0 00:00:00,000 --> 00:00:00,989 My what 1 00:00:03,449 --> 00:00:09,030 Jorge Pullin: Okay so speaker to this mother and mother engine will speak about space time covariance on propagation and canonical Congress 2 00:00:10,980 --> 00:00:17,279 madhavan: Okay hello everybody in this very strange surreal time I, at least, I find it comforting to come back to something 3 00:00:18,630 --> 00:00:30,660 madhavan: Is is is shared between all of us, so welcome. This is the title of my talk space time covariance and propagation canonical new quantum gravity and 4 00:00:31,590 --> 00:00:44,970 madhavan: Basically, this talk is to give a kind of broad overview of the work I've been involved in over the last many years, either in collaboration, or on my own and all this work concerns. 5 00:00:46,050 --> 00:01:00,270 madhavan: The open problem of the quantum dynamics for quantum gravity, namely trying to fix the various ambiguities, which are present. When one tries to define the Hamiltonian constraint operator. 6 00:01:02,670 --> 00:01:05,190 madhavan: So this is the rough plan of my 7 00:01:07,170 --> 00:01:13,050 madhavan: First just review what the basic problem is with the quantum Hamiltonian constraint. 8 00:01:14,490 --> 00:01:26,490 madhavan: Is that you have infinitely many choices in its construction and therefore the strategy, which I have been trying to follow is to constrain the choices by imposing 9 00:01:27,150 --> 00:01:36,900 madhavan: non trivial physical requirements on the quantum dynamics, which results from these choices and the two requirements, I said, is 10 00:01:38,010 --> 00:01:49,410 madhavan: One as one is that of space time covariance and the other is that of propagation softer basically reviewing what problems are and why they occur. I will then 11 00:01:50,730 --> 00:01:52,290 madhavan: Translate to the second part of 12 00:01:53,550 --> 00:01:58,440 madhavan: Space Time covariance and by spacetime covariance. I mean, 13 00:01:59,580 --> 00:02:06,510 madhavan: The existence of an anomaly free representation of the quantum constructive. 14 00:02:08,340 --> 00:02:17,250 madhavan: I will start with some general remark about space and coherence in energy, and then I'll translate to a particular setting. 15 00:02:17,760 --> 00:02:28,890 madhavan: Of a model which I call the yuan Q model. And this is a model which is obtained by replacing the tribe rotation as you to grow. 16 00:02:29,310 --> 00:02:41,550 madhavan: Lydian gravity by three copies of you one and this model can also in by a novel Newton's constant going to see on demand of Ukrainian gravity, as was shown by me a long time ago. 17 00:02:42,240 --> 00:02:48,870 madhavan: For our purposes. This is a very useful to our model in order to study these issues for FaceTime parents and propagation 18 00:02:49,380 --> 00:03:02,760 madhavan: And then pending on time available, I will go to the third part which is an account of propagation. Again, I'll first make some general remarks and then to illustrate things are in the context of the you want to model. 19 00:03:04,260 --> 00:03:13,230 madhavan: So let me start before going to the Hamiltonian construct. Let me with at least what my graph general viewpoint is and 20 00:03:14,190 --> 00:03:32,730 madhavan: I think each of us has a slightly different point towards what Luke quantum gravity is and maybe we share most of the things which I'm going to say except perhaps for the last, last point, the very last sentence of the sponsor. Let me say what my viewpoint is my viewpoint is that 21 00:03:33,870 --> 00:03:50,340 madhavan: He is a non productive generally Kobe and quantization of general relativity, the quantum quantum kinematics has very well understood and an understanding of the quantum dynamics is still in progress. The three properties of flow quantum gravity. 22 00:03:51,360 --> 00:04:04,230 madhavan: Our first that the representation of the tribe operators picture of districts patient geometry with area contest with a not smallest nonzero eigenvalue, which is approximately the blank area. 23 00:04:04,950 --> 00:04:09,810 madhavan: Second do to spatial data from autism invariance, as we'll see in the next slide. 24 00:04:10,560 --> 00:04:17,970 madhavan: The local connection operator simply does not exist, and only exponential functions of connections exist as operators. 25 00:04:18,420 --> 00:04:34,770 madhavan: And so the deep ultraviolet degrees of freedom are therefore not local connection fields, but there are some new strange degrees of freedom, which are discreet and non local graphical excitations really quantum dynamics dynamics of these degrees of freedom. 26 00:04:36,210 --> 00:04:44,490 madhavan: So, to your point is that the fundamental theory, some degree and we approving it through continuum tools and canonical energy 27 00:04:44,910 --> 00:05:03,510 madhavan: And at some stage one will have to jump and confirm the discussion is on its own terms, I feel that there is still a lot to learn from the continuum structures before make this jump. So that is the viewpoint, which will pervade what I'm trying to do. Let me go to the next slide. 28 00:05:05,490 --> 00:05:19,740 madhavan: So let me now come to the problem of the Hamiltonian constraint, the quantum dynamics of look quantum gravity in its canonical form is driven by the Hamiltonian constraint operator and the following problem arises in its construction. 29 00:05:20,880 --> 00:05:25,800 madhavan: The classical Hamiltonian constraint depends on local feels like the curvature of the connection. 30 00:05:26,970 --> 00:05:38,700 madhavan: But of course the basic content operators are non local economies and one would like to write down an operator version of this constraint. And so, one would like to write down an operator version of the curvature 31 00:05:39,750 --> 00:05:49,650 madhavan: So classically, there is no problem which can be extracted from the whole army of coordinate size delta Lou to a shrinking limit. So the limit 32 00:05:50,850 --> 00:05:57,630 madhavan: as delta goes to zero or minus one divided by the coordinate area of which causes delta squared. 33 00:05:58,140 --> 00:06:01,530 madhavan: So this limited exists in the classical theory and there's no problem. 34 00:06:01,950 --> 00:06:10,500 madhavan: But if one wants to replace these classic quantities like the whole on me by operators and quantum mechanically this limit does not exist. 35 00:06:10,740 --> 00:06:17,430 madhavan: And it does not exist for a very good reason it does not exist because our Hilbert space is background independent 36 00:06:17,760 --> 00:06:32,490 madhavan: And the human space norm cannot distinguish between the smaller and the still smaller still smaller loops which occur in this process. So this operator limit does not exist and one proceeds therefore as follows. 37 00:06:33,540 --> 00:06:42,660 madhavan: One first replaces all the local connection dependent fields in the Hamiltonian constrained by alanna means of small loops of coordinate size del 38 00:06:43,950 --> 00:06:51,450 madhavan: Some of expression to the Hamiltonian constrained which agrees with it as we take delta going to see no 39 00:06:53,190 --> 00:06:58,230 madhavan: One looks at non zero delta one replaces the 40 00:06:59,700 --> 00:07:12,330 madhavan: Enemies and trials and this approximate each delta of n by the corresponding operators and one gets an operator valued approximate age hacked delta. And when is the laps. 41 00:07:13,020 --> 00:07:23,070 madhavan: And one then attempts to take the Delta going to zero limit of this operator approximately in some sense. And the hope is that even though individual 42 00:07:23,670 --> 00:07:32,370 madhavan: Bits and pieces which make up this operator do not have delta going to zero limits the conglomeration of approximately 43 00:07:32,910 --> 00:07:42,630 madhavan: Together, which make up this approximate operator to the Hamiltonian in this conglomeration actually has a limit as delta going to zero. So, that is the hope. 44 00:07:43,260 --> 00:07:54,540 madhavan: And toes. See whether such a limit exists or how such a limit exists, it is useful to do the following exercise. So I will go now to the next slide. 45 00:07:56,910 --> 00:08:05,700 madhavan: So I call this exercise as counting overall factors overall explicit factors of delta in the approximate 46 00:08:06,540 --> 00:08:27,420 madhavan: Age delta. And so in order to do this, of course, the Hamiltonian constraint itself is an integral. And when one writes down an approximation then typically this is what happens DQ decks, which is the coordinate measure is supposed as a factor of delta Q, the electric field. 47 00:08:28,680 --> 00:08:35,520 madhavan: Can be replaced by Fox operator, which has a well defined action on the kinematics but space divided by delta square 48 00:08:36,090 --> 00:08:52,140 madhavan: Square root of Q operator by volume of a small region divided by delta q its volume and the curvature as a horror on me minus one divided by the coordinate area of the law, delta square. So, when one puts all these things together. 49 00:08:53,400 --> 00:08:59,070 madhavan: One can find out what the overall factor of delta is one just cancel various factors of delta 50 00:08:59,580 --> 00:09:07,860 madhavan: And what one finds for the density wait one constraint is that there is no overall factor at all. And what you are left with our 51 00:09:08,520 --> 00:09:18,840 madhavan: What I call finite operators, they may be parameters by some loops of size delta or some surfaces of sight of size delta squared, etc. 52 00:09:19,530 --> 00:09:32,280 madhavan: But they all have a well defined action on the kinematics space and one can then try and deal with this. The operator, which has no explicit factors of delta and 53 00:09:32,790 --> 00:09:41,010 madhavan: As Thomas showed a long, long time ago, one can show that in a suitable sense the limit of this operator does exist in a certain topology. 54 00:09:41,370 --> 00:09:51,390 madhavan: And as you rip and model showed the limit of your of the operator can also be shown to exist, almost suitable vector space, which they call the habitat. In any case, there is a certain 55 00:09:52,560 --> 00:09:58,530 madhavan: There are ways in which one can take the letter going to zero limit of these are my office and get some action. 56 00:09:59,130 --> 00:10:08,400 madhavan: Unfortunately, the operator action which one gets them continuum limit depends really on the infinite choice autonomy approximates do I take 57 00:10:08,640 --> 00:10:24,390 madhavan: Triangular loops or circular groups or do I take some groups which entangle with the spin networks take their acting upon in some particular way each choice one, one makes them gives you a different choice of continuum limit action. 58 00:10:25,920 --> 00:10:30,570 madhavan: So the problem is that the action of the Hamiltonian constraint is infinitely ambiguous. 59 00:10:31,590 --> 00:10:40,650 madhavan: And therefore, the idea of strategy, which one would like to follow is to try to constrain these choices by extremely non 60 00:10:41,250 --> 00:10:57,810 madhavan: Physical requirements. The two requirements. I will be talking about that of space time covariance and of propagation. Some now transiting to the second part of my talk, which is that on quantum space time covariance. And I will start with some general remarks. 61 00:10:59,250 --> 00:11:12,720 madhavan: So classical space time covariance is encoded in the characteristic form of the constraint algebra. This was shown a long time ago in classic PayPal, by which man who cash and try to avoid 62 00:11:13,770 --> 00:11:25,170 madhavan: And therefore, given this this particular form of the constraint algebra, which encodes spacetime covariance in the classical theory. 63 00:11:25,800 --> 00:11:36,240 madhavan: One x as a definition of quantum space time covariance its implementation in quantum theory without anomalies. So one would in our words. 64 00:11:37,140 --> 00:11:46,320 madhavan: One has various pause on brackets and one would like to implement them in some technical sense as committed as in the quantum theory. 65 00:11:47,010 --> 00:11:57,450 madhavan: Comes non triple points on bracket, which is the one I will pick for the purposes of this talk is that between the Hamiltonian constraints. And that, of course, gives a different monetarism constrain 66 00:11:57,960 --> 00:12:06,300 madhavan: Which is smeared with the shift which depends on the lapses but also on the metric, which is a dynamic variable. 67 00:12:07,590 --> 00:12:14,550 madhavan: So their structure functions in the algebra, and that is what makes this very complicated in the stock. I will call 68 00:12:15,120 --> 00:12:22,620 madhavan: The Hamiltonian constraint commentators, the left hand side and this different more ism, which is dependent on the metric as the right hand side. 69 00:12:23,400 --> 00:12:27,810 madhavan: So in order to implement this FaceTime forbearance and quantum theory, I would like to 70 00:12:28,020 --> 00:12:40,680 madhavan: Look for a choice or a class of choices of the Hamiltonian constraint operator which yields and normally free commentators where the left hand side operator committed to is equal to Irish bar times the right hand side operator. 