Abstract
Phys. Rev. D 98, 025019 (2018) 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.