I Relativistic hidden variable quantum mechanics?

[A. Neumaier]

I just came across the following paper:

Gisin, N. (2011). Impossibility of covariant deterministic nonlocal hidden-variable extensions of quantum theory. Physical Review A, 83(2), 020102.

proving that, under sensible hypotheses, nonlocal hidden variable theories for relativistic quantum mechanics are impossible. Discussions of the assumptions can be found in

Laudisa, F. (2014). Against the ‘no-go’philosophy of quantum mechanics. European Journal for Philosophy of Science, 4(1), 1-17.

This paper criticizes Gisin's assumptions.

Oldofredi, Andrea. "No-Go Theorems and the Foundations of Quantum Physics." Journal for General Philosophy of Science(2018): 1-16.

This paper rebuts the criticism of Laudisa.

Nikolic, H. (2013). Time and probability: From classical mechanics to relativistic Bohmian mechanics. Unpublished.

This paper shows on p.45 and example of relativistic Bohmian trajectories for a system with 3 particles. Clearly, the trajectory of the third particle violates causality. Thus something is really wrong with the relativistic Bohmian dynamics proposed there. This does not inspire much faith in the credibility of the criticism of Gisin given on p.46.

 

[Demystifier]

Why do you think that violation of causality is necessarily wrong? If that violation occurs at a level of hidden variables, it is not in conflict with current observations.

 

[A. Neumaier]

If that violation occurs at a level of hidden variables, it is not in conflict with current observations.

...but with the basic principles of cause and effect. If you are prepared to sacrifice the latter, it is no better than sacrificing the basic principle of reality. At least to my taste.

 

[Demystifier]

...but with the basic principles of cause and effect.

It isn't in conflict with this basic principle. This principle is still valid, but in a slightly modified form. As explained in Sec. A.1.4, the concept of "before" does not longer mean "at smaller coordinate time $x^0$". It means "at smaller proper time $s$".

 

[A. Neumaier]

It isn't in conflict with this basic principle. This principle is still valid, but in a slightly modified form. As explained in Sec. A.1.4, the concept of "before" does not longer mean "at smaller coordinate time $x^0$". It means "at smaller proper time $s$".

So it is not the ''before'' relevant in relativity theory - your figure 1 shows that your world lines are not future directed.

 

[Demystifier]

So it is not the ''before'' relevant in relativity theory - your figure 1 shows that your world lines are not future directed.

It is different from the usual definition of "before" in relativity theory, yet it is relevant in relativity theory because the theory is Lorentz covariant, in the sense that equations take the same form in all Lorentz frames. The world lines are "future" directed, provided that you define future with respect to the proper time $s$.

 

[A. Neumaier]

The world lines are "future" directed, if you define future with respect to the proper time sss.

But future in relativity is defined by the spacetime metric, and the particle momenta have to point into the future cone. Only because of that is proper time an appropriate time coordinate. With your definition, there is no Lorentz frame in which proper time of the third particle is an actual time in the sense of Lorentz.

 

[PeterDonis]

As explained in Sec. A.1.4, the concept of "before" does not longer mean "at smaller coordinate time $x^0$". It means "at smaller proper time $s$".

In section A.1.4 I see the following, in the discussion of spacelike hypersurfaces:

"In an attempt to define a preferred hypersurface which connects all points with the same value of s, one immediately recognizes that there is an infinite number of such hypersurfaces for any given value of s."

I'm not sure I see how this follows from Fig. 1; in fact, from looking at Fig. 1 it seems to me that the issue is that there might not be any spacelike hypersurface at all that includes all events with a given value of $s$, because there might well be pairs of events with the same value of $s$ that are not spacelike separated.

Is there any more detailed mathematical model given in the paper, or in one of the references, that corresponds to Fig. 1?

 

[martinbn]

As explained in Sec. A.1.4, the concept of "before" does not longer mean "at smaller coordinate time $x^0$". It means "at smaller proper time $s$".

How do you set up an initial/boundary value problem for this theory?

The world lines are "future" directed, provided that you define future with respect to the proper time $s$.

What do you mean by proper time if the lines are not time-like?

