 You can hear me now, no? OK, we can start. So this is the last day of the Winter College. I hope you enjoyed it. I guess it was a rather long time, these two weeks, and you're tired. So these are the two last hours of this college. And I will try to be a little bit entertaining. And I'll try to also to involve you in the discussion, to go slightly towards the end. But I hope also to give you what many people also I think is an important message about understanding quantum mechanics. So the goal of the Winter College, as you've seen in these past two weeks, was to introduce you or to review, for those of you already knew, some basics about quantum optics and quantum information, then slowly drifting towards quantum technologies, in particular, related to optics, quantum communication, and all the possibilities, both research-wise and also technology-wise, with a secure cryptography. But the core was quantum, the core. And then I want to go back, basically, to the foundations, which is my job, and to begin of it and discuss. And then I will ask you a question. So what does it mean that something is quantum? So you give it more or less for granted, probably, by now. But if I ask you, what does it mean that a particle, a microscopic particle, is quantum? What would you answer to such a question? Anyone has the courage to give an answer to that? And speak up. It's a fact, if you mean, from an experiment that it is quantum, but that doesn't answer the question. So that would be the answer to the question. Is a particle quantum or not? I'm talking about microscopic. And then the answer is, yes, it is quantum, because experiments say so. Everybody speak up. Loud, otherwise I cannot hear. Sorry? It is small. But not necessarily small means quantum. If you leave the Newtonian world, according to Newton, size didn't matter so much. Either a small or a big particle would behave classically, according to Newton's laws. So the fact that something is small is not sufficient for it to be quantum. It happens. It is nature experiments, and nature tells us that it is quantum. Okay, we are going a little bit closer. So a particle is quantum because it can be in a coherent state. Do I get it correctly? And then it's good, but then I can shift the question. What does it mean that a quantum particle can be in a coherent state? Because it's protected, not because it doesn't want to, because if you have a coherent state in a strong environment, you lose, it interacts. So in an isolated, so there's something also that we learned that a quantum system is fragile against the environment. So if you want to keep the quantum state, you have to protect it from the rest. Good, but still, this is not the point. What does it mean that a particle is quantum or that a particle can live hopefully for a long time or for a short time, whatever, in a coherent state? That's okay, so we can observe correlation, we can entangle, but about the particle, what does it mean for the particle to have the possibility of reacting that way towards experiments? Okay, everything okay, so what does it mean? So what does it mean that a particle can be prepared in superposition or what does it mean that a particle has a wave aspect? What does it mean? That's another one, so obey the Schrodinger equation. Obey the Schrodinger equation is a more sophisticated way to say that it is quantum. It's the same story. That is true, but I don't like explanations by negation because if I tell you what does it mean that space-time is according to general relativity? You cannot answer to me. It means that it doesn't obey Newtonian physics. It obeys Einstein's physics. So it is true, but this is not the answer. It should be something what does it mean that it obeys all, sorry, louder? Well, not necessarily because you can have quantum systems that have continuous, no? Continuous, so it started like that with the discreteness of something. That is true, but it is not always like that. For example, a free quantum particle can be delocalized, but there is nothing discreet in it. So there are Eisenberg's principle, there is a superposition principle, there is interferences, all these things, but there is nothing discreet in this case. So discreteness was important at the beginning. It was somehow sold as the characteristic feature of quantum mechanics, but nowadays we know that it is an aspect of it, but not the whole aspect of it. So if you wish, and then probably I can come to my next slide, what does it mean that the particle is in a superposition? Which at least for me is another way of asking the question, what is quantum in a way? I mean, going to the essence, of course, quantum is many things, but going down to the essence. Yes, that is correct, but I'm talking about the state of the particle, so. If I tell you, I don't want to, I don't want to necessarily, I'm a theoretical physicist, so I don't have a strong necessity of doing measurements. But I'd like to understand the state of systems, conceptually. So if it is in a superposition, what? So it's a particle here, now in a superposition. Eventually, I will measure to confirm or deny what I'm expecting from it. But what does it mean that the particle is now in a superposition? Or think, if you do an interference experiment, you prepare your measure. What happens to the particle in between? What does it mean that the particle behave? I'm sorry, then we will go on. But just to provoke you, what does it mean that a particle behaves like a wave? What does it mean that a particle plus particle gives no particle? How can a particle, if it is a particle, interact with the structure itself? Is it a wave or a particle? Is it both? How can something be both a particle away? So particle, when it comes, the name, comes from the Greek, I suppose, I'm not sure. It means a part, a part of. So a delocalized particle, for me, means that it is like, I don't know, like a jelly that when the temperature is a little bit high, it melts. That would be the picture that I would have of a delocalized particle. So it's not a particle anymore, at least in the usual Newtonian sense, no problem. But it is a jelly that delocalizes. But then when we make measurements, what happens? So it's localized. So that is, OK, let's stick to this picture. Otherwise, we will not. This discussion went on for 100 years, so we will not certainly give the final answer there. But let's stick for the moment of this picture that the particle is not a rigid object, but it is something that delocalizes, for whatever reasons, according to the Schrodinger equation, it's like diffusion. When you put some ink in water, it diffuses. But then it has a weird property. Well, when you try to look at it, it focuses back in more or less one point. I'm talking as a theoretician, so it's not really always a point. OK, we stick to this picture now, I go on. So everything, all these questions come about from the possibility of creating superpositions, which mathematically is the physical way of saying that the Schrodinger equation, which governs linear processes, is a linear equation. So if two states are a possible solution, any linear combination is a possible solution. Oh, by the way, the answer that we more or less came to was more or less exactly the picture that Schrodinger had in mind when he formulated the Schrodinger equation. He thought that the wave function, or the square of the wave function, if you wish, was giving the mass distribution of del, at the time, the important issue was the hydrogen atom, among other things. The wave function was giving the mass distribution of the electron around the nucleus. So it's not a point that rotates around the nucleus, but it's a mass that surrounds, a mass density that surrounds the nucleus in different configurations. And the different configurations are stable when you have the orbitals, or not stable when it's a superposition of orbitals and then it would collapse to one of them possibly to the ground state. OK, so superposition, and you can have many superpositions. You can have typically now, especially if you are exposed to quantum information theory, typically you think of spin, spin up and spin down, because it makes presentation simpler. But from the foundational point of view, what is stronger, in a sense, is the superposition in space, here and there. So formally, it's more or roughly the same thing. But then conceptually, or at least psychologically, it makes a difference, so something that can be in two locations at the same time. And I mean, why is it conceptually different? Because this goes really to our ability of understanding things in nature. The Greeks philosophers themselves were basically, when they were giving, were trying to understand what is the essence