0 00:00:08,590 --> 00:00:10,660 Mehdi Assanioussi: Uh, thank you. Can you hear me? 1 00:00:11,480 --> 00:00:12,320 Yes, 2 00:00:12,460 --> 00:00:14,490 Jerzy Lewandowski: very good. Very well. 3 00:00:37,440 --> 00:00:51,800 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 4 00:00:51,810 --> 00:01:04,579 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. 5 00:01:05,730 --> 00:01:21,000 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, 6 00:01:21,690 --> 00:01:44,639 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 7 00:01:44,650 --> 00:01:47,860 Mehdi Assanioussi: uh open uh, 8 00:01:48,170 --> 00:01:52,020 Mehdi Assanioussi: but in the verse that I' to be answered and worked on, 9 00:01:52,660 --> 00:02:06,830 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, 10 00:02:06,860 --> 00:02:36,800 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 11 00:02:37,520 --> 00:02:44,420 Mehdi Assanioussi: have been present for a long time in in the work of the community, and there were 12 00:02:45,090 --> 00:02:56,769 Mehdi Assanioussi: several, many uh directions followed and investigated, and uh, here I would chose to uh focus, particularly on uh, 13 00:02:57,090 --> 00:03:08,770 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 14 00:03:09,550 --> 00:03:18,480 Mehdi Assanioussi: So before that uh, this in the canonic loop quantum gravity, we know how to basically quantize uh 15 00:03:18,830 --> 00:03:36,110 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 16 00:03:36,120 --> 00:03:39,360 Mehdi Assanioussi: and other way you impose into dynamics. 17 00:03:39,500 --> 00:03:49,680 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 18 00:03:50,080 --> 00:04:08,910 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 19 00:04:08,920 --> 00:04:14,910 Mehdi Assanioussi: uh get to the point where we can make actually predictions that can be verified. 20 00:04:15,580 --> 00:04:21,690 Mehdi Assanioussi: So here I gathered uh just uh, more or less, it's a collection of uh 21 00:04:22,000 --> 00:04:39,310 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 22 00:04:39,320 --> 00:04:55,469 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 23 00:04:55,910 --> 00:05:03,130 Mehdi Assanioussi: uh, we have the complexity file coherent state. I want to, I think, safely say that this one of the 24 00:05:03,370 --> 00:05:20,579 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 25 00:05:20,590 --> 00:05:37,390 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 26 00:05:37,400 --> 00:05:50,269 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 27 00:05:50,300 --> 00:06:04,320 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 28 00:06:04,860 --> 00:06:06,000 Mehdi Assanioussi: Uh 29 00:06:06,140 --> 00:06:24,369 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. 30 00:06:24,600 --> 00:06:25,720 Mehdi Assanioussi: Uh, 31 00:06:27,070 --> 00:06:41,949 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 32 00:06:41,960 --> 00:06:49,000 Mehdi Assanioussi: to answer or try actually to answer the question, What would happen if we want to construct some States 33 00:06:49,220 --> 00:06:55,579 Mehdi Assanioussi: which behave nicely under the action of change uh graph changing operators. 34 00:06:55,590 --> 00:07:12,979 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 35 00:07:12,990 --> 00:07:29,450 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. 36 00:07:30,430 --> 00:07:45,790 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 37 00:07:45,840 --> 00:08:05,469 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 38 00:08:05,480 --> 00:08:25,299 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 39 00:08:26,000 --> 00:08:42,419 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, 40 00:08:42,429 --> 00:08:55,519 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 41 00:08:55,900 --> 00:09:07,610 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, 42 00:09:16,160 --> 00:09:23,420 Mehdi Assanioussi: and from there one is able to actually somewhat construct some mapping of the of the fox States 43 00:09:23,470 --> 00:09:37,619 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. 44 00:09:38,090 --> 00:09:43,760 Mehdi Assanioussi: So here I drew on this slide, I through some examples, concrete examples of what I was talking about. 45 00:09:43,770 --> 00:09:59,200 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. 46 00:09:59,980 --> 00:10:16,199 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. 