How to extract physical predictions from a diffeomorphism invariant quantum field theory

0:00 Baudelaire, 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 the people who know me believe that Chris 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 the speakers, that Chris was not my thesis advisor. But there is a reason for which a lot of people think, my friends think that, because indeed, certainly, Chris is the person who had the most influence on my life, I would say, 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 about the world that I found in the development of classical mathematics and field theory, especially in mathematical and quantum mechanics. There was at least an aspect of modern physics that I saw around me in which this was lost in some sense, there are 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 the 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 basically all my life. Go in that direction, and when I was a PhD student and he came here, in a sense my life as a student started in this building, I came to see Chris, I remember one day I arrived, it was absolutely nobody, people told me it's a bank holiday, you're not from this country, you don't understand why it's a bank holiday or physics holiday. And then things started from there. The first time I thought I did something interesting in physics, what I did with my collaborator was first to go see a bio, and then immediately take a plane and come to London to see whether Chris would say or not, and so on. So Chris, thanks for all, thanks for your advice. I sort of apologize to everybody because this is a talk that I prepared in a sense thinking about it. It's not a big review thing about past work. It's not a big review about quantum gravity which is a theory which has most of the work.

2:30 First, because I thought, please, know all of that. Second, because Abhayashikar is speaking after me, and I believe he will explain the basics of the theory of quantum gravity and many of the consequences that have been worked out for it. And third, because I wanted to present a number of ideas, suggestions. That I hope could interest Chris. So in a sense I'm giving for granted everything that I will say for the next talk. I will assume that you know everything about nucleotidization and I will discuss a problem and so I will begin by carefully explaining what is the problem that I will discuss which is how to extract physics from the theory And then, a large part of my talk will be just point one, which is sort of present a sequence of ideas, which I hope... All of this could help us to the point of a general technique for extracting physics from theory, and then 2 and 3 will be rather fast. 2 is just a simple model, a toy model, in which these ideas are implemented, and 3 is just a suggestion, it's not much of a result, it's just an equation I will present on the fly, in which I'll say, okay, here is a way I think calculations could be done. The last problem of this lecture was using all the material in this lecture to find the consistent, intervalent, finite, and the devolves invariant. Background-independent quantum theory of gravity, and in a sense, many of us have been working on that problem all these years, and to some extent, I think that there is a theory, which is super-violet-violet, or at least is most presumably super-violet-violet, and it is certainly background-independent, and it is a quantization of general relativity, plus meter. So, in a sense, we have a theory of quantum gravity, but of course there's a but.

5:00 So what about? First of all, we don't know if this is correct, it might be all wrong, but one of the reasons we don't know if this is correct is that the actual theorem exists in a number of different versions, and in one way or the other they are all incomplete, and... Why can't we compare these two versions of the theory? Why can't we completely choose between them? Basically because we can do certain calculations with the theory. We can compute spectral parameters, we can discuss the whole entropy, we can discuss cosmology, we can do a number of things. But this sort of specificity, what we cannot do, what we don't have, is what we have in QE, just a machine for computing systematically. 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, there is no background space time. How do we compute scattering? How do we concretely compute, if I have two particles that interact, what comes out? So this is all I'm saying about loop quantum gravity. If you know loop quantum gravity, just to recognize what I'm talking about, if you don't, it doesn't matter because I'm not really then using any of this, you know, just hammering back to this last component here. What I'm saying, just pick a theory, pick a version of the theory and... There is a version of 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 through geometries, or through geometries which are discrete on the Planck scale. And in the formalism there are certain area and volume operators 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 the area and volume of the spaces.

