The Feynman propagator for quantum gravity: spin foams, proper time, orientation, causality and timeless-ordering

TL;DR

This paper explores causality in quantum gravity using spin foam models, introducing proper time as a variable to distinguish between causal and acausal transition amplitudes, and relating these to Feynman propagators.

Contribution

It introduces a generalized spin foam model with proper time, differentiating causal and acausal models, and relates them to the Feynman propagator in quantum field theory.

Findings

Different proper time integration ranges yield causal and acausal models.

Causal models incorporate a notion of timeless ordering.

Current models are a-causal transition amplitudes.

Abstract

We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the spin foam models currently studied, that are independent of the underlying orientation/causal structure and are therefore interpreted as a-causal transition amplitudes; the second corresponds to a general definition of causal or orientation dependent spin foam models, interpreted as causal transition amplitudes or as the Quantum Gravity analogue of the Feynman propagator of field theory, implying a notion of ''timeless ordering''.