11th UK Foundations of Physics Conference — Quantum Histories

0:00 So I will come from the University of Calabria and to talk about the comprehensive Thank you. is a foundational problem of concept and history approach to quantum theory, the approach composed by Dreyfus in 1984. The problem was raised by Bassi and Girardi, which was that some assumptions which may be necessary for some reputations of the theory lead to logical contribution in the constant constant historic approach. The work of Bassett-Gerardi can be considered as a generalization of the criticism raised by Kant which could that contrary differences in the consistent approach leads to contrary differences. Griffith answered test criticism by claiming that they are the result of Ames' leading understanding of this theory. But the critics of consistent approach are not convinced by the arguments of this, so that the subject, the problem, cannot be considered possible today. Now, here I present an approach to the problem, which starts from formalizing just the genuine concept of Protestant history of college's briefings. In such a way, it's possible to a natural interpretation of problematic aspects of Protestant history of our college as

2:30 contra-differences at the problem when I said by Vassin and Girardi. And this interpretation is free from contradiction and of conceptual difficulties. I begin my talk by applying a consistent standard, a consistent visual approach, so that I can start the problem and at the end I present my approach and do this with the interpretation. Now, constant history approaches make use of the same space used by standard quantum theory to discover the physical system. Now I shall adopt the estimate picture and assume that the dimension of this space is finite. end. The basic concept of consciousness approach is the concept of family of history. To foster a family of history, we start by fixing a finite and modern sequence of times, P1, P2, end. Then we assign to each time a decomposition of the event. A decomposition of the event is a final set of projection vectors such that they are mutually parallel with each other and such that the sum is the identity of the event. Then each time we have assign at the composition of the event. If for each time we take a projection operator which is the sum of a number of projection operators from the corresponding the composition we get a sequence of a sequence of events and such a sequence is called history. A family of history is just the set of all possible histories we may construct starting from a given decomposition of the identity.

5:00 A particular subset of histories in a family is the subset of elementary history. An elementary history is a sequence in which every event is just one projection operator is corresponding to the composition of the identity. The texts are elementary histories. And in the center of all elementary histories is noting the Cartesian product of the decomposition of the identity. Now, to each history we consider the bound operator we obtain as the product of the events form the history in the reversal order and give an history, this product is a band of retron we denote by c h. This is the formalism of constant history approach, in short, and And now we view the interpretation. According to the constant history approach, an history occurs. The fact that an history-age occurs means that all events which form this history objectively occur. Then, according to this theory, the occurrence of a history is an objective fact, independent of the performance of a measurement which reveals such an occurrence. The basic principle of consistent approach establishes that when a family C satisfies a criterion of consistency, then the set of ordinary histories of C is a simple space of mutually inclusive elemental events one no matter which occurs

7:30 then when the family is consistent we have to think that a single elementary history occurs or other do not occur and the criteria for establishing whether a given family is constant establishes that a family is consistent if it is weakly decoying, which means that this holds. And when this happens, this formula gives the probability of occurrence of every history in the family. What is an elementary history? What is the difference between a history and an elementary history? Yes, yes. I am interested in this point. Elementary history is a sequence in which every projection operator is just one element of the corresponding composition. while history, in the history, every event is the sum or sum of a number, it's a coarser grinding, it's more fine, more elementary. Now, so, if we have a constant family, C, we have a probability on the history, and we can have some empirical data, our information about the fiscal system. By using the logical reasoning, which makes use of these ingredients, we can get a set of conclusions about the fiscal system, which, according to his theory, are valid statements. Now, the problem arises because we can have another family which is also constant, C dash. And we can use the same empirical data with this family and we can get another set of conclusions.

10:00 and this new conclusion may conflict in general with the conclusion corresponding to family sin. To avoid such conflicts, Griffiths give the single-family view. He says that, or valid physical predictions, are those obtained by using only a single constant family at once. The prediction obtained by using different constant families, in general, does not hold together. This is the good of conflicts for avoiding conflicts which may occur using different constant families. A dramatic example of this conflict are just the contrary inferences of Kant. Kant was able to find two different constant families, C1 containing E0, P2 and P1, C2 containing E0, P2 and F1, which can be considered history or events as you want, and E2 that using the rule of constant history approach may prove that in C1 if E0, E2 occur then also E1 must occur. but in C2 can be proved that if E0 it will occur, then F1 occurs, where E1 and F1 are mutually exclusive because they are orthogonal projection operators, mutually orthogonal projection operators. There is this problem which, according to Kant, makes the constitutional approach not a fundamental theory. He rejected this theory or this reason. Reapit replied, let's speak this easily, arguing that the contradiction, the conceptual difficulty

12:30 arises because single family rule is violated, because we use together reasoning from C1 and reasoning from C2. And another argument, more recently, is that the conceptual difficulty gets away if one supposes that the two incompatible frameworks are being used to describe distinct fiscal systems. Now, these arguments have not convinced the critics of constant historical approach. And let me attempt to explain the reason of such an argument by using just counter-references of care. Suppose that we have a concrete physical system, just a single physical system, as... We know that E0 and E2 occurs from the physical system, from this particular physical system. These are the empirical data. And a physicist, A, uses family C1 because it is consistent, a single family, and he concludes that E1 must occur because this is a theory. So, another physicist, B, uses family C2 for the same content sample of the physical system, and then he must conclude that F1, of course, if the use of consistent approach or then physics, the real reality of A is different from the reality of B, to the theory the occurrence of a history then of p1 and f1 when established by the theory must be considered as an objective fact then this cannot be dependent from from the physicist's choice

