Flow of time during energy measurements and the resulting time-energy uncertainty relations

TL;DR

This paper explores how energy measurements affect the flow of time in quantum mechanics, revealing fundamental limits on measurement duration and non-unitary evolution during measurements within the Page and Wootters framework.

Contribution

It introduces a unified approach to time-energy uncertainty relations using the Page and Wootters framework, deriving new bounds on measurement duration and internal clock evolution.

Findings

Energy measurement duration cannot be arbitrarily short from the internal clock perspective.

During energy measurements, the internal clock evolution becomes non-unitary.

The framework recovers and extends previous results on time-energy uncertainty relations.

Abstract

Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time" between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.