0 00:00:03,090 --> 00:00:04,920 Okay, sorry. Speaker today's Vulcan villain. 1 00:00:04,980 --> 00:00:08,220 will speak about quality local energy from El que je bond remotes 2 00:00:09,540 --> 00:00:11,639 Thank you very much and 3 00:00:12,750 --> 00:00:36,900 Good morning everybody and thanks for being here. So just a brief introduction brief comments So originally I thought, I thought to speak about some new results in 3D gravity with a positive cosmological constant which is a project with student. He from PGi 4 00:00:37,350 --> 00:00:48,000 However, I decided not to talk about that, but rather give an overview about past results and some perspectives on how to treat 5 00:00:49,020 --> 00:00:53,160 Cause a local observers in Luke quantum gravity. So this will be 6 00:00:54,210 --> 00:00:58,590 Partly past results, but it will also should give some 7 00:00:58,620 --> 00:01:07,140 Future perspectives and also explain how everything relates to other developments in the 8 00:01:15,600 --> 00:01:16,920 Technical program. 9 00:01:27,720 --> 00:01:29,430 Okay, I was just using the wrong. 10 00:01:31,260 --> 00:01:33,360 So let me go to the first 11 00:01:36,240 --> 00:01:37,800 Question. Can you hear me. 12 00:01:41,100 --> 00:01:42,180 Yes. Okay. Sorry. 13 00:01:45,840 --> 00:01:58,410 So first let's go to the basic motivation, Emily, what is why we are why we should be interested in considering quasi local observers general relativity well 14 00:01:59,460 --> 00:02:14,940 In any and that has basically two motivations. So in any realistic experiment in in physics we are considering certain class of coarse grained microscopic observers typically 15 00:02:16,080 --> 00:02:24,780 Which are characterized typically by regions in in face face that are characterized by a few coarse grained observers. 16 00:02:25,560 --> 00:02:43,860 Now, one of the main technical and conceptual difficulties in classical and general relativity is that there is no obvious such course grading on the vast ADM face face of general relativity and this issue has 17 00:02:44,370 --> 00:02:50,880 Manifestations both in classical and quantum general relativity in classical general relativity 18 00:02:52,170 --> 00:02:53,220 Maybe the most 19 00:02:54,780 --> 00:02:55,530 Famous 20 00:02:57,180 --> 00:03:04,200 Manifestation of these issues that are emerging problem in cosmology, which is how to relate 21 00:03:06,480 --> 00:03:10,650 In homogeneous solution of Einstein's equations to 22 00:03:10,950 --> 00:03:12,450 Logic and solution and not 23 00:03:16,230 --> 00:03:19,530 This issue becomes way more violent in 24 00:03:20,520 --> 00:03:22,230 Quantum Gravity, because in quantum 25 00:03:22,230 --> 00:03:30,660 Gravity, we are not dealing with one particular solutions of Einstein field equations, but we are interested with the gravitational 26 00:03:31,020 --> 00:03:44,820 Face face or a gravitational and States face in its entirety and then we need to know how to characterize observers, not just for one particular solution of Einstein's equations, but in 27 00:03:46,260 --> 00:03:55,260 In in regions of faces. So the question is how to do this, how to construct gravitational observers and 28 00:03:56,340 --> 00:04:07,200 The main strategy here is to reverse the logic. So instead of starting out with the vast ADM phase, phase for us, in particular, let boundary conditions. 29 00:04:07,470 --> 00:04:20,970 At infinity. The idea is to construct a gravitational faceplates from a bottom up approach approach where we start out with the gravitational face face in finite domains and then 30 00:04:22,110 --> 00:04:26,280 Learn how to go to ever larger regions. 31 00:04:30,270 --> 00:04:30,960 If we 32 00:04:32,850 --> 00:04:45,840 If we are considering this a certain difficulty will always appear, namely how to separate the gravitational subsystem from its environment. 33 00:04:46,410 --> 00:04:55,800 And this has a bit to do both with technical reasons, as well as physical reasons the technical reason has to do with how we treat 34 00:04:56,220 --> 00:05:07,740 The Hamiltonian time evolution in the covariance face face approach, which was so I won't go through the technical details of how to define the 35 00:05:08,520 --> 00:05:22,830 Kabbalah and face face because this was already well considered in the one of the last it je SS by Richie here I oh I only want to give you a brief. 36 00:05:23,490 --> 00:05:32,100 Introduction to the to the show. So consider a finite dimensional Hamiltonian system, which consists of 37 00:05:32,910 --> 00:05:41,760 The system itself copied to its environment. So the action in the first order formerly formerly small in me 20th formalism. 38 00:05:42,330 --> 00:05:52,170 Consists of the syntactic structure for the system and its environment capital letters correspond to the degrees of freedom of the environment. 39 00:05:52,710 --> 00:06:04,470 And then a Hamiltonian which consists of the Hamiltonian of each individual system cast coupling that determines how the system copies to the environment. 40 00:06:05,040 --> 00:06:23,610 And now the question is, how do we separate the degree. How do we separate the two. How do we consider the Hamiltonian formalism just of the subsystem alone, and then a obvious difficulty appears in the How To Treat the degrees of freedom of the environment. 41 00:06:24,690 --> 00:06:33,330 So if you look at the equations of motion derived from this action. It's clear that this electric structure for the system is just PDQ 42 00:06:33,900 --> 00:06:43,650 That determines the face face of the system. But now if you look at the Hamiltonian equations of motion. It's also clear that the Hamiltonian 43 00:06:44,040 --> 00:06:59,490 Of the subsystem. We consist both of the interaction term plus the Hamiltonian of the that only depends on the faceless variables of the system itself. 44 00:07:00,630 --> 00:07:11,790 But if we then look at the variation of the Hamiltonian in the Cabal Ian face face approach, we will see the days violation of the Hamiltonian 45 00:07:13,020 --> 00:07:27,990 Of Hamilton's of the interoperability condition, simply because now we have to take into account variations of the interaction term with respect to the degrees of freedom of the environment. 46 00:07:29,250 --> 00:07:41,190 The solution to this program, namely two separate to to guarantee integral reality is to realize that something very obvious in a way that the degrees of 47 00:07:41,550 --> 00:07:50,700 Freedom of the environment must be treated as external degrees of freedom that have to be have to be fixed from the onset as 48 00:07:51,630 --> 00:08:16,260 Classical background field or as a see number on face to face. And in that way, the Hamiltonian of the of the subsystem itself will be implicitly time dependent because P, Q, the time dependence of key and Q needs to be determined externally. 49 00:08:17,640 --> 00:08:27,510 Now how does this issue show up in gravity well to determine what we mean by reputation and subsystem. 50 00:08:28,380 --> 00:08:40,860 Two points need to be clarified. So, first of all I want to say that the degrees of freedom, of course, of a gravitation and subsystem. 