How to Extract Physical Predictions from a Diffeomorphism Invariant QFT

0:00 For the day, let me start. Chris, thank you very much. And actually, I just want to take one second to say something about the reason of this. Thank you. All of you who know me believe that Chris Mott was my thesis advisor, or that I started working with Chris, or that all this is false. Renata said this morning she's the only student of Chris among his speakers, that Chris was not my physicist. But there is a reason for which a lot of my friends think that, because indeed, certainly Chris is the person who had the most influence on my life, I would say, and my life in physics, certainly. And I will say why, at least some bits of it, because I think it's the same for many people here. When I started studying physics, what fascinated me is the effort to find a new way of thinking above the world that I found in the development of classical science and field theory, especially in the future of physics. But then somehow there was at least an aspect of modern physics that I saw around me in which this was lost in some sense. Their basic rules were there. and then it is through Chris writing and papers and his description of the problem of quantum gravity which I found excitement about here is a world we don't understand it, we have to find new ways of understanding it that has inspired me, motivated me to go into physics, to go into and basically all my life has gone that direction and when I was a PhD student and he came here in a sense my life as a student I came to see Chris. I remember one day I arrived and there was absolutely no idea that people told me it was a bio-commitary. You are not from this country, you don't understand why it's a bio-commitary. And then things started from then. The first time I thought I did something interesting in physics, what I did with my collaborator was first to go to Syracuse and then immediately take a plane and come to London to see whether Chris would say and so on so Chris thanks for all thank you for your advice and in preparing this talk I sort of I sort of apologize to everybody

2:30 because I this is a talk that I prepared in a sense thinking about it's not a big review thing about past work it's not a big review about which is a theory in which I have most of the work. First, because I thought, please know all of that. Second, because Avayashika is speaking after me, and I believe he will explain the basics of the theory of incremental gravity and many of the consequences that have been worked out of it. And third, because I wanted to present a number of ideas, suggestions that I hope could interest the proof. So in a sense I'm giving for granted everything that I will say in the next talk. I will assume that you know everything about democratization and value. And I will discuss a problem and so I will begin by carefully saying what is the problem that I discussed, which is how to extract physics from the theory. and then the large part of my talk will be just point one, which is sort of present a sequence of ideas which I hope could get us, could help us to the point of general technique for extracting physics from theory and then two and three will be rather fast, two is just a simple model, a toy model in which these ideas are implemented And three is just a suggestion, because it's not much results, it's just an innovation I would present with an on-flight in which I'll say, okay, here's a way I think calculations can be done. So what is it that I... What's the problem? Now, before you jump up, let me push this. the last problem of Chris's lecture was using all the material in this book in this lecture find a consistent divided and finite and general develops environment and remember the expression that produces the background independent of quantum theory of gravity and in a sense many of us have been just working with that for all these years and to some extent I think that there is a theory which is good quantum gravity which is unprovided by it or at least is most presumably unprovided by it and it is certainly background independent and it

5:00 is a quantization of general activity plus matter so in a sense we have a theory of quantum first of all, well we don't know if it's correct, it might be wrong, but one of the reasons we don't know it is correct is that the actual theory exists in a number of different versions, and in one way or the other they all are incomplete, and why can't we compare the versions of the theory, why can't we choose between them, And basically because we can do certain calculations with the theory, we can compute spectrum parameters, we can discuss the whole entropy, we can discuss cosmology, we can do a number of things. But this sort of specifically works. What we cannot do, what we don't have, is what we have in QE, just a machine for computing systematically, or most predictions, for instance, class of prediction of the theory. So, it seems to me that it is certainly interesting to do what a lot of people are doing in this field, which is to compare the different versions, find reasons for which one is better than the other. But the key problem is to address three, and forgetting whether the theory is right or wrong, suppose we have a theory which is formulated in a different way, which there's no background space-time, how do we compute scattering? How do we concretely compute, if I have two particles that interact, what comes out? And so this is all I'm saying about loop quantum gravity. If you know loop quantum gravity, you just want to talk about what I'm talking about. If you don't, it doesn't matter, because I'm not really using any of this. Just hand it back to this last component. What I'm saying is just pick a theory, pick a version of the theory, and there is a version of low quantum gravity in which you have a certain Hilbert space which is spanned by a basis labeled by abstract spin networks, which roughly are quantized to three geometries, or three geometries which are discrete on the blank scale, and in the formulas there are certain area volume operators to such that we sort of have an interpretation of these things in which the nodes are chunks of spaces and the labels are the quantum numbers of area volume of

