--- In [EMAIL PROTECTED], Julia Thompson <[EMAIL PROTECTED]> wrote:
>
>Some of your tables are wider than 80 characters, making them a lot >harder
to understand.
>
> Julia
Sorry about that, I thought I had that fixed. I think this is <80 wide.
Dan
Consider two spin 1/2 particles created from a spin zero particle.
In the center of mass of these particles, they will travel with equal
energy and momentum in opposite directions. They will go through two
detectors stationed a significant distance away from the original event.
The spin of these two particles is then measured. There are only two
possible values for spin for a spin 1/2 particle: +1/2 and -1/2.
Fractional values of spin are never measured. We call +1/2 spin
"spin up" or |u>. We call -1/2 spin "spin down " or |d>. We proceed
to measure a significant number of pairs of particles in our
experimental setup. In the first step, detector 1 and detector
2 measure spin in the same direction. In this case, whenever
the particle measured by detector 1 is up, the particle measured by
detector 2 is down. The reversal is also true. We express this in
quantum mechanical terms by the expressions: |u1d2> & |d1u2>.
We can separate the detectors far enough so that we will make
spacelike measurements. By spacelike, we mean that the distance
between the detectors: x is greater than c*dt, where c is the speed
of light and dt is the time between the measurements. When x >c*dt,
there is no chance for a signal to travel from one detector to the other
without violating special relativity. Thus, if there are hidden
variables that determine the spin of a particle in every direction that
a particle will be measured, they have to be set when the two
particles are close enough for a signal to pass between them.
For the sake of convenience, this is assumed to be the moment of
creation of the two particles. While spin 1/2 particles show no
intrinsic spin until they are measured, we will use "having spin X
in direction Y" as a short hand for "the hidden variables are so
arranged that when the spin is measured in direction Y, the value
will be X. We place no limit on the number of hidden variables,
allowing both a continuous and a discrete infinity of variables.
Having established a 100% anti-correlation between measurements
in the same direction, we now consider measurements in different
directions. Three directions are considered: 1, 2, and 3. They all like
in planes perpendicular to the direction of flight of the two particles.
Direction 1 is set at angle 0 deg, direction 2 is set at angle 18.44
deg, and direction 3 is set at 36.88 deg. We find that there is not 100%
correlation when pairs of the particles are measured at these different
angles. Rather, we get the following probabilities:
____________________________________________________
|u1> | |d1>
----------------------------------------------------------
|u2> 90% | |u2> 10%
------------------------------------------------------------
|d2> 10% | |d2> 90%
and
___________________________________________________________
|u2> | |d2>
----------------------------------------------------------------
|u3> 90% | |u3> 10%
----------------------------------------------------------------
|d3> 10% | |d3> 90%
-----------------------------------------------------------------
We are now ready to predict the correlation between the spins measured
in direction 1 and direction 3. If spin is determined by hidden variables,
the particles "have" the spin from the moment the separate. While
it is not exhibited macroscopically, the value to be measured is
determined by a number of variables that we just haven't gotten to yet.
So, we can use the following logic to determine the correlation between
spin in direction 1 and direction 3. We know the correlation between
1 and 2, and between 2 and 3. We know that when we measure 1 and 3,
2 is not measured, but it is either up or down. This produces the
following table:
_________________________________________________________________________
|u1> | |d1>
--------------------------------------------------------------------------
|u2> 90% | |u2> 10%
---------------------------------------------------------------------------
|u2>& |u3> 81%, |u2> & |d3> 9% | |u2>& |u3> 9% | |u2> & |d3> 1%
---------------------------------------------------------------------------
|d2> 10% | |d2> 90%
----------------------------------------------------------------------------
|d2>& |u3> 1% |d2> & |d3> 9% | |d2>& |u3> 9% | |d2> & |d3> 81%
----------------------------------------------------------------------------
-
TOTAL
----------------------------------------------------------------------------
-
|u3> 82% | |d3> 18% | |u3> 18% | |d3> 82%
----------------------------------------------------------------------------
-
This analysis assumes that there is not a specific three way correlation
betwen |u1> |u2> and |u3>. Assuming the anticorrelaton between |u1>
and |u3> is maximized, we still do not get 64% correlation
To see that, let us look at an example. Lets assume that there are
100000 pairs of spin one particles produced. We know that 90000
(give or take very small statistical uncertainty of 0.3%) of these
have correlations between direction 1 and direction 2
(either |u1u2> or |d1d2>) and that 10% have anti-correlation
(either |u1d2> or |d1u2>). We also know that 90000 of these have
correlations between direction 2 and direction 3
(either |u2u3> or |d2d3>) and that 10% have anti-correlation (either
|u2d3> or |d2u3>). Let us try to obtain the minimal correlation
possible between 1 and 3 with this setup.
This case occures if there is 100% correlation between directions 2 & 3
whenever 1 & 2 anti-correlate, and 100% correlation between directions
1 & 2 whenever 3 & 3 anti-correlate. This gives us the following table:
____________________________________________________________________________
_
1 & 2 correlate | 1 & 2 anti-correlate |
2 & 3 anti-correlate | 2 & 3 correlate | 1 & 2 & 3 correlate
----------------------------------------------------------------------------
-
10000 | 10000 | 80000
____________________________________________________________________________
_
Thus, at a minimum, there is 80% correlation.
In order to construct this table, we used the following.
1) The spin in direction 2 is either up or down. That is to say the
hidden variables are so arranged that up will be measured or
down will be measured.
2) The distributive law is true:
If
A and (B or C)
then
(A and B) or (A and C)...
Statistical analysis of finite numbers of random events is valid. We
can establish experimentally that the spin measured for one pair of
particles is independent of the spin of any other pairs of particles.
Thus, if we measure the spin of one particle in direction 1 as up
100,000 times and measure the spin of its partner in direction 2 as up
10,000 times, the correlation is 10% ( 0.1%. From statistical
analysis, we know that the probability of observing < 9.5% or > 10.5%
correlation in the next 100,000 pairs of particles is less than 0.001%.
This type of analysis in absolutely essential to any work with random
events. To question this technique would be to question all work in
nuclear physics and most work in solid state physics and thermodynamics.
4) The empirically confirmed laws of physics are valid.
One of the four assumptions listed above must be removed. If we get rid
of the distributive law, we remove the foundation of most of
mathematics. If we assume that statistics are not really valid for
finite numbers, we eliminate a key area of math. If we assume the
laws of physics are invalidated in a "hidden manner," we drop one of
the basic assumptions of science. If we get rid of the first
assumption, we undermine realism, but not science or math.
Since realism is a philosophy that states that we can know the world
as it is, it is hard to have a realistic worldview that states that the
world is other than scientific measurements indicate. Thus, realism
has a difficult time explaining quantum mechanics.
