New Insights into Quantum Time of Arrival through Moyal Deformation

In a recent paper titled "Moyal deformation of the classical arrival time," authors Dean Alvin L. Pablico and Eric A. Galapon explore the quantum time of arrival (TOA) problem. This issue involves determining the statistics of measured arrival times based on the initial state of a particle. Traditionally, this problem requires finding a quantum representation of the classical arrival time, typically expressed in operator form.

The authors approach this problem using the phase space formulation of quantum mechanics. They propose a new quantum image, denoted as ( \mathcal{T}_M(q,p) ), which is a real-valued and time-reversal symmetric function. This function is expressed as a formal series in terms of ( \hbar^2 ), with the classical arrival time serving as the leading term. The authors derive this representation directly from the Moyal bracket relation with the system Hamiltonian, interpreting it as a Moyal deformation of the classical TOA.

The paper discusses the properties of this new quantum image and demonstrates how it circumvents known challenges in quantization. Specifically, it shows an isomorphism between ( \mathcal{T}_M(q,p) ) and a previously constructed TOA operator in rigged Hilbert space, which adheres to the time-energy canonical commutation relation (TECCR) for arbitrary analytic potentials.

Furthermore, the authors examine TOA problems for specific cases, including a free particle and a quartic oscillator potential. This research could have implications for understanding quantum systems and their behavior, particularly in contexts where time of arrival is a critical factor.

The full paper can be accessed at arXiv:2309.00222.