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Abstract
In this paper we present a local realistic model for the polarisation of photons that violates the inequalities, suppost so far as a proof to reject local realism. The model is proposed as a genotype – phenotype interpretation of objects and assumes unconscious conspiracy. The genotype refers to local hidden variables of the objects; the phenotype to the methods applied to them to give outcomes of their behaviour during interaction with its environment. It is as well shown to violate Bell’s inequality as CHSH inequality.
By showing the existence of such a model, we prove that local realistic models are possible by the grace of unconscious conspiracy.
 
The cause
The basis idea to come to a model with local hidden variabels was induced by the observation that Malus’ law describes the passing of linear polarized photons through two filters, i.e. I = ½ I₀ cos²b, where I is the intensity of light after the two filters having an angle b rotated polarization direction between them, and I₀ is the original intensity of ligth before passing the two filters. The factor ½ is due to the fact that only fifty percent of all photons will pass the first filter. Similar, for the “intensity” of coincidenses for two entangled photons is given by I = I₀ cos²b.
 
Genotype – phenotype model of polarized photons
Let v₀ and v₁ be two unit vectors, perpendicular to the propagation direction of the photon, with an angle a between v₀ and v₁, while the vector addition v₀ + v₁ is uniform distributed. Among the population of photons, the angle a between v₀ and v₁ has probability P(a) = ½ sin(a), and the two unit vectors are hidden variables, i.e. local hidden variables [1], which we will call the genes for polarization of the photon. An interaction with a polarization filter device with direction v is obtained from the following process description:
 
     func polfilpass(v: unit vector): boolean;
     hidden var: v₀, v₁: unit vector;
     begin
           polfilpass:= (sign(v.v₀) = sign(v.v₁))
     end
 
or in communicating guarded command language [2]
 
     polfilpassin?v ® polfilpassout!(sign(v.v₀) = sign(v.v₁))
 
The distributon P(a) = ½ sin(a) is choosen to satisfy Malus’ law for the passing of two polarization filters, while for one polarization filter the passing probability is ½ due to the uniform distribution of the direction of v₀ + v₁.  
 
Proof:
Let f be the filter function defined by
 
               f(a,b) = 1 Û (sign(v.v₀) = sign(v.v₁))
               f(a,b) = 0 Û (sign(v.v₀) ¹ sign(v.v₁))
 
where a is the angle between the two genes for polarization, and b is the angle between v of the polarisation filter in question and (v₀ + v₁). Malus’ law is obtained from
 
               (1/4p)(òa: 0 £ a < p: (òb: 0 £ b < 2p: sin(a) f(a,b-b_A) f(a,b-b_B)))
= ½ cos²(b_B – b_A),
 
where b is the angle of v₀ + v₁, and b_A and b_B the angles of the filters.
We do disregard the undefined value of the filter function f if v is perpendicular to either v₀ or v₁, due to the fact that left and rigth limits for the integrals have the same result.
 
Passing a filter X (= A,B) is simply obtained from f(a,b-b_X) = 1, which holds for the angles satisfying
 
((-p/2 < b - b_X - ½a < p/2) Ù (-p/2 < b - b_X + ½a < p/2)) Ú
((p/2 < b - b_X - ½a < 3p/2) Ù (p/2 < b - b_X + ½a < 3p/2))
 
From entangled photons, the model predicts the experimental results violating Bell’s inequallity [3] by the assumption that by generating entangled photons, both photons inherit the same hidden variables. So the coincidence on an Alice and a Bob filter passing is similar to Malus’ law and is given by
 
               P(A Ù B) =
               (1/4p) (òa: 0 £ a < p: (òb: 0 £ b < 2p: sin(a) f(a,b-b_A) f(a,b-b_B))))
 = ½ cos²(b_B – b_A),

               P(ØA Ù ØB) =
               (1/4p) (òa: 0 £ a < p: (òb: 0 £ b < 2p: sin(a) (1-f(a,b-b_A))(1- f(a,b-b_B)))))  
= ½ cos²(b_B – b_A),
 
and the coincidences of such an experiment are C₌(b_A, b_B) = cos²(b_B – b_A).
The contracoincidence on an Alice and a Bob filter passing are obtained from
 
P(ØA Ù B) =
               (1/4p) (òa: 0 £ a < p: (òb: 0 £ b < 2p: sin(a) (1-f(a,b-b_A)) f(a,b-b_B)))
 = ½ sin²(b_B – b_A),
 
P(A Ù ØB) =
               (1/4p) (òa: 0 £ a < p: (òb: 0 £ b < 2p:sin(a) f(a,b-b_A)(1- f(a,b-b_B))))
= ½ sin²(b_B – b_A),
 
so C_¹(b_A, b_B) = sin²(b_B – b_A).
 
