The Problem of Time in Quantum Gravity

0:00 I want to talk first about general relativity, then about quantum theory, and then about the business of putting them together. And to survey the philosophical problems or conceptual problems that it confronts, at the end of this first talk, I will discuss in just a slide or two what the problem of time is. In the second talk, which I'll immediately go on to, I imagine, I'll say more about the problem of time. The curious thing is that the very content of some of my slides matches without deliberate collusion between me and Troy the content of some of his. So actually his graphics are, as you'll see, better than mine. So in any case, in general relativity, needless to say, in this audience, you have a theory of space, time, and gravity, a manifold M, a metric gamma, Greek letters for indices 1 to 4, defining length, time, and causal connectability of points, an equation relating the geometry and the configuration of matter, and boldly saying, as Slides force one to do in elementary talks, so one sort of risks oneself, broadly speaking, an unproblematic theory, a state being an assignment of values and quantities, and determinism on analogy with position and momentum in classical mechanics. So, on the other hand, in quantum theory, a theory of matter in the absence not of gravity you can model gravity as Harvey was just stressing across the street but not gravity a la GR I can write down gravitational potential in quantum theory but quantum theory doesn't conceive gravity as it is conceived in GR the state is a function defined on a configuration the position of a particle or the values throughout space of a and I will write square brackets when I have a function when I have a function whose argument is itself a function and so one has this complex number, this amplitude states are represented by operators, they'll be given a hat

2:30 and the probability as we say in elementary courses regardless of foundational issues of finding a value in a given set of possible values is controlled by this mod squared number The theory is mysterious. The state doesn't assign values, and they seem not to exist before being found. And Copenhagen's answer, or the answer with the Copenhagen interpretation, is to assume a classical background, and it is said that this will provide a framework for avoiding mystery. And the question that we will be preoccupied by, and we're already since 2 o'clock preoccupied by, is whether this classical background should include, needs to include, something about the space-time manifold. So, turning to the movement of the program of putting these things together, it is worth stressing it's something I'm in particular learned from Chris Isham that there is such a variety of programs and there's such a variety of programs partly because there's a variety of motivations we know that in some sense we have to revise the Einstein field equations to allow for quantum the quantum nature of matter on the right hand side another motivation and this goes back of slides, is that there are apparent deficiencies of classical general relativity arising from the existence of singularities generically in the theory. And it is suggested that this small tiny length, constructible from some fundamental constants of nature, and a corresponding tiny time, which is the time it takes light to travel that tiny length, are tiny lengths in very short times, at which classical GR should break down in a way so as to avoid singularities. Though I would perhaps ask Joy for a Xerox of the wonderful inflationary limit last slide with which he made these numbers more split than I did. Okay, so not now, Joy, but for personal, private study later.

5:00 So the other thing, and this is Joy's main theme of his last third of his talk especially, was, of course, the idea that there are deficiencies in standard quantum theory. And one point that is made in connection with this tiny length is the idea of avoiding the infinities that you meet in this short-distance regime of quantum field theory. Quantum gravity also, as is Joy's theme, and Penrose's theme, might give a mechanism for the collapse of the wave packet. A fourth quite different motivation would be that, and this was a small picture on the top right-hand corner of one of Joy's slides, was the idea of gravity being necessary for the unification of the fundamental force system. So, various motivations. given the motivations and given that there is as I say here no unequivocal data there are various styles with which the subject can be undertaken different aims and assumptions, different conceptual assumptions so two examples of that are should you start with the classical theory and quantize it or should you start with the quantum theory and then look for the classical limit of it in particular should the low energy limit of some distinctively quantum theory that you don't get by starting with a classical theory? Or alternatively, do you start with general relativity and quantize it? And in most of our discussion, both in Joy's talk and mine, the way of quantizing GL that we'd be mostly concerned with would be the so-called canonical in which the split is made from between 3 and 1, That is to say, space-time is decomposed into space and space-plus time. Another example of these various cross-cutting assumptions is how much classical space-time structure is needed for quantum gravity. Although one could similarly query standard quantum theory, most current programs do not. And in this sense, the state reduction discussion is not, in my sense, of the main current programs. That's not meant to be disparaging. I think it's just a kind of head count. One way in which one can make this question more precise is to look at a chain of

7:30 ever richer mathematical structures that one can impose on a bare point set in order to end up with the structure you are eventually using in classical general relativity and ask at what stage in this chain should one draw a line having those things to the left be classical and those things to the right be in some sense quantum. And one must be aware that if you initially assume a certain amount of structure putting it to the left of your line it can be hard to remove later on. That pair of assumptions, that pair of questions shows a bit of a variety of assumptions. There are some conceptual hurdles which are going to be faced on almost any collection of assumptions about how to do the subject. And this is the hurdle that Joy also had with the wonderful red and green light cones turning one over another talk this afternoon uh so without the good graphics but at least with a bit of color let me say it again in case you weren't there at two uh there is a principle micro causality which is fundamental to quantum field theory which says that if x and y are two space-time points that can't be connected by a signal uh far as fast as light or slower than light then uh a certain property holds between fields associated with those two points, which has the implication that they are simultaneously measurable. And in quantum gravity, with an operator metric, we are going to expect that all points will be potentially time-like and microcausality will be lost. And in this sense, the idea of time and the idea of the evolution of a system in time becomes very unclear. It sounds clear what to make of it. Okay, so that's part B, some styles of quantum gravity and some common hurdles that confront the different styles. I want to now give two examples of choosing the amount of background

10:00 in the strategy where you quantize general relativity. And the first is a sort of simple space-time approach which, in effect, isn't... I call it a space-time approach because it's not canonical. There isn't a split of 3 plus 1. This is the kind of thing done late 60s, early 70s, in which you use standard quantum field theory to quantize classical general relativity. And the idea is to have a background structure which remains classical and which is quite rich, Because, although there will be a hat on a metric, gamma hat, it will be decomposed into a background, in effect classical metric, which is in fact one of special relativity, plus another bit, the h hat, which is genuinely quantum. And when you do this, you obtain, indeed, with some, so this is a good success, a particle in the sense in which quantum field theory talks of particles called a graviton because it transmits gravitational force just like the photon transmits electromagnetic force. And in that sense, a standard, let's quantize the metric approach from quantum field theory delivers what you want. the rich assumption of a background structure means that the conceptual hurdles about the interpretation of the failure of microcausality you overcome them but in ways that you might find dubious I mean microcausality and the notion of time are all defined with respect to this however this theory faces recalcitrant infinities. The theory is, in the jargon, not renormalizable. And when this was realized, it in effect led to two things. One is, from the mid-70s onwards, the canonical program to treat the qualization of general relativity using a 3 plus 1 split. And it also led to the idea of killing off these recalcitrant infinities

