Meaning of Spacetime Foliations

[stglyde]

Only in the simplest case of a single particle. When multiple particles are present, the space the wave function "lives" in is a tensor product of multiple copies of spacetime, one for each particle. So realistically, you can't think of a wave function even in Bohmian Mechanics as propagating in the actual spacetime we perceive.

(Actually, for standard Bohmian Mechanics, which is non-relativistic, even the single-particle wave function isn't a function on spacetime, it's a function on space that evolves in time; time is a parameter, not a coordinate. Multiple-particle wave functions are functions on a tensor product of N copies of space, one for each particle, which evolve in time. I'm assuming that a relativistic version of Bohmian Mechanics would work as I said above, since otherwise it wouldn't be able to match the predictions of standard quantum field theory; I haven't read through the paper linked to earlier that claims to develop such a relativistic version.)

If you have time. Please try to read through the paper which is only 10 pages long and half of it is in simple question and answer and the following is the last paragraph of half of it:

"O: Isn’t it shown that the Bohmian interpretation requires a preferred Lorentz frame?
R: That is true in the usual formulation of the Bohmian interpretation based on the
usual formulation of QM in which time and space are not treated on an equal footing.
When QM is generalized as outlined in 2) above, then the corresponding Bohmian inter-
pretation does not longer require a preferred Lorentz frame.
O: I think I’ve got a general idea now. But I’ll not be convinced until I see the technical
details."

The technical details are summary of what are presented in peer reviewed Physics Review Journal. It's written by he who called himself Demystifier who is here to defend himself.

The contents are radical.. It's perhaps like when Minkowski changed the world when he said "The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality". Demystifier attempted to change the shadows into actual, and even more radical and deserves a Nobel (if he were right).

Anyway if you still don't have time to read it. What do you make of the following where he said time and space should be treated on an equal footing? Don't we treat space and time as equal footing now? Time is in imaginary axis while space is in real axis. Perhaps what he did is make time another space too? (what don't we and if not why do we not do it in the first place?)

"By 2) I mean that time and space should be treated on an equal footing. Note that in the usual formulation of QM, time and space are not treated on an equal footing. First, for one particle described by the wave function psi(x,t), the infinitesimal probability in the usual formulation is |psi|^2d^3 x, while from a symmetric treatment of time and space one expects |psi|^2 d^3 x dt. Second, for n particles the wave function in the usual formulation takes the form (x1, . . . , xn, t), while from a symmetric treatment of time and space one expects (x1, t1, . . . , xn, tn). I formulate QM such that fundamental axioms involve the
expressions above in which time and space are treated symmetrically, and show that the usual formulation corresponds to a special case."

http://xxx.lanl.gov/abs/1002.3226

 

[harrylin]

Have you not read Maudlin article called "Non-Local Correlations in Quantum Theory: How the Trick Might Be Done".

? That's the article of this thread to which I referred after I read it; I don't understand the new interpretations that are mentioned there.

Anyway. Tumulka stuff was a topic in Scientific American March 2009 edition called "Was Einstein Wrong?: A Quantum Threat to Special Relativity"

http://www.scientificamerican.com/article.cfm?id=was-einstein-wrong-about-relativity
Here's sample of what's being said: [..] Just read the rest from Sci-Am. I don't like Tumulka's theory. If it can't even describe particles that attract or repel. I'd say it's such a long way.. much worse than Bohmian Mechanics.

Thanks for the sample; I don't have access to that article. Note that I can understand why my university dropped SciAm. Maybe that article is OK, but maybe not... Do you understand Tumulka's theory? [Edit] I see that now in post #66 more links are provided - will have a look at those.

 

[stglyde]

? That's the article of this thread to which I referred after I read it; I don't understand the new interpretations that are mentioned there.

Thanks for the sample; I don't have access to that article. Note that I can understand why my university dropped SciAm. Maybe that article is OK, but maybe not... Do you understand Tumulka's theory? [Edit] I see that now in post #66 more links are provided - will have a look at those.

It's easy to understand.

Maudlin said Bohmian Mechanics needs Preferred Foliation
Original GRW with wave function needs Preferred Foliation too.

But Tumulka GRW with flash doesn't need Preferred Foliation anymore. Somehow Alice and Bob or Entangled Pair doesn't have the correlations occurring at the same time.. the flash or beable wave function collapse (which is physical compared to Copenhagen version) is not simultaneous... but choreographed by the wave function. Something like that.

Maudlin was very interested in Tumulka because it was the only non-local theory he encountered that doesn't require Preferred Foliation and needs only the lorentz metric. Maybe he missed Demystifier's paper on "Making nonlocal reality compatible with relativity" at http://xxx.lanl.gov/abs/1002.3226

Anyway. Maudlin ended his "Non-Local Correlations in Quantum Theory: How the Trick Might Be Done" with the following observations:

"It has been a constant complaint against Bohmian mechanics, from its inception, that it “has no Relativistic version”. The reason that the theory is hard to reconcile with Relativity is clear: it is because of the way the non-locality of theory is implemented. In fact, the easiest way to extend that implementation to a space-time with a Lorentzian metric is to add a foliation, as we have seen. There may be some other way, but no one has discovered it yet.

But the non-locality of Bohm’s theory is not a peculiar feature of that theory: it is instead a feature of the world. Any theory adequate to explain the phenomena will have to have some non-locality. As we have seen, Ghirardi’s version of GRW, which takes a rather different approach to the non-locality than Bohmian mechanics, also is most easily adapted to a Lorentzian space-time by adding a foliation. One can create a fully Relativistic version of GRW, as Tumulka has shown, but only by a very different choice of local beable than Ghirardi made. So these are real, physical problems, with implications for the physical ontology, not merely matters of rewriting a theory in one way or another.