7:30 the spaces so there is a there is this mathematics and besides it we can we can have the way of defining amplitudes associated to those states here using for instance a a Barrett-Craymore, BCC comes from, is the terminologies, one particular Barrett-Craymore, the terminologies is from my book, just from the book, so this is just a one minute advertisement about my book, it's coming out next week, when you use the best. so this amplitude is roughly the functional integral on four geometries of the action integrated in all four geometries which are bounded by a three geometry described by this pinetro here this pinetro is a three geometry so intuitively can be thought in this way but using this model in particular using the group theory formulation this can be computed order by order in a certain parameter as a finite expansion so this is a finite expression we have this now in principle the calculation is complicated by order by order in lambda in one of these models and the way it's actually computed is using an auxiliary theory of theory as an expectation value of auxiliary theory of theory Again, it's not important for what I'm saying, but what I want to stress is that there is a little space with a basis of states, with a roughness of geometry, with a way of associating an amplitude to it, which, if the theory was right, could be thought as the functional integral. Okay? So suppose we have that. Suppose we are given this . How do we continue? In particular, how do we look scalable? Now, if I take the derivative generativity, quantum generativity, I can compute scattering. The thing is normalizable, which does not mean that I cannot compute anything. It only means that when I compute things, I have three parameters that enter in my scattering output. So if I believe that this is correct in some sense, what the finite theory of the Planck scale should give me is to fix those parameters.

10:00 So in a sense, I can say that my norm-patermity theory of Planck scale, I understand it, I control it, if it is a way to give me a way of computing the three parameters of the normalization. And where is the difficulty? Well, everybody who has worked with norm-patermity quantum gravity knows it. This is just one way of proving it. The key thing about these general arguments is just the core of generativity, background dependence. Because if, in computing, for instance, one way of computing scatter is to use n-point functions. But if this theory here, this action and this measure, have different multi-environments, this one is independent of x1 over xn. In other words, I don't know where my endpoint function sit, where are the legs, right? Because I don't have a background geometry to locate points. So I don't even have the first elements I need to do well. So that's a problem addressing. And I am going into this problem through a lot detour by presenting a number of ideas, Thirdly, a lot of information goes from Figueroa's sum of the geometries, and from gene parts of general lifespan to the canon Calcene, which certainly connects to it. So let me start from the very beginning, because I think the problem is in the classical theory, right? I can do classic AGR of physics, but I think canonically, how Chris told me to think first. And I say, well, I want gage invariant direct observable in the theory. I don't know any general activity. I don't know quantities that commutes with a few models this way. It is sort of the same problem from another point of view. So let me go back to the beginning and say, how can I think about the general covariant theory? How can we think of general quantum mechanics? Please follow me through a number of steps and I'll get back to the question. Suppose I have a system that I want to describe, a super simple system. So what do I do? I have to measure this angle up here, and we call it q, and make a list of values. But of course that's not enough. I don't do physics this way.

12:30 have a clock and measure, say, the activity of the times, the angle between the hand and the clock and the 12. And so I have two quantities, which I am considering, Q and D. And then I make a table. I have to measure the two together. I'll make a table of various couplets of the measurements and a series about those tables. Now, what I want to do, what we usually do is say, well, What I want to do is to get out from this way of thinking and consider Q and T on the same ground, from day one, and then work on the way to the end. So let me just define, let me just code Q and T partial observables and a couple, a combination or an event, and the space of the couple, so the state of this point, a relativistic configuration space. Of course, the configuration space of this system is one dimension squared. So, the relativistic configuration space is two-dimensional. Now, if I make an assembled measurement, I get a set of points, there's a set of couples, a set of points here, which, generically, will sit on the sub-manifold of C, which I'm not making any assumption of any sort. So I call this line a motion. And this is defined by just a function of c being equal to zero. Now so far I'm not doing physics. Where does physics sound like this? This is mainly because if I now push this thing, and I let it move differently, and I repeat the experiment, and then I push it again, and then I repeat the experiment, every time I get a different motion, every time I get a different line here, but a priori the number of personal lines is infinite, it, in reality, what I find, the spirits tell me that the number of possible motions is very small. In fact, it's a two-dimensional space out of a two-dimensional space of motions. So, this is what I was saying. It's so dependent on an experiment that I observe a equal motion, but the number of possible motions is small, and then we call gamma the space of possible In this case, the motion, the space, of course, is beaten two-dimensional and is labelled, for instance, by the amplitude and the phase.