12:30 by having compensating terms. This was first done in a program called supergravity, and that led into superstrings, which is the main, in terms of a head count, program for quantum gravity today. However, I will not say a word about superstring since know-nothing, and the problem of time is not much discussed in it. So the problem of time is much discussed in the canonical program, and therefore, broadly speaking, I'm going to concentrate on the canonical program. And my second example of choosing the background, when you say it's not much discussed would you say that's because they haven't got that that's my understanding or they don't consider conceptualists at all feeling about it is that as a community they don't but as that you can interpret it as the time is not right That's how Chris Eichmann interprets it. Just a comment. I asked Michael Green about it in the 6th episode just after he speaks back to him and what's his theory is very much of his analogy. And he stopped at track and think at all that when he thought of that. Just a comment. There's someone thinking about it. So, thank you so much. Well, as we all know in the philosophy of physics there's a lot to think about so in a sense it's no discredit to Green that he hadn't thought about it. But in any case, from now on, nothing to say. So here, within the format of an example of the amount of background structure to assume, an example of how much to assume, I want to take the example of the canonical program where, according to the 3 plus 1 split and according to the strategy of quantizing general relativity, starting with a classical theory, spatial geometry. And a four-dimensional space-time n is therefore in effect going to correspond to a history of a three-space, which I'll call sigma, and I'll be using little Roman

15:00 letters, little a, little b, for those indexes running only from one to three for three geometries. And therefore, you'd expect to have states like this and operators like G had. Now there's no background three geometry, nor is there a space-time metric assumed once and for all, but there is a manifold structure being assumed namely that the manifold is decomposed as sigma cross the time dimension represented by the real numbers. Furthermore, assumptions about causal structure lurk and one main approach in particular imposes this condition right at the beginning but the justification for this insofar as it has one is by analogy with ordinary quantum field theory where points lying upon this thing sigma are space-like according to the ambient space-time metric that you've got in mind but there isn't now an ambient classical space-time metric that you've got in mind to justify the problem In the three spaces, does it not assume that the tangent spaces are included? No, I don't think it is. Which is that a main approach, generally? Well, this is on the way to Wheeler-DeWitt equation in the sort of time after quantization, putting hats on every on every one of the six metric components independent metric you put this in and then also you would naively say this and the one for the P's and then the CCR and then you're off so in any case And that is a point, that there is early on the imposition of this, which is an assumption about causal structure, even though, in fact, no background metric is assumed, there's a shadow there of the appeal to classical causal structure. Okay, so now before we talk specifically about the problem of time, there is a kind of philosophical punchline, so it's a rehearsal report for a general philosophy audience, namely, the kind of thing we've been discussing already this afternoon, is space-time really a continuum?

17:30 or is the very successful continuum model of space-time that we're using all the time in physics a phenomenological construct? Or in other words, what is the minimal background structure for quantum theory to work? And given that minimal background structure, what are the details of explaining the rest of the manifold structure as emergent? A less commonly pressed question, this corresponds to my comment that in this game gets kept as standard in its general relativity ideas that are put under pressure in the community as a whole, one could press this question about making standard quantum theory, both formalism and interpretation, be somehow emergent from something else. Okay, so that's that. Let us try to make this a little more specific in terms of now that lasts the problem of time. So, in general terms, I want to say there's time being treated very differently in our two ingredient theories, and there are various approaches for mending this mismatch. treated very differently in GR and quantum theory in the sense that in GR, time is internal to the system. It is various in a certain sense, and it can be a physical quantity. But what I mean by this is that the space-time, with its space-time metric gamma, can be given various foliations into space-like slices. Each defines a time, although rather an unphysical one, you, how on earth would you measure it? And furthermore, gamma defines more physical times like the so-called proper time, so that's the sense in which time can be a quantity. One can find genuine quantities in the theory representing time. On the other hand, in quantum theory, time is external to the system. It's univocal, and it's not a quantity, where univocal

20:00 is intended to embrace the relativity of simultaneity that you find in special relativity. It's essentially treated as a background parameter. It's not represented by an operator, but it is, however, crucial to the formalism and interpretation. It's very hard to think of quantum theory without the concept of time acting as a parameter. So that's the general contrast. And the strategies for solution, well, there are four that I pick out, This is not an original taxonomy. First of all, you can fix a background structure which is strong enough to give a time for quantum theory. And here's two technical examples, which I won't explain because time is short. Asymptotically flat space times will restrict them to a certain kind of preferred foliation. The next two strategies both try to find what's called an internal time, which is to locate events by functionals of the gravitational and the other fields. And, well, here we are in a, I'm afraid, in a very crude and questionable two-liner summarizing delicate discussion of the whole argument that George did so well, so with pictures. But the lesson of Einstein's whole argument being taken to be the slogan in quotes motivates, one might, well, say, the idea that one has to make, one has to find functionals of the gravitational and other fields which can locate events. Now, this common idea is implemented in two different ways. First of all, there's finding an internal time within the classical GR before quantizing. and in effect you then have hopefully a Schrodinger equation with respect to this internal time that you found and this involves this involves so called solving constraints to eliminate so called redundant variables before you quantize this strategy can you push it down a bit more? the next two strategies are number two and number three and they have in common the idea of an internal time which is going to involve

22:30 these functionals it's motivated by this historical controversy and the first is internal time found already in the classical theory alternative to before is obviously after you can find an internal time only after quantizing and here have in mind that you postulate states to be functionals of the three geometry where this includes redundant variables and this approach involves the constraints being the content of the they are now, in fact, equations in terms of operators, and that means that they restrict the allowed quantum states, and time is to be somehow recovered from solutions to these equations. And the fourth and more radical, and this is a mixed bag of very various approaches, of course, is to try and have a quantum theory that is timeless. This is putting pressure on the quantum theory. there are very various approaches all obviously must plan to provide a notion of phenomenological approximation in which ordinary ideas of time are recovered. So the first three strategies are especially prominent in the canonical program where most of the work on the problem of time has been done. Now that, my dear expert friends is actually about half and more 25 minutes, I think, 22 minutes. We could stop now and discuss this, or you could hear, if you like, about more details about the problem of time and constraints and gauge theory. Harvey is a good judge of these matters. Yeah, a bit more. I don't want to wait even more. Okay, okay. So, let me just shove off these to the side. So what I want to do now is say again, freshly, but with a bit more jargon what the problem of time is, then to talk a bit about Hamiltonian system and constrained Hamiltonian system, because we'll be involved in those, then to talk about interpretive issues about those kinds of theory

