Measurement in Classical & Quantum Mechanics

0:00 For me it is a great pleasure. Last year I was here and it's been a pleasure to be, I hope to sponsor and to be as antagonistic in last year. To start with, I will tell you... You don't have Richard Keel this year. I noticed, I noticed, but I'm sure that's how a lot of people will be here. I will want to talk about quantum measurement and mostly as an asymptotic bridge from the to classical. Physics, the history of measurement is as old as civilization. If you look at the old geography of science, then you have the question of how do we know what we know? And this is a very weak form of the measurement. Then people realized that there was a difference between classical and quantum mechanics. Einstein and Bohr is just a milestone. There were several discussions after them. Usually one speaks of the formalism of von Neumann, which is correct. I like to speak of Lüders, which is a version of von Neumann. And Wigner introduced as a religion, something that von Neumann said very simply, is the fact that depending how you do, you want to measurement, you need an apparatus to measure the measuring apparatus, you need an apparatus to measure that, and this is the icon of Wigner's friend, friend

2:30 which is never shown the theory of classical measurement there is no such thing in classical measurement so my point in this conference is to say that if i can reduce the product bring it to the classical level i can invite wigner span for the beer but to talk about fear and not about wonderland. There were a flow of papers in the middle 70s, 25 years ago, and then almost every year, there were papers, I mentioned a few here. The theory changed name and became decoherence theory after Zee and Zurich and many other people last year Belafkin okay so he gave a lecture on quantum mechanics and it was an inspiration for me to come back to this and just to make sure that you don't think that the problem is solved appeared about a few months ago a paper of Adler which the title of which is why decoyance has not solved the Benjamin problem and it is in response to the paper of Anderson and other people so subject is still alive this conference is about reconsideration so let us reconsider there was a problem The first thing is that you have a system, a quantum system, and you have a state on that system, this is the algebra of observables, and you measure one observable of the function of the observables, and the function are generated by the polynomials, so this is what you measure. I'm sorry, but it's B, and you have an Abinion algebra here, and then once you have done the

5:00 measurement, and that is the first main difference between quantum and classical theory, the effect of measurement cannot be made small. In classical theory, if you want to measure the distance from the moon to here, you send the forward moon and you wait it to come back. naturally that change a little bit the optics the orbit of the moon but you can make that change as small as you want this is not the case in quantum theory change the state changed drastically these days this is called decoherence for no man call it coherent collapse these are not quite the same thing and if I have time I will discuss at the end of the lecture what is the difference and why I talk about decoherence and not about collapse and naturally this is not within the standard form of quantum mechanics you can do quantum mechanics and do all of it with shredding the equations there is only one field which is one of our theory in which shredding the equation doesn't cover things because the state here is a pure state and it can be a pure and this state can be a mixture even if you start from a pure state so the Schrodinger equation cannot do it now you can interpret this mixture as an apparent picture with a true mixture it's an apparent picture if this system is not quite existing in itself but is coupled to the environment and then you could have a Schrodinger evolution and the whole state would still be pure if the state of the apparatus of the environment were pure it would not be and this state were pure you could still have a pure state evolving to a pure state but if you look on the partial system that you have apparent big state or one of the system which is again a quantum phenomena which is not true in classical mechanics, if you have two systems in classical mechanics, if the two system is in a pure phase and each system is in a pure phase. Decoherence nowadays is seen as an effect of the environment. Now, in the measurement theory you have to be a little bit more precise on that because

