Jorge Pullin: So our speaker to this Norbert for the North fork or speak about normalization for cosmological LPG models North yeah.
Norbert Bodendorfer: Okay, thank you very much.
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.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: We hope are suitable to describe cosmological space times, and this will simplify things as we will see later.
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.
Norbert Bodendorfer: And for the prerequisites that you need for this talk it turns out that everything can be done actually quite simple terms.
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.
Norbert Bodendorfer: However, we will see that the lessons should also apply for more complicated settings in particular also for full quantum gravity theories.
Norbert Bodendorfer: So what is new, here in this talk as opposed to other works that you may.
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.
Norbert Bodendorfer: and in doing so we are able to compute for a normalized hamiltonian.
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.
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.
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.
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.
Norbert Bodendorfer: Applying this in the context of accuracy and first we will use these as you want one.
Norbert Bodendorfer: group techniques to derive first quantum system that we will discuss and then we will hear if the core screening.
Norbert Bodendorfer: And later we will use these techniques that we discovered here in section two and we will implement them.
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.
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.
Okay.
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.
Norbert Bodendorfer: it's relevant for the smooth space time limit, where we want to get a continuum GR.
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.
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.
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.
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.
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.
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.
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.
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.
Okay.
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.
Norbert Bodendorfer: So now, if you consider construction of a cosmological model.
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.
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.
Norbert Bodendorfer: And now, this is a powerful tool that you can use in cosmology because it turns.
Abhay Vasant Ashtekar: out, he was shaving space, time is homogeneous or just.
Norbert Bodendorfer: A story, I mean i'm assuming that the spatial slice.
Is homogeneous thank.
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.
Norbert Bodendorfer: And you use that in cosmology in order to get kind of where you find faith based description, so you will.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: And then we compute the kind of course physics of this a few set of this many small cells here.
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.
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.
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.
Norbert Bodendorfer: So, in what sense it is so so this I mean this refined arrow.
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.
Norbert Bodendorfer: Find theory, maybe I should not have put this refine arrow here but.
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.
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.
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.
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.
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.
simone: it's done.
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.
simone: Oh let's just think about it classically as a classical question I want to just.
Norbert Bodendorfer: classically it's correct what you're saying, but quantum mechanical it's not.
Norbert Bodendorfer: And there is because we, at some point he did a sweetness scale on the right.
simone: Okay, so, even if the state is the same you mean that there are additional operators.
Norbert Bodendorfer: That I can just, for example, say you have some.
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.
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.
Abhay Vasant Ashtekar: yeah so.
Norbert Bodendorfer: If you want to get volume three here they actually two possibilities to get volume three on this side is the.
Norbert Bodendorfer: volume to Volume one and Volume one volume two.
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.
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.
Norbert Bodendorfer: audio about the physical volume.
Abhay Vasant Ashtekar: So you're refining your physical volume.
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.
Abhay Vasant Ashtekar: So therefore your boxes that fixed in time.
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.
Abhay Vasant Ashtekar: Right, so the corner sizes fixed.
Abhay Vasant Ashtekar: But, therefore, this is division, referring to the coordinates size or is it different referring to the physics.
Norbert Bodendorfer: Vision referring to the coordinate sighs.
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.
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.
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.
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.
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.
Norbert Bodendorfer: possibilities to get a certain course of volume if we.
Norbert Bodendorfer: want to discuss this course of volume on the left hand side.
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.
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.
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.
Norbert Bodendorfer: and also what we.
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.
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.
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.
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.
Norbert Bodendorfer: where you can show that you can construct certain full quantum gravity theories which are very similar to quantum gravity.
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.
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.
Norbert Bodendorfer: spin at work, you would have one copy of the loop quantum cosmology here but space.
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.
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.
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.
Norbert Bodendorfer: And you want to extract the course dynamics of that quantum state yeah.
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.
Norbert Bodendorfer: course going into quantum cosmology has also this a full theory embedding aspect that makes it again sensible to ask this question.
Okay.
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.
Norbert Bodendorfer: And here we just mentioned a few things about the quantum cosmology model that we want to use, and this is a.
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.
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.
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.
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.
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.
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.
Abhay Vasant Ashtekar: is just one time, is it kept classical.
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.
OK.
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.
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.
Norbert Bodendorfer: And that sets your quantum geometry scale, and that is related to the Bavarian music parameter like that, if you.
Norbert Bodendorfer: For example, thing of the standard derivations of quantum cosmology and then you will also multiply this by some integer end here.
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.
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.
Okay.
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.
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.
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.
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.
Norbert Bodendorfer: And what you need from these faith based functions as set is that their points separating meaning.
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.
Norbert Bodendorfer: Now this of course leaves a lot of arbitrariness, to pick these functions and you need some good guiding principle.
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.
Abhay Vasant Ashtekar: yeah.
