Jorge Pullin: Okay, So our speaker today is A. By, we'll speak about probing space like singularities with quantum fields.
Abhay Vasant Ashtekar: But as was intended, this is a topic which is kind of closely related to
Abhay Vasant Ashtekar: look on gravity, but not the quantum privilege of proper.
Abhay Vasant Ashtekar: So these will work
Abhay Vasant Ashtekar: that is done as continued collaborations over the last couple of years with the Lorenzo uh Tomaso de Lorenzo, Adrian del Rio and Mark Schneider,
Abhay Vasant Ashtekar: and there are several papers which are already uh up here. Uh, there's one in preparation on Schwarzschild and some mathematical structures were taken from earlier work with, uh Alex Kurichi
on the
Abhay Vasant Ashtekar: Okay, So let me begin with a preamble,
Abhay Vasant Ashtekar: so space, like singularities, are taken to be back here, beginning or the end. Absolute end of space, time in general relativity, and geodesics of test particles basically in there, and the title of forces between them become infinite.
Abhay Vasant Ashtekar: But what happens if we you start using classical test particles if we use quantum probes.
Abhay Vasant Ashtekar: Now it has been long argued that singularities may be tamer for physically more realistic probes.
Abhay Vasant Ashtekar: For example, uh, you know already in one thousand nine hundred and ninety-five uh Gary Cotter is on down model. I considered a certain static space times which are motivated by some string theory considerations which have time like singularity. So there's static and time like singularities, and they consider dynamics of test particles.
Abhay Vasant Ashtekar: And there's you know the the space is well defined because it is static space time, and this they showed that the dynamics of this case particle is well defined in spite of that kind of Make it time like to you
Abhay Vasant Ashtekar: across certain time, like singularities in particular. In the second paper they just look at the rise to nosdom and look at this unity of pricing now is done, which is uh
Abhay Vasant Ashtekar: just time. That and then, uh, more recently, uh uh Hoffmann's an item they consider shorts in space like singularity, and they pro with test on the field, they concluded that these are the same.
Abhay Vasant Ashtekar: However, the arguments in this this work are rather formal uh, because they use storage representation, don't. Specify regularly what the measure is, and infinite number of degrees of freedom, therefore, did not receive enough care, and that is why uh, So I take this uh to this uh qualification.
Abhay Vasant Ashtekar: So the goal of this talk and goal of this in the papers was to revisit the issue of for the physically most important dynamical singularities with procedure, and required to handle the infinite number of degrees of freedom. Of course, of field theory. Carefully
Abhay Vasant Ashtekar: now, because of I. I want to convey the main ideas and not all the details, and therefore what i'll do is to focus on the Big bang of the curve. Singularity. The short and singularity is a work which is in progress. I mean,
Abhay Vasant Ashtekar: we understand most of it, but not completely. Um and uh and i'm going to consider, therefore, in this talk the the freedom and the method of I was to walk up on,
Abhay Vasant Ashtekar: and the main question is the following
Abhay Vasant Ashtekar: that we have in these models the big bang of the big crunchingularity. But supposing I have a quantum field, which is uh
Abhay Vasant Ashtekar: uh! Which is propagating on the spacetime. Then does the propagation come to an abrupt halt at the singularity, and all the observables constructed from them, like, for example, the what it's called as two point function normally, and then the renormalized operator product
Abhay Vasant Ashtekar: uh and renormalized on the intensive. Do they actually remain singular in the sense of quantum field theory at the Big Bang. Uh, or they are the regular there. So that is a question that we would like to understand here.
Abhay Vasant Ashtekar: Then, as you all know, the quantum field can be expanded out in the I'm looking especially flat case. Therefore I can just go to the Spatial Fourier modes
Abhay Vasant Ashtekar: linear differential. I mean the linear ordinary differential equation. And the problem is that these motor functions, of course, they diverge at the big back.
Abhay Vasant Ashtekar: But you call that already in casket space for a hat
Abhay Vasant Ashtekar: is not an operator, but is an operator value distribution.
Abhay Vasant Ashtekar: Now, there are two kinds of distributions, ordinary distributions and temperatures, and that has to do with
Abhay Vasant Ashtekar: You know they, what these distributions, what you do is to take the distribution object, and then you submit it out with test function
Abhay Vasant Ashtekar: and the nature of the test function tells you if it is an ordinary distribution or um as a as a tempo distribution, and the automated distributions are the ones in which the test functions are taken to be smooth functions of compact support,
Abhay Vasant Ashtekar: and the shorts and the temper distributions are taken to be a space
Abhay Vasant Ashtekar: which is a larger space, and this is the space of It's called the short space. And this is the test functions. They don't have to a compact support. But there's some that the the value should decay at infinity faster than any polynomial, and in this
Abhay Vasant Ashtekar: I mean normally in spacecraft got a field theory. When you use a stamped distribution in short, short space for distribution functions. Because so I, because the short space is stable under this to be under Fourier transform.
Abhay Vasant Ashtekar: That means that if I have a test function of X,
Abhay Vasant Ashtekar: which is, uh uh in the short space, then in fact, it's Fourier transform is also in the short space, and that is very convenient. That is not true for c. Zero infinity functions so seen free from some contact support that is not true. So very often one talks about temperature distributions, and since uh every shots,
Abhay Vasant Ashtekar: this function
Abhay Vasant Ashtekar: which is decaying. Um
Abhay Vasant Ashtekar: sorry if if if you I got a test function, which is, c. Zero. Infinity, then in particular, decays at infinity, because it faster than any polynomial, because we just have a contact, support, and therefore a temper distribution is always in particular that audited distribution. So I mean, there's a technicality you don't have to pay much attention to it. Think of it. It's an ordinary situations if you want. I just want you to mention it, because in mean casket space one uses can for distribution.
Abhay Vasant Ashtekar: Now, the main results that I would like to tell you about as the holiday
Abhay Vasant Ashtekar: that in all cosmological freedom and models
Abhay Vasant Ashtekar: as operator rather distributions. So this off this object, this object up here is, in fact, completely buildifying, in spite of the fact that these, as functions blow up because this is to be duplicate as operated by a distribution and not an operator. It has to be with a test function
Abhay Vasant Ashtekar: and summarize the results in more general cases, and in particular, the expectation values uh, uh, these these bind objects, that well defined object by distributions, the distributions in X and X prime, and in in the extended space time very way extended the space time across the big back,
Abhay Vasant Ashtekar: and I will see that interestingly correlations between fields evaluated at spatially and temporarily separated points exhibit in our symmetry that is reminiscent of the Vkl behavior from classical General Qt. This is not related to the main part of the talk, but this might be interesting to some people who are looked at the detail behavior, the renormalized up products like the Phi squared operator and this session you can for up. Okay.
Abhay Vasant Ashtekar: They also remain well-defined distributions, and I want to emphasize that this is not because,
Abhay Vasant Ashtekar: uh, we're using conformal coupling. I mean these models freedom and the myth robbers or worker models I can formally flag, and if you just use, for example, a Maxwell field, or it can form a couple of scalar field, one might think that. Well, the dynamics is the same as a M. In casket space, and everything is the same.
Abhay Vasant Ashtekar: But that's not the case. I mean. The statement is that conformal coupling is not really necessary for these considerations to hold. So everything is well defined, even though we do not have and forward coupling. Thus, when pro with observers associated with corner fields. The Big Bang and big current singularities are quite harmless.
Abhay Vasant Ashtekar: So many of you are probably sitting there and saying all Okay, So there's a big bang, and the current fields are perfectly well-defined. Uh, maybe we can perfectly identify. But why do we care about this?
Abhay Vasant Ashtekar: We know that non perturbative content. Geometry effects will resolve the space like singularity, and therefore this issue, in some sense, will not arise in condon gravity.
Abhay Vasant Ashtekar: This is absolutely right.
Abhay Vasant Ashtekar: But there are two points that we should look at. The first is that common field theory in her Space Times has a wash domain of applicability, and but usually one just says that the this theory fails in the plank.
Abhay Vasant Ashtekar: But how exactly does Lqg. Cure send the classical gravity that comes that arises out of a quarter field in cur space time.
Abhay Vasant Ashtekar: This is what you like to understand. And to understand this, we are to understand how exactly corner field theory fails, and then we can hope to see.
