 01:15:10.260 --> 01:15:12.770 around one of these little defects, 01:15:12.770 --> 01:15:15.060 it's actually very complicated close to the defect, 01:15:15.060 --> 01:15:17.910 but as you get far away it has a very simple form. 01:15:30.860 --> 01:15:33.742 The gradient goes as 1 over r, it 01:15:33.742 --> 01:15:35.200 tells you the field itself actually 01:15:35.200 --> 01:15:36.380 goes something like log. 01:15:44.370 --> 01:15:51.950 That means your energy looks something like, R squared, 01:15:51.950 --> 01:16:00.070 1 over R squared dr which goes as R, which diverges as you 01:16:00.070 --> 01:16:02.050 make the sphere bigger and bigger and bigger. 01:16:10.209 --> 01:16:11.500 So, what's the mistake we made? 01:16:14.400 --> 01:16:16.719 Well, the Higgs field doesn't always 01:16:16.719 --> 01:16:18.260 just sit there and operate on itself. 01:16:18.260 --> 01:16:21.030 The Higgs field actually couples pretty strongly 01:16:21.030 --> 01:16:23.340 to all of our vector bosons. 01:16:23.340 --> 01:16:25.160 Particularly, it couples pretty strongly 01:16:25.160 --> 01:16:27.390 to electric and magnetic fields. 01:16:27.390 --> 01:16:34.360 So, we have to repeat this calculation including 01:16:34.360 --> 01:16:46.050 the interaction of the Higgs field with the E&D field. 01:17:04.500 --> 01:17:06.500 And in Alan's notes he gives you some references 01:17:06.500 --> 01:17:08.791 on this because this is not the kind of calculation you 01:17:08.791 --> 01:17:10.330 can really sketch out very easily 01:17:10.330 --> 01:17:12.590 in an undergraduate class. 01:17:12.590 --> 01:17:17.140 To make this integral convergent, 01:17:17.140 --> 01:17:21.170 the only way it can be done is if that little nugget 01:17:21.170 --> 01:17:24.410 of Higgs field is endowed with magnetic charge. 01:17:24.410 --> 01:17:29.780 You need to have a monopolar magnetic field that ends up 01:17:29.780 --> 01:17:31.920 putting in interaction terms, that 01:17:31.920 --> 01:17:34.270 make the divergence of this integral go away. 01:18:02.830 --> 01:18:09.470 So, I at last get to the punchline of all this, 01:18:09.470 --> 01:18:17.000 we are left inevitably, if we accept the whole foundation 01:18:17.000 --> 01:18:20.130 story of particle physics that the different interactions were 01:18:20.130 --> 01:18:23.360 unified in some high energy scale and then froze out. 01:18:32.959 --> 01:18:34.500 We are driven inevitably to the story 01:18:34.500 --> 01:18:46.995 that defects in the Higgs field create magnetic monopoles. 01:18:55.670 --> 01:18:58.660 Now, I realize I'm out of time, so let me just quickly sketch 01:18:58.660 --> 01:19:00.317 a few interesting facts about this 01:19:00.317 --> 01:19:02.650 and there's a few exercises that you guys are apparently 01:19:02.650 --> 01:19:06.640 going to look at in your homework assignment. 01:19:06.640 --> 01:19:11.410 When we do this calculation, one which 01:19:11.410 --> 01:19:13.200 is I believe just referenced in the notes 01:19:13.200 --> 01:19:16.150 that Alan has for the class, we learn a couple 01:19:16.150 --> 01:19:17.910 of things about this magnetic charge. 01:19:24.300 --> 01:19:27.020 One of them is that if you work in the fundamental unit, 01:19:27.020 --> 01:19:31.470 say CGS units, the value of the magnetic charge, 01:19:31.470 --> 01:19:42.350 we'll call that g, is exactly 1 over 01:19:42.350 --> 01:19:44.270 2 alpha where alpha is a fine structure 01:19:44.270 --> 01:19:46.540 constant, times the electric charge. 01:19:49.980 --> 01:19:52.070 So if you have two magnetic monopoles 01:19:52.070 --> 01:19:53.570 they attract each other with a force 01:19:53.570 --> 01:19:58.720 that is-- so 1 over 2 alpha is approximately 68.5 I think-- 01:19:58.720 --> 01:20:02.420 and so it would be 68.5 squared times the force of two 01:20:02.420 --> 01:20:04.570 electric charges at that same distance. 