1 00:00:00,080 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,550 Your support will help MIT OpenCourseWare continue 4 00:00:06,550 --> 00:00:10,160 to offer high quality educational resources for free. 5 00:00:10,160 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:21,112 at ocw.mit.edu 8 00:00:21,112 --> 00:00:23,070 PROFESSOR: As you know, Professor Guth is away. 9 00:00:23,070 --> 00:00:27,250 I'm substituting for today, he didn't leave me 10 00:00:27,250 --> 00:00:29,230 with a particularly coherent game plan, 11 00:00:29,230 --> 00:00:33,340 so I'm going to begin with where he thinks we should start. 12 00:00:33,340 --> 00:00:36,350 Please jump in if I am just repeating something 13 00:00:36,350 --> 00:00:39,259 that he has already described to you guys, 14 00:00:39,259 --> 00:00:41,550 or if there's anything you like me go over a little bit 15 00:00:41,550 --> 00:00:43,310 more detail, I will do my best here. 16 00:00:43,310 --> 00:00:48,470 So, I'm working off of a fairly rough plan. 17 00:00:48,470 --> 00:00:50,450 But let me just quickly describe what-- 18 00:00:50,450 --> 00:00:55,100 based on what Alan has explained to me --what we're planning 19 00:00:55,100 --> 00:00:57,686 to talk about today, and if there's 20 00:00:57,686 --> 00:00:59,102 any adjustments you think I should 21 00:00:59,102 --> 00:01:00,351 be making that would be great. 22 00:01:00,351 --> 00:01:02,800 So, the game plan for today. 23 00:01:05,780 --> 00:01:07,616 What I want to do very quickly is 24 00:01:07,616 --> 00:01:09,240 hit on a couple of the key points which 25 00:01:09,240 --> 00:01:11,330 I believe you talked about last week, 26 00:01:11,330 --> 00:01:16,040 which is a quick review of the essential features 27 00:01:16,040 --> 00:01:21,669 of symmetries of the gauge fields 28 00:01:21,669 --> 00:01:22,960 the make up the standard model. 29 00:01:33,690 --> 00:01:35,490 Now, I believe you guys did in fact talk 30 00:01:35,490 --> 00:01:37,400 about this last week, at least briefly. 31 00:01:37,400 --> 00:01:39,483 And you talked about how you can take these things 32 00:01:39,483 --> 00:01:43,141 and embed them in a larger gauge group, the group SU(5). 33 00:01:43,141 --> 00:01:44,890 I'm not going to talk about that too much, 34 00:01:44,890 --> 00:01:47,560 but I want to just quickly hit on a few elements related 35 00:01:47,560 --> 00:01:49,240 to this before we get into that. 36 00:01:49,240 --> 00:01:54,034 From this we'll then talk about the Higgs mechanism-- really 37 00:01:54,034 --> 00:01:55,700 I'm going to talk about the Higgs field, 38 00:01:55,700 --> 00:01:57,658 I'm not going to talk about the Higgs mechanism 39 00:01:57,658 --> 00:02:02,870 quite so much as motivate why it is necessary-- 40 00:02:02,870 --> 00:02:05,550 and then talk about how the Higgs field behaves 41 00:02:05,550 --> 00:02:09,870 and why it's important for the next problem, which 42 00:02:09,870 --> 00:02:15,230 is what is called the cosmological monopole problem. 43 00:02:19,166 --> 00:02:21,685 To be more specific magnetic monopole problem. 44 00:02:33,350 --> 00:02:35,560 I confess I feel a little bit awkward talking 45 00:02:35,560 --> 00:02:39,650 about this problem on behalf of Alan. 46 00:02:39,650 --> 00:02:42,370 This would be kind of like if you were planning 47 00:02:42,370 --> 00:02:45,840 on studying Hamlet and there was this guy W. Shakespeare who 48 00:02:45,840 --> 00:02:49,200 was listed as the instructor and you walk in 49 00:02:49,200 --> 00:02:51,890 and discover there's this guy Warren Shackspeare, who's 50 00:02:51,890 --> 00:02:54,098 actually going to be teaching or something like that. 51 00:02:54,098 --> 00:02:56,850 I kind of feel like Warren here. 52 00:02:56,850 --> 00:03:02,290 This stuff really is Allen's thing, so it's sort of, 53 00:03:02,290 --> 00:03:05,057 I'm probably going to leave this at the denouement of all this 54 00:03:05,057 --> 00:03:06,890 when you actually get into inflation to him. 55 00:03:06,890 --> 00:03:09,500 I may have a little bit of time at the end to just motivate 56 00:03:09,500 --> 00:03:14,030 it a little bit, but the grand summary will come from him. 57 00:03:14,030 --> 00:03:26,230 OK, so, as discussed by Alan the standard model 58 00:03:26,230 --> 00:03:28,480 describes all the fundamental interactions 59 00:03:28,480 --> 00:03:31,510 between particles via gauge theories. 60 00:03:53,460 --> 00:03:57,400 OK, and these gauge theories all have a combined symmetry 61 00:03:57,400 --> 00:04:06,620 group that is traditionally written in a somewhat 62 00:04:06,620 --> 00:04:17,990 awkward form, SU(3) cross SU(2) cross U(1). 63 00:04:17,990 --> 00:04:20,529 U(1) could be an SU(1) for reasons 64 00:04:20,529 --> 00:04:23,070 which I'll elucidate a little bit more 65 00:04:23,070 --> 00:04:24,450 clearly in just a moment. 66 00:04:24,450 --> 00:04:27,210 There's really no point in putting the S on that one. 67 00:04:27,210 --> 00:04:30,130 So each of these things essentially 68 00:04:30,130 --> 00:04:34,170 labels the particular symmetry group. 69 00:04:34,170 --> 00:04:45,910 So, the "S" an element of SU(n) is 70 00:04:45,910 --> 00:05:04,436 a matrix that is n x n, that is unitary-- that's the U. 71 00:05:04,436 --> 00:05:07,380 Unitary just means that the inverse and the transpose 72 00:05:07,380 --> 00:05:09,610 of the matrix at the same, actually the Hermitian 73 00:05:09,610 --> 00:05:13,169 conjugate because they can be complex, in fact, 74 00:05:13,169 --> 00:05:13,960 they generally are. 75 00:05:16,840 --> 00:05:18,955 And it has determinant of 1. 76 00:05:24,580 --> 00:05:28,460 That's what the special refers to, special, the S in SU(n) 77 00:05:28,460 --> 00:05:30,880 stands for special unitary n. 78 00:05:30,880 --> 00:05:32,990 So, the S means that the determinant 79 00:05:32,990 --> 00:05:35,640 is one-- that's what's special about it-- unitary 80 00:05:35,640 --> 00:05:38,730 is this idea that the inverse Hermitian conjugate are 81 00:05:38,730 --> 00:05:41,430 the same, and then n refers to all these things. 82 00:05:41,430 --> 00:05:45,656 So, that tells us that the gauge degrees of freedom 83 00:05:45,656 --> 00:05:47,530 are related by a symmetry that looks like a 3 84 00:05:47,530 --> 00:05:50,620 by 3 matrix with these properties, 85 00:05:50,620 --> 00:05:54,480 as listed there for the SU(3) piece of the symmetry. 86 00:05:54,480 --> 00:05:57,120 SU(2) means it's a 2 by 2 matrix. 87 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? 88 00:06:01,290 --> 00:06:03,392 It's a number, its a complex number. 89 00:06:03,392 --> 00:06:04,850 And that's why we don't really need 90 00:06:04,850 --> 00:06:06,230 to put an "s" in front of it. 91 00:06:06,230 --> 00:06:08,740 If it's a complex number its determinant 92 00:06:08,740 --> 00:06:11,480 is 1 if it's just a complex number whose modules is one. 93 00:06:11,480 --> 00:06:13,860 That's why we don't bother with the S on the U(1). 94 00:06:16,720 --> 00:06:19,229 So, I think you've already hit on some of this 95 00:06:19,229 --> 00:06:21,270 but this is sort of useful to review because it's 96 00:06:21,270 --> 00:06:23,910 going to set up why we need to introduce a Higgs 97 00:06:23,910 --> 00:06:26,050 mechanism in a little bit. 98 00:06:26,050 --> 00:06:29,820 Let me just quickly hit on what the details 99 00:06:29,820 --> 00:06:33,110 structure of this looks like for you want to think 100 00:06:33,110 --> 00:06:46,150 is the easiest one understand, So, as I just said, 101 00:06:46,150 --> 00:06:49,540 a one by one matrix is just a complex number. 102 00:06:49,540 --> 00:06:54,820 So that means that any element of this group 103 00:06:54,820 --> 00:07:11,390 is a complex number, which we can write in the form z 104 00:07:11,390 --> 00:07:14,990 equals ei theta, where theta is a real number. 105 00:07:18,620 --> 00:07:20,460 Now, the thing which is I want to hit on 106 00:07:20,460 --> 00:07:22,960 in this, the reason I want to describe this a little bit is, 107 00:07:22,960 --> 00:07:26,140 this may not smell like the gauge symmetry 108 00:07:26,140 --> 00:07:28,620 that you're used to if you study classical E&M. Some of you 109 00:07:28,620 --> 00:07:30,190 here are in 807 with me right now, 110 00:07:30,190 --> 00:07:32,300 and we've gone over this quite a bit recently. 111 00:07:32,300 --> 00:07:36,160 How is this akin to the gauge group that we are normally 112 00:07:36,160 --> 00:07:38,799 used to when we talk about the gauge 113 00:07:38,799 --> 00:07:40,340 freedom of electricity and magnetism? 114 00:07:56,092 --> 00:07:57,550 Well, it turns out there's actually 115 00:07:57,550 --> 00:08:01,520 a very simple relationship between one and the other, 116 00:08:01,520 --> 00:08:04,880 rather between this view of it and the way we learn 117 00:08:04,880 --> 00:08:11,095 about it when we study classical E&M. It's simply 118 00:08:11,095 --> 00:08:13,720 that we use a somewhat different language, because when we talk 119 00:08:13,720 --> 00:08:16,340 about it in this group theoretic picture we're doing it 120 00:08:16,340 --> 00:08:20,310 in the way that is sort of tuned to a quantum field theory. 121 00:08:20,310 --> 00:08:22,800 So, the way we have learned about electromagnetic gauge 122 00:08:22,800 --> 00:08:43,739 symmetry in terms of the fields sort of goes as follows. 123 00:08:43,739 --> 00:08:46,030 We actually work with the potentials, and so what we do 124 00:08:46,030 --> 00:08:54,860 is we note that the potentials Amu, which you can write 125 00:08:54,860 --> 00:08:59,760 as a four vector, whose time-like component is 126 00:08:59,760 --> 00:09:02,710 the negative of the scalar potential, 127 00:09:02,710 --> 00:09:04,320 and whose spatial components are just 128 00:09:04,320 --> 00:09:08,160 the three components of the vector potential. 129 00:09:08,160 --> 00:09:34,030 So, this potential and this potential-- --okay 130 00:09:34,030 --> 00:09:40,179 this is possibly module of factor of c somewhere in here 131 00:09:40,179 --> 00:09:41,970 but I'm going to imagine the speed of light 132 00:09:41,970 --> 00:09:44,490 has been set equal to 1. 133 00:09:44,490 --> 00:09:47,760 Both of those potentials generate the same E&B fields. 134 00:10:09,970 --> 00:10:12,589 OK, again you still should be looking at this 135 00:10:12,589 --> 00:10:14,380 and thinking to yourself what the hell does 136 00:10:14,380 --> 00:10:18,200 this have to do with the U(1) as we presented it here. 137 00:10:18,200 --> 00:10:19,960 I've given you a bunch of operations 138 00:10:19,960 --> 00:10:23,060 that involve some kind of a scale or function of time 139 00:10:23,060 --> 00:10:23,680 and space. 140 00:10:23,680 --> 00:10:26,490 And I've added particular components of this four vector 141 00:10:26,490 --> 00:10:29,790 in this way, what does that to do with this multiplication 142 00:10:29,790 --> 00:10:32,887 by a complex number? 