12th UK Foundations of Physics — Quantum Gravity & Structural Realism

0:00 So my aim really is not to kind of solve the problem of time or understand really the ways of being solved, but just to kind of understand the issues of what's going on in the topic. So, I'll start by talking about an example of Buret, Buret and Das. I don't know if any of you are familiar with this open idea. I should point out that the title of the talk was going to be, it's a shame to let this go too late, was going to be, is the problem of time and quantum gravity just a pain in Buret and Das? And if you know what that is, but my answer is yes, more or less. If you don't know what it is, I'm going to explain it. So after introducing you and giving the extent of the existence, I'll show how the kind of reasoning that goes on in that argument lead on to Leibniz's shift argument, and it corresponds to one or two in a class. The central difference is that it's really the same kind of reasoning going on in each case, or the use of the principle of sufficient reason, because Leibniz adds an extra big aspect, and and how our idea is kind of akin to my mixing relationship following that um The idea is to move on to the frozen form-lism, which is a classical problem, rather than a quantum problem at this day, but it's very similar to the quantum problem of time, which I then move on to. So, I suppose to begin with, some general remarks then.

2:30 So I suppose it's still quite a new and trendy thing, the philosophers, quantum gravity, and generally when I say for other philosophers or even some physicists that are interested in quantum gravity and that's what I do my work on, the general response is kind of a groaning of the eyes or a kind of get you with the physics kind of response. I mean, as far as the physics goes, of course, it's a bit of an old timer, clearly. And it's... I mean, I guess I was thinking about it in 1916, since, in general theory, relativity was completed. Heisenberg was thinking about it shortly after. A guy called Bornstein was thinking about it in quite modern terms shortly after that. And then, of course, you've got Rosenberg, which was in 1930, where he discussed the idea. And one of the things that makes quantum gravity so difficult to deal with philosophically is the fact that there isn't just one way of approaching the problem, there is a vast range of ways of approaching the problem. These are considered as distinct theories and distinct ways of approaching the problem, as in the case of Hamiltonian quantization or path-integral quantization. So we really should be seen as distinct theories. Now you can generally discern, well you could previously discern quite a major ideological divide between the various factions in the approaches of quantum gravity. So on the one hand we have the particle physicists where the motto is to basically do into gravity as you do to any other force. So you quantise it in much the same way that you quantise the electromagnetic field. On the other hand, you've got the general relativists, who think that gravity is somehow special, it's not like the other fields. And that if you were to get a quantum theory of gravity, then you would have something tantamount to a quantum theory of space-time.

5:00 Now, as far as philosophy goes, most of the attention is going to focus on canonical approaches, and these are primarily approaches aligned to the general notice of this side of the ideological divide. And, well, I mean, what are the reasons for this? Well, firstly, the problems of the kind of particle physics type approaches tend to be the problems of quantum field theory, general problems to do with the framework of quantum field theory, rather than anything to do specifically with gravity. That obviously falls from the fact that they treat gravity no different to any other force. On the other hand, the general relativists have primarily philosophical predilections, really. Okay, so we've got philosophical motivations in making a quantum theory of gravity. We believe that gravity is special, we believe that it says something special about the world that gives us the theory of space-time. and the belief that we should respect what general relativity is saying about space, namely that gravity isn't just a field sitting on top of a flat lantern bowl, gravity actually is an aspect of the curvature of the lantern bowl. So, like my philosophy-minded colleagues who are interested in quantum gravity, I'm going to focus on the canonical approaches, because theory is more transparent from an interpretive point of view. But, I mean, that's not to say that there are no interesting issues in the other approaches. Certainly in extreme theory, extreme theory clearly possesses a high degree of symmetry, and we have certain kind of gauge symmetry, called dualities, which certainly you need some kind of philosophical, which should, which deserve philosophical interest, some philosophical attention really. But what's special about the problems associated to the canonical approaches is that they say very, very profound things, very deep things about

7:30 nature of time and change in space. So in the classical theory of general relativity even, there's an argument that says that change that doesn't exist in general relativity. If by change we mean possession, by change we mean, well let's see, about an observable having one value at one time and then another incompatible value at another time okay that kind of change appears to be ruled out by general relativity why because as we've seen observables have to be data in various quantities the quantum version of the problem kind of feeds off this idea of gauging variance but it applies it to the states instead so the states have to be gauging variance in this case so no states change over time so well i think that will do with a general graph so i'll move on to the idea of an indifference argument The basic idea of an indifference argument is to have an indifference premise. An indifference premise generally will contain, will concern some kind of symmetry, symmetry between objects or symmetry between alternatives. So another idea is to find out whether this formal set of alternatives match on one-to-one or main-to-one physical reality. So you should already see the connection with gauge theories going on. So I suppose the most famous example then, the example of And in this theory, we'll have to consider, this is supposed to be an identical shape of A. How the hell am I going to draw a donkey? Imagine this thing here. It's a very symmetrical donkey.

10:00 Okay, so B1 is taken to be quantitatively identical to B2, distances between the donkey and the failed to be identical and implied with a tree or a tree of donkey. Now the point of this example was to, it was an argument against puridom really. So puridom has an idea that actions or decisions require an underlying reason. Okay, so the Puritan adversary, whoever it was, came up with a case where, according to Guridan's theory, there would be no action or no decision, because there would be no reason to privilege one decision over the other. So if we ask which bale of hay does the donkey eat from, according to, say, the law, the donkey will eat from the best bale of hay where we cash out best in terms of whatever. Biggest, brightest, nicest smell. It's not going to be able to decide on any basis really. It's not going to have a basis for moving to one, to be four, to be one. So Björn's adversary concluded that, well, since surely Dante can move, Björn's theory must have just been wrong. Well, what can we say about that? We might say that Björn's theory doesn't necessarily have to be wrong, because actually we agree with the argument. I mean, after all, if we can maybe consider something like an atom in a spherically symmetric magnetic field, and we place it at the centre of this field, then it's going to be stationary. Why is it moving to one side? Well, it's got no reason to move one way or the other, because it's spherically symmetric. But, I mean, through a variety of intuitive options, which kind of mesh quite nicely with the options we can get in the case of Leibniz, Schindler's world, there's a crucial difference. I mean, these two veils tend to be simultaneously related, and it's one of the same strange types.

12:30 Leibniz, Schindler's world are. So what could we possibly say? Well, we could do, I suppose, that align it, so we can impose the principle of identity and say that, well, actually this doesn't really represent a possible situation, because if these two share exactly the same properties, including obviously the same relational properties, then we're going to have to identify them. There really isn't two bales of hay, there really aren't two bales of hay, there's just one bales of hay. What amounts to the same kind of thing, I suppose, in this case, is that we might say, well, if B1 is a possible object, then that means that B2 can't be a possible object. And that's kind of like maybe an alternative way of putting a peaceful identity in the circle in this case. Or we can say that, well now they are in fact distinct and then you get the problem that the donkey doesn't move, the donkey can't decide. So let's kind of imagine the situation where we have lots of bales of pain, bales of pain in the universe. Some are better than others. But there are two which are exactly the same. And the donkey's motion is guided by the law, deep from the best bales of hay. So the donkey's going around for the universe, and it's chomping on the bales of hay, and then lo and behold finds itself in a situation where there are just these two bales of hay like this, exactly between the two. And we can see that the dynamics, which was previously fine, we had a nice trajectory, are suddenly frozen, okay? The donkey's frozen still, according to the law. And the law kind of cannot determine an outcome in this case, or rather the outcome is determined that there's no relationship anymore. OK, so that's the example of Buridan. Now, in the Leibnizian case, we consider the case where these two are not simultaneously existing things.

15:00 Okay. This is taken to be a world. This is taken to be one in the same space-time which which the two worlds exist in. And, well, and we call, actually, let's call the whole thing world one and world two. Now, instead of Durian's act, we now have to consider that God is sitting and God has to make some kind of decision. Which one of these worlds does he choose to act to on this? Rabbi Van Krasen kind of also saw the similarity between the two cases. He considered that Leibniz is God, the Purians, as magnified. So he's simply the same kind of rational being, perfectly rational being because it only acts for the best possible reasons okay so say that this the universe in this world is the actual world and then this one is alive in its shifted world where so you just shifted the material content of the universe five meters to the east. The example Leibitz used was reflection, was rotation, what was it saying, east into west. But this will be as good as that. So now why did Leibitz consider it? What was he using this for? Well, he was using it as a reductive argument against And according to Newtonian Substantivalism, the points of space-time exist independently of any material contents that happen to be within that space-time. They have identities given prior to the existence of any material content.

