Jorge Pullin: Okay, sorry speaker to this machine hand will speak about pathological formulation semi classical lemon cosmological perturbation theory.
Muxin Han: Hi, thanks for joining and thanks everybody for coming.
Muxin Han: Here I'm going to talk about some very reason resolved, we get with my collaborator, homeboy, Leo and high Valley, so I'm
Muxin Han: Sorry I and
Muxin Han: I probably can't see my mouth right
Muxin Han: Away. So yeah, let's go back to the regular more. Okay, so here
Muxin Han: The most part of the work and most part of the talk is going to be based on several papers and two of them has been published and they are another two papers and which hopefully coming very soon.
Muxin Han: So here it is an outline of my talk. So firstly at beginning I'm going to reveal or introduce the new pass integral formulation coming from reduce based on gravity. So this is
Muxin Han: A great reveal because and this is the result we opened last year and I have talked about this past into a formulation in the LDS talk
Muxin Han: Last year. So then, secondly, I'm going to talk about the semi classical limit the right from this past integral formulation and it's a standard way to obtain
Muxin Han: Very using my original principal to get equation motion. And then part solutions of the equation motion and we're
Muxin Han: Going to show you. And the result of that semi classical limit give you correct semi classical result and
Muxin Han: Certainly, I'm going to compare this new path integral formulation with the existing inform formulation and show some advantage of it and firstly I'm going to derive from these past Indira formulation, a cosmological perturbation theory from food Groupon gravity.
Muxin Han: And in and I'm also going to briefly talk about the relation with numerical relativity
Muxin Han: OK, so now let's come to the introducing these past integral formulation from reduced place based upon gravity. So let me first a break, very briefly.
Muxin Han: Review the reduced reduced by space formulation of gravity. So, usually in this formulation will have to couple gravity to some matter field and those Netherfield our so called talk fields.
Muxin Han: And three, there are three popular scenarios and you can couple gravity to so called Bronco cash. Does this is the playground.
Muxin Han: Of Bronco cash does. So here you got some fields t and as they are skaters and or we call them. Does the field.
Muxin Han: And and similarly, you can also have a modified Lagrangian which is we call golden dust.
Muxin Han: And the structure is very similar also depend on this clock do TNS and also you can have you can build a model with just the regular matter with
Muxin Han: For example, the real just a real skater field. So in this talk, I'm going to many focus on first two models. And although our, our work also did calculation with mass with Skylar field.
Muxin Han: Okay, so then this is standard and construction of the direct observable using the clock fields. So here we say these
Muxin Han: These scholar field equals towel.
Muxin Han: Corresponds to the data time or physical time variable and these as today. The these there another three figure fields, you could see my J and the value of those fields are our so called fixed physical space variable.
Muxin Han: So basically we build a task the frame in the space time. And the idea is that, and we construct the observable.
Muxin Han: By permit rising gravity variables with value of that field. Namely, we say this, we construct the field as he the gravity conjugate variables.
Muxin Han: The they are in terms of task time and Dustin space sigma and how then equals to the field evaluated at space time point x
Muxin Han: We are those tasks field equals toe and see. So this is the relatively define those York very direct of durables yes and they are eating variant.
Muxin Han: And then we find nicely. If we compute the opposing bracket, they just give the standard closing bracket, very similar to the usual wasn't bragging between A and E and the only difference is that here. Those delta functions are in the so called Dallas to space using sigma sigma prime
Muxin Han: So here from these we we see that they are actually the conjugate pairs. They are firstly dr, dr absorb oils. Secondly, the account, you get pairs and they they are actually the canonical coordinates in the reduced bass, bass.
Muxin Han: So we build up the bass, bass, which is reduced bass, bass and remove all constraints and gauge redundancy and, secondly, because we coupled matter field, we can solve those constraints and we can, right. So, these are the equivalent
Muxin Han: Hamiltonian constraint anthropomorphism constraint, you can say, you can see that
Muxin Han: It's it's linear to pee and Pj and here P is a momentum variable or tea and Pj is moment variable for Friday. So here it's all of those constraints, it comes up here and PJ.
Laurent Freidel: Sorry maxing. Yeah. Yeah. Could you just go full screen so we can see your screen better
Like, like, oh,
Muxin Han: Yes, about the thing is that you can
Muxin Han: Yeah, you can see my mouse anymore. I don't know why.
Muxin Han: Well, which one you prefer.
Muxin Han: I don't know why. So before usually can
Muxin Han: You prefer go to full screen was
Laurent Freidel: Not the only one.
Abhay Ashtekar: I think the mouse is important. I mean, the pointer is important because otherwise, we don't know.
Muxin Han: What you're talking about, okay, yeah. Sorry, I, I don't know why.
Jorge Pullin: Maybe you can close the side panels.
Muxin Han: Yes, I can go do this record. Yeah, I can.
Carlito2: Up right top right, the greenish thing closes right one.
Carlito2: Right up top left next to it next next next week.
Carlito2: Yeah.
Muxin Han: This one, this one,
Muxin Han: Yes. Yeah, okay. Oh, you got it. Okay, thank you. Thank you. Now,
Muxin Han: Okay, so, so this should be better.
Muxin Han: All right, we got. Okay, so, so, so we get these okay so we solve the constraint and then
Muxin Han: So because we are in the retail space based. And when we talk about the dynamics of the system.
Muxin Han: We find that the dynamic is generated by so called physical Hamiltonian and these physical time time Tony and is just a week we consider this a towel evolution and this, how is the variable that is the past the time does the clock.
Muxin Han: And and these physical Hamiltonian is just a space into raw of this age, this little he is just the
Muxin Han: Quantity of extract from the Iberian is the Hamiltonian constraints. Now, so this is a physical Hamiltonian and you can compute physical Hamiltonian with any faith based function. It just keeps the Tao evolution.
Muxin Han: Of this function. And here I show is basically the physical Hamiltonian correspond to Bronco cashless and gulshan does
Muxin Han: Here you see that the bronco catch it involves a square root, and these see you have c square and more for see a square the square. It is just the usual Hamiltonian constraint for the gravity. Gravity part Hamiltonian constraint.
Muxin Han: But now here's all the quantities are, are those a a stigma. A an easy one. The idea observable. They are composed by using your service, and here we also include a Cosmo your constant term and hear all the work is is is all the work assumes there's constant
Muxin Han: All right. And here the CIA is is the TV movie don't constraint. Yeah.
Muxin Han: So the gulshan does it looks a simpler. It is just that the integration is just the Hamiltonian euro Hamiltonian constraint rabbit.
Muxin Han: By the way, here, here is it's very important to remark that the are some now Holloman constraint.
Muxin Han: Or this radio space base in the in this tool does models and firstly the Hamiltonian constraint here this see has to be negative reason is is very simple, because the total Hamiltonian constraint is this guy.
Muxin Han: Plus the, the metal part and the metal part is always positive. And, and so the Hamiltonian constraint must be negative because the total Hamiltonian constraint goes to zero.
Muxin Han: And, and, secondly, and these these now. Hello. Mommy constraint is essentially for Bronco cash, does it comes from when you saw a billion is
Muxin Han: The Hamiltonian constraint. What you got is and these quantity equals to some task density square. Can you post something square and then there's some where it must be positive. And so this guy. Matthew positive to any question.
Abhay Ashtekar: Yeah and we shouldn't look when you say that you have responsive constant, do you mean that you cannot do it by setting lambda is equal to
Muxin Han: 00 definitely, I can you
Abhay Ashtekar: Allow
Muxin Han: Play I allow. Yeah. Yeah. Thank you.
Muxin Han: Okay. So here are two physical requirements.
Muxin Han: Which are now Holloman constraint classically has to be imposed on the face now, but it is these two guys call some subtleties in the inequality as we we will see
Muxin Han: Okay, so let's come to the quantum theory and the colonization is is carried out in in a standard LTE manner. And so here, our organization is carried out on a fixed lattice on a pizza to be ladies.
Muxin Han: Can be graph comma karma and and here in this world. We don't consider the boundary terms. So we assume the space.
Muxin Han: Has no boundary is contact has no boundary, for example, and because some from some of the work is related to cosmology and customer information theory we consider this is karma is the coup de gras petitioning three Taurus.
Muxin Han: And then
Muxin Han: Just the standard procedure we construct the Holloman variable and flax variable associated to all the edges of the of the lattice of this lattice.
Muxin Han: And. And now, here they are all Dr observable because they are constructed by using direct observable a he now.
Muxin Han: And and the human space is just a constructed by using way functions of Harlem is able to space Apollo enemies.
Muxin Han: And so all the previous construction and so I didn't talk about goals constraint. So all the one we solve those constraint class will be we are actually solving honey Tony constraint and the movement of constraints and and the goals constraint has has to be solved on mechanically
Muxin Han: So when we find the gauge invariant we function of holidays and we construct these finally physical space.
Muxin Han: Age karma and this is already a physical space because we solve because constraint on mechanically and all the classical constraint all the Hamiltonian consuming and you can move it up a spring has been solved.
Muxin Han: Sorry, we saw girls can sprinkle mechanically and all the other constraint has has been solved classic and these these final physical space is is coming from upon ization of reduce basis. And that's why we call this radio play space loop on Bing.
Muxin Han: All right. Okay. And, and then we come to the colonization of physical Hamiltonian is this central part
Muxin Han: And we want to study dynamics and as a result of the Hamiltonian
Muxin Han: Is a number of changing Hamiltonian build on the lattice. And the result is also is positive operator and self a joint
Muxin Han: Yeah, and and so here I want to put a Bronco cash danced and golden dust model in a uniform manner. So I introduce private alpha can be one or zero.
Muxin Han: And if you see this alpha intercept formula and alpha equals one, it corresponds to Bronco cash just an alpha equal to zero. It costs money to go from us. So here you can see this is a
Muxin Han: Quantum analog of c square Hamiltonian customer square and taking what we don't constrain square, so you can see it is a quantum analog
Muxin Han: Of this is the quantum analog of that.
Muxin Han: And so we construct age is is just the square of these operators, because this operator is not manifested itself would want. And so we have decorate and put them together and then take a
Abhay Ashtekar: Lost you
Muxin Han: Oh, sorry.
Muxin Han: Can you hear me.
Jorge Pullin: Yes, we can hear.
You
Muxin Han: Sorry what objects that you got lost.
Jorge Pullin: Might be a local problem for him. Go ahead.
Jorge Pullin: Okay, so
Muxin Han: So here is
Abhay Ashtekar: Our question since you asked me.
About
Abhay Ashtekar: Changing Antonia
Huh.
Yeah.
Muxin Han: Hello.
Abhay Ashtekar: Yes, but is that something that is. I mean, is it there is, there's some fundamental obstruction in using a graph ending Hamiltonian. Like, for example, which, like the one that Alessi uses or is there some i mean
Abhay Ashtekar: Is this something that you're not looked at, or is that an obstruction or
Muxin Han: Yeah yeah yeah yeah there's object. And the reason is that I'm so late on our procedure and doing semi classical analysis and and for the
Muxin Han: The semi classical limit for the number of changing Hamiltonian. It is correct has been proved to be correct, but for the abrupt ending Hamiltonian. The semi Kathleen is is not clear.
Abhay Ashtekar: But
It's funny.
