From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: Re: hidden variables
Date: 10 Jan 93 21:14:07 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <1993Jan10.141141.7849@oracorp.com>, daryl@oracorp.com (Daryl McCullough) writes:
> [...] However, I don't see how in the world you can
> claim that quantum mechanics predicts nonlocal effects, regardless of
> whether one assumes a hidden-variables completion. The form of Bell's
> proof is the following:
> 
>    1. Assume a hidden-variables completion of quantum mechanics.
>    2. Assume that all interactions are local (travel at light-speed
>       or lower).
>    3. Assume classical probability theory.
>    4. Assume the probabilistic predictions of quantum mechanics are
>       correct.
>    5. Show that all this leads to a contradiction.

Eberhard has derived Bell's theorem without *any* reference to
hidden variable theories. See "Bell's Theorem without Hidden Variables",
P. H. Eberhard, Il Nuovo Cimento, 38 B 1, p 75, (1977). In the light
of Eberhard's result it is misleading to tie Bell's theorem to hidden
variable theories. Eberhard proved that any theory that violates Bell's
inequality violates locality.

Let me quote Eberhard (page 76):
	
	Let us consider two measuring apparatus located in two different places
	A and B. There is a knob 'a' on apparatus A and a knob 'b' on
	apparatus B. Since A and B are separated in space, it is natural to
	think that what will happen in A is independent of the setting of
	the knob 'b' and vice versa. The principles of relativity seem to
	impose this point of view if the time at which the knobs are set and
	the time of the measurements are so close that, in the time laps,
	no light signal can travel from A to B or vice versa. Then, no signal
	can inform a measurement apparatus of what the knob setting of the
	other is. However, there are cases in which the predictions of quantum
	theory make that independence assumption impossible. If quantum theory
	is true, there are cases in which the results of the measurements in A
	will depend on the setting of knob `b' and/or the results of the
	measurements in B will depend on the setting of the knob a.

This is a clear statement of why quantum mechanics is not a local theory
PERIOD.

> Assumption 1 is necessary (in Bell's proof, anyway) to get a
> contradiction. Therefore, I don't know how you can say that quantum
> mechanics is nonlocal, regardless of whether one assumes a
> hidden-variables completion.

It is not used in Eberhard's simple and elegant proof. Do you think
there is an error in Eberhard's result?

> Also, I don't know how you can say that collapse of the wave function
> is a necessary assumption in the derivation of the contradiction. Bell
> didn't mention collapse of the wave function, he simply used the
> quantum mechanical prediction for probabilities of measurements. In
> the case of the EPR experiment, there is a compound measurement, where
> the spin of two different particles is measured (with respect to two
> different axes). Since spins of different particles are commuting
> observables, the results don't depend on the order the measurements
> are made. There is no need to assume collapse of the wave function
> between measurements.

You are mistaken here. You no doubt know a lot more about quantum mechanics
than I do, but you are glossing over the implicit assumption that you must
make to prove that Bell's inequality is violated. When you analyze the
two measurements you must assume that the first measurement (I agree
it makes no difference which is first) changes the probability
density function used in estimating the probability of detection at the
second detector.  If it does not you will not be able to prove that Bell's
inequality is violated. There is a space-like separation between the two
measurements. The only law of physics that will permit an observation at
one location to influence a probability density at a space-like separated
event is some version of the collapse postulate. This is the mechanism
in quantum mechanical calculations that permits the setting of a knob
to influence a space-like separate measurement.

Again I mention that at one point I was convinced that Bell's result was
false because all the laws of physics are local. I wrote Eberhard a letter
expressing my doubts. He explained to me that the nonlocal physical
principle that allows one to prove Bell's inequality is violated is the
collapse postulate.

Paul Budnik