Implementing causality in the spin foam quantum geometry

TL;DR

This paper develops a causal spin foam model for four-dimensional quantum gravity, integrating causality into the Barrett-Crane framework and connecting it with various quantum gravity approaches.

Contribution

It introduces a covariant causal spin foam model that incorporates causality and links multiple quantum gravity frameworks, including loop quantum gravity and causal sets.

Findings

Constructed causal transition amplitudes as Feynman propagators.

Reformulated the model within quantum causal sets.

Linked spin foam models with Lorentzian Regge calculus.

Abstract

We analyse the classical and quantum geometry of the Barrett-Crane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transiton amplitudes between quantum gravity states, i.e. realising in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories).