Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time

TL;DR

This paper develops a quantum Bohmian mechanics framework on shape space, showing how it relates to traditional absolute space theories and how different gauge choices influence the dynamics and probability interpretations.

Contribution

It introduces a quantum motion formulation on shape space, connecting relational and absolute descriptions, and analyzes gauge-dependent emergence of dynamics and probability.

Findings

Quantum motion on shape space can be defined using its metric structure.

Lifting free motion on shape space to configuration space results in an interacting theory.

Different gauge choices correspond to different lifts, affecting the dynamics.

Abstract

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows one to straightforwardly define a quantum motion, a Bohmian mechanics, on shape space. We show how this motion gives rise to the more or less familiar theory in absolute space and time. We find that free motion on shape space, when lifted to configuration space, becomes an interacting theory. Many different lifts are possible corresponding in fact to different choices of gauges. Taking the laws of Bohmian mechanics on shape space as physically fundamental, we show how the theory can be statistically analyzed by using conditional wave functions, for subsystems of the universe, represented in terms of absolute space and time.