of a particle, they came to the conclusion that the essence of a particle is to be somewhere. The real essence, the particle is somewhere. So what is the difference? If you take two identical particles, forget the identity of quantum mechanics. Let's go back to the Greek time. What is the difference between two otherwise identical particles? Is there a difference? The answer was yes, because one is here and the other one is there. They cannot be on top of each other, never. Very close, if you wish, but not on top of each other in space. So the essence, the real essence of a particle is to be somewhere. And then now, somewhere, not somewhere, somewhere in a point, in a real point of space. And then quantum mechanics is challenging this view that a particle can be, has to be always somewhere and not nowhere. So it goes back to ancient philosophy and it's something that is rooted in our understanding. So you don't know, especially if you are children, you're not anxious so much. But if you are parents, you want to know where your children are and you expect that they are somewhere, not that they are nowhere. So it is really something that is in our understanding of nature. So the point is, so what does it mean? That's the question, that's the provisional answer. So what does it mean that a particle can be in a superposition? And so if I have to find a core answer that you gave me, it means that it can be delocalized. It can be here and there in some form at the same time. Now, there was a discussion. This was the core of the discussion when quantum mechanics popped up. So probably you know that in a long debate, since the very beginning, between Einstein and Bohr and the Copenhagen School and the people about this discussion, what it means that a particle is in a superposition, what it means that it is delocalized, what it means that it can be here and there. Can a particle be really here and there at the same time? That is something that went on. And so we start with a bit of history and we come back to, and I will end with something that you know very well, which are very inequalities and which is the foundation basically entanglement, the foundation of modern quantum technologies. Well, the story starts far apart. So I will not tell you so much about the modern quantum technologies because you've heard already. I will tell you about this story that goes there. So there were two positions at that time about understanding the meaning of a quantum superposition, which I summarize, even if the story is much more complicated than this, as the position of Einstein versus the position of Bohr. Just to keep it simple. So Einstein didn't believe. So Einstein was, in a sense, from the philosophical point of view, was a very traditional physicist. So he was a revolutionary as a physicist. So he invented, or gave a completely new, different flavor to special relativity. If you wish, somehow he invented, he contributed to inventing quantum mechanics with a photon. He certainly invented general relativity. He gave a huge push forward to Brian motion into the microscopic descriptions of statistical physics. So he was a revolutionary. But philosophy, philosophy, philosophy why? He was a very down to earth, so to say, and simple-minded person in a positive way, I mean, not a silly person. Simple-minded in a good way. So for him, a particle, if it is a particle, it is a particle. And it is always somewhere. So it cannot be in a superposition because it is a particle, by definition of particle. If a little ball, a point, a square, whatever it is, something localized, it is always localized in some point of space. So for him, being in a superposition was only, it was something, so to say, I'm a bit brutal in this discussion. Philosophers would be much more careful, but it's not a point here. It would be like a subjective knowledge of the state of the system to some extent. The particle is somewhere. I just do not know where it is. More or less, not completely, and we will see how it goes, more or less like in towing costing, when you toss a coin, you do not know whether it will end up heads or tails. You only can give an expectation, the probability, about the outcome. It is not that the coin is in a superposition of being head or tail, classically speaking. It is always head or tails. We just do not know. And then we have to see if we want to do something, or we keep completely silent, or if you want to say something, we can give the probability that after at the end of the tossing, the coin will end up head or tails. So we knew somehow that the thing is not that simple, but that's somehow the intuition that he had about the state of the system. Yes? That is true, intrinsic. Yes, that is. So I just tell you something that I didn't want you to write. And I tell you something that is actually the outcome of a long process of understanding that you cannot fully take this subjective point of view. You cannot make the claim that the wave function, in a specific sense, there is always freedom of understanding, of making models, but in some specific sense, you cannot claim that the wave function is fully subjective, is fully in our mind as our ignorance about the state of the system. It is completely unrelated to the physics of the system, in a sense. So the probability in coin tossing, the probability somehow is not in the dynamics of the system. It is, in a sense. But it's not that it governs the evolution of the system. If we are Newtonian, the evolution of a coin is given by Newton's laws. It's not given by probabilistic laws. In quantum mechanics, this analogy doesn't hold completely. The wave function says something about the state of the system. But the issue of Einstein is, does it say only something and not everything? Or does it say everything about the state of the system? So the incompleteness position of Einstein was not that the wave function is completely unrelated to the state of the system. But it is an incomplete description about the state of the system. So the typical, classical example would be temperature in a gas. The temperature in a gas is not something subjective that it is in my mind. Because if the gas is hot and you put your finger, you burn. That's something really, something that happens really. It's not that happens in our minds. So temperature is not a subjective description of the state of the system. It's something that describes the state of the system. But it is not the full description of the state of the system according to classical mechanics. Because the full description of the state of the system is the position and velocities of all the particles of the gas at some time. So that would be, so I don't know so much about what Einstein really had in mind. But if you want to make a classical analogy or what I think Einstein had in mind is that it was considering the wave function somehow like temperature in a gas. Something that describes something that happens really. But it doesn't capture everything we can know about the state of the system, in particular the position of the state of the system. The other position is in studies completeness, is the opposite one, is the position of Bohr, more or less in the Copenhagen school. One again, I want just to stress that everyone in the Copenhagen school had a slightly different view about quantum mechanics. It was not fully homogeneous. But you cannot spend days here in discussing all the scientists that contributed to this. So we just make this rough claim. It is completeness. The wave function contains everything there is to know about the state of the system, nothing else. Forget about even in principle to hope that there is something more that we can know about the state of the system. So the answer to the question, what does it mean that the particle is in superposition? The answer to the question would be, the question is meaningless. It doesn't, you cannot ask what it means. It means it's in superposition. If you make an experiment, you will see something. It is a mathematical tool to describe outcome of measurement. So in that sense, it has a meaning, of course. But without a measurement observation, it doesn't make sense to ask the question, what it means that it is in superposition. So if we were, if we made a classical analogy, it would as if we had thermodynamics, classical thermodynamics. But Newton was not born. So we didn't have classical mechanics. And then people were claiming that thermodynamics is the complete description of a state of a gas. There is nothing else that we can know because there's nothing else. In a strict sense, a gas is a fluid. It is. And the fluid has some properties. And the properties are pressure, temperature, and whatever you want. Then Einstein would