WEBVTT 1 00:00:00.670 --> 00:00:01.370 Jorge Pullin: Cloud. 2 00:00:09.320 --> 00:00:11.470 Western: So, Diego, you have the floor. 3 00:00:15.010 --> 00:00:16.830 Western: Please unmute yourself. 4 00:00:19.690 --> 00:00:21.010 Western: It's frozen. 5 00:00:21.270 --> 00:00:22.020 Western: No. 6 00:00:24.270 --> 00:00:27.810 Thiago Lucena: Sorry. I'm somewhat nervous. It's my 7 00:00:28.050 --> 00:00:36.639 Thiago Lucena: second time overall presenting to the loops community. And 1st time here also. Thank you for the invitation. 8 00:00:36.910 --> 00:00:37.595 Thiago Lucena: And 9 00:00:38.709 --> 00:00:42.200 Thiago Lucena: well, this is a joint work, as Francisco Chip said. 10 00:00:42.450 --> 00:00:49.209 Thiago Lucena: It involves myself my supervisor, Marcus Muller, Bilemo Mina, Marugan, and Francesco bidotto. 11 00:00:49.480 --> 00:01:11.899 Thiago Lucena: and, as the title says itself, we decided to look at graph changing dynamics and loop quantum gravity, which is something that is often not really looked at. And I think that raises the question, why, looking at graph changes, I think that the 1st reason is because it hasn't been looked at. 12 00:01:11.900 --> 00:01:30.390 Thiago Lucena: And it's a really interesting problem in general. So the partition used in Tieman's derivation of the Hamiltonian constraint. It allows to express the curvature locally in terms of holonomies over small loops with sizes that are, of course, subpartition sized. 13 00:01:30.940 --> 00:01:36.340 Thiago Lucena: and these loopholeonomies they couple to this ping network around the nodes 14 00:01:36.480 --> 00:01:44.114 Thiago Lucena: overall, changing the graphs. If one accounts for the possibility that the Hamiltonian is 15 00:01:44.730 --> 00:01:57.999 Thiago Lucena: permission, then one can either add or remove links from these ping networks, and that, of course, drastically increases the number of, perhaps, that one has to account for when doing calculations. 16 00:01:58.400 --> 00:02:14.719 Thiago Lucena: And I think that's 1 of the main things that scared people so the result was that few graph changing copulations have been published, actually, at least, as far as as we know, overall. 17 00:02:14.740 --> 00:02:22.510 Thiago Lucena: I found 3 derivations for the graph changing effects of the constraint on 3 balance 18 00:02:22.780 --> 00:02:29.970 Thiago Lucena: intertwiners, but 2 of them used the temporary leap tangle algebra. 19 00:02:29.990 --> 00:02:51.780 Thiago Lucena: which actually leads to different results, and only one of them actually addressed fourvalent intertwiners. But it was only a part of the revision, and it actually contained some mistakes which led to incorrect results. So we looked at that, and we decided to try to fill this gap. 20 00:02:52.150 --> 00:03:09.329 Thiago Lucena: And of course, that also offers a reference point to, let's say, Judge, the commonly use coarse, grained graph, preserving approximations that are preferred because they make things easier. Of course, one just has to look at one single graph. 21 00:03:09.550 --> 00:03:15.899 Thiago Lucena: But it's not entirely clear, at least not quantitatively clear, as far as we know. 22 00:03:15.920 --> 00:03:29.379 Thiago Lucena: whether this is well justified, and once one is able to do to perform graph changing calculations. Then one will probably ask whether one can generate 23 00:03:29.380 --> 00:03:51.260 Thiago Lucena: dynamics from that. Following these ideas of Rosenberger and Rovelli, for example, or how this compares to covariant loop quantum gravity where this concept of expanding a unitary is actually one of the pillars of the thing. And then what happens to observables once you're able to evolve speed networks. 24 00:03:51.750 --> 00:04:03.970 Thiago Lucena: And we're gonna try to answer all these questions, at least to some extent. But before that I think it's more important that we all stay on the same page regarding recoupling theory. 25 00:04:04.471 --> 00:04:09.769 Thiago Lucena: Because I'm pretty sure that not everyone knows the entire thing. 26 00:04:09.990 --> 00:04:28.049 Thiago Lucena: And of course there is a large variety of different conventions. In the literature. In particular, people used to work with temporally tangle algebra at first, st and then there was a change in conventions. 27 00:04:28.470 --> 00:04:31.679 Thiago Lucena: and some results are not interconvertible. 28 00:04:31.710 --> 00:04:37.659 Thiago Lucena: as I just said, but the idea is that the coupling theory allows us to 29 00:04:37.960 --> 00:04:44.969 Thiago Lucena: represent elements of Su. 2 and certain representations in a graphical manner. 30 00:04:45.310 --> 00:04:52.090 Thiago Lucena: And these elements are generally represented as big name matrices. 31 00:04:52.110 --> 00:05:08.389 Thiago Lucena: which are basically these big hollow arrows with the Su 2 element here inside and the representation on top. And some elements are particularly important, like the identity, so that they deserve their own symbols. Just a straight line 32 00:05:08.610 --> 00:05:22.169 Thiago Lucena: or an error. In this case, the anti-diagonal, invariant tensor, which is basically the simplest intertwiner. And I call it a trivial intertwiner because it just has 2 connection points 33 00:05:22.540 --> 00:05:47.589 Thiago Lucena: and one can use these relations to do some manipulations as well. So one can invert an element by basically squeezing it from both sides with this anti-diagonal tensor, or one can pretty much use the anti-diagonal tensor through its cage invariance to create 2 identical vigna elements on the 2 sides of this arrow. 34 00:05:47.850 --> 00:05:51.305 Thiago Lucena: or one can also combine these errors and get an overall 35 00:05:51.810 --> 00:06:13.499 Thiago Lucena: phase factor. That's just twice the spin and an identity on top. And of course the generators of the algebra also have a graphical representation, and that's what people sometimes call grasps. So these are these little things here with one extra link that is related to the actual 36 00:06:15.060 --> 00:06:17.399 Thiago Lucena: index of the generator. 37 00:06:17.962 --> 00:06:25.380 Thiago Lucena: And once one knows how to manipulate single elements, then it's important to handle several elements together. 38 00:06:25.400 --> 00:06:32.260 Thiago Lucena: And we couple these elements through intertwiners. These are invariant tensors of Sv. 2. 39 00:06:32.330 --> 00:06:41.231 Thiago Lucena: And they are usually represented, or they composed into Vigna 3 J symbols. So that's the minimal non-trivial unit. 40 00:06:41.770 --> 00:06:46.340 Thiago Lucena: and from that you can build higher valency intertwiners. 41 00:06:46.660 --> 00:06:51.929 Thiago Lucena: The idea is that we basically have these 3 spins and the 3 magnetic 42 00:06:52.080 --> 00:06:56.560 Thiago Lucena: numbers, and they can be represented graphically as these 43 00:06:57.213 --> 00:07:08.339 Thiago Lucena: nodes here with a plus or a minus that. Just tell you how these spins are assigned. Either anti-clockwise quantity, clockwise or clockwise. 44 00:07:08.450 --> 00:07:27.569 Thiago Lucena: and you can convert between them. That's what sometimes people call braiding, and that gives you a phase of minus one to the power of the sum of these things, and as these things are invariant, you can pretty much absorb or create su. 2 elements, the same element on all of the 3 legs. 45 00:07:27.880 --> 00:07:37.900 Thiago Lucena: So you can use this to actually absorb elements from a bigger spin network into the nodes and leave some degrees of freedom in terms of elements there. 46 00:07:38.110 --> 00:07:40.940 Thiago Lucena: And there is an important relation. 47 00:07:41.180 --> 00:07:51.289 Thiago Lucena: That is the connection between the 3 Valence and the 2 Valence intertwiners, which basically are the same up to one small 48 00:07:51.710 --> 00:08:00.060 Thiago Lucena: prefactor. And this D here just represents the dimension of this representation, and of course, one anti 49 00:08:01.578 --> 00:08:03.129 Thiago Lucena: until diagonal tensor here. 50 00:08:03.940 --> 00:08:30.209 Thiago Lucena: and one can couple 2 wigner matrices as well. And these coupling have respects the Klavich quantum condition. So that's implemented basically here on the nodes by the wigner, 3 j. Symbols. So if you come here with 2 wigner matrices in different representations, you can couple them forming this structure here with 2 connection points 51 00:08:30.210 --> 00:08:36.289 Thiago Lucena: where you have 2 wigner, 3 j. Symbols, and this will enforce that the sum here only runs 52 00:08:36.309 --> 00:08:40.590 Thiago Lucena: over spins that are allowed by clavisch coordinate inequalities. 53 00:08:40.960 --> 00:08:51.990 Thiago Lucena: Of course, you can also couple different elements. In this case, you just have to factor out this element here, and then you will have the the rest here hanging on the side. 54 00:08:52.350 --> 00:08:58.980 Thiago Lucena: and this relation is accompanied by another relation, which is a partner to to move. 55 00:08:59.160 --> 00:09:17.599 Thiago Lucena: This is actually used to change bases of intertwiners. As I said, you can compose higher valence intertwiners from 3 valent intertwiners, and there are multiple ways through which you can perform this combination. So this extra level of redundancy means that you can have different bases. 56 00:09:17.600 --> 00:09:30.159 Thiago Lucena: and you can convert from one into another, using the partner moves. And in specific, this is one partner move. That's very useful, and I found it hard to find in the literature. 57 00:09:30.160 --> 00:09:33.189 Thiago Lucena: so I had to re-derive it from scratch. 58 00:09:33.220 --> 00:09:39.209 Thiago Lucena: and I was somewhat surprised with this. Let's say a symmetry here in the indices 59 00:09:39.480 --> 00:09:51.014 Thiago Lucena: which even led me to discuss this with ilka it's an interesting thing, but it really has to be like this, because there is an orientation in this move here. So you see that the arrow goes from 60 00:09:51.350 --> 00:09:59.219 Thiago Lucena: left to right to bottom to top. So if you perform this 4 times left to right, bottom to top, right to left, and 61 00:09:59.360 --> 00:10:15.459 Thiago Lucena: top to bottom, and then you recover this. Here you will see that you will get 4 minuses that will mutually cancel some indices and lead to this thing being unitary, or you can be just smarter and just flip this thing and apply the 2 2 move again. Then you recover the initial structure. 62 00:10:16.040 --> 00:10:18.369 Thiago Lucena: But that's an important relation to look at. 63 00:10:19.076 --> 00:10:27.659 Thiago Lucena: And, as I said, you can assemble these prevalent intertwiners into higher valence intertwiners. 64 00:10:28.020 --> 00:10:51.859 Thiago Lucena: as I show here, and you can, of course, implement the resolution of the identity using these things. So for any desired valency, you can pretty much introduce the resolution of the identity in this form. So you basically sum over all intermediate spins in the connecting links. So these are virtual links, actually. 65 00:10:51.860 --> 00:10:58.260 Thiago Lucena: And this is basically the direct bra, and this is the direct cat. And I will not represent the big 66 00:10:58.410 --> 00:11:02.134 Thiago Lucena: brackets here. I will just keep the free 67 00:11:02.790 --> 00:11:04.920 Thiago Lucena: legs to opposite sides. 68 00:11:05.460 --> 00:11:20.560 Thiago Lucena: just for simplicity, and one can see a simple or the simplest and most useful implementation of it. In this example, here you have this fraction of a spin network, and basically you break it here at these almost not visible green points. 69 00:11:20.710 --> 00:11:26.469 Thiago Lucena: And then I introduced the resolution of the identity with 3 balance intertwiners 70 00:11:26.590 --> 00:11:33.519 Thiago Lucena: which allow me to basically contract these 3 ends of this triangle here, giving me this structure. 71 00:11:33.730 --> 00:11:50.789 Thiago Lucena: and leave another one simpler intertwiner here on the side which is just the 3 external legs. And this thing here. It has actually a very important meaning, which is the vignus. 6 j. Symbol. So you can see it's a contraction 72 00:11:51.370 --> 00:12:10.409 Thiago Lucena: or 4 3 j's, with some anti-diagonal tensors in between, and this thing here it has some interesting symmetries, namely, you can swap any columns, or you can swap the upper and lower entries of 2 selected columns. It will lead to the same result in the end. 73 00:12:11.330 --> 00:12:11.930 Thiago Lucena: And 74 00:12:12.820 --> 00:12:37.840 Thiago Lucena: overall, one can think about these things and think, okay, that's everything I need to know to study the Hamiltonian constraint. But, as I just said at the beginning, it actually changes the speed networks, and one needs to know how to normalize these things after they have been changed several times, and it is not as trivial. So we know how to normalize intertwiners normalize intertwiners. 75 00:12:38.472 --> 00:12:44.750 Thiago Lucena: As I said, it's just this term here, like actually the inverse of 76 00:12:44.780 --> 00:12:57.440 Thiago Lucena: the norm of the intertwiner is basically given by the inverse of the dimension of these spins here, as you can see for this case of a covalent intertwiner. So if I contract it with itself. 77 00:12:57.470 --> 00:13:02.950 Thiago Lucena: here, have a bra, and here have a cat, and the contraction point is shown in red. 78 00:13:03.030 --> 00:13:07.359 Thiago Lucena: and of course here I had to do some braiding to connect them properly. 79 00:13:07.700 --> 00:13:18.979 Thiago Lucena: but the idea is that you can use recoupling theory and basically factor out this bubble, giving me this factor. And then this remaining thing is just the identity in the end. 80 00:13:19.140 --> 00:13:26.460 Thiago Lucena: So I can easily derive this. And I can generalize this result to show that the normalization just depends on the virtual 81 00:13:27.745 --> 00:13:28.500 Thiago Lucena: spinks. 82 00:13:28.790 --> 00:13:57.709 Thiago Lucena: But in the more general case when I start to change the speed networks through the Hamiltonian constraint. And I create all these inner loops. Things get a little bit more complicated. And that's why we use this thing here that I call no local speed network, because it's not just the intertwiner, but it has some stuff, some actually already coupled inner loops there. So it's the intertwiner and the neighborhood. But it doesn't really 83 00:13:57.760 --> 00:14:07.183 Thiago Lucena: cover the other intertwiners. It's still like a modular structure. But the idea is that you need to keep these loops here with these free 84 00:14:07.950 --> 00:14:26.230 Thiago Lucena: su, 2 elements, because these are key to enforce the autonormality of spin networks, and these are actually the structures that will be formed and coupled to by the Hamiltonian constraint, as I will show later. But the point is that you need to use this relation here 85 00:14:27.058 --> 00:14:30.569 Thiago Lucena: to actually prove the ophthalmology. And if you 86 00:14:30.610 --> 00:14:40.710 Thiago Lucena: omit this element which I've seen in a few papers, then you can't really claim that your speed networks are orthogonal. 87 00:14:40.950 --> 00:14:51.750 Thiago Lucena: Basically, the demonstration is simpler and simple in this case. For these 3 valence structures. You can contract 2 of them. And you form this structure up to topological deformations. 88 00:14:51.790 --> 00:14:59.640 Thiago Lucena: And since one is the Bra, it means it's it's conjugate. So you actually get the inverse element. Then you have to 89 00:14:59.740 --> 00:15:22.529 Thiago Lucena: convert it back into the original element, using your coupling theory relations. And then you use this formula here, and you get this thing. You can factor this out. And then you get this nice normalization factor which you can through induction, show how it works for infinitely many inner structures. So you can form these spider web, like structures 90 00:15:22.530 --> 00:15:31.399 Thiago Lucena: which is what comes out of multiple applications of the Hamiltonian constraint. And you have a well-defined formula for what's its normalization? 91 00:15:31.840 --> 00:15:40.359 Thiago Lucena: But you really need to keep some free elements there so that you can apply this formula and have things actually being orthogonal. 92 00:15:41.630 --> 00:15:45.857 Thiago Lucena: and only then we can actually go to the study of the 93 00:15:46.430 --> 00:15:50.049 Thiago Lucena: to the Hamiltonian constraint. 94 00:15:50.590 --> 00:15:56.909 Thiago Lucena: We investigated 1st the Euclidean constraint, because that's, of course, the 95 00:15:57.150 --> 00:16:07.347 Thiago Lucena: simplest thing to start to start with. And and there wasn't really so much to to start from. So we basically began almost from scratch 96 00:16:08.110 --> 00:16:12.230 Thiago Lucena: and the idea is that there are 3 97 00:16:12.800 --> 00:16:38.380 Thiago Lucena: classes of constraints that one has to fulfill, one being the gauge constraints, the other one, that the film of these constraints. And then there's the Scalar constraint, and ideally, one would address the Lorentzian version of it. We're here just talking about the Euclidean version. But the Lorentzian version is actually ready in the plans, and we know how to extend these things. 98 00:16:38.670 --> 00:16:41.860 Thiago Lucena: and the gauge invariance is naturally implemented 99 00:16:41.900 --> 00:16:44.429 Thiago Lucena: by the Vigna. 3 J. Singles. 100 00:17:03.700 --> 00:17:06.900 Western: I'm afraid there was a problem with the connection 101 00:17:07.089 --> 00:17:08.130 Western: diagno. 102 00:17:09.230 --> 00:17:10.599 Western: Are you still there? 103 00:17:12.500 --> 00:17:15.059 Cong Zhang: Yeah. So I think the problem. The speaker. 104 00:17:15.290 --> 00:17:16.319 Thiago Lucena: Perfect. 105 00:17:17.280 --> 00:17:19.199 Thiago Lucena: Sorry for that. 106 00:17:20.770 --> 00:17:27.560 Thiago Lucena: So basically, the idea is that gauging variance is naturally implemented by the vignet 3 J symbols. 