WEBVTT 1 00:00:02.280 --> 00:00:06.500 Jorge Pullin: Okay, so our speaker today is San Shander, who will revive quantum geometer dynamics. 2 00:00:07.670 --> 00:00:09.789 Susanne Schander: Yeah, thank you very much 3 00:00:09.870 --> 00:00:30.040 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. 4 00:00:30.400 --> 00:00:43.950 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. 5 00:00:44.080 --> 00:00:47.129 Susanne Schander: Let me start by 6 00:00:47.270 --> 00:00:53.049 Susanne Schander: oops. Going back and asking what is actually quantum geometrodynamics. 7 00:00:53.710 --> 00:01:05.379 Susanne Schander: So this was actually the earliest approach of quantizing Gr very closely related to Einstein's formulation, with metric variables. 8 00:01:05.440 --> 00:01:12.489 Susanne Schander: and most prominently promoted by David and Wheeler in the sixties and seventies. 9 00:01:12.860 --> 00:01:21.850 Susanne Schander: So the theory starts from a classical Hamiltonian perspective. So you would perform an adm split 10 00:01:22.000 --> 00:01:30.179 Susanne Schander: and find that your canonical variables are the special metric and their conjugate momenta 11 00:01:30.700 --> 00:01:47.560 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. 12 00:01:48.710 --> 00:01:57.739 Susanne Schander: And then the idea by Wheeler and Dewitt was to take this classical Hamiltonian field theory and to quantize it. 13 00:01:57.750 --> 00:02:10.739 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 14 00:02:10.759 --> 00:02:14.490 Susanne Schander: and their conjugate momenta as a derivative operator. 15 00:02:14.600 --> 00:02:22.499 Susanne Schander: and then, following Dirac's approach to quantizing constraint system, you would 16 00:02:22.510 --> 00:02:33.030 Susanne Schander: request that the constraints, the set of constraints vanishes on a set of appropriate physical states. Psi. 17 00:02:33.860 --> 00:02:57.110 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. 18 00:02:57.370 --> 00:03:03.589 Susanne Schander: So you have ill-defined expressions. And you cannot even make sense of 19 00:03:03.950 --> 00:03:06.190 Susanne Schander: these quantum objects. 20 00:03:07.140 --> 00:03:32.920 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 21 00:03:32.920 --> 00:03:58.969 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 22 00:03:59.020 --> 00:04:04.339 Susanne Schander: in order to keep the causal structure of the theory. 23 00:04:05.080 --> 00:04:08.330 Susanne Schander: And so with all these problems, 24 00:04:09.140 --> 00:04:23.969 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 25 00:04:24.020 --> 00:04:29.669 Susanne Schander: an extensive summary of all the problems. But I guess these ones are 26 00:04:29.860 --> 00:04:31.769 Susanne Schander: the more important ones. 27 00:04:32.980 --> 00:04:46.060 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. 28 00:04:46.420 --> 00:04:56.259 Jerzy Lewandowski: I think that at some point it was not pointed out by that. Actually, we can consider global polymerization of 29 00:04:56.360 --> 00:05:10.270 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 30 00:05:10.310 --> 00:05:21.320 Jerzy Lewandowski: that's that's makes sense. However, it was not looking as so sophisticated as we didn't try, I think. 31 00:05:21.510 --> 00:05:29.150 Susanne Schander: okay, yeah, thank you very much for this comment. Yeah, I'd be interested to go back to this reference. 32 00:05:29.160 --> 00:05:43.469 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. 33 00:05:43.520 --> 00:05:58.710 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 34 00:05:58.810 --> 00:06:03.470 Susanne Schander: also probably due to the failure of quantum geometrodynamics. 35 00:06:03.540 --> 00:06:11.210 Susanne Schander: and hundreds of people were working on developing these approaches. 36 00:06:11.440 --> 00:06:26.509 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. 37 00:06:26.660 --> 00:06:43.909 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. 38 00:06:45.040 --> 00:06:54.769 Susanne Schander: but it actually appears that a lattice regularization in the original adm variables has never been tried. 39 00:06:54.840 --> 00:07:05.410 Susanne Schander: and that's actually our motivation to go back to this original quantum geometry, dynamics approach and 40 00:07:05.480 --> 00:07:12.299 Susanne Schander: try to make progress by adopting a lattice regularization in an yeah. 41 00:07:12.560 --> 00:07:26.879 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 42 00:07:27.080 --> 00:07:38.789 Susanne Schander: this regularization scheme and then present a method for how to realize positive, definite metrics in the quantum theory. 43 00:07:39.150 --> 00:07:51.160 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. 44 00:07:51.330 --> 00:07:56.590 Susanne Schander: And then I'll shortly touch upon the continuum limit in Section 4, 45 00:07:57.420 --> 00:08:07.309 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. 46 00:08:08.440 --> 00:08:18.139 Susanne Schander: Okay? So then, let's get into the solutions. So we start with the regularization scheme. 47 00:08:18.320 --> 00:08:24.420 Susanne Schander: and just to outline the general idea that we have. 48 00:08:24.530 --> 00:08:31.050 So we would start with an Ir regularization. We actually use a torus 49 00:08:31.090 --> 00:08:33.880 Susanne Schander: to second one piece. 50 00:08:34.090 --> 00:08:43.349 Susanne Schander: I are divergencies. for the special manifold. And then, on the other hand, we also 51 00:08:43.419 --> 00:09:05.180 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. 52 00:09:06.100 --> 00:09:19.360 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. 53 00:09:19.670 --> 00:09:25.359 Susanne Schander: then quantize the letters theory and study the continuum limit. 54 00:09:26.750 --> 00:09:43.770 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. 55 00:09:44.630 --> 00:09:54.819 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. 56 00:09:54.830 --> 00:10:01.039 Susanne Schander: And so we introduce a letter spacing called epsilon. 57 00:10:01.130 --> 00:10:05.569 Susanne Schander: and we require the fields. So, for example, the special metric 58 00:10:05.650 --> 00:10:07.170 Susanne Schander: to 59 00:10:07.210 --> 00:10:11.870 Susanne Schander: be constant over these Hypercubes. 60 00:10:12.350 --> 00:10:14.719 Susanne Schander: and we do the same for the momentum. 61 00:10:15.460 --> 00:10:30.790 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. 62 00:10:31.840 --> 00:10:43.639 Susanne Schander: Then the Poisson algebra has the same form as the continuum algebra. So it's just inherited from the continuum fields 63 00:10:43.990 --> 00:11:01.970 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. 64 00:11:03.340 --> 00:11:27.269 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. 65 00:11:27.770 --> 00:11:37.749 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. 