17:30 And remember, in these cases, this is a world, and this is a world, and they possess exactly the same space. So, according to Larnitz, the substantivalist is committed to accepting that the actual world located where it is, and this possible world located five metres to the east have to represent a distinct state of the planet. In fact, just to keep it easier, let me shift that one up a bit. Okay, so here, you say, the piece of the universe that occupies point X. Connect the tip of this universe. that. Well, the same point in here is not on the tip. In fact, the point Y is on the same bit of the universe. In this world, so a substandard of this would have to say that these represent physically distinct possibilities, physically distinct situations. Okay? The points have different properties in each of these cases. In this case it's occupied, In this case, it's unoccupied. Now, for Leibniz, that really wasn't, hmm, these kind of possibilities for Leibniz should be ruled out because using this principle of specific reason, using God as the kind of implementation of the principle of specific reason, If it were the case that these representatives have been a possible world, then God couldn't have made a decision as to which one to actualise. Do I actualise the one here, or do I actualise the one here? And the point is that the points of space and the instance of time are homogenous, so there's no way to distinguish between these worlds. There's nothing in this one that could make God want to actualize that one as opposed to the other one. Right, and, well, there's an alternative to the PSR, the principle of sufficient reason. We can impose the principle of identity of indiscernible, and say, well, look, if this one has all the qualitative properties,

20:00 all the qualitative properties of this world, which it does, all the qualitative properties, the observable relations between the objects in the world, okay, they're all exactly the same in both worlds, and if that's so then we should identify those worlds. Okay, we should say that w1 equals w2. Okay, clearly the subsantile is committed to this option, w1 does not equal w2, the world is extinct, because it has different properties, it's probably the localisation. now there is our bird option again you can say that well w1 is a possible world and w2 isn't a possible world because this maybe because the properties of this world are somehow essential to it and if we shifted the bits around in this world we've got rid of some its essential properties and change what it is. So it's no longer that one and the same object. No longer one and the same space-time, no longer one and the same world. So that's the Leibniz shift argument. Remember in this case, these two things are not simultaneously existing in space-time, that represents the possible world. Now, in the case of the Hull argument, the Hull argument was originally put forward by Einstein as an argument against general covariance. And remember the general cullerians is the idea that if this thing is a solution of space-time, And so is this thing, where M is a force, or manifold, which is a space-time, G is a metric

22:30 tensor, space-time, giving distances and angles, and T is the stress-energy tensor, representing the distribution of matter and energy on the space-time. Okay, D is a dipheomorphism, and remember that general relativity dipheomorphism invariant. So general pro-variant property, but this is a model, just in case this is a model or a solution for any range of the amorphism. Now, it's possible to choose particularly a law of examples. Well, first of all, let's take space time, the monocle M, and let's consider a little region, M, we call H for the whole. And within this region, the stress energy tends to baggage. Outside, well, doesn't have to baggage. Now we can choose we can choose the differential markers in D so that D is the identity map outside of the hole but D isn't the identity map inside the hole and we have to say that the two points join, the two pieces join smoothly on the boundary. Well, since the stress, well, let's apply it to the metric first. Clearly, it acts as the identity on the metric outside of here, and it acts as the identity on the stress energy outside of here. Since the stress-energy tensor isn't defined in that hole, it trivially doesn't really act, so it acts as the identity really in the hole as well.

25:00 Now, the metric tensor can't vanish inside the hole. That's the only field that's defined inside the hole. And we set people not equal to the identity within the hole. So there are distinct metric tensors going into that hole, but not outside of the hole. So with this choice of metric tensor, with this choice of diphtheromorphism, sorry, we can see that, well, So, this is a solution, and this is a solution, but also, because, hang on, but also that is a solution. Now the point of this is that it shows that we can have the same stress energy tensor, but different metric fields, compatible with one another. But this, for our answer, is a violation of one of the fundamental things about general relativity, which, well what he thought then was one of the most fundamental things, which was the causal law, which was the stress agent answer was causally related to the idea of the geometry of space punch. So I ruled out Sherlock O'Leary, an account of that. Now of course, Eminem Norton, following a spatial rediscovery of his argument, had rejigged it as an argument against substantivalism. Then we say, well, let's assume sub-santarism. Well, that amounts to the two models related by the theory of organisms of these distinct things, representing distinct multiple worlds.

