Quantum Observables and Recollapsing Dynamics

TL;DR

This work develops a method to construct gauge-invariant quantum observables for time-reparametrization invariant systems by integrating densities over the time-label manifold, yielding a complete, densely defined set of operators on a physical Hilbert space with a well-defined inner product. The central construction, $[A]_{Q=\tau}$, is shown to commute with the Hamiltonian constraint under appropriate convergence, and its Hermitian properties are analyzed, with explicit treatments for already deparametrized systems and the relativistic free particle. Applying the framework to separable semi-bound minisuperspace models (notably LRS Bianchi IX and Kantowski–Sachs) demonstrates convergence of the observables and reproduces classical recollapse behavior in the quantum theory, via strong decay of observables as $\tau\to\infty$. The results provide a robust route to define observables and inner products in finite-dimensional quantum gravity-like systems and point to potential extensions to full quantum gravity, while highlighting technical challenges such as convergence criteria and spectral properties.

Abstract

Within a simple quantization scheme, observables for a large class of finite dimensional time reparametrization invariant systems may be constructed by integration over the manifold of time labels. This procedure is shown to produce a complete set of densely defined operators on a physical Hilbert space for which an inner product is identified and to provide reasonable results for simple test cases. Furthermore, many of these observables have a clear interpretation in the classical limit and we use this to demonstrate that, for a class of minisuperspace models including LRS Bianchi IX and the Kantowski-Sachs model this quantization agrees with classical physics in predicting that such spacetimes recollapse.