Transcription
Thank you. Uh yes, so thanks whoops to the organizers for inviting me to this uh lovely conference in lovely city. Um so Griffith University for those of you don't know is here uh on the East Coast of Australia. Uh that's where I am and that's where Michael Hall is. Uh the other co-author on this work uh and I should say that I I I guess these co-authors uh just refer to the last part of uh my talk. Uh but the other one is uh Dirk André Deckert who's currently at University of California, Davis.
Okay, so um as you can see I'm am ambitiously trying to cover a lot uh in my talk and a lot of things that uh have come up elsewhere uh and will continue to come up elsewhere in in the conference. Uh and so I'm I'm going to basically try to cover the first two topics fairly briefly. Uh that is weak values uh and their relation to bohmian-like trajectories. And then for the second half of my talk uh hopefully uh talk mainly about this topic of many interacting worlds.
Okay, so weak values have been introduced by uh or been mentioned I should say by a couple of people, but um I'm not sure there's been much of an introduction. So this is uh uh just a couple of slides explaining what weak values are. So they're introduced in this uh paper from 1998 by Aharonov, Albert, and Vaidman. Uh and uh so this is uh my explanation of what they did. So essentially they're considering a system uh coupled to a probe. So the probe is just going to be treated as a particle with a position and momentum uh prepared initially in a minimum uncertainty state. So uh that means that the probe state is defined by just three numbers. The uh mean initial momentum, the mean initial position of the probe, and the uncertainty in the momentum of the probe uh because being a minimum uncertainty state the the uncertainty in the position is just uh like the reciprocal of that essentially. Uh and so yes, so this system is going to be coupled to some operator in particular of the system, some arbitrary operator A, uh by a von Neumann style of of coupling like this. Uh so that couple is basically causes a force uh on the uh probe here so that the after the interaction, the momentum of the probe uh has shifted. So this is these are Heisenberg equations uh relating just before and after uh and saying basically that the the momentum of the probe shifts by an amount exactly equal to the operator A for the system. Uh so what that means is that we can use this probe to estimate that operator A for the system. And in particular, I can simply measure the final momentum of the probe here uh and take off the I can't take off Ideally, if I could take off the initial operator for the momentum of the probe, then I would get exactly A. But of course, I can't, you know, uh in in the lab subtract an operator from a measurement result. Uh all I can do is subtract the mean of that operator, which I know, the mean of the initial momentum, and use that as my estimate for uh the this observable A for the system.
Okay, so if I do this experiment over and over again, uh every time preparing the system in the same initial state psi in, uh then what you'll find is the the expected value of that estimate is exactly what quantum mechanics gives for uh an an average value for an operator. Okay, that that the Born rule uh expression there. Okay, so there's nothing particularly interesting in that. Uh but it becomes more interesting when we also consider uh not just making this momentum in here, but we have the system has gone through this interaction. Uh we can still do something to the system. Uh so for example, making a final measurement here onto projecting onto the state phi f. Okay? Uh and so then we can also say, well, when I say projecting, what I mean is we make a measurement and it may or may not project the system onto state phi f. But let's consider the sub-ensemble where it does. Okay, where I get the result uh phi that yes, it was in state phi f at that point. Uh and so then I can consider So I still have this information down here on by which I'm estimating uh A. Uh but now I can consider that that expectation value on the sub-ensemble where not only did I prepare in the state psi in, but I projected on the state phi f finally. Uh And so what uh Aharonov, Albert, and Vaidman sh and Vaidman showed uh was that in the limit of a weak measurement, which is sigma p going to infinity, that expectation value, the pre- and post-selected expectation value, uh is equal to uh the this expression here, the real part of this uh ratio. So the point about that is that uh this ratio can be arbitrarily large, okay? Because on the bottom uh is this this uh inner product between the initial and final states. that's small, then this can be very large. Uh and that's why, and I should have pointed it out uh two slides ago, that's why uh they could write a paper with the title like this of how the result of a measurement of a component of the spin of the spin half particle can be turn out to be 100. Okay, so what they meant by that was that the the weak value the these expectation values defined here could be uh 100. It could be anything you like.
