Meaning of Spacetime Foliations

*stglyde*

Quote by PeterDonis

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*

Quote by 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/p.../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

Quote by 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.

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*

Quote by PeterDonis

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 http://www.physicsforums.com/showthr...=366994&page=6 where I learnt 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/p...007.4946v1.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 [tex]r=(x,y,z)[/tex]. Consider two particles with space positions [tex]{\bf r}_1[/tex] and [tex]{\bf r}_2[/tex] at a given time [tex]t[/tex]. There is a Cartesian frame (given by a rotation of the original Cartesian frame) in which [tex]{\bf r}_1[/tex] and [tex]{\bf r}_2[/tex] have the same value of [tex]z[/tex]. 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 occured 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*

Quote by harrylin

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

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