51 00:08:41,130 --> 00:08:58,110 Correspond to the degrees of freedom that belong to a partial cushy hyper surface. But then the question is, how in space time the boundary of this partial cushy hyper surface extends into a virtue in space time 52 00:08:59,670 --> 00:09:09,270 Now, one possibility is to say that this hyper surface in space time should be a time like cylinder that follows. For instance, a minimal 53 00:09:10,620 --> 00:09:31,950 Surface that the trace of extrinsic curvature vanishes. But doing this it becomes a bit tricky to distinguish the degrees of freedom that the gravitational and radiation that may fall into the system from the degrees of freedom that fallout that leak out into the environment. 54 00:09:33,360 --> 00:09:37,230 And therefore it is kind of natural in gravity to consider. 55 00:09:39,000 --> 00:09:40,560 A situation where the 56 00:09:41,850 --> 00:09:52,710 This virtue is virtue is simply a normal hyper service. So we want to extend the boundary of our partially cushy hyper surface into 57 00:09:54,240 --> 00:10:01,440 Not into another hyper surgeons, the further advantage of this treatment is that then 58 00:10:02,280 --> 00:10:24,930 They say way cleaner characterization of the gravitational flux that may cross the external the null hypothesis. And the idea here in the following is that this that the degrees of freedom of the gravitational radiation that processed in our hyper surface will be treated as an external 59 00:10:26,580 --> 00:10:44,040 Source, namely pack found field or in other words, see number on face face and it will have will play the very same role that in this time model the degrees of freedom of the environment play for the evolution of the 60 00:10:45,210 --> 00:10:51,810 Degrees of freedom of the subsystem. So what I treated as a classical source was the background field. 61 00:10:52,650 --> 00:11:11,550 Is gravitational flux that may cross the hyper surface. And then the question is, what kind of degrees of freedom will survive in the Hamiltonian formalism that we attach to this partial cushy hyper surface and this is in my opinion. 62 00:11:13,440 --> 00:11:26,730 Contribution to an actually an emerging field in loucon gravity. But now, more and more people are interested in this program from various directions, especially here at a meta Institute. 63 00:11:27,900 --> 00:11:36,330 Hello, hello aneela pathetic, but also members of the faculty and many other people in our community and are working on this. 64 00:11:37,530 --> 00:11:53,010 Now, the choice here is to identify the gravitational subsystems with the degrees of freedom with the gravitational degrees of freedom in domains that are bounded by not hyper surfaces. 65 00:11:54,450 --> 00:12:02,010 And how do we then characterize the degrees of freedom that may leak out into the environment and this can be found in 66 00:12:03,150 --> 00:12:13,380 The literature and the construction is as follows. If you take the pullback of the four dimensional space time metric to the boundary. This 67 00:12:14,730 --> 00:12:32,910 To the not hyper surface, then this metric is to generate it has fun. The general direction, which, of course, the knowledge, it generates and this metric can always be diving allies in a, in terms of spinner in terms of diabetic one forms m&m bar. 68 00:12:34,170 --> 00:12:36,960 That that diagonal lies. 69 00:12:38,670 --> 00:12:51,300 The three dimensional degenerate metric on the null hypothesis. Notice that Ma ma ma Kovacs just for Co Baxter's intrinsic to the north boundary, they don't 70 00:12:52,500 --> 00:12:58,980 I don't give a prescription, how to extend them into the interior. The gravitational 71 00:13:00,120 --> 00:13:14,460 The free data. So the reputation radiation is then characterized by engaging equivalence class of the static one forms. So these phases in cotangent space. 72 00:13:16,560 --> 00:13:23,700 And the free data is the H equivalents class under this from former rescaling 73 00:13:25,230 --> 00:13:34,050 To then source the Einstein equations, one has an higher hierarchy of of of equations to be 74 00:13:35,130 --> 00:13:35,880 Specified 75 00:13:37,260 --> 00:13:38,790 In fact, they 76 00:13:40,200 --> 00:13:48,360 In addition to the free data on the novel hyper surface as additional data that needs to be characterized on the cross sections. 77 00:13:49,500 --> 00:13:55,590 And these are conditions on the consumer factor. 78 00:13:56,700 --> 00:14:06,030 In addition to the in and outgoing expansion, the outgoing cheer plus one additional spin coefficient 79 00:14:07,110 --> 00:14:18,540 To integrate the Einstein and create ancient equations along the novel generator additional gauge conditions need to be satisfied, namely 80 00:14:19,350 --> 00:14:34,170 What which basically tell you what this parameter you means geometrically, namely that, for instance, it's an effing power meter long the null hypothesis says to set up. 81 00:14:36,960 --> 00:14:55,950 The Hamiltonian formalism, and I want to speak about the space of history's. In other words, the thing content of the bark last boundary theory in conducting the interior using a first order formalism is quite clear. It's given by the tetra 82 00:14:57,150 --> 00:15:02,040 And self tour connection, out of which we can construct the 83 00:15:03,630 --> 00:15:10,380 Curvature to that. So to see value search for curvature to form. In addition to the urbanski to fall. 84 00:15:11,760 --> 00:15:17,310 But now we also need to do specify the field content at the boundary 85 00:15:18,540 --> 00:15:27,810 And to identify the field content, the boundary. Consider first the pullback of the tetra to the hyper surface. 86 00:15:29,070 --> 00:15:44,730 And then it turns out that one can always fix spin frame. So K and L or normalized spinners on the novel hyper surface such that the pullback of the tetra to the novel hyper surface. 87 00:15:45,210 --> 00:15:57,360 Assumes the following form. So the statement is that I can always choose k and l such that this equation is satisfied. In addition to the 88 00:15:57,840 --> 00:16:17,010 The skill set the boundary. We also want to know the Kabbalah and derivatives and the Cabal Ian derivative of this Venus can be simply taken by considering the pullback of the connection to the boundary. Notice that the pullback in the following 89 00:16:18,240 --> 00:16:23,640 Has a somewhat prefer role in constructing and identified 90 00:16:24,180 --> 00:16:39,900 Identifying the boundary degrees of freedom, simply because there's no other way of projecting fields to the to the null hypothesis on a spatial hyper surface. We have a preferred normal with respect to which we can 91 00:16:40,410 --> 00:17:01,710 Decompose 10s of fields into their spatial and temporal components on an all hyper surface. This is not so. However, they still this yet a natural way of projecting certain kinds of fields, namely those types of fields that define differential forms. 92 00:17:03,240 --> 00:17:14,850 The boundary conditions that fix the treat the flux of gravitation radiation as fixed Ektron field can be now summarized as follows. 