WEBVTT 1 00:00:03.160 --> 00:00:13.240 Hal Haggard: Welcome everyone. It's my pleasure to introduce Bianca Dietrich who will speak on the continuum limit of spin foams. Thank you, Bianca. 2 00:00:14.090 --> 00:00:17.119 Bianca Dittrich: Thank you, Han, and thank you for 3 00:00:17.680 --> 00:00:28.490 Bianca Dittrich: cargo organizing this. And as a committee to invite me to talk about the continuum limit of spin forms. It probably should be refinement limit. 4 00:00:29.250 --> 00:00:37.740 Bianca Dittrich: That is the limit of having many degrees of freedom. Many bodies like you have many fish, but I'm 5 00:00:37.880 --> 00:00:47.289 Bianca Dittrich: possibly using continuum limit just because people are used to that so to give you a plan. 6 00:00:47.510 --> 00:00:58.940 Bianca Dittrich: I will motivate why we should be looking into the refinement limit, and my motivation is to film off in symmetry. We will then look at various ways to 7 00:00:58.990 --> 00:01:08.239 Bianca Dittrich: construct discretizations which have different morphine symmetry as perfect discretization, and the consistent boundary formalism 8 00:01:08.710 --> 00:01:20.069 Bianca Dittrich: which motivates certain cost training algorithms. I will then comment on peculiarities on the renormalization flow in different environment theories. 9 00:01:20.460 --> 00:01:26.769 Bianca Dittrich: And hopefully, I have time to actually speak about some results on the continuum limit 10 00:01:26.840 --> 00:01:29.830 Bianca Dittrich: for spin forms. And so 11 00:01:29.900 --> 00:01:32.770 Bianca Dittrich: where this work goes back 12 00:01:33.660 --> 00:01:48.690 Bianca Dittrich: a couple of years, and there have been a number of people involved in that. If you want to have a recent review, there was an article we put out last year with Ses and Sebastian 13 00:01:49.210 --> 00:01:51.189 Bianca Dittrich: for the handbook of quantum gravity. 14 00:01:52.500 --> 00:02:02.580 Bianca Dittrich: So the femoral symmetry we would like to use as a guiding principle for quantization. and indeed, it shows the correct number of propagating degrees of freedom. 15 00:02:03.190 --> 00:02:04.519 Bianca Dittrich: and by that 16 00:02:04.730 --> 00:02:15.929 Bianca Dittrich: the correct dynamics at larger scales it, and also ensures constrained implementation. So the constraints come from having the femorphin symmetry. 17 00:02:17.270 --> 00:02:22.059 Bianca Dittrich: and, as we will see, if we have took him off in symmetry, in the discrete. 18 00:02:22.240 --> 00:02:25.420 Bianca Dittrich: it actually ensures discretization, independence. 19 00:02:25.810 --> 00:02:30.849 Bianca Dittrich: and that not only means it's independent of a choice of triangulation 20 00:02:31.040 --> 00:02:37.310 Bianca Dittrich: or any other discretization. But it does resolve discretization, ambiguities, and artifacts. 21 00:02:38.690 --> 00:02:44.259 Bianca Dittrich: And in particular, we will see that it basically trivializes to take the continuum limit. 22 00:02:47.620 --> 00:02:56.950 Bianca Dittrich: But with all these new kind of wonderful things, it's possibly not surprising that discretizations typically break the film morphine symmetry. 23 00:02:57.030 --> 00:03:08.890 Bianca Dittrich: And so the exceptions we know of are topological. Q of t's and 0 plus one dimensional systems. If we choose to go with appropriate discretization. 24 00:03:12.030 --> 00:03:15.020 Bianca Dittrich: And so to explain like these. 25 00:03:15.040 --> 00:03:29.559 Bianca Dittrich: 0 plus one dimensional systems. So that just means usual mechanical systems which you can make carry a notion of different morphism, invariance, or reparametisation. If you do at time as a variable. 26 00:03:30.440 --> 00:03:38.879 Bianca Dittrich: and in that case, in the continuum, you have a a gauge symmetry which is busily a free choice of time function. 27 00:03:40.540 --> 00:03:44.379 Bianca Dittrich: And if you look at a reparameterized particle in a potential. 28 00:03:44.400 --> 00:03:48.890 discretize it in kind of by choosing piecewise 29 00:03:49.020 --> 00:04:11.130 Bianca Dittrich: linear pieces for the trajectory, you will find basically such an action. such a discrete action. which actually looks the same as for the non-parametrized particle. But the differences is that you would vary this action with respect to both variables. QN, that is a position and tn for all TN. 30 00:04:11.940 --> 00:04:18.389 Bianca Dittrich: Now, typically, for if you have any non trivial potential, you will find a unique solution. 31 00:04:19.269 --> 00:04:31.270 Bianca Dittrich: So this is different from the continuum where, from which you would basically expect you have a free choice of T, and that would determine the queues. Here you find a unique solution. But at the same time 32 00:04:31.880 --> 00:04:49.410 Bianca Dittrich: you find that if you look at the Hessian evaluated on the solution, it has basically a large eigenvalue and a very small eig value. So it's almost 0. And so you have almost a symmetry, but not really a symmetry. But overallmatization! Evidence of 33 00:04:49.510 --> 00:04:52.060 Bianca Dittrich: of the continuum is portion. In this case. 34 00:04:53.110 --> 00:04:59.360 Bianca Dittrich: however, you know, you see already the system kind of might wants to be invariant. 35 00:04:59.810 --> 00:05:01.270 Bianca Dittrich: and so, indeed. 36 00:05:01.330 --> 00:05:09.739 Bianca Dittrich: in this case there is a choice of an action which carries this enviance, and we call it the perfect discretization. 37 00:05:09.940 --> 00:05:11.420 Bianca Dittrich: Oh, perfect action! 38 00:05:12.430 --> 00:05:18.849 Bianca Dittrich: This happens if you take the Hemidakobi action, so the action evaluated on a solution 39 00:05:19.130 --> 00:05:35.730 Bianca Dittrich: for given boundary data and use it for the discretization. In that case you can easily produce solution by taking really the continuum solution and choosing any subdivision of this continuum trajectory. 40 00:05:36.280 --> 00:05:38.970 Bianca Dittrich: By your your point. 41 00:05:39.740 --> 00:05:48.839 Bianca Dittrich: so you can move these points along the trajectory in any way you want. And so this shows that you have a gauge symmetry. 42 00:05:48.900 --> 00:05:56.190 Bianca Dittrich: And indeed, if it now checks the Hessian, it has not direction. So reporter reparitization, invariance is restored. 43 00:05:57.240 --> 00:05:58.000 And 44 00:05:58.190 --> 00:06:04.440 Bianca Dittrich: and so you basically to reproduce the continuum trajectory. 45 00:06:04.650 --> 00:06:06.959 Bianca Dittrich: And so you see see that 46 00:06:07.200 --> 00:06:08.500 Bianca Dittrich: this perfect 47 00:06:08.640 --> 00:06:11.630 Bianca Dittrich: action perfectly miller's 48 00:06:11.650 --> 00:06:25.170 Bianca Dittrich: so continuum dynamics at arbitrary scales. So you can also choose the distance between these points arbitrarily large, and you will still get to continuum the same result. As for the continuum. 49 00:06:25.300 --> 00:06:28.049 Bianca Dittrich: so in some sense the continuum limit becomes trivial. 50 00:06:29.030 --> 00:06:33.440 Bianca Dittrich: Now you might ask, what is the case for Vecchi? 51 00:06:33.760 --> 00:06:36.780 Bianca Dittrich: Gravity which underlies spin forms. 52 00:06:37.030 --> 00:06:45.860 Bianca Dittrich: And so for 3D. Reggae and vanishing cosmology constant. They, you know that all solutions are flat 53 00:06:46.830 --> 00:06:56.330 Bianca Dittrich: and effect any triangulation of lead space gives a solution. Particularly, you can move again the vertices around as much as you want. 54 00:06:57.450 --> 00:06:59.580 Bianca Dittrich: So these vertex translations 55 00:06:59.640 --> 00:07:11.690 Bianca Dittrich: acts are basically a representation of different morphine symmetry in the discrete. and again they allow to probe solution at any scale, so he can kind of move all what he says 56 00:07:11.970 --> 00:07:14.359 Bianca Dittrich: in one corner. 57 00:07:14.370 --> 00:07:28.090 Bianca Dittrich: and have very large fields in the other corner, or very large edge lengths in the other corner, and you can even move vertices on top of each other, and in effectly cause screen your triangulation. 58 00:07:30.160 --> 00:07:34.350 Bianca Dittrich: But in 40, Reggie. you can have curvature. 59 00:07:36.620 --> 00:07:40.189 Bianca Dittrich: And so A while ago we 60 00:07:40.320 --> 00:07:46.520 Bianca Dittrich: computed explicitly a family of solutions with and without curvature. 61 00:07:47.990 --> 00:08:00.780 Bianca Dittrich: And so we evaluated the hashing on the solutions. And here you see a plot on the of the lowest ein value, and so at 0 curvature. You indeed have a 0 ein value. 62 00:08:00.960 --> 00:08:15.789 Bianca Dittrich: meaning you have a symmetry. So that was also kind of a reduced triangulation where you only had expected a dipmorphine symmetry in one direction. But if you turn on curvature by changing the boundary data. 63 00:08:15.840 --> 00:08:24.609 Bianca Dittrich: you find that this lowest eigenvalue goes. And so you see that you get something quadratic in curvature. 64 00:08:25.760 --> 00:08:30.370 Bianca Dittrich: But as an example, you also see that this lowest eigenvalue, the size 65 00:08:30.780 --> 00:08:37.990 Bianca Dittrich: here is basically the difference between the lowest and the next one is kind of 7 orders of magnitude. 66 00:08:38.600 --> 00:08:42.300 Bianca Dittrich: So it is a broken symmetry. 67 00:08:43.500 --> 00:08:50.830 Bianca Dittrich: And here we would expect that we have. Well, there are only solutions for small edge lengths. 68 00:08:50.840 --> 00:08:53.059 Bianca Dittrich: approximate values of continuum. 69 00:08:53.440 --> 00:09:08.789 Bianca Dittrich: If you think about what we do in the past, integrated along quantum theory. We integrate over configurations with arbitrary large at edge lengths. And so that means we take into account also the configurations and also solutions 70 00:09:08.890 --> 00:09:14.569 Bianca Dittrich: which could be quite far away from continuum solutions. 71 00:09:17.260 --> 00:09:21.350 Bianca Dittrich: So how can we construct discretizations with different symmetry? 72 00:09:21.680 --> 00:09:27.759 Bianca Dittrich: And the first clue I showed already. That's a hundred. And Jacobi function 73 00:09:29.830 --> 00:09:33.390 Bianca Dittrich: that it's that is the action evaluated on solutions. 74 00:09:34.860 --> 00:09:40.689 Bianca Dittrich: And I mentioned one notion that you get discretization independence. 75 00:09:41.180 --> 00:09:44.400 Bianca Dittrich: which is that you move 1 point on top of 76 00:09:44.600 --> 00:10:02.299 Bianca Dittrich: another point. And so you effectively coarse grain. Your triangulation gets basically inviance. So the same thing. But also, if you explicitly integrate out or solve for the corresponding variable, you do find that you get again the Hemidobi action. 