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Jorge Pullin: Okay. So our speaker today is that we'll speak about the Lq. Of Black holes.
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Beatriz Elizaga Navascués: Thank you. Perfect. So yeah, I was. Thank you for suggesting me for giving this talk.
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Beatriz Elizaga Navascués: I was
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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.
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Beatriz Elizaga Navascués: I
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Beatriz Elizaga Navascués: most of my research career. I've been working in the quantum cosmology aspect of cosmological space time.
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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
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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
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Beatriz Elizaga Navascués: to the issue, and then only Indian talk about the recent progress.
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Beatriz Elizaga Navascués: So
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Beatriz Elizaga Navascués: in any case, i'll start with some background on this topic.
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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
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Beatriz Elizaga Navascués: in the context of lu quantum gravity and quantum cosmology.
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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
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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.
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Beatriz Elizaga Navascués: So
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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
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Beatriz Elizaga Navascués: control, mathematical or robust control over it, and we can make predictions satisfactory predictions out of it.
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Beatriz Elizaga Navascués: What the most or many of the investigations do in order to attain this objective
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Beatriz Elizaga Navascués: is to focus on a spherical symmetry. So the simplest type of Black Hole Space Times.
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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
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Beatriz Elizaga Navascués: cosmology techniques can be applied in that.
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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
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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
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Beatriz Elizaga Navascués: our
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Beatriz Elizaga Navascués: just
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Beatriz Elizaga Navascués: they.
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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.
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Beatriz Elizaga Navascués: So
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Beatriz Elizaga Navascués: there are some quantization or ambiguities in the choice of that regularization of that affected Hamiltonian. and the
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Beatriz Elizaga Navascués: even though all of the effective models resulting from that has succeeded in resolving the singularity to some the
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Beatriz Elizaga Navascués: 3 or another, and most of them can
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Beatriz Elizaga Navascués: mit ctl, and be pictured as describing a transition from a kind of black hole. Roughly speaking, to a Whitehall.
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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
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Beatriz Elizaga Navascués: quantum geometry effects appearing at low curvature, depending on depending on if you do, and so on and so forth.
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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.
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Beatriz Elizaga Navascués: attend to
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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
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Beatriz Elizaga Navascués: is independent of the mass of the back home, and only diverges when the
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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.
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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.
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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
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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
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Beatriz Elizaga Navascués: of this approach
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Beatriz Elizaga Navascués: in particular, what are the general properties of the
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Beatriz Elizaga Navascués: And if this kind of effectively regimes can be derived actually from a quantum theory, actually there's
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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.
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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.
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Beatriz Elizaga Navascués: So they did. Your of a black hole in general relativity admits of a foliation
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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.
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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
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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.
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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
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Beatriz Elizaga Navascués: measures physical areas of the spheres of symmetry of the interior of the Black Hole.
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Beatriz Elizaga Navascués: The classical Hamiltonian takes this form. namely, it with a wise choice of laps function that absorbs this pre-factor. Here
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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
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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
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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
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Beatriz Elizaga Navascués: which is proportional to the Atmos. Once we extend the solution to the exterior.
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Beatriz Elizaga Navascués: so remember both partial Hamiltonians are equal to the mass of the black Hole.
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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.
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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.
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Beatriz Elizaga Navascués: so what one usually do, at least in in traditional approaches. To
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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.
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Beatriz Elizaga Navascués: keeping the technical details. 1 1
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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.
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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
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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.
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Beatriz Elizaga Navascués: So I will still call this parcel Hamiltonian, because for
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Beatriz Elizaga Navascués: many interesting choices of this Delta
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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
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Beatriz Elizaga Navascués: the the dynamics on each of the all sectors and the the the resulting.
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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.
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Beatriz Elizaga Navascués: Okay, so
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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
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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
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Beatriz Elizaga Navascués: geometry that one obtains when one takes these regularized Hamiltonian seriously and compute the evolution generated by it.
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Beatriz Elizaga Navascués: and then studies the the result in the the properties of the results. In geometry
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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.
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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
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Beatriz Elizaga Navascués: are not constants of motion and parallel. The analysis is done in standard quantum cosmology, and the
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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
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Beatriz Elizaga Navascués: is a model that I will be calling. Aos came into into consideration
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Beatriz Elizaga Navascués: in the in this model the
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Beatriz Elizaga Navascués: delta B, and that the c parameters are actually fixed
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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.
