Towards a Hartle-Hawking state for loop quantum gravity

TL;DR

This paper explores the construction of a Hartle-Hawking-like state within loop quantum gravity, identifying challenges in satisfying the Hamiltonian constraint and proposing modifications that align with known quantum states.

Contribution

It demonstrates the difficulties in formulating a Hartle-Hawking state in loop quantum gravity using Ashtekar-Barbero variables and proposes a modified approach that satisfies the Hamiltonian constraint.

Findings

The constructed wave function does not satisfy the Lorentzian Hamiltonian constraint.

A modified proposal can satisfy the constraint at least formally.

The resulting state resembles the Ashtekar-Lewandowski state in certain conditions.

Abstract

The Hartle-Hawking state is a proposal for a preferred initial state for quantum gravity, based on a path integral over all compact Euclidean four-geometries which have a given three-geometry as a boundary. The wave function constructed this way satisfies the (Lorentzian) Hamiltonian constraint of general relativity in ADM variables in a formal sense. In this article we mimic this procedure of constructing an initial state in terms of Ashtekar-Barbero variables, and observe that the wave function thus constructed does not satisfy the Lorentzian Hamiltonian constraint even in a formal sense. We also investigate this issue for the relativistic particle. We finally suggest a modification of the proposal that does satisfy the constraint at least in a formal sense and start to consider its implications in quantum cosmology. We find that for certain variables, and in the saddle point approximation, the state is very similar to the Ashtekar-Lewandowski state of loop quantum gravity.