Homogeneous-isotropic sector of loop quantum gravity: new approach

TL;DR

This paper introduces a new approach to loop quantum gravity by defining a homogeneous-isotropic subspace within the vertex Hilbert space, enabling reduction and potential numerical analysis of scalar constraint operators.

Contribution

It proposes a homogeneous-isotropic subspace invariant under new scalar constraint operators, facilitating their reduction and analysis in loop quantum gravity.

Findings

Invariant subspace under scalar constraint operators identified

Reduced operators allow for potential numerical diagonalization

Eigenstates of reduced operators correspond to full operator eigenstates

Abstract

Recently, a new class of scalar constraint operators has been introduced in loop quantum gravity. They are defined on a space of solutions to the Gauss constraint and partial solutions to the vector constraint, called a vertex Hilbert space. We propose a subspace of the vertex Hilbert space formed by homogeneous-isotropic states, which is invariant under the action of the new scalar constraint operators. As a result, the operators can be reduced to our homogeneous-isotropic subspace. The (generalized) eigenstates of the reduced operator are eigenstates of the full operator. We discuss the feasibility of numerical diagonalization of the reduced scalar constraint operator.