A Lorentzian Signature Model for Quantum General Relativity

In [1], we proposed a model for quantized discrete general relativity with a Euclidean signature. The model was constructed by combining the structure of a certain tensor category and of a full subcategory of it with the combinatorics of a triangulated 4-manifold. The category was the representations of Uqso(4), for q a 4n-th root of unity and the subcategory the representations which are called balanced or simple. The main technical tool was the introduction of spin networks for these balanced representations, which we called relativistic spin networks. The adjective relativistic is appropriate because the relativistic spin networks are related to four-dimensional geometry whereas the original SU(2) spin networks of Penrose [3] are related to three-dimensional geometry. The name anticipated the development of relativistic spin networks for the physically realistic case of the Lorentz group, SO(3, 1). The purpose of this paper is to supply the analogous concepts to our previous work for the Lorentz group and its q-deformation, UqSL(2, C). The steps of the construction of the model in [1] were as follows: