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Explicit Calculations with a Hidden- Variable Spin Model

Barut

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

The basic problem we study here is whether the two-spin correlation experiments together with the theoretical Bell inequalities have already excluded the possibility of introducing hidden variables into quantum theory, as is often concluded. This question is answered negatively by explicitly reproducing the quantum-mechanical two-spin correlation function by a classical model, where the spin is associated with a classical dipolemoment vector. We then study the behavior of single events in classical and quantum models and conclude that the detector efficiency may be a fundamental limitation rather than just a technical problem to be overcome by better techniques. We further show that the assumptions underlying the derivation of Bell inequalities involve statements about single events which are consistent with neither the classical nor quantum models. It is important therefore to work with explicit physical situations rather than with abstract assumptions.

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The basic problem we study here is whether the two-spin correlation experiments together with the theoretical Bell inequalities have already excluded the possibility of introducing hidden variables into quantum theory, as is often concluded. This question is answered negatively by explicitly reproducing the quantum-mechanical two-spin correlation function by a classical model, where the spin is associated with a classical dipolemoment vector. We then study the behavior of single events in classical and quantum models and conclude that the detector efficiency may be a fundamental limitation rather than just a technical problem to be overcome by better techniques. We further show that the assumptions underlying the derivation of Bell inequalities involve statements about single events which are consistent with neither the classical nor quantum models. It is important therefore to work with explicit physical situations rather than with abstract assumptions.
Bell derived the given inequalities on the basis of one rather forceful assumption that was supposed to hold in the hidden variable theory. However, this assumption has been so strong that it has corresponded only to the classical physics; any dependence on spin orientations of individual objects (i.e., on quantum characteristics) having been practically excluded. And consequently, the hidden-variable theory has not been refused by corresponding experimental data. On the other side, Einstein's criticism of Copenhagen quantum mechanics based on ontological approach refusing nonlocality has been entitled as a much higher limit value corresponds to Bell's probability combination in such a case. And there is not any reason to believe in direct interaction between matter objects at greater distances as shown recently also by K. Hess on the basis of another approach.
Abstract The squares of the three components of the spin-s operators sum up to s(s + 1). However, a similar relation is rarely satisfied by the set of possible spin projections onto mutually orthogonal directions. This has fundamental consequences if one tries to construct a hidden variable (HV) theory describing measurements of spin projections. We propose a test of local HV models in which spin magnitudes are conserved. These additional constraints imply that the corresponding inequalities are violated within quantum theory by larger classes of correlations than in the case of standard Bell inequalities. We conclude that in any HV theory pertaining to measurements on a spin one can find situations in which either HV assignments do not represent a physical reality of a spin vector, but rather provide a deterministic algorithm for prediction of the measurement outcomes, or HV assignments represent a physical reality, but the spin cannot be considered as a vector of fixed length.
Bell’s inequality can be traced back to a classical probabilistic formula that is invalid clearly, while spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Bell’s two misunderstandings about Bell’s theorem are pointed out: Firstly, the hidden variable theory Bell’s used, which is a special form of such theories, is regarded as a general one. Secondly, there are two theses obtainable from Bell’s hypotheses; one is compatible with quantum mechanics, the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G.Lochak has revealed the first thesis, and thereby he pointed out that Bell’s inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually without regard to Bell’s inequality.
The experimental violation of the Bell inequality establishes necessary but not sufficient conditions that any theory must obey. Namely, a theory compatible with the experimental observations can satisfy at most two of the three hypotheses at the basis of Bell's theorem: free will, no signaling, and outcome independence. Quantum mechanics satisfies the first two hypotheses but not the latter. Experiments not only violate the Bell inequality, but show an excellent agreement with quantum mechanics. This fact restricts further the class of admissible theories. In this work, the author determines the form of the hidden-variable models that reproduce the quantum mechanical predictions for a spin singlet while satisfying both the hypotheses of free will and no signaling. Two classes of hidden-variable models are given as an example, and a general recipe to build infinitely many possible models is provided.
A local hidden variable model exploiting the detection loophole to reproduce exactly the quantum correlation of the singlet state is presented. The model is shown to be compatible with both the CHSH and the CH Bell inequalities. Moreover, it bears the same rotational symmetry as spins. The reason why the model can reproduce the quantum correlation without violating the Bell theorem is that in the model the efficiency of the detectors depends on the local hidden variable. On average the detector efficiency is limited to 75%.