1 00:00:00,080 --> 00:00:02,440 The following content is provided under a Creative 2 00:00:02,440 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,610 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,610 --> 00:00:17,265 at ocw@mit.edu. 8 00:00:21,290 --> 00:00:23,470 PROFESSOR: OK, let us start. 9 00:00:23,470 --> 00:00:25,480 So maybe start by reminding you what 10 00:00:25,480 --> 00:00:29,900 we did in the last lecture. 11 00:00:29,900 --> 00:00:32,920 So you take the black hole metric 12 00:00:32,920 --> 00:00:34,740 so we explain that you can actually 13 00:00:34,740 --> 00:00:39,680 extend-- so this is the other black hole horizon-- so you can 14 00:00:39,680 --> 00:00:42,760 actually extend the region outside the horizon 15 00:00:42,760 --> 00:00:47,100 to actually four total regions for the four black hole 16 00:00:47,100 --> 00:00:47,600 spacetime. 17 00:00:50,470 --> 00:00:55,680 And then this [? tag, ?] this is a singularity. 18 00:00:55,680 --> 00:00:58,870 And of course, this is only the rt plane, 19 00:00:58,870 --> 00:01:06,440 and then you also have [? s2. ?] So now [INAUDIBLE], 20 00:01:06,440 --> 00:01:08,230 let me show that in order for the metric 21 00:01:08,230 --> 00:01:11,050 to be regular at the horizon into this when you go 22 00:01:11,050 --> 00:01:15,910 to Euclidean signature then you want to identify Tau to be 23 00:01:15,910 --> 00:01:18,860 periodic with this period. 24 00:01:18,860 --> 00:01:23,990 So in essence, when you go to Euclidean signature, 25 00:01:23,990 --> 00:01:29,430 then this rt plane becomes, essentially, a disc. 26 00:01:29,430 --> 00:01:34,860 And this Tau is in the angular direction, and then the rho, 27 00:01:34,860 --> 00:01:38,040 this r is in the radial direction. 28 00:01:38,040 --> 00:01:41,980 And then the region of this disc is the horizon. 29 00:01:41,980 --> 00:01:45,160 The region of the disc is the horizon. 30 00:01:45,160 --> 00:01:51,590 So when you say, yeah, so this Tau has to be periodic. 31 00:01:51,590 --> 00:01:56,040 So that means if you fit any theories in this spacetime, 32 00:01:56,040 --> 00:02:00,050 then that theory must be at a finite temperature 33 00:02:00,050 --> 00:02:03,560 with inverse of this, which gives you this [? H ?] 34 00:02:03,560 --> 00:02:04,760 kappa divided by beta. 35 00:02:04,760 --> 00:02:08,400 And the kappa is the surface gravity. 36 00:02:08,400 --> 00:02:11,220 So you can do a similar thing with the [INAUDIBLE]. 37 00:02:11,220 --> 00:02:16,660 And the [INAUDIBLE] is just 1/4 of a Minkowski, which 38 00:02:16,660 --> 00:02:18,570 [? forniated ?] in this way. 39 00:02:18,570 --> 00:02:20,920 This is a constant of the rho surface, 40 00:02:20,920 --> 00:02:24,070 and this is a constant of the eta surface. 41 00:02:24,070 --> 00:02:30,220 And, again, [INAUDIBLE] in eta to Euclidean. 42 00:02:30,220 --> 00:02:33,660 And in order for the Euclidean need [? congestion ?] 43 00:02:33,660 --> 00:02:37,560 to be regular, the region, then the theta 44 00:02:37,560 --> 00:02:39,710 have to be periodical in 2pi. 45 00:02:39,710 --> 00:02:42,900 And then this theta essentially, again, 46 00:02:42,900 --> 00:02:44,810 becomes an angular direction. 47 00:02:44,810 --> 00:02:49,330 And then this implies that the local observer 48 00:02:49,330 --> 00:02:52,690 in the window of spacetime must observe a temperature given 49 00:02:52,690 --> 00:02:54,210 by this formula, OK? 50 00:02:56,970 --> 00:02:58,400 Any questions regarding that? 51 00:03:14,070 --> 00:03:26,041 Good, so today we will talk about 52 00:03:26,041 --> 00:03:28,040 the physical interpretation of this temperature. 53 00:03:47,291 --> 00:03:47,790 OK? 54 00:03:51,680 --> 00:03:55,840 So let me just write down what this temperature 55 00:03:55,840 --> 00:04:02,830 means in words, so I will use black hole as an example. 56 00:04:02,830 --> 00:04:07,040 We can stay exactly parallel [INAUDIBLE]. 57 00:04:07,040 --> 00:04:10,880 It says if you consider a quantum fields 58 00:04:10,880 --> 00:04:26,290 theory in the black hole spacetime, then 59 00:04:26,290 --> 00:04:30,540 the vacuum state-- so when you put the quantum fields 60 00:04:30,540 --> 00:04:35,375 theory, say, in the spacetime, always ask, what is the vacuum? 61 00:04:35,375 --> 00:04:45,950 Then the vacuum state for this curve t 62 00:04:45,950 --> 00:05:03,860 obtained via the analytic continuation procedure 63 00:05:03,860 --> 00:05:10,720 where the analytic continuation procedure from Euclidean 64 00:05:10,720 --> 00:05:24,340 signature is a thermal equilibrium 65 00:05:24,340 --> 00:05:40,400 state with the stated temperature, OK? 66 00:05:52,460 --> 00:05:57,930 So emphasize when you want to talk about temperature, 67 00:05:57,930 --> 00:06:03,830 you first have to specify what is the time you use. 68 00:06:03,830 --> 00:06:09,640 So this temperature refers to this particular choice of time, 69 00:06:09,640 --> 00:06:12,560 and so this is the temperature corresponding to an observer 70 00:06:12,560 --> 00:06:15,300 while using this time. 71 00:06:15,300 --> 00:06:18,620 And then this is the time, as we said, corresponding to observer 72 00:06:18,620 --> 00:06:20,510 leaving at infinity. 73 00:06:20,510 --> 00:06:22,690 And so this would be the temperature 74 00:06:22,690 --> 00:06:25,920 observed by the infinity. 75 00:06:25,920 --> 00:06:35,110 So similarly, this temperature is the temperature 76 00:06:35,110 --> 00:06:38,140 for given [? with ?] the observer 77 00:06:38,140 --> 00:06:40,370 at some location of rho. 78 00:06:40,370 --> 00:06:41,770 So this is for local observer. 79 00:06:41,770 --> 00:06:43,960 If you just talk about the eta, then the temperature 80 00:06:43,960 --> 00:06:48,460 is just h divided by 2pi because theta is periodical in 2pi. 81 00:06:48,460 --> 00:06:50,610 And if you ask what is the temperature associated 82 00:06:50,610 --> 00:06:53,230 with eta, it's just h pi divided by 2pi. 83 00:06:59,686 --> 00:07:01,435 So any questions regarding this statement? 84 00:07:08,350 --> 00:07:12,040 So now let me [INAUDIBLE] is going 85 00:07:12,040 --> 00:07:15,347 to be abstract, so let me elaborate a little bit, 86 00:07:15,347 --> 00:07:16,555 and let me make some remarks. 87 00:07:25,380 --> 00:07:36,850 So the first is that the choice of vacuum 88 00:07:36,850 --> 00:07:50,740 for QFT in the curved spacetime is not unique. 89 00:07:56,340 --> 00:07:57,640 So this is a standard feature. 90 00:07:57,640 --> 00:07:59,880 This is a standard feature, so if you 91 00:07:59,880 --> 00:08:02,734 want to define a quantum fields theory in the curved space time 92 00:08:02,734 --> 00:08:04,650 because, in general, in the curved space time, 93 00:08:04,650 --> 00:08:08,080 there's no prefer the choice of time. 94 00:08:08,080 --> 00:08:10,910 So in order to quantize here, you have to choose a time. 95 00:08:10,910 --> 00:08:14,210 So depending on your choice of time, 96 00:08:14,210 --> 00:08:17,190 then you can quantize your theory in a different way. 97 00:08:17,190 --> 00:08:20,800 And then you have a different possibility 98 00:08:20,800 --> 00:08:23,660 of what is your vacuum. 99 00:08:23,660 --> 00:08:26,840 So in other words, in the curved spacetime, 100 00:08:26,840 --> 00:08:28,965 in general, the vacuum is observer dependent. 101 00:08:31,620 --> 00:08:37,360 So of course, the black hole is the curved spacetime, so 102 00:08:37,360 --> 00:08:42,539 the particular continuation procedure 103 00:08:42,539 --> 00:09:05,580 we described corresponds to a particular choice, 104 00:09:05,580 --> 00:09:14,890 to a special choice, of the vacuum. 105 00:09:14,890 --> 00:09:20,110 In the case of a black hole, this 106 00:09:20,110 --> 00:09:32,340 has been called to be the Hartle-Hawking vacuum 107 00:09:32,340 --> 00:09:36,090 because the Hartle-Hawking first defined it 108 00:09:36,090 --> 00:09:39,750 from this Euclidean procedure. 109 00:09:39,750 --> 00:09:45,430 And for the Rindler spacetime, time 110 00:09:45,430 --> 00:09:49,962 so we see a little bit later in today's class 111 00:09:49,962 --> 00:09:55,370 that this is actually-- the choice of vacuum 112 00:09:55,370 --> 00:10:00,190 actually just goes one into the standard of the Minkowski way 113 00:10:00,190 --> 00:10:00,690 vacuum. 114 00:10:04,640 --> 00:10:10,320 reduced to the Rindler patch. 115 00:10:25,889 --> 00:10:27,430 So of course, you can actually choose 116 00:10:27,430 --> 00:10:30,860 the vacuum in some other way. 117 00:10:30,860 --> 00:10:50,550 For example, say if for a black hole, 118 00:10:50,550 --> 00:10:54,870 instead of periodically identify Tau with this period, 119 00:10:54,870 --> 00:11:00,090 we can choose not to identify Tau at all. 120 00:11:00,090 --> 00:11:13,390 Suppose for a black hole, in Euclidean signature 121 00:11:13,390 --> 00:11:18,550 we take Tau to be uncompact. 122 00:11:24,160 --> 00:11:26,022 So we don't identify. 123 00:11:26,022 --> 00:11:30,520 So this, of course, runs into a different Euclidean manifold. 124 00:11:30,520 --> 00:11:33,570 Then that corresponds to a different Euclidean manifold. 125 00:11:33,570 --> 00:11:39,322 And, again, you can really continue the Euclidean theory 126 00:11:39,322 --> 00:11:42,240 on this manifold back to [INAUDIBLE] 127 00:11:42,240 --> 00:11:45,150 to define our vacuum. 