15:00 If I fix the amplitude and fix the phase, I fix the motion. So let's call A and P, the amplitude. So all the information about the system is for any choice of a motion, amplitude and phase, I have a relation of infinity than a function that I have. So why is the function n, which is from the example of the initial phase, the physical phase, times the phase phase of R, and the vanishing of it is all the information I have about the system and for any more. Now, what I find is that the structure is general. You can think any physical you can only see that you have a list of partial circles which you view as co-ordinates on a relativistic oscillation space and then you have a set of states and your physics is defined by the dimension of a partial object. I have not chosen a particular variable as time, I'm not viewing evolution of the system as evolving in time, general that these motions don't come back in time for instance and so on and so forth. Okay this is a way of thinking about systems and why do I want to use it because generally systems will sort of fit naturally in this way of thinking and I want to argue that you can view mechanics, quantum mechanics, field theory and quantum field theory from this perspective, and sort of things becomes easier, nicer, at least from some point of view, and practicable when using what we write. So let me start by Hamiltonian theory. First of all, in general, Hamiltonian the analysis can be defined by Hamiltonian function on the cotangent space of the relativistic. So this is a more function of position, time, momentum, and energy. And you can, in general, say that F, emotions, are given by an initial principle, which is the minimization of an action

17:30 where what you're realising is just the map with no specific dynamical information on the australian solace, on this space here, even by the heterotonic, by the naturalistic heterotonic. So this, what's remarkable is that all systems that you can think about can be formulated in this way if you want. This is a heterotonic constraint. So, the normal Hamiltonian system will have an age of the form energy-minus Equivalently, there is a Hamilton-Giacobbi formulation, but what I'm going to focus on is one particular structure that is there is a Hamilton function. The Hamilton function is a solution of the Hamilton-Giacobbi. By the way, the Hamilton-Jokovian equation here, this formulation, the characteristic Hamilton-Jokovian equation, the principal Hamilton-Jokovian equation, are the same order. We don't have a special time variable. The area is a time variable. It's not necessary, it's not needed to separate it from the rest. So the radical function is a function on two points in the contributing space, so it's a function of space C plus C, which are called boundary space, which is defined as the minimum of the action given the initial and final position. So for a free particle, for instance, is a function of x and t, x prime and t prime, just the action of the physical motion that goes for one to the other. If you know that, you know the motions. Just by derivation, you can compute all the motions. Or you can compute anything about the system. You just write this, so these are constant. And this is a function from the configuration space to the page space, which is made by those, and you have the entire future. Hamilton insisted on the importance of this function. If you read Hamilton's beautiful, it goes over and over how this is the correct way of thinking about mechanics and so forth. Here I'm using it in this sort of covalent language in which I am mixing the dependent and dependent variables. And just to see what is happening in some particular . Let me just show a few examples. a relativistic particle has

20:00 coordinate x and t in c so the Hamiltonian is just this and the Hamilton function is installed here and you can get a partial portion a relativistic particle to formulate it in this way you have a completely low-end invariant Hamiltonian formulation of the particle where the tau parameter plays normal at all when I use this in general at 18 it would be the x and t coordinate that plays normal at all And this is just a simple example which shows that the formulation is genuinely more general than ordinary Hamiltonian mechanics because you can see that there is no standard Hamiltonian system that is equivalent to that because this is a system whose organs are closed, whose is no parallel to the whole. Now let me make the first, somehow, step here by suggesting a way of thinking about quantum theory in terms of that. So the suggestion here is to formulate quantum theory on a header space, which is a header space on the function of C, the function of X and T, okay, and the solution in the function of C, I'm viewing this in both of the people in the solution, so it's the most compact support of functions. The Hamiltonian, if I choose an order, gives a Hamiltonian constraint, which on the state equals zero, will be the will and will equation. And if I have this operator self-rejoint here, I can consider another logic, and of course it's not a projector, but it always comes from every other point of projectors, which is sort of the projectors zeroing in value, that's a thing, which is a well-defined operator from here and from here, because it's the kernel of H that is here, and we need to not say from here to there, in the form of the graph to this way, because it is a projector, zero is a distinct value of H. and the matrix element of this between eigenstates of partial observable operators which I'll define here I call them

