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Okay. Our speaker today will be machine Han who speak effective dynamics from lukewarm, Robin.
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Okay, so, so first Sam Cooke a for letting letting me have this opportunity to present our written work and this work is in collaboration with someone new. And, and the result will be published by this
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Week or next week or
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So,
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Okay, so
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So many perspective of this talk will be about the relation between loop on cosmology and gravity. So let me first in the first lies briefly review.
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The scheme of advocacy. So on cosmology. One Star Force star from classical gravity and then classically
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Reduce the degree of freedom of, if any, many of you have freedom because gravity to homogeneous and as a child, the degree of freedom.
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So then the resulting model is so called the classically symmetry reduce model.
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And then the after reduction, the faith based on the reduced model is only two dimensional. It's only two dimensional space because you only have one degree of freedom, which is the scale better
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And focus model and then people study the quantum theory, corresponding to on quantization of this reduce model and it's like quantum mechanics. And the result in quantum theory is called a look on cosmology, or else you'll see
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Okay. And then there was one on study about coherence state evolution in in the framework of our currency and turns out that you can you can use so called effective dynamics to describe
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Your state.
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And this effective behind this, you can extract the effective equation and this equation is a modified pretty much equation and freedom and question and
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Ask them topically
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In the large distance, it's reduced to freedom my equation but modify for the manipulation. Here the singularity. In the end, you solve this equation you find
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Nicely these singularity. The Big Bang singer. He got resolved and the similarities replaced by a boss and there's a finite
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Create for volume and finite about large critical density. Yeah. So this is a very nice and remarkable results from Groupon Groupon and cosmology. But from this connotation in
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It is clear that it has some issue and those issues are our motivations for this work. So firstly, and it is well known that it is not clear how to relate
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Luke Pong cosmology and this result of singularity resolution to the pool. I'm ready. So I'm the loop on quality is compound.
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Asymmetry reduction classically and then on ization so then when you ask how to relate them how to relate these reviews the quantum theory to the for quantum theory. Okay, and then
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Secondly, because because of this corner. This first reduction Dan Kennedy scheme.
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Is not clear how to compute quantum fluctuations beyond
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The public sector. So, of course, you can compute the fluctuations within the framework of LTC then you only got one of fluctuation within the
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Homework homogeneous and such public sector then suppose you want to quantify. You want to find the quantum fluctuation beyond the sector, you have to embed this theory to follow.
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Alright, so before we come to our idea. So I want to mention there's a reason interesting result by the other four and and the cloud listener.
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What they do is, and their, their work is interesting and definitely inspiring. Our, our work. So what they do is they they take
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Full upon grabbing Hamiltonian and compute the coherence dating citation value and the coherent state is picked at a homogeneous and isotopic data. So this is the formula.
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The result.
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Of this computation of the computation. And the bigger the Hamiltonian
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The expectation value of the full Hamiltonian upon gravity, but the resulting the result of the calculation is a function of cosmology data C and P
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Which is which are the coherent state labels and then these that what they do. They do these resulting Hamiltonian. These resulting Hamiltonian function.
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As a function of CMT as the effective Hamiltonian of Lupin cosmology and and these Hamiltonian actually is also coincided with earlier proposal by by using drama and since young and then they
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They, they define the effective equation by the Empire Hamiltonian evolution equation from this Hamiltonian and they reproduce the they get again. Then he got singularity resolution and they find a symmetric bonds from this Antonia
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And however, suppose we asked how to embed the result into food theory.
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And then these relation to the full column balance of gravity relies on a conjecture whiskey, the conjecture on the existence of dynamically stable coherent state in in
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Gravity, because if you want to assume that the you can try this effective equation which are busy describing the coach the location of the current state in the face face.
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Then you have to assume that and you'll always have these coherence dating all the dynamics.
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Okay but but as you can in fact that it's very hard to prove this conjecture, because for in general for interacting quantum field theory is very hard to show there is a there. I
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Find the that I'm ask a question. Sure. Yeah, so I i first a minor comment that are on the last transparency when he said that
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We don't know how to I mean that that was the markets that you said that we wanted using only the colon states.
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The transparency before maybe in lukewarm cosmology, but that is not quite true. You want can use widely spread out states and a very wide class of states in a lukewarm cosmology and they still are effective questions of course they're not the same.
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Okay, yeah.
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And the second thing, and there's a minor comment but the main question is really about the current transparency, which is the next one.
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Next one. Yeah, this one. So, am I correct in saying that in this transparency, your equations refer to sort of say the me not scheme as opposed to me about scheme.
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Um, yeah, this is
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The formula right this community.
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So, so, and also, I think, in this formula is, if I understand correctly.
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A graph is fixed. Right. Well,
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Just one graph.
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One doesn't have a whole number of jobs one just as well.
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So I'm generalizing. Are you keeping these two features, that is to say, are you trying to keep up.
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And I think this keeping the graph is what really leads to this may not scheme as opposed to be asking is that
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You're trying to
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Yeah, we are always keeping yeah so in. And so the way we keep the fixed graph and the derivation. We result in again the musical skin, the skin and but in the end I will comment on how to relate this to me about how this motivator to me. Thank you.
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Okay. All right. So, so now I'm so these are the old motivation. So let me come to our idea. So as I said, and it is usually difficult too.
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Difficult to to find dynamical stable coherence states. So then we
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Both the, the result by Andrea Klaus. The really inspiring us to think about how to somehow bypass this is conjecture and and then the the
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The way to bypass it we find is passed into it so. So our idea is to use passive role. So we will formulate a full of gravity as a passing to roll like that. And then, as we know, once we take
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A spa goes to zero in the past integral we got motion and it is the standards semi classical analysis using quantum mechanics and on easier. And then our proposal is
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That the the cosmology effective dynamics should be solution of equation emotion were full of gravity from the passenger
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And this equation emotion can be either classical or upon them. So classical means that it it is a variation of principle from the classical
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Action inside the interrupt if it is quantum quantum equation motion, then it means this is a variation of principle from quantum
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effective action of the entire capturing entire quantum corrections. So I mean most of this talk. I'm all work is exploring the first the situation.
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Where we assume we are, we tried to get the classical you could emotion I realized to cosmological effective dynamics, but in the end we will comment on
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The other possibilities.
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Okay, so. So our idea is that is trying to bypass these
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Difficulty to find the chemical coherence technical stable coherent state. And this is also a lesson from interacting quantum field theory.
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Because in quantum field theory and most of time we we studied effective dynamics, not by coincidence that because it's very hard to find. Again, stay there. But passing the role is the most powerful, powerful tool in in in quantum fields.
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And more. And once we have a solution of those efficient motion we can, in principle, compute all quantum fluctuations know whether
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Well its body. We have freedoms and because we are using passing through a plugin solution in the past into a computer productive expansion. We, in principle, capture all all quantum fluctuations.
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Okay, so this is a working follow our procedure. So on the left, it was, it was a standard procedure for loop loop on cosmology and on the right.
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It is our procedure and so again we started from classical gravity, but now we we quantifies classical gravity by by the framework of Hulu look on gravity and, more specifically, what we are, we are going to do is so called reduce the faith based organization.
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And so in the retail space based on ization we can have a physical Hamiltonian and we use this physical Hamiltonian to perform a coherent state passing through
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So we derive a coherence, they passed into a formula from the food framework of the gravity and then
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In this past integral, we take the semi classical me. He goes to zero and derive the equation motion.
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And those equations emotions are can be viewed as an effective equations are full of gravity those equations are valid for all space time
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Or degree of freedom is is valid for everything and then we can we solve those equations and we specifically we apply those equations cosmology.
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In the sense that we we plug in homogeneous isotopic us and find solutions. It's just like we have Einstein equation and and we plug in certain towns are satisfying certain symmetry. And we saw
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Dynamics of course models. It's the same scheme. And then we find solutions and the solution again give a similarity resolution and give you a box.
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So this is our
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Framework and there are some remarks, so. So firstly, as I said, I'm over our scheme is based on reduced basically quantization and the are three popular scenarios for for these quantities.
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For the framework. So because based on Sunday penetrates models and the schemes are there are three popular models and so firstly Bronco cash task model. And then there's also a Gaussian data model.
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And also a. You can also do a massively scale or field. So they are three scenarios and, secondly, the colonization is based on the fixed graph.
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And we used a number of changing Hamiltonian to perform perform passing on ization and passing through and and here this specifically karma is a finite cubic lattice and the spiritual background is a three tours. So comma is a cubic like is petitioning sweet Horace
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And second, moreover, and they are two possible popular choice of physical Hamiltonian. So there are standard Hamiltonian by Christina and Thomas and also there's a reason
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Hamiltonian proposed by by the company of Europe, and I will call these Hamiltonian Wurzels Hamiltonian, because all the
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Other authors are from Warsaw.
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And so then our studies actually consider the opposite with all those possibilities. So, I mean, they are in total six plausible scenarios and our discussion cover all of them. So they are very similar and and we can discuss all
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Alright, so now let's come to firstly reduce basic foundation. So here I only give you some key points and because those real face conversations, the framework of reduce bass, bass, bass monetization is here for more than 10 years already.