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.
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.
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.
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.
Abhay Vasant Ashtekar: This is very beautiful and very natural, as you say that my equation, but long time ago.
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.
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.
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.
Norbert Bodendorfer: OK, so now, maybe say a few words we now set we need.
Norbert Bodendorfer: A set of faith based functions.
Norbert Bodendorfer: So that the person either bra size, a morphic to Su one one yeah and that can be done.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: Now one possibility to get an SE one one advice given here in this box, so the Jay Z.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: Okay, and this.
Abhay Vasant Ashtekar: But you would not send us at vm equal to zero.
Norbert Bodendorfer: You could.
Abhay Vasant Ashtekar: call this formula so far doesn't seem to.
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.
Abhay Vasant Ashtekar: Okay, so bigger than zero is not necessary vm is bigger than to conditions in one.
Norbert Bodendorfer: So.
Norbert Bodendorfer: We will see later in this talk that in the quantum model vm will be zero.
Norbert Bodendorfer: airborne and an analog of vm will be zero.
Norbert Bodendorfer: So your fundamental correct dynamics would just be signed squared times V.
Norbert Bodendorfer: And this from getting the very end, the fundamental dynamics, but we will see.
Abhay Vasant Ashtekar: That.
Norbert Bodendorfer: The coarse grained dynamics will produce such.
Norbert Bodendorfer: small volume corrections, so this would be.
Norbert Bodendorfer: Electric.
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.
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.
Norbert Bodendorfer: Okay, thank you.
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.
Norbert Bodendorfer: mean curvature of the time speed.
Norbert Bodendorfer: And that is what you would call the complexity fire in the complexity fire coherent states.
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.
Norbert Bodendorfer: And rearrange it a little, then you can write decided muscle that CV and H our map to these three generators.
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.
OK.
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.
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.
Norbert Bodendorfer: And this now tells you that this vm this is encoded in the coming year of the group.
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.
Okay.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: yeah, this means that m this quantum number m, this is proportional to the volume after this factor to Lambda.
Norbert Bodendorfer: And therefore em if you want to later take analogies to full energy is similar to an energy spin.
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.
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.
Norbert Bodendorfer: And since the lowest possible m is J K minus acting on the State jj test to kill the state.
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.
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.
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.
Okay.
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.
Norbert Bodendorfer: A perspective few, but it will become very clear in the next section and once we are in the quantum theory.
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.
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.
Norbert Bodendorfer: To Lambda J yeah at leading order, meaning that you get the kind of second order nj correction incorrect, if you take.
Norbert Bodendorfer: This precise description here yeah so this Kazimierz matched only two leading order, but this is something acceptable because.
Norbert Bodendorfer: You know this just a quantum direction, and again in the next section, you will see that there is a precise way.
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.
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.
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.
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.
Norbert Bodendorfer: That representation in which we constructed and a set of too complex numbers zero nz one.
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.
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.
Norbert Bodendorfer: And intuitively how the state is constructed, well, it has to dependencies said.
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.
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.
simone: yeah sorry, just to clarify, I suppose a priori only need the one complex number to.
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.
simone: And their override on the resolution of the interview just take action wait on the Norman face of the.
simone: spinners I suppose.
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.
simone: Yes.
Norbert Bodendorfer: Good, so now we have those states, and now we can compute something, and we can for example compute expectation values.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: So going back then to this computation.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: And the spinner label we just leave as it is, we don't change that.
Norbert Bodendorfer: Because the Z compute intensive quantities which are ratios of extensive quantity so, for example, the ratio of.
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.
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.
Norbert Bodendorfer: Of so one, one that we had up here, but with the higher representation J.
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.
Abhay Vasant Ashtekar: Can I just ask a question, this is very, very, very pretty structure here.
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.
Abhay Vasant Ashtekar: And they're also a notion of course grading.
Abhay Vasant Ashtekar: And I could have done this because you know the same vehicles to four times we etc.
Abhay Vasant Ashtekar: And I got some quantum numbers and so.
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.
Norbert Bodendorfer: As far as I see this is purely technical tool to get to the next section.
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.
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.
Abhay Vasant Ashtekar: Okay, but so basically you're saying that well there's no loss of generality.
Abhay Vasant Ashtekar: in choosing not only group structure, but the specific representation that you're choosing, or is there.
Norbert Bodendorfer: A loss in general that you have right now we're defining a certain quantum theory yes okay.
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.
Norbert Bodendorfer: may get an.
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.
Abhay Vasant Ashtekar: Exactly exactly.
Abhay Vasant Ashtekar: i'm completely with you, I just want to know what the freedom isn't Thank you this is good.
Norbert Bodendorfer: Okay, so now.
Norbert Bodendorfer: You obviously wanted to check to which extent this calculation you generalize this so to which operators and states.