47 00:10:16,460 --> 00:10:34,080 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 48 00:10:35,170 --> 00:10:49,639 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 49 00:11:08,920 --> 00:11:10,290 Mehdi Assanioussi: Um. 50 00:11:10,400 --> 00:11:23,759 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 51 00:11:23,870 --> 00:11:37,920 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 52 00:11:38,680 --> 00:11:56,009 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. 53 00:11:56,020 --> 00:12:13,209 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. 54 00:12:13,750 --> 00:12:25,409 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 55 00:12:25,460 --> 00:12:43,240 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 56 00:12:43,250 --> 00:13:03,189 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? 57 00:13:03,710 --> 00:13:18,649 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, 58 00:13:18,660 --> 00:13:34,879 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. 59 00:13:34,890 --> 00:13:47,639 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. 60 00:13:47,650 --> 00:13:55,739 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. 61 00:13:57,080 --> 00:13:58,330 Mehdi Assanioussi: Uh, 62 00:13:59,020 --> 00:14:10,879 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, 63 00:14:10,950 --> 00:14:19,749 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. 64 00:14:34,040 --> 00:14:48,019 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. 65 00:14:49,020 --> 00:15:01,500 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. 66 00:15:01,590 --> 00:15:04,189 Mehdi Assanioussi: There are. This loop has a prescription, 67 00:15:04,480 --> 00:15:10,819 Mehdi Assanioussi: and this prescription is encoded also in the definition of those uh canonical pairs, 68 00:15:11,600 --> 00:15:27,400 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 69 00:15:28,090 --> 00:15:48,659 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. 70 00:15:48,670 --> 00:16:04,740 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. 71 00:16:04,750 --> 00:16:11,229 Mehdi Assanioussi: Just take this annihilation operator and define the coherent state as the eigenvectors basically 72 00:16:30,920 --> 00:16:49,360 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, 73 00:16:49,660 --> 00:16:52,350 Mehdi Assanioussi: and then you you take the inverse and the square root. 74 00:16:52,830 --> 00:16:57,919 Abhay Vasant Ashtekar: I think we should go on because I don't understand the notation. So I should. 75 00:16:58,100 --> 00:17:14,570 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, 76 00:17:14,859 --> 00:17:21,779 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? 77 00:17:22,329 --> 00:17:36,269 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. 78 00:17:37,510 --> 00:17:40,709 Mehdi Assanioussi: That's how I would do it in this case. 79 00:17:47,910 --> 00:17:57,169 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 80 00:17:57,480 --> 00:18:15,859 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 81 00:18:15,870 --> 00:18:35,330 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. 82 00:18:35,340 --> 00:18:39,819 Mehdi Assanioussi: So yeah, it's a lot of details that I'm: on meeting here. Yeah, 83 00:18:40,280 --> 00:18:54,380 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, 84 00:18:54,400 --> 00:19:02,860 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 85 00:19:02,870 --> 00:19:18,379 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, 86 00:19:18,390 --> 00:19:28,079 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, 87 00:19:28,090 --> 00:19:40,289 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 88 00:19:40,300 --> 00:19:54,020 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. 89 00:19:58,530 --> 00:19:59,760 Mehdi Assanioussi: And then 90 00:19:59,770 --> 00:20:18,620 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, 91 00:20:19,470 --> 00:20:34,920 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 92 00:20:35,190 --> 00:20:54,900 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. 93 00:21:02,130 --> 00:21:09,810 Mehdi Assanioussi: And the next thing was just to basically once you have uh this representation. Uh, 94 00:21:10,580 --> 00:21:28,059 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. 95 00:21:28,070 --> 00:21:31,919 Mehdi Assanioussi: And then that's how you try to solve this equation 96 00:21:31,930 --> 00:21:52,089 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. 97 00:21:52,100 --> 00:21:53,360 Mehdi Assanioussi: Um, 98 00:21:53,550 --> 00:22:07,639 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. 