Bell's Inequality

[Tommi]

So, it's terribly obvious to me that a, say, 45 degree angle between the two orthogonal ones is going to let you get more light through, the same applying to all polarized waves. Which is why the inequality is violated often, as a non-90 degree angle changes the polarization.

What is less obvious to me is what happens if you have a fourth direction of measurement, say again, at 45 degrees to the two orthogonal ones, but which is also orthogonal to the third, and no measurement is taken on a shared direction?
What I mean is a second pair of orthogonal directions at an arbitrary angle to the "original" pair, say z & x & q & w, and just as it is pointless to measure z twice, it should be pointless to measure q twice, so you measure w instead.
What does this do to the inequality?

Or is this a pointless question and why?

[joigus]

The waves that are used in Bell's gedanken are completely un-polarised. The key to Bell's scenario is that entangled pairs of particles are described with fewer variables than those needed to describe two independent particles, so the particles are sharing some internal physical reality, so to speak. The reason for the angle 45º is a bit technical: It's the angle at which a certain projection of spin for a 2-state spin differs the most from two states that are perpendicular in ordinary 3-space. Keep in mind perpendicularity --orthogonality-- in the internal space is different from perpendicularity in ordinary 3-space.

I don't see anything terribly obvious in any of that. But maybe it's just me.

[Tommi]

Would you believe it's the first time I hear internal space being specified? Because I can't remember hearing about that before.
Still, wouldn't measuring the particle along a specific axis effectively polarize it in ordinary space as well? 

The obvious part I meant in a more general sense, I'm afraid.
I'm thinking about the polarization demonstrations with rope, where a circular wave is converted by a slit and negated by the next which is at 90º to the first.
It seems perfectly intuitive to me that if a non-orthogonal slit is placed between the two, it's going to change the wave that passes through and let more through the next.

What I'm asking is whether or not this interaction with the 45º is something relevant, or that should be compensated for with it's own orthogonal pair.

[joigus]

20 hours ago, Tommi said:

Would you believe it's the first time I hear internal space being specified? Because I can't remember hearing about that before.

Field variables always can be seen to "inhabit" a space of their own. In the case of the electromagnetic field this internal dimension is the angle along a circle. You can picture the EM field as an entity that, for every point in space time, is given by an angle in this internal dimension. That's why Kaluza and Klein were able to model the EM field as a curly dimension (circle) superimposed on every space-time point.

20 hours ago, Tommi said:

Still, wouldn't measuring the particle along a specific axis effectively polarize it in ordinary space as well? 

 

Yes, you're right. But that happens after the measurement has taken place.

[Prof Reza Sanaye]

I don't think this is a pointless question. Under the function of the vector space, a subspace is closed. In this case, it must be a number if you add two vectors to the space. The formal description of a subspace is the following: the zero vector must be used. So if, by definition, you take any vector in the space and add it to be negative, sum is the zero-vector. An object created by perpendicular intersections of lines drawn from points on the object to a projection plane may not generally be represented in two dimensions, as orthography, by the arbitrated reference system or by being in or outside. Original pairs may thus translate further back   OR  further on to intensified bifurcations in arbitrary-axes subspace.

[studiot]

6 hours ago, joigus said:

Field variables always can be seen to "inhabit" a space of their own.

I dunno about always, but this is an incredibly profound yet often overlooked statement.  +1

[joigus]

2 hours ago, studiot said:

I dunno about always, but this is an incredibly profound yet often overlooked statement.  +1

Well, you're right. Maybe not always. In continuum mechanics that may not be the case. In field theory it certainly is.