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So so sad. They are they are three different scenarios.
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Bronco cash fast, it's horseman gravity couple to Bronco CASH BASIS Gaussian dust and the mass new scanner field. And so, firstly, this is action for the bronco cash does. And that's the field. There are two different type of data to T and S It corresponds to the time like
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It will be the time physical time variable, it will be this ass will be a physical space variable. So, so we say this though, Stansfield our material frame or physical freedom and once we have those styles that we can we can
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Define the observer goals by prioritizing gravity variable with value of that. So, so we, as I said, is that the scale to tee will be the physical time variable. And that's the value of tasks of yield as will be the physical space variable and then the gravity of your variable is
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Is a field as gravitational field depend on the physical space and physical time and it is the field evaluate at the point where the stars, the field equals towel and sigma
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So, so everything is defined religion only and and that's how we define this Dr absorb oils and they are impartial observer partial doorbells which are KG market.
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Then the Gaussian does is, is the story is very similar and also the the action is a little bit different. Still, and they are T dust and dust field.
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And but for for a massive scale airfield. So this is a standard action for scanner field. And there's only a single skater field here. So the singles career field only department tries time. So the field value is the physical time variable.
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So the, the gravitational field Dr adorable. They are they are not competing against humanity. We only dependent Tris in time direction. So it equals to
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Evaluate as a spacetime point we are these scholar field equals two top. So, which means we leave the spatial direction and the parameters. So as you only consider the physical time variable.
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Then the nice feature of this framework is that the canonical structure of those Dr observable is just the same as the standard canonical structure.
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you preserve the procedure preserved possum bracket. And by the way, here the indices. I use a and b in this is for ask you to grow in disease. Here I am J for the spatial indices and
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Not the other way as the reason is that here I am, J. They are special indices for for the dots polka dots space because when we
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Department tries spatial direction. So we have a so called that space. And this is the space of sigma physical space variable and or you can also view this as, as it's just a time slides with constant physical time top
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And then once we couple gravity with those tasks field or skater field, you can solve those constraints explicit
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Aware of so called a billion idols constraint. So these are the equivalent Hamiltonian constraint and equivalent
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Up more with them constraint and here p and PJ. They are they are momentum of tea and momentum of s.
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And then the once you solve the Hamiltonian constraint, you can extract so called the physical Hamiltonian. The physical Hamiltonian. It's just that the integral of this age function over the
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Past the space S and these physical Hamiltonian generate the physical how evolution. So you can you can ride the Hamiltonian evolution equation for for town just by putting bracket H
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And here I list all possible scenario, the physical Hamiltonian. The opposite scenario. And these for the bronco cash, does it is square root of Hamiltonian constraint. You see, is just the standard Hamiltonian constrained by replacing all the variable.
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By by those the observable. And so this is the Hamiltonian constraints square and minus 104 and this is the DP more rhythms constraints.
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And so here I least the formula for the Hamiltonian constraint and more freedom comes constraint and for Gaussian does the formula is slightly simpler.
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There is no different, more freedom constraint. There's only having 20 constraint integrated over the vast space.
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And for the master list skater is seems a little more complicated and teamwork double square roots. But in this case.
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For the massive scale or field because we only dependent ties in time. So we have still the different algorithm constraint we have still need to impose those difficult within constraints.
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And also, I mean, all these frameworks and the Gaussian constraint is not the parameters are in the quantum theory, we still need to impose at the quantum level of the gospel to ensure, as you to get
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All right, so, so in the quantum theory.
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So we perform the pronunciation on a fixed lattice. So these lattices hybrid could be lattice and with pure all the boundary condition and we assume this latics karma is petitioning three Taurus because so I will talk about cosmology. So sweet policies and excellent apology for that.
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Then, classically, we can we can promote the connection and that's fine too. Hello. Me and flax, just a standard way standard construction, these, these plaques is a cage covariance flux, but the BlackBerry when you vote, he he posed agent.
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And then the quantization is is just a standard and we can define the Kenya magical, we can define the mobile space and impose the girls constraint.
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And the Gaussian strange has to be imposed on clinically and advocating married, who was based on the graph. But now this this service based it's not in the medical space but but already physical space because it is constructed a company with the rock of the roles.
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Okay, so then once we have to go space, we can define Hamiltonian we quantify those Hamiltonian. So these are the operators.
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For all all scenarios and so firstly for Bronco cash and Gaussian dust. So this is a physical Hamiltonian operator. So here I I I put the formula for for the two different past models in the same formula by introducing company constant
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Alpha and in case of a equals one. It corresponding to Bronco cash does alpha equal to zero, it corresponds to else industry. So here, remember the
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Bronco cash, does it has square root of c square and CJ square
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But these so here it is the dagger see and CJ CJ, but these these operators, not necessarily self a joint. So I'm for taking square root, what I do is I'm put these operator and dagger of this operator and taking the power over
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And here and CJ is what see as Hamiltonian constraint operator and CJ is if you move with them constraint operator. So here, for those operators and it is convenient to define
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These CVO CMU operator I following Convention by Thomas and Christina this CMU for the senior operator C zero is just the Euclidean Hamiltonian constraint operator and CJ is just the different moving them constraint operator you you plug in here.
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And technical
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So I agree that
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That you have to take a fourth power. Sorry, one fourth power because the object under consideration is not self a joint is not was definitely like in square root, and so on.
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But
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Is that a factoring issue here or is does am dagger commute with them and Matt, I mean, you don't do that.
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Oh, I don't really know that
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You don't know that. So you just chose one time.
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Yeah yeah this is 111 factor older. Yes. Okay. Thank you. Yeah.
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I'm
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Okay.
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Question.
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When you said in both spaces. Now physical space. So on our pics graph. And I was one
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All this context. Also some notion of cylindrical consistency or something that allows you, then, to take an interactive limited to get the full physical labor space or is it more complicated. How does it work to
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Recall
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So, so here I firstly, I can see only a single graph. So I don't really consider inductive limit so well in principle you can also generalize the story to
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To all the graphs. But the thing is is that I'm here here all the construction is rely on the non graph changing Hamiltonian
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And because I mean we use coherent state. So, you will see it doesn't work for a graph tempting me, Tony. But apart from that you can you can generalize those operators to to to to all grass.
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Yeah.
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Can you hear me, we had some
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Oh yeah, I can't, I can't hear you.
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Okay.
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Well then, if it's gonna grab some, I just wanted to understand the physics of it is like, then some physical space for our final number of Degrees of Freedom House. That's right.
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But should we think of them in some discreet way, or should we think of them in some distributional way. What is the
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Is embarrassing about indiscreet way. I will say,
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Okay, thank you.
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Alright, so, so, yeah. So, so we come to the Hamiltonian. So as I said, there are two different voices of Hamiltonian and and firstly, there's a union by
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Christina and Thomas and this is the more standard Hamiltonian coming from Thomas regularization
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And adapt to a non Grafton the operator.
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And and this is a formula if you can see this is Thomas construction for writing K in using fossil bracket and commentators and. But now, but recently there's also another Hamiltonian, bye bye Wurzels company and the you can see these these Hamiltonian looks simpler.
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The law and the end part is it runs warranty and part of the Hamiltonian becomes the so called Taylor is a scholar commercial operators.
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So I will come to this Hamiltonian at some point when we did the exclusive formula for for the coverage operator at some point.
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Later discussion and and then I we also can see the scenario of single mastering skater appealed. And this is the Hamiltonian
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But these Hamiltonian is not defined on each comma, but you have to define on the TV movie them you Marion states because
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You have to impose the DP Mormonism constrained automatically because I'm there was only a single scatter fuel.
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Only department tries time direction. You didn't solve it anymore. We don't have to impose did you move it on constraint quantum technically and you got to move within the United States and and these operators will be imposed on if you move into the current state. That's the difference.
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Alright, so now so we finished a review of reduce faith based organization and let's come to go, here's the past
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So, um, before we come to pass in grow. Let's again review a little bit about coherent state, just to set out to notation. I'm pretty sure most of you know about coherence.
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Coherence dating on remedy, but just for set up some notations. So, so here we we use heavily these so called complex fire coherent state.
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Defined by studied by handles our man musty man and Oliver Winkler
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And so this is a formula for the coherent state one ash and the coherence state label the sky and he is so to see group element, it is a hollow morphic coordinate
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For the acuity face face look on the face face and so does it contain information about flux and hold on Ami both blocks and how long this little p is the flux I showed you before. And this he is financial Sita is a whole army. So the seat is both P and C are real vectors.
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And so yeah, so this permit rising both blocks and how long and, you know, homophobic manner.
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And also these coherence, they depend on a parameter t and this T is the so called semi
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Semi classical at parameter. And this is the diamond dimension is parameter equals to LP square divided by A square is a is a lens unit. For example, you can take a to be like 111 centimeter and the semi classical limit corresponding to te goes to zero or LP much smaller than your lunch.
