Proof of a Conjecture by Lewandowski and Thiemann

TL;DR

This paper proves a conjecture by Lewandowski and Thiemann, showing that degenerate labelled webs are strongly degenerate for certain groups, which ensures the proper behavior of diffeomorphism invariant operators in quantum gravity models.

Contribution

It establishes the conjecture that degenerate labelled webs are strongly degenerate for compact, connected, semisimple groups, aiding the development of diffeomorphism invariant operators.

Findings

Degenerate labelled webs are strongly degenerate.

Diffeomorphism invariant operators preserve web decomposition.

Supports well-defined operators on diffeomorphism invariant states.

Abstract

It is proven that for compact, connected and semisimple structure groups every degenerate labelled web is strongly degenerate. This conjecture by Lewandowski and Thiemann implies that diffeomorphism invariant operators in the category of piecewise smooth immersive paths preserve the decomposition of the space of integrable functions w.r.t. the degeneracy and symmetry of the underlying labelled webs. This property is necessary for lifting these operators to well-defined operators on the space of diffeomorphism invariant states.