Measuring the metric in (2+1)-dimensional quantum gravity

Gravitational fields are ordinarily measured by observing their effects on test particles, from whose trajectories one can reconstruct the geometry of spacetime. In 2+1 dimensions, spacetime geometry is encoded in the holonomies of a flat ISO(2,1) connection, and the quantum mechanical observables that correspond most closely to particle trajectories are Wilson lines. The author discusses the extraction of geometric information from such Wilson line observables, and shows that a quantum description of geometry can be recovered. In the process, the gravitational Hilbert space is investigated in some detail, and an interesting relationship to Thurston's theory of geometric structures (1979) is discovered.