have hoped for Newton to come and give the microscopic description of a gas out of which thermodynamics quantities emerge as suitable average description. OK. So Einstein, so it was a stubborn man. Einstein, he didn't give up very easily. And so he devised the following thing. So take a box. Take a particle in a box. Take. And we have the wave function. So the particle will kick the round scatter. The wave function will diffuse. So more or less, it will become homogeneous inside the box. So this is the pink color somehow captures the idea of a uniform wave function inside the box. Remember now the answer that you gave me at the beginning that we could think of a delocalized particle like a jelly that diffuses in space like an onion. I think there are octopuses, not that they are. When you touch them, they, or whatever, also some flowers. When you measure, you scare them. And then it collapses to one point. So Einstein said, take a particle, take. Split the two particles. Split the box. Put a divide. And you separate the box into two parts. You separate. And you bring them arbitrarily. You don't make a measurement. You bring them arbitrarily far apart. So you have a superposition wave function here. Last wave function there. The question is, where is the particle? So we have a particle, no? Because we took it, we put it there. So you can actually, if you really believe in quantum mechanics, the particle could be arbitrarily big. Because at the moment, there is no boundary to the validity of quantum mechanics. So it could be also a big particle. Then the Schrodinger diffusion, it's very slow, the Schrodinger diffusion. But we are theorists, so we have as much time as we want. We can wait for years and centuries. We don't have any problem for the spread to become uniform in the box, or uniform enough. And then we split. And the two in one box is here, and the other box is there. So the question is, where is the particle? What do you answer? That's not, I understand. But I didn't ask him which state is the particle. I asked, where is the particle? So it is like a jelly. Half of it is here, and half of it is there. So we have one option. The option is that in the sense like a jelly, half is here, half is there. The other answer is that it is nowhere. Is it correct? Because I didn't make a measurement. I'm a theoretician and don't make measurements. Is that OK? So we have these two answers. Any other answer? Either here or there. I think that we exhausted the logical possibilities in this way, I think. So just again, it's the last day we just for the fun of it. Let's make a poll. So who votes for the particle being half here and half there, half here and half there? Just to don't be ashamed, I will not record who answered what. This is not a political election, so no problem. So raise your hands just to count how many. 1, 2, 3, 4, 5, 6, 7, 8, 9, more or less. So who votes that the particle is nowhere because we didn't measure it? So 1, 2, 3, 4, 5, 6. And who votes that the particle is either here or there? 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Can you raise your hands again? Sorry. No, wait, wait, no, sorry. Wait a minute. Can you raise your hands? Can I make an interesting, so the majority said it is either here or there. And I noticed something that, if I'm correct, that this answer was preferred mainly by female colleagues. And there is a, and then there is, I think there, so I hope, I mean, no, this proved me if I'm wrong, but I think that there is a very simple reason to that. That typically women are more bound to earth in a positive sense. I mean, they don't, at the minute, in a very positive sense. So there should be something simple in the end, which is what Einstein's position, by the way, which was. So you are in a good company, you are. So it's more or less uniform with a small majority towards it's either here or there. So either here or there is Einstein's position. So we'll comment a little bit on three of them. Either here or there, it is Einstein's position. It is. So that means incompleteness. It means, because the wave function, sorry, the particle is somewhere, but the wave function does not tell me where it is. So the description given by the wave function is incomplete. Now it is interesting, because I probably, those who voted this third option would say, yeah, so what? OK, I don't see a big problem with that. The theory is incomplete, we can live with that. But then those times were time of ideology, where not time of reasonableness, where people were confronting opposites. So we are in the 20s, we are in the 30s, so you know what happened, especially in Europe, unfortunately, in those times. And that was the spirit of the time. And so telling board and the Copenhagen school that the quantum mechanics is incomplete, is it was like telling your children, your children is nice, but there is something not right about your children, your children. Then, especially women, would kill that person. How dare you tell me that my child is something that is missing? My child is perfect. But it was really like that. It was not a, it became ideologically very much. But just a comment of it. So I also would take that position personally, but it's not important at the moment. So the particle is either here or there. But that means that the theory is incomplete. And that means that there is more work to do to complete the theory. It's an open research project. I see nothing wrong against it, but then ask many colleagues of mine, they would shoot at me if I say this. The other position, so it is neither here or there because we didn't make a measurement. Now, one has to be careful about this. So this is a standard position, which is subscribed by many people. So full respect for that. But they want us to be careful about it because what does it mean that it is nowhere? Did it disappear from space? Because before we put it in the box, then did it magically disappear from space? That would be probably the better way of putting the question. That would be more the Copenhagen way. So it is not that it is nowhere, but you cannot ask the question. And then my reaction to this, so are you an obscurantist ideologist that you forbid me as a physicist to ask question that it is not legitimate? Is it really not legitimate to ask where is the particle? Is it really outside physics? I don't want to measure. I'm theoretical physics. I want to understand the world. But you're putting a limitation to my knowledge. Why? Are you doing that? I'm not talking about emotional states. I'm talking about a very objective thing about being somewhere. That's the Copenhagen thing, but I would say that being somewhere, so I know that now I put it here. I think that as a down-to-earth physicist, I can expect that a second later will be somewhere. No, you cannot just hold it. That's a different attitude towards physics, but I would find it very curious that the very fact that I put a particle there and I close the box forbids me to think that the particle is somewhere. That would be, so I think it's a very strong position, I think, and I would say that I don't fully understand why one would take such a position, but it's legitimate. The first position that is half here and half there, now we have to face something. We have, now, and I'll go on and then I will see that that position that half is here and half is there is not so easy to take. It can be done, but we have to pay a price. So there is always, and then I will summarize, there is always, oh wait, no, I can tell you now, then I go on. If you say that the particle is half here and half there, then when you measure, you find the particle either here or there. Meaning that, suppose I find the particle there, meaning that the half that was there instantly vanished and recombined with the other half there to make a whole thing. Could be, but you have to pay the price. So if you take the third option that the particle is either here or there, you pay the price of incompleteness. You have to work more to make a better theory. If you take the price that you cannot ask the question, so if you take the position that the particle is nowhere in the sense that you cannot ask the question where the particle is before making a measurement, then you are paying the price, the price of limiting your knowledge. And personally, why would you do that? Usually other people limit, want to put constraints on you. Why do you want to put such constraints on yourself to limit your possibility of knowing things? If you take the position, the particle is half here and half there, you pay the price that you have to understand or to accept at least that when you make the measurement, half of it magically disappears from there and magically