From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: hidden variables
Organization: ORA Corporation
Date: Fri, 22 Jan 1993 14:19:44 GMT

paul@mtnmath.UUCP (Paul Budnik) writes:

>> >When we observe the first photon in a test of Bell's inequality we must
>> >assume that the singlet state *pair* has become *exactly* aligned with the
>> >polarizer that photon just traversed. It is this assumption that implicitly
>> >involves collapse.
>> 
>> I agree. But that assumption is not necessary; all that is necessary
>> is that the probability of observing photon 1 pass through a polarizer
>> at angle A *and* of observing photon 2 pass through a polarizer at angle
>> B is ... [whatever the quantum prediction is].
>
>The question is: what are the principles of quantum mechanics necessary
>to derive that joint probability? If you do not use quantum collapse to
>generate such an alignment how can you derive the joint probability
>distribution?

Paul, you will have to tell me what you think would resolve this
impasse. It is mathematically possible to write down a set of
standard axioms for quantum mechanics, delete the axiom referring
to collapse of the wave function, and still derive the predictions
for joint probability of detection in twin particle experiments.
The collapse hypothesis in not mathematically necessary.

Perhaps you are saying that there is no way to get the system in
the appropriate initial condition (a singlet state of the twin
particles) without the collapse hypothesis?


>Keep in mind that before the observation of either photon
>there is a particular wave function at each site.

No, I don't think that there is. There is a probability distribution
in phase space, but there is no unambiguous probability distribution
at each site.

>The probability of observing either photon based on this wave
>function does not give the correct answer for joint probabilities. The
>distant wave function has to be changed based on the local observation
>to get a wave function at the distant site that will produce the
>correct probabilities.

You can certainly go from a two-particle wavefunction to two
one-particle wave functions by simply integrating over the unwanted
coordinates. However, by doing such an integration, you are throwing
away information, you are averaging over variables that you don't care
about. Whether or not you view the full wave function as physical,
there is no absolutely no reason to think of the results after
averaging to be a physical quantity. Quantum mechanics does not say
*anything* about one-particle wave functions; they are always simply
approximations to a full, many-particle wave function.

>We have a singlet state wave function that implies a correlation between
>some observable in two distant particles. I agree that you can derive
>the correlations from this principle by simply arguing that since
>these properties are correlated the observations must have a certain
>correlation. You can argue that collapse is not needed to reach this
>conclusion. I think this prediction outside the wave function model
>is a specious argument. The only model QM supports is the wave function
>model (or an equivalent formalism) and I think all QM arguments
>implicitly depend on that model. But all this does not matter
>because you cannot prove that Bell's inequality
>will be violated without using collapse.

Paul, I have derived the violation of Bell's inequality myself, and
I didn't assume collapse---I only assumed the quantum prediction for
probabilities of observations. (To get the probability of measuring
observable A to have value a, write the wave function |Psi> as
a supersition |Psi> = c1 |Psi_a> + c2 |Psi'>, where |Psi_a> is
an eigenfunction of A with value a, and |Psi'> is orthogonal to
|Psi_a>. In the EPR experiment, if you let A be the operator that
returns the pair of spins for the two particles, then the correlations
that violate Bell's inequality follows.)

>To get that you need more than
>correlations. It is easy using local processes to construct a system that
>will violate Bell's inequality as a mathematical relationship. To get
>a violation of Bell's inequality you must show that there is a space-like
>separation between when you manipulate some experimental parameter,
>for example a polarization angle, and when this has an effect on the
>probability for joint detections.

Bell's inequality is simply a statement about correlations. I think
what you mean is that the violation of Bell's inequality is only
surprising in the circumstances you are talking about.

>Strictly speaking you need to have
>two polarizers that are manipulated and both these manipulations must
>be space-like separated from the more distant detector to rule out any
>possibility of a local hidden variables model. To predict this timing
>relationship requires the assumption that the wave function changes
>instantaneously when an observation is made. If the wave function changes
>in a local fashion when an observation is made you will still get the
>correlations but you will not get a violation of Bell's inequality.

I don't think that you have that completely right. It is impossible
for the correlations to reproduce the predictions of quantum mechanics
without violating Bell's inequality. If quantum mechanics were false
(if your theory that the wave function is a physical quantity that
propagates at light speed were correct), then you would find that for
very distant measurements taken close together in time the quantum
predictions for correlations would be wrong. There is no way to preserve
the correlations without violating Bell's inequality.

Daryl McCullough
ORA Corp.
Ithaca, NY