0 00:00:08,720 --> 00:00:17,709 Thomas Thiemann: Thank you very much. Um. So I was uh given the task by the organizers of the seminar to somehow summarize 1 00:00:24,690 --> 00:00:30,129 Thomas Thiemann: um. There's a paper that somehow explains a kind of new strategy. 2 00:00:30,370 --> 00:00:42,160 Thomas Thiemann: Um! Two papers that are discussed, models which somehow support the strategy. And then another set of two papers which contain some mathematical tools. 3 00:00:43,260 --> 00:01:02,809 Thomas Thiemann: So um! Here's the table of contents of my talk. First uh the first part concerns the the motivation. Um, uh, So I will discuss the interplay between the hyper surface, the information algebra, quantum non degeneracy, and no notion 4 00:01:02,820 --> 00:01:10,569 Thomas Thiemann: of a non degeneracy of quantum metrics, anomalies, and density weights that have been. 5 00:01:15,730 --> 00:01:16,820 Um, 6 00:01:18,520 --> 00:01:36,000 Thomas Thiemann: then uh, I will um have a look at uh lease molens uh model the u one cubed truncation of uh Euclidean vacuum quantum gravity, and show that here in this model all these notions are nicely fitting together. 7 00:01:37,410 --> 00:01:45,790 Thomas Thiemann: And uh, this then breaks me to um uh the last part of the talk where I discuss renormalization 8 00:01:45,920 --> 00:01:51,720 Thomas Thiemann: uh, as a possibility to arrive at this quantum non degeneracy 9 00:01:51,910 --> 00:01:58,119 Thomas Thiemann: which will see it is probably a key notion that has been much overlooked in the past. 10 00:01:59,280 --> 00:02:03,680 Thomas Thiemann: So let me come to the first part of the talk. Um! So what is the 11 00:02:04,300 --> 00:02:09,500 Thomas Thiemann: uh? I should. I will say at the beginning that this is a lot of 12 00:02:24,740 --> 00:02:36,010 Thomas Thiemann: so um. What is the classical hyper surface? The formation algebra will let me denote by capital. Q. And what follows: The square of the determinant of the spatial metric 13 00:02:36,440 --> 00:02:39,959 Thomas Thiemann: capital. D is the density we wait W. 14 00:02:40,010 --> 00:02:46,360 Thomas Thiemann: Equal to one spatial different of them. Constraint and Cw. Is the density weight W. 15 00:02:46,440 --> 00:02:48,760 Thomas Thiemann: Uh Hamiltonian constraint, 16 00:02:49,290 --> 00:02:53,800 Thomas Thiemann: Then, as is of hopefully well known, 17 00:02:53,820 --> 00:03:11,699 Thomas Thiemann: uh in in January, Covent theory, the quas on bracket between two Hamiltonian constraints is given by a spatial different office and constraint. However, it's not the but the so called open algebra, because the this quantity queue 18 00:03:11,710 --> 00:03:21,519 Thomas Thiemann: appears on the right hand side as A. So in the definition of this vector field which this spatial de humorphism is smeared 19 00:03:22,000 --> 00:03:23,130 Thomas Thiemann: uh 20 00:03:25,320 --> 00:03:35,780 Thomas Thiemann: uh two observations with kind of two observations is that uh, this algebra makes only sense. Uh, if the metric is not the generate, 21 00:03:36,370 --> 00:03:40,100 Thomas Thiemann: unless you have w bigger than equal to two, 22 00:03:40,390 --> 00:03:49,939 Thomas Thiemann: and it's, of course, to real. If the integral of this spatial, different office and constraints has somehow the back manager, Zero support. 23 00:03:51,820 --> 00:03:55,400 Thomas Thiemann: The reason why I mentioned this will become clear, and what follows. 24 00:03:57,540 --> 00:04:09,990 Thomas Thiemann: So then, come, let us come to this choice of density, weights, and which which choices are made in in quantum new quantum gravity in particular, is somehow dictated by the dynamics. 25 00:04:10,100 --> 00:04:26,120 Thomas Thiemann: So, first of all, every single term in the Hamiltonian constraint, be it the vacuum, constant contribution, the Ransen or Euclidean one the cosmological, constant, or any matter contribution. It couples to the electric field 26 00:04:31,200 --> 00:04:35,050 Thomas Thiemann: to look for. Uh Hilbert space representations, 27 00:04:35,330 --> 00:04:46,339 Thomas Thiemann: uh which, of course, should be such that the Hamiltonian constraint be densely defined in the space, and to pick a vacuum which is killed by the electric field. 28 00:04:48,610 --> 00:04:56,639 Thomas Thiemann: Uh. But this, of course, has the uh media problem that then this vacuum is a zero eigenvector of queue. 29 00:04:56,800 --> 00:05:01,649 Thomas Thiemann: So the vacuum itself is, of course, quantum degenerate in this sense 30 00:05:04,730 --> 00:05:09,650 Thomas Thiemann: uh and here are two little limits which you can show uh 31 00:05:09,910 --> 00:05:24,219 Thomas Thiemann: This single one line uh observation already fixes a so-called non-off, or for turing type of representation of which the Ashtaker level Doski representation is a particular example, 32 00:05:25,320 --> 00:05:35,939 Thomas Thiemann: and this is a representation in which one of the variables is a discontinuous cannot be defined, they can only define exponentials, while the other is continuous. 33 00:05:36,980 --> 00:05:46,909 Thomas Thiemann: And the second observation is that um whenever there is at least a quadratic piece in uh for the um uh 34 00:05:47,420 --> 00:06:01,930 Thomas Thiemann: the potential of your gauge field, which is the case for uh young bills, and also for the clean piece of of uh general relativity. And this is such a quadratic piece. 35 00:06:01,940 --> 00:06:08,030 Thomas Thiemann: And if you want the uh hamil and the volume operator to be well-defined, which is kind of 36 00:06:08,090 --> 00:06:12,379 Thomas Thiemann: necessary in uh, in order, for example, to define 37 00:06:12,450 --> 00:06:16,340 Thomas Thiemann: matter terms of the cosmological, constant, and so on. 38 00:06:16,710 --> 00:06:19,810 Thomas Thiemann: Uh and rem on some 39 00:06:20,050 --> 00:06:21,270 Thomas Thiemann: um 40 00:06:22,080 --> 00:06:25,650 Thomas Thiemann: regularization of the Hamiltonian constraint 41 00:06:25,900 --> 00:06:36,350 Thomas Thiemann: over tetrahedral cells of forward and volume absent to the power of D is densely defined if and only if the density rate is equal to one, 42 00:06:36,710 --> 00:06:43,390 Thomas Thiemann: and if a and e are smeared in one and two miles, one the minutes respectively. So in two dimensions. If 43 00:06:43,490 --> 00:06:48,560 Thomas Thiemann: if You' in for space-time dimensions, which is, of course, what we do? 44 00:06:52,690 --> 00:06:59,560 Thomas Thiemann: Um. So let us go back now to the hyper surface deformation. It is well-defined 45 00:06:59,920 --> 00:07:09,809 Thomas Thiemann: if and only if the classical one is well-defined, if and only if uh who is bit? Q. Is bigger than zero for the density weights w equal to one 46 00:07:13,790 --> 00:07:15,020 Thomas Thiemann: um 47 00:07:17,050 --> 00:07:25,369 Thomas Thiemann: so we in the classical view. We want Q. To be bigger than what bigger than zero, so the the matrix should be should be no degenerate. 48 00:07:25,820 --> 00:07:33,379 Thomas Thiemann: On the other hand, in the loop quantum gravity, Herbert Space. We have the spin network functions as a basis 49 00:07:33,690 --> 00:07:37,460 Thomas Thiemann: uh which define a domain capital D, 50 00:07:37,830 --> 00:07:48,269 Thomas Thiemann: labeled by Graphs late finite graph, with final number of vertices, and therefore the quantumatic in this sense is actually 51 00:07:49,700 --> 00:07:50,700 oops. 52 00:07:55,840 --> 00:08:03,210 Thomas Thiemann: The quantum magic is in this sense uh completely degenerate, so it has zero volume, the back almost everywhere. 53 00:08:05,970 --> 00:08:24,449 Thomas Thiemann: And that's that explains why the absent to zero limit in our we month some approximation of the Hamiltonian constraint is actually a quite delicate thing to do, and then knife uh the knife limit would be would be infinite if Q is uh allowed to be. Zero 54 00:08:32,370 --> 00:08:45,289 Thomas Thiemann: can be done in particular by uh the to shawn off our regularization, or using some for some bracket identities which are being used in uh blue quantum cosmology all over the place. 55 00:08:46,830 --> 00:08:48,380 Thomas Thiemann: So um 56 00:08:49,380 --> 00:09:04,810 Thomas Thiemann: how to take this limit. And there have been basically two proposals in the literature. Um. One is using an operator topology that exploits the fact that, uh, we have at our disposal the notion of, If you're more in advance, 57 00:09:04,940 --> 00:09:15,060 Thomas Thiemann: so uh early ideas among those lines would be uh expelled by, and then later also by myself. 58 00:09:15,500 --> 00:09:34,360 Thomas Thiemann: And uh, the second option is to go on to the distributional dual of these uh dense domain of uh, the span of spin network functions, and this has been called habitat kind of spaces first introduced by with all. For 59 00:09:34,570 --> 00:09:36,430 Thomas Thiemann: uh all go 60 00:09:36,620 --> 00:09:39,430 Thomas Thiemann: um you work, and uh, 61 00:09:43,300 --> 00:09:58,830 Thomas Thiemann: but in both cases one gets uh um one why one gets closure. So there's no mathematical consistency in consistency. But there are still anomalies. So we get in the a option we get non-trivial, but wrong quantum structure functions, 62 00:09:59,090 --> 00:10:05,870 Thomas Thiemann: and in the B road we get to real and therefore also wrong quantum structure functions. 63 00:10:08,360 --> 00:10:11,089 Thomas Thiemann: So what is the origin of this T. 64 00:10:12,260 --> 00:10:28,359 Thomas Thiemann: This, or the origin of this tension becomes obvious when you look at the um right hand side of the Poisson bracket of two Hamiltonian constraints which I brought here. Top line again. One more time for W. For density, general density rate. W. 65 00:10:28,510 --> 00:10:31,480 Thomas Thiemann: Uh, and here we see that? Uh, 66 00:10:37,100 --> 00:10:38,869 Thomas Thiemann: you see that, uh, 67 00:10:39,640 --> 00:10:45,520 Thomas Thiemann: the classical Riemann sum contents an order of absent to the miles. Deep terms 68 00:10:45,910 --> 00:10:49,849 Thomas Thiemann: of size, absolute power to our D. 69 00:10:51,680 --> 00:10:58,780 Thomas Thiemann: The quantum sum, all the contents and gamma terms where and gamma is a number of vertices of a spin over function 70 00:10:59,380 --> 00:11:08,660 Thomas Thiemann: and of size, Epsilon, because this derivative here, which in the class in the casting here is actually a derivative becomes a difference one hundred and fifty, 71 00:11:08,900 --> 00:11:11,379 Thomas Thiemann: and the differences of all the epsilon. 72 00:11:11,700 --> 00:11:24,469 Thomas Thiemann: And now what happens is, of course, is that in the classical theory these two powers of Epsilon just cancel each other, and you get an order, One contribution. So when you get to classical, integral, which is where you find a non vanishing. 