Quantum and Classical Integrability: A Geometric Perspective

This podcast reviews one possible unified geometric framework for understanding the relationship between classical and quantum mechanics, particularly within integrable systems. It uses contact and symplectic geometry to connect classical trajectories on a cosphere bundle with the localization of quantum eigenfunctions. Symplectization bridges the "quantum boundary" and "classical bulk," establishing a holographic correspondence. The authors incorporate symplectic field theory to provide refined algebraic invariants capturing this interplay, suggesting potential connections to quantum gravity. The framework offers a novel perspective on the links between classical and quantum dynamics and suggests avenues for future research.