0 00:00:02,910 --> 00:00:07,310 Jorge Pullin: Okay. So our speaker today is that we'll speak about the Lq. Of Black holes. 1 00:00:08,980 --> 00:00:17,020 Beatriz Elizaga Navascués: Thank you. Perfect. So yeah, I was. Thank you for suggesting me for giving this talk. 2 00:00:17,160 --> 00:00:18,460 Beatriz Elizaga Navascués: I was 3 00:00:18,560 --> 00:00:26,790 Beatriz Elizaga Navascués: toll to, maybe talk about Lucont and cosmology of Black Horse in the condom gravity. Well for those of you that don't know me. 4 00:00:27,190 --> 00:00:28,390 Beatriz Elizaga Navascués: I 5 00:00:28,400 --> 00:00:35,050 Beatriz Elizaga Navascués: most of my research career. I've been working in the quantum cosmology aspect of cosmological space time. 6 00:00:35,160 --> 00:00:49,130 Beatriz Elizaga Navascués: so I have only been working in black holes over the last year. So by no means. I i'm a certified or experienced, you know, expert to give a review talk. So this is not going to be a You talk. I'm going to talk about some recent progress 7 00:00:49,770 --> 00:01:01,100 Beatriz Elizaga Navascués: in the genuine quantum features of black holes when they are Qantas following look on them cause more of the techniques, and in order to do that, I will provide a very 8 00:01:01,380 --> 00:01:07,080 Beatriz Elizaga Navascués: to the issue, and then only Indian talk about the recent progress. 9 00:01:07,330 --> 00:01:08,260 Beatriz Elizaga Navascués: So 10 00:01:08,280 --> 00:01:12,070 Beatriz Elizaga Navascués: in any case, i'll start with some background on this topic. 11 00:01:12,190 --> 00:01:28,730 Beatriz Elizaga Navascués: I think we would all be in this audience that black holes nowadays, especially as we are leaving the rise of gravitational wave. Astronomy are very promising observational windows to eventually test the quantum nature of gravity 12 00:01:29,460 --> 00:01:32,930 Beatriz Elizaga Navascués: in the context of lu quantum gravity and quantum cosmology. 13 00:01:32,960 --> 00:01:43,630 Beatriz Elizaga Navascués: These issues have been investigated already for almost 20 years. There's a long history of works, and many, many people have collaborated to the 14 00:01:43,660 --> 00:01:59,370 Beatriz Elizaga Navascués: investigation of quantum aspects of black holes within these frameworks. I'm sorry I apologize if some contributors not here. I Well, it's such a big an active field of research. 15 00:01:59,830 --> 00:02:01,120 Beatriz Elizaga Navascués: So 16 00:02:01,600 --> 00:02:19,870 Beatriz Elizaga Navascués: eventually there seems to be yet not a full consensus of of which one should be a satisfied, satisfactory, yet a simple enough description of quantum black holes in the contest of root quantum gravity, and the simple in the sense that we have some 17 00:02:19,870 --> 00:02:25,480 Beatriz Elizaga Navascués: control, mathematical or robust control over it, and we can make predictions satisfactory predictions out of it. 18 00:02:25,960 --> 00:02:34,410 Beatriz Elizaga Navascués: What the most or many of the investigations do in order to attain this objective 19 00:02:34,440 --> 00:02:39,910 Beatriz Elizaga Navascués: is to focus on a spherical symmetry. So the simplest type of Black Hole Space Times. 20 00:02:39,990 --> 00:02:57,220 Beatriz Elizaga Navascués: and among the all the contributors, many of them have actually used Luke quantum cosmology techniques to describe the interior of the Black Hole. Since the in general relativity this interior can be described as an an isotropic cosmology, so look on the 21 00:02:57,360 --> 00:03:00,690 Beatriz Elizaga Navascués: cosmology techniques can be applied in that. 22 00:03:02,880 --> 00:03:14,820 Beatriz Elizaga Navascués: In fact, in the beginning, those Atqc Oriented studies we're focusing on the pure quantum aspect of in the interior black Hole space time. But 23 00:03:14,900 --> 00:03:28,840 Beatriz Elizaga Navascués: over the years most of the studies recent studies have been studying what are called the effective models which, beyond the the context of computers and isotropic cosmology couple to a massless 24 00:03:28,900 --> 00:03:30,140 Beatriz Elizaga Navascués: our 25 00:03:30,160 --> 00:03:31,820 Beatriz Elizaga Navascués: just 26 00:03:31,970 --> 00:03:33,060 Beatriz Elizaga Navascués: they. 27 00:03:33,320 --> 00:03:51,370 Beatriz Elizaga Navascués: If you want a geometry, the the dynamics of a geometry that is described by a Hamiltonian. That is a regularization following group techniques of the classical one. So that's what people now what is usually called effective models with the the most simple cosmology. 28 00:03:51,930 --> 00:03:53,490 Beatriz Elizaga Navascués: So 29 00:03:53,680 --> 00:04:03,980 Beatriz Elizaga Navascués: there are some quantization or ambiguities in the choice of that regularization of that affected Hamiltonian. and the 30 00:04:04,080 --> 00:04:12,870 Beatriz Elizaga Navascués: even though all of the effective models resulting from that has succeeded in resolving the singularity to some the 31 00:04:12,940 --> 00:04:16,190 Beatriz Elizaga Navascués: 3 or another, and most of them can 32 00:04:16,329 --> 00:04:23,540 Beatriz Elizaga Navascués: mit ctl, and be pictured as describing a transition from a kind of black hole. Roughly speaking, to a Whitehall. 33 00:04:24,790 --> 00:04:34,010 Beatriz Elizaga Navascués: there are many caveats that one can raise. There are some and physical features that appear in this model, such as a big 34 00:04:34,190 --> 00:04:40,430 Beatriz Elizaga Navascués: quantum geometry effects appearing at low curvature, depending on depending on if you do, and so on and so forth. 35 00:04:40,700 --> 00:04:57,930 Beatriz Elizaga Navascués: So Within this context a special heightenite is deserved by 2,019 extension of crystal space, and proposed by as the car. On my side, it has it, at least in what concerns the Black Hole interior it, Your. The the resulting in effective space time. 36 00:04:59,130 --> 00:05:01,020 Beatriz Elizaga Navascués: attend to 37 00:05:01,670 --> 00:05:18,960 Beatriz Elizaga Navascués: provide many satisfactory physical properties, all together, from which I will just remark that the carvature invariance attain a Ha! Are bounded from above, and that bound which lies at the transition surface between a black hole and a white whole kind of space. Time 38 00:05:19,070 --> 00:05:25,140 Beatriz Elizaga Navascués: is independent of the mass of the back home, and only diverges when the 39 00:05:25,340 --> 00:05:37,140 Beatriz Elizaga Navascués: area cap in the condom gravity goes to 0. It's a very strong result. That parallel is the situation found in cost knowledge. This is quite satisfactory. It has many other features, but I will not go into that. 40 00:05:38,380 --> 00:05:49,220 Beatriz Elizaga Navascués: So, as I said, even though in the beginning many approaches were looking at the conversation, at the quantum aspect of the interior of the Black Hole recently. Not. 41 00:05:49,700 --> 00:06:06,720 Beatriz Elizaga Navascués: and mostly it's been effective models, including this extension, of course, kind of space, and and even the original authors of this model stated as a limitation of it, that it, the effective dynamics was not really derived from a quantum theory, but rather as 42 00:06:06,740 --> 00:06:14,430 Beatriz Elizaga Navascués: as a as a hypothesis, so it would be interesting to see what is the quantum quantization, the actual quantum properties of 43 00:06:14,440 --> 00:06:17,060 Beatriz Elizaga Navascués: of this approach 44 00:06:18,220 --> 00:06:21,660 Beatriz Elizaga Navascués: in particular, what are the general properties of the 45 00:06:21,850 --> 00:06:29,000 Beatriz Elizaga Navascués: And if this kind of effectively regimes can be derived actually from a quantum theory, actually there's 46 00:06:29,160 --> 00:06:35,250 Beatriz Elizaga Navascués: at at least a couple of groups of people working in this top if nowadays. so it's becoming more more active. 47 00:06:35,740 --> 00:06:47,520 Beatriz Elizaga Navascués: So, as I said, I want to give a rather pedacogical introduction to the topic, so I will start with the basics black hole in theory or in general relativity, and what i'm calling effective at Qc. Motors. 