73 00:11:24,480 --> 00:11:35,310 Thomas Thiemann: But in the in the quantum theory we get an order of epsilon to the power. One times the finite and constant number of vertices which goes to zero in the limit absent, goes to zero. 74 00:11:35,400 --> 00:11:43,219 Thomas Thiemann: And this is exactly what those people that introduced the habitat found. So in some sense they were completely correct. 75 00:11:43,670 --> 00:11:45,260 Thomas Thiemann: Uh, because 76 00:11:45,420 --> 00:11:56,229 Thomas Thiemann: what we do in the quantum here is somehow to replace the the argument of this stuff, especially if you m offers, and constraint by something which has support on a Lebeck measures you will set, 77 00:11:58,040 --> 00:12:03,850 Thomas Thiemann: so that is of course not satisfying, and therefore, uh, one can 78 00:12:03,930 --> 00:12:14,880 Thomas Thiemann: one can think what what to improve. So one one option is to go non degenerate to non-degenerate back here. In fact, there has been a proposal to 79 00:12:14,890 --> 00:12:29,900 Thomas Thiemann: long time ago by Um, Tim Koslowski and Hano Zaman. It's for the consider uh a vacuum representation. There's a vacuum that has a a condensate. So the queue expectation value is different from zero, 80 00:12:30,930 --> 00:12:33,060 Thomas Thiemann: or one can try to 81 00:12:33,130 --> 00:12:52,579 Thomas Thiemann: get there out this problem that uh, you have this Epsilon to the power one contribution Times and Gamma going to zero. So we want to try to cancel this one. This Epsilon, by something which is one over absent. This forces you to go to non-standard density weight between two and one. This is 82 00:12:52,840 --> 00:12:57,099 Thomas Thiemann: uh one of the aspects of the recent work by by mother one. 83 00:12:58,020 --> 00:13:10,640 Thomas Thiemann: Um! So when you when you try to do this uh uh, in the first option, you will unfortunately find that the backup is not in the domain of the Hamiltonian constraint. 84 00:13:11,130 --> 00:13:13,119 Thomas Thiemann: Um. And also 85 00:13:13,170 --> 00:13:30,160 Thomas Thiemann: in order to get something close to the right hand side of the possibility of two Hamiltonian constraints you would like still to uh get, not an constant number of terms, but an infinite number of terms, in order to get uh something that uh resembles the classical interval. 86 00:13:34,200 --> 00:13:35,350 Thomas Thiemann: Um, 87 00:13:35,710 --> 00:13:40,030 Thomas Thiemann: With respect to the changing of the um 88 00:13:40,170 --> 00:13:42,240 Thomas Thiemann: and density. Wait. This is, 89 00:13:42,280 --> 00:13:48,790 Thomas Thiemann: or this electric shift strategy by a by and another one, 90 00:13:48,980 --> 00:14:02,379 Thomas Thiemann: which it seems to be to at least to me at the moment very much tailored to Euclidean, back from quantum gravity. Possibly you can extend it to the Lorentz in case using a big transform. 91 00:14:08,920 --> 00:14:13,810 Thomas Thiemann: Uh, I don't see how this could be uh uh transferred 92 00:14:28,720 --> 00:14:31,390 Thomas Thiemann: the for W equal to one. 93 00:14:32,170 --> 00:14:39,950 Thomas Thiemann: The classical hyper surface deformation. Algebra requires that the metrics be non-degenerate, 94 00:14:40,640 --> 00:14:47,600 Thomas Thiemann: how unfortunately, our look quantum gravity. Spindle functions are quantum degenerate almost everywhere 95 00:14:49,030 --> 00:14:57,510 Thomas Thiemann: and that's the reason why, in quantum representations of the hyper surface deformation, algebra, we meet severe difficulties. 96 00:14:58,810 --> 00:15:05,919 Thomas Thiemann: So somehow our quantum representation violates the necessary assumption about the very definition of age, 97 00:15:06,390 --> 00:15:17,290 Thomas Thiemann: and therefore a a a strategy which seems to be promising is to try to find new representations which are quantum non degenerate. 98 00:15:17,870 --> 00:15:27,180 Thomas Thiemann: So, in in other words, it might be a a good idea to make quantum degeneracy a part of the definition of an normally freeness. 99 00:15:28,110 --> 00:15:34,550 Thomas Thiemann: And this um um rents the further plan of my talk. So 100 00:15:34,760 --> 00:15:37,399 Thomas Thiemann: in the second part um I 101 00:15:42,390 --> 00:15:45,640 Thomas Thiemann: Lee, I hold your one cube model 102 00:15:46,160 --> 00:16:03,920 Thomas Thiemann: and show that in a non digitized uh representation, which is slightly different from the I could be at Qg. Representation, but still closely related. You get uh, you can implement all of this. You get an exact, unnormally free 103 00:16:03,930 --> 00:16:10,969 Thomas Thiemann: quantum nondegenerate quantization of this model, which is very close to being quantum integrable, 104 00:16:13,010 --> 00:16:23,659 Thomas Thiemann: and in the second or last part of the talk, then I will go beyond this model, because this model is still much simpler than uh one hundred and fifty 105 00:16:23,880 --> 00:16:31,660 Thomas Thiemann: full quantum gravity. And here I suggest that we we use methods of randomization 106 00:16:31,720 --> 00:16:39,330 Thomas Thiemann: to um derive a systematic constructions of uh quantum non-generated representations. 107 00:16:40,340 --> 00:16:41,430 Thomas Thiemann: So 108 00:16:41,830 --> 00:16:52,660 Thomas Thiemann: let me first go to the you one cube model, then I'll show uh how these features, which I mentioned before nicely fit to each, are here. 109 00:16:52,790 --> 00:16:54,350 Thomas Thiemann: So what is the 110 00:16:54,430 --> 00:16:58,690 Thomas Thiemann: what is the uh you want? Cube model for those who who don't know it. 111 00:16:59,210 --> 00:17:18,100 Thomas Thiemann: So take Euclidean vacuum, so you can. You can define it Both Hamiltonian and Lagrangian way. Hamiltonian definition is the the one that has been uh given by the in the first place. So you take you played in vacuum uh general relativity in 112 00:17:18,109 --> 00:17:22,190 Thomas Thiemann: and drop the a squared term from the density one 113 00:17:22,220 --> 00:17:24,030 Thomas Thiemann: Hamiltonian constraint, 114 00:17:24,810 --> 00:17:38,550 Thomas Thiemann: or, if you prefer the uh lagrangian definition, you do it almost the same thing. But you start with the self to a back from general relativity, of course, in our Euclidean signature, 115 00:17:38,690 --> 00:17:43,730 Thomas Thiemann: and again drop the A squared from from from the lagrangian. 116 00:17:44,150 --> 00:17:51,720 Thomas Thiemann: So this is almost you believe in vacuum gravity, but with a building structure group. So instead of su two, you have v one cubed. 117 00:17:53,900 --> 00:18:08,169 Thomas Thiemann: Um. What is nice about this model? Is that uh the hyper surface deformation algebra is completely unchanged. Um, in particular, we still have non non-polynomial structure functions, 118 00:18:08,850 --> 00:18:15,650 Thomas Thiemann: and therefore it is an ideal test laboratory. For many technical and conceptual issues of quantum gravity. 119 00:18:15,760 --> 00:18:33,839 Thomas Thiemann: Uh, and there's been a lot of work by about um Madavan in the past. Um! And he had. He, he was able to show that uh, you can go much further with this uh opinion more than in uh, and within the actual quantum we have to 120 00:18:33,850 --> 00:18:36,720 Thomas Thiemann: to you, because it is simpler, 121 00:18:39,210 --> 00:18:50,100 Thomas Thiemann: but it still has this uh not toable hyper-surface deformation either. And therefore it it enables us to shed light on the uh set of questions that I was was raising before. 122 00:18:50,500 --> 00:19:02,290 Thomas Thiemann: So now i'm going to quantize this model, but not using charge network states uh that have been used in the past in particular in the world by uh 123 00:19:02,360 --> 00:19:09,229 Thomas Thiemann: another one. But i'm using a slightly more general uh set of functions, 124 00:19:09,330 --> 00:19:17,380 Thomas Thiemann: i'm using uh a more general form factor, as people call it. So i'm smearing the the connection, not with some 125 00:19:17,830 --> 00:19:25,989 Thomas Thiemann: singular uh functions that are supported on one-dimensional lines. But i'm using more general speaking functions. 126 00:19:26,370 --> 00:19:30,870 Thomas Thiemann: Um, I'm still using a vacuum which is killed by the electric field 127 00:19:31,820 --> 00:19:50,939 Thomas Thiemann: and W. Of f here denotes so-called generalize for now always hold on if you want, and this is still a a quite uh discontinuous representation, which you can see from the fact that the expectation value of a generalized hold on me, 128 00:19:50,950 --> 00:20:05,589 Thomas Thiemann: this uh one, if smearing function is equal to zero and otherwise. Thomas I'm. Confused in the last uh integral is the three X. What? What's one dimensional there? I I got something wrong. 129 00:20:05,840 --> 00:20:13,910 Thomas Thiemann: So in in the um, if you want to, if i'm not sure if you are familiar with charge network functions. 130 00:20:15,050 --> 00:20:31,769 Thomas Thiemann: Okay, charge network functions are somehow the abuse in analog of uh, the spin network functions with the support on the one line is that the same? Exactly. Okay. Got it? Yeah, thank you. This is the that's really a one-dimensional integral at the end of the day. 131 00:20:31,780 --> 00:20:42,520 Thomas Thiemann: So if I want to. If I want to have a singular sphereing functions corresponding to a spin network or a charge network functions I would have f support on one-dimensional lines one hundred. 132 00:20:43,100 --> 00:20:50,970 Thomas Thiemann: But what i'm doing here is I'm. I'm. Looking at more general experiences which do not necessarily have that property. 133 00:20:55,120 --> 00:21:00,689 Thomas Thiemann: So, as I said, this in this can be viewed as generalized for long ways. If you want 134 00:21:00,810 --> 00:21:17,349 Thomas Thiemann: erez Agmoni and his s in in in in the quantum gravity, these w of F is how long we are discontinuous. So the the operator, corresponding to the connection cannot be defined by fluxes, are continuous, or the electric field can be defined one hundred and fifty 135 00:21:20,440 --> 00:21:30,759 Thomas Thiemann: um geometrical operators, like the volume or the the error error operator uh have the property that they are diagonal in this representation, so i'm 136 00:21:31,390 --> 00:21:46,529 Thomas Thiemann: trivial, almost trivial, to compute the eigenvalue of the volume operator for the region R. In this representation it's just the classical integral over the square of the modulus of the determinant of F. 137 00:21:47,060 --> 00:21:52,240 Thomas Thiemann: So it's also quite easy to spot those States, so 138 00:21:53,690 --> 00:21:57,540 Thomas Thiemann: it's quite easy to spot those states um which 139 00:21:57,920 --> 00:22:03,849 Thomas Thiemann: no degenerate. So in this case, of course, this determinant of F should be not zero. 140 00:22:03,930 --> 00:22:11,809 Thomas Thiemann: It is also quite easy to solve. The Gauss constraint is to address those smell functions which have vanishing divergence. 