22:30 of course if you're working for a particle this is a probability this is a factor of x and tx'y and t' and what you find is just a probability so what's the idea The idea here is that my operators, my part of the mechanical operators, I'll take several joint operators up from the state K. If I want, I can define the physical space in the kernel, but I don't need it. So the operators are operators there. I want to read the spectra as kinematical predictions, because when my devices don't measure complete observables, don't measure observables, they measure partial observables. They associate self-adjustment with partial observables. They sit up there. And the common-legged states of the partial observables are the states, and I have transition I have a quantum event, a measure of q and a t together, and I have transition I have Now, this is just a sense of the writing of quantum mechanics, very much inspired by, but it's very tricky because it treats x and t totally on the same ground it connects remarkably nice to the classical theory because you can show that the chemical function is nothing but the first only approximation in one of each part of the propagator. And of course, if you can find a particular formation, you can write this as a parallelism of appropriate action on all the paths in C that go from here to here. And just a small comment that I will read at the end. if my two graders have this good spectrum, this is always true. In quantum mechanics, the argument of the propagator is not a master variable. It's a label of the eigenstate. If you want the propagator of a particle say in the energy basis,

25:00 if, for instance, a monocidator with a perturbation, the propagator is not WEE prime, it's EN prime N, prime, so from one NJN state to another NJN state. So the argument of W if this is a continuous vector is equal to another, not a class vector. I'll use this later on so let's see. Okay, so far you could say, well, we'll just say this slightly twisted way of common ideas, but let me use this in field theory as post, and this is one of them. The propagator and the Hamilton function are functions of the boundary space, Q and Q prime. So what is the boundary space? The boundary space is just, you have a trajectory, you just cut a final piece of it, and you take the value here and here in the relativistic of the linear space. What the hell are we going to confuse here? Well, why don't we take a solution of the field in all space time? Let me get the final piece of it. So a four-dimensional finite box. And look at the boundary of this finite portion called boundary sigma. And now I want to interpret the boundary space Q to prime as the space of possible value of sigma and phi. the position of this box and the value of the field on this box. Position by the time of the value of the field, this is the value of the field. So the value of the values are now in this case here. And so I can define the chemical function as a function of sigma and phi defined as the value of the action of the field inside, which is a solution equation of motion which is given by, which takes the value p on the value sigma. And in terms of this as partial result, p and sigma . Now all this works, I'm not going into all the mass of that, this is not important. For instance you get all the field equations, sorry, all the solutions to the field equation if you know specifically this here. And the idea here is that I'm treating initial, final, and spash any boundary values in the field uniformly.

27:30 All or equal to. What is the corresponding object in quantum think theory? Well, it's, you fix a, sorry, first let me do filter in Gorski space, but you do, say, 3 and 3, you don't see it, but it's like 3, it's not the space. And I can compute this function in the 3, I guess that this is multiplied and I can compute it. And this is what I would call the field propagator or the field projector. It's a function of phi and sigma, which is sort of a function in the long bulk with phi and value on the value. Now, this quantity satisfies a sort of generalized formulaic string equation, and it can be shown to contain all the information you want about field theory. Without going through that, if you want to compute this particular form or function, you can just compute that in some appropriate sense. If you know this, the value of this function will appear on an appropriate boundary, you contract it with the proper initial final state, and you get that. This is for real in a three-year-old theory, and interacting theory is sort of for real in the perturbation theory. Okay, finally we get to the issue. What about if it was invariant? I've just gone from the endpoint functions to this thing. Now you say, well, you're lost as before, because, obviously, from the definition of this object, if the action and the measure are invariant and if you want, exactly for the same reason from which the end-point functions turn out not to depend on where we put the points, these things turn out not to depend on where we put the box, namely not depend on the shape of the box. So what I have for me, for my definition, is that if the theory is different box environment, this function is a function of the value of the field on the surface but not on the shape of the surface, So I'm lost, because this is supposed to contain information on...

