Mehdi Assanioussi: Uh, thank you. Can you hear me?
Yes,
Jerzy Lewandowski: very good. Very well.
Mehdi Assanioussi: So there won't not be a lot of new stuff in my presentation. But uh, it's a subject uh the subject of our folk representation. Shadows say the subject I got uh interested in uh uh lately, and
Mehdi Assanioussi: um with you reck we started doing some work in this direction uh revisiting this one, all the work that has been done, trying to see whether we can push forward in this direction.
Mehdi Assanioussi: So naturally this uh goes in the context uh more related to um. The coupling of gravity with matter fields, as we know already, the classical level, gravity and matter fields are intertwined,
Mehdi Assanioussi: and also from a um. Our quantum theory experience. We know that uh matter feeds eventually would need uh quantum gravity, but at high energies, for instance. And the thing that we know also is that there are many open questions in this in this regard, and I just threw some uh questions here, and some
Mehdi Assanioussi: uh open uh,
Mehdi Assanioussi: but in the verse that I' to be answered and worked on,
Mehdi Assanioussi: and namely, could, for instance, quantum gravity also need a quantum matter field, and can quantum gravity actually restrict the matter. Contact at some point, or in certain regimes,
Mehdi Assanioussi: erez agmoni, and of course. Uh the question uh, that comes somewhat uh every time when one is talking about uh quantum gravity is emergent regime of uh quantum magic fees on fixed classical background, and that thing includes several um several points and elements. I noted here, for instance, the treatment of symmetries, and eventually the course graining of gravity coupled to matter. So all those questions, two hundred and fifty
Mehdi Assanioussi: have been present for a long time in in the work of the community, and there were
Mehdi Assanioussi: several, many uh directions followed and investigated, and uh, here I would chose to uh focus, particularly on uh,
Mehdi Assanioussi: well, on the some construction, but also States. Uh that we have the like that have been constructed in the framework of loop quantum gr the couple to matter fees, which I would briefly review
Mehdi Assanioussi: So before that uh, this in the canonic loop quantum gravity, we know how to basically quantize uh
Mehdi Assanioussi: all kinds of matter feeds so far present in the standard model, and also uh, but at least you find the kinematical uh framework, and then try to impose certain dynamics, whether it's canonical or covariant or
Mehdi Assanioussi: and other way you impose into dynamics.
Mehdi Assanioussi: So that was uh, more or less uh uh done and achieved. Then everybody was happy about that. And uh question uh that
Mehdi Assanioussi: erez agmoni still in a way uh about the result, is somewhat what are the physically relevant uh classes of States? Uh: in order to be able to understand, for instance, the semi-classical uh limit and the continuum limits of our theories, and possibly one hundred and fifty
Mehdi Assanioussi: uh get to the point where we can make actually predictions that can be verified.
Mehdi Assanioussi: So here I gathered uh just uh, more or less, it's a collection of uh
Mehdi Assanioussi: erez agmoni. There's many ah construction of kinematical states that have been present in the literature. But I focus particularly on those three, because they are somewhat despite that they came from different Ah, let's say, from different perspectives, but also motivations one hundred and fifty
Mehdi Assanioussi: Erez agmoni uh they have somehow carried certain similarities in particular. It was a a great focus, or say that there is a significant role played by the graphs in the loop framework. In these constructions one hundred and fifty
Mehdi Assanioussi: uh, we have the complexity file coherent state. I want to, I think, safely say that this one of the
Mehdi Assanioussi: Erez Agmoni states that have been used in the loop cotton gravity, framework, and heavily studied. They are based on this construction introduced by Thomas Semen, based on the complexifier, which has to satisfy a certain set of properties. One hundred and fifty
Mehdi Assanioussi: erez agmoni, and uh, usually uh, what we end up working with, and it was actually quite successful is one when when one restricts the complexity fire to be dependent only on the electric feed, or say the fluxes, for instance, in the context of one hundred and one
Mehdi Assanioussi: Erez agmoni gravity. And that's because it makes the, for instance, the access to the spectral decomposition of the complexifier easier as well as one hundred and eighty
Mehdi Assanioussi: several properties technical properties, such as evangelist and positivity, and so on, and so forth. And those properties are important in the construction, as they push forward to obtain this sorts of
Mehdi Assanioussi: Uh
Mehdi Assanioussi: operators which are identified with points in the face space in the classical faith space. And then from that those operators can be viewed as some sort of annihilation operators, and those coherent States are simply defined as being the eigenvectors one.
Mehdi Assanioussi: Uh,
Mehdi Assanioussi: Then the next thing was the I wanted to mention here is about those graph coherence states which were introduced rather recently, and they come from a specific motivation, which was this one
Mehdi Assanioussi: to answer or try actually to answer the question, What would happen if we want to construct some States
Mehdi Assanioussi: which behave nicely under the action of change uh graph changing operators.
Mehdi Assanioussi: So the graph change. Here was the notion that the that was important. And uh, there was, uh, as we know, when we introduced graph changing operators, the questions and the understanding uh properties of those operators becomes very complicated because one erez agmoni
Mehdi Assanioussi: erez agmoni uh the set of graphs is infinite and uncountable, and so on. And so forth. So one had the ideals there. Can we try to impose certain control over this graph change and and impose certain ah, nice behavior in this sense coherence, one hundred and fifty.
Mehdi Assanioussi: And the idea there was basically the introduction of uh, some sort of a canonical structure where you have some inhalation on oper operators and creation operators which has, which are associated directly to this graph change
Mehdi Assanioussi: Erez agmoni. And naturally this graph change has certain impacts. Because, for instance, If you are in an onability gauge theory, such as gravity, then you have also the role that is played by uh, the intertwiners or the spins, and so on and so forth, and that also need to be taken into account, which makes the construction more complex. Two hundred and fifty
Mehdi Assanioussi: erez agmoni. But at the end of the day. The depending on the graph change that one is interested in one is able in many cases to uh construct such a canonical structure. And then again, here, similarly to the complexifier case, one defines sort of coherent states that are eigenvectors of those inhalation operators. One
Mehdi Assanioussi: and last uh class of States that probably I will be focusing on the most here in this talk was this uh for Shadow States, and they are actually from this perfectly out of this three they are more or less the oldest to to be introduced,
Mehdi Assanioussi: and the motivation was entirely different. As I will go to. And here I just through the elements that led to a derivation. And there were many works about this around Twenty years ago
Mehdi Assanioussi: erez agmoni, and the idea was based uh motivation started with trying to find a connection between the fork representation and the loop representation one hundred and fifty,
Mehdi Assanioussi: and from there one is able to actually somewhat construct some mapping of the of the fox States
Mehdi Assanioussi: uh the to the the roof space, and then from there, uh obtaining some sort of states uh that in principle describe uh matter fees propagating on.