7:30 So there is this mathematics and besides it we can have the way of defining amplitudes associated to those states here using for instance a A Barrett-Cray model, BCC comes from, is the terminology, one particular Barrett-Cray model, the terminology 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. So this amplitude is roughly The functional integral on four geometries of the action integrated all four geometries which are bounded by a three geometry described by this spin network here. This spin network 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 expansion. We have this now in plain form. The calculations are complicated by order by order in lambda in one of these formulas. And the way it's actually computed is using an auxiliary field theory as an expectation value of an auxiliary field theory. Again, it's not important for what I'm saying, but what I want to stress is that there is a human space with a basis of states, with a roughness of geometry, with a way of associating an amity to it, which, if the theory was right, could be thought as the foundational integral. Okay? So suppose we have that. Suppose we are given this answer. How do we continue? In particular, how do we look at it? Now, if I take the derivative generativity, quantum generativity, I can do, I can compute scattering, right? The theory 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. So if I believe that this is correct in some sense, what the finite theory of the Planck scale should give me is fix those parameters.

10:00 So in a sense, I can say that my Non-paternity theory on a scale, I understand it, I control it, if it gives me a way of computing the three parameters of the normalization of the theory. And where is the difficulty? Well, everybody who has worked with non-paternity quantum reality knows it. This is just one way of putting it. The key difficulty is general coordinates. It's just the core of generativity, the background in penance. Scanner used to use them for functions, but if this theory here, this action and this measure have different environments, this one is independent of x1 or xn. In other words, in natural motion environment theory, I don't know where my endpoint function sits, 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 problem addressing. And I am going into this problem through a long detour by presenting a number of ideas Certainly, a lot of these durations go from some of the geometries and from a gene part of the life of the mechanical thing which is connected to it. So let me start from the very beginning, because I think the problem is in the classical theory, right? And in classical GR, of course, canonically, Chris told you to think first, right? Gauge invariant, Dirac observable, and the theory. I don't know any general relativity. I don't know quantities that commute with a cumulus display, which 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 I think about the general covariant mechanics? And please follow me through a number of steps and I'll get back to the question and answer. 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, I'm going to call it Q, and make a list of values, but of course that's not enough, I don't do physics this way, I also have to have a clock, and measure the time, the angle between the hand of the clock and the 12, and so I have two quantities, which I am

12:30 Consider Q and D. And then I make a table. I have to measure the two together. I'll make a table of various couples of the measurements and a series about those tables. Now what I want to do, what we usually do is say, well, T is my external time parameter that involves this static Q as an initiative. 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. To the end. So let me just define, let me just call Q and T partial observables and a couple, a correlation 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, right? So the relativistic configuration space is two dimension. Now, if I make an ensemble of measurements, I get a set of points, a set of capitals, a set of points here, which generically will sit on the submanifold of C, on 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 stand with this? This is the answer 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 possible lines is infinite, in reality what I find... Spirits tell me that the number of possible motions is very small. In fact, it's a two-dimensional space of motions. So, this is what I'm saying, if I disturb the pendulum and experimentally observe a liquid motion, but the number of possible motions is small, then we call gamma the space of the possible motions. In this case, the motion, the space of course is being two-dimensional. And is labeled, for instance, by the amplitude and the phase.

15:00 By the amplitude and the phase, I fix the motion. So I call A and B the amplitude. So all the information about the system is for any choice of the motion, amplitude and phase, are in relation to the function f. So why is the function f, which is... From the example of the spatial space, the physical space is nice, the phase space is warm, and the vanishing of it is its own information I have about the system and for anyone of course. Now, what I claim is that this structure is general. You can think any physical system in this way. You can always think that you have a list of partial circles. Which you do as coordinates on a relativistic or linear space, and then you have a set of states. And your physics is defined by the management of a function on it. I have not chosen a particular variable as time, I'm not viewing evolution of this system as evolving in time, I'm not requiring in general that these motions don't come back in time, for instance, and so on and so forth. A way to think about systems, and why do I want to use it, because generally these 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 become easier, nicer, at least... from some point of view, and practical 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, this Hamiltonian, which is a function of the cotangent space of the relativistic wave. So this is one function of position, time, momentum, and energy. And you can in general say that F motions are given by a variation of principle, which is the minimization of an action, where what you're minimizing is just the math, with no specific dynamical information, on the constraints of this space here, given by the anatomistic anatomical.

