0 00:00:02,760 --> 00:00:10,320 Jorge Pullin: So our speaker to this Norbert for the North fork or speak about normalization for cosmological LPG models North yeah. 1 00:00:10,349 --> 00:00:11,550 Norbert Bodendorfer: Okay, thank you very much. 2 00:00:11,580 --> 00:00:23,670 Norbert Bodendorfer: And thank you very much for the possibility to give this talk here and, as you can see, this will be something like an overview talk over several results that have been obtained in the past five years or so. 3 00:00:24,330 --> 00:00:35,790 Norbert Bodendorfer: with varying co authors and of the co authors, you may not know Fabian Anita, who was a veteran a master student in my group and ring spork and Dennis while he was a master student also here. 4 00:00:37,620 --> 00:00:44,940 Norbert Bodendorfer: Okay, so let me give you a very brief overview of this talk here what we're going to do and what we hope to achieve. 5 00:00:45,810 --> 00:01:01,470 Norbert Bodendorfer: and technically what we're doing is of course screening or normalization and we will do this using certain a group theoretic techniques that are based on the group so one one and we will see in detail later why it makes sense to use such techniques and how they come into play here. 6 00:01:03,690 --> 00:01:16,080 Norbert Bodendorfer: So, first of all, why this is interesting to do, we all know that re normalization is necessary in quantum field theory, however, it is largely ignored at the moment in quantum gravity and groupon cosmology. 7 00:01:16,980 --> 00:01:26,940 Norbert Bodendorfer: For several reasons, but mainly because it's very complicated yeah so with this state of affairs it's very useful to have some analytically tractable toy models. 8 00:01:27,300 --> 00:01:35,910 Norbert Bodendorfer: where you can do some explicit computations and see how things may play out in a more complicated setting yeah, and this is what we hope to achieve here. 9 00:01:37,350 --> 00:01:50,160 Norbert Bodendorfer: So we do this in a certain a simplified setting and we're going to do this in a cosmological settings we're doing a core screening of four states that we call a cosmological states So these are quantum states that. 10 00:01:50,820 --> 00:01:58,260 Norbert Bodendorfer: We hope are suitable to describe cosmological space times, and this will simplify things as we will see later. 11 00:01:59,280 --> 00:02:07,410 Norbert Bodendorfer: And then, again, the technical basis will be this representation theory of the group Su one, one that we need to discuss later. 12 00:02:07,860 --> 00:02:17,070 Norbert Bodendorfer: And for the prerequisites that you need for this talk it turns out that everything can be done actually quite simple terms. 13 00:02:17,370 --> 00:02:32,970 Norbert Bodendorfer: Meaning that everything we do here today works on copies of the quantum cosmology healer space, so if you know about the simplest version of the loop quantum more legitimate space, we will of course review that later, then you can already follow along everything we're doing here today. 14 00:02:34,920 --> 00:02:42,090 Norbert Bodendorfer: However, we will see that the lessons should also apply for more complicated settings in particular also for full quantum gravity theories. 15 00:02:43,890 --> 00:02:48,000 Norbert Bodendorfer: So what is new, here in this talk as opposed to other works that you may. 16 00:02:48,870 --> 00:02:57,240 Norbert Bodendorfer: have come across so what we can do here is, we can do an analytic computation will be core screen many small spins into few large spins. 17 00:02:57,750 --> 00:03:02,280 Norbert Bodendorfer: and in doing so we are able to compute for a normalized hamiltonian. 18 00:03:02,730 --> 00:03:11,310 Norbert Bodendorfer: That is now a hamiltonian that X on large screens or few quantum numbers why capturing the physics of many small spins so. 19 00:03:11,670 --> 00:03:23,010 Norbert Bodendorfer: This is something that is really interesting because it tells you how you can efficiently describe complicated system in terms of few degrees of freedoms, while still keeping track of the underlying physics, of the. 20 00:03:23,700 --> 00:03:37,140 Norbert Bodendorfer: Final degrees of freedom Okay, and after having a computed this renewal estimate tune in, we can also give an example of how large the errors that you're making if you're neglecting this normalization effects. 21 00:03:39,720 --> 00:03:53,640 Norbert Bodendorfer: Okay, so then for brief overview of the talk, we will start with some preliminaries where we introduce what we mean by core screening in cosmological setting and after having done that, we will dive into. 22 00:03:54,930 --> 00:03:59,850 Norbert Bodendorfer: Applying this in the context of accuracy and first we will use these as you want one. 23 00:04:00,870 --> 00:04:07,200 Norbert Bodendorfer: group techniques to derive first quantum system that we will discuss and then we will hear if the core screening. 24 00:04:07,740 --> 00:04:12,600 Norbert Bodendorfer: And later we will use these techniques that we discovered here in section two and we will implement them. 25 00:04:12,960 --> 00:04:22,980 Norbert Bodendorfer: Now, on the accuracy hibbert space with a standard htc operators, yes, we will have a hamiltonian in terms of the volume operator and the point autonomy, so the exponential of the mean curvature. 26 00:04:23,460 --> 00:04:39,750 Norbert Bodendorfer: And they are, we would have this phenomenal 10 and 20 and in terms of the standard xc operators Now then, as promised, we will do this error estimation of this non anomalous dynamics and discuss where are these results that we derive your should apply and comfort. 27 00:04:41,730 --> 00:04:42,090 Okay. 28 00:04:43,590 --> 00:04:56,970 Norbert Bodendorfer: So first, a few words about a core screening or reorganization, so this is a key open problem in Luke quantum gravity and, in particular, this is a problem that is connected to many other key open problems so first of all. 29 00:04:57,540 --> 00:05:02,310 Norbert Bodendorfer: it's relevant for the smooth space time limit, where we want to get a continuum GR. 30 00:05:03,540 --> 00:05:12,600 Norbert Bodendorfer: And in this limit, then we also want the correct hamiltonian that X on some you know course degrees of freedom that would describe such a small space time limit. 31 00:05:13,110 --> 00:05:24,480 Norbert Bodendorfer: That is important because we want to kind of extract the correct dynamics from our theory and then, in turn, is of course important to go extra mental tests, because we want to have some reliable prediction support theory. 32 00:05:26,610 --> 00:05:33,390 Norbert Bodendorfer: know, at the same time, of course, screening why being important it's a very hard problem in a particular subtle problem in quantum gravity. 33 00:05:33,960 --> 00:05:48,810 Norbert Bodendorfer: And Much has been written about this by many authors and I hope I have captured the most important ones, but certainly I probably also forget some of them so sorry for that, but what I want to recall from this works is that. 34 00:05:49,950 --> 00:05:58,380 Norbert Bodendorfer: In kind of recent time people have started looking into this in detail and made some good progress, but in general, the problem is conceptually difficult. 35 00:05:58,920 --> 00:06:08,130 Norbert Bodendorfer: Which is because there's no natural energy scale which you can use in romanization here and also typically you find some democratization dependence and the types of models that we use in here. 36 00:06:08,700 --> 00:06:17,370 Norbert Bodendorfer: yeah and, on top of that, of course, it's computation involved because it requires you to some extent to solve the dynamics, or at least integrate out the five degrees of freedom. 37 00:06:19,200 --> 00:06:28,290 Norbert Bodendorfer: And now, here, given the state of affairs, it is useful now to try to gain some insights from point models and, as I said, this is what we want to do in this talk here. 38 00:06:31,440 --> 00:06:31,740 Okay. 39 00:06:33,240 --> 00:06:42,090 Norbert Bodendorfer: So now, I want to try to explain what I mean by core screening in a cosmological setting and why this makes sense yeah. 40 00:06:43,290 --> 00:06:46,680 Norbert Bodendorfer: So now, if you consider construction of a cosmological model. 41 00:06:48,180 --> 00:06:57,570 Norbert Bodendorfer: You know you start from 4G or, at least at the classical level, and then you making some surgery assumptions so today, we will make the strongest possible symmetry assumptions. 42 00:06:57,990 --> 00:07:04,590 Norbert Bodendorfer: But, in particular, we are we're demanding that our space, time is homogeneous now it just means that it looks the same everywhere. 43 00:07:06,120 --> 00:07:09,630 Norbert Bodendorfer: And now, this is a powerful tool that you can use in cosmology because it turns. 44 00:07:10,260 --> 00:07:12,720 Abhay Vasant Ashtekar: out, he was shaving space, time is homogeneous or just. 45 00:07:12,780 --> 00:07:15,210 Norbert Bodendorfer: A story, I mean i'm assuming that the spatial slice. 46 00:07:15,240 --> 00:07:16,290 Is homogeneous thank. 47 00:07:19,140 --> 00:07:31,500 Norbert Bodendorfer: You so we're assuming that the spatial slice is homogeneous and that means that we can track that physics, in a certain patch of the spatial slice and knowing that physics, we can extrapolate to the whole space of slice yeah. 48 00:07:32,340 --> 00:07:38,640 Norbert Bodendorfer: And you use that in cosmology in order to get kind of where you find faith based description, so you will. 49 00:07:39,060 --> 00:07:46,860 Norbert Bodendorfer: fix certain part of your space or slice that people then called a fuel cell and that may be this black box here on the left hand side yeah. 50 00:07:47,850 --> 00:07:55,890 Norbert Bodendorfer: And then, in that for you to sell you compute everything, so you integrate your homogeneous quantities over that cell you get some finite quantities. 