1 00:00:00,090 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,030 under a Creative Commons license. 3 00:00:04,030 --> 00:00:06,880 Your support will help MIT OpenCourseWare continue 4 00:00:06,880 --> 00:00:10,740 to offer high quality educational resources for free. 5 00:00:10,740 --> 00:00:13,360 To make a donation or view additional materials 6 00:00:13,360 --> 00:00:17,237 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,237 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:21,570 --> 00:00:24,000 PROFESSOR: Good afternoon. 9 00:00:24,000 --> 00:00:27,440 Welcome to April fool's day. 10 00:00:27,440 --> 00:00:32,409 But we'll be serious about atomic physics. 11 00:00:32,409 --> 00:00:34,540 I hope you enjoyed your spring break. 12 00:00:34,540 --> 00:00:35,380 At least I did. 13 00:00:35,380 --> 00:00:39,530 It's just a little bit change of pace. 14 00:00:39,530 --> 00:00:44,410 So let me remind you what we have been discussing. 15 00:00:44,410 --> 00:00:47,550 We have been talking about interactions 16 00:00:47,550 --> 00:00:49,930 between neutral atoms. 17 00:00:49,930 --> 00:00:55,320 And that can be trivial and very profound. 18 00:00:55,320 --> 00:01:00,360 Trivial because you've heard about van der Waal's forces. 19 00:01:00,360 --> 00:01:03,060 Polarization forces between neutral atoms. 20 00:01:03,060 --> 00:01:05,519 And if you describe them phenomenologically, 21 00:01:05,519 --> 00:01:08,730 you just have 1 over r to the 6 or whatever power 22 00:01:08,730 --> 00:01:10,740 of an interaction. 23 00:01:10,740 --> 00:01:13,980 On the other hand, if you take the position 24 00:01:13,980 --> 00:01:17,780 that everything which happens between neutral objects 25 00:01:17,780 --> 00:01:25,240 has to be mediated, has to be created by the photon field, 26 00:01:25,240 --> 00:01:30,030 then it becomes a question of the vacuum, the vacuum which 27 00:01:30,030 --> 00:01:32,350 surrounds the two neutral atoms. 28 00:01:32,350 --> 00:01:35,800 And I've tried in this chapter to show you 29 00:01:35,800 --> 00:01:40,298 both sides, to similar or the same physics from two 30 00:01:40,298 --> 00:01:41,256 different perspectives. 31 00:01:43,850 --> 00:01:46,540 We talked mainly about the van der Waal's interaction 32 00:01:46,540 --> 00:01:49,590 between two neutral atoms. 33 00:01:49,590 --> 00:01:53,890 I don't want one to-- we pretty much finished it, 34 00:01:53,890 --> 00:01:55,530 so I just give you a bird's view. 35 00:01:55,530 --> 00:01:57,640 I don't really repeat a lot of things. 36 00:01:57,640 --> 00:02:01,910 Because today we want to talk about the forces which 37 00:02:01,910 --> 00:02:05,950 happen between neutral atoms in metal plates and then two metal 38 00:02:05,950 --> 00:02:08,639 plates, which is the Casimir force. 39 00:02:08,639 --> 00:02:13,720 But just to remind you that we showed 40 00:02:13,720 --> 00:02:17,450 that the forces between neutral objects 41 00:02:17,450 --> 00:02:22,150 are actually created by quantum fluctuations. 42 00:02:22,150 --> 00:02:25,580 Classical objects would not create fluctuating dipole 43 00:02:25,580 --> 00:02:28,540 moments, would therefore not create-- 44 00:02:28,540 --> 00:02:32,060 would not give rise to any force between neutral atoms. 45 00:02:32,060 --> 00:02:37,340 And I showed you that there are two aspects where 46 00:02:37,340 --> 00:02:42,380 quantum physics comes in, namely through the fluctuations 47 00:02:42,380 --> 00:02:48,070 or the zero point motion of the atoms as oscillators. 48 00:02:48,070 --> 00:02:50,870 And we even had a circuit model for that. 49 00:02:53,530 --> 00:02:56,700 But then I also said that, well, assume 50 00:02:56,700 --> 00:02:58,750 that you have two neutral atoms and you 51 00:02:58,750 --> 00:03:01,080 have the vacuum fluctuations of the field. 52 00:03:01,080 --> 00:03:04,200 This vacuum fluctuation is driving 53 00:03:04,200 --> 00:03:08,210 coordinated synchronized dipole moments between the two atoms, 54 00:03:08,210 --> 00:03:10,640 and then the two atoms can attract each other. 55 00:03:10,640 --> 00:03:13,620 So we have seen the dual role which quantum physics plays 56 00:03:13,620 --> 00:03:16,820 here in terms of quantum fluctuations 57 00:03:16,820 --> 00:03:20,550 of the atomic oscillators and quantum 58 00:03:20,550 --> 00:03:23,150 fluctuation of the electromagnetic field, which 59 00:03:23,150 --> 00:03:25,880 is another harmonic oscillator. 60 00:03:25,880 --> 00:03:30,790 And at least when I use very simple pictures, 61 00:03:30,790 --> 00:03:34,360 it seemed that the 1 over r to the six potential, 62 00:03:34,360 --> 00:03:37,560 the standard van der Waal's potential at short range, 63 00:03:37,560 --> 00:03:43,000 comes from the vacuum-- from the fluctuations of the oscillator. 64 00:03:43,000 --> 00:03:45,540 I didn't do the derivation, but the simplest derivation 65 00:03:45,540 --> 00:03:50,635 for the long range Casimir Polder retarded potential, 66 00:03:50,635 --> 00:03:54,150 1 over r to the 7, focused on the fluctuations 67 00:03:54,150 --> 00:03:56,620 of the electromagnetic field. 68 00:03:56,620 --> 00:03:58,090 So these were some preliminaries. 69 00:04:01,430 --> 00:04:04,960 Then I thought I want to show you the power of diagrams 70 00:04:04,960 --> 00:04:09,900 by showing to you that there are two different kinds of diagrams 71 00:04:09,900 --> 00:04:13,620 dominant at short range and at long range. 72 00:04:13,620 --> 00:04:18,959 And with that we realize diagrammatically that, because 73 00:04:18,959 --> 00:04:22,555 of the nature of virtual photons which are exchanged by the two 74 00:04:22,555 --> 00:04:25,640 neutral atoms, there is a change in power law. 75 00:04:25,640 --> 00:04:29,705 The power-- the inverse power of the distance between the two 76 00:04:29,705 --> 00:04:33,250 neutral atoms is one higher at long range. 77 00:04:33,250 --> 00:04:37,250 And long range meant-- that also became clear diagrammatically. 78 00:04:37,250 --> 00:04:45,150 Long range means the distance is longer than the wavelengths 79 00:04:45,150 --> 00:04:46,819 of the resonant radiation for the atom. 80 00:05:01,010 --> 00:05:02,070 Any questions about that? 81 00:05:06,750 --> 00:05:17,140 So the goal for the first hour today is to go from two atoms 82 00:05:17,140 --> 00:05:20,830 to many, many atoms, which then form a metal plate, 83 00:05:20,830 --> 00:05:23,970 and make the transition from forces between neutral atoms 84 00:05:23,970 --> 00:05:26,280 to forces between two metal plates, which 85 00:05:26,280 --> 00:05:28,250 is the famous Casimir force. 86 00:05:28,250 --> 00:05:32,020 And I want you to sort of hold the thought about what 87 00:05:32,020 --> 00:05:34,150 is really the quantum aspect here? 88 00:05:34,150 --> 00:05:36,390 Is it the vacuum? 89 00:05:36,390 --> 00:05:38,230 Is it the zero point fluctuations 90 00:05:38,230 --> 00:05:39,510 of the electromagnetic field? 91 00:05:44,020 --> 00:05:47,700 Or are the atoms the zero point fluctuations 92 00:05:47,700 --> 00:05:50,750 of the atomic oscillators responsible for those forces? 93 00:05:50,750 --> 00:05:54,110 So we'll come back to it after we have discussed the Casimir 94 00:05:54,110 --> 00:05:56,374 interaction between two neutral plates. 95 00:06:12,310 --> 00:06:16,410 So maybe one highlight for you today is that maybe in an hour 96 00:06:16,410 --> 00:06:19,430 you have should at least have an expert opinion 97 00:06:19,430 --> 00:06:24,420 whether zero point fluctuations of the electromagnetic field 98 00:06:24,420 --> 00:06:29,660 are real or just a convenient way to describe some physics. 99 00:06:29,660 --> 00:06:31,980 But let's get there. 100 00:06:31,980 --> 00:06:37,590 So the way how I want to make the transition from two 101 00:06:37,590 --> 00:06:40,100 neutral atoms to two metal plates 102 00:06:40,100 --> 00:06:47,340 is by reminding you that the potential was proportional 103 00:06:47,340 --> 00:06:53,590 at long range to 1 over R to the 6, at short range at 1 over R 104 00:06:53,590 --> 00:06:54,700 to the 6. 105 00:07:02,450 --> 00:07:05,770 And if you had the model that the long range interaction 106 00:07:05,770 --> 00:07:09,390 comes, because the vacuum fluctuations drive 107 00:07:09,390 --> 00:07:14,240 dipole moments in the two atoms, then we 108 00:07:14,240 --> 00:07:16,640 know that the potential is proportional 109 00:07:16,640 --> 00:07:19,039 to the polarizability of atom one times 110 00:07:19,039 --> 00:07:20,330 the polarizability of atom two. 111 00:07:24,290 --> 00:07:29,270 So let's now extend that to the interaction 112 00:07:29,270 --> 00:07:34,090 between an atom and a wall. 113 00:07:34,090 --> 00:07:42,850 Well since I don't know-- at least not 114 00:07:42,850 --> 00:07:46,900 yet-- how to describe the wall. 115 00:07:46,900 --> 00:07:48,830 I take the position that, well, we 116 00:07:48,830 --> 00:07:51,020 have an atom come close to a wall, 117 00:07:51,020 --> 00:08:02,490 but now I extend the wall to a sphere of radius z. 118 00:08:02,490 --> 00:08:06,930 And it's not obvious that if an atom interacts with the wall, 119 00:08:06,930 --> 00:08:09,650 if the wall were flat, it is of course this part 120 00:08:09,650 --> 00:08:11,480 of the wall which is most important. 121 00:08:11,480 --> 00:08:13,910 So we should actually get a quantitative or semi 122 00:08:13,910 --> 00:08:18,180 quantitative result out of that. 123 00:08:18,180 --> 00:08:23,770 But now I can use a nice result from electrostatics, 124 00:08:23,770 --> 00:08:31,920 which I also discuss in 8421, in the [? MO1 ?] course, namely 125 00:08:31,920 --> 00:08:40,683 when we have a conducting sphere of radius c. 126 00:08:40,683 --> 00:08:47,410 Do you know what the polarizability of a sphere is? 127 00:08:47,410 --> 00:08:49,350 You must have also had this problem 128 00:08:49,350 --> 00:08:51,700 in classical electrodynamics. 129 00:08:51,700 --> 00:08:53,430 You apply an electric field. 130 00:08:53,430 --> 00:08:57,070 You calculate with the boundary condition of the sphere 131 00:08:57,070 --> 00:08:58,700 some spherical harmonic function. 132 00:08:58,700 --> 00:09:00,740 You calculate what the dipole moment is, 133 00:09:00,740 --> 00:09:02,750 and you find the proportionality constant. 134 00:09:02,750 --> 00:09:05,960 AUDIENCE: It's like z squared or [INAUDIBLE]. 135 00:09:05,960 --> 00:09:06,790 PROFESSOR: Yes. 136 00:09:06,790 --> 00:09:11,470 The one thing I want is that it's proportional to z cube. 137 00:09:11,470 --> 00:09:13,940 It's the volume of the sphere. 138 00:09:13,940 --> 00:09:17,040 And the reason why I discuss that in atomic physics 139 00:09:17,040 --> 00:09:20,040 is when we talk about the dipole moment 140 00:09:20,040 --> 00:09:23,170 at the polarizability of the hydrogen atom, 141 00:09:23,170 --> 00:09:26,460 we find that the polarizability of the hydrogen atom 142 00:09:26,460 --> 00:09:28,926 is the pole radius cube. 143 00:09:28,926 --> 00:09:31,880 So it seems that when it comes to electric fields 144 00:09:31,880 --> 00:09:37,920 and polarizability, an atom, a hydrogen atom behaves exactly 145 00:09:37,920 --> 00:09:40,990 like a conducting sphere, and the size of the sphere 146 00:09:40,990 --> 00:09:42,285 is now the size of the atom. 147 00:09:48,260 --> 00:09:59,315 But that means now that the long range potential will now 148 00:09:59,315 --> 00:10:01,670 be proportional the polarizability of the atom. 149 00:10:04,190 --> 00:10:09,550 And we lose three powers in the power law, 150 00:10:09,550 --> 00:10:12,090 because we have three powers of z 151 00:10:12,090 --> 00:10:13,830 in the polarizability of the sphere. 152 00:10:18,880 --> 00:10:23,220 OK, if that worked so well once, we 153 00:10:23,220 --> 00:10:27,190 can now use that at least as a bridge 154 00:10:27,190 --> 00:10:30,360 to discuss what happens when we have two metal walls. 