Regarding Bell -

**we know that nature violates his inequalities, so we need to find an erroneous assumption in his way of thinking.**Let's look at a simple proof from

http://www.johnboccio.com/research/quantum/notes/paper.pdfSo let us assume that there are 3 binary hidden variables describing our system: A, B, C.

We can assume that the total probability of being in one of these 8 possibilities is 1:

Pr(000)+Pr(001)+Pr(010)+Pr(011)+Pr(100)+Pr(101)+Pr(110)+Pr(111)=1

Denote by Pe as probability that given two variables have equal values:

Pe(A,B) = Pr(000) + Pr (001) + Pr(110) + Pr(111)

Pe(A,C) = Pr(000) + Pr(010) + Pr(101) + Pr(111)

Pe(B,C) = Pr(000) + Pr(100) + Pr(011) + Pr(111)

summing these 3 we get Bell inequalities:

Pe(A,B) + Pe(A,C) + Pe(B,C) = 1 + 2Pr(000) + 2 Pr(111) >= 1

Now denote ABC as outcomes of measurement in 3 directions (differing by 120 deg) - taking two identical (entangled) particles and asking about frequencies of their ABC outcomes, we can get

**Pe(A,B) + Pe(A,C) + Pe(B,C) < 1 what agrees with experiment ... so something is wrong with the above line of thinking ...****The problem is that we cannot think of particles as having fixed ABC binary values describing direction of spin.**We can ask about these values independently by using measurements - which are extremely complex phenomena like Stern-Gerlach.

Such measurement doesn't just return a fixed internal variable.

Instead, in every measurement this variable is chosen at random - and this process changes the state of the system.

Here is a schematic picture of the Bell's misconception:

The squares leading to violation of Bell inequalities come e.g. from completely classical Malus law: the polarizer reduces electric field like cos(theta), light intensity is E^2: cos^2(theta).

http://www.physics.utoronto.ca/~phy225h/experiments/polarization-of-light/polar.pdfTo summarize, as I have sketched a proof, the following statement is true:

**(*): "Assuming the system have some 3 fixed binary descriptors (ABC), then frequencies of their occurrences fulfill**

Pe(A,B) + Pe(A,C) + Pe(B,C) >= 1

(Bell) inequality"**Bell's misconception was applying it to situation with spins: assuming that the internal state uniquely defines a few applied binary values.**In contrast, this is a probabilistic translation (measurement) and it changes the system.

Beside

**probabilistic** nature, while asking about all 3, their values would

**depend on the order** of questioning - ABC are definitely not fixed in the initial system, what is required to apply (*).