Abhay Vasant Ashtekar: And also second thing is that in L curious scenarios of Black Hole evaporation
Abhay Vasant Ashtekar: there is a large portion of the transition surface. The The place which which which was a big uh, which was a big crunch. Singularity in the Financing Black Hole. Singularity. There's a large portion of it in which geometry the Adiabatic basic in these Well,
Abhay Vasant Ashtekar: for example, i'm looking at what the by modeling the in falling, you know, in falling, fucking uh radiate in falling more from hawking radiation as as kind of a null fluid. And therefore, looking at the white that geometry.
Abhay Vasant Ashtekar: And then the question is, anyway, what happened to the tens of of the boards. Of these modes, which failed to the dynamical horizon, as when I approaches the planned per vision.
Abhay Vasant Ashtekar: What is there about that battery action likely to be in on quantum geometry? And so we would like to sort up again and make a bridge between semi-classical gravity and food look on privacy.
Abhay Vasant Ashtekar: So the organized. So this is kind of a long interaction. And the organization is that I would like to.
Abhay Vasant Ashtekar: I'd like to first tell you very briefly about Minkowski's space itself. I know more. I might, most of you know, but it's good to sort of focus on conceptual issues that deal with in casket space and then turn to freedom of the microbes and walker Space times, and then somebody at a broader perspective.
Abhay Vasant Ashtekar: Okay, So Nature of
Abhay Vasant Ashtekar: uh in Milkowski space time. Sorry. But you know the this is not an operator distribution. We have an operative added distribution fire, hat
Abhay Vasant Ashtekar: and um,
Abhay Vasant Ashtekar: and that operate a value distribution. Yeah.
Abhay Vasant Ashtekar: And that operator value distribution is just given as usual by this E to the Ik dot x, and then you to I omega t minus of make it t to the omega t. And then these are the normalized uh basis uh the boards that I report you before in the task in space, and they satisfy this operative value. Distribution satisfy the the field equation.
Abhay Vasant Ashtekar: What this means is that if I take the operator value distribution, and i'll create it basically and give it by pass. I'll operate this off this object up here, since it is a second, derivative I don't have to take me twice on f I doesn't change sign. So if if, in fact,
Abhay Vasant Ashtekar: block minus m squared times, that
Abhay Vasant Ashtekar: which is of the type block minus. I'm scared of F,
Abhay Vasant Ashtekar: but it is conceptually important.
Abhay Vasant Ashtekar: I mean very often we just call it a field operator.
Abhay Vasant Ashtekar: The fact that it is not an operator, but the operator value distribution is really important. Of course we see that our immediately first, if you take the commuting of that of these operators. The right side right hand side is the grease functions.
Abhay Vasant Ashtekar: Um, A are genuine distributions. For example, in the zero right last case, this distribution is sharply peaked on the light calls.
Abhay Vasant Ashtekar: So so this really is a distribution up here.
Abhay Vasant Ashtekar: And similarly, if I take for hat up, find the the the the problem from them, and I can spec the uh the product and take the explication value in Minkowski Vacuum. Then you obtain a quantity which is usually one just Don't. What too much about this? One does not worry too much about this Epsilon, but that at selling is very important. We get this result in the cost of space,
Abhay Vasant Ashtekar: and the right right-hand side is are genuine distributions in particular. The meaning of this I excellent prescription is that where to first wear this out would test functions F. Of X.
Abhay Vasant Ashtekar: G. Of X. Prime.
Abhay Vasant Ashtekar: I've integrated, with respect to D. Fourx. D. Fourx. Prime
Abhay Vasant Ashtekar: you value the answer. And then we take the right-hand side and do the same thing, and after it evaluating it, then you take the limit as goes to zero. This is a sense, in which this is the quality is true. So, more importantly, i'll break this like like objects like fire, hatch, square, and discussion you can set
Abhay Vasant Ashtekar: so this this time about terminology can be quite misleading if they can. Literally So the question is, Do fields for hack effects continue to be verified as operator value, distribution in freedom and models. For example, in three dimensions, one up on R is singular function, but it's seen infinity as a distribution.
Abhay Vasant Ashtekar: One of the facts about stores, distribution, or ordinary distribution is that if you give me a distribution, then I can differentiate it infinity many times, and I would again get a distribution, or the shares distribution depending on what the character of the initial institution.
Abhay Vasant Ashtekar: Okay. So this is just very quick.
Abhay Vasant Ashtekar: Uh I'm: A recalling what happens in main custom, space, time. And now we go to the freedom and space time up here. So the question is of extending space time beyond the big back. So this is not kind of feels yet just ordinary space time. So
Abhay Vasant Ashtekar: now recall that every Friedman space-time is confirmedly flat and if it's especially if I look at the spaces reflect ones. Then we can write them out as an informal factor Times a flat metric,
Abhay Vasant Ashtekar: and this is the contamble time, the time, and then a Ip attack can be taken to the I'm. Going to consider a feta which are polynomial, which is a A. B times each of the beta. So this is. The beta is just bigger than or equal to zero, is an exponent, a constant up here
Abhay Vasant Ashtekar: and in the feedable models we are taking it to be positive, because that the E. T. I go to zero. This vanishes, and that is going to be the big crunch on the Big Bang. Singularity.
Abhay Vasant Ashtekar: Um! So you I go to Zeros a similarity.
Abhay Vasant Ashtekar: Now we can expand, we. We can extend a square
Abhay Vasant Ashtekar: across because a is zero, a vanishes, but it's perfectly fine. It's a continuous function there, and therefore we can actually extend it to negative values. And thus, if I consider the full negative value, I will get him in casting manifold. And not
Abhay Vasant Ashtekar: this only the I mean, half of them. Task is space with positive time, and now I can extend it to all times up here, when it is going from minus infinity to infinity. I can extend this metric as a continuous as a as a continuous t of field. But of course it is djend that at it I could at eta zero, and therefore the curvature actually blows up.
Abhay Vasant Ashtekar: And then, similarly, I can do this for a transformation, and this is pointed out the nice paper. But uh for the
Abhay Vasant Ashtekar: oh, so now the this is kind of just duly behind. But there is, in fact, a systematic rational behind this, and that comes from using the connection variables in traditional Adam Hamiltonian framework for the initial value problem of full general utility.
Abhay Vasant Ashtekar: It's based on three metrics of that moment, that and this breaks down if the three metrics become.
Abhay Vasant Ashtekar: But the questions satisfied by the connection variable? Do not
Abhay Vasant Ashtekar: the two are. These two equations are equivalent to the three. Metric is non degenerate, but the connection varies. Variables. Allow for digital metrics
Abhay Vasant Ashtekar: in the freedom elementary. So this is true in general. This is not in symmetric release model. This is just a much more general statement, and and a formulation of this was given in this papers with the
with Adam and and Flaw,
Abhay Vasant Ashtekar: and and so in female models, as well as um in the Yankee models and shorts in solution.
Abhay Vasant Ashtekar: This procedure enables one to evolve across the singularity and ambiguity,
Abhay Vasant Ashtekar: and this has been shown in a whole bunch of by a whole bunch of people. So the statement is the answer that you get up here is just this description that we talk to told you about. But this is a systematic, rational way. I'm not just saying that, but let's just do it. But, in fact, you can use the these field equations to evolve across the singularity and extend space time.
Abhay Vasant Ashtekar: But anyway, the final answer is the simple one in which you will just extended this as a computers function. If beta is fractional. This will be a continuous function. It beta is a positive integer, then. In fact, Um, this should be a It would be a differentiable function and smooth function. So uh, if if so, the metric will actually be smooth as the can of really be guys, it gives you a positive
Abhay Vasant Ashtekar: Okay. Now, we want to do on a field theory in this space
Abhay Vasant Ashtekar: for definiteness, because you are massless scale of you
Abhay Vasant Ashtekar: and i'm, i'm going to talk about generalization with that. Now, since Eta A. F is of this particular type
Abhay Vasant Ashtekar: um in presence of a universal time dependent potential, so the curvature that is implicit in this operator is transform into the flat, dull version operator and the and the potential and the potential in this is your Beta times, beta minus one divided by Ita Square.
Abhay Vasant Ashtekar: So eternal dependence is universal, and does not depend on Beta. It is just one upon it as well. Now the d guy is equal to zero. Then, of course, we just have then casket space. The spacetime metric is this: It's the scale tackle. It's a it's that way. Beta is equal to one. Then I get eight times it,
Abhay Vasant Ashtekar: you top here, and in that case
Abhay Vasant Ashtekar: it is radiation, free universe.
