Coarse Grained Quantum Dynamics

TL;DR

The paper develops a framework to coarse-grain quantum systems by partitioning into infrared and ultraviolet sectors with a hierarchical energy gap, and derives the reduced-density-matrix dynamics for the IR sector via perturbation theory to ${\cal O}(\lambda^2)$ combined with time averaging to reflect finite experimental resolution. It shows that the IR dynamics is non-Hamiltonian and non-Markovian at higher orders, organized as a Born-Oppenheimer–like expansion in $\Delta E_{IR}/\Delta E_{UV}$, and validates the approach through three explicit examples: coupled spins, linearly coupled oscillators, and a cubic scalar QFT. The work connects this open-system dynamics to holographic Wilsonian renormalization by identifying the logarithm of the Feynman-Vernon influence functional as the correct Wilsonian analogue for time-dependent processes, highlighting UV–IR entanglement and nonlocal effects on the cutoff scale. These results illuminate how coarse-graining induces entanglement and memory effects, with potential implications for real-time holography and gauge–gravity duality.

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 $ΔE_{UV} > ΔE_{IR}$, 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 $ΔE_{IR}/ΔE_{UV}$, 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.

Figures (1)

• Figure 1: Cartoon of band structure: large jumps correspond to UV quanta, small jumps to IR quanta.