A quantum kinematics for asymptotically flat spacetimes

TL;DR

This paper develops a quantum kinematic framework for asymptotically flat spacetimes using the KS representation, extending Loop Quantum Gravity to include background electric fields and analyzing asymptotic behaviors and symmetries.

Contribution

It introduces a generalized quantum kinematic construction incorporating background electric fields, enabling analysis of asymptotic conditions and symmetries in quantum gravity.

Findings

Supports a unitary action of asymptotic rotation and translation groups.

Realizes KS states as wave functions on a generalized connection space.

Contains states exhibiting fermionic behavior under 2π rotations.

Abstract

We construct a quantum kinematics for asymptotically flat spacetimes based on the Koslowski-Sahlmann (KS) representation. The KS representation is a generalization of the representation underlying Loop Quantum Gravity (LQG) which supports, in addition to the usual LQG operators, the action of `background exponential operators' which are connection dependent operators labelled by `background' $su(2)$ electric fields. KS states have, in addition to the LQG state label corresponding to 1 dimensional excitations of the triad, a label corresponding to a `background' electric field which describes 3 dimensional excitations of the triad. Asymptotic behaviour in quantum theory is controlled through asymptotic conditions on the background electric fields which label the {\em states} and the background electric fields which label the {\em operators}. Asymptotic conditions on the triad are imposed as conditions on the background electric field state label while confining the LQG spin net graph labels to compact sets. We show that KS states can be realised as wave functions on a quantum configuration space of generalized connections and that the asymptotic behaviour of each such generalized connection is determined by that of the background electric fields which label the background exponential operators. Similar to the spatially compact case, the Gauss Law and diffeomorphism constraints are then imposed through Group Averaging techniques to obtain a large sector of gauge invariant states. It is shown that this sector supports a unitary action of the group of asymptotic rotations and translations and that, as anticipated by Friedman and Sorkin, for appropriate spatial topology, this sector contains states which display fermionic behaviour under $2\pi$ rotations.