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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.
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Beatriz Elizaga Navascués: And this is actually
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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
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Beatriz Elizaga Navascués: in which this model is supposed to be valid. They don't assume the validity of this model for all possible
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Beatriz Elizaga Navascués: rather massive, much so
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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.
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Beatriz Elizaga Navascués: and, in fact.
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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.
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Beatriz Elizaga Navascués: Okay. So this is fixing the it seems that Fix in Black Hole models fixing the
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Beatriz Elizaga Navascués: the
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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.
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Beatriz Elizaga Navascués: Okay, so. and naturally
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Beatriz Elizaga Navascués: step next step in the analysis
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Beatriz Elizaga Navascués: of these models would be to consider
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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.
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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
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Beatriz Elizaga Navascués: has been a question of debate
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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
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Beatriz Elizaga Navascués: on solutions.
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Beatriz Elizaga Navascués: So I come back to that sort that I told you to hold.
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Beatriz Elizaga Navascués: Okay, so the effective dynamics
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Beatriz Elizaga Navascués: is the As I said, the Ios effective dynamics is generated by this Hamiltonian. The difference of the 2
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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
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Beatriz Elizaga Navascués: that, or do you need to treat the Delta parameters as pure Const.
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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
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Beatriz Elizaga Navascués: for large black holes that the ones that I was showing previously.
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Okay.
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Beatriz Elizaga Navascués: Now.
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Beatriz Elizaga Navascués: what many people realized as soon as this model was suggested was that
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Beatriz Elizaga Navascués: wait.
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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.
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Beatriz Elizaga Navascués: This was was some concern that was raised.
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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.
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Beatriz Elizaga Navascués: then the equations of motion, that result do not describe the Aos model. That is, it is a different geometry so
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Beatriz Elizaga Navascués: well. In fact, even in the original works, even with the you know
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Beatriz Elizaga Navascués: paper that proposed this model. They
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Beatriz Elizaga Navascués: argued that the equations of motion
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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
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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.
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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
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Beatriz Elizaga Navascués: fix on shell? What is the relation between the Delta parameters and the mass of the black on each solution?
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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
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Beatriz Elizaga Navascués: that that the constraints are functions we both are part of.
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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.
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Beatriz Elizaga Navascués: But other than that, this is quite general. So you could actually consider a wide variety of models here.
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Beatriz Elizaga Navascués: So
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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.
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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.
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Beatriz Elizaga Navascués: So
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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
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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
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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.
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Beatriz Elizaga Navascués: So, in fact, this the fact that the reduce space is not equivalent to the general relativistic. One is
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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.
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Beatriz Elizaga Navascués: even though they are a constant of motion.
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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.
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Beatriz Elizaga Navascués: So well. Okay, so we want to work with just this Hamiltonian.
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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
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Beatriz Elizaga Navascués: is the one from general relativity.
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Beatriz Elizaga Navascués: So, as a quantization strategy.
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Beatriz Elizaga Navascués: It seems quite clean.
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Beatriz Elizaga Navascués: and for the constructive purposes to consider they
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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.
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simone: Hi! Can I ask a question
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simone: on this nice picture? Maybe I grew in the joke. Sorry!
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Beatriz Elizaga Navascués: Oh.
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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
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simone: a fundamental classical algebra one could use deltas that Don't depend on the atmos
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simone: and get the singularity free model.
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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
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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.
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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.
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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
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Beatriz Elizaga Navascués: that model had in the in the in its effective formulation, so to say
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Beatriz Elizaga Navascués: so. If another
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Beatriz Elizaga Navascués: or one
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Beatriz Elizaga Navascués: motivated from similar arguments, would collect all of those properties
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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
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Beatriz Elizaga Navascués: I? Well, I
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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?
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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.
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simone: Okay, Thank you.
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Beatriz Elizaga Navascués: Yeah.
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Beatriz Elizaga Navascués: Okay. So again, I I continue at the pedagogical philosophy. We want our strategy in this talk.
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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.
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Beatriz Elizaga Navascués: This is, I mean.
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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
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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.
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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
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Beatriz Elizaga Navascués: information of the connection that comes through a complex exponential of it.