143 00:10:32,887 --> 00:10:34,470 Well, where it comes from is that when 144 00:10:34,470 --> 00:10:37,930 we study E&M, not as a classical field theory 145 00:10:37,930 --> 00:10:40,250 but as a quantum field theory, we 146 00:10:40,250 --> 00:10:43,800 have a field that describes the electron. 147 00:10:43,800 --> 00:10:47,540 So, where it comes from is that when 148 00:10:47,540 --> 00:10:50,360 you examine the Dirac field, which is the quantum field 149 00:10:50,360 --> 00:10:54,270 theory that governs the electron, when you change gauge 150 00:10:54,270 --> 00:11:00,300 the electron field acquires a local phase change. 151 00:11:35,060 --> 00:11:36,520 So in particular, what we find is 152 00:11:36,520 --> 00:11:46,292 that if we have a field 5x, which those of you who 153 00:11:46,292 --> 00:11:48,250 have taken a little bit of quantum field theory 154 00:11:48,250 --> 00:11:50,240 should know this is actually a spinner field, 155 00:11:50,240 --> 00:11:53,140 but for now, just think of it as some kind of a field that 156 00:11:53,140 --> 00:11:57,970 under the field equations of quantum 157 00:11:57,970 --> 00:12:01,544 electrodynamics-- the Dirac equation or high order 158 00:12:01,544 --> 00:12:03,710 ones that have been developed by Feynman, Schwinger, 159 00:12:03,710 --> 00:12:06,860 and others-- under a change of gauge 160 00:12:06,860 --> 00:12:14,290 this goes over to si prime of x, which 161 00:12:14,290 --> 00:12:17,800 equals e to the-- terrible notation 162 00:12:17,800 --> 00:12:24,030 I realized-- 1e is obviously the root of natural logs, 163 00:12:24,030 --> 00:12:27,240 e sub 0 is the fundamental electric charge. 164 00:12:38,510 --> 00:12:40,200 OK, can everyone read that? 165 00:12:40,200 --> 00:12:42,870 I didn't block it too badly here I'm not used to this classroom. 166 00:12:42,870 --> 00:12:48,240 So, here's the thing to note, is that this field lambda, which 167 00:12:48,240 --> 00:12:52,425 we learned about in classical E&M 168 00:12:52,425 --> 00:12:58,165 directly connects to the phase function of the Dirac field 169 00:12:58,165 --> 00:12:59,290 in quantum electrodynamics. 170 00:13:04,030 --> 00:13:07,730 So, our gauge symmetry is simply expressed 171 00:13:07,730 --> 00:13:11,700 in the quantum version of electrodynamics 172 00:13:11,700 --> 00:13:18,660 by a function of the form e to the i real number, where 173 00:13:18,660 --> 00:13:21,840 that real number is the fundamental electric charge 174 00:13:21,840 --> 00:13:23,630 times the classical gauge generator. 175 00:14:03,930 --> 00:14:07,780 So, this is what is meant when people say that electrodynamics 176 00:14:07,780 --> 00:14:09,325 is a U(1) gauge theory. 177 00:14:26,080 --> 00:14:29,190 Now, I'm not going to go into this level of detail 178 00:14:29,190 --> 00:14:31,760 for the other two gauge symmetry that 179 00:14:31,760 --> 00:14:33,680 are built into the standard model. 180 00:14:33,680 --> 00:14:36,380 But, what I want you to understand 181 00:14:36,380 --> 00:14:39,690 is that the root idea is very, very similar. 182 00:14:39,690 --> 00:14:47,770 It's just now, instead of my gauge functions looking like e 183 00:14:47,770 --> 00:14:53,430 to the i, some kind of a local gauge phase of x multiplying 184 00:14:53,430 --> 00:14:57,820 my functions, my quantities which 185 00:14:57,820 --> 00:15:00,020 generate the gauge transformation are going 186 00:15:00,020 --> 00:15:02,340 to become complex value matrices. 187 00:15:02,340 --> 00:15:04,390 So that makes them a lot more complicated, 188 00:15:04,390 --> 00:15:05,830 and it's responsible for the fact 189 00:15:05,830 --> 00:15:07,860 that the weak and the strong interactions 190 00:15:07,860 --> 00:15:10,420 are non-abelian paid which order you perform the gauge 191 00:15:10,420 --> 00:15:11,620 transformation in matters. 192 00:15:11,620 --> 00:15:12,690 Question. 193 00:15:12,690 --> 00:15:14,440 AUDIENCE: What's the physical significance 194 00:15:14,440 --> 00:15:16,625 of them being non-abelian? 195 00:15:16,625 --> 00:15:17,250 PROFESSOR: Yes. 196 00:15:17,250 --> 00:15:18,470 So, what is a physical significance 197 00:15:18,470 --> 00:15:19,553 of them being non-abelian? 198 00:15:23,877 --> 00:15:29,790 I'm trying think of a really simple way to put this, 199 00:15:29,790 --> 00:15:36,510 it's-- Alan would have an answer to this right off the top 200 00:15:36,510 --> 00:15:38,120 of his head, so I apologize for this-- 201 00:15:38,120 --> 00:15:40,920 this isn't the kind of thing that I work on every day so I 202 00:15:40,920 --> 00:15:43,790 don't have an answer right at the very top of my head, 203 00:15:43,790 --> 00:15:44,920 unfortunately. 204 00:15:44,920 --> 00:15:46,604 Let me get back to you on that one, OK, 205 00:15:46,604 --> 00:15:48,770 that's something I can't give you a quick answer to. 206 00:15:48,770 --> 00:15:55,180 It's an excellent question and it's an important question. 207 00:15:55,180 --> 00:15:58,500 Any other questions? 208 00:15:58,500 --> 00:16:04,210 OK, so, here's a basic picture that we have. 209 00:16:10,160 --> 00:16:21,230 So, we find is that the strong interactions have 210 00:16:21,230 --> 00:16:29,680 a similar structure where my need to e to the i factor 211 00:16:29,680 --> 00:16:47,380 goes over to a 3 by 3 matrix, and the weak interactions 212 00:16:47,380 --> 00:16:57,070 in a similar structure with my e to the i factor going over 213 00:16:57,070 --> 00:16:58,760 to a 2 by 2 complex matrix. 214 00:17:07,865 --> 00:17:09,740 OK, what does this have to do with cosmology? 215 00:17:14,234 --> 00:17:16,400 In fact, as an enormous amount to do with cosmology, 216 00:17:16,400 --> 00:17:19,079 as we'll see over the course of the rest of course. 217 00:17:19,079 --> 00:17:24,240 Part of the thing which is interesting about all this 218 00:17:24,240 --> 00:17:28,600 is that we have strong experimental reasons, 219 00:17:28,600 --> 00:17:31,110 and theoretical reasons to believe, 220 00:17:31,110 --> 00:17:35,230 that the different symmetries that these interactions 221 00:17:35,230 --> 00:17:37,380 participate in, the different symmetries that we 222 00:17:37,380 --> 00:17:42,880 see them having, that isn't the way things have always been. 223 00:17:42,880 --> 00:17:45,580 So, in particular when the universe was a lot hotter 224 00:17:45,580 --> 00:17:49,550 and denser these different symmetries actually 225 00:17:49,550 --> 00:17:50,880 all began to look the same. 226 00:17:50,880 --> 00:17:53,395 In particular the one which is particularly important, 227 00:17:53,395 --> 00:18:04,410 and you guys have surely heard of this, is that the SU(2)-- 228 00:18:04,410 --> 00:18:15,360 if we just focus on electric and the weak piece of this-- 229 00:18:15,360 --> 00:18:18,530 SU(2) cross U(1). 230 00:18:18,530 --> 00:18:23,100 So, this is associated with the gauge boson that 231 00:18:23,100 --> 00:18:26,620 carry the weak force, OK, the z boson, the w plus, and the w 232 00:18:26,620 --> 00:18:27,460 minus. 233 00:18:27,460 --> 00:18:31,181 And your U(1) ends up being associated with the photon. 234 00:18:31,181 --> 00:18:32,680 In many ways, when you actually look 235 00:18:32,680 --> 00:18:34,730 at the equations that govern these things, 236 00:18:34,730 --> 00:18:36,830 they seem very, very similar to one 237 00:18:36,830 --> 00:18:42,025 another except that the-- here's partly 238 00:18:42,025 --> 00:18:43,400 an answer to your question I just 239 00:18:43,400 --> 00:18:49,280 realized-- the gauge generators of these things 240 00:18:49,280 --> 00:18:51,185 have a mass associated with them. 241 00:18:51,185 --> 00:18:52,560 That mass ends up being connected 242 00:18:52,560 --> 00:18:56,380 to the non-abelian nature of these things. 243 00:18:56,380 --> 00:18:59,400 That's not the whole answer, but it has a connection to that. 244 00:18:59,400 --> 00:19:01,940 That's one thing which I do remember, 245 00:19:01,940 --> 00:19:03,450 like I said I feel this is really 246 00:19:03,450 --> 00:19:07,140 Alan's perfect framework here and I'm just 247 00:19:07,140 --> 00:19:11,640 a posture in bad shoes. 248 00:19:11,640 --> 00:19:14,050 So if we look at this thing, what we see 249 00:19:14,050 --> 00:19:16,690 is that these symmetry groups, what's particularly interesting 250 00:19:16,690 --> 00:19:23,740 is that U(1) can be regarded as a piece of SU(2). 251 00:19:23,740 --> 00:19:29,530 And we would expect that in a perfect world 252 00:19:29,530 --> 00:19:31,900 they would actually be SU(2) governing 253 00:19:31,900 --> 00:19:34,430 both the electric and the weak interactions. 254 00:19:34,430 --> 00:19:36,350 Whereby perfect I mean everything 255 00:19:36,350 --> 00:19:40,090 is a nice balmy 10 to the 16th GeV throughout all of space 256 00:19:40,090 --> 00:19:43,470 time, and all the different vector bosons happily 257 00:19:43,470 --> 00:19:47,569 exchange with one another, not caring with who is who. 258 00:19:47,569 --> 00:19:49,110 It's actually not very perfect if you 259 00:19:49,110 --> 00:19:51,568 want to teach a physics class and have a nice conversation, 260 00:19:51,568 --> 00:19:54,050 but if you are interested in perfect symmetry among gauge 261 00:19:54,050 --> 00:19:58,520 interactions it's very, very nice. 262 00:19:58,520 --> 00:20:12,130 So, the fact that these are separate 263 00:20:12,130 --> 00:20:15,870 is now-- I was about to use the word believed but it's stronger 264 00:20:15,870 --> 00:20:17,780 and that, we now know this for sure thanks 265 00:20:17,780 --> 00:20:21,590 to all the exciting work that happened at the LHC 266 00:20:21,590 --> 00:20:26,440 over the past year or two-- the fact that these symmetries are 267 00:20:26,440 --> 00:20:31,650 separate is due to what is called spontaneous symmetry 268 00:20:31,650 --> 00:20:32,150 breaking. 269 00:20:45,610 --> 00:20:51,100 So, let's talk very briefly about what 270 00:20:51,100 --> 00:20:53,590 goes into this spontaneous symmetry breaking. 271 00:21:09,220 --> 00:21:11,560 So SU(2) turns out to actually be 272 00:21:11,560 --> 00:21:15,204 isomorphic to the group of rotations on a sphere. 273 00:21:15,204 --> 00:21:16,620 So, when you think about something 274 00:21:16,620 --> 00:21:19,120 that has perfect SU(2) symmetry it's 275 00:21:19,120 --> 00:21:22,010 as though you have perfect symmetry when you move around 276 00:21:22,010 --> 00:21:24,624 through a whole host of different angles. 277 00:21:24,624 --> 00:21:26,790 OK, so you move through all of your different angles 278 00:21:26,790 --> 00:21:31,050 and everyone looks exactly identical to all the others. 279 00:21:31,050 --> 00:21:32,820 If you break that symmetry it may 280 00:21:32,820 --> 00:21:35,530 mean you're picking out one angle as being special, 281 00:21:35,530 --> 00:21:39,090 and then you only retain a symmetry with respect 282 00:21:39,090 --> 00:21:42,250 to the other angle. 