128 00:11:45,150 --> 00:11:47,310 And that gives you a different vacuum 129 00:11:47,310 --> 00:11:51,520 from the one in which you identified Tau. 130 00:11:51,520 --> 00:11:53,910 So this gives you a so-called Schwarzschild vacuum. 131 00:12:03,390 --> 00:12:05,520 Or sometimes people call it Boulware vacuum 132 00:12:05,520 --> 00:12:12,550 because Boulware had worked on it in the early days 133 00:12:12,550 --> 00:12:13,490 before other people. 134 00:12:16,290 --> 00:12:20,120 And, in fact, this Schwarzschild vacuum 135 00:12:20,120 --> 00:12:21,245 is the most natural vacuum. 136 00:12:27,600 --> 00:12:33,720 In fact, this is the vacuum that one 137 00:12:33,720 --> 00:12:55,150 would get by doing canonical connotation 138 00:12:55,150 --> 00:13:05,490 in the black hole in terms of this Schwarzschild time t. 139 00:13:14,980 --> 00:13:19,585 So let me just emphasize, so this is a curbed spacetime. 140 00:13:19,585 --> 00:13:20,960 You can just put a quantum fields 141 00:13:20,960 --> 00:13:23,630 theory in this spacetime, and then there's 142 00:13:23,630 --> 00:13:29,110 a time here which, actually, the spacetime is time dependent. 143 00:13:29,110 --> 00:13:32,360 So you have a time-independent Hamiltonian [? so ?] respect 144 00:13:32,360 --> 00:13:33,527 to this time. 145 00:13:33,527 --> 00:13:34,943 And so, in principal, you can just 146 00:13:34,943 --> 00:13:38,150 do a straightforward canonical connotation. 147 00:13:38,150 --> 00:13:41,520 And then that connotation will give you a vacuum. 148 00:13:41,520 --> 00:13:45,560 And that vacuum would corresponding to the same 149 00:13:45,560 --> 00:13:48,040 as you do analytic continuation to the Euclidean. 150 00:13:48,040 --> 00:13:53,364 And don't compact Phi Tau because there's 151 00:13:53,364 --> 00:13:54,030 nothing special. 152 00:13:56,817 --> 00:13:58,650 Just like when you do a 0 temperature fields 153 00:13:58,650 --> 00:14:03,400 theory in the standard Minkowski spacetime. 154 00:14:03,400 --> 00:14:05,877 But when you compactify Tau, then you 155 00:14:05,877 --> 00:14:06,835 get a different vacuum. 156 00:14:06,835 --> 00:14:10,750 It is what would be now called a Hartle-Hawking vacuum, 157 00:14:10,750 --> 00:14:14,890 and it's what you get a finite temperature. 158 00:14:14,890 --> 00:14:15,540 Is this clear? 159 00:14:18,700 --> 00:14:19,846 Yes? 160 00:14:19,846 --> 00:14:22,095 AUDIENCE: So in what sense are these different vacuums 161 00:14:22,095 --> 00:14:23,110 physical or not? 162 00:14:23,110 --> 00:14:26,990 PROFESSOR: Yeah, I will answer this question in a little bit. 163 00:14:26,990 --> 00:14:30,450 Right now, I want you to be clear. 164 00:14:30,450 --> 00:14:33,830 Right now we have two different vacuum here 165 00:14:33,830 --> 00:14:36,475 corresponding to two different ways 166 00:14:36,475 --> 00:14:38,710 of doing analytic continuation. 167 00:14:38,710 --> 00:14:41,625 And one way you compactify Tau, and one way 168 00:14:41,625 --> 00:14:43,440 you don't compactify Tau. 169 00:14:43,440 --> 00:14:45,680 And the way you don't compactify Tau 170 00:14:45,680 --> 00:14:48,080 is actually the most straightforward way. 171 00:14:48,080 --> 00:14:50,090 So if you just do a straightforward canonical 172 00:14:50,090 --> 00:14:53,400 connotation using this time, then that 173 00:14:53,400 --> 00:14:54,870 will be the vacuum you get. 174 00:14:54,870 --> 00:14:58,526 So that's why this is called a Schwarzschild vacuum, OK? 175 00:14:58,526 --> 00:15:03,390 AUDIENCE: So do you not have a similarity in writing? 176 00:15:03,390 --> 00:15:05,030 PROFESSOR: I will talk about that. 177 00:15:05,030 --> 00:15:13,280 So similarly for the Rindler, again, 178 00:15:13,280 --> 00:15:16,590 if you do straightforward canonical connotation, 179 00:15:16,590 --> 00:15:20,080 in this spacetime with eta has your time, 180 00:15:20,080 --> 00:15:21,900 then when you go to Euclidean signature, 181 00:15:21,900 --> 00:15:24,780 you want compactify the Euclidean version of eta. 182 00:15:24,780 --> 00:15:26,120 Then you won't compactify theta. 183 00:15:28,630 --> 00:15:42,824 So if you take the theta to be uncompact, 184 00:15:42,824 --> 00:15:44,740 then you get what is so-called Rindler vacuum. 185 00:15:51,150 --> 00:15:54,520 So it just, again, can be obtained. 186 00:16:26,410 --> 00:16:32,430 So then you ask them, why do we bother to define those vacuums? 187 00:16:32,430 --> 00:16:37,810 Why do we bother to make this identification given that this 188 00:16:37,810 --> 00:16:39,190 is the most straightforward thing 189 00:16:39,190 --> 00:16:42,010 to do, say, when we consider this quantum fields 190 00:16:42,010 --> 00:16:45,850 theory inside curved spacetimes. 191 00:16:45,850 --> 00:16:49,050 So here is the key, and so that's 192 00:16:49,050 --> 00:16:50,820 where the geometric consideration becomes 193 00:16:50,820 --> 00:16:52,260 important. 194 00:16:52,260 --> 00:16:55,940 So, again, let me just first use the black hole 195 00:16:55,940 --> 00:16:58,090 as an example case. 196 00:16:58,090 --> 00:17:15,839 So in the Schwarzschild vacuum, the corresponding Euclidean 197 00:17:15,839 --> 00:17:28,990 manifold is singular add to the horizon. 198 00:17:32,630 --> 00:17:38,760 So we explained last time, when we do this [INAUDIBLE], 199 00:17:38,760 --> 00:17:43,970 the spacetime is regular only for 200 00:17:43,970 --> 00:17:46,260 this periodic identification. 201 00:17:46,260 --> 00:17:48,880 For any other possibilities, then you 202 00:17:48,880 --> 00:17:54,980 will have a conical singularity add to the horizon. 203 00:17:54,980 --> 00:18:04,310 So this implies the Euclidean manifold singular horizon. 204 00:18:04,310 --> 00:18:06,470 Say, if you cut computer correlation function 205 00:18:06,470 --> 00:18:12,161 [INAUDIBLE], then you can have single behavior at the horizon. 206 00:18:12,161 --> 00:18:13,660 In particular, when you analytically 207 00:18:13,660 --> 00:18:15,895 continue to [INAUDIBLE] signature, 208 00:18:15,895 --> 00:18:29,180 then that means that physical observables often 209 00:18:29,180 --> 00:18:41,660 are singular after the horizon. 210 00:18:47,230 --> 00:18:50,310 So we need to continue to [INAUDIBLE] signature. 211 00:18:50,310 --> 00:18:54,630 For example-- which actually I should have maybe the 212 00:18:54,630 --> 00:18:58,240 [INAUDIBLE] problem but then forgot earlier. 213 00:18:58,240 --> 00:19:00,890 For example, you can check yourself 214 00:19:00,890 --> 00:19:04,610 that just take a free scalar fields theory. 215 00:19:04,610 --> 00:19:08,910 You compute the stress tensor from this free scalar fields 216 00:19:08,910 --> 00:19:13,020 theory, and you find the stress tensor 217 00:19:13,020 --> 00:19:20,640 of this theory blows up after the horizon in the Minkowski 218 00:19:20,640 --> 00:19:21,140 vacuum. 219 00:19:26,360 --> 00:19:29,470 And if this is not the case, for the vacuum 220 00:19:29,470 --> 00:19:35,080 we obtained by doing analytic continuation this way-- 221 00:19:35,080 --> 00:19:44,850 so this is not the case for a Hartle-Hawking vacuum. 222 00:19:50,430 --> 00:19:53,247 So Hartle-Hawking vacuum just is [? lame ?] 223 00:19:53,247 --> 00:19:55,580 for the vacuum we obtained by this analytic continuation 224 00:19:55,580 --> 00:20:08,410 procedure for which all observer was regular just, 225 00:20:08,410 --> 00:20:11,095 essentially, by definition, by construction. 226 00:20:24,510 --> 00:20:27,360 So essentially by construction because Euclidean manifold 227 00:20:27,360 --> 00:20:31,410 is completely smooth, and if you want to compute any correlation 228 00:20:31,410 --> 00:20:34,350 function-- for example, you can compute the Euclidean signature 229 00:20:34,350 --> 00:20:37,360 and then I need you to continue back to [INAUDIBLE] signature 230 00:20:37,360 --> 00:20:39,860 because it's the smooth side of the horizon of the Euclidean 231 00:20:39,860 --> 00:20:40,030 signature. 232 00:20:40,030 --> 00:20:41,238 I need the continuation back. 233 00:20:41,238 --> 00:20:44,565 We also give you a regular function at the horizon there. 234 00:20:48,230 --> 00:20:53,520 So similar remarks apply to the Rindler space, 235 00:20:53,520 --> 00:21:02,870 and this Rindler vacuum will be singular at this Rindler 236 00:21:02,870 --> 00:21:04,200 horizon. 237 00:21:04,200 --> 00:21:07,600 It will be singular at Rindler horizon. 238 00:21:07,600 --> 00:21:15,800 So since from the general relativity, 239 00:21:15,800 --> 00:21:17,570 the horizon is a smooth place. 240 00:21:21,110 --> 00:21:25,397 The curvature there can be very small 241 00:21:25,397 --> 00:21:27,105 if you have a big black hole [INAUDIBLE]. 242 00:21:32,120 --> 00:21:35,270 So physically, we believe that these Hartle-Hawking vacuums 243 00:21:35,270 --> 00:21:38,390 are more physical then, for example, the Schwarzschild 244 00:21:38,390 --> 00:21:40,130 vacuum. 245 00:21:40,130 --> 00:21:44,140 But Schwarzschild is still often used for certain purposes. 246 00:21:44,140 --> 00:21:47,400 But if you want to consider these physical observables, 247 00:21:47,400 --> 00:21:50,050 then the Hartle-Hawking vacuum should be the right one 248 00:21:50,050 --> 00:21:53,460 to use if you don't want to encounter singularities 249 00:21:53,460 --> 00:21:56,310 at the horizon. 250 00:21:56,310 --> 00:21:58,980 Any questions? 251 00:21:58,980 --> 00:21:59,591 Yes? 252 00:21:59,591 --> 00:22:00,507 AUDIENCE: [INAUDIBLE]? 