Abhay Ashtekar: Because Allison says that is fine. So, yeah.
Jerzy Lewandowski: There is
Jerzy Lewandowski: There is another obstruction. So, as far as I understand, here we are talking about the cooker cooker model. So in the case of cooker model all the constraints are solved.
Jerzy Lewandowski: And add a few more feelings are not anymore gauge transformations. So, so in this case our Hubert spaces is this space is the key is the usual kinematics Hubert space for Luke quantum gravity and all the operators have to add this operator has to be more physical barrier.
Jerzy Lewandowski: There is no difficulties invariant operator that changes graphs graphs are preserved by these are those those sectors preserve super selection sectors preserved by all the pure monotheism variant so that during operators. So that is the reason why in this
Jerzy Lewandowski: In this so called algebraic quantum gravity Thomas
Jerzy Lewandowski: introduces a new one. The big Rockies, and all operators preserve Islam.
Muxin Han: Yeah.
Abhay Ashtekar: Thank you very much.
Muxin Han: So yeah, I will come come to this point in
Later.
Muxin Han: Alright so so okay so so here. So this is the Hamiltonian operator enter for the invitation. So you see, I introduced this see new
Muxin Han: The new at vertex be and these new can goes from zero to one, two, and three. So here, this new the new equals one, two, and three. This is this quantity is just the the CA that you can walk in the JP Morgan and constraint on that is
Muxin Han: So when C zero and new equals to zero. This is just a Hamiltonian as a Ukrainian part of the Hamiltonian constraint.
Muxin Han: Okay, and then the Hamiltonian constraint part is what we use the standard guilty man's Hamiltonian. So this is the European Hamiltonian C zero and the Lawrence part is also the standard form of guilty months I'm Tony so
Muxin Han: So in this calculation, we also carry out analysis for for the Hamiltonian used by I was old school.
Muxin Han: Yeah, so, but it caused some problems. So, Linda, I will also briefly comment on that.
Muxin Han: Alright, so, so now most of the discussion is on this gives documents one. Yeah.
Muxin Han: Okay, so here. Here it is. Some remark also relates to a previous subtlety about know Holloman constraint. So the. You see, you can see that our organization is not exactly the the colonization of what we call it zero, the continuum.
Muxin Han: The classical musical Hamiltonian. So, what we are monetizing is actually the square root square with an absolute value version of of HDL. So I have so
Muxin Han: The colonization somehow. So, so the colonization is actually you have hidden procedure that you first extend you the face face to the entire
Muxin Han: Face based upon gravity and ignoring those now Holloman constraint. But, but then the problem is that under the square root, this quantity is not any more.
Muxin Han: Money possibly positive. So, then, to, to have a well defined operators. So actually we insert a absolute value for the quantity inside the square root. And so, and this quantity definitely can extend to the entire face face and and it can be squared and can be carried out.
Muxin Han: So, which means that I'm this so we didn't really impose the. Now, who knows. Now how gnomic constraint those physical conditions on the
Muxin Han: at the quantum level, but instead we modified the physical Hamiltonian. So these consistent with the classical Hamiltonian
Muxin Han: In in a portion of the of the faceless okay and these costs, some subtleties, we, we will talk about it in a moment.
Muxin Han: Okay, so once we have these positive language changing self joined physical Hamiltonian and we can just have a unitary time evolution just a standard on mechanics or quantum field theory.
Muxin Han: So we can define the transition between two to physical states in the mobile space and to hear because we are interested in semi classical physics. So we introduced the
Muxin Han: Semi classical stays as the initial state and final state. And they are gauging variant coherent state and labeled by gauge orbit orbit of quantity G and these so these
Muxin Han: These gauging very cogent state is just a group leveraging of language environment in state i think most of you are familiar with that.
Muxin Han: So here, this G label G is a hormone a parameter ization of Groupon gravity Facebook and it relates to both flat variables and then hold on. So here are usually parametric alone me as exponential Sam cedar.
Muxin Han: Okay and and here it also depends on I semi classical parameter, what we call, usually called T
Muxin Han: This is LP square divided by A square A is just a length unit to make this T dimension is to make LP dimensional is you can view this T IS JUST A LP
Muxin Han: Evaluated at a certain unit. So for example, this a can be like one centimeter and and obviously when we take semi classical emit t equals to zero.
Muxin Han: And then this is what we have done is just a standard procedure of coherence, they pass into raw and so we're desensitized. These unitary
Muxin Han: Time evolution operator in in in it has more steps steps. So we with arbitrarily large and and the each step of the time evolution is arbitrarily small
Muxin Han: Okay, and what we insert is the over company's needs relation of coherent state were normalized cohesion stateside widow and insight Widow is normalized in state and these G is a cell to see as an integral over so to see
Muxin Han: Okay, so this is all very standard. And so here I'm going to just escape all the intermediate steps and it has been taught last year.
Muxin Han: In LTS talk. And you can also find out details in our paper and the result is is we got we got the past integral formula.
Muxin Han: And this is a discrete passed into law and a four dimensional hydrophobic lattice.
Muxin Han: And is hybrid to be like is because this is a gamma, which is a cubic lattice times discrete time. So we got a hybrid cubic lattice and it's it's integrated over intermediate
Muxin Han: Firstly, it integrates over intermediate coherent state labels G for all the intermediate time and there's another folder integral coming from the group everything
Muxin Han: Of cooking state of gauging Murray State. So here, because the initial state and final state. The dependent on gauge orbit. This th is just integrating over all the transformations
Muxin Han: All right and and you can easily see that we can extract and classical action would be class and the exponent is as divided by T T is a semi classical or I mean it's like each bar.
Muxin Han: Here it is the expression of classical action. So the first term. You can see it's analog up the kinetic term and second term is a relates to the matrix element of the Hamiltonian and it's a, it's a, the past interval is the analog of
Muxin Han: Facebook's policy integral component mechanics.
Muxin Han: And this quantity and there's in the integration is another major factor is magnified to click on new and it is some function of G. And here it is independent of
Muxin Han: Some classical parameter to this one is the independent semi classical property so and so therefore it doesn't really involve in too much in our discussion about semi classical analysis.
Muxin Han: So here they are some remarks, firstly, these past into raw formulation is really rigorously derived from canonical formulation. So it's we what we have done is just taking
Muxin Han: Reduced bass, bass quantization and taking this unitary time evolution defined the transition amplitude and then the standard coherence, they pass into a procedure and we get these paths in zero
Muxin Han: And secondly, this past into raw computer, of course, computer loop on the transition period between certain boundary state. This is very similar poverty with steam form MTG
Muxin Han: And also, importantly, these past integral is manifested finite and manifestly unitary and because
Muxin Han: The these transition amplitude is clearly finite and he just well defined unit operator well defined state and its transition on PTO finite
Muxin Han: And transition amplitude equal to the integral integral must be finite. And it's manifesting unity, because this is derived from a unitary time you will know. Okay, so this is the path, the integral formulation
Abhay Ashtekar: Sorry. So what happens to all the large J.
Abhay Ashtekar: divergences that one normally has oh
Muxin Han: Here, there's no divergent anymore so large rate.
Muxin Han: So here there's no luxury. So it's the integral is expressed in in in coherent state variables. So, so there is no I'm not anymore something over. Jay, I don't
Muxin Han: I'm not, it should realize to something over j. Now if I changing the representation from convince a presentation tool tools be networks representation. For example, it should give a semi over j. But yeah, it seems, it's just a changing of representation. So it doesn't change.
Abhay Ashtekar: But then why can't we say that in the order that he spent forms, I can just do the same change of representation and there is no there were just with large chains that are looking for a divergence
Why cannot say no.
Muxin Han: No, no, no. So, so for us being forms, if you somehow changed through the coherence date representation, definitely the divergence is still there. Because, because, at the beginning, it is divergent
Muxin Han: So here is really that you have no divergent. It's just the derived from we can find that quantity now. And so it's it's
Carlito2: It's being able to produce products of delta functions in the integral and you can write everything in terms of integral service you too, but the integration is the product of delta functions in the same point that's
Muxin Han: Yeah, that's right.
Abhay Ashtekar: Thank you. Okay.
Muxin Han: All right, so, so having this path into raw so we can consider semi classical image.
Muxin Han: So this is really just a standard
Muxin Han: Computation for the stationary face approximation because we have written this into rolling in a standard past integral manner and
Muxin Han: Semi classical limit is particles to zero or in our case s t equals to zero and and then what we need to do is stationary Facebook automation and this integral is dominated by by so called critical points and those who weren't satisfied classical equation emotion or the original principal
Muxin Han: And and those equation motion we have derive it last year.
Muxin Han: I have talked about it last year in the in the talk already. And so just a brief.
Muxin Han: Review. Is that what we have done is, is the calculation is we did the whole more make their defamation. So to see to compute this variation
Muxin Han: And we got two equations relates to the variation respect to G and the variation respect to the complex conjugate of G. So we got these two equations. These are relates to time evolution equation. So you can see it.
Muxin Han: Relates to delta top. The more time, more time step. And you can see the right hand side is a whole morphic
Muxin Han: The derivative of matrix element of the physical Hamiltonian and here this is under homomorphic derivative
Muxin Han: Of the matrix element and left hand side it is you have a delta t tower in the denominator and numerator is a difference of two terms and the second second equation is very similar.
Muxin Han: And it's this difference relates to the faith based variables at to neighboring timestamps to two different concepts.
Muxin Han: And the last equation is coming from the variation of things with respect to this age, and the result of the variation. It's just the closure condition and disclosure condition. It's imposed as a constraint on the initial on the initial condition on the initial data.
Okay.
Muxin Han: And then when we analyze those equations we find some interesting and very useful properties as following
Muxin Han: So firstly, so there is a being to be the coherent state has introduced feature is that this past interval is dominated at the neighborhood, we are
Muxin Han: If you have two different timestamps and the faceplates variables has to be in a short distance has to be the distance between to face face variables has to be order square root of tea or square HR and this is coming from Golden State passing through also
Muxin Han: At least in the near the semi classical regime. The past interview is dominated by in this in this neighborhood. We are nearby steps the faith based variables is cannot be
Muxin Han: Of long distance and the distance has to be very fraught controlled by the order of square root of it.
Muxin Han: And then once we come into this this small neighborhood and we find that these, these two functions, the one and the to the you can find the have unique route.
Muxin Han: If you solve this equation you want equal to zero and d two equals zero. You can find isolated root of this equation, and they are they are precisely
Muxin Han: GI plus one equals two GI GI equals two t minus one. So, which means the face point has to be coincide, in order that you want to be zero. Okay.
Muxin Han: And it relates to the limit that delta top equal to zero as I am going to see right now and I'm going to talk about it right now. So, so let's consider this equation and
Muxin Han: And the solution of an any solution of this equation. And of course, this is a discrete solution. And because the equation motion is this great
Muxin Han: And and here's the observation is that the right hand side is always finite because this is just a matrix elements of the info Hamiltonian and the derivative and they are finite and therefore, for any solutions. The left hand side must be finite to. Yeah. However,
Muxin Han: The standard how he is arbitrary small yeah it's totally arbitrary small because the timestamp and is arbitrary art.