283 00:21:42,250 --> 00:21:45,920 And essentially, that is what happens when SU(2) breaks off 284 00:21:45,920 --> 00:21:47,600 in a U(1) piece of it. 285 00:21:47,600 --> 00:21:50,280 Something has occurred that picked out 286 00:21:50,280 --> 00:21:52,220 one of these directions. 287 00:21:52,220 --> 00:21:55,142 And by the way, you have to think very abstractly here. 288 00:21:55,142 --> 00:21:57,350 This is not necessarily a direction in physical space 289 00:21:57,350 --> 00:21:58,725 we're talking about here but it's 290 00:21:58,725 --> 00:22:01,450 a direction in the space of gauge fields. 291 00:22:01,450 --> 00:22:04,714 So, if we imagine that all of these, my gauge fields in some 292 00:22:04,714 --> 00:22:06,630 sense the different components of them defined 293 00:22:06,630 --> 00:22:08,840 in some abstract space direction, 294 00:22:08,840 --> 00:22:11,120 initially these things are completely 295 00:22:11,120 --> 00:22:12,970 symmetric with respect to rotations 296 00:22:12,970 --> 00:22:15,400 in some kind of an abstract notion of a sphere. 297 00:22:15,400 --> 00:22:17,570 And then something happens to freeze 298 00:22:17,570 --> 00:22:19,825 one of the directions and only symmetries 299 00:22:19,825 --> 00:22:21,950 with respect to one of the angles remains the same. 300 00:22:27,180 --> 00:22:29,390 Let's just write that out, when SU(2)'s symmetry 301 00:22:29,390 --> 00:22:52,830 is broken so one of the directions in the space 302 00:22:52,830 --> 00:23:02,090 of gauge fields is picked out as special. 303 00:23:07,790 --> 00:23:09,980 That direction then ends up being 304 00:23:09,980 --> 00:23:12,740 associated with your U(1) symmetry. 305 00:23:39,400 --> 00:23:44,570 So, what is the mechanism that actually breaks the symmetry 306 00:23:44,570 --> 00:23:46,720 and causes this to happen. 307 00:23:46,720 --> 00:23:50,490 Well, this is what the Higgs field is all about. 308 00:23:50,490 --> 00:23:57,120 The idea is there is some field that fills all of space time. 309 00:24:00,720 --> 00:24:05,490 It has the property that at very high energies 310 00:24:05,490 --> 00:24:10,310 it is extremely symmetric, with to respect all these gauge 311 00:24:10,310 --> 00:24:13,330 fields, all directions and sort of gauge field space 312 00:24:13,330 --> 00:24:14,980 look exactly the same. 313 00:24:14,980 --> 00:24:19,370 And then as things cool, as the energy density goes down 314 00:24:19,370 --> 00:24:21,580 by the temperature of the expanding universe, cooling 315 00:24:21,580 --> 00:24:26,190 everything off, the Higgs field moves to a particular place 316 00:24:26,190 --> 00:24:30,460 that picks out some direction in the space of gauge fields 317 00:24:30,460 --> 00:24:31,470 as being special. 318 00:24:34,514 --> 00:24:36,430 So let's make this a little bit more concrete. 319 00:25:13,070 --> 00:25:13,570 OK. 320 00:25:13,570 --> 00:25:15,903 You guys have probably heard quite a lot about the Higgs 321 00:25:15,903 --> 00:25:17,710 field over the past couple years, 322 00:25:17,710 --> 00:25:21,510 months-- what actually is it? 323 00:25:21,510 --> 00:25:38,410 Well, the field itself is described by a complex doublet. 324 00:25:45,340 --> 00:25:49,000 So, if you actually see someone write down a Higgs field what 325 00:25:49,000 --> 00:25:53,080 they will actually write down is h, 326 00:25:53,080 --> 00:26:00,900 being a two components spinner, whose components are 327 00:26:00,900 --> 00:26:06,450 h1 of x, h2 of x-- where x really 328 00:26:06,450 --> 00:26:08,840 stands for space time coordinates, so that's time 329 00:26:08,840 --> 00:26:10,690 and all of your spatial coordinates-- 330 00:26:10,690 --> 00:26:15,940 and both h1 and h2 are complex fields. 331 00:26:44,850 --> 00:26:47,760 The thing which is particularly key to understanding 332 00:26:47,760 --> 00:26:53,040 the importance of this thing is that h transforms, 333 00:26:53,040 --> 00:26:59,300 under gauge transformations, with elements of SU(2). 334 00:27:06,495 --> 00:27:08,120 So, if you want to change gauge the way 335 00:27:08,120 --> 00:27:13,060 you're going to do it is you're going to have some new Higgs 336 00:27:13,060 --> 00:27:13,560 field. 337 00:27:16,610 --> 00:27:30,930 So remember, if U(2) is an element of SU(2) 338 00:27:30,930 --> 00:27:34,250 we call it the two by two matrix. 339 00:27:34,250 --> 00:27:43,150 This is what they look like in a new gauge OK-- pardon me 340 00:27:43,150 --> 00:27:45,380 a second I don't see a clock in this room, 341 00:27:45,380 --> 00:27:47,920 I just want to make sure I know the time, thank you. 342 00:27:50,750 --> 00:27:54,266 OK, so, what are we going to do with this? 343 00:27:54,266 --> 00:27:56,390 Well, there's a couple features which it must have, 344 00:27:56,390 --> 00:28:01,330 so the Higgs field fills all of space time 345 00:28:01,330 --> 00:28:04,780 and it has an energy density associated 346 00:28:04,780 --> 00:28:07,680 with it, which we will call just the potential energy. 347 00:28:07,680 --> 00:28:13,612 It's really an energy density, but, whatever. 348 00:28:13,612 --> 00:28:15,820 The energy density that is associated with this thing 349 00:28:15,820 --> 00:28:16,915 must be gauge invariant. 350 00:28:39,660 --> 00:28:41,870 OK, even when you're working with strong fields 351 00:28:41,870 --> 00:28:46,390 and weak fields, the lesson of gauge invariance from E&M 352 00:28:46,390 --> 00:28:47,190 still holds. 353 00:28:47,190 --> 00:28:50,450 OK, one of the key points was that the gauge fields affect 354 00:28:50,450 --> 00:28:54,760 potentials, they allow us to manipulate our equations 355 00:28:54,760 --> 00:28:57,800 to put things into a form where the calculation may be easier. 356 00:28:57,800 --> 00:29:00,425 But at the end of the day, there are certain things it actually 357 00:29:00,425 --> 00:29:03,080 exert forces that cause things to happen, 358 00:29:03,080 --> 00:29:05,770 those must be invariant to the gauge transformation. 359 00:29:05,770 --> 00:29:07,410 Energy density is of those things. 360 00:29:07,410 --> 00:29:10,016 If you were to get into your spaceship 361 00:29:10,016 --> 00:29:11,390 and go back to the early universe 362 00:29:11,390 --> 00:29:13,640 and actually take a little scoop of early universe out 363 00:29:13,640 --> 00:29:17,590 and measure the energy density, A, that would be cool, but B 364 00:29:17,590 --> 00:29:20,350 it would be something that couldn't actually 365 00:29:20,350 --> 00:29:22,417 depend on what gauge you were using 366 00:29:22,417 --> 00:29:23,500 to make your measurements. 367 00:29:23,500 --> 00:29:26,024 That is something that is a complete artifice of how 368 00:29:26,024 --> 00:29:27,440 you want to set up the convenience 369 00:29:27,440 --> 00:29:28,770 of your calculation. 370 00:29:28,770 --> 00:29:36,300 So, in order for the energy density to be gauge invariant 371 00:29:36,300 --> 00:29:38,570 we have to find a gauge invariant quantity that 372 00:29:38,570 --> 00:29:42,160 is constructed from this, which is the only thing the energy 373 00:29:42,160 --> 00:29:44,060 density can depend on. 374 00:29:44,060 --> 00:29:47,360 This means, let's call our energy density V, 375 00:29:47,360 --> 00:29:49,150 it's the potential energy density. 376 00:30:04,560 --> 00:30:11,380 So, it can only depend on the following combination 377 00:30:11,380 --> 00:30:23,050 of the fundamental fields Pretty much just what you'd expect. 378 00:30:23,050 --> 00:30:27,330 This is sort of the equivalent to saying 379 00:30:27,330 --> 00:30:31,220 that if you're working in spherical symmetry 380 00:30:31,220 --> 00:30:33,220 the electrostatic potential can only 381 00:30:33,220 --> 00:30:35,200 depend on the distance from a point charge. 382 00:30:35,200 --> 00:30:37,130 This is a very similar kind of construct 383 00:30:37,130 --> 00:30:40,160 here, where I'm taking the only quantity that 384 00:30:40,160 --> 00:30:43,150 follows in a fully symmetric way, of calling 385 00:30:43,150 --> 00:30:47,447 the fact that this is a special unitary matrix that I 386 00:30:47,447 --> 00:30:48,780 can construct from these things. 387 00:30:52,480 --> 00:30:55,670 So then, where all the magic comes 388 00:30:55,670 --> 00:31:01,890 in is in how the Higgs field potential energy density 389 00:31:01,890 --> 00:31:06,980 varies as a function of this h, this magnitude of h. 390 00:31:40,240 --> 00:31:42,857 So, as I plot v as a function of h, 391 00:31:42,857 --> 00:31:44,940 in order to get your spontaneous symmetry breaking 392 00:31:44,940 --> 00:31:49,070 to happen what you want is for the minimum of V, 393 00:31:49,070 --> 00:31:53,540 the minimum potential energy, to occur somewhere 394 00:31:53,540 --> 00:32:21,310 out at a non-zero value of the Higgs field H. 395 00:32:21,310 --> 00:32:24,135 Now, why is that so special? 396 00:32:38,850 --> 00:32:41,030 The thing that is so special about that 397 00:32:41,030 --> 00:32:46,320 is that when I constructed this magnitude of h, 398 00:32:46,320 --> 00:32:48,350 I actually lost a lot of information 399 00:32:48,350 --> 00:32:51,000 about the Higgs field. 400 00:32:51,000 --> 00:32:52,860 OK, let's just say for the sake of argument 401 00:32:52,860 --> 00:32:56,120 that this minimum occurs at a place where 402 00:32:56,120 --> 00:33:01,580 the Higgs field in some system of units has a value of 1. 403 00:33:01,580 --> 00:33:05,990 So, all I need to do is as my universe cools 404 00:33:05,990 --> 00:33:08,100 what I'm going to want is energetically, 405 00:33:08,100 --> 00:33:11,160 my potential is going to want to go down to its minimum. 406 00:33:11,160 --> 00:33:14,620 So, that just means that as the universe is cooling, maybe 407 00:33:14,620 --> 00:33:17,080 at very, very early times when everything is extremely 408 00:33:17,080 --> 00:33:21,020 hot and dense, I'm up here where the potential energy is 409 00:33:21,020 --> 00:33:23,020 very high. 410 00:33:23,020 --> 00:33:24,960 As the universe expands, as everything cools, 411 00:33:24,960 --> 00:33:26,459 it moves over to here, it just moves 412 00:33:26,459 --> 00:33:29,780 to someplace where the Higgs field takes on a value of 1. 413 00:33:29,780 --> 00:33:33,200 And that's exactly correct, that is what ends up happening. 414 00:33:33,200 --> 00:33:48,340 But remember, the minimum occurs at some value 415 00:33:48,340 --> 00:33:52,360 in which the magnitude of this field 416 00:33:52,360 --> 00:33:56,140 does not equal zero, but given that value-- where again let's 417 00:33:56,140 --> 00:33:58,110 just say for this for sake of specificity 418 00:33:58,110 --> 00:34:00,340 that we set it equal to the magnitude of this thing 419 00:34:00,340 --> 00:34:03,370 equal to 1 in some units-- there's actually 420 00:34:03,370 --> 00:34:07,090 an infinite number of configurations that correspond 421 00:34:07,090 --> 00:34:10,380 to that because this is a complex number, 422 00:34:10,380 --> 00:34:12,159 this is a complex number. 