The Complete Barrett-Crane Model and its Causal Structure

The causal structure is a quintessential element of continuum spacetime physics and needs to be properly encoded in a theory of Lorentzian quantum gravity. Established spin foam (and tensorial group field theory (TGFT)) models mostly work with relatively special classes of Lorentzian triangulations (e.g. built from spacelike tetrahedra only), obscuring the explicit implementation of the local causal structure at the microscopic level. We overcome this limitation and construct a full-fledged model for Lorentzian quantum geometry the building blocks of which include spacelike, lightlike and timelike tetrahedra. We realize this within the context of the Barrett-Crane TGFT model. Following an explicit characterization of the amplitudes via methods of integral geometry, and the ensuing clear identification of local causal structure, we analyze the model's amplitudes with respect to its (space)time-orientation properties and provide also a more detailed comparison with the framework of causal dynamical triangulations (CDT).