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