 

[Demystifier]

But future in relativity is defined by the spacetime metric

That's the standard definition, yes, but sometimes progress involves a change of old definitions.

 

[Demystifier]

it seems to me that the issue is that there might not be any spacelike hypersurface at all that includes all events with a given value of s

hypersurface $\neq$ spacelike hypersurface

 

[Demystifier]

How do you set up an initial/boundary value problem for this theory?

Essentially, this is the initial value problem for the system of first order ordinary differential equations given by Eq. (117).

What do you mean by proper time if the lines are not time-like?

See Eq. (108).

 

[Demystifier]

I can answer many additional particular questions, but it is almost impossible to understand it intuitively if you don't get the following: In the standard relativity theory, space coordinates $x^i$ and time coordinate $x^0$ are treated on a similar footing. In my paper above, they are treated on an even more similar footing. All the differences between relativity in my paper above and standard relativity stem from the idea that in my paper one must take the 4-dimensional view of spacetime very seriously, much more seriously than standard relativity theory does. Once you digest this hard conceptual idea, the rest is easy. Very much like the standard relativity becomes easy once you replace Einstein's 1905 form of relativity with the Minkowski-spacetime formulation of relativity.

 

[Demystifier]

The related chronological development of relativity can be summarized as follows:
1) Lorentz symmetry (1904) -> makes Maxwell equations "easy"
2) Einstein special relativity (1905) -> makes Lorentz symmetry "easy"
3) Minkowski spacetime (1908) -> makes relativity "easy"
4) Einstein-Podolsky-Rosen (1935) -> non-locality, in apparent conflict with relativity
5) Stuckelberg (1941) / Horwitz and Piron (1973) spacetime formulation of relativistic QM -> makes non-local relativity "easy"

But more recently, I turned back to 1) with a denial of 2), 3) and 5) at the fundamental level. In this approach Lorentz symmetry is emergent, which makes Lorentz symmetry hard again. The gain is that non-locality, Bohmian mechanics and emergent QFT are "easy".

 

[stevendaryl]

I'm only partially following this discussion, but it seems to me that

The related chronological development of relativity can be summarized as follows:
1) Lorentz symmetry (1904) -> makes Maxwell equations "easy"
2) Einstein special relativity (1905) -> makes Lorentz symmetry "easy"
3) Minkowski spacetime (1908) -> makes relativity "easy"
4) Einstein-Podolsky-Rosen (1935) -> non-locality, in apparent conflict with relativity
5) Stuckelberg (1941) / Horwitz and Piron (1973) spacetime formulation of relativistic QM -> makes non-local relativity "easy"

But more recently, I turned back to 1) with a denial of 2), 3) and 5) at the fundamental level. In this approach Lorentz symmetry is emergent, which makes Lorentz symmetry hard again. The gain is that non-locality, Bohmian mechanics and emergent QFT are "easy".

The original post is about covariant hidden variables. Bohmian mechanics is not covariant, right?

 

[Demystifier]

The original post is about covariant hidden variables. Bohmian mechanics is not covariant, right?

The standard version of Bohmian mechanics isn't covariant, but some versions are covariant.

 

[PeterDonis]

That's the standard definition, yes, but sometimes progress involves a change of old definitions.

I don't understand this. You are claiming to have developed a relativistic Bohmian mechanics. How can that claim be justified if you are changing the definition of "relativistic"?

 

[martinbn]

Essentially, this is the initial value problem for the system of first order ordinary differential equations given by Eq. (117).

I see, so the theory only deals with particles, no fields.

See Eq. (108).

This doesn't clarify things for me. What you call proper time has square that is a multiple of the square of the line element in relativity. For the world lines in that figure there will be parts with a negative value, parts with a positive value and parts where it is zero. Not sure how that is interpreted as proper time.

All the differences between relativity in my paper above and standard relativity stem from the idea that in my paper one must take the 4-dimensional view of spacetime very seriously, much more seriously than standard relativity theory does.

I haven't read the paper so I am probably wrong, but it doesn't seem that you take the space-time view seriously. You take it at liberty.

 

[Demystifier]

I don't understand this. You are claiming to have developed a relativistic Bohmian mechanics. How can that claim be justified if you are changing the definition of "relativistic"?