141 00:22:14,090 --> 00:22:22,700 Thomas Thiemann: But this was, of course, already known. Now we come to the interesting part which is spatial diffumor, and the Hamiltonian constraint; and since we, 142 00:22:29,230 --> 00:22:42,660 Thomas Thiemann: uh, we cannot directly define these these operators corresponding to them, because the a operator does not exist in this representation, just as in blue quantum gravity. 143 00:22:42,670 --> 00:22:50,009 Thomas Thiemann: Therefore, uh, we cannot directly talk about representation of the hyper-surface, the information algebra of course. 144 00:22:50,690 --> 00:23:01,030 Thomas Thiemann: But And now this is the uh, the the new insight, and something that really shocked me when I saw this is that you can actually exponentiate them. 145 00:23:02,410 --> 00:23:08,489 Thomas Thiemann: So for the U one cube quantum gravity model of you. Uk and um 146 00:23:08,570 --> 00:23:15,329 Thomas Thiemann: vacuum gravity. You can actually define unitary operators corresponding to both, 147 00:23:15,440 --> 00:23:19,699 Thomas Thiemann: especially if you more often consent, and the Hamiltonian constraint, 148 00:23:19,900 --> 00:23:21,180 Thomas Thiemann: using 149 00:23:21,530 --> 00:23:23,570 Thomas Thiemann: the exponential of both, 150 00:23:24,200 --> 00:23:27,899 Thomas Thiemann: and I can, even in one line, just write down how 151 00:23:28,050 --> 00:23:30,430 Thomas Thiemann: this operate the acts 152 00:23:30,900 --> 00:23:33,090 Thomas Thiemann: it acts on. 153 00:23:34,070 --> 00:23:54,049 Thomas Thiemann: You may call it a generalized spin network function, or a generalized Halloween, by changing the label of the generalized, follow me in the following way: So if this is denoted by the this thing here, which is E. To the X. W. U. And M. You, is this smearing function of the spatial different offence? Constraint? 154 00:23:54,060 --> 00:23:59,530 Thomas Thiemann: M. Is the smearing function of the Hamiltonian constraint and W. Is the density weight, 155 00:24:00,030 --> 00:24:01,860 Thomas Thiemann: And what is K. 156 00:24:02,110 --> 00:24:18,870 Thomas Thiemann: K. Is the uh is a function on the face space of this model which selects from uh, from G and F. Where G is the uh configuration variable, and F is the momentum variable the momentum variable. 157 00:24:22,730 --> 00:24:37,839 simone: Uh, that was a question. But I didn't get it. 158 00:24:37,960 --> 00:24:40,470 simone: Three dimensions. How does that work? 159 00:24:42,110 --> 00:24:43,290 Thomas Thiemann: Um! 160 00:24:43,400 --> 00:24:49,529 Thomas Thiemann: So the the definition of the inner product is in the top line, just the first entry. 161 00:24:52,140 --> 00:25:03,129 Thomas Thiemann: So all you need to all, All you need to know is that the expectation value of a generalized follow me, is the chronic or Delta. Everything else follows from that. 162 00:25:08,870 --> 00:25:10,290 Thomas Thiemann: I can do that. 163 00:25:10,890 --> 00:25:12,299 Thomas Thiemann: But I don't need to. 164 00:25:12,610 --> 00:25:15,759 Thomas Thiemann: This is uh, this is the Gns construction, 165 00:25:17,150 --> 00:25:25,010 simone: so maybe it would be nice if you. Could you explain, then, remind us why, in the usual case we need to do that then. 166 00:25:34,440 --> 00:25:42,640 simone: So in the usual case, if you want a background, independent uh scalar product. 167 00:25:43,180 --> 00:25:52,200 simone: It would be nice if you remind us why, in the usual case, we are forced to do that. And what are you? 168 00:25:52,330 --> 00:26:11,039 Thomas Thiemann: You're not I can. I can. Also I can also, in the usual case, define uh, what I mean by hold on to me and uh spin that work functions. And then i'm just saying that the expectation value of the connect or function is zero unless it's the through the zoom, connect, function. Everything else follows from that 169 00:26:11,430 --> 00:26:12,710 Lee: questions. 170 00:26:13,330 --> 00:26:31,309 Lee: So I I think what we're wanting to here is something about when you're in the so real quantum gravity, then these functions are still too singular, and it's because they're not higher in order terms, because we're in. You want to keep gravity that we can do this 171 00:26:31,320 --> 00:26:34,750 Lee: is that is, that the start of a correct argument. 172 00:26:35,040 --> 00:26:45,999 Thomas Thiemann: So the in in in the nominal building case, of course, if you want a nice or let's say a covariant behavior under one hundred 173 00:26:46,280 --> 00:27:04,430 Thomas Thiemann: uh of functions under the uh. As for under Gauss gauge transformations, then you may wonder, uh, what kind of functions do such have such properties, and then you uh you uh um motivated to go into these uh singular functions, 174 00:27:04,540 --> 00:27:13,380 Lee: right? Which you end up trying to do is smear loops. May, or a lot of things over something like the exponential of the 175 00:27:13,420 --> 00:27:14,400 Thomas Thiemann: right. 176 00:27:14,990 --> 00:27:18,810 Lee: And and that's to singular in the 177 00:27:21,520 --> 00:27:33,079 Thomas Thiemann: yeah. But I think that uh um! The the previous question was was just asking whether one can actually define the representation. Also, you're just using this uh um 178 00:27:33,230 --> 00:27:37,800 Thomas Thiemann: expectation values, and that is, that is indeed the case. 179 00:27:38,390 --> 00:27:50,789 simone: No, The previous question was really uh is a singular distribution, a singular meeting necessary in order to have a background independent uh measure. And you. Your answer is no 180 00:27:50,980 --> 00:27:58,209 simone: including in the 181 00:27:58,620 --> 00:28:05,539 Thomas Thiemann: So suppose we do not want to have an explicit solution of the Gauss constraint. 182 00:28:06,090 --> 00:28:07,150 Thomas Thiemann: Okay. 183 00:28:07,220 --> 00:28:09,659 Thomas Thiemann: So I'm: I'm: I'm looking at 184 00:28:09,890 --> 00:28:20,030 Thomas Thiemann: now three constraints, the the goals constraint, the spatial different, often consent, and the Hamiltonian concerned. I'm. Just want to represent them on some Hilbert space. 185 00:28:20,270 --> 00:28:23,210 Thomas Thiemann: Then I can do exactly what I'm doing here 186 00:28:23,570 --> 00:28:24,820 Thomas Thiemann: in the top line. 187 00:28:25,070 --> 00:28:26,150 Thomas Thiemann: This 188 00:28:26,390 --> 00:28:28,030 Thomas Thiemann: generous meeting functions. 189 00:28:40,380 --> 00:28:41,380 Thomas Thiemann: Okay, 190 00:28:41,560 --> 00:28:56,430 simone: Yeah. Yeah. So your answer is that the background dependent measure does not require singers meeting even in the case. It's only finding a simple solution to the Uh Gauss constraint that requires a singulars meeting. That's what you say? Okay, Thank you. 191 00:28:56,700 --> 00:29:07,820 Thomas Thiemann: I mean, maybe it does not even require that. But we we we choose to do that uh, because it gives you a nice way to solve the galaxy, but maybe it's not the only way to do it. I just don't know. 192 00:29:10,180 --> 00:29:18,430 Thomas Thiemann: But here in this model, of course, one uh one can much simpler. So the cost consent by just using It's meeting functions, Which are, 193 00:29:18,600 --> 00:29:20,629 Thomas Thiemann: we have banishing diversions. 194 00:29:23,760 --> 00:29:25,180 Thomas Thiemann: Can you hear me? 195 00:29:25,510 --> 00:29:43,750 Thomas Thiemann: Yes, okay. Yes. So I I was trying to explain how the Hamiltonian constraint acts on these Uh States in the about space and at that. So by using the uh exponential of the Hamiltonian vector field 196 00:29:43,960 --> 00:29:49,799 Thomas Thiemann: of the some of the different was from often and Hamiltonian constraint. 197 00:29:51,150 --> 00:29:52,510 Thomas Thiemann: And um! 198 00:29:53,170 --> 00:30:03,800 Thomas Thiemann: If you think about it, this is exactly the way. One can also discover how the spatial, different morphine constraint acts on the loop quantum gravity about space. 199 00:30:04,200 --> 00:30:07,819 Thomas Thiemann: And here in this model it turns out that the same can be done 200 00:30:07,850 --> 00:30:09,999 Thomas Thiemann: for the Hamiltonian constraint. 201 00:30:11,440 --> 00:30:12,789 Thomas Thiemann: And um 202 00:30:14,400 --> 00:30:21,969 Thomas Thiemann: to me this is the first quantum realization of what one what people call the Bergmann Quotmar roof 203 00:30:22,840 --> 00:30:36,569 wolfgang wieland: and it you you can, of course, just right. Excuse me. Pardon me, um! Can I just can uh both of you. Could we just jump back to the last slide. I have a quick question. 204 00:30:37,180 --> 00:30:50,920 wolfgang wieland: Maybe it could also clarify some things. So um! When you explain how the exponential of the Hamiltonian and vector constraint acts on the smearing function. 205 00:30:50,960 --> 00:30:59,300 wolfgang wieland: I am kind of puzzled because I I would think of this meeting function as something that is 206 00:30:59,310 --> 00:31:20,550 wolfgang wieland: only dependent on littlelex like the coordinates on uh three space. But uh the Hamiltonian and the diff your constraint. They are um. They act on field space, on on the configuration variables classically. So I don't quite understand how they how this action is defined. 207 00:31:20,560 --> 00:31:25,420 Thomas Thiemann: Could you please clarify, Sure. So the uh thing of 208 00:31:25,880 --> 00:31:29,169 Thomas Thiemann: so K. Of of zero. And F is basically F: 209 00:31:29,810 --> 00:31:30,860 Thomas Thiemann: Okay, 210 00:31:30,900 --> 00:31:43,910 Thomas Thiemann: and uh, what the Hamiltonian vector fuel does is it acts? It acts on on the electric field. And in order that you you get something which is again just depending on the electric field 211 00:31:51,880 --> 00:31:53,180 Thomas Thiemann: in the connection. 212 00:31:55,430 --> 00:32:03,260 Thomas Thiemann: So if you think about a constraint which is linear momentum. So i'm thinking of a as a momentum 213 00:32:03,800 --> 00:32:11,550 Thomas Thiemann: time, something which is arbitrary, complicated, and in the configuration variable. The Hamiltonian vector field preserves the 214 00:32:11,650 --> 00:32:14,320 Thomas Thiemann: functions which depends only on 215 00:32:14,480 --> 00:32:17,390 Thomas Thiemann: uh on the. 216 00:32:17,760 --> 00:32:19,130 Thomas Thiemann: This is how 217 00:32:19,340 --> 00:32:22,899 Thomas Thiemann: this is why this um action is well-defined. 218 00:32:27,120 --> 00:32:41,460 Thomas Thiemann: Okay, thank you. This is also the reason why this simple solution does not work for actual quantum gravity, because there, uh, we have not a linear dependence of the constraints on a but a quadratic dependence. 