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Ivan Agullo: Okay. The speaker today is Luca Jim Elliot. And he's gonna talk about synthetic synthetic analysis of null, Ricky Durian. Take it away, Luca.

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Luca Ciambelli: Okay, thank you very much, Ivan and Jorge for the invitation. It's a pleasure to give this talk. I'm gonna talk about a recent paper. Now I should be able you should be able to see the laser pointer. Let me know if something goes south.

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Luca Ciambelli: So this is based on this paper that appear on the Archive last year. Something that we're finishing up. The second part of this talk would be about a paper that we're finishing up now, and some future working progress together with Laurent Rob Lee. And it is about the sympathetic analysis of on arbitrary. Now, hypersurface. So in particular, we're gonna study the geometry.

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Luca Ciambelli: The dynamics

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Luca Ciambelli: and the synthetic structure induced by gravity on hypersurfaces. As it is very well known Einstein equations projected on another surface gives rise to intrinsic equations that are the now rich. More constraints. And in this talk we're gonna use as much as possible an intrinsic approach to now physics. So we're gonna try to reinterpret all these features that gravity induce on our hypersurface is from a an intrinsic point of view

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for an observer on the now hypersurface.

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Luca Ciambelli: And in particular, what we're gonna see is that the rich. Other equation is many facets we will interpret it as a conservation equation based those on all the results on the membrane paradigm. But also we will interpret it as a stress tensors balance law which will give us some clue about the quantum properties of this equation.

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Luca Ciambelli: We will then focus on the sympathetic analysis and the understand the face space intrinsically, and in particular, from the classical simplistic analysis. We will find a positive and monotonic boost charge in a particular notion of time, which is not the fine time on genetic analysis. It is what we doubt the dressing time, and I will explain that carefully

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Luca Ciambelli: later. I will talk about more undergoing studies. And we're gonna show that in a perturbative geometric setting that I've called perturbative quantum geometry. Who's perturbation schema will explain carefully. There is an emergent spin 0 beta gamma safety. So there is a Beta gamma safety sector related to this

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Luca Ciambelli: been 0 gravitational degrees of freedom on null hyper surfaces. And this is very interesting, because, it opens the possibility to a quantization directly from the face base of the geometric degrees of freedom.

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Luca Ciambelli: And at the very end I will explore some more futuristic developments of our topic. And I'm gonna talk about so-called atomic quantization, and how this fits perfectly with the audience of this seminar. Alright.

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Luca Ciambelli: please. Yes, exactly. I was. Gonna say, interrupt me at any moment, and there was a question, but I couldn't hear it. Please, Beta Gamma! Oh, right so the Beta Gamma Cft is a particular cft, which is a chiral sector of a 2 dcft I'm gonna explain it later. But it appears

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Luca Ciambelli: it appears in some wealth

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Luca Ciambelli: it's a cft per se, but it was useful in some strings or ethical construction for the Ghost Associated super streams. We're gonna just take from there the properties that we need. It's a particular cft given by the OP. Between the primary field Beta and Gamma.

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Luca Ciambelli: And I'm gonna I'm the. And I'm gonna show. Thanks for the question. I'm gonna show precisely how these Beta gamma fields that come from a complete related study to Nile surfaces are actually associated to spin, 0 geometric degrees of freedom. So like the area for

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Luca Ciambelli: of the surface.

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Luca Ciambelli: alright. Okay. So let me start. Let me review very quickly the genometersurfaces. This is a very, very long story, and I'm gonna just revisit in a modern language what has been done in the last 60 years or so.

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Luca Ciambelli: So I'm gonna consider a 3D. For simplicity, the non manifold, because I have in mind for the Einstein gravity in the bath. And I'm gonna work in a topology which cuts away the caustics has written here so that you know, my LA null generator is move that there's no proper, no problems.

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Luca Ciambelli: and the a null manifold of this form is defined by what is nowadays call the carole instructor. This is nothing but a a null vector field and the degenerate metric. So this is given by the condition that L is in the kernel of the metric and degenerate metric and L is nowhere vanishing.

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Luca Ciambelli: Okay, so this define a carole instructor. I'm gonna end, though, with with the father property, which is this ruling Ka, this comes about because I want to talk about the horizontal

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Luca Ciambelli: sub manifold or cuts that I will refer generically at C at this currently see. And so I'm gonna introduce, if you want the tangent bundle dual tool, which is always defined by having this property this guy. Come, of course with the huge you know, gauge the generacy because I can shift it with anything that is in the kernel of this contractual, and they will still be defined

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Luca Ciambelli: ruling for me. And so I'm gonna take these and keep this symmetry with me. And this Ka is sometimes referred to as an Irishman connection, because these 3D, not manifold, can be thought of as a fiber bundle where the now generated la is the typical fiber. And therefore you can imagine, then, that Ka plays the role of the horizontal connection here in the fiber bundle.

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Luca Ciambelli: Okay? So these are my defining properties. Instructors I've introduced. I'm gonna further introduce the expansion tensor. This is the second fundamental form of now. Yes, yes, you like, please.

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Jerzy Lewandowski: So is the vector field. L, also, given.

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Luca Ciambelli: I mean the ambiguity in rescaling L by a function. Right? So indeed, what really defined the theory is a conformal class of Ls defined by a rescaling. And I'm gonna same way as the conformal class of Ka is defined by shifting Ka

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Luca Ciambelli: with something which is orthogonal. As I was discussing. As you reck said, that is perfectly right. There is also an ambiguity in defining L, which is a a boost what I will call a boost, and I'm gonna keep this boost ambiguity and use it as a symmetry for the face space. So this symmetry for KAI just mentioned it here, because it will not appear in the rest, as I will be focused on each other. But this

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Luca Ciambelli: a boost symmetry. I will call on the boost will appear later on, and I will talk about it extensively. So yes, there is a inherent ambiguity in defining. L,

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Luca Ciambelli: okay, so I gonna define the expansion tensor in this way, this is the usual second fundamental form, except that here it should be stress that, given the natural surfaces is something that is intrinsic to the null geometry. And I'm gonna define the expansion tensor with an index down and an index up to contain the so-called expansion and the traceless part which is the shear.

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Luca Ciambelli: Okay? The sheer would play a fundamental role later on.

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Luca Ciambelli: This has been made possible, thanks to the introduction of the resultant projector. And these are risant projector as being introduced using the caroleam vector, and the ruling arrangement. K. In this way it projects to the space or togona to K and L. So essentially projects to the cut C written here.

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Luca Ciambelli: And finally, there's no let me give it a trm, I'm gonna specify a connection given by the properties it has when he talks on the geometric data. So the volume form Epsilon, N. Has been, made possible to be introduced in a shift. Invent way this way.

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Luca Ciambelli: using K. And the Caroleian connection is defined to have this property, so it does not preserve the volume form and it makes appear one form omega, which contains 2 tensor, a Scalar Kappa, called in affinity, and a or one form pi, called ag connection. From the intrinsic geometry that appears through the Caronian connection.

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Luca Ciambelli: from the extrinsic geometry. These are the usual in affinity of L, as it is, expand through the bulk, and they are to check connection again, associated to the bulk. Geometry. Okay? So, as I was saying in introduction, I'm trying to be as intrinsic as possible. So I'm seeing how these quantities that are defining the bulk are defined intrinsically on the not hyper surface.

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Luca Ciambelli: Okay. Another reason for introducing this Caroleian connection is that I want to talk about the I stand equation projected, and it is instrumental in writing down the conservation law of the Nile brown York stress test. The Nile brown York tensor is defined in this way. It can be regularly shown to be defined through the Riemann properties of this Carole connection.

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Luca Ciambelli: and it has now the property that if I project gravity on the null hypersurface they give rise to this conservation equation

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Luca Ciambelli: with this corona and connection previously defined, and this stress tensor here, here the matter play is the gravitational bulk coupled matter that plays a role as a source from the intrinsic point of view. Okay? And in particular, indeed, if I project

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Luca Ciambelli: to the vertical. I get the famous now, Richard equation written here, and if I project to the horizontal, I get the as well famous. The more equation written here which are now intrinsic evolutionary equations for the data of the null geometry.

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Luca Ciambelli: Okay, so let me tell you another thing, I've introduced a quantity here. Mu. This mu is this particular combination of in affinity and expansion it is called surface extension. The reason why I couple expansion and in affinity in this way is because this new object will play a fundamental role later on. So I prefer to ready define it from the beginning.

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Luca Ciambelli: Okay, so let me recap this first, a very brief introduction to geometry and dynamics of nala surfaces. If I take Einstein equation before the end I projected to the Nile upper surface. It gives rise to the dabur and the rich childhood equation. II wrote here already, the Richard, because that's what I will be focusing on later on. And what we have seen is an intrinsic way to define this equation as a conservation law for a Carolinian stress tensor.