17:30 What's remarkable is that all systems that you can think about can be formulated in this way. If you want, this is a Hamiltonian constraint. So, the normal Hamiltonian system will have an age of the form energy minus normal characteristic Hamiltonian. Equivalently, there is a Hamilton-Jacobi formulation, but what I'm going to focus on is One particular structure that is there is the Hamilton function. The Hamilton function is a solution of the Hamilton-Jokowi equation. By the way, the Hamilton-Jokowi equation here, in this formulation, the characteristic Hamilton-Jokowi equation and the principal Hamilton-Jokowi equation are the same object, right? Because you don't have a special time variable. The idea is that the time variable is not necessary, is not needed to separate it from the rest. So the angle function is a function on two points in the contiguous space. So it's a function on the space C cross 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, it is a function of x and t, x prime and t prime, just the action of the physical motion that goes from one to the other. If you know that, you know all the motions. Just by derivation you can compute all the motions. Or you can compute anything about the system. And you have the entire future. Now this, of course, Hamilton insisted on the importance of this function. If you read Hamilton's beautiful, he 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, it's not a particular topic here, it's kind of, let me just show a few examples. A relativistic particle has coordinate x and d, a, and c, so the Hamiltonian is just this.

20:00 And the Hamilton function is solved in here, and you can get the ratio portion. A relativistic particle, if you formulate it in this way, you have a completely low-end invariant Hamiltonian formulation. And this is just a simple example which shows that the system is just the formulation is genuinely more general than ordinary mechanics because we can you can see that this is there's no standard system That is equivalent to that because this is a system whose motions are closed, so there is no kind of argument for it. Now let me make the first 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 the... There is a Hilbert space, which is a Hilbert space of every function in C, that's a T, okay, and as you can see, I'm doing this in both of the people with these most compact support functions, the Hamiltonian, if I choose an order, gives a Hamiltonian constraint, which on the state equal to zero will be the Willett-Witten equation. And if I have this operator self-adjoint here, I can consider an object, which I call a projector, which is sort of the projector of the zeroing-in value of the thing, which is a well-defined operator for here and here, because it's the curl of H that is here, and the unit is not the same for me. The matrix is an element of these, between agent states of the partial observable operators, which I'll define here, I call them propagator.

22:30 Of course, if you work it for a relativistic particle, this is the operator. This is the function of x and tx, y, and t prime, and what you find is just... So what's the idea here? The idea here is that my operators, my quantum mechanical operators, affect several joint operators up from the state k. If I want, I can define the physical unit space as the kernel, but I don't need it. So the operators there... I want to read this picture as kinematic contradictions because when my devices don't measure complete observances, don't measure observable things, they measure partial observances. Common eigenstates of the partial observables are the states and I have transitional amplitude between those. So the common eigenstates, I can call them a quantum event, a measure of q and a t together, and I have transitional amplitude between the two. Now, this is just a sense of the writing of quantum mechanics, very much inspired by, if you want, Feynman-Hipp's pages. But it's very pretty 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 Hamilton function is nothing but a sort of first-order approximation in one of each part of the... And of course, if you try to find a particular formulation, you can write this as a parallelism from the action on all the paths in C that go from here to here. And just a small comment that I will leave at the end. My cube vectors have discrete spectrum. This is always true. In quantum mechanics, the argument of the propagator is not a vast environment. This is a label of the other states. If you want a propagator of a particle, say in the energy basis, for instance a harmonic oscillator with a perturbation, the propagator is not W of E E prime, it's E N prime N, or N N prime, so from one energy state to another energy state. So the argument of W of thesis of continuous spectra is a quantum number, not a class number. I'll use this later on.

