Dear Aleks,
On one hand I like very much your example and discussion, but on the
other hand I do not think that your example can be used to illustrate
the situation with Bell�s inequality.
Aleks:> Bell\'s formulation implies that the hidden variable is
independent of> the configuration of the detectors. On the other hand,
if we allow > for > the fact that the hidden variable is dependent on
the mutual > configuration, then there is no need to abandon the
classical> framework.
Andrei:I completely agree with this viewpoint.
> To some extent this does imply some sort of non-locality. But it is
> the > same type of non-locality as in a perfectly classical context.
Let me> describe an experiment.
Andrei: I completely agree with this viewpoint.
Andrei: But regarding the experiment -- see my comment after your
description of the experiment.
Aleks:> Imagine a bathtub. Imagine primitive measurement technology that
> involves putting person A in the water so that his nose is aligned
> with
> the water. He is measuring the incoming waves, and outputs 1 when a
> wave
> splashes into his eyes, and 0 otherwise. There is another person B
> measuring in the same way on the other side of the bathtub. You can
> change the orientation of both people, and this will clearly affect
> the
> frequency of water splashing into their eyes. Now throw a stone in
> the > bathtub, creating an entangled pair of wavefronts propagating in
both> > directions. Measure the correlation of splashes into person A\'s
eyes> and > person B\'s eyes.
> There are three interpretations of the resulting correlations of
> splashes:> *
(QM) The waves are in a superposition of splashing and
> not-splashing,
> until person A experiences a splash or not-splash, and collapses the
> wave function. The collapse telepathically
\"informs-at-a-distance\"
> the > other entangled wavefront whether person B will experience a
splash> or not.
> * (New Age) The two detectors are not independent - they are an
> entangled pair in the aether of global consciousness, with the
> incredible ability of transmitting thoughts at a distance.
>
> * (classical) The fact that two people are in a bathtub with their>
big > measuring apparatuses (bodies, noses and eyes) will affect the
joint > distribution of splashes. The position of one person will affect
the > measurements experienced by the other person, because it affects
the > shape of the body of water in which the wavefronts propagate.
Andrei: Unfortunately, you would not be able to violate Bell\'s
inequality in this way. Although you use waves, these are classical
waves. The crucial difference between the classical wave mechanics and
quantum wave mechanics is that the system of two classical waves is
still described by the wave on the same three dimensional physical
space, but in the quantum case (and this was first time emphasized by
Schrodinger) the corresponding equation for the the system of two
quantum waves, <<quantum particles>>, is written on the R^6 and not on
the physical space R^3. Essentially this is responsible for a special
quantum correlations and the violation of Bell\'s inequality. So if you
like the source of nonlocality is already in Schrodinger equation.
Therefore I am not much excited by the Bell inequality.
However, one may say: well we have a rather strange description of
quantum waves-particles, namely, by using the tensor product of Hilbert
spaces to describe composite systems, but it may be, nevertheless, a
purely classical and local model behind this? Of course, Bell would say
you: not at all.
But I say: it seems yes. And here I would like to follow precisely your
argument:>if we allow > for > the fact that the hidden variable is
dependent on the mutual > configuration, then there is no need to
abandon the classical> framework.
There is one thing that couple both detectors. This is nothing else than
time. In all experiments there is such a thing as TIME WINDOW and
experimenter identify two clicks as belonging to an entangled pair
if these clicks are inside the time window. There is no other way to
identify a pair. We do not know times of emissions from the cristal.
Moreover, particles (considered as entangled and belonging to the same
pair) can be emited by different atoms.
This time window couple two detectors or I would like to say determine
unification of two local contexts of measurements. If there are two
fixed orientations A and B in lab1 and lab2, respectively, then by using
the time window we identify a special series of clicks in labs, so we
extract a special ensemble S_AB of particles. If we choose another
orientations, say C and D, then through the time window we shall get
another ensemble S_CD. If the situation is really such and ensembles are
really statistically different, then this gives us purely classical
explanation of the violation of Bell\'s inequality. However, it is common
to say that S_AB has the same statistical properties as S_CD. This is so
called FAIR SAMPLING ASSUMPTION. I thing it is not justified, see
http://www.arxiv.org/abs/quant-ph/0309010
[Experimental Scheme to Test the Fair Sampling Assumption in EPR-Bell
Experiments]
The main problem for me: to find such a dependence of statistics on
experimental settings in other domains of science. May be somebody
could come with some ideas?
With Best Regards,
Andrei Khrennikov
Director of International Center for Mathematical Modeling in Physics,
Engineering, Economy and Cognitive Sc.,
University of Vaxjo, Sweden
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Received on Sat May 27 22:24:13 2006