219 00:32:45,780 --> 00:32:48,530 Lee: Question. Yes, yes. 220 00:32:48,800 --> 00:32:54,979 Lee: Is there something like a dual volume? That is, if you take the linear the 221 00:32:55,150 --> 00:33:05,049 Lee: and treated like the E. Exactly and right to state which is a function of P, which is the integral over the square root of it. Is that a good state for you? 222 00:33:06,630 --> 00:33:10,639 Thomas Thiemann: Um! So you mean kind of a Jo and Simon's did. 223 00:33:10,780 --> 00:33:16,650 Lee: No, no, it's it's. It's just if I pretend that my is 224 00:33:16,910 --> 00:33:22,939 Lee: that's the determinant of my A, Because I can still take this determinant, even if it is only one queue. 225 00:33:23,050 --> 00:33:26,080 Lee: Um that I can write a volume with that. 226 00:33:26,310 --> 00:33:32,659 Lee: And this is the volume of the exponential of the volume. A good stage for you of the dual volume 227 00:33:32,990 --> 00:33:37,190 Thomas Thiemann: in the to determine on of B 228 00:33:37,220 --> 00:33:48,310 Thomas Thiemann: will be so. That would be something which is, uh uh higher than linear in in a, so that would not be well-defined. 229 00:33:49,480 --> 00:34:03,730 Thomas Thiemann: Um! So you can. Actually, you can, of course, just write down this this model and check that um hyper surface. The information algebra is well represented, but can you can also use it? Just 230 00:34:10,730 --> 00:34:21,659 Thomas Thiemann: a in terms of sinus of A, and so on, and you you derive this representation uh, and this uh action of the Hamiltonian constraint, using, 231 00:34:21,690 --> 00:34:25,470 Thomas Thiemann: uh polymerization and standard tools of your continuity 232 00:34:36,860 --> 00:34:42,930 Thomas Thiemann: one. But you can be more general, and it but it also works for the natural, then, that you wait 233 00:34:43,070 --> 00:34:44,709 W. Equal to one. 234 00:34:45,460 --> 00:34:55,780 Thomas Thiemann: The operators corresponding to Hamiltonian and uh sp spatial, different, often consent, act, in fact, unitarily on the Hilbert space. 235 00:34:56,090 --> 00:35:07,520 Thomas Thiemann: And, as I said before, uh, when you, when you set m equal to zero, this returns for you the action of the spatial. If you'm off in that, we know and laugh from two quantity 236 00:35:08,650 --> 00:35:18,919 Thomas Thiemann: Um, you can implement this uh uh algebra directly on the on, the on the Herbert space. So we do not need to go to this uh dual space, 237 00:35:19,480 --> 00:35:30,609 Thomas Thiemann: and the anomalyfulness is manifest. So the quantum algebra is exactly encoded in the Hamiltonian flow of the classical constraints 238 00:35:30,910 --> 00:35:44,509 Thomas Thiemann: and um. The The fact that we need to be known to Janet is also spelled out quite nicely here, because this uh classical algebra, of course, is again still defined. If the Hamiltonian vector fields 239 00:35:44,570 --> 00:35:48,490 Thomas Thiemann: um uh do not exist, which is the case for 240 00:35:53,790 --> 00:35:54,930 Thomas Thiemann: um 241 00:35:55,290 --> 00:36:01,109 Thomas Thiemann: also nice is that it? It makes contact to a recent uh 242 00:36:01,180 --> 00:36:10,869 Thomas Thiemann: uh discoveries. For example, there is this electrification covariance, and then, if you mobile perspective by a by 243 00:36:23,510 --> 00:36:25,910 Thomas Thiemann: um, you can compute 244 00:36:25,980 --> 00:36:29,569 Thomas Thiemann: the Hamiltonian floor of this uh 245 00:36:29,740 --> 00:36:33,500 Thomas Thiemann: Hamiltonian constraint for arbitrary density. Weight 246 00:36:48,160 --> 00:36:49,740 Thomas Thiemann: of this constraint 247 00:36:49,760 --> 00:37:00,549 Thomas Thiemann: uh delivers that's in order for the numbers of n plus one in F, depending on spatial derivatives of n of so of the 248 00:37:00,900 --> 00:37:03,629 Thomas Thiemann: uh of of of order, and 249 00:37:05,430 --> 00:37:17,000 Thomas Thiemann: erez agmoni. Um. Now let me try to get as close as possible to the loop quantum we are. These states to spend a lot of functions, So it's been that we functions would correspond to this kind of singular one. 250 00:37:17,030 --> 00:37:20,130 Thomas Thiemann: Smearing functions which are supported on the graph, 251 00:37:35,580 --> 00:37:48,729 Thomas Thiemann: so to say, second out spin network functions is that the action X. The actions along the whole graph, not only at the vertices, there is no abrupt loop attachment 252 00:37:49,210 --> 00:37:59,169 Thomas Thiemann: and the action is very non-polynomial on the charges. So, in in other words, the in the obedient analog of the spin that spin labels 253 00:38:02,570 --> 00:38:19,629 Thomas Thiemann: you can also use uh these habitats that were introduced uh um by with all, for at our um, anyway, and then have access also the to the hyper-surface uh the information algebra, not to only to it's exponential, and they are also. 254 00:38:19,640 --> 00:38:22,710 Thomas Thiemann: Uh, we get an normally freeness by construction. 255 00:38:24,190 --> 00:38:31,000 Thomas Thiemann: Um! You can go even further. Uh, so we can try to solve this model, and uh um 256 00:38:31,100 --> 00:38:39,460 Thomas Thiemann: compute the Iraq observers, and so on, so forth. Go to the physical Herbert Space, compute the physical Hamiltonian, 257 00:38:39,620 --> 00:38:51,150 Thomas Thiemann: and and this is, in fact, possible in this model. So group averaging versus reduce face-based quantization, we can. You find relational observvers and the physical Hibbert Space 258 00:38:51,470 --> 00:38:59,340 Thomas Thiemann: Um and the physical expert space and Hamiltonian lead to nonlinear self-interesting electrodynamics 259 00:38:59,680 --> 00:39:14,249 Thomas Thiemann: and self interacting means that uh we can compute the analog of endpoint functions and these are not determined by the two point functions. So the model, the physical Hamiltonian is interacting in that sense. 260 00:39:25,460 --> 00:39:28,400 Thomas Thiemann: Uh, listen to in the similar 261 00:39:33,400 --> 00:39:34,910 Thomas Thiemann: and um. 262 00:39:36,500 --> 00:39:49,799 Thomas Thiemann: Finally, one can also try to define a spin form model from this model, uh, from from starting from the Hamiltonian interior. And here, rather than having formal lead back measures, 263 00:39:49,830 --> 00:39:56,760 Thomas Thiemann: um, we can derive the model um, using discrete and board measures from first principles, 264 00:40:04,420 --> 00:40:05,679 Thomas Thiemann: which I think, 265 00:40:05,900 --> 00:40:07,549 uh, what should you get 266 00:40:12,980 --> 00:40:24,709 Thomas Thiemann: erez agmoni? So the you want cute quantum, all by by introduced by B. Lee is is close to being quantum integrable, at least in this nano for tearing type of representation one hundred and fifty. 267 00:40:26,600 --> 00:40:32,689 Thomas Thiemann: Uh, there's a convergence of ideas, canonical cover, and relational observables 268 00:40:32,960 --> 00:40:34,689 Thomas Thiemann: nicely fit together. 269 00:40:35,700 --> 00:40:45,580 Thomas Thiemann: I would consider this model as a paradigm model, for now on has the harmonic oscillator of loop quantum gravity at which you should test all our ideas. 270 00:40:45,690 --> 00:40:50,790 Thomas Thiemann: Um, It highlights the importance to implement quantum degeneracy. 271 00:41:01,050 --> 00:41:06,749 Thomas Thiemann: Um. The lqg techniques that have been introduced in the past decades work 272 00:41:06,940 --> 00:41:07,979 Thomas Thiemann: um 273 00:41:08,360 --> 00:41:12,950 Thomas Thiemann: particular for density, way to unity, and without using habitats. 274 00:41:22,300 --> 00:41:29,020 Thomas Thiemann: The Hamiltonian vector field of both constraints preserved the momentum polarization of the face base. So, in other words, 275 00:41:35,810 --> 00:41:45,130 Thomas Thiemann: and of course the full con to grab is much more complicated. So here the constraints are no longer polarization preserving, 276 00:41:45,440 --> 00:41:49,400 Thomas Thiemann: and therefore on uh and get needs to go. 277 00:41:49,710 --> 00:41:59,549 Thomas Thiemann: All the techniques have to go beyond that uh, one interesting aspect or idea that has been mentioned by Uh 278 00:41:59,910 --> 00:42:04,689 Thomas Thiemann: in one of his recent papers is that, uh, now that we have 279 00:42:04,760 --> 00:42:09,699 Thomas Thiemann: a kind of integrable model, and it might be a good idea to actually 280 00:42:33,740 --> 00:42:35,680 Thomas Thiemann: and the other uh 281 00:42:36,060 --> 00:42:38,289 Thomas Thiemann: yeah, the uh idea is, 282 00:42:38,330 --> 00:42:52,680 Thomas Thiemann: it brings me to the last part of my talk. So um, going beyond perturbation to you around an international model going trying to define the ue uh non-perturbatively 283 00:42:52,920 --> 00:42:58,160 Thomas Thiemann: it may be a good idea to uh have a look at um renormalization 284 00:42:58,690 --> 00:43:09,289 Thomas Thiemann: in order to uh define representations or find representations constructively, which are which have this property of being a quantum non-degenerate. 285 00:43:09,800 --> 00:43:25,199 Laurent Freidel: So there was two questions, please. Uh hello, uh, Thomas! Uh, It's about the previous slide uh something you you mentioned, so can you explain it a bit more? The fact when you're saying that. Uh, 286 00:43:25,210 --> 00:43:43,000 Laurent Freidel: let's say, yeah me, you know they may usually give me to the constraints on E, you know, generates the excessive over here, which is kind of the connection, and that's that's the basics of the of the quantum dynamics, right? So uh the action of the 287 00:43:43,010 --> 00:43:51,320 Laurent Freidel: uh. So here can you explain why that's not happening. Why, why, the the time derivative of the frame field is not the excessive curvature? 288 00:43:51,790 --> 00:44:07,120 Thomas Thiemann: Yeah. So if you so, why do you get uh? You dot equals to to the extent the curvature. That's because, uh in in the in the actual quantum gravity model uh, we have uh su two gauge group. 289 00:44:07,130 --> 00:44:12,769 Thomas Thiemann: So f the the curvature has. It has a linear part in the quadratic part. 290 00:44:12,960 --> 00:44:18,700 Thomas Thiemann: Uh, in this you you one cube truncation. You drop the ace, the the quadratic part. 291 00:44:19,560 --> 00:44:24,029 Thomas Thiemann: So F is just uh the exterior derivative of a 292 00:44:30,520 --> 00:44:41,319 Laurent Freidel: that. That's what I call this a polarization preserving okay and and geometrically. Do you understand what it means that it is not the excess of curvature. What kind of geometry does that 293 00:44:41,940 --> 00:44:43,509 Laurent Freidel: correspond to? 294 00:44:44,150 --> 00:44:45,479 Thomas Thiemann: Hmm. 295 00:44:45,620 --> 00:45:02,860 Thomas Thiemann: It's difficult to uh to um derive um. A geometric intuition about this is just uh the the dynamics. Dynamics is such that, uh, in fact, you can. You can first solve uh a first order equation just for you. 296 00:45:20,450 --> 00:45:21,620 Laurent Freidel: Okay, 297 00:45:22,300 --> 00:45:28,260 Laurent Freidel: Yeah, it's a it's a it's a it's not it's, not. It's not a 298 00:45:28,570 --> 00:45:41,980 Thomas Thiemann: but the nice feature is that it has compared to other models which are also So it has this problem: property that we have a non-trivial and open hyper-surface deformation, algebra. 