Quantum mechanics for classical transport equations

TL;DR

This paper demonstrates that classical transport equations with probabilistic initial conditions can be reformulated as quantum systems, capturing quantum features like superposition and interference within a classical probabilistic framework.

Contribution

It introduces a complex functional integral representation for classical transport equations that reproduces key quantum mechanical phenomena.

Findings

Classical transport equations can be viewed as quantum systems with wave functions.

Quantum features such as superposition and interference are realized in classical probabilistic transport.

A functional integral formalism describes the quantum-like behavior of classical transport equations.

Abstract

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schr\"odinger equation. Statistical observables are represented by operators which do not commute with the ones associated to classical observables. Examples are functions of the quantum energy or the quantum angular momentum. They are important conserved quantities. We construct a complex functional integral for the quantum system which describes the probabilistic classical transport equation. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations.