Jorge Pullin: Okay, so let's begin today's been a different city who speak about Edge Mode mojo gravity.
Daniele Pranzetti: Alright, so thank you very much for joining.
Daniele Pranzetti: Today and learn present
Daniele Pranzetti: Some new ideas about
Daniele Pranzetti: Different way to think about Greg both of the classical and a quantum level.
Daniele Pranzetti: This is work which has been done in collaboration with several authors this around and
Daniele Pranzetti: Started with a paper with Andrew, which they are continuing with that data and, more recently, we started a series of papers with Mark and run
Daniele Pranzetti: In title elements of gravity and unfortunately they managed to put the first two papers of the series online in time, but they should appear soon and
Daniele Pranzetti: I think this is a place where we are. We have reached the clear, clear, is understanding how the new proposal that we were suggesting, but let me start with some perspective so
Daniele Pranzetti: In our quest for quantum gravity and essential task is to reach a problem the standing of degrees of freedom and the seamlessness of gravity associated with
Daniele Pranzetti: Local sub regions. Now there are free interlaced insights around this. One of the questions which are quite important for the rest of the talk and
Daniele Pranzetti: On one hand, we have the insight about the local graphic behavior of generativity and this stems from the idea that black course carry geometrical entropy proportion to the area.
Daniele Pranzetti: On the other hand, there is a deep experience with doing canonical gravity that jar is a constraint system. Now, this means that the total energy associated to any specific region, for instance, for any arbitrary time flow is entering quarter in this boundary
Daniele Pranzetti: And then there is a third ingredient which is more recent and it's the understanding that we can assign traditional edge modes. When the boundary surfacing space.
Daniele Pranzetti: Of course, this requires identify your boundary symmetry algebra associated with any surface and understand and agree additional modes as its representation states.
Daniele Pranzetti: So I'm going to be, of course, much more specific about what I mean this is just give you general framework so
Daniele Pranzetti: They interesting these insights best leads naturally good picture that the relevant geometrical degrees of freedom leave on boundaries.
Daniele Pranzetti: That the dynamics and the fabric of space itself is encoding to the entanglement and the central point is that disentangle is derived from the fusion properties of the traditional boundaries in yogic
Daniele Pranzetti: So this is some of the new perspective that we're trying to propose and trying to develop through the talk. And I want to say that this renewed interest in understanding
Daniele Pranzetti: The nature and the dynamics of degrees of freedom fighting boundaries as being very active recently in the world. Not so easy. In fact, it's been a few years already. And there are quite a few papers is just a sample of work and, in particular, there's a lot of TV along on the slide.
Daniele Pranzetti: Okay, so it's good to slay on the free now let me
Daniele Pranzetti: Make a point for regiment. First of all, with Eastern Edge mode. So you can see an animal inch waist. You can see an Edge mode as
Daniele Pranzetti: So from my perspective. So yeah, you can see from him as a contact in momentum to a boundary church.
Daniele Pranzetti: And at the same time as was mentioned for you can see a service official states of the boundaries in the algebra. Now these white is somehow
Daniele Pranzetti: Can be understood as the usual burritos, the correspondents, I will again I will be more specific and clear understanding of this definition but
Daniele Pranzetti: Not from the perspective of canonical framework Punjab, which, as well as the describe the propagation pollution time with the geometry Edge Mode leap on corners.
Daniele Pranzetti: That is given to the boundary surfaces. Are we free region of space like upper surface and in the context of holography they are supposed to reflect and represent in degrees of freedom of the 3D geometry.
Daniele Pranzetti: Now,
Daniele Pranzetti: Since classical spacetime is fully as a manifold, which can be described as an atlas of charts or if you want physical damage as a Union have been estate space can also be model as the union of free new regions grew together for the interfaces.
Daniele Pranzetti: Is somehow this this vision of winner of a quantum form. If you want and then one can emission gravity to be written best guess a theory of Edge Mode leaving on the boundary surfaces of these patches of pretty geometry.
Daniele Pranzetti: And this is somehow similar philosophy towards the gap approach is based on and also just by looking at recently. And then this just lets best into picture or space as a network of bubbles, which should eventually need to quantum gravity is a dynamic whole network of quantum edge months.
Daniele Pranzetti: Now this picture, picture is naturally compatible. We love holography and it's the same to offer. If you want a perfect segue into sorry corresponding of gravity, both of the classical the country levels. Now the main reason why I want to spec good course winning property is that
Daniele Pranzetti: Because we propose to label the states by boundary symmetry charges. And then, of course, bringing all
Daniele Pranzetti: Of these charges of symmetry is a more rigorous selection process, in a sense, because it can be implemented as a fusion of symmetry algebra.
Daniele Pranzetti: And there is a lot of work in this direction for instant by Bianca and operators and what I'm going to present the rest is not a reformulation at the macro level of gravity along those terms.
Daniele Pranzetti: So let me go to Slide number for now.
Daniele Pranzetti: So why edge modes. Okay, let's look at more from your system. We know that in quantum field theory, we need to introduce a cut off.
Daniele Pranzetti: Which means basically to democratize it and then we need to learn how to remove this cut off without breaking the cemeteries.
Daniele Pranzetti: So one resolution is to put the system on the lattice. Now the lattice can be understood as the partition of space into sub regions, if you want.
Daniele Pranzetti: And the boundaries of the Rockies corresponds basically to the building. So the bond is corresponds to the building block of the law is what charges can be defined and the gauge the meeting.
Daniele Pranzetti: Gets the form the sense that it's replaced by a lot is version of the gauge cemetery, which is necessary to the presence of the cutoff.
Daniele Pranzetti: Is this a clear example of this is in the meals, where the continuum, the electric field community itself and its continued momentum is a
Daniele Pranzetti: Connection field. And then when you go to the descriptor case, the electric field gets me a lot edges of the lattice and becomes on community and the gauging valiant in the continuum then is replaced by largest portion
Daniele Pranzetti: Where are the some of the, let me let me practice as to bash
Daniele Pranzetti: So this is something separate from your picture and effect. Now go to Slide number five.
Daniele Pranzetti: I could you sum up
Abhay Ashtekar: He said, I asked you a question.
Abhay Ashtekar: I'll just the general philosophy. I just want to make sure I don't miss something so
Abhay Ashtekar: So this, this idea that you so nicely presented in the last few slides that you know that the edge morsel living on this kind of these two surfaces. Right.
Daniele Pranzetti: Yes, yes.
Abhay Ashtekar: But so what is different from what everybody has been has been doing right any fun just consider city district ization like for example, tetra hydro, we always said that the area leaves on this boundary and
Abhay Ashtekar: So this is same, or is there some new new idea.
Daniele Pranzetti: That no no no. It did.
Daniele Pranzetti: In fact, I'm gonna, I want to take this idea very seriously and the different is that I'm going to, I want to
Daniele Pranzetti: Develop this idea in the continuum picture not in the discrete setting, but it is along the same lines. Absolutely.
Abhay Ashtekar: Okay, thank you.
Daniele Pranzetti: So in fact,
Daniele Pranzetti: A now. Okay. It's like number five.
Daniele Pranzetti: So, energy is basic, basic set a formulation of gravity elegant meets now.
Daniele Pranzetti: The key medical sector theory provides an elegant description can be geometry and where the dramatic operators admin spiritual states can be defined in terms of
Daniele Pranzetti: A fundamental flexibility in these eventually leads to the speaking of the sweetness of the spectrum, which is one of the mirror sets of will continue it it's it's
Daniele Pranzetti: Very exciting great result. And now I want to emphasize one thing that is that classically the knowledge of the flux in the buck.
Daniele Pranzetti: Allows to reconstruct different field fully or however in the quantum theory, the flax becomes a smear great honor to the surface.
Daniele Pranzetti: And so what it means is that we only have access to boundary flux, if you want. So, from this point of view, there is a suitable speed quantum numbers that we call those natural with includes information about abundant flux charge
Daniele Pranzetti: So from this perspective, then
Daniele Pranzetti: There is two questions, which
Daniele Pranzetti: One can ask and one is can we understand the flux non committal activity in grabbing it from the continuum point of view, we know that there is a an understanding of this community from the discrete point of view.
Daniele Pranzetti: And there are these paper or regionally by bike equations apart and then Morrison throat So Alejandro and
Daniele Pranzetti: Pride new perspective on this problem, but we really would like to understand the design community from the continuum. And if you want to pull it
Daniele Pranzetti: Has wrong with face it. We're not sort of quantum gravity without solving first $1 million problem, of course, in reference to the Millennium price question of the massive gap between your meals and their confinement, because we see that this I probably read the questions. In fact,
Daniele Pranzetti: And. And then the second question is, so if the we only have access to the flux and demand theory, then is there to choose pin point number enough to reconstruct tangential frame field. This is a crucial question that we want to address.
Daniele Pranzetti: In the continuum.
Daniele Pranzetti: And so
Daniele Pranzetti: Another motivation for business is the fact that we know it's been from approach.
Daniele Pranzetti: was constructed to tackle the issue of or the nomics which we saw that it's a it is some ambiguity is in the sense of a there is a question of each covariance that
Daniele Pranzetti: One is paint very well in the future. Could you and also definitional simple quantity using activity. So what at what speed from does is they provide a transition and Peter between boundary states where Lauren's empire is manifest in the park.
Daniele Pranzetti: In the center is implemented from the get go the same time, we saw that the economic quantization program refugee provides basically the physical Kiera setting in which informs our summer to be interpreted
Daniele Pranzetti: So recalling the canonical could use a space where a set to seize broken down to so true. When was the time engage as the boundary spaces infamous been
Daniele Pranzetti: Of course a goal of crucial interest and there's been a lot of work along these lines, which eventually led to this not celebrated
Daniele Pranzetti: In many map where covariance is recovered from embedding of a switcheroo into SF DC and. Now, one question though is,
Daniele Pranzetti: That is still not clear some is the compatibility between the canonical me time constraint and this information.
Daniele Pranzetti: In this sense, can we derive a spin for model from economic quantization. This is still an open question. If you want
Daniele Pranzetti: And along those lines. Then from from an economic point of view, maybe what we really need is to construct the key management space where all the Lawrence generators can be represented as direct reports. So as answer the second question could
Daniele Pranzetti: Probably important insights into the first issue.
Daniele Pranzetti: So let me bring another piece of evidence which
Daniele Pranzetti: Another if you want it is another feature of refugee which supports this holographic understanding of the flat charges.
Daniele Pranzetti: And nearly. This is why did they actually got better connection arises from the introduction of a topological turn the action that does not change the classic official motion but eating uses economical transformation across space.
Daniele Pranzetti: Now, the introduction of the connections are crucial ingredient to reveal the non legal representation of the charges.