71 00:12:42,630 --> 00:12:44,580 madhavan: Okay, so let me then go to them. 72 00:12:45,630 --> 00:13:01,200 madhavan: And let's see what happens with density with one constraints, because these are the constraints which was shown to admit nice can continue on limits. So let me look at my hand side and do this counting of powers of delta. So, 73 00:13:02,430 --> 00:13:17,130 madhavan: One has DQ X over here than one as the laps typical lap combination and then one has these face face functions. So one is delta q from the two decks and then from the two 74 00:13:17,610 --> 00:13:29,040 madhavan: Electric fields. One has a flux over delta square hole square from the Q A delta Q by volume hold squared. Then from the ear of flux over delta squared and then from the curvature Ilana me minus one. 75 00:13:30,840 --> 00:13:50,340 madhavan: So if you put all the factors of delta together, what you find is that you get an overall factor of delta times a finite operator. And so, more or less, no matter how you regulate the right hand side as delta goes to zero, you are guaranteed to get a zero answer and this 76 00:13:51,600 --> 00:14:10,890 madhavan: Simplified in in a calculation, a long time ago by Jorge Rodolfo Don and Jurek and indeed the writer is equal to zero. Therefore, if one wants to avoid anomalies. The left hand side must also manage. However, it turns out, as shown in 77 00:14:12,090 --> 00:14:20,760 madhavan: In in dawn and and eurex habitat paper. This can manage for many different actions of the Hamiltonian constraint and 78 00:14:21,330 --> 00:14:28,950 madhavan: The question is how can we discriminate between these various different actions. And as a side note, I would like to also say that 79 00:14:29,670 --> 00:14:48,930 madhavan: The left hand side as is in Thomas's constraint, maybe it is vanishing due to the wrong reasons. So for example, if the second Hamiltonian constraint does not add on the spin network deformations created by the first time tuning in space. So let me say that again. 80 00:14:49,950 --> 00:14:50,850 madhavan: In more detail. 81 00:14:52,110 --> 00:15:00,210 madhavan: That the Hamiltonian constraint, because of the factors of the determinant of metric x only at vertices of spin network states. 82 00:15:00,570 --> 00:15:07,800 madhavan: So let's say we act by the first Hamiltonian constraint default to spin at some particular vertex of the sprint network. 83 00:15:08,790 --> 00:15:19,500 madhavan: And then the second Hamiltonian constraint x on it. So the first Labs is evaluated at the initial vertex we and the second Hamiltonian constraint now has to act. 84 00:15:20,010 --> 00:15:41,640 madhavan: No, supposing it does, it cannot act or does not act on the deformations created by the first time constraint, then the labs, which is the second lab is also evaluated at the first word x where x and the whole expression for the complicated than simply vanishes by anticipated. 85 00:15:42,660 --> 00:15:52,680 madhavan: So I think that this sort of an action then of the Hamiltonian constraint hides an anomaly, because the typical term which is the M one derivative 86 00:15:54,240 --> 00:16:03,810 madhavan: Should come really from the first Hamiltonian constraint labs acting at vertex evaluating at somebody. 87 00:16:04,590 --> 00:16:22,320 madhavan: And when the second level to knit constraint x, I'd like it to act at some different vertex or some deformation of this vortex created by the first time constraint, so that the labs argument is v plus delta and that is kind of a discrete analog of this one. They'll enter 88 00:16:23,400 --> 00:16:31,530 madhavan: In any case, the question is how can we use the constraint algebra discriminate between choices. 89 00:16:33,090 --> 00:16:37,710 madhavan: And so the idea is to somehow not let the right hand side trivialize 90 00:16:38,970 --> 00:16:54,060 madhavan: One way in which to do it is to use high density Hamiltonian constraints. So we can use high density Hamiltonian constraints by scaling them up by powers of square it of cues. So you take the Hamiltonian constraint density or 91 00:16:55,470 --> 00:16:59,940 madhavan: Density one and you multiply by appropriate power scarab square root of three. 92 00:17:00,390 --> 00:17:12,480 madhavan: If you do this you can compute the pause on bracket between two Hamiltonian constraints and you find that the metric in the right hand side is then scaled also by appropriate powers of square few 93 00:17:13,140 --> 00:17:21,690 madhavan: Cents in accounting square root of Q is like the volume divided by delta cute, you see that every power of square root of Q gives you 94 00:17:22,140 --> 00:17:33,300 madhavan: One over delta q power. And so if you adjust the pause appropriately. You can bring the right hand side approximate operator from triviality into non triviality 95 00:17:34,380 --> 00:17:45,750 madhavan: Because you can arrange for this overall factor delta to go away. So if the idea is that if I could do this if one could get a non trivial, right hand side approximate operator. 96 00:17:46,140 --> 00:18:01,200 madhavan: Then one could now try and see whether one could implement space time covariance in a non trivial manner that is the left hand side right hand side would then both be non trivial and one more time, see whether they were equal or not. 97 00:18:02,100 --> 00:18:12,120 madhavan: So that is the general strategy. But recall. Then when one look at the density with one Hamiltonian constraint on the factors are delta cancel. 98 00:18:12,660 --> 00:18:26,610 madhavan: So the moment one is going to reschedule by any power of square root of Q then already the Hamiltonian constraint itself will have overall factors of one over delta 99 00:18:27,300 --> 00:18:40,680 madhavan: And at looks very singular in the Delta going to zero limit. So the question is, when confronted with such a problem that is a first problem which says, What can we do 100 00:18:41,880 --> 00:18:54,360 madhavan: So in order to answer this question. Let me make a brief digression, and talk now about not the Hamiltonian constraint, the spatial different more physical constraint. 101 00:18:56,280 --> 00:18:58,560 madhavan: So I'll go to the next, right. 102 00:19:02,250 --> 00:19:02,700 madhavan: Now, 103 00:19:04,650 --> 00:19:17,310 madhavan: In Luke quantum gravity, only the finite different morph isms are represented as a unit finite unit three operators on on the kinematics about space. 104 00:19:17,970 --> 00:19:26,250 madhavan: And the generators, namely the different more physical constraints smeared with appropriate ship these operators are not defined. 105 00:19:26,880 --> 00:19:39,810 madhavan: Kinematic in that space. And we somehow makes sense of this generator can we construct this generator can we convert the different more physical constraint operator itself by following the methods which 106 00:19:40,470 --> 00:19:53,820 madhavan: Team and develop from the Hamiltonian constraint. So that's what to briefly recall for you. So let me do accounting of powers of delta for the different Marxism constraint. 107 00:19:54,540 --> 00:20:05,190 madhavan: smeared with shift. And so again we have over here, dq X and shift and trial and the curvature. So we have an overall factor of 108 00:20:05,670 --> 00:20:11,250 madhavan: Delta cube coming from the coordinate measure than a flux over delta square from the electric field. 109 00:20:11,850 --> 00:20:26,580 madhavan: And roughly speaking, all over me minus one divided by delta score from the curvature. And so you can see that this is one over delta force and a delta cubed in the numerator. So you get a finite operator divided by delta which is a singular operator. 110 00:20:27,900 --> 00:20:38,220 madhavan: Despite this, one can construct a satisfactory continuum limit. So this is what I talked about in long time ago in the in the Madrid loops conference. 111 00:20:39,720 --> 00:20:42,780 madhavan: And so one secondly follows strategy. 112 00:20:44,280 --> 00:20:52,920 madhavan: Tries to be more flexible with it as what what Thomas and before him, other workers in the field lay down. So what constructs approximates 113 00:20:53,370 --> 00:21:11,850 madhavan: Me dump of Salamis and plexus and I just remind want to remind you that one can do this in such a way that the approximate operator acting on a swing network access follows acts as the unitary operator. 114 00:21:13,110 --> 00:21:16,890 madhavan: For the final different more physical generated by the shift. 115 00:21:17,610 --> 00:21:30,150 madhavan: And I find parameter delta. So, that is the first thing over here, minus the identity divided by delta with some factor of mine side. So this is how the operator acts on a spin network state. 116 00:21:30,630 --> 00:21:40,710 madhavan: And of course it's in complete accordance with accounting of delta. This is a single operator. These two operators are finite or it isn't can mannequin but space. 117 00:21:41,010 --> 00:21:55,890 madhavan: And therefore, of course, there is no continuum limit on the kinematic space. Nevertheless, this of course admits continually limit on the Lewandowski Mirage habitat. So let me just give you a flavor of how that goes. 118 00:21:57,090 --> 00:22:02,040 madhavan: The habitat states are labeled by default prison. 119 00:22:08,940 --> 00:22:16,920 madhavan: Transfers in network, then what was causing the different class of this spin network. Can you hear me still. 120 00:22:19,050 --> 00:22:20,820 Jorge Pullin: Yes, you brought up briefly but 121 00:22:24,210 --> 00:22:31,530 madhavan: OK. So, did you hear me when I talked about the different Marxism operator being represented as you minus one by delta 122 00:22:32,730 --> 00:22:32,970 madhavan: Yes. 123 00:22:34,080 --> 00:22:50,400 madhavan: Okay, so this object doesn't admit a continuum limit or on the cinematical but space, but it does admit one on the Lewandowski morale habitat. So the habitat state is labeled by a different, more efficient processing networks and what they call a vertex mode function. 124 00:22:51,540 --> 00:23:05,880 madhavan: So given a different, more efficient class of networks. One is embedded characteristic is the vertices of the sprint network. So let it have n burgesses so every member of this classes in bonuses. 125 00:23:06,600 --> 00:23:15,120 madhavan: And then this is the state, which is a state in the algebra dual, namely, this is a state which is a 126 00:23:15,930 --> 00:23:24,750 madhavan: Linear function on the finite span of spin network states to the complexes and it can be formally expanded in terms of these bra. 127 00:23:25,440 --> 00:23:38,940 madhavan: Spin networks. So the, the coefficient of the bra S is them. The is obtained from this vertex mode function. So the word smooth function is a function smooth. 128 00:23:39,420 --> 00:23:55,470 madhavan: And which is a function on MPs of the spatial manifold into the complexes and we simply evaluate this function at the end vertices of this network state. And then we some over all the elements of this different Marxism class and we get this algebraic state. 129 00:23:57,810 --> 00:24:05,460 madhavan: And then it what one can show quite straightforwardly is that when you act, find the dual action through 130 00:24:06,900 --> 00:24:17,400 madhavan: This expression of the approximate what one gets is a new habitat state which is labeled by a new function g of delta 131 00:24:18,000 --> 00:24:33,630 madhavan: Which is obtained by taking to evaluations of this of the Vertex mode function one on slightly move vortices of the spin network state simply they are moved by inverse of this defeat modernism. 132 00:24:34,680 --> 00:24:39,570 madhavan: FINAL FEW MORE system generated by the shift that is this toma year 133 00:24:40,020 --> 00:24:56,100 madhavan: Minus the evaluation at this original spin network vortices. So this is quite easy to see because this guy over here moves the spin network state over here and in exactly this manner. This is what the result is. This is the minus one contribution. 134 00:24:56,670 --> 00:25:02,370 madhavan: And if you take the Delta going to zero limit this give some sort of a derivative of the 135 00:25:02,700 --> 00:25:13,920 madhavan: Vertex mode function over here with the shift over here. So, that is also not difficult to see the main point is that the Delta actually gives you a derivative and that's completely well defined. 136 00:25:14,370 --> 00:25:20,250 madhavan: And in the limit delta going to zero, you get a new Vertex mode function, which has given us 137 00:25:21,150 --> 00:25:22,290 So let me make some 138 00:25:23,490 --> 00:25:23,880 madhavan: Yes. 139 00:25:24,030 --> 00:25:25,710 Abhay Ashtekar: Just, just a quick look. 140 00:25:26,340 --> 00:25:37,200 Abhay Ashtekar: At just to finish. Also, just to state which is level by this House, saying that as a result of this operation, you get a new column state which will be labeled by G. Is that correct, 141 00:25:37,980 --> 00:25:38,610 madhavan: That's correct. 142 00:25:39,960 --> 00:25:40,950 Abhay Ashtekar: Okay, so 143 00:25:43,140 --> 00:25:45,270 Abhay Ashtekar: Is that for the operation. 144 00:25:47,280 --> 00:26:01,380 madhavan: Yes, yes. So I'm not hearing you completely. Well, but I could make out what you were saying, I guess I'm not able to access the controls. Okay, so in future trend when when you are talking of it. 145 00:26:01,740 --> 00:26:03,660 madhavan: But yes, that is correct. Okay. You 146 00:26:04,200 --> 00:26:05,130 Will get a very defined 147 00:26:07,980 --> 00:26:11,430 madhavan: Okay, so let me make the following comments when 148 00:26:14,520 --> 00:26:19,860 madhavan: I'm sorry, the second time using zoom, so please bear with me. I am not yet really comfortable with it. 149 00:26:21,840 --> 00:26:26,040 madhavan: He saw the following okay I make sense now. So, one can show 150 00:26:27,060 --> 00:26:43,830 madhavan: As is not difficult to guess from the form, which I wrote down that this continuum operator action provides a representation of the constraint algebra is really algebra, the different prism group because what one got on the previous slide was very much like 151 00:26:45,090 --> 00:26:45,450 madhavan: A 152 00:26:46,620 --> 00:26:56,910 madhavan: Lead racking up this early derivative of this function in each of its arguments. So if you compute the commentator two different offices, you will get the right the algebra representation 153 00:26:58,290 --> 00:27:07,800 madhavan: But not triviality of this representation clearly rests on the cinematically singular nature of the constraint operator, which has a limit on a different 154 00:27:08,400 --> 00:27:27,570 madhavan: Space, not the kinematics but space, but the Lewandowski model habitat. So I'll just go back over here. And what I want to save this delta was not there. All then you would simply get zero in the live data went to zero. So you just get a trivial, you get the action of the 155 00:27:28,590 --> 00:27:44,190 madhavan: Constraint of the defeat modernism constraint operator to just be zero. And of course, then the the the algebra would also completely be trivialized. So this delta is extremely important in getting everything to be non zero. And again, when one looks at 156 00:27:45,330 --> 00:28:04,020 madhavan: The Algebra one one takes two streams. Again, it's important that one gets second derivatives and all this has to do with this factor of delta in each of the most operators, so this this niche is important to get something on preview. 