Um okay, but why is this the we I just called it the weak measurement limit uh and I called this thing a weak value, uh which is what they called it. Uh but why is this the weak measurement limit? Uh well, it's because there are two ways of understanding and they're both in a sort of necessary to understand. Uh so first it's weak in the sense that you get very little information in any measurement that you make. Uh and that's weak can see by looking at uh the operator for this estimate of A. Uh and so that the operator is the actual observable A plus this uh term, which is the the difference between the initial momentum of the probe and its mean value. So in the limit where uh the uncertainty in the moment initial momentum of the probe goes to infinity, clearly this operator here, this this contamination, this noise term here uh has a very large mean square value uh which goes to infinity uh in the absolutely weak limit. Uh but it's also weak, and this is crucial, not only uh So this is necessary to to have this to call it a weak measurement, but it's also necessary to for it to be weak in the sense of causing uh very little disturbance to the system. Okay, and that's a consequence of the assumptions we made, because I can look at an arbitrary system operator. I can look at its final value for that system operator. Turns out to be given by this expression here. Uh and so the it involves these initial position of the probe. Uh and precisely in the limit where sigma p goes to infinity is where this q the mean square value of q in goes to zero, because it's simply the reciprocal of that um because it's a minimum uncertainty state. Okay, so that that's why it's called a weak measurement. Uh and and that's why we call this thing a weak value. Uh and so the uh So the point is that yeah, it's but it's still a a value obtained in the same ways we obtain expectation values in any experiment. We have to repeat the experiment many times. And in fact, it because we're dealing with a weak measurement, we have to repeat the the experiment even more times than you normally would, because every individual result has a lot of noise in it. So essentially uh you have to do uh even more work, but the basic idea is the same for for getting the ensemble average.
Okay, uh so so weak values have been mentioned a number of times in this conference. Um so this is just some general remarks I said about, you know, what why they're coming up. You know, what what's at least what's my opinion about why weak values turn up in these sort of discussions of fundamental questions in quantum mechanics. Um and so I think I think the for me the answer is that uh there are a lot of questions that we can ask classically we'd have no trouble um understanding, but in quantum mechanics, there just is no answer, or at least there maybe there are it's an ambiguous answer. Uh and so what do we do? And and so what I think is the advantage of weak values is that they they do offer answers. Um Uh in some cases it may be a completely new answer to the question. In other cases, it might uh single out one of the answers which has already been proposed. Um but in in either case, I think that this gives new insights into the phenomenon and can prompt new research. Uh even if it doesn't, at least what it can do is enable an experiment to be done. And that's a good thing in itself, uh not least because it uh typically brings the the issue in the foundational issues to the attention of a much broader audience than theoretical papers typically do. And so there are many examples of of of this. I'm not going to go through them. Some of them you've heard about already. Some of them I'm sure uh you'll be hearing about um later today. Um and yeah, I'll just go on that. But what I'm going to talk about is the application uh to measuring bohmian-like trajectories.
Okay, so so bohmian mechanics has also been introduced um today, um or maybe even yesterday. Uh so this is the way I'm going to describe it is that, what we have is a So this is a I'm going to call it a a world configuration. So I'm considering a system, could be many particles. Uh and so bohmian mechanics says that you have a definite configuration, which evolves according to some uh velocity, which is determined by the wave function. Okay, that's broadly um what bohmian mechanics is about. Now the point is that there are actually infinitely many velocity functions of the velocities which I could choose which will satisfy this continuity equation. So, so why do we why choose the particular one that that is the standard one in Bohmian mechanics? Well, what I suggested a few years ago now in this paper in New Journal of Physics was that we can motivate a particular expression for the velocity in terms of weak values by considering Well, what would it mean to to measure the velocity of a particle if you were a completely naive experimentalist who didn't know anything about quantum mechanics? Well, you might discover that in general making measurements seems to do something nasty to your system, but nevertheless you could also make make the measurements weaker and then they seem to be less nasty. And so then you could say, "Okay, so the definition of velocity is the difference between two positions divided by the time elapsed. And I want the velocity at a particular point, so I'm going to take that that time elapsed to zero. And but I want to make the first measurement not disturbing, okay? So, I'm going to make a a weak measurement at time t, then a strong measure We're having a disturbing disturbing measurement going on there. A weak measurement at time t, a strong measurement a short time later, look at the difference between those those two results conditioned on the strong measurement showing that the particle is at point x, which is the point at which we want to know the velocity. And so in terms of weak values I can write it like that. And when I work that through, it turns out that this gives you the standard Bohmian expression for for the velocity. But more more interestingly, as I pointed out in this paper, this is as I was saying something you can actually go and do an experiment for. And so we'll be probably seeing this again. But so this was the experiment done in the lab of Ephraim Steinberg, the Kocer paper published in Science a couple of years ago. So, they you probably hear they actually did things slightly differently from what I just described, but nevertheless the basic idea is is is similar.
