12th UK Foundations of Physics — Fuzzy Probability & Unsharp Quantum Reality

0:00 I'm afraid we've got time for one quick question, and an even quicker answer. There's three hands up. Do you want to? You're happy. Okay. Well, I'm sorry, last question, you talked beautifully. The problem in terms of the homochrome complexity is, as you said, defined in terms of the sequence. So you give me a sequence, I can determine its homochrome complexity. On the other hand, the homochrome complexity is determined in terms of a probability measure. So, in that case, it looks like you give me a probability measure, but a particular sequence is of no use to me. Because it's not functional on sequences, it's a probability measure. The point is exactly, they give you the probability for an inevitable event, for having symbols such and such coming out at times such and such. Yeah, but that doesn't tell me enough, right? Because in order to calculate that, that could be an average over technical histories and so on. So I really wonder how, you mentioned a theorem that says that the two are a problem. But I don't understand that. If one is defined on sequences, the other, on individual sequences, and the other is defined on probability measures. The Brunan theorem is not... It's considered... So, what's the emphasis of your question? It's the individual sequence. One is defined on a particular sequence. The product of a certain source. The other is defined on probability methods. These are distinct, conceptual... I was covering a long story short, what Brutner theorem really says you pick any sequence out of the space of all possible sequences

2:30 and it will almost... the probability that this has non-zero entropy is one. This means almost all possible sequences in a given dynamical system, they will have positive elements. That's what the theorem says. So it's going to be a source-like characterization of how products type like letters are in chapter. Sorry, I misrepresented that, just for the sake of privity. There'll be other opportunities to think over lunch for people to pursue Roman over the sandwiches. Let's thank you again for the next meeting. Our next speaker is a neighbor of ours, Paul Bush, whose title is Fuzzy Procurement Theory and Unsharp Fontaine Reality, a tribute to S. Ugaistki. Ugaistki. Ugaistki. Ugaistki. trying to get us half-ways away from classical mechanics to a bit of quantum, but only half-ways because talking about the ways to phrase, to capture the relationship between the two. Part of that piece of work got inspired by the line of research of the late Swatek Bugajski, whose relevant papers you'll find listed on my web page, that's my global reference. Brugaisky rediscovered or resurrected a way of embedding quantum states into classical states that was discovered in the 1970s by Misra and others, but he, I think, was the

5:00 first to see the virtues of that particular representation in various respects, and I'm going to talk about that a bit. What I'm going to add to it is an argument or an outline of an argument that in a specific way this is perhaps the only good way of embedding and representing quantum mechanics in classical terms. And I have to spell out what I see And to that end, I'm going, I'm afraid I'm going to throw even more mathematics at you. When the previous speaker was I hold it with him that I show the tools. I hold it more or less to the mathematical line of argument down there, but I'll talk about something else anyway. So you don't have to read all this. Yeah, not all of it. Well, the basic structure that I'm concerned with, or that one should be concerned with in such a comparison of classical and quantum mechanics, is the mathematical structure that captures the essence of what physicists are doing, carrying out experiments, preparing objects, preparing systems, or ensembles thereof, and making measurements, taking statistics. And the upshot is that the mathematical model of all that is known as the statistical duality. And that works for all kinds of physical theories, for classical statistical mechanics as well as quantum mechanics. So what we need is a structure of preparations and needed represented by states, measurements represented by observables, and the two are structures are married to predict the statistics of experiments. We take as given that states, preparations, have a convex structure, and the idea of an observable kind in general is to assign frequencies, probabilities to the outcomes of an experiment. And then mathematically captured by a mapping that sends the states, the preparations, to probability

