11th UK Foundations of Physics Conference — Classical and Quantal Structures

0:00 Thank you. Welcome to our speaker for this session, this is Jeffrey Sewell, and I'm going to talk about the individual to investigate a clean structure of science. I'm going to thank the organisers for giving me this opportunity to express myself. Okay. Let me start by reminding you that, as I've decided by law in particular, by the early days of quantum mechanics, quantum theory has got any significant classical components. I'm going to see this in at least two ways. If you consider, for example, the continuation of the process, then the measuring apparatus and its actions have got to be described in purely classical terms. That's necessary in order, in those words, or the English version thereof presumably, in order that we may tell our friends what we've seen. Secondly, if we look at macroscopic systems in general, then one sees that these exhibit such classical properties as those of thermodynamics, hydrodynamics, etc. So you have underlying quantum mechanics looking eventually to classical behaviour at the macroscopic level. And these two, say, many particle systems, the underlying quantum dynamics and the resultant gross phenomenological classical dynamics, the interplay between these two, is really crucial to an understanding of quantum mechanics. Now, may I just say that when one is talking about macroscopic systems, or enormous disabilities of particles, one might ask, does one need a model in which you think of these systems as quasi-mechanical, even at the microscopic level? Why not, for example, use classical statistical mechanics?

2:30 The answer, I will give, there are many answers, but to my mind, one clinching argument is that without quantum mechanics, matter is unstable against collapse due to electromagnetic attractions. This was well established in the 1970s in a great theorem by Eliot-Leib and Walter-Tilling that in fact the Heisenberg and Pauli principles between them are both necessary and sufficient for stability against electromagnetic interactions. Therefore, we owe it to quantum mechanics at the microscopic level that the very stability of matter, and therefore of all of us, prevails. That's the argument that impresses me most. Now, what are my objectives in this talk? Just to give you a sketch of how a, in quotes, I'll say what I mean by that presently, of quantum mechanics can lead to a picture of both quantum and classical processes interplaying with one another, and in a way which is free of Bohr's dualism. The point of the dualism that I'm referring to is that in which you consider a system of interest on which you're going to apply measurements, which you describe quantum mechanically, and in both terms, a classical system that you have as the measuring apparatus. There is a dualism in that description. In the one that I am going to present, we do not have that. Now, we may ask what is the generic model, what is this version of quantum mechanics that avoids this dualism, and that I'm going to describe in a bit of a sketchy way.

5:00 This is an idealized model, and I emphasize, particularly in a philosophical gathering, that this is an idealization, an idealized model in which macroscopic systems and systems of n interacting particles with n huge, typically of the order of 10 to the greater fourth, is represented as an infinite system. In other words, what passes to a description in which end becomes infinite. Now, why adopt such lunacy? I mean, anybody in his right mind knows that this pen, for example, is finite. We're all finite. Why describe these objects, including ourselves? Well, we're going to come into the game. As infinite. The essential purpose of this is to exhibit sharply properties of large but finite systems that are otherwise masked by boundary and other finite size effects. A canonical example is that of phase transitions. If you apply standard statistical mechanics, it's rigorously proved that the thermodynamic potentials that you gain there from are all perfectly small functions of thermodynamic parameters, temperature, chemical potential, and so forth. You may have enormous gradients, but they won't have singularities. Therefore, if you want to characterise phase transitions by singularities, which represent sharp, qualitative properties rather than mere quantitative ones like large gradients, then you need to pass to a limit in which the number of particles of the system goes to infinity. But more than that, the passage or the formulation of the system

7:30 an infinite one or one with an infinite number of degrees of freedom has another huge gain and I can describe it as follows. If you look at a finite system in quantum mechanics then according to a theorem of There is only one irreducible representation of its observable, up to unitary equivalents, and therefore, the algebraic structure of the observables, say, governed by the canonical computation relations, or anti-computation, if you like, compare me on it, is faithfully represented by this unique Hilbert Space description. That state of affairs does not prevail when you go to infinite systems. What happens there is that you have an infinity, even an uncountable infinity, of inequivalent representations, inequivalent irreducible representations, etc. And these different representations carry qualitatively different properties. These different irreducible representations can, for example, exhibit such things as spontaneous symmetry breakdown that you could never get for a finite system. In other words, say for a ferromagnet, you can have equilibrium states with different orientations of the polarization in the absence of the magnetic field. Quite impossible for a finite system where there is a unique Gibbs state. So, you have these structures sharply revealed by the idealized infinite system model. Also, I would point out that if you're interested in irreversible processes, Then, when you pass to the infinite system description, you can eliminate quantum eyesight in a perfectly straightforward way. And so you get a decent description of the irreversibility. So, the model, the generic model that we have called sigma, it's a system of particles that occupies an infinitely extended space X.

10:00 See? This might be a Euclidean continuum, it might be a lattice. We denote by L the family of bounded open regions, lambda of X. And so for each lambda, each of these bounded regions, you have a set of particles inside lambda, which would be a subsystem of sigma, although this would interact with all the rest. And I call this system sigma lambda. Let us say that system, sigma lambda, would be here. The set of particles in lambda discounting interactions between the interior and the exterior of that region. Let me now sketch how to formulate a model and now let us say, let me take it for granted that we all know how to formulate a quantum, the model of a finite quantum mechanical system say, included, say, contained in lambda. So, for each such lambda, I formulate the algebra of, I formulate the observables, the states, and the dynamics of the system. This we know how to do, by standard methods. So, what we do is produce a lambda and formulate an algebra, a lambda, of the observables. I say the algebra of observables because the observables have got to get in an algebra. The structure in general is gathered by canonical or anti-computation or anti-computation relations. And the whole set of these algebras, the different bounded open regions, are constructed in a standard way, all this is given in these first two references, for example, well these are not the original works, and these are satisfying the obvious conditions

