Geometric Phases for Classical and Quantum Dynamics: Hannay Angle and Berry Phase for Loops on a Torus

Abstract

In this paper we have considered closed trajectories of a particle on a two-torus where the loops are noncontractible (poloidal and toroidal loops and knots embedded on a regular torus). We have calculated Hannay angle and Berry phase for particle traversing such loops and knots when the torus itself is adiabatically revolving. Since noncontractible loops do not enclose any area Stokes theorem has to be applied with caution. In our computational scheme we have worked with line integrals directly thus avoiding Stokes theorem.(1) Introduction: Classical limit to quantum phenomena is much more than a naive → 0. In this context the relation between two geometrical objects, Berry phase [1] in quantum regime and Hannay angle [2] in classical domain is indeed fascinating and has generated a lot of interest in theoretical and experimental physics. Quite interestingly, contrary to common occurrences, Berry phase (the quantum effect) was discovered first which was followed by its classical counterpart, Hannay angle and the delicate issue of identifying the latter as a semi-classical limit of the former was rigorously settled by Berry [3]. As a quantum system traverses a closed loop in parameter space the anholonomy, (in Hilbert space where the system lives), is responsible of the appearance of Berry phase. On the other hand, Hannay angle is generated by anholonomy in the trajectory of the classical system in physical configuration space (or more generally in the abstract action-angle manifold).An important generalization to multi-dimensional systems, (consisting of more than one degrees of freedom), where the quantum system in classical limit can show chaotic behavior, was performed by Robbins in [4] (hereafter referred to as JR). Precursors to this work are [5] where physical effects of (the classical limit of) geometric 2-form manifests in a Lorentz force form of "geometric magnetic" force 1 . However, an interesting question, as pointed out in [7], was whether the classical limit 1 This turned out to be the anti-symmetric partner of "deterministic friction", a dissipative force proposed by 1 arXiv:1905.03491v1 [quant-ph]