48 00:06:48,980 --> 00:06:54,250 Beatriz Elizaga Navascués: So they did. Your of a black hole in general relativity admits of a foliation 49 00:06:54,380 --> 00:07:13,040 Beatriz Elizaga Navascués: where the metric is of a Kantowski Sax type, namely, this homogeneous and an isotropic cosmology with the spacial hyper services that have the topology of an interval in the real line. This is just to avoiding interactive divergences and they 2 sphere. This is the Metric on Tuesday. 50 00:07:13,780 --> 00:07:27,570 Beatriz Elizaga Navascués: So the symmetry that this metric can be written exclusively in terms of the labs, and 2 point like degrees of freedom that qualify all of the point particle, degrees of freedom that qualify the information 51 00:07:27,610 --> 00:07:44,520 Beatriz Elizaga Navascués: about the density. On the other hand, after it, when when I poses the symmetry reactions of this system. they ask. The caravano connection is fixed in terms of this, to point particle variables again. B and C. 52 00:07:44,920 --> 00:07:59,120 Beatriz Elizaga Navascués: The bees with the piece form a canonical pairs. and I will just say as a as a physical interpretation. so that you remember that PC. Actually 53 00:07:59,460 --> 00:08:05,510 Beatriz Elizaga Navascués: measures physical areas of the spheres of symmetry of the interior of the Black Hole. 54 00:08:07,630 --> 00:08:17,670 Beatriz Elizaga Navascués: The classical Hamiltonian takes this form. namely, it with a wise choice of laps function that absorbs this pre-factor. Here 55 00:08:17,740 --> 00:08:30,030 Beatriz Elizaga Navascués: it is the difference between 2 Hamiltonians that I will call parcel Hamiltonians, because they only depend on the B sector of Face space, and the C sector of face face that are or often 56 00:08:30,030 --> 00:08:42,820 Beatriz Elizaga Navascués: refer to as the radial sector for B, because the Pb. Is the only appears in the radial part of the metric, and this angular sector see, because it completely fixes the 57 00:08:43,909 --> 00:08:53,330 Beatriz Elizaga Navascués: now on solutions. The Hamiltonian must be 0. So the 2 parcel Hamiltonians must coincide, and they are, in fact, a constant of motion 58 00:08:53,390 --> 00:08:58,970 Beatriz Elizaga Navascués: which is proportional to the Atmos. Once we extend the solution to the exterior. 59 00:08:59,420 --> 00:09:04,630 Beatriz Elizaga Navascués: so remember both partial Hamiltonians are equal to the mass of the black Hole. 60 00:09:06,410 --> 00:09:20,030 Beatriz Elizaga Navascués: Right? So something that is, despite all the rest of competition and migration, something something that seems to be a universal, at least up to date in the quantum cosmology type of representation. And it's actually this from the full theory. 61 00:09:22,320 --> 00:09:36,290 Beatriz Elizaga Navascués: is the fact that they there is a lack of strong continuity in the representation of the canonical algebra of a variable, so the connection variables are what are not well defined as so pretty. 62 00:09:36,570 --> 00:09:41,020 Beatriz Elizaga Navascués: so what one usually do, at least in in traditional approaches. To 63 00:09:41,190 --> 00:09:56,030 Beatriz Elizaga Navascués: look onto cosmology is to use the symmetries on the, to simplify the form of the Hamiltonian as much as possible, and then rewrite the curvature of the as the Caribbean better connection in terms of an allonomy secret that encloses a minimal area. 64 00:09:56,400 --> 00:09:59,990 Beatriz Elizaga Navascués: keeping the technical details. 1 1 65 00:10:00,020 --> 00:10:11,490 Beatriz Elizaga Navascués: performs this procedure that is called the regularization, usually of the Hamiltonian. One gets this results on. Now I can up to it what wise choice of Laps function. 66 00:10:11,690 --> 00:10:21,020 Beatriz Elizaga Navascués: All that one needs to do is to replace in the each of the partial Hamiltonian the connection by a sign of the connection 67 00:10:21,280 --> 00:10:36,790 Beatriz Elizaga Navascués: times the over a parameter that. for each of the 2 sectors represents the coordinate length of the minimum idea plackets along which we construct them all on a mystery with it in closing the minimum area. 68 00:10:38,500 --> 00:10:45,080 Beatriz Elizaga Navascués: So I will still call this parcel Hamiltonian, because for 69 00:10:45,470 --> 00:10:48,970 Beatriz Elizaga Navascués: many interesting choices of this Delta 70 00:10:49,000 --> 00:11:02,950 Beatriz Elizaga Navascués: parameters this mean, coordinate minimum links. They are actually also dynamically decoupled. So we only so. All we commute with with Oc. And they generate the 71 00:11:03,030 --> 00:11:08,490 Beatriz Elizaga Navascués: the the dynamics on each of the all sectors and the the the resulting. 72 00:11:08,540 --> 00:11:21,930 Beatriz Elizaga Navascués: A partial Hamiltonians are also a constant of motions in the in the classical theory. If one takes very seriously so, and I and I will simply call them also V and Oc. You know all the following: now, the rest of the talk. 73 00:11:23,270 --> 00:11:24,830 Beatriz Elizaga Navascués: Okay, so 74 00:11:24,990 --> 00:11:38,150 Beatriz Elizaga Navascués: erez agmoni. The quantization ambiguity that I was referring to in the introduction regarding the different choices of effective models for the description of the interior Black Hole actually is related to how 150 75 00:11:38,200 --> 00:11:55,980 Beatriz Elizaga Navascués: defines the Delta parameters of quantum origin in terms of the minimum area gap of. So there is a lot of freedom here. The what I, what i'm refer to as effective models is the 76 00:11:56,080 --> 00:12:05,640 Beatriz Elizaga Navascués: geometry that one obtains when one takes these regularized Hamiltonian seriously and compute the evolution generated by it. 77 00:12:05,740 --> 00:12:10,210 Beatriz Elizaga Navascués: and then studies the the result in the the properties of the results. In geometry 78 00:12:10,260 --> 00:12:29,680 Beatriz Elizaga Navascués: all solutions. This is a again, a constraint system in the effective model. So on solutions all B is equal to the mass, and it for wise choices of this Delta parameters. This mass, what I will call Hamiltonian Mass. In analogy with the gr case is a constant of motion of the system. 79 00:12:30,130 --> 00:12:42,070 Beatriz Elizaga Navascués: So there's been many proposals to fix this parameters first, that they were chosen, as, for instance, then they were chosen as functions on face space that 80 00:12:42,140 --> 00:12:48,710 Beatriz Elizaga Navascués: are not constants of motion and parallel. The analysis is done in standard quantum cosmology, and the 81 00:12:49,700 --> 00:13:09,520 Beatriz Elizaga Navascués: afterwards they start. People started to concede that that these delta parameters could be defined in terms of constants of motion. In that case the the model nice kind of nice properties, in fact, in this context is where this quantum configuration of crus call black holes. Where? I asked. The current medium in 2,019 82 00:13:09,570 --> 00:13:15,760 Beatriz Elizaga Navascués: is a model that I will be calling. Aos came into into consideration 83 00:13:16,840 --> 00:13:20,380 Beatriz Elizaga Navascués: in the in this model the 84 00:13:20,430 --> 00:13:23,950 Beatriz Elizaga Navascués: delta B, and that the c parameters are actually fixed 85 00:13:24,300 --> 00:13:33,860 Beatriz Elizaga Navascués: in terms on each solution. And I will explain what this means in very briefly. So how all that thought on each solution. 86 00:13:33,860 --> 00:13:46,900 Beatriz Elizaga Navascués: These delta parameters are fixed in terms of the constant of motion that i'm. Calling the Hamiltonian Mass. So on the coincident value of of Ob an Oc. And they are fixed in this very precise way. 87 00:13:47,120 --> 00:13:49,480 Beatriz Elizaga Navascués: And this is actually 88 00:13:50,110 --> 00:14:00,480 Beatriz Elizaga Navascués: for this. Okay. So this is here. There is a total. This means that this they are fixed in this way for large black hole masses, which is the regime 89 00:14:00,500 --> 00:14:08,220 Beatriz Elizaga Navascués: in which this model is supposed to be valid. They don't assume the validity of this model for all possible 90 00:14:08,410 --> 00:14:10,810 Beatriz Elizaga Navascués: rather massive, much so 91 00:14:11,800 --> 00:14:25,440 Beatriz Elizaga Navascués: the form of this, the asymptotic form, if you want of these parameters is behind the nice properties that were found in this model in particular, the but upper bounds for the curvature invariance that do not depend on the mass of the black hole at dominant. 