Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time
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.
KeywordsShape space dynamics Quantum mechanics Bohmian mechanics Typicality analysis on shape space
We are grateful to Florian Hoffmann for his input to a very early draft of this paper and to Antonio Vassallo for his insights. We thank Sahand Tokasi for stimulating discussions. We thank Eddy Chen and Roderich Tumulka for a careful reading of the manuscript and useful suggestions. The many discussions with Julian Barbour are gratefully acknowledged, especially for sharing with us in his well known enthusiastic way his ideas on shape dynamics. N. Zanghí was supported in part by INFN.
- 1.Anderson, E.: The problem of time and quantum cosmology in the relational particle mechanics arena (2011). arXiv:1111.1472
- 19.Marques, F.C.: Scalar curvature, conformal geometry, and the Ricci flow with surgery. In: Proceedings of the International Congress of Mathematicians 2010, pp. 811–829. World Scientific, Singapore (2010)Google Scholar
- 21.Small, C.G.: The Statistical Theory of Shape. Springer, New York (2012)Google Scholar