Symmetry & Structures in Physics — Noether & GR / Everett

0:00 I'm sorry, Thursday. Thursday is 4.30. And the other announcement is, if you haven't paid Jeremy Butterfield three or six pounds, and he'll determine which one of those numbers applies to you, you should see him, please, at the last break. Our first speaker is Harvey Brown, who will talk to us about aspects of symmetry. Some of you will be aware that last year, in the first of the Archie of Princeton workshops, one of those spoke in Princeton, and there was interest in certain quarters for these talks to be re-given here. Unfortunately, half the people in those quarters can't make it to their first quarter, I should say eights. So I'm going to try to reproduce at least some of that talk that I gave to Princeton last year. I suspect, I had a little bit longer, so I suspect I won't be able to get through it all, but what I would really like to do, first of all, is make some remarks on Neuter's first theorem, in particular, to look at some applications in quantum, some very simple applications in quantum mechanics and electromagnetism. Well, it says new here, this is, of course, a dangerous word. it often means that you haven't seen it in the literature which is not quite the same thing and in fact yesterday I was speaking to Ian Acheson in theoretical physics and he pointed out to me that one of these applications may in fact be reasonably well done at least in relativistic quantum field theory anyway these applications have hopefully some interesting foundational implications so that's really And then I'd like to make some remarks about general covariance in general activity. And in particular, how to read Einstein's 1918 reply to Kretschmann? Because you remember that Kretschmann raised this sort of acuity charge. I mean, Kretschmann pointed out any theory, any physical theory, with a bit of sweat, can be made to be pros and general covariance. So what's the big deal about general covariance? Why was Einstein taking general covariance so seriously as some sort of physical principle in 1916?

2:30 And there we'll have a look at the second theorem, which really came out of Hilbert's work in trying to understand a series of issues related to general covariance, the Cauchy problem, the problem of underdetermination, and also the problem of conservation principles in general relativity. And we'll see if Einstein's own work that sort of shadowed Hilbert's work has something to do with a change that occurred between 1916 and 1918 on Einstein's thinking about children's invariance. Well, there's another issue. Incidentally, this first work, of course, I should mention, that's in collaboration with Peter Holland, in the second collaboration with Captain Brady. This work in particular is on the Pestberg Filsai archive. This work is still in progress. Some years ago I was doing some work with Guido Bacigalupi and Muratio Suarez and again for how one transforms operators in quantum mechanics in relation to standard symmetries, space-time symmetries. But I suspect I won't have time to get to that today. In relation to Neurta's first theorem, well, the first term of Neurta, 1918, of course, is famous for systematizing the connection, the deep connection of physics between symmetries and conservation principles. And I should say, by symmetries, I mean global continuous symmetries, symmetries that depend on arbitrary constant parameters, not functions of space and time. Now, in 1954, Wigner warned us about the dangers of what he called a façade identification of symmetry and conservation laws. Because he pointed out that you can have interesting dynamics that's not Lagrangian. Of course, Nuremberg's work all had to do with physics can be given a Lagrangian formulation. So that it's equations of motion that follow. It's all Lagrangian equations from Hamilton's. But, of course, most interesting physics is Lagrangian, but we could perhaps, in a sense, extend Vignes' warning slightly and say, well, notice that there are situations where the symmetry itself may not even lead to a continuity equation, let alone an order of charge or

5:00 a concerted quantity. When I say charge, I mean, for example, I mean, this is a point, for example, that Troutman stressed in his early papers, I think it was from 1962, actually, on nervous periods of consolation principles and their relativity. Even if there is a continuity equation, it doesn't necessarily lead to an concerted charge, or at least not in all cases, because it depends on certain boundary conditions. The continuity equation itself may be trivial in the light of the equations of motion. In fact, we're going to look at cases where the continuity equation coincides with the equations of motion. They're completely trivial given the equations of motion. The charge in certain quantity need not be real valued. Even the Lagrangian need not be real valued. There are cases where you have complex Lagrangians. and we'll look at a case where the known asymmetry itself, the transformation may not carry well-defined dynamical states into states in other words, it may lift say in the case of quantum mechanics it may lift the wave function out of the open space so let me just remind you quickly of how this phrenodon stress theorem goes let's think, let's imagine a situation where so generic situation we have an action which is the integral of some Lagrangian density which is a functional of certain fields a system of fields psi i and their first derivatives I'm going to restrict this to dependence on first derivatives but it's going to be high derivatives I'll probably generalize it suppose we introduce specific infinitesimal transformations on both the independent variables, just to see what happens. Well, then we want to calculate the variation in S, which is defined in this way, and you'll know, particularly if you're familiar with Hamilton's principle, which is a special case of this, that this is going to be generically equal to, the integrand will be a sum of, it's a linear combination of Euler expressions, and here, a total divergence term.

7:30 Now, the Euler expression, again, this is really the variational derivative of the Lagrangian with respect to the field in question. We're summing over the i's of the various fields. And now the question is, what conditions must hold? In other words, what must be a constraint in the form of the Lagrangian such that this variation originally not to look at the case where this variation strictly matches I should say by the way this is the first-order variation because we're dealing with infinitesimal transformations we need to consider the first-order variation. Nerter raised the question what happens, what conditions must hold if delta S is equal to zero? The first variation vanishes. That was that was Nerter's idea of the symmetry but of course now Nerter wasn't aware of the fact that there are many examples in physics, of symmetries that do not leave the action stationary, when in fact stationary up to itself a divergence term here. We'll see examples of this a little bit later on. So the question nowadays, the nervous problem is posed in a more general way, and even this isn't the most general, perceivable way, but it's the sort of standard way nowadays. What conditions must hold if the first variation in the action under these transformations is equal to a surface term, in other words, the integral of a total divergence? Here we're using I-sine's convention, the sum of these is mu and the z's. This is sometimes called quasi-variant. If these transformations satisfy this condition, it's easier to show that the Euler-Lamrange equations, the equations of motion once you apply Hamilton's Principle, for each of the individual fields, in other words the EI is equal 0, most of the equations of motion they are form invariant under these transformations and that's what we mean by symmetry. Equations of motion form invariant co-variant, under the transformation. So solutions are mapped into solutions, the same equation. So this is really a sufficient condition, not a necessary, but a sufficient condition for a symmetry to hold. Well, we're going to show, in a very general way, that if the transformation is global,

10:00 depending on these parameters, there'll each one of these parameters, you get an equation that looks like this. Again, it's a linear combination of the Euler expressions, equal to some specific total divergence. Now, and if we assume that the equations of motion hold for all the fields, in other words, all the fields are dynamical, they all satisfy equations of motion, then this left-hand side vanishes, and we end up So Narza's first theorem says that if you have a global symmetry and you apply the equations of motion, in other words, you're on the shell for all the fields, all the fields are dynamical. Incidentally, that's not always the case. I mean, we have plenty of interesting examples in physics where we have background fields, absolute fields if you like, in which case you will not get a continuity equation. This was a point that was nicely stressed by Charlie and others. But generally, the fields are dynamical, so at least often the fields are dynamical, all dynamical, so we get a continuity equation. And this term here takes a simpler formula that would be the case of if it was a local symmetry, depending on space and time. Now, let's just go quickly to the Schrodinger equation, and here I'm going to look at the free Schrodinger equation. Here it is. There's no external potential acting on the quantum particle, the Schrodinger particle. And the standard Lagrangian density for this field is given by this expression here, which involves psi, this complex conjugate psi star, and the first derivatives. incidentally it should strike you as being interesting that you have first derivatives in time first derivatives in space but you only get first derivatives in time in the equation whereas you get second derivatives in space this Lagrangian clearly contains redundant variables you can find other forms of Lagrangian that are equivalent to this which is by adding a divergence in which that redundancy is reduced but we'll stick with this Lagrangian here Now let's consider the following internal continuous symmetries. Which means that we're not going to be interested in any transformations,

12:30 we're not going to be interested in any transformations in the space-time coordinates, just in the fields. And it's the simplest thing you can imagine. You just add to the field a complex number. It's like a constant complex field in space. And similarly, in the case of the conjugate variable, now the Lagrangian is not invariant under this transformation, but it is quasi-invariant. In other words, it's invariant up to total divergence. If you run through the standard Nortar expressions, remember the Nortar theorem in the non-relativistic case gives you a continuity equation. There it is, d rho by dt plus divergence of j. and if you run through the standard Noether formula for these expressions the continuity equation out pops the Schrodinger so this is a case where the continuity equation given by the Noether formula for a particular symmetry, global symmetry is the Schrodinger equation itself you've learned absolutely nothing interestingly close-analog in electromagnetism. There again is our quantum mechanical abrangian density, there's the Schrodinger equation, and here's the symmetry we've just been looking at. which is just the same as the Schrodinger equation. And then you play the usual game you play in Lerner's first theorem. From the equation of continuity, if you take the integral over the whole space and you assume that psi goes to zero at infinity, spatial infinity, then you can show the d by dt of the wave function itself, the integral of the wave function, zero to zero. In other words, the neurons have conserved charge associated with this funny symmetry, which incidentally doesn't preserve the state because it doesn't preserve square integral

15:00 relative to the wave function. There's no reason against using the machinery of neurons there. Notice there are never claims that the transformations on the fields carry, in this case, states into states. So you end up with d by t of the integral of the wave function, not the integral of the wave function squared. Now, there's a very familiar application of Newton's first theorem in quantum mechanics, which has to do with another internal symmetry, which is just the global phase transformation on the wave function. This is the U1 group. So, if you have a global phase transformation on the wave function, that is the symmetry of the Schrodinger equation, of course, indeed even of the Schrodinger equation with potentials. And the neural charge in that case of the modulus squared of psi, the integral of the modulus squared of psi, so it's in a sense of probability of finding the particle is constant over time. That's to say, the particle, there's no creation and violation of the particle. But here we get something that's rather odd, we get the integral of psi, so it's considered over time, in relation to this dynamics. Now, if we think of Maxwellian electrodynamics, dynamics. Here's the standard Lagrangian density for the maximum field, where F mu, of course, is defined in terms of the first derivative of the four-vector potential. And here's the Euler-Lagrange equation associated with variation with respect to the A mu's using Hamilton's principle. But notice, this here takes the form of a divergence, an equation of continuity. Maxwell's equations themselves take a form of a continuity equation. So you might think, wait, if that continuity equation is normally associated with symmetry. So what's the node of symmetry associated with the very equations themselves? Then it's just this one. It's this very simple one. It's the analog of the translation of the wave function. Here it's just the translation of the vector potential. so we add a constant four vector field to the vector potential we end up, again, we just go through the standard nurture machinery we plug in all the

17:30 again, the electron the levongeon in this case is strictly invariant not quasi-invariant, strictly invariant you can see this just by inspection and you end up with an equation of continuity which is just And here, if the magnetic field is assumed to vanish to infinity, then the concerned charge is just the integral of E. Now, this is not a surprising result, because just think of this Maxwell equation here, written in three vector forms, this familiar Maxwell's equation. Obviously, if b vanishes to infinity, and this is a non-trivial condition, generally not true, but if b would have vanished to infinity, and you take the integral of this, you're going to get the d by dt of b, the integral of b is equal to 0. which we, Peter Hall and I were unaware of until very recently, literally this week, which looks like this. Suppose instead of adding to the wave function, a piece of free shredding of dynamics, a constant complex field, we now add c dot x, where again c is a complex vector, and we do the same with a complex conjugate. Lagrangian is quasi-invariant, so this is a symmetry of the Schrodinger equation. And here's our equation of continuity, where this is the row and this is the J. Notice that it involves first derivatives of psi, multiplied by a special coordinate, and then there's something linear inside here. But when you actually calculate this this just reduces to this equation here, which is just the Schrodinger dynamics multiplied by the spatial coordinate, which cancels on both sides. So we're just back again to the Schrodinger equation. So this is another example of inertia symmetry whose continuity equation coincides with the original equations of motion.