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And then you can define the normalized coherent state by this coherent state divided by the norm. And you're right, it says by these nice over companies relation
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It is over company basis. And if you integrate with back to some some integration matter of as well to see and cable companies completeness relations and what we as you can spend doing patentable, you will be heavily use this over companies relations is very useful property.
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Then on the graph, so that these these is stayed on the single ash and on the graph, we simply make sense of all that all the other states and we put the title.
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Means normalized TensorFlow that have normal lives coherent state. And here we also use the occasion invariant coherent state and discotheque invariant coherent state is given by a group averaging of
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The coherence data nonagenarian competent state on the graph. And so the gauging invariant coherent state is labeled by the cage orbit. It's not labeled by a single g a single cell to scope element but but the cage orbit of these assaults equal balance.
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Okay, so now let's come to pass into the now we well maybe I forgot to mention the key point of those physical Hamiltonian. They are all cell phones right they're all policy van cell phones right so that's that's very important because
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Deriving past integral rely on a cell, but one Hamiltonian
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So, so now let's consider this none of changing policy myself to our fiscal Hamiltonian and we can define the transition amplitude getting to gauge invariant coherent state.
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And this time evolution operator. Sorry, is you have capital T equals to just exponential of the Hamiltonian operator.
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And because this is a gauge in Marion coherent state. So you, if you, if we relate this formula to get non Asian American compensate, which is easier to compute, we have to
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Add a integration, which is group averaging relating language immersion coherent state to state by state. So when we consider
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Gravity scanner model because remember black gravity scanner model should be defined on the sci fi movies me, Mr into space. So you have to add another average which is four different moving average.
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So now let's focus on these these empty job. So then this is standard discrete ization procedure and you just discreet is this
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Time evolution operator into an steps and then you insert and plus one over companies relation to this to this, you know, product and then you, you get these into raw
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Over n plus one.
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coherent state variables.
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And then those the amplitude is factor is the into smaller steps and the staff is labeled by delta towel and his daughter Tao is clear to divide by one. And since n is arbitrary large. So this is official really small.
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And so I try to make everything rigorous and I put out all the corrections to the formula. So I'm all the question will be qualities or why do some automation, I will, I will mention I will mention it.
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So now this. This is the transition me to to add a little step towards the top. And because these are you need to evolution operators strongly continuous delivery group. So you can perform this standard calculation in quantum field theory.
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So you're right you linear eyes is exponential. And of course, there's a reminder correction to the formula and and then you can write this linear is
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One plus Hamiltonian to be exponential again and and then the pre factor is overlap function overlap and amplitude between two coherence.
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Normalize a coherent state and aren't exponential is the matrix elements of of the Hamiltonian. Again, there's a correction reminder here which is of course very small
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So house more are they so here I define to excellence in late at this is this is called a bar axon, I think. And this is called the standard apps on the output is different.
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So this bar epsilon is
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Is the operator. So, can I ask a question I want the last slide. Sorry, because I could not answer that.
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So, I mean, since you mentioned explicitly that you're trying to be rigorous here, but we don't really know that us
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human operator. Operator,
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Yes, they are self would want
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To start, so did you actually look at the domain and the domain of that joint
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So day
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One. Yeah, you can you can you can do because it is just a policy. So firstly is positive as a metric you can definitely find a self to an extension is freely extension.
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So you're saying that is because it's like a finite dimensional spaces that one.
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Dimensional space, although it is even dimensional space. And so I choose to ordering in this way so that these under the one over for ideas manifestly positive and symmetric operator.
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Is a summit.
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But, but he went to take the one port power your to show that the domain of this operate the M tiger and
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Is the same so I shown that I mean that the
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So, I mean, under well under this search. So firstly, these operator.
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Including you, taking the power of our for
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Our for unless you're already showed that
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Yeah, yeah. These operate yourself again because this is manifesting
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Dimensional space.
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So to look at the domain.
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Here, um, there's a serum called which is called affinity extension.
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So once you have an operator, which you can throw
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It is a symmetric his permission and positive and then you can define a quadratic before policy definitely let me definitely correct form and then you can define the these these so called free the extension and this is self, to an extent.
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So you are using specifically you're using this have federal, state, and everywhere.
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Yes. Okay.
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Start that you're showing them to join, but he was using the fragile states.
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That's one. Yes. Okay. Thank you. That's right.
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Um, okay. So, then those what they are just the technical details and for those excellence and he asked more because firstly you define a operator by epsilon square absolutely hat. And this is just the, the, the operator, which
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You approve that these operator convert strongly to zero because you the Unity operators strong strongly continues units regroup and all that he goes to zero and this operator just convert strongly to to zero and and then strongly convergent implies weekly convergent.
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And then
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Yeah.
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I mean, I have a question to this to this formula or
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Evolution of the Caribbean states are you assuming that the Korean states are stable and the information because most Korean states well
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No.
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I'm not. I'm that's that's that's the point I want to be away from
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So know the those. Can you say that are definitely not stable under evolution and but they are they all satisfy over completing these relations and those steps are infinitesimal those steps are very small.
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OK. And then nice so because strong convergence implied week emergence and then if you take the
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matrix element of of this epsilon bar excellent operators, what, that's what I call bar epsilon
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I plus one, I and these quantity definitely commercial zero. If you take delta toggles here. Okay, so which means and inaction. Well, which means this, this term is far epsilon is very small.
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And secondly, this standard epsilon and in the exponent. This is a formula, it's, it's not that complicated actually
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So bad. But you can check it easily that you take the other to the other toggles to zero. These
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Epsilon standard epsilon goes to zero as well. So, which means in the action. So, this term is really something. If you take out the towel to be to be small but does it. How is indeed arbitrary Islam.
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Okay and and then. So there's a you can compute this, this factor of inner Kodak have to coherent state. So this is just the standard computation. And the nice thing is that right, these result as a financial and these exponential K MS k will be the Kinect, the term of action.
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So here just for set up some notation. So this K depend on pee pee. It was just the flux P here with no vector index.
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Norm of the flux and so it depends on some political see the III minus one, or the one to two one. So this is our code of some variable called x and this x variable is a one half of the trace Cheatwood dagger Angie one
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X is the symbol combination of two to coherent state labels.
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Alright, so now we can plug in these formula back into the past integral, but back into the transition amplitude
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And we got the past integral and this passing the raw is exponential have some action.
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And so here, here this new is some macro factor is coming from the overlap of coherent state and there's, there are two types of integrals. One is integral over coherence labels.
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Coming from over coming from the overlay overlap the company's relation and then there's another integral of as you to group elements and they are coming from the group average to make gauging art make it to engaging
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And for this action. Sorry.
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There are many, many so every vertex has has
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No, no.
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Marion it commute with gay transformations
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So you did the resolution of the identity into a bigger space.
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Yeah, you can use resolution, I didn't even a bigger space. But if you for the initial state you you made a group, everything is fine for all steps.
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Okay.
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All right.
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So yeah, so. So now this is our action. So I mean, call it a effective action for for
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For a loop on gravity. And so, the first term is is the the connected interim I mentioned in the last life coming from the overlap of coherent state and the second term is coming from you this potential term or the Hamiltonian term is coming from the matrix element of tommy. Tommy
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And then there's a term. Excellent. So I will just neglect this term, I will do the approximation and neglect this time because because there are opportunities more
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Time steps.
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Alright, so, so now. So from this passing through, there's a, there's one standard fact is, is that these passing the ROI is actually a dominant it by
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The support in is that you have a group element G. You have a coherent state label G and i plus one, you have another g. So the passing there is dominant by these, these two G's has to be very cross has to be closed roughly like older one over to a tee is the same Cosco priming.
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And the reason is that every step. And I even it has more Tata Tata steps. This is the formula and without it all goes to zero and this is approximately
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This exponential is like one. And so these amplitude at every step is
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Dominant by the contribution of overlap of the amplitude and overlap of the coherent state and the overlap function is behave behave like a Gaussian
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So it peaked at the coconut state. We are these two are these two g's are the same. And it dumped on like in in the gulshan fashion. So yeah, so this passing toys. Amanda by these two g's are our calls
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So now let's once we have the past integral, we can derive the equation motion and taking the semi classical limit and t.
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Is just a standard like pass into in quantum mechanics, quantum theory and take tea or each bar goes to zero.
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And use a stationary phase of automation. So here I again, these, these passing through at the district level.
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When, when they take and we finite although uncanny arbitrary large but and it's finite. And this. This interview is just financial
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Company rules. So unless you take really take and goes to infinity. And I don't know. I don't know how to define this passing but so at the moment so far all these TV. I tried to make it as rigorous as possible.
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Okay, so now let's derive the emotional level.
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The parameter of the states. Right. Yeah, that's right. So this is a tea.
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Means you're making them very squeezed. So they're very plugged in the whole on Ami and very spread in the flexes, is it that
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No, that's not, no.
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Please no.
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No, it's just like each bar goes to zero. So, I mean, you use Heisenberg
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Quantum quantum to classical
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Yeah.
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Again, in one of the slides.
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I'm sorry, can you show again the definition of the coherence states. Oh definitional cultures and this is just a standard
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Cost a
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Cent fee to zero. That's just a delta function. She are making it very sharply peak yummy.
