Interpretation & Reconstruction of Quantum Mechanics / Holism & Non-Separability

0:00 Thank you, Henry. Can you hear me this way? Sure. so let's see five minutes ago I discovered in the back room here there's a small back room there there are about 50 copies of this book about Dostoevsky and it starts with this phrase in Spanish which I will not read in Spanish which is that the essence of our times is to live in the middle of a contradiction so I think one of the mechanics is the essence of our times And I will try to argue for a way out of the contradiction, if you like. Or maybe that's too ambitious. I will present you several remarks on the change, which I think is a paradigmatic change, on the kind of work we do in the foundations of physics. Whereby we move from interpreting quantum theory to doing something which I call reconstructing quantum theory. And there will be more details coming so you will see how these notions are related and also different. So the outline of this tool. First of all, just in a few words very quickly, I will tell you what is wrong And then I will introduce, in a sort of dry theoretical way, the notion of proof destruction. And then very quickly, and by passing on certain theoretical aspects, I will quickly go to the examples and focus on the examples for you to see what is really different in these new pieces of work that have been done in the last maybe 10 or 15 years. The last section is called something really unique. So let's see, what's really wrong is that Oxford and Rutgers don't agree.

2:30 They are in the contradictions. So we do not have a consensus on the meaning of quantum theory. Well, that's something which we can live with, as author of this book said. But we could also try to maybe shift our focus of attention from living with this contradiction, from doing one interpretation, yet another interpretation, yet another interpretation, and try to do something that would be valid whatever philosophical interpretation, whichever philosophical interpretation you wish to adopt. will be something that remains valid independently of one's philosophical stance, you know, like reality or just a systemic error of one theory or whatever else. So this will become more clear, I think, in the next few minutes. So interpretation is, to me, an ill-conceived task. Now what would a better task be? Here is a quote from Ravelli, and there is a number of such quotations, not just from Ravelli, but from various people. quantum mechanics will cease to look puzzling only when we will be able to derive the formalism from a set of simple physical assertions postulates or principles about the world therefore we should not try to append an interpretation to the quantum mechanical formulas but rather to derive the formulas from a set of experimentally motivated principles to a postulate so I will not open a big debate about how this relates to what Einstein did with relativity in grounded special relativity in a set of simple principles. Rather, I would probably limit myself on this talk to giving you this schema, these three stages of reconstruction, and then showing how this works independently of one's ontological or other commitments. Reconstruction would be a piece of mathematics. And the most important part of it, so the most important part of the reconstruction

5:00 is this number three, rigorously divided formalism of a theory. And of course the most problematic part is given set of physical principles. So as such, this may not sound to you very new, But let me, before telling you what is really new, let me still develop a little bit of philosophical theory, if you like, of pure philosophy about reconstruction. So, unlike interpretation, reconstruction now possesses supplementary persuasive power due to this use of mathematical derivation. So, instead of attaching something to the formalism, we actually derive the formalism, is given by the principles and by the mere fact, the sheer fact of mathematical derivation. So why is quantum mechanics the way it is? Well, because we derived it. That's something which is a surplus of which the significance should not be underestimated. This is something that, in principle, is capable of bringing consensus among the various schools of interpretation. So mathematics is the only way to bring this consensus. Now first, a few remarks about the choice of first principles, because in the examples we'll see a lot of these various axiomatics, various postulates. So the choice of axioms is, again, they only allow a degree of freedom for life and every that whatever comes after the choice of axioms is a mathematical derivation. So how do we select plausible axioms? Well, one thing here is that axioms should not be taken as truth, as utmost true propositions about the world. Axioms are axioms here and now, in the sense that in one reconstruction, there can be such and such axioms, And in another reconstruction, these axioms may actually, in order to understand why these things are fundamental principles, we may actually want to derive them from some other principles, some other axioms. So there is an interplay of these views of what is a theorem and what is an axiom that

7:30 to me, which to me is beneficial. This is something that allows us, at the end of the day, to understand better the structure of a theory. So this may sound quite surprising, but let me spend most of my time on giving you examples, and then if you have questions on the theoretical content, just come back to these ideas. So here are some examples. This is what I do not call a reconstruction. This is Maccae's work. Now, why is this not a reconstruction? Apparently, this has mathematical axions. And as we know, Maccae derived a theory from these axioms. Now, the meaning of these axioms is completely non-transparent. So if you look at these things, you have no idea what that means. that means in terms of physical, in terms of physics, in terms of physical assertions about the world. Now, this is almost a reconstruction, except that there is no motivation for these axioms. Now, here's something else. Here are, listen, art as axioms. In their initial forth, let's say. There are other versions of hardy's axioms. So hardy's axioms are as you see, dramatically different from hardy's axioms. Here we have certain principles that have names. I don't know if you can see the names. Well, hopefully you can. Probability, subspaces, composite systems, and continuity. So we have an idea about what the basic principles are And then there is a mathematical formulation of these principles, and then there is a derivation. So this is one change that occurred, let's say, between the past of mathematical derivations, which is both Maggie's school, with, you know, Beuron's school, if you like, and Ludwig's school, all of which were doing mathematical derivations. and something which occurred probably in the early 90s,

10:00 let's say, some words of course get back from the 70s, and so where people on purpose would not limit themselves to mathematical formulations, but would try to find a principle, an easily understandable principle in the 90s. Here are Aureli's axioms, and I've done some work on trying to derive theory from these axioms. things are even easier to comprehend. These are some fundamental principles that still need translation into mathematics, but that comes up to words. So here are some more examples. Here I said the CBH, the Eclipse and Blue Halverson ISEMs. It's of course debatable whether what we can derive from these things, but still the meaning of the ISEMs is very which I would derive from these axioms will not raise any questions about what it means, because here are the axioms, here's what it means. So there is a whole body of work, and these are not all axioms, which is on the market, there is a whole body of work moving from pure mathematical formulations to something which is formulated in the language of physical principles, of hopefully directly comprensible physical principles. Now, let me talk about something really new, something which did not exist in the times of Mackey or even Lutheran, and this to me is really a sign that the area of the foundations has moved to something, to do something else, not to interpreting or deriving quantum theory from pure early mathematical axioms as Mackey did. So these new things are philosophically called, if you want this, the high philosophical parlance. We can call them intentionally complete reconstructions. Or better, call them toy modems, which is a simple way to refer to these modems. There are other names. Sometimes they're called fantasy quantum mechanics or quantum mechanics light or whatever you like. And these are not all references, there are more references. So they succeed in making a whole new point.

