From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: hidden variables
Organization: ORA Corporation
Date: Wed, 6 Jan 1993 17:44:55 GMT

rkoehler@author.gsfc.nasa.gov (Bob Koehler) writes:

>Would someone be so kind as to summaries the argument that hidden
>variable theories violate causility?  It's been a long time since grad
>school, and this has been scratching at the front of my mind recently,
>while I know the answer lies burried deep in the back.  I just haven't
>had the time to wade through my books.

The simplest inequality to derive was given by Wigner in the American
Journal of Physics, Vol 38, page 1005, 1970. (There are also several
derivations in the book _Speakable and Unspeakable in Quantum Mechanics_
by J.S. Bell.) A simple derivation of a "no hidden variables" theorem
goes as follows:

Let A,B, and C be three events. Then the joint probabilities obey the
following inequality:

    P(A and not-B) + P(B and C) >= P(A and C)

To see this, simply note that

   P(A and C) = P(A and not-B and C) + P(A and B and C)

the first term is less than or equal to P(A and not-B), and
the second term is less than or equal to P(B and C).

Now, let's apply this inequality in the case of measuring spins of a
spin-1/2 particle such as an electron. Assume that there are hidden
variables that determine, for each direction, whether the spin will be
measured to be up or down in that direction.

Let a, b, and c be three unit vectors in the x-y plane such that the
angle between b and a is 60 degrees, and the angle between a and c is
also 60 degrees (so that the angle between b and c is 120 degrees).
Let A be the event "The particle has spin-up in the a direction", B be
the event "The particle has spin-up in the b direction", and C be the
event "The particle has spin-up in the c direction".

Now, it is only possible to measure the spin of the particle in one
direction at a time. However, in a twin-pair experiment, two spin-1/2
particles are produced with exactly anti-correlated spins; whenever
one particle is measured to have spin up in one direction, the other
particle is measured to have spin down in that same direction.

Therefore, it becomes possible to measure the values of the spin in
two different directions, by measuring the spin of one particle in
one direction, and measuring the spin of the other particle in the
other direction.

The predictions of quantum mechanics for such an experiment are:

     P(A and not-B) = 1/8
     P(B and C) = 1/8
     P(A and C) = 3/8

This violates the derived inequality:
       P(A and not-B) + P(B and C) >= P(A and C)

This implies that our assumption was wrong; it can't be the case that
the result of a spin measurement simply tells us the value of a
pre-existing property of the particle.

The other possibility is that the value of spin measured in a
particular direction depends not only on the hidden variables
associated with the particle, but also on the choices of orientations
of the measurement devices. A slightly more complicated derivation
shows that the following hypothesis is incompatible with the predicted
results of quantum mechanics:

   The value of the spin of a particle depends only on (1) hidden variables
   associated with the particle, plus (2) the choice of orientation of the
   measurement device that the particle passes through.

The only way to get a hidden variables theory that is consistent with
quantum mechanics and probability theory is if the value of the spin
measured in a particular direction for a particle depends on the
choices of directions of *both* measurement devices (the one for
measuring the particle, and also the one for measuring its partner).
In other words, the only way a hidden variables theory can work to
explain the twin pair experiment is if the orientation of a distant
measurement device instantaneously affects the measured properties of
a particle here.

Daryl McCullough
ORA Corp.
Ithaca, NY
Newsgroups: sci.physics,alt.sci.physics