# The Stochastic-Quantum Correspondence

**Authors:** Jacob A. Barandes  
**arXiv:** [2302.10778](https://arxiv.org/abs/2302.10778)  
**Date:** 2023-02-20  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2302.10778v2/

## Key points

- Exact correspondence between a general class of stochastic systems and quantum theory
- New framework for formulating non-Markovian types of stochastic dynamics
- Reconstruction of quantum theory from physical models with trajectories in configuration spaces undergoing stochastic dynamics
- Insights into interference, decoherence, entanglement, noncommutative observables, and wave-function collapse
- Identification of previously undescribed forms of gauge invariance
- Analysis of the measurement process and exploration of implications for dynamical symmetries and formal enlargements or dilations of a system's Hilbert space
- Revisiting questions related to nonlocality in quantum theory using a concrete model of stochastic physical variables
- Bidirectional relationship established between generalized stochastic systems and quantum systems in configuration spaces
- Definition of generalized stochastic systems and differentiation between divisible and indivisible dynamics
- Construction of Hilbert-space representations for these systems
- Formulation of a dictionary connecting generalized stochastic systems with quantum systems
- Introduction of new classes of gauge transformations
- Reviewing unistochastic matrices, introducing division events, and highlighting characteristic features bridging the gap between stochastic dynamics and quantum phenomena
- Contribution to understanding the interplay between stochastic processes and quantum mechanics
- Opening up avenues for further research into complex dynamics within both fields
- Shedding light on fundamental aspects of quantum theory that were previously unexplored

## Summary

This paper presents an exact correspondence between a general class of stochastic systems and quantum theory. It offers a new framework for formulating highly generic, non-Markovian types of stochastic dynamics and allows for the reconstruction of quantum theory from physical models consisting of trajectories in configuration spaces undergoing stochastic dynamics. This novel formulation provides insights into phenomena such as interference, decoherence, entanglement, noncommutative observables, and wave-function collapse. The paper also identifies previously undescribed forms of gauge invariance, analyzes the measurement process in detail, and explores the implications for dynamical symmetries and formal enlargements or dilations of a system's Hilbert space. By utilizing a concrete model of stochastic physical variables, it revisits important questions related to nonlocality in quantum theory. The stochastic-quantum correspondence presented here establishes a bidirectional relationship between generalized stochastic systems and quantum systems in configuration spaces. The paper is structured into sections that define generalized stochastic systems and distinguish between divisible and indivisible dynamics. It constructs Hilbert-space representations for these systems, formulates a dictionary connecting them with quantum systems, and introduces new classes of gauge transformations. The establishment of this correspondence involves reviewing unistochastic matrices, introducing division events, and highlighting characteristic features that bridge the gap between stochastic dynamics and quantum phenomena. Overall, this paper significantly contributes to our understanding of the interplay between stochastic processes and quantum mechanics. It opens up avenues for further research into complex dynamics within both fields and sheds light on fundamental aspects of quantum theory that were previously unexplored.