423 00:34:12,159 --> 00:34:15,539 I could put it all into little h1, 424 00:34:15,539 --> 00:34:18,080 and I could set into the value where that thing is completely 425 00:34:18,080 --> 00:34:22,540 real, or I could put it all into little h2 being completely 426 00:34:22,540 --> 00:34:27,310 imaginary or all on to h1 being all imaginary, halfway into h1, 427 00:34:27,310 --> 00:34:28,800 halfway into h2. 428 00:34:28,800 --> 00:34:31,460 There are literally an infinite number of combinations 429 00:34:31,460 --> 00:34:34,429 that I can choose which are consistent 430 00:34:34,429 --> 00:35:20,958 with this value of the magnitude of H. So, yeah-- 431 00:35:20,958 --> 00:35:22,704 AUDIENCE: So, I don't know if I'm 432 00:35:22,704 --> 00:35:24,870 putting too much physical significance on the gauge, 433 00:35:24,870 --> 00:35:27,480 but with the other cases of spontaneous symmetry, 434 00:35:27,480 --> 00:35:29,640 briefly, that we discussed you can always measure. 435 00:35:29,640 --> 00:35:32,140 OK, I've broken my symmetry, and now it's lined up this way, 436 00:35:32,140 --> 00:35:34,280 or there's something measurable. 437 00:35:34,280 --> 00:35:35,896 Now, the field has to be physical 438 00:35:35,896 --> 00:35:38,736 because the fact that you have gauge symmetry 439 00:35:38,736 --> 00:35:41,350 gives you some concerned quantity, right? 440 00:35:41,350 --> 00:35:44,710 But, how can I measure what direction in gauge space 441 00:35:44,710 --> 00:35:46,300 that I picked out? 442 00:35:46,300 --> 00:35:47,960 PROFESSOR: So, that is, let me talk 443 00:35:47,960 --> 00:35:51,090 about this just a little bit more. 444 00:35:51,090 --> 00:35:53,390 I think answering your question completely 445 00:35:53,390 --> 00:35:55,450 is not really possible, but there 446 00:35:55,450 --> 00:35:58,406 is a residue of that is in fact very interesting, 447 00:35:58,406 --> 00:36:01,030 and let me just lay out a couple more facts about what actually 448 00:36:01,030 --> 00:36:02,550 happens with this gauge symmetry, 449 00:36:02,550 --> 00:36:04,500 and it's not going to answer your question 450 00:36:04,500 --> 00:36:06,666 but it's going to give you something to think about. 451 00:36:06,666 --> 00:36:10,130 OK, so that's an excellent and very deep question, 452 00:36:10,130 --> 00:36:12,120 and there are really interesting consequences. 453 00:36:12,120 --> 00:36:15,645 And this is a case where my failure 454 00:36:15,645 --> 00:36:17,520 to answer the previous one is because there's 455 00:36:17,520 --> 00:36:19,160 details I can't remember, in this case, 456 00:36:19,160 --> 00:36:21,160 I think it's because there's details we actually 457 00:36:21,160 --> 00:36:22,260 don't understand fully. 458 00:36:24,709 --> 00:36:27,250 Research into the mechanism of electroweak symmetry breaking, 459 00:36:27,250 --> 00:36:29,194 which is what this is all about, is 460 00:36:29,194 --> 00:36:31,360 one of the hot topics in particle physics right now. 461 00:36:39,492 --> 00:36:41,200 AUDIENCE: I was just wondering if gravity 462 00:36:41,200 --> 00:36:43,660 has any gauge symmetry associated with it. 463 00:36:43,660 --> 00:36:47,390 PROFESSOR: It does, but it fits in a very, very different way, 464 00:36:47,390 --> 00:36:54,210 and with the exception of the fairly speculative framework 465 00:36:54,210 --> 00:36:56,810 of string theory-- which I think is very, very promising, 466 00:36:56,810 --> 00:36:58,280 but it's just sufficiently removed 467 00:36:58,280 --> 00:37:00,290 from experimental verification that I'm 468 00:37:00,290 --> 00:37:03,050 going to have to label it speculative-- it doesn't quite 469 00:37:03,050 --> 00:37:04,700 tie in in the same way. 470 00:37:04,700 --> 00:37:07,870 And that's the best I can say right now. 471 00:37:07,870 --> 00:37:10,030 The gauge symmetries of general relativity 472 00:37:10,030 --> 00:37:13,110 are, at the classical level, they correspond 473 00:37:13,110 --> 00:37:16,790 to coordinate transformations, at a quantum level, 474 00:37:16,790 --> 00:37:22,450 there's not such a simple way to put it. 475 00:37:22,450 --> 00:37:24,530 All right, where was I, OK, sorry 476 00:37:24,530 --> 00:37:25,780 I didn't get to your question. 477 00:37:25,780 --> 00:37:29,540 So, the point we made here is that we have spontaneously, 478 00:37:29,540 --> 00:37:32,400 when we actually choose which one of these infinite number 479 00:37:32,400 --> 00:37:35,040 of values we're going to have, we just 480 00:37:35,040 --> 00:37:36,500 randomly break the symmetry. 481 00:38:08,514 --> 00:38:10,180 OK, and you guys apparently have already 482 00:38:10,180 --> 00:38:12,900 talked a little bit about spontaneous symmetry breaking. 483 00:38:12,900 --> 00:38:14,730 The analogy that people often make 484 00:38:14,730 --> 00:38:17,430 is to the freezing of water, OK, prior 485 00:38:17,430 --> 00:38:21,620 to the water entering its solid phase its completely 486 00:38:21,620 --> 00:38:26,480 rotationally symmetric, then at a certain point 487 00:38:26,480 --> 00:38:28,310 crystalline planes start to form, 488 00:38:28,310 --> 00:38:30,850 the water forms, all the molecules 489 00:38:30,850 --> 00:38:32,990 get set into a particular orientation, 490 00:38:32,990 --> 00:38:35,940 you lose that rotational symmetry. 491 00:38:35,940 --> 00:38:41,130 In this case, we started out with a theory, 492 00:38:41,130 --> 00:38:44,740 with a set of interactions that were completely symmetric 493 00:38:44,740 --> 00:38:46,590 in sort of gauge field space. 494 00:38:46,590 --> 00:38:49,290 And now by settling down and picking 495 00:38:49,290 --> 00:38:52,730 a particular special value of h1 and h2 496 00:38:52,730 --> 00:38:55,435 we have at least nailed down one direction. 497 00:38:55,435 --> 00:38:58,110 It's like we've defined a crystalline plane, 498 00:38:58,110 --> 00:39:02,330 and so now things, suddenly, aren't as symmetric. 499 00:39:02,330 --> 00:39:06,770 And we start to pick out preferred directions 500 00:39:06,770 --> 00:39:12,150 in our gauge fields. 501 00:39:16,620 --> 00:39:20,610 What we can do with this is really 502 00:39:20,610 --> 00:39:23,200 a topic for a whole other course, 503 00:39:23,200 --> 00:39:25,550 and that course is called quantum field theory, 504 00:39:25,550 --> 00:39:27,539 but I will sketch a couple of the consequences 505 00:39:27,539 --> 00:39:29,705 and this gets directly to the answer your questions. 506 00:39:40,180 --> 00:39:42,740 So, one of the consequences of this 507 00:39:42,740 --> 00:39:48,950 is that once we have picked out a particular direction, 508 00:39:48,950 --> 00:39:55,210 electrons and neutrinos are different. 509 00:40:00,630 --> 00:40:02,780 When the Higgs field is equal to zero 510 00:40:02,780 --> 00:40:06,350 there is no difference between an electron and a neutrino. 511 00:40:06,350 --> 00:40:08,810 They obey exactly the same equation, 512 00:40:08,810 --> 00:40:11,450 there's literally no difference between them. 513 00:40:11,450 --> 00:40:14,940 Once we have actually settled on an h1 and an h2 514 00:40:14,940 --> 00:40:17,680 some combination of the fundamental underlying fields 515 00:40:17,680 --> 00:40:21,100 comes together, acquires a mass, acquires an electric charge, 516 00:40:21,100 --> 00:40:25,780 and we say A-HA thou beist an electron. 517 00:40:25,780 --> 00:40:31,093 It wasn't like that in the original unbroken symmetry. 518 00:40:31,093 --> 00:40:32,420 AUDIENCE: Also, [INAUDIBLE]? 519 00:40:32,420 --> 00:40:33,919 PROFESSOR: Presumably, but I'm going 520 00:40:33,919 --> 00:40:35,680 to stick with just these for now, 521 00:40:35,680 --> 00:40:40,490 but I've I'm pretty sure that's the case, yeah. 522 00:40:40,490 --> 00:40:42,800 That gets into even more complications of course 523 00:40:42,800 --> 00:40:44,770 because the additional generations are actually 524 00:40:44,770 --> 00:40:47,680 consequence presumably of some broken higher level symmetry, 525 00:40:47,680 --> 00:40:51,540 which is even poorly, more poorly understood. 526 00:40:51,540 --> 00:40:54,280 But you raise a good point. 527 00:40:54,280 --> 00:40:56,870 So, that's one partial answer your question. 528 00:40:56,870 --> 00:40:59,360 How one can actually walk that backwards 529 00:40:59,360 --> 00:41:01,730 to understand this thing about the initial state? 530 00:41:01,730 --> 00:41:05,410 That's hard to say. 531 00:41:16,760 --> 00:41:18,260 I actually think this particular one 532 00:41:18,260 --> 00:41:20,745 is one of the profound and interesting aspects 533 00:41:20,745 --> 00:41:27,440 of this, in part because we now know the neutrino has a mass. 534 00:41:27,440 --> 00:41:31,270 We have no idea what that is, and in fact we only really 535 00:41:31,270 --> 00:41:35,460 have bounds on the mass, such that we know it is non-zero, 536 00:41:35,460 --> 00:41:37,150 and we have upper limits that are 537 00:41:37,150 --> 00:41:39,450 set by very indirect measurements. 538 00:41:39,450 --> 00:41:41,190 But the actual values of the mass 539 00:41:41,190 --> 00:41:42,810 are very, very poorly constrained. 540 00:41:45,790 --> 00:41:48,590 Within the standard model you just 541 00:41:48,590 --> 00:41:50,610 take the electroweak interaction, 542 00:41:50,610 --> 00:41:52,780 introduce a Higgs coupling and allow the symmetry 543 00:41:52,780 --> 00:41:55,880 to be spontaneously broken, the neutrino mass is zero. 544 00:41:55,880 --> 00:41:57,494 Full stop zero. 545 00:41:57,494 --> 00:41:59,160 So something's not right, we're actually 546 00:41:59,160 --> 00:42:00,950 missing something here. 547 00:42:00,950 --> 00:42:03,120 People have kind of jury rigged the standard model 548 00:42:03,120 --> 00:42:06,920 to put in the masses by hand, and it works OK, 549 00:42:06,920 --> 00:42:09,770 but it's not completely satisfying. 550 00:42:09,770 --> 00:42:12,000 And a lot of experiments going on right now 551 00:42:12,000 --> 00:42:14,060 to explore the neutrino sector are hopefully 552 00:42:14,060 --> 00:42:17,000 going to open us up to a deeper understanding of this 553 00:42:17,000 --> 00:42:21,330 and may say a lot about all this physics, which is at present, 554 00:42:21,330 --> 00:42:24,140 pretty poorly understood. 