107 00:17:27.630 --> 00:17:36.259 Thiago Lucena: So they basically enforce that these beams of incoming and outgoing links in one node form singlets. 108 00:17:36.670 --> 00:17:40.859 Thiago Lucena: And the film of his invariance is somewhat trickier. 109 00:17:41.070 --> 00:17:56.030 Thiago Lucena: And basically one needs to consider equivalence classes of dual spin networks where you just superpose all the spin networks that are related through the philomorphisms. But we will not account for that here. 110 00:17:56.290 --> 00:17:57.510 Thiago Lucena: although this. 111 00:17:57.610 --> 00:18:07.989 Thiago Lucena: as far as I see, depends a lot on the adjacency relations that the structure that you choose for your speed network. So for some choices. It will not really affect much. 112 00:18:08.050 --> 00:18:30.209 Thiago Lucena: And then there's a scalar constraint which is what we focus on, and it's built from holonomies and from the volume. And and this theeman derivation. And basically, you have 4 different arrangements of these things. The volume is a problematic operator on its own, because you need to basically select 113 00:18:30.270 --> 00:18:39.939 Thiago Lucena: subspaces of spin networks. It acts on a lot of people think even that it acts. I mean it. It's going to span an infinite by infinite matrix, which is not the case. 114 00:18:40.000 --> 00:18:41.699 Thiago Lucena: you can basically. 115 00:18:41.910 --> 00:18:55.709 Thiago Lucena: it's actually block diagonal in the end. So you can select these blocks. But then you have to account for flexible sizes, and you will have to diagonalize it. Then apply the absolute value, then the square root, and then the diagonalize. 116 00:18:56.310 --> 00:19:02.317 Thiago Lucena: which is, of course, costly, and these holonomies, especially these 2 here, that 117 00:19:03.010 --> 00:19:12.375 Thiago Lucena: are inherited from the curvature. They introduce these loops that modify the spin network, and you have to account for these things in 4 different 118 00:19:12.920 --> 00:19:24.460 Thiago Lucena: all the rings, in a way, because these here is an anti-commutator, and the Aigji visibly accounts for loops being coupled clockwisely, or contact clockwisely. 119 00:19:25.212 --> 00:19:28.830 Thiago Lucena: These squares are just a representation. 120 00:19:29.120 --> 00:19:47.480 Thiago Lucena: The partitioning of your manifold into little cubes or polyhedra in general your choice of Polyhedra will affect the prefector, but not really change. What happens here. And these things are the lapses, and we will only consider one 121 00:19:47.770 --> 00:20:05.190 Thiago Lucena: spin network node in a sense, so we don't really have a sum. So this lapse is the only parameter, and that's a parameter we will use as a perturbation parameter later, but it just dictates how you translate things from one foliation to another. 122 00:20:07.980 --> 00:20:15.710 Thiago Lucena: And here is a simple example of how it works for the simplest 3 valent node local spring network. 123 00:20:16.170 --> 00:20:30.650 Thiago Lucena: You 1st introduce this inverse holonomy. And it's basically coupled here vertically. And I basically just couple the identity tail of this matrix to this B network. 124 00:20:30.730 --> 00:20:41.153 Thiago Lucena: then you apply the volume and the volume here for for the 3 valid case, and also for the partially 4 dimension, 4 valid case. It's just 125 00:20:42.470 --> 00:20:50.270 Thiago Lucena: diagonal. And one thing that I must say, I keep here just as as a scalar thing. But this 3.5 means that 126 00:20:50.410 --> 00:20:54.839 Thiago Lucena: because of this holonomy I partially extend the valiancy 127 00:20:55.120 --> 00:20:57.189 Thiago Lucena: of these intertwiner. So 128 00:20:57.290 --> 00:21:19.100 Thiago Lucena: what the volume will see is actually 4 legs, these 2 and these 2, but it cannot grasp this leg here, so it will let grasp this leg, this and this, and it will only feel this extended valency through the attachment here. So that's why I write 3.5. Maybe not the best notation. But still, that's how we kept it. 129 00:21:19.100 --> 00:21:28.110 Thiago Lucena: And then you apply the inverse Holonomy here, I mean the the direct Holonomy. And these 2 things cancel out. You end up with these 2 free 130 00:21:28.765 --> 00:21:34.129 Thiago Lucena: legs here, which are basically at the same point of the manifold, and you close this 131 00:21:34.170 --> 00:21:52.979 Thiago Lucena: with the loopholeonomy and the trace, of course, which I do not represent. So basically it will attach its sense to these 2 legs here, and also go around here coupling to these pings in the spin network forming this structure here. That's 1 of the possibilities. There are actually 4 cases, as I said, for the 3 valid case. 132 00:21:53.190 --> 00:22:07.350 Thiago Lucena: And these things here, they can be factored out using the resolution of the identity. So 1st you factor out these, these and these one, and then you can factor out this one and that results in this big expression 133 00:22:07.540 --> 00:22:13.409 Thiago Lucena: where the factored out term. Just give you weakness. XJ symbols and some phase factors. 134 00:22:13.480 --> 00:22:35.489 Thiago Lucena: And of course you have to account for all the 3 possibilities of coupling of loops. So not only this one, but also here and here. And if you want to do this over and over, then there are some unspoken rules that one must be careful with, which is to say, if you apply things here, and then you apply it again. 135 00:22:35.490 --> 00:23:04.620 Thiago Lucena: Then you can, of course, couple to the same loop here, and that's why we kept it, general, in fact, that just shows how the Hamiltonian is Hermitian. So you can start with spin 0 here and then get spin one half, but you can equally start with one half and get 0, so you can see how you can recover the intertwiner, or you can go from the intertwiner to something with the loop attachment, and in general you can have any spin here, and here you will have the same spin plus minus one half. 136 00:23:04.830 --> 00:23:18.620 Thiago Lucena: But if you want to attach something here, for example, you have to go deeper inside. You can't just start here and attach something to here, but you have to go deeper. It basically translates to the idea that you will need a new 137 00:23:18.860 --> 00:23:29.229 Thiago Lucena: partitioning where you go closer to the intertwiner, because now you have some extra intertwiners here, so you will go closer, and then you will introduce a new loop that is deeper inside. 138 00:23:29.260 --> 00:23:39.159 Thiago Lucena: and then, once you introduce a new loop deeper inside. If you introduce now a new loop here, you will have to go even deeper inside. So this this loop here is locked now. 139 00:23:39.390 --> 00:23:42.387 Thiago Lucena: So it's an interesting thing. There's this kind of 140 00:23:42.990 --> 00:23:54.720 Thiago Lucena: chain rule that is not really talked about anywhere. So we had to figure this out ourselves, and it only gets more complicated when one goes to the prevalent case. 141 00:23:55.511 --> 00:24:11.838 Thiago Lucena: Here they are in total 12 possibilities. So you can attach an inner loop here, or here, or here, or here, or along the diagonals, and of course, because you have the loopholemies and one extra 142 00:24:12.980 --> 00:24:19.059 Thiago Lucena: support polynomy, so you can use either this or this edge as support when you introduce this loop here. 143 00:24:19.160 --> 00:24:37.720 Thiago Lucena: giving a total of 12 possibilities, and each of them gives you a term that's basically this here it's an ugly term, but mostly simple things, just long phase factors, a lot of big chase. The real problem is in the volume matrix. I mean, this is a matrix. Actually. 144 00:24:37.720 --> 00:24:54.860 Thiago Lucena: So you have to build these things along the computation. And these square roots are just normalization factors because we start with no normalized spin networks and we have to normalize them before we apply the volume. Then we denormalize them again. And then we keep going. 145 00:24:55.100 --> 00:24:59.340 Thiago Lucena: and the volume overall is a problem in these computations, both 146 00:24:59.360 --> 00:25:06.349 Thiago Lucena: for the calculation of the constraints and for the calculation of its expectation. Values are as an observable 147 00:25:06.560 --> 00:25:20.329 Thiago Lucena: that actually leads to the volume appearing in 2 different ways, or even in 3 different ways throughout the calculations. So for the 3 valent spin networks. When you raise the valency to 4, 148 00:25:20.880 --> 00:25:28.680 Thiago Lucena: then you have one expression for the volume, which is just diagonal. So it's it's easy, nice. It's a function and the volume. 