Daniele Pranzetti: And this would be clarified shortly. But I think it's most people are aware of this fact. Now what I want to point out here is the fact that the discussion on the transformation can be encoded in the appearance of a boundary free space there.
Daniele Pranzetti: And these can be understood as an extension of the boundary space, which basically allows us to define boundary flat charges and to compute decimate the algebra.
Daniele Pranzetti: In a very from the consumer point of view. And so what we witness is how the passage based interconnection formulation is intimately related
Daniele Pranzetti: To the appearance of unknown vanishing turnover charges which is satisfying to me the algebra or reading the continuum theory and another aspect I want to point out, which might be a part of the obvious, but he can have important implication is that
Daniele Pranzetti: The origin of the non competitive of the field.
Daniele Pranzetti: Which
Daniele Pranzetti: makes this interesting constraint on community is a direct consequence of the discrimination procedure that we use in a spin forms.
Daniele Pranzetti: And it seemed to know community with the flag system canonical energy. So with what this suggested that it contained description of the flexes and community.
Daniele Pranzetti: Will allow an understanding of human community structure of the same physical strength as well. From my point of view, the continuum theory and this can provide crucial insight to on how to reconcile the lorenzen environments with the presence of a fundamental left in theory.
simone: Are you now doing these only in the time gauge or in an arbitrary
Daniele Pranzetti: Yeah, so I'm gonna start reviewing the time gets case and then I will relax the time which
simone: But the equation you wrote here.
Daniele Pranzetti: This is why this is not enough time to engage this is true in a in general.
simone: Right, okay.
Daniele Pranzetti: Yeah, I'm going to review. First, the case on the language to see how this boundary charges appear and then I would go to the food court case.
Laurent Freidel: Sorry, could you use your pointer to show which equation you talking about when you yeah
Daniele Pranzetti: Yeah, yeah. Okay.
Daniele Pranzetti: Right, so yeah.
Daniele Pranzetti: Then Lend me an important thing in the rest for the for notation what by beta, what we mean is gamma minus one. So, I know it's a bit
Daniele Pranzetti: Unusual convention. It's just 2% Muslim equations, be that will be upstairs to simplify that we we define it as the inverse of the meaty part
Daniele Pranzetti: OK, so now
Abhay Ashtekar: Just as a side to mark when you write lines open source over the reason why in many, many years did not use beta was because it was getting confused with the thermodynamics, right, the temperature and with the beta as it
Abhay Ashtekar: Which copywriting hawking temperature and such and
Abhay Ashtekar: Such and since this all the surfaces and horizons and such to relate to that temperature, you have to be really careful about the use of data. This is a very minus side point
Daniele Pranzetti: Yeah, absolutely. Yeah.
Daniele Pranzetti: Okay, so, um, so, from an organic perspective that we're proposing.
Daniele Pranzetti: If the boundaries in theology is not big enough, then we cannot will account for all the degrees of freedom and we cannot hope for a buck reconstruction from the knowledge of the boundary charges so
Daniele Pranzetti: It is clear, and once you look them up for a formulation that has the biggest one we seem to group. Now, the smallest and this extension corresponds basically to add the edge modes and it's encoded into the choice of the simplicity boundary potential. So the more extensive
Daniele Pranzetti: Sorry.
Daniele Pranzetti: Yes, the question yet highly
lsmolin: High. So are you really claiming that for these situations with finite boundaries. He animals are enough to return to be given an equivalent formulation of dynamics to the boat and
Daniele Pranzetti: That is the perspective, we're trying to propose, yes, making them with something
lsmolin: Analogous to the strong around the same injection. That's about it is a fair amount entirely reasonable bite is a field on the edges on these finite
Daniele Pranzetti: Yeah, I mean,
Daniele Pranzetti: Philosophically is similar proposal, but yeah, we're not using specific boundary conditions and we're talking about local
Daniele Pranzetti: sub regions not nothing topic. Yes. Okay.
lsmolin: So you have a dynamics for these Edge mode which is equivalent to the dynamics in which your Gravity Forms for advocating and so forth.
Daniele Pranzetti: That would mean i. So, in this token. Just going to present a documented medical case. And eventually, one would like to be able to recover or today dynamics of course strong from the buck from the boundary dynamics, but in in the rest of the book. And I'm going to focus mainly on them.
Daniele Pranzetti: On the case with the yang mills in the dissertation to be more visible.
Thank you.
Daniele Pranzetti: So,
Daniele Pranzetti: So best yet. So the more extended formulation. The bigger this the boundary algebra in the more we can reconstruct the bucket degrees of freedom. And then I'm extremely inspired
Daniele Pranzetti: Now this bucket construction follows from the fact that the conservation of boundary charges.
Daniele Pranzetti: Gives us an expression for tobacco strings.
Daniele Pranzetti: In fact, I mean, this is going to be a one. One thing I'm gonna try to emphasize that we can understand that at least the canonical level constraint in the park as conservation low for boundary charges and so since
Daniele Pranzetti: The dynamics of gratis more or less entire formulated in terms of constraints we need the maximum amount of material boundary seem interesting only to encapsulate of information about the economics.
Daniele Pranzetti: And another reason why we should look for Maximus. Any theories forest from our experience with the composition of each theories, as was mentioned before.
Daniele Pranzetti: That starts from the transition of bankruptcy or boats and the representations. So if you don't have a proper complete set of non tree or banners several then we don't really have a handle on on position and in this talk, we're going to see the benefits of this
Daniele Pranzetti: Boundary for space extension by edge months in particular, what we're going to see is that the internal normal to a cushy surface becomes a dynamical Bible on the boundary free space, which can be understood as a for momentum of a new element upon carry algebra.
Daniele Pranzetti: And the tangential metric becomes the canonical field which is necessary to reconstruct the tangential framer
Daniele Pranzetti: Moreover, would see that the boundaries NPC constraint satisfy a second class algebra or reading the continuum and the correct position in terms of a first class piece and the one master constraint piece where it done is to break contrary Dr. Lawrence.
Daniele Pranzetti: And then the the boundary area corresponds to the pond Craig's been cast me. So this means that its environment and the Lord and transformation and this reconciled is quickness of the spectrum, which is going to be derived from the content theory with the manifest Lawrence compare environments.
Daniele Pranzetti: And finally, after position the constraints, the manifest space would be permit tries by a set of miracle server boats, which form, it's a module Pontic given by such Lucy.
Daniele Pranzetti: Plus the search lot of the tangible metric, which basically means that we have two new quantum numbers on the boundary here space. So these are all the men restarted. I'm gonna industry in the rest of the book and you
simone: Could you say a little bit more about the the first pointing the slide because like natively, one could think, think of the case of spatial infinity.
simone: If one wants to have more than just the tempo encourage generators. One way would be to have stronger full of condition so that more charges are finite. But then this will lead to less solutions in the bank.
simone: Can you say more on
simone: So what we need to have more bulk dynamics from more from a larger boundary algebra.
Daniele Pranzetti: So what I'm
Daniele Pranzetti: Trying to say is that in order if you pre sense, look. Think of the switch recharges of quantum gravity, you should don't reveal those charges.
Daniele Pranzetti: You will not be able to introduce an echo states right so that's what I'm trying to say that if you have a bigger boundary algebra, and then you're able to reveal more concrete numbers which can be used for the quantization of their to reconstruct the degrees of freedom.
Daniele Pranzetti: Is that like gauging violence in the back, the Gauss law can be coded in the the some of the flags on the boundary to vanish and
Laurent Freidel: And I
Laurent Freidel: Yeah. Can I say something about that because we have a work without doing that shows that if you want or an extended the bigger Andrew brought infinity. In fact, that you don't need stronger boundary conditions you need weaker boundary conditions, but it's kind of where we can I
Said,
simone: I don't know what I meant to them, usually you would require stronger fall off. No.
Laurent Freidel: No you you have to require less less full of condition.
Abhay Ashtekar: Yeah, I think someone that is right. Right. Every week a boundary conditions gives you recharge.
simone: You don't require the fields, go to zero faster in order to have more than just the
Laurent Freidel: Not just, oh no, you require the exact opposite. But maybe that's
Laurent Freidel: Another question we we cannot
Abhay Ashtekar: Know i i agree with Laura completely. So maybe we can talk about this. The end. Yes.
Laurent Freidel: Yeah, it's
Daniele Pranzetti: Okay.
lsmolin: Let me ask very nice question. So the claim. Now is that the boundary area corresponds to a chasm near the park array and therefore is boosting very
lsmolin: So, so it doesn't correspond. I mean, I think in the older formulations, like the isolated rising picture was pretty clear that the boundary area we're accounting or spaceflight.
lsmolin: And was not a Tasmania with that invariant, but the transform to somebody named Bruce pantry. So this is a different quality.
Daniele Pranzetti: Not is there.
Daniele Pranzetti: Is the same quantum is the same quantity. It's just that.
Daniele Pranzetti: It's embedding a bigger
Daniele Pranzetti: Symmetry.
lsmolin: So here's my problem. If I have a city.
Daniele Pranzetti: And it's it's normal to have vector. Can I make a difference transformation maker and spoon said that factor changes and therefore the to surface.
Daniele Pranzetti: Area corresponds to changes. Yeah. You see the key point is that not the internal normal respect which you define that surface is dynamic. Also, it also gets altered rotated by the by the booster and that's why you restore its importance.
Daniele Pranzetti: It's not they can what you're referring to is when a case in which the normal is a kinematics quantity that doesn't rotate
Daniele Pranzetti: So,
lsmolin: We're saying the opposite. But let's come back to this.
Daniele Pranzetti: Yeah, I mean,
Abhay Ashtekar: It's really one way to say this is that
Abhay Ashtekar: What was done it, I sort of horizon etc etc is something that will happen in a fixed time gauge. Is that what you're saying.
Daniele Pranzetti: Yeah, exactly.
Daniele Pranzetti: Engaged yeah if you if you fix it engage. Then you recover the usual picturing which of course you don't have Lawrence him audience, right.
Daniele Pranzetti: Okay, yeah, let's keep going. Yeah, this will, it will become. These are just the main reasons, man. It would be clear thinker, as we keep going. Yeah, so, um,
Daniele Pranzetti: Now, let me first industry this pattern of teaching to breaking from the familiar case of data GST to flux. Now the main points that in the presence of the boundary, they may go slow is no longer
Daniele Pranzetti: Differentiable and when you barely respect to the covering field. So in order to make a differential, you need to add the boundary there which defines the generator in this this expression here where sigma high is there, which are the two fields.
Daniele Pranzetti: Now, if we impose a geographical zero then this is going to be true if alpha sigma
Daniele Pranzetti: Vanish on the surface, which means basically that the gauging bias is broken in the presence of abundance. Now, there are several options to deal with this issue.
