Quantum constraints, Dirac observables and evolution: group averaging versus Schroedinger picture in LQC

TL;DR

This paper investigates quantum constraints in Loop Quantum Cosmology, comparing group averaging and Schrödinger picture approaches, revealing how different spectral assumptions lead to distinct quantum dynamics and observable structures.

Contribution

It introduces a framework for analyzing quantum constraints with both discrete and continuous spectra, connecting group averaging with Schrödinger-like quantum theories in LQC.

Findings

Discrete spectrum case yields Schrödinger-like quantum mechanics.

Continuous spectrum case results in a family of quantum theories.

Relational observables can mix different members of the family.

Abstract

A general quantum constraint of the form $C= - \partial_T^2 \otimes B - I\otimes H$ (realized in particular in Loop Quantum Cosmology models) is studied. Group Averaging is applied to define the Hilbert space of solutions and the relational Dirac observables. Two cases are considered. In the first case, the spectrum of the operator $(1/2)\pi^2 B - H$ is assumed to be discrete. The quantum theory defined by the constraint takes the form of a Schroedinger-like quantum mechanics with a generalized Hamiltonian $\sqrt{B^{-1} H}$. In the second case, the spectrum is absolutely continuous and some peculiar asymptotic properties of the eigenfunctions are assumed. The resulting Hilbert space and the dynamics are characterized by a continuous family of the Schroedinger-like quantum theories. However, the relational observables mix different members of the family. Our assumptions are motivated by new Loop Quantum Cosmology models of quantum FRW spacetime. The two cases considered in the paper correspond to the negative and, respectively, positive cosmological constant. Our results should be also applicable in many other general relativistic contexts.