I guess Maudlin was wrong when he mentioned above "There may be some other way, but no one has discovered it yet.". He may not have known about Demystifier paper on "Making nonlocal reality compatible with relativity" at http://xxx.lanl.gov/abs/1002.3226 which is based on peer reviewed Physical Review Journal paper. If Peterdonis or others have read the paper shared in the link, please let us know if it is Viable, or Not Even Wrong. Thanks.

 

[PeterDonis]

He may not have known about Demystifier paper on "Making nonlocal reality compatible with relativity" at http://xxx.lanl.gov/abs/1002.3226 which is based on peer reviewed Physical Review Journal paper.

I still haven't read through the papers completely, but from skimming them I certainly don't think they are in the "Not Even Wrong" category.

One item I think is missing from the paper linked to above is that there are no specific references given for this statement on p. 1: "it is often argued that no model of nonlocal reality can be compatible with relativity". I'm not sure this is right; what I understand to be "often argued" is that no *local* model of reality can be compatible with quantum mechanics. I also have seen it said that nonlocality *appears* to be incompatible with the "spirit" of relativity, but that is clearly not a rigorous claim as it stands; and AFAIK nobody has actually tried to make such an argument rigorously (which would be very difficult since standard quantum field theory predicts correct results for EPR experiments but is explicitly relativistically covariant).

The statement is not central to the paper, which is mainly about giving a relativistically covariant version of Bohmian mechanics, so it's not a crucial point, but I would still be interested to see specific references in relation to it.

 

[stglyde]

? That's the article of this thread to which I referred after I read it; I don't understand the new interpretations that are mentioned there.

Thanks for the sample; I don't have access to that article. Note that I can understand why my university dropped SciAm. Maybe that article is OK, but maybe not... Do you understand Tumulka's theory? [Edit] I see that now in post #66 more links are provided - will have a look at those.

Searching for GRW and Tumulka stuff in the net. I found out this blog by physicist Lubos Motl, an extreme Copenhagenist who considers Bohmian Mechanics, Many Worlds Interpretation as Pseudo-science. He also considers Collapse Theories like GRW as pseudo-science. In the following, he is describing the essence of GRW and it may give an good introduction to you or others who don't have idea what it is.

http://motls.blogspot.com/2011/06/ghirardi-rimini-weber-collapsed.html

Ghirardi, Rimini, Weber: a real collapse

But I want to discuss another approach that hasn't been described on this blog yet: the GRW approach. The basic 1986 paper in PRD has 1217 citations as of today which is just gigantic if you realize that the paper is complete crackpottery:

Much like in the other anti-quantum approaches, this approach is trying to make some "classical reality". Unlike the Bohmian pseudoscience, it doesn't add any sharp positions of the particles. Instead, it keeps Schrödinger's equation only and adds some nonlinear "flashes" into the evolution that are meant to squeeze the state vector in the mantinels that the authors consider "appropriate".

It's very easy to describe what their proposal is - even though you may have a hard time to extract this basic point from the dozens of useless pages of the paper above. Imagine Schrödinger's equation for N non-relativistic particles - like the Bohmian pseudoscience, the formalism is linked to particles of the non-relativistic type, so all attempts to apply it to fields are inevitably awkward.

It is evolving according to Schrödinger's equation but GRW don't like that it's spreading because they want to imagine that the wave function is a "real object" and "real macroscopic objects" are not spreading - a classical misinterpretation of the wave function by all the anti-quantum "thinkers". Well, it obviously is spreading and there are many outcomes that have various probabilities - which doesn't hurt - but GRW just don't like it. So they decide that the wave function shouldn't freely spread! How do they ban the spreading? Well, that's easy for GRW.

They say that every 10^{15} seconds, which is a randomly chosen new bureaucratic constant of Nature (whose value is of course completely fabricated and has nothing to do with any justifiable laws of physics or any observations) each particle is obliged to prove to the census officials that it has a rather well-defined location. So there is a Poisson process running for each particle that once per 10^{15} seconds in average, it says "flash" to each particle. The more particles you have, the more flashes you obtain.

So what do or others think about this? Motl didn't mention about Tumulka which is a newer addition (I think Tumulka paper is like Lisa Randall RS1 and RS2 paper in popularity in the physics community) but it has the same GRW essence.

It seems that by trying to do away with Preferred Foliation. They have to de-linearized Quantum Mechanics and violate the superposition principle. So what's the lesser of the two evil?

 

[bohm2]

It is evolving according to Schrödinger's equation but GRW don't like that it's spreading because they want to imagine that the wave function is a "real object" and "real macroscopic objects" are not spreading - a classical misinterpretation of the wave function by all the anti-quantum "thinkers".

But the new PBR theorem suggests (or so it is claimed), that if one wants to take a scientific realist stance then the wave function must be ontological and not epistemic/statistical; unless you want to take a purely instrumental approach with respect to the wave function. In that case it is irrelevant.

Papers:
The quantum state cannot be interpreted statistically (this is the original paper)
http://lanl.arxiv.org/abs/1111.3328
Generalisations of the recent Pusey-Barrett-Rudolph theorem for statistical models of quantum phenomena
http://xxx.lanl.gov/abs/1111.6304
Completeness of quantum theory implies that wave functions are physical properties
http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.6597v1.pdf

Popular:
Quantum theorem shakes foundations
http://www.nature.com/news/quantum-theorem-shakes-foundations-1.9392

Blogs:
http://mattleifer.info/2011/11/20/can-the-quantum-state-be-interpreted-statistically/
http://www.scottaaronson.com/blog/?p=822
http://www.fqxi.org/community/forum/topic/999

The first blog above by Matt Leifer does a real good descriptive job of hi-liting which interpretations of QM have been eliminated as a result of this new theorem. The second one is Scott Aronson's and in there Lubos Motl gives his take on it and he appears to have misinterprated it accoring to Leifer.

