Vague objects and identity in quantum theory

0:00 Thank you. Is that OK? Can I put it closer? Uh, that looks fine. I don't think we've got time as well. Yeah. So we're going to take this one. So we're going to take this one. Ladies and gentlemen, I'm sorry about those delays, so from now on I hope the college gets the ball in. And I'm also bothered about the late start to our series, not the first week in term. It's a great pleasure to have a local friend and colleague to many of us giving our talk. I would like to say out loud, many of you will know this, it's also a very sad week in analytic philosophy, because one of the great men in the subject, David Lewis died on Sunday in Princeton, and I will send you, if you're on the email list, a URL for Princeton University's account of his life, which is on their website, including wonderful things said by Mark Johnston and Paul Nassar, about what a wonderful person he was. And as Johnston puts it, the greatest systematic metaphysician since life, which I think a very strong case could be made. So it's a sad week in arithmetic philosophy. That said, let's turn to philosophy. And welcome, David. Thank you very much, David, for giving a talk on the emergence of localised particles in QFP. Thank you. Okay, quick apology before I start. If you're talking about a nation which is a field theory, there's, as a lot of people know, there's a sort of range you can go, you can cover in terms of how careful and how ambiguous you are in mathematics.

2:30 The mathematical rigor in this talk is probably best described as virgin or non-existent. so there are going to be infinities left right at the centre which I'm not really going to draw attention to anyone who thinks that that brings up relative finations and I think we'll have to talk about that more in the discussion at the end I think it's easier to leave that sort of thing out of the way we talk anyway what is a quantum theory of fields let's talk about that in the context of what's a quantum theory of particles put up my hand draft sorry if you turn and close that on the window behind you on both sides that's a lot better if you've got a classical particle theory it's going to be something where the system's configuration is going to be described by some point in three dimensional space and the space of all such configurations is just going to be more of three dimensional space in going from a classical to a quantum theory we go from describing the system by a point in the space to describing the system by a complex function on the space, the wave function. And how do we make sense of that wave function? Well, in a number of ways but one of the most direct ways to get a grip on it is, imagine we try to measure the location of the particle then the probability of us finding it in some given region D of space is the integral of the squared modulus of the wave function over that region D. Is this readable? And maybe so you could close that up because in the course of time it will get drift across. Okay. So, as I say, staying very nice in mathematics, essentially we can say pretty much the same thing is going on in constructing a quantum theory of fields. In the classical theory, instead of the state of the classical system being, or configuration being, a pointed space, it's some sort of field, some sort of real function of space, maybe required to be smooth in some way, but basically a function of the whole space, therefore carrying a hell of a lot more information than a single particle point. And similarly, in doing quantum mechanics, we need some sort of wave function, and the wave function will be a function on all the possible configurations, that is to say, it's a function of all the possible functions in space. So, it's a function on a very large dimensional space, and clearly a much larger, more complicated entity than the quantum state of quantum mechanics. And

5:00 again, how do we make sense of that state? There are a number of ways, and because the system is back to a complex, there are many more ways of interpreting it than it's possible again it does make sense at least conceptually to say we imagine making some measurement to pick up some properties of what the field configuration is we might imagine selecting some subset d of possible field configurations and again the probability of finding the system to be in one of those configurations is the integral over this sort of potentially interdimensional subset of possible field configurations of the mod squared wave function to give an example of what I meant about being mathematically fairly relaxed, this interval written here is a very difficult thing to define technically, and trying to define and get into all sorts of issues of renormalisation theory and such like which I'm glossing over the first of the two are. So anyway, notionally speaking, we've got a picture of quantum field theory which is very closely known as quantum mechanics, and also makes it fairly transparent why we say it's a quantum field theory. It's a quantum field theory in the same way that quantum mechanics is a quantum particle theory. It's that the configuration space of the system is a configuration space not of particle points but of fields. And that seems conceptually theoretically okay. What's interesting is it contrasts rather sharply with the sort of phenomenological way we tend to think about relativistic quantum mechanics. And that's not generally a picture of fields and field configurations and configuration measurements. It's a picture of protons and electrons and antimatter and particles basically interacting attack forces, and are being created, destroyed, and scattering off each other, and trying to face C, that's a very different language indeed. There's always a temptation to say, well, we should just totally rethink what we mean by quantum field theory. In that case, it's not really a theory of fields at all, it's a theory of particles, and the fields may be just some sort of building block to take you to particle theory. The general way discussions tend to be done in quantum field theory at the moment would be to deny that, the theory of fields. In which case, how is it logically possible that we can have this particle description? And a way of seeing why it's perfectly possible is to go to genetic physics, which is often noted as quite a clear synagogist to quantum fields in a number of ways. If I consider I've got a solid block, say a crystal or something, in one sense the ontology of that crystal is a pretty clear cut. It's made up of various crystal atoms

7:30 and crystal ions, and various electrons linking together, the whole thing into a rigid object, and all the particles in that system are extremely strongly interacting, it would really get you nowhere at all to say that a crystal is deferred to approximation a gas of atoms and electrons. The interactions are so strong in that picture isn't helpful. And yet, despite the fact that it seems fairly clear that it really is made up of atoms and electrons and so on, that's often not the language to be used when describing condensed atoms and I take my crystal and I hit it, or I heat it up or something, then in both cases there's a lot of sort of vibrational energy going on and vibrational waves moving through the system. And quantum mechanically we can often talk about that in terms of quantised vibrations or in terms of sort of particle of quantised vibrations which we call a phonon. And a phonon isn't, it's not an atom or an electron, it's not sort of two or three atoms of an electron, and it's a collective excitation of all the atoms and electrons in the crystal. And the description in terms of phonons is often dramatically easier than the description we'd have in terms of the particles that make up the crystal. The reason for that is that while those particles interact very strongly, the phonons generally speaking are interacting pretty weakly, so you can treat phonons as basically free particles with small amounts of perturbation. and the peak just becomes a great deal simpler when you make this move. Does that justify thinking of thermals as real? Well, you can create them, you can destroy them, you can create them in certain places, you can bounce them off each other, you can bounce them off quote-unquote particles. In some situations, you can measure how long they take to get made to be. So they've got a lot of the properties that we'd expect particles to have. So in some sense, yeah, we're going to treat them as real. In another important sense, they're not real, because they're not fundamentally written into the theory. Anyone who was at my rather different talk in January talked quite a lot about quite how ontologically we should think about that, I'm not going to cover that material again here, just as sufficient to say that there's a pretty good sense in which we can think of the phonons as actual things, despite the fact they're not fundamentally written into the ontological theory. And essentially that's the same as a picture we expect to occur in quantum field theory, conceptually the same and actually mathematically the same in a number of ways. So we'd expect the particles of quantum field theory just to be certain states of the field

10:00 which have properties that make them behave an awful lot like, quote, ordinary, unquote, particles. So that's conceptually the sort of story we'd expect to be told. And why they'll be doing this talk is spelling out just sort of how that story goes and the way in which that particle concept comes out, from a certain angle. I mean, this has been addressed in various places and from various angles, so I would say this is the only story that can be told. It's complementary to a number of accounts you can give, most particularly the sort of thing you can get in a lot of the work of the algebraic quantum field theory, which I'm not going to touch on directly. So, it's been broken into two parts. I'm going to start in the first half by talking about conceptually what states ought to be thought of as particles. And in doing that, I'm going to have to digress quite extensively onto actually what states ought to be thought of as localized, which quantum field-driven states could be legitimately said to be over here or be over there. It's not a sufficient condition for a particle that's localized, it's not a necessary condition either, but clearly there's going to have to be a lot of interplay between the idea of locality and the idea of particle-ness. So that's the third part. In the second part, I'm going to develop those concepts. I'm going to try to answer the question of what is it about quantum fields that makes particle states in them. I don't mean how do we mathematically prove it, I mean that's relatively straightforward, I mean why should we not be surprised by this? What is it about those theories that leads from having particle behaviour? Because primarily basically that description there doesn't suggest that it has this radically different particle talk attached to it. It would be quite nice to get the most sort of heuristic account we could manage and point why that's happening. So I'm going to make what I think is a fairly modest contribution towards what the heuristic really is there. Anyway, moving into that first part of the talk, the conceptual structure of the story. As I said, I'm going to start with the idea of what it means for a quantum field theory state to be localised. So here's some quantum localisation. We might start by thinking of what it means for a classical field state to be localised. And there's a pretty straightforward sense we can make of that. classical fields localised in this region here if it just vanishes outside the region and we might want to say it should better vanish outside this region if it better stay vanished outside that region it wouldn't be acceptable for the field right with zero here

12:30 if its derivative was a million billion or so or if its derivative was non-zero here so we'd sort of want to say the field vanishes here, the derivative vanishes here or equivalently the field vanishes in some way we can set in time over here and that's equivalent to saying I mean, as it says there, the classical field is localised in some space-time region here if it vanishes outside the sort of forward-like and backward-like kind of this region here. So classically, there's no problem. There's a nice straightforward definition of what we mean. And that suggests a nice, comforting, straightforward version of that for quantum field. We could say that a quantum field state is localised here, a quantum state is localised here if it vanishes outside here. and we might imagine it can make sense of it vanishing by saying, well, it's expectation valuable, the operators vanishes. Okay, clearly that's rubbish. It's analogous to saying we could think about certain particle states as being those states which were perfectly localised in position and perfectly localised in momentum. They were at the origin and they weren't moving. There's no point characterising that particular set of states because it's empty. No states have those properties. The connotation relations between the various observables of any remotely interest in quantum theory or quantum theory mean that that condition is much, much too strong to impose. Another way of seeing that is it's to do with what gets called vacuum fluctuation sometimes in a lot of semi-popular discussions of quantum theory. Even in the vacuum there's a lot of surging stuff happening, and that means that this sort of very quiet behaviour outside age is just unrealistic. I think it's worth just saying briefly here that when we're thinking from a field point of view, thinking of the field as primary, there's no reason to be particularly surprised about all this vacuum excitement, excitement in more than one sense, because if we're thinking in particle terms, then the vacuum is nothing. There ain't anything there, so we'd be a bit surprised to find activity. If you're thinking field terms, the vacuum is just another way of saying the ground state of the field, the lowest energy quantum state, and there's no reason why the lowest energy state shouldn't still have plenty going on in it. in, say, a sympharmony oscillator, the lowest energy state is still a sort of finitely extended package around what would be the classical lowest energy state. So there's nothing surprising or odd about the fact that the vacuum has all these properties in it, but nonetheless it spoils that previous idea of localization. My localization is based on a definition for localization of the observed, isn't it?