15:00 Then, this is the reason for which the critics of the consistent approach does not accept the reference in a synthetic way. Now I present my approach to the problem, which starts, as we said, from the genuine concept of constant history approach. The first concept I use is that of occurrence of a history. Accordingly to consistent approach, the occurrence of a history is an objective fact when established by the theory. Then, this means that if we have a family sea of history, we can introduce a set BC of physical system, where a physical system belongs to BC, if and only if one and only one elementary history in C occurs for this particular S, then the introduction of this set is a direct consequence of the notion of occurrence of history. In this approach, then, it is very easy to establish when a family is consistent. A family is consistent even if the set BC is different from the empty set, because there are physical systems for which the history of C makes sense. Of course, if we accept this approach, we have to accept, we have also to accept that statement drawn from a family ceiling, all for a given fiscal system

17:30 if this fiscal system belongs to ZBC. Otherwise, if B do not belong to BC, there is no reason S must obey to the rule of C. It is meaningless. Now, another consequence of this approach is that this assumption must be independent of the family, see to which H belongs. This is one of the two assumptions of Bassi and Girardi. then we have in our approach we say two axioms essentially one is this one it's assumption and the other is this axiom C which says that the occurrence of history cannot depend on the basis, depend only on the fiscal system and on the history age and of the times. Now, using these basic ideas, We have proved, we reached the forum, conclusion.

20:00 The conclusion is that if we have as data about the fiscal system, a given subset X of history, in the sense that we know that the history in X occurs for the fiscal system, Then E, X is a set of history, which means now that occurs for the physical system, S. Then we may conclude that the family generated by such subset X is such that all statements drawn from Cx are true statements, are objective physical facts. On the contrary, if C is a refinement of C, Cx, also if it is consistent, the statement down by from C cannot be considered to statement because it may occur that if s belongs to b of cx does not imply in general that s belongs to the B of the refinement C. A refinement is characterized by the fact that it includes C, C, or X. The fact that C is a refinement does not imply that X belongs also to the refinement. then there is a problem difference between the family generated by the data and the

22:30 of this CX. Now, let me go to the theorem of Bassi and Girardi. Thank you. When we take in account the two assumptions, one is C, the axis of Algerian history cannot dependent on the coerent family one is considered. This is also our assumption and it is also an assumption of Griffiths, this is inversely accepted. And we have this other assumption which establishes that any coherent family can be chosen for this kind of system, then we get a contradiction. This is the theorem of Vassin and P. Rav. Now, we have proven that, actually, the second assumption does not hold in our approach, because the only family which we are sure get a valid statement is the family generated by the empirical data. while all other refinement may generally, may, generally, start with such a whole impossible.

25:00 We don't know if they are valid or not. Now, if we consider college inferences on the basis of this approach, we have the following explanation without contradiction and without conceptual distinguities. We have two constant families and the sentences. the pair p0 p2 is a history h0. Then if we take as empirical data the part that this history occurs What we know, what we can conclude, is that all sentences drawn from CH0 are varied sentences. In this situation of positive effects, what happens is the following thing. Can we prove that VH0 and the set of physical system which X0 occurs or does not occur contains both BC1 and BC2. And the contradiction arises only if there is some system in this intersection. Because if there is S in here, which belongs also to B1, in the sense that

27:30 H0 occurs. Since S belongs both to BC1 and BC2, then the sentences of C1 and C2 must be two statements. And the sentences say that B1 occurs because S belongs to C1. And says also that F1 occurs because S belongs to Then we have a contradiction if there is some physical system in the set of theoretical intersection. But this is not necessarily the case of contra-inference. Contra-inference does not imply that this set of theoretical intersection must be different from the empty set. It is perfectly compatible that this set of theoretical intersection is empty, so that no contradiction occurs. In other words, in contrary inferences, what happens is the following. the following. When H0 occurs, we have this possibility. Or E1 occurs, in this case F1 cannot occur because S belongs only to BC1 or S belongs to BC2 and then F1 of course because C2 contains F1 or S does not belong neither to BC1 nor to BC2 and this is This is not a mysterious paradox. The suggestion of Griffiths is right. In the sense that the different family refers to two distinct... Man refers to two distinct peaceful systems.

30:00 What is true for a non-paradox horizon, no conceptual difficulties, but some concept of the deep theory should be the society, not the system. Thank you. I think there's one question. Sorry, just before we have questions. One of the speakers from the opposition has just cancelled. So what I would like to suggest is that after this talk, And then for Michael Sebel to talk, the other upstairs. So, is there a question? Thank you. Thank you. And now I have Dr. Dean Brickle from that call. Okay, well, um, unfortunately, um, I didn't set a picture on Monday morning, and I forgot my original set, so I, um, I tried to rewrite them, but I wouldn't remember most of the equipment, what the quote is, and it turned out that I ended up with about half of the content, so this might actually ugly a bit. Okay, so what I want to do in this talk is, well I click on my typing first, I don't know if you don't know the records.