149 00:25:28.710 --> 00:25:35.819 Thiago Lucena: as an observable of 3 valence intertwiners is just 0. So you don't really have to to do much effort there 150 00:25:36.020 --> 00:25:52.349 Thiago Lucena: for the 4 valent case. You have to derive a volume for actually 5 valence intertwiners when they happen inside of the constraint. And then, as the observable, it's for 4 valent, right? So we will start looking at the 3 valent case. 151 00:25:52.690 --> 00:25:56.600 Thiago Lucena: And, as I said, there are 2 ways, one that goes into 152 00:25:56.630 --> 00:26:05.680 Thiago Lucena: the the constraint, and that happens after the Holonomy coupling. So it will see erased valency, and the one that we actually use as observable. 153 00:26:05.700 --> 00:26:20.180 Thiago Lucena: and the volume is defined by grasping all trios of edges for this 3 valent case. That becomes 4 valent. It's actually trivial, because you can only grasp 3 edges. So you end up with this structure here. 154 00:26:20.720 --> 00:26:22.880 Thiago Lucena: And that's the formula. Basically. 155 00:26:22.970 --> 00:26:40.710 Thiago Lucena: you have these operators here with 3 grasp operators. They are connected. And you generate this extra structure. Here it grasps this P network. You basically use recoupling theory to factor out these things and these prefactors also come from these grasp operators. 156 00:26:41.470 --> 00:26:56.453 Thiago Lucena: If you have more possibilities. Then you have some sums over all possible ways. You grasp 3 legs of your speed network, and there was some. There are some weights here, Kappa, they are somewhat tricky. 157 00:26:57.080 --> 00:27:07.320 Thiago Lucena: and once you build this thing, you basically take its absolute value and square root, which in a matrix language, means that you have to diagonalize it before. 158 00:27:07.540 --> 00:27:11.609 Thiago Lucena: Then you calculate the volume of the of of the speed networks. 159 00:27:12.030 --> 00:27:40.349 Thiago Lucena: and it's important to say that there are different definitions. So there is one definition where you take the absolute value of these things here. First, st but then, computationally speaking, that would be much more costly, because you would have to diagonalize it several times, and it doesn't even make much sense here, because for the cases we studied it surprisingly, turned out that all the contributions here, they give the same matrix. So you just sum the same matrix 4 times for the full balance case. 160 00:27:41.920 --> 00:27:48.440 Thiago Lucena: and that's basically the case I'm talking about here. You're looking at this stricture, you you're gonna grasp 161 00:27:48.660 --> 00:27:58.419 Thiago Lucena: these 4 things 3 at a time, and they are, of course, 4 ways of doing it. And it's important to say that when you grasp this 162 00:27:58.570 --> 00:28:09.119 Thiago Lucena: this link here, you're gonna have a different prefactor that's related to the way how the grasps attach to to vigna matrices. 163 00:28:09.560 --> 00:28:14.480 Thiago Lucena: And one important thing is the choice of this Kappa here. 164 00:28:14.530 --> 00:28:24.859 Thiago Lucena: because it turns out that there are choices that allow you to mutually cancel all the terms, and you will always get 0 for any speed network you act on. 165 00:28:24.960 --> 00:28:28.140 Thiago Lucena: In particular, we set 166 00:28:28.220 --> 00:28:47.929 Thiago Lucena: for every choice of links. We set this to 6, because basically, if you permute the arguments here, it doesn't really give much because you're going to get a minus factor from here. But you also get a minus factor from here. So it just makes computations more complicated if you allow for every possible permutation which is 167 00:28:48.780 --> 00:28:52.410 Thiago Lucena: agglomerate. Everything into equivalence classes. 168 00:28:52.560 --> 00:28:57.229 Thiago Lucena: and the idea is that if we take the 2 1st entries. 169 00:28:57.360 --> 00:29:05.049 Thiago Lucena: we take the wedge product, then take the dual, and if it's perpendicular to the 3rd entry. Sorry if it's 170 00:29:05.230 --> 00:29:10.749 Thiago Lucena: parallel to the 3rd entry, then it gets a plus. If it's anti-parallel, it gets a minus. 171 00:29:10.960 --> 00:29:15.890 Thiago Lucena: and that's how we set these things, and we calculate the volume. And, of course, for the volume 172 00:29:15.940 --> 00:29:24.270 Thiago Lucena: acting as an observable on these things, we just take this being to 0. And then you get a reduced matrix. That is the observable volume. 173 00:29:25.260 --> 00:29:27.559 Thiago Lucena: Then we put this to practice. 174 00:29:28.051 --> 00:29:32.370 Thiago Lucena: And it raises a lot of questions. Because 175 00:29:32.450 --> 00:29:46.800 Thiago Lucena: from the beginning it was not really clear whether one could do much numerics with it. But there were just too many too lengthy equations to handle, so I didn't think one would be able to do much with it just by hand. 176 00:29:46.820 --> 00:29:57.189 Thiago Lucena: And that was when we figure out that there are some language loopholes. For example, in Bofram the the language used in Mathematica 177 00:29:57.190 --> 00:30:20.690 Thiago Lucena: that allow you to work with functions without really defining them. So you never really gave a functional form to them. But all that matters is their arguments. That's very similar to that very early stage of learning quantum mechanics. When the book just tells you that you have a psi, and this psi has some indices there, for example, the angular momenta, and 178 00:30:20.690 --> 00:30:32.270 Thiago Lucena: all that you need are these angular momenta to calculate all the observables, and you never really care about what's the shape of psi. You can just perform all your calculations by knowing the indices. 179 00:30:32.310 --> 00:30:38.709 Thiago Lucena: The same can be done here. So you store all the information you need about the locations and the spins of the spin network 180 00:30:38.910 --> 00:30:41.120 Thiago Lucena: in the argument of the function. 181 00:30:41.470 --> 00:30:44.820 Thiago Lucena: And you can manipulate these functions. So these are called. 182 00:30:45.000 --> 00:30:46.860 Thiago Lucena: we call them ghost functions. 183 00:30:47.000 --> 00:31:08.049 Thiago Lucena: And they have very large arguments. These are basically lists. And these lists include a set of coordinates and spins and a very large pool of zeros. Because since we're going to modify the spin network, we need more entries. So we're going to fill these zeros with extra entries as we modify the spin network. 184 00:31:08.170 --> 00:31:18.130 Thiago Lucena: And the only property that defines psi is that we create a functional that acts on them, and it will give you one whenever the arguments match. 185 00:31:18.170 --> 00:31:31.809 Thiago Lucena: This functional is, of course, anti-linear in the 1st argument, and and linear in the second. And then you can pretty much do everything as you usually do. You really create an effective 186 00:31:32.010 --> 00:31:40.180 Thiago Lucena: vector, space out of these things. And then the scalar constraint is implemented as a linear functional. 187 00:31:40.290 --> 00:31:53.090 Thiago Lucena: and it will basically read the entries of this ghost function and manipulate them according to the equations that we derived, and then give you linear combinations of ghost functions with 188 00:31:53.310 --> 00:31:53.995 Thiago Lucena: properly 189 00:31:55.360 --> 00:31:57.530 Thiago Lucena: chosen arguments. 190 00:31:57.770 --> 00:32:08.400 Thiago Lucena: And the idea is that graph changes are handled by using the pool of zeros. As I just said, you will have the old information about the lower depth loops 191 00:32:08.410 --> 00:32:10.290 Thiago Lucena: being shifted to the right. 192 00:32:10.790 --> 00:32:29.570 Thiago Lucena: and then the new information will be encoded in this space created. So you have here an example of I mean, it might not really make much sense looking at it just like that. But the 1st 4 entries are the outermost spins of the 4 legs, and then the 4 next ones are the innermost spins. 193 00:32:29.600 --> 00:32:34.629 Thiago Lucena: Then the 9th entry is just the central virtual spin. 194 00:32:34.690 --> 00:32:43.910 Thiago Lucena: Then here is a location entry which will tell you where you have loop insertions. And here's Location 2. We have 1, 2, 3, 4, 5, and 6, 195 00:32:44.060 --> 00:32:46.579 Thiago Lucena: and it will have your 196 00:32:47.405 --> 00:32:50.424 Thiago Lucena: the spin of this thing. And then the adjacent 197 00:32:51.100 --> 00:32:57.509 Thiago Lucena: link spins, and then the zeros right? And when we, for example, create a new 198 00:32:58.050 --> 00:33:19.250 Thiago Lucena: loop here and use a new link here, then I don't have a drawing for it here in this slide I will have in the next one. But the idea is that you get these last 4 entries. You shift them to the right by 4. And then you create new information with the data about the new loop created at location one, and that basically translates into this pseudo code here. 199 00:33:19.450 --> 00:33:37.069 Thiago Lucena: So the idea is that if you start with just this 4 valent intertwiner, there are 6 ways in which you can change it by adding links at 6 different locations. 