25:00 that use a Hamiltonian or constrained Hamiltonian system, And then to talk about GR as such a constrained Hamiltonian system. And then finally, I'll talk about the strategy number two. That is to say, find an internal time before quantizing. In discussion, we can talk about the others. But I'm going to stop at that most conservative and thereby perhaps comprehensible proposal to find an internal time in the classical theory. So that's my plan for this. So, now the problem of time, as we're now going to discuss it in talk number two, so to speak, is something that, the very phrasing of which will require some of the technicalities of classical GR and quantum theory, I'm afraid. But I agree with my mentor, Aisha, actually, that there's a sense in which, although the literature is preoccupied by precisely the problem of time in that technical sense, it's just undiscussable without quite a bit of framework, technical framework. you know ultimately the problems are exactly the issues of how to understand superposition of geometries and how to understand the breakdown of micro causality and other planks crucial planks in this uh ship called quantum theory so i don't mean to say that one that the discussion after this must be at about this i think there are issues about emergence or what would it mean which one can raise without talking about gauge theories or constrained Hamiltonian systems. In any case, that said, launching on, the problem in a nutshell is a tangle of issues related to GR and quantum gravity. And concerning GR, one has really four comments. But once you formulate GR in a way that makes it suitable for quantization, according to this program, namely as a constrained Hamiltonian system, then there are going to be no gauge invariant quantities whose value changes over time. Furthermore, I'll explain what gauge invariance is and so on.

27:30 You're not meant to know that yet. I'm just giving you the punchline to orient you. Gauge invariance is often considered necessary for a quantity to be physically real. Okay. Not sufficient, perhaps, but necessary. Sorry, what I mean is, I think you'd be nuts to say it isn't sufficient, right? I think widely agreed that sufficiency, it's sufficient to be physically real, that you be gauge-invariant, but the question is whether it's necessary. It is often considered to be necessary. Now, this situation, that there are no gauge-invariant quantities whose value changes over time, seems to suggest that there's no change if only gauge-invariant quantities are physically real. But of course there's change, as we look around. So, in GR, the situation is normally considered to be tolerable in that you can fall back on other formulations of GR which don't use this stuff. and maybe actually you can deny too, you can deny that physical reality implies gauge invariance final comment on this slide this result in Little Roman 1 reflects the general covariance of GR that is to say the invariance of the theory under the group of diffeomorphisms of the manifold and this is closely related to the debate about substantivalism and relationism And I will be arguing in subsequent slides that substantivists have to deny that physical reality implies gauge invariance, but relationists need it. So that's the problem in a nutshell. As a tangle of issues about classical GR, now, as a tangle of issues about quantum gravity, I have to admit my slides don't utterly perfectly interdigitate. This slide has words on it that you've already seen on other slides. So I'm now in two minds about whether to make it more confusing by showing you words, which you already have in the front of your mind, which I'm now doing, or alternatively, to hide those and then infuriate you with your feeling of curiosity about what might be behind the sheet of paper. But all that you really need on this slide, because it's fresh, it's not on previous slides, is that we're going to see that finding an internal time

30:00 before quantization is actually a matter of finding a non-gauge-invariant quantity. That's what finding an internal time is going to amount to. It will be quantity that is not gauge-invariant. And to the extent that you are going to have to take it as physically real, you will be going against the doctrine that gauge-invariance is necessary for physical reality. what you'll be doing will be motivated by substantivalism. It will not require philosophical substantivalism but if you were a substantivalist that would motivate you because a substantivalist, I will be arguing believe in the physical reality of non-engage-invariant quantities and the hunter for internal time before quantization must believe in them and so there's a confluence of motivation. That said, we've now said what the problem is in a nutshell. Let me now go to the second section, which is to introduce the Hamiltonian and constrained Hamiltonian systems. and this is like the bird's eye view of the expert and craftsman-like details that so many of you are familiar with. So the Hamiltonian system is essentially the idea that there's a certain set of instantaneous states equipped with a certain kind of structure, call it omega, H for Hamiltonian. And this set gamma of instantaneous states supports a representation of quantities as real value functions on gamma. The classic case would be the point particle in three spatial dimensions, with the instantaneous state specified by its three components of position and its three components of momentum. The main thing we need about omega is that by its nature, defines for any such real value function on big gamma a vector field, call it x sub f. And doing that to h is itself such a function.

32:30 Sorry, this is not on the slide. h is itself a real value function on gamma. So defining x sub h, we get a vector field on big gamma. And H, in this sense, determines a unique trajectory, a unique curve, through every point little s in big gamma. And this uniqueness represents determinism, because H isn't just any old function. It really is true that it controls the physics of the system through the famous Hamilton's equations. So that is actually a picture of the history of the system. It's not just any old vector field. Hamiltonian system, I'll call it delta sigma range, we begin, in effect, with a Hamiltonian system, but we then cut down to a subspace of gamma, which is called delta. So given a gamma and an omega, you're to imagine that there are some equations, C, A equals 0, called constraints, which define a sub-manifold delta. And I'm then going to define sigma as just a restriction of omega to delta. And what one finds when you spell out the details is that, please imagine now that the three-dimensional space gestured at in this picture, the ambient space of this beautiful perspectival diagram I've got, is gamma, written in green. But it has a two-dimensional sub-manifold, delta, pictured in blue, but the existence of omega and sigma means that delta itself is partitioned up. It's got fibres. Right. Now, jargon. If you have a map on delta, that's to say sending points of delta to points of delta, if you have such a map, and it preserves those orbits, it's called a gauge transformation. So if it just moves things in the red lines along. On the other hand, a real value function on delta, a quantity, which on any orbit is constant, so it takes some constant value on any orbit, I don't mean it's constant throughout all of delta, I mean that along a red line it's constant. Take any two arguments that are on the same red line on the same orbit, they give the same value. Such a quantity is called gauge invariant.

