An Overview of Quantum Gravity and Its Mathematical Difficulties

0:00 ...involved, his graduate work is officially within the quantum computation group at the University of Oxford. David's interests are very, very wide, and apart from quantum gravity, he's interested about the physics and philosophical software. And I might mention just two further topics that David has worked on, namely the natural worlds and structure and the other interpretations of quantum mechanics. This is going to dovetail a bit medium well with Jeremy. I want to basically look at the same sort of general programs in quantum quantum gravity. It's a slightly more mathematical So hopefully a slightly complementary look at the vanatial issues, and I'm going to see a slightly more how this works in the vanatial breakfast. So Stanley and Pilch's style with a lot of tough shaves, and sort of quote minority approaches to what you're going to get, something like in Russian French or the pathogeny approach, or anything like that. The strategy I want to use is basically going to be to draw a comparison between the methods we might use to qualify for gravity and the methods we've used are much more successful a lot of the mathematics of general relativity is minorly difficult and often where it's not difficult it's extremely long complicated and often you can see the structure of mathematical points better with something sort of intermediate between just toy models and the book of general relativity and electromagnetism sort of provides that role so I'm going to be doing mathematics in some of the details like rankings and then just the other reason some of the opposite of that by sort of seeing this analogy between how will I try to work gravity works we get a better field for just one point what's so difficult about him activity and why So, we start with electromagnetism. We usually start with that, the electromagnetic action,

2:30 so the usual trick, integrate that thing in space-time, externalise it, and that gives you the natural curves. And, as everyone knows, patronising people, that's invariant under-set transformation in that form, and we call those gauge transformations. And what's special about gauge transformations, as well, there's a symmetry of the action. Find a laser symmetry of the action, a space translation of the action, rotation symmetry of the action, they're not gauge transformations. What makes a gauge transformation is broadly this picture, a normal, non-gauge symmetry. You've got your trajectory through configuration space, so here's time, here's the regulations, this should be the dimension. And in all symmetry, you take a path and you move the whole path, and you can't generally get the end-coids fixed, which means, say this curve extremises the action, well it's the symmetry, so that curve extremises the action as well, but it's totally different, it doesn't start the same point, it doesn't finish the same point. And when I go to symmetry, because I can sort of vary this thing locally, I can make a symmetry that leaves the end-coids fixed and move the curve in between. And, again, because it's a symmetry moves to a curve that extremises the action. So what that says is that when you've got a gauge symmetry, you've got multiple curves from given the starting end points, all of which extremise the action, which is to say all of them are equally selected by the principle to be valid solution curves. So you've got a quite dramatic indeterminance here. It's like we don't have any rule or principle for saying which of these is equal to the solution. And by and large, we're not system they have with either. So where we tend to understand this is to say, well, this curve and this curve describe the same space-time, so in some sense our theory has got redundancy in it, and it might be useful to remove redundancy or have a description that doesn't engage with the redundant variables, although here it happens as basically as a species in the second so if we don't want to say that's it I think joy's example that I'm trying to complice things straight from statements for you it's crazy it's like refer to it to a Hamilton point. And that's just as straightforward as a field theory

5:00 as to a mechanical theory. We'll just slide again in seconds to remind people. We've got our action, from the action we make it as a splitter space and time, which people think that's a lot too, to throw up one's hands and hold up for making that step, obviously. But then we get an ordinary space and the ground should define, this is a function of all the fields on a given slice. And then we have an ordinary ground should create, which is exactly the same as the ordinary mechanical wand, because it just happens to be in these dimensions. Okay, play that game of electromagnetism. Ground should again, and we get out this gorgeous looking expression. So again, we're making the space and time very manifest, so the sort of four, the electromagnetic four vector here is split up into a scale of vector potentials. And then if you want to take that from a Lagrangian-Halopanian point of view, it's the same sort of trick, the momentum is defined by the variation of Lagrangian with the base change of the original coordinates. So you get a sort of momentum conjugate to the vector potential, which looks like that, surprise-surprise at the electric field, and then you find the slight surprising will be the momentum conjugate to the skeleton, which actually disappears. It's not an angle there, but it just drops out of the theory. Which is some characteristic that tells us we've got some gauges, the gauges in here, who wasn't likely the obvious and made it from symmetry. And we then work out how Etonian, we find it's a classical form. So that's the classical, this is not just the energy, it's the classical energy to turn here, which is related to the fact that it's a gauge of symmetry. Straightforward observation, that thing, that thing, are all gauge-dependent, in that you've got a gauge of symmetry you can carry out. But when the edge of the thing is you're carrying out the symmetry at a single moment of time. So we've gone from that space-time symmetry I drew up earlier to a symmetry at each moment of time set for this. And otherwise, that indeterminacy of the history of the drawing, where the history is redundant, actually drops down to a statement that at each moment of time the description is redundant. And that might seem obvious, but it's not really the case in general, I think. Anyway, we push on through this description.