51 00:07:56,220 --> 00:08:03,780 Norbert Bodendorfer: You have some well you find description in terms of them and you're doing your physics yeah and then you again extrapolate it to the rest of the universe by homogeneity. 52 00:08:06,360 --> 00:08:13,140 Norbert Bodendorfer: So this is fine and at the classical 11 you can show immediately that there is no problem in making this cell larger or smaller. 53 00:08:13,650 --> 00:08:21,210 Norbert Bodendorfer: However, at some point if you're doing a quantum theory and you may have some disagreements in your geometrical operators or geometrical quantities. 54 00:08:22,200 --> 00:08:31,020 Norbert Bodendorfer: You may, if you do not find a description of this way if you make yourself smaller or sub the white this big sell into many small cells like it's written on the right hand side here. 55 00:08:31,410 --> 00:08:39,420 Norbert Bodendorfer: And at some point, you may encounter the discrete and escape of your theory with the cells and then interesting things may happen yeah. 56 00:08:40,140 --> 00:08:53,250 Norbert Bodendorfer: And this is, in particular, where we will see later that interesting things will happen yeah so what we're going to discuss in this talk is we're going to take a fuel cell in the cosmological setting and we're going to stop divided into many fuel cells. 57 00:08:54,420 --> 00:09:02,280 Norbert Bodendorfer: And we are asking whether the dynamics that we compute on one side is equivalent to the dynamics that we compute, on the other side yeah. 58 00:09:02,850 --> 00:09:10,890 Norbert Bodendorfer: And, in particular, we would like to define kind of a true dynamics or the correct dynamics to be the one on the right hand side here, where we take many small sales. 59 00:09:11,400 --> 00:09:17,190 Norbert Bodendorfer: And then we compute the kind of course physics of this a few set of this many small cells here. 60 00:09:17,550 --> 00:09:26,670 Norbert Bodendorfer: And that will be the physics of this large cell and we wanted to obtain a description in this large scale that captures the physics of his many small cells on the right hand side here. 61 00:09:27,210 --> 00:09:34,620 Norbert Bodendorfer: yeah So this is the goal of this talk, and this is a perfectly well defined question that you can ask now if you have some quantum cosmological model. 62 00:09:35,340 --> 00:09:50,460 simone: Sorry, no but can I ask a naive question, yes, this here you put the refined and course green, but then you also see that the States are the same in all sales, yes, so it looks like the amount of information on the Left and the Right, it will be the same them. 63 00:09:51,600 --> 00:09:55,020 Norbert Bodendorfer: So, in what sense it is so so this I mean this refined arrow. 64 00:09:55,080 --> 00:10:00,690 Norbert Bodendorfer: You can strike it in the sense that this is of course not an exact error, I mean once you have the course theory, you cannot get the. 65 00:10:01,230 --> 00:10:05,130 Norbert Bodendorfer: Find theory, maybe I should not have put this refine arrow here but. 66 00:10:05,430 --> 00:10:15,630 Norbert Bodendorfer: What you are what you are doing, you have the theory and the right inside and you coarse grain and you get fewer observers on the left hand side here, in particular, you only get the total volume of all of these slices. 67 00:10:16,080 --> 00:10:21,510 Norbert Bodendorfer: yeah so the of the quantity, that you would have on this side would, for example, be the volume of the spatial slice of this big box. 68 00:10:22,260 --> 00:10:29,370 Norbert Bodendorfer: and clear the quantity that you consider you consider the volumes of these small boxes, but you sum them up, you know, so you have this equation down here. 69 00:10:29,730 --> 00:10:37,500 Norbert Bodendorfer: So the volume on the right hand side will be the sum of the volumes on volume on the left hand side will be the some of the volumes on the right hand side. 70 00:10:38,010 --> 00:10:54,870 simone: Fine, but I mean to to there be a real course Green, I would like to allow the individual states to be mean to why they keep this data, all the same, in the final picture all i'm losing when I was Green is the scaling factor really so. 71 00:10:54,870 --> 00:10:55,320 simone: it's done. 72 00:10:55,380 --> 00:11:02,400 Norbert Bodendorfer: yeah that is about, that is not correct yeah there's only correct and the what you would call the semi classical limit where quantum numbers are large. 73 00:11:03,210 --> 00:11:06,780 simone: Oh let's just think about it classically as a classical question I want to just. 74 00:11:06,780 --> 00:11:09,600 Norbert Bodendorfer: classically it's correct what you're saying, but quantum mechanical it's not. 75 00:11:11,160 --> 00:11:14,310 Norbert Bodendorfer: And there is because we, at some point he did a sweetness scale on the right. 76 00:11:14,520 --> 00:11:19,470 simone: Okay, so, even if the state is the same you mean that there are additional operators. 77 00:11:19,560 --> 00:11:21,780 Norbert Bodendorfer: That I can just, for example, say you have some. 78 00:11:21,870 --> 00:11:35,940 Norbert Bodendorfer: Some state here that has some superposition of volumes yeah and now you make a measurement on this sense in this right inside yeah now say you measure Volume one in this box volume two in this box and say we just have two boxes and then your volume three in. 79 00:11:35,940 --> 00:11:42,690 Norbert Bodendorfer: Total yeah, but you will also measure sometimes volume two in this box and Volume one in this box and it still is volume three. 80 00:11:43,080 --> 00:11:43,620 Abhay Vasant Ashtekar: yeah so. 81 00:11:44,250 --> 00:11:48,570 Norbert Bodendorfer: If you want to get volume three here they actually two possibilities to get volume three on this side is the. 82 00:11:48,630 --> 00:11:50,820 Norbert Bodendorfer: volume to Volume one and Volume one volume two. 83 00:11:51,480 --> 00:11:56,850 simone: yeah yeah more operators on one side and the others if you need the states are all the same, you can okay understand, thank you. 84 00:11:59,310 --> 00:12:09,870 Abhay Vasant Ashtekar: I also quick question which is you talk about defining scale and so on, I imagine you're talking about como escape rather than the physical body right because in. 85 00:12:09,930 --> 00:12:11,430 Norbert Bodendorfer: audio about the physical volume. 86 00:12:11,910 --> 00:12:14,400 Abhay Vasant Ashtekar: So you're refining your physical volume. 87 00:12:14,670 --> 00:12:26,160 Norbert Bodendorfer: Yes, so say say I have voice physical volume for here yeah, then I could have I could represent the state here, which is physical volume for by having physical Volume one in each of these boxes. 88 00:12:26,850 --> 00:12:29,280 Abhay Vasant Ashtekar: So therefore your boxes that fixed in time. 89 00:12:32,610 --> 00:12:40,230 Norbert Bodendorfer: know the box is fixed in time in the physical size of the box is not fixed in time yeah the coordinate size of the fusion centers fixed. 90 00:12:40,260 --> 00:12:42,000 Abhay Vasant Ashtekar: Right, so the corner sizes fixed. 91 00:12:42,030 --> 00:12:46,800 Abhay Vasant Ashtekar: But, therefore, this is division, referring to the coordinates size or is it different referring to the physics. 92 00:12:46,920 --> 00:12:48,810 Norbert Bodendorfer: Vision referring to the coordinate sighs. 93 00:12:48,810 --> 00:12:59,370 Abhay Vasant Ashtekar: yeah that's what I was my question, so the division because it's a finding of scale you're talking about dividing this into this part is really referring to the to the core moving gorgeous to the audience size. 94 00:13:00,420 --> 00:13:12,180 Abhay Vasant Ashtekar: Yes, yeah Okay, so of course the the volume of the physical volume of the box, even on the left hand side is getting smaller if we call it the past and the future Okay, yes. 95 00:13:13,290 --> 00:13:25,980 Norbert Bodendorfer: Exactly so then Okay, so I hope I have taught you what I mean by course grading now in this cosmological setting now I still want to capture some of these bullet points that I missed and then. 96 00:13:26,880 --> 00:13:38,550 Norbert Bodendorfer: say some other important analogy here, so one thing is, we set the states are the same in every cell, meaning there will be a quantum state in this fine cell, that is, some superposition of volumes eyeing states. 97 00:13:39,120 --> 00:13:51,990 Norbert Bodendorfer: And if we have a quantum state in this, so we put the same quantum state in all of the other cells yeah but that still means if we make a measurement we may measure one certain volume, I can sit here and one certain volume I sit here now, so we have to consider all. 98 00:13:52,500 --> 00:13:55,050 Norbert Bodendorfer: possibilities to get a certain course of volume if we. 99 00:13:55,050 --> 00:13:57,540 Norbert Bodendorfer: want to discuss this course of volume on the left hand side. 100 00:13:58,710 --> 00:14:06,600 Norbert Bodendorfer: And this idea to put the same quantum said everywhere, that is something that, for example, you also find in the govt condensates when you try to derive cosmology there. 101 00:14:08,490 --> 00:14:20,430 Norbert Bodendorfer: Now, also this idea of putting several a good legacy Petrus next to each other that has been investigated before by Edward and he was imposing some interactions between the cells. 102 00:14:21,810 --> 00:14:32,670 Norbert Bodendorfer: And that is something interesting that you can do in cosmology but he will do know interactions between the cells, so this is a simplification that we're doing here that will allow the computations that we're doing. 103 00:14:34,440 --> 00:14:36,450 Norbert Bodendorfer: and also what we. 104 00:14:38,010 --> 00:14:46,230 Norbert Bodendorfer: This question here that were like discussing here, in the context of course grading is a question that you can discuss also without mentioning call screening, and this is. 105 00:14:47,280 --> 00:14:54,000 Norbert Bodendorfer: equivalent to what people call influence of the video for cell size in new quantum cosmology and that has also been studied before. 106 00:14:56,580 --> 00:15:04,230 Norbert Bodendorfer: OK now The last point I want to mention here is here what I said is we discussed everything in the context of quantum cosmology. 107 00:15:04,680 --> 00:15:12,630 Norbert Bodendorfer: But you can now go back to a series of papers from 2015 and 16 and also to an IQ GS talk that I gave back then. 108 00:15:13,170 --> 00:15:19,320 Norbert Bodendorfer: where you can show that you can construct certain full quantum gravity theories which are very similar to quantum gravity. 109 00:15:19,740 --> 00:15:27,510 Norbert Bodendorfer: But they are built from variables and gauge fixings that you would do in Locarno cosmology now so you're trying to build to quantum gravity. 110 00:15:28,380 --> 00:15:44,250 Norbert Bodendorfer: as similar as possible to look on cosmology and when you do, that this picture changes as follows yeah So if you do that, you find that you again have something like spin it works as the quantum states in your theory and for every vertex have such a. 111 00:15:45,300 --> 00:15:49,410 Norbert Bodendorfer: spin at work, you would have one copy of the loop quantum cosmology here but space. 112 00:15:49,920 --> 00:16:01,500 Norbert Bodendorfer: yeah so in the sense if you would consider a single vertex or a quantum state with a single vertex in the theory, you would have one patch of loucon cosmology or one for us to sell of cosmology. 113 00:16:02,130 --> 00:16:09,720 Norbert Bodendorfer: And now, if you would consider more vertices or a finer state than you would have more of these patches yeah so in these full theory and readings. 114 00:16:10,080 --> 00:16:20,700 Norbert Bodendorfer: This question is precisely looking like that, and this is the question that you would usually ask in blue quantum gravity when you would discuss a core screen now, so you want to have a fine quantum state. 115 00:16:21,060 --> 00:16:24,900 Norbert Bodendorfer: And you want to extract the course dynamics of that quantum state yeah. 116 00:16:25,740 --> 00:16:34,110 Norbert Bodendorfer: But the details of these fulfilling readings, they go beyond the timeframe of this talk, so I will not say anything more about that, except that this question of. 117 00:16:34,680 --> 00:16:42,420 Norbert Bodendorfer: course going into quantum cosmology has also this a full theory embedding aspect that makes it again sensible to ask this question. 118 00:16:43,800 --> 00:16:44,070 Okay. 119 00:16:47,370 --> 00:17:00,450 Norbert Bodendorfer: So that is all about this kind of initial strategy that we want to follow, and now we can dive into some version of quantum cosmology and try to you know work out the details of the problem that we asked. 120 00:17:02,670 --> 00:17:09,090 Norbert Bodendorfer: And here we just mentioned a few things about the quantum cosmology model that we want to use, and this is a. 121 00:17:09,570 --> 00:17:16,530 Norbert Bodendorfer: promise, it will be the simplest possible, or at least the one that I consider as simple as possible, so we consider a space time. 122 00:17:17,010 --> 00:17:36,210 Norbert Bodendorfer: That has spatial slices which are flat, so the three dimensional richie skater is zero there homogeneous and there is a topic yeah so we can capture the physics completely by knowing the volume of the spatial slice and the mean curvature be and they have the standard canonical record. 123 00:17:38,040 --> 00:17:49,590 Norbert Bodendorfer: Now it's well known that if you just have vacuum GR with the symmetry assumptions This is equivalent to mean kosky space so it's not very interesting, so you need to add some form of matter to make or to get interesting dynamics. 124 00:17:50,310 --> 00:17:58,830 Norbert Bodendorfer: And there are various choices possible in here what we take is non rotating dusk dust, which is nice, because it gives you. 125 00:17:59,220 --> 00:18:12,900 Norbert Bodendorfer: A true rotation hamiltonian that has just the form of the gravitation Anatolian without the necessity to put any square roots or absolute values and it's last point that you don't even need absolute values is this customer paper by choose esky. 126 00:18:14,550 --> 00:18:29,490 Norbert Bodendorfer: Okay, so we have a true hamiltonian that because hga so it's a rotation part of the hamiltonian constraint, but now it's non vanishing because we have this the prioritization with respect to dust and it generates the dynamics of these wii and wii variables in this dust time. 127 00:18:30,510 --> 00:18:32,640 Abhay Vasant Ashtekar: is just one time, is it kept classical. 128 00:18:32,730 --> 00:18:42,690 Norbert Bodendorfer: it's cut classic as we, the parameter is at the classical level, and then we just so we have this classical theories vs vs this face face, and we have this hamiltonian and that we want to want us. 129 00:18:46,140 --> 00:18:46,500 OK. 130 00:18:48,060 --> 00:18:54,480 Norbert Bodendorfer: So now, there are several ways to get a quantum theory out of that in the standard htc way would be to promote. 131 00:18:54,960 --> 00:19:06,000 Norbert Bodendorfer: The volume to an operator and also the exponential of the mean curvature operators and you just need to put the kind of correct numbers here, so you will put one number that is some real number. 132 00:19:06,450 --> 00:19:12,540 Norbert Bodendorfer: And that sets your quantum geometry scale, and that is related to the Bavarian music parameter like that, if you. 133 00:19:13,350 --> 00:19:21,750 Norbert Bodendorfer: For example, thing of the standard derivations of quantum cosmology and then you will also multiply this by some integer end here. 134 00:19:22,110 --> 00:19:29,040 Norbert Bodendorfer: And the fact that you just need integers here comes from the typical properties of the hamiltonian constraints that it's super selects certain letters in. 135 00:19:29,310 --> 00:19:38,910 Norbert Bodendorfer: volume is States now you could also multiply that by another real number use of warcraft commodification as a habit space, but it's sufficient to use integer here. 136 00:19:40,080 --> 00:19:40,320 Okay. 137 00:19:41,640 --> 00:19:49,170 Norbert Bodendorfer: And this is something that we will do later in Section three, but before we do the standard quantization that we do another quantization which. 138 00:19:49,650 --> 00:19:57,840 Norbert Bodendorfer: Simply uses the idea that you're not required to quanta is directly V and the explanations of be but you're also allowed to quantify other faceless functions. 139 00:19:59,100 --> 00:20:07,350 Norbert Bodendorfer: And this, I want to make clear again on the next slide because it's an important point, and for this we just recall what does quantization need. 140 00:20:07,860 --> 00:20:21,840 Norbert Bodendorfer: yeah so quantization means the following you promote a point separating set of faith based functions to linear operators and inhibit space, so that the commentator of these operators reflects the classical person record structure. 141 00:20:23,550 --> 00:20:27,690 Norbert Bodendorfer: And what you need from these faith based functions as set is that their points separating meaning. 142 00:20:28,230 --> 00:20:39,600 Norbert Bodendorfer: That, if you know the values of these functions, you know the point where you are on Facebook yeah, for example, if your vm be you know where you are in faith based but you could also use some other function could use the D amp D plus beat for. 143 00:20:42,300 --> 00:20:48,750 Norbert Bodendorfer: Now this of course leaves a lot of arbitrariness, to pick these functions and you need some good guiding principle. 144 00:20:49,410 --> 00:21:03,390 Norbert Bodendorfer: And one of them is given to you by group quantization now, so the idea of group quantization now is to choose these functions in such a way that the form a plus on otter bra with a pulse on bracket so that this idea rise as a morphic to add algebra. 145 00:21:04,740 --> 00:21:05,010 Abhay Vasant Ashtekar: yeah. 146 00:21:05,700 --> 00:21:13,950 Norbert Bodendorfer: So assume you can do that as soon as possible, address as a morphing to suddenly algebra, then the problem of quantization trivializes. 147 00:21:14,550 --> 00:21:24,000 Norbert Bodendorfer: yeah because you can just take a math book and look up the representation theory of deadly algebra and then you already have all the representations and you know how your quantum theory looks like. 148 00:21:26,250 --> 00:21:32,460 Norbert Bodendorfer: And this is precisely what we're going to do today and we're going to do it by using the algebra se one one. 149 00:21:33,540 --> 00:21:40,860 Norbert Bodendorfer: And we will see later y Su and one is a good choice for that, because it has many properties that make it suitable and to link into quantum cosmology. 150 00:21:42,300 --> 00:21:47,370 Abhay Vasant Ashtekar: This is very beautiful and very natural, as you say that my equation, but long time ago. 151 00:21:49,230 --> 00:21:54,810 Abhay Vasant Ashtekar: But this is a choice, and so you could have chosen some other proposals, and you might get the correct answers yeah. 152 00:21:55,440 --> 00:22:03,510 Norbert Bodendorfer: Exactly and we're doing this year for purely technical reasons, so we will see that the group structure that we have allows us to do a course grading operation. 153 00:22:04,200 --> 00:22:11,070 Norbert Bodendorfer: And later, we want to relate that to the standard accuracy operators because they are we really can gain some insight how a normal estimate Tony may look like. 154 00:22:15,000 --> 00:22:20,160 Norbert Bodendorfer: OK, so now, maybe say a few words we now set we need. 155 00:22:20,250 --> 00:22:21,840 Norbert Bodendorfer: A set of faith based functions. 156 00:22:22,110 --> 00:22:27,300 Norbert Bodendorfer: So that the person either bra size, a morphic to Su one one yeah and that can be done. 157 00:22:27,690 --> 00:22:40,710 Norbert Bodendorfer: And I want to present this on this slide here and there will be several formulas on the slides, but the details of the formulas here are not important, and whenever something will be important for later it will come up again and that was saved in detail yeah. 158 00:22:42,000 --> 00:22:55,500 Norbert Bodendorfer: So just to start up here so so one one has three generators, and that you would call jeezy K plus and a minus yes and they're very similar to the generators of seo tool which will be jeezy J plus injury mice yeah. 159 00:22:55,860 --> 00:23:06,210 Norbert Bodendorfer: So particular this Jay Z would be something like an object that you soon magnetic quantum number it's diagonal in the standard way of writing on the representations and K plus and a minus they are letter operators. 160 00:23:07,110 --> 00:23:12,750 Norbert Bodendorfer: And the difference to the Su two operators are now some factors of I which come from the non compactness of this group. 161 00:23:14,490 --> 00:23:22,590 Norbert Bodendorfer: And they have an answer right now that is written here in this person record for, and will you can look at it, if you want, but it's not very important. 162 00:23:23,910 --> 00:23:35,190 Norbert Bodendorfer: What is little important that I want to mention, of course, this K plus minus they can be written in a different way as objects that he would call it K X and a y and this just the usual way in which you would construct letter operators. 163 00:23:37,890 --> 00:23:44,100 Norbert Bodendorfer: Now one possibility to get an SE one one advice given here in this box, so the Jay Z. 164 00:23:44,520 --> 00:23:52,830 Norbert Bodendorfer: That would just be the volume up to some constant where this Lambda is the same London that we will use to define new quantum cosmology in the standard way. 165 00:23:53,610 --> 00:24:03,330 Norbert Bodendorfer: And the K plus minus I given by this function here, so they contain exponential of the mean curvature know that there's a factor of two years so it's not just standard exponential. 166 00:24:03,750 --> 00:24:19,290 Norbert Bodendorfer: And they have this function of the volume here yeah and this vm for now, that is a free constant of that you can put here and whatever value of the end you put you still get this algebra up here and we will see in a second why it's important that you have a frequency. 167 00:24:22,590 --> 00:24:31,890 Norbert Bodendorfer: Now then, it said, you could instead take the objects JC K X and ky the algebra would then look slightly different and the. 168 00:24:32,580 --> 00:24:47,610 Norbert Bodendorfer: If you rewrite these functions here with this X and y you get these three functions here and we just want to take a brief look at them, so the Jay Z is still the same object now okay X has two terms up to Constance there's a volume here. 169 00:24:48,690 --> 00:25:01,770 Norbert Bodendorfer: And there is a hg here, so this will be a gravitational Antonia now, and this is something like a classically polarized version of the previous gravitation hamiltonian yeah so it's written down here. 170 00:25:02,310 --> 00:25:12,030 Norbert Bodendorfer: And it has the following contents, so it has assigned squared of London be overlander squared, so this is what we would typically substitute B squared with if you write it in. 171 00:25:18,600 --> 00:25:27,690 Norbert Bodendorfer: All this via very large, as opposed to this, and the end here, then this will be proportionate to the volume, so this object here would be the standard, I mean Tony. 172 00:25:28,530 --> 00:25:41,130 Norbert Bodendorfer: And here, if you take the very large this is proportional to V and this subtracts me you get zero here yeah so in something like a classical limit where these very large and B is much smaller than. 173 00:25:41,820 --> 00:25:59,340 Norbert Bodendorfer: One over Lambda you would get the classical hamiltonian back yeah so this hg in this pay X is something like polymerase version of your Antonia containing something that you would call a small volume correction now or an analogy to an inverse tried correction. 174 00:26:02,280 --> 00:26:03,510 Norbert Bodendorfer: Okay, and this. 175 00:26:03,540 --> 00:26:05,850 Abhay Vasant Ashtekar: But you would not send us at vm equal to zero. 176 00:26:06,870 --> 00:26:07,560 Norbert Bodendorfer: You could. 177 00:26:08,070 --> 00:26:10,590 Abhay Vasant Ashtekar: call this formula so far doesn't seem to. 178 00:26:10,650 --> 00:26:18,090 Norbert Bodendorfer: Nothing sinister grandson yes it is you could set it to zero, it is a free constant and we will see in a second what you should set it to. 179 00:26:18,810 --> 00:26:24,000 Abhay Vasant Ashtekar: Okay, so bigger than zero is not necessary vm is bigger than to conditions in one. 180 00:26:25,770 --> 00:26:26,310 Norbert Bodendorfer: So. 181 00:26:27,420 --> 00:26:33,120 Norbert Bodendorfer: We will see later in this talk that in the quantum model vm will be zero. 182 00:26:34,020 --> 00:26:36,660 Norbert Bodendorfer: airborne and an analog of vm will be zero. 183 00:26:36,870 --> 00:26:40,980 Norbert Bodendorfer: So your fundamental correct dynamics would just be signed squared times V. 184 00:26:41,490 --> 00:26:45,570 Norbert Bodendorfer: And this from getting the very end, the fundamental dynamics, but we will see. 185 00:26:46,920 --> 00:26:47,220 Abhay Vasant Ashtekar: That. 186 00:26:47,250 --> 00:26:49,830 Norbert Bodendorfer: The coarse grained dynamics will produce such. 187 00:26:49,890 --> 00:26:51,630 Norbert Bodendorfer: small volume corrections, so this would be. 188 00:26:52,980 --> 00:26:53,640 Norbert Bodendorfer: Electric. 189 00:26:54,000 --> 00:27:04,140 Abhay Vasant Ashtekar: Just a quick reminder that this algebra also arises of course it two plus one dimension gravity financial reason because it, and that was analyzed long time ago and the cashmere sweater. 190 00:27:05,040 --> 00:27:11,250 Abhay Vasant Ashtekar: associated with the observers of two plus one gravity basically but mass and angular momentum so just wanted to let you know. 191 00:27:12,060 --> 00:27:12,720 Norbert Bodendorfer: Okay, thank you. 192 00:27:15,360 --> 00:27:30,330 Norbert Bodendorfer: I just want to say one thing here this or Okay, thank you, so another remark about this algebra, so there is a see here, so the sky is an object that because see that is written here that, if you would take this classical limit now is something like volume times. 193 00:27:30,630 --> 00:27:32,220 Norbert Bodendorfer: mean curvature of the time speed. 194 00:27:32,820 --> 00:27:37,860 Norbert Bodendorfer: And that is what you would call the complexity fire in the complexity fire coherent states. 195 00:27:38,160 --> 00:27:49,590 Norbert Bodendorfer: yeah so when this address typically discussed in the literature, you would call it the ch ar D Russell complexity fire volume and hamiltonian algebra so, at least if you take the classical limit here. 196 00:27:50,610 --> 00:27:57,630 Norbert Bodendorfer: And rearrange it a little, then you can write decided muscle that CV and H our map to these three generators. 197 00:27:57,930 --> 00:28:08,880 Norbert Bodendorfer: If you have this polymerization here, then you need to change that a little and have these combinations that are right here yeah but this seems to be good comes from complexity fire, so this is like the polymerase version of the complexity. 198 00:28:11,370 --> 00:28:11,670 OK. 199 00:28:13,020 --> 00:28:25,620 Norbert Bodendorfer: So now, one thing you may worry about is that before we envy envy, which are two independent functions, but now we have three functions yeah so there should be some relation between JC K plus and a minus. 200 00:28:27,030 --> 00:28:36,900 Norbert Bodendorfer: And to get this relation you compute the cosmic operator, so you take the cosmic operator insert this classical quantities and what you get as a vm squared over four Lambda squared. 201 00:28:37,680 --> 00:28:44,880 Norbert Bodendorfer: And this now tells you that this vm this is encoded in the coming year of the group. 202 00:28:45,420 --> 00:28:58,110 Norbert Bodendorfer: yeah and now, since when you quantity quantity in a certain representation of your group, this vm will determine which representation you're in, or in other words, if you are in a certain representation of the mst turn. 203 00:29:01,050 --> 00:29:01,350 Okay. 204 00:29:04,020 --> 00:29:12,990 Norbert Bodendorfer: Good so now let us believe that the calculations here are correct and the SE one one algebra is reproduced and then we can go directly to the quantum theory. 205 00:29:14,790 --> 00:29:22,410 Norbert Bodendorfer: So now we need to discuss the representations of Su one one and, once you open your math book you find that there are actually many of those representations. 206 00:29:23,730 --> 00:29:33,690 Norbert Bodendorfer: And we don't need all of them today, we just need to have them yeah so there's one set of representation that is called the unitary discrete series. 207 00:29:34,260 --> 00:29:41,220 Norbert Bodendorfer: And that set of representations is very similar to the standard as you do representations and they're labeled by a half integer Jay. 208 00:29:41,910 --> 00:29:53,700 Norbert Bodendorfer: Jay is one half one three half and so forth, and within such a representation, you have magnetic quantum numbers that started Jay and then go to G plus one G plus two, and all the way up to infinity. 209 00:29:54,270 --> 00:30:04,470 Norbert Bodendorfer: yeah so these representations, they are infinite dimensional, and this is necessary because we have a non compete group, and if you have a unitary representation, has to be infant dimensional. 210 00:30:07,920 --> 00:30:16,590 Norbert Bodendorfer: Now there's another representation that we will use today, and that is the defining representation of this group, so this is a representation on see tool. 211 00:30:16,980 --> 00:30:25,440 Norbert Bodendorfer: yeah so on the vectors containing to complex numbers and that turns out to be a non unitary representation which again has to be the case because it's finite dimension. 