35:00 and this is less important jargon so feel free to snooze for the next half of the slide well, third of the slide there's a jargon for a constraint having a certain property a jargon is first class a constraint is first class if defining X sub of CA upstairs in the ambient big gamma when you restrict When you consider the restriction of this vector field to delta, is it everywhere tangent to the gauge orbit? If it is, it's called first class. Now, it being tangent to the gauge orbit means that being a little vector, it's pointing in a certain direction, and starting at a given point in a certain gauge orbit, you could follow your nose. You could follow that vector, go a little step, look at this vector field again at the new point, which is everywhere tangent to this gauge orbit, and follow your nose again for a little step. And in this sense, the definition of this vector field generates a map along a gauge orbit. And one sees this in this picture. The green dotted lines are gauge transformations moving along a gauge orbit, because now problems of picturing dimensions on a two-dimensional slide to a kind of three-dimensional visual imagination. The red rectangles are now the gauge orbits. They correspond to the previous red lines. So the ambient space of this diagram is delta. Okay. This mathematical structure has the property that H will determine for any S in delta many trajectories. But there is nevertheless determinism of the system in a limited sense, of which orbit at a given time t the system is in. And therefore, there is determinism of the values of the gauge-invariant quantities. And you see that in the picture. There's t equals 0, there's t equals 1, there's t equals 2. There are many, many trajectories coming out of a given point here. There are many coming out of here too, but I only drew one. are many, but nevertheless, of course, the three points that you get to along the three curves all lie in this same gauge orbit and therefore match on all gauge

37:30 invariant quantities. Yes. What is delta? Delta is indeed essentially a phase Space 2. Yeah, right. And here's the absolutely standard example. Sub-manifold. Exactly. Yeah, sub-manifold. It's the sub-manifold that is relevant because of the existence of the constraints. So in any case, here's an example. Point particle, three elementary point particles in three spatial dimensions. We had a six-dimensional phase space, three components of momentum, now set P1 equal to 0, which is to say, I like to think of this in terms of P1 being not the x-axis, but the z-axis, so it's the vertical axis, really. P1 is 0, there is zero momentum in the vertical direction, that is to say the little guy is confined to the plane. How does this work out in the maths? What it works out as, that the gauge orbits are these kinds of set. That is to say, if one is thinking that the physically real corresponds to a gauge orbit, and the variety within an orbit is a redundancy of one's formalism, the physically real instantaneous states correspond to the x and y components of momentum and the x and y components of position. guy that's confined to a plane, right? However, notice in the maths that there is this compulsory consequence of the idea of specifying constraints in sub-manifolds and orbits, that we don't just have literally four numbers here. We have a whole class of quintuples which vary in this other fifth component, okay? So the maths represents this, its inner plane, as freedom about the spatial coordinate Z, or as I call it, Q1. That's quite an important fact about understanding what's going on. this dimension, this degree of freedom, Q1,

40:00 is called a gauge degree. That is not on the slide, but it is a quite common terminology. Okay, so never mind GR, We can already say some things about interpretive issues here. First of all, let's go back to that question I had as a headline at the start. Is it true that physically real only if gauge invariant? If so, then certainly gauge systems, that's, sorry, I should have really constrained Hamiltonian systems. That is a species of gauge system. in the jargon uh the gauge systems with which this entire discussion is concerned are always constrained hamiltonian systems so should have really written constraint hamiltonian systems if only gauging varying quantities are physically real then gauge systems do have determinism of the physically real so not of every variable you could write down like that variable r of q1 of the previous slide um and anything function that depends upon the value q1 or r but there are enough gauge-invariant quantities which are physically reasonable for one to be satisfied that you fully interpret the theory, then a philosopher would say well, there's a deterministic interpretation of this theory. And the standard example of this is electromagnetism in a simple case electromagnetism in a vacuum the physical space nothing fancy the physical space in which the electromagnetism is being studied is simply connected there are no special flux tubes or anything naughty like that to give one strange effects, then E and B, the electric and magnetic field, are regarded as a sufficient set of deterministically involving gauge-in-variant quantities. Yet, in a very general sense, you might well want to reformulate the theory so as to avoid redundant variables, argument of the so-called reduced phase space, that is to say, the points of this phase space will be the very gauge orbits we had before. So they would be labeled, in our example, by quadruples of real numbers, P2, P3, Q2, Q3. Those would label, as they did on the

42:30 slide, the class of quintuples that vary in that fifth component. So that's... the one way of taking it, if you believe that only the gauge invariant quantities are physically real, you may well think we should have a clean formalism using a reduced phase space. On the other hand, if you actually hold that distinct phase space points represent distinct physical possibilities, even when they are in the same gauge orbit, then you will surely hold that for any pair of points quantity that takes different values at two points, because presumably a distinction of physical possibilities in classical physics should imply distinct values for some physically real quantity or other, right? So you want to get a separating or discriminating physical quantity, right? And obviously this is going to have to be a quantity which is, for points in the same gauge orbit, it's going to have to be a quantity that is not gauge invariant, or in the jargon, sorry about that, it doesn't commute with those first class constraints. That's a bit of unnecessary jargon. Setting up then that general discussion is one thing, let's now just connect it with GR, and then we'll do the details of GR as a gauge theory, but what is the connection of that general discussion about physical reality and gauge invariance short, I think the connection is with the grand debate between substantivalism about space-time and relationism. I want to take the substantivalist as what some people call simple, which could be rude, could be complementary. I take it as complementary, i.e. it means comprehensible, as against fancy doctrines. So I want to take the substantivalist as simple, straightforward. forward, she takes isomorphic models of general relativity, even if they have the same base set of points, to represent distinct possibilities, because of the variety amongst them about which point has which role, which geometric properties and which material events occur at them. Now, once GR is formulated as a gauge theory, that's the next section, we will see that such a pair of models involves a pair of phase-space points that are in the same gauge orbit.

45:00 So this substantivalist is committed to their being physically real but non-gauge invariant quantities. And, of course, you could undertake as a piece of physics to identify them. And that is what I had in mind earlier when I spoke about substantivalists. A substantivalist has a motivation by being a substantive timelist to find the time before you quantize to find the non-gauge invariant internal time in the classical theory. Okay, so on the other hand, a relationist might deny that there are such quantities and therefore want to reformulate GR with its reduced phase space without redundant variables. Well, now I've got on the slide what I just said out loud, this contrast that I've set up in classical GR between substantivalism and relationism, whether or not physical reality implies being gauged invariant, that contrast lines up with issues in the problem of time and quantum gravity because substantivalism's commitment to physically real but not gauged invariant quantities motivates finding an internal time before you quantize. And if you deny that there are such quantities, then you will go for the quantize and then find time or try to have a theory without time at all. Okay. So, we're now really two sections to go. And the first is GR as a gauge theory, and then there is finding an internal time before you quantize. I think it might be good to ask our revered leaders, Simon and Robert, whether we should stop now for questions, because I have gone on for 50 minutes, haven't you? How much more do you think? How much more? I think there are about six slides. Yeah, there are eight, actually, but we could maybe skip one, too. So, you're very quiet and patient. I hope you're managing to remember all the errors I'm making so you can miss them.