Daniele Pranzetti: The first one is to simply which is someone most traditional one is to simply accept this fact and to promote the numbers, he charges to the status of a symmetry generators
Daniele Pranzetti: Then there is the
Daniele Pranzetti: Option, which basically one imposes that Jaffa vanishes only on a set of measured zero on the boundary which is represented by the punchers in this way, then the best tangential flux becomes
Daniele Pranzetti: If you want senior in a sense that that the only charges left are the loop gallery flexes, which are associated with a with a puncher's
Daniele Pranzetti: Now then, the third option that we want to follow is
Daniele Pranzetti: Which leads to basically this loop run the picture we produce what a goal is to introduce Edge mode to restart the gauge symmetry.
Daniele Pranzetti: And in this case, engage in violence is restore basically by extending the face face Emily by defining engagement stringent the data, which involves
Daniele Pranzetti: The terms, depending on depart field that before the in black in this formula here.
Daniele Pranzetti: But then it's also component, depending on the Edge nodes which I use it. I did not mean in a different font to make clear that these are
Daniele Pranzetti: Boundary these features, edit, which are related.
Daniele Pranzetti: To the boundary friendfield by the purity. These are independent fields that won't happen in order to restore gauging violence. In fact, then
Daniele Pranzetti: One can see that the imposition of the this constraint to vanish imposes the gal from the bar class is increased equals three on the surface name is telling us that there's a continuity condition between the edge modes and the pullback of the offering field from the bug.
Daniele Pranzetti: Which can be understood as a boundary simplicity constraint, if you want. So in this way, we can
Daniele Pranzetti: We can define number two charges. So we're still getting violence. But the key point is that we can still define non vanishing sync recharges that act on the edge, what fields.
Daniele Pranzetti: So these are the signatures and defining here and they basically
Daniele Pranzetti: They in the coffin field is on the boundary remains dynamical and the letter G modes carry their own petty potential
Daniele Pranzetti: In particular, they, they tried feel become some communities and we have the data non trivial representation of the boundary symmetry algebra.
Daniele Pranzetti: So the key difference between this picture and some of the nice thing to bring teachers, they now become straightforward. Why does seem to charges the flexes become Norco community to satisfy the current algebra.
Abhay Ashtekar: I'm confused about the notation here. So, yes.
Abhay Ashtekar: Edge edge mortar. But the thing that they're tangential is that
Daniele Pranzetti: Yes, yes.
Abhay Ashtekar: Yes. So, therefore, that diet right no picture.
Abhay Ashtekar: Died right
Laurent Freidel: Well, yeah.
lsmolin: tires on the to surface.
Abhay Ashtekar: Yeah, he's a diet to surface. Yeah, so I'm confused about this index i
Daniele Pranzetti: Not, I mean this space, the space time index is put back on the surface but internal index.
Daniele Pranzetti: Is like the difficult for him field.
Abhay Ashtekar: So it's not a diet. Okay.
Daniele Pranzetti: No, no, no, no, it's
Abhay Ashtekar: Not about it. Just a pullback in terms of the thing, but there are three into a good
Daniele Pranzetti: Idea. Yeah.
Sorry.
lsmolin: It's not that the segment is that these I thought live on the on the boundary. And that's the point. Yeah.
Daniele Pranzetti: Under but internal index is still expanding that
lsmolin: Too. Yeah.
Daniele Pranzetti: It is a one form on the boundary. Yes.
Daniele Pranzetti: Yeah, yeah. Then they are so yeah well that means that they are related to the pool back of the tried on the market through the Gulf slow the biggest constraint, but they'd only live on the boundary. And that's why they
Daniele Pranzetti: They only act on the they seem to charge a different here on the x on this element Bibles on the boundary
Daniele Pranzetti: So,
Daniele Pranzetti: Yes. And then I think that we can recover their non community. We have the flux algebra, but we're also going to see that this Edge Mode contains more information than just the flux that is one of the key point.
Daniele Pranzetti: Now,
Daniele Pranzetti: This idea of same logic of basic expressing kinematics constraint in the back so conservation looking also be extended to them to the momentum to the difficult string to defeat the difficult strength.
Daniele Pranzetti: And but then we can see that in the present a boundary. We need to in order to have the day from a very simple straight to be differentiable, we need to include a boundary piece.
Daniele Pranzetti: And and so what we can do is to replace the standard difficult thing with them momentum constraint which I wrote down here, which is equivalent to the imposition of the default constraint and
Daniele Pranzetti: Then this constraint, then a contains a bad piece, this piece I G which are basically implements the different or constraint in the park and then there is a boundary piece, which again is, it's a symmetry. It's a charge they found the boundary
Daniele Pranzetti: And this API is really defined here in the box is the best is the portion of the hashtag or a better connection. So this is the chart associated to three different phases.
Daniele Pranzetti: And then we can see the different posts the bar component. This defines the the translation engagement stream which is equivalent to be from a physical by taking this say hi to be the construction of the vector of excite tangent to the surface with with their co free. Yeah.
Daniele Pranzetti: And so in this way one can see that 50 different can be written as a conservation law for this church, which we call the sutra moment in charge, and they will be clear on the moment the physical meaning of this church.
Daniele Pranzetti: Once a week, compared with the metric case. But this is just to say that this philosophy can be standing, not just to the gauge in the internet age minds, but also to feed the difficult strength.
Abhay Ashtekar: But in this case is the philosophy or slightly different from the previous one. Because previously you added a new degree of freedom and the boundary
Abhay Ashtekar: Yes. Your and then only kind of equations emotion told you that the new degree of freedom was related to the sigma by pullback right
Abhay Ashtekar: Yes. Whereas, you're, you're not adding a new degree of freedom and you could add a new degree of freedom.
Abhay Ashtekar: I could answer that the question emotion relate the new degree of freedom to be equal to what you call the I'm the
Abhay Ashtekar: Equation below right but you're not doing that is there is
Daniele Pranzetti: No, there is no
Daniele Pranzetti: We haven't done that, in fact, and I was gonna say this in the next slide.
Daniele Pranzetti: Yeah, it's fine. I mean, in fact, we can go to the next slide that maybe we can can so
Daniele Pranzetti: We could have had those Edge mode for the momentum charges as well and they want, you could study how the imposition of the constraint leads to
Daniele Pranzetti: Basically these identification lynch mobs with the bulk of the pushback from the market. The momentum, what we did instead was to follow or more energy approach for those charges, namely to introduce
Daniele Pranzetti: Local sources of momentum and portion on the surface and then to quantifies the the the boundary
Daniele Pranzetti: Degrees of freedom you using those boundary condition. So what we did so far is to treat it if you want curvature, especially curvature sources in as classical background structure.
Daniele Pranzetti: And the momentum sources as a distribution or as you would do in quantum gravity for their for the flux. So, it was. It's an elaborate on decision that we need in order to
Daniele Pranzetti: As the first step to the quantization of this boundary
Daniele Pranzetti: Boundary face space, but the philosophy. Yes, will be the same that you keep adding Edge mode and then you can contest in from a content point of view.
Daniele Pranzetti: The boundary degrees of freedom. But then, so the ideas and then will be the user with this Bible we bonded by surface which will be described the stream that can be understood. So
Daniele Pranzetti: Some quantum jump to the log or subsystem and with this expectation of quantum space localized at the pictures were which includes some stranger charges.
Daniele Pranzetti: And the states then associated with the region or representation states of the boundaries community group and each bubble of space is represented by adding the pointer.
Daniele Pranzetti: Which represents the conservation law, the vertex of this boundary charges of course would get a generalized interpersonal justice between the webinar.
Daniele Pranzetti: And then the this chunks of quantum space can be patched together by basically first flattening if you want this.
Daniele Pranzetti: Ball to create the speed restructure you cylinders around the curvature defects which propriety propagate then by matching the quantum numbers of the punchers when this happens.
Daniele Pranzetti: And then in a further step to achieving a more discreet picture is too coarse grain basically this algebra and free someone can choose to
Daniele Pranzetti: Keep only the dual mode sector of this algebra, and we saw that if you do this, for instance, from this picture here, you end up with this point what we call our current networks which contain the control numbers bouncing additional momentum label associate exactly this.
Daniele Pranzetti: Different more feasible or constraint charge. I was talking about before. So this is
Daniele Pranzetti: This will be the philosophy of how to eventually glued together this local sub regions in order to a constructor state of quantum geometry for all the surface.
Daniele Pranzetti: But so there were two key ingredients, as was noted already in this construction, which was the use of time gauge personal and the use of some discrete oneness in the center of this devotional
Daniele Pranzetti: audition for the charges of momentum and
Daniele Pranzetti: And from portion and the curvature. So what we want to do now is to somehow step back
Daniele Pranzetti: And relax the assumption you can gauge where only the electric elements, corresponding to the rotation of part of the law or Lauren city were introduced and
Daniele Pranzetti: We want to. Oh, like
Daniele Pranzetti: Trying to unravel the Edge Mode structure of gravity from the continue on to you.
Daniele Pranzetti: To have a clear understanding of what the food boundary simply algebra from the computer is and this eventually will allow us to derive all the physical direct server boats and
Daniele Pranzetti: Eventually, to understand how the tangential frame can be reconstructed from this miracle server boards, both in the classical ending the quantum case.
Daniele Pranzetti: And a first point that will show in a mom is that the different components of the symmetry group are revealed by different gravity formulations. But before getting there me just introduce a bit of meditation, so
Daniele Pranzetti: Yeah.
lsmolin: Just to clarify, because there are several things going on in the previous slide.
Daniele Pranzetti: Yes.
lsmolin: One of them is the choice to and I think this is a choice of boundary conditions to take the punctures two circles.
lsmolin: And the edges two cylinders in a way that lets you introduce the category algebra or the, I guess the Lawrence plus Parker. A into which is what you mean by this little circle. So you go from one puncture where you represent some charge of the park or algebra to representing its
Daniele Pranzetti: Current algebra, there were, there are now a circle and two degrees of freedom. Yeah.
lsmolin: So I have a stage right what what what part of this movement.
Daniele Pranzetti: Yeah, this is a, it's a binary condition. If you want is the distribution organization of curvature and ocean defects around punctures and then yet this regularization if you want of
Daniele Pranzetti: These defects in terms of discs, which leads to this that's more the charges but
Abhay Ashtekar: There is no sound. Can we can people hear
Daniele Pranzetti: Can you, can you hear me or
lsmolin: It was I hear me.
Laurent Freidel: Yes, we can. I can hear. Okay.
lsmolin: So my question is, why, why did you choose to do this. Is this a choice.
lsmolin: Because we're the last find you take it away and you say, then truncate down to the final dimension algebra and I'm left wondering why is into that in the first place.
