Jorge Pullin: Okay, so our speaker today is San Shander, who will revive quantum geometer dynamics.
Susanne Schander: Yeah, thank you very much
Susanne Schander: for this introduction. And first of all. I would like to thank the organizers for this invitation and the opportunity to present our work here. This is a joint project with Charleston lang and is based on these publications here.
Susanne Schander: Yeah. So why do we want to revive quantum geometry, dynamics? We think there are a number of reasons for doing that, and to motivate a bit our approach.
Susanne Schander: Let me start by
Susanne Schander: oops. Going back and asking what is actually quantum geometrodynamics.
Susanne Schander: So this was actually the earliest approach of quantizing Gr very closely related to Einstein's formulation, with metric variables.
Susanne Schander: and most prominently promoted by David and Wheeler in the sixties and seventies.
Susanne Schander: So the theory starts from a classical Hamiltonian perspective. So you would perform an adm split
Susanne Schander: and find that your canonical variables are the special metric and their conjugate momenta
Susanne Schander: and then performing a Dirac analysis, you'll find that you get a first class systems of Hamiltonian and diffiomorphism constraints as they are written down here, so your theory is fully constrained.
Susanne Schander: And then the idea by Wheeler and Dewitt was to take this classical Hamiltonian field theory and to quantize it.
Susanne Schander: In a straightforward manner. But yeah, so the idea is to just take the metric, the special metric operator as a multiplication operator
Susanne Schander: and their conjugate momenta as a derivative operator.
Susanne Schander: and then, following Dirac's approach to quantizing constraint system, you would
Susanne Schander: request that the constraints, the set of constraints vanishes on a set of appropriate physical states. Psi.
Susanne Schander: but of course, that's not so easy. And that's where, like the problem event of the this program eventually start. So how can you even make sense of the nonlinear functions that appear in the Hamiltonian constraint? For example, if you want to pass to to a quantum formalism.
Susanne Schander: So you have ill-defined expressions. And you cannot even make sense of
Susanne Schander: these quantum objects.
Susanne Schander: Another question is, like, where would these possible quantum operators act on so? You cannot even build Hilbert space with enough physical States to make sense of a theory like of an associated theory of gravity. But, on the other hand, if you put too much States, then you wouldn't be able to find an inner product, and in the end
Susanne Schander: nobody has ever come up with a suitable suggestion for a Hilbert space. And finally, this is, of course, a question of debate. But if you're coming from the classical theory and a Lorentzian setup, you would actually like to have the metric operator justice in the classical theory to remain positive, definite
Susanne Schander: in order to keep the causal structure of the theory.
Susanne Schander: And so with all these problems,
Susanne Schander: the abandonment of quantum geometrod dynamics happens. So I didn't go into more detail about other problems, such as direct consistency or the problem of time. There are great references that give
Susanne Schander: an extensive summary of all the problems. But I guess these ones are
Susanne Schander: the more important ones.
Susanne Schander: and so actually, while we think it is very sad that this approach was abandoned. It led to the birth of yes, please.
Jerzy Lewandowski: I think that at some point it was not pointed out by that. Actually, we can consider global polymerization of
Jerzy Lewandowski: of those variables just by taking exponents to I times Q or something like this. And then we can obtain huge keyboard space very non separable. But
Jerzy Lewandowski: that's that's makes sense. However, it was not looking as so sophisticated as we didn't try, I think.
Susanne Schander: okay, yeah, thank you very much for this comment. Yeah, I'd be interested to go back to this reference.
Susanne Schander: yeah. But II guess our approach might be a bit different. So yeah, as you will see in the coming slides, we actually have a a procedure that we propose to make progress.
Susanne Schander: So yeah. But thanks for this comment. okay, so yeah. As I was mentioning, there were lots of very appealing other approaches that were invented
Susanne Schander: also probably due to the failure of quantum geometrodynamics.
Susanne Schander: and hundreds of people were working on developing these approaches.
Susanne Schander: And it's, of course, hard to like. Of course all these approaches have like very distinct differences, but at the same time, in a very naive way, you could say that these approaches take Gr.
Susanne Schander: Perform a reformulation of the theory, and then adopt some kind of lattice regularization to gain non-perturbative control over the UV. Divergences, for example, that appear due to this ill-defined multiplication of operators.
Susanne Schander: but it actually appears that a lattice regularization in the original adm variables has never been tried.
Susanne Schander: and that's actually our motivation to go back to this original quantum geometry, dynamics approach and
Susanne Schander: try to make progress by adopting a lattice regularization in an yeah.
Susanne Schander: And this brings me now to the overview, so I guess my motivation should be clear now. We then propose in Section 2 some solutions as I was mentioning. I first go into details about
Susanne Schander: this regularization scheme and then present a method for how to realize positive, definite metrics in the quantum theory.
Susanne Schander: In section 3. I will briefly talk about how to represent gauge transformations in this quantum theory, at least for the diffiom constraints.
Susanne Schander: And then I'll shortly touch upon the continuum limit in Section 4,
Susanne Schander: and yeah. So of course, feel free to ask questions whenever you want or make any comments. We are happy for, all the feedback we can get.
Susanne Schander: Okay? So then, let's get into the solutions. So we start with the regularization scheme.
Susanne Schander: and just to outline the general idea that we have.
So we would start with an Ir regularization. We actually use a torus
Susanne Schander: to second one piece.
Susanne Schander: I are divergencies. for the special manifold. And then, on the other hand, we also
Susanne Schander: implement a UV regularization by replacing derivatives, by finite differences in the constraints, and also by restricting the phase space of classical geometrodynamics. To piecewise constants fields on these cubic lattices. But I'll show this more in detail.
Susanne Schander: and in to implement this regularization. We would, of course. implement these ideas into the constraints. then compute the constraint algebra, and see what modifications we get.
Susanne Schander: then quantize the letters theory and study the continuum limit.
Susanne Schander: So here everything is in 2 dimensions, just for illustrative purposes. But you can actually everything I'm telling you. You can do it in any dimension you want. So this is just really for making things easier to to picture.
Susanne Schander: So let us consider here just a regular special lattice. So we are actually regularizing the special degrees of freedom, not the time variable.
Susanne Schander: And so we introduce a letter spacing called epsilon.
Susanne Schander: and we require the fields. So, for example, the special metric
Susanne Schander: to
Susanne Schander: be constant over these Hypercubes.
Susanne Schander: and we do the same for the momentum.
Susanne Schander: So we end up with a finite number of decrease of freedom, because one queue has only The associated number of degrees of freedom for the metric and the momentum.