157 00:28:06,150 --> 00:28:12,660 madhavan: Secondly, in order to get the different offices minus one over delta action of this operator. 158 00:28:13,830 --> 00:28:28,590 madhavan: To choose the economy flux approximates in such a way that they are a tune structure of the sprint network state. So not only are they are tuned to structure the graph on the line display network, but also to the label of 159 00:28:29,670 --> 00:28:36,150 madhavan: The momentum labels. So this is actually this is a statement that that is what 160 00:28:37,380 --> 00:28:51,450 madhavan: They should I mean expected in hindsight, because, for example, supposing one wants to move G H along the orbits of the shift we somehow need to cancel the enemy in the general presentation. 161 00:28:51,930 --> 00:28:57,030 madhavan: And then create a new law know me with this displaced also in the JIRA presentation. 162 00:28:57,600 --> 00:29:09,720 madhavan: And it turns out there for that need to know where the sages line so that you want to cancel it and you also need to know its representation so that you produce a new edge which exactly has the same representation able J. 163 00:29:10,260 --> 00:29:15,720 madhavan: So the switch you use our Taylor, who bought the spin and the graph labels. 164 00:29:16,650 --> 00:29:26,670 madhavan: So it's not a constant spinoff representation of anonymous or justice think David J. Fixed representation better tailor it to stay. You're acting upon 165 00:29:27,330 --> 00:29:39,090 madhavan: And the point I want to make is that it would have been impossible for to guess the correct approximates if we did not understand what the classical constraint gender gets 166 00:29:40,500 --> 00:29:45,480 madhavan: You pretty a wall. We do the curvature pronouncements depend 167 00:29:47,400 --> 00:29:48,930 madhavan: On the flux is as well. 168 00:29:50,040 --> 00:30:11,370 madhavan: So we really understand union vector field of the classical function well before trying to convert the operator into quantum and so I'm going to try and use these lessons for the Hamiltonian constraint. So if there any questions. I can take them. Now at this point. 169 00:30:16,170 --> 00:30:19,470 madhavan: Okay, so let me proceed, since there are no no questions. Yeah. 170 00:30:22,590 --> 00:30:30,450 madhavan: So let me then outline based on this a strategy for time to implement the constraint as an operator. 171 00:30:32,100 --> 00:30:39,930 madhavan: And trying to get a non trivial commentator, which has the right properties know quantum gravity. 172 00:30:40,830 --> 00:30:51,150 madhavan: So first I want to get an overall factor of one over delta. This is just mimicking what happens for the different Martin and spring and therefore I will scale. 173 00:30:51,780 --> 00:31:02,850 madhavan: Density wait one and unconstrained density by Square to to the one by three, which gives me to the one but over delta nine gives me the overall pattern one over there. 174 00:31:04,380 --> 00:31:15,600 madhavan: The strategy. I'd like to follow is to look for approximates to the Hamiltonian constraint again since there are factors or volume. If the Hamiltonian constraint Brockman 175 00:31:16,710 --> 00:31:27,390 madhavan: Follow the same choice or three monitors and it will pick out only vertices, the skin network state. So there'll be some overt is this will be the evaluation of the labs. 176 00:31:28,170 --> 00:31:40,350 madhavan: And then what I'd like to ask this expression is into a form where there is some finite operator I you which deforms the vertex we on which is x 177 00:31:41,070 --> 00:32:02,700 madhavan: deforms it in some neighborhood delta have quiet neighbors delta of this vortex minus one divided by delta. And then there are some more efficient which level kinds of informations are involved. So that is the structure. I want to make the Hamiltonian constraint have 178 00:32:04,110 --> 00:32:12,630 madhavan: The structure, I would like to find defined the Hamiltonian constraint approximate on a suitable space of option states. 179 00:32:13,950 --> 00:32:23,970 madhavan: So an option state will be again like Lewandowski models habitat state, so roughly very roughly speaking apps label by a function f. 180 00:32:24,780 --> 00:32:45,930 madhavan: On suitable copy or one or sorry unsuitable copy and sigma and we will refine this much more as we come to the you want to theory. But basically, again, some grass with some efficiency which depend on suitable evaluations of this vertex moves function. 181 00:32:47,100 --> 00:32:47,670 madhavan: And 182 00:32:48,930 --> 00:33:00,390 madhavan: These what I will then do is I will evaluate the dual action of this constraint operator and schematically, what I would like to happen is that I would like 183 00:33:01,500 --> 00:33:15,000 madhavan: fumbling combination and you had minus one over delta to yield a contribution to this complex number which is up the form of the evaluation of the laps at the vertex at which it's 184 00:33:16,110 --> 00:33:17,940 madhavan: The Hamiltonian constraint is acting 185 00:33:19,890 --> 00:33:32,070 madhavan: Up would like the function to be evaluated at roughly speaking, a displaced works which is obtained by the action of you had on s displacing vertex 186 00:33:32,820 --> 00:33:42,750 madhavan: Minus the one we just gives me the evaluation of the function at the vertex itself by by death, and some of them take that are going zero 187 00:33:43,500 --> 00:33:54,720 madhavan: Which would get given no function and then St. Louis by and derivative of f. So this is just schematics will see sort of how 188 00:33:55,410 --> 00:34:07,800 madhavan: I will not go to implement it such a nice, but this is kind of what I want to happen and that spiritual lesson I think will be implemented and what I will do in in detail in the you want to carry 189 00:34:09,210 --> 00:34:29,250 madhavan: I would like to happen when I get the second Hamiltonian constraint acting is similar. I will just get an M derivative of this, and the left when I take the commentator, then I will get this nice and minus and and and turn which what I and it's essential, therefore, 190 00:34:30,300 --> 00:34:46,020 madhavan: That, again, the second Hamiltonian constraint actually act on deformations created by the first one in this, although I don't want to right over here, but that is what must happen but this this this thing to actually have this form. 191 00:34:48,180 --> 00:34:55,380 madhavan: So that is the rust on how to get the left hand side which is non trivial, let me now go to the right hand side. 192 00:35:00,450 --> 00:35:00,900 madhavan: Yes. 193 00:35:01,950 --> 00:35:02,370 madhavan: You 194 00:35:04,950 --> 00:35:05,760 Abhay Ashtekar: Like that. 195 00:35:08,730 --> 00:35:09,120 Abhay Ashtekar: I'm 196 00:35:09,480 --> 00:35:10,950 madhavan: Not able to get my 197 00:35:13,140 --> 00:35:13,560 Sorry. 198 00:35:15,450 --> 00:35:16,710 madhavan: Not able to access 199 00:35:18,360 --> 00:35:19,860 Zoom controls. 200 00:35:21,420 --> 00:35:24,270 madhavan: Now I'm trying to mute it or not able to mute my 201 00:35:27,630 --> 00:35:30,450 madhavan: Okay, maybe I will not. I'm not able to mute it away, but 202 00:35:30,480 --> 00:35:32,550 Abhay Ashtekar: Please. Please continue. Oh yeah, it is. 203 00:35:35,250 --> 00:35:35,400 It. 204 00:35:40,200 --> 00:35:41,880 madhavan: OK, one moment of I'm just want 205 00:35:43,980 --> 00:35:45,720 Abhay Ashtekar: You don't need anything just 206 00:35:45,840 --> 00:35:46,290 This 207 00:35:49,920 --> 00:35:50,430 Abhay Ashtekar: So, 208 00:35:56,010 --> 00:35:57,630 Abhay Ashtekar: I just wanted you to explain 209 00:35:59,040 --> 00:36:01,050 Abhay Ashtekar: That you have law. 210 00:36:02,820 --> 00:36:04,770 Abhay Ashtekar: Mesa know V. V. 211 00:36:06,600 --> 00:36:11,280 Abhay Ashtekar: I think that is you deformation minus that we have 212 00:36:12,510 --> 00:36:18,570 Abhay Ashtekar: Visions I understand, particularly, there is a index he paid for me. Delta. What does he 213 00:36:24,270 --> 00:36:24,480 Abhay Ashtekar: That 214 00:36:26,700 --> 00:36:30,060 madhavan: Yeah. Okay so that is over here, right. 215 00:36:31,380 --> 00:36:34,890 Abhay Ashtekar: Yeah. What a stand for before we go. 216 00:36:35,820 --> 00:36:37,380 madhavan: So these coefficients. 217 00:36:37,440 --> 00:36:57,300 madhavan: Would be made up of various labels of of this network state which. So this is a complex coefficient and it would just wait. Each of these operator deformations by some numerical coefficients and perhaps since which depend on the jays etc. 218 00:36:58,740 --> 00:36:59,910 madhavan: So I do not 219 00:37:01,410 --> 00:37:05,850 madhavan: Put into the you have because I'm on the you had minus one form over here. 220 00:37:09,540 --> 00:37:13,260 Abhay Ashtekar: The Windows is arbitrary is complete. 221 00:37:14,280 --> 00:37:14,850 Abhay Ashtekar: Freedom in 222 00:37:15,060 --> 00:37:15,720 madhavan: Terms of a 223 00:37:16,530 --> 00:37:35,850 madhavan: Understand. Understand, so the point is this is a general strategy, I would like to bring the hammer constraint approximately into this form. And when I try to do so, of course, these will be completely fixed by the particular approximates I'm choosing 224 00:37:37,110 --> 00:37:47,820 madhavan: In order to implement and make this phone. So I'm to be completely determined. Once I tell you what particular proximate, I'm going to be using for 225 00:37:48,420 --> 00:37:59,280 madhavan: Constraint. But what I want to be guided by is this general form. I want to be able to get a general form when I have this particular structure acting 226 00:38:01,620 --> 00:38:01,980 Abhay Ashtekar: In this 227 00:38:04,350 --> 00:38:05,760 Abhay Ashtekar: Okay, so 228 00:38:05,790 --> 00:38:07,050 madhavan: I will go now to the 229 00:38:07,350 --> 00:38:13,980 madhavan: Next slide, let me now go to the left hand side to the right hand side. This was about the left hand side. 230 00:38:17,910 --> 00:38:19,620 madhavan: The right hand side approximate 231 00:38:21,150 --> 00:38:25,530 madhavan: Of course side, which was just to be prison, which was mirrored on 232 00:38:27,720 --> 00:38:29,400 madhavan: Independent shift. 233 00:38:31,470 --> 00:38:43,800 madhavan: So it looks at that right hand side, it's the corresponding operator who is single operator would have tries delta which I would have to take to zero. 234 00:38:45,240 --> 00:39:10,140 madhavan: However, if it so happens that the classical right so it could be written as opposed to bracket between objects which were some on to the formalism, then the items. Don't approximate would like the left hand side also have to amateurs delta and delta prime, which would have to be a zero. 235 00:39:11,430 --> 00:39:11,940 madhavan: And 236 00:39:13,020 --> 00:39:25,050 madhavan: Non zero delta delta prime, it would be much simpler to compare the left hand side and the right hand side operators and try to find ways in which to make the two again. 237 00:39:25,740 --> 00:39:36,870 madhavan: So it would be nice if this could happen. And indeed, this happens do miraculous classical points on racket identity, which 238 00:39:38,310 --> 00:39:46,920 madhavan: CASEY Casey Tom and I found in 2012, a long time ago and the identity is as follows. 239 00:39:48,720 --> 00:39:57,270 madhavan: Remember the right hand side there to Hamiltonian constraints. So we have two lapses. So what one can do first is take apps and 240 00:39:58,320 --> 00:40:02,220 madhavan: Multiplied by the densities tried he 241 00:40:03,390 --> 00:40:12,210 madhavan: And divided by some factor in this case of due to the one by three, when one puts in the various density weights of the labs. 242 00:40:12,960 --> 00:40:24,300 madhavan: And the, the electric field and whatever comes from here, one finds that this object behaves like an enterprise vector field. 243 00:40:24,780 --> 00:40:43,560 madhavan: So there are three shifts for I call to 123 and I call these shifts electric chips because they involve this electric field dependence. If you take the electric shift and smear aphorism constraint to get an option, which I will corn d of em. 244 00:40:45,090 --> 00:40:54,540 madhavan: Similarly, you could instead of n, you could use the laps em and you could get d of em i and then if I take this poison bracket. 245 00:40:54,960 --> 00:41:10,590 madhavan: And I some from, I do want to three with some numerical coefficient, then this turns out to be exact. Pause on bracket identity, where the minus sign goes for the Euclidean theory and the plus sign from 246 00:41:12,780 --> 00:41:24,630 madhavan: An empty one can actually choose any density wait one once and get a similar identity. The only thing which changes factor over here, except 247 00:41:25,770 --> 00:41:42,600 madhavan: For the is when the Hamiltonian contained is of density way one. So this is somewhat intriguing because due to some other reasons about materiality of the right hand side, I'd argue that it might be good to look at 248 00:41:43,620 --> 00:42:00,510 madhavan: Higher density wait constraints and therefore not a density wait one and so classically here what one finds density wait one this right hand side just identically vanishes. One does not have a useful identity. So that's kind of an integration. 