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Yeah, so, because these these current state is a heat Colonel evolution from delta function.
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But then you said that wasn't the case. So you're making them in this limit, you're making them very squeezed right so very peaked in one variable, but very unpicked very spread in the dual viable.
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It was debate about it but but I think, well, I mean, from the
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Like operator expectation value calculation or the matrix element of follow me and flux and like the techniques that work by Oliver Winkler and and Thomas about picnics.
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Poverty of these coherent state what I can see is, is that it is picked out a face, and you take te goes to zero if you compute expectation value of operators and then you just recover a classical value of both H amp D.
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Of course you do. But the point is spring this year.
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Very, very small and spread the very large
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I mean, you just look at anyways. Anyways,
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I think we are clear about that. It's a standard result.
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Yeah, I mean,
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Okay. Yeah, you can. Yeah, I know, I know what you mean. But yeah, I think. Yeah. For me, you can be like that. Yeah.
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That was a question.
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That you're doing here right parameter, I suppose, so that your story. And so it really is like what originated right so
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So, in that case what is what is doing is really defining the time interval. So unfortunately you're calling it something, but the tea is not the time to build that
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Time. Right.
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That's right. Yeah. Yeah.
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And so in financing description. Of course, you know, the committee to get the storage equation to get the transition. I've
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Yet to take this limit delta t goes to zero, which meant that the number of artists has become innocent, which means an infinity. Hmm. So you're not doing that. Is that correct,
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At the moment, well I mean at the past integral level. I'm not doing that because I'm the for example the equation emotion. I'm going to derive first is, is a
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Discrete equation. The equation emotion I come I derive those equation just as a district level, even description time
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But then when we look, look at the solution of equation motion. So I will look like look at the continuous approximation of the equation or motion. So, which means at the level of the equation motion, I take the continuum limited in time, but not at the passing
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Of the physics of this is right. I mean, because it's not really the what you are scrutinizing and keeping finite is this fictitious time variable of like the deeper dive into is time variable and
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Keeping it discreet I'm looking at
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The end to be and not then you're keeping that time step to be something right. Yeah. And so what is the meaning of that. I mean, we don't know the meaning of that is
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There's no just no physical meeting just mathematically. I just don't. I mean, passing the role because my pic angles community this passing that will become unified, I just don't know. Okay.
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So, so this
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This mathematical step and you don't know the we're not worrying about the physical meaning of this
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So we should not
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Take this dispute equation. Seriously. I mean, you're not taking them seriously as a as telling us any physics we should just
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Know, I mean,
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And so we should take the description seriously because
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Because I
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To write it in companies continues and you will see those things I will formulate those equation in district level, but
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When you look at solutions you you you take continuum. It just, I mean, though it is very hard, I think, well, it may be
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Not that nice the equation won't look at Nice if you if you take the take the continue limit of the equation or at continually meat of the action for them to say action me. You take the continually middle of the action.
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And
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I don't know what will be so it won't be looks at Nice anymore, you know, but
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I know that for from just order the partial differential equations, even for the harmonic oscillator. Something we know that
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Yeah, for
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The District ization of the equation.
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Is not the same as this I mean that you can look at a discrete eyes equations and then look at their solutions and then take the continuum limit that does not compute the other way around. That is to say,
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I'm so, so, I mean, we know that I mean that that you know this is a standard fact about differential equations. I mean,
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But yeah, well I'm
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just ignoring that each I'm happy to ignore it. But I mean, unless you have some deep reason to think that
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So, I mean, you can you can view that this is just one procedure I derive the equation.
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In the district level.
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So I because I don't know how to take the continuum limits.
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For this action on it.
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I'm happy to get another medical procedure to opt in and some pilot solutions. And then the question is open about what happens if you change the procedure.
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That's right.
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Thank you. Okay.
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I'm sorry machine. I have a short comment about singleness question. So I think that what happens with this call you and states is that
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The expectation value of the flexor also depends on the little tea. And so the relative dispersion not, of course not. The absolute inspiration, but the relative dispersion of both Rolando me and flux
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Are small when you send it to zero. So, in the sense, they are picked on both variables.
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Okay, yep.
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Alright, so now let's come to the original principle of this action. So I mean, specifically, I tried all kinds of variations for those variables and I find if I do the following variations that the equation of ocean looks nice. The nicest
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So what I do is to firstly, and they are two different type of integration variables and one is the Solomonic
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G and and also there's a su to variable. Ah, so for the hollow morphic variable G.
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So I do this homomorphic that formation. So just G multiply exponential epsilon and how, how is
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The algebra generator and these extra long is a complex. So, so it is hollow morphic defamation variable and for the for the SU to I make the real defamation. Ah, multiply exponential ITAR timestamp and here it is. And then we make plug in the defamation and make derivative
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Of the action and then you can see the left hand side of this. So those are those three are the equations.
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From the variation. So this first equation first two equations are coming from the hollow morphic derivative and unravel working with you about the action and the left hand side comes from the connected term, of course, from Ohio State and up and down V and X and Z and X.
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Is I I am precise definition of Zeke, D is our coach as x x is the trace of geography.
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And then the right hand side is here, you can see exclusively that what I'm what what we are doing is may 1 make them defamation of tea and then make the dirty. So this is a variation of each of the matrix elements of each
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Yeah, and the second equation is just anti for a morphic derivative
344
00:54:10,080 --> 00:54:21,060
The third equation coming from variation respect to as YouTube element, it gives them or just a well known equation which is closer condition.
345
00:54:21,870 --> 00:54:34,890
So this is just a kosher condition for Q and of course we have know for those equation, we have to use our initial condition coming from the initial state and the final condition coming from final states.
346
00:54:36,960 --> 00:54:48,480
And let let me. Let's look more about discuss more about these equations. And yeah, so these are the relation between CL X and X and geography.
347
00:54:49,320 --> 00:54:58,170
So, so, um, what you can throw is is that, so firstly you can notice from the right hand side of the equation when so
348
00:54:58,770 --> 00:55:07,350
This part this derivative of the Hamiltonian is is really finite. So, if you will, if you calculate. You can see the are really finite
349
00:55:07,800 --> 00:55:19,830
And so which means I for for this equation. If you take the continuum limit for the equation or are you looking for solutions. Yes, which are approximately continuous in tall.
350
00:55:21,330 --> 00:55:31,320
Well, if you thought. So it's, we're looking at continuum limits, does it all goes to zero, and then the right hand side for any solutions and the right hand side.
351
00:55:31,770 --> 00:55:38,520
Approach to zero. So, which means for for for any solution. If you take the other toggles to zero the
352
00:55:39,360 --> 00:55:45,900
Left hand side, the right hand side of proof of zeros and the left hand side must also be zero.
353
00:55:46,890 --> 00:55:55,140
And then the left hand side approaches zero. Actually, you can prove that. And suppose we are in the neighborhood GI plus one.
354
00:55:55,740 --> 00:56:07,890
Close to G close to ti for because this neighborhood is where the past indigo and got dominant contribution and in this neighborhood and you have to force GI plus one approach to GI
355
00:56:08,610 --> 00:56:21,600
So, which means that if you take the other half goes to zero, your solution gee i is a discrete time step, then you can replace it to Tao, and the solution or continuous in top if you take
356
00:56:22,080 --> 00:56:31,050
Total tacos to do with. So, which means that for the solution of those equations, although this equation has this great, you can use a continuous approximation.
357
00:56:31,410 --> 00:56:41,670
If you agree that this does it, how is it sure is more so I'm this is what we are going to find is the continuous approximation of discrete solution. So, I mean, for
358
00:56:42,180 --> 00:56:50,700
For the poverty of the quantity in the left hand side. So we can prove a dilemma. So you can show that. So, suppose a lot of insight.
359
00:56:51,780 --> 00:57:09,450
In this equation are all zero and and you can find the root of those two equations and and we've seen the neighborhood and you only find a isolated route which are GI plus one equals two g i n g i n g i n minus one. So, which means
360
00:57:10,530 --> 00:57:15,060
Well, when does it all goes to zero and your solution must be continuous Intel
361
00:57:18,180 --> 00:57:24,930
And so once we know these poverty, so we taking the continuous approximation of the
362
00:57:26,010 --> 00:57:37,080
Cheese. So we sending GI approach Richie i plus one and Ti minus one approach to GI and then the right hand side of this equation can greatly simplified.
363
00:57:38,010 --> 00:57:48,540
And so the matrix element of age of the Hamiltonian operator becomes expectation value of Hamiltonian operator. So for the matrix elements of Hamiltonian
364
00:57:48,900 --> 00:57:57,240
I don't know how to compute it, so it will be because at the moment I'm pulling on the operators like boredom boredom operators and and so on.
365
00:57:58,140 --> 00:58:10,890
So it's very hard to compute the matrix element, but for the expectation value of Hamiltonian. It's much easier. I know how to compute it at least prohibitively. So there is the technique developed by by Christina and Thomas
366
00:58:12,240 --> 00:58:19,560
So, so called semi classical perturbations theory what one control is that in the semi classical approximation, the leading order.
367
00:58:20,880 --> 00:58:28,050
In the semi gospel limit is just the classical discrete Hamiltonian on the graph, at least for the for the
368
00:58:29,190 --> 00:58:32,130
Hamiltonian by by Christina Thomas
369
00:58:33,210 --> 00:58:40,620
So you will have this nice poverty and then we know that equation motion is only sensible if you, you know, a spark right
370
00:58:42,000 --> 00:58:51,300
So, so you can plug in his expectation value replace expectation value by classical Hamiltonian and you just
371
00:58:52,080 --> 00:59:02,220
reflect poorly in classical Hamiltonian for this equation and then this is the final version, our of our equation emotion and you can see that those equation, the
372
00:59:03,030 --> 00:59:10,920
Computer computer. Well, so this is just classical Hamiltonian and these are simple formula simple expression from the state.
373
00:59:11,850 --> 00:59:23,970
And so we have three equations to time evolution equation and closure conditions. And again, initial and final and we are you comfortable and solve those questions and so
374
00:59:24,870 --> 00:59:32,460
What I propose is that they are actually effective equation of Hulu Kong gravity. So they are they are
375
00:59:33,240 --> 00:59:42,810
They have very much. They have low vantages advantages. So firstly, it is for really all the real freedom food pool can remedy. So it's valid for all space times
376
00:59:43,650 --> 00:59:54,690
And secondly, it is solvable. So all the quantity in this equation or computer will it can be solved analytically or numerically. So definitely have
377
00:59:55,620 --> 01:00:03,120
To have connection to record relativity, because this is just described the equations for time evolution for me conclusions.
378
01:00:04,110 --> 01:00:17,940
And certainly, so once we have a solution of those equation we can immediately parking pass integral and computing, quantum corrections systematic computation by productive EXPANSION, OR PASS integral for all solutions of the equation.
379
01:00:21,030 --> 01:00:22,740
Your understanding because
380
01:00:24,060 --> 01:00:31,350
Cuba, just because there are 6 million vertices right but here the labels don't describe groups. They could be whatever
381
01:00:31,560 --> 01:00:32,100
Yeah, it could be
382
01:00:32,610 --> 01:00:33,150
All
383
01:00:33,210 --> 01:00:35,010
It can be everything. Yeah, that's right.
384
01:00:37,980 --> 01:00:41,730
I'm sorry, I don't understand something very basic. If you just go back one slide back
385
01:00:42,210 --> 01:00:51,660
And one slide back, you said that. Well, if I take the limit delta t goes to zero, then right hand side goes away and therefore the left hand side equals zero. So I said that that is a question.
386
01:00:52,200 --> 01:01:00,150
So why do you still. I mean, you use the fact, in some ways, the limit that it goes to zero that goes to zero.
387
01:01:01,320 --> 01:01:01,560
But
388
01:01:01,800 --> 01:01:03,000
You're still keeping that term. So,
389
01:01:04,890 --> 01:01:07,950
I'm just not gonna understand the logic here at all. So can you explain
390
01:01:08,100 --> 01:01:18,660
Oh yeah, well I mean because I'm was from passing through, you have to be when you're in your individual motion. We have to divide the top side. So the left hand side will become a time
391
01:01:22,170 --> 01:01:27,960
Yeah. So that, that's good. So it will become time derivative. Therefore, then there's no argument, right, which says that, I mean,
392
01:01:28,620 --> 01:01:37,050
When you take, take the limit delta goes to zero, you also take the limit and goes to infinity which you're not doing you're keeping and fixed and and talking about all these
393
01:01:38,730 --> 01:01:39,960
higher order terms in
394
01:01:41,940 --> 01:01:54,330
The higher order terms are different. I mean, still the number of terms itself will grow. So I'm confused about your delta goes to zero limit, but keeping and fixed and
395
01:01:54,450 --> 01:02:08,070
No, no, no, no, no. I'm definitely keeping keeping so as you can do in this way. So, so let me add the equation emotional level. I'm looking for solutions. Yeah. So then I I always
396
01:02:08,730 --> 01:02:20,670
Looking for the continuous approximation. I mean those equation adequate equations, but for this will allow this equation are pretty typical saw because it involves matrix element of age, which I don't know how I feel.
397
01:02:21,330 --> 01:02:29,580
But what I can do is looking for a continuous approximation of those disparate solutions which means I take the other tacos to zero.
398
01:02:30,300 --> 01:02:31,140
I know but i mean
399
01:02:31,950 --> 01:02:40,080
That that's not a continuous approximation of those equations because because if then then and had to also go to infinity, right, because after all, downtown
400
01:02:41,760 --> 01:02:43,170
Yes, yes I take angles.
401
01:02:44,880 --> 01:02:45,510
And the equation.
402
01:02:45,990 --> 01:02:54,330
At the level of equations, you're taking the limit and goes to infinity, but some are not. So I say, so not not in the action itself, but in the the
403
01:02:54,540 --> 01:02:56,850
Wisdom of the equation. I see.
404
01:02:57,570 --> 01:03:01,920
I see that you're taking and goes to infinity in this equation emotion. What are you doing to take that limit.
405
01:03:02,640 --> 01:03:14,820
Oh, and because that's how he is his capital T d by by a capital T is fix the total time and space but well angles and beauty means the equivalent to dollar tacos to do
406
01:03:16,380 --> 01:03:16,770
So,
407
01:03:17,940 --> 01:03:21,300
So there's no some or n which is take going to infinity.
408
01:03:23,100 --> 01:03:25,320
Samples what happened in the action. I mean,
409
01:03:27,240 --> 01:03:28,500
In the morning, there's no
410
01:03:32,460 --> 01:03:33,030
Okay, great.
411
01:03:36,660 --> 01:03:37,350
Okay, yeah.
412
01:03:40,770 --> 01:03:50,280
Alright, so those are equation emotion of the food theory. So now let's go to the last step and apply those equations to cosmology.
413
01:03:51,330 --> 01:04:11,670
Until. These are the set up for for cosmology. So we are taking the lattice. Again, the hybrid cubic lattice. There are three directions and the edges are labeled by 1123 at a given vertex me and
414
01:04:12,960 --> 01:04:28,650
So then we we have the equation for for the real freedom and we want to find the solution relates to cosmology and what we do, we we we plug in the US. So these are around us and respecting the homogeneous and isotopic
415
01:04:30,630 --> 01:04:37,650
Symmetry, just like with everyone do when solving an equation. And so these arms us looks
416
01:04:38,460 --> 01:04:53,490
As follows. So these the hollow me a exponential of Sita time style but tall only so so this whole army along the either direction i from one to three. So on the either direction only depend on time.
417
01:04:54,150 --> 01:05:06,750
in either direction. And these flux vector in either direction along the edge in either direction only has component in either direction.
418
01:05:07,440 --> 01:05:20,550
Yeah, so that's the setup for FOR HOW LONG FOR HOW hello me and flux and then of course you're plugging horny and flux in in the harmonic variable, see what you got this one and these these said is exactly the same as
419
01:05:20,940 --> 01:05:29,730
What Andrea and close the India. India work and here this see don t they are constant on on every time slice
420
01:05:31,050 --> 01:05:34,320
The label. I just the timestamp labels, the timestamp.
421
01:05:35,850 --> 01:05:49,530
And and again there's so we have the closure condition and but if you plug in this owns us and disclosure condition satisfy automatically. So do you close continue that in the polls and constrained to to our honors
422
01:05:50,820 --> 01:05:53,160
And then the only non trivial.
423
01:05:54,540 --> 01:05:57,300
Work is for those two evolution equation.
424
01:05:58,380 --> 01:06:05,760
And if you plug in that you find those evolution equation that greatly simplify the especially the left hand side, the left hand side.
425
01:06:07,110 --> 01:06:16,800
Then you can see the left hand side reduced to just the, I mean, of course, you need to divide the total from right to left and then you see
426
01:06:17,280 --> 01:06:28,230
The left hand side is manifestly a time derivative of Sita plus time do you over to our key. And here's the second equation is time derivative of Sita minus items time you will
427
01:06:29,250 --> 01:06:35,460
Have P. Yeah. And you get this expression is not surprising, because you are always doing coherence that
428
01:06:36,630 --> 01:06:37,170
Calculation.
429
01:06:38,640 --> 01:06:50,820
And but but also importantly you find proof factor this adult AI because you plug in these on us for for key you only have AI components in that number I
430
01:06:51,840 --> 01:06:57,270
Yeah, to equation. Both have prefect adult AI. So, which means these two equation has two parts.
431
01:06:57,840 --> 01:07:08,910
So the there's a diagonal equation. There is also have the equation. So the diagonal equation. They are really the time evolution equation. So, the right hand side will be the
432
01:07:09,390 --> 01:07:19,500
Solo morphic derivative of h respect to the accident I to defamation parameter I on an AI in either direction.
433
01:07:20,040 --> 01:07:30,990
Yeah, and those I will call those epsilon. Our, our longitude perturbations along the homogeneous and so public sector because those animations of tea.