Doing such experiments for filter settings b’_A , b’_B and b”_A , b”_B, we obtain the coincidences C₌(b’_A, b’_B) = cos²(b’_B – b’_A) and  C₌(b”_A, b”_B) = cos²(b”_B – b”_A)  and contracoincidences C_¹(b’_A, b’_B) = sin²(b’_B – b’_A) and  C_¹ (b”_A, b”_B) = sin²(b”_B – b”_A). From these we obtain Bell’s inequality:
 
               C₌(b’_A, b’_B) £ C_¹(b’_A, b’_B) + C₌(b”_A, b”_B)
               Û
               cos²(b’_B – b’_A) £ sin²(b’_B – b’_A) + cos²(b”_B – b”_A)
 
For b’_B – b’_A = p/6 and b”_B – b”_A = p/3 we obtain the violation of Bell’s inequality ¾ £ ½ + ½. 
 
The CHSH inequality [4] is given by,
 
               -2 £ C₌(b’_A, b’_B) – C_¹(b’_A, b’_B)  – C₌(b’_A, b”_B) + C_¹(b’_A, b”_B) +
C₌(b”_A, b’_B) – C_¹(b”_A, b’_B)+ C₌(b”_A, b”_B) – C_¹(b”_A, b”_B) £ 2
               Û
               -2 £ cos²(b’_B – b’_A) – sin²(b’_B – b’_A) – cos²(b”_B – b’_A) + sin²(b”_B – b’_A) +
cos²(b’_B – b”_A) – sin²(b’_B – b”_A) + cos²(b”_B – b”_A) – sin²(b”_B – b”_A) £ 2
 
For b’_A = 0, b”_A = p/4, b’_B = p/8 and b”_B = 3p/8 we obtain the correlations
 
for b’_A = 0 and b’_B = p/8: cos²(p/8) – sin²(p/8) = 1/Ö2
for b’_A = 0 and b”_B = 3p/8: cos²(3p/8) – sin²(3p/8) = -1/Ö2
for b”_A = p/4 and b’_B = p/8: cos²(-p/8) – sin²(-p/8) = 1/Ö2
for b”_A = p/4 and b”_B = 3p/8: cos²(p/8) – sin²(p/8) = 1/Ö2
 
and CHSH is violated from 4/Ö2 £ 2.
 
Previous attempts for finding local hidden variables did not consider the possibility of regarding a particle from a computing science view, i.e. an object oriented view, with hidden variables for an object class, and the behaviour for instance for interaction with a filter as a process description by applying a method on those variables.
 
The behaviour is described by methods on the hidden variables. A down-conversion of a photon with energy E is a method, creating two photons of energy E/2, with the hidden variables inherited from the original photon, together with a destroy of the original photon. The behaviour for interaction with a polarization filter is described above by the boolean valued function polfilpass.
 
Remarks on conspiracy
There are two types of conspiracy, which are indistinguishable from a mathematical point of view. So, let me explain it from a human point of view. Mostly, if we talk about conspiracy, we mean a consciously making of appointments on how to behave in certain circomstances. It is obvious that we have to reject such a type of conspiracy to explain the behaviour of particles like photons [1]. However, there is an other kind of conspiracy, which is unconsciously. This is our genetic inheritance. Having a twin, they act almost the same in similar social circumstances, due to there common genetic inheritance. This hidden or unconscious conspiracy is exactly what is proposed above. If a twin has inherited a sensitivity for schizophrenia, both will have coincidental consequences if there social environments are similar, but they can be contracoincidental if the social environments are different. As such, schizophrenia consequences can violate Bell’s inequality or CHSH inequality as well. It is from this type of conspiracy to denote the proposed model as a genotype – phenotype model of local hidden variables.
On the other hand, the Stockholmer interpretation of quantum mechanics must be seen as consciously conspiracy, with a communication channel given by the common wavefunction of the entangled particles.
 
Final remarks from a metaphysical point of view
In the view of genes for polarisation, the genotype of the photon was created during the big bang, or after that from any “social” interaction, supposing that all interactions in the universe from that moment preserve the distributions of all types of genes. As such, it is a predestinated view, where only the ”creator” knows the hidden variables of each object. And from this point of view “God does not play dies”.
From the Stockholmer interpretation of quantum objects the drawing is from the wave function at the moment of interaction. As such, it is a probabilistic behaviour at the moment of interaction where the entanglement of the wavefunctions is needed to predict Bell-like experiments.
The genotype - phenotype view is consistent with relativity theory due to the fact that no communication channel is needed, which implies that a predestinated view of the universe is preferable.
 
References:
[1] A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777-780 (1935).
[2] C.A.R. Hoare, Communicating Sequential Processes, CACM 21 666-677 (1978).
[3] J.S. Bell, On the Einstein–Poldolsky–Rosen paradox, Physics 1 195-200 (1964).
[4] J. Clauser, M. Horne, A. Shimony and R. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Physical Review Letters 23 880-884 (1969).

Ik neem aan dat dit kopie-pasta is van ergens anders. Zo is het echter niet te lezen. Kun je een linkje geven naar het originele document?

Eigen tekst denk ik, afgesplitst van een topic in natuurkunde, zie o.a. deze post.

Inderdaad, eigen tekst, ingebracht hier naar aanleiding van draad in de natuurkunde. Formules zijn goed leesbaar op mijn computer, maar bij mij ook niet op de mobiel.