25:00 Okay, so far you could say, well, we were just saying slightly twisted way of common ideas, but let me use this in few serious forms, and this is one of them. The propagator and the Hamilton function are functions of the boundary space, right, q and q prime. So what is the boundary space? The boundary space is just, if you have an ejector, you just cut a finite piece of it. Then we take the value here and here in the relativistic model. What else are we going to confuse here? Well, when we take the solution of the field, norm space, let me add a final piece, so a four-dimensional finite box and look at this boundary of this finite portion, we call the boundary sigma, and now I want to interpret the boundary space Q to time as the space of the possible values sigma and phi, so of the position of this box and the value fulfilled on this box, the position of the time and the value fulfilled. So the boundary values are now the space here, and so I can define the Hamilton function as a function of sigma and phi defined as the value All the action of the field inside, which is a solution of the equation of motion, which is given by, which takes the value of P on the value of sigma. And in terms of this as a partial term, P and sigma, you want to . Now all this works, I'm not going into all the math of that, this is not important. For instance, you get all the field equations, sorry, all the solutions of the field equation if you look specifically at this one here. And the idea here is that I'm tweaking initial, final, and boundary values of fields uniformly, if it's all or equal to.

27:30 And what is the corresponding logic in quantum field theory, right? It's, you fix a surface, and then you filter it across the space, and you do, say, Euclidean-free field theory. And I can compute this function literally in the Euclidean case. 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 of integral bulk with given value on value. Now, this quantity satisfies a sort of generalized combinator 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 for a function, you can just compute it out of that in some appropriate sense. If you know the value of this function here on an appropriate boundary, you contract it with the proper initial final state, and you get that. This is for real in a theory, and interacting theory is sort of for real in perturbation theory. Finally, we get to the issue of what about if it was invariant. I've just gone from the endpoint functions to this field thing. Now you say, well, we are lost as before. Because, obviously, from the definition, we want to fulfill this object. If the action and the measure are invariant and if you're off, isn't it? Exactly for the same reason from which the endpoint functions turn out not to depend on where to point. These things turn out to not depend on where I put the box, namely not depend on the shape of the box. So what I have, formally, from the definition, that if the theory is different from the environment, this function is a function of the value of the field on the surface, but not on the shape of the surface, not on the geometry of the surface. So I'm lost, because this is supposed to contain information on...

30:00 Suppose I do a scattering experiment, right? I measure the incoming field, I measure the outgoing field at a certain time. The shape of the box is the time that has lapsed from the beginning to the end. So, at first sight, the fact that this does not have a 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 it's better. 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 it flows inwards, comes back through the window, because I'm considering things at a given boundary. This correlation, this is a function of all the fields, including the incremental identification between theoretical measurement and field measurement, which is the core of generativity. Let me say it a little bit more seasonally. 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 incoming particles, which is sort of field measurement on a surface. And outgoing particles, filmed by the energy of the field in a certain region, assuming that it can be bounded. But that's not enough. I also have to measure the distance between my detector and the time-lapse and the clock that give me the time-lapse from the beginning to the end of my experiment. The first measurement is matter field measurement. The second measurement is geometry, namely gravitational field. All fields on the surface of a closed region. That's the idea. And to this, I want to add just one thing. 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 loop quadrogram, which I mentioned at the beginning,

32:30 On the surface, it's quantized and its eigenstates are labeled by the C-network. Therefore, the argument of W for the gravitational field is not the classical gravitational field, but the C-network. So this is where I expect the information, this is how I expect the information to be coded in order to apply physics to the gravitational field. And that's what the theory gives me in the second transparancy. So this leads me to a suggestion, so we can take that in view. In terms of this numerical state of the boundary state, and we have a very volume of great context in how we think it can be, in terms of the spectra of this directly estimative deletions. And interpret the amplitude as a relative probability amplitude for any certain boundary value. Now to formulate this roughly, and this probably needs more thought, in terms of gene formulation, what I'm suggesting here is to take, I can skip back to the region of transparency, to take the spin force 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-graining all the sum of the interior, all the set of histories which have the same boundary, and interpret this as the measure. Let me make this complete, and then I go to point 2, which is a simple term model. I'm shifting gear here. Presenting a concrete example of that super-simple. The idea is the following. Take four-dimensional quantum gravity, which is what we generally teach, which you probably don't want to do. Let's go to three-dimension. I think it's a very simple thing. Let's take a rigid triangulation, still very simple, so it's traveling in space-time in terms of theta, eta. Let me take the simplest rigid triangulation, which is a single theta, eta.

35:00 So this is a dynamical theory, which is a single tetrahedron, and the only variables are set to the length of it, but it's just to say roughly how this can be viewed. Now let me just begin by one flight of geometry. Moreover, I'm taking a tetrahedron which is equilateral. So it has one at length a, one side at length b, and all the other four the same length. If I work a little bit with geometry, I make this equation, which relates the length with the three angles, which are the dihedral angles. This is just from Euclidean geometry, namely, from assuming that this is sitting in a flux space. There's no dynamics here. But I can think with the dynamic by doing what I said. So I take generativity, I go 3D to Euclidean, Boundary, regi-action, or the same of the gradient, when this is the Hamilton function, because this is the action of something which is that boundary, then I do all the calculations I get for this thing. 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 that there is this bit of spacetime, you can measure spacetime, how long is this, how long is this, and the length, and also the angles. And there are relations between these, and these relations are determined by this thing. If I use this, and I use the machine to compute the equation of motion, I get exactly this geometric equation, where the momenta are related to the others, to the diagonal. And here, we also have the Combe equation, and the Denham tutorial. So, I have three variables. With their momentum, equations that relate the two, which are contained in this Hamilton equation, but if I want, I can read it differently by choosing one of the variables, say, z, and think of it as time. Imagine that I'm viewing this as something which I view at different times, because if I thought that this would cost space, this is exactly the proper time of order.

37:30 So out of the, so let me call c, which is the length of the side b to the t, out of the Hamilton function, and I reinterpret now this as a Hamilton function that involves some time t from the value of p to the value of a, and I compute just by standard formulas the energy, and if I write the Hamilton equations, I do get my geometric equations. But why do I get the geometric equations? Because the field equation here and the Einstein equation say the space-time is flat, and the geometric equation emphasizes the condition of that for space-time being flat. Now this is a quantum theory thing, I'm not going to do that because I don't have much time, I'll just say what the point is that I can define the Hilbert space, k, in the... These are the ways I take the distribution of the geometric relativistic variables from this boundary. In this space A I can find the projector, my projector operator. This is the next page. Before doing that I have operators that correspond to the partial observables, which are the lengths of this thing. So I write these operators. And I found out that this is a spectrum, this is a spectrum, so the theory tells me, on a mathematical level, that the length of this thing is quantized to those values, and when the analysis is obtained here, again, I don't want to go to that, but the projector projects on a state, which, if I do the computation, turns out to be the six J symbols of this state here, the ones who feel more familiar with the... The Poinzano-Ranger model is made for formalists who recognize that this is precisely the amplitude used by Poinzano-Ranger in the formulation of 3D quantum gravity on each tetrahedron. This is okay if you're going to feel it. Again, this is sort of corporeality, but nobody prevents me to say, okay, but now I want to think that the c-direction is time. I interpret C as a discrete time variable, so I can interpret this as the amplitude for going from the state jp to the state ja, where these are states only that knows about the top and the bottom lengths.

40:00 And I interpret this often 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 Hilbert space that knows about all these qualities here, and to call the dynamics in a unique propagator W, which is a function of these qualities here, and inside here there is anything we need hidden. And in particular in the validation of field, one of the side variables that I want, I can interpret it as a time. Because that was the idea. 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'm over, just a couple of times. This is just a repetition of what I said before. In the case of flat-space field theory, a four-point fraction by picking up a sufficiently long box here, And the W associated with this box, which depends on the field of all the mathematics, and of course, it depends on the geometry of this box, which is, say, log r and... And finally, T, where T is the time separation between the first two points and the second two points here. And I just had 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 things that we do not know how to do in reality. I want to suggest, and this is only a suggestion, only to be completely implemented, so I will just change the level of rigor dramatically with the last transparency. I mean, decreasing them, not increasing them.

