Beyond Fock space in three dimensional semiclassical gravity

Quantization of relativistic point particles coupled to three-dimensional Einstein gravity naturally leads to field theories living on the Lorentz group in their momentum representation. The Lie group structure of momentum space can be traced back to the classical phase space of the particles coupled to topological gravity. In this work we show how the non-trivial structure of momentum space leads to an unusual description of Fock space. The latter is reflected in a deformed algebra of creation and annihilation operators which reduces to the ordinary algebra when momentum space "flattens" to Minkowski space in the limit in which the three-dimensional Newton's constant vanishes. The construction is covariant under the action of relativistic symmetries acting on the Lorentz group-momentum space. This shows how it is possible to build a Fock space on a group manifold momentum space in a way consistent with the underlying (deformed) relativistic symmetries.