155 00:10:33,780 --> 00:10:41,510 Well, we can now say we have two spheres, 156 00:10:41,510 --> 00:10:48,000 and the polarizability of each sphere 157 00:10:48,000 --> 00:10:50,030 is proportional to z cubed. 158 00:10:53,280 --> 00:10:56,520 Well when we have two walls, we want 159 00:10:56,520 --> 00:10:59,260 to use the potential per area. 160 00:11:01,890 --> 00:11:06,950 So what we had is we had 1 over R to the 7. 161 00:11:06,950 --> 00:11:10,920 We lose three powers of R or z, because of one sphere. 162 00:11:10,920 --> 00:11:14,760 Another three powers of the other sphere, so it's 1 over z. 163 00:11:14,760 --> 00:11:20,190 But when we normalize by area, it becomes 1 over z squared. 164 00:11:20,190 --> 00:11:25,330 And now we find that the potential is 1 over c cubed. 165 00:11:25,330 --> 00:11:28,300 And at least with it I have-- I wouldn't 166 00:11:28,300 --> 00:11:32,550 say derived, but motivated for you the famous Casimir 167 00:11:32,550 --> 00:11:37,670 potential, which is 1 over c cubed. 168 00:11:37,670 --> 00:11:40,740 And I want to give you an exact derivation of the Casimir 169 00:11:40,740 --> 00:11:45,150 potential in the next few minutes. 170 00:11:45,150 --> 00:11:48,420 But before I do that-- yes? 171 00:11:48,420 --> 00:11:51,950 AUDIENCE: Why did you choose the spheres to be 172 00:11:51,950 --> 00:11:54,664 the same size as the distance between them. 173 00:11:54,664 --> 00:11:55,590 That seems kind of-- 174 00:11:55,590 --> 00:11:56,965 PROFESSOR: It's all just a trick. 175 00:11:56,965 --> 00:11:58,945 [LAUGHTER] 176 00:11:58,945 --> 00:12:06,260 You know what would happen is-- let me just go back here. 177 00:12:06,260 --> 00:12:08,760 If I would choose this sphere smaller, 178 00:12:08,760 --> 00:12:11,330 I would not have given the atom the whole exposure. 179 00:12:11,330 --> 00:12:13,590 I have to make it at least larger 180 00:12:13,590 --> 00:12:16,870 than c to make sure that the curvature of the sphere 181 00:12:16,870 --> 00:12:19,330 doesn't dominate and what the atom sees 182 00:12:19,330 --> 00:12:23,380 is it sees there is a wall. 183 00:12:23,380 --> 00:12:28,010 But then you can see when I make this sphere larger than c, 184 00:12:28,010 --> 00:12:30,810 then I delocalize the dipole moment. 185 00:12:30,810 --> 00:12:35,100 And the bigger sphere has a big polarizability, 186 00:12:35,100 --> 00:12:37,040 but the atom which is so close to the sphere 187 00:12:37,040 --> 00:12:38,600 doesn't care about it. 188 00:12:38,600 --> 00:12:40,640 So you would say the right choice 189 00:12:40,640 --> 00:12:44,350 is just when you think either picture gets corrupted. 190 00:12:44,350 --> 00:12:46,900 But that's often what it is. 191 00:12:46,900 --> 00:12:49,530 So if you think about it, it's the only choice. 192 00:12:49,530 --> 00:12:53,547 Both other choices would lead to an unphysical result, 193 00:12:53,547 --> 00:12:55,130 and you would immediately realize why. 194 00:12:55,130 --> 00:12:59,260 In one case it's the curvature, and the other case you're not 195 00:12:59,260 --> 00:13:01,760 really sampling the dipole of the sphere 196 00:13:01,760 --> 00:13:06,980 because you want to do a dipole field, 197 00:13:06,980 --> 00:13:12,730 and if you have a dipole of size z, the field with the dipole 198 00:13:12,730 --> 00:13:15,990 starts to be an approximation if you go distance z away. 199 00:13:19,780 --> 00:13:21,678 Other questions? 200 00:13:21,678 --> 00:13:23,160 Yes? 201 00:13:23,160 --> 00:13:26,618 AUDIENCE: Can you explain where the first term [INAUDIBLE] 202 00:13:26,618 --> 00:13:29,582 z to the [INAUDIBLE] come from? 203 00:13:32,145 --> 00:13:33,020 PROFESSOR: We had it. 204 00:13:36,720 --> 00:13:39,380 Last class we had the result that the Casimir-- 205 00:13:39,380 --> 00:13:42,720 that the van der Waal's-- the retarded van der 206 00:13:42,720 --> 00:13:45,490 Waal's potential is 1 over R to the 7. 207 00:13:45,490 --> 00:13:47,680 But with the physical picture in mind 208 00:13:47,680 --> 00:13:50,700 that the van der Waal's potential at long range, which 209 00:13:50,700 --> 00:13:54,680 is 1 over R to the 7 comes because a vacuum fluctuation 210 00:13:54,680 --> 00:13:59,070 drive to dipoles, and the two dipoles attract each other. 211 00:13:59,070 --> 00:14:02,640 When each dipole is alpha 1 times a vacuum field, 212 00:14:02,640 --> 00:14:05,260 the other one is alpha 2 times a vacuum field. 213 00:14:05,260 --> 00:14:10,270 But now for this sphere, alpha is z cubed. 214 00:14:10,270 --> 00:14:12,680 So therefore three powers in the denominator 215 00:14:12,680 --> 00:14:15,400 cancel, and I get z to the 4. 216 00:14:15,400 --> 00:14:18,550 So you can say the alpha 2, I absorbed-- 217 00:14:18,550 --> 00:14:22,350 I used the kind of result that alpha 2 is z cubed, 218 00:14:22,350 --> 00:14:25,130 and then the alpha 2 cancelled with three powers 219 00:14:25,130 --> 00:14:27,175 of the distance in the denominator. 220 00:14:30,290 --> 00:14:32,950 Other questions? 221 00:14:32,950 --> 00:14:34,570 I have a question for you now. 222 00:14:34,570 --> 00:14:38,780 If you have two metal plates, what about short range and long 223 00:14:38,780 --> 00:14:41,340 range? 224 00:14:41,340 --> 00:14:44,320 We expect something under the rock. 225 00:14:44,320 --> 00:14:48,970 If we assume a metal plate, what is the vessel-- 226 00:14:48,970 --> 00:14:53,000 we had the distinction between long range and short range, 227 00:14:53,000 --> 00:14:55,320 which came from the resonant radiation 228 00:14:55,320 --> 00:14:57,860 from the resonant excitation of the atom. 229 00:14:57,860 --> 00:15:00,075 And when we are closer than the wave lengths, 230 00:15:00,075 --> 00:15:01,700 we're in short range, if you're further 231 00:15:01,700 --> 00:15:04,530 than the wavelengths of the resonant transition, 232 00:15:04,530 --> 00:15:07,970 we're in long range. 233 00:15:07,970 --> 00:15:10,480 Between an atom and a plate, well, we still 234 00:15:10,480 --> 00:15:12,220 have the resonance of the atom. 235 00:15:12,220 --> 00:15:16,740 But what happens when we have two plates? 236 00:15:16,740 --> 00:15:20,230 There's clearly no resonance, so therefore there 237 00:15:20,230 --> 00:15:23,910 is only one power law, 1 over z cubed. 238 00:15:23,910 --> 00:15:27,660 But I want you to sort of think about it for a moment 239 00:15:27,660 --> 00:15:31,120 whether this is now-- whether everything 240 00:15:31,120 --> 00:15:33,350 is long range or everything is short range. 241 00:15:33,350 --> 00:15:38,870 In other words, you say effective resonant radiation, 242 00:15:38,870 --> 00:15:41,730 you say effective wavelengths of the plate lambda 243 00:15:41,730 --> 00:15:43,490 equals 0 or lambda equals infinity. 244 00:15:54,205 --> 00:15:54,705 Any opinion? 245 00:16:02,505 --> 00:16:04,606 AUDIENCE: [INAUDIBLE] ability, so it 246 00:16:04,606 --> 00:16:06,342 must be infinite [INAUDIBLE]. 247 00:16:19,238 --> 00:16:22,875 Take the two metal plates, the vacuum energy 248 00:16:22,875 --> 00:16:24,528 between those plates, and compare 249 00:16:24,528 --> 00:16:29,158 it to vacuum energy at the same volume free space 250 00:16:29,158 --> 00:16:32,630 and subtract the [INAUDIBLE] energy difference. 251 00:16:32,630 --> 00:16:37,900 And high frequency components cancel out, so you have the-- 252 00:16:37,900 --> 00:16:43,730 PROFESSOR: OK, we'll actually do exactly what you suggest, 253 00:16:43,730 --> 00:16:48,210 to compare vacuum energies, in the next few minutes. 254 00:16:48,210 --> 00:16:50,080 It's not quite obvious, but I want 255 00:16:50,080 --> 00:16:53,660 to tell you now we have made-- by assuming 256 00:16:53,660 --> 00:16:56,180 the metallic boundary condition, by assuming 257 00:16:56,180 --> 00:16:59,050 that we have idealized metal plates which fulfill 258 00:16:59,050 --> 00:17:02,200 the boundary condition that the electric field is 259 00:17:02,200 --> 00:17:04,839 0 at the surface. 260 00:17:04,839 --> 00:17:07,420 The parallel component of the [INAUDIBLE] at the surface. 261 00:17:07,420 --> 00:17:10,859 We've made an assumption here, and the assumption is actually 262 00:17:10,859 --> 00:17:18,980 that the resonant wavelengths goes to 0, 263 00:17:18,980 --> 00:17:23,280 and resonance frequency goes to infinity. 264 00:17:23,280 --> 00:17:28,010 And the power law we obtain is actually the long distance 265 00:17:28,010 --> 00:17:31,380 power law so to speak, and for two metal plates, 266 00:17:31,380 --> 00:17:34,510 there is no short distance. 267 00:17:34,510 --> 00:17:37,410 I'm not sure if I can give you a simple explanation 268 00:17:37,410 --> 00:17:42,010 because the boundary condition is so idealized that you don't 269 00:17:42,010 --> 00:17:45,372 recognize, you don't smell any atomic resonance anymore. 270 00:17:45,372 --> 00:17:46,830 But maybe the one thing I would say 271 00:17:46,830 --> 00:17:49,770 is that you say the boundary condition, metallic boundary 272 00:17:49,770 --> 00:17:52,680 condition for infinitely high frequency. 273 00:17:52,680 --> 00:17:55,340 And sometimes that would also mean 274 00:17:55,340 --> 00:17:58,150 that the plasma frequency of the metal is pushed to infinity. 275 00:18:00,850 --> 00:18:02,800 We come back to that in the next chapter. 276 00:18:24,947 --> 00:18:25,530 Any questions? 277 00:18:27,824 --> 00:18:28,740 AUDIENCE: [INAUDIBLE]? 278 00:18:35,470 --> 00:18:37,680 AUDIENCE: Excellent question. 279 00:18:37,680 --> 00:18:41,840 I will go for you now through-- its mainly classical e and m. 280 00:18:41,840 --> 00:18:45,990 This is why I use pre-written slides. 281 00:18:45,990 --> 00:18:49,910 We sum up over all modes with a boundary condition, 282 00:18:49,910 --> 00:18:51,680 with a metallic boundary condition. 283 00:18:51,680 --> 00:18:55,960 If you have in general a dielectric or semiconductor 284 00:18:55,960 --> 00:18:59,420 which is not an ideal metal, has different boundary conditions, 285 00:18:59,420 --> 00:19:01,320 the Casimir force will change. 286 00:19:01,320 --> 00:19:04,720 And this is actually a pretty outstanding difficult problem 287 00:19:04,720 --> 00:19:06,930 where people have really chewed on hard. 288 00:19:06,930 --> 00:19:09,830 I think Lifschitz has some famous solution on it. 289 00:19:09,830 --> 00:19:15,760 What is the Casimir force for an arbitrary dielectric constant 290 00:19:15,760 --> 00:19:17,720 of the surface material? 291 00:19:17,720 --> 00:19:19,060 But absolutely yes. 292 00:19:19,060 --> 00:19:21,260 If you don't have a metallic boundary condition, 293 00:19:21,260 --> 00:19:22,385 your Casimir force changes. 294 00:19:26,640 --> 00:19:29,400 I think this is sort of an interesting point, 295 00:19:29,400 --> 00:19:31,260 and it may trigger immediately discussions 296 00:19:31,260 --> 00:19:32,342 what does it really mean. 297 00:19:32,342 --> 00:19:34,550 But let me go now through school the derivation where 298 00:19:34,550 --> 00:19:36,800 we have metallic boundary condition, 299 00:19:36,800 --> 00:19:40,700 and this actually leads to a quantitative derivation 300 00:19:40,700 --> 00:19:41,690 of the Casimir force. 301 00:19:41,690 --> 00:19:46,720 Quantitative in the way that we get-- 302 00:19:46,720 --> 00:19:48,370 we're not just getting the power law. 303 00:19:48,370 --> 00:19:49,530 We get the prefector. 304 00:19:49,530 --> 00:19:52,320 We everything 100% of this derivation. 305 00:19:52,320 --> 00:19:55,075 Collin? 306 00:19:55,075 --> 00:19:58,012 AUDIENCE: So what is the mechanism for the Casimir force 307 00:19:58,012 --> 00:19:58,512 changing? 308 00:19:58,512 --> 00:20:01,949 Because if think of it in the picture of two metal plates, 309 00:20:01,949 --> 00:20:05,386 the lower density and modes inside and then do outside. 