555 00:42:24,140 --> 00:42:26,540 The consequence, which has received the most popular 556 00:42:26,540 --> 00:42:28,920 press, and what you guys have certainly 557 00:42:28,920 --> 00:42:31,860 seen about in newspapers, given the results that came out 558 00:42:31,860 --> 00:42:40,330 from the LHC over the past year is that quarks and leptons have 559 00:42:40,330 --> 00:42:45,000 mass, or put more specifically, rest mass. 560 00:42:50,930 --> 00:42:52,810 To understand what this actually means 561 00:42:52,810 --> 00:42:55,440 I think you really need to ask yourself what is mass 562 00:42:55,440 --> 00:42:56,000 meant to be. 563 00:43:01,560 --> 00:43:06,470 Well, the idea is you calculate the spectrum of oscillations 564 00:43:06,470 --> 00:43:36,560 associated with the fields of your theory, 565 00:43:36,560 --> 00:43:39,660 and then if your theory predicts a discrete spectrum 566 00:43:39,660 --> 00:43:42,180 of oscillations, it doesn't even have to be discrete 567 00:43:42,180 --> 00:43:44,560 but predict some spectrum of oscillations, 568 00:43:44,560 --> 00:43:47,790 then for every oscillation frequency omega 569 00:43:47,790 --> 00:43:53,340 there's an associated mass that is just 570 00:43:53,340 --> 00:43:58,190 H bar omega over c squared. 571 00:43:58,190 --> 00:44:03,630 If your omega has some lower bound that 572 00:44:03,630 --> 00:44:10,030 is greater than zero, then your theory 573 00:44:10,030 --> 00:44:14,310 has particles with nonzero rest mass. 574 00:44:56,082 --> 00:44:58,040 Without going into the details-- and this again 575 00:44:58,040 --> 00:44:59,970 is something which those of you who 576 00:44:59,970 --> 00:45:02,930 are going to go on to study this in more detail in a higher 577 00:45:02,930 --> 00:45:05,730 level course, which is fairly standard stuff is done 578 00:45:05,730 --> 00:45:09,897 in probably the first or maybe late in the first or early 579 00:45:09,897 --> 00:45:12,230 in the second semester of a typical quantum field theory 580 00:45:12,230 --> 00:45:16,250 course-- what you'll find is that when the Higgs field is 581 00:45:16,250 --> 00:45:19,470 zero then quarks and leptons have, 582 00:45:19,470 --> 00:45:22,890 the field that describes quarks and leptons-- and yes including 583 00:45:22,890 --> 00:45:25,210 mu and tau, so including all the leptons, this one 584 00:45:25,210 --> 00:45:27,530 I'm very confident on-- the spectrum 585 00:45:27,530 --> 00:45:30,550 goes all the way to zero if the Higgs field is zero. 586 00:45:58,150 --> 00:46:01,260 But when the Higgs field becomes non-zero, roughly speaking, 587 00:46:01,260 --> 00:46:04,500 it shifts the spectrum over for these particles. 588 00:46:04,500 --> 00:46:08,740 There's an interaction between the things like the electron 589 00:46:08,740 --> 00:46:12,880 field in the Higgs field or the up quark field and the Higgs 590 00:46:12,880 --> 00:46:16,400 field, which shifts the spectrum over just enough 591 00:46:16,400 --> 00:46:18,470 so that the frequency is never allowed 592 00:46:18,470 --> 00:46:20,598 to go below some minimum. 593 00:46:20,598 --> 00:46:22,590 AUDIENCE: Going back a bit, I'm confused 594 00:46:22,590 --> 00:46:25,950 about how picking a specific value to the Higgs field 595 00:46:25,950 --> 00:46:29,562 is breaking SU(2) symmetry and not U(1), 596 00:46:29,562 --> 00:46:32,060 because it seems like we're fixed on a circle, right? 597 00:46:32,060 --> 00:46:34,559 PROFESSOR: That's right what U(1) is a symmetry on a circle, 598 00:46:34,559 --> 00:46:37,354 SU(2) is kind of like symmetry on a sphere, essentially. 599 00:46:37,354 --> 00:46:40,763 AUDIENCE: Right, so how are we not picking a specific value 600 00:46:40,763 --> 00:46:42,932 [INAUDIBLE] circle [INAUDIBLE]? 601 00:46:42,932 --> 00:46:45,390 PROFESSOR: Well, what we're doing is, think of it this way, 602 00:46:45,390 --> 00:46:48,590 imagine SU(2) is a symmetry on a sphere, 603 00:46:48,590 --> 00:46:51,670 and then when we break the SU(2) symmetry 604 00:46:51,670 --> 00:46:54,460 it's like we're picking some circle on that sphere. 605 00:46:54,460 --> 00:46:57,370 So, we've broken one circle, we've picked one circle, 606 00:46:57,370 --> 00:46:58,870 but now we're allowed to go anywhere 607 00:46:58,870 --> 00:47:02,770 on that remaining circle, which is a U(1) symmetry. 608 00:47:02,770 --> 00:47:04,160 Does that help? 609 00:47:04,160 --> 00:47:05,145 Yeah, OK good. 610 00:47:09,110 --> 00:47:11,890 And it comes down to the fact if you sort of count up 611 00:47:11,890 --> 00:47:15,110 your degrees of freedom, it has to do with the fact 612 00:47:15,110 --> 00:47:20,930 you you've got four, you have two complex numbers, 613 00:47:20,930 --> 00:47:23,650 so there's four real parameters associated with this thing, 614 00:47:23,650 --> 00:47:28,060 and they are isomorphic to sort of rotations in a three space 615 00:47:28,060 --> 00:47:29,435 and you're adding one constraint. 616 00:47:33,220 --> 00:47:35,960 OK, so let me just finish making this point here again. 617 00:47:35,960 --> 00:47:41,240 So, when h does not equal zero, spectrum 618 00:47:41,240 --> 00:48:00,340 get shifted for the quarks and leptons, 619 00:48:00,340 --> 00:48:03,280 so everything picks up a little bit of a mass. 620 00:48:03,280 --> 00:48:06,025 And the final one, final consequence 621 00:48:06,025 --> 00:48:11,400 which we're going to talk about today, 622 00:48:11,400 --> 00:48:15,765 is that the universe is filled with magnetic monopoles. 623 00:48:25,550 --> 00:48:28,040 We all remember studying Maxwell's equations 624 00:48:28,040 --> 00:48:31,140 learning that del dot b is equal to 4 pi 625 00:48:31,140 --> 00:48:32,750 times the density of magnetic charge-- 626 00:48:32,750 --> 00:48:35,520 this all makes perfect sense, right? 627 00:48:35,520 --> 00:48:42,000 Well, this is actually something that when it first sort of came 628 00:48:42,000 --> 00:48:45,570 out and people begin to appreciate this thing with sort 629 00:48:45,570 --> 00:48:50,510 of a "Um, well everything else works so well, 630 00:48:50,510 --> 00:48:55,340 maybe we're just not looking hard enough. " So, 631 00:48:55,340 --> 00:48:57,085 it was a bit of a surprise. 632 00:49:06,940 --> 00:49:09,460 So, where do these magnetic monopoles come from? 633 00:49:31,580 --> 00:49:34,120 And essentially, the magnetic monopoles 634 00:49:34,120 --> 00:49:38,010 are going to turn out to be a consequence of the fact 635 00:49:38,010 --> 00:49:41,770 that when spontaneous symmetry breaking happens 636 00:49:41,770 --> 00:49:44,715 it doesn't happen everywhere simultaneously. 637 00:50:26,210 --> 00:50:28,319 So, think again about-- yeah? 638 00:50:28,319 --> 00:50:30,110 AUDIENCE: Doesn't that bring up possibility 639 00:50:30,110 --> 00:50:32,068 that the symmetry could break in different ways 640 00:50:32,068 --> 00:50:32,940 in different places? 641 00:50:32,940 --> 00:50:34,780 PROFESSOR: That is in fact exactly what this 642 00:50:34,780 --> 00:50:35,750 is going to be. 643 00:50:35,750 --> 00:50:38,430 Magnetic monopoles are in fact exactly a consequence 644 00:50:38,430 --> 00:50:42,052 of this, yes. 645 00:50:42,052 --> 00:50:43,510 Give me a few moments to step ahead 646 00:50:43,510 --> 00:50:46,517 to fill in a couple of the gaps, but you're basically 647 00:50:46,517 --> 00:50:47,100 already there. 648 00:50:50,160 --> 00:50:56,480 So, think about crystalline crystal formation again. 649 00:50:56,480 --> 00:51:03,030 Imagine you have, we could do ice 650 00:51:03,030 --> 00:51:05,604 if you like or choose something that's got a little bit more 651 00:51:05,604 --> 00:51:07,270 of an interesting crystalline structure. 652 00:51:07,270 --> 00:51:12,420 Imagine you have a big bucket full of molten quarts, OK. 653 00:51:12,420 --> 00:51:14,550 So, if you have a big thing of quartz 654 00:51:14,550 --> 00:51:18,740 that you want to sort of freeze into a single gigantic crystal, 655 00:51:18,740 --> 00:51:21,380 what you typically do if you'd like to do this is 656 00:51:21,380 --> 00:51:23,710 you actually seed it with a little bit 657 00:51:23,710 --> 00:51:26,340 of a starter crystal. 658 00:51:26,340 --> 00:51:29,060 So, you put a little bit of crystal into this thing, 659 00:51:29,060 --> 00:51:33,390 and what that does is it sort of defines a preferred orientation 660 00:51:33,390 --> 00:51:37,570 of the crystal axes, so that as things 661 00:51:37,570 --> 00:51:39,920 start to cool in the vicinity of that they have 662 00:51:39,920 --> 00:51:42,420 a preferred orientation to grab on to. 663 00:51:42,420 --> 00:51:44,330 And that seed then gradually gets 664 00:51:44,330 --> 00:51:47,560 bigger and bigger and bigger, and all the little crystals 665 00:51:47,560 --> 00:51:50,120 as they form near it tend to latch 666 00:51:50,120 --> 00:51:52,350 onto the preexisting crystalline structure, 667 00:51:52,350 --> 00:51:54,010 and that allows you to grow actually 668 00:51:54,010 --> 00:51:55,819 extremely large crystals. 669 00:51:55,819 --> 00:51:58,360 I don't know if anyone here is doing a year off with the LIGO 670 00:51:58,360 --> 00:52:02,800 project but these guys have to make these sort of 100 kilogram 671 00:52:02,800 --> 00:52:07,990 mirrors of very pure either Sapphire or silicon dioxide, 672 00:52:07,990 --> 00:52:10,574 and when you make 100 kilograms of crystal 673 00:52:10,574 --> 00:52:12,490 you need to build it really, really carefully. 674 00:52:12,490 --> 00:52:14,531 It's extremely important for the optical purposes 675 00:52:14,531 --> 00:52:17,150 that all the axes associated with the crystal 676 00:52:17,150 --> 00:52:18,970 will be pointing in the right direction. 677 00:52:18,970 --> 00:52:20,780 Otherwise you spend $100,000 on this thing 678 00:52:20,780 --> 00:52:25,070 and it ends up being the world's prettiest paperweight. 679 00:52:25,070 --> 00:52:28,880 So, similar things happen when the Higgs field cools. 680 00:52:28,880 --> 00:52:34,150 Let's imagine that we've got our universe, 681 00:52:34,150 --> 00:52:39,730 time going forward like this, and at some point over here 682 00:52:39,730 --> 00:52:45,390 the universe cools enough that's the Higgs field condenses 683 00:52:45,390 --> 00:52:46,900 into some particular direction. 684 00:52:46,900 --> 00:52:52,370 And symmetry is spontaneously broken right at this one point 685 00:52:52,370 --> 00:52:54,199 over here. 686 00:52:54,199 --> 00:52:55,990 So, I'm going to draw my diagram over there 687 00:52:55,990 --> 00:52:57,198 and put some words over here. 688 00:53:18,622 --> 00:53:20,330 I shouldn't say Higgs field cools enough, 689 00:53:20,330 --> 00:53:22,590 the universe cools enough so that the Higgs 690 00:53:22,590 --> 00:53:23,905 field breaks the symmetry. 691 00:53:43,230 --> 00:53:49,370 So, just to be concrete, let's imagine that at 0.1 over here 692 00:53:49,370 --> 00:54:00,476 it takes on a field of the value one for h1 and I for h2. 693 00:54:00,476 --> 00:54:02,600 So just for concreteness imagine it looks something 694 00:54:02,600 --> 00:54:05,297 like this at this point. 695 00:54:05,297 --> 00:54:07,380 And so what happens is as the university continues 696 00:54:07,380 --> 00:54:11,020 to expand other areas are going to cool off. 697 00:54:11,020 --> 00:54:12,620 The bits that are closest to it are 698 00:54:12,620 --> 00:54:15,010 going to see that there is already 699 00:54:15,010 --> 00:54:18,650 a preferred orientation defined by the Higgs field. 