Local hidden variables for double slit

[atyy]

The double slit experiment is not known to violate any Bell inequality, and thus may have a local hidden variables description. Does Bohm-Dirac theory provide a local hidden variables description for the double slit? Are there other local hidden variables descriptions for the double slit?

(If I understand correctly, a local variable is (among other things) a variable defined on ordinary space, whereas wave functions for more than one particle are necessarily nonlocal, since they are defined on Hilbert space. A wave function for one particle is potentially (but not necessarily) local, since it can be considered a wave in ordinary space.)

 

[Demystifier]

Wave function (without a collapse) for one particle is local. In this sense, Bohm's theory of a double slit experiment is local, as long as the experiment can be viewed as a one-particle process.

However, if you take also into account the quantum nature of detector which consists of many particles, then the theory is not longer totally local. Nevertheless, nonlocalities in this case reduce to very short distances related to the entanglement of electrons within molecules.

 

[atyy]

When you say Bohm's theory of the double slit is local one considering just one particle, do you mean the usual Bohm theory with Schroedinger equation, or do you mean the Bohm-Dirac theory?

I can understand how Bohm-Dirac theory is local for one particle, but I don't see how Bohm (Schroedinger) theory is local. To my understanding, only a theory defined on Minkowski spacetime can be local, since to set up a Bell test, one needs the notion of a past light cone, which doesn't exist in a Galilean spacetime.

 

[Demystifier]

When you say Bohm's theory of the double slit is local one considering just one particle, do you mean the usual Bohm theory with Schroedinger equation, or do you mean the Bohm-Dirac theory?

Both.

To my understanding, only a theory defined on Minkowski spacetime can be local,

That's not true.

since to set up a Bell test, one needs the notion of a past light cone, which doesn't exist in a Galilean spacetime.

You seem to have a very narrow definition of locality in mind. For example, the contemporaries of Newton criticized his gravitation theory for being non-local, but they certainly did not think that a more satisfying theory should be defined on Minkowski spacetime. They thought that the gravitational force should be transmitted by some material objects having a finite velocity (which would mean that the theory is local), but they did not think that there should be a maximal possible velocity or an observer independent velocity.

Or in more contemporary terms, the tests of Bell inequalities do not necessarily need the notion of a past light cone. They need it if they want to close the causality loophole, but not all tests of Bell inequalities close the causality loophole.
http://en.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments#Communication.2C_or_locality

 

[atyy]

You seem to have a very narrow definition of locality in mind. For example, the contemporaries of Newton criticized his gravitation theory for being non-local, but they certainly did not think that a more satisfying theory should be defined on Minkowski spacetime. They thought that the gravitational force should be transmitted by some material objects having a finite velocity (which would mean that the theory is local), but they did not think that there should be a maximal possible velocity or an observer independent velocity.

Or in more contemporary terms, the tests of Bell inequalities do not necessarily need the notion of a past light cone. They need it if they want to close the causality loophole, but not all tests of Bell inequalities close the causality loophole.
http://en.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments#Communication.2C_or_locality

Well, I guess that I need at least a notion of a speed of light or some sort of maximum speed so that I can define a past light cone, otherwise I can only prove "non-factorizability", where factorizability is that the measurement outcomes $A,B$, measurement settings $a,b$ and hidden variables $\lambda$ satisfy $P(A,B|a,b,\lambda) = P(A|a,\lambda)P(B|b,\lambda)$. In order to go from non-factorizability to non-locality I think I need something like a past light cone if I understand correctly the discussion of http://www.scholarpedia.org/article/Bell's_theorem.

Perhaps I don't need Minkowski spacetime, and Galilean spacetime with a maximum speed would be good enough. But still it seems I need a maximum speed, and I think Schroedinger's equation has an infinite propagation speed, so it would seem to be a nonlocal equation.

Edit: OK, I think I understand. By "local", you mean the the speed of a single Bohmian particle is always finite?