01:20:04.570 --> 01:20:06.100 We also end up learning the mass. 01:20:11.010 --> 01:20:19.381 It turns out to be 1 over alpha times the scale of GUT symmetry 01:20:19.381 --> 01:20:19.880 breaking. 01:20:30.945 --> 01:20:33.320 Anyone recall what the scale of GUT symmetry breaking is? 01:20:38.230 --> 01:20:40.080 10 to the 16 GeV. 01:20:40.080 --> 01:20:43.400 So, this is a particle, 1 over alpha is approximately 10 01:20:43.400 --> 01:20:45.670 to the two, so this is a particle that 01:20:45.670 --> 01:20:52.348 has a mass of about 10 to the 18th GeV, 01:20:52.348 --> 01:20:54.730 in other words it's a single particle with a mass of 10 01:20:54.730 --> 01:20:57.700 to the 18th protons. 01:20:57.700 --> 01:21:00.260 This is approximately one microgram. 01:21:04.160 --> 01:21:10.490 If you put one of these things on a scale it could measure it, 01:21:10.490 --> 01:21:12.420 that's bloody big. 01:21:12.420 --> 01:21:16.250 So getting to the last bit of the class, which I am just 01:21:16.250 --> 01:21:19.776 going to very basically quote the answer. 01:21:19.776 --> 01:21:23.270 The question becomes how often do these things get created 01:21:23.270 --> 01:21:25.290 and here I'm going to refer to Alan's notes. 01:21:25.290 --> 01:21:28.470 What you'll find is that, remember 01:21:28.470 --> 01:21:30.640 when we sketched our original picture of this thing 01:21:30.640 --> 01:21:33.310 we looked at regions of the universe 01:21:33.310 --> 01:21:34.860 where the Higgs field was initially 01:21:34.860 --> 01:21:37.780 seeded with different values. 01:21:37.780 --> 01:21:40.200 In order for the Higgs field to take on different values, 01:21:40.200 --> 01:21:43.620 initially, these regions had to be out of causal contact 01:21:43.620 --> 01:21:45.530 with one another. 01:21:45.530 --> 01:21:54.030 So we are going to require that the initial seed areas be 01:21:54.030 --> 01:22:01.540 separated by a distance, which is the correlation 01:22:01.540 --> 01:22:04.600 length, which has to be less than or of 01:22:04.600 --> 01:22:09.260 order the horizon distance. 01:22:09.260 --> 01:22:12.910 You can get a lower bound on this thing by imagining that 01:22:12.910 --> 01:22:16.110 it's-- sorry let me say one other thing. 01:22:16.110 --> 01:22:18.910 If you do that, then you can estimate that the number 01:22:18.910 --> 01:22:27.140 density associated with these things, 01:22:27.140 --> 01:22:29.530 the number density of these monopoles 01:22:29.530 --> 01:22:34.435 will be 1 over the correlation length cubed. 01:22:34.435 --> 01:22:36.690 To get a lower bound on the number 01:22:36.690 --> 01:22:39.960 density of these things, set the correlation length exactly 01:22:39.960 --> 01:22:46.927 to the horizon distance, and then do the following exercise. 01:22:46.927 --> 01:22:48.510 So first, let's set up the correlation 01:22:48.510 --> 01:22:52.000 length equal to the horizon distance. 01:22:52.000 --> 01:23:01.140 Set the density in monopoles equal to the mass of a monopole 01:23:01.140 --> 01:23:06.670 over rH cubed, normalized to the critical density. 01:23:19.330 --> 01:23:21.270 If you do this, you will find that 01:23:21.270 --> 01:23:27.630 just due to magnetic monopoles alone, 01:23:27.630 --> 01:23:30.866 the density of the universe. 01:23:30.866 --> 01:23:32.240 PROFESSOR 2: Excuse me, professor 01:23:32.240 --> 01:23:34.750 PROFESSOR: Yes, I'm wrapping up right this second. 01:23:34.750 --> 01:23:36.083 PROFESSOR 2: It's seven minutes. 01:23:36.083 --> 01:23:37.980 You were supposed to end at 10:55. 01:23:37.980 --> 01:23:39.896 PROFESSOR: I'm substitute teaching, I'm sorry. 01:23:44.350 --> 01:23:48.430 OK, so this tells us that we are at 10 01:23:48.430 --> 01:23:51.335 to the 20 of the critical density. 01:23:54.680 --> 01:23:56.370 And a consequence is that the universe 01:23:56.370 --> 01:23:58.800 is approximately two years old. 01:24:01.430 --> 01:24:04.280 I will let Alan pick it up from there.