249 00:42:02,640 --> 00:42:13,680 madhavan: Okay, very good. So now let me tell you what the status of this program is what I would like to do is to him Hamiltonian constraints, so that it has a particular form. 250 00:42:14,070 --> 00:42:27,030 madhavan: And then pose this identity as, as in some suitable fashion as a commentator identity, the quantum theory. And of course, I can't do it. Even for Euclidean gravity. Yet that is the direction I'm going to 251 00:42:27,930 --> 00:42:44,310 madhavan: Focus. You want you model where we replace su to try rotations by you want to transformations, one finds the model as a constraint Algebra I saw more free to that of utility gravity's one has identical structure functions for them. 252 00:42:46,590 --> 00:42:51,990 madhavan: And as I said, the more sense. Lee's humble Newton constant were given limited up 253 00:42:53,070 --> 00:42:56,010 madhavan: So for this model, one can implement this strategy. 254 00:42:57,210 --> 00:43:02,460 madhavan: One can construct continuum limits of these operators age and deep 255 00:43:03,540 --> 00:43:09,150 madhavan: Suitable space and many technicalities I am not able to go into 256 00:43:09,930 --> 00:43:24,420 madhavan: But in some suitable way one can define a limit of this commentator, which is non trivial and which agrees with this right hand side in Islam paper which I had in 2018 257 00:43:24,930 --> 00:43:43,950 madhavan: So, before going to a few assorted details of the you want to model. Let me make some general comments and and then a prognosis of where this is all all heading and then I'll come back and talk a little bit more in detail about the you 258 00:43:46,380 --> 00:44:00,810 madhavan: So let me make the following comments, then the strategy works and you want to model because the analysis of the classical Hamiltonian vector fields shows that the evolution of classical evolution of the connection. 259 00:44:01,440 --> 00:44:12,750 madhavan: Can be written in terms of combinations or electric dependent mature morph isms electric field dependent gauge transom 260 00:44:14,490 --> 00:44:33,690 madhavan: Since we have a very good understanding of face face independent Detroit office hours and H transformations and can use this understanding to try and write not operator correspondence of peace. Peace dependent different modes engage transformation. 261 00:44:36,420 --> 00:44:48,930 madhavan: Secondly, the detail calculations which I did are on a very small so options states the states have someone's which 262 00:44:50,430 --> 00:45:11,670 madhavan: Spin network summons have only one vertex with a half to train constraint. So these are single vertex distributions. Now it. It's already a long calculation, but I am quite sure, looking at how thing that it should be straightforward to extend this 263 00:45:13,260 --> 00:45:16,260 madhavan: Result of family free non trivial. 264 00:45:17,550 --> 00:45:22,800 madhavan: Constraint algebra to the multi vertex case as well. There are some 265 00:45:23,430 --> 00:45:32,460 madhavan: Other technical issues. I can't go into them technical issues there more about the nature of particular condition choices one makes which I would have liked to debate. 266 00:45:32,880 --> 00:45:43,440 madhavan: But there is no time for that. The broad picture is that yes, or I've done it for single vertex states should be straightforward to extend it to multiple states as well. 267 00:45:44,490 --> 00:45:57,270 madhavan: But the main problem is my action of the constraint which I used in this long paper does not support production and propagation. I'll come to hopefully in the next part of my talk. 268 00:45:57,900 --> 00:46:16,410 madhavan: And therefore, I have not been in this in more detail what I have done is that with a slight modification of the choice of constraint action nevertheless has this same you had minus one form for changes are these different coefficients which are assigned to 269 00:46:17,430 --> 00:46:28,110 madhavan: And, you know, some technicalities one actually does get rigorous propagation. So, this I showed in a paper in 2019 and I will tell you what I mean by propagation 270 00:46:29,250 --> 00:46:37,800 madhavan: And normally freedom for this choice remains to be shown, but over the last few months, I've been working and I think that it's quite 271 00:46:38,400 --> 00:46:52,560 madhavan: Promising that one will be able to show anomaly freedom as well, in a certain sense. So that is the status. Let me go to a brief prognosis of happen in future. 272 00:46:53,670 --> 00:47:04,500 madhavan: And then I'll come to the you want to see where it will become little bit technical. So I hope at least this first and broad part of top you can take away with you. 273 00:47:05,670 --> 00:47:15,690 madhavan: So the next step, which I would like to do, or I think we should be done by by people is to to go to Euclidean gravity. 274 00:47:16,260 --> 00:47:33,750 madhavan: And again, as I will come at the end of the top recent analysis of classical equations shows that the tenement equations can be written in terms of electric field dependent different more fizzles as an electric field depends gauge transformations 275 00:47:35,370 --> 00:47:36,780 madhavan: And roughly speaking. 276 00:47:37,950 --> 00:47:51,750 madhavan: All the graph related problems already arise in the you want to model and in the Euclidean case one will need to confirm, of course, all the non trivial problem from nature of a city. 277 00:47:52,830 --> 00:48:01,950 madhavan: So, at present, I think the domain open issues are just some for you want you to normally free action consistently propagation. I think this should be possible. 278 00:48:02,640 --> 00:48:14,010 madhavan: Then to generalization of this and I'm optimistic that progress can be made the lawrenson case however is wide open. I don't know how to do this. 279 00:48:14,940 --> 00:48:28,080 madhavan: One possibility which I discussed in a more speculative paper was to try and map directly Euclidean solutions to long term solutions are using the team and complex. So far, this is 280 00:48:28,530 --> 00:48:38,280 madhavan: Which was introduced by my mom beautifully long time ago. And then also advocated by eBay, to go from Florida incentive to Euclidean 281 00:48:39,600 --> 00:48:55,980 madhavan: So that is the rock prognosis, let me know. Come in more detail to the you want to model. So the plan any general questions I could take number two is I will go. Do you want you model and present. Just a few details because back time 282 00:49:03,930 --> 00:49:04,290 madhavan: Okay. 283 00:49:05,580 --> 00:49:16,860 madhavan: So let me then go to the you want to module and give you some asserted detail. So this is to just give you a flavor of various things which go on in the you want to model. 284 00:49:17,940 --> 00:49:20,880 madhavan: Pertaining to space time covariance and anomaly free 285 00:49:22,560 --> 00:49:23,490 Constraint algebra. 286 00:49:24,750 --> 00:49:36,450 madhavan: Okay, so let's look at the model itself. The face face variables are a triplet of you want connection is an electric fields which are economically candidate. So these API's and he is for ICO 2123 287 00:49:37,380 --> 00:49:44,880 madhavan: The consumer just the kind of a billion obvious you want two versions of the astute Euclidean constraints. So you have 288 00:49:45,900 --> 00:49:53,190 madhavan: Three God's law constraints which are just the opinions of constraints. Remember here, the electric field is completely gauging barrier. 289 00:49:54,240 --> 00:50:01,620 madhavan: Then you have the different more physical constraint which is just a similar combination of the vector constraint and the gospel constraint. 290 00:50:02,820 --> 00:50:17,190 madhavan: And then the Hamilton time constraint which is exactly the same. So it's excellent i j k where this doesn't have any is just the alternating simple, it doesn't have any structure constant interpretation, then you E. F. 291 00:50:18,360 --> 00:50:29,820 madhavan: In accordance with a strategy, the same content works. So one puts in a factor of 10 to the minus one by three and the labs then becomes a density weighted object or minus one by three. 292 00:50:30,990 --> 00:50:46,380 madhavan: The curvature is just the ability of curvature over here. And what I mean she was just one would take two G's. So in the initial 14 times to add and Q is just the determinant of 293 00:50:47,880 --> 00:50:48,810 madhavan: You know me. 294 00:50:50,760 --> 00:51:03,570 madhavan: So let me know different the key objects which will be present in whatever I talk about, namely the electric shifts. So there are three of them. And I've already said what they are and times eight times goodness one by three. 295 00:51:04,140 --> 00:51:14,250 madhavan: And then one can smell the different autism constraints. So there is a summation with Jay over here. So you are E dot f. And when you implement the gospel, then 296 00:51:14,670 --> 00:51:20,550 madhavan: This part just goes away. So you have the different Marxism constraint which is this number gospel constraints office. 297 00:51:20,910 --> 00:51:33,510 madhavan: And then you have this identity as usual for the astute also holds for you want you can have the pause on record. So the Hamiltonian constraint is equal to this. And this is what we'd like to implement in the quantum theory. 298 00:51:35,610 --> 00:51:38,310 madhavan: So let me know. Good. The quantum kinematics. 299 00:51:42,780 --> 00:51:51,660 madhavan: So, one has just the you want to analog of the sprint network states, which I will call charge network states because the integer value charges which 300 00:51:52,740 --> 00:52:12,210 madhavan: Label the representations of one cube there three of them to want you to acutely and you have states again label by colored rocks and just products of the edge salamis over these various edges and each edges color with these three into the new charges. 301 00:52:13,740 --> 00:52:23,460 madhavan: Since it's a billion theory. These are exact I can state. So the electric clocks operators electric clocks operators just communicate with each other. There is none of the issue non quantitative at 302 00:52:24,210 --> 00:52:31,410 madhavan: Engaging variance simply means that the sum of the outgoing triple of charges at every vertex vanishes. 303 00:52:33,630 --> 00:52:48,960 madhavan: Let me now come to the quantum shift greater which is the key object. And what I want to do. So the quantum shift is, as I said, the labs, times the electric field operator times the inverse matrix over here. 304 00:52:50,880 --> 00:53:04,140 madhavan: These are all completely electric field dependent objects even once when even if we perform the team trick. So, this turns out to be completely electric field dependent and therefore the 305 00:53:05,460 --> 00:53:11,550 madhavan: Child network states are eigen states of quantum shift in one can extract an item value. 306 00:53:13,230 --> 00:53:30,000 madhavan: So this is what what happens, since subject a mantra is non zero only at vertices of the quantum of the talent network, the quantum shift eigenvalues also only non zero charge network workspaces. 307 00:53:31,110 --> 00:53:35,400 madhavan: So let me go in a little more detail to the next slide about the quantum shift. 308 00:53:40,200 --> 00:53:53,730 madhavan: I'll give you more details. I think maybe on the slide. After this, so let me just have some quantum shift. And now let me go to the schematics of how I will use the quantum shift to define the action of the Hamiltonian constraint. 309 00:53:55,290 --> 00:54:08,730 madhavan: So this is going to be heuristics just schematically how things happen and then I will tell you in detail just give you the final results because there is due to essentially due to lack of time. 310 00:54:09,690 --> 00:54:24,690 madhavan: So the Hamiltonian constraint is he, he, an F, and what are done and with this due to the one by three to get like density rate. And so what I've done in red is to isolate the part which is the classical electric shift. 311 00:54:26,490 --> 00:54:32,550 madhavan: Now, when I look what I want to do is I want to quote the action of the Hamiltonian constrained in terms of 312 00:54:32,970 --> 00:54:43,050 madhavan: Different more isms generated by this electric shift. And for that I'm going to use an identity, which is which we see all the time. For example, in the t plus one, the composition 313 00:54:43,650 --> 00:54:56,700 madhavan: Which is if you take a shift vector field and.it to the curvature of the a billion curvature. Now, then you can write it as the leader derivative of the connection minus 314 00:54:58,830 --> 00:54:59,880 madhavan: This object kill 315 00:55:01,200 --> 00:55:10,080 madhavan: This identity also works in the IBM theory with the electrician. So instead of this chip. You just placing by the Electric Sheep over here. 316 00:55:11,070 --> 00:55:19,350 madhavan: And the Hamiltonian constrain them looks like when looks like this lead derivative 317 00:55:19,800 --> 00:55:38,640 madhavan: Coming from here and then this total derivative, one can do an integration by parts. And this of delve deep comes and hits the Eb over here. So you get something proportion to the gospel on those on the gospel surface where the gospel song I can forget about this town. 318 00:55:39,720 --> 00:55:48,930 madhavan: So let me now go to the heuristics for the quantum theory. So the quantum theory with reasonable choice of operator ordering 319 00:55:49,620 --> 00:56:08,250 madhavan: What happens is that we get the he over here to Atlanta directional derivative with some type items each each bar Richard one then one gets this Absalom i j k l m one gets a lead derivative of the connection with respect to the shift. 