12:30 that if lambda is a subset of lambda prime then correspondingly a lambda is a subset of a lambda prime this is the isotonic condition and further on the other hand if lambda and lambda prime are disjoint by bounded open regions then the elements of those two algebras intercommutes. Now, with that set-up, I can define the union of all these local algebras. I call that A lambda. I might say that in order to make this a decent mathematical object, we equip it with a certain topology and I won't go into details about it, and we close it with respect to that topology. Let me not go into technical details about this. It's the Algebra of Observables, it's also equipped with a group of space translations, these are defined in a rather canonical way. A to AX, where X is displacement in the space capital X, whether a continuum or lattice, and this sends A lambda, the algebra A lambda, to its space translate A or O, the A of its space translate lambda plus X. So that's how we construct the observables. We just blend together the observables for each finite feature. What about the states? Please forget all the stuff you've ever heard about vectors and density matrices in the Hilbert's bit, at least for a moment. The raison d'être, the state, is always to give you expectation values of observables. And So, in this generalised form of quantum mechanics, we define state as positive, linear, normalised functionals on the algebra of observable. So, if rho is a state and A is an observable, rho of A is a number, which corresponds to the expectation value of A.

15:00 I might regard rho as a non-commutative probability measure, if you like, or quantum probability measure. But moreover, in order to make contact with quantum mechanics of a finite system, we demand that the restriction of rho to the observables in lambda does indeed correspond to a density matrix for this subsystem sigma lambda. Incidentally, there's more than just convention in this. It's necessary, at least for systems in a continuum, in order that you do not have a collapse of an infinite number of particles in a finite region. This is a classic theorem. They are not all Doplican and Lyon. So, it's more than just convention. There is real physics in this. So, locally, you have finiteness. Okay, now the dynamics. This is constructed always for the infinite system as a natural limit of that of finite system. So we constructed as follows. For each finite region, lambda, we construct the evil root at time t, at, lambda, for the state sigma lambda. This would be a Heisenberg operator for this local system, sigma lambda, and then we pass to a limit as lambda goes to, as lambda expands to cover the whole space, the infinite volume limit, and that would take to be the eva root of A. Why do I insist here on the functional description of states? Let me go back to this. As I said earlier, for an infinite system you have inequivalent representations. I don't know offhand what the Hilbert space is, so I couldn't define a state of the infinite

17:30 system as a density matrix or a vector in a Hilbert space, because I don't know which state it is, which space it would live in. In fact, in order to have, in order to nail my representation, I first read a global description, which really is induced by the state, and they, this is a famous theme of Gelfand, Neumann and Siegel, once I have the state, then I get the representation, And I make the remark here, that whereas for a finite system, one describes the system exclusively, the states exclusively, and the representation includes it exclusively in macroscopic terms, that situation does not prevail for influence systems. If you need a macroscopic and a microscopic description, the microscopic description is needed for your representation. The microscopic description is needed for the vector or density matrix within the representation. In fact, each representation carries just a family of states, if you like, functionals on the algebra, which are globally equivalent, macroscopically equivalent, but globally different ones. Now, I want to say something about macroscopic observables. I start by talking about global macroscopic observables. And these observables, typically, one can generalise this, but just for simplicity, let's just take the most natural ones. These are observables such as the space average, the full space average of the space translates to the local observables, defined in an obvious way. I would point out that these are classical observables, all of them.

20:00 This follows from local commutivity, which you remember, because if I have two observables for disjoint regions, then they inter-commute. And because of this, it's easy to prove that these are classical observables. They commute with one another and with all the local observables. So we have some classical stuff here already. I also want to say a word to you about hydrodynamical observables. Remember that when you formulate hydrodynamics, you talk about hydrodynamic points, but you know these are not points. A hydrodynamic point, so-called, is really a cell in space, which looks small, if you like, point-like, from the gross point standpoint, from the macroscopic standpoint, but is huge from the macroscopic standpoint. it's big enough to include an enormous number of particles. So, bearing that in mind, to describe these observables, resolve the space X into cells of sign L. Each of these cells large enough to contain enormous numbers of particles. A hydrodynamical point I take then to correspond to the cell CL of X, I put an underline of this X, centred at this X. Now, the hydrodynamical observables at the hydrodynamic point X are then the space average O and CX of the locally conserved quantities. these would be observables q of x which would be densities of mass charge etc. Let me see for the benefit of the cognoscente these would be operator value distribution without I won't go into details of this and we then look at these space averages in the limit where L looks to infinity this represents the fact that the second limit represents that they're huge from the macroscopic standpoint,

22:30 small from the microscopic standpoint, because L is of an order less than the size of the system. So, I have within this framework a scope for correlating with the microscopic and the macroscopic problems. Let me now just state some results that have been obtained about the picture you get from this. First thing I want to point out is the emergence of classical thermodynamics with phase structure. Face structure characterised not only by singularities in the thermodynamic potential, but also changes in symmetry. And I might say, in this new book of mine, I've also formulated the scheme for non-equilibrium thermodynamics, which is a non-linear generalisation of the famous Ansarge relations. So, one can see thermodynamics above both the equilibrium and the non-equilibrium species emerging from this picture by this interplay between the microscopic and the macroscopic observables. Secondly, I pointed out that the infinite system model gives you, or liberates you from the cycles and then offers you, thereby offers you the chance of getting irreversibility. From a description in which at the microscopic level you have reversibility. And family conditions, initial conditions are at the same point, I might say. But when I say one has irreversibility, I mean this. That if I look at the evolution of a system from a state at t equals nought,

25:00 and I look at the suitable subset of observables and look at their expectation values, Then, under appropriate viable conditions, these observables or their expectation values thereof, evolved according to an irreversible law. It can look like a diffusion law, etc. And lastly, I would like to point out that as shown in references 4 and 5, some Eulerian hydrodynamics has been derived for systems of interacting fermions. Now, in reference four, I prove this for a plasma model, the Coulomb interactions, fermions with Coulomb interactions, moving against and utilizing a positive background. In reference five, that's just a preprint by Nachachala and Yao, The Eulerian hydrodynamics was approved for systems of panrions with short-range interactions. That was quite a tool of a force, I might say. And now I want to go into a bit more detail about a scheme which I discuss in my new book, which exhibits the interplay between classical and quantum dynamics. Let me say that for certain models which we've analysed in some detail, you'll have the following set. Firstly, there's a classical space M, subset of a Euclidean space, whose points M, that I call them, represent values of macroscopic observables. And these rules, according to a deterministic law like this, dm dt is a function of mt, which can be integrated to a form mt's dtm,

27:30 where TTs form a one-parameter group, generally non-linear, of transformations of this macrospace. The star, equation star, is the semi-group property. Secondly, the local microscopic observables evolve according to a dynamics in which they're piloted That should be macroscopic, by the macroscopic ones. So the AT by BT is L of MT, linear operator depending on MT. MT is QTMA, and this integrates to a form AT is V of T and MA, and this V is also a linear transformation of the opposite. How much time have I? Now, well, I hope that you're a little bit generous, and it follows from this, that the BT has got to satisfy what is termed the co-cycle condition, which is this. I don't ask you to commit those two to memory, but let me just say there's the semi-group condition and there's the prosanical condition. Now, I want to finally look at the fruits of that setup, because this gives us a new picture, which in a recent paper with Ratty and Sen will say more about this, we showed the problem. This picture that I've just described can be expressed in terms of fibre bundles. Writing will tell you more about fibre bundles, but let me just give you a simple idea of it. If I define the space B, which is just the Cartesian product of N, the macrospace, and A, the microspace, if you like, you think of N as the base space, a classical space. And at each point M of this script M, you have what is termed a fibre, which is A.

30:00 And you mustn't think of it as a one-dimensional thing like a real fibre. It can be of infinite dimensionality, like the algebra of observables. It's a big one. But each point of this base space carries a fibre. Now, according to the results that I quoted on the previous transparency, if I call the points of this bundle, as it's called B, B is a pair of objects, a macro-observable, a value for the macro-observables, and the micro-observable. And it's EvoLoop takes this form, and I call ST, this defines transformation ST of B. Now I make this one. It follows from those two formulae that I've written down, stars and two stars, that this S is also a semi-group, as far as this semi-group problem. And therefore, I conclude that the macro-microdynamics, which is given by this bundle dynamics, is just given by this one-parameter semigroup of transformations of this fiberbundle. I apologise for running over time. Thank you for your attention. I understand that you have a general framework, but you know it is difficult to have an example which goes beyond the key field. Therefore, to what extent do you have a model which realises this structure? A good model is a laser model, and this is a highly non-linear model, and it is a minimal generalization of the old Dicke model, of the Dicke-Hep-Lied model.

32:30 It's a generalization that was formulated in terms of quantum dynamical semigroups that's by Giovanni Alibi and myself. And so, do I have more questions? Thanks. Thanks, Peter. So the next slide is by Alton Sennett and by the Omniversum. We'll find all of them. Jeffrey's hearing reform is a hard act to follow, so let me begin with the Kamotsara remark that my thought will not be quite as bad as this initial condition would make it seem. Now, if we saw these two equations and in a different notation, they describe one, an x-substitute t on the space x, which we call the A space, and the motion y, so it's yt, on capital Y, that is piloted by the flow x-based t on x. Now, the state space of this dynamic system may be... Could you stand at the other side of the screen? Oh, yes, I could. I'm sorry, I'm... I read my language. Now, the state space of this dynamic system may be due as fibrobundle. I would like to explain briefly for those who don't know, and with apologies to those who do, what a fibrobundle is, because in an earlier talk, for example, we heard about super-selection rules, and you may remember that in the very first

35:00 paper, the paper of which introduced the there was a statement about continuous superselection rules being the most perhaps the most interesting but also the most difficult to handle fiber bundles among other things provide a very natural way of handling continuous superselection rules and they may have other nice properties as well what I would like to do is explain what a fiber bundle is, and I will not try to explain it by this notion that it will be useful in physics in a safe time. And let's illustrate this above by describing elementary excitations in superfluidemia in terms of connections on fiber bundles. Now, Jeffrey's talk concentrated basically on the dynamics, and what I'm going to do will concentrate more on the symmetries. Now, this separation of aspects of the physical system into the dynamical aspects and the symmetry aspects may be artificial but it's well known to everybody why it is done, what its limitations are and so on. So, I will not go into this. Now, what I should like to add for those who are interested in the mathematical aspect of the question, that there exists a fairly satisfactory and general theory of symmetries with group actions on vector bundles. The corresponding theory of dynamics on vector bundles is still in its infancy. In fact, apart from this observation made by Jeffrey, there is nothing. And the reason is that if you approach the problem generally, you'll see that you have to find the infinitesimal generator of one parameter proof of maps of a fiber bundle onto itself. Now, supposing this fiber bundle is a Hilbert bundle, a bundle of Hilbert spaces, then this infinitesimal generator will have to act as a unitary, as a separate joint operator on the fibers and a vector field on the bases. And so far, there is no mathematical receptor in which these two can be combined in one.

37:30 The result will be a combined stone-written theorem which nobody has done. Now, well, the notion of a fiber bundle arose in mathematics as the generalization of the topological product. You see, a fiber bundle is locally, but not globally, not necessarily globally a product. And, well, the spaces involved may also carry additional structures, other than the topological ones. And, of course, when we talk about maps between subjects, these objects, they will have to preserve these extra structures as well. Now, I'll give you two examples. One is a cylinder. This is just a product of S1, the circle in one dimension, times the interval, real interval minus 1 to 1. And the other is the merpheus band, which is locally the product of a real interval times this close interval. But globally, it differs from S1, from this cylinder. Now, I tried to show it. I tried to make a couple of pictures. My intentions were honorable. Alright, I'll explain in a moment. See, the top picture is just a strip of paper. To get a cylinder from this strip of paper, you just identify it. It's clear what you're doing. Now, the last, I tried to represent a Möbius strip, and when I looked at this miserable object that I had produced, I remembered that 58 years ago, I had to choose between two years of art, Sanskrit, or Persian, and wisely, I chose two years of Sanskrit. However, I had forgotten that when I started launching this enterprise, my apologies. Everybody knows what a new industry looks like. Now, to describe the class of maps of interest called bundle maps, we introduce some relations and they, well, I will try to, well, some stuff is written on the transparency, but I'll try to be a little bit more intuitive.

40:00 There is basically a total, something that is called global space, which is a bundle, which is a bunch of fighters, whatever else it's called. And the interesting thing is that there's also another space which comes together with it. little picture that Jeffrey had drawn. He had drawn the base space and probably one fiber was trying to stick out, you see. And then there is a map called the projection which projects every fiber onto one single point of the base space, you see. Now, okay, this fiber, oh yes, it's a very important distinction. The point is that in fiber bundles every fiber is required to be homeomorphic to every other fiber. Okay, now, the point is that the gluing together, how do you glue together these various fibers, you can ask for, to form this total space, using the base space. That's where the trick comes in, and that's, I will skip over some of the details that are written here, because those who are familiar with it will not know it, where will not need it, and those who are not familiar will probably not be able to absorb it geographically enough. So what happens is that a group comes in. You see, in the case of the movie strip, just imagine that. Imagine the fibers as, say, the real light, and the upper direction being positive, the upper direction being negative. And when you twist it and glue it together, You see, what you're doing is that you're gluing the negative axis onto the positive axis and vice versa. And that requires a little bit of trickery and that is managed with what is called the group of the bundle. However, it's because of this trickery that the mathematicians are interested in it. If the fiber bundle is locally a product, but globally it is essentially a different product. However, in the cases that we are interested in, our fibers will be either input spaces or C-star algebras,

42:30 and our bases will be specific majors, and because of good mathematical reasons that I will not go into, all these bundles will turn out to be products. You may think of my Hilbert bundles as a Hilbert space, as a topological product of a Hilbert space and a base space. Now, so therefore, what we are interested in is not so much the departure from the global product structure, very interesting is that the different fibers, essentially distinct inverse spaces, we might call distinct super selection sectors, are woven together, glued together in a very nice way by this topological product. So, I'm moving this rather fast. Now, here, I think I would have to take time to be a little bit more precise. So, what I want is construct this object which I call B, which is the topological product of a separable interdimensional Hilbert space over the complex numbers, and a certain base space, which is a, let us say, a finite dimensional dimensional manifold, and the fact that every Hilbert bundle can be, or almost every Hilbert bundle can be brought down in this form, I'm not going into that. What is, what I will speak to is this decomposition. I'll assume that this work with objects of this kind. Now, so what I want to do, you see, in ordinary group representations, you have a Hilbert space and you want to represent the group on it. What I have here is instead of the Hilbert space, I have a product of a Hilbert space with a base manifold and I want to represent the group on it. So, well, let's g, big g is a group element, and let me write, the little phi is a vector in the Hilbert space, and x is a point on the base manifold, so, and little b is, of course, this pair, the inner point of the bundle. So, g acting on b is g acting on x and phi, g is, of course, an element of the group.

45:00 Now, how will it act? Well, first of all, you put it in, it acts on G, it's a manifold, it's a standard action on the manifolds, which you write as GX. And then, you pull it through, it'll have to act as a unitary operator on the file, but this unitary operator will now depend not only on the base, not only on the group element, but also on the point X through which it is pulled, you see. Now, after that, what you do is, I mean, the source of all information in group representations, practically all algebraic information, is the associativity of multiplication. And from that, you, we get this second information, which is what Jeffrey called the cocyclic rendition. And then there are essentially some division renditions, which I'll just skip over. Okay, the first is that it is obvious that it, and the second is that it comes from the associativity of multiplication, and then they are consuming ambition effects. All right. Now, the point is that we have essentially to solve that co-cycle condition. You see, find a way of determining the little u that I've shown on the u which depends on x and g, so that it satisfies the requirement that follows from the associative multiplication. Now this construction was given by Wiener in his paper on the infant eventual representation of the Lawrence Group, I think in 1939, right? And interestingly enough, it was rediscovered about two or three years later by Hasler Whitney in his basic papers on differential manifolds and fiber bundles. So, of course, Whitney had no idea who Wigner wanted to do this. So, now this, look, I'm just going to flash it through, this Kiddler construction again, because those who know it, I mean, will recognize it from the NGC construction of Wigner,

47:30 and those who are not familiar with it will not see it. But anyway, there is a standard way in which we can define, what is the term that Wigner used? The little group method or the stabilizer. This is just the little group method. We solve this equation by the little group method and then choose the representation of the little group and stop at that stage. that the full inducing construction of Wigner or Mackey afterwards, then you fold the whole thing over into a Hilbert space again, and then you get the induced representation. But if you stop at that one mile short, you get a representation, or you get an action on a bundle, on this bundle that I'm still. Okay, there are some mathematical conditions that I think are, Now, this is the general formalism which is actually only very slightly different from the inducing construction. And now, what I'm going to do is to give a particular case. Now, although I'm mentioning for simplicity, I'm talking about, I'll talk about the LA group. The construction whole, you've done this for a semi-group as well, and Jeffrey was talking about a semi-group. It's just that I had already written these things down, so I was not able to adapt it in real time to a semi-group in this case. It's in the paper there. So now again, this is a bit messy, but the formula are complicated, or somewhat complicated, so I won't try to explain that in detail. But what happens is the following, that, well, consider a situation that is roughly described as spontaneous breakdown of symmetry, you have an infinite system and you want to boost it, how can you do it? You can't do it on a Hilbert space because in order to implement a boost on an infinite system, you'd have to give it an infinite amount of energy and the loose physical argument grants that there's no unitary operator that will do the job for you. Well, we can give a better argument, but that's more or less it.

50:00 So what you want to do is, if you wanted to describe this as the action of the Galilee group on an infinite system, you can no longer work with the Hilbert space. But what you can implement on the Hilbert space are time translations, space translations, rotations, but not the boosts. So what you do is you choose a Hilbert Bantu, in which you decide to implement the the entire translations, base translations, and the replications on the fibers, and you implement the boosts on the base base of the Bantu. Now, this is the multiplication law for the Galen A group, written out in full. V is a space translation, sorry, A and V are vectors, they're space translation, then velocity boost, and R is a rotation. And this is the minimum multiplication law. Now, you see, you may ask, why should I try to represent this thing on a bundle and why not just represent the group on a Hilbert space and see what happens. Well, what happened, this is what Emanuel and Wigman tried in 1952, and what they found that in an irreducible input space representation of this group, p squared is a constant. This is one of the Casimir invariants. And this, roughly speaking, means that there are not enough delta functions available to form, you know, there are not enough momenta available to form a delta function in position states. So you cannot have any localizable state and we went one step further, they tried to form localizable states of localizable velocity, of constant velocity as in the case of photons and found that in that could not be done. So they said that you cannot do anything about it, I mean these representations are useless. The only representations of the Galilei that are of any interest are those that were discussed by the department named the projective representation, the mask. Ok, however, if you look at Bandhu representation, the situation changes.

52:30 So, this is the space of the boosts and you can parameterize by this boost vector and the gallery group will act upon the boost in this way. You see W, only the V and the rotation, the boost itself and the rotation will have any effect on this part and the whole thing Now, forget the rest of the stuff, the processes and things like that, the details can be done in the paper. It's very straightforward. Now, what you can do is you have these processes. But choose a representation of the little group, here the rotations, the time translations and the space translations, on the Hilbert space of the following form, you see. Now, and this representation has certain properties. So let us list these properties. First of all, it is reducible. It contains every e squared with the same weight. Now, if we use this representation, we obtain this particular bundle representation with a formula, and where this capital lambda is again given by another formula, fairly complicated, and to see what it really means physically, we look at the rest fiber W to 0 and apply just a boost and a time translation. So what happens, you see, then you see that the boost will of course take the rest fiber to the fiber V, and the time translation will give you this factor experimental phase. Now, this essentially shows that under a boost V, the energy and the momentum, well, use the term energy and momentum for lack of better ones for the moment, transform in the following way. You see, E goes to E plus P dot P and P goes to states constant. Now these are

55:00 precisely the transformation properties that Landau postulated for his exudation in superfluity So the conclusion is that under these conditions what we have really obtained a non-relativistic zero-mass system, which describes pretty well Landau's excitations in superfluity airport. Now, of course, they will immediately say, some of them will say, but you know, these excitations, As you go away from the peak, the excitation becomes highly unstable and essentially in the neutron scattering experiments it's here only a little bit. Well yes, fair enough, we can rectify that by going over to the semi-group instead of the group. You know, we don't represent the entire type translation in both directions, only one direction. And then you have a width and then that part is also taken care of. Now, observe that a dispersion law is allowed because the boosts are not implemented on the fibers. They are implemented on the base states. You see, imagine this, this situation. No, okay, leave it, sorry. Right, from this, the conclusion is a non-relativistic zero-mass quantum mechanical system can be defined on a integral bundle, but it cannot be defined on a... Oh, I'm sorry, I have a bit more. I have forgotten that I have also written this thing out. Sorry, I am in the throes of, not really a menopause, but almost an anopause. And therefore, in some sense, I don't know what I'm doing. Well, you see, I've written down the representation of the same group over here, you see that, and the last relation properties and the boosts, and that's the thing of interest, you see. It's just the width is unchanged.

57:30 There are some references over here, as you see, they are pretty ancient, these references are quite readable. And this is a reference number three is a basic, essentially the full mathematical theory is brought together in a fairly elegant way in this reference. And reference number five is a very good reference for the Galilei rule and its representation whatever you want to do and leave it across. It's for 2002 or 2001. It's 2002. I was going to correct it. I made a mistake there. We did the work in 2001 but it was published in 2002. Well, thank you very much. I mean, I hope I'm not, I mean, my job was not as good as Jeffrey. Just two related questions, really. I wasn't quite understanding. What is the base manifold M? Is a point of it a set of values for observables, as in Jeffrey Sewell's talk, and related question, you seem to implement rotations and space and time translations up in the fibre, but is there a physical reason not to implement them at all in the base manifold? It looks as if macro observables might be altered in value by a spatial rotation. Well, I mean, in these references, reference number two, I mean, the whole zoo is listed. The reason why I picked this one up is it seems physically one of the most interesting, you see. And here, of course, what is the observable on the base manifold is the velocity of the reference plane of the board. Only that. Only that. In this model.

