The Koslowski-Sahlmann representation: Quantum Configuration Space

TL;DR

The paper explores the properties of the Koslowski-Sahlmann (KS) representation in Loop Quantum Gravity, showing it supports similar algebraic structures and topological properties as the standard LQG representation, but with additional background exponential functions.

Contribution

It demonstrates that the KS representation's algebraic and topological features mirror those of LQG, extending the framework to include smooth geometries and background exponentials.

Findings

The KS algebra can be completed to a $C^*$ algebra.

The space of semianalytic $SU(2)$ connections is dense in the spectrum.

A measure exists making the KS Hilbert space isomorphic to square-integrable functions.

Abstract

The Koslowski-Sahlmann (KS) representation is a generalization of the representation underlying the discrete spatial geometry of Loop Quantum Gravity (LQG), to accommodate states labelled by smooth spatial geometries. As shown recently, the KS representation supports, in addition to the action of the holonomy and flux operators, the action of operators which are the quantum counterparts of certain connection dependent functions known as "background exponentials". Here we show that the KS representation displays the following properties which are the exact counterparts of LQG ones: (i) the abelian $*$ algebra of $SU(2)$ holonomies and `$U(1)$' background exponentials can be completed to a $C^*$ algebra (ii) the space of semianalytic $SU(2)$ connections is topologically dense in the spectrum of this algebra (iii) there exists a measure on this spectrum for which the KS Hilbert space is realised as the space of square integrable functions on the spectrum (iv) the spectrum admits a characterization as a projective limit of finite numbers of copies of $SU(2)$ and $U(1)$ (v) the algebra underlying the KS representation is constructed from cylindrical functions and their derivations in exactly the same way as the LQG (holonomy-flux) algebra except that the KS cylindrical functions depend on the holonomies and the background exponentials, this extra dependence being responsible for the differences between the KS and LQG algebras. While these results are obtained for compact spaces, they are expected to be of use for the construction of the KS representation in the asymptotically flat case.