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Holonomy and vortex structures in quantum hydrodynamics

Published online by Cambridge University Press:  10 May 2024

By
Michael S. Foskett  and
Cesare Tronci
Edited by
Albert Fathi ,
Philip J. Morrison ,
Tere M-Seara  and
Sergei Tabachnikov
Albert Fathi
Affiliation:
Georgia Institute of Technology
Philip J. Morrison
Affiliation:
University of Texas, Austin
Tere M-Seara
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Sergei Tabachnikov
Affiliation:
Pennsylvania State University
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Summary

We consider a new geometric approach to Madelung’s quantum hydrodynamics (QHD) based on the theory of gauge connections. Our treatment comprises a constant curvature thereby endowing QHD with intrinsic nonzero holonomy. In the hydrodynamic context, this leads to a fluid velocity which no longer is constrained to be irrotational and allows instead for vortex filaments solutions. After exploiting the Rasetti–Regge method to couple the Schrödinger equation to vortex filament dynamics, the latter is then considered as a source of geometric phase in the context of Born–Oppenheimer molecular dynamics. Similarly, we consider the Pauli equation for the motion of spin particles in electromagnetic fields and we exploit its underlying hydrodynamic picture to include vortex dynamics.

Keywords

quantum hydrodynamics
Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
 , pp. 173 - 214
Publisher: Cambridge University Press
Print publication year: 2024

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