434
01:07:31,950 --> 01:07:44,610
Will stay in the homogeneous and isotopic sector because you can see another defamation. It doesn't change the form of your arms, as you just change, see the value of SI p
435
01:07:45,840 --> 01:07:59,130
But the of the ag know you got not A, you, you, you, you get what you get is not the time you will drink equation are some constraints because delta aim and a not equal size. Left hand side just equals to zero.
436
01:07:59,730 --> 01:08:06,930
So you get some right hand side is something equals to zero. And those something our
437
01:08:07,410 --> 01:08:24,240
Production perturbation. We are transfers perturbations. Well, those courses on a naughty host it is epsilon those affirmations are transfers perturbation those perturbations are away from homogeneous and isotopic sector, you can see physically that I'm those perturbation breaks, you're
438
01:08:25,770 --> 01:08:36,630
On that change your own us to something else. Now, and you definitely have this equation because our previous equation is very for all the real freedom you have to definitely take into account those patients.
439
01:08:39,180 --> 01:08:42,180
So definitely, you need to consider perturbation away from from
440
01:08:43,380 --> 01:08:48,990
The homogeneous so topics. And so whether those equations give you some new constraint.
441
01:08:50,370 --> 01:09:11,070
Alright, so now let's more specific more specific and plugin Hamiltonian. So firstly, let's talk about Tony and by Christina and Thomas and you see those Hamiltonian are a bit complicated and anymore. Many closing brackets and we're commentators and then, however, a good thing about this.
442
01:09:12,750 --> 01:09:14,130
I mean, Tanya, is that
443
01:09:15,480 --> 01:09:20,460
You can we compute the expectation married because I'm so sad.
444
01:09:21,720 --> 01:09:33,420
You reduce the matrix elements of Hamiltonian to values, but exactly for this Hamiltonian, the expectation value of the leading order of expectation value reproduce the classical
445
01:09:33,990 --> 01:09:43,140
Hamiltonian. Yeah, because you can make the superpower expansion of volume operator in case the power of warning is positive.
446
01:09:43,680 --> 01:09:50,070
Which is important. So you can you can do this expansion and get the classic special
447
01:09:51,060 --> 01:09:58,260
OK. And now what we only do do is get a classical, especially the Hamiltonian plugin is equations and solve them.
448
01:09:58,950 --> 01:10:04,530
Yeah. And again, they are two equation time evolution to type of equations time evolution equation.
449
01:10:05,070 --> 01:10:13,860
From off from the diagonal art and the art constraint equation of tearing apart and our strategy is just a brute force the computer right hand side.
450
01:10:14,340 --> 01:10:24,480
Yeah, so the right hand side. Well, that's the most, and I'm sure you're part of this work. So to compute the right hand side. And what we do is just make the motivation and inspiration.
451
01:10:24,840 --> 01:10:32,610
At the linear level and flat conditions and security. Okay. And this Bruce wall computation is carried out and company analytically, you must match.
452
01:10:33,090 --> 01:10:42,360
So is there is no way to compute by half. So, I mean, the linear expansion and what we did is use parallel computational environment of Mathematica with 30
453
01:10:42,900 --> 01:10:51,930
Parallel kernels. So what an honor CPU, GPU server. So it has to be done. But what we do is we do it on the server.
454
01:10:52,710 --> 01:11:07,020
With certain unit or parallel computation. So the computation, although the computation only use CPU doesn't use GPU and while the competition because they are so many terms and those competition evolved kinds of manipulation and cancellations about
455
01:11:08,340 --> 01:11:18,630
300,000 terms and I'm the comes to the entire competition last maybe about two hours. So, you know, for any intermediate steps. If you want to extract some
456
01:11:19,620 --> 01:11:30,270
Some output at the intermediate step. I mean, the total file will be a few gigabytes. So I'm I am an intermediate step will be very long. So definitely, I'm not able to show it here.
457
01:11:32,730 --> 01:11:41,910
So that's why I'm you can't really compete by hand. We have to do something like how to do this competition, but the output and the final output. It's really simple.
458
01:11:43,260 --> 01:11:45,390
You won't you won't need
459
01:11:46,470 --> 01:11:56,430
harddrive tool to store those result. So the result you see the results for the dynamical equation is just a, like a Hamiltonian evolution equation.
460
01:11:57,150 --> 01:12:15,660
And also, remarkably, the off diagonally questions. Those constraint equation you can check the are satisfied automatically. So all those 300,000 terms they they will just cancel for those for those 10 for the diagonal equations, you just cancel. You got just exactly zero
461
01:12:17,160 --> 01:12:29,550
So for the evolution equation, it's like Hamiltonian evolution equation. And those are Hamiltonian. The results for for all cases and the Hamiltonian. It's just like a loop on cosmology.
462
01:12:31,020 --> 01:12:35,910
effective Hamiltonian. Okay, so it's reduced to very, very simple expression.
463
01:12:37,860 --> 01:12:49,050
So I'm they are a few remarks. So firstly, I'm the cancellation and the especially if they have that both diagonal part of that part. So the more canceling
464
01:12:49,620 --> 01:12:55,920
Many, many terms and those cancellation RBC contribution from different vertices, they are not like canceling
465
01:12:56,640 --> 01:13:09,090
Contribution from a single versus vertex, but it came up cancellation from neighboring versus and especially and also we have to use a pure all the boundary condition to ensure this cancellation, otherwise we will get some boundaries.
466
01:13:11,370 --> 01:13:19,650
And secondly, for the for the bronco cash and Gaussian dost, you will see it gives and gives the same result is because
467
01:13:20,970 --> 01:13:23,070
The for the homogeneous as a topic.
468
01:13:25,710 --> 01:13:41,190
On that the team over them constraint equal to zero. So the after defamation. It's only have the leading order contribution. Come on, linear epsilon, but for ground Bronco cash task it up and down correctly.
469
01:13:42,030 --> 01:13:52,530
These different moving them constraints. So these term only contribute at Foursquare for the derivative of epsilon won't see this term. These are some technical details or dust.
470
01:13:53,610 --> 01:14:03,780
And then for, for we can both the solution. Those effective equation, they become simplified, we can do for solution of those equation.
471
01:14:04,170 --> 01:14:17,430
And committed conveniently we perform a channel variable we have p and Sita, we know how we depend on p. So, well, we can change your variable and use
472
01:14:18,120 --> 01:14:37,290
The traditional be read variable in the loop on quantum cosmology. And here he is just a Sita and be to pronounce it empowers three three half hour of tea and there is no time evolution, there is a concern quantity which is the physical Hamiltonian
473
01:14:38,460 --> 01:14:45,540
Or or divided by number of vertices it find a scripted he to be a energy per production site.
474
01:14:47,130 --> 01:15:03,180
And then we can provide a solution. Yeah. And we got your standard and symmetric bounce from the solution and here as importantly, when t becomes it becomes large. It reduced to Friedman worker Friedman vision.
475
01:15:04,740 --> 01:15:06,870
And there's a critical volume and density
476
01:15:08,730 --> 01:15:16,800
Here you can see that is critical volume and density exclusively depend on depend on the console quantity conserve energy
477
01:15:17,430 --> 01:15:26,190
So I'm the suggested that we are actually in the new zero scheme, the corresponds to mosque in oncology
478
01:15:26,700 --> 01:15:34,680
And you can also demonstrate it more efficiently by changing variable. If you set Sita equals c times new and P equals
479
01:15:34,980 --> 01:15:48,030
New square p capital P divided by A square, right, so here I always use capital C and capital P for for cosmology CFP, just because a little key has been used for the flat.
480
01:15:48,870 --> 01:16:01,980
OK, and then then you plug in this channel variable and you reduce your having Tony n to the standard Hamiltonian well known haven't shown yet for for cosmology.
481
01:16:02,370 --> 01:16:14,670
It is the same Hamiltonian. This derived by by using command and and and Andrea and cross. Yeah. So here is really coincide with musical scale.
482
01:16:15,750 --> 01:16:16,650
You can see
483
01:16:18,000 --> 01:16:24,540
And another remark is is interesting. We have recently, there is a work by by clouds and prom.
484
01:16:25,770 --> 01:16:42,030
What they do is is making a new channel variable but corresponding here is the new channel variable and they use the covariance covariance flux and plug in cosmology variable CMT and then they get a modified
485
01:16:43,920 --> 01:16:53,670
Flat variable which dependent both PNC right and and then it will corresponding to here is a different channel variable. And of course, we have
486
01:16:55,050 --> 01:17:13,020
Put these trends are viable into our effective equation, but we find the result is different from there and effective equation. But anyway, and our, our result use is completely independent of can do very well because I'm using all all of them are just different channel variable.
487
01:17:14,070 --> 01:17:27,030
Change are very fact visits. So, so we can, yeah. As I said before, we can just do the channel variable ones. Whoa, I'm changing Pete, we sent out to be. And then we get these
488
01:17:31,140 --> 01:17:31,650
Okay.
489
01:17:32,700 --> 01:17:34,890
So then let's come to. Yep.