From that blog in the comments section (with respect to the PBR), Lubos Motl writes:

You couldn’t have possibly read the paper or you’re unfamiliar with basic facts about QM or basic terminology used by those who want to replace QM by something else, if I avoid the term crackpot. Quantum mechanics is definitely probabilistic. It means that it can only make probabilistic predictions of measurements which can’t be made more unambiguous, not even in principle, and it says nothing about the “real state” of a system prior to the measurement. These are completely basic and universal properties of any quantum mechanical theory that are not open to any interpretation. The probabilistic framework of quantum mechanics is undoubtedly valid. Even the very abstract of the Pusey paper presents these things very clearly and argues – totally absurdly – that the statistical interpretation (by Max Born) is wrong. On the other hand, when you say that the paper is trying to resolve the battle between the “ontic” and “epistemic” camps, what you fail to understand is that, as you can see in the first paragraph of e.g.

http://arxiv.org/abs/0706.2661
both of these two camps are composed of advocates of hidden-variable models, if I avoid the term crackpots for the second time. To say the least, the hidden variable models have been falsified for more than 40 years and they are by definition incompatible with the basic principle of quantum mechanics: both of these adjectives are anti-quantum-mechanics. So the paper surely can’t prove that ontists or epistemists are right because both of these groups have been known to be wrong for 40+ years (and I would really say for 85+ years).

But Leifer writes:

First up, I would like to say that I find the use of the word “Statistically” in the title to be a rather unfortunate choice. It is liable to make people think that the authors are arguing against the Born rule (Lubos Motl has fallen into this trap in particular), whereas in fact the opposite is true. The result is all about reproducing the Born rule within a realist theory. The question is whether a scientific realist can interpret the quantum state as an epistemic state (state of knowledge) or whether it must be an ontic state (state of reality). It seems to show that only the ontic interpretation is viable, but, in my view, this is a bit too quick.

 

[stglyde]

As the year draw to a close, I realize I learn more things in these few weeks here than I had the last years.. especially key conceptual stuff. I wonder what life would be like without Physicsforums and the quality advices that we get. I learned everything about LET and FTL and Causality. I learned about Preferred Foliations and realized something about Gauge Transformation which I hadn't consider before. So I can understand papers better now than before. Thanks to Peterdonis and Dalespam and others. I learned that without bringing out what's bothering one. It couldn't be resolved. So before we all forgot about this thread and since this thread is about Preferred Foliation. Let me inquire something more about it before we all forgot about Maudlin stuff.

In Tumulka theory, instead of Preferred Foliation. Tumulka used another device that can produce non-locality but without any Preferred Foliation and it is relativistic. Peterdonis, do you understand the following? Hope you can rephrase them in your own words because I'm not exactly sure how it occurs or how to picture it in my mind (refer to Lubos Motl intro about "flashy GRW" but let's just focus on the Foliation thing (which is independent of the flash GRW):

When we turn to Tumulka’s theory, we find that the punctate ontology permits definition of a surrogate for the foliation theory’s universal clock. Given a point in a relativistic space-time, there are well-defined regions of constant invariant relativistic intervals from the point. The loci of zero intervals are, of course, the light cones, and regions of fixed time-like intervals form hyperbolae that foliate the interiors of the light cones. Since the ‘‘particles’’ are not allowed to reappear at space-like intervals from their previous location, these hyperbolae can serve the role of the universal time function for this particle: the likelihood of the next flash being at a certain location is a function of the time-like interval between the last flash and that location. Such a ‘‘flash-centered clock’’ can be defined using only the Lorentzian metric.

Furthermore, given a locus of constant time-like intervals from a given flash, the Lorentzian metric on space-time will induce a pure spatial metric on that locus (which will be a space-like hypersurface). This pure spatial metric can in turn be used to define the Gaussian used to specify the GRW collapse of the wave function, conditional on the next flash occurring at a specified location on the hypersurface. So the basic tools used in the original GRW theory can be adapted to this milieu using only the Lorentzian metric and the location of the last flash. In the multi-time formalism, the spatio- temporal structures needed to calculate both the conditional probability of the next flash and the new (post-collapse) wave function can be constructed using only the Lorentzian metric and the location of the last flash.

Since only the relativistic metric has been used in the construction, we have a completely relativistic theory.

Note when he referred to "flash", just replace it with "physical wavefunction collapse"
So without Preferred Foliation, he can produce the same effect of non-locality by replacing Preferred Foliation with something else that produced the same effect, do you understand what he is doing Peterdonis?

 

[PeterDonis]

Note when he referred to "flash", just replace it with "physical wavefunction collapse" So without Preferred Foliation, he can produce the same effect of non-locality by replacing Preferred Foliation with something else that produced the same effect, do you understand what he is doing Peterdonis?

What you quoted seems pretty clear, yes. Instead of foliating the entire spacetime with lines of constant t, he's foliating the future light cone of a given event with hyperbolas of constant t^2 - x^2, i.e., constant timelike interval from the chosen event. Since those hyperbolas are taken into themselves by a Lorentz transformation (unlike lines of constant t), I can see how this would allow a Lorentz covariant description of the dynamics of a single particle.

What the quote doesn't describe (but maybe it's elsewhere in one of the papers that are still on my list to read) is how this scheme is extended to the case of multiple particles. The problem is that a wavefunction for N particles is not just N copies of a single-particle wavefunction with some interactions thrown in; it's a single wavefunction of N variables, one for each particle. In non-relativistic mechanics, where time is a parameter, not a coordinate, the single time parameter is "shared" by all the particles, i.e., the multi-particle wavefunction evolves in "time" the same way the single-particle one does. But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact, but you also can't just have a "shared" time coordinate among all of them because they may be in relative motion. So I'm not sure how you would come up with a Lorentz-covariant analogue of the above hyperbolas of constant t^2 - x^2 for a multi-particle wavefunction.

The reason this is important is that nonlocality is not an issue for a single particle; it only becomes an issue at all if at least two particles are involved, so an EPR-type experiment can take place.

 

[stglyde]

What you quoted seems pretty clear, yes. Instead of foliating the entire spacetime with lines of constant t, he's foliating the future light cone of a given event with hyperbolas of constant t^2 - x^2, i.e., constant timelike interval from the chosen event. Since those hyperbolas are taken into themselves by a Lorentz transformation (unlike lines of constant t), I can see how this would allow a Lorentz covariant description of the dynamics of a single particle.

Hyperbolas? Honestly.. I'm not so verse in spacetime diagram to know what hyperbolas are. But I have this book Spacetime Physics by Edwin Taylor in one of the boxes of books in the attic. I'll try to find it tomorrow and read again. But if you think it's as dumb down as Brian Greene books, let me know.

I https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20

I'm sure there are software where one can input some parameters and the spacetime diagrams and hyperbolas can be shown in the screen. Do you know such program? There ought to be one given how useful spacetime diagram is.

What the quote doesn't describe (but maybe it's elsewhere in one of the papers that are still on my list to read) is how this scheme is extended to the case of multiple particles. The problem is that a wavefunction for N particles is not just N copies of a single-particle wavefunction with some interactions thrown in; it's a single wavefunction of N variables, one for each particle. In non-relativistic mechanics, where time is a parameter, not a coordinate, the single time parameter is "shared" by all the particles, i.e., the multi-particle wavefunction evolves in "time" the same way the single-particle one does. But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact, but you also can't just have a "shared" time coordinate among all of them because they may be in relative motion. So I'm not sure how you would come up with a Lorentz-covariant analogue of the above hyperbolas of constant t^2 - x^2 for a multi-particle wavefunction.

The reason this is important is that nonlocality is not an issue for a single particle; it only becomes an issue at all if at least two particles are involved, so an EPR-type experiment can take place.

Is "extended to the case of multiple particles" related to "interaction Hamiltonian" of some kind "between the various families" or is the latter a separate problem? (see context below) But then it is mentioned "The multi-time wavefunction is defined for a fixed collection of “particles”.. it means Tumulka has solved your issues above? The following is the context of what I'm saying in Mauldin description:

What are the essentials of Tumulka’s theory that allow it to live within the narrow means of the Lorentz metric? It seems essential that the local beable be point-like: the point is a locus that privileges no reference frame. The point also has the virtue of determining, in conjunction with the Lorentz metric, a set of hyperplanes that can be used in place of an imposed foliation. It seems essential that the dynamics be stochastic: we have seen how the determinism of Bohmian mechanics forces a distinction between the two sides of the singlet experiment: one measurement or the other must “come first”. And it seems essential that the local beable be intermittent, or flashy, so that experiments done on one side need not immediately register on the local beables on the other side, as they do in Ghirardi’s theory. Perhaps some of these features can be eliminated, but it is not obvious to me how this could be done.

There are some technical shortcomings of Tumulka’s theory at this point. The multi-time formalism can only be employed when there is no interaction Hamiltonian between the various families, so a lot of existent physics cannot be done in this form. The multi-time wavefunction is defined for a fixed collection of “particles”, so particle creation and annihilation phenomena are not covered. And there seems to be an initiation problem: the calculation of probabilities depends on specifying the “most recent” flash for each “particle”, but presumably right after the Big Bang most of the “particles”had had no flashes at all. It is not clear to me how to get the whole calculation of probabilities off the ground in this circumstance.

But these shortcomings are vastly overshadowed by Tumulka’s accomplishment: he has figured out how to construct a theory that displays quantum non-locality without invoking anything but the Lorentz metric. For years it has been known that no one has proven that the trick can’t be done, and many people suspected that somehow or other it could be done, but Tumulka has shown at least one way to do it."

If you have time in the weekend holidays. The following is Tumulka full paper about it:

A Relativistic Version of the Ghirardi–Rimini–Weber Model

http://arxiv.org/PS_cache/quant-ph/pdf/0406/0406094v2.pdf

 

[stglyde]

Hyperbolas? Honestly.. I'm not so verse in spacetime diagram to know what hyperbolas are. But I have this book Spacetime Physics by Edwin Taylor in one of the boxes of books in the attic. I'll try to find it tomorrow and read again. But if you think it's as dumb down as Brian Greene books, let me know.

I https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20

I'm sure there are software where one can input some parameters and the spacetime diagrams and hyperbolas can be shown in the screen. Do you know such program? There ought to be one given how useful spacetime diagram is.




Is "extended to the case of multiple particles" related to "interaction Hamiltonian" of some kind "between the various families" or is the latter a separate problem? (see context below) But then it is mentioned "The multi-time wavefunction is defined for a fixed collection of “particles”.. it means Tumulka has solved your issues above? The following is the context of what I'm saying in Mauldin description:



If you have time in the weekend holidays. The following is Tumulka full paper about it:

A Relativistic Version of the Ghirardi–Rimini–Weber Model

http://arxiv.org/PS_cache/quant-ph/pdf/0406/0406094v2.pdf

Reflecting on what you were saying that it can't work on multiple particles. Slowly the "multi-time wave function" came ringing. I guess Tumulka was proposing some kind of multiple Spacetimes as when he said that: (??)

"The technical apparatus Tumulka uses is a bit unfamiliar- for example, he uses a multi-time wavefunction defined over N copies of space-time (for N families of flashes), and he models the “collapse of the wavefunction” as a change from one multi-time wavefunction defined over the whole multi-time space to a different wavefunction defined over that whole space- but these refinements need not detain us. What we can already see is how Tumulka’s choice of local beable aids in rendering the theory completely Relativistic."