Um So, I I do want to stress and and I guess the other people as well that this is not actually following the motion of individual particles. This is patching together little bits of velocities measured locally at at individual points, okay? So, these are reconstructed velocities because the theory doesn't say that you can actually follow the particles. It just says you can measure the velocity at any given point. Okay, so I haven't yet shown you the expression for that velocity, and those who are familiar with Bohmian mechanics will know what it is. But what we could ask a broader question is in general, what does that formula give that I wrote up here? What does this give for the velocity? And the answer is this expression here. So, here I've made no assumptions about the about any about the Hamiltonian about dynamics at all. So, everything is just written in terms of the Hamiltonian and the position operator or position um states. So, the question you could ask is is this always actually consistent with quantum mechanics? Does this form of the velocity in fact propagate an initial probability distribution for the the configuration to a final probability distribution which still agrees with quantum mechanics if the initial one does? And and the answer is it does if and only if the Hamiltonian here is at most quadratic in operators which are canonically conjugate to the position operator or the configuration operator more general. And and so the the you could say, "Well, is this isn't this a limitation of the approach?" Because it it only has limited validity. And I'd say no, because it turns out that well, of course as we know that all physical Hamiltonians are so constrained. Basically all the Hamiltonians we know of are quadratic or well, at most quadratic in the momentum variables, and that's why this the formalism all works. Of course, that works as long as we take the hidden variable as long as we take x to be the configuration, to be the position of all the particles in the world. So, rather than being a limitation on the theory, I actually see this as as a a boon in the sense that it tells you that you have to choose the hidden variable to be the configuration. You can't choose it to be the momentum, for example, and say that the reality of the world lies in the momentum of particles. If this theory is going to work, the reality has to lie in the positions of the particles because of the nature of the Hamiltonian, and that's a an objective fact of the world as far as we know that what the Hamiltonians are like.
Okay, so does this actually prove then that Bohmian mechanics is correct and the positions of the particles must be the reality? So, so I want to be clear I'm absolutely not claiming that, okay? Because this whole thing is predicated on the idea that I have to choose some hidden variable, and of course not everybody believes with that. But what I and and certainly even if you do, you don't have to believe that its velocity is governed by weak value, which can be experimentally measured. But what I think it shows is that that it's not it doesn't show the theory is correct, but I think it shows that it's self-substantiating in the sense that Bohmian mechanics, well, if you choose this, then you get a theory in which if you go out and do an experiment according to that theory, you would measure a weak value which would be in accord with the velocity in the theory, and so that makes it I think a very attractive and natural theory. But it doesn't prove that it's necessarily correct.
Okay, so going on then to to the last topic in my talk. So, so I've just Here's a picture again of those trajectories. Here's another one which is this that's purely theoretical one. So, you've see everyone's seen pictures like this before. Um If you if you didn't know anything about quantum physics or whatever, what you might look at these pictures at and and think is that this is describing a a bunch of particles which repel one another because you can see the trajectories don't cross. So, if you were completely Yeah, if you didn't know anything about streamlines or anything, if you you just might say, "Oh, I've got a whole bunch of particles here and and something they've got to an interaction repelling them so that they don't cross." So, of course, you know, that's what we'd normally say is we we don't take this literally. There's no actual bunch of particles, there's only one particle. I'm just plotting the different possible paths it could be taking. But why? Why not take this literally? Um and so this in fact So, we heard from Bill Poirier this morning. I'm very glad to to admit Bill now. Uh on this idea of taking literally the idea that you have an ensemble of particles all obeying Bohmian mechanics or something like Bohmian mechanics rather than just an individual one. And so so as you said, the work was published last year. So, so actually around the same time this was published and not knowing about his work, I with and Michael Hall and and Dirk André Deckert um had a very similar idea, um but it's different in in I think one crucial way at least in in that for us at least it's difficult to imagine what a continuous ensemble actually means. And so what we're trying to do is to develop a theory for a finite but extremely large ensemble of particles. Well, I shouldn't say ensemble of worlds I should say describing, for example, in this case even a single particle. So, so let me jump to the the one-dimensional case just to show what what I'm thinking about. So, I'm describing a a single particle here just as those pictures are meant to be describing, but it's a many worlds theory of that single particle, okay? So, so a single world in this in this nomenclature is a single non-relativistic particle with certain mass. It could be in some potential. But what I'm going to consider is many worlds, okay? So, so I have therefore many positions that I have to consider. It's in one dimension, so I can order those positions. So, xn + 1 is greater than xn. And I'm going to assume here that those that those worlds are going to be arranged with a slowly varying interworld separation. So, they're fairly evenly I mean, they don't have to be evenly spaced, but if when they become the spacing itself only changes slowly. So, in other words, the difference between spacings between neighboring worlds is changes what Yeah, the difference is about order n minus n to the minus one where n is the number of worlds. I'm also going to assume that the initial velocities of those worlds smoothly vary in the same sense, okay? That the if you have two worlds which start off close together, their velocities are going to be close as well.
