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Register a new userHolonomy and vortex structures in quantum hydrodynamics
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In this paper we consider a new geometric approach to Madelungs quantum hydrodynamics (QHD) based on the theory of gauge connections. Unlike previous approaches, 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 Schrodinger 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.
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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-Poincare 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-Poincare 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.
The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noethers conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called `collective. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite $hbar$ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the $hbar e0$ dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called `Bohmions, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.
Whereas it is easy to reduce the translational symmetry of a molecular system by using, e.g., Jacobi coordinates the situation is much more involved for the rotational symmetry. In this paper we address the latter problem using {it holonomy reduction}. We suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection. Using the fact that for the special case of the three-body problem, the holonomy group is SO(2) (as opposed to SO(3) like in systems with more than three bodies) we obtain a holonomy reduced configuration space of topology $ mathbf{R}_+^3 times S^1$. The dynamics then takes place on the cotangent bundle over the holonomy reduced configuration space. On this phase space there is an $S^1$ symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method. Using a theorem by Arnold it follows that the resulting symmetry reduced phase space is again a natural mechanical phase space, i.e. a cotangent bundle. This is different from what is obtained from the usual approach where symplectic reduction is used from the outset. This difference is discussed in some detail, and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established.
Based on Koopmans theory of classical wavefunctions in phase space, we present the Koopman-van Hove (KvH) formulation of classical mechanics as well as some of its properties. In particular, we show how the associated classical Liouville density arises as a momentum map associated to the unitary action of strict contact transformations on classical wavefunctions. Upon applying the Madelung transform from quantum hydrodynamics in the new context, we show how the Koopman wavefunction picture is insufficient to reproduce arbitrary classical distributions. However, this problem is entirely overcome by resorting to von Neumann operators. Indeed, we show that the latter also allow for singular $delta-$like profiles of the Liouville density, thereby reproducing point particles in phase space.
Based on the Koopman-van Hove (KvH) formulation of classical mechanics introduced in Part I, we formulate a Hamiltonian model for hybrid quantum-classical systems. This is obtained by writing the KvH wave equation for two classical particles and applying canonical quantization to one of them. We illustrate several geometric properties of the model regarding the associated quantum, classical, and hybrid densities. After presenting the quantum-classical Madelung transform, the joint quantum-classical distribution is shown to arise as a momentum map for a unitary action naturally induced from the van Hove representation on the hybrid Hilbert space. While the quantum density matrix is positive by construction, no such result is currently available for the classical density. However, here we present a class of hybrid Hamiltonians whose flow preserves the sign of the classical density. Finally, we provide a simple closure model based on momentum map structures.
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