Time Fisher Information associated with Fluctuations in Quantum Geometry

TL;DR

This paper investigates the limitations of using the Hamiltonian constraint in quantum gravity to analyze time evolution, proposing Fisher information as a tool to identify meaningful operators that reflect passage of time.

Contribution

It introduces Fisher information as a criterion for meaningful quantum gravitational operators related to time, highlighting the limitations of the Hamiltonian constraint.

Findings

Hamiltonian constraint lacks Fisher information about time.

The Hamiltonian operator generates time but does not observe change.

An operator on the string world-sheet contains Fisher information about time.

Abstract

As time is not an observable, we use Fisher information (FI) to address the problem of time. We show that the Hamiltonian constraint operator cannot be used to analyze any quantum process for quantum geometries that are associated with time-reparametrization invariant classical geometries. This is because the Hamiltonian constraint does not contain FI about time. We demonstrate that although the Hamiltonian operator is the generator of time, the Hamiltonian constraint operator can not observe the change that arises through the passage of time. This means that the problem of time is inescapably problematic in the associated quantum gravitational theories. Although we explicitly derive these results on the world-sheet of bosonic strings, we argue that it holds in general. We also identify an operator on the world-sheet which contains FI about time in a string theoretical processes. Motivated by this observation, we propose that a criteria for a meaningful operator of any quantum gravitational process, is that it should contain non-vanishing FI about time.