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Okay, sorry. Speaker today's Vulcan villain.
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will speak about quality local energy from El que je bond remotes
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Thank you very much and
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Good morning everybody and thanks for being here. So just a brief introduction brief comments So originally I thought, I thought to speak about some new results in 3D gravity with a positive cosmological constant which is a project with student. He from PGi
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However, I decided not to talk about that, but rather give an overview about past results and some perspectives on how to treat
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Cause a local observers in Luke quantum gravity. So this will be
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Partly past results, but it will also should give some
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Future perspectives and also explain how everything relates to other developments in the
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Technical program.
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Okay, I was just using the wrong.
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So let me go to the first
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Question. Can you hear me.
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Yes. Okay. Sorry.
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So first let's go to the basic motivation, Emily, what is why we are why we should be interested in considering quasi local observers general relativity well
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In any and that has basically two motivations. So in any realistic experiment in in physics we are considering certain class of coarse grained microscopic observers typically
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Which are characterized typically by regions in in face face that are characterized by a few coarse grained observers.
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Now, one of the main technical and conceptual difficulties in classical and general relativity is that there is no obvious such course grading on the vast ADM face face of general relativity and this issue has
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Manifestations both in classical and quantum general relativity in classical general relativity
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Maybe the most
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Famous
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Manifestation of these issues that are emerging problem in cosmology, which is how to relate
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In homogeneous solution of Einstein's equations to
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Logic and solution and not
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This issue becomes way more violent in
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Quantum Gravity, because in quantum
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Gravity, we are not dealing with one particular solutions of Einstein field equations, but we are interested with the gravitational
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Face face or a gravitational and States face in its entirety and then we need to know how to characterize observers, not just for one particular solution of Einstein's equations, but in
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In in regions of faces. So the question is how to do this, how to construct gravitational observers and
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The main strategy here is to reverse the logic. So instead of starting out with the vast ADM phase, phase for us, in particular, let boundary conditions.
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At infinity. The idea is to construct a gravitational faceplates from a bottom up approach approach where we start out with the gravitational face face in finite domains and then
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Learn how to go to ever larger regions.
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If we
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If we are considering this a certain difficulty will always appear, namely how to separate the gravitational subsystem from its environment.
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And this has a bit to do both with technical reasons, as well as physical reasons the technical reason has to do with how we treat
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The Hamiltonian time evolution in the covariance face face approach, which was so I won't go through the technical details of how to define the
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Kabbalah and face face because this was already well considered in the one of the last it je SS by Richie here I oh I only want to give you a brief.
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Introduction to the to the show. So consider a finite dimensional Hamiltonian system, which consists of
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The system itself copied to its environment. So the action in the first order formerly formerly small in me 20th formalism.
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Consists of the syntactic structure for the system and its environment capital letters correspond to the degrees of freedom of the environment.
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And then a Hamiltonian which consists of the Hamiltonian of each individual system cast coupling that determines how the system copies to the environment.
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And now the question is, how do we separate the degree. How do we separate the two. How do we consider the Hamiltonian formalism just of the subsystem alone, and then a obvious difficulty appears in the How To Treat the degrees of freedom of the environment.
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So if you look at the equations of motion derived from this action. It's clear that this electric structure for the system is just PDQ
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That determines the face face of the system. But now if you look at the Hamiltonian equations of motion. It's also clear that the Hamiltonian
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Of the subsystem. We consist both of the interaction term plus the Hamiltonian of the that only depends on the faceless variables of the system itself.
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But if we then look at the variation of the Hamiltonian in the Cabal Ian face face approach, we will see the days violation of the Hamiltonian
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Of Hamilton's of the interoperability condition, simply because now we have to take into account variations of the interaction term with respect to the degrees of freedom of the environment.
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The solution to this program, namely two separate to to guarantee integral reality is to realize that something very obvious in a way that the degrees of
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Freedom of the environment must be treated as external degrees of freedom that have to be have to be fixed from the onset as
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Classical background field or as a see number on face to face. And in that way, the Hamiltonian of the of the subsystem itself will be implicitly time dependent because P, Q, the time dependence of key and Q needs to be determined externally.
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Now how does this issue show up in gravity well to determine what we mean by reputation and subsystem.
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Two points need to be clarified. So, first of all I want to say that the degrees of freedom, of course, of a gravitation and subsystem.
