Relative QM measurment and the ontology of space-time structure (contd.)

0:00 To do quantization of electrodynamics in a manifestly gauged invariant polarity form, it proceeds as follows. Here are the Maxwell equations. This piece is classically known. I introduce that particular combination of the 4-gradient. This is the partial with respect to X-mu and then I subtract off A-mu, A-lambda, partial lambda, where A-n is of course referring to a particular hyperplane orientation. Then I explicitly build an A mu of x and eta, which is now not a local field, it becomes a hyperplane-dependent field by construction, by explicit construction. Here's the F nu mu of x prime. I apply this d, but a differential operator now with respect to x prime. I do an integration over all of spacetime, but I confine that integration to a particular space like hyperplane of orientation eta passing through this pre-variable point x. I divide by that Cauchy denominator and you can show very easily that that constructed beast satisfies the equations that we normally associate with the vector potential for its relationship with the field tensor. The field tensor is the local field, the immediately physically significant local field. Hyperplane Dependent Vector Potential explicitly constructed algorithmically moves and is very non-locally related to it, but also there is no gauge dependence in it. That AMU satisfies the matrix equation with its significance is clear, it's just the application of that truncated meaning A by ring, but can be explicit.

2:30 Quantization goes through a function on that part of X which is part of A, and the eta guarantees the operator The satisfaction of this constraining way, in fact, of the explicit construction of Phi and A, manifestly covariant, albeit the price you pay, is hyper-plane dependent. My contention is, of course, not something unacceptable. It is something which it seems useless to cover and line all of the basic ideas of my lovely pale. So, those are the civil things, my friends.

5:00 Again, very quickly, I'm going to give a better entry. Some of the tools we've presented are much, I think, clearer to the font and the time of this question, that you do have causal evolution in a higher dimensional space, as the causal connection in the lower dimensional visual values, and in a more general way, I think I can state with some of your tools, on the issue of causal evolution in the case of physics. Well, in as much as that, it appears to be a response to Simon rather than a question to Max. Michael? I have a question about how you get these two different values into a four-point function, in your Indian case. It seems to me, just to kind of be intuitive about it, the problem is going on, is that you kind of snuggle in, not just ordinary type of evolution, done by some social theory or some other kind of equation, But you have to start with the projection hypothesis. You haven't ever mentioned the quantum projection, but we're going to cut that now. Because when you go from psi to this psi-i, psi-i is not what you would get by deforming your hyperplane in accordance. I mean, imagine the dependence on the actual configuration of the hyperplane, if it was a local field there, it would satisfy the Schlinger-Thomann algorithm. I think the best way to... I've got a clear question. What I don't feel would happen is that if you, if you wouldn't have signed up, you'd have a long and tangled date which would sign up. That's right. Well, now my own work would be the information of a hybrid bed which just tilted you which the bed wound its way down under the object. You're thinking of a hypersurface. That's right. It allows you to discuss local information.

7:30 If you did that, then my hunch would be that they wouldn't get a different answer, except insofar... Well, I would answer that in two stages. The first stage will be an answer in which the von Lehmann cut is kept on this side of the apparatus. The second stage will be an answer in which I put the von Lehmann cut on the other side of the apparatus and incorporate the apparatus into the quantum mechanical description. So nothing like a state vector reduction occurs at all. But nevertheless, I will still get the discrepancy. And the way that will work is as follows. First of all, keeping the von Leumann curve on this side of the apparatus, my claim will be, admittedly a claim in the absence of any accompanying calculation, because I have only looked at flat space-like hyperplanes, not curvilinear hypersurfaces, but on the basis of the way they behave, my claim is that if you were to look at a curvilinear hypersurface that wrapped itself around this, you would get the kind of result that you get. In an analytic function theory, when you deform contours, so long as you keep a little curve going around the singularity, however small an extent, all the necessary structure will still appear in the integrals you do. And this, what's important is that these are particle-like measurements, the things that I was calling particle-like measurements. Particle-like measurements you do, and even if you graph the space-like actor surface around there, the effect of this measurement on these portions of the space-like actor surface is the only thing you need. Particle-like measurements are local. That's right, that's right. I think it would work if you were talking about local quantities. Having a four-point function couldn't be better. No, you're absolutely right. It's because of the local character of particle measurements. That's right. In the present, we retain a fundamental status in the absence of measurements, or in the presence of the asymptotic measurements we use in scattering theory. But in the presence of finite time measurements, I think the thoroughgoing...

