A self-adjoint arrival time operator inspired by measurement models

TL;DR

This paper presents a new self-adjoint arrival time operator linked to measurement models, providing a distribution that aligns with physical intuition across different momentum regimes.

Contribution

It introduces a novel self-adjoint arrival time operator with a spectrum related to measurement models, bridging the gap between theoretical constructs and practical measurement outcomes.

Findings

The operator's spectrum yields an arrival time distribution similar to the Kijowski distribution at high momentum.

At low momentum, the distribution is proportional to the kinetic energy density.

Comparison with existing operators highlights differences in self-adjointness and physical interpretation.

Abstract

We introduce an arrival time operator which is self-adjoint and, unlike previously proposed arrival time operators, has a close link to simple measurement models. Its spectrum leads to an arrival time distribution which is a variant of the Kijowski distribution (a re-ordering of the current) in the large momentum regime but is proportional to the kinetic energy density in the small momentum regime, in agreement with measurement models. A brief derivation of the latter distribution is given. We make some simple observations about the physical reasons for self-adjointness, or its absence, in arrival time operators and in the momentum operator on the half-line and we also compare our operator with the dwell time operator.