Time, dynamics and probability

Jincheng Zhang

Probability Space-Time Geometry (PSTG) presents a fundamentally new framework for modeling probabilistic spacetime, offering a pathway towards exploring temporal dynamics and potential time travel scenarios through the integration of quantum probability fields. The core concept revolves around embedding time as a parameter, dynamically evolving based on probability distributions, thereby circumventing the limitations of linear time models. This investigation details the mathematical foundations, operational mechanisms, and potential implications of PSTG, highlighting its novelty through the introduction of the 'time probability' concept and its ability to address the inherent paradoxes associated with temporal manipulation.

Győző Egri, Marton Gomori, Balazs Gyenis +1 more

The probabilistic description of the time evolution of a physical system can take two conceptually distinct forms: a trajectory of probabilities, which specifies how probabilities evolve over time, and a probability on trajectories, which assigns probabilities to possible histories. A lack of a clear distinction between these two probabilistic descriptions has given rise to a number of conceptual difficulties, particularly in recent analyses of stochastic-quantum correspondence. This paper provides a systematic account of their relationship. We define probability dynamics and stochastic process families together with a precise notion of implementation that connects the two descriptions. We show that implementations are generically non-unique, that every probability dynamics admits a Markovian implementation, and characterize when non-Markovian implementations are possible. We expose fallacies in common arguments for the linearity of probability dynamics based on the law of total probability and clarify the proper interpretation of ``transition matrices'' by distinguishing dynamics-level maps from the conditional probability matrices of implementing processes. We further introduce decomposability as the appropriate general notion of stepwise evolution for (possibly nonlinear) probability dynamics, relate it to divisibility in the linear case -- showing that the two can come apart -- and disentangle both notions from Markovianity and time-homogeneity. Finally, we connect these results to what we call statistical dynamics, in which linearity is indeed physically motivated, and contrast the framework with quantum mechanics.

Boyarsky, A., Paweł Góra

We present a simple dynamical systems model for the effect of invisible space dimensions on the visible ones. There are three premises. A: Orbits consist of flows of probabilities [P].which is the case in the setting of quantum mechanics. B: The orbits of probabilities are induced by (continuous time) differential or partial differential equations. C: Observable orbit are flow of marginal probabilities where the invisible space variables are averaged out. A theorem is presented which proves that under certain general conditions the transfer of marginal probabilities cannot be achieved by continuous time dynamical systems acting on the space of observable variables but that it can be achieved by discrete time dynamical systems.

Lazar Mayants