Solving the Problem of Time in Mini-superspace: Measurement of Dirac Observables

TL;DR

The paper tackles the problem of time in reparametrization-invariant quantum systems by focusing on evolving constants of motion $[B]_{c=\tau}$ as relational Dirac observables. It employs refined algebraic quantization with a group-averaging rigging map $\eta_0$ to construct the physical Hilbert space and observables of the form $[B]_{c=\tau}$, using clock-related operators $F_\tau$. A perturbative framework with a clock-detector interaction $\Delta H = F_{\tau_0} B D$ shows that the detector memory $\Delta P_D$ faithfully measures $[B]_{c=\tau_0}$ in the small-$g$ limit, with accuracy enhanced by large clock momentum $P_c$. The results connect to von Neumann-style measurement and offer a concrete operational interpretation for evolving constants in a minisuperspace setting, while outlining paths to higher resolution and extensions to field theory and gravity under further study.

Abstract

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.