Marc Geiller: Say I'm
Jorge Pullin: Yeah.
Jorge Pullin: Can you hear me okay so speaker to this mark.
Jorge Pullin: Yes.
Jorge Pullin: Go ahead, Mark.
Marc Geiller: Thanks.
Marc Geiller: Hi, Jorge, it is like maybe I hope, everything's fine.
Marc Geiller: So hi, everybody. Thank you for tuning in. And so first of all, I hope everybody is fine and healthy and doing well. So yes, I'd like to tell you about
Marc Geiller: Aspects of quantum gravity at the corner. So I have decided to revive this old title used by ordinary humble and hopefully this will become clear. The reason for this title. And so this is a recent work.
Marc Geiller: In collaboration with Raphael, and Daniel upon city. And so I will try to go through these three papers chronologically and review some of the materials inside of these papers.
Marc Geiller: So our motivations for this work is of course quantum gravity. As always, and
Marc Geiller: The questions we want to ask in quantum gravity is
Marc Geiller: What type of degrees of freedom. We want to describe what are the fundamental degrees of freedom of quantum gravity. Where do these degrees of freedom live
Marc Geiller: What is the role of matter in constructing quantum gravity.
Marc Geiller: And
Marc Geiller: We have received many different answers from different approaches to to these questions. What is clear is a first point here is that a
Marc Geiller: central role in quantum gravity is played by the notion of cemeteries. And so what I would like to ask here is the question of which symmetries, we should use to construct and describe the theory of of quantum gravity.
Marc Geiller: So for this, what we want to present is a new proposal for an approach to quantum gravity based on the notion of so called local or cause a local holography
Marc Geiller: And the region. The reason for this name is that we would like to focus on the symmetry contents of local but arbitrary sub regions and corners associated to the sub regions.
Marc Geiller: What you want to do, in turn, is to associate Hilbert spaces states and quantum numbers to these corners based on their algebra of symmetries.
Marc Geiller: And we want, in a sense, to describe degrees of freedom, which are described and organized by her presentations have a notion of corner symmetry algebra.
Marc Geiller: Then in this picture. Once we understand the corner symmetry algebra, and it's a presentation. We want to build space as an untangling and fusion of these corner degrees of freedom.
Marc Geiller: And then the dynamics and spacetime evolution and the constraints as some notion of charge conservation.
Marc Geiller: So just before I continue, let me make clear once and for all that by a corner here. I mean code. I mentioned two regions of space time. So in three plus one dimensional four dimensional space, time, this would be
Marc Geiller: Two dimensional manifolds, which you can think of as boundaries of spatial sub regions.
Marc Geiller: So if you look at this, the points of this new proposal here it should maybe come kind of
Marc Geiller: apparent that this actually has lots of ingredients of what we are used to in a look quantum gravity and it's been four models.
Marc Geiller: So I have just presented here this artist rendering of these little bubbles chunks of space democratized and then there is this collection of
Marc Geiller: Building blocks of quantum geometry glued together connected by maybe some speed network or some, you know, fundamental quanta of geometry.
Marc Geiller: And then we want to describe the evolution in time of these little bubbles is little building blocks so schematically, this will look a lot like quantum gravity. But what we want to do, actually. You will see
Marc Geiller: What we naturally discover is that there are some built in ingredients of Luke quantum gravity.
Marc Geiller: But we want to push the logic of look quantum gravity actually further. So of course the quantum gravity was initiated.
Marc Geiller: Based very strongly on the, the idea that we should focus on the symmetries of general relativity and what we have done very successfully so far in Luke quantum gravity was to understand the role of
Marc Geiller: Different Marxism symmetry and also internal as YouTube confirmations. But the question we want to ask you is, why not go further. If we can associate type of symmetries to other
Marc Geiller: Structures of spacetime, namely here corners of our because the sub regions, maybe we should focus on trying to understand the physical content of these cemeteries and set aside for the moment before more systems and issue or whether
Marc Geiller: extend them and try to understand what lies beyond different morph isn't an issue.
Marc Geiller: And again, we want to maybe rediscovered this notion that space is a network of some quantum geometry building blocks, but we want also will see to discover or some kind of generalization of this notion
Marc Geiller: And also recently. The third point is that there's been a lot of work into quantum gravity trying to think in terms of course grading of degrees of freedom.
Marc Geiller: And also thinking in terms of topological theories and defects and this brings us to this idea of kind of enlarging the theory space that we're working with. So the question we want to ask is,
Marc Geiller: Are we actually describing the proper fundamental degrees of freedom are we actually describing the proper quantum numbers which we should coarse grain and
Marc Geiller: Build space with and build space time with. So that's the
Marc Geiller: Sorry, the care. OK, so the further motivation for this work is that we will see that this feeling naturally enable us to start reconciling different approaches.
Marc Geiller: So here I just mentioned approaches which are holographic like ADHD and more general notions of bolognesi which in a sense of the property of focusing more on what happens at the boundary of so usually a synthetic boundaries at infinity for space time
Marc Geiller: And we want to reconcile this with approaches, more like energy. So, energy type approaches which focus more so far on the structure of the book.
Marc Geiller: Also our motivation and you will see that this is what we will kind of nicely discover in this work is that we can actually consistently resolve and clarify some tensions which has
Marc Geiller: subsisted so far in new quantum gravity. So, namely to list a few of them. We want to clarify the interplay between the world of these collectivization and quantization.
Marc Geiller: Will also clarify the whole play by the multi parameter and also affiliated issue is, of course, that after the sweetness of the spectrum and internal line sometimes
Marc Geiller: Will also clarify the whole of the simplicity constraints which are of course very important in the construction of the dynamics of quantum gravity.
Marc Geiller: And also say something you would see in a minute about non committal activity of the fluxes, and also a key point is to have access for the first time now to the frame operator.
Marc Geiller: And the hope in the long term, is that this will enable us to have a new look into the dynamics of quantum gravity and maybe the way to include matter, in theory,
Marc Geiller: So these were the motivations. Now the roadmap for this proposal, and for this construction. So let me tell you where this construction comes from and what the idea is. So the starting point is just the organization that
Marc Geiller: We work with the gauge theory imaging volatility gravity and when we have boundaries in a gauge theory, the boundaries actually turn
Marc Geiller: Gauge symmetries into physical symmetries. This means that to boundaries, you can associate non trivial charges and analogy, poverty is charges.
Marc Geiller: So if we think about little corner which are boundaries of special regions so called I mentioned two corners will have non trivial charges and algae was associated to these corners.
Marc Geiller: Now it turns out as a mathematical tool that these charges and the algebra of these charges is best studied you know formalism, which is called the combined faith based formalism.
Marc Geiller: So the starting point of the Covenant faith based formalism. It's just like an engine for your theory and by taking as usual variations of the technology and
Marc Geiller: If you want to isolate the equations of motion. We have to integrate by parts. And this produces a boundary term with an object called Sita here and we can see that the syntactic potential. So this electric potential will be the, the central
Marc Geiller: The central object of this of this construction and of this talk that will focus on
Marc Geiller: From now on, so what you want to achieve is to actually find some classifying criterion for different formulations of gravity.
Marc Geiller: And we want this to get us to a notion of we'll see is the notion of corner symmetry eligible. So what I want to motivate and argue is that
Marc Geiller: This notion of corner symmetry algebra is able to distinguish between different formulations of of gravity. So what we want to focus on in order to do this, is that we want to make some
Marc Geiller: Not simplifying assumptions, because these are not made for simplicity, but these are made for clarity, because in studying charges and algebra of charges.
Marc Geiller: It's not possible to if you mix together boundary conditions and different locations of boundaries and regions we completely lose control over the structures that were constructing. So what we want to do here is
Marc Geiller: Build everything with a very bottom up approach. And for this, what we want to do is consider only
Marc Geiller: entangling spheres. So this means that these are Cornell's of regions which do not evolve in time. These are locations have a spatial manifold, which
Marc Geiller: Do not change under time evolution. We also want to postpone for the moment the discussion of boundary conditions.
Marc Geiller: Postpone the discussion of time evolution or they just said, and only focus will come back to this in a minute on attention different systems.
Marc Geiller: And if we start doing this. There's already enough material to extract physical content and extract some structures and then from there on, we can understand how to generalize this and how to go, how to go forward.
Marc Geiller: So now what we can do with this very basic setup so just working in the Quran faith based formalism is to realize a very simple but important and deep result.
Marc Geiller: And this simple but important result is that if we focus on any formulation of gravity, which I will call f here.
Marc Geiller: We can look at the syntactic potential of this chosen formulation of gravity and then realize that
Marc Geiller: Genetically, we can write this potential as a sum of two contributions. One of them is a universal peace, namely about these
Marc Geiller: Which is associated for a reason that will become clear on the next slide with a canonical ADM general relativity. So we were just the note this both contribution by GR
Marc Geiller: So for canonical GR and we will see that the contribution of this bulk piece of the syntactic structure when we look at charges additional morph isms. And there are jabbar will contribute universal peace which is just do Mark isms of the corner sphere.
Marc Geiller: Then in addition to this universal bulk piece there isn't read a corner piece which can itself depend on the formulation
Marc Geiller: And this corner piece as I will show adds extra charges to
Marc Geiller: The Caribbean face face and contributes to extract components of the corner of symmetry Antoine, in addition to the different morph isms of the sphere which is the universal piece of this corner symmetry algebra.
Marc Geiller: So the formula is just that the syntactic potential data F of any formulation is the sum of the universal bulk piece data GR plus a corner term used to describe terms of total derivative
Marc Geiller: Plus potentially a total variation and it'll tell valuation. Later on we can kind of forget about it. Because when we go from this eclectic potential to the syntactic structure this total domination will actually drop out.
Abhay Ashtekar: Mark and ask your question.
Marc Geiller: Yes, sure.
Abhay Ashtekar: I mean throughout here is a key point and you keep referring to those impacted potential for the Cobain face face. I mean, it's
Abhay Ashtekar: Unbelievable emphasize that there is no such thing as the synthetic potential the sibling extraction is canonical ultimately when you get on the spatial solutions oriented Facebook, but there is a freedom of choice of this and potential so
Abhay Ashtekar: I mean, I hope that you'll tell us what you mean by Bill synthetic potential and how you are somehow imposing extra conditions to select a preferred one
Marc Geiller: Yes, exactly. Thank you for the question. So that's indeed a very important point. And I'm saying maybe I'm jumping ahead of myself a little bit, but I'm saying this eclectic potential because
Marc Geiller: So as we argue in the in the first paper. And as I also want to argue here there is in fact a way to assign a canonical and unique simplex potential to theory and this is by kind of
Marc Geiller: Forgetting about this this ambiguity. So I know that many people have called
Marc Geiller: This ability to play with this corner term an ambiguity and that because of this the syntactic potential is actually ambiguous. What we want to argue here is that this
Marc Geiller: This corner PCs are not actual ambiguities, but they are just features of a given formulation of gravity and we should actually focus on these features and try to consider this kind of contributions for what they are and try to extract physics out of them.
Abhay Ashtekar: Right in this
Abhay Ashtekar: Is your viewpoint. I mean, maybe you're going to explain it, in which case I stopped. But if not,
Abhay Ashtekar: This is important for most of us to understand is your viewpoint, then that the theory by itself does not have
Abhay Ashtekar: The synthetic potential or the quarter term.
Abhay Ashtekar: But each choice. I mean, they love freedom and then you're going to supplement it with some principles. Or perhaps you're going to say that choice of a return this corresponds to different physics.
Abhay Ashtekar: International and therefore God has this unique some great potential, but
Abhay Ashtekar: As supplemented with the synthetic potential has a has a unique physical interpretation.
Marc Geiller: Yes, exactly. Thank you. So that's exactly what I want to get to
Marc Geiller: And that's my last point here, what I want to argue is that different formulations of the same theory. Maybe that's the important vocabulary parts we talking about the same.
Marc Geiller: Theory in the sense that we're talking about gravity, but there are different formulations of descriptions descriptions of gravity.
Marc Geiller: Or the metric one or the tetrad one that we're using look quantum gravity and what I want to argue is that different formulations of the same theory, namely gravity.
Marc Geiller: Have different kind of syntactic potentials and have different physics associated to these kind of simplistic potentials and this is the difference that we should be careful about and acknowledge
lsmolin: Can I ask a question.