310 00:20:05,386 --> 00:20:09,314 Even with the arbitrary boundary condition, I [INAUDIBLE]. 311 00:20:09,314 --> 00:20:10,787 But what is the physical mechanism 312 00:20:10,787 --> 00:20:13,615 for changing the form? 313 00:20:13,615 --> 00:20:15,240 PROFESSOR: To the best of my knowledge, 314 00:20:15,240 --> 00:20:18,830 it is the density of modes. 315 00:20:18,830 --> 00:20:20,215 But it's very subtle. 316 00:20:20,215 --> 00:20:22,750 I suggest, since not everybody is probably 317 00:20:22,750 --> 00:20:24,850 familiar with the way how I will count 318 00:20:24,850 --> 00:20:27,710 all the modes in the metal plate, let me just do that, 319 00:20:27,710 --> 00:20:30,170 and then we should-- and I hope then it's obvious, 320 00:20:30,170 --> 00:20:32,570 or we should then discuss the boundary conditions being 321 00:20:32,570 --> 00:20:34,210 modified. 322 00:20:34,210 --> 00:20:35,370 And you have a question? 323 00:20:35,370 --> 00:20:37,619 AUDIENCE: Yes, but maybe we can discuss it after this. 324 00:20:41,510 --> 00:20:44,140 PROFESSOR: So the idea is the following, 325 00:20:44,140 --> 00:20:49,320 if the Casimir force is only due to zero 326 00:20:49,320 --> 00:20:53,030 point fluctuations of electromagnetic field-- 327 00:20:53,030 --> 00:20:55,280 let me say up front this is a valid assumption. 328 00:20:55,280 --> 00:20:56,510 I get a valid result. 329 00:20:56,510 --> 00:20:58,540 But I tell you in half an hour that you 330 00:20:58,540 --> 00:21:00,460 can take another perspective and say 331 00:21:00,460 --> 00:21:03,480 this is just a holistic description. 332 00:21:03,480 --> 00:21:06,520 There are other derivations of the Casimir force 333 00:21:06,520 --> 00:21:10,195 which do not use the concept of zero point energy at all. 334 00:21:10,195 --> 00:21:11,570 So in other words, I'm presenting 335 00:21:11,570 --> 00:21:15,330 you one presentation of the problem, one solution 336 00:21:15,330 --> 00:21:18,380 of the problem, but there's a whole different formulation. 337 00:21:18,380 --> 00:21:21,780 But here I assume-- mean I can easily modulate. 338 00:21:21,780 --> 00:21:23,920 OK, well here we have two metal plates. 339 00:21:23,920 --> 00:21:26,810 We want to understand if there is potential energy 340 00:21:26,810 --> 00:21:29,040 and interaction between the two metal plates. 341 00:21:29,040 --> 00:21:32,800 And let's just calculate what is the total energy of the system. 342 00:21:32,800 --> 00:21:34,400 And if I have two metal plates, which 343 00:21:34,400 --> 00:21:36,440 are two mathematical boundary conditions, 344 00:21:36,440 --> 00:21:39,680 all I have to calculate is the ground state 345 00:21:39,680 --> 00:21:41,950 of the electromagnetic field in between. 346 00:21:41,950 --> 00:21:44,800 And that means I have to find all of the modes, 347 00:21:44,800 --> 00:21:48,810 and each mode contribute h bar omega over 2. 348 00:21:48,810 --> 00:21:50,600 So this is our agenda now. 349 00:21:50,600 --> 00:21:53,530 We say the metal plates are not real objects. 350 00:21:53,530 --> 00:21:56,580 They're mathematical boundary conditions. 351 00:21:56,580 --> 00:21:57,980 Looks like a good assumption. 352 00:21:57,980 --> 00:22:00,440 And now based on those boundary conditions 353 00:22:00,440 --> 00:22:03,935 we calculate the zero point energy of the world, 354 00:22:03,935 --> 00:22:06,320 at least of the electromagnetic world. 355 00:22:06,320 --> 00:22:10,390 And then we ask if the plates are a little bit closer, 356 00:22:10,390 --> 00:22:14,290 tool has the total zero point energy changed? 357 00:22:14,290 --> 00:22:16,520 And when it has changed that means 358 00:22:16,520 --> 00:22:18,270 there is a force between the atoms because 359 00:22:18,270 --> 00:22:20,570 of potential changes as a function of distance. 360 00:22:20,570 --> 00:22:25,000 So this is the agenda I went to execute now. 361 00:22:25,000 --> 00:22:29,190 Any questions about what we want to do? 362 00:22:29,190 --> 00:22:32,770 As you know when you write out modes in e and m 363 00:22:32,770 --> 00:22:36,990 and you want to do it exactly the many, many indices. 364 00:22:36,990 --> 00:22:39,140 But this is a result of which could actually 365 00:22:39,140 --> 00:22:41,490 come straight out of Jackson. 366 00:22:41,490 --> 00:22:43,940 For your convenience, this is also 367 00:22:43,940 --> 00:22:47,330 discussed-- a wonderful discussion by Serge Haroche. 368 00:22:47,330 --> 00:22:51,630 In some these summer school notes from the '70s or '80s. 369 00:22:51,630 --> 00:22:53,610 I've posted those on the web. 370 00:22:53,610 --> 00:22:58,880 So if there is a mathematical detail you want to look up, 371 00:22:58,880 --> 00:23:01,400 this is posted on the web. 372 00:23:01,400 --> 00:23:05,898 So anyway, we have two metal plates with a distance l. 373 00:23:05,898 --> 00:23:10,500 And we have metallic boundary conditions. 374 00:23:10,500 --> 00:23:18,000 That means that when we calculate the modes, we get TE 375 00:23:18,000 --> 00:23:20,030 modes and TM modes. 376 00:23:20,030 --> 00:23:23,620 TE modes means there is no longitudinal electric field. 377 00:23:23,620 --> 00:23:28,460 TM modes mean there is no longitudinal magnetic field. 378 00:23:28,460 --> 00:23:42,350 And the frequency of each mode is discrete in the z direction. 379 00:23:42,350 --> 00:23:45,710 In the z direction we have standing waves, 380 00:23:45,710 --> 00:23:49,460 which are labelled with an intentional value of m, 381 00:23:49,460 --> 00:23:56,050 whereas parallel to the plate the modes can have an arbitrary 382 00:23:56,050 --> 00:23:58,620 transverse wave vector k. 383 00:23:58,620 --> 00:24:01,280 And when we look at the dispersion relation, 384 00:24:01,280 --> 00:24:04,630 the frequency of those modes is in quadrature. 385 00:24:04,630 --> 00:24:06,960 You can see the component k squared c 386 00:24:06,960 --> 00:24:09,230 squared of the transverse propagation 387 00:24:09,230 --> 00:24:11,658 plus the component of the standing wave. 388 00:24:14,590 --> 00:24:17,940 So for each m, for each integer value, 389 00:24:17,940 --> 00:24:22,680 we have a TE and TM mode, with the exception of m equals 0. 390 00:24:22,680 --> 00:24:25,360 For m equals 0 we don't have a TE mode. 391 00:24:25,360 --> 00:24:28,900 We only have a TM mode. 392 00:24:28,900 --> 00:24:31,245 So that's just fresh out of Jackson. 393 00:24:34,020 --> 00:24:36,057 But it's very important now-- and I 394 00:24:36,057 --> 00:24:37,890 want to spend a little bit more time with it 395 00:24:37,890 --> 00:24:40,180 so it's just a purely classic result. 396 00:24:40,180 --> 00:24:43,680 What is the density of states? 397 00:24:43,680 --> 00:24:48,510 Well, let's assume we have a given m. 398 00:24:48,510 --> 00:24:51,990 So we have a standing wave perpendicular 399 00:24:51,990 --> 00:24:54,310 to the capacitor plates, and therefore 400 00:24:54,310 --> 00:25:01,840 now the density of modes only comes-- the density of modes 401 00:25:01,840 --> 00:25:07,420 only comes from the transverse. 402 00:25:07,420 --> 00:25:09,620 Let me just get my toolbar back. 403 00:25:13,690 --> 00:25:16,060 So it is from this k squared. 404 00:25:16,060 --> 00:25:19,780 And you know that the number of modes 405 00:25:19,780 --> 00:25:23,520 is proportional to the volume in k space. 406 00:25:23,520 --> 00:25:31,640 And the important part is that I can write this as dk squared. 407 00:25:31,640 --> 00:25:35,480 But k squared is d omega squared. 408 00:25:35,480 --> 00:25:38,240 And this gives me omega d omega. 409 00:25:38,240 --> 00:25:43,120 So for each n value, for each discrete mode perpendicular 410 00:25:43,120 --> 00:25:45,500 to the plates, we have now a density 411 00:25:45,500 --> 00:25:51,030 of modes which goes as omega d omega. 412 00:25:51,030 --> 00:25:54,440 Before I show you the mathematical expression, 413 00:25:54,440 --> 00:25:59,730 let me just sort of plot that, what that means. 414 00:25:59,730 --> 00:26:07,230 The density of modes was-- the number of modes was omega 415 00:26:07,230 --> 00:26:08,320 d omega. 416 00:26:08,320 --> 00:26:13,690 That means the density of modes was proportional with omega. 417 00:26:13,690 --> 00:26:19,570 That means for m equals 0 we started omega equals 0, 418 00:26:19,570 --> 00:26:21,430 and we get a straight line. 419 00:26:21,430 --> 00:26:28,340 For m equals 1-- m equals 1 shown in green, 420 00:26:28,340 --> 00:26:32,060 and m equals 2 shown in purple. 421 00:26:32,060 --> 00:26:40,070 The frequency omega-- we have omega squared 422 00:26:40,070 --> 00:26:47,730 is sick the quadrature sum k squared and m squared. 423 00:26:47,730 --> 00:26:53,090 So therefore if you want to ask what 424 00:26:53,090 --> 00:26:56,620 is the density for modes for m equals 1 or me equals 2, 425 00:26:56,620 --> 00:27:00,020 you have to start at the lowest frequency for m equals 1, 426 00:27:00,020 --> 00:27:02,230 at the lowest frequency for m equals 2. 427 00:27:02,230 --> 00:27:06,030 But then as we just derived, the density of modes 428 00:27:06,030 --> 00:27:09,590 is proportional to omega. 429 00:27:09,590 --> 00:27:11,530 So what you we to do in the end, we 430 00:27:11,530 --> 00:27:14,820 want to sum those modes all up. 431 00:27:14,820 --> 00:27:19,070 And what we will observe is that here we 432 00:27:19,070 --> 00:27:21,630 have a line with slope 1. 433 00:27:21,630 --> 00:27:24,400 Here we have to sum two lines with slope 1. 434 00:27:24,400 --> 00:27:27,680 Here we have to sum three lines with slope 1. 435 00:27:27,680 --> 00:27:32,460 So progressively, the slope gets higher and higher. 436 00:27:32,460 --> 00:27:36,350 But the slope, which increases linearly with the coordinate 437 00:27:36,350 --> 00:27:37,070 axis. 438 00:27:37,070 --> 00:27:41,830 If it were continuously, would just be a parabola. 439 00:27:41,830 --> 00:27:44,670 So in other words, if I would [INAUDIBLE] 440 00:27:44,670 --> 00:27:47,470 the density of states which comes from the exact mode 441 00:27:47,470 --> 00:27:49,500 spectrum, if I would [INAUDIBLE], 442 00:27:49,500 --> 00:27:51,400 it would be parabola. 443 00:27:51,400 --> 00:27:56,180 And parabolic density of state, the number of modes 444 00:27:56,180 --> 00:27:58,160 is omega squared d omega. 445 00:27:58,160 --> 00:28:00,870 This is three dimensional space. 446 00:28:00,870 --> 00:28:04,030 So in other words, what we observe is what exactly 447 00:28:04,030 --> 00:28:07,220 the effect of the boundary condition is. 448 00:28:07,220 --> 00:28:10,770 Instead of getting this smooth parabola omega squared, 449 00:28:10,770 --> 00:28:13,960 we have sort of broken up the parabola 450 00:28:13,960 --> 00:28:15,880 into piecewise linear pieces. 451 00:28:18,420 --> 00:28:23,030 And the fact that we have a boundary condition, 452 00:28:23,030 --> 00:28:26,620 that we plates in free space, and not just the space, 453 00:28:26,620 --> 00:28:30,260 is actually the difference between the two. 454 00:28:30,260 --> 00:28:33,220 So it is the difference between the two, which 455 00:28:33,220 --> 00:28:37,030 leads to the fact that a boundary condition modifies 456 00:28:37,030 --> 00:28:37,990 the vacuum energy. 457 00:28:41,027 --> 00:28:41,526 Questions? 458 00:28:47,500 --> 00:28:52,070 OK with that the rest is mathematics, 459 00:28:52,070 --> 00:28:54,380 but I want to show you the steps because it 460 00:28:54,380 --> 00:28:57,350 is ingenious mathematics. 461 00:28:57,350 --> 00:28:59,120 It makes a few approximations. 462 00:28:59,120 --> 00:29:01,680 It creates a few infinities and gets rid of it. 463 00:29:01,680 --> 00:29:06,612 It's actually rather wild, but amazing that, in the end, 464 00:29:06,612 --> 00:29:09,050 we have a simple exact result. 465 00:29:09,050 --> 00:29:14,740 OK so the most important thing is we have a density of states. 