77 00:10:02.320 --> 00:10:06.760 Bianca Dittrich: So in this sense the handed Nakubi action is 78 00:10:07.010 --> 00:10:10.159 Bianca Dittrich: invariant under changes of the discretization. 79 00:10:10.430 --> 00:10:21.660 Bianca Dittrich: And the same holds actually for the propagator. So for the past, integral amplitude itself, if it's a continuum pass integral amplitude. 80 00:10:22.320 --> 00:10:28.099 Bianca Dittrich: so that means that the handwriting Jacobi function is a fixed point of a core screening flow 81 00:10:29.590 --> 00:10:42.790 Bianca Dittrich: where you start with some guests for your Hematnacobi function or for your action. you construct a new one by integrating out points. And so in this case you can just do that with any pair 82 00:10:42.960 --> 00:10:55.910 Bianca Dittrich: or any triple of points, integrate out the middle point, and you get a new action. And if you repeat that you expect to flow to this fixed point, and so this can be applied in practice 83 00:10:56.240 --> 00:11:02.930 Bianca Dittrich: on a perturbative level, and also in the quantum theory for the pass integral to solve the path integral. 84 00:11:03.170 --> 00:11:07.840 Bianca Dittrich: and it does amount to solving the theory and steps 85 00:11:08.670 --> 00:11:22.489 Bianca Dittrich: but it actually seems to also give you a tool, because, in fact, we used it for the unharmonic oscillator to, in fact, solve the path integral. 86 00:11:22.630 --> 00:11:29.870 Bianca Dittrich: Why are fixed point equations? So you can actually use the fixed point equations to solve the continuum pass integr 87 00:11:30.790 --> 00:11:38.710 Bianca Dittrich: and put it provided with a perfect discretization. Another example where it's headful 88 00:11:39.280 --> 00:11:42.740 Bianca Dittrich: is, for instance, for the Ratchipars integral in 3D. 89 00:11:42.770 --> 00:11:54.260 Bianca Dittrich: You can use these fixed point equation, and these cases they involve. Partner moves to construct a unique triangulation in my end, one loop measure. 90 00:11:54.820 --> 00:12:05.510 Bianca Dittrich: and interestingly, it even includes a pforce phase shift which you have, which you get in the asymptotics of the Ponzano Ritchie model. 91 00:12:05.520 --> 00:12:16.039 Bianca Dittrich: where it's usually explained by the sum of orientations. But here you don't need any sum of orientation. So you just get that by a triangulation. Environment's requirement set 92 00:12:16.470 --> 00:12:22.130 Bianca Dittrich: is quite interesting. So you can it? It shows you that you can fix 93 00:12:22.600 --> 00:12:27.989 Bianca Dittrich: parameters and your discretization uniquely. And so, Zissa. 94 00:12:28.160 --> 00:12:33.390 Bianca Dittrich: examples, where these principles are useful to construct 95 00:12:33.640 --> 00:12:37.610 Bianca Dittrich: discretizations, or even solutions to the continuum and the discrete. 96 00:12:38.590 --> 00:12:44.909 Bianca Dittrich: If you go to higher dimensions or more complicated systems. Well, then, if you go to 4 d. 97 00:12:45.090 --> 00:12:50.920 Bianca Dittrich: And try to construct a measure which is invariant and a 5 one moves. So these are particular. Partner moves 98 00:12:51.200 --> 00:13:03.699 Bianca Dittrich: and the action, the rich action and 4 d. Is actually invariant under these partner moves because the 5 one moves, takes the simplex and subdivides it into 5 simplices. But the new solution is also just flat. 99 00:13:04.050 --> 00:13:05.980 Bianca Dittrich: but it turns out that to measure. 100 00:13:06.340 --> 00:13:21.600 Bianca Dittrich: So one, if you want to have it to be invariant to one loop order only has to be non-local. So you can actually prove that. Furthermore, if you construct these perfect discretization via kind of coarse graining 101 00:13:22.410 --> 00:13:29.369 Bianca Dittrich: for 3 letters field series, with or without gauge symmetries, they turn out to be non local. 102 00:13:29.600 --> 00:13:37.430 Bianca Dittrich: So with kind of in principle, infinite couplings at infinite far away sides which decay exponentially. 103 00:13:38.590 --> 00:13:47.099 Bianca Dittrich: And we have also done this construction for 4 d linearized gravity. Again, the principle is non local because his construction involves free transform. 104 00:13:48.050 --> 00:14:03.620 Bianca Dittrich: So one way to deal with this non-locality is to risk to go to restricted space of configurations, and this has been done by Benjamin Ban Sebastian Steinhaus, for instance. And there is context, and there you've 105 00:14:03.670 --> 00:14:11.930 Bianca Dittrich: kind of again, can fix face weights and also find, for instance, a phase transition, and this restricted, restricted context. 106 00:14:13.170 --> 00:14:14.970 Bianca Dittrich: But in general 107 00:14:16.950 --> 00:14:19.700 Bianca Dittrich: I will comment on later. 108 00:14:19.970 --> 00:14:28.070 Bianca Dittrich: These non-localities are very inconvenient, and they basically go against our philosophy philosophy to have local amplitudes. 109 00:14:29.130 --> 00:14:37.530 Bianca Dittrich: so why do? Non topologically, epimorphism, symmetry, discretizations have to be non local 110 00:14:37.600 --> 00:14:41.699 Bianca Dittrich: so far I should claim that you can prove it. 111 00:14:41.760 --> 00:14:48.749 Bianca Dittrich: For the case of Yf. One moves and the results show that it's not local. But there's also kind of a simple argument 112 00:14:48.960 --> 00:14:52.370 Bianca Dittrich: as a choice view that they have to be non-local. 113 00:14:53.850 --> 00:14:57.199 Bianca Dittrich: And so, if you consider a discretization 114 00:14:57.530 --> 00:15:02.910 Bianca Dittrich: here, for instance, of a cylinder, it's a one plus one dimensional model. 115 00:15:03.430 --> 00:15:07.050 And you have a number of discretization points. 116 00:15:07.210 --> 00:15:12.730 Bianca Dittrich: So if your amplitudes are local, that means you can go from T. One to T. 2. 117 00:15:13.010 --> 00:15:19.149 Bianca Dittrich: Why, I completely look at amplitudes which only uses data at T, one and T. 2, 118 00:15:20.050 --> 00:15:22.949 Bianca Dittrich: and it's the same for T. 0 2 t, one. 119 00:15:23.390 --> 00:15:27.730 Bianca Dittrich: and so if you have different symmetry. 120 00:15:27.840 --> 00:15:34.769 Bianca Dittrich: then you can basically move the points around, for instance, at T one. and again effectively co-screen them. 121 00:15:35.530 --> 00:15:45.320 Bianca Dittrich: So you would have in principle less data. But if it's the femor office menu, and you would expect the same answer. If you go from T. 0 to T. 2. 122 00:15:46.090 --> 00:15:53.770 Bianca Dittrich: But if you insist on having local data. it would mean that you lose information going to T. One. 123 00:15:54.360 --> 00:16:06.000 Bianca Dittrich: and you cannot kind of reconstruct everything you had at T. 2, if all the degrees of freedom are propagating. Oh, if you don't have a TQ of t and so, in fact. 124 00:16:06.240 --> 00:16:11.250 Bianca Dittrich: that tells you that you need a non-local action for this to be 2. 125 00:16:12.380 --> 00:16:23.979 Bianca Dittrich: So you, because the non local action allows you to refer not only to the data of kind of 2 slices, but more than 2 slices to construct your dynamics. 126 00:16:27.920 --> 00:16:32.969 Bianca Dittrich: So these non-local amplitudes, as I mentioned, are very cumbersome. 127 00:16:34.320 --> 00:16:52.820 Bianca Dittrich: Some people even kind of put us one of our axioms that we have to be local. So that is maybe another question. But if you think of non local amplitudes would have to revamp the entire formalism and kind of redoing your canonical analysis. 128 00:16:53.040 --> 00:16:58.619 Bianca Dittrich: for instance, or thinking in a completely new framework about boundaries. 129 00:16:58.880 --> 00:17:03.910 Bianca Dittrich: And so, in fact. But I will introduce boundaries now 130 00:17:03.990 --> 00:17:07.889 Bianca Dittrich: as one method to avoid these non local 131 00:17:08.270 --> 00:17:17.149 Bianca Dittrich: actions. and that's a consistent boundary formalism, which kind of was introduced around 2,012. 132 00:17:18.790 --> 00:17:22.789 Bianca Dittrich: So, and you can understand that as a shift of perspective. 133 00:17:23.020 --> 00:17:29.970 Bianca Dittrich: so usually we take, like the simplest gluey building blocks carrying Min and my boundary data 134 00:17:31.000 --> 00:17:35.300 Bianca Dittrich: like simply says, and glue them together to construct our amplitudes. 135 00:17:37.090 --> 00:17:42.410 Bianca Dittrich: But and we want to shift the perspective, and instead, think of 136 00:17:42.600 --> 00:17:48.320 Bianca Dittrich: more generally boundaries, or, you know, more general boundary data. 137 00:17:48.740 --> 00:17:51.320 Bianca Dittrich: an effect we need a partially ordered set 138 00:17:51.660 --> 00:18:01.519 Bianca Dittrich: of building blocks, or you could say of of boundaries, and the partial order should be with respect to the amount of boundary data. 139 00:18:02.410 --> 00:18:11.550 Bianca Dittrich: and what we will need is consistency relation between amplitudes for these building blocks. And that will give us basically the normalization flow. 140 00:18:13.160 --> 00:18:24.720 AAipad2022: Is, was that supposed to be a strategy to avoid non localities or strategies to to to live in in, in presence of non localities. 141 00:18:25.140 --> 00:18:36.769 Bianca Dittrich: It's a strategy to avoid non localities. In the end it will lead to a renormalization flow which looks local. Still it might evolve with more boundary data. 142 00:18:36.820 --> 00:18:40.569 Bianca Dittrich: We will get some locality. And 143 00:18:40.730 --> 00:18:48.600 Bianca Dittrich: anyway, the the crucial point in these, in in this, in this, having this family of boundaries and partially ordered set 144 00:18:49.010 --> 00:18:54.890 where basically the idea is borrowed from from a Qg. And the cylinder consistent constructions, there 145 00:18:55.100 --> 00:19:02.760 Bianca Dittrich: is that we need an embedding map which tells us how to regain the course data from the fine data. 146 00:19:03.480 --> 00:19:06.910 Bianca Dittrich: So in this example, you can just imagine 147 00:19:07.090 --> 00:19:14.519 Bianca Dittrich: well, you can prescribe how the date, how you have to choose the lengths for this more complicated sphere triangulation 148 00:19:14.570 --> 00:19:24.950 Bianca Dittrich: to regain basically something which looks like a tetrahedron but in quantum theory. There's much more choices for choosing these embedding maps. 149 00:19:25.340 --> 00:19:28.149 Bianca Dittrich: And so in principle, you can kind of. 150 00:19:28.450 --> 00:19:33.260 Bianca Dittrich: and that's a kind of tetrahedron and an arbitrarily fine 151 00:19:33.370 --> 00:19:35.489 Bianca Dittrich: apparently. 