reappears over there. Yes, that would be, yes, I'm okay, particle is a field, but I'm saying that that field is half here and half there. Then when you make a measurement, it vanishes from here like this and everything appears over there. No, information didn't flow. There is no flow on anything, but the objective fact that according to the first position is that before the measurement, half was here and half was there. After the measurement, the whole particle is there because that's what happens. That simply means that the other half vanished like this from there and reappeared over there. Unless you can come up with something else. So all three positions require a big sacrifice about our usual way of understanding physics because that's quantum mechanics is challenging our understanding. I leave it to you which one you prefer. So I don't want to take a specific chance about it here. So let's go back to Einstein. So where is the particle? So the position of Einstein was very simple. The particle is somewhere. So the way function describes whatever it describes, the particle is somewhere. I make a measurement. I don't find, so half of the box is close to me. The other half is far away. So I look in the left box where I am. I don't see the particle. That means that the particle is in the other box. The other half is in the other half of the box. Nothing dramatic for the description of physics. The particle was there even before. So this is option number three. Very down to earth, very simple answer. And usually simple answers are close to truth. And just my knowledge was incomplete before. And then the collapse of the way function is just an update of my knowledge. Now I know where the particle is so I have to change the way function. Then Einstein says, take the completeness point of view. So what would be option number two? So the standard Copenhagen way of understanding things, that it doesn't make sense so that the way function is something objective, but it doesn't tell me where the particle is because I cannot even ask the question where the particle is before I make a measurement. So it doesn't make sense, so it is not. So understand the constraint that you are putting that why Einstein was against it. The fact that I cannot ask the question means that it is not part of physics claiming that the particle is somewhere but I do not know where. It is not part of physics. You cannot think it would be like, to make a strong claim it would be like religion, it would be. I believe that God is round, I believe that God is square. That's my belief. It's not physics. It's not that God is here as an object moving as a square or round. It's complete, it's another thing. It's another level of knowledge which you can accept or deny. That's another story that is. Physics is not, there is nothing, no physical fact about the particle being somewhere. So it's a really a strong, very strong claim. But let's take it. So Einstein said, okay, take the position of the particle. There is no physical fact about the position of the particle. There is only the wave function. The only objective fact is the wave function that it is as a mathematical object up here and up there. We make the measurement. We are always, we make the measurement in one side. On the left, we do not find the particle. The particle is on the other side. So now the particle has become the position of the particle has become part of physics because we made the measurement. And then he says that, look, if you really take this position, you have a non-local effect. Because before there was, there were, the description involved, included somehow in the objective description of the system which was the wave function and only the wave function was in both books. Now it is only in one of the two boxes. So for him, this is non-locality. So he says, so he was trying, so we are before EPR. Now we come to EPR. So for him, the completeness, so he was unhappy about the bore or the Copenhagen position. He was unhappy because he thought that it is our right as human beings as physicists to ask all possible questions and also the question where the particle is. So for him, the theory was incomplete. People were telling him, shut up, you're old, go away. And then he was answering, look, let's accept your completeness position. Then your position implies non-locality because you can, completeness means that the wave function describes everything there is to know and nothing else. And you can change instantly the wave function in separate, in far apart regions of space. And then, so he was throwing the ball on the other side. So do you really want to deny special relativity? Are you saying really that the world is non-local? Of course the answer would be no because relativity was a successful theory by that time. And so for him, there was an issue. Completeness is non-local. I believe in locality because relativity is a successful theory. So this non-locality cannot be true. And therefore completeness has to be abandoned. Can you speak louder, sorry? Okay. No, he was not so much worried about information if you mean it in the modern sense. He was worried about, it was, so it was, yeah, it was not so much worried but necessary about information possibly being sent super luminarily. We know now that those quantum mechanics satisfies the no-faceted light signaling theory. It was more fundamentally, so independently from the possibility to exchange information was just a description that if I do something here, I change something there independently. So if you want to exchange information, you have to do repeated measurements, so it's a different thing. It was just a do one experiment, do. So I make a measurement here. Before making my measurement, that colleague has the possibility to find or not find the particle. But if I make the measurement and find the particle, I take away from my colleague the possibility of finding or not finding the particle because he or she will not find the particle. So my measurement here changed instantly things over there. I don't want to communicate with my colleague. So the exchange of information is a subsequent thing. I don't care about communicating. I'm only saying, he's only saying that if I do something here, I change things over there. For him, this is no locality for him, and which is unacceptable because he strongly believed in relativity because he invented relativity. So the bottom line that I hope I will go at the end of this lecture is that quantum mechanics is no local, but it doesn't allow for superluminal signaling. But yes, your reaction is the correct one. What does it mean? No one knows. No one knows, really, it's true. There is no answer to that. Or at least no official answer. People have different point of view, but there is no answer on which a large enough part of the community would agree. That will be the bottom line of my today's talk. You were, that could be, I'm happy with that. But then I would call the extra dimension your hidden variables. So it's incompleteness. So you have to, you want to make a theory that could be viable. But then you have to make a theory about that. That would be, actually, this is something that you know, I don't know if you saw the paper ER equal, EPR equal ER by Maldassin and Saskin. They wanted to explain entanglement and non-locality via, not via extra dimension, but via Einstein-Rosen bridges. So they went two particles far away. They are not really far away because they can communicate. So this is possible, but first it would fall under the category of incompleteness. So there is something more to what the way function says. And second, you have to make a theory, which is difficult. But it's a research project, so to say. Okay, so this is the point of view of Einstein. Okay, so if, so for Einstein, things were that easy in a sense from the conceptual point of view. So if it is so easy, then the most reasonable position is Einstein's, if you think. So coming back to the two options, coming back, the three options, sorry. Option one was that the particle is half here and half there. Option two is that I cannot ask, the question is meaningless, I only have to, I have only to talk about outcome of measurements. Option three is that the particle is either here or there. Option one and two imply non-locality in that strict sense. Option three, not necessarily. And so the question is, so let's take option three end of the story. It's not so easy, again. Yes. Again, sorry. On curved space, yes, but you can do physics. So I'm not expert about quantum field theory and curved space time. But certainly you can also, you can level down your discussion to the case, to the, basically to the Newtonian case. You don't have to, so to discuss