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Correspond to the degrees of freedom that belong to a partial cushy hyper surface. But then the question is, how in space time the boundary of this partial cushy hyper surface extends into a virtue in space time
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Now, one possibility is to say that this hyper surface in space time should be a time like cylinder that follows. For instance, a minimal
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Surface that the trace of extrinsic curvature vanishes. But doing this it becomes a bit tricky to distinguish the degrees of freedom that the gravitational and radiation that may fall into the system from the degrees of freedom that fallout that leak out into the environment.
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And therefore it is kind of natural in gravity to consider.
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A situation where the
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This virtue is virtue is simply a normal hyper service. So we want to extend the boundary of our partially cushy hyper surface into
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Not into another hyper surgeons, the further advantage of this treatment is that then
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They say way cleaner characterization of the gravitational flux that may cross the external the null hypothesis. And the idea here in the following is that this that the degrees of freedom of the gravitational radiation that processed in our hyper surface will be treated as an external
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Source, namely pack found field or in other words, see number on face face and it will have will play the very same role that in this time model the degrees of freedom of the environment play for the evolution of the
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Degrees of freedom of the subsystem. So what I treated as a classical source was the background field.
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Is gravitational flux that may cross the hyper surface. And then the question is, what kind of degrees of freedom will survive in the Hamiltonian formalism that we attach to this partial cushy hyper surface and this is in my opinion.
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Contribution to an actually an emerging field in loucon gravity. But now, more and more people are interested in this program from various directions, especially here at a meta Institute.
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Hello, hello aneela pathetic, but also members of the faculty and many other people in our community and are working on this.
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Now, the choice here is to identify the gravitational subsystems with the degrees of freedom with the gravitational degrees of freedom in domains that are bounded by not hyper surfaces.
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And how do we then characterize the degrees of freedom that may leak out into the environment and this can be found in
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The literature and the construction is as follows. If you take the pullback of the four dimensional space time metric to the boundary. This
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To the not hyper surface, then this metric is to generate it has fun. The general direction, which, of course, the knowledge, it generates and this metric can always be diving allies in a, in terms of spinner in terms of diabetic one forms m&m bar.
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That that diagonal lies.
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The three dimensional degenerate metric on the null hypothesis. Notice that Ma ma ma Kovacs just for Co Baxter's intrinsic to the north boundary, they don't
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I don't give a prescription, how to extend them into the interior. The gravitational
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The free data. So the reputation radiation is then characterized by engaging equivalence class of the static one forms. So these phases in cotangent space.
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And the free data is the H equivalents class under this from former rescaling
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To then source the Einstein equations, one has an higher hierarchy of of of equations to be
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Specified
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In fact, they
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In addition to the free data on the novel hyper surface as additional data that needs to be characterized on the cross sections.
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And these are conditions on the consumer factor.
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In addition to the in and outgoing expansion, the outgoing cheer plus one additional spin coefficient
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To integrate the Einstein and create ancient equations along the novel generator additional gauge conditions need to be satisfied, namely
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What which basically tell you what this parameter you means geometrically, namely that, for instance, it's an effing power meter long the null hypothesis says to set up.
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The Hamiltonian formalism, and I want to speak about the space of history's. In other words, the thing content of the bark last boundary theory in conducting the interior using a first order formalism is quite clear. It's given by the tetra
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And self tour connection, out of which we can construct the
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Curvature to that. So to see value search for curvature to form. In addition to the urbanski to fall.
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But now we also need to do specify the field content at the boundary
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And to identify the field content, the boundary. Consider first the pullback of the tetra to the hyper surface.
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And then it turns out that one can always fix spin frame. So K and L or normalized spinners on the novel hyper surface such that the pullback of the tetra to the novel hyper surface.
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Assumes the following form. So the statement is that I can always choose k and l such that this equation is satisfied. In addition to the
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The skill set the boundary. We also want to know the Kabbalah and derivatives and the Cabal Ian derivative of this Venus can be simply taken by considering the pullback of the connection to the boundary. Notice that the pullback in the following
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Has a somewhat prefer role in constructing and identified
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Identifying the boundary degrees of freedom, simply because there's no other way of projecting fields to the to the null hypothesis on a spatial hyper surface. We have a preferred normal with respect to which we can
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Decompose 10s of fields into their spatial and temporal components on an all hyper surface. This is not so. However, they still this yet a natural way of projecting certain kinds of fields, namely those types of fields that define differential forms.
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The boundary conditions that fix the treat the flux of gravitation radiation as fixed Ektron field can be now summarized as follows.