152 00:19:37.770 --> 00:19:40.460 Bianca Dittrich: and so the consistency 153 00:19:41.400 --> 00:19:55.230 Bianca Dittrich: condition is again the same as you are used in loop on gravity. So if you have amplitudes for this entire family. If you pull back the amplitude from a fine one to a coarse one. 154 00:19:55.290 --> 00:19:59.540 Bianca Dittrich: you want to find again the amplitude for the course one 155 00:19:59.600 --> 00:20:05.870 Bianca Dittrich: from the amplitude for the fine one. So that means that basically, you get always the same answer. 156 00:20:05.880 --> 00:20:07.770 Bianca Dittrich: independent of 157 00:20:07.900 --> 00:20:12.369 Bianca Dittrich: how fine a triangulation you use to compute your answer. 158 00:20:16.090 --> 00:20:19.800 Bianca Dittrich: And so that allows us also an explicit algorithm 159 00:20:19.970 --> 00:20:33.150 Bianca Dittrich: and to construct such consistent amplitudes. So we start with an amplitude for the simplest building block. We define amplitudes for more complicated building blocks. 160 00:20:33.570 --> 00:20:40.550 Bianca Dittrich: for instance, why are gluing, gluing these building blocks to the more complicated one? 161 00:20:42.490 --> 00:20:47.629 Bianca Dittrich: what we then need to do is to construct a truncation. 162 00:20:48.980 --> 00:20:53.709 Bianca Dittrich: And for this truncation, however, again, the choice is crucial. 163 00:20:53.750 --> 00:21:06.429 Bianca Dittrich: and you could say that can be done by this embedding map. I will comment that you can actually find a dynamically preferred embedding map which is basically defined by the amplitudes themselves. 164 00:21:06.570 --> 00:21:11.189 Bianca Dittrich: And in practice it's really important to use these dynamically preferred embedding maps. 165 00:21:11.310 --> 00:21:22.139 Bianca Dittrich: because if you don't use these dynamically preferred embedding maps and just an arbitrarily chosen embedding map and and practice, for instance, for instance, if you just choose something which sets 166 00:21:22.300 --> 00:21:24.860 Bianca Dittrich: autism final degrees of freedom to 0, 167 00:21:25.040 --> 00:21:31.239 Bianca Dittrich: you basically force a system to flow to this vacuum state, where all the degrees of freedom 168 00:21:31.400 --> 00:21:32.630 or set possible 169 00:21:33.650 --> 00:21:43.469 Bianca Dittrich: but if you do that, you do get improved amplitude. It's a fixed point for this iterative flow. for the simplest building block. 170 00:21:44.140 --> 00:21:55.050 Bianca Dittrich: And you can basically repeat these steps and between building blocks which are more complicated. And so you would get improved amplitudes on all States. 171 00:21:56.520 --> 00:22:05.469 Bianca Dittrich: And so you basically iteratively, would construct consistent amplitudes with more and more for more and more complicated boundaries. 172 00:22:05.730 --> 00:22:12.920 Bianca Dittrich: An important point is here that the amplitudes in principle, you can expect that they change across all scales. 173 00:22:14.430 --> 00:22:20.580 Bianca Dittrich: So for the meaning, the scale now is basically the amount of boundary data 174 00:22:21.940 --> 00:22:27.279 Bianca Dittrich: and the renormalization trajectory is actually encoded in this consistent family of amplitudes. 175 00:22:27.820 --> 00:22:30.530 Bianca Dittrich: So 176 00:22:31.140 --> 00:22:38.730 Bianca Dittrich: if you have said, then constructing the continuum limit is a small part of this construction again, because you would know that 177 00:22:38.910 --> 00:22:47.350 Bianca Dittrich: using kind of a coarse triangulation, or using a very fine triangulation, would give you the same answer. 178 00:22:47.410 --> 00:22:56.660 Bianca Dittrich: at least for the, for the questions you can ask at using the coarse triangulation. So for sufficiently scars, observance 179 00:23:00.850 --> 00:23:08.379 Bianca Dittrich: and tools, Us. Ideas actually implemented in in tensor network anomalization methods. 180 00:23:10.120 --> 00:23:15.869 Bianca Dittrich: So here you have building blocks which are these squares with amplitudes 181 00:23:15.880 --> 00:23:25.890 Bianca Dittrich: and say, you have 4 variables and you glue these building blocks to a larger square which has more boundary data, which basically are these blue edges. 182 00:23:25.900 --> 00:23:29.889 Bianca Dittrich: And Louis means that you sum over these shared boundary data. 183 00:23:31.500 --> 00:23:43.270 Bianca Dittrich: and the next step. So now we have a basically building block with more boundary data. But to have something iterative, you need again, the same amount of boundary data you started with. 184 00:23:43.410 --> 00:23:45.030 Bianca Dittrich: And so to do that 185 00:23:46.750 --> 00:23:49.280 Bianca Dittrich: to do the following. 186 00:23:50.180 --> 00:24:01.870 Bianca Dittrich: you find basically the truncation of this boundary data, which as best as possible approximates the the gluing between these bigger 187 00:24:02.120 --> 00:24:03.870 Bianca Dittrich: blocks 188 00:24:05.310 --> 00:24:08.409 Bianca Dittrich: to basically a new building block. 189 00:24:08.920 --> 00:24:12.780 Bianca Dittrich: So and there's a serm out there that 190 00:24:12.810 --> 00:24:14.280 Bianca Dittrich: basically says 191 00:24:14.730 --> 00:24:18.739 Bianca Dittrich: the best approximation you can find in this way 192 00:24:18.940 --> 00:24:22.410 Bianca Dittrich: is by a singular value, decomposition. 193 00:24:23.600 --> 00:24:28.420 Bianca Dittrich: So what you do is you takes these 194 00:24:28.560 --> 00:24:34.930 Bianca Dittrich: these building blocks or these amplitudes, which have a certain amount of boundary data, you 195 00:24:35.580 --> 00:24:39.939 Bianca Dittrich: partition them an in and out boundary data. And this you can 196 00:24:39.950 --> 00:24:47.890 Bianca Dittrich: and quote in a matrix. And for this matrix, you do a singular value decomposition. And what you then do is just to carry 197 00:24:48.140 --> 00:24:57.890 Bianca Dittrich: the Eigenvalues with basically a num, the largest number, the largest Eigenvalues. So you choose a truncation size. 198 00:24:58.020 --> 00:25:02.730 Bianca Dittrich: say 4 and you take just the 4 largest Eigenvalues. 199 00:25:04.620 --> 00:25:15.850 Bianca Dittrich: And this singular value decomposition in particular, the unitary matrices which kind of make the spaces transform. You can understand as fields redefinition. 200 00:25:16.580 --> 00:25:23.219 Bianca Dittrich: And so what you do is to order your fields into more and less relevant for this course training. 201 00:25:24.410 --> 00:25:27.729 Bianca Dittrich: So these basically 202 00:25:27.760 --> 00:25:35.249 Bianca Dittrich: unitary Max. they're basically represented here by these maps, which have 2 203 00:25:35.500 --> 00:25:38.520 Bianca Dittrich: 2 in ongoing and one outgoing edge. 204 00:25:38.740 --> 00:25:44.919 Bianca Dittrich: and these you can use us embedding maps. So you glue them to your larger building block 205 00:25:45.320 --> 00:25:55.900 Bianca Dittrich: block, and you get again something a new building, I mean a building block which now has the same amount of boundary data you started with. But the amplitude changed. 206 00:25:56.380 --> 00:26:02.260 Bianca Dittrich: And so this allows you to iterate this procedure. And that's basically the tensor network algorithm. 207 00:26:03.110 --> 00:26:05.919 Bianca Dittrich: How much freedom is there in this? I mean 208 00:26:06.020 --> 00:26:10.680 AAipad2022: to find this embedding mapping right when it's trying to minimize the error. 209 00:26:10.860 --> 00:26:14.349 AAipad2022: But you might find one way to minimize error. I might find another. 210 00:26:14.370 --> 00:26:24.470 Bianca Dittrich: Yeah. So so in this context, Sscm, that you know, single value. Decomposition is really the best 211 00:26:24.800 --> 00:26:28.969 Bianca Dittrich: choice. Given this problem, you know, which I've thought. 212 00:26:29.460 --> 00:26:30.560 Bianca Dittrich: And 213 00:26:30.590 --> 00:26:33.470 Bianca Dittrich: but different algorithms kind of 214 00:26:33.630 --> 00:26:36.319 Bianca Dittrich: differ in how large 215 00:26:36.350 --> 00:26:46.919 Bianca Dittrich: a triangulation, or in the details of of what you want exactly to approximate. No. So that come goes into more the specifics. So 216 00:26:47.310 --> 00:26:59.260 Bianca Dittrich: a criticism could be that this is kind of too much of a local thing to consider this approximation. So you want to do it with larger building blocks. And so there are different algorithms 217 00:26:59.450 --> 00:27:02.110 Bianca Dittrich: to do that which, however, are more costly. 218 00:27:02.580 --> 00:27:07.880 Bianca Dittrich: and so that's how how these things sometimes differ. 219 00:27:08.580 --> 00:27:14.299 Bianca Dittrich: But there's always a step of of finding 220 00:27:14.650 --> 00:27:29.690 Bianca Dittrich: by, typically of using a singular value decomposition, you can still try something else. But hopefully, the details would not be too much. I mean, you can type from minimizing the error of and comparing that explicitly. 221 00:27:29.770 --> 00:27:36.950 Bianca Dittrich: But also, I believe, the singular value. Decomposition is basically the most cost friendly one. 222 00:27:37.710 --> 00:27:58.200 AAipad2022: Okay. But I mean, there's a difference between what one can do computationally and what is kind of the theoretically proved that something exists. And then, you know, then one might be able to. One might try to find the best way to get there, and there may be a lot of freedom, computational freedom. But here the 2 things seem to be a little bit mixed up in the sense that 223 00:27:58.330 --> 00:28:04.209 AAipad2022: as you're saying, you know, one might have to decide as to what exactly 224 00:28:04.940 --> 00:28:09.460 AAipad2022: what aspects of the of the system should be better approximated. 225 00:28:09.500 --> 00:28:13.779 AAipad2022: or something like that. So I'm not clear about whether there's a clear organism. 226 00:28:13.990 --> 00:28:20.450 AAipad2022: that computation may be difficult to to impose or to carry out. 227 00:28:20.720 --> 00:28:24.289 AAipad2022: or it's kind of one has to do it by trial and error. 228 00:28:25.040 --> 00:28:33.009 Bianca Dittrich: where Zisa. Zisa is a kind of most popular kind of tensor network anomalisations. 229 00:28:34.840 --> 00:28:38.710 Bianca Dittrich: which use a singular value decomposition. 230 00:28:40.790 --> 00:28:42.800 Bianca Dittrich: Their more complicated version. 231 00:28:43.030 --> 00:28:51.510 Bianca Dittrich: which uses some concept called entanglement filtering. But, for instance, they have only been in 232 00:28:52.650 --> 00:28:55.830 Bianca Dittrich: developed in 2 dim for 2 dimensional systems. 