these things, you don't have to go necessarily to curved space times. You can level it down to the Newtonian world where there is no problem about time. And the problem is still there. So I wanna say that this is a problem that does not appear only in curved space time and then it evaporates somehow. It appears also at the level of Newtonian gravity, Newtonian, I mean, ordinary quantum mechanics where there is no issue about time. No, no, no, you mean that because of special relativity? Oh, okay, no, sorry. Well, I mean, no, no, I mean that something very simple. Take a frame. So being instantaneous is not any invariant concept. It changes from frame to frame. But in one frame, so if you stick a frame, I can speak about something being instantaneous. In one frame. Well, but forget about the other frame. We can, I can do, so if you want to put it in a relativistic way, make, take two measurements that happen in space-like separated region of space-time. If you want to be that, that would be an invariant statement. So I have A and B that are roughly located in two places that make measurements that last a specific amount of time and the whole thing is space-like. So that's a relativistic, that's an invariant statement. The claim is that measurement that A makes influences B. So that's non-locality. That is, because the usual singlet EPR set up. Before making the measurement, B could have gone, got as an outcome, spin up or spin down. But if A, so, but then the measurement of A has to be correlated to B. So depending on the way you look it, one makes the measurement collapse and decides also the outcome on the other side or vice versa, depending on, but you have perfect correlations while which you have to explain. How can these space-like separated measurements have perfect correlations? Correlation means usually non-locality unless you go to two. So okay, so I just, we come back to question. Let me follow the thing. So in a naive way, so I would say, so why do you fight against all these things? Let's take the incompleteness point of view. The particle was somewhere, no big deal about that. Quantum mechanics is a successful theory. But what should it be? The final theory of everything. We know that theories come and go. Quantum mechanics as well will come and go. It's the theory we use at the moment. Tomorrow we will use another theory. Well, I mean, there is a problem about incompleteness. So by taking the simple, the idea that the particle is somewhere, we just don't know that we function encodes our ignorance about the state of the system. And it is the following one. I mean, one can express in many ways, but it's again interference, it is. We always go back to that. So suppose that you have this kind of, this simple experiment, that you have the double slit experiment where you randomly close one of the two slits. So in reality, it is a single slit experiment. It is repeated experiments. And after each repetition, you change randomly which slit to close. If the particle is always somewhere, the particle, then each particle sees one slit, either the upper slit or the lower slit, depending which one you close. It will pass or you can hit the screen and it is stopped or it passes through the slit. If it is open, perhaps it hits the wall. It lets the edges of the slit, therefore it is a little bit deviated. But more roughly, doing that experiment would be equivalent to doing the double slit experiment because the particle, even if you keep the two slits open, the particle will pass through one slit. It's a particle. It is always in one point in space. So whether you close one of the slits or you keep them both open, nothing should change. The only thing that could change is that you block more particles. But the particles that pass through, for example, the lower slit, would have passed independently of whether you closed or you didn't close the upper slit because it was prepared so to pass through the lower slit. But you know that this is not the case from quantum mechanics, there is interference. So if you do the first, the left part of the experiment, the particle will end up either in the lower or in the upper part of the screen. If you randomly close one of the two slits, but if you keep them both open, you have the interference pattern. So that is why Einstein's position was not so simple. It's not just, okay, let's take it. The particles are somewhere end of the story. It's not so simple because at least ideally, if you have the double slit experiment and the particle goes through one slit in whatever this means, so don't take it literally because it cannot be like that. So whatever it means, the particle feels, knows, particle have no feelings, particles don't know. So in some way that has to be understood, the particle feels or knows that this, whether this slit is open or not. And reacts differently depending on whether the other slit is open or not. If it is closed, it reacts classically, so to say. It will end up behind the slit. If it is open, it reacts wave-like and will do the interference pattern. This is another way of saying non-locality because experiments are difficult, but theoretically you can make the two slits far apart. So when the particle is here, if it is here, it feels whether what is happening here far away. Again, non-locality. So it is not so easy to claim, okay, particles are somewhere, we just do not know where, so it's classical physics, a little bit sophisticated classical physics. It's not, that's simple. I want to move a little bit, so I do 50 years of histories in one minute. So I hope I guess that some of, most of you have heard about the EPR argument. I hope I don't want to repeat it, but basically the EPR argument is a more convincing way or a more sophisticated way of claiming the same thing that Einstein was claiming with the box. Things, measurements here change instantly, things over there. Properties here can change properties there unless you accept that this property pre-existed. So that's the EPR paper, is the title of the paper, is can the quantum mechanical description of reality be considered complete? So the EPR paper is about completeness or incompleteness of quantum mechanics. In the Einstein box, there was no entanglement. With entangled state, the thing became even more evident, more clear, but it's the same concept that measurement here can influence properties over there instantly, therefore no locality, unless you accept that they pre-existed. So the conclusion is of the paper is more or less while we have proved that the quantum mechanics is incomplete, more or less the last paragraph of the paper, the task of finding its completion is an open problem, roughly, the paper. So for them, incompleteness is clear, then the next job is to find a new theory of nature, a big theory. Do you know the answer of Bohr? Everything died away because it was dismissed as useless philosophy for somehow for 15, 20 years. Meanwhile, in the 50s, David Bohm rediscovered Bohmian mechanics, the idea of the Broglie-Boy, the Broglie of the pilot wave theory, and turned this idea of the Broglie into a better formulated theory, which is now called Bohmian mechanics, the way you wanna call it, which is an example of a way to complete quantum mechanics by attaching positions to particles. And I won't just open one, make one comment, and then I will not go further with Bohmian mechanics, that when you hear, so Bohmian mechanics is at a non-relativistic level, fully compatible with everything we know about quantum physics. So when you hear claims saying that experiments prove that systems do not have reality or that particles do not have position, that is a false statement. Experiments do not prove this because you can take the Bohmian way of reading experiments and in the Bohmian way of reading experiments, particles always have positions. Certainly, experiments prove that microscopic systems do not behave naively as Newtonian systems. But experiments prove that systems do not have properties, do not have position before. This is not what experiments prove. In that sense, experiments prove nothing. It's our way of reading the experiments. Then you can be Copenhagen people and I'm very happy with that, but the claim that experiments prove that is a false statement. It is the way that experiments are interpreted in the standard way. It's not a unique way. End of the story. Yes. In the relativistic regime, there is a problem with Bohmian mechanics in the sense that a consistent formulation of relativistic Bohmian mechanics is a satisfactory. So it's consistent with the experiment, but it's a kind of a trick. It exists, but not another thing. There is no fully consistent quantum theory because when you're relativistic quantum