12:30 And let's see how the point is. Now, intentionally complete constructions. Start again with some principles. And instead of deriving quantum mechanics as it takes, as we know it, potentially complete reconstructions derive something which is formed consciously and purposefully not quantum mechanics. And then we can analyze the difference between what we have derived and quantum mechanics. And by doing so, by putting quantum mechanics into a whole landscape of various theories that are somehow different, almost quantum mechanics, but not quite. We finally see why quantum mechanics, or we are trying to see why quantum mechanics, why the true quantum mechanics, has this feature and not that other feature, has, let's say, this much non-locality and not more non-locality or not less non-locality. So these reconstructions, which I intentionally complete, teach us something completely new, which we have not yet known about quantum theory, let's say, 20 years ago or so. So time models focus on physical principles, certain physical principles which are held while other possibilities are modified and lead to a better understanding of the structure. examples which I think most of you know of, but let me present these examples in the way that you see that this is a reconstruction of something which is not one inferior, it still teaches us an important lesson. So here's Rob Spagans' model. Rob has one foundational principle. If one has axiomal knowledge, then for every system at every time, the amount of knowledge one possesses about the state, about the actual state, the wanting state of the system must be equal to the amount of knowledge one lacks so basically this gives us this email instead of the full block sphere in quantum mechanics we have a discrete state space with only these states allowed so you see every time we know

15:00 as much as we don't know so we know that the state of the system can be somewhere in the blue do what? So this is dramatically not quantum mechanics. You know, a discrete state space which has nothing to do with a continuous block sphere. And still, this model can accommodate non-commutativity, interference, multiplicity of convex decompositions of mixed states, no floating, teleportation, and a bunch of other properties of the true, cool, and complete quantum mechanics. And of course there are some other properties which are absent. So there is no belt here, there is no textuality, there are some other things which lack. Now from just doing this high mode we learn something about quantum mechanics. We learn that the appearance of these features, of these properties, can be unconnected, can go unconnected with the presence or not of these things. So this is something which before we couldn't see just by doing quantum mechanics as such, by interpreting quantum mechanics. We couldn't really get such clear lessons about what the structure of a zero-gaze. So here's another example, very quickly, but I guess most of you know the example, so let me just give a definition and then focus on what is different in this example. These are the best overall for non-local boxes, if you like. So non-local boxes are boxes with two inputs, which can be a zero and one, and two outputs, which can be zero and one. And there is only one rule. This is the rule about probabilities of outcomes if there are certain mean rules. I will show you the state space so it will be easier to understand. So this is all definition. All that is in the definition. What is PR? That's it. PR is Papasko and Rolik, who first invented them in 1993 or so, maybe 1922, something like that. Oh, here's the reference, 1994, excuse me. Yeah, so they are not local because one cannot signal.

17:30 There is no superluminal, like, signaling. Now, PR bosses are, this is the main feature, it's explored by the best-wing role. They are more than local in quantum mechanics. but more than low. So in the club that we're in Shimoni inequality, pure bosses give us valid form, which is something quantum mechanics cannot achieve. So you clearly see that this is not really quantum mechanics. Let me show you this. Here's a state space. So parts of the circle here are the state space of quantum theory. It's called Q. So quantum theory, as usual, has a sphere, has a state space. Now this, here, this polynum inside is classical mechanics. This polythode, with lines connected called L, is the state space of classical theory. Now, non-local boxes allow for these parts over here which are not in the state space of quantum mechanics so we can get more. Now with non-local boxes, here is a list for you. Here are several, well here are two different, a little bit different, slightly different theories with non-local boxes. And here is a list of certain properties or features of quantum mechanics which you see sometimes are repeated in non-local boxes, sometimes there's more, sometimes there's something to quantum mechanics which is not repeated in other theory, et cetera, et cetera. So the question of why, let's say, for example, the question of why there's as much non-locality in quantum mechanics, and not more, and not less, which has not been answered to this we don't know the answer is a new question that we can only ask when we are doing reconstruction. So this is something really new about reconstructing quantum theory instead of interpreting. In order to understand quantum theory, we want to put it in a landscape and in context of various other alternatives

20:00 which will reproduce most of quantum theory but not quite all of it. we will try to better understanding why we have this quantity rate and not something completely different. So here's, to finish this, I think I'm about to, yeah, 20 minutes exactly, so here's a reference to the paper in BJPS where I discuss all these things. And thanks for your attention. thank you very much comments question if you look at the the axioms of some of the systems you describe to us they contain words that somebody like bell would have been upset by things like measurement of information and for someone who's a more unreconstructed interpreter that worries me that the project is really going to give us that's what we want so emery emery is before we post the access, before we formulate the access, we have to worry about the language. What is our language then to be, which we're getting to formulate these things. Now there is a variety of languages for formulating the first principles. And there is of course a debate open about which of them is, there is an informational language. There can be a language of, let's say, physical space or we can also use different mathematical languages. Certain people would translate the same principle into something which would use one logical language and other people would translate it into something which uses the algebraic language or C star algebra or some other language. So the question of the choice of our fundamental notions is a question which I haven't read it yet because I couldn't treat all of that, but it's discussed in the paper, and that's something which is important. So my claim would be that there is no a priori argument for saying this language is better than some other language. You know, try with this language, formulaic principles, see what you can do. And if, at the end of the day, you manage to do something really good and important, That means that there is an important lesson to get from this work.