233 00:28:56.980 --> 00:28:59.850 Bianca Dittrich: And so indeed. 234 00:29:00.620 --> 00:29:09.569 Bianca Dittrich: maybe to emphasize that the main step or the main. So step is this truncation step is, and this is kind of really choosing the embedding map 235 00:29:11.200 --> 00:29:14.399 Bianca Dittrich: in some sense is connected to choosing. 236 00:29:14.430 --> 00:29:15.530 Bianca Dittrich: What is 237 00:29:16.290 --> 00:29:22.389 Bianca Dittrich: I mean? It's connected to choosing the to constructing a vacuum state. 238 00:29:23.860 --> 00:29:25.370 Bianca Dittrich: But 239 00:29:26.200 --> 00:29:32.049 Bianca Dittrich: this kind of algorithm has been tested on on at least many examples in 2 2 dimensions. 240 00:29:32.390 --> 00:29:34.879 Bianca Dittrich: And we have done all those 3 dimensional systems. 241 00:29:36.230 --> 00:29:39.400 Bianca Dittrich: Okay. yeah, just go ahead. 242 00:29:39.490 --> 00:29:47.839 Bianca Dittrich: So well, this, this allows you to iterate and find fixed points. So the entire 243 00:29:48.190 --> 00:29:56.419 Bianca Dittrich: algorithm, of course, depend on how many boundary data you have. So that's crucial in in basically estimating the cost and what is doable. 244 00:29:57.760 --> 00:29:59.810 Bianca Dittrich: And so 245 00:30:00.860 --> 00:30:19.379 Bianca Dittrich: indeed, where you could in principle compute it for a given amount of boundary data, and then repeat for more boundary data. But this is technically where basically, the the constraints come from on applying this, this algorithm is how much boundary data you want to handle. 246 00:30:21.290 --> 00:30:31.690 Bianca Dittrich: And so we have worked with these tensor networks quite a lot and developing also new algorithms. So we have done a lot in 2D. 247 00:30:32.280 --> 00:30:37.880 Bianca Dittrich: These models which were supposed to physically act as underlocks of spit forms. 248 00:30:40.500 --> 00:30:44.239 Bianca Dittrich: But we also develop new 249 00:30:44.540 --> 00:30:50.190 Bianca Dittrich: versions of these models to be able to treat with a gauge series. 250 00:30:50.430 --> 00:30:56.949 Bianca Dittrich: There. The main difficulty is again, that you want to save on some out of boundary. But if you have gauge series 251 00:30:57.160 --> 00:31:01.959 Bianca Dittrich: you have redundant data. So you kind of want to only work with cage and my aunt data. 252 00:31:02.910 --> 00:31:15.529 Bianca Dittrich: And these models, these algorithms have all always been tested and also applied with 3D. Spin forms. Again, where there is some notion of simplicity, constraints 253 00:31:16.110 --> 00:31:26.210 Bianca Dittrich: which we constructed. and the extracted phase, diagram, and phase transitions in principle. So 254 00:31:27.240 --> 00:31:38.310 Bianca Dittrich: we also constructed a new tensor network algorithm with a fusion basis which the fusion basis is basically a basis which is dual to the spin network basis 255 00:31:38.700 --> 00:31:51.390 Bianca Dittrich: and is much more suited for for coarse graining. It's basically already in the name. Fusion means that you fuse physically the charges in the spaces. 256 00:31:52.150 --> 00:32:06.790 Bianca Dittrich: and in fact, it's more suited for coarse graining, because it also captures torsion, which you do get by a just coarse graining gauge series. In particular, the spin networks. If you coarse grain them, you do get structures which are more 257 00:32:07.340 --> 00:32:09.180 Bianca Dittrich: generous and spinetworks. 258 00:32:10.480 --> 00:32:15.910 Bianca Dittrich: And this is still the leading algorithm. And in the field. 259 00:32:17.140 --> 00:32:22.109 Bianca Dittrich: and Sebastian also considered coupling matter to interd finorlets 260 00:32:22.910 --> 00:32:24.190 Bianca Dittrich: in particular. 261 00:32:25.450 --> 00:32:31.509 Bianca Dittrich: trying to figure out what is a what is exactly the the function or the kind of 262 00:32:31.580 --> 00:32:33.069 Bianca Dittrich: coupling constant 263 00:32:33.350 --> 00:32:41.030 Bianca Dittrich: for these meta models. And that brings me to the to the next point, which is 264 00:32:42.300 --> 00:32:49.359 Bianca Dittrich: what is different between a series where we have different symmetry and series. 265 00:32:49.420 --> 00:32:53.589 Bianca Dittrich: like letter Sketch series, where we have just a background lattice. 266 00:32:53.890 --> 00:33:06.920 Bianca Dittrich: And so in Letter Skate series, you usually start with a given regular letters which has a lattice constant, a. You define the model at the scale. And so you compute the action at larger scales. 267 00:33:07.280 --> 00:33:15.179 Bianca Dittrich: And particular find basically is a beta function, which is how the coupling changes as a function of the scale. 268 00:33:16.850 --> 00:33:20.689 Bianca Dittrich: however, with a dynamic lattice. So if you have different morphine symmetry 269 00:33:21.390 --> 00:33:27.989 Bianca Dittrich: or just gravity. That means you. These edges can have any lengths. 270 00:33:28.560 --> 00:33:34.850 Bianca Dittrich: and it's not only that they can globally have any lengths, but that 2 different edges can have also 2 different lengths. 271 00:33:35.050 --> 00:33:38.689 Bianca Dittrich: So also I've drawn this, let this very regular. 272 00:33:41.120 --> 00:33:43.809 Bianca Dittrich: I copy a highly irregular letters. 273 00:33:45.250 --> 00:33:49.670 Bianca Dittrich: and so if you want to define like 274 00:33:49.850 --> 00:34:00.909 Bianca Dittrich: any any coupling to matter, for instance, on these letters. or any system on this letters. that we need to know the physics, and these couplings 275 00:34:01.060 --> 00:34:11.759 Bianca Dittrich: for all length, scales, and for all possible inhomogeneous length scales. So we need to already know the Beta functions to define the consistent dynamics. 276 00:34:12.449 --> 00:34:14.960 Bianca Dittrich: So that is a lot to ask for. 277 00:34:15.550 --> 00:34:17.940 Bianca Dittrich: And so instead. 278 00:34:18.420 --> 00:34:23.229 Bianca Dittrich: indeed, what we should see is that we have initial seed amplitudes. 279 00:34:24.310 --> 00:34:38.770 Bianca Dittrich: I wouldn't say they are fundamental, because these amplitudes are defined on all length scales, but we wouldn't trust them on arbitrary large length scales. So what we have we should see as a guess of the dynamics over all length scales. 280 00:34:38.909 --> 00:34:48.269 Bianca Dittrich: Then we do this iterative course, training procedures, and to get improved amplitudes. Then again, the amplitudes on all length, scales could be effective. 281 00:34:49.110 --> 00:34:50.250 Bianca Dittrich: affected. 282 00:34:53.199 --> 00:35:05.279 Bianca Dittrich: And so here, maybe, is another example to illustrate the challenges. to just write down actions or amplitudes which are consistent. 283 00:35:05.850 --> 00:35:09.890 Bianca Dittrich: and that's something. You know, I just called letters form factors. 284 00:35:11.220 --> 00:35:18.939 Bianca Dittrich: So here's just an example. You can take, for instance, the harmonic oscillators. But so this will be a 3 field theory. 285 00:35:19.230 --> 00:35:26.159 Bianca Dittrich: But then you have 2 terms, and you have to come up with basically finding functions of the lengths 286 00:35:26.530 --> 00:35:29.859 Bianca Dittrich: around these vertices you are considering. 287 00:35:31.070 --> 00:35:32.870 Bianca Dittrich: So that 288 00:35:33.340 --> 00:35:41.990 Bianca Dittrich: we get the consistent dynamics which basically is reproduced by choosing arbitrary lettuces. 289 00:35:43.420 --> 00:35:48.019 Bianca Dittrich: And so even for the one dimensional system of the harmonic oscillator. 290 00:35:48.050 --> 00:35:55.460 Bianca Dittrich: where these functions would be ratios of trigonometric functions, so not necessarily easy to guess. 291 00:35:55.920 --> 00:35:59.920 Bianca Dittrich: But you can kind of compute them by using the Hamidniac obi function. 292 00:36:01.680 --> 00:36:07.590 Bianca Dittrich: So these functions already have to take the normalization flow into into count. 293 00:36:08.240 --> 00:36:11.230 Bianca Dittrich: And so that's what you need to compute. 294 00:36:14.250 --> 00:36:22.190 Bianca Dittrich: So my claim is kind of impossible to guess all these consistent amplitudes. but you need to construct them via an iterative course gain in flow. 295 00:36:23.220 --> 00:36:26.679 Bianca Dittrich: and the consistent boundary formalism allows to do so. 296 00:36:27.000 --> 00:36:32.049 Bianca Dittrich: And identifies a dynamically preferred truncation scheme. 297 00:36:34.900 --> 00:36:35.870 Bianca Dittrich: Okay. 298 00:36:36.860 --> 00:36:44.269 Bianca Dittrich: so other questions to this, because I slightly change 299 00:36:45.410 --> 00:36:46.290 Bianca Dittrich: topic. 300 00:36:51.790 --> 00:36:52.790 Bianca Dittrich: Okay. 301 00:36:53.920 --> 00:37:03.810 Bianca Dittrich: so let me comment on some more recent work, and that's actually involving for these spin forms 302 00:37:03.870 --> 00:37:05.099 Bianca Dittrich: cartoon. Yep. 303 00:37:05.380 --> 00:37:08.000 John Barrett: just 304 00:37:08.720 --> 00:37:15.510 John Barrett: in general terms, what you've done is, your iterative procedure ends up with some sort of 305 00:37:16.890 --> 00:37:23.710 John Barrett: presumably limiting dynamics in which all the equations that you wanted satisfied? Exactly. 306 00:37:23.990 --> 00:37:31.060 John Barrett: It was consistent. So can you just sort of think of that as being the like the Hamilton Jacobite 307 00:37:31.130 --> 00:37:37.900 John Barrett: flow that you talked about? In the first place, I mean, does it sort of formerly have the same properties? Or is it something sort of different? 308 00:37:38.260 --> 00:37:39.910 Bianca Dittrich: But 309 00:37:41.870 --> 00:37:50.519 Bianca Dittrich: yeah, it would be kind of this, the same as a Hamid Kobe flow. Just was the same Kobe flow example 310 00:37:51.070 --> 00:37:54.109 Bianca Dittrich: that I presented it in one dimension. 311 00:37:54.470 --> 00:38:04.140 Bianca Dittrich: And then, when when I mentioned, you have this cool, very convenient properties. That's a boundary. The amount of boundary data does not change under course training 312 00:38:04.250 --> 00:38:05.240 John Barrett: right? 313 00:38:06.080 --> 00:38:12.430 Bianca Dittrich: Whereas you know. If you do, if you go to higher dimensions. If you go to 2D. Already 314 00:38:12.640 --> 00:38:17.900 Bianca Dittrich: you have Cisco difficulties that you're 315 00:38:18.150 --> 00:38:31.039 Bianca Dittrich: that amount of boundary data changes, and that's where you need this embedding map and the like, choosing a truncation to allow for this iterative procedure. 316 00:38:31.830 --> 00:38:40.239 John Barrett: but you've computed something like the quantum effective action. Is that is that how you think of it, or I mean, do you also have a picture for this limiting? 