An analysis of the theta sectors of quantum gravity

Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)




Rafael D. Sorkin


unitary irreducible representations, geon, topology

Subject Categories



It is well known that the distinct unitary irreducible representations (UIR's) of the mapping class group G of a 3-manifold ${\cal M}$ give rise to distinct quantum sectors in theories of quantum gravity based on the product manifold $\IR \times {\cal M}.$ In this thesis we study these irreducible representations of G in an attempt to understand the physical implications of these quantum sectors. We see that the mapping class group of a 3-manifold which is the connected sum of $\IR\sp3$ with a finite number of irreducible primes is a semi-direct product group. Following Mackey's theory of induced representations, we provide an analysis of the structure of the general finite dimensional UIR of such a group. In the picture of quantized primes as particles (topological geons), this general group-theoretic analysis enables us to draw several qualitative conclusions about the geons' behavior in different quantum sectors, without requiring an explicit knowledge of the irreducible representations corresponding to the individual primes. An important general result is that the classification of the UIR of the so called particle subgroup is reduced to the problem of finding the irreducible representations of the internal diffeomorphism groups of the individual primes. Moreover, this reduction is entirely consistent with the geon picture, in which the UIR of the internal group of a prime determines the species of the corresponding quantum geon, and the remaining freedom in the overall UIR of G expresses the possibility of choosing an arbitrary statistics (bose, fermi or para) for the geons of each species. For irreducible representations in which the slides are nontrivially represented, we do not provide a complete classification, but find some new types of effects due to the slides, including quantum breaking of internal symmetry and of particle indistinguishability. In connection with the latter, a novel kind of statistics arises which is determined by representations of proper subgroups of the permutation group, rather than of the group as a whole. Finally, we observe that for a generic 3-manifold there will be an infinity of inequivalent UIR's and hence an infinity of "consistent" theories, when topology change is neglected.


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