7:30 measures. And that mapping has to be affine in order to reflect, to respect the convex structure. So this map in itself is already setting up a relationship between quantum structures and classical structures, but that's just a way of reflecting how we proceed in physics. So if we had states and observables in that sense, then what comes out of it naturally are effects, namely dissociation of a state with the probabilities particular outcome, here represented as a subset of a sigma algebra over the outcome space. Those mappings are defined functions from the states to the zero one interval, and they are called effects by this tool. In quantum mechanics we know them as the positive operators that appear in POVMs, the positive operator So all this flows naturally and trivially from the idea of the statistical duality which describes quantum experiments. So we can go backwards then. We have to define an observable as an applied map from states of probability measures over the outcome space that gives rise to effects, That gives rise to effect-valued measures, and as a result, we have one-to-one correspondence between those affine maps between state and probability measures, which we refer to as observance, and the effect-valued measures, which people now in quantum mechanics know as positive-operated-valued measures. And that's summarized here in set-theoretic ways. This is the subset, sigma-algebra of outcome subsets of an experiment. There is the POVM that leads to the set of effects over the set of states. The state sends each effect to a probability, associates a probability with the effect, that's the outcome probability for the experiment under consideration. and all that induces then, well, probability may not be true.

10:00 So that's a crash course in QVMs in an abstract manner which works, as I said, for quantum as well as classical mechanics. Let me just focus on two or three bits on this slide. The set of states being a convex set is naturally, well, spans naturally a vector space that can be defined in a minimalist set if you like. The associated set of effects over the set of states, well, is then nothing but all affine functions from the set of states to, well, probabilities, numbers between 0 and 1. Again, there is a lot of structure to the set of effects that comes out of this. In particular, there is an order structure. Effects can be ordered by way of the ordering of their associated probabilities over all states. there is a convex structure on a set of effects, which is quite nice, and rather than the last decade, the structure of an effect algebra has attracted a lot of attention. I'm not going into this here. The essential bit for me is that starting with a set of states as a subset of a vector space, we naturally get a set of effects as a subset of a dual vector space. So there is a duality in the sensor linear algebra, and there is more, there is creeping up on the set of states and the dual norm on the set of effects and all this comes essentially for free or for very mild mathematical assumptions. So I'm now making the transition from classical to quantum to begin with but only by starting with the classical example. So the structure I just showed you works for of classical systems, you can think of probabilities of a phase space of a dynamical system. So that's gamma with the Borel algebra of subsets. The gamma is equipped with a suitable apology to make that possible. And the classical states are then, well, subsets of all probability

12:30 measures, sometimes one takes them all, sometimes one just is interested in probabilities or densities. There is a notion of measurable effects, and they are the outside functions on space defined as integrals over, well, functions on phase space. So, just as quantum mechanics has effects and good effects which are projections, so has classical mechanics all that structure as well. Observables, remember, are these functions A from the set of states affine maps to the probability measures. Now I introduce an outcome space. Let's say we have phase space, position and momentum, and we project position. That would be the position observable in this description, and that corresponds, actually, in a way to the spectrum measure, which we know better in the context of quantum science. here, I look at the ordinary case. Ordinary observables are just functions from phase space to some other value space. I can capture that in terms of the affine map picture by using characteristic functions or delta as it were here. And that idea gives rise to a generalization which introduces, well, kind of POVMs in classical probability theory. And that is the observation that Bugeisti made, that you can do this. He was led to it, coming from the Interfestivit classical and quantum mechanics with that embedding of Miesra. It turns out that in that embedding, you do need those classical, as he calls them, fuzzy random variables. What we do is, in this affine embedding, we replace the CRISP function, the characteristic function, point measure, with a Markov kernel. And then you get a totally analogous formula for your embedding, every state, every probability