Susanne Schander: Then the Poisson algebra has the same form as the continuum algebra. So it's just inherited from the continuum fields
Susanne Schander: and due to the regularization we get periodic boundary conditions. So that's all like, very easy. And yeah, easy to visualize. I guess because we are just, we just have this very regular special letters.
Susanne Schander: So we then considered the constraints, and what we would do is to replace the integral by Riemann sums here, so we have a weighted sum, so to say. Weighted by the lattice constant, and here we sum over all possible lattice points. And here we have XY, because we are just in 2 dimensions.
Susanne Schander: And so actually, all these quantities here depend on x and y, it's just to make the notation easier that I've put it outside the brackets here.
Susanne Schander: But this is yeah. Just taking the standard continuum constraints and putting them on a letter, so to say, and of course we are replacing
Susanne Schander: the derivatives by finite differences.
Susanne Schander: and I have to mention that this includes a choice, because the chain rule for the finite differences has an extra term compared to the
Susanne Schander: chain rule for the in the continuum. So
Susanne Schander: you would get additional terms that are proportional to Epsilon, but we chose to represent the constraints in the most easiest way possible, and so we we made this choice here.
Susanne Schander: We then continued to compute the constraint. Algebra, and what we get looks as the following, so in blue, here we just recover something that looks like the continuum algebra. But of course we get additional terms here, the anomalies.
Susanne Schander: because we break the general covariance by putting the theory on the lattice. So the first class property is broken. We get a second class algebra which also leads to unphysical degrees of freedom. So if you're propagating your degrees of freedom. With respect to the constraints, you would quickly move out of the constraint surface.
Susanne Schander: But actually, since these terms here are proportional to Epsilon, and as long as you have
Susanne Schander: smearing fields, n and m. that tend to well-defined continuum fields.
Susanne Schander: these expressions are also finite. So, taking the limit, epsilon to 0 would actually lead to these additional terms going to 0.
Susanne Schander: And this is an important point, as, yeah, as you well see.
Susanne Schander: And yeah, here's the hint for the continuum limit. So you can actually, you can always tune the continuum limit in such a way that for long time evolutions.
Susanne Schander: You have to ensure that the lattices get finer, even quicker, so that you would
Susanne Schander: suppress these unphysical decrease of freedom and the anomalies. And that's the idea for the continuum limit later on.
Susanne Schander: Okay, so yeah, this was actually the introduction to the lattice theory. I'll come back to that a bit later, when we talk about representing the gauge transformations. But for now this is just the the general setup for regularizing the theory.
Susanne Schander: So then, in a next step before considering the continuum limit, we would want to quantize this lattice theory.
Susanne Schander: and we found a very intuitive way to do that. So first let me come back to the standard Schrodinger representation.
Susanne Schander: You would if you follow the shooting approach, you would just represent the metric as a multiplication operator and the momentum operator as the derivative operator, and then let them act on on a simple l. 2 Hilbert space. Note that here. This is possible because we are on the lattice. So we really have a finite number of degrees of freedom.
Susanne Schander: And of course we can satisfy the standard commutation relations. No problem here. That we would inherit from the continuum theory again.
Susanne Schander: But States can have support on non positive, definite metrics, and the causal structure would be lost.
Susanne Schander: So you can understand this in a way that so given, you have a state that is, has only a support on positive, definite metrics.
Susanne Schander: Then you act with the momentum operator on it, or with the exponent the exponent ponient version.
Susanne Schander: and you could be kicked out of this region, and also get state, a state that has support on non positive, definite metrics.
Susanne Schander: And so our idea to solve this issue. is to introduce a new representation that actually ensures positive definiteness. On the one hand.
Susanne Schander: So there were other approaches before, like implemented an affine gravity, for example, that are able to ensure positive definiteness, but they actually come at the expense of
Susanne Schander: introducing non-canonical commutation relations.
Susanne Schander: And so our approach is actually able to keep the standard commutation relations.
Susanne Schander: The basic idea is to represent the metric Q, which is positive, definite
Susanne Schander: as a product of an upper triangular matrix u, that has positive diagonal elements.
Susanne Schander: So u transpose times u. and this decomposition is actually unique.
Susanne Schander: And these u matrices belong to the group UT plus here. So the group of upper triangular matrices with positive diagonal elements, and they formally group. And this comes in very handy because we can then use the leak group properties to easily build a Hilbert space.
Susanne Schander: So we built the Hilbert space on this group and then used the left hand measure row of U. To define the measure for this space.
Susanne Schander: We can then represent the metric queue on this new Hilbert space.
Susanne Schander: In 2 dimensions. It's very easy. So this is just a relation on the slide I showed before, but written down explicitly. and this manifestly realizes the positive definiteness of the special metric. But now the important question is how to represent the momentum operator.
Susanne Schander: and our approach is not to first ask about the momentum operator, but first to define generators of shifts in positive Q direction.
Susanne Schander: So we were actually able to find such generators. U. Of S. That shift the quantum metric by a certain positive number. S.
Susanne Schander: And this looks as follows.
Susanne Schander: so just to explain how it comes to this formula, so we actually shift the state psi, like you would also do for In the standard Schrodinger representation. So Gs is just the shift in use space.
Susanne Schander: But then we need an additional prefactor here to get certain properties for these use, namely, we can show that the the group of these transformations is actually a contraction semigroup
Susanne Schander: and is strongly continuous. And this is very important to define the momentum operator
Susanne Schander: because, using so if you have strongly continue.
Susanne Schander: Oh, sorry row is again the ha! Measure. I'm I'm sorry if I got over this too quickly. So row is here. The how measure that is the measure associated with our Hilbert space?
AAipad2022: Thank you.
Susanne Schander: Yeah. Thanks for the question.
Susanne Schander: and J, is the Jacobian matrix of this map? U,
Susanne Schander: yeah. So thanks for asking. I wasn't very explicit about this formula. So yeah, Jacobian matrix. And this is the the how measure?
Susanne Schander: Okay. So since we have this we can now define the momentum operator. Given our transformations view.
Susanne Schander: Oh, I'm sorry.
Susanne Schander: by using the so it states that as long as you have the the strong continuous contraction semi-group, you can actually define an infinitesimal generator. With the following form.
Susanne Schander: And this gives our momentum operator. And just for illustrative purposes, I wrote down the concrete formula for 2 dimensions.
Susanne Schander: And we also did the same for 3 dimensions. You can do it for whatever dimension you would like to have.
