Solving the problem of time in mini-superspace: Measurement of Dirac observables

One solution to the so-called problem of time is to construct certain Dirac observables, sometimes called evolving constants of motion. There has been some discussion in the literature about the interpretation of such observables, and, in particular, whether single Dirac observables can be measured. Here we clarify the situation by describing a class of interactions that can be said to implement measurements of such observables. Along the way, we describe a useful notion of perturbation theory for the rigging map $\eta$ of group averaging (sometimes loosely called the physical state “projector”), which maps states from the auxiliary Hilbert space to the physical Hilbert space.