15:00 distribution on phase space is sent to a probability distribution on our outcome space, which can be represented then with that mark as an integral of our marked kernel. That replaces the, generalizes the crisp observable, which is characterized by characteristic functions. So that is the basics of the first part of my title, which is on Fuzzy Probability Theory, and that is the first line or outflow of Bugajski's observation of the fruitfulness of this Mishra map, maybe that it gives rise to a new extended classical probability theory that he refers to as Fuzzy Probability Theory. This was then taken up by Gada and others to develop in a fully-fetched mathematical theory, and I'm not going into that line. The second tutorial type of transparency here is the quantum statistical model, or duality, where the set of states is a convex set of density operators on the Hilbert space. the effects are just operators, linear operators bounded between the 0 and 1 operator, and the linkage between states and circles, I should have shown that earlier in the duality, I showed it and said it in words, the duality gives rise then to, well, the view being that A acts as a linear punctual on this random state all the other way round, and with quantum mechanics we know that all vibrate, that that works by way of the trace formula. Here are some observations, namely that, again, the set of quantum states is known, of course, to be convex. It has its set of pure states, which are the one-dimensional projections. The set is not a simplex, that is, any convex decomposition of non-pure states will be non-unique. And on the side of the effects, we know that the convex set and the extreme elements are all projection operators. So these structures are well known and they fit the scheme of things that I'm interested in. So this is the general framework which helps me to say what I mean by a good representation of quantum mechanics in a classical setting and that is one where I can carry over all state or quantum states and all quantum effects and all probabilities

17:30 between them and can give a classical expression for them that is what I mean by a good I think that's on sufficiently innocent to be acceptable as a criterion. Well, here's a little further tutorial. Sheet, how am I time-wise? Oh, you have two things. Okay, you can then briefly go over this. Well, one side further, the classical side. Take an outcome space of four elements, or a classic space space of four elements, four events, and then you can construct a set of convex set of states as a subset of a four-dimensional vector space, four tuples of numbers, p1, p2, p2, p3, p4, all positive adding up to one, the other probabilities for the outcomes, one, two, three, or four. The effects are all affine functionals on those states into the 0-1 interval and if you run through the construction you find they're all also represented as four elemental vectors with a constraint saying that the numbers, the components have to be 3, 0, and 1. So this is a classical simplex of states, the extreme points being the 0-1 probability assignments. And I think what I'm doing here is a reduction of dimensions and I see there's a three dimensional simplex down there with an origin in the vector space, that represents the abetting of this structure into a four- actually, as it were, four-dimensional vector space, where this convex structure arises here, with one dimension suppressed. A set of effects, again, we call it subject to the dimensional restructure, comes out as a hybrid view. On the other hand, the smallest possible quantum system matches this classical example dimensionally, but that's all, and here's our usual Klein or Stoker or Bloch or Stokes or Poincaré representation

20:00 of quantum states in terms of four-dimensional vectors, density operators written up in terms of the power matrices, and the condition on the parameters is that, well, suppress the R-norm, which is known here, 1, and the tree vector R, which has norm less than 1. Effects of two-dimensional separate spaces can be written in a similar way, and there is a constraint on the parameters in terms of the eigenvalues, which have to be between 0 and 1. So if we mimic the graphical representation, the set of states, quantum states we all know is the Poincare sphere, the solid sphere, unit sphere, and if a suppressed one-dimension I can show that this is embedded into a four-dimensional setting with that space, with that set coming in an item plane as indicated here with one-dimension suppressed. The set of effects also has a nice representation. One-dimensional suppressor looks like a diamond structure. The zero effect being here, the one effect being there, the extreme effects being, well, the zero-one, and that circular circumference. So these are the classical and quantum statistical realities in the simplest possible representations or realizations. The next step will be to say a bit about how to relate to statistical realities. And that is by way of embeddings. are given two statistical models, two physical systems represented by those, or two ways of describing one system, who knows? How do we relate them naturally to each other? Well, what we are up to is, let's say that is the given structure and we want to represent it in terms of another structure. So we want to take states and effects and find them over there, represented over there, such that the probabilities between a state and an effect also is given by the natural reality over here as appropriate probability. So that is