466 00:29:14,740 --> 00:29:23,320 And the density of states for one m is proportional to omega, 467 00:29:23,320 --> 00:29:25,639 but now we have to multiply-- when 468 00:29:25,639 --> 00:29:27,930 we want to know what is the density of state in a given 469 00:29:27,930 --> 00:29:31,830 omega, we have to multiply with the integer 470 00:29:31,830 --> 00:29:37,330 number of discrete modes which are available at this omega. 471 00:29:37,330 --> 00:29:41,580 So this is because there is only one m equals 0 node. 472 00:29:41,580 --> 00:29:42,470 We have a one here. 473 00:29:42,470 --> 00:29:46,240 And then for m equals 1 and higher, we have two [INAUDIBLE] 474 00:29:46,240 --> 00:29:49,160 between TE and TM. 475 00:29:49,160 --> 00:29:50,930 So this is just counting the modes. 476 00:29:54,260 --> 00:29:57,850 And this is now a mathematical trick 477 00:29:57,850 --> 00:29:59,830 that, instead of counting the modes-- 478 00:29:59,830 --> 00:30:04,070 it's an integer-- we can sum up heavy side step functions 479 00:30:04,070 --> 00:30:05,800 and sum over all m. 480 00:30:05,800 --> 00:30:07,780 This is just an [? exact v, ?] right? 481 00:30:07,780 --> 00:30:12,820 But this is something which helps us in the next step 482 00:30:12,820 --> 00:30:14,400 to do an integral. 483 00:30:14,400 --> 00:30:17,480 So now so far we haven't made any approximation. 484 00:30:17,480 --> 00:30:20,960 We've just reminded ourselves what 485 00:30:20,960 --> 00:30:23,890 is the exact result of classical e and m. 486 00:30:26,870 --> 00:30:31,990 OK, but now we want to calculate the zero point. 487 00:30:31,990 --> 00:30:33,940 So now it's getting interesting, because we 488 00:30:33,940 --> 00:30:37,990 take the classical result, and take it into quantum physics. 489 00:30:37,990 --> 00:30:39,990 The first step is very innocent. 490 00:30:39,990 --> 00:30:45,060 We say each mode contributes h bar omega over 2. 491 00:30:45,060 --> 00:30:49,070 And then we integrate with the density of states. 492 00:30:49,070 --> 00:30:52,290 And the next line is also an exact result, 493 00:30:52,290 --> 00:30:56,360 just using the density of modes from the previous page. 494 00:30:56,360 --> 00:30:59,640 But of course, now we are asking for trouble. 495 00:30:59,640 --> 00:31:03,120 We are integrating with omega squared d omega, 496 00:31:03,120 --> 00:31:04,810 and there is uv catastrophe. 497 00:31:08,440 --> 00:31:11,450 Now I can say I can put on my mathematical head 498 00:31:11,450 --> 00:31:15,400 and say let's add a convergent term, something 499 00:31:15,400 --> 00:31:19,420 which exponentially decreases with omega, which is just 500 00:31:19,420 --> 00:31:22,120 cutting off the high frequency part. 501 00:31:22,120 --> 00:31:25,670 And I hope that in the end-- and this will come out too-- 502 00:31:25,670 --> 00:31:28,820 that I get a result which is independent of the cutoff. 503 00:31:28,820 --> 00:31:35,570 So by choosing this cut of parameter lambda to zero, 504 00:31:35,570 --> 00:31:38,320 I have-- if this is zero, I have no cut off. 505 00:31:38,320 --> 00:31:41,920 So maybe I can introduce a cut off, and then at the end 506 00:31:41,920 --> 00:31:43,180 let lambda go to got to zero. 507 00:31:43,180 --> 00:31:45,270 And if this gives me a physical result, 508 00:31:45,270 --> 00:31:48,070 I've sort of mathematically tricked around. 509 00:31:48,070 --> 00:31:51,800 But I also like the physical justification for that. 510 00:31:51,800 --> 00:31:55,060 I mean you know that a metal plate tennis has 511 00:31:55,060 --> 00:31:59,110 not metallic boundary conditions at infinitely high frequency. 512 00:31:59,110 --> 00:32:01,730 If you go deep to the x rays and gamma rays, 513 00:32:01,730 --> 00:32:03,770 it becomes transparent. 514 00:32:03,770 --> 00:32:07,100 So therefore, we would argue that, once we go to very, very 515 00:32:07,100 --> 00:32:12,130 high frequencies, the modes do not 516 00:32:12,130 --> 00:32:15,500 feel that there is a boundary condition, 517 00:32:15,500 --> 00:32:20,720 and therefore we may be allowed to do some physical cut off. 518 00:32:20,720 --> 00:32:27,000 OK so we do this cut off by-- so now 519 00:32:27,000 --> 00:32:29,720 we have to find the zero point energy. 520 00:32:29,720 --> 00:32:33,920 The zero point energy has a sum over m. 521 00:32:33,920 --> 00:32:37,230 And I call each integral I sub n. 522 00:32:37,230 --> 00:32:40,030 And this integral is the omega square 523 00:32:40,030 --> 00:32:43,240 from the density of states, but now multiplied 524 00:32:43,240 --> 00:32:45,510 with a convergence term. 525 00:32:45,510 --> 00:32:48,260 Whether it's mathematical or physical. 526 00:32:48,260 --> 00:32:56,350 And by introducing this cut off, I 527 00:32:56,350 --> 00:32:58,190 can now exactly solve the integral. 528 00:32:58,190 --> 00:33:01,130 The omega squared can be taken out of the integral 529 00:33:01,130 --> 00:33:03,480 by taking the second derivative. 530 00:33:03,480 --> 00:33:05,620 And this is just an exponential function 531 00:33:05,620 --> 00:33:07,930 which can be integrated. 532 00:33:07,930 --> 00:33:10,910 And therefore we have a mathematically exact result. 533 00:33:10,910 --> 00:33:12,936 So we are now able to do the integral 534 00:33:12,936 --> 00:33:14,560 because we have avoided the divergence. 535 00:33:17,450 --> 00:33:18,460 OK. 536 00:33:18,460 --> 00:33:22,240 Now we come back and say we are really interested if the cutoff 537 00:33:22,240 --> 00:33:24,740 is pushed to higher and higher frequency. 538 00:33:24,740 --> 00:33:27,780 And this means lambda equals 0. 539 00:33:27,780 --> 00:33:34,050 So therefore, we want to take this expression 540 00:33:34,050 --> 00:33:36,970 and expand it around lambda equals 0. 541 00:33:39,860 --> 00:33:44,720 Well, you could say that's physically very well motivated. 542 00:33:44,720 --> 00:33:46,660 But if I take this function, which 543 00:33:46,660 --> 00:33:50,770 is lambda e to the lambda over x times e to the x, 544 00:33:50,770 --> 00:33:55,620 and expand it, I get divergence is 1 over x squared. 545 00:33:55,620 --> 00:33:57,810 So you would say how can you expand it. 546 00:33:57,810 --> 00:33:59,040 Let's see what happens. 547 00:33:59,040 --> 00:34:00,100 Let's just do it. 548 00:34:00,100 --> 00:34:03,611 Let's say we are interested around lambda or x equals 0. 549 00:34:08,159 --> 00:34:11,139 We have some divergencies now, but we 550 00:34:11,139 --> 00:34:14,630 have to find a way to deal with it. 551 00:34:14,630 --> 00:34:17,929 So we get rid of some of the divergencies 552 00:34:17,929 --> 00:34:19,780 in the following way. 553 00:34:19,780 --> 00:34:23,540 By saying, well, if you simply calculate the zero point 554 00:34:23,540 --> 00:34:26,090 energy between those two capacitor plates, 555 00:34:26,090 --> 00:34:28,570 and then when, let's say, if there 556 00:34:28,570 --> 00:34:35,909 is-- if the plates come closer and there is less zero point 557 00:34:35,909 --> 00:34:38,989 energy, we would see the plates attract each other. 558 00:34:38,989 --> 00:34:42,179 But what happens is the plates are not moving, 559 00:34:42,179 --> 00:34:44,690 and there is no world behind them. 560 00:34:44,690 --> 00:34:48,250 The world is sort of as it is. 561 00:34:48,250 --> 00:34:50,520 And that means we have to sort of now 562 00:34:50,520 --> 00:34:54,239 look more carefully at the zero point energy of our capacitor 563 00:34:54,239 --> 00:34:56,050 plates where l is small. 564 00:34:56,050 --> 00:34:57,900 And the rest of the world. 565 00:34:57,900 --> 00:35:01,460 So we are now representing the world 566 00:35:01,460 --> 00:35:04,440 as a big capacitor of size l 0. 567 00:35:04,440 --> 00:35:06,390 And then our plate is just moving 568 00:35:06,390 --> 00:35:10,900 within that, because we are not assuming that then the two 569 00:35:10,900 --> 00:35:12,830 capacitor plates attract each other 570 00:35:12,830 --> 00:35:15,995 and move towards each other that the size of the universe 571 00:35:15,995 --> 00:35:17,430 is changing. 572 00:35:17,430 --> 00:35:21,620 So we want to move a boundary condition within the universe. 573 00:35:21,620 --> 00:35:25,240 So therefore, the correct expression for the zero point 574 00:35:25,240 --> 00:35:28,940 energy is not the zero point energy of capacitor l. 575 00:35:28,940 --> 00:35:31,440 It's the zero point energy of a capacitor size 576 00:35:31,440 --> 00:35:35,210 l plus of a capacitor l0 naught minus l. 577 00:35:38,240 --> 00:35:41,680 And so we take the result and simply add 578 00:35:41,680 --> 00:35:46,340 the expressions for l and l0 minus l. 579 00:35:46,340 --> 00:35:51,450 And what is now a nice is that the first highly divergent term 580 00:35:51,450 --> 00:35:54,560 with 1 over lambda to the 4-- reminder 581 00:35:54,560 --> 00:35:56,810 that we want to let lambda go to zero. 582 00:35:56,810 --> 00:35:59,810 It becomes independent of l. 583 00:35:59,810 --> 00:36:02,700 So we would say, well, if there is an infinity, 584 00:36:02,700 --> 00:36:04,700 it doesn't change with l. 585 00:36:04,700 --> 00:36:06,480 It doesn't provide a potential. 586 00:36:06,480 --> 00:36:11,130 It's just one of those infinities which is constant, 587 00:36:11,130 --> 00:36:13,220 which is not affecting the physics. 588 00:36:13,220 --> 00:36:17,940 Also here, this is independent of l. 589 00:36:17,940 --> 00:36:24,670 And then the next term gives us the famous Casimir potential, 590 00:36:24,670 --> 00:36:28,820 1 over l cubed, and it is independent of lambda. 591 00:36:28,820 --> 00:36:31,585 So we have the result which was independent of the cut 592 00:36:31,585 --> 00:36:33,020 off we introduced earlier. 593 00:36:35,570 --> 00:36:40,070 So therefore, we have derived, with a little bit 594 00:36:40,070 --> 00:36:43,130 of hand waving to navigate about infinities, 595 00:36:43,130 --> 00:36:50,290 that the potential between two metal plates is 1 over l cubed. 596 00:36:50,290 --> 00:36:55,130 a squared was the area, the size of the capacitor plates. 597 00:36:55,130 --> 00:36:56,560 And this is the exact prefector. 598 00:36:59,250 --> 00:37:01,690 Or if you want to figure out what 599 00:37:01,690 --> 00:37:03,820 is the pressure off the vacuum. 600 00:37:03,820 --> 00:37:07,007 Pressure is, of course, force per unit area. 601 00:37:07,007 --> 00:37:09,090 So you take the derivative of the potential that's 602 00:37:09,090 --> 00:37:11,990 the force, that's the force per unit area, 603 00:37:11,990 --> 00:37:14,250 and this is now the vacuum pressure. 604 00:37:16,800 --> 00:37:18,730 Sort of funny now you can put in the unit. 605 00:37:18,730 --> 00:37:21,971 It's 10 to the minus 5 millibar at a distance 606 00:37:21,971 --> 00:37:22,720 of one micrometer. 607 00:37:26,530 --> 00:37:30,740 So anyway this is, I think, the classic deviation 608 00:37:30,740 --> 00:37:36,100 of the Casimir potential by, I would say, 609 00:37:36,100 --> 00:37:38,990 exactly summing up the zero point 610 00:37:38,990 --> 00:37:42,210 energy within a certain boundary condition. 611 00:37:48,161 --> 00:37:48,660 Good. 612 00:37:56,570 --> 00:38:00,140 Maybe to address Collin's question. 613 00:38:00,140 --> 00:38:05,840 What would happen if the metal plates were maybe 614 00:38:05,840 --> 00:38:07,840 semiconductors or had a different dielectrical 615 00:38:07,840 --> 00:38:09,120 constant. 616 00:38:09,120 --> 00:38:10,880 The mode spectrum would change. 617 00:38:10,880 --> 00:38:13,090 AUDIENCE: Like if you had mixed boundary conditions. 618 00:38:13,090 --> 00:38:15,562 You don't have these nicely spaced-- 619 00:38:15,562 --> 00:38:17,270 PROFESSOR: I don't know how to calculate. 