317 00:38:40.910 --> 00:38:44.240 Bianca Dittrich: so I 318 00:38:45.060 --> 00:38:53.100 Bianca Dittrich: don't have a picture, because in practice we do the right, basically, for I mean, we did it in practice using these tensor network algorithms. 319 00:38:53.280 --> 00:38:57.140 Bianca Dittrich: And the highest examples we went to are 3D 320 00:38:57.700 --> 00:39:04.469 Bianca Dittrich: and did it for a certain amount of boundary data on, for instance. 321 00:39:04.600 --> 00:39:05.620 Bianca Dittrich: cubes. 322 00:39:07.540 --> 00:39:17.170 Bianca Dittrich: But then, at least, if you interpret that in the in the lattice gauge series, then, so the tensor network algorithms it can interpret. 323 00:39:18.220 --> 00:39:25.429 Bianca Dittrich: like both in the kind of spin form systems and lattice gauge theory sense. These, the 324 00:39:25.890 --> 00:39:31.250 Bianca Dittrich: fixed points there, indeed, would give you information about the continuum limit. 325 00:39:32.370 --> 00:39:47.210 Bianca Dittrich: And so you can kind of understand it as a. as yeah, as a quantum, effective action. If you want. If you want to have information over all scales, you would have to tune the system to the phase, transition. 326 00:39:47.670 --> 00:39:50.859 Bianca Dittrich: and need to involve more and more boundary data to 327 00:39:51.870 --> 00:39:55.959 Bianca Dittrich: have to not suffer from truncation artifacts. 328 00:39:58.120 --> 00:39:59.770 John Barrett: Okay? Great thanks. 329 00:40:00.980 --> 00:40:19.930 Daniele Oriti: sorry, Bianca. Can I? Just one more question on the Amazon Jacobi case. In the example you gave you had to evaluate on the solution. And you said that corresponds to you know, the evaluation at a given fixed point. 330 00:40:20.550 --> 00:40:35.319 Daniele Oriti: How does the situation change when you, when you have several solutions? And this will so this solution is not unique. You evaluate Theampton Jacobi on. So you sum over all the possible evaluations of different solutions, or or what? 331 00:40:35.720 --> 00:40:47.190 Bianca Dittrich: Yeah, I wanted to avoid this question. I think in one series that is discussed for some morning oscillator, so that you might have to take into account 332 00:40:47.590 --> 00:40:50.060 Bianca Dittrich: and are solutions which do a full period. 333 00:40:50.920 --> 00:40:51.729 Daniele Oriti: Huh! 334 00:40:51.910 --> 00:40:55.209 Bianca Dittrich: Classically, I haven't stopped about it. 335 00:40:55.280 --> 00:41:02.460 Bianca Dittrich: but I think one can work it out with a harmonic oscillator. What one should do. Okay, thanks. 336 00:41:02.540 --> 00:41:09.110 Bianca Dittrich: But but indeed, in that case is kind of okay. What happens if you choose, like your discretization, step 337 00:41:09.220 --> 00:41:13.150 Bianca Dittrich: larger than at the time for one oscillation? 338 00:41:13.320 --> 00:41:14.200 Daniele Oriti: Yes, indeed. 339 00:41:14.520 --> 00:41:31.440 AAipad2022: just following up on John Barrett's questions. I mean this about possibly effective or something. I mean, you got this form factors. And you know here, I mean, it does look very much like, you know, we have some effective action. So is there any. What is the relation between this 340 00:41:31.530 --> 00:41:33.589 AAipad2022: strategy to find 341 00:41:33.760 --> 00:41:35.989 AAipad2022: the refinement or the continuum limit. 342 00:41:36.380 --> 00:41:44.190 AAipad2022: and the strategy of say, for example, the asymptotic safety in which one has this renormalization group flow. 343 00:41:44.340 --> 00:41:52.010 AAipad2022: and then, you know, one is also trying to find some 344 00:41:52.730 --> 00:41:56.980 Bianca Dittrich: in the final theory I can. So what what is the Mo? What is the relation? 345 00:41:57.650 --> 00:41:59.220 Bianca Dittrich: But 346 00:41:59.650 --> 00:42:06.529 Bianca Dittrich: one relation is an asymptotic safety kind of 347 00:42:07.170 --> 00:42:13.310 Bianca Dittrich: where famously runs renormalization flow backwards from from infrared to UV 348 00:42:15.020 --> 00:42:22.170 Bianca Dittrich: whereas for letters models, one would usually say, one runs from UV to infrared 349 00:42:22.900 --> 00:42:33.449 Bianca Dittrich: but actually, my point of saying this year was that it's not very clear which direction you run, and in in fact, and and Gr. And with 350 00:42:33.660 --> 00:42:39.950 Bianca Dittrich: those being become independent where we have difficulties in determining what a scale. 351 00:42:40.560 --> 00:42:45.309 Bianca Dittrich: And you could argue that we rather use infrared dynamics to 352 00:42:45.340 --> 00:42:51.629 Bianca Dittrich: start with our seat amplitudes. and then use the iterative, coarse, graining procedure to 353 00:42:51.700 --> 00:43:01.839 Bianca Dittrich: find to try to find a consistent family of amplitudes over all scales. So you could say that indeed, what we should do is also to 354 00:43:01.870 --> 00:43:05.449 Bianca Dittrich: find a consistent UV dynamics at all. 355 00:43:05.740 --> 00:43:07.430 Bianca Dittrich: But, 356 00:43:07.690 --> 00:43:20.980 Bianca Dittrich: I would say a priori the well, it could happen that we improve over all scales. But, in fact, what we wouldn't want to change too much is at least semi classical infographics, which is pure. 357 00:43:21.590 --> 00:43:23.529 Bianca Dittrich: So in this sense 358 00:43:23.670 --> 00:43:25.980 Bianca Dittrich: we are similar to asymptotic safety 359 00:43:26.910 --> 00:43:49.449 AAipad2022: did this procedure. There's no guarantee that, you know, when you do this going back, this loops that, in fact, your internal dynamics will not be altered. Guarant, you know. I mean, that will be one of the criteria of successful course. Grading procedure 360 00:43:50.190 --> 00:43:54.940 Bianca Dittrich: offer of a successful series. 361 00:43:54.970 --> 00:44:01.580 Bianca Dittrich: I mean, it's yeah. So you know, I can, of course, start with this 362 00:44:01.720 --> 00:44:07.630 Bianca Dittrich: initial amplitudes which are not suited for this exercise, or many things can move on. 363 00:44:07.880 --> 00:44:12.000 Bianca Dittrich: But indeed that it's 364 00:44:12.400 --> 00:44:15.179 yeah. I think, in general, if we start with some 365 00:44:16.110 --> 00:44:22.539 Bianca Dittrich: attitudes, and I think we should be open because it's a criticism that we kind of people say, are we just? 366 00:44:22.780 --> 00:44:34.369 Bianca Dittrich: I don't guess these amplitudes and claims they are fundamental. but in fact. we kind of should expect that the change over oil scales. 367 00:44:34.590 --> 00:44:42.050 Bianca Dittrich: And so that is implementing the the consistent amplitudes and finding a dynamics consistent of our skills, which, of course. 368 00:44:42.140 --> 00:44:54.989 Bianca Dittrich: is super super difficult, and is is a dream which you know we try has never be, has never been achieved in physics. So is is not an easy 369 00:44:55.300 --> 00:45:01.139 Bianca Dittrich: program. But then, at least this iterative course training procedure 370 00:45:01.850 --> 00:45:21.750 Bianca Dittrich: you do. You can improve in steps. So you will. And and you kind of improve in steps. So the liability of your of your calculations. So in some sense you can hope, maybe that's the difference with asymptotic safety. There another criticism is the choice of truncation. 371 00:45:21.910 --> 00:45:36.289 Bianca Dittrich: and which typically is kind of dictated by locality. whereas here you have these procedures, which is kind of dynamically informed truncations. and maybe also more flexibility in how you choose your truncations. 372 00:45:37.430 --> 00:45:40.510 Bianca Dittrich: okay. 373 00:45:41.790 --> 00:45:44.480 Western: there's another question from Western 374 00:45:45.970 --> 00:45:57.119 Western: yeah. Moment. I have a question about different variants. And I just want to understand what? 375 00:45:57.170 --> 00:46:14.040 Western: What exactly do you mean by this? And the the question is actually about by about your first slide when you when you when you say that if a most invariance is broken, I want to understand what what this mean in in the following sense. 376 00:46:14.170 --> 00:46:29.040 Bianca Dittrich: maybe do that after I do the last part of my talk, I'll wait for the end. That's fine. okay, well. 377 00:46:29.220 --> 00:46:34.090 Bianca Dittrich: maybe. Yeah. So so let me say a few words 378 00:46:34.470 --> 00:46:43.679 Bianca Dittrich: on on actual results, on, on the continuum limit of 4 d spin forms. So it's actually not connected so much to the first part. 379 00:46:43.710 --> 00:46:52.430 Bianca Dittrich: realizing all these issues with consistent amplitudes. 380 00:46:53.030 --> 00:47:01.680 Bianca Dittrich: it's a perturbative result. So for this we have to assume that spin forms a metaphase which at larger scales does lead to smooth geometries. 381 00:47:02.650 --> 00:47:10.320 Bianca Dittrich: Baby knows that we have specified kind of at least 2 different phases. This is a Ashika Lewandowski vacuum. 382 00:47:10.480 --> 00:47:22.890 Bianca Dittrich: where basically, you set all the spins to 0. So it's a very degenerate geometry on the other hand, you have the Bf vacuum where you have bf. or be a homogeneously curved geometry 383 00:47:23.660 --> 00:47:28.699 Bianca Dittrich: still, for spin forms where you import some simplicity constraints. 384 00:47:28.810 --> 00:47:33.339 Bianca Dittrich: You might wish something intervene, at least, for the constraints 385 00:47:33.480 --> 00:47:41.639 Bianca Dittrich: have been set to 0 or set to some cautions, as we will see and still, you have kind of geometrically flatness. 386 00:47:42.950 --> 00:47:59.109 Bianca Dittrich: And so and basically need to make this assumption that to construct the perturbative continuum limit, I mean, I can construct a perturbative continuum limit. But it does rely on this assumption that one can actually do that. 387 00:47:59.910 --> 00:48:02.420 Bianca Dittrich: And the main question is, do we obtain linearized 388 00:48:03.050 --> 00:48:07.470 Bianca Dittrich: gravity? Lennox? That's a perturbative order. I will look at 389 00:48:08.080 --> 00:48:13.799 Bianca Dittrich: and to do that. I will use xactive spin forms. 390 00:48:14.550 --> 00:48:15.320 Okay? 391 00:48:15.430 --> 00:48:19.520 Bianca Dittrich: Because that allows to give me an answer and actually even more information. 392 00:48:21.020 --> 00:48:26.239 Bianca Dittrich: And so the effective spin forms there reconstruct them, because they're much more inevitable to 393 00:48:26.410 --> 00:48:31.970 Bianca Dittrich: calculations, and the dynamics is encoded in a quite transparent way. 394 00:48:32.170 --> 00:48:35.379 Bianca Dittrich: So we directly have this oscillating factor 395 00:48:35.550 --> 00:48:41.409 Bianca Dittrich: which is given by the exponential of the ratchet of the area. Rachi action. 