93 00:17:15,240 --> 00:17:40,320 As follows. So we take only we consider only those tangent vectors in face face that satisfy or only those few variations that satisfy his condition, which means that we keep fix and gauge equivalents class of this gigantic one phones. In addition to that, I treat 94 00:17:41,670 --> 00:17:50,400 You as a fix coordinate and what is now a variation, how do we write down now an action for which 95 00:17:52,830 --> 00:17:55,680 The variation or problem is elephant. 96 00:17:57,870 --> 00:18:19,860 For this given boundary conditions. Well, this action consists just of the usual bulk action, which is the pitches Kayla expressed in this first order barriers, but then we need an additional boundary term and the only in the boundary term that that satisfies 97 00:18:21,090 --> 00:18:27,300 The our boundary conditions and be then written in the following form. If you go on shell. 98 00:18:28,410 --> 00:18:38,820 So if we impose all equations of motion. This is, this term is just a certain combination of the non affinity and expansion of the null hypothesis. 99 00:18:39,270 --> 00:18:54,030 But at the level of the action. We are at a more primitive level because these action can be evaluated on or field come field configurations of the bark and boundary fields. Notice that 100 00:18:54,930 --> 00:19:15,030 The delivery of the histories, the field content in the interior and the field content at the boundary are detached. In fact, only if we were we divert the action which was quick. Perfect boundary conditions which has been specified on the last slide. 101 00:19:16,590 --> 00:19:28,950 We get the equations of motion in the park. These obviously just Einstein equations, but by very the film content at the boundary with respect to the 102 00:19:29,760 --> 00:19:42,930 Forgiven boundary conditions for we find other boundary field equations. The first of them tells us that the pullback of the self tool to form. 103 00:19:43,980 --> 00:19:58,740 To the not hyper surface can be expressed in terms of the boundary fields alone and this gives the linking between the degrees of freedom at the boundary to those in the bark and in addition to that, we find an additional 104 00:20:00,240 --> 00:20:18,570 Equation of motion at the boundary that gives meaning to the coordinate view, namely that it is and a field parameter along the knowledge generators. So, only if you go on shell in this variation principle, it turns out that you is enough in parameter 105 00:20:20,220 --> 00:20:22,740 From the first variation of the action. 106 00:20:24,270 --> 00:20:39,840 For given boundary conditions, again we find the box press pound weeks, you'd like to structure which consists of have a term in the interior, which just tells you that the service to a connection is canonical conjugate to the 107 00:20:41,310 --> 00:20:48,180 To the search tool to form a plus an additional boundary syntactic structure. If you go to 108 00:20:50,970 --> 00:21:01,950 To those creations that are characterized as you equal constant hyper surface is the boundary, really, the second term survives, which has a very simple structure. 109 00:21:03,330 --> 00:21:05,940 Just, just a question to yes 110 00:21:08,220 --> 00:21:08,730 So the 111 00:21:09,930 --> 00:21:11,040 Space, like in 112 00:21:12,150 --> 00:21:12,960 In your setup. 113 00:21:16,140 --> 00:21:17,520 One form the you 114 00:21:20,190 --> 00:21:20,850 Do you 115 00:21:22,770 --> 00:21:28,170 So do you is one form intrinsic to the boundary. It corresponds 116 00:21:28,470 --> 00:21:30,090 Okay, yeah. 117 00:21:30,330 --> 00:21:32,190 So it does not 118 00:21:32,850 --> 00:21:38,340 I'm not extending it into space. It's a coordinate only intrinsic to the null hypothesis. 119 00:21:41,130 --> 00:21:45,150 To proceed with the Hamiltonian formalism, we want to 120 00:21:45,630 --> 00:21:46,170 Me. 121 00:21:48,630 --> 00:21:55,140 It seemed to me that so far, everything you said is something that you had said before, or is there something new, so far. 122 00:21:58,140 --> 00:22:05,460 In terms of your previous talks about the boundary term synthetic structures and null boundaries and so on. 123 00:22:06,600 --> 00:22:11,370 So I would like to understand what is new. When something new is a new happens 124 00:22:12,000 --> 00:22:23,640 Uh huh. Yes. So what is new is that a more so it is true that many of these things I've said before, but what is new is 125 00:22:25,980 --> 00:22:26,430 That 126 00:22:28,650 --> 00:22:40,800 Different that. What a treat differently now the degrees of freedom at the boundary, namely, did I treat them as so no. So normally we would say, well, 127 00:22:43,050 --> 00:22:50,340 On a partially cushy hyper surface. There are two degrees of freedom to gravitational degrees of freedom point 128 00:22:51,810 --> 00:22:52,230 And 129 00:22:53,460 --> 00:23:04,080 I just want to evolve this Archie cushy hyper surface forward in time. What is now different is that older radiative degrees of freedom, instead of 130 00:23:04,890 --> 00:23:20,340 treating them as dynamic variables on this Archie cushy hyper surface and now swapped or translated into this boundary into the flux long the not hyper surface such that 131 00:23:22,350 --> 00:23:26,940 We're better now treated as a spectrum fields as 132 00:23:28,230 --> 00:23:47,400 Which are not part of the face face that I assigned to a partial cushy hyper surface single it's it's it's only a subtle shift of perspective, I would say, where the radiative degrees of freedom and now put on the 133 00:23:49,590 --> 00:24:06,630 Yeah, the radiative degrees of freedom of gravity and not put as or not or not viewed as the two degrees of freedom point on the scoresheet cushy hyper surface, but they're treated as an external source on the north hyper surface. 134 00:24:07,200 --> 00:24:11,220 It's only a very indeed it's a subtle point but 135 00:24:12,420 --> 00:24:14,670 It's a shift of perspective. 136 00:24:16,380 --> 00:24:21,420 In addition to that, you're right. Many of the things that I'm 137 00:24:23,010 --> 00:24:27,570 That I am telling you about today. I have already 138 00:24:28,800 --> 00:24:36,510 Given in previous talk talks, but there was also a bit perspective you to give a bit of an overview, but 139 00:24:36,570 --> 00:24:44,070 I'm saying that the hook. How can this new prospect to be valued, which is something I don't understand, because you cannot really trade them. I mean, 140 00:24:44,460 --> 00:24:54,450 That radiate your degrees of freedom are also on sigma, as well as on an M. I mean, if I have a gravitational wave in the space time region you're considering then 141 00:24:55,590 --> 00:25:06,750 Also radio degrees of freedom on sigma which, for example, will go away if in fact you are continued your space time region, they will go out to infinity, infinity. So it's not 142 00:25:08,760 --> 00:25:11,130 That I'm fixing too much know 143 00:25:11,430 --> 00:25:13,290 You're taking too much. I'm just saying that 144 00:25:14,580 --> 00:25:24,420 You cannot just say say that the radio. There are no ready to degrees of freedom on sigma you I just traded them all to this null hypothesis, unless you go to school. 