The Feynman propagator for spin foam quantum gravity

TL;DR

This paper establishes a connection between causality and orientation in spin foam quantum gravity, introduces a proper time variable to differentiate models, and constructs a new class of causal spin foam models analogous to the Feynman propagator.

Contribution

It introduces a proper time framework into spin foam models, enabling the distinction between causal and non-causal models, and constructs a new causal class of models in quantum gravity.

Findings

Current spin foam models are orientation-independent.

Different integration ranges of proper time yield distinct model classes.

A new causal spin foam model analogous to the Feynman propagator is proposed.

Abstract

We link the notion causality with the orientation of the 2-complex on which spin foam models are based. We show that all current spin foam models are orientation-independent, pointing out the mathematical structure behind this independence. Using the technology of evolution kernels for quantum fields/particles on Lie groups/homogeneous spaces, we construct a generalised version of spin foam models, introducing an extra proper time variable and prove that different ranges of integration for this variable lead to different classes of spin foam models: the usual ones, interpreted as the quantum gravity analogue of the Hadamard function of QFT or as a covariant definition of the inner product between quantum gravity states; and a new class of causal models, corresponding to the quantum gravity analogue of the Feynman propagator in QFT, non-trivial function of the orientation data, and implying a notion of ''timeless ordering''.