A Relativistic One-Particle Time of Arrival Operator for a Free Spin-1/2 Particle in (1 + 1) Dimensions

TL;DR

This paper develops a relativistic Time of Arrival operator for a free spin-1/2 particle in one dimension, extending previous spin-0 models and analyzing its properties and symmetry considerations.

Contribution

It constructs a TOA operator for a relativistic spin-1/2 particle, including analysis of its components and symmetry properties, advancing the quantum measurement framework for relativistic particles.

Findings

The TOA operator includes a term similar to the spin-0 case and an additional parity-breaking term.

Imposing parity symmetry leads to discarding the parity-breaking component.

The resulting TOA operator is consistent with relativistic quantum mechanics for spin-1/2 particles.

Abstract

As a follow-up to a recent study in the spin-0 case [J. Bunao and E. A. Galapon, Ann. Phys. 353, 83-106 (2015)], we construct a one-particle Time of Arrival (TOA) operator conjugate to a Hamiltonian describing a free relativistic spin-1/2 particle in one spatial dimension. Upon transformation in a representation where the Hamiltonian is diagonal, it turns out that the constructed operator consists of an operator term $\mathcal{\hat{T}}$ whose action is the same as in the spin-0 case, and another operator term $\mathcal{\hat{T}}_0$ which commutes with the Hamiltonian but breaks invariance under parity inversion. If we must impose this symmetry on our TOA operator, then we can throw away $\mathcal{\hat{T}}_0$ so that the TOA operator is just $\mathcal{\hat{T}}$.