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Luca Ciambelli: Okay? And this is as I was saying, go back to papers by the Moorethorn Price and Mcdonald associated to the membrane paradigm. And of course it's a framework that has been growing in recent years and reach this sort of universal structure very recent.

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Luca Ciambelli: Okay, so this is the very first part.

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Luca Ciambelli: Some questions before I go on and please interrupt. Of course, if you have

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Luca Ciambelli: questions

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Luca Ciambelli: alright, let me move on to the sympath structure. Now again. Intrinsically, we can introduce a canonical sympathetic potential. This is given by first of all,

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Luca Ciambelli: expanding the Carolyn structure extensor in terms of the momenta tau and tau a

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Luca Ciambelli: B, and then right? The canonical simpleity, potential as essentially the momenta delta, the sources so delta the geometry, the geometry of a Caroleian, manifold, as I was saying before is given by La and Qab. And so I'm gonna regard these to be the null analog of writing team Delta Gmu, new for a time, like, or space like Hypersurface.

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Luca Ciambelli: Okay? And indeed, indeed, while one can introduce it as I did here intrinsically, this can be shown to be the right to be. What one derives when studied the bulk. Einstein uber sympathetic

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Luca Ciambelli: a. And this is also something very, very well known. But it has a lot of history and a lot of works is very significant. The paper of hyword 31 years ago. But okay, many of the people in the audience and the colleagues have worked on this. So the this is the familiar result that Britain in this form it appeared in the paper by Chandra, Sagana, Chandra, Secaran apologies, Flanagan Esperanza in 21.

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Luca Ciambelli: But it's an old result. Now let me study the charges, so I define the canonical sympletic 2 form, and I just contract with the diffomorphism defined on the null hypersurface. This is a different morphism of n the null hyper surface, and one gets this beautiful result, which is just the equations of motion, plus a coordinate term or a term on the cut, which is the usual current contracted.

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Luca Ciambelli: Okay?

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Luca Ciambelli: And let me go even one step further. Now I wanna talk about the constraints, and I wanna be able to now define what I refer before as this PIN zeros be one spin, 2 sectors. What do I mean by that? By doing to do so? I will do the unimodular decomposition. So I will just write this, the generate metric as an area form for the cut, plus a determinant one

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Luca Ciambelli: Q. Bar a. B, and then I will decompose the measure in this way, and this is, of course, given that we are in a here and 2D. Here. This is, of course, that decomposition that just make Epsilon 0 C, for instance, the coordinatized measure. So if I introduce some coordinated would be just the DC. Watch DC. Bar measure on the cuts, for instance.

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Luca Ciambelli: and therefore given all these ingredients, I can define the spin, 0 spin one and spin 2 data, this being one data given by this orthogonal add check and the vector field spin 2 is the shear and the unimodular metric, while spin 0 is the aerial form and the surface tension, which is this combination of affinity. And this function I've introduced before.

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Luca Ciambelli: Okay, now I wanna go, II wanna go a step farther. And given this symplectic structure, my goal is to compute the kinematic up some brackets. So I want to try to invert this symplet structure. And I wanna do so by focusing on the rich childhood sector. Because, as you can see from the evolutionary equation that the move sector is an evolutionary equation for this

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Luca Ciambelli: degrees of freedom, the Rgc. And La. But as Richard, we doesn't make the Rgc. Connection appear so, I want to divide the sector in 2 pieces because I want to. Essentially start from the simplest one, which is the rich child. And then as work in progress. And we have some results on that. We are now focusing on the demo.

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Luca Ciambelli: Okay, so let's focus on the Rachagi equation.

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Luca Ciambelli: And to do so, the first step is to consider a different morphism. That is, just the composing vertical and horizontal directions and focus only on time rep parametrization. We're gonna show that the constraint is the constraint associated to time rep parametrization. So we're gonna set Y and to see

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Luca Ciambelli: we can do a bit less. We can focus on where this Ya is a functionality on the base, on the cut variables. But for simplicity, let me just really focus on Fla.

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Luca Ciambelli: And here comes the the symmetry that Urec asked me about the L that we introduced before is defined modulo. These internal boosts right any of such an internal boost we give rise to the same structure.

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Luca Ciambelli: and if I want to require that there is no speed one, the way to do so is to require the delta and the variational bell on the face. Space is proportional to L itself.

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Luca Ciambelli: And then we have that the parameterization just duties. L of F and the boost. Just shift by lambda. Right. Lambda is the ghost on 40. If you want

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Luca Ciambelli: the dressing time I'm gonna focus on is a way to consistently throughout the face base. Impose that delta and vanishes. So the cup of this P. One, and instead of reducing my set of fields, I'm gonna make the symmetry store to each other. So every time I do a timing parameterization I will do an accompanying internal boost, so that if alpha was 0 before it is 0 after.

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Luca Ciambelli: and the way it goes is that the internal boost should have this specific form.

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Luca Ciambelli: and notice that this is by for the tech. For the experts in the audience. Notice that this is a field independent. Now, because I've assumed that dial equal to 0.

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Luca Ciambelli: Okay, so this prime phase space I've been focusing on is a way to focus or recharge the sectorality by imposing that that equal to 0 and exploiting my senators to do so.

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Luca Ciambelli: Okay, now, as I said, that, that is 0, and therefore we can also gauge. Fix it. And I'm gonna gauge fixing to be partial D, so I'm gonna just introduce a coordinate V for the non generators. And since this is constant throughout face, I'm gonna keep it partial D, and this is very useful to define the Poisson brackets.

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Luca Ciambelli: Now that each other constraint written in this time at this very specific form.

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Luca Ciambelli: And actually, what's remarkable about this equation now is that you can immediately see the spin. They sectors here. This is the spin 0 sector. This is the spin, 2 sector, and this is the matter tensor, the sector. In this precise way. So this is the constraint written with respect to this specific time.

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Luca Ciambelli: and the charge that we compute on the prime phase space turn out to be the charge under time. Reparametrization for the dress face base. That's this prime here. Turn out to be just the reach audio constraints, plus a corner term, which is the charge that I'm gonna

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Luca Ciambelli: talk about later.

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Luca Ciambelli: Okay.

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Luca Ciambelli: very good. Now, we can compute the kinematical Poisson bracket for this prime phase space. So first of all, the Omega canonical that I've written before. That was just. It was simply Delta! Of this data, canonical that was written as momenta times delta of the sources take this form

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Luca Ciambelli: when I now assume del equal to 0, went to the L equal partial D

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Luca Ciambelli: and the to compute the Poisson bra. Sorry to compute the Poisson bracket. What we introduce is the Beltrami differential. So this Q. Bar a. V is a unique modular. 2 dimensional metric.

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Luca Ciambelli: Okay, although it's not the generic part is 2 dimensional, and therefore I can generically parameterize it as a function of Zeta and Zeta Bar, where Zeta Zeta bar are the Beltrami differentials. Okay, so this is a convenient way to explode to web. The is imp.

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Luca Ciambelli: To have explicitly the 2 independent degrees of freedom in this metric.

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Luca Ciambelli: Okay? So one should think of the zeta zeta bars as the sources for this PIN. 2 sector.

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Luca Ciambelli: I'm going to introduce another quantity, which is a bit ugly! And apologies about it. There's no better way to write it. It's called a propagator. In a it would be clear in a moment. Why, I call it propagator, and it has this exponential dressing here that has this specific function that resembles an Olonomy in this into sector. It is local on the base, and it contains a heaviside step function on V,

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Luca Ciambelli: and why this propagator is useful, because now I can write down the brackets, and in particular the spin 2 brackets. Remember, here. P. Is just modular dellonomy proportional to this the spin 2. But Beltrami fields. Brackets is just proportional to this propagator I've introduced.

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Luca Ciambelli: Okay, we can go a step further.

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Luca Ciambelli: compute the bracket between the spin 0 field Mu and the spin, 2 field zeta, and this now is not 0, and it is given by the V derivative of zeta, which contains information about the shear. So whenever I have sheer full. Situation! Instances which from the bulk are associated to some gravitational perturbation arriving to the surface, then the spin, 0 sp. 2 sector mix in this way.

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Luca Ciambelli: There's been 0 data satisfies just an Eisenberg algebra there. Just an Eisenberg pair throughout him.

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Luca Ciambelli: and very importantly, new, is a no commutative field among itself on the face space. And again, this is due to the fact that there is this radiation

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Luca Ciambelli: arriving. If there's no radiation, of course Mu becomes a commutative field.

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Luca Ciambelli: So let me see, stop some of the results.

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Luca Ciambelli: There's just a delta here. So on z bar. So on the cart. This is ultra local.