How could Einstein call his theory relativistic, if he changed the definition of relativity set by Galileo?
How could Heisenberg call his theory quantum, if he changed the definition of quantum set by Bohr in the Bohr model of atom?

 

[Demystifier]

For the world lines in that figure there will be parts with a negative value, parts with a positive value and parts where it is zero.

There will not, due to the denominator in (108).

 

[Demystifier]

I haven't read the paper so I am probably wrong, but it doesn't seem that you take the space-time view seriously. You take it at liberty.

What I take seriously is the 4-dimensional view of spacetime.

 

[stevendaryl]

What I take seriously is the 4-dimensional view of spacetime.

That's the beauty of General Relativity. It forces you to take the 4-dimensional view seriously. But then armed with that idea, we can go back to previous theories, such as Newtonian physics, and realize that it is a 4-D theory, as well. The difference is that there is much more symmetry between the dimensions in GR, or at least, they are treated much more symmetrically.

 

[Demystifier]

That's the beauty of General Relativity. It forces you to take the 4-dimensional view seriously. But then armed with that idea, we can go back to previous theories, such as Newtonian physics, and realize that it is a 4-D theory, as well. The difference is that there is much more symmetry between the dimensions in GR, or at least, they are treated much more symmetrically.

Exactly! And in my approach to reconcile relativity with quantum non-locality, I make a further step by treating the dimensions even more symmetrically than classical SR and GR do. To add this extra symmetry some relativistic concepts must be somewhat generalized and hence redefined. Some people don't like it, they want to reconcile classical relativity with quantum physics by retaining classical relativity intact. But there is no doubt that quantum physics forces us to reformulate some of the ideas of classical physics. From that perspective, reformulation of classical relativity seems like a viable possibility.

 

[PeterDonis]

How could Einstein call his theory relativistic, if he changed the definition of relativity set by Galileo?

Actually, I think he called it the theory of invariants. But you're right that the word "relativistic" isn't the point.

Here's a better way to ask the question: you are relying on a notion of "proper time" that, in standard relativity, has a precise meaning: it's arc length along a timelike curve. But in your model, if I'm understanding your responses correctly, there don't seem to be any timelike curves. There are just curves. But in our actual world, the distinction between timelike, spacelike, and null curves seems to be real; so how does that distinction appear in your model? Or if it doesn't, why do you think that's ok?

 

[martinbn]

There will not, due to the denominator in (108).

I still don't understand. The QM "proper time" is a multiple of the relativistic space-time interval. You write it explicitly in (110). You have $ds_{QM}^2=(something)\times ds_{rel}^2$. Even if the dynamics of the theory makes $(something)$ have the correct sign to avoid imaginary QM proper time (which would require a proof), you still have a problem with portions of the world line where it is null. If $ds^2_{rel}$ is zero you cannot avoid that $ds^2_{QM}$ is also zero.

What I take seriously is the 4-dimensional view of spacetime.

Everyone takes this seriously, there is nothing radical here about your position. But in relativity space-time is a pair $(M,g)$, and the metric is important for several reasons, one is that it gives the causal structure of the space-time. You completely ignore that part. I wouldn't say that you take the 4-dimensional view of spacetime seriously, but that you take the rest not seriously enough.

p.s. There is till the question raised by PeterDonis about the hypersurfaces, even if you don't need them to be space-like it is still not obvious that you can foliate the whole space-time in that manner.

 

[A. Neumaier]

To add this extra symmetry some relativistic concepts must be somewhat generalized and hence redefined. Some people don't like it, they want to reconcile classical relativity with quantum physics by retaining classical relativity intact. But there is no doubt that quantum physics forces us to reformulate some of the ideas of classical physics. From that perspective, reformulation of classical relativity seems like a viable possibility.

But then you need to show that the standard results from relativity, based on the old notion, are recovered by your modification, at least to the accuracy that these are corroborated by experiment. Thus to be taken seriously you need at least to recover the Lamb shift and the anomalous magnetic moment of hydrogen!

 

[Demystifier]

But in your model, if I'm understanding your responses correctly, there don't seem to be any timelike curves.