22:30 our picture of a good representation, good being a meaning that we want to represent every state, every effect here, in a good way over there. Now you can just count the ways in which to do that. One is to start with what I call an S embedding, the states get embedded one way round, S to S dash, and I want this to be a fine in order to preserve the linear structures, otherwise I couldn't keep up that sort of relation between the probabilities, the probability formulas. This naturally induces a dual map, we are here settings, and therefore I get an induced map from the effects of the prime structure to the effects of the original structure. Well, what I want is that all my original effects are captured in this way, and that requires the original map of states to be injected it, purely mathematically. Another way of going about it is that of an effect embedding. You take the original effects and represent them in the new structure. Again, this gives rise to a boomer which goes down the front. So basically we are saying we can represent one structure in terms of another by going from the first state to the second set of states, or the other way around. That is all I need for the time period. To show you a few examples of such embeddings, the first one isn't one. It's the Wigner map, which sends the quantum states to Wigner functions, which are functions on phase space, which functions even on phase-based, but unfortunately, most of them are not non-negative, they are not probability densities. So, there is a map that I find is injective, which is needed in order to have a comprehensive embedding, but it's not a positive mapping, so it doesn't map states to probabilities, which is kind of bad. Nevertheless, this has established itself as a very useful technical tool.

25:00 Second example is what we started with. Well, here I take quantum mechanics, the quantum space, and map the infinite I have to an outer space, probability measures over an outer space. That is what we defined to be an observable in the general sense of UVM. What's remarkable is that if I require the injectivity, then the effects have to be unsharmed. They have to be, they can't be crisp, they can't be projections. But in that representation, they get represented by classical crisp properties effects. It's rather trivial, but worth to note anyway. So the injectivity corresponds to the POVM being informationally complete, it's good enough to detect, to identify states through the statistics. That's the idea of injectivity of that map of an observable. Do we get the subjectivity of the dual map? Do we get to represent all quantum effects? Well, in terms of the linear maps, yes, but there's a glitch here, as was with the Wigner map. The images under that rule map of the classical effects are really a strict subset of the quantum effects. That means that not all quantum effects can be represented as classical effects. They have to be represented by classical functions which are not even positive all the time. And certainly, I can't capture the projections, the sharp effects, in any way with such an injective or information-decomplete embedding. Bringing together the two examples I showed you beforehand here, you see what's happening actually. You take the quantum states, embed them into that tetrahedron classical state of that four-dimensional outcome space. that turns the sphere into an ellipsoid in general at being a time mapping and there is a dual map that sends your classical effects into the set of quantum effects and there's it's absolutely impossible to construct these maps so that classical would include

27:30 enclose the quantum effects so that's simply impossible That's the generic picture. So, let me turn to the mapping that I'm really interested in. And that's the one very briefly here. The B-Strom map is the one that extends or reduces the, well, as it were, classical states to quantum states in a surjective way. That guarantees that the effects, the quantum effects are injectively mapped to classical effects. And that can be the only, well, I'll show does work, turns out to be the only way of getting a picture of all quantum states as well as all quantum effects, as classical states and classical effects. Don't need that example, Billeroux. Well, I also skipped the proof sketch that really one can do this in a minimalist way and get the classical phase space which is, if you require minimality, you can see and show it's equivalent to the set of extreme points of the quantum states. And that is what the canonical classical extension is about, or the MISRA map. It's based on this probability formula, rho labeled by a classical probability measure, which lives on the pure quantum states times the fact that the trace thereof, that's the probability, that can be written as an integral of a phase-phase function on that phase-phase over a measure. So each measure gives you a density operator, each measure mu on the omega, the quantum state gives you a density operator in the way that you integrate all the pure states with that measure.