Daniele Pranzetti: Or that it will recover. Like if you want to simplify things, and to do some cross training that show how the ecology limit can be recovered, but
Daniele Pranzetti: To show that at the same time, if you do this truncation, you still end up with summit and one extra one to number which is this momentum charge
lsmolin: Right, so I understand that. But when you resolve the singularity of the puncture is that
lsmolin: That choice or is that somehow first time you buy some
Daniele Pranzetti: Notes. If you want it. If you want to simplify your life and want to work just with the introduction of the algebra. That's how you can envision it but you don't have to course you can.
lsmolin: simplify my life. I wouldn't be doing this but
Daniele Pranzetti: Yeah, well, wait.
Daniele Pranzetti: This is
Laurent Freidel: Going to come. No, because
Daniele Pranzetti: Yeah.
Laurent Freidel: You know what you're going to do next is not doing any of these choices and redirecting to continue
Laurent Freidel: You don't have to do these tours is, it's just these choices. The show how you go from do full continuum algebra to do energy
Yeah.
Okay.
Daniele Pranzetti: Yeah, so
Daniele Pranzetti: Meditation. So yes, we, we introduce
Daniele Pranzetti: Space that man, which has a rental metric. And then we we do the free body composition with a dimension one slice sigma here. And then we have the
Daniele Pranzetti: The normal one for n from which we can resolve the internal normal vector, and I would take. So this is a time like vector
Daniele Pranzetti: And then they could do tangential co frame field can be a snotty with you still there. So all the qualities within the basically means a pullback on this slide, sigma
Daniele Pranzetti: And so we have the news medical sigma with GTA. And then they induce metric on the boundary of sigma the corner would be denoted by a cute baby.
Daniele Pranzetti: And the last bit of notation is this killed epsilon, which is basically used to define an inner product and internet space. So these are the quantities and I'm gonna use in the following
Daniele Pranzetti: And OK, so now let's go to the next slide.
Daniele Pranzetti: Now, so let's start with the metric formulation of gravity. And so the static points that the construction of the quality space feminism is this precipitous potential
Daniele Pranzetti: So we call it a jar, the potential that includes basically they disappear destruction contained me Tanya. The composition of GR
Daniele Pranzetti: And this is press the one in fact that best the momentum densities conjugated to the induce metric. So it's eliminating to rewrite this in the first hundred variables to show or the fundamental nature of this potential which takes except listen for what they are not good now have this
Daniele Pranzetti: Mental entity is the momentum charge. I was showing you before the API, which is this for momentum aspect.
Daniele Pranzetti: And it's related again to them to the portion of the adventure entangled. This relates portion of the actual better connection and the country barber is the, the owner of the math, which is there, the friend field.
Daniele Pranzetti: And in fact, one can think of these as the reducing what potential caption the bat canonical degrees of freedom, which are common to end formulation GR
Daniele Pranzetti: And starting from this combo piece. Any other formulation of gravity can be understood as being built basically from this job potential plus some band director
Daniele Pranzetti: And we will speak to show this in the center of gravity for a reason. But this can be this short facing the Iceland Hebrew formulation and and formulation
Daniele Pranzetti: And then in the absence of matter. So the destruction. I'm showing here is that we tend to have any formulation can read as the buck job class some boundary piece.
Daniele Pranzetti: Then of course in Absalom market, especially for more efficient takes the form of the conservation equations for the momentum density. So we see that this conservation law of this momentum. It's just the first order. If you want equivalent or what happens metric, grab the case.
Daniele Pranzetti: And
Daniele Pranzetti: And then they so we can write, basically, especially from a threesome as conservation of this for momentum aspect now.
Daniele Pranzetti: If you look at them. If any charges then Lucy did the look at tangent. He feels and
Daniele Pranzetti: We do is just usually we contract this entity structure than the derivative alone. The vector feel excited, which is tangent
Daniele Pranzetti: And what went up with some Italian generator which is about peace and the boundary piece. So the but pieces just, again, it's this may have vector constraint of canonical GR in the boundary piece is
Daniele Pranzetti: Again, this is exactly this charge. I was showing you before that you issue right in the first order from is exactly the same quantity which is nothing has the brown your charge. So this brown. Your child is this which are meant to charge. I was
Daniele Pranzetti: I was talking about earlier. And this shows as the common piece of the boundaries in the algebra graduates basic game by disintegrated presentation.
Daniele Pranzetti: Of the different are facing tangent to the surface. So this is a piece which is common to any formulation which comes from this data jr component of their potential.
Daniele Pranzetti: Now if we go to the tender from nation.
Daniele Pranzetti: So we introduce a keyframe field and Lawrence connection which our country as independent Bibles, then the buck.
Daniele Pranzetti: So this is the ice and Katana grandjean be field contains the two parts is the usual pieces star emoji. And then there is a
Daniele Pranzetti: Very meets the perimeter component. And so the synthetic potential is given by this. But expression here in terms of be an omega
Daniele Pranzetti: But now if we produce it will care for free plus one. The composition of all the geometrical quantities that appear in data gravity.
Daniele Pranzetti: We can show that basically this potential can be become post again in the bud piece which is exactly the data jar that
Daniele Pranzetti: Was Australia before. Then they said peace which vanish if you go and share the gospel as this increasing constraint.
Daniele Pranzetti: There is another piece which is just the population, which doesn't affect the synthetic form which is given talking to her, and then there is the boundary component of ice and Captain formulation which I'm not going to show you
Daniele Pranzetti: Explicitly I'm going to do in the next slide.
Daniele Pranzetti: This is just to really show the structure of the synthetic potential of this I think enough formulation which follows the same pattern I was mentioning
Daniele Pranzetti: And then they saw the lot the different charities are as before. Nothing changes the new thing is now that we have known when she Lawrence charges for CC to launch transformations and you see that basically the boundary charges are given by the boundary be field now.
Daniele Pranzetti: And this contains two pieces. There is a boost part with in terms of this
Daniele Pranzetti: Big which from shell of the simplicity takes this form here. And then there is the rotation part of the usual Titian part to that I was showing you on the time each case. So this for
Daniele Pranzetti: Now just, I'm not going to talk about the formulation, but there is an expression of the synthetic addition to that gravity in the work of the morning
Daniele Pranzetti: And the poly and I will if you want to talk about there. I would suggest that we do at the end of the talk because it's not really going to be discussing longer between the rest of their of
Daniele Pranzetti: The talk. But again, when can show that this structure of a bad piece of the job, plus a brand new piece can be recorded.
simone: Right to just do in the case of Anthony, but also there is a boundary term, but it is right. He's just that is different from day one.
Daniele Pranzetti: Yes.
simone: Sorry about the noise.
Yeah.
lsmolin: So the this Lorenz charge of vanishing is equivalent to the boundaries and
lsmolin: Constraints boundary simplicity constraint.
Daniele Pranzetti: Yes. Once you have detriments
lsmolin: Yeah. So what's the interpretation of the charges the failure of the boundaries in place, you
Daniele Pranzetti: Know it's so easy. That's what I was trying to say the first area is that you can see recover engage in violence in the sense that you can set the gate charges zero by you're still left with symmetry charges which are just the one given now, but the Edge mode.
Daniele Pranzetti: Component. So you have the 3D representation of the boundaries in the algebra given by the fact that you have included edge modes which
Daniele Pranzetti: Apulia independent from this pullback have resolved the boundary value of the field coming from the bike.
Daniele Pranzetti: Maybe this is going to be clear.
Daniele Pranzetti: In a moment, if one include the Edge mode for the Lawrence second largest but yeah the charges are basically
Daniele Pranzetti: Is there again, but the Edge Mode which corresponds to if you want killing this Lawrence charge here.
Abhay Ashtekar: So is a point that it's only on shelf that the edge more so related to sigma right
Abhay Ashtekar: Yes, yes. And so therefore, but here you are not on shelf.
Daniele Pranzetti: Know, yeah.
Daniele Pranzetti: Yeah, I've included the Edge Mode yet yeah but
Abhay Ashtekar: And you are not on shelves. So there is more independent determined by what is happening, the ball.
Daniele Pranzetti: Yeah. Yes.
Abhay Ashtekar: Except for and therefore you can be God as the bulk rotations as gauge, but the boundary rotations, not as case is that the pork.
Daniele Pranzetti: Yes, absolutely. Yes, that's a point. Okay, thank you.
Daniele Pranzetti: Okay, so now we can add electric which months.
Daniele Pranzetti: And then we can follow the same strategies we didn't have the case of the time, engage and we can resort to violence and Lawrence invites by extending wonderful space. And so the edge months now. Are these
Daniele Pranzetti: Fields here, there is the this big. EI. You notice I'm using different font exactly to stress the fact that these are pure independence on the boundary from the back fields these internal normal nine. Then there is the frame field.
Daniele Pranzetti: And these are 16 degrees of freedom. And this is the boundary suppression of the synthetic potential that I didn't show you before now here use BF
Daniele Pranzetti: To set you stress the fact that this big AI is not a priority related to it doesn't, it's not as pressing in terms of the of the frame field. It's a. These are independent fields to begin with and in n
Daniele Pranzetti: So if we want now to
Daniele Pranzetti: Record the gallery space. We have to impose extra constraint. And then there's the first set of constraints which is killing magical which are given by the fact that the new ization
Daniele Pranzetti: Of this internal normal is speaks to minus one and then that basically the pulled back of the normal on the surface vanishes.
Daniele Pranzetti: So these are the main constraint and if you use the person brackets, including this potential here to complete the algebra this constraint, you realize that the first piece here is first class.
Daniele Pranzetti: And then this n equals zero is a second class component of this kinematic constraint. So what we can do then is to
Daniele Pranzetti: derive a complete set of direct observable the commute with these constraints. And this basically give us 12 observable. It's the commutes with these constraints.
Daniele Pranzetti: And these attributes are these the normal which is three degrees of field and they said boosts operator is bi, which is can be reading them. So the region of fields in this form. And then there is a tangential frame which is this ETF there which
Daniele Pranzetti: Also is expressed in terms of the regional variables.
Daniele Pranzetti: In this form and you can show that this free observable this preset observable now commute.
Daniele Pranzetti: With the with this constraints with any magical constraints. So these are really acceptable in this, in the sense that they they go commit with this clinic module constraints and a parameter price.
Daniele Pranzetti: They face based on Shelby for one of these constraints.
Daniele Pranzetti: And particularly seated moose operator is conduit to this internal normal, why not required to because they're also an automation for some brackets. But as this awesome ragged, we've been phenomenal and day to day or to satisfy the analog of there.
Daniele Pranzetti: Was some brackets for the boundary frame that I was showing you before.
simone: Sorry, these are capital E. Yes, me, just the unit normal time like unit normal to the surface. So inside
simone: Sigma engines piece, the sigma the 3D cushy ever surface.
Daniele Pranzetti: This this country here a purity is independent on any of the structure when you go and share of the simply single string is because basically the flux discovery.
