On Berry’s phase∗

We discuss geometro-algebraic aspects of the Berry’s phase phenomenon. In particular we show how to induce parallel transport along states via Kaluza-Klein mechanism in infinite dimensions. 1 Classical and Quantum Worlds In recent times, an increasing amount of work is being devoted to developing a new mathematical framework of noncommutative geometry, also called quantum geometry. As a result, we have: • quantum groups (Connes, Drinfeld, Ocneanu, Woronowicz,...) • quantum nets (Finkelstein,...