27:30 because the points in this solution are different geological properties than they do in that solution. And I have to say that even though the two models cannot be extinguished from one another by any physical means the way the argument goes is well, take first of all you have to polliate space-like hyper-surfaces. Now take one of these slides here and assume that you completely, well, remember we're using the whole dpomorphism, which is this thing I defined over here. D equals identity in the whole, not the identity everywhere else. Okay, so models agree all over here, but just disagree within the whole. So we assume that we've got a complete specification of the state on sigma, on this slice at C from zero. and surely according to a reasonable definition if we completely specify the state at this time here we should be able to find the state at any future time okay but given any kind of trajectory that gets here we're going to have different trajectories corresponding to the different metrics inside the hole okay so here's g and here's d of g and properties have been shuffled around points being shuffled around so there are at least two distinct possibilities future evolutions for it from this slice okay and that seems like a massive violation of determinism. Now, of course, I mean, this is very similar to the kind of thing that comes in gauge theory, you know, it's just a general problem of gauge

30:00 the point the point is that all the metrics in here related by these actually correspond to an equivalence class of metrics and then the idea is to say that members, since members of this equivalence class are physically indistinguishable if you treat them as representing one of the same physical state. So the talism is recovered, but at the price of that's substandardialism. Substandardialism has to go after talism stays. Okay, and that kind of approach will generally correspond to the So that kind of interpretive option, where you say that the two distinct metrics are equivalent, or the two distinct models are equivalent. Okay, I mean there are a variety of ways to avoid Evelyn Newton's, Evelyn Lawton's conclusion. We might say that, well, since it's only this kind of indeterminism, where the determinism concerns hexatistic differences, differences where the only difference concerns which objects what play which role so this one this this point is playing the role of being at that point in the trajectory now forget that I can't explain that properly. Oliver considers a lot of these responses it. Possibly all of the responses that have been used to death itself are very likely to achieve these, so I would recommend you go on to that if you want to go on to that or buy it when it comes out in book form.

32:30 OK, so that's the whole article. Moving on, hopefully you can see how it relates to the One thing I should have said was that the position whereby members of an equivalence class of metrics or where models related by different morphisms are identified or seen as representing one the same physical state is known as what I have to do with this. Okay, now let's... Put the bones in my pocket. Okay, so suppose we've got a fake space, and I'll be ambivalent about what the fake space is going to be the base-fasor geometry dynamics, or we could use connection dynamics where we used to value connection and it's called you good mentor so the idea is that the point of this space represents a state either a state like mass or a state like that. Clearly this face-face is going to be either the cotangent bundle of the electric or actually two value connections, depending on which one of these approaches I'm using. Okay, so a point in face-face represents a physical state, but we know that since

35:00 general relativity is a constrained Hamiltonian system, when it's put into this form, we get the word Grandjean has singularities. When we do a Lejean transformation, this appears in the presence of constraints, which is to do with the four-dimensional diphtheromorphism invariant of theory. So these constraints appear in some kind of distortion. So we can't pick any old, any old states, any old pair of variables. We have to pick some those which satisfy the constraints. And imposing this condition picks out a a sub-manifold in the fake space and this sub-manifold is defined to be that bit of the manifold where the constraints are solved where the constraints are satisfied rapidly running out steam yeah okay um so an observable length on the full phase space is just defined to be a function from our course our current phase phase camera function from gamma the real numbers because that's what we normally take to be Or we can engage invariant observables that come off this constraint surface, which we write down the tilde. And in this case, our observables are given by this one. Okay, and these are the observables that we're interested in, these are the physical observables. Okay, and on the surface, the physical observables are found to be those observables that keep a constant long gauge orbit, or those that commute with all the constraints. And of course since, in general relativity, one of these constraints is the Hamiltonian