92 00:14:25,470 --> 00:14:26,620 Beatriz Elizaga Navascués: and, in fact. 93 00:14:26,780 --> 00:14:41,620 Beatriz Elizaga Navascués: those when the mass goes to infinity, the upper bounds remain finite, and they only diver when Delta goes to 0. So in that sense it's when the quantum corrections go to 0, when we will recover the classical singularity. 94 00:14:42,150 --> 00:14:47,660 Beatriz Elizaga Navascués: Okay. So this is fixing the it seems that Fix in Black Hole models fixing the 95 00:14:47,680 --> 00:14:48,760 Beatriz Elizaga Navascués: the 96 00:14:48,850 --> 00:14:58,340 Beatriz Elizaga Navascués: regularization parameters in terms of constant promotion, seems to be a good idea in the sense of physical properties that result in physical properties of them. Effective mode. 97 00:14:59,410 --> 00:15:03,050 Beatriz Elizaga Navascués: Okay, so. and naturally 98 00:15:03,090 --> 00:15:06,450 Beatriz Elizaga Navascués: step next step in the analysis 99 00:15:06,480 --> 00:15:09,960 Beatriz Elizaga Navascués: of these models would be to consider 100 00:15:11,230 --> 00:15:19,100 Beatriz Elizaga Navascués: what is actually the quantum theory from which they arise. So so I said. Even the authors of the aeros model considered the limitation of it. 101 00:15:19,200 --> 00:15:37,680 Beatriz Elizaga Navascués: The fact that they were assuming that effective dynamics. So one would want to see what is the quantum counterpart of this this effective model? What is the full underlying quantum theory from which it may emerge, and for that one meets a consistent Hamiltonian formulation of the of the model. And this 102 00:15:37,680 --> 00:15:40,320 Beatriz Elizaga Navascués: has been a question of debate 103 00:15:40,530 --> 00:15:49,680 Beatriz Elizaga Navascués: pretty much since the model was first proposed, and this is related to the fact that the deltas, the quantum parameters, are fixed in terms of the mass of the Black Hole 104 00:15:49,780 --> 00:15:51,420 Beatriz Elizaga Navascués: on solutions. 105 00:15:52,960 --> 00:15:56,200 Beatriz Elizaga Navascués: So I come back to that sort that I told you to hold. 106 00:15:58,230 --> 00:16:01,570 Beatriz Elizaga Navascués: Okay, so the effective dynamics 107 00:16:01,670 --> 00:16:11,000 Beatriz Elizaga Navascués: is the As I said, the Ios effective dynamics is generated by this Hamiltonian. The difference of the 2 108 00:16:11,220 --> 00:16:23,560 Beatriz Elizaga Navascués: I I will call this from now on the eighth effective of of the a. Us. Model. And if you want to derive the the equations that rule the model from this Hamiltonian, you need to assume 109 00:16:23,580 --> 00:16:27,590 Beatriz Elizaga Navascués: that, or do you need to treat the Delta parameters as pure Const. 110 00:16:28,090 --> 00:16:47,280 Beatriz Elizaga Navascués: Then on each solution we have the constant of motion. M. Equal to the 2 partial Hamiltonians function, and on its solution the more than the nice space and geometry in the interior is obtained. If you declare that the Delta are functions of that constant of motion 111 00:16:47,290 --> 00:16:50,880 Beatriz Elizaga Navascués: for large black holes that the ones that I was showing previously. 112 00:16:51,020 --> 00:16:51,660 Okay. 113 00:16:52,280 --> 00:16:53,760 Beatriz Elizaga Navascués: Now. 114 00:16:54,600 --> 00:17:00,260 Beatriz Elizaga Navascués: what many people realized as soon as this model was suggested was that 115 00:17:00,470 --> 00:17:01,110 Beatriz Elizaga Navascués: wait. 116 00:17:01,230 --> 00:17:19,740 Beatriz Elizaga Navascués: M. Is actually a function of phase space right in general, because it is a constant of motion. So it's a not constant, as you change from one solution to another. So then, why are we treating the parameters in the derivation of the equations here? Why do we have to treat the parameters as pure, constant. 117 00:17:19,740 --> 00:17:22,260 Beatriz Elizaga Navascués: This was was some concern that was raised. 118 00:17:22,540 --> 00:17:35,980 Beatriz Elizaga Navascués: and, in fact, if you don't treat them as constants, and you introduce the dependence of the Delta, on the on the face these face space functions which are actually indistinguishable from each other. So in general, you can introduce a dependence of both of them. 119 00:17:36,200 --> 00:17:46,120 Beatriz Elizaga Navascués: then the equations of motion, that result do not describe the Aos model. That is, it is a different geometry so 120 00:17:46,410 --> 00:17:52,040 Beatriz Elizaga Navascués: well. In fact, even in the original works, even with the you know 121 00:17:52,090 --> 00:17:58,070 Beatriz Elizaga Navascués: paper that proposed this model. They 122 00:17:59,260 --> 00:18:02,270 Beatriz Elizaga Navascués: argued that the equations of motion 123 00:18:02,320 --> 00:18:13,320 Beatriz Elizaga Navascués: from which they, their nice black hole interior, the space time emerges can be derived from a Hamiltonian, that is, an extension of the regularized one in general relativity 124 00:18:13,460 --> 00:18:22,810 Beatriz Elizaga Navascués: on a on an extended phase space that includes the parameters delta as a free space variables canonical values together with moment. 125 00:18:24,020 --> 00:18:39,800 Beatriz Elizaga Navascués: So what the one does is one considers an extension of phase space, and then a a an extension of the Hamiltonian, in which 2 new constraints come into play. These Lambda s a Lagrange multipliers, and these 2 constraints all that they do is they actually 126 00:18:39,800 --> 00:18:47,550 Beatriz Elizaga Navascués: fix on shell? What is the relation between the Delta parameters and the mass of the black on each solution? 127 00:18:47,760 --> 00:19:07,320 Beatriz Elizaga Navascués: Originally this this functions of the of the partial Hamiltonians. We're just like in the case of Delta, be just a function of phobia, and in the case of Delta, C, just a function of all, c. But in general, because they are in distinguishable. The most genetic situation in which you could recover that model would be to consider 128 00:19:07,420 --> 00:19:11,480 Beatriz Elizaga Navascués: that that the constraints are functions we both are part of. 129 00:19:12,800 --> 00:19:30,510 Beatriz Elizaga Navascués: Of course, if you want to actually derive the Aos space time from this extended phase, space formulation. You need to require that in the large and limit these functions behave in this way. 130 00:19:30,510 --> 00:19:36,060 Beatriz Elizaga Navascués: But other than that, this is quite general. So you could actually consider a wide variety of models here. 131 00:19:37,490 --> 00:19:38,280 Beatriz Elizaga Navascués: So 132 00:19:38,340 --> 00:19:47,680 Beatriz Elizaga Navascués: now a natural question to ask is, what happens with if I fix the the gates associated with this constraints and videos the phase space. 133 00:19:47,780 --> 00:20:03,490 Beatriz Elizaga Navascués: So this can be done in a consistent way by in a dynamically stable, and actually it with a good gauge fixing conditions. If this land up this plugins multiplier set equal to 0. 134 00:20:04,040 --> 00:20:04,850 Beatriz Elizaga Navascués: So 135 00:20:05,160 --> 00:20:16,550 Beatriz Elizaga Navascués: let's one can check what happens. And skipping the details. The result, which was, it seems it was unnoticed until the actual computation was performing. Detail 136 00:20:16,570 --> 00:20:31,160 Beatriz Elizaga Navascués: is that the reduced phase Space is not symptomorphic to the general relativity. One so the kantoski's access to knowledge, they connection and try it. Algebra is not the Drl there in fact, you have non-banishing 137 00:20:31,230 --> 00:20:41,880 Beatriz Elizaga Navascués: brackets, or we'd use for question brackets between different components of the trial, and different components on the connection, and the B and C sectors also get mixed. 