396 00:48:42.280 --> 00:48:51.610 Bianca Dittrich: So here we have areas which kind of give you much more general configurations and lengths, variables. 397 00:48:51.820 --> 00:48:54.940 Bianca Dittrich: And so what you also have are Gaussian factors. 398 00:48:55.040 --> 00:48:59.650 Bianca Dittrich: which imposes constraints reducing areas to length. 399 00:48:59.790 --> 00:49:02.439 Bianca Dittrich: but not sharply, but as cautions. 400 00:49:02.620 --> 00:49:11.499 Bianca Dittrich: And so the this of these kind of these co options is determined by the commutation relations between these constraints. 401 00:49:11.610 --> 00:49:14.540 Bianca Dittrich: which, for instance, involves the barbarian music parameter. 402 00:49:16.260 --> 00:49:29.120 Bianca Dittrich: and so the main part is a reti action. And then you have basically, if you want to write it and kind of an action, an imaginary part which basically involves the logarithm of these cautions. 403 00:49:29.790 --> 00:49:38.800 Bianca Dittrich: And so you can basically ask, what is the lattice continuum limit for the area redj action. And that's an open fashion. Since the 90 s. 404 00:49:39.340 --> 00:49:46.919 Bianca Dittrich: And so. and my understanding was, it's widely assumed to not to lead to gr. 405 00:49:47.060 --> 00:49:53.519 Bianca Dittrich: Also. John taught me at some point he was not the one assuming that it does not leak to Gr. 406 00:49:53.830 --> 00:49:57.539 Bianca Dittrich: And you can claim that this is the original flatness problem. 407 00:49:57.770 --> 00:50:09.690 Bianca Dittrich: because if you look at the equation of motion for area Reggae. It does say it does basically say that some quantity which looks like the deficit angle. But it's actually not the deficit angle has to vanish. 408 00:50:11.200 --> 00:50:20.389 Bianca Dittrich: And so this that's the first question we will answer and But you have also the constraints. So you can ask, do the constraints term change anything. 409 00:50:22.150 --> 00:50:24.780 Bianca Dittrich: and so I won't tell you 410 00:50:25.150 --> 00:50:27.619 Bianca Dittrich: much about how this calculation is done. 411 00:50:28.120 --> 00:50:34.220 Bianca Dittrich: So you do take the action and define it on an infinite 412 00:50:34.440 --> 00:50:35.880 Bianca Dittrich: regular letters. 413 00:50:36.040 --> 00:50:42.530 Bianca Dittrich: and but I have taken also various lattices, and the result don't change. 414 00:50:43.360 --> 00:50:44.060 Okay. 415 00:50:44.190 --> 00:50:50.540 Bianca Dittrich: What you have sent to do is to classify the variables according to their scaling, and how how they behave! 416 00:50:50.660 --> 00:51:01.010 Bianca Dittrich: What is the scaling in the momenta in the Hessian momenta? Here means Fourier labels. So the regular letters allows you to do a Fourier transform. 417 00:51:01.200 --> 00:51:07.380 Bianca Dittrich: And then basically, you can compute a serious expansion of the effective action in the lattice constant. 418 00:51:09.880 --> 00:51:15.860 Bianca Dittrich: And so what is basically the slightest continuum limit for the linearized? Have you a rich action? 419 00:51:16.520 --> 00:51:27.020 Bianca Dittrich: So here is a crucial point is, despite having an enormous amount of variables. So for various discretizations, it's 50 or 100 or 300 420 00:51:27.120 --> 00:51:29.700 Bianca Dittrich: variables pair let aside 421 00:51:30.640 --> 00:51:36.990 Bianca Dittrich: so only degrees of freedom which are massless, are really corresponding 422 00:51:37.110 --> 00:51:38.959 Bianca Dittrich: to actually the lengths matrix. 423 00:51:39.650 --> 00:51:52.009 Bianca Dittrich: So you have only always 10 degrees of freedom which are massless. On top of that you always have this linearized dipmorphine symmetry. so always 4 will be 424 00:51:52.430 --> 00:51:53.350 Bianca Dittrich: each. 425 00:51:54.210 --> 00:52:00.890 Bianca Dittrich: And so this is more or less the reason that in the continuum limit, if you just look at the 426 00:52:01.370 --> 00:52:15.869 Bianca Dittrich: I guess. Now lowest order, and the lattice constant. The continuum limit of this action is given by the linearized ancient heard that action. but we could also compute the next to leading order term, when the lattice constant 427 00:52:16.140 --> 00:52:19.360 Bianca Dittrich: and that was given by a Y square term. 428 00:52:20.680 --> 00:52:25.729 Bianca Dittrich: an effect we could show that it arises from an effective area matrix. 429 00:52:25.760 --> 00:52:34.420 Bianca Dittrich: This array matrix we could construct for each hypercube but also Joseim 430 00:52:34.890 --> 00:52:41.390 Bianca Dittrich: has recently shown that you can construct an area metric for each simplex. 431 00:52:42.470 --> 00:52:54.860 Bianca Dittrich: an area matrix has 20 components. So it has more components. If you integrate, grade out the 10 additional components of the area matrix set is what leads to this Y squared term. 432 00:52:57.050 --> 00:53:03.240 Bianca Dittrich: And so here, what is happening is that you have a retic calculus. It has more degrees of freedom. 433 00:53:03.270 --> 00:53:11.889 Bianca Dittrich: In fact, macroscopically, you can extend, understand that as an extension of the configuration space from lengths to area matrix. 434 00:53:12.910 --> 00:53:23.469 Bianca Dittrich: but only the length, metric degrees of freedom are massless. and all the additional degrees of freedom have Planck, mass. and that's the reason why you do get gr on the limit. 435 00:53:25.400 --> 00:53:30.710 Bianca Dittrich: Do these constraints, terms change anything? And so the answer is 436 00:53:31.090 --> 00:53:35.590 Bianca Dittrich: actually no, they don't seem to be essential in this continuum limit. 437 00:53:36.480 --> 00:53:41.740 Bianca Dittrich: Say so. The constraints only affect the massive degrees of freedom. 438 00:53:41.830 --> 00:53:53.819 Bianca Dittrich: There are these degrees of freedom which are in the area metric, but not in the length, matrix, and so they add something to the mass, which is imaginary and but they are parameter dependent. 439 00:53:54.690 --> 00:53:57.380 Bianca Dittrich: But that doesn't change the overall picture. 440 00:53:58.150 --> 00:54:06.599 Bianca Dittrich: So you kind of still gets a YI squared term as the next correction. But it's a coupling constant. There changes. This is complex parameter. 441 00:54:07.290 --> 00:54:12.540 Bianca Dittrich: So that's actually good news, because it means a universality. 442 00:54:12.630 --> 00:54:15.010 Bianca Dittrich: a kind of universality inside. 443 00:54:15.130 --> 00:54:18.600 Bianca Dittrich: because the different models kind of change. 444 00:54:18.830 --> 00:54:23.490 Bianca Dittrich: or they differ in how the constraints are implemented in details. 445 00:54:24.130 --> 00:54:28.040 Bianca Dittrich: But that doesn't seem to matter too much for the continuum limit. 446 00:54:28.580 --> 00:54:31.010 Bianca Dittrich: So I've got a question here. Yeah. 447 00:54:31.270 --> 00:54:39.110 John Barrett: so you you start with area variables. And then you, you want to say that in the limit, basically, only the length variables 448 00:54:39.250 --> 00:54:40.350 John Barrett: survive. 449 00:54:40.610 --> 00:54:43.589 Bianca Dittrich: Actually, it's a length matrix variables. 450 00:54:43.750 --> 00:54:52.499 John Barrett: Yeah, yeah, exactly. So, how non local is that is it? Is it local in each simplex, or does it need neighboring simplexes, or or what 451 00:54:53.040 --> 00:54:56.960 Bianca Dittrich: it needs? A, I think it's it's it's local on each hypercuper. 452 00:54:57.380 --> 00:55:04.959 Bianca Dittrich: or it's hypercube, right? So even in the in even in the cases where I refine, yeah, I refine these Hypercubes. 453 00:55:07.060 --> 00:55:13.720 Bianca Dittrich: it was still just 10 length variables per 10 length matrix degrees of freedom per hypercube. 454 00:55:14.240 --> 00:55:16.870 John Barrett: Right? So it's not local in each simplex. 455 00:55:17.860 --> 00:55:28.310 John Barrett: No, on each simplex actually, area, we have 10 areas and 10 links. So it doesn't mean that those links are the ones you get from those areas. 456 00:55:29.100 --> 00:55:34.190 Bianca Dittrich: No, it's really a construction on the level of each wipe equipment. 457 00:55:34.480 --> 00:55:37.050 John Barrett: I see. So it's hypochond 458 00:55:37.180 --> 00:55:46.549 Bianca Dittrich: like there's a whether something is massless or or not. You need to lose a Fourier transform for that. You need kind of many Hypercubes? 459 00:55:48.660 --> 00:55:53.109 Bianca Dittrich: So, yeah, so, so, wh? What is? I would like to understand the relation between this 460 00:55:53.170 --> 00:56:15.830 AAipad2022: various part various parts of the talk? So in the last part you sort of had this this this cylinder to consistency requirement, you know the flows that went on. So is this, what was exactly done here, or this is something this part I didn't use cylindrical consistency right? 461 00:56:16.260 --> 00:56:20.709 Bianca Dittrich: So now this, you know, they started from a different corner. 462 00:56:20.800 --> 00:56:21.970 Bianca Dittrich: and 463 00:56:22.170 --> 00:56:35.340 Bianca Dittrich: from effective spin forms which you can see may be in general as an effort to make spin forms more computable and more amenable to one day apply this framework. 464 00:56:36.030 --> 00:56:44.839 Bianca Dittrich: but the result is that well, at least, if you assume that none pert well, that there is a suitable non perturbative 465 00:56:45.190 --> 00:56:48.410 Bianca Dittrich: limit for spin forms so that you get 466 00:56:48.620 --> 00:56:52.469 Bianca Dittrich: a suitable face, and you can do on top of it 467 00:56:52.480 --> 00:56:59.289 Bianca Dittrich: a perturbative construction. Then at least, we know that we can get GR. Out of the limit. 468 00:57:00.240 --> 00:57:15.789 AAipad2022: No, I agree. This is this is very strong, but on that, on the other hand, once we have the answer, can I look at the answer and reinterpret it as saying that? Well, somehow this is not what I did, but in the in retrospect I can think of this as being obtained from cylindrical consistency. 469 00:57:16.630 --> 00:57:38.370 Bianca Dittrich: yeah, I would need to sit down and take a story up with that right now, because that's what we need to patch that. Otherwise it's sort of very disparate ideas. And we don't know. So so basically, what I need to specify is kind of what kind of truncations I use. So 470 00:57:38.660 --> 00:57:44.130 Bianca Dittrich: right? And so basically here III said hopefully. 