Mehdi Assanioussi: So here I drew on this slide, I through some examples, concrete examples of what I was talking about.
Mehdi Assanioussi: So in the first column this is the case of the complexify coherent state, and this is actually the most famous example in a sense where one takes the uh, usually the quadratic Casimir. If you want in the case of gravity, it's just the square of the area.
Mehdi Assanioussi: And then, uh using this complexifier, since the area is just depends on the fluxes. So there is no uh, no graph change present there. So one basically rates the question or the construction to graph by graph.
Mehdi Assanioussi: And uh, at the end of the day. The result is, basically uh, you obtain those coherent states as icon factors of some annihilation operator, and they look pretty much like some sort of Gaussian uh for each uh for each edge of the graph on over the group
Mehdi Assanioussi: erez Agmoni and and the second class of States. As I was mentioning this uh tries to focus on the notion of the graph change, and he, I gave the example of, for instance, the well, it's basically the simplest example One
Mehdi Assanioussi: Um.
Mehdi Assanioussi: And here, as I was, I said earlier, one things about uh introducing some sort of canonical uh structure, which is very simple Uh, one can think of it as a borderline mimicking uh one
Mehdi Assanioussi: harmonic oscillator. In some cases, in the simplest cases. And basically uh, as you can see here. Uh, you can take, for instance, the trace of the Holonomy that is this uh, associated to this loop
Mehdi Assanioussi: Erez agmoni, and there make uh, basically the identification with some operators that uh, uh, basically either create or uh annihilate uh such. Let's say, a graphic cetacean in a sense, Can you? Can you explain a few things? I mean two hundred and fifty.
Abhay Vasant Ashtekar: What is the subscription subscription? Gamma? Stand for just a loop, or what? Yeah, Just the loop, just the loop in this case, just the loop. So this is a you should think of this Uh, in the case of uh an ability engage here. So think about it as you one. So you don't have any to run your structure.
Mehdi Assanioussi: Ah, that's why I put it here because it was one of the simplest case to treat. When you go to not have been engaged here, there are more degrees of freedom to take into account one hundred and fifty
Mehdi Assanioussi: and uh yes, and that's something that one also can uh can do It's just that. The the the instruction is much more complex. And you Don't, have, for instance, here I should just only one set, but actually and usually you have a a collection or set
Abhay Vasant Ashtekar: of uh of such canonical pair of operators. You don't have only one. Uh but so. But the thing is that this. So this level gamma refers to the loop that we had in that is correct. But the original state, gamma, that is our gamma plan that that we had to which you had with the loop in arbitrary. So what happened if I had?
Mehdi Assanioussi: Yeah, because here I I suppressed the I suppressed a lot of the heavy notation, actually. Uh. But the truth is, it goes a bit similarly to one one, the to what one does uh with the constraint operators,
Mehdi Assanioussi: namely, uh, say, for instance, I consider just a graph with one vertex, as it says shown here. Right So here, for instance, one of the ways to do it is to consider uh, to associate a pair of canonical operators for each wedge at the vertex.
Mehdi Assanioussi: So each pair of edges basically uh you you associate somewhat uh in this opinion case, you can think of it as basically associated in a space of a harmonic oscillator basically one.
Mehdi Assanioussi: And you have this inhalation creation operators which create those loops for an inhale those loops for associated to this specific pair of wedge.
Mehdi Assanioussi: Uh,
Abhay Vasant Ashtekar: and i'm still confused, because I mean gamma is supposed to be an accident that is given to me how and here the yeah. So the gamma is an abstract graph. I mean, if you are talking about the State, Yes,
Mehdi Assanioussi: and here this this is basically you give me. You give me a graph Uh: I would start with a graph which doesn't have any loop structure.
Mehdi Assanioussi: And then I I built uh, I build this uh canonical pair of operators by choosing, selecting a set of operators that are defined in terms of. For instance, here I give you the simplest example, which is a holonomy associated to the loop.
Mehdi Assanioussi: So if if you think about the constraint operator that we have one of the regularizations that each time we add, uh, we add a closed loop at the vertex. However, we don't add it arbitrarily.
Mehdi Assanioussi: There are. This loop has a prescription,
Mehdi Assanioussi: and this prescription is encoded also in the definition of those uh canonical pairs,
Mehdi Assanioussi: and that's how you So in a sense, this this this loop you could say. It's defined. With respect to the graph you are given me, it's defined as a very specific sense, so that basically in the same, and it's a very similar way to what we do for the constraint operator. One
Mehdi Assanioussi: so is, Yeah. And this is actually very important, because many questions arise one one. And through this such canonical structure, namely, uh, what we mean by uh distinguishability. And one of the things is that uh, here what? No no matter how many loops you create, they are the few more quickly equivalent, for instance.
Mehdi Assanioussi: So this is notion how we introduce notion of distinguishability. And uh, yes, and one has to show that actually this, this canonical structure actually exist, and everything is well defined in order to be able to uh at the end of the day.
Mehdi Assanioussi: Just take this annihilation operator and define the coherent state as the eigenvectors basically
Mehdi Assanioussi: So it's not. It's not that the action of this operator on the State. It's a it's a It's a another operator Also, in terms of this inhalation case. It's actually just. Uh you can write in terms of the number of oper operator. It's just number operator shifted, and then you take the inverse shift it so that actually doesn't have any kernel,
Mehdi Assanioussi: and then you you take the inverse and the square root.
