Manufacturing time operators: covariance, selection criteria, and examples

TL;DR

This paper develops the most general covariant and normalized quantum time operators, explores their physical interpretations, and provides criteria for selecting unique, optimal operators in various quantum contexts.

Contribution

It introduces the most general forms of quantum time operators, discusses their physical meanings, and establishes criteria for choosing unique, optimal operators.

Findings

Multiple possible quantum time operators are discussed.

Criteria for defining unique, optimal time operators are provided.

Applications include quantum clocks and arrival time measurements.

Abstract

We provide the most general forms of covariant and normalized time operators and their probability densities, with applications to quantum clocks, the time of arrival, and Lyapunov quantum operators. Examples are discussed of the profusion of possible operators and their physical meaning. Criteria to define unique, optimal operators for specific cases are given.