Susanne Schander: And it's a straightforward procedure just using this formula.
Susanne Schander: and what we
Susanne Schander: succeed. To do with this procedure is to satisfy the standard commutation relations on the lattice, of course
Susanne Schander: and still having a positive, definite metric in the sense that this operator here is a positive operator for any S.
Susanne Schander: And I'm sorry I actually forgot a one over Epsilon squared here. So this should be it. I'm sorry.
Susanne Schander: Yeah. So this is our idea of how to implement a positive, definite metric in the quantum theory.
Susanne Schander: Okay, so
Susanne Schander: this kind of closes the more introductory part in the sense that we now have the the lattice regularization on the one hand, and we are able to define a quantum theory on any lattice.
Susanne Schander: That also has a positive, definite metric operator.
Susanne Schander: So then, let me come to the question of how to represent gauge transformations in this theory.
Susanne Schander: It looks like you may have a question about oh, I'm sorry I didn't see that. Please feel free. Maybe that
AAipad2022: you had.
AAipad2022: I mean, morally, you know, you is like the is like a diet. Right? It's because square of U is a metric.
AAipad2022: And so, you know you use like a
AAipad2022: like in that sense. So in 3 dimensions it would be like a trial.
AAipad2022: Somehow you so you you did not. But for some reason you sort of want to
AAipad2022: think of Q as the basic object and not you. You could have taken you as a basic object, and constructing this canonical conjugate momentum, which will be very straight, you know.
AAipad2022: Very nice, because they both. And then you could have thought of Q and P as being composite of objects.
AAipad2022: Why is it that I mean, that would be more natural to me?
And I was wondering if there's a reason why you didn't want to do that.
Susanne Schander: Yeah. So I'd say, the main reason is that we wanted to stay close to the original approach by Wheeler and the wit.
Susanne Schander: which is not to say that this is like something we adhere to. In a strict manner. It's just our way to starting this program and staying as close as possible to the original Wheeler divid approach.
Susanne Schander: But it is actually possible to to also implement triad fields, which are, of course, necessary to couple fermions.
Susanne Schander: So I can actually come back to that later. But so thank you for this question. So yeah. So I mean.
Susanne Schander: II guess you're asking this because of, yeah, it's actually also natural to go to the try it feel representation. And we have this in mind. But yeah, just to briefly answer your question your question. It's more like for the time being, we wanted to stay close to the original approach.
AAipad2022: Yeah, so just the comment then, and this is continuation of what you like was saying. So there's a paper I wrote in 2,009 in honor of your mailers who decide
AAipad2022: about that. If you mob with zoom invariance and you know and everything. And then I there's a explicit result there that if you use the Heiser mug algebra in geometry, dynamics, as you're doing.
AAipad2022: then there would not be, maybe few moves of any value in state on it.
AAipad2022: So that's really that is really a quite important of obstacle in the continue.
AAipad2022: So you may want to, you know. Think that because you might do a lot of work. And then there's this obstacle that you might come across. So
AAipad2022: there is no yeah, there's no state contrary to the common intuition. So okay, I just wanted to say that and that's why the clients
AAipad2022: and I'd rather you representation might be you. And this cannot be going to get momentum might be
Susanne Schander: something that might be. You might want to consider.
Susanne Schander: okay, so yeah.
Susanne Schander: with this, let me come back to the representation of gauge transformations. And so here I'll actually restrict to theories whose constraints form lie algebra.
Susanne Schander: So this would be the case for the subgroup of the diffumorphism, constraints, and gravity. But we make it even simpler here, and for illustrative purposes we just consider Scalar field theory. But of course, we we always have in mind that
Susanne Schander: these results also apply to the diffiomorphism constraint. We would just need to extend
Susanne Schander: the one degree of freedom to the higher number of degrees of freedom in for the metric field, in whatever dimension you would like to consider.
Susanne Schander: So just as a recap. So we can consider here the classical continuum theory. The general form of the constraint would look as follows.
Susanne Schander: so this is actually just a recap of what we have seen before, for the diplomatism constraints.
Susanne Schander: We assume that. Oh, yeah. And so Phi is the scalar field, and pi is conjugate momentum. And this is the constraint of the theory or the constraints.
Susanne Schander: And then we can compute that these constraints form a first class Poisson algebra in the following sense.
Susanne Schander: And yeah, this is just what we would also get for the the form ofism constraints.
Susanne Schander: So then, in the next step, we actually proceed in the same way as we have already done for the case of gravity. So we use a lattice discretization for the Phi and the Pi fields.
Susanne Schander: such that they are constant over cubes of Hypercubes in the respective dimension.
Susanne Schander: and we can represent them with the characteristic function.
Susanne Schander: And then in the second step, we can represent the letters constraints in the following way. So again, we would replace integrals by weighted sums. So here's the sum, and there's the the weight factor.
Susanne Schander: and then replace derivatives by finite differences. Delta and yeah. So just the same as we did before.
Susanne Schander: And similarly, we also assume that the algebra on the lattice has
Susanne Schander: a similar form, as in the continuum. But we get extra terms.
Susanne Schander: That are proportional to epsilon.
Susanne Schander: and again, we are assuming that both FN so FN would go to the to F in the continuum, and GN would go to some bounded function such that when we send epsilon to 0, so we take the limit of final lattices.
Susanne Schander: This entire term goes to 0 as well. And this is what we observe for the diffomorphism, constraints, and even the Hamiltonian constraints
Susanne Schander: in the case of gravity.
Susanne Schander: Okay? And then, in the next step, we can solve Hamilton's equations of motion on the lattice.
Susanne Schander: So we evolve Phi with respect to the constraint.
Susanne Schander: And in order for this flow the Hamiltonian flow. Phi, s. To have a solution. It is important that
Susanne Schander: the constraints are only linear in the momentum.
Susanne Schander: So this is one important assumption for this part of the talk. So that's why we are only considering the different constraints, because we can only incorporate a theory. For what we consider here. If the constraints are linear in the momentum.
Susanne Schander: and then again, the Hamiltonian flow can be interpreted as an approximate gauge transformation. Of course we would move out of the
Susanne Schander: The constraint surface with longer time intervals.
Susanne Schander: but by making the lattices smaller and smaller you can actually counteract
Susanne Schander: and ensure that the flow remains very close to the constraint surface.
Susanne Schander: Okay? And then, as before, we would go to the quantum theory and define approximate gauge transformations on the letters.