42:30 Here's the procedure of the following formula. So, the formula is the following. Imagine that you have a box again, and on this box I want to consider a skin network, 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, Which I keep fixed. And then I sum the W of this common exponential here with this folding 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. Time, proper time from here to here, which is toxic. Fixing one problem by this eigenstate is a challenge. There's a given size L from here to here. And I know what I mean by it 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 perturbation expansion, and I fold it with states which I know also how to write, which correspond, which I can obtain from the generalized theory, which gives me the same two-particle state in this case, viewed as a function of the sort of initial and final spin-in. Now this is not 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 this formula is going to work, it's not that there's a problem, but this is a formula that should allow us this sort of multiplication of the idea, given the background independent formulation, to extract Scattering up to at least an ingredient, one of the main ingredients of scattering up, 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 in front of you, that some of us report to reality like some of us like a lot, could be used to get to scattering up.

45:00 Let me summarize. So, I've thrown in many ideas. First of all, the main idea is that thinking of classical dynamics not in terms of time evolution of observables, but in terms of relation between partial observables all on the same ground, I've shown that one can do high performance in that sphere 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 a finite region of space-time, in the quantum theory, instead of trying to solve the electron, instead of trying to look for the physical in the space, the suggestions stay at the sort of bundle straight space. And look at the ways of computing matrix elements of the projector between the eigenstates of the quantum surface and then directly us. In gravity, if you play this game, the argument of this object does not depend on the background geometry, but it does matter. Because you have control of the known background geometry, which is what you need. It's the physical thing that you measure when you measure time in an experiment in which you use a clock. And that if you believe quantum gravity, the argument is not the gravitational field but is the spin form. You can attach to this a temporal interpretation by using side time, but temporal interpretation is not necessary. This is my take towards computational scattering after it started from Euclidean.

47:30 Can you pick this box that you drew in space-time? 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, those geometries are Euclidean. And the induced geometry on the side surfaces is Lorentzian, particularly because you want to assign some kind of proper time between the initial and final surfaces. And then you have this spin network S, which is all over the boundary. To make the connection with the box of space-time and to allow you to give this proper time, you know, to, is that what? This is the question? Yeah, well, yeah, I finished this question because I've been on purpose vague on Lorentz and versus Euclidean formulation of this. You can play this game in two different ways. One way is what you just exactly described. And in order to do that, you have to have a sort of clean notion of correctness. You can play it in a different game, which is to take a Euclidean theorem, and then you don't have that problem, you can just go ahead with it. In the theorem I referred to, you can believe Stephen that the only sensible way of doing it If you do that, you still have a T, which is a length down from the side, and you can presumably ask yourself whether or not C allows you an analytical opportunity. I'm not saying in any way that I'm addressing that problem. What I'm saying is somehow the spirit of this is just that all of us use a video on how exactly to think. Even a theory of how to actually get to W's. So, I'm suggesting a machine that could be used in different ways, and could be used either in a Euclidean context or in a Lorentzian context.

50:00 So, of course, with possible problems, it's a standard problem, is what it is, really. Well, as you know, the rigid formulation of what the problem is, as I used to put it... The great problem of all quantum theory is the probability of the attitude because of process. The process, you do something, you create something here and now and then you want to know when you go there and then, what can you, what will you find? And therefore the here and the now, the there and the then are part of the question in all the rest of physics. In general relativity, the here and the now... I'm part of the answer, not part of the question. Yes, that's the problem. You can't disentangle the two in that way. And you'll formulate that in terms of the box. The job of the box is now part of the answer, not part of the question. It's a brilliant way of putting it. 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 it up, as you yourself mentioned, with measurements. So the question is, How to create the conditions which produce this bottom top of the box, let's say, the initial spin network and the final spin network. How do we do that? First of all, what physical process do we do it? And how do we isolate the sides 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, but 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 did, When they set up a measuring problem for the electromagnetic 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 the middle of the test body by non-automatic 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 it out, not to say there is no influence. But in general relativity, we have no shielding, and it's not clear to me how we can control the influence that we can't prevent from leaking in.