47:30 So, GR is a gauge theory. A little bit of notation on the one hand and then a restriction of discussion follow in the second half of the slide well this first half you will be familiar with just saying that I want to use the square brackets for the function to indicate that the argument is itself a function I'm going to use the large x for a point in a four dimensional manifold small x for a point in a slice capital sigma I'm going to limit discussion in three ways. First of all, a condition called globally hyperbolic, and secondly, talk about vacuum, and thirdly, talk about the slice capital sigma being spatially compact. And this is, without knowing the technicalities of the subject, a restriction that I understand enables us to report a few truths about what the results are in the study of GR as a gauge theory or as a constrained Hamiltonian system this is a substantial restriction this is perhaps is less substantial but I think I've a little backtracked a little bit backtracked on this slide's claim that it's not substantial I think it is in a way that we'll see in the final slides, but it's not as substantial as A. The interpretive issues are similar, if we include matter, yes, but the results of the details may be different if we include matter, in a way that I'll say at the very end. By left, you just mean left in between two. Right inside of zero. Yes, exactly, yes. And a third comment, spatially compact lies between these extremes. Not having the spatially compact would allow certain things, for example, the use of asymptotically flat spacetimes to give one a background structure in terms of which to make sense of time. Okay, so the GR as a gauge theory involves, first of all, let's just remind ourselves

50:00 it involves what we already discussed, a manifold with a metric gamma obeying the Einstein field equations, I call them EFE. As I said even in the first half, it involves foliating M into Cauchy surfaces, which I've written with the green slices here, and now saying new things. If we start in the space-time way of thinking with M, and we then think about the structure induced on a slice sigma, we have that gamma induces on sigma a Riemannian metric, G, that's good old plus plus plus ordinary kind of geometry and an so-called extrinsic curvature and the fundamental idea from now on to permeate all the subsequent slides is that g comma k these two things that are induced or rather something very closely related to k written p is going to represent the instantaneous state of the gravitational field where g is like the q of elementary classical mechanics Can you say post-post-post journey? Can we do to a equivalent tangent space? Oh, well, this is classical theory. Sorry. Yes. In the previous discussion, we were quanta already, but now I'm purely classical. So, the main comment, crucial thing on this slide, is that not any pair, g, p, is thus induced by a gamma. They must obey two constraint equations called vector and scalar. This has got three components. This has got one. At each point x. This is also called super momentum. This is also called super Hamiltonian. And leave it there for the moment, but we'll come back to that. That's crucial. On the other hand, going the other way, Embedding an arbitrary spatially compact three-dimensional manifold equipped with an arbitrary three-geometry with a suitably related P as a Cauchy surface of some M comma gamma

52:30 will, of course, require this triple to obey the constraints. After all, having been embedded, it must be such that the gamma that's ambient induces the G and P, and therefore the constraints satisfy. That's fair enough. But there's a surprise around the corner, or more specifically at the next slide. Namely, the dynamics is in the constraints in a certain sense. M gamma satisfies the vacuum EFE, the outside field equations, if and only if the constraints hold on every space-like hypersurface. So those two little equations with the two H's, HA and H per, catch the dynamics. They catch EFE, in a sense. It's worse than that, or it's more surprising than that. History, in a sense, lies in an orbit. That is to say, the dynamical trajectories lie in an orbit. And this, as we'll see, it's almost immediate, vindicates my initial statement when I'm saying the problem in a nutshell. The problem in a nutshell is that all the gauge-invariant quantities are constant in time. That's the same point as I'm now making. around. History lies in an orbit. This really follows from two things that we can say in words in which we've already said. Well, number one, we've already said the word. We haven't already said it, but we've got the notions to hand. The constraints are first class. Those constraints I had, HA and H per, both zero, they're first class in the maths of this. They generate gauge transformations. And the second point is just to report, the Hamiltonian of the theory is actually a linear combination of them, and so it generates gauge transformations. So starting at a point in the state space, age tells you shove along, make, evolve, and where does it send you? It sends you somewhere within the gauge orbit you're in. In fact, a bit more is true. The orbit consists entirely of slices of such histories for a single space-time. Or to put it more precisely in the red box, a gauge orbit consists of all the pairs G, comma, P that can be obtained from Cauchy surfaces of some specific Einstein vacuum solution.

55:00 and therefore really punchline from the first slide of this half of the talk any gate invariant function on phase space must be a constant of the motion it can't change along the evolution so that's a very surprising fact about classical general relativity suggest that you haven't got the equipment to talk about change If I may just say something, is this really general relativity? You weren't doing damage, right? It was flipping sigma. Ah, yes. Pardon me? No, you're right. That is a very, you could say, well so much the worse for formulating general relativity a 3 plus 1 way, and in particular thinking in terms of gauges and thinking of it as a gauge theory. That's right. So that's... Okay, so on the way now to the final four slides to go. We're now on the way to this conservative but comprehensible attempt to deal an internal time within the classical theory. And we're going to talk about finding the internal time in the classical theory right now. Obviously, this first comment is a very general comment applying to any theory. That you've got more variables in your theory than are really physically significant. And in order to solve the equations of motion, to specify an evolution in any theory, you need to obviously crunch out eventually a number not some red line on a diagram saying this is the orbit made, you're in there somewhere. So you need to fix a gauge you need to remove these redundant variables in order to make a prediction if you will. How many do we have to remove? This slide counts

57:30 how many you need to remove. namely, the punchline there are eight non-dynamical degrees of freedom which I've abbreviated to DOF per point x in sigma and the reason is that actually, this is physics that I don't understand but it seems to be completely agreed that the gravitational field involves two dynamical degrees of freedom per space-time point sorry, per point x in sigma, okay? And that means that there are four times, there are four space-time points, let me say again. This report, first of all, do it in the order of the slide, this comment that there are two configurational degrees of freedom per point x in sigma is often written like this, two times infinity cubed, just to remind you that it's per point little x in sigma. that will mean that there are at every point little x in sigma four phase space variables there's a configuration one and then there's a momentum one if you were to choose a coordinate system on sigma with a g and a p there would be 6 and 6 equals 12 variables per point little x in sigma 6 because it's 3 plus 2 plus 1 symmetric matrix is the idea. In principle, we have four constraint equations, a scalar and a three-vector one, h sub little a. They could remove four variables per space-time point, but that still leaves eight, which we just calculated, therefore, four times four at each space-time point of too many. And these spurious variables the diff-m invariance of the original theory. Now, the plan is to remove these degrees of freedom by finding internal space and time coordinates which will be the values of some functionals, curly x or chi. There'll be four of them. And they're functionals, notice of g and p. They're not just associated with the point x. P, the whole function. This will be important in my punchline.