Daniele Pranzetti: Is I'm going to show you, they're
simone: Not nothing depends on the
simone: eyeballs on the normal to s that Nandi nothing depends on the normal that that our minds as inside the see
Daniele Pranzetti: A purity enn so we have n which is the one of the normals, the sort of internal normal and he is independent on and take the
Daniele Pranzetti: Conjugate to each other.
simone: Right. But this is only the second normal to us.
Marc Geiller: Yeah, two minutes, asking if the second number two is it
simone: Should be just the second number two is, if I remember correctly.
Laurent Freidel: No, no, it will come. It will come there.
Laurent Freidel: Is no, wait a minute.
Daniele Pranzetti: Yeah, I mean, that what. Yeah, that would be a diff mean something which is related to it, but it's not it's not really that I will, I will show you in a second what it is just one side. OK.
Daniele Pranzetti: OK, so now
Daniele Pranzetti: The information about the tangential frame can be expressed in terms of speed operator. So is this this si
Daniele Pranzetti: Simoni that it's really what the second moment is and
Daniele Pranzetti: Actually mentioned engine tangential to all the information in this at that can be encoded that we're almost for us. We will see chemical in this
Daniele Pranzetti: New quantities. This has been operator and there is a tangential metric and the true or not. They are not independent.
Daniele Pranzetti: In the sense that there is a geometrical relation between these two that relates the su casa me with the determinant of this search bar.
Daniele Pranzetti: Of these, the metric here which you've seen a second satisfied etc symmetry.
Daniele Pranzetti: But the spin density generates unless you truly algebra, which preserves the internal normal. So this is the, these are the brackets.
Daniele Pranzetti: And then one can show that the such large enterprise generated by this isn't a surcharge are generated by this kind of method with me, which I give you here. So this really is a relation between the switch accosted me, and such lot Casimir come from this observer boats. Yeah.
Daniele Pranzetti: And
Daniele Pranzetti: And the medical be of course also commutes with the generators. Big sigh Oh, the surgeon generators that I introduced before. So there is
Daniele Pranzetti: This balance equations. And now what I'm going to do is finally to introduce the funk array.
Daniele Pranzetti: So, one can see that, basically, if we introduce the starter on the moment and generators JJ, which are built from the other
Daniele Pranzetti: Variables to introduce a five day boost the internal normal and the spin. So the first part can be understood as the angular momentum. The second part is the spin, then we see that this generator satisfy, together with the with the internal normal in algebra, which is upon carry algebra.
Daniele Pranzetti: So the first line here is basically the such as see sector.
Daniele Pranzetti: Which generates large transformation. The boundary space. And these are related, basically, to the boundary cows law.
Daniele Pranzetti: Basically by the button. You see that is generated, Jr. But if the back of the field by the boundary gap slow. So this is what I was, was asked before. And so when you post these the charges vanish. But this now you have the SF DC symmetry generated by this Jay Jay Jay
Daniele Pranzetti: And
Daniele Pranzetti: And now.
Daniele Pranzetti: You see that the balance equation or basically what it does that actually before release day one, Christina Casimir to the boundary medic determinant
Daniele Pranzetti: So what this equation means is that, then the element is given by this elemental conquering and spin and is therefore Lawrence importance quantity and this is really due to the fact that the normal now is has become the angle.
alejandroperez: That nearly one question in the previous slide, the equation you showed looks similar to something we had with the
alejandroperez: Long in the paper.
alejandroperez: quandary that the corner, what
Daniele Pranzetti: You're the first reference here.
Daniele Pranzetti: It's
alejandroperez: The same thing
Daniele Pranzetti: It's, it's the, what was the tango version of this basically
alejandroperez: I see. Okay.
Daniele Pranzetti: This image of is just we all the time. He
alejandroperez: Okay, thank you. Yeah.
Daniele Pranzetti: Yeah, I mean this this is this essential as a long history. Somehow it just that.
Daniele Pranzetti: It was never really taken seriously, to a certain extent what we're, what we're seeing here is that it's a fundamental observable in order to reconstruct the tangential frame is not something that you can just, you know, kill innocuously
Daniele Pranzetti: And
Daniele Pranzetti: So we see that
Daniele Pranzetti: The, the surface element is a learning environment and
Daniele Pranzetti: What's very nice consistency check is the fact that as we can understand this area man autos and etc.
Daniele Pranzetti: Casimir, then it's not obvious that you will have a discrete representation for the for the spectrum, but the fact that they're basically that the surface is based like selects the discrete a search or presentation.
Daniele Pranzetti: So there is this compatibility between the sweetness Lawrence inviting that shows up in this balance equation and between essential are passing MDA on careers Pincus me
Daniele Pranzetti: Now if you want to have a better understanding of this elemental punk reality rap, we can try to introduce some particle like parameter ization of the algebra.
Daniele Pranzetti: In terms of those two letters, where we have this excited, which is a particular position, we have the momentum bi the particle moment and then we have some internal degrees of freedom, which is a nice pair of oscillators.
Daniele Pranzetti: And then the total the moment is in Mario under translation generated by this generators PA defined like this and the p square and this this generators, Jenna.
Daniele Pranzetti: Define for the ice and pre algebra. So now we can we can fix this ambiguity, Issue one by choosing going to the nursery where we said to equal to zero and then
Daniele Pranzetti: We see that this condition basic breaks, they seem to group them to center.
Daniele Pranzetti: And then the dramatic dramatically. They seem to break in Mexico ensures that the moment our normal to the sphere and additional terms often position of discuss
Daniele Pranzetti: Initial from constraint, what we see is basically said the flax corresponds to a relativistic positional potato and the frame correspond to the spinner simulator.
Daniele Pranzetti: So, and, as I was saying before, is the is the momentum and this this risk of the mass which is an interesting to prediction, which basically if you introduce the party basket vector
Daniele Pranzetti: You can see Bessie from the special of the Casimir the second customer that the mass of this one career, a particle particle is really related to the mid see parameter
Daniele Pranzetti: So,
Daniele Pranzetti: What this is saying is that basically
Daniele Pranzetti: once a semester just interpretation, the spinner to link as a particle of quantum space which cares. It's been s and the mass given by the Missy parameter
Daniele Pranzetti: Or then one quote something for instance of the gamma zero limit and then you see in this picture. The gamma gluten to limit Betsy corresponds to the medical face corresponds to the merciless
Daniele Pranzetti: Limit of the select from this elementary punk and metal punk or a point of view. So this is just to give maybe a bit more of a physical interpretation of this boundary variables here.
Daniele Pranzetti: OK, now there is a second set of some pieces, which is really where we're interested in is the boundaries and physical strength and
Daniele Pranzetti: This is the best relate the booster operator to this innovator. Now this is these are free. So conditions because there is this
Daniele Pranzetti: Wanting the medical relation satisfy that only few of these four components are independent and one can show using the space that we introduced that there is they don't be satisfied is algebra.
Daniele Pranzetti: From the continue or we can see that this is the algebra dissatisfied. We're not is that this the first piece here is that this is different from the discrete touch with one of things from disquisition producing the phones.
Daniele Pranzetti: But again, these other shows that basically these constraints are second class and
Daniele Pranzetti: So what this does basically he's constraint killed four degrees of freedom because
Daniele Pranzetti: Since they are free constrain means that one has to be first class. In short, there has to be second class. So they call they can choose degrees of freedom. And when that Dan is with 800 year cups.
Daniele Pranzetti: So these are the ones that now we want to recover eventually Lisa understand what are the state, you know,
Great.
Muxin Han: Hi done yet. So about the previous life. So if you include cosmological constant does this outer bark at the form somehow
Daniele Pranzetti: Possibly, yes. We haven't really looked into that.
Daniele Pranzetti: But that's possible.
Daniele Pranzetti: That. Okay. Okay. Yeah.
Muxin Han: Yeah.
lsmolin: That sounds really interesting
Daniele Pranzetti: Right, so
Daniele Pranzetti: Okay, now we can so we can use. Now, the set of
Daniele Pranzetti: Fields and I si N et di it's not to talk on basis. So you see this is
Daniele Pranzetti: Someone was asking this aside plays the role of this economic one. And we can decompose this constraint and see that there is a second class pair which are given by the tangential part if you want of the constraints and they're satisfied this algebra here.
Daniele Pranzetti: And then and now we can isolate the first class term and which commute strongly with the second task constraint and turns out the best. It's a combination of two possible candidates, which are see square and see the. So today, the pistons are the potential to us. And this is where and
Daniele Pranzetti: In
Daniele Pranzetti: One can also see the basically this first class constraint is a function of the central seekers image only
Daniele Pranzetti: Which is going to be useful. Eventually, for the implementation of a string. So now this expression here is the first gospel singer, which basically
Daniele Pranzetti: Corresponds to the content version of the district diagonal simplicity constraint, as it was a returning recently in the paper of Canon Law and smaller.
Daniele Pranzetti: There should be some special is introduced us the Lawrence inviting part of the Indian and the speed, simplicity can see which we pleased, they have they have the young CPC constraint in the
Daniele Pranzetti: FK piano model. We're over. Speaking of this in physical symptoms of first class second class was also done. For instance, in the spirit my work done.
Daniele Pranzetti: In this sprint from formulation
Daniele Pranzetti: Let me just point out that, however, the first place where this expression appear is in this paper or whether and then yell already. He doesn't true so way before the new models. In fact, they, they were trying to recover the Berkeley.
Daniele Pranzetti: Berkeley quantum constraints on the representation in the case in which one include includes the music or music to be affection and I think it's quite interested in this quote
Daniele Pranzetti: What they say that
Daniele Pranzetti: One should be
Daniele Pranzetti: It's not easy to construction system. For example, using non simple presentation but this should be investigated. In fact, a few years later, we know what the
Daniele Pranzetti: History for the path. But anyway, so this first class components. It's not new. It's a as a PA before you need to. But the second class, especially here it's I think it's
Daniele Pranzetti: It's quite a new
Daniele Pranzetti: Especially that we found.
Daniele Pranzetti: Now the Cubs ever boats.
Daniele Pranzetti: So it's best to me to see that.
Daniele Pranzetti: The JJ the generators of the Lawrence decimation they are all basically the old commute with
Daniele Pranzetti: With this kinematic constraint and with the same physical same retreat in terms of first class second class piece, which means basically they taught on the momentum.
Daniele Pranzetti: Form 16 strong Dirac upset boats and again this is affect the Lauren symmetry reflection inside the learning team is preserved by the imposition the simplistic constraint and they can the magical constraint.
Daniele Pranzetti: And however this is no longer true. If we look at the boundary medical appointments. They don't sound. The commutes with the second guessing PCT and to fix this one can for the commerce strategy in
Daniele Pranzetti: In order to deal with segment has constraints in form which is basically replaced it. The second class component of this MPC will still be the Master Splinter
Daniele Pranzetti: And you can continue then basic. The second class nature of this constraint. It's affected by the fact that the drug embed doesn't appear in respect of the master constraint.
