On probability theory and probabilistic physics?Axiomatics and methodology

Andrei Yu. Khrennikov

The situation in probability theory is more or less analogous to the situation in geometry in the 19th century. Many scientists begin to understand that the Kol-mogorov axiomatic approach [1], 1933, to the modern theory of probability cannot describe all probabilistic phenomena observed in nature. Hence, the theory of probability based on this axiomatic approach is not a unique and universal probabilistic formalism. It is only one model, namely, the Kolmogorov probabilistic model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Lazar Mayants

A. Hartkämper, Heinz–Jürgen Schmidt

Raoul Bianchetti

Hilbert’s Sixth Problem asks for an axiomatization of physics, especially the probabilistic foundations of mechanics and the passage from microscopic dynamics to macroscopic laws. Despite foundational progress in probability theory and statistical mechanics, a persistent conceptual gap remains: probability appears as a primitive or as an epistemic tool, rather than as a quantity derived from the structure of physical evolution itself. This paper proposes a Viscous Time Theory (VTT) reformulation of Hilbert 6: physics is axiomatized by admissibility (what transformations can occur while preserving identity), coherence (the measurable structure of informational organization), and recoverability (which collapses are irreversible independent of observer perspective). We present a minimal axiomatic system in which (i) time emerges as informational viscosity, (ii) probability emerges as an induced measure over admissible trajectories, and (iii) the micro→macro transition becomes a geometric scaling limit of coherence fields rather than an ad hoc coarse-graining step. We do not claim a completed proof of Hilbert 6 in the classical sense. Instead, we provide a coherent, falsifiable program that reframes the target: the objective is not “derive probability from mechanics,” but “derive probability from the geometry of admissible informational evolution,” with classical mechanics and statistical mechanics emerging as distinct regime limits.

Andrei Yu. Khrennikov

We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be defined by classical Bayes' formula. We also use the approach based on Bell-type inequalities in the conventional probabilistic approach, Kolmogorov model. We prove an analog of Wigner's inequality for conditional probabilities and by using this inequality show that predictions of the conventional and quantum probability models disagree already in the case of noncomposite systems (even in the two dimensional case).

Kaumudibikash Goswami, Archana Barvar, Manabendra Nath Bera

Quantum states over time were introduced, in part, to extend Bayesian probability to quantum theory and have since found applications across diverse areas, ranging from quantum causality and open-system dynamics to tests of macroscopic realism. While axiomatic characterizations of these states exist for Markovian dynamics, a general framework accommodating non-Markovian dynamics is still lacking. Moreover, extending this framework beyond quantum mechanics to general physical theories also remains elusive. In this work, we develop an axiomatic framework for multi-time states within a class of operational probabilistic theories (OPTs) that includes quantum mechanics as a special case. Based on three assumptions---compositionality, reconstructibility, and spatio-temporal marginal consistency---the framework accommodates both Markovian and non-Markovian dynamics, and provides sufficient conditions to certify signaling and temporal memory within OPT. As a concrete application of our OPT framework, we fully characterize the unique multi-time state in boxworld and develop a method for Bayesian retrodiction of physical transformations there. Our general framework is consistent with quantum theory, reproducing several existing results. In the non-Markovian quantum dynamics, we establish a correspondence between multi-time states and the process-matrix framework, and fully characterize the multi-time states compatible with general dynamics. This correspondence also lets us infer quantum causal models from multi-time states, and as a proof of principle, we propose a semidefinite program to infer causal order from quantum multi-time states. These results provide a unified framework for studying spatio-temporal states, causal relations, and Bayesian reasoning within OPTs, with implications for both fundamental and applied research.