Coarse Grained Quantum Dynamics

TL;DR

This paper develops a framework for understanding coarse grained quantum dynamics by deriving non-Hamiltonian, nonlocal equations of motion for long-distance observables, inspired by holographic renormalization and effective field theory.

Contribution

It introduces a perturbative approach to coarse graining in quantum systems, deriving master equations that account for non-Hamiltonian effects and non-Markovian behavior at higher orders.

Findings

Derived a hierarchy of effective equations of motion for coarse grained quantum systems.

Identified conditions under which dynamics deviate from Hamiltonian and Markovian behavior.

Applied the framework to models including spins, harmonic oscillators, and scalar QFT.

Abstract

Inspired by holographic Wilsonian renormalization, we consider coarse graining a quantum system divided between short distance and long distance degrees of freedom, coupled via the Hamiltonian. Observations using purely long distance observables are described by the reduced density matrix that arises from tracing out the short-distance degrees of freedom. The dynamics of this density matrix is non-Hamiltonian and nonlocal in time, on the order of some short time scale. We describe this dynamics in a model system with a simple hierarchy of energy gaps $\Delta E_{UV} > \Delta E_{IR}$, in which the coupling between high-and low-energy degrees of freedom is treated to second order in perturbation theory. We then describe the equations of motion under suitable time averaging, reflecting the limited time resolution of actual experiments, and find an expansion of the master equation in powers of $\Delta E_{IR}/\Delta E_{UV}$, after the fashion of effective field theory. The failure of the system to be Hamiltonian or even Markovian appears at higher orders in this ratio. We compute the evolution of the density matrix in three specific examples: coupled spins, linearly coupled simple harmonic oscillators, and an interacting scalar QFT. Finally, we argue that the logarithm of the Feynman-Vernon influence functional is the correct analog of the Wilsonian effective action for this problem.