20:00 And in this case, the conserved quantity is the interval of psi times the spatial coordinate. Whatever that means. Well, again, there's an abusing analog in the case of electromagnetism, which is just that these two transformations are essentially equivalent. You either add an antisymmetric constant tensor field to second-rank tensor field to F mu mu, or equivalently, you add this object here to A mu, which is the analog of that dot product. Again, you go through the neural machinery in the relativistic case, you end up with a continuity equation assuming the equations of motion are true, and now this four current is given by this formula here, which is very, it's a very analogous to the current here because it involves derivatives of the field, the first derivatives of the A, the new field, times the spatial component, just like here and here, and something linear in the field. At the moment, we're in the process of figuring out what the conserved quantity looks like in terms of the three-vector formula. We're not quite there yet. But this is just to point out that there are applications of Nürnberg's theorem, but quantum mechanics and electromagnetism, where one sees that the continuity equation is trivial in the light of the equations of notions, I think they coincide, and where the conserved quantity itself might not even be something real, real value. So much for the first theorem. Now I want to move on very briefly to the second part of the talk, which has to do with a local symmetry. Of course, Nernin's work, 1918 work, was really much, much more about general relativity and local symmetries than global symmetries, even though she's famous for the connection between consummation principles of symmetries in the case of global symmetries.

22:30 But really her work was designed to understand the mathematical role of general covariance and general relativity. That's where she was following in the footsteps of Hilbert. And that's why she was employed by Hilbert to work on this, because it was really an issue about what is the role of general covariance and general relativity? Why does it lead to underdetermination of field equations? Why does it lead to complications of the Cauchy problem, in other words? and also why doesn't it lead to standard conservation principles I mean if you have a local symmetry that's to say a symmetry that depends on functions of space and time and not just constant parameters very often a global symmetry will be a subgroup of the local symmetry so if you start with a global symmetry that has a non-trivial global subgroup you then apply this first theorem subgroup. You should get out conserved quantities. But you don't have conservation principles in general activity in the same way that you do, say, electromagnetism. You just don't get them. Because all the continuity equations involve covariant derivatives and not partial derivatives. This is already fairly well understood in 1915-16. The question is, what's going on? Why are you not getting interesting conservation principles given general covariance? But why general covariance in the first place? Well, Einstein, if you read his famous paper on introducing the field equations in 1916, I mean, his review paper has to say in 1916, Einstein introduces four separate arguments, four independent arguments of general covariance. I'll say what they are in just a second. It's clear that Einstein's instincts are outstripping his ability to say why this thing is important. Then, of course, there's the whole problem in life as equivalence. How to remove the underdetermination? This is the thing that this is... In this company, I know I don't have to say very much about this. Everyone's aware of this problem. This is a problem, of course, that held Einstein back for a couple of years from developing the general government and the field equations. And then how do I overcome...

25:00 Now, in the case of the whole problem, of course, Einstein realized that the level of suppose was the solution. Hillman himself was interested in the problem from the point of view of the mathematics of partial differential equations and the Cauchy-Koblitz-Skaia theorem and so on, the Cauchy problem, and what role general covariance was playing in complicating the issue. And that, of course, led to Nürnberg's work in 1918. But then, next question, I want to concentrate more, in fact, Catherine will say something about this in her lecture next. I want to say something a little bit more about Kretschmann, because there is a clear change in attitude from 1916 to 1918, variants. And I want to say that this is really, this does have to do with the question of conservation principles. And it's when Einstein was kind of shadowing the Nerta type, in fact, he was anticipating, in part, the Nerta type analysis of general relativity. So let me just say, Einstein's justification of general covariance in 1916, he applies to Marx's principle, the weight equivalence principle the non-operational significance of Gordon's and rotating frames and the point coincidence argument which is just like this equivalent which is how we solve the whole problem which of these is the most important argument who knows is the collection any better than any single argument again this is I think a place where the great man Unsure is the best way to defend or to justify the introduction of general covariance in GR. And notice that Einstein originally thought general covariance was a generalization of the relativity principle. Galilean, Newton, Einstein, relativity principle. And again, this is highly questionable. But notice that what I want to highlight is the change of attitude between 1960 and 1918. Those of you who are familiar with Einstein's early papers on GR are aware that he constantly used a particular gauge, in other words, he chose a particular blurb of a set of coordinate systems, which are this volume-preserving gauge, where you choose your coordinates in

27:30 such a way that the square root of minus g, g being the determinant of a metric, is equal to 1. Notice what he says here. had an important simplification of the laws of nature and considerable, and I later in the paper, considerable simplification of formulae interpolations provided by this choice. I say clearly things, despite all of the arguments in favor of general covariance, that there are coordinate systems that simplify life. He says this explicitly in two places. But by 1918, in replying to Kretschmann, can be written generally covariant. So what's the significance of this principle? Einstein says something that anticipates a line of thought that runs all the way through the literature from Einstein to the present day, which is something like general covariance means the absence of absolute objects or something of that type. Namely, it's true that every theory can be written generally covariantly, but the ones that you are interested in are the theories which, when written generally simplest form. That's the issue now. And Einstein says that the general covariate formulation of the theory should be the simplest and most transparent one available to it, and that holds for GR. It does not hold for the Newtonian theory of Brahe. Now, it seems that there is a significant shift in his thinking from here to here. In other words, if he'd been arguing along these lines, it's hard to see how he could have said this in 1918. So what's I don't know for sure, and maybe this is a question that should be related to the real historians. But let me just mention a letter that Einstein wrote in the 1960s, a file, where he says, I belatedly came to the view that GR becomes more perspicuous when Hamilton's scheme is applied. In other words, we apply Hamilton's principle. I mean, in that sense, we would say the Lagrangian formulation is applied. Now, when no restrictions are put on the choice of the frame of reference, So he's now saying, I'm backing away from that special gauge that gives you the square root of minus g is equal to 1. Special coordinates. I'm backing away from this. It's true that the formulas then become somewhat more complicated, but more suitable for applications. For it appears that the free choice of the reference system is advantageous in the calculations.

30:00 But this is the point I wanted to draw your attention to. The connection between the general coherence requirement also becomes clearer. Why is that? Well, I have just two or three minutes left, and I'll run quickly through the second theorem, and what Einstein did in relation to the second theorem. In a sense, it's sort of an anticipation. Remember, in Nourdes' theorem, we're looking at the problem of the quasi-stationarity of the action under certain transformations. If the transformations are local, In other words, they depend on certain functions of space-time instead of parameters. And the variations in the field themselves only depend on these functions and their first derivatives. Then we end up with what I sometimes refer to as Bianchi. I think this is a slightly unfortunate terminology. But nonetheless, we would prefer, Catherine and I would prefer to call them neurotic conditions. you end up with off-shell, that's to say certain relationships that hold independently of the equation of the motion and they take a form again of a linear combination of the other expressions being equal to a total divergence of again another linear combination of field equations where these coefficients depend on the nature these transformations. Now, what happens when you apply this in general relativity? Many textbooks deal with this case in general relativity, where they just put in the Hilbert action, run through all the machinery, and out called what contract on the Yankee companies, which were not any of them, because they just mathematically did it. But Dirac, in his lectures on general relativity, does something a little bit different. He says, let's apply the neurotic machinery. He doesn't refer to the neurotic, but he's saying, let's apply the consequences of general covariance for arbitrary matter fields and for a situation where you don't even know what the form of the Lagrangian for the gravitational field is. In other words, he just leaves open the form of the Lagrangian. And then he gets out an interesting result. It's not a mathematical idea because he hasn't specified

32:30 It's just the reciprocal response equation, it's the statement that T, the covariant divergence of T mediums equal to zero, in other words, it's the equation that tells you how the matter fields are being affected by gravity. Now, Catherine and I struggled to understand Dirac's proof, so we thought we'd do it ourselves, It's very straightforward. And the way it works is you simply assume the quasi-invariance, and this is the trick, of both the gravitational Lagrangian density and the Lagrangian density of the matter fields. So you assume each one of them is quasi-invariant. In other words, they're both generally covariant. You give rise to field equations that are generally covariant, independent of each other. Now, again, if you put in, suppose we don't know what the form of this is. We won't have to specify the form of the gravitational branch. We will not assume it's the same as the Hilbert action. Then, if we apply the norm to machinery, outprops an equation which is independent of the equation of motions, which is a divergence equation, where this term here is the variational derivative of the total Lagrangian with respect to g minu. In other words, it's the Euler expression for the gravitational field. And this term here is the variational derivative of the matter Lagrangian with respect to g minu, which is by definition proportional to t minu, the matter of stress-energy tensor. Now, if we assume that the gravitational field equations are hold, in other words, we assume whatever the field equations are that they hold, we don't know what they are because we haven't specified the Lagrangian, then we end up getting a response equation, a familiar response equation. In other words, we end up getting the covariant divergence of the stress-energy tensor relations, which means we know how, in principle, gravity is acting back on the matter. And Einstein didn't know this in 1915. Nobody knew it in 1915, but this is what you know as a very geodesic principle itself.