15:00 How would you define that? Oh, by fiat. I'd say, in setting up quantum field theory, I'll go to the technicalities of this a little later, but the observables are going to be built from field observables, that is operators like phi and pi of x at a given point of space. observables are a bore attached to the spatial points, so that's the initial point at which spatial localization enters the picture. The problem is how do we extend that talk of operational localization to state localization, and I'll talk slightly more about what those observables are when I get into the tentatively after the talk. How do you look at this? Is it in connection with Hegel first? I don't think it directly comes up here. It's actually this definition is too naive so we can connect with that. But again, those sort of issues, they've indirectly related to the reshade of problems with localisation, which again, I'm going to come up to. Anyway, so the vacuum's property is spoiled, naive localisation. Obvious generalisation of that, let's say a state is localised in A, you can't tell it to part from the vacuum unless you're in A or the forward back would like O of A. So, whereas we've got the vacuum as some sort of reference state, localisation means it's only locally different from the vacuum. OK, that begs the question, what does it mean for a state to be differing only locally from the vacuum? That's a fiddler question than superficial here, because classically it makes loads of sense, because you can imagine you've got one state, one classical field state here, another one here, and you just have a look, do they coincide in a given subspace? But because of quantum entanglement, because if I've got some sort of field configuration, some sort of quantum state, then I'm going to be able to say, well, what we get if we just looked at this region of space is going to be a subsystem of the global normal space, and what we get if we just looked at this bit of space is going to be another subsystem of the global normal space. But if we look at both regions at once, then we can have a hell of a lot of entanglement between these bits, and trying to say what the state of this subsystem is, is both complicated and . So, that's the end of the problem. So, just abstractly moving away from what we feel to, what we say is we've got

17:30 the big little space H, which is a product of HA and HB. we've got two entanglement states sine and omega and we want to set their identical respect to H-A but they're different to H-B here's four ways you could do it firstly you could do it by construction you could say well suppose I start with my state omega and I do something to it but the something is localised in B only then clearly it can't be made to differ inside A that again comes up to this point the idea of an operator being localised is sort of happening by a fiat here, we just, by an operator being localised in B, we just mean it's open form 1 cross UB, so you with some unitary operator acting in space B only, and one is their entity on space A. So that would be so, we don't really know what it means to talk about the local state of a system, but on general grounds we can say, well, whatever it means, it must be the case that if we only do something in section B, then there must then, two states that started being the same everywhere must be the same in A. Second possibility, instrumentalist Shutterman Calculate approach to it. If we can't tell them apart with the spectra observable on HA, they must be the same on HA. Third approach, let's try actually making sense of the idea of the state of system A alone. One way to try to make sense of that is to consider the reduced density operator on that space, so trace out subsystem B. We started with a pure state, but we're now left with a mixed state. I'm not going to go into detail this a lot, but most people should know it from general quantum mechanics. So that leaves a density operator for each of our two states. If it's the same density operator, then they're the same. If we take that idea of what's meant by the state of a subsystem, then we're committed to a sort of intrinsic non-locality of quantum states. Plenty of people would agree quantum states are local. Fairly recently on the computation that was challenged by David Deutsch and Patrick Hayden they came up with a rather different definition where they sort of developed what's known in the Heisman picture and developed the concept of really associating something local to each region to describe the state. That's really for anyone who knows the Deutsch-Hayden definition of locality in the states. But in any case If you do, then you can see there's another fairly straightforward way you could define locality there.

20:00 OK, so the conceptual... I'm just coming to that. The conceptual conundrum of deciding which of these definitions most centrally captures the whole notion of locality is somewhat smooth over the fact that they're all exactly the same. They're all mathematically equivalent. So, and in a certain sense, you can convert that sort of support each one of them as a reasonable cashing out of what we could use to mean by localisation in a sense. So I'm going to take that as a working definition of localisation, and I'm not going to mention them doing so, and the technically easiest one to use tends to be that first definition. So a state that's localised in A is then any state I can make from a vacuum by the action of a unitary operator localised in A. So that's the working definition. To the best of my knowledge, it was first introduced by, I was about to say by the first name of a guy called Knight, but I have no idea what his first name is anyway. I'm calling that Knight's localisation, and there it is. now probably most people know that going along that direction defining localisation leads into a rather nasty can of worms in Richelieu's theorem on the can Richelieu's theorem says that suppose I have some localised region of space time as small as you like and I consider all the operators inside that region of space time and I let all of those operators act upon the vacuum and it asks how big is the space of states I get as a consequence and we might naively suppose well it's going to be the space of all states localised in A because they're the states I've made for more operations within A and instead that includes the non-unitary operations so I take every operator I've got localised in this space every field operator, every mental operator in this small region I let them act on the vacuum, how many states do I get? and the most surprising answer is I get the entire hill of space Mathematically, why that is, is because the vacuum is a very entangled state, and you can sort of explore those entanglements to create any state you like. So, that seems at first completely paradoxical. We tried to sap off some of the paradox. The reasoning to theorem only works if you consider every operator localised in A. If you restrict the unitary operators, you don't get the whole Hilbert space, but you get a set of states and spans of the Hilbert space.

22:30 You get a basis for the whole Hilbert space. What it says is, if I consider the set of all states localized, or might localize, in a region A, then it's not a subspace, and if you make it a subspace by closing it and taking linear combinations of those states, then you get the entire hilt of space. Now, clearly any definition that says every single possible configuration is localized over there is not terribly helpful. is this paradoxical conceptually no it's not it doesn't muck up the reason to think 9th localisation is a good way of understanding localisation all it says is the statement localised in A isn't a property conserved under linear sequins so it says if a state is local if state 1 is localised in A and state 2 is localised in A that doesn't automatically tell you anything about the localisation of state 1 plus state 2 perfectly reasonable properties of quantum states that have that property, consider being a non-entangled state, being a product state. I might have, if state 1 is a non-entangled state and state 2 is a non-entangled state, clearly linear superpositions of those can be entangled as all sorts of entangled means. Similarly, property of being an eigenstate of energy. If I have an eigenstate of energy, energy of E1, and another eigenstate of energy of E2, a superposition of blatantly not an eigenstate of any sort. It just makes it unlike a lot more familiar properties, having energy E, for instance, is conserved under the superposition, a superposition of two eigenstates with the same eigenvalue is obviously still an eigenstate of the same eigenvalue. And this is about the problem. In non-relativistic quantum mechanics, spatial localization works like that. The non-relativistic concept of localizing Vj, which is now, is conserved under superposition. If I've got a wave function here and a wave function here that they've localised in this region, then clearly their superposition is also localised in this region. So we don't have as much of a conceptual problem as, sorry, it's almost an empirical problem. This concept of localisation doesn't seem to gel properly with the extremely robust well-tested, well-dealt-with concept of localisation we've got in that much of the problem in Phoenix. But as soon as we acknowledge that, the problem becomes a lot less serious because we're now, it's not that we've got a conceptual problem with this We've just got a fitting problem. It doesn't fit properly with non-multivistic quantum mechanics.