30:00 It's more like you're a scouting experiment, right? I measure the incoming field at a certain time. The shape of the box is the time that has lapsed from the beginning to the end. Okay? So, at first sight, the fact that this does not depend on sigma takes me to the standard problem. But, that's not the case. Because if I'm doing gravity, the field I'm considering is the gravitational field, therefore the method. And if I know the metric on the boundary of the box, I know the shape of the box, I know exactly how much time has lasted from here to there. I know the distance from here to there, because that's what the gravitational field tells me. So, the information about the location that goes out of the door, because most of us come back from the window because I'm considering things at a given boundary. In other words, this correlation, this is a function of all the fields, including the of the field on balance, we implement the identification between geometrical measurement, which is the core of general activity. Let me say it a little bit more seizingly. If I make an experiment, I'm at CERN. I make an experiment, a scatter experiment. What do I have to do? I have to measure cutting particles, which is sort of field measurement on a surface, right? An outgoing particle, the energy of the field in a certain reach time, right, and assume that nothing can be found in the world. But that's not enough. I also have to measure the distance between my detector and the time lapse in the plot that gives me the time lapse from the beginning to the end of my experiments. The first measurement is matter field measurement. The second measurement is geometry, namely gravitational field. So I can capture both information by simply giving all fields on the surface of a closed region. So that's the idea. And to this, I want to add just one step. Remember that I said that the argument of the propagator is not the classical object, it's the quantum numbers of the corresponding gradient. Now, if I believe the theory of luper-mogravity which I mentioned at the beginning, this regionally on the surface is quantized and its eigenstates labelled by the synethoid.

32:30 Therefore, the argument of W for the limitational field is not the classical limitational field, but the synethoid. So this is where I expect the information. information. This is how I expect this information to be coded in order to attract physics from meditation and this multi-variant. And that's what it really gives me in the second transparacy. So this leads me to a big suggestion so we can take that in terms of of these genital states as boundary states, and we have every volume of frequency in our thinking of the physiognomy. It tells the spectra of this directly a single act of deletions, and interpreted the amplitude as a relative probability amplitude for having a certain boundary value. Now, to formulate this roughly, and this probably needs more fork, in terms of gene formulation, what I'm suggesting here is to take, to take the spin forms from which I'm signing to define w of s as the histories, and use as coarse graining the spin networks on the boundary. So coarse grain or all the sum of the material, all the set of histories which have the same boundary, interpret this as the nature. Let me make this complete and then I go to point 2 which is a simple toy model. I'm shifting gear here, I'm now presenting a concrete example of that super-simple. The idea is the following, take four-dimensional quantum gravity, which is what generative equations you want to do, let's go to three-dimensional, thinks everything is very simple, let's take So let me take a regi-tragulation, I think it's still much simpler, so it's traveling space-time in terms of theta-a. Let me take the simplest regi-tragulation, which is a single tetraegle.

35:00 So this is a dynamical theory, which is a single tetraegle, and the only variables are sort of the length of it. But it's just to say roughly how this can be viewed. Now let me just begin by one slide of geometry. Moreover, I'm thinking of the gradle which is equilateral. So it has one side of length A, one side of length B, and all the other four are the same length C. If I work a little bit of geometry and make this equation, we should make the length with the three angles, which are the dihido angles at the end. Euclidean geometry, namely from assuming that this is sitting in a flat space time. There's no dynamics here. But I can think by doing what I said. So I take generativity, I go 3D to a regi theorem, this is Euclidean, and just compute this is the Hamilton function, because this is the action of something which is like . So here I have a dynamical model, which is defined by this Hamilton function, which is a function of a, b, and c. So imagine it is this bit of space-time, you can measure How long is this? How long is this? And the length, and also the angles, and the relation between these, and these relations are determined by this thing. As I use this, and I use the machine to compute the equation of motion, I get exactly these geometrical equations, which were in the terms of the parts, where the momenta are related to the angles, the diagonal values. and here we have the whole equation, the Hamiltonian so I have three variables with the momenta equations that relate the two which are contained in this Hamiltonian or in this Hamiltonian equation but if I look, I can read it differently by choosing one of the variables say c and think of it as time Imagine that I'm viewing this as something which I view at different times, because if