Marc Geiller: Yes. Link is
lsmolin: Yes, there's a couple of related points you freeze the lab said at the boundary, if I understand it. And that gets you out of having to answer the question of what are the physical conditions that define
lsmolin: an esoteric boundary. And so that leads me to ask in this situation where you're tapping everything that would be dynamical infinity and you're not
lsmolin: Being responsible to tell us what somehow what physics this boundary condition as matter how much of what you're showing us remains true if you put those complications back in
I
Marc Geiller: Don't really want to answer this, because I don't know why could tell me that
Marc Geiller: Yeah, I don't know how much of this actually remains true, but I think that's kind of a red herring, because maybe it's not the good question to ask because maybe
Marc Geiller: If you know taking again the viewpoint of quantum gravity that resume.
Marc Geiller: Instead of, you know, quantities and space time as a holistic object and looking at what happens for boundary conditions at infinity. If instead zoom in on Space Time.
Marc Geiller: Then start chopping it up and try to understand what are the summit fees associated to these arbitrary smaller building blocks.
Marc Geiller: Then we can forget about the question about what happens when we push the boundaries to infinity, because what we want to point is, is the algebra and the degrees of freedom associated to little cause a local chance
lsmolin: Now, I don't. I also don't want to push the boundaries.
lsmolin: But I want you to tell me the physical conditions, whether express in terms of external fields or better feel for the gravitational fields that define these boundaries that My fear is that if you don't tell me that I don't know how to measure the client is you're having charges.
Laurent Freidel: It can I can I help there. I think lead. This is a question for further took you need to
Laurent Freidel: Think is that
Laurent Freidel: Maybe your
Marc Geiller: Talk, we can go back to
lsmolin: Okay, I'm happy justice.
Laurent Freidel: Yeah, once you have to charge you to to look at dynamics as evolution charge evolution.
Laurent Freidel: But the keys to understand what is, what are the child's first
Marc Geiller: Yeah, exactly. We have to kind of be a bit more
Laurent Freidel: You cannot talk about evolution. If you don't know what other kinds of charges for their quality construction.
Marc Geiller: Okay, so to continue the first natural thing to do is to actually give an example of this statement that there is this split of the simplicity potential between about peace on a corner piece.
Marc Geiller: And this example is a very familiar example actually because it's just starting from the familiar Einstein Hilbert like on Jenna for metric gravity.
Marc Geiller: So this is our familiar Einstein Hibbert like Ron channel or is he scale or curvature and exciting and here is the space time volume square root of G.
Marc Geiller: And now the simplistic potential for this formulation. So I've been taking variations of this like engine and isolating the Einstein equations of motion, we get this simplistic potential, which is written here in terms of the normal n to
Marc Geiller: cushy slices and excitement tilda is doing used volume element on the slices
Marc Geiller: So the first line is what comes out of the calculation and then just a few lines of algebra enables you to actually rewrite this Einstein Herbert potential
Marc Geiller: Into precisely about peace and kind of these. So it's total derivative here in red and a total variation which we can forget for the moment.
Marc Geiller: And now you will see that this
Marc Geiller: Bulk piece of the syntactic potential I have in blue. I have called it feat. A GR because the statement is that this is the the canonical universal peace come on to our formulations of gravity and the corner contribution, I have called it
Marc Geiller: Ah, what it is to general relativity, because I want to introduce the notion of qualities syntactic potential. So we have a syntactic potential for one theory, which is
Marc Geiller: So what for one formulation of gravity. So, which is the answer here but formulation
Marc Geiller: And we have the potential for another formulation of the same theory, namely
Marc Geiller: The canonical ADM formulation and the difference to go from one formulation to the other. The reason for that is simplistic potential and these qualities and electric potential is just a corner term and its displayed here in red.
Marc Geiller: Now why these names. Well, this result is actually familiar result because it comes to us from the well known.
Marc Geiller: Geometrical fact, namely using the gospel. That's the equation that we can rewrite the Einstein Herbert like engine as the some of the
Marc Geiller: ATM or GR like engine, which is this piece here and the boundary term which is essentially using the the extrinsic curvature. So the kind of given talking term and the acceleration here.
Marc Geiller: So this is just the family that girls could that see the composition of the engine Herbert like engine. But now there is a little subtlety, which is how to precisely go from this statement about like engines to a statement about
Marc Geiller: What is made above a statement about the different simplistic potentials and this is due to a less known facts and this fact is to actually acknowledge that once we
Marc Geiller: Go from one initial theory, namely
Marc Geiller: Here canonical general relativity given by the EDM like an engine to the answer and Herbert like Engine using a boundary like engine, we have to acknowledge that the boundary like engine itself has a simplistic potential
Marc Geiller: And you see that what is written in the first part here that so this unless you see the pointer. When I moved here you see them.
Marc Geiller: Yes. Okay. So this first statement here is just in fact a rewriting of the fact that if you take the valuation of this collective like engine, which is the given talking term.
Marc Geiller: This is exactly this result here. So the variation of the holiday like engine gives us the understand here but simplistic.
Marc Geiller: Simplistic potential minus the canonical simplicity potential and there is a total derivative which is this qualitative simplistic potential
Marc Geiller: So this just means that boundary like engines, may I say maybe because maybe you consider to formulations, for which the bundle and gunjan does not have derivatives. And in this case, this isn't trivializes
Marc Geiller: But like here when the boundary like auction contains derivatives, you can compute variations of the boundary like engines and this gives you naturally a common law contribution.
Marc Geiller: To the syntactic structure of the theory. So this is again the statement that discount of times they're not in the US, but they're just features of whatever formulation of the theory, namely here gravity, you're trying to describe
Marc Geiller: And this is the reason for which looking at different formulations of the same theory gives you a different boundary charges and different challenge algebra, and this is what we want to investigate.
Marc Geiller: So to illustrate this further. Why this actually has important implications, we can actually dig a bit deeper and look at the structure of the charges and be eligible for metric gravity.
Marc Geiller: So now that we have this eclectic potential, we can construct the syntactic structure. And as always, if we look at the charges coming from the syntactic structure.
Marc Geiller: We get the bulk piece, this piece is just a special different markets and constraints and the statement is that
Marc Geiller: Because this is the bulk piece which is the same universal contribution, both in the canonical GR formulation and in the engine Hubert formulation. This gives us, of course, the same
Marc Geiller: The same chart. So the same constraints which is just the DM
Marc Geiller: Momentum constraint.
Marc Geiller: Now, if you look at the algebra of these constraints, of course, we will find the algebra of different markets.
Marc Geiller: But now the tricky question is how exactly is this a bunch of different mechanisms represented
Marc Geiller: To answer this question, we have to now look more closely at the charges that live on the corner.
Marc Geiller: And because these two formulations of the same theory differ precisely by a corner contribution to the potential and to the syntactic structure. They have different charges and this we can see very easily, because if you compute the charges for these two formulations of of gravity.
Marc Geiller: For canonical GR we find the familiar upon your charge and for the action here but formulation we find the familiar so called calm of charge.
Marc Geiller: Now this makes an important physical difference because as you can see the barnyard charge is actually vanishing if the if the vector field generating the different markets and you're looking at vanishes on the surface.
Marc Geiller: Whereas the coma charge actually contains derivatives of this vector field. So it may be non zero or even if the vector field is vanishing
Marc Geiller: On the surface,
Marc Geiller: And this is important physical implications which were actually explained nicely in despair for Bye bye, no one really only 2016 and if you just allow me to skip over the details here because
Marc Geiller: We want to really move on to protect gravity, but really refer you to this paper.
Marc Geiller: If you actually perform a two plus two decomposition. So the composition adapted to the fact that we have a cold. I mentioned to corner.
Marc Geiller: This will actually revealed that the eligible of this comet charges has actually an extra components, which is an SM to our components.
Marc Geiller: So the edge of the barnyard charges in the jar formulation is just the S, but the edge of the different more fees and charges in the ashram Hubert formulation. So, namely the comet charges.
Marc Geiller: Is the S and an SMS or component. So, we see a stated that these two formulations of the same theory has a different symmetry contents at corners.
Abhay Ashtekar: So can I just ask. I mean, usually when one says brown York and there are seven boundary conditions that are imposed on the boundary
Abhay Ashtekar: Yeah, so, but here you are not imposing any boundary conditions at all because
Marc Geiller: Yes, exactly. Here, we don't need any boundary conditions to the
Abhay Ashtekar: Reddish brown y'all came in your version of brown. We are in which there is no boundary condition imposed. Is that right,
Marc Geiller: Yes, I don't want to say my version, but it's a very clinical results here. It's just taking the syntactic structure and
Going
Abhay Ashtekar: Down, because the original brown. We are very clear, you know, the fixing the metric and so on. And the boundary and you're not doing any of that.
Marc Geiller: Yeah, this is not required here. Yeah. So now we can just take these two results and just put it in this little table here where here we put the type of content symmetries that we get
Marc Geiller: And again, jumping a little bit ahead of ourselves. I have already put what will happen next. And what is of interest for us because we want to focus on tech hub gravity.
Marc Geiller: So here we have just the two results, we just derived for canonical GR and for the engine here but formulation in one case we have defense. And in the other case, we also have our next time.
Marc Geiller: I sent to our component. Now what I want to explain is what happens in the case of tech hub gravity. So if you just go back for a moment to our
Daniele Pranzetti: Motivations
Marc Geiller: We see that
Marc Geiller: These shows that indeed different formulations of the same theory me gravity reveal different components of this notion of kind of symmetry algebra.
Marc Geiller: So this already tells us where to go ahead now because if indeed this kind of symmetry algebra plays a role in the quantization of gravity. We better ask the question of what is the full summit algebra, when do we stop. When do we know if we have
Marc Geiller: The maximum any extended notion of symmetry algebra.
Marc Geiller: Maybe it's desirable to have some notion of bigger symmetry algebra, because if we have a bigger algebra that we can monetize this means that we have more quantum numbers or more handles.
Marc Geiller: To reconstruct from the knowledge of this corner charges the bulk degrees of freedom. And then the dynamics of quantum general relativity
Marc Geiller: Now I want to keep arguing that this is indeed the case, and continue. So justifying that this has to do with quantum gravity and that indeed by looking at
Marc Geiller: Extra components of the corner symmetry algebra we can squeeze interesting physics out of this and this, I will do by jumping now to the tetrad formulation of general relativity and showing that if we just apply the same ingredients with
Marc Geiller: Zero set of assumptions we find natural anti back key ingredients of loop quantum gravity.
Marc Geiller: So to do this, the strategies to now study tetra gravity.
Marc Geiller: The strategy which we are familiar with.
Marc Geiller: Working in quantum gravity is to start from this topological theory me the BF like engine. So just be which F, which is the curvature of the connection omega
Marc Geiller: This has a very simple simplistic potential, namely be which does the omega
Marc Geiller: And then once we understand BFD we want to impose simplicity constraints and this simplicity constraint is just sending us again this familiar result that
Marc Geiller: These two form be is the wedge product of Ted had one forms II and we have added also beta, so the the topological house contribution. What we want to do and what I want to illustrate is that if we take this point of view of looking at the corner symmetry algebra we can
Marc Geiller: Work completely without fixing the time gauge and therefore access to boosts. This is already something which was very difficult and ambiguous to do in
Marc Geiller: Let me say maybe you were the traditional usual approaches to look quantum gravity focusing on the bulk and you will see that this realized very crucial and heavily on the use of the internal normal. And so, just like in the study of metric gravity.
Marc Geiller: We had the normal to the
Marc Geiller: Affiliation that was entering crucially, the syntactic structure also on the corner here a similar role will be played by the the internal normal and I, this is something that was acknowledged already by many authors before and which we push here to the
Marc Geiller: Tree take very seriously here.
Marc Geiller: So now to go forward. The first thing to do is to actually take this be field of BF theory and just do something very simple, namely decompose these be field into boost and quotation parts. So you will see
Marc Geiller: That it is very convenient to actually call the quotation part the spin part actually so we decompose these to form into a boost part behind and spin part SK here.
Marc Geiller: And then, in turn, what we can do is write these two forms, be an S in terms of two one forms. So two frames so boost friend little be and spin frame little less
Marc Geiller: So this is just a legitimate decomposition of this to form of BF theory on a hyper surface sigma and this decomposition. As you can see, uses the internal normal and
Marc Geiller: So this is the composition of the the field, but now be a few. He has also another few memory, the connection. So we also have to decompose the connection.
Marc Geiller: And the natural decomposition of the connection is to write it as a connection gamma, which is the analog of the spin connection and tensor K which is essentially the derivative with respect to omega of the normal and
Marc Geiller: Now just plugging this decomposition into the BF syntactic potential gives us the first results which we explain with or gory details in paper to have our series, which is that
Marc Geiller: The syntactic potential of BF theory also splits in the son of a bulk piece and external contribution. But you see that
Marc Geiller: We can already expect that the FDA is not gravity is a topological fury and you see that as we can expect the bulk piece here as actually to contributions.
