[Paper Review] The Imaginary Time in the Tunneling Process

Zhong Chao Wu|ArXiv.org|Apr 1, 2008

Scientific Research and Philosophical Inquiry2 references3 citations

TL;DR

This paper proposes that tunneling particles spend purely imaginary time within a potential barrier, explaining apparent superluminal tunneling as an illusion caused by non-causal, stochastic behavior in imaginary time. Using quantum mechanical and instanton methods, it shows that real time is zero during tunneling, preserving relativity while accounting for experimental observations of fast tunneling in quantum optics.

ABSTRACT

By using techniques developed in quantum cosmology, it is found that a tunneling particle spends purely imaginary time on a barrier region. The {\it imaginary} time is associated with the stochastic acausal behaviour of a state, while the {\it real} time is associated with the deterministic causal evolution of a state. For the tunneling case the nonzero imaginary time is associated with the transmission rate of the tunneling process, which is related to the thickness of the barrier. The physical meaning of the zero real time is that the particle instantly jumps from one side of the barrier to the other regardless of the thickness. This leads to the illusion that tunneling particles could actually travel faster than light. The results of recent experiments in quantum optics concerning tunneling time can be thought of as the first experimental confirmation of the existence of imaginary time. Relativity is not violated.

Motivation & Objective

• To resolve the paradox of apparent superluminal tunneling in quantum mechanics and quantum optics.
• To clarify the role of time in quantum tunneling, particularly distinguishing real time from imaginary time.
• To show that causality and relativity are preserved despite experimental observations suggesting faster-than-light tunneling.
• To interpret the physical meaning of imaginary time in tunneling processes using quantum cosmology techniques.
• To connect tunneling behavior with broader concepts such as instantons, unitarity, and the Wheeler-DeWitt equation.

Proposed method

• Uses the WKB approximation to derive the transmission coefficient for tunneling through a potential barrier.
• Applies the analytic continuation to imaginary time (τ = it) to transform the Schrödinger equation into a Euclidean form, enabling instanton-like solutions.
• Reinterprets the tunneling transmission rate as the exponential of the negative action of a classical bounce in imaginary time.
• Derives the imaginary time spent in the barrier region via t = ∫(mdx)/p, where momentum p is imaginary within the barrier.
• Uses the Kramers-Kronig relations and information theory to argue that information velocity never exceeds c, even when group velocity does.
• Applies the Larmor precession method as a thought experiment to measure tunneling time, showing it aligns with imaginary time interpretation.

Experimental results

Research questions

• RQ1 Can the apparent superluminal tunneling of particles be reconciled with special relativity?
• RQ2 What is the physical meaning of time spent by a tunneling particle in a classically forbidden region?
• RQ3 How does the use of imaginary time in quantum tunneling preserve causality and unitarity?
• RQ4 Why do experiments in quantum optics observe tunneling photons arriving before those traveling freely through vacuum?
• RQ5 To what extent is stochastic behavior in quantum systems a manifestation of imaginary time?

Key findings

• Tunneling particles spend zero real time traversing the barrier, leading to an instantaneous jump from one side to the other.
• The time spent in the barrier region is purely imaginary and proportional to the integral of mdx/p, where p is imaginary within the barrier.
• The transmission coefficient D ≈ exp[−2∫√(2m(U(x)−E))dx] is equivalent to exp(−A), where A is the action of a bounce solution in imaginary time.
• The apparent superluminal behavior in experiments is an illusion caused by the non-deterministic, acausal nature of imaginary time evolution.
• Imaginary time manifests through exponential decay of the wave function in the barrier, consistent with the Sokolovski and Connor calculation.
• The framework is consistent with relativity and causality, as information velocity never exceeds c, even when group velocity does.