490
01:17:34,950 --> 01:17:47,550
I mentioned, can I please ask a question on previous slide, so you, you mentioned that in the in the plot, the, the, the right hand side of it agrees with gra symbiotically
491
01:17:48,120 --> 01:17:55,680
What about in the Intermediate Region after the bounce has happened, do we understand like how quickly it approaches the classical GR curve.
492
01:17:57,600 --> 01:17:59,340
Which region you in this region.
493
01:17:59,520 --> 01:18:03,030
In the post bounce on the, on, on the right hand side. Yes.
494
01:18:03,750 --> 01:18:04,320
This one.
495
01:18:04,620 --> 01:18:06,000
Yes, this one. Oh, you
496
01:18:06,030 --> 01:18:07,260
How fast union.
497
01:18:07,710 --> 01:18:13,950
Yeah, how fast it approaches the GR trajectory. You mentioned it only a purchase does it only approaches a sim totally or
498
01:18:15,690 --> 01:18:26,160
Because getting picture in the picture in the model which young gay man and claws and angry, etc. Have the approach is very fast to the GR trajectory like an elk UC
499
01:18:27,240 --> 01:18:32,640
Yeah. Um, so I'm our result is exactly the same as as calls and Leah.
500
01:18:34,170 --> 01:18:45,480
Yeah claws and real results because ECM. The, the equation just to reduce the same equation as them and and yeah for for yeah the channel variable doesn't change anything.
501
01:18:46,710 --> 01:18:56,400
Okay, so that those expressions are not true, like in some limit, but they are. Those are the true Hamiltonian physical Hamiltonian throughout evolution.
502
01:18:58,500 --> 01:19:01,530
And the true. Yeah. The other two Hamiltonian. Yeah.
503
01:19:02,160 --> 01:19:11,070
Okay, and you mentioned that your result when you take the case over influxes differs from what claws and I did like can you briefly tell us how they differ.
504
01:19:12,480 --> 01:19:19,320
Oh, it's just because I'm the equation is the same. It's just a tangible variable. For example, if I
505
01:19:20,370 --> 01:19:33,330
Change with these two variables. You see, and I was, you know, I was expecting. I should get the same result as you and clause because I was using copy Korea blacks. Right. But I was also surprised.
506
01:19:33,990 --> 01:19:45,120
That I didn't get the same result as you and cross. The result is not as nice as here because here I can extract the Hamiltonian is d AMP T. But if I change.
507
01:19:45,930 --> 01:19:59,130
Change variable as you did you on cloth. Did I won't extract Hamiltonian for tmp. I said, Yeah, yeah. Definitely. And I mean we discussed about it.
508
01:20:00,120 --> 01:20:15,630
Sure. But, and just one final question you're concerned you define this conserve energy like you have some like I missed the part, apart from the bronco crash test or caution does. You have probably a master skill or field is it in the model.
509
01:20:16,530 --> 01:20:18,600
It's valid for all cases.
510
01:20:18,810 --> 01:20:23,400
So the Hamiltonian is always concerned the physical Hamiltonian is always concerned.
511
01:20:24,570 --> 01:20:34,440
So I'm just slightly the very, very naive question. So your the way you define conserve energy seems like the square root of energy density in cosmology.
512
01:20:36,240 --> 01:20:38,880
You know, late and then
513
01:20:38,940 --> 01:20:44,220
But then you are defining our OC near the figure again dividing it by the volume.
514
01:20:45,630 --> 01:20:46,140
Hmm.
515
01:20:47,370 --> 01:20:50,310
That slightly inconsistent to me.
516
01:20:53,100 --> 01:20:59,970
See ya. Bye. Critical William. So you mean these formula. So this is the energy
517
01:21:00,960 --> 01:21:04,170
Yeah, and this is the critical volume.
518
01:21:04,200 --> 01:21:11,400
At the if you're if you're conserve energy is the square root of what cosmologists to find us the energy density
519
01:21:12,000 --> 01:21:13,560
Then you will see should be just a
520
01:21:13,560 --> 01:21:14,400
Scare of he
521
01:21:15,990 --> 01:21:20,400
Just was just a square. Yes. He not
522
01:21:21,540 --> 01:21:23,610
Sure. And we're dividing by another VC there.
523
01:21:27,930 --> 01:21:39,150
If I bought another VC. So here right now. I'm not completely clear about. I mean, so I probably see your point. So and so, because here I take these concerns. This is actually the energy per site.
524
01:21:40,050 --> 01:21:40,410
Yes.
525
01:21:41,430 --> 01:21:47,520
Well, I think it should be yeah issue, right, because this is a critical warning. You can also view. This is the volume per site.
526
01:21:48,540 --> 01:21:53,430
If you have. That's what I'm doing, if you will. This is volume per site. Yeah.
527
01:21:53,550 --> 01:22:00,600
I'm a bit confused. The way you're defining Rosie, Kevin. What is the definition of quote unquote conserve energy you're using
528
01:22:01,410 --> 01:22:13,320
Uh huh. Yeah. Yeah. And this is energy, energy divided by volume than this density right at at at at at a given vertex right that's what I
529
01:22:14,760 --> 01:22:15,060
Think
530
01:22:15,510 --> 01:22:20,250
Okay, yeah, my, my confusion is, is your consultancy really energy or something else.
531
01:22:20,670 --> 01:22:22,230
Is energy is energy
532
01:22:25,050 --> 01:22:26,370
Yeah, okay.
533
01:22:29,760 --> 01:22:34,050
Um, alright, so let's come to work those Hamiltonian
534
01:22:35,550 --> 01:22:43,800
Yeah, and and these Hamiltonian. Well, I'm here, I give you the expression of the entire Hamiltonian. So the first part is just Euclidean Hamiltonian
535
01:22:44,100 --> 01:22:59,760
Known and the second part is, is a curvature operator, the scanner commercial operator and how did you find these commercials skater operator. So you define it like ready calculus three and it is lens. Some overall the
536
01:23:01,110 --> 01:23:04,620
Hinge and here's a lens lens of the kings, and there's a
537
01:23:05,670 --> 01:23:22,380
Definite angle of the hinge and and then this is the lens operator and this stuff is no opportunity and down lots operators and and this lens operator and what what used by by Amanda was the lens operate defined by
538
01:23:23,610 --> 01:23:24,450
Virginia Yankee
539
01:23:25,470 --> 01:23:32,850
Involved and what a square root of some flax and and the one involved in this universe volume operate.
540
01:23:33,780 --> 01:23:44,820
More humorous more awkward, but the problem is is that once you invoke this numerous volume operator. Although, and you can. There's a way to define it instance that you can make it once you define on people's face.
541
01:23:46,380 --> 01:24:01,440
But the the 7000 expansion, use the technique used by Christy and Thomas in the US and mechanical ventilation theory seemed to me is not working anymore for this negative power of volume. So they are expansion for volume operator and
542
01:24:02,580 --> 01:24:09,630
Fraction the power of volume operator is only work for positive power, it doesn't work for negative power. So, which means that
543
01:24:11,310 --> 01:24:32,010
So the, the expected value. I'm Hamiltonian equals to the classical Hamiltonian. Here is the conjecture is not I'm not able to prove it rigorously, but I do believe this conductor to be true because man you did the data and his collaborators, deep, deep, a lot of a numerical checks.
544
01:24:33,780 --> 01:24:42,240
To check the semi Cosco of these scanner coverage operator should be always should be correct. But, I mean, I just don't know a rigorous proof.
545
01:24:43,080 --> 01:24:54,510
Of these formula but well assume let's let's say I'm supposed these conjecture is true. Yeah, what's going to happen. So of course the equation is going to be simplified now and
546
01:24:55,710 --> 01:25:05,220
Again, you reduce your equation to be diagonal part of octagonal part and and then you can perform a computation not I'm your competition and not necessarily all server.
547
01:25:05,670 --> 01:25:25,410
You can do your own personal computer to run them as medical and to check that these abdominal part is again automatically satisfied and the time evolution kind of dial apart again reduced to Hamiltonian evolution equation and the Hamiltonian. You just just reduce the standard
548
01:25:26,550 --> 01:25:34,740
So the Hamiltonian just reduced to the standard Hampton is different from the Hamiltonian. Bye bye. Thomas and Christina there's additional term.
549
01:25:35,340 --> 01:25:48,870
But yeah, so the for the for the worst time Tony and just reduced to a standard Hamiltonian, it just because, well, there's a simple reason for that. And these curvature, because the scale of cooperative becomes becomes zero.
550
01:25:50,040 --> 01:25:51,540
For the homogeneous so Toby.
551
01:25:54,060 --> 01:26:03,540
Yeah, and then the effective dynamics just people use the standard effective than me. See, again, a new zero scheme and the balance is symmetric. Now,
552
01:26:04,770 --> 01:26:16,440
Okay, so, um, yeah. And finally, we can, once we have the solution, we can plug back to the action to evaluate the actual action and get some politics.