99 00:22:08,570 --> 00:22:11,849 Mehdi Assanioussi: Okay, So uh, 100 00:22:11,930 --> 00:22:17,709 Mehdi Assanioussi: no. This red line. But okay, uh, next thing is um. 101 00:22:18,860 --> 00:22:29,139 Mehdi Assanioussi: Well, the question that is, uh, always present in the context of quantum gravity theory in particular, in the loop quantum gravity 102 00:22:29,180 --> 00:22:35,320 Mehdi Assanioussi: framework is how to connect the low energy physics uh to the, to the our theory 103 00:22:35,420 --> 00:22:44,629 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. 104 00:22:44,670 --> 00:22:59,690 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 105 00:23:00,290 --> 00:23:18,859 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, 106 00:23:18,870 --> 00:23:28,499 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, 107 00:23:29,400 --> 00:23:45,879 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. 108 00:23:46,120 --> 00:23:55,820 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, 109 00:23:55,830 --> 00:24:06,580 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. 110 00:24:07,670 --> 00:24:15,549 Mehdi Assanioussi: And this scale actually basically measurements at a given scale, you can always find 111 00:24:15,940 --> 00:24:31,970 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. 112 00:24:32,720 --> 00:24:50,520 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 113 00:24:50,840 --> 00:24:58,310 Mehdi Assanioussi: in here, and that's what, in a sense, motivated us is to construct to have this mapping between the Fox States 114 00:24:58,950 --> 00:25:17,870 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. 115 00:25:18,720 --> 00:25:29,619 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. 116 00:25:29,670 --> 00:25:34,360 Mehdi Assanioussi: And then you define this algebra that is, on the left 117 00:25:34,860 --> 00:25:43,060 Mehdi Assanioussi: of the smeared hol on the electric field, and as I was mentioning, the first step is to smear the connection to 118 00:25:43,300 --> 00:25:50,380 Mehdi Assanioussi: uh with some uh, it's actually Schwarz function which satisfy this property here on top corner. 119 00:25:51,260 --> 00:26:01,909 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 120 00:26:01,920 --> 00:26:13,520 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 121 00:26:13,640 --> 00:26:28,649 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. 122 00:26:29,750 --> 00:26:39,989 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 123 00:26:40,060 --> 00:26:57,479 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 124 00:26:58,280 --> 00:27:03,170 Mehdi Assanioussi: algebra, the loop algebra. I feel the standard autonomy and 125 00:27:03,420 --> 00:27:07,510 Mehdi Assanioussi: the smeared electric field. Here the smeared electric field is 126 00:27:07,890 --> 00:27:13,689 Mehdi Assanioussi: over three dimensions, but it still works. That's one of the miracles of the Avenue gauge theory 127 00:27:13,900 --> 00:27:27,579 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 128 00:27:27,590 --> 00:27:33,940 Mehdi Assanioussi: the folk representation, or any representation, for that matter of one, provides a representation for the other, 129 00:27:34,230 --> 00:27:53,740 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. 130 00:27:54,340 --> 00:28:01,369 Mehdi Assanioussi: But that was basically, the core ideas in this, in this construction, 131 00:28:01,450 --> 00:28:07,989 Mehdi Assanioussi: and it goes. That's the details. Basically. Go Follow that 132 00:28:08,420 --> 00:28:17,779 Mehdi Assanioussi: at some point you can consider. Uh, you consider the uh, the fog vacuum expectation values of the smeared Holonomy operators, 133 00:28:17,800 --> 00:28:24,379 Mehdi Assanioussi: and you identify that with the expectation value of the standard holiday operators in some 134 00:28:24,900 --> 00:28:26,240 Mehdi Assanioussi: new vacuum. 135 00:28:26,550 --> 00:28:43,089 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 136 00:28:43,130 --> 00:28:53,180 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. 137 00:28:53,720 --> 00:29:01,580 Mehdi Assanioussi: And basically, and I put that the out for representation is just cyclic representation generated by this new vacuum. 138 00:29:02,250 --> 00:29:08,690 Mehdi Assanioussi: One can carry on, of course, with the calculations and figure out actually what's the new measure, 139 00:29:08,720 --> 00:29:22,270 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. 