Tunneling time problem

It was previously shown in Phys. Rev. Lett. 108, 170402 (2012) that a time-of-arrival (TOA) operator approach to the quantum tunneling time problem predicts that only above-barrier energy components of an incident wavepacket contribute to the barrier traversal time. On the other hand, the contribution of the below-barrier components vanishes which implies an instantaneous tunneling time. This now raises the question on whether the supposed instantaneous tunneling time is a consequence of using a non-relativistic TOA-operator, that is, could there be a fundamental difference if one uses a relativistic TOA-operator?

We address this question in my Dissertation entitled Theory of quantized relativistic time-of-arrival operators for spin-0 particles and its application in the quantum tunneling time problem. The entire approach is based on our proposed formalism of quantized relativistic TOA-operators which was done by inverting the equation of motion of special relativity to obtain an expression for the TOA, i.e.,

wherein, \(x\) is the arrival point, \(q\) is the initial position, \(\mu_o\) is the rest mass, \(c\) is the speed of light in vacuum, \(\text{sgn}(z)\) is the sigunum function, \(V(q)\) is the interaction potential, and the Hamiltonian is \(H(q,p)=\sqrt{p^2 c^2 + \mu_o^2 c^4} + V(q)\).

The main result of my Dissertation is summarized in this Letter while the full details are outlined in this preprint. Moreover, a detailed study of the free case can be found here.

I was also able to present my research during the poster session of the conference Time in Quantum Theory 2022 which was held at the Technical University of Vienna.