22:30 Am I the old one? No, no. It's just a very insignificant comment. Your notion of deconstruction reminds me of what said about how one should proceed in physics if one wishes to proceed axiomatically. The first two elements in your pre-lected procedure is for my memory. Give the physical axioms and then the mathematical representation and the attitude that the physical axiom should be strong enough to determine in principle, ideally, to determine the mathematical representation completely. Now he realized immediately, of course, possible. Therefore, this axiomatic procedure cannot really be followed, literally. You're right. So you can look at the paper, on page two of the paper, I have a quote from the 1927 paper, like Monod and Hilbert, and discussing all this. So the paper has all of those, yes. The paper has Monod and Hilbert. I was wondering about the overall motivation of this reconstruction of quantum theory. First, you alluded to a case of Einstein, if I understood correctly. I'm not sure if you did something like this. Wasn't he rather wondering about contradictions, maybe he was wondering about magnets going through a wire and that there is some inconsistency? And then the second point is about the overall logic of such reconstructions. Just a naive question would be, I can derive things from false assumptions. that does follow. So how much do you really gain if you can derive something from a set of premises? The premises can be false. Yeah, okay, two questions. Now, the first question, by the parallel with what Einstein did, there's been a big debate about that, and I think there will be papers coming out soon. And there have been a few papers, but for Stimson and other people. so let me summarize this by saying without explaining all the details

25:00 whether Einstein wanted this to be so whether he believed that principal theories are better than constructive theories or worse there is some extent to some extent, not truly to some extent, this idea of deriving, of reconstructing is analogous to what Einstein Then Einstein went further on. He actually wasn't happy with principal theories. He wanted to have ultimately constructive theories. So there's a long historical story, which I didn't want to go in here, but we can probably talk about it later if you want to hear the full story. So there is some analogy with Einstein, but not a complete analogy. Now about deriving from axioms, which are supposedly false. this is exactly the point. Instead of, so in order to know what is false and what is true, which axioms are called, axioms, not theorists, but axioms, you have to have an interpretation. in the way that, you know, here are my fundamental axioms, which I believe, and here I'm derogating something. Now, this whole program is trying to be epistemologically modest, and I even have this slide here, which is called epistemological modesting, which says, you know, we don't really know what the good language is, what the true formulations of axioms or whatever are, but let's try with these axioms that we have here and now and derive a theory, compare it with what we have with quantum mechanics, and see whether we can learn something about the structure of quantum mechanics from doing this. So maybe we start with axioms, you know, like Rob Speckens' term of this, or some other term of this, start with axioms because it's not like this in the world. But they still manage to derive a theory that which, when compared with quantum mechanics, gives us some new insights about quantum mechanics. Not about this time, but a little bit about quantum mechanics. So we can start with something which we will know is incomplete, or not true in the real world, and still learn something about quantum mechanics. That's the whole point of this work.

27:30 Yeah, I don't know if I just told you, but I think that there is two different things in your talk. One is the reconstruction with the three stages you mentioned. But this other thing of making up new theories closer to quantum mechanics and comparing in order to know how non-locality or how different features emerge, It seems to me that it's something different from the first three stages you mentioned. And I want to know which is the connection with these two projects. And regarding the second project, of just making up new theories closer to quantum mechanics, just to see how certain features emerging come in quantum mechanics, I want to ask you if you don't think that this project is also suitable or worth doing it with relativity or with even classical mechanics, this kind of play, don't we? Well, the last question and the answer is, I don't really know because I haven't tried, so maybe, who knows. To the first question, why am I putting together these things, why complete reconstructions, analogous to complete reconstructions, in a way? This is because they still start, so the whole idea, the basic, if you like, the fundamental idea, is that you start with a principle. So here you start with a principle, you can know one half of what can be ideal in the other. And this is, if you compare it with, let's say, Orvelius, or Sorvelius, here, so if you compare this Speckensys principle with Eurelis principle, with Eurelis principles, there exists a maximum amount of relevant information that can be extracted from a system which is always possible to obtain new information about the system. So you see that there is an analogy in how we formulate these axioms, these principles. So we start with principles that are comprehensible that we, which we understand the meaning, and we're trying to derive something from it. So then we can have different ambitions. We can have an ambition of deriving the actual point of mechanics, which one can actually not do from just

30:00 these two Revelli's axioms. So you need to add stuff to that. Or we can have an ambition of deriving something modus, something which is not quantum mechanics, but still trying to learn something about quantum mechanics. So what is common is the approach, the way we work. We start with principles and we do right things. Okay, we have time for one last very quick comment, please. I just wondered what your conception of theory was. Theory. So is it that the theory is the actions and the closure? when people talk about something following from the theory, that's really a pattern to violate because nothing follows from the theory. Yeah, so, no, no, well, my idea of what is a theory is more complex, if you like, because, of course, I'm aware of the fact that, and there was a short discussion of this, probably I went to ask. The axioms themselves need to be somehow motivated. We need to know why these things are, you know, why do we select them as axioms? And this is done by changing, if you like, changing your skin by going to another theory. So to me, there is an interplay of theories in which things that are axioms in one theory can be derived or theorems in another theory. So I would claim that this whole program does not limit itself to this reductionist view of the theory as an axiom and axioms and closure of those, because there's a problem of motivating axioms of where they come from, et cetera, et cetera. So I have about what I call a systemological circles, like how theories will relate to each other, and that's a whole different story. Okay, thank you very much again.

32:30 So our next speaker will be Eric, please, from Athens, who will speak to us about holism and non-separability by analogy, if you can get that. If you can get the PowerPoint out of that one. Thank you. It is certainly not original to maintain that the major value of analogy in science and philosophies and enhances our understanding of one state of affairs by pointing out its similarities to another better understood state of affairs. I thought I'd be original in this talk by pointing out the similarities of a not well understood state of affairs doing even less here is the plan of the talk but let me go directly to content holism is usually understood as a metaphysical thesis that has been elaborated by Richard Healy of course I'm paraphrasing here and simplifying the bit One system exhibits physical property holism if it can possess qualitative intrinsic properties

35:00 that do not supervene on qualitative intrinsic properties and relations of its components of systems, the parts. Recently, Michael Sieving has advocated instead an epistemological criterion for characterizing theories as holistic. The criterion again reads roughly like this. it is impossible for a set of local agents each having access to a single subsystem only to infer the global properties of a compound system by using the resource basis and by this he means at least local operations in classical communication available to them. The global properties of the compound system as assigned by the theory. Related to policy is also the issue of state non-separability that a physical system exceeds state non-separability if it admits states that do not supervene upon the states assigned under the same conditions to its components of systems. Now it's not my aim here to explore the philosophical subtleties in those approaches to policy non-separability in physics. My only aim is to use them as a springboard in order to probe an analogy between the typical system that is taken to exhibit holies of non-separability in ordinary quantum mechanics and a system treated by relativistic quantum field theory. I shall argue that the quantum field theoretic system does instatiate physical property holies in the metaphysical sense, as well as that there are possible grounds to deem the describing theory holistic in the epistemological sense. I shall also argue that due to the intricacies of the quantum theory of systems having infinite number of degrees of freedom, the quantum field theoretic model does not exhibit state non-separability, albeit in a rather weak epistemological sense. namely the compound system admits states such that no agent confined to a single subsystem can determine the state she is in or retrieve knowledge about the state from any complete knowledge about objects intrinsic to the subsystem she has access to.