700 00:54:18,650 --> 00:54:21,280 And so it's energetically favorable for those regions 701 00:54:21,280 --> 00:54:23,750 of the universe to fall into the same alignment 702 00:54:23,750 --> 00:54:26,390 and so there'll be a region in space times 703 00:54:26,390 --> 00:54:31,080 that grows here as the universe cools, in which the Higgs 704 00:54:31,080 --> 00:54:34,640 field all falls into this configuration, which 705 00:54:34,640 --> 00:54:35,660 I will call h1. 706 00:54:44,920 --> 00:54:49,000 But suppose somewhere over here at 0.2, 707 00:54:49,000 --> 00:54:51,920 and the key thing is that initially 0.2 708 00:54:51,920 --> 00:54:55,580 is going to be so far away from 0.1 that these points are 709 00:54:55,580 --> 00:54:57,210 out of causal contact with one another. 710 00:54:57,210 --> 00:55:01,410 I can not send a message from event one to event two. 711 00:55:01,410 --> 00:55:03,450 The Higgs field also reaches a point 712 00:55:03,450 --> 00:55:07,220 that the universe cools enough that at 0.2, just you know, 713 00:55:07,220 --> 00:55:10,717 it's a system that's not in thermal equilibrium. 714 00:55:10,717 --> 00:55:12,550 So, some places are going to be a little bit 715 00:55:12,550 --> 00:55:13,604 hotter than others, some are going 716 00:55:13,604 --> 00:55:14,687 to be a little bit cooler. 717 00:55:14,687 --> 00:55:18,040 And so, at these two points it just so happened 718 00:55:18,040 --> 00:55:19,850 that the Higgs field got to the point 719 00:55:19,850 --> 00:55:22,250 where it could spontaneously break the symmetry. 720 00:55:22,250 --> 00:55:31,215 So at 0.2 the Higgs field also got to the point 721 00:55:31,215 --> 00:55:33,215 where it could spontaneously break its symmetry. 722 00:56:00,295 --> 00:56:01,920 And the only thing that's got to happen 723 00:56:01,920 --> 00:56:04,140 is, remember the only constraint we have is 724 00:56:04,140 --> 00:56:05,762 that the magnitude of the Higgs field 725 00:56:05,762 --> 00:56:08,220 be equals to some value-- I should normalize that to root 2 726 00:56:08,220 --> 00:56:09,720 in units I want to use but whatever. 727 00:56:09,720 --> 00:56:15,510 Let's say on this one my h1 is equal to y, 728 00:56:15,510 --> 00:56:17,890 and h2 is equal to minus 1. 729 00:56:21,265 --> 00:56:22,640 So, it's basically the same thing 730 00:56:22,640 --> 00:56:25,727 but all the fields are multiplied by i. 731 00:56:25,727 --> 00:56:27,310 It's the same magnitude, so it's going 732 00:56:27,310 --> 00:56:29,370 to have the same potential energy. 733 00:56:29,370 --> 00:56:31,150 So that's cool. 734 00:56:31,150 --> 00:56:36,370 Clearly this is allowed, and now all the regions 735 00:56:36,370 --> 00:56:38,311 in the universe that are close to the this 736 00:56:38,311 --> 00:56:40,560 are going to sort of smell this particular arrangement 737 00:56:40,560 --> 00:56:42,101 of the Higgs field and say OK, that's 738 00:56:42,101 --> 00:56:45,410 preferred arrangement I want to go into. 739 00:56:45,410 --> 00:56:48,740 So, we have two separate values of the Higgs field 740 00:56:48,740 --> 00:56:53,320 that are happily swooping out space time here. 741 00:56:53,320 --> 00:56:55,180 This gets to the excellent question 742 00:56:55,180 --> 00:57:00,670 I was just asked a moment ago-- what happens when they collide? 743 00:57:00,670 --> 00:57:04,190 As the universe expands and gets cooler, all of it 744 00:57:04,190 --> 00:57:07,060 is going to end up getting swooped 745 00:57:07,060 --> 00:57:09,810 into either the field that was seeded at event one, 746 00:57:09,810 --> 00:57:11,754 or the field that was seeded in to event two, 747 00:57:11,754 --> 00:57:13,170 but at a certain point we're going 748 00:57:13,170 --> 00:57:15,890 to get the bits where they're smashing into one another. 749 00:57:55,810 --> 00:57:58,630 So what happens when these different domains come 750 00:57:58,630 --> 00:58:00,530 into contact with one another? 751 00:58:05,640 --> 00:58:08,210 The absolutely full and probably correct 752 00:58:08,210 --> 00:58:11,390 answer is we don't know. 753 00:58:11,390 --> 00:58:14,470 The reason is that we don't really, to be perfectly blunt, 754 00:58:14,470 --> 00:58:20,150 fully understand every little detail about the symmetry 755 00:58:20,150 --> 00:58:24,580 breaking, or about the structure of whatever 756 00:58:24,580 --> 00:58:26,880 grand unified theory brings all these things together 757 00:58:26,880 --> 00:58:28,310 at the temperatures at which this is happening. 758 00:58:28,310 --> 00:58:30,268 Because this is happening when the universe has 759 00:58:30,268 --> 00:58:32,960 a temperature of like 10 to the 16th GeV. 760 00:58:32,960 --> 00:58:34,730 And so it's way beyond the domain 761 00:58:34,730 --> 00:58:36,830 of where we can push things. 762 00:58:36,830 --> 00:58:41,000 But we can, as physicists are fond of doing, 763 00:58:41,000 --> 00:58:43,570 we can paramaterize our ignorance, 764 00:58:43,570 --> 00:58:45,960 and we can ask ourselves, well what 765 00:58:45,960 --> 00:58:49,480 happens if these various parameters that characterize 766 00:58:49,480 --> 00:58:52,770 my grand unified theory take on the following plausible kinds 767 00:58:52,770 --> 00:58:54,000 of parameters. 768 00:58:54,000 --> 00:58:56,216 And what we find is that generically, 769 00:58:56,216 --> 00:58:58,340 when you have two different domains where the Higgs 770 00:58:58,340 --> 00:59:01,480 field takes on different values like this, when 771 00:59:01,480 --> 00:59:03,560 these domains come into contact you 772 00:59:03,560 --> 00:59:05,345 get what are called topological defects. 773 00:59:35,880 --> 00:59:39,299 The topological defects come in three different flavors. 774 00:59:39,299 --> 00:59:41,090 To understand something about those flavors 775 00:59:41,090 --> 00:59:42,756 you have to know a little bit about what 776 00:59:42,756 --> 00:59:47,430 happens in general when you have phase transitions, 777 00:59:47,430 --> 00:59:51,410 and different regions of your medium 778 00:59:51,410 --> 00:59:53,690 go through a phase transition with different values 779 00:59:53,690 --> 00:59:55,070 of the parameters. 780 00:59:55,070 --> 00:59:57,100 So, it's a general case that whenever 781 00:59:57,100 --> 01:00:02,250 you have some kind of a phase transition 782 01:00:02,250 --> 01:00:04,560 and you have domains of different phase 783 01:00:04,560 --> 01:00:07,930 that come into contact with one another, 784 01:00:07,930 --> 01:00:29,300 your field will attempt to smoothly match itself 785 01:00:29,300 --> 01:00:30,235 across the boundary. 786 01:00:35,790 --> 01:00:37,110 But that can be very difficult. 787 01:00:37,110 --> 01:00:38,980 So if you imagine these particular two cases 788 01:00:38,980 --> 01:00:41,150 that I have here, that's essentially 789 01:00:41,150 --> 01:00:42,690 saying that when these two domains 790 01:00:42,690 --> 01:00:44,530 coming to contact with one another 791 01:00:44,530 --> 01:00:46,830 there's going to be sort of a transition zone 792 01:00:46,830 --> 01:00:49,590 where the field is attempting to rotate 793 01:00:49,590 --> 01:00:52,130 from one value of the Higgs to the other. 794 01:00:52,130 --> 01:00:54,630 And it's going to pick some value that is in some sense 795 01:00:54,630 --> 01:00:56,240 intermediate to those two things. 796 01:00:56,240 --> 01:00:58,390 So that, let's say we continue these up here, 797 01:00:58,390 --> 01:01:03,890 so that the collision is occurring right in this place 798 01:01:03,890 --> 01:01:06,920 here, in this little locus of events in space time. 799 01:01:06,920 --> 01:01:10,830 I have Higgs field 2 over here, Higgs field 1 over here, 800 01:01:10,830 --> 01:01:13,232 and I've got some crazy intermediate field that 801 01:01:13,232 --> 01:01:14,690 goes between the two of them, which 802 01:01:14,690 --> 01:01:19,760 is trying to sort of force itself to smoothly transition 803 01:01:19,760 --> 01:01:21,570 from one to the other. 804 01:01:21,570 --> 01:01:25,010 In so doing, I might end up pushing my field away 805 01:01:25,010 --> 01:01:27,120 from the minimum, in which case there will then 806 01:01:27,120 --> 01:01:31,320 be some energy trapped in that layer. 807 01:01:31,320 --> 01:01:34,920 And there's a reason we do this level the class 808 01:01:34,920 --> 01:01:36,540 in a bit of a hand wavy way, I mean 809 01:01:36,540 --> 01:01:39,490 it's very, very complicated to get the details right. 810 01:01:39,490 --> 01:01:43,770 But the key thing we see is that in doing this match, 811 01:01:43,770 --> 01:01:46,782 the field has to do some pretty silly shenanigans order 812 01:01:46,782 --> 01:01:48,240 to make everything kind of match up 813 01:01:48,240 --> 01:01:52,430 and we can be left with odd observable consequences 814 01:01:52,430 --> 01:01:55,110 from the energy associated with the Higgs field getting 815 01:01:55,110 --> 01:01:57,210 pinned down at that boundary here. 816 01:01:57,210 --> 01:02:02,480 Now, the details of the forms of this boundary 817 01:02:02,480 --> 01:02:05,730 vary a lot depending upon to the specific assumptions 818 01:02:05,730 --> 01:02:08,330 you make about your underlying grand unified theory. 819 01:02:36,230 --> 01:02:38,720 OK, so I should back up for a bit. 820 01:02:38,720 --> 01:02:41,270 I'm sort of assuming here when I discuss all this that there 821 01:02:41,270 --> 01:02:45,750 is some underlying SU(5) theory which describes 822 01:02:45,750 --> 01:02:48,590 the strong weak and electromagnetic interactions 823 01:02:48,590 --> 01:02:51,114 are very, very high temperatures as one gigantic thing. 824 01:02:51,114 --> 01:02:52,530 And we're getting to the point now 825 01:02:52,530 --> 01:02:54,060 where all the different interactions are beginning 826 01:02:54,060 --> 01:02:55,643 to just sort of crystallize out of it. 827 01:02:59,620 --> 01:03:02,920 There's a lot of different ways you can pack your underlying, 828 01:03:02,920 --> 01:03:05,470 fundamental, what we now think of as our standard model, 829 01:03:05,470 --> 01:03:10,010 into SU(5) grand unified theories. 830 01:03:10,010 --> 01:03:11,560 And so the ways in which we can get 831 01:03:11,560 --> 01:03:13,640 different topological defects depend 832 01:03:13,640 --> 01:03:15,580 upon how we choose to do that. 833 01:03:15,580 --> 01:03:26,580 So defect flavor one is you get something called a domain wall. 834 01:03:34,870 --> 01:03:37,600 When we do this the fields attempts 835 01:03:37,600 --> 01:03:42,170 to make itself smoothly match from one region of Higgs field, 836 01:03:42,170 --> 01:03:44,850 say from Higgs 1 to Higgs 2. 837 01:03:44,850 --> 01:03:46,510 It succeeds, but you end up with kind 838 01:03:46,510 --> 01:03:52,070 of a two dimensional structure-- a wall-- in which there's 839 01:03:52,070 --> 01:03:55,570 some kind of anomalous field that is just pinned down there. 840 01:03:59,290 --> 01:04:01,145 And so we end up with a big sheet. 