37:30 and the generated tiny dilution, these observables have to commute with the Hamiltonian. OK, well, of course what that means is that no observable is actually going to be changing. the observables become constants of motion on this scheme. So the value of an observable is going to be the same along the trajectories of the gauge orbits. And this is the direction in which motion is generated. It's a very long gauge orbit, it's a motion, time of motion is gauge transformation, it doesn't make sense, so we're closing out the dynamics. And then that's the problem with the classical theory. Following the quantum theory, I should say incidentally, you could consider quantising this kind of system, where you go into the reduced space, where you take each one of corresponding to a point in a new space where you take the gauge equivalent classes corresponding to these new states. And that would seem to be quite a nice approach to take, it's kind of in line with the gauge theory way of doing things. But the reduced space of general relativity is rather a tricky thing, and you still don't understand it particularly well. Is that still right? You still don't understand the reduced space of particular? It's certainly a right to most people. There's somebody who understands it well now or not. That's another question. Okay, I mean, so those would be quite nice if I was just to follow, but constructing a quantum theory of love and love is very, very difficult. So the other alternative, if you still go either observable gauge invariant or observable gauge invariant, is to perform the Dirac procedure, where we impose the constraints.

40:00 we don't solve the constraints beforehand and then solve the kind of Hamiltonian system that results from that we impose the constraints and operate the constraints on states okay so for example a Hamiltonian constraint acting on a state Well, we can take these states to be a function of free geometries or connections or something. It's quite nice. So, imposing that as an operating constraint for the quantum states in your theory should wield a RIP question. Okay, now, that essentially encodes the problem at times. Of course we can see that it's all followed from the assumption of gauging variance. Whenever two things are a gauge transformation, whenever two things are physically equivalent, you should identify them. okay now it's quite now we can take something similar to what Ehrman said of substandardism okay but in relation to determinism we're going to say that we're going to hear determinism fighting should at least try and if determinists were fighting out of a chance So it shouldn't be ruled out for reasons of that. I think it was methodical, methodical is a methodical type of time. Well, we can see that essentially what we've done here is we've taken a position which is essentially relational. And we've got out the conclusion that time and change don't exist, but you might want to say, are going to be given a final chance, which is due to not for the reason of the simple interpretation.

42:30 So I mean, just to go back to the kind of examples we started off with, to make it transparent how these, how these more modern problems, more complex problems, are analogous to the kind of experience as a black issue. We can imagine that, well, in the case of Buren's past, if we kind of show our little What we're going to see along these days of it is this kind of distinct state consisting of these little systems, two bales and a donkey and maybe permutations of these. In the case of the widening shift, we focus on magnifying glass, what we're going to see is a universe where there's a world where the material contents of the universe are placed on one location in space, and then in another, then in another, then in another, rotating itself. In the case of the whole argument, we're going to see different kind of shifts in the metric, different models, different hyper surfaces. So, I think I should really stop there and finish ahead of time. So that, well, all I hope is that maybe you've got a little bit more intuition for what's going on in the problem of time. I've just got a really naive question going back to the to Buridan's ass and the Leibniz Leibniz has always got in a way he's got an easy way out because he's got a theory of possibility you see this in the debate Clark. Clark is saying, look, it's possible that there could be two things absolutely indiscernible. And Leibniz insists, no, that's not a genuine possibility, because for Leibniz,

45:00 genuine possibility has to conform to the principle of sufficient reason. And so he can reject these indiscernible options. In the case of Buretz and Zass, he can simply say, well you claim that you say the following first of all he'd say look you know that's not what happens in the real world in the real world there would be some subtle difference one blade of a different from the other and then Clark says ah but it's possible that the two bales could be absolutely indiscernible and Linus will then invoke his theory of possibility and say I don't recognize that as a genuine possibility. So the Leibnizian can take that option. Can he or she, it's difficult to see how he or she can take the same option, the same line in this case. For one reason, you know, in the case of gauge and variance, or even in the whole argument, it's one reason we don't have anything like a similar principle that will tell us what counts as a genuine possibility or not because we're not you know we don't have the principle sufficient reason reason whether or not you take that as theologically informed you don't have to take it as theologically informed but we don't have anything similar to that so it's the line i'm just in a sense of sort of questioning whether there's there is sort of parity of responses Yeah, just to make yourself very clear. Yeah, I know what you mean. I mean, when, kind of, just to give an example, when Dirac kind of talks about quantum statistics, and he says that, well, the two systems define some kind of operation, they're indistinguishable, than being a physicist, we should identify them. I mean, I see what you mean there. There's no intuition that's underlying that. It's not a theory of what kind of general impossibility or anything else. Well, in Dirac's case, it's just, you know, Dirac's sort of, and let me put it crudely, you know, I do like it, but it's a crude positivism. They're essentially observational indistinguishable, therefore we'll simply identify the two and of course it doesn't take you know