66 00:11:38.610 --> 00:11:48.759 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 67 00:11:48.820 --> 00:11:52.080 Susanne Schander: the derivatives by finite differences. 68 00:11:52.370 --> 00:12:03.730 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 69 00:12:03.950 --> 00:12:10.200 Susanne Schander: chain rule for the in the continuum. So 70 00:12:10.280 --> 00:12:25.969 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. 71 00:12:27.480 --> 00:12:44.909 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. 72 00:12:45.460 --> 00:13:10.439 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. 73 00:13:11.540 --> 00:13:19.380 Susanne Schander: But actually, since these terms here are proportional to Epsilon, and as long as you have 74 00:13:19.420 --> 00:13:28.180 Susanne Schander: smearing fields, n and m. that tend to well-defined continuum fields. 75 00:13:28.490 --> 00:13:40.829 Susanne Schander: these expressions are also finite. So, taking the limit, epsilon to 0 would actually lead to these additional terms going to 0. 76 00:13:40.950 --> 00:13:45.729 Susanne Schander: And this is an important point, as, yeah, as you well see. 77 00:13:46.560 --> 00:13:57.830 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. 78 00:13:57.970 --> 00:14:07.309 Susanne Schander: You have to ensure that the lattices get finer, even quicker, so that you would 79 00:14:07.490 --> 00:14:17.889 Susanne Schander: suppress these unphysical decrease of freedom and the anomalies. And that's the idea for the continuum limit later on. 80 00:14:20.030 --> 00:14:36.940 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. 81 00:14:38.290 --> 00:14:46.779 Susanne Schander: So then, in a next step before considering the continuum limit, we would want to quantize this lattice theory. 82 00:14:46.960 --> 00:14:59.829 Susanne Schander: and we found a very intuitive way to do that. So first let me come back to the standard Schrodinger representation. 83 00:15:00.050 --> 00:15:27.019 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. 84 00:15:28.310 --> 00:15:39.919 Susanne Schander: And of course we can satisfy the standard commutation relations. No problem here. That we would inherit from the continuum theory again. 85 00:15:40.440 --> 00:15:46.900 Susanne Schander: But States can have support on non positive, definite metrics, and the causal structure would be lost. 86 00:15:47.090 --> 00:15:57.809 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. 87 00:15:57.850 --> 00:16:06.180 Susanne Schander: Then you act with the momentum operator on it, or with the exponent the exponent ponient version. 88 00:16:06.470 --> 00:16:16.180 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. 89 00:16:16.880 --> 00:16:32.360 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. 90 00:16:32.520 --> 00:16:48.080 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 91 00:16:48.180 --> 00:16:52.210 Susanne Schander: introducing non-canonical commutation relations. 92 00:16:52.220 --> 00:16:58.090 Susanne Schander: And so our approach is actually able to keep the standard commutation relations. 93 00:16:58.780 --> 00:17:06.290 Susanne Schander: The basic idea is to represent the metric Q, which is positive, definite 94 00:17:06.460 --> 00:17:14.930 Susanne Schander: as a product of an upper triangular matrix u, that has positive diagonal elements. 95 00:17:15.050 --> 00:17:20.980 Susanne Schander: So u transpose times u. and this decomposition is actually unique. 96 00:17:21.430 --> 00:17:43.760 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. 97 00:17:43.880 --> 00:17:53.650 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. 98 00:17:54.860 --> 00:18:01.419 Susanne Schander: We can then represent the metric queue on this new Hilbert space. 99 00:18:01.620 --> 00:18:22.340 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. 100 00:18:23.200 --> 00:18:33.890 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. 101 00:18:34.200 --> 00:18:47.849 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. 102 00:18:48.880 --> 00:18:51.470 Susanne Schander: And this looks as follows. 103 00:18:51.480 --> 00:19:09.730 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. 104 00:19:10.180 --> 00:19:28.059 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 105 00:19:28.140 --> 00:19:34.760 Susanne Schander: and is strongly continuous. And this is very important to define the momentum operator 106 00:19:35.340 --> 00:19:43.230 Susanne Schander: because, using so if you have strongly continue. 107 00:19:43.460 --> 00:19:56.729 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? 108 00:19:58.150 --> 00:19:58.920 AAipad2022: Thank you. 109 00:19:59.810 --> 00:20:03.480 Susanne Schander: Yeah. Thanks for the question. 110 00:20:03.580 --> 00:20:11.410 Susanne Schander: and J, is the Jacobian matrix of this map? U, 111 00:20:11.440 --> 00:20:20.240 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? 112 00:20:21.260 --> 00:20:33.860 Susanne Schander: Okay. So since we have this we can now define the momentum operator. Given our transformations view. 113 00:20:34.020 --> 00:20:36.890 Susanne Schander: Oh, I'm sorry. 114 00:20:36.910 --> 00:20:54.189 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. 115 00:20:54.920 --> 00:21:07.639 Susanne Schander: And this gives our momentum operator. And just for illustrative purposes, I wrote down the concrete formula for 2 dimensions. 116 00:21:08.210 --> 00:21:14.949 Susanne Schander: And we also did the same for 3 dimensions. You can do it for whatever dimension you would like to have. 117 00:21:15.010 --> 00:21:20.090 Susanne Schander: And it's a straightforward procedure just using this formula. 118 00:21:20.840 --> 00:21:23.110 Susanne Schander: and what we 119 00:21:23.160 --> 00:21:33.229 Susanne Schander: succeed. To do with this procedure is to satisfy the standard commutation relations on the lattice, of course 120 00:21:34.170 --> 00:21:44.910 Susanne Schander: and still having a positive, definite metric in the sense that this operator here is a positive operator for any S. 121 00:21:45.430 --> 00:21:51.989 Susanne Schander: And I'm sorry I actually forgot a one over Epsilon squared here. So this should be it. I'm sorry. 122 00:21:52.430 --> 00:22:01.309 Susanne Schander: Yeah. So this is our idea of how to implement a positive, definite metric in the quantum theory. 123 00:22:02.850 --> 00:22:06.389 Susanne Schander: Okay, so 124 00:22:06.440 --> 00:22:20.129 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. 125 00:22:20.310 --> 00:22:26.160 Susanne Schander: That also has a positive, definite metric operator. 126 00:22:27.200 --> 00:22:34.650 Susanne Schander: So then, let me come to the question of how to represent gauge transformations in this theory. 127 00:22:35.950 --> 00:22:46.860 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 128 00:22:47.390 --> 00:22:49.050 AAipad2022: you had. 129 00:22:49.130 --> 00:22:56.989 AAipad2022: I mean, morally, you know, you is like the is like a diet. Right? It's because square of U is a metric. 