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Luca Ciambelli: Every generator talk only to itself, to its future. And it pass. But it does not talk to the next generator. Okay, so it's ultra local on the base of the cut. Okay.

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Luca Ciambelli: so this is because everything has a delta. 2.

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Luca Ciambelli: However, there is no commutativity on the non generator. Okay, so each non generator contains a intrinsic reach structure, but they are all split.

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Luca Ciambelli: split in the algebraic sense.

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Luca Ciambelli: And let me insist again about the fact that no, perturbatively in G. Newton here. I'm not doing any sort of perturbation. A mu is a no community field that mix the spins. Even spin 2 sectors in this way.

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Luca Ciambelli: and this is gonna be very important. And finally, as I already mentioned, let me further stress the wither locality of carole and physics, so on now, like our surfaces each generator can be treated as independent. This is very different, radically different for what happened on the time, like on the space like upper surface and the time like evolution.

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Luca Ciambelli: Okay, this term, this ends the sympathetic part. Are there questions?

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Luca Ciambelli: Okay, let me move on, then, and let me talk very quickly about the boost charge. This is something that I'm gonna just talk ambassad, because I want to focus on the most now

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Luca Ciambelli: new things that we're talking about about the perturbation, quantum perturbation and atomic representations which I think they're perfect for this audience. I'm just gonna mention that if one wants to decouple this being 0 and spin 2 sectors, of course, one way to do it is to set me equal to 0. However, mu is very important, dynamical, variable in the tier, so we cannot just set it to 0. This will be a restriction on the face

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Luca Ciambelli: space. So the way to proceed is to trade it with something else. Okay? And actually, I'm going to show that it can be traded by a the notion of time which becomes now part of the face space. So I consider a different morphism, which is just a time requirementization.

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Luca Ciambelli: And then I'm gonna just address all my field with this different morphism. I am. Yes, sorry.

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Luca Ciambelli: Yeah. Sorry. Sorry I actually did it. Thank you. Thank you. Bye. So Sigmar. The coordinates on the count.

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Abhay Vasant Ashtekar: Okay.

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Luca Ciambelli: coordinates on C. Yes.

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Luca Ciambelli: I call them sometimes it does it a bar. So that's why so yeah. So as I was saying, Sigma Ab, now, this guy is the share. And now I'm considering a d feel, but it's always on the prime phase space. So whenever I do it, if we are do also a boost to compensate. And that's why you see this boost the way it's appearing here.

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Luca Ciambelli: and but also, very importantly, the way it is Mu is defined. Or if you want K itself, Kappa itself to an affinity transform as a connection under this. And I'm gonna exploit this fed. So the trading become is really about the fact that Mu is a connection with respect to this. And of course a cross check is that the Rachadic constraints is a a boost to

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Luca Ciambelli: covariant expression, right? And then as I was saying, I'm gonna go on the Pr. On a different phase space where I assume new Tilde equal to 0 and to do so. As I just said, I gonna have to write Mu as a function of D, and now, since Delta Mu is not 0,

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Luca Ciambelli: whenever I see Delta Mu, I'm going to use this expression to rewrite it as a function of Delta B, and that's the trading I was mentioning. Ok. Now let me make a very important comment.

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Luca Ciambelli: It is true that mu and mu is just kappa plus title.

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Luca Ciambelli: Now, if I am on a non expanding horizon. Tita is 0.

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Luca Ciambelli: Mu is equal to Kappa, but Kappa is the in affinity. But for non-expanding operon this is equal to the surface gravity.

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Luca Ciambelli: and then saying that

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Luca Ciambelli: saying that Kappa is 0

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Luca Ciambelli: is the same. Sorry. Say that viewing 0. So this stressing time I'm using for non-expanding horizon is exactly the same as the find time. So dressing time is the same as a fine time for non-expanding horizons, and this is important, because all the beautiful results that have been developed in the past in gr for non-expanding horizon in a fine time supply to everything we're doing here.

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Luca Ciambelli: The difference arise when I consider now Hypersurface, which is not in on expanding horizon, then this is a new time, which is different from the find time in that framework.

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Luca Ciambelli: Okay?

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Luca Ciambelli: So, as I was saying, I trade the dressing. I trained the surface tension with this dressing time, and therefore I'm essentially implementing in this theory a a dynamical frame in the sense of these references here, and then I can just take my canonical sympathy, structure, and rewriting it as a function of Delta V.

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Luca Ciambelli: If I do that systematically. The final result is that. And I'm not very careful about the cut here, because it's important for me to focus on the constraints on the full upper surface is that there's no more delta mu which delta omega in the bulk

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Luca Ciambelli: that has been completely trade off with Delta V.

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Luca Ciambelli: Wedge delta of the constraint itself.

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Luca Ciambelli: So the dynamical dress in Time V is conjugate, sympathetic, conjugate to the rechaddy constraints.

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Luca Ciambelli: and that's beautiful, because we know that time requirementization is associated to enforcing the rich other constraints. And now we see that the level of the face space.

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Luca Ciambelli: Okay, this is what I just said in words here

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Luca Ciambelli: and now these 2 fields are dressed with respect to this.

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Luca Ciambelli: Okay.

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Luca Ciambelli: now, one of the consequences is that in dressing time we can show that the boost charge is monotonic, and let me do that very quickly before moving on. So I remember for you that the dressing time. Richard, equation is this one because the usual deter each other. Equation in any times will be this one. And I just said mutila to 0.

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Luca Ciambelli: And then let me consider the charge associated to the parametrization.

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Luca Ciambelli: Now, I'm gonna focus on a cut, and I'm gonna focus on the generator, which is a boost who's fixed point is the cut C sub t here.

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Luca Ciambelli: and that's introduced by the specific time reparametrization. V minus t partial d.

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Luca Ciambelli: okay, so this is a boost to fixed point is the Cut city. This is an intrinsic picture from the bulk perspective. I have something like this. Okay, this picture here is this, this cut here.

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Luca Ciambelli: Okay? And I'm gonna talk about boost because it's monotonicity and the properties of modular flows has been historically used by con and con and robe to study the cross product, and most recently reused again in this framework, to show the the reduction from type 3 phonom algebra to type 2. Affinity in the presence of gravity constraints.

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Luca Ciambelli: Okay, that's why I'm interested in the boost. And what I'm going to show is that it's 2 properties. First of all, add the cut Ct.

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Luca Ciambelli: the boost charge is just positive, because it's just proportional to the entire area of the gap.

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Luca Ciambelli: So the boost charge is positive, and the boost charge, if I rewrite it with the boost charge aspect, as the property that is first derivative is just v minus t

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Luca Ciambelli: times the spin, the complete spin, 2 plus matter sector

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Luca Ciambelli: here in the middle. At this equality I have used Rach Audrey in dressing time.

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Luca Ciambelli: and therefore interesting time. The boost charge for a future directed boost. So for B bigger than T. Assuming that matter satisfied, the null energy condition is monotonic. Its first derivative is positive.

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Luca Ciambelli: Okay.

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Luca Ciambelli: so this boost charge now has been derived completely from the phase space and realize an idea that has been and appeared already later literature that has been under discussion these days, which is the notion of dynamical entropy.

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Luca Ciambelli: We want to generalize to a generic analog per surface. Essentially, the notion of thermodynamics that people have done for the horizons. And to do that we use the property of this boost generator as dynamical entropy, and we write down the so-called Dynamical Cycle, though this has been introduced by the Jacobson and Parentani

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Luca Ciambelli: in all 3, and there's been workout recently by some collaborators and friends. And there's actually 2 works in progress by a world and collaborators and months these certain collaborators on this notion of dynamic calendar.

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Luca Ciambelli: Now, while one can introduce this dynamical entropy, that there's this general icycle, low framework

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Luca Ciambelli: in the language of evolutionary law. Here we have done it directly from the phase space using the boot charge. And this phrase in this statement is true in dressing time. So we've shown why the dressing time is also useful in this language

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Abhay Vasant Ashtekar: is is the boost the same as the one that was given by Cfp. The prabhu

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Luca Ciambelli: so it is the boot. Okay, let me go back. What do I write it down? I write it down over here. So this

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Luca Ciambelli: is the so the right hand side. Yes.

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Luca Ciambelli: but what it correspond to to the left-hand side. No.

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Luca Ciambelli: so it's not for them. Is not the reparative. The time reparametrization in dressing time for the prime dipmorphisms.

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Luca Ciambelli: This has not been done by them, they have considered a boost only so, indeed, if you do an internal boost. So this is a dipyomorphism boost. I should have specified, of course, a dye morphism boost, but you can also do the internal boost I talked before. Which is this symmetry?

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Luca Ciambelli: And then you can compute Q. Lambda. and if you compute Q. Lambda, essentially, it becomes the area. a bombing.

