Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme

TL;DR

This paper develops a combined semiclassical, timeless, and histories approach to the Problem of Time in Quantum Cosmology, extending it to models with nontrivial linear constraints and constructing Dirac observables.

Contribution

It introduces a novel application of Halliwell's approach to models with linear constraints, explicitly resolves Kuchar observables for 1- and 2-dimensional RPMs, and proposes methods for constructing Dirac observables.

Findings

Successfully applies the approach to relational particle models with linear constraints.

Explicitly solves the Kuchar observables problem for 1- and 2-dimensional RPMs.

Proposes indirect methods for constructing Dirac observables in theories with known Kuchar observables.

Abstract

I approach the Problem of Time and other foundations of Quantum Cosmology using a combined histories, timeless and semiclassical approach. This approach is along the lines pursued by Halliwell. It involves the timeless probabilities for dynamical trajectories entering regions of configuration space, which are computed within the semiclassical regime. Moreover, the objects that Halliwell uses in this approach commute with the Hamiltonian constraint, H. This approach has not hitherto been considered for models that also possess nontrivial linear constraints, Lin. This paper carries this out for some concrete relational particle models (RPM's). If there is also commutation with Lin - the Kuchar observables condition - the constructed objects are Dirac observables. Moreover, this paper shows that the problem of Kuchar observables is explicitly resolved for 1- and 2-d RPM's. Then as a first route to Halliwell's approach for nontrivial linear constraints that is also a construction of Dirac observables, I consider theories for which Kuchar observables are formally known, giving the relational triangle as an example. As a second route, I apply an indirect method that generalizes both group-averaging and Barbour's best matching. For conceptual clarity, my study involves the simpler case of Halliwell 2003 sharp-edged window function. I leave the elsewise-improved softened case of Halliwell 2009 for a subsequent Paper II. Finally, I provide comments on Halliwell's approach and how well it fares as regards the various facets of the Problem of Time and as an implementation of QM propositions.