253 00:22:03,920 --> 00:22:04,583 PROFESSOR: No. 254 00:22:04,583 --> 00:22:05,499 AUDIENCE: [INAUDIBLE]. 255 00:22:05,499 --> 00:22:07,350 PROFESSOR: Yeah. 256 00:22:07,350 --> 00:22:12,994 Yeah, because, for example, Euclidean Tau is uncompact. 257 00:22:12,994 --> 00:22:16,264 AUDIENCE: So the singularity of the physical variables 258 00:22:16,264 --> 00:22:21,720 [INAUDIBLE] to become singular to Hartle-Hawking? 259 00:22:21,720 --> 00:22:23,444 PROFESSOR: Yeah, that's right. 260 00:22:23,444 --> 00:22:24,360 AUDIENCE: [INAUDIBLE]? 261 00:22:30,990 --> 00:22:35,550 PROFESSOR: I would say this is the physical reason. 262 00:22:35,550 --> 00:22:41,750 If I think about it, I might other not-so-essential reasons. 263 00:22:41,750 --> 00:22:44,926 Yeah, but at the moment, I could not think of any others. 264 00:22:44,926 --> 00:22:45,782 Yeah? 265 00:22:45,782 --> 00:22:47,407 AUDIENCE: So if I understand correctly, 266 00:22:47,407 --> 00:22:49,365 when you quantize a theory in curved spacetime, 267 00:22:49,365 --> 00:22:51,875 you have to choose a space-like foliation of your spacetime, 268 00:22:51,875 --> 00:22:54,370 then you quantize it on that foliation basically. 269 00:22:54,370 --> 00:22:57,590 So the problem is that I still don't understand. 270 00:22:57,590 --> 00:22:59,300 So the different ground suits that you 271 00:22:59,300 --> 00:23:01,680 get for different possible quantizations based 272 00:23:01,680 --> 00:23:02,680 on different foliations. 273 00:23:02,680 --> 00:23:05,750 Like, to what degree are they compatible versus 274 00:23:05,750 --> 00:23:07,730 incompatible with each other? 275 00:23:07,730 --> 00:23:10,430 Are they at odds with each other? 276 00:23:10,430 --> 00:23:12,170 Like is one more real than another one, 277 00:23:12,170 --> 00:23:14,160 or is it just sort of an artifact of your coordinates, 278 00:23:14,160 --> 00:23:15,210 and there are things that we can do 279 00:23:15,210 --> 00:23:17,330 which are independent of which one we end up with. 280 00:23:17,330 --> 00:23:22,630 PROFESSOR: Yeah, so first, in general, they 281 00:23:22,630 --> 00:23:25,165 are physically inequivalent to each other. 282 00:23:25,165 --> 00:23:28,170 For example, in this case, the Schwarzschild vacuum 283 00:23:28,170 --> 00:23:32,340 is physically inequivalent to the Hartle-Hawking vacuum. 284 00:23:32,340 --> 00:23:37,080 And actually, with a little bit of work, 285 00:23:37,080 --> 00:23:40,360 one can write down the explicit relation between them. 286 00:23:40,360 --> 00:23:44,885 And so from the perspective of the Schwarzschild vacuum, 287 00:23:44,885 --> 00:23:46,760 the Hartle-Hawking vacuum is a highly excited 288 00:23:46,760 --> 00:23:50,550 state and vice versa. 289 00:23:50,550 --> 00:23:52,140 So vice versa. 290 00:23:52,140 --> 00:24:03,285 And so, yeah, for local observable, say in the local, 291 00:24:03,285 --> 00:24:04,490 it depends. 292 00:24:04,490 --> 00:24:18,550 I think the vacuum is more like-- sometimes 293 00:24:18,550 --> 00:24:21,070 certain vacuums are more convenient for a certain 294 00:24:21,070 --> 00:24:22,740 observer and a certain vacuum is more 295 00:24:22,740 --> 00:24:24,170 convenient for that observer. 296 00:24:24,170 --> 00:24:26,270 Sometimes it depends on your physical [INAUDIBLE], 297 00:24:26,270 --> 00:24:26,770 et cetera. 298 00:24:29,990 --> 00:24:31,470 And other questions? 299 00:24:31,470 --> 00:24:36,290 AUDIENCE: So we cannot treat these two vacuums as like some 300 00:24:36,290 --> 00:24:37,852 transformation? 301 00:24:37,852 --> 00:24:40,310 PROFESSOR: Yeah, there is some transformation between them. 302 00:24:40,310 --> 00:24:44,170 AUDIENCE: So they stay just the same? 303 00:24:44,170 --> 00:24:47,710 Like do [INAUDIBLE]? 304 00:24:47,710 --> 00:24:49,420 PROFESSOR: No, they are different states. 305 00:24:49,420 --> 00:24:51,324 They are completely different states. 306 00:24:51,324 --> 00:24:52,990 You can relate to them in a certain way, 307 00:24:52,990 --> 00:24:54,769 but they're different states. 308 00:24:54,769 --> 00:24:56,060 Yeah, they're different states. 309 00:24:56,060 --> 00:24:58,750 AUDIENCE: If they are different, than why can 310 00:24:58,750 --> 00:25:02,080 we only say one is wrong? 311 00:25:02,080 --> 00:25:06,600 Because, for example, if it's singular, it's not physical, 312 00:25:06,600 --> 00:25:09,475 why don't we just discard it? 313 00:25:09,475 --> 00:25:10,350 PROFESSOR: I'm sorry? 314 00:25:10,350 --> 00:25:11,990 AUDIENCE: Why we-- 315 00:25:11,990 --> 00:25:14,390 PROFESSOR: Yeah, physically we do discard it. 316 00:25:14,390 --> 00:25:18,730 Physically we don't think the Schwarzschild vacuum is-- 317 00:25:18,730 --> 00:25:20,840 so here there's assumption. 318 00:25:20,840 --> 00:25:25,110 So the assumption is that we believe that the physics is now 319 00:25:25,110 --> 00:25:26,290 single at the horizon. 320 00:25:26,290 --> 00:25:29,260 So this is a basic assumption. 321 00:25:29,260 --> 00:25:32,730 And if that's the basic assumption, 322 00:25:32,730 --> 00:25:35,710 then you should abandon Schwarzschild vacuum 323 00:25:35,710 --> 00:25:39,160 as the physical choice of the vacuum. 324 00:25:39,160 --> 00:25:44,160 Then you say, then if we really do experiment in the black hole 325 00:25:44,160 --> 00:25:45,830 geometry, then what you will discover 326 00:25:45,830 --> 00:25:50,060 is the property of the Hartle-Hawking vacuum. 327 00:25:50,060 --> 00:25:53,730 But if our physical assumption is wrong 328 00:25:53,730 --> 00:25:56,500 that actually a black hole horizon is singular, 329 00:25:56,500 --> 00:25:59,490 then maybe Schwarzschild horizon will turn out to be right. 330 00:25:59,490 --> 00:26:02,127 But in this case, we actually don't know. 331 00:26:02,127 --> 00:26:04,210 We cannot really do experiments in the black hole, 332 00:26:04,210 --> 00:26:07,400 so we cannot really check it. 333 00:26:07,400 --> 00:26:09,990 AUDIENCE: But we can measure black hole temperature 334 00:26:09,990 --> 00:26:12,240 from infinitely far away, and the Schwarzschild 335 00:26:12,240 --> 00:26:13,480 vacuum it would be 0. 336 00:26:13,480 --> 00:26:15,560 PROFESSOR: No, but you cannot measure it. 337 00:26:15,560 --> 00:26:17,625 We have not been able to measure it. 338 00:26:17,625 --> 00:26:19,370 AUDIENCE: And what about the [INAUDIBLE]? 339 00:26:22,020 --> 00:26:24,790 PROFESSOR: Yeah, first produce the black hole first. 340 00:26:24,790 --> 00:26:27,280 [LAUGHTER] 341 00:26:27,280 --> 00:26:29,570 Yeah, we will worry about it after we 342 00:26:29,570 --> 00:26:30,779 have produced the black hole. 343 00:26:30,779 --> 00:26:32,319 AUDIENCE: But if they didn't radiate, 344 00:26:32,319 --> 00:26:33,544 they would not decay at all. 345 00:26:33,544 --> 00:26:35,460 PROFESSOR: Yeah, then you may also not see it. 346 00:26:41,070 --> 00:26:47,820 Anyway, so this can considered as a physical interpretation 347 00:26:47,820 --> 00:26:51,270 of the temperature inside the regularity at the horizon 348 00:26:51,270 --> 00:26:54,010 force us to be in this Hartle-Hawking vacuum. 349 00:26:54,010 --> 00:26:55,860 And then the Hartle-Hawking vacuum 350 00:26:55,860 --> 00:26:57,210 is like [? a similar ?] state. 351 00:26:57,210 --> 00:27:02,160 OK, so now they may explain why [? they ?] say, 352 00:27:02,160 --> 00:27:04,970 they will behavior like [? a similar ?] state, OK? 353 00:27:04,970 --> 00:27:08,490 So let me go to the second thing I will explain today. 354 00:27:08,490 --> 00:27:11,194 It's the physical origin of the temperature. 355 00:27:22,640 --> 00:27:25,390 So, again, I can use either black hole or the Rindler 356 00:27:25,390 --> 00:27:25,970 example. 357 00:27:25,970 --> 00:27:28,780 The mathematics are almost identical. 358 00:27:28,780 --> 00:27:30,310 But I will use the Rindler example 359 00:27:30,310 --> 00:27:35,270 because the mathematics are slightly simpler. 360 00:27:35,270 --> 00:27:37,760 So I will use the window of spacetime. 361 00:27:37,760 --> 00:27:49,250 So I will explain using the Rindler example. 362 00:27:55,360 --> 00:28:04,710 So I will explain at A that this choice of theta 363 00:28:04,710 --> 00:28:08,800 to periodically identify theta plus theta plus 2pi [INAUDIBLE] 364 00:28:08,800 --> 00:28:11,600 into the choice of a Minkowski vacuum. 365 00:28:23,140 --> 00:28:25,960 Then the second thing I will do is 366 00:28:25,960 --> 00:28:41,439 I will derive this temperature-- derive the temperature using 367 00:28:41,439 --> 00:28:42,230 a different method. 368 00:28:50,300 --> 00:28:52,950 So that's I will do today. 369 00:28:52,950 --> 00:28:56,380 So here, when me say, this temperature, 370 00:28:56,380 --> 00:28:59,890 we just directly read the temperature 371 00:28:59,890 --> 00:29:09,480 from the period of the Tau, OK but I will really 372 00:29:09,480 --> 00:29:13,880 derive with the thermal density matrix. 373 00:29:13,880 --> 00:29:17,270 And then you will see that this temperature is indeed 374 00:29:17,270 --> 00:29:19,745 the temperature with appears in the density matrix. 375 00:29:27,362 --> 00:29:27,945 Any questions? 376 00:29:32,470 --> 00:29:37,640 So these two together also amounts 377 00:29:37,640 --> 00:29:39,304 to the following statement. 378 00:29:39,304 --> 00:29:40,720 So this is an important statement, 379 00:29:40,720 --> 00:29:46,660 so let me just write it down, so this A and B also 380 00:29:46,660 --> 00:29:48,472 amounts to be the following-- also 381 00:29:48,472 --> 00:29:50,180 is equivalent to the following statement. 