Wrong.

But in our actual world, the distinction between timelike, spacelike, and null curves seems to be real; so how does that distinction appear in your model?

The same as in standard theory of relativity.

Or if it doesn't, why do you think that's ok?

I introduce a new notion of proper time, but I don't reject the standard notion of proper time. They are two different physical quantities, which, according to the model, describe different properties of nature.

 

[Demystifier]

I still don't understand. The QM "proper time" is a multiple of the relativistic space-time interval. You write it explicitly in (110). You have $ds_{QM}^2=(something)\times ds_{rel}^2$. Even if the dynamics of the theory makes $(something)$ have the correct sign to avoid imaginary QM proper time (which would require a proof), you still have a problem with portions of the world line where it is null. If $ds^2_{rel}$ is zero you cannot avoid that $ds^2_{QM}$ is also zero.

The $(something)$ changes the sign precisely when the $ds_{rel}^2$ changes the sign. Therefore it should not be surprising to you that $(something)^{-1}=0$ precisely when $ds_{rel}^2=0$, in such a manner that in the appropriate limit $0^{-1}0\neq 0$.

I wouldn't say that you take the 4-dimensional view of spacetime seriously, but that you take the rest not seriously enough.

I would say that you didn't actually read the paper.

p.s. There is till the question raised by PeterDonis about the hypersurfaces, even if you don't need them to be space-like it is still not obvious that you can foliate the whole space-time in that manner.

The existence of foliation is not essential for my theory. That indeed is a part of taking 4-dimensional view seriously, because insisting on foliation reflects the desire to have a 3-dimensional view. What is important is that there is no preferred foliation into spacelike hypersurfaces, which reflects relativistic covariance of the theory. What is seen from the picture is that if there is a foliation at all, then it obviously isn't a foliation into spacelike hypersurfaces.

 

[Demystifier]

Thus to be taken seriously you need at least to recover the Lamb shift and the anomalous magnetic moment of hydrogen!

That's recovered because the standard equations of QFT are not rejected in this theory. As in other formulations of Bohmian mechanics, the standard quantum equations are not rejected, but only augmented by additional equations describing particle trajectories.

 

[A. Neumaier]

the standard equations of QFT are not rejected in this theory. As in other formulations of Bohmian mechanics, the standard quantum equations are not rejected, but only augmented

But the quantum equations for QED are mathematically meaningless because they use ill-defined bare coefficients. How do your particle trajectories survive renormalization when not even the number operator survives it?

 

[Demystifier]

But the quantum equations for QED are mathematically meaningless because they use ill-defined bare coefficients. How do your particle trajectories survive renormalization when not even the number operator survives it?

By a cutoff. Now you will say that cutoff ruins Lorentz invariance, after which I will reply that cutoff should also be used without trajectories, so my theory is not less Lorentz invariant than standard QFT.

 

[PeterDonis]

They are two different physical quantities, which, according to the model, describe different properties of nature.

Which one (i.e., which proper time) describes the time shown on my wristwatch? And what property of nature does the other one describe? How would I measure that property?

 

[A. Neumaier]

By a cutoff. Now you will say that cutoff ruins Lorentz invariance, after which I will reply that cutoff should also be used without trajectories, so my theory is not less Lorentz invariant than standard QFT.

It is with cutoff never Lorentz invariant, and ugly, ugly, ugly...

Then you need to explain why you need Lorentz invariance in the first place to set up the equations (before you know where to introduce the cutoff) and to define renormalizability.

You also get a fine-tuning problem because the couplings must be very large or tiny and precisely matched to get quantitative agreement with experiments.

Moreover, all predictions become dependent on the precise way of defining the cutoff.

You get a whole class of mathematically ugly theories with an infinite number of adjustable constants in the cutoff prescription, none of which is favored more than any other by experiment.

The only distinguished choice is the renormalization limit, which depends for QED on 2 parameters only. And particles don't survive this limit, except asymptotically, at times $\pm\infty$, where the scattering interpretation applies.

cutoff should also be used without trajectories

This is never done in the standard textbook treatments, which define the meaning of QED. Everything there is Lorentz invariant.