So it's like saying one particle uses an entire separate Spacetime? And the wave function is for a collection of them? Do you think this is as outrageous as Many Worlds? Any problems it produced?

I know. If it's true. I think we must not be serious with Tumulka because it is far behind Bohmian Mechanics.

 

[stglyde]

What you quoted seems pretty clear, yes. Instead of foliating the entire spacetime with lines of constant t, he's foliating the future light cone of a given event with hyperbolas of constant t^2 - x^2, i.e., constant timelike interval from the chosen event. Since those hyperbolas are taken into themselves by a Lorentz transformation (unlike lines of constant t), I can see how this would allow a Lorentz covariant description of the dynamics of a single particle.

What the quote doesn't describe (but maybe it's elsewhere in one of the papers that are still on my list to read) is how this scheme is extended to the case of multiple particles. The problem is that a wavefunction for N particles is not just N copies of a single-particle wavefunction with some interactions thrown in; it's a single wavefunction of N variables, one for each particle. In non-relativistic mechanics, where time is a parameter, not a coordinate, the single time parameter is "shared" by all the particles, i.e., the multi-particle wavefunction evolves in "time" the same way the single-particle one does. But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact, but you also can't just have a "shared" time coordinate among all of them because they may be in relative motion. So I'm not sure how you would come up with a Lorentz-covariant analogue of the above hyperbolas of constant t^2 - x^2 for a multi-particle wavefunction.

The reason this is important is that nonlocality is not an issue for a single particle; it only becomes an issue at all if at least two particles are involved, so an EPR-type experiment can take place.

It's New Year's eve in my country and I'm reflecting on all what you wrote because I feel it is important. How about QFT, it's relativistic quanfum field theory.. how do they interact in the context of what you mentioned about about time being coordinate here and not a parameter?

Happy New Year, the year 2012 will be a year of breakthrough that will chance things for long time. And thanks for your assistance.

 

[stglyde]

What you quoted seems pretty clear, yes. Instead of foliating the entire spacetime with lines of constant t, he's foliating the future light cone of a given event with hyperbolas of constant t^2 - x^2, i.e., constant timelike interval from the chosen event. Since those hyperbolas are taken into themselves by a Lorentz transformation (unlike lines of constant t), I can see how this would allow a Lorentz covariant description of the dynamics of a single particle.

What the quote doesn't describe (but maybe it's elsewhere in one of the papers that are still on my list to read) is how this scheme is extended to the case of multiple particles. The problem is that a wavefunction for N particles is not just N copies of a single-particle wavefunction with some interactions thrown in; it's a single wavefunction of N variables, one for each particle. In non-relativistic mechanics, where time is a parameter, not a coordinate, the single time parameter is "shared" by all the particles, i.e., the multi-particle wavefunction evolves in "time" the same way the single-particle one does. But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact, but you also can't just have a "shared" time coordinate among all of them because they may be in relative motion. So I'm not sure how you would come up with a Lorentz-covariant analogue of the above hyperbolas of constant t^2 - x^2 for a multi-particle wavefunction.

The reason this is important is that nonlocality is not an issue for a single particle; it only becomes an issue at all if at least two particles are involved, so an EPR-type experiment can take place.

Analyzing what you wrote above for several hours reading other refereces. I think what you meant is that for non-relativistic wave function. It is N copies of spacetime for each particle. But for relativistic. It is lorentz covariant and one interactive equation with changing reference frame. So since Tumulka define one family of flash as one particle and he has no Interaction Hamiltonian analogue even for 2 particles. Then it seems Tumulka stuff is not entirely relativistic. He was only able to make "before" and "after" disappear at the non-locality.. but this doesn't make it truly relativistic. This is exactly the same (I guess) in Demystifier paper where he was able to make "before" and "after" disappear at the non-locality but the equations not truly relativistic.

If I mentioned before that "Deep Down Things" book mentioned Yang and Mills cheated while no one was looking, producing the cheating terms in Gauge Transformation. Well. This time. Tumulka and Mr. Demys really cheated.. because it wasn't truly relativistic. Lol.

Thanks a lot. This Tumulka stuff made me learn many things. It makes the problems bare (which made me aware of it) which I haven't thought of before.

 

[PeterDonis]

I think what you meant is that for non-relativistic wave function. It is N copies of spacetime for each particle.

No. I probably should have given more details before; here they are.

In non-relativistic QM, there is no "spacetime". Time is a parameter, not a coordinate, and there is only one time parameter. Wave functions are functions in a 3N-dimensional space, where N is the number of particles. You can sort of think of this space as N "copies" of a 3-dimensional space, one per particle, but you have to be careful. For one thing, this 3N-dimensional space has different representations; the "configuration space" representation, in which the coordinates in the space are related to spatial positions, is only one such representation. (Another is the "momentum space" representation, where the coordinates in the space are related to spatial momenta.) For another, most wave functions are not "separable", meaning you can't interpret them as saying something like "one particle is at spatial point A and another is at spatial point B"; the factors attached to the different particles mix together in a way that has no straightforward interpretation in terms of "particles at spatial points".

Wave functions on the 3N-dimensional space at one time then evolve into wave functions on the same 3N-dimensional space at another time according to the Schrodinger Equation, for non-relativistic QM.

 

[stglyde]

No. I probably should have given more details before; here they are.

In non-relativistic QM, there is no "spacetime". Time is a parameter, not a coordinate, and there is only one time parameter. Wave functions are functions in a 3N-dimensional space, where N is the number of particles. You can sort of think of this space as N "copies" of a 3-dimensional space, one per particle, but you have to be careful. For one thing, this 3N-dimensional space has different representations; the "configuration space" representation, in which the coordinates in the space are related to spatial positions, is only one such representation.