32:30 A while back, the philosopher, what the blind guard wrote in the paper, told the philosopher looks at string theory, presented at the PSA. And what I want to do in this paper is just reconsider a number of issues that Weingard considered in that paper, in the light of recent developments in string theory, so developments occurring in any theory, but not just in any theory. So Weingard wrote that paper in 1988, which was just a few years after what we had called the first SuperString Revolution. And the theory then was essentially perturbative, and well since then quite a lot has changed in the world of string. And in 1995 we had the second SuperString Revolution, which was sparked by, well, two things really. first Witten's mapping of the various dualities between the five previously known string theories, and also Joseph Placzynski's Debrains, which are a new type of object, an extended object, on which open strings have an end point, supposedly. So Weingart, I think, was essentially concerned with three main issues. So I'd consider a few more along the way, at least for the main ones. Firstly, the problem of the variety of the dimensions that the space-time has to take for consistent quantisation to go through. And then the related problem of the compactification down to four dimensions of these dimensions. And then there's the problem of background dependence, which is the problem that string theory fears to require a fixed metric manifold in order for the quantization to go through. So it's defined with respect to a fixed metric. And also there's a related problem to do with the conservative nature of string theory. And finally, business of unification of string theory and the ontologically-economical way which does it. So, I should just point out, I'm taking a slight liberty with the value to you, because Weingard was more concerned with the kind of unification is more about our synthesis of the tree, and the idea that there's really just one type of particle that can sometimes appear as a boson, sometimes

35:00 as a folio. What I want to consider is, I think, a more obvious way in which string So that's in its reduction of the entire spectrum of elementary particles and forces to the spectrum of string states and the interactions of strings, the splitting and joining of strings. Okay, so that's the kind of unification amongst the particle colonial we consider. So, okay, so I'll go through these three issues in time, but I should first say something about string theory, the kind of picture that Weingart would have been acquainted with, i.e. the kind of string theory you get in the two volumes by Green, Schwartz and Witton in 1988-87. So you're probably not going to use to any of these, but I should just go through it. What time do I finish, by the way? So you've passed the discussion for today. The basic idea, as you all know, is to go from zero-dimensional point particles to one-dimensional curves of strings. So we go from point particles to strings. These strings can be open, in which case they contain endpoints, or closed, in which case they don't contain endpoints. And at the end points are crucial, but D grains. I don't know if I can go into that, or maybe. Okay, then the world lines of the old point particle theory, which would have been parametrized by just a coordinate t, get transformed into world sheets. where we have proper time, but we also have an extra degree of freedom, an extra coordinate called sigma, which came close from zero to pi, and what we consider, the bit that we that points from the world sheet, that's world sheet really, to the full space on the manifold.

37:30 So it's an embedding map and we contact this embedding map and we get out of that a bosonic stream. Mathematically, you can see you've got an action, you get the string equation of motion. This equation of motion basically can be split up into two parts. You've got a holomorphic part and an anticholomorphic part, which are the left and right movers, and these are the oscillations. And then you quantize the coefficients of these left and right movers, and then these give you an infinite teller of modes corresponding to different string states and different states of matter and spin. So we build up the spectrum from the vibrational nodes of the string. If we want fermions in our theory we have to put a spinner at each point on the world sheet, on the string world sheet. This gives us supersymmetric theory and that's where super stream theory comes from. So in the quantization we get various states, but in the massless sector, the low-energy sector, which with the calendar that we will be able to unterm, we get a mass of spin 2 state, and we get vector-closin states. Obviously the mass of spin 2 state corresponds to the graviton, so it looks like a quantum theory of gravity. Another reason why it looks like a good candidate for a quantum theory of gravity is that the interactions in string theory, which are just modelled by splitting and joining of strings, actually reproduce the perturbative expansion of general relativity. OK, so we've got the wiring reduction going on, and we've also got the math that's going to be possible. We don't only use a The theory of gravity is theory of all the forces, because we've also got the next boson as well. The consistent quantisation imposes a constraint on the dimension of space. So in the case of the bosonic string, the dimension might be 26, and in the case of, well, they're further older than the super-string to be a dimension of 10. The old trick is to use the Calusa-Kline compactification. The compact is about D-4 dimensions down to some compact manplug.

40:00 This way of doing things gives us five consistent super stream theories, i.e. it equals ten theories. I think consistent in this case because the bosonic stream actually is inconsistent because it contains an unstable vacuum. you get tachyons and all sorts of other things. I have a lot more on how the Second Revolution altered this picture, but I just couldn't remember. But there are two ways in which it changes that picture. Firstly, there's a new fundamental practical object called the D-brain, the open string end points are frozen, meaning the internal part of the string needs to move but the end points are frozen on some private surface. So it's all to do with boundary conditions, we're going from Newman boundary conditions where the end points are able to go around 3L to reach those boundary conditions. The main thing, and the big thing about n-theory is the duality symmetries. So a duality is basically, well two theories are dual if their physical spectra and their dynamics are equivalent. So they're really just describing the same physics. We have various dualities in string theory, but we even get them at the Paternity level, one called T-duality, which relates a closed string theory on a circle of big radius, compact value on a circle of big radius, to one on a small radius. That's T-duality, target space duality. There's also another duality which connects the weak coupling of one theory to the strong coupling of another theory. Okay, so what this allows us to do, really, is to prove the non-perturbative these string theories using a weak, weaker couple of an alternative theory. That was the big move forward. What is n-theory? Well, n-theory is a theory that has 11 dimensional super gravity, that's its low energy limit, and it also subsumes the five string theories that we have.

42:30 So we can view these five string theories as points in the moduli space of n-theory, where we modded out these various dualities and things. So it's like a reduced phase base. So that's what N theory is, basically. And it's the whole thing. And briefly, it says there are 11 dimensions, and it's a substantive theory of 10 dimensions. So what happens here? I mean, the reason why it contains a 10-dimensional teeth-screen theory is to do with D-brain. So you can wrap a D-brain around the cylinder. So consider that you wrap around that cylinder. Well, what happens when you take the radius down to zero? Well, you recover a string in a dimension one less, so it contains it in that sense. I mean, that's another aspect of it than the dualities, I think. So, come to the first issue then, which is the large number of dimensions. Okay, so we said we had a constraint saying that dimension of space time has to be 26 on 10. And the problem is, well why don't we observe these extra d minus 4 dimensions? Well there are a number of solutions. One and two were available in the old super-stream theory, the first revolution picture. The first solution is to, it's called dimensional reduction. And we just consider all of the physical fields to be independent of all but four of the dimensions. Independent in the sense that they don't propagate in any of those dimensions. So really this, the extra dimensions are just kind of circle structure. And we don't, we needn't attribute any physical reality to these extra dimensions. That's how people viewed the kind of reduction going on pre-super-Graphic Theory. But I think it was Kramer, Scher, and somebody else, who decided to take the extra dimensions as real.

45:00 is completely real. And the idea is to compactify these actual dimensions onto a compact manifold. And then we make a radius of the manifold very small so that we can't preserve them. So the old example is Kalouza-Klein theory. The problem is to unify gravity and electromagnetism, and Kaluza suggested that we should consider five-dimensional general relativity. So consider GI five dimensions. So we write five-dimensional Einstein-Hilbert action. That's the determinant of the bi-inectric. We make M, we do a split We want to consider the massless sector of this theory, to do that we have to consider only those bits of the metric that are dependent on the coordinates on this on this Bormerfeld. In the metric tensor that we get out, that we get out of it, is written by what? What matrix? So that's a 4 by 4 matrix, and that's just the standard metric sensor. From that we get a spin 2 field, which figures gravity. So we've got general relativity in that, we've got gravity in that. This is a 2 by 2 matrix, these are 2 by 2 matrices.

47:30 Sorry? Zero components. A0 alpha and A. You have mismatched on the left hand side and right hand side. Sorry, never mind. I think it's not important. Go ahead. Okay. Okay, so yeah, these are 2x2 matrices and these give us our vector find. So we've got, they give us a U1 gauge group. So we've got electromagnetism in there as well. So we've general relativity and electromagnetism. This thing, now that's just a scalar, a grand dig scale. You generally don't give that any reputation. It pops up again in string theory, incidentally, that scalar. So that's a kind of thing. So what we did, we compactified a fifth dimension down into a circle, consider the metric tensor only on some of those coordinates. We get out a theory that describes So it's essentially the topological structure of a manifold that we compactify down to, which gives out our U1 gauge of circles. So that's the idea of compactification. And that's the kind of compactification that Weingart considers. And, well, the writer points out that it's not an ad hoc thing. It's not an ad hoc way of getting rid of these electrification engines. Because what it's actually giving us the kind of compactification we choose determines the kind of physical theory we get out. Okay, so the idea is, maybe if we compactify on slightly, my hopes were slightly different in compact topologies, maybe we can get our extra gauge groups and get our stronger re-forces as well. Which you can do by factifying all these Calumni-Down-Banicals which are probably ahead of them. And clearly gives us an extra alternative solution. Which is that the observable part of the universe is a hyperservice embedded in a high dimensional space type. So I suppose the idea is, just like you get the now by slicing through space-time, in that sense, what you get by slicing through the high-dimensional space-time, called the

50:00 bulk, is the whole four-dimensional universe. So it's like a hyperservice. It's a hyperservice living in this big high-dimensional space-time. So basically we live on a free brain is the idea. A free brain is, you can geometrically visualise, well I can't visualise it but it's given by free form. And then we can find the strong, weak, and electro-magnetic forces to the brain, and their reality can propagate in the book. Now this is a proposal by Vandell and Sondrum, it's called The Alternative to Compactification, and I can't remember where it's going. Actually I'm just thinking about that. There's a number of interesting questions you can If that is what's going on, there's all sorts of questions you can ask about time. So it looks like here you have got a case where the universe is, you've given you four areas of a block sitting in this big-time elemental space. So two problems that we might consider. First, there's a long distribution between the three solutions. Well, it kind of looks like there is, so there's a problem for the realist there. Another problem, one for Oliver. Can we tell a relational story about impactification? I'm not quite sure how we get the properties. Required for that, out of a relational story. So the second issue, factor dependence. There's a lot more on this. So the problem is that in string theory we have to assume a fixed metric for our action. The related problem is that we only have a first quantized theory. We have a theory of a single perturbatively quantized free relativistic string, quantized in respect to a fixed metric. Weingard considers the suggestion of Witten, string field theory, whereby he thinks by solving two, he can solve one. Witten's idea is that what you need is a two-dimensional field theory, a field theory on a world sheet, on a world surface.

52:30 And you don't need an ambient space pattern. The reason for this is that all you need to do streamy diagrams is this two-dimensional field theory. So Witten's idea. I mean, there's a problem with this. That's absolutely incredibly short-sighted. I mean, to set up a problem we still need, we still put a metric in the action, okay, to actually get our two-dimensional field theory. So there's the action. Now this is just, that's the cutman constant, the string cutman constant. The metric's put in my hand, but you don't have to do this in the action. And then for each different metric we're putting, we get a different two-dimensional field theory. about the metric beforehand, in order to know what kind of field theory to get out as well, I think we've only made this point against wind. So, I don't know, one way you could get out of it is to say that, well, this kind of action is like a ladder which you get, you use to get to your two-dimensional field theory and then you kick it away and say, well, it's not physically relevant. I don't know about that. An alternative is to use a matrix theory, which is an implementation of M-theory. It seems to be a step away from background dependence. Because what we have is a theory of D-brains, essentially. And the coordinates of the D-brains become non-commutative. So the locations of D-brains are non-commutative. So in a sense, space-time is becoming a bit more, well, fuzzy. So the idea is, I suppose, a quick diagram. Okay, you've got your x-coordinate, you've got your y-coordinate. Say you try and measure precisely x, well, you lose your position on y. Same with y, measuring y precisely position on X, that's going to go all the way around. So you're going to have some kind of cells, solar structure in space. So that looks like a step away from the kind of rigid fixed background in stream theory. I'm not sure, there's still really a metric underlying all that.

55:00 As I said at the beginning, we have a case where a spectrum of particles are forces that produce a spectrum of stream states and interactions. We've still got the dualism because we've still got this fixed metric, and how we to view this metric field, and so on, and that can be reduced to stream states. An extra point which comes from M-theory is that this nice picture gets spoiled anyway by adding new forms of fundamental objects. So we have a super membrane and a super 5-brane, these are new extended objects, and Paul Townsend suggested that there's no reason why it quite generalises the P-branes. So we've almost got a reverse picture of what we have in elementary particle physics where we're getting lower, smaller and smaller, less and less. In this case we're getting going to higher and higher dimensional objects. And so it looks like we're at an even worse position in the screen case without our ontological parser. Three possible solutions. Well, the first one isn't really a solution but it makes the problem slightly faster. And the reason that supersymmetry restricts the possible dimension of supersymmetry for extended objects. So you can only have certain dimensions within a certain space-time dimension. Certain dimensions of objects within a certain space-time dimension. Since supersymmetry puts a cap of 11 dimensions on the space-time, then the number of fundamental objects is going to be capped down to a minimum. They need to 2. But you've still got more than one type. Now the second solution, I need to take solatonic interpretation, allows you to get back to fields. So the idea is, taking grains and extending objects, we've all similar solutions to the classical field equations. So what we've got are lumps of field energy

57:30 I'm not sure what to make of that in this case, I'm not sure what's up to make of the clear language here. Anyway, third possible solution is to construct p grains from p-1 grains. So if you follow So all the way back to its logical conclusion, the nought brain appears to become a fundamental thing. So it looks like we've gone back to point particles, because the nought brain is essentially a zero-dimensional function. So we've gone back down to the idea of points again, but we have really, this time, got one type of thing, just the zero brain, from which we construct all the other high-dimensional frames. The first issue why we are considered remains. We've still got the large dimensions, we've got alternatives for compatification, but there are still problems on the next one. The background dependence issue seems to be slightly ameliorated by the matrix theory, but actually I should have said, with the matrix theory example, it still requires a preferred frame of reference in a sort of momentum frame so you need to actually break covariance to set the theory up, that's actually a more serious issue, I think. Then finally M-theory appears to spoil the ontologically pathogenonious communication stream theory by autoimmunities, extra high-dimensional objects. You said that this whole program has to do with brand unification of all of it, or not. Where is gravity in this? I see more gravity anyway. Consume theory? Yeah, the whole program. Well, what do you mean by gravity? Do you mean a spin to real volume? I don't know what that is. All I know is the best theory of gravity we have is Einstein theory. Where is it there? We can get the same results as Einstein's in general relativity, but we just don't have the same picture. What kind of same results? Can it explain the pulsar decay rate?

1:00:00 I mean, you can get out Einstein equations from student theorem. It reduces the general relativity. I can't understand that. You already discussed the background problems. background mechanics. How can you get an Einstein equation out when you already have a metric fixed there? Well, you get a series and you can get a solution. But each one will have become a fixed metric. Which I know you really hate. Ok, I'll stop that. No, I mean, I've got no grief to defend string theory, can't get me wrong, I mean, I think the background independent problem is a really, really big problem, and I mean, it's what all the little quantum gravity people are really excited about, because it's the only thing they've got, they've got on string theory. I guess I'm going to come at you from the other direction, regarding background independence, you've asserted that it's a problem, but you have not shown why, unless we know in advance that backgrounds are bad, and I guess I don't know that. Why would we expect that? In the traditional particle physics approach to general relativity, you have two metrics. You've got the flat background, you've got the effective curve one, and the ladder gets planetized where the former doesn't, but that doesn't seem problematic in any obvious way I don't know why I would expect that. There are two. There are two metrics. well you don't have to make the perturbation expansion necessarily well that's exactly what you're doing when you're getting these electric I'm not sure there's a quire well in that case you don't get too much Just a short comment to some of this, I mean, one of the things that I would think would

1:02:30 be interesting to look at is how good strength theory actually is on this score about deriving lower energy physics. I mean, one of the big things is to try to see if you can get all this unification, but the truth is that there are very little results about getting general relativity out, getting anything. It's worth mentioning, but keep on doing it because it's good to get it done. Yes, from there? Yes, from there. I'm going to change the problem a little bit. We call it multi-partheid design, which is the first multi-partheid model. And you should note that parts were done together with those I think in Georgia. Those are things also in Utah, Georgia, and Brazil. So, this is my multi-partheid model. First of all, I'd like to clarify the relationship between lovely environment theories, infrastructure, and the most important part I'd say. And I can quote by Werner, who did some pioneering work, whereas in the late 80s there was hardly indifference between independent states and states violating about inequality, with a much more subtle discrimination nowadays. So further, from the fundamental point of view, most people working in a particular form of representation, multiplied by the fundamental states, the question is do we have two n-part sides, or n-part, or quantum entanglement, and not just fast combinations of quantum entanglement of a smaller number of parts? How can you distinguish between the both cases? and to say for a more fundamental point of view perhaps nature somehow limits the number of particles that can be fully entangled and to say whether or not some form of partial separability this partial separability will be more and more defined throughout the talk could you use the pointer? great let's start the plan

1:05:00 let's start the plan, great okay so we've had the motivation now as the top of these four first consider entanglement and no locality in my product systems this could also be a warm-up for upcoming talks on relevant qualities especially on correct Thursday morning so we live at quantum mechanics the future entanglement is and locally in the variable theories, we have factorizability and the related bell inequalities. Next, we take from the extension a multipartite setting and again a quantum mechanics. If you look at full and or partial entanglement, this is being what we define. And locally in the variable theories, we have full factorizability and partial factorizability. and just as in the bi-particle case we have some related inequalities called the solution inequalities for clarity only the two-particle cases we treated multi-particle extensions have been performed but they just increase notation and no new consequences and I'll end with a conclusion at the moment Before presenting a lot of formulas, which I will not by the way, but I'll still say a little bit about the formalism, ideas, and limitations. So in quantum mechanics, I will consider spin-half particles on these Hilbert spaces, and I'll consider mixed or pure quantum mechanics space represented by density operators. And I will only consider orthodox measurements, that is, observables represented by self-adjoint operators. in the hidden variable theories part of the talk I will consider hidden variable space pure states which are points on that space mixed states the distributions on the hidden variable space and observables are then real-building functions or disfunctions from the space a set of real numbers in the back part of that case Well, we're all familiar with bi-partite intelligence. They're equivalent to the so-called non-separable states. And these are states which can be composed into a convex sum of product states.

1:07:30 And a well-known example, of course, is the Siemens state. And also in the bi-partite case, in a variable setting, the concept of factorizability and locality are relevant. So in the hidden variable model, you'd like to predict certain correlations or probabilities for, say, measuring observable A and B. You'd like to predict the probabilities to get out from small a and small b. And the hidden variable model assumes there exist probability functions dependent on the hidden variable lambda such that these correlations are found in this way. It's an integral of the hidden variable states. And I guess we're all familiar with the localistic conditions, both for outcomes and parameters, so-called outcome independence and parameter independence. These probability functions are factorized. So they're not local settings and the outcomes of the other side. And this is called factorizability, they're completely statistically independent, items A and B. And you can say that locality implies factorizability of correlations. Okay, so we have the concepts of entanglement and factorizability. How do they relate to each other in the bipartite case? Well, the Senate for S in 1996 proved that all pure entangle states can be made to violate the Bell method, such as the CXSH. And we all know that from the work of Bell in 1964 that there's no load within the variable model for quantum mechanics. Because he considered, well, the well-known, single state, and it showed that it did not violate the Bell McCauley. It did violate the Bell McCauley, sorry. However, Werner, in Acts 89, in a pioneering article, he considered mixed states that are entangled, but nevertheless allow for a local linear variables model for orthodox measurements, he explicitly constructed one. And for Pesco in 1994, he looked at this It's a subtle hidden monocality. Hidden because it's entangled. It's non-local because you feel it's entangled. It's nevertheless hidden. You cannot reveal it using better than the quality.

1:10:00 You nevertheless revealed it when you looked at more generalised measures, notably the POPM measures. further on, a little more. Recently, Zukovsky will give a talk on this same topic, and independently Myrna and Bool, take a day apart, gave necessary and also sufficient sets of element qualities for measurements of two dichotone observables. It was not only done for the back part of the And perhaps to summarize the results in the bipartite case, I'm going to make a little diagram. On the right, we have the intervariable concepts, on the left, the quantum-mechanical relevant concepts. So the Bell inequality, if you have a local intervariable theory, it implies that Bell inequality will be satisfied. The opposite direction only holds, that is, the bound points are also sufficient for the case that you have two particles, two observables, and two outcomes. So when you look at the dichotomic observables, two different dichotomic observables. And as I said, the first two could be any number, so it holds for any number of particles. Brother, if the state is separable, if your quantum state has a separate description of it, it implies that you can have a local hidden variable theory for that fact. But the opposite does not hold. That was the example discussed by Werner. So you have an entangled state, but nevertheless have a local hidden variable model. And because of all the implications, we find these as well. So if it's made a separate mold, it will not violate a bell in full, but the opposite does not mold. Again, the example I learned. This was in the barquartite case. The extension to the multipartite case gives us a more self-distinction, as Werner already said in a quote. First of all, the relative concepts in the multipartite case, as said, for simplicity and clarity, are considered the three-particle case.

1:12:30 I've made a few little diagrams of certain physical systems, where in this case, the The taglines represent un-entangled correlations, so this system is given by un-entangled, its states are un-entangled states, so it's a sum of product states of the three independent the particles. But in the second case, one of the, well you can have a subsystem of two particles of which each of them could be entangled, whereas the third particle could be disentangled from the other two. So you could have a partially entangled state. That is, for example, a convex sum of the first state entangled and the other state disentangled. These are all the possible permutations. In the third case, you can have a fully entangled state. That is a state which cannot be decomposed with a complex sum of density operators, where all these density operators rely, are separable into product states of less than three parties. And the is if you would like to exclude the fully separable states or biseparable states and other such permutations. And a well-known example is the GZ state. A simple position of the three-particle spin state of all three spins up plus all three spins down. So, what are the relevant concepts in the local intervariable case? Again, I consider the same physical systems. In the first case, the first system, the straight lines represent factorizable correlations. That is, this system can be described by a fully factorizable intervariables model where all the probabilities to obtain happen to ABC by measuring large A, large B, large C are given by a factorisal expression of the marginals. In this case

1:15:00 you have a subsystem consisting of two particles of which its internal states could be anything. There's no assumption on its correlations. So, for example, these probabilities would be obtained by results on the third particle times the joint probability on the two other particles. And this is just one of the examples of that partial factorizability. But the definition is that partial factorizability refers to a state of a composite system, These subsystems can be partitioned into groups of which the internal states can be correlated in any way, for example entangled. But the groups themselves behave factorizable or independent with respect to each other. And, of course, the third option, you'd have no factorizability at all. The sun is shining a little. Let me stand in between the side. So, in that case 2, you know the context of different factorizations, say A, B against C, D, C against A. Yes, this is one of, for example, this is one of them. But you could also add a, that, that would be that one. Yeah. So it could be A, B, F subsystem, or A, C, O, and C. And in the third case, there's no factorizable in the whole. In the hybrid case, there's only options in the one and three. They're either factorizable or non-factorizable. And now this second option, a new option, allows for a new type of invariant theory, which I call a partial-loaded invariant theory. It's a hybrid model with a partial factorizable to requirements, such as this one.

1:17:30 Okay, in the bipartite phase, the factorizability requirement led to Bell inequalities which were experimentally testable and testable. Here again we see that partial factorizability will lead to so-called tradition inequalities. Do you find conditions that are sufficient to determine whether or not you'd have a partially factorizable model for the outcome to be considered. And again, in the tri-file case, by the way, this was done already in 1989, only recently been picked up. So you have the assumption of partial factorizability. Again, only one of the possibilities where you consider AB as a subsystem, but you allow for all permutations of a, b, and c. And you look at expectation values in the usual way. And you want inequalities, linear sum, linear inequalities actually, linear sums of expectation values, and you want them to be non-trivial bound. We're looking only at two different possible measurements on each particle. So the trivial bound would be 8, because you have 8 different possibilities of these observables. The digital proved that a certain sum of these 8 expectation values is bounded by 4. We actually found 2, and this is one of them, and because of the way it derives that these inequalities are for all three particle partial-level invariable theories. So if you violate one of these, you know that it will not be such a model possible. The data you're considering only allows a fully non-factorizable model. Just as in a bar-partic case, there are quantum violations. can be shown that these sufficient inequalities are for all such states and for convex sums of these bioseptical states that's for all states which are not three-partible examples and you can conclude through that

1:20:00 that the violation of these inequalities is a sufficient condition for three-partible examples and these can be or these can be experimentally tested or can be applied to experiment. And the maximum violation is by the G-Z state, and equal to 4 square root of 2, a value of square root 2 higher than the table by the local invariables model. So it's quite analogously analogous to the bipartite. Here's a short little note on On multipartite generalizations, me and another group by phone looked at multipartite partial factorization and showed that they also implied prediction inequalities. And again, these inequalities are a necessary condition for any partial local hidden variables model to hold, and the violation will exclude all multipartite partial factorization. To divide with those inequalities, a fully non-factorized model must apply. But, as in the black pilot case, there were sufficient and necessary set inequalities, at least for dichotomic observables, two dichotomic observables per party. In this case, it's not clear whether or not there is a sufficient set as well. These are necessary, but not sufficient for a particular theory to hold. and well I can check that those exist and we'll try and see if I can find them this is in the back part that case where we're in a found state so-called hidden non-local states we have another state in this case so we have a family of three particle states, the fraction of the fully entangled GHS state, and the fraction of the identity. And these are the coefficients for the fractions. The Kim Shun at least has a fully entangled, so they cannot be decomposed in a convex zone, bi-separable, or two-partum entangled states. But furthermore, the Kim Shun, they cannot violate or any possible set of local orthodox measurements I should say. So we see that partial factorizability, the partial separability, so this is in the invariable context, in the quantum mechanical context, they're distinct notions and this will lead to the conclusion of my talk.

1:22:30 The requirement of having a partially separable quantum description of three or more subsystems is at least as stringent a condition as a requirement of admitting any possible partial local in a variable model. And further, it's conjectured to be a strictly more stringent condition. I'd like to illustrate these results and also the conclusion and the conjecture part by a little diagram, or two actually. The second figure is the negation of the first figure. So if there is a partial level in the variable field, so these inequalities are not violated. The opposite does not hold, that's the conjecture part, but as you have not a sufficient set, only necessary set if the state is partial or fully separable, that is not fully entangled it's not fully entangled, you see that a partial other than variable theory can be constructed for that for those states however the opposite is conjecture not to hold where the question mark here comes from This question mark is reflected here. The conjecture part makes this a conjecture as well. And if we have a partially or fully separable quantum state, we know that it will not violate these switching inequalities. But the example I just showed shows that there are fully entangled states which nevertheless do not violate these switching inequalities. not to be true. And we go to the negation of this figure. We see that if you violate all solution inequalities, you cannot construct a partial local invariance theory. But if there's no such theory, you can, because of the lack of sufficient sets, you cannot imply the opposite. And if you violate inequalities for a certain problem state, you know that