7:30 Again, all this is just ordinary classical mechanics just played out in viable dimensions than we used to. So you might use your dynamic equations. This is the analogue of sort of q dot equals dh to p and q dot equals minus dh to q. So you get h dot is the variation and you take the curl of that, you get one of the Maxwell equations, e dot, the variation of A, and you get the other Maxwell equation. And then this is the interesting thing that comes up. When you want to vary the Hamiltonian with respect to the scalar potential to get the variation of this thing, remember that the relevant term there is going to be this term in the Hamiltonian. So when we vary with respect to V there, we find we drop out the equation dV is naught. Okay, so it says there's no surprises that's on massive equations, but what's slightly odd about turning up is this isn't the dynamic equation, it doesn't tell us how the momentum and the position's change of time, it's a constraint on the momentum themselves, and this is the constraint equation that Jeremy was talking about in the last talk. So this is saying not all possible values are being allowed, and the only ones allowed are the ones that satisfy that considering equation. And in a certain sense we shouldn't be dramatically surprised to find something of that form in theory, because we started when we got the Hamiltonian picture notionally with, you know, V's dropped out because it's not identical. We've got this vector field A and the conjugate to have got a vector field E, so that's like six degrees of freedom plus base point, but this one doesn't really have three degrees of freedom because of the gauge transformations, so even once we sort of allow for the fact that we've thrown away B, we've still got transformations like A goes to A plus F, and because of that this has only really got two degrees of freedom, in other words, if you imagine you've got three degrees of freedom to work, the small variations one of them one set of areas just look like that so in effect then to specify all the aids which have got so then you have a rather mis-shapen sort of three of them enter 22 positions there's some of them kill these there please better mental time and that's what gives us

10:00 they're left it's got two degrees it's got two or two four degrees bigger to And that is how you go about quantising, or sorry, Hamiltonianising, you could say, a classical field with this sort of gauge symmetry in it. And in analysing that theory, we've sort of got two ways we can play it. We can either understand it as talking about sort of equivalence classes of gauge tensors. So we can say it's actually a theory of entities of the form A plus grad f for any function f, so that works. These equivalence classes will be able to look for players. Or we can say, I'm not going to do that, I'm going to put some specific gauge condition on A. So I'm going to say, for instance, the divergence of A vanishes, or maybe the third component of A vanishes, or something of that nature. And with electro-animation, because it's a nice, friendly theory, either of those steps That expression is pretty straightforward to understand, and it's really pretty easy to tell whether two A's are the same or different from one another, and it's also pretty easy to specify gauge choices. I mean, I just made two apples off my head, but there's loads of other ones you can just write down. So, either approach is the same, basically, but it's easily doable, and either approach is a basic group of energy. Okay, so that's some of the electromagnetism was to stop. Now, look at classical relativity. Notionally speaking, this is another field theory. In a sense, it notionally looks the same as electromagnetism, you can number the grunge and functional space time. Actually, it's wildly different. Two ways. The first way is electromagnetism's linear, which makes it very easy to construct solutions. General relativity is massively non-linear, so sums of solutions are not solutions. Financiation equations is made that much harder. We find this in other non-linear theories, like sort of Young-Mills theory. or harder to work on for that reason. But even in something like Yang-Mill's theory, so like theory of colour or something, you've still got this picture, the fields are taking values on a nice friendly background space-time. So you, in sort of conceptualising your solutions and finding them, you can imagine, I've got a lump of colour field here and another one here, and I want to know whether I can find stable chunks of colour that are moving around like that or something. You've got some framework that's all created. And that just isn't on the cards in general relativity, because general relativity, as a theory of space-time, just doesn't give

12:30 you any back, doesn't give you any place to stand. I mean, in that way, it makes it really quite, quite, almost unique among the theories of modern physics. So, in finding solutions like that, you've always got to, like, do the whole commute at once, and we don't have any remotely general ways of doing it. So, if you look at the solutions we have, they sort of fall into two classes. So, I'm the one to be found through imposing draconian levels of symmetry of the theory. So, we'll say it must be slightly symmetric or actually class of those but clearly that's only scratching the surface of what's possible or we find them by basically taking the solution we know about and the obvious one's just flat space time and then some wiggling in it and the way you can imagine thinking about that is um if the windows are small enough and you can think of the space time still being there and the wheel being something else stuck on the top let's take an analogy if you imagine you've got a mill pot you've got a little bit of crock of it because the surface of the mill pot is perfectly easy to think of the mill concert is there a little bit on it but you know if you drop dynamite into the middle continent then it's going to be fairly difficult to say which bits the way and which bits the background so that all those sort of things going on general activity and that's what's making it even classically a fantastic people theatre outlet it's one of the most beautiful theatre you've come across and it's also one of the most piggish things to work with i don't know about it so um And for example, if you want to be relevant, at least part of the GR, it's going to be understanding the gauge symmetries in general relativity, comparing them to the gauge symmetries in electromatism. Now, in one sense, as everyone knows, the gauge symmetries are linked to this whole lack of background, the fact that we can use any sort of coordination system like for manifold. But I want to look at it in a slightly different way. So I want to look at it in terms of the way we foliate space-time and the way we reach those forward. So you can see this by now to a previous theory. In Newtonian space-time, there's an absolute, well-given way of designating slices of continuity. You're stuck with it, the fixing is defined, it's given, that's it. So given any two events, you can say, you need to have a very conventional picture of time. And special relativity leases the game of it. We've got the wrenth transformation of it, so this is sort of space, special relativity, sort of prior to modern ways of thinking about it, just as it was understood close to its inception. So we can slice it up in a parallel way, and we can then slice it up in a