140 00:29:22,430 --> 00:29:40,790 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. 141 00:29:42,210 --> 00:30:00,079 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. 142 00:30:00,260 --> 00:30:08,690 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. 143 00:30:09,490 --> 00:30:16,400 Mehdi Assanioussi: No, no, no. Well, it's. Uh if you want it's equivalent. 144 00:30:19,190 --> 00:30:34,499 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 145 00:30:34,550 --> 00:30:37,380 Mehdi Assanioussi: it's it's different. It's just different. 146 00:30:37,670 --> 00:30:43,979 Mehdi Assanioussi: One can think of the folk representation as some sort of singular limiting case. 147 00:30:44,040 --> 00:30:46,310 Mehdi Assanioussi: That's how I would say it. 148 00:30:46,740 --> 00:30:48,370 Abhay Vasant Ashtekar: I mean, is it you on the 149 00:30:49,230 --> 00:30:52,479 Abhay Vasant Ashtekar: I the way you're presenting it. I'm a little confused, but 150 00:30:53,600 --> 00:30:56,019 Abhay Vasant Ashtekar: it looks like there are two different algebra right? 151 00:31:01,540 --> 00:31:03,550 Abhay Vasant Ashtekar: And then the next slide 152 00:31:03,920 --> 00:31:06,040 Mehdi Assanioussi: uh next. Okay, 153 00:31:06,240 --> 00:31:11,890 You know the top equation on the left-hand side. You've got your 154 00:31:12,270 --> 00:31:16,110 Abhay Vasant Ashtekar: on the right hand side. You I got. 155 00:31:18,410 --> 00:31:19,360 Mehdi Assanioussi: Yes, 156 00:31:20,450 --> 00:31:21,660 Abhay Vasant Ashtekar: so 157 00:31:22,510 --> 00:31:24,129 Abhay Vasant Ashtekar: you are defining. 158 00:31:25,480 --> 00:31:31,880 Abhay Vasant Ashtekar: So the The algebra themselves are quite different. Right. So the question of territory equivalent doesn't even arise that right? 159 00:31:32,730 --> 00:31:40,069 Mehdi Assanioussi: Uh well, the algebra, if you refer to the algebra of the smith, hold on isn't. There always they are isomorphic 160 00:31:40,750 --> 00:31:42,320 Mehdi Assanioussi: in the billion kids. 161 00:31:43,220 --> 00:31:47,140 Abhay Vasant Ashtekar: Um. But the question is whether this as a mark is Um, 162 00:31:47,390 --> 00:31:48,330 Abhay Vasant Ashtekar: um. 163 00:31:50,290 --> 00:32:08,239 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 164 00:32:19,240 --> 00:32:22,419 Abhay Vasant Ashtekar: No, you always be right. Connection is not there. 165 00:32:26,350 --> 00:32:29,990 Abhay Vasant Ashtekar: So she always Yeah. So what do you mean by most meeting? 166 00:32:30,990 --> 00:32:33,469 Mehdi Assanioussi: No, I'm just talking about just 167 00:32:33,720 --> 00:32:40,600 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. 168 00:32:40,700 --> 00:32:48,980 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. 169 00:32:49,600 --> 00:32:52,709 Abhay Vasant Ashtekar: But it is it not? Can I not do that? 170 00:32:53,340 --> 00:32:54,410 Mehdi Assanioussi: I'm sorry 171 00:32:54,790 --> 00:32:58,320 Abhay Vasant Ashtekar: it is not equivalent if I just consider you to hold on on this 172 00:32:59,180 --> 00:33:08,739 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. 173 00:33:09,320 --> 00:33:16,900 Mehdi Assanioussi: So you the the standard for quantization it provides you. So you have a folk representation that 174 00:33:22,450 --> 00:33:26,389 Abhay Vasant Ashtekar: yeah, there is, there is a single fork representation with both. We are 175 00:33:27,050 --> 00:33:29,159 connections. Is your home exact? 176 00:33:34,550 --> 00:33:38,099 Abhay Vasant Ashtekar: But now, in the or for representation 177 00:33:38,360 --> 00:33:40,550 Abhay Vasant Ashtekar: only I 178 00:33:41,150 --> 00:33:44,940 Mehdi Assanioussi: Yes, 179 00:33:45,540 --> 00:33:46,630 Abhay Vasant Ashtekar: so 180 00:33:47,270 --> 00:33:55,610 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, 181 00:33:55,970 --> 00:34:05,099 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? 182 00:34:05,930 --> 00:34:17,569 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. 183 00:34:19,440 --> 00:34:21,179 Abhay Vasant Ashtekar: Yeah, I'm talking about 184 00:34:21,610 --> 00:34:25,739 Abhay Vasant Ashtekar: mathematical physics at the moment. I'm not. Don't. All get into issues about measurement theory. 