299 00:45:42,190 --> 00:45:46,890 Thomas Thiemann: And this is what I wanted that what I was interested in in testing. 300 00:45:47,330 --> 00:45:50,129 Laurent Freidel: Okay, Thanks. Sure. 301 00:45:50,330 --> 00:45:52,549 Lee: Uh. Comment. 302 00:45:52,650 --> 00:46:02,409 Lee: Yeah, I i'm sorry if you should tell with this at the beginning. But are you aware of? This? Is a whole community that studies this model under a different name? 303 00:46:13,760 --> 00:46:22,870 Lee: They call it self to gravity, and they attribute it to Palestine. Um to i'm sorry, Townsend 304 00:46:23,710 --> 00:46:37,079 Lee: Johnson, The The town that was that is known for his work on super membranes right, 305 00:46:37,180 --> 00:46:39,630 Lee: and they all point back to Thompson 306 00:46:41,500 --> 00:46:43,969 Thomas Thiemann: while the model was introduced by you 307 00:46:44,710 --> 00:46:47,900 Lee: way way Prime? No, I was way prime. 308 00:46:52,410 --> 00:46:56,339 Lee: So, anyway, I think actually it's an opportunity. 309 00:47:06,490 --> 00:47:07,859 Thomas Thiemann: Thanks. 310 00:47:08,390 --> 00:47:14,390 simone: Sorry. And can you say something about kitchen? But then, because, if understood the previous point, 311 00:47:14,440 --> 00:47:20,109 simone: now we are going to look at quantum states which are not charge networks. 312 00:47:20,480 --> 00:47:29,980 simone: So then, we have this nice representation of the algebra of these algebra. There's a quantum operators. But then, what happens when we post gauge in audience? 313 00:47:30,210 --> 00:47:32,499 simone: You mean for this model. Right? 314 00:47:33,280 --> 00:47:34,510 Thomas Thiemann: I solved it. 315 00:47:41,070 --> 00:47:43,750 Thomas Thiemann: So it's the middle of this slide. 316 00:47:43,880 --> 00:47:48,239 Thomas Thiemann: You just use functions which have vanishing divergence. 317 00:47:49,400 --> 00:47:51,410 simone: So in a 318 00:47:51,960 --> 00:48:11,099 simone: e, even if it's a even if it's in a video model with some lines, it could be still good observable. But here you're not using those as observable to What are you using apart from formulas, Just in words, just to give us intuition of what is going on because it's A, B. And you could say, use the electric field and the magnetic field that they gauge invariant. Here, 319 00:48:12,780 --> 00:48:17,090 Thomas Thiemann: you mean just uh of those with respect to the goals constraint 320 00:48:17,130 --> 00:48:26,109 simone: right? Because if you're If the way to solve the Gauss constraint is by using the electric field, then I think I understand the the what's going on. But I just needed to 321 00:48:27,070 --> 00:48:29,720 simone: confirm if that's what's going on or not. 322 00:48:30,140 --> 00:48:41,689 Thomas Thiemann: Yeah. So in in particular, a Wilson kind of a generalized Listen: Function Will's loop function, which is, be it with a function that has vanishing divergence. It's as if, 323 00:48:44,510 --> 00:48:53,779 Thomas Thiemann: and the electric field, of course, is also gauge environment with respect to the Gauss gauge transformations. 324 00:48:55,430 --> 00:49:13,829 wolfgang wieland: But in the uh, in the non-abelian case, and even more so in the gravitational case, the gauge symmetries are highly nonlinear. So the smearing function, or what? What would it mean for this mirroring function, Now that it is it's a field dependent. Or 325 00:49:14,000 --> 00:49:16,190 wolfgang wieland: could you comment on that, 326 00:49:16,550 --> 00:49:20,420 Thomas Thiemann: please? So, as I was mentioned before. One 327 00:49:20,440 --> 00:49:25,179 Thomas Thiemann: One can define the same representation also in the non-obeling case. 328 00:49:25,360 --> 00:49:27,669 Thomas Thiemann: But then it is much 329 00:49:28,160 --> 00:49:33,830 Thomas Thiemann: less natural. Let's say, to try to solve the Gauss constraint in terms of those variables. 330 00:49:34,370 --> 00:49:35,490 wolfgang wieland: Mhm 331 00:49:38,440 --> 00:49:41,450 Thomas Thiemann: but the representation itself works the same 332 00:49:42,230 --> 00:49:44,209 Thomas Thiemann: also in the normal building case, 333 00:49:44,740 --> 00:49:58,670 Western: unless since there are some some general question before you go to to about the setup. Uh there, there'll be questions about the the the, the the gauge constraints. But what about the different Also for you. It seemed to me that they going to to um. 334 00:50:04,490 --> 00:50:15,340 Western: It's it's main tools that allows us to work with the most invariant things. So here I mean for me that they exist. I don't see why it shouldn't, but 335 00:50:20,010 --> 00:50:35,720 Thomas Thiemann: we we use actually not. I mean the the nice thing is that um uh. So the these uh Hamiltonian vector. Fields. Let let's. Let's take the case that M. Is equal to zoom. Just. We just look at the spatial different offers. And now, 336 00:50:36,180 --> 00:50:37,180 Thomas Thiemann: then, 337 00:50:37,290 --> 00:50:52,889 Thomas Thiemann: the action of this Hamiltonian flow corresponding to a spatial Dishimorphism is exactly the action of a spatial decomorphism. Of course it's not down to one that's how to solve them, how we find that the the basis of states which are the environment 338 00:50:53,700 --> 00:50:56,560 I and they they pretty well defined. I see it. 339 00:50:57,320 --> 00:51:03,290 Thomas Thiemann: Oh, I go in much work, much more beyond that, because i'm going to the reduced phase Space 340 00:51:11,390 --> 00:51:19,929 Thomas Thiemann: It it I've missed it, as it already happens, or you. It has happened. Yes, in my paper. 341 00:51:20,250 --> 00:51:39,910 Thomas Thiemann: Oh, it's not a not on this slides. Oh, not that That's why I was saying in the beginning. I'm sorry for, because the the home this model alone deserves to talk by it by itself. But I have. I was supposed to summarize three or four papers. So it's not here. Okay, that's all right. But then uh, 342 00:51:39,920 --> 00:51:49,229 wolfgang wieland: why, why uh going back to the last slide, perhaps. Why, to the previous slide on me. Um! 343 00:51:49,500 --> 00:52:02,130 wolfgang wieland: Why, then, choose such a singular state for defining Omega. Why not? Uh do the Gns construction with the Gaussian state in F? 344 00:52:02,520 --> 00:52:15,190 Thomas Thiemann: Because you're stopped in line zero uh the operator that you've tried you will find is badly, badly. Uh no, ill-defined and it's not certainly not tensely defined 345 00:52:15,830 --> 00:52:23,209 Thomas Thiemann: that forces you. Because that that's why I was saying in the in the beginning, Um, 346 00:52:23,420 --> 00:52:25,000 Thomas Thiemann: These kind of 347 00:52:25,720 --> 00:52:35,279 Thomas Thiemann: the non-ho Fartherian type of representations are natural when you, Hamiltonian, is such that the electric field is part of every single term, 348 00:52:35,510 --> 00:52:38,009 Thomas Thiemann: so that, uh, 349 00:52:38,170 --> 00:52:42,130 Thomas Thiemann: and a vacuum which is killed by the electric field as a natural choice. 350 00:52:43,860 --> 00:52:46,149 wolfgang wieland: Okay, I see. Yes, 351 00:52:50,650 --> 00:52:52,699 Thomas Thiemann: okay. Should I actually go on? 352 00:52:56,480 --> 00:52:59,069 Jerzy Lewandowski: Yes, please go on. Okay. 353 00:53:04,330 --> 00:53:06,090 Thomas Thiemann: Hmm. So 354 00:53:06,450 --> 00:53:10,269 Thomas Thiemann: yes, in the last couple of minutes. Then we go to um 355 00:53:11,180 --> 00:53:31,000 Jerzy Lewandowski: uh as I try to motivate uh Hamilton as a normalization. So actually an organizational e an announcement. So, eh? Eh? The The chairman is gone. I'm replacing the chairman, and I suspect that even Abai is not with us today, so probably there are no adults just us. 356 00:53:31,010 --> 00:53:33,169 Jerzy Lewandowski: We can do whatever we we want. 357 00:53:33,650 --> 00:53:35,129 Jerzy Lewandowski: Okay, Go, go. 358 00:53:35,700 --> 00:53:41,379 Lee: I I'm definitely not an entail. So 359 00:53:43,150 --> 00:53:44,990 Thomas Thiemann: okay, so um. 360 00:53:45,330 --> 00:53:51,369 Thomas Thiemann: The idea for for this line of line of work comes from constructive quantum theory 361 00:53:51,560 --> 00:53:56,420 Thomas Thiemann: uh as initiated by this gentleman that i'm listing here in the first line. 362 00:53:56,530 --> 00:53:57,680 Thomas Thiemann: Um, 363 00:54:01,300 --> 00:54:10,389 Thomas Thiemann: and uh, i'm trying to cut a very long story short. So uh the script. And here is a set of resolution. Skills! 364 00:54:10,520 --> 00:54:11,750 Thomas Thiemann: Um 365 00:54:11,940 --> 00:54:16,490 Thomas Thiemann: uh! This is supposed to be a partially ordered and direct set, 366 00:54:18,020 --> 00:54:24,950 Thomas Thiemann: and uh, we have at our we're supposed to have at our to dispose of this um 367 00:54:25,160 --> 00:54:33,289 Thomas Thiemann: uh triples of objects. The first entry is a Herbert space; the second is a vacuum, and the third is a Hamiltonian, 368 00:54:33,380 --> 00:54:38,949 Thomas Thiemann: and these are labeled by elements of this label space of this resolution skills. 369 00:54:40,310 --> 00:54:56,550 Thomas Thiemann: And uh, suppose that you have what people call consistent systems of isometric injections. So these are isometric injections of Herbert spaces into each other from course, resolution to fine resolution, 370 00:54:56,610 --> 00:55:06,880 Thomas Thiemann: which are such that if m one is smaller than m two, it's more than three that uh, some of these things work together in the expected way. 371 00:55:07,070 --> 00:55:08,120 Thomas Thiemann: Uh, 372 00:55:08,820 --> 00:55:20,200 Thomas Thiemann: then uh you get, you get a consistency condition also for the Hamiltonian, which says that if your call screening uh fine resolution, Hamiltonian and you get 373 00:55:20,240 --> 00:55:21,250 Thomas Thiemann: yeah 374 00:55:21,370 --> 00:55:24,390 Thomas Thiemann: uh low resolution, Hamiltonian 375 00:55:25,630 --> 00:55:40,940 Thomas Thiemann: and Um. If such a system of of Uh theories labeled by resolutions is given, then you can, uh you get an equivalent definition of a continuum theory, in which there are two 376 00:55:40,990 --> 00:55:42,109 Thomas Thiemann: uh 377 00:55:42,760 --> 00:55:48,399 Thomas Thiemann: the continuum Herbert spaces what people call the so-called inductive limit of a good spaces, 378 00:55:48,740 --> 00:55:51,799 Thomas Thiemann: and the Hamiltonian is kind of a projection 379 00:55:52,120 --> 00:56:03,650 Thomas Thiemann: uh, or the the The continuum Hamiltonian has projections to uh find out the resolution which cons co coincide with the Hamiltonian defined at 380 00:56:03,750 --> 00:56:06,160 Thomas Thiemann: uh find that resolution. 