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Luca Ciambelli: which is what I write down below here.

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Luca Ciambelli: So this is the boost considered by Cfp.

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Luca Ciambelli: So now.

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Luca Ciambelli: is that right? Thank you.

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Luca Ciambelli: Thank you for the question before actually, I move on, are there other questions on this? I'm gonna switch gear completely.

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Laurent Freidel: Maybe just just

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Laurent Freidel: to add to a buy comments.

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Laurent Freidel: I mean this equation you you have here shows on the right hand side that if there's no flux

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Laurent Freidel: now? Sorry. No. The question that was later

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Luca Ciambelli: later.

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Abhay Vasant Ashtekar: yeah. Dv, of S. So if you look at the right hand side, dvos. Can you comment on?

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Laurent Freidel: You know, Omega sigma square plus TV. This object is 0.

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Luca Ciambelli: Yeah.

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Luca Ciambelli: that that's the main point.

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Luca Ciambelli: If I don't have spin 2. And I have no matter. Of course. This is Jazir.

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Luca Ciambelli: So what I should have said and thank you bye. And Lauran, is that not only this boost chart is monotonic, but the inequality is saturated of flat space.

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Abhay Vasant Ashtekar: Yeah.

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Luca Ciambelli: okay, thanks a lot. Thank you very much.

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Luca Ciambelli: Not just flat space. It can be stationary space. And yeah, yeah, exactly. Whenever there's no spin to a matter degrees of freedom, the boost

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Luca Ciambelli: is saturates the inequality. Yes, yeah, that's very. That's very important, and that's also one of the property that I know, these 2 collaborations are requiring on this notion of dynamical entropy

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Luca Ciambelli: because it has to measure something dynamic if you want. So that's why your question was extremely relevant.

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Luca Ciambelli: Okay. let me other questions. I can move on.

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Luca Ciambelli: Okay, so let me talk a bit about work in progress. So I hope we will able to report with the paper next month about it. But the there are some parts of it that are actually for the respiration either. So now, wanna talk about some quantum properties, because so far it was completely classic. And the first thing I wanna talk about is how to set up this quantization scheme. Now, of course, one can do.

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Luca Ciambelli: Well, first of all. Let me plot.

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Luca Ciambelli: The 2 dimensional plane where I put the Newton Constant on the Y axis and H. Bar on the X axis, and what we have done so far is just imposing. The Rachado reconstrain in a point which is just here right. There was no H bar. It was all classical. The matter itself was assumed to be classical so far, and we just assume Rachaduri. For finite G,

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Luca Ciambelli: and we've noticed the properties that we've seen before for the kinematical Poisson bracket.

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Luca Ciambelli: Now, of course, the the goal here, if we want to talk about quantum gravity, is to arrive here, is to arrive to this point here, somewhere in the middle of the graph, for finite value of the Newton and finite value of H. Bar. Not infinitesimal.

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Luca Ciambelli: and the only way as preaching to the choir again. This would make sense is indeed, if we can write this dynamical equation as an operatorian state. So putting ahead.

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Luca Ciambelli: Hand-waving. Putting a hat.

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Luca Ciambelli: Okay, however, doing this line. So setting up a perturbation in this way as some drawbacks that we've just because of what we've just seen. First of all, the sectors mix so the spins even spin 2 sector meets. It's very hard now to set up a perturbative scheme in H bar, where you don't have a

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Luca Ciambelli: split of the sectors. Precise.

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Luca Ciambelli: Okay, this is challenging, even even just to set up this part of the perturbation. Let aside all the way here.

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Luca Ciambelli: The other feature is that it would require to quantize the geometry in a non-perturbative way, because, even if I am perturbative in H bar. I am in a situation here where I am strong gravitationally

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Luca Ciambelli: so. I have a geometric setup where I have some spin, 2 gravitational radiation spin 0 degrees of freedom, all coupled together. No commutative, so it would be going directly from no quantum gravity to quantum gravity with quantum geometry.

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Luca Ciambelli: and of course we know a way to go there. This is something that people at the audience have discovered, but we want to try to set up a perturbative scheme to arrive there.

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Luca Ciambelli: and the last feature is the fate of symmetries.

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Luca Ciambelli: If we do it this way, it's unclear how the constraints will act on the different morphism invariance, but it should commute quantumly with the different morphism invariant objects of the tier.

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Luca Ciambelli: So what we're gonna propose is rather a detour to arrive to this final and this detour. I'm gonna call it perturbative quantum geometry, because I'm gonna show that one can talk about quantum geometric operators in a perturbative set.

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Luca Ciambelli: Okay, so what is the teacher? The teacher is forced to forget H. Bar. And just consider a classical with grab, weak gravity. This just means setting G bar small in whatever we have done so far, there's no H bar. Everything is classical, and we just have one dimensionful constants. G, and we just set it small.

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Luca Ciambelli: Okay, small compared to the liberal lead, sorry, small, compared to the typical scale of the system.

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Luca Ciambelli: should celebrate. And this is just telling you that the constraints will develop with expand as a leading order, constraint, sub leading and further sub leading. Okay, am I gonna stop to see 2 for the purposes of the discussion today.

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Luca Ciambelli: Okay, what we're gonna see explicit in the moment is that the sectors? And by sectors I mean spin 0. Sorry spin 0 to and mother

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Luca Ciambelli: completely. The couple want to each other perturbatively. And this is one of the reason why we want to go down this road is because we have fully control of the geometric data, the radiative data and the matter data one after the other.

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Luca Ciambelli: Furthermore, we can talk in this framework about back reaction on geometry in the perturbative way. So this is another way to avoid directly talking about nonperturbative quantum geometries.

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Luca Ciambelli: and importantly related to the comment about symmetries before the face space is fully under control. So we can just ask that order by order. Our childly constraints infer the symmetry constraints on the physical, observable.

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Luca Ciambelli: Okay, but so far this is something well known. It's just the weak gravity. But what we're proposing to do later on is to simpletically quantize the phase space for finite H bar. So we're not gonna assume H bar small. In doing that, we're just taking the simplest structure and simpletic poisson bracket, and we're just quantized them canonically.

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Luca Ciambelli: This bring us to this

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Luca Ciambelli: bottom right corner of the diagram, where now, g neuter remains small, because this is just a second order constraint in genuine. But H. Bar is fine. So I put this point directly below the endpoint. I want to go. Okay.

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Luca Ciambelli: What we are. Gonna see doing this first of all, as I just said, is that we can do it non perturbatively in H bar. We don't have to assume that the quantum effects are small.

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Luca Ciambelli: and we're gonna be able to talk about quantum geometric operators for the spin 0 sector in this perturbative setting.

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Luca Ciambelli: And though, furthermore, as I already said, this is just a consequence of having a controlled phase space in the week rabbit assumption on the left. The face space is under control. So symmetries are under control.

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Luca Ciambelli: I insist again that in this framework here.

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Luca Ciambelli: due to the decoupling of the sectors, we're going to be able to talk about spin 0 operators.

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Luca Ciambelli: Okay?

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Luca Ciambelli: And then, finally, of course, we want to go back to the endpoint. So at the beginning we had the perturbation sector we were talking about going there. And we're just taking A. D 2.

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Luca Ciambelli: Okay? And I want to see that this is different from the semiclassical regime. The semi classical regime that people talk about is quantum gravity, where we take the perturbative scheme of

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Luca Ciambelli: keeping G. Finite and send h bars more.

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Luca Ciambelli: which would be this now, since the scale of the system is anyway defined by the plank led, divided by the De Broglie of the problem, the Bolly

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Luca Ciambelli: land of the problem.

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Luca Ciambelli: This could be seen as taking G. 5G. Small or H. Finance. But this is actually what

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Luca Ciambelli: the perturbative quantum gravity scheme is typically implemented to be in this framework. The Einstein equation that people would write just to make an aside comment is that Jim, you knew, is 8. By G, the expectation value on a specific state of the quantum matter operator team. So this now is interpreted as A. C number

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Luca Ciambelli: in this setup. Okay, this is not the framework we are using. Of course, the endpoint at the end of the day is the same. So we're gonna end up on this line, no matter what. But the way to quantize for us is to go down and then up here.

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Luca Ciambelli: Okay? And I think I said everything. So let me start doing that.

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Luca Ciambelli: Okay? So first, the first step is the week gravity dispatches. So it's just taking genu to small. There's no other there's no h bar, and this is be is done by talking about this typical scale. 32, by G is a framework that people should be familiar with from the the dollar gauge analysis of gravitational waves. Except here we do it for the geometry.

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Luca Ciambelli: Okay?

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Luca Ciambelli: So we expand the aerial form in this way, and then we include gravitons or radiatives in 2 degrees of freedom inside the Beltrami differential.