47:30 much just as you have to sort of move away from that naive positivism and say well actually no there's more to it than that I'm just saying in this case you see what you see what I mean one one one can get out of certain situations one can do a move that life is does against class by saying well look what I mean by from you guys. You can think that as a clever move or as a sneaky move, but it has to be underpinned by some principle. Can't see that move as being even an option in this case. Sorry, Oliver. Count Leibniz, I don't see why he currently uses the same move in the case of the Halal Archimant. I mean, these models are physically in this case, but this is what you do, this is how sound about that God wouldn't have a sufficient reason to do more than I could go really over. No, that's fine if you're liveness and you believe in God and the principle of sufficient reason. Is anyone here going to split up their hand and believe in the principle of sufficient reason? I mean, in our case, I mean, one would like to say, would we like to say, even, these do not count as genuine possibilities. If you're going to say that, fine, but then give me a framework that tells me what counts If you want to introduce some heavy and abnormal metaphysics, you can do a bunch of fields and use your countable theory and say look, this can't represent a genuine possibility because I only exist, I'm a world bound individual. These things represent a world bound individual. Oh, I'll do more of it. The points in this space-time, this is not a physical possibility, because you've changed the essential properties in this space-time. There's no one who's got the same relations, you're using the same thing. We'll talk some more. I think if you talk about this Burgen's S problem, then you have to accept that there is not one grain of paint more on one blade than on the other, because otherwise, of course, it's no problem, it's just a problem. And you have the same, but then if it is so, is this a specific artificial effect of modeling?

50:00 You have very similar things in game theory, where you arrive at very suboptimal equilibrium, and then just some weird guy playing an absurd strategy helps you to be driven out of the much better equilibrium. And also here one could, in such a Buridan case, one could think, well, I mean, isn't there some effect that's not connected with the setup of the problem that just acts as, now I'd put it, an auxiliary motive? And one point one might say, in such a situation, well, I mean, in evaluating all those theories, at the end of the day we would need some perturbative methods or so to get out our numbers, and so we shouldn't be too much shocked by having such a buridam situation. That would be one track. Or the other would be, yes, these things are really equivalent, and then we learn a lot about this equivalence. I mean, that's like interpreting in engaged theory. But then we what does it then mean to fix a gauge, or what does it mean to single out and back the workroom state? Sometimes it works smoothly, and sometimes you cannot impose the symmetry. So then you would make, so to say, work more virtue of necessity, and not introduce this necessity of choice, which is the setup of the Buridan problem. Yes, well, no one's going to disagree, in your case, in Europe's case, that if you've got some influence you're going to get things moving again but the way the problem is set up is that you've got perfect bilateral symmetry that includes having soon if there is a bit of universe out there having bilateral symmetry with the universe or having a completely empty you know apart from this system and in that case you don't get according to that long he isn't going to move unless There are so many fluctuations in the world. No, in the Buridan world. No, but that's to say the way to get out of the Buridan problem might be to say, yeah, there are some effects that are not in the setup of the problem, that are not in the theory at all, that just would act as a motive without trans-rational value.

52:30 So that's one of the solutions you could have, and the solution which means the evocations would get around that. Would that apply, how would that apply to the complex? Exactly because it's a matter of complexity and you might have, you might expect from the model that it's, or from the theory that the final theory determines everything, or so they might. So you take this as an effective build theory? Oh, or even more fundamental, but still by fluctuations around. We're saying that we don't have to do anything with the theory, and we don't expect the theory to talk about that. Yeah, but that's kind of you sharing your theory in a way that most people wouldn't agree more. I agree too. I'm willing. When I started making potential for the whole article, it was, how much did you present it? I mean, you make certain that T in zero is making a whole. Yeah. I mean that's just me to do that. You can have T-shirt. Yeah, it's really important thing. Everything for say now. Really, really. The T's are the people. A couple of things. First of all, pure and glass. It just seems to be pure and glass is unstable. colourful example of an unstable equilibrium, and it just seems to be physically observationally true that, well, not quite observationally, but if you could really hit yourself exactly in an unstable equilibrium, you really wouldn't fall. Most of us would agree, if you really could balance the pen on its tip, then it really wouldn't fall. And you can check this by trying to get as close as you can, and by noticing that the better you do, the longer it takes to fall down. None of us would excise that piece of baseball. None of us would say, well, the perfect pen on its tip.

