Shape space in terms of Wasserstein geometry and application to quantum physics

by Bernadette Lessel
Doctoral thesis
Date of Examination:2018-06-28
Date of issue:2018-11-26
Advisor:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Thomas Schick
Referee:Prof. Dr. Dorothea Bahns
Referee:Prof. Dr. Anja Sturm
Referee:Prof. Dr. Ingo Witt
Referee:Prof. Dr. Madeleine Jotz Lean
Referee:Prof. Dr. Karl-Henning Rehren
Persistent Address: http://dx.doi.org/10.53846/goediss-7084

 

 

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Abstract

English

This thesis offers a mathematical framework to treat quantum dynamics without reference to a background structure, but rather by means of the change of the shape of the state. For this, Wasserstein geometry is used. The so called Shape space, then, is defined as the quotient space of a Wasserstein space modulo the action of the isometry group of the background space. On Shape space we find a natural metric distance, the Shape distance, of which we investigate topological, metric and geodesic properties. Canonically mapped solutions of the Schrödinger equation turn out to be naturally nice curves in Shape space and some even constitue geodesics there. To also be able to speak about infinitesimal change of shapes, a definition for a tangent space at a point in Shape space is defined and applied. Finally a notion of differentiability for maps between Wasserstein spaces is proposed.
Keywords: Wasserstein geometry; Quantum physics; Shape space; Mathematical physics; Quantum foundations
 
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