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.
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.
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.
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.
simone: yeah, so this is a leading order you're.
simone: interested in, I suppose largest believe it.
Norbert Bodendorfer: No, this is exactly correct.
Norbert Bodendorfer: This is exactly correct for all J and for all orders.
Norbert Bodendorfer: So so for this, you first have to I mean there are several subtleties in the computation, for example.
Norbert Bodendorfer: Here we see that you know.
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.
Norbert Bodendorfer: And you get all of them exactly correct.
Norbert Bodendorfer: By this prescription here.
Norbert Bodendorfer: And if you want to see the details, I have an extra slide for that.
simone: No, I understood Okay, thank you, yes.
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.
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.
Abhay Vasant Ashtekar: just said that if I do square then i'm not going to get it right.
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.
Abhay Vasant Ashtekar: Okay, so I have the same question okay we'll talk later okay.
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.
Abhay Vasant Ashtekar: And then, this last one more quick question is there a factor ordering in one linear polynomials.
Abhay Vasant Ashtekar: Of course, so so therefore you're saying that for some specific factory you get the same answer.
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.
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.
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.
Norbert Bodendorfer: Natural hector ordering, it is the.
Norbert Bodendorfer: kind of the trivial one that you would get just by this substitution.
Abhay Vasant Ashtekar: Okay, thank you okay.
Norbert Bodendorfer: So now as.
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.
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.
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.
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.
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.
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.
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.
Norbert Bodendorfer: and, more specifically it's related to the property.
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.
Norbert Bodendorfer: In particular, this would remove unwanted representations if you kind of.
Ivan Agullo: And then you.
Norbert Bodendorfer: are having this in mind, you can see some other details, for example, if you take a product of.
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.
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.
Norbert Bodendorfer: So this is an important thing that we're going to use and.
Norbert Bodendorfer: The other thing which is a very nice.
property.
Norbert Bodendorfer: You derive this as a path into go.
Norbert Bodendorfer: Is that if you act with your.
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.
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.
Norbert Bodendorfer: yeah So this is the deeper reason why the computations that someone somewhat surprising he worked actually have to work out.
Okay.
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.
Norbert Bodendorfer: Action of a group element on this coherent state is absorbed in the spinner.
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.
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.
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.
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.
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.
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.
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.
Ivan Agullo: X and JC.
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.
Norbert Bodendorfer: And of course you can also transfer this now to the spinner.
Norbert Bodendorfer: You can arrive in this equation from both sides, if you want, and you can.
Completely this this operation.
Norbert Bodendorfer: So now.
Norbert Bodendorfer: At this point.
Norbert Bodendorfer: If you want, so this coherent States they give you complete or give you and overcome the basis of your heart space.
Norbert Bodendorfer: You have a soft the time evolution explicitly if you just do this computation here, which is quite simple.
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.
Norbert Bodendorfer: kind of cells.
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.
Norbert Bodendorfer: renomination, so this is what we would like to understand.
Norbert Bodendorfer: and for this we have to go now to Section three and we have to do some other computation.
Norbert Bodendorfer: So again, up to now, we quantization a classical Su one one course on algebra, meaning that we have.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Okay.
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.
yeah.
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.
Abhay Vasant Ashtekar: But is there a uniqueness statement as well, I mean the middle.
Norbert Bodendorfer: Here, the only statement is that these operators reproduce so one one.
Norbert Bodendorfer: and
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.
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.
Okay.
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.
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.
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.
Norbert Bodendorfer: yep and now we come come again to this previous question of our by.
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.
Abhay Vasant Ashtekar: twiddle vanishes yeah.
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.
Okay.
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.
Norbert Bodendorfer: yeah, this is just a consistency check that Jay is giving you the lowest possible magnetic quantum number yeah.
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.
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.
Okay.
Abhay Vasant Ashtekar: So yeah.
Abhay Vasant Ashtekar: So I.
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.
Abhay Vasant Ashtekar: Which is to simply all the standard equation of groupon cosmology yes okay good Thank you.
Norbert Bodendorfer: OK, so now.
Norbert Bodendorfer: Before presenting you the renewal as operator.
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.
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.
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.
Norbert Bodendorfer: and
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.
Norbert Bodendorfer: So for me go to one half we would for quite some time have that I can state Lambda in accuracy.
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.
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.
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.
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.
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.
Norbert Bodendorfer: And this is precisely this embedding structure of these seo and one representation spaces in the secret space.
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.
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.
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.
OK.
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.
Norbert Bodendorfer: And now we need to choose which one is the correct quantum theory in some sense the most fundamental one.
Norbert Bodendorfer: And we would like to choose it such that the fundamental quantum theory has the highest possible volume resolution yeah and that.
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.
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.
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.
Norbert Bodendorfer: And since this Jay we said is an rg scale in our case it.