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As follows. So we take only we consider only those tangent vectors in face face that satisfy or only those few variations that satisfy his condition, which means that we keep fix and gauge equivalents class of this gigantic one phones. In addition to that, I treat
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You as a fix coordinate and what is now a variation, how do we write down now an action for which
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The variation or problem is elephant.
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For this given boundary conditions. Well, this action consists just of the usual bulk action, which is the pitches Kayla expressed in this first order barriers, but then we need an additional boundary term and the only in the boundary term that that satisfies
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The our boundary conditions and be then written in the following form. If you go on shell.
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So if we impose all equations of motion. This is, this term is just a certain combination of the non affinity and expansion of the null hypothesis.
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But at the level of the action. We are at a more primitive level because these action can be evaluated on or field come field configurations of the bark and boundary fields. Notice that
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The delivery of the histories, the field content in the interior and the field content at the boundary are detached. In fact, only if we were we divert the action which was quick. Perfect boundary conditions which has been specified on the last slide.
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We get the equations of motion in the park. These obviously just Einstein equations, but by very the film content at the boundary with respect to the
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Forgiven boundary conditions for we find other boundary field equations. The first of them tells us that the pullback of the self tool to form.
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To the not hyper surface can be expressed in terms of the boundary fields alone and this gives the linking between the degrees of freedom at the boundary to those in the bark and in addition to that, we find an additional
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Equation of motion at the boundary that gives meaning to the coordinate view, namely that it is and a field parameter along the knowledge generators. So, only if you go on shell in this variation principle, it turns out that you is enough in parameter
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From the first variation of the action.
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For given boundary conditions, again we find the box press pound weeks, you'd like to structure which consists of have a term in the interior, which just tells you that the service to a connection is canonical conjugate to the
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To the search tool to form a plus an additional boundary syntactic structure. If you go to
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To those creations that are characterized as you equal constant hyper surface is the boundary, really, the second term survives, which has a very simple structure.
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Just, just a question to yes
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So the
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Space, like in
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In your setup.
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One form the you
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Do you
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So do you is one form intrinsic to the boundary. It corresponds
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Okay, yeah.
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So it does not
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I'm not extending it into space. It's a coordinate only intrinsic to the null hypothesis.
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To proceed with the Hamiltonian formalism, we want to
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Me.
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It seemed to me that so far, everything you said is something that you had said before, or is there something new, so far.
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In terms of your previous talks about the boundary term synthetic structures and null boundaries and so on.
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So I would like to understand what is new. When something new is a new happens
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Uh huh. Yes. So what is new is that a more so it is true that many of these things I've said before, but what is new is
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That
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Different that. What a treat differently now the degrees of freedom at the boundary, namely, did I treat them as so no. So normally we would say, well,
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On a partially cushy hyper surface. There are two degrees of freedom to gravitational degrees of freedom point
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And
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I just want to evolve this Archie cushy hyper surface forward in time. What is now different is that older radiative degrees of freedom, instead of
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treating them as dynamic variables on this Archie cushy hyper surface and now swapped or translated into this boundary into the flux long the not hyper surface such that
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We're better now treated as a spectrum fields as
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Which are not part of the face face that I assigned to a partial cushy hyper surface single it's it's it's only a subtle shift of perspective, I would say, where the radiative degrees of freedom and now put on the
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Yeah, the radiative degrees of freedom of gravity and not put as or not or not viewed as the two degrees of freedom point on the scoresheet cushy hyper surface, but they're treated as an external source on the north hyper surface.
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It's only a very indeed it's a subtle point but
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It's a shift of perspective.
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In addition to that, you're right. Many of the things that I'm
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That I am telling you about today. I have already
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Given in previous talk talks, but there was also a bit perspective you to give a bit of an overview, but
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I'm saying that the hook. How can this new prospect to be valued, which is something I don't understand, because you cannot really trade them. I mean,
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That radiate your degrees of freedom are also on sigma, as well as on an M. I mean, if I have a gravitational wave in the space time region you're considering then
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Also radio degrees of freedom on sigma which, for example, will go away if in fact you are continued your space time region, they will go out to infinity, infinity. So it's not
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That I'm fixing too much know
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You're taking too much. I'm just saying that
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You cannot just say say that the radio. There are no ready to degrees of freedom on sigma you I just traded them all to this null hypothesis, unless you go to school.
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But for a finite boundary. There are already degrees of freedom of sigma, as well as on em.