37:30 Now this is all the standard argument in quantum mechanics. It's familiar stuff so I'm going to go We consider a pair of spin one-half particles if they are prepared in size zero, this is a pure state, it gives a definite value, namely the value zero to the total spin, however the partial states of the single particle states are mixed states undescribed by the density operators decaying and these states really do not assign any definite value to the spin in fact they can be seen as states that describe completely if you unpolarized or disorganized spin and in these states really the formoyman entropy is maximal it takes the maximum value it can take for a two-dimensional different space, so it takes a value of logarithms too. So here we have physical property holders and in order to establish state non-separability, one just says that consider again the state side 2 really. So side 0 and side 2 are distinct states However, the partial states are the same that is, are again described by D-K. Okay? Now, to construct an analog group of the capital pairs of P-1 half-particles in quantum theory, we can start from a mixed state now, in fact a thermal state, and purify by doubling the system. Bernard K has done this mathematical work. So here's some preliminary definitions. What is a quantum dynamical system? It is just a sister algebra that contains the observables of the system, and U, and one parameter group of automorphins algebra alpha t t ranges over the whole real line and we are interested in a particular kind of

40:00 states over dynamical systems the so-called kms states i'm not going to give the precise definition of the kms state but suffice it to say that there are good reasons to say that kms states are states of thermal equilibrium of the system okay if you want we can talk about that sort of for good reasons there are. Now, a double quantum dynamical system, so one can really have a quantum dynamical system and then make it twice as large, which because we are talking about infinite systems, twice as large means equal as the previous one, in terms of carbonated binary, make it twice as large, and make a double quantum dynamical system. really has this structure, the bar items or the undersigned items refer to the double quantum dynamical system. There is a systal algebra which in turn is the tensor product of two systal algebras, the left and the right algebras I'm going to call them, and these are commuting. The dynamical automorphions really alpha-t now map the left to the left right onto the right and there is also j which is an anti-voluntary sorry the involutive anti-automotions that maps the left algebra to the right algebra the right algebra to the left algebra and commutes with the dynamical automotions group so the strategy now of purification is to pass from a kms state a thermal state on a quantum dynamical system to a pure state on the associated double quantum dynamical system. And the key concept here is the concept of a double KM state. Here is the definition. A double KM state at some value beta, which is positive value, over a double quantum dynamical system is invariant other than dynamical automorphism, is also invariant with respect to the involuntary anti-eutomorphism state over the double algebra and the GNS triple of it has several characteristics anyway I'm going to point out a couple of them only

42:30 the vector representing the cyclic vector representing the KMS state is highly in fact for both the left and the right presented algebras there is a self-adjoint operator which i'm going to call the total hamiltonian who generates the unitary implementable dynamics and satisfies uh really is an angular vector of the thought of hamiltonian who i can value zero and there is also an anti-unity operator anti-unitary operator sorry that that it limits the involuted anti-automotor. Now, the thing is that a simple computation can verify that the restriction to the right subalgebra of a WKMS state at value beta is a KMS state in the union, so a term of state. Lucy speaking, that is, a double KMS state of value beta on the compound system is a state of thermal equilibrium at inverse temperature beta, that is, temperature one over beta, when you stick it to each of the components of systems. The important point here is that the double KMS state itself is a pure state. And this is a content, the content part of the content in the way of the theory, again, by Bernard Kaye. So we consider the double KMS state over the double quantum dynamical system. The associated GNS tripled. Then the left and the right von Neumann algebras are commutants of one another. And furthermore, if this condition that is written over there is satisfied, which is satisfied in the concrete quantum theoretical model that I'm going to talk about, then the left and the right Homoimann algebra are factors and in fact omega V is a pure state over the double algebra the double chemist is a pure state over the double algebra and the assumption that I'm going to use from the Norwebs is that the Homoimann factors are these two which are through the complete quantum the Homoimann factors the left and the right algebra

45:00 common algebras are type 3 and in fact they are type 3-1 and this has an importance for what I'm going to say here are some consequences even at this level of the abstract system a double KM state is a pure state on the compound system but its restrictions to the components of systems are states of thermal equilibrium KM states and I'm going to say in order to, with some caution here, I mean there is a lot that these are states really of maximal entropy okay there are very much intricacies when trying to really define entropy density in continuous systems but let me just extrapolate from from other systems now since omega can be shown omega beta can be shown to be cycling and separating for each one of the two following algebras and this allows for the application of the theory to the right algebra. J-Vita is the modular conjugation. Delta-Vita is the modular operator, and really the modular Hamiltonian, the one that generates the modular or the modular group, is what I call the double Hamiltonian H-Vita. Here are some other consequences. Since the elect and the right for Neumann algebras are hackers and commutants of one another, they are sister-independent. This means that any state on the one is combined with any state on the other. And this is important when it comes to investigating issues of polism and non-separability. Really sister independence of the pair guarantees is that no preparation of a specific state for the one subsystem precludes the preparation of any state on the other subsystem. And this is important because really since we are talking about state-dependent properties, we see that there are no if like logical restrictions of other sort of state-dependent properties that are going to be assigned to the two subsystems. The fourth consequence is that since the Ith-Permarman algebra is a type 3 factor, the unitary operators that really define the modular automorphic group do not belong to that algebra.