381 00:56:12,730 --> 00:56:15,440 Thomas Thiemann: And how does it help to find 382 00:56:15,490 --> 00:56:18,140 Thomas Thiemann: quantum non-degenerate representations? 383 00:56:19,820 --> 00:56:28,070 Thomas Thiemann: So here I was had prepared a a technical slide about what people call multi-resolution analysis and wavelet ue, 384 00:56:28,210 --> 00:56:29,620 Thomas Thiemann: and uh 385 00:56:29,800 --> 00:56:42,320 Thomas Thiemann: erez agmoni did Three most important ingredients of this are what actually is a multi-resolution analysis uh associate scaling functions and associated scale uh wavelets one hundred and fifty. 386 00:56:42,330 --> 00:56:57,469 Thomas Thiemann: But I will skip that slide uh, and maybe come back to it. Uh, if there's interest later, let me just say for the moment that the multi-resolution announce is a nested system of uh vector spaces, actually her good spaces, 387 00:56:57,540 --> 00:57:06,929 Thomas Thiemann: uh part of which we are part of the to space over the Sigma and Sigma are the spacious license of our system, 388 00:57:07,170 --> 00:57:15,449 Thomas Thiemann: and you can think of them of these silver spaces as one particle in that spaces and coding the resolution that we are looking at. 389 00:57:15,510 --> 00:57:22,979 Thomas Thiemann: So you can think of them as many functions, form factors, and so on depending on the theory that you are interested in. 390 00:57:23,830 --> 00:57:41,040 Thomas Thiemann: And uh, the second ingredient is a is a scaling function. Wh. What people in wavelengths you you call a scaling function, and such as mother scaling function. There's only one of them, or several few of them, but certainly a finite number of them are such that res scaleings and translates of them 391 00:57:41,050 --> 00:57:47,790 Thomas Thiemann: provide an orthonormal basis of this final resolution uh smelling function. Hilbert space 392 00:57:50,450 --> 00:58:03,179 Thomas Thiemann: uh And organization. With respect to such a multi-resolution analysis uh was um considered by people working in constructive quantum fe three before um, 393 00:58:30,340 --> 00:58:39,889 Thomas Thiemann: and we this use this um These sky functions, these uh um scaling functions as an off-normal basis. 394 00:58:41,180 --> 00:58:58,920 Thomas Thiemann: And it turns out that because of uh, the way these things work together, this uh combination I am I am dagger is a projection in that space of smearing functions which will become a little bit important in in what follows. 395 00:58:59,360 --> 00:59:08,049 Thomas Thiemann: And using such a Mrr structure, mri structure, you can you find a a call course grading system. 396 00:59:08,760 --> 00:59:25,059 Thomas Thiemann: So if if you don't know to understand the details of this construction, that's not important, but all of that is important is to just keep in mind that such an Mrr. A structure is an organizational principle. Did you find what is a course screening map? 397 00:59:27,040 --> 00:59:33,100 Thomas Thiemann: And uh, of course, there's lots of choices of such course grading structures. And 398 00:59:33,210 --> 00:59:48,170 Thomas Thiemann: Erez Agmoni people. So this is a whole discipline in mathematics. Uh Wavelet theories. So people are interested in in such functions which are, or or scaling functions which are somehow local, both in position and momentum space one hundred and fifty, 399 00:59:48,230 --> 00:59:55,440 Thomas Thiemann: and have some scooter's properties, and there's lots of lots of works by by these people here 400 00:59:55,560 --> 01:00:00,039 Thomas Thiemann: uh and for zoom, notice, or even a field metal for 401 01:00:01,950 --> 01:00:04,710 Thomas Thiemann: that has been given to Ingred to Pages 402 01:00:06,140 --> 01:00:07,279 Thomas Thiemann: um 403 01:00:13,970 --> 01:00:27,629 Thomas Thiemann: do this in practice. So we pick an Mr. A structure, so which I said, As I said, it's just nothing else, and a nice way to organize your uh course training um procedure. 404 01:00:28,410 --> 01:00:30,550 Thomas Thiemann: And then i'm supposed to give 405 01:00:31,110 --> 01:00:34,069 Thomas Thiemann: be equipped with uh a system. 406 01:00:34,220 --> 01:00:35,339 Thomas Thiemann: Um 407 01:00:36,100 --> 01:00:47,379 Thomas Thiemann: uh which consists of a Hamiltonian together as a phase space. And I can use this uh, this, Mmr: A structure or course training tools in order to discretize my my face space. 408 01:00:47,640 --> 01:00:51,609 Thomas Thiemann: I can use them to discretize my Hamiltonian 409 01:00:52,710 --> 01:00:54,000 Thomas Thiemann: in this way, 410 01:00:54,310 --> 01:00:56,549 Thomas Thiemann: and this is, of course, uh 411 01:00:57,880 --> 01:01:00,419 Thomas Thiemann: kind of an arbitrary choice, 412 01:01:00,740 --> 01:01:15,679 Thomas Thiemann: and the whole out of the game consists in showing that, uh, no matter what kind of choice you do at the beginning uh you're defining a organization flow which drives you into a kind of unique theory, 413 01:01:18,190 --> 01:01:19,270 Thomas Thiemann: and 414 01:01:35,900 --> 01:01:42,159 Thomas Thiemann: you you find a H about space, a vacuum annihilated by this Hamiltonian. 415 01:01:42,440 --> 01:01:49,180 Thomas Thiemann: You define, for example, while elements corresponding to this discretized harm. Um! 416 01:01:53,650 --> 01:01:55,750 Thomas Thiemann: And then you start 417 01:01:56,060 --> 01:01:58,550 Thomas Thiemann: computing the random. 418 01:01:59,260 --> 01:02:09,000 Thomas Thiemann: So you are supposed to compute a vacuum from from vacuum. So you have some old vacuum at high resolution 419 01:02:09,370 --> 01:02:10,810 Thomas Thiemann: uh at 420 01:02:11,140 --> 01:02:13,589 Thomas Thiemann: uh little, and and 421 01:02:19,310 --> 01:02:21,139 Thomas Thiemann: of your um 422 01:02:28,790 --> 01:02:40,500 Thomas Thiemann: consisting of a vacuum, a Herbert space, and a Hamiltonian at each resolution. And Here's the formula that allows you to compute the new vacuum from the old vacuum 423 01:02:41,610 --> 01:02:51,029 Thomas Thiemann: where M. Is August, smaller than the resolution labels by M. Is more, it's always cooler than the roof, resolution, and prime, 424 01:02:52,300 --> 01:03:00,390 Thomas Thiemann: and the last step then consists in looking for after you have computed this flow looking for fixed points. 425 01:03:02,500 --> 01:03:18,610 Thomas Thiemann: And why does this help to define or to to construct quantum non-degenerate representations? Because, uh at each resolution this wild states are are non degenerate because they are excited everywhere at that resolution, 426 01:03:18,670 --> 01:03:26,870 Thomas Thiemann: and then you then go to the continuum. You expect that this bone degeneracy is inherited also at the 427 01:03:26,890 --> 01:03:29,799 Thomas Thiemann: Uh. It had infinite resolution, so to say. 428 01:03:32,390 --> 01:03:34,689 Thomas Thiemann: And now we want to test this, 429 01:03:35,000 --> 01:03:49,620 Thomas Thiemann: and uh, as you may have seen or realized, um, all of this in constructive quantity is made not for constraint systems, not even for gravity, is actually made for for a nice one. 430 01:03:49,740 --> 01:04:00,050 Thomas Thiemann: Systems with a well-defined Hamiltonian. So not other. So so for for consensus, and has never been defined so my 431 01:04:00,520 --> 01:04:02,919 Thomas Thiemann: well. Our proposal was here to 432 01:04:03,640 --> 01:04:23,150 Thomas Thiemann: never the less apply this kind of kind of ideas, also a to a constraint system. To each of these constraints let's say, for the to to each of the spatial different from some constraints, as if it was a Hamiltonian. So just get copy paste the formulas, but do it for all the smelling functions. 433 01:04:23,160 --> 01:04:33,609 Thomas Thiemann: That's the proposal. And then, of course, the questions that I immediately arise. What about the vacuum? Is there a common vacuum for all the constraints? 434 01:04:34,930 --> 01:04:38,700 Thomas Thiemann: Is it necessary to have a vacuum for all the constraints? 435 01:04:40,110 --> 01:04:43,940 Thomas Thiemann: Should we also discretize the labs and shift functions. 436 01:04:44,650 --> 01:04:46,270 Thomas Thiemann: If, yes, how 437 01:04:46,950 --> 01:04:53,150 Thomas Thiemann: and how does the constraint? Algebra and the anomaly structure react to the renormalization flow? 438 01:04:53,950 --> 01:05:08,109 Thomas Thiemann: And the this this kind of uh techniques have been applied so far only to, and the most uh simple and more complicated the theory that we could think of 439 01:05:08,150 --> 01:05:23,220 Thomas Thiemann: um. It's uh Carl Cookers parameters to be into them two dimensions, which is, uh, classically. So Google. And therefore we um. You look at this model from a quantum perspective, and from the perspective of renormalization. 440 01:05:25,490 --> 01:05:39,229 Thomas Thiemann: So this is now parametized. V. Three on the cylinders or the Time access cross uh uh circle. It's boundary boundary conditions. Oops 441 01:05:40,650 --> 01:05:41,810 Thomas Thiemann: um, 442 01:05:45,170 --> 01:05:47,209 Thomas Thiemann: and i'm just listing 443 01:05:47,250 --> 01:05:58,019 Thomas Thiemann: erez agmoni what we found. So the first lessons that some's degree of smoothing smoothness of this creating function is mandatory in order to have a well-defined one hundred 444 01:05:58,090 --> 01:06:01,009 Thomas Thiemann: algebra of constraints. 445 01:06:01,640 --> 01:06:08,289 Thomas Thiemann: The racket decrease of the Fourier transform of uh these very functions is also necessary, 446 01:06:08,820 --> 01:06:14,720 Thomas Thiemann: i'm mentioning this because people like to work It's a so called ha 447 01:06:14,970 --> 01:06:28,809 Thomas Thiemann: multi-resolution analysis. This is the classic blocks and course grading uh transportation that people use, for example, in in the uh renormalize the ising model. So this would not work in this case. 448 01:06:29,600 --> 01:06:30,700 Thomas Thiemann: Um! 449 01:06:30,890 --> 01:06:34,789 Thomas Thiemann: What work? For example, is the so-called Jewishly 450 01:06:34,920 --> 01:06:50,379 Thomas Thiemann: multi- resolution analysis again. I'm not going to into D. Thirty. I'm just flashing how the uh Orthonormal basis at Resolution Capital M. Looks like, and M. There's a little m denotes the lattice point on on the circle. 451 01:06:51,370 --> 01:06:52,509 Thomas Thiemann: Um. 452 01:06:52,730 --> 01:06:57,739 Thomas Thiemann: One can show that the flow has the unknown continuum theory as a fixed point 453 01:07:16,330 --> 01:07:19,680 Thomas Thiemann: uh the classical algebra is centrally extended, 454 01:07:19,910 --> 01:07:25,519 Thomas Thiemann: and therefore we get to be with all algebra, and you recover that we was over algebra. And this, uh, 455 01:07:25,720 --> 01:07:28,840 Thomas Thiemann: uh, in this normalization procedure, 456 01:07:30,090 --> 01:07:34,589 Thomas Thiemann: and therefore there is no come back from in this case. But it's not necessary. 