Loop quantum gravity coupled to a scalar field

TL;DR

This paper advances loop quantum gravity by constructing a Hilbert space for gravity coupled with a scalar field, defining a new scalar constraint operator, and developing methods to solve it, revealing partial Dirac algebra structure.

Contribution

It introduces a novel Hilbert space and scalar constraint operator for gravity coupled to a scalar field, addressing degrees of freedom lost in deparametrization.

Findings

Constructed a Hilbert space for gravity-matter system.

Defined and analyzed a new scalar constraint operator.

Developed methods for solving the coupled system's constraints.

Abstract

We reconsider the Rovelli-Smolin model of gravity coupled to the Klein-Gordon time field with an eye towards capturing the degrees of freedom of the scalar field lost in the framework in which time is deparametrized by the scalar field. Several new results for loop quantum gravity are obtained: (i) a Hilbert space for the gravity-matter system and a non-standard representation of the scalar field thereon is constructed, (ii) a new operator for the scalar constraint of the coupled system is defined and investigated, (iii) methods for solving the constraint are developed. Commutators of the new constraint do not vanish, but seem to reproduce a part of the Dirac algebra. This, however, poses problems for finding solutions. Hence the states we consider -- and perhaps the whole setup -- still needs some improvement. As a side result we describe a representation of the gravitational degrees of freedom in which the flux is diagonal. This representation bears a strong resemblance to the BF vacuum of Dittrich and Geiller.