35:00 The fact that test bodies belong in geodesics in general relativity follows from this equation. And notice we've derived this equation without assuming a form of Einstein's equations. Just by assuming general covariance of both these bits. Because the neural theorem, generally speaking, tells you when you have a local symmetry, the equations of motion are not all independent of each other. So you're going to get little bits of dynamics being essentially determined by other bits of dynamics, roughly speaking. To our surprise, when Catherine and I were reading Einstein's 1916 paper on applying Hamilton's Principle in General Relativity, this is probably a lesser known paper, we found what looked like exactly this argument, or pretty close to this argument. In other words, Einstein was showing, based on general covariance, how the response equations could be derived without even necessarily putting in the form of the field equations. so this might be one of the factors, I'm just putting this forward as a speculation this kind of derivation might have been one of the factors that made Einstein realize how potent general covariance was of course with other things added such as the non-trivial assumption that both bits of the LeBron are independently of general covariance and there are other assumptions as well of course well I won't go into them at the moment Einstein clearly was impressed by the power of general covariance in establishing some kind of connection between symmetry and... Well, the closest thing you get to a conservation principle in general activity is the vanishing and covariant divergence of TBNU. That's what he was getting. So maybe that was one of the factors in the shift in Einstein's thinking between 1960 and 1918, when he says to Kretschmann, the issue is not general covariance per se the issue is whether or not the theory when written generally covariant we, finds its most natural expression I'll give him a back thank you Thank you.

37:30 Thank you. for example that very fascination that's not going to present spectacability what happens there what is the space of what is the space certainly the solutions of the Schrodinger equation have to be You have to belong to an open space for the thing to make sense from a problemistic point of view. But no, there's no role in this. Yeah, but what is the space that's defined or that transformation's defined? Well, it's the space of complex-valued functions on space, the three-space. Differentiable, same-liantical? They have to be differentiable, I imagine. For all energy machinery to work. They have to be well-behaved, but not necessarily starting to work. Don't you need to actually say what space is? Don't you need to publish what space? It certainly won't be the hills of space. No, that's right. But as I say, there's no reason in principle why you can't use the machines. It's just a mathematical exercise. And I was struck by the fact, I think Peter and I was struck by the fact, that here's a case where the continuity question didn't arrive, but it's just a question of emotion. I don't see any reason or principle why you can't apply nervous theory in this case.

40:00 I'm sure you can. I mean, I think the question is that if you get to rigorously what we prefer to be, then it's just a question. You're right, I suppose the space in this complex function should be specified, but as I say, we won't be able to do this. You sort of raised the question when you moved on to the topic of general covariance, which was why don't you have conserved charges because surely the global administrations are just a substitute to localize and you seem to have made that hanging. Well, I mean, isn't it the case that for a generic space-time, you're not going to have any sort of global transformation, particularly when it's compact or something like that? And in cases where you do have the right sort of space-time to have a genuine global transformation, so you've got a well-defined global transformation over all the space-time, then in those cases, you don't have some sort of concern. No, no. Thank you. Thanks, Ari. understanding of it is this. You're worried about the fact that generic spacetimes won't have, there won't be a single coordinate system in the whole of spacetime. But suppose you just take a patch of spacetime that does have a coordinate system. Now, you can't coordinate with a single coordinate system. You can still define global versus local transformations in that patch, and that's really what's going on in this thing. That's my understanding of it. And secondly, what happens is that generically, I don't know whether I'm anticipating something the captain will say, generically, what Nerdist showed was something like this. In those cases where you have a local symmetry that gives rise to a non-trivial global subgroup, and you expect Nerdist's first theorem to kick in and give you conserved charges, it turns out that the continuity equations are trivial because they take the form of div curl is equal to 0. That's just an identity. In other words, you can show that when you combine the first and second theorems

42:30 in these special cases where you have non-trivial global subgroups, the continuity equations are not interesting from the physical point of view. You get continuity equations, they're just not interesting because they take the form of div curl is equal to 0. And that's what's going on in GR. Of course, you get non-trivial equations like, you know, the vanishing of the covariant divergence is highly non-trivial. There's a lot of physics in it. He's telling you that the matter fields have to be dynamical and these other things. He said that they don't take the form of a standard conservation system because they involve covariant of the mud. We started a couple minutes later, so let's have one more question. I have more a comment on Simon saying, surely the answer to Simon is that the theory is not just a set of equations. In quantum mechanics, it's clearly one hell of a lot more than just a stripping equation, the whole business of that. And above all, it's got the interpretation of the theory. I mean, one thinks that general relativity is a completely different theory, but it isn't, absolutely crucial is the chronometric significance of the metric, that is actually describing what actually what are quantum beasts quantum atomic clocks and rods which are solid state objects how they are related to the theory, so the theory is in many ways an empty shell without all this extra superstructure of quantum mechanics the third of all the business of the Hildred space and in general precise physical interpretation but I just I completely agree with that I mean I certainly agree with the thrust of both of them but I would just say notice there is a theorem that applies to a certain I don't know part of the theory and there it functions perfectly satisfactorily I just don't see of course there's more to quantum mechanics Schrodinger equation. But nonetheless, I don't see that there's anything legitimate in applying the Nodot theorem in this case, or whether we have. I could just illustrate what could happen. I mean, if you go to an LP space, which means like two, okay, so, and you're completing that topology, and you've got to complete, you know, it's about much space and everything. You've got some control over it, but then it wouldn't be the board interpretation that would be the probabilistic rule of that. It

45:00 I do think maybe you can't mess around with this without altering physical interpretation but there's no physical interpretation necessarily associated with the symmetry that's what I'm trying to say it's just a mathematical fact it's only a mathematical fact given that we've got a function space which is well defined but I would say that you can make the space well defined and still have the symmetry it just doesn't have an act of interpretation Is it not like a boost or a translation space? The space that would contain that symmetry contains the space of spirit. So in that bigger space, you can derive that identity and then be sure that that identity holds throughout the Roman physical space, which is a smallest object thereof. I know you might have mentioned extending the classical spaces between the physical extensions prove the symmetry is out there, I'd love to keep going but we better stop now. Thank you. Our next speaker is Catherine Green who will talk about some variations on another murder of the people. Thank you. The main thing that I want to talk about is the reason why Nerter came to write this 19th paper that has these two theorems in it, the famous one connecting global symmetries with constellation doors, and also the second theorem, and I'll be talking about the end of this talk there. I thought I would start by just briefly reminding people about the variational problem that

47:30 theorems are that are in her paper, and I'm afraid I missed the beginning of Harvey's talk, so I don't know if you already did that. Or maybe I'll do it very quickly just to make sure that the things I need for later have been said. And then I'll talk a little bit about what Gilbert and Klein and Einstein were discussing that led Nerter to prove the results that she created in her paper and what she thought the significance of them was. So, very briefly, the problem that she posed in her paper with a small extension here is if the first-order functional variation in the action vanishes after a surface term, that's the extra bit that wasn't in her paper, but arbitrary region of integration, what general conditions have to hold and what conditions does that impose on your theory or variables here expressing your theory in terms of how that she derives this general result and then she makes two specialisations and the first one is two clonal symmetries and that's her first theorem and the second one is to talk about local symmetries and that's how she derives her second theorem so the first theorem we consider depending on constant parameters, and from that, you get this relation here, from which you get the continuity equation, which, so this is the familiar first theorem, so we integrate that together to consider quantity, and it's this first theorem that gives you the connection between global symmetries and consider quantities, and this is the theorem that her 1918 paper is remembered for, but really it's just the first step in her argument trying to do in her paper. So very quickly again, just to go over what Harvey's already talked about. Her second theorem applies to local symmetries, by which she means symmetries that depend on arbitrary functions of space and time rather than not constant parameters. And then the reasoning that she follows is to say, since these are arbitrary functions, these functions are arbitrary, we have to allow for the possibility that they and their derivatives

50:00 vanish on the boundary of this arbitrary region of integration. And because of this, the boundary and interior contributions to the general expression that she proved, to say that, so. Here, we've got a divergence, interior, this is in the interior of the region of integration, we can convert it to a surface term, so we've got two terms here, and each of those we're going to say, in the case of local symmetries of arbitrary functions, that they vanished independently of one another. And there's a second theorem that helps from considering what happens when we require that we look at the vanishing of the interior contribution to this variational problem that Jesus poses and what we get is these relations between the different Euler expressions and the various fields appearing in the theory that we're talking about. This is what Harvey mentioned, the Bianchi identity and general relativity and so on. But there are various other things that you can do with Nertes' second theorem, and Harvey's talked about some of those, and I want to briefly mention one other. and then come to this historical point of what it was, what Nert herself used this for, why she thought this was an important result. So the first variation on this Nerti theme that I want to mention very briefly is something that Herman Weill did. and here's something I've talked about before there were quite a few people here who've only talked about it before so I'm just going to mention it briefly in his 1918 theory where he was trying to produce a rival theory to general relativity for him he thought one of the key things in favour of his theory was that he managed to get out the formulation of charge and he wrote that we shall show that just as according to the researchers of Hilbert, Durant, Einstein, Klein and the author All conservation laws of the data, the energy momentum tensor, are connected with an invariance of the action with respect to coordinate transformations expressed through four independent functions. The electromagnetic conservation, or so just as in this way for conservation of energy

52:30 so for the electromagnetic conservation law is connected with a new scale invariance expressed to a fifth arbitrary function. The manner in which the latter resembles the energy momentum principle seems to me to be So what he's saying is that we've looked at how, in general relativity, how we get constellation of energy from being connected to general covariance. Similarly, in his theory, you're going to get constellation of electric charge by looking at the local symmetry by looking at his newly introduced local scale invariance. It's local because it depends on an arbitrary function. So his proof of conservation of electric charge in his theory doesn't go by a global symmetry. It doesn't go by Nertes' first theorem, which is the one that's famous for connecting concerned quantities with symmetries. It instead goes by a local symmetry, and it turns out that the way that he gets to this conservation law is by something that's the same as what Nert is doing in the general case, and then an extra step by making use of some equations of motion, or the assumption that the equations of motion are satisfied. So he uses some reasoning that's very much the same reasoning as Nert with respect to her second theorem in order to get at a conservation law. So that's one thing you might do that she didn't do in her paper with the same kind of reasoning. But why did she herself prove these two theorems? What was the significance of the second theorem as far as she's concerned? And to answer that, we need to look at what Hilbert and Klein discussing with respect to energy conservation and generally covariant theories. Merter was invited to go to Göttingen where Klein and Hilbert were in order to help with the various mathematical things that they were working on. One of the things they asked for her help with was this issue of the status of energy conservation in generally covariant theories. So I've just got a few quotes here from various papers and letters from Klein, Milton, Einstein.