212 00:30:29,100 --> 00:30:37,710 Norbert Bodendorfer: And what we will start now with this this first bullet point here, so we will quantifies the previous the system from the previous page using such a representation. 213 00:30:39,960 --> 00:30:54,660 Norbert Bodendorfer: Now, if you do that Jay Z will act like the typical gc in Su tool, so you will have quantum states, they will buy the representation Jay and the magnetic quantum number em and Jay Z we're just multiply them by m yeah. 214 00:30:55,800 --> 00:31:04,890 Norbert Bodendorfer: And now, this means that jeezy classically that was volume over to Lambda so that volume over to London operates diagonally on the States. 215 00:31:05,910 --> 00:31:12,630 Norbert Bodendorfer: yeah, this means that m this quantum number m, this is proportional to the volume after this factor to Lambda. 216 00:31:13,020 --> 00:31:18,840 Norbert Bodendorfer: And therefore em if you want to later take analogies to full energy is similar to an energy spin. 217 00:31:19,230 --> 00:31:30,360 Norbert Bodendorfer: Because an ekg spin, for example, sets the size of an area for like EG, and here the analog is the volume, there is a geometric quantity that we track, and this is tracked by this quantum number m. 218 00:31:33,720 --> 00:31:41,070 Norbert Bodendorfer: OK, so now, you also have the letter operators K plus minus year they were just changed this magnetic quantum number of a plus minus one. 219 00:31:41,910 --> 00:31:49,470 Norbert Bodendorfer: And since the lowest possible m is J K minus acting on the State jj test to kill the state. 220 00:31:50,100 --> 00:32:01,170 Norbert Bodendorfer: As similar to what happens in Su to, and therefore there has to be a smallest possible resolved volume here yeah so this J sets the smallest possible result volume in your theory. 221 00:32:01,560 --> 00:32:09,870 Norbert Bodendorfer: And this is what we will later consider as an organization scape so this gave the smallest possible geometric scale that resolve is our our escape. 222 00:32:11,280 --> 00:32:22,440 Norbert Bodendorfer: yeah so Therefore, we should always keep in mind that the JCR they labeled the representation, but what is really analogous to negligee spin something that sets the size of some geometric object that is this me here. 223 00:32:24,930 --> 00:32:25,200 Okay. 224 00:32:26,220 --> 00:32:32,850 Norbert Bodendorfer: So now we can come back to this question what this vm should be, and that is a little subtle once you come from this classical. 225 00:32:33,180 --> 00:32:38,370 Norbert Bodendorfer: A perspective few, but it will become very clear in the next section and once we are in the quantum theory. 226 00:32:38,790 --> 00:32:45,660 Norbert Bodendorfer: So here we would just say, we take the cosmic operator and the cosmic operator, and so one one gives you J amp J minus one. 227 00:32:46,320 --> 00:32:57,030 Norbert Bodendorfer: And the you know compare that with the cosmic operator, on this side here, and when you do that, you will just find that this minimum volume so should be to Jay. 228 00:32:57,570 --> 00:33:06,150 Norbert Bodendorfer: To Lambda J yeah at leading order, meaning that you get the kind of second order nj correction incorrect, if you take. 229 00:33:06,660 --> 00:33:13,740 Norbert Bodendorfer: This precise description here yeah so this Kazimierz matched only two leading order, but this is something acceptable because. 230 00:33:14,490 --> 00:33:19,590 Norbert Bodendorfer: You know this just a quantum direction, and again in the next section, you will see that there is a precise way. 231 00:33:20,490 --> 00:33:33,180 Norbert Bodendorfer: of how to determine this vm appearing in these objects here in the quantum theory and there are things are more subtle, because we need to get the ordering of these operators correct and then we will see that everything works out exactly. 232 00:33:38,820 --> 00:33:49,140 Norbert Bodendorfer: Good so yeah, so now we have this quantum theory defined we know what the volume is and we could do, in principle, some computations now for some competitions, we need some quantum states. 233 00:33:50,610 --> 00:34:03,750 Norbert Bodendorfer: And there's a very convenient set of quantum States here which are given by period room of coherence states, and these are some this a set of coherence, states that has very nice properties and they are derived using group theoretic techniques. 234 00:34:05,280 --> 00:34:12,210 Norbert Bodendorfer: Now I don't want to give the details here, I just want to mention that the state this paranormal coherent State that is labeled by two numbers. 235 00:34:12,960 --> 00:34:19,680 Norbert Bodendorfer: That representation in which we constructed and a set of too complex numbers zero nz one. 236 00:34:20,100 --> 00:34:26,520 Norbert Bodendorfer: That We grew up together into one object Z that you may call the spinner, and that is typically referred to as a spinner the literature. 237 00:34:27,180 --> 00:34:40,920 Norbert Bodendorfer: And once you have those numbers, you can write down the state like this, so you have an expansion in this magnetic quantum numbers, we have some over here, and then there are some factors depending on me and Jay and then there are some factors depending on this complex numbers. 238 00:34:42,510 --> 00:34:46,860 Norbert Bodendorfer: And intuitively how the state is constructed, well, it has to dependencies said. 239 00:34:47,220 --> 00:34:58,470 Norbert Bodendorfer: One of them be terms of representation you're in so you can take the representation you're taking the lowest weight states, the lowest volume state in that representation and you act on that with a group element of so one one. 240 00:34:59,100 --> 00:35:11,100 Norbert Bodendorfer: And this action on the roof element, we will see later can be transferred to the spinners so the spinners here they capture the group element that you were acting on the last wait state with. 241 00:35:12,750 --> 00:35:18,390 simone: yeah sorry, just to clarify, I suppose a priori only need the one complex number to. 242 00:35:19,770 --> 00:35:34,230 Norbert Bodendorfer: record a new one complex number it's more convenient to use to, because then you can show that, on this too complex numbers, the group will act in its fundamental representation in yeah and it's different so in its defining representation. 243 00:35:35,130 --> 00:35:40,680 simone: And their override on the resolution of the interview just take action wait on the Norman face of the. 244 00:35:41,790 --> 00:35:42,930 simone: spinners I suppose. 245 00:35:43,740 --> 00:35:52,590 Norbert Bodendorfer: You put I guess it's correct what you say, so you put a data function, you have four dimensional space and you put a data function Okay, thank you. 246 00:35:54,090 --> 00:35:54,390 simone: Yes. 247 00:35:56,340 --> 00:36:02,220 Norbert Bodendorfer: Good, so now we have those states, and now we can compute something, and we can for example compute expectation values. 248 00:36:04,230 --> 00:36:20,250 Norbert Bodendorfer: And this will lead us to the idea of the core screening and when we develop this back in 2018 I think this was just you know the tribe of luck and we later understood why this works and on the next slide I will tell you why all of this works. 249 00:36:21,720 --> 00:36:37,350 Norbert Bodendorfer: So what you compute now is you compute the expectation value in these coherence states have an operator G with G is just a linear combination of your generators now so some you have some efficiency and you might apply them with your generators, and you just take the expectation back. 250 00:36:39,360 --> 00:36:45,510 Norbert Bodendorfer: And this, you can look up in you know many previous papers, but this expectation that it is, and it comes out like this. 251 00:36:45,900 --> 00:36:59,880 Norbert Bodendorfer: yeah and the important thing here is that effect arises in terms in fact arises in Jay which multiplies the expression, and then effect arises into some function of this spinner Z that depends on how you choose your coefficients here. 252 00:37:03,330 --> 00:37:18,030 Norbert Bodendorfer: And now, remembering this we go back to our picture of course painting let's go back to this picture and now say we have a quantum state in the fine sell here and it's the same quantum state in every cell. 253 00:37:19,260 --> 00:37:35,730 Norbert Bodendorfer: And we now measure the expectation value of the volume in one of the cells, then we get some number, and now we are supposed to measure the expectation values of the volume in all of these cells and some of them up to get the course quality yeah, but we know, since the states in every. 254 00:37:37,170 --> 00:37:45,270 Norbert Bodendorfer: cell are the same this expectation value will be the same every time, so instead of summing them four times we can just multiply one of them by four. 255 00:37:48,630 --> 00:37:51,810 Norbert Bodendorfer: So going back then to this computation. 256 00:37:53,070 --> 00:38:03,720 Norbert Bodendorfer: We see that if we want to increase the value of g by four and say gee is something that we call screens, like the volume, then it would be sufficient to multiply J by four. 257 00:38:05,010 --> 00:38:15,120 Norbert Bodendorfer: So if you increase J by effect or four and the result will be increased by a factor of four and then we would have the course quantity, instead of the fine quantity referring to this previous picture. 258 00:38:17,790 --> 00:38:26,820 Norbert Bodendorfer: Now, for this to work all of these operators have to be intensive have to be extensive operators, meaning that they scale with a volume now let's go back. 259 00:38:29,820 --> 00:38:37,890 Norbert Bodendorfer: To this page here, we cannot check that everything your scales, with the volume or the generator scale with volume, yes, when jeezy certainly scales, with the volume. 260 00:38:39,000 --> 00:38:54,180 Norbert Bodendorfer: This K plus minus also skates with the volume, because this exponential does not scale with the system size, the volume here scales quite radically, but then the square root makes that into linear scaling and now the only thing we have to check us at this vm also scales, with the volume. 