382 00:29:53,450 --> 00:30:13,111 It says, the vacuum inside the Minkowski-- 383 00:30:13,111 --> 00:30:14,860 the standard of the Minkowski vacuum where 384 00:30:14,860 --> 00:30:20,350 you do your quantum fields theory, 385 00:30:20,350 --> 00:30:28,390 the Minkowski vacuum appears to be in a similar state. 386 00:30:33,420 --> 00:30:43,940 This temperature which is pi divided by 2pi or T 387 00:30:43,940 --> 00:30:47,690 equal to hbar divided by 2pi in terms of eta. 388 00:30:52,470 --> 00:31:05,291 OK, it depends on which time you use to a Rindler observer 389 00:31:05,291 --> 00:31:06,540 of constant acceleration rate. 390 00:31:20,870 --> 00:31:28,684 So a says, actually, this choice [INAUDIBLE] a periodic of 2pi, 391 00:31:28,684 --> 00:31:30,600 actually we are choosing the Minkowski vacuum. 392 00:31:33,140 --> 00:31:36,660 And then the second statement you said, 393 00:31:36,660 --> 00:31:38,910 but the stand of the Minkowski vacuum, which 394 00:31:38,910 --> 00:31:44,130 appears to be 0 temperature to ordinary Minkowski observer 395 00:31:44,130 --> 00:31:49,450 than appears to be a similar state to a Rindler observer. 396 00:31:49,450 --> 00:31:51,860 OK, so that's the physical content 397 00:31:51,860 --> 00:31:55,470 of this things which we will show. 398 00:31:55,470 --> 00:31:56,450 Any questions on this? 399 00:32:06,130 --> 00:32:06,807 Yes? 400 00:32:06,807 --> 00:32:07,723 AUDIENCE: [INAUDIBLE]? 401 00:32:15,709 --> 00:32:17,500 PROFESSOR: It's actually a tricky question. 402 00:32:21,800 --> 00:32:24,390 In some sense, they don't really belong to the same Hilbert 403 00:32:24,390 --> 00:32:25,474 space. 404 00:32:25,474 --> 00:32:27,640 Yeah just when you talk about quantum fields theory, 405 00:32:27,640 --> 00:32:28,560 it's a little bit tricky when you have 406 00:32:28,560 --> 00:32:29,851 an infinite number [INAUDIBLE]. 407 00:32:33,060 --> 00:32:35,368 Yeah, but one can write down the relation between them. 408 00:32:35,368 --> 00:32:37,760 AUDIENCE: [INAUDIBLE]. 409 00:32:37,760 --> 00:32:40,260 PROFESSOR: Yeah, one can write down a relation between them, 410 00:32:40,260 --> 00:32:42,737 and then if you take the [? modulus ?] of that vacuum. 411 00:32:42,737 --> 00:32:44,570 Then you'll find that it's actually infinite 412 00:32:44,570 --> 00:32:46,780 because you have an infinite number that [INAUDIBLE]. 413 00:32:49,940 --> 00:32:54,350 It's not possible to normalize that state. 414 00:32:54,350 --> 00:32:57,300 OK, so that's what we are going to show. 415 00:32:57,300 --> 00:33:01,560 Hopefully, we will reach it by the end of this hour. 416 00:33:01,560 --> 00:33:10,300 But before that, we need a little bit of preparation 417 00:33:10,300 --> 00:33:11,550 to remind you of a few things. 418 00:33:14,360 --> 00:33:17,140 So once we go through these preparation, 419 00:33:17,140 --> 00:33:24,500 then final derivation only takes less than 10 minutes. 420 00:33:24,500 --> 00:33:28,865 So first, he said, they are actually two descriptions 421 00:33:28,865 --> 00:33:30,110 of a similar state. 422 00:33:37,815 --> 00:33:38,690 So let me remind you. 423 00:33:38,690 --> 00:33:40,565 They are two descriptions of a similar state. 424 00:33:49,690 --> 00:33:53,880 So we will use harmonic oscillator as an example. 425 00:34:01,290 --> 00:34:06,470 So now let's consider a single harmonic oscillator 426 00:34:06,470 --> 00:34:09,600 a finite temperature. 427 00:34:09,600 --> 00:34:13,039 So the standard way of doing it is 428 00:34:13,039 --> 00:34:14,830 that if you want to compute the expectation 429 00:34:14,830 --> 00:34:17,889 value of some operator, of some observable 430 00:34:17,889 --> 00:34:21,389 at finite temperature, you just do 431 00:34:21,389 --> 00:34:26,659 the standard canonical average. 432 00:34:32,239 --> 00:34:35,150 So the H would be, say, the Hamiltonian 433 00:34:35,150 --> 00:34:37,400 of this single harmonic oscillator, 434 00:34:37,400 --> 00:34:41,120 and the z is the partition function. 435 00:34:41,120 --> 00:34:46,635 So can also be written as trace x and as a thermal density 436 00:34:46,635 --> 00:34:47,135 matrix. 437 00:34:51,080 --> 00:35:07,590 So the thermal density matrix is 1/z minus beta h, 438 00:35:07,590 --> 00:35:11,340 and the z is just the sum of all possible states. 439 00:35:18,440 --> 00:35:21,390 So this is a standard way you would do, say, 440 00:35:21,390 --> 00:35:24,430 the finite temperature physics. 441 00:35:24,430 --> 00:35:28,240 So this, of course, applies to any quantum systems 442 00:35:28,240 --> 00:35:31,890 causal quantum fields theories, et cetera. 443 00:35:31,890 --> 00:35:35,570 But actually, this alternative way 444 00:35:35,570 --> 00:35:42,630 to do thermal physics-- so this was realized by Umezawa 445 00:35:42,630 --> 00:35:52,900 in the 1960s. 446 00:35:52,900 --> 00:35:57,790 So he said, instead of considering 447 00:35:57,790 --> 00:36:01,730 this thermal density matrix, let me just consider 448 00:36:01,730 --> 00:36:03,339 two copies of the same. 449 00:36:03,339 --> 00:36:05,380 Let's consider two copies of harmonic oscillator. 450 00:36:21,280 --> 00:36:30,840 Let's double the copy, and then now we have H1. 451 00:36:30,840 --> 00:36:35,660 Then the foreseeable space for these two copies 452 00:36:35,660 --> 00:36:40,150 will be the H1 of one system tensor product of H2 453 00:36:40,150 --> 00:36:43,500 and the Hilbert space of the other. 454 00:36:43,500 --> 00:36:50,100 And then you will have H1 H2, the Hamiltonian H1 H2 455 00:36:50,100 --> 00:36:52,020 associated with each of them. 456 00:36:52,020 --> 00:36:53,960 But, of course, these two H's are the same. 457 00:36:58,350 --> 00:37:10,210 Then the [INAUDIBLE] system in a typical state 458 00:37:10,210 --> 00:37:18,610 would be, say, of the form sum mn, amn, m1 [? canceled ?] 459 00:37:18,610 --> 00:37:22,950 with n and 2. 460 00:37:22,950 --> 00:37:26,170 So the general state, say, of this doubled system 461 00:37:26,170 --> 00:37:26,920 will be like this. 462 00:37:29,980 --> 00:37:32,520 So this is two copies of the same system with no interaction 463 00:37:32,520 --> 00:37:33,210 with each other. 464 00:37:36,182 --> 00:37:38,390 So now he says, in order to consider thermal physics, 465 00:37:38,390 --> 00:37:42,900 let's consider special state defined 466 00:37:42,900 --> 00:37:50,868 by the following, 1 over square root of z sum over n. 467 00:38:04,560 --> 00:38:07,850 And he said, now let's consider the following states. 468 00:38:10,570 --> 00:38:14,350 So this is an entangled state between the two systems. 469 00:38:14,350 --> 00:38:15,870 So this is an entangled state. 470 00:38:25,520 --> 00:38:33,140 So the key observation is that if you 471 00:38:33,140 --> 00:38:36,980 want to consider the thermal physics, 472 00:38:36,980 --> 00:38:48,370 it says, for any observable, say this x-- so [INAUDIBLE] 473 00:38:48,370 --> 00:38:53,270 1, which only acts in one of the systems. 474 00:39:01,450 --> 00:39:05,720 [INAUDIBLE] act on the first system. 475 00:39:05,720 --> 00:39:08,830 So let's consider any such observable. 476 00:39:08,830 --> 00:39:10,420 So now, if you to take the expectation 477 00:39:10,420 --> 00:39:43,950 value between the sides, the x between the sides, 478 00:39:43,950 --> 00:39:45,770 so you essentially have a side squared. 479 00:39:45,770 --> 00:39:48,540 Then you have 1/z. 480 00:39:48,540 --> 00:39:52,596 Then also Tau will become better En, 481 00:39:52,596 --> 00:40:08,130 and then this just becomes 1/z sum over n [INAUDIBLE] and x n. 482 00:40:08,130 --> 00:40:10,110 So this is the thermal average. 483 00:40:15,140 --> 00:40:16,080 This is sum average. 484 00:40:23,520 --> 00:40:27,720 So if you understand the why, you can also 485 00:40:27,720 --> 00:40:34,000 consider because if we are interested only in system 1, 486 00:40:34,000 --> 00:40:36,700 then you can just integrate our system 2. 487 00:40:36,700 --> 00:40:39,630 We can trace our system 2. 488 00:40:39,630 --> 00:40:42,530 Say, suppose we trace our system 2 489 00:40:42,530 --> 00:40:53,550 of this state, then, of course, what do you find? 490 00:41:04,750 --> 00:41:09,880 Which is the thermal density matrix of the system 1. 491 00:41:12,740 --> 00:41:19,290 So in other words, the thermal density matrix in one system 492 00:41:19,290 --> 00:41:23,800 can be considered as [? entangled ?] pure states 493 00:41:23,800 --> 00:41:26,870 of a doubled system. 494 00:41:26,870 --> 00:41:30,450 And because we know nothing about the other system, 495 00:41:30,450 --> 00:41:36,550 once we trace the l to the other system, then 496 00:41:36,550 --> 00:41:40,060 you get a density matrix. 497 00:41:40,060 --> 00:41:43,610 And this density matrix comes from our insufficient knowledge 498 00:41:43,610 --> 00:41:44,570 of the other system. 499 00:41:47,620 --> 00:41:50,310 So this is another way to think about this thermal behavior. 500 00:41:53,761 --> 00:41:55,530 Do you have any questions on this? 501 00:42:01,640 --> 00:42:22,730 So the temperature arises due to an ignorance of system 2. 502 00:42:28,160 --> 00:42:32,060 So because if you now have full knowledge of one of the two, 503 00:42:32,060 --> 00:42:36,060 then this would be just the [? size ?] of a pure state. 