Dispersion Relations Across the Unitarity Boundary

Abstract

Kramers-Kronig (KK) relations rest on a binary premise: a response function is either analytic in the upper half-plane or it is not. We show that a single reduced-state transform organizes both outcomes into a sharp dichotomy controlled by microscopic unitarity. One closed-form function carries, simultaneously, a zero and a pole in the upper half-plane; the spectral abscissa alpha of the reduced propagator decides which is realized. For alpha < 0 (unitary reduction) the upper-half-plane object is a protected zero: KK holds, yet the zero is directly measurable from a finite-time coherence record by a damped Fourier transform (no analytic continuation), obeying a closed law Im(zeta) = 0.3092 g. For alpha > 0 (gain-driven non-unitary reduction) the zero is replaced by a genuine pole, the Blaschke winding number jumps from 0 to 1, and KK acquires a Lorentzian residue correction scaling as a power law with negative exponent nu ~ -1.08, peaking at threshold. The protected zero is not inert: any scalar single-channel kernel extraction is forced to reproduce a phantom resonance -- a refractive feature with no absorptive origin, at a protocol-independent frequency -- without any initial system-bath correlation. We give the closed-form criteria, a measurable terahertz signature (31-1391 GHz), and the solvable dimer and Jaynes-Cummings models that realize both sides of the boundary.

4 pages, 2 figures. Supplemental Material included (3 pages, Sections A-D). v2: major revision --- extended from single-side topological charge to the full unitary/non-unitary dichotomy; adds measurable protected zeros, phantom resonance (one-condition), and the sharp unitarity boundary. Title changed from 'Topological Charge of Causality at a PT-Symmetric Exceptional Point'