130 00:22:57.280 --> 00:22:59.679 AAipad2022: And so, you know you use like a 131 00:22:59.960 --> 00:23:03.929 AAipad2022: like in that sense. So in 3 dimensions it would be like a trial. 132 00:23:04.050 --> 00:23:09.759 AAipad2022: Somehow you so you you did not. But for some reason you sort of want to 133 00:23:09.910 --> 00:23:18.760 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. 134 00:23:18.770 --> 00:23:24.180 AAipad2022: Very nice, because they both. And then you could have thought of Q and P as being composite of objects. 135 00:23:24.220 --> 00:23:28.340 AAipad2022: Why is it that I mean, that would be more natural to me? 136 00:23:28.410 --> 00:23:31.290 And I was wondering if there's a reason why you didn't want to do that. 137 00:23:31.960 --> 00:23:40.640 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. 138 00:23:40.880 --> 00:23:56.510 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. 139 00:23:56.530 --> 00:24:08.940 Susanne Schander: But it is actually possible to to also implement triad fields, which are, of course, necessary to couple fermions. 140 00:24:09.210 --> 00:24:21.539 Susanne Schander: So I can actually come back to that later. But so thank you for this question. So yeah. So I mean. 141 00:24:21.540 --> 00:24:39.229 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. 142 00:24:40.130 --> 00:24:50.210 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 143 00:24:50.320 --> 00:25:01.870 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. 144 00:25:01.890 --> 00:25:05.419 AAipad2022: then there would not be, maybe few moves of any value in state on it. 145 00:25:05.650 --> 00:25:11.299 AAipad2022: So that's really that is really a quite important of obstacle in the continue. 146 00:25:11.410 --> 00:25:19.109 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 147 00:25:19.160 --> 00:25:31.530 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 148 00:25:31.670 --> 00:25:37.960 AAipad2022: and I'd rather you representation might be you. And this cannot be going to get momentum might be 149 00:25:38.060 --> 00:25:46.610 Susanne Schander: something that might be. You might want to consider. 150 00:25:48.020 --> 00:25:52.319 Susanne Schander: okay, so yeah. 151 00:25:52.380 --> 00:26:08.130 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. 152 00:26:08.190 --> 00:26:25.659 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 153 00:26:25.680 --> 00:26:31.790 Susanne Schander: these results also apply to the diffiomorphism constraint. We would just need to extend 154 00:26:31.800 --> 00:26:43.599 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. 155 00:26:45.040 --> 00:26:57.500 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. 156 00:26:57.610 --> 00:27:05.520 Susanne Schander: so this is actually just a recap of what we have seen before, for the diplomatism constraints. 157 00:27:06.500 --> 00:27:19.000 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. 158 00:27:19.910 --> 00:27:29.730 Susanne Schander: And then we can compute that these constraints form a first class Poisson algebra in the following sense. 159 00:27:29.840 --> 00:27:37.590 Susanne Schander: And yeah, this is just what we would also get for the the form ofism constraints. 160 00:27:39.310 --> 00:27:54.000 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. 161 00:27:54.370 --> 00:28:04.120 Susanne Schander: such that they are constant over cubes of Hypercubes in the respective dimension. 162 00:28:04.230 --> 00:28:07.950 Susanne Schander: and we can represent them with the characteristic function. 163 00:28:08.990 --> 00:28:23.629 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. 164 00:28:24.030 --> 00:28:35.860 Susanne Schander: and then replace derivatives by finite differences. Delta and yeah. So just the same as we did before. 165 00:28:36.710 --> 00:28:44.789 Susanne Schander: And similarly, we also assume that the algebra on the lattice has 166 00:28:45.200 --> 00:28:49.780 Susanne Schander: a similar form, as in the continuum. But we get extra terms. 167 00:28:49.840 --> 00:28:54.080 Susanne Schander: That are proportional to epsilon. 168 00:28:54.750 --> 00:29:14.880 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. 169 00:29:14.890 --> 00:29:24.459 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 170 00:29:24.550 --> 00:29:27.130 Susanne Schander: in the case of gravity. 171 00:29:28.740 --> 00:29:36.130 Susanne Schander: Okay? And then, in the next step, we can solve Hamilton's equations of motion on the lattice. 172 00:29:36.180 --> 00:29:43.000 Susanne Schander: So we evolve Phi with respect to the constraint. 173 00:29:43.880 --> 00:29:54.589 Susanne Schander: And in order for this flow the Hamiltonian flow. Phi, s. To have a solution. It is important that 174 00:29:54.670 --> 00:29:58.700 Susanne Schander: the constraints are only linear in the momentum. 175 00:29:58.840 --> 00:30:17.019 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. 176 00:30:18.100 --> 00:30:28.070 Susanne Schander: and then again, the Hamiltonian flow can be interpreted as an approximate gauge transformation. Of course we would move out of the 177 00:30:28.180 --> 00:30:33.770 Susanne Schander: The constraint surface with longer time intervals. 178 00:30:33.800 --> 00:30:41.010 Susanne Schander: but by making the lattices smaller and smaller you can actually counteract 179 00:30:41.080 --> 00:30:48.219 Susanne Schander: and ensure that the flow remains very close to the constraint surface. 180 00:30:50.710 --> 00:30:59.690 Susanne Schander: Okay? And then, as before, we would go to the quantum theory and define approximate gauge transformations on the letters. 181 00:30:59.850 --> 00:31:06.170 Susanne Schander: The procedure is just the very same as before. So 182 00:31:06.550 --> 00:31:14.220 Susanne Schander: we have a state. Psi n of our letter syllab space. We can 183 00:31:14.290 --> 00:31:39.890 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. 184 00:31:39.980 --> 00:31:55.000 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. 185 00:31:56.230 --> 00:32:02.699 Susanne Schander: so this is the part where we implement the transformation that we want to see. 186 00:32:02.710 --> 00:32:17.150 Susanne Schander: And this is again this factor, including the Jacobian determinant that gives us the right property. Of these transfer gauge transformations. U, 187 00:32:18.170 --> 00:32:24.629 Susanne Schander: and, in fact, here we can show that these U form a unitary one parameter group. 188 00:32:24.770 --> 00:32:27.220 Susanne Schander: in fact. It is 189 00:32:27.440 --> 00:32:41.450 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. 190 00:32:41.750 --> 00:32:48.290 Susanne Schander: and similar to the Hilausita theorem for the contraction semigroup case. 191 00:32:49.080 --> 00:33:06.359 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 192 00:33:06.490 --> 00:33:10.820 Susanne Schander: for other. 193 00:33:11.090 --> 00:33:15.420 Susanne Schander: Purposes. But yeah, the idea is quite similar. 