Continuously distributed holonomy-flux algebra

TL;DR

This paper develops a method for constructing a continuously distributed holonomy-flux algebra along smooth links, improving the regularization of gravitational constraints in loop quantum gravity using a refined connection representation.

Contribution

It introduces a new approach to approximate the holonomy-flux algebra continuously along links, enabling more precise gravitational Hamiltonian constraint regularization in loop quantum gravity.

Findings

The continuous distribution approximates the algebra arbitrarily well.

The improved connection representation eliminates correction terms.

Poisson brackets match those of smeared variables at nodes.

Abstract

The procedure of the holonomy-flux algebra construction along a piecewise linear path, which consists of a countably infinite number of pieces, is described in this article. The related construction approximates the continuous distribution of the holonomy-flux algebra location along a smooth link arbitrarily well. The presented method requires the densitized dreibein flux and the corresponding operator redefinition. The derived result allows to formulate the gravitational Hamiltonian constraint regularization by applying the Thiemann technique adjusted to a piecewise linear lattice. By using the improved Ashtekar connection holonomy representation, which is more accurate than the one used in canonical loop quantum gravity, the corrections related to the redefined densitized dreibein flux vanish. In this latter case, the Poisson brackets of the continuously distributed holonomy-flux algebra along a link between a pair of nodes are equal to the brackets for these smeared variables located at the nodes.