30:00 And then you see that each effect gives rise to a quantum effect, gives rise to a classical effect, a function on that face-to-face. So that is all there is. And the important observation of the Bugajski was that this map is indeed good in every respect, and one can show it's the only map that is good in the way described. Alright. Well, there are some interesting observations here that come with it, and I conclude with that essentially. As I said, the map from the probability matrix over that phase space to the quantum space is subjective. It does actually reflect all possible complex decompositions of each quantum state. So there are many mu's giving the same quantum states, and there are as many classical measures as there are the compositions of that quantum state. So that is actually a nice feature here, this many quantum states. All quantum effects are represented by classical effects, that is essential. Well that comes to the price, it almost looks like an invariable model of quantum effects, but it isn't. The fact is that all quantum effects, including the sharp ones, including the projections, are necessarily represented by fuzzy classical effects. So, any quantum observable that I carry over into that classical setting will be a fuzzy random variable. And that shows that in the only available, apparently available, good representation of quantum mechanics in classical terms, All quantum observables are to be seen classically as fuzzy random variables. There is no use classically for no quantum interpretation of the classical CRISPR observables that are also there. And that has been used by Lugaisky and Bertrametti to explore the Einstein-Bodolsky-Rosen and the Bell, the experiment, and the Bell inequalities, which get carried over here into a classical setting. And that is a Bell phenomenon in classical systems, as they call it. Quite a curious observation, and it makes you think about what it means that classical observables in that setting are compatible, are jointly measurable.

32:30 There's a range that is hidden in that, which I just mentioned to get your appetite for that. I guess that is about all I wanted to say. Yeah, I emphasized repeatedly that I think this is actually the one good representation of quantum mechanics, and it's due to misrepresentation. Thanks for your attention. One very quick question, Jeremy. Well, thank you for the talk. Can you remind me, the fuzzy random variable will send a point in classical phase space to a probability distribution over its values rather than to a single point? that's yeah right it's presenting the characteristic function by a smear dog right you can regard it yeah so the question is then is it or what would naturally think of that as a disturbing measurement a noisy measurement if you're thinking classically is it possible to represent this kind of classical canonical extension with its fuzzy random variables in terms of noise from a classical environment in a useful way. So one could have a picture that you only see a quantum world because you're actually, it's the classical canonical extension in operation but there's a large classical universe and you don't know much about that environment and so the smearing is actually environmental noise. I haven't looked into this for a while but I think it's true that what you have here is an analogue or a complete analogue to what we know for quantum theorems there is a Neumark theorem which says that you can, for a given POVM that is not sharp, you can get an extended description in which that turns out to be a projection of a sharp one. What you're missing, there are states which can discover that actually in a bigger world, there are more states which can see the crispness better. And you have a similar, you have similar results in the classical case.

35:00 If you have these stochastic kernels, then you can get some extended information. I think Bugajski has looked into it also and had some examples. So, to some extent, one could maintain this speculation. Yeah. We'll have to stop there and thank Paul again. The next speaker, the final one for this session is Thomas Dirk from Belgium, who will be talking on the Bonne de Broglie Interpretation for Quantum Field Theory, completion of John Bell's program. Yes, actually I shall present the work that was not done by me, but by the physicist Tom Poulard-Nerve, Samuel Poulard, and he obtained interesting results in the implementation of the proposal of John Bell, It was written in 84, so we find it in 2nd, 3rd, and he wanted to develop a Bohmian model for problem theory and for fermions. And I shall present quickly the first program, its motivation, the model itself. Tanasha presents one of the results of his new program. He obtained during his master's in 2000, results for the discrete and continuous was an economic future, and it was more recently a third-field aid program for this free and continuous family. And what is interesting is that if one follows birth prescription and that one considers that family charge is to be able, It depends really to be a privileged observable and that's related to the superception of the charge. And it is also consequences in relation to the general.

37:30 So, I propose this model. always for the same reasons. He didn't like the Copenhagen interpretation. For instance, because the division between a race and a cellar was something vague, according to him, was not quite defined and professional. Often, Bohm's interpretation, the answer to this question was criticized as being non-relativistic, and this is why Bell wanted to apply Bohm's interpretation to quantum field theory for fermions, where you can formulate a relativistic expression for the theory. And we did this also because Bohm's interpretation for bosonic quantum I was interested in these results because, actually, I don't understand very well quantum You have many things that become super, you have super wave function, you have super Hamiltonian. But for me it's super abstract and I really don't understand what it means. And it's good to have a picture and at least a Boolean picture is a picture. So I think it's interesting for everybody to learn how to use it. And the proposal of that was actually very simple, he proposed as a diagram, the location of fermionic charges, the upside is the direct field, and in this paper, it was more simple, it did it with a discrete model of continuity, so it was a lattice approach. And it's not enough to specify this, you must also consider other variables such as spin, something like that, in order to get a behavior that corresponds to a complete absorption.

