From: bhv@areaplg2.corp.mot.com (Bronis Vidugiris)
Subject: Hidden Variable Theories
Organization: Motorola, CCR&D, CORP, Schaumburg, IL
Date: Tue, 15 Sep 1992 17:33:25 GMT

I've been thinking some about how to express my views on the topic
of hidden variable theories more clearly, and I came up with the following
idea.

Let us imagine that we are trying to write a computer simulation that models
the behavior of the universe - actually, a very small subset of the
universe, the behavior of a pair of particles in the 'singlet state'.  This
seems to me to be a worthwhile goal, in that if we can write a program that
gives the same results as what we observe, we have some understanding of how
it could work.  The particular  implementation may or may not be 'correct' -
the issue is to come up with _a_ model that explains all observaable
phenomenon.  This scheme does seem to implicitly involve the idea of hidden
variables, though.  Frankly, I must admit that I don't really understand how
to address the issue conceptually without using 'hidden variables'.


What we have to work with in our model is a particle, which has some state
information.  We will express the randomness of this state information as a
single 'random number' from a random-number generator.  We will not worry
about the implementation of this generator - that's a minor detail at this
level of abstraction.  We can use several random numbers if we feel the
need.

We also have two analyzers which measure the spin of the particles.  These
analyzers have axes 'A' and 'B' which can be set by the experimenter.
We will assume that the experimenter has 'free will' and reject explanations
that require the experimenter to specify certain settings in advance.

Rather than deal with a continuous model of the system, we will 'quantize'
the variabiles 'A' and 'B' (the settings of the measurement units) to some
number of arbitrary but discreete settings.  This will simplify the analysis.
We will also 'quantize' the state information, also for simplicity.  (Any
'real' computer will have only finite precision in any event, so we don't
loose much ultimately by this qunatitization, which we can make as
fine as we like.)

We can then express the behavior of the system with the following model.
We generate a single random number.  We then lookup the result of the
experiment in a lookup table - actually a 3 dimensiona array - given the
random number, the setting of measurement device A, and the setting of
measurement device B.  (Results can probably be limited to two or three
categories, dependent on the experiment.  Results could be 
'detection/absence', or 'up detection, down detection, no detection'
depending on the experimental setup).

Conceptually, this is saying that we have some probability functions PA and
PB which are function of the random number (the 'hidden' state information,
call it s), the setting of measurement unit'A' (call it a), and thet setting
of measurement unit 'B' (call it b).  This is PA(a,b,s) and PB(a,b,s).
The value of PA when it is looked up in this table gives us the result
measured at measurement unit 'A', and PB is a table of the results at
measurement unit B.  This scheme does allow us to model the system
accurately.

Now - we would like to simplify this program.  This is a really big table.
Isn't it overkill?  Can't we use *two* lookup tables of smaller dimension? 
Can we get the result for measurement unit 'A' from a table given a and s,
and the result for measurment unit 'B' from a table given b and s.

Conceptually, wouldn't:

PA(a,s) and PB(b,s) be a better formualtion of the problem?

The very surprising answer turns out to be NO!  There is a mathemtatical
condition, called 'Bell's inequality', which prevents any such model from
yielding the proper results.

No such model can explain the results predicted by QM.  

Newsgroups: sci.physics,sci.math