47:30 In technical terms, the modular automorphic group is not a group of inner automorphic groups. And in this sense, I mean, this is one conclusion we can derive here, the total dynamics of the double quantum dynamical system is not described by objects belonging to the components of systems. Now, I'll go to the typical, I mean, so far it's kind of been in the abstract, this is a realization in Poundfield theory, you simply take the Klein-Gordon Poundfield on Minkowski space-time, And we are talking about the left L and R, the left and the right wringler wedges that are defined in that way in a global initial coordinate system. What you see there as C A is a time-like vector field. In fact, it's a time-like vector field that corresponds to the row-relibuses, okay? and HA and Hb are now hypersitruses. I'm going to go a bit more quickly. This is how HA looks like. And from then onwards, if you have this setting, you can apply a standard procedure for quantizing a linear classical system using some techniques of algebraic conflict theory. Namely, you define or then you define the symplectic group that is giving the dynamics. It is important to notice here that the dynamics really correspond to the Lorentz boosts, all right? As you see here, the top of T corresponds really to a Lorentz boost. So you have the classical linear dynamical system, then you pass to the Weyl communication relation algebra over that, and so forth. Let me skip the details, and this is how you define the subjective evolution, and you define the involuntary atom of anti-photomorphism.

50:00 I already thought I was going to pronounce this. by each action on the Weyl operator, so you get that end up with a double quantum dynamical system. Now the final technical detail, and then I'm going to come to conclusions. Bernard K. constructed a ground particle, a regular ground particle structure, defined explicitly, sorry, for the right words. and on the basis of this you can define explicitly a double KMS state for every value beta. The GNS representation now associated by the United States acts on the tensor product of space and notice this is how the H beta which is I mean, one is expressed in that way, okay. All right, now, let me, no, so I propose, and we can talk about, whether, I mean, this presupposition. I propose to regard the double quantum dynamical system of the Klein-Rodder field as a compound system, okay, with component subsystems, it's a restriction to the two wedges, L and R. So let's talk about polys in a metaphysical sense. Eraborated by Richard Healy, the question to ask here is this, are there states in which the compound system possess intrinsic properties that do not supervene on intrinsic properties which are subsystems? The answer is yes. any double KMS state assigns a definite value to the total Hamiltonian in the associated, in the sense that in the associated TMS representation, the vector representing the double KMS state is an idea vector with a total Hamiltonian value 0. So the double KMS state assigns however if we restrict now the double KMS state to the one subsystem the GNS representation of the right algebra

52:30 used by the restriction to the right wedge of a double KMS state is provably unically equivalent I'm not going to read that to that And the part of the total Hamiltonian, if you like, that is contributed by the right wedge, is not even expressable in it. Formally, one can show that the norm of D gamma H omega beta becomes 16. Okay. Now, if this is by interpret the situation, thus, when the double quantum dynamical system is in a double KMS state and hence has zero energy, the expectation value of the fluctuations of the energy for the well subsystems become infinite. the restriction of a WKMS state does not assign any definite energy property if you like to the wedge quantum dynamical systems therefore there are qualitative increasing properties of the compound system that do not supervene on qualitative increasing properties which components of systems that is we do have instantiation of physical property always. Okay, there is a question here whether this should be understood as a class, a quantum or a class in the sense of, say, thermostatistical fluctuation phenomenon. I have a few things to say about that, but I'm not going to say, but perhaps they come up in the discussion. Let's now talk about holism in the epistemological sense in Sivins. So, come to agents, Lisa and Rob, here I'm trying to be original over quantum theorists, quantum formation theorists, I'm not talking about Alice and Bob, I'm talking about Lisa and Rob. Okay? So, come to agents, Lisa and Rob, having access only to L and R, respectively, infer the global properties of the compound system by using the resources available to them. well I don't have a definitive answer but the answer is this no, it would demand that every local operation performed by Roe on a global state to turn into an anger state or some observer who will produce a state that is empirically distinguished group

55:00 from the original one by Lisa this cannot be done and it cannot be done I show here the argument because really the lock, the right, was from the first time three. In short, let me give you, I mean, you can look at this, but let me give you what is in words. The distinctability of states by local agents, in this case, is delimited by the possibility be appointed by type three factors to change any state into a state of a projection by a local operation in a region without disturbing the expectation value of any observables in the region's space-like community. So really, Roe has turned the global state to an active state of an observable in this region. And there are going to be no changes to the expectation values in the space that can be injected by laser. In the sense of state non-separability, I'm just going to make a statement and then we may discuss because I'm running out of time. Here's how we're going to state these other distinct states of the compound system to restrictions to the components of systems are identical in the analogy to the spin case? The answer is no when it comes to double KMS states, because the restrictions of two double KMS states are at different values, are KMS states at different temperatures of the quantum dynamical system, and there's a mathematical theory that says that these are disjoint, all right, given that the phenomena or sometimes three, this means that the restriction of the one WK state is not even expressible as a density matrix into the GNS representation of the restriction of the WK state. I take this to be the presentation problem and aspect of the presentation problem for the quantum theory of systems with different number of degrees of freedom i have some remarks that

57:30 i've only really to tone down this difference in the spin example but let's come up they come up in this discussion and what i was planning to uh talk about the is really to try to justify the idea that, I mean, if we are trying seriously to talk about do these, here are the questions there is, do these left and right wage systems, if not that really has parts of a compound, of a whole. All right, and can we really talk about increasing properties there, okay? I have some things to say about that, but let me just save time, and if they come up, they come up. Okay, thank you very much. We have some time for discussion. Questions, comments? say more about oh yeah okay I'm not going to say that in a company in a theory the system in a region can be considered as a subsystem in any region can be considered as a subsystem of any larger one. I'm not going to defend that. I'm going to talk about this to the left and right. Now I take it responding to some of your work. What we need here is this. Can we think of the double quantum dynamical system as wholly composed of these two parts during the dynamical evolution? And the answer is really yes, both at the classical level and at the quantum level because the dynamics give the wages invariant. Now, the second question is can we really talk about increasing property? Now, in order to talk really about intrinsic properties, I guess we should know exactly what an intrinsic property is.

