Real Tunneling Solutions and the Hartle-Hawking Wave Function

TL;DR

This paper investigates real tunneling solutions as instantons in the Hartle-Hawking path integral, showing they correspond to extrema of the wave function and may be maxima under certain conditions, advancing understanding of quantum cosmology.

Contribution

It introduces the concept of real tunneling solutions as instantons with fixed boundary momentum and analyzes their role as extrema of the Hartle-Hawking wave function.

Findings

Real tunneling solutions are instantons with fixed boundary momentum.

These solutions correspond to extrema of the Hartle-Hawking wave function.

They may be maxima of the wave function at a fixed time.

Abstract

A real tunneling solution is an instanton for the Hartle-Hawking path integral with vanishing extrinsic curvature (vanishing ``momentum'') at the boundary. Since the final momentum is fixed, its conjugate cannot be specified freely; consequently, such an instanton will contribute to the wave function at only one or a few isolated spatial geometries. I show that these geometries are the extrema of the Hartle-Hawking wave function in the semiclassical approximation, and provide some evidence that with a suitable choice of time parameter, these extrema are the maxima of the wave function at a fixed time.