Susanne Schander: The procedure is just the very same as before. So
Susanne Schander: we have a state. Psi n of our letter syllab space. We can
Susanne Schander: move it, so to say, with the Hamiltonian flow just as before. So we take the Phi and evolve it along the Hamiltonian flow, and this is actually the part where it is very important to only have the constraint being linear in the momentum, because if it was not the States would also depend.
Susanne Schander: On the momenta or sorry. We couldn't define the flow, only depending on the configuration variables. And so this whole procedure actually wouldn't work out in this case.
Susanne Schander: so this is the part where we implement the transformation that we want to see.
Susanne Schander: And this is again this factor, including the Jacobian determinant that gives us the right property. Of these transfer gauge transformations. U,
Susanne Schander: and, in fact, here we can show that these U form a unitary one parameter group.
Susanne Schander: in fact. It is
Susanne Schander: now unitary, because we are not restricting to positive, definite metrics. So here we don't have the contraction semigroup properties, but indeed a unitary one-parameter group.
Susanne Schander: and similar to the Hilausita theorem for the contraction semigroup case.
Susanne Schander: We would get the we would have a generator, and now given by the stone theory. And yeah, let me just mention that also Thomas Teamon was using a similar approach for
Susanne Schander: for other.
Susanne Schander: Purposes. But yeah, the idea is quite similar.
Susanne Schander: Oh, I'm sorry. So again, we can compute the the generator here and get
Susanne Schander: The the constraint as a quantum operator for the lattice theory.
Susanne Schander: Okay, so that actually closes the part on the representation of gauge transformations. This is, of course, only for the letters theory.
Susanne Schander: but I guess it's quite remarkable that it's possible to represent the gauge transformations and the constraints in this way on the lattice.
Susanne Schander: in and in a strongly continuous way.
Susanne Schander: Okay, so with this, let me come to the continuum limit.
Susanne Schander: This is more like an outline. There are still things that need to be proven in a rigorous way, but I just would like to outline the general idea of how we intend to take the continuum limit here.
Susanne Schander: So first of all, we can define a while. Algebra a c star, while algebra of
Susanne Schander: of the exponential exponentiated canonical variables just by taking the span and the completion of the while elements here
Susanne Schander: and then taking the inverse limits of these letters while algebras
Susanne Schander: and to take this inverse limit we need kind of an id identification between the different lattices.
Susanne Schander: and this is given by the following identification, so Phi hat n plus one and then labeled by some, some k
Susanne Schander: would be the quantum operators
Susanne Schander: for the files on the N plus first letters, and we identify them with the files on the n-th letters. Although this formula looks a bit complicated, it's actually just a center of mass identification. So if you would.
Susanne Schander: divide this entire formula by this factor here, you would see that this is just the center of mass formula kind of.
Susanne Schander: okay, so this is the identification to get the inverse limit that we propose. And then in the next step, we would choose a sequence of states on every lattice.
Susanne Schander: Again. This is very easy. Here we just have an L. 2 Hilbert space on the 5 variables.
Susanne Schander: and with this state we can compute the inner product of the often of a while element. and compute or define. An algebraic state associated with the psi N States.
Susanne Schander: And then, in the next step, we actually suggest to take the continuum limit in the following way.
Susanne Schander: So we we say that Omega, which is now the continuum state of
Susanne Schander: the inverse limit of this, while element
Susanne Schander: is defined as
Susanne Schander: the limit of this cauchy sequence, or I mean it, it is still, it still need to be shown that this is, in fact, the cauchy sequence. But this term here you can actually compute it just by taking the definition here with the States that we defined before.
Susanne Schander: So the idea is to show that this object here defines the Koshi sequence.
Susanne Schander: For so a cauchy sequence of algebraic states.
Susanne Schander: And then this would give us actually a vacuum state omega in the continuum Hilbert space, and then we can use the gns construction to obtain the entire continuum Hilbert space.
Susanne Schander: So this is the broad. Why do you call it vacuum? Because it is
AAipad2022: I mean size, random state. No, it's not a preferred state
AAipad2022: the science where some random some states in the
AAipad2022: Nth lag is right. I mean?
AAipad2022: what is preferred about science? I
AAipad2022: I mean, it's a vacuum only in the mathematical sense of gns construction. Or is it the
like?
AAipad2022: Is is your science and Omega supposed supposed to satisfy constraints.
Susanne Schander: yeah, in the sense that. so of course, we want to have the Cochi sequence property. So, so we would tune them in a way to get the kushi sequence.
Susanne Schander: On the other hand, no, I mean the the constraint which is the
AAipad2022: the, the physical constraints to the city, like the different models of constraint, Hamiltonian constraint, and so on. Are the science satisfying any constraints?
Susanne Schander: No, except for I mean that you would choose a state that satisfies the properties you have to. You want to have at the end
Susanne Schander: you. You are basically very free to choose these dates, and that's actually, I would say, a strong point of this approach, because
Susanne Schander: you don't have to make any assumptions on the States, but you're really free to choose them. And this is the gives you the opportunity to actually find a continuum state.
Susanne Schander: Because you have this freedom to, to tune this trajectory, and to choose this state quite freely.
AAipad2022: But will the continuum state satisfy that if you margin constraint, for example, that you had in the last that last section.
Susanne Schander: Yeah. So we haven't computed this for now we only have the computations for this scalar field here.
Susanne Schander: But as I said, the deform office, I mean, doing the same for gravity is, yeah, it's actually so here, this is just. I'm sorry if I went over this to quickly. So I was so this is still for the the scale of you theory that I was introducing for section 3.
Susanne Schander: So just to make it simpler at the beginning, we actually restricted our computations to a simple scale of field theory.
Susanne Schander: So Phi and pi would be the canonical variables.
Susanne Schander: But of course we ha! We always have in mind the different constraints. And so we actually asked the Scalar field theory to have similar properties. To what? A theory of like the different constraints would have. But we haven't done actual as in part 3.
AAipad2022: And then the diffumologism constraint on the scale of field theory also
AAipad2022: and you are looking at representations
AAipad2022: off of that that would be good.
Thorsten Lang: I think my understanding of the questions is that you ask whether we are the
Thorsten Lang: trying to look for the physical or the the kinematical Hilbert space. Right. This is only for the and then in the second step, we want to represent the the group of gauge transformations
Thorsten Lang: on on it. And then so, of course, that for the
AAipad2022: so it's kinematical, and therefore
AAipad2022: your state Omega does not satisfy any constraints.
Susanne Schander: Yeah, yeah, I'm I'm sorry I didn't get the question, thanks for the the clarification. Great thanks.
