The Complete Barrett-Crane Model and its Causal Structure

TL;DR

This paper develops a comprehensive Lorentzian quantum gravity model incorporating all types of tetrahedra, explicitly encoding causal structure at the microscopic level within the Barrett-Crane tensorial group field theory framework.

Contribution

It constructs the first full Lorentzian Barrett-Crane model including spacelike, lightlike, and timelike tetrahedra, explicitly encoding local causal structure.

Findings

Explicit characterization of amplitudes via integral geometry.

Analysis of space-time orientation properties.

Comparison with causal dynamical triangulations (CDT).

Abstract

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).