Abhay Vasant Ashtekar: I think we should go on because I don't understand the notation. So I should.
Mehdi Assanioussi: Yeah, it's it. But I try to minimize the the the notation here, because actually, if I want to go to the dates will be much more heavy. But it was not my the point of focus uh, of my talk. So that's why I didn't give it a lot of details here. But,
Abhay Vasant Ashtekar: associate, I give you a grab, gamma, and associated with that. Yeah, I got two operators called A and a dagger. Is that correct?
Mehdi Assanioussi: Uh, if you give me a graph, gamma uh depending on the graph. So for this I want an interesting loop. No, I would. I would associate actually uh uh a pair of operators of canonic operators to each wedge of the graph.
Mehdi Assanioussi: That's how I would do it in this case.
Mehdi Assanioussi: Yeah. So you will have actually depending on the graph depending on how many verses are there, and how many edges are there. I'll get to some finite set of such operators
Mehdi Assanioussi: in the case of the Abelian case. Uh they have been engaged here because there are no intertwiners, for instance, then the attachment of the loop is much simpler, and I don't have any problems with the commutations among those canonical structures, so they will all again commute. So when I have an eigenvector
Mehdi Assanioussi: four, one uh for all of them, it's basically the eigenvector for the whole corresponding to the whole graph would be just the tensor product uh among those eigenvectors. But in the non ability gay sticks get more complicated and one has to be careful. That's Why, there were modifications that uh, that followed from this. But yes, it's uh.
Mehdi Assanioussi: So yeah, it's a lot of details that I'm: on meeting here. Yeah,
Mehdi Assanioussi: Okay. And uh. So yeah, basically probably I I've talked a lot about this. So it's just that at the end of the day, once uh one obtain this uh canonic structure. One defines the coherent States as the eigenvectors,
Mehdi Assanioussi: and uh, and then, of course, they will satisfy uh all the properties with with respect to this operators, and that was nice, because
Mehdi Assanioussi: in the context of the graph change in operator, and then one can take the part that is actually responsible for the graph change, and one can actually see that uh, one can have some coherent properties uh for this elements. So uh,
Mehdi Assanioussi: but this is this this: this: this is rather recent. So there wasn't a lot of progress in this direction when it comes to the semi classical studies, and so on. But uh,
Mehdi Assanioussi: erez agmoni. Yeah. And the third part uh both uh those of our folks shadow States. And here the example that we give here, and which i'll discuss in more detail in a moment. One hundred and one
Mehdi Assanioussi: Erez agmoni uh was to uh introduce. This algebra of smear Autonomy is an electric field which is basically the smeared hunonomy is just a holonomy uh, where you take the the the connection, and you smear it one hundred and one.
Mehdi Assanioussi: And then
Mehdi Assanioussi: once you have this algebra. Uh, thanks to the fact uh, basically uh, maybe first step would be that the standard for quantization of the eligible of the connection and the electric field provides also uh sort of a representation of this algebra. Sm: me and an electric field,
Mehdi Assanioussi: and then uh, there. The The next point is that the presence of the existence of this uh correspondence, which actually it's an isomorphism uh, between the of Smear and Holonomy, the electric field, and actually the
Mehdi Assanioussi: and the algebra of the Holonomy and the smeared electric field, which is the one that we usually use in the loop quantization, And this is of often is what plays a role in the in providing thanks. Uh, uh, thanks to this for quantization and a new representation for the Holonomy.
Mehdi Assanioussi: And the next thing was just to basically once you have uh this representation. Uh,
Mehdi Assanioussi: you can basically you take basically the the eigenvectors of the annotation uh operators in the focal presentation. You try. Well, first you write what the the equation in terms of the of the new Uh operators, which are those uh holonomies.
Mehdi Assanioussi: And then that's how you try to solve this equation
Mehdi Assanioussi: in the space of Uh C star, which is the dual. So space of symmetrical functions. And that basically gives you the correspondence, or let's say uh it gives you to produce a certain States that I will be talking about in a moment, and that project well, the States turn out because they are in your style, but they are actually distribution and not normalizable, and so on and so forth.
Mehdi Assanioussi: Um,
Mehdi Assanioussi: one is naturally interested in the uh sort of projection into the kinematical space. Because then, if you obtain a normalizable states, then you can uh proceed to a semic classical analysis, and so on.
Mehdi Assanioussi: Okay, So uh,
Mehdi Assanioussi: no. This red line. But okay, uh, next thing is um.
Mehdi Assanioussi: Well, the question that is, uh, always present in the context of quantum gravity theory in particular, in the loop quantum gravity
Mehdi Assanioussi: framework is how to connect the low energy physics uh to the, to the our theory
Mehdi Assanioussi: and uh particular, there was this question. That was rates long ago, which is how to relate the for quantization of matter, fields and the loop quantization.
Mehdi Assanioussi: And uh, that's uh when Madavan I was first to propose uh this approach uh, which is was later going to the out for representations, and it was in the context of you on uh gauge theory
Mehdi Assanioussi: Uh, the original motivation was actually to understand the relation between quantum states of linearized gravity and states in loop quantum gravity. Then you ask gravity. In that case, basically the gauge theory becomes a. You want three page theory for it's a billion. And ah, one can proceed uh,
Mehdi Assanioussi: also with one can perceive one. You have a fixed background, which is, uh Minkowski, and we can proceed with the fog quantization. The question, What is the relation between the two,
Mehdi Assanioussi: and what one ends up with is that to sort of presentations are basically representations for uh, this matter field propagating with coffee space done, and it connects basically the standard for representation with the background, the dependent one.
Mehdi Assanioussi: Uh, Later on it was given a certain interpretation, because uh, this, uh, there is this. It's called this outfoot representation, because there is this: our parameters sitting there,
Mehdi Assanioussi: uh which i'll mention later. But basically uh, this to a stated that this our parameter actually is some sort of can provide some sort of scale.