194 00:33:15.970 --> 00:33:24.329 Susanne Schander: Oh, I'm sorry. So again, we can compute the the generator here and get 195 00:33:24.370 --> 00:33:31.239 Susanne Schander: The the constraint as a quantum operator for the lattice theory. 196 00:33:32.210 --> 00:33:43.229 Susanne Schander: Okay, so that actually closes the part on the representation of gauge transformations. This is, of course, only for the letters theory. 197 00:33:43.390 --> 00:33:52.630 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. 198 00:33:52.770 --> 00:33:57.520 Susanne Schander: in and in a strongly continuous way. 199 00:33:58.940 --> 00:34:04.220 Susanne Schander: Okay, so with this, let me come to the continuum limit. 200 00:34:04.230 --> 00:34:20.070 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. 201 00:34:21.260 --> 00:34:31.109 Susanne Schander: So first of all, we can define a while. Algebra a c star, while algebra of 202 00:34:31.239 --> 00:34:42.750 Susanne Schander: of the exponential exponentiated canonical variables just by taking the span and the completion of the while elements here 203 00:34:43.560 --> 00:34:52.220 Susanne Schander: and then taking the inverse limits of these letters while algebras 204 00:34:52.909 --> 00:35:01.430 Susanne Schander: and to take this inverse limit we need kind of an id identification between the different lattices. 205 00:35:01.650 --> 00:35:12.359 Susanne Schander: and this is given by the following identification, so Phi hat n plus one and then labeled by some, some k 206 00:35:12.680 --> 00:35:18.720 Susanne Schander: would be the quantum operators 207 00:35:18.980 --> 00:35:40.069 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. 208 00:35:40.250 --> 00:35:48.029 Susanne Schander: divide this entire formula by this factor here, you would see that this is just the center of mass formula kind of. 209 00:35:49.620 --> 00:36:02.329 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. 210 00:36:02.380 --> 00:36:09.089 Susanne Schander: Again. This is very easy. Here we just have an L. 2 Hilbert space on the 5 variables. 211 00:36:09.880 --> 00:36:25.019 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. 212 00:36:26.270 --> 00:36:35.159 Susanne Schander: And then, in the next step, we actually suggest to take the continuum limit in the following way. 213 00:36:35.240 --> 00:36:45.820 Susanne Schander: So we we say that Omega, which is now the continuum state of 214 00:36:46.120 --> 00:36:49.690 Susanne Schander: the inverse limit of this, while element 215 00:36:49.870 --> 00:36:52.380 Susanne Schander: is defined as 216 00:36:52.460 --> 00:37:11.969 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. 217 00:37:12.010 --> 00:37:18.310 Susanne Schander: So the idea is to show that this object here defines the Koshi sequence. 218 00:37:18.550 --> 00:37:23.960 Susanne Schander: For so a cauchy sequence of algebraic states. 219 00:37:25.000 --> 00:37:41.000 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. 220 00:37:41.430 --> 00:37:46.019 Susanne Schander: So this is the broad. Why do you call it vacuum? Because it is 221 00:37:46.100 --> 00:37:50.699 AAipad2022: I mean size, random state. No, it's not a preferred state 222 00:37:51.350 --> 00:37:55.629 AAipad2022: the science where some random some states in the 223 00:37:55.900 --> 00:37:58.689 AAipad2022: Nth lag is right. I mean? 224 00:37:59.150 --> 00:38:02.109 AAipad2022: what is preferred about science? I 225 00:38:02.480 --> 00:38:07.640 AAipad2022: I mean, it's a vacuum only in the mathematical sense of gns construction. Or is it the 226 00:38:07.820 --> 00:38:08.560 like? 227 00:38:08.910 --> 00:38:13.350 AAipad2022: Is is your science and Omega supposed supposed to satisfy constraints. 228 00:38:13.550 --> 00:38:26.520 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. 229 00:38:26.960 --> 00:38:32.099 Susanne Schander: On the other hand, no, I mean the the constraint which is the 230 00:38:32.990 --> 00:38:40.680 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? 231 00:38:41.580 --> 00:38:52.389 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 232 00:38:52.550 --> 00:39:03.209 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 233 00:39:03.320 --> 00:39:16.430 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. 234 00:39:16.610 --> 00:39:23.980 Susanne Schander: Because you have this freedom to, to tune this trajectory, and to choose this state quite freely. 235 00:39:25.580 --> 00:39:32.739 AAipad2022: But will the continuum state satisfy that if you margin constraint, for example, that you had in the last that last section. 236 00:39:33.680 --> 00:39:43.639 Susanne Schander: Yeah. So we haven't computed this for now we only have the computations for this scalar field here. 237 00:39:43.900 --> 00:40:11.829 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. 238 00:40:11.870 --> 00:40:21.110 Susanne Schander: So just to make it simpler at the beginning, we actually restricted our computations to a simple scale of field theory. 239 00:40:21.480 --> 00:40:26.199 Susanne Schander: So Phi and pi would be the canonical variables. 240 00:40:26.780 --> 00:40:51.760 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. 241 00:40:51.940 --> 00:40:56.150 AAipad2022: And then the diffumologism constraint on the scale of field theory also 242 00:40:56.860 --> 00:40:59.049 AAipad2022: and you are looking at representations 243 00:40:59.400 --> 00:41:01.980 AAipad2022: off of that that would be good. 244 00:41:02.700 --> 00:41:07.740 Thorsten Lang: I think my understanding of the questions is that you ask whether we are the 245 00:41:08.250 --> 00:41:23.109 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 246 00:41:23.520 --> 00:41:26.660 Thorsten Lang: on on it. And then so, of course, that for the 247 00:41:27.420 --> 00:41:29.269 AAipad2022: so it's kinematical, and therefore 248 00:41:29.500 --> 00:41:32.560 AAipad2022: your state Omega does not satisfy any constraints. 249 00:41:32.570 --> 00:41:43.250 Susanne Schander: Yeah, yeah, I'm I'm sorry I didn't get the question, thanks for the the clarification. Great thanks. 250 00:41:43.390 --> 00:42:02.899 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 251 00:42:03.190 --> 00:42:18.750 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 252 00:42:18.840 --> 00:42:37.070 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. 253 00:42:38.580 --> 00:42:41.530 Susanne Schander: Alright. So 254 00:42:41.920 --> 00:42:49.210 Susanne Schander: this actually concludes, or, yeah, let me conclude with a summary and an outlook. 255 00:42:49.290 --> 00:42:56.289 Susanne Schander: So at the beginning I have introduced our let us regulation 256 00:42:56.460 --> 00:42:59.079 Susanne Schander: of quantum geometry, dynamics. 257 00:42:59.180 --> 00:43:07.020 Susanne Schander: with the motivation that in the original approach a lattice regularization has never been introduced. 258 00:43:07.130 --> 00:43:18.209 Susanne Schander: Then, for this lattice theory we were able to find non standard representation of the canonical commutation relations. 259 00:43:18.420 --> 00:43:37.929 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. 