15:00 different power of the way here. As James was saying, it just means we don't have any unique rules to say where the two points lie in the same slice, but we still have relatively rigid rules for how we're to evolve the physical state forward in time. So in Newtonian physics we can mention, if you want to specify the state of the world, there's only one way to do it is to specify the state of one of these slices. And then to find what happens next, you move to the next line and the next slice and so on. And in special relativity, you to make any flat slice you like. Having done it, though, there's dynamics then fairly naturally tell you to think about it in these ways. I mean, I'm not saying you can't formulate special relativity in ways to get round that, but that's the natural thing to do if you didn't know about general relativity. And general relativity all help, basically, because to make this sort of slicing, you're taking advantage of strong symmetry properties of the space-time, and of course those symmetry properties just aren't there in general relativity, or not generically. So you can pretty much slice it any way you like. Providing all the points on the slice are space-like from one another, then you can just make pretty much any slice you like, and then you revolve it forward any way you like. You can push these wiggles forward and these ones back. You can even move this one forward and then relieve it and then move this bit. So you've just got the freedom to do that any way you like. And that's at least a way of thinking about the gauge freedom in general relativity. And there's sort of an analogy in the step here to here with what's going on it to managers of so as everyone knows the electrosis and we start with an idea of so it's a gate rich let me start the idea that we can make them reach a phase of nations that's field or what my function then we imagine like to make a different phase transformation in point and in doing so we have to we if we make it set that assumption we have to end making the sub sub sub sub underlying space dynamical and in a sense of negative similar game here we want to make this slice of the principle local then we end up having a dynamical space-time of the line. I wouldn't want to push that as any more than an analogy if you're trying to make that mathematically rigorous then all power rates loose but you can sort of get the h of what's going on that way. Now this sketch shows from image symmetry rather is of a very different nature to the one we're just talking about in electromagnetism because of the way it's linked up in time I mean, because I can choose my slicing arbitrarily and because my gauge transformation sort of moves one slice to another slice, so imagine I've got some slicing like that and then I move it to, you know, if the gauge transformation would maybe move this slice to that one and it moves this slice to that one and it moves this slice to mark one or something like that.

17:30 And that's one obvious sort. But then a much simpler sort of sounds looks like something like this. Here's my calculation. Then my gauge transformation just moves this slice to that one, and moves this slice to that one. that's just saying all the stuff on this slice moves forward to the stuff on this slice, which is a better way of putting it, if that slice is time nought, then suddenly this slice becomes time nought, so the data of time nought is now actually whatever this stuff was doing one unit of time in the future. So, inside that group of gauge transformations lies all the dynamics contained today. So that's a, if you like, a physical way of seeing the point that Jerry was making mathematically, that because the gauge transformation is linked to the way time moves, then the set of what you might call sort of ordinary evolution, which just moves from one slice to another, is contained in a more gauge group. Presumably there is a relation between that factor field and that factor field, because there are two different observers, two different slices. Yeah. And that would be contrary. What was that? That would be related to the contrary invariance or whatever. Yeah. Okay, so that's the classical picture of general relativity. So that might start to incline some of the problems we're going to come across. but I'm going to step back in classical general relativity again and return to electromagnetism and now return to quantum magnetism. And there are sort of two basic ways that you tend to think about quantum magnetism. The first is sort of what you tend to see in any textbook approach that wants to use electromagnetism for itself and the second tends only to be seen as a way of a sort of precursor to understanding quantization about the fields but they're actually formally equivalent. And the first approach is what you might call a particle approach. And this basically involves trying to interpret the quantized field as a theory of particles or a theory of photons particularly in this case. So the process goes broadly like this. We fix the gauge right

20:00 at the beginning. So now we've got a theory of this vector potential here, but it's constrained to only have two main degrees of freedom. We then do a plane wave expansion, your plane waves solve for three particle equations, and the field theory is linear, so any random thing of this form could be expanded in plane waves and expansion and something like that. And then if we might count anything like that, we get this sort of form here, which we sort of recognise as telling us that classically it's the sum of homogosylator terms. Again, this is probably the same thing people are probably familiar with. And then for a mixture of the sort of theoretically moving from the fact that it's a classical monogosylator and sort of wanting to make connections with experimental ways to look at it, and basically to cut it on the story short, we want to interpret these things as operators which are to be understood as creating and annihilating particles, i.e. photons, and then the an ontology of photons, so we create any given state by operating on these things, and by operating successfully we create more than more photons. So we've gone from a classical theory to a quantum many-particle theory, although it's a sort of many-particle theory where particles can be created in the story and do all sorts of non-classical things. Then having done all this, which is all in terms of this free field, if we've then got, say, a matter of fields we're interacting with, then we stick some interaction terms into our opinion, and we analyse those paternitically. So we're sort of going from a free field to an interactive field. In those steps we quantise freely and then we add the interactions in afterwards. And of course handling those sort of paternitory interactions is something that businesses have developed incredibly powerful tools to do, and the main tools are the microdiagrams. I didn't think I got about 12 hours to spare, so I'm not going to go through the derivation of how that works, I don't think we have much anyway, but broadly speaking, formally these are very matters of analyzing what happens when you have a perturbative term, and they reveal, they clearly, one of the things that happens when you have a perturbative term is already tracking the infinite. This is sort of a little bit embarrassing, but we have fairly well developed ways of renormalizing those infinities. I'm really not going to go into the whole normalization program basically if we we sort of somehow truncate the theory to