1, 2, 3, 4, 5, 6. Of course you will change all these things. They are located in between. 200 00:33:37.590 --> 00:33:41.920 Thiago Lucena: and that all has to be performed according to Klebbish golden inequalities. 201 00:33:42.822 --> 00:33:51.900 Thiago Lucena: So the idea is that our function now will look at the inputs, be networks. If it's a linear combination, I will just split them and look at each of them. 202 00:33:52.090 --> 00:34:01.609 Thiago Lucena: and it will ask if there are any inner loops present. If no, as is the case here, then it will add all of the 6 in the loops. 203 00:34:01.790 --> 00:34:05.549 Thiago Lucena: otherwise it will ask, Where is this? In a loop? 204 00:34:05.680 --> 00:34:11.520 Thiago Lucena: And namely, it will ask, Where is the innermost loop, because there can be many. 205 00:34:12.449 --> 00:34:19.149 Thiago Lucena: We're gonna look here at the case of location one and the other cases are similar. So they're just represented by the dash line. 206 00:34:19.270 --> 00:34:25.330 Thiago Lucena: And if there's something here at location one, for example, it will just couple a new loop 207 00:34:25.469 --> 00:34:27.910 Thiago Lucena: to this thing, and that might just 208 00:34:28.139 --> 00:34:31.879 Thiago Lucena: change these pins here, or completely remove this extra link. 209 00:34:32.179 --> 00:34:36.435 Thiago Lucena: It will also add the loops to the all the locations here. 210 00:34:36.860 --> 00:34:43.519 Thiago Lucena: that do not cross with these edges. Sorry that do cross with these edges. So these 4 locations 211 00:34:43.940 --> 00:34:47.020 Thiago Lucena: they all will also couple to 212 00:34:47.210 --> 00:34:52.320 Thiago Lucena: links that belong to this loop, so they have to be inserted for the deeper inside. 213 00:34:52.330 --> 00:35:10.229 Thiago Lucena: So if you had something here already, the next one here will be deeper inside. I don't know if it's visible. But the idea is that since they share one common link, you have to go deeper inside. So your partition of your Mini fold has to focus further into the intertwiner. 214 00:35:10.820 --> 00:35:22.880 Thiago Lucena: And for this part, for the particular case of the opposite link loop, right? If you're if you have something here at location one, and then you try to attach something to location 3. 215 00:35:22.950 --> 00:35:28.849 Thiago Lucena: The point is that they are completely independent from each other, and they don't interfere with each other. So you can just 216 00:35:29.010 --> 00:35:35.660 Thiago Lucena: couple something to the location 3. Without even caring about the location one, which means 217 00:35:35.840 --> 00:35:44.509 Thiago Lucena: it can either change the structure by adding a loop or not, change it all. Just shuffle these things, or it can completely remove the loop. 218 00:35:44.980 --> 00:35:54.750 Thiago Lucena: So there is an entire hierarchy of possibilities that one has to account for, and that has actually to be implemented well, in a 219 00:35:55.150 --> 00:36:00.570 Thiago Lucena: cascaded way. If one tries to do perturbative calculations, which is what we do in the end. 220 00:36:00.900 --> 00:36:17.379 Thiago Lucena: So we basically built this unitary. And and here redundantly, I factored out the lapse, even though it was defined as being included in the constraint, but just to highlight, that it is used as our perturbative parameter. And then we expand this. 221 00:36:17.420 --> 00:36:23.980 Thiago Lucena: and we go up to 3rd order for the graph changing case. And up to 4th order in the graph preserving case. 222 00:36:24.540 --> 00:36:32.559 Thiago Lucena: Why? Well, because we're not really able to go to 4th order in a graph changing case. The computational times are just too absurd. 223 00:36:33.550 --> 00:36:34.330 Thiago Lucena: And 224 00:36:35.670 --> 00:36:46.360 Thiago Lucena: we basically do this evolution of the spin networks. We consider basically these 2 node, local spin networks, and and here they are concretely just intertwiners 225 00:36:47.110 --> 00:36:57.680 Thiago Lucena: with the smallest spins on the outer legs, and the inner legs are either 0 or one with 2 different colors. I mean, the whole thing is color coded. 226 00:36:58.130 --> 00:37:22.399 Thiago Lucena: and then we evolve the speed networks, and then we calculate expectation values for the volume as a function of the lapse. One interesting thing is that when I 1st showed these results at loops, I had a somewhat different profile for the graph preserving case which is blowing up. The point is that I try to be very general and keep the whole thing. 227 00:37:22.660 --> 00:37:24.590 Thiago Lucena: The whole analysis. 228 00:37:24.720 --> 00:37:51.260 Thiago Lucena: constraint to one single intertwiner which is not really possible. So the idea is that when you do graph preserving, you really have to fill the entire space between intertwiners. When you add these loopholeonomies, and then, if you break this thing again to isolate one single intertwiner, even though you started with a Hermitian constraint, we will end up with the behavior that's not characteristic of a hermitian constraint. 229 00:37:51.260 --> 00:38:09.219 Thiago Lucena: And that's what I had. So there was a blow up of possibilities because I was basically breaking up the thing and allowing for everything to happen. So I realized that I would have really to stick to particular choices of spin networks to account for the graph preserving case. 230 00:38:09.220 --> 00:38:24.189 Thiago Lucena: and in particular, I want it to be modular so to have little strictures that I can build together, because that's what I do with a graph changing case. So I look at this thing here, and I can do all the changes I want, and then I can glue such things together and create speed networks out of it. 231 00:38:24.580 --> 00:38:27.620 Thiago Lucena: So I looked here 1st at this case. 232 00:38:28.110 --> 00:38:32.519 Thiago Lucena: where I introduce only holonomies along these 2 triangles 233 00:38:33.340 --> 00:38:37.640 Thiago Lucena: so that I can connect these structures together and form sort of rings 234 00:38:37.730 --> 00:38:52.120 Thiago Lucena: of these things. Of course I do not allow for holonomies being added from the sides, which would then lead to autumns in in hours of the lapse, and that will break the symmetry of the plot 235 00:38:52.510 --> 00:38:59.740 Thiago Lucena: from that still. So these constraints choices, we see that for the graph changing case. 236 00:39:01.050 --> 00:39:11.679 Thiago Lucena: we have here basically the solid profiles. And interestingly, for the 2 choices here the profiles are the same. So the coincide entirely. I just made one curve 237 00:39:12.010 --> 00:39:14.509 Thiago Lucena: thicker than the other, so that we could see the 2 238 00:39:14.760 --> 00:39:23.720 Thiago Lucena: for the graph preserving cases. We have these dash profiles, and they depart both from each other and from the graph changing cases. 239 00:39:24.791 --> 00:39:32.959 Thiago Lucena: So they are actually smoother and well, not smoother, but like they changed last thing. They have a smaller second derivative. 240 00:39:33.500 --> 00:39:53.459 Thiago Lucena: and here are the variances for these cases as well. We see that there is a bounce in the variances for the graph preserving cases. That's because of the 4th order term. Also for the volume. It's gonna start increasing so surprisingly, it decreases. But the fact is that the 4th order terms will make it finally increase, and I believe the same holds for the graph changing case. 241 00:39:53.900 --> 00:39:57.909 Thiago Lucena: So as a more complete analysis of the graph preserving 242 00:39:58.647 --> 00:40:06.420 Thiago Lucena: situation. I actually compared 6 different choices. Actually, 5. These data is not completed yet. 243 00:40:06.760 --> 00:40:24.459 Thiago Lucena: but the idea is that I can have, for example, these small speed networks, and then attach the holonomies here or here, and same here, and these are actually flat profiles, so they don't change under the constraint. At least, the changes are probably just numerical error, because they're 244 00:40:24.610 --> 00:40:26.540 Thiago Lucena: 10 to minus 245 00:40:26.580 --> 00:40:50.259 Thiago Lucena: 60 something and then these 2 so shown here as dash lines, and this is shown as a dotted line in red. And you see that there's basically no difference between these 2. So it seems to me that basically increasing the number of connections here all around this intertwiner. If you don't allow for 246 00:40:50.600 --> 00:41:03.539 Thiago Lucena: holonomies being added from the sides, it will not really change much. The volume profile. They really have the same profile up to numerical disagreement, and the same can be seen also in in the variance. 