185 00:34:27,060 --> 00:34:29,480 Abhay Vasant Ashtekar: So I just can't complete any one. 186 00:34:36,020 --> 00:34:37,509 Abhay Vasant Ashtekar: Uh. So. 187 00:34:38,239 --> 00:34:39,149 Mehdi Assanioussi: Yep. 188 00:34:48,840 --> 00:34:51,699 Abhay Vasant Ashtekar: On the one hand, you have one algebra, 189 00:34:52,070 --> 00:34:56,480 Abhay Vasant Ashtekar: namely, of. On the other hand, you've got a number of 190 00:34:56,630 --> 00:34:59,140 Abhay Vasant Ashtekar: the two algebra are kind of distinct 191 00:34:59,350 --> 00:35:03,149 Abhay Vasant Ashtekar: Yes, and all the default representations near. How long is 192 00:35:04,240 --> 00:35:08,450 Abhay Vasant Ashtekar: they define expertise in values, but unsmared along this do not. 193 00:35:09,030 --> 00:35:09,890 Mehdi Assanioussi: No, 194 00:35:10,030 --> 00:35:13,960 Abhay Vasant Ashtekar: so one algebra is is represented the other you right not representing, 195 00:35:15,150 --> 00:35:18,449 Abhay Vasant Ashtekar: but on the last transparency you are saying that the two are I to 196 00:35:19,490 --> 00:35:32,720 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. 197 00:35:32,890 --> 00:35:33,740 Abhay Vasant Ashtekar: Okay, 198 00:35:34,030 --> 00:35:37,839 Mehdi Assanioussi: just by using the isomorphism between the two algebra. 199 00:35:38,180 --> 00:35:39,169 Abhay Vasant Ashtekar: Okay, 200 00:35:45,230 --> 00:35:48,519 Mehdi Assanioussi: A non-focked representation of the smeared, homogeneous 201 00:35:48,780 --> 00:35:53,629 Mehdi Assanioussi: you're holding algebra that our fault representation, which is not that 202 00:35:53,730 --> 00:35:55,560 Abhay Vasant Ashtekar: standard for representation? Right? 203 00:35:57,030 --> 00:36:00,439 Mehdi Assanioussi: No, it is not the standard for it. It's just derived from it. 204 00:36:01,410 --> 00:36:04,129 Abhay Vasant Ashtekar: So what does is the Iso? 205 00:36:04,420 --> 00:36:11,650 Mehdi Assanioussi: It's exactly that is, that investigator allows you to derive a new representation for the ordinary holonomy. 206 00:36:18,210 --> 00:36:28,660 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. 207 00:36:29,260 --> 00:36:31,149 Abhay Vasant Ashtekar: That's correct. 208 00:36:36,120 --> 00:36:40,870 Abhay Vasant Ashtekar: The statement is that 209 00:36:42,950 --> 00:36:45,430 Abhay Vasant Ashtekar: that just different algebra Is that right? Or 210 00:36:47,320 --> 00:36:49,270 Mehdi Assanioussi: well? Um. 211 00:36:50,900 --> 00:36:56,110 Abhay Vasant Ashtekar: Why, why don't you go on? I don't just talk to you because I just because uh, 212 00:36:57,670 --> 00:37:12,029 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. 213 00:37:12,390 --> 00:37:23,910 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 214 00:37:25,330 --> 00:37:32,850 Mehdi Assanioussi: because of this is more than they should agree. You just need to know how to translate one, 215 00:37:33,800 --> 00:37:36,260 Mehdi Assanioussi: translate basically, one into the other, 216 00:37:36,430 --> 00:37:39,060 Mehdi Assanioussi: and that's what this is. A zoom thing does. 217 00:37:40,480 --> 00:37:42,639 Abhay Vasant Ashtekar: What is it? I some option between. 218 00:37:43,520 --> 00:37:46,699 Mehdi Assanioussi: So first it's the it's between the algebra. 219 00:37:49,690 --> 00:38:01,300 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. 220 00:38:01,320 --> 00:38:05,379 Mehdi Assanioussi: If you do want to on one side, you should be able to recover it on the other side. 221 00:38:18,710 --> 00:38:29,899 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 222 00:38:49,230 --> 00:39:02,150 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 223 00:39:02,380 --> 00:39:07,150 Mehdi Assanioussi: and uh, she is basically uh some sort of uh, 224 00:39:08,290 --> 00:39:12,069 Mehdi Assanioussi: the function which basically couples the edges of the graph 225 00:39:12,720 --> 00:39:14,049 Mehdi Assanioussi: Um. 226 00:39:14,480 --> 00:39:29,050 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, 227 00:39:29,060 --> 00:39:39,360 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 228 00:39:39,380 --> 00:39:40,939 Mehdi Assanioussi: uh 229 00:39:40,980 --> 00:39:44,550 Mehdi Assanioussi: in terms of the harmonomies and the and the electric field. 230 00:39:44,730 --> 00:40:01,779 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, 231 00:40:01,980 --> 00:40:06,169 Mehdi Assanioussi: some factor which, of course, depends on 232 00:40:06,380 --> 00:40:10,980 Mehdi Assanioussi: it, depends on this on this function. G 233 00:40:18,320 --> 00:40:20,160 Mehdi Assanioussi: and uh, 234 00:40:20,210 --> 00:40:36,950 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. 235 00:40:36,970 --> 00:40:56,029 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. 