37:30 I thought that this would cost the space, this is exactly the problem I would think. So out of the, so let me call c, which is the length of the side page p, out of the handle function, I can reinterpret all this as a handle function that evolves in time and p from the value of p to the value of a and I compute just by standard formless energy and if I write the Hamilton equations I do get my geometrical equations. But why do I get the geometrical equations? Because the field equations here are the Einstein equations, say the spacetime is flat. The geometrical equations are precisely the conditions that for spacetime being flat. Now this is a one thing, I'm not going to because i don't have much time i'll just say what the point is that i can define the hidden space k in a reasonable way by taking a distribution of the genetic variables in this boundary in this space k i can find the projector this is the next stage. Before doing that, I have operators that correspond to their partial observables, which are the length of these things. So I write these operators and I found out that this is the spectrum, this is the spectrum, so the theory tells me, at the mathematical level, that the length of these things is quantized to those values and the dynamics is obtained here. Again, I don't want to go through that, but the projector projects on a state which turns out to be the six-stage symbols of this state here the ones of things were familiar with the tonzano-range model this is precisely the amplitude used by tonzano-range in the formulation of of 3d quantum gravity on each titrator This is OK if you want to go to phi. Again, this is sort of co-variating, but nobody prevents me to say, OK, but I don't want to think that the c direction is time. And so I want to post this to the . I interpret c as discrete time variables.

40:00 So I can interpret this as the amplitude for going from the state jb to the state ja, where these are states only that knows about the top and the bottom lengths. And I turned the propagator as a propagator from this state to this state. And this is just a little more. The point is that the suggestion here is to use the boundary values to mix initial state, final state, and boundary values, to consider the space that knows about all these quantities here, and to call the dynamics in the unique propagator W, which is a third of these quantities here. mean hidden. And in particular, the gravitational field, one of the side variable which I want to kind of interpret it as a time, right? Because that was it. The field on the boundary of the box, and the field itself, the gravitational field, tells me how much time has passed. So let me get to the last point, and then I'll go over just a couple of first points. And This is just a repetition of what I said before. In the case of flat space field theory, I can compute a four point fraction by picking up a, this is in the case of the V of P, there's a sufficiently long box here and the W associated to this box which depend on the field all around it. And, of course, you have the geometry on this box, which is, say, not long bar and t, where t is the time separation between the first two points and the second two points here. And I just have to fold in these two states, which don't need to know anything about the dynamics. Now, the question is that this is the kind of states that we did not know how to do about gravity. I want to suggest, and this is only a suggestion, I want to be completely to be implemented, so I will be just changing the level of rigor dramatically in the last transparency. I mean, decreasing, not decreasing.

42:30 Here's the procedure for the formula. So, the formula is the formula. Imagine that you have a box again, and on this box I want to consider this connector, which is, I can think in terms of various components upper component, which I call SF, the final bottom component, which I call SI, initial and then a side component, which I call SI which I keep fixing and then I sum the W of this commons connection here with this folding in with these two states here what is the idea? The idea is that each one of this ensemble of S's here correspond to a gravitational field around the box, and people are to a geometry around the box. I keep the side geometry fixed, such that there is a given time, proper time, from here to here, which starts fixing one problem by this higher state of the geometry. There's a given size, L, from here to here. And I know what I mean by being a certain distance from the boundaries this quantity I use the quantity that the spin for models give me explicitly I know this quantity explicitly in a certain configuration expansion and I fold it with states which I know also how to write which I can obtain from the generalized theory which gives me the same particle state in this case viewed as a function a sort of initial and final experiment. Now, this is known up to some things. So in principle, more or less, this is something that could be computed. Now, what I'm saying is not that the particular theory I mentioned is the correct one. It's not that these formulas will work. It's not that there's a problem. But this is a formula that should allow us, this sort of modification, given the backhanding to extract scouting out to at least an ingredient, one of the main ingredients of scouting out of course from that is some work. So all the ingredients, more or less under control, can we go ahead and compute it and see whether this sort of beautiful thing, and some of us like a lot, could be used to get scouting out.

45:00 Let me summarize. So I've shown many ideas. First of all, the main ideas that think about classical dynamics, not in terms of time evolution of observables, but in terms of relation between partial observables, all on the same ground. So I've shown that one can do Hamilton formulas in that theory sort of easily, and in the second suggestion that in the case of field theory, the partial observance can be taken on the boundary of the finite region of say, in the quantum theory, instead of trying to instead of trying to look for the physical in the space, the suggestion is to stay at the sort of angle straight space, where the wind equation is defined, and look at the ways of computing matrix elements of the projector between the eigenstates of the partial observance and then the rectumius probability here. In gravity, the third idea that in gravity, if you play this game, the argument of this object does not depend on the background geometry, but it doesn't matter, because you have control of the non-background geometry, which is what you need. It's a physical thing that you measure when you measure time in an experiment in which you use a clock. And that if you believe the argument is not the gravitational field but is a spin-form. You can attach to this a temporal interpretation by using side time, but temporal interpretation is not necessary. And then I've given this formula that, hopefully, my take toward computational scattering activity starting from . Thank you. So, you have to pick this box that you drew in space-time.