1:00:00 And we don't know exactly, but let's say this, that an intrinsic property is a property an object can possess independently of the existence of any other contingent object or state. okay now so can we talk about increasing properties in the two wedges we can in the following sense okay because we are talking about state dependent properties really what we're talking about here is say energy okay the two So the question is now whether there can be independent state ascriptions to the two questions. And there can be independent state ascriptions to the two questions just because they are so far. May I just jump? Sure, sure. Yes, they are, but they are not . That's true. So therefore, this is a stronger argument, even. This is an additional argument in theory that they're non-unitent, so to say. Yes, you are absolutely right. And of course, yeah, they are sister-independent, but they're not W are sister-independent. Notice here that my informative answer says, yes, but. which is reading smaller letters and let me give you one more so this leads me to my question on you to what extent to do these non separability features you have discussed depend sensitively on the features of these WP and I say I just don't see this play but I want one has the impression that some of the features of holism both in the matter, physical and the other,

1:02:30 academic standards, do or pain, even if you don't refer to these specific features of the WMS case, I have the impression. Not all, probably, I don't see which ones do, and which ones don't. If you have a comment on this, that would go back. I would also have a very specific question concerning page 13, when you discussed the, one of the non-local, I'll be able to recall exactly your claim that the Hamiltonian you said something like the Hamiltonian and assigns some definite value. This one, thank you very much. Yeah, any double KMSK assigns a definite value to the total Hamiltonian in the first GMS representation. What is meant by this? This is a KMSK and the Hamiltonian is probably an unvalued operator. What do you mean by assigning a definite value okay yeah I just didn't understand that all right wait yeah I see black and just very specific issue I mean when we don't came state introduced these work has been done by Bernadette in 1985 the purification of states I responded to the first question, yes, some of these double KMS states are really familiar states. In fact, double KMS state was depending on the normalization of the group, I mean the the double-game state at temperature 2.5 is in costumacrum. So, but the idea is here that we start from the subsystem and we purify the state.

1:05:00 So you're right, it's not peculiar about this, but I think that, you know, this shows really very nicely the analogy with the spin system. Now, the second is this, okay, surely the Hamiltonian is unbounded, when I say that the WKM state assigns a definite value, I'm just meaning this, alright, if you have, if you take the GNS representation of the double algebra, then you can define it there. the total Hamiltonian and the vector that represents, the scientific vector that represents the the dark became a state is an eigenvector of value zero. Nothing more than that. Perhaps I should say since the associated GNS representation. Do you have a... I don't know now okay so we don't know anyway I can just take one very short anyway what I wanted to say is that I mean the last is a black part of the talk about and this has to do that okay the two states are disjoint but the agents that they are confined to the subsistence they cannot know which they vary and this has to do again with it and the other thing is that even if they are even if they are the joint you cannot really distinguish payment states at different temperatures in the smoke there has been some recent work about a thermal field theory where you can say that to find yourself in a very small region, then you can have any other time temperature there and in the space light won't be maintaining even temperature, all right, but this is just it. Okay, thank you very much. No, no, no, it's okay. I'm all right.

1:07:30 To take notice of the philosophy of physics, if people are going to invoke quantum mechanics in contemporary metaphysics debates, then we need to be sensitive to the terms in which metaphysics is framed there, framed there to base. So there's a basis about vagueness. I'm interested in these four pages where people thought that quantum mechanics has something to tell us about vagueness. So traditionally we think of language as being vague, it's indeterminate whether the walls are yellow, so kind of a shade of yellow, kind of a shade of white. And our use of language doesn't kind of fit an extension, a precise extension for all the terms. And some people have thought, well, it might not just be language, but the world. And other people thought that's kind of incoherent. And some people have thought, well, look at quantum mechanics, isn't there indeterminacy there? Maybe that'll have something to do with this debate. So I'm going to have a few puzzles about the way that's done. is the first lecture in a maintenance course is always about sororities series I think sororities is green for heat or something because you keep adding grains of sand when you start with one grain of sand well that's not a heap, two is that a heap, three is that a heap, at some point we get a heap it's kind of not clear where it is there must be something wrong with this, oh right a zero degree bath is cold well the ten degrees are cold must be cold because cold can't draw a boundary precisely somewhere along the series but obviously a hundred degree bar is not cold okay so there must be something on the side and the process is diagnosed what the fallacy is now not all of these series come in argument for i claim that that the left end of that bar is light gray and the other end of it is not like It's kind of indeterminant. It's unclear where along that line, light gray turns to dark gray. So the series is one thing having the vague in this literature. Indeterminacy is the other thing to have. So that's captured by people working in medical visits and philosophy of language and so on, using this sentential operator.

1:10:00 because a D means definite, it's definitely not like gray. I think it's definitely not like gray at the other end. In the middle, it's not definitely like gray. It's not definitely not like gray. I guess we'll abbreviate that by this del simple, which means indeterminately. It's like contingency in modal logic. The reason this seems to be important is that people wonder about what kind of logic that D operator obeys. The reason they're worried about that is because origin determinacy. So it seems kind of implausible to say there's a sharp cut-off between light gray and not light gray. So we replace that with a distinction between the definitely light grays, the definitely not light grays, and the indeterminately light grays. That seems equally implausible because now you've just got a three-fold sharp distinction. So we want to be able to iterate these indeterminacy operations. that's no good so now we want a sharp however many policies, you just kind of keep going up and up and up. Then we get to what's called radical higher-order indeterminacy where there are streams of indeterminacy operators. Now, so now there's a question of what logic is operating the base. If the logic's too strong then you get modal collapse. So in S4, a whole stream of these is just equivalent to 1T, and then you can't draw these kind of distinctions. The only reason I'm bringing that up is that it's important, it seems to me, that that's the local operator. Okay, so you can put it into indeterminacy operator in front of any sentence. It's indeterminate whether the C is blue, which is kind of a green color. It's indeterminate whether Europe is identical to the European Union because it's not clear on there's a pebble say on the beach somewhere it's not clear whether that's part of Europe it's not clear whether it's part of this thing that we call the European Union. You don't know that example just in Europe. So vagueness has to do with indeterminacy, it has to do with this variety of series. Some people think it doesn't have to do with series indeterminacy is enough. Everyone thinks more or less that vagueness is featured of language, but terms fail certainly if you make sharp distinctions, and there's no reason why this should.