Abhay Vasant Ashtekar: Um! Now there is a rigorous reason
Abhay Vasant Ashtekar: that because the form of the potential that I got up here one can introduce a canonical, positive, and negative frequency decomposition, or a canonical Kaylor structure or canonical um state um on the algebra of while operators
Abhay Vasant Ashtekar: on the space of classical. So we can use positive negative frequencies. Solution on the space of classical solutions. Um
Abhay Vasant Ashtekar: the basis functions
Abhay Vasant Ashtekar: which are this positive. This is the positive frequency basis function that is selected uh times. Some uh some coefficients up here, and we're going to take this coefficients to belong to short space. So these are all in short space. I'm integrating this in the in the
Abhay Vasant Ashtekar: d three K dependence is, uh i'm in integrating it with respect to the K. Up here, and therefore I get here a function which depends on x, the the spatial x as well as E to.
Abhay Vasant Ashtekar: And now, as I've already said, these are the positive frequency Board and G. Of K. Are regular coefficients, which are in short space.
Abhay Vasant Ashtekar: Now, then, the you operate. A value. Distribution will be obtained by taking the solution and replacing the the keeping, the the the the more functions, and E to the Ikx, and replacing this coefficients by creation and then by inhalation and creation operators.
Abhay Vasant Ashtekar: Now, the more functions that's basically known
Abhay Vasant Ashtekar: and generically the more function diverse at eta equal to zero. For example, my best feel in us. We calculate two. They are given by this, this, this particular factor.
Abhay Vasant Ashtekar: So these board functions diverse,
Abhay Vasant Ashtekar: and there's also one up on eta setting it outside, because Phi Zero was equal to five a of eta times five. Therefore five is equal to five, not div by eta, so I get five not divided by a up here. So therefore I get here
Abhay Vasant Ashtekar: some discontinuity vanishes, and furthermore, these board functions also diverse. Therefore I got two types of diversions up here
Abhay Vasant Ashtekar: Guys functions.
Abhay Vasant Ashtekar: These board functions vanished, and there is this addition factor.
Abhay Vasant Ashtekar: Question: Yeah.
Abhay Vasant Ashtekar: Uh,
Abhay Vasant Ashtekar: as a result. But what we can show is we can
Abhay Vasant Ashtekar: that there is a unique complex structure on the space of this, on the space of the solution and the space of solutions of this object up here. Um.
Abhay Vasant Ashtekar: And that unique complex structure basically tells to the mill cask and complex structure in that infinite future and in the past.
Abhay Vasant Ashtekar: And then interesting thing is that but this complex structure could be selected by just looking at what happens in the infinite future, for example. But then we find that, in fact, that complex structure is also the one which agrees with the casket space complex structure in the infinite parks.
Abhay Vasant Ashtekar: Uh. So that is the statement that there's a canonical complex structure. So you're right that the more functions themselves are not canonical, because I can always multiply each of these, for I can multiply them by a phase factor, for example. But the positive negative frequency decomposition is canonical.
Ivan Agullo: Thank you that. Does it imply that that back you may also back you, because you said that the the same complex structure agrees with the standard one in the past and future.
Abhay Vasant Ashtekar: Um, All right. So let me just think about this for a second.
Abhay Vasant Ashtekar: I got a complex structure.
Let me just come back to that a little bit later. Is that okay? Are you on
Abhay Vasant Ashtekar: you want?
Ivan Agullo: I'll come back with that. I was. I was mute. It's for of course we can't come back because it's half an hour or so, so I I didn't really notice that it.
Abhay Vasant Ashtekar: So
Abhay Vasant Ashtekar: what about the K.
Abhay Vasant Ashtekar: This is what I would now like to look at. And so the statement is that I just look at Phi. Hat of five five is given by Default V. Times. This. So this is my We are operated by a distribution.
Abhay Vasant Ashtekar: The question that we want to ask is is this well-defined and satisfied the planning question on the entire space time.
Abhay Vasant Ashtekar: So now, on the initial field question. Data is equal to one, and if I looked at the the definition up here, it beta is equal to one. This actually vanishes, and therefore I do not have any um
Abhay Vasant Ashtekar: uh any uh
Abhay Vasant Ashtekar: to be again as the same thing I protect, find out to be a of e to the I hat. Then it it now just satisfies the the trying, the client, or the question which would be in cost, because there is no potential here at all, and has the mode functions uh are the same as the new customer space, and that trivially regular
Abhay Vasant Ashtekar: for um in in in the entire class of space, and it extends space time. But the physical feel of the freedom of Robertson Walker Space time has this extra one up on a factor up here, and this vanishes at theta, equals zero, and therefore the question is, uh, the more functions are giving us is the fox space. I mean task space.
Abhay Vasant Ashtekar: And, on the other hand, five of us has this extra factor. So how could this be a regular in the casket? And the answer is that it is because this is regular with respect to the for hat effects. Uh, it's going to be a regular um
Abhay Vasant Ashtekar: distribution. We can split the volume element of the Friedman model, so that is given by D, four, d, four V times a to the four times before X. One here. Hence, if I take now the smear object up here
Abhay Vasant Ashtekar: I take for hat, and I swear you do with I, for here. But now, if I had as a one up on a in it, but d four V has a A to the four in it, and therefore I just get a q times F. Of here. But type. F is a test function in short space, then a cube of eta
Abhay Vasant Ashtekar: times times. This is also in short space, because it is just a polynomial, you know, and so this in the in the
Abhay Vasant Ashtekar: But this is again in a short space up here, and therefore five or five is, in fact, a well-defined operator on the whole, because the fox space, and therefore this object is a well-defined operator valid distribution on the external of space time uh the free month of myth, robust over walker space time. Next look at the expertise, the value of the product of fields,
Abhay Vasant Ashtekar: and if I look at the expectation value of out of field. I get here. I got two fields.
It's like
Abhay Vasant Ashtekar: I got.
Abhay Vasant Ashtekar: Yeah, I've got two fields up here, and each of these fields brings me a a futa, which is to say, eternal. So I get one up on E to the prime up here. Times the expectation value in the custom space, and this is something that we already know perfectly well, and I get, therefore, this object, every type of right
Abhay Vasant Ashtekar: and just like in the casket space. Now this is something that I've been expecting already, mates and grounds, because what can appear on the right hand side or the mason ground is, it has is proportional to the scalar curvature, and the scalar curvature vanishes in this particular case
Abhay Vasant Ashtekar: but the stationary tension. On the other hand, it is non-trivial. There are two different in independent parts, because of because of the space of a majority. And so we can just have such terms up here, and each of the coefficients here vanishes, diverges as one upon eight out of the six, and one upon eight out of the six,
Abhay Vasant Ashtekar: but now d, four V has
Abhay Vasant Ashtekar: a one times e to the four. Therefore, if I even take the product of this times A to the four.
Abhay Vasant Ashtekar: If I multiply this by a to the fourth, then I will obtain still a divergence, which is one up on Eta Square, because I got eight hundred and six and met four here nonetheless. Um! It is a well defined distribution in the extended space time. So that is a question, is it nonetheless, like well-defined distribution.
Abhay Vasant Ashtekar: So I got is one upon Eta square going to the well defined distribution. Now that two basic properties of distributions I already told you one
Abhay Vasant Ashtekar: name, the second one, namely, that the distributions are infinitely differentiable. If you give me a distribution, I can differentiate it if and then again, I get that distribution is the origin of what she wants. The reserve is all. Again, I show up Every local integral function is a Schwarz distribution.
Abhay Vasant Ashtekar: Now this is the second property. If you give me a function which is integral on any compact interval, then it's a source distribution in particular.
Abhay Vasant Ashtekar: So now, Log of Eta is a locally integral of a function, because integral log of it times as this, and if I just integrate it between if I evaluate between A and B, and even if a happens to be zero, this is still a well-defined object, so some local integral function. Hence all derivatives of this function are locally integrable.
Abhay Vasant Ashtekar: The first day review will give me one up on it and hire a derivative will give me one of E to the minus.
Abhay Vasant Ashtekar: So eta to the minus n, so underbar means. This is a distribution as a distribution. It is defined to be this particular uh M. Derivative of log of Beta, or, if you like, given a test function, it sends it to one upon N. Minus N factorial times, the log of Eta types. Dm: F: By data
Abhay Vasant Ashtekar: this this one upon uh this fact, only the same here, this. This sign here disappears because I have to do M. Integration by parts, and i'm left with this with the minus sign.