Muxin Han: And this delta x goes to zero and but the left hand side must be finite, for any solution. It forces and that he won, and he to this to function equals has to goes to zero in together. We started out, you
Muxin Han: Know, and because we have seen that there's a these two equations has unique route which are key i plus one equals two GI
Muxin Han: Okay, so when when these two quantity are forced to approach and zero, then it forces GI plus one approaches. Yeah. So it means that you get when we take the
Muxin Han: What we call time continues limit until the top goes to zero, it forces that she i plus one approaches Jia. Yeah. And so what we have in congruent, is that
Muxin Han: All all the disparate solutions because all the assumption is that for any solution of the question motion. So, so all these great solutions to meet a
Muxin Han: Continuous time approximation, as all the different solutions can be approximated arbitrarily well by by using continuous function see up top.
Muxin Han: And these functions continues in top and then in this in this limit and from this formula, it's it's it's suggest that and the left hand side is just some kind of a time derivative
Muxin Han: Yeah, it's it's a it's a similar to the standard equation motion on
Muxin Han: The left hand side is just some time derivative and, indeed, so we can so it shows that whenever we look at the solution of those equation motion. We can take the time continuously to study those solutions.
Muxin Han: Yes. And now when we take this time continually mid for those two equations and we see those equation get quickly simplified so firstly, the right hand side here, it relates to the matrix elements.
Muxin Han: Of the Hamiltonian, but we may take the time continuously mid July goes to GI plus one.
Muxin Han: It reduced just the matrix element. And as we know that because this age is highly not normal, and the matrix element is different, very hard to compute
Muxin Han: So, but when it reduced to the the expectation value and it is relatively easier to compute and at least we can use the semi classical perturbation theory developed by Christina Tomas
Muxin Han: To expand this quantity as a power seriousness bar and the leading order is just a classical discrete Hamiltonian. So here it is also
Muxin Han: Considering a bias comments about the graph changing operator and these this procedure cannot be done anymore. So this semi classical perturbations theory developed by Thomas and it only works for for a number of changing operator.
Muxin Han: Alright, so, so then the second simplification is that the left hand side, indeed, really reduced to some derivatives. They are derivatives of homey and flags up to a linear combination
Muxin Han: Okay, then this. So when we carry out the reduction and you can see the left hand side reduced to the time derivative of EP and the Sita Sita is
Muxin Han: Exponent me and and this equation reduced to these
Muxin Han: first order differential equation tell
Muxin Han: And there is a matrix T and this matrix T. It contains very long formulas, all these calculations dummy Mathematica and and this T matrix, it has some long formula. Although, actually. The university has not so many long formulas that you will see in a moment.
Muxin Han: So, so here and this those long formulas. These T is a six by six matrix and and the formula, but you can download those formulas from from GitHub. So we have posted all the mathematical code and and result you get up so you can download everything over there.
Muxin Han: So here from this formula can see that the the solutions have a and b equation A and B.
Muxin Han: Not only can be approached by continuous functions. But actually, it must be differentiable function because because the time it contains time. You ready
Muxin Han: OK, so now
The question.
Abhay Ashtekar: This is again going back to this graph changing what is not gloves.
Abhay Ashtekar: I mean you argued that what is getting a good classical limit.
Abhay Ashtekar: If I just have fixed the graph. And for example, if it is some simple cosmological any situation but sake cosmological situation in which the universe is expanding very rapidly.
Abhay Ashtekar: Then in the wall me changing enormously and a fixed graph. The only way are the areas are changing enormously. And the only way that this can happen.
Abhay Ashtekar: Is going to be in in this graph, I'm going to get continuously the jays changing enormously right because the number of intersections will be remain the same, because of graph is completely fixed number of intersection even surface of the graph is
Abhay Ashtekar: And so it's this. I mean, on the one hand, formerly I see that you know keeping this fixed graph arguments you and Thomas and then
Abhay Ashtekar: You sell give that, in fact, you'll get this similar equation, but how do I visualize it. What is happening in the quantum mechanical situation jays of just changing enormously. Is that what is happening. I mean, to get the classical limit correctly.
Muxin Han: Well, I'm here. Everything is so so here we are not using J variable. So here, everything is classical faceplates variable P and Sita so
Muxin Han: So here is that, in the end, when we do semi classical limit and we see those variables changes quickly support down for expanding universe. So those variable will changes.
Muxin Han: Very large, but these are semi classical variables. And these are just like electricity in face base so so when you talk about Jay which which changes radically rapidly.
Muxin Han: I guess what I mean is at the quantum level.
Abhay Ashtekar: Yeah, I mean I got some. I mean, your, your calculation is really relating the quantum theory, the classical theory, you've got the size and you know you're saying that well as the newest was
Abhay Ashtekar: Classically, there's a trajectory and the size of following that trajectory. That's what you're saying. Right.
Muxin Han: That's right.
Abhay Ashtekar: That's right in the limit. Yeah, but then to follow that trajectory. I mean I you say that this classical label the label of the query status changing
Abhay Ashtekar: But that just means that the peak.
Abhay Ashtekar: Of the coin stayed in the JIRA presentation is changed enormously.
Abhay Ashtekar: Yeah, I mean the same thing in the quantum geometry is very right here in this sentence, I didn't know that there was a there are finite number of links, but each of them has an absolute huge area right
Abhay Ashtekar: Right, right. So,
Abhay Ashtekar: I mean I mathematical, no problem. But I don't understand. I'm uncomfortable physically about it. I just wanted to express this you can go
Muxin Han: Suppose we somehow take the continuum limit but but suppose, and as well what we had. We can were able to do is is taking the continually me and semi classical level.
Muxin Han: And and then
Muxin Han: In case of the lattice spacing is infinitesimally small
Muxin Han: And I think it is just approximately approximately like continuous theory.
But
Abhay Ashtekar: Limit you're taking it just in time.
Abhay Ashtekar: It's nothing.
Muxin Han: Like it so just just a moment. And at some point, I will talk about also the company meeting space.
Abhay Ashtekar: So I can talk about later.
Abhay Ashtekar: Okay, but you know just time that's that's
Abhay Ashtekar: Not limited just in time. Yes.
Muxin Han: Yeah, yeah, yeah, that's right.
Jerzy Lewandowski: But if I can ask you, you said that something is wrong about the classical discrete Hamiltonian equation which is in the middle of this slide, if we can see that graph changing operators. So what would be wrong about this equation.
Muxin Han: So, suppose this age is cropped changing. And then this in Nepal, I will just give you a zero, because it changes squat.
Jerzy Lewandowski: Oh, but this is only if you consider states like this, but if you consider some states, which also consists of many different grabs, then it may be not zero.
Abhay Ashtekar: Right.
Abhay Ashtekar: Right, this is
Abhay Ashtekar: The states like what the elastic Container Service. Right. I mean, in which is a protein distribution that are all possible possible
Muxin Han: Yeah, probably. But, but
Muxin Han: For that states and that that is not a status identity matrix, right, in principle, that identity matrix. Yeah, well, but that I don't know how to do.
Muxin Han: Coherence they pass into so I'm so at least for this work. So maybe I in the future I will try to sing about the other way. But, but, at least for this work because it's coming from coherence, the past in Toronto and use it this comprehensive
Abhay Ashtekar: So you'd agree. I agree that if it is pure states that it is, it may be difficult to make the argument you're making is that right
Jerzy Lewandowski: Yes, I understand. I mean, I understand that that there is no obstacle to do it in a different way. But in in. We just don't know how to do it in a different way, for the time being.
Abhay Ashtekar: Thank you.
Jerzy Lewandowski: Is it correct
Muxin Han: Yeah.
Muxin Han: That's right, that's right. Correct, yes.
Muxin Han: All right. Um,
Muxin Han: Okay. Okay, so, so now I'm when when we find the the semi classical inclusion motion coming on passing the role reduced to this form. So we are wondering that. Maybe it's related to just the Hamiltonian time evolution in the reduce
Muxin Han: Reduce reduce
Muxin Han: Okay, and indeed we are trying to show this week we agree we are able to show this. So now we are able to transform this equation into a
Muxin Han: Hamiltonian equation and this is they are some detail calculations and with the only thing we need to compute is is
Muxin Han: Expanding the person bracket between Hamiltonian and PNC to variables into a linear combination of derivatives and and those coefficients. You can see that the artists, p, p, cause in brackets PC de
Muxin Han: possum bracket and so on and so here we can define the matrix P which is just containing those posts and bracket and this is zero because Sita Sita percent bracket equals zero.
Muxin Han: And when we carry all this computation computer all the matrix elements and we find that these p is just the inverse of this matrix T what we
Muxin Han: Extract from the semi finals week which emotion and this is awkward this society is is it's a long matrix is a big matrix.
Muxin Han: But if you let her run for for for a while, you will you will just simplify the equation, you find that this key metric is it precisely the inverse of the matrix T. OK. So now you can see that if you if you plug in this relation in the
Muxin Han: Semiconductor equipment motion, what you get is, is just the Hamiltonian. The Hamilton's equation.
Muxin Han: For the, for the physical Hamiltonian. Now, and this is the equivalent. This is just the equivalent form of the semi classical equation motion coming from our past integral
Muxin Han: And
Muxin Han: You can also see that and this equation motion escaping Marianne, because this is just the standard Hamilton's equation and the cost of the closure condition GT this G is just our culture condition it is
Muxin Han: The time evolution of closure condition is zero because he is gauging Mario. So this quarter condition is a concern on it.
Muxin Han: Okay, so here's three remarks, we have seen that the semi classical than me. It's just given by the Hamiltonian flow generated by physical this great Hamiltonian
Muxin Han: Okay. And, and now the good thing about it is that the semi classical dynamics is becomes just a initial value problem in in the basement, because we set up certain
Muxin Han: Point in the face face as the initial condition and then you can let the Hamiltonian flow and it's generate solution. This is just really the standard
Muxin Han: Classical mechanics in the face and then the initial conditions uniquely usually the initial condition uniquely determines the solution.
Muxin Han: But of course it has some conditions and if you are in the regime, we are this age is regular and then the initial condition uniquely determines a solution. So, here the regular. It just means that
Muxin Han: The, the uniqueness of the solution is controlled by the serum, the fundamental serum of
Muxin Han: Of first order differential equation and and then this age. And so when when program the Hamiltonian flow the Hamiltonian vector field is
Muxin Han: Is satisfying the elections condition certain continuity of the first dollar differential. Then you have unique solution. Even the
Muxin Han: Initial condition. So that's the case when when when some weird when he is regular so indeed, this age is not completely regular. It has some irregularities, as we will see in a moment.
Ivan Agullo: So what one wishes just re understand. So this is you're just getting the emotion that you would get
Ivan Agullo: From the initial classical discrete Hamiltonian is a true
Muxin Han: So this is the equation motion derived from passing through. And it's the same as it so it's just the same as the classical equals emotion from a disgrace.
Ivan Agullo: But is that they, they, the classical discrete Hamiltonian that you started with.
Muxin Han: It is, yes, yes.
Ivan Agullo: Right. But then, you know, Mike. My question is,
Ivan Agullo: One really needs to do all this calculation to reach this result because you know in web the mechanics. When we started popping out
Ivan Agullo: You know, you start with a classical Hamiltonian you promote it on operator, you put it in the bathroom.