1:12:30 The reason we have it in terms of densities is that the names are very, very, very set. Some people think that it might not just be the world, it might not just be the language to play, but the world, whatever that means. So it's a very, maybe a property is a set Europe's a vague object, so it's not just the term Europe. You might refer to this region, might refer to this slightly different region. No, there's just one thing, Europe, which is a vague object, kind of fuzzy boundaries. Fuzzy boundaries. So then we have these inter-term sense of statements, not because the terms are false, not because the terms are vague, but because there are these vague objects. lots of people think that's crazy obviously it's just a picture of action the world being determined so at this point people say oh well the quantum mechanics isn't there supposed to be indeterminacy that I'm sure someone once told me so E.J. Love has met a position of Stephen French and Dessier Krauss physics, have kind of taken this up and said, right, quantum objects are vague objects. That's a type of steam and deserts. What they're doing here is replying to the things like Garrett Evans, which goes down to this. Start out with your indeterminate identity statement, because indeterminate whether these two things are identical. But it's not determined whether something's identical to itself, supposedly. Because we have these two things, A and B, and it's indeterminate whether A is identical to B, well then B has a certain property, namely being such this identity to A is indeterminate, but A doesn't have that property because A isn't identical to itself, indeterminate B, so by sort of diversity of this is similar, right, a country positive environment, one of them has a property, the other one doesn't have, so they must be distinct after all. in fact they're distinct, they can't be indeterminate when they're identical, but they're just distinct. So Evans thinks that long-term makes vague objects are impossible. So Lowe brings this up because he thinks he has an example

1:15:00 of an indeterminate identity in quantum mechanics, so there must be something wrong with them, so they can kind of go and say what it is. What I'm interested in is how it motivates this diagnosis of a fallacy in evidence. It's the stuff people were talking about yesterday with the non-individuals. I'm sure you all know that much better than I do. The idea is that, given permutation symmetry, it doesn't make any sense to think of two states as having a particle of this side, that swaps around, because the particles of non-individuals, that just doesn't, those states are the same. Okay, now this is a very minor, I wish I'd skip because I've got a lot of time. My main concern is this is not an example of an indeterminate identity. So the way French and Crouse capture non-individual theory is to say that, okay, they have what they call a Schrodinger logic, a two-sorted logic, we have species of terms for the macro-objects, like human eye and so on, and terms for the micro-objects, the non-individuals. And for the non-individual, terms of example, non-individuals, the identity is not well-booked, it's not a book. So the reason that I was so keen to say that the definition of the operation is a mobile operation was because I was having the impression and standard logic have rules that tell you what the formulas are, so they tell you what the atomic formulas are and then they say you get a formula, you get a disjoint formula, you get a formula, you quantify these formulas, you get a formula and if you're doing modal logic, you take the formula and you attach your modal operator to it, then you get a formula and that's all the formulas. So it seems to me that if you take something that's not a formula in the tantrum modal operation to it you don't get a formula just by doing that right so if I okay that's not a formula that's just meaningless so if I say definitely that's also meaningless okay I don't get a formula just by putting modal operations from some point something's meaningless um okay there's the reason I thought that's important because if you're trying to give reply to the Evans argument, so a case where you do have an indeterminous

1:17:30 identity which you're going to capture using that indeterminous operator, you can't do that by making meaningless what it is you're trying to defend. So it doesn't seem to me to have a counter-example to Evans so much as just kind of making being completely unsaid for this argument. So it seems to me it's not indeterminacy of identity, but just outright in-applicability, and in-applicability and indeterminacy seems to be different. So compare these two cases. Seriously, that's meaningful, it's just indeterminacy. That's why we use this operator. This is a case, I think it's Shoemaker's example you take, I think it's Mr. Brown's brain gets transplanted into Mr. Robinson's body and then supposedly it's indeterminate whether Mr. Brown is identical to this, the guy who comes out of the uber, it's Mr. Brown's. it seems meaningful at least to us are these people identical even if it's indeterminate whereas French and Krauss encountering this idea of Schroding as the identity for the funniest of things in quantum mechanics is means, just completely means doesn't make any sense to say that so it seems that we don't have a reply to evidence so I was wondering how we can get out of that Well, there's a sense of Bay in which he's out of the Bay context. Sure, but it's not the sense that people in the Bay's literature use. That's why it doesn't seem to be right. That's why I thought it was important that we should engage with metaphysical issues on the metaphysicians' terms. because you could say you could kind of add something to the syntax so that del a equals b got to be a formula even though a equals b was not but I don't see how that's going to help that just doesn't capture what evidence is after in any way he thinks it is a formula because it started out with del a equals b and then got to the not a A equals B is in a form and then not A equals B is in a form, and that doesn't seem to work to me. Because they might say the way to capture the non-individuals is to use its in-determincy operator.

1:20:00 But that doesn't seem, so I thought I kind of understood this non-individuals thing. identity didn't mean anything for the particles in quantum mechanics. That was why it kind of didn't make any sense to say you swapped them around, and that's why you got the quantum statistics, you have three possibilities instead of fours. I'm not sure that indeterminacy would give you the same thing. So I kind of don't really see that you have vague objects in quantum mechanics. It sounds like something kind of different. So the last paper on that list was by Silvio Chimeli. So he said, well, let's not go to the vague objects route, let's think of another example. So the idea would be, we have a very old cat, it's kind of been determined whether it's alive. Maybe it's hard to stop beating, but the electrical activity in its brain hasn't quite gone to zero yet. well it seems to tell whether it's a lie but that's just because the term alive is vague it's not clear where we draw the lie but it might be indeterminate whether the cat's alive because you let you go to schrodinger to look after and he put it in one of his boxes and now it's kind of in a superposition of being dead in the lie maybe that is the kind of thing the metaphysicians are after So I kind of managed to convince myself that maybe it was, so it's not like the non-individuals case, because it is meaningful to say a counsellor, it's just indeterminate, so you can pre-fix your dependency operator now. Okay, it has to kind of go deep in the notes of the philosophy of economics, because I found someone using it determinately, but I found it in the air. So you can kind of, okay, this is where you just have a feeling wrong, but people say that what's strange about the super positions, you can't say that the system's in this state, you can't say it's in the other, you can't say it's in both, because that's crazy. But it didn't go through both states. It has to go through, it didn't go through neither. It didn't go through this

1:22:30 go through this one kind of sounds like we can we can formalize it like this okay so definitely not definitely this one not definitely this one and that is just what the people in the vagus literature used that indeterminacy operator for that's how it's defined this is all right So the last thing I was thinking you might do would be to draw a parallel between terms in the superposition and precisification in the supervaluational approach to vainness, let's say. skip that a bit. The way vain sentences are treated in super-evaluations and is to say we have lots of different ways we might make our language precise. So perhaps a child is anyone under 14, perhaps a child is anyone under 50, maybe it's anyone under 60, maybe it's anyone under 70. So there's lots of different ways we could fix the reference of all our terms. If something gets to be true, of simplists are super true of it. True on all of those ways. So no matter how you understand a child, any legitimate way of understanding a child, a 65 year old is not a child. In some ways of understanding a child, a 14 year old is a child. In some ways of understanding a child, a 14 year old is not a child. That's why it's indeterminate whether a 14 year old is a child. Now some people in the ontic vagus debate would like to, because that was kind of a semantic way of understanding linguistic vagus, but lots of ways of understanding a term, lots of ways of making a language precise, sentence comes up trill and some involves on others. And so some of the people that advocate ontic behaviors would like to say they're kind of really different possible worlds. They're kind of all competing to be actualized. And some of them the cats are alive and some of them the cats are dead. That's why it's indeterminate.

