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Groups of flux-like transformations in Loop Quantum Gravity

The flux variables in LQG correspond to infinitesimal transformations in the quantum configuration space, and the one-parameter groups of transformations generated by each flux variable are well understood. However, the set of all such transformations does not form a group. We present a group of transformations that contains the set of transformations generated by the flux variables. This group is labelled by certain SU(2)-valued functions on the bundle of directions in the spatial manifold. A further generalization is obtained by considering functions that depend on germs of analytic curves, rather than just on directions. All these transformations leave the Ashtekar-Lewandowski measure invariant.

Uniqueness of the quantization of a scalar field on $S^1$ with time dependent mass: a generalization of the case of Gowdy comologies

Motivated by the interest in the quantization of symmetry reductions of GR with two commuting spacelike Killing vectors, we consider the quantization of a scalar field on the circle with a time dependent mass. We prove that the representation of the CCR corresponding to the case of zero mass provides a unitary implementation of the dynamics for all (regular) time dependent mass terms. Also, this representation is uniquely specified, among the class determined by $S^1$-invariant complex structures, as the only representation allowing a unitary dynamics. These conclusions can be extended to fields on the two-sphere with axial symmetry. This generalizes uniqueness results previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.

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