Moyal deformation of the classical arrival time

TL;DR

This paper introduces a Moyal deformation of the classical arrival time in quantum mechanics, providing a phase space approach that overcomes quantization obstructions and aligns with existing TOA operators.

Contribution

It develops a real-valued, time-reversal symmetric quantum arrival time function using Moyal brackets, extending classical concepts within phase space formulation.

Findings

The Moyal-deformed TOA function is isomorphic to the rigged Hilbert space TOA operator.

The approach satisfies the time-energy canonical commutation relation for various potentials.

Explicit analysis for free particle and quartic oscillator demonstrates the method's applicability.

Abstract

The quantum time of arrival (TOA) problem requires the statistics of measured arrival times given only the initial state of a particle. Following the standard framework of quantum theory, the problem translates into finding an appropriate quantum image of the classical arrival time $\mathcal{T}_C(q,p)$, usually in operator form $\hat{\mathrm{T}}$. In this paper, we consider the problem anew within the phase space formulation of quantum mechanics. The resulting quantum image is a real-valued and time-reversal symmetric function $\mathcal{T}_M(q,p)$ in formal series of $\hbar^2$ with the classical arrival time as the leading term. It is obtained directly from the Moyal bracket relation with the system Hamiltonian and is hence interpreted as a Moyal deformation of the classical TOA. We investigate its properties and discuss how it bypasses the known obstructions to quantization by showing the isomorphism between $\mathcal{T}_M(q,p)$ and the rigged Hilbert space TOA operator constructed in [Eur. Phys. J. Plus \textbf{138}, 153 (2023)] which always satisfy the time-energy canonical commutation relation (TECCR) for arbitrary analytic potentials. We then examine TOA problems for a free particle and a quartic oscillator potential as examples.