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I mean, because
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I'm because guy Brazil. Brazil escape from Sigma as well. Right. I mean, it's not just from. I mean, if you just continue, you know, hyper surface a little bit to the future, then whatever the and there'll be nude.
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Radio degrees of freedom. If you like going out of the
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Surface and those are deciding on sigma
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So I understand what you mean what you're saying. But what I mean is that so I'm topological this this
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The, the cylinder extends all the way up to up to infinity and soda soda my prescription of
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Of the of the flux. So me skin.
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And long it or the equivalent class me skipping all along the cylinder, but
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What is sigma really call it goes to infinity, there is still sigma left or not if sigma is not left or if in fact
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Then I agree with you. But if, in fact, the sigma is left or there's still radius or dirty degrees of freedom on sigma which will go out right i'm not i'm not clear what is happening here.
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personas en is really a litecoin have appointed infinity, somehow, yes. Then I mean that point at infinity will be i plus and you are enrolled prescribed and then we understand what is happening.
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The new element here is really that you don't want to extend it all the way to infinity, you want to consider a finite portion of space time as you explained in your first part of the transparencies, because your local thing.
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And then there are ready to degrees of freedom on sigma, as well as on and I agree with you that we cannot ignore that add degrees of freedom on, but I don't think it is correct to say that you are just transferred all the ready to begin from Sigma to add
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I agree that I need for
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I would need some more clarification on that point, but I
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In my opinion, in my view, it will
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At this point, we just have to turn the run the Hamiltonian
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homilies meant it will tell us what are better down degrees of freedom left on sigma or not.
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And well, we will
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I will come back to them.
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So what I want to understand now is
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How to separate gage degrees of freedom from actual physical degrees of freedom.
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And
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Now,
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And now there are several different
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Directions on face face. First of all,
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Those different offices in the interior that when you set the boundary
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So they they vanished. The entire knowledge hypothesis.
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And it's quite easy to see that these are gauge directions. So they are the general direction of this lecture to form. Similarly, we now have also
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H trans H transformations
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That only appear as pink variations of the boundary degrees of freedom of the boundary fields.
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So these are you one rotations of the scenario frame plus corresponding rotations of the diabetic one form.
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Similarly, there are simultaneous SL to see gauge transformations of the fields in the interior and the same frame at the boundary and when can easily show that all three of them are on physical gauge directions.
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Now, for as far as if you malfeasance are concerned. Now subtlety arises in fact
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A tangent vector
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To the Caribbean face face associated to this region sigma must satisfy several conditions. Well, first of all, it must
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Be a linear I solution of the Einstein's equation in the in the interior must also be a linear right solution of the boundary
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Of the boundary degrees of the boundary equations of motion, but in addition to that, it must also satisfy the boundary conditions and it's quite obvious the generic such a few more of his
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will not satisfy this condition. In particular, it will not satisfy this condition. If this if this shield. So if there's gravitational radiation that crosses the boundary
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How do we then define
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How do how can we then Cindy speak about the few more finite if you malfeasance that are not vanishing at the boundary. And here, the idea is for the procedure is to use simply projection to say that
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So what we do is that
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A generic search lead derivative will or generic such a few more efficient will not satisfy the got engaged condition, but we can always project down by the complexity to form those components of the field variation that lie within face face by saying that we that
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The fields variation on the volume face face is defined by its projection with respect to any other tangent vector on the Kevorkian faces. And if we do that, we can see we can
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Integrate any different, more isn't that preserves the boundary and we get a finite charge for any such if you more efficient that lies tangential to
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To the boundary of the partial pushy hyper surface. Similarly, we can speak about delectation relations, namely those fields variations on the client face space that only we scale the
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Spin frame at the boundary
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And the corresponding charge for this boundary. The locations is very simple. It's in fact nothing but the area to form.
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Their oriented area of the
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Cross section.
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We can, it is 10 OC instructive to evaluate his charge. So the charge for the final
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Tangential diffuse the preserve the boundary of the partially cushy hyper surface on those vector fields that are area preserving and by taking a partial integrated. So what is sending you is essentially a coefficient of the scene connection so
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Christopher
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Symbol so to say. And if you now use this
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Now use those.
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Area preserving extra fields that can be written themselves as the differential of some function and then by taking a partial derivative on can express this charge in terms of side to subside to being
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A specific component of the
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Vile cancel on the you require constant cross sections.
