We revisit the computation of correlations of spin projections onto unit vectors for spin-1/2 particles in Quantum Mechanics. We then choose one of the Boole inequalities that, as we recall, must be obeyed by collections of sequences of normalized spin-1/2 projections onto unit vectors that belong to, say, the plane that is orthogonal to the classical trajectory or to the three dimensional space. Next we define the concept of being a-p for i-indexed sequences of spin-1/2 particles: it turns out that a sequence of spin-1/2 particles is a-p if and only if the particles behave as if they had indeed been prepared (in the usual sense) to have definite spins +1 or -1 along the axis a, the sequence of signs being pre-determined. The a-p property is formulated in a way abstract enough to let one generalize it and we define the concept of being a-p and b-p. However, using our chosen Boole inequality we prove that no particle sequence can be both a-p and b-p when a and b are not co-linear. We next consider two modifications of Quantum Mechanics: - The first one by assuming both Weak Realism, the weakest form of Realism needed to develop any main theorem in Bell's Theory, and Locality. - The second one by assuming Weak Realism and an hypothesis that turns out to be none else than (relativistic) Causality extended to cover as well the observable values that make sense only if one assumes Weak Realism. In both cases we arrive at the conclusion that in the EPRB setting where successive pairs of spin-1/2 particles are prepared in the so-called singlet state, one of these particles sequences is both a-p and b-p and a and b are the opposite of the vectors respectively used to project the spin of the other particle and to get the Realism-based alternate value for a projection. Thus the usual discussions of Bell's inequalities under such hypotheses do not even make physical sense.
We revisit the computation of correlations of spin projections onto unit vectors for spin-1/2 particles in Quantum Mechanics. We then choose one of the Boole inequalities that, as we recall, must be obeyed by collections of sequences of normalized spin-1/2 projections onto unit vectors that belong to, say, the plane that is orthogonal to the classical trajectory or to the three dimensional space. Next we define the concept of being a-p for i-indexed sequences of spin-1/2 particles: it turns out that a sequence of spin-1/2 particles is a-p if and only if the particles behave as if they had indeed been prepared (in the usual sense) to have definite spins +1 or -1 along the axis a, the sequence of signs being pre-determined. The a-p property is formulated in a way abstract enough to let one generalize it and we define the concept of being a-p and b-p. However, using our chosen Boole inequality we prove that no particle sequence can be both a-p and b-p when a and b are not co-linear. We next consider two modifications of Quantum Mechanics: - The first one by assuming both Weak Realism, the weakest form of Realism needed to develop any main theorem in Bell's Theory, and Locality. - The second one by assuming Weak Realism and an hypothesis that turns out to be none else than (relativistic) Causality extended to cover as well the observable values that make sense only if one assumes Weak Realism. In both cases we arrive at the conclusion that in the EPRB setting where successive pairs of spin-1/2 particles are prepared in the so-called singlet state, one of these particles sequences is both a-p and b-p and a and b are the opposite of the vectors respectively used to project the spin of the other particle and to get the Realism-based alternate value for a projection. Thus the usual discussions of Bell's inequalities under such hypotheses do not even make physical sense.
We review the computation of correlations of successive projections of the spin onto axes for spin-1/2 particles and EPRB pairs (in the Singlet state). We assume forms of Realism (at least as general as the Predictive Hidden Variables in the classical Bell's Theory), to review one particle correlations before we further assume Locality and analyze EPRB pairs. To our surprise, we find that the Abstract Bell Inequalities with three or four axes cannot have physical meaning i.e., two spin projections cannot belong to one particle or some correlations that one would need cannot be evaluated. A recent version of Bell's Theory without Locality turns out to be unaffected by our results that refocus the open issues on "realism" rather than "local realism", inviting us to leave locality alone so to speak.
