Times of arrival and gauge invariance

Abstract

We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov–Bohm–Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski’s result is pointed out, along with an inescapable consequence of the ‘negative arrival times’ inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time set-up featuring a charged particle moving in a constant magnetic field. A natural generalization of the ABK distribution is in this case shown to be critically gauge-dependent. A direct comparison to the QF distribution, which does not exhibit this flaw, is drawn (its acknowledged drawback concerning the quantum backflow effect notwithstanding).

When does a particle arrive?

“…The results of Kijowski reproduce the TOA distribution that Aharonov and Bohm obtained through the quantization of the classical time of arrival, by means of the correspondence principle [4]. See [29] for a review and a comparison between these two results.…”

“…In [28] Leavens discusses some paradoxical behaviours arising from the Kijowski's TOA: for a particle that propagates towards an infinite potential barrier the distribution predicts non-vanishing probabilities even in prohibited regions. In this regard, further comments are given in [30,31] while other paradoxes and issues are analysed in [29].…”

“…The TOA probability distribution of Eq. ( 8) can also be derived in the framework of Bohmian mechanics [2,3,29]. In [2] the authors discuss the existence of free particle states with definite momentum sign for which the probabilistic interpretation of the quantum flux is still forbidden.…”

Diffraction and Interference with Run-and-Tumble Particles

“…An implication concerns the possibility for measuring arrival times. Predicting or even defining arrival times for quantum particles are subject of on ongoing discussion [41,42]. Arrival times can however straightforwardly be considered in terms of the trajectories.…”

Questioning the adequacy of certain quantum arrival-time distributions

“…2 Another well-known example applicable for freely moving particles, V .x; t/ D 0, is the Aharonov-Bohm [22,Sec. 3] and Kijowski [23] arrival-time distribution [16,Sec. 2], which is typically indistinguishable from … QF .…”