457 01:07:34,810 --> 01:07:40,630 Thomas Thiemann: It is possible to discretize also labs and shift functions, but it's not necessary 458 01:07:41,790 --> 01:07:44,589 Thomas Thiemann: and the most interesting. 459 01:07:44,700 --> 01:07:47,359 Thomas Thiemann: Our lesson is maybe the last one here. 460 01:07:58,370 --> 01:07:59,439 Thomas Thiemann: Um, 461 01:08:00,680 --> 01:08:12,830 Thomas Thiemann: and it's physically correct that there are finite resolution artifacts here. In this case we can actually compute them quite in close form. So what we do is we take the known one 462 01:08:12,860 --> 01:08:18,599 Thomas Thiemann: quantizations of the continuum constraints which have a central extension 463 01:08:18,939 --> 01:08:20,090 Thomas Thiemann: um, 464 01:08:21,040 --> 01:08:25,670 Thomas Thiemann: we project them to find out resolution. We compute the algebra of these guys. 465 01:08:25,700 --> 01:08:38,370 Thomas Thiemann: You get the finite resolution analog of the commutator of the x-axis commentator, plus the central extension that should be there, because it's central 466 01:08:39,109 --> 01:08:41,590 Thomas Thiemann: at Resolution, M. And 467 01:08:41,790 --> 01:08:43,409 Thomas Thiemann: out of which 468 01:08:43,500 --> 01:08:49,990 Thomas Thiemann: looks like an anomaly, but which is more or less just an artifact that comes from finite resolution, 469 01:08:50,069 --> 01:08:53,290 Thomas Thiemann: which you can see from the fact that um 470 01:08:53,439 --> 01:08:55,109 Thomas Thiemann: the week 471 01:08:55,350 --> 01:09:03,209 Thomas Thiemann: operator topology limit of this artifact is going to zero as you turn up the resolution to infinity. 472 01:09:23,290 --> 01:09:24,309 Thomas Thiemann: Um! 473 01:09:24,529 --> 01:09:35,129 Thomas Thiemann: The natural density of one is not necessarily an obstacle to our uh to close the algebra in at least not in non degenerate representations. 474 01:09:35,680 --> 01:09:36,790 Thomas Thiemann: Um, 475 01:09:51,939 --> 01:09:56,929 Thomas Thiemann: um! The exponential Hamil exponentiated Hamiltonian constraint, 476 01:09:57,650 --> 01:09:59,989 Thomas Thiemann: at least in the you one two model 477 01:10:00,100 --> 01:10:07,830 Thomas Thiemann: has a very different action from what was guessed so far. Uh because it's so nonlinear acting. 478 01:10:08,180 --> 01:10:09,559 Thomas Thiemann: And um, 479 01:10:09,710 --> 01:10:17,280 Thomas Thiemann: it's very nice to see that this can be worked out in a lease you one cube model, and so explicitly, 480 01:10:18,420 --> 01:10:25,279 Thomas Thiemann: and uh, it's. This is called the Hamiltonian, or, in fact, uh general, is minimization. 481 01:10:25,370 --> 01:10:33,039 Thomas Thiemann: Um. It gives you a systematic tool to give to look for quantum non-degenerate representations, 482 01:10:33,170 --> 01:10:37,130 Thomas Thiemann: and it gives you a tool to disentangle uh 483 01:10:37,270 --> 01:10:43,920 Thomas Thiemann: we are discretization artifacts from two anomalies at least in the model that we have 484 01:10:44,120 --> 01:10:45,940 Thomas Thiemann: been able to look at so far. 485 01:10:46,180 --> 01:10:48,220 Thomas Thiemann: Thank you very much for your attention. 486 01:11:09,720 --> 01:11:13,570 Jerzy Lewandowski: Hello! And can you confirm that you can hear me? 487 01:11:13,670 --> 01:11:17,640 Thomas Thiemann: I can. I can hear you. You're okay. Can anybody hear me? 488 01:11:19,620 --> 01:11:22,519 Jerzy Lewandowski: I can hear you? 489 01:11:23,370 --> 01:11:24,540 Jerzy Lewandowski: Okay? 490 01:11:31,590 --> 01:11:33,289 Jerzy Lewandowski: Hmm. Hello! Hello! 491 01:11:37,480 --> 01:11:49,950 simone: Well, maybe for questions. 492 01:11:50,870 --> 01:11:54,940 simone: What do you suggest is the way for the starting to meet in the quantum gravity. 493 01:11:55,470 --> 01:12:02,479 Thomas Thiemann: Um. So at least on my on my computer there was a strong interference. Can you repeat the questions, please? 494 01:12:09,260 --> 01:12:13,050 simone: What do you see as the way forward? 495 01:12:13,460 --> 01:12:14,559 Thomas Thiemann: Um, 496 01:12:14,650 --> 01:12:15,869 Thomas Thiemann: Yeah. So 497 01:12:16,020 --> 01:12:17,030 Thomas Thiemann: hmm 498 01:12:17,110 --> 01:12:33,310 Thomas Thiemann: to me. Um. The The lesson is is the following: I mean if I look at the first part of my my talk, then the the tangent comes from the fact that, uh, on the one hand, we are strongly interested in 499 01:12:33,460 --> 01:12:41,910 Thomas Thiemann: representations in which we want the volume to act only at the vertices, 500 01:12:41,930 --> 01:12:50,650 Thomas Thiemann: in order that, for example, the Hamiltonian constraint is not blowing up on our States of the Herbert Space. 501 01:12:51,640 --> 01:12:59,790 Thomas Thiemann: On the other hand, then, we get a problem to close the algebra, because a classical algebra wants us to have an infinite number of terms 502 01:13:00,610 --> 01:13:02,999 Thomas Thiemann: need for a number of contributions. 503 01:13:03,340 --> 01:13:06,690 Thomas Thiemann: This is where where this tension comes from, 504 01:13:06,730 --> 01:13:09,179 Thomas Thiemann: and the tension can be resolved 505 01:13:09,370 --> 01:13:17,219 Thomas Thiemann: hopefully. If you look for representations in which the geometry is not so degenerate, 506 01:13:20,670 --> 01:13:33,310 Thomas Thiemann: and that maybe it's it requires to have a fresh look at what our smearing functions should be, Maybe we should uh So the Gauss constraint in a different way, 507 01:13:33,360 --> 01:13:35,290 Thomas Thiemann: and so on, so far, so. 508 01:13:35,800 --> 01:13:43,749 simone: Oh, basically just to I think I understood what you say, but just to make it more explicitly, what you're saying is that the way forward should be to stop using the speed network basis 509 01:13:44,810 --> 01:13:45,889 Thomas Thiemann: um 510 01:13:47,630 --> 01:14:02,390 Thomas Thiemann: erez agmoni. That could be one possibility or one one one could um try to find in the same Herbert space a different uh space of of functions which are closer to being non-degenerate one hundred and one. 511 01:14:05,640 --> 01:14:20,509 simone: So so you're saying the Apr space may be the same, but still we should stop using spin network states, or the inverse space is completely different. 512 01:14:27,960 --> 01:14:37,170 Thomas Thiemann: These these are much closer to being a quantum non-degenerate and spin network. Functions. Maybe we should like all the weed States, for instance, 513 01:14:52,030 --> 01:14:55,499 Thomas Thiemann: while one had would have to look at super positions of those. 514 01:15:06,940 --> 01:15:07,969 simone: Okay, 515 01:15:09,570 --> 01:15:10,570 simone: Thank you. 516 01:15:10,930 --> 01:15:11,870 Sure. 517 01:15:18,600 --> 01:15:31,599 Western: Um, Thomas, this is Carla. Uh, I I I found very convincing your the the the part of the of the top where you say that the exponentiation of the misconceptions it's a 518 01:15:31,640 --> 01:15:35,220 Western: it's presumably the the same one has to look at. 519 01:15:35,400 --> 01:15:38,749 Western: Uh. Your main point, however, is not. This one is about 520 01:15:39,120 --> 01:15:47,199 Western: the non sequential no degeneracy. And um uh, I mean. I hear what you say, and I see I see the argument. It's it's general. I would 521 01:16:02,740 --> 01:16:07,849 Western: we couldn't couldn't the problem be the algebra itself. I mean It's not focusing so much 522 01:16:08,050 --> 01:16:09,660 Western: on having 523 01:16:10,190 --> 01:16:19,450 Western: a uh cosine algebra will defined. Is that right? Because at the light of all this, because uh um, 524 01:16:20,090 --> 01:16:25,450 Western: in a sense uh the the overall physical 525 01:16:26,490 --> 01:16:32,360 Western: message of so many years of refugees. That is perhaps a reasonable theory. 526 01:16:32,580 --> 01:16:45,310 Western: Um, which is discrete. So it's a we're used in in like, you see, in in anything to the fact that because of this discreetness you just cannot take derivatives. 527 01:16:45,710 --> 01:16:55,779 Western: Uh, that has been a strength of the full construction, not the weakness. So if you cannot take the reasons, why should you have a well defined uh 528 01:16:55,940 --> 01:17:01,220 Western: uh constraint algebra, which are which are communities, 529 01:17:02,440 --> 01:17:14,980 Western: so I mean this W. All this is going to see, but it isn't isn't that sort of could couldn't but one look also this in in terms of Well forget the the 530 01:17:15,080 --> 01:17:24,980 Western: forget, the constraints algebra, and they find it here in a different manner of we. We have already a Hilbert space that seems to work very well for many things. 531 01:17:25,040 --> 01:17:33,340 Western: Um, we have a number of different ways of thinking of the dynamics. Why? Why? Insisting so much on the constraint algebra. 532 01:17:35,900 --> 01:17:41,770 Thomas Thiemann: So um uh you. You're saying that uh you want uh 533 01:17:42,150 --> 01:17:49,459 Thomas Thiemann: you use. I mean what this set of works um suggests is to look at the exponentiated 534 01:17:49,610 --> 01:17:53,680 Thomas Thiemann: uh objects, not at the objects themselves. Is that, 535 01:17:54,350 --> 01:18:02,660 Thomas Thiemann: Yeah. So that works quite well for the Abelian theory for the non-abelian theory. It's much more complicated 536 01:18:02,810 --> 01:18:04,010 Thomas Thiemann: um 537 01:18:11,600 --> 01:18:20,530 Thomas Thiemann: like this Hamiltonian remarization program, for example, where um we have to define um 538 01:18:34,220 --> 01:18:37,709 Thomas Thiemann: uh invented so far and in the in the 539 01:18:37,880 --> 01:18:41,360 Thomas Thiemann: none of you in case i'm afraid that's not sufficient, 540 01:18:43,650 --> 01:18:48,319 Thomas Thiemann: even if you want to define. Finally the the exponentiated version 541 01:19:02,560 --> 01:19:10,270 simone: Wait. But if I understand your message, if I had to you. If I used the standard refugee methods in this U. One model, 542 01:19:10,370 --> 01:19:25,570 simone: I will not end up in a a corrective presentation of the open surface information on to the right. That was your point. You still need not to use charge network states, but at least in your combinations or whatever you are arguing for, 543 01:19:25,810 --> 01:19:34,290 simone: so that it is not as distribution as in the usual setup right. So what is important is that at the end of the day i'm using two, 544 01:19:34,490 --> 01:19:43,719 Thomas Thiemann: not uh, not new functions which are just uh excited on on graphs, but they must be excited somehow everywhere, 545 01:19:45,670 --> 01:19:46,519 simone: Right? 