22:30 be finished we can't find out it and then we gradually let the truncation go to infinity and we didn't change the values the cramps at the same time then you can get nice well-defined answers out and then it might be certain families which were normally normalizable but luckily electromagnetism plus some realistic jacks the matter is a matter of theory so we get better well-defined quantum theory from the future. Okay, that's the third method, and that's what you practically would tend to do if you're doing any sort of atomic physics problems that you would need wrong. However, it doesn't bear a resemblance to the way we tend to quantify some particle systems in non-validistic quantum theory, and if you want to make a more direct link in this thing, that takes its field quantization. So this talk is full of analogies, because I say as we're doing mathematics in grade dev, if you think about what we're doing in a We have, say, a configuration space, say, an individual space, and a particle in an instantaneous state is a position in that space. And for a quantum state of the system, we consider some sort of complex function, which takes values on every single configuration particle we have, and we make sense of that complex function by saying, well, suppose we may have to see the particles in this region here, then we take the multiple square of the function, we integrate it over that region, and then, To get dynamics out, we take the sort of classical Q and P, and we can make the draw operators, which we find like this, so multiplication and differentiation, all this stuff even next is kindergarten, but having done that, we can then find the Hamiltonian in terms of them, we can sort of blithely ignore any terrifying problems in the operation of their commute, and we get out of a quantum dynamic for the system. Now we can actually play at least formally precisely the same game in the field theory. Classical state of a field is a field, actually. We can measure the space of all possible fields, which I'm calling f. And this is obviously a very large, inter-dimensional field. And again, formally by analogy, the quantum state is a complex function on the space of all fields. In other words, give it a field, not just a field as a point, but the whole field across space, and it returns to a complex number. And we again make sense of this by some sort of integrals. We say we've got some subset of all possible fields, call it R, we integrate over all fields containing R, the not squared wave function, and that gives us probability. That thing's hopelessly ill-defined mathematically, but what hell physicists? I mean, formally speaking, the ill-definedness there is equivalent to the whole renormalisation of them, so we have a partial description.

25:00 And then we define operators the same way, so there's some field configuration operators to multiply as well as field value, and some field momentum operator takes as a variation derivative in this value. So we've played a whole game for a field theory just as we can for a particle theory. There's nothing new that comes in in that picture, other than some influential technicalities. What's more interesting, and this is again making content of Jeremy's book, is when we consider a theory that's got gate variation, like electromagnetism. Because then we've somehow got to put not only put in Hamiltonian, but put in the constraint. And that's the constraint of electromagnetism. And the obvious naive thing to do is to impose that thing. But that's way, way, way too strong to imply. The reason why it is strong is because E is something like an momentum, so we're already stuck pretty much with what that operator is, it's something in the form of D by DA, and if we impose a relation like that, then that's an imposition of the operator itself, we're given the operator already. Are you staying in electromagnetism? Yeah. So why keep Gaussian constraint? You have an equalist class of potentials that you mentioned. That means that gets rid of the Gaussian. Oh, sorry, I'm assuming that we are taking the theory quantized and we're going to apply the constraint in the calculation. You're right that we could go to a reduced version of electromagnetism and get straight down. You're right that I could have gone that way instead. So when we did this, we can't impose that, that's hopeful, that that's simply mathematically false unless we do something totally unreasonable to the space of functions we're considering. So what that's analogous to is trying to impose this as a restriction on all vector fields, not just the dynamically possible vector fields, and clearly that's just not possible, it's physically, it's mathematically false. So we need to get some sort of analogy and what we do to that is we say, this equation, this thing is going to rely on the physical states. So there's a whole watch of possible states, and they're all sort of functionals on the space of fields. And on those wave functions there will be some which we call physical that are actually realisable states, and we'll say that the realisable states are just those which rely on these web constraints.

27:30 And if we do that, we get a slightly surprising bonus because we find that this operator is actually the infinitesimal generator of the gauge transformation. In other words, if we exponential fasting, sort of apply the operator e to the i to the e times the lambda, that generates gauge transformations. So if we're saying that this operator ignites all the are invariant under gauge transpirations. In other words, when we've gone to the quantum situation, we've actually already removed the gauge dependence. The only physical states left are the gauge independent states, which is something that came along for free. And this is a demonstration of mathematics trying to respect the fact that actually that apparent indeterminacy was just some bad choice of coordinates and actually it can all be got bit of, and we can get back to a nice, friendly description in terms of unique dynamics. Again, there is an obvious way stronger next to the way the generalist's describing is. Okay, so we've got these two apparently different descriptions, one in terms of field wave functions and one in terms of photons. But of course, those are equivalent. There's a unity transformation that just links to people talk about a singularity between field and particle concepts, which holds in this case. Unfortunately, it doesn't hold wildly well that's a picture of the state of play in quantization of gravity. If we try to apply field quantization methods to gravity, we get the canonical quantization program. We try to apply particle quantization methods to gravity, we get the conveying quantization program, and they're really very different to the way they analyze things. And the sort of two major programs currently in play, sort string theory in the loops-based approach to quantum gravity are essentially descendants conceptually of these programs and lie separated sorry also lie separated by this massive gap of approaches so just discuss again by analogy these are the two methods geberian quantization is going to be the analogy of the particle way of quantizing So we want to say somehow the quantum version of the gravitational field is going to be a collection of particles, what should we call them, because it's very magical, gravatons, and somehow gravity is going to be a sort of mass of gravatons floating around propagating

