Don’t get me wrong. What Bell did was extremely clever, and
he showed how to close an important philosphical loophole in the argument of
local realism. But what people don’t seem to realize is that for all practical
purposes, local realism was already in a shambles before anyone thought of
varying the angle of the crossed polarizers.
I wrote last week about how Einstein pointed out the
philosophical problem with two particles shooting off from each other with
opposite momenta. According to the theory, the actual trajectory of either
particle was completely indeterminate up to the surface of a sphere, until the
moment when one of them was detected. At that moment, the wave function of the
second particle collapsed simultaneoulsy: the measurement of one had affected
the properties of the other.
This was the clear and unmistakeable implication of the
theory, and it should have been extremely troubling. However there was a catch:
no conceivable experiment could distinguish this philosophical nightmare from
the more prosaic explanation that the particles had simply been endowed with
their complementary momenta at the moment of separation; that the randomness in
their detection was merely a lack of information on our part. To be sure, the
theory was clear on the distinction: but that distinction remained, so it seems,
only theoretical.
People seem to think that this happy state of affairs came
crashing down at the moment Bell proposed his experiment with the crossed
polarizers in 1964. That’s what I don’t understand. For me the disaster occurred
in 1950 when Bohm proposed the experiment with spin states. The disaster didn’t
depend on varying the angle of the polarizers: it should have been evident from
the get-go.
We know how the experiment must look if two electrons are
created with opposite spins: let one be in the positive z direction and the
other in the negative z direction. If we set up two Stern-Gerlach detectors at
opposite ends of the lab, we know what must happen. When one detects an “up”
electron, the other must detect a “down” electron. There is nothing mysterious
about this.
(Yes, I know the Stern Gerlach apparatus does not work on
charged electrons but only on neutral atoms: but the theory is the same and for
all I know, modern experimenters are able to adapt Stern Gerlach to charged
species.)
Where the problem occurs is when we get a stream of
electrons at each end of the lab, randomly up and down: what happens is each
time detector A clicks “up”, detector B clicks simultaneously “down”. That’s a
real problem.
But how is this different from the first situation, I hear
you ask, where I said there was nothing mysterious? Sometimes the particle
detected at A is up, so B must be down. And vice versa. Each detection stream
appears random, but compare the streams and you get perfect anti-correlation. It’s
all very ordinary, isn’t it?
No it isn’t! It is indeed possible to prepare pairs of
electrons in complementary states, one up and one down (or at least it’s
possible to write down an expression for the wave function!) and it’s very clear
what must happen when we detect them: if we detect one of them at A with spin
up, the other must be detected spin down at B…with a 75% probability! This is
the result we get if the particles are endowed with opposite spins at the
moment of creation: the detection streams are anticorrelated but not to the
extent of 100%. It is only a 75% correlation.
Why is it only 75%? Because in practise, there is no source
which prepares electrons in states of alignment only along the z axis: any real
source must produce pairs of electrons anti-aligned along a random axis. So, for example, if they
are aligned along the x axis, the z polarizer at A will detect its electron up
or down with 50% probability; likewise, and completeley independently, the
polarizer at B. You can see there is a 25% chance that both polarizers will
detect up coincidences, and another 25% chance they will detect both down
coincidences. You simply cannot get perfect anti-correlation with this kind of
setup.
Unless, that is, you prepare the electrons in a very
different type of state. It’s the quantum state we’ve alluded to already
whereby the two electrons have opposite spins but there is no actual axis along
which the spins are defined, until the moment of detection. That’s the
mysterious entangled state that leads to all the philosophical headaches about
local realism, and you don’t need to tilt your polarizers to 22.5 degrees to
come face-to-face with it. It was, or should have been, obvious from the moment
Bohm wrote his spin-modified version of EPR in 1950. There was no need to wait
for Bell to come along in 1964 to set off all the excitement.
What I don’t understand about the whole history is why I don’t
read anywhere about experiments to detect the perfect anti-correlations
predicted by Bohm in 1950? Why does everyone only talk about the
crossed-polarizer results motivated by Bell’s Theorem? To be sure, the later
experimenters record their data at a variety of different angles, so the Bohm
correlations are part of the record. But why does nobody ever talk about them
as being significant?
4 comments:
This is an excellent explanation that is difficult to find as clearly anywhere else.
Thanks, anonymous. I find the level of discourse in the world of physics to be very distressing. It's a bunch of know-it-alls repeating what someone else has already said a hundred times, and if you try to inject a different perspective into the discussion, they just call you a crackpot.
Hi Marty,
Came to this post after I had had the same realization as you, and -- amazed that no one had noticed it before -- decided to look for it more thoroughly on the web...
However, I have since (the realization happened two days ago...) understood our error: it is possible to explain the perfect correlation between the two measured spins when the measuring axes are parallel using local variables. You can see an example of that in section III of Bell's original paper. It is exactly those further correlations at other angles of measurement that lead to Bell's theorem, namely that no local-variable theory can account for the predictions of quantum mechanics.
So those physicists are so dumb after all ;)
Tal
Hi Tal
Yes there is a loophole whereby the 100% correlation could be explained, but it is such an unphysical contrivance that no one could take it seriously. I give Bell credit for closing the logical loophole, but that's all.
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