260 00:43:39.130 --> 00:43:52.380 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. 261 00:43:52.910 --> 00:44:01.569 Susanne Schander: And then so the last slide was to show that there is a procedure to 262 00:44:01.850 --> 00:44:13.510 Susanne Schander: Consider the continuum limit of this like the theory of a certain subgroup and gravity. So the the deformorphism constraints 263 00:44:13.960 --> 00:44:17.000 Susanne Schander: and their Li algebra. 264 00:44:17.090 --> 00:44:38.309 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 265 00:44:38.460 --> 00:44:40.240 Susanne Schander: continuum theory. 266 00:44:40.510 --> 00:44:54.940 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. 267 00:44:55.030 --> 00:45:11.210 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. 268 00:45:11.420 --> 00:45:15.390 Susanne Schander: But we also have 269 00:45:15.430 --> 00:45:28.620 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. 270 00:45:29.310 --> 00:45:45.789 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. 271 00:45:46.000 --> 00:45:47.939 Susanne Schander: So yeah, thank you very much. 272 00:45:48.690 --> 00:45:55.769 Hal Haggard: Thank you, Suzanne. Thank you for the presentation. Yurik. I believe your hand was up first. Please go ahead. 273 00:46:00.110 --> 00:46:02.700 Hal Haggard: Eric, are you? Did you have a question? 274 00:46:03.270 --> 00:46:10.410 Jerzy Lewandowski: Yes, sorry I was muted. I must have slept through the moment when you introduced those approximates 275 00:46:10.450 --> 00:46:17.250 Jerzy Lewandowski: gauge transformation. So can you explain again what? Why they are approximating? 276 00:46:17.660 --> 00:46:34.489 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. 277 00:46:35.220 --> 00:46:39.020 Jerzy Lewandowski: I see. So they don't have any continuum limit. 278 00:46:39.070 --> 00:46:49.660 Susanne Schander: Not yet. So in section 4, I was referring to the continuum limit. But we haven't done the explicit like. 279 00:46:50.070 --> 00:46:55.170 Susanne Schander: I mean, there we were, referring to the scalar field a simple scalar field 280 00:46:55.350 --> 00:46:59.439 Susanne Schander: and I mean, as I was mentioning, you can 281 00:46:59.480 --> 00:47:18.020 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 282 00:47:18.030 --> 00:47:20.329 Susanne Schander: in quantum geometron dynamics. 283 00:47:21.580 --> 00:47:24.309 Jerzy Lewandowski: I see. Okay, thank you. 284 00:47:24.570 --> 00:47:26.210 Hal Haggard: Bye. And then, Lee. 285 00:47:27.470 --> 00:47:30.009 AAipad2022: yeah, I have a comment and a question. So 286 00:47:30.480 --> 00:47:32.530 AAipad2022: the comment is, really that? 287 00:47:34.250 --> 00:47:44.010 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 288 00:47:44.190 --> 00:47:57.509 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. 289 00:47:57.840 --> 00:48:01.759 AAipad2022: The the main problem was that 290 00:48:04.500 --> 00:48:13.050 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 291 00:48:13.150 --> 00:48:29.140 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 292 00:48:29.210 --> 00:48:30.510 AAipad2022: radius 293 00:48:30.570 --> 00:48:31.880 methods from 294 00:48:33.000 --> 00:48:45.060 AAipad2022: context, whereas we could not make any. 295 00:48:45.130 --> 00:48:53.229 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 296 00:48:53.360 --> 00:48:58.300 AAipad2022: about, you know what would happen to continue, that you might 297 00:48:59.010 --> 00:49:05.360 AAipad2022: ultimately find some nice kinematics. But then it will be maybe difficult to form with the Hamiltonian on that. 298 00:49:05.390 --> 00:49:08.839 AAipad2022: So just just to keep that in mind. The question is. 299 00:49:10.920 --> 00:49:13.870 but there is a real difference between scale of Field and them. 300 00:49:14.030 --> 00:49:22.470 AAipad2022: the metrics and the in the scalar field. You know. You can just divide this space in 3 dimensions into this 301 00:49:22.790 --> 00:49:31.400 AAipad2022: into the squares, and then the scalar field is constant in each of the squares, and that is the point. Use of freedom. 302 00:49:32.020 --> 00:49:33.550 AAipad2022: But 303 00:49:34.000 --> 00:49:39.459 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 304 00:49:39.670 --> 00:49:42.080 as a lattice on the 94, 305 00:49:42.120 --> 00:49:48.659 AAipad2022: I said that I consider constant. It doesn't make any sense unless I specify some coordinate system. 306 00:49:49.370 --> 00:49:55.209 AAipad2022: and so your continuum limit will be then very, very terrible to certain coordinate system in which things were constant. 307 00:49:55.550 --> 00:50:03.679 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 308 00:50:03.840 --> 00:50:06.010 AAipad2022: the connection along 309 00:50:06.210 --> 00:50:11.779 AAipad2022: along the lines, and that is coordinating value. 310 00:50:12.360 --> 00:50:20.279 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. 311 00:50:20.330 --> 00:50:35.640 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. 312 00:50:36.450 --> 00:50:42.829 Susanne Schander: Yeah. So of course, you would need to choose a chart so I guess 313 00:50:42.920 --> 00:50:51.480 Susanne Schander: you have to be careful with that. That's what I can say to that point, and I guess so in general. 314 00:50:51.610 --> 00:50:55.700 Susanne Schander: I mean, our approach is to just 315 00:50:55.890 --> 00:51:01.640 Susanne Schander: or our idea was to just introduce. like a new approach that 316 00:51:01.740 --> 00:51:11.899 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 317 00:51:11.940 --> 00:51:16.030 Susanne Schander: you should really see it as a as a new way of understanding. 318 00:51:16.070 --> 00:51:19.589 Susanne Schander: the problem of quantum gravity. 319 00:51:19.600 --> 00:51:36.200 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 320 00:51:36.300 --> 00:51:48.370 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 321 00:51:48.480 --> 00:51:51.840 Susanne Schander: to have a theory that 322 00:51:52.170 --> 00:51:58.130 Susanne Schander: is a coordinate, invariant. 323 00:51:58.410 --> 00:51:59.800 Susanne Schander: But 324 00:51:59.930 --> 00:52:04.800 Susanne Schander: for us the important point is that you can restore 325 00:52:04.940 --> 00:52:17.229 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. 326 00:52:17.270 --> 00:52:20.789 Susanne Schander: Like quantum gravity, for example. 327 00:52:20.980 --> 00:52:22.570 Susanne Schander: So it's just 328 00:52:22.580 --> 00:52:24.799 Susanne Schander: that we are not 329 00:52:24.860 --> 00:52:33.049 Susanne Schander: restricting to the case of having a theory that is coordinate, invariant on the lattice. 330 00:52:33.420 --> 00:52:41.519 Susanne Schander: So we see these letters theories as a tool to get to a well-defined continuum theory, but 331 00:52:41.750 --> 00:52:54.419 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 332 00:52:54.490 --> 00:52:57.820 Susanne Schander: the rationale. I don't know if that answers their question. 333 00:52:59.200 --> 00:53:06.979 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 334 00:53:07.500 --> 00:53:09.999 AAipad2022: explains that all it like is. 335 00:53:10.030 --> 00:53:17.540 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 336 00:53:17.660 --> 00:53:23.100 AAipad2022: feels what is really using jocks. coordinate systems 337 00:53:23.540 --> 00:53:28.240 AAipad2022: that the job dependent disappeared. 338 00:53:28.770 --> 00:53:40.990 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 339 00:53:41.120 --> 00:53:51.330 AAipad2022: had to be very, very careful, you know, if you chat that, nothing depends on chat is not because almost always chat leaves 340 00:53:51.630 --> 00:54:00.930 AAipad2022: the limit. Chart is signature. Things tend to depend on it. Okay, II just wanted to point that out. It's not. 341 00:54:02.090 --> 00:54:02.840 AAipad2022: okay. 342 00:54:03.000 --> 00:54:13.189 Susanne Schander: maybe just a common. So if you consider the continuum limit of the algebra. Then actually, the chart dependents would 343 00:54:13.470 --> 00:54:21.750 Susanne Schander: go away right? So I don't know if that plays a role, or if that's important for the continuum limit. 344 00:54:21.840 --> 00:54:28.000 Susanne Schander: But yeah, I mean, we can discuss about that later as well. I just wanted to make this quick. Comment. 345 00:54:28.430 --> 00:54:39.280 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. 346 00:54:39.460 --> 00:54:41.830 Lee Smolin: So I wanna consider, 347 00:54:42.070 --> 00:54:51.770 Lee Smolin: I'm on s 3 or s. 3. And I have, manifold, and on this manifold I have some loops 348 00:54:51.970 --> 00:54:57.069 Lee Smolin: to have all the loops. and I define a representation 349 00:54:57.220 --> 00:55:03.360 Lee Smolin: of the Diffie Morphisms in the space of loops as follows, if I 350 00:55:03.430 --> 00:55:07.100 Lee Smolin: map a dipymorphism in 2, 351 00:55:07.810 --> 00:55:14.129 Lee Smolin: I lap map a sorry, a loop into another loop, and I trace 352 00:55:14.250 --> 00:55:15.210 Lee Smolin: that. 353 00:55:15.370 --> 00:55:27.400 Lee Smolin: and I diffuse that to define a new representation, which is basically one if the loop went to itself and 0 otherwise. 354 00:55:27.680 --> 00:55:29.069 Lee Smolin: do you see what I'm saying? 355 00:55:29.880 --> 00:55:32.680 Susanne Schander: Yeah, I guess so. So you would have 356 00:55:32.690 --> 00:55:41.140 Susanne Schander: space of loops. And then you would want to map one of the loops into another loop. Right? I want to map 357 00:55:41.250 --> 00:55:51.169 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. 358 00:55:52.010 --> 00:55:57.770 Lee Smolin: So when I so I get a big representation stays of the roofs 359 00:55:57.860 --> 00:56:00.319 Lee Smolin: on themselves, and then I act. 360 00:56:00.610 --> 00:56:02.670 Lee Smolin: We could do a few more things. 361 00:56:03.400 --> 00:56:11.560 Lee Smolin: and it seems to me that makes a big representation of different morphisms. That's not very easy 362 00:56:12.040 --> 00:56:14.990 Lee Smolin: to to construct your construction. 363 00:56:17.100 --> 00:56:23.459 Susanne Schander: Yeah. So I can't really tell what would happen if we considered loops? 364 00:56:23.490 --> 00:56:26.820 Susanne Schander: because so our theory, 365 00:56:27.080 --> 00:56:37.930 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. 366 00:56:37.960 --> 00:56:50.329 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 367 00:56:50.410 --> 00:57:00.490 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 368 00:57:00.530 --> 00:57:03.890 Susanne Schander: still, if you want to consider it as a lattice theory. 369 00:57:04.000 --> 00:57:19.770 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 370 00:57:19.960 --> 00:57:26.199 Susanne Schander: a bit easier than considering loops. Right? So i. 371 00:57:26.720 --> 00:57:35.110 Susanne Schander: i don't know, if like, so i mean the basic idea here is to to use this regular lattice 372 00:57:35.390 --> 00:57:41.339 Susanne Schander: and not the loops. So I don't know, if 373 00:57:41.380 --> 00:57:51.820 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 374 00:57:52.190 --> 00:58:02.220 Lee Smolin: carries a representation. but there's a few more feasible than as well. The space of lattices on a manifold, does not 375 00:58:03.140 --> 00:58:13.050 Lee Smolin: in any smooth way containing a representation of the Duchy morphisms, so as probably how it would break in the second case. 376 00:58:13.260 --> 00:58:18.710 Lee Smolin: and not break in the first place, but breaks means you don't get to arrive. It's smooth. 377 00:58:20.180 --> 00:58:22.549 Lee Smolin: infinite, dimensional representation. 378 00:58:23.370 --> 00:58:27.000 Susanne Schander: Yes, so 379 00:58:27.220 --> 00:58:29.669 Susanne Schander: okay. So we don't have the loops. 380 00:58:29.720 --> 00:58:33.580 Susanne Schander: But I so the thing I 381 00:58:33.970 --> 00:58:36.380 Susanne Schander: always 382 00:58:36.780 --> 00:58:39.770 Susanne Schander: like. So in in loop quantum gravity. 383 00:58:40.120 --> 00:58:42.890 Susanne Schander: You define your objects. 384 00:58:42.930 --> 00:58:45.579 Susanne Schander: On these loops. 385 00:58:45.790 --> 00:58:59.359 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 386 00:58:59.420 --> 00:59:03.069 Susanne Schander: we want to have a representation that is strongly continuous. 387 00:59:03.130 --> 00:59:07.490 Susanne Schander: And this is the reason why we are not using loops. 388 00:59:07.540 --> 00:59:18.159 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. 389 00:59:18.770 --> 00:59:25.299 Susanne Schander: But of course it's at the expense that we lose general covariance on the lattice. 390 00:59:25.530 --> 00:59:32.459 Susanne Schander: And this is yeah, as I was like, 391 00:59:32.540 --> 00:59:35.410 Lee Smolin: you don't have a representation. 392 00:59:36.900 --> 00:59:50.979 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 393 00:59:51.100 --> 00:59:58.349 Susanne Schander: very plausible to have them starting from the scalar field theory that we already have written down. 394 00:59:58.450 --> 01:00:08.490 Susanne Schander: And then we take the continuum limit in the way I suggested. In Section 4 here. So this is actually the way. 395 01:00:08.570 --> 01:00:14.749 Susanne Schander: We would also get the representation of the deformivism constraints 396 01:00:14.940 --> 01:00:18.609 Susanne Schander: for both, like on the letters and also in the continuum. 397 01:00:18.730 --> 01:00:28.080 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. 398 01:00:29.540 --> 01:00:37.430 Thorsten Lang: yeah, just wanted to comment on that. So so I think from our perspective, the ideas that the break the diffumorphism 399 01:00:37.620 --> 01:00:39.250 Thorsten Lang: group on the letters. 