261 00:38:55,620 --> 00:39:05,520 Norbert Bodendorfer: And it does yeah because it is to London J and J, we want to scale with the volume yeah and this vm in particular would be something like. 262 00:39:06,030 --> 00:39:13,290 Norbert Bodendorfer: A minimum of volume, he has since it's to Lambda J it's the minimum value that the volume can take, and that should obviously scale with the system size. 263 00:39:13,740 --> 00:39:20,520 Norbert Bodendorfer: Because, if you have a minimum volume in each of the cells than the minimum of volume in the course cell is four times the minimum volume in these cells. 264 00:39:22,290 --> 00:39:30,870 Norbert Bodendorfer: yeah so the fact that all of these generators are extensive quantities is consistent with this course gaining idea that we have here. 265 00:39:32,910 --> 00:39:50,940 Norbert Bodendorfer: So, to put this in formula again so if we start with some coherent State that is labeled by a J, not a representation J not and some spinner Labor Z record screen it by changing the representation to n times J not where we coarse grain and sales together. 266 00:39:51,960 --> 00:39:55,290 Norbert Bodendorfer: And the spinner label we just leave as it is, we don't change that. 267 00:39:56,430 --> 00:40:03,630 Norbert Bodendorfer: Because the Z compute intensive quantities which are ratios of extensive quantity so, for example, the ratio of. 268 00:40:04,020 --> 00:40:15,660 Norbert Bodendorfer: The hamiltonian integrated over yourself by the volume of yourself yeah there is something like the energy density is there, and that is invariant and we see here Jay would cancel out of this calculation. 269 00:40:18,120 --> 00:40:31,290 Norbert Bodendorfer: And that's for operators, I said before, if we want the total volume, we would sum up the volume operators in each of the fine sales and in the course sell we know take again the volume operator or the generator of. 270 00:40:32,550 --> 00:40:36,450 Norbert Bodendorfer: Of so one, one that we had up here, but with the higher representation J. 271 00:40:38,460 --> 00:40:48,300 Norbert Bodendorfer: Okay yeah So this is the idea of this course screening, and this is suggested by looking at this computation which no deals just with these operators three zeke a plus and a minus. 272 00:40:50,880 --> 00:40:54,210 Abhay Vasant Ashtekar: Can I just ask a question, this is very, very, very pretty structure here. 273 00:40:56,490 --> 00:41:05,370 Abhay Vasant Ashtekar: But supposing I had not done as you went on at all or not, then group a vertical quantization but just stuck with the original things that you gave us, which has been V. 274 00:41:06,030 --> 00:41:08,070 Abhay Vasant Ashtekar: And they're also a notion of course grading. 275 00:41:09,210 --> 00:41:13,320 Abhay Vasant Ashtekar: And I could have done this because you know the same vehicles to four times we etc. 276 00:41:14,490 --> 00:41:15,870 Abhay Vasant Ashtekar: And I got some quantum numbers and so. 277 00:41:17,100 --> 00:41:29,820 Abhay Vasant Ashtekar: My question is really whether this is using these 211 and the specific representations on fsu on one is that just streamlining the process or dancers can depend on. 278 00:41:30,090 --> 00:41:35,670 Norbert Bodendorfer: As far as I see this is purely technical tool to get to the next section. 279 00:41:36,510 --> 00:41:44,040 Norbert Bodendorfer: Okay, so, as you say that the question is where do you find the corresponding questions completely when you find without any appeal to group structure here yeah. 280 00:41:44,850 --> 00:41:51,870 Norbert Bodendorfer: However, when you try it, you will run into difficulties that you cannot do it analytically and this group structure here helps you to do it analytically. 281 00:41:53,010 --> 00:41:57,150 Abhay Vasant Ashtekar: Okay, but so basically you're saying that well there's no loss of generality. 282 00:41:58,200 --> 00:42:02,670 Abhay Vasant Ashtekar: in choosing not only group structure, but the specific representation that you're choosing, or is there. 283 00:42:03,060 --> 00:42:07,410 Norbert Bodendorfer: A loss in general that you have right now we're defining a certain quantum theory yes okay. 284 00:42:07,920 --> 00:42:13,650 Abhay Vasant Ashtekar: Okay, so therefore if I just stuck with dmv and coarse grain and so on, I might get any good answers I mean I. 285 00:42:13,740 --> 00:42:14,310 Norbert Bodendorfer: may get an. 286 00:42:15,240 --> 00:42:23,250 Norbert Bodendorfer: Answer unless you pick the hamiltonian in terms of D amp D and precisely such a way that it produces this as one one structure. 287 00:42:23,340 --> 00:42:24,480 Abhay Vasant Ashtekar: Exactly exactly. 288 00:42:25,620 --> 00:42:29,400 Abhay Vasant Ashtekar: i'm completely with you, I just want to know what the freedom isn't Thank you this is good. 289 00:42:30,870 --> 00:42:32,520 Norbert Bodendorfer: Okay, so now. 290 00:42:33,150 --> 00:42:40,350 Norbert Bodendorfer: You obviously wanted to check to which extent this calculation you generalize this so to which operators and states. 291 00:42:41,130 --> 00:42:48,690 Norbert Bodendorfer: And what I can tell you, and I have a detailed extra slide if you're really interested in the details, but this computation seems to be. 292 00:42:49,080 --> 00:42:57,270 Norbert Bodendorfer: Correct for arbitrary coherent state matrix elements, meaning that you don't need the same spinner on the two sides, here we can take different ones. 293 00:42:57,750 --> 00:43:04,800 Norbert Bodendorfer: And it holds for arbitrary polynomials in the generators, so you could do something like Jay Z to the power 10 J X, to the power five and. 294 00:43:05,430 --> 00:43:16,440 Norbert Bodendorfer: Okay excellent a man's heart 17 something like that yeah so for all of these said holds and we have proven that and that is a long computation to prove that but it works out. 295 00:43:16,860 --> 00:43:18,810 simone: yeah, so this is a leading order you're. 296 00:43:18,810 --> 00:43:20,940 simone: interested in, I suppose largest believe it. 297 00:43:21,150 --> 00:43:23,310 Norbert Bodendorfer: No, this is exactly correct. 298 00:43:24,750 --> 00:43:28,560 Norbert Bodendorfer: This is exactly correct for all J and for all orders. 299 00:43:31,530 --> 00:43:36,810 Norbert Bodendorfer: So so for this, you first have to I mean there are several subtleties in the computation, for example. 300 00:43:37,710 --> 00:43:39,870 Norbert Bodendorfer: Here we see that you know. 301 00:43:41,070 --> 00:43:55,590 Norbert Bodendorfer: Extensive operators or something that scales, with the volume is linear in J now this, for example, does not mean that the square of the volume is linear in J squared yeah it would be J squared at leading order and then it will have something in corrections and so forth. 302 00:43:56,610 --> 00:43:58,320 Norbert Bodendorfer: And you get all of them exactly correct. 303 00:43:59,640 --> 00:44:00,720 Norbert Bodendorfer: By this prescription here. 304 00:44:02,580 --> 00:44:05,520 Norbert Bodendorfer: And if you want to see the details, I have an extra slide for that. 305 00:44:05,550 --> 00:44:07,710 simone: No, I understood Okay, thank you, yes. 306 00:44:07,740 --> 00:44:13,470 Abhay Vasant Ashtekar: yeah So those are the basic point is that you're restricting yourself to a set of operators and for them, this is exactly. 307 00:44:14,010 --> 00:44:23,460 Norbert Bodendorfer: Yes, but the operators, I mean the course operators, we have our you know more or less as general as you can get you an arbitrary putting your meals in this in a full set of operators. 308 00:44:25,980 --> 00:44:28,500 Abhay Vasant Ashtekar: just said that if I do square then i'm not going to get it right. 309 00:44:28,680 --> 00:44:36,120 Norbert Bodendorfer: No, you get it correct you get it correct for all you get it correct for the linear once and you get it correct for all of the polynomial send them. 310 00:44:38,280 --> 00:44:41,220 Abhay Vasant Ashtekar: Okay, so I have the same question okay we'll talk later okay. 311 00:44:41,670 --> 00:44:57,300 Norbert Bodendorfer: So there's an extra slide which shows you the formula for all the higher powers, but you really get it correct for all polynomials not just not just an eating order anything Okay, and this came surprising to me as well, and now I will tell you why this has to be precisely like that. 312 00:44:58,200 --> 00:45:01,830 Abhay Vasant Ashtekar: And then, this last one more quick question is there a factor ordering in one linear polynomials. 313 00:45:02,010 --> 00:45:06,300 Abhay Vasant Ashtekar: Of course, so so therefore you're saying that for some specific factory you get the same answer. 314 00:45:07,800 --> 00:45:16,590 Norbert Bodendorfer: Yes, so, for example, if you if you have the operator, you know JC K plus and jeezy and K plus they don't commute, then you know. 315 00:45:17,910 --> 00:45:24,570 Norbert Bodendorfer: Jay Jay Z and you add all of those operators for jeezy and you have K plus and add all of those and you put them in the same ordering of course. 316 00:45:27,030 --> 00:45:32,010 Abhay Vasant Ashtekar: So there is a fact, yes, so that is a factor or in for polynomials so that is a structure that lighting. 317 00:45:32,910 --> 00:45:34,410 Norbert Bodendorfer: Natural hector ordering, it is the. 318 00:45:34,470 --> 00:45:37,350 Norbert Bodendorfer: kind of the trivial one that you would get just by this substitution. 319 00:45:38,040 --> 00:45:39,990 Abhay Vasant Ashtekar: Okay, thank you okay. 320 00:45:40,710 --> 00:45:42,660 Norbert Bodendorfer: So now as. 321 00:45:42,750 --> 00:45:51,240 Norbert Bodendorfer: As we just saw that may come as a surprise that this works out, but you know with sufficient hindsight, it is not a surprise that because of group theory. 