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And
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You're now free beyond the denominator service when you define side to or you still be entering suggested a three dimensional knowledge or service. It's
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Its intrinsic so you don't need to extend the spring frame away from the normal hyper surface simply by this
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By this equation. I mean the the speed of the violence.
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spinners, or the
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The
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Irreducible components of divided tensor. They are just contractions of
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The vile tensor with certain all vectors and these four to evaluate these contractions. I don't need to know the the extra fields in a nice neighborhood of denial hyper surface. It is completely enough to just know it on the hyper surface itself. So you don't
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Have just picked up one transverse vector on the surface and you're not extending. Okay. That's right.
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About an ocean of energy.
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Namely cause a local Hamiltonian that would generate on that face face that I've prescribe
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Translations in you.
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The idea is to take our normal
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Petra to our normal frame. And now we do have indeed to extended into the interior such that say is a vector field or seeing this up inside of our space time region, such that
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If you're restricted to the boundary, it becomes tangential to the non generators
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And now we can define the following field variations of the fields in in Korea and the fields. The boundary. The park translations are simply the gate. So to see gauged if you malfeasance with respect to this vector field.
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The boundary translation is also a few more fishing, but now weighted with an additional rescaling of the novel frame, actually there's a typo. So this is a
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LA. And so the this
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The dinar frame is just
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Translated by default, more efficient, but also we scale, according to this.
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Rule. Similarly, the diabetic one forms just lead right
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And now we can ask whether this Hamiltonian is integral over
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On the face, but can I ask a question with the last class, per se. So the last class, principally
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Yeah, so the top line.
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With the spinner La la la bar, a pioneer defined just the boundary
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That's right. No.
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Smoothly
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Okay, so any old extension you're taking. There's no equation that is being satisfied by
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By
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By the spinner.
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You're not using something like with Nicholas it like to extend the spin. No.
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No, that's right. So you can use. You can use any extension.
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Simply because we know already that any different monotheism that finishes at the boundary is the general direction of is just gauge. So you can just extend it in whatever way you want. But, and this will not
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And any any to such extensions will yield the same charge the same Hamiltonian, simply because it can only differ differ by the few more facing
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That
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vanishes at the boundary, but that is just to gauge transformation. OK.
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And then
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So then we can then consider them the variation there arbitrary and arbitrary field variation of this Hamiltonian. So, delta is not necessarily a tangent vector on the covenant face face that I've constructed, then you find a syntactic term. So, which corresponds to
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To the, to the Hamilton to the Hamilton equations on face face. But then there's also leakage similarly to what I prescribed on the first slides.
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That describes flux of degrees of freedom outside that leak out into the environment, but now I said that on the covenant face face in this finite region.
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Only on a tangent vector to the covenant face face in this final region must satisfy the linear is calculus foundry.
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Field equations, but it must also satisfy the boundary conditions or the boundary conditions, just tell you that this term vanishes. So on the pace pace should have introduced this Hamiltonian is in fact into rubber and it just generate translations along the knowledge generators
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In addition, on an isolated horizon, they will the expansion Valley vanishes and this whole expression vanishes, which would mean, which means that
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Isolated right just
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As or that this was a local energy has to property that it vanishes isolated for us. So it's maybe
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It doesn't characterize the closet locally mass of the of the black hole solution.
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Now, what does this. How do we then
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Make contact to
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What we know in Luke on gravity.
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And again, I stress this point before. But the point is to stress is that the boundary degrees of freedom that lead
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On the boundary of this partially cushy hyper surface. They are just the very spin oil.
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foundry the release of the same spin oil variables that appear in the scenario representation of spin networks as introduced by some owners for Charla my pay flow. And generally, it's even and many others.
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Now the point is, in order to make context to look quantum gravity, we have to work with the different action, Emily, an action that corresponds that contains the mercy parameter. In fact, in the first order formalism. We have always freedom to add
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This expression, this contraction of the remark tends to the launch in the interior of space time without changing the equations of motion.
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But if we do that, then we have to also modify now in our prescription of not hyper surfaces because funding foundry term just gets multiplied with the same complex valued coupling constant. Now, indeed, this term does not change the feet. The physics in the interior. It doesn't change.
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It does not change the Einstein recreate it does not affect dance and equations, but it has an effect on the film he we at the boundary. In fact, the boundary equations of motion as I studied them or introduced them in the
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One of the slides about
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An hour difference. So the, the field theory that you have at the boundary is not anymore. The same as in the case of gamma goes to infinity.