theory, because when you do relativistic quantum theory, you can open any book on quantum field theory, you will find pages and pages, which is the scope of the group book in relativistic equations, the Dirac equation, or the Maxwell equation, but no one tells you about the collapse of the wave function. In the orthodox quantum mechanics, the collapse of the wave function is one of the axioms of the theory that is orthodox quantum mechanics. Whatever that means, I don't want to enter into the meaning of the collapse of the wave function, but if you take a book, not modern books, forget also about that, but if you take old books about quantum mechanics, the list of axioms, that is the collapse of the wave function. Quantum field theory on purpose, neglect that. So because, why do they do that on purpose because they don't know how to make it relativistic? Why they don't know how to make it relativistic because it cannot be made relativistic? Why can't it be made relativistic because of no locality? Well, now we're going along debate. My answer would be no, because a Copenhagen for me means also the collapse of the wave function. So you have to tell me, do you collapse or no? If you do not collapse, you are a many world guy. Be aware of that. If you collapse, then the collapse is part of the description and that has to be made relativistic. They have to choose which of the two and then we can debate and you will have problems with, so again, you have to pay a high price, a very high price, whichever solution you take, but really high. Okay, so quantum mechanics. So, quantum mechanics would be, in a sense, at the dream of Einstein, it completed a theory and particles are somewhere, but it's not local. It is clearly, plainly, non-local. If you play with the theory, you have a particle here, a particle there, entangled wave function, you kick the particle here, also the other one reacts instantly at a distance. 60s, John Bell came. John Bell was unhappy with the orthodox quantum mechanics, with Copenhagen interpretation, he discovered Bohmian, so he saw the papers by Bohmian mechanics. Oh, at least something new. Let's play with it. It's non-local, no way, it's horrible. Let's try to make it local. It didn't succeed. He was trying to make it local. It didn't succeed. And then he asked the question, but am I limited that I cannot make Bohmian mechanics? Yes, it is, so it depends what you mean. So there is no uncertainty in the sense that the particle objectively has, each time, a perfect, or at least theoretically, a perfectly fine position and a perfectly fine momentum. But the uncertainty principle is recovered because if you make measurements, so if you interact with the system with the particle, you scare the particle. So if you want to make a measurement that tries to understand the real position, then any other measurements that wants to measure the velocity will have a huge uncertainty. So the Eisenberg principle comes, is there. It is still fulfilled, but it is not anymore an intrinsic limitation to our knowledge as people would claim. It is just a practical limitations because measurements are always invasive. Every time that you make a measurement, you disturb the system. That would be, so in that sense, it's fully recovered and consistent with the experiment, but not at the fundamental level. Okay, and so Bell was not capable of making Bohmian mechanics compatible with relativity, and then he asked the question, but is it me, or is it that it is intrinsically impossible to make a quantum theory that is compatible with relativity? And then he came up with Bell inequalities. Bell inequalities say that if a theory is local in a precise sense of the measurements, space-like measurements don't influence each other, are completely uncorrelated because they are space-like, then the inequalities are satisfied. If measurements, if instead the inequalities are violated, then these space-like measurements are not uncorrelated. So what I do here as influence, whatever it is influences things over there. That's no locality. Personally, nothing, no big deal. Okay, so there are two great merits in Bell inequalities. The first thing was to have brought what was up to then a philosophical discussion down to a mathematical equation. So a philosophical question has turned into a physical question. Second merit is that that physical question was not just a conceptual question, but could be tested experimentally. So this is a double merit of John Bell to have brought the discussion back to physics but accessible to experimental physics. In the 80s, Bell inequalities were tested by aspect. You know, they were violated. So the predictions of quantum mechanics were against Bell inequalities. The nature now violates Bell inequalities. Conclusion, the theory is no local. That is the bottom line. The theory is no local and then one has to do something with that. But now let's try to see Bohmian mechanics. So now the game starts. We have, yes, more or less half an hour. So you know what we have. So I want to again to provoke you about the meaning of Bell inequalities. I will not go through the derivation. So now we make this game. We have Alice and Bob. You know, you have experience to that. But now they run a show. They are one in, let's say, here in Trieste. The other one is in New York by a city chosen randomly in the world. And now they claim the following thing. So there is audience. There are two audiences, if you wish. People like you running the show and you give them a card. So it's kind of a silly game because I want to connect them by inequalities. But you give them a card and in this card you can freely, it's up to you. You can write one of the three numbers, one, two, three. So there is here in Trieste, I'm Bob, not Alice. I'm Bob and you give me a card with one of the three numbers. My partner Alice is in New York. There is someone else giving a card, one, two, three. The game is rather simple. I, Alice and Bob write either yes or no. And they give the card back. So you imagine, you are here doing the experiment with me. You give me a piece of paper with the number and then you repeat this. So one time you may pay one, two, three. I give it back to you with the yes and no. As simple as that. Try to imagine really to watching TV, whatever. Same thing in New York, whatever. Okay, we repeat the measurement and this is the outcome. So there are different rounds. Rounds one, two, three, four, five. I listed eight of them. So in the first round you gave me, I'm Bob. You gave me a card with the number three and I gave it back to you with the answer no. Simultaneously, more or less, or space like separated. Let's say as physicists. In New York, Alice was given the card with the number one and she answered yes. And so number two, third round. So what happens? And that's the promise that Alice and Bob made. Outcomes are random. But each time you give us a card with the same number, we will give the same answer. So that is the reason why they are touring the world and becoming famous because you are the audience and you decide which card with which number to give. But the promise and they earned money out of that because the people pay to go to the show is that each time you give us the card with the same number, we give the same answer, yes or no. And the yellow thing is the cases where Alice and Bob were given a card with the same number. So this is the show, this is the show that they run. Probably you cannot go to all shows so you watch it on TV. So what is your reaction to this? And the claim is that we are telepathic. So we are a first example of, we are so in love with each other. Alice and Bob, we are telepathic, we can communicate among ourselves and take decisions in this sense. Are you happy with that? Do we buy telepathy that comes to that? But my question is, are you happy with that? In a sense? Do you believe telepathy means non-locality? Are we happy with non-locality? Remember relativity. That means that we dismiss special relativity. No, I'm okay with that. But that means that we are, so if it's non-local, we are dismissing relativity, special relativity, we are dismissing general relativity, we are dismissing one half of physics, at least. Which I would be very happy if you are happy with that because I think that's the right spirit. I think that to destroy current physics and make new physics, I'm fully happy with that. But just be aware of what it means. That no, I'm not talking about, I mean, non-locality means that something, some things are correlated in that sense. That means non-locality means. Yes, don't go mathematics because entanglement is mathematics. I'm talking about physics, I'm talking, I'm telling that. If Alice, if one of the two