504 00:42:36,060 --> 00:42:40,370 It's just a very special pure state. 505 00:42:40,370 --> 00:42:44,040 But if you don't know anything about system 2 506 00:42:44,040 --> 00:42:46,882 and if you consent of only about system 1, 507 00:42:46,882 --> 00:42:48,590 then when you trace out system 2 then you 508 00:42:48,590 --> 00:42:53,180 get the thermal density matrix for 1. 509 00:42:53,180 --> 00:42:55,790 So let me make some additional remarks on this. 510 00:42:58,119 --> 00:42:59,410 I'll make some further remarks. 511 00:43:04,200 --> 00:43:09,300 First, this, of course, applies to any-- 512 00:43:09,300 --> 00:43:13,440 even though I'm saying I'm using the harmonic oscillator, this, 513 00:43:13,440 --> 00:43:19,551 of course, applies to any quantum systems including 514 00:43:19,551 --> 00:43:20,050 the QFT's. 515 00:43:25,620 --> 00:43:29,250 So the second remark is that this side is actually 516 00:43:29,250 --> 00:43:31,560 a very special state. 517 00:43:31,560 --> 00:43:47,110 This size is invariant under H1 minus H2, H1 minus H2. 518 00:43:47,110 --> 00:43:48,980 So you can see it very clearly from here. 519 00:43:52,750 --> 00:43:59,030 It says, if you add H1 and H2 on this state, H1 minus H2 520 00:43:59,030 --> 00:44:03,420 because both n's have the same energy, so they just cancel. 521 00:44:03,420 --> 00:44:11,640 So this is invariant under this-- yeah, 522 00:44:11,640 --> 00:44:25,730 I should say, more precisely, it's [? annihilated ?] 523 00:44:25,730 --> 00:44:28,570 by H1 minus H2. 524 00:44:28,570 --> 00:44:33,080 And it's invariant under any translation 525 00:44:33,080 --> 00:44:35,864 created by H1 minus H2. 526 00:44:35,864 --> 00:44:36,780 AUDIENCE: [INAUDIBLE]? 527 00:44:42,444 --> 00:44:44,110 PROFESSOR: Yeah, just double the system. 528 00:44:44,110 --> 00:44:45,832 It's two copies of the same system. 529 00:44:57,150 --> 00:45:08,130 So the third remark now relies on the harmonic oscillator. 530 00:45:08,130 --> 00:45:18,750 For the harmonic oscillator, we can write 531 00:45:18,750 --> 00:45:21,250 this psi in the following form. 532 00:45:26,539 --> 00:45:27,830 So this you can check yourself. 533 00:45:27,830 --> 00:45:31,010 I only write down the answer. 534 00:45:31,010 --> 00:45:34,690 You can easily convince yourself this is true. 535 00:45:34,690 --> 00:45:37,481 Some of you might be able to see it immediately just 536 00:45:37,481 --> 00:45:38,230 on the blackboard. 537 00:45:50,150 --> 00:45:52,540 As you actually write this in the-- 538 00:45:52,540 --> 00:45:56,550 if you can write this as some explanation of some-- so a1 a2 539 00:45:56,550 --> 00:46:02,310 are creation operators respectfully for two systems. 540 00:46:02,310 --> 00:46:06,280 And then the 0 1 and 0 2 are the vacuum of the two 541 00:46:06,280 --> 00:46:09,350 harmonic oscillator systems. 542 00:46:09,350 --> 00:46:11,210 And you can easily see yourself, when 543 00:46:11,210 --> 00:46:15,250 you expand this exponential, then you essentially 544 00:46:15,250 --> 00:46:18,130 just take the power of n. 545 00:46:18,130 --> 00:46:21,350 And then that will give you n. 546 00:46:21,350 --> 00:46:25,900 Yeah, when you act on 0 0, then that will n a times n, OK? 547 00:46:29,310 --> 00:46:32,980 So this is normally called a squeezed state. 548 00:46:32,980 --> 00:46:36,050 So this tells you that this psi is 549 00:46:36,050 --> 00:46:39,280 related to the vacuum of the system 550 00:46:39,280 --> 00:46:42,900 by some kind of squeezed-- so this is a squeezed state 551 00:46:42,900 --> 00:46:46,480 in terms of the vacuum. 552 00:46:46,480 --> 00:46:51,015 So this form is useful for the following reason. 553 00:46:53,710 --> 00:47:01,740 It's that now, based on these three, 554 00:47:01,740 --> 00:47:11,880 one can show it's possible to construct 555 00:47:11,880 --> 00:47:23,420 two oscillators, which are b1 and b2, which [INAUDIBLE] 556 00:47:23,420 --> 00:47:24,280 the psi. 557 00:47:27,690 --> 00:47:33,270 So the b1 and b2 are constructed by the following. 558 00:47:33,270 --> 00:47:36,360 So, again, this you should check yourself. 559 00:47:36,360 --> 00:47:39,540 It's easy to do a little bit of algebra. 560 00:47:55,280 --> 00:48:03,670 So cos theta is equal to 1 over 1 561 00:48:03,670 --> 00:48:07,860 minus the exponential minus beta omega. 562 00:48:07,860 --> 00:48:12,240 So the omega is the frequency for this harmonic oscillator 563 00:48:12,240 --> 00:48:14,840 system and the [INAUDIBLE]. 564 00:48:35,960 --> 00:48:40,267 So this b1 b2 are related to a1 a2 565 00:48:40,267 --> 00:48:41,516 by some linear transformation. 566 00:48:44,770 --> 00:48:54,730 So what this shows is that [? while ?] 0 1 and the 0 2 567 00:48:54,730 --> 00:49:05,920 is a vacuum for a1 and a2, and this side 568 00:49:05,920 --> 00:49:11,370 is a vacuum for b1 b2. 569 00:49:14,182 --> 00:49:15,640 So then maybe you can see that this 570 00:49:15,640 --> 00:49:17,514 goes [? into ?] a different choice of vacuum. 571 00:49:20,340 --> 00:49:21,990 So as we'll see later, the relation 572 00:49:21,990 --> 00:49:23,490 between the so-called Hartle-Hawking 573 00:49:23,490 --> 00:49:26,900 vacuum and the Schwarzschild vacuum is precisely like this. 574 00:49:30,149 --> 00:49:32,190 They just, of course, run into a different choice 575 00:49:32,190 --> 00:49:33,274 of alternators. 576 00:49:33,274 --> 00:49:34,690 And, in particular, their relation 577 00:49:34,690 --> 00:49:38,810 is precise over this form because, say, 578 00:49:38,810 --> 00:49:41,099 if you consider some [? three ?] series 579 00:49:41,099 --> 00:49:43,390 because the [? three ?] series essentially just reduces 580 00:49:43,390 --> 00:49:44,390 to harmonic oscillators. 581 00:49:46,726 --> 00:49:50,170 OK, so this is the first preparation. 582 00:49:50,170 --> 00:49:52,140 And those things are very easy check yourself 583 00:49:52,140 --> 00:49:56,710 because this is just a single harmonic oscillator. 584 00:49:56,710 --> 00:49:57,710 Any questions on this? 585 00:50:12,230 --> 00:50:18,076 Oh, by the way, this kind of transformation 586 00:50:18,076 --> 00:50:19,825 is often called Bogoliubov transformation. 587 00:50:32,220 --> 00:50:39,560 So the [INAUDIBLE] you assume by this transformation 588 00:50:39,560 --> 00:50:43,040 is that in the expression for b1 b2, 589 00:50:43,040 --> 00:50:46,390 a [? dagger ?] appears here. 590 00:50:46,390 --> 00:50:52,610 If there is only a that appears here, then b1 b2, a1, a2, 591 00:50:52,610 --> 00:50:55,470 they will have the same vacuum because they will 592 00:50:55,470 --> 00:50:57,831 [INAUDIBLE] the same states. 593 00:50:57,831 --> 00:51:00,330 So the [? nontrivial ?] thing is because of the [? dagger ?] 594 00:51:00,330 --> 00:51:05,750 appearing here, so now the state which they are [INAUDIBLE] are 595 00:51:05,750 --> 00:51:07,980 completely different and [? graded ?] by this kind 596 00:51:07,980 --> 00:51:11,760 of squeezed-state relation. 597 00:51:11,760 --> 00:51:12,470 Yes? 598 00:51:12,470 --> 00:51:15,330 AUDIENCE: So it's related to the H2 minus H2? 599 00:51:15,330 --> 00:51:16,240 PROFESSOR: I'm sorry? 600 00:51:16,240 --> 00:51:18,185 AUDIENCE: The operators like b1 b2 601 00:51:18,185 --> 00:51:20,114 are somewhat similar to the H1 minus H2? 602 00:51:20,114 --> 00:51:21,030 PROFESSOR: Not really. 603 00:51:23,720 --> 00:51:26,250 You can write down H1 H2 in terms of b1 and b2. 604 00:51:26,250 --> 00:51:28,140 You can certainly do that. 605 00:51:28,140 --> 00:51:30,157 Yeah, but H1 H2 is also very simply. 606 00:51:30,157 --> 00:51:31,490 You write in terms of a1 and a2. 607 00:51:34,225 --> 00:51:40,140 AUDIENCE: If a letter transformation [INAUDIBLE] 608 00:51:40,140 --> 00:51:45,002 we get-- never mind. 609 00:51:45,002 --> 00:51:47,520 PROFESSOR: Yeah, you can find the b1 [INAUDIBLE]. 610 00:51:47,520 --> 00:51:50,520 Just take the conjugate. 611 00:51:50,520 --> 00:51:55,620 Good, so this is the first preparation, 612 00:51:55,620 --> 00:52:02,490 which is just something about the harmonic oscillator, which 613 00:52:02,490 --> 00:52:10,480 actually can be generalized also to general quantum systems. 614 00:52:10,480 --> 00:52:16,730 So the second is I need to remind you a little bit 615 00:52:16,730 --> 00:52:27,060 the Schrodinger representation of QFT's. 616 00:52:32,420 --> 00:52:35,635 So normally when we talk about quantum field theory, 617 00:52:35,635 --> 00:52:37,320 you always use the Heisenberg picture. 618 00:52:37,320 --> 00:52:40,780 We don't talk about wave function. 619 00:52:40,780 --> 00:52:44,190 But this equivalent formulation of, of course, quantum 620 00:52:44,190 --> 00:52:47,970 fields theory, which you can just talk about wave functional 621 00:52:47,970 --> 00:52:53,160 and talk about it in terms of the Schrodinger picture. 622 00:52:53,160 --> 00:52:55,591 So for example, let me just consider a scalar field's 623 00:52:55,591 --> 00:52:56,090 theory. 624 00:53:11,710 --> 00:53:23,800 Then the Hilbert space, the configuration space, 625 00:53:23,800 --> 00:53:30,200 of this theory of this system is just phi x. 626 00:53:30,200 --> 00:53:33,370 You validate it at some given time. 627 00:53:33,370 --> 00:53:37,340 Say, you validate it at t equals to 0, for example. 628 00:53:37,340 --> 00:53:38,670 So let me just write it as 5x. 629 00:53:45,850 --> 00:53:48,040 The configuration space is just all possibles 630 00:53:48,040 --> 00:53:55,850 of phi defining the spatial slice. 