On the Stochastic-Quantum Correspondence
Sami Calvo
TL;DR
This work formalises a Stochastic Axiom from which the six textbook quantum mechanics axioms are derived, via a unistochastic transition matrix $\Gamma_{ij}=|\langle i|U(t)|j\rangle|^2$ that yields $p(t)=\Gamma(t)p(0)$ and a Schrödinger-type evolution with $\hat{\mathcal{H}}=i(\partial_t U)U^{\dagger}$. It critically examines the extension to continuous bases, revealing divergences that suggest space (and other variables) may be fundamentally discrete, and extends the framework to classical and quantum fields, including scattering where the $S$-matrix coincides with $\Gamma$. The practical discussion argues that, while the approach clarifies the measurement problem by reinterpreting collapse as conditional probability and provides a unified view of classical and quantum regimes, it does not immediately simplify problem-solving and motivates further exploration in QFT and relativity. Overall, the paper offers a cohesive ontological and mathematical bridge between stochastic processes and quantum dynamics, with explicit treatments of environments, measuring devices, and the emergence of classical behaviour via the Ehrenfest equation.
Abstract
This paper aims to first explain, somewhat more clearly, the Stochastic-Quantum correspondence put forward in by Barandes in 2023. Specifically, the quantum-mechanical bra-ket notation is used, illuminating some results of previous results. With this, we prove the six axioms of textbook quantum mechanics from a single axiom: every physical system evolves according to a, generally indivisible, stochastic law. Afterwards, we generalise the treatment to continuous bases, which showcases a problem with them, indicating that space (and other physical variables) may be discrete in nature. Some concrete examples are also given, including the generalisation to classical and quantum fields. Then, we treat some practical issues of this new stochastic approach, regarding the solving of problems in physics, which turns out to still be most tractable in the traditional way. Finally, we explain the classical limit, where a system of many particles is found to behave classically according to Newton's second law. Along with that, we present a way of solving the measurement problem, characterising what is an environment and a measuring device and explaining how the wavefunction collapse comes about. Specifically, it is found that what distinguishes an environment is its number of degrees of freedom, while a measuring device is a low-entropy type of environment.