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Beatriz Elizaga Navascués: Like, on the other hand, flexes of the density is that are fully fixed by
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Beatriz Elizaga Navascués: the TV and PC.
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Beatriz Elizaga Navascués: Point particle degrees of freedom. So the oversimplified version of the Alonomy Flax Algebra. In this case takes this form.
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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.
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Beatriz Elizaga Navascués: So we can apply directly the techniques that we know very well from
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Beatriz Elizaga Navascués: the quantization of homogeneous and isotropic model. Just to
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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.
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simone: Sorry, can I ask you a question about this.
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Beatriz Elizaga Navascués: Yes. so here the flax is commute right.
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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.
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Beatriz Elizaga Navascués: I think.
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simone: but a are in these. So maybe
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Beatriz Elizaga Navascués: i'm i'm i'm integrating over squares that are adopted to the directions of symmetry.
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Beatriz Elizaga Navascués: No.
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simone: So
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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
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Beatriz Elizaga Navascués: the coordinate
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Beatriz Elizaga Navascués: squares or rectangles in each of the of the
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Beatriz Elizaga Navascués: coordinate directions. So yeah, they are. So I mean, they intersect if you extend them. Yes, they
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simone: right. So usually when the intersect we expect
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simone: no competitivity. But here you don't see it. I see.
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Beatriz Elizaga Navascués: Okay, because of this, because of how much it I I believe.
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simone: Okay.
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simone: can I make a remark about that. This is this is John. So if remember.
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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
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FAU (Florida): what is it this
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FAU (Florida): anyways? But this is one of these anomalies in quantum gravity. So
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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.
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Beatriz Elizaga Navascués: Yeah, thanks. I I mean.
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Beatriz Elizaga Navascués: I would have to think a bit more about it that for the moment I would.
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Beatriz Elizaga Navascués: My
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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.
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Beatriz Elizaga Navascués: Thank you.
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Beatriz Elizaga Navascués: Okay. So yeah, they commit. So in the quantum theory they would also commit
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Beatriz Elizaga Navascués: the 2 sectors.
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Beatriz Elizaga Navascués: So i'm going to call the basis with this. It's an orthonormal basis
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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.
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Beatriz Elizaga Navascués: So the if you're
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Beatriz Elizaga Navascués: they are again of the
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Beatriz Elizaga Navascués: So yeah period operators that I, by multiplication and the allonomy operators is the translate this an act by translation.
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Beatriz Elizaga Navascués: Okay. So
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Beatriz Elizaga Navascués: for the
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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.
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Beatriz Elizaga Navascués: so we will.
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Beatriz Elizaga Navascués: It is the simplest
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Beatriz Elizaga Navascués: thing that one can think of was a as
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Beatriz Elizaga Navascués: treating them as a matter in the standard of quantum cosmology, to adopt the continuous representation for them.
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Beatriz Elizaga Navascués: and then the total Hilbert space is the tensor product of these 2 continuous.
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Beatriz Elizaga Navascués: so did the kilograms of square integral functions associated with these 2 continuous representations, and then the kinematical.
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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.
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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
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Beatriz Elizaga Navascués: the angular part is quite simple. It's just the up to constants the same.
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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.
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Beatriz Elizaga Navascués: We can write the sign in terms of follow elementary. and
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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.
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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.
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Beatriz Elizaga Navascués: then the action of the is Omega squared. This basic operator for the Oc.
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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.
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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.
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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.
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Beatriz Elizaga Navascués: In super selection sectors
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Beatriz Elizaga Navascués: that have support on.
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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
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Beatriz Elizaga Navascués: the super selection sector, we can identify it with the closure of the nonseparable inverse space for B
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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.
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Beatriz Elizaga Navascués: So because recall that. But actually, of all
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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.
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Beatriz Elizaga Navascués: Now the regular part is more complicated.
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Beatriz Elizaga Navascués: I will not give a specific prop like suggestion for its quantum representation.
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Beatriz Elizaga Navascués: I will just require some minimal properties on it.
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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
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Beatriz Elizaga Navascués: it.
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Beatriz Elizaga Navascués: I require that the presets the separate selection sectors of the Omega, P.
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Beatriz Elizaga Navascués: Which allow us to separate the non-separable part of the piece sector of the kinematical in that space.