553
01:26:17,220 --> 01:26:29,310
Of the amplitude and suppose a given the initial and final condition. So our past injuries faith based pass into rock and suppose a unique. The solution is unique, then
554
01:26:30,150 --> 01:26:44,400
These attitudes just have a single exponential exponential on show action or the actual value at solution. However, I'm to me. It is not hundred percent clear up to the solutions unique
555
01:26:45,960 --> 01:26:53,700
In for for the for the effective you pushing or fall upon grabbing the reason is just, you have a non linear equation.
556
01:26:54,960 --> 01:27:10,620
By depend on Hamiltonian case. Yeah, it depends a complicated one yet. So I'm not 100% clear whether the solutions meeting the full theory, but I'm hundred percent clear that the solution is unique in the sector of homogeneous isotope effect.
557
01:27:11,790 --> 01:27:28,800
So yeah, and then you can evaluate the actual action and you can show that this thing especially for the actual action is the same in in both toys of I'm Tony, you can take either portals Hamiltonian or I'm Tony by Thomas Christina and and you got exactly the same expression.
558
01:27:30,900 --> 01:27:43,440
OK, so now this is we finish the last step, and we recover the loop on cosmology effective dynamics in our scheme, but this is really a top down the relation from fall upon gravity.
559
01:27:44,520 --> 01:27:51,930
On ization and derive a fool equation motion of all degree of freedom and then
560
01:27:52,950 --> 01:27:56,670
Reduced to cosmology by by looking for solutions.
561
01:27:58,290 --> 01:28:00,570
Alright, so as I mentioned earlier,
562
01:28:01,650 --> 01:28:02,430
There is a
563
01:28:03,510 --> 01:28:14,100
Difference between zero schema new bar scheme. And so, and then the so we asked ourselves that. And so how we get a little scheme because
564
01:28:15,750 --> 01:28:24,570
So they're, they're the people say, there, there, there was some problem about musical scheme and and we just seeing how to relate new Baskin
565
01:28:25,980 --> 01:28:39,180
So, but let's look at the problem of musicals paint scheme again and from the perspective of food here. So I mean, firstly, we have conserve energy. So, I mean, all the
566
01:28:40,200 --> 01:28:56,850
The, the problem of musical scheme is that those critical volume and density depend on the conserve energy yeah the the volume is third power of energy and the critical density is an energy
567
01:28:58,260 --> 01:28:59,220
Power minus two.
568
01:29:00,330 --> 01:29:12,240
Right, so, so in case these energy to conserve energy is large. Yeah, then it would imply a large because your volume and some more critical density
569
01:29:12,870 --> 01:29:23,430
Yeah, because then it means that you will have bounced at low density or large because you're a woman. We're yeah but but from who's your point of view.
570
01:29:24,660 --> 01:29:32,130
We find it is it is not really true. It's because these conserve energy is not really the total energy but but the energy precise.
571
01:29:33,120 --> 01:29:46,290
Its energy its total energy divided by the number of vertices and edges so so ultimately you have to take continuum limit. So, which means and number of sites and goes through each
572
01:29:46,830 --> 01:29:56,820
I'm to remove the lattice dependence of your theory. The theory is defined on the things that is. But ultimately, in the end, you have to take remove your regulate
573
01:29:57,270 --> 01:30:06,540
Take the lattice refine latics yeah and then effectively send this energy is conserved. The energy to zero.
574
01:30:07,110 --> 01:30:26,580
And VC goes to zero and Rosie goes to infinity. So, which means I'm pretty sure you shouldn't have a large, large critical volume and it's more critical density, conceptually, it shouldn't be. You shouldn't have a lot I pounds happening at loaders.
575
01:30:27,660 --> 01:30:38,520
However, theoretically, if you take volume goes to zero. So it means that the minimal gap area gardening coupon code.
576
01:30:39,270 --> 01:30:56,250
From gravity should take into effect for which means and the critical volume goes to zero this limit is not really shouldn't be true in the theory of the loop on gravity because you have meaningful Eric and me know what you got. So we should make which may be, um,
577
01:30:57,750 --> 01:31:05,460
Well you fact every well maybe touristic way to do it to, for example, you can pull the cut off for the critical volume.
578
01:31:06,870 --> 01:31:11,880
Yeah, like a clunky and cut off. And maybe that's a good effective cereal ready
579
01:31:14,250 --> 01:31:24,420
But having to systematically understand those quantum effect and definitely need an need to take into account calling back and now especially non productive on facts.
580
01:31:24,720 --> 01:31:33,480
Because those minimal Erica is a definitely a number of years on back. So we have to we have to understand how those number to do that on the back.
581
01:31:34,320 --> 01:31:57,720
manifested in in the passage. So these addresses that the maybe the ultimate effective equation to be a quantum effective equation, it should be derived from these quantum effective action well in quantum theory, we know how to compute common factor of action. Right. And so once we
582
01:31:58,740 --> 01:32:08,250
Systematically got this quantum factor of action and we performed the variation of things for for the quantum effect of action. The results should be a ultimate the best
583
01:32:09,420 --> 01:32:10,560
Sort of, say, the
584
01:32:12,630 --> 01:32:19,230
Best the fact that equation for customers now at that equation might have some relation to the new blocky
585
01:32:19,830 --> 01:32:30,960
And and here I have to mention that I'm there was work by Vanya and you see and collaborator, they, they sort of relates to the company mimic
586
01:32:31,710 --> 01:32:50,400
What they do is they they got a new boss came from a random average of Hamiltonian position value of Hamiltonian over keeper likenesses so although I mean I think it's very, it's very interesting. I should mention it. Also, I don't know how to relate to do work on from passing
587
01:32:53,130 --> 01:33:00,930
OK, so now the last outlook. Yes. So, so we have a effective equation from Groupon right
588
01:33:01,560 --> 01:33:13,950
So those effective equation capturing all the real freedoms. So, so this is our set of interesting dynamic evolution equation for for general relativity and it is naturally democratized by by
589
01:33:14,640 --> 01:33:20,610
Grabbing and those equations and our example shows that you can solve them analytically.
590
01:33:21,120 --> 01:33:35,070
And it may be better to solve them numeric is is those equation already for numerical simulation. So it can be applied to all the ankle scenarios and can be applied for all space times when you just need to plug those equations supercomputer.
591
01:33:36,270 --> 01:33:55,230
So then, of course, there's many to do this. And there is something we are doing at the moment is to relate to the cosmology perturbations theory because our best Michael Cole is just ready for this study, and hopefully compare with own work by Christina seven hope and told us, Oliver.
592
01:33:56,250 --> 01:34:07,830
And definitely this equation time you apply to black holes and binaries and and also gravitational waves. So this is interesting perspective. Yeah, I think that's all I want to say thank you.
593
01:34:14,340 --> 01:34:15,660
Any final questions.
594
01:34:17,910 --> 01:34:20,310
No. Yes, close to the machine.
595
01:34:21,390 --> 01:34:27,510
THANKS VERY NICE DOG. I think it's very good. I have a question about the last comment to the continuum limited
596
01:34:28,440 --> 01:34:29,190
Namely,
597
01:34:29,400 --> 01:34:45,270
When you want to send the volume at each vertex to zero. I think the first message. My understanding, we could get in trouble with using the geezer team and volume operator, because this requires that T is the smallest quantity so that your regular
598
01:34:46,950 --> 01:34:49,740
Suggestions how to overcome this caveat.
599
01:34:51,690 --> 01:34:52,170
Oh,
600
01:34:53,610 --> 01:35:03,090
No. So, so these and sending VC to zero is definitely not allowed in the quantum regime, and that's why I say this is a heuristic.
601
01:35:03,720 --> 01:35:11,280
Argument. So it just indicates that and I mean this just a suggestion suggest you that you won't get a definitely you won't get a large
602
01:35:11,670 --> 01:35:16,500
Critical warning you. The critical woman has to be small and crystal critical density has to be large.
603
01:35:17,340 --> 01:35:32,100
And but how large they are and how small they are. And I don't know. So you have to really go into the number of active on impact of the passing the role and and really look at the volume and then that semi classical approximation definitely break up over there.
604
01:35:34,410 --> 01:35:35,400
Okay, thank you.
605
01:35:38,520 --> 01:35:39,030
Any other
606
01:35:40,170 --> 01:35:47,760
I have a question. And I want to apologize if this is something that had been discussed in the seminars before
607
01:35:48,900 --> 01:35:58,500
It's a little bit more general. My feeling is that I'm glad to see that basically cosmology could be recovered symbiotically
608
01:35:59,160 --> 01:36:15,090
Now, the natural question is, is there because vertical constants. And if it's not, I'm expecting to see it in like first or second or the in perturbation. If you try to expand the equations and do the solutions there.
609
01:36:17,160 --> 01:36:26,400
Um, well, so far in this calculation, we didn't really put in cosmological counseling, but I'm definitely we can put in, it won't be hard.
610
01:36:28,080 --> 01:36:33,300
And Cosmos perturbations, you add something ongoing so I'm right now. I'm not able to focus on
611
01:36:35,010 --> 01:36:35,820
Okay, thank you.
612
01:36:42,330 --> 01:36:44,700
Okay, there are no further questions that thank the speaker again.