620 00:38:17,270 --> 00:38:18,860 That's a hard problem, and a lot of people 621 00:38:18,860 --> 00:38:20,610 have really worked very, very hard on it. 622 00:38:20,610 --> 00:38:24,190 It seems for idealized boundary condition you can do it easily, 623 00:38:24,190 --> 00:38:26,020 but it has been clearly a challenge 624 00:38:26,020 --> 00:38:28,170 to mathematical physics. 625 00:38:28,170 --> 00:38:31,080 And I'm not even sure if it has been solved in all generality 626 00:38:31,080 --> 00:38:34,110 or only in special examples to extend that 627 00:38:34,110 --> 00:38:37,410 to every type of surfaces. 628 00:38:37,410 --> 00:38:41,260 But we can, for instance, assume if you have a conductor, which 629 00:38:41,260 --> 00:38:46,780 is not-- conductor which has not infinite conductance, 630 00:38:46,780 --> 00:38:50,350 0 resistivity, then the electric field 631 00:38:50,350 --> 00:38:52,790 can penetrate a little bit into the conductor. 632 00:38:52,790 --> 00:38:55,180 And if the electric field penetrates into it, 633 00:38:55,180 --> 00:38:58,150 it changes the frequency of the mode. 634 00:38:58,150 --> 00:39:01,377 So it's clear that the boundary conditions have an effect here. 635 00:39:13,570 --> 00:39:15,670 But let's now have some discussion. 636 00:39:22,790 --> 00:39:28,670 We had focused before on the force between two atoms. 637 00:39:28,670 --> 00:39:33,370 And if we use a microscopic model for the metal plates, 638 00:39:33,370 --> 00:39:39,170 it should be, it must be possible to obtain the Casimir 639 00:39:39,170 --> 00:39:43,770 potential by just summing up all the forces between atoms. 640 00:39:43,770 --> 00:39:46,430 Because ultimately if you have no metal plates, 641 00:39:46,430 --> 00:39:47,560 you don't have any force. 642 00:39:50,510 --> 00:39:53,950 Where s-- in the last few minutes, 643 00:39:53,950 --> 00:39:56,590 we focused on the derivation which was simply summing up 644 00:39:56,590 --> 00:39:59,380 the zero point energies. 645 00:39:59,380 --> 00:40:01,900 And there are two papers which I've 646 00:40:01,900 --> 00:40:09,035 posted on the wiki which argue that both results are 647 00:40:09,035 --> 00:40:09,535 equivalent. 648 00:40:12,090 --> 00:40:14,120 So I would really recommend to you-- 649 00:40:14,120 --> 00:40:17,350 he expresses it much better than I can. 650 00:40:17,350 --> 00:40:19,170 Read the introduction and the conclusion 651 00:40:19,170 --> 00:40:21,030 from Bob Jaffe's paper. 652 00:40:21,030 --> 00:40:23,700 He clearly says in the introduction 653 00:40:23,700 --> 00:40:27,190 that there are a lot of books and references by famous people 654 00:40:27,190 --> 00:40:30,170 who have said the Casimir force is 655 00:40:30,170 --> 00:40:32,920 the manifestation of the zero point energy. 656 00:40:32,920 --> 00:40:36,030 The fact that people have observed the Casimir force 657 00:40:36,030 --> 00:40:40,290 means that h bar omega over 2 is real. 658 00:40:43,640 --> 00:40:45,950 Those authors take the opposite approach. 659 00:40:45,950 --> 00:40:49,060 They say this is just heuristic. 660 00:40:49,060 --> 00:40:49,980 It's convenient. 661 00:40:49,980 --> 00:40:51,830 You can get the correct result. 662 00:40:51,830 --> 00:40:55,130 But there is an equivalent derivation 663 00:40:55,130 --> 00:40:58,680 which gives exactly the same result, which focuses only 664 00:40:58,680 --> 00:41:01,620 on pair wise interactions between atoms, 665 00:41:01,620 --> 00:41:07,090 and it does not include any-- it doesn't make any reference 666 00:41:07,090 --> 00:41:10,450 to the zero point energy of the electromagnetic field. 667 00:41:10,450 --> 00:41:13,800 So therefore, one, I would agree with those authors 668 00:41:13,800 --> 00:41:17,450 that the conclusion is there is absolutely 669 00:41:17,450 --> 00:41:23,260 no observational evidence for the zero point energy, 670 00:41:23,260 --> 00:41:25,560 that the zero point energy is sort of not just 671 00:41:25,560 --> 00:41:29,450 convenient counting, but that it is real energy. 672 00:41:29,450 --> 00:41:32,420 And the Casimir force, of course, also 673 00:41:32,420 --> 00:41:35,550 it has often been quoted as a counter example 674 00:41:35,550 --> 00:41:37,400 that this is a direct observation, 675 00:41:37,400 --> 00:41:39,650 is not-- should not be regarded as such. 676 00:41:43,510 --> 00:41:48,850 Now people at MIT probably know Bob Jaffe. 677 00:41:48,850 --> 00:41:51,560 He's a high energy physicist. 678 00:41:51,560 --> 00:41:54,310 And why is he interested in atomic physics? 679 00:41:54,310 --> 00:41:57,015 Why is he interested in the Casimir force? 680 00:41:57,015 --> 00:42:00,680 Well, the question of zero point energies 681 00:42:00,680 --> 00:42:05,060 is very, very important in these days for dark energy. 682 00:42:05,060 --> 00:42:08,550 I mean, you know that 80% of the energy of the universe 683 00:42:08,550 --> 00:42:11,200 is dark energy. 684 00:42:11,200 --> 00:42:13,930 Then there's dark matter, and then there is baryonic matter. 685 00:42:13,930 --> 00:42:16,690 But this dark energy is really mysterious. 686 00:42:16,690 --> 00:42:18,810 And one candidate for dark energy 687 00:42:18,810 --> 00:42:23,062 is the dark energy simply comes from the zero point energy. 688 00:42:23,062 --> 00:42:24,520 So therefore, people are now really 689 00:42:24,520 --> 00:42:27,280 interested is the dark energy real. 690 00:42:27,280 --> 00:42:31,380 And if it is real, does it have a gravitational effect? 691 00:42:31,380 --> 00:42:33,320 Does it change the metric of space? 692 00:42:33,320 --> 00:42:35,955 And can it accelerate the expansion of the universe? 693 00:42:42,820 --> 00:42:46,140 Of course, you have a problem if you calculate the zero point 694 00:42:46,140 --> 00:42:48,530 energy, you don't introduce a cut off. 695 00:42:48,530 --> 00:42:52,230 You have a spectacular uv divergence. 696 00:42:52,230 --> 00:42:57,590 You can maybe now concatenate the divergence 697 00:42:57,590 --> 00:43:03,570 by saying, OK, eventually if the wavelengths become shorter 698 00:43:03,570 --> 00:43:06,120 and shorter and shorter, there is a cut off. 699 00:43:06,120 --> 00:43:09,740 And if you have no idea at all, you 700 00:43:09,740 --> 00:43:12,000 would introduce a cut off at the Planck lengths, 701 00:43:12,000 --> 00:43:14,770 because beyond the Planck lengths 702 00:43:14,770 --> 00:43:17,280 all physical descriptions break down. 703 00:43:17,280 --> 00:43:19,870 But if you cut it off at the Planck lengths, 704 00:43:19,870 --> 00:43:23,520 you find that the zero point energy of all modes 705 00:43:23,520 --> 00:43:28,570 is larger than the observed dark energy in the universe 706 00:43:28,570 --> 00:43:31,060 for acceleration of the cosmic expansion 707 00:43:31,060 --> 00:43:34,970 by 124 orders of magnitude. 708 00:43:34,970 --> 00:43:37,920 And if you think you want to introduce a cut off which 709 00:43:37,920 --> 00:43:41,340 is maybe the classical electron radius. 710 00:43:41,340 --> 00:43:43,540 The classical electron radius is the radius 711 00:43:43,540 --> 00:43:47,980 of a charged sphere for which the electrostatic energy would 712 00:43:47,980 --> 00:43:51,590 just be the rest energy of the electron. 713 00:43:51,590 --> 00:43:55,310 Well, then you're off by 43 orders of magnitude. 714 00:43:55,310 --> 00:43:57,940 So some people have called those discrepancies, 715 00:43:57,940 --> 00:44:00,960 which are completely unresolved as of now, 716 00:44:00,960 --> 00:44:05,490 the biggest failure ever of theoretical physics. 717 00:44:05,490 --> 00:44:09,410 So anyway, that's why there is renewed interest in the Casimir 718 00:44:09,410 --> 00:44:10,180 force. 719 00:44:10,180 --> 00:44:13,140 And at least in electromagnetics, 720 00:44:13,140 --> 00:44:14,940 we have a very, very simple model. 721 00:44:14,940 --> 00:44:17,450 And we have experiments which measure the Casimir force, 722 00:44:17,450 --> 00:44:22,010 but, as I've said, it cannot be regarded as a direct 723 00:44:22,010 --> 00:44:24,017 observation of the zero point energy. 724 00:44:27,840 --> 00:44:32,230 OK, let me now come to a question 725 00:44:32,230 --> 00:44:40,660 we started discussing earlier, namely if we-- here we 726 00:44:40,660 --> 00:44:41,550 have atoms. 727 00:44:41,550 --> 00:44:44,100 Here we have a metalized boundary condition. 728 00:44:44,100 --> 00:44:48,990 So maybe let's just talk for one minute about it. 729 00:44:48,990 --> 00:44:52,940 What kind of assumptions do we make about atoms 730 00:44:52,940 --> 00:44:55,900 when we replace them by a metallic boundary condition? 731 00:45:00,680 --> 00:45:02,770 From the whole derivation of the Casimir force 732 00:45:02,770 --> 00:45:08,480 it's not obvious that there are no atomic properties which 733 00:45:08,480 --> 00:45:13,090 have entered the calculation of the Casimir potential. 734 00:45:17,350 --> 00:45:21,350 What Bob Jaffe argues in this paper-- and again, 735 00:45:21,350 --> 00:45:22,580 it's wonderful reading. 736 00:45:22,580 --> 00:45:25,470 I really recommend Just if you want to enjoy some physics, 737 00:45:25,470 --> 00:45:29,000 open up the document and read the first and the last page. 738 00:45:29,000 --> 00:45:33,160 The Casimir force from an idealized boundary condition 739 00:45:33,160 --> 00:45:36,540 is completely independent of atomic properties. 740 00:45:36,540 --> 00:45:41,290 So therefore, it must be a limit where-- well, we 741 00:45:41,290 --> 00:45:43,560 talked about the resonant wavelengths, 742 00:45:43,560 --> 00:45:46,950 but we can also talk about the fine structure constant, 743 00:45:46,950 --> 00:45:52,140 because the fine structure constant in a hydrogenic model 744 00:45:52,140 --> 00:45:56,300 provides what that resident transitions in hydrogen are. 745 00:45:56,300 --> 00:45:58,510 So you can argue if something is completely 746 00:45:58,510 --> 00:46:00,760 independent of atomic properties, 747 00:46:00,760 --> 00:46:05,310 somehow by sweeping things under the rug we must have set 748 00:46:05,310 --> 00:46:08,400 alpha to either zero or infinite. 749 00:46:08,400 --> 00:46:10,340 Because if it is set to any value, 750 00:46:10,340 --> 00:46:13,370 it would correspond to some version of the hydrogen 751 00:46:13,370 --> 00:46:15,970 atom which has a resonant radiation. 752 00:46:15,970 --> 00:46:18,340 So the question is, is alpha set to zero 753 00:46:18,340 --> 00:46:22,900 or is alpha set to infinity when you derive the Casimir force? 754 00:46:22,900 --> 00:46:24,755 And I think this is now easy to discuss, 755 00:46:24,755 --> 00:46:27,670 because the fine structure constant alpha 756 00:46:27,670 --> 00:46:30,870 is e squared over h bar c. 757 00:46:30,870 --> 00:46:33,050 If you set alpha to zero, you do it 758 00:46:33,050 --> 00:46:36,640 by setting the charge of the atoms to zero, 759 00:46:36,640 --> 00:46:39,130 and then you have no interaction at all. 760 00:46:39,130 --> 00:46:41,680 So therefore, if you want to have the Casimir 761 00:46:41,680 --> 00:46:44,360 force in a limit which is independent of atomic 762 00:46:44,360 --> 00:46:48,290 properties, the alpha equals zero limit gives no force. 763 00:46:48,290 --> 00:46:57,160 So therefore, the equivalence this picture, 764 00:46:57,160 --> 00:47:01,610 summing up over the electrons and obtaining the Casimir force 765 00:47:01,610 --> 00:47:03,760 for the metallic boundary condition, 766 00:47:03,760 --> 00:47:08,520 this equivalence happens when you do this derivation 767 00:47:08,520 --> 00:47:11,585 and set the fine structure constant to infinity. 768 00:47:17,090 --> 00:47:20,780 How big is the fine structure constant? 769 00:47:20,780 --> 00:47:24,940 1 over 137. 770 00:47:24,940 --> 00:47:27,400 In some expansions of alpha and alpha squared, 771 00:47:27,400 --> 00:47:29,000 it's actually a smaller parameter. 