1:00:00 So we're intending that setting these things equal to some fixed functions, imposing that extra condition, is going to, in effect, restrict the paths in phase space. This is a picture of a history that for every time T there should be a certain path, a point in phase space, in such a way that each gauge orbit contains just one path. That's the whole point of the internal time. Okay. Oh, by the way, just to... The X through the words at the bottom of that slide was because they're wrong, not just skipping over them. It's not the way to say it. So this is the joke, I think, invented by Carol Kukacs to put it in Latin. Time before you quantize. So the idea is to quantize in the usual way the formalism of canonical GR once you have found your internal time. And this will be done especially in a version where you have a kind of foliation independence. And this is called the internal Schrodinger interpretation. we don't need this comment. You get something that looks a bit like a Schrodinger equation. You have ih times the derivative of the state function equal to an h-hat hitting And this is a functional derivative in the technicality, and it is with respect to the internal coordinates that we were just discussing and hoping to discover in classical GR. The chi is not subject to quantum fluctuations. It's a C-number thing. This enterprise faces many, many technical problems. For example, how do you find internal time? And there are many studies in simple models like many superfaces. But now, final slide, anticlimactic, like the entire enterprise this afternoon, especially after Joy's talk, but final slide, live anticlimactic. Make the audience do the work in the discussion period by solving the problem of time before DNA, if possible. As to interpretive problems, you would hope That because we've been conservative, seeking an internal time in canonical GR,

1:02:30 we would hope to only face the interpretive problems of familiar quantum theories. But I don't think we are this lucky. In particular, I want to pick out that it's very difficult to see how we could ever apply this theory in the way I've been discussing it, developed in the way I have been discussing it, because the theory is meant to be saying at the end of the day that the square of this mod is the probability density for the field to be in the state phi, that is now the representative of the gravitational field. It's corresponding to G. There was, so to speak, 20 years of hard effort by brilliant people to go from G to phi, a change of variables, But all that was done in the internal time in classical GR. So that's what that phi is. It is representative now of the gravitational field. And this number is to be the probability density for this field to be in the state phi for a given chi. That's saying where to look to find the gravitational field at a certain value. And chi is, in general, I said it was a functional of G and P. It's non-local functional. It would be some massive, hairy integral across all of sigma, across the whole of the universe, of the state G, P, on that slice. So, punchline, even if you solve all the technical problems, and if you mathematically manage to identify, in particular, this internal time chi as a functional of G and P, And if, furthermore, chi is foliation dependent, you can't make a kind of foliation independent choice of chi, then let's suppose that you've somehow got access to the foliation that you're working with. Nevertheless, this non-locality will, in general, prevent you confirming a probabilistic prediction because you would never know G and P to shove into the integral of this integral expression of chi. Well, this quandary prompts moving to, in particular it's prompted Kukat, to move away from vacuum to matter variables which hold out the hope of giving you enough material to make chi turn out to be a space-time scalar, like, as it might be, the scalar curvature at a point.

1:05:00 The final philosophical comment about this is that, first, it seems in a sense represents a reconciliation of substantivalism and relationism, because I talked about the internal time before quantization program as motivated by philosophical substantivalism. It's clear that in order to work, it may well have to fill space-time with matter in order to surmount this problem. So there's a sort of promise of a reconciliation of substantivism and relationism in this new version of the program. So thank you very much. Let me invite questions. Thank you. just a quick question in relation to the very last point what is the intuition behind the expectation of introducing matter variables maybe don't know actually truly and probably Harold just believes in time Harold has been trying to have to be taken out for 30 years, and he hasn't actually done anything yet. Well, I mean, there's a certain, I mean, the problem just comes up somewhere else. He does actually get some quite nice simple mathematical equations to come out. but it means that he imagines that he's got, for example, dust particles that are moving around that are armed with clocks that read proper time, and he's actually using that. Now, this is, to me, a complete fiction, I'm afraid. I mean, Carroll is a great master of canonical conservation and a tremendous thing, but he is determined somehow or other to keep time, I think, and he will do it however he can. Now, if you allow yourself in the fiction that you've got dust particles that you can recognize, so they give you the coordinates, you have dust particles that give you the coordinates,

1:07:30 and he imagines that they're carrying clocks which measure proper time. Now, how alert a particle is meant to do that, I do not know. But if you allow yourself that fiction, then you can actually formally write down these things, and it comes out remarkably simple. But you've put in one hand on a lot. Sorry, yes. So there is a tremendous amount that's gone into this, and I just... I think it's just not going to work Naively, why couldn't one as they say, counterfactually suppose that these dust particles were actually little clocks developed by some sort of nanotechnology then what properly adjusted clocks then what would they register in the line of proper time Isn't that a particular thing? Let me ride my whatever it is my hobby The root problem in all of this is that Einstein only addressed one out of two problems of what is made for time. There are two issues to find. How do you define simulconality? And what is duration? What does it mean to say a second day? And Einstein never addressed that second question. That's the root idea. I think that's the second question. He never addressed the second question. What is the meaning of duration? He never asked what is a clock? but what a clock is is a very, very complicated issue it's far more sophisticated Einstein himself says in two or three times in his writings later on that it was a gap in the work he deliberately introduced clock phenomenologically he never discussed the issue of what a clock is and this is an absolutely fundamental issue and I think it's bedeviling the whole thing that people just love the best I guess I have three things to say let me just make one mark I mean people say that an atom is a clock but it's not true, one atom alone it's not a clock, just look at what a cesium beam clock is it's actually, first of all there's a solid state thing there, you have to have an oscillating magnetic field, you have to have a microwave field and you send one atom after another through it and they have to come in and if you get if they sort of made a transition then they come in one place and they haven't servo-feedback mechanism. All of this has to be done in an inertial frame of reference, which is determined by the universe of the whole. It's a huge, complicated thing, what clock is. And because Einstein never addressed this question, theoretical physicists don't ask what a clock is. You're not happy with an Aeol's Koranian shield lay out or something like that?

1:10:00 No, it is not adequate. There's a real issue there. I have talked to Ehlers quite a bit about this and he admits that I've got some point. Whether he agrees with everything that I would say. Because I'm trying to get around the problem of poking out of someone high and so on and build the clocks into it. The other thing is that Einstein never thought about how he defined the interaction between the rest of them. He always said, who's given? He said, if you've got one, then you're away. But he never asked how it's given. that's the root of all the problems I'm sure yeah I mean two tiny things I'd say at this stage because there's much that Julian said that people may want to respond to quite apart from their own questions I would say just that the intuition in answer to Harvey's original question what is the basic intuition that matter will save the day I should have said more dynamical in your formalism because there's all the matter degrees of freedom enabling you giving you structure with which to build scalar functionals that's the short but the other thing that of course to be you're quite right with that area because the fact is if you've got the comparative thing the crucial thing is these self-partum are the different carrying thoughts now I ask you do that part of the girls carrying thoughts they don't no they don't just in a joking way, I mean, to say you're quite right, as we often discuss with Harvey, too, the way in which Einstein admits intermittently rightly that it's a temporary theoretical posit, the basic clock, like the autobiographical notes, he says. But, I mean, Kukash could pull the Einstein gambit and say temporary theoretical posit. I think, just to compare to the terrible Kukash, I think his motivation is that, to summarize in one sentence, he said, I see things evolving. That's what he said. I see them, I need to explain. I think that's fair, yes. Whether it's possible or whatever. Yes, well that's one of the reasons. But he wants to explain the illusion of, illusion of matter, or whatever, how the subject is. That's what he wants to explain. I mean, they all do these concepts, I think that's what you say, actually I would defend