Susanne Schander: Okay. Yeah. So as I said, this is just like, rather associated with this sub group of different constraints. We don't make any statements here for the the Hamilton constraint, Hamiltonian constraint, and how to
Susanne Schander: consider its continuum limit because, we have the problem that it is quadratic in the momenta. So this will necessarily include other methods. So for example, we proposed
Susanne Schander: a method for generalized, while quantization scheme which we could use for the Hamiltonian constraint on the lattice. And then we certainly need renormalization group methods in order to define the continuum limit.
Susanne Schander: Alright. So
Susanne Schander: this actually concludes, or, yeah, let me conclude with a summary and an outlook.
Susanne Schander: So at the beginning I have introduced our let us regulation
Susanne Schander: of quantum geometry, dynamics.
Susanne Schander: with the motivation that in the original approach a lattice regularization has never been introduced.
Susanne Schander: Then, for this lattice theory we were able to find non standard representation of the canonical commutation relations.
Susanne Schander: With inherently positive definite metric. So as I said, there had been proposals by, for example, Aisham, Kakas, and Clouder, but they all had non canonical canonical communication relations. And with this new representation that's actually possible to realize.
Susanne Schander: In section 3, I've shown a way to represent the approximate lattice gauge transformations on the lattice especially in the in the quantum theory.
Susanne Schander: And then so the last slide was to show that there is a procedure to
Susanne Schander: Consider the continuum limit of this like the theory of a certain subgroup and gravity. So the the deformorphism constraints
Susanne Schander: and their Li algebra.
Susanne Schander: Yeah. So this is what we've achieved so far, and we are looking forward to continue our program. So of course, we need to explore. The converging sequences of the letters, theories. The the letter states, in order to be able to define a faithful
Susanne Schander: continuum theory.
Susanne Schander: and of course, also the the limit of the approximate age transformations. And our goal is to find a strongly continuous representation of the diffumorphism group.
Susanne Schander: As I was mentioning, like the the proposals I showed in slides in section 3 and 4 are only applicable to gravity for the case of the different morphism constraints.
Susanne Schander: But we also have
Susanne Schander: found a generalized well quantization to represent the lattice Hamiltonian constraints. And of course, the ultimate goal is to study the continuum limit of the Hamiltonian constraint.
Susanne Schander: And yeah, so thank you very much for your for your attention and your interest. We are happy about comments and feedback, and please feel free to reach out to us also after this talk.
Susanne Schander: So yeah, thank you very much.
Hal Haggard: Thank you, Suzanne. Thank you for the presentation. Yurik. I believe your hand was up first. Please go ahead.
Hal Haggard: Eric, are you? Did you have a question?
Jerzy Lewandowski: Yes, sorry I was muted. I must have slept through the moment when you introduced those approximates
Jerzy Lewandowski: gauge transformation. So can you explain again what? Why they are approximating?
Susanne Schander: Oh, this is because they are still on the letters. So we didn't take, you know. So we first regularized the theory, then quantize it and then take the continuum limit and all. What I said in section 3 was for the letters theory. So this is why they are approximate.
Jerzy Lewandowski: I see. So they don't have any continuum limit.
Susanne Schander: Not yet. So in section 4, I was referring to the continuum limit. But we haven't done the explicit like.
Susanne Schander: I mean, there we were, referring to the scalar field a simple scalar field
Susanne Schander: and I mean, as I was mentioning, you can
Susanne Schander: apply these techniques to the to gravity, at least to the different myth. Different morphism, constraints. But the actual computations here are for the scale of field, but of course we always had in mind that we want to use it to consider the different morphism constraints in
Susanne Schander: in quantum geometron dynamics.
Jerzy Lewandowski: I see. Okay, thank you.
Hal Haggard: Bye. And then, Lee.
AAipad2022: yeah, I have a comment and a question. So
AAipad2022: the comment is, really that?
AAipad2022: I mean, I think the way one way to say it is that you're the part 4, you introducing Kilimani grammatical, something about space, and and then
AAipad2022: one would be able to say that one would be able to hopefully write down the continuum an hour of the constraint. The question that Yurik was just asking in this kinematical space, and then, you know, solve it. And such thing.
AAipad2022: The the main problem was that
AAipad2022: the main problem was really with this came up. And so, as you accept one, you know, one can do the exactly the same thing with just the
AAipad2022: geometry, dynamical variables in the full theory, not using lattices just in the continuum but then the main main problem came that the Hamiltonian constraint is very difficult in the in the connection variable, you know we could import. Let
AAipad2022: radius
methods from
AAipad2022: context, whereas we could not make any.
AAipad2022: At least I could not make much progress with the Hamiltonian constraint. I think some progress at one stage. So I think that there is a big issue here
AAipad2022: about, you know what would happen to continue, that you might
AAipad2022: ultimately find some nice kinematics. But then it will be maybe difficult to form with the Hamiltonian on that.
AAipad2022: So just just to keep that in mind. The question is.
but there is a real difference between scale of Field and them.
AAipad2022: the metrics and the in the scalar field. You know. You can just divide this space in 3 dimensions into this
AAipad2022: into the squares, and then the scalar field is constant in each of the squares, and that is the point. Use of freedom.
AAipad2022: But
AAipad2022: if it is any tensor field, I don't know what that means to say. Tensor field is constant. If I just drew up
as a lattice on the 94,
AAipad2022: I said that I consider constant. It doesn't make any sense unless I specify some coordinate system.
AAipad2022: and so your continuum limit will be then very, very terrible to certain coordinate system in which things were constant.
AAipad2022: Now, what does not do that in latest gauge series. I just want to emphasize that is gauge series. You know, one is always integrating
AAipad2022: the connection along
AAipad2022: along the lines, and that is coordinating value.
AAipad2022: Integral doesn't depend on coordinates because I'm integrating one form or on 1 one index. And similarly, the spinner fields are just sitting at the vertices.
AAipad2022: There's no problem with coordinating guys. But I I'm very disturbed by keeping tens of fields constant on some little little patch here, a little patch here, because I don't know what that means. So I just wanted to ask you what you meant by that.
Susanne Schander: Yeah. So of course, you would need to choose a chart so I guess
Susanne Schander: you have to be careful with that. That's what I can say to that point, and I guess so in general.
Susanne Schander: I mean, our approach is to just
Susanne Schander: or our idea was to just introduce. like a new approach that
Susanne Schander: might be able to to tackle problems in quantum gravity, and that goes back to this all formulation of Wheeler and Tibet. So
Susanne Schander: you should really see it as a as a new way of understanding.