 

[Demystifier]

Which one (i.e., which proper time) describes the time shown on my wristwatch?

It's a classical apparatus, so this time is described by the standard classical proper time.

And what property of nature does the other one describe? How would I measure that property?

Unfortunately, it's related to hidden variables which, so far, we don't know how to measure. (Now if you point out that unmeasurable things are physically meaningless, then why did you enter this discussion in the first place, where it was clear from the beginning that it is about hypothetical hidden variables?)

 

[Demystifier]

It is with cutoff never Lorentz invariant, and ugly, ugly, ugly...

Physics is not a beauty contest.

 

[A. Neumaier]

Physics is not a beauty contest.

Wth regard to theories about unobservable things (as hidden variables are), the only sensible selection criterion is in terms of beauty, simplicity, and the like. This is the content of Ockham's razor.

Thus the physics of hidden variables is a beauty contest.

 

[Demystifier]

Thus the physics of hidden variables is a beauty contest.

How about an intuition contest? For me cutoff is indeed ugly, but very intuitive.

 

[martinbn]

The $(something)$ changes the sign precisely when the $ds^2_{rel}$ changes the sign. Therefore it should not be surprising to you that $0(something)^{-1}=0$ precisely when $ds_{rel}^2=0$, in such a manner that in the appropriate limit $0^{-1}0\neq 0$.

Is that shown in the paper?

I would say that you didn't actually read the paper.

Well, I wrote that in post #18

The existence of foliation is not essential for my theory. That indeed is a part of taking 4-dimensional view seriously, because insisting on foliation reflects the desire to have a 3-dimensional view. What is important is that there is no preferred foliation into spacelike hypersurfaces, which reflects relativistic covariance of the theory. What is seen from the picture is that if there is a foliation at all, then it obviously isn't a foliation into spacelike hypersurfaces.

This brings me back to my first question about the initial value problem. You need a hypersurface for the initial conditions. For some equations, if it isn't space-like, it may lead to an ill posed Cauchy problem. There may be no solutions at all. You said that the equations for the other fields are the same, so there is some justification to be done here. Even if you ignore them and only treat particles you still have the quantum potential or the wave function or whatever it is in Bohmian mechanics that guides them. You will have some PDEs and the need of some foliation.

 

[Demystifier]

Is that shown in the paper?

Not explicitly, it's left as an exercise for the reader, which should be easy for a reader who read the paper carefully.

This brings me back to my first question about the initial value problem. You need a hypersurface for the initial conditions. For some equations, if it isn't space-like, it may lead to an ill posed Cauchy problem. There may be no solutions at all. You said that the equations for the other fields are the same, so there is some justification to be done here. Even if you ignore them and only treat particles you still have the quantum potential or the wave function or whatever it is in Bohmian mechanics that guides them. You will have some PDEs and the need of some foliation.

This objection is too general, you don't direct your objection to specific equations in the paper. I could reply by writing down specific equations and explaining why your general objections do not apply in this case, but it wouldn't make much sense if you didn't read the paper.

If you don't have time to read the whole paper, I would suggest you to read only Sec. 3.3 of https://arxiv.org/abs/1002.3226
In this older paper I have not yet been recognizing that $s$ can be interpreted as a generalized proper time, but mathematics relevant to the Cauchy problem is essentially the same.

 

[A. Neumaier]

How about an intuition contest? For me cutoff is indeed ugly, but very intuitive.

It is not intuitive at all why you assume Lorentz invariance to be violated (through the cutoff), but require it to hold in the first place to define the eligible class of Lagrangians.

 

[Boing3000]

Wth regard to theories about unobservable things (as hidden variables are),

Is it a formal axiom in BM that hidden variables are unobservable ? Or is it simply the case that we cannot "yet" figure a way to test them (more directly than a EPR experiment, which is pretty much for me a direct observation of non-local hidden variable)
Isn't it the case that in classical QM particle trajectories take an infinite number of "unobservable path (or even virtual)", because as soon as we try to observe them, they "disappear" ?
Is classical "temperature" observable ? Or is it not some kind of useful fiction that we "invented" because we cannot observed the momentum of millions of particles ?

the only sensible selection criterion is in terms of beauty, simplicity, and the like. This is the content of Ockham's razor.