Abhay Vasant Ashtekar: So this object is perfectly well defined on our appetite, and then these distributions for the most satisfied, individually expected properties. I can take the derivatives, and then I will up there as though they were functions, and I can multiply them by but to function e to, and then the power of this is one, even when
Abhay Vasant Ashtekar: um
Abhay Vasant Ashtekar: these objects a to the four of eta, and for a heck of it it's not in this case, but in general. In in the relation field case is just zero. But in general, even when these objects actually where to diverse,
Abhay Vasant Ashtekar: if they die, word like one upon recounts.
Abhay Vasant Ashtekar: I'm sorry about this.
Abhay Vasant Ashtekar: What do I get back,
Abhay Vasant Ashtekar: and you, even in the already I mean casting space observable of of of the fields, a temple distribution, not functions, so that you cannot ask better at singularities.
Abhay Vasant Ashtekar: So what about K. The more complicated case? The dust field model. Now, in this case the scale factor
Abhay Vasant Ashtekar: behaves like Beta Square up here, and hence uh one up on the air that was is faster, for that was that e to the one up Anita Square, and the more function themselves. That was
Abhay Vasant Ashtekar: um. Now the one product available space is built out of solutions, for Ifx, which is given by just like that,
Abhay Vasant Ashtekar: and they are diverse at each. I equals zero.
Abhay Vasant Ashtekar: So how can there be a well-defined fox space?
Abhay Vasant Ashtekar: Can there be a well defined for space? This is a question we are asking because the the one particle Hilbert. Space is made out of out of the positive frequency solutions to forget about this. Forget about this part, if you like, and these are positive frequency solutions, these diverges, and these also diverse. So how can there be, uh, how can they define one particle
Abhay Vasant Ashtekar: Well, in fact, the one particle norm of them of these objects is perfectly fine and non zero at theta equal to zero. Because, again, because the divergence in the value of our fax is cancel. It's compensated precisely by the by, the vanishing of the three dimensions.
Abhay Vasant Ashtekar: So you can check that very explicitly. It's not very surprising. It's it's at all. It has just to do. The fact that simply the structure uh is is is completely identifying and does not depend on. Is it time independent? That is really what is happening up here,
Abhay Vasant Ashtekar: and you can calculate a norm, and that normally just given by the the Z squared times Dtk: upon this. So. In fact, the coefficients which are in short space define the norm, and this is in short space. Therefore this is perfectly verified.
Abhay Vasant Ashtekar: So the phone, the function itself diverges that it defined
Abhay Vasant Ashtekar: leads to one well-defined one particle spin and this is really similar to what happens if quantum mechanics, ordinary quantum mechanics, even uh in order to the quantum mechanics in three dimensions. If it's not a it's to be a mechanis. If I take the wave function to be one upon our times, E. To the minus of our.
Abhay Vasant Ashtekar: Then it replaces the state in this simple space, because the volume element goes to zero as r squared, even though it's a size staff side that is like one up on our screen. There's a cancellation, and exactly the same thing happens
Abhay Vasant Ashtekar: now since D. Four V. It has a A. A to the Eeta in it. This is, in fact, a well defined or operate operated distribution, but it turns out that because the more functions have this one upon square root of T, and again one upon K. Up here.
Abhay Vasant Ashtekar: So this is nothing to do with ultraviolet problems. We are the Big Bang that you are talking about. We just the infrared problem problem. And that was pointed out by and and Ford and Park a long time ago, and this, of course, already is in order that you know, away from a big bang, but he are bigger than zero.
Abhay Vasant Ashtekar: So you have to introduce an infrared regulator to make the theory verify, and once it is like regulated, then there is no further problem at Big Bang at all. If you have this it right cut off. Then there is no for the problem, and this cut off has nothing to do with the big back at all.
Abhay Vasant Ashtekar: Um, as I just said, If you you make this operator, well define or operate a distribution for it. A positive does in fact, in this can continues to be well-defined for e time uh equal to zero and less than zero by the and yeah, these for massless fields right, that this the
Abhay Vasant Ashtekar: Yes, it out. But it's from the
Abhay Vasant Ashtekar: because i'm talking about the master experience.
Abhay Vasant Ashtekar: That's good.
Ivan Agullo: Thank you.
Abhay Vasant Ashtekar: And the the the the the expectation value of this fuel operators in place. It will be just given by this and this. But now I got this extra factors up here coming in. And next of fact, this up up here coming in, and these are divergent up here. These are the export, and there's also a lot term that is coming in here.
Abhay Vasant Ashtekar: No,
Abhay Vasant Ashtekar: The point is that as we just saw that this is a well-defined opportunity distribution; that for this is actually a well-defined by distribution.
Abhay Vasant Ashtekar: And again, it's because the four V has e it e to the eight of the power. It here and I got two volume elements, and they
Abhay Vasant Ashtekar: now for space like in time, like separate points. One interprets this object as a correlation function
Abhay Vasant Ashtekar: in the casting space Correlation, Dk: as one upon distance square. This would be just the correlation, and that decays as one upon distance, square, both for space like and time like that.
Abhay Vasant Ashtekar: But now for for the feedback model there is an interesting space versus time. Our symmetry as you can approach the singularity.
Abhay Vasant Ashtekar: Consider the points that are space like, or time, like separated by fixed, proper judicial distance, so I can have big done similar in here, and I can consider space like separated points, or
Abhay Vasant Ashtekar: like that, or I can consider time like separated points like that,
Abhay Vasant Ashtekar: as an approach to the Big Bang. The space, like correlations, dominate for the time.
Abhay Vasant Ashtekar: So if I take the limit, eternal goes to zero. But on the top X and X. Prime are different. So this is Space Station Dep. What? That points are spatially displaced at the same mention of time, and it on that is not equal to zero. I'm. Just taking the limit of that. And here is the opposite. I got X X in the same point, and it does are different here,
Abhay Vasant Ashtekar: and this diverges as one up on it.
Abhay Vasant Ashtekar: There are strong correlations.
Abhay Vasant Ashtekar: Now. Strong correlation means that small variations as I go from one point to another. This is heuristic completely. This whole idea of correlation is also physical and heuristic. Um. And so
Abhay Vasant Ashtekar: that means that I got small variations, and therefore there are smaller derivatives.
Abhay Vasant Ashtekar: So, therefore, since the spatial correlations dominate, or the time, time, separation, correlations, we can say the time derivative is dominant, or space derivatives, as in the detail be able.
Abhay Vasant Ashtekar: This is a remark, and i'm not saying that there's something that is related to the but it's a useful thing to keep in mind, and somebody might be able to use it in other works.
Eugenio Bianchi: Question:
Abhay Vasant Ashtekar: Yeah,
Eugenio Bianchi: yeah, hi, Mike, at the top of the previous slide you have this correlation function. The question is, what is the state that you're considering here?
Abhay Vasant Ashtekar: Uh, what we saw was that uh,
Abhay Vasant Ashtekar: but that's beta is grow too. Therefore this factor up here is two times beta minus one, and therefore Beta is equal to.
Abhay Vasant Ashtekar: That means that this factor is just also to.
Abhay Vasant Ashtekar: But if Beta is equal to minus one,
Abhay Vasant Ashtekar: then this factor will be minus two, and this will minus one. So this is again two. So this potential is the same for for dust as it is in in the sitter.
Abhay Vasant Ashtekar: So if you like, I'm just using the budget name is back in.
Eugenio Bianchi: I see the back in that is chosen. It's just a bunch of. But the question was was really uh, Does the remark apply to a large class of states, or is it specific of this state,
Abhay Vasant Ashtekar: so that I mean because the the the precise way that it goes up like one of Anita Square, etc. Is is really a lot. Is that is to this our our working just refers to this particular.
Abhay Vasant Ashtekar: So then, finally, for dust, three reverses, we can look at operator products. So far half of X is a dimension, one operative added distribution. Five squared is dimension, two and five, and especially it's the mix and four
Abhay Vasant Ashtekar: so priori. The fact that I have been well defined across the big bank does not mean that these operators will be wellified. This products should be verified, are they where they are older works by Budget Davis, and others. They imply that in fact, I get, for a hat of X is given by
Abhay Vasant Ashtekar: the formula like that, and this L. Is the infrared that I talked about just a while ago.
