Exact Non-Markovian Master Equation for Entangled Coherent States — E8 Intelligence Research

FINDING: Exact non-Markovian master equation for entangled coherent states via Feynman-Vernon influence functional yields time-dependent memory coefficients; other sources are pedagogical (PSD definitions, spectral approaches to friction). | MATH: The core result (arXiv:0705.2472v3) derives an exact master equation of the form: \[ \frac{d\rho}{dt} = -i[H_S(t),\rho] + \gamma(t)\mathcal{L}[\rho] + \Delta(t)\mathcal{D}[\rho] \] where \(\gamma(t)\) and \(\Delta(t)\) are non-local, time-dependent coefficients encoding the spectral density \(J(\omega)\) of the vacuum environment. Specifically, these coefficients involve integrals over \(J(\omega)\coth(\hbar\omega/2k_BT)\sin(\omega t)/\omega\) and \(J(\omega)\cos(\omega t)\) — the memory kernel. For a Lorentzian spectral density \(J(\omega) = \lambda \gamma_0^2/((\omega-\omega_0)^2+\gamma_0^2)\), the coefficients become exponential/oscillatory: \(\gamma(t) \propto e^{-\gamma_0 t}\cos(\omega_0 t)\), \(\Delta(t) \propto e^{-\gamma_0 t}\sin( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com