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And this has done physical effects and these effects show up in the conversation of the area spectrum. In fact, if you look at this complex structure we get, again, the same
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The same
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Over
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Expressions, namely a term interior. The corresponding simplistic structure for the spin all your boundary fields.
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And Bora. Now the canonical variables. So if it now only consider the conversation of the
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Boundary degrees of freedom at the quarter mentioned to two surfaces, what are, what is the corresponding simplistic structure. What are economically conjugate fields. Well,
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If I, if you're working with. If you're working with this boundary conditions with this former boundary conditions for the fix the flux of gravitation and radiation and cost and are hyper surface.
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You can always our price this diabetic one form by a fiduciary repressed representatives times conforming factor and the corresponding fiduciary volume element. It's just I am
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Not
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Economically conjugate variables of the boundary theory or then the knowledge slack or this spinoff la rated by this
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Control factor and it's economically conjugate expression which is now a to form on the quarter mentioned to surface.
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And for these variables to correspond to to gravity certain conditions need to be satisfied, namely a reality condition that implies that the
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That the contraction of k with L is will which then corresponds to the reality of the area to form on the two dimensions process in the spinners pie and l became construct
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The simplest generators, which are just simply by Linnaeus of the boundary fields, namely the generator of relations and have you want flag rotations. Now, in the case of
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Where we removed the music meter, we saw that the you you want frame rotations are just to gauge transformation, but by adding the
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The music parameter that is not any more the case, but the age transformation that we now have to impose is generated by a specific linear combination of the two generators. In fact, so the statement is that came. I know some L is now a generator and this generator
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Generates transformations on the Cobra and face to face that lie in the corner of the corresponding simplistic to form. And now we have to impose and if we now.
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Want to impose his condition on the quantum lever immediately see that it will yield the sweetness of area of the areas faction loucon gravity and that's simply because whatever
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Whatever we do or whatever representation of the quantum theory, we will choose the orbits of L of the you want generator unnecessarily complex unless unnecessarily.
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Complex, which means that the spectrum of the operator must be discreet. But if we impose this constraint now at the quantum level, it means that only certain eigenvalues of K will be realized in the quantum theory, Emily, those that are related to the discrete
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Eigenvalues or a corresponding equations. In fact, if
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If we use
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Glue quantum gravity inspired quantization of the boundary
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So poorly man conversation of the boundary modes, we will proceed as follows. We would say that a bay Bay functional
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Is now a complex valued functional of the configuration variable but using a polymer quantization, the US only we are only considering those functions of the configuration variable that depend on the value of the configuration variable which is
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The national flag of the boundary in a number of points in the number of punctures
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For any such only merit based function. It's an easy to cause to introduce the corresponding
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conjugate momentum, which now which is introduced by smearing the spinner value to form, which I introduced in here over puncture over a neighborhood around this punctures and this operator. Now, x, just as an ordinary derivatives.
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The constraint. On the other hand,
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Becomes now quite immediate quantifies simply because these operators.
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Are
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Are easy to diagnose. In fact, just deletions complex with killings and
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And this so that i can i can functions of these operators are just homogeneous functions which also carry unitary irreducible representations of the Romans, who, which then in turn means that only some eigenvalues of by imposing the constraint, only some eigenvalues of the irreducible.
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The of this i i can values will be realized mainly those bear the continuous labor is related to the district labor by this
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Restriction and these are the same type of faith functions that we use in industry. Oil representation of Groupon and gravity, which was introduced by an altogether.
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altogether different construction, namely by starting out with the discrete face face and then realizing that the district.
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Face face that we assigned to a given spin network graph can be permitted tries in terms of scenario variables.
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Here I've done exactly the opposite. I just started out with the classical face face in the presence of boundaries and then realized that the same variables that appear the discreet level also appear for general relativity in the continuum. But as reputation and Edge mode.
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On a coder mentioned to boundary of a null hypothesis.
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And maybe now we have some time for
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Questions.
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Okay, so questions.
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There's a question.
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So the Archibald.
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Pizza that okay
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So the question that Daniela onset onset is asking is better. I have considered also the algebra of the boundary operators Emily K and L.
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And the answer to this question is is relatively simple simply because these these generators can tell they are ready, causing me so they they commute among themselves.
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They are just relations and
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So these operator commutes
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So these two operators just community came on.
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So these has Kayla operators, they are not saying that you may be confused by the
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By how we treat this constraints in this film foam formula isn't where we think of them as factorial constraints here. These are Kayla's sectors Kayla's under the law school if they could just correspond to the two
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To the to cause a means of the loans and so they commute and
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More questions.
