Loop Quantum Cosmology IV: Discrete Time Evolution

TL;DR

This paper presents a framework in loop quantum cosmology where time is discrete, leading to difference equations for evolution, and demonstrates that physical volume spectra match kinematical spectra.

Contribution

It introduces a dynamical interpretation of the Hamiltonian constraint as a discrete evolution equation using a dreibein representation and internal time.

Findings

Time evolution is discrete, leading to difference equations.

Physical volume spectrum equals the kinematical volume spectrum.

Evolving observables enable spectral analysis of geometrical quantities.

Abstract

Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity also time turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed in this framework which enables an investigation of physical spectra of geometrical quantities. In particular, the physical volume spectrum is proven to equal the discrete kinematical volume spectrum in loop quantum cosmology.