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Luca Ciambelli: except that, of course, we make them started order root G. This is the usual way to add, therefore, the Hamiltonian for the spin, 2. Which is just the canonical one in perturbation theory.

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Luca Ciambelli: and indeed the sheer starts now at order. G,

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Luca Ciambelli: okay, so I have a background geometry and some sheer at second daughter.

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Luca Ciambelli: The Richard constraints has this specific perturbative expansion, and let me just write down the C 0 order. Now, the C 0 order is this way, because the matter the way the matter couple, remember, had an a pi, G.

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Luca Ciambelli: So therefore, it couples a total epsilon square, and that as we're gonna see couples in C 2. Okay, that's why the C 0 constraints. Sorry in the setup I have established. Just sorry. The C. 0 constraints I set up I've established just contains being 0 degrees of freedom.

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Luca Ciambelli: and there are 2 ways to solve this one way is to go into stationary backgrounds. So we just have a specific solution, which is just that theta 0 is 0 and Mu bar, therefore, remains completely free.

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Luca Ciambelli: and the other way is to go on a generic background and solve Mu bar as a function of Omega 0. And notice that this solution is exactly the same framework that we did in the dressing time before.

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Luca Ciambelli: Was there a question? I hear a glitch?

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Luca Ciambelli: No, okay.

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Luca Ciambelli: very good. So for the purposes of this talk, I'm gonna just assume a a stationary background to begin with. Okay? And therefore also there is another issue appearing, which is that if we just expand the addressing time as well in all powers of Epsilon.

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Luca Ciambelli: actually, one can show, and I spare you the details is quite technical that there is a phenomenon called proliferation of face-based variables. And therefore the face space is not uniquely determined perturbatively.

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Luca Ciambelli: So what we're gonna do is that we're gonna start

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Luca Ciambelli: with the non nonzero values of Mu. But in the perturbations order, and we just stop at Order 2. Here, we're gonna reach the dressing time.

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Luca Ciambelli: Okay, by doing that. The C one constrain. You see, there's no new one appearing here is very simple. It has the exact same shape as the C. 0 constrain. There's no external sources. So say that Theta changes that forced order would be a completely pure gauge assumption. So we just assume that theta one is 0 as well. And then, finally, we reach the C 2 constraints where things get interest.

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Luca Ciambelli: The C 2 constraint display this being 2 degrees of freedom. These are the perturbative Beltrami gravitons is being 0 sector and the matter sector like this.

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Luca Ciambelli: And now, if we introduce the spin, 2 stress tensor defining this way and the spin 0 stress tensor defining this way, we reach a shape of the each other equation, which, as advertise, is a stress tensors balance law, English.

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Luca Ciambelli: so that each other equation at Order 2 is just written as the sum of the stress tensor of the check of each sector to bunch.

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Abhay Vasant Ashtekar: Let me reiterate that because this is an important.

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Abhay Vasant Ashtekar: the one before. Yeah, yeah, no. Okay. So here. So I mean, if I were to think of this bill to part as being, you know. radiative and spin 0 on spin one subset

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Abhay Vasant Ashtekar: then, in the stationary case, of course, there's no radio part of this ping tool.

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Abhay Vasant Ashtekar: and but that could not be possible. So now here I want out. What I would say is that you are.

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Abhay Vasant Ashtekar: Are you keeping terms which are of order, 2 in the colobic part, that it's been 0 and spin one part.

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Luca Ciambelli: Yes, but but yes, but only one. So let me just show you explicitly the question I wrote before. So I'm gonna keeping sorry this. This, I should have stressed that I called the order 2 of the area for gamma. So you see, it appears here and there. But that's what I was saying when I did the perturbative dressing time. This is mu 0

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Luca Ciambelli: plus order, epsilon 3. That's the sense in which I'm keeping the spin 0 coulombic data. But only half of it at that expansion order.

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Luca Ciambelli: And that's why there is this coupling.

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Luca Ciambelli: Thank you. Thank you for the question and clarification

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Laurent Freidel: additional comments. I don't think you you really say it, but it's very important to appreciate that because of the Utah locality.

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Laurent Freidel: right? What you're doing here is that there's a one Cft paranoia. So now you

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Laurent Freidel: it will come. It will come surface. You really need only to quantize one dimensional system. It will come in the next slide. Yes.

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Luca Ciambelli: Sorry. No, no, no, no problem even better creates anticipation.

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Luca Ciambelli: Okay, so I have. I have 10 min more or less. Right? So I think I'm gonna be able to touch about it. As as now. As Lauren was saying, I'm gonna focus on the spin 0 sector. And I wanna make some comments. I'm gonna reiterate what we've just found. So we found that the reach out to order 2 is a stress tensors balance law. That is this far.

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Luca Ciambelli: We found that perturbatively, or maybe I am just stating it for you now, but it can be show fully, rigorously given. The assumptions I've made perturbatively. This pins even, has been 2 sector decouples. Now

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Luca Ciambelli: so Mu becomes a commutative field. First of all.

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Luca Ciambelli: up to the order of of perturbation. We're working these zeros means up to subletting orders. It commutes with the spin 2 sector, and therefore the stress tensors of each sector form an algebra. But commute with each other sector.

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Luca Ciambelli: Okay, now let me insist on this PIN. 0 geometric stress tensor. It has this specific form

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Luca Ciambelli: and interpreting the geometric things degrees of freedom. As a stress tensor, as carrying a stress tensor means, for instance, that one can talk about this in 0 energy now, or one can talk about this being 0 algebra. And that's what I'm gonna do next. And indeed, in another way of saying is that it brings it an exactly equal footing as the other sectors.

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Luca Ciambelli: And now, therefore, we have a framework where not only we can talk about perturbative quantum matter or perturbative back reaction, but we can also talk about perturbative quantum geometry.

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Luca Ciambelli: And indeed, I'm gonna show that distrust as interpretation allows to interpret Gamma as an operator. Now.

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Luca Ciambelli: okay, and that's the next sector. And it has to do again with the quest, with the observation that Lauren stressed and should be stressed throughout all these lines, which is this fantastic power of ultra-locality of Caroleian degrees of freedom.

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Luca Ciambelli: Or, if you want an all library surfaces of freedom. So let me return to the spin. 0 again the spin. 0 sector is again an eisenber up to the order we want. Where now I'm talking about Gamma, not Omega 0 Omega. 0 became background. Structure is no more part of the phase space.

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Luca Ciambelli: and the new. As I said, commute. Now

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Luca Ciambelli: we can canonically quantize this symplet structure. It's just an Eisenberg pair. So we can replace this by a commutator. And if we consider Weitman function and the OP. E. Therefore we just say, you know, we just take a positive frequencies. We get an OP. Between the field, mu and gamma, which is just this form here.

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Luca Ciambelli: Okay? And notice again, the ultra-lucality delta, 2 in Z, minus Z, one minus 2. Ok, now, I'm finite in H bar. This is not H. Bar corrected.

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Okay?

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Luca Ciambelli: And then remember that the spin 0 stress tensor is this. Now, these 2 equations are exactly the one satisfied per point with our locality on the cut by a curved Beta Gamma system. What I mentioned in the introduction, and what have I asked about. It's indeed seen by these 2 properties, the OP. E. And the stress tensor. Indeed, let me write the opes of a Beta gamma system. This is a chiral cft on the well, here on the null generators

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Luca Ciambelli: per point, it has this form

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Luca Ciambelli: per point on the base

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Luca Ciambelli: and the stress tensor as this form and the analogy between the 2 suggest to identify mu with Beta. And this is good because Beta in the Beta Gamma system is a connection. And Mu was a connection, as we shown. And this particular twisting, this is kind of a technical detail with E to the 2 gamma.

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Luca Ciambelli: Okay, so this now is a bitagama safety. It can be. It's completely quantum. We know. All these properties is whenever I don't talk about this being to a matter sector, which I can not talk about perturbatively. This can be fully quantized on its own is a beautiful, simple tier.

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Abhay Vasant Ashtekar: Is this not a small extrapolation? Because to talk about positive frequencies. And then this informal field theory, you really need V to run from minus infinity to infinity.

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Luca Ciambelli: that in your case it would not, because of, you know, costing something like that. That is a a very, very beautiful comment. And II have an answer to that. I'm gonna just make this point here finishes live and reply to that. So I was, I was just gonna insist again on the power of ultra-locality for this null theory for this Carolinian cfts. This PIN. 0 geometry, as I said, is a Beta gamma safety on 9 lines, but each of them independent.

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Luca Ciambelli: Okay? And let me just go back and comment on a buy question. Slash comment he he's perfectly right. There is this issue, which is that all these should be defined in a situation where L goes from minus infinity to plus infinity. And this will not generically happen.