55:00 Here's what Roe, I'm sorry, I didn't get the full reference, so this is a contemporary paper, Roe wrote about what Hylbert, what Klein, let's say with respect to Hylbert, Klein's main technical achievement was to give a simplified and much clearer derivation of what Hylbert called the invariant energy equation, and what Hylbert actually said with respect to this invariant energy equation of Hilbert's work is that Hilbert's energy conservation law is a mathematical identity in contrast to the energy conservation laws of mechanics. So Klein said this to Hilbert about Hilbert's theory and then Hilbert writes back to Klein saying, with your considerations on the energy theorem, I'm going back to agreement. With Eminetta, we found accordingly that the energy components set up by me can by means of Lagrangian differential equations, my first contribution, that's his first note, into expressions in divergence, identically, that is, without reference to the Lagrangian equations, vanishes. On the other hand, so in contrast, so this is his theory, which is a generally covariant theory. By contrast, he says, on the other hand, the energy equations of classical mechanics are fulfilled only as a consequence of the Lagrangian differential equations of these problems. certainly I maintain that for the general relativity that is in the case of general invariance of the Hamiltonian function such energy equations do not exist and might designate this circumstance as a characteristic trade of general theory of relativity so Hilbert's putting forward this proposal that the status of conservation of energy in any general eco-variant theory is different from that in the theories that we're used to prior to general relativity such as classical mechanics And the claim is that conservation of energy becomes a mathematical identity in these theories. Klein passed these observations that he made with respect to Hilbert on to Einstein, and not surprisingly, Einstein reacted negatively to this because he felt that it undermined the significance of conservation of energy and the physical significance of that in general relativity. And he said, he wrote to Klein,

57:30 I do not find your remarks about my formulation of the conservation law appropriate. It is by no means an identity. And Klein writes back to Einstein, saying that Einstein's conservation law is the divergence of a term which itself vanishes by the field of equations, and a term whose divergence vanishes identically. And so the vanishing of the divergence itself isn't a consequence of differentiation. The divergence relation is physically meaningless. And Einstein writes back to Klein, I do not conceive that the divergence relations are a different content. What they contain is a part of the content of the field of agents. So there's a disagreement here about what the status of constellation energy is in general relativity. And Klein is saying, for these reasons, the divergent relation expressing conservation of energy is without any physical content and Einstein is of course it's got physical content and it's so Klein and Nerter published in papers in 1918 and it's through those papers that this disagreement between Einstein and Klein gets resolved so what I want to have a look at is what Klein and Nerter did how this discrete would get sorted out, which is Einstein and Klein. So this brings us to the second variation on the next theme that I want to talk about, something that Harvey and I have called the boundary theorem, and in the spirit of the title of today's meeting, the theorem that there's very tight restrictions on the possible form that a theory can take. Any theory that contains a local symmetry will have tight restrictions on the structure that that theory can take. And there's a second theorem I would say, just very briefly, I said, comes from considering the vanishing of the interior contribution to the variational problem that she posed. And the other question you can ask is, contribution, what do you get from considering the banishing of that part, and hence the boundary theorem from this. And what you get is three identities, and I don't want to spend time going through what they say. Today I just want to think back on the first of these

1:00:00 identities and use that one to look at what Nerter and Klein were doing. These identities relate directly, this reasoning and these identities relate directly to what Klein was doing in this 1918 paper. And he was working with Nerter and he acknowledges the discussions that they were having. So these three theorems are very much the two in Nerter's paper and lines paper very much prolonged together to see that they came together from the work that these two were doing. So if we just take the first identity that you get from the boundary theorem, on the right-hand side, you've got the divergence of the net of current that appears in net is the first theorem. And on the left-hand side, the different side, the straight curled brackets, it's just all your expressions, so when these field equations are satisfied, that goes to zero. If you rearrange that just to get everything onto one side, inside the divergence, then you get yourself a divergence expression that holds identically in the sense that you haven't used any of the field equations to get there. And that's the sense of identity operating with. And from this, we infer the existence of what in the liquid are called superpotentials. These things here, where this holds identically. Having just made this stipulate down with this notation, then we can rewrite the noted current, just rearrange that thing up there using this. We can rewrite the noted current so that it consists of two term which vanishes when the equations of motion are satisfied and another term is divergence vanishes automatically this vanishes in virtual mathematical form of listening here and this is exactly and what Klein was saying to Einstein about the form that conservation the conservation law takes in general relativity, saying, look, it's physically empty, because to say, the divergence, the vanishing of the divergence of this thing here, the left-hand side, is physically empty,

1:02:30 because it's not the taking of the divergence of that that gives you zero. This thing vanishes on a shell automatically, and the divergence of that holds identically, so there's no physics going in to making divergence of the current equal to zero. On the other hand, just as Einstein said, you only get this result when you assume that the field of convergence satisfies in some sense that some physics is going in. So you can see why each other thought they were right about this. So that's what's going on in the disagreement between them. But what did NERTA do to try and clarify this? So I began to have first and second theorems. So the first part of her paper is the first theorem and then the second theorem is the one tendency here. And what she showed in her paper is that only when the global symmetry appearing in the first theorem is a subgroup of a local symmetry, can we rearrange the current in the form that Klein puts it in his paper. So both theorems appear for what she wants to show, because she wants to support Hilbert in his claim that conservation laws, that there's a distinction between conservation laws and theory, that there's only global of constellation laws that appear in the theories of global symmetries, and that distinction can be captured by what Klein was talking about, by showing that when you've got a global symmetry group and it's a subgroup of a local symmetry group, then in that case we can always rewrite the noted parent, which appears in the first theorem, in this form such that it vanishes in the way that Klein said, and that's what Nerta was trying to do. She was trying to support Gilbert's speculation that there's something distinct and different about conservation laws that are associated with living symmetries. I could go on and talk about other things, but I think we'll probably stop there. Okay.

1:05:00 It just occurred to me when you were talking, Catherine, I wonder whether part of what the Confucian was and might still be a sort of a metaphysical pre-assumption that must have been very widespread that there must be conservation of something of substance. You know, going right back to Aristotle or the medieval period and then this sort of colossal significance of the energy period and conservation of energy acquired in the 19th century. And that perhaps, in fact, I often feel that the case is that people are not actually just looking at what the theory is actually saying. The theory is a certain set of equations. I mean, that's the mathematical core of it anyway. And that people are trying to put on metaphysical presuppositions, which may not always be appropriate in these things. It's certainly the case that Klein and Hilbert were trying to investigate the mathematical structure of possible theories. whereas Einstein was trying to do physics and he held on to conservation of energy as a way of getting him to general relativity so he had a very different significance for him I just have a historical question are you saying that it is explicitly in Nerta's paper that her intention is to show that the courage can be written Yes, more or less, but you need the background to understand what she means when she makes these remarks. She says things like, you know, thus Hilbert's conjecture is supported. You need to dig around and find out what that conjecture was. It's structurally impossible to have a to be fully conserved at quantity like energy and energy. Now, that in case, that seems to undermine a lot of the logic of how we think about energy. And that in case, what's the picture of energy terms like?

1:07:30 I mean, in these areas of physically relevant thinking, because I mean, what was it? in the world, which was part of the business. What was it? Einstein himself? Yeah. But he, of course, he introduced a coordinate-dependent connection of energy and energy that was non-tensorial. And this is rather obscure. I mean, there's many of these things in the literature. There's many ways you can do it. But Einstein himself was defending a non-tensorial object, which was but it's the trouble was being not too short locally it can be arbitrary locally it can always be reduced to zero and broke the coordinate transformation so it's curious I mean he doesn't I sent himself with one stage remarks with a letter I think it was something like 1918 I do not see why physical objects must be reduced representation from the tensile world in prominence. So there's something going on here that's his views are certainly outstanding interests. Yeah, John. Yeah, just taking up on that one. I mean, clearly energy momentum and conservation energy was hugely important. One of the things that really struck me, having done my PhD thesis four years ago, not quite probably, on the flat space approach to general relativity assuming a linear gravitational field, and then you say, well, it's energy, momentum tensor should be the source of the field, and, you know, just by applying ideas of all energy creating gravitational field. In fact, Einstein was incredibly close to getting that. The only thing he lacked in about 1912, 1913, to have got it generally relatively that way, was the idea of a unique spin representation, which came with the Howley-Fiertz paper in 1940. sort of part of Einstein's thinking and that's very understandable in view of the results he got on special relativity for the connection between energy and inertia and things like that certainly he had a tremendous physical intuition I think which was driving him a lot of the time where he perhaps wasn't never properly succeeded in getting it absolutely crystal clear what the logic of the argument

1:10:00 I obviously have a false understanding of it because There's not this answer to your question, which is that one has a local notion of ancient preservation of people in the back of the... What is that? Well, I mean, one thinks in terms of some pre-forming frame that locally, the world's not and nobody more has learned to sort of train what they want to do. That means, but that means that you're short. I think this means what does it not come out. Right, I mean it's... No, but that means that you're not responsible for it. It just means you're ignoring gravity. You're putting gravity out of the picture, so it's a partial story. No, that's right. You want to know about distinction between love. It's just that we're not talking about the energy of non-gravitation in directions. Because Einstein was wanting a conserved quantity. I mean, I still like Roger Penrose today. He wants the feeling that there's some energy that he can get his hands on and really sort of see and feel that it's there. It's the global conservation that's the important thing. There's no doubt that there's the local conservation, I mean, they're there. It's trying to somehow squeeze a globally conserved thing so that you can say, I know exactly how much energy is sitting inside that pharmacy. As I said, when you say something, you mean the whole thing, then you're not saying anything about that. But that's an experience. It's amusing that the electromagnetic context, the potential to belong to our students as a kind of non-physical object,