1:25:00 you'll have a fully entangled state because the violation of the inequalities is a sufficient condition for full entanglement full multi-partite entanglement but the opposite does not have to evolve certain fully entangled states an example I gave do not violate the Swedish inequalities and again this conjecture gets rise to this conjecture that if you have full entangled states it could be, or I conjectured it not to be Partially locally in Barrel's theory exists. I'll end with an outlook. We saw the hidden entanglement. And we know from the bi-partite case, through the work of Professor, that when you consider more general measurements, example we presented by POVMs you can reveal the intelligence you can nevertheless you can violate an inequality although it will not violate using only the POVM measures the orthodox mission so in this case considering more general measures my example would not go through and further in order to close the conjecture what is a necessary and sufficient set to build up The character is all possible partial factorisability. So anybody has any ideas? I'd like to hear. Thanks. Are there any questions? This is just a very little question. When you're talking about Pesca and hidden on locality, He said he was using POVMs but I thought what was crucial in his argument was to use two successive projective measurements and I thought the two weren't equivalent because it was crucial that you had one then the other and that's not equivalent to a single POVM, or am I wrong? I think the POVM structure is rich enough to model those. But indeed, yes, he considers sequential measurement and the first acted as sort of a pre-selection for the second. I thought Jonathan Ballett had actually extended from projected measurements to POVM measurements,

1:27:30 which does not include the pestilence series of measurements. So I believe the same is true for the M measurements, but for a series of measurements, it's a different situation at times. The point being that you're allowed to change the second measurements, depending on what happens. Yes, the reselection. can I also make a comment about your pair of diagrams they're not negating to each other they're equivalent to each other one is the one so I will start with a kind of disclaimer because those inequalities that I had luck to discover they are sufficient to create a local realistic model for n-particle correlation function. It does not mean that all the lower correlation functions would be a different model. So let's say it's a challenge for the others. It's not a really close question. and the other thing is let's say I don't know whether it's a question comment or let's say a conjecture that something still can be done in that picture is because something like that as the local result you have this compound system AB and the third systems and you treat this AB system as one system and then you take the correlation functions and as the local result you take A times B in the coloration function and the results are A times in the expectation value so if A and B are plus minus 1 in that way you do not resolve all the possibilities because the result minus 1 times minus 1 is the same result as 1 and 1 So I think there is something more that you can most probably derive still stronger in equalities, which would resolve those cases.

1:30:00 Just to remind you of the generality, so to say. Thank you. There's a question. I was wondering if the consideration in this multi-partite context says anything, to your knowledge, about the detector efficiency loophole? I don't know if you're familiar with it. does it affect the minimum efficiency to rule that out for example I don't know anything about efficiency but I do know that in general when you increase the number of particles and look at bell type inequalities for them the ratio violation exponentially increases so you take the so called microscopic limits instead of decreasing I think the local realistic inter-variable prediction mechanical prediction converge, they diverge. But that's not about efficiency. The duration violation increases. I'd like to add a comment to that. I think that the experimenters would say that the difficulty of doing those experiments increases as an exponential of an exponential. As a rider to that, I'd like to ask you a more general question which really follows the previous question. Are you in a position to suggest an experimental programme theories to rival the experimental program which tests inequalities of the bell type. I would consider this a bell type. No, I mean suggest experiments. Clauser-Horn, Shimoni, and suggested an experimental program, and then the experiment started. And I want to know if you're in a position to do something which was equivalent in this context of that paper. I haven't thought of it yet. You can test this. There are literally inequalities. These inequalities can be readily attested. Because they're very similar to Bell type inequalities. The GHZ is theoretically very much better than the Bell inequalities. But the experimenters continue to work on Bell type of actually realizing a GHZ-type to test local hidden variables. I think, Mark, can I answer it? I'm surprised, because there were

1:32:30 GDC experiments, okay, and so perhaps on Thursday one of my, of the things that I will show you will be a four-particle experiments. We have a violation of the generalized that inequality. Okay, sorry, sorry. There are experiments. It seems to me that Popescu's results really showed that the Berner states aren't consistent with the local invariable theory. Because if you tell me, oh, you know, you can do a measurement, and then after you've gotten a certain outcome, go and do these measurements, and you'll find a no local invariable theory explains it, then I would say, well, that initial state you had can't be described by local invariable theory. And another way of saying that, I think the modern way of saying it, is that we can do a protocol where we distill some pure, maximally entangled state, if we can distill some maximally entangled state, then we say that that violates the local invariable theory, because we know that the maximally entangled state violates the local invariable theory. So, what I was wondering is whether it might be the case that you could say, look, the situations when there's no partial local invariable theory are the situations where I can't distill out any full tripartite Maxwell intended space and that seems like a logical argument. You know you can't distill, if there's no PLHV then you can't distill. Right. I agree, because the best scheme could be considered as the first distillation protocol. Okay, well thank you very much. Thank you very much. Excuse me, just one word please. I just want to remind you that the dream speaks are at 9 o'clock in the old dining room, which is just on the left when you come into the quiet, 9 o'clock. And secondly, could I just ask anyone staying on to do further, to do using the internet or the computers here? It would be better if you could leave well before 7, if possible, because there's a chance that the Harry Potter game will turn off.