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Luca Ciambelli: However, here I am, considering something very simple. I have under set of mild assumptions. So if you assume that the background is stationary, which is what I've done here, I've assumed. If you, Omega 0 0, a solution of the 0 order.

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Luca Ciambelli: Then you can think yourself of being on an event horizon. Of course this is not the only case, but if you are on an event horizon, then you know that the stability of horizons imply that there are no caustic in this time.

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Luca Ciambelli: Okay, so in this very simple situation. It it it can work what I'm doing here, and of course, more generically. It's very hard. But I want to make another comment about the more generic case. And this is something that I've been thinking about these days, and it is the fact that when you write the Richard equation with Tita.

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Luca Ciambelli: So if you write the Richard equation as Dv. T. Theta plus theta square over 2 equal dot dot dot tta goes through minus infinity through acoustic.

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Luca Ciambelli: However, thanks to our time and our prime phase space, we can rewrite this equation as d omega d square omega

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Luca Ciambelli: equal omega times the rest.

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Luca Ciambelli: Now these 2 equations are the same equation. But what this equation has is that omega goes to 0 at acoustic.

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Luca Ciambelli: So this equation is fully regular, namely, you have a 0 equal to 0 left and right and sides.

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Luca Ciambelli: This I'm not saying, I'm not saying this. So on on my issues, but it gives me a hope of being able to see caustics is just as Junction point where I can glue together each portion of the Nullipersurfaces which is delimited by the caustic.

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Luca Ciambelli: Okay, it's again. It's not an answer. It's not a solution of the problem I'm just hoping upon. I'm just hoping there's a possibility there.

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Abhay Vasant Ashtekar: No, I think that unless you have non expanding horizons this really will not work. I really don't think it will work. I mean, your doctor talked about the horizons themselves. and that are no costing. But there, no cost is precisely because of bifurcations, you know. 2 black folks come together. It's not going to be a single world. 2

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Abhay Vasant Ashtekar: this this map.

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Luca Ciambelli: I think we should just go. II mean, I think we have. I'm I'm just being very greedy here to say that you can, you know. Perhaps you can do more. But I really feel ahead because I don't want to take too much.

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Luca Ciambelli: Okay, so yeah, let me move on and let me just recap what we've done here. So we've seen the problem of a fine time I call it the problem of a fine time to make a plan with respect to the gravitational problem of time. I'm just saying that Miu is crucial in what we've been doing. It's related to what we actually just discussed the technical discussion we had with the buyer. And now and the we've seen that this picture now give us some hope, because this been 0 is easy to quantize.

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Luca Ciambelli: Another very important aspect I didn't insist on. I will do in the next 3 min for the atomic sector is that we do not need to discretize the null generators. The whole point is that the cuts

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Luca Ciambelli: should be, and that's what I'm gonna propose later should be discretized or will have to be discretized. But the null generators contains in the spin 0 0 sector, just a beta gamma, CFT.

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Luca Ciambelli: So this is a continuous theory, and it does not give rise due to ultra locality. It does not give rise to UV divergences. The continuity of the null generator is not an issue for the quantization of this theory. In this framework

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Luca Ciambelli: this is very important, and indeed what I was saying for the cat appears here. There are universal UV divergences, nonetheless in energy fluctuations.

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Luca Ciambelli: because if I now take the stress tensor tooling functions, the Tt, if you want OP. E. Not to pay function? I did. I don't want to talk about stays. Just the Tto. P.

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Luca Ciambelli: It contains a double delta as central charge. So one can see this feature as having. This is an explicit computation as having a central charge that goes to infinity, due to the infinitely many degrees of freedom on the cuts.

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Luca Ciambelli: Okay? And if can, I have 2 min more, I see that it's almost 10.

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Ivan Agullo: But to me, that's

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Luca Ciambelli: okay. Thank you. So if that's fine in these 2 min, I will briefly talk about that. So

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Luca Ciambelli: in the same vein of something that has been discussed by this community, for since many, many years, and some of the historic paper are cited here, there are many. There are infinitely many degrees of freedom on the cut. And this is something that whatever situation setup you set up with it will lead to a UV divergences, just because for the just, because of the infinitely many degrees of freedom.

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Luca Ciambelli: These, as we just show, implies that the central charge in the Tt. Obe is infinite.

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Luca Ciambelli: just because there was a Delta square.

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Luca Ciambelli: and if they set up we've been doing, therefore we post too late that quantum gravity, then point here should have a finite central charge, and the way. And now we introduce a way to re realize that in our setup.

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Luca Ciambelli: So the way to realize that is that we have to entertain the idea that the aerial form operator omega heads omega hat in the non perturbative setting, or gamma head in the perturbative setting, must have representation with discrete supports what does it mean that I'm gonna write an equation later on and

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Luca Ciambelli: stressing against something we've already shot. Talk about utter localities fundamental. This can be done preserving symmetries, because everything the symmetries realize is null time re parameterization. It's all about the null generator is never about the first neighborhood on the count.

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Luca Ciambelli: and this is really something completely different from time, like a space like entities. If you want the ultra power of now.

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Luca Ciambelli: Okay, so what do I mean by the area entertaining that the area of discrete representations. I just mean that as a function of the continuous function of the. So here there are no restriction about

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Luca Ciambelli: continuous Cv.

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Luca Ciambelli: We can entertain the idea that on the cut there are all these certain points that are turned on by this operator. Okay, this sum goes from one to capital N, and we just have the sum through the points in the cut.

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Luca Ciambelli: This is a property of the representation as just, I just said, as I just already written, is continuous in the null generator is very important that I'm not discretizing those, and it leads to a finite center central charge. Because if I just now use that in what I've been doing, so by saying that I just go again perturbative and talk. Come ahead.

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Luca Ciambelli: I get here. Of course, it's very simple that the deltas were come just in.

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Luca Ciambelli: Okay.

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Luca Ciambelli: And this is just a very concrete way of starting from a classical face space under control and quantized, this being 0 sector in a way that leads to discretization of the area operator, which is something that should be very familiar for a lot of you.

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Luca Ciambelli: Okay, so again. And therefore here, just one final comment, and then I'll stop with some conclusion. This being 0 central charge, therefore, really counts the geometric degrees of freedom in this setup.

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Luca Ciambelli: Okay, and thank you for giving me the 2 extra minutes. I conclude this one slide is very quick. I just recoup what we've seen. That each other constrained is can be interpreted in 3. Second is a Caroleian conservation law

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Luca Ciambelli: the sympathetic analysis reveals the spin. Zero's been to mixing. So a radiative and coulombing sector, mixing in a non-trivial way for finite G. Newton.

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Luca Ciambelli: We've shown that. Nonetheless, if we go interesting time, we can prove boost monotonicity, which is a very important properties for the discussion of dynamical entropy going on these days. And we have essentially argued that this perturbative quantum geometry set up we've been talk about is a viable bottom up approach to the quantization of geometry.

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Luca Ciambelli: So we have seen, indeed, arising as PIN 0 safety geometry.

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Luca Ciambelli: And finally, we have just humbly commented about how atomic quantization may lead to UV finatness in the spin 0 sector

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Luca Ciambelli: future directions. Of course, the more this is under development it will not take us long, I hope

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Luca Ciambelli: modular theory and quantum focusing contractor is something, I think, in these days. The generalization of this quantum information properties of horizons to genetic. Now, hypersurfaces. And of course, the relationship between these atomic quantization and loop quantization that has been doing by

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Luca Ciambelli: some of the audience here. Okay, thank you very much. I'll stop here.

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Ivan Agullo: Thank you again. This was a very, very interesting talk, and clear.

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Ivan Agullo: and questions to look at.

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Abhay Vasant Ashtekar: if if there are questions I have some comments, but I think I should. Other people should talk. Ask first.

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Daniele Pranzetti: Can I ask a question, Ivan?

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Ivan Agullo: Yes.

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Daniele Pranzetti: Hi, Luga, thank you. I'm not sure it was very nice talk. I wanted to ask you if I take like a non infinity limit, that this be 0 degrees of freedom, would they correspond to some like the Goldstone through translation mode? Or is there a way in which I can connect with what we know about columbic degrees of freedom and non infinity.

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Luca Ciambelli: so okay, so this is a a beautiful question. And I was, I was gonna II was gonna comment about it, can I? Oh, yes, I was gonna comment about it. Here when I talk about Richard recon conservation law. And of course this

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Luca Ciambelli: this Sector Karen conservation law for nulliber service is started in the framework of flatallography.

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Luca Ciambelli: So we have seen these Carolyn conservation laws as dynamical laws for Einstein gravity at null infinity, not at finite distance.

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Luca Ciambelli: So it's a very natural question. The one Daniel is asking on how to connect what we've been down here for finite distance plus in topic.