1:12:30 Carol in all sorts of ways, I mean I've learned more from Carol than anybody else, I don't know, but he certainly believes very deeply in time, and I think he takes the evidence of his sight, he's seeing things at its face value, but I mean, when we look at a cinema film, we see it as a continuous film, but we know absolutely truly that that is something our mind has deceived us into seeing, because the cinema thing is in fact it's a movie a succession of films, and our mind is tricking us into something that is continuous, so Carol won't accept that what we see with our eyes is the intersection. That's a little bit. Simon. Did you want to respond? I'm having trouble in understanding quite why this is on 65 reasons it would be clear to me that it would the gauge invariance the constancy of gauge invariant quantities I don't know you would say this in particular these gauge orbits are not and what one means by gauge invariance and clearly one has different multidisciplinary quantities which are different in the class of GR and are different at different times. And so insofar as it bears a total on relations and so on, it's got to be because this new form of gaiting bounds, or 18 years to engage transmissions that you've got it's formally similar to gauge transmissions as they figure anywhere else in physics and there's a standard for this article which is the relation on the source of substantiveness debate and I take it that's roughly why you are saying the source of substantiveness and the source of dimension I think so though I'm not getting some of your question now do you want to say something extra so far, because if not, then I'd like to try to rephrase it. I mean, relationalism versus substantivalism is a debate that you can play out wherever you've got a gauge transformation which

1:15:00 takes you from physically to physically, which transforms among physically equivalent states of affairs. Okay, now this is not my normal jargon, actually. I would say substantivalism and relationism is distinctively about the existence of space-time points object, but you would like to use that contrast in another case. I'm prepared to see that when you generalize the debate I was not trying to do that. So let me say what I was going to say which I think might help. Namely, I had on the slide about a simple-minded substantivist who will say that two models of GR that even may have the same base set of points which paint these points in different colours in terms of the matter fields and the geometric properties represent different physical states of affairs. And that's, of course, basic substantivalist simple position. And then I said, we're going to see that a pair of such models is going to involve phase space points that lie on the same gauge orbit. Now, what did I mean there? I mean, let me spell out what I meant, and that will in some way coordinate the substantivalist normally cast in four-dimensional terms, as I just have, with the discussion of a three-plus-one conception of GR, which is, of course, how these gauge approaches to GR operate. What I had in mind, really, was that the simple-minded substantivalist won't just say these two total cosmoses are distinct possible worlds. she will also say there are all distinctions amongst their instantaneous states she will be saying for example I think that a slice with a G and a P on it in the one is a different instantaneous state physically distinguished from a certain slice with a G and a P on it a different G and P in the other G and P perhaps because of the way P codes information about the situation outside sigma being like a derivative. But the upshot, I think, is that the simple-minded subtle timers

1:17:30 will contemplate a G, P and a G dash comma P dash lying in the same gauge orbit representing physically distinct states. That at least for quite separate reasons in a way, namely the Kukash intuition as reported by Julian and Joy is something that one would like to say because people want to say there's a reality to temporal evolution, to the non-constancy of physical quantities over time. So there must be physically distinct states. But there was a slide there in my discussion definitely from a four-dimensional way of normally discussing substantivism versus relationism to three-plus-oneness. So is that helpful? It is and it isn't. I mean, look, I think what I'm trying to get at is a simple... Look, here's maybe a parallel. By all means, identify different representations of a piece of physics insofar as you prove that physics is invariant when you transform from one to the other. But don't do that when you haven't got any decent physics there. I can see that when it's clearly physics is invariant on the transformation there is then a philosophical method that kicks in typically like the reciprocal relations opposed by the sexual environment. But it seems to me that that method should only kick in once you do have a satisfactory physics, which is invariant on a certain group of transformations. Now, it seems to me that in this situation, we just don't have a satisfactory physics invariant on a group of transformations. We have something invariable, right, but it doesn't give us a temporary change in the world. Now, of course, you may say that's okay. For those of us who would like to see that, we would say, we don't have and therefore there isn't as yet a relation in terms of substantive debate. Right, and your point, of course, I really concur with that, and let me just put it on the icing on the cake. I mean, your point, as I take it, is valid for classical GR too. It is the fact we don't really understand classical GR is showing up in the fact that when you do something natural from other areas of physics with a constrained Hamiltonian system,

1:20:00 gauge orbits containing the trajectory, which you wouldn't expect. It seems to be just not successful. I think this one has a kind of formalism. What shows us is that we don't get more. Which means... I mean, the ADN formalism is not for the underrated. It's just an extraordinarily beautiful thing and I think it reflects that in many ways I believe that there isn't really any time in classical TR, and that there is a theory of what time is, is hidden within G.R. without Einstein ever knowing anything about it it's all coded in the Ricci tensor Einstein went into a shop called Mathematics and just bought the Ricci tensor off the shelf and he checked that it did the helion advancing light reflection, that was splendid and he never, as far as I can see, gave any theory thought of what actually does that Ricci tensor the vanishing of the Ricci tensor mean and when you look at what the vanishing of the Ricci tensor means, it means actually that G.R. contains a beautiful theory of what is. It is actually the solution. It actually does contain the theory of what crops are and what time is and what duration is. But that isn't saying classical GR contains no time. I mean, I think there are two things you could be saying. The one you just said was that GR unknown to Einstein has got the materials to explain the sameness of duration of the second today and the second tomorrow. but the other thing to say is in what I wrote down as a shocking thing that the trajectories, dynamical trajectories lie in a gauge orbit is a sign that there is no time even in classical GR. That's exactly what I'm saying, I'm saying the constant theory is exactly concerning my statement the constant theory comes out, the normal constant theory comes out final and that's just a reflection of the classical theory it hasn't got any independent thing in it that one would call time it's all done with With three-dimensional contigualism, it's only a theory of time. Can I?

