Non-linear equation of motion for Page-Wootters mechanism with interaction and quasi-ideal clocks

TL;DR

This paper derives a non-linear equation of motion for a quantum system interacting gravitationally with a quasi-ideal clock within the Page-Wootters framework, incorporating gravitational decoherence effects.

Contribution

It introduces a novel non-linear equation of motion for a mixed state system with gravitational interaction and quasi-ideal clocks in the Page-Wootters mechanism.

Findings

Derived a non-linear equation of motion dependent on initial conditions.

Incorporated gravitational decoherence into the Page-Wootters framework.

Extended the mechanism to interacting, quasi-ideal clock systems.

Abstract

Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, such a mechanism explored a stationary bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge as an equation of motion for a quantum particle conditioned to the measurement of the state of the clock. After the seminal proposal, many variations of it were considered, and different aspects of the mechanism were elucidated. Our contribution to these investigations is that we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the system's initial conditions.