Traveling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notion of time, where space and time have the same hierarchy, essential in describing events in spacetime. On the other hand, in quantum mechanics, time appears as a classical parameter, meaning that it does not have an uncertain relation with its canonical conjugate. In this work we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space: the time parameter t and the position operator ̂ \^X in one subspace, and the position parameter x and time operator T in the other subspace. Time as an operator is better suited for describing tunneling processes. We then solve the $1/2$-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that for wave packets well resolved in time, the expectation value of the operator T is the energy average of the classical time Tclass=∂S/∂E, where S is the classical action, which can be real or imaginary. The imaginary classical time does not contribute to the traveling time. Furthermore, we apply our results to a Gaussian energy distribution and compare them to previous works. This work is a correction of a previous paper [Phys. Rev. A 107, 052220 (2023)].