Feynman Rules for Simplicial Gravity

TL;DR

This paper develops Feynman rules for lattice quantum gravity, establishing a link between continuum and lattice formulations, and applies these rules to compute the conformal anomaly in two dimensions.

Contribution

It introduces a formalism for perturbative expansions in simplicial lattice gravity and details Feynman rules for 2D Regge calculus, aiding non-perturbative studies.

Findings

Established a correspondence between continuum and lattice gravity quantities.

Derived explicit Feynman rules for 2D simplicial gravity.

Computed the conformal anomaly for scalar fields in 2D perturbation theory.

Abstract

We develop the general formalism for performing perturbative diagrammatic expansions in the lattice theory of quantum gravity. The results help establish a precise correspondence between continuum and lattice quantities, and should be a useful guide for non-perturbative studies of gravity. The Feynman rules for Regge's simplicial lattice formulation of gravity are then discussed in detail in two dimensions. As an application, the two-dimensional conformal anomaly due to a $D$-component scalar field is explicitly computed in perturbation theory.