Abhay Vasant Ashtekar: And so at the Big Bang
Abhay Vasant Ashtekar: the scale occur which of blows like one up on E to the six, and therefore it's divergent as a function. But but as we just saw it's perfectly fine as the distribution.
Abhay Vasant Ashtekar: But in any case, since the olive elements goes like each other, the eight and this is just being like eight out of six. This object, in fact, is a seat is is a C two function, and, unlike the the relation field case, it does not vanish because
Abhay Vasant Ashtekar: um r is not equal to zero, so it's a perfectly fine function, in fact, not only distribution, but it's also a function across the big back. Now the order was also provide expression for for the stationary tensor
Abhay Vasant Ashtekar: being a dimension for operator valid distribution. It involves products, and second, there it is of curvature tenses. The express expression is long, but it has a simple form Again, it is because of symmetry. There is a coefficient times gravita, another coefficient, and the metric up here,
Abhay Vasant Ashtekar: and the most diverted term here goes like E to the minus eta times log of Rita
Abhay Vasant Ashtekar: default v. As E. To the eight, and therefore this product goes like log of vita, which is locally integral function. It's divergent, but it's locally integrable. Therefore it's a perfectly well defined See if you need that for distribution
Abhay Vasant Ashtekar: to the summaries that dynamics of I had is much more non-trivial in this case,
Abhay Vasant Ashtekar: so that is somebody a little bit summarized
Abhay Vasant Ashtekar: There's a long history of probing classical generally duty, singularities with classical fields and condem particles.
Abhay Vasant Ashtekar: But most analysis, where for conformity, that expense that so either started spacetime or the reason,
Abhay Vasant Ashtekar: and formerly start for space time up here with time lexing you like
Abhay Vasant Ashtekar: um.
Abhay Vasant Ashtekar: Here we consider
Abhay Vasant Ashtekar: here we consider time dependent space stance with space licensing varieties which are also physically far more interesting, because there. There's a big bang and the big crunch kind of singularities, but time dependent forces. You want to consider kind of fields, of course, pros, and some are surprising that the big ban on the big crunching united is a remarkably Ted. But table, when pro with observers associated with corner fields,
Abhay Vasant Ashtekar: when we keep in mind that they are operated distributions,
Abhay Vasant Ashtekar: classical fields, five or five that define the one particle Space do divers at the Big Time singularity, but they're not in the one particle Space is finite, because the shrinking of the volume element exactly compensates for the divergence as we saw.
Abhay Vasant Ashtekar: And again, I've mentioned about this wave function in quantum mechanics is diversion, but it's a perfectly well-defined element of the about space
Abhay Vasant Ashtekar: similarly The more functions that enter the expansion of I have diverge. But this object is a well-defined operator value distribution. Uh, because the we are offered is identified and the by distribution, and you want the product of operating operators
Abhay Vasant Ashtekar: uh the normalized power of operators
Abhay Vasant Ashtekar: value distributions are perfectly identified.
Abhay Vasant Ashtekar: Just that in the past the space
Abhay Vasant Ashtekar: now generalization, the main result on the same behavior for linear tests on the field is A has been extended to other feedman. Uh, let me Rob Roberts and Walker models. K. Equal to zero with beta bigger than zero, so fractional power, and beta, for example. Uh, but it turns out that then the
Abhay Vasant Ashtekar: something that becomes technically much more complicated, and I cannot use a simple argument. I give saying that well, one up on e to the M. Is a well-defined distribution, because I can just define it by taking derivatives of the log of it.
Abhay Vasant Ashtekar: But there's a notion of homogeneous distribution which is much more complicated. But for all purposes, you know, it always suffices to say that that again well defined temporal distributions, and furthermore, they satisfy the algebraic property that I told you about, namely, if I take the derivative. Of the distribution, then,
Abhay Vasant Ashtekar: or if I multiply the distribution by Vita, then the these operations are very exactly milling. What happens to functions, one upon each to the and
Abhay Vasant Ashtekar: if I take this directly i'll update something. If i'm a deployed by it, I up there something exactly the same thing happens for the distributions.
Abhay Vasant Ashtekar: So
Abhay Vasant Ashtekar: Uh, then, this is a non-trivial generalization, and this is done with a work with Eddie and up here, which is k equal to plus minus one feedman. Lemme the robust and the models up here
Abhay Vasant Ashtekar: it is Alpha scattered, for example, in the spatially closed universe It is just going to be given by Einstein's new us, which is to say, she's here metric cross with time,
Abhay Vasant Ashtekar: and so that is not a flat match. The spacetime appears the ultra-static man.
Abhay Vasant Ashtekar: Uh, therefore, while the K equal to zero procedure goes through, the conceptual procedure goes through. The spatial dependence of the basic function is no longer exponentially to the R. K dot x. They are not playing base, but they are much more complicated. And in fact,
Abhay Vasant Ashtekar: uh, then, we're doing this One Adrian found that in the literature. It's got a field in the space-time. All the properties of this basis, functions, and so on. Had not been the systematically worked out, There are gaps, and therefore in the in the paper we just appear in Trd, there's a long appendix which gives you a systematic treatment which feels in all the gaps.
Abhay Vasant Ashtekar: Now, second difference up here from the K equals zero case is that now we are to use just this, these test functions, because, for example, the specially come back case.
Abhay Vasant Ashtekar: The you know you cannot get short distribution is not well-defined, because you say that the test functions actually fall off faster than any polynomial, as X goes to infinity, but in a specially compared case in this person, Direction X. Does not go to infinity, and therefore where to use this, this, this function. But again,
Abhay Vasant Ashtekar: everything is an ordinary distribution, or all of operator value distributions rather than tempered ones. But other than that, everything is the same
Abhay Vasant Ashtekar: now, And the interesting thing is that there is a simplification of these models now being for a divergence that we saw in the dust case, and also for higher value, the beta do not arise,
Abhay Vasant Ashtekar: and the reason is because the K. The the spatial curvature, provides an in in for a cut off, and so the close and the open Friedman Morals, in fact, on a field setting, is in this sense simpler than the spatially platform.
Abhay Vasant Ashtekar: The by distribution continues to have a so exactly like in K equal to zero. If you are away from
Abhay Vasant Ashtekar: the type of zero surface.
Abhay Vasant Ashtekar: Uh, that's true. But you and when, in fact, the two points are on the other side of you type with a zero surface.
Abhay Vasant Ashtekar: You even in that case the notion of Jud, this distance up here is okay, and you can find that everything is fine. Of course it's the two points longer. Stuff, Is it? Said that there's no no, I'm. Jur: this this doesn't make sense, because the metric is, is is regenerate there. Uh, therefore, that motion of Haraman,
Abhay Vasant Ashtekar: so it doesn't make any sense. But apart from that, The statement is that it is as if you like, as it could be given that the matrix is each of the big back and the expresses of P. Hat and Pb. Are much more complicated, but again, eight to four times five square continues to be a regular function, as we saw in the before,
Abhay Vasant Ashtekar: as in the K. Equals zero case
Abhay Vasant Ashtekar: uh highest sense. Since female events Robertson are, and Family Flat
Abhay Vasant Ashtekar: Maxwell Fields are trivially regular across the big back and big crunch. You don't have to worry about it, and the results are massless. Scalar Field also implant that it's time that's possible for spin to uh linearize very for the field. I should confess that this is something that we looked at in detail in the K. Equals zero case. It is also true. K. Equal to plus and minus case. But we are not looked at this in detail.
Abhay Vasant Ashtekar: What are black holes in your life?
Abhay Vasant Ashtekar: Well, there's a work in progress with Andrew and Schneider, and earlier part of the work was done with the with the the Lorenzo, and in the in the case of the and it we're going to focus on, as we know, on the Nigeria region, and task is sax metric,
Abhay Vasant Ashtekar: and the nice thing is that we do have analytical expressions of more functions as an infinite Conversion series. So this is because Mark could find some beautiful literature very old data check Um, which actually can be used for on Tv is to say that in time the series is called Virgin,
Abhay Vasant Ashtekar: and the work is in progress to identify the States, which in the interior region correspond to the Andrew, or how to walk hawking back in. We're very close, but it's not completely finished up here, and uh, so these are the states that synchronic region we can identify them. And now what we want to do is to extend them,
Abhay Vasant Ashtekar: However,
Abhay Vasant Ashtekar: Generic Black Hole singularities. I'm not going to be. Space like this is true for all the Black Hole evaporation problems that people look at. Um, the generate black hole. Singularities are going to be knocked because basically they are going to the cauchy of horizons of code and the rice on all some tapt of
Abhay Vasant Ashtekar: you know, solutions up here, this case seems much more difficult technically, but there are no conceptual difficulties in in in in extending this kind of work, but technically is much more difficult, because now, at the fortiation that I got Space Law will not be true, but, for example, this world but the foremost to extend the
Abhay Vasant Ashtekar: or in the Einstein's equation I mean. Put the he puts perturbation on right north of now on Space Times, and wants to
Abhay Vasant Ashtekar: nonlinear perturbations and extend the spacetime in a distribution set across the and you can do it using a C not Metric, but not a C one Metric. And so, therefore this is something that you know could actually be done. Just that. We had to look at music questions much more, you know.