Ivan Agullo: And then, you know, just sales consistency of the batting developed tells you that they're stationary face approximation should give you the classical equation of motion. So it's not what you
Muxin Han: Want. So this is Luke on gravity. So it's not completely yes dramatically, it should work. Yeah, this is, but it's not so obvious for the congress it because
Muxin Han: You see, the state is is different from the standard
Muxin Han: Fields theory, the quantum mechanics coherent state and Hamiltonian is also complicated. You see the intermediate step is is non trivial. It's, it's very much rely on the semi classical perturbations theory.
Muxin Han: Of Hamiltonian and also also it relies on the structure of the Oregon State and and that's why you can take the time continues limit.
Muxin Han: Yeah, and and it's not is schematically, yes, you are right but but to work it out in detail. It's. Is it reliable rely on non trivial properties of current state or physical Hamiltonian
Ivan Agullo: Think, but then, then this equation has nothing when the money is. Yes, a result of the initial discovery session you know the consequences that you get from these equations has no nothing went on.
Muxin Han: Well, dear, dear us a little bit
Muxin Han: That, that's it. That's the regularity of these Hamiltonian which is not exactly the same as classical I will talk about the right now.
Ivan Agullo: Okay, thank you.
Muxin Han: Yeah, but you're right it's most mostly it is the same as the classical and that's why I say I see it is our theory is semi classical a consistent
Abhay Ashtekar: Can, can I just add to the statement that you want made it's completely right, there's a finite number of degrees of freedom system. So because you had fixed graph.
Abhay Ashtekar: So you don't have to go to filter at all. And so you just order the quantum mechanics, but it is a car or any kind of mechanics on a manifold because configuration spaces as you too, and not a, not an
Abhay Ashtekar: Extra two times and and that is why you have to be careful that the police station case you do have enough st have the same prop share
Abhay Ashtekar: enough number of properties with standard query states on our end for this to go through and that is what I think we should is saying is that right mission.
Muxin Han: Yeah, that's exactly
Muxin Han: Correct.
Muxin Han: Yes. Alright, so, so now let's let's come to the continuum limit, as I promised, so this is a lattice continuum limit of those 10 casco equals emotion.
Muxin Han: Yeah. So here, this is our lattice and I consider there's a coordinate lattice spacing accordingly lens of the lattice at what I called you.
Muxin Han: And then the lattice continuum game. It is a new equal to zero, but at the same time, the number of ladies vs equals infinity. This is the total number of
Muxin Han: Vertices in the lattice. We said that goes to infinity, such that new cube times the number of vertices speaks sort of the total coordinate size of the lattices fix, but we will have total, total number of vertices goes up.
Muxin Han: Okay. And, but here.
Muxin Han: Rigorously speaking. So, what we are doing is taking the continuum limit after the semi classical me so. So because we derive our sand castle equation motion coming from the semi classical image.
Muxin Han: And so we are taking new equals zero, after this. So it means that we are actually in this region that LP is much more than you and much more than eight. So how to understand this, so
Muxin Han: You can view that we picked a is just a unit like once in a mirror one centimeter and then we'll send LP goes to zero faster than musicals to do so. Both LPN new goes to zero, but it's an LP faster and new slow, but both of them.
Muxin Han: Has to send to zero.
Muxin Han: In this limit. And then in this limit we are taking continuously need for this equation emotion. Yeah, and then
Muxin Han: You can you can expand the holiday variable and flux variable and follow me expand in terms of museum order is just a connection field located at point the vertex we
Muxin Han: And and flux give you the density. I tried to do it vertically and up to higher order new and and then it is a some tedious calculation.
Muxin Han: To to expand the the derivatives of physical Hamiltonian and and changing variable and tool to continue. And then after the computation and awkward this competition is tedious. A lot of computer is down Mathematica and running on the server or for a few days.
Muxin Han: And then
Muxin Han: The result is is nice and simple. A nice. The result is that just the so it's very similar to the ordinary
Muxin Han: Hamilton's equation for fields in theory the ordinary Hamptons equation theory.
Muxin Han: In in the continuum theory of gravity does the system and and this is time derivative, it relates to the functional derivative of
Muxin Han: Of the physical Hamiltonian on the continuum and up to higher order in new okay but here, this is not as I said before, and there are subtle body.
Muxin Han: And this is not precisely the classical theory in the continue the different is that you got a absolute value under the square root
Muxin Han: And it's a consequence is, it's very simple. It's just because at very beginning for the quantum theory we have extend the
Muxin Han: Face face from the Costco allowed region to the entire face based on gravity and then to quantify this Hamiltonian. So we have to
Muxin Han: Taking a absolute value because the quantity and of the square root is not any more positive in the entire face, at least for the bronco cash.
Muxin Han: That our bike was one is correspond to Bronco cash does so. So here the you see this is, as I said, is not complete different company, the same as the classical theory in the continue to hear you got extra absolutely
Muxin Han: However, is indeed the same if we have the physical requirements. So here's the physical requirement for the for the dust.
Muxin Han: Is that the quantity and the square root is positive. So, here the semi classical promotion indeed class coincide with a classical theory of gravity and asked when these
Muxin Han: Inequality satisfied and here you can also see the regularity of the physical Hamiltonian is precisely the point we are this quantity goes to zero because that's that's the point we are
Muxin Han: This this physical Hamiltonian is not an it's not differentiable anymore. It's because of this absolute value. It's not debatable anymore. So, so which means at this point.
Muxin Han: The equation motion is really unified is is Hamilton's equation is not well defined. So this is the regularity of the semi classical dynamics.
Muxin Han: OK, and then how to understand this, but it is the nice part of it.
Abhay Ashtekar: Can I can ask question over the last slide please. Yeah. So. So just to go back to what you want was asking. So this irregularity is really not because of the difference
Abhay Ashtekar: In the kinematics of this theory that you're looking at an ordinary part integrals in order the parts integral, the kinematics. Is that the configuration spaces are and here it is issue two to the power number of ages.
Abhay Ashtekar: That is difference that is not the main point. The main point is just that the Hamiltonian. That's right. Yeah. Okay. Is is different. And because you just had to do this thing in diameter.
Abhay Ashtekar: And the second point that was that the two things will be as you said it already, that, you know, if I don't just restrict myself to the case to the classical part
Abhay Ashtekar: Where that in that integration is positive. That's what classical constraint asked you to do, then there is perfectly fine, except that it could be that the singularity, or something like that, then
Abhay Ashtekar: The classical evolution will just stop right
Abhay Ashtekar: On this branch, whereas it could happen, that in fact you might be able to continue to the other branch.
Abhay Ashtekar: Which is classical. It was not there, because that quantity was negative there but but you are
Abhay Ashtekar: Modified Hamiltonian might allow you to do such a such a transition. So have you looked at that, that something like that does happen in this man yes singularities.
Muxin Han: Know, so that's the thing I'm going to talk about immediately.
Muxin Han: Everything is can everything is actually determined by initial condition. So, most of the time you're just going to avoid this.
Muxin Han: So, so this is something I'm going to talk about next slide.
Abhay Ashtekar: Thank you.
Muxin Han: Okay, yes. So now, now here we got a puddle that and you see the arson is the equation emotion is not precisely the same as the classical theory.
Muxin Han: Continuum and the only difference is this irregularity and this absolute value. However, we, we find that, and these are the different is completely controlled by the initial condition.
Muxin Han: And so, so this summer kinds of consistency and regularity are determined by the initial condition and unit or the initial state of the past, you've got the initial state of hospital pick that
Muxin Han: certain basic one as the initial condition and the here. I suppose the initial condition in the faith based are are within the classical allowed regime. Suppose the initial condition satisfies
Muxin Han: These novel gnomic constraint and both of them. It turns out, both of them both of these two now Holloman constraints are going to be preserved by the time you're supposed to total time emotion is finite.
Muxin Han: And the reason is falling. The reason is that for Bronco cash dust in the continuum that this the
Muxin Han: The quantity under the square root, and also that you can walk in and constraint are our conserved are concerned quantities in the classical theatre in the continuing series. Yeah. So here we are not exactly in the continuum. But you can see that continue limit really
Muxin Han: Really relates to the release the same equation.
Muxin Han: So, you find that the, the, they are not. These are not exactly concerned quantity in this great theory, but they are conserved approximately up to older up to corrections of autoimmune because because the equation motion is correct it up to older me in the semi classical soon so
Abhay Ashtekar: I'm sorry, I don't. So why are the concert in the continually isn't isn't the sequel to the matter I Newtonian and yes
Muxin Han: That's right. Yeah, so
Abhay Ashtekar: Why isn't it just a property abroad cash.
Muxin Han: Yeah, this is just
Abhay Ashtekar: A matter Hamiltonian is constantly time
Muxin Han: Independently and also
Muxin Han: Read young and you are mostly right but but the the immediate reason is that if you compute the
Muxin Han: So, so the quantity and as the square root, if you compute the person bracket with Sufism Hamiltonian zero and also that if you move them constraint.
Muxin Han: CJ certain certain certain moving them constraint, you can compute the bottom bracket with which is zero. It's you just get zero. This is the poverty of the bronco gosh we can't even. Okay.
Abhay Ashtekar: Um, but that you love to the property of a scale of field, for example, because it's not true.
Abhay Ashtekar: Hamiltonian his words.
Muxin Han: Were skillful. Let me, let me see for for scanner field you you
Muxin Han: I don't remember, but I think something similar should happen.
Abhay Ashtekar: No, no, no. So
Abhay Ashtekar: We know that skill level. I've been told is not covered under the, you know, order the evolution in general relativity. It is only the total I
Abhay Ashtekar: Think zero if it changes on the
Abhay Ashtekar: Fly, which are also changes so
Abhay Ashtekar: So torn between general okay
Muxin Han: So, so yeah I
Muxin Han: Think I was interviewed I not company. Sure, yeah.
Abhay Ashtekar: No, I'm saying that is not true.
Muxin Han: Okay.
Abhay Ashtekar: I'm saying
Carlito2: Is that, is that not true. Also, sorry. Just to clarify, is that also also in the special gays engagement, which uses the scale a field itself as a time because that's a question is not obviously they they they manage those kind of finish off conservative change with Graham.
Muxin Han: Yeah, so
Okay.
Muxin Han: A special gift. Yes.
Muxin Han: Yeah, yes, it is a special game. So, so I'm not. But if Christina is here. She should be able to answer this question.
Muxin Han: So I'm
Abhay Ashtekar: Not Gallo you're right but it's absorbed that gauge does break down in the classical face face.
Muxin Han: Yeah.
Carlito2: Yeah, just, I was just mentioning that is a very cool. You're not sure about the
Abhay Ashtekar: Greater not. I think the scale of field is is constant spatially right, then let's let's talk about later.
Abhay Ashtekar: So the spacer derivatives will be valued and where do worry about whether their time derivative and the square to the Met the metrics or the census quarter the metric would again be times time dependent where you actually consider the
Abhay Ashtekar: The total Hamiltonian. So I think that he would, in that case, it is likely that
Abhay Ashtekar: The total energy density minority concept but let will. Okay. I think about. Thank you. Thank you.
Muxin Han: Thank you.
