The generator of spatial diffeomorphisms in the Koslowski- Sahlmann representation

Understand

A generalization of the representation underlying the discrete spatial geometry of Loop Quantum Gravity, to accomodate states labelled by smooth spatial geometries, was discovered by Koslowski and further studied by Sahlmann.

• We show how to construct the diffeomorphism constraint operator in this Koslowski- Sahlmann (KS) representation from suitable connection and triad dependent operators.
• We show that the KS representation supports the action of hitherto unnoticed "background exponential" operators which, in contrast to the holonomy- flux operators, change the smooth spatial geometry label of the states.
• These operators are shown to be quantizations of certain connection dependent functions and play a key role in the construction of the diffeomorphism constraint operator.