Given a sequence of pairs (pi, pi) of spin-1/2 particles in the singlet state, assume that Alice measures the normalized projections ai of the spins of the pi's along vector $\mathbit{a}$ while Bob measures the normalized projections bi of the spins of the pi's along vector $\mathbit{b}$. Then quantum mechanics (QM) lets one evaluate the correlation ⟨a,b⟩ as $\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{\mathbit{a}}\ensuremath{-}{\ensuremath{\theta}}_{\mathbit{b}})$ where ${\ensuremath{\theta}}_{\mathbit{v}}$ is the angle between the vector $\mathbit{v}$ and a reference vector chosen once and for all and in a fixed plane. Assuming classical microscopic realism (CMR) there exist also normalized projection pairs (${a}_{i}^{\ensuremath{'}}$, ${b}_{i}^{\ensuremath{'}}$) of the spins of the pairs (pi, pi) along (${\mathbit{a}}^{\ensuremath{'}}$, ${\mathbit{b}}^{\ensuremath{'}}$) so that $⟨{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}⟩=\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{{\mathbit{a}}^{\ensuremath{'}}}\ensuremath{-}{\ensuremath{\theta}}_{{\mathbit{b}}^{\ensuremath{'}}})$. Since all projections are in ${\ensuremath{-}1,1}$, $|⟨c,d⟩+⟨c,e⟩|+|⟨f,d⟩\ensuremath{-}⟨f,e⟩|\ensuremath{\le}2$ for c, d, e, and f in ${a,b,{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}}$. Assuming locality (the impossibility of any effect of an event on another event when said events are spatially separated) beside QM and CMR, Bell's theory lets one deduce various violations of this inequality at some choices of quadruplets $Q\ensuremath{\equiv}(\mathbit{a},\mathbit{b},{\mathbit{a}}^{\ensuremath{'}},{\mathbit{b}}^{\ensuremath{'}})$. Our main result is the existence of such Q's where at least one of the above inequalities is violated if one only assumes QM, CMR, and some very mild further hypotheses that only concern the behavior of correlations that appear in these inequalities near special Q's.
While it is widely agreed that Bell's theorem is an important result in the foundations of quantum physics, there is much disagreement about what exactly Bell's theorem shows. It is agreed that Bell derived a contradiction with experimental facts from some list of assumptions, thus showing that at least one of the assumptions must be wrong; but there is disagreement about what the assumptions were that went into the argument. In this paper, I make a few points in order to help clarify the situation. Introduction Different authors have expressed very different views about what Bell's theorem shows about physics. The disagreement concerns particularly the question of which assumptions go into the argument. Since we have to give up one of the assumptions leading to the empirically violated Bell inequality, knowing what the assumptions were is crucial. For example, if author X believes that the argument requires only one assumption, A 1 , while author Y believes that it requires two, A 1 and A 2 , then X will conclude that A 1 ¬ must be abandoned, while Y will conclude that either A 1 or A 2 must abandoned, so A 1 may well be true in our world if A 2 is false. In this paper, I consider several assumptions that have been mentioned in connection with Bell's theorem, and I look into their roles in the proof of Bell's theorem. While I do not say anything here that has never been said before, I hope that my remarks can nevertheless be helpful to some readers. For the sake of definiteness, the version of the relevant experiment that I will consider involves two spin-1\2 particles, initially in the singlet spin state ψ = 2 −1/2 (|↑↓ ⟩ − |↓↑ ⟩). Two widely separated experimenters, Alice and Bob, have one particle transported to each of them. At spacelike separation, they each choose a direction in space, represented by the unit vectors a and b , and carry out a quantum measurement of the spin observable in this direction on their particle, ( a · σ ) ⊗ I and I ⊗ ( b · σ ), with I the 2 × 2 unit matrix and σ the triple of Paulimatrices. The possible outcomes, A and B , are±1 on each side. Bell's theorem is the statement that under the assumptions we are about to discuss, Bell's inequality holds; quantum theory predicts violations of Bell's inequality, and experiments have confirmed these predictions.
Why ψ is incomplete indeed: a simple illustration
With the Nobel Prize attributed to Aspect, Clauser, and Zeilinger, the international scientific community acknowledged the fundamental importance of the experimental violation of Bell's inequalities. It is however still debated what fails in Bell's hypotheses, leading to these inequalities, and usually summarized as "local realism", or maybe more appropriately "classical local realism". The most common explanation is "quantum non-locality", that remains however fully compatible with relativistic causality; this makes wondering whether any non-local phenomenon is really involved in these experiments. Here we want to recapitulate another option, sometimes called "predictive incompleteness", closely related to the idea that the usual state vector ψ is incomplete indeed, as it was claimed by Einstein, Podolsky and Rosen. However, the right way to complete ψ has nothing to do with hidden variables, but requires to specify the measurement context, as it was claimed by Bohr. Here we will consider the simple case of two spin 1/2, or two qubits, in order to keep the argument simple, but it does apply generally in quantum mechanics.
A new pure quantum state, isotropic in spin variables, is defined in an extended spin phase space beyond standard quantum mechanics. It allows to represent the entangled singlet state in separable form. The statistical correlations between Alice and Bob measurements become self correlations between hidden spin values for each particle, together with perfect anti correlation between spin values on the pair. Alice determines through measurement on her particle the value of spin in some direction. Spin in another direction is inferred from Bob measurement on the companion. Bell's inequalities are violated because of the wave like behaviour of quantum systems. In full analogy with the two slit experiment, interference terms between spin field components appear determining the contextual character of quantum distributions of probability.
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,