From: paul@mtnmath.UUCP (Paul Budnik)
Newsgroups: sci.physics
Subject: A proof that quantum mechanics is an incomplete theory
Date: 5 Jan 93 17:54:00 GMT
Organization: Mountain Math Software, P. O. Box 2124, Saratoga. CA 95070

In article <1993Jan5.000032.4518@cs.wayne.edu>, atems@igor.physics.wayne.edu (Dale Atems) writes:
> [...]
> >I have posted the proof before. It is not complicated.
> 
> Would you be willing to post it again for the benefit of those of us
> who missed it the first time? [...]

The following is an edited version of my posting in November.

Quantum mechanics can be thought of as consisting of a linear and
nonlinear theory. The linear theory describes how the wave function
evolves according to the Schrodinger equation or equivalent formalism.
The nonlinear theory describes the quantum jumps that the wave function
undergoes when, for example, particles are created or destroyed. The
linear theory is what most physicists use most of the time. The nonlinear
theory is almost never needed in practical problems in QM. It would be
surprising if there is a fundamental logical flaw in the linear theory
given its wide spread regular use by so many physicists.

In contrast the nonlinear theory is known to be vague and the source of
60 years of confusion about quantum theory and measurement. The
nonlinear theory consists of nothing more than the assumption that
the wave function changes instantaneously when a measurement is made
to become consistent with the results of that measurement. It sounds more
like a philosophical principle and for the most part it can be regarded
as such. However there are certain quantum mechanical calculations in
which this principle *must* be used. It is in these calculations that
QM is vulnerable to attack.

The nonlinear theory is crucial to the claim that QM is a complete theory.
We know these nonlinear changes happen because we know particles are
created and destroyed. If the nonlinear changes do not occur
instantaneously they must have a space-time structure. This structure
is not part of any current theory.

The following argument shows that the vague assumption that wave function
changes instantaneously with an observation is not adequate to predict the
delays that can be measured experimentally in tests of Bell's inequality.

The standard approach to analyzing tests of Bell's inequality treats
the polarizer angle and the photon detection as a single measurement of
the state of the photon. The state is detected at one site and the wave
function is changed in accord with that observation. This restricts the wave
function at the other site in a way that influences the probability
of a joint detection. This works only if the polarizers are not changing.
However, an effective test of Bell's equality requires that the polarizer
angles are changing.

In an experiment with changing polarizers, the critical element is the delay
between when the angle changes and when this affects the probability of
joint detections. Unless this delay is less than the time it takes light
to travel from either polarizer to the more distant detector one cannot
claim to have shown that Bell's inequality is violated. Yet this delay
must be as long as the time it takes light to travel from either polarizer
to the detector *at each local site*.  If this were not true one could
use this affect for superluminal communication. All you need to is
locate the polarizers close to the photon source and redirect the photons
so that instead of moving apart they travel to detectors that are
closely spaced. You can then vary the relative angle of the two
polarizers and this will superluminally change the probability of a
joint detection. Because the two detectors are close together you can
use this experimental setup as a communications channel.

Once you start considering the distance between the polarizer and detector,
using the assumption that the wave function changes instantaneously becomes
problematic. In the usual geometry the detectors are equally distant from
the photon source. By the time you make an observation and change the
wave function at either detector, the angle of the polarizers *at that time*
can no longer influence the results. Thus the *instantaneous* change must
proceed *backwards* in time so that the state of the distant wave function
is consistent with the observation just made and the position of the distant
polarizer at some *previous* point in time. You might be tempted to change
the wave function when it traverses the polarizer, but this violates rules
of quantum mechanics and would lead to false predictions in other
experiments. If the polarizers are changing there is no basis for
deciding how to do the calculation, i. e. for deciding what polarizer
angle to use at either site.

You might try to build a wave function for the probability of a joint
detection and thus do the computation without using the nonlinear theory.
This is not possible. The nonlinear theory is the *only* nonlocal physical
theory.  Without it or some equivalent formulation you cannot obtain the
standard results for the correlations observed in tests of Bell's inequality.

The theory you need to predict these delays would give the probability for the
state of the photon if it is detected at a given time. The problem is that
quantum mechanics has no mechanism for computing the probability of a state
that is not an *observable*. It is central to quantum mechanics that the
state is determined by the observation and did not preexist. If the
polarizers are fixed, then the detection of the photon is a direct
observation of the polarization state. If the polarizers are changing
their angle, there is nothing in the detection to suggest the
polarization state. The probability of detection at either location
is unaffected by the angle of either polarizer.

We can contrast tests of Bell's inequality with experiments in which such
delays are predictable. Consider an experiment where a single photon
passes through two successive polarizers.  After the wave function
passes through the second polarizer the probability of a later detection
will be determined by the relative angle between the polarizers.
Changing the angles will influence this probability of detection.
This effect is local and predictable because it is *mediated* by the
propagation and structure of the wave function. No such mechanism
exists for computing the delays between changes in polarizer angles
and joint detections that are influenced by these changes in tests
of Bell's inequality. The nonlocal nature of the nonlinear part of
QM precludes the *possibility* of such a mechanism.

Quantum mechanics, as currently formulated, is an incomplete theory.
We need to measure these delays experimentally and extend quantum mechanics
based on these results. My expectation is that such experiments
will confirm that nature is local and that quantum mechanics as currently
formulated is not only incomplete but also incorrect.

Paul Budnik
Newsgroups: sci.physics