Quantum Arrival Time Formula from Decoherent Histories

TL;DR

This paper derives a quantum arrival time formula using decoherent histories, showing it aligns with the standard probability in semiclassical cases but fails with backflow states where decoherence does not occur.

Contribution

It demonstrates how the quantum arrival time formula emerges from decoherent histories and clarifies conditions for its validity, especially regarding backflow states.

Findings

The arrival time probability formula is consistent with decoherent histories for certain initial states.

Backflow states prevent decoherence, invalidating probability assignment.

The standard formula is positive and semiclassically correct for states without backflow.

Abstract

In the arrival time problem in quantum mechanics, a standard formula that frequently emerges as the probability for crossing the origin during a given time interval is the current integrated over that time interval. This is semiclassically correct but can be negative due to backflow. Here, we show that this formula naturally arises in a decoherent histories analysis of the arrival time problem. For a variety of initial states, we show that histories crossing during different time intervals are approximately decoherent. Probabilities may therefore be assigned and coincide with the standard formula (in a semiclassical approximation), which is therefore positive for these states. However, for initial states for which there is backflow, we show that there cannot be decoherence of histories, so probabilities may not be assigned.