Denseness of Ashtekar-Lewandowski states and a generalized cut-off in loop quantum gravity

TL;DR

This paper demonstrates the density of Ashtekar-Lewandowski states in the space of all states and analyzes a modified cut-off procedure in loop quantum gravity, focusing on vector states.

Contribution

It establishes the denseness of Ashtekar-Lewandowski states and introduces a new cut-off approach tailored for vector states in loop quantum gravity.

Findings

Ashtekar-Lewandowski states are dense in the state space.

A modified cut-off procedure for vector states is proposed.

The weak convergence properties of the new cut-off are analyzed.

Abstract

We show that the set of states of the Ashtekar-Isham-Lewandowski holonomy algebra defined by elements of the Ashtekar-Lewandowski Hilbert space is dense in the space of all states. We consider weak convergence properties of a modified version of the cut-off procedure currently in use in loop quantum gravity. This version is adapted to vector states rather than to general distributions.