1:00:00 Some people like HEP, for example, advocate the use of classic observables to solve the measurement problem, if you take a classic observable to be the point observable. Now, in that case, it seems that it would be necessary that the macrodynamics is somehow piloted by the micro state of the system to be observed. obviously or it seems that something like that would not fit into your framework because in your framework the micro dynamic is piloted by the macro dynamic would there be any models or any generalization of the framework where these macro dynamics are possible yes I don't see that any basic difficulty you see just even when the mid It's conjectural. If I can find slide number two. Well, you see, you can ask the, you know, reverse the role of the fibers and the base space in this case, mathematically, you see, and let the motion on the fibers pilot that on the base space. Now, again, I mean, you see, the reason why we can speak with some confidence at this level is because Germany has this very good model which works, and there you can see what is going on. Now, if you try to do what I said just now, you will either have to produce a convincing model or a general theory, a general Stone-Whitney theorem, generalized Stone-Whitney theorem. Now, if you can produce a generalized Stone-Whitney theorem, well, you'll have a mathematical theorem so that these things can be done, whether they are of physical interest is a different question. You see? That I cannot answer. But before, or at the very least, you'll have to produce a convincing, a good, I mean, sufficiently complicated model. I mean, that's all I can say. Okay, thanks. I was wondering, is there any possibility of either raising the screen or dispensing

1:02:30 with it and using the wall to benefit if those of them sit right here and can't see the bottom of the transferences? The wall goes higher. Can I see stuff on the wall? I don't think. I don't see what I'm on. Please take a check. the chairman of the session here tomorrow. It's probably all ready. In this last session, there's more space at the front, so you might want to... If you take the screen away, the bottom already comes down. The last speaker of the session is David from the University. He takes the screen away. The bottom line of transparency will be even lower. Thank you. First question, Copenhagen interpretation holds, the extension is not anywhere beyond that. So quantum measurement theory is the usual, and the extension that I want to talk about is involved with this one piece of jargon, POVM, Positive Operator Value Measures. Normally when you do quantum mechanics, when we're talking about analyzing the properties of observables in a given state, you use spectral theory and in general you will have to use the spectral theorem which will involve projection value measures whose expectation values are measures which give you the probability distribution in any given state now for various reasons some people have suggested that you could generalize this by looking at measures which come from positive operators positivity is necessary for probabilities but the projection is done away with. There is a mathematical theory which deals with that.

1:05:00 It's involved with what's called a non-ethical resolution of the identity. In this particular case, because of the phenomenon I want to show you, a strange, curious phenomenon, outcome, we will be talking about systems on the circle t. Work is in progress about extending it to r. But a number of technical issues are bypassed in doing this. So we want to resolve the identity operator on functions in L2 of t. So a resolution of identity is a collection of maps f of t. t belongs to... So each ft is bounded, the difference will be bounded, and this difference is positive when t is greater than s. This, by the way, would be similar for the ordinary or phagonal resolutions of the identity if you added in this last condition. This last condition is not required. The projection or a phagonality condition. So we do away with that. So, correspondingly, we can do this in terms of measurements. Now, V, V sub n, and such are subsets of T, or L-substance to detect it. The first condition is positivity. The second condition is the measure theory. The last condition is normalization, which will preserve probability. And the connection between the resolution of the identity and a positive operator value measure is that you take the difference, Ft minus Fs, and that is F confined to the interval ST. This is standard measure theory. And you don't have to know very much about that to know what follows.

1:07:30 But I thought it would be probably as well to write down all these conditions. So if you know spectral theory, then this is what you use without the projection value of measure. without the projection problem. So, where do the quantum operators come from? Here is an operator formulation. If you have a projection value, a positive operator value measure, you can find its first moment. And that will be, in this particular case, is A, it will define a symmetric operator. Which, because everything is bounded in this case, because T is a closed and bounded interval, will be self-injoint. You can define, in this case also, all the higher moments. But because you have done away with it, the orthogonality, the projection property, the k-th moment is not equal to the k-th power of the operator. So you do not have a spectral calculus. Everything is well-defined, but they're not equal. So you can define variance in the usual sense for a sub f, and you can define variance in an obvious sort of way using the measure. and the interpretation of what is going on comes from essentially looking at that. The uncertainty that you obtain from a positive operator value measure is always greater than the uncertainty for the operator that comes from its first moment. So in a sense, by relinquishing orthogonality, you have a non-optimal measurement scheme. less quantum information. You can imagine this positive operator value measure as being an arrangement to measure a sub f and it's an unoptimal arrangement in general. So if you if you use a positive operator value measure you lose some

1:10:00 sort of quantum information. I use the word information in an ordinary sense not anything technical it's just some things you don't know an operator has many positive any operator can have many positive operator value measured representations this is due to the theory of Neumann you take this operating symmetric operator you put it in lots of different Hildred spaces by extending the Hildred space you could double it up you could treble it up you could tensor it to something else you do spectral theory in the larger space you can check original one, and for every different thing you've done, you get a different positive operator value measure. But the usual spectral measure, because A is self-adjoint, is unique. It is the only projection value measure of the unitary equivalence. So you have the usual quantum mechanical scheme involving spectral theory, and you have others which are non-optimal measurements obtained by relinquishing orthogonality. Yet some people said, including Brian Davison might add, that a positive operator value measure is an observable. Now in some cases, in a sense, you have to do that. If you had a torus, an inner tube, and you had a particle bouncing around reflecting off the walls inside this tube, there would not be a single operator You would not have a spectral field, but you could have a positive operator value measure inside. It might be possible to break up this inner tube in a manifold sense through a number of vertical and flat regions and piece them together. No one is looking for that. My, well Mark Hennings and I, as opposed to our two other authors, believe in operators. our two other authors believe in positive operator value measures and in a sequence of papers and which we were referees for each other came to this this example which we present to the community at large it pays your money it takes your choice we will show you something interesting I think and unexpected that comes out and it has to do with putting symmetry in. The number operator in ordinary quantum mechanics generates what was called gauge

1:12:30 transformations. I forget of which kind, Dune, which suggests a reference line And you could talk about positive operator value measures for which you have this translation, this covariance. So I mean, not all positive operator value measures, when you took the questions, all the function information associated with the region v plus a are nicely related to those in v by such an equation. So this pulls out a class, a subclass, of POVMs, which are gauge covariant. And the set of all of them I call gc. And this is modulo 2 pi, of course, plus n, plus n, the number operator, by 0, 2 of t, i d d theta, for example. You could do this in R, by the way, with support in a compact interval and get the same results. For those of you who are followers of the work of Mackie, this is a system of imprimitivity, but a generalized one because this is not a rejection, it's a POV one. A generalized system of imprimitivity. It is discussed in not precisely this way in the book of Haleva and in the book on open systems by Brian Davis. The point about gauge covariant POVMs is that you can characterize them in a rather neat way. So, I mean, I don't tell you where this comes from. It comes from taking major settlements. I won't go through it. but this is where the mechanization comes from, so this looks rather ad-hoc. pdf is positive definite. Why there's a dot between the d and f? I don't know, maybe. Positive definite sequences of a particular kind. You might call them matrices, but I want to stay away from them because I'm not actually doing any matrices. It's just doubly indexed real sequences, indexed from zero. The diagonals are one, and every minor, so that much matrix, is a positive definite matrix.

1:15:00 That's a particular class of doubly-indexed sequences. And it turns out that every gauge covariant, POVM, is equivalent to such a sequence, or matrix if you will, conversely. This was essentially the work of our two other authors. You can have an explicit formula for going from a sequence to a positive operator value measure to gauge covariance and conversely and their inverse. And if you put in topology it's continuous transformation. These two things are all the same. So if you have an F, you have this formula to find the sequence. HM and HN are the E to the IM theta. If you're doing it on L2 of T, they would be the Hermite Gauss functions on L2 of R. And going the other way, you have a CMN, you use HM, I don't know what this came out funny, but HM, HM, this Ket bar quantity. this is the Fourier transform, the characteristic function of the volume V and so there is a technical formula to go from one to the other. You don't have to know that to understand what's happening but we have a good angle on gauge covariant positive operator value measures. Now, the other thing is that these things, whichever way you do it, can be stated in

1:17:30 operator terms and it's for those who know the usual quantum mechanics it would be interesting to know. So, if you start with an operator A and it satisfies this commutator identity, so it's really the commutator of A with N, but N is unbounded so you have to take smooth wave functions F, so F of C infinity on the problem of T, and it satisfies this technical rule which you get from playing around with the pencil paper, but then it turns out that there is a positive operative value measure, which is gauge covariant, whose first moment would be A, but it misses by a certain function, an arbitrary function of A, which is bounded. So it's almost. However, and then you can relate it also to the sequence. However, if A is such that its diagonal matrix elements a is equal to a f and there are lots of those. This just seems like a mathematical toy. However, think of a as a quantum mechanical quantity being measured. On the one hand you have a and you do the usual spectral theory and you get all the quantum information. On the other hand you look at the positive operating value, the positive operating value measure f which has a higher uncertainty and yet because of this by measuring F and mathematically constructing AF you get all the quantum information so if you're prepared to do your mathematics A and F are equivalent in terms of the quantum information you can extract from experiments which is not... which is new, I would say, and very surprising. And the reason it happens is because you have covariance. That is a strong condition. So for a large class of operators, you have you have a positive operating value measure, not quantum information. Why pi? Why pi? Why pi? Why pi? Why pi? Why pi? Why pi? Why pi? Why 2 pi? Why 2 pi and the Fourier transforms?

1:20:00 That's where it comes from, more or less. It's a normalization. Now, a curious example, says he. As curious as Alice, we're curious enough. And this is not obvious, it just happened that we fell over this because the sequence is the simplest one you can think of. You choose two positive integers and a complex number of modulus less than or equal to one and you fix it. And from that you form a positive operating value measure which is gauge covariant as so. take basically the size of V, for V, and P of S and T, remember, are the Dirac-Ket-Bras for HS and HT. So you form this. And this is a definite positive operator value measure. You can show that it's gauge covariate, you can show it's not a projection, and you can show that it's uncertainties are non-optimal. At the same time you can do the mathematics as before and construct its first moment which turns out to be the right and its first moment is rather interesting he has three eigenvalues only that's his full spectrum here's one is the other is the third these are certain one-dimensional projections so these are non-degenerate which are formed from PS and PZB the complicated formula and And this is the rest. This is the identity minus the sum of those, and so it's infinitely degenerate. So what do you have? You have an observable with three eigenvalues, which has an equivalent, a quantum equivalent representation in terms of a continuum. That's what spectra can mean. Cage covariant. Isn't that strange? Let's, let's, let's see, let's figure this down, if we can. The quantum data from AF and F are equivalent and optimal. Yet their spectral properties are rather different. one is zero two pi and the other is three eigenvalues one has symmetry and the

1:22:30 other doesn't you would think if I got you a coffee and I said if a system shows symmetry in an experiment does it have symmetry say that's a tautology that obviously so and strong and conversely but it's not so this is the one quality which in every state shows symmetry in one way of making an experimental arrangement and not in another and the quantum information you extract from both turns out to be identical. Moreover AF, because it has three eigenvalues, you can, subject to technology, measure it perfectly. Three isolated eigenvalues. There's no question of continuum. The bell rings if you your restaurant, a good enough machine instrument. If it's in one cell or something, the bell rings, the other, perfect measurement. On the other hand, if you measure it by F, you have the continuum, zero to two pi, the usual sorts of problems. There's always a little bit of an error in measurement in principle. Since, because of the simplicity of A, it's perfect, it's strictly repeatable. If you measure it, it collapses. Sorry, I shouldn't say collapse here, it's dangerous. But you keep on repeating the experiment, after the first one you keep getting the same answer. But as is well known, if there's a continuous spectrum, that's not true for the F measurement. So you have two completely different kinds of measurements which give you identical results for quantum matter. You couldn't distinguish from the answers you got when you ordered your data. So we at least found this rather strange. So it brings us back to the question of what you believe is observable in quantum mechanics. Which is more fundamental? POVMs are operators. I believe in operating, but here's a POVM which gives me the same results. Some people believe in POVMs and they'd say that this system has gauge symmetry, but I could measure it without gauge symmetry. Much simpler and more accurate. Question, is there an ideal quality, platonic I guess you could say, observable,

