Complex-Time Parametrization of Evanescent Tunneling and Transmission Delay in non-Hermitian Scattering

Quantum tunneling is one of the most counterintuitive phenomena in physics. It describes a situation where a tiny particle, like an electron, manages to pass through a solid barrier that it should not be able to cross according to the laws of classical physics. Imagine a ball rolling toward a hill; if the ball does not have enough energy to reach the top, it will roll back down. In the quantum world, however, particles can sometimes appear on the other side of that hill without ever having climbed over it. For nearly a century, physicists have accepted this as a fact of nature, but the mathematical description of how it happens has always contained a troubling paradox. To make the equations work, scientists have traditionally been forced to treat the particle's momentum as an imaginary number—a concept that exists in mathematics but has no direct physical counterpart in our real world. This creates a disconnect between the math and the physical reality, leaving a lingering question about what is actually happening to the particle while it is inside the barrier.

A new study by researcher Daia Iulia-Maria at the West University of Timișoara proposes a fresh way to resolve this long-standing puzzle. Instead of accepting that the particle's momentum becomes imaginary, the author suggests that we need to change how we view time itself during the tunneling process. The paper introduces a model where time is not just a single, straight line moving forward, but a complex plane that includes both a standard, real time and an auxiliary, imaginary component. In this framework, the "imaginary" part of the particle's journey is not a mathematical trick, but a representation of a specific type of spatial decay. By treating time as having two dimensions—one that flows forward like a clock and another that relates to how the particle's presence fades away—the author shows that the particle never actually breaks the rules of physics or travels faster than light.

The core of this work is a geometric reimagining of the tunneling event. The researcher divides the journey of the particle into five distinct stages, tracking how the relationship between real time and this new auxiliary time shifts as the particle approaches, enters, and leaves the barrier. Before the particle hits the barrier, it behaves normally, moving through standard time. As it begins to penetrate the barrier, its motion enters a hybrid state where real time and the auxiliary time coexist. Once the particle is fully inside the barrier, the model shows that the flow of real, thermodynamic time effectively pauses. During this pause, the particle's existence is described entirely by the auxiliary time component, which corresponds to the exponential decay of its wave. This shift is not a sudden jump but a smooth, continuous rotation of the particle's state. By the time the particle exits the barrier, the process reverses: the auxiliary time fades, real time resumes its flow, and the particle emerges on the other side.

This approach solves the paradox of the imaginary momentum by showing that the particle does not acquire a strange, non-physical property. Instead, the particle simply transitions into a state where its movement is governed by a different geometric signature. In the language of relativity, the particle shifts from a "timelike" state, where it moves through time, to a "spacelike" state, where its presence is defined by spatial decay, before returning to the timelike state. Crucially, the paper demonstrates that this entire process remains strictly bounded by the speed of light. Even though the particle seems to traverse the barrier instantly or in a way that defies classical intuition, the model proves that its total velocity never exceeds the universal speed limit. The apparent superluminal behavior is an illusion created by the way real time and the auxiliary time interact, much like how a shadow can move faster than the object casting it without the object itself breaking any speed limits.

To support this theoretical model, the author draws a parallel with experiments involving microwave scattering. In these experiments, scientists send microwave pulses through complex, open systems and measure the time it takes for the signal to pass through. They have observed that the transmission delay is not just a simple number; it has both a real part and an imaginary part. The study suggests that the imaginary part of this delay, which has been difficult to interpret physically, corresponds directly to the auxiliary time component described in the new model. When the microwaves encounter a resonance that mimics the conditions of a quantum barrier, the real time delay drops while the imaginary component spikes. This behavior mirrors the theoretical prediction that real time pauses while the system dives into the auxiliary domain. While the paper does not claim to have observed quantum tunneling directly with this method, it provides a strong phenomenological analogy, showing that the mathematical structure used to describe the tunneling particle is consistent with measurable data from microwave experiments.

The significance of this work lies in its ability to restore a sense of physical continuity to a process that has long been viewed as discontinuous. By replacing the concept of an instantaneous quantum jump with a continuous geometric rotation, the model offers a more coherent picture of how particles navigate the impossible. It suggests that the strange behavior of quantum particles is not a violation of the laws of physics, but a consequence of a more complex structure of time that we have not fully accounted for in standard descriptions. The research does not overturn the established results of quantum mechanics, such as the probability of a particle tunneling, but it provides a new, rigorous geometric language to describe the journey. It resolves the anomaly of imaginary momentum by showing that the particle is not doing something mathematically impossible, but is simply moving through a different dimension of time that is inextricably linked to its spatial decay.

Ultimately, this paper offers a bridge between the abstract mathematics of quantum mechanics and the concrete reality of physical measurement. It proposes that the "imaginary" aspects of quantum tunneling are not just mathematical artifacts, but reflections of a real, albeit hidden, temporal dimension. By mapping the tunneling process onto a complex-time plane, the author demonstrates that the particle's traversal is a smooth, causal event that respects the fundamental limits of the universe. The work stands as a theoretical proposal that aligns with existing experimental data from microwave scattering, suggesting that the key to understanding the deepest mysteries of the quantum world may lie in rethinking the very nature of time itself.