J.
S. Bell responded to the problem raised by the EPR Paradox by showing that it is, in fact, the
realist position that is untenable. The
realists believed that something else determined the things left undetermined
by quantum mechanics- they didn’t say that quantum mechanics itself was
wrong. They held that the theory was
incomplete, that something else determined the positions, velocities, etc that
were ambiguous under quantum mechanics.
This “something else” could be an as-yet-undiscovered force, butterflies
beating their wings in China (or some other undiscovered planet), space dust,
alien ray guns, some attraction to particles we haven’t discovered yet, or
anything else. This type of variable was
called a hidden variable, because no one had any idea what it is, how to
calculate it, or how to measure it, and I’m going to call it λ.
In
the EPR argument, the spins of the electron and positron are measured in the
same direction- the only options are up and down. However, space (at least macroscopically) has
three orthogonal dimensions.
If we looked at a table of the
measurements of the decays, it might look like this:
Electron
Spin Positron Spin Product
-1 +1 -1
+1 -1 -1
+1 +1 +1
-1 -1 +1
and
so on…
For
arbitrary directions, quantum mechanics predicts that P(a,b)= -a ∙ b.
This basically comes from some math involving the wave equation and
spins, and is rather unenlightening and so I will not include it here. The idea is to give a conceptual basis for
why the orthodox interpretation is right, and not an infallible argument. The problem is rather simple, though, and the
result is provable from quantum mechanics.
Now
suppose the complete stat of the electron/positron system is characterized by
the variable λ, which is a
hidden variable. Also, suppose your
experiment is set up such that the direction that the positron’s spin is
measured in is independent of the direction the spin of the electron is
measured in- for example, have the distances long enough and the times short
enough that no information can be communicated by the experimenters running the
experiments, according to locality. Now,
even though we have no idea what causes λ, or how it varies from one experiment from another, or even if we
can quantify it numerically, we can create a function E(a,λ), to describe the result of the electron measurement, and a
function P(b,λ), to
describe the result of the positron measurement. These functions have values of +/- 1. When the detectors are aligned, again the
results are exactly opposite: E(b,λ)=- P(b,λ), for all λ, because the spins are always
opposite regardless of what λ is doing to
determine them.
From
math, we know that the average of the product of the measurement is:
where ρ(λ) is the probability density for the hidden variable. Like all probability densities, it is
non-negative, and if you put all the probabilities together, you get 1, written
mathematically as:
This is a property of probability
densities.
Since E(b,λ)=- P(b,λ), for all λ, we can write:
If c is any
other unit vector, then:
Remember that E(a,λ)=+/-1, we know that E(a,λ) squared is 1, so then:
Again, since E(a,λ)=+/-1 and E(b,λ)=+/-1, this means that -1≤ E(a,λ)E(b,λ)≤+1.
Similarly,
ρ(λ)[1- E(a,λ)E(c,λ)]≥0
All this
means that:
By simplifying, we get the Bell
Inequality:
This inequality holds for ANY local
hidden variable, with conservation of angular momentum and quantization of spin
(a central tenet of quantum mechanical theory that NO ONE disagreed with- not
even Einstein.)
This
does disagree with quantum mechanics.
For example, let a and b be perpendicular to one another, and let c be at a 45° angle between the two, in the same
plane. Then, quantum mechanics
(remembering that P(a,b)= -a ∙ b=|a||b|cos(θ) ), predicts that P(a,b)=0 and P(a,c)=P(b,c)=-.707.
But then
.707≤ 1-.707=.293, which is
clearly NOT true.
So,
since realism and quantum mechanics actually DO predict different results for a
measurement, experimenters did the experiment…
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