Marc Geiller: This is kind of consistent with the fact that this is not yet gravity for the moment he says the next how canonical parent, the bulk, namely the game of S which is here. And sorry, I realized that I forgot the delta is this as here should actually be second should be delta is
Marc Geiller: Now, once we have this decomposition of the BF simplistic potential is very straightforward to go to tetra gravity and this is just by imposing the simplicity constraints.
Marc Geiller: Now this form of the simplicity constraints here can be written in terms of these objects be these boost to form and the spin frame little s. This is just a statement that
Marc Geiller: The boost to form is just now he so the two form coming from
Marc Geiller: The wedge product of tetrapods, and the spin frame is proportional to the tetrad frame. He and now by just plugging this in the decomposition of the BF simplistic potential
Marc Geiller: We get the decomposition of the Einstein cast on host synthetic potential. Yes. So he CH here stands for engine kappa host
Marc Geiller: And I'm sorry, do you also see this black
Marc Geiller: Thing when I
Philipp Hoehn: Yes.
Marc Geiller: Oh yeah, I realize this maybe stop scrolling down there, because it's a bit annoying so sorry. OK.
Marc Geiller: So now this is kind of the result that we wanted to announce, and that was expected and me that this electric potential of the tech hot formulation of gravity with
Marc Geiller: Host term is again given by Universal bulk piece which is here in blue and a contribution which is only pushed to the corner.
Marc Geiller: Now I just have to finish by arguing that this piece in blue E which data K is indeed what I called earlier, the Sita G RPC. So the canonical piece coming from canonical general relativity
Marc Geiller: And this is just a simple rewriting introducing this for momentum capital P which is k across he, you can actually show or with a simple rewriting so this p
Marc Geiller: New new delta G, this is indeed the simplistic potential of the ATM formulation of general relativity
Marc Geiller: And this is just written up to a total derivative as he watched into k, which is what we had in the previous slide.
Marc Geiller: So this shows as expected that also for Ted have gravity, you have this decomposition into a universal bulk piece which is Tita GR and external contribution which is specific to tetra gravity because it has the tech house on the corner and also the internal moment.
Marc Geiller: So we see here, but that by doing this, we're already kind of departing from what we're used to in Luke quantum gravity because we're shifting emphasis from the bulk to the corner and this is due to this.
Marc Geiller: To this went on vacation, which is a rewriting of the highest contribution to this eclectic potential. So there is this beta term here which by the way is the inverse of the usual before me as a parameter
Marc Geiller: It turns out that when omega is the spin connection preserving E on Chen, this can actually be written as a as a as a total derivative. So this is itself a corner term.
Marc Geiller: So instead of keeping this House contribution in the bulk and constructing a national carbon better connection with this term we push this term to the corner.
Marc Geiller: And keep the universal general relativity simplicity potential in the book. So if we do this, we could be worried that we completely lose contact with the
Marc Geiller: With the ingredients of quantum gravity. But what we want to show now is that this is not the case, actually. And that just by focusing on the corner we recover in a completely and ambiguous way or not so familiar and cadence of quantum gravity.
Marc Geiller: So now to go further. Once we have this decomposition of this electric potential, we can ask again about the fate of the generators of transformations and the charges. So the first thing to do is to look at look at different Mark isms.
Marc Geiller: So as usual, when looking at different reasons, we find that
Marc Geiller: There is a bulk contribution which is just the momentum constraints which here is written in first order formulation as a very elegant conservation equation.
Marc Geiller: So, this we should put in parallel with the statement that I made in the introduction that we want to reconstruct the constraints of the theory.
Marc Geiller: As conservation laws for the boundary for the corner charges. So here, indeed, we see that the constraints in the spatial hyper surface is nicely and the canonical written as a conservation low for p and we can also write the corner of charge in terms of this momentum up
Marc Geiller: Now precisely focusing on this charge we get the usual first order charge for four different Marxism.
Marc Geiller: And now we also know that in touch with gravity. In addition to the for more freedoms we have internal organs transformations and for these internal organs transformations we get
Marc Geiller: As usual, a bulk piece which is just the gas constraints and this is again a conservation equation, but now for this for this flex to form yeah i j and we have logins charges on the boundary which are the integrals over the corner of
Philipp Hoehn: Alpha times
Marc Geiller: He and now what is important is that this charge contains both boost and rotations and this is because we have not fixed.
Marc Geiller: The normal we have not chosen the language.
Marc Geiller: So we have access to the boost part and the rotational part of these two for me. And now we see that again we can go back to the statement about the corner algebra formed by these charges.
Marc Geiller: And we can see here that in the tetrad formulation this college algebra has so far, two components, namely the diff S which is formed by default more fees and charges and an ultra local SL to see algebra, which is found by the Lawrence charges.
Marc Geiller: Now to continue what I want to do is to actually
Marc Geiller: Forget about the bulk and zoom on the corner, simply click on the corner simplicity potential because the statement is that by zooming
Marc Geiller: On the corner we can extract lots of interesting and useful physical information. So let's do this once and for all. Now zoom on the corner syntactic potential and see what it contains.
Marc Geiller: To do this, it's actually very useful to step back a little bit and go back momentarily to the BF formulation of the theory. And the reason for this is that if we go back to the BF formulation we can actually
Marc Geiller: Discover lots of very nice.
Marc Geiller: Geometrical in algebraic structures which are specific to this kind of simplistic potential
Marc Geiller: So to do this we can go back to the BF counterfeit space. So this is just to rewrite what we had before. We have this be field, the normal n
Marc Geiller: And we have this frame in here and maybe just to hear the confusions you notice that I'm this is BF theory, but I have the frame field here in the corner potential
Marc Geiller: This is because in BF theory. What is important is that this be field here. These two form is not defined in terms of this one from
Marc Geiller: The simplicity constraint is the constraint that relates these to form here be to the one for me. If we don't impose this condition is BF theory if we impose this condition. This becomes gravity.
Marc Geiller: Now, it turns out that on this BF corner face space there is a very nice geometrical structure which is an analogy with the face base of a massive particle. And in this analogy with the face of a massive particle, the role of the momentum.
Marc Geiller: is played by the internal normal. So, this is this condition here that the momentum p of this particle is essentially the body or emails, the parameter Ghana, times the internal for normal n
Marc Geiller: And
Marc Geiller: So this massive particle face face which
Marc Geiller: Is I will argue has a plaque asymmetry.
Marc Geiller: Has also notion of a pony dubinsky vector like when looking at the college algebra and this probably Lansky vector for this bunker here Gemma is actually the barbell immunity parameter gamma times this spin to form, which I introduced a bit earlier. So the spin to farm.
Marc Geiller: When our when we can maybe come back to this spin to formula me but it's been to farm is essentially just the cost product of these frames of these frames. He
Abhay Ashtekar: Might just a quick quick lesson.
Abhay Ashtekar: Then stop lying.
Abhay Ashtekar: On the left hand side you'll go to WTF, and then you explain to us that the FDA does not a big he taught
Marc Geiller: Yes, yes.
Abhay Ashtekar: So, I mean, I'd be expressing the synthetic potential for the BF theory in terms of some ingredients which are not in that theory.
Marc Geiller: Yes, thanks. Thanks for the question. Let me just try this right away. So let me just go back to this slide here, as I said,
Marc Geiller: In BF theory, we can take these two form and the composite in terms of boost to form and it's been to form.
Marc Geiller: And these boost and spin two forms can themselves be written in terms of one forms. So you see that in a sense, BF theory has to
Marc Geiller: To friends or be frame or it will be frame and the little s frame the expression of the simplicity constraint is just the matching of these two frames. It's just the proportionality between these two frames.
Marc Geiller: So tetra gravity only has one frame which we traditionally called E and F theory as we explain here and in the paper has actually natural in two friends, the little be frame and the return is fine.
Marc Geiller: But now in the rest of the talk just for convenience and sorry. I should have made this more explicit just for convenience. We can use a kind of a notation in each recall one of the frames already
Marc Geiller: So say this frame little so we can already call it and still take me to be independent frame.
Abhay Ashtekar: I said,
Abhay Ashtekar: That we wrote down for the for the synthetic potential to be a theory. Yes, that's what you mean by eat, you don't mean
Marc Geiller: Exactly.
Marc Geiller: What I mean by he is one of the frames and be self has another frame. Okay, so in this simplistic potential there are two frames one frame is used to construct be and one frame is the frame. And this is just the matching of these two frames.
Abhay Ashtekar: Okay, since I'm just asking. So up to slide nine I felt that everything you said was, what was done before.
Abhay Ashtekar: Because, I mean, you, you, you don't download the app to the slide. I mean, this, this, this is the charges that you wrote down. So, I mean,
Abhay Ashtekar: Routine using look calm or, you know, dynamic horizon frameworks are look quantum gravity inspired frameworks and so on.
Marc Geiller: Yes, yes, yes.
Abhay Ashtekar: So I so I just want to make sure that I want to know where the new things happen. So am I right in saying that up to now.
Abhay Ashtekar: This new viewpoint, or the importance of the corner term and so on. It's not crucial, you said it that way, but it can be, they were obtained the charges were obtained
Marc Geiller: Yes, except that now we're giving these these terms and these potential crucial formulation, the definition of the whole theory and not only for isolated horizons or for some specific
Abhay Ashtekar: But I think Dyson horizon. It was just application. The general result that people knew what the results are applied for isolator prizes, I would like to know that exactly the new ago
Abhay Ashtekar: Some beautiful inside that you have where exactly it comes in.
Abhay Ashtekar: So I IT SEEMS TO BE THAT IS. IN THE NEXT TIME spices that are starting from the next chapter.
Abhay Ashtekar: We knew. I feel like we knew
Marc Geiller: When the because this, this thing that I underlined here is that we're shifting completely viewpoint from the boat to the corner, which we don't usually do in quantum gravity.
Abhay Ashtekar: As a writer. I get it. But we don't you think that emphasis, we can still get the same challenges that you wrote down to 44 million road before
Abhay Ashtekar: But I haven't told you I stand Carta and Einstein, you know, the ESF those we could update
Marc Geiller: Record thing but we never use them into quantum gravity also because we're working to engage
Marc Geiller: So I agree that lots of these ingredients for unknown, but they were not used consistently together. So indeed, these formulas appeared here and there.
Marc Geiller: But they're not used consistently to show what we can actually squeeze out of them, which will come, which I will come to you know in a few slides. I agree.
Abhay Ashtekar: Thank you.
Marc Geiller: But this is to kind of gather these results and put them in perspective and see that this is what we want to do.
Marc Geiller: Like it already here. As I said, we use this normal which is not fixed. So we're not in the time gauge. We put the holster on the corner, instead of keeping it in the bulk. So this is very different from usual new quantum gravity.
Construction
Norbert Bodendorfer: So maybe let me add this point to the question about by tomorrow if you can go back to slide number 10
Yeah.
Marc Geiller: I'm sorry 10 yes
Norbert Bodendorfer: I agree what i mean i agree with up i said before, so for for me to clarify this question. So, to the best of my understanding what is new in this work. So first of all, in the first line this theater BF simplistic structure this be delta n minus
Norbert Bodendorfer: Beta, which does he mean, of course, people have contest that before and they have also the difference here is that before you either quanta is the second part, or you want is the first part, but in this work both parts are contest together.
Norbert Bodendorfer: And that will later lead to this edition. So who are either way you can derive
Norbert Bodendorfer: quantization of the area, even though you may use an S one, three,
Norbert Bodendorfer: Symmetry to point is the first part, which was funders before and in such a way. So this is the main thing that is different to previous work here.
Abhay Ashtekar: Thank you. Yes.
Marc Geiller: And also, yeah.
Marc Geiller: And also, let me just restate that this decomposition even have the BF simplicity potential into about peace and a quarter piece and using these two frames. This in itself is also is also a new ingredient.
Marc Geiller: So what is important in this
Marc Geiller: In this analogy. So there is another aspect that I won't cover here, but maybe for questions or for the discussion on happy to go back to is that thinking in terms of these geometrical energy by instructors.
Marc Geiller: There's actually a very nice geometrical interpretation of these BF corner office space which is in terms of a point of asymmetry.
Marc Geiller: And this is kind of also expected because it's known that BF has an underlying as an underlying point asymmetry and here is made manifest by looking at the corner face face.
