Generalized Probabilities in Statistical Theories

We discuss different formal frameworks for the description of generalized probabilities in statistical theories. We analyze the particular cases of probabilities appearing in classical and quantum mechanics and the approach to generalized probabilities based on convex sets. We argue for considering quantum probabilities as the natural probabilistic assignments for rational agents dealing with contextual probabilistic models. In this way, the formal structure of quantum probabilities as a non-Boolean probabilistic calculus is endowed with a natural interpretation.

The undecidable dynamics generate quantum probabilities

We conjecture a new approach to quantum mechanics that, if confirmed, will explain the wave function from a fundamentally deeper level.

On the Connection Between Quantum Probability and Geometry

We discuss the mathematical structures that underlie quantum probabilities. More specifically, we explore possible connections between logic, geometry and probability theory. We propose an interpretation that generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories. We stress the relevance of developing a geometrical interpretation of quantum mechanics.Quanta 2021; 10: 1–14.

Quantum Cognition and the Mind

Clues from psychology indicate that human cognition are not only based on classical probability theory as explained by Kolmogorov’s axioms but additionally on quantum probability. Quantum probabilities lead to the conclusion that our brain adapted to the Everett many-worlds reality trough the evolutionary process. The Everett many-worlds theory views reality as a many-branched tree in which every possible quantum outcome is realized. In this context, one of the cognitive brain functions is to provide a causally consistent explanation of events to maintain self-identity over time. Causality is related to a meaningful explanation. For impossible explanations, causality does not exist, and the identity of the self breaks. Only in meaningful causal worlds may personal identities exist.

From Quantum Probabilities to Quantum Amplitudes

The task of reconstructing the system’s state from the measurements results, known as the Pauli problem, usually requires repetition of two successive steps. Preparation in an initial state to be determined is followed by an accurate measurement of one of the several chosen operators in order to provide the necessary “Pauli data”. We consider a similar yet more general problem of recovering Feynman’s transition (path) amplitudes from the results of at least three consecutive measurements. The three-step histories of a pre- and post-selected quantum system are subjected to a type of interference not available to their two-step counterparts. We show that this interference can be exploited, and if the intermediate measurement is “fuzzy”, the path amplitudes can be successfully recovered. The simplest case of a two-level system is analysed in detail. The “weak measurement” limit and the usefulness of the path amplitudes are also discussed.

Locality and Non locality

Aim: Whether, under all the circumstances considered, a relativistic concept of locality and non-locality may fully reproduce the quantum probabilities for outcomes of experiments, is re-investigated.Methods: The usual methods and rules of statistics, probability theory and quantum mechanics were used. Results: The interior logic of the variance has been re-investigated. A relationship between the Pythagorean theorem and the variance has been established. A n-dimensional Pythagorean theorem has been derived. The problem of locality and non-locality and the relationship to the variance has been analysed.Conclusion: It may no longer stay an open question how to deal with the notions of locality and non-locality.

Quantum Probability’s Algebraic Origin

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.