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Yes, question. This is really just a request. Can you elaborate more find the connections to the twister theory in the content modern theory of evolution is based on as we speak.
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So there. Thank you.
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Or that
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So there. If you go back
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You may want you may have so
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To those of you who have seen similar talks of mine before you may have wondered why slightly changed the boundary term.
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And in fact, what you see here is is
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The following if you now we organize, you know, you can introduce a new object.
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By saying I consider
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I consider
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We cannot see anything on the blackboard that you're right.
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Okay, sorry, sorry, sorry.
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So the point is now that
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So you see my cursor.
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Yes.
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So the point now is that if you
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reorganize so LM k can be reorganized in into a twister but into a three star that is now.
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One form. So, it is a one form value.
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keylock spinner. That is
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However, homogeneous in embark so this, this is just set
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Its upper and lower components are L and
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K bar times Mr. And
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This is the canonical variable at the boundary
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The question now to S as to amplitude and all that.
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This
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I would
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Consider in the in the following way.
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If we consider the the boundary action by itself.
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Then we can revise it in terms of the set and said bar variables.
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But now we have to also impose an additional so if we view them as the prime objects of the theory that we need to impose additional constraints which are which will be added to the boundary action by
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Gosh multiplier and this. The resulting action is an action that is fully written or is can can be completely written in terms of set and set alone plus additional constraints. And this is a higher dimensional analogs analog if you wish of
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Of a two dimensional action that appears that appeared in a matrix to string theory which can be which which is
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which underpins
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The
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Recent developments in Trista theory.
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In fact, the, the idea is to
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To now integrate out the bark completely and write down a few key we only for this boundary
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For for for these fields as the bound
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And the proposal for
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Such a
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Few theory I developed in
357
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One of my last papers JUST THAT WILL APPEAR SOON. In classical quantum gravity.
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The idea is that this
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Boundary fields are then coupled to SF to CH and Simon's theory at the boundary and the corresponding field equations correspond to the pullback of the Einstein equations to the north boundary
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Well, just to ask question whether this activity. The for Einstein equations continued in from the boundary or just itself to search and between you know the full Einstein equations.
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So the underlying action will have a similar structure is here that it's always a part of the self tour process on deserves to our part weighted by context value coupling.
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Right you right here, then that the whole action is also the fancy term with the spin to feel
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That's right.
364
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Thank you.
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Yes.
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This is a journey. I have a question about what is the perspective of this use of not surfaces I you try to identify a subsystem. It is finite, like a castle domain or infinite
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Indeed, it's the first to the perspective is to
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Join the center quantization of finance subsistence subsystems in domains with boundaries that final distance and quantifies
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Those and then the limit to infinity would only be obtained within the quantum theory for instance by selecting an appropriate family of career and states that in the limit to infinity.
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That is that can then describe the limit to infinity.
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At the level of the
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Of the quantum theory.
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So, is it correct
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Yes, I'm sorry to interrupt. Please it correctly interpret your results on the quantization of the area as saying that there's an area gap for the
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Core for the area of the corner of a casa de mon, there's a minimal nonzero area for the corner or capsule diamond.
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That's right. And the point that I want to make is that this is a Dr. Cobb server, it is a Dr. Cobb server if for for for the face face that we assigned to the to the diamond itself.
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It and and this and
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That are we have any such cross section.
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evolves and with respect to a true Hamiltonian, a true Hamiltonian which is the which is the one that are prescribed, namely the expansion times the
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Times the area to form integrated over the
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Over you equals constant crossing
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The key.
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Point is that by dividing
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By restricting attention to a subsystem.
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Degrees of Freedom become observer which read cage degrees of freedom before
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But I mean this about again. So I'm troubled about your Hamiltonian because you emphasize again Hamiltonian evolving and so on so forth area.
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I mean, normally you know when people write down cause a local content is the right put some criteria and this criteria typically that well you know on the horizon. You should get the
388
01:01:16,710 --> 01:01:30,270
Hawking mass at infinity should get the bond, the mass media mass and your bought your Hamiltonian blows up at infinity is zero on the, on the, the black hole horizon.
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And so I sort of, I don't see
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It seems to miss Mrs key physics. And so why are you not troubled by this
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So in the case of so it is true that this Hamiltonian blows up as we go to infinity.