471 00:57:44.190 --> 00:57:55.340 Bianca Dittrich: hopefully, the interpretation of my microscopic interpretation of what we have of lengths and areas does survive in the continuum limit, so that I can do these perturbative constructions 472 00:57:56.890 --> 00:58:07.519 AAipad2022: that but even within pertinent theory, right? Even like we can talk about. Enormous group flows in the peripatetic theory so similarly here in this, assuming that but should be still able to say that well. 473 00:58:07.680 --> 00:58:16.219 AAipad2022: there is still is really is implementing the idea of cylindrical consistency. And just that that's not how it was arrived at. But that's what it did. 474 00:58:16.230 --> 00:58:19.200 AAipad2022: That's, I think would be really satisfactory. I mean, there's some. 475 00:58:19.420 --> 00:58:26.069 Bianca Dittrich: Yeah, III agree. but I would have to think kind of how to 476 00:58:26.100 --> 00:58:29.859 Bianca Dittrich: how to construct the the kind of embedded map. 477 00:58:30.000 --> 00:58:33.110 AAipad2022: Correct, exact, exact. 478 00:58:33.420 --> 00:58:34.330 Bianca Dittrich: okay. 479 00:58:35.810 --> 00:58:37.430 Bianca Dittrich: David Yamulla. 480 00:58:38.140 --> 00:58:52.310 Bianca Dittrich: as a further consistency result. Just let me mention that you can ask, can I get the same effective action or same action which we got, like Einstein Hilbert, plus, while square directly from the continuum. 481 00:58:53.350 --> 00:58:55.980 Bianca Dittrich: And the short answer is, Yes, you can. 482 00:58:56.360 --> 00:59:08.370 Bianca Dittrich: And for that we kind of use modified Plabanski's theory. So that was proposed by Kirill. And basically the idea is that you take Lebansky, but replace the simplicity constraints. 483 00:59:08.420 --> 00:59:10.060 By Mosomes! 484 00:59:10.360 --> 00:59:34.880 Bianca Dittrich: So we modified it further by saying, we don't replace all constraints by mass terms, but we impose some of these constraints strongly, and these should agree exactly with the same, which which are imposed strongly in the continuum, and some of them we impose weakly or by mass terms. And so we made this choice. And then, indeed, directly from the continuum, we basically found 485 00:59:34.990 --> 00:59:38.500 Bianca Dittrich: the same action that is Einstein Hilbert, plus y squared. 486 00:59:38.890 --> 00:59:49.779 Bianca Dittrich: But here we found a momentum dependent prefactor, which we couldn't see on the lattice, because we only went to the kind of fourth order and derivatives. 487 00:59:50.730 --> 00:59:56.359 Bianca Dittrich: So here we find this prefactor. And surprisingly, it makes this action. Actually. 488 00:59:57.010 --> 01:00:00.250 Bianca Dittrich: it makes this go stream. 489 01:00:01.070 --> 01:00:11.839 Bianca Dittrich: You don't have any higher ports in the propagator. as as a further result. Further remark I will throw out here is that. 490 01:00:12.160 --> 01:00:15.509 Bianca Dittrich: but in principle you have now 491 01:00:15.640 --> 01:00:19.629 Bianca Dittrich: or to gamma in your cyst, in your coupling parameters. 492 01:00:19.740 --> 01:00:25.340 Bianca Dittrich: And so if you introduce a cosmos constant. we have 3 coupling constants. 493 01:00:26.400 --> 01:00:38.500 Bianca Dittrich: and the same in some sense holds another product safety. They have 3 relevant couplings and Cdt. And an edit. 494 01:00:38.930 --> 01:00:55.620 Bianca Dittrich: And it seems you need really at least 3 coupling constants to be able to find a suitable continuum limit. Moreover, in CD. T's additional constant is also an unisotropy parameter. It's a difference between its effector between space like and time, like 495 01:00:56.250 --> 01:01:05.840 Bianca Dittrich: edges. And the same holds also in new common gravity, because Gamma appears in the space like area spectrum, but not in the timelike form. Okay? 496 01:01:06.780 --> 01:01:12.400 Bianca Dittrich: So indeed, where I had basically, these 2 different parts of the talk. 497 01:01:12.510 --> 01:01:21.540 Bianca Dittrich: So the first I was introducing 498 01:01:21.650 --> 01:01:34.410 Bianca Dittrich: kind of a framework for for doing coarse graining and renormalization and background independent theories. And so we see that this notion of tip amorphous symmetry in the discrete and truest discretization. Independence. 499 01:01:34.450 --> 01:01:40.709 Bianca Dittrich: in some sense the continuum limit becomes trivial. You can use perfect actions which mirror 500 01:01:40.870 --> 01:01:52.450 Bianca Dittrich: exactly the continuum dynamics, but are non lucal for non topological theories. The consistent boundary formalisms allows you to avoid these non localities. 501 01:01:52.910 --> 01:02:02.380 Bianca Dittrich: but to construct a renormalization trajectory us, a family of consistent amplitudes. Basically. this dynamically preferred truncation scheme. 502 01:02:02.930 --> 01:02:08.080 Bianca Dittrich: And you can use tensor networks to try to construct that 503 01:02:08.150 --> 01:02:13.440 Bianca Dittrich: we have kind of pushed to 3D. And so they need to be also pushed, too. 4 d. 504 01:02:14.780 --> 01:02:22.409 Bianca Dittrich: And the second part of the talk. We looked at the perturbative continuum limit for effective spin forms. 505 01:02:22.530 --> 01:02:25.379 Bianca Dittrich: But there we have very good news, because. 506 01:02:25.770 --> 01:02:33.140 Bianca Dittrich: we find general relativity at leading order and inspection. It's actually very surprising, because 507 01:02:33.230 --> 01:02:34.820 Bianca Dittrich: an eye for 508 01:02:35.170 --> 01:02:40.479 Bianca Dittrich: I particularly thought that every wretch action would not lead to GR. 509 01:02:40.720 --> 01:02:50.199 Bianca Dittrich: And I think many people did believe the same. And so you can actually see that as a source of the original flatness problem. 510 01:02:51.160 --> 01:03:00.179 Bianca Dittrich: but surprisingly, it does lead to general relativity, and that means that, for instance, the Bell chain model is still might lead to general relativity. 511 01:03:00.550 --> 01:03:14.140 Bianca Dittrich: and, moreover, we could go one order further in the continu in in the expansion and find the Y squared term as a correction. And basically that arises from the extension from length to every matrix 512 01:03:14.440 --> 01:03:23.949 Bianca Dittrich: and this seems to be really a strong signature of spin forms in general that areas are the more fundamental degrees of freedom. 513 01:03:24.720 --> 01:03:26.170 Bianca Dittrich: Thank you. 514 01:03:27.410 --> 01:03:29.029 Hal Haggard: Thank you, Bianca. 515 01:03:31.480 --> 01:03:34.230 Hal Haggard: Western had their hand up with a question. 516 01:03:37.940 --> 01:03:41.490 Bianca Dittrich: though that was the notion of diplomophysymmetry, I guess. 517 01:03:41.730 --> 01:03:50.889 Western: Yeah. Hi, first of all, let me say that. I think the second part that there's this 518 01:03:51.150 --> 01:03:58.440 Western: second talk that you gave on the on the area 519 01:03:58.730 --> 01:04:07.550 Western: action. And it's very good. It's super good, and and makes me very happy, of course. So it's great work, I think. 520 01:04:07.700 --> 01:04:23.420 Western: But my question is about the the first. yeah, thank you for putting that slide. And it's about different. That's something that confuses me. So I want to understand what you mean exactly by different movies. For for let me tell you what confuses me. 521 01:04:23.610 --> 01:04:35.209 Western: The the question you have there is the discretization of a formulation of a particle in a potential. 522 01:04:35.580 --> 01:04:59.389 Western: And of course, the continuous version of this has a different symmetry or symmetry whatever, which is the continuous, which is very clear, because what start with a queue as a function of a parameter, Tau and T as a function of parameter Tau, and as an action which is invariant under implementation of Tau which means that if you, if you plot 523 01:04:59.570 --> 01:05:04.520 Western: queue is a function of Tau in T is a function of Tau, you have a lot of 524 01:05:05.060 --> 01:05:21.270 Western: trajectories which are gauge related. That's my understanding of of of different variants, namely, if you, if you make a different tau, you you have gauge in a gauge equivalent trajectory. 525 01:05:21.390 --> 01:05:23.879 Western: Now, I would say, and maybe 526 01:05:24.190 --> 01:05:30.890 Western: maybe that's what it's it's a terminology different. I would say that if you want to capture the 527 01:05:31.240 --> 01:05:40.950 Western: this, a morphism invariant part of Q. Of Tau, T. Of Tau. You have to look at queue queue as a function of T. You have to solve Tau away. 528 01:05:41.190 --> 01:05:45.869 Western: So if you plot in the Qt. Space a trajectory. 529 01:05:46.130 --> 01:05:48.839 Western: all the different moves invariant 530 01:05:49.050 --> 01:06:02.090 Western: solutions trajectories are all mapped to the same quantity. So given a plot in Qt. Space, that's a difficult environment quantity. Because if I now make a different office, nothing changes. 531 01:06:02.620 --> 01:06:09.650 Western: Now you seem to be saying that that's not the case, that once we go down to the Qt. Space. 532 01:06:09.670 --> 01:06:15.580 Western: still, there is something that you still want to call different, and that's what escapes me. 533 01:06:19.480 --> 01:06:22.609 Bianca Dittrich: They don't know. Yeah, I mean, it's 534 01:06:23.430 --> 01:06:37.019 Bianca Dittrich: I could. You could say it's on this level that indeed, if you choose. If you choose the teas to be differently. In the continuum as a function of Tau 535 01:06:39.120 --> 01:06:41.809 Bianca Dittrich: you you find 536 01:06:41.890 --> 01:06:50.649 Bianca Dittrich: kind of the same solution, but you find the same solution. But it's the same continuum solution. Here you don't find the same solution. 537 01:06:51.040 --> 01:06:56.849 Western: That's what I don't understand. What what. So how form of that blue curve. 538 01:06:57.560 --> 01:06:58.560 Bianca Dittrich: Sorry! 539 01:06:58.790 --> 01:07:06.289 Bianca Dittrich: What is the gauge transformation of that blue curve? I would say it remains itself. But you say, doesn't. 540 01:07:06.410 --> 01:07:17.410 Bianca Dittrich: What is a gauge? Does I mean, first of all, gauge symmetry is broken. So if I do change the blue points here, I do get a slightly different curve 541 01:07:17.640 --> 01:07:22.400 Bianca Dittrich: connected by these piecewise linear, piecewise linear pieces. 542 01:07:22.560 --> 01:07:38.140 Bianca Dittrich: but it would be a different depend dependence of Q and T. So that's I would call it physical. Why would call it a change in no, I mean, we can. Yeah. So so again, this notion of different one. 543 01:07:38.180 --> 01:07:40.399 Bianca Dittrich: which then leads to constraints. 544 01:07:41.550 --> 01:07:43.949 Bianca Dittrich: And so, if you want to have this relation. 