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Luca Ciambelli: There are work in progress that are not published on this. What I can say is, One thing is that if one thing just bold phase, constraint.

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Luca Ciambelli: What it leads to asymptotically is the equation. Dv. If sorry, let me call it u, as is more traditional for Scribe.

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Luca Ciambelli: do you? Or it's even married I should use

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Luca Ciambelli: DU, because it's just a

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Luca Ciambelli: shareless part, which is DU plus du that

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Luca Ciambelli: Q. Capital a. B. Acting on Q. Capital a. B equal to 0. This is, as we know, is the first is the leading order of Einstein equation on Scribe.

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Luca Ciambelli: It doesn't tell you, and it doesn't tell us anything dynamical. But it's important that because this means that the radiation degrees of freedom affect the sub-leading orders of the metric and not the leading order.

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Luca Ciambelli: On the other hand, as you go sub leading now in Rechad, there is the bondi Maslow's formula, so the bondi Maslow's formula.

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Luca Ciambelli: minus is a loss

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Luca Ciambelli: is realized from Richardi as one goes up leading

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Luca Ciambelli: in the expansion, you know.

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Luca Ciambelli: So our to infinity. The leading order is this is leading it that so the interpretation of gold stones at infinity for this being 0 sector is kind of hidden in these separation of orders. I don't wanna say too much, because I'm not a hundred percent sure. And for instance, I Loren had some comment before, so if he's still there, he might have something more to say. And you himself, you yourself, Dana, you might have more to say about it.

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Luca Ciambelli: but for the moment. And it is us actually something we're working on. The the chatary constraints give rise to the bondi Maslow's formula in the right Reg.

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Luca Ciambelli: that's what I can say.

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Daniele Pranzetti: Okay, thank

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Laurent Freidel: right.

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Laurent Freidel: Maybe I can. I can. I can add the I mean there, there's a upcoming work where we show this discounting feature is is extremely important, because you can view as long as infinity as a as a kind of a knowledge surface, you know, you know, come formally, compactify space time. You need that contacts you can.

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Laurent Freidel: you can view Assembly infinity as a

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Laurent Freidel: what we call a stretch Caroleans.

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Luca Ciambelli: Sorry

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Laurent Freidel: gallery instructure, and then and then what it means it means that this analysis, done at finite surface or infinity. You know, they are part of the same

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Laurent Freidel: conceptual and technical framework. And and that's going to be very nice. So then we can compare, you know, quantization in the bulk with infinity and etc. But hopefully, that's coming very soon.

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Luca Ciambelli: Yeah, yeah, on the same vein, following up with that comment, indeed. So this Caroleian framework in

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Luca Ciambelli: was born to understand that flat allography. So infinity. So now we have extrapolated to find a distance in a beautiful way. It has been done already by 2019 for the horizon. Now it's done forever in a library surface, and we are hoping some how to reconnect. Now the dots to infinity, because it's the same. Ca, from the Carolinian perspective is the same framework

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Luca Ciambelli: again, just echoing what to? Lauren said

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Ivan Agullo: direct.

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Jerzy Lewandowski: yeah, thank you. So for to to define charges and and and some flexes, the most difficult part is to

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Jerzy Lewandowski: it, it to restrict to some symmetries. So what are exactly your symmetries? How? How large is this symmetry group?

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Luca Ciambelli: Okay, that's a a good good question. It allows me to recap a bit. So at the beginning we started. and we had this set of cement. So we have dip amorphism of the null.

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Luca Ciambelli: So this is just an element of defense

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Luca Ciambelli: where N is just a non manual. We had boost that are the symmetry. L. Prime is equal to E to the lambda S.

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Luca Ciambelli: And we and we had sheets, which is the symmetry K. Prime is K minusetta. We set a total equal to 0 this one of the beginning, then.

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Luca Ciambelli: we have consistently decoupled this sector of the theory because it has to do with Demo.

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Luca Ciambelli: This is the way we reach Richard. So at the whole beginning these are all the symmetry of the theory. Let's see, when I say symmetry, I should specify. I'm not asking whether they are pure gauge or not. I have the answer to that. I'm just saying, what are the symmetries as a generic?

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Luca Ciambelli: A

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Laurent Freidel: can. I can. I just say there's one just sorry. I'm sorry to interject, but the answer to the question is that the group of symmetry that has nonzero charges called Bmsw. It's the same group symmetry that exists at the same. To take infinity. Bmsw is the is the ultimate symmetry group.

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Luca Ciambelli: Let me get there, Lauren, let me get. Yeah. So that's what I was gonna go. So

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Luca Ciambelli: Lauran is spoiled for me. The answer I mean, I can. I can give you the old story if you have time later you like. There was the direction of was going to give a more consistent way to arrive to be Ms. W. But the end of this story is that the symmetries that are physical at the single cut at a subgroup of these guys

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Luca Ciambelli: that are generated by vector fields that I can write down

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Luca Ciambelli: as being a function of the cut. I call the cut coordinate sigma

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Luca Ciambelli: partial v

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Luca Ciambelli: plus v times another function. This is just a boost in my language, partially

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Luca Ciambelli: plus diplomorphism of the plane that are independent of it.

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Luca Ciambelli: Partial. So this is as a index, a partially

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Luca Ciambelli: okay.

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Luca Ciambelli: This is the end of the story.

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Jerzy Lewandowski: Hmm, I see. That's thank thank you. That's that's that's interesting. How you achieve this on a generic. No, no surface. Yeah. So I have to say that there's a paper I've written in

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Luca Ciambelli: collaboration with the Lee Marto

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Luca Ciambelli: Pope and Petropulos

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Luca Ciambelli: in 19, where we have shown that these are natural in 19, where we have shown that these are actually the natural symmetries of any null manifold per se. They just arise. Naturally.

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Laurent Freidel: there's also a very nice paper by Pabu and Flanagan, where they identify Bmsw as a generic symmetry group.

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Luca Ciambelli: I mean, there's the most generous, not from the canonical analysis, but from the geometric code analysis.

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Luca Ciambelli: Yeah, I should stress that this paper I'm mentioning of myself is also geometric. It's not

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Luca Ciambelli: okay, I see. So so in. So here V is the is a fine parameter.

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Jerzy Lewandowski: Right?

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Luca Ciambelli: So, yes.

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Jerzy Lewandowski: and and yes. So so this is pretty. Actually, this is pretty natural group. This is. This is a large group. And do you have any notion of positivity here? For if you try to to identify energy.

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Luca Ciambelli: Well, in in in which sense? So this is what I was chat talking about here. Right? So let me just go back. Maybe I focus on the boost, but it's since you have ultra locality. This is a boost per cart

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Luca Ciambelli: right here.

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Luca Ciambelli: So this object that I wrote here I should stress is exactly one of the generators that was writing before. Right is the one that you I had an F of sigma partial d plus say, v times g of sigma, partial d plus

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Luca Ciambelli: YA of sigma partial a. So this is just any a specific instance that is here for a specific value of G right? And then you can do the same for this guy.

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Luca Ciambelli: Okay? And if you take this guy where where F is a constant, you just get partial. V,

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Luca Ciambelli: right? So if you get partial D, you can see from the charge we're computing here. Q. Prime of partial d, using this equation is one over 8, by G, minus one over 8. By G,

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Luca Ciambelli: the integral of partial V of Omega.

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Luca Ciambelli: Okay, so this is just data or minus. Okay. Now, Tita, if you are on an horizon and you're on your fine time as a specific sign.

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Luca Ciambelli: So this charge as well, which is the energy as a specific sound.

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Jerzy Lewandowski: Okay? So so actually, the question on positivity is about flexes rather than

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Luca Ciambelli: I mean under the flux. The flux is this question.

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Laurent Freidel: okay? You need to show again this equation for Dv of S, that's the positive. Yes, here, this is positive. Yeah.

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Luca Ciambelli: exactly.

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Luca Ciambelli: Yeah. So this is the flux. What I said, sorry I before I should have be more precise when I said that Tita is a positive sign. This is a statement made about the flux.

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Luca Ciambelli: and just because the flux

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Luca Ciambelli: is positive, because it's just the flux is the right-hand side of this equation.

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Luca Ciambelli: Does that help?

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Jerzy Lewandowski: Yeah, okay, thank you.

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Jerzy Lewandowski: So maybe Abi makes his comments now

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Ivan Agullo: haven't happened.

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Ivan Agullo: and then we will come back to you.

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Abhay Vasant Ashtekar: Yep.

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Western: Hi! I look at thanks for the I was wondering whether you say that? Let me sit. Is there anywhere? Are there anywhere I normally coming in? And do you have any comment on that?

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Luca Ciambelli: Oh, yeah, I didn't say it. So you didn't. You didn't miss it. There could be

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Luca Ciambelli: absolutely

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Luca Ciambelli: there could be. And the way the way here. the way there could be adhere. Right?