772 00:47:29,000 --> 00:47:31,780 And now I'm telling you when you have two metal plates 773 00:47:31,780 --> 00:47:33,420 and measure the Casimir force, this 774 00:47:33,420 --> 00:47:36,820 corresponds to the infinite alpha limit. 775 00:47:36,820 --> 00:47:38,940 So this is now getting complicated 776 00:47:38,940 --> 00:47:44,330 and I can refer here only to Bob Jaffe's paper. 777 00:47:44,330 --> 00:47:49,250 But he shows that when you go to distances of 0.5 micron 778 00:47:49,250 --> 00:47:52,200 that people have measured the Casimir force. 779 00:47:52,200 --> 00:47:55,415 You are already the alpha equals infinity limit. 780 00:47:58,460 --> 00:48:03,060 If you are-- I wish I could read my handwriting. 781 00:48:03,060 --> 00:48:05,700 I think that's minus 5. 782 00:48:05,700 --> 00:48:09,730 If alpha is larger than 10 to the minus 5. 783 00:48:09,730 --> 00:48:16,020 So therefor, alpha equals 1 over 137 is for that purpose 784 00:48:16,020 --> 00:48:18,670 already very, very well in the large alpha limit. 785 00:48:22,030 --> 00:48:23,490 But I just hate this result here. 786 00:48:23,490 --> 00:48:27,160 This is rather complicated. 787 00:48:27,160 --> 00:48:31,560 Bob Jaffe says that to get the Casimir potential by summing up 788 00:48:31,560 --> 00:48:34,189 over atoms is actually a quite challenging 789 00:48:34,189 --> 00:48:35,105 mathematical exercise. 790 00:48:40,860 --> 00:48:43,106 Questions? 791 00:48:43,106 --> 00:48:44,010 Yes? 792 00:48:44,010 --> 00:48:47,243 AUDIENCE: If these calculations of summing over atoms 793 00:48:47,243 --> 00:48:51,878 has been [INAUDIBLE], then what is it-- we can just 794 00:48:51,878 --> 00:48:54,373 talk about the plates being dielectric from that atom 795 00:48:54,373 --> 00:48:57,367 [INAUDIBLE]? 796 00:48:57,367 --> 00:49:00,940 PROFESSOR: There are solutions for dielectric plates. 797 00:49:00,940 --> 00:49:05,160 And sometimes when people do experiments, 798 00:49:05,160 --> 00:49:10,342 the best surfaces you can get is a silicon surface. 799 00:49:10,342 --> 00:49:12,550 Because they want you to do qualitative measurements, 800 00:49:12,550 --> 00:49:15,545 they're discussing corrections due to the dielectric constant. 801 00:49:19,350 --> 00:49:22,220 I know the problem has been worked out 802 00:49:22,220 --> 00:49:24,080 for dielectric surfaces. 803 00:49:24,080 --> 00:49:29,430 I assume not by doing the transition from summing up atom 804 00:49:29,430 --> 00:49:31,890 by atom, but by rather looking at the boundary 805 00:49:31,890 --> 00:49:34,340 condition for electromagnetic fields 806 00:49:34,340 --> 00:49:36,380 in the presence of a dielectric. 807 00:49:36,380 --> 00:49:39,240 But I don't know more details about that. 808 00:49:44,440 --> 00:49:45,060 Yes? 809 00:49:45,060 --> 00:49:46,810 AUDIENCE: This question may not make sense 810 00:49:46,810 --> 00:49:48,393 because I haven't thought [INAUDIBLE]. 811 00:49:48,393 --> 00:49:50,530 But again, when we consider two plates, 812 00:49:50,530 --> 00:49:53,624 if you have room for two atoms, we get the two 1 over R 813 00:49:53,624 --> 00:49:57,802 to the 6, 1 over R to the 7 cases. 814 00:49:57,802 --> 00:49:59,802 Once of which is the retarded Casimir potential, 815 00:49:59,802 --> 00:50:04,046 but for two plates-- again, we conclude that it was more like 816 00:50:04,046 --> 00:50:05,180 the short range-- 817 00:50:05,180 --> 00:50:06,346 PROFESSOR: Well, long range. 818 00:50:06,346 --> 00:50:07,543 We are always retarded. 819 00:50:07,543 --> 00:50:10,898 AUDIENCE: OK, but then I'm a bit confused. 820 00:50:10,898 --> 00:50:12,856 Because then if you say alpha goes to infinity, 821 00:50:12,856 --> 00:50:17,200 don't you have an h bar or c to zero? 822 00:50:17,200 --> 00:50:17,700 Oh wait. 823 00:50:17,700 --> 00:50:19,910 That makes sense. 824 00:50:19,910 --> 00:50:23,290 PROFESSOR: Alpha infinity is-- we 825 00:50:23,290 --> 00:50:27,440 know that this is the vastness of the electron. 826 00:50:27,440 --> 00:50:31,690 The [? Whitbeck ?] constant is alpha squared times smaller. 827 00:50:31,690 --> 00:50:33,800 And the [? Whitbeck ?] constant-- well, 828 00:50:33,800 --> 00:50:36,790 a quarter of the [? Whitbeck ?] constant 829 00:50:36,790 --> 00:50:39,707 is alignment alpha radiation. 830 00:50:39,707 --> 00:50:41,290 So anyway, the [? Whitbeck ?] constant 831 00:50:41,290 --> 00:50:43,880 is a scale factor in the transition frequencies 832 00:50:43,880 --> 00:50:45,050 of the hydrogen atom. 833 00:50:45,050 --> 00:50:46,710 And so therefore, it is consistent 834 00:50:46,710 --> 00:50:50,440 that, if you make alpha large, we actually 835 00:50:50,440 --> 00:50:55,780 get-- we push the frequency of the atomic transition 836 00:50:55,780 --> 00:50:59,230 to very, very high frequencies. 837 00:50:59,230 --> 00:51:02,870 And that would mean the resonant wavelengths goes to zero. 838 00:51:02,870 --> 00:51:06,250 And therefore any finite distance between the capacitor 839 00:51:06,250 --> 00:51:08,920 plate is, in the long range, is longer than 840 00:51:08,920 --> 00:51:10,480 the wavelengths of the transition. 841 00:51:10,480 --> 00:51:11,972 AUDIENCE: I think equivalently you 842 00:51:11,972 --> 00:51:13,930 could say if the speed of light is really slow, 843 00:51:13,930 --> 00:51:17,074 you always will have [INAUDIBLE]. 844 00:51:17,074 --> 00:51:17,990 PROFESSOR: Yeah, sure. 845 00:51:17,990 --> 00:51:20,240 You can do the transition in many ways 846 00:51:20,240 --> 00:51:23,530 by tuning the speed of light, or tuning the charge, 847 00:51:23,530 --> 00:51:25,605 but it all has the same implication. 848 00:51:28,314 --> 00:51:28,980 Other questions? 849 00:51:31,660 --> 00:51:32,608 Lindsay? 850 00:51:32,608 --> 00:51:34,978 AUDIENCE: Is there any other observable reason 851 00:51:34,978 --> 00:51:37,640 why we should constantly be thinking 852 00:51:37,640 --> 00:51:40,622 about Casimir forces and dark energy? 853 00:51:40,622 --> 00:51:43,610 Like experimenting [INAUDIBLE]. 854 00:51:43,610 --> 00:51:46,420 PROFESSOR: As far as I know, this question about dark energy 855 00:51:46,420 --> 00:51:49,100 and whether it is related or not to zero 856 00:51:49,100 --> 00:51:54,670 point energy of the field is completely open. 857 00:51:54,670 --> 00:51:57,740 I think if you had an idea how to do a decisive-- if any 858 00:51:57,740 --> 00:52:02,280 of you had an idea how to make a decisive experiment measuring 859 00:52:02,280 --> 00:52:05,430 whether zero point energies are real or not, 860 00:52:05,430 --> 00:52:06,820 that would have a big impact. 861 00:52:06,820 --> 00:52:09,010 As far as I know, it's completely open. 862 00:52:09,010 --> 00:52:10,670 Nobody has really good idea. 863 00:52:13,950 --> 00:52:18,460 So some people feel, mathematically, the zero point 864 00:52:18,460 --> 00:52:23,810 energy has the right, correct physically characteristics. 865 00:52:23,810 --> 00:52:27,920 Zero point energy could explain the extra term 866 00:52:27,920 --> 00:52:30,360 in Einstein's field equation, which is often 867 00:52:30,360 --> 00:52:32,530 called the cosmological constant term. 868 00:52:32,530 --> 00:52:37,910 But the quantity, it's so many orders of magnitude 869 00:52:37,910 --> 00:52:41,680 off that people don't even know if that is correct or not. 870 00:52:41,680 --> 00:52:44,360 One could also say, well, maybe zero point energies 871 00:52:44,360 --> 00:52:47,500 do not give rise at all to any gravitation potential. 872 00:52:47,500 --> 00:52:49,710 And therefore maybe if energy doesn't give rise 873 00:52:49,710 --> 00:52:52,600 to gravitational potential, maybe the energy is not real. 874 00:52:52,600 --> 00:52:55,570 And what we find, what is responsible for the expansion 875 00:52:55,570 --> 00:53:10,084 of the universe is gravity-- so Einstein's-- what's the 876 00:53:10,084 --> 00:53:11,000 constant called again? 877 00:53:11,000 --> 00:53:11,912 I just mentioned it. 878 00:53:11,912 --> 00:53:12,870 AUDIENCE: Cosmological. 879 00:53:12,870 --> 00:53:14,260 PROFESSOR: Cosmological. 880 00:53:14,260 --> 00:53:16,420 So that is just the cosmological constant, 881 00:53:16,420 --> 00:53:19,120 but the cosmological constant reflects something else 882 00:53:19,120 --> 00:53:21,000 in the universe and is not related at all 883 00:53:21,000 --> 00:53:21,930 to zero point energy. 884 00:53:21,930 --> 00:53:26,190 To the best of my knowledge, these are the possibilities. 885 00:53:31,710 --> 00:53:37,180 OK, so let's go back to atoms. 886 00:53:37,180 --> 00:53:44,120 Our next big chapter 12 is on resonant scattering. 887 00:53:57,520 --> 00:53:59,510 Just that you see the structure of the course. 888 00:54:02,020 --> 00:54:05,660 I wanted to introduce sort of some mysteries and subtleties 889 00:54:05,660 --> 00:54:09,280 of electromagnetic interactions to you by using diagrams, 890 00:54:09,280 --> 00:54:12,440 by using an exact formulation of QED. 891 00:54:12,440 --> 00:54:14,510 And then I gave you one example where 892 00:54:14,510 --> 00:54:16,720 we could use those diagrams, and these 893 00:54:16,720 --> 00:54:18,570 were van der Waal's forces. 894 00:54:18,570 --> 00:54:20,890 So we got a little bit side tracked with the vacuum, 895 00:54:20,890 --> 00:54:22,800 with van der Waal's and Casimir forces, 896 00:54:22,800 --> 00:54:25,670 but now we want to go back to the diagrams. 897 00:54:25,670 --> 00:54:29,250 And there is one aspect of the diagrams which 898 00:54:29,250 --> 00:54:33,190 is sort of fascinating, and this is where we really 899 00:54:33,190 --> 00:54:37,280 need a treatment which goes beyond perturbation theory. 900 00:54:37,280 --> 00:54:40,820 And this is when we have resonant radiation. 901 00:54:40,820 --> 00:54:46,680 When resonant light interacts with atoms. 902 00:54:46,680 --> 00:54:48,810 Because, as I'm going to show you, 903 00:54:48,810 --> 00:54:51,910 we have then in any perturbative expression 904 00:54:51,910 --> 00:54:55,800 zero in the energy denominator, and we would have divergence. 905 00:54:55,800 --> 00:54:58,750 So I want to show you now, partially 906 00:54:58,750 --> 00:55:01,180 also for the beauty of the physical theory, 907 00:55:01,180 --> 00:55:05,790 how can we deal-- how can we fix those divergencies? 908 00:55:05,790 --> 00:55:07,750 How can we fix those infinities we 909 00:55:07,750 --> 00:55:10,750 encounter in perturbation theory? 910 00:55:10,750 --> 00:55:13,850 The answer will be we get rid of these infinities 911 00:55:13,850 --> 00:55:17,670 by summing up an infinite number of diagrams. 912 00:55:17,670 --> 00:55:19,630 So I'm going to show you now-- it actually 913 00:55:19,630 --> 00:55:22,470 sounds so much more harder than it is. 914 00:55:22,470 --> 00:55:25,860 How, with rather modest mathematical effort, 915 00:55:25,860 --> 00:55:30,100 we can perform an infinite sum over diagrams. 916 00:55:30,100 --> 00:55:32,230 And I want to show you what those diagram are. 917 00:55:32,230 --> 00:55:35,560 And the result of that is that the divergencies disappear, 918 00:55:35,560 --> 00:55:37,510 and we have a valid description of what 919 00:55:37,510 --> 00:55:40,270 happens when atoms encounter resonant radiation. 920 00:55:44,970 --> 00:55:53,850 So I'm following here, almost religiously, 921 00:55:53,850 --> 00:55:57,080 the atom photon interaction book. 922 00:55:57,080 --> 00:56:01,860 So this in production is a summary of those pages. 923 00:56:06,380 --> 00:56:13,440 So when we discussed diagrams, we 924 00:56:13,440 --> 00:56:20,180 wanted to figure out what is the amplitude for a system 925 00:56:20,180 --> 00:56:23,730 to go from an initial to a final state. 926 00:56:23,730 --> 00:56:27,120 And those were the matrix elements 927 00:56:27,120 --> 00:56:31,260 of the t matrix, the transition matrix. 