545 01:07:44.380 --> 01:07:50.420 Bianca Dittrich: You you! You come to call it the femoral symmetry. 546 01:07:50.530 --> 01:07:58.430 Bianca Dittrich: If you want to have really a notion of where, in the continuum, I really have one degree of freedom which is physical, which is Q. 547 01:07:58.750 --> 01:08:03.960 Bianca Dittrich: Whereas here, if you kind of ignore that, that it is a broken symmetry. 548 01:08:04.060 --> 01:08:07.340 Bianca Dittrich: then you have 2 degrees of freedom, which are q and T. 549 01:08:07.870 --> 01:08:15.259 Bianca Dittrich: So from this point of view of having the same number of degrees of freedom, and the continuum and the discrete. 550 01:08:15.420 --> 01:08:17.369 Bianca Dittrich: or having constraints. 551 01:08:18.640 --> 01:08:23.499 Bianca Dittrich: or having gauge equivalent set of solutions which you get on the right hand side. 552 01:08:25.200 --> 01:08:35.159 Bianca Dittrich: you should call it the phimorphic symmetry. and so in the and in the continuum again. you can choose t arbitrarily as a function of Tau. 553 01:08:35.939 --> 01:08:43.759 Bianca Dittrich: and in the discrete you want to do the same, and on the right hand side you can indeed choose T to be arbitrarily. 554 01:08:44.689 --> 01:08:54.650 Western: No, I understand that what comes after I'm questioning so what would be a gauge 555 01:08:55.300 --> 01:09:07.550 Bianca Dittrich: gauge symmetry. You had a paper where you say, basically, oh, actually choosing the discrete values of Tau is a gauge symmetry. Now, the Taos are actually not even variables. 556 01:09:07.800 --> 01:09:18.679 Bianca Dittrich: And they do drop out of this discretization. That's why you say it's a gauge parameter. But the tows are not even variables. So for me. That is not a symmetry, because the towers are just arbitrary parameters. 557 01:09:19.890 --> 01:09:27.230 Bianca Dittrich: and what are variables? Are Q. And T. And you are asking, then at least me. I am asking 558 01:09:27.330 --> 01:09:32.580 Bianca Dittrich: what is a symmetry which kind of maps solutions to solutions. 559 01:09:34.109 --> 01:09:41.659 Bianca Dittrich: and leave the action in my aunt. And so the answer is, is. while in that case you go to perfect actions. 560 01:09:44.520 --> 01:09:56.940 Western: Okay? I'm also understanding. I mean, I understand, in the case of the perfect action, as you say in the head to Jacobi, but I, and understanding the case of the in the left case, what would be a gauge transform of the blue curve? 561 01:09:57.310 --> 01:10:01.269 Bianca Dittrich: What is? Hello! I'm I'm I'm saying there is no gauge. 562 01:10:01.870 --> 01:10:06.510 Bianca Dittrich: Sorry the minute I am saying there is no gauge symmetry, because the symmetry is broken. 563 01:10:11.750 --> 01:10:22.930 Bianca Dittrich: There would be a gauge symmetry if the Hessian, evaluated on a solution has a 0 eigenvalue. But there's only a small eigenvalue, and that's why I say it's actually a broken symmetry. 564 01:10:25.720 --> 01:10:26.970 Bianca Dittrich: And so. 565 01:10:28.000 --> 01:10:29.509 Bianca Dittrich: but for me. 566 01:10:29.700 --> 01:10:34.699 Bianca Dittrich: age symmetries have to also do something non-trivial on the ladder of of C. Action 567 01:10:35.200 --> 01:10:38.000 Bianca Dittrich: of on the level of the of the variables. 568 01:10:39.170 --> 01:10:43.479 Bianca Dittrich: whereas your your notion doesn't seem to do anything on, on. 569 01:10:43.960 --> 01:10:46.330 Bianca Dittrich: on the action, on the variables. 570 01:10:50.540 --> 01:10:58.710 Bianca Dittrich: I mean, you are basically saying there is a notion of dipmorphine symmetry, because I define dipmorphic symmetry to leave this curve invariant. 571 01:11:01.050 --> 01:11:07.500 Western: Yeah, yeah. So I don't understand what is broken. It's a II don't see the symmetry which is broken. 572 01:11:08.670 --> 01:11:24.979 Bianca Dittrich: Well, first, it suggests the the mathematical effect that this is very small. Eigenvalue. Okay, that's what you mean the existence of a smaller value. That's but then you can even go in a in a that if you do the renormalization, of course, training 573 01:11:25.240 --> 01:11:28.240 Bianca Dittrich: you do that, you? You restore the symmetry 574 01:11:29.990 --> 01:11:30.670 Bianca Dittrich: P, 575 01:11:32.210 --> 01:11:33.030 Bianca Dittrich: and 576 01:11:33.140 --> 01:11:36.779 Bianca Dittrich: kind of that's that's a behavior of many other gauge symmetries. 577 01:11:41.390 --> 01:11:42.310 Western: Okay. 578 01:11:43.000 --> 01:11:48.320 John Barrett: aren't you just saying that the action doesn't depend on the intermediate values of T. 579 01:11:50.120 --> 01:11:52.089 Western: So tell me what you say 580 01:11:52.460 --> 01:11:57.050 Bianca Dittrich: in the in the I mean here on the left hand side, or on the right hand side. 581 01:11:57.100 --> 01:11:58.699 John Barrett: on the left hand side. 582 01:11:58.850 --> 01:12:03.379 Bianca Dittrich: on the left hand side. It does depend on the intermediate values of T, 583 01:12:03.440 --> 01:12:06.000 John Barrett: yeah. So on. On the right hand side. It, doesn't 584 01:12:06.470 --> 01:12:10.670 Bianca Dittrich: it? It still does. But what the inviance is is where it's 585 01:12:11.160 --> 01:12:14.530 Bianca Dittrich: what what else I mean. If you if you 586 01:12:15.200 --> 01:12:16.609 Bianca Dittrich: well, let me see him all 587 01:12:16.930 --> 01:12:23.399 Bianca Dittrich: where there is an inviance. But you have to change t and Q, 588 01:12:23.470 --> 01:12:26.510 Bianca Dittrich: and the one thing it's not just tea. 589 01:12:28.770 --> 01:12:30.080 Bianca Dittrich: I don't. 590 01:12:30.260 --> 01:12:31.480 Bianca Dittrich: You have to change 591 01:12:31.620 --> 01:12:33.349 t and Q 592 01:12:33.520 --> 01:12:37.690 Bianca Dittrich: along the trajectory, for instance, said Sir Wright in my arms. 593 01:12:38.900 --> 01:12:41.200 Bianca Dittrich: Right? Okay, not just he. 594 01:12:42.700 --> 01:12:47.629 John Barrett: Yeah. So in the quantum model you would integrate over queue. So it's just 595 01:12:48.110 --> 01:12:50.959 John Barrett: the remaining action is just a function of the teas. 596 01:12:51.590 --> 01:12:54.820 Bianca Dittrich: hmm, yeah. 597 01:12:55.300 --> 01:12:58.600 John Barrett: And you are just saying it's independent of the intermediate teas. 598 01:12:59.400 --> 01:13:05.190 Bianca Dittrich: Yeah, I mean in the end, that is, you could. That's just translate, like vertex translation symmetry. 599 01:13:06.130 --> 01:13:06.840 John Barrett: Yep. 600 01:13:07.190 --> 01:13:15.690 Bianca Dittrich: whereas you know, on the left hand side. Your result would depend on how you have chosen the intermediate values of your teeth. Exactly. 601 01:13:16.000 --> 01:13:20.109 Bianca Dittrich: And okay, I mean, I would say, it's it's possibly not physical. 602 01:13:24.120 --> 01:13:24.840 Bianca Dittrich: Yeah. 603 01:13:25.750 --> 01:13:27.920 Hal Haggard: shall we proceed to advise? Question. 604 01:13:27.930 --> 01:13:36.810 AAipad2022: yeah. So this is almost at the end last end of their talk? Yeah, it has to do with this, this effective spin models that you talked about. 605 01:13:36.900 --> 01:13:39.569 AAipad2022: So, as you pointed out. Now 606 01:13:39.960 --> 01:13:59.980 AAipad2022: that you know there are various indications that we need 3 relevant coupling constants. And we've got G and lambda and some. And you like to think of this gamma as being the third relevant coupling. Constant. Okay. But we also sort of know that from this modern perspective effective field series, and particularly the things which have to do with this. 607 01:14:00.380 --> 01:14:07.880 AAipad2022: this form factors? This idea of form factors, and so on, that the GN. Lambda. 608 01:14:08.730 --> 01:14:10.369 AAipad2022: physically they don't run. 609 01:14:10.800 --> 01:14:16.410 AAipad2022: And and so here Gamma also does physically does not run. Is that the statement? 610 01:14:17.370 --> 01:14:24.529 Bianca Dittrich: I mean, it's the same visa. 611 01:14:25.890 --> 01:14:36.199 Bianca Dittrich: I think it. It might run, I mean in the con. Yeah. So so in Cdt and Eddp. you need 3 couplings constants 612 01:14:36.250 --> 01:14:42.759 Bianca Dittrich: to be able to find you. And so indeed, G and lambda you form into kind of an 613 01:14:44.440 --> 01:14:46.179 Bianca Dittrich: a combination of 614 01:14:46.740 --> 01:15:04.600 Bianca Dittrich: and and then you you need another com, another parameter to to find zoom. The question is that they always run in the theory space as John don't know who explain to us in the theory space that are. But physically they don't run. 615 01:15:05.260 --> 01:15:22.970 Bianca Dittrich: Yeah, Jean Jean Lamna don't run. I mean, I'm talking. I'm talking that these statements. It's in 2 different contexts one is an asymptotic safety, and and one is in Cdt and Ed and I wanted to see in C. Dt. Edt. They're still working on a notion of running coupling constants. So I 616 01:15:23.110 --> 01:15:32.449 Bianca Dittrich: it's it's hard for me to spontaneously say what happens to this Unizol? But with grammar. Yeah, I well. 617 01:15:32.490 --> 01:15:37.959 Bianca Dittrich: I mean, there's one paper by Simona and Andario, where they make gamma running. 618 01:15:39.690 --> 01:15:53.240 AAipad2022: No, but I think there's a difference between running in the theory space and running physically right. And so that that was the big difference that John has been pointed out, pointing out to us last few months. 619 01:15:53.400 --> 01:15:58.169 Simone SPEZIALE: but by running physical you mean a running that you would see with asthmatics, I suppose right 620 01:15:59.180 --> 01:16:19.670 Simone SPEZIALE: log in regularization instead of cut off. 621 01:16:20.110 --> 01:16:39.130 Simone SPEZIALE: then, if you add the cosmod, your constant lambda, and you have fermions, then Gamma would run in that sense. But if you have only fermions, or only it goes logical, constant, then Gamma will not run in that sense. I'm talking 622 01:16:39.480 --> 01:16:49.040 Simone SPEZIALE: pure gravity. So lambda. Gmg, but no form. Yeah. Okay. Good. Good. No, no running that you would see in this matrix. 623 01:16:49.750 --> 01:16:52.650 AAipad2022: Good, good. Good. It's consistent with my understanding. Thank you. 624 01:16:56.860 --> 01:16:59.480 Hal Haggard: Simone, did you have an independent question? 625 01:17:02.050 --> 01:17:06.849 Simone SPEZIALE: No, it was really just to comment on this. Sorry I can lower my hand. No problem. 626 01:17:08.380 --> 01:17:09.160 thank you. 627 01:17:09.360 --> 01:17:11.080 Hal Haggard: Are there other questions? 628 01:17:16.890 --> 01:17:19.520 Hal Haggard: If not, let's thank Bianca again. 629 01:17:21.850 --> 01:17:24.989 Bianca Dittrich: Thank you for listening. 630 01:17:27.350 --> 01:17:28.359 Hal Haggard: Bye, everyone.