Mehdi Assanioussi: And this scale actually basically measurements at a given scale, you can always find
Mehdi Assanioussi: range for this parameters such that, uh your measurement uh in the that you would obtain from a theory with the fork representation would be indistinguishable from a measurement that you would make in this uh off of representation.
Mehdi Assanioussi: And the role was basically to sum it up in a nutshell was to provide new measures for uh, for the loop States, because uh, one is able basically to construct a measure Uh, thanks to that. And uh, the second thing which I find actually more important
Mehdi Assanioussi: in here, and that's what, in a sense, motivated us is to construct to have this mapping between the Fox States
Mehdi Assanioussi: and uh and the loop space, and uh one can in principle it at least for the I'll be engaged to this, as I will show uh one, is able to actually do the produce this mapping, and can have uh sort of a presentation of the fox States in the in the loop space.
Mehdi Assanioussi: So how does it work. Uh: in the Abelian case basically you start from the algebra that we have, which is the connection and the electric field.
Mehdi Assanioussi: And then you define this algebra that is, on the left
Mehdi Assanioussi: of the smeared hol on the electric field, and as I was mentioning, the first step is to smear the connection to
Mehdi Assanioussi: uh with some uh, it's actually Schwarz function which satisfy this property here on top corner.
Mehdi Assanioussi: And basically one has to keep in mind that actually you start with a connection which is a uh temper distribution that you end up with some which is
Mehdi Assanioussi: with an object which is also a connection. It's just that it's actually at the intersection of the spaces of um temper distributions. And
Mehdi Assanioussi: let's say, discontinuous connections, the ones actually the configuration space that we use in the quantum gravity for which the Holonomy is actually well-defined, and that's why we end up with this phenomenon that we call the smear and holonomy because it's an origin, and it comes from a smeared connection.
Mehdi Assanioussi: And this R. Parameter that I was mentioning earlier is basically this parameter that controls some sort of a peak in this. So to make it easy, one can imagine simply a Gaussian
Mehdi Assanioussi: Uh. And this on is basically that controls uh, the parameter which controls the width. And that was actually part of the original formulation uh of this Um, yes. And on the other side, on the right side, we have uh this standard uh
Mehdi Assanioussi: algebra, the loop algebra. I feel the standard autonomy and
Mehdi Assanioussi: the smeared electric field. Here the smeared electric field is
Mehdi Assanioussi: over three dimensions, but it still works. That's one of the miracles of the Avenue gauge theory
Mehdi Assanioussi: and uh, the observation. The key observation in this case was this: the the isomorphism between those two and the one on the right and decided for motion basically leads to the statement that
Mehdi Assanioussi: the folk representation, or any representation, for that matter of one, provides a representation for the other,
Mehdi Assanioussi: and in particular, the focal representation of this algebra on the left side, which is induced from the standard for representation of a ae gives uh induces this out for clip that we call the Alpha representation for the holonomy flux.
Mehdi Assanioussi: But that was basically, the core ideas in this, in this construction,
Mehdi Assanioussi: and it goes. That's the details. Basically. Go Follow that
Mehdi Assanioussi: at some point you can consider. Uh, you consider the uh, the fog vacuum expectation values of the smeared Holonomy operators,
Mehdi Assanioussi: and you identify that with the expectation value of the standard holiday operators in some
Mehdi Assanioussi: new vacuum.
Mehdi Assanioussi: This is what we call the out for vacuum. And here this can be calculated explicitly. And here I was given the result. And to this uh, this uh function X tilde is basically just uh uh, the Fourier transform
Mehdi Assanioussi: of Function X that I'm giving you here, which is basically just the integral of this meeting function along a part that is associated to the Holonomy here.
Mehdi Assanioussi: And basically, and I put that the out for representation is just cyclic representation generated by this new vacuum.
Mehdi Assanioussi: One can carry on, of course, with the calculations and figure out actually what's the new measure,
Mehdi Assanioussi: and even get the relation between this new measure that we call the outfoot measure, because it's defined also on the same uh space and the Ashika Lewandowski measure.
Mehdi Assanioussi: And this is here in the case of the opinion what you one gauge, cheering it just to explain the notation. So those ends are basically what you would call the spin network in the context of gravity. But this here is there you want. So gamma, or the graphs and and are the colors.
Mehdi Assanioussi: And uh, yeah, And we can see here, this is basically uh, it's a sum over all graphs and all colors, and so on and so forth. So you what you get is basically uh, if you have states which are normalizable with respect to one measure they are not normalized, but with respect to the other, at least not necessarily.
Abhay Vasant Ashtekar: And this factor can you just tell us a little few things here. First of all, shop is our fork representation. You literally equivalent to the standard for representation.
Mehdi Assanioussi: No, no, no. Well, it's. Uh if you want it's equivalent.
Mehdi Assanioussi: I just wanted to make sure. No, it's not really equivalent. I mean, there is an equivalence here, of course, between these two here, because of the isomorphism, but it's not really equivalent to the standard for representation. No, the answer would be, No, because
Mehdi Assanioussi: it's it's different. It's just different.
Mehdi Assanioussi: One can think of the folk representation as some sort of singular limiting case.
Mehdi Assanioussi: That's how I would say it.
Abhay Vasant Ashtekar: I mean, is it you on the
Abhay Vasant Ashtekar: I the way you're presenting it. I'm a little confused, but
Abhay Vasant Ashtekar: it looks like there are two different algebra right?
Abhay Vasant Ashtekar: And then the next slide
Mehdi Assanioussi: uh next. Okay,
You know the top equation on the left-hand side. You've got your
Abhay Vasant Ashtekar: on the right hand side. You I got.
Mehdi Assanioussi: Yes,
Abhay Vasant Ashtekar: so
Abhay Vasant Ashtekar: you are defining.
Abhay Vasant Ashtekar: So the The algebra themselves are quite different. Right. So the question of territory equivalent doesn't even arise that right?
Mehdi Assanioussi: Uh well, the algebra, if you refer to the algebra of the smith, hold on isn't. There always they are isomorphic
Mehdi Assanioussi: in the billion kids.
Abhay Vasant Ashtekar: Um. But the question is whether this as a mark is Um,
Abhay Vasant Ashtekar: um.