928 00:56:31,260 --> 00:56:43,890 And if we do it in second order-- well, 929 00:56:43,890 --> 00:56:49,520 remember the structure is in second order. 930 00:56:49,520 --> 00:56:53,200 We sort of emit a photon, virtual or real. 931 00:56:53,200 --> 00:56:54,290 We have another photon. 932 00:56:54,290 --> 00:56:56,280 This gives us second order. 933 00:56:56,280 --> 00:57:06,120 And in between the system propagates 934 00:57:06,120 --> 00:57:07,280 with its eigen energy. 935 00:57:17,220 --> 00:57:20,930 So this was the structure of the diagrams we had. 936 00:57:20,930 --> 00:57:26,640 And this term here, let's expand it 937 00:57:26,640 --> 00:57:31,115 in one wave functions of the unperturbed Hamiltonian. 938 00:57:35,780 --> 00:57:37,680 And then we obtain this following structure. 939 00:57:41,460 --> 00:57:45,420 So this is just rewriting in a very suggestive way 940 00:57:45,420 --> 00:57:48,330 what we had discussed before. 941 00:57:48,330 --> 00:57:55,190 And the case I focus on now is the interaction 942 00:57:55,190 --> 00:57:58,240 of atoms with resonant light, because this 943 00:57:58,240 --> 00:58:01,070 is when we have divergencies. 944 00:58:01,070 --> 00:58:03,730 If we have state a, state b. 945 00:58:03,730 --> 00:58:07,760 We have a laser and we scatter light. 946 00:58:07,760 --> 00:58:18,190 If the frequency is approximately resonant, 947 00:58:18,190 --> 00:58:27,960 then those perturbative results have a divergency. 948 00:58:27,960 --> 00:58:32,000 So for resonant light scattering, the typical matrix 949 00:58:32,000 --> 00:58:35,410 element we are interested in, we start and end of course 950 00:58:35,410 --> 00:58:36,370 in the ground state. 951 00:58:36,370 --> 00:58:37,300 We scatter photon. 952 00:58:37,300 --> 00:58:41,110 We go up to an excited state and go back to the ground state. 953 00:58:41,110 --> 00:58:45,410 But we started initially with a photon of wave vector k 954 00:58:45,410 --> 00:58:48,060 and polarization epsilon. 955 00:58:48,060 --> 00:58:50,270 And this may be different for the scattered light. 956 00:58:54,720 --> 00:59:04,780 The interaction happens by the interaction operator, 957 00:59:04,780 --> 00:59:09,950 which, for the purpose of this discussion, 958 00:59:09,950 --> 00:59:15,480 can either be the A dot p term or the d dot E term. 959 00:59:15,480 --> 00:59:17,230 We've discussed about the special role 960 00:59:17,230 --> 00:59:18,730 of the a squared term. 961 00:59:18,730 --> 00:59:20,180 We discussed it earlier. 962 00:59:20,180 --> 00:59:23,090 I don't want to include it in the discussion right now. 963 00:59:28,400 --> 00:59:33,395 And then you would say-- just want 964 00:59:33,395 --> 00:59:34,770 to make sure they do not confuse. 965 00:59:34,770 --> 00:59:36,610 In second order perturbation theory, 966 00:59:36,610 --> 00:59:39,230 you always have an energy denominator. 967 00:59:39,230 --> 00:59:42,110 But as I also explained to you, the energy denominator 968 00:59:42,110 --> 00:59:45,340 comes from the free propagator. 969 00:59:45,340 --> 00:59:48,010 You had a dot propagate and a dot-- 970 00:59:48,010 --> 00:59:52,030 and what is the time evolution of this free propagation when 971 00:59:52,030 --> 00:59:53,580 you integrate it with time? 972 00:59:53,580 --> 00:59:55,500 This gave rise to the energy denominator. 973 00:59:55,500 --> 00:59:56,547 Just a side remark. 974 00:59:56,547 --> 00:59:58,130 So you should be very familiar that we 975 00:59:58,130 --> 00:59:59,830 have an energy denominator. 976 00:59:59,830 --> 01:00:03,260 But now this energy denominator-- 977 01:00:03,260 --> 01:00:06,570 the initial energy is the initial atomic state 978 01:00:06,570 --> 01:00:09,570 plus the photon. 979 01:00:09,570 --> 01:00:13,360 And then if you want to sum over all states, 980 01:00:13,360 --> 01:00:17,090 I can just put in here the operator H0. 981 01:00:17,090 --> 01:00:20,110 But you may immediately think about it, the only state which 982 01:00:20,110 --> 01:00:23,570 really matters is the state b. 983 01:00:23,570 --> 01:00:28,450 And what matters is only when we go through intermediate state 984 01:00:28,450 --> 01:00:32,860 b, then H0 is Eb, and then we have a divergence 985 01:00:32,860 --> 01:00:34,720 because the denominator is 0. 986 01:00:37,240 --> 01:00:41,380 So let's see how we can fix it. 987 01:00:41,380 --> 01:00:44,250 But let me first give you the perturbative result, 988 01:00:44,250 --> 01:00:49,040 which you have seen many, many times. 989 01:00:49,040 --> 01:00:54,020 It is the product of two matrix elements. 990 01:00:54,020 --> 01:00:56,580 We have an intermediate state. 991 01:00:56,580 --> 01:00:59,720 The excited state without photon. 992 01:00:59,720 --> 01:01:06,880 We started with a photon, and we end up with a photon. 993 01:01:06,880 --> 01:01:12,770 And here we have our dipole or A dot p term. 994 01:01:12,770 --> 01:01:20,170 And our energy denominator diverges 995 01:01:20,170 --> 01:01:24,890 when the frequency of the photon approaches the resonance 996 01:01:24,890 --> 01:01:25,390 frequency. 997 01:01:28,410 --> 01:01:32,260 Well, how can we fix it? 998 01:01:32,260 --> 01:01:37,150 You know that in many cases you're 999 01:01:37,150 --> 01:01:43,100 just adding an imaginary part to it, 1000 01:01:43,100 --> 01:01:48,480 which reflects that, due to the coupling with the vacuum 1001 01:01:48,480 --> 01:01:52,590 of the electromagnetic field, the excited state is broadened. 1002 01:01:52,590 --> 01:01:56,460 And it gives its energy an imaginary part. 1003 01:01:56,460 --> 01:01:59,240 This of course is very phenomenological, 1004 01:01:59,240 --> 01:02:07,580 but at least technically it avoids the divergence. 1005 01:02:07,580 --> 01:02:13,070 Now what I want to show you, in the next 15 minutes 1006 01:02:13,070 --> 01:02:17,320 and we continue on Wednesday, is how 1007 01:02:17,320 --> 01:02:21,060 I can get this result of a very rigorous approach. 1008 01:02:21,060 --> 01:02:22,900 I want to use the diagrammatic approach. 1009 01:02:22,900 --> 01:02:25,510 I want to sum an infinite number of diagrams. 1010 01:02:25,510 --> 01:02:30,680 And will tell you what are the approximations to get this. 1011 01:02:30,680 --> 01:02:35,880 For instance, as you know, when you have an imaginary part, 1012 01:02:35,880 --> 01:02:38,460 when you have just an imaginary part of the energy, 1013 01:02:38,460 --> 01:02:41,940 the physics of it is exponential decay. 1014 01:02:41,940 --> 01:02:45,040 Well I hope, not today but on Wednesday, 1015 01:02:45,040 --> 01:02:50,360 you realize what assumptions lead to exponential decay, 1016 01:02:50,360 --> 01:02:54,350 and that, in principle, non exponential decay is rather 1017 01:02:54,350 --> 01:02:57,810 the standard than the exception. 1018 01:02:57,810 --> 01:02:59,870 But the effects are small. 1019 01:02:59,870 --> 01:03:01,700 But I want to show you. 1020 01:03:01,700 --> 01:03:05,130 So in some sense I want to give-- 1021 01:03:05,130 --> 01:03:07,200 we have to go through a half an hour of work 1022 01:03:07,200 --> 01:03:08,770 to derive this result. 1023 01:03:08,770 --> 01:03:10,210 So you'd say, what's about it? 1024 01:03:10,210 --> 01:03:11,930 You just get the phenomenological result. 1025 01:03:11,930 --> 01:03:15,870 Well, we have put it on a rigorous basis. 1026 01:03:15,870 --> 01:03:19,630 And number two is we do understand what assumptions 1027 01:03:19,630 --> 01:03:22,050 really go into this phenomenological result. 1028 01:03:25,490 --> 01:03:31,680 So the way how we will actually obtain this result 1029 01:03:31,680 --> 01:03:49,510 is that in-- the second order diagram is that you start 1030 01:03:49,510 --> 01:03:52,980 in state a, you go in state b, and you're back in state a, 1031 01:03:52,980 --> 01:03:56,190 and you exchange photons. 1032 01:03:56,190 --> 01:04:03,650 So the way how we-- the physical picture we want to use now 1033 01:04:03,650 --> 01:04:12,540 is after the photon is absorbed right here, 1034 01:04:12,540 --> 01:04:18,855 the system is not propagating with the Hamiltonian H0. 1035 01:04:18,855 --> 01:04:24,220 It will propagate with the Hamiltonian H. 1036 01:04:24,220 --> 01:04:29,010 If we do that, then we avoid the divergence. 1037 01:04:29,010 --> 01:04:32,230 We get a result which can be applied toward-- the formalism 1038 01:04:32,230 --> 01:04:34,080 can be applied to many situations. 1039 01:04:34,080 --> 01:04:37,910 And in one limiting case, we obtain that. 1040 01:04:37,910 --> 01:04:41,840 So the physics behind it is the following. 1041 01:04:41,840 --> 01:04:45,260 The divergence which we encounter in resonant physics 1042 01:04:45,260 --> 01:04:50,705 is only a divergence because we assume the atom in state b 1043 01:04:50,705 --> 01:04:52,210 is naked. 1044 01:04:52,210 --> 01:04:55,530 It's an excited state of an isolated atom. 1045 01:04:55,530 --> 01:04:59,290 But in reality, the state b interacts 1046 01:04:59,290 --> 01:05:01,530 by virtual photon emission, interacts 1047 01:05:01,530 --> 01:05:03,900 with all the modes of the vacuum. 1048 01:05:03,900 --> 01:05:08,320 And once we throw that in by saying that between two-- 1049 01:05:08,320 --> 01:05:10,640 between absorbing and emitting a photon, 1050 01:05:10,640 --> 01:05:14,630 the state b propagates not with a free Hamiltonian, 1051 01:05:14,630 --> 01:05:19,640 but with the total Hamiltonian, we avoid the divergence. 1052 01:05:19,640 --> 01:05:22,910 But this is no longer a perturbative result. 1053 01:05:22,910 --> 01:05:26,820 And let me illustrate that to you. 1054 01:05:30,580 --> 01:05:38,590 If you take the energy denominator, omega minus 1055 01:05:38,590 --> 01:05:47,980 and omega 0, and we add the imaginary part to it, 1056 01:05:47,980 --> 01:06:00,560 I can Taylor expand it, assuming the imaginary part is small. 1057 01:06:06,970 --> 01:06:10,220 Well, that's sort of a trivial expansion, 1058 01:06:10,220 --> 01:06:13,210 but now you should know that gamma, 1059 01:06:13,210 --> 01:06:15,400 the rate of spontaneous emission, 1060 01:06:15,400 --> 01:06:17,990 the simplest expression is obtained 1061 01:06:17,990 --> 01:06:20,470 in [INAUDIBLE] is called null in the second order 1062 01:06:20,470 --> 01:06:23,450 in the interaction Hamiltonian. 1063 01:06:23,450 --> 01:06:25,310 Spontaneous emission is proportional 1064 01:06:25,310 --> 01:06:28,370 to the atomic dipole matrix element squared. 1065 01:06:28,370 --> 01:06:30,840 So it's second order in the Hamiltonian. 1066 01:06:30,840 --> 01:06:35,525 But that means that this harmless addition 1067 01:06:35,525 --> 01:06:40,230 of an imaginary part means that if you look at the Taylor 1068 01:06:40,230 --> 01:06:45,590 expansion, that we are using now infinite orders. 1069 01:06:45,590 --> 01:06:47,835 Infinite orders in this expansion. 1070 01:07:00,370 --> 01:07:07,070 So the phenomenological addition of [INAUDIBLE] gamma 1071 01:07:07,070 --> 01:07:10,980 to the excited level b is actually highly nontrivial, 1072 01:07:10,980 --> 01:07:13,290 because it's corresponds technically 1073 01:07:13,290 --> 01:07:17,600 or physically to a non perturbative result which 1074 01:07:17,600 --> 01:07:21,455 involves infinite order in the interaction matrix element. 1075 01:07:30,980 --> 01:07:33,850 Question about it? 1076 01:07:33,850 --> 01:07:35,627 So this is more setting up the agenda. 1077 01:07:35,627 --> 01:07:37,710 I wanted to remind you of the perturbative result. 1078 01:07:37,710 --> 01:07:40,090 I want to sort of show you how we can fix it. 1079 01:07:40,090 --> 01:07:41,930 I want to show you what it means. 