7:30 So every environment, if you believe the apostle of the environment theory, every environment will lead to a decoherence. When you do a measurement process, you want to measure something, so you have to measure with a measuring apparatus which is adapted to the problem you want. To measure, if you want to measure an electric current, you don't take a hammer. So, you have to take an apparatus which is adapted to what you want to measure, and this apparatus is in a certain state, and there are pointers, and now, I already mentioned one of the difficulties, which is to pass from possibly pure to possibly mixture, that is resolved by this, considering a compound system and noticing that the pure state of the compound system is not necessarily a pure state in H, so that was part of the way to solve the problem. The other part is that if you believe, as von Neumann suggested and believed, and suggested to Wigner particle and Wigner believed it, you can do that, then in the course of time, this is time evolution, this could happen, you could have a measuring apparatus which which, when you measure it with the point of M, you get lambda k, which is this number here, which characterizes the state you're interested in. However, there are several problems with that. One is that if you do that, then you have to say, in your head, time, when the measurement is completed. Which is not what you want. What you want is to say that when the measurement has been achieved to a certain degree of precision if you continue the allowance and you get better precision it works so in other words what you want to do is an asymptotic for a long time now if you want to do long-time asymptotics then you cannot do that with a finite apparatus for the finite apparatus has vacuolences so if you have vacuolences

10:00 So, you have to take the thermodynamic limit of a limit for a large apparatus. This was past Wigner time. Wigner was still alive but he was not active anymore when these kind of things were developed. The thermodynamics of the infinite system, which is, in fact, the asymptotics in which statistical mechanics turn into thermodynamics. This is concentrating on the scientific behavior. Now, there is one thing which happens there, which is something which did not occur in the usual formulaic of quantum theory. If you have an infinite system, be it quantum theory or classical or statistical mechanics, you may have infinite energy. In fact, if you have an equilibrium state, is then the energy of the apparatus is infinite locally is finite and so you may not have the usual representation of the canonical communication relation and that was a discovery of the 50s they are in equivalent representation in particular the whole presentation which are dependent on the situation which you are this is something which is not intrinsic it is ending it is contextual the representation which you deal with depends on what kind of problem you are looking for what kind of system you are looking for and then the representation is not necessarily irreducible. This was discovered in the context of field theory by super selection rule by Wigner, Weitman, and Bick. Wigner never made the connection with country theory I know because they don't ask him and by that time but somebody they are situation in which particular ones are charged according to people that you favor in which you

12:30 that make coherent superposition of states. They are superselection rules. That means that not all observables, not all operators, self-adventing operators in trigon space are observables. In other words, they are either symmetry or observables which commute with all other observables and which are not trigon. These are superselection rules. And this happens in statistical mechanics fact it is a significant condition for you to find the purpose to find phase transition and the contention was when I was starting thinking about these things that after all are the same kind of things as phase transition so maybe we should have the presentation which are not irreducible representation and I was led directly to this kind of thing by considering an apparatus which is my prospect now let us this is the problem let us be a little bit specific the point is that i want to show that this is not a dream but there is a model for it so the simplest thing that you can want to measure is a particle spin half particle and you measure the sigma z and what you want is to measure the sigma z, so you see lambda k, the probability is that the state speed is up or down, and you measure this lambda k, which are somewhere above, yes, rho is here, and you make the trace, and this lambda plus or minus of the expectation and then you want that at the end of the operation piece of the end of time this reduce to see a no man or leaders in this case there is no difference because the projection of dimension one you want this to reduce to this state and even if the state originally is represented by a pure state the final state is not so this is what you would like to have and now you want to have an apparatus the apparatus

15:00 which written and I designed was to have two chains of spin-half particles and Now there is one model in dynamics of spin, which is very nice, which is called the xy model. It's a model in which instead of having coupling sigma z, sigma z, like in the easing model, which incidentally should be called the lens model. Instead of having a sigma z, sigma z, you have a sigma x, sigma x, plus sigma y, sigma y. It's not quite the Eisenberg, which would be sigma x, sigma x, sigma y, sigma y, sigma z, sigma z, it's only x and y. This system has very strong erratic properties. For instance, if you take a chain, cut the piece, which is in equilibrium, cut the piece, finite piece of it, do anything you want from that piece, glue it back if it comes to equilibrium. So it's very strong erratic properties, which are vague, but this one doesn't happen. So, we take these two chains, but instead of considering the linear chain, I change it into an accordion. And I call the bottom one chain, and the top the other chain. And the interaction will go from here to here, to here, to here, to here, to here, So it is a coupling between two chains. And I will define the magnetization of each chain and the function of the magnetization of the odd and even chain. The coefficient here are simply to make the mathematics simple. So we put the difficulty in the beginning. The constant which makes things fine. So you have a magnetization here, pointer and the other pointer here, and the magnetization of the two chains, defined as usual as the magnetization in the same direction. Now, at time t equals zero, you switch off the magnetic field which was given you the equilibrium state of your system.

17:30 So at the beginning you had your chain, no interaction, but every spin in a strong magnetic field is the same direction. You let that go to equilibrium and at time t equals 0 everything is in equilibrium and at that point you switch off the magnetic field and switch on the interaction which is a XY model which I sent. And then you define a compound evolution, which is the model here, and then a part which has to do with the system. So the Hamiltonian is here, and the component evolution is just the thing which is generated by that. And in principle, you can compute everything. And, in fact, it's more than a principle. You can actually compute everything and show that the first result, the first result is that if the size of the apparatus goes to infinities, then the compound evolution, this is the initial state of the system, which you don't know, it can be anything you want. This is the equilibrium state of the apparatus. This is the joint system. You let it evolve with the compound evolution and let that go to infinity. The size of the system goes to infinity. Then immediately, the state dedutes a collapse or decoding. Now, this is not very realistic, it happens because the apparatus is very big and it drives the system very fast, and now comes the first asymptotics. If the system is not infinite but finite, so the evolution is that of equalities, there is a difference between the two, and it is given by that and you can control it. So, you have an asymptotic behavior, which is what you want, and you can control it. See, saying it naturally, when n goes to infinity, you have here a product of single charge smaller or equal to 1. It becomes the infinite product. So if you have an infinite product of single charge smaller than 1, the product is 0.

20:00 So this comes from that, so it's essentially how the apparatus drives the system very fast. And that is in coherence, or in decoherence studies, this is known as the fact that decoherence happens very quickly. And so in this model, it happens essentially, it's something else that heals the thermodynamically. Now, what happens on the apparatus side? The apparatus side is a little bit less cooperative. You have again the sinus evolution of the compound system, and you get it from the pointer, and this I call the left-hand side because it takes too much time. And then what you do is to look at the infinite size limit, and then the infinite time limit. That time is not for all time, but for the infinite time limit. And you find that indeed, the pointer plus will give you lambda plus, and minus will give you lambda plus. So, the apparatus behaves as you would expect. Now, for large but finite T, but still infinite temperature, infinite volume, so you have this, but now the time is finite, and you have here that the difference between the result which you want and the result that you have at time T is something which is proportional to a Bessel function of order of zero. function so this thing goes this is naturally nothing which is nice In other words, measurement of the Q to arbitrary precision in finite time, you tell me what precision you want and I tell you how long you have to wait, and if you wait longer than that you don't destroy your experiment. It's asymptotics, it's not finite time, it's asymptotics.

22:30 Now, if you don't like infinite apparatus, if you don't like asymptotics, then you have something which happens for finite and we can compute since the product is exactly soluble and we can find this majorization and again you have a kind of asymptotic behavior. So, you don't need to take the asymptotic. Now, this is the mathematics which goes to the model. As I said, the model is exactly the same. So, there is no approximation once you have to set the Hamiltonian, you tell the crank, and it works. The limit which I wrote exists in the mathematical sense of the world, and I teach calculus, I explain to the students they exist in that sense, there is no approximation, however, as very often in physical science, the limit is not what you want, what you want is the asymptotic behaviour. For instance, if I look at a cup of coffee, you don't want to consider the fact that there are 10 glucose 23, 24, 22, depending on how big your appetite is. Molecules, if you do that, you have no baseline issue. And you know perfectly well that if you have water, and you look at water, it exists in liquid and phase. In liquid phase and in vapor phase. And if you could throw it in the head of the room over there, there are even ice, which is not why you've got the topic, so you see yourself makes the ice. Phrases occur in Indian system, and this was a great problem in nature. In fact, for those of us who like to look at the Jewish Bible, there is the story of Trump, if you look at chapter 58, the story of Trump, he marvels about his power. God explained to him, look, how are you going to explain that in the morning you have ice, In the day you have water, and in the evening you have favor.