138 00:20:42,430 --> 00:20:50,900 Beatriz Elizaga Navascués: So, in fact, this the fact that the reduce space is not equivalent to the general relativistic. One is 139 00:20:50,980 --> 00:21:00,610 Beatriz Elizaga Navascués: a a actually permits or allows one to solve the tension between the fact that the one was treating the parameters as constants when deriving the equations of motion. 140 00:21:00,640 --> 00:21:03,210 Beatriz Elizaga Navascués: even though they are a constant of motion. 141 00:21:03,610 --> 00:21:15,160 Beatriz Elizaga Navascués: because if one considers that these deltas are actually functions from phase space on this Hamiltonian and uses this modified as you have to derive the equations of motion, the Aos model for us. 142 00:21:15,280 --> 00:21:21,070 Beatriz Elizaga Navascués: So well. Okay, so we want to work with just this Hamiltonian. 143 00:21:21,130 --> 00:21:32,830 Beatriz Elizaga Navascués: We need a very complicated. This is the lesson here connection and algebra. But actually, we know how to do Luke quantum cosmology, and, in fact, the quantum gravity we have the triad and connection i'll give that to 144 00:21:32,900 --> 00:21:36,740 Beatriz Elizaga Navascués: is the one from general relativity. 145 00:21:36,870 --> 00:21:39,930 Beatriz Elizaga Navascués: So, as a quantization strategy. 146 00:21:40,110 --> 00:21:42,810 Beatriz Elizaga Navascués: It seems quite clean. 147 00:21:43,000 --> 00:21:48,680 Beatriz Elizaga Navascués: and for the constructive purposes to consider they 148 00:21:48,740 --> 00:21:56,610 Beatriz Elizaga Navascués: a full extended Hamiltonian on an extended phase space, and then use the Qc. Tools to find a quantum representation of the constraints. And Microsoft. 149 00:21:57,190 --> 00:22:00,080 simone: Hi! Can I ask a question 150 00:22:00,390 --> 00:22:05,430 simone: on this nice picture? Maybe I grew in the joke. Sorry! 151 00:22:06,230 --> 00:22:07,610 Beatriz Elizaga Navascués: Oh. 152 00:22:08,160 --> 00:22:23,660 simone: so the the paper by but in door, for they decided, and then they started this discussion that you so nicely summarized, also proposed an alternative that doesn't require to do any of these. They just showed that by polymerizing a different type of 153 00:22:23,890 --> 00:22:33,650 simone: a fundamental classical algebra one could use deltas that Don't depend on the atmos 154 00:22:33,890 --> 00:22:38,090 simone: and get the singularity free model. 155 00:22:38,630 --> 00:22:47,010 simone: This is the you also residing. So there is this alternative construction which is just based on a different choice of. Can you comment on that, or 156 00:22:47,890 --> 00:22:58,830 Beatriz Elizaga Navascués: Well, Yes, but they at least I, as far as I know, there is the the bounce that is not a symmetric as in the area as modern. 157 00:22:59,420 --> 00:23:09,890 Beatriz Elizaga Navascués: I am not sure about the curvature invariance whether they they've displayed the same nice properties in the area as modeling as in the sense that the upper bounds do not depend on the mass. 158 00:23:10,310 --> 00:23:25,670 Beatriz Elizaga Navascués: But the the reason why I would I was interested in studying a it. A Hamiltonian framework that leads to an Aos type of model was the combination of organized features that the 159 00:23:26,080 --> 00:23:31,690 Beatriz Elizaga Navascués: that model had in the in the in its effective formulation, so to say 160 00:23:31,960 --> 00:23:33,900 Beatriz Elizaga Navascués: so. If another 161 00:23:35,270 --> 00:23:36,210 Beatriz Elizaga Navascués: or one 162 00:23:36,310 --> 00:23:41,630 Beatriz Elizaga Navascués: motivated from similar arguments, would collect all of those properties 163 00:23:42,160 --> 00:23:51,490 Beatriz Elizaga Navascués: which I think, at least from the perspective of the of not having a mass amplification, or or after the bounce. This one does not 164 00:23:51,830 --> 00:23:54,700 Beatriz Elizaga Navascués: I? Well, I 165 00:23:55,220 --> 00:24:03,920 Beatriz Elizaga Navascués: I I think it would be equally interesting to to investigate that. But i'm the motivation here was to see what are the quantum? 166 00:24:04,140 --> 00:24:11,330 Beatriz Elizaga Navascués: What is the quantum theory? If there is a quantum theory underline and model that, at least in the interior seems to be quite nice from the physical. 167 00:24:12,050 --> 00:24:13,220 simone: Okay, Thank you. 168 00:24:13,460 --> 00:24:14,050 Beatriz Elizaga Navascués: Yeah. 169 00:24:15,360 --> 00:24:24,360 Beatriz Elizaga Navascués: Okay. So again, I I continue at the pedagogical philosophy. We want our strategy in this talk. 170 00:24:25,690 --> 00:24:36,190 Beatriz Elizaga Navascués: I i'm gonna talk about the kinematics, the quantization. So, starting from this standard phase space, i'm going to talk about the kinematics of the conversation. 171 00:24:36,290 --> 00:24:38,110 Beatriz Elizaga Navascués: This is, I mean. 172 00:24:38,150 --> 00:24:49,360 Beatriz Elizaga Navascués: this can be attributed to many, many people. This is a standard Lqc techniques, so I will just summarize and try to give it a so 173 00:24:49,990 --> 00:24:57,860 Beatriz Elizaga Navascués: in a look, quantum gravity, or look on the cosmology, we try to capture the degrees of freedom of the connection and all economies. 174 00:24:57,960 --> 00:25:05,130 Beatriz Elizaga Navascués: In this highly symmetric case it is enough to consider all on a miss on the polar and radial directions, to qualify all of the 175 00:25:05,160 --> 00:25:10,180 Beatriz Elizaga Navascués: information of the connection that comes through a complex exponential of it. 176 00:25:10,920 --> 00:25:17,290 Beatriz Elizaga Navascués: Like, on the other hand, flexes of the density is that are fully fixed by 177 00:25:17,300 --> 00:25:20,050 Beatriz Elizaga Navascués: the TV and PC. 178 00:25:20,620 --> 00:25:29,410 Beatriz Elizaga Navascués: Point particle degrees of freedom. So the oversimplified version of the Alonomy Flax Algebra. In this case takes this form. 179 00:25:30,190 --> 00:25:39,460 Beatriz Elizaga Navascués: It's a very simple, and up to constant factors. It is actually 2 copies of the algebra that we have in a homogeneous, and is a tropical Qc. 180 00:25:39,510 --> 00:25:45,440 Beatriz Elizaga Navascués: So we can apply directly the techniques that we know very well from 181 00:25:45,450 --> 00:25:49,650 Beatriz Elizaga Navascués: the quantization of homogeneous and isotropic model. Just to 182 00:25:50,010 --> 00:25:59,130 Beatriz Elizaga Navascués: so the kinematical Hilbert space for the geometry in this case is then the space of a square, some of all functions on R. 2 on the real plane. 183 00:25:59,340 --> 00:26:01,860 simone: Sorry, can I ask you a question about this. 184 00:26:01,930 --> 00:26:05,540 Beatriz Elizaga Navascués: Yes. so here the flax is commute right. 185 00:26:06,170 --> 00:26:16,610 Beatriz Elizaga Navascués: if I understand correctly, whereas in the full t, or usually they don't, is there some intuitive reason as to why this is. 186 00:26:17,400 --> 00:26:19,160 Beatriz Elizaga Navascués: I think. 187 00:26:19,550 --> 00:26:26,510 simone: but a are in these. So maybe 188 00:26:26,800 --> 00:26:38,380 Beatriz Elizaga Navascués: i'm i'm i'm integrating over squares that are adopted to the directions of symmetry. 189 00:26:39,850 --> 00:26:40,550 Beatriz Elizaga Navascués: No. 190 00:26:41,000 --> 00:26:42,170 simone: So 191 00:26:42,230 --> 00:26:53,280 Beatriz Elizaga Navascués: cause I thought maybe one. It's a radial direction. Well, yeah, well, yeah, they like No, but it it's so. Do you consider 192 00:26:54,590 --> 00:26:56,060 Beatriz Elizaga Navascués: the coordinate 193 00:26:56,090 --> 00:27:00,880 Beatriz Elizaga Navascués: squares or rectangles in each of the of the 194 00:27:00,990 --> 00:27:07,750 Beatriz Elizaga Navascués: coordinate directions. So yeah, they are. So I mean, they intersect if you extend them. Yes, they 195 00:27:07,870 --> 00:27:11,250 simone: right. So usually when the intersect we expect 196 00:27:12,260 --> 00:27:16,020 simone: no competitivity. But here you don't see it. I see. 197 00:27:16,030 --> 00:27:20,050 Beatriz Elizaga Navascués: Okay, because of this, because of how much it I I believe. 198 00:27:20,560 --> 00:27:21,300 simone: Okay. 199 00:27:21,330 --> 00:27:26,630 simone: can I make a remark about that. This is this is John. So if remember. 