Susanne Schander: the problem of quantum gravity.
Susanne Schander: And I totally understand that. You are worried about like these normally set up here so that coordinate invariants is broken. But we see it in a way that you actually
Susanne Schander: so it's just natural to think of these anomalies appearing in the letters theory, because, of course, you introduce a letter, so you wouldn't even expect
Susanne Schander: to have a theory that
Susanne Schander: is a coordinate, invariant.
Susanne Schander: But
Susanne Schander: for us the important point is that you can restore
Susanne Schander: the first class algebra in the continuum, and not so much that you have to keep it on the letters. I guess that's the main rationale which differs from many approaches.
Susanne Schander: Like quantum gravity, for example.
Susanne Schander: So it's just
Susanne Schander: that we are not
Susanne Schander: restricting to the case of having a theory that is coordinate, invariant on the lattice.
Susanne Schander: So we see these letters theories as a tool to get to a well-defined continuum theory, but
Susanne Schander: def like having the first class property is an important feature of the continuum limit, and not so much of the atse. So I guess that's the
Susanne Schander: the rationale. I don't know if that answers their question.
AAipad2022: Yeah, no, I understand that. Thank you. Just that, I think in the latest you are going to be choosing. It's not just a matter of
AAipad2022: explains that all it like is.
AAipad2022: but it is not just a matter of the constraints big volume that is, but really to make sense of what you mean by constant
AAipad2022: feels what is really using jocks. coordinate systems
AAipad2022: that the job dependent disappeared.
AAipad2022: So it's not clear to me that the chat dependence can just disappear and continue. I mean mother one, when he was trying to do these things and for look on gravity showing that algebra is is abnormally free
AAipad2022: had to be very, very careful, you know, if you chat that, nothing depends on chat is not because almost always chat leaves
AAipad2022: the limit. Chart is signature. Things tend to depend on it. Okay, II just wanted to point that out. It's not.
AAipad2022: okay.
Susanne Schander: maybe just a common. So if you consider the continuum limit of the algebra. Then actually, the chart dependents would
Susanne Schander: go away right? So I don't know if that plays a role, or if that's important for the continuum limit.
Susanne Schander: But yeah, I mean, we can discuss about that later as well. I just wanted to make this quick. Comment.
Lee Smolin: Okay, I'd like to introduce a very weird representation and ask you to if you can do your find your invariance on it.
Lee Smolin: So I wanna consider,
Lee Smolin: I'm on s 3 or s. 3. And I have, manifold, and on this manifold I have some loops
Lee Smolin: to have all the loops. and I define a representation
Lee Smolin: of the Diffie Morphisms in the space of loops as follows, if I
Lee Smolin: map a dipymorphism in 2,
Lee Smolin: I lap map a sorry, a loop into another loop, and I trace
Lee Smolin: that.
Lee Smolin: and I diffuse that to define a new representation, which is basically one if the loop went to itself and 0 otherwise.
Lee Smolin: do you see what I'm saying?
Susanne Schander: Yeah, I guess so. So you would have
Susanne Schander: space of loops. And then you would want to map one of the loops into another loop. Right? I want to map
Lee Smolin: the loops, the space of loops, the loops into the space of loops in such a way that the diffumorphisms are carried along.
Lee Smolin: So when I so I get a big representation stays of the roofs
Lee Smolin: on themselves, and then I act.
Lee Smolin: We could do a few more things.
Lee Smolin: and it seems to me that makes a big representation of different morphisms. That's not very easy
Lee Smolin: to to construct your construction.
Susanne Schander: Yeah. So I can't really tell what would happen if we considered loops?
Susanne Schander: because so our theory,
Susanne Schander: like. is much more simple in the sense that you only define. So you have lattice points in the theory, and you formally define your theory.
Susanne Schander: on these lattice points. So I mean, the idea of this approach is actually to to not have the loops, but to make it simpler in the sense that
Susanne Schander: you only have this regular letters and your lattice variables are defined on the on the points formally, although we have these piecewise, constant fields. But
Susanne Schander: still, if you want to consider it as a lattice theory.
Susanne Schander: then you would have only variables defined on the points. And that's actually the thing that makes it like from the conceptual point of view, or like, just imagining how the space looks like
Susanne Schander: a bit easier than considering loops. Right? So i.
Susanne Schander: i don't know, if like, so i mean the basic idea here is to to use this regular lattice
Susanne Schander: and not the loops. So I don't know, if
Lee Smolin: your question applies to our approach, or maybe I didn't get it, yeah, maybe I didn't say it clearly. The space of loops on a manifold
Lee Smolin: carries a representation. but there's a few more feasible than as well. The space of lattices on a manifold, does not
Lee Smolin: in any smooth way containing a representation of the Duchy morphisms, so as probably how it would break in the second case.
Lee Smolin: and not break in the first place, but breaks means you don't get to arrive. It's smooth.
Lee Smolin: infinite, dimensional representation.
Susanne Schander: Yes, so
Susanne Schander: okay. So we don't have the loops.
Susanne Schander: But I so the thing I
Susanne Schander: always
Susanne Schander: like. So in in loop quantum gravity.
Susanne Schander: You define your objects.
Susanne Schander: On these loops.
Susanne Schander: But then, like the representation of the I mean, you can define the representation of the diffios. but these are not strongly continuous right? And so I guess our approach comes from the perspective that
Susanne Schander: we want to have a representation that is strongly continuous.
Susanne Schander: And this is the reason why we are not using loops.
Susanne Schander: But of course, I mean, these are different approaches. And we just focused on this on this property of having a strongly continuous representation.
Susanne Schander: But of course it's at the expense that we lose general covariance on the lattice.
Susanne Schander: And this is yeah, as I was like,
Lee Smolin: you don't have a representation.
Susanne Schander: Why, so I mean we would so on the lattice we have the representation of the different constraints, or I mean, it's
Susanne Schander: very plausible to have them starting from the scalar field theory that we already have written down.
Susanne Schander: And then we take the continuum limit in the way I suggested. In Section 4 here. So this is actually the way.
Susanne Schander: We would also get the representation of the deformivism constraints
Susanne Schander: for both, like on the letters and also in the continuum.
Lee Smolin: Okay, now, I'll confuse. But let's let's cut it here. Yeah. So, Thorston, if you have a related comment, maybe keep it brief. Western has a related question.
Thorsten Lang: yeah, just wanted to comment on that. So so I think from our perspective, the ideas that the break the diffumorphism
Thorsten Lang: group on the letters.