Marc Geiller: And in this in this bunker a parameter ization the punk and momentum is played by the word of the momentum is played by the internal normal
Marc Geiller: Up to the value of the parameter the power to bond sky vector is played by this spin to form again proportional the Babylonians a parameter
Marc Geiller: What is important is that the chasm Ian's of this of this one, Korea, Java, namely the momentum squared and the positive urbanski vector square are related to, as you can see the the momentum square
Marc Geiller: Identify is actually the mass of dysfunctional present presentations with the Babylonian see parameter some this point Gary analogy.
Marc Geiller: The mass of this one can do presentations is a display the one of the masses paid by the way the inverse for the Babylonians a parameter
Marc Geiller: And the total angular momentum of this support cardiology by is given by this boost and spin part terms of BNS
Marc Geiller: Now, why this is important because the whole point of our construction was to show explicitly that imposing the simplicity conditions on this BF face base breaks, actually this bunker Asymmetric and Symmetric
Marc Geiller: Which is very different from what usually happens in the manipulation of simplicity constraints in informs where we traditionally used to engage
Marc Geiller: And this gives the impression that the simplicity constraints break law and symmetry to an issue to circle here taking the point of view of this
Marc Geiller: BFA space and looking at the corner face face. We can make very manifest and explicit the fact that the simplicity constraints, they just break out into your hands. So at the end of the day, full lawrenson violence is completely preserved. The other important thing is that
Marc Geiller: The area elements at the corner is actually proportional to a quantity which is evidently the handsome violent, which is the pocket is in Kashmir.
Marc Geiller: And this is essentially the reason we've seen a minute for which we get quantization of the corner area in the continuum. And, you know, internal organs in violent manner.
Marc Geiller: And this is related to the last point that i right here because the element is proportional to the spunk is in Kashmir and because
Marc Geiller: This kind of surface this space like this selects the discrete a central hub presentations and this in uses a quantization of area and I will come back in the next slide now will show why this isn't to our structure actually appears
Marc Geiller: And this is just a bit the summary, because we have many violence and things happening. So I have just gather at this here and say that we have an extensive description in this in this work and in these papers of these different permutations of the space.
Marc Geiller: I would just hearing grey this point. It has an index structure, which I think I will skip because
Marc Geiller: We're running a little late. But the important thing is that we start from this point cafe space, which has 12 degrees of freedom parameters by the normal n the be the boost to form and the frame. He
Marc Geiller: Now what you want to do is actually decompose this frame. He on the corner in terms of the the spin operator s a tangential metric QA be and an angle feta.
Marc Geiller: And the pilot algebra, which I just talked about is actually formed by the internal normal and the boost and spin operators and the boost and spin operators. Together they are repackaged in the total angular momentum.
Marc Geiller: So the statement is that this part of your job is formed by these total angular momentum and the internal normal and
Marc Geiller: And now, again, the expression of the simplicity constraint is just the fact that the spin and the boost frames are proportional to each other. And once we do this we get back to tech gravity and we have identified on the space, space of gravity. The ad hoc observable.
Marc Geiller: Which parameters. This face of gravity and this is given by the total angular momentum and JJ, but also the tangential metric your ad and an angle theta.
Marc Geiller: And I will explain. Now, why this gives us. And so to see structure found by the total angular momentum tangential so door algebra fun by the tangential metric. And there's a you want a component from by these twist angle.
Abhay Ashtekar: Being treated differently from the space in this is like creating the space index right
Abhay Ashtekar: So when you have momentum.
Abhay Ashtekar: It really is the internal angular momentum, not
Marc Geiller: Yes, yes.
Abhay Ashtekar: Rotations and so yes it does.
Marc Geiller: Yes, thank you.
It's
Marc Geiller: So now, where does the sweetness of area comes from this comes from the fact that now that we focus
Marc Geiller: On the corner potential. And now that we have moved the horse contribution from the belt to the corner we have access to the corner frame.
Marc Geiller: At the corner and this tangential corner frame actually contains very important physical information.
Marc Geiller: To extract this physical information where we have to do is actually decompose all these components of the tangential corner frame. So the I. A. So as just a two dimensional index on the corner. We can decompose all this violence in terms of these spin operator.
Marc Geiller: The tangential metric QA be and then there is a next component which is a twist angle which is the part of he which is not captured by neither spin operator and all the tangential metric
Marc Geiller: And then it turns out that the parcel bracket on this kind of space, we can compute the algebra of the tangential metric and we find an asset to our eligible
Marc Geiller: But now the important
Marc Geiller: Property and the really surprising is that
Marc Geiller: There is actually a relationship between the chasm. Yes, of the values geometrical objects which are involved here, namely the Academy of this SL to are eligible, which is skew. So the square of the of the area of the corner.
Marc Geiller: And the SU to kazimi so the because me of this podcast has been operator, which is an NES you to continue
Marc Geiller: And it's balanced relation is actually relating these two entities and this is just an algebraic identity coming from the definition of the spin operator, you see that issue, take a square we get that this is beta squared times the determinant of this SLR metric
Marc Geiller: And now, because of the space like nature of these kind of surface. This gives us immediately automatically in the continuum, an internal organs environment quantization of the area. So I just mentioned that this is a bit similar in spirit to think what that Vulcan was doing
Marc Geiller: And he still pursuing accent that he was working with the spinners. And I think focusing on not surfaces, but it would be interesting to kind of put these things together and to and to compare this works a bit more
Marc Geiller: Now, the new important thing. Now that we have access to the frame on the corner. So, as we have already seen just in this point before is that access to the frame gives us access to the tangential metric
Marc Geiller: This tangential metric satisfies a message to our algebra and from this follows this intolerance and volume quantization of the area around the corner in the continuum. Now the other thing which is provided by the corner frame is actually a complex structure.
Marc Geiller: This is understood if we think about where this twist angle comes from this twist angle comes from the fact that we have an ambiguity in performing such rotations of the frame.
Marc Geiller: So with this angle feta, where this star operation here. So he's defined so so here on sling and the star operator actually turns out to be a complex structure because it's whereas to minus one.
Marc Geiller: The star structure as we show is also related to
Marc Geiller: The area operator and
Marc Geiller: To the fact that the area and this kind of faith based is actually conjugated to this twist angle feta.
Marc Geiller: Now from the knowledge of this complex structure star using the Giacobbe identity, we can actually show that this corporate structure is compatible with the person records and this is very important for us because it will enable us to actually
Marc Geiller: Study properly for the first time the simplicity constraints on this corner faceplates
Marc Geiller: So now that we have understood where these sent to our algebra and the sweetness of area comes from let us look at the simplicity at the corner face base.
Marc Geiller: So to look at this simplicity constraints. Once again we go back to the BF potential with two independent frames or frame used to construct be and the frame.
Marc Geiller: And the simplicity condition the simplicity constraint is just a simple organization between the two frames, you can do to two forms BNS so the boost to form and the spin to form and as usual the proportionality is given by the interest of your emails a parameter
Marc Geiller: Now that we have a corner face base.
Marc Geiller: We're able to compute
Marc Geiller: For some brackets of these simplicity constraints and find that the simplicity constraints, they're actually second class with themselves.
Marc Geiller: And you see that this comes from the fact that, well, this is just directly rooted in the form of this BS electric potential at the corner. So the difference with what happens in a in usual studies of simplicity constraints in in spin for models.
Marc Geiller: Usually what happens is that the simplicity constraints become second class with themselves because of this pinfold quantization, which assigns the algebra elements to the field.
Marc Geiller: But there was always a tension here because the beef in itself in the bulk of the theory commuted itself.
Marc Geiller: So here we understand that the simplicity constraints already second class with themselves because we should think of them as constraints on the corner face space and on the common have a space.
Marc Geiller: The field. Does not compute with itself. And this is the reason in using this bracket here between the simplicity constraints.
Marc Geiller: Now, to understand and study further these constraints we have to realize that there are actually three constraints here.
Marc Geiller: So the index is four dimensional, but whenever you have something with one index i should contact it with the normal, you get zero.
Marc Geiller: So this is actually only three independent constraints. And if you have three constraints which do not commute, they cannot all be a second class, there are, we have to split them somehow into a first class piece and second class pieces.
Marc Geiller: Now we show in the papers that the first class piece can be extracted as follows you you construct these two scanners. The first one is that you square the simplicity constraint c squared.
Marc Geiller: And then you connect it with the the spin operator and you construct this quantity courtesy and you can show that this is actually the first class piece of the simplicity constraints. And this is actually the continuing version of the diagonal simplicity constraint which was identified
Marc Geiller: Back in the days in the studies of spin for models and all the second class piece of these constraints is actually obtained by contacting the simplicity constraints CI.
Marc Geiller: With the frame. So once again, we see that knowledge of the frame on the corner is essential to construct the second class piece of these simplicity constraints.
Marc Geiller: Now the first class piece we can kind of treat
Marc Geiller: Want to say in standard ways. The question is how to treat the second class constraints, especially in the quantum theory.
Marc Geiller: And now, again, we are saved here by the fact that we're focusing on these kind of face face.
Marc Geiller: On which we have just seen that you have access to the frame field. And because we have access to reframe field we have access to a complex structure.
Marc Geiller: And we can now use this complex structure to build automatic and anti hello mafia components of a constraint, but just defining C plus minus
Marc Geiller: And then using C plus, minus, we can do a good job learner quantization an imposition of the second class constraints.
Marc Geiller: Maybe what you want to do is replace the second class constraints ca by automatic first class constraints which is C minus emulating the states.
Marc Geiller: And alternatively what one can also do and what we explained in the paper Atlanta, is that we can also use the master constraint which is just the square of the second last piece of the constraints.
Marc Geiller: You can see that classically using the master constraint is equivalent to imposing C minus because this master constraint can be just written as two times C plus QC minus
Marc Geiller: at the quantum level. This is more complicated because there are commentators involved here which are not changing the quantum theory.
Marc Geiller: And so if you want to rewrite the master constraints in terms of C minus, you get actually what is called an anomaly due to the quantum commentators that this can be kind of studied very explicitly
Marc Geiller: So this is just to emphasize the fact that by focusing on the corner we have access to the from field to field gives us an SM to are eligible for the tangential metric
Marc Geiller: This is responsible for quantization of ally in the continuum. It gives us a complex structure. It enables us to identify the first class second class piece of the simplicity constraints and then to impose the second class simplicity constraints with the group who formalism.
Marc Geiller: Now we can
Simone: Mark
Simone: And ask a question.
Marc Geiller: Yes.
Simone: Well, it seems to me like most of these things.
Simone: Can be done and have been done for the API model. Can you comment on what results would expect to be different.
Marc Geiller: Well, I think that the. So there are many differences. The first thing is that I'm not sure that this form of the second class simplicity constraint was identified in the model.
Simone: It's like I am thinking of the user spinners that we had for instance we Wolfgang we had exactly this meeting and identification of
Simone: Extraction or to split the second class constraints that we used to call to, it seems to me, and the result was just a standard API model at the end of the day.
Marc Geiller: Yes, what it seemed to me that the second class constraint, you are getting was not actually equivalent to to do this one, but
Simone: So it's a classical difference or it's a quantum difference
Marc Geiller: Where there's already a classical difference. And then there will be of course quantum difference
Simone: So you. So you're saying that the classical second class constraints are using different than the ones that were used to get the model.
Marc Geiller: Yes, yes.
Simone: Yes. And what is it different, a decade now proportional to any longer.
Laurent Freidel: The difference is that n is part of the face space.
Marc Geiller: You're here.
Marc Geiller: K is not what I mean. You mean by can and if you mean
Marc Geiller: So you see, usually defined K and L with respect to some kinematics vector that you have chosen and you can show here that imposing these kind of
Marc Geiller: Usual epl simplicity constraints capable optional to L does not imply these covariance simplicity constraints that we have on the corner here. So they're not equivalent. Yeah.
Abhay Ashtekar: So the
Simone: Keys like equal gamma hell.
Simone: Can be written in any free right
So did
Simone: You mean by client here.
Simone: By going to mean that you're a large in the pace pace and by working with a larger pace pace, you're getting a slightly different definition of the second class constraints.
Simone: And that's what you mean by the fact that there are different radio the classical because they find on the large base, base.
Marc Geiller: Yes. And also another crucial differences that here. You know, usually to in the construction of the epl model, you have this wine that that embeds the ST two states into into the Lawrence and why it's important to us and to see
Marc Geiller: But here what we are not explaining this here on the slides because we don't have enough fine, but what this enables to do here, the crucial differences that you can actually hear
Marc Geiller: embed the northern states into one cafe and the whole construction is environments under all law and transformations here which is not the case.
