Quantum mechanics for 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 Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. 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. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.
Related
- Finite-step statistical mechanics of scale-invariant learning: spectral equations of state and curvature transport
- A polynomial-time classical sampler for noisy quantum circuits from statistical mechanics
- What is Physics For? Why Classical Physics is not the Limit of Quantum Mechanics
- Quantum-Classical Neural Networks for Solving Lindblad Master Equations
- Indeterminism and classical mechanics
BibTeX
@misc{wetterich2026quantum,
title = {{Quantum mechanics for classical transport equations}},
author = {Christof Wetterich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.15969},
eprint = {2605.15969},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15969},
}
RIS
TY - UNPB
TI - Quantum mechanics for classical transport equations
AU - Christof Wetterich
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2605.15969
UR - https://arxiv.org/abs/2605.15969
AB - 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 Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. 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. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.
ER -