25:00 And then we can say, well, in that case, all we really need is to say, in certain regimes of quantum field theory, for certain sorts of states, even if localisation doesn't work like non-multivistic localisation, it works pretty much like non-multivistic localisation. It works up to 10 to the minus 10, or it works on scales big compared to 10 to the minus 12 metres, or concepts like that. it becomes, given we don't think of non-baltivistic quantum mechanics as the last word, or, I mean, if we're seeing insofar as it's true it's going to be derivative from quantum field theory, then it doesn't matter if we slightly get quantum mechanics wrong. In fact, we'd expect to. I mean, there are plenty of more mundane ways in which we'd expect a non-baltivistic theory to have small errors, even in its domain validity. I mean, if I throw a ping-pong ball across the room, then ordinary class mechanics are going to predict it pretty well, but technically they're small relative to the corrections. So, provided those corrections and problems are pretty small, we don't have any problem here. And that means that the problem we're dealing with is a lot smaller and a lot more modest. We don't need a conceptual resolution of the Richelieu paradox. We need a regime in which we can pretty much do what Richelieu stops us doing. I'm a bit confused, really. Maybe you can tell us what it means to be a material point of what lies in A Okay, could I add to that question? In the original analogies on the first slide, we would have had it all in the space-time. The quantum field theory analogy would have been a ball in the space of fields, and so a localization that would be a direct analogy to the space-time analogy would be a ball in the space of fields. That's why I don't understand the introduction of space-time in the context of the quantum field theory. I think that's his question also. Okay, so, in a sense, and you like that analogy, the problem there is, in a sense, in my first slide's analogies, the analogy of spatial localisation wouldn't be, as you say, wouldn't be spatial localisation, it would be sort of field localisation. and that's fine, we can have a perfectly good picture of that but what we actually need is direct spatial localisation the very problem in a sense is it's disanalogous to the idea of localisation in quantum mechanics we can't just pull the concept across, the analogy takes us in the wrong direction so to speak which is why we find ourselves in a very different concept that we've somehow got to make and manage with

27:30 as for what it means for an observable to be localised in a given region Basically, I'll skip ahead slightly and run through mathematics about an example and show people it. If you start, imagine you've got a classical field equation like the Klein-Gordon equation, it's a paradigm case. That's the dynamic equation for Klein-Gordon. We can construct a Lagrangian flat theory, which means just the same as in non-relativistic classical mechanics. some function of the field and the field's way to change from which we can get the dynamics by Lagrangian, by the Orgallian equations even. And then just as in ordinary classical mechanics, we can go from that Lagrangian picture to Hamiltonian picture by defining a conjugate momentum to the quote position. and in ordinary classical mechanics the conduit meant of P is a greater change of L with respect to DQT in classical field theory where it's an unconventional problem the analogy of the coordinates like Q1, Q2, Q3 which together are coordinates for the configuration space are going to be the value of the field at every point in the same way if I tell you all the components of a particle position, then I've told you where the particle is. If I tell you the value of a field at every given point, then I've told you exactly which field it is. So the 5x functions are, the individual 5x for each x are equivalent to the qi, and the label x is equivalent to the label i on those coordinates. And then I can define a momentum pi of x, which is analogous to P, I, and I define the same way, I look at how L varies with respect to, as I verify, there should actually be a dot in there, there should be a dot, but I look at how L varies with respect to d phi to t. Put the dot in here. And all this looks scary, so I can look scary because it's interdimensional, but basically it's the same game we're very used to in all the classical particle mechanics. that game, and you get a Hamiltonian, which you then go on and quantise, again, in a way

30:00 very analogous to how you quantise a classical mechanical theory. I think I'd like to come back to the technicalities of that, but the point of understanding is that the observables that correspond to things like Q1, Q2, Q3 are all the objects pi of x for different x, so the observables come with a label which tells you their spatial location, and then And that's where the idea of localisation arrives for the observables. And when we quantise, the observables go up to operators, but they carry their labels with them. So the classical observables localised at a point or in a small region generate quantum mechanical operators, which, again, by definition, are localised in that same region as what the label tells you. Does that sound special? Environment, yeah. So going back, what we said is we need a sort of, we need a de facto concept of localisation that basically works pretty much as non-multivistic localisation works, at least for some family of states, and even work for the whole space because there can be plenty of states that just aren't remotely well analysed by non-multivistic quantum mechanics or remotely well analysed by this notion of opposition, but we still need some regime in which it holds. Here's a candidate. I don't know if they're qualitatively. You can make it precise, but I don't want to get into technicalities again. There's night localisation again. A state's night localised in A if it's indistinguishable from the vacuum anywhere space-like separated from A. And I want to say a state's effectively localised in A if it's pretty much indistinguishable from the vacuum may. So that's to say, it's not that all observables, all expectation values are the same outside A, it's that they're nearly the same. And of course you need to make some good sense of that, you want to have some idea of them dropping off exponentially fast as you go outside A or something, and you can provide those precise definitions, but the specifics of them aren't particularly important here. All we're saying is that if you measure, say you measure the field strength operator phi of X when X is inside A, then the expectation value of that compared to the vacuum is going to be massive relative to what I get if I measure it over here, and then I just cast off the same result for the vacuum.

32:30 So that would be a concept of effective localisation. And then we can say, we could define something called the effective localisation principle, and this is something that's defined not for an individual state, but for a whole space of states. and the effective localisation principle what we're saying is for this space of state S imagine we have some length scale L set and it can be quite small but whatever the length scale L is we're saying inside this space S and only inside S if I have two states which each of which is effectively localised in some region here and if that region is being compared to L then the superposition effectively localised inside L. So in other words, what it's saying is a space of states obeys the effectual localisation principle if inside that space localisation is effectively a linear property, provided I don't push my luck, provided I don't look at the scale small compared to this defining scale L. So then if I was in such a space and if L was microscopically small, then with respect to that space, localization would behave pretty much as it would behave for non-optimistic product mechanics. So finding spaces satisfying that property is sort of saying we're finding spaces in which localization works as we'd be connected to. Note that S couldn't be as big as the whole Hilton space because clearly among the that's effectively localised in A, other states that are actually not localised in A, and we know that the space spanned by all states like that is the whole Hilder space, that's what the Recheath actually says. So, this is only ever going to apply for certain subspaces of the whole Hilder space. Okay, so there's the concept of localisation, and that's a necessary building block which I need to talk about the concept of particle. But now I want to address the question, what now that we've got this localisation question out of the way. I might start again with the classical analogy. If we thought the world really was a classical field theory, but we observed particles localised blobs of matter, then the obvious definition of a particle would be, well, a particle is a localised blobs of matter. It's a classical field configuration, rather,

35:00 that is large here and small, or very nearly small outside that region. So, you know, a very tightly concentrated wave package particles. And that would seem to be okay as a classical definition. It's no use as a quantum definition. It's too weak and it's too strong. The reasons for that, firstly, it's intrinsic in the nature of quantum particles that they've got a certain amount of stability and robustness. They can suddenly decay, you can bash them into things and you can make other particles from them, but they at least somehow keep their particle-ness for long enough for us to preserve it and are used to it. Whereas in classical field theories, at least fairly linear ones, which are the ones that tend to be treating the project theory, packets tend to spread out fairly rapidly. There's a sort of smooth and reasonably quick transition between particles and non-particle states, and there wouldn't be like any sort of radical decay or interaction, just gradually things are spotting particle in. So that seems to be unsatisfactory. The other problem is that requiring particles to be localized is just way too strong for quantum mechanics. The whole, well not the whole paradox of quantum mechanics, but a hell of a lot of it is bound up in particle number counting. If you think about the two-slit experiment, the whole nature of the two-slit experiment is that the particle, the electron or photon, isn't here or here, it's here and here at the same time. So any notion of locality which requires a particle to be in a particular place is just way too strong for the problem, of course. But the two-slit experiment also gives us some insight into how we get around that problem. The insight goes like this. Although in the two-slit experiment the particle is delocalised, it's still the case that whenever I measure it, I'll only find it in one place. And again, even in the two-slit experiment, you could find the particle here and here at the same time, then the two-slit experiment wouldn't be a paradox. It would just demonstrate that the matter is fundamentally wavy-like. It's a paradox because it picks up the wavy and particle behaviour. And the thing that's particulate about it is that you always find the particle in a specific place of where you look at it. And we can ensure that mathematically if we require not that all particle states are localized, but that the space of all one-particle states has a basis of localized states in it. So in other words, a given one-particle state is allowed to be non-local, but it's got to be a superposition of states, each of which is local. And if you have a classical field that can sometimes seem obvious, because you think of your state as being a wave like this,

37:30 and you can say, well, it's really that, plus that, plus that, plus that, plus that, so it seems trivial. It's not trivial. I mean, a simple counter example, consider a 30,000 particle state where the particles are all over the place. So it's an eigenstate of position, but the position is that of all 30,000 particles in lots of different places. So it is actually quite a strong condition to require that of a quantum particle. I should say that that concept of what it means to be a particle is not particularly new. You can see it in a much more operational sense in algebraic quantum field theory, where people say an n-particle state is a state that can trigger n different particle detectors, but n more than n. So it's the same type of concept going on there. I wouldn't want to claim an internet for that. Nonetheless, if that's the concept of particle locality we've got, or particle mystery we've got, I could say, write down a reasonably clean definition of what the one particle space is, and the definition I want to use is something like that. So we've got this one particle subspace, and our conditions are, firstly, well, even before, firstly, it's a space, it's a subspace, not a subset, so it's closed under the limited superposition, any superposition of particles is still a particle. And the properties of that subspace that I have, firstly, it's spanned, as I said, spanned by these local states. So there's bases of states, each of which is localized, not necessarily perfectly localized, but localized inside some small region. I haven't put a bound for the size of L, but clearly this is going to work for number L to visit particles if it's actually small. And in fact, what it actually turns out for massive particles is L is the constant wavelength, so it tends to be down at 10 to the minus 15 meters from up to the protons and so forth. But secondly, the space would better obey the effective localization principle. a particle state effectively localised here and another one effectively localised here then there's two positions effectively localised here so a particle space a one particle space is going to be an example of the effectively localisation principle spaces we talked about earlier it's going to be a space in which the non-relativistic idea of localisation works pretty well and thirdly it has to be fairly stable it better be the case if you start with a one particle state it remains a one particle state for quite a while I don't need to do it ever. In relatively quantum mechanics, quantum field theory tends to be the case

40:00 that the particles remain saved for quite a while, but they don't need to, they can decay. It's necessary, I don't think that a pion, for instance, that pion behaves in a particle-y way for a bit and then breaks up, so that's okay. And again, there's a bit of, there's a certain amount of vagueness in this definition, you know, how long, how long is appropriate time scale, how long is good enough, and that's not really a problem, because again, particles are part of the foundation stones in nature or something as highfalutin as that, we're saying that particles are a great convenient way of analysing certain states of the field theory. What happens when we make time scales shorter, it starts getting a less and less convenient way of analysing it. And eventually there's going to come to a point where it's so useless as a way of analysing it, you might as well forget it and go back to the direct field description. I need to go now, let me ask you, what's wrong with weakness definition of the particles how does that link to the idea of the particle as a localized object so that's the idea you want to yeah, I mean of course those turn out in the particles and of course that definition is a lot going for it but from a sort of point you start with localization there's no obvious connection, I mean for instance if I have a in a certain sense the planet Saturn the central mass of the planet Saturn of the concullion group, and yet it's not a particle in this sense, so of course that's very bound up in particle, although I'm not going to make much contact with it, but it's not, it's not, if you're starting with a localisation, it takes more of you, you don't need to come across it. Anyway, so I'll start with that as a definition of particle. that more or less concludes the first half I want to talk about the same question I want to come on to is why should we expect to be able to find particle states in quantum theory so I now want to address that question why are those states there here's what I'm not going to do I'm not going to write down the states which anyone who knows any of T knows down well are the particle states and then prove to have these properties that's easy anyway that's a straightforward calculation that gives us very much insight into why there are parts there in the first place. We sort of need some reason to look at that space. So what I actually want to do is give some sort of heuristic motivation

42:30 as to why we should expect to find that space there in the first place. So I don't want to claim that there aren't much more direct ways to prove that the one particle subspace has these properties. I just want to try to motivate its existence. So let's come on to that. I already talked about what the classical Klein-Gordon equation looked like, and I sort of very briefly touched on what happens when you quantize it. Let me do this in more detail now. back to my analogy. At the top, that's what happens in quantum mechanics. The states become wave functions on the configuration space. To the observables, you map operators. The position observable becomes multiplication by position. The momentum observable becomes differentiation times the square root of minus one. We play exactly the same game in quantum field theory. The analogous operator to position is particle configuration, so the operator pi of x, and that acts by multiplying by the particle, by the field strength at x. The analogous operator to momentum is field momentum pi of x, that acts by doing this variational derivative. Again, don't let the interdimensional stuff hide away the fact that this is technically as you play in non-optimistic quantum mechanics, look at the technicalities and all sorts of infinite disasters happen, which take a normalisation and so forth. But conceptually speaking, at least formally, it's the same trick. And you get to write down a Hamiltonian. The Hamiltonian, again, looks hideous, but it's really just similar in structure to a Scherlinger-type operator. There's something where you're differentiating the wave function, there's something we're just multiplying the wave function by stuff so clearly it doesn't commute with the field observable pi it doesn't commute with the mental observable pi but it's a quadratic it's a fairly simple combination of bits the way you can see quite rapidly that it's just the dimensional version of the ordinary number of fistic games, you write down the commutation of these things and again gone to the Kronika delta, you know, if I sort of have x comma py, it's 0, but x comma

45:00 px is i. So here I've replaced that Kronika delta with a delta function because I've got a continuum of labels, but basically it's the same game. I haven't felt anything deep or distinctly. Nonetheless, although that's conceptually about the same, it's technically rather hard to make sense of, and what one actually does in making sense of it is playing a trick, and the trick is called modal analysis. Again, anyone who is familiar with quantum theory theory, I apologize because it's getting quite basic, but you remember through this or aren't. It's called what? Well, I'm calling it modal analysis. I wouldn't claim that, but that's the one true name for it. Go back to the... You've had to code my handwriting together. The mathematics got too much, but I can bother for it to keep in mind. Classical Klingon equation is a linear Note that that linearity is rather special in classical mechanics. The fact that Schrodinger equation linear is a general property of quantum mechanics, the fact that the Lagoon equation linear is a specific property of a certain, very specialised, somewhat unrealistic class of things, and an approximate property of a much larger and more manageable class of things. So it's linear, and what that allows us to do is write any solution as a sum of modes, and the modes are sort of sine and cos functions, or mathematical convenience by using exponential functions. So, e to the i, kx minus 0, and t satisfies the Klein-Gordon equation. Therefore, any linear combination of those things satisfies the Klein-Gordon equation. Cc is complex conjugate because I said by fiat that I'm using real Klein-Gordon equations. So, I've got to make sure that's real, but that doesn't make much difference to it. So, any solution can be expressed in this way. It's not just that you can sum field configurations at a time. it's that if I have this field configuration and it evolves over time to this, and I've got this field configuration and it evolves over time to this, then the sum here will evolve to the sum there. So again, that's not general property field theory, this is probably pretty obvious. But for linear theories and for the Klein-Gordon theory, this is a totally general expression for a solution. And that means the alpha k's are complex numbers which are effectively coordinates for the spaceable solutions, and that's analogous to say they're coordinates for space space. Any face-to-face operator can improve to be quantized. If I just use the quantization I've already developed enough,

47:30 what does that say about quantizing these things? Then what I find is they go from these alpha-k things to operating them calling AK. The Hamiltonian now has this really nice friendly form, and the commutators of the AKs have this friendly form, and everyone should recognize that is the Hamiltonian of a collection of human costumators. so blindingly obvious well well known fact number one in a mathematical sense the Klein-Gordon equation and fairly generally any linear equation in this form quantises to a sum of half of the oscillators so we need that technically to do some mathematical work and we'll light us up to that 10th half so I can go back yeah so I'm afraid I didn't look probably at least 10 times Okay, so that mathematically is a lot easier to make sense of, though it's important not to lose track of the connection back through the field of circles to locality. Each of these AKs is made up of a very non-local, it's a very non-local combination of the original field operators, so if we want to understand localisation, then this form has to be treated with a certain amount of caution. Anyway, let's get back to the question we're addressing, how are we going to find particle states? Well, we've established that locality is not remotely a good enough condition of particles, but it's quite a good place to start. So let's start by trying to find quantum field theory states that are localised. Now, remember, the space of states of this quantum field theory is staggeringly huge. The space of states for classical field theory is pretty big. I mean, it's every single possible function you can write down in this room. but the quantum wave functional is a function by every possible function you can write down in our space so it's an incredibly complicated mathematical object and very hard to analyse very hard to ask in principle in advance which such states are going to count as localised because there's going to be plenty of scope for all sorts of interferences between the value of it on this field configuration or on that field configuration and this is a consequence of what I was saying of localization, of space localization, being field localization. Field localization is straight forward to understand, position localization is really hard to understand. So rather than trying to do anything as ambitious as saying, in general, how do we characterize the localized states, let's just start by building some. And the general strategy used by physicists, time-wise tradition, when dealing with the complicated quantum theory, it's trying to pretend it's classical or nearly classical.

50:00 we know pretty well which classical field states are localised so to construct quantum field fields states are localised we'll just try to approximate the quantum mechanics those will have a wave function in the space of all fields which is very tightly localised around a particular point in that space in other words a particular classical field and will require that classical field is itself really very localised it's quite difficult to keep in mind we've got two sets of functions here It's not good enough for the quantum wave packet to be very tightly localised in field configuration space if the thing's localised around is itself very spread out as a classical field. So it's important to keep both of these aspects in mind. Anyway, because this thing is a sum of monoc oscillators, there's a very straightforward trick we can do to construct good, robust quantum approximations to classical states, and that's called the coherent state method. and probably most people are familiar with it, but I'm going to run over it just quickly for the case of just one harmonic oscillator, just partly to establish the mechanical room and partly to the refresher. So, coherent state is a concept initially from ordinary non-multivistic quantum mechanics, and it's considering a harmonic oscillator. So, Hamiltonian is half Q squared plus P squared, plus some constants if you're much more authentic than that. And again, we can specify the location phase based by the numbers Q and P. They're both real, equivalent, but we can specify location phase based by the complex number Q plus IP, and there's a factor in there for mathematical convenience. Okay, quantising that system in the obvious way, and look at it's Hamiltonian, we find the ground state, the micro ground state is omega by analogy to the quantum vacuum. And everyone knows the ground state is a Gaussian state, and the Gaussian is centred on the classical ground state, the particle at the orbit if you apply this operator, E to 1 root 2 Q, A joint A is the creation operator so it creates a contour of the of energy of the monobosal space so you apply A to the ground state you get the first A joint grip to the ground state you get the first excited state, apply it again you get the second excited state, etc, etc, etc so if I apply not A, but E to the start times A, it's not obvious, although it's pretty straightforward to prove, what that actually does is it moves the wave packet, it displaces it rigidly away