Simone: Of the one so that the quantum the quantum counterpart of these large classical face pieces defense and instead of working with unitary representations of the learnings group, it would work. We need a representation of the one Gary group right or something like that.
Marc Geiller: Well, you will invite them. Yes.
Simone: Thank you. Yes.
Okay.
Marc Geiller: Yeah. So here I'm just glossing over real quickly over the what we would get if we solve just classically the simplicity constraints. So the first class piece and in the master constraint.
Marc Geiller: setting them to the to zero, then we can rewrite these constraints in terms of the of the values Kazimierz which are a play here. And then the, the two lines Kazimierz he went to jail.
Marc Geiller: And these podcasts in customer as to which has already appeared before
Marc Geiller: And then the simplicity constraint is basically a relationship between the law Hanson weights and no one can explain. But the crucial thing maybe to go back to someone is comment is that this is very different from the usual gauge fixed energy picture which we fix
Marc Geiller: Reference normal which would be here because one control that. So, here what I have explained before, is that this the norm of the spin operator is proportional to the corner area.
Marc Geiller: And the spread operator has a quantum number, which is this little s here, but the energy areas that you get by fixing
Marc Geiller: Nabi timeframe are actually boosted areas with respect to these this kind of hires. So the spin j that you would get an energy is always bigger because of these boost than this one guy quantum number is
Marc Geiller: And as I mentioned, we are completely shown by computing
Marc Geiller: All the necessary for some records and all sorts of
Marc Geiller: Factors that we get these ad hoc observable on the face face after we have imposed the simplicity constraints and these ad hoc observable or this total
Marc Geiller: Internal angular momentum JJ, which is just the this Lawrence charges at the corner the tangential metric your ID and the angle feta.
Marc Geiller: And so because of this, we see that the algebra that we have to focus on in the case of data gravity.
Marc Geiller: Is actually has a different markets and peace. There's an incentive to CPS coming from the total angular momentum. So the Lawrence charges. There's an Excel to RP is coming from the tangential metric. And there's a you want peace coming from this twist angle.
Marc Geiller: And just as a comment. You can notice that this the SLT our part in this angle. These are components of the college algebra, which are not actually coming and arising as charges occasional formations.
Marc Geiller: This is something that you cannot really actually studying the usual coverage Facebook picture but this you can only get if you focus and look squeeze all the physical content out of the corner syntactic structure.
Marc Geiller: So now we can go back to the stable and we'll be
Simone: Sorry, one, one final question about this enlarge face to face.
Simone: And do you know what is the orbit generated by the first class part of the simplicity constraints in a large space in the. Can I see in the older one in the standard one for the PRL. It was the boost the hero angle you no worries here.
Marc Geiller: Why it's the way you say it's the booster heater angle in the
Simone: When it was the result of a calculation. I mean, we just looked at the algebra that we had on bass, bass.
Simone: The one that we use the in terms of the history of myself to see a covenant a space. You don't know if you remember
Simone: With this description of the Covenant with Krista geometries. Yes. And they're there. We could look at the orbits generated by the first class part of the constraints and we could identified in terms of the if you want the boost SL to see equivalent of the
Simone: Angle of twisted geometries. I don't know if
Simone: Yeah, I think, which for us was very nice at the time because he had a very clear analogy with what happens in the continuum. So I was wondering if something like this is still happening.
Laurent Freidel: Here is going to continue
Laurent Freidel: To go to to
Laurent Freidel: Continue here. So there's no died. Remember the stage.
Simone: Okay, it should be.
Simone: Part of the relation
Laurent Freidel: You know,
Simone: Okay. So forget about the word that
Simone: You should still be some parts of the connection, I suppose.
Marc Geiller: Yes, it should be some yeah but then I don't know that.
Simone: You don't know. You don't know yet.
Marc Geiller: Not yet, but I will go back to reschedule in a minute.
Marc Geiller: So now we can just go back to this.
Marc Geiller: To this table and then filling the blanks. And again, look at these different formulations.
Marc Geiller: And realize that by working with the
Marc Geiller: Einstein capital formulation with the holster and not using the time gauge. We have access to the next heart. So I said to our as YouTube and the boost components. So all of us all to see.
Marc Geiller: Now, of course, this begs the question of what happens if you go to probe the two other directions, if you will find other formulations of gravity.
Marc Geiller: Which have may be other types of Connor symmetries. And what is the what is the organizing structure behind this, but at least. Here we see now we understand clearly that
Marc Geiller: There is a sense in which we can do the split between bulk and Connor contribution and once you do the split. We can unambiguously start to build such entries in this table.
Marc Geiller: And so now I just make a comment about the possible quantization of this college algebra. So again, what what is important here is that we have recovered these ingredients of
Marc Geiller: Of quantum gravity, like the the sweetness of the sweetness of the area spectrum and also a proper understanding and imposition of the simplicity constraints, but completely in the continuum.
Marc Geiller: Now to stay in the continuum. What we have to do is to push further the quantization of the theory. So this means that we have to study.
Marc Geiller: States which are serving the first class implicitly constrained courtesy and the holographic constraints C minus also studied the quantum anomaly. A and also what this enables us to do for the first time. This opens the possibility of enticing the friendship, which is something
Marc Geiller: I think that would be very promising and first look at this actually tells us that to have a proper quantization of the French field, we have to actually use tensor operators which, as we know, are actually related to the structure of entrepreneurs.
Marc Geiller: In the fury. So these would be here. I said to our times. As you to transfer operators.
Marc Geiller: And the fact that they are potentially related to entrepreneurs tells us that this has potentially something to do with the construction of bulk information.
Marc Geiller: From the knowledge of what happens, what happens in the corner, and more generally speaking, the logic would be on the next
Marc Geiller: Natural step for us is to try to actually quantifies or final presentations of this corner symmetry algebra, which, as we have seen is the S and there is a component which is
Marc Geiller: I put here really algebra. Ah. So, hello, can be eligible each at each point on the corner. But the important thing is that
Marc Geiller: This is not an completely unreasonable and daunting task because we don't have to only point is
Marc Geiller: Just different Marxism. Of the two sphere. This is the for more seasons of the year, but subject to the condition that we have
Marc Geiller: A discrete area element, square root of view as we have shown and these discrete area elements.
Marc Geiller: Is also giving us a relationship with on the Kazimierz of the of the algebraic structures which are involved here.
Marc Geiller: So we have to quantify the Scar Symmetry algebra, but we have the knowledge with the crucial information that all the Kazimierz are determined and related to yada yada element cube with these relationships which are given
Marc Geiller: Here on the side. So there is some work hasn't yet there is the point guard has been kazmir and the to SF DC Kazimierz yes
Abhay Ashtekar: So since I mean especially nice very, very beautiful, but I want a clarification year so you're, you're not continue as you enlarge
Abhay Ashtekar: Emphasize and that's great.
Abhay Ashtekar: But then all these things operators are talking about in your, your corner are all in the corner Hilbert space.
Abhay Ashtekar: Yes. So, therefore, when you say things like, when like eat you say you say that, well, for the first time you got conversation on this evening, but that he is still refer to the Parliament not to the buck.
Marc Geiller: Yeah, but it's better than having not yet. All right.
Abhay Ashtekar: No, no, no, no, I just look but then when you say that, well, therefore, you'll get bulk information that is a hope is it. I mean, why is that such a hope.
Marc Geiller: Yes, so it's so indeed, thanks. So it's true that in the DC is the is the corner it and it's not the it's not the bulky and
Marc Geiller: So the goal here eventually to to push this is to show that we can reconstruct bulk information by size. I've already shown the for the expression of the bulk constraints takes a very simple form as just conservation laws for the count of charges.
Abhay Ashtekar: So, in principle, that's the action of the bulk constraints. That's not the expression, but that's okay let's maybe we should go on.
Abhay Ashtekar: Now I would like to understand this very much because this is coming to the heart of the issue about whether really there is a philosophy or not.
Abhay Ashtekar: And so far, what you're done is very, very nicely constructor this corner face space and told us about structures there, and perhaps some ideas or conversation, but I don't think so far. I don't see her and have a
Laurent Freidel: Good is going to show up. Is that what would you call the BC. The new quantum gravity.
Laurent Freidel: Is, in fact, just the corner BC, so you know there you're going to be in the same position as you're a look at that your recollections of BC, except that now you have extra information, especially the French. So the challenge.
Abhay Ashtekar: Is part of the information, but
Abhay Ashtekar: The field is not the full bulk information as everybody's emphasized on
Laurent Freidel: Oh yeah we do it you don't have access to the today said to see charge to the site will charge into the Frankie and that's why you cannot click on Submit and constraints.
Laurent Freidel: So now that you have this access to this information is the hope that the missing information is there. So you can start over again with with a more complete understanding of the underground operators.
Abhay Ashtekar: Okay.
Norbert Bodendorfer: And let me may interject at this point also because it's a good point to address again the question that I had earlier about previous work.
Norbert Bodendorfer: So to connect to a previous work, which is the previous two what is cited here so
Norbert Bodendorfer: What was done before is again. You were quantifying this boundary algebra. When this parameter beta was set to zero, then you can see that
Norbert Bodendorfer: The first, second and fourth object in this equation. See, are there some p zero and just the third one gives to the area density. Now back then.
Norbert Bodendorfer: We were working not Lawrence sad to see that we were working with the so for so with the Euclidean version, which you can do by change your variables.
Norbert Bodendorfer: Which would give you a discrete areas spectrum. But what's not changing here. What's the new ingredient. You can also work with. So one three. So with a with a non compact group.
Norbert Bodendorfer: And then you can match this sad to see or this asset, who are causing. So the first expression there and it will give you this discreet spectrum in the area by mentioning this kazmir from so to our and mentioning the cosmic from so to see
Abhay Ashtekar: Okay. I mean, since you know what has brought brought up this point. So what I'm confused about is in the previous transparency, I think it was like number
Abhay Ashtekar: 13 where where it was said that some of this this Wooster areas l is equal to jail because any progress.
Abhay Ashtekar: But area just area. Right. I mean, in other words, you are in a continuum. You're gonna fix the boundary and we're talking about the area of the boundary, as far as I couldn't stand
Abhay Ashtekar: So I don't see why something is boosted because in fact I insisted that as an answer to leave earlier question that the boundary is completely fixed. They're not allowing your laps to become zero, etc. So what does Bousquet area mean
Marc Geiller: No, I mean, I mean boosted you know it's an internal boost right on these internal in this is
Abhay Ashtekar: Right. But I mean,
Laurent Freidel: I think
Abhay Ashtekar: Any of the surface, right.
Abhay Ashtekar: Internal booster internal anything
Laurent Freidel: With
Laurent Freidel: That showing the paper that you just been is not the area of the surface. So they are able to assist her face. Here is the speed
Laurent Freidel: Low point guys been and where people you before we are computing is not the area, the surface, but it's a boosted area because it was not taking into account the fact that n is like a momentum. It's part of the things
Laurent Freidel: That's kind of technical things be nice. Okay.
Marc Geiller: Let's say I agree with you about you're completely right. Indeed, there is one I should have clarified, it's there is only one a given area and the area is the area right there are there are no two different areas are
Marc Geiller: So there is only one area and then this area of course is completed in terms of the of the metric on the corner the metric on the corner is constructed from
Marc Geiller: The frame field on the corner and what we show here is that the Franklin on the corner.
Marc Geiller: Information about the IRA is encoded in this point guard has been causing me. So this just tells us that what we said what we previously identified in the quantum theory as the area, Nikki G was not the area because it was
Marc Geiller: It was corresponding to L and this has been s
Marc Geiller: And
Abhay Ashtekar: Then I will you ask for the spectrum of the area and that spectrum baby is the same. That is what Robert has been sort of intersection. Right.
Abhay Ashtekar: It's true that this all comes in some mathematical framework, but then if I just give you the surface and asked for one of the eigenvalues of area of the surface, what you're getting is the same as what we were doing before, because that was like
Marc Geiller: Well, the same in the sense, yes, in the sense that
Abhay Ashtekar: Everyone operator, you're going to surface physically to ask you for eigenvalues of ad operator.
Abhay Ashtekar: You can say that there's some of you do some calculation and other people did some calculations and the two are not quite the same as fine with me. But both of the people are claiming that the spectrum is appalling and the both of them are agree, is that kind
Laurent Freidel: Of a square. It's a square, this
Abhay Ashtekar: Area operator in any other area. Yeah, it's the same spec car.
Laurent Freidel: Coming from the same thing.
Abhay Ashtekar: That that thing that is what matters. And most of
Laurent Freidel: The same spec for the new thing is the reconciliation of the loop with Lawrence.
