#1.2. SEPRB delivers EPRB-correlated particles. Detectors D are polarizer-analyzers. The principalaxis of Alice’s dichotomic linear-polarizer is oriented a in 3-space, Bob’s b. The respective analyzeroutputs are A± and B±. Q∗ denotes experiments here; eg, EPRB [Q 1 2 ], Aspect (2002) [Q1]. Based on Bell (1964): it’s a matter of indifference whether λ denotes a single variable or a set, or whether the variables are discrete or continuous. (I associate spin s with λ. In (2), pristine λi and λi are correlated by the conservation of angular momentum.) Given Einstein-locality, local particle/detector interactions yield local results. Ai is thus determined by a and λi alone, Bi by b and λi alone.
#1.2. SEPRB delivers EPRB-correlated particles. Detectors D are polarizer-analyzers. The principalaxis of Alice’s dichotomic linear-polarizer is oriented a in 3-space, Bob’s b. The respective analyzeroutputs are A± and B±. Q∗ denotes experiments here; eg, EPRB [Q 1 2 ], Aspect (2002) [Q1]. Based on Bell (1964): it’s a matter of indifference whether λ denotes a single variable or a set, or whether the variables are discrete or continuous. (I associate spin s with λ. In (2), pristine λi and λi are correlated by the conservation of angular momentum.) Given Einstein-locality, local particle/detector interactions yield local results. Ai is thus determined by a and λi alone, Bi by b and λi alone.
2014.06.03 – the 25th anniversary of my phone-call re Mermin (1988) – 1989.06.03 Abstract: Generalizing Bell 1964:(15) to realizable experiments, CHSH (1969) coined the term “Bell’s theorem”. Since the results of such experiments (eg, see Aspect 2002) contradict Bell’s theorem: at least one step in his supposedly com-monsense analysis must be false. Using undergraduate maths and logic, we find a mathematical error, a false equality, in Bell (1964). Uncorrected, and therefore continuing, this error undermines all of Bell’s EPR-based analysis and many later variants, rendering them false. We can therefore predict with certainty that all loophole-free EPRB-style experiments will also give the lie to Bell’s theorem. “It is a matter of indifference... whether λ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, [Bell writes] as if λ were a single continuous parameter, ” Bell (1964:195). λ may denote “any number of hypothetical additional complementary variables needed to complete quantum mechanics in the way envisaged by EPR, ” Bell (2004:242); from his final essay. #1. Re Bell (1964) (which, like other key essays here, is available online; see References): Let the unnumbered equations between Bell’s (14)-(15) be (14a)-(14c). Let a,b, c replace his ~a,~b,~c. Given Bell’s (2004:242) λ specification above (from his final essay), let A(a, λi) and B(b, λ′i) denote the respective outcomes (±1) when Alice and Bob test the i-th particle-pair. #2. NB: Primes (′) distinguish λs in Bob’s domain from those in others. With i = 1, 2,.., N (and j similarly), there is no requirement here that any two particle-pairs should be the same. #3. Let expectation 〈A(a)B(b) 〉 replace Bell’s equivalent term P (~a,~b); etc. Let P (. | Z) denote a probability conditioned on Z: where Z is shorthand for EPRB, the experiment based
Adjusting EPR (to accord with Bohr's insight), and accepting Bell's principles (but not his false inferences), we:— (i) complete the QM account of EPR correlations in a classical way; (ii) deliver Bell's hope for a simple constructive model of EPRB; (iii) justify EPR's belief that additional variables would bring locality and causality to QM's completion; (iv) refute key claims that such variables are impossible — including CHSH, Mermin's three-particle always-vs-never variant of GHZ, and this: in the context of Bell's theorem ‘it's a proven scientific fact that a violation of local realism has been demonstrated theoretically and experimentally,' (Annals of Physics Editors, 2016). In short: we refute Bell's theorem and endorse Einstein's locally-causal Lorentz-invariant worldview.
In the debate of the foundations of quantum theory (QM), Bell’s theorem [1] is an important milestone. Based on Einstein’s criticism of completeness [2], Bell formulated an expression for the correlation between distant spin measurements such as described by Bohm [3]. In Bell’s expression, hidden variables to restore locality and causality to the theory (LHV’s) are introduced through a probability (mass) density function and through their influence upon the elementary measurement functions in the two separate wings (denoted by the Aand the B-wing) of the experiment. Many experiments and theoretical developments arose from Bell’s original paper. The most important experiment was performed by Aspect [4]. Aspect’s results were interpreted as a confirmation of the completeness of quantum mechanics. From that point, QM was considered a non-local theory. In a previous paper [5] the present author argued that there was insufficient ground for this conclusion. The author would like to point that merely an appeal to the simplicity and apparent logical truth of Bell’s theorem is advanced to support the exclusive non-locality interpretation. Beyond that, there is no proof that experimental results must exclusively be interpreted in this manner. According to the author the inequalities of Bell were given the status of theorem without sufficient supportive evidence save simplicity. In this paper it will be demonstrated that such a conclusive supportive argument of Bell’s theorem does not exist because an approximate classical model is possible. The principles of the proposed model can also be expressed in numerical terms. This enables numerical simulation of the experiments on a computer. Let us shortly describe a typical idealized Bell experiment. In such an experiment, from a single source, two particles with opposite spin are sent into opposite directions. For instance, we could think of a positron and an electron arising from para-Positronium that are drawn apart by dipole radiation. Subsequently, in the respective wings of the experimental set-up, the spin of the individual particle is measured with a SternGerlach magnet. The measurements are found to be correlated with the, unitary, parameter vectors of the magnets, a = (a1, a2, a3),b = (b1, b2, b3). It is well-known that the QM correlation for singlet state electron and positron is equal to,
EPR (1935) famously argue that additional variables will bring locality and causality to QM's completion; we show that they are right. More famously, Bell (1964) cried 'impossible' against such variables; we give the shortest possible refutation of his theorem. With EPR-based variables -- and no QM -- a thought-experiment delivers common-sense locally-causal accounts of EPRB and GHZ in 3-space. We then find the flaw in Bell's theorem: Bell's 1964:(14a) does not equal Bell's 1964:(14b). Thus, at odds with EPR (and us), Bell's unrealistic theorem and its many variants (eg, Mermin, Peres) miss their mark. In short, mixing common-sense with undergrad math and physics in the classical way so favored by Einstein, we interpret QM locally and realistically. Long may EPR rule OK we say.
The Bell theorem [103] can be seen in several different ways, as the EPR argument. Initially, Bell invented it as a direct logical continuation of the EPR theorem: the idea is to take the existence of the EPR elements of reality seriously, and to push it further by introducing them explicitly into the mathematics with the notation λ; one then proceeds to study all possible kinds of correlations that can be obtained from the fluctuations of one or several variables λ, making the condition of locality explicit in the mathematics (locality was already useful in the EPR theorem, but not used in equations). The reasoning develops within determinism (considered as proved by the EPR reasoning) and classical probabilities; it studies in a completely general way all kinds of correlations that can be predicted from the fluctuations of some classical common cause in the past – if one prefers, from some random choice concerning the initial state of the system. This leads to the famous inequalities. But subsequent studies have shown that the scope of the Bell theorem is not limited to determinism; for instance, the λ variables may determine the probabilities of the results of future experiments, instead of the results themselves (see Appendix B), without canceling the theorem. We postpone the discussion of the various possible generalizations to §4.2.3. For the moment, we just emphasize that the essential condition for the validity of the Bell theorem is locality: all kinds of fluctuations can be assumed, but their effect must affect physics only locally.