47:30 So, there's an initial surface and a final surface and these side surfaces. And presumably, the geometry on the initial and final surfaces are Euclidean. and the induced geometry on the side surfaces is Lorentzian, particularly because you want to assign some kind of proper time to the, you know, to the, to the, between the initial and final surfaces. And then you have this spin network, S, which is sort of all over the boundary. So the side S, the side spin network, are you imagining that somehow a Lorentzian spin network on the side to make the connection with the box and space line and to allow you to give this proper time to calculate a proper time is that what this is a question yeah, I finish with this question because I've been on purpose vague on Lorentzian versus Euclidean formulation of this in two different ways. One way is what you just exactly described, and in order to do that you have to have a clean notion of Lorentz as we did. You can play it in a different game, which is to take a Euclidean theory and see what happened with Euclidean theory, and then you don't have that problem, you can just go ahead. In the theory I referred to, you can believe Stephen that the only sensible way of doing portal gravity is to do your killing and then if you do that um you still have a t which is a length now from the side and you can presumably ask yourself using t bar on the side you can ask yourself whether or not serialized you have an analytical motivation in this i'm not saying in any way that i'm addressing that form the analytical What I'm saying is somehow the spirit of this is just all of us disagree on how exactly this thing would be. And all of us know, in a theory, how to actually get to Ws. So I'm suggesting a machine that could be used in different ways. And it could be used either in the Euclidean context or in an enzyme positive.

50:00 So, of course, there's possible problems. It's a standard problem, is what it is again. John? I think, as you know, you've read the formulation of what the problem is. As I used to put it, the great problem of all quantum theory is how to do a probability amplitude for some process. The process, you create something here and now, and then you want to know when you go there and then, and therefore the here and the now and there and the then are part of the question in all the rest of physics. In general relativity, the here and the now are part of the answer, not part of the question. That's the great problem. We can't disentangle the two in that way. And you'll formulate that in terms of the box, and the box is now part of the answer, not part of the question. It's a brilliant way of putting this. And therefore, my questions about this are not meant to demolish what we do, but to hopefully find a way to overcome these problems because this is a formalism and obviously we want to connect that up as you yourself have to begin with measurements so the question is how to create the conditions which produce this bottom top of the box let's say the initial screen network and final screen network how do we do that first of all what physical process to do it, and how do we isolate the size of the box from all other influences, particularly when we realize that to do this, we actually have to use some physical system. And every physical system that you know in general has a stress-energy tensor attached to it, and therefore we can't prevent it from leaking into the box, so to speak, because there's no shielding for gravity. I mean, if you look at what Paul Rosenthal do, when they started of the measuring problem for the right-wing magnetic field, sure, you have to introduce test bodies to see how the field reacts on those, but you can, first of all, you can turn this, you can manipulate the test body by non-alcoholic magnetic processes, and you can separate positive and negative charge and then rejoin them in such a way that you can turn on and off the field. You can control the influence they have on the box and subtract that, but not to say there is no influence of the body to use within the main, but you have to control it in the German sense, living, know how to take the input now. But in general relativity, we have no shielding, and it's not clearly how we can control the influence that we can't prevent from leaking

52:30 in. So in a sense, can we just do it on a three-dimensional slice, or do we need some sort of four-dimensional approach to subvention? That's my fundamental problem. I have a second question, and I'll leave that aside. We have actually a way of cleaning asymptotes, which is much better than the perturbation method, maybe what I've got here is a kind of quantization. You can do a clean definition of gravitons there as representations of the BMS group, but we have to work on no hyperservices. And therefore, I think if you want to connect this up with something where we now have to do graviton cleanly, you have to find a way of treating your initial and final hyperservices as no hyperservices, ultimately taking a stride. So I think this form must be generalized with space-like to know. Yeah, the second question is sort of easy at once, Yes, that's possibly one direction to go. I mean, that's where we answer. The first question, I see the problem that you're saying. Of course, this is a problem in sort of measuring between quantum gravity. I see the point that you're raising about four-dimensional sphere and three-dimensional sphere, but you're also raising your text, which I don't have anything to say except that yes, it's a problem in a quantum system. Here, the idea is, suppose I idealize a scattering experience, thinking that between the initial and the final time, there was a total time, T made by some clock and so on, that the initial state is a Minkowski plus two particles, with two granities of the new world. There's a second part, but I think the first way I handle scattering is to the null. Right, so in that, under those assumptions, which are very heavy, and big approximation of reality, how can I put those data in my theory, in my theoretical apparatus, and really a piece of something I compute to compare it with, so. Chris? Yeah. I'm not trying to fully follow this, but suppose you want to apply this to quantum cosmology, where you didn't have a scattering situation at all, so it's a compact three-manacole with no boundary. Can you do that? Is it just a trivial problem? Yeah. So there's no initial, it was just a three-manacole, that's all.

