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

Abstract

Causality -- that a response cannot precede its cause -- is among the most universal constraints in physics. Yet when a unitary microscopic theory is reduced to an open-system memory-kernel equation, causality is not inherited for free: an upper-half-plane pole of the kernel forces exponential growth of the reduced propagator and is incompatible with any completely positive trace-preserving (CPTP) reduced dynamics. We examine three objects with different causal structure under projection -- the Nakajima-Zwanzig (NZ) memory kernel $\tilde{\mathcal{K}}(z)$, the reduced-state Laplace transform $\tildeσ(z)$, and the force-fit effective kernel $\tilde{\mathcal{K}}_\text{eff}(z)$ -- using the Kramers-Kronig (KK) relations as diagnostic. Under a real-axis spectral hypothesis on the projected generator, $\tilde{\mathcal{K}}(z)$ lies in a vector-valued Hardy space and obeys (subtracted) KK relations, giving a CPTP-consistency criterion, a passivity-analyticity statement, and a Carleman diagnostic. We prove $\tildeσ(z)$ analytic in the upper half-plane for any initial state -- unitarity bounds $\|σ(t)\|_{\mathrm{op}} \leq 1$, so acausality cannot be blamed on the state alone. Yet the force-fit kernel can develop upper-half-plane poles at simple zeros of $\tildeσ(z)$: passive baths sit in a robust regime where these zeros stay real, while near-resonant systems enter a fragile regime in which coherence-channel zeros migrate into the upper half-plane, an intrinsic symmetry property already present for factorized states. We verify the full operator-valued KK relation on the extracted $4\times4$ NZ memory kernel of the Jaynes-Cummings model, the relative $L_2$ residual decreasing under refinement ($3.8\% \to 0.95\%$), consistent with exact matrix-valued KK in the continuum limit.