320 00:56:09,990 --> 00:56:16,230 madhavan: So let me write it. Now, if I'm going to write it down approximate at finite data. 321 00:56:17,430 --> 00:56:30,510 madhavan: Then what I would try to do is to write this lead derivative in terms of finite different more fun. So what we what the first step is to basically take this 322 00:56:31,080 --> 00:56:41,760 madhavan: Wave function side of the connection. And what this tells you to do is to evaluate it at a displaced argument. So you have a j over here. 323 00:56:42,570 --> 00:56:52,860 madhavan: Plus this displaced argument minus i at AJ. So this is where you get the u minus one structure. This is the scene of the yo minus one structure. 324 00:56:53,280 --> 00:57:05,820 madhavan: And the displacement looks almost like a different than generated by the quantum shift it would have been from autism, except for this epsilon i j k. So, for example, 325 00:57:06,900 --> 00:57:22,740 madhavan: If it was exactly a different office on then I supposing I looked at the G equal to one component over here, I would add an appropriately derivative also equal to one component of the connection but 326 00:57:24,930 --> 00:57:33,000 madhavan: If I'm going to take the one component here. Then I'm going to add the lead derivative of the second component of the electric ship. 327 00:57:33,330 --> 00:57:45,540 madhavan: Of the third component of the connection, minus the derivative with respect to the third component of the electric shock of the second component of the connection. Take that multiplied by delta and then add it over here. 328 00:57:46,110 --> 00:58:01,740 madhavan: So it is kind of some twisting of internal is also going on over here. So that's what the heuristic structure looks like, but we also have to confront the fact that, like in the classical 329 00:58:03,000 --> 00:58:09,960 madhavan: Where the electric shift is adopted in the quantum theory, the electric shift is no longer a smooth object. 330 00:58:11,040 --> 00:58:25,050 madhavan: And then the you want to carry one can really see what is happening because we know that the elect the electric flux or the charge network states. 331 00:58:27,360 --> 00:58:43,920 madhavan: They can be interpreted really visually as want have electric trucks along the edges. So these are these contests electric lines of force as Asian and therefore you can see at the word is that you don't even have 332 00:58:44,940 --> 00:59:01,800 madhavan: The electric field at all, something very similar and one will have to make sense of this object. So the quantum theory, but the main message of this few logistics is that, oh, the Hamiltonian constraint at five o'clock delta 333 00:59:03,000 --> 00:59:03,510 madhavan: See 334 00:59:04,530 --> 00:59:15,960 madhavan: The charge network see it at each of its vertices and some combination of singular defeat more physical lack of any other words and 335 00:59:17,610 --> 00:59:18,150 madhavan: There is some 336 00:59:19,740 --> 00:59:23,280 madhavan: Some start flipping in the neighborhood of 337 00:59:24,570 --> 00:59:25,110 madhavan: Copy. 338 00:59:26,760 --> 00:59:29,250 madhavan: So let me go a little 339 00:59:31,050 --> 00:59:32,970 madhavan: Quantum shift itself. 340 00:59:35,250 --> 00:59:35,970 madhavan: So the point 341 00:59:37,860 --> 00:59:38,460 madhavan: Is this 342 00:59:39,630 --> 00:59:40,050 madhavan: On 343 00:59:43,980 --> 00:59:48,330 madhavan: And because of operator here. It's not the reporter says, and 344 00:59:51,510 --> 00:59:53,970 madhavan: One can easily see that the 345 00:59:55,290 --> 00:59:57,960 madhavan: Contributing factors because it has a 346 00:59:59,400 --> 01:00:06,540 madhavan: Planet Index it contributes actor or national the tangent at the word XV. 347 01:00:08,550 --> 01:00:13,110 madhavan: So the final result for this item is as follows. Do 348 01:00:14,370 --> 01:00:24,360 madhavan: I value contribution from took you to the one by three operator and then the electric field gives you, because it's all 349 01:00:25,590 --> 01:00:29,850 madhavan: Coming from trucks in the flux eigenvalue Q over 350 01:00:31,470 --> 01:00:35,040 madhavan: That is where the index i comes comes from this is a 351 01:00:36,810 --> 01:00:41,910 madhavan: Be more and then you better Long the 352 01:00:45,360 --> 01:00:46,500 madhavan: Evaluation. 353 01:00:49,860 --> 01:00:51,360 madhavan: Enough to actually do this. 354 01:00:52,980 --> 01:00:54,600 madhavan: Because laps in 355 01:00:55,980 --> 01:00:56,460 madhavan: Dependent 356 01:00:57,570 --> 01:01:11,520 madhavan: Waited all j, the density rate minus one by three objects. I have to choose a regulating coordinate patch around vertex charge network at Valley doing eigenvalue. 357 01:01:13,200 --> 01:01:17,010 madhavan: Isn't that coordinate patch that this lapses evaluate and it is 358 01:01:18,540 --> 01:01:20,880 madhavan: A look at you for 359 01:01:22,080 --> 01:01:26,400 madhavan: Vectors on edge and then as some of what the 360 01:01:28,230 --> 01:01:32,070 madhavan: What the Eigen looks like the vertex me 361 01:01:33,210 --> 01:01:42,870 madhavan: So what it does is it adds the sales and mechanics graph deformations alone he tangent down by 362 01:01:45,570 --> 01:01:47,610 madhavan: Each one gives you permission. 363 01:01:53,160 --> 01:01:53,910 madhavan: So that is 364 01:01:55,770 --> 01:01:56,550 madhavan: Give you a 365 01:01:59,940 --> 01:02:01,350 madhavan: Representation of what 366 01:02:02,520 --> 01:02:07,980 madhavan: The final constraint operator does this is building up to that. 367 01:02:12,780 --> 01:02:38,550 madhavan: One who's had city with this one just opens up a box of new technical problems which one is Paris laboriously one by one. And the first one and the most is that one of us needs this patch to regulate and to define the quantum shift. And so, this turns out to be coordinate dependent 368 01:02:42,120 --> 01:02:44,730 madhavan: And I need to choose one minute. 369 01:02:46,590 --> 01:02:48,000 madhavan: Of every judgment. 370 01:02:50,280 --> 01:03:04,950 madhavan: And the immediate question is that, well, what happens when two different variants, because I'm not going to be worried about the pot on racket between the Hamiltonian and I'm on him, but also between me and the different offers and constraint. 371 01:03:07,050 --> 01:03:18,570 madhavan: Want to be around, or is this going to be possible. I'm going to get an anomaly feed representation also have the few more physical constraint constraint bracket. 372 01:03:21,060 --> 01:03:26,700 madhavan: It turns out that this is actually possible and technically complex complicated 373 01:03:27,870 --> 01:03:29,400 madhavan: But no one has to me. 374 01:03:32,760 --> 01:03:33,600 madhavan: Up ending this 375 01:03:34,710 --> 01:03:45,480 madhavan: Isn't constrained commentators in other choices must be consistent with different more and and very roughly due to lack of time. 376 01:03:46,530 --> 01:03:52,650 madhavan: This can be done by choosing patches for one charge. And it's the morphic image. 377 01:03:54,030 --> 01:03:55,200 madhavan: Different systems. 378 01:03:56,820 --> 01:04:04,980 madhavan: What does this and there is again a lot of freedom in doing this, there are amateurs which tell you that this freedom. Doesn't matter. 379 01:04:06,240 --> 01:04:23,820 madhavan: This then it turns out your office and pool variance and then we achieved by tailoring the action of the constraint, a two properties of the option to the ball. Let's say this again. 380 01:04:25,410 --> 01:04:43,830 madhavan: These various constructions of them constraint we have been seeing one Taylor's the Hamiltonian constrained direct stay active with grass structure one one looks at all the Burton isn't one Taylor's attention. 381 01:04:45,060 --> 01:04:45,750 madhavan: This 382 01:04:47,970 --> 01:04:51,240 madhavan: State approximate to the constraints, so that every time. 383 01:04:53,520 --> 01:04:55,290 madhavan: I need things to be it is 384 01:05:00,450 --> 01:05:02,670 madhavan: What I'm saying is that one also. 385 01:05:03,990 --> 01:05:04,440 madhavan: Which is 386 01:05:06,210 --> 01:05:12,240 madhavan: Interested in the actual state which is infinitely the states. 387 01:05:13,500 --> 01:05:15,810 madhavan: And approximate really x 388 01:05:21,240 --> 01:05:21,570 madhavan: This 389 01:05:23,940 --> 01:05:26,190 madhavan: Has to be tailored now. 390 01:05:29,430 --> 01:05:33,630 madhavan: Labels have this option dual state being acted on. 391 01:05:35,430 --> 01:05:37,080 madhavan: In an appropriate way. 392 01:05:38,970 --> 01:05:47,130 madhavan: Taking advantage of the different over and choice of GORDON SAYS, one can show if you're Marxism variance or 393 01:05:49,530 --> 01:05:50,520 madhavan: Constraint algebra. 394 01:05:52,650 --> 01:05:56,520 madhavan: So I just want to a new 395 01:05:57,660 --> 01:05:58,440 madhavan: Which has been 396 01:06:02,580 --> 01:06:04,440 madhavan: Now go to the next slide. 397 01:06:09,690 --> 01:06:11,010 madhavan: Okay, because 398 01:06:18,060 --> 01:06:20,190 madhavan: Of lack of time I'm James and questions. 399 01:06:22,290 --> 01:06:22,830 madhavan: For point 400 01:06:27,390 --> 01:06:27,630 madhavan: And 401 01:06:30,270 --> 01:06:30,750 madhavan: I think 402 01:06:32,460 --> 01:06:36,930 madhavan: The Discourses to give you what is happening and water. 403 01:06:38,430 --> 01:06:40,770 madhavan: One can learn from this exercise. 404 01:06:41,910 --> 01:06:51,510 madhavan: Questions to take them. Now later or talk individually or schedule as well in more detail. 405 01:06:52,560 --> 01:06:53,190 madhavan: So, 406 01:06:55,380 --> 01:07:02,670 madhavan: The calculations are what the number is this. So let me describe what these linear vertices are 407 01:07:04,110 --> 01:07:07,770 madhavan: Because the structure of the constraint which are like to 408 01:07:14,370 --> 01:07:17,100 madhavan: Of the lab, some information. 409 01:07:20,190 --> 01:07:21,150 madhavan: Structure over here. 410 01:07:24,750 --> 01:07:25,770 madhavan: Off shell states. 411 01:07:42,840 --> 01:07:43,110 madhavan: Yes. 412 01:08:16,950 --> 01:08:17,610 madhavan: Go ahead. 413 01:08:30,690 --> 01:08:32,160 Jorge Pullin: Why don't you go ahead 414 01:08:35,790 --> 01:08:36,270 madhavan: Go ahead. 415 01:09:00,390 --> 01:09:01,920 madhavan: Okay. Can anyone hear me. 416 01:09:02,970 --> 01:09:03,210 Jorge Pullin: Yeah. 417 01:09:06,930 --> 01:09:08,490 madhavan: Okay, so I'm going ahead right 418 01:09:10,890 --> 01:09:11,490 Jorge Pullin: Yes. 419 01:09:14,700 --> 01:09:15,660 madhavan: Let me come to 420 01:09:16,890 --> 01:09:18,300 madhavan: These linear says 421 01:09:19,560 --> 01:09:22,560 madhavan: Sorry, as the transparencies side. 422 01:09:24,210 --> 01:09:26,190 madhavan: I have the structure of the constraint. 423 01:09:27,450 --> 01:09:31,830 madhavan: The Hamiltonian constraint, roughly speaking, this is it's 424 01:09:33,720 --> 01:09:37,740 madhavan: How do I get this, I get this object by comparing 425 01:09:42,690 --> 01:09:43,800 madhavan: All the child. 426 01:09:50,460 --> 01:09:52,500 madhavan: And come bearing the 427 01:09:56,130 --> 01:09:57,120 madhavan: Please for text. 428 01:09:58,140 --> 01:10:11,790 madhavan: And the word the evaluation of the function at the vertex of the parent, the parent vertex and the child word exact compare the two, I take the difference in that is what goes into this derivative. Your and I have a multiplicative 429 01:10:14,520 --> 01:10:24,660 madhavan: Commentator I wanted to know this can constraint. He had em on the new actual state. So again, this the approximate operator. 430 01:10:28,770 --> 01:10:33,300 madhavan: State in Malta parish and of times this partial derivative of 431 01:10:35,520 --> 01:10:39,390 madhavan: X of the tile and at the vertex of the parrot. 432 01:10:41,610 --> 01:10:46,740 madhavan: But em over here is a lapse of data being evacuated and 433 01:10:49,320 --> 01:10:54,480 madhavan: So I need to evaluate it and at the coordinate match of the child vertex 434 01:10:55,980 --> 01:10:56,460 madhavan: Coordinates. 435 01:10:59,580 --> 01:11:01,230 madhavan: And these because I'm 436 01:11:02,370 --> 01:11:03,300 madhavan: Requiring the few 437 01:11:06,570 --> 01:11:07,740 madhavan: distinct patches. 438 01:11:09,570 --> 01:11:18,360 madhavan: Complete the competition have to transfer from the child coordinate attached to the parental court for have to 439 01:11:20,130 --> 01:11:21,540 madhavan: Have this transformation. 440 01:11:23,280 --> 01:11:30,150 madhavan: And these Jake opens appear on the left hand side and similarly frightened side, we have to electric different offers and constraints. 441 01:11:31,620 --> 01:11:38,340 madhavan: And it turns out the DJ the calculation only works in a trance and the peons 442 01:11:39,540 --> 01:11:42,360 madhavan: Through this derivative operator over here. 443 01:11:44,040 --> 01:11:44,220 madhavan: The 444 01:11:45,960 --> 01:11:50,730 madhavan: Constant Jacob eons and cons means arise. 445 01:11:52,470 --> 01:12:00,690 madhavan: From linear coordinate transformations. If the quarter transmission between the child and the parent is laid out there on can get a constant Jacoby 446 01:12:03,930 --> 01:12:17,070 madhavan: Is technicalities, to tell you the due to some very abstruse go reason I need an element of linearity in whatever I do. And this linearity. 447 01:12:25,050 --> 01:12:33,000 madhavan: Some items on have vertices which are the property of being linear and that is what am 448 01:12:37,650 --> 01:12:47,040 madhavan: I know it's a bit opaque. But what I want to tell you from this slide is that the competition involved because coordinate which is on board. 449 01:12:47,610 --> 01:13:05,940 madhavan: And the competition comes for really simple and trans. If one has Jacob Ian is there has to be some element of linearity in these coordinate transformations and this in turn districts, the brass summons. 450 01:13:07,020 --> 01:13:10,800 madhavan: All these off shelves dates to have vortices, which are 451 01:13:12,390 --> 01:13:15,600 madhavan: All linear versus to define on the next slide. 