Geometry of quantum hydrodynamics in theoretical chemistry

TL;DR

This thesis develops geometric frameworks for quantum hydrodynamics to enhance theoretical quantum chemistry, introducing new models, regularizations, and connections that incorporate vortex solutions and quantum geometric effects.

Contribution

It introduces novel geometric formulations of QHD, including regularized Lagrangians, mixed quantum-classical models, and holonomy-based approaches with non-Abelian connections.

Findings

Introduction of Bohmions as singular solutions in QHD.

Development of a geometric model with non-trivial circulation and vortex filaments.

Application of geometric QHD to quantum chemistry models with new insights.

Abstract

This thesis investigates geometric approaches to quantum hydrodynamics (QHD) in order to develop applications in theoretical quantum chemistry. Based upon the momentum map geometric structure of QHD and the associated Lie-Poisson and Euler-Poincar\'e equations, alternative geometric approaches to the classical limit in QHD are presented. These include a new regularised Lagrangian which allows for singular solutions called 'Bohmions' as well as a 'cold fluid' classical closure quantum mixed states. The momentum map approach to QHD is then applied to the nuclear dynamics in a chemistry model known as exact factorization. The geometric treatment extends existing approaches to include unitary electronic evolution in the frame of the nuclear flow, with the resulting dynamics carrying both Euler-Poincar\'e and Lie-Poisson structures. A new mixed quantum-classical model is then derived by considering a generalised factorisation ansatz at the level of the molecular density matrix. A new alternative geometric formulation of QHD is then constructed. Introducing a $\mathfrak{u}(1)$ connection as the new fundamental variable provides a new method for incorporating holonomy in QHD, which follows from its constant non-zero curvature. The fluid flow is no longer irrotational and carries a non-trivial circulation theorem, allowing for vortex filament solutions. Finally, non-Abelian connections are then considered in quantum mechanics. The dynamics of the spin vector in the Pauli equation allows for the introduction of an $\mathfrak{so}(3)$ connection whilst a more general $\mathfrak{u}(\mathscr{H})$ connection is introduced from the unitary evolution of a quantum system. This is used to provide a new geometric picture for the Berry connection and quantum geometric tensor, whilst relevant applications to quantum chemistry are then considered.