Baby Universes and Worldline Field Theories

TL;DR

This paper explores the structure of quantum gravity path integrals, highlighting how different choices in constructing the Hilbert space lead to an abelian algebra of observables, contrasting with the non-abelian algebra in quantum field theory, using 1D models for clarity.

Contribution

It clarifies how to construct Hilbert spaces from quantum gravity path integrals and distinguishes the algebraic structures from those in quantum field theory, especially in one-dimensional models.

Findings

Quantum gravity path integrals define an abelian algebra of observables.

Different Hilbert space constructions lead to different algebraic structures.

Comparison with worldline QFT reveals key differences in algebraic properties.

Abstract

The quantum gravity path integral involves a sum over topologies that invites comparisons to worldsheet string theory and to Feynman diagrams of quantum field theory. However, the latter are naturally associated with the non-abelian algebra of quantum fields, while the former has been argued to define an abelian algebra of superselected observables associated with partition-function-like quantities at an asymptotic boundary. We resolve this apparent tension by pointing out a variety of discrete choices that must be made in constructing a Hilbert space from such path integrals, and arguing that the natural choices for quantum gravity differ from those used to construct QFTs. We focus on one-dimensional models of quantum gravity in order to make direct comparisons with worldline QFT.