452 01:13:20,220 --> 01:13:43,320 madhavan: So let me go places and the nature of these deformations, which will happen to an end constraint actually makes, so I will say that the vertex is linear. If there is a coordinate patch in which all edges at the vertex appear as straight lines. So if you're familiar with this paper or 453 01:13:45,450 --> 01:13:53,340 madhavan: Oh, this is saying that all higher per module I banish at these workspaces. 454 01:13:54,660 --> 01:13:59,190 madhavan: So this is a different, more physical environment concept, the linearity of this vortex. 455 01:14:01,110 --> 01:14:24,300 madhavan: And the coordinate patch patches in the exclusive or as linear, namely in which the the edges explicitly appear a straight lines I will call as Linda or not. It is not true that when a linear, what unique linear coordinate patch. There are many, many of the 456 01:14:25,320 --> 01:14:27,240 madhavan: There is a way of, I mean, 457 01:14:28,320 --> 01:14:44,520 madhavan: choosing one of them and evaluating the quantum shift in in in that and making a choice so that finally everything is different office and corner and I will not go into these details, I will take a linear coordinate patch for the purpose talk 458 01:14:48,840 --> 01:14:51,150 madhavan: Now with this type of these coordinates. 459 01:14:54,660 --> 01:14:55,140 Is the 460 01:14:57,180 --> 01:14:57,840 madhavan: Shift 461 01:14:59,340 --> 01:15:05,070 madhavan: As them about pulling on the vertex structure along the I knew edge. 462 01:15:07,020 --> 01:15:14,400 madhavan: Shift was a some over age tangents and coordinate edge tangents in the corner. I was using 463 01:15:15,690 --> 01:15:32,790 madhavan: And the an important part of the deformation generated by the Hamiltonian constraint was related to a lead derivative structure with respect to this quantum we have lead derivatives along the edges of 464 01:15:34,080 --> 01:15:40,050 madhavan: Edge engines leader returns with respect to the sentence of the words and 465 01:15:41,310 --> 01:15:49,380 madhavan: On the road because the quantum shift you was zero almost everywhere except at the vertex said 466 01:15:50,580 --> 01:16:05,940 madhavan: One can then visualize quantum theory or the relevant different Marxism at parameter delta to be something like an abrupt all of the text structure along the edge and then some over all these 467 01:16:07,350 --> 01:16:07,830 madhavan: For 468 01:16:10,530 --> 01:16:12,210 madhavan: What one has 469 01:16:13,560 --> 01:16:25,680 madhavan: Political conical deformation, which is well defined because of the linearity of the world as well, then the remaining n minus one edges and vertex which has an violin. 470 01:16:26,880 --> 01:16:27,480 madhavan: Okay. 471 01:16:30,600 --> 01:16:41,760 madhavan: On the page. And so, one gets this sort of a structure where one has these kinks now because was made an abrupt and as delta two zero. 472 01:16:43,260 --> 01:16:55,590 madhavan: Approach the original work so much faster than later. So in the dental going to zero in abroad along the edge and then I will some of these deformations for each of the 473 01:16:56,520 --> 01:17:13,020 madhavan: I equal to one, two images and they will also be charged flips, which I will come to in the next slide here, I'm just giving you a flavor of things, the displaced vertex has the same as the parent will bond because I'm pulling on this along 474 01:17:14,730 --> 01:17:17,130 madhavan: And because I'm using Lynette 475 01:17:18,150 --> 01:17:26,970 madhavan: And because cones are linear structures. It turns out that this Burton new child vertex is also linear as I needed to 476 01:17:29,250 --> 01:17:42,570 madhavan: Do my point. I want to want to want to say is that the the calculation. Because of all this underlying linearity hints. Finally, at some Soto, he smiles linear discrete structure. 477 01:17:43,890 --> 01:17:44,340 madhavan: This 478 01:17:47,010 --> 01:17:55,680 madhavan: truce and technical really making a case of with the simplest spin form also use some 479 01:17:57,480 --> 01:17:58,050 madhavan: Lynette 480 01:18:02,220 --> 01:18:06,660 madhavan: Put to the next slide spawn to wrap up and give you 481 01:18:08,070 --> 01:18:24,840 madhavan: Should have these look like if you're interested in much more technical details I given an earlier top which was much more detailed only to for a and Rodolfo and and that recording is available for people who are interested 482 01:18:26,070 --> 01:18:32,430 madhavan: So this is what the different deformations created by the Hamiltonian constraint look like 483 01:18:33,450 --> 01:18:45,150 madhavan: So be on the phone. I'm just focusing at a particular vertex. So the books, then you have deformations alongside the edge which look like this. 484 01:18:46,170 --> 01:18:52,140 madhavan: And because of various charge flips etc etc finer deformation says, Jay. 485 01:18:54,060 --> 01:19:05,580 madhavan: Looks like this. This is for a particular charge flip and a particular choice of edge tangent along which to pull you get a conical deformation. And it turns out that in this deformation 486 01:19:06,930 --> 01:19:16,110 madhavan: This word is which you create is generically non degenerate and if a second Hamiltonian constraint x, it can act on this vortex. 487 01:19:16,680 --> 01:19:25,230 madhavan: It also turns out due to the details of the flipping that this vertex. The original vertex comes degenerate in the child. 488 01:19:26,190 --> 01:19:44,790 madhavan: So to what happens is that this is a delta neighborhood or the word test outside this delta enable nothing else happens, similar to what Thomas is constrained while Thomas's constraint, but only put one extraordinary edge over here, this difference the entire vertical structure. 489 01:19:46,290 --> 01:20:02,370 madhavan: In a particular way, and what form giant and B form and display vertex with coordinate distance delta where the second number two on a constraint can act and provides the scene for LM and the 490 01:20:03,900 --> 01:20:04,530 madhavan: Structure. 491 01:20:06,660 --> 01:20:19,110 madhavan: One can have what I call this is a downward pointing Co. So in the theory, one can have downward deformed children, one cannot operate the form children as well. 492 01:20:19,500 --> 01:20:31,080 madhavan: Where one extends the graph. And one of the upward pointing code where this is the phone access and it looks like there's only to give you a flavor of what these things look like. 493 01:20:31,800 --> 01:20:42,570 madhavan: This is for what the Hamiltonian constraint does what the different more efficient constraint does is electric different more physical changes is very simple. It just deforms the graph only 494 01:20:43,590 --> 01:20:56,010 madhavan: Have these electric shift type of deformations, with no clips at all. So you have exactly the original charges as before, but you have a deformed graph on which the child six 495 01:20:57,510 --> 01:21:14,220 madhavan: But what you can see is that the electric deformations are very much related to the Hamiltonian deformation and this is what makes it easier to compare the left hand and the right hand sides, and that is why this identity which Casey and iPhone important 496 01:21:15,960 --> 01:21:17,940 madhavan: So this is, again, just to give you a flavor. 497 01:21:20,040 --> 01:21:21,690 madhavan: So let me now. 498 01:21:22,950 --> 01:21:31,290 madhavan: So this was really to give you a flavor of the kind of deformations, which are involved and the kind of manipulations with going to finding what 499 01:21:32,040 --> 01:21:40,080 madhavan: The hammer to an extent action should be the material is there's a lot of technicality. And so I will, I will just stop over here. 500 01:21:40,800 --> 01:21:56,220 madhavan: And then now go to the issue of it's a bit of an abrupt stop. But basically, to give you a flavor of what happens in the US and the expectation is that a student radium theory also very similar things will happen. 501 01:21:57,690 --> 01:22:04,230 madhavan: So any questions. I can take them now. Otherwise, I'll go on to the to the propagation part of my talk. 502 01:22:16,410 --> 01:22:17,520 madhavan: Okay, so 503 01:22:19,470 --> 01:22:21,900 madhavan: Let me then go on to the provocations. 504 01:22:22,950 --> 01:22:30,000 madhavan: Over here, please I'm losing track of time, but please continue. Please let me know when when I'm running out of time. 505 01:22:31,080 --> 01:22:34,620 Jorge Pullin: While you're gone for an hour and 2030 minutes so 506 01:22:37,290 --> 01:22:38,970 madhavan: What, what would you, what would you like 507 01:22:38,970 --> 01:22:40,620 Jorge Pullin: Anything you're not 508 01:22:40,740 --> 01:22:41,850 Jorge Pullin: Too long though. 509 01:22:42,540 --> 01:22:44,850 madhavan: Okay. Okay, so let me do that. 510 01:22:46,980 --> 01:22:50,040 madhavan: I think I should I should be able to finish in 10 minutes 511 01:22:53,040 --> 01:22:54,630 madhavan: Okay, so let me now go to the show. 512 01:22:56,460 --> 01:23:05,220 madhavan: So Hamiltonian constructs constructions in LPG, as we've seen, lead to constraint actions at vertices or spin mats. 513 01:23:05,880 --> 01:23:12,060 madhavan: And these actually obtain the zero continuum actions of the Hamiltonian constraint approximates 514 01:23:12,750 --> 01:23:30,210 madhavan: The Hamiltonian constraint approximate is order again so that you have only vertex contributions and he action whether it isn't what I told you about, or whether it isn't Thomas's action or any other action demands the structure in a delta size neighborhood of the vertex 515 01:23:33,060 --> 01:23:44,370 madhavan: Vertex is completely independent. We haven't done constraint action another vertex. So this is what is called ultra locality of the action of the Hamiltonian constraint. 516 01:23:45,270 --> 01:24:04,830 madhavan: In a very influential and very beautiful paper in the night, we provided. I think the first clear articulation of what proper be in Luke quantum gravity and its potential tension with ultra locality of the action of the constraint. 517 01:24:06,180 --> 01:24:08,850 madhavan: He also argued that article LPG 518 01:24:10,020 --> 01:24:17,220 madhavan: One system with propagation of quantum perturbations from one vertex, or the spin it to another. 519 01:24:22,290 --> 01:24:23,220 madhavan: However, 520 01:24:25,560 --> 01:24:36,180 madhavan: If one looks at that paper. Unfortunately, his dictation arguments are based on almost his claims in the very fresh at that time. 521 01:24:36,720 --> 01:24:46,800 madhavan: He was April, which were only available in pre printed form, there is a lot of tangled history to this which which I will go into if there are questions later. 522 01:24:47,430 --> 01:25:00,210 madhavan: But the main point is that once the fork of confusion on everything is cleared. It turns out that the conclusion and the ensuing folklore that ultra locality of constraint action. 523 01:25:00,810 --> 01:25:19,830 madhavan: precludes property incorrect and this showed in models in parameters field theory in 2017 and then in the US to model in 2019 and there is a loophole in these arguments with the promise. 524 01:25:21,090 --> 01:25:23,370 madhavan: Which is in preparation. 525 01:25:25,320 --> 01:25:34,500 madhavan: The folklore seems to be based on the following fact that the action while the action of the Hamiltonian constraint can create a spirit. 526 01:25:34,920 --> 01:25:48,390 madhavan: Of spin mixtape for a small enough delta, it cannot merge tortoises. So if you have the Hamiltonian constraint that gentleness network vertex. He over here can create the mission over here. 527 01:25:48,840 --> 01:25:59,490 madhavan: And then one can use a few more efficient to transfer it to the neighbor of a second vertex, but there is no way in a delta no sense that you can ever make this vertex 528 01:26:00,090 --> 01:26:08,250 madhavan: This because there will always be a smaller delta where you can only move this to a small data and you know in the limit dental going to zero. 529 01:26:08,640 --> 01:26:22,620 madhavan: You can't run into different more efficient class of this to this, and that really seems to be the problem with electronic collection over here, you cannot do this action and then how could you ever take this perturbation and move it beyond is the question. 530 01:26:27,750 --> 01:26:37,860 madhavan: A little bit more location, the engine to notion of propagation. I've been talking about can be made only just a little bit more precise as follows. In terms of physical states. 531 01:26:38,670 --> 01:26:45,630 madhavan: So any physical state lies in the kernel of constraints and we can view it as an element of the algebraic dual and 532 01:26:46,440 --> 01:26:54,450 madhavan: It admits an expansion, like the option states in terms of spin network brass. There's some coefficients to some more. You get a physical state. 533 01:26:55,260 --> 01:27:09,180 madhavan: Let me you use the nomenclature. We call the set of all these brands summons to be on the brass it and they can't analogues to be the cat set. So, I will say physical state has an associated get set 534 01:27:10,350 --> 01:27:16,290 madhavan: And the elements of this respondents. They are someone's over here with non zero coefficients. 535 01:27:16,980 --> 01:27:21,360 madhavan: And I'll say that the physical state in quotes propagation in the elements 536 01:27:21,630 --> 01:27:40,740 madhavan: Gets it are related by propagation, that is there a subset of these elements which form of propagation sequence where you have a get as a perturbation. I'd say one of it, versus giving you a top stay propagation of this perturbation do a neighboring vertex of as 537 01:27:42,510 --> 01:28:01,980 madhavan: Which is time and then absorption of this perturbation by overtakes we do giving you a better state than an emission of this propagation past vertex we to at cetera so in this, get set, which underlies this physical state has many, many sequences or long sequences. I will say the 538 01:28:03,660 --> 01:28:05,910 madhavan: Biggest populations we 539 01:28:09,360 --> 01:28:21,570 madhavan: shimmer propagation. So this notion of propagation in terms of physical states is lovely distinct from that deriving from repeated constraints landscape state. 