Geometric phases enable real-world models of classical and quantum dynamics in Hall effects and in three-body molecular dynamics

Almost sixty years ago, Aharonov and Bohm pointed out that electrons could be affected by vector potentials without an external magnetic field. They described an ad hoc phase shift required for wave functions in vector potentials, e.g., representing magnetic fields. The phase shift exemplifies a geometric phase (or Berry’s phase). In classical and quantum dynamics, vector potentials producing coupled overall rotation lead to geometric phases. Instead of neglecting it, now the coupling is used to create a frame with decoupled overall rotation and vanishing classical and quantum geometric phases. A general formulation of classical dynamics describes both the dynamics of topological matter, such as Hall effects, and three-body molecular dynamics. A quantum extension describes the quantum dynamics of topological matter and the three-body molecular dynamics in the Born-Oppenheimer approximation. Real-world models with or without magnetic fields contribute to developing optoelectronic and photonic devices. Real-world models for three-body dynamics contribute to developing optimal molecular reaction dynamics.