Comment on "An Arrow of Time Operator for Standard Quantum Mechanics" - a sign of the time!

TL;DR

This paper clarifies the relationship between the recently proposed arrow of time operator and the canonical time observable, providing a physical interpretation and discussing potential generalizations.

Contribution

It demonstrates that the arrow of time operator is directly related to the sign of the canonical time observable, offering new insights into its physical meaning.

Findings

< M_F > decreases monotonically over time

< sgn T > increases monotonically with time

Provides a physical interpretation of M_F < 1

Abstract

It is shown that the `arrow of time' operator, M_F, recently suggested by Strauss et al., in arXiv:0802.2448v1 [quant-ph], is simply related to the sign of the canonical `time' observable, T (apparently first introduced by Holevo). In particular, the monotonic decrease of < M_F > corresponds to the fact that < sgn T > increases monotonically with time. This relationship also provides a physical interpretation of the property M_F < 1. Some further properties and possible generalisations are pointed out, including to almost-periodic systems.