540 01:28:22,830 --> 01:28:42,300 madhavan: So therefore it is conceivable even if the latter notion, namely repeated action does not generate propagation duty ultra due to a locality propagation can still put it in physical states and I will be state based notion, actually. He also uses in his analysis. 541 01:28:43,950 --> 01:28:55,770 madhavan: Of course, the operators determines there are no that is determines the structure, the physical states. So, the importance of propagation in that sense is tied to the form that constraint operators. 542 01:28:56,280 --> 01:29:07,020 madhavan: Because the physical state depends on the structure that constraint operate it. Isn't this sense that demanding propagation constraints, the available choices of the Hamiltonian constraint or 543 01:29:07,830 --> 01:29:25,320 madhavan: No one possible route to propagation is as follows. Let the action or the Hamiltonian conceit parents Wynette create or set of different children are they doing almost this case on my case let the structure of the cat said be such 544 01:29:28,020 --> 01:29:36,480 madhavan: That if a parent is in a cat set is in the children. In other words, if you have a particular spin network state in the case 545 01:29:37,470 --> 01:29:51,630 madhavan: Then if you act upon it by the Hamiltonian constraints, their children and all these children are also months in the brand some which defines them become a regional parent was in 546 01:29:53,940 --> 01:30:04,830 madhavan: Similarly, if a child is in the cat set, then all possible so also indicates. So if you have a particular a headset. 547 01:30:05,760 --> 01:30:15,690 madhavan: And if you have state which when acted upon by the Hamiltonian constraint gives you this particular state. I will call that state of possible 548 01:30:16,470 --> 01:30:38,040 madhavan: Maybe many possible parents for your child. And all these possible parents are also the cancer. So, suppose the caps at underlying physical state has this property then it turns out that propagation can ensue to the existence of these possible parents as follows. 549 01:30:40,110 --> 01:30:48,510 madhavan: These conclusions basically pertain to the formation generated by the constraints that is to all of a given pair. 550 01:30:49,770 --> 01:30:56,100 madhavan: And these do not include propagation sequences. Because of the possibility of the vertex merging operation as we saw 551 01:30:57,600 --> 01:31:03,360 madhavan: The get set, namely the summons also include all possible parents 552 01:31:04,410 --> 01:31:16,770 madhavan: One can have possible proper one application sequences as follows. So again, you have a Hamiltonian constraint acting on a Penn State is your child over here. 553 01:31:18,360 --> 01:31:22,950 madhavan: The state is Marxism and manner. So all different morphic images of this child or in the cat. 554 01:31:24,420 --> 01:31:35,640 madhavan: This is where takes over here by default Morpheus. And here, and of course not merging action American strange constraint, but you can ask, is there a state as prime 555 01:31:35,970 --> 01:31:51,510 madhavan: Such that when the Hamiltonian constraint act on this next. It gives me this for this particular child. If it does, then this is a portable parent this child over here. 556 01:31:52,470 --> 01:32:12,390 madhavan: In other words, this child has known unique parent. It has a distinct parent year but both of them occur in the cat said, and then give you a proper sequence. So that is the basic idea. And this idea is implemented in Parliament price field theory. 557 01:32:13,920 --> 01:32:15,330 madhavan: You lose some technicality. 558 01:32:18,390 --> 01:32:36,900 madhavan: And in order to bypass these arguments. It turns out that the existence of these possible parents that is the existence of nominate parentage is crucial. Go back to his paper, then there's an implicit assumption based on the Q st prepayment that 559 01:32:38,220 --> 01:33:00,630 madhavan: Any deformation, which is obtained by the Hanford constraint can be uniquely associated particular parental vertex and Morrison I show is that this is not true. You can have non unique parentage. And then once they are non unique parentage many arguments. 560 01:33:05,760 --> 01:33:14,790 madhavan: How very quickly. It turns out that the end to enter formation is not consistent with propagation and play because if you do form. 561 01:33:15,420 --> 01:33:19,290 madhavan: an embodiment vertex, you get an imbalance child or text. 562 01:33:19,980 --> 01:33:38,160 madhavan: And if you have a neighboring vertex of some other violence, it could never produce this child because it will only purchase a child or four different violence. So these sort of deformations really cannot give you a vigorous propagation and it is possible to define a slightly modified 563 01:33:39,570 --> 01:33:48,660 madhavan: Construction wearing a lot is people who, for the purposes. The children and environment vertex 564 01:33:49,920 --> 01:34:05,760 madhavan: Something like this where you have a fall for balance because I'm running out of time. I wouldn't be able to tell you exactly how this happens. But basically, only three of these edges are pulled at one particular chooses to some over the choice of these three edges. 565 01:34:06,840 --> 01:34:08,550 madhavan: Very similar. 566 01:34:09,720 --> 01:34:15,510 madhavan: To one of the possibilities which Thomas consider in the early days. That is what most tells me 567 01:34:17,010 --> 01:34:38,520 madhavan: One can build a constraint action based on these n before vortices and all, with regard to the section one can show that you get propagation and work in progress, shows that I think I normally freedom also might be possible with these constraint action so that 568 01:34:40,260 --> 01:34:45,420 madhavan: finishes the technical talk. I'm just going to show you a few pictures and then wrap up. 569 01:34:46,770 --> 01:34:55,590 madhavan: So in the US tube theory, one needs to show propagation sequences and I'm just going to show you one certain sequence over him. 570 01:34:56,760 --> 01:35:11,820 madhavan: So here we have a parent stay with two verses of defiance the Hamiltonian constraint x at the first vertex and all three of these edges along the fourth one here gives you a for violent edge over here. 571 01:35:13,020 --> 01:35:18,540 madhavan: Then using the different morph isms we drag this to the vicinity of the next word 572 01:35:20,790 --> 01:35:26,940 madhavan: We can modify this vertical picture by the action of different comes 573 01:35:29,160 --> 01:35:31,110 madhavan: In this direction, sorry. 574 01:35:32,400 --> 01:35:37,470 madhavan: And then it turns out that this particular chart. 575 01:35:39,030 --> 01:35:39,540 madhavan: Which has 576 01:35:41,130 --> 01:35:48,840 madhavan: Been coming from here to here. So remember, these kinks or form but pulling by the Center for deformation 577 01:35:49,290 --> 01:35:59,550 madhavan: These now are directly connected to vertex vs overtakes me as a much higher balances balances in my three older and it turns out my taking this vertex 578 01:36:00,300 --> 01:36:21,810 madhavan: Default of the Hamiltonian by the action of the electric defeat modernism constraint and get exactly this. And all these states are contained in the tech side so in a way you can get a propagation sequence. And let me show you putting all these things together in a pictorial day 579 01:36:23,370 --> 01:36:27,630 madhavan: Or if I put the propagation sequence then really doesn't 580 01:36:34,320 --> 01:36:43,110 madhavan: Look as go upward. See, and finally lands up at sea, and then it can go beyond. So this is just to show that this is 581 01:36:46,860 --> 01:36:55,230 madhavan: And then this was just trying to visualize this is what could simply spin form in some on analog centering 582 01:36:56,610 --> 01:36:58,200 madhavan: And then let me summarize 583 01:36:59,220 --> 01:37:03,450 madhavan: In the last slide, and really sorry to Yvonne so overboard. There's a lot of material. 584 01:37:04,140 --> 01:37:14,760 madhavan: I'm sorry. I'm so over the last decade or so I tried to use the requirements of anomaly free constrict algebra of propagation to home in on to the physically correct 585 01:37:15,390 --> 01:37:23,940 madhavan: concentrator progress has been possible in toy models of increasing complexity certain parameters field theory. 586 01:37:24,420 --> 01:37:33,090 madhavan: And those are just two different more physical constraint, I talked about in Madrid Conference. And then finally, in the you want to model. 587 01:37:33,990 --> 01:37:46,320 madhavan: And this progress is due to the fact that classical evolution can be understood in terms of social reform of this particular spatial different systems in the you want tube theory which I electric field and 588 01:37:47,520 --> 01:38:01,470 madhavan: In Euclidean theory we are hopeful progress because the evolution equations can again be written in very and full form. This is new to me. This is my by actually a few 589 01:38:02,850 --> 01:38:04,950 madhavan: In to visit India. 590 01:38:05,970 --> 01:38:14,730 madhavan: Is denied something on the blackboard for the triad field and then with a little help from me looking at the connection fee equations and 591 01:38:15,630 --> 01:38:32,700 madhavan: The equations can be written in really a very nice form where he is some epsilon i j k. These are the electric Shipley derivatives of he similarly for the magnetic field which is just from the kitchen a curvature and 592 01:38:34,380 --> 01:38:42,390 madhavan: Script L objects are living ordinary literary terms, except that in the ordinary Lee. 593 01:38:43,440 --> 01:38:54,420 madhavan: Are you see an audit of operator you replace it by gauge covariance Arabic and then there's similar revolution equation for the connection. 594 01:38:54,900 --> 01:39:04,230 madhavan: And if you use the cell dual were also these equations are true. I've just shown them for the Euclidean theory and they're so beautiful. There must be some 595 01:39:04,830 --> 01:39:19,860 madhavan: Bundle interpretation for this, which will hopefully help in the analysis for quantum theory. So, so thank you so much. Wherever have stayed on to listen to the longer version of this talk. I'm sorry. I went on. 596 01:39:20,880 --> 01:39:25,230 madhavan: And I thank you all for listening, patiently to see. Thank you. 597 01:39:32,310 --> 01:39:33,180 Jorge Pullin: Any questions. 598 01:39:39,240 --> 01:39:39,510 Abhay Ashtekar: So, 599 01:39:40,710 --> 01:39:42,570 Abhay Ashtekar: If you can tell for me. 600 01:39:45,420 --> 01:39:45,990 Abhay Ashtekar: Can you hear me. 601 01:39:47,880 --> 01:39:50,490 madhavan: Yes, I can hear you on mute. 602 01:39:52,440 --> 01:39:55,320 Abhay Ashtekar: Okay, so the 603 01:39:57,540 --> 01:40:00,540 Abhay Ashtekar: Father, just in general for, you know, not 604 01:40:01,950 --> 01:40:11,130 Abhay Ashtekar: Getting every detail right but just broad picture, I would like to understand the following issue you began by that, then I'll have 605 01:40:12,270 --> 01:40:13,410 Abhay Ashtekar: A problem is that 606 01:40:15,300 --> 01:40:26,010 Abhay Ashtekar: A couple of problems. One is too much studying Hamiltonian constraint that there has been too much freedom and the second was about the the 607 01:40:28,170 --> 01:40:36,660 Abhay Ashtekar: Right, so don't or do a custom there with the concert with me on the north end 608 01:40:38,730 --> 01:40:49,170 Abhay Ashtekar: And then you showed us various ideas and various you know enormous technical progress that has happened at the end of the day, then I go back to the main question about ambiguities. 609 01:40:50,370 --> 01:40:57,750 Abhay Ashtekar: But what is the status and and i know similar very general question, but maybe just some first 610 01:41:00,600 --> 01:41:04,680 madhavan: Okay, thank you. So, um, it's, it's a difficult 611 01:41:05,700 --> 01:41:19,410 madhavan: question to answer, or because it certainly on suicide started it. But, but I find my financing this so firstly classifying ambiguities in in a 612 01:41:20,220 --> 01:41:37,380 madhavan: Way itself is a big job, so I can't say. But, you know, technically, there was a whole function space of ambiguity is which are so much and now there is only so much what I can say is qualitatively. What do I believe 613 01:41:38,820 --> 01:41:39,480 Are 614 01:41:40,830 --> 01:41:42,480 madhavan: The things which may have a chance of 615 01:41:43,980 --> 01:41:47,310 madhavan: And I one thing which which seems to be more 616 01:41:48,510 --> 01:42:10,650 madhavan: Strongly from the calculation is the fact that one has this kind of a linearity or vertices and that underlying whatever structure we have or perhaps there is some want to take advantage, not worry too much about the smooth different offices, but also lean into 617 01:42:11,700 --> 01:42:19,260 madhavan: Into some picture where where you can take advantage of these wise linear kind of structures. 618 01:42:20,310 --> 01:42:21,960 madhavan: That general comment, which I think 619 01:42:23,160 --> 01:42:31,620 madhavan: Probably people have been doing all along, gives additional sort of meat or have to was doing this. 620 01:42:32,700 --> 01:42:47,070 madhavan: I think the fact that, given this linearity, one can focus on clinical deformation, which actually move vertices. I think that is a big takeaway for me on so 621 01:42:48,120 --> 01:42:54,960 madhavan: I think that what seems to work. Are these kinds of conical deformations and 622 01:42:56,220 --> 01:43:09,030 madhavan: Not say that. Okay. Other things would certainly not work. And so I cut down a whole lot of ambiguity by telling you that uniquely one has these conical deformations, which are to work. I can't really say that 623 01:43:09,990 --> 01:43:18,990 madhavan: However, after working on and hard on the problem, at least in the US military and looking at the structure of the theory. 