30:00 balances off each other. It's going to be the same sort of picture we tell for the propagation of electromagnetic fields, so two particles emit a graviton, so a particle emit a graviton and it's another particle and they're going to get that sort of picture. And then we've run the whole sort of interaction page with finite matrix. And again, this basically requires us to take a free field picture of gravity, so we might like to say, we'll just consider this a bit of a test with weak gravity waves propagating on the background, and then introduce the interactions that make the field numb through. And that seems like it comes with something to do in a way, because general relativity is such a wild phenomenon in theory, it's to appreciate it, to squash it into this linear form and then expand it out again. But we've got precedent for this working moderately well in other contexts. It works okay in sort of common-credited lags where, again, you've got a very non-linear classical theory, and the wild theory is a certain thing that works quite well. And all that would probably be fine if a theory could be made to work, so often how physicists work very pragmatically is a scramble to fill in the conceptual gap there behind. And similarly with the other obvious conceptual gap here, gravity is a theory of space-time, yes, so if space-time itself is in some sense this set of gravitons, what do they mean? So it turns into the constant of particle here that the particles are propagating in space-time, at least primary phase it appears to be. So there seems to be a conceptual gap there, which again we could brush which of course it doesn't. It's not be normal as well. It's a wildly not be normal as well theory. That means that we've no need to suppose we can neatly remove the infinities in our final diagrams. Quite contrary, we've already spent infinities doing it worse and worse and worse. And we can try some tweaking the theory to try to remove the infinities, and that's what takes us the super gravity and algebra takes us the stream theory. But by that stage, we're then very far from the start point here, and I think it's a strictly like beyond scope, whatever we know about. In any case, the sort of issues of the gauge literature here are being ignored. They're being assumed in some... They're either being... We're not going to address them, or we're saying we're explicitly going to ignore them. We're not going to regard general relativity, which is a theory, we're going to consider it on this nice, flat background space. So it's to some extent, the virtue of the to some extent a major problem of the other equation, non-clontisation, is that it faces up much more to the gate issues. So, non-clontisation, we're gonna have a theory

32:30 of complex functionals on the space of geometries. So, this is the analogy of field-clontisation, where electromantism has a wave function on the space of electric vectors. Here, we've got a wave function in the space of electromantism geometries. And, again, just as there was a gauge constraint in electromantism, so there's a gauge constraint gravity, and that's the Hamiltonian constraint, the group called the Weyland-DeWitt equation, which I'm just writing about that. H is a really hideous mess of G and DeWitt to the vector G and so forth. But in any case, you've got an equation of that sort of problem. So all that Steve's notion could be going. Technically, it's ludicrously badly defined and difficult to make sense of. I think it's probably worse even than the Codarian approach in a lot of ways. But conceptual problems are what I want to focus on here, which I'm going to get in touch with my Jeremy, are linked to the whole notion of a gauge transformation. This thing is, just by analogy of electromagnetism, is the generator of gauge transformations from the classical theory and going through to the quantum theory. So if this holds, it must hold to get a physical state, then the physical states are gauge invariant. But we've already established in this development that one set of gauge transformations are perfectly or if you come on a garden time translations and time revolutions. So what this is saying is the states do not evolve in time. So time has just completely dropped out of the theory. And it's sort of interesting to stop from there and to ask what happened, who authorised that. What's happening here is quite an important disanalogy. between the gauge concept in electromagnetism and in general activity. So remember our basic picture of a gauge theory, classically, we've got this curve in configuration space, so here is time, and the gauge transformation takes this curve to another curve, and because these both solutions and equations, so in some sense we've got to say these are the same solutions. Okay, and when you run that picture for the Hamiltonian picture, the quantum Hamiltonian picture, what if it actually achieves that goal by saying individual states are actually the same state. So it's not just the solution, which is sort of objective space and time is the same solution,

35:00 state by state we match it up. And that works just fine in electromagnetism, because we can actually, as I say, we can actually match up state by state and say, actually, individual vector potentials related by that transformation are exactly the same state. But in gravity, we can't do that. I mean, the change in curves for gravity reflects a different foliation, and if we make a foliation change, then none of the foliation slices could match up. So, there's no sense in which we can make a point-by-point identification of this thing, which is a solution by solution identification. So, in a sense, the whole Hamiltonian programme has done something we can ask to do in giving this to us, and that makes it very difficult to understand what these states are meaning, and you sort of want us to try and imagine seeing it as the state is, in some sense, describes a whole history rather than, again, a physical state, but then trying to identify how to make physical sense of that state becomes next to impossible. You can do it whole as you're having to define timeless observables that came up in Jeremy's talk. And although that program looks conceptually okay in theory, it's actually incredibly difficult to implement as a practical theory. So that's essentially the state of the play, and most times I'm presumpting to stop there. Except to say that although that picture is relatively old, I think conceptually the advance here is not very dramatic. The canonical programme has given way to the loose-based quantisation programme, which people like Ashdikar and Sperling have pushed, and the Covarian programme has given way to string theory, to brain theory, to M-theory, which I wish I would have pushed. and both cases those are trying to really see if we get to go to the technical problems with I think some success will be according to their defendants but I think the conceptual problems at least on Facebook come through more or less unchanged in that situation. I'll start there. APPLAUSE APPLAUSE Many questions?