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Beatriz Elizaga Navascués: in order to study the dynamics.
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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.
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Beatriz Elizaga Navascués: and the point spectrum is the script. I will also assume that
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Beatriz Elizaga Navascués: so basically.
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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.
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Beatriz Elizaga Navascués: so it can be purely discrete or purely. I will not go and tell me many more.
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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.
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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
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Beatriz Elizaga Navascués: those additional constraints.
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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
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Beatriz Elizaga Navascués: recent produce
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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.
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Beatriz Elizaga Navascués: So
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Beatriz Elizaga Navascués: again
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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
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Beatriz Elizaga Navascués: algebraic spaces to a dense subset of the kinematical inverse space, because this is where we have defined the constraint operators.
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Beatriz Elizaga Navascués: So we enlarge the habit of possible physical states because they may not leave in the
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Beatriz Elizaga Navascués: in the kinematical one.
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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.
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Beatriz Elizaga Navascués: This means
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Beatriz Elizaga Navascués: right, that ob
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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.
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Beatriz Elizaga Navascués: which means
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Beatriz Elizaga Navascués: immediately under a caveat. that I will comment in a moment that the wave function must be 0
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Beatriz Elizaga Navascués: when it supports it. Doesn't belong
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Beatriz Elizaga Navascués: 2.
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Beatriz Elizaga Navascués: The spectrum of the OP operator that I call it, may depend on on the Delta
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Beatriz Elizaga Navascués: as a difference with the Oc. Which doesn't depend on. Okay. So
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Beatriz Elizaga Navascués: what the the caveat that I was talking about is that for this second line to follow from the first one.
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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
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Beatriz Elizaga Navascués: by the eigenfunctions and eigenstates of Ob.
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Beatriz Elizaga Navascués: If we take that. so then subspace the this need not follow.
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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.
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Beatriz Elizaga Navascués: But as long as we look for physical state, which is what we usually do in look onto cosmology. In fact.
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Beatriz Elizaga Navascués: using the analogy between the operators that we have there, we look on that to one.
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Beatriz Elizaga Navascués: So additionally, we have
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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
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Beatriz Elizaga Navascués: the spectrum of Ob and Oc. Do not depend
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Beatriz Elizaga Navascués: on a delta, c. But the spectrum of for being general will depend on delta B.
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Beatriz Elizaga Navascués: So
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Beatriz Elizaga Navascués: it is not hard to realize, by imposing this constraint that the physical States will be in one-to-one correspondence
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Beatriz Elizaga Navascués: with wave functions of the mass that have support
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Beatriz Elizaga Navascués: on at this implicit relation set. So what this means is that
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Beatriz Elizaga Navascués: take a a value of them in the real line, construct
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Beatriz Elizaga Navascués: the Delta, B, compute the Delta B associated with it with this formula.
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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
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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
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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.
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Beatriz Elizaga Navascués: Okay. So this is a rather formal characterization of the physical States.
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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.
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Beatriz Elizaga Navascués: I will talk about possibilities that
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Beatriz Elizaga Navascués: for the actual form of this physical state, so possibilities for the spectrum of.
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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
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Beatriz Elizaga Navascués: algebraic dual for looking at so looking for physical states in a different habitat and the physical consequences of
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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
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Beatriz Elizaga Navascués: having a discrete set of mass with gaps to a continued set of masses without caps.
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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
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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.
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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.
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Beatriz Elizaga Navascués: So in imagine that we can so look at this this regularize
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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.
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Beatriz Elizaga Navascués: But now this means that on any State on the domain of the OP. Squared
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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
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Beatriz Elizaga Navascués: Pd. Squared.
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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.
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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.
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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
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Beatriz Elizaga Navascués: so
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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.
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Beatriz Elizaga Navascués: It's actually consistent with recent works
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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.
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Beatriz Elizaga Navascués: And they
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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.
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Beatriz Elizaga Navascués: Now this is well definitely interesting. It has physical consequences. Having actually in the platform, as
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Beatriz Elizaga Navascués: I would
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Beatriz Elizaga Navascués: as a caveat, I would only say
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Beatriz Elizaga Navascués: first.
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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.
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Beatriz Elizaga Navascués: We have to introduce a Dirac delta that has.
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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.
