Relational observables in 2D quantum gravity

Local observation is an important problem both for the foundations of a quantum theory of gravity and for applications to quantum-cosmological problems such as eternal inflation. While gauge-invariant local observables cannot be defined, it has been argued that appropriate relational observables approximately reduce to local observables in certain states. However, quantum mechanics and gravity together imply limitations on the precision of their localization. Such a relational framework is studied in the context of two-dimensional gravity, where there is a high degree of analytic control. This example furnishes a concrete example of some of the essential features of relational observables.

Figure 1

As $p^0$, $R$ are increased with ${\ell }_s$ held fixed, a string-scale area on the world sheet becomes smaller relative to the world-sheet area.