 

[Demystifier]

In order to go from non-factorizability to non-locality I think I need something like a past light cone if I understand correctly the discussion of http://www.scholarpedia.org/article/Bell's_theorem.

Even if this is true, the point is that non-factorizability is a NECESSARY (if not sufficient) condition for non-locality in a Bell sense. If your wave function is factorizable, then you cannot have non-locality. But for a single particle (relativistic or not) the wave function is trivially factorizable, so in the single-particle case you always have locality.

 

[Demystifier]

Edit: OK, I think I understand. By "local", you mean the the speed of a single Bohmian particle is always finite?

No. If I meant this, then even the many-particle Bohmian mechanics would be local.

By locality I mean absence of non-locality. As I said in the post above, there is no non-locality (in the Bell sense) without non-factorizability.

 

[atyy]

Even if this is true, the point is that non-factorizability is a NECESSARY (if not sufficient) condition for non-locality in a Bell sense. If your wave function is factorizable, then you cannot have non-locality. But for a single particle (relativistic or not) the wave function is trivially factorizable, so in the single-particle case you always have locality.

But if I require non-factorizability in order to have non-locality, then wouldn't the wave function of a single particle be local even with collapse?

I guess it's like Many-Worlds - MWI is not nonlocal in the sense of Bell, since by assuming all outcomes occur, we cannot talk about locality/nonlocality in the sense of Bell. However, this doesn't mean that MWI is not nonlocal in some other reasonable sense.

Thus single particle QM is local, or at least not nonlocal, in the sense of Bell. However, it may be nonlocal in other senses like the Schroedinger equation having infinite propagation speed, or the wave function collapsing.

 

[akhmeteli]

The double slit experiment is not known to violate any Bell inequality, and thus may have a local hidden variables description. Does Bohm-Dirac theory provide a local hidden variables description for the double slit? Are there other local hidden variables descriptions for the double slit?

(If I understand correctly, a local variable is (among other things) a variable defined on ordinary space, whereas wave functions for more than one particle are necessarily nonlocal, since they are defined on Hilbert space. A wave function for one particle is potentially (but not necessarily) local, since it can be considered a wave in ordinary space.)

The reasonably realistic local models of my article http://download.springer.com/static/pdf/480/art%253A10.1140%252Fepjc%252Fs10052-013-2371-4.pdf?auth66=1410186002_cee4d7e712b389fec894c0344abc7b64&ext=.pdf (published in the European Physical Journal C, free access) might be of interest for you. The models contain electromagnetic field only (and external conserved currents, if necessary), but describe the matter field (scalar or spinor) as well. They contain, say, the Dirac equation, so they describe electron diffraction as well. Furthermore, the models can be embedded into quantum field theories. The Bell inequalities are not violated in these models, but there has been no loophole-free experimental evidence of the violations so far.

 

[Demystifier]

But if I require non-factorizability in order to have non-locality, then wouldn't the wave function of a single particle be local even with collapse?

No. Non-factorizability implies non-locality, but factorizability does not imply locality. Factorizability is a necessary condition for locality, but not a sufficient one.

I guess it's like Many-Worlds - MWI is not nonlocal in the sense of Bell, since by assuming all outcomes occur, we cannot talk about locality/nonlocality in the sense of Bell. However, this doesn't mean that MWI is not nonlocal in some other reasonable sense.

I disagree. Perhaps MWI is local in some sense (*), but not in the Bell sense. Bell locality is locality in the 3-dimensional physical space, while MWI cannot even be formulated in that space. Since it cannot be formulated in that space, it cannot be local in that space.

But one must add that it cannot be non-local in that space either. So I would say that MWI is neither local nor non-local in the Bell sense. The set of local and non-local theories considered by the Bell theorem does not include MWI.

Thus single particle QM is local, or at least not nonlocal, in the sense of Bell. However, it may be nonlocal in other senses like the Schroedinger equation having infinite propagation speed, or the wave function collapsing.

That's true.

(*) MWI is local in the configuration space, but Bohmian mechanics is also local in that space. So if MWI can be considered local, then so can Bohmian mechanics.