Holonomy and vortex structures in quantum hydrodynamics
Michael S. Foskett, Cesare Tronci
TL;DR
This paper introduces a geometric framework for quantum hydrodynamics (QHD) that incorporates holonomy via a constant-curvature $U(1)$ connection, allowing nonzero hydrodynamic vorticity and vortex-filament structures without relying on multi-valued phases. By rewriting the wavefunction as $oldsymbol{ heta}$ with an explicit phase connection, it derives modified QHD equations and shows Schrödinger dynamics emerge through minimal coupling to an internal vector potential $oldsymbol{oldsymbol{ abla} imesoldsymbol{eta}}$ (denoted $oldsymbol{ ext{Λ}}$), with external fields added via standard vector potentials. The framework is then extended to coupled vortex dynamics using the Rasetti–Regge approach, and applied to Born-Oppenheimer molecular dynamics and exact wavefunction factorization, including spin via the Pauli equation and two-level electronic systems, before outlining non-Abelian generalizations. The results provide a unified, holonomy-centered perspective on geometric phases in quantum dynamics, offering regularized treatments of conical intersections, improved modeling of nonadiabatic processes, and new avenues for spin and gauge-generalized quantum chemistry. Overall, the work anticipates practical benefits for simulating molecular dynamics with geometric-phase effects and for exploring non-Abelian gauge structures in quantum systems.
Abstract
We consider a new geometric approach to Madelung's quantum hydrodynamics (QHD) based on the theory of gauge connections. In particular, our treatment comprises a constant curvature thereby endowing QHD with intrinsic non-zero 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.