Abhay Ashtekar: Yes, exactly.
Laurent Freidel: But I think the same spec.
Abhay Ashtekar: Exactly, exactly. Thank you.
Daniele Pranzetti: And Jessica if vacation but also maybe the most important thing is the fact that there are choose p. Now isn't in a new country number in the picture. So this Jamia
Daniele Pranzetti: boosie area to give you some more physical dramatic interpretation, but the real new piece of information that we have a next one to number now in the game.
Daniele Pranzetti: That enters the quantization Davis. Yeah, yeah.
Abhay Ashtekar: That is something that came also because you've got an extended face face and
Abhay Ashtekar: On the CD yoga extended Ignite. It's very, very nice as marketing and and Laura emphasize, you know, is the pieces of this, we are all over, but some are having a clear picture is what when big progress can happen. So I think having this clarity is extremely nice. Thank you.
Simone: Just a quick question. When you say that the spectral the same
Simone: I mean you know continuity to defend the spectra. You want to know whether the puncture is like going tangential to the surface or
Simone: Can you go to the surface, whether there's a know that the surface of the spectrum could change of the area operator, whereas here everything is continuous. So when you say that the spectrum. The same you mean with the case in which there's a single puncture
Simone: We're going through the surface, right. So they might differ if one considers different type of punctures, I suppose.
Abhay Ashtekar: No, no, no, not the Spectrum. Spectrum.
Abhay Ashtekar: The question is how are the eigenvectors disconnect
Simone: The Specter. Yes, yes, absolutely. Yes, absolutely.
Simone: Yeah. This was the question. So okay, so we're not saying that the eigenvalues are this week, decision making process is the same are just making the algebraic statement about the spectrum.
Abhay Ashtekar: The same writing the spec is the same as a certified values. So the state of eigenvalues. The same then hold that I can back. This will come is something that will have to
Simone: Write your novel talking about states and
Simone: What are they
Okay.
Abhay Ashtekar: Going back this is knowing the extent that you must be extended face face and and then extended but specially the extra random number. So the description is going to be different, but the specter of the same that is it.
lsmolin: Yes.
lsmolin: Can I ask something here.
Marc Geiller: I see or just
lsmolin: Go with it. I agree. It's very beautiful and it's clarifying but with the simpler structures we used to have it didn't need so much clarification.
lsmolin: One of the basic ideas in your early work about surfaces at night at night as entire services boundaries is fixed position was that there was
lsmolin: One of the constraints, the gas law constraint link the puncture which had the spectra that we're talking about to an edge which went into the interior into the bowels of the manifold of the quantum space time. Is that still true.
Laurent Freidel: I think it's the same. Next slide. The
Marc Geiller: The twist. Yeah, maybe.
Marc Geiller: Yeah, so maybe Leah, please let me go back to maybe your question, you know, either on the next slide, or after the because I still have a few slides. Maybe they retire. If I in fact that you
lsmolin: Sure. Now, but you're saying, the answer is yes, you're going to
Laurent Freidel: When we
Laurent Freidel: Give you know okay
Marc Geiller: But let me just so indeed, as you mentioned, the link with the and also by mentioned the link with a what what we're used to in in in quantum gravity. This can now be recovered from this fully
Marc Geiller: Coherent and continuing picture by constructing by kind of simplifying the study of this whole presentation of this
Marc Geiller: Kind of similar to Gemma, we can kind of simplify this by trunk eating it. And for me, discreet about algebra.
Marc Geiller: And this is just by putting no extra structure on this corner, for example, partitions of the corner and considering some earrings, which are piece wise constant, for example, for the Lawrence charges.
Marc Geiller: And this enables you to go from an old article SM to see algebra to local SM to see or su to in the time gauge. Our job as per patches which is the usual MTG picture.
Marc Geiller: And alternatively you can also have introduce kind of defect like pictures like we did with the Ganga and also in the work of a lower than you, they are you know and
Marc Geiller: So this enables you to kind of define different types of
Marc Geiller: Scripts about Japan.
Marc Geiller: So now the last. The last point. What I want to talk about is the role of the whole of edge notes because as you notice the name of the of the papers has elements. And this was also the initial title of the of the talk, and I haven't really talked about this yet.
Marc Geiller: Because this is the last ingredient. I want to present now maybe quickly because it's running out of time.
Marc Geiller: But to understand what we mean by Edge mode, we have to now go back to the questions that brought us here in the, in the very first place
Marc Geiller: And this this question in the simple fact was the fact that boundaries break actually engage in violence. So once we are faced with the statements we can have to
Marc Geiller: Go to basic possibilities. One of them is to restore gauging violence by actually
Marc Geiller: Playing with what we think are actually corner and you it is and tuning these ambiguities to suppress the charges in order to start gauging volumes. This is what was done by
Marc Geiller: Example in I'm in Somalia, but as I have tried to argue this kind of goes against the systematic treatment of these qualities potentials and the statement that is called contributions that actually features and we should not
Marc Geiller: kind of try to suppress them. But if there are here, we should keep them and try to squeeze information of them. And also, if we suppress some charges, we might
Marc Geiller: You know, lose information about Connor observable and important quantum numbers which should potentially play an important role in the construction of the of the quantum theory.
Marc Geiller: So, at age modes enabled us to do at least what what we mean in the series of papers by by Edge mode because sometimes it means something different.
Marc Geiller: Different works. What we mean here is a structure and mechanism which enables you to restore gauging violence on the boundary while having non trivial symmetry charges.
Marc Geiller: So the way to do this was actually
Marc Geiller: initially introduced by law and William and then subsequent work by many people studying the inflection of these animals in many different theories. But essentially, if I if you allow me to skip a review for this. The idea is to extend the potential of the theory.
Marc Geiller: By now adding on the corner. So see that here I have written again they have used this canonical split of the potential into universal bulk contribution beta sigma
Marc Geiller: And the corner contribution is but now we want to consider the fields on the corner of to be independent fields from the bulk fields, which is why I didn't know them with bold letters now.
Marc Geiller: Now, once we have promoted these fields to independent fields from the bulk fields. We also want to a place in this corner syntactic structure deviation delta by kind of shifting deviation for
Marc Geiller: Japan reasons which I'm happy to, to explain after if it's not clear. But the idea is that we want to include us here, a group element chi.
Marc Geiller: Which suggestively here. I say leads in SM to see times as into our because these are the two components of the symmetry algebra, which you have uncovered before when this group elemental response to a choice of gauge frame on the corner.
Marc Geiller: Now just doing this working with this extended potential, we can take this as a definition we take the potential of our theory we make the fields on the corner of independent from the fields on the back.
Marc Geiller: And we translate the variation on the corner by adding this group element sky. Now, once we do this, we get to consequences of this construction.
Marc Geiller: The first consequences that again. Forgive transformations. There is a bird component and the corner components, the bulk components is the usual gals low
Marc Geiller: And the corner components is now kind of a corner constraint or equation of motion.
Marc Geiller: Which is imposing the continuity condition of relating the bulk fields to the boundary issues and this relationship between the bulk fields and the boundary fuse is exactly through this
Marc Geiller: Gauge frames. So this group elements which you have introduced in this extended simplicity potential
Marc Geiller: So this is what they have written in these three equations here. The beefy in the bulk gets identified with the beefy on the corner up to an SM to see consolidation.
Marc Geiller: Same thing for the normal and the frame fill in the bulk gets identified with the frame field at the corner of to listen to seek information and an asset to our information.
Marc Geiller: Now, to understand what is the content of this of this edge mode and this group elements. We can also mention, and I will conclude with this.
Marc Geiller: The, the interpretation of this continuum structure in terms of twisted geometries.
Marc Geiller: So just to recall the initial inclination of Christie geometries, as they were introduced by behind semana.
Marc Geiller: Is actually as a map between the face face of Luke quantum gravity on a single link or a product of this to get the faith based on the graph, which is the star of St to on a single link.
Marc Geiller: And twisted geometries are just a nice amorphous and between the space, space and some geometrical data which is geometrical data have some
Marc Geiller: Some notion of discrete geometries and its geometric on data is basically the knowledge of
Marc Geiller: Of a vector associated to
Marc Geiller: The source and target triangle. So here every presented on this picture a little
Marc Geiller: An edge of a usual as the network and it has a source know that I target node. And there's a triangle that leaves. Both of these nodes and then a scene from the source frame and the target frame this triangle has a normal
Marc Geiller: And these triangles have also an area given by J. And there's also a twist and go cheetah, which is conjugated to this beam j, which is the ability to kind
Marc Geiller: Of twist, these, these triangles like this and the usual familiar Helena means of reporting gravity in this picture there is because there are just a couple and rotating the normal from the source frame to the target for him.
Marc Geiller: But in this construction of twisty geometries. It was actually pointed out, and notice that there is a discontinuity of the metric between
Marc Geiller: Between the two triangles because matching the metric actually requires too much the frame field instead of just the flexes
Marc Geiller: Whereas in the Trista geometry picture with this group elements were transporting the flexes, which is a casar information than transporting the frame field itself.
Marc Geiller: And to actually much the metrics we need an area preserving different Marxism, which is an SL to Athens formation. This is nice because we have just now argue that there is this software component
Marc Geiller: And now I can present this construction at the corner of this data of twisty geometries, but from the continuum information that we have
Marc Geiller: naturally derived in this work.
Marc Geiller: So we get this a discrete geometrical data by information which is again in this point. Got a spin operator is
Marc Geiller: The metric and the twist angle.
Marc Geiller: So what now plays the all of these normals. Is this for vector which is the point counting spin operator. There's one associated to the source node.
Marc Geiller: And want to target node. This comes with the quantum number of pitches s and there is again this twist angle which, as have argued before is a
Is
Marc Geiller: Is what is necessary when we want to
Marc Geiller: So it's the extra information about the frame which is captured neither by the tangential metric, nor by the point is been operator. This is this twist angle theta.
Marc Geiller: And are there is an SL to see confirmation, which is taking the spin operator in one frame has been operator in the other frame and. Now the nice thing is that this transformation.
Marc Geiller: Actually comes from these Edge Mode fields which I have introduced on the slide just before. So you can think of these editable fields as just have fallen dummies.
Marc Geiller: And we can see what is helpful on them is do if we think again about this book to kind of continuity condition.
Marc Geiller: So the bird to Connor continuity condition which is on this extended face face with edge modes tells us that the bulk field which is here, the blue.
Marc Geiller: Belt Franklin is identified with the governor Franklin add to the action of a group element chi.
Marc Geiller: Now if you match from both sides mentioned that we have 222 bubbles of space which identify around the corner.
Marc Geiller: And you have a source Frankie and and the target for infusion. If you want to match the source from field and the target French is coming from the bulk
Marc Geiller: In terms of the corner frame fields. This is exactly this bold relation of the bottom here, the corner from field on the left is identified with the corner framed on the right up to the action of this of this of this group element dysautonomia chi st which is is minus one times right
Marc Geiller: So this illustrates how this these Edge Mode which we have now introduced at the very end actually encode some information analogous to
Marc Geiller: Pilot HubSpot into two columns. So this kind of completes the picture and this expensive be divination sheet between
Marc Geiller: This is continuous data and how we can, how we can, if we want kind of these criticized is continuous data and get something that looks like a generalization of twisted geometries.
Marc Geiller: So to conclude, what I wanted to show is that the starting point was to just argue that for any given formulation of gravity.
Marc Geiller: Is the canonical splits of this eclectic potential and of the syntactic structure into a bulk component and a common components and that by just focusing on the counter component we can extract important and
Marc Geiller: Useful physics. And what we have seen on what I think is very nice aspect is that with this very general statement which I feel is applicable to any
Marc Geiller: To any theory of gravity to any formulation theory of gravity, we see that when we apply this to tetra gravity. We are naturally and unavoidably brought to features of the quantum gravity.
Marc Geiller: But we some kind of generalizations and some subtle differences, namely one is that the internal normal now is part of the face face and present very important all
Marc Geiller: The other one is that we have clarified that the common metric is non competitive because we have added the bar before me as a parameter
Marc Geiller: Is also enabled us to show that the second class. The simplicity constraints are second class with themselves already in the continuum.
Marc Geiller: We have also recover all the discrete areas spectrum with manifest in town alliance and violence.
Marc Geiller: And argued that once we take the point of view of working in the corner we get access to new quantum numbers and maybe you notions of
Marc Geiller: These twisty geometries and it is now focusing on the corner face face kind of pave the road and tells us where we can go next, because it tells us, naturally, how we can generalize this
Marc Geiller: And what kind of structure, we can expect to extract to extract out of this. So I think that this shows that these features of quantum gravity are a bit impossible to avoid if we take this point of view, seriously, once we look at the corner, these features are just here.