Abhay Vasant Ashtekar: Step back a little bit, and look at this.
Abhay Vasant Ashtekar: Now let me just finish the the broader perspective here.
Abhay Vasant Ashtekar: Now the broader perspective is that
Abhay Vasant Ashtekar: um
Abhay Vasant Ashtekar: that we can extend corner field theory in cur space times,
Abhay Vasant Ashtekar: because normal corner field theory is Kurds by Stan. In recent years there have been extremely nice progress that was made,
Abhay Vasant Ashtekar: and this progress. And we're using very front sets and
Abhay Vasant Ashtekar: um and and and and the scaling arguments, and this these ideas, therefore, could be applied. It could be extended here. The point is that all that work, however, relies heavily on globally hyperbolic spacetime, and of course, this space time, which are extended about globally hyperbolic because we've got the metric is degenerate along the big band surface up here, and therefore it is a but now we could extend contours. So this is a nice door
Abhay Vasant Ashtekar: uh send a classical gravity. It is a theory which the matter is quantum mechanical. The metric is classical, and that couple We have these equations, but t hat Ab is no longer smooth, but it is a genuine distribution forcing us to seek distribution of semiclassical solutions.
Abhay Vasant Ashtekar: Existence is not obvious, because Einstein's equation on nonlinear. So the target distribution on that I can side left, right. I can sign up here where that I can find a distribution. Metric is not obvious, because then the products of metrics and their derivatives coming in here. But I just said the
Abhay Vasant Ashtekar: um. And but now one can extend that that actually did this to the classical guy, we can extend it, and that is quite possible. Examples are known, as I said, and possibilities are this possibility opened up because
Abhay Vasant Ashtekar: this is our investigation pinpointed? What exactly is different in kind of field theory on cur space time at the Big Bang. What is different is that this there's any transfer is a genuine distribution rather than of a function,
Abhay Vasant Ashtekar: and that is what is different. Now
Abhay Vasant Ashtekar: in full quantum gravity. The plank regime excitation of quantum geometry has support in two space and dimensions. We know that from spin in the phones very, very well, and but this happens in many approaches, for example, clips to review up here,
Abhay Vasant Ashtekar: and our concrete example is provided by the distribution nature of the between the gravity and spin forms, and therefore the interesting challenge now is the follow up.
Abhay Vasant Ashtekar: Can we systematically show
Abhay Vasant Ashtekar: that the semi-classical distribution of geometry is an approximation to the lqg distribution geometry.
Abhay Vasant Ashtekar: So first of all, who should be able to solve this problem. To say that there is a sense in the classical Um, well defines in the classical distribution challenge. And then, with the statement, he said, Is that really a good approximation to the and advances along this direction will provide concrete bridge.
Abhay Vasant Ashtekar: As I began by saying that well, this is supposed to be talked from outside the field, and i'm just filling in at the last minute because of planned speaker. Um was not available, and therefore
Abhay Vasant Ashtekar: uh, the statement, here is again building bridges, and we can. We can do this as as usual. You know you saw other people's problem, or
Abhay Vasant Ashtekar: then that boarding concept. Thank you.
Hal Haggard: Any questions.
Abhay Vasant Ashtekar: Even your question was not directly related to this this my talk, so i'll rather take it at the end, if possible.
Abhay Vasant Ashtekar: Yes, we we can discuss later. Yeah, we'll discuss it later. I think that's not. I mean it is. It is discussed in the paper.
Ivan Agullo: So so so. So let me ask a a A. All concrete results You show where for the massless case, uh, is there any embedded in for using a mass for the quantum field, or what is different.
Abhay Vasant Ashtekar: Uh this. The beta is fractional. Beta is one half for that. Yes,
Abhay Vasant Ashtekar: and so they it's not impediment. But, as I mentioned before, that what we had to use is over genius distributions, because what happens in that case is really
Abhay Vasant Ashtekar: um
Abhay Vasant Ashtekar: uh, that team is a little bit more complicated. Uh, but I think you know, if you want, call into your results, and they establish them. And this is summarized in the appendix. Uh, I think the second appendix in the first paper. Uh, with the from also and mark
Abhay Vasant Ashtekar: uh, but other than that, there is no difficulty of principle.
Ivan Agullo: I I am confused. What What is the relation between that M. And the mass of the field.
Abhay Vasant Ashtekar: Okay, Sorry. Thank you.
Abhay Vasant Ashtekar: The is is is a fractional. So if you are in that case,
Abhay Vasant Ashtekar: because, like E to the one half.
Abhay Vasant Ashtekar: But but but but the a of it is a scale factor. Uh, And why this can factor uh the form of, because what we have is all that it's a scale factor. And so on, that come up will be delivered with the scale factor. Right?
Ivan Agullo: Okay.
Abhay Vasant Ashtekar: So I think it's very. It's expressed well various expressions of this, and so on. Instead of getting individual powers of E time, the denominator, I will get happy to the power of We talk to the um, three by two, for example, or five by two,
Abhay Vasant Ashtekar: so I can give meeting to each of the five by two, and so on
Abhay Vasant Ashtekar: as a as a distribution
Abhay Vasant Ashtekar: minus one, which is supposed to correspond to the one of Anita. To that. Now my job is going to be. If there is a mass. Yeah, that must have nothing to. Then, if I got a mask, then
Abhay Vasant Ashtekar: M. Is replaced by a fraction of power. So here I would get fractional powers,
and I will have to.
Ivan Agullo: Is that clear? Yes, I I just don't understand why a mass of the field a give you a fraction of power to the scale factor. Oh, that's just because i'm just like a dust is a
Abhay Vasant Ashtekar: Oh, i'm! I'm so sorry you're talking about the the of the All right. I thought that you're talking about the source.
Thomas Thiemann: Okay, so much as it's on race. Uh. So I want to come back to uh the thing that you just uh discussed,
Thomas Thiemann: uh, which is differentiable.
Abhay Vasant Ashtekar: I I I think if you
Abhay Vasant Ashtekar: send it some references I would love to look at this. It's It's in a written Simon. Okay, uh second volume, one
Abhay Vasant Ashtekar: uh first volume. Actually, I'll look at video and Simon first of all. Yeah. So then, I think thank you very much. And then maybe that's it would be easier. So it's good that I misunderstood the question because I learned something. You're nice. You have to discuss what happens when you extend the space time
Thomas Thiemann: from a positive to the negative free line,
Abhay Vasant Ashtekar: but probably that can also be done with.
Abhay Vasant Ashtekar: So so you write down a to be equal to absolutely to the Beta Beta. Is that fraction of power up here, and then you extend it so again it will remain continuous, but it will not be differentiable there. Yeah. But then, this should be a applicable this year. Thank you very much, because I
Abhay Vasant Ashtekar: I was not aware of this. And so Um:
Abhay Vasant Ashtekar: yeah,
Jorge Pullin: okay, Good morning.
Abhay Vasant Ashtekar: Absolutely. So. I mean, we know what the extension is, right. So on the one line, I mean, at least in the
Abhay Vasant Ashtekar: I think. Yeah, I, at least in the in the in the Friedman models like we have an extension like that. You are a function which is sharply peaked. At late times we have a extension which is, you want about the effective equations like,
Abhay Vasant Ashtekar: uh, and so there is an extension there. But of course, in that extension uh the corner field will just evolve, and that's what we do all the time that you would study for the patients.
Abhay Vasant Ashtekar: The challenge is basically what is the exact relation
Abhay Vasant Ashtekar: between what we do there and what kind of filtering cur space time uh does.