Mehdi Assanioussi: So that's the market. Okay, so that you think of the our Fork representation, you'd also use the representation all for all. Exactly. Exactly. Exactly. Yes. So the only question. Yeah. So the only question, Yeah. So the only question that arises is when you compare the outlook representation to the foc quantization of the
Abhay Vasant Ashtekar: No, you always be right. Connection is not there.
Abhay Vasant Ashtekar: So she always Yeah. So what do you mean by most meeting?
Mehdi Assanioussi: No, I'm just talking about just
Mehdi Assanioussi: you want. You can define. Of course you you would need to smear. But if you want to, you don't have to.
Mehdi Assanioussi: If uh to define the fork representation, you don't have to to go through the Holonomy or smear the Holonomy, for that matter.
Abhay Vasant Ashtekar: But it is it not? Can I not do that?
Mehdi Assanioussi: I'm sorry
Abhay Vasant Ashtekar: it is not equivalent if I just consider you to hold on on this
Abhay Vasant Ashtekar: a presentation. Is it Isn't, that representation the same as the standard representation? It is what you carry that sort of presentation to the smear. Hold on so. Yes.
Mehdi Assanioussi: So you the the standard for quantization it provides you. So you have a folk representation that
Abhay Vasant Ashtekar: yeah, there is, there is a single fork representation with both. We are
connections. Is your home exact?
Abhay Vasant Ashtekar: But now, in the or for representation
Abhay Vasant Ashtekar: only I
Mehdi Assanioussi: Yes,
Abhay Vasant Ashtekar: so
Abhay Vasant Ashtekar: I mean, In what sense are you saying that? Can you even compare the two representations? This is my question, because I got on the one hand,
Abhay Vasant Ashtekar: and i'll give up all of the wrong On this, On the other hand, I got algebra of I believe that all but they're still distinct from each other. Right?
Mehdi Assanioussi: Yes, but you can. Okay, If you want to compare, then uh, you may. What you mean by that is comparing the sense of, for instance, uh measurement of observables.
Abhay Vasant Ashtekar: Yeah, I'm talking about
Abhay Vasant Ashtekar: mathematical physics at the moment. I'm not. Don't. All get into issues about measurement theory.
Abhay Vasant Ashtekar: So I just can't complete any one.
Abhay Vasant Ashtekar: Uh. So.
Mehdi Assanioussi: Yep.
Abhay Vasant Ashtekar: On the one hand, you have one algebra,
Abhay Vasant Ashtekar: namely, of. On the other hand, you've got a number of
Abhay Vasant Ashtekar: the two algebra are kind of distinct
Abhay Vasant Ashtekar: Yes, and all the default representations near. How long is
Abhay Vasant Ashtekar: they define expertise in values, but unsmared along this do not.
Mehdi Assanioussi: No,
Abhay Vasant Ashtekar: so one algebra is is represented the other you right not representing,
Abhay Vasant Ashtekar: but on the last transparency you are saying that the two are I to
Mehdi Assanioussi: uh, and therefore something. And and that's what I do. Yes, no, I said, the I said. They are isomorphic, which means that a a representation of one, a Java, should provide you a representation of the other.
Abhay Vasant Ashtekar: Okay,
Mehdi Assanioussi: just by using the isomorphism between the two algebra.
Abhay Vasant Ashtekar: Okay,
Mehdi Assanioussi: A non-focked representation of the smeared, homogeneous
Mehdi Assanioussi: you're holding algebra that our fault representation, which is not that
Abhay Vasant Ashtekar: standard for representation? Right?
Mehdi Assanioussi: No, it is not the standard for it. It's just derived from it.
Abhay Vasant Ashtekar: So what does is the Iso?
Mehdi Assanioussi: It's exactly that is, that investigator allows you to derive a new representation for the ordinary holonomy.
Abhay Vasant Ashtekar: So I got a smear algebra, and I got an unsmail algebra, and that just distinct algebra, and the order of representation is for smeared, and you are constructing.
Abhay Vasant Ashtekar: That's correct.
Abhay Vasant Ashtekar: The statement is that
Abhay Vasant Ashtekar: that just different algebra Is that right? Or
Mehdi Assanioussi: well? Um.
Abhay Vasant Ashtekar: Why, why don't you go on? I don't just talk to you because I just because uh,
Mehdi Assanioussi: because what the way I see it is that actually you have those uh this, this new representation for the the ordinary Holonomy right? And this representation actually uh, it's.
Mehdi Assanioussi: If on in principle, everything that you that you should be able to do on the other side, and this folk representation of the smeared hold on, you should be able to do it. And actually they should agree
Mehdi Assanioussi: because of this is more than they should agree. You just need to know how to translate one,
Mehdi Assanioussi: translate basically, one into the other,
Mehdi Assanioussi: and that's what this is. A zoom thing does.
Abhay Vasant Ashtekar: What is it? I some option between.
Mehdi Assanioussi: So first it's the it's between the algebra.
Mehdi Assanioussi: Yes, but then it provides you with the It's basically it gives you the relation between. If you do something on one side, in the, in, the, in, the in the quantum theory.
Mehdi Assanioussi: If you do want to on one side, you should be able to recover it on the other side.
Mehdi Assanioussi: Yeah. So I was thinking about uh talking about. It was uh this relation between the measures, and what I was saying is that, uh, basically one can construct this relation explicitly
Mehdi Assanioussi: uh to the to this new measure and the uh. This. This relation is defined by this uh, by this meeting function basically, which is sitting inside this uh, G
Mehdi Assanioussi: and uh, she is basically uh some sort of uh,
Mehdi Assanioussi: the function which basically couples the edges of the graph
Mehdi Assanioussi: Um.
Mehdi Assanioussi: And from there, as I was mentioned in earlier. If you have, uh, you can define on the one side basically and the busy. That's actually one of the things here related to a vice question. Which is it I define, uh,
Mehdi Assanioussi: say, my coherent states in the folk representation as being eigenvectors of my inhalation operator. So I translate this equation in terms of one
Mehdi Assanioussi: uh
Mehdi Assanioussi: in terms of the harmonomies and the and the electric field.