624 01:43:20,220 --> 01:43:29,220 madhavan: It seems that on this lead simply because of the action of the classical Hamilton and Butterfield's and the requirement. 625 01:43:29,700 --> 01:43:50,100 madhavan: Or creating displaced workspaces, together with the linearity of these vortices, more or less, roughly speaking into these chronic formations, not whether these clinical deformations are these of these n to four or n to n or some mixture. I do not know and i i think 626 01:43:52,050 --> 01:44:01,410 madhavan: So, so I I really cannot answer your question. Precisely. But I think, given all the implications of getting the constraint algebra to work. 627 01:44:03,090 --> 01:44:11,160 madhavan: And the propagation to work strongly moves me towards a consideration of these end to for deformations. 628 01:44:12,330 --> 01:44:13,830 madhavan: And again, the 629 01:44:15,330 --> 01:44:23,280 madhavan: At every step one would have to show actually that there is an anomaly freedom there is propagation. Maybe there are other requirements which are written 630 01:44:23,790 --> 01:44:37,680 madhavan: I have not yet gone into important technical choices which make things look not as good as I did them perhaps and and I'm working to modify those things. But they're all within the general 631 01:44:39,990 --> 01:45:02,460 madhavan: Domain of these kind of conical deformations of linear vertices. So I would say that the takeaway is that perhaps what one should look at. Are these kind of deformations which involve literate linear structures and which roughly speaking, have kind of 632 01:45:03,510 --> 01:45:08,940 madhavan: Kind of a political nature and which displays vertices, that is that is my feeling 633 01:45:09,390 --> 01:45:25,620 madhavan: From what I've been, I cannot classify ambiguous and they look at so many ambiguity is before and now now we have only so many back that I don't think I would be able to do but but I think kind of what we were doing before or probably Thomas would also agree. 634 01:45:26,790 --> 01:45:35,670 madhavan: That would not be satisfactory, from the point of view of the constraint algebra, it's time to look at new things and pretty much. I think if one 635 01:45:37,530 --> 01:45:43,530 madhavan: In more detail. This is where one has led to and and just one more point to add, I was talking to Thomas when I was 636 01:45:43,860 --> 01:45:56,490 madhavan: In London, and he said, especially this into for kind of thing from was one of the things he had thought, thought about when when he looked at what it would do to a vertex and, you know, but it would 637 01:45:57,390 --> 01:46:14,760 madhavan: You know how we define tetrahedral at each vertex and one could one could make use of these and perhaps try to actually get these sort of deformations maybe naturally also in the suitcase. So basically, that is to say, 638 01:46:18,240 --> 01:46:19,650 Abhay Ashtekar: Okay, so I think that the 639 01:46:21,150 --> 01:46:29,040 Abhay Ashtekar: Hormones that there is no wrong in just looking at peace with Genia category and it and I think because during our time. 640 01:46:29,610 --> 01:46:38,100 Abhay Ashtekar: If you have some brief comments make it. But otherwise, you can go to the next point, namely, I think the question is really, whether in the 641 01:46:39,000 --> 01:46:54,060 Abhay Ashtekar: Category of that from the beginning. I don't think anything wrong with our district along that continuum limit we just our words, look at peacefully crops that is a theory. And it seems to me that the procedure is much more streamlined and one may have more 642 01:46:57,180 --> 01:47:09,270 Abhay Ashtekar: To be retractable we show that, and they do these are very, very few are completely controllable, but you can either agree or disagree, but don't quote me 643 01:47:12,120 --> 01:47:22,890 madhavan: Um I yeah I yeah i i would like to talk to you more about this because I haven't. I mean, to, to actually do it at initial from the piece wise linear theory. 644 01:47:23,940 --> 01:47:26,010 madhavan: I, yeah, I would like to talk. I'm not 645 01:47:27,600 --> 01:47:30,090 madhavan: I haven't used because you know 646 01:47:31,110 --> 01:47:37,800 madhavan: Now, we should we should talk a little bit more about this, it would be great. I think if actually it's true, then that would be marvelous 647 01:47:38,370 --> 01:47:41,010 Abhay Ashtekar: Okay, so the last quick question was about 648 01:47:41,310 --> 01:47:41,580 Abhay Ashtekar: This 649 01:47:42,240 --> 01:47:43,260 Abhay Ashtekar: Or about propagation 650 01:47:43,920 --> 01:47:45,480 Abhay Ashtekar: And I really easy. 651 01:47:46,590 --> 01:47:49,650 madhavan: One for the comment of I'm sorry to interrupt just one small part 652 01:47:51,510 --> 01:47:53,070 madhavan: So I also wanted to say that 653 01:47:54,360 --> 01:48:02,550 madhavan: Again you are familiar with some of you with some of the technicalities but someone in this whole business, there is a there is a 654 01:48:03,120 --> 01:48:07,530 madhavan: There is a St. Lucia coordinate patch and a traditional vertex structure. 655 01:48:08,040 --> 01:48:30,600 madhavan: And that permeates things, but there is a way to finally get Babs get rid of this or maybe do it as undesirable and integrate over all choices, etc. But if one does this then at the end of the day, one is led to a picture from here, where to look at the physical states then then as 656 01:48:32,100 --> 01:48:48,540 madhavan: Carlo and and and Norbert or someone else in one of these papers speculated, you could just not worry about these various higher order modulation and I wanted to, I just wanted to make that point that this this emerges. 657 01:48:50,250 --> 01:48:55,170 madhavan: Probably quite nicely from the sort of calculations that was the only item I want to thank you 658 01:48:56,550 --> 01:48:57,210 madhavan: I will mute if 659 01:48:57,450 --> 01:49:02,220 Abhay Ashtekar: I just correct that. Can I still ask one more question, because I'm asking to model as 660 01:49:03,660 --> 01:49:03,840 Jorge Pullin: Well, 661 01:49:04,710 --> 01:49:23,670 Abhay Ashtekar: So the last question was about propagation and the, I mean you you sort of have very nice somebody in I think it will then slide 25 but maybe not about you know you wrote down these two conditions about when you get a bus propagation. Right. 662 01:49:24,780 --> 01:49:28,110 Abhay Ashtekar: And Mike Yeah, I think it was like 23 or 25 663 01:49:29,820 --> 01:49:39,000 Abhay Ashtekar: If a parent is in the in the cat said then, so all the children and the child is and get set them up. So a proper possible parents. So what we would like to understand is really 664 01:49:40,230 --> 01:49:42,810 Abhay Ashtekar: I mean, how easy is it to get this. I mean, can I just 665 01:49:44,160 --> 01:49:53,550 Abhay Ashtekar: Can I just want to start with the graph and look at all these tests and my ex say that, well, let me look at all these children and look at all these 666 01:49:54,390 --> 01:50:11,280 Abhay Ashtekar: Parents of the children of the resulting thing. And can I say that this procedure actually converges. And therefore, of course, a presumably, I will just have an infinite number of what is this. I mean, I'll get a golf Arbitrary Arbitrary lines to borrow vertices 667 01:50:12,930 --> 01:50:15,240 Abhay Ashtekar: And so the question is, can I just 668 01:50:17,550 --> 01:50:26,010 Abhay Ashtekar: Buy some general mathematical construction arrive at such get sets which are admissible in there is robbers dynamical propagation 669 01:50:27,120 --> 01:50:32,850 Abhay Ashtekar: And secondly, if that is the case in and out of this. Is this set of of 670 01:50:34,080 --> 01:50:42,720 Abhay Ashtekar: distributional in the sense of physical state that come from this sketch sets are they going to be rich enough. Do you have any idea about 671 01:50:45,900 --> 01:50:48,720 madhavan: Okay, so let me 672 01:50:50,550 --> 01:50:58,020 madhavan: Let me let the mountain seven seven stages. So I put these two points. 673 01:50:59,310 --> 01:51:05,790 madhavan: Because this is what the structure is in parameters field area as well as 674 01:51:07,110 --> 01:51:24,210 madhavan: What can happen in you want cute. Well, the reason this, this happened. So let's let me go very briefly into the reason why this happens in in you want cube theory and excuse me, and it's simply because of this you minus one over delta kind of structure. 675 01:51:25,350 --> 01:51:26,550 madhavan: Now, it turns out that 676 01:51:27,780 --> 01:51:29,610 madhavan: Father kind of 677 01:51:30,870 --> 01:51:33,510 madhavan: regularization, which I'm using and you want cubed. 678 01:51:35,400 --> 01:51:52,770 madhavan: Solutions are easy to find. Because what one really needs is that forgetting about the coefficients. It turns out that one can forget about the questions and at each vertex, one can only required that this you minus one. 679 01:51:56,610 --> 01:52:00,630 madhavan: Actually vanishes when when you evaluate it with the option state. 680 01:52:01,290 --> 01:52:06,330 madhavan: So, one can easily see what are the different missions, or what are the children and one can 681 01:52:06,630 --> 01:52:15,690 madhavan: Make the statement that I'm making because of the u minus one. The you gives you information about the child and the one kind of tells you that the parent should be there. 682 01:52:16,380 --> 01:52:27,960 madhavan: However, the you want your regular issues regularization I use maybe a little questionable due to some technical points which which one can talk about later. 683 01:52:29,010 --> 01:52:30,120 madhavan: It might be that 684 01:52:32,550 --> 01:52:41,580 madhavan: This is not so clean or that all the parents and the one and two mean we may not be able to show it so cleanly. Even with this structure. 685 01:52:44,310 --> 01:53:01,980 madhavan: So that is the first statement. The second statement. So I'm saying, even in a favorable case of you want you. If I were to do a slightly better treatment than it could be that one cannot infer this structure, one cannot info one and two immediately. That's the first statement down to me. 686 01:53:03,810 --> 01:53:10,980 madhavan: The second, the second statement is that what is really important for propagation is that 687 01:53:13,110 --> 01:53:15,090 madhavan: It is not necessary that 688 01:53:16,500 --> 01:53:23,850 madhavan: Number to actually hold. Okay, so in order to to say this a little bit better. 689 01:53:25,620 --> 01:53:30,420 madhavan: Let me look at the propagation sequence. I'm still sharing the screen and 690 01:53:31,650 --> 01:53:40,920 madhavan: When I look at this propagation sequence from s to s perturbation here and to S, Brian. The reason I wanted 691 01:53:41,610 --> 01:53:49,890 madhavan: Ends is that that I wanted this is prime itself to be there, but it could be that s crime itself is not there. 692 01:53:50,250 --> 01:53:58,950 madhavan: But if you imagine that this vertex structure went on to the right hand side, the right hand side, the right hand side that 693 01:53:59,490 --> 01:54:13,380 madhavan: This deformation actually appeared directly on the right hand side without something appearing or which which had an absorption interpretation. In other words, one could only have children, all the way 694 01:54:13,950 --> 01:54:21,720 madhavan: But what was very important is that non unique parentage be the, namely that you had 695 01:54:22,680 --> 01:54:38,610 madhavan: That you had some parent, which could, in principle, generate this child and could generate another child which had a deformation, to which right the parent itself need not be there. So what happens in Thomas's constraint. 696 01:54:39,240 --> 01:54:50,220 madhavan: Is that one can actually show model again because the equations become technically very complicated, but one can show that there is reasonable grounds. 697 01:54:50,520 --> 01:54:59,430 madhavan: For believing that solutions are of the type that they may not contain the parents themselves so they don't have the u minus one structure. 698 01:54:59,910 --> 01:55:07,650 madhavan: But the container of children, which do have non unique parentage, so you can view sequences as 699 01:55:08,310 --> 01:55:16,800 madhavan: As something as a perturbation of one vertex moving here another one and then jumping across to the next one. So, this is 700 01:55:17,370 --> 01:55:29,940 madhavan: Also propagation. It may not be the strict propagation. Which one is looking at, but it could also be so propagation. So what what I'm trying to say is that the very notion of propagation needs to be a little bit fine tuned 701 01:55:30,390 --> 01:55:35,730 madhavan: And then looking at the structure of the Hamiltonian constraint itself. 702 01:55:36,930 --> 01:55:49,290 madhavan: Does not really Gannon, it's not a quick answer as to whether what you will get is propagation or not. I think what is a quick answer or what would be very quick to conclude 703 01:55:49,740 --> 01:55:59,970 madhavan: Is if your constraint does not display propagation. I think one would be able to see this quite quickly that it displays no propagation at all. 704 01:56:00,930 --> 01:56:08,460 madhavan: However, to show that it displays vigorous propagation, as opposed to only a little bit of proposition. I think that 705 01:56:09,030 --> 01:56:21,420 madhavan: You know, what are the kind of propagation sequences in the physical state space that I think would be a little harder to to show, but I'm optimistic that once one one gets various 706 01:56:21,840 --> 01:56:39,030 madhavan: You know, if one can show that the constraint Algebra one walk on one should be able to very quickly info propagation, or actually happens or doesn't happen. So I'm so long answer, but but I didn't know how else to you actually answer the question. 707 01:56:40,230 --> 01:56:40,650 Abhay Ashtekar: Thank you. 708 01:56:44,070 --> 01:56:45,030 Jorge Pullin: Any other questions. 709 01:56:54,150 --> 01:56:55,110 madhavan: Okay, thanks a lot.