37:30 In which sense this is an equation? I'm sorry? In which sense this is an equation? Let me spell it out. I don't believe that it's an equation. It's immediately the way I say it's a division. You mean because it's two of the time? It's completely defined. If you call it an equation, it's a functional differential. Which is completely defined in the sense that you have all sorts of operators ambiguity is only to try to find a small practice. What that physically means is it's completely ambitious. So you can put any meaning you like into that so-called equation. I have what to say. I think I agree with that. I mean, I suppose I said it seems to be like, as long as it's almost born out by trying to play the same game in just two 2 plus 1, then it should be 2, 2, 8, and then it's equal. I'm not protecting you, but it's... No, I agree to you again. It's just the whole... I mean, okay. The practical experience is both these things are not theories, they're programs, and the claim of the... the claim of the program is an equation like that will pop out. If the whole... if the loop-space conversation program or any other approach to solve these essential problems through mathematical tour de force, contour de force, contour de force, actually comes up with a perfect, well-defined, clean equation like that. We have no clue about what to do with it and what it means, and that's the problem of time. And there's exceptional problems which will exist even if, and I think it's big, if the technical problems are completely solved. I thought the technical problems were solved with the Ashtachar variables. I thought you got an internet. Solved it. Solved it. It'll do in the resolutions. I think solved is much too strong. They make some progress. I mean, I think the way constraint in the F2-Hash algorithm is saying that we're not going to use the metric as a variable, we're going to use the connection and various complex functions, and it's going to say, they're saying that their version of the hamiltonian constraint is a much less well, a much better behaved object than the one we get from the ordinary metric approach, and it's true that it's still a horrifically difficult thing to wear. I think you have to believe that they're extremely smart people and they keep on trying. And when they do, one problem is solved, but it appears in a different guy somewhere else. So it's not that it's smart.

40:00 This is an interesting point I've never said this way, so I just want to make sure I understand. with these symmetries, are these symmetries of the solution or of the reaction? These are symmetries of the action. So, coordinate symmetry, you mean like mobile symmetry? Yeah, some of the translation, and the point is the translation is hitting the end points as well. So, it's symmetry fine, but it's moving on to the solution curve. It's a way of generating a new solution small, but it doesn't give you Why doesn't the game so much change elements? Essentially because you can choose this thing I'm trying to, so you choose the function here, here and here and here and here and here and here and here and here and here. You could use it at any point if you wanted to. Oh yeah, I mean I wouldn't say these are the only bottom game, just these happen to be a subset and the amateur is not the bigger set. And those do, some of them will generate, you know, some of them will generate new solutions. The theory of always treat gauge and good solutions to the same solution, but the motivation just comes from it much smaller than us. There's a small technical section as well, if you consider gauge solutions, which can't be reached from the analytics, which can't be reached from the small ones, then you do some sketchy ideas that keep coming out of those. And again, it turns out to be similar reasons, because it doesn't mean anything's fixed. solution so in a sense you this ordinary symmetry changes what I specify I mean it's slightly the holes of action means we need to think about it's bound to value because it's equivalent here specify some initial values I'll fix it I guess the way you want to think about this here you actually specify your point here, then you try a whole range of end points, you find the curves for each end point, and you then look at what the way to change at the beginning, and it's there at the end point until you find one that matches up, so you can get between those views, I suppose. Yes, because you've got a one-to-one match between the data here and the data there.

42:30 I'm just going to repeat a bit of clarification about what the approach is with the Ashdakar program and so on, because, I mean, even these guys are very good propagandists, or, you The technical problem concerning the animal training constraint is all consult, and it is now just a good question, you know, I mean, they say, they would go so far as to say they have a theory, it's just that it's not one that can, you know, be tested, because the predictions it makes are on a scale which is beyond what's observed. No, I think what they tell them is that they've got a complete kinematics, of course, but the problem is just with the dynamics at the moment. So, this is totally how much time you consume. I think that's one thing you could say there. I mean, one thing I'm totally not bothered mentioning in this whole description is what you call the vector constraint, the spatial constraint. I mean, as well as, and I don't know where to go, as well as my, when I'm drawing you see pictures of different filiations, as well as changing the filiation, you can imagine sort of changing the coordinates within a filiation, and that's the sensation within all of this. Now, that means the symmetry at an instant in time, it doesn't affect the temporal nature, it's purely kinematic, because we don't take the dynamics. Conceptually, at least to me I didn't say there's anything that a deep line in the special of the tree but technically it's still a picked handle and I think one thing is to cut the whole new space break it was done I think it makes a hell of a progress on that problem I think that might be one of them for very kids yes so it is but so the different ones constraint doesn't really get me from I mean I sort of got the impression that two or three years ago people were saying that yeah they have no problem but actually you know there's more and more I mean it makes the spirits the spin phones which is supposed to be the solution that's how it's supposed to solve it to be temporal if you're I stopped at string theory because that just goes back in my head, I mean, the joy in my head.