Mehdi Assanioussi: And I try to solve this in the in this and space you start, and the result gives you something which is uh a non-normalizable state, which is defined uh over all the graphs and all the color in, and you have just some uh,
Mehdi Assanioussi: some factor which, of course, depends on
Mehdi Assanioussi: it, depends on this on this function. G
Mehdi Assanioussi: and uh,
Mehdi Assanioussi: and yes, and basically the states this what we call the let's say I call them here the out for States. So, as I was mentioned, they are in a mental system or normalizable. But what is for sure certain is that they in code mean costs and geometry because they come from that.
Mehdi Assanioussi: So the way they were defined, for instance, as when we say that they are eigenvectors of the annihilation operator. We are talking about, uh poncari, a symmetry, for instance, for the vacuum on my impose that the the vacuum is in he latent uh by uh, by all inhalation operator.
Mehdi Assanioussi: Uh.
Mehdi Assanioussi: And this is, and this this geometry is exactly encoded in those coefficients which are basically this X and G.
Mehdi Assanioussi: Those those functions actually that that that that define this uh, those States. And naturally, as you can see, There is a non local spatial correlation in this in this coefficients, because
Mehdi Assanioussi: here, when you look at it, there is some over all edges of a graph. If you look at one graph, you have some of all the edges of the graph, and then you have some factor which depends on both edges, independently of whether, uh they are meeting at the vertex or not.
Mehdi Assanioussi: It's basically entirely on local over the graph. So it couples uh in a specific way. Of course it's couples the all the edges of uh of the graph on which the state is defined,
Mehdi Assanioussi: and this was an interesting property, because it didn't it wasn't uh it
Mehdi Assanioussi: erez agmoni. It was, basically I think, so far uh the only state uh I know where this there is. Such correlation is actually present and defined in a specific way, One hundred
Mehdi Assanioussi: Um. The problem, of course, here is because they are not normalizable. There is not much to do if you want to to use the tools from the loop quantum gravity side when you have the homony and the like and the the fluxes. So you need somewhat to to, to
Mehdi Assanioussi: to make, to make at least, or to to break this state in some controllable pieces pieces that you one can study separately.
Mehdi Assanioussi: Uh, and that's Basically, what the notion of shadow States comes in. Um. Basically, one has to take the this. This
Mehdi Assanioussi: fox states art folk states, and somewhat project them on some ah so separable sub Hilbert space so parable. I think he uses a parable, because, in fact, in the original Ah, the original work! It was defined just for a fixed graph
Mehdi Assanioussi: erez agmoni. And in this case it works as shown here in this equation, but in principle one can do that, even if it's not a fixed graph, it's just one has to have a control over those over those graphs in such a way that one
Mehdi Assanioussi: subsets of glass and then try to see uh to study uh the the shadow States in this context, and that already gives you some information about the original state, and then the idea is to infer from this that all information or properties of the Fox States. Uh
Mehdi Assanioussi: I'm: Basically, to yeah, that's that's basically the idea.
Mehdi Assanioussi: And that was the what was developed in the context of a video gauge theory.
Mehdi Assanioussi: And then, of course, that systematically brings the question: Can we go beyond
Mehdi Assanioussi: the the thing about going beyond in this very system like it was done in the case of having theories that many things don't work, and many miracles actually do not occur.
Mehdi Assanioussi: Uh,
Mehdi Assanioussi: for this is there is no I zoom office between the algebra, even classically. So. There is no such. Uh I zoom office. There is no such a relation. Um! But there are other details that are actually uh
Mehdi Assanioussi: relevant, and they they do not. They are not present in the context of uh, none of any engage theory.
Mehdi Assanioussi: So one of the proposals that that appeared in the literature was to actually take the expression of those Shadow States that one obtains in the in the Abelian case one
Mehdi Assanioussi: since one can see that basically here what I marked it right at the beginning were the colors.
Mehdi Assanioussi: So if one go goes uh try to write standards of bread, and naturally one is electorate just basically that it's the electric field operator acting on the on the color state, producing basically just the color here in the I believe in case it's simple, because uh, basically um,
Mehdi Assanioussi: it's diagonal. And the electricity is the age invariant. So you what you get is it's just the color,
Mehdi Assanioussi: and once you do have this expression in terms of the operators, you just replace the operators of the obedient theory with the corresponding ones in the non-obin in theory.
Mehdi Assanioussi: But you can, for instance, here the case of su two, and one can write it as such with taking the generators basically of the algebra.
Mehdi Assanioussi: And the thing what happens that once you run down this expression, you can notice that the the State that one produces not gauge invariant, but one, of course, can solve that by saying that one can
Mehdi Assanioussi: and proceed with a group of averaging and taking out the long as you've any part. But it's not clear whether the resulting state would be still
Mehdi Assanioussi: in a in a sense, because here, because of this operator expression. One doesn't exactly have the relation with respect to some fox state, yet
Mehdi Assanioussi: erez agmoni. But and when one does, the the average in one eliminates some components, so it's not clear whether it would be relevant. But in general, actually, those States have not been studied Ah! Extensively. So actually, The answer is just: We don't know what happens. One hundred and fifty
Mehdi Assanioussi: um
Mehdi Assanioussi: erez agmoni. Next thing is the this actually relates back to this to the States. I mentioned earlier the complexifier, coherent state, and it was shown that in the Abelian case, actually those Shadow States can be cast in the form of complexifier, coherent state, one hundred and fifty.
Mehdi Assanioussi: The complexifier. There is rather simple. It's not difficult. It's more complicated than the one uh used in general. That is the area square. Uh, but uh, it's not very difficult either. However.
Mehdi Assanioussi: Um
Mehdi Assanioussi: one of them, for this is gauge invariance, the other promised to reduce consistency and
Mehdi Assanioussi: other properties that actually a complexifier should satisfy. But in this case, at least, the generalizations that were considered so far did not fail to
Mehdi Assanioussi: to contain all the necessary properties.