236 00:40:56,240 --> 00:40:57,509 Mehdi Assanioussi: Uh. 237 00:40:58,200 --> 00:41:05,629 Mehdi Assanioussi: And this is, and this this geometry is exactly encoded in those coefficients which are basically this X and G. 238 00:41:06,260 --> 00:41:20,679 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 239 00:41:20,690 --> 00:41:37,430 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. 240 00:41:37,440 --> 00:41:50,070 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, 241 00:41:50,660 --> 00:41:56,969 Mehdi Assanioussi: and this was an interesting property, because it didn't it wasn't uh it 242 00:41:57,110 --> 00:42:09,080 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 243 00:42:09,320 --> 00:42:25,159 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 244 00:42:25,580 --> 00:42:35,040 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. 245 00:42:35,130 --> 00:42:43,729 Mehdi Assanioussi: Uh, and that's Basically, what the notion of shadow States comes in. Um. Basically, one has to take the this. This 246 00:42:43,740 --> 00:42:59,259 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 247 00:42:59,810 --> 00:43:13,640 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 248 00:43:36,420 --> 00:43:55,519 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 249 00:43:55,910 --> 00:43:59,829 Mehdi Assanioussi: I'm: Basically, to yeah, that's that's basically the idea. 250 00:44:00,810 --> 00:44:07,189 Mehdi Assanioussi: And that was the what was developed in the context of a video gauge theory. 251 00:44:07,320 --> 00:44:12,110 Mehdi Assanioussi: And then, of course, that systematically brings the question: Can we go beyond 252 00:44:12,520 --> 00:44:24,149 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. 253 00:44:24,330 --> 00:44:25,600 Mehdi Assanioussi: Uh, 254 00:44:26,530 --> 00:44:40,049 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 255 00:44:40,350 --> 00:44:47,199 Mehdi Assanioussi: relevant, and they they do not. They are not present in the context of uh, none of any engage theory. 256 00:44:47,730 --> 00:45:00,680 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 257 00:45:05,240 --> 00:45:10,970 Mehdi Assanioussi: since one can see that basically here what I marked it right at the beginning were the colors. 258 00:45:11,420 --> 00:45:29,899 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, 259 00:45:29,910 --> 00:45:37,419 Mehdi Assanioussi: it's diagonal. And the electricity is the age invariant. So you what you get is it's just the color, 260 00:45:37,740 --> 00:45:47,870 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. 261 00:45:47,950 --> 00:45:56,600 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. 262 00:45:57,620 --> 00:46:09,479 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 263 00:46:09,810 --> 00:46:20,290 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 264 00:46:21,120 --> 00:46:32,389 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 265 00:46:32,770 --> 00:46:45,769 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 266 00:46:46,080 --> 00:46:47,399 Mehdi Assanioussi: um 267 00:46:47,620 --> 00:47:04,400 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. 268 00:47:04,410 --> 00:47:20,389 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. 269 00:47:26,500 --> 00:47:27,529 Mehdi Assanioussi: Um 270 00:47:28,000 --> 00:47:34,479 Mehdi Assanioussi: one of them, for this is gauge invariance, the other promised to reduce consistency and 271 00:47:35,150 --> 00:47:45,360 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 272 00:47:45,980 --> 00:47:49,430 Mehdi Assanioussi: to contain all the necessary properties. 273 00:47:49,930 --> 00:47:56,129 Mehdi Assanioussi: Uh, another thing was done which is the the construction of Africa representation for Scalar Field, 274 00:47:56,680 --> 00:47:59,359 Mehdi Assanioussi: and in this case it was 275 00:48:04,450 --> 00:48:19,189 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, 276 00:48:19,200 --> 00:48:38,140 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, 277 00:48:38,540 --> 00:48:47,670 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 278 00:48:48,550 --> 00:48:57,850 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. 279 00:48:57,920 --> 00:49:09,690 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. 