200 00:27:26,720 --> 00:27:40,130 FAU (Florida): some like from the full theory, the only reason why you have non-community of the flux is is yeah, I mean if you, if you look at the continuum in theory, the fluxes commute with each other, the non-communativity just comes about in order to make sure that the 201 00:27:41,040 --> 00:27:42,980 FAU (Florida): what is it this 202 00:27:43,050 --> 00:27:47,250 FAU (Florida): anyways? But this is one of these anomalies in quantum gravity. So 203 00:27:48,320 --> 00:27:59,640 FAU (Florida): so. But but here I so here it's not so surprising that the flux is a commute with each other. But it but you're right. It's a very interesting question about how those 2 can be consistent with each other, because the point is to represent what's happening in the full theory. 204 00:28:03,870 --> 00:28:06,330 Beatriz Elizaga Navascués: Yeah, thanks. I I mean. 205 00:28:07,760 --> 00:28:11,050 Beatriz Elizaga Navascués: I would have to think a bit more about it that for the moment I would. 206 00:28:12,230 --> 00:28:13,380 Beatriz Elizaga Navascués: My 207 00:28:13,750 --> 00:28:19,580 Beatriz Elizaga Navascués: intuition would say that this is because of the homogeneity of the model that I will think a bit more about that. 208 00:28:20,980 --> 00:28:22,880 Beatriz Elizaga Navascués: Thank you. 209 00:28:24,420 --> 00:28:29,400 Beatriz Elizaga Navascués: Okay. So yeah, they commit. So in the quantum theory they would also commit 210 00:28:29,550 --> 00:28:30,840 Beatriz Elizaga Navascués: the 2 sectors. 211 00:28:31,080 --> 00:28:36,660 Beatriz Elizaga Navascués: So i'm going to call the basis with this. It's an orthonormal basis 212 00:28:38,280 --> 00:28:45,730 Beatriz Elizaga Navascués: of the of the kinematical feedback space. These Muse labeled the basis, and they will each of them belong to the to the real line. 213 00:28:46,390 --> 00:28:50,530 Beatriz Elizaga Navascués: So the if you're 214 00:28:50,890 --> 00:28:53,820 Beatriz Elizaga Navascués: they are again of the 215 00:28:54,050 --> 00:29:05,920 Beatriz Elizaga Navascués: So yeah period operators that I, by multiplication and the allonomy operators is the translate this an act by translation. 216 00:29:07,110 --> 00:29:08,880 Beatriz Elizaga Navascués: Okay. So 217 00:29:10,060 --> 00:29:12,470 Beatriz Elizaga Navascués: for the 218 00:29:13,960 --> 00:29:24,760 Beatriz Elizaga Navascués: the one to paramet, the parameters of one to Mor. Again, this tells us that we were used to regularize the Hamiltonian. They are related to the Black Hole mass in the end on shell. 219 00:29:24,900 --> 00:29:26,580 Beatriz Elizaga Navascués: so we will. 220 00:29:27,080 --> 00:29:29,070 Beatriz Elizaga Navascués: It is the simplest 221 00:29:29,090 --> 00:29:31,870 Beatriz Elizaga Navascués: thing that one can think of was a as 222 00:29:32,000 --> 00:29:37,940 Beatriz Elizaga Navascués: treating them as a matter in the standard of quantum cosmology, to adopt the continuous representation for them. 223 00:29:38,000 --> 00:29:44,610 Beatriz Elizaga Navascués: and then the total Hilbert space is the tensor product of these 2 continuous. 224 00:29:44,660 --> 00:29:53,710 Beatriz Elizaga Navascués: so did the kilograms of square integral functions associated with these 2 continuous representations, and then the kinematical. 225 00:29:54,970 --> 00:30:06,990 Beatriz Elizaga Navascués: Now we are doing quantum mechanical quantum cosmologies. So in the end we will want to search for physical states that are annihilated by the constraint. So we need a a quantum representation for this constraints. 226 00:30:07,010 --> 00:30:25,960 Beatriz Elizaga Navascués: So we'll start with Oc. Because it's the simplest one. I'm not going to go into the details of of a factor or the in ambiguities. This can be this could lead to a whole new talk, and I think results can be generally enough, regardless of this factor or the in the so 227 00:30:26,610 --> 00:30:31,080 Beatriz Elizaga Navascués: the angular part is quite simple. It's just the up to constants the same. 228 00:30:31,100 --> 00:30:37,470 Beatriz Elizaga Navascués: The po quoted did sign like the quoting between the sign and the and the Delta parameter times, the the try it. 229 00:30:37,770 --> 00:30:42,700 Beatriz Elizaga Navascués: We can write the sign in terms of follow elementary. and 230 00:30:43,560 --> 00:30:56,670 Beatriz Elizaga Navascués: let me disarrow encodes all the factor, ordinary ambiguities. Let's say that we choose one, and we get an operator of on a dense subset of of the kinematical. 231 00:30:57,300 --> 00:31:09,970 Beatriz Elizaga Navascués: So now, if we change the for it's fixed pair of delta parameters, we make this simple relabeling of the basis. 232 00:31:10,620 --> 00:31:18,330 Beatriz Elizaga Navascués: then the action of the is Omega squared. This basic operator for the Oc. 233 00:31:19,480 --> 00:31:30,700 Beatriz Elizaga Navascués: Partial Hamiltonian has the square of it has the same exactly the same maxim as that team, at least simplified versions of homogeneous and isotropic cosmology. 234 00:31:31,640 --> 00:31:48,120 Beatriz Elizaga Navascués: When one replaces the volume there with the PC. Here. So it's, replacing volume with areas in this case. So this operator is very well known. This is very good. This is a very good property, because this operator has very nice. 235 00:31:48,270 --> 00:32:01,920 Beatriz Elizaga Navascués: It's very well known. It has very nice properties on the Hilbert space, because it's a self at joint, and it allows one to separate the non separable part. The associated with the sector. You see the sector, c. 236 00:32:01,990 --> 00:32:04,240 Beatriz Elizaga Navascués: In super selection sectors 237 00:32:04,770 --> 00:32:08,550 Beatriz Elizaga Navascués: that have support on. 238 00:32:09,550 --> 00:32:19,390 Beatriz Elizaga Navascués: They got India spectra properties. It has an absolutely continuous spectrum equal to that real line, and it is independent of this parameter. So actually the closure of 239 00:32:19,410 --> 00:32:29,400 Beatriz Elizaga Navascués: the super selection sector, we can identify it with the closure of the nonseparable inverse space for B 240 00:32:29,740 --> 00:32:41,190 Beatriz Elizaga Navascués: and an L. 2 space for that. It's just the written. It's just the the spectral like it's a constructed using the spectral decomposition of the omega. 241 00:32:41,290 --> 00:32:44,300 Beatriz Elizaga Navascués: So because recall that. But actually, of all 242 00:32:44,480 --> 00:32:55,020 Beatriz Elizaga Navascués: we call that all on shell in the classical theory is proportional to the mass. So here I will just call the elements of the spectrum of how that's in little end. 243 00:32:55,780 --> 00:32:58,140 Beatriz Elizaga Navascués: Now the regular part is more complicated. 244 00:32:59,960 --> 00:33:06,820 Beatriz Elizaga Navascués: I will not give a specific prop like suggestion for its quantum representation. 245 00:33:06,840 --> 00:33:10,850 Beatriz Elizaga Navascués: I will just require some minimal properties on it. 246 00:33:11,480 --> 00:33:25,490 Beatriz Elizaga Navascués: The first one is that we build it exclusively in terms of this Omega B, which is completely analogous to the Omega c. Previously, and the triad rescaled triad operator. Okay 247 00:33:26,020 --> 00:33:28,000 Beatriz Elizaga Navascués: it. 248 00:33:28,070 --> 00:33:32,250 Beatriz Elizaga Navascués: I require that the presets the separate selection sectors of the Omega, P. 249 00:33:32,280 --> 00:33:39,010 Beatriz Elizaga Navascués: Which allow us to separate the non-separable part of the piece sector of the kinematical in that space. 250 00:33:39,050 --> 00:33:41,240 Beatriz Elizaga Navascués: in order to study the dynamics. 251 00:33:41,460 --> 00:33:49,700 Beatriz Elizaga Navascués: and I would also require that it is essentially self at your end. It has no singular spectrum, so the continuous part is absolutely continuous. 252 00:33:49,890 --> 00:33:54,510 Beatriz Elizaga Navascués: and the point spectrum is the script. I will also assume that 253 00:33:54,580 --> 00:33:56,240 Beatriz Elizaga Navascués: so basically. 