give the same card, one of the two answers one way, the other one will answer exactly the other way. This is non-locality. Alice and Bob are two human beings, sir. No, no, I'm telling you that don't deny the game, otherwise we don't go, the game is like that. And there are two human beings, real human beings that exist and there is an audience here that are looking at the person. So the person exists and there is another audience in New York that is looking at the person and the other person, the person exists. That is, we cannot go so far to deny the existence of people. That's too much. So they are running the show and they are, and I mean, don't get used to quantum mechanics too much. I think it's something really astonishing, I think, that you arbitrarily choose which card with which number to give, arbitrarily choose. And then they systematically give the same answer if the card chosen by you happens to have the same number. I find it something astonishing, I find it. There is a simple answer to that, actually, there is, without invoking entanglement, which is Einstein, which is the EPR thing. There is a very simple answer to the story. They agreed beforehand how to answer the question. They had a book, they had a huge book, and they decided before starting the show that if they get a card with number one, in the first round, if they get the card with number one, for example, they both will answer yes, if they get a card with number two, they both answer yes, if they get a card with number three, they both answer no. In the second round, if they get a card with number one, they will answer no, blah, blah, blah, and so on and so forth. Very simple, down-to-earth explanation. No need of non-locality, no issue about entanglement. Very simple explanation. Like? Yes, like that. So Einstein was very naive in a good way, down-to-earth person. There is no need to be amazed by that. And actually, people that are not experienced with quantum mechanics, they always give me this answer. Because it's a simple answer. It is really, at this stage, it is the most simple, the simplest possible answer. They agreed beforehand. There is nothing. So it's clear that if you do not know about the book and you look only at what they answer, the answer will be random. But obviously, if you give the two cards with the same number, they will give the same answer. So correlations are classical correlations. It's like the case of a ball. The ball. You take a ball, which is two balls, a blue and a red ball. You separate one is here and one is there. I open the box. I open the red ball. Then I instantly know that the other one will be blue. No magic. No entanglement. Nothing. It's something very simple, very physical. The color was there before. Now, Einstein was saying nothing more than that. So he was deeply rooted into a relativistic description of spacetime where non-locality cannot be, in any sense of survival, that cannot be non-locality, things can propagate slowly at the speed of light. So if there are correlations, it is because property pre-existed, not because they came out of magic. So I insist on that. Properties, correlated properties, instantly, or also of superluminal happening in nature, are a form of non-locality that doesn't imply that there is an interaction among the two, that doesn't imply that you can do communication with that. But it is non-local. Something that happens here and there at the same time, or that happens in space, like separate the region, is non-locality. It's something non-local. Nothing to be worried about, but things have to be called with their names, non-locality. So Einstein, so no telepathy, so basically the request that non-local things cannot happen, so locality, so the request of locality, implies that the answers of Alice and Bob were decided beforehand. So they existed before the games. This is what he did, the variables. So incompleteness. This is the EPR paper presented as a game. The request of locality implies incompleteness. Properties, so you can talk about position, momentum, you can talk about spin. Properties pre-exist. But then John Bell comes. John Bell tells, and that's the essence of very inequalities, so OK, let's look at the table. You see this, let's take round number one. So what are the possible options? So you can give Alice card number one and Bob card number one. They will give the same answer. Yes, they will agree. You can give Alice card number two and Bob card number one. They will still agree. Because according to the book, they will both answer yes. Only if you give to Alice card number three and to Bob card number one, they will disagree. So A means agreement, D means disagreement. Because Alice will answer no, and Bob will answer yes. And then you can check all other nine possibilities. So what do you see? So Bell says, basically, Bell tells Einstein, so to say, don't look, don't be amazed only. Don't consider and be amazed only for the case when they are given the card with the same number. Let's look at the full statistics, also when they are given cards with different numbers. And we see that five cases out of nine, they should agree. And four cases out of nine, they should disagree. For round number one, round number two is C and three and four are similar. You will have the same situation. In case number in round number five, even worse, they will agree any case. So in round number five, they will certainly agree. So the number of agreements is greater than the number of disagreements. So if we look at the book that is the prediction, we give randomly cards with numbers and not by a selection, of course, of numbers to write in the card, the agreements will be greater than the disagreements. That is translated for this game by inequalities. A greater than D. Let's count the statistics. You can count them for the eight rounds. Of course, you should go for a longer statistics. And you find that A is equal to D, up to statistical uncertainties, which can keep smaller and smaller by running the game longer and longer. That is aspect, if you wish. Bell was the slide before. Now it is aspect. So that is the first, if you wish, version of Bell in equalities, which is violated by evidence in running the show, the inequalities are not fulfilled. They are violated. That would be a typical example. So this is the logic that I have outlined. The Bell CRM is made of two parts, actually. The first part is EPR, that locality implies incompleteness, implies hidden variables in the modern language. Then Bell came, incompleteness implies the inequalities. So the full logic is, and then I will show you, locality implies inequalities. Evidence, experiments, say that the inequalities are violated. Now, before making the comments, you will read often in the literature that Bell's theorem is not this one. Locality implies incompleteness that implies inequalities. You will always often read that locality and incompleteness implies completeness, which was called determinism in the papers of Bell, which is called realism in modern paper. So locality and realism imply nonlocality. Therefore, Bell inequalities, therefore, if the inequalities are violated, you have two options, either to deny locality or to deny realism. This is wrong. And Bell was clear about that. Bell's theorem is not locality and call it realism, call it hidden variables, call it incompleteness, imply nonlocality. Bell's theorem is, in this first formulation, locality implies not locality and locality implies incompleteness that implies inequalities, that thesis. So this is classical logic that goes back to the Greeks. It's not nothing more than that. If the thesis is wrong, then the hypothesis is wrong. Bell inequalities are violated, and therefore, you have no locality. And in fact, in subsequent papers and in modern formulation, this is the way that you can present Bell's theorem. You can skip the middle step. You can. And go directly from locality to locality and get a summation by Bell. It implies inequalities. You can skip the middle part. Violation of Bell inequalities implies, so what I said before you mean. So the paper, I will write it here. So Bell's theorem is loc in the first formulation. There are different formulations. The first formulation, because it was reasoning on the EPR, is basically, if you wish. And that's the EPR part, which implies the inequality. And that's Bell's contribution to that. What now is presented in some books is, how is N? N is like this, no N. That is the way they present it. Locality and realism imply inequalities. These are two different things. These are two completely different statements. So this is Bell's theorem. No, I'm saying that this is just an intermediate step, which is unnecessary for the theorem to