Muxin Han: Okay. Okay, so. So, because of this conservation poverty and so to this is, it works for both Bronco casters and gouging us
So,
Muxin Han: So then, and these at least this not automatic constraints are going to preserve, although in this case theory. These two quantity that I'm not precisely get preserved. But, but at least those equalities will be get preserved if the total time tease is finite.
Muxin Han: And. All right, so then so, consequently, suppose we we are imposing a semi classical and allowed
Muxin Han: Initial condition if the initial condition are within the semi classical a medium and then these seminars of inclusion motion is complete regular you avoided the regularity.
Muxin Han: And and the solution is uniquely suited solution is unique and and is determined by the initial condition right and. However, the other way around the other
Muxin Han: Way around and suppose initial state relates to a classical forbidden region in the bass, bass and and in that I'm which was some initial condition which violate and those two
Muxin Han: Qualities and you can use that. But the thing is that these initial conditions this initial state is not a more semi Cosco. Okay, so these are not sending passwords.
Muxin Han: Although they are coherent state but but because it peaked at some classical forbidden some some Facebook point which is cosplay forbidden. So then the state is not some Costco.
Muxin Han: And then of course the time evolution of
Muxin Han: Of the state, the time evolution of the
Muxin Han: Of the of the of the place by point is is not going to give a semi classical trajectory, you may get some solution and but it's not going to realize to classical theory.
Muxin Han: Yeah, so this is some kind of analog of negative energy state in quantum field theory and and the early paper by Christina and Thomas has expected these kind of
Muxin Han: You know what happens and it's probably also relates to untie space time because 99 of the states. It's a particle and and it's probably relates to enhance this time mentioned. Bye bye. Hello, Mario on an order.
Muxin Han: But here's the point that things are controlled by by initial stage of passing. So you have the classical
Muxin Han: Semi classical state in the semi classical allowed regime, and then the evolution, listen to classical and and otherwise is not semi classical so here we find and then semi classical analysis is successful and and we got some a cosplay consistent result.
Muxin Han: Okay, so now we can also compute the politics of the transition amplitude as we didn't inform
Muxin Han: The similar method is important because we did the calculation is just the station or a phase of automation and in this approximation, the integral is dominated by the semi classical trajectory. And now, suppose
Muxin Han: We assume the initial state and final state. They are they are both semi classical
Muxin Han: And suppose the are connected by a trajectory in bass, bass surgeons by equation motion. OK. And now we have the solution and what we need to do is simply plug in the solution. The classical action.
Muxin Han: And we got this awesome how the formula we have the US and how the approximation of the integration of G.
Muxin Han: And and this age is we just need to compute and principal compute the actual matrix. And we just get the additional expansion for the integration of G. So here, we still have a
Muxin Han: Integration of age, because we still need to integrate over gateway. So because initial final condition they are anyway really there any way to gauge or it
Muxin Han: So here the, the interesting the good poverty of this formula is that the equation emotion is really unique or the semi classical boundary state. Suppose the boundaries. They are semi classical, the solution is unique. You don't have any regularity.
Muxin Han: And and then the result is g into raw behave as syntactically same single accessory.
Muxin Han: So here, you only have a single explanation because the solution is unique and that now you have to each interval. So he into his overall the game transformation of the initial condition.
Muxin Han: And otherwise, I suppose the initial and the final state. They are they are not connected by any trajectory. So he's fine. Question emotion and then these empty to justice repressed and suppressed exponential because you don't have a solution and
Muxin Han: Okay, so, so this is our semi classical result so
Ivan Agullo: Damn little this essentially zero or one, whether these two populations are classically connected
Muxin Han: Is zero know it's a is so suppose they are connected by a solution, then it's also the auxiliary. This is exponential
Muxin Han: Sheila oximetry and otherwise is is is decaying faster than any polynomial UT
Muxin Han: It's a like the exponential decay is it's like exponential
Minus one over to something like that.
Carlito2: Is standard quantum mechanics are connected by solution, you should be the exponent of the Hamilton function of the
Carlito2: Reactor the trajectory, they're connected to
Muxin Han: Yeah, yeah. And this is the Hamptons function and
Abhay Ashtekar: I think that some of the confusion is coming because of missing eyes or some kind of venture about it.
Muxin Han: Oh, OK. I see. Yeah. So here, somehow I didn't really expect I
Muxin Han: Mean from them. So, so, so this is purely imaginary is as experience.
Muxin Han: OK, so now next let's compare with speakerphone formulation. Yeah.
Muxin Han: Okay, so. So here on the left hand side is the simple formulation and right hand side is our, our new passing through. And so here
Muxin Han: We have some similarities and both of them describes the transition amplitude between on web networks days and. And here our population mostly use use coherent state, but it's just a matter of changing representations, you can calculate the
Muxin Han: Network they continue to build
Muxin Han: And and these hour and the speed formulation is defined on the triangulation where our formulation is defined is a disgrace passing the ball hybrid to be goddess and and more. And it's I find it's just a
Muxin Han: It can be reformulated as a hybrid to be involved that. So because this model is closing relates to the easy model derived by by matching philosophy and your retirement ASCII
Muxin Han: Over there. They couple gravity to us the most killer field and the derive a simple model for this and and and it's equivalent to a department riding gravity using scale of view and then transition to. And so our model is closely related to what they are doing.
Muxin Han: More and both of them relates to customer gravity. So Slim form relates to read your calculus and and I have shown in previous lives and our as integral relates to grab the tasks here.
Muxin Han: Okay. And so these are the similarities and they are differences. So here at least some open issues for some form formulation and these are these open problem has been reviewed several weeks ago.
Muxin Han: In the talk, or four simple and so firstly, it has well known cosine problem but but here's a cosine phone
Muxin Han: It is just a problem after nine unique solution even waste Space. Space initial conditions. Yeah, so the cosine problem, it just
Muxin Han: Is the problem that even you specify the boundary condition as
Muxin Han: Three boundaries three metric and also boundaries, we experience the culture. You can specify both metric and experience their culture on the boundary
Muxin Han: The solution of the equation motion is still not unique is not unique. So, so the 99 uniqueness come from different orientations. Right.
Muxin Han: And that's the origin of the cosine form. Well, I'm you sometime you get this cosine problem resolved only at a, at a level or single force. In fact, there was work by Virginia and Claudio, meaning
Muxin Han: And they study in the single person blacks amplitude and single posting like amplitude and imposing boundary condition.
Muxin Han: Goals extremely curvature and metric and over there. Indeed, you don't have cosine. You only have an angel. But in general, suppose you can see their multi versus 90 versus those more physical situations.
Muxin Han: Yeah, you can't avoid cosine. So, so then the initial condition cannot cure.
Muxin Han: Other solution. The solution is not unique. That's, that's origin of of cosine problem okay and and then secondly, you have also
Muxin Han: Classical flatness problem and and we are some evidence showing that in large a limited amplitude is seems dominant by flat space time
Muxin Han: Right and and then or in my opinion it, it actually means that the larger limit is probably not the right limit so
Muxin Han: We need some or modify the larger limits and to look at the semi classical problem for what's being bombed. So, so therefore the for for these problems, the statement is it's not clear. The semi classical and poverty is not complete clear or
Carlito2: Machine. Sorry. Since you're mentioning that. And since many people listening. There's some strong new results coming from not from me from
Carlito2: The people doing the American in my say on that. So I just want to announce it will be
Muxin Han: Now, okay.
Carlito2: So, so those are some clarity coming in soon.
Muxin Han: I see, I see. Okay. I'm looking forward to that. Yes, thanks.
Muxin Han: All right. And so, Sir. Sir, Polonius is relation relation to canonical upon gravity is of course now not clear.
Muxin Han: And there's also a mild problem about divergence and the if you consider model with no Cosmo, the constant, it is divergent and a new model which cost model costume is not divergent
Muxin Han: And then lastly, there's a university degree it's it's not clear for or personal forms and in these covariance approach. It's not so clear about what the sense of integrity.
Muxin Han: And alright so so now in our population and all those problems getting improved.
Muxin Han: So I'll show you that in case that you have a semi classical initial states and in the same in classical out regime, the solutions unique and the complete consistent with classical
Muxin Han: Classical continues theory and you have no cosine and we just get a single central
Muxin Han: Yeah. And so the key point is that if we don't have a cosine is because of this time continuously because the solution of we have better control of the
Muxin Han: Equation emotion. Yeah. And we know that the other solution of equals emotion has time continuously. Yeah. And because of this time continue doing it kill all the other
Muxin Han: Solutions and and you can do this. You might suppose you have some analog of course is but but because of this continuous this
Muxin Han: Those solutions just got killed. So you can imagine that suppose in speed form. You can also take sort of this kind of time continues limit, then you won't have cosine anymore. And because you see those different solution orientation is kind of junk.
Muxin Han: It's kind of junk of the configuration of the jump of the configuration and going from one place to the other. And suppose you have time continuous limit and all the
Muxin Han: Variables are continuously evolve in time, then you don't have these kind of jump. Yeah. And then means that you don't have because I know but but unfortunately these new forms we have, we don't have central control.
Muxin Han: V motion.
Muxin Han: And secondly, really, we don't have flatness problem because we semi classical human just reduced to the
Muxin Han: Gospel equation of of gravity does the system and we can have all interesting corrupted space times and and we can study so. So last year we have studied cosmology and and later on I will talk about cosmological perturbations.
Muxin Han: Yeah, and and the relation with canonical loop on gravity. Yeah, this is the point here that it is derived from canonical gravity. So it has clear relation with economic theory and
Muxin Han: Moreover, we have no divergence and because it's just the derived from a finite quantity and it is a tradition amplitude. It is finite and and this island is irrelevant to cosmological constant, you can have model with no cost model constantly still finite and of course we also I night.
Muxin Han: Also is also manifestly unitary because it just equals to unitary foundation. So there is a sense of integrity with more
Muxin Han: More awkward. The are open issues and the development of our new formulation. It's still, you know, early stage.
Muxin Han: So, so, but in both formulation of steam form and and and our passing formulation and you have these issues. So firstly you have complete certain computational complexity.
Muxin Han: So, as far as I know, for the numerical calculation and the complexity grows pretty fast equals Rajai and also you know when number of vertices grows and but but Carlos mentioned that there is some new strong results probably
Muxin Han: They are they are a lot of improvement with all this perspective and for our formulations. The computational complexity comes from the non polynomial property of Hamiltonian operator.
Muxin Han: But when we take the time continue steam it and when these matrix element reduced to expectation value so
Muxin Han: The perturbation cost prohibitive calculation are are lots. We can use a semi classical percussion theory compute is although this is something we are going to do in the future and and the also the last poverty is is
Muxin Han: Is last, the issue is also important. It's a both of them are triangulation dependent. Both of them are lightest dependent and inform depend on triangulation, and our model depends on
Muxin Han: To be Gladys. And so we need a better understanding both of the models and to understand how we refine the lattice at the quantum level. So I would have thought it was the continuum living in the classical level at the classroom level. So here, what more
Muxin Han: Difficulty with more challenges is the company, maybe at the quantum level.
Ivan Agullo: To machine just to make sure I, I understand.
Ivan Agullo: Okay, so the unitary the property of dimension in the previous slide rests on the absolute value that you introduce in the initial Hamiltonian, isn't it.
Muxin Han: Um, yes, yes.