841 01:04:36,430 --> 01:04:38,240 So in a theory like this, it would 842 01:04:38,240 --> 01:04:40,600 predict that somewhere out in the universe if there were 843 01:04:40,600 --> 01:04:42,350 regions in which the Higgs field had taken 844 01:04:42,350 --> 01:04:43,725 on a different value than the one 845 01:04:43,725 --> 01:04:45,969 that we encounter around us right now, 846 01:04:45,969 --> 01:04:48,260 it could be somewhere out gigaparsecs away, essentially 847 01:04:48,260 --> 01:04:50,477 a giant sheet of some kind. 848 01:04:50,477 --> 01:04:52,310 And there would be weird, anomalous behavior 849 01:04:52,310 --> 01:04:53,620 associated with it. 850 01:04:53,620 --> 01:04:55,940 People have really looked long and hard 851 01:04:55,940 --> 01:04:58,500 to try to find things like this and in fact it 852 01:04:58,500 --> 01:05:01,380 would be expected to leave interesting residuals 853 01:05:01,380 --> 01:05:03,630 in the cause of microwave background. 854 01:05:03,630 --> 01:05:05,230 My understanding of the literature 855 01:05:05,230 --> 01:05:07,290 is that there are actually now very 856 01:05:07,290 --> 01:05:10,187 strong bounds on the possibility of having 857 01:05:10,187 --> 01:05:12,270 a grand unified theory that leads to domain walls. 858 01:05:12,270 --> 01:05:15,110 And so this kind of a topological defect 859 01:05:15,110 --> 01:05:17,055 is observationally disfavored. 860 01:05:19,730 --> 01:05:23,820 So this, I should mention, only occurs 861 01:05:23,820 --> 01:05:27,210 in some grand unified theories. 862 01:05:38,360 --> 01:05:48,920 Basically, As we move on to the other flavors of defects 863 01:05:48,920 --> 01:05:53,950 we end up just going down a step in dimensionality associated 864 01:05:53,950 --> 01:05:55,845 with the little kinks that are left over when 865 01:05:55,845 --> 01:05:58,220 the different domains come into contact with one another. 866 01:06:02,550 --> 01:06:08,070 Flavor two, we would get what's called a cosmic string. 867 01:06:11,150 --> 01:06:12,850 Some of you may have heard of this. 868 01:06:12,850 --> 01:06:18,920 This is essentially, at its core, just a one dimensional, 869 01:06:18,920 --> 01:06:22,684 it could be gigparsecs long, but one dimensional, truly 870 01:06:22,684 --> 01:06:24,100 one dimensional-- essentially just 871 01:06:24,100 --> 01:06:28,590 a point in the other two dimensions-- string of mismatch 872 01:06:28,590 --> 01:06:31,150 Higgs field with some kind of an energy density associated 873 01:06:31,150 --> 01:06:32,995 with it when the different domains get in contact. 874 01:06:32,995 --> 01:06:34,390 AUDIENCE: Do we have any estimate 875 01:06:34,390 --> 01:06:38,110 of how close in actual space these different regions would 876 01:06:38,110 --> 01:06:39,050 have started? 877 01:06:39,050 --> 01:06:42,430 PROFESSOR: We do and I'm actually going to get to that. 878 01:06:42,430 --> 01:06:45,390 So, let me give you two answers to that. 879 01:06:45,390 --> 01:06:48,870 One of them is you are going to estimate that apparently 880 01:06:48,870 --> 01:06:52,760 on PSET 10, according to the notes that Alan left for me. 881 01:06:52,760 --> 01:06:55,440 But I'm going to spell out for you the arguments that 882 01:06:55,440 --> 01:06:58,320 go into it in the last 10 minutes of a class. 883 01:06:58,320 --> 01:07:01,350 But yeah, so let me just quickly finish up this one because this 884 01:07:01,350 --> 01:07:04,100 again-- so a cosmic string is sort of like a one dimensional 885 01:07:04,100 --> 01:07:05,570 analog of a domain wall. 886 01:07:14,596 --> 01:07:16,470 And because it would be this sort of long one 887 01:07:16,470 --> 01:07:20,140 dimensional structure, that has actually up a lot of energy 888 01:07:20,140 --> 01:07:22,560 sort of pinned down to it by the fact it has a Higgs 889 01:07:22,560 --> 01:07:25,974 anomaly associated with it, it would be strongly gravitating 890 01:07:25,974 --> 01:07:28,140 and so it would leave really interesting signatures. 891 01:07:28,140 --> 01:07:30,910 It was thought for a while that cosmic strings might 892 01:07:30,910 --> 01:07:33,490 have been the sort of original gravitational anomalies 893 01:07:33,490 --> 01:07:35,510 that seeded some of the structures we see 894 01:07:35,510 --> 01:07:36,700 in the universe today. 895 01:07:36,700 --> 01:07:38,960 Again, it's now pretty highly disfavored. 896 01:07:41,690 --> 01:07:43,910 If cosmic strings exist, they don't 897 01:07:43,910 --> 01:07:49,944 appear to contribute very much to the budget of mass 898 01:07:49,944 --> 01:07:50,610 in our universe. 899 01:07:53,266 --> 01:07:54,890 I should also mention that this is only 900 01:07:54,890 --> 01:07:57,120 predicted by some grand unifying theories. 901 01:08:03,244 --> 01:08:04,660 If you guys are curious about this 902 01:08:04,660 --> 01:08:06,590 I suggest when Alice back you ask him 903 01:08:06,590 --> 01:08:08,650 what the difference between these sums, 904 01:08:08,650 --> 01:08:10,150 why some predict a domain wall, some 905 01:08:10,150 --> 01:08:11,275 predict the cosmic strings. 906 01:08:14,820 --> 01:08:18,680 Flavor three is where you end up with the Higgs 907 01:08:18,680 --> 01:08:21,609 field essentially being able to smoothly transition 908 01:08:21,609 --> 01:08:25,000 without leaving any defect anywhere except at a zero 909 01:08:25,000 --> 01:08:27,840 dimensional point. 910 01:08:27,840 --> 01:08:42,029 So you end up with just a little knot in the Higgs field. 911 01:08:45,300 --> 01:08:51,500 And for reasons that I will outline very soon, 912 01:08:51,500 --> 01:08:55,604 it turns out that this little not must carry magnetic charge, 913 01:08:55,604 --> 01:08:57,479 and so it must be a magnetic monopole. 914 01:09:03,420 --> 01:09:06,290 The domain walls and the cosmic strings 915 01:09:06,290 --> 01:09:11,460 are, as I've emphasized, only predicted 916 01:09:11,460 --> 01:09:14,609 by certain specific grand unified theories. 917 01:09:14,609 --> 01:09:17,290 Magnetic monopoles are actually predicted by all of them. 918 01:09:17,290 --> 01:09:18,020 Question. 919 01:09:18,020 --> 01:09:21,490 AUDIENCE: What does it mean to have a one dimensional domain 920 01:09:21,490 --> 01:09:24,440 wall, because there's no different region separated 921 01:09:24,440 --> 01:09:25,451 by one [INAUDIBLE]. 922 01:09:25,451 --> 01:09:26,450 PROFESSOR: That's right. 923 01:09:26,450 --> 01:09:27,779 So what ends up happening, and this 924 01:09:27,779 --> 01:09:29,320 is where I think you're going to have 925 01:09:29,320 --> 01:09:31,850 to ask Alan to sort of follow up on this a little bit. 926 01:09:31,850 --> 01:09:35,330 So, as the domains come into contact with one another. 927 01:09:35,330 --> 01:09:38,970 The fields do their best to smoothly transition from one 928 01:09:38,970 --> 01:09:39,880 to the other. 929 01:09:39,880 --> 01:09:42,810 And grand unified theories that predict a cosmic string, 930 01:09:42,810 --> 01:09:44,852 they succeed pretty much everywhere. 931 01:09:44,852 --> 01:09:47,060 They're able to actually smoothly make it all go away 932 01:09:47,060 --> 01:09:49,018 so you don't end up with feel being pinned down 933 01:09:49,018 --> 01:09:53,290 anywhere, except in a little one dimensional singularity that 934 01:09:53,290 --> 01:09:57,020 is somewhere along where the two dimensional services originally 935 01:09:57,020 --> 01:09:58,050 met. 936 01:09:58,050 --> 01:10:00,224 And that is-- there's details there 937 01:10:00,224 --> 01:10:01,890 that I'm not even pretending to explain. 938 01:10:05,190 --> 01:10:07,120 And as I say, those are only predicted 939 01:10:07,120 --> 01:10:09,100 by certain kinds of grand unified theories. 940 01:10:09,100 --> 01:10:11,475 All of them will then predict that even if you don't have 941 01:10:11,475 --> 01:10:14,730 that, that cosmic string will then shrink itself down 942 01:10:14,730 --> 01:10:17,270 and it'll just be left with a little knot of Higgs field, 943 01:10:17,270 --> 01:10:20,380 where there's a little bit of residual mismatch 944 01:10:20,380 --> 01:10:23,041 between the two regions. 945 01:10:23,041 --> 01:10:24,915 AUDIENCE: Do all three types of defects carry 946 01:10:24,915 --> 01:10:26,560 a magnetic charge, or only the knots? 947 01:10:26,560 --> 01:10:28,230 PROFESSOR: I think only the knots. 948 01:10:28,230 --> 01:10:29,950 They do carry other kinds of fields, 949 01:10:29,950 --> 01:10:32,830 though, in particular the other ones gravitate, in fact 950 01:10:32,830 --> 01:10:34,240 all them gravitate, and so that's 951 01:10:34,240 --> 01:10:35,400 one of the ways in which people have 952 01:10:35,400 --> 01:10:37,546 tried to set observational limits on these things. 953 01:10:37,546 --> 01:10:38,920 In particular there have recently 954 01:10:38,920 --> 01:10:40,740 been a fair amount of work of people trying 955 01:10:40,740 --> 01:10:43,920 to set limits on cosmic strings from gravitational lensing, 956 01:10:43,920 --> 01:10:46,274 and there was really a lot of excitement 957 01:10:46,274 --> 01:10:48,440 because people thought they discovered want a couple 958 01:10:48,440 --> 01:10:50,436 years ago. 959 01:10:50,436 --> 01:10:54,342 And they saw basically two quasars 960 01:10:54,342 --> 01:10:56,550 that looked absolutely identical, that were separated 961 01:10:56,550 --> 01:11:00,904 or scale that was just right to be a cosmic string. 962 01:11:00,904 --> 01:11:03,320 And then people actually looked at with better telescopes, 963 01:11:03,320 --> 01:11:05,778 and saw they had absolutely nothing to do with one another. 964 01:11:05,778 --> 01:11:09,580 They were not cosmic, they were not lenses, it's just every now 965 01:11:09,580 --> 01:11:12,260 and then God is screwing with you. 966 01:11:12,260 --> 01:11:18,590 OK, so without going into some of the details what you have, 967 01:11:18,590 --> 01:11:24,500 these little point like defects-- and I'm short on time 968 01:11:24,500 --> 01:11:27,490 so I'm going to kind of go through this a little bit 969 01:11:27,490 --> 01:11:31,090 in a sketchy way enough so that I can pay for you how 970 01:11:31,090 --> 01:11:33,250 to do some calculations you're going to need to do. 971 01:11:36,230 --> 01:11:46,250 So the point like defects end up being regions, 972 01:11:46,250 --> 01:11:55,850 where at that point the Higgs field actually takes 973 01:11:55,850 --> 01:11:57,480 the value zero. 974 01:11:57,480 --> 01:12:00,150 So remember I was describing how when you have two regions where 975 01:12:00,150 --> 01:12:07,890 the Higgs fields are both taking on values such as there 976 01:12:07,890 --> 01:12:10,597 at the minimum of the Higgs potential energy, 977 01:12:10,597 --> 01:12:12,180 and they come in to match one another, 978 01:12:12,180 --> 01:12:13,721 and what we have a boundary condition 979 01:12:13,721 --> 01:12:16,770 that very far away the Higgs field has values 980 01:12:16,770 --> 01:12:19,190 such as the energy is minimized. 