Thorsten Lang: But this analytic way
Thorsten Lang: of how we define it gives us the opportunity to implement it in the continuum by this limiting procedure. So we want it to arise in the limit
Thorsten Lang: we were aware that it's broken on the letters, but it it gets less broken as you
Thorsten Lang: make the letters finer and finer. So because of these epsilons that go to 0,
AAipad2022: I want to intervene. This is exactly the point I was making. That may be true classically, that is not generally true quantum mechanically, because we get products of operator value distributions other in the in the terms which are
AAipad2022: which are going to go to. You want to go to 0.
AAipad2022: And that is exactly the reason why, mother, one just 1 s mother, one had to do
AAipad2022: very very careful analysis. And, mother, one does have a continuous representation of the diffum modelism group. He does have
AAipad2022: the the generators of different Zoom group in the continuum. Sorry. Go ahead.
Susanne Schander: Yeah. So I guess our point is that you? I mean, you're pretty free to tune this limit with the the anomalies we get, and with the States that you can choose on the lattice and their continuum limit.
Susanne Schander: And so as long as you have the smearing fields, m and n going to
Susanne Schander: a continuous version of it. So you have a continuous, you have continuous smearing fields. These additional terms here actually go to
Susanne Schander: a bounded function.
Susanne Schander: And if you send
Susanne Schander: epsilon to 0, you can make sure that this term also goes to 0. I mean. Of course we haven't done the computations for the quantum case, I agree. But like, just from looking at these terms, in the classical case, it
Susanne Schander: it's very plausible, or it seems very plausible that we can end up with a representation.
Susanne Schander: That's
AAipad2022: that is where all the problem comes when you try to take the continuing limit. But I think this is.
AAipad2022: it's just because we talked about it before. I wanted to make this comment. Other people have to other comments. So I should stop. Let's pause that discussion there, and we'll come back to it if if needed. That's fine
Western: approach. In order to solve
Western: all the problems you want to gravity, and I don't understand exactly what. What's the problem that you are addressing here. So I hear a lot about the continuously meant. And indeed, it seems to me that what you're doing is a very nice way to compare what happens when you use
Western: group variables with the data. And what you do, what you have instead, when you use the metric, all that. But and then all the focus is about the continuous limit is about finding conditions that needs to be satisfied with the but let me. And then we're starting with a lot of beginning
Western: nicely, saying that there are all these approaches that share having starting point. But II share with the the feeling the idea that in approaches like loop loops.
Western: And, in fact, without having to worry about all the problems that arises.
Western: The thing that I don't understand is what you're doing is that of course you lose in this framework, with the information that you get when you have the group variables, and also the fact that that's somehow, in the moment in which you have group variables is not so harmful, because in the end the space time is discrete.
Western: So can you? So maybe okay, I have horizon, my concern, my confusion. So maybe maybe there is more so. Can you tell us more about what are the problems that you have in mind? What are your goals beside? So I guess.
Susanne Schander: I mean, I'm aware of all the work that has been done in Lukeon gravity, and also that you can find a a representation of the constraints in a continuum, and that you have a Hilbert space.
Susanne Schander: But so our point of view is that we want to take like this new formulation and this new approach with like more freedom in the sense that
Susanne Schander: to define a Hilbert space that is actually separable on the one hand, and, on the other hand, the I guess the important point that drives this program is that we want to find
Susanne Schander: strongly continuous representations of the constraints.
Susanne Schander: And I know I mean I know very well that it has advantages to work with the loose variable, like the one you mentioned, and that also Lee was addressing before, and also a buy
Susanne Schander: but it seems to me that it might be like our approach focuses on the fact to start.
Susanne Schander: even like, even though we have to break general covariance on the lattice. This is just a tool to be able to fine tune the continuum limit in such a way that we get, on the one hand, the separable Hilbert space and the continuous, like a strongly continuous representation of the constraints in the continuum. So I guess this is our main motivation for this approach.
Susanne Schander: And of course it differs from what has been done in, for example, in lukewarm gravity.
Susanne Schander: and yeah, so maybe another point is also that our representation is less singular.
Susanne Schander: So here we have so, as I was mentioning at the beginning, let me just go back to the slide. So actually, our fields are defined everywhere in space-time.
Susanne Schander: Whereas in loop quantum gravity you would have the fields defined on these links or loops.
Susanne Schander: And this was also one motivation.
Susanne Schander: so
Susanne Schander: the way I view it is that this kind of more singular regularization leads to the fact that you get a non strongly continuous representation of the constraints.
Susanne Schander: And that's exactly where we wanted to suggest a new formulation where you have
Susanne Schander: less singular definitions of your basic variables and smearing, and whatsoever.
Susanne Schander: And from this perspective, trying to find a new representation that is strongly continuous.
Susanne Schander: I don't know if that helps, but I guess I mean, you were asking what what problems we want to address. And these are the 3 ones, I guess, that are important for us. So we want to have
Susanne Schander: non-singular regularizations.
Susanne Schander: a separable Hilbert space, and strongly continuous representation of the constraints. And this goes, of course, at the expense of breaking general covariance on the letters.
Western: Thank you, Susan. Of course. I understand better, even though I still have a a a a. So somehow a a difficulty in appreciate what you're doing, maybe. And I think that did. Difficulty comes from a A and not your quantity. So
Western: some space time, but actually to to start will be the.
Western: So I think that's what makes for me difficult to do. Yeah, I totally understand. Like, from the loops perspective, it's maybe harder to crash.
Susanne Schander: Yeah. So II see. I see these comments and I understand them. But so
Susanne Schander: maybe it's yeah. It's best to see it as a new approach to
Susanne Schander: to that maybe comes at the expense of certain features that have been implemented in the quantum gravity. But, on the other hand, we might be able to to get certain features of a continuous quantum theory that are certainly desirable. In the end. So yeah.
Susanne Schander: thanks for the question.
Susanne Schander: Europe's been waiting patiently. You're sorry.
Jerzy Lewandowski: So the discussion is very interesting, so I don't have to.
Jerzy Lewandowski: You use all my patients, and perhaps you discuss this already. But
Jerzy Lewandowski: how do you
Jerzy Lewandowski: manage the coordinate transformation? Is this lattice related to some coordinates, or are coordinates variants independent on the lattice?
Susanne Schander: So so the latest is on the chart.
Susanne Schander: so you would have coordinates and but so
Susanne Schander: you have a chart, and your lattice is on this chart, so to say. But for now we are working with one chart only.