Mehdi Assanioussi: Uh, another thing was done which is the the construction of Africa representation for Scalar Field,
Mehdi Assanioussi: and in this case it was
Mehdi Assanioussi: erez Agmoni, in a way similar to the to the Abbey in case there was no problem. Two others, other points which are rather more recent. One of them is that we are still working on it. It's that for the for the Fermions one,
Mehdi Assanioussi: and that also we don't expect a lot of departure from the construction of this in the case of the scalar field. But we we try to do first, was to look at the if you engaged theory. And here, of course, as I was mentioning, there are many things that break down,
Mehdi Assanioussi: and it's much more complicated, and that's I think, what I will be talking on talking about. Uh for the rest of the talk. So
Mehdi Assanioussi: if one one goes back to uh to the U, one case on notices that actually basically in this equation, this new measure.
Mehdi Assanioussi: Ah is defined just as the expectation value of the sphere. H. Anonomy is in the vacuum state, the foc vacuum state, and naturally because we are in diabetes here, it is generalizes to any cylindrical function.
Mehdi Assanioussi: This is basically just a consequence of bundle, some identities, if you want,
Mehdi Assanioussi: uh,
Mehdi Assanioussi: and that actually this this observation, one can say
Mehdi Assanioussi: that if one is to
Mehdi Assanioussi: uh consider construction of a measure. For as you engaged theory, then natural generalization would be to uh follow uh this uh mundle stem identities for this un, and consider defining this measure as an expectation. Values of some one
Mehdi Assanioussi: uh, say, with the loop operators
Mehdi Assanioussi: in uh the vacuum they interacted to.
Mehdi Assanioussi: Now, the idea Here,
Mehdi Assanioussi: let's let's say that the original motivation and our interest is to eventually get to somehow a way to construct
Mehdi Assanioussi: uh shadow States for no not being engaged to the laser, for that is that we are interested in studying properties of such state that may uh somehow encode uh properties of uh fox States. That is matter, if it's propagating, I mean, it costs space. Now,
Mehdi Assanioussi: uh, however,
Mehdi Assanioussi: before getting there it's actually It's that would be basically the end of the road in some sense, to get to those States. So we thought we were less ambitious. And we said, we asked ourselves, Is it even possible to define a measure in this in the context of Not a bit English tears,
Mehdi Assanioussi: and naturally it's also a bit ambitious, because if we follow this way of calculating this expectation value, then basically we can't,
Mehdi Assanioussi: and so be cheating
Mehdi Assanioussi: uh where we will look at not the expectation value that is on the left side. But we will define some expectation value some new operators uh that are noted on the right side. And basically
Mehdi Assanioussi: um,
Mehdi Assanioussi: one could think about it basically as some sort of U one to n gauge theory. What we ignore basically interactions from this U. N case one
Mehdi Assanioussi: in addition. Uh, but we keep this uh, we define those operators somehow uh, as some sort of quantization of some uh su and group element.
Mehdi Assanioussi: So basically if you compare to a a you one end gauge theory, the difference lines in this uh term that I circled here in blue, which is basically a trace of a product of generators uh of the Su engaged group.
Mehdi Assanioussi: If you had the you one to end gauge theory that you have just uh one basically Um. But instead, here we put in the generators of as you engaged theory, and this expansion of a with the loop,
Mehdi Assanioussi: and what we try to do is basically we say, Okay, this is our Pullman's version. Can we define a measure
Mehdi Assanioussi: uh using this expectation value basically in this? In this uh simple way, and the answer of that Well, it's a so simple one.
Mehdi Assanioussi: We didn't know whether it actually even would work. But it turns out that actually does.
Mehdi Assanioussi: So one starts by defining some Uh linear functional which is basically given in terms of those expectation values of this operators. And this uh
Mehdi Assanioussi: vacuum of the free theory.
Mehdi Assanioussi: And basically this this, this linear functional acts on basically it's algebra,
Mehdi Assanioussi: uh, which I can write in terms of actually with the loops,
Mehdi Assanioussi: and define that the the result would be related to the expectation value of the corresponding operators in the free vacuum here. Um!
Mehdi Assanioussi: What I wanted to mention. Yes, so the product here is also a consequence of Mundell's time identities in the sense that I can write every um
Mehdi Assanioussi: cylindrical function as an product of N. Minus one, a linear combination of products of N minus one with the groups.
Mehdi Assanioussi: Once we have this functional, then to get to the measure, there are certain elements to be shown. First, one is actually the definiteness. Is this well defined? On the whole, algebra? H. A. And that goes by showing that it converges. Uh, in general, it converges that this expansion,
Mehdi Assanioussi: which is here, basically which we always use to compute to this expectation value on the right hand side always converges. So that was the first thing,
Mehdi Assanioussi: and that actually can be shown.
Mehdi Assanioussi: The positivity of this uh, uh of the Selena functional can be somewhat shown by using Mandel some identities for this, what we call smeared with the loop operators. This one has to show, first, that those properties are satisfied by those operators,
Mehdi Assanioussi: and then proceed with the with the calculation. And one can show that actually, that also holds. And finally, the last point for the existence of the measure is to prove this continuity with respect to the C star known
Mehdi Assanioussi: the algebra. But in this case it actually doesn't differ much
Mehdi Assanioussi: from what happens in Diabetes case. And uh, it's basically relies on the fact that this meeting, which is core and the construction of the Africa representation for that video engage theory actually is, it's quite powerful. And it's role is basically this: that it maps the
Mehdi Assanioussi: uh temperature distribution to the space uh of uh,
Mehdi Assanioussi: discontinuous connections.
Mehdi Assanioussi: Once we have that, then basically uh, the measure can be defined, and once one finds is that you can write to the relation between the two that is, between the Ashika Levinowski measure and this sort of new measure defined via this linear functional that we introduced.
Mehdi Assanioussi: And basically uh, it can be expressed in this way, those are basically sort of the equivalent of speed networks on this side. And this is the image of uh of this uh cylindrical function.