Ivan Agullo: So it's a consequence of that, if you don't lose that absolute value. Yeah.
Muxin Han: I don't have a union operator and myself, June 20
Muxin Han: Yeah, that's a good point. Thanks.
Muxin Han: Okay, so now let's apply all these series to to cosmological perturbation theory. Let's Let's derive cosmological perturbation theory from top down and get some feelings.
Muxin Han: Okay, so now let's. This is our record. This is our equation motion from the past into raw from quantum theory and derived from passing the ball from semi classical analysis and the sisters Hamiltonian Hamilton's equation emotion and now we are just going to solve this equation by inserting
Muxin Han: Arms as of cosmological perturbations here. So, so here is our own that's for the hollow me and flux and and or exponent Apollo me and it equals to the cosmology.
Muxin Han: part as a leading water and plus some perturbations. So, similarly for flux and so the leading order the zeros order is just a homogeneous and isotopic cosmology.
Muxin Han: And here's a script X and sweep the why they are perturbations and different lattice has all different different vantage point has all different perturbations.
Muxin Han: And now when we plug in this
Muxin Han: plugin is around us into this solution and we can we get a first
Muxin Han: Order equation motion and also the first order linear it Krishna motion for the perturbations and the zeros older perturbation and it is just our work last year.
Muxin Han: And the result is just the fact the dynamics of cosmology. He knew not skin, new, new zero scheme and the result is a suppose museum museum is not zero, then you get a symmetric bounce.
Muxin Han: And now the first order equation emotion is a linear equation motion now perturbations and the background on on a new not scheme effective cosmological background so it will be just a linear ization motion on these kind of background.
Muxin Han: Alright, so, so now for
Muxin Han: For comedians I introduced some shorthand notation I called V role.
Muxin Han: As a function of vertices. And these are, it's a vector of all the provisions and in total, you have 18 components and because you have nine components or horny probation and nine components for flux perturbations.
Muxin Han: And then the calculation is done in the for your space we can make the standard lattice Porter transform and usually
Muxin Han: It's a standard for transforming that is field theory and the interest is over the first of all, you're in, zoom, and because we have the screen is the lattice. It gives you we cut off with momentum pie divide by new new is a coordinate 90 spacing.
Muxin Han: And now once we plug in the US and get the first order linear equation motion and we get a standard, we can put the equation, the standard form.
Muxin Han: So all those most of those perturbations are coupled and you get a relatively complicated complicated matrix you is 18 by 18
Muxin Han: And this this again contain long form and all this calculation has to be down in Mathematica and the calculation last few days, one or two days to get this equation because the reason is that we use. These are two months Hamiltonian has complicated expression if you expand
Muxin Han: But, but you can download all those. So, so those are those forming all those calls will appear in this GitHub can download, so here we also make a assumption but but this assumption with without losing generality this k is had only the zero, the x component
Muxin Han: Okay, and to make the formula simple
Muxin Han: And moreover, we also have a linear is a closure condition. So this is just a kosher condition linear it perturbations and zeros order, you don't have sales order because zero order and satisfying closure automatic and we are going to just dissolve those equation you mark.
Ivan Agullo: What do you mean by closure conditions. Sorry.
Muxin Han: Oh closure condition is the dispute is God's constraint.
Muxin Han: So this is a closure condition of the queue.
Ivan Agullo: So you have to have it.
Muxin Han: All the vertex is six violent and you have to be closure condition.
Ivan Agullo: And this is the only constraint that you have for perturbations didn't have any other word
Muxin Han: That only constraint. Yes, that's the only constraint. Well, I mean, some constraint that dumb Hamiltonian constraint now is transformed into consideration.
Muxin Han: Transforming the conservation. So the exam. The Hamiltonian constraint. Now in the in the paradise remark, it becomes the conservation of the physical i'm tony
Abhay Ashtekar: But I think just to clarify, if you're a lot. Apply the previous general framework to the to this particular setting of cosmology, then you still have the brown cost model so that it is still at some deep parameters in some
Abhay Ashtekar: Sense. And so this is not what normally people will call us political participation. Right.
Abhay Ashtekar: Because the normal course Walter perturbation theory, people would say that there are constraints genuine constraints on the cosmological perturbations themselves and for the on the background metric, which, for you have disappeared because you are just reduces the system.
Abhay Ashtekar: So from that perspective, this is not what people normally would call the cosmological perturbation t this some extended version of cosmology and which one is embedded in dust variables and eliminating the constraints. And so, and got to reduce space.
Muxin Han: That's right, yes. So, so this is at this moment. This is not a standard formulation. It is the department tries formulation using Bronco cash last but it's closely related related. I'm going to talk about in the next briefly mentioned the next slide. So this is a walk down by Christina.
Abhay Ashtekar: But this is the thing that I was just, this is why he was asking this question about other other constraints. And I think the reason why there aren't is just because of the the parameter ization
Muxin Han: That's right. Exactly.
Muxin Han: Okay, so here we are going to solve those to actually solve those equation we need numeric and I talked about in a moment.
Muxin Han: But, but before the result. Let me first look at the continuum limit to see the consistency of those equations with the some with some existence with some existing results.
Muxin Han: And so now we have 18 linear equation motion for cosmological perturbations. So now we are just we can take continually me but take New Jersey zero and then it's reduced simplifies these big matrix you
Muxin Han: And also we can take in enclosure condition we take new goes to zero, it just reduced to the linear, it goes
Muxin Han: OK, and now we can build a three metric perturbations, we can construct the metric perturbations using using productive flux variables.
Muxin Han: And we get a metric perturbation Tata and this does he can see that it's just a linear combinations of those perturbations be and these are all flux perturbations and we can also make the standard as VTT Composition, Writing those perturbations into deeper
Muxin Han: To get a tensor mode skater mode and better most, by the way. So, this equation to be in the form of space between these are I case.
Muxin Han: On to i times k
Muxin Han: So once we plug in these decomposition into the continuum limit of our provision provision series.
Muxin Han: In production equations.
Muxin Han: And what we got. We can decompose equation in in three different kinds of most scholar mode Record mode and tangible and we get the equations.
Muxin Han: With those equations, we got the complete reproduce the classical gauging variant cause model cosmological provision theory developed by Christina.
Muxin Han: 7129 and all your link or service or what their what the study is for Brown cooker dust and the the study the classical gauging variant cosmological perturbations theory.
Muxin Han: Yeah, and we get these kind of equations over there. And so here you can see that when sort of more precisely here when alpha equals one. So here alpha equal to zero, it would correspond to
Muxin Han: Couch and asked for the alpha equal to one and those equations and company will produce
Muxin Han: The equations and and India paper. They also have some. We also have a discussion that and those equations are closing relates to the Standard Cosmological perturbations you
Muxin Han: Know, and here also interesting remark is that for tensor mode, you see those exclusively the dispenser mode reduced to the familiar
Muxin Han: Equations motion for for gravitational wave and an Indian from this equation, we just reproduce gravitas skin to particle emergent
Muxin Han: Right, so he all these televisions on full of gravity some cash flow analysis and we got this equation and we got gravity. Right. And so you
Muxin Han: can grab the we have gravity gravity. Tell us into particle excitation, by the way they are. They are and several there are a bunch of group of condensed matter physicists that they can really be
Muxin Han: Focused on revenue. You can reproduce that's been two articles and those people they they they they try to emergent single particles from biblical orders and and so on. And they don't really believe one gravity has into particles, but here you see that we indeed have
Muxin Han: We have seen two particles well to Congress.
Ivan Agullo: But this is the linear new going to shrink.
Muxin Han: Yeah, this is a continuing this is in the country.
Ivan Agullo: Right, you
Muxin Han: Know in the district level, you got some corrections of product dispersion relation, this is in the next life.
Ivan Agullo: Okay, thank you.
Muxin Han: Alright, so there's a small comment about Warsaw's Hamiltonian usually use by was asked group and with all these TV was down using these are two months I'm Tonya
Muxin Han: But we also try to use was Hamiltonian. But unfortunately, at least from our calculation is that the that Hamiltonian doesn't really give the right answer.
Here.
Muxin Han: At least in the continually. So that's just a comment.
Muxin Han: All right.
Ivan Agullo: Right machine, but I can you come back to the previous slide, because I am a bit confused with the logics because, you know, you start with a modified Hamiltonian modified by a distribution parameter is by new
Ivan Agullo: You go to the passenger and then take the semi classical limit and you obtain as one would expect that the equation suggest the classical equations obtained from your modified Hamiltonian
Ivan Agullo: And now, if you take the leaving New go into zero to recover GR. So, one should shouldn't be too surprised that you recover the types of perturbations from GR because you are just copy back to Jerry. Is that accurate.
Muxin Han: Yeah, of course, this is
Muxin Han: Well, this wasn't so trivial and because you see for spin bombs and for the previous passing to formulation know we haven't achieved that before.
Muxin Han: So it wasn't so trivial but but indeed you're right in the end you recover. We from the calculation we recovered. Dr. And then, yes, the cosmological perturbations theory in this limit also recovered, er,
Muxin Han: But, but the non trivial, of course, the non trivial consequence is the how these discrete news new relates to correct those equations right and it's a non trivial parties correction of those equations right that's
Muxin Han: Correct. You're interested
Ivan Agullo: And here.
Abhay Ashtekar: So that
Abhay Ashtekar: Might be that, you know, younger this put specific Hamiltonian gap preserving Hamiltonian constraint and it could have been wrong. Right. I mean, there's no guarantee that you're going to give you the correct
Abhay Ashtekar: classical limit, of course, as one is saying is that if we know that you give the correct classical limited full Jia that of course you should get the correct classical emitting in cosmology and data, I agree that is a consequence. No.
Muxin Han: Yes, yes, this is a consequence. But then of course the interesting part is the corrections. So this is something here.
Javier: I before. Before you go, I have a question. So
Javier: I agree that these resellers non trivial and maybe the example is that the you use the Hamiltonian by the Alice ski ski and so on. You mentioned that he seems that he doesn't good continue limit right
Muxin Han: Right as far as our calculation. It doesn't seem to
Javier: Me.
Javier: Can you explain a little what or why
Javier: Did
Javier: A good
Muxin Han: I don't have a very precise answer, but my feeling is that this is this is a feeling, maybe the, the, the problem is is that is appalling. So all the
Muxin Han: deregulate regularization is and they take the usual up a part of Hong Kong. And the last part of Hamiltonian, they, they use us a three quarter three converter skater and then the three commercial skater and regularized using using Reggie calculus.
Muxin Han: But my feeling is that if you use regular calculus. As for the second term, but but the continuum limit of the radical goes and the continuum limits of the Ukrainian Hamiltonian is is not precisely the same because ready calculus is is a distributional regularization so
Muxin Han: Continue millionaire up the regular calculus, it is much more subtle than, than the Ukrainian part of the Hamiltonian so so
Muxin Han: I wonder, maybe the reason is that because these two terms has two different ways of cutting limit, one has to be extremely careful about
Muxin Han: About how to take the company me, at least for for the standard way, just like the spacing goes to zero is my reason, no problem. But I, I don't have a precise answer for for for for this, but at least for for the company. We use it's completely fine. Okay, thank you.
Muxin Han: Alright, so now let's come to some some consequences.