Jerzy Lewandowski: and the second question is little, also little technical. So did those
Jerzy Lewandowski: characteristic functions of some
Jerzy Lewandowski: sets. Actually they are not so nice from the distribution of point of view.
Jerzy Lewandowski: So you have to. You multiply distributions by step functions. So we know that usually it
Jerzy Lewandowski: leads to some ambiguities, and in particular, in
Jerzy Lewandowski: yeah, in luo quantum gravity. We also use those flux flexes, but
Jerzy Lewandowski: I usually prefer to consider some smearing function which has compact support. And then II feel saved. And they
Jerzy Lewandowski: I don't have any boundaries, or any
Jerzy Lewandowski: issues with when I multiply distribution by by some discontinuous function. So do you think that in your case it's not
Jerzy Lewandowski: yeah.
Susanne Schander: So
Susanne Schander: maybe. So, my perspective on this is that
Susanne Schander: we actually do nothing than integration theory. So we actually take the continuous theory and then make it discrete and
Susanne Schander: represented in the way that that one would also consider in you, if you like. Take integration theory and want to consider its continuum limit. So it's just like
Susanne Schander: like, if you want to go into like one dimension. You would just consider these little queues and have constant values over certain intervals.
Susanne Schander: So that's why I think them. Yeah, it's well motivated in this sense. On the other hand, it is, of course, a strong approximation or simplification of the real problem.
Susanne Schander: And we already like, it's actually quite easy to go. Beyond this strict
Susanne Schander: a simplification. The only important point is that you have a finite number of degrees of freedom in on each of the cubes. But you could also consider a continuous function, so you would have a couple of number more decrease of freedom, but you still have a finite number of decrease of freedom.
Susanne Schander: So
Susanne Schander: I mean, you can consider that you make them like continuous. Then you could even assume that the face space of field variables is
Susanne Schander: has a well defined derivative. So you just have to add one set more of parameters.
Susanne Schander: So it wouldn't be too fixed on this piecewise, constant functions thing. It's just the first step to have
Susanne Schander: a finite number of decrease of freedom. But you can easily lift this restriction and make them differentiable across the the borders of these Hypercubes.
Susanne Schander: Does it make sense?
Jerzy Lewandowski: Yes, II hope that it? It can work, I guess.
Susanne Schander: Yeah, II mean, we're quite optimistic. But yeah, II see your point and I guess my main
Susanne Schander: my main answer to that is that it's easy to lift this like piecewise constant restriction and make it even differentiable.
Jerzy Lewandowski: So so in mathematics, people often use partition of unity
Jerzy Lewandowski: in terms of small fractions. But probably it is not consistent with your latest experience.
I mean, if you consider smooth functions. Then, of course, you would get an infinite number of degrees of freedom again.
Susanne Schander: which goes against the lattice idea of this approach. And I mean it's all work in progress we can try to to take. I don't know the first 5 derivatives and make it as smooth as as this.
Susanne Schander: but for now we are just considering the simple case to get started, so to say.
Jerzy Lewandowski: So. So, so, so actually there, and and partition of unity can can be, can consist of finite many
Jerzy Lewandowski: functions if your lattice is finite, so maybe a little more than than cubes in the lattice, but not but but also financing.
Jerzy Lewandowski: But but but then you have some smeared variables instead of just
Jerzy Lewandowski: integrated against the constant.
Susanne Schander: Yeah, yeah, that's certainly something that we would be interested to consider in in the future.
Susanne Schander: yeah, thanks.
Hal Haggard: But did you have further comments?
AAipad2022: I just wanted to say that the 2 remarks. Maybe that too many questions. Where about, you know? But we don't do this in the point of gravity.
AAipad2022: I mean, I think that's the relevant right? I mean, if there's another approach, just works. That's perfectly fine. And so I'm I'm completely with design. And
AAipad2022: across them about this, I mean this, they don't have to oblige. They're not obliged to follow methods of whatsoever.
AAipad2022: But I should say that you know, long before
AAipad2022: I mean not. But in this paper, 2,009. I did consider this continuum series not. Not. That is discretization, that is, that is new. But in the continuum limit, whatever way that you obtain the theory, the continuum, the bit. There really is obstruction
to producing
AAipad2022: majest to to to to have different invariants. And so, in fact, II don't remember no other. The title of the paper had some certainty associated with different invariants.
and so it might be worth looking there.
AAipad2022: because I really feel that there is too much intuition seems to be based on the fact that in the classical theory
AAipad2022: this extra terms disappear as epsilon goes to 0. But the whole point about the conduct theory is that they don't, because
AAipad2022: classical fields, continuous content fields are distribution. Okay?
AAipad2022: And so I think that you know that that that faculty is that
AAipad2022: just in the continuum limit by itself, you might want to look at from this 2,009 paper before you spend, you know.
lot of time
AAipad2022: on the, on the
AAipad2022: on the program. I mean, you know, it may all work and so on. But just to make sure that. In fact, these obstructions are not really obstructions for you.
Susanne Schander: Yeah, yeah, yeah, thank you for this comment. And
Susanne Schander: again. So our hope is that because we have all this freedom now in
Susanne Schander: In defining the state sequences and the way we represent the theory that this will help us to
Susanne Schander: to do the trajectory towards a continuum in such a way that these problems can be circumvented.
AAipad2022: yeah. So I mean, this is my, but I just I want to say that even all for there is no 4 dimensional product fee
AAipad2022: right even for scalar fields
AAipad2022: that you can put on a lattice and take the continuing limit and get a well-defined, continuous theory.
AAipad2022: There's no interacting theory. So I mean, so hope that's something like this is true for quantum gravity without something.
AAipad2022: you know most subtlety is is is legal.
AAipad2022: It's too ambitious. Perhaps. Normally, I wanted to keep that in mind that
AAipad2022: but classically, things happen and continue to limit. But kind of mechanically, we don't know how to do it, even for laptop out of the phone, right?
AAipad2022: Yeah. Yeah. So we completely. I mean, I acknowledge that's a very difficult problem.
Susanne Schander: But
Susanne Schander: yeah, I guess one should. I mean, even though it's a hard problem we. It's we should think about it right? So
Susanne Schander: that's how.
Susanne Schander: Yeah. So thank you.
Hal Haggard: Are there any further questions or comments?
Hal Haggard: If not, let's thank Suzanne again.
Susanne Schander: It's always a bit tricky on Zoom. But thank you very much, Susan. Yeah, thank you. Thank you for the opportunity. And yeah, getting all these comments and questions. That's great.
Hal Haggard: wonderful!
Susanne Schander: Hmm.