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.
Comments
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'
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.
Retarded support, upper-half-plane holomorphy, and Hardy boundary control are distinct properties of a memory kernel. We give sufficient conditions linking them for the Nakajima--Zwanzig kernel of an open system with finite-dimensional system Liouville space. If the projected kernel has an absolutely continuous real-axis representation with density w in L1 intersect Lp0, no singular part, and p0 > 1, its transform lies in the operator-valued Hardy class Hp(B) for 1 < p <= p0. A finite-dimensional componentwise argument yields the corresponding principal-value and once-subtracted Kramers--Kronig (KK) boundary formulas; the H1 endpoint is separate. Microscopic unitary evolution gives holomorphy and a trace-norm bound for reduced-state transforms for arbitrary trace-class initial states, without that spectral hypothesis. On a common analytic domain, a perturbative force-fit equation has a first-order quotient pole at a simple baseline state-transform zero zeta only when the first-order inhomogeneous numerator I_tilde^(1)(zeta) is nonzero; at finite perturbation the actual zero and numerator must be checked, while matrix reconstructions additionally require adjugate non-cancellation. For rational reconstructions, the printed local or global clearing and no-cancellation hypotheses convert a rational kernel pole into a genuine upper-half-plane propagator pole. Such a pole is incompatible with a uniformly bounded CPTP family when the transforms agree on an open set, with Vieta's formula giving the imaginary-part budget. We formalize these paper-specific implications in Lean 4 from named classical inputs. A finite Jaynes--Cummings calculation provides a shifted, distributional KK check with a 0.024 percent fixed-truncation residual.
The memory kernel of an open quantum system obeys Kramers--Kronig (KK) relations if and only if its Laplace transform is analytic in the upper half-plane -- a property known as Hardy-space analyticity. Here we show that non-unitary exchange statistics, the defining property of paraparticles, intrinsically breaks Hardy-space analyticity. The metric η that guarantees a real closed-system spectrum for these particles necessarily differs from the physical Born inner product (∥η−I∥F/∥I∥F=0.51) -- a mathematical consequence of the R-matrix's non-unitarity, not a parameter choice. This metric is a "shadow metric": Schur's lemma forces it to commute with every bilinear observable, making the distortion physically invisible in the closed system. But when the paraparticle is coupled to a bath, any coupling operator that lies outside the symmetry algebra -- that is, any interaction that sees the internal flavour structure -- exposes the distortion. The memory kernel then develops upper-half-plane poles at coupling gc≈0.1, breaking standard dispersion relations before the closed-system spectrum complexifies. Fermions and bosons, whose exchange is unitary (η=I as an analytic fact of the canonical anticommutation algebra), are immune at any coupling, because there is no distortion to expose. The violation is intrinsic: it distinguishes non-unitary exchange statistics from ordinary particle statistics at the level of the memory kernel's analytic structure.
Kramers-Kronig (KK) relation between the dispersion and absorption responses of a signal field can be mapped from the frequency domain into the space domain via the dipole-dipole interactions between a homogeneous sample of target atoms and a control atom. This is achieved by establishing an effective two-level configuration for the three-level target atoms in the single-photon far-detuned driving regime while maintaining a high Rydberg excitation for the three-level control atom in the single-photon resonant driving regime. We find in particular that it is viable to realize a dynamically tunable spatial KK relation supporting asymmetric and even unidirectional reflection for appropriate signal frequencies in a controlled range. Taking a periodic lattice of target atoms instead, multiple Bragg scattering can be further incorporated into spatial KK relation to largely enhance the nonzero reflectivity yet without breaking the asymmetric or unidirectional reflection.