Daniele Pranzetti: And we can position the casino in a month to selecting mean we're getting value instead. But we, we've seen a moment that I mean there is this approaches vehicle with some subtleties, but
Daniele Pranzetti: For the moment, let's just follow the path and replace the master conspiracy. The second class with the massive constraint and this is the expression that we find
Daniele Pranzetti: And it can be checked that this expression here now is such that the boundary medical mute strongly with with the master constraint. So they ought to be free tangential medical points are
Daniele Pranzetti: The represent the three remaining vehicles circles. So what we did then is revealed at the boundaries in the group is of ground in the teller formulation is given by a citrusy cross, etc. Now there is
Daniele Pranzetti: A downward partner seeming to breaking that has been going on. So first we started with this chart cross the Heisenberg
Daniele Pranzetti: Algebra and then posting the initial friend constraints basically breaks the same three down to set to our and then the same thing happens for the other
Daniele Pranzetti: Click on the CPC was saying, we start with a concrete group and imposing the database constraint breaks it down to a central see so that the final symmetry algebra. It's a central processor
Daniele Pranzetti: Now let me let's grind them more dramatic insight into the search bar sector.
Daniele Pranzetti: Now, what we saw is that the physical boundaries basis parameters financial troubles. However, I was mentioning that not all of them are independent. In particular, there is a geometrical bands equation, which relates the pancreas been
Daniele Pranzetti: To the asset to our customers. So what this means is that we still need one more valuable basically to reconstruct the boundary frame and this Bible is given by an angle which represents the information in the frame that is not capturing SI and into a big
Daniele Pranzetti: One can check the sap basically left in by by just permission.
Daniele Pranzetti: Parameters, but this angle. So we introduced this one here. So if you do this information, you can see that these companies are left environment, in particular, one can show that this angle is contribute to the area and
Daniele Pranzetti: In fact, so this condition relational of tea, then the reform shows the best Tita is the analog of the priests angle enters the definition of the twisted geometry and this angle is necessary in order to reconstruct along along the edge relating to frames.
Daniele Pranzetti: On the two sites. So you have two premises to the triangle shaped by Twitter data in order to reconstruct the Delano me that basic release these two frames you need this angle. This was shown by someone around
Daniele Pranzetti: And this is the job description.
Daniele Pranzetti: Of geometry. So, the, the joint is the description
Daniele Pranzetti: Of discrete geometries associated can manifest based of gravity. And in general, the metric is this contains across the triangle shaped by the edge deployed and
Daniele Pranzetti: Basically basin etc transformation which is needed to match the geometry in the two triangles.
Daniele Pranzetti: And the simple way to see this. For instance, if to reduce the coordinate system that covers the
Daniele Pranzetti: Shooter Phaedra bombing the training will face and then just for Michigan be in this to this linear transformation of coordinates. Basically the sensor condition three angle on one side. So from the conditions for angle to the other side and then
Daniele Pranzetti: The boundary metric basically is exactly at this generators. So it's the generators of this ship matching transformation that codifies in the continuum. This information about the connection triangulation on the edge of the graph board is to drive downloads.
Daniele Pranzetti: So also an interesting quote at the end of this was this assault charges from issuer shown in this nice paper here.
Daniele Pranzetti: And the quote there at the end of the paper was that there's a reason for this increase equal string to suppress twisting and fail. We're seeing that this thing is absolutely compatible with the with the simplicity constraints effect.
Daniele Pranzetti: Is necessary to reconstruct information a tangent frame.
Daniele Pranzetti: And notes, let me
Daniele Pranzetti: Point out that this church quality defines a complex structure on the surface, which is very important. In fact,
Daniele Pranzetti: This established basically the default source the archdiocese compared with the person structure.
Daniele Pranzetti: And this is based in Portland renting position or second class constraint, as I'm going to share in a moment. My more Jeff in this project of the originator of this area affected the air is a generic have this opportunity can have important implications of quantum theory.
Abhay Ashtekar: I just have a quick question. I mean, until the last slide, you're always in the continuum and
Abhay Ashtekar: Gary. Oh, yes. Now you're making this transition in this this diagram that you're drawing about the twist angle and so on for some time.
Daniele Pranzetti: Before. Now I'm trying to
Daniele Pranzetti: To show that this continuous variables are continue analog of quantities that have been produced in the literature.
Abhay Ashtekar: Good. Okay. Thank you. Bye bye. But yet, you're not going to be working with a with
Abhay Ashtekar: Always going to be working in the continuum is that
Daniele Pranzetti: Right. Yes. Is
This just
Daniele Pranzetti: To clarify some passive existing the twisted jaunty about the compatibility between the twisty angle and this NPC constraint.
Abhay Ashtekar: And
Abhay Ashtekar: Okay.
Daniele Pranzetti: Thanks. So yeah, OK, so
Daniele Pranzetti: Just me summarize what we saw is that the the edge of astronomy go observe is generated by this
Daniele Pranzetti: JJ and the Qb. And for the boundaries in algebra is different formulations game at the essential see processor to are you today.
Daniele Pranzetti: Eric observable and the boundary space. And here is a big roadmap of the path, but I think I'm going to need, so I'm not gonna go through this. Let me just now the last part of the talk. It's a body position of the
Daniele Pranzetti: Simplicity was strange.
Daniele Pranzetti: So a way to pro feminism has consumed one to level is basically still rely on the Gutenberg prescription
Daniele Pranzetti: But this requires a specific choice of a complex structure on the face base, which has to be compared to the clustering algebra.
Daniele Pranzetti: Now, as we've seen before. This has been provided this article ID is available, do they affect the frame field on the boundary space is private message that the frame. Feel the benefit space and then. So the strategy is to impose second class simplicity
Daniele Pranzetti: Two steps. First, we use the compatibility of the door with a bracket to establish that we can basically we can define your market constraints, the pluses and minus, such as they compete with each other.
Daniele Pranzetti: And then we replace the second class constraints to do with the first class condition that c minus eight equals zero. Word. Now, this can be imposed strongly on
The quantum level.
Daniele Pranzetti: But, as was mentioned earlier.
Daniele Pranzetti: It's often convenient to replace these conditions equals zero, with the master constraint.
Daniele Pranzetti: Now the classical level. The two things are equivalent at the quantum level one has to be a bit careful because messy. What happens is that when you read the master constrain
Daniele Pranzetti: You get this quote the same expression, but you also get an anomaly which takes everyone down here, but it doesn't matter. Exactly discretion is just, it's an anomaly, of course, to the the commentators now.
Daniele Pranzetti: So,
Daniele Pranzetti: If you post strongly ca zero c minus two equals zero, then you have this relation for the between the massive constraint anomaly on the steak.
Daniele Pranzetti: And it's only
Daniele Pranzetti: means basically only if the anomaly commutes with the this sometimes constraint that you can really
Daniele Pranzetti: Identify the meaning and value of the master constraint with a good value of the anomaly evaluated on the state that they have analyzed both C minus na
Daniele Pranzetti: And one can check that this is the case in the foreign debt financing that we wrote by basically
Daniele Pranzetti: You can use a semi classical evaluation of the anomaly gives you this expression for the anomaly and then you can check that this anomaly commutes
Daniele Pranzetti: Basic this weekly if you want with this emancipated enough because the mind is can be imposed strongly now. So if you can meet weekly with the the constraint. It's means that
Daniele Pranzetti: You can say to zero this bracket. And so the the computer. Yes. So this is just a consistency check that the massive constraint approach can be is complete, it's consistent with the weekend position of the second tax constraints.
Daniele Pranzetti: Okay, so let's find an important this constraints. First we need to admit the bundle space.
Daniele Pranzetti: And what to do. So it's important to remember that the Lawrence algebra emerges from the elements of concrete algebra.
Daniele Pranzetti: Once this is increasing constraints are introduced. So it's, it would be useful to introduce first upon carry algebra at the bunker representation
Daniele Pranzetti: So introduce introduce me to your presentation where we live by the various of the customer and square which is the momentum square. It's minus one and then they disappear customer takes this label by this
Daniele Pranzetti: half integer s now as usual can introduce ecumenical were to TI and define a kidney magical rotation generated in terms of it and then
Daniele Pranzetti: It's important to keep in mind that the SI this paint the bookcase penis enjoy different from this kind of magical or generator. It's only when you go to the time engage that the choo.
Daniele Pranzetti: Choo are the same. So basically, this means that
Daniele Pranzetti: There is no germination between j and s
Daniele Pranzetti: See the classical it's increasing constraint that restricts further than the fact that J squared will be bigger than a square but
Daniele Pranzetti: And this is going to hold on to the quantum theory, but in Singapore to keep into account, it s NJ two separate one two numbers. So then what we can do is to
Daniele Pranzetti: Introduce the cut down by basis which is adapted to the merge them this learning symmetry. And this is formed by a set of commuting emission generators that can all be diagonal is at the same time.
Daniele Pranzetti: So using it for you so nice. And you can see that the set the set of generator approved by the checkpoint and Kerri Kasem is the true Lawrence Casimir screen cookies. And then there's the kingdom article spinner J squared and its component Jeffrey, so what what we have are basically
Daniele Pranzetti: Five quantum numbers because Esquire, it's been fixed them and Swan with us. We have Brian K, which are labeled today. Lawrence Casimir and then we have Jay and m which
Daniele Pranzetti: Corresponds to this magical spin quantum numbers and these are the the egg and values of this observable to. So now on this space, we can increase the constraint we start with the first class, which can be strongly and this gives us a tradition of right as a function of key.
Daniele Pranzetti: Again, we have two branches, but only one. It's as the correct some classical limit. So it's the give us the usual erotically retardation.
Daniele Pranzetti: And now we can impose the second class constraint. Again, we're looking for demand innovative the master constraint and by using this solution or plus we found before Bessie, one can see that the main minimum value is given by the condition that case will be equal to us.
Daniele Pranzetti: So what this shows is that the physical vessel in order to solve some cross constraint.
Daniele Pranzetti: We can we can label the physical space by to spin. There's the pancreas been s and economical spin j, which is fenestration just to be bigger than S and
Daniele Pranzetti: Lauren cemetery is fully preserved or to the quantum level, we have a representation of the assets juicy algebra at the quantum level and then
Daniele Pranzetti: This means the best at the quantum level this in physical restraints can be implemented.
Daniele Pranzetti: By embed and embedding a bit later if you want the maps their presentation. So to see into the bunker representation which is if you wanted generalization of the many map. I will show you at the beginning.
Daniele Pranzetti: Okay, so let me conclude
Daniele Pranzetti: So we see basically the edge for formalism capture the nature of the local news feed on which are involved.
