Geometric Spectral Foundations of Non-Markovian Open Quantum Dynamics

This a presentation of a self-contained derivation of a geometry-driven fractional master equation (FME) for open quantum systems and its finite-dimensional Markovian embedding via an augmented Lindblad (AL) model. Starting from a microscopic Hamiltonian in which the environment is a quantized Laplace-type field on a compact manifold, Weyl spectral asymptotics give power-law spectral densities. These, in turn, generate algebraic bath correlations and long-time kernels. Under standard Born/Nakajima--Zwanzig assumptions and a low-frequency scaling limit we show how the convolutional master equation reduces to a fractional-in-time generator. To restore complete positivity and numerical tractability, we provide an explicit constructive mapping from the geometry-derived correlation \(C(t)\) to a positive sum-of-exponentials (SOE) approximation and then to an augmented Lindblad equation on a system-plus-auxiliaries Hilbert space.The SOE-based embedding transforms what was once a numerical convenience into a physically grounded approximation: every exponential component corresponds to a damped auxiliary oscillator mediating memory. Unlike ad-hoc kernel fits, the positivity-constrained SOE ensures a thermodynamically consistent extension whose parameters can, in principle, be engineered or measured. The framework thus elevates kernel fitting from a heuristic to a model-building principle.