52:30 from the origin to the point Q. So what you've essentially done there, if a fairly tiny Gaussian is a good approximation to a classical point, then what we've done is we've moved from this being a good approximation to the classical point of the origin, to being a good approximation to some other classical point. And this thing isn't an idea of state of energy, so it doesn't stay still. It does something nearly as good. It moves backward and forward. It doesn't dispersion into its shape. It oscillates backward and forward, and it does so in such a way as to follow the movement of the classical state. So it's not just a good approximation to a classical state at an instant. It stays being a good approximation. It's effectively a good approximation to a whole classical solution, which is actually quite remarkable. As a digression, I think Schrodinger found that the wave function did this oscillator, and Imab hoped this would be a general property so that we could understand wave particle duality that way, but it turns out it's a very special property of the symphalmonic oscillator. If you use most Hamiltonians, the thing would spread out and become less good approximation at the time. Anyway, it works beautifully here, and generalizing this slightly to allow for approximating the non-zero momentum states as well. If you find this is why I use this complex form, if you apply e to the alpha age, or the alpha is this complex number characterised in a state, then that gives you an approximation, a quantum approximation to the classical point, to that classical point. So it's really a remarkably clean, simple way of doing it. And what you can do for one monoconstillator, you can do for two or three or continuously, and that's what we do. Going to quantum field theory, we know that Hamiltonians are put along to a sum of monoconstillators, so we know that any given state can be written by specifying all of these complex numbers specify the amplitude of each mode in the expansion so here's the form, I've added this factor here again that's just mathematically to make the numbers work out otherwise you always get some awkward factors somewhere I'm trying to admit to them openly like the beginning so if I want to approximate this state well mathematically they're all independent hominocosylators so I approximate the hominocosylators one at a time I apply e to the alpha k n a k n a joint to approximate the nth one and then I just go through and I apply it all in order, it's an infinite sequence in principle but that's, again, just a technical matter. So I find myself with this state that is now a good quantum approximation to any classical field solution that I feel like. And mathematically, because the probability of exponentials

55:00 is equal to the exponential of the sums, and because that happens still to work when I'm using operators when they commute, that I can equally write in that form there. So this state's what I call the C-phi is a good quantum approximation to the classical state of phi. Is a particle? Hell no. These states do not do as a particle. They're localised, or the ones which are coherent approximations to localised classical states are localised, but that is not good enough. Firstly, the classical localised states will gradually spread out, which means that the quantum states, although the way packet remains tied to configuration space, it's tied around something that's itself less and less local. so you've got this gradual sidious movement from being a particle to not being a particle which we didn't want, we wanted some degree of robustness to the particle space. Secondly, it's not closed on the linear superpositions if I have a coherent state C1 and another coherent state C2 and I suppose them, then that's not a coherent state that's easy to prove mathematically you can sort of see why as well, I mean say I've got a classical field that's over here, I've got a classical field that's over here, a coherent state that's approximated that and a coherent state approximated that that, if I sum those, I certainly don't get a coherent state to approximate that plus that. I get a superposition of something that basically has a quantum null locality. And clearly, any good approximation, one thing we'd require in a good approximation of classical states is better not have quantum null locality in it. So, what that means is the set of coherent states is not closed under superposition, which was one of our defining quantities of particles. However, we can imagine that maybe we'll close it, maybe we'll say we'll take the span of all the coherent states. That's no use because the span of the coherent states is the entire health space, and clearly we don't understand the impossible quantum field stage in one particle stage. So they're no use. It's a shame because they have got the localisation properties we want. You can prove that if I've got a classical solution that is here, then the quantum approximation to it is effectively localised here. you've got a little bit of spillover outside that area you've got exponentially small chances to find field strength over here but basically it stays localized so it's effectively localized in the sense it requires not micro-localized but it's nearly not localized so it's can we solve that problem of course we can, otherwise we're going to talk sorry David, let me just ask before you say the answer you just pointed out that the sum of two co-hearing states wouldn't be co-hearing and I see that

57:30 but then you made a going back to the idea of spanning the whole Hilbert space and not wanting the whole Hilbert space to be one particle can you just say that bit again? oh just if you want you need to see if we wanted the human space to be particles it wouldn't be good enough because by theat the space of particles are going to be a space and the space of the human space is enclosed in a superposition you might imagine we could just try closing it, we'll take the span of all the human space the coherent states form the basis of the Hilbert space. So the span of the coherent states is the whole space. Okay. What's going wrong? Why doesn't this work? And the way to see why it goes wrong is something like this. Suppose I start with some localized classical states labeled by N. I've written this in terms of, well, phase-based points actually, which is probably unhelpful, but phi comma pi is disrepresenting a given solution. So what I could do with those N low-class states and then I could take a coherent approximation to that superposition. I get some state down here. That's the coherent approximation to the state of blob plus blob plus blob. Or I could do it the other way. I could approximate each of these things independently to get n localized coherent states, and I could take their superposition, and that would be a non-local combination of blob, blob, blob, blob. So the reason why this isn't working is this diagram doesn't commute. What I actually want is some notion whereby those two linear structures interconnect in a much friendlier way and if I could have that I would have a nice manageable size space which was closed on a linear super position and that was potentially much better can than the particles so is there any way of modifying the coherent state approach in order to get states which have that property? Yes it is and it works something like this to start with our classical state it's localised in some region A Let's build a coherent state around it. So here, stage two, that's effectively left, like I said. Again, there's this exponential tail off, but we've established that doesn't really matter. Try instead of proximate the state phi, let's proximate the state lambda phi. Well, it's pretty straightforward to see from the definition that a proximate the state lambda phi is like a proximate the state phi, but adding a sort of lambda in the exponential here. so conversely what that's saying is the state if the state e to this thing omega is low twice in a then the state e to the lambda of this thing

1:00:00 omega is also low twice in a if you make lambda smaller then this exponential is well approximated about power series and well it's always possible for power series but make it smaller then I can throw away all of the few terms of power series so I can make lambda armature only small without spoiling the effect of representation so I find that at the sort of strength limit, then this stage, vacuum plus small bit, this thing acting on vacuum, is also low times an A. And in the last stage, which I stress as a conjecture, or at least from this perspective as a conjecture, because the vacuum is translation invariant, we might suppose that we could throw the vacuum bit away from here without spoiling the fact it's low times an A. I stress that doesn't follow logically here. We've already established that in general, locality is not a linear property. However, when I say I think it's a conjecture, I don't think it's like a fermat conjecture, it's a conjecture you've proven two lines. So it turns out that this statement is true, and again, this comes into my philosophy for doing this, I'm not trying to just establish a posteriority that the particle space is the particle space, I'm trying to motivate the particle space. So here, I'm saying there's a very strong motivation for the conjecture being, for making that conjecture, and it's also very straightforward to check the conjecture is right. so now we've found we've moved from the coherent states which were created by sort of exponentials of creation operators acting on the vacuum being localised in A to these new states where we've just got a bit of a coherent state localised we've just got a bit of a coherent state construction but it's also localised in A, we can prove it's localised in A and there's something that's rather convenient come down with that because although superpositions of coherent states aren't coherent and superpositions of these bits of them are still superpositions of this form. I haven't really shown that well, but the point is broadly this. A adjoint phi, this comes to follow this statement, phi 1 plus phi 2 is equal to A adjoint phi 1 plus A adjoint phi 2. So there's a linearity there which isn't true for the operators up here. For this, if you have e to the a-adjoint phi 1 plus e to the a-adjoint phi 2, it's certainly not equal to e to the a-adjoint phi 1 plus phi 2. Sorry, I've gotten confused with it. Is i a subspace of the full quantum field theory over space, or is i a region of space?

1:02:30 It's a region of space. So how does A get into the definition of that A5 dagger per meter, or pi is like an A than A? That's right, 5 is like an A, so it's gone to the top of the page. Okay, so we've chosen our alpha k to get 0. Absolutely, yeah. But anyway, the choosing isn't that difficult, all we do is we pick a wave function here and that's it. So the point again is that the locality is dark and mysterious in the quantum picture and it's open and transparent in the classical picture so we try to steal the classification of the quantum picture. So the point is by this slight sight of hand by effectively going to the limiting case of very weak coherent state and then throwing away the vacuum piece we've actually introduced linearity into the picture and that now means the superposition of the superposition of the superposition principle in the classical domain and in the quantum domain effectively commute. If I take a classical superposition and then make this sort of quantum approximation to it then it's the same as going the other direction, it's the same as quantum approximating each state separately and then taking the quantum superposition. And that leaves us with this diagram which is pretty much where I'm going to finish. I've lost the title, but I don't know. What's going on in this? Basically, classical mechanics is here, quantum mechanics is here. We imagine starting with the modes, we could try doing the classical superposition, or we could try putting the modes together to make a classical state, and then approximating that by the Gehielm method. My D is looking at the makes a Gehielm state. So this is equal to what I was called C fine earlier. So we could do that trick. Going down the diagram from here to here, basically the states stay the same. There's a bit of quantum blurring that they're still basically classical states. And because they're still basically classical states, this root doesn't commute. a linear process. If I try doing it linearly, then I go back to the problem I had before. Yet, if I now make a further move, going down the diagram again, I restrict to the first

1:05:00 order bit, I throw away all the high order terms, I throw away the vacuum, then I get states where the combination remains linear. So if I go all the way down the diagram, it meets. And I do that at the price of losing the sense of which these are good approximations. this state approximates that one pretty well this one approximates this one only in the sense they're localised in the same way this one approximates this one only in the sense that they have the same localisation properties and you can see that because if I'm the idea of supposition means totally different things if I think about this classical state and then I consider a quantum state as a supposition of this and this, that means something really very different and yet the mathematical structures are intertwined in such a way that I could pictures. So they're mathematically quite equivalent in a way that are conceptually wildly different from each other. And from this diagram, you can now read off the fact that we've got particles, because going down the diagram preserves locality the linear structures commune. So since in the classical picture we know that localization is a linear property, if I have a classical wave packet here and other ones with here and I superpose them classically, I get the stables, the localization and because localised the account is going down the diagram and because the linear structure is commute I now know that I've got a quantum particle state localised here and I see a place of another localised then I get one that's also localised I say again effectively localised there's a certain amount of exponential drop off but again we decide that doesn't matter so you just find it by saying take a coherence I think coherence is going to have corresponding I've tried to do it without supposing in advance that we won't think that they're going to do it. And that does bring up my point, everyone knows these particles at the beginning are not saying they're much faster when it's proving this, it's just a case of motivated people to know about it. So I think what I'd like to finish on, really, is the point that for a quantum field theory, what's allowing us to do this whole game is the presence of two different linear structures. There's a linear structure in quantum mechanics, which is always there.