45:00 Can I talk? About propaganda, there are much less propagandists than the string theories, but there are definitely propaganda. If you read Ashton Karras' papers, he's very honest with you. He tells you what he's doing. He tells you that some of the time, most of the time, he's doing effective things. He's not doing the fundamental theory. which is, you can't say that about the string fields, claims that they've got the whole thing. About the Willow-Devitt equation and the kinematics and all that, which is, in a sense, true that they've got something, that they made some progress, if you call it a problem. But the difficulties appear in a different guise. They appear in now what is known as Barbaro-Uritsi Parameter. Now, I'll have something to say about this in my talk, so there you can see the detail of what it is about. But in the end, they end up with unitarially inequivalent theories, which is the same thing as the ambiguity that I pointed out about the real-requivalent theory. So it's just the same sort of unitarially inequivalent theories. Can I ask a question that's just related to this business of primatrization of parents? Evolution emerges as just a gauge transformation. Take any theory into this. it's coming. Express it generally occurring. No, you can always do it. Now, why doesn't a similar problem arise? I suppose you'd have to say because it's more fun. If you could see this on the space time, here's what's better about GR. You've got given solutions to space-time. You can't from that read-off a chord that's times high description. So, to get something of that nature, you're going to have to reduce variation, and that's going to give you a theory for which you find a measure. Muck with the anything generically, and all you think will survive, generally it will survive. It's not a terribly satisfactory answer, but I think that's... But where is that written into the... Maybe we should come back to this. It's not clear. I guess that... I take your point that GR is a relation. What seems to be a play here is simply the fact that it's just something you can

47:30 gauge invariance, simple coordinates. Here's another one. Suppose you've got a theory of a single particle moving in space. You make a parameter relation independent, so now you can use a parameter, however you like, and you can make a gauge. You've got the particle, here's its trajectory through space, and I can make a gauge transformation and move from there to there, or move like that, or move like that, but for a while. Then I can't quantise it. In quantising it, I have made a parameter between all the constraints of my space. I have to introduce something like Revelli, Revelli-Vold constantly describe it. I can't happen to do it because the problem is simply now, but it's still more complicated. Why have I had this much complexity? Because I was stupid enough to push them in the parameters instead of all at the beginning. And if you could throw GR into an isopendory parameter, in theory, with a fixed inter-fixed linear parameter, then you wouldn't have much trouble. You could argue about what that was physically sensible or appropriate thing to do, but you'd have a much better defined translation. So it's really, it's a combination of general covariance plus the atmosphere for the structure. Yeah, well I'd say anything that, I mean you don't say that's a structure, you need, as a structure may be a very natural thing to do to define a particular parameterisation, but you don't need it. A New York time is an attempt to provide a nice Basically, a choice of gain is what you need. You need a particular way of specifying a variation that you can take, which will say, give me any manifold with a geometry on it, and I'll tell you how to foliate it. Yorktime tries to do that. Yorktime doesn't work in full geometry. Play this trick in 2 plus 1 dimensional geometry. Whenever you have a certain symmetry, a certain existence of a thing like a field of some kind, which allows you to do it. Yeah, but you could read that point as you want to see if you can reply to any fault-based time out of the hat. So, if you could find something like that that would work nicely with EGI, condolation is slightly more tractable, but it's not a method of logic. That's Jeremy's method 2, I guess. Sorry, what are you saying about 2 plus 1? Yeah, so one of the main ways which is that you can do it is you can find your time and then play this whole game. But the point about 2 plus 1 dimensional GR is it's dynamic, in fact, it's a dynamic and trivial.

50:00 There's no, the graduation field doesn't have any local degrees of freedom. So, while it's a nice toy in some respects, in other respects doesn't really tell you very much. So, Ed Witton in 1980, yeah, explicitly stored completely in that time. Well, one other question I'll just throw in in the absence of a take is, what's the strongest argument we have that gravity should be quantised in the first instance? Yeah, yeah. Well, to you, in the first instance. Okay, well my answer is, not necessarily that gravity has got the quantile in it, but why we need to have a single overarching theory that includes quantum field theory and classical GR in special cases is, well for me I'd say the need to do cosmology, the need to have a universal theory, I mean, on a conceptual level I don't see how it makes sense to understand, you know, these three conventional theories that don't describe, so, that don't, And on a practical level, if you want to study topology, we need a theory that combines these factors. There are two different...quantization is not the same as... Right. So then the second point is why do we quantize in general relativity? That's the best thing you can think of to do. It's very hard to pull a brand new theory out, perhaps. I think we've done Christy Wickles, I forget if it's Willard or someone calls this radical conservatism, you take a theory where he pushes people who are absolutely responsible to manage. I think one thing that characterises that that sort of great divide over time is to some degree sociological, people in particle physics, like normalisation issues, like non-sourced theories, like climate diagrams, like gramification, couldn't care less about people, about different relativists waffling on about space-time and stuff and so forth, so they get to the end of it when they find themselves a stream of it. Relativists, like the structure of geography, aren't particularly bothered about the other fields that have been later, are generally not quite so focused on the whole realisation issue, so they get down to the realisation

52:30 Can I just ask a question on that chasm? I mean, the impression you get is that there is now more talking between the two communities. But also, I mean, presumably, for example, well, I just don't know, in string theory, although they're sort of doing quantum theory against the fixed space-time structure of some sort or another. I mean, presumably they see that as an interim thing to do, and at some stage they've got to get rid of it. I think they'd have to face up to that at some point. I guess you could, if you wanted to really go ahead, you could say, I don't think even most Australian theorists want to go that far in their knowledge and developments of raccolds and cosmological solutions and stuff like that, but it's not clear when they're going, because their theory is, again I'm not an Australian theorist, The theory is, by its nature, perturbative. It's beginning to try to get into the starting assumptions of perturbative, you take a background with them all, and there's always the hope with this sort of thing that somehow the theory transcends its starting point, and then you look back at it and realise that you just sort of clawed your way out by some rather engaging move, and actually you now see the small structure of it. But it seems that they have played for that sort of objectivity to come across something, because if the theory starting assumptions to survive, then it's not, by its nature, they see it to me, and in some ways that's quite over-generous when I talk about problematics, as an example, he does work, he works fine in the high-energy regime, where it's asymptotically free, you can think about quark-quark scattering. We are doing really all that well in analysing the low-energy regime where the force is strong, and there the whole concept of these available quark particles isn't really a good description what's going on what you've got is different states of the quark field really don't have a particle description control so in certain year you get a picture of a proton and a neutron of the three quarks together and that doesn't feel very much very simplified on to the quark being asleep in Mexico you've got some neutron three-calorie excitation of the quark field they're talking about this

