The Leggett-Garg inequality and Page-Wootters mechanism

TL;DR

This paper demonstrates that the Page-Wootters mechanism operates within the quantum domain by violating the Leggett-Garg inequality, and explores implications for distinguishing graviton types via conditional probabilities.

Contribution

It establishes the quantum nature of the Page-Wootters mechanism through LGI violation and analyzes conditional probabilities for entangled photons under various measurements.

Findings

LGI violation confirms quantum behavior in the Page-Wootters setup

Conditional probabilities increase with entanglement for both ideal and unsharp measurements

Potential application in distinguishing massless and massive gravitons

Abstract

Violation of the Leggett-Garg inequality (LGI) implies quantum phenomena. In this light we establish that the Moreva \textit{et al.} \cite{moreva} experiment demonstrating the Page-Wootter's mechanism \cite{wootters} falls in the quantum domain. An observer outside a 2-photons world does not detect any change in the $2-$photons state,i.e. there is no time parameter for the outside observer. But an observer attached to one of the photons sees the other photon evolving and this means there is an "internal" time. The LGI is violated for the clock photon whose state evolves with the internal time as measured by the system photon. Conditional probabilities in this 2-photons system are computed for both sharp and unsharp measurements. The conditional probability increases for entangled states as obtained by Page and Wootters for both ideal and also unsharp measurements. We discuss how the conditional probabilities can be used to distinguish between massless and massive gravitons. This is important in the context of gravitational waves.