Malus-Law Models for Aspect-Type Experiments

The inequalities of Bell, Clauser-Horne-Shimony-Holt (CHSH) and others are shown to be inconsistent with the Fundamental (Universal) Model of probability theory when combined with physics laws of the Malus-type. This combination permits the modeling of all results of quantum theory related to CHSH-Aspect-type experiments, while respecting Einstein’s separation principle


Introduction
Einstein-Podolsky-Rosen (EPR) [1] have suggested a Gedanken-experiment for correlations of spatially separated measurements, proposing elements of physical reality as the cause of distant quantum-correlations. They intended to provide a firm logical framework for their discussions with Bohr about the nature of physical reality and the "entanglement" of quantum entities. Their framework of thought was, at least in principle, confirmed by Kocher and Commins [2] in experiments that involved especially prepared (entangled) photon pairs.
Bell [3] and Clauser-Horn-Shimony-Holt (CHSH) [4] presented mathematical models and inequalities related to these discussions which together with subsequent experimental results of Aspect and others [5] appeared to rule out the existence of Einstein's elements of physical reality (also referred to as Einstein's elements or just elements). The new twist of CHSH, Aspect and others beyond the Kocher-Commins experiments was the switching of the polarizers involved in the measurements between four different polarizer-angle pairs. The measurement results involving these specific polarizer-angle pairs, together with the Bell-CHSH theories, seemed to favor influences at a distance instead of Einstein's elements.
The 2022 Nobel Prize in physics has led to waves of articles in the popular press that approve of and support instantaneous influences at a distance, based on the original theories of Bell-CHSH [3] [4] and confirmations of these theories by well-known scientists and mathematicians including Leggett [6], Mermin [7] and Gill [8]. These latter works appear, at first sight, entirely incontrovertible, understandable even for the non-expert and have never been refuted in a self-contained and condensed way, although numerous elaborate counterarguments have been published [9] [10] [11] [12].
It is the purpose of this paper to present a concise and self-contained refutation of Bell-CHSH-type inequalities, based on the following facts: 1) None of the well-known proofs of Bell-CHSH including [3]- [7] have modeled Einstein's elements by the Fundamental Model [13] of probability theory (choosing a real number between 0 and 1 at random and uniformly), which is recognized as being universal and emphasizes the possibility of all different elements. All Bell-CHSH proofs emphasize instead the repeated appearance of identical elements. Leggett [6], Peres [14] and many others have used counterfactual reasoning, Bell has used countable numbers of elements (Bertelmann's socks), while Mermin has used even small numbers of elements; all designed to model the repeated appearance of identical elements and all less general than the Fundamental Model.
2) The correlation of CHSH-Aspect-type pair-measurements may only be understood through a consistent evaluation of distant pair-events. One measurement must establish how Einstein's elements are being evaluated and the distant measurement needs to recognize the elements also and evaluate them with global consistency. Therefore, only the outcomes of such pairs relative to each other may indicate a physical law (such as the Malus law), not the single outcomes by themselves. Bell-CHSH have taken great efforts to guide the theory toward the single outcomes, because of Einstein's separation principle. However, their inequalities deal exclusively with the relative judgement of equal versus not-equal of the distant pair-events.
I present below a model based mathematically on the Fundamental Model of probability theory and physically on analogs to the Malus law. My model violates Bell-CHSH-type inequalities, agrees with quantum theory and does not involve instantaneous influences at a distance.
Aspect-type experiments [5] feature the fast switching of the polarizer angles j and j'. For simplicity, we consider only angles in the plane that is perpendicular to the photon propagation. The angle j is randomly switched in station S A between a and a', while j' in station S B is randomly switched between b and b'. According to Einstein, this switching separates the two stations S A and S B and guarantees the stochastic independence of the emanated photon-pairs from the chosen polarizer angle-pairs, because it ensures that no signal can be of influence as long as it propagates slower than or equal to the speed of light c in vacuum.
The two detectors are mounted after the polarizers in each station. Click of detector 1 in station S A is registered as Mermin [7], Gill [8] and many others have taken this path, except that they did not use the Fundamental and Universal Model of probability theory and they did not consider any physical law in addition to Einstein's separation principle.
In order to stay as close as possible to the Bell-CHSH notation and to put this notation into a one-to-one correspondence with the Aspect-type experimental data, I use the model-notation in λ for the photon pairs that are measured at time in t (which also may represent a pair of times ( )

Inequalities of the Bell-CHSH-Type
Instead of theoretical expectation values that Bell originally considered, we con- where j and j' represent a given value of the polarizer angles: a or a' in station S A and b or b' in station S B , respectively. Consider further the absolute value for the following combination of the four values of μ (corresponding to the four sets Bell-CHSH and supporters have claimed that their EPR-model is exclusively based on Einstein's separation principle and assumptions that appear selfevident in Einstein's world of physical reality. The procedure of Bell-CHSH and supporters (see particularly [7]) to derive a constraint for Q , is equivalent to neglecting the index i in the sums corresponding to Equations (1) and (2). This neglect is incommensurate with the Fundamental Model and implies that in n λ λ = and that the sets i L are, therefore, identical and independent of i, which results immediately in the Bell-CHSH-type inequality: As is well known, this inequality is violated by Aspect-type experiments that use the CHSH angles i θ .
Leggett [6], Gill [8] and many others have arrived at the same result without using any λ or in λ at all in their equations. Gill [8]