280 00:49:10,160 --> 00:49:15,910 Mehdi Assanioussi: This is basically just a consequence of bundle, some identities, if you want, 281 00:49:15,940 --> 00:49:17,200 Mehdi Assanioussi: uh, 282 00:49:17,650 --> 00:49:22,779 Mehdi Assanioussi: and that actually this this observation, one can say 283 00:49:22,810 --> 00:49:25,200 Mehdi Assanioussi: that if one is to 284 00:49:25,540 --> 00:49:43,590 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 285 00:49:43,600 --> 00:49:46,330 Mehdi Assanioussi: uh, say, with the loop operators 286 00:49:46,910 --> 00:49:49,990 Mehdi Assanioussi: in uh the vacuum they interacted to. 287 00:49:50,280 --> 00:49:52,220 Mehdi Assanioussi: Now, the idea Here, 288 00:49:52,410 --> 00:50:00,729 Mehdi Assanioussi: let's let's say that the original motivation and our interest is to eventually get to somehow a way to construct 289 00:50:00,760 --> 00:50:17,729 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, 290 00:50:18,310 --> 00:50:20,819 Mehdi Assanioussi: uh, however, 291 00:50:20,910 --> 00:50:38,510 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, 292 00:50:38,870 --> 00:50:50,749 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, 293 00:50:58,580 --> 00:51:00,510 Mehdi Assanioussi: and so be cheating 294 00:51:00,610 --> 00:51:15,230 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 295 00:51:15,270 --> 00:51:16,700 Mehdi Assanioussi: um, 296 00:51:17,230 --> 00:51:26,830 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 297 00:51:27,240 --> 00:51:38,810 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. 298 00:51:39,440 --> 00:51:57,549 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. 299 00:51:57,590 --> 00:52:13,650 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, 300 00:52:15,450 --> 00:52:23,750 Mehdi Assanioussi: and what we try to do is basically we say, Okay, this is our Pullman's version. Can we define a measure 301 00:52:24,120 --> 00:52:35,530 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. 302 00:52:41,040 --> 00:52:45,799 Mehdi Assanioussi: We didn't know whether it actually even would work. But it turns out that actually does. 303 00:52:46,020 --> 00:52:56,770 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 304 00:52:56,800 --> 00:52:58,759 Mehdi Assanioussi: vacuum of the free theory. 305 00:52:59,270 --> 00:53:06,790 Mehdi Assanioussi: And basically this this, this linear functional acts on basically it's algebra, 306 00:53:07,300 --> 00:53:11,390 Mehdi Assanioussi: uh, which I can write in terms of actually with the loops, 307 00:53:11,820 --> 00:53:22,400 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! 308 00:53:22,610 --> 00:53:33,439 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 309 00:53:33,620 --> 00:53:43,170 Mehdi Assanioussi: cylindrical function as an product of N. Minus one, a linear combination of products of N minus one with the groups. 310 00:53:44,460 --> 00:54:02,949 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, 311 00:54:02,960 --> 00:54:12,640 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, 312 00:54:12,660 --> 00:54:14,720 Mehdi Assanioussi: and that actually can be shown. 313 00:54:14,970 --> 00:54:31,049 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, 314 00:54:31,060 --> 00:54:44,100 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 315 00:54:44,670 --> 00:54:48,440 Mehdi Assanioussi: the algebra. But in this case it actually doesn't differ much 316 00:54:48,500 --> 00:55:06,719 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 317 00:55:06,730 --> 00:55:12,189 Mehdi Assanioussi: uh temperature distribution to the space uh of uh, 318 00:55:12,560 --> 00:55:14,839 Mehdi Assanioussi: discontinuous connections. 319 00:55:16,690 --> 00:55:34,099 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. 320 00:55:34,110 --> 00:55:48,660 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. 321 00:55:48,720 --> 00:55:50,080 Mehdi Assanioussi: Um, 322 00:55:51,240 --> 00:55:52,529 Mehdi Assanioussi: uh 323 00:55:52,960 --> 00:55:56,409 Mehdi Assanioussi: via this, this linear uh linear map, 324 00:55:58,020 --> 00:56:10,969 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 325 00:56:11,000 --> 00:56:16,330 Mehdi Assanioussi: when one uses a big winner, one is already working on the gauge invariance level. 326 00:56:16,500 --> 00:56:21,039 Mehdi Assanioussi: So, in a sense, one obtain something uh 327 00:56:21,300 --> 00:56:36,710 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 328 00:56:36,750 --> 00:56:37,859 Mehdi Assanioussi: space 329 00:56:38,170 --> 00:56:44,520 Mehdi Assanioussi: of non gauge, invariant and basically protecting you every time down to the gauge invariant part. 330 00:56:45,750 --> 00:56:50,459 Mehdi Assanioussi: So here I just wanted to give you the case of su two. It looks 331 00:56:50,550 --> 00:57:08,670 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, 332 00:57:08,890 --> 00:57:16,979 Mehdi Assanioussi: which uh depends on the permutations uh of the levels in this uh, in this result. 