The Koslowski–Sahlmann representation: quantum configuration space

Abstract

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