Paradoxical Consequences of Equation (3)
The disregard of the index i and the avoidance of any explicit use of the symbols in λ , has far-reaching and paradoxical consequences. Consider four different Aspect-type experiments performed at four different places, Paris, Vienna, Urbana and the Canaries, respectively, and assume similar photon-pair sources.
We use the N model-data for polarizer angles ( )

Cause and Resolution of the Paradox
This paradox raises the suspicion that more than self-evident assumptions have been used in addition to Einstein's separation principle in order to derive the inequalities. In the nascent status, the photon pairs and the in λ must be independent of the polarizer settings (because of the fast switching). However, only interaction with a given polarizer angle provides meaning of what may be regarded e. g. as horizontal or vertical polarization of photons. Careful distinction of Einstein's elements before passing the polarizers and when actually detected, is definitely necessary and leads to possible resolutions of the paradox [10] [11].
Here, I present are solution based on the differences between data averages and theoretical expectation values with respect to the cardinality of the set of Einstein's elements versus the cardinality of the set of measurements (actual or model). All well-known proofs of Equation (3) use assumptions with respect to these cardinalities that are not self-evident at all as, for example, Mermin's addition of a "well known sampling theorem". Mermin [7] derives the virtual identity of the sets i L using Einstein's separa- larger than that of any countable sets of measurements and, therefore, M N  and the reasoning of Mermin and all others does certainly not apply for the quadruple data averages Q . From a mathematical point of view, one must realize that, in general, one just cannot express probabilities that are defined on the interval [0, 1] by countable elements. From the physics point of view, one must realize that Bohr's ideas of complementarity certainly do not exclude the relation of physical entities to both a continuum and countable characteristics. It is the mathematical subtlety involving the cardinality of the involved sets of Einstein's elements versus the cardinality of measurement numbers that probably was not understood by Bell-CHSH, Mermin and others, although it had been noticed in reference [12].
In all of the Bell-CHSH-type proofs including [3]- [8], there exists a basic identification of the sets i L or equivalent sets. That identity may not be deduced from locality considerations alone and is a non-sequitur for the averages of the model-data if the cardinality of Einstein's elements is that of a continuum.

Aspect-Type Experiments and the Fundamental Model of Probability Theory
The Fundamental Model [13]  As an aside, the use of the Fundamental Model also invalidates the counterfactual reasoning of Peres [14], Leggett [6] and others, who have argued as fol- We, therefore, may use the randomly and uniformly chosen in λ to search for a more complete model that obtains the experimental averages in agreement with quantum theory.

A Model Obeying Einstein's Separation Principle and Violating Bell-CHSH
It is important to realize, before starting with a more detailed modeling-approach, Theoreticians developing a model must further be able to use a coordinate system and the corresponding macroscopic equipment configurations in space and time; physical models have not yet exorcised spook in any other way. Theoreticians must also agree on a globally consistent meaning of the measurement outcomes. For example, the polarizer angle a together with a click of a designated detector in both experimental wings, means that the measurements indicate a global value of (for example) "horizontal" or "right-circular" or just otherwise. Such a possibility contradicts the Bell-CHSH inequality and has already previously been considered (see e. g. [11] and [15]). Now, however, the inequality is invalidated to start with by the use of the Using all above model-conventions and elementary trigonometry, we obtain: that once we switch the polarizer to an angle a' different from the angle a in station S A , we must not use the same detector-outcomes for the definition of horizontal (+1) or vertical (−1) in station S B , without risking logical contradictions regarding the global physical or geometric characterization of the photon pairs. In order to avoid this problem, we rotate the coordinate system around the axis of photon propagation such that a' turns into a and postulate that such a rotation permits the evaluation of the in λ exactly as before, but now using the rotated polarizer angles. With this new coordinate system, we may use the same model that we have developed for D 1 , D 2 now for D 3 , D 4 . Aspect's polarizer switching has no effect at all within the so described model. Indeed, Aspect and many others have emphasized the very fact that the switching has no influence on the data averages. The author is, of course aware that the above model is reverse engineered and does not necessarily identify any actual physical "machinery". However, the model certainly does relieve us from the necessity of instantaneous influences and encourages the search for the machinery related to Einstein's elements.

"Freedom" and the Sets L i
Gerard 't Hooft's suggestion [16] that "freedom" and "free will" do not apply to models of CHSH-Aspect-type experiments is, thus, mathematically confirmed by the use of the Fundamental Model of probability theory [13] and without any "conspiracies". We only need to permit that the cardinality of Einstein's elements be that of a continuum. Counterfactual reasoning, on the other hand [6], [14], is refuted, because we have no freedom to demand identical elements for different experiments; model or actual. Along the same line one finds that the "Bell-Game" [8] cannot be played by Alice (in station S A ) and Bob (in S B ), who are asked to predict the possible outcomes for the other station, given only one value in λ for four polarizer angles.
It is not possible to obtain 4 consistent outcomes that obey Malus-type laws for one given in λ . Therefore, the Bell-Game cannot be played, independent of any considerations of locality and of what Alice and Bob may or may not know.

Conclusion
I have proven the existence of violations of Bell-CHSH inequalities and the possibility to derive experimental averages close to quantum theory by a model that respects Einstein's separation principle, applies Malus-type laws and uses the Fundamental Experiment of probability theory.