322 00:45:51,870 --> 00:46:07,110 Norbert Bodendorfer: And the point is that typically higher representations meaning representations with larger J they build from products have lower representations and then just picking out some of the representations in the product yeah. 323 00:46:07,620 --> 00:46:16,380 Norbert Bodendorfer: And, for example, typically you can get like the highest representation in such a product and by taking the total symmetry ization of your. 324 00:46:18,150 --> 00:46:28,590 Norbert Bodendorfer: Nor representations yeah and this total symmetry ization happens because of our homogeneity assumption, more specifically, that the States are the same in all of the cells. 325 00:46:29,700 --> 00:46:39,240 Norbert Bodendorfer: So we have a symmetric product of identical states, so we have this representation J, not with a spoon or Z which is just you know encoding group element, acting on this. 326 00:46:39,600 --> 00:46:53,670 Norbert Bodendorfer: lost weight state, we have a tensor product of n times of them and that now maps into the state, with the same group element here, but with the higher representation yeah, and this may be familiar to you, because this is how you construct higher representations. 327 00:46:56,010 --> 00:47:03,630 Norbert Bodendorfer: And now the fact that all of the other computations work out is then related to the fact that we use this parallel of coherent states. 328 00:47:03,870 --> 00:47:06,300 Norbert Bodendorfer: and, more specifically it's related to the property. 329 00:47:06,540 --> 00:47:17,580 Norbert Bodendorfer: That they are constructed by acting with the group element on the lowest weight vector so this lowest weight vector is important, and if you go through the gory details, you will see that it's important that is that, because then some terms ash. 330 00:47:18,630 --> 00:47:23,550 Norbert Bodendorfer: In particular, this would remove unwanted representations if you kind of. 331 00:47:29,520 --> 00:47:30,870 Ivan Agullo: And then you. 332 00:47:31,620 --> 00:47:38,640 Norbert Bodendorfer: are having this in mind, you can see some other details, for example, if you take a product of. 333 00:47:39,780 --> 00:47:48,810 Norbert Bodendorfer: Last week states in your find description and the Act on each of them with a group element that is now a group element g of se one one. 334 00:47:49,470 --> 00:48:04,770 Norbert Bodendorfer: In representation J not and you act on each of these states in the find representation, and that is the same as acting on the course representation, where this is no the lowest weight state in the course representation, with the limit in the course representation yeah. 335 00:48:05,850 --> 00:48:09,120 Norbert Bodendorfer: So this is an important thing that we're going to use and. 336 00:48:09,240 --> 00:48:10,740 Norbert Bodendorfer: The other thing which is a very nice. 337 00:48:10,800 --> 00:48:11,370 property. 338 00:48:13,080 --> 00:48:14,910 Norbert Bodendorfer: You derive this as a path into go. 339 00:48:15,630 --> 00:48:18,930 Norbert Bodendorfer: Is that if you act with your. 340 00:48:21,900 --> 00:48:36,240 Norbert Bodendorfer: Some cheesy state and you act on that with the SEC one run representation, and you can completely absorbed that action as an action on this spinner variable we're now this spinner representation is a two dimensional defining representation of se one one. 341 00:48:37,770 --> 00:48:46,620 Norbert Bodendorfer: So in this is why we have to complex numbers here so that we have this action on this in this fundamental representation or defining representation here. 342 00:48:47,940 --> 00:48:54,720 Norbert Bodendorfer: yeah So this is the deeper reason why the computations that someone somewhat surprising he worked actually have to work out. 343 00:48:56,820 --> 00:48:57,060 Okay. 344 00:48:58,410 --> 00:49:08,160 Norbert Bodendorfer: And now, with them, we can conclude with another nice property of this course mining operation, which is based on this property we just vote on the previous slide so this. 345 00:49:09,270 --> 00:49:12,750 Norbert Bodendorfer: Action of a group element on this coherent state is absorbed in the spinner. 346 00:49:14,100 --> 00:49:24,540 Norbert Bodendorfer: And now, what happens is the gravitation hamiltonian we didn't talk about it, yet, but well, it was hidden in these variables like this yeah so we have the K X. 347 00:49:24,990 --> 00:49:35,340 Norbert Bodendorfer: That contains experimentation and tuning and the volume and you cannot obtain this reputation and Tony by subtracting the volume from the kayaks and That just leaves you with this hg are two factors. 348 00:49:38,970 --> 00:49:49,350 Norbert Bodendorfer: So therefore hg is given by a sum of generators like this, and that means that if you exponential that hamiltonian to obtain the finite. 349 00:49:50,430 --> 00:50:01,350 Norbert Bodendorfer: Time of illusion and the final time evolution is an seo one one group element and by this equation here at X on the spinner or you can completely put the time evolution onto the spinner. 350 00:50:02,010 --> 00:50:10,380 Norbert Bodendorfer: And then time evolution is very simple, because the two dimensional representation of seo one is very simple, you just need to exponential a two dimensional matrices. 351 00:50:12,960 --> 00:50:25,140 Norbert Bodendorfer: And this, in turn, means that dynamics commutes with a core screening yeah in the following sense, you can act with your hamiltonian in the fine representation on each of your sub states. 352 00:50:26,520 --> 00:50:36,450 Norbert Bodendorfer: And that is the same as acting with the hamiltonian in the course representation yeah, the only thing you need to change then it's a representation office hg meaning the representation of pay. 353 00:50:36,450 --> 00:50:37,380 Ivan Agullo: X and JC. 354 00:50:38,970 --> 00:50:45,120 Norbert Bodendorfer: To act on this state yeah so you can either in the final four in the course and you always get the same answer. 355 00:50:46,410 --> 00:50:49,140 Norbert Bodendorfer: And of course you can also transfer this now to the spinner. 356 00:50:49,680 --> 00:50:53,160 Norbert Bodendorfer: You can arrive in this equation from both sides, if you want, and you can. 357 00:50:54,660 --> 00:50:56,040 Completely this this operation. 358 00:51:04,020 --> 00:51:04,650 Norbert Bodendorfer: So now. 359 00:51:04,710 --> 00:51:05,520 Norbert Bodendorfer: At this point. 360 00:51:05,610 --> 00:51:11,310 Norbert Bodendorfer: If you want, so this coherent States they give you complete or give you and overcome the basis of your heart space. 361 00:51:12,030 --> 00:51:16,740 Norbert Bodendorfer: You have a soft the time evolution explicitly if you just do this computation here, which is quite simple. 362 00:51:17,490 --> 00:51:32,310 Norbert Bodendorfer: And you are essentially done with your quantum theory now you have an explicit core screening here which just tells you that you change this, so one one representation in which you have this theory and that gives you the coarse grained version of many smaller. 363 00:51:33,750 --> 00:51:34,590 Norbert Bodendorfer: kind of cells. 364 00:51:36,060 --> 00:51:49,380 Norbert Bodendorfer: But now, this is not satisfactory, for us, because we would like to understand how a typical loucon cosmology hamiltonian, and that is an operator, that is built from volume and the point alone to me is. 365 00:51:50,910 --> 00:51:53,400 Norbert Bodendorfer: renomination, so this is what we would like to understand. 366 00:51:54,900 --> 00:51:59,400 Norbert Bodendorfer: and for this we have to go now to Section three and we have to do some other computation. 367 00:52:01,650 --> 00:52:09,030 Norbert Bodendorfer: So again, up to now, we quantization a classical Su one one course on algebra, meaning that we have. 368 00:52:09,990 --> 00:52:27,150 Norbert Bodendorfer: You know perfectly fine quantum operators for all of these se one one generators, but we do not know how they are built from volumes and and point anonymous as quantum operators now because we don't know that there's no there is not contained in this description. 369 00:52:29,640 --> 00:52:41,010 Norbert Bodendorfer: So the goal now is the following, we want to do this now on n copies of the standard look on cosmology here that space where we start with a volume and the point alone to me as operators. 370 00:52:42,120 --> 00:52:56,310 Norbert Bodendorfer: And the task is now that we find around parameter family of operators built from these here that reproduce so one one jira on this htc about space in the representation J. 371 00:52:57,450 --> 00:53:06,930 Norbert Bodendorfer: yeah, because if we have this one parameter family of operators yeah and they they satisfy this as one one either been representation J. 372 00:53:07,320 --> 00:53:24,060 Norbert Bodendorfer: Now, then we can just take some small J here's a good one half to be the standard a QC hamiltonian yeah and then we can simply read off the coarse grained hamiltonian by just taking these operators in the higher representation so representation with larger g. 373 00:53:25,620 --> 00:53:32,940 Norbert Bodendorfer: yeah, and that is kind of the task, so this is a well defined task, and that is a task that has been done yeah. 374 00:53:34,860 --> 00:53:45,810 Norbert Bodendorfer: And the result is given now here in this box, so the details are not important to remember here, we will see the important things, we will see again on a later slide. 