55:00 There's no such thing. We just acknowledge that it's not very likely to get to it. So, there's no contradiction in saying that if that picture was perfectly symmetric, the ass wouldn't eat. We've got particular examples of perfectly symmetric situations where the thing has to put up with, at least, you know, if we could get it perfectly symmetric, a thing would just put up with not being able to find its equaler. Okay, and we wouldn't do a lot of those extra possibilities. No, in the experimenters, they have to try very hard sometimes to set them up because they need them for a reason or another. When they try to hold it and it's still, they're supposed to be in fact experiments or whatever, they're trying to keep it in an understanding. I do agree, all these things, these are, you can easily say that these really are real possibilities. if you're not going to reduce the bulk of the pale and pale space. Thank you. But, I mean, if that's the case, you're going to have to show that if you do take these kind of arguments and why would it shift everything to be similar to what's going on with these kind of arguments, you'd do them in the same way. I wouldn't say that they're so similar, which makes me on to my head, I think. OK, the shift everything to the left and you can't tell the difference. is an example of a global symmetry where you do seem to have this option to say well i can't tell the difference so i will think that these are different representations or i can think that there are genuinely different studies it just happens but i can't tell the difference but you have that option because it's a global symmetry the problem with the gauge symmetry is that if shifting the thing a bit to the left is a gauge symmetry and you decide that these really are physically distinct possibilities, that this variable that you're shifting it along really does exist. It just happens that you can only observe that variable when it changes. So in these two cases, you can't tell the difference Because, you know, like the simple example that I gave, you can acknowledge that the

57:30 center of mass coordinate exists, it's just that we can't figure out exactly what the initial value of it was. So, if the two, if the whole two parts are translated, you can't have a difference. But in the gauge symmetry, if you're going to acknowledge that the center mass coordinate exists, but that if two parts are transferred a bit, you can't stop the difference. Well, now you've got the problem that you don't just have the option of transferring two parts, you have the option of starting from one point and drifting off after and from there. So, you lose determinism unless you conclude that that's what brings them to the whole life yeah it's not just that they're physically indistinguishable so we want to identify it's that we just determine unless we acknowledge that that variable is okay really if you're going to think of an allergist case of the light it's stupid as you say to the world argument just the I mean, it's certainly a bit of a debate where the global cemeteries map, whether we're going to state something different from the local state or not, but with the gay cemeteries, we've certainly got the choice I do, we use determinism, or, say, or, again, I don't know. to disagree precisely because I mean there's nothing wrong with inter-terminism per se I mean we can imagine perfectly sensible physical sensible inter-minister theories what's bad about the intern is that in this case is that it just seems intuitively be unphysical and that same intuition is what underrides your thinking well look this difference between these two worlds that we've done a global shift of everything over the space-time manifold can't come to us one thing physical that we've seen it um okay that's well it doesn't contradict my claim that you've got the truth between determinism you're just saying that you had the option of

1:00:00 I think I might get them. If I quote, there's my reason to come. My concern about that, sorry, you always have the option of taking other reasons. Pretending it dependent on, you know, varying it with respect to variables that weren't in it. And the variables that weren't in it will always correspond to gauge symmetry. You can vary them however you like and it won't change the environment of function. So if you don't say that you have to identify gauge symmetry, you can say you know oh god you just changed whilst you always put the fifth coordinate you know oh oh that was a bit of a jolt you can always go arbitrary and if you don't have this rule to just take it out of its aperture and apply it all the time then it seems you're giving yourself a Right. So, we thank you. I'm a bit surprised to find myself talking in a session on quantum gravity because quantum gravity is not the business I'm in, but for reasons best known to themselves, the organisers decided that they would like me to tell you something about some a little bit of work, an isolated piece of work that I did quite a long time ago now, and let me say with a significant amount of thought from Richard Epp, that you would like to see what I'm going to tell you in more detail, and that's the record story published a long time ago. And I'm going to be talking about the problem of time, but in the context of a toy model. I don't really know whether it has anything to do with them in general or other parts.