333 00:57:17,120 --> 00:57:33,330 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. 334 00:57:33,340 --> 00:57:41,200 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, 335 00:57:41,230 --> 00:57:43,879 Mehdi Assanioussi: And that's where you keep the the result. 336 00:57:45,770 --> 00:57:56,869 Mehdi Assanioussi: So to summarize uh, basically here, the idea behind this talk is this: Uh: So our interest actually and uh 337 00:57:56,980 --> 00:58:03,420 Mehdi Assanioussi: pushing this idea of uh constructing shadow states that somehow can 338 00:58:03,810 --> 00:58:11,990 Mehdi Assanioussi: bridge, the the kinematical space of the loop theory, and the 339 00:58:12,030 --> 00:58:14,169 Mehdi Assanioussi: and the fox space in general. 340 00:58:15,170 --> 00:58:28,510 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, 341 00:58:28,590 --> 00:58:47,160 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 342 00:58:47,170 --> 00:58:49,529 Mehdi Assanioussi: which is this mean of skin geometry. 343 00:58:50,650 --> 00:59:07,360 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. 344 00:59:08,180 --> 00:59:11,600 Mehdi Assanioussi: The extensions beyond the opinion case are still, 345 00:59:18,310 --> 00:59:25,259 Mehdi Assanioussi: but I think there are also, maybe several ways to get something out of it. 346 00:59:26,580 --> 00:59:37,780 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, 347 00:59:37,810 --> 00:59:44,980 Mehdi Assanioussi: they come actually from states which are not normalizable in C Star, and we Don't have really 348 00:59:45,410 --> 00:59:53,310 Mehdi Assanioussi: dynamic to impose dynamics to impose there. So can one thing, some sort of effective dynamics 349 00:59:53,330 --> 01:00:06,260 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, 350 01:00:07,160 --> 01:00:22,089 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. 351 01:00:25,320 --> 01:00:28,589 Mehdi Assanioussi: Do you have any questions, 352 01:00:41,310 --> 01:00:42,509 no questions. 353 01:00:43,970 --> 01:00:55,209 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 354 01:00:55,460 --> 01:01:14,229 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 355 01:01:14,240 --> 01:01:19,529 Wolfgang Wieland: care of um of the internal gauge symmetry. And how how does this? 356 01:01:20,230 --> 01:01:27,419 Wolfgang Wieland: Um: Yeah. Feed together with the Uh. S. U. N spin network representation. 357 01:01:27,840 --> 01:01:37,649 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 358 01:01:43,960 --> 01:01:47,940 Mehdi Assanioussi: in sense. How this gauge invariance at the end of the day comes about it. 359 01:01:48,070 --> 01:01:51,380 Mehdi Assanioussi: Uh, we just know that it's there at the very end. 360 01:01:51,570 --> 01:01:56,840 Mehdi Assanioussi: Our suspicion is the fact that we use those somewhat those 361 01:01:57,330 --> 01:02:03,470 Mehdi Assanioussi: they are somewhat modified with the loops, But the presence of this trace somehow takes care, 362 01:02:03,590 --> 01:02:14,729 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, 363 01:02:14,740 --> 01:02:29,930 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. 364 01:02:30,220 --> 01:02:37,020 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. 365 01:02:37,670 --> 01:02:43,969 Mehdi Assanioussi: And in truth, we did not know whether it's. We will end up with something which is gauge invariant. 366 01:02:44,350 --> 01:02:47,369 Mehdi Assanioussi: It's just that it happened to be at the end 367 01:02:47,390 --> 01:02:56,160 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. 368 01:02:56,720 --> 01:02:57,810 Mehdi Assanioussi: So 369 01:02:58,340 --> 01:02:59,220 Mehdi Assanioussi: yeah, 370 01:03:02,910 --> 01:03:04,330 any other questions 371 01:03:05,900 --> 01:03:08,429 Abhay Vasant Ashtekar: do you have any idea about 372 01:03:08,600 --> 01:03:12,380 Abhay Vasant Ashtekar: but what this Tv represents? I mean, is it I can find phase? Is it A. 373 01:03:13,180 --> 01:03:26,630 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. 374 01:03:27,930 --> 01:03:46,470 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. 375 01:03:49,480 --> 01:03:50,370 Abhay Vasant Ashtekar: Okay, Thank you. 376 01:03:54,150 --> 01:03:55,100 Jorge Pullin: Any other. 377 01:04:01,250 --> 01:04:03,149 Okay, Let's thank the speaker again. 378 01:04:03,600 --> 01:04:04,509 Mehdi Assanioussi: Thank you.