Complex-Time Parametrization of Evanescent Tunneling and Transmission Delay in non-Hermitian Scattering
The quantum tunneling process is a fundamental concept that presents the unconventional nature of quantum dynamics. The main anomaly of this effect revolves around the choice of imposing the classical 4-momentum to become imaginary during the barrier traversal. This paper addresses this paradox by introducing a complex-time parametrization, which uses a complex temporal coordinate, $\tau = t + i\kappa$. Here, the real axis $(t)$ represents standard thermodynamic time, while the imaginary axis $(i\kappa)$ is correlated to spatial evanescence. We demonstrate that evaluating this extended complex-time framework yields a hyperbolic section of the causal lightcone, proving that the traversal is strictly bounded by the speed of light and preserves causal disconnection. Applied to quantum tunneling, we resolve the anomaly of imaginary momentum through a continuous geometric signature flip of the effective complex-time momentum norm from a timelike to a spacelike state. This theoretical model finds a physically phenomenological analogy in microwave scattering experiments using open, non-Hermitian systems, which link complex transmission time delay to spatial wave decay.