Abhay Vasant Ashtekar: So. What do you think? It's space time? So far we'll just say that. Well, I come to a plan for gym. I throw my hands up, and I don't do anything, so I don't know you, you you.
Abhay Vasant Ashtekar: But now what we see is that, in fact, you don't have to throw your hands up. You just have to make both the space-time geometry distribution, and this all the fields also distribute the
Abhay Vasant Ashtekar: space time dancing distribution right? That's what we do.
Oh,
Abhay Vasant Ashtekar: yeah, Maybe I don't need them. So I I need to do the space-time community distributional and um
Abhay Vasant Ashtekar: uh and the point of feel, of course, to it, to also to be taken as a distribution. And so now what we would like to do is on the left hand side. You can have the kind of feel free description with the space-time job which is distribution. The usual sense of the work right.
Abhay Vasant Ashtekar: If you're kind of sloppy with look for them. Gravity. Uh, then we'll get the distribution of geometry that is given that's given in this extension that the live extension up here. And then I also feels propagating on that night extension. So what happens with those fields in the in the in the description?
Abhay Vasant Ashtekar: So we we should be able to match.
Abhay Vasant Ashtekar: You know we've got these two things, but they are sort of standing here and There's the usual thing about trying to make contact with Low in the physics
Simone: right? I I guess. What I'm really wondering is whether, with this type of approach you can learn something about the validity of the symmetry reduced description that one is taking for the gravitational sector.
Simone: So
Abhay Vasant Ashtekar: yeah, I I think that's a I I I mean, I think that we will learn much more about the value you have minutes of this first. Uh, I think, just coming from the the kind of world that various people have been doing right in the last couple of years last few years with us to do it,
Abhay Vasant Ashtekar: taking some States and look for them, drive with the full loop cotton drive through which are isotropic, homogeneous in some sense, and so that, I think is likely to be
Abhay Vasant Ashtekar: okay. I see. Okay, so. But I think it's more likely to be much more useful for us to see. You know how much to trust this, but this is more to make a bridge to contribute to the Space time people, and me if we do something non-trivial, right? I mean, we said, Well, look you guys, were saying this.
Abhay Vasant Ashtekar: I think that we're not going to learn tremendous thing about. Look at why we came from here. It's more about the bridge.
Jorge Pullin: I don't
Hanno Sahlmann: erez agmoni thanks for the nice talk. Um! There's a point about this work, where it's very fascinating, but I'm. Also slightly confused. So I wanted to ask you to comment on this one hundred and fifty,
Hanno Sahlmann: and that is that um the Space Times, as you said in the talk, they are not globally hyperbolic, because there is a singularity. So there is no a priori no well-defined evolution across.
Hanno Sahlmann: On the other hand, you show that with this, with this mapping where you map the the to another equation, to show that there is a very natural
Hanno Sahlmann: evolution across, in fact. And so I was wondering if there is something that can replace global hyperbolicity
Hanno Sahlmann: little bit more generally that captures this, that although on the nose this this thing is not globally hyperbolic, in some, in some
Abhay Vasant Ashtekar: in some sense, it is that because there is a natural E extension across the singularity. Exactly exactly. That, I think, is the main. I mean to me this is very, very interesting, so I I I agree with you completely. That that is that. That is why I think this is interesting, and I think
Abhay Vasant Ashtekar: so. This So how most questions are the following: that if I take normally um
Abhay Vasant Ashtekar: on the fields in Thursday night, I would say, i'm sorry if I take normal the kind of fields in cur space that then you get here five five of X for Ip. Why do you want to start beginning writing the equations here? And that is view, and by ig. Bar times with advance
Abhay Vasant Ashtekar: needs function.
Abhay Vasant Ashtekar: Um, But these don't exist
Abhay Vasant Ashtekar: by themselves.
Abhay Vasant Ashtekar: You start looking at this by head of X, and for I had a y right.
Abhay Vasant Ashtekar: What we did was to consider
Abhay Vasant Ashtekar: field operators associated with solutions, classical solutions so like we like classical solution. This is something else.
Abhay Vasant Ashtekar: Feel like that
Abhay Vasant Ashtekar: capital of I, also the classical solutions.
Abhay Vasant Ashtekar: So we also have this. These are algebra.
Abhay Vasant Ashtekar: Um,
Abhay Vasant Ashtekar: absolutely every classical solution. You are going to feed the operator. Another solution feel operator, and they committed is given a synthetic structure.
Abhay Vasant Ashtekar: This doesn't make sense, but this makes sense, because the simplic structure is time independent. So I got here at the equal to zero, and then I can just evolve this solution
Abhay Vasant Ashtekar: across, and then the the supply that is perfectly well defined here.
Abhay Vasant Ashtekar: So somehow this kind of population, algebraic formulation based on this or the while Operator: you to the I for hat of this feels up here. This is perfectly well defined.
Abhay Vasant Ashtekar: So somehow, I just say that there is
Abhay Vasant Ashtekar: global, hyperbolic city was kind of a
Abhay Vasant Ashtekar: straightforward machine which enable you to do things. But it's not necessary. So that, but to everything in way, front set, as you know, much better than I do, depends on global hyperbolicity.
Abhay Vasant Ashtekar: And so I think that one should be able to. Uh, I mean, since you can do it, then I think one should revisit everything that was done using global and publicity and see where it can be dropped.
Hanno Sahlmann: Did I answer your question? I mean, did what? That? That is very helpful. Yeah. Yeah. So so you you pointed out where that where that magic happens in that
Hanno Sahlmann: maybe one one can use this exactly to to to generalize uh several things. Yes, thank you.
Yeah. Sure.
Abhay Vasant Ashtekar: I think there was a something in the chat box, but I think that I might have already answered that question.
Ivan Agullo: So so her hair has left. So I am the new. Okay. So next question is, by dinga gia,
Ding Jia: I have a um
Ding Jia: nice talk. Uh, i'm curious about one of the questions you raised uh towards the beginning.
Ding Jia: Uh, there's a slide uh uh what we're, what what what exactly fails in. So you you remember that's like right? It's It's how right Why, we should care now, you said the um.
Ding Jia: Well, to understand how i'll i'll Qgq. Or semi-class will probably we need to understand, but it's like a fails in capture. So what is the what is your view on this question? Do we have an answer right already?
Abhay Vasant Ashtekar: If you look at on a field theory into space time as well,
Abhay Vasant Ashtekar: you know, in a mathematical way which what we want to look at, then nothing else but on the
Abhay Vasant Ashtekar: If you want to look at it from the point of view of Uh,
Abhay Vasant Ashtekar: uh,
Abhay Vasant Ashtekar: but
Abhay Vasant Ashtekar: yeah, here we go.
Abhay Vasant Ashtekar: So if we want to say that, in fact,
Abhay Vasant Ashtekar: G is a classical metric. By that we means good metric, that of course, this object has to be smooth in order for this, to this equation, to make sense
Abhay Vasant Ashtekar: to that first,
Abhay Vasant Ashtekar: that this object is really a generic distribution.
Abhay Vasant Ashtekar: So what it is telling us is that
Abhay Vasant Ashtekar: if you wanted to handle singularities, you
Abhay Vasant Ashtekar: then it you had to do it in such a way that
Abhay Vasant Ashtekar: the metric is a
Abhay Vasant Ashtekar: it's tricky to be. It's a lot to be distribution. I mean it's. It's it's. It's moved everywhere except at the zero. But if you want it to, including type, one zero, there is a distribution.
Abhay Vasant Ashtekar: Okay? So that is what it is.
Abhay Vasant Ashtekar: So it's not a failure of
Abhay Vasant Ashtekar: want to feel very proper, but it's a failure of the way. Semi classical gravity has been used, or as the thought of in the literature.
Ding Jia: It seems like they're they're saying that
Ding Jia: the the traditional way of treating some in classical gravity needs to be generalized to distribute
Abhay Vasant Ashtekar: exactly, And The second thing is what I was talking about on those.
Abhay Vasant Ashtekar: So the second thing is that
Abhay Vasant Ashtekar: what fails is that if you are the if you have the uh, the singularity then retarded. Not. It's not globally hyperbolic, But for many of the techniques people have been using
Abhay Vasant Ashtekar: and back and forth. I'm: That is a global I have a policy is really a solid building block. I mean, It's really solid and rock on which
Abhay Vasant Ashtekar: everything is built in on a field in cur space time. But if you don't have global hyperbolic city, then people will just code through the hands up. For example, the first step that they law write down is this day, and then this doesn't exist. If, in fact, you have done, do not have an open hyperbolic setting.