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Western: Yeah.

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Luca Ciambelli: this is a classical statement. and at the quantum level there could be a right hand side. That is not 0.

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Western: That could be a quantum to interpret it as a quantum anomaly in this.

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Western: So here's probably

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Laurent Freidel: charge, this charge is right? So

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Luca Ciambelli: yeah, yeah, yeah. But it's a different story. So let me just comment back on this. So, Lauran, you saying the right thing that is the central charge. What is it?

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Luca Ciambelli: A bit later? But you might here well, in this case is infinite, but in the atomic is finite. But you might still say that the 0 plus 2

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Luca Ciambelli: sorry.

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Luca Ciambelli: plus 2 plus matter that I wrote before that the question in blur is still 0

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Luca Ciambelli: completely 0. Okay, well, here, this is infinite. So it's a very silly case, but in atomic later on. And I'm just saying that even this can be not true at the quantum at the quantum level.

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Luca Ciambelli: Okay?

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Western: And you were, and you had the question about the flux. I'm sorry I didn't understand which flexible interpretation for what would be the solution? That case?

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Luca Ciambelli: No.

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Luca Ciambelli: actually.

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Luca Ciambelli: no, I don't have a very yeah.

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Laurent Freidel: so maybe. But it seems you you say this like they are normally.

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Laurent Freidel: maybe it's a different question. Do you normally is proportional to the number of degrees of freedom.

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Laurent Freidel: I mean. So there's there's a conformal and normally on the one dimensional line, which means that the right hand side of the thing is controlled by the course cycle which can be computed exactly. And it's going to be proportional to the

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Laurent Freidel: to the yeah. I mean, essentially, the purpose of quantum gravity is to show that this I normally coefficient is finite. If you postulate, it's finite, then quantum gravity, and it's not line exist.

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Luca Ciambelli: But it is a no matter, because it's the number of degrees of freedom. Yeah. So what? I'm what I'm saying. So, okay, so wh, okay, this is the intuition I would have for this being 0 sector. But in the, if you want to extrapolate it for all that's what we are saying right? So what Laura was commenting about is the fact that each of these guys can carry his own charge. And now the question is, how do you make the right hand side?

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Luca Ciambelli: Behave right? And you can consider that as counting the total degrees of freedom number.

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Luca Ciambelli: and that will be the interpretation of the quantum.

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Luca Ciambelli: Now, the reason why I say I didn't have intuition for that for the whole sectors is that I've been focusing on this being 0 only.

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Luca Ciambelli: and therefore II don't want to make over claims. But for the spin 0 this was clear right for the spin 0. This was this result

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Luca Ciambelli: that the N appearing up here is the same N appearing down here right in this sense. This is clear in this sector.

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Western: I see. Thank you.

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Ivan Agullo: Bye, bye, do you want to go ahead?

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Abhay Vasant Ashtekar: Okay, II just wanted to make a couple of comments. Something one is that.

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Abhay Vasant Ashtekar: I mean, I think you know the specific formula that I've been used they have been in the literature, I mean, as Luca pointed out since

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Abhay Vasant Ashtekar: by paper, since 1,978 to 2,004. But I think this interpretation of you know

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Abhay Vasant Ashtekar: certain terms has been spin 0 parts in one part, etc., is is very interesting, and I mean, that does throw light and make contact with confirmal field theories. And I think that's very, very nice.

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Abhay Vasant Ashtekar: The second second comment is that

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Abhay Vasant Ashtekar: I mean, since last summer I've been thinking about just well, infinity. And this is in part

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Abhay Vasant Ashtekar: the the question that was asked about video, and just a while ago.

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Abhay Vasant Ashtekar: And then I think it is very similar, namely, that

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Abhay Vasant Ashtekar: II mean as as as so emphasized several times, you know, both by Luca and Dora. The one dimension is very, you know.

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Abhay Vasant Ashtekar: this continuous one dimension is something that is demands continuous is the other directions in some ways.

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Abhay Vasant Ashtekar: I, that is really out for locality, that that is something that is also important, that infinity. If you just look at, for example, the synthetic structure, and so on. It really involves only one direction

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Abhay Vasant Ashtekar: and that level infinity.

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Abhay Vasant Ashtekar:  and then the

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Abhay Vasant Ashtekar: I mean just both. From a point of view of Maxwell theory. I can. One can look at Maxwell theory at infinity, and from the point of view of

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Abhay Vasant Ashtekar: point of view of gravity.

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Abhay Vasant Ashtekar: In fact, it is very natural to look at spin networks on these 2 spheres.

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Abhay Vasant Ashtekar: And really and then there are this this polyony lines as running up and depositing area, and so on. So I there's a really very close kind of nice relation. III was thinking about this in the context of this one. I'm on algebra and entropy and various, you know, people starting with Don Marlow. But then, later on, people in India who have been thinking about

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Abhay Vasant Ashtekar: accounting for entropy by looking at infinity. You know, there's

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Abhay Vasant Ashtekar: so I think that this is very, very nice and very promising, and in in that general direction.

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Abhay Vasant Ashtekar: and also in that, in the case of somebody asked this question about

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Abhay Vasant Ashtekar: about the columbic and the and the spin 0 spin one, if you like, Coulombic and the

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Abhay Vasant Ashtekar: and the and the radio aspects of non infinity. And that's exactly what happens, namely, that you can actually do 2 2 different things at the classical level. One can look at just the radiative phase space.

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Abhay Vasant Ashtekar: I mean, it's complete. What can can give rigorous meaning to this idea? Even, you know.

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Abhay Vasant Ashtekar: Just general code space times for matter, fields, but certainly for

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Abhay Vasant Ashtekar: null infinity. You can actually give meaning to local face spaces which are really constructed out of, you know, degrees of freedom, which, deciding some finite portions of non infinity.

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Abhay Vasant Ashtekar: And those degrees of freedom are all purely

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Abhay Vasant Ashtekar: radiative. And you can actually calculate fluxes of Dms generators across

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Abhay Vasant Ashtekar: such finite surfaces, using these radio local phases.

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Abhay Vasant Ashtekar: But in order to bring in the polemic degrees of freedom you have to really look, make contact

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Abhay Vasant Ashtekar: with.

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Abhay Vasant Ashtekar: we are constrained basically between the radiative degrees of freedom and the coulombic degrees of freedom and their full lifestyle sequences are needed.

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Abhay Vasant Ashtekar: And so I think there's a fair understanding of how all these things come about, and you know.

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Abhay Vasant Ashtekar: and Simon and I are writing this up, so should be out pretty soon.

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Luca Ciambelli: Figures of middle desk.

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Laurent Freidel: Yeah, thank you, if I can. I? If I can. I mean, II agree, 100%

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Laurent Freidel: what you just said.

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Abhay Vasant Ashtekar: In fact, most of these studies where where you know

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Laurent Freidel: I don't know. Inspired by what people were able to do with your, you know, at infinity, with your local structure. I would add something else to those things. So not infinity. There is also something which is very important for for us is that it's the same symmetry group. So game as the W.

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Laurent Freidel: Somehow, you know, in the work with Simone we realize it as a group of symmetry, as infinity. But it's also a group of symmetry at finite surface. So I think that extra link between the 2 is also maybe essential.

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Abhay Vasant Ashtekar: Yeah, I think, okay, I think we shouldn't

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Abhay Vasant Ashtekar: perfect, because I think, in my view, we have slightly different views about this symmetry group. I mean to be. There is a space time symmetries, and the Hamiltonian symmetries, and I agree that this extension of Bms group are very interesting

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Abhay Vasant Ashtekar: Hamiltonian cemeteries. like

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Abhay Vasant Ashtekar: like the lens vectors, if you like, in the in the in the Coulomb problem. But I do not want to consider them as space time symmetries. Because I mean.

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Luca Ciambelli: yeah, maybe maybe one thing that is universal is this language of describing these surfaces in the carole way, which again goes back goes back eras to the original world that I levy blow on the market. Now I've been doing. But the way we are formulated also the presence of radiation, of finite distance, hyper surfaces in terms of corona connections

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Luca Ciambelli: is now bring. It even find that distance, not a per services on equal footing that what about discussed many years ago? Not infinity. Right?

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Luca Ciambelli: Right? So it's there's a

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Luca Ciambelli: there's a you know, universality of frameworks that I think is playing a useful role in this story.

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Abhay Vasant Ashtekar: Yeah. And also, I think this idea about our spin networks they really do extend to

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Abhay Vasant Ashtekar: in an appropriate manner. So I think that will be something that will be exciting to see, and also and they will not, as you say, they will naturally give you this finite, not not finite normally would be met finite in that case. So.