254 00:33:56,260 --> 00:34:08,940 Beatriz Elizaga Navascués: depending on the properties in in a priori, it may be that it has some part of the spectrum. The spectrum will be real. It may be that some part of it is a continuous, some part of it is discrete, or one of them can be empty. 255 00:34:09,260 --> 00:34:14,580 Beatriz Elizaga Navascués: so it can be purely discrete or purely. I will not go and tell me many more. 256 00:34:16,040 --> 00:34:24,219 Beatriz Elizaga Navascués: Okay. So because all B and O. C are by that Ob. My assumption we see by construction are essentially self at joint. 257 00:34:25,870 --> 00:34:35,370 Beatriz Elizaga Navascués: They and the rest of the constraints. I just relate the delta parameters with a function of these 2 parcel Hamiltonians, we can use the spectral theorem to define 258 00:34:35,409 --> 00:34:37,280 Beatriz Elizaga Navascués: those additional constraints. 259 00:34:37,530 --> 00:34:43,780 Beatriz Elizaga Navascués: So now we have a representation of the constraints. We're ready to look for physical state, and this is the part of 260 00:34:43,929 --> 00:34:45,810 Beatriz Elizaga Navascués: recent produce 261 00:34:47,969 --> 00:34:56,610 Beatriz Elizaga Navascués: that I would like to talk about this at least, as I said, a couple of different groups of people working on this or that have recently worked on this. 262 00:34:57,490 --> 00:34:58,910 Beatriz Elizaga Navascués: So 263 00:35:00,530 --> 00:35:01,470 Beatriz Elizaga Navascués: again 264 00:35:01,560 --> 00:35:20,230 Beatriz Elizaga Navascués: we follow the Dirac approach. So we actually want to look for states that are annihilated by the constraints. In fact, in general, these States will not leave the habit that will not be the kinematical universe space. We have to look them in bigger spaces that can be understood as dual 265 00:35:20,230 --> 00:35:29,580 Beatriz Elizaga Navascués: algebraic spaces to a dense subset of the kinematical inverse space, because this is where we have defined the constraint operators. 266 00:35:29,840 --> 00:35:35,880 Beatriz Elizaga Navascués: So we enlarge the habit of possible physical states because they may not leave in the 267 00:35:35,930 --> 00:35:37,840 Beatriz Elizaga Navascués: in the kinematical one. 268 00:35:38,410 --> 00:35:46,810 Beatriz Elizaga Navascués: So this is the when we talk about the Hamiltonian constraint. This is simply the constraint equation. The 2 parts of the Hamiltonians must be equal to each other. 269 00:35:48,380 --> 00:35:49,970 Beatriz Elizaga Navascués: This means 270 00:35:50,150 --> 00:35:53,180 Beatriz Elizaga Navascués: right, that ob 271 00:35:53,610 --> 00:36:04,180 Beatriz Elizaga Navascués: acting on the wave function of the physical states must be equal to the mass for each value. Real value of the mass multiplying the wave function of the physical state. 272 00:36:04,310 --> 00:36:06,150 Beatriz Elizaga Navascués: which means 273 00:36:07,550 --> 00:36:16,550 Beatriz Elizaga Navascués: immediately under a caveat. that I will comment in a moment that the wave function must be 0 274 00:36:16,970 --> 00:36:20,580 Beatriz Elizaga Navascués: when it supports it. Doesn't belong 275 00:36:20,730 --> 00:36:21,970 Beatriz Elizaga Navascués: 2. 276 00:36:22,020 --> 00:36:27,520 Beatriz Elizaga Navascués: The spectrum of the OP operator that I call it, may depend on on the Delta 277 00:36:27,570 --> 00:36:34,180 Beatriz Elizaga Navascués: as a difference with the Oc. Which doesn't depend on. Okay. So 278 00:36:35,200 --> 00:36:42,350 Beatriz Elizaga Navascués: what the the caveat that I was talking about is that for this second line to follow from the first one. 279 00:36:43,220 --> 00:36:55,230 Beatriz Elizaga Navascués: We are implicitly assuming that we are looking for physical estates in the algebraic tool off. Essentially, or roughly speaking. they dance, subset, spanned 280 00:36:55,320 --> 00:36:59,470 Beatriz Elizaga Navascués: by the eigenfunctions and eigenstates of Ob. 281 00:37:00,910 --> 00:37:07,180 Beatriz Elizaga Navascués: If we take that. so then subspace the this need not follow. 282 00:37:07,930 --> 00:37:19,950 Beatriz Elizaga Navascués: And this is actually quite a quite an important message that will be clear at the end of the talk, I think I hope, and it is. It has actually has physical consequences. So. 283 00:37:20,010 --> 00:37:25,760 Beatriz Elizaga Navascués: But as long as we look for physical state, which is what we usually do in look onto cosmology. In fact. 284 00:37:26,630 --> 00:37:31,030 Beatriz Elizaga Navascués: using the analogy between the operators that we have there, we look on that to one. 285 00:37:31,670 --> 00:37:35,390 Beatriz Elizaga Navascués: So additionally, we have 286 00:37:35,530 --> 00:37:46,070 Beatriz Elizaga Navascués: the constraints that relate the the for the regularization parameters in terms of the partial Hamiltonians. In position of this one is straightforward, because the 287 00:37:46,110 --> 00:37:49,910 Beatriz Elizaga Navascués: the spectrum of Ob and Oc. Do not depend 288 00:37:50,020 --> 00:37:55,500 Beatriz Elizaga Navascués: on a delta, c. But the spectrum of for being general will depend on delta B. 289 00:37:55,620 --> 00:37:57,120 Beatriz Elizaga Navascués: So 290 00:37:58,620 --> 00:38:06,150 Beatriz Elizaga Navascués: it is not hard to realize, by imposing this constraint that the physical States will be in one-to-one correspondence 291 00:38:06,320 --> 00:38:10,780 Beatriz Elizaga Navascués: with wave functions of the mass that have support 292 00:38:12,080 --> 00:38:19,290 Beatriz Elizaga Navascués: on at this implicit relation set. So what this means is that 293 00:38:19,580 --> 00:38:23,660 Beatriz Elizaga Navascués: take a a value of them in the real line, construct 294 00:38:23,850 --> 00:38:28,190 Beatriz Elizaga Navascués: the Delta, B, compute the Delta B associated with it with this formula. 295 00:38:28,600 --> 00:38:42,870 Beatriz Elizaga Navascués: and then ask whether that in belongs to the spectrum of for B associated with that value of delta? If the answer is yes, then that M. Is physical mass. If the answer is no, then that time is not a physical mass, so it's not in the in the support 296 00:38:42,980 --> 00:38:54,480 Beatriz Elizaga Navascués: of of the the wave functions. Finally, a following this procedure a natural way, because we have relied all the time on the on the 297 00:38:54,610 --> 00:39:07,600 Beatriz Elizaga Navascués: spectrum Ofobi, and after all way of of completing the the Hilbert, the physical fever spaces to introduce a an inner product on this wave functions coming from the spectrum as you of the open it. 298 00:39:07,900 --> 00:39:13,630 Beatriz Elizaga Navascués: Okay. So this is a rather formal characterization of the physical States. 299 00:39:17,570 --> 00:39:25,320 Beatriz Elizaga Navascués: and once this has been done, following again a specific procedure looking for a States in a specific, as if I do one. 300 00:39:26,450 --> 00:39:30,380 Beatriz Elizaga Navascués: I will talk about possibilities that 301 00:39:31,010 --> 00:39:36,920 Beatriz Elizaga Navascués: for the actual form of this physical state, so possibilities for the spectrum of. 302 00:39:37,180 --> 00:39:45,690 Beatriz Elizaga Navascués: And then I will talk about an alternative possibility for looking for in order to look for physical spaces that relies on a different 303 00:39:47,090 --> 00:39:57,380 Beatriz Elizaga Navascués: algebraic dual for looking at so looking for physical states in a different habitat and the physical consequences of 304 00:39:57,420 --> 00:40:09,260 Beatriz Elizaga Navascués: so the the the physical situation that we are going to found in each of the 2 cases. So, looking at for physical system. 