Trajectories for the Wave Function of the Universe from a Simple Detector Model

TL;DR

This paper develops a simple quantum cosmology model with detectors to derive probabilities for trajectories of the universe's wave function, showing they align with classical paths and are minimally disturbed by measurement.

Contribution

It introduces a detector-based framework for quantum cosmology that yields well-defined trajectory probabilities consistent with classical behavior.

Findings

Probabilities are peaked around classical trajectories.

Wave functions satisfy Wheeler-DeWitt with small measurement corrections.

Results align with Hartle's earlier findings for modified boundary conditions.

Abstract

Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we consider a simple model for quantum cosmology which includes, in the total Hamiltonian, model detectors registering whether or not the system, at any stage in its entire history, passes through a series of regions in configuration space. We thus derive a variety of well-defined formulas for the probabilities for trajectories associated with the solutions to the Wheeler-DeWitt equation. The probability distribution is peaked about classical trajectories in configuration space. The ``measured'' wave functions still satisfy the Wheeler-DeWitt equation, except for small corrections due to the disturbance of the measuring device. With modified boundary conditions, the measurement amplitudes essentially agree with an earlier result of Hartle derived on rather different grounds. In the special case where the system is a collection of harmonic oscillators, the interpretation of the results is aided by the introduction of ``timeless'' coherent states -- eigenstates of the Hamiltonian which are concentrated about entire classical trajectories.