How do deterministic Bohmian/Pilot Wave Theories Handle These?

[Ilja]

So, do you find something worse with http://xxx.lanl.gov/abs/0811.1905 ?

I do not understand how one can hope for a really Lorentz-invariant Bohmian theory. There is the trivial way - take absolute time and use a relativistic-looking equation for it, \square t = 0. But this is the same preferred frame.

If we have BI violations for arbitrary pairs of events A, B, Bell proves that they cannot have a common cause, thus, A->B or B->A. If there is some causality without closed causal loops, only one of the claims can be true. And for all pairs of events, this gives easily a preferred foliation. Where is the fault in this proof?

 

[Demystifier]

I do not understand how one can hope for a really Lorentz-invariant Bohmian theory. There is the trivial way - take absolute time and use a relativistic-looking equation for it, \square t = 0. But this is the same preferred frame.

If we have BI violations for arbitrary pairs of events A, B, Bell proves that they cannot have a common cause, thus, A->B or B->A. If there is some causality without closed causal loops, only one of the claims can be true. And for all pairs of events, this gives easily a preferred foliation. Where is the fault in this proof?

My point is that you don't need an a priori universal preferred foliation. In my approach the equations of motion do not involve any preferred foliation. Instead, an effective "preferred foliation" occurs dynamically, due to a specific choice of initial conditions. Other initial conditions lead to other effective "preferred foliations". Hence there is no fundamental preferred foliation at all.

 

[Demystifier]

Hi Demystifier,

I just had a look through this paper. In section B you attempt to
set up an inf-dim space relying on a measure of the form:
<br /> Dx ~:=~ \prod_{n=1}^\infty \; d^4x_n<br />
(I'm just paraphrasing your equations 28,30,31, etc.)

If I've understood what you're trying to do (forgive me if I've
got it wrong), you're taking the ordinary Lebesgue measure dx
on a 1-particle configuration space, then the corresponding Lebesgue
measure on a N-dimensional product of these spaces (which is fine
so far), and then assuming that this remains valid for N\to\infty.
However, it's a theorem that a non-trivial Lebesgue measure on an
inf-dim space of this kind does not exist. See, e.g.,

http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure

The essence of the proof is that if we demand the measure be translation-invariant,
(roughly, that dx = d(x+c) for c constant), then the measure is trivial (zero).

Exactly the same problem occurs in path integrals, forcing people to adopt
Wiener (Gaussian) measure instead, which is not translation-invariant.
(This issue is also related to the stuff currently being discussed over in the
other thread about "unbounded operators".)

Strangerep, thank you for this technical remark. I have no doubts that ideas in my paper can also be put in a much more rigorous framework. You will certainly understand that mathematical rigor was not my primary concern.

 

[Ilja]

My point is that you don't need an a priori universal preferred foliation. In my approach the equations of motion do not involve any preferred foliation. Instead, an effective "preferred foliation" occurs dynamically, due to a specific choice of initial conditions. Other initial conditions lead to other effective "preferred foliations". Hence there is no fundamental preferred foliation at all.

But this looks like an unnecessary complication for me. The classical absolute space - absolute time scheme with absolute causality is preferable because of its simplicity. To make it dynamical is not an advantage. Many preferred foliations seem as unnecessary to me as many worlds.

 

[Demystifier]

But this looks like an unnecessary complication for me. The classical absolute space - absolute time scheme with absolute causality is preferable because of its simplicity. To make it dynamical is not an advantage. Many preferred foliations seem as unnecessary to me as many worlds.

You missed my point. In my approach, there are no preferred foliations at all. Instead, a particular solution makes an illusion that a preferred foliation exists, even though it does not.

You can compare it with the gravitational field of planet Earth that makes an illusion that there is a preferred foliation of 3-space into equipotential 2-surfaces, even though a preferred foliation of 3-space does not exist.

