Relativistic free motion time of arrival operator for massive spin-0 particles with positive energy

TL;DR

This paper extends the relativistic time of arrival operator for spin-0 particles using rigged Hilbert space, providing new insights, distributions, and consistency with relativity.

Contribution

It develops a rigged Hilbert space extension of Razavi's relativistic time of arrival operator, enabling detailed analysis and distribution construction.

Findings

Time of arrival eigenfunctions exhibit unitary arrival.

Particles can arrive earlier or later than classical predictions.

The distribution and expectation value are consistent with special relativity.

Abstract

A relativistic version of the Aharonov-Bohm time of arrival operator for spin-0 particles was constructed by Razavi in [Il Nuovo Cimento B \textbf{63}, 271 (1969)]. We study the operator in detail by taking its rigged Hilbert space extension. It is shown that the rigged Hilbert space extension of the operator provides more insights into the time of arrival problem that goes beyond Razavi's original results. This allows us to use time of arrival eigenfunctions that exhibit unitary arrival to construct time of arrival distributions. The expectation value is also calculated and shown that particles can arrive earlier or later than expected classically. Lastly, the constructed time of arrival distribution, and expectation value are shown to be consistent with special relativity.