52:30 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 some measurements? That's my fundamental problem. I have a second question, I'll leave that aside. We have actually a way of treating isotopes, which is much better than the perturbation method, namely what Abhay gives, asymptotic quantization. You can do a clean definition of gravitons there, as representations of the BMS rule, but we have to work on no-hyperservices. And therefore, I think if you want to connect this up with something where we don't have to do gravitons cleanly, you have to find a way of treating your initial and final hyperservices as no-hyperservices, ultimately taking a scribe. So I think this form must be generalized to a 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 a great start. The first question... I see the problem you're saying, of course this is a problem in sort of measurement of quantum gravity, I see the point that you're raising about four measures greater than three measures greater, that you're also raising in your text, which I don't have anything to say except that yes is a problem in quantum physics. Here, the idea is, suppose I idealize a scattering experiment, thinking that between the initial and the final time, there was a total time t measured by some block, etc., that the initial state is a Minkowski plus two particles, the two gravitons of the new world. Right, so in that, under those assumptions, which are very heavy, and big approximation of vectoriality, how can I put those data in my theory, in my theoretical apparatus, and create a piece of something I compute to compare it with? Suppose you want to apply this to quantum cosmology, where you didn't have a scattering situation at all, so you say a compact 3-manifold with no boundary. Can you do that? Is this a trivial thing? So there's no initial discussed 3-manifold, that's all.

55:00 In that case, I will be in the... I don't have the box, in a sense. In the case in which my boundaries, 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, no? I want an initial one, you're directly responsible for creation. Alright, or I just have a given spatial, then this is a standard, how do you say, for hard-locking. Yeah, absolutely. Yeah. If you want... Karl, isn't that what that is? I mean the thing that you've drawn? No, no, no, no. In the Euclidean framework, if everything is Euclidean, all the interior things that you sum over are Euclidean, your boundary data is Euclidean, that is the... I mean 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 perturbation starting from the small. And I have high energy particles that interact, and I want to walk away from that. So I want a small region of space-time and to view two particles coming into part of the world. Now I don't want to think in terms that this is the entire universe, but I want to think of the entire universe through a calculation of random gravity. I have nothing against thinking in terms of the entire universe, but this is a correction of time. You want to separate the problems of quantum gravity and quantum cosmology. Yeah, this is, in a sense, no quantum cosmology, only because it's a different subject. There are two scalar problems, or neutrals, that interact only gravitationally, right? They come very close, they come out. So what do I want? What is quantum gravity? I can compute it in perturbation theory, but perturbation theory has three parameters. Can I get those parameters? I want to make a box, very, very small, just a few blank sides. 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, and then to do computation, I hope to be able to do it in perturbation theory, so I'm not thinking of the universe as a whole, I'm thinking of a small region in space-time, hoping to be able to compute scattering evidence from this.

57:30 You feel what 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 coordinate geometry, lose the path integral over all internal coordinate geometries holding the boundary fixed, and call that some amplitude associated with the boundary geometry. Are you proposing a discrete version of that? The related question is then... That's exactly it. Are you in effect, in relation to loop quantum gravity, dropping the Hamiltonian constraint? Because you've been talking about spin networks as possible states on the boundary. The status of them with respect to them is not settled, because one doesn't know what it is either. So what role is that playing in this boundary? I am replacing the information in the Hamiltonian constraint with the information in W. That's what I meant. So you're doing is saying forget the Hamiltonian constraint and use a spin. The W. Yeah, but the W can be thought as the delta of the Hamiltonian constraint. If all stayed together the way we would like, you could compute the amplitude by exfoliating the internal space. It's like, I'm saying, it's just finally for things, right? You can forget the Newtonian and use the Yule, the evolution of Newton. And then you don't have to think about the Newtonian anymore. And the evolution of Newton goes from initial time to final time. So I'm not changing the structure of quantum mechanics in any sense, I'm just rewriting it slightly. But I think you're reinterpreting the Hamiltonian state in a different way, instead of trying to realize it from an algebra. No, no, if you apply this to a theory, if you apply this to a normal theory, it becomes a normal thing.

1:00:00 It doesn't need a Hamiltonian state, any mechanism that gives it the sum of all the processes in between. Gilson W. would satisfy the Hamiltonian-Australian relationship. Yeah, that would satisfy the Hamiltonian-Australian relationship. Well, do you want to continue? Okay, let's thank Carl.