Symmetries and time operators

TL;DR

This paper derives covariant time operators using symmetry principles, generalizes arrival time distributions for particles, and introduces conditional distributions for particles in potentials, linking quantum operators to classical currents.

Contribution

It provides a unique derivation of covariant time operators and extends the arrival time distribution concept to particles in potentials, connecting quantum and classical descriptions.

Findings

Derived all covariant time operators with normalized distributions.

Generalized the arrival time distribution for free particles.

Introduced conditional arrival-time distributions for particles in potentials.

Abstract

All covariant time operators with normalized probability distribution are derived. Symmetry criteria are invoked to arrive at a unique expression for a given Hamiltonian. As an application, a well known result for the arrival time distribution of a free particle is generalized and extended. Interestingly, the resulting arrival time distribution operator is connected to a particular, positive, quantization of the classical current. For particles in a potential we also introduce and study the notion of conditional arrival-time distribution.