981 01:12:19,190 --> 01:12:25,140 And there is a theorem, which in his notes Alan-- the way 982 01:12:25,140 --> 01:12:26,930 he describes it is he gives you a figure 983 01:12:26,930 --> 01:12:30,090 and outlines the various things that 984 01:12:30,090 --> 01:12:32,310 are necessary for the theorem to be true, 985 01:12:32,310 --> 01:12:34,410 and invites you to think deeply for a moment 986 01:12:34,410 --> 01:12:40,330 and until insight comes to you, I guess. 987 01:12:40,330 --> 01:12:43,340 And when you put this ingredient that the Higgs field has 988 01:12:43,340 --> 01:12:46,020 this asymptotic, very far away value that drives you 989 01:12:46,020 --> 01:12:49,580 to the minimum of the field, and yet it must change value 990 01:12:49,580 --> 01:12:51,810 somewhere in the middle, the theorem 991 01:12:51,810 --> 01:12:56,089 requires that there be one point at which H equals 0. 992 01:12:56,089 --> 01:12:57,630 And apparently, this is a consequence 993 01:12:57,630 --> 01:13:00,520 in all grand unified theories. 994 01:13:00,520 --> 01:13:02,640 So, recall, H equals 0. 995 01:13:02,640 --> 01:13:14,990 This is a point at which the potential energy 996 01:13:14,990 --> 01:13:16,580 density can be huge. 997 01:13:29,760 --> 01:13:32,470 So, when you have a little point like defect like this, 998 01:13:32,470 --> 01:13:36,440 it looks like a massive nugget, little massive particle. 999 01:13:47,640 --> 01:13:51,170 You can in fact calculate the total amount 1000 01:13:51,170 --> 01:13:53,620 of energy associated with this particle. 1001 01:13:53,620 --> 01:14:16,897 If you do so just including the influence of the Higgs field, 1002 01:14:16,897 --> 01:14:19,440 the calculation basically goes like this. 1003 01:14:19,440 --> 01:14:22,110 It's very similar to the way we calculate the energy associated 1004 01:14:22,110 --> 01:14:25,850 with electric and magnetic fields in electrodynamics. 1005 01:14:25,850 --> 01:14:33,010 Ask yourself, how much energy is contained 1006 01:14:33,010 --> 01:14:39,800 in a sphere of radius, capital R, 1007 01:14:39,800 --> 01:14:44,520 centered on this little knot of Higgs field. 1008 01:14:44,520 --> 01:14:48,380 Well, it's going to look like 4pi times 1009 01:14:48,380 --> 01:14:53,460 an integral of the gradient of the Higgs field 1010 01:14:53,460 --> 01:15:06,080 squared r squared dr It turns out, 1011 01:15:06,080 --> 01:15:10,260 when you calculate the [INAUDIBLE] of the Higgs field 1012 01:15:10,260 --> 01:15:12,770 around one of these little defects, 1013 01:15:12,770 --> 01:15:15,060 it's actually very complicated close to the defect, 1014 01:15:15,060 --> 01:15:17,910 but as you get far away it has a very simple form. 1015 01:15:30,860 --> 01:15:33,742 The gradient goes as 1 over r, it 1016 01:15:33,742 --> 01:15:35,200 tells you the field itself actually 1017 01:15:35,200 --> 01:15:36,380 goes something like log. 1018 01:15:44,370 --> 01:15:51,950 That means your energy looks something like, R squared, 1019 01:15:51,950 --> 01:16:00,070 1 over R squared dr which goes as R, which diverges as you 1020 01:16:00,070 --> 01:16:02,050 make the sphere bigger and bigger and bigger. 1021 01:16:10,209 --> 01:16:11,500 So, what's the mistake we made? 1022 01:16:14,400 --> 01:16:16,719 Well, the Higgs field doesn't always 1023 01:16:16,719 --> 01:16:18,260 just sit there and operate on itself. 1024 01:16:18,260 --> 01:16:21,030 The Higgs field actually couples pretty strongly 1025 01:16:21,030 --> 01:16:23,340 to all of our vector bosons. 1026 01:16:23,340 --> 01:16:25,160 Particularly, it couples pretty strongly 1027 01:16:25,160 --> 01:16:27,390 to electric and magnetic fields. 1028 01:16:27,390 --> 01:16:34,360 So, we have to repeat this calculation including 1029 01:16:34,360 --> 01:16:46,050 the interaction of the Higgs field with the E&D field. 1030 01:17:04,500 --> 01:17:06,500 And in Alan's notes he gives you some references 1031 01:17:06,500 --> 01:17:08,791 on this because this is not the kind of calculation you 1032 01:17:08,791 --> 01:17:10,330 can really sketch out very easily 1033 01:17:10,330 --> 01:17:12,590 in an undergraduate class. 1034 01:17:12,590 --> 01:17:17,140 To make this integral convergent, 1035 01:17:17,140 --> 01:17:21,170 the only way it can be done is if that little nugget 1036 01:17:21,170 --> 01:17:24,410 of Higgs field is endowed with magnetic charge. 1037 01:17:24,410 --> 01:17:29,780 You need to have a monopolar magnetic field that ends up 1038 01:17:29,780 --> 01:17:31,920 putting in interaction terms, that 1039 01:17:31,920 --> 01:17:34,270 make the divergence of this integral go away. 1040 01:18:02,830 --> 01:18:09,470 So, I at last get to the punchline of all this, 1041 01:18:09,470 --> 01:18:17,000 we are left inevitably, if we accept the whole foundation 1042 01:18:17,000 --> 01:18:20,130 story of particle physics that the different interactions were 1043 01:18:20,130 --> 01:18:23,360 unified in some high energy scale and then froze out. 1044 01:18:32,959 --> 01:18:34,500 We are driven inevitably to the story 1045 01:18:34,500 --> 01:18:46,995 that defects in the Higgs field create magnetic monopoles. 1046 01:18:55,670 --> 01:18:58,660 Now, I realize I'm out of time, so let me just quickly sketch 1047 01:18:58,660 --> 01:19:00,317 a few interesting facts about this 1048 01:19:00,317 --> 01:19:02,650 and there's a few exercises that you guys are apparently 1049 01:19:02,650 --> 01:19:06,640 going to look at in your homework assignment. 1050 01:19:06,640 --> 01:19:11,410 When we do this calculation, one which 1051 01:19:11,410 --> 01:19:13,200 is I believe just referenced in the notes 1052 01:19:13,200 --> 01:19:16,150 that Alan has for the class, we learn a couple 1053 01:19:16,150 --> 01:19:17,910 of things about this magnetic charge. 1054 01:19:24,300 --> 01:19:27,020 One of them is that if you work in the fundamental unit, 1055 01:19:27,020 --> 01:19:31,470 say CGS units, the value of the magnetic charge, 1056 01:19:31,470 --> 01:19:42,350 we'll call that g, is exactly 1 over 1057 01:19:42,350 --> 01:19:44,270 2 alpha where alpha is a fine structure 1058 01:19:44,270 --> 01:19:46,540 constant, times the electric charge. 1059 01:19:49,980 --> 01:19:52,070 So if you have two magnetic monopoles 1060 01:19:52,070 --> 01:19:53,570 they attract each other with a force 1061 01:19:53,570 --> 01:19:58,720 that is-- so 1 over 2 alpha is approximately 68.5 I think-- 1062 01:19:58,720 --> 01:20:02,420 and so it would be 68.5 squared times the force of two 1063 01:20:02,420 --> 01:20:04,570 electric charges at that same distance. 1064 01:20:04,570 --> 01:20:06,100 We also end up learning the mass. 1065 01:20:11,010 --> 01:20:19,381 It turns out to be 1 over alpha times the scale of GUT symmetry 1066 01:20:19,381 --> 01:20:19,880 breaking. 1067 01:20:30,945 --> 01:20:33,320 Anyone recall what the scale of GUT symmetry breaking is? 1068 01:20:38,230 --> 01:20:40,080 10 to the 16 GeV. 1069 01:20:40,080 --> 01:20:43,400 So, this is a particle, 1 over alpha is approximately 10 1070 01:20:43,400 --> 01:20:45,670 to the two, so this is a particle that 1071 01:20:45,670 --> 01:20:52,348 has a mass of about 10 to the 18th GeV, 1072 01:20:52,348 --> 01:20:54,730 in other words it's a single particle with a mass of 10 1073 01:20:54,730 --> 01:20:57,700 to the 18th protons. 1074 01:20:57,700 --> 01:21:00,260 This is approximately one microgram. 1075 01:21:04,160 --> 01:21:10,490 If you put one of these things on a scale it could measure it, 1076 01:21:10,490 --> 01:21:12,420 that's bloody big. 1077 01:21:12,420 --> 01:21:16,250 So getting to the last bit of the class, which I am just 1078 01:21:16,250 --> 01:21:19,776 going to very basically quote the answer. 1079 01:21:19,776 --> 01:21:23,270 The question becomes how often do these things get created 1080 01:21:23,270 --> 01:21:25,290 and here I'm going to refer to Alan's notes. 1081 01:21:25,290 --> 01:21:28,470 What you'll find is that, remember 1082 01:21:28,470 --> 01:21:30,640 when we sketched our original picture of this thing 1083 01:21:30,640 --> 01:21:33,310 we looked at regions of the universe 1084 01:21:33,310 --> 01:21:34,860 where the Higgs field was initially 1085 01:21:34,860 --> 01:21:37,780 seeded with different values. 1086 01:21:37,780 --> 01:21:40,200 In order for the Higgs field to take on different values, 1087 01:21:40,200 --> 01:21:43,620 initially, these regions had to be out of causal contact 1088 01:21:43,620 --> 01:21:45,530 with one another. 1089 01:21:45,530 --> 01:21:54,030 So we are going to require that the initial seed areas be 1090 01:21:54,030 --> 01:22:01,540 separated by a distance, which is the correlation 1091 01:22:01,540 --> 01:22:04,600 length, which has to be less than or of 1092 01:22:04,600 --> 01:22:09,260 order the horizon distance. 1093 01:22:09,260 --> 01:22:12,910 You can get a lower bound on this thing by imagining that 1094 01:22:12,910 --> 01:22:16,110 it's-- sorry let me say one other thing. 1095 01:22:16,110 --> 01:22:18,910 If you do that, then you can estimate that the number 1096 01:22:18,910 --> 01:22:27,140 density associated with these things, 1097 01:22:27,140 --> 01:22:29,530 the number density of these monopoles 1098 01:22:29,530 --> 01:22:34,435 will be 1 over the correlation length cubed. 1099 01:22:34,435 --> 01:22:36,690 To get a lower bound on the number 1100 01:22:36,690 --> 01:22:39,960 density of these things, set the correlation length exactly 1101 01:22:39,960 --> 01:22:46,927 to the horizon distance, and then do the following exercise. 1102 01:22:46,927 --> 01:22:48,510 So first, let's set up the correlation 1103 01:22:48,510 --> 01:22:52,000 length equal to the horizon distance. 1104 01:22:52,000 --> 01:23:01,140 Set the density in monopoles equal to the mass of a monopole 1105 01:23:01,140 --> 01:23:06,670 over rH cubed, normalized to the critical density. 1106 01:23:19,330 --> 01:23:21,270 If you do this, you will find that 1107 01:23:21,270 --> 01:23:27,630 just due to magnetic monopoles alone, 1108 01:23:27,630 --> 01:23:30,866 the density of the universe. 1109 01:23:30,866 --> 01:23:32,240 PROFESSOR 2: Excuse me, professor 1110 01:23:32,240 --> 01:23:34,750 PROFESSOR: Yes, I'm wrapping up right this second. 1111 01:23:34,750 --> 01:23:36,083 PROFESSOR 2: It's seven minutes. 1112 01:23:36,083 --> 01:23:37,980 You were supposed to end at 10:55. 1113 01:23:37,980 --> 01:23:39,896 PROFESSOR: I'm substitute teaching, I'm sorry. 1114 01:23:44,350 --> 01:23:48,430 OK, so this tells us that we are at 10 1115 01:23:48,430 --> 01:23:51,335 to the 20 of the critical density. 1116 01:23:54,680 --> 01:23:56,370 And a consequence is that the universe 1117 01:23:56,370 --> 01:23:58,800 is approximately two years old. 1118 01:24:01,430 --> 01:24:04,280 I will let Alan pick it up from there.