Which is why my favorite "interpretation" is BM. There is no un-observable infinities or Wolds (if they can at least be defined), nor an unobservable ledger where the "ensemble" of past probabilities are recorded so the next one behave somewhat correctly...

Thus the physics of hidden variables is a beauty contest.

I agree, that's why I will dig further into your thermal interpretation of QM.

 

[A. Neumaier]

in classical QM particle trajectories take an infinite number of "unobservable paths''

There is no classical QM. You seem to refer to some version of the Copenhagen interpretation.

Is it a formal axiom in BM that hidden variables are unobservable ?

If they were observable in principle, the predictions would be different from quantum mechanics. But the claim BM makes is that the observable predictions are exactly the same. Thus they have to be unobservable...

 

[Boing3000]

If they were observable in principle, the predictions would be different from quantum mechanics. But the claim BM makes is that the observable predictions are exactly the same. Thus they have to be unobservable...

Different or just more precise ? As every new theory try to reproduce the previous one "as a limit", I don't understand your logic.
As QM cannot even predict where one photon/electron/whatever will end up on that plate (whatever the number of slits), I always supposed the goal of BM was to be able to predict at least with reasonable precision the destination position/value given that we knew with reasonable precision the hidden variables values
Now if those BM trajectories are totally chaotic (small variation in hidden variable values => huge variation in destination position), then the whole endeavor still seams to me to be more than pointless, because I would understand how nature "implement" itself, using some very simple tricks.

 

[Demystifier]

It is not intuitive at all why you assume Lorentz invariance to be violated (through the cutoff), but require it to hold in the first place to define the eligible class of Lagrangians.

Cutoff is intuitive, Lorentz invariance is beautiful. In this way, like in a marriage, one retains both, or loses both, depending on whether one is optimist or pessimist.

When I was writing this paper I was an optimist. In the meanwhile I became a pessimist, so now my preferred version of Bohmian mechanics is different, the one in which fundamental Lorentz invariance is completely rejected. In this way the unnatural marriage is divorced, so Bohmian mechanics can be fully intuitive and ugly.
For more details on the evolution of my thoughts see https://www.physicsforums.com/insights/stopped-worrying-learned-love-orthodox-quantum-mechanics/

 

[PeterDonis]

It's a classical apparatus, so this time is described by the standard classical proper time.

This would mean that a classical apparatus cannot follow the spacelike trajectories shown in Fig. 1, correct? Those trajectories can only be followed by the unobservable Bohmian particles?

 

[Demystifier]

This would mean that a classical apparatus cannot follow the spacelike trajectories shown in Fig. 1, correct? Those trajectories can only be followed by the unobservable Bohmian particles?

Yes.

 

[PeterDonis]

Yes.

So what in your model prevents a classical apparatus from following a spacelike trajectory, if the individual Bohmian particles can?

 

[martinbn]

Not explicitly, it's left as an exercise for the reader, which should be easy for a reader who read the paper carefully.

Well, you have established that I am not a reader of the article. Can the writer supply the proof?

This objection is too general, you don't direct your objection to specific equations in the paper. I could reply by writing down specific equations and explaining why your general objections do not apply in this case, but it wouldn't make much sense if you didn't read the paper.

Well, let's say equation $(94)$ or if you prefer $(99-100)$ with the equations for the particles $(117)$. How do you set up the IVP?

 

[A. Neumaier]

if those BM trajectories are totally chaotic

They are, so the detailed predictions are untestable. That's the whole point. BM is not supposed to predict in any real sense but to solve the interpretation problem by introducing some sort of reality into the picture.

 

[A. Neumaier]

now my preferred version of Bohmian mechanics is different, the one in which fundamental Lorentz invariance is completely rejected.

But then the problem of how to motivate and explain the Lagrangian of QED, QCD and the standard model, which are all based on Lorentz invariance, becomes even more pressing. Just assuming that there is a miraculous way these appear by coarse graining from some unspecified and unknown noninvariant theory requires a lot of faith, and few would follow your faith.