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But
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Again,
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The it blows up in in a universal manner, namely as one as are linearly in in in our being the distance or the size of the
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as as as we go to infinity. But now the point is that in in this face face description or is it tough part of phase, phase. So, it is a. It's not a. It's not a classical. It's not a parameter, but it's a
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It's a Q number. It's part of phase space. And so to take the limit to
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To take the limit to infinity, and then remove the divergence. You have to do that. I would argue at the level of the
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Of the quantum theory. So not prior to quantization, but only
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After within the quantum theory.
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Now that may seem strange because we think of our justice coordinate, but our determine sort of radius determine determines the, the size of our sphere as it as this sphere or the boundary of the partially cushy hyper surface goes to infinity. But in this cause a local
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01:03:34,500 --> 01:03:40,380
Description The radius is related to the area of the
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01:03:41,580 --> 01:03:49,260
Of the boundary, but the area is itself an observer, as we know very well in Luke on gravity, where we
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Where we
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01:03:53,250 --> 01:04:08,580
Identify very, very identified it with area operator. It has a discrete spectrum. So I want to make sense of of this limit at the level of the quantum theory and that requires to
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To use
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To use this Hamiltonian
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Question.
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01:04:17,190 --> 01:04:20,370
Is does your Hamiltonian doesn't have a guy.
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Um,
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I don't know at the moment.
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Is, is I think that's something to worry about. That's a different way of stating advice.
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01:04:36,600 --> 01:04:36,960
Well,
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Maybe I related question which I find someone
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The find equally important is better. The resulting better the ground state of his Hamiltonian is degenerate. So whether there is just one back home.
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In the case of synthetically flat boundary conditions then corresponding to the enclosed space or whether days and unexpected far from the generous.
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Yes, that's
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01:05:19,410 --> 01:05:28,830
Why can't you look at the question, classically, I mean you got a Hamiltonian framework, I can look at it. Classically, and you will say that the classical the energies just infinite way everything
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And therefore the, I mean, there's no question on the ground state. All the question is just not well defined. So somehow, you're saying that
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You know we got all these beautiful results in general relativity about finiteness autonomy. Find that is but positivity of energy and so on and and
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01:05:47,880 --> 01:06:03,120
Drastically. That is what tells us that there's a calm state and some are, you're saying that that is about you, relevant, we should just think about it in the quantum theory and not is wrong to think about it in the classical theory at all. So I'm just gonna confuse about the viewpoint.
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01:06:08,280 --> 01:06:10,830
Um no I wouldn't say that it's
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01:06:17,580 --> 01:06:17,970
So did
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That divergence has a universal
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Universally in terms of our so in
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01:06:30,330 --> 01:06:30,900
Emphasize
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It's not magical quantity
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01:06:36,930 --> 01:06:39,990
No, I want to emphasize the disease, an observer.
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Always not always not
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01:06:48,360 --> 01:06:51,450
Always have is an observer quantity
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01:06:51,810 --> 01:06:57,060
Right so so you know what SALLY DOESN'T MEAN very much, right, because, I mean, if it was something that would
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01:06:57,810 --> 01:07:07,110
Normally when you subtract sort. For example, you know, something blows up on some tracks or a quantity in flat space or something or something, which doesn't, it's not opposite right and that's why us
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01:07:08,580 --> 01:07:14,160
Now you're okay. I mean, I think this is the long discussion. So I don't think I want to continue
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01:07:15,450 --> 01:07:19,590
Because I just taking everybody's time so we should talk about Bobby. Thank you.
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01:07:22,470 --> 01:07:24,330
Any more questions. Yes.
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01:07:24,600 --> 01:07:26,460
You hear me, yes.
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01:07:27,330 --> 01:07:42,810
About the adding the music parameter. This part here. I was wondering you then move to the quantum part but at the classical level do the generators in charges corresponding to the few more features that depend on gamma now in your construction or they don't.
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01:07:44,130 --> 01:07:45,720
So the charges of
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01:07:48,660 --> 01:07:49,770
Tangential deeper, more
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01:07:50,970 --> 01:07:56,880
Assertive charges of the deployment business to not depend on gamma. That's because
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01:07:58,050 --> 01:07:58,740
The
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01:07:59,820 --> 01:08:06,180
corresponding term that would would introduce a gamma dependence turns out to be
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Complete derivative that is integrated over to serve, which then to surface, which then vanishes by Stokes your perfect. Okay. That's very clear.
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Any more questions.
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01:08:32,310 --> 01:08:33,030
Speaker again.