1:07:30 It's part of the conceptual foundations of quantum mechanics that is linear. And there's a linear structure in classical field theory, which isn't usually there. it's an mathematical approximation to suppose it's there any interest in theory has interactions, any interest in theory is non-linear but it's because of the interplay between those two structures that particles are occurring here, and in fact if you go further although I won't, technically speaking that interplay extends to pulling the complex structure not created by i back from here to the classical picture, and the inner products pull back between them, and they also play crucial roles in the mathematical details in the picture. So I think it's at least interesting to see you've got this interplay between these two very different objects between quantum superposition, which is weird and wonderful, and classical superposition, which is banal in a certain sense, and a mathematical trick. And I think it sort of fits also with the way we think about particles. We use particles essentially in scattering situations, insofar as we can't talk about scattering, a weak perturbative approximation to free particle motion, then particle language generally doesn't work for us. If we do, for instance, quantum field theory of quarks, and we look at... Then for high-energy situations where they can be chosen to cross the tree, we're talking in quark particle language. For low-energy stuff, we want to look at the issues of confining of things. We tend to play lattice games where the whole particle picture drops out of view. And I think a way to understand that is that the whole... It's not just dynamically scattering involves, sorry, it's not just that dynamically we can handle particles in scattering situations because the interactions are weak, it's that even kinematically the very nature part of the concept is emerging from the interplay between the linear structure in field space and the linear structure on classical field space. I think I'll stop there, and I just want to say at the end that I want to stress that this picture applies in a limited context. It applies to linear fields, that's not a problem I said, I think there's a good rationale on that. It applies to massive fields, and there are interesting problems when you take the mass turn away. The problem with taking the mass turn away is that the effect of localisation starts breaking down, I haven't gone into 10 calories and that, but it stops being the case, you've got this exponential drop-off. I've been looking for real fields, that's not a serious restriction, it goes through perfect straight-forwardly, the complex fields, SU2 fields, and so on.

1:10:00 and most importantly my whole talk has been in terms of the phosonic field the entire picture is different potentially in the phoenix field and I have not got a clear view where that goes so in that sense it's a work in progress thank you, I'll stop now applause applause applause applause applause Well, I guess my first thought is could you guess upon a little bit more of the dropping of the vacuum in that last step? And, well, it's probably fairly trivial pre-field theory can you tell us yeah sure my conjecture step to me right in a sense there's a couple ways to specify one is in a pre-field theory ultimately the proof that in the meeting if you want you want just actually write down the expectation to prove the conjecture just write down the expectation values of that and check that it actually is legalized if you want to say what's justification for saying, for making the step from a heuristic way, why do you think it works? I'm not sure I can very much, actually. It sort of seems, it seems highly reasonable to suppose that, given the vacuum has totally smooth localisation properties, that its presence in the superposition is not going to materially affect the localisation of the remaining components of the superposition. But I don't, I couldn't go further than that, and in a certain sense If I tried to justify that, I'm not sure where I could stop short of just doing a calculation. Wouldn't you just, I mean, you'd have to put a bound on whatever that second state is, superposition of the vacuum, it has to be close to 1, if it's like the vacuum. Well, you mean the lambda parameter? Right, so you'd have to put a bound on what you mean by close, and then you would just fall out. I don't think I see that. Can you just... Well, that's what I was concerned about, because if you want to define that as being close to the vacuum, typically you would take an expectation value to the vacuum, and that should be fairly close to 1. But if you throw away the vacuum bit, you've lost your 1.

1:12:30 And so that's why I was wondering what you mean by close to the vacuum. I guess except for the localized parts. I would say you mean in the sense of my definition of actual explanation. So what's a more formal definition? My more formal definition of vector localisation is effectively exponential drop-off. So you've got this label scale L which we've needed. So in a more formal definition, I put that into a vector localisation straight away. What I need is that the differences drop off exponentially as you leave the area A. So then we just have to look at that state after you've dropped the vacuum. Yeah, it's a calculation matter. It's a calculation. So you're saying that with the vacuum or without the vacuum in that term it is a localised particle state? That's right, yeah. In the first case it's not a local particle state. I'm sorry, yeah. Yes, that's right. The fact that it's localised with the vacuum follows from the fact that the hearing states are good approximations to classical local states. Moving the vacuum, as I say, is, I think at the level of this discussion, it's conjectural. I don't want to stress, but the calculation takes three lines in three things, so it's not difficult at all. I have a question as well. You seem to avoid at any point talking about a position operator or a projection operator onto a volume. It does appear you could have gotten to a fairly similar result by taking the projection operator or the projection operator out of a volume set and quantizing it and requiring it that it be zero or a localised product? I think that will work. I hope it will pick up two practical tasks by doing a localised product as well. Well, that's one reason why it won't work. My fundamental reason for each problem is we know from the Ritchie's theorem that that must fail at a conceptual level. There must be some sense. if there was a localisation operator of that sort then it would follow that it would have item states which were themselves states localised inside that region and it would follow that their seat positions would also be localised in that region and that would imply the localisation of the property and we know if there is anything that is I mean defining position operator you can define position operator in this picture but sort of only after the fact

1:15:00 you've got the one part of the subsystem you can then find a position operator you've got to find quite a few position operators that basically do the right job in that sector provided you don't mind a bit of fuzz the most standard one is the need for bigger construction and in a certain sense the bit of fuzz there is it's not going to be a variant and that doesn't matter provided we understand it in an approximate term we understand that None of these things are exactly legalised, they're just roughly legalised. And in a sense that... So you're talking about something being woke-like at a time, rather than woke-like in a region of space-time. So you're already presupposing that you're in somebody's reciclame or something like that. You would hope it doesn't seem too much on who's supposed to be reciclame. Yeah, and you can try to defend the new people of pathologies on those grounds, so I guess on that subject. My defence is a bit more banal, to be honest. This is all sufficient about the many. We've established from a way to go that the concept of localisation we expect to get out is any bit of an approximate. So any position operator is only going to approximately track the localisation of states. So if that approximation shifts a bit when you train reference frames, it doesn't really matter. It doesn't get in the way of usefulness. and we've sort of abandoned right from the beginning the idea of a conception fundamentally position it's quite interesting the order of priority changes the way we tend to develop positions we tend to start the position operation and then construct an icon state and this is more saying we start with a set of states that we think of good claim to counter the icon state position and then we construct an operation can I just ask Could you say something about the relation between the space you went up to find and the definition you gave earlier on where you said that these various things were in the way you could get the overhead. My part of the definitions? Well, you had a definition of a one particle space. Yeah. So the space space for which represent you? Yeah, I'll go through the definition check, I guess. you said that was vague

1:17:30 the thing that you end up defining must still be vague and is that because the states have to be they're all superpositions which involve these different momentum modes such that they correspond to a single part of a coherent state or the first term is coherent state established? Definition? The space I end up with is actually precisely defined. Well, that's what I was... That's how you go from this vague... I suppose the point is, it's a vague definition. What it says is, it sort of says, give me a subspace, and if the subspace meets the definition, then it's a particle space. So I can give this definition a very precise space, but what it can only tell me is vaguely, is that a particle space. so in the perfect field theory I developed it fits that definition absolutely precisely and there's no problem, it's quite obvious you can imagine if you had a more interactive theory that it might become a bit less obvious whether the precise defined space satisfied for a theory for instance and it would start getting less and less clear whether it deserved to be called a particle but the space itself can do it to be not defined When you say that the space that you end up defining I mean, isn't it only precisely to find up to the extent to which you're taking localised classical field conclusions? I don't know if I follow, sorry. Oh, I see what you mean, sorry, yes. Space, let me put the duality trend here. The space I defined, let me define the intercourse, let me not clean up. The space I defined is, it doesn't just include the local states. I should have stressed that slightly more. it's, all of the one particle space is local or non-local is in that space, but the important thing about it is because the classical space has a basis of localized states and localized configuration I should say, and because the, because locality is preserved when you move Trinity pictures, it then follows that the quantum space also has a basis of localized states. So we don't mind the fact that a wild-tool-localized particle is better than a wild-tool-localized particle to satisfy an amortistic quantum mechanics and we get that from the fact that there's a base of localized states in the classical case. Okay, but the thing you just had up there was all the states that were localized?

