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Saturday, March 26, 2005

Another locally realistic violation of Bell?s inequality ?

On December 9 2003 I was thinking about the EPR experiment. I am actually thinking about the
experiment as proposed by E P and R, not the experiment as performed. There are
four basic but very weird points to remember in order to understand the proposed
experiment. One is that electrons spin around sort of like little globes but that the
absolute value of the angular momentum is always hbar/2 no matter from which
direction you look at them. This is not like, say, the earth which spins around the
North-South axis so the angular momentum is less if you look down another axis.
The second strange fact is that you can?t measure the angular momentum of one
single electron around two different axis (one version of the Heisenberg
principal). This can be understood as measuring around one axis changes the spin
around another axis. This makes sense in terms of comprehensible things like
macroscopic magnets because the way to measure spin is with a magnet and a
magnet does change spin. The third strange fact is that sometimes you know that
the spin of two electrons around any axis is opposite. This is true if the two
electrons are in what is called a singlet state. This means that even if you can?t
know which way (clockwise or counterclockwise) each electron is spinning
around both the North South axis and the East West axis you can know that the
two are spinning in opposite directions around each axis. The fourth strange fact
is that, according to quantum mechanics the correlation between ?spin is
clockwise around the North South axis? and ?Spin is clockwise around the North-
East South-West axis? is greater than one !!! that is the probability both are
clockwise is greater than the probability that the spin around the North South axis
is clockwise and the spin around the other axis is whatever and unmeasured. I
won?t be able to explain this so that it makes sense. It is clearly crazy. It is also an
experimental result not just a theory. I am sure I will never understand this, so I
can?t explain it. Trying to grasp this fact, and it appears to be a fact (see below) is
like trying to pound a square peg into a round hole.

Now the point of EPR is that Heisenberg shmeizenberg you can use the singlet
state fact to test the quantum mechanical correlation greater than one by
measuring the spin around N-S of one of the electrons and the Spin NE-SW of the
other. They were very sure that the experimental result would disprove quantum
mechanics. After some decades the experiment was finally performed (with light
not electrons) and quantum mechanics was confirmed. I gave up.

Then I heard about the work of Luigi Accardi and Massimo Regoli. They have an
argument that you can reconcile the experimental result with a locally realistic
theory (one that makes sense as correlations are less than or equal to one) if the
singlet state is a statement about measuring spin around one magic axis (NS say).
The quantum craziness comes from treating electron 2 is counterclockwise around
NS as equivalent to electron 1 is clockwise around NS AND electron 2 is
counterclockwise around NE-SW as equivalent to electron 1 is clockwise around
NE-SW. What if the singlet state had to do only with the N-S axis ?

An objection is that the singlet state fact has been tested for many directions and
always works.

I have a slightly modified version of the Accardi Regoli story (which is probably
in one of their working papers which I haven?t read). Here the experiment is we
have electrons that were in the signlet state flying out of a source each one
through an electromagnet which can be oriented NS or NE-SW. If both
electromagnets are oriented the same way, the spins are opposite. If the
experimenter moves the electromagnets so they are not oriented the same way, the
electrons see this and don?t act like singlet state electrons at all anymore. This
means that the EPR experiment result is not equivalent to a correlation greater
than one.

OK so the little electrons are spying on the experimenter. There are two things.
Which way does he point the electromagnets and are both turned on (measuring)
or not. If they are oriented on the same axis then the measurement of electrons
going through magnet 1 does not depend on whether magnet two is turned on and
the measurement of electrons going through magnet 2 is always opposite. If they
are oriented in different directions, then the measurement of electrons going
through magnet 1 depends on whether magnet two is turned on. This is an effect
of an event which took place a long time ago (by flying electron terms) so there is
nothing impossible about it.

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