55:00 Yeah. Yeah, in that case I will be in the, yeah, I don't have the box in a sense, I have, I am in the, in the case in which my boundary is, you presumably want to compute the, to worry about the entire spatial universe, right? Yeah, in that case I will have an initial geometry and a final geometry, yeah. I'll have an initial one, in fact you're already responsible for creation. Or I'll just have a given spatial, then this is the standard, how do I say, for hard-locking. Yeah, sort of thing. Yeah, yeah. If you want... Carl, isn't that what that is? I mean, the thing that you've drawn? No, no, no, no, no, no, no, no, because... In the Euclidean framework, if everything is Euclidean, all the interior things that you sum over in Euclidean, your boundary data is Euclidean, that is the... Yes, and I have a machine, a theory that works the Planck scale. So interesting what happened in a small region of Planck scale, where I can do perturbations starting from the small. And I have high energy particles that interact, and I want to walk away from that. and to view two particles coming in and two particles coming out. Now, I don't want to think in terms that this is the entire universe. I don't want to think of the entire universe through a calculation of quantum gravity. I have nothing against thinking in terms of the entire universe. This is a correction of time. You want to separate the quantum gravity and the quantum cosmology. Yeah, this is, in a sense, not quantum cosmology, only because it's a different thing. there are two scalar problems the mutual that interact only they come very close, they come out so what do we have? I can compute it in perturbation theory but perturbation theory has two parameters I want to make a box very very small just a few blank size at the minimum needed to compute it like QCD people do when they do their lattices which are as small as possible to keep the problem inside then to do computation, I hope to be able to do it in perturbation theory, so I'm not thinking of the universe at all, I'm thinking of a small region in space, and I'm hoping to be able to compute scouting out from this side.

57:30 One last question? This is too closely related, so first, is there some difference between you've done in the following, except that you have a discrete version, just choose any boundary geometry you like, fill in with an arbitrary core geometry, lose it half integral over all internal core geometries holding the boundary fixed, and call that some amplitude associated with the boundary geometry. You're proposing a discrete version of that. The second, so the related question is that, that's exactly the rule, is that, are you in relation to loop quantum gravity dropping the Hamiltonian constraint? No. Because you've been talking about spin networks as possible states on the boundary, and they're dipheomorphism invariant, but they're not... The status of them with respect to Hamiltonian constraint is not settled, because one doesn't know what it is, even. So what role is that playing in this boundary? I am replacing the information the Hamiltonian constraint with the information the W. That's what I meant, yeah. So you're doing, you're saying, forget the Hamiltonian constraint, and use a spin, using a spin phone or something, instead of... Yeah. Well, the W can be thought as the delta of the Hamiltonian constraint. So, if all stayed together the way we would like, you could compute the answer by ex-associating the internal expression. It's like, I'm saying, it's just fine that it's for things, right? You can forget the internal and then use the UO printer, the evolution printer. And then you don't have to think about the internal anymore. And the evolution later goes from initial time to final time. I'm just rephrasing this in language in which the time is part of the environment. So I'm not changing the structure of quantum mechanics in any sense. I'm just providing it slightly. But I think you're re-interpreting the Hamiltonian state in this different way. Instead of trying to realize it from an algebraic. No, no. If you apply this to a normal theory, it becomes the normal thing.

1:00:00 It doesn't need a Hamiltonian straight, any mechanism that gives him the sum over all the processes in between. You'll say W would satisfy the Hamiltonian straight. Yeah, W would satisfy the Hamiltonian straight. Let's thank Carl. Thank you.