1:20:00 It's not, it's spanned by states that were localized. Oh, okay. That's very crucial, it's too strong to require all states localized, but it may be the case that if I check where a particle is, it better not be here and here at the same time. in a certain sense of course it is but it can't be the case that I find it here and here so you're now saying that the previous coloured slide where the conceptions were in the way you said localised in red A that localised is the classical straightforward notion of just supporting red A it is here this is classical Yes, and so the idea is that this construction, well, it gives a very precise part of the Hilden space of the bottom, and that precisely specified Hilden space meets the vague definition of a one-partible state that we had before. when you allow the width of red A to correspond to your L effective length L in your I just want to clarify in the picture I've got here if you equally imagine rather than the classical modes here you put some classical localised states in then you'd have a classical basis here in which you make any state you liked any state you liked here, and the localised ones would give localised particles here, and because the linear structure is the same at the top as it is at the bottom, then the states that were a classical basis would become a quantum basis that would satisfy the particle of that. Yes, in a sense, the part analog of this momentum-ish right column would correspond closer to the position to the other's lines. Yes, yes, that's true. Or the spirit of your For a classical weight packet, which is a solution to the classical Klein-Gordon equation, then if it's localised in small region L at some time,

1:22:30 then at some later time it's going to be isolated only within the forward light cone. So I don't see that the localisation is preserved case either. It's not and it shouldn't be either. Oh I thought one of the principles of the space as a whole needs to be preserved but the position I can say don't need to be preserved. And in fact they may not be preserved because again in the multivistic quantum mechanics if I take a very localized state it's going to be fast. So the point is although it will spread out it will spread out into a superposition of localized state. And again that contrasts with what you could have happening. That's not an obvious fact. It need to be the case and it occurs because the The reason it's closed is because the space is closed dynamically. The classical space is closed. And therefore the constant state. Because that whole diagram can be used to time as well. I mean, this is done on solutions, no matter what time I do it. Because this space is closed, trivially closed, because it's the whole space, then this space is also dynamically closed. And that's much more, that's much less trivial, of course, because it's a proper subspace. so that's a satisfying space strongly in that requirement that the mechanical space is closed forever in a sense it's saying appropriate timescales is a luxury that was slightly why I'm going to put myself back in the queue at the end this is a slightly different direction, I guess it's two related questions, the lead question is in dealing with three fields that you, if you went to an interactive field, you wouldn't have this construction in one particle states because then you would have one in many particle states. Exactly because you're not the company. Instead, you wouldn't have them clean in there. Right. But I guess that would be one concern if you... Well, it looks to me like your construction is very similar to thinking in terms of asymptotic states. In the, you know, the old quantum field theory picture when the first comes out, Mathematically has always been very difficult, especially in interacting field theory. But another problem there, aside from the particle accounting problem, is that the vacuum in the interacting field theory would presumably be quite non-local. So how do the definitions carry over? OK, well, several half-hands to that.

1:25:00 First answer, yes, it's very much in the asymptotology. In the paper I've got, I've got this then, I'd say explicitly, you can run this quite carefully in the asymptotic picture, insofar as you buy in the asymptotic picture. that's not totally satisfactory because I think we actually, we don't just need a concept of asymptotic particle in relativistic collision theory we'd be happier with the asymptotic we might be perfectly happy to say while the collision's going on there's a lot of facts about particles, there's just some junk but clearly in non-relativistic physics there's got to be a good sense of non-scattering situations where the particles are still beginning at the moment, that glass that silicon atoms are not freely moving but they're also fairly linear that's silicon atoms, so we've got to get further on that In terms of the mathematical definitions, it's true that in the algebraic on a field theory perspective, it's very difficult to do that and to make that's not a bit of work. That's not a problem for page-to-date renormalization, seriously, because the problem occurs because of things like Harkspin, Harkspin occurs because of degrees of freedom. renormalisation and by imposition cut-off truncates those degrees of freedom and I think if you're prepared to accept the cleanliness and foundational relevance of the normalisation programme then I think you can do this safely but I don't want to get into the defence of that because I guess it's a long way off topic. I agree that in AQFT there's a problem but in a certain sense it is AQFT's problem Well but isn't that an assumption there's an assumption that another formulation of the field theory would present a reasonable foundation for mapping this picture onto the asymptote. Well, I think there's a problem, you have to say, does the, what you might call, mainstream formulation of quantum field theory, the actual calculation one, is that, does that deserve to be called a proper formulation, or is it just a heuristic data? Well, there's no mathematical foundation for that, I don't I don't agree with that statement, but I think if I get this, that could go on for a long time. So I think I'll bear out of that. Can I come just and talk about the counter-accounting vacuum point? Yes, the vacuum is going to be unlocal. The vacuum in pre-QFT is also unlocal. Do you mean that the vacuum is going to be more wild and unlocal? That, I don't know. That's not obvious to me. Well, it would have to account for things like entanglement in the field theory as well,

1:27:30 states in the asymptotic regions. Okay, but even, I mean, the free vacuum is still highly entangled. It's not clear to me that the interacting vacuum would be a more entangled than free vacuum. I mean, at least if we're thinking about the term to free common field theory. And again, that's assuming it can make sense of these things. I guess I'm worried about the definition of the localized states when you're dealing with such a non-local vacuum. That's really what I'm worried about. Okay, I wouldn't fear it marks the definition out enough, because the vacuum, if the vacuum, the pre-vacuum has exponentially dropped a lot of correlations in this sort of theory. If I imagine my toy models, let's say it's interacting with my thought, and then again, as far as I understand it, you can calculate the turbotipity of the interaction, that vacuum, and I think again it will drop off exponentially, mouse, not the bear mouse, but it's still going to have the same problem. What we're not going to have, what would cause problems, would be a situation where, for instance, there's the correlations between the vacuum here and in Andromeda are just as bad as the correlations between the vacuum here and around Jeremy. So, I mean, that sort of behavior would cause problems, but I don't think that would be a problem here. Go into a wild amount to appeal to it, and I think, yes, all hell breaks loose, and I don't know what would happen to this little picture. I'm not clear what work conceptual work the coherent state is doing for you, this middle line it seems to me you can step from the classical wave packet to the localised particle state perfectly happily without introducing the idea of the coherent state at all yeah, you can, but you have to think that's a good thing to do if you know in advance what the particle space is it doesn't take long to check that it is in fact particle space and the method you use to check that doesn't engage with the given space at all so you're quite right what I'm trying to do is motivate why in advance of being given that space we should perfect climate and this is one route to it I could show you another route to it that, where again it would be a very natural construction to make

1:30:00 which wouldn't engage with the human space which again would be very indirect I mean it's sort of dependent, you can debate the relevance of playing this sort of game at all it's certainly, it's straightforward to say well look one particle space is made by the action of the created is spanned by the action of each single creation operator on the vacuum it takes very little time to check that's right, to me at least that gives you rather a little insight into why you would expect that particular space to have the process it does. So that's essentially what I'm doing. It's a heuristic, it's pedagogical. I mean, the difficulty I had with the current period of state was that the process of approximation is not analytically clear what's going on there at all, although, you know, one can wave one's hands around that process, and I'm happy with the end product, but that step from the middle line to the bottom line, It leads to one quite a concern. This is my conjectural step in the slide. Yes, conjectural, yes. Well, I mean, yeah, it's unclear, and you can put it in the way you want to, and there's some justice in that. Essentially, I suppose, I would say pedagogia is in the way to some extent. A sensible derivation, a sensible understanding of where something comes from has got to be a combination of seeing the hard maths with getting a good intuitive understanding of why you did the mathematical steps you did. Now, clearly, if that conjecture was an incredibly complicated mathematical transformation or worth it, it was quite wrong, then it would have to be worth nothing. But given we have mathematical steps, it's a place to try and see what the motivation is. And the motivation you tend to have in QFT textbook level is just to say, well, it's to go straight from the... that it's that, the Hamiltonian looks like that, you say that Hamiltonian is the same Hamiltonian as a collection of harmonic oscillators, or it's the Fox-based quantization of the second quantization theory, and you say therefore they're particles. Now I find that kind of satisfactory because in making that step on losing track of locality entirely, locality enters the picture a priori in a very different way in Fox-based second quantization and in So it's trying to do a heuristic in which locality is always the form. I think you come to this thing again. There's no one true heuristic.

1:32:30 This is a heuristic focused on position. Jeeva's heuristic was focused on the key representations. The QFT heuristic tends to be via fox space. Ultimately, at the end of the day, your heuristic gives you a subspace and you check it's got the property you're supposed to have. is based via history of his locality, always inside. Localization, I suppose. Localization, I suppose. Going back to your computation diagram, David. The final one. What's the trouble would we be in in terms of boson phenomenology if we couldn't find such a computation? Well, I think in the absence of such computation relation or in the absence of an even approximation to it, I think we'd have a field theory that didn't have a particle subspace. I think we'd have a field theory which couldn't be analysed in particular language. There's nothing wrong with that conceptually. What I think is pretty much standard in classical physics these days is, I have to say particle is somewhere between in a fashion of a parallel and an emerging concept. I guess that's as much a philosophy question as anything else. But if there aren't any particles, it doesn't matter. It would be sort of like finding that, I don't know, if the planet didn't support life, it might be a bit depressing in various aspects, but it wouldn't be a paradox. Whether that actually occurs, I think it's possible that there are regimes of field theories in which particle language doesn't do anything good. I think there's some reason to think that lattice quantum field theory is such a regime. In quantum chromodynamics, there's a regime which quarks, you can think about direct quark, quark scattering, and it's particulate. There's another regime where you've got protons and neutrons, and it's particulate. I'm not aware of anyone having found a direct particle language way of talking about this. Quark can find that seems to do much good, you tend to be talking directly in terms of fields there. But, I mean, I'm out of my depth with those questions. There's no experiment involving this, which would... which would suggest the existence of such a connotation for a figure. I mean, this is not being pushed by experiment. I don't think it is. I think there's no, I mean, I suppose the experimental evidence