1:12:30 although they appear in the energy expression, you need to know that the energy is uncertain by the transformation of the potential. I mean, different medical assumptions are being made about things to create the same equation. I suggest that we take since I see no other questions take maybe five more minutes or so since both of these talks were very closely related perhaps we can open it up to questions generally about is that okay Catherine? because we have actually by the books we have quite a few more minutes to go so let's just open it up on other questions earlier. I think you were right. Well, then I have a question for Harvey. The answer of Einstein to the Kretschmann objection is, of course, many times repeated in the literature. Stephen Weinberg says so, and Meissner, Thorne, and Wheeler, when they discuss it, they say so. And we usually well, except that physicists talk about things like than something else but for a philosopher of physics we should put standards a little bit higher can you say it can you say anything more about what is natural and unnatural yes I would like to say when Einstein introduces when Einstein introduces this gauge in his early papers it clearly simplifies something that's why he did it now there's another gauge that's become well known in the literature called harmonic coordinates and again this will be something that's very familiar to you and this is a gauge in which um that relation holds this gauge is indispensable in all the literature So much so that James York once remarked, maybe there's something about these coordinates

1:15:00 that we haven't really understood yet. There's some really deep significance about these coordinates we haven't understood yet. And Falk, in his textbook on general relativity, makes a great deal about these conditions. He says these really are special coordinates in general relativity. They play a special role. again he gives a whole lot of reasons for this these give rise to what Walt called reduced vacuum there is a kind of reduction or simplification that takes place in terms of the vacuum equations when you use these Gordon systems what therefore do we mean when we say general relativity is an interestingly generally covariant theory because it takes its simplest form and express generally covariantly And I just don't know. I mean, it seems to me that's a very imprecise statement. It depends on what you want to do. And this is a subject that philosophers of physics I didn't need to spend more time on. What is wrong with, for example, Fox defense, that these things play a special role in general? I mean, there's a quick discussion of this issue of the Troutman's base, but it seems very dismissive. I mean, it seems almost perfunctory. So I do take your question. Einstein was picking up this intuition that goes back to the work that was picked up particularly by Troutman and Anderson, and their writings on general relativity, which is that general relativity is a theory that has no absolute objects. and because it has no absolute objects you can't, we normally take this to mean you can't find a subset of the coordinate systems with respect to which the equations of motion take particularly simple form as is the case for example in Minkowski spacetime obviously any well defined theory, any familiar theory in Minkowski spacetime takes a simplified form when you relate it to the inertial coordinate systems nothing like that seems to be happening in GR with this qualification I don't think this is necessarily a very clear cut issue so I'm I feel it needs more more discussion myself Jeremy well this is a quick follow-up in these harmonic coordinates is there any single object

1:17:30 that spins by all transformations between any two harmonic coordinates because that's part of part of this familiar tradition that you're now leaning against a little bit is along the lines of there are no absolute welfare so someone who knows that tradition would say well there may be some special coordinates but show me something that's fixed by the transformation if you take these coordinates the transformation is between the Lorentz transformations Yeah, it looks like, it's almost as if you've got a Minkowski space-time and something on top of it, and the appropriate transformation is the Lorentz transformation, which is the Minkowski structure. I don't fully understand this yet but Fock certainly had non-trivial things to say about why these coordinates are important he even used by the way the following argument which I leave it to you to decide how good it is is Copernicus right or wrong well if all coordinate systems are equally good what difference does it make but if in some sense you want to say that Copernicus was right then you better specify privileged coordinate systems because this is exactly what Newton did this is why Newton turned the Copernican issue into something real, something really dynamical because he pointed out that there are special coordinate systems, there are inertial coordinate systems and with respect to our lows the sun is essentially approximately regressed so again Fogg was pointing out which respect of these things is right linearly strictly sorry the sun moves linearly in one of the four I mean it can move yes that's right but not accelerating very much he's a two minutes of sketch to answer that question, which probably won't work, it might be interesting. There are a hell of a lot you can do in the monitor audience, but you can't do anything. There are a lot of arguments in, particularly modern general activity, or semi-modern, that

1:20:00 need a specific order-free formulation, because you want to look at the document comparing most results on, obviously, generalist information, or the proof, I think I've been approved about that asymmetric spacetimes have very restricted forms, that sort of thing, then you're not really going to be useful in harmonic correspondence. So if you say the theory is harmonic, then you've got to do a lot of, you know, fundamental forms of harmonic, but there's a lot of really needed to justify the design. Conversely, if you say the theory is about fundamental form is 1.3, then you can have that harmonic argument more or less than 3, but you can you can also get the natural factor that doesn't mean that one. So that's a question which probably doesn't mean that it's the primary position, because something like, when we say the most natural one, it's the least general one you can find, well, if you still do anything without success. Yeah, but again, this is a question of convenience, because, I mean, after all, you can do everything given a single coordinate system. I mean, if you couldn't, we wouldn't know what we mean by coordinate systems in place to place. I mean, it's just that it's more inconvenient. What you're saying is that there are certain results in modern general relativity that are more conveniently derived using arbitrary coordinates. Yeah, and... It's not that you can't get them by any way. Right. So, Fred's question just simply is reapplied, right? But you can't do anything with a coordinate system if you, say, have exposure to the stairs. Modular effect, yeah, but I mean a type of coordinate system. Let's just restrict ourselves to regions of space and we can't be co-ordinatized by a single coordinate system. It's not that you can't, I suppose, I suppose York's point is that everything becomes so much simpler. This is recent work of York. He made this point, I think, back in the 70s. it's not an essential condition in his solution of the initial value constraints that doesn't play any role in his work that I'm aware of in getting the solutions

1:22:30 to the initial value I think it may be I think he's emphasizing these sort of things more recently to do with the American calculations and propagation things I'm not certain about this I'd like to see if you're interested I'm going to take the chairperson's privilege to ask after a question. Could you give us more of a sense of the nature of the collaboration, or maybe that's not even the right word, between Noether and Klein? So you showed, so we have Noether's two theorems, and then you showed us this Klein paper which has a theorem that might almost be called Noether's third theorem in the sense of its content. to do next as you pointed out and what i mean so i guess i i guess that raises the question in my mind what was the what was the nature of the collaboration between them did she just sort of leave it as an open problem and he did it or was he talking about it actively and he just happened to guys wrote it down or what it's difficult to tell from the records because they were speaking to each other so there's not a great deal it's written down to get there and have a look at but it seems clear that he and Hilbert were discussing things to do with general relativity which he per se wasn't particularly interested in but they asked for her help and then she would get her and they would get certain mathematics to do and she would go off and do things and come back and she was the one who found a way of formulating the problem in order to be able to get And then what client does in his imagination paper is to do this boundary work for the specific case of general activity. So he's really interested in what he's doing. It's the general mathematical framework within which client can then go on and do the stuff he wants to do with respect to G.L. Okay, well let's thank Catherine for her great virtue in giving us all that time for discussion. Our final slot before lunch is to be given by David Wallace, and his title is Quantum Probability and Decision Theory Revisited.

1:25:00 Okay, this is a much wealth in progress, and it's progressed to be too much since May. I want to talk about the probability problem in the Everett interpretation, which I think is probably the big outstanding problem in the interpretation of these things, but briefly how it would go is that the main problems have always been how do you select a preferred basis and how do you make sense of the probabilistic interpretation of splitting, and the mixture of technical results from decoherence and conceptual results from various results of and make the third basis problem much more tractable than 20 years ago, which leaves the probability problem fairly much up there as the thing that needs to be able to come to take the ever-interpretation seriously. And the objective of ever-interpretation is always going to be get a formulation that you can take seriously in all regards except the ontological extremity of it, and then consider whether the ontological extremity means or not in a cleaner context. In particular, I want to talk about the approach to the probability problem by David Deutsch, which effectively involves considerations of the kind of classical decision theory. I'm not actually going to follow Deutsch's own presentation very closely here, but this is all strongly motivated by his work. Here's one way of decomposing the problem. So I've got some paradigmatic quantum broadcasting events. I've got Schrodinger's cats or let's say I've got my own observation and I'm going to make this statement of a lot of particles where I know the particles that are not equal to position of up and down, and so I believe the evidence interpretation, I know the maths of quantum mechanics, and so I know that following the measurement there'll be one branch with one amplitude in which I observe a spin-up measurement, and another branch with a different amplitude where I observe a spin-down measurement. So three stages of the problem. Firstly, how can I possibly regard that process as uncertain? I know if the physics is deterministic, there there are no facts I'm lacking, there's no stochastic processing back, how can I think of this in terms of uncertainty? Stage two, I suppose I can understand the uncertainty, why should, what justifies quantifying that uncertainty by something formal like assigning you both approvals to each branch, that step might seem a little pedantic for those reasons

1:27:30 And step three, even assuming I justified assigning probabilities to the branches, why are those probabilities the ones given by quantum mechanics? Why does the probability assigned to the at-branch equal the square, the altitude of the at-branch? And I take it that a satisfactory answer to one, two, and three is a satisfactory answer to the probability problem. And for part one of that is arguably the philosophically most interesting step of the problem. and for reasons of time, I'm not really going to go into it. I think it's been answered pretty satisfactorily by a lot of Cyber Saunders' work, basically by analogy with considerations of classical splitting like a teleporting machine or a plastician split-brain experiment. And I think if you don't like that justification, there are alternative ones available where they're liable, which must be less satisfactory, but you're still not. So I'm going to assume without further ado that we've got some justification for regarding subjectively uncertain. In other words, the right attitude I should have to quantum splitting is, I expect to C spin up, or I expect to C spin down, and I'm not sure which one I'm going to see. So, I mean, we can talk in the discussion necessarily about what those justifications are, but it's not going to be the core of my work, and it's certainly not what George talks about. Okay, values of 2 and 3, and 3 is obviously a quintessentially quantum mechanical problem, but two is a very general problem in the philosophy of decisions of decision-making and so forth, and it's just a general problem. I've got some situation where I am uncertain about whether A is going to occur or B is going to occur or C is going to occur. How do I go about quantifying that uncertainty or putting more agrationalist terms? How do I decide between possible action, given the outcomes of those actions depend on the result of the uncertain process. And that's one of the problems in which long classical decision theory is out there to answer. And, broadly speaking, the way decision theory answers it is it says, well, suppose I've got quite a rich class of possible actions I could take that depend on uncertain outcomes. So, to make it concrete, suppose I've got a wide variety of possible bets I could make on uncertain processes. And suppose I've already got some order of preference between those possible bets. And suppose further, that my preference satisfies certain assumptions which seem intuitively reasonable for a rational agent.