Abhay Vasant Ashtekar: But then you can bypass it.
Abhay Vasant Ashtekar: That says that. Well, this is not essential, anyway. This is what was the same shape, then, people would say, But with this, can you still talk about
Abhay Vasant Ashtekar: for some operators uh five square would be normalized,
Abhay Vasant Ashtekar: and we see that. Yes, if you can, and these are perfectly well-defined distributions. In fact, for five square. It is, in fact,
Abhay Vasant Ashtekar: they that, in fact,
Abhay Vasant Ashtekar: a genuine distribution.
Abhay Vasant Ashtekar: So out of the corner field three people would say that well
Abhay Vasant Ashtekar: or in the kind of field you would say that that distribution is perfectly fine,
Abhay Vasant Ashtekar: but they don't know how to do kind of field theory if you, if it's not global. You're hyperbolic because I don't want to advance them. They can't even function. And what we are saying is, No, you can bypass that difficulty.
Abhay Vasant Ashtekar: Um. So and again here they would say, Okay, it's a distribution. That's perfectly fine. But then we will come back and said, Okay, then, let's see if in fact, we can solve Einstein's equations, semi classical equations, with distribution of right hand side, and
Abhay Vasant Ashtekar: sometimes you can. And this is easy enough case, because the distribution of character is just on one surface, so that you know
Abhay Vasant Ashtekar: methods that were introduced by the and and others to talk, to evolve. And there is actually a distribution on geometry along a surface like that uh, may be useful, or the methods that the foremost is using may be useful.
Abhay Vasant Ashtekar: So to summarize again globally, hyperbolicity fails. And the way that semi classical gravity is not when he used that phase.
Eugenio Bianchi: Yes, yeah, this is very nice of right. So I have a question. Uh yeah, You consider the specific states that one could do all the calculations again, and the question is, Do you have an intuition on what to expect if there's any.
Abhay Vasant Ashtekar: You are you talking about Dkl: injection or a big deal behavior in general, or just this general deal that everything is remaining. Yeah, everything is is is up to down there. Um!
Abhay Vasant Ashtekar: I mean anything which is in the same vogue of our class will be perfectly fine, of course,
Abhay Vasant Ashtekar: but you are probably asking about what is not in the same locally about class. Is that what you are saying?
Eugenio Bianchi: Yeah. So so, just to make it more precise at the state they are considering, especially in many ways. But in one way it's like the class of uh idiomatic,
Abhay Vasant Ashtekar: uh Adiabatic. That will be fine.
Abhay Vasant Ashtekar: Okay, I get. I get experience. Calculation will probably be possible. So But what? What? What? We just have to give
Abhay Vasant Ashtekar: in general.
Thanks
Ivan Agullo: so. And and I have a sort of date. Quick question. Uh: So my my understanding is that, you know, even though the space time is not globally hyperbolic at the classical level, you can still
Ivan Agullo: evolve the distribution of character of even two point functions across the singularity. But um uh, but in some situations you mentioned that the metric is, See not only is not, uh doesn't have a smooth derivatives, and you know, this reminds me of these discussions we have in the past that when, whenever the metric is not at at least seat, you,
Ivan Agullo: uh, so to speak, infinitely, infinitely many quant are created, and you know the evolution doesn't it's not unitary, et cetera, et cetera. And And have you thought, if that that also happens in your situation?
Abhay Vasant Ashtekar: Yeah. So then, there's a question about metric. I mean we is a metric is not C two. We sort of normally mean metric, and it's in. So this what is happening up here is something.
Abhay Vasant Ashtekar: Yeah. So already. The cases that we looked at you see, the reverse is not C two right? Because it was goes up. Okay, it's not. You know it doesn't exist,
Abhay Vasant Ashtekar: and and involved in this across the boundary.
Abhay Vasant Ashtekar: Hmm. This is maybe we should discuss more, because this is related to my previous question about the the complex. I think I I I also already discuss it. But I think it just uh
Abhay Vasant Ashtekar: It's It's a lot more. It's longer discussion. Yeah,
Abhay Vasant Ashtekar: the two of us. Can I assume a lot of things? But if I cannot,
Yeah,
Muxin Han: um, I have a question. Um. So it seemed to me that the the calculation. Um depend on the choice of extending the space time from from a positive time to negative time. So it is. Am I right?
Muxin Han: So it's. So basically you choose that? Um the they depend on it, by this absolute value we put absolutely on either
Abhay Vasant Ashtekar: the fraction of that.
Abhay Vasant Ashtekar: If you beta is not fractional. Then there is no problem.
Muxin Han: Because,
Muxin Han: yeah, I have. I have a question because I'm: we know you. You see, they are um different scenarios uh for bounce, so they are symmetric bounds. So I mean, It's so in in the limit that um
Muxin Han: the uh Delta, where where the area gap goes to zero, is, it's like With un symmetric bound you connect F. Rw. To to what the scissors space the spacetime. So then, um i'm wondering whether I and then seems to me that these two different scenarios it might give a different um
Muxin Han: five of distributions for for one field.
Abhay Vasant Ashtekar: Yeah, yeah, it's two of them. Okay,
Abhay Vasant Ashtekar: um!
Muxin Han: Are you continuing so that the metric is continuous there and there as a tensor field, it's continuous there or not, it's the method. It's not continuous. Then I think we have to think a lot, but it is about in in case of the bones in in case of what a non-zero delta non-zero um area gap.
Muxin Han: Then then the easy metric is continuous because it is a bounce.
Abhay Vasant Ashtekar: We can just take the test case and evolve that there's no problem at all. In that case, I mean, there's a problem of finding which state to use. But apart from that, there's no problem of you worrying at all.
Abhay Vasant Ashtekar: I think I think the the the question is in the limit.
Muxin Han: Yeah, in the limit that i'm the whole. I'm contacting the the The Lqc result to the funnel field theory in that space time result in the link that we're You You um. You remove this every gap. Yeah. So what The My question is in the limit. Is the metric going to be continuous or not?
Muxin Han: I would assume it still continues, maybe even knows better.
Ivan Agullo: And um not on the top of my head. But but but are you are you, eh, eh, eh? Asking Why? What is the justification for a bite to choose a symmetric metric? Because we have examples in Lc. Where the metric is not symmetric, and uh across the would be singularity.
Muxin Han: Um, Well, partly yeah, that's one of the question. Maybe another question is, how upon and field here it may, may, may change. The distribution, may change the behavior in case a different choice of the continuation of space time.
Abhay Vasant Ashtekar: Um! So I mean the State is that one had to use some continuation. So this was the continuation I I mean, I gave the motivation. But
Abhay Vasant Ashtekar: there's there's something simple you can do. But more than that, If I just use Einstein's question the connection variables I mean, generalize. I's a question kind of connection variables which Don't break down with the metric becomes non-d generated.
Abhay Vasant Ashtekar: If the metric is continuous, then I would imagine qualitatively, the behavior will be similar.
Abhay Vasant Ashtekar: But uh, I mean
Abhay Vasant Ashtekar: the precise
Abhay Vasant Ashtekar: I mean, yeah, the precise mode functions may well have
Abhay Vasant Ashtekar: um different kind of uh behavior at it. I call zero right, and they will be single, not like one up on each type of the end, but as some other power.
Abhay Vasant Ashtekar: Okay, so that's that's that's good possible. So. No, I think I will. W. One has to look at in case my case I don't, I mean, here is a broad class of cases, and this. What happens in this case was motivated because of the evolution to
Abhay Vasant Ashtekar: connection dynamics of ordinary Einstein's equations uh, but we can take other cases, and then we'll have to look at it case by case. So my statement is that if the if the field is continuous, I will be able to evolve. But, On the other hand,
Abhay Vasant Ashtekar: the details will depend on how I do it.
Muxin Han: Okay. So so you're okay. Your feeling is that it? It might be the the behavior of the and the behavior of the distribution. For example, the the precise nature of the distribution will be different. I mean it could be, I mean, from Thomas's remark. It seems that you know we'll almost always get distribution, you know, along the lines that he was suggesting.
Abhay Vasant Ashtekar: But I think
Abhay Vasant Ashtekar: you good to
Abhay Vasant Ashtekar: for me to understand that
I see. Thank you.