Comment on the transition amplitude for toy universe models

TL;DR

This paper explores the quantization of toy universe models using path integrals, focusing on transition amplitudes and the effects of topology, with potential applications to more complex minisuperspaces.

Contribution

It introduces a method to quantize ideal clock models and applies it to analyze transition amplitudes in empty and more general minisuperspace models.

Findings

Path integral quantization with canonical gauges is feasible for toy universe models.

Topology of the constraint surface influences transition amplitude calculations.

Generalization to models with true degrees of freedom is discussed.

Abstract

The parametrized system called ``ideal clock'' is turned into an ordinary gauge system and quantized by means of a path integral in which canonical gauges are admissible. Then the possibility of applying the results to obtain the transition amplitude for empty minisuperspaces, and the restrictions arising from the topology of the constraint surface, are studied by matching the models with the ideal clock. A generalization to minisuperspaces with true degrees of freedom is also discussed.