1:30:00 So things like, if I prefer A to B, if I prefer B to C, I'd better prefer A to C. And if A always gives me more money than B, regardless of the outcome of the uncertain process, I should go for any things like that. from those sort of assumptions you can prove what you might call a representation theorem which says any individual's preferences are represented by an allocation of probabilities to possible outcomes and by allocation of numerical values of possible cash rewards you might get such that I prefer A to B even though the expected value of A with regard to those probabilities is larger than the expected value of B And in philosophical terms, this is operationalist and subjectivist. It's operationalist because what I mean to say A has probability one-half is that I'll bet on A due to a loss. And it's subjectivist in the sense that all it gives us so far is a notion of probability which depends on an individual's assessment of how likely some individual outcome is. It doesn't tell us uniquely what those probabilities are. they're going to do different rational preferences which give wildly different probabilities to A.D. So it's not all that standard classical decision theory, standard philosophy probability, makes no recollection to one of the things. But what David Deutsch did, and what I'm going to be to pitch later on before, is to apply those ideas to one of the things and specifically to some quantum on certain processes and see what results we get. And the results are surprisingly strong. Okay, here's the framework I'm going to do that too I want to consider a certain special set of bets I could take, which are of course a walking game Again, the idea of which is that the formula is rather more precise that this is was attacked in the literature or being very ambiguous in a certain way, so this is almost pedantically precise to try to get around that So by definition, a quantum game is a trigger like that Si is a state operator on the space in which that state lives, and p is any multinational function from the spectrum of the operator to the numbers. Now, I've been quite pedantic here, what we always have to find is technically a mathematical object, which in some way represents a physical process, but the reason for the planetary is going to become obvious. But here's the intended interpretation of the mathematical object. I've got some problem

1:32:30 system, I prepare it in stage psi, I then shove it into a measuring device, it was a class and it outcomes a measurement of x, and then if the result of a measurement of x is some particular value of little x, then I get p of little x in cash, so that's a power sign that's not very readable. So incidentally for decision theory in Fiscianado, I'm simplifying a lot already by assuming I've already got that sort of cash value is its own value, I value 10 pounds, 10 times a month, so I value 1 pound and so forth, that's implausibly strong, it's implausible about your lots of money and the amount of bets and that's more. Okay, so that's the framework I've got, some example, I've got, a trivial example, I've got a spin-off particle, sine, substitute position, up and down, x is a measurement of spin in that up-down basis, and p says I get 20 quid if it's been up, I lose 10 quid if it's been down. Here's the decision theory, how many it is. This slide is pure decision theory, but the quantum context is very inessential here for the movement. So we assume for every game, there's some fp engine, there's some function v of g on the game, and the function assigns to each game its cash value. And what I mean by the cash value of the game is, this is how much money, such that if I could be indifferent between having that money outright and playing the game, or another way of looking at it, if anyone offered me the opportunity to play the game at anything less than its value, so if the value of a game is 5 quid, and somebody says, well, you can play at 2 quid, I'll do it. If they say you can't, you'll play at 6 quid, I won't do it. So assuming that function is defined as all the game, of course it follows from that, that if game 1 has a higher value than game 2, then I prefer to play game 1, game 2, give me the choice. That's the sort of minimal structure, and on top of that I'm going to have a little bit more structure. And the structure representation of these two axioms, these are, again, axioms of pure decision theory. They're axioms of a fairly strong decision theory, to be fair. We can talk about an axiom after the discussion, if necessary. Okay, I think they're in personal order, actually. Dominance is the obvious one. Suppose game one always gives at least as much money as game two, regardless of the outcome. Then I should assign a value to game one that's at least as much as the value of game two. So it's pretty obvious, I think, regarding you.

1:35:00 If regardless of what my measurement device turns out, I get more money playing more than two. better than they want. Additivity is a little less obvious. What this is, I mean, formally it's saying, well, I mean, formally it's saying, it's pretty obvious from the slide, and informally what it's saying is, suppose I'm interested in whether to make a certain bet on a given quantity measurement, how much I'm prepared to pay to make that bet is independent of what bets I've already got placed on the measurement. So if I've already bet 10 quid then that's not going to affect how much money I'm going to pay for a possible bet that it's going to come up and spin down. Now, effectively, what that's equivalent to is the assumption that I value money linearly so that, again, I'll take a 10% chance of money 10 quid over once, but for a certain, equally with a certain difficulty. Now, that's not plausible in general. I mean, I imagine that the value of the sale value of the year You said that the amount you'd be willing to place on any bit is independent of any bits you've already placed, which seems to suggest that your bits will always be 0 or infinity, since if you suppose you're willing to bet 10 quid on it being spit out, having bet 10 quid on it being spit out, you assign that I'm willing to bet another 10 quid on it being spit out, independently of the fact that I won't really place that exact bit. I mean, if the, if what we say, suppose the bet is I get nothing when it's been held, I get 10 when it's been up. Well, if somebody says, will you take that bet for free, then absolutely I'll do it. And in fact, I'll cancel it, I'll take a bit more of them. If I'm, what I'm actually going to say is that the value, there's some value, say five pounds, that I'm prepared to pay instead of making a 10-foot bet. then it's saying, well, I'm also, having made that bet, I'm happy, and I'm indifferent to paying another £5 to make another £10 bet. So, if I'm indifferent between £5 absolutely and having a £10 bet, I'm also indifferent between having £5,000 to make absolutely and making £10,000 to make a bet. That's true, but it doesn't, it doesn't spiral you up to, because these are indifferent situations. If, in fact, I prefer to make £10 to make £5,000, it does say I prefer to have £10 million than to have £5 million.

1:37:30 it only applies to the indifferent cases. Yeah. If you're indifferent, then you don't care how much you're making. Effectively, yeah. That's right. And, yeah, this is sort of only plausible if you can find yourself turned to small amounts. Like I say, the value of the resale value is more than 1,000 times the value being was mortgage and it was prepared to pay their mortgage on a 1,000 chance to get on with a year. But if the amounts are sort of pennies or small amounts of cash, then it's a reasonable assumption. Okay, that's all pure decision theory. I mean, when I say it's all purely the same thing, what I mean is we haven't used quantum mechanics in the game at all. All we've basically used is the fact that the outcome of the uncertain process is definitely going to be an element in the spectrum of the operative. For all this, we can just replace cycle of X just with the spectrum of X, for all the sake of this form, doesn't it? Okay, so what can we do within that framework? Reasonably, I think not enough. Basically we can use that framework to define probabilities. We can say that suppose I've got some outcome X and, excuse me, that we're talking about suppose I'm prepared to pay £5 to make a 10 greater than X then by definition the probability of X is going to be taken to half so the probability of a given outcome in the observed process is by definition prepared to pass on it. And if you combine that with the additivity and the dominant assumptions, then you can prove effectively an expected utility theorem, this is one of the representation themes I'm talking about, you can prove that any given professional agent makes decisions as if he's assigned some number to each bottom outcome such that the numbers are up to one and such that the cash value that you gave is given by averaging the cash value to the payoffs with respect to those numbers. That's quite good. That's essentially the answer to step two of the process we were talking about. We've now shown that uncertainty cashed out qualitatively must actually have a cashing out equal quantitative problems determined. But it's not step three, because this is what we want to get. We want to get something that would be similar. We want to say that the cash value of the game is the average of the possible payoffs weighted by the mod squared aptitudes, not just by any old numbers. So we actually want that to be forming in place. And obviously we can't do that with the mechanism

1:40:00 we've so far put together because we made no relevant use of the quantum mechanical structure at all. And at that point it's made by a whole lot of people who have been trying to do it in the literature, Okay, but there's one step further we can take, which don't you take, although not terribly obviously, which allows us to sort of break out a little bit and get as far as this next stage. And you might call that assumption physicality, which is that if two given games actually describe the exact same physical process, then they have the same value. Which in this mathematical formula is a substantial assumption, because we'd be taking two different triplets and saying they're both of the same value. But obviously, if this is supposed to represent a physical decision theory, it's completely trivial. Whether I decide to bet to play this physical game sitting here on my desk can't depend on how I choose to describe it. And why that assumption isn't trivial is because, and this is why I would be fanatic about games being these mathematical triples, is that it turns out, not like a lot of situations in physics, that a single physical process is actually described by many triples. And that's the core of what I'm going to be talking about, and the core of Deutsch's argument. But to show that, we've got to actually get clear what physical processes in the average interpretation actually count as measurements. Because what we're going to have to say is The physical press is going to be some physical mechanism whereby the state is prepared, the measurement is made, and the payoff is given out. Now, the first and third of those are easy, and in some interpretations, what it means is the payoff of a state is complicated and operational, and the effort of interpreting is dead easy, however you do it might be complicated, how you come to know it is complicated, but to say the system is prepared in the state of science is literally to say that the objective of the physical state of that subsystem of the mass versus of science. And the payoff bit is pretty easy, that means what it means classically, it just means that you put some gadget, such that if the gadget sees X on the screen, it gives you $10, sorry $22. The measurement bit is the thinner bit. And the definition I'm going to take looks something like this. And this is definitely short, it's not commercial, but this will be going over it. The measurement, any physical predecessor, might be the state, such that

1:42:30 if the state's an eigenstate of X, then the measurement returns the eigenvalue of the Another way to see how that works is it's going to be some gadgets such that if I input an eigenstate of x, if I input a state of a definite value of x, I get an output of that very definite value. And I think that's that value, that's certainly at least the standard definition. In addition, I think it's basically going to be the only definition in town because what is a measurement process if it's not something that measures the definite result if it's out there to measure it? it's a definite result, if we've got a superposition of possible results, then we're going to get branching, but that's the best we can do in the end of the day. So formally, what does that look like? It's a process that's defined this way, little xa is one of the eigenstates of mig x, like with the value little xa, m0, the initial state of the metric divided into which I shove it, is an analytical process, outcomes a phonetic bit either a single state displaying XA or potentially a single position over lots of states displaying XA. In fact, the latter is much more likely to be the real situation because if you imagine, say, it's a diagram to read or something, you go up from the micro state to the macro state through a lot of stochastic subject matter, stochastic stuff inside the measurement device, you're actually going to have an awful lot of possible outcome states, all of which correspond to the micro-state of the XA. But the important thing is that all the outcome states coming out actually physically display XA in some way, which could just mean it's on the dial or it comes up on the LCD or something. It's relevant to say that what makes a state display in XA is a combination of its physical state and some convention in the mind or on the paper of the observer, because suppose F is a one-to-one map from the spectrum of sigma X into the reals, then a state could display XA or could display F of XA, because I could just say by convention a state with a needle here means XA or a state with a needle here means F of XA. Okay, so that's the definition of measurement, but what are we going to do with that? Basically this, so this is, I'm using the board, but that's the definition structure

1:45:00 at the start. So I've got some observable x, some use transformation, some other observable y I get from the use transformation, and little y a is the, where the little x a I get from So consider this process. One, I pair the system in some state, sum of a, lambda a, and xa. Secondly, I operate with u, and I end up in the state, little xa, sorry, sum of a, lambda a, little ya. And then for me, I measure this observer from y. I should add in the measurement device here. I make the measurement, and now I get out some gray, by definition, the measurement process. I'm going to assume for simplicity there's only one measurement outcome state per state, so in the definition of measurement I'm going to ignore that lambda a term if I take my word for it, I don't think that makes any substantive difference, it just makes a scripted an easier. So what I need to get out, I need to get some of it, some of it is nice, and revealing the eigenstate of y a well, the eigenstate of y a is actually alexa And then I pay. So that's the process under visitation. And that instantiates the following This instantiates the game u u psi, because that's the state up there. and it measures y, which is u, x, and u, a joint on it, and u, p.

1:47:30 the pair of the stage each side, is the measurement of the stage, is the pair of the stage. However, suppose in fact all of this stuff is put in a black box. Then how do I know, is that my third process, where I start with the state pair of the state lambda A xA, and then I end up, across the interstate measurement device, and I end up with this state here, lambda A m xA. And again, by definition, that process is just a measurement of the operator x on the state psi, because if you put in an eigenstate xA of x here, out comes the result xA. It's the definition of measurement process. So in fact, this self-same instantiates a different game, sine x e. So those different triples actually describe the very self-same physical presence, and the physicality assumption says they ought to be given the same value. And they call that measurement difference. It basically says this, for any u you like, the value of psi x e equals the value of u psi, u x, u an edge, e dot. And that's another similar example, which is even similar. However, I consider for this game, psi x p dot f, for f is some function of the spectrum. That game says prepare the state of psi, measure the operator of x on it, and give it a payout p dot f as a consequence. Well, suppose it's made that for a different operation. I changed the convention, as I say, you can just do this after a while, so the measurement state n comma x actually well x but f of x but they also changed my rule for the payout so instead of giving a payout p of f of x I just give a payout p of x with everything's out. I think it's clear to see that what's happened there again we've just all we've done is we've changed the description we've decided to apply the function f to the measurement outcome rather than to the payout given but that's only a matter of how we slice up the physical process going on. Are you assuming f is the objective here?

1:50:00 Assume f is nice as you like, it can be a one-to-one map or a permutation of the spectrum, it's a simple description, that's all I'm going to need for the rest of it. I think the neatest way to do it is probably that p is going to be a functional spectrum of f of x, f is then fairly arbitrary. So because of that, we've actually got this problem as well, f of x. And those two results are between them extremely powerful because they allow you to move between different descriptions of the same day and prove results about the next three. I'm not going to go through the whole of what we can do with that, but I'm going to show you the central result, which is that from this we can prove that a superposition of equal global states must have equal probabilities assigned to them. So, I've got some other generators over my measuring x. It's got n i, considering the values. And I'm going to let pi be in the order of communication that I've got. So, by definition, u pi of x a is pi of x a. So I'm just going to swap c and make space around and do that. Now, take any order of sign-like, and consider this some generic game on it. Well, by the measurement equivalence result, it's actually just this measurement. This game can be physically realised by the same physical process that realises this game. So those must be equal by measurement equivalence, like this one here. And similarly, any process that realises this game also realises that game, because whether I apply pi to the measurement outcome or to the payoff is different. So I've got this equivalence between those results there. But, and this is in connection to the symmetry topic of the workshop, I suppose, suppose that size is actually an equally weighted z-position on all of the NL outcomes. Then, of course, it's invariant on permutations. And so I've got this result here. But this is now where we can engage the decision theory, because now I've proved an equivalence between different payoffs applied to the same measurement process. And that's what the decision theory of maximums earlier, additivity and dollars, were actually coming into play for. And here's how, if I sum over all the permutations possible, and I use additivity to bring the sum inside the bracket, and if I just observe that summing over all the permutations pi

1:52:30 over here pi just gives you n factorial times the average outcome of p, then I prove that the value of this game must be that very average, so that the, so which by giving the definition of the probability of an outcome here is the dot I bet on it, is say that in this EPS position is going to have to assign probability 1 over n to each of the outcomes, on pain of not being rational as ever after all. Which is pretty striking because we've actually shown a connection between the quantum symmetries and what you might call symmetry of probability allocation. So we've shown not that there's a symmetry there so it's sort of nice to observe the symmetry of assigning probabilities, And in effect, from that, using more of this stuff and using the activity assumption, then you can prove any sort of rational weighted superposition, you better obey the model model and using dominance as a continuity assumption, which is sort of not essential but it's dead easy, then you can prove it applies for any other superposition you like. So this sort of structure together with these assumptions is enough to acknowledge the of what the probability problem was, but to give you a statement as well, to show the models there, which would seem to be a solution for a probability problem with that setback, which it is, with one flying the orbit, which I was going to mention briefly at the end of the talk. Here's how in mathematical physics we generally think about the connection between physics and physics and maths. mathematical models, here are the physical models, here are the sort of physical situations, and here's some sort of one-to-one correspondence. Now the reason why all this stuff that George has done to our copy of the end works is because actually the maths is in many-to-one correspondence with physics. But if we're going to consider that, we've got a lot of the possibility that maybe the physics is in many-to-one correspondence with maths. And of course it is. There are many different as a measurement of observable X on state side. And so it's an implicit assumption, it's well given before this, but it's an implicit assumption that a rational observer doesn't care

1:55:00 which of those processes he chooses to perform. And I think I'd call that assumption in press measurement neutrality, the assumption that all I need to know to make a bet is what's the state being measured and what's the observable being measured. Now that's sort of quite obvious, and I think it can be justified, that it's not completely trivial, because in a sense, this is what you have to get by moving to the evidence interpretation. If we stay in a sort of operational situation where a measurement is a primitive thing, which is by definition specified by the state measure that they observe and measured on it, then we wouldn't be able to do any of the arguments I've given to prove the probability rule, But conversely, we get for free, in fact, by definition, a measurement is one that is given by these things, and I don't care about two measurements. What we get by physicalizing the whole thing by going to Everett is the right to observe that many different measurements count as the self-same physical process, and therefore prove these far-reaching connections between the values of different games. But the price we have to pay is we better also accept that we can't just help assemble to the fact that, well, I've spent a measurement to an actual kind, if there's awful things. or probabilistic in actual kind, you might say. And so that, I think that assumption of measurement neutrality is one that needs to be looked at and justified quite carefully in order to let the Deutsche Programme go through. I think it probably can be justified, but that would be going from the time I got. So that's not fair. All right, I have a question. How do you know what state you've got? X-hypothesis. This is the decision theory you can apply if in fact you have that but we're in a situation where we're trying to drive and you're telling me I know what the state is and I'm saying to you I don't know what the state is I mean, you've got to have some operational procedure that will tell you what that state is I should be introducing to the effect that, well, I know it's this particular state because if I were to measure it in a certain manner, I would get a certain way to resolve it. And I'm just trying to understand operation, what you mean by this state preparation. Point, point, okay. Well, two or three responses to that. I mean, the first is that we're sort of on all fours with the board, as generally given.

1:57:30 I mean, the board will assess if, in fact, the state is this and these are the probability. So the board, too, assuming the state is. But in terms of the operation process, well, two ways you might help the state is one of them is some sort of theoretical connected probability, so that might be something like if it's not in state, then you can just measure it and establish it, or it might be that you can calculate what it is from physical considerations starting on stuff you already know Would vision neutrality already come in at that point? I don't know, good question I don't think it would, it wouldn't for the first third at least because there's no control about It's a preparation process that's certainly proving it's measurement neutrality. It measures some non-degenerate observable in a tiny one way that means I've still got the state again with the measurement. So that state is reliably aligned today. But the second possible way I'm more realistic is the sort of research process where I'm going to say I don't know what the state is, but I've got some prior credence to what the state is. I've got a device that manufactures them. I manufacture 30 trillion of them. I can visualise all those credences, and out of that I get virtually certain credence at the stage of this form. So you've got to have some feedback between you. This is sort of what David calls inferential and synthetic principles. I mean, there's a kind of inferential principle play, which is going to sort of play up, but I guess in a fake, you know, I do think it might help to try to find what some of them they know would say, and I do I think that's one of the things that I've said. Legiment neutrality is no good, because it begs the question. I'm going to say that you're going to be very worried about the precise nature of the apparatus in the literature review, because you might find yourself facing and branching into a thousand groups with all of the identical copies. and that won't be a good mention of the advice because maybe government isn't down with the number of branches. So I think that would be the sort of response that Chris doesn't know what we're getting. And I'm not sure whether, is it wise to try to make a principle clear

2:00:00 that the principle being that what is neutral about fishing is we know that the outcome is the same. I think it's a bit of a bit of a bit of a bit, I mean, so the problem is sort of here, here's the, this getting in the way of a lambda thing, this is saying that actually my assumption with little loops could include branching is highly non-optimous, and if A, it's more crucial in the proof as I gave of the equal suppositions, but it's very crucial in some of the latest stages of a discipline for the proof, but I think, so, and I think obviously it's true, little neutrality implies branching of difference, but actually branching of difference is implied even by a satisfactory solution or step one of our probability problem, if we've already justified quantum branching as an uncertain, as correctly regarded objectively as an uncertain event, that already implies indifference to branching. Because, I mean, suppose that I've got two possible situations. In one of them, I'm definitely going to be given 50 quid. And in the other one, a quantum random number is going to generate a number between 1 and 10 to the 12. It's going to display, and I'm going to see it. In the former case, there's no branching. In the latter case, there's a branching to 10 or 12 columns. But the subjective perspective is, in the former case, I'm certain that I'll get 50 quid. In the latter case, I'm certain I'll get 50 quid. I'm also uncertain what number I'll see. But what number I see is a supreme indifference. It's just my self-philosophia needs, in fact, that I want to see this number before I get the catch. That's fine about me because I just don't care what the number is. And the reason I don't care is because I'm always committed by virtue of exclusion to this stage to just thinking about it as a subjectively uncertain process. So I think if we assume, as I think we have to assume, to some extent, that we need an understanding of measurement of subjectively uncertain before we even crank out the position of political mechanism, then we've already, in some sense, justified indifference to Ransky, and so it's not an additional assumption of measurement neutrality that requires. Jeremy. I just had a comment on the same topic, and so it can be very short now. It does seem to me that this measurement of equality is something that you shouldn't be too apologetic about. Because in ordinary decision theory context, it's often remarked that the way action options have their values summed normally assumes that no value attaches to which action you do intrinsically.