145 00:25:25,650 --> 00:25:31,200 But for a finite boundary. There are already degrees of freedom of sigma, as well as on em. 146 00:25:32,280 --> 00:25:33,540 I mean, because 147 00:25:33,600 --> 00:25:45,930 I'm because guy Brazil. Brazil escape from Sigma as well. Right. I mean, it's not just from. I mean, if you just continue, you know, hyper surface a little bit to the future, then whatever the and there'll be nude. 148 00:25:47,100 --> 00:25:49,170 Radio degrees of freedom. If you like going out of the 149 00:25:52,050 --> 00:25:55,080 Surface and those are deciding on sigma 150 00:25:56,820 --> 00:26:07,050 So I understand what you mean what you're saying. But what I mean is that so I'm topological this this 151 00:26:10,170 --> 00:26:19,950 The, the cylinder extends all the way up to up to infinity and soda soda my prescription of 152 00:26:21,300 --> 00:26:25,500 Of the of the flux. So me skin. 153 00:26:27,570 --> 00:26:35,100 And long it or the equivalent class me skipping all along the cylinder, but 154 00:26:35,130 --> 00:26:43,530 What is sigma really call it goes to infinity, there is still sigma left or not if sigma is not left or if in fact 155 00:26:44,010 --> 00:26:54,270 Then I agree with you. But if, in fact, the sigma is left or there's still radius or dirty degrees of freedom on sigma which will go out right i'm not i'm not clear what is happening here. 156 00:27:01,980 --> 00:27:14,310 personas en is really a litecoin have appointed infinity, somehow, yes. Then I mean that point at infinity will be i plus and you are enrolled prescribed and then we understand what is happening. 157 00:27:15,150 --> 00:27:25,590 The new element here is really that you don't want to extend it all the way to infinity, you want to consider a finite portion of space time as you explained in your first part of the transparencies, because your local thing. 158 00:27:26,370 --> 00:27:39,630 And then there are ready to degrees of freedom on sigma, as well as on and I agree with you that we cannot ignore that add degrees of freedom on, but I don't think it is correct to say that you are just transferred all the ready to begin from Sigma to add 159 00:27:47,730 --> 00:27:50,460 I agree that I need for 160 00:27:51,630 --> 00:27:57,540 I would need some more clarification on that point, but I 161 00:28:00,120 --> 00:28:02,460 In my opinion, in my view, it will 162 00:28:05,280 --> 00:28:09,990 At this point, we just have to turn the run the Hamiltonian 163 00:28:11,550 --> 00:28:17,850 homilies meant it will tell us what are better down degrees of freedom left on sigma or not. 164 00:28:20,850 --> 00:28:22,500 And well, we will 165 00:28:25,710 --> 00:28:26,760 I will come back to them. 166 00:28:31,440 --> 00:28:35,100 So what I want to understand now is 167 00:28:38,490 --> 00:28:44,400 How to separate gage degrees of freedom from actual physical degrees of freedom. 168 00:28:45,480 --> 00:28:45,960 And 169 00:28:49,170 --> 00:28:49,650 Now, 170 00:28:53,640 --> 00:28:55,860 And now there are several different 171 00:28:57,090 --> 00:28:59,880 Directions on face face. First of all, 172 00:29:02,550 --> 00:29:07,890 Those different offices in the interior that when you set the boundary 173 00:29:09,540 --> 00:29:12,960 So they they vanished. The entire knowledge hypothesis. 174 00:29:14,250 --> 00:29:26,370 And it's quite easy to see that these are gauge directions. So they are the general direction of this lecture to form. Similarly, we now have also 175 00:29:27,600 --> 00:29:29,970 H trans H transformations 176 00:29:31,140 --> 00:29:38,340 That only appear as pink variations of the boundary degrees of freedom of the boundary fields. 177 00:29:39,090 --> 00:29:50,250 So these are you one rotations of the scenario frame plus corresponding rotations of the diabetic one form. 178 00:29:50,670 --> 00:30:07,080 Similarly, there are simultaneous SL to see gauge transformations of the fields in the interior and the same frame at the boundary and when can easily show that all three of them are on physical gauge directions. 179 00:30:07,890 --> 00:30:15,990 Now, for as far as if you malfeasance are concerned. Now subtlety arises in fact 180 00:30:17,490 --> 00:30:18,960 A tangent vector 181 00:30:20,040 --> 00:30:29,580 To the Caribbean face face associated to this region sigma must satisfy several conditions. Well, first of all, it must 182 00:30:30,810 --> 00:30:39,870 Be a linear I solution of the Einstein's equation in the in the interior must also be a linear right solution of the boundary 183 00:30:41,100 --> 00:30:54,450 Of the boundary degrees of the boundary equations of motion, but in addition to that, it must also satisfy the boundary conditions and it's quite obvious the generic such a few more of his 184 00:30:55,080 --> 00:31:09,540 will not satisfy this condition. In particular, it will not satisfy this condition. If this if this shield. So if there's gravitational radiation that crosses the boundary 185 00:31:10,830 --> 00:31:12,510 How do we then define 186 00:31:13,710 --> 00:31:30,570 How do how can we then Cindy speak about the few more finite if you malfeasance that are not vanishing at the boundary. And here, the idea is for the procedure is to use simply projection to say that 187 00:31:34,410 --> 00:31:36,090 So what we do is that 188 00:31:37,860 --> 00:32:01,950 A generic search lead derivative will or generic such a few more efficient will not satisfy the got engaged condition, but we can always project down by the complexity to form those components of the field variation that lie within face face by saying that we that 189 00:32:03,270 --> 00:32:18,000 The fields variation on the volume face face is defined by its projection with respect to any other tangent vector on the Kevorkian faces. And if we do that, we can see we can 190 00:32:18,840 --> 00:32:32,280 Integrate any different, more isn't that preserves the boundary and we get a finite charge for any such if you more efficient that lies tangential to 191 00:32:33,930 --> 00:32:50,490 To the boundary of the partial pushy hyper surface. Similarly, we can speak about delectation relations, namely those fields variations on the client face space that only we scale the 192 00:32:52,200 --> 00:32:54,060 Spin frame at the boundary 193 00:32:55,080 --> 00:33:02,850 And the corresponding charge for this boundary. The locations is very simple. It's in fact nothing but the area to form. 194 00:33:05,490 --> 00:33:08,220 Their oriented area of the 195 00:33:09,630 --> 00:33:10,470 Cross section. 196 00:33:12,900 --> 00:33:18,600 We can, it is 10 OC instructive to evaluate his charge. So the charge for the final 197 00:33:19,920 --> 00:33:41,820 Tangential diffuse the preserve the boundary of the partially cushy hyper surface on those vector fields that are area preserving and by taking a partial integrated. So what is sending you is essentially a coefficient of the scene connection so 198 00:33:42,990 --> 00:33:44,010 Christopher 199 00:33:45,030 --> 00:33:50,610 Symbol so to say. And if you now use this 200 00:33:51,750 --> 00:33:53,340 Now use those. 201 00:33:54,960 --> 00:34:12,720 Area preserving extra fields that can be written themselves as the differential of some function and then by taking a partial derivative on can express this charge in terms of side to subside to being 202 00:34:14,640 --> 00:34:17,460 A specific component of the 203 00:34:19,410 --> 00:34:24,450 Vile cancel on the you require constant cross sections. 