Time Operators and Time Crystals

TL;DR

This paper explores the mathematical structure of time operators in quantum time crystals, revealing their connection to topology, PT-symmetry, and uncertainty relations in ring systems.

Contribution

It introduces a class of self-adjoint time operators for quantum time crystals using the generalized weak Weyl relation, linking them to topology and PT-symmetry.

Findings

Derived self-adjoint time operators for ring systems.

Connected time operators to topology and PT-symmetry.

Established energy-time uncertainty relations.

Abstract

We investigate time operators in the context of quantum time crystals in ring systems. A generalized commutation relation called the generalized weak Weyl relation is used to derive a class of self-adjoint time operators for ring systems with a periodic time evolution: The conventional Aharonov-Bohm time operator is obtained by taking the infinite-radius limit. Then, we discuss the connection between time operators, time crystals and real-space topology. We also reveal the relationship between our time operators and a $\mathcal{PT}$-symmetric time operator. These time operators are then used to derive several energy-time uncertainty relations.