Okay, so today we are going to talk about Bell's inequalities and the question is simply

this, if you take this EPR experiment, for example the boom spin version of the EPR experiment,

can you find some simple underlying microscopic theory in the sense of Einstein Podolsky

Hosen, some realistic theory where you can think of properties even before the measurement,

can you make that work?

This is the question.

This is not the answer.

Now there's a word, people would call it such a theory, a hidden variable theory, because

it would contain more variables than quantum mechanics and they would be hidden in the

sense that in experiments then again you can't measure both position and momentum, but they

would be there and they would be part of the theory.

Also notice this is relatively modest, we don't even want to find a hidden variable

theory to explain all of quantum mechanics, we just want to find a simple theory that

explains this specific experiment, only a single Gedanken experiment to be explained.

We don't worry about whether it's a general description or not.

So when you face such a question you don't try to approve something general at first,

rather you would just try the first simple model that comes to your mind and if that

fails you will try the next one, if that fails again maybe you will start thinking of why

it fails and then in the end you might arrive at something general.

So we start just like Bildit with a very simple model and what could be the simplest model

you have in mind?

Well remember we just have these two particles with a spin and if we measure the spin it

can be either up or down regardless of the direction we measure, it can only have the

two values up or down and so what would be your picture, your classical like picture

of such a situation, well at least I imagine a spin is like a little arrow, a little vector

associated with a particle and probably the singlet state then should be like one arrow

pointing in one direction and the other arrow pointing in exactly the opposite direction.

So let's just draw this situation and let's call this our first attempt.

So we have these two particles that are emitted from the same source and then I would just

pick a spin direction for particle number A and the opposite spin direction for particle

B because we already know the singlet state has this property that if one is up the other

is down.

Now this cannot yet be quite the correct model because we also know that the singlet state

is isotropic, there is no preferred direction, I can measure in the z direction both of the

particles get opposite spins, I can measure the x direction also get opposite spins so

I should also make a model that is isotropic.

Now unfortunately this particular situation of course has a preferred direction, namely

this one which I picked but I can just claim that the source in here will then randomly

pick such a direction each time it emits this pair of particles and it's randomly distributed

uniformly over the sphere so that in the end all directions are equal statistically speaking,

that will be the idea.

So just pick a direction but pick it randomly each time and pick the reverse direction for

the other particle.

So our spin is just represented by a direction which is a unit vector and we would call it

lambda, vector lambda.

So that would be lambda and this here obviously minus lambda.

Okay and lambda should be randomly distributed.

So this is our best approximation that at the moment we can come up with for the singlet

state.

Then there's something else we have to talk about the measurement.