204 00:34:25,620 --> 00:34:26,040 And 205 00:34:29,910 --> 00:34:40,590 You're now free beyond the denominator service when you define side to or you still be entering suggested a three dimensional knowledge or service. It's 206 00:34:41,190 --> 00:34:49,080 Its intrinsic so you don't need to extend the spring frame away from the normal hyper surface simply by this 207 00:34:51,240 --> 00:34:56,700 By this equation. I mean the the speed of the violence. 208 00:34:58,050 --> 00:34:59,250 spinners, or the 209 00:35:00,360 --> 00:35:00,720 The 210 00:35:02,100 --> 00:35:07,440 Irreducible components of divided tensor. They are just contractions of 211 00:35:08,790 --> 00:35:31,650 The vile tensor with certain all vectors and these four to evaluate these contractions. I don't need to know the the extra fields in a nice neighborhood of denial hyper surface. It is completely enough to just know it on the hyper surface itself. So you don't 212 00:35:33,060 --> 00:35:38,670 Have just picked up one transverse vector on the surface and you're not extending. Okay. That's right. 213 00:35:43,980 --> 00:35:46,050 About an ocean of energy. 214 00:35:49,080 --> 00:35:56,820 Namely cause a local Hamiltonian that would generate on that face face that I've prescribe 215 00:35:57,930 --> 00:35:59,790 Translations in you. 216 00:36:01,980 --> 00:36:05,100 The idea is to take our normal 217 00:36:06,180 --> 00:36:23,910 Petra to our normal frame. And now we do have indeed to extended into the interior such that say is a vector field or seeing this up inside of our space time region, such that 218 00:36:24,960 --> 00:36:31,230 If you're restricted to the boundary, it becomes tangential to the non generators 219 00:36:32,370 --> 00:36:48,480 And now we can define the following field variations of the fields in in Korea and the fields. The boundary. The park translations are simply the gate. So to see gauged if you malfeasance with respect to this vector field. 220 00:36:49,770 --> 00:37:01,650 The boundary translation is also a few more fishing, but now weighted with an additional rescaling of the novel frame, actually there's a typo. So this is a 221 00:37:03,270 --> 00:37:05,040 LA. And so the this 222 00:37:07,650 --> 00:37:09,300 The dinar frame is just 223 00:37:10,920 --> 00:37:15,780 Translated by default, more efficient, but also we scale, according to this. 224 00:37:16,860 --> 00:37:23,730 Rule. Similarly, the diabetic one forms just lead right 225 00:37:25,080 --> 00:37:28,590 And now we can ask whether this Hamiltonian is integral over 226 00:37:28,620 --> 00:37:35,220 On the face, but can I ask a question with the last class, per se. So the last class, principally 227 00:37:39,150 --> 00:37:40,920 Yeah, so the top line. 228 00:37:42,120 --> 00:37:46,560 With the spinner La la la bar, a pioneer defined just the boundary 229 00:37:47,490 --> 00:37:48,960 That's right. No. 230 00:37:49,470 --> 00:37:49,980 Smoothly 231 00:37:53,730 --> 00:37:58,320 Okay, so any old extension you're taking. There's no equation that is being satisfied by 232 00:37:59,970 --> 00:38:00,630 By 233 00:38:02,580 --> 00:38:03,270 By the spinner. 234 00:38:05,430 --> 00:38:09,330 You're not using something like with Nicholas it like to extend the spin. No. 235 00:38:09,420 --> 00:38:15,000 No, that's right. So you can use. You can use any extension. 236 00:38:16,260 --> 00:38:37,110 Simply because we know already that any different monotheism that finishes at the boundary is the general direction of is just gauge. So you can just extend it in whatever way you want. But, and this will not 237 00:38:38,730 --> 00:38:50,670 And any any to such extensions will yield the same charge the same Hamiltonian, simply because it can only differ differ by the few more facing 238 00:38:50,910 --> 00:38:51,330 That 239 00:38:51,660 --> 00:38:56,190 vanishes at the boundary, but that is just to gauge transformation. OK. 240 00:38:58,980 --> 00:38:59,550 And then 241 00:39:01,380 --> 00:39:23,580 So then we can then consider them the variation there arbitrary and arbitrary field variation of this Hamiltonian. So, delta is not necessarily a tangent vector on the covenant face face that I've constructed, then you find a syntactic term. So, which corresponds to 242 00:39:25,230 --> 00:39:36,840 To the, to the Hamilton to the Hamilton equations on face face. But then there's also leakage similarly to what I prescribed on the first slides. 243 00:39:37,530 --> 00:39:51,660 That describes flux of degrees of freedom outside that leak out into the environment, but now I said that on the covenant face face in this finite region. 244 00:39:53,340 --> 00:40:03,960 Only on a tangent vector to the covenant face face in this final region must satisfy the linear is calculus foundry. 245 00:40:04,800 --> 00:40:24,060 Field equations, but it must also satisfy the boundary conditions or the boundary conditions, just tell you that this term vanishes. So on the pace pace should have introduced this Hamiltonian is in fact into rubber and it just generate translations along the knowledge generators 246 00:40:25,890 --> 00:40:37,050 In addition, on an isolated horizon, they will the expansion Valley vanishes and this whole expression vanishes, which would mean, which means that 247 00:40:38,430 --> 00:40:41,040 Isolated right just 248 00:40:42,570 --> 00:40:50,820 As or that this was a local energy has to property that it vanishes isolated for us. So it's maybe 249 00:40:54,090 --> 00:40:59,910 It doesn't characterize the closet locally mass of the of the black hole solution. 250 00:41:02,160 --> 00:41:05,040 Now, what does this. How do we then 251 00:41:06,900 --> 00:41:08,340 Make contact to 252 00:41:09,570 --> 00:41:11,310 What we know in Luke on gravity. 253 00:41:12,990 --> 00:41:26,580 And again, I stress this point before. But the point is to stress is that the boundary degrees of freedom that lead 254 00:41:27,150 --> 00:41:34,680 On the boundary of this partially cushy hyper surface. They are just the very spin oil. 255 00:41:35,310 --> 00:41:54,150 foundry the release of the same spin oil variables that appear in the scenario representation of spin networks as introduced by some owners for Charla my pay flow. And generally, it's even and many others. 256 00:41:55,680 --> 00:42:14,280 Now the point is, in order to make context to look quantum gravity, we have to work with the different action, Emily, an action that corresponds that contains the mercy parameter. In fact, in the first order formalism. We have always freedom to add 257 00:42:15,810 --> 00:42:25,260 This expression, this contraction of the remark tends to the launch in the interior of space time without changing the equations of motion. 258 00:42:26,190 --> 00:42:48,360 But if we do that, then we have to also modify now in our prescription of not hyper surfaces because funding foundry term just gets multiplied with the same complex valued coupling constant. Now, indeed, this term does not change the feet. The physics in the interior. It doesn't change. 