631 00:53:55,850 --> 00:54:00,100 OK, so this your configuration space for the quantum 632 00:54:00,100 --> 00:54:01,750 fields theory. 633 00:54:01,750 --> 00:54:11,030 And the Hilbert space of the system 634 00:54:11,030 --> 00:54:18,080 just given by all possible y functionals 635 00:54:18,080 --> 00:54:20,520 of this configuration space variables. 636 00:54:26,520 --> 00:54:30,220 So if you feel this is a little bit too abstract to you, 637 00:54:30,220 --> 00:54:36,420 then just think of space is discrete, 638 00:54:36,420 --> 00:54:41,390 and then you can just write this as some discrete set 639 00:54:41,390 --> 00:54:44,530 of variables that them become identical 640 00:54:44,530 --> 00:54:46,300 to the ordinary quantum system. 641 00:54:52,580 --> 00:54:58,490 So two things to remind you, two more things. 642 00:54:58,490 --> 00:55:02,080 First, just as in quantum mechanics, 643 00:55:02,080 --> 00:55:10,760 if you ask the value of, say, a time t2, 644 00:55:10,760 --> 00:55:13,080 you are in the [INAUDIBLE] state of this phi, 645 00:55:13,080 --> 00:55:16,390 this [INAUDIBLE] value phi 2 and the 646 00:55:16,390 --> 00:55:24,200 overlap we said t1 in the [INAUDIBLE] state of phi, 647 00:55:24,200 --> 00:55:26,500 this [INAUDIBLE] value phi 1. 648 00:55:26,500 --> 00:55:31,030 This is given by a path integral [? d ?] phi. 649 00:55:40,700 --> 00:55:45,420 You integrate with the following boundary condition. 650 00:56:05,720 --> 00:56:08,500 And actually, just [INAUDIBLE] for this theory. 651 00:56:15,610 --> 00:56:21,599 In particular, by taking a linear of this formula, 652 00:56:21,599 --> 00:56:23,640 one can write down a path integral representation 653 00:56:23,640 --> 00:56:28,150 of the vacuum wave function. 654 00:56:34,530 --> 00:56:45,950 So the vacuum wave function, in this case, 655 00:56:45,950 --> 00:56:49,790 will be very functional. 656 00:56:49,790 --> 00:56:53,850 [INAUDIBLE], so if you have a vacuum state, 657 00:56:53,850 --> 00:56:55,670 then you just consider the overlap 658 00:56:55,670 --> 00:57:00,210 with the vacuum to your configuration space variable. 659 00:57:00,210 --> 00:57:06,110 So I will denote it by phi 0 phi x. 660 00:57:06,110 --> 00:57:07,730 This is the vacuum. 661 00:57:07,730 --> 00:57:14,910 And this has a path integral [INAUDIBLE] 662 00:57:14,910 --> 00:57:18,620 is that you compactify your time. 663 00:57:18,620 --> 00:57:22,280 OK, so it's where this is your real time, 664 00:57:22,280 --> 00:57:26,796 and this is your imaginary time, which I call t e. 665 00:57:29,390 --> 00:57:33,090 You compactify your time. 666 00:57:33,090 --> 00:57:40,795 And then integrate the path integral or back 667 00:57:40,795 --> 00:57:43,790 to imaginary time. 668 00:57:43,790 --> 00:57:50,160 So this can be written from the path integral as t phi t e x. 669 00:57:50,160 --> 00:57:57,880 So you go to Euclidean, the but integrate all t's more than 0 670 00:57:57,880 --> 00:58:02,536 with the boundary condition te equal to 0x. 671 00:58:02,536 --> 00:58:07,590 You can do phi x, and your Euclidean action. 672 00:58:12,770 --> 00:58:16,732 OK, I hope you're familiar with this. 673 00:58:16,732 --> 00:58:20,160 This is where we obtain the vacuum [INAUDIBLE] function. 674 00:58:24,470 --> 00:58:26,950 So if this is not for me to [INAUDIBLE] say, well, 675 00:58:26,950 --> 00:58:31,340 how do you get to the wave function 676 00:58:31,340 --> 00:58:35,390 of a harmonic oscillator in the vacuum from path integral? 677 00:58:38,160 --> 00:58:39,124 Yes? 678 00:58:39,124 --> 00:58:40,040 AUDIENCE: [INAUDIBLE]? 679 00:58:48,210 --> 00:58:50,739 PROFESSOR: You will see it in a few minutes 680 00:58:50,739 --> 00:58:52,405 because I'm going to use those formulas. 681 00:58:59,840 --> 00:59:00,630 Yes? 682 00:59:00,630 --> 00:59:02,180 AUDIENCE: So one question is that 683 00:59:02,180 --> 00:59:06,582 what is the analog of the wave equation for the wave 684 00:59:06,582 --> 00:59:07,760 functional? 685 00:59:07,760 --> 00:59:09,630 Like what is the [INAUDIBLE] equation 686 00:59:09,630 --> 00:59:11,430 for the wave functional? 687 00:59:11,430 --> 00:59:13,150 PROFESSOR: The same thing. 688 00:59:13,150 --> 00:59:17,486 i [? partial ?] t phi equal to H phi. 689 00:59:17,486 --> 00:59:19,571 AUDIENCE: OK. 690 00:59:19,571 --> 00:59:21,820 PROFESSOR: Yeah, it just has more [INAUDIBLE] freedom. 691 00:59:21,820 --> 00:59:24,298 The quantum mechanics works the same. 692 00:59:24,298 --> 00:59:26,567 AUDIENCE: OK. 693 00:59:26,567 --> 00:59:28,150 PROFESSOR: So is this familiar to you? 694 00:59:28,150 --> 00:59:31,660 If not familiar to you, I urge you 695 00:59:31,660 --> 00:59:34,449 to think about the case of a harmonic oscillator. 696 00:59:34,449 --> 00:59:36,490 For the harmonic oscillator, that will be the way 697 00:59:36,490 --> 00:59:42,090 you obtain the ground state wave function from the path integral 698 00:59:42,090 --> 00:59:43,530 is that you first need to continue 699 00:59:43,530 --> 00:59:46,120 the system to the Euclidean. 700 00:59:46,120 --> 00:59:50,380 And they integrate the path integral all the way from minus 701 00:59:50,380 --> 00:59:55,530 Euclidean time equal to infinity to Euclidean time equal to 0. 702 00:59:55,530 --> 00:59:59,540 Let me just write down the-- so the standard of quantum 703 00:59:59,540 --> 01:00:04,830 mechanics if you want to right down to the ground state wave 704 01:00:04,830 --> 01:00:09,710 function, so that's what you do. 705 01:00:09,710 --> 01:00:12,240 Again, you go to the path integral. 706 01:00:12,240 --> 01:00:20,250 You go to go to Euclidean, and you really 707 01:00:20,250 --> 01:00:25,150 integrate of all tE smaller than 0. 708 01:00:25,150 --> 01:00:27,850 And with the boundary condition that x [? evaluated ?] 709 01:00:27,850 --> 01:00:33,590 at tE equal to tE equal to 0 equal to x. 710 01:00:33,590 --> 01:00:36,331 So that gives you the wave function, 711 01:00:36,331 --> 01:00:37,497 gives you the wave function. 712 01:00:40,700 --> 01:00:46,290 OK, now with this preparation, that's 713 01:00:46,290 --> 01:00:49,220 just some ordinary quantum mechanics. 714 01:00:49,220 --> 01:00:51,370 And this is a generalization of that 715 01:00:51,370 --> 01:00:54,675 to quantum fields theory just replace that x by 5. 716 01:01:00,090 --> 01:01:02,150 Good, so now let's go back to prove 717 01:01:02,150 --> 01:01:07,960 the statement we claim you're going to prove, this one, OK? 718 01:01:07,960 --> 01:01:10,611 So now let's come back. 719 01:01:10,611 --> 01:01:13,070 So we finished our preparation. 720 01:01:13,070 --> 01:01:15,620 Now come back to Rindler space. 721 01:01:25,060 --> 01:01:27,575 So it is considered scalar field theory, for example. 722 01:01:30,670 --> 01:01:34,200 So let me just remind you, again, this Rindler-- so 723 01:01:34,200 --> 01:01:37,890 write down some key formulas here-- 724 01:01:37,890 --> 01:01:48,820 so Rindler is the right quadrant of the Minkowski time, which is 725 01:01:48,820 --> 01:01:51,220 separated by this right column. 726 01:01:51,220 --> 01:01:57,450 Suppose this is X. This is T. Then Minkowski [INAUDIBLE] 727 01:01:57,450 --> 01:02:01,591 is minus dT squared plus dX squared. 728 01:02:01,591 --> 01:02:08,060 And then the Rindler path is minus rho squared to eta 729 01:02:08,060 --> 01:02:13,120 squared plus the rho squared. 730 01:02:13,120 --> 01:02:15,165 And so this is the rho equal to constant. 731 01:02:18,700 --> 01:02:20,505 And this is the eta equal to constant. 732 01:02:23,410 --> 01:02:27,210 And here is eta equal to minus infinity. 733 01:02:27,210 --> 01:02:30,530 So here is eta equal to plus infinity. 734 01:02:30,530 --> 01:02:33,400 So spacetime foliates like this, OK? 735 01:02:36,410 --> 01:02:50,240 And when we go to Euclidean signature for the Minkowski, 736 01:02:50,240 --> 01:02:54,654 of course, it's just T goes to minus iTE. 737 01:02:54,654 --> 01:02:57,420 So I call this Euclidean time. 738 01:02:57,420 --> 01:03:03,765 For the Rindler, I will do minus i theta. 739 01:03:03,765 --> 01:03:04,890 This is my notation before. 740 01:03:21,630 --> 01:03:35,080 So in the Euclidean signature, the standard Minkowski 741 01:03:35,080 --> 01:03:39,010 just becomes TE squared plus the dX squared. 742 01:03:39,010 --> 01:03:45,470 And then this Rindler is just rho squared 743 01:03:45,470 --> 01:03:47,670 [? to ?] theta squared plus the rho squared. 744 01:03:52,550 --> 01:03:58,550 In particular, this theta identified 745 01:03:58,550 --> 01:04:00,805 to be plus period 2pi. 746 01:04:04,530 --> 01:04:19,220 The Euclidean, under the continuation 747 01:04:19,220 --> 01:04:26,920 of Minkowski and Rindler, are actually identical. 748 01:04:36,740 --> 01:04:44,926 Both of them are the full two-dimensional Euclidean 749 01:04:44,926 --> 01:04:45,425 space. 750 01:04:48,020 --> 01:04:50,000 So if you do the standard of the Minkowski, 751 01:04:50,000 --> 01:04:51,945 just replace X by TE. 752 01:04:54,510 --> 01:05:03,240 But for the Rindler, you just replace it by rho theta. 753 01:05:08,010 --> 01:05:11,130 So they just go one into a different foliation, 754 01:05:11,130 --> 01:05:14,270 and this goes one into the [INAUDIBLE]. 755 01:05:14,270 --> 01:05:18,170 And this one is bound for the Cartesian coordinate. 