Marc Geiller: So the next natural thing to do is to kind of know investigate this question of what is the biggest cemetery algebra. So Luke potentially at other topological terms, aside from the horse term.
Marc Geiller: Popularly constructor quantization, and the whole presentation theory of this kind of symmetry algebra.
Marc Geiller: And then think about the the proper entangling infusion of this corner states, think about the dynamics in terms of some notion of conservation of charges and think about the whole of matter and the role of
Marc Geiller: Maybe potential relationship between matter and the effects of all of these geometries.
Marc Geiller: And with this, I conclude, thank you.
Ivan Agullo: Thank you, Mark. For this beautiful talk and a lot of questions during the talk and bad if there are any other question, please go ahead
lsmolin: Yes, it's a very beautiful picture I what happens when you turn back the cosmological tension.
Marc Geiller: So indeed, a good question. So here, if we had included the cosmological constant
Marc Geiller: Seems that nothing would
Marc Geiller: Change at this level, because as you know the the syntactic potential there's not
Marc Geiller: There's not actually depend
lsmolin: It depends on the boundary conditions if you
lsmolin: If you choose self to boundary conditions. It does depend
Marc Geiller: It depends on the boundary condition, but here you see that we have phrased and approach the study in such a way that we post on to end the study of boundary conditions because we don't want to be
Marc Geiller: kind of confused by the use of boundary conditions and the hive results which hold with one boundary condition is not with another one.
lsmolin: Right but well anyway. It's something to investigate.
Marc Geiller: Yes, in fact, to. So when I talked about the introduction of the elements here. I did it in a simplified setup and you see that the only introduced Edge mode for these
Marc Geiller: SL to see an ascent to Athens formations. So these kind of
Marc Geiller: I didn't talk about different more films. So the statement as shown in work by
Marc Geiller: By an internist bonanza and also really am enjoying myself is that if you also add the edge modes for different washes and confirmations.
Marc Geiller: You have a nonzero cosmological constants. There are kind of subtleties to take into account and this will actually give you structure is on the corner, which depends on the cosmetic held constant
lsmolin: defamation of the Lorenzo with the
Marc Geiller: We have not seen this yet but potentially but at least I think that now we have kind of the tools and the framework to investigate this and to
Marc Geiller: See what happens. Okay.
Ivan Agullo: More questions.
Abhay Ashtekar: Coming. I mean this continuation of what leasing something I think what has happened was that, you know,
Abhay Ashtekar: Most people who are interested in the initially in the US in particular flag boundary condition or, you know, some article in the sitter or anti the sitter boundary conditions and so on so forth.
Abhay Ashtekar: So, people didn't really look carefully enough. I mean, there are scattered results, but not a systematic investigation of what happens
Abhay Ashtekar: With just finite boundaries and as you say, dropping the boundary conditions, and this has brought up very rich structure and put a lot of order.
Abhay Ashtekar: But I think at some stage, you would have to look at appropriate boundary conditions. For example, what do you mean by gravitational waves. We don't have it until we have the correct boundary condition that infinity.
Abhay Ashtekar: So those kind of degrees of freedom and have it's kind of bridge physics as we have learned over the last five years.
Abhay Ashtekar: Will have to be they make. I mean, it's a complimentary question almost is no it's not a
Abhay Ashtekar: It's not clear to me that will fit in naturally with this particular program to put those kinds of us in a flat or was invited to sit the boundary conditions.
Abhay Ashtekar: But that's something that has to be kept in mind, I think at some stage because there's also rich physics there.
Abhay Ashtekar: Initially, we're all drawn to that physics. And what you're saying is that in doing that, we ignore a lot of other physics, which is in some sense microscopic and even more important by looking at this local thing and
Abhay Ashtekar: Dropping the boundary conditions. But I think at some stage a generalization and matching. We did two things will activate that
Marc Geiller: Yeah, I completely agree things. Yes.
Norbert Bodendorfer: I have a question about the degrees of freedom here and this is
Norbert Bodendorfer: Quite naive, but I'm not really understanding it. So in earlier formulation. Someone was looking at this problem of monetizing the boundary degrees of freedoms. One was working with
Norbert Bodendorfer: With zero Lawrence charger. So you were working in this formulation, where we don't have this additional internal momentum. The boundary that you declare now as new observable.
Norbert Bodendorfer: And I was always happy with that because it seems that you were not enlarging the amount of degrees of freedom that you have in GR which are two per point
Norbert Bodendorfer: With no you say
Norbert Bodendorfer: Instead of two per point I have eight pro boundary point. And do you understand how this goes together.
Marc Geiller: So let me just already see
Marc Geiller: Yes, there is eight per boundary points in the sense that we have eight the lock observable. It's
Per
Laurent Freidel: per barrel. Can I say something about that.
Laurent Freidel: It's a very good question. So there is a pair boundary point but I six of them are really related to gauge charges. So when you when you
Laurent Freidel: When you look at the interface between a bank should be seen as an interface between the left side and the right side. And when you put that interface, in some sense, you have to gauge out these six Lawrence charges and you're left with the two, which are the asset to our children's
Laurent Freidel: But that has to be done in in proper discussion and thats related to the question of our by which is also, you know, people understand now much better our sympathetic symmetries are related to find it symmetries, and all of that.
Laurent Freidel: But I hope that that counting helps you. So out of these eight six tolerance and to our to our
Norbert Bodendorfer: Okay, I understand that. But the question is then, why do you introduce these additional six degrees of freedom in the first place. If you want to engage them out later.
Why
Laurent Freidel: Why, why do you include us up to speed in energy. And so, so it is part of the labeling of the degrees of freedom.
Right.
Laurent Freidel: I think there's a difference between what are the degrees of freedom and then somehow the memory effect attention to that. So gravitational waves.
Laurent Freidel: The way they enter in these charges. You have to look now at what happens when you take your sphere and you move it right and and got the charges if there's no gravitational waves. The charges. I mean, be conserved. And that's what Lee and and
Laurent Freidel: By according boundary condition. But if you want to go beyond you know how boundaries with no boundary condition you will have to include the ideation, which is the flow of charges and
Laurent Freidel: The degrees of freedom is not really encoded into the, you know, these two physical degrees of freedom. Our other to handle that you have to change the charges. Does that make sense.
Laurent Freidel: Is a collection of charges at fix sphere. And then there's one other the Garvey to already the two degrees of freedom that allow you to change these charges.
Okay.
Laurent Freidel: But it's you know it's not done. Okay, so this is
lsmolin: Can we
Simone: Can we still
lsmolin: Go ahead.
Simone: Uli first
lsmolin: Okay, can we put it in a very simple physical way. And maybe you can help clarify you we were in a space time with one of your boundaries as its boundary in the bulk of the space time we shake. Something we produce a quantum gravitational wave of spin trust to
Laurent Freidel: 3D. Can I, can I precise we were talking about corner. So a boundary, you know, time development of a corner.
lsmolin: Yes, I know. I know. Presumably you can develop at a talk about the development. Are you going to be eventually able to tell me what happens when that final gravitational wave hits the boundary
Laurent Freidel: Yes, exactly. That's the charge conservation. So how does the church change when you move the boundary. That's how
Laurent Freidel: It is, in fact, this is what you know this angel group is is doing right now in terms of a synthetic infinity labeling or the notion of as metrics and scattering process in terms of charge evolution.
lsmolin: Which has to do with the celestial sphere and their picture because they're in there.
Laurent Freidel: No cosmological. Yeah. And here we don't need the sphere to speed. So last year, this year, can be anywhere. It's going to be true no matter where does he hear that when you move it a gravitational wave of a to change the charges.
Laurent Freidel: Or a nonlinear generalization of it, of course.
lsmolin: Yes, I'm interested in that. Definitely.
Abhay Ashtekar: Except that is very unlikely that you will be able to talk about guy little waves or anything. Simple it finite boundaries, you
Laurent Freidel: Know, I think I've taken away the have to go. But what will stay is the notion of
Laurent Freidel: Charge conservation charged change. So it's going to give you a new handle about what is supposed to replace the national gravitational wave
Laurent Freidel: If you look at the work of art is Tom injury. This is exactly what they're doing. They're starting from the gravitational waves picture and then they are rephrasing it everything in terms of the cells to whatever changes and
Abhay Ashtekar: Challenges as about that seriously phrasing was no
Abhay Ashtekar: refreezing there were like
Nearly
Abhay Ashtekar: momentum conservation law.
Abhay Ashtekar: I mean,
Abhay Ashtekar: That's all they're talking about right.
Laurent Freidel: Yeah, yeah, they're there. They're refreezing with the refreezing opens a new door where you don't have
Laurent Freidel: So you know where you can you can say now what you would mean by a nonlinear gravitate towards you want to call it like that.
Abhay Ashtekar: Right now that you want that was called before. In fact, exactly.
Laurent Freidel: Whatever it
Abhay Ashtekar: Is and does exactly that way, but
Abhay Ashtekar: I'm just saying that that is not
Abhay Ashtekar: Really good at Pro light on this finite boundaries and I'm just saying that I think we're on the same wavelength that we were just
Abhay Ashtekar: Talking about some small differences. We're on the same wavelength line and month
lsmolin: Do you know
Simone: Mark. Oops.
lsmolin: Sorry.
lsmolin: Do you know what boundary, what
lsmolin: Conditions correspond to your boundary being a horizon.
Marc Geiller: Or if you want it to be an isolated horizon. You can use the usual
lsmolin: Yes, I'd be very happy, which is right.
Marc Geiller: Yeah, but then this study will also have to be revisited because potentially you could
You could
Marc Geiller: unveil a new also new quantum numbers there as
lsmolin: Well, but that would be very interesting. And that would be very surprising and too many people that there are extra quantum numbers under horizon, besides the ones that people talk about
Abhay Ashtekar: No, I think because of the extra horizon boundary conditions which right now. You're not talking on boundary conditions. Once you put in isolate horizon boundary conditions as Laura was saying before, I mean this this extra charges will go away. I don't think that
Abhay Ashtekar: There will be not to the charges in this particular case.
Abhay Ashtekar: But I think similarly has been
Laurent Freidel: Sent to our charges will go away. But there's a new type of child that has made its appearance that Mark didn't
Laurent Freidel: Talk about which is the discharge. So the default more fees and charge which controller. This one will not go away. And this one as a as a very interesting and important vacation for understanding the geometry of arising.
Abhay Ashtekar: And exactly those will give you multiple moments, which is what
Abhay Ashtekar: You know,
Abhay Ashtekar: Yes, the
Abhay Ashtekar: Characterization of that will be precisely the multiple moments that are either multiple but I think the morning has been off time to ask a question for quite some time.
Simone: Thank you. Okay. Yeah. Mark you stress very much that this is a very, it's a continuous approach. And so I was wondering if in this road, you want to pay for you still see a place for speed networks to spin forms.
Simone: And in case. What is the main message is it something that one is to use the punk or a group or sometime you seem to have hinted that these or there isn't any financial risk informs to be seen in this road think yeah
Marc Geiller: Yeah, I think so. This tells us two things. Now he tells us that some I think aspects of accuracy are just unavoidable because they're just beating features of gravity and we have to consider them.
Marc Geiller: And now, it could be indeed that for the purposes of actually need the quantization, we have to resort to some organization. And as I mentioned on some slide and you some
Marc Geiller: Thank you. The highest description of the of the covenant of the charges living on the corner and this could be done through the use of networks.
Marc Geiller: But then, at least, this will tell us that we need maybe to label these networks with x quantum numbers like
Marc Geiller: Law, and then you'd have suggested we did this one guy networks or this bubble geometries which carry extra information which could be useful for you know purposes, of course, raining and building's been for models and building the dynamics.
Marc Geiller: So there is this possibility that we have to acknowledge the space of
Marc Geiller: What we mean by SP network with a noxious definition.
DJ: Of Mark have a question related on this slide. When you say continuum quantization. What precisely do you mean
Marc Geiller: Which means I'm in a position of unknown college algebra, which is, is if you're in the sense of continue magic so quantization, which does not resolve to any dispute I structure.
DJ: So would you can this be networks and
DJ: Spin formal structures as described structure here.
Marc Geiller: Yes, yes.
DJ: So I'm a bit confused. Then the other slide at least you're not
DJ: Imagining using these structures.
Marc Geiller: So I'm not, I'm not imagining i'm i'm postponing the use of democratization structures that can be networks and spin forms for the moment.