1080 01:07:41,930 --> 01:07:47,530 But with that motivation I want to now back 1081 01:07:47,530 --> 01:07:51,440 to our diagrammatic approach, our formal solution, 1082 01:07:51,440 --> 01:07:55,080 our exact formal solution of the time evolution of a [INAUDIBLE] 1083 01:07:55,080 --> 01:07:58,500 system, and show you one how I can implement 1084 01:07:58,500 --> 01:08:04,800 this agenda in particular by not allowing the atom to propagate 1085 01:08:04,800 --> 01:08:07,380 with a naked Hamiltonian H0, but sort 1086 01:08:07,380 --> 01:08:10,310 of staying dressed with photons and propagating 1087 01:08:10,310 --> 01:08:15,310 with the Hamiltonian H. Questions about that? 1088 01:08:23,550 --> 01:08:31,300 OK, so the derivation is now focusing 1089 01:08:31,300 --> 01:08:34,139 on two chapters in API. 1090 01:08:49,529 --> 01:08:54,455 I need one result which I hope you remember from about a week 1091 01:08:54,455 --> 01:08:58,140 and a half or two weeks ago. 1092 01:08:58,140 --> 01:09:03,340 We formulated the time evolution of the system using the time 1093 01:09:03,340 --> 01:09:06,109 propagation operator u. 1094 01:09:06,109 --> 01:09:10,420 And we found an exact expression. 1095 01:09:10,420 --> 01:09:13,350 I'm dropping all of the arguments here, t and t prime. 1096 01:09:13,350 --> 01:09:15,140 You can look them up either in the book 1097 01:09:15,140 --> 01:09:16,890 or in the previous lecture notes. 1098 01:09:16,890 --> 01:09:20,870 We found that the propagator in the interaction representation 1099 01:09:20,870 --> 01:09:24,399 is the unperturbed operator u0. 1100 01:09:30,540 --> 01:09:32,380 Now I need, actually, the terms. 1101 01:09:32,380 --> 01:09:36,600 It was a time integral over some time 1102 01:09:36,600 --> 01:09:48,120 t1, which involved u0, v, and u. 1103 01:09:48,120 --> 01:09:50,649 And I introduced it to you as you could plug that 1104 01:09:50,649 --> 01:09:53,439 into whatever is the equivalent of Schrodinger's 1105 01:09:53,439 --> 01:09:55,270 equation for the time evolution operator, 1106 01:09:55,270 --> 01:09:58,890 and you really find this is a solution for u. 1107 01:09:58,890 --> 01:10:00,880 But well, it's not really a solution, 1108 01:10:00,880 --> 01:10:04,760 because it's a solution of u expressed in terms of u. 1109 01:10:04,760 --> 01:10:08,960 But then I showed you that it allows an iterative approach, 1110 01:10:08,960 --> 01:10:10,190 and this is why we did it. 1111 01:10:10,190 --> 01:10:12,050 If you have a first order result, 1112 01:10:12,050 --> 01:10:14,100 the zero order result is u0. 1113 01:10:14,100 --> 01:10:16,610 When you plug the zero order result in here, 1114 01:10:16,610 --> 01:10:18,262 you get the first order correction. 1115 01:10:18,262 --> 01:10:19,970 When you put the first order result here, 1116 01:10:19,970 --> 01:10:21,640 you get the second order correction. 1117 01:10:21,640 --> 01:10:26,030 So you get a recursive formula which solves the problem. 1118 01:10:26,030 --> 01:10:28,020 And this is now our starting point. 1119 01:10:28,020 --> 01:10:30,630 This is an exact formulation of QED. 1120 01:10:30,630 --> 01:10:33,040 And this is now our starting point 1121 01:10:33,040 --> 01:10:36,760 to address the problem of divergencies. 1122 01:10:36,760 --> 01:10:40,390 We want to describe what many of you do in the lab, 1123 01:10:40,390 --> 01:10:43,740 namely have resonant laser light interact with atoms. 1124 01:10:43,740 --> 01:10:47,880 And I want to show you how can you describe it with QED 1125 01:10:47,880 --> 01:10:49,460 with the formalism we've introduced. 1126 01:10:57,260 --> 01:11:08,665 OK, now for this chapter we want to have one big simplification. 1127 01:11:08,665 --> 01:11:10,290 Things aren't only getting complicated. 1128 01:11:10,290 --> 01:11:11,290 They're getting simpler. 1129 01:11:13,830 --> 01:11:20,140 If you have a time integral which is a convolution, 1130 01:11:20,140 --> 01:11:25,180 and you can often simplify it by Fourier transforming it, 1131 01:11:25,180 --> 01:11:28,530 because the Fourier transform of a convolution 1132 01:11:28,530 --> 01:11:33,080 is simply the product of the two Fourier transforms. 1133 01:11:33,080 --> 01:11:34,240 There is one glitch. 1134 01:11:34,240 --> 01:11:38,220 So one subtlety I want to feed in a moment. 1135 01:11:38,220 --> 01:11:41,480 But let's just assume for a moment, 1136 01:11:41,480 --> 01:11:43,140 we go from time before the Fourier 1137 01:11:43,140 --> 01:11:46,160 transform to frequency or energy. 1138 01:11:46,160 --> 01:11:56,180 So G of E is the Fourier transform 1139 01:11:56,180 --> 01:12:00,940 of-- after filling a few corrections of u. 1140 01:12:00,940 --> 01:12:06,760 That would mean, if that were true, 1141 01:12:06,760 --> 01:12:11,850 the equation above would now be an equation 1142 01:12:11,850 --> 01:12:19,080 where the integral, V sort of constant, 1143 01:12:19,080 --> 01:12:22,340 instead of a convolution of U0 and U, 1144 01:12:22,340 --> 01:12:27,772 we get simply the product of the two Fourier transforms. 1145 01:12:27,772 --> 01:12:28,855 That's much, much simpler. 1146 01:12:28,855 --> 01:12:31,040 It's an algebraic equation instead 1147 01:12:31,040 --> 01:12:32,710 of an integral equation. 1148 01:12:32,710 --> 01:12:34,510 And remember, when we get into nth order, 1149 01:12:34,510 --> 01:12:37,200 we get n integrals with some time ordering. 1150 01:12:37,200 --> 01:12:39,817 We were able to do it, but it's mathematically 1151 01:12:39,817 --> 01:12:40,525 more complicated. 1152 01:12:48,270 --> 01:12:50,815 The problem is that this is not a convolution. 1153 01:12:50,815 --> 01:12:56,480 It would be a convolution if the time limits were infinite. 1154 01:12:56,480 --> 01:13:02,610 So what we can do instead is we can define G of E. 1155 01:13:02,610 --> 01:13:07,240 Not the Fourier transform of U. If we 1156 01:13:07,240 --> 01:13:15,120 say it is the Fourier transform of k, which uses a step 1157 01:13:15,120 --> 01:13:24,540 function, then if you insert now k in the formula above, 1158 01:13:24,540 --> 01:13:31,130 you see that because of this step function 1159 01:13:31,130 --> 01:13:35,700 we can now take the integral limits from minus infinity 1160 01:13:35,700 --> 01:13:37,840 to plus infinity. 1161 01:13:37,840 --> 01:13:42,200 So therefore, when I say G, G is called the resolvent. 1162 01:13:42,200 --> 01:13:44,280 It's really the mathematical quantity 1163 01:13:44,280 --> 01:13:46,100 we have to work on now. 1164 01:13:46,100 --> 01:13:48,360 And I wanted to tell you very simply hey, 1165 01:13:48,360 --> 01:13:53,370 it's just a Fourier transform of the time evolution operator. 1166 01:13:53,370 --> 01:13:54,080 Not quite. 1167 01:13:54,080 --> 01:13:57,850 It's the Fourier transform of a time evolution operator 1168 01:13:57,850 --> 01:14:02,380 with a theta function, just to make sure that formally we 1169 01:14:02,380 --> 01:14:05,845 can extend the upper and lower integration limits to infinity. 1170 01:14:08,770 --> 01:14:11,480 Or this is also a name-- what I'm really 1171 01:14:11,480 --> 01:14:15,290 kind of discussing are Laplace transformations. 1172 01:14:20,690 --> 01:14:23,860 But I want to convey to you the physics. 1173 01:14:23,860 --> 01:14:27,580 It would take many weeks to do the mathematical accurately, 1174 01:14:27,580 --> 01:14:31,530 but you can get the idea and really profound feeling 1175 01:14:31,530 --> 01:14:34,100 for how QED works and what we are 1176 01:14:34,100 --> 01:14:37,660 doing with not so much mathematical rigor. 1177 01:14:37,660 --> 01:14:40,190 So with that grain of salt we have 1178 01:14:40,190 --> 01:14:41,990 Fourier transformed equation. 1179 01:14:41,990 --> 01:14:44,870 We have the Fourier transform of the time evolution operator. 1180 01:14:44,870 --> 01:14:47,080 But if you look at it mathematically more exact, 1181 01:14:47,080 --> 01:14:50,330 it's a Laplace transform, and we are little bit navigating 1182 01:14:50,330 --> 01:14:52,225 in the complex place. 1183 01:14:52,225 --> 01:14:55,660 The Fourier transform does not exist for real values. 1184 01:14:55,660 --> 01:14:58,680 We always have to add a small imaginary part 1185 01:14:58,680 --> 01:14:59,860 to make things converge. 1186 01:14:59,860 --> 01:15:02,310 But you've seen that in many other places, I assume. 1187 01:15:06,000 --> 01:15:12,850 OK, so the quantity which allows us 1188 01:15:12,850 --> 01:15:24,170 to describe resonant interaction in a very straightforward way 1189 01:15:24,170 --> 01:15:25,480 is called the resolvent. 1190 01:15:28,630 --> 01:15:31,732 I can simply define the resolvent. 1191 01:15:31,732 --> 01:15:33,190 All I gave you was some motivation. 1192 01:15:33,190 --> 01:15:35,390 But if I had wanted, I could just 1193 01:15:35,390 --> 01:15:44,250 say the resolvent is defined by this operator equation. 1194 01:15:44,250 --> 01:15:46,470 z is the complex number. 1195 01:15:46,470 --> 01:15:50,720 H is the Hamiltonian including interaction 1196 01:15:50,720 --> 01:15:54,940 between atoms in electromagnetic field. 1197 01:15:54,940 --> 01:16:07,470 And actually if you would take that-- if you take this 1198 01:16:07,470 --> 01:16:22,450 and plug it into the equation above-- so if you plug this 1199 01:16:22,450 --> 01:16:27,300 into the equation, you find this expression solves the equation. 1200 01:16:27,300 --> 01:16:29,746 So I just want to say you can forget 1201 01:16:29,746 --> 01:16:32,120 about all the subtleties with Laplace transform and such. 1202 01:16:32,120 --> 01:16:35,360 You can just say let me define some interesting object. 1203 01:16:35,360 --> 01:16:37,450 But it is connected to what I said earlier, 1204 01:16:37,450 --> 01:16:39,880 because this solves this equation. 1205 01:16:39,880 --> 01:16:43,860 And also I like to sort of show you the mathematical structure. 1206 01:16:48,830 --> 01:16:53,660 If you write the time evolution operator as e 1207 01:16:53,660 --> 01:16:59,890 to the minus the Hamiltonian, and now you Fourier 1208 01:16:59,890 --> 01:17:01,700 transform it. 1209 01:17:01,700 --> 01:17:04,460 If you Fourier transform it, you multiply it 1210 01:17:04,460 --> 01:17:11,039 with e to the I omega t, and you integrate-- 1211 01:17:11,039 --> 01:17:14,890 let me integrate from 0 to infinity. 1212 01:17:14,890 --> 01:17:17,750 There are some issues with 0 or infinities. 1213 01:17:17,750 --> 01:17:23,190 But if you simply solve this integral with time, 1214 01:17:23,190 --> 01:17:29,240 you have sort of an exponent e to the i omega minus h. 1215 01:17:29,240 --> 01:17:32,780 So if you Fourier transform it, what 1216 01:17:32,780 --> 01:17:41,860 you obtain is nothing else than the energy minus h. 1217 01:17:41,860 --> 01:17:46,882 And this looks very different. 1218 01:17:46,882 --> 01:17:48,840 This looks very, very similar to the definition 1219 01:17:48,840 --> 01:17:50,530 of the resolvent. 1220 01:17:50,530 --> 01:17:53,400 The only mathematical thing I'm not fully discussing here 1221 01:17:53,400 --> 01:17:56,980 is z real, or has it a small imaginary part? 1222 01:17:56,980 --> 01:17:59,140 The small imaginary part-- actually, 1223 01:17:59,140 --> 01:18:01,710 you have to add a small imaginary part here 1224 01:18:01,710 --> 01:18:05,090 to make sure the Fourier transform exists. 1225 01:18:05,090 --> 01:18:08,900 But you should clearly see how things are connected. 1226 01:18:08,900 --> 01:18:12,220 So OK, I'm focusing now on the resolvent. 1227 01:18:12,220 --> 01:18:15,310 It is equivalent to the time evolution operator. 1228 01:18:15,310 --> 01:18:17,950 It is a sort of Fourier transform of it. 1229 01:18:17,950 --> 01:18:21,540 And we want to formally write down 1230 01:18:21,540 --> 01:18:27,700 a diagrammatic solution for the resolvent. 1231 01:18:27,700 --> 01:18:29,830 And with that I think I should stop now. 1232 01:18:29,830 --> 01:18:31,150 We start on Wednesday? 1233 01:18:31,150 --> 01:18:33,395 Any questions to that point? 1234 01:18:36,680 --> 01:18:39,060 OK, then see you on Wednesday.