Last time we talked about Bohmian mechanics, a possible alternative to Quantum mechanics.
One of the appeals of Bohmian mechanics is that we can always, at any time, say this
is the position of this particle.
On the other hand, with Quantum mechanics, the superposition principle forces to say
things like, the particle is in many different states at the same time.
With Bohmian mechanics, it's as though reality properly exists regardless of whether we've
measuring or not.
Or does it?
Today we're going to look at a potentially big problem with Bohmian mechanics.
The issue is something called Contextuality.
Let me explain it
With any sensible theory you want measurement to reveal a property that a particle already
had, not create or change it.
I should point out, under this criterion, Quantum mechanics is clearly not sensible.
But from what we was last time, it seems like Bohmian mechanics is.
So if you measure, say the spin of a particle in the y direction, and you find it to be
spin up, in Bohmian mechanics, you want to be able to say, then that particle had spin
up before hand.
But what about the case where there is more than one way to measure the spin of a particle?
Another sensible sensibility condition is that different ways to measure something should
agree, so both those measurements should give the same answer, right?
We'll call this noncontextuality- because the result of the measurement shouldn't depend
on the experimental context.
This is where the problem is though.
In Bohmian mechanics this doesn't always happen.
So instead of being always able to say statements like "this particle has spin up in the y direction"
you are instead are reduced to saying things like, "this particle has spin up in the y
direction if you measure it this way, but spin down if you measure it another way".
It's still perfectly deterministic, there is no randomness but... it's just not that
nice.
This is what we call contextuality.
Let me show you an example of this happening.
This example is going to be about the spin of an electron.
Don't panic if you've forgotten all about spin, I'm only going to use some very basic
facts about it.
If you want a refresher on spin though, you can go watch this video about it.
First remember that every electron has it's spin pointing in one particular direction.
However, it's also possible to measure the spin of the electron in any direction and
this is where it gets weird.
Whatever direction you pick, there are only two possible values you can get, which we
call spin up and spin down for reason that you'll see soon.
If you're feeling uncomfortable about spin at this point, feel free to watch that video
and ask me questions.
Back to spin though: There is a pretty standard way of measuring spin of a particle; it's
called the Stern- Gerlach machine.
You set it up to measure spin along a certain axis, say the y axis.
Then what happens is electrons goes through this passage.
I won't go into how this works but if your particle happened to be one that already had
spin up in the y axis, it will get deflected up and get measured around here.
If the particle started with spin down in y it will get deflected down.
How does Bohmian mechanics explain this?
Remember that in Bohmian mechanics there is that quantum force that depends on the wavefunction.
Well, spin is one of the things that influence the wavefunction so the quantum force takes
spin into account, and then pushes the electrons in the correct directions in these cases.
Of course, most electrons have their spin in other directions.
What about if a bunch of electrons happened to have their spin in the x direction which
is perpendicular to y?
As we saw in the spin video, when you measure these particles in the y direction, half the
time the electron will have spin up and half the time it will have spin down.
So in the experiment half these particles travel up and half down.
Again, Bohmian mechanics needs to explain this.
First we need to deal with something.
There is no real randomness in Bohmian mechanics, only randomness from ignorance, so we need
to be able to explain why half the particles go one way and half the other way.
This is how it's done.
Let's say all the electrons start off somewhere inside this little sphere, but we don't know
exactly where inside.
If one of these particles starts in the top half, you can calculate by Bohmian mechanics
that it will deflect up, and similarly for the bottom half, giving exactly the right
result.
No issues so far.
Now let's look at sensibility condition number 1 again.
We measure that this particle that starts here has spin up in the y direction, so our
criteria tells us, it must have had spin up in that direction along.
Again, that seems fairly innocuous.
The problem is that there's another way to measure the spin of that particular particle.
Why not set the Stern-Gerlach machine to measure spin in the negative y direction instead?
What should happen?
Well, we've already decided that this particle has spin up in the y direction so by the properties
of spin it must have spin down in the negative y direction.
Sorry, that's the last fact that I'll use from that spin video.
So that means that our particle is spin down in the negative y direction so it's deflected
down now right?
Nup, that's not what Bohmian mechanics says.
It still goes up, which forces us to conclude it has spin down in the y direction.
And now we see how our 2nd sensibility criterion is broken.
If we measure the first way, our particle will look like it had spin up, but if we measure
the second way, it will look like it had spin down.
Notice that I'm not saying Bohmian mechanics makes the wrong experimental prediction, because
in the second case we also expect that half the electrons deflect up and half deflect
down, and that's exactly what happens.
Bohmian mechanics can give you the right predictions on any stern-gerlach experiment...
the issue is much more subtle than that.
If we can't say our particle has property X with out adding the proviso "so long as
we measure it this way", then what property does the particle have before anyone decides
how to measure it?
I mean this seems like a quantum mechanics level philosophical problem.
So, should we scrap Bohmian mechanics and look for some other, better alternative to
quantum?
Not so fast.
In fact, it's possible to prove that for any theory that has sensibility criterion 1: That
experiments reveal preexisting properties, then our second condition, that properties
don't depend on the type of experiment, will always be broken- at least whenever you are
talking about spin.
In other words, there are no completely sensible theories
There's a lot of hype about this contextuality, but I'm not sure how seriously we should take
it.
You see, this issue doesn't come up at all when we're talking about most variables in
Bohmian mechanics, like position, momentum and energy.
It's perfectly legitimate to say a particle has this position, regardless of how you measure
it.
The problem only comes when you try and say statements like that about spin and other
similar variables.
That's why some people say the theory isn't broken, only the way we think about spin is.
They say spin is not a real property of a particle like position and momentum, instead
it only comes into play in an indirect way: the spin affects the wavefunctions which affects
the quantum force which changes the direction of the particle.
It doesn't make sense to say "a particle has spin up", instead, what you mean is the particle
experiences some extra quantum force upward because of spin.
I'm pretty conflicted about all this, and I'd love hear your opinions about all this.
Does Bohmian mechanics escape, or is this really the final blow for all sensible alternatives
to quantum mechanics?
I'd like to leave you with one deep thought though: Spin... is weird.