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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
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Beatriz Elizaga Navascués: this limit is either obtained on shell
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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.
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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.
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Beatriz Elizaga Navascués: so at least the masses should pack
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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
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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?
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Beatriz Elizaga Navascués: I just leave that caveat open, and I go to the second possibility.
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Beatriz Elizaga Navascués: which prices again.
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Beatriz Elizaga Navascués: because
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Beatriz Elizaga Navascués: we
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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
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Beatriz Elizaga Navascués: This is an open question. So
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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.
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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.
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Beatriz Elizaga Navascués: then, the constraint equation is equivalent to this one.
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Beatriz Elizaga Navascués: Okay. the the Hamiltonian constraint equation. And the
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Beatriz Elizaga Navascués: In fact.
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Beatriz Elizaga Navascués: if we take this operator here to the left hand side. we are a.
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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
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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.
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Beatriz Elizaga Navascués: imagine that we take. So we project this equation on in eigenstate in the kinematical finger space of Pv.
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Beatriz Elizaga Navascués: So I, a polymeric element and an element of the polymeric.
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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.
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Beatriz Elizaga Navascués: So not only for discrete ones for any values of them. So it's. Actually, this is this is a solvable.
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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
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Beatriz Elizaga Navascués: a an operator with a continuous spectrum. So this, at least from that that perspective that is a
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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.
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Beatriz Elizaga Navascués: And why are we able to solve this constraint equations? Why are we able to find physical, prospective physical states
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Beatriz Elizaga Navascués: for continuous values of the not? It seems intensive with the previous result right
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Beatriz Elizaga Navascués: things that now we are looking on the algebraic dwell of the domain of Ov, which is the linear span
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Beatriz Elizaga Navascués: of the polymer.
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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
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Beatriz Elizaga Navascués: And so.
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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
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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.
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Beatriz Elizaga Navascués: Maybe you want. But yeah.
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Beatriz Elizaga Navascués: very different physical scenario depending on the dual. So this is the
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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
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Beatriz Elizaga Navascués: very evident that the the physical consequences are important. The choice of the world space to construct the physical space
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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
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Beatriz Elizaga Navascués: cosmology things for interior is that we get
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Beatriz Elizaga Navascués: operators
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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.
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Beatriz Elizaga Navascués: formulation, or strategy as a starting point for the conversation. I think
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Beatriz Elizaga Navascués: Christina Gizel and Hong Kong. Do you have been actually been working recently on seminar.
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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.
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Beatriz Elizaga Navascués: I a to explore. Oh.
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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
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Beatriz Elizaga Navascués: sufficiently picked states?
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Beatriz Elizaga Navascués: Can we actually recover the Ios dynamics. I think those
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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
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Beatriz Elizaga Navascués: and all that.
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Beatriz Elizaga Navascués: That's that's all I wanted to say. Thank you
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questions.
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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
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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
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FAU (Florida): trick here, and I I think that's very clever.
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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.
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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
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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
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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.
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Beatriz Elizaga Navascués: We focus on the 2 1.
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Beatriz Elizaga Navascués: So but this I mean, I think this should be investigated with care.
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Beatriz Elizaga Navascués: We we actually yeah.
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Beatriz Elizaga Navascués: this should provide.
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Beatriz Elizaga Navascués: If it doesn't, then this relation, then
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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
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Beatriz Elizaga Navascués: County absolutely continues. That's all I can say.
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Beatriz Elizaga Navascués: I don't know if that helps or not.
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FAU (Florida): Yes, it helps. Thank you.
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Beatriz Elizaga Navascués: Okay.
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Here.
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Jorge Pullin: Okay, for no more questions to thank the speaker again.
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Adrià Delhom i Latorre: Thank you.
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I have a question.
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Jorge Pullin: Go ahead.
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Adrià Delhom i Latorre: Okay, so.
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Adrià Delhom i Latorre: And I don't expect you to have an answer or something. But have you thought about
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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
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Adrià Delhom i Latorre: you that question that I
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Beatriz Elizaga Navascués: I would point you to to to the work I, Christina, and the on one for house.
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Beatriz Elizaga Navascués: No, I don't know exactly to what extent, but at least they they are taking
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Adrià Delhom i Latorre: alright, Thank you.
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Jorge Pullin: I was talking to Speaker again.