247 00:41:04.860 --> 00:41:10.629 Thiago Lucena: and of course it doesn't make sense to talk just about volumes and not talk about 248 00:41:10.920 --> 00:41:30.460 Thiago Lucena: one of the elephants in the room, which is, the Eigenstates of the constraint, and we use this function, of course, to to try to find eigenstates, and we indeed found some. Some. They are very simple, so we found 2 families, one that's just given by a certain type of intertwiner 249 00:41:30.460 --> 00:41:48.309 Thiago Lucena: and another one that is a more complicated construction that requires summing infinite sums over compositions of spin networks that are generated by the constraint. So this is the most interesting one, because you can really kind of start with a spin network and then 250 00:41:49.320 --> 00:41:52.688 Thiago Lucena: construct an eigenstate out of this big network? 251 00:41:54.380 --> 00:41:57.150 Thiago Lucena: As long as this network fulfills some conditions. 252 00:41:57.340 --> 00:42:03.140 Thiago Lucena: And then the last thing is, how long does it take to compute all these things? 253 00:42:03.230 --> 00:42:22.010 Thiago Lucena: So 2 things that I had to say is that the computational times increase with the spins that one looks at. So if I consider these structures here, for example, with the 4 outer spins and one virtual spin in between, this thing here doesn't change the computational time. 254 00:42:22.030 --> 00:42:44.319 Thiago Lucena: What changes the computational time is what I choose here, and as I increase these spins, you see that the times really increase drastically, you start with just 3 min for the 1st application of the constraint, and then it goes to 10 min, and then it rapidly goes all the way to almost 2 h when you have spins, one everywhere. And of course, if you go to spins 5 255 00:42:44.740 --> 00:42:48.399 Thiago Lucena: that really goes out of hand, if you don't have a more 256 00:42:48.640 --> 00:42:52.010 Thiago Lucena: refined code than mine. 257 00:42:52.810 --> 00:43:15.119 Thiago Lucena: and this is one of the reasons why applying the thing perturbatively takes too much time. And of course, on top of that you're going to have very large superpositions. So you have to act on each of these terms in the linear combination, and that makes the time scale really well, apparently super exponentially, for the graph changing case. 258 00:43:15.130 --> 00:43:23.849 Thiago Lucena: I have here some estimates. These are not really updated results, but for the graph changing and graph preserving. There is a comparison 259 00:43:24.579 --> 00:43:28.969 Thiago Lucena: for one application of the constraint, 2 applications, 3 applications. 260 00:43:29.100 --> 00:43:37.670 Thiago Lucena: This data here is missing because I run them when I was on vacation, and I forgot to set the timer. So I just let the computer run for one entire month. 261 00:43:38.517 --> 00:43:41.562 Thiago Lucena: But that's totally over a week. 262 00:43:42.570 --> 00:43:58.489 Thiago Lucena: and these are some of the functionals that I use. So the collector basically cleans up the mass created by the Hamiltonian. So it basically collects all the coefficients and organizes them so that they're tidy. And then the normalizer pretty much normalizes the speed networks before I apply 263 00:43:58.540 --> 00:44:01.681 Thiago Lucena: either the volume or I just calculate 264 00:44:03.204 --> 00:44:10.479 Thiago Lucena: expectation values in general, and these are fairly rapid, like they're fairly fast even for these extreme cases. 265 00:44:12.060 --> 00:44:16.540 Thiago Lucena: But the the real problem here is how the Hamiltonian constraint 266 00:44:16.570 --> 00:44:19.969 Thiago Lucena: consumes time when one considers graph changes. 267 00:44:20.830 --> 00:44:28.739 Thiago Lucena: So basically, that was most of what we did. We updated the derivations of the action of the pleadian 268 00:44:29.010 --> 00:44:32.520 Thiago Lucena: scalar constraint. On 3 valent and 4 balanced speed networks 269 00:44:33.220 --> 00:44:40.749 Thiago Lucena: we encoded graph changing and graph preserving dynamics numerically, and we also have a volume operator calculator 270 00:44:40.760 --> 00:44:48.214 Thiago Lucena: which one can use either as a matrix generator or is it functional? And we studied the changes that 271 00:44:49.010 --> 00:44:52.960 Thiago Lucena: the graph changes provoke and the volume, expectation, value. 272 00:44:53.910 --> 00:44:57.261 Thiago Lucena: And we look for adding states, of course. 273 00:44:57.720 --> 00:45:12.320 Thiago Lucena: and that's probably the 1st step towards a more systematic study of graph. Changing Hamiltonians. We already have a few directions to to look at in different ways. One is going Laurentian or the one is 274 00:45:12.340 --> 00:45:16.899 Thiago Lucena: is implementing computations in a different perspective. 275 00:45:16.930 --> 00:45:25.030 Thiago Lucena: not gonna spoil it. But there is actually more stuff coming, including this one, which should be ready by the end of the month on archive. 276 00:45:25.710 --> 00:45:33.067 Thiago Lucena: And yes, thank you so much for your attention. I hope the interruption was not that brutal? 277 00:45:33.880 --> 00:45:37.329 Thiago Lucena: And please let me know if you have questions. 278 00:45:37.330 --> 00:45:39.760 Western: Thank you so much. Carlo. 279 00:45:41.730 --> 00:45:42.720 Western: we need to 280 00:45:44.250 --> 00:45:53.229 Western: thank you so much. Thank you also for the introductory part. That was very clear, and I hope it would be useful for the younger 281 00:45:54.290 --> 00:45:57.960 Western: field to get to know. What does it take? A concrete bit. 282 00:45:57.960 --> 00:46:01.880 Wrocław Gravity Group: I'm sorry, Francesca, you are away from the mic. Please speak to the mic. 283 00:46:03.640 --> 00:46:31.290 Western: But I was just thinking, Thiago, for giving such a clear introduction that maybe is going to be useful for the newer people to the field to get to know what does it take to compute with spin networks? So I leave the floor to comments and questions. Please use the rise, your hand up function in order to speak. 284 00:46:50.984 --> 00:46:55.820 FAU Florida Gravity Team: I I we couldn't figure out how to use the raise hand thing, but I have a question. 285 00:46:55.820 --> 00:46:57.270 Western: Go ahead and tell it. 286 00:46:57.800 --> 00:47:16.099 FAU Florida Gravity Team: So you mentioned that you studied you calculated eigenstates of the Hamiltonian constraint. Of course, in vacuum. The most interesting eigenvalue is 0, because it's a constraint. And so I was wondering if you had anything more specific. 287 00:47:16.100 --> 00:47:22.350 Thiago Lucena: Yeah, it's eigenstates for the 0 eigenvalue to normal eigenstates. That's what we were looking for. 288 00:47:23.020 --> 00:47:24.540 FAU Florida Gravity Team: Oh, okay, okay. Got it. 289 00:47:25.100 --> 00:47:27.239 Thiago Lucena: To your actual physical states, right? 290 00:47:27.770 --> 00:47:28.730 FAU Florida Gravity Team: Yeah. 291 00:47:28.730 --> 00:47:33.210 Thiago Lucena: Yeah. So so this thing here, for example, if you apply the the constraint on that, you get 0. 292 00:47:34.320 --> 00:47:35.619 FAU Florida Gravity Team: Okay, got it. Thank you. 293 00:47:38.480 --> 00:47:57.770 Cong Zhang: Yeah. So I actually also have a question on this on this eigenstate of this Hamiltonian constraint. So so have you checked like the magnitude. I mean the the magnitude of this coefficient. It is like exponentially blow up or. 294 00:47:57.770 --> 00:48:04.450 Thiago Lucena: Yeah, that's the point. I I don't know. I couldn't prove that you can normalize this thing. So it might be that 295 00:48:05.220 --> 00:48:10.950 Thiago Lucena: the the magnitude of this this eigenstate is, is just infinity. 296 00:48:11.400 --> 00:48:28.930 Cong Zhang: Yeah. But but the point is like like even like that. For instance, if this, if this, this, this magnitude of this is like like oscillating with respect to some, to some to this N, or something like that. So there should also be this like 297 00:48:29.040 --> 00:48:33.370 Cong Zhang: the amplitude of this of of Yeah. 298 00:48:34.110 --> 00:48:35.060 Cong Zhang: All the. 299 00:48:35.060 --> 00:48:38.170 Thiago Lucena: Didn't check that. So we were. 300 00:48:38.460 --> 00:48:40.980 Thiago Lucena: I mean, there was already. Let's say. 301 00:48:41.270 --> 00:48:52.929 Thiago Lucena: a lot to chew. So we just drop this. And then we were like, okay, we're gonna look at this later. And that's also in the plans to look at this later. But it's just one of the results that we got. 302 00:48:53.230 --> 00:49:00.570 Thiago Lucena: but we didn't really go into depth, into exploring it, because it was so much more to look at. 303 00:49:01.080 --> 00:49:05.560 Thiago Lucena: So we're still trying to process things bit by bit. But 304 00:49:05.610 --> 00:49:25.719 Thiago Lucena: it's it's a lot. Because basically, it's an entire phenomenology that was mostly closed right like people didn't quite have access to it. So we started to get things and and see strange things like the volume going down first, st and it was a lot to process. And then when we figure out this actually generates 305 00:49:26.270 --> 00:49:33.060 Thiago Lucena: eigenstates, we we were like, Wow, that's nice. And at 1st I didn't even want to include it. 306 00:49:33.260 --> 00:49:43.450 Thiago Lucena: And then I talked to some people, and they were like, why not include it there in the paper? Right? And just put it there and then, if you want later, you can come back to it and and investigate. Or maybe someone can do it as well. 307 00:49:45.090 --> 00:49:49.099 Cong Zhang: Okay, okay, yeah. So so the point is like, I ask this because 308 00:49:49.120 --> 00:49:58.799 Cong Zhang: because there is one thing is like, you can find something. But you need to prove this. 0 is really in the spectral of this of the of this operator. 