259 00:42:50,070 --> 00:43:06,930 It does not change the Einstein recreate it does not affect dance and equations, but it has an effect on the film he we at the boundary. In fact, the boundary equations of motion as I studied them or introduced them in the 260 00:43:07,200 --> 00:43:08,430 One of the slides about 261 00:43:08,880 --> 00:43:18,600 An hour difference. So the, the field theory that you have at the boundary is not anymore. The same as in the case of gamma goes to infinity. 262 00:43:19,560 --> 00:43:33,330 And this has done physical effects and these effects show up in the conversation of the area spectrum. In fact, if you look at this complex structure we get, again, the same 263 00:43:34,740 --> 00:43:35,190 The same 264 00:43:36,300 --> 00:43:36,990 Over 265 00:43:38,340 --> 00:43:46,770 Expressions, namely a term interior. The corresponding simplistic structure for the spin all your boundary fields. 266 00:43:48,810 --> 00:43:56,070 And Bora. Now the canonical variables. So if it now only consider the conversation of the 267 00:43:56,580 --> 00:44:08,220 Boundary degrees of freedom at the quarter mentioned to two surfaces, what are, what is the corresponding simplistic structure. What are economically conjugate fields. Well, 268 00:44:08,880 --> 00:44:22,080 If I, if you're working with. If you're working with this boundary conditions with this former boundary conditions for the fix the flux of gravitation and radiation and cost and are hyper surface. 269 00:44:22,560 --> 00:44:36,450 You can always our price this diabetic one form by a fiduciary repressed representatives times conforming factor and the corresponding fiduciary volume element. It's just I am 270 00:44:37,500 --> 00:44:38,310 Not 271 00:44:39,480 --> 00:44:50,940 Economically conjugate variables of the boundary theory or then the knowledge slack or this spinoff la rated by this 272 00:44:52,020 --> 00:45:00,330 Control factor and it's economically conjugate expression which is now a to form on the quarter mentioned to surface. 273 00:45:02,580 --> 00:45:16,080 And for these variables to correspond to to gravity certain conditions need to be satisfied, namely a reality condition that implies that the 274 00:45:16,890 --> 00:45:33,930 That the contraction of k with L is will which then corresponds to the reality of the area to form on the two dimensions process in the spinners pie and l became construct 275 00:45:35,040 --> 00:45:49,530 The simplest generators, which are just simply by Linnaeus of the boundary fields, namely the generator of relations and have you want flag rotations. Now, in the case of 276 00:45:50,580 --> 00:46:01,020 Where we removed the music meter, we saw that the you you want frame rotations are just to gauge transformation, but by adding the 277 00:46:02,940 --> 00:46:25,530 The music parameter that is not any more the case, but the age transformation that we now have to impose is generated by a specific linear combination of the two generators. In fact, so the statement is that came. I know some L is now a generator and this generator 278 00:46:26,790 --> 00:46:41,160 Generates transformations on the Cobra and face to face that lie in the corner of the corresponding simplistic to form. And now we have to impose and if we now. 279 00:46:42,090 --> 00:46:57,930 Want to impose his condition on the quantum lever immediately see that it will yield the sweetness of area of the areas faction loucon gravity and that's simply because whatever 280 00:46:58,590 --> 00:47:11,520 Whatever we do or whatever representation of the quantum theory, we will choose the orbits of L of the you want generator unnecessarily complex unless unnecessarily. 281 00:47:11,970 --> 00:47:31,320 Complex, which means that the spectrum of the operator must be discreet. But if we impose this constraint now at the quantum level, it means that only certain eigenvalues of K will be realized in the quantum theory, Emily, those that are related to the discrete 282 00:47:32,340 --> 00:47:37,710 Eigenvalues or a corresponding equations. In fact, if 283 00:47:39,300 --> 00:47:40,110 If we use 284 00:47:41,430 --> 00:47:45,930 Glue quantum gravity inspired quantization of the boundary 285 00:47:47,190 --> 00:47:56,160 So poorly man conversation of the boundary modes, we will proceed as follows. We would say that a bay Bay functional 286 00:47:56,880 --> 00:48:16,710 Is now a complex valued functional of the configuration variable but using a polymer quantization, the US only we are only considering those functions of the configuration variable that depend on the value of the configuration variable which is 287 00:48:18,600 --> 00:48:24,540 The national flag of the boundary in a number of points in the number of punctures 288 00:48:26,850 --> 00:48:35,640 For any such only merit based function. It's an easy to cause to introduce the corresponding 289 00:48:36,390 --> 00:48:57,150 conjugate momentum, which now which is introduced by smearing the spinner value to form, which I introduced in here over puncture over a neighborhood around this punctures and this operator. Now, x, just as an ordinary derivatives. 290 00:48:58,530 --> 00:49:00,300 The constraint. On the other hand, 291 00:49:02,010 --> 00:49:07,050 Becomes now quite immediate quantifies simply because these operators. 292 00:49:08,250 --> 00:49:08,760 Are 293 00:49:11,490 --> 00:49:20,010 Are easy to diagnose. In fact, just deletions complex with killings and 294 00:49:23,550 --> 00:49:49,740 And this so that i can i can functions of these operators are just homogeneous functions which also carry unitary irreducible representations of the Romans, who, which then in turn means that only some eigenvalues of by imposing the constraint, only some eigenvalues of the irreducible. 295 00:49:53,340 --> 00:50:06,720 The of this i i can values will be realized mainly those bear the continuous labor is related to the district labor by this 296 00:50:07,620 --> 00:50:18,600 Restriction and these are the same type of faith functions that we use in industry. Oil representation of Groupon and gravity, which was introduced by an altogether. 297 00:50:19,350 --> 00:50:27,930 altogether different construction, namely by starting out with the discrete face face and then realizing that the district. 298 00:50:28,530 --> 00:50:36,780 Face face that we assigned to a given spin network graph can be permitted tries in terms of scenario variables. 299 00:50:37,050 --> 00:50:56,130 Here I've done exactly the opposite. I just started out with the classical face face in the presence of boundaries and then realized that the same variables that appear the discreet level also appear for general relativity in the continuum. But as reputation and Edge mode. 300 00:50:57,300 --> 00:51:01,950 On a coder mentioned to boundary of a null hypothesis. 301 00:51:04,110 --> 00:51:07,500 And maybe now we have some time for 302 00:51:11,250 --> 00:51:12,030 Questions. 303 00:51:25,080 --> 00:51:26,010 Okay, so questions. 304 00:51:27,510 --> 00:51:28,620 There's a question. 305 00:51:35,640 --> 00:51:36,690 So the Archibald. 