25:00 Say that in another language. It is that. So, for understanding that, Job did not know about the thermodynamical limit, so he could not understand God. These days we can. A little bit better. yes so we're going to bridge camp is a thermodynamical limit for the moment process in exactly the same way you take the thermodynamical limit one knows how to control that when you get is that you get a representation which is in equivalent to your original representation. As long as you have a finite system, all self-agrant operator on your Bittock space. The Bittock space is fixed. I mean, you are talking about matrices, so you see the end, because it is mixed. All self-agrant operator are observables. It may be difficult to describe which observable was there, the principal observable. When you take the thermodynamical limit, you lose your Hilbert space. C infinity doesn't exist. What you have is a Hilbert space, which is called the ,, that is the Hilbert space, which is contextual. It corresponds, the representation corresponds to the state you start from. two representations with two different temperatures are very different. In fact, they are disjoint in the mathematical sense, they are not irreducible, but they are disjoint in the sense that no subrepresentation of one appears as a subrepresentation of the other. That's very different. Right? Do you remember when you learned group theory when you were doing spectroscopy, like we were all doing, We all did spectroscopy and learn about the group and past and representation of all that you propose in that presentation, irreducible representation, etc. For infinite system, the representations are more complex.

27:30 And that, in fact, is building block number. Irreducible, the primal, in fact, of representation. As I approach, you can't be comfortable. A representation which can be irreducible. But, representation that you obtain from equilibrium state, which is what I started on. These representations are not irreducible, and in fact, they are not even practical. In other words, they have a center, which is the center of representation. The center of representation is the center of all elements of representation, which communes with every other. So these are the essential observables. That is, why are they called essential observables? The world was hooked up by Wick and Dignac and Weissmann. They are essential for, well let me still be here, they are essential for the simple, for the common reason. We all agree that the measurement is done on a billion set, a beautiful set of observables. And the complete measurement is a complete set of commuting observables. Now, there is an easy theorem from the management which tells you that the center of a fundamental management is contained in every complete set of commuting observables. So whenever you do a complete set of observable, when you look at the complete set, like what they are called, CS, C-Pose, whenever you do that, you measure the observable in the center. Whatever the set is, the complete set, the center is in it. So, these are essential observables. That is what the situation was in the 50s, then when with the advent of proper statistical mechanics of infinite systems, it was recognized that the macroscopic observables are in the center. And this is a consequence of locality. If you have an infinite system, and I take it one-dimensional because it is easier for me to describe, although I hold two functions that relate into two dimensions,

30:00 some of us more than others. And I don't know. It is one-dimensional. So, you have the one infinite system. Now if I take something here, local, it commutes with everything which is out of that region, if it is space-like. Now, if you take bigger, you just have to move it bigger. So, everything which is taken locally, if you make an average over space, it will smooth out and commute with everything in the underground. because this commute with everything which is outside is finite so most of the time it is outside of a region in which you have an observable which you are interested in so most of the time so when you do the average of a local observation this is a global observation and this global observation commute with everything else This is called asexceptic abelian mass. This is a proper consequence of locality. In particular, the global observables occur as space averages, and the magnetization is nothing but the space average of the spin. But that's how we constructed our pointers. So our pointers in the thermodynamic limit are macroscopic observables. If you want to be sneaky about it, in which sense does the limit exist? If you are a mathematician, it is again a context you have seen. It depends. It is a weak operator topology result, it depends on the representation, but the weak operator limit exists, and it is in the fundamental algebra, and it is in the center. So, macroscopic observables, point first in particular, are macroscopic observables, are essential observables. So, in other words, what has it done?