I guess you are just describing Hilbert Space above.. so why didn't you mention Hilbert Space. Unless things can be explained in non-relativistic QM without referring to Hilbert Space?

(Another is the "momentum space" representation, where the coordinates in the space are related to spatial momenta.) For another, most wave functions are not "separable", meaning you can't interpret them as saying something like "one particle is at spatial point A and another is at spatial point B"; the factors attached to the different particles mix together in a way that has no straightforward interpretation in terms of "particles at spatial points".

Wave functions on the 3N-dimensional space at one time then evolve into wave functions on the same 3N-dimensional space at another time according to the Schrodinger Equation, for non-relativistic QM.

 

[PeterDonis]

I guess you are just describing Hilbert Space above.. so why didn't you mention Hilbert Space. Unless things can be explained in non-relativistic QM without referring to Hilbert Space?

"Hilbert space" is a general term for a type of abstract vector space in mathematics. The 3N-dimensional space I was describing for the wave function is a Hilbert space, yes.

 

[stglyde]

No. I probably should have given more details before; here they are.

In non-relativistic QM, there is no "spacetime". Time is a parameter, not a coordinate, and there is only one time parameter. Wave functions are functions in a 3N-dimensional space, where N is the number of particles. You can sort of think of this space as N "copies" of a 3-dimensional space, one per particle, but you have to be careful. For one thing, this 3N-dimensional space has different representations; the "configuration space" representation, in which the coordinates in the space are related to spatial positions, is only one such representation. (Another is the "momentum space" representation, where the coordinates in the space are related to spatial momenta.) For another, most wave functions are not "separable", meaning you can't interpret them as saying something like "one particle is at spatial point A and another is at spatial point B"; the factors attached to the different particles mix together in a way that has no straightforward interpretation in terms of "particles at spatial points".

Wave functions on the 3N-dimensional space at one time then evolve into wave functions on the same 3N-dimensional space at another time according to the Schrodinger Equation, for non-relativistic QM.

Earlier you wrote "But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact". You sounded as if in non-relativistic theory, one can have separate "copy" of 4-D spacetime. But then, spacetime doesn't exist in non-relativistic theory. So what is the scenerio where one can have "a separate "copy" of 4-D spacetime for every particle"? I guess none. Anyway. How does relativistic quantum field theory implement it all? But Smolin mentioned particles still move in fixed background even in QFT.. so you agree that we still don't have a true and pure relativistic formulation even in relativistic QFT?

 

[stglyde]

Earlier you wrote "But for a relativistic theory this doesn't work; you can't just have a separate "copy" of 4-D spacetime for every particle because they have to interact". You sounded as if in non-relativistic theory, one can have separate "copy" of 4-D spacetime. But then, spacetime doesn't exist in non-relativistic theory. So what is the scenerio where one can have "a separate "copy" of 4-D spacetime for every particle"? I guess none. Anyway. How does relativistic quantum field theory implement it all? But Smolin mentioned particles still move in fixed background even in QFT.. so you agree that we still don't have a true and pure relativistic formulation even in relativistic QFT?

Mulling it. I think what Smolin was saying was our QFT was not background independent (GR metric) although it uses the Minkowski metric (SR). And we know the Dirac Equation is the equation of relativistic QM. The following passage in the Tumulka paper we saw before would have many good food for thoughts:

http://arxiv.org/PS_cache/quant-ph/pdf/0406/0406094v2.pdf

A Relativistic Version of the Ghirardi–Rimini–Weber Model

The wavefunction is a multi-time wavefunction, i.e., it is defined on the Cartesian
product of N copies of space-time. We use the Dirac equation as the relativistic version
of the Schr¨odinger equation determining the evolution of the wavefunction apart
from the collapses (but we will mostly not worry whether the wavefunction lies in the
positive energy subspace, except in Section 3.7). More precisely, we use the multi-time
formalism with N Dirac equations. For the consistency of this set of equations, we
cannot have interaction potentials. To avoid discussing the question of interaction in
relativistic quantum mechanics, we will assume non-interacting particles. Interaction
can presumably be included by allowing for particle creation and annihilation, which
however is beyond the scope of this paper. In any case, the difficulty of including interaction
that we encounter here does not stem from the spontaneous collapses but rather
from the mathematics of multi-time equations, and is thus encountered by every kind
of relativistic quantum mechanics.

You mentioned that non-relativistic QM doesn't have spacetime. Since the paper mentioned it has "N copies of space-time", then it is relativistic after all. It uses the Dirac equation which is completely relativistic. However, there is something I can't quite completely understand. It avoided "interaction potentials" to avoid inconsistency of equations. And in the last sentence, it mentioned that relativistic quantum mechanics has difficulty with interactions. Maybe it's referring to QFT as being a theory where interactions occur and QM having no interactions, and since it uses the latter, interactions are temporarily avoided? But I still can't understand how it is able to use relativistic equations for the hyperbolas of t^2-x^2 as you mentioned earlier. Anyway. For those who do. Please let us know how. Thanks.

 

[stglyde]

I just realized something silly. We have been discussing about Spacetime Foliations for more than a week. All along I was thinking it was for Bohmian Mechanics. But Peterdonis emphasized there was not even spacetime for nonrelativistic theory. And since the BM as we knew it is nonrelativistic. Then it doesn't even need spacetime foliations. It simply needs Newtonian absolute simultaneity. So our spacetime foliations are for other theories with Beable like features. But since Tumulka flashing GRW is relativistic, and so is Demystifier BM, then there is no use of the spacetime foliations. And considering Copenhagen doesn't need any foliations because it's nonlocality is in the equations and no way to send signal. Then there is absolutely few or use of the preferred foliations.