306 00:51:37,680 --> 00:51:39,180 Pizza that okay 307 00:51:39,360 --> 00:51:49,020 So the question that Daniela onset onset is asking is better. I have considered also the algebra of the boundary operators Emily K and L. 308 00:51:49,710 --> 00:52:05,400 And the answer to this question is is relatively simple simply because these these generators can tell they are ready, causing me so they they commute among themselves. 309 00:52:06,660 --> 00:52:09,750 They are just relations and 310 00:52:11,160 --> 00:52:13,530 So these operator commutes 311 00:52:16,530 --> 00:52:20,130 So these two operators just community came on. 312 00:52:22,860 --> 00:52:27,900 So these has Kayla operators, they are not saying that you may be confused by the 313 00:52:30,420 --> 00:52:45,660 By how we treat this constraints in this film foam formula isn't where we think of them as factorial constraints here. These are Kayla's sectors Kayla's under the law school if they could just correspond to the two 314 00:52:46,980 --> 00:52:50,610 To the to cause a means of the loans and so they commute and 315 00:52:55,680 --> 00:52:56,400 More questions. 316 00:52:57,270 --> 00:53:08,580 Yes, question. This is really just a request. Can you elaborate more find the connections to the twister theory in the content modern theory of evolution is based on as we speak. 317 00:53:09,630 --> 00:53:11,190 So there. Thank you. 318 00:53:12,540 --> 00:53:13,590 Or that 319 00:53:14,910 --> 00:53:18,630 So there. If you go back 320 00:53:20,160 --> 00:53:23,370 You may want you may have so 321 00:53:24,540 --> 00:53:32,190 To those of you who have seen similar talks of mine before you may have wondered why slightly changed the boundary term. 322 00:53:33,510 --> 00:53:38,310 And in fact, what you see here is is 323 00:53:39,660 --> 00:53:47,280 The following if you now we organize, you know, you can introduce a new object. 324 00:53:48,660 --> 00:53:51,720 By saying I consider 325 00:53:59,340 --> 00:54:00,180 I consider 326 00:54:01,980 --> 00:54:04,020 We cannot see anything on the blackboard that you're right. 327 00:54:04,680 --> 00:54:06,720 Okay, sorry, sorry, sorry. 328 00:54:08,040 --> 00:54:10,110 So the point is now that 329 00:54:14,490 --> 00:54:15,900 So you see my cursor. 330 00:54:17,910 --> 00:54:18,240 Yes. 331 00:54:19,980 --> 00:54:23,580 So the point now is that if you 332 00:54:25,230 --> 00:54:33,930 reorganize so LM k can be reorganized in into a twister but into a three star that is now. 333 00:54:35,640 --> 00:54:38,760 One form. So, it is a one form value. 334 00:54:40,470 --> 00:54:43,830 keylock spinner. That is 335 00:54:45,240 --> 00:54:49,680 However, homogeneous in embark so this, this is just set 336 00:54:51,270 --> 00:54:54,930 Its upper and lower components are L and 337 00:54:56,130 --> 00:55:00,360 K bar times Mr. And 338 00:55:01,440 --> 00:55:04,410 This is the canonical variable at the boundary 339 00:55:05,850 --> 00:55:11,460 The question now to S as to amplitude and all that. 340 00:55:12,750 --> 00:55:13,440 This 341 00:55:15,210 --> 00:55:15,810 I would 342 00:55:19,980 --> 00:55:21,660 Consider in the in the following way. 343 00:55:23,010 --> 00:55:27,480 If we consider the the boundary action by itself. 344 00:55:29,460 --> 00:55:34,110 Then we can revise it in terms of the set and said bar variables. 345 00:55:35,070 --> 00:55:49,410 But now we have to also impose an additional so if we view them as the prime objects of the theory that we need to impose additional constraints which are which will be added to the boundary action by 346 00:55:49,920 --> 00:56:09,630 Gosh multiplier and this. The resulting action is an action that is fully written or is can can be completely written in terms of set and set alone plus additional constraints. And this is a higher dimensional analogs analog if you wish of 347 00:56:10,710 --> 00:56:18,510 Of a two dimensional action that appears that appeared in a matrix to string theory which can be which which is 348 00:56:23,880 --> 00:56:25,140 which underpins 349 00:56:28,920 --> 00:56:29,580 The 350 00:56:30,900 --> 00:56:33,540 Recent developments in Trista theory. 351 00:56:38,790 --> 00:56:41,610 In fact, the, the idea is to 352 00:56:43,410 --> 00:56:51,330 To now integrate out the bark completely and write down a few key we only for this boundary 353 00:56:52,530 --> 00:56:55,530 For for for these fields as the bound 354 00:56:56,940 --> 00:56:58,650 And the proposal for 355 00:56:58,710 --> 00:56:59,550 Such a 356 00:56:59,610 --> 00:57:02,850 Few theory I developed in 357 00:57:04,830 --> 00:57:10,440 One of my last papers JUST THAT WILL APPEAR SOON. In classical quantum gravity. 358 00:57:10,740 --> 00:57:11,910 The idea is that this 359 00:57:12,090 --> 00:57:26,520 Boundary fields are then coupled to SF to CH and Simon's theory at the boundary and the corresponding field equations correspond to the pullback of the Einstein equations to the north boundary 360 00:57:29,820 --> 00:57:40,470 Well, just to ask question whether this activity. The for Einstein equations continued in from the boundary or just itself to search and between you know the full Einstein equations. 361 00:57:45,810 --> 00:57:58,830 So the underlying action will have a similar structure is here that it's always a part of the self tour process on deserves to our part weighted by context value coupling. 362 00:58:00,840 --> 00:58:06,660 Right you right here, then that the whole action is also the fancy term with the spin to feel 363 00:58:07,950 --> 00:58:08,310 That's right. 364 00:58:11,670 --> 00:58:12,120 Thank you. 365 00:58:15,780 --> 00:58:16,050 Yes. 366 00:58:17,730 --> 00:58:31,350 This is a journey. I have a question about what is the perspective of this use of not surfaces I you try to identify a subsystem. It is finite, like a castle domain or infinite 367 00:58:32,940 --> 00:58:38,670 Indeed, it's the first to the perspective is to 368 00:58:41,160 --> 00:58:53,010 Join the center quantization of finance subsistence subsystems in domains with boundaries that final distance and quantifies 369 00:58:53,700 --> 00:59:07,890 Those and then the limit to infinity would only be obtained within the quantum theory for instance by selecting an appropriate family of career and states that in the limit to infinity. 370 00:59:08,940 --> 00:59:12,750 That is that can then describe the limit to infinity. 371 00:59:13,830 --> 00:59:15,300 At the level of the 372 00:59:17,220 --> 00:59:18,330 Of the quantum theory. 373 00:59:19,650 --> 00:59:20,760 So, is it correct 374 00:59:23,520 --> 00:59:34,410 Yes, I'm sorry to interrupt. Please it correctly interpret your results on the quantization of the area as saying that there's an area gap for the 375 00:59:35,640 --> 00:59:41,760 Core for the area of the corner of a casa de mon, there's a minimal nonzero area for the corner or capsule diamond. 376 00:59:42,570 --> 00:59:57,540 That's right. And the point that I want to make is that this is a Dr. Cobb server, it is a Dr. Cobb server if for for for the face face that we assigned to the to the diamond itself. 377 01:00:01,620 --> 01:00:04,860 It and and this and 378 01:00:05,910 --> 01:00:08,310 That are we have any such cross section. 