Symmetry groups of state vectors in canonical quantum gravity

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Abstract
In canonical quantum gravity, the diffeomorphisms of an asymptotically flat hypersurface S, not connected to the identity, but trivial at infinity, can act nontrivially on the quantum state space. Because state vectors are invariant under diffeomorphisms that are connected to the identity, the group of inequivalent diffeomorphisms is a symmetry group of states associated with S. This group is the zeroth homotopy group of the group of diffeomorphisms fixing a frame of infinity on S. It is calculated for all hypersurfaces of the form S=S3/G‐point, where the removed point is thought of as infinity on S and the symmetry group S is the zeroth homotopy group of the group of diffeomorphisms of S3/G fixing a point and frame, denoted π0 DiffF(S3/G). Before calculating π0 DiffF (S3/G), it is necessary to find π0 of the group of diffeomorphisms. Once π0 Diff(S3/G) is known, π0 Diffx0(S3/G) is calculated using a fiber bundle involving Diff(S3/G), Diffx0(S3/G), and S3/G. Finally, a fiber bundle involving DiffF(S3/G), Diff(S3/G), and the bundle of frames over S3/G is used along with π0 Diffx0(S3/G) to calculate π0 DiffF(S3/G). The groups π0 DiffF(S3/G) are comprised of SU(2) coverings of SO(3) crystallographic groups, the product of these with a cyclic group, cyclic groups, and the product of two cyclic groups.

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