Foundational Structure of Local Amplitudes in Quantum Gravity

TL;DR

This paper develops a theory-independent framework for local amplitudes in quantum gravity using light-like hypersurfaces, emphasizing causal diamonds and symmetry principles to understand local quantum gravitational dynamics.

Contribution

It introduces a top-down construction of local amplitudes in causal diamonds based on null slabs, Hilbert space factorization, and symmetry considerations, applicable across various quantum gravity approaches.

Findings

Local transition amplitudes satisfy Ward identities.

The framework ensures charge conservation for symmetries.

A general method to construct local quantum gravity amplitudes is proposed.

Abstract

There has been recently renewed interest in the quantisation of gravity by considering local subsystems on light-like hypersurfaces. The main purpose of this paper is to present a theory-independent perspective on these developments assuming only basic knowledge of quantum theory and general relativity. In addition, we present a top-down approach to constructing local amplitudes in causal diamonds. The fundamental building block is a slab of light-like geometry (e.g. a segment of a light cone embedded into spacetime). Each slab is a three-dimensional light-like hypersurface bounded by two cuts, its past and future corners. After briefly reviewing the timeless formalism of quantum theory, we equip each null slab a with a kinematical Hilbert space that factorizes into constituents associated to the three-dimensional interior and its two corners. Upon assuming the existence of vacuum states for the bulk and boundary symmetries and a fundamental projector onto physical states, we explain how to introduce local amplitudes by contracting boundary states according to the pattern of a causal diamond. Finally, we show that the resulting local transition amplitudes satisfy Ward identities and charge conservation for the underlying symmetries. The paper closes with a summary and discussion for how different approaches to quantum gravity can realise our proposal in practice.