Muxin Han: And compare the discrete equation motion and continuous equation motion. So firstly we this is a plot of the scanner more power spectrum.
Muxin Han: So it's, by the way. So this is you my opinion. So this is currently just a toy model because I'm here, we only can see that the only major contribution is the dust the bronco cash does
Muxin Han: And and we don't consider radiation matter. And also, we don't really, we didn't really take into account the inflation. We don't. We didn't couple influence and this is
Muxin Han: A few gravity coupled gas and and we extract the skater mode and this cloud. His word his body in potential by and this project is done is a for for cosmetic constraint equal to 10 to the minus five and the lighting spacing new equals to 10s of mine is too. And this is for Bronco casual
Muxin Han: And we set up certain initial conditions for budding potential and other quantities and and we let let it evolve and those dashed lines are the continuum Siri power spectrum or a skater mode and those solid curve. They are discrete
Muxin Han: Equation emotion coming from Luke on grabbing. So you can see that it is similarly the similar quality behavior as
Muxin Han: As the dress the metric approach and that when when for for large momentum and these two result coincide, but for more momentum. They got pretty different. So you got difference that's more momentum. This is
Ivan Agullo: But here you have your considering a balance with inflation after the bounce etc or
Muxin Han: Or what. No, no, there's no inflation. No, we didn't really consider inflation and the, the initial condition is is imposed after the bath at t equals to what those bounds is like t equals zero. So the initial condition is imposed at equals one.
Muxin Han: So this is my, this is the poverty of the body and potential
Abhay Ashtekar: So it's more like looking at what was done in local cosmology for the background, you start at a later time, and you will towards the singularity service and the statement is that you get
Muxin Han: So we didn't really evolved towards singularity. We still evolve to the future. So t is t equals zero. It's a singularity and he goes to one after the singularity. And then we evolved to the to the future.
Abhay Ashtekar: Okay, so therefore, at any given time. So I'm just going to confuse our body so
Abhay Ashtekar: You want to
So,
Abhay Ashtekar: For for a fixed time. Like, for example, equal to two.
Abhay Ashtekar: You still find that there is a departure.
Abhay Ashtekar: From classical behavior.
Muxin Han: Right for for small moment that the reason for that.
Muxin Han: Formula is not here. I think we figure out the reason for that, for that is because the definition for the bad in potential
Abhay Ashtekar: OK. OK, I will talk later because it's really late now. So we should do, what did you finish.
Muxin Han: You mustn't and it's because the bottom potential is
Muxin Han: The bottom financial is anti proportional to decay is is one is partial one overcame. Yeah. So then, then when the cables to small those discrepancies get amplified.
Muxin Han: Is good the effect of the screen is done and amplify. Yeah, but we can discuss later so. So here, this is a scanner more power spectrum. And when we come to a tensor mode. So we be computed easily the dispersion reaction.
Muxin Han: Or for this great situation and Adelaide time anytime the extremes ecology goes to zero and this is formula in case that cosmic cosmic equal to zero as well and and we get
Muxin Han: This dispersion relation and this dispersion relation is the same as the one opt in by earlier by Andrea DePaul and and calls
Muxin Han: And over there. The, the, the study questions gravitational perturbations on the flat space time and they get the same
Muxin Han: Dispersion relation and which the leading order to speed up the gravitational wave is just the speed of light, and the second order, get some corrections.
Muxin Han: Or new new square
Muxin Han: Feet is a little bit smaller. You know, like
Ivan Agullo: I just find it confusing that that you you find a corrections for tensor most for high k because these extra term in the square bracket is important for
Ivan Agullo: Dominance. Okay, but
Ivan Agullo: You know better relations do you find corrections for small k find that. Right, right.
Muxin Han: So these are so as I said, this is a relates to the definition of gravitational potential. So if you just look at the
Muxin Han: time evolution of them are Holloman flux and you don't really see you only got height, a
Muxin Han: Corrections. Yeah, there's a reason for for these kind of departure is because of this button potentially to contract that it is proportional to one. Okay.
Muxin Han: It's been a while. Okay, okay, goes to smoke small, then the description is because they are in the numerator is proportional to me. Yeah. And when k equals to zero, effectively making this new law.
Muxin Han: So that's the reason for this department.
Muxin Han: Okay, and also particular mode. This is also there are some other difference is that it relates to backer mode. So to distance or multiple this great case, it interferes with vector more
Muxin Han: Spectrum of view. So it's interfered by the tensor mode at this grade level or this interference is appears at the continued so
Muxin Han: Now this is a difference between pitching our result. And the result and the report and over there. Yeah, so it's all working is coming from because our work is coming from the
Muxin Han: Fools theory, taking into account and all different kinds of perturbations. So we find the interference between tensor mode and make them all.
Abhay Ashtekar: But if you don't have this test, but we just don't scale a field like inflation. Then there are no victims. And so there are no
Muxin Han: Yes. Maybe I I don't really so we didn't really do calculation for skater more probably
Muxin Han: May be right but but for skater was a couple to scatter feels so over there, you'll still have the demo villain constraints. So you still have another equation for the constraint. So probably got it got better, more back
Muxin Han: Button. So, probably we need to look into that.
Muxin Han: Okay. And, okay, so. So let's come to the last little bit about relation with numerical relativity. So, so
Muxin Han: Okay, so here, here it is logic. So all the previous calculation is that we have these equations motion for equation motion.
Muxin Han: On the gravity form passed into law and will have what we have done before is that first
Muxin Han: We simplify this equals emotion by by plugging some owns us and the sounds as relates to certain symmetry or linear ization
Muxin Han: Of the variable respect to some background and somehow and get these equations simplified and then we solve the simplified equation. This is a logic.
Muxin Han: In a previous life right and however there's a different approach. So we could just to solve the equation. Right. And we have the numerical tool to solve the
Muxin Han: Equation right and put the entire equation and numerical package and but the symmetry of the solution is imposed by initial conditions. Now we just the truth, all different kinds of initial conditions and to evolve numerically.
Muxin Han: The entire equation and this is more in the logic of numerical relativity and and, moreover, and this equation and the equation motion of the coupon, where he is indeed of the tag.
Muxin Han: As it has been studied in numerical activity because left hand side is sometimes the relative and on the right hand side is some some function on Facebook.
Muxin Han: Yeah, so therefore a given we could. What we can do what we can do is that we put the entire equation.
Muxin Han: On the computer and then giving various initial conditions and then the numerical method should create all different kinds of space times from semi classical look on gravity. So not only
Muxin Han: cosmology and Cosmos division theory inputs for all the
Muxin Han: Different kinds of interesting space times will be created from this equation, and indeed so so my collaborator one Leo is
Muxin Han: He's a person who many, many make these progress and he has made a c++ package.
Muxin Han: For numerically evolving the equation motion and we have running various tests for for this code or gamut of the simplest, the test is just a revisit
Muxin Han: The homogeneous as a topic cosmology, we post the initial condition which is as a topic and homogeneous in space, but we plug in the time evolution of the equation and you see here is the plot.
Muxin Han: The blue curve is the
Muxin Han: Is the expected result which come from the first approach.
Muxin Han: Simplify equation and solve simplified equation and then those purple dots are coming from the numerical relativity come from numerical
Muxin Han: Come from the full equation new record involving the full equation and and only imposing
Muxin Han: So probably homogeneous initial condition and here the red dot are so here is basically putting to plot in in the same plot.
Muxin Han: So those red dots are arrows. But when you look at those arrows, you have to use the axis right
Muxin Han: And you can see that those arrows are all bonded to the 10 to the minus six. So Dr. Numerical areas which are well controlled
Muxin Han: And on the right hand side is a numerical computation of physical Hamiltonian right and and consider this physical Hamiltonian. We can we can numerically relatively precise precisely to compute this.
Muxin Han: Physical Hamiltonian to error is 10 to the minus lab.
Muxin Han: Some consistency test for the mark.
Muxin Han: Okay, so, so, at last, let me conclude. So here we have present a new path integral formulation of loop on whether the transition amplitude and
Muxin Han: So this is the past in zero. This is the face face.
Muxin Han: integrating over an intermediate state and you have seen that
Muxin Han: Semi classical limit of these past integral reproduces the classical gravity does to Siri on the continuum.
Muxin Han: And we have compare the discrimination with inform formulation we find this new formulation has advantages, including the finiteness integrity and relations can call upon gravity.
Muxin Han: And episodes of cosine platinum sponsor or we can derive from the the spoon theory. The cosmological and derived cosmological
Muxin Han: Cosmological perturbation theory from the EU complexity theory we got scanner more than more than sensor mode and we can find the power spectrum which are used for a phenomenal logical. Those are useful to compare with observation in the future poverty.
Muxin Han: And also for tensor mode, we see these elite that rabbit town spin to cetaceans ball Looper loop on revenue and so everything is somewhat costly consistent. So on revenue is a theory has the correct semi classical
Muxin Han: Moreover, the semi classical dynamics of Hulu com where you can realize to numerical relativity
Muxin Han: Alright, so now let's make outlook. So firstly, the next thing we should do is of course do more computations for the gravity transition amplitude, especially the computation.
Muxin Han: quantum level at the quantum level because of the company's vision here is semi classical
Muxin Han: And then there's the thing we need to do first is to compute the matrix elements or the expectation value of the Hamiltonian and at least for the expectation value of hunting me should be able to compute at least productive.
Muxin Han: And then we should be able, some more computations. We may help us to understand the behavior of the lightest refinements at the quantum
Muxin Han: More work for cosmological perturbations theory, we have to in the future, coupled with info towns and relates to phenomenology. So here in our form and another advantage of this formulation is that we can we can cover all kinds of matter.
In this
Muxin Han: In the framework and because in canonical theory, the matter company is it's relatively easy. So we can couple all kinds of matters is it's just, it's really 341 of my student is actually working on this direction and
Muxin Han: We should be able to couple inflict on to the theory and do the cosmological perturbation theory to, you know, in a more physical situation.
Muxin Han: And also we new record activity we we we should be able to compute create more semi classical space times using the Mac nicer and we can, we should study black holes and other more generic space times
Muxin Han: And also, in the end, we should, in the future, we next thing we should do is that you should study the cosmological perturbations theory from new Baskin on web the Arctic Siri to there. There was a work.
Muxin Han: Last year that we can generalize.
Muxin Han: The new bar scheme in cosmology to the food theory. So, we can write down a new bar scheme.
Muxin Han: For Hamiltonian of gravity and and enrich the equation motion renewals back to the new Baskin loop on cosmology.
Muxin Han: From the equation emotion when when we insert the founders of homogeneous isotopic
Muxin Han: Solutions.
Muxin Han: So, so, I mean, that's the next thing we should do is doing cosmological perturbation theory you asking that remark. Okay, I think that's all I want to see. I want to say.
Jorge Pullin: Any questions left
Harold Haggard: Quick comment, it's, it's very much tangential to your talk, but relates to what Carla was mentioning
Harold Haggard: BIANCA Dietrich's at the Santa. And I also have a paper coming out this week on the flatness problem.
Harold Haggard: And it goes in the in the same direction that you were mentioning, where, where we think about large J limit but also a small barbero emergency parameter limit. So it very much goes in that direction that you were mentioning, and does so in a very simple way.