be proven. So now I'm talking about mathematics, so to say. So this is a middle step, which was used in the first formulation of the theorem, but can be removed in a refined formulation of the theorem. So a theorem can be proved in different ways, the same theorem. It doesn't exist. Realism means that there are microscopic properties. What Einstein would call incompleteness. So the hidden variables. So that the particle, so in the case of spin, that the spin exists. Even in a superposition, there is a definite property about spin. Or the particle is somewhere, if you want to talk about the wave function space. Realism means reality, so the existence of properties. In a kind of technical sense. So realism, this is equivalent to hidden variables. So to something extra, some extra variable. Well, no, not at all. It's independent properties that are independent of the observer. Yes. Well, again, it's like the example would be a gas with the temperature. The hidden variables, which you do not have access to, practically, are the positions. You can call them predetermined, if you wish. Position. There's nothing more than that. Louder, sorry? No, no, realism means no, we can have superpositions. But the superposition is not a full story. There is also something more, the position of the particle. The particle has a definite position, but it's not the entire story. It is in a definite position, but also in a special state, which is the superposition. So the Bohmian analog would be like the surfer on the water. So the surfer has a definite position. But then, depending on the wave, it moves in different ways. So if you know the position of the surfer, you cannot predict the future motion of the surfer. You need also to know the wave. So in that sense, that would be a rough analogy. So the wave goes, but there is realism. There is the surfer. And the surfer goes with the wave. You need to know both the surfer and the wave to predict the dynamics. It's deterministic. Bell's theorem proves that this is the wrong way of expressing the theorem. So go back to the right way of expressing the theorem. That means that any local theory has, in the sense of Bell, locality. You can express many localities, but in the sense of Bell, which is a very weak condition, any local theory has to satisfy Bell inequalities. Bell inequalities are violated by nature. Nature is non-local. Any local theory you make is not a theory of nature. That's in the sense of Bell, in the sense that the sense of locality of Bell is that space-like measure. I don't want to talk about predetermined properties about whatever. I'm talking about measurements. I can talk. Space-like, separated measurements are uncorrelated. Completely uncorrelated. Now you are used to entangling, especially if you do quantum information. You're used to entanglement and these kind of things. But really take out your habits of working with quantum information. It's something amazing it is that via entanglement, of course, if you make a measurement on the moon and on earth, instantly in some sense, space-like if you want to be relativistic, but instantly these two measurements are correlated in a, not in a classical way, like the blue and the red ball, are non-local correlated. This is against relativity. So now I go to, and then I just take questions, because this is the main slide. Nature is non-local. And that poses a problem with relativity. When you hear, when people do quantum field theory, when people do string theory, quantum field theory in curved space time, only now the thing is emerging because the quantum field theory community and quantum information community start in talking to each other. But it didn't come to this problem yet. But all these people are skipping the problem of correlations and measurements. They don't talk about it. Take any book on quantum field theory. There is nothing like that. Because the program of a quantum field theory is incomplete until you say something about the collapse of the wave function. So why quantum mechanics is non-local? I make a measurement. I collapse the wave function instantly everywhere. This is the Copenhagen interpretation. And then I influence the experiment over there. And that's how I have correlations. So if you think how correlations happen in EPR setups, I make a measurement. You collapse. You change the system also over there. This is an instantaneous thing. It's really an instantaneous thing that happens. That was what bothered Einstein. If you want to do that in quantum field theory, you have to accept that this has to be instantaneous. Because if you say that I make the measurement here and the effect of the measurement propagates at the speed of light and eventually reaches over there, you will get a theory that is falsified by experiments. Because all the Bell experiments tells you that there is the correlation. The correlation doesn't propagate at the speed of light. The correlation is superluminal, possibly instant. So we don't have the experimental proof that is instantaneous, of course. But it's certainly faster than the speed of light. This is not information. That's another story. Because information, the way you mean information is encoding the qubit in the state and repeating the measurement. This is not what I'm talking. Here I'm talking of the fact that I measure here. And according to the standard theory, I collapse the wave function that collapse is instantaneous or at least superluminal. And if you want to have a complete description of a complete relativistic theory, a fully relativistic theory, you have to explain also this thing. You don't find it in books in quantum field theory. So in that sense, quantum field theory is as incompatible with relativity as it is with mechanics. It agrees with the experiments. There is truth in that, but this is not the point. The point is that we want to have a consistent, coherent theory of nature. And there is a tension, a clear problem, between quantum physics, relativistic quantum physics, and relativity because of value inequalities. And the thing that there is nothing against that, we should simply admit that we are human beings, nothing more than that. We are not understanding things instead of being not human symbols. Defending fathers. Instead of being arrogant about quantum physics, now we have the true theory of nature. Einstein was wrong. Newton was wrong. Galileo was wrong. Maxwell was wrong. Now we are right. No. It is, we are not under, in a deep way. Of course, in some phrenological way, we understand nature. Newtonian mechanics is the theory of engineers that build bridges and high rises. So these are all good theories. But in a deep sense, we are simply not understanding nature. It's an open question. Like the same as quantizing gravity, we are not understanding how it works. As simple as that. People are working on that. That will come one day, but not today. Okay, so what about incomplete? So the bottom line is non-locality. So I showed you that this slide here, the end of the story, you see that now incompleteness vanished from the slide. There is no mention of that. Locality implies inequalities, inequalities are violated, therefore the world is non-local. So what happens to incompleteness? That remains an open question. So is the wave function the full description of the state of the system or only an incomplete description of the state of the system? There is no answer to the question at the moment. There are possibilities. One possibility is the many-worlds interpretation where the wave function is the complete description of the state of the system. It is a program. It's not a box you can buy. It's not take the many-worlds interpretation and then everything is happy and you can live a happy life and do something else. There are open issues to make it a really a theory of nature, the emergence of probabilities with one of them, understanding the many branches. It's an option. People have to work at the moment. It's not a fully worked out alternative. It's an incomplete in the sense not fully developed theory, not incomplete in the sense of Einstein. It is not fully developed. You can take collapse models. You can take also there. The wave function is the complete description but you change the dynamics. The Schrodinger equation is not correct. Or you can go on the incompleteness side. The wave function is not everything. There is something else. Bohm's mechanics is an attempt to do that. But perhaps there is something better than Bohm's mechanics. Something smarter that can be done. It is an open question and so which is related to the measurement problem of quantum mechanics. But this is another story which will take hours. So I finish here. I take your questions if you have.