756 01:05:18,170 --> 01:05:22,100 So the remarkable thing is that even 757 01:05:22,100 --> 01:05:25,915 in the Lorentzian signature, the Rindler is only part 758 01:05:25,915 --> 01:05:27,210 of the Minkowski spacetime. 759 01:05:30,130 --> 01:05:37,010 But once you go to Euclidean, if you do this identification 760 01:05:37,010 --> 01:05:41,030 theta cos theta plus 2pi, they have exactly the same Euclidean 761 01:05:41,030 --> 01:05:43,920 manifold, just identical. 762 01:05:48,950 --> 01:05:50,800 So this immediately means one thing. 763 01:05:54,800 --> 01:06:08,370 It's that all the Euclidean observables just 764 01:06:08,370 --> 01:06:10,120 with the trivial coordinate transformation 765 01:06:10,120 --> 01:06:17,040 from the Cartesian to the polar coordinates, 766 01:06:17,040 --> 01:06:25,580 all the Euclidean observables are identical in two theories. 767 01:06:30,450 --> 01:06:32,770 AUDIENCE: What is that down there, "are identical?" 768 01:06:32,770 --> 01:06:35,580 What's R sub E squared? 769 01:06:35,580 --> 01:06:38,370 PROFESSOR: Yeah, this just means the Euclidean two-dimensional 770 01:06:38,370 --> 01:06:40,620 spacetime. 771 01:06:40,620 --> 01:06:41,577 AUDIENCE: Oh, I see. 772 01:06:41,577 --> 01:06:43,160 PROFESSOR: Yeah, let me just right it. 773 01:06:43,160 --> 01:06:48,830 This is R subscript E squared 2. 774 01:06:48,830 --> 01:06:51,940 So this is just a two-dimensional Euclidean 775 01:06:51,940 --> 01:06:52,440 space. 776 01:07:02,390 --> 01:07:05,020 So from here, we can immediately lead 777 01:07:05,020 --> 01:07:14,586 to the conclusion we said earlier because for Minkowski-- 778 01:07:14,586 --> 01:07:16,460 say, if you compute the Euclidean correlation 779 01:07:16,460 --> 01:07:20,280 functions, and then [INAUDIBLE] to the Lorentzian signature, 780 01:07:20,280 --> 01:07:23,710 what you get is that you get the correlation 781 01:07:23,710 --> 01:07:26,660 function in the standard Minkowski vacuum. 782 01:07:29,730 --> 01:07:39,320 So for Minkowski and then back to Lorentzian, 783 01:07:39,320 --> 01:07:48,700 so the typical observables-- so Euclidean, say, 784 01:07:48,700 --> 01:07:58,550 correlation functions just goes to correlation functions 785 01:07:58,550 --> 01:08:02,330 in the standard Minkowski vacuum, OK? 786 01:08:07,170 --> 01:08:14,503 Yeah, this is just trivial QFT in your high school. 787 01:08:14,503 --> 01:08:20,279 [LAUGHTER] 788 01:08:20,279 --> 01:08:27,140 But with this top statement, we reach a very [INAUDIBLE] 789 01:08:27,140 --> 01:08:35,149 conclusion is that for Rindler, when you go back to Lorentzian 790 01:08:35,149 --> 01:08:41,090 signature, then that tells you for the Rindler 791 01:08:41,090 --> 01:08:42,880 when you do that [INAUDIBLE] continuation 792 01:08:42,880 --> 01:08:46,370 and back to the Lorentzian signature, what do you get? 793 01:08:46,370 --> 01:08:56,970 It's that you get a correlation function 794 01:08:56,970 --> 01:09:00,000 in the standard Minkowski vacuum. 795 01:09:06,750 --> 01:09:24,560 But for observables, restricted to Rindler 796 01:09:24,560 --> 01:09:26,740 because it's the same thing. 797 01:09:26,740 --> 01:09:29,640 It's just the same function when we do a [? continuation, ?] 798 01:09:29,640 --> 01:09:31,689 just you do it differently. 799 01:09:31,689 --> 01:09:34,000 It's just on the continuation procedure, 800 01:09:34,000 --> 01:09:35,180 it's a little bit different. 801 01:09:35,180 --> 01:09:37,680 And a way to go into Rindler, you just 802 01:09:37,680 --> 01:09:41,260 get observable restriction to the Rindler path. 803 01:09:41,260 --> 01:09:44,560 So the Euclidean things are exactly the same. 804 01:09:44,560 --> 01:09:49,660 So that tells you that the corresponding Lorentzian 805 01:09:49,660 --> 01:09:53,700 Rindler correlation function is the same as the correlation 806 01:09:53,700 --> 01:09:56,180 function in the standard Minkowski vacuum, 807 01:09:56,180 --> 01:10:00,120 but you just restrict to the Rindler patch. 808 01:10:00,120 --> 01:10:03,020 OK, now is our final step. 809 01:10:06,590 --> 01:10:09,950 So now let's talk a bit more about the structure 810 01:10:09,950 --> 01:10:10,855 of the Hilbert space. 811 01:10:14,920 --> 01:10:17,015 So now I have to derive the temperature, 812 01:10:17,015 --> 01:10:20,904 but somehow in the Minkowski, there's no temperature. 813 01:10:20,904 --> 01:10:22,070 Here there's no temperature. 814 01:10:22,070 --> 01:10:25,210 It's T equal to 0 from your high school. 815 01:10:25,210 --> 01:10:28,830 And here, we must see a temperature. 816 01:10:28,830 --> 01:10:31,900 So where does this temperature come from? 817 01:10:31,900 --> 01:10:35,580 So now let's look a little bit at the structure of the Hilbert 818 01:10:35,580 --> 01:10:37,000 space. 819 01:10:37,000 --> 01:10:40,620 So using that picture-- so let me write here-- so the Hilbert 820 01:10:40,620 --> 01:10:45,170 space of the Rindler-- so this is all not in the Lorentzian 821 01:10:45,170 --> 01:10:47,870 picture. 822 01:10:47,870 --> 01:10:49,680 So the Hilbert space in the Rindler 823 01:10:49,680 --> 01:10:54,580 is essentially all possible square integrable wave 824 01:10:54,580 --> 01:11:02,740 functional of the phi, which defines in the right patch. 825 01:11:02,740 --> 01:11:06,210 So when we define this wave functional, 826 01:11:06,210 --> 01:11:08,569 we verify at the single time slice. 827 01:11:08,569 --> 01:11:10,360 So that's evaluated at a slice which is eta 828 01:11:10,360 --> 01:11:11,850 equal to 0, which is just here. 829 01:11:14,830 --> 01:11:28,640 So with phi R is essentially just phi for x greater than 0 830 01:11:28,640 --> 01:11:29,715 and the T equal to 0. 831 01:11:32,760 --> 01:11:41,630 This is the right half of the real axis. 832 01:11:41,630 --> 01:11:46,150 So this is a Hilbert space of the Rindler. 833 01:11:46,150 --> 01:11:49,060 And, of course, we can also write down the Hamiltonian 834 01:11:49,060 --> 01:11:55,490 for the Rindler respect to eta. 835 01:11:58,360 --> 01:12:01,630 So this is called the Rindler Hamiltonian. 836 01:12:01,630 --> 01:12:05,630 And as I said before, you can quantize this here unit, 837 01:12:05,630 --> 01:12:11,316 this Hamiltonian, then you will get-- say, 838 01:12:11,316 --> 01:12:12,690 you can quantize this Hamiltonian 839 01:12:12,690 --> 01:12:15,640 to construct all the excited states, so to me just label 840 01:12:15,640 --> 01:12:18,080 them by n. 841 01:12:18,080 --> 01:12:34,180 So I say, this complete set of eigenstate for HR, 842 01:12:34,180 --> 01:12:37,750 so the second value will say, En. 843 01:12:37,750 --> 01:12:40,570 OK, you can just quantize it, then 844 01:12:40,570 --> 01:12:42,000 you can find your full state. 845 01:12:42,000 --> 01:12:46,002 In particular, the ground state, when you do that, 846 01:12:46,002 --> 01:12:47,460 is what we call the Rindler vacuum. 847 01:12:50,590 --> 01:12:54,140 We said before that if you just to straightforward 848 01:12:54,140 --> 01:12:58,820 canonical connotation, you get the Rindler vacuum, which 849 01:12:58,820 --> 01:13:02,190 is different from the vacuum which we identify theta by 2pi. 850 01:13:02,190 --> 01:13:08,870 And this we just straightforward quantize respect to eta. 851 01:13:08,870 --> 01:13:10,570 So now let's go back. 852 01:13:10,570 --> 01:13:14,290 So this is a structure of the Hilbert space 853 01:13:14,290 --> 01:13:17,590 in some sense for this Rindler. 854 01:13:30,030 --> 01:13:34,430 So now let's look at the Hilbert space for the Minkowski. 855 01:13:37,310 --> 01:13:40,310 So Minkowski is defined-- by Minkowski, I mean, 856 01:13:40,310 --> 01:13:43,180 the theory defined for the whole Minkowski 857 01:13:43,180 --> 01:13:47,900 spacetime is this T. Quantize with respect to this T. 858 01:13:47,900 --> 01:13:53,555 So this would be just standard psi phi x. 859 01:13:56,960 --> 01:14:02,990 Now this x can be anything, and then 860 01:14:02,990 --> 01:14:05,940 we also have a Minkowski Hamiltonian defined 861 01:14:05,940 --> 01:14:11,810 with respect to T, to capital T. OK, 862 01:14:11,810 --> 01:14:14,700 so this is the standard of the Minkowski Hamiltonian. 863 01:14:14,700 --> 01:14:18,070 And then the vacuum, which I denote to be the Minkowski 864 01:14:18,070 --> 01:14:33,140 vacuum is M. And then vacuum functional phi x is just phi 865 01:14:33,140 --> 01:14:42,800 x 0 M. 866 01:14:42,800 --> 01:15:01,870 So now the key observation here is 867 01:15:01,870 --> 01:15:10,940 at phi x, which this phi x is for the full real axis, 868 01:15:10,940 --> 01:15:12,850 for the full horizontal axis. 869 01:15:12,850 --> 01:15:15,200 So this is in some [? contained ?] 870 01:15:15,200 --> 01:15:20,450 phi Lx and the phi Rx. 871 01:15:20,450 --> 01:15:22,460 This goes one into the variable of phi. 872 01:15:22,460 --> 01:15:25,730 You [? variate ?] it to the right patch, 873 01:15:25,730 --> 01:15:29,725 and the value you've added to the left patch. 