400 01:00:39.420 --> 01:00:43.290 Thorsten Lang: But this analytic way 401 01:00:43.500 --> 01:00:51.830 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 402 01:00:51.950 --> 01:00:57.579 Thorsten Lang: we were aware that it's broken on the letters, but it it gets less broken as you 403 01:00:57.630 --> 01:01:02.300 Thorsten Lang: make the letters finer and finer. So because of these epsilons that go to 0, 404 01:01:04.070 --> 01:01:18.190 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 405 01:01:18.200 --> 01:01:21.230 AAipad2022: which are going to go to. You want to go to 0. 406 01:01:21.500 --> 01:01:26.799 AAipad2022: And that is exactly the reason why, mother, one just 1 s mother, one had to do 407 01:01:26.810 --> 01:01:33.960 AAipad2022: very very careful analysis. And, mother, one does have a continuous representation of the diffum modelism group. He does have 408 01:01:34.040 --> 01:01:38.969 AAipad2022: the the generators of different Zoom group in the continuum. Sorry. Go ahead. 409 01:01:39.280 --> 01:01:54.460 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. 410 01:01:54.600 --> 01:02:04.170 Susanne Schander: And so as long as you have the smearing fields, m and n going to 411 01:02:04.220 --> 01:02:19.850 Susanne Schander: a continuous version of it. So you have a continuous, you have continuous smearing fields. These additional terms here actually go to 412 01:02:20.360 --> 01:02:23.040 Susanne Schander: a bounded function. 413 01:02:23.110 --> 01:02:24.749 Susanne Schander: And if you send 414 01:02:24.850 --> 01:02:39.150 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 415 01:02:39.360 --> 01:02:45.279 Susanne Schander: it's very plausible, or it seems very plausible that we can end up with a representation. 416 01:02:46.010 --> 01:02:48.810 Susanne Schander: That's 417 01:02:48.820 --> 01:03:10.700 AAipad2022: that is where all the problem comes when you try to take the continuing limit. But I think this is. 418 01:03:10.920 --> 01:03:24.380 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 419 01:03:24.380 --> 01:03:49.370 Western: approach. In order to solve 420 01:03:49.370 --> 01:04:10.749 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 421 01:04:10.750 --> 01:04:34.670 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 422 01:04:34.910 --> 01:04:54.199 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. 423 01:04:54.210 --> 01:05:19.099 Western: And, in fact, without having to worry about all the problems that arises. 424 01:05:26.410 --> 01:05:49.479 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. 425 01:05:49.600 --> 01:06:13.919 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. 426 01:06:14.000 --> 01:06:25.879 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. 427 01:06:26.000 --> 01:06:39.650 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 428 01:06:40.050 --> 01:06:53.750 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 429 01:06:53.820 --> 01:06:58.269 Susanne Schander: strongly continuous representations of the constraints. 430 01:06:58.450 --> 01:07:11.870 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 431 01:07:11.960 --> 01:07:21.979 Susanne Schander: but it seems to me that it might be like our approach focuses on the fact to start. 432 01:07:22.260 --> 01:07:46.349 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. 433 01:07:46.460 --> 01:07:53.649 Susanne Schander: And of course it differs from what has been done in, for example, in lukewarm gravity. 434 01:07:54.210 --> 01:08:01.369 Susanne Schander: and yeah, so maybe another point is also that our representation is less singular. 435 01:08:01.540 --> 01:08:13.690 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. 436 01:08:14.170 --> 01:08:22.549 Susanne Schander: Whereas in loop quantum gravity you would have the fields defined on these links or loops. 437 01:08:22.850 --> 01:08:27.350 Susanne Schander: And this was also one motivation. 438 01:08:27.470 --> 01:08:28.939 Susanne Schander: so 439 01:08:29.270 --> 01:08:43.900 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. 440 01:08:44.000 --> 01:08:50.000 Susanne Schander: And that's exactly where we wanted to suggest a new formulation where you have 441 01:08:50.270 --> 01:08:56.729 Susanne Schander: less singular definitions of your basic variables and smearing, and whatsoever. 442 01:08:56.930 --> 01:09:02.690 Susanne Schander: And from this perspective, trying to find a new representation that is strongly continuous. 443 01:09:02.840 --> 01:09:14.410 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 444 01:09:14.500 --> 01:09:18.660 Susanne Schander: non-singular regularizations. 445 01:09:18.770 --> 01:09:33.460 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. 446 01:09:34.020 --> 01:09:59.009 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 447 01:09:59.010 --> 01:10:07.319 Western: some space time, but actually to to start will be the. 448 01:10:07.320 --> 01:10:16.790 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. 449 01:10:16.790 --> 01:10:26.260 Susanne Schander: Yeah. So II see. I see these comments and I understand them. But so 450 01:10:26.590 --> 01:10:30.910 Susanne Schander: maybe it's yeah. It's best to see it as a new approach to 451 01:10:31.090 --> 01:10:50.489 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. 452 01:10:51.090 --> 01:10:52.600 Susanne Schander: thanks for the question. 453 01:10:53.980 --> 01:10:57.550 Susanne Schander: Europe's been waiting patiently. You're sorry. 454 01:10:57.770 --> 01:11:01.879 Jerzy Lewandowski: So the discussion is very interesting, so I don't have to. 455 01:11:02.380 --> 01:11:09.790 Jerzy Lewandowski: You use all my patients, and perhaps you discuss this already. But 456 01:11:09.800 --> 01:11:12.809 Jerzy Lewandowski: how do you 457 01:11:12.900 --> 01:11:23.230 Jerzy Lewandowski: manage the coordinate transformation? Is this lattice related to some coordinates, or are coordinates variants independent on the lattice? 458 01:11:24.190 --> 01:11:35.340 Susanne Schander: So so the latest is on the chart. 459 01:11:35.390 --> 01:11:40.370 Susanne Schander: so you would have coordinates and but so 460 01:11:40.470 --> 01:11:48.389 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. 461 01:11:48.680 --> 01:11:54.540 Jerzy Lewandowski: and the second question is little, also little technical. So did those 462 01:11:54.650 --> 01:11:57.760 Jerzy Lewandowski: characteristic functions of some 463 01:11:57.820 --> 01:12:04.239 Jerzy Lewandowski: sets. Actually they are not so nice from the distribution of point of view. 464 01:12:04.260 --> 01:12:12.790 Jerzy Lewandowski: So you have to. You multiply distributions by step functions. So we know that usually it 465 01:12:12.880 --> 01:12:17.259 Jerzy Lewandowski: leads to some ambiguities, and in particular, in 466 01:12:17.470 --> 01:12:23.960 Jerzy Lewandowski: yeah, in luo quantum gravity. We also use those flux flexes, but 467 01:12:24.250 --> 01:12:32.490 Jerzy Lewandowski: I usually prefer to consider some smearing function which has compact support. And then II feel saved. And they 468 01:12:32.720 --> 01:12:35.680 Jerzy Lewandowski: I don't have any boundaries, or any 469 01:12:35.950 --> 01:12:46.660 Jerzy Lewandowski: issues with when I multiply distribution by by some discontinuous function. So do you think that in your case it's not 470 01:12:46.860 --> 01:12:47.660 Jerzy Lewandowski: yeah. 471 01:12:48.060 --> 01:12:49.800 Susanne Schander: So 472 01:12:50.140 --> 01:12:53.869 Susanne Schander: maybe. So, my perspective on this is that 473 01:12:54.000 --> 01:13:05.019 Susanne Schander: we actually do nothing than integration theory. So we actually take the continuous theory and then make it discrete and 474 01:13:05.120 --> 01:13:17.709 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 475 01:13:17.880 --> 01:13:29.299 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. 