Daniele Pranzetti: In the division and the scene of the systems in the subsystems. Now this process can be in the switcher brightly as the fusion of states associated with boundary symmetry algebra.
Daniele Pranzetti: And this understanding Hopkins away to a new description on geometry where features of photographing economic approaches can be combined.
Daniele Pranzetti: Now the country on it that we perform the same physical strain basically shows that the boundary symmetry group is the correct group which is generated by this
Daniele Pranzetti: Lawrence and the moment that antenna normal n and before the imposition of the CPC constraints. Now, the presence of discernment upon k
Daniele Pranzetti: Which is purely associated to the geometry. There's no matter that we introduced so far HERE. IT'S QUITE SURPRISING FEATURE OF THE CONSTRUCTION by sorts of their recent guy on this nice property with the with arriving in particular, we saw that
Daniele Pranzetti: The time like direction becomes dynamical Bible, please. The momentum and gets routed by the Lord to traders and, most importantly, we saw that missing piece. He was only big the transitional subgroup of conquering but preserves the learnings in violence.
Daniele Pranzetti: And this is compatible with a with a discrete specter of the Arabic
Daniele Pranzetti: And the boundary symmetry algebra contains or two. This is such
Daniele Pranzetti: A component which is necessary to reconstruct the tangential frame.
Daniele Pranzetti: And this means the best. We need to look into numbers, North Korea constructed tangential frame operator is the pancreas DNS and there's a quantum shape if you want
Daniele Pranzetti: Number, which corresponds to an integer. And that label states in reducing position or the presentation or central which I didn't really have the time to talk about this representation here, but it's a familiar episode number in the representation cetera
Daniele Pranzetti: So basically showing how the suspended boundary algebra provides the continuum of structure which were introduced in recent geometry description and the imposition of this increasing constraints does not imply the position should match conditions.
Daniele Pranzetti: And particular among many directions and now this offense this quantum numbers can play an important role in understanding for isn't the flatness problems in forms which has been a topic of
Daniele Pranzetti: Edge that's generated a lot of activity and also this morning there was an Uber about this and we find that this understanding, could that could help solving the contrast between different approaches. Okay, thank you very much for listening, sorry for being so late.
Abhay Ashtekar: I understand now the general philosophy from the point of view of going beyond kinematics.
Abhay Ashtekar: Mm hmm. So
Abhay Ashtekar: On slide 18 you had this
Abhay Ashtekar: This fictitious particle description basically X MP and some internal degrees of freedom and so on so forth. So are you thinking the dynamics could be simple in terms of these political like variables or
Daniele Pranzetti: Well, one thing that can simplify dynamics is the fact that the different field is available now in the sense that we don't have to rely on the Fluxus and the volume to describe the different field, but we have quantum numbers that allow us to reconstruct
Daniele Pranzetti: Them in a more direct way and this could provide the
Daniele Pranzetti: A new definition of time constraint.
Abhay Ashtekar: Okay, but I mean at the moment that you sort of really have restricting yourself to the constraint. And so, I mean, you said there are observable that really kinematics.
Daniele Pranzetti: Yes.
Abhay Ashtekar: Okay, so, but is a viewpoint that you're still trying to get Hamiltonian constraint and then you will get a kind of a canonical theory or is this supposed to give you more.
Abhay Ashtekar: I mean in economic theory in which then the fundamental thing would be just submission Hamiltonian constraint and it is not clear what time evolution means and so on so forth or
Abhay Ashtekar: Yeah, I'm just
Abhay Ashtekar: Trying to understand the direction I'm thinking, or is it more considered to be more promising that even though you're using this canonical language YOU REALLY WANT TO GO ALONG PARTS integral, for example, or on the lines of something. I mean, maybe not lying, but
Abhay Ashtekar: Something like what she was saying, and so on.
Abhay Ashtekar: So I'd like to understand your views about dynamics in general. I mean, anybody not just you, but around anyway.
Daniele Pranzetti: Yeah, so
Daniele Pranzetti: One could i mean the the motivation for this world was truthful. If you want the sense that they were both trying to understand how to reconcile certain forms with the canonical approach and the sense of maybe
Daniele Pranzetti: Trying to construct them for modern eventually from the canonical point of view, and at the same time to to define probably economical dynamics. So one way to one path that we have is to consider now now boundaries.
Daniele Pranzetti: Because in the case if you will contain outbound this then you can you have a notion of time if you want to you can risk press dynamics as a true time evolution and what we will need to understand then is how the boundary dynamics of this edge modes encodes the buck dynamics.
Daniele Pranzetti: But the idea would be basically to yes to use the boundary dynamic of this degree so further on, and not surface to go to define the dynamics.
Laurent Freidel: You can either. Yes.
Laurent Freidel: Thank you. I like very much that question. I think that's such a hard question for me, I would you know what i see what we're
Laurent Freidel: Proposing right now is, like, in some sense, we're proposing a new playing field and I would, I don't know, I think, because it's a new playing field, we should we should keep an open mind.
Laurent Freidel: And look at both ways of doing it. That is, you know, we should know. You should be able to do it can equally and n by informing kind of evaluate which one is the more efficient and ultimately
Laurent Freidel: Do it in such a way that there's no differences between the two approach that would be a bigger consistency check, but depending on your knowledge and your
Laurent Freidel: You know, I think, I think, or do you know it's a new playing field, especially because now the quantum frame fee, the frame Philippines or quantum operating
Laurent Freidel: So that does
Laurent Freidel: It's not nothing. It's easy or done but but i think you know it's it's like a reciting of
Laurent Freidel: You know, a que je if you want instead of looking on the issue two variables which are not enough to because for the operators. Now we ever he said to see times they sell to our absorbent which may be enough. We
Abhay Ashtekar: Thank you that that's good. So I understand it is a beautiful playing field. So I think it's worth definitely looking at it in great detail.
Abhay Ashtekar: But since you also mentioned that nearly about now surfaces that he seems to be
Abhay Ashtekar: There should be some close relation with what Oregon has been doing either you are or gang or anybody else can comment on that. I mean, in that case also you know he and I were very excited about the fact that
Abhay Ashtekar: All can also had some particle like interpretation, right, and some some particles in fictitious mccaskey space.
Abhay Ashtekar: And so we thought that that is going to be a very, very important way perhaps us have no understanding dynamics, but I haven't seen too much development along that lines and so on. So I would like anybody to comment on all these points.
Daniele Pranzetti: Yeah. So yeah.
Daniele Pranzetti: Just wanted it. In fact, I mean, there are several connection with the work of what can and if you want it. This the frame shield in our formalism plays the role of this point or in the approach of welcome and the sense of freedom prize a complex structure and
Daniele Pranzetti: Fundamental degrees of freedom.
Daniele Pranzetti: It's, it would be the we, it would be important to understand how to recast back dynamics and terms of
Daniele Pranzetti: Bandwidth degrees of freedom. Yeah, the food in the food, the case. And that's, I think, what what it's been a mistake. Getting already
Daniele Pranzetti: Quite a bit. And then, yeah, maybe this new insights that we have now environments which are more somehow familiar to them to G framework could help to clarify how to study boundary dynamics in terms of this particle like parameter ization of
Daniele Pranzetti: Degrees of freedom, but
Daniele Pranzetti: I don't think we haven't really looked too much into that at the moment. Okay.
Abhay Ashtekar: But you're you're struggling in terms of
Abhay Ashtekar: I sold to our and as you do all that I think is all already there in the spin up a prayer framework of. Am I right about that.
Laurent Freidel: Future about decent well though but
Yeah.
Laurent Freidel: Anyway,
Abhay Ashtekar: Yeah, because I sell to our because of this.
Abhay Ashtekar: Freedom, having to do with it. So maybe we should answer.
Marc Geiller: And we lost him. Unfortunately, he's
Abhay Ashtekar: Okay, so I mean so to our was because there was this
Abhay Ashtekar: Freedom in rescaling the spinners.
Abhay Ashtekar: There was a it was not completely against you know it is not fixed. You could reschedule each of the spinners keeping to the normalization condition and so on, but okay.
Laurent Freidel: Maybe, maybe that's the boost component of the software which is
Abhay Ashtekar: Yeah, that is exactly the most important visitor.
Laurent Freidel: You know that I would that I would agree.
Laurent Freidel: I mean, you're in some sense, because you are poor cat Andrew, but we focused on the massive representations, because it's very hard work.
Laurent Freidel: But, but, of course, you can think about the mass last presentation. So, and, and, and as we know there's two masters limits right of like a presentation. One is the usual mastery summit.
Laurent Freidel: Which would correspond to go into another surface and the other one is like the continue speed limit which you which corresponds going to beta gamma equal zero. So there is again, there's a lot to be done in that direction. But, but now that we have a coven formulations of all cases.
Laurent Freidel: You know, it's clear that
Laurent Freidel: They shouldn't be a disconnection between the chemical energy isn't space likes her face faces.
Laurent Freidel: But it's not
Abhay Ashtekar: Yeah, I agree. It's very, very beautiful. Thank you.
lsmolin: You are going to introduce matter.
lsmolin: Matter show up in these boundary observable where if supposing you had some now radiation Maxwell field crossing the boundary which, where would you see that in the extension of these equation.
Daniele Pranzetti: So there would be recent sources for the momentum. I was referring to, because that is really them. It's really like that is the real punk or a
Daniele Pranzetti: Algebra, the momentum society from our system. So for instance, if you have a particle, you can now talk about particle momentum and the source these charges with Nike doing three dimension basically good
Laurent Freidel: Maybe
Laurent Freidel: Yeah clarification. Is that your ear. The newly talked about an elemental quick 100 which is the only associated geometry.
Laurent Freidel: And this one should not be confused with this particular like momentum. Right. Forget algebra and the relationship between the two is beautiful is that the elemental
Laurent Freidel: Momentum is associated with the frame shielded say but the particle momentum. If you look at the formula is energy to the differential of the frame.
Laurent Freidel: So, so in some sense what we're hoping for is that the physical moment the physical political momentum becomes from a deeper structure, you know, by, by taking differences. It's a moment of element and momentum between two ages.
lsmolin: Yes, it's supposed to correspond to something like a distortion or disturbance in the
Laurent Freidel: Change in your element or momentum. So
lsmolin: The defect, a defect as well.
Laurent Freidel: Yeah, yeah, the defect in the conservation. Yes.