1:25:00 The reason I was interested in this is because maybe we can kind of make contact with their approach here. So I kind of want to expand on this kind of light, because that was a little bit wishy-washy. So I want to kind of squeeze it into this equation. So we need a set of points, so we need a set of points where we cancel out, which is the points where we cancel out. And terms and superpositions are supposed to do that. This will say, here's the cast-line term, here's a possible spin-up term, here's a spin-up in the edge direction term, here's a spin-down in the edge direction term. That's why it's interterm, whether it's spin-up or spin axis, or whether it's spin-up or spin axis, or whether it's spin-down or spin-down. But then we need a different set of points for each thing we want to evaluate. If you want to know whether it's definitely spin up in the y direction, you'll be able to take a different set of points. Whereas the vaguest people want a safe set of points to evaluate all their sentences. So the final kind of idea is to say, let's have a point for each. kind of each point will represent a complete possibility. So it'll say whether our class is enough in the x-direction, it's enough in the y-direction, and so on for every direction it's been in the hat. So we'll do that for all the observables. So we'll have just what they have in classical super-valuations. And we have every point in this space space will have a depth and value bravely observable. So bravely member of a bassist will pick out just one element of that bassist. But you can't do that, I hope, because that's what the quotient spectrum did, it says. So it's just impossible. That makes it sound like indeterminacy for guessing quantum counts isn't quite what you have down in the same way, because you can't give this kind of supervaluation model to grow. If you can't do that, it kind of seems to me that what you have in quantum mechanics isn't what the people who said they might be on to

1:27:30 have had in mind, in which case it's kind of a very negative answer. Quantum of values isn't values at all, despite what the friendship graphs and so on said. I kind of hope I can tell you that I'm wrong because I'm here to support. Thank you very much. Do we have some time for questions? Since it's super true that I'm in Madrid, I feel challenged to say something. The term vagueness is also used in sort of different branching quantum mechanics in that we usually say that physical magnitudes have sharp values or not, so your length is some specific number. what I just said that's an example but then there's also this this whole development in quantum mechanics which is based on the notion of all the positive value of measure and they assign intervals to physical magnitude so for instance I assign much one value to your length by an interval How would that qualify in your terminology? I would have thought that would be indeterminate. People tend to use vague and indeterminate interchangeably. Some people insist that you have to have the series kind of features, but that's not compulsory. So I would have thought that would count as indeterminate. Perhaps it's very depending on it. Just one sentence is that if we take the statistical interpretation of one sentence, then this is obviously true that one sentence has nothing to do with weakness in a physical sentence. Because everything has a definite value, you just, but sure, right, you have to have the right sentence. This is another sentence from my side. This is the only possible interpretation of one mechanic that's...

1:30:00 Okay, fine, but then you can't use that in the metaphysics space, so here's our agent, because there isn't anything else, right? Psychophagy, you agree with me. So it seems, so really what I was saying was, even if you don't have a statistical interpretation, have something where there's genuine interpretation, but still it doesn't look really like what the metaphysicians wanted. I'm struggling to see how you fit it into their picture. You said that French and Krauss identity, so A equals B where A and B are, terms of non-individuals, is not even a formula. Yeah. I mean, is there a very simple thing you can take to explain that is how their logic works because and if you've got the identity credit in the language and you've got these terms i mean is it just a syntactic form that allows a yeah um it's right so i think that it's the syntax just doesn't right you're told what the atomic form is and that's not one here's an attempt to give you face objects I don't think it will work but let's try it ordinarily one thinks that and if we were more precise we could instead of talking about heat we could say well there's such and such but then we might get fussy about what constitutes a grain because it has fuzzy edges too but there might be an intuition that the ultimate objects themselves are not vague and how we decide to construct ordinary objects out of ultimate objects is where linguistic vagueness gets in on the act but that sort of presupposes that we construct everything from ultimate objects which I'm on a full way And then perhaps you could say it's hard to find ultimate objects in quantum mechanics. So there are no ultimate objects? Well, I've got you a bit more cautious. It's hard to find them. I mean, it's difficult to substantiate the point that, let's say, electrons would be good candidates for ultimate objects if you wouldn't want to be a theory.

1:32:30 right so what would the ultimate object be after which ordinary objects sort of slippery slope argument the world and you can't build ordinary objects out of the world so it's too big so what would be the vague object any ordinary object where all the objects now includes electrons hydrogen so if you can get objects at all and the sense in which they're vague is it vague is in terms of identity is it kind of in terms of whether two of these things are identical let's see I think it's going to be because to say what these objects are, you're not going to be able to say that, in precise terms, this object is composed exactly, you don't want to look in such a way, because you don't have anything else to look in such a way. I didn't expect a quick answer to get you thinking about something else. Well, that's the only thing I'd say there is that that really sounds like we're now a very long way from the Evans arc. I really don't see how that's going to connect with the anti-Begas thing. It might be kind of farther than I would expect to make objects than they have. I think it was satisfied with that, which was comfortable. One last question? A quick response also to Arthur Pooley. What logicians frequently do is that, and I think you can do this very easily in such a certain logic, You can simply allow for A equals B and add an axiom by saying for every A and B, A is unequal to B. So you can allow for a formula which was previously not well formed, but simply add an axiom that it's always false.

1:35:00 That is just a logical trick. So the argument should not depend on that. So it's always false, still there's no in-determinacy, I'm sure. But then it is not officially... Thank you.