A symmetric scalar constraint for loop quantum gravity

TL;DR

This paper introduces a new Hilbert space in loop quantum gravity that simplifies the analysis of scalar constraints, enabling straightforward investigation of their properties and solutions.

Contribution

It defines a novel Hilbert space of states solving many diffeomorphism constraints and constructs a family of scalar constraints within this space, advancing the understanding of quantum gravitational dynamics.

Findings

A new Hilbert space of solutions is constructed.

A family of scalar constraints preserving the Hilbert space is obtained.

Spectral decomposition is used to define the space of solutions.

Abstract

In the framework of loop quantum gravity, we define a new Hilbert space of states which are solutions of a large number of components of the diffeomorphism constraint. On this Hilbert space, using the methods of Thiemann, we obtain a family of gravitational scalar constraints. They preserve the Hilbert space for every choice of lapse function. Thus adjointness and commutator properties of the constraint can be investigated in a straightforward manner. We show how the space of solutions of the symmetrized constraint can be defined by spectral decomposition, and the Hilbert space of physical states by subsequently fully implementing the diffeomorphism constraint. The relationship of the solutions to those resulting from a proposal for a symmetric constraint operator by Thiemann remains to be elucidated.