Muxin Han: Okay, yeah, I'm looking forward to that. Yeah.
Abhay Ashtekar: Thanks. Yeah, I just got a couple of quick comments. I mean, the first was that
Abhay Ashtekar: God will raise this issue about whether in this if you have a scale of field. And if I looked at as you rightly said, if you look, look at the scale of field equal constant for creation. Is it the case that
Abhay Ashtekar: That the
Abhay Ashtekar: The Hamiltonian
Abhay Ashtekar: The scanner feel Hamiltonian is preserving time or is not preserving time and this really the first. This is a direct impact to our
Abhay Ashtekar: Mission was saying.
Abhay Ashtekar: In is the discussion about why there is no transition. So to say from the part where the geometrical part of the scale of constraint is positively to the negative and so
Abhay Ashtekar: So the statement that I'm I would like to make is that if you're taking count. You ready for taking into account the final constant
Abhay Ashtekar: For radiation, then the Hamiltonian of the scale of view is not constantly time and therefore the argument that machine made will have to be generalized be modified.
Abhay Ashtekar: And the second comment I would make was so
Abhay Ashtekar: We should do want to say something. Oh, yeah.
Muxin Han: Yeah, so, so no i i remember that there is actually a I think it's still conserved and and I can remember there's a formula in in
Muxin Han: In the paper by Turek and Christina and other collaborators.
Muxin Han: But the title of paper is gravity on highs over there. The, the study dependent tries gravity boots skater is a single scale appealed and over there.
Muxin Han: They indeed I sink. And I remember that there is a formula, showing that the physical hands on the integration of the physical Hamiltonian are mutually community mutually community.
Muxin Han: So,
Abhay Ashtekar: When I just look at the equation emotion I
Abhay Ashtekar: Will talk about
Muxin Han: Yeah, so I want to use the formula.
Abhay Ashtekar: Right. But I can also tell you what, if you just take the evaluate the Hamiltonian without going to handle and without good framework, just take the Newtonian take the equation emotion and evolve evolve it in time and then see what happens. And I'm saying that it will not
Abhay Ashtekar: be concerned if the total volume is changing in time.
Abhay Ashtekar: Okay. So was this something that we should sort out first, because that is important for this.
Abhay Ashtekar: Transition
Jerzy Lewandowski: But indeed, moving is right that this is this phenomenon is that if we integrate this Hamiltonian which we obtain when we modify it constraints, then then actually does commute.
Abhay Ashtekar: That the matter Hamiltonian and the
Abhay Ashtekar: Concept.
Jerzy Lewandowski: Where this is effective Hamiltonian which we, which is used as we
Abhay Ashtekar: Thought so. It's not the matter Hamiltonian. I was talking about the math. Okay, let's do this. Most people are not interested in this. So let's, let's not kid ourselves.
Abhay Ashtekar: The. The second thing that I just wanted to mention was that
Abhay Ashtekar: That the
Abhay Ashtekar: Inner in the last but one transparency, you had
Abhay Ashtekar: Issues about
Abhay Ashtekar: The
Abhay Ashtekar: Comparison with the standard spin form right and the cosine problem and so on so forth. Yeah.
Abhay Ashtekar: Yeah, right. So the cosine problem and so on so forth. And I think that what you're on the right hand side is exactly what happened in the time model that
Abhay Ashtekar: Anderson completely and I are considered where we consider just started with the Hamiltonian framework of lukewarm cosmology.
Abhay Ashtekar: And just
Abhay Ashtekar: Wrote down the corresponding
Abhay Ashtekar: Spin falls flat. That's right. And then for for the corresponding spin falls for that really had all the features that you are talking about here that, first of all, there was no cosine problem that everything was
Abhay Ashtekar: Was finite. There are no diversion whatsoever and everything was unitary and this is very nice. But what you're saying is that it is, it can be generalized completely to the full lukewarm gravity context provided one was for the fixed lackeys
Muxin Han: Yes.
Abhay Ashtekar: But I still feel that working with a fixed flat is is not is not really the right thing to do because I think physically
Abhay Ashtekar: As you as the university walls and more and more. What is it should be created and the kind of mixtapes that were considered, for example by LSC and so on so forth in which
Abhay Ashtekar: One of the probability distribution of the number of vertices and and then you know the number of bodies is the change as time time evolves in in the in the quantum theory.
Abhay Ashtekar: That I think is something that you might want to consider seriously, then you're also looking at the new basking because that thing naturally does lead you to Baskin, so I just wanted to pick a destination.
Francesca Vidotto: And yes, I want to comment on these vehicles.
Francesca Vidotto: There's been a prejudice in quantum cosmology against the use of the interchanging graph, I'm etonian exactly because of the difficulties of implementing the member scheme.
Francesca Vidotto: So I think here there is a so I am very much in favor of using a notion geographically fun in cosmology, in particular, because there is a very straightforward interpretation here is that the physics that is more clear. I think
Francesca Vidotto: So the point is not to think of the sides of the lattice as representing a really quantum space time and but rather as
Francesca Vidotto: It truncation of the theory. So in cosmology, we wanted to we describe it location of food general activity in the same way we adopted the
Francesca Vidotto: graph that we use to this particular approximation. So what I think is very nice here is not using the changing of me, Tanya, is the fact that that. So the graph represents to education.
Francesca Vidotto: And by evolving our I miss Tanya and we keep the same approximation. So in particular, so if you start with a homogeneous needs to tropic Hamiltonian geometry.
Francesca Vidotto: You evolve into a homogeneous among them into in and these are trapped in German, and I think this is very nice. Of course, it would be good to see the full implementation of the new bar scheme.
Francesca Vidotto: There. And this was a comment. I know you have a question for machine. And because I would like to see
Francesca Vidotto: I was thinking, okay, it's easy to add matter and to other an influx of potential, but can we just do this work with this model in a kind of a matter bounces. And are you using the cash as as a dominant matter for the bounce.
Muxin Han: A little So firstly, it is just a straightforward to to to to add matter. So that's why I gave it to one of my PhD, so
Muxin Han: So this is really straightforward to add met and do calculations.
Muxin Han: So, so the bounds of his own it. So here, indeed it has bounced and
Muxin Han: So, probably because here we don't really have the, the other kinds of matter. So this is the bounce, which is
Muxin Han: In the, in the past, the dominant regime. Is that what you asked me.
Francesca Vidotto: Well, the thing is that if you have a matter bouncer the gas to the matter you have is dominating the contracting face and therefore when you studied the
Francesca Vidotto: Corresponding the emerging power spectrum where you would find the almost create environment, the power spectrum for free. So that could be nice and could be
Francesca Vidotto: Something to
Francesca Vidotto: The by itself as interesting by itself.
Muxin Han: So is that what you mean to here for for the for the continuum theory for the continuum.
Muxin Han: Cosmological perturbations theory. Indeed, the power spectrum is flat. So it has no cave and those dashed lines we are continuing series. Yeah. So I'm here. So, this is this great theory with finite new and we got yeah yeahs for large K is the same but but for small k you get some departure.
Muxin Han: Is that what you mean.
Francesca Vidotto: Well, yeah, of course, the devil monster scanning variance concerned at the most. We we do which we have access at the moment. So yeah, so it's open for what happens for the very energetic modes. Yeah. Okay. I want to think about this.
Okay.
Muxin Han: Yeah, thanks. So I about an advice comment about the these lattice. So, so to me. And so, of course, this question is always
Muxin Han: Two different questions. Conceptually, or practically conceptually yes and i by Stephanie. Correct. And so they are issues that always depend on lattices. Always good to to move from that is dependent small to some that is independent model. That would be nice.
Muxin Han: Yeah, and this is something we need to look into that, in the future, but on the other hand,
Muxin Han: Practically, so to my opinion is that I'm to actually do calculation. This is much more convenient.
Muxin Han: To the conclusion because this is just something like lattice case here, and this is
Muxin Han: Just the same structure us and that is chaos theory and. And the only difference is that, well, of course.
Muxin Han: You got different action and but here you get, the more complicated action, then that escapes the little worth. And so therefore, and lots of calculation in that use case theory should be able to apply to you.
Muxin Han: For example, the first thing one we have to try is the largest perturbations here at the quantum level, the largest perturbations here.
Muxin Han: So, so, I mean, from the practical computation know perspective. Probably this lattice.
Muxin Han: Siri is more convenient to compute and and then take the understanding continually or refinement.
Muxin Han: Of the opportunities.
Muxin Han: Yeah, so there are two different approaches. That's, that's what I you
Abhay Ashtekar: Know, I completely agree that, you know, once you get a lot of infusion. The technical structure of the theory using the simplest possibility, which is just fixed like it's I completely agree with that, but I feel that one is really missing out physics in
Abhay Ashtekar: In terms of
Abhay Ashtekar: Work to the fixed slide is not just because
Abhay Ashtekar: The lattice, not just because of the
Abhay Ashtekar: Kind of the
Abhay Ashtekar: The crop scale is fixed ISIS was as as Francesca of saying that is, OK, but what is not okay, I think, is that
Abhay Ashtekar: That is really that
Abhay Ashtekar: The description of the quantity of description of the dynamical nature of geometry is not properly.
Abhay Ashtekar: Incorporated unless we allow addition creation and subtraction removal of focuses I think that if you have a fixed number of vertices and fixed number of ages.
Abhay Ashtekar: That just will not give you a good column description of the of the levels.
Abhay Ashtekar: Even that late stages of evolution. So there is a conceptually shares machine or something. There's a practical issue and but you definitely do this fixed lot is to get as much information as possible.
Abhay Ashtekar: In the practical issue of activity computations and see technically what is true. What is not
Muxin Han: Right. Right. Yes.
Jorge Pullin: And nothing else.
wolfgang wieland: And ask a question machine.
So,
wolfgang wieland: So at some point or when you introduce deeper read recession, you were using the bond cooker cooker formalism and
wolfgang wieland: Well, you had there was read time coordinates. Think you call them capital key space coordinates, a bit less, and later on when we were going to the when you were going to the quantum theory.
wolfgang wieland: I understood that you had a notion of time evolution or so it's time, Hamilton. A true Hamilton operator that creates tension translations in key. But what I was wondering about this. But what about as. So have you also studied I
wolfgang wieland: Have a few more physicians
Muxin Han: And so, so the human translations generating sigma
Muxin Han: Exactly example so support Bronco cash dust and it is generated by
Muxin Han: By these different over them. This quantity is not not not not precisely. It's not see a but
Muxin Han: But
Muxin Han: But CJ, there is a linear transformation of that. So this is not precisely
Muxin Han: This is not precisely conserved, but if you contract with this guy with a tetra and and this will be a concert quantity in this
Muxin Han: In this Bronco cash that model and and these generate the space translation. See this this so to hear this small age and these see it gives you a symmetry algebra and these age give you can generate a time translation and these guys generate the space translation.
wolfgang wieland: So just a small follow up, if possible. So the amplitude Steve will finally depend both on T and S or sigma
Muxin Han: The amplitude. No, no. The, the amplitude is just to depend on t is because the spatial what the it is a state and the initial aptitude, which is Hamiltonian operators time translation and the with initial and final state. So the final MP God depend on state.
Muxin Han: But and and the time translation.
Okay, thanks.
Jorge Pullin: Any other questions.