2 different and algebraic tools is that it is as drastic as 305 00:40:09,660 --> 00:40:15,570 Beatriz Elizaga Navascués: having a discrete set of mass with gaps to a continued set of masses without caps. 306 00:40:15,590 --> 00:40:29,770 Beatriz Elizaga Navascués: So this is actually something that I think is even relevant for the full theory. Where do we look for solutions To the Hamiltonian constraint? This is an important question, and I think this is. It's very remarkable that this has actual 307 00:40:30,100 --> 00:40:44,570 Beatriz Elizaga Navascués: physical consequences, even in symmetry, reduces more reduced models such as this one. So I think we can learn a lot from this. I'm not going to state any strong results here. I just want to contemplate that possibilities. Okay. 308 00:40:44,640 --> 00:40:51,960 Beatriz Elizaga Navascués: So now, let's look about. This is the look on on on the situation where we can get the discrete match for a black hole. 309 00:40:52,480 --> 00:40:58,150 Beatriz Elizaga Navascués: So in imagine that we can so look at this this regularize 310 00:40:58,550 --> 00:41:07,120 Beatriz Elizaga Navascués: form of the OP. The person coming to me. If you take it, it's square. It is not unreasonable that there will be many representations such that they square will take this form. 311 00:41:08,360 --> 00:41:15,700 Beatriz Elizaga Navascués: But now this means that on any State on the domain of the OP. Squared 312 00:41:15,870 --> 00:41:26,780 Beatriz Elizaga Navascués: the square of this operator, this relation, that there will be that it would be always greater than equal; that the expectation, essentially the expectation value of 313 00:41:26,790 --> 00:41:28,200 Beatriz Elizaga Navascués: Pd. Squared. 314 00:41:28,210 --> 00:41:37,380 Beatriz Elizaga Navascués: and this is unbounded from above, and has a discrete spectrum. So this actually standard theorems in functional analysis tells us that the spectrum of the Ob squared. 315 00:41:37,520 --> 00:41:49,000 Beatriz Elizaga Navascués: at least for delta, be different from 0. So from Delta B equal to C or this relation doesn't tell us anything. But from Delta we we tell when Delta B is different from 0. The spectrum is going to be discrete. 316 00:41:49,010 --> 00:41:55,050 Beatriz Elizaga Navascués: so the spectrum of the square root, if we define the square root of to assign it, will we also be discrete 317 00:41:55,350 --> 00:41:56,120 Beatriz Elizaga Navascués: so 318 00:41:56,260 --> 00:42:05,900 Beatriz Elizaga Navascués: immediately from the discussing before. This means that there's gonna be gaps in the physical platform. Us. So the physical things are going to be supported only on discrete values of the message. 319 00:42:06,410 --> 00:42:10,250 Beatriz Elizaga Navascués: It's actually consistent with recent works 320 00:42:10,560 --> 00:42:29,330 Beatriz Elizaga Navascués: that address the quantization of this different. The situation of the Hamiltonian. without considering a standard phase, space, formulation, and where Delta B only depended on Oc. Which, if you don't consider the extension of phase space. If you consider the Delta V depends on OP. 321 00:42:29,430 --> 00:42:30,340 Beatriz Elizaga Navascués: And they 322 00:42:30,390 --> 00:42:45,440 Beatriz Elizaga Navascués: the quantization becomes very, very complicated. In fact. So in this work they only consider the fact that the the possibility that delta we will depend on on all, c. And they also arrive at physical states with support and discrete values of the black. 323 00:42:45,690 --> 00:42:52,550 Beatriz Elizaga Navascués: Now this is well definitely interesting. It has physical consequences. Having actually in the platform, as 324 00:42:52,660 --> 00:42:54,080 Beatriz Elizaga Navascués: I would 325 00:42:54,630 --> 00:42:57,680 Beatriz Elizaga Navascués: as a caveat, I would only say 326 00:42:57,710 --> 00:42:58,420 Beatriz Elizaga Navascués: first. 327 00:42:58,800 --> 00:43:09,770 Beatriz Elizaga Navascués: that in the actual construction of physical states this, because we are taking the intersection between the spectrum of policy that is continuous and the spectrum of all that is discrete. 328 00:43:09,870 --> 00:43:12,700 Beatriz Elizaga Navascués: We have to introduce a Dirac delta that has. 329 00:43:12,980 --> 00:43:20,920 Beatriz Elizaga Navascués: that is at the same time defied on a on a functional space with a discrete mission and with a continued measure. And this is mathematically a bit strange. 330 00:43:21,220 --> 00:43:30,770 Beatriz Elizaga Navascués: and, on the other hand, I still would like to understand what happens in the Delta B going to 0 limit. Recall that 331 00:43:31,120 --> 00:43:34,740 Beatriz Elizaga Navascués: this limit is either obtained on shell 332 00:43:34,840 --> 00:43:48,710 Beatriz Elizaga Navascués: either for the gap in loop quantum gravity, the area cup going to 0, or very massive black holes, so very massive black holes are. I mean, what we would consider to be a classical platform. So in the in the must go into 0 limit. 333 00:43:50,110 --> 00:44:02,690 Beatriz Elizaga Navascués: This operator becomes essentially delta B. I, omega 3 Sorry. And Omega B. Has a continuous spectrum, so we need somehow to jump from a discrete set of masses to our continue in the limit. 334 00:44:03,010 --> 00:44:05,290 Beatriz Elizaga Navascués: so at least the masses should pack 335 00:44:05,510 --> 00:44:15,190 Beatriz Elizaga Navascués: as as the as the mass go goes to very high values, they gap between the masses to become smaller and smaller. It seems that there is a 336 00:44:15,350 --> 00:44:22,540 Beatriz Elizaga Navascués: numerical evidence from these works that this happens, but I still, theoretically, I I I mean it is a bit strange what is happening here? 337 00:44:22,830 --> 00:44:27,680 Beatriz Elizaga Navascués: I just leave that caveat open, and I go to the second possibility. 338 00:44:28,190 --> 00:44:32,190 Beatriz Elizaga Navascués: which prices again. 339 00:44:32,310 --> 00:44:34,120 Beatriz Elizaga Navascués: because 340 00:44:34,860 --> 00:44:35,940 Beatriz Elizaga Navascués: we 341 00:44:36,070 --> 00:44:45,120 Beatriz Elizaga Navascués: are in principle allowed to change the dense subspace of the kinematical space on whose dual research for physical the state this is 342 00:44:45,190 --> 00:44:47,090 Beatriz Elizaga Navascués: This is an open question. So 343 00:44:48,300 --> 00:45:00,390 Beatriz Elizaga Navascués: yeah, this is what I mean here the standard habitat. By standard I mean the the dual of the dense that's self-p is spanned by the eigenfunctions, and I guess it's so for me, maybe too small. 344 00:45:00,670 --> 00:45:16,840 Beatriz Elizaga Navascués: So, for example, there are other ways in which one can obtain this results. So this is just an example. I I told you that I think many of these results have quite robust against one decision, and because it is in the factor of that instance. So imagine that we just put cuts on all of the operators Here. 345 00:45:16,850 --> 00:45:19,550 Beatriz Elizaga Navascués: then, the constraint equation is equivalent to this one. 346 00:45:20,900 --> 00:45:25,770 Beatriz Elizaga Navascués: Okay. the the Hamiltonian constraint equation. And the 347 00:45:30,020 --> 00:45:31,620 Beatriz Elizaga Navascués: In fact. 348 00:45:33,160 --> 00:45:38,480 Beatriz Elizaga Navascués: if we take this operator here to the left hand side. we are a. 349 00:45:38,840 --> 00:45:45,860 Beatriz Elizaga Navascués: It can be shown again with similar procedures as before, that the spectrum of the whole operator. On the left hand side 350 00:45:45,880 --> 00:45:56,910 Beatriz Elizaga Navascués: is the script, and if we would be looking for States on the original, on the standard algebraic dual. Then we would get a discrete spectrum for the mass. However. 351 00:45:57,250 --> 00:46:08,340 Beatriz Elizaga Navascués: imagine that we take. So we project this equation on in eigenstate in the kinematical finger space of Pv. 352 00:46:08,600 --> 00:46:13,680 Beatriz Elizaga Navascués: So I, a polymeric element and an element of the polymeric. 353 00:46:13,930 --> 00:46:29,980 Beatriz Elizaga Navascués: This is a finite difference operator, because the member that is built from a sign that contains Solonomy that act as translation. So it's a finite difference equation that can be solved for any values of N. 354 00:46:30,250 --> 00:46:36,460 Beatriz Elizaga Navascués: So not only for discrete ones for any values of them. So it's. Actually, this is this is a solvable. 355 00:46:36,500 --> 00:46:50,670 Beatriz Elizaga Navascués: and, in fact, at least from just the perspective of the constraint equation and the properties of the spectrum of Delta V. It it has to consistent delta. We going to see the limit because we're going to see that these reduces to 356 00:46:50,670 --> 00:46:57,270 Beatriz Elizaga Navascués: a an operator with a continuous spectrum. So this, at least from that that perspective that is a 357 00:46:57,340 --> 00:47:09,790 Beatriz Elizaga Navascués: so, there is not no puzzling such. Again. I'm. I'm. Not stating anything too too strongly against or in favor of against. But at least this is something I should be commended. 