2:02:30 It's all about the consequences that might occur in various states of nature. And you don't, after all, need to assume that measurement outbreaks as they typically occur have this feature. It's enough that they could have. It's enough that the measurement device doesn't slap you in the face or give you an ice cream the little x-a, in general, we are pretty easy, because they don't slap us in the face of the inside screens while, in the course of delivering the little x-a, so it seems to me you can be pretty robust on this one, having done the hard metaphysical work that you just did in answer to science. I think I'm quite keen on that, but I think, as we saw in the metaphysics, it's very much linked to this uncertainty, because if I can justify things in terms of subjective uncertainty, at least, can say, well, my indifference to the details of the measurement device is not some extra possible, it's just that I am in fact indifferent to whether the device is red or green or something. But I guess one does need to do that work and there's certain different ways for these reasons. I had a correspondence with somebody just before Christmas who was saying well, suppose I prefer getting a red ice cream to getting a green ice cream but because I'd like to have variety in my future copies, I'd prefer 50% chance of red and 50% Of course, that violates decision-democratic assumptions, in particular, it violates dominance. And that person would be somebody who has in-form knowledge of quantum mechanics, its objectives of getting to observe the perspective, and sort of transcended it and said, well, I'm going to look at things in a different perspective. And probably all you can do to somebody like that is say, well, that's a different choice. It's rather like a sort of perfect decision in the face of understanding this personality. I'll have a sort of Zen-like view of the world. And that's a step you can make. It's a step you can only make in having already accepted the framework. I think if we're not here trying to... It's actually, we're trying to see if everyone is true or unfensible, should we say. Then we've got to see where we start. And the way we start with is the fact that we can see these things. That's the actual answer. If having become determined to them, they'd like to change our view. Then we can introduce something. I had a question. At the beginning, I was pointing out that talking about the value of tax in terms of money, we've only made stocks for small values and became problematic in the top,

2:05:00 I was wondering whether that affected the explanation might have to give a quantum Russian collect games for any effects of the life or death situation. Is there a problem with that explanation of that sort of situation? I think there's not a deep problem. The reason there's a deep problem is that I made the assumption that it can actually transcend it. in a sketch is how in standard classical decision theory, a lot of early work would take a probability of being the definition of the odds, so you take the cash values, get it permissible. Now, savages are taking on the chalky of that, and you can formulate it in a much more qualitative way. Basically, what's going on is somehow, if you've got the utilities and outcomes given, you can define probabilities in terms of bets. Conversely, if you have the probabilities given, you could define utilities in terms of, by definition, A is twice as valuable as B, if I'm indifferent between 50% chance of getting A and 70% chance of getting B. And the way established approach goes is to say that A is twice as probable as B is to say I'm indifferent between a bet on A and the same bet on either B or something else, same probability of B. So you've got a structure, and then you can do some quite strong structural assumptions to assume you can grade up the space of events such that you can always form, say, any events that are going to get global to another event. And then that way you can get around that assumption at all. Now, you can mirror that proof in the quantum situation completely. Basically, if you just assume transitivity preferences, if you assume dominance in a qualitative sense, and if you assume non-geniality, either it's what they do or what such that they prefer one to the other, then you can prove there's a unique probability, and you assume that physicality is the image of two elements itself, then there's a unique probability assumption, such that our best on A is above the B, if and only plays a problem with B, and that probability assumption is the one given by the moral rule, And then once you've got that, you basically take savages, other stuff across, return to pretendies, and you prove expectation values, which might not really, you just might say that the value that is already in you is less than a thousand times the value of you already does that for now. So I think they're relatively shallow. The exposition of that is rather more complicated, and it's then a very clean break from what Donch has done with H2J, which is why I haven't said it here, but it could be done. David, will that happen to you?

2:07:30 I can see that if you know what you're facing is the superposition of equally weighted elements, then you should check on them equally, but the worry that arises here is, after the interpretation, all outcomes always happen, so all sequences outcomes always happen, So if we're estimating the weights from observed frequencies, there are going to be people who see road sequences all through their history and will not have one that's going to be comparable to different elements. Now, they won't be observing frequencies conditionalizing in just the way you suggested and getting different weights. Now, I have two questions. you say about beings which by our rights have got the wrong weights that are conditionalising and then acting on the weights are they being objectively mistaken are they being making misguided decisions the other question was is there any of this promise of trying to get the weights from yet further symmetry considerations without appealing to the frequencies. At the end of your long paper, I was rather getting the impression that that's what you might be hoping for. Okay, well, applying in order. What we should say about those beings is, I think I would not say they're just objectively wrong. There's not a real relativity between them, they're just wrong. Why can I live with that? it's rational for me to assume I'm not one of those four seconds. So there's no stability here. It's rational for me to assume that I am somebody in a in a normal situation. It's also rational to assume that I'm going to get a set of sequences which are going to be reasonably high weight, or not going to be enormously low weight. So if I've got my preparation device and it pairs a state it pairs a trillion cost of state and I make measurements and I'm about to make measurements of them. It's rational for me to assume approvably, that unless my credence, my initial credence is completely nuts, then it's rational to assume that I'm going to get rough in the right state. If there's going to be something like that, we're going to get completely wrong state, that's very unfortunate, and they're

2:10:00 going to get, and they're going to think the wrong theory, but nonetheless, they're going, as rational to me, I'm not going to be one of them. I mean, furthermore, those guys out there are not going to get, they're not going to get the evidence of reputation because has, they've been very unlucky in the data they've got, but if for some reason they still believe the other interpretation, they'd still agree with me, they'd still say that they, well, they'd either just be very mistaken about the state they've got, or if they happen to know the state independently from, say, an eigenvalue consideration, then they'd agree with me that they've just been very unlucky. So that's the first statement. The Probabilities without distribution of frequencies. Epistemologically, no. Epistemologically, you're often going to be getting the probabilities from the frequencies, although not always, because, again, if you pair an Einstein, if you pair an Einstein, spin Z up, which I could do in a non-problemistic way, and then I'd measure it in the spin X direction, then the theory is predicted straight off that I should give frequency half to each of the possible outcomes. It depends on whether I've ever seen such a device before. so generally I'm usually going to get at what the amplitudes are via frequencies but the frequencies are not in a bit of a Lewis sense the ontologically prior thing that determines what the probabilities are, the probabilities are actually determined purely by the amplitudes I suppose indirectly there by the symmetries of the amplitudes if I but knew what those were independent of frequencies is that So, I can't see how you can make that assumption at the beginning of your argument. The argument, as I present it, entirely assumes that I happen to know the physical state, for some, let's say for the sake of argument, for example, then it's saying, in fact, if I know the state and I also believe the effort of interpretation in toto,

2:12:30 then I must assign these probabilities, and that's all, and that's tripping it down. Where there is the question begging for the uncertain situation, if I'm uncertain of what the state is, I can presume you put some credences on what it might be, more or less after travelling, you can get into the question there, and then I can say, well, for each of those possible situations, I know what it would be highly national for me to expect, but again, by this argument, because I have personal knowledge, and then I'm just going to, it's amazing, if I see the statistics that correspond to state one, then I infer that it's probably state one. I'd like to ask one question, because I'm still not clear, I think it's a question that others have asked in different forms. I'm going to ask it in a simple-minded way. Okay, so I present you with two different measurement situations. One, there's a preparation device which is, for all practical purposes, identical in the two cases. And then in the one case, it's going to spit out some spin-out particles. And in one case, I'm going to measure spin with my Stern-Gerlach magnets oriented thus. And the other one, it spits out the particle, and then I've got this sort of complicated magnetic device here that processes the spin through 90 degrees. I fit the image with my Stringer-Lock magnets thus. Do you mind the way the fashion was? They're 90 degrees apart. I've applied the Unitary Transformation to both... I've done something to the state and I've done something to the vertical. It's the case that you wrote down. But what I want to know is, why do I treat those the same? Why should I treat those the same? They're not physically the same at all. I mean, it's not the case where I have two instances of devices that are otherwise physically equivalent, right? It's too, I mean, me, ignorant old me coming here, I look at these things and I say, that one doesn't look anything like that. Why should I treat them the same? Why should I make the same bets? Well, that's a very nice way of putting a whole measurement reality problem. Those are two devices which, arguably the least, are the same measurement device described by the same Psi-X pair. I'm not completely sure I think if I can treat the, here's a very constructive end of it if I can treat the measurements objectively uncertain then I know these devices agree on

2:15:00 eigenstates so I know if I can eigenstate an uncertain what I'll see if you look at the way in the interpretation you sort of move between the objective deterministic to get to the inter-terministic stage, effectively that's happening when a measurement happens or when it gets magnified up to a macro level, but there's no, and at the point when you're doing that you can start saying I'm simply ignorant of what's taken and I'm supposed to find out rather than it's an ejection position, but there isn't going to be any clean way of saying where you put that cut, and it's not like some sort of classic theory where I don't know where to put the cut, it's that as a matter of principle there's no way to put it, so there's sometimes never a bit less game-pagan here, but it does show in a way I mean, you see why when you consider that situation, why various people have suggested that there's a question-baking element here. I don't think there is necessarily, but the natural response to my question is, same probabilities, of course. But that's not an answer that's available to you, right? Absolutely, yeah. Well, I think that's it. I'm optimistic on that resolution, but it's almost what I've tried to do, it's trying to show it's a problem, because as you say, these that you've got to bring out. And I think in the dialectic of discussion of this problem, the question-beggy assumption that Deutsch was accused of making in the initial work has very little to do with this problem, I think. It's much more explicitly, you're assuming they're pro and say that... Oh, sure. So I think... Sorry, that's not... I think it's... So what I can... I'm trying to say is that that's not that way. It is the case that measuring neutrality... If you've got measuring neutrality, that's all we need to get the more the plus A reasonable decision But it's also the case that neither Deutsch's paper nor the reply to it really brings up these other questions in, I hope not a question I can really show how to answer. Okay, well we've gone a little bit over mostly my fault, so let's thank David.