General covariance, loops and matter

TL;DR

This paper introduces a four-dimensional covariant field theory coupling non-dynamical geometries with scalar fields, revealing an infinite set of observables constructed from matter loops, with implications for quantum gravity quantization.

Contribution

It presents a novel covariant theory with loop-based observables that mirror the algebra of 3+1 gravity loop variables, extending loop quantum gravity concepts to matter loops.

Findings

The theory has an infinite number of conserved quantities from scalar loops.

The Poisson algebra of observables matches that of Ashtekar's 3+1 gravity variables.

Solutions to Hamilton-Jacobi and Dirac quantization are expressed via matter holonomies.

Abstract

A four dimensional generally covariant field theory is presented which describes non-dynamical three geometries coupled to scalar fields. The theory has an infinite number of physical observables (or constants of the motion) which are constructed from loops made from scalar field configurations. The Poisson algebra of these observables is closed and is the same as that for the 3+1 gravity loop variables in the Ashtekar formalism. The theory also has observables that give the areas of open surfaces and the volumes of finite regions. Solutions to all the Hamilton-Jacobi equations for the theory and the Dirac quantization conditions in the coordinate representation are given. These solutions are holonomies based on matter loops. A brief discussion of the loop space representation for the quantum theory is also given together with some implications for the quantization of 3+1 gravity.