Yes.
simone: Hello.
simone: Yes. Well, first of all I wanted to say that the fact that you have continue operator. The corresponds to
simone: The twist tangled sigh of the twisted geometries. I think it's really cool. And I'm looking forward to seeing the details in the papers you said are coming out soon. And then I had a curiosity about these elemental porn curry point of view, you're pushing
simone: For instance, and in their conclusions. The third point was, this is embedding in
simone: In the punk era group and then I remember one of the things that I looked at was a similar perspective, but we did embedding in
simone: The conformal group so forth to which is very natural may look like a just academic question, but it's very natural, from the point of view of the two story a parameter ization
simone: Of the start of a cell to see. And there are two, you can see that the simplicity constraints.
simone: Break. They can form a group down to Lauren to with one of the simplicity constraint being interpreted as the delegation operator.
simone: And I was wondering what one interest in Google form a group is that is semi simple and like they conquer a group. And I was wondering if these is
simone: A further enlargement that may be of some hypothetical using these bigger playground or whether you have reason to suspect that these elemental conquer is the end of the story. And there's no need to go larger
Daniele Pranzetti: I'm
Daniele Pranzetti: Free in order for lawrenson violence to be restored. This is enough now.
Daniele Pranzetti: We will know if this will be at me. We will probably. These are not all the edge modes that are necessary in the sense that we know for instance that we need Edge mode for for the different constraint to be included and
Daniele Pranzetti: If there is an interpretation on those lines of, I don't know, but it's
Daniele Pranzetti: Maybe once you try to really work out. They twisted john eviction more and they're trying to reconstruct their autonomy from this
Daniele Pranzetti: Body bolts, maybe it would be more clear if what you're suggesting, actually. Yeah, it can be implemented at the moment, maybe Lauren as their idea or no.
Laurent Freidel: Yeah, yeah. No, I agree with you. It's not. It is not the maximum maximally extended algebra. That's clear.
Laurent Freidel: Because we don't have to. We need an extension to get the como charge for instance that he hasn't talked about. But that's a very interesting suggestion Simoni so
Laurent Freidel: Yeah, we should look into, you know, whether that maximally extended algebra is the one that leads to Twister. For me, I mean that's an interesting conjecture.
Laurent Freidel: That would be amazing.
simone: If you want to get the coma algebra. Another comment is that you might want to look at these exact three for my the alpha does the job for you.
Laurent Freidel: Rarely, we have he has a slide about that.
simone: We have ongoing discussion about
Laurent Freidel: The need to control the next slide.
Daniele Pranzetti: Yeah, so just to
Secret
Daniele Pranzetti: Nights.
A week okay
Yeah.
Daniele Pranzetti: So yeah, you can
Daniele Pranzetti: You see basically the the coma charge can be attained by adding an extra term of this form here is this TED is so if you introduce the space time normal in your face base.
Daniele Pranzetti: Then you can you can show that basically if you add this to the one that we already have the boundary potential that we use so far that you can get the comet charge
Daniele Pranzetti: Which is an animal to there is a some exactly the same. Another way to write the potential that you wrote The, the, I think in Taiwan that we use plus this extra piece here that would give the boundary potential in fact is this original here that I worked on you.
Daniele Pranzetti: So introducing these extra edge months here, then these two notes spaces and normally that that would be the next step to
Daniele Pranzetti: Recovery.
Daniele Pranzetti: The common challenge.
simone: So you're listening to respect is that I should really think of these additional alpha, the way used to call it or if you want to call it, it is there as an Edge Mode effect.
Daniele Pranzetti: Yeah, I mean, it's
Daniele Pranzetti: That's the perspective that we're trying to
Get
Daniele Pranzetti: That, in fact, you can see that that charger. You try to recover recover your potential generates this as an extra piece of the boundary symmetry which is not really the same one that
Daniele Pranzetti: Is not a tangential metric is more like the normal metrics. So it's the metric in the normal plane to the surface, which in a sense, it's kind of dual to their search for that we have. It's, I suspect that there is a traditional and once we implementing PC constraint, but
Daniele Pranzetti: It's, it's, it's a very different piece of the algebra that you reveal
simone: So we pause it at the term.
simone: By hand but you're actually revealing a deeper understanding behind it. Basically, yeah.
Daniele Pranzetti: Yeah, it's, it's in any can be inverted it your way in this way as a sum of all these different terms then you convert it, you have a clear face be structure, you know, which competes with what you can compute brackets.
Abhay Ashtekar: But you did not because certainly here a little bit, because you're getting Hamiltonian to be the coma charged with a specific numerical factor, but the numerical factors are quite different for momentum and angular momentum for the energy and angular momentum.
simone: Presumably is calmer, just for the tangential
simone: Yeah yeah
simone: To get the additional I excited term.
Daniele Pranzetti: Yeah yeah yeah
Abhay Ashtekar: So, so these are just engine, that is to say just rotations.
Daniele Pranzetti: Yes, yes, yes, yes, yes. Okay.
Abhay Ashtekar: Okay, good.
Jorge Pullin: Other questions.
alejandroperez: Yeah, I have a question.
alejandroperez: At the end, you showed this quantum numbers associated to to your
alejandroperez: semen tree generators. And so, so which evokes you know quantum states in a healer space so are
alejandroperez: So you were calling everything continue until the very end. So, but I suppose when you talk about this quantum numbers you're thinking of
alejandroperez: The X on your boundary punctures as you as you mentioned at the beginning. Is that right, and if it's so mean. Do you, do you know already how to construct the hero space without of, I mean,
alejandroperez: You talked about these quantum states, but do you know what the hero spaces where do they live. I mean, the analog of the, you know, speed networking
Or do something to be
Daniele Pranzetti: What. So yeah, it's true. I mean, this would be the if you want this state side would be like edge lingo hyperspace in the quantum theory.
Daniele Pranzetti: Now the possibility to construct the hyperspace would be to use the twisted john three approach in a sense of reconstructing the autonomy of the door graph from the data of the twisted geometry.
Daniele Pranzetti: reconstructing the connection from this list angle the normals, the frames year associate today two faces.
Daniele Pranzetti: We haven't really looked into that yet. So this is the, the idea that one possible way to, we call it the healer space that you're
Daniele Pranzetti: The you're asking
Daniele Pranzetti: Because I mean there's no direct etc. See connection to associate with this, such as the algebra, if you if you just start from the content theory. So one possibility is to yet we use the trees, the geography philosophy to construct that
May be fighting effectiveness.
Laurent Freidel: At something about that. So yeah, this one on numbers are about. Now, do you think you have to appreciate is that all these are surprising to continue our local so that is there is proportionality of delta x, y.
Laurent Freidel: And you follow called algebra as a very singular to quantify in the continuum. Now the important thing here is that the full algebra contains a diff s
Laurent Freidel: Okay, which which showcase that the prime and create some charges. So anyway, so I think you know it's not done but but
Laurent Freidel: It is possible to go. I think beyond a loop cavity.
Laurent Freidel: Quantization if you include into the mix your presentation of the different offers but that's that's that's something that will come in the future.
Laurent Freidel: That anybody can think of. But that's a very important piece of the of the information. If you think about look at
Laurent Freidel: The different Sufism group is just treated cinematically whereas now we do have a generator a canonical generator that moves to point around and and and that changes a lot, the way we think about quantization.
Abhay Ashtekar: Thank you. So then also to Alex, this question is that decides at the moment of formal thing. You don't know what it is, is that correct
Laurent Freidel: They are doing the same size as the size of. Look at the size you would suppose you take your sphere and you discreet is Ethan You just say that only a finite number of cells are activated, then these other loop gravity. Right.
Abhay Ashtekar: No, but I mean I just said, we are in the continuum. So we should discuss anything with. So we just have some graphs and then the size of those those networks of
Abhay Ashtekar: And then I thought the answer to the next question was going to be that you've got this big networks that is the workspace and then you got this punches.
Abhay Ashtekar: Which is the sections of the spin networks to the boundary
Abhay Ashtekar: But the question is about interpreting the cell to see or something that a lot are saying
Laurent Freidel: Her ya know what you're saying is right. So one route is just to do the interview, je type of conversation where you
Abhay Ashtekar: Okay.
Laurent Freidel: Right. So that's one route and that's in that sense that up. Sighs What Sighs What sighs
Laurent Freidel: Space.
Laurent Freidel: That you would assigned to the edges of the spinner took, although the another spin the scenes of the spear more spins. Okay.
Laurent Freidel: So, and then everything we do in Spain low gravity, you can do there with the extended okay what I was saying is that there is something deeper lying beneath is that
Laurent Freidel: The loop gravity.
Laurent Freidel: Does not yet completely allowed to have a full representations of the form of isms acting on the boundary of the sphere, right, and taking that into account can provide regularization ski. I know what I'm saying sounds very theoretical, but believe me, this is this is a
Daniele Pranzetti: Me if you want a proxy for these intermediate step are this tubular structure that I was showing you before, where you can have a part of this defense to represented in terms of this, the formation of the circles.
Daniele Pranzetti: Around the puncture so it's not the full defense by it's you know it's a proxy for for defense which captures some parts of the of the symmetry algebra.
Daniele Pranzetti: Which is a generalization of if you understand speed networks where you extend their this the links to cubes.
Daniele Pranzetti: You can understand this new control numbers in terms of the formation of the circles around the punctures so that's a proxy towards what we're on
Daniele Pranzetti: Suggested
Laurent Freidel: In fact, it goes back to a very old problem that you know very well I buy a new gravity there is when you we know
Marc Geiller: We lost by I think he
Marc Geiller: It will. Yeah.
Laurent Freidel: Anyway, Christmas will not hold on, which is that when you go to the boundary, you know, a puncture space or presentation of gravity election simultaneous equations or what is the
Laurent Freidel: What is the statistics. So what is the action. How does the D from officer groupies represent represented by the exchange of on the exchange of these functions.
lsmolin: Now. But actually that's given by the level. Okay.
Laurent Freidel: No, we don't know. There's several iPod is like some papers, take them as distinguishable some papers takes they might indistinguishable the diversion of
Laurent Freidel: Which is the most accepted is that there are distinguishable either of these things spin and indistinguishable. They are the same speed. And there's a collections of different statistics.
lsmolin: I agree.
Laurent Freidel: But it means. It means that what we know is we know that if you permit the point. The point opinionated. But what phases appear is not known. It's pretty specific. And now these phases we have the possibility to derive them, not just randomly selected.
Laurent Freidel: And accounting depends crucially under absolutely crucial and you can get any number for the entropy, depending on what statistics you give these punctures
lsmolin: But this is another discussion, it's outside of Daniel is talk, but let's have it some time either because I've always suspected that there's only one right answer to this, and it's the one given by the onion.
lsmolin: And sharing, but
Laurent Freidel: Maybe that's not the one that is used to
Laurent Freidel: Shares most popular what most people use just
lsmolin: Just here classical stuff.
Laurent Freidel: Where they use a mix like friends do anyway. The, the most popular one is a mix, where there are sometimes distinguishable sometimes in distribution.
lsmolin: Yes.
lsmolin: The direction the evolution of the discussion.
Laurent Freidel: But if it is an open question. What is the spin of these functions know those statistics.
Jorge Pullin: Any other questions.
Jorge Pullin: Again,