From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: Re: hidden variables
Date: 23 Jan 93 18:36:22 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <1993Jan22.141944.7214@oracorp.com>, daryl@oracorp.com (Daryl McCullough) writes:
> paul@mtnmath.UUCP (Paul Budnik) writes:
> [...] 
> >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.

There is no impasse here. I have agreed that you can go outside the
wave function model and derive these probabilities. I consider that
argument specious because the wave function model is the only model
QM supports but I do not question that you can do this derivation.
By saying that you are outside the wave function model I mean you
are not describing how the wave function changes as a function of
time in your derivation. You are using higher level abstractions and
you are masking the role that collapse in some form must play to
get these correlations. I claim you need collapse in some form because
the axioms of QM without collapse only talk about probability distributions.
Proofs that Bell's inequality is violated refer to macroscopically
observable events. The only connection between these two is the collapse
postulate. You do not need to explicitly reference collapse because
you only need to talk about joint probabilities in a derivation that
ignores questions of timing.

The impasse between us seems to rest on the question of whether you
can claim Bell's inequality is violated without talking about timing.

> >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.

One can compute the probability that one will observe either photon assuming
no observation is made of its paired partner. That is an unambiguous
probability distribution at each site.

> 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.

I am not proposing to go to two one-particle wave functions. I am saying
to get timing information you need some model of how the wave function
changes over time. You do not have to assume that this is what is 
`really happening', but you need a mathematical model that describes
how the wave function changes and how this change relates to four different
macroscopic events: the two changes in polarizer angles and the two
detections. The way to do this is compute the probability of observing
the first photon, change the wave function based on this observation
and compute the probability for observing the second photon based
on this changed function. Do you have some other way to model the timing
relationship between these macroscopic events?

> >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.

This seems to be the central point that we are add odds about. Let me
point out that you do no have a violation of locality and you cannot
rule out the possibility of local hidden variables model accounting for the
observed results unless you include constraints on timing. I think you
will readily agree to this if you think about it. If not it is easy
to go through the exercise of constructing an appropriate local hidden
variables model that violates the mathematical relationship of Bell's
inequality.

> 
> >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.

When you talk about very distant measurements taken close together in time
you seem to be talking about the observed correlations. You are completely
wrong if you think this is true in general. Well I do not think there
is a local hidden variables model for QM that has this property there
certainly could be. Unless you talk about the timing relationship between
when polarizer angles change and when this effects the probability of joint
detections you cannot rule out a local hidden variables model regardless
of what the distances and timings are for measuring the detections.

If you are talking about the correct timings that must be measured in
tests of Bell's inequality then you are almost, but not quite correct.
You are correct that my local version of QM will almost certainly violate
the mathematical relationship specified in Bell's inequality for an
experiment conducted over sufficiently large distances and I have made
that point several times. However this is not necessarily true.
All that a local theory need do is to keep increasing the
delays as the geometry of the experiment gets larger. There is plenty of
evidence against this possibility for a model that deals with singlet state
particles. However, in general one can have a local hidden variables model
in which the mathematical relationship of Bell's inequality is violated
at arbitrarily large distances. The delays associated with that
violation simply need to increase linearly with distance.

I ask the critical question once again. What does QM predict about these
delays? I do not think you can compute these without using collapse and I do
not think you can compute them unambiguously using collapse.

Paul Budnik
Newsgroups: sci.physics