546 01:19:48,550 --> 01:19:55,780 Thomas Thiemann: Still, they can be deep. They can be quasi-excited on the graph, because you can always take a graph 547 01:19:55,880 --> 01:20:10,969 simone: distribution and and smell it out. Some all and make it almost look like a one dimensional right. I guess that would be for me. That would be the next step here, mainly figuring out whether one can obtain the same results. Starting from a charge network theory, 548 01:20:11,030 --> 01:20:14,789 simone: finding a suitable meeting over the graphs. 549 01:20:14,810 --> 01:20:24,979 simone: That would mean that's been. Networks could still be used also in on a billion case. But if you prove some no go, or whatever, some very unnatural 550 01:20:33,950 --> 01:20:35,759 simone: something that one needs to know. 551 01:20:37,610 --> 01:20:52,829 Thomas Thiemann: Yeah, all I can say is that in the you one cube model uh one can play this game uh. So you take you, take some whole um. This case of the smearing function, which is as as as you want to. Uh uh a a a graph, 552 01:20:52,940 --> 01:20:57,359 Thomas Thiemann: a graph steering function which is just excited on the edges of a graph 553 01:21:05,620 --> 01:21:08,670 Thomas Thiemann: from what we thought it should act. 554 01:21:14,210 --> 01:21:27,200 Thomas Thiemann: But of course one keeps. Keep in mind that this is the finite action of the Hamiltonian. It's not just a single action, so it's somehow the the result of of many, many actions. So it's uh it's not that easy to to compare, 555 01:21:29,200 --> 01:21:33,110 simone: but still you said it can also be represented as painful model. 556 01:21:34,470 --> 01:21:38,590 Thomas Thiemann: Yes, uh. So what I'm saying is that um you can. 557 01:21:38,670 --> 01:21:39,860 Thomas Thiemann: You can 558 01:21:50,990 --> 01:21:52,030 Thomas Thiemann: uh, 559 01:21:52,150 --> 01:21:59,280 Thomas Thiemann: as is contained in many, many textbooks, and then you define first of all the path undergo 560 01:21:59,840 --> 01:22:11,270 Thomas Thiemann: on what people call the reduced phase space with the with the physical Hamiltonian, and then you unfold the uh, the reduced phase, space to the kinematical phase space. 561 01:22:11,570 --> 01:22:20,809 Thomas Thiemann: And finally, after many, many, many steps, you end up with something that looks like the some measure times the exponential of the the action 562 01:22:22,660 --> 01:22:25,119 Thomas Thiemann: that can be can be done In this model. 563 01:22:25,970 --> 01:22:37,100 Mehdi: I can see several race hands, but maybe some of them are already. It is so already satisfied. So what about Western? Have you asked your question? 564 01:22:38,650 --> 01:23:06,419 Western: Okay, What about? Mostly 565 01:23:06,430 --> 01:23:21,570 Western: uh the gradient sector, and maybe the Lorentz one or the three D and the forty in the sense that there there is this quantum reduction that connects with the full theory. Is there anything we can learn and import uh from there here, 566 01:23:22,160 --> 01:23:31,680 Thomas Thiemann: so that and their desk can be there. Some kind of interaction between this and uh, what a Manuela was doing in the sense that um 567 01:23:31,830 --> 01:23:34,470 Thomas Thiemann: he is somehow um 568 01:23:34,650 --> 01:23:40,060 Thomas Thiemann: as far as our recall, he's somehow engaged fixing the 569 01:23:40,230 --> 01:23:48,699 Thomas Thiemann: at least some partial gauge facing is going on, which enables him to somehow reduce the gauge group. 570 01:23:48,770 --> 01:23:51,310 Thomas Thiemann: Uh, also to an a little one. 571 01:23:52,420 --> 01:24:00,329 Thomas Thiemann: Uh, in that sense uh his procedure and the one called cube model are quite similar, but of course the 572 01:24:06,580 --> 01:24:09,929 Thomas Thiemann: is ah such that there is an a squared. 573 01:24:10,130 --> 01:24:13,220 Thomas Thiemann: So the one cube model is still simpler in that sense, 574 01:24:31,400 --> 01:24:44,920 Thomas Thiemann: erez 575 01:24:59,330 --> 01:25:16,150 Thomas Thiemann: uh, and this is this is the deformation of of gate groups which is called, uh consistent in the mathematical, The literature, and what he he points out is that? Suppose that one one of the members of this two Uh theories, 576 01:25:16,160 --> 01:25:21,380 Thomas Thiemann: which are consistent deformations of each other, is such that one of them is integrable. 577 01:25:22,010 --> 01:25:35,339 Thomas Thiemann: Then it looks like a good idea to to to quantize the into global model and perform perturbation to you for the full model for the for the full month of theories around that. Uh, in the global model. 578 01:25:35,710 --> 01:25:37,960 Thomas Thiemann: So I think that's something that 579 01:25:38,160 --> 01:25:40,760 Thomas Thiemann: might might be worth looking into. 580 01:25:40,970 --> 01:25:43,700 Western: That's very interesting, Thank you, 581 01:25:43,960 --> 01:25:47,070 Mehdi: and would you like to 582 01:25:47,120 --> 01:25:48,550 Muxin Han: thank you? 583 01:25:48,700 --> 01:26:00,329 Muxin Han: So So first the thanks for the talk. Um! So I I think there was one slice on, and one line about the relation uh with renewal space based organization. So for the you, one queue theory, 584 01:26:00,340 --> 01:26:10,609 Muxin Han: it's a and it's. Are they company equivalent to operator and the the constraint colonization and reduce basic from that is um organization. Are they company equivalent? 585 01:26:10,710 --> 01:26:12,209 Thomas Thiemann: So um 586 01:26:12,720 --> 01:26:15,540 Thomas Thiemann: uh, let me see whether I have something 587 01:26:15,850 --> 01:26:18,100 more specific to stay here. 588 01:26:18,200 --> 01:26:26,839 Thomas Thiemann: No, unfortunately, there's no details about this. But, as you know very well from uh your Phd: this is work. Uh 589 01:26:27,170 --> 01:26:38,610 Thomas Thiemann: yeah, What What one has to do is the following: Um. So here we are dealing with It's U. We whose constraint algebra is not the algebra, 590 01:26:39,080 --> 01:26:49,040 Thomas Thiemann: and therefore uh, one cannot use the rigging map techniques per se. One has to first 591 01:26:49,550 --> 01:26:57,400 Thomas Thiemann: uh reformulate uh the classical theory in such a way that the constraints form a linear algebra, 592 01:26:58,130 --> 01:27:06,829 Thomas Thiemann: and this can be done for this, for this model for the one cube model. And starting from this Abelianized 593 01:27:07,100 --> 01:27:18,579 Thomas Thiemann: version of the consent algebra, and you can show that reduce face-based quantization and the rigging, map, or group averaging construction are equivalent. 594 01:27:20,890 --> 01:27:23,189 Muxin Han: Okay, so um 595 01:27:23,410 --> 01:27:27,679 Muxin Han: But then um. The question is because you know, the 596 01:27:27,870 --> 01:27:32,080 Muxin Han: um we'll, we'll, we'll 597 01:27:32,130 --> 01:27:48,999 Muxin Han: um it doesn't rely on uh the the um uh quantum normally free uh construction of uh constraint algebra, but of course it's um and it's relates to the uh algebra. So i'm. I think my question is um 598 01:27:49,480 --> 01:27:57,070 Muxin Han: whether um so? The the reduced face for the organization is Um, It's more advantage has more advantage. 599 01:27:57,350 --> 01:28:00,329 Muxin Han: Um, then the um 600 01:28:00,390 --> 01:28:04,499 um than the constraint quantization, because um! It seemed to me that 601 01:28:04,540 --> 01:28:09,420 Muxin Han: the red deals by facebook have reduced Space-based colonization is always correct. 602 01:28:09,700 --> 01:28:10,900 Muxin Han: Um! 603 01:28:11,110 --> 01:28:20,339 Muxin Han: And it doesn't see the the um um fundamentally coming from the constraint algebra! And do you think it's uh it's right way to understand. 604 01:28:20,790 --> 01:28:21,969 Thomas Thiemann: Um, 605 01:28:22,270 --> 01:28:42,259 Thomas Thiemann: yeah, yes, and no um. So looking at the details of reduce face face quantizations, one has to take this as a grain of sort. For example, if you, if you, if you're trying to solve your constraints for for a set of momenta, you may have to solve quadratic equations, and then there are 606 01:28:42,270 --> 01:28:47,729 Thomas Thiemann: sign issues. So there are always several branches in your face space, and you have to pick one 607 01:28:57,050 --> 01:29:00,729 Thomas Thiemann: quantizing at the end. Only half of this phase. Space. 608 01:29:00,930 --> 01:29:02,040 Thomas Thiemann: Um. 609 01:29:02,840 --> 01:29:11,620 Thomas Thiemann: Also there are assumptions that you have to make, and when you, when you solve for this momenta then you have to assume that certain matrices are non-degenerate 610 01:29:12,290 --> 01:29:23,699 Thomas Thiemann: uh having made that assumption. And of course, the reduced phase space quantization is more convenient, because you never need to worry about this anomalies. 611 01:29:25,940 --> 01:29:45,100 Thomas Thiemann: But uh, if if you, if you're interested in in in the kinematical quantization and the constraint, quantization, and how disconnect with other than uh, so it can be also the operator quantization over the to operate the uh constraint Conversation is, it's interesting, 612 01:29:45,150 --> 01:29:58,400 Thomas Thiemann: but I agree that uh from from an economical point of view. The reduced face-based quantization for very complicated systems is is probably the most uh promising route to go. 613 01:29:58,930 --> 01:30:05,450 Muxin Han: Okay, is there. Some new license. Um, you inside, coming from this, you one two phone because it's easy to compute. 614 01:30:06,550 --> 01:30:08,969 Thomas Thiemann: Yeah lesson in this sense. 615 01:30:09,620 --> 01:30:14,549 Muxin Han: Um. I mean the advantage of one or the other 616 01:30:14,980 --> 01:30:18,689 Muxin Han: um reduce space based, or or um constraint out of pop 617 01:30:18,740 --> 01:30:20,529 Muxin Han: constraint. Organization. 618 01:30:20,920 --> 01:30:38,860 Thomas Thiemann: Um. So not not not really, because here it was not so difficult to um to arrive at the reduced phase space. The description also is coming from the constraint description. But if I try to extrapolate to a more complicated systems, then 619 01:30:38,970 --> 01:30:44,130 Thomas Thiemann: let's say, Euclidean vacuum general relativity, 620 01:30:44,470 --> 01:30:45,559 Thomas Thiemann: I mean, 621 01:30:53,990 --> 01:30:59,319 Thomas Thiemann: uh approach to define the physical habit. Space is probably a hopeless task. 622 01:31:01,580 --> 01:31:02,910 Muxin Han: Okay, thank you. 623 01:31:14,410 --> 01:31:15,450 Mehdi: Hmm. 624 01:31:16,530 --> 01:31:17,690 Mehdi: If not 625 01:31:17,760 --> 01:31:21,280 Mehdi: with a single speaker again, 626 01:31:25,620 --> 01:31:26,620 and 627 01:31:27,050 --> 01:31:29,790 Mehdi: thank you. 628 01:31:30,280 --> 01:31:33,200 Thomas Thiemann: Thank you. Bye, bye, 629 01:31:34,250 --> 01:31:35,429 Western: right. 630 01:31:35,450 --> 01:31:36,719 Lee: Thank you. 631 01:31:54,200 --> 01:32:01,990 Western: Now you like we will over your shirt, and we hope to have you soon investor with us.