358 00:47:09,840 --> 00:47:17,910 Beatriz Elizaga Navascués: And why are we able to solve this constraint equations? Why are we able to find physical, prospective physical states 359 00:47:18,060 --> 00:47:24,290 Beatriz Elizaga Navascués: for continuous values of the not? It seems intensive with the previous result right 360 00:47:24,610 --> 00:47:30,980 Beatriz Elizaga Navascués: things that now we are looking on the algebraic dwell of the domain of Ov, which is the linear span 361 00:47:31,100 --> 00:47:34,370 Beatriz Elizaga Navascués: of the polymer. 362 00:47:34,710 --> 00:47:51,150 Beatriz Elizaga Navascués: and this actually turns out to be much bigger than the previous to on space. It allows for a very divergent wave functions in the sense that the the the inner product on the on the kinematical field space diverges faster than 363 00:47:51,630 --> 00:47:53,890 Beatriz Elizaga Navascués: And so. 364 00:47:54,390 --> 00:48:07,600 Beatriz Elizaga Navascués: yeah, the question is, Well, of course we we're not gonna be able to to to construct an inner product in the original kinematical, but nothing prevents one to construct an inner product appropriate for this 365 00:48:07,600 --> 00:48:15,470 Beatriz Elizaga Navascués: wave functions in the physical space. The question is which one would be a natural one. Well, that's an open question for the moment. Working progress. 366 00:48:15,500 --> 00:48:18,620 Beatriz Elizaga Navascués: Maybe you want. But yeah. 367 00:48:18,880 --> 00:48:24,470 Beatriz Elizaga Navascués: very different physical scenario depending on the dual. So this is the 368 00:48:24,580 --> 00:48:37,370 Beatriz Elizaga Navascués: important part message from the work in progress of this talk. and that, I think, is relevant beyond the the the symmetry reviews model the but in this case it is 369 00:48:38,470 --> 00:48:45,410 Beatriz Elizaga Navascués: very evident that the the physical consequences are important. The choice of the world space to construct the physical space 370 00:48:45,420 --> 00:48:56,210 Beatriz Elizaga Navascués: can have as drastic consequences as you, getting from a discrete set of black or masses to the continuous one. But a good thing of the following look: techniques, Look what 371 00:48:56,290 --> 00:48:59,350 Beatriz Elizaga Navascués: cosmology things for interior is that we get 372 00:48:59,550 --> 00:49:00,800 Beatriz Elizaga Navascués: operators 373 00:49:00,940 --> 00:49:13,200 Beatriz Elizaga Navascués: that are well known from as a topical. Qc. So that's a a good thing. Also, I would like to emphasize on the extended face, space. 374 00:49:13,940 --> 00:49:21,810 Beatriz Elizaga Navascués: formulation, or strategy as a starting point for the conversation. I think 375 00:49:22,070 --> 00:49:27,090 Beatriz Elizaga Navascués: Christina Gizel and Hong Kong. Do you have been actually been working recently on seminar. 376 00:49:27,110 --> 00:49:40,970 Beatriz Elizaga Navascués: you know, similar, at least along the similar philosophy to try to justify the improved dynamics of schemes. It's just an extended face, based formulation. So I well, I think this is quite interesting. 377 00:49:41,080 --> 00:49:43,870 Beatriz Elizaga Navascués: I a to explore. Oh. 378 00:49:44,120 --> 00:49:55,520 Beatriz Elizaga Navascués: and so we'll open questions. What would be the we? We'll, we'll, we'll, we'll, we'll we'll we'll we'll! We'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll, we'll have the with limit of the physical wave functions. Can we define our relation on dynamics and using that and 379 00:49:55,770 --> 00:49:57,600 Beatriz Elizaga Navascués: sufficiently picked states? 380 00:49:57,650 --> 00:50:01,140 Beatriz Elizaga Navascués: Can we actually recover the Ios dynamics. I think those 381 00:50:01,240 --> 00:50:07,550 Beatriz Elizaga Navascués: I would give a lot of robustness to the to this model, which has at least in the interior of nice physical fix 382 00:50:08,220 --> 00:50:09,280 Beatriz Elizaga Navascués: and all that. 383 00:50:09,470 --> 00:50:13,040 Beatriz Elizaga Navascués: That's that's all I wanted to say. Thank you 384 00:50:19,630 --> 00:50:20,360 questions. 385 00:50:22,120 --> 00:50:30,510 FAU (Florida): I have a question I just want to make sure that I understand. So one of the originally when I saw the Aos 386 00:50:30,520 --> 00:50:48,680 FAU (Florida): model, I. The difficulty I saw in defining the Hamiltonian constraint operators what you call Omega B. The inverse defining it on the board hill. But it seems you've sidestep this by only worrying about the action on the distributional states. I just want to make sure if I understand that that's like the main 387 00:50:48,740 --> 00:50:51,070 FAU (Florida): trick here, and I I think that's very clever. 388 00:50:52,570 --> 00:51:07,420 FAU (Florida): You mean they they only that. My! Do you mean they can. You? Can you repeat the question? So the I think if i'm right, the the omega be operated there's this omega b inverse that if you were to try to find it's action on the board Hilbert Space, it would be ill to find. 389 00:51:07,420 --> 00:51:18,430 Beatriz Elizaga Navascués: but you avoid that issue by only focusing on its action on distributional states. Is that correct? Yeah. Well, we look at the dual. But, on the other hand, the spectrum of the 390 00:51:18,620 --> 00:51:28,860 Beatriz Elizaga Navascués: all makeup is con continues. So if I mean difficulties in the definition of it in there, maybe could be avoided by just defining it's actually in a 391 00:51:29,010 --> 00:51:38,840 Beatriz Elizaga Navascués: in a non-standard way, but because it that the Theido is not in the like. Nothing is in the the spectrum. But but yes, we we. We focus on the action. 392 00:51:38,930 --> 00:51:40,890 Beatriz Elizaga Navascués: We focus on the 2 1. 393 00:51:41,110 --> 00:51:45,590 Beatriz Elizaga Navascués: So but this I mean, I think this should be investigated with care. 394 00:51:49,050 --> 00:51:53,230 Beatriz Elizaga Navascués: We we actually yeah. 395 00:51:53,440 --> 00:51:55,350 Beatriz Elizaga Navascués: this should provide. 396 00:51:55,370 --> 00:52:00,160 Beatriz Elizaga Navascués: If it doesn't, then this relation, then 397 00:52:00,220 --> 00:52:08,470 Beatriz Elizaga Navascués: we would have to be a little bit more careful, but at least the spectrum of the operator from which we are trying to take the Inverses 398 00:52:08,580 --> 00:52:12,200 Beatriz Elizaga Navascués: County absolutely continues. That's all I can say. 399 00:52:14,900 --> 00:52:17,020 Beatriz Elizaga Navascués: I don't know if that helps or not. 400 00:52:17,340 --> 00:52:18,620 FAU (Florida): Yes, it helps. Thank you. 401 00:52:18,680 --> 00:52:19,300 Beatriz Elizaga Navascués: Okay. 402 00:52:24,040 --> 00:52:24,700 Here. 403 00:52:36,500 --> 00:52:39,190 Jorge Pullin: Okay, for no more questions to thank the speaker again. 404 00:52:40,270 --> 00:52:41,900 Adrià Delhom i Latorre: Thank you. 405 00:52:41,940 --> 00:52:44,290 I have a question. 406 00:52:45,960 --> 00:52:46,920 Jorge Pullin: Go ahead. 407 00:52:47,640 --> 00:52:49,580 Adrià Delhom i Latorre: Okay, so. 408 00:52:50,600 --> 00:52:57,560 Adrià Delhom i Latorre: And I don't expect you to have an answer or something. But have you thought about 409 00:52:57,590 --> 00:53:07,880 Adrià Delhom i Latorre: how many people would it be to be able to? I mean, Tony, and without that, somehow or this is an extended one in in 410 00:53:07,970 --> 00:53:09,730 Adrià Delhom i Latorre: you that question that I 411 00:53:10,570 --> 00:53:15,250 Beatriz Elizaga Navascués: I would point you to to to the work I, Christina, and the on one for house. 412 00:53:15,610 --> 00:53:27,480 Beatriz Elizaga Navascués: No, I don't know exactly to what extent, but at least they they are taking 413 00:53:28,940 --> 00:53:30,580 Adrià Delhom i Latorre: alright, Thank you. 414 00:53:36,210 --> 00:53:37,760 Jorge Pullin: I was talking to Speaker again.