Um Okay, so I can think of these initial spacings and initial velocities as defining a wave function according to the usual Bohmian If I take those velocities and one over the separations as being some representation of density, then I can write down a wave function which corresponds to these initial conditions, okay? But the crucial point here is that I'm talking about a theory in which there is no wave function, okay? There is only the ensemble of of worlds with their initial positions and their initial velocities, okay? So, that's the crucial thing. There is no wave function in the ontology of our theory. Our ensemble of worlds is a real ensemble, not a virtual ensemble, Uh and it's necessary that they all be real because we're going to postulate that those worlds interact. Okay? So, this is quite different from the Bohmian approach. Um we're actually postulating then that we have this equation of motion. So, uh this is the Newtonian force and then this is a non-Newtonian force. Extremely non-Newtonian in the sense that it's actually interacting worlds, interacting parallel universes, if you like. Uh and so, what it's saying is that this is a a I call it a three-body uh interaction, but of course, that's you know, three the the it's a interaction between three nearby parallel universes uh in this case. Uh and so, when I call the local potential, it's local only in the sense that uh nearby parallel universes uh interact and they but they don't interact with far away parallel universes. But, it's obviously not local in any uh ordinary sense of that word. So, the specific uh three-body potential which we consider uh looks like this one here. So, it involves H bar, as you can see. So, it's a quantum uh effect. Uh and it it it Well, I I won't go into details. You can stare at that for a couple of seconds if you want and and see if you can make anything of it. Um but but it's I it's not that obvious what what you can do. But any in any case, it's clearly Well, I suppose one thing you can take from it is you can see that it blows up uh if the particles get too close together. Okay? So, if if nearby universes, parallel universes, nearby worlds, if the if the particle uh position gets too close together, then this potential blows up. And so, of course, this is going to prevent that from happening.
Um and what we believe is that with those initial conditions I talked about, if we take the limit N N goes to infinity, uh that we believe that this potential will actually result in an ensemble which is equivalent to the the virtual Bohmian ensemble uh for that uh for that wave function. And so, the here's some um actual solutions of that equation. Uh so, I hope you can see that looks reasonably good from here. Uh so, this is just solving those dynamical equations for 100 worlds, so not very many at all. Uh and and what you can see, so I should point out so the the the overall shading is just actually the mod squared of the wave function, where the wave function was defined as I said before. Uh the red curves, the nice smooth one red curves, is the Bohmian ensemble. But, this is just a this is just picking, I don't know, 10 12 how many worlds there are here. Just some sub-ensemble of the actual 100 which we uh which we generated, which which is these blue curves here. So, what you see is that after some break time, at least, which is around about here, uh those solving those uh that many interacting world equations does give you good agreement with the Bohmian equations guided by the wave function. In our theory, there is no wave function. It's just the worlds interacting with neighboring worlds. Uh after that break time, things go haywire. Um but but I'm told Dirk actually is one who did this uh these simulations that uh as you increase N, that break time becomes longer. Uh so, it doesn't seem to be any sort of fundamental thing uh as far as we can tell uh at the moment. So, basically, it seems like it it's promising. The idea seems to work. Uh at least, it seems to be able to reproduce uh to some extent the interference pattern uh from a double-slit experiment. Okay. So, I think I I'm probably getting near to Yeah. So, I'll rush I'll Oh, okay. Just so you point out So, there's a bunch of things which we can show analytically uh from our our many interacting world uh dynamics. Uh we can show Ehrenfest theorems. We can show uh various things which are common in classical and quantum mechanics. Uh we can explain uh effects like non-classical barrier penetration. Uh um and we can solve some simple systems like harmonic oscillators and it all seems to uh work with them. Uh I I just say uh very briefly that we've thought about how you generalize this beyond the simple one particle in one dimension. Uh and it becomes a lot more complicated. Uh we don't have any explicit form of the of the potential uh that we know would work. Um but the basic idea would be is captured by this equation here that every world again obeys a Newtonian This is This is just Newton's equation. Uh so, the only thing which we're doing is adding some quantum force, which is exactly the force from the Bohm's quantum potential. But, we're imagining that that quantum force is determined by some sort of local averaging of the the density of worlds uh in in the region for the where that particular world is.