Marc Geiller: Because the goal would be to quantify what the theory just gives you and what the theory naturally gives you is just a somewhat natural leaves at these corners are actually coming to us. So I'll Palo Alto boss, which are very different from
Marc Geiller: From
Marc Geiller: From global NGOs of cemeteries.
Marc Geiller: And this is what one should quantities.
Abhay Ashtekar: Me I intervene dispensing I mean one pass this also question for Mark, but like to try to answer this question that was asked. I mean what one could have is really this
Abhay Ashtekar: Boundary Hilbert spaces with, you know, some quantum numbers on them and so on. And some states and operators and and eigenvalues eigenvectors, etc.
Abhay Ashtekar: And then there is another question about how do we represent them concretely, you know the differences between actually having abstract Hilbert space and representation for example.
Abhay Ashtekar: Representation of the point Gary group you know it for the years of represented upon can group. And the other thing is to really write them as
Abhay Ashtekar: Really coming from a Muslim scholar feel a message delafield maximum field and so on so forth conclusion about space. So what they're giving us is really kind of an abstract Hilbert space.
Abhay Ashtekar: But the nice thing is that that obstacle but space is the results from that extra space. I completely in agreement with the with some of the fundamental results from new quantum gravity and therefore spin networks, to me at least would be would provide
Abhay Ashtekar: And have a complete realization, but with possibly adding quantum numbers we spend with network did not have before.
Abhay Ashtekar: So it will be concrete realization, just like for example the you know the solutions to client garden equation and then Fox space constructed out of that.
Abhay Ashtekar: Gives us a concrete realization of the unitary representation of the irreducible unit represent is a point. Can I grew up with MySQL as humans can equal to zero. So, it is that that kind of philosophy mark and not on that end.
Marc Geiller: Yeah, I think.
Daniele Pranzetti: Yeah, I just exactly what by said, I mean, the hope is that since we have some our discrete measure now which is given by the area, the
Daniele Pranzetti: Area of the corner, then the hope is that even if you start with a continuum representation, some sense you can end up with the Hebrew space which is funny dimensional
Daniele Pranzetti: And so that the coarse grained in that speed networks represented or with a generalization end up being a non no longer course grading might not show an exact representation of this algebra.
Daniele Pranzetti: The hope here is given by the fact that we saw that the measure is going to be discreet on the corner. So even if you have a continued representation and that could
Daniele Pranzetti: could provide a
Daniele Pranzetti: Hebrew space which is funding dimension or even if it's so that will be the the the jump to prove lotto. Robert, if you want in a, in a concrete sense right that there is a finite number of degrees of freedom. The corner that capture all the buck degrees of freedom.
Daniele Pranzetti: Through that were describing themselves be networks before
Abhay Ashtekar: The dynamical level.
Abhay Ashtekar: Right, because I mean this this picture was done the covariance so
Abhay Ashtekar: Yes, that would correspond to dynamical pitcher not magical magical pitch.
Daniele Pranzetti: Yeah, from what at least for recent for the for the spatial different constraints.
Daniele Pranzetti: You know if you call that dynamical can him article but already there. You could try to to to to show this in the level of specialty from official
Abhay Ashtekar: Right, I mean on shelf versus
Daniele Pranzetti: Yeah, yeah. Oh sure, yeah, yeah, yeah, yeah, absolutely. Yeah.
Laurent Freidel: Maybe if I can say a word about that. So, as Mark was saying this is a loop algebra. So the study of loop. Other Brian 3D. For instance, the first one time something and he gives you BMS group.
Laurent Freidel: So, which contains some of the house over symmetry. So this year has organizations is an example of a continuous quantization now.
Laurent Freidel: We are starting the you know the theory of nobody has really fully developed with your presentation of this loop algebra is
Laurent Freidel: The key element in the stupider Bravo is which I think we will bring the new ingredients compared to spin form is that it's not just about this the the supervisor semi direct product, right. It contains a D factor beginning as a rotation and in a
Laurent Freidel: You know,
You Caliban factor.
Laurent Freidel: It focuses on entirely on these rhetorical factor. And it's kind of fixing or jelly out the diff so presentation. And now we understand much more of these two interplay. So it will contain some element of of loop, but it will be more like
Laurent Freidel: You know, like a fried picture of looping issue more like, you know, that includes seamlessly the difficult and that's not that that that is essentially the next upgrade.
Laurent Freidel: And as Mark was saying, or they were saying once the diff. The semi direct product presentation. They are they are classified by the choice of measure you put your queue. And there's really different presentation was just curious, is a continuous was a little bit measure or discrete
Abhay Ashtekar: I think there's a hand raised by linking Chen. Maybe she was a question.
Lin-Qing chen: Hi, Mark SEND YOU SO MUCH FOR very nice talk. Have a question about, so usually different choice of the boundary lager Andrea is important for well defined a variation or principal
Lin-Qing chen: And so, so it's not a freedom to choose. So, but here I also understand that this boundary Lagrangian will become this corner piece of simulating potential also plus the exact form on the face base, which isn't important. Your formulation. So I'm wondering
Lin-Qing chen: So how, how to consider this a well defined the variation or principle in in the framework you are discussing.
Marc Geiller: Yes, the question because indeed I didn't talk about vibrational principle.
Marc Geiller: Because I said we want to postpone this to later. And I think the reason which also kind of unlock the possibility to do this work for us is that, indeed, as we know, usually to
Marc Geiller: To to fix specific boundary conditions like we do in general relativity, we have to use a Banda he like Origen
Marc Geiller: To change the colonization on the boundary and to be able to work with the boundary conditions that we
Marc Geiller: That we desire but what was under appreciated before was that once we put the boundary, like on Jan, it actually does two things. One thing is that it enables us to choose the boundary conditions.
Marc Geiller: But the other thing is that it also changes, these kind of contributions to the syntactic structure. Yeah.
Marc Geiller: I think that these two aspects were kind of mixed and a bit confused together. And there's lots of discussions where we don't actually
Marc Geiller: Really understand clearly whether what is the algebra is being discussed is a arises because of choices of boundary conditions or because of Connor contributions to the syntactic structure.
Marc Geiller: So I think that's why it was important to disentangle these two issues and to first acknowledge that the boundary like Hungarian
Marc Geiller: Gives the corner contribution to this eclectic structure with no extra choice to be made.
Marc Geiller: And then on top of these were free to discuss boundary conditions if you want add another abandoned like on Jan to change their boundary conditions, but this would be done separately, you know, in a separate step, but it can be done, but it's extra
Lin-Qing chen: Oh, I see.
Lin-Qing chen: Purely discussing this corner piece of simplicity potential which will be changed if one. Consider adding different boundary Lagrangian but that can be
Lin-Qing chen: Used to be a separate question because
Lin-Qing chen: Yeah, I understand. Okay.
Lin-Qing chen: Thank you.
lsmolin: I tried to summarize, you have a few more physicians on the boundary
lsmolin: There are three possible fates of these different more physics you freeze them, they become gauge
Marc Geiller: Then
lsmolin: You might apply them or they become physical degrees of freedom they become Edge mode.
lsmolin: Do we have a
lsmolin: Two questions. Do we have a free choice as to which of those three, we believe is a nature and what, if any, to those to be efficient, have to do with gravitational waves and rather times to
lsmolin: Why you may have created some degrees of freedom that we we have no experimental head.
Marc Geiller: Yes, thank you. I would say that, you know, so for different morph isms. I don't think we know that there are other theories for which we know that these
Marc Geiller: Extra degrees of freedom that are revealed on the boundary actually play an important role in this is for example in a in comments matter and in
lsmolin: Theory quantum halt.
Marc Geiller: Monty Hall effect. Exactly, yes, sure. I don't know what he's done have additional analog of this, but it's kind of following the same expectation and the same the same logic.
lsmolin: So what would you say if somebody said, our job is to understand quantum general relativity and we should just freeze out Austin is a feeling
Laurent Freidel: Good luck.
lsmolin: No. No, I'm serious. I'm really serious. Good luck, man.
Abhay Ashtekar: I think, let me just say something that is that may help
Abhay Ashtekar: I think I mean there's a point that Laura has been somehow hinting at, which is that if I look at infinity, right, and the degree of freedom isn't really in the bond in us. I mean, that is what categories and waves.
Abhay Ashtekar: But what I can look at is really in some ways, you know what I call the mass in the conversation various vacuum, you know, this
Abhay Ashtekar: In other words, so configurations which are there, which are mapped to do each other by the in that case by super translations.
Abhay Ashtekar: Yes, but but you know when that part of the Facebook and that's and then the statement is here that something like what they're done is really
Abhay Ashtekar: In some sense, looked at this basic elements which are there and they're change is going to be captured by kind of two degrees of freedom or something like that in, like, for example, if it was a sufficiently far away, then it will be captured by gravitational waves.
Abhay Ashtekar: Exactly. Though there is some subsidiary, I mean there is some
Abhay Ashtekar: These are not kind of directly physical degrees of freedom but changing them either directly physical
lsmolin: Physical beautiful
lsmolin: Beautiful and
lsmolin: Bread.
Laurent Freidel: It's like, it's like
Laurent Freidel: Like a ground and beating the basement before you, you'd be the world's
lsmolin: Know, so you're talking about the very infrared platinum degrees of freedom and you
Laurent Freidel: Have to be infrared, they are true. That's the point. They are true for any surface. So they are they are the these degrees of freedom are the same, whether you're an infrared
Laurent Freidel: Or in the movie. They are just more of it in infrared because the area is what are the Casimir so the Casimir grows. There's no states but but apart from that they saw the same degrees of freedom.
Laurent Freidel: Okay, think about the area is one of the Casimir and in fact in this industry by this is the only one which is depending on the scan. There's an infinite number of other cars music, all of them are scaling dependence.
Abhay Ashtekar: Know, but i think i think a lot of what I haven't had maybe putting words in his mouth. But I think one can understand what he's saying in the following terms, namely that they will find in the sense that they themselves don't carry quote unquote energy
Abhay Ashtekar: Yeah okay that changing the changing them carries energy
Abhay Ashtekar: Beautiful. So in that sense that that they themselves that says that, I mean, yeah, I'm not in front of in the sense of long wavelength or anything like that, is that that I don't know what
Abhay Ashtekar: We know if you use that sense totally true. They're there they are.
Laurent Freidel: I don't know if I would call them for it because they're everywhere but but they are they are. I mean, they are they are the basement. They are the building blocks until you don't know the building blocks, you cannot decide what you mean by gravitation radiation or
Abhay Ashtekar: Your dad like like connection young new see right and
Abhay Ashtekar: The if you're just please a connection and instead of time doesn't have kind of
Abhay Ashtekar: He has physical base and yet he doesn't talk single information.
lsmolin: They something like the following and complex manifold theory, you have pieces of see
lsmolin: Which could patch to each other by
Abhay Ashtekar: Yeah.
He
Abhay Ashtekar: Is the transition functions which are which are the physics.
Abhay Ashtekar: Yes, yeah.
Abhay Ashtekar: So that's the
Abhay Ashtekar: Analogy, I mean, from that wonderful Rogers all on construction right that any one patch. It looks like Flash space, but is the way that the two patches complex holographic patches are glued together.
Abhay Ashtekar: That, that, that, that is what the, you know, has the, has the physical information, right. So I'd love to thrive in the sense of long wavelength or anything like that but that, in fact, in the sense that they themselves don't have energy momentum. Angular momentum, etc. But genuine care is there.
lsmolin: Has to be fixed before you define as mentality.
Laurent Freidel: Yeah, if you want to define change. You know, you need to know what you mean by static or
Laurent Freidel: Your country. Yeah.
Abhay Ashtekar: These are that kind of analog of what I call the vacuum. I mean, I'm not
Abhay Ashtekar: Local I don't remember what I call you back here.
Laurent Freidel: And
Abhay Ashtekar: Change them is what is the
Abhay Ashtekar: Dynamics.
Abhay Ashtekar: I think that were going on very long and maybe he was very patient with us.
Marc Geiller: I just may want to before we chose because I just wanted to make an announcement on the extra slide which is about the loops.
Marc Geiller: Next summer a kid just wanted to flush this and to just announce that you can book in your calendar that it will be July 19 to the 23rd and also summer school in Macedo the week before and we just hope that we can do this in person.
Marc Geiller: Yeah, we'll announce it soon and just hope we can all meet in person.
Ivan Agullo: Thank you very much, Mark, and thank you everybody for the thing that I think discussion that
Ivan Agullo: We are will be on the time. So let's thank marketing.