I have to write this down so I can refer to this thread in the future for review and as note. Journey entry New Year's Day: Worrying about foliations and non-existent spacetime in nonrelativistic theory. Gee.

 

[harrylin]

I just realized something silly. We have been discussing about Spacetime Foliations for more than a week. All along I was thinking it was for Bohmian Mechanics. But Peterdonis emphasized there was not even spacetime for nonrelativistic theory. And since the BM as we knew it is nonrelativistic. Then it doesn't even need spacetime foliations. It simply needs Newtonian absolute simultaneity. So our spacetime foliations are for other theories with Beable like features. But since Tumulka flashing GRW is relativistic, and so is Demystifier BM, then there is no use of the spacetime foliations. And considering Copenhagen doesn't need any foliations because it's nonlocality is in the equations and no way to send signal. Then there is absolutely few or use of the preferred foliations.

I have to write this down so I can refer to this thread in the future for review and as note. Journey entry New Year's Day: Worrying about foliations and non-existent spacetime in nonrelativistic theory. Gee.

I'm now far behind with looking at the interesting references of this thread, that's for later. Consequently I can't really comment on your conclusions. My first impression is that those "flashing" foliation theories are pseudo science, and I think that the Copenhagen interpretation is a pseudo interpretation (effectively a physical non-interpretation). Anyway, the subtle thing about "spacetime" is that of course space-time is (and has been from the start) an important part of classical mechanics. Only Minkowskian (as well as post-Minkowskian) Spacetime is a new concept that corresponds to a popular interpretation of relativistic mechanics.

 

[stglyde]

I still haven't read through the papers completely, but from skimming them I certainly don't think they are in the "Not Even Wrong" category.

One item I think is missing from the paper linked to above is that there are no specific references given for this statement on p. 1: "it is often argued that no model of nonlocal reality can be compatible with relativity". I'm not sure this is right; what I understand to be "often argued" is that no *local* model of reality can be compatible with quantum mechanics. I also have seen it said that nonlocality *appears* to be incompatible with the "spirit" of relativity, but that is clearly not a rigorous claim as it stands; and AFAIK nobody has actually tried to make such an argument rigorously (which would be very difficult since standard quantum field theory predicts correct results for EPR experiments but is explicitly relativistically covariant).

The statement is not central to the paper, which is mainly about giving a relativistically covariant version of Bohmian mechanics, so it's not a crucial point, but I would still be interested to see specific references in relation to it.

Peter, I searched Demystifier paper at Physicsforum archives and there are many hits but only few comments. The most is the one by Maaneli in the thread https://www.physicsforums.com/showthread.php?t=366994&page=6 where I learned quite a few things and in one of the papers referenced there. I found out the following which is Nikolic claiming time as parameter in Newtonian space is somehow carried to relativistic coordinate time! Yes. So before I start a new thread on this. Hope to get your comment on the following since it's related to our recent discussions about parameter vs coordinate time (and in the following Nikolic said the former is retained in the latter explaining non-locality??!):

http://xxx.lanl.gov/PS_cache/arxiv/pdf/1007/1007.4946v1.pdf

sec 2.1

For nonrelativistic particle systems with conserved energy, the forces do not have an explicit dependence on time t. The only quantities that have a dependence on t are particle trajectories Xⁱa (t), i = 1, 2, 3. Thus, the parameter t has a physical meaning only along trajectories; time without trajectories does not exist! In this sense, t is only an auxiliary parameter that serves to parameterize the trajectories in 3-dimensional space, not a fundamental physical quantity by its own. Yet, a “clock” can measure time indirectly. Namely, a “clock” is nothing but a physical process described by a function Xⁱa (t) periodic in t. One actually observes the number of periods, and then interprets it as a measure of elapsed time.

The theory of relativity revolutionized the concept of time by replacing the parameter t with a coordinate x0 treated as a 4th dimension not much different from other 3 space dimensions. Yet, it does not mean that an auxiliary Newton-like time parameter is completely eliminated from relativistic mechanics. Such a parameter can still be introduced to parameterize relativistic spacetime particle trajectories in a manifestly covariant manner. This parameter, denoted by s, can be identified with a generalized proper time defined along particle trajectories of many-particle systems [23]. The parameter s can even be measured indirectly by a “clock” corresponding to a physical process periodic in s, in complete analogy with measurement of t in nonrelativistic mechanics. As discussed in more detail in [23], this makes the parameter s appearing in (6) a physical quantity, very much analogous to Newton time
t. With this physical insight, the relativistic-covariant equation (7) is to be interpreted as physical probability conservation during the evolution parameterized by the evolution-parameter s"

Comment anyone? Could it be true?

 

[stglyde]

The following are important exchanges between Demystifier (Nikolic) and Physicforums only challenger (Maaneli). The contents are important because what is at stake is the soul of physics and the great debate about non-locality.

Selected vital quotes from the thread "Re: Pilot wave theory, fundamental forces":

Maaneli: the relativity of simultaneity is nevertheless a consequence of the metrical structure of Minkowski spacetime

Demystifier: No, this is not true. What is true is that the metrical structure of Minkowski spacetime implies relativity of simultaneity IF THERE IS NO ANY OTHER STRUCTURE. But in the case we are considering there is another structure. And this additional structure is not the parameter s (as you might naively think), but the non-local wave function. (Or the scalar potential in the classical setting discussed in http://xxx.lanl.gov/abs/1006.1986.)

And yet, you can see that this nonlocal wave function (or the scalar potential) is compatible with the metrical structure of Minkowski spacetime and does not introduce a foliation-like structure.

So, does it mean that you agree that WITH example of my theory there IS a known dynamical structure that is consistent with the metrical structure of Minkowski spacetime, and yet violates the relativity of simultaneity?