309 00:50:00.140 --> 00:50:01.429 Thiago Lucena: You can use a what. 310 00:50:01.870 --> 00:50:12.560 Cong Zhang: And you need to prove that this 0 0 is like this. Eigenvalue you choose is belonging to the spectrum of this operator. 311 00:50:14.840 --> 00:50:21.219 Thiago Lucena: Yes, I mean, if you apply the the constraint on it, you you see that it actually gives 0. 312 00:50:21.850 --> 00:50:26.639 Thiago Lucena: But it could just be that this entire sum keeps 0 in the end. 313 00:50:26.850 --> 00:50:28.890 Thiago Lucena: I don't think that's the case, though. 314 00:50:29.180 --> 00:50:30.489 Cong Zhang: Okay. Okay. Okay. 315 00:50:30.840 --> 00:50:33.299 Western: Think the next line is Thomas. 316 00:50:35.473 --> 00:50:52.990 Wrocław Gravity Group: Hi, I have, maybe very basic question, namely, if we consider graph changing prescriptions, there are quite a few ways of doing so so have you considered the original proposal by team, and essentially adding this 317 00:50:53.290 --> 00:50:59.199 Wrocław Gravity Group: triangular loops? Or is there some modification to it which you used. 318 00:50:59.200 --> 00:50:59.890 Thiago Lucena: Yeah, we can see 319 00:51:01.100 --> 00:51:05.129 Thiago Lucena: we considered Theman's original version of it. 320 00:51:05.580 --> 00:51:06.949 Wrocław Gravity Group: Okay, thanks. A lot. 321 00:51:07.170 --> 00:51:07.880 Thiago Lucena: Yes. 322 00:51:10.230 --> 00:51:10.750 Thiago Lucena: Okay. 323 00:51:10.750 --> 00:51:11.720 Western: Eugenia. 324 00:51:13.150 --> 00:51:22.707 UP: Whitmore 320: Hey? Yeah, it's impressive that you can do all this numerically. This is really interesting. I have a few questions about 325 00:51:23.840 --> 00:51:26.260 UP: Whitmore 320: possible checks 326 00:51:27.053 --> 00:51:36.349 UP: Whitmore 320: so, for instance, now that you have a you have a framework. Have you tried to use it for 2 plus one 327 00:51:36.940 --> 00:51:59.760 UP: Whitmore 320: gravity where you know the answer, where? It's a topological theory. It seems to me that it's it's something that you can compare to. There are these results of Alejandro Peretz of some years ago, but this would be like a way of testing that you can solve the Hamiltonian constraint with your formula for the for the Eigenstates. 328 00:51:59.800 --> 00:52:01.810 UP: Whitmore 320: You have thoughts, comments on that. 329 00:52:03.090 --> 00:52:08.415 Thiago Lucena: Actually, I wasn't aware of this work. Because I'm I'm not a loop person entirely. 330 00:52:08.780 --> 00:52:16.780 Thiago Lucena: So what we did were all basic checks that that quantum mechanics allows right? So, whether 331 00:52:17.780 --> 00:52:23.289 Thiago Lucena: at no point you violate unitarity or sofa jointness. 332 00:52:23.390 --> 00:52:32.869 Thiago Lucena: and then, if you apply the constraint on the on the bra or on the cat, you get the same thing, and whether you apply it twice here, once a year, you get the same. 333 00:52:33.280 --> 00:52:42.589 Thiago Lucena: whether the volume always gives positive expectation values and so on. So forth. All the basic checks we did. 334 00:52:43.000 --> 00:52:53.153 Thiago Lucena: Everything was fine but but we didn't really find any any reference point there, which is actually good that you told us now. 335 00:52:54.390 --> 00:52:57.639 Thiago Lucena: I'd actually love if you could send me the paper. 336 00:52:58.120 --> 00:53:21.890 Thiago Lucena: because then I can look at this because the the idea here is that we allow for every possibility of creation of inner loops, and that would, of course, be a topological kind of interpretation of the theory, but I can, of course, fine tune things and remove some of these loops by assuming some specific constraints on the triangulation. 337 00:53:22.290 --> 00:53:25.617 Thiago Lucena: and then we can pretty much check whether this fits 338 00:53:27.710 --> 00:53:31.550 Thiago Lucena: some predictions from 2 plus one, or even 3 plus one 339 00:53:32.041 --> 00:53:36.990 Thiago Lucena: it. It really just depends on what I can shut down here, and and and I can 340 00:53:37.240 --> 00:53:41.139 Thiago Lucena: to take my functional and and turn off some of the terms. 341 00:53:41.860 --> 00:53:42.992 Thiago Lucena: of course, manually. 342 00:53:43.780 --> 00:53:46.809 Thiago Lucena: and then see what comes out in the end. 343 00:53:49.200 --> 00:53:55.199 UP: Whitmore 320: Thank you. And I was also curious about your formula for the Against State. Can you go back to the page. 344 00:53:55.350 --> 00:53:56.140 Thiago Lucena: Yeah. 345 00:53:57.060 --> 00:54:04.480 UP: Whitmore 320: Yeah, okay, so this is fully written in terms of matrix elements or the constraint. Yeah. 346 00:54:04.480 --> 00:54:04.990 Thiago Lucena: Yes. 347 00:54:04.990 --> 00:54:07.920 UP: Whitmore 320: There are a few things that I'm missing. 348 00:54:08.280 --> 00:54:11.060 UP: Whitmore 320: Is this one unique state 349 00:54:11.120 --> 00:54:18.050 UP: Whitmore 320: out of many are the coefficients all real? Can you say a bit more about this. 350 00:54:18.410 --> 00:54:24.980 Thiago Lucena: Yes. So this is basically you start with one speed network. 351 00:54:25.420 --> 00:54:36.469 Thiago Lucena: It can also be a linear combination of screen networks that are not related to each other by loop insertions, because if they are related to each other, then you apply the the constraint, and they kind of 352 00:54:36.610 --> 00:54:48.369 Thiago Lucena: flip into each other. That's not nice. So it has to be such that it has 0 overlap with the the outcome of applying the constraint on itself right? 353 00:54:48.470 --> 00:55:01.250 Thiago Lucena: And from that you have the matrix elements of the constraint. And the the 0 is what you start with. The 2 is what you get. After 2 applications of the constraint and then normalizing it. 354 00:55:01.730 --> 00:55:10.270 Thiago Lucena: And then the full 4 applications you normalize. And actually, it's not just normalization. But you have to factor out overlaps with these other terms. 355 00:55:11.260 --> 00:55:16.200 Thiago Lucena: And then, by composing all these things, which is actually not so easy 356 00:55:17.146 --> 00:55:21.180 Thiago Lucena: you get this thing here. And the interesting thing is that? 357 00:55:21.390 --> 00:55:25.128 Thiago Lucena: Well, it's it's actually something that you also see in context, matter 358 00:55:25.670 --> 00:55:42.090 Thiago Lucena: where you you generate also big matrices. But the point is that when you apply the constraint on this, it will generate a term, and then, when you apply the constraint on this, it will generate the same term, but with the opposite coefficient, and they mutually cancel each other from the 2 opposite sides. 359 00:55:45.970 --> 00:55:47.400 UP: Whitmore 320: Very nice. Thank you. 360 00:55:51.390 --> 00:55:52.679 Western: Any more questions. 361 00:55:53.010 --> 00:55:54.510 Western: any further question. 362 00:55:55.590 --> 00:56:06.650 Cong Zhang: Yeah, just a small question. So when you consider this, you can see that the Hamiltonian operator plus is stagger. Or just consider the Hamiltonian operator, the initial one. 363 00:56:07.800 --> 00:56:09.340 Thiago Lucena: Was it? Plus what. 364 00:56:09.770 --> 00:56:12.210 Cong Zhang: Plus a tagger. So the adjoint. 365 00:56:13.510 --> 00:56:16.410 Thiago Lucena: Yes, it's it's it's so called joint. 366 00:56:16.870 --> 00:56:17.520 Cong Zhang: Okay. 367 00:56:17.520 --> 00:56:23.119 Thiago Lucena: This. This whole thing here is self a joint, and that's a key feature for this actually to hold. 368 00:56:23.180 --> 00:56:26.593 Thiago Lucena: If you do not account for for this 369 00:56:27.180 --> 00:56:41.179 Thiago Lucena: permission character it will not work exactly because I need the whole process to be reversible, so that, for example, when I apply it. Here I will generate some speed networks, and then I apply it. Here I will generate the same spin networks with the opposite sign. 370 00:56:41.510 --> 00:56:45.779 Thiago Lucena: and then they mutually cancel each other, and then you just have this chain effect 371 00:56:45.790 --> 00:56:47.873 Thiago Lucena: where you keep cancelling them 372 00:56:48.930 --> 00:56:58.910 Thiago Lucena: pair wise, so each term will span 2 terms right? One that is modified by removing loops, and one that's modified by adding loops. 373 00:56:59.150 --> 00:57:05.909 Thiago Lucena: and then, when I remove loops from here is the same as adding loop on the previous term, and then they mutually cancel each other. 