32:30 You have lifted the quantum problem to a classical problem. This is an abelian system. These are the set of all macroscopic observables. And you remember, maybe you don't remember, but those of us who were old enough, or at least Rosenfeld, would explain to us that a measurement should be an operation by which you live in the microscopic world to the macroscopic world. And, well, this is it. You lift things to macroscopic absorbables, the macroscopic absorbables are classical absorbables, and you have achieved what Bohr wants. Now you have achieved more because there are two more theorems which you can prove on this model. The first one is that all the measuring apparatus, you really measure things nicely in the sense that the state, when you have taken your limit, the state that you receive from your system as looked from the pointers is dispersion free. So you don't get a distribution, you get a point. Well, that's not a distribution. That is, when I see the thing, that is not in what Bohr and Einstein is now. But it comes free with the model. And I think it can be with any reasonable model, but that I cannot prove because I don't know what a reasonable model is. Now, there is another part which I put under the heart so far. I said the state of the system, of the measuring apparatus. Now, one thing which is very bad in the usual formulation, and this is why we had to reconsider, The first reconsideration was the Introduction of Microscopy to Solibut. The second is this one. If you have a large system, if you want to determine the state of the system, you have to do a lot of measure. You don't want to do that. You don't want to measure all the observance. In other words, the state of your system is not, certainly not a pure state,

35:00 but it's not well defined the state of your apparatus you cannot control the state of their apparatus it is an infinite apparatus and there is not nice your end which is the following suppose that you believe that you have had an homogeneous magnetic field in the beginning and have the state of the apparatus being a canonical equilibrium state from that and suppose that your My wife would not watch the kids that day and one of the kids came in the laboratory and started playing with the experiment. Kids can do a lot of damage because they are only local damages. And it changed the state by a local perturbation. So you have a new state and the new state gives you the service. In other words, you have stability of the system against local perturbation. This is out of question for a violent system. Okay, now comes the problems which are not solved. Fortunately, I don't have much time for those. I still have ten minutes, right, according to the statement, but there is part of the discussion. There are two problems which are not solved. The first one is the following. I told you that there is stability now in mathematics and the mathematics of dynamical systems are two kinds of stability. The first stability is the stability against perturbation of the initial conditions. If I put this thing on the top here, then this is a stable equilibrium state. But if I move a little bit away from this equilibrium, right? This is stability against initial state. That's the ordinary type of stability. That's the stability which I was talking about with my PR4. stability against local perturbation of the initial condition. Now, in the theory of finishing the system, there is another of the thermical systems, there is another type of stability, which is called structural stability, which is that if you change a little bit the amicronium, what happens?

37:30 Now, there are a lot of things which we believe, which we like to believe, are certainly stable, and which are not. For instance, Hamiltonian dynamical system is a category of dynamical system, are not generic. It's a little generic. Moving around it, making the system as little as you want. So a little system is not generic. Now, worse than that, ergodic properties, even in the category of Hamiltonian system, are not generic. So, proving any generic reason is bound to be very hard. In particular, it took quite a bit of work to prove the theorem. And in fact, the theorem about the erotic system not being generic in the class of dynamical system. This theorem is not well known. And I mentioned it to Sinai, and Sinai said it can be owned, if it had to be owned. So, that is the first reason. You want to prove that not only the reasons of experiment doesn't depend on the initial state of the apparatus, but it doesn't depend also on perturbation of the dynamics of the apparatus. the second and I suspect that it may not be cool but I don't know because what makes the system nice and being exactly sorry for you can prove that it's That is my nice equity property and nice equity property are not in the current state and so on. That is not a sort of problem. It should be proven. The second problem which I alluded to, I don't know if you noticed it, I mentioned that the state after the measurement is decoheered,

