On the Measure in Simplicial Gravity

TL;DR

This paper evaluates various functional measures in lattice quantum gravity, confirming the standard simplicial measure's validity and dismissing recent non-local proposals due to their failure in the weak field limit.

Contribution

It demonstrates that the standard simplicial measure aligns with continuum measures in the weak field limit, while recent non-local measures do not, establishing criteria for acceptable discrete measures.

Findings

Standard simplicial measure satisfies continuum limit requirements.

Recent non-local measures fail in the weak field expansion.

Non-local measures are unsuitable as discrete functional measures.

Abstract

Functional measures for lattice quantum gravity should agree with their continuum counterparts in the weak field, low momentum limit. After showing that the standard simplicial measure satisfies the above requirement, we prove that a class of recently proposed non-local measures for lattice gravity do not satisfy such a criterion, already to lowest order in the weak field expansion. We argue therefore that the latter cannot represent acceptable discrete functional measures for simplicial geometries.