10:00 Let me make a comment on the way you would describe what's going on if you put a diamond cut on the other side of the apparatus. If you put it on the other side of the apparatus, then what you're talking about is not the Heisenberg picture, you're talking about the Dirac picture. In which, any state, whether pure or mixed, for the subsystem of interest, evolves from hyperplane to hyperplane only in the presence of coupling to the apparatus. It's a giraffe picture. So what you find in that case is the state vector, if it were a direct product state for the apparatus and the subsystem of interest prior to any coupling. Then the state vector here for the subsystem doesn't change at all as the hyperplane approaches either one of these red regions, but then the hyperplane crosses over this and in a continuous way the state, which would now have to be represented by a density operator, The state of the subsystem of interest continuously changes into a mixed state as you go through that region, so that the density operator for the subsystem of interest would be pure on these hyperplanes and mixed on those hyperplanes as a consequence of the coupling of the apparatus. Once again, if you calculated the four-point function, now by taking the trace of the subsystem density operator here for those hyperplanes, And comparing it with the trace of the subsystem density operator over here, for those hyperplanes, you once again get the inequality, even though it's now a continuous causal evolution with no state reduction taking place. ...deluge, even if it will make all the interaction. That would be a very central problem for students. Yes, yes. And I think for doing relativistic quantum measurement theory, it is utterly essential. I have tried to do... I'm not sure I'm completely absorbed. I really know I've heard this three years in a row.

12:30 But anyway, I'm just thinking of the EPR. And you make a measurement just on one system, and then just on one, and I'm wondering how the states are represented in hyperplanes, or what's the hyperplane that collapsed, occurred, and is that the right way to even think about it? And that's related to the question, do you think the correlations would occur in, are independent of what, is your point in there? Yes, yes, yes. Let me ask you the second part first. The correlations, whatever they are, are independent of the frame of reference in the following sense. Whatever they are, the description in any frame of reference from the description in any other frame of reference by the standard unitary transformation from frame of reference to frame of reference. The correlations, however, to be specified must refer to particular space-like hyperplanes, which are invariant geometrical constructs described differently in different frames of reference but done in order to be always talking about the same hyperplane. Now, and now having answered that, I certainly forgot the first part of your question. Well, what I'm thinking of, of course you were thinking in terms of spin, I'm thinking in terms of, let's say, spatial position. Let's think of the original EPR. Now, let's say, where, in what hyperplanes are the correlated predictions taking place? If I localize the product over here, it doesn't get localized and all the other ones are just in certain hyperplanes. I just want to see how you think of that. As a matter of fact, that's a very important point. Here we go. One of the things that made Newton and Wigner very upset about their position operator having discovered it and its eigenvectors was their recognition... That if you localize the particle at a point, x0 at the time t0 in the black frame, therefore from our point of view on that hyperplane, and then you perform the unitary transformation to a relatively moving inertial frame, Newton and Wigner, quite naturally, but alas, quite naively, expected that that unitary transformation of this eigenvector would give a position eigenvector at that time in the transformed prime frame. It didn't at all.

15:00 What they found was that the unitary transform of this position eigenvector at that point was a complete smear. It wasn't a localized state at all. And they, in effect, took out their hands and said, well, I don't know, this position vector, eigenvector, and the position operator came from perfectly plausible hypotheses, and it's uniquely defined, but it has uninterpretable physical properties. But it is not an interpretable physical property because what happens as one goes from that hyperplane to that hyperplane is a dynamical question. And the kinematical question, which can be answered by simply performing a unitary transformation from one inertial frame to another, is the question of how the description of localization on this hyperplane changes as you go to another frame of reference, but you stay on that hyperplane. And it turns out that what this unitary transform of this position eigenvector is, is that it is the position eigenvector that would be used by the blue frame of reference, but not for describing localization at that point at a definite time, but for describing localization at a definite point on this same black hyperplane that the localization originally occurred upon. And that's my reason for emphasizing the absolutely critical necessity for specifying the hyperplane orientation passing through the point as well as the point upon which localization takes place. And once you take that into account, the transformation properties, there are still bizarre counterintuitive properties displayed by these eigenvectors and the position operators. The transformation properties are not among them. The dynamic or evolution properties are. My claim, for which I do not have an airtight proof, is that those properties are only counterintuitive, they are not incompatible with the principle of relativity, they are not incompatible with anything we empirically know, but they are not what we naively expected. So, let me see if I can hear you. In certain frames whose simultaneities like correspond to, you know, to certain hyperplanes, the correlations would be manifest. But in other frames, moving with respect to them, they might not see, they wouldn't see the correlations. They see the correlations in those same hotspots on the Canadian Plains, but they might not see the correlations.

