Relativistic hidden variable quantum mechanics?

[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.

 

[Demystifier]

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

A classical apparatus is a macroscopic object, the trajectory of which is defined by the motion of the non-empty wave packet. See Fig. 2 in https://arxiv.org/abs/1309.0400 for an idea how macroscopic trajectory may look very different from the microscopic one.

 

[Demystifier]

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.

"The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man."
- George Bernard Shaw

 

[A. Neumaier]

"The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man."
- George Bernard Shaw

Well, Shaw didn't achieve anything in physics. Progress in physics depends on finding something adapted both to the world and to one's vision! This characterized the work of Planck, Einstein, Heisenberg, Dirac, Feynman ...

 

[Demystifier]

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?

One first writes down the general solution of (94), parametrized by an infinite number of free constants. Then one picks a particular solution by picking some specific values of those constants. In this way (94) is solved without choosing an initial value. Finally, one solves (117) by an initial condition at, say, $s=0$.

 

[Demystifier]

Well, Shaw didn't achieve anything in physics. Progress in physics depends on finding something adapted both to the world and to one's vision! This characterized the work of Planck, Einstein, Heisenberg, Dirac, Feynman ...

How about string theorists like Maldacena, Witten, etc? Would you say that they achieved something in physics?

 

[Demystifier]

Can the writer supply the proof?

He can, which doesn't mean that he will.
You have to be a better motivator. You have to say something like "Wow, you explained that so well, it would be great if you could just fill this little detail, so that my understanding of your great ideas can be complete."

But I am in a good mood these days, so I will give you a sketch of the proof for $n=1$. Suppose that $V^{\mu}V_{\mu}=0$, but $V^{\mu}\neq 0$ and $V^{\mu}\neq \infty$. Then
$$\frac{dX^{\mu}}{ds}=V^{\mu}\neq 0, \infty \;\;\;\; (1)$$
Therefore on a line segment on which $dX^{\mu}\neq 0$ we have $dX^{\mu}dX_{\mu}=0$, but (1) implies that $ds\neq 0$. $\Box$

If you don't accept a proof using infinitesimals, I live it to you to reformulate this in the $\epsilon$-$\delta$ language or in an integral form.

 

[A. Neumaier]

How about string theorists like Maldacena, Witten, etc? Would you say that they achieved something in physics?

The future will tell about the contribution of string theory to physics.

But Witten also contributed to ordinary QFT and to pure mathematics. In mathematics he obtained the fields medal, at the time the highest distinction.

 

[Demystifier]

The future will tell about the contribution of string theory to physics.

Likewise the future will tell whether Lorentz invariance is only emergent at a coarse grained level.

 

[A. Neumaier]

Likewise the future will tell whether Lorentz invariance is only emergent at a coarse grained level.

But we are discussing the present state of affairs.

 

[martinbn]

One first writes down the general solution of (94), parametrized by an infinite number of free constants. Then one picks a particular solution by picking some specific values of those constants. In this way (94) is solved without choosing an initial value. Finally, one solves (117) by an initial condition at, say, $s=0$.

It is still not clear to me. What do you mean by the general solution with infinite number of constants? And how do you pick their values? What information is given?

Here is a simple example. Take the usual wave equation in $1+1$ dimensions. It has a general solution $u(x,t)=f(x-ct)+g(x+ct)$ for arbitrary functions $f$ and $g$. How do you pick your solution? And what happens in higher dimensions, what is the general solution?