374 00:57:07.260 --> 00:57:14.540 Thiago Lucena: and for that to be possible, I have to allow for both removal and addition of of loops in these PIN networks. 375 00:57:15.310 --> 00:57:17.059 Cong Zhang: Okay. Okay. Thank you. Thank you. 376 00:57:20.870 --> 00:57:26.660 Western: Okay. So if there are no further questions before thanking Tiago. 377 00:57:28.060 --> 00:57:30.030 Western: fantastic Ika, please go ahead. 378 00:57:39.510 --> 00:57:42.430 Ilkka Mäkinen: For inserting the loops. 379 00:57:43.010 --> 00:57:48.189 Western: So, Ika, can you repeat? Because at the beginning your voice was not very clear. 380 00:57:49.830 --> 00:57:54.689 Ilkka Mäkinen: I have a question about the slide where you showed this algorithm for. 381 00:57:55.450 --> 00:57:58.030 Ilkka Mäkinen: or how insert the loops. 382 00:57:59.650 --> 00:58:00.300 Ilkka Mäkinen: Yes, yes. 383 00:58:00.300 --> 00:58:08.039 Thiago Lucena: Yeah. 1st of all, thanks for for the help back then, I mean, it's been almost one year since that. But, like our discussions were pretty pretty cool. 384 00:58:09.030 --> 00:58:12.340 Thiago Lucena: And yeah, please ask your question. 385 00:58:12.340 --> 00:58:17.020 Ilkka Mäkinen: Yeah. So the the this boring technical question. But 386 00:58:18.000 --> 00:58:19.690 Ilkka Mäkinen: so I, 387 00:58:20.940 --> 00:58:28.990 Ilkka Mäkinen: wondering about this case which you have where you say that there is an innermost loop in in location one. 388 00:58:30.100 --> 00:58:37.369 Ilkka Mäkinen: So then, then, you say that you update the spin on the innermost loop, I'm wondering why 389 00:58:37.720 --> 00:58:41.769 Ilkka Mäkinen: why you update the spin. Do you assume that the edge 390 00:58:42.190 --> 00:58:50.320 Ilkka Mäkinen: coming from the new loop is somehow placed on top of the existing edge, and then you recouple the spins. 391 00:58:50.970 --> 00:59:02.019 Thiago Lucena: Yes, and it has to be like this for for the imagenity to work right, because when you couple it again there, then you allow for the removal of the extra link. 392 00:59:02.770 --> 00:59:18.870 Thiago Lucena: because if you couple it again, then this extra link will have its being shifted by plus minus one half. So if it was at one half it might become 0, and then it disappears. And that's how you you have this reversibility, and that's only possible. If you match the 2 things. 393 00:59:19.890 --> 00:59:23.090 Thiago Lucena: but you can only match. If this is the innermost. 394 00:59:23.260 --> 00:59:35.289 Thiago Lucena: then I I put something else again, and then you can either remove the link or raise it spin if there is something in between, then I cannot really do it again. 395 00:59:35.560 --> 00:59:39.470 Thiago Lucena: But that's what enforces the the graphical version 396 00:59:39.490 --> 00:59:42.020 Thiago Lucena: of hermitianity. 397 00:59:44.550 --> 00:59:46.779 Ilkka Mäkinen: I see I'm just 398 00:59:47.780 --> 00:59:50.989 Ilkka Mäkinen: I I'm not sure if if these 3 months 399 00:59:51.280 --> 00:59:55.900 Ilkka Mäkinen: prescription, or if 3 months prescription would make just a new 400 00:59:56.080 --> 00:59:57.569 Ilkka Mäkinen: smaller loop. 401 00:59:58.360 --> 00:59:59.020 Ilkka Mäkinen: so. 402 00:59:59.020 --> 01:00:06.730 Thiago Lucena: Gmail's prescription isn't so isn't really her mission right? He? He actually advocates against this, as far as I remember. 403 01:00:06.930 --> 01:00:19.890 Thiago Lucena: because of the the commutators of the constraints. I don't precisely remember what was the argument, but he was against using a Hermitian version. The point is. 404 01:00:20.120 --> 01:00:27.348 Thiago Lucena: if you take the thing, and then you create. So you add its conjugate right to create the emission constraint 405 01:00:27.860 --> 01:00:44.579 Thiago Lucena: it works as a formula. But when you go to the graphical language. If you don't do this, of putting the new loop inside of the previous one, you will never be able to come back to the original Sp network which will ultimately lead to violation of hermitianity. 406 01:00:46.010 --> 01:00:51.210 Thiago Lucena: So it's really necessary to be this way. Otherwise, you can't really implement this graphically. 407 01:00:51.370 --> 01:00:57.169 Thiago Lucena: because you really need. If you go from spin network A to speed network B, you must be able to go from B to A. 408 01:01:00.240 --> 01:01:01.373 Ilkka Mäkinen: Okay, I 409 01:01:02.150 --> 01:01:15.019 Ilkka Mäkinen: I I'll think about what you said. I. I agree with you that Beaman's Hamiltonian is not her mission. I thought this for a different reason. But maybe it's just the same reason. But you 410 01:01:15.260 --> 01:01:18.140 Ilkka Mäkinen: you expressing it in a different way. 411 01:01:18.760 --> 01:01:25.660 Thiago Lucena: Yes. So yeah, that's a correction to your previous question, because, someone asked me if I use team A. Yes, I use it. But I made it 412 01:01:25.670 --> 01:01:26.355 Thiago Lucena: permission 413 01:01:27.440 --> 01:01:31.080 Thiago Lucena: because I just thought it makes more sense it. It's actually 414 01:01:31.350 --> 01:01:36.680 Thiago Lucena: easier to understand and interpret things if they actually follow the basic rules of quantum mechanics. 415 01:01:37.810 --> 01:01:39.540 Ilkka Mäkinen: Okay. Thank you very much. 416 01:01:39.960 --> 01:01:40.800 Thiago Lucena: Thank you. 417 01:01:41.990 --> 01:01:42.730 Western: Done. 418 01:01:43.390 --> 01:01:59.810 FAU Florida Gravity Team: Yes, so I realized when you said you calculated the eigenstates of the of the constraint. Of course the is that there's a question of what the lapses. So I is that that's for a constant lapse, I assume. Just is that the only case you've solved for. 419 01:01:59.810 --> 01:02:03.670 Thiago Lucena: Or or constant lapse. Yes, you you fix the lapse there. Yeah. 420 01:02:03.670 --> 01:02:04.649 FAU Florida Gravity Team: Okay. So in some sense. 421 01:02:04.650 --> 01:02:07.158 Thiago Lucena: It's principle for one single 422 01:02:08.260 --> 01:02:20.800 Thiago Lucena: intertwiner I mean a node local speed network, as I like to call it. I'm gonna brand this thing. But for one such node local thing, you. 423 01:02:21.130 --> 01:02:28.990 Thiago Lucena: the the whole lapse factors out right? So unless you say it's 0, there's no problem here, and in fact, you can just really factor out before 424 01:02:29.050 --> 01:02:31.730 Thiago Lucena: and and and it will not affect anything. 425 01:02:32.990 --> 01:02:33.649 FAU Florida Gravity Team: And then the Laptop. 426 01:02:33.650 --> 01:02:34.670 Thiago Lucena: When you had, because. 427 01:02:34.670 --> 01:02:35.880 FAU Florida Gravity Team: Constant, right. 428 01:02:36.030 --> 01:02:36.720 Thiago Lucena: Huh! 429 01:02:37.040 --> 01:02:44.040 FAU Florida Gravity Team: The lapse only factors out if it's constant right, I mean, if it differs, or is this only a calculation for one node at a time? 430 01:02:44.230 --> 01:02:46.950 Thiago Lucena: One single intertwiner doesn't really matter. 431 01:02:48.670 --> 01:02:49.580 Thiago Lucena: Okay. 432 01:02:49.580 --> 01:02:50.100 FAU Florida Gravity Team: Got it. 433 01:02:50.100 --> 01:02:56.569 Thiago Lucena: Because it will fuck 2 h from. So you see it, it's here. But it's also here. Right? So it just factors out everywhere. 434 01:02:57.750 --> 01:02:58.440 FAU Florida Gravity Team: Okay. 435 01:02:59.647 --> 01:03:06.729 Thiago Lucena: So you you can. You can say it changes, but it really it doesn't, doesn't. The result is the same in the end. 436 01:03:06.760 --> 01:03:09.099 Thiago Lucena: This is not dependent on the on the lapse. 437 01:03:11.820 --> 01:03:20.970 FAU Florida Gravity Team: All right. So this this solves, because in some sense I mean the Hamiltonian constraint is many constraints. It's a constraint for every choice of lap. So this is a solution to all of the Hamiltonian constraints. 438 01:03:22.297 --> 01:03:27.719 Thiago Lucena: As far as you're concerned with a single intertwiner. I believe so. 439 01:03:27.790 --> 01:03:34.829 Thiago Lucena: The problem is different intertwiners, each of them having a different lapse then it gets more complicated. 440 01:03:35.420 --> 01:03:36.820 FAU Florida Gravity Team: Okay, got it? 441 01:03:44.500 --> 01:03:45.949 Western: Any further question. 442 01:03:49.710 --> 01:04:04.009 Western: If not, I would like to thank everybody for the questions and the comments, and please don't hesitate to contact us. If you have any idea about how to use this new computational tool. 443 01:04:04.360 --> 01:04:09.960 Western: And at this point I thank Thiago again. Thank you very much for your presentation. 444 01:04:10.380 --> 01:04:17.390 Thiago Lucena: Thank you, Francesca, and thank you. Everyone for the attention. And yes, I hope you guys contact us. We really 445 01:04:18.390 --> 01:04:21.380 Thiago Lucena: well, happy to to talk about these things. 446 01:04:22.200 --> 01:04:22.970 Thiago Lucena: Hey? 447 01:04:22.970 --> 01:04:23.730 Western: Thank you.