1:02:30 And since what I've got to say overlaps to some extent with what Carl was telling us the other day, the top 13 has just been telling us and I think I will skip some of the introductory remarks that I might otherwise it's made, except to look at my one over head on general relativity, with a few just to look at some notation that I'm going to use, to remind you that there is an initial value formulation of general relativity in which you foliate space-time manifold relations. and that if you do that, then the space-time metric decomposes a three-dimensional metric which is supposed to tell you about the distances inside one of the hyper-surpieces and a set of functions, one of which is the lapse function which tells you about the amount of proper time that elapses from the normal between one hyper-surface So, according to what Dean tells me, or told me in his last talk, and everybody knows this stuff already, so I'm going to sort of skip to details unless that will seriously inconvenience everybody. I've just got you to know that the N that will appear on the number of my overheads is this NUPS one. And so, for the model, I can't understand that. Well, if I want to make a connection with general relativity, then I want to say that I'm thinking about a one-line and a one-fold, and I'm talking about a one-fold T, so in a sense, space is just a single point. The only surviving component of the metric tensor in this model is NUPS. And so here is one version of the model that I want to think about, and it is almost simply

1:05:00 simply a simple harmonic oscillator, except that I have written here a lapse on the horizon. And because of the way in which this function appears in the regression, this model is invariant under timely gravitation, so that if I replace time by some function of another time coordinate, and that's Halston transformed by one form, and the X is the oscillator, the variable here is the scalar of this oscillation, and then the original X and the original expansion, that's exactly the new one. So if we have this model, then we can try to follow our noses and see what is the mechanical information of this. and you vary the two functions that you have, the x and the n, you get two order-of-range equations. And the first one that you get by varying x is an equation of motion that you can find solve for x. And when you vary the lapse function, you get another order-of-range equation, but it is not an equation of motion, it gives you a constraint of solutions to the first equation. The model for high blocks is sufficiently simple that I can actually solve that first equation. The solution is the one that I just found here. Since it's a harmonic oscillator, you are not surprised to see that it's a linear combination of a sine and a cosine. But what is important is what is the argument that it's inside the sine and the cosine. And the argument is only that I talk to the cycle psi of t, which is the integral of the lapse function. And according to what I would tell my students while I lecturing on general relativity, this psi is the proper time. And, well, I'll say it's a bit more like that, so I'll make it wrong. And if I plug this solution into the second equation, it gives me a constraint, but the constraint is on these two constants of integration. What it tells me, in fact, is that the energy of this oscillator has to be equal to the constant E law that I put to the higher energy.

1:07:30 So I can solve the Euler-Lagrange equations and can straight make some sort of sense, so if I have been planning some problem, I start planning a problem and I ask what I recommend to the mentor, the momentum conduit to x is 1 over n times x dot, x dot is the end of t, the momentum conduit to the lapse function is just 0 because the ligandian doesn't have I can first figure out a Hamiltonian, and the answer that I get is typical of the answers that you get in this kind of system. The Hamiltonian is lax on two times the constraint on two. And that means in particular that when the coordinate of momentum for x, I know something right, when they satisfy the equation, then the halitone is like x to 0. If I try to quantise this system, of course I run into a couple of well-made problems. and the first one is that since the momentum of conjugate collapse is identity zero, then I cannot impose a logical commutator that I would like to impose a positive sense, and the most important problem is ag-I-P that is straight. And so I'm going to have in mind what I think is potentially a description of that, plus. The first thing I'm going to say is that if I think in terms of the wave function, and I think that the physical momentum comes against the lapsed function is minus IFRD by the n, and what this tells me is that when I act on wave function, with that I get zero, wave function is independent of n, and so n should not really be appearing in my content theory. And so now the way that I can divide the quantum of the quantum is that I can forget about the constraints and I can just pointise the system of x in the usual way and that will give me a Hilbert space which is spanned by the particular low-end star space.

1:10:00 I'm going to identify a physical subspace, if I'm familiar with this physical, and sub-family vectors in the subspace satisfy this constraint. So the Hamiltonian, they're acting on any one of the physical vectors, which is zero. And that, from the point of view of the constant theory, is what leads us to the constant time, because I then have to ask, what are the observable quantities, what are the operators in my theory that represent the observable quantities? They must be operators on physical. What that means is that if I act on one of the physical vectors with some that operate so that I'm not representative of a solution, obviously the answer must be another vector that will be right as well, it's physical. Well, H acting on one of these vectors would be 0, H acting on the new vector that I get must be so 0, and that means then that the commutator of A of H when it acts on any vector See ya.