A Noether Theorem for Markov Processes

TL;DR

This paper extends Noether's theorem to Markov processes, showing that an observable's commutation with the Hamiltonian ensures constant expected value and standard deviation over time, revealing new symmetry-conservation relations.

Contribution

It introduces a version of Noether's theorem for Markov processes, linking symmetries to conserved quantities beyond quantum mechanics.

Findings

Observable commutes with Hamiltonian iff expected value and standard deviation are constant

Provides a new symmetry-conservation relation for Markov processes

Bridges concepts between quantum mechanics and stochastic processes

Abstract

Noether's theorem links the symmetries of a quantum system with its conserved quantities, and is a cornerstone of quantum mechanics. Here we prove a version of Noether's theorem for Markov processes. In quantum mechanics, an observable commutes with the Hamiltonian if and only if its expected value remains constant in time for every state. For Markov processes that no longer holds, but an observable commutes with the Hamiltonian if and only if both its expected value and standard deviation are constant in time for every state.