The Jaynes-Cummings (JC) model, introduced in 1963 and central to cavity quantum electrodynamics, describes a two-level system coupled to a single bosonic mode under the rotating-wave approximation. When the mode is projected out via the Nakajima-Zwanzig (NZ) formalism, the memory-kernel generator QLQ is manifestly non-Hermitian -- yet we prove analytically that its spectrum is strictly real at every coupling and every finite truncation, for both vacuum and thermal baths. For the vacuum bath the characteristic polynomial factorises completely; the nonzero eigenvalues reproduce the JC dressed-state ladder for n >= 2, while the lowest mode is suppressed by exactly sqrt(2) relative to the bare-Hamiltonian prediction. For any thermal Gibbs state, the squared generator reduces to an asymmetric rank-one perturbation symmetrised by a closed-form diagonal metric, with eigenvalues guaranteed real by Cauchy interlacing. We construct a positive-definite metric eta_osc intertwining QLQ with its adjoint on the nonzero spectral subspace, proving hidden pseudo-Hermiticity. A classification theorem extends these results to the full U(1)-conserving single-photon-exchange class with arbitrary complex couplings. The phase boundary under counter-rotating deformation is mapped analytically at resonance and numerically across the coupling-truncation plane, revealing a weak-coupling protected wedge, re-entrant real-spectrum bubbles, and N-universal plateaus organised by a three-family band catalog with closed-form level spacings. The full phase structure is proved well-defined in the N -> infinity thermodynamic limit. The sqrt(2) suppression provides a platform-independent experimental falsification target spanning seven orders of magnitude in coupling strength.
A theoretical and numerical study of when the anti-holomorphic (conjugate-variable) dependence of a two-point transfer kernel is forced by the underlying open-system dynamics rather than removable by gauge, basis, or preparation. For two counter-winding mode ladders coupled unidirectionally (reservoir-engineered, Lindblad-type, per-pair Jordan double-pole structure), the transfer kernel sums to a closed form involving the Lerch transcendent in the product of conjugated coordinates. Reciprocity is shown to be equivalent to the existence of an exchange-Hermitian mirror partner of the kernel; unidirectionality removes that partner. All symbolic claims are certified by exact Wirtinger-derivative computation. A recirculating-fiber-loop implementation and a time-domain protocol are proposed, with noise/imperfection analysis and explicit falsifiability criteria. Version 3, additive over v2 — the boundary arc (Sections 3–5): the kernel's intrinsic boundary at |x| = 1 with its generic-existence statement (the wall belongs to the class; its location is set by the tail of the weights); the chiral phase law of the branch-cut jump (machine-exact ratio, linear drift at rate kappa/2delta, family-independent); the log-periodic interference calendar on the log-scale coordinate, with prior-art positioning against discrete scale invariance (Sornette; Derrida–Itzykson–Luck); the Fourier floor theorem with full proof (the residual relief of large kernel sums equals the window-contrast of the Fourier transform of the weighted winding measure; finite atomic ladders keep the calendar forever by Bohr almost-periodicity, continua forget by Riemann–Lebesgue); closed-form floors (Ci/Si integrals, cardinal sine); and an inverse spectroscopy reading the hidden weighted distribution of winding speeds from the measured relief, with proven limits (exact mirror blindness of modulus-only data; window = aperture). Non-Hermitian point-gap topology references added (Wanjura et al. 2020; Zirnstein et al. 2021). Title-date correction (July 10, 2026). No physical claim is added; all results are machine-verified structural characterizations. Note added at deposit time: subsequent to the sealing of this version, the mirror-blindness limit of the inverse spectroscopy has been lifted by a reference-needle heterodyne protocol (demodulation law, asymmetry law, quadrature recovery of the full complex transform), machine-verified and judge-certified; traces and harnesses are in the public repository below. This upgrade will be the subject of the next linked deposit. All FINDINGS and machine harnesses backing every number in Sections 3–5, plus an independent external from-scratch reproduction audit, are public: https://github.com/antparis/oneway-kernel (MIT).
Kramers-Kronig or dispersion relations that arise from the analytic behavior of electromagnetic and other classical waves are here adapted to quantum waves. In the framework of theories that view the wave packet (wp) collapse as a time-continuous process, we postulate analytic properties for the component-amplitudes in a wp as functions of a complex time. We then construct a model which embodies the removal of the non-selected components in the aftermath of the measurement, to be ultimately followed by a thermalequilibrium like superposition state of the system. Conjugate relations hold between component-moduli and phases and these show that a non-selected component (one that vanishes in the measurement) acquires within the duration of the collapse a fast oscillating phase factor. Thus, by virtue of mathematical properties, both phase-decoherence and amplitude-decay have to occur in a collapse process, indifferently to the physical mechanism.