Fractional integrodifferential equations and (anti-)hermiticity of time in a spacetime-symmetric extension of nonrelativistic Quantum Mechanics

Abstract: 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, which is 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 uncertainty relation with its canonical conjugate. In this work, we use a recent proposed spacetime-symmetric formalism~\href{https://doi.org/10.1103/PhysRevA.95.032133}{[Phys.~Rev.~A {\bf 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 $\hat{X}$ in one subspace, and the position parameter $x$ and time operator $\mathbb{T}$ in the other subspace. Time as an operator is better suitable for describing tunnelling processes. We then solve the novel $1/2$-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunnelling time through a rectangular barrier. We compare to previous works, obtaining pure imaginary times for energies below the barrier and a fast-decaying imaginary part for energies above the barrier, indicating the anti-hermiticity of the time operator for tunnelling times. We also show that the expected time of arrival in the tunnelling problem has the form of an energy average of the classical times of arrival plus a quantum contribution.