Kramers-Kronig Relations and Causality in Non-Markovian Open Quantum Dynamics: Kernel, State, and Effective Kernel

Kramers-Kronig (KK) relations are usually invoked for causal response functions, but their precise status for non-Markovian quantum memory kernels is less explicit. We separate three Laplace-domain objects: the Nakajima-Zwanzig memory kernel $\tilde{\mathcal K}(z)$, the reduced-state transform $\tilde{\sigma}(z)$, and the force-fit effective kernel $\tilde{\mathcal K}{\rm eff}(z)$. Under a real-axis spectral-representation hypothesis for the projected generator, with a coupling-weighted spectral density in $L^1 \cap L^p$, we show that $\tilde{\mathcal K}(z)$ belongs to the operator-valued Hardy space $H^p+$ and obeys KK or subtracted KK relations. This gives a Hardy-space consistency criterion for CPTP reduced dynamics, a passivity-analyticity compatibility statement for passive bosonic baths, and a finite-truncation Carleman diagnostic for moment-based kernel reconstructions. In contrast, $\tilde{\sigma}(z)$ is analytic in the upper half-plane for any initial system-bath state, including correlated states, because microscopic unitarity gives $|\sigma(t)| \leq 1$. Apparent acausality can therefore enter only through the force-fit object: in scalar channels, uncancelled zeros of $\tilde{\sigma}(z)$ can generate upper-half-plane poles of $\tilde{\mathcal K}_{\rm eff}(z)$. Numerically, we verify the full matrix-valued KK relation for an extracted $4 \times 4$ Jaynes-Cummings memory kernel. The measured integrated relative residual, $3.8$, lies below the calibrated noise floor of the circular FFT-Hilbert protocol, about $5$, and is therefore consistent with exact KK within numerical accuracy. We also present Born-order and correlated-state diagnostics showing how discarded inhomogeneous terms can contaminate force-fit kernels without violating microscopic causality.