40:00 like a number of people collapse, when I said it's not collapse, it's decoyons. And there is a difference between decoyons and collapse. The difference between decoyons and collapse is very good. Briefly stated, it is the fact that I told only that the state at the end is a density matrix, which is a mixture of two states up and down And you may be tempted to ask, what is the result of a single experiment? If I measure in a single experiment, M plus is the state of the system. Ah, now, this is a very natural question if you believe in the traditional interpretation of probability, namely the sequential, from Maize's description of probability. And von Neumann, when he was doing his quantum mechanics book, said for quantum mechanics it is the same thing as for classical mechanics, and he put that in a footnote and left it alone. What he did not know, even von Neumann can make mistake, and what did von Neumann not know at the time, is that von Neumann was not mathematically directed. It took another work, certainly 20 years, of work for people to determine what the collective is, and in fact it is the birth of the mathematical theory of recursive function. The mathematical theorem here the function is a part of logic which was developed by logician like Church and also Church and this is extremely delicate and it makes it fairly difficult to check that you have a collective. Now this approach is a very nice aspect in it, that's the only approach in which I know how to define a random number.

42:30 And this is the 100th anniversary of Kolmogorov. Perhaps this is important to point out that this theory, not recursive function theory, but complacency theory, which is using many of the reasons that collect the recursive function theory, that means this theory of complexity is Kolmogorov. Now, depending on whether you are satisfied with decoyance, or whether you want to look at the collapse of the wave back at the distance, depends, I think, on whether you are satisfied with the definitive approach to probability or, if you insist, on a fundamental interpretation, sequential interpretation of quantum mechanics. As I've said, since Windermar was a second authority, everybody believes that, well, they can be in sequence and they are . Right, things are not fine. It's very difficult. and even you cannot pull out even in the nicest form you cannot pull out working gain by which the limit is approached by the same segment is approached on the same side so you should bet you always win but you win less and less so even this time so this is And this problem is still, I think, one of the open problems in quantum theory, not last in measurement theory, which is a proper interpretation of quantum probability, or probability in general, quantum or not. Is it a sequential interpretation, or is it an interpretation which is, like DFINITY, based on, essentially, information? Now, DFINITY has a main theorem which shows you that, in some sense, if you believe it is, you can recover from me.

45:00 This is the so-called Exchangibility Theorem. It's not quite that, but it is. Fellow presented, things like that, the Exchangibility Theorem. For quantum theories, there is an Exchangibility Theorem. Actually, there are several Exchangibility Theorem. Some of them are good, some of them are less good. And I think one of the speakers, I don't know if it is here, has talked about the Exchangibility Theorem. Okay, so we will listen to that. I spoke a little bit about the DFINITY program last conference. I must say that my own inclination as I can be told is to move more and more into the community approach to probability theory. It seems to me the one which satisfies the criticism of the old time when I want to say that something is probable and don't talk about the significance. I'm told about what I expect. And that's what the credibility is what I expect, is based on the knowledge that you have. And, well, this is what we're going to show now. Essentially, is about, I don't want to say that it is that, but it is just about this thing. And for the woman, therefore, we have a model which shows that in the usual form of statistics quantum mechanics, without having the cause quantum theory, like Belafkin did in the last conference. Belafkin had also a system in which you had this kind of measurement, but in the framework of the stochastic, his type of stochastic quantum mechanics in Belafkin. And I have not seen Robin Hudson, we don't know who to go to him, but I have not seen him either. to house about stochastic quantum mechanics so I don't use that the usual

47:30 framework is at the quantum mechanics with one distinction which is that I want to consider infinite system but since the 70s we know what statistical mechanics of infinite system is and we know that it describes thermodynamics, the aspects of thermodynamics, in particular pure thermodynamic double phases, and the coexistence of different thermodynamic double phases. And my impression is that this goes to the heart of the measuring process. Thank you. Thank you. Thank you. Thank you. Thank you.

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