17:30 Yes, they probably would. But if one adopts this point of view, that wouldn't be particularly puzzling. They would recognize that the appropriate hyperplane orientation for describing the correlations are the same hyperplanes that were used by the other observers. Sure. I think it has to be that in a relativistic quantum mechanics, if things collapse or the wave functions seriously, the collapse has to be a long definite hyperplane. It can't, it can't depend on your frame and deal with all whatever you assign it to. They said frame independent. That's right. There is, there is this sense in which there is a profound difference. If the concept of localization was indeed not frame independent, then there would be a sense in which one had violated the principle of relativity. But the concept of localization is frame independent. So long as you specify all the necessary parameters that are needed to characterize the kind of localization that took place, namely on a particular hyperplane. I completely agree with that, and then my only claim would be that when that vector you're introducing is associated with an invariant geometrical structure in spacetime, I would say it's obvious that you haven't violated the principle of relativity, but when you carry around in your head the notion that somehow that vector is picking out a frame... Even in a state-specific way, one gets nervous. I remember a long, long time ago, many years ago, I had a very brief conversation with Professor Wigner. We were both in a hurry, so it couldn't last. And I said to him, a position operator could be salvaged, and it was a very useful concept, and here is how. What we mean by relativistic is independent. I don't think...

20:00 As an operator-valued constraint, it can be in the case of a free particle or a closed system for which the position operator is a global operator designating the center wherever you like, is trivially satisfied that it's not a weak constraint. ...doing its localization across the radial physicals. And the kind of non-localization that is thereby induced on the blue hyperplane passing through that same point is of an exponentially damped type. All of these terms may be used to describe a system of mathematics and physics, but they are not the only things that are used in mathematics.

22:30 The central property of the macroscopic system, the center of mass and the like, the exponential damping, the exponentially damped smearing of localization on a filter type of plane would be utterly unobservable. It's probably only in the case of microscopic particles that one may be able to see this sort of thing, unless people like Leggett get very lucky. These are the fields of observation, which is the deadline. Yes, this is part of the question. Yes, I understand well. Through each point in ordinary space-time, you can pass one hyperplane. Through each point in space-time, you can pass an infinite number of hyperplanes. Well, I mean, at least one. At least one, yes, yes, yes, yes. But if we consider a point, an event going on in a quarter of an instant, then we actually move from going to a region, you know, spacetime, to another region. So actually, how does the result of the evolution of the system depend on... Is that something like... I think I understand your request to go back to this. I wanted to start off the description of the dynamic evolution on this hyperplane. For the time being, let's assume that none of these measurement interaction regions were there. So I'm just interested in a nicely, causally evolving quantum mechanical system in the relativistic domain. No measurements are taking place. Then if I'm in the Heisenberg picture for that system, of course the same state vector gets associated with all of these hyperplanes. There's no change from hyperplane to hyperplane in the state vector assignment. However, if I want to look at a state function, which means I pick out a particular complete set of commuting observables and I look at the complete set of eigenvectors for those commuting observables and I take the inner product of the state vector and the eigenvectors so that I look at something like this.

25:00 I find that that state function, the a's of the eigenvalues of the complete set of commuting observables on the a-thal hyperplane and a-thal specified hyperplane, this state function causally and continuously evolves from one hyperplane to another. Now your question is, what do I have to say initially in order to pin down the dynamical evolution of this state function? And the answer is that on one of these hyperplanes, and it doesn't really matter which one, I've got to give a complete set of Cauchy data for satisfying what in this case would be the Schrodinger-like equations of motion. There are two of them. There's a Schrodinger-like equation of motion for the tau dependence, and there's a Schrodinger-like equation of motion for the eta dependence. They must satisfy interoperability conditions to make them compatible with one another. But I specify a complete set of Cauchy data and I solve partial differential equations and I watch the system evolve. Well, mind you, if I take a given set of Cauchy data and put it on one hyperplane, I'll get one dynamical evolution. If I take that same set of Cauchy data and start the system out on a different hyperplane, that's like the ordinary method of starting the system out at a different time, and perhaps spatially displaced or rotated. I'll get a different dynamical evolution. My question was about specifying additional. Is it not equivalent to saying specifying some bibliographies? Ah, no, not hidden, because the point, in other words, explicating something that prior to my explication might have been called a hidden variable. I don't think so. I think these eta variables, the tau is essentially an alias to t, the eta variables completing the specification of the conditions. Under which you stipulate Cauchy data or the conditions under which you execute measurements. I don't think that that will play any kind of role in either moving us towards a more deterministic understanding of quantum mechanics or playing any of the roles that the hidden variable enthusiasts wanted their hidden variables to play.

27:30 Any kind of, because in the non-relativistic case it would be the same. That's right. The non-relativistic limit of this is the limit in which all the hyperplanes collapse onto an instantaneous slice. Thank you for your attention.