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arXiv:2304.05696 · Foundations of Physics

On the Representations of Bell’s Operators in Quantum Mechanics

Silvio Paolo sorella

Published May 27, 2023• 3 citations
arXivHTMLPDF
DOI
10.1007/s10701-023-00699-6

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In this thesis, entanglement under fully relativistic settings are discussed. The thesis starts with a brief review of the relativistic quantum mechanics. In order to describe the effects of Lorentz transformations on the entangled states, quantum mechanics and special relativity are merged by construction of the unitary irreducible representations of Poincaré group on the infinite dimensional Hilbert space of state vectors. In this framework, the issue of finding the unitary irreducible representations of Poincaré group is reduced to that of the little group. Wigner rotation for the massive particles plays a crucial role due to its effect on the spin polarization directions. Furthermore, the physical requirements for constructing the correct relativistic spin operator is also studied. Then, the entanglement and Bell type inequalities are reviewed. The special attention has been devoted to two historical papers, by EPR in 1935 and by J.S. Bell in 1964. The main part of the thesis is based on the Lorentz transformation of the Bell states and the Bell inequalities on these transformed states. It is shown that entanglement is a Lorentz invariant quantity. That is, no inertial observer can see the entangled state as a separable one. However, it was shown that the Bell inequality may be satisfied for the Wigner angle dependent transformed entangled states. Since the Wigner rotation changes the spin polarization direction with the increased velocity, initial dichotomous operators can satisfy the Bell inequality for those states. By choosing the dichotomous operators taking into consideration the Wigner angle, it is always possible to show that Bell type inequalities can be violated for the transformed entangled states.
A parameterized operational representation of an EPRB-like experiment is presented that facilitates direct comparisons among mathematical models that characterize the concepts of local hidden variables, the factorization condition, remote context independence, remote outcome independence, local causality, determinism, predictability, and quantum mechanics. The different models and conditions are derived from a common template, differentiated by parameter adjustment only. This perspective enables new and easier proofs of some existing results. It also clarifies the relationships among the various notions of locality and their relationship to the Bell-CHSH inequalities. Included are tables that make proofs short and intuitive, and figures that dramatize the differences between sets of correlations of quantum mechanics (QM) and those of realistic, local, deterministic, or predictable models. In addition, a novel proof of the no-communication theorem for QM provides a new perspective on why it is true.