476 01:13:30.070 --> 01:13:43.780 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. 477 01:13:43.790 --> 01:13:53.070 Susanne Schander: And we already like, it's actually quite easy to go. Beyond this strict 478 01:13:53.220 --> 01:14:13.509 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. 479 01:14:13.780 --> 01:14:15.280 Susanne Schander: So 480 01:14:15.400 --> 01:14:27.910 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 481 01:14:27.940 --> 01:14:35.029 Susanne Schander: has a well defined derivative. So you just have to add one set more of parameters. 482 01:14:35.070 --> 01:14:43.069 Susanne Schander: So it wouldn't be too fixed on this piecewise, constant functions thing. It's just the first step to have 483 01:14:43.150 --> 01:14:53.850 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. 484 01:14:54.720 --> 01:14:56.409 Susanne Schander: Does it make sense? 485 01:14:57.480 --> 01:15:00.879 Jerzy Lewandowski: Yes, II hope that it? It can work, I guess. 486 01:15:01.970 --> 01:15:10.060 Susanne Schander: Yeah, II mean, we're quite optimistic. But yeah, II see your point and I guess my main 487 01:15:10.550 --> 01:15:18.570 Susanne Schander: my main answer to that is that it's easy to lift this like piecewise constant restriction and make it even differentiable. 488 01:15:19.670 --> 01:15:25.629 Jerzy Lewandowski: So so in mathematics, people often use partition of unity 489 01:15:25.830 --> 01:15:35.430 Jerzy Lewandowski: in terms of small fractions. But probably it is not consistent with your latest experience. 490 01:15:37.040 --> 01:15:44.189 I mean, if you consider smooth functions. Then, of course, you would get an infinite number of degrees of freedom again. 491 01:15:44.330 --> 01:15:58.340 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. 492 01:15:58.470 --> 01:16:04.090 Susanne Schander: but for now we are just considering the simple case to get started, so to say. 493 01:16:05.380 --> 01:16:12.770 Jerzy Lewandowski: So. So, so, so actually there, and and partition of unity can can be, can consist of finite many 494 01:16:12.990 --> 01:16:24.570 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. 495 01:16:25.090 --> 01:16:30.169 Jerzy Lewandowski: But but but then you have some smeared variables instead of just 496 01:16:31.220 --> 01:16:34.119 Jerzy Lewandowski: integrated against the constant. 497 01:16:34.780 --> 01:16:41.050 Susanne Schander: Yeah, yeah, that's certainly something that we would be interested to consider in in the future. 498 01:16:41.250 --> 01:16:44.869 Susanne Schander: yeah, thanks. 499 01:16:45.270 --> 01:16:47.459 Hal Haggard: But did you have further comments? 500 01:16:49.200 --> 01:16:55.979 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. 501 01:16:56.170 --> 01:17:05.679 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 502 01:17:05.980 --> 01:17:13.170 AAipad2022: across them about this, I mean this, they don't have to oblige. They're not obliged to follow methods of whatsoever. 503 01:17:13.480 --> 01:17:17.460 AAipad2022: But I should say that you know, long before 504 01:17:17.810 --> 01:17:34.120 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 505 01:17:34.620 --> 01:17:35.890 to producing 506 01:17:35.950 --> 01:17:46.869 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. 507 01:17:47.060 --> 01:17:49.360 and so it might be worth looking there. 508 01:17:49.550 --> 01:17:57.330 AAipad2022: because I really feel that there is too much intuition seems to be based on the fact that in the classical theory 509 01:17:58.060 --> 01:18:05.840 AAipad2022: this extra terms disappear as epsilon goes to 0. But the whole point about the conduct theory is that they don't, because 510 01:18:05.940 --> 01:18:10.009 AAipad2022: classical fields, continuous content fields are distribution. Okay? 511 01:18:10.300 --> 01:18:15.000 AAipad2022: And so I think that you know that that that faculty is that 512 01:18:15.100 --> 01:18:23.829 AAipad2022: just in the continuum limit by itself, you might want to look at from this 2,009 paper before you spend, you know. 513 01:18:23.940 --> 01:18:25.249 lot of time 514 01:18:27.320 --> 01:18:29.060 AAipad2022: on the, on the 515 01:18:29.090 --> 01:18:36.860 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. 516 01:18:38.010 --> 01:18:42.260 Susanne Schander: Yeah, yeah, yeah, thank you for this comment. And 517 01:18:42.760 --> 01:18:50.580 Susanne Schander: again. So our hope is that because we have all this freedom now in 518 01:18:50.630 --> 01:19:01.720 Susanne Schander: In defining the state sequences and the way we represent the theory that this will help us to 519 01:19:01.780 --> 01:19:14.190 Susanne Schander: to do the trajectory towards a continuum in such a way that these problems can be circumvented. 520 01:19:14.290 --> 01:19:21.009 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 521 01:19:21.070 --> 01:19:22.730 AAipad2022: right even for scalar fields 522 01:19:22.990 --> 01:19:27.860 AAipad2022: that you can put on a lattice and take the continuing limit and get a well-defined, continuous theory. 523 01:19:28.890 --> 01:19:35.939 AAipad2022: There's no interacting theory. So I mean, so hope that's something like this is true for quantum gravity without something. 524 01:19:35.960 --> 01:19:38.860 AAipad2022: you know most subtlety is is is legal. 525 01:19:38.950 --> 01:19:42.840 AAipad2022: It's too ambitious. Perhaps. Normally, I wanted to keep that in mind that 526 01:19:43.000 --> 01:19:49.650 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? 527 01:19:49.860 --> 01:19:55.259 AAipad2022: Yeah. Yeah. So we completely. I mean, I acknowledge that's a very difficult problem. 528 01:19:55.620 --> 01:19:57.560 Susanne Schander: But 529 01:19:57.670 --> 01:20:14.169 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 530 01:20:14.180 --> 01:20:28.620 Susanne Schander: that's how. 531 01:20:28.640 --> 01:20:31.300 Susanne Schander: Yeah. So thank you. 532 01:20:32.700 --> 01:20:35.660 Hal Haggard: Are there any further questions or comments? 533 01:20:38.150 --> 01:20:40.869 Hal Haggard: If not, let's thank Suzanne again. 534 01:20:41.740 --> 01:20:51.910 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. 535 01:20:52.250 --> 01:20:53.330 Hal Haggard: wonderful! 536 01:20:56.020 --> 01:20:56.500 Susanne Schander: Hmm.