Negative Dwell Times as Resolvent Derivatives: A Structural Operator-Theoretic Explanation of 𝑻𝑻=𝒕𝒈

Abstract

A structural operator-theoretic account of negative group delays and post-selected excitation times in resonant atomic media. Within a scalar Lorentzian propagation model, the excitation time and the optical delay are shown to be parameter derivatives of the same frequency-resolved transmission amplitude — the first with respect to the atomic transition frequency, the second with respect to the optical frequency. Because the internal resolvent R = (z − ω)⁻¹, z = ω₀ − iΓ/2, depends on these only through their difference, the two are linked by an exact identity. For a fixed detected frequency, T_T(ω) = Re[i ∂_{ω₀} ln t_med(ω)] (Theorem A); its difference from the local Wigner delay is exactly A Re R(ω), which vanishes at line centre (Corollary B). For an unresolved detector, coarse-graining gives T_T = t_g^det whenever the detected spectral weight is even in detuning — a condition of symmetry, not of bandwidth (Theorem C). Negativity is treated separately from the equality. Causality is established from the analyticity of the retarded response and the full two-pole susceptibility: negative delays are properties of the pulse centroid, and the front velocity remains c exactly. Appendices cover departures from the idealised model, and a Python script reproduces all quoted numerics. All identities are exact within the stated first-order propagation model. Version note. Supersedes the May 2026 deposit. The central derivation has been replaced: the earlier conditional-probability argument is withdrawn. Keywords: post-selected excitation time, weak value, group delay, resolvent, Feynman–Hellmann identity, Lorentzian resonance, anomalous dispersion, front velocity, causality, quantum optics