It is of great physical interest to construct a canonical quantization of asymptotically flat spacetimes. The classical phase space variables are subject to delicate boundary conditions at spatial infinity and the first challenge is to construct a quantum kinematics which carries an imprint of these boundary conditions.
This work is one of a series of papers which seeks to construct such a quantum kinematics based on a recent generalization of Loop Quantum Gravity (LQG). While LQG exhibits a microscopically discrete (and degenerate) spatial geometry, this generalization by Koslowski and Sahlmann (KS) allows for non-degenerate quantum spatial metrics. Since asymptotically flat spatial triads approach a non-degenerate flat metric at infinity, the KS representation is ideally suited to address the spatial metric boundary condition.
In order to probe the asymptotic behaviour of the conjugate phase space variable, our strategy is to (a) realise the KS representation as one in which operators dependent on the conjugate variable act by multiplication on square integrable functions on a suitable `quantum configuration space’ so that elements of this space can be thought of as quantum analogs of the classical conjugate variable (b) understand the asymptotic behaviour of the elements of this quantum configuration space to see if they carry an imprint of the classical boundary conditions. In this work we carry out step (a) in the spatially compact case as a prelude to a similar construction for the
asymptotically flat spatially non-compact case .
With the standard cautionary caveats applicable to unpublished work, our results in 
demonstrate the success of our strategy. While the quantum spatial geometry satisfies the exact classical asymptotic conditions, elements of the quantum configuration space satisfy a slightly weakened version of the classical asymptotic conditions.
The resulting kinematics supports a unitary representation of spatial diffeomorphisms
generated by the diffeomorphism constraint.
This allows us to construct spatially diffeomorphism invariant states which support a unitary representation of the transformations generated by the total spatial and angular momenta at infinity thus exhibiting the desired features of a satisfactory kinematics. As in the spatially compact space the quantum dynamics remains a key open issue which we hope to tackle using the kinematics we have developed.
 M. Campiglia and M. Varadarajan “The Koslowski-Sahlmann representation: Asymptotically flat kinematics”, in progress.
Read the full article in Classical and Quantum Gravity:
The Koslowski–Sahlmann representation: quantum configuration space
Miguel Campiglia and Madhavan Varadarajan
2014 Class. Quantum Grav. 31 175009
ArXiv preprint: 1406.0579
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