Mehdi Assanioussi: Um,
Mehdi Assanioussi: uh
Mehdi Assanioussi: via this, this linear uh linear map,
Mehdi Assanioussi: and of course, because we decided to go through, and that was the pretty much as I was mentioned in terms of this generalization. V. Among the some identities we had to go through this, whilst the loops one
Mehdi Assanioussi: when one uses a big winner, one is already working on the gauge invariance level.
Mehdi Assanioussi: So, in a sense, one obtain something uh
Mehdi Assanioussi: a measure which is defined in the space of gauge invariant, uh uh gang invariant space, and to basically to extend it. It's just by uh, by saying that, uh introducing some group averaging over the whole uh
Mehdi Assanioussi: space
Mehdi Assanioussi: of non gauge, invariant and basically protecting you every time down to the gauge invariant part.
Mehdi Assanioussi: So here I just wanted to give you the case of su two. It looks
Mehdi Assanioussi: horrible. Uh, But basically. If you want uh this, this tweak that we introduced, which is to put in the generators of su and gauge theory is basically what here I market in blue because it results in this uh, in this factor,
Mehdi Assanioussi: which uh depends on the permutations uh of the levels in this uh, in this result.
Mehdi Assanioussi: And this is not present in the I believe, in case you get something which is symmetric with respect to permutations. That's why it gets dropped. Basically. Uh, in this case it's not. And that's why it stands. It remains uh basically there. That was basically just for.
Mehdi Assanioussi: And the C two example, we have. Just we need just to know what's the image of one single one with a loop,
Mehdi Assanioussi: And that's where you keep the the result.
Mehdi Assanioussi: So to summarize uh, basically here, the idea behind this talk is this: Uh: So our interest actually and uh
Mehdi Assanioussi: pushing this idea of uh constructing shadow states that somehow can
Mehdi Assanioussi: bridge, the the kinematical space of the loop theory, and the
Mehdi Assanioussi: and the fox space in general.
Mehdi Assanioussi: And here I didn't present a little new stuff, except this last part, but that was like sort of our baby step, or in understanding better how those things work. Uh in the general case. And uh,
Mehdi Assanioussi: I know our interest is justified by this uh the nature of the States. Somehow, those properties, although it's not yet investigated totally what they display. Is there no local correlations, as well as somewhat encoding a geometry in them, one
Mehdi Assanioussi: which is this mean of skin geometry.
Mehdi Assanioussi: Ah! And as I mentioned, there are a lot of things to to explore, including this non-locality, which maybe can be reflected in some sort of entanglement between different uh parts of the States, as well as the classic semi classical properties of those shadow States two hundred and fifty.
Mehdi Assanioussi: The extensions beyond the opinion case are still,
Mehdi Assanioussi: but I think there are also, maybe several ways to get something out of it.
Mehdi Assanioussi: And further, there are questions about what would be, for instance, the the dynamics that one could associate to the Shadow States, because, as we know, or, as I mentioned,
Mehdi Assanioussi: they come actually from states which are not normalizable in C Star, and we Don't have really
Mehdi Assanioussi: dynamic to impose dynamics to impose there. So can one thing, some sort of effective dynamics
Mehdi Assanioussi: uh for those States, from the perspective, also that in some sense they are approximate physical states, because they they come from state with, describe method, propaganda, Minkowski space, time,
Mehdi Assanioussi: and one open question uh, or curiosity is whether it's such state actually could play any role in the construction of a continuum limit for loop quantum gravity. And uh, yeah, and that would be all. Thank you.
Mehdi Assanioussi: Do you have any questions,
no questions.
Wolfgang Wieland: I have a question. Um! Hi! There! Hi! Hi! I'm a messy um! So my question is, i'm a bit um
Wolfgang Wieland: uh I i'm a bit confused how you define the Uh fork representation in already just in the su n non Abelian case, without ever introducing an actual gauge, fixing or ghost ghost fields that would take
Wolfgang Wieland: care of um of the internal gauge symmetry. And how how does this?
Wolfgang Wieland: Um: Yeah. Feed together with the Uh. S. U. N spin network representation.
Mehdi Assanioussi: So there are two things that I maybe one thing that I didn't mention here is, or maybe I did. But the the measure that we end up with is gauge-invariant one hundred and fifty
Mehdi Assanioussi: in sense. How this gauge invariance at the end of the day comes about it.
Mehdi Assanioussi: Uh, we just know that it's there at the very end.
Mehdi Assanioussi: Our suspicion is the fact that we use those somewhat those
Mehdi Assanioussi: they are somewhat modified with the loops, But the presence of this trace somehow takes care,
Mehdi Assanioussi: erez agmoni of uh, of the issue of gauge invariance, to come back to the question of the for it. It's not really a focal point, because we don't have a folk representation for s un one hundred and fifty,
Mehdi Assanioussi: so one one can do is just as I said, this poor man version is in some sense just thinking about uh the non-interacting uh limit. So it will be a very uh course approximation in this case one hundred and fifty.
Mehdi Assanioussi: However, here we took it from the perspective that what we want is to have a map in for with which we can construct the measure.
Mehdi Assanioussi: And in truth, we did not know whether it's. We will end up with something which is gauge invariant.
Mehdi Assanioussi: It's just that it happened to be at the end
Mehdi Assanioussi: that it's manifestly gauge invariant. But we we cannot say that we had some out, some control on the technicalities to how we got there.
Mehdi Assanioussi: So
Mehdi Assanioussi: yeah,
any other questions
Abhay Vasant Ashtekar: do you have any idea about
Abhay Vasant Ashtekar: but what this Tv represents? I mean, is it I can find phase? Is it A.
Mehdi Assanioussi: Uh. So I would say, Yeah, that's what I would call it something like a Maxwell face like um. It's a face where the the the interaction is totally ignored.
Mehdi Assanioussi: That's what I mean. The the technical meaning to this thing about what happens to this can do. Is it grow like in the Does it go like That's a good question. Actually, we we we still didn't investigate that. So it's a good question, but I I really don't have an answer for it at the moment.
Abhay Vasant Ashtekar: Okay, Thank you.
Jorge Pullin: Any other.
Okay, Let's thank the speaker again.
Mehdi Assanioussi: Thank you.