By the way, one can introduce such a structure even in classical local relativistic mechanics. Consider two twins who initially have the same velocity and same position, and their clocks show the same time. After that, they split apart, and each has a different trajectory, independent of each other. Yet, one can consider pairs of points on two trajectories which have THE SAME VALUE OF PROPER TIME (showed by a local clock on each trajectory). Such a structure (defined at least mathematically, if not experimentally) also can be said to violate relativity of simultaneity, in a way very similar to that of my theory. Of course, there is a difference, but the similarity may be illuminating too.

Demystifier: Or let me use an analogy with nonrelativistic BM. A point is space is denoted as $$r=(x,y,z)$$. Consider two particles with space positions $${\bf r}_1$$ and $${\bf r}_2$$ at a given time $$t$$. There is a Cartesian frame (given by a rotation of the original Cartesian frame) in which $${\bf r}_1$$ and $${\bf r}_2$$ have the same value of $$z$$. In this frame, we say that the interaction between these two particles is z-taneous. Does it lead to any paradoxes? Does it mean there is a preferred z-coordinate? Does it mean there is a preferred foliation of space into 2-surfaces? Whatever your answer is, the same answer applies to analogous questions in relativistic-covariant BM. And if you still don't get it, then look at the equations of relativistic-covariant BM again.


Maaneli: ... by virtue of the fact that you have to synchronize the initial (spacetime) positions of the particles at a common time s,

Demystifier: The parameter s is not time.

Maaneli: But as a "joint parameter", it plays precisely the role of a universal time parameter for the evolution of the particle spacetime coordinates. Yes, I realize that the wavefunction on configuration spacetime doesn't depend on s, but that doesn't mean that s cannot also be interpreted as a time parameter (even if it is a fictitious one).

Demystifier: You are right. The parameter s can be interpreted as a sort of time. However, this is more like Newton absolute time, note like Einstein relativistic time [which I already implied by saying it is a UNIVERSAL time parameter]. OK, that's clear enough. And you are right, nonlocal signaling violates the relativity of simultaneity. Yet, in the next post I explain why it is NOT in contradiction with metrical structure of Minkowski spacetime.

(Maaneli in other message mentioned Nikolic theory has foliation like structure)

Demystifer: No, there is no foliation-like structure. The synchronization parameter is NOT something additional to the SR metrical structure, just as time in nonrelativistic BM is NOT something additional to the 3-space rotational-symmetry structure.

My point is that relativistic-covariant BM in 4-dimensional spacetime is ANALOGOUS to nonrelativistic BM in 3-dimensional space. I am just trying to make you understand this ANALOGY, because when you do, you will suddenly say: "Oh, THAT is what you meant. Now I get it. In fact, it is trivial." But it is essential that you see this analogy by yourself, while I can only guide you in the right direction. And at the moment, it seems to me that you don't have a clue what I am talking about, because you are not able to see the analogy. And that is probably because you are unable to think of time as just another "space" coordinate.

To help you think in the correct way, let me suggest you a mental trick. For a moment, FORGET that the spacetime metric has the form (+---). Instead , think of metric as just any metric, which can be (++++), (++--), or whatever. In fact, simply don't think about metric at all. Just pretend that you have a 4-dimensional space with some unspecified metric. Or if it is easier for you, just pretend that the metric is (++++). And forget that one of the coordinates is called "time". (Who cares about names, anyway?) And NOW try to understand again what equations of relativistic-covariant BM are actually saying. This trick works for many physicists, so it could work for you as well.

(Note: the last exchange between Demystifer and Maaneli occurred in Jun 22, 2010)

Maaneli: Of course, by covariant, you must mean "fundamentally covariant", because anyone can construct a covariant particle dynamics on a preferred foliation. I think it might be best for us to resume our discussion on the old thread, "Pilot waves, fundamental forces, etc.", regarding whether your proposal of using a synchronization parameter and treating time and space on equal footing is truly fundamentally covariant or not, and whether or not it does have the condition of equivariance. We never got to finish that discussion, mainly because I became too overwhelmed with deadlines and work and kept forgetting to reply to the thread. My apologies about that.

Demystifier: I would like to continue the discussion there. But I will wait for your first step.

(But they never continued. And since June, 2010. There is no other challenger to Demystifier. And he continued to share his papers which are referenced in peer reviewed journals but unfortunately, no one read it much. Therefore let us continue and settle it once and for all whether Demystifier is fundamentally correct. Before I make a new thread out of this. Just want some comment from you PeterDonis on what you think about all this. Thanks)

 

[stglyde]

I'm now far behind with looking at the interesting references of this thread, that's for later. Consequently I can't really comment on your conclusions. My first impression is that those "flashing" foliation theories are pseudo science, and I think that the Copenhagen interpretation is a pseudo interpretation (effectively a physical non-interpretation). Anyway, the subtle thing about "spacetime" is that of course space-time is (and has been from the start) an important part of classical mechanics. Only Minkowskian (as well as post-Minkowskian) Spacetime is a new concept that corresponds to a popular interpretation of relativistic mechanics.

Look at the nice graphic illustration (even in motion) of the flashes in Tumulka's presentation:

http://www.math.rutgers.edu/~tumulka/talks/penn09.pdf

Mulling at them would give a good work out in our final search for Quantum Gravity.

 

[Demystifier]

I've just noticed this thread, in which some of my papers have been discussed a lot.

If someone is still interested, now I can answer any related questions.

In addition, I would like to mention my very recent paper in which I propose a different relativistic-covariant version of Bohmian mechanics, in which a "preferred" foliation exists, but is determined in a covariant way by the relativistic-invariant wave function:
http://xxx.lanl.gov/abs/1205.4102