55:00 in the heart of the business of crying out in some way to handle non-poterbative non-poterbative non-poterbative creeps slowly towards it just to add just a bit to what David said it's true that in the end they claim that they should be able to which is fine but the problem is that the starting point is graviton now graviton does not make sense outside linear approximation So, where are we going to go with that? That's right, sorry. You can't go anywhere else. Yeah. So, you take another way of looking at this, this is the string theory. Maybe from grabbing the flat base, there's been string theory in flat space. You have looked even in string theory in curse-based time. But ultimately, one, I suppose, like, political way of being in that road, is after all that effort, you move, you start with a problem of particle in curse-based time to problem of probability, and now it's the same, you've just got strings out of particles. Well, there's absolutely nothing stopping you replacing the cost of energy. Oh, absolutely, but that doesn't help. That's my point, you don't get away with doing that. I mean, I think David Dewey just said, because he did this stuff ages ago, he, we studied down quantum fields in curved space time, but we hope it wants to be understood there properly. We don't understand quantum gravity. Now we understand quantum fields in curved space are really, really well. There's some sort of danger that will take me there. But I just want to say, yeah, there's always nothing about string theory, so I don't want to be in a really strong position to it. Well, in an earlier slide, you were suggesting that the way to think of fogeation invariance is a sort of neutralisation of Lorentz invariance. Now, I'm sure I've come across the idea that actually the fogeation invariance is more radical than that, but considered as an action principle on the configuration space of the theory, you can actually generalise it to semi-humanian 3 geometries and foliate across with not necessarily always everywhere space like hyperservices. I don't know about that. I think if you were to do that, I mean, of course, you can foliate, just as a piece of pure mathematics, you can foliate a metric amount of gold, correct you feel like, and foliation just means a, you know, split it up into four miles across sigma, so sure that's possible, I'm not quite sure why that would take you with your Hamiltonian picture of theory, I mean, there's an underlying principle of using space-like slices, it's a space-like slice, it's a maximum specification,

57:30 on what you can know about the theory. You specify the data on all points which are mutually independent and then everything else is in the thought of the past or the future, like having that slice. Now if you've got a slice that's time-like, then you can feel you want to have some of the points and have to have some of the course dependent on other points. So, I'm not saying that can't be done, and I'm sorry, it has been done, but I think that you have to look quite carefully at the nature of that theory to be happy. It's sort of like light front or light care motivation tricks. They work, but there is something planning on going on. It's not the natural way to... David, I thought your point, the point that I was referring to in your talk, was you were making sort of an oblique reference to this view of general relativity as a gauge theory, in the sense that you generate a non-trivial gauge field by the local free and memory sort of rotation. In other words, you do local arrangements. You make them all independent of each other. And then, for invariance, you introduce this compensated gauge field that happens to be a curvature field. I think that's what you were referring to, not so much polarization invariance. Well, I am sort of a fan to it that I'm If I rule the world, then I will get rid of the word gauge because it's incredibly ambiguous. Sometimes when you mean gauge, you mean . there is some sort of engage in a young mills kind of analogy. Now, there is, both GR and young mills are precisely informed gaiters of this type of inscription signature. A lot of people spend an awful lot of time there that you think on trying to establish young relativity as exactly engaged area. And as far as I can see, the reason it failed are got the issue of the metric on space time and that that just doesn't have a direct analogy

1:00:00 there's a more direct even if we're just dealing with a connection of space time a connection attention bank is a mathematical object to connect for a connection you've got you've got a sort of well you can do you think you've got a young mills derivative it is going to be something like grad x, v, v is something else, and if you've got a tantrum up next, you can write things like grad x, y, and that will actually define things like, because those narratives are the same nature, so you can define things like grad y, x, you can sum over this, it's just a different map of this project. So I think most of the that tends to push that interesting step analogy beyond the analogy ultimately don't really get in the fiber-bundle language the basement for the space time man which is not connected in the face that it's only part which closely connects it which makes all the difference in the world. But conceptually, it's much more simple. All these gauge transformations that you define are defined at a point. How are we going to do that to GR? And in a sense, that's what makes the quantization gauged industry straightforward. Those transformations are totally kinematic. What's weird here is the way in which the kinematics and dynamics is really meshed together in the Hamiltonian constraint. That's why I mean that the space if you're always constrained to get too worried about conceptually it's the sense of objects with the description of a state of the given instance it's the the fact that you've got phenomenal things thinking it is i mean i suppose you're right even a spatial if your office isn't happening at a point um but that's so that's so that you can then maybe mathematically make an analogy but um conceptually the conception you can make an as well as that's. Any further questions? Well, if not, I would like to thank David for an absolutely splendid introduction to the .

1:02:30 Thank you.