Aspects of spin and statistics in generally covariant theories

In this paper, we explore the consequences of diffeomorphism invariance in generally covariant theories. Such theories in two and three dimensions are known to admit topological excitations called geons. It is shown by specific examples that a quantum state of two identical geons may not be an eigenstate of the geon exchange operator, which means that a geon may have no definite statistics. As shown before by Sorkin and as discussed further here, it may also happen, for instance, that in 3 + 1 dimensions a tensorial (spinorial) geon obeys Fermi (Bose) statistics, while in 2 + 1 dimensions an "integral-spin" geon can obey "fractional statistics". Thus ideas on spin and statistics borrowed from Poincaré invariant theories are not always valid in quantum gravity, at least without further physical inputs. Previously, it was shown by Friedman and Sorkin that pure gravity in three space dimensions may admit spinorial states. This result is extended to two dimensions where pure gravity is shown to admit "fractional spin" geons