Adiabatic driving, geometric phases, and the geometric tensor for classical states

• Published in: Annals of physics Vol. 468; p. 169728
• Main Author: Bermúdez Manjarres, A.D.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-09-2024
• Subjects: Adiabatic gauge potential; Classical integrability and chaos; Geometric tensor; Koopman–von Neumann mechanics; Adiabatic gauge potential; Classical integrability and chaos; Koopman–von Neumann mechanics; Geometric tensor
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2024.169728

We use the Hilbert space formulation of classical mechanics, known as the Koopman–von Neumann formalism, to study adiabatic driving, geometric phases, and the geometric tensor for classical states. In close relation to what happens to a quantum state, a classical Koopman–von Neumann eigenstate will acquire a geometric phase factor expiΦ after a closed variation of the parameters λ in its associated Hamiltonian. The explicit form of Φ is then derived for integrable systems, and its relation with the Hannay angle is shown. Additionally, we use quantum formulas to write an adiabatic gauge potential that generates adiabatic unitary flow between classical eigenstates, and we explicitly show the relationship between the potential and the classical geometric phase. We also define a classical analog of the geometric tensor, thus defining a Fubini–Study metric for classical states, and we use the singularities of the tensor to link the transition from Arnold–Liouville integrability to chaos with some of the mathematical formalism of quantum phase transitions. While the formulas and definitions we use originate in quantum mechanics, all the results found are purely classical, no classical or semiclassical limit is ever taken.
•Information geometry approach the transition from classical integrability to chaos.•Definition of a Fubini–Study metric for Koopman–von Neumann states.•Holonomy of Liouville eigenfunctions and Hannay angles.