1:22:30 Well, that's a kind of, yes. That's a waste. Just continuing my earlier remark, and continuing what you had said, why are we talking about this as GR? I can't understand it, because, after all, all the lessons that we talked about, the whole article, is all about the humorism of four manifolds. This is a topological space. When you say sigma plus R, you are putting the R in a straight jacket, and then, of course, also to problemise it. This is a good tool, we can think about it, but when you're talking about quantizing in the G-R, this is not G-R. Well, this is like Isham, as many of you will know from reading review papers of his, like I have, he calls it in the summary histories he might give of the last 30 years he always refers to the grand canonical versus covariant debate and that's exactly what we're about to descend into Perhaps not that's not the direction because covariant would be atamnibus, the 360 background, plus the perturbation. That covarian distribution is carried on by extreme periods. They start with some sort of background. That's what I am saying. I mean, there also you are doing a type. You are doing different kinds of types. You are imposing a different kind of topology. I am so excited for something else. See, there is a covarian problem in this canonical problem. Both of them are doing damage to original GR, which does not require any a priori topology position on the structure of space. You have a four-dimensional medical arbitrary topology, and especially when you consider it one time ago, and Hawking would point out that why should we a priori assume that a topology is one topology than that? It could be all sorts of quantum functions. Yeah, right. Well, obviously as an outsider, I have to be pluralistic and liberal, like any sensible person looking at the field as it's developed over 40 years would. I mean, it would be nuts to be dogmatic about which of half a dozen main programs is good and the kind of, well, effectively you're saying things like topology change in space-time require things like functional integral methods. Well, how do you do black hole things, for example? Yeah. You've seen us see a month by a time.

1:25:00 I don't know. Yeah. I mean, wouldn't an outsider with no axe to grind, say, observing, surveying all these different approaches, and say, which is the one to go for, the answer would be none. And it would be much along these lines, because every approach either reduces a four-dimensional manifold to a product of a three-dimensional and a one-dimensional, or on the other hand it assumes a perfect space-time background. I mean, it may well be that one can make some useful progress, and actually, for example, both the canonical program and the extreme theory or perhaps M-theory as we're supposed to call it now have both come up with something with what looks like the right answer to the Bekenstein balance they're obviously doing something right but nevertheless is the question are either of these programs leading us to a satisfactory unification of quantum mechanics and relativity and how could they give them as it were the restrictive assumptions that they make in the first place I agree with what Joe has said And there is a serious issue there which I'm not quite sure what the answer is, but there's another way of looking at this story which I think you would agree with, that there's these famous guys, I mean this is again from Krupa, that you can think of space time, if you have a space time in which the Gauss-Kodazzi embedding conditions hold for all space-like hyper, for all hyper-services, doesn't it have to be space-like, then that is a reachy-flat space-time as a solution. And these Gauss-Codazzi embedding conditions are really just those constraints in another guide. And that's what makes me, and I think also I'd like to say that this would be in my bonnet about let's talk about general covariance of things, let's actually say specifically what a theory does. Now, I would say the essence of what one normally thinks of as the general covariance of the TR is precisely this fact that in a reach of that space-time the Gauss-Codazzi embedding conditions hold for any three-dimensional hyper-surface, and those really are the constraints, and when I say GR is timeless it's really in the existence of those constraints that those do it there, and I think, I would say that's got to be better than this this idea of a background metric, the ATNU I think, I won't say that there aren't an awful lot of problems, and I think you've got

1:27:30 to be prepared to say that it isn't just necessarily Riemannian three geometries, you could have you could have two plus one things like that, I mean, and this is very much also in the Hawking line, where he talks about Euclidean, that, you know, I mean, general relativity doesn't make any statement about the signature of space-time, the field equations. I think the reality, I believe, and the wheel of the width equation is the same for any signature of space-time, it doesn't say anything about it. It's the same, you have the same quantum wheel of the width equation, can give you space-times with... What is the quantum wheel of the width equation? But so the Wheeler-Dewitt equation is the same... If you tried to quantize Euclidean GR, you would get exactly the same Wheeler-Dewitt equation. But it's a mean-related equation. It doesn't have any two different concepts. It's a symbol, mathematical symbols you can write down. All sorts of functional differentiations which have no meaning. And I think it's very apt to point out. It's because of the regularization problem. but conceptually it has meaning well I think this is something I this is something I decided not to talk about I think there are real conceptual problems about the feeling of doing I mean this is something that I can say because it follows on from the slide about this this isn't really picking up Michael's last comment about a sensible outsider wouldn't bet on anything to which I would broadly agree it's just that there's no outstandingly good candidate of course, so you have to fund all the candidates rather than them, that would be my hunch being an intellectual sort of a guy but I mean I would say in answer to this last thing that came up between Joy and Julian it's worth saying that when we had the constraint P1 false nought, and we got that Q1 as a gauge degree of freedom. All that was hunky-dory in a classical setting. The Wheeler-DeWitt equation arises in this approach where you have the time after you quantize, and this means that you adopt an approach due to Dirac, in which... No, no, no, no, no, no, no. Oh, of course. We've had one actually. Oh, I see that. Is that all that was on my slides? this is the kind way I was writing a constraint before and Dirac

1:30:00 suggests that in order to have a quantum theory corresponding to one of these constrained Hamiltonian theories it's a good idea to turn this into an operator and to say that it should annihilate any physically possible quantum state this is the root idea and the way this will work out, so in the case of which we have on the slide if this is represented in terms of an ID by this will now mean that psi is independent of Q1 because its derivative with respect to Q1 is 0. So it's rather neat, right? So the idea that you carry from a classical constraint to a quantum constraint way, uses the idea, to justify it really, of taking a constraint which is like this, or more generally is linear in momentum. And there isn't really, as I understand it, a conceptual justification for imposing this when the constraint is, say, quadratic in the momentum. And that thing, h per, is quadratic in the momentum. And so there is an underlying, that's just one of four I would say main conceptual problems about even getting going on but that's a different but it fits into what I just to put a poster I have to be a relation of which one is the best at all I would of course put unification and unification and part of the co-operative I assume the background I'm being asked about it. It is terrible. At least from that perspective I think the answer is far back. But compared to either of this, Hawking of course, I'm not saying Hawking of course, but this is called phone problems of study for the process, but at least he doesn't need more than 8-hour restrictions, he doesn't use the time to try something to do, whereas the other two approaches do the time to try something to do. So I would put the gradation in that there. or, I mean, of course some would say, thinking of Julian's cinema analogy maybe it just looks continuous but it's actually discrete or in some way really weird

1:32:30 that even Hawking with topology variable is too too much structure too much structure, you know but that's, this is going back to my first talk where you say well, you know, it's all totally open not looking at specific The question boils down to whether one is obsessed by black hole physics and big band theory or not. If you are concerned about these two things, and only there is where quantum gravity facts are expected to be significant in the first place. If you are obsessed by these two bands, singularity in general, then I would say that without them, flexibility in topology, you would have no chance of making a difference. but the joy