Contextual Approach to Quantum Formalism

Philippe Grangier

Andrei Yu. Khrennikov

Andrei Yu. Khrennikov

In this chapter we develop the contextual approach to quantum mechanics.This approach matches with the views ofBohr who emphasized that the quantum description represents complexes ofexperimental physical conditions, in the modern terminology – experimentalcontexts. In this chapter we formalize the contextual approach on the basisof contextual probability theory which is closely connected with generalizedprobability theory (but interpretationally not identical with it). The contextualprobability theory serves as the basis of the contextual measurementmodel (CMM). The latter covers measurements in classical, quantum, andquasi-classical physics.

Andrei Yu. Khrennikov

Jaskaran Singh, Rajendra Singh Bhati, Arvind

The two-state vector formalism (TSVF) proposed by Aharonov, Bergmann, and Lebowitz [A. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964)] allows a counterfactual assignment of probabilities of outcomes of contemplated (but unperformed) measurements on quantum systems. The probabilities assigned by the Aharonov-Bergmann-Lebowitz (ABL) rule and the associated weak values have been used to provide insights into quantum situations and to unearth underlying quantum contextuality. We apply the principle of exclusivity on ABL probabilities which are assigned to mutually orthogonal projectors to define paradoxical and nonparadoxical scenarios. Given a pre- and a postselected pair of states, we consider the nonparadoxical sector with a view to explore the demonstration of quantum contextuality. For the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenario, we numerically show that the ABL probabilities of the TSVF in the nonparadoxical sector do not offer any contextual advantage. Our approach can be easily generalized to other contextual scenarios as well. We thus argue that several previous proofs of the emergence of contextuality in pre- and postselected scenarios are only possible if the principle of exclusivity is violated and are therefore classified as paradoxical.

Diederik Aerts, Marek Czachor, Sandro Sozzo

Population ecology is mainly based on nonlinear equations of the Lotka-Volterra type, which provide mathematical models for describing the dynamics of interacting species. However, for many interacting populations, these equations entail complex dynamical behavior and unpredictability, generating such difficulties and problematical situations as illustrated by the 'paradox of the plankton' and the 'paradox of enrichment', for instance. A careful analysis shows that an ecosystem is a fundamentally contextual system, hence any formalism describing such systems should incorporate contextuality from the very beginning. But existing approaches are based on classical physics and probability theory, and introduce contextuality as an external effect, so that they cannot generally explain the main peculiarities of ecosystems. Basing ourselves on a contextual formalism elaborated to study microscopic systems in quantum mechanics and including appropriate nonlinear equations, we construct a generalization of the Lotka-Volterra equations for contextual systems, apply these equations to discuss some paradoxical situations encountered in ecology, and propound alternative solutions to those currently existing in the literature.