375 00:53:46,410 --> 00:53:54,600 Norbert Bodendorfer: But what I want to say here is these and other operators JC K X and y that we saw before they're just in a suitable ordering now. 376 00:53:55,170 --> 00:54:12,540 Norbert Bodendorfer: Yes, for example, you see Jay Z so the volume by to Lambda and there is no ordering choices necessary here, and this again tells us that the volume is values are to London N, where am is not an element starting a Jay and then you add some unnatural number or zero. 377 00:54:14,550 --> 00:54:24,870 Norbert Bodendorfer: Okay, so now let's look at K X X previously contained this reputation hamiltonian you see that here so there's a sign volume sign. 378 00:54:25,380 --> 00:54:34,500 Norbert Bodendorfer: That is typically your rotation hamiltonian but it now has again a free constant that'd be now called vm twiddle, and this will be slightly different than the previous vm as we will see. 379 00:54:36,090 --> 00:54:48,510 Norbert Bodendorfer: And then, it has again terms that you can interpret as small volume corrections so terms that are relevant when when this volume is close to this minimum volume, but they are irrelevant when the volume is very large, as opposed to this minute volume. 380 00:54:51,990 --> 00:54:52,290 Okay. 381 00:54:53,460 --> 00:55:06,660 Norbert Bodendorfer: Now ky this was this complexity fire which was volume times sign of to Lambda classically and the sign of to Lambda is now split into cosine and sine and you have this peculiar ordering where the course and this one time on the left and one on the right. 382 00:55:08,340 --> 00:55:08,550 yeah. 383 00:55:09,630 --> 00:55:19,290 Norbert Bodendorfer: So this is complicated computation to check off these operators reproduces as you want one, but it can be done and it turns out to be a correct. 384 00:55:20,160 --> 00:55:22,860 Abhay Vasant Ashtekar: But is there a uniqueness statement as well, I mean the middle. 385 00:55:24,870 --> 00:55:28,800 Norbert Bodendorfer: Here, the only statement is that these operators reproduce so one one. 386 00:55:29,220 --> 00:55:29,670 Norbert Bodendorfer: and 387 00:55:29,970 --> 00:55:39,840 Norbert Bodendorfer: At least you know, by doing a basis transformation, as you want one, there should be other operators, which also do that yeah so you have you have at least basis transformations here that you could do. 388 00:55:43,020 --> 00:55:53,190 Norbert Bodendorfer: But this is a convenient choice for this course screening that we want to do in particular it's very convenient that this cheesy proportionate to the volume and you have volume eyeing states, they would buy this quantum numbers. 389 00:55:55,170 --> 00:55:55,500 Okay. 390 00:55:56,640 --> 00:56:09,060 Norbert Bodendorfer: So now we again determine the representation were in were just computing the cosmic operator so jeezy squared minus X squared minus K y squared acting on any volumizing state in the accuracy limited space. 391 00:56:09,690 --> 00:56:17,670 Norbert Bodendorfer: By to Lambda in here, and this is not supposed to be equal to James James on this one, because they are forming the S one on algebra. 392 00:56:18,540 --> 00:56:27,420 Norbert Bodendorfer: And for this to be correct, it follows that this vm twiddle appearing these equations here, it has to be to Lambda times J minus one half and then this is exactly correct. 393 00:56:29,550 --> 00:56:34,770 Norbert Bodendorfer: yep and now we come come again to this previous question of our by. 394 00:56:36,060 --> 00:56:43,770 Norbert Bodendorfer: The smallest possible volume gap, you have it in the representation G one to one half and in project with one half this vm. 395 00:56:43,800 --> 00:56:45,630 Abhay Vasant Ashtekar: twiddle vanishes yeah. 396 00:56:46,290 --> 00:56:56,760 Norbert Bodendorfer: So in some sense your fundamental correct theory that results your volume as much as possible there the square would simply reduced to the volume yeah them theories much simpler. 397 00:57:00,300 --> 00:57:00,630 Okay. 398 00:57:02,190 --> 00:57:12,900 Norbert Bodendorfer: So now we can do another consistency check, we can act with K minus on the volume it states and demand that this is zero, and this will tell you that this Ms given by J. 399 00:57:13,530 --> 00:57:20,520 Norbert Bodendorfer: yeah, this is just a consistency check that Jay is giving you the lowest possible magnetic quantum number yeah. 400 00:57:21,180 --> 00:57:29,730 Norbert Bodendorfer: This now means that again the lowest possible volume that we can have in your theory that's what we call the gap in representation J, that is to lumber J. 401 00:57:30,330 --> 00:57:43,410 Norbert Bodendorfer: And this is what we call the minimum volume classically yeah, so this is a suitable definition of the minimum volume is the volume gap, whereas this vm twiddle here that appears in these expressions, this has this gentleman spent half start. 402 00:57:45,990 --> 00:57:46,290 Okay. 403 00:57:48,150 --> 00:57:49,350 Abhay Vasant Ashtekar: So yeah. 404 00:57:50,490 --> 00:57:50,940 Abhay Vasant Ashtekar: So I. 405 00:57:52,080 --> 00:58:06,240 Abhay Vasant Ashtekar: So, if that is the case, then you know if I want to know something kind of really real economy can in the plank regime, I maybe I don't want to coarse grain, I want to look at the fundamental theory and that I could just look at your equations with vm set equal to zero. 406 00:58:09,360 --> 00:58:15,960 Abhay Vasant Ashtekar: Which is to simply all the standard equation of groupon cosmology yes okay good Thank you. 407 00:58:18,390 --> 00:58:19,890 Norbert Bodendorfer: OK, so now. 408 00:58:19,950 --> 00:58:22,290 Norbert Bodendorfer: Before presenting you the renewal as operator. 409 00:58:22,590 --> 00:58:34,560 Norbert Bodendorfer: I want to draw a picture here which tells you about the embedding of the Su one one representation spaces into the legacy habits space now, because what I told you here's there are operators on the accuracy here but space. 410 00:58:35,190 --> 00:58:51,930 Norbert Bodendorfer: That reproduce the seo on it Brown and do so in a representation J, yes, so therefore if you act with these operators on the htc here but space, they have to preserve certain subspace of the accuracy of that space that you identify with this, so one one representation space. 411 00:58:53,580 --> 00:59:09,750 Norbert Bodendorfer: And that looks as follows, so here, in this first line we right the htc states So these are volume eyeing states, so we have zero volume and you have volume Lambda you have volume to Lambda three Lambda and so forth, let me also go into a negative direction. 412 00:59:11,130 --> 00:59:11,700 Norbert Bodendorfer: and 413 00:59:13,050 --> 00:59:23,610 Norbert Bodendorfer: Now you look at this relation that up here, which I see the relation between the accuracy states to learn the M and the Su one one states, so this m maps like this. 414 00:59:24,030 --> 00:59:28,740 Norbert Bodendorfer: So for me go to one half we would for quite some time have that I can state Lambda in accuracy. 415 00:59:29,340 --> 00:59:37,920 Norbert Bodendorfer: yeah so for this, so one one representation with jake would one half, we have the state one half one half that is identified with the state Lambda and accuracy. 416 00:59:38,460 --> 00:59:44,820 Norbert Bodendorfer: And then one half three half will be identified the three lumber and so forth yeah so we always are hopping of tool. 417 00:59:45,180 --> 00:59:54,390 Norbert Bodendorfer: which you may recall, also from the standard htc hamiltonian because that object always has assigned squared yeah so due to the science squared you always have a hopping have to will not have one. 418 00:59:56,790 --> 01:00:05,520 Norbert Bodendorfer: And now, if you increase your representation say to jake to one then your lowest possible volume state is not increased, and here. 419 01:00:06,030 --> 01:00:24,420 Norbert Bodendorfer: Looking at quantum number one of correspond to volume to London yeah and then so force you continue, and now we just can increase J and once you have your minimum volume eyeing state tool under Jay you just increase the volume by to Lambda yeah and then it goes, all the way up to infinity. 420 01:00:25,950 --> 01:00:31,710 Norbert Bodendorfer: And this is precisely this embedding structure of these seo and one representation spaces in the secret space. 421 01:00:33,780 --> 01:00:43,710 Norbert Bodendorfer: And here this nominee again makes sense for the cross training say you have four cells, where you are in the one half representation of the minimum volume is Lambda. 422 01:00:44,100 --> 01:00:52,140 Norbert Bodendorfer: If you patch them together, then you are in the two representation and then your new minimum volume needs before number, you know if if all of the cells in the. 423 01:00:52,800 --> 01:01:02,580 Norbert Bodendorfer: or fine sales take the minimum volume then also the core cell takes its minimum volume and that's why this embedding structure is precisely what you would expect from such a corresponding operation. 424 01:01:04,950 --> 01:01:05,220 OK. 425 01:01:06,870 --> 01:01:16,260 Norbert Bodendorfer: So now we can read off the renominated hamiltonian, as promised, we again remember that the orientation hamiltonian is just K X minus jeezy up to a factor. 426 01:01:17,010 --> 01:01:23,070 Norbert Bodendorfer: And now we need to choose which one is the correct quantum theory in some sense the most fundamental one. 427 01:01:23,430 --> 01:01:30,210 Norbert Bodendorfer: And we would like to choose it such that the fundamental quantum theory has the highest possible volume resolution yeah and that. 428 01:01:30,630 --> 01:01:45,660 Norbert Bodendorfer: means that chase one half, because then we have the smallest possible volume, and that means the gravitation hamiltonian that defines the true quantum dynamics is the standard accuracy hamiltonian sign volume sign now it's just a symmetric ordering of this and corny. 429 01:01:47,700 --> 01:01:57,270 Norbert Bodendorfer: But now I said, if you know read off this hamiltonian in a representation with larger J, which means you have coarse grained to some extent consider many, many small cells. 430 01:01:57,690 --> 01:02:14,970 Norbert Bodendorfer: Then this one looks as follows, so it has again sign in sign, but now the volume, instead of just being linear in V has this squared structure where you now have this corrections depending on this representation J, which tells you how many cells, we have coarse grained. 431 01:02:17,250 --> 01:02:21,570 Norbert Bodendorfer: And since this Jay we said is an rg scale in our case it.