379 01:00:11,160 --> 01:00:23,550 evolves and with respect to a true Hamiltonian, a true Hamiltonian which is the which is the one that are prescribed, namely the expansion times the 380 01:00:25,860 --> 01:00:28,650 Times the area to form integrated over the 381 01:00:30,120 --> 01:00:32,160 Over you equals constant crossing 382 01:00:34,620 --> 01:00:34,920 The key. 383 01:00:37,170 --> 01:00:38,910 Point is that by dividing 384 01:00:40,710 --> 01:00:44,130 By restricting attention to a subsystem. 385 01:00:46,110 --> 01:00:51,780 Degrees of Freedom become observer which read cage degrees of freedom before 386 01:00:54,930 --> 01:01:04,110 But I mean this about again. So I'm troubled about your Hamiltonian because you emphasize again Hamiltonian evolving and so on so forth area. 387 01:01:04,650 --> 01:01:15,300 I mean, normally you know when people write down cause a local content is the right put some criteria and this criteria typically that well you know on the horizon. You should get the 388 01:01:16,710 --> 01:01:30,270 Hawking mass at infinity should get the bond, the mass media mass and your bought your Hamiltonian blows up at infinity is zero on the, on the, the black hole horizon. 389 01:01:31,320 --> 01:01:34,860 And so I sort of, I don't see 390 01:01:36,090 --> 01:01:40,560 It seems to miss Mrs key physics. And so why are you not troubled by this 391 01:01:49,170 --> 01:01:55,260 So in the case of so it is true that this Hamiltonian blows up as we go to infinity. 392 01:01:56,520 --> 01:01:56,880 But 393 01:01:58,260 --> 01:01:58,980 Again, 394 01:02:01,530 --> 01:02:15,570 The it blows up in in a universal manner, namely as one as are linearly in in in our being the distance or the size of the 395 01:02:16,500 --> 01:02:35,610 as as as we go to infinity. But now the point is that in in this face face description or is it tough part of phase, phase. So, it is a. It's not a. It's not a classical. It's not a parameter, but it's a 396 01:02:37,320 --> 01:02:42,690 It's a Q number. It's part of phase space. And so to take the limit to 397 01:02:44,130 --> 01:02:53,610 To take the limit to infinity, and then remove the divergence. You have to do that. I would argue at the level of the 398 01:02:54,750 --> 01:02:59,940 Of the quantum theory. So not prior to quantization, but only 399 01:03:03,840 --> 01:03:06,720 After within the quantum theory. 400 01:03:08,190 --> 01:03:33,090 Now that may seem strange because we think of our justice coordinate, but our determine sort of radius determine determines the, the size of our sphere as it as this sphere or the boundary of the partially cushy hyper surface goes to infinity. But in this cause a local 401 01:03:34,500 --> 01:03:40,380 Description The radius is related to the area of the 402 01:03:41,580 --> 01:03:49,260 Of the boundary, but the area is itself an observer, as we know very well in Luke on gravity, where we 403 01:03:50,280 --> 01:03:51,360 Where we 404 01:03:53,250 --> 01:04:08,580 Identify very, very identified it with area operator. It has a discrete spectrum. So I want to make sense of of this limit at the level of the quantum theory and that requires to 405 01:04:09,810 --> 01:04:10,350 To use 406 01:04:11,370 --> 01:04:12,840 To use this Hamiltonian 407 01:04:15,810 --> 01:04:16,530 Question. 408 01:04:17,190 --> 01:04:20,370 Is does your Hamiltonian doesn't have a guy. 409 01:04:23,430 --> 01:04:24,030 Um, 410 01:04:26,940 --> 01:04:28,710 I don't know at the moment. 411 01:04:29,220 --> 01:04:33,600 Is, is I think that's something to worry about. That's a different way of stating advice. 412 01:04:36,600 --> 01:04:36,960 Well, 413 01:04:43,830 --> 01:04:46,980 Maybe I related question which I find someone 414 01:04:50,580 --> 01:05:02,220 The find equally important is better. The resulting better the ground state of his Hamiltonian is degenerate. So whether there is just one back home. 415 01:05:02,880 --> 01:05:13,770 In the case of synthetically flat boundary conditions then corresponding to the enclosed space or whether days and unexpected far from the generous. 416 01:05:15,510 --> 01:05:16,200 Yes, that's 417 01:05:19,410 --> 01:05:28,830 Why can't you look at the question, classically, I mean you got a Hamiltonian framework, I can look at it. Classically, and you will say that the classical the energies just infinite way everything 418 01:05:29,490 --> 01:05:38,070 And therefore the, I mean, there's no question on the ground state. All the question is just not well defined. So somehow, you're saying that 419 01:05:38,520 --> 01:05:47,190 You know we got all these beautiful results in general relativity about finiteness autonomy. Find that is but positivity of energy and so on and and 420 01:05:47,880 --> 01:06:03,120 Drastically. That is what tells us that there's a calm state and some are, you're saying that that is about you, relevant, we should just think about it in the quantum theory and not is wrong to think about it in the classical theory at all. So I'm just gonna confuse about the viewpoint. 421 01:06:08,280 --> 01:06:10,830 Um no I wouldn't say that it's 422 01:06:17,580 --> 01:06:17,970 So did 423 01:06:19,080 --> 01:06:22,260 That divergence has a universal 424 01:06:25,470 --> 01:06:28,860 Universally in terms of our so in 425 01:06:30,330 --> 01:06:30,900 Emphasize 426 01:06:31,950 --> 01:06:33,420 It's not magical quantity 427 01:06:36,930 --> 01:06:39,990 No, I want to emphasize the disease, an observer. 428 01:06:44,310 --> 01:06:45,990 Always not always not 429 01:06:48,360 --> 01:06:51,450 Always have is an observer quantity 430 01:06:51,810 --> 01:06:57,060 Right so so you know what SALLY DOESN'T MEAN very much, right, because, I mean, if it was something that would 431 01:06:57,810 --> 01:07:07,110 Normally when you subtract sort. For example, you know, something blows up on some tracks or a quantity in flat space or something or something, which doesn't, it's not opposite right and that's why us 432 01:07:08,580 --> 01:07:14,160 Now you're okay. I mean, I think this is the long discussion. So I don't think I want to continue 433 01:07:15,450 --> 01:07:19,590 Because I just taking everybody's time so we should talk about Bobby. Thank you. 434 01:07:22,470 --> 01:07:24,330 Any more questions. Yes. 435 01:07:24,600 --> 01:07:26,460 You hear me, yes. 436 01:07:27,330 --> 01:07:42,810 About the adding the music parameter. This part here. I was wondering you then move to the quantum part but at the classical level do the generators in charges corresponding to the few more features that depend on gamma now in your construction or they don't. 437 01:07:44,130 --> 01:07:45,720 So the charges of 438 01:07:48,660 --> 01:07:49,770 Tangential deeper, more 439 01:07:50,970 --> 01:07:56,880 Assertive charges of the deployment business to not depend on gamma. That's because 440 01:07:58,050 --> 01:07:58,740 The 441 01:07:59,820 --> 01:08:06,180 corresponding term that would would introduce a gamma dependence turns out to be 442 01:08:07,980 --> 01:08:19,290 Complete derivative that is integrated over to serve, which then to surface, which then vanishes by Stokes your perfect. Okay. That's very clear. 443 01:08:28,080 --> 01:08:28,890 Any more questions. 444 01:08:32,310 --> 01:08:33,030 Speaker again.