874 01:15:29,725 --> 01:15:34,440 So the space of phi x, it's the combination 875 01:15:34,440 --> 01:15:38,370 of space of phi L x for the left part and space for the phi 876 01:15:38,370 --> 01:15:39,127 right part. 877 01:15:42,011 --> 01:15:48,936 In particular, this tells us the Minkowski Hilbert space, 878 01:15:48,936 --> 01:15:53,740 which is defined as a functional of this full phi x, 879 01:15:53,740 --> 01:15:58,270 should be the tensor product of the Rindler 880 01:15:58,270 --> 01:16:05,700 to the right tensor product of the Rindler to the left. 881 01:16:05,700 --> 01:16:08,980 Notice that we talk about this right Rindler, 882 01:16:08,980 --> 01:16:13,550 but there's a similar Rindler to the left, 883 01:16:13,550 --> 01:16:18,180 but the structure of the Minkowski space or Hilbert 884 01:16:18,180 --> 01:16:21,610 space, it's the tensor product of the two. 885 01:16:21,610 --> 01:16:28,110 In particular, this ground state wave functional, 886 01:16:28,110 --> 01:16:31,610 you can write it as phi Lx and phi Rx. 887 01:16:35,220 --> 01:16:37,510 So this ground state way functional 888 01:16:37,510 --> 01:16:42,450 should also be understood as the functional for phi L. 889 01:16:42,450 --> 01:16:46,620 OK, so now here is my last key. 890 01:17:06,042 --> 01:17:07,410 And here is last key. 891 01:17:10,270 --> 01:17:15,180 So remember, the ground state wave function 892 01:17:15,180 --> 01:17:23,870 will be obtained by doing path integral on the Euclidean half 893 01:17:23,870 --> 01:17:24,380 plane. 894 01:17:24,380 --> 01:17:26,360 So T is smaller than 0, OK? 895 01:17:30,260 --> 01:17:38,930 So let me just draw the Euclidean space again, X, TE. 896 01:17:38,930 --> 01:17:47,530 So the wave function can be written 897 01:17:47,530 --> 01:18:10,140 as define [? tE ?] x [INAUDIBLE] minus SE Euclidean action. 898 01:18:10,140 --> 01:18:14,420 And the lower half plane of the Euclidean space 899 01:18:14,420 --> 01:18:21,700 with the boundary condition-- OK, 900 01:18:21,700 --> 01:18:26,540 remember from path integral, that's 901 01:18:26,540 --> 01:18:29,910 how you obtain the-- so that means you integrate over 902 01:18:29,910 --> 01:18:33,400 all this region, path integral over all of this region. 903 01:18:33,400 --> 01:18:36,590 Then with boundary condition fixed at here, 904 01:18:36,590 --> 01:18:41,576 we do this path integral, you get the Minkowski vacuum wave 905 01:18:41,576 --> 01:18:42,075 function. 906 01:18:48,420 --> 01:18:51,670 Now here is the key. 907 01:18:51,670 --> 01:18:56,360 So this half space, when I write in this form, 908 01:18:56,360 --> 01:19:01,470 you treat this T as the time, but now let's 909 01:19:01,470 --> 01:19:03,630 consider a different foliation. 910 01:19:03,630 --> 01:19:06,060 I don't have colored chalk here. 911 01:19:06,060 --> 01:19:08,920 I can consider a different foliaton. 912 01:19:08,920 --> 01:19:10,670 It's that foliation in terms of the theta. 913 01:19:14,650 --> 01:19:18,080 Then you just have this foliation, 914 01:19:18,080 --> 01:19:23,320 so for each value of theta, you have a rho. 915 01:19:23,320 --> 01:19:27,810 So integration for minus theta to theta. 916 01:19:27,810 --> 01:19:31,190 So if you think about from this foliaion-- so 917 01:19:31,190 --> 01:19:33,810 think about from this foliation, then this path integral 918 01:19:33,810 --> 01:19:42,430 can be written as a following, D phi theta 919 01:19:42,430 --> 01:19:52,850 rho, which you fix phi theta equal to minus pi rho. 920 01:19:52,850 --> 01:19:57,110 Rho here when theta equal to minus pi becomes x, 921 01:19:57,110 --> 01:19:58,115 you could do phi Lx. 922 01:20:04,860 --> 01:20:09,220 And the phi theta equal to pi-- theta 923 01:20:09,220 --> 01:20:22,790 equal to 0rho equal to phi Rx and exponential minus S, 924 01:20:22,790 --> 01:20:26,000 so these two should be the same. 925 01:20:26,000 --> 01:20:28,370 I just chose the different foliation 926 01:20:28,370 --> 01:20:29,500 to do my path integral. 927 01:20:34,210 --> 01:20:42,130 But now if you think from this point of view, 928 01:20:42,130 --> 01:20:51,100 this a precisely like that because theta 0 and theta 929 01:20:51,100 --> 01:20:54,320 minus pi are just two different times in terms 930 01:20:54,320 --> 01:20:57,015 of this Euclidean Rindler time. 931 01:20:59,900 --> 01:21:06,700 So this you can write it as phi, so let me write one more step. 932 01:21:06,700 --> 01:21:09,560 So this can written as phi2 [INAUDIBLE] 933 01:21:09,560 --> 01:21:14,770 minus iH t2 minus t1 phi1. 934 01:21:21,870 --> 01:21:26,750 So this last step can be now written 935 01:21:26,750 --> 01:21:42,901 as I can think in terms of the time 936 01:21:42,901 --> 01:21:49,140 within the Rindler of time, this can 937 01:21:49,140 --> 01:21:59,140 be written as phi R exponential minus i minus i 938 01:21:59,140 --> 01:22:06,940 pi because we have a Euclidean time H Rindler Hamiltonian phi 939 01:22:06,940 --> 01:22:08,790 L, OK? 940 01:22:15,160 --> 01:22:23,606 So this tells me that Minkowski wave functional 941 01:22:23,606 --> 01:22:26,650 can be written as the following-- can 942 01:22:26,650 --> 01:22:40,310 be written as phi R exponential pi HR phi L. 943 01:22:40,310 --> 01:22:51,790 So now let me expand this in the complete set of states of HR. 944 01:22:51,790 --> 01:22:57,335 Then this is sum of n, so if you expand 945 01:22:57,335 --> 01:23:00,905 in terms of a complete [INAUDIBLE] to HR, 946 01:23:00,905 --> 01:23:02,530 then this is, [? say, ?] wave function. 947 01:23:15,240 --> 01:23:24,440 So with this chi n equal to chi n phi equal to phi. 948 01:23:24,440 --> 01:23:29,040 So remember, this n is defined from this Rindler Hamiltonian. 949 01:23:34,840 --> 01:23:36,270 Now let me erase this. 950 01:23:41,380 --> 01:23:44,580 So I can slightly rewrite this because I don't 951 01:23:44,580 --> 01:23:47,310 like this complex conjugate. 952 01:23:47,310 --> 01:23:48,886 I can slightly rewrite this. 953 01:23:52,550 --> 01:24:03,170 It's thi0 phi x is equal to sum over n [? and then ?] minus pi 954 01:24:03,170 --> 01:24:11,690 En chi n phi R and the chi and tilde 955 01:24:11,690 --> 01:24:25,790 phi L. So chi and tilde define to be phi and star phi L. 956 01:24:25,790 --> 01:24:31,130 So this chi and tilde can be considered-- 957 01:24:31,130 --> 01:24:35,430 just like in ordinary quantum mechanics, 958 01:24:35,430 --> 01:24:42,160 this can be considered to belong to a slightly different Rindler 959 01:24:42,160 --> 01:24:55,770 Hilbert space whose time duration is opposite. 960 01:24:55,770 --> 01:25:01,140 So when you switch the direction of time, 961 01:25:01,140 --> 01:25:05,550 you put a complex conjugate. 962 01:25:05,550 --> 01:25:07,140 So this complex conjugate can also 963 01:25:07,140 --> 01:25:11,910 be thought of just as a wave function in the Hilbert space 964 01:25:11,910 --> 01:25:14,750 with the opposite time direction, 965 01:25:14,750 --> 01:25:19,930 with opposite to the Rindler we started with, 966 01:25:19,930 --> 01:25:25,398 to the right patch Rindler. 967 01:25:35,090 --> 01:25:39,940 So now this form is exactly the form we have seen before. 968 01:25:39,940 --> 01:25:43,175 In this [INAUDIBLE] way to think about thermal state. 969 01:25:46,690 --> 01:25:56,700 So this is exactly that because this tells you 970 01:25:56,700 --> 01:26:00,310 that the Cartesian in the Minkowski vacuum 971 01:26:00,310 --> 01:26:04,490 is the sum of expression minus pi 972 01:26:04,490 --> 01:26:12,550 En with n Rindler of 1 patch [? tensor ?] and Rindler 973 01:26:12,550 --> 01:26:14,962 of the other one. 974 01:26:14,962 --> 01:26:16,420 So this or the other Rindler should 975 01:26:16,420 --> 01:26:19,510 be considered Rindler of the left 976 01:26:19,510 --> 01:26:23,790 except we should quantize the theory with the opposite time 977 01:26:23,790 --> 01:26:30,745 direction, which gives you this Rindler tilde Rindler left. 978 01:26:33,880 --> 01:26:40,020 So this is just precisely the structure we saw before. 979 01:26:40,020 --> 01:26:42,860 In particular, if you are ignorant 980 01:26:42,860 --> 01:26:46,670 about the left Rindler, you can just 981 01:26:46,670 --> 01:27:00,850 trace it out, trace out the left part of this ground state wave 982 01:27:00,850 --> 01:27:03,310 function. 983 01:27:03,310 --> 01:27:10,720 Then, of course, you just get rho Rindler, 984 01:27:10,720 --> 01:27:14,320 the thermal density matrix in the Rindler. 985 01:27:14,320 --> 01:27:22,920 And this is equal to, of course, 1 over Rindler exponential 986 01:27:22,920 --> 01:27:24,350 minus 2pi HR. 987 01:27:28,470 --> 01:27:35,110 And we see that the beta is equal to 2pi, 988 01:27:35,110 --> 01:27:37,120 but this is the beta associated with eta. 989 01:27:41,680 --> 01:27:45,210 As we said, when you change to a different observer, 990 01:27:45,210 --> 01:27:48,360 a different rho, then you have to go to the local proper time. 991 01:27:48,360 --> 01:27:50,240 But this is precisely what we found before. 992 01:27:59,720 --> 01:28:06,810 So now we understand that the thermal nature of this Rindler 993 01:28:06,810 --> 01:28:12,190 observer is just precisely that because this observer cannot 994 01:28:12,190 --> 01:28:17,590 access the physics to the left patch. 995 01:28:17,590 --> 01:28:21,910 But the vacuum state of the Minkowski spacetime 996 01:28:21,910 --> 01:28:26,140 is an entangled state between the left and the right. 997 01:28:26,140 --> 01:28:28,590 And when you trace out the left part, 998 01:28:28,590 --> 01:28:31,575 then you get a similar state on the right part, 999 01:28:31,575 --> 01:28:33,075 so that's how the temperature rises. 1000 01:28:36,561 --> 01:28:38,810 And the same thing can be said about black hole, which 1001 01:28:38,810 --> 01:28:40,640 we'll say a little bit more next lecture. 1002 01:28:40,640 --> 01:28:42,487 And let's stop here.