 

[Dmitry67]

Do the commonly accepted relativistic BM theory exist?
I mean, if even BM proponents can not agree on some basic things...

 

[Ilja]

You missed my point. In my approach, there are no preferred foliations at all. Instead, a particular solution makes an illusion that a preferred foliation exists, even though it does not.

You can compare it with the gravitational field of planet Earth that makes an illusion that there is a preferred foliation of 3-space into equipotential 2-surfaces, even though a preferred foliation of 3-space does not exist.

Ok, let's compare it. Assume that our world is globally flat and the gravitational field homogeneous. In this case, we can use two theories: One where a preferred foliation of height exists, and one where you have some solution-dependent potential V(x,y,z) approx z.

As the absolute preferred height, as your solution fulfill the equation Delta V = 0.

In our real world, the first theory is simply false. But if above theories would be viable, the first would be preferable because of its simplicity.

The case for absolute time is clearly very much stronger, because we move only forward in time, which defines a qualitative difference to spatial directions with no analogy in this flat world analogy.

 

[Ilja]

Do the commonly accepted relativistic BM theory exist?
I mean, if even BM proponents can not agree on some basic things...

There exist various relativistic pilot wave theories. Which of them are preferable for which metaphysical arguments is discussed among pilot wave proponents. There is nothing wrong with it.

Problems with recovering empirical predictions of relativistic quantum theories do not exist. One rather small loophole related with possibly significant overlaps of macroscopic states in pilot wave theories with field ontology I have recently closed in http://arxiv.org/abs/0904.0764" .

But there exists enough metaphysical disagreement about the preferable version. Which is, in some sense, one aim of hidden variable theories: It is the disagreement about the special form of the hidden variables which allows, finally, to reject some versions and to get in this way some insight which would not be possible without it.

 

[Ilja]

If falling observer's time is slowed down in the same ratio as the star, then for that observer star won't look frozen. So what would he see?

He would hit a very hot surface with extreme speed. After this, he would be unable to see anything else.

 

[Demystifier]

The case for absolute time is clearly very much stronger, because we move only forward in time, which defines a qualitative difference to spatial directions with no analogy in this flat world analogy.

OK, I admit that it is a valid argument.

But the approach that I prefer has also other advantages. Take for example fermions. As you know, it is difficult to construct field beables for fermions. Thus, fermions are more likely to be particles. But then, how to describe particle creation and destruction of fermions? There are some stochastic approaches, but they are very inelegant. Besides, if you take field ontology for bosons and particle ontology for fermions, this difference makes the whole theory even more inelegant. But my approach, that treats time on an equal footing with space, automatically describes creation and destruction of fermionic and bosonic particles in the same way, without adding artificial stochastic elements to the theory.

 

[Ilja]

OK, I admit that it is a valid argument.

But the approach that I prefer has also other advantages. Take for example fermions. As you know, it is difficult to construct field beables for fermions. Thus, fermions are more likely to be particles. But then, how to describe particle creation and destruction of fermions?

Good question. My answer is part of my paper http://ilja-schmelzer.de/papers/clm.pdf" . The quantization procedure used there for fermions is canonical quantization. Thus, one can use canonical quantization for fermionic degrees of freedom.

But with canonical quantization, the corresponding pilot wave theory is already straightforward.

 

[Demystifier]

But with canonical quantization, the corresponding pilot wave theory is already straightforward.

For bosons, yes. But not for fermions.

 

[Ilja]

For bosons, yes. But not for fermions.

No, my canonical quantization proposal is for fermions, and done in a way that standard pilot wave theory can be applied.

The first step is to use a potential with two minima to obtain an effective Z_2-valued field theory for low energies from an R-valued field theory.

The second step transforms the algebra of commuting operators into an algebra of anticommuting ones. This is the really nontrivial part.

I do all this on a spatial lattice, so I also have to take into account species doubling, and obtain two Dirac fermions (electroweak doublets) from a single R-valued field.