40:00 And for instance, in this case, there are four other degrees of freedom. If he has n thermions, then in every location we have four extra behaviors that we find. for instance it could be related to the sign of the energy and to the spin and so the theory corresponds to what it calls configuration which is intuitively for each position of the site you define how many particles you have there essentially what is the energy and what is the speed and things like that and then you derive a stochastic process that we have exactly the same results as quantum field theory So, this is the Schollinger equation and then, I won't give the details, we introduced the stochastic process in such a way that if you choose the group distribution of initial time in order to fit this quantum statistics Then for all the times, the two probabilities will coincide. Then there is a rule made by the quantum state which is like a . So here is a stochastic process for all the configurations of the feminine field. What was new is that a theory, the evolution process is stochastic in principle. If you consider Bohm theory for the wave function, it is deterministic. Stochasticity is just present at the initial time, but here from the theory it was necessary to introduce the stochasticity.

42:30 And Rome suspected that in the limit of the continuum, the stochasticity would disappear. And this was not proven, this was derived by Samir Khurana. So what he did first in 2000 is that he applied this stochastic process for bosulic stellar fields that was described by Glenn Gordon and Amgen with an interaction, an active interaction. And if I look at the usual quantization procedure we consider the local number of bosons in the Fourier transform of alpha so to disoperate the wave and you can show very easily that this is where the label all these observables come in with each other So you can use them to define the theory. And after this, it discretized the theory. So it worked on a lattice. And it considered non-volatilistic limits. It was only considered in other places. And this non-volatilistic limit corresponds to the situation no bosom is created, the energy is too low, so the number of bosoms is constructed in the process. In that case, it is possible to associate an effective wave function with the process.

45:00 I don't give the details, I'll just compare it. And Aster is then a wave function, the behavior is deterministic, also for the discrete points. That was the result for the lattice, yet in this limit we have a deterministic regime. And similar results are also obtained for fermions by a separate type of context. And you could also generalize this to one particle. The first treatment was one dimension, but you can have a lattice with two dimensions. A priori it doesn't seem to be deterministic, but if you turn the lattice along the right direction, you can show that deterministic reason. So this is a general result. You can also obtain a string result for two particles. And the reason for this is that in this regime, when you can apply the Zin-Justine procedure, you can derive actually an effective wave function and an effective Hamiltonian which corresponds to a given amount of particles and there is a conservation equation for this equation and this is why you can actually derive directly the Bohm interpretation for wave functions. It's totally equivalent. That's the reason of determining that region. But in other cases in general stochasticity is still there. So that was its first result for bosons, but then more recently we applied also these ideas to fermions and that was the original idea of Bell, and you can find this work in the archives and first what he did was that he considered discrete termionic part of field theory

47:30 and it's not very easy to treat the direct theory on a lattice you must follow lines of discrete termionic part of field theory it's something complicated very technical but he could do it and he also could consider the limit of the continuum and once again the result is the same the stochastic process of John Bell is deterministic so you recover determinism at all times and you have a corresponding current And now, the explanation for this determinism of the process is that in this case it is well known that charge is conserved And this is actually related to invariance and change of phase, something very deep and very general. And this is also true even if you introduce interaction with an external electromagnetic field. So, in virtue of the conservation of the charge that is very general and well-accepted, if you consider the field, you can restrict yourself to sectors of the fog space where the number of permeants is constant, so you will be given the number of charges. and then you can also, following the procedure of Zinjistar, introduce an effective wave function responding to the sector with this amount of charge and because of super selection there is no interaction between all these sectors

50:00 So this is how you obtain the effective expression. And it can be shown that the Bohmian velocity that you derive from this is the same that you will get from the lattice treatment. and this result was more or less accepted it also needed minor changes to the physical data scale and so that's for me the main result is that this confirms the first intuition of John Bell that it's possible to realize problems with thermions and in the continuum that we stick and everything is already fine. Now, there were other approaches, other attempts For instance, there was an attempt where the was not the location of the particles, but rather was the configuration of the field. This was known for the electrical field. The idea was that the little variable was the configuration of the wall field everywhere in space at that time. This is totally different from the particular approach of Jantel. There are many people who have worked that way. So there is a problem with this, and this was shown as an explicit example by Sander-Geran is that it's possible to find that certain initial configurations that will correspond at time zero to a particle will explode in time, will explode. So there are analytical solutions that correspond to this kind of models initially you have located the particle and then it will simply spread with time and that's the problem because when we make experiment usually we see clits in detectors so it's very difficult

52:30 to give an interpretation for this variable you could measure the field in some places this is not really compatible with the fact that we see clits in detectors It's difficult to explain, when the field is like this, why all the energy gets concentrated in one detector. So for me, this kind of approach is not consistent. There were also attempts to realise the first field best problem with other operators. For instance, there was an attempt with psi-liopside instead of psi-diopside. And there were also attempts where instead of considering the charge number, people considered the particle number. That's slightly different, because charge corresponds to electrons of positive and negative energy, but for particles, you must consider electrons and anti-electrons. Actually, it's not the same. I'll give here a sketch. So if you follow the normal quantization procedure, you express the direct field in this way, and then you have operators associated positive energy at 20 minute energy at 20, and then you can fulfill all the problem of them. that if you consider particles instead of charges, so in this other approach, then you must replace, you must permute creation and restriction operators for negative energy electrons. And what is important is that you must define a vacuum that is different. So This is the DRAG-C with all the negative energy levels that are filled. And the vacuum that is associated to the charge interpretation is this one, it differs from the DRAG-C.

55:00 And for this reason the two operators, the two behaviors are not the same, they don't compute, and they need different results. But the advantage of working with the charge is that you have this sequence reduction tool that I already explained and so the charge is strictly on self-quantity and this is why you have data limits Now it's not very convenient technically to work with a vacuum that is not a physical vacuum but nevertheless you can do it, just define state to make the computation, but it's not really important. So technically it's not easy, but you have determined. For the other approach, working with particles, it's convenient to work with the good vacuum, which is the physical vacuum, but you have an avoidable stochasticity and it's not easy also to formulate a super-selection rule in this context. Okay, we're going to do it too many times. I shall talk a little bit more about this. If you assume that the charge is a concert quantity and we know that different sectors of the full space do not interfere, then super selection is a consequence of the Bohmian interpretation. If you make the assumption that the behavior is the localization of the charges, this explains Because when we make a measurement in the Bohm's approach, what we do is that we measure the privilege of some level, in this case it will be the charge.

57:30 And eventually, we have the initial state to make a unitary evolution and then we make this measurement again. And actually, if we do that, what we measure is equivalent to the vehicle, but translated backwards in time, up to the unitary evolution view. So this is the way that you can measure different observables in both interpretation. Now if all the unitary separations, all the physical separations are such that you remain in the sector of the fog space with a given charge number, then the separator will do Therefore, it is impossible to observe interference or superpositions of states that would differ by the charge. Even if they existed, it would be impossible to observe. Because the behavior and the observation of charge forbid this possibility. So super-selection is just a consequence of the choice of the behavior and of well-known properties of the approach theories and so on. So in this case, and that's something new, super-selection is a prediction of the Bohmian approach. It is not something that you must impose beside the theory. And in general, it is criticized by Bohmian theory, it's an ad hoc theory, but in this case, well, this is a little bit theoretical, but it allows you to make a physical prediction of what the superscription could do, to explain the superscription. And if you choose another variable, for instance, the field configuration of the particle number, then you must postulate the selection from the beginning. And here again, there's two possibilities. Either you say that certain superpositions to the composition of states with different charges, that those are forbidden, but then you cannot explain why, so you must postulate it from the beginning. Or you could also say that we, the observers, we are not able to see such things because

1:00:00 of the properties of the measurement devices. So there will be a physical limitation on the apparatus. But this will introduce a distinction between the system and apparatus, and this is precisely what the Bohmian implementation wanted to avoid, so I think that this is not very consistent in the Bohmian approach and so we remain with this question that there is nothing that explains the classification rule from this kind of view. So I think I will finish here. Thank you very much. We have time for questions. One simple question. You mentioned that to reproduce the results of the conventional we have to increase the stochastic element. How do you choose that stochastic element? How is it governed? So this is the Bell model. On your model? Yes, I went quickly on it because it's a little bit technical. Is it something that is outside, is it within quantum field? Yes, it's outside of quantum field. This is the model. So what you do is that from quantum field theory you have the evolution of the state. And then, you make the choice of a behavior, of a privileged observation, and you look at how evolves the probability to see this and that, so I move further to the behavior. You can use this also from the state and from the Schrodinger version. And what Bell did is that he introduced a stochastic process that mimics this distribution from all times. So actually this is done in such a way that this G and M, they are like transitions between different configurations. And they are directly used in such a way that they will give the good probability of transitions.

1:02:30 It's nothing else, you just project Schrodinger equation on the latent states of the behavior and you look at the evolution of the polarities. And there are different, actually there is more than one process that you do that. That is one choice but there are different prescriptions that you can solve. That's one choice, is it kind of, you know, multiple pages and specific choices for a study, is there a difference between situations? Yes, there's one choice. All people usually follow their description, which is on the other page. in orthodox quantum field, we don't need to skew the chance of selection to thin. How does that interact with what you were saying about the various ways in which sometimes it's that odd and sometimes it's... Could you repeat the question? Sure. In an orthodox quantum field with interactions, then I assume selection is a theorem, it's not so much as humans that can be proved. Now, that seems to sit slightly and easily with the discussion where we consider where we can't be proved. I don't consider this problem in general. For me it's multi-theorem. For instance, there could be of surveillance of which eigenstates will be states that are not, that are not pure states of the charge, that are superpositions of states with different charges, in principle. Well, on the assumption that... Why don't they exist? If it's only on the assumption that the observable is built up a label field operators, then you won't be able to build up an observable, which has standards across certain sectors. I mean, you can prove that forming an algebraic probability, but you can make a pretty good informal marking for it. Yes, I don't know what I said about the consistency of making this choice for the development. So for me, a priori, super-selection charge is a mystery.

1:05:00 Super-selection charge is a mystery. Well, let me do that later. I think we're going to have to call it all. Yeah. Thank you. I'd like to beg the indulgence of this afternoon's speakers and propose that instead of trying to rush back here to 1.30, that we kick off at 2.00. Instead, we just shift everything up the time after this half an hour. We'll go from 2.00 to 3.30, break at 3.30, come back. Is that OK with the speakers? because it seems to make sense to me. And then we'll finish at 5.30 instead of 5. Also, secondly, we've got a change to the programme. I should have mentioned this before. Pan, hello, you're here, we're free. Has agreed to commute with Thomas Conrad. So, Pan will kick off the Saturday afternoon at 2. I don't know if that's OK. will break at 334T, again in the foyerie, on 334. So where is Thomas? Sorry? It's a straight permutation. Pan and Thomas, or wherever Pan is, I think it's Thursday at 4.30. Right, and lunch is ready in the foyerie. I'm going to call Leslie Hall and see if I can... Oh, that's extremely fine. Yeah, because in my head, I bet you, yeah, but it's a question of, I have to tell the accommodation to not take the money for your accommodation out, because they'll just do it right away,

1:07:30 and that'll be weeks before the people who visit to give me the money to pay, and then Thank you.