1:35:00 is indirect. It's basically there is ample experimental evidence. Modulo, the incoherent, the increase of projections that for the quantum field is we can hear. If you're prepared to ignore that, then there's ample experimental evidence quantum field theory is basically right for a very wide range of phenomena and all this is analytic from that starting point. So it's pretty indirect but I think I don't know how it makes sense of testing it. I mean a test could basically be to invalidate the entire usage. I think it's more the case that if you could think of such a hypothetical test the test would destroy the foundation because both of them are going to use an act. That's it. Oh, sorry. In high-end experiments, I'm not sure the localized part of the Do I think that your picture is an account of how such a real nice contrast between the rise in the quantum field? And again, I mean, high energy is a good example because that's the picture in which my description works most cleanly. It applies to free-field theory, it applies modulating the validity of the asymptotic I suppose this is a technical question, and it comes from one of your first overheads. In fact, the first time I asked a question, I said, for now. all different definitions all of which oh yes you'll probably want to put that out as soon as I can and I was confused about what you mean by an operator that was like a blind operator operator was going to be some function of field operators and the field operators I should just say quickly again, that picture is not necessarily that friendly to the AQT

1:37:30 perspective either, so, again, that's within the framework of what you've got to do. Just because your operator has to be some function of the fields doesn't mean it has to be a local function of fields that functions with an organizational kernel and fields expressed in lots of different positions. In fact, it usually would have been a genetic model. A genetic model would be something like a model. So in that setting, does localize in A mean all of the integrals that would occur in your kernel are restricted to A, or just at least one of them? If you're integrating over A, a correlation between fields, and you're looking at correlations I don't quite understand what you mean by inside and outside integrals. If you, if you have an integral, the qubits, the quid, the quid, the quid, the quid, the quid, the quid operator at the x and the quid operator at y is the best part of an integral. Oh, I see. So it's like some sort of quadratic operation here. No, I don't expect you like that. I don't want you to have to support it. I only like that. I can strictly like that in the sense that no matter how many integrals you like to be in A, or you're only allowed one integral. no matter how many integrals basically I'm allowing a lot of wiggle room for approximate locality mistakes not allowing any for locality operators in a certain way it's at some point you have to put your foot down and require precision in your definition you don't need to have a ground point which is very precisely but you need to have a ground point that's very precisely defined before you start approximating it for locality is operator localization. Operator localization is precisely fine, and we're not allowing real room here. Can I repeat that? Please. Now, the purpose of my previous question, which was to, not my previous, the one before

1:40:00 that, which was saying that localization to a three-dimensional region only isolates curve was, and your answer came back and I was entirely satisfied, but now your answer to Harvey's question makes me worried again because it seems to me that the kind of localization you're talking about doesn't address the kind of localization we have in the table where the future of the table is not constrained to its future like curve. The future of the table is constrained to more or less this three-dimensional region just move forward in a particular time-like direction and the kind of localization you've been talking about doesn't address that at all and of course in classical physics a linear equation for something like the table at time t would always, would precisely result in the table being dispersed through all of the forward light count but in non-linear equation you would end up with what you can in something like KBV equations and so on and so forth and indeed a soliton equation solutions which move forward in time without dispersing. Now, it seems to me that one can't get that kind of behavior without introducing the analog of linearity in a classical equation, such as, at the very least, that you have in QED. And it seems to me, in fact, that you need QCD kind of phenomenology in order to actually result in the kind of localization that Harvey was talking about. And specifically, it seems to me we never actually observe an electron that is localised, except when it's localised in bi-hadronic matter or something, or, well, hadronic matter. Although, obviously, you're like technically solutions of a linear theory will be found everywhere in the forward-like curve, there won't be found very much in it. if I consider a nice, friendly slow moving wave packet in a fairly well-spreaded wave packet of the type of equation and I watch where it goes although notionally we spread across the whole for light code, it will mostly overwhelmingly, in fact, be still in the same place you expect if you extrapolate it to the classical trajectory so I think it's much too strong to say that a three-field case will get dispersal across the whole for light code the fact that there's a small amount of I think some of them are not directly in quantum field theory.

1:42:30 So I'd say, secondly, that level is enough for Harvey's picture because that's a picture of, leaving aside the mechanism of detection mode, the localisation of detected objects like electrons, whatever, in quantum field theory, is precisely that sort of localisation. Sorry, in scattering experiments. It is that you get past moon trajectories of essentially... So it's the same sort of localization we get in a broadly free or mildly perturbitant number of quantum mechanics. That's enough of that picture, leaving aside the methods of detection. Point three, yes, agreed that that's not strong enough to cope with the solid matter. I don't agree that it's necessary to go to anything as baroque as a non-linear field equation. What we've just got is a non-particular regime which isn't neatly handled by this gravity. purely linear quantum mechanics handle solid matter perfect well and I say 0.4 QED is fine in that picture provided you a QED with or photon charged matter interactions are perfectly fine to explain the table provided you postulate nuclei in a stable game. That's not fundamental but it's still a stable description. Does that answer all the time? That answers I'd agree that just non-linear equations is a particular way of organizing for non-locality. It's not necessarily the only way it might be. But I'd have a problem with the very first thing, I suppose, and that is it seems to me if you say given initial conditions which have support on a region A and you say, well, okay, all possible initial conditions, I don't believe that it's going to be a large subset of those are going to result in only that move forward through time. Dynamically, the vast majority of initial conditions in there are going to end up spread out through the whole lifetime. I'm sure that's true, but this isn't a standard of prepared state. It's not a standard of prepared state. The scattering experiment is a fairly carefully prepared fact. I think the initial state of the scattering experiment is deliberately prepared in a state that does have exactly those properties. And then the fact the state is coming out and also have those properties just consulence essentially

1:45:00 consulence of momentum so I mean certainly none of it's a scatio sure a random localised quantum state is going to spread rapidly across some of the states but the states you throw into a number of the scatio's terms are not random states they need to be carefully selected to do what you want do you want to ask why should those states arrive that's getting more complicated and start talking about the adherence and all that stuff I said I was just saying the fact that my data reconstruction of solid matter is part and parcel of what I was saying about a good complete derivation of particles out of QFT ought to be able to handle the full range of non-relativistic phenomenology and not just the situation where we've got that sort of freedom. The problem with that is that you won't end up wanting to have contact forces, so contact forces are going to be essentially a non-particulate manifestation of the electromagnetic field, that if he's some non-partisan state it's going to manifest as a look at so it gets quite complicated way beyond what I'm going to boldly grab the last question because I don't see anyone I'm a bit puzzled as I sit here about some aspects which relate to the obtaining, broadly about the role of taking a non-relativistic limit And I guess the two places that I'd like to give my comments on are that early comment where you said, on the slide, something like, all the non-relativistic states are superpositions of localised. And that led us to require of the one particle space that it be spanned by the localised. and later on when we were examining the coherent states as a candidate for the quantum analog for the beam of the one-party state quantum analog of classically left-wise field they failed because the sum of coherent states around phi 1 and phi 2 classical configurations wasn't itself a coherent state it was a quantum superposition Right. Now, in a sense, my question is, why should we especially see that if we're just looking for some kind of localized quantum state or quantum cousin of a localized classical state,

1:47:30 why should we care about the sum the interaction of that representation with sum with quantum sum I think your heuristic or pedagogy is very fascinating the interplay between these two classes the classical and quantum class but there isn't anything a priori in the concept of localization that in my opinion just off the top of my head that forces us to the desideratum that all our non-relativistic states must be superpositions of localised. I agree that's actually true in this act, but it doesn't have to be, and it doesn't on the face of it have to be about quantum analogues of classical localised states that quantum sums of such quantum analogues must be themselves analogues. And I think, in a sense, it's true, because one particle says they're not really big quantum analogues of classical way-packing. They're the same accounting properties, but even much else is a problem. So the quantum one particle? The one particle is not very good analogues. They're analogues rather than approximations. Okay, I suppose the response would be, we required that superposition to hold by fiat, because we wanted, by fiat, localisation to be a linear property. In other words, we wanted superpositions of localised state to remain localised. That's over time? No, to be localised immediately. We required a subspace that got around the achievement here. So we required that it had a state localised here, another localised here, but there's some. So, that property is by its nature a property of space estates. So, if we're looking for a space of, if we're looking for a non-meltive subspace, then we need a problem, I think, if we don't have, if the coherent states don't really move us towards getting that sort of subspace. I should say I've heard cases of coherent states in the light surface as far themselves actually are also perfectly good, heuristically useful

1:50:00 devices, I mean, my desiccation of particles, what particles again are are ordained, it's because particles are analytically powerful way of feeling certain situations, now, there are some situations in coherent states that also do that in the electromagnetic field I mean, the best way of understanding, I would say, classical electromagnetic waves is asking, from a proper perspective, is that in the States. So that's, they're probably given as an approximation, but I suppose I was looking quite specifically for the particle sector. And it is a problem, as I said, it is a problem in particles that they have these, this multiplication. there's no reason why it should be particles there's no reason why non-relativistic physics should be articulate in that way I think it's a more natural way to implement non-relativistic quantum physics but it's no more than that it's a slightly disordered answer to that sort of address I don't see If I may, I'd like to ask everybody to thank them very much.