Almost Ideal Clocks in Quantum Cosmology: A Brief Derivation of Time

TL;DR

The paper tackles the problem of time in quantum gravity by deriving the external-time quantum-mechanics framework from the almost local QORD formalism for time-reparametrization-invariant systems, using an almost ideal clock to induce clock-time evolution.A detailed adiabatic, semiclassical construction is developed: a clock described by a pendulum coupled weakly to a counter yields a self-adjoint clock Hamiltonian and observables that reproduce the canonical external-time algebra in the ideal-clock limit.Key contributions include an explicit map between the physical Hilbert space ${\cal H}_{phys}$ and the external-time space ${\cal H}_{ext}$, a unitary evolution law in clock time, preservation of the equal-time algebra, and a Klein–Gordon/Newton–Wigner structure emerging in the ideal limit; it also clarifies limitations when clocks measure metric-time near cosmological singularities.The work provides a concrete resolution to how external time can emerge from a time-reparametrization-invariant quantum system, with implications for quantum cosmology and the study of Bianchi-type models, while highlighting that metric-based clocks have breakdowns in highly curved regimes.

Abstract

A formalism for quantizing time reparametrization invariant dynamics is considered and applied to systems which contain an `almost ideal clock.' Previously, this formalism was successfully applied to the Bianchi models and, while it contains no fundamental notion of `time' or `evolution,' the approach does contain a notion of correlations. Using correlations with the almost ideal clock to introduce a notion of time, the work below derives the complete formalism of external time quantum mechanics. The limit of an ideal clock is found to be closely associated with the Klein-Gordon inner product and the Newton-Wigner formalism and, in addition, this limit is shown to fail for a clock that measures metric-defined proper time near a singularity in Bianchi models.