1:25:00 which is represented by a self-adjoint operator independent of the way it is measured? In fact, some do, some don't. Or, must we include the experimental arrangements in the definition of observable as POVM theorists would have us? Because when you say the observable is F, it corresponds to an experimental arrangement to measure an operator imperfectly. In that sense, if a PLPM is the observable, and it's more than just language, if there's meaning in there, then they are including the experimental arrangements in their definition of the observable. So these are the questions that individuals who work with the meaning of quantum mechanics have to answer for themselves. And this example is interesting, because in spite of the loss of variance, in one case, and the loss of symmetry, the two different possibilities, you get the same information. I mean I completely share your feelings about Larty and Helen Parr and Bush and so on in just identifying nice guns but I mean I absolutely agree with your criticism of them you identified yourself as an operator man and I'm wondering just what you'd say about the concept You have to make sure what you want. If by quantum phase you mean the phase of a laser and you don't mean it heuristically but you mean it dynamically, you have to take the electromagnetic field and have to go into a cavity and do its magic and have coherent radiation come out and nobody can do that in full generality but the laser model which Jeffrey referred to you can actually do that you can trace through the various candidate phase operators in actually constructing coherent radiation from a finite number of loads of the field and you discover something interesting there is only one of those candidates which after the magic is done

1:27:30 Depends only on, determines the phase of the emergent classical radiation. And that is the angle, the vital quantization of the angle function of phase space. All the others, the so-called phase operators, when you put it through this operational formalism, end up depending on the intensity of the beam as well. although there's a theorem which says as the intensity goes to infinity it converges. That's what I think. I think you can actually prove that these things the Turpin's operator, the Bogdan Siegel operator, all those are not strictly phase operators. Just because you have an angle zero to two pi doesn't mean it's the phase of a laser. You're saying there is an operator identical. It's not canonical. It's nearly Yes, it's the quantization, the vile quantization of the angle function phase-space. It's about as canonical as you can get, and in the one model which comes close to describing the formation, if you might say, the generation of coherent radiation, it gives the phase of the radiation, and it's the only one which does. These are like gauge transformations, not phases. What does phase mean? No, no, it's a very good question. Sorry, I don't want to take that. I'll say something else later. I just wonder what kind of a physical quantity would have the number operator as its generator of its covariance equation that was the beginning. What did I have to say? Well, it started off with these POVM people looking at, I guess following from the work of Brian, that you should look at groups acting on the support space of these POVMs and the natural groups of translations. And so if it's supported on a compact interval like 0, 2, 5, you would have that. And then it turns out that if you do it on L2 of t, or you did that on L2 of r, the generator is the number operator.

1:30:00 Physically, you're asking me what this is? I wish I knew. The thing that prevents you from knowing precisely what that is, by the way, is that nobody knows how to pull a Turplitz operator from the hardy space of the circle back to L2 of R. Way back when people started looking at the phase problem again and realized that you couldn't identify what the Turplitz operator was as a quantum quality, which is really associated with this question, We don't know what those things are. They're just self-adjoint operators and so they're bona fide quantum observables and that's but what kind of a lab arrangement would correspond to this? I don't know. Do you know how many operators we know the lab arrangements to measure? And conversely was it three or four? position momentum some energy and momentum after that you know things things get very difficult so I can't answer your question but I'd be very surprised if anyone knows if they do I would be extremely pleased to know it can I just short corollary why call them gauge covariate rather than just covariate history because history because the transformation you know the original the meaning of gauge is Hermann Weyl introduced this in general relativity and then there was this is an Einstein said that the clock will depend on the path and so everybody got rid of that and then some years later in terms of wave functions Fritz London pointed out that e to the i theta acting against the wave function would lead to the in modern language lead to the same state had the corresponding diagonal matrix that would receive the I think this would cancel. And he called that gauge transformation. So you have to ask him why you call that gauge transformation. Yeah, but I mean, the general idea now is that it's a transformation that doesn't alter the physical state. Ah, this is not invariant, this is covariant. Right.

1:32:30 So, I mean, it seems to me that it would be clearer if you simply call it covariant as some, right. Rather than gauge covariant? The reason that I insisted on calling the Gage Cobert when we took our work and we wrote a didactic paper which is in press, the Gage was I wanted to ambush the possibility of calling it a phase as in phase operator without saying what phase it is. If you're going to use a word, an old quantum word, you might have Gage was earlier. What it is, it's translations, it's translations, and you support it on 0, 2, 5, it's no more, no less, just a group, a particular group. But there isn't a gauge field, that's... That's the... It's not what you would call multiple gauge transformations. Oh, one would love to do e to the i theta of x properly. That's the thing that... Yes, oh yes, oh yes. Big prizes await you if you can do that. Maybe that the origin of the UNLOG is irrelevant from the distant point of view. So this is the reason why we keep calling that gates. There is no physics. It's just a word. So we understand what we're actually talking about, but we might be looking for the proper way to say it. I would be perfectly happy with translationally and simply, I would agree with you, with COVID-19, yes. OK, we'll close this morning.