Uh, okay. So, it's probably important to say something about how we understand this theory. So, I want to as I stressed before, like that we are considering a real ensemble of worlds. So, all worlds are equally real. Uh your consciousness supervenes on only one of those worlds. Uh and if you think that's odd, well, you should just remember that of course, in this world, your consciousness only supervenes on one part of it as well, which is you. Uh so, I don't think that that's uh that's particularly surprising. Uh so, there's no wave function. So, of course, there's no wave function collapse. Uh so, the effective wave function collapse is just a consciousness on one world updating its idea about which world it is likely to be supervened upon, if we put it that way. Uh and so so, yeah, we in the same way that Bohmian mechanics showed that the regular many uh many worlds interpretation uh is believed to agree with standard quantum mechanics, we think that our theory would be the same. Uh I'll leave that thing. Okay. So, let's move on to the conclusion. So, I've talked about weak values and and uh how they can shed light on fundamental questions, in particular, uh picking out a unique Bohmian velocity. Uh and when you look at these pictures, this suggests, at least to me, uh this this idea of uh an ensemble of worlds with mutual repulsive interaction between nearby ones. And what we've done is shown that that intuition can be made precise, at least in the one-dimensional case with an explicit uh three-body interaction. Um and clearly, there's a lot of work remains to be done to see if this is a viable theory. Thank you. Thank you for your clear presentation and sticking to the time so well. So, we still have time now for a few questions. Just please. Uh only two posts last speakers. May I ask you your slits or when you draw these trajectories, they are Gaussian slits or have you rectangular slits? Or is there a difference or what would be the difference on that? Yeah. This So, this simulation, I think, is Gaussian slits. Uh I think the back here, these uh probably rectangular slits in um Actually, I'm not 100% Yeah, this case is probably also Gaussian in the experiment. Um Do you want to comment, Boris? Sure. I mean, this the these are essentially Gaussians, but they're truncated by some physical object. So. Okay. So, um So, it sounds like the answer to your question is basically Gaussian in most cases. Would there be a strong difference or you would not say too much? Okay. Okay. Um Yes, so I will ask my question. Um I'm not so much pleased by your use of the word worlds uh in general. I have to deal with one world, but um Let me ask you the When you are in one world, but you have to deal with me. to stay here. Um Uh now, let me ask my question in the following way. Uh suppose you will describe the air molecules in this uh in this room, this ensemble of uh particles. Would you also then use the words many worlds Absolutely not. No. No, because because an air molecule is not a world. Um So, the point is that a a world is uh something extremely rich. Um So, the this is a core the what I talked about here is a toy model, of course, of a world of the one-dimensional particle. I mean, the real theory, of course, is that uh a world corresponds to this configuration, which is the configuration of all the particles in the world. So, all the air molecules in all, you know, We take this We take this room. That's enough for me. Yeah. Okay. Sure. So, all the air molecules and all the molecules in the world. So, a nearby world will be one in which all of those particles are in almost the same positions. This is called the state or so in in phase space in classical phase space. well, it's in configuration space rather than phase space here, but but yeah, sure. I mean, I called it a configuration, but I I call it world because this is the term that's used in the many worlds interpretation. And I think that it has it's basically a similar idea to that. Exactly for that reason I'm unhappy about it, but okay. Now, regarding the issue of the of the slit, yes, there there are important differences. I mean, it's not the same to compute with a square. I mean, just total um transmission than with with a with a Gaussian because the function I mean, diffraction or function itself is going to cause you a modulation of the interference. So, for example, in the case of um a full transmission, what do obtain is a very fast flow of a I mean quantum flow flow of slits. So it makes that the that in between the two slits the emergence of the of the pattern I mean the this non-crossing appears before. With the Gaussian you get a very fast attenuation so you you don't see this sort of secondary maximum and so on. In